declaration
stringlengths 27
11.3k
| file
stringlengths 52
114
| context
dict | tactic_states
listlengths 1
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|
|---|---|---|---|
theorem Nat.Prime.sq_add_sq' {p : ℕ} [h : Fact p.Prime] (hp : p % 4 = 1) :
∃ a b : ℕ, a ^ 2 + b ^ 2 = p := by
rw [← div_add_mod p 4] at h ⊢
rw [hp] at h ⊢
let k := p / 4
apply sq_add_sq_of_nonempty_fixedPoints
have key := (Equiv.Perm.card_fixedPoints_modEq (p := 2) (n := 1) (obvInvo_sq k)).symm.trans
(Equiv.Perm.card_fixedPoints_modEq (p := 2) (n := 1) (complexInvo_sq k))
contrapose key
rw [Set.not_nonempty_iff_eq_empty] at key
simp_rw [k, key, Fintype.card_eq_zero, card_fixedPoints_eq_one]
decide
|
/root/DuelModelResearch/mathlib4/Archive/ZagierTwoSquares.lean
|
{
"open": [
"Set",
"Function",
"Zagier"
],
"variables": [
"(k : ℕ) [hk : Fact (4 * k + 1).Prime]",
"(k : ℕ)",
"[hk : Fact (4 * k + 1).Prime]"
]
}
|
[
{
"line": "rw [← div_add_mod p 4] at h ⊢",
"before_state": "p : ℕ\nh : Fact (Prime p)\nhp : p % 4 = 1\n⊢ ∃ a b, a ^ 2 + b ^ 2 = p",
"after_state": "p : ℕ\nh : Fact (Prime (4 * (p / 4) + p % 4))\nhp : p % 4 = 1\n⊢ ∃ a b, a ^ 2 + b ^ 2 = 4 * (p / 4) + p % 4"
},
{
"line": "rewrite [← div_add_mod p 4] at h ⊢",
"before_state": "p : ℕ\nh : Fact (Prime p)\nhp : p % 4 = 1\n⊢ ∃ a b, a ^ 2 + b ^ 2 = p",
"after_state": "p : ℕ\nh : Fact (Prime (4 * (p / 4) + p % 4))\nhp : p % 4 = 1\n⊢ ∃ a b, a ^ 2 + b ^ 2 = 4 * (p / 4) + p % 4"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "p : ℕ\nh : Fact (Prime (4 * (p / 4) + p % 4))\nhp : p % 4 = 1\n⊢ ∃ a b, a ^ 2 + b ^ 2 = 4 * (p / 4) + p % 4",
"after_state": "p : ℕ\nh : Fact (Prime (4 * (p / 4) + p % 4))\nhp : p % 4 = 1\n⊢ ∃ a b, a ^ 2 + b ^ 2 = 4 * (p / 4) + p % 4"
},
{
"line": "try (with_reducible rfl)",
"before_state": "p : ℕ\nh : Fact (Prime (4 * (p / 4) + p % 4))\nhp : p % 4 = 1\n⊢ ∃ a b, a ^ 2 + b ^ 2 = 4 * (p / 4) + p % 4",
"after_state": "p : ℕ\nh : Fact (Prime (4 * (p / 4) + p % 4))\nhp : p % 4 = 1\n⊢ ∃ a b, a ^ 2 + b ^ 2 = 4 * (p / 4) + p % 4"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "p : ℕ\nh : Fact (Prime (4 * (p / 4) + p % 4))\nhp : p % 4 = 1\n⊢ ∃ a b, a ^ 2 + b ^ 2 = 4 * (p / 4) + p % 4",
"after_state": "p : ℕ\nh : Fact (Prime (4 * (p / 4) + p % 4))\nhp : p % 4 = 1\n⊢ ∃ a b, a ^ 2 + b ^ 2 = 4 * (p / 4) + p % 4"
},
{
"line": "with_reducible rfl",
"before_state": "p : ℕ\nh : Fact (Prime (4 * (p / 4) + p % 4))\nhp : p % 4 = 1\n⊢ ∃ a b, a ^ 2 + b ^ 2 = 4 * (p / 4) + p % 4",
"after_state": "p : ℕ\nh : Fact (Prime (4 * (p / 4) + p % 4))\nhp : p % 4 = 1\n⊢ ∃ a b, a ^ 2 + b ^ 2 = 4 * (p / 4) + p % 4"
},
{
"line": "rfl",
"before_state": "p : ℕ\nh : Fact (Prime (4 * (p / 4) + p % 4))\nhp : p % 4 = 1\n⊢ ∃ a b, a ^ 2 + b ^ 2 = 4 * (p / 4) + p % 4",
"after_state": "p : ℕ\nh : Fact (Prime (4 * (p / 4) + p % 4))\nhp : p % 4 = 1\n⊢ ∃ a b, a ^ 2 + b ^ 2 = 4 * (p / 4) + p % 4"
},
{
"line": "apply_rfl",
"before_state": "p : ℕ\nh : Fact (Prime (4 * (p / 4) + p % 4))\nhp : p % 4 = 1\n⊢ ∃ a b, a ^ 2 + b ^ 2 = 4 * (p / 4) + p % 4",
"after_state": "p : ℕ\nh : Fact (Prime (4 * (p / 4) + p % 4))\nhp : p % 4 = 1\n⊢ ∃ a b, a ^ 2 + b ^ 2 = 4 * (p / 4) + p % 4"
},
{
"line": "skip",
"before_state": "p : ℕ\nh : Fact (Prime (4 * (p / 4) + p % 4))\nhp : p % 4 = 1\n⊢ ∃ a b, a ^ 2 + b ^ 2 = 4 * (p / 4) + p % 4",
"after_state": "p : ℕ\nh : Fact (Prime (4 * (p / 4) + p % 4))\nhp : p % 4 = 1\n⊢ ∃ a b, a ^ 2 + b ^ 2 = 4 * (p / 4) + p % 4"
},
{
"line": "rw [hp] at h ⊢",
"before_state": "p : ℕ\nh : Fact (Prime (4 * (p / 4) + p % 4))\nhp : p % 4 = 1\n⊢ ∃ a b, a ^ 2 + b ^ 2 = 4 * (p / 4) + p % 4",
"after_state": "p : ℕ\nh : Fact (Prime (4 * (p / 4) + 1))\nhp : p % 4 = 1\n⊢ ∃ a b, a ^ 2 + b ^ 2 = 4 * (p / 4) + 1"
},
{
"line": "rewrite [hp] at h ⊢",
"before_state": "p : ℕ\nh : Fact (Prime (4 * (p / 4) + p % 4))\nhp : p % 4 = 1\n⊢ ∃ a b, a ^ 2 + b ^ 2 = 4 * (p / 4) + p % 4",
"after_state": "p : ℕ\nh : Fact (Prime (4 * (p / 4) + 1))\nhp : p % 4 = 1\n⊢ ∃ a b, a ^ 2 + b ^ 2 = 4 * (p / 4) + 1"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "p : ℕ\nh : Fact (Prime (4 * (p / 4) + 1))\nhp : p % 4 = 1\n⊢ ∃ a b, a ^ 2 + b ^ 2 = 4 * (p / 4) + 1",
"after_state": "p : ℕ\nh : Fact (Prime (4 * (p / 4) + 1))\nhp : p % 4 = 1\n⊢ ∃ a b, a ^ 2 + b ^ 2 = 4 * (p / 4) + 1"
},
{
"line": "try (with_reducible rfl)",
"before_state": "p : ℕ\nh : Fact (Prime (4 * (p / 4) + 1))\nhp : p % 4 = 1\n⊢ ∃ a b, a ^ 2 + b ^ 2 = 4 * (p / 4) + 1",
"after_state": "p : ℕ\nh : Fact (Prime (4 * (p / 4) + 1))\nhp : p % 4 = 1\n⊢ ∃ a b, a ^ 2 + b ^ 2 = 4 * (p / 4) + 1"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "p : ℕ\nh : Fact (Prime (4 * (p / 4) + 1))\nhp : p % 4 = 1\n⊢ ∃ a b, a ^ 2 + b ^ 2 = 4 * (p / 4) + 1",
"after_state": "p : ℕ\nh : Fact (Prime (4 * (p / 4) + 1))\nhp : p % 4 = 1\n⊢ ∃ a b, a ^ 2 + b ^ 2 = 4 * (p / 4) + 1"
},
{
"line": "with_reducible rfl",
"before_state": "p : ℕ\nh : Fact (Prime (4 * (p / 4) + 1))\nhp : p % 4 = 1\n⊢ ∃ a b, a ^ 2 + b ^ 2 = 4 * (p / 4) + 1",
"after_state": "p : ℕ\nh : Fact (Prime (4 * (p / 4) + 1))\nhp : p % 4 = 1\n⊢ ∃ a b, a ^ 2 + b ^ 2 = 4 * (p / 4) + 1"
},
{
"line": "rfl",
"before_state": "p : ℕ\nh : Fact (Prime (4 * (p / 4) + 1))\nhp : p % 4 = 1\n⊢ ∃ a b, a ^ 2 + b ^ 2 = 4 * (p / 4) + 1",
"after_state": "p : ℕ\nh : Fact (Prime (4 * (p / 4) + 1))\nhp : p % 4 = 1\n⊢ ∃ a b, a ^ 2 + b ^ 2 = 4 * (p / 4) + 1"
},
{
"line": "apply_rfl",
"before_state": "p : ℕ\nh : Fact (Prime (4 * (p / 4) + 1))\nhp : p % 4 = 1\n⊢ ∃ a b, a ^ 2 + b ^ 2 = 4 * (p / 4) + 1",
"after_state": "p : ℕ\nh : Fact (Prime (4 * (p / 4) + 1))\nhp : p % 4 = 1\n⊢ ∃ a b, a ^ 2 + b ^ 2 = 4 * (p / 4) + 1"
},
{
"line": "skip",
"before_state": "p : ℕ\nh : Fact (Prime (4 * (p / 4) + 1))\nhp : p % 4 = 1\n⊢ ∃ a b, a ^ 2 + b ^ 2 = 4 * (p / 4) + 1",
"after_state": "p : ℕ\nh : Fact (Prime (4 * (p / 4) + 1))\nhp : p % 4 = 1\n⊢ ∃ a b, a ^ 2 + b ^ 2 = 4 * (p / 4) + 1"
},
{
"line": "let k := p / 4",
"before_state": "p : ℕ\nh : Fact (Prime (4 * (p / 4) + 1))\nhp : p % 4 = 1\n⊢ ∃ a b, a ^ 2 + b ^ 2 = 4 * (p / 4) + 1",
"after_state": "p : ℕ\nh : Fact (Prime (4 * (p / 4) + 1))\nhp : p % 4 = 1\nk : ℕ := p / 4\n⊢ ∃ a b, a ^ 2 + b ^ 2 = 4 * (p / 4) + 1"
},
{
"line": "refine_lift\n let k := p / 4;\n ?_",
"before_state": "p : ℕ\nh : Fact (Prime (4 * (p / 4) + 1))\nhp : p % 4 = 1\n⊢ ∃ a b, a ^ 2 + b ^ 2 = 4 * (p / 4) + 1",
"after_state": "p : ℕ\nh : Fact (Prime (4 * (p / 4) + 1))\nhp : p % 4 = 1\nk : ℕ := p / 4\n⊢ ∃ a b, a ^ 2 + b ^ 2 = 4 * (p / 4) + 1"
},
{
"line": "focus\n (refine\n no_implicit_lambda%\n (let k := p / 4;\n ?_);\n rotate_right)",
"before_state": "p : ℕ\nh : Fact (Prime (4 * (p / 4) + 1))\nhp : p % 4 = 1\n⊢ ∃ a b, a ^ 2 + b ^ 2 = 4 * (p / 4) + 1",
"after_state": "p : ℕ\nh : Fact (Prime (4 * (p / 4) + 1))\nhp : p % 4 = 1\nk : ℕ := p / 4\n⊢ ∃ a b, a ^ 2 + b ^ 2 = 4 * (p / 4) + 1"
},
{
"line": "refine\n no_implicit_lambda%\n (let k := p / 4;\n ?_)",
"before_state": "p : ℕ\nh : Fact (Prime (4 * (p / 4) + 1))\nhp : p % 4 = 1\n⊢ ∃ a b, a ^ 2 + b ^ 2 = 4 * (p / 4) + 1",
"after_state": "p : ℕ\nh : Fact (Prime (4 * (p / 4) + 1))\nhp : p % 4 = 1\nk : ℕ := p / 4\n⊢ ∃ a b, a ^ 2 + b ^ 2 = 4 * (p / 4) + 1"
},
{
"line": "rotate_right",
"before_state": "p : ℕ\nh : Fact (Prime (4 * (p / 4) + 1))\nhp : p % 4 = 1\nk : ℕ := p / 4\n⊢ ∃ a b, a ^ 2 + b ^ 2 = 4 * (p / 4) + 1",
"after_state": "p : ℕ\nh : Fact (Prime (4 * (p / 4) + 1))\nhp : p % 4 = 1\nk : ℕ := p / 4\n⊢ ∃ a b, a ^ 2 + b ^ 2 = 4 * (p / 4) + 1"
},
{
"line": "apply sq_add_sq_of_nonempty_fixedPoints",
"before_state": "p : ℕ\nh : Fact (Prime (4 * (p / 4) + 1))\nhp : p % 4 = 1\nk : ℕ := p / 4\n⊢ ∃ a b, a ^ 2 + b ^ 2 = 4 * (p / 4) + 1",
"after_state": "No Goals!"
}
] |
example : ¬ LucasLehmerTest 2 := by norm_num
|
/root/DuelModelResearch/mathlib4/Archive/Examples/MersennePrimes.lean
|
{
"open": [],
"variables": []
}
|
[
{
"line": "norm_num",
"before_state": "⊢ ¬LucasLehmerTest 2",
"after_state": "No Goals!"
}
] |
example : (mersenne 2).Prime := by decide
|
/root/DuelModelResearch/mathlib4/Archive/Examples/MersennePrimes.lean
|
{
"open": [],
"variables": []
}
|
[
{
"line": "decide",
"before_state": "⊢ Nat.Prime (mersenne 2)",
"after_state": "No Goals!"
}
] |
example : (mersenne 3).Prime :=
lucas_lehmer_sufficiency _ (by norm_num) (by norm_num)
|
/root/DuelModelResearch/mathlib4/Archive/Examples/MersennePrimes.lean
|
{
"open": [],
"variables": []
}
|
[
{
"line": "norm_num",
"before_state": "⊢ 1 < 3",
"after_state": "No Goals!"
},
{
"line": "norm_num",
"before_state": "⊢ LucasLehmerTest 3",
"after_state": "No Goals!"
}
] |
example : (mersenne 5).Prime :=
lucas_lehmer_sufficiency _ (by norm_num) (by norm_num)
|
/root/DuelModelResearch/mathlib4/Archive/Examples/MersennePrimes.lean
|
{
"open": [],
"variables": []
}
|
[
{
"line": "norm_num",
"before_state": "⊢ 1 < 5",
"after_state": "No Goals!"
},
{
"line": "norm_num",
"before_state": "⊢ LucasLehmerTest 5",
"after_state": "No Goals!"
}
] |
example : (mersenne 7).Prime :=
lucas_lehmer_sufficiency _ (by norm_num) (by norm_num)
|
/root/DuelModelResearch/mathlib4/Archive/Examples/MersennePrimes.lean
|
{
"open": [],
"variables": []
}
|
[
{
"line": "norm_num",
"before_state": "⊢ 1 < 7",
"after_state": "No Goals!"
},
{
"line": "norm_num",
"before_state": "⊢ LucasLehmerTest 7",
"after_state": "No Goals!"
}
] |
example : (mersenne 13).Prime :=
lucas_lehmer_sufficiency _ (by norm_num) (by norm_num)
|
/root/DuelModelResearch/mathlib4/Archive/Examples/MersennePrimes.lean
|
{
"open": [],
"variables": []
}
|
[
{
"line": "norm_num",
"before_state": "⊢ 1 < 13",
"after_state": "No Goals!"
},
{
"line": "norm_num",
"before_state": "⊢ LucasLehmerTest 13",
"after_state": "No Goals!"
}
] |
example : (mersenne 17).Prime :=
lucas_lehmer_sufficiency _ (by norm_num) (by norm_num)
|
/root/DuelModelResearch/mathlib4/Archive/Examples/MersennePrimes.lean
|
{
"open": [],
"variables": []
}
|
[
{
"line": "norm_num",
"before_state": "⊢ 1 < 17",
"after_state": "No Goals!"
},
{
"line": "norm_num",
"before_state": "⊢ LucasLehmerTest 17",
"after_state": "No Goals!"
}
] |
example : (mersenne 19).Prime :=
lucas_lehmer_sufficiency _ (by norm_num) (by norm_num)
|
/root/DuelModelResearch/mathlib4/Archive/Examples/MersennePrimes.lean
|
{
"open": [],
"variables": []
}
|
[
{
"line": "norm_num",
"before_state": "⊢ 1 < 19",
"after_state": "No Goals!"
},
{
"line": "norm_num",
"before_state": "⊢ LucasLehmerTest 19",
"after_state": "No Goals!"
}
] |
example : (mersenne 31).Prime :=
lucas_lehmer_sufficiency _ (by norm_num) (by norm_num)
|
/root/DuelModelResearch/mathlib4/Archive/Examples/MersennePrimes.lean
|
{
"open": [],
"variables": []
}
|
[
{
"line": "norm_num",
"before_state": "⊢ 1 < 31",
"after_state": "No Goals!"
},
{
"line": "norm_num",
"before_state": "⊢ LucasLehmerTest 31",
"after_state": "No Goals!"
}
] |
example : (mersenne 61).Prime :=
lucas_lehmer_sufficiency _ (by norm_num) (by norm_num)
|
/root/DuelModelResearch/mathlib4/Archive/Examples/MersennePrimes.lean
|
{
"open": [],
"variables": []
}
|
[
{
"line": "norm_num",
"before_state": "⊢ 1 < 61",
"after_state": "No Goals!"
},
{
"line": "norm_num",
"before_state": "⊢ LucasLehmerTest 61",
"after_state": "No Goals!"
}
] |
example : (mersenne 89).Prime :=
lucas_lehmer_sufficiency _ (by norm_num) (by norm_num)
|
/root/DuelModelResearch/mathlib4/Archive/Examples/MersennePrimes.lean
|
{
"open": [],
"variables": []
}
|
[
{
"line": "norm_num",
"before_state": "⊢ 1 < 89",
"after_state": "No Goals!"
},
{
"line": "norm_num",
"before_state": "⊢ LucasLehmerTest 89",
"after_state": "No Goals!"
}
] |
example : (mersenne 107).Prime :=
lucas_lehmer_sufficiency _ (by norm_num) (by norm_num)
|
/root/DuelModelResearch/mathlib4/Archive/Examples/MersennePrimes.lean
|
{
"open": [],
"variables": []
}
|
[
{
"line": "norm_num",
"before_state": "⊢ 1 < 107",
"after_state": "No Goals!"
},
{
"line": "norm_num",
"before_state": "⊢ LucasLehmerTest 107",
"after_state": "No Goals!"
}
] |
example : (mersenne 127).Prime :=
lucas_lehmer_sufficiency _ (by norm_num) (by norm_num)
|
/root/DuelModelResearch/mathlib4/Archive/Examples/MersennePrimes.lean
|
{
"open": [],
"variables": []
}
|
[
{
"line": "norm_num",
"before_state": "⊢ 1 < 127",
"after_state": "No Goals!"
},
{
"line": "norm_num",
"before_state": "⊢ LucasLehmerTest 127",
"after_state": "No Goals!"
}
] |
example : (mersenne 521).Prime :=
lucas_lehmer_sufficiency _ (by norm_num) (by norm_num)
|
/root/DuelModelResearch/mathlib4/Archive/Examples/MersennePrimes.lean
|
{
"open": [],
"variables": []
}
|
[
{
"line": "norm_num",
"before_state": "⊢ 1 < 521",
"after_state": "No Goals!"
},
{
"line": "norm_num",
"before_state": "⊢ LucasLehmerTest 521",
"after_state": "No Goals!"
}
] |
example : (mersenne 607).Prime :=
lucas_lehmer_sufficiency _ (by norm_num) (by norm_num)
|
/root/DuelModelResearch/mathlib4/Archive/Examples/MersennePrimes.lean
|
{
"open": [],
"variables": []
}
|
[
{
"line": "norm_num",
"before_state": "⊢ 1 < 607",
"after_state": "No Goals!"
},
{
"line": "norm_num",
"before_state": "⊢ LucasLehmerTest 607",
"after_state": "No Goals!"
}
] |
example : (mersenne 1279).Prime :=
lucas_lehmer_sufficiency _ (by norm_num) (by norm_num)
|
/root/DuelModelResearch/mathlib4/Archive/Examples/MersennePrimes.lean
|
{
"open": [],
"variables": []
}
|
[
{
"line": "norm_num",
"before_state": "⊢ 1 < 1279",
"after_state": "No Goals!"
},
{
"line": "norm_num",
"before_state": "⊢ LucasLehmerTest 1279",
"after_state": "No Goals!"
}
] |
example : (mersenne 2203).Prime :=
lucas_lehmer_sufficiency _ (by norm_num) (by norm_num)
|
/root/DuelModelResearch/mathlib4/Archive/Examples/MersennePrimes.lean
|
{
"open": [],
"variables": []
}
|
[
{
"line": "norm_num",
"before_state": "⊢ 1 < 2203",
"after_state": "No Goals!"
},
{
"line": "norm_num",
"before_state": "⊢ LucasLehmerTest 2203",
"after_state": "No Goals!"
}
] |
example : (mersenne 2281).Prime :=
lucas_lehmer_sufficiency _ (by norm_num) (by norm_num)
|
/root/DuelModelResearch/mathlib4/Archive/Examples/MersennePrimes.lean
|
{
"open": [],
"variables": []
}
|
[
{
"line": "norm_num",
"before_state": "⊢ 1 < 2281",
"after_state": "No Goals!"
},
{
"line": "norm_num",
"before_state": "⊢ LucasLehmerTest 2281",
"after_state": "No Goals!"
}
] |
example : (mersenne 3217).Prime :=
lucas_lehmer_sufficiency _ (by norm_num) (by norm_num)
|
/root/DuelModelResearch/mathlib4/Archive/Examples/MersennePrimes.lean
|
{
"open": [],
"variables": []
}
|
[
{
"line": "norm_num",
"before_state": "⊢ 1 < 3217",
"after_state": "No Goals!"
},
{
"line": "norm_num",
"before_state": "⊢ LucasLehmerTest 3217",
"after_state": "No Goals!"
}
] |
example : (mersenne 4253).Prime :=
lucas_lehmer_sufficiency _ (by norm_num) (by norm_num)
|
/root/DuelModelResearch/mathlib4/Archive/Examples/MersennePrimes.lean
|
{
"open": [],
"variables": []
}
|
[
{
"line": "norm_num",
"before_state": "⊢ 1 < 4253",
"after_state": "No Goals!"
},
{
"line": "norm_num",
"before_state": "⊢ LucasLehmerTest 4253",
"after_state": "No Goals!"
}
] |
example : (mersenne 4423).Prime :=
lucas_lehmer_sufficiency _ (by norm_num) (by norm_num)
|
/root/DuelModelResearch/mathlib4/Archive/Examples/MersennePrimes.lean
|
{
"open": [],
"variables": []
}
|
[
{
"line": "norm_num",
"before_state": "⊢ 1 < 4423",
"after_state": "No Goals!"
},
{
"line": "norm_num",
"before_state": "⊢ LucasLehmerTest 4423",
"after_state": "No Goals!"
}
] |
theorem calculation (n k : ℕ) (h1 : k ∣ 21 * n + 4) (h2 : k ∣ 14 * n + 3) : k ∣ 1 :=
have h3 : k ∣ 2 * (21 * n + 4) := h1.mul_left 2
have h4 : k ∣ 3 * (14 * n + 3) := h2.mul_left 3
have h5 : 3 * (14 * n + 3) = 2 * (21 * n + 4) + 1 := by ring
(Nat.dvd_add_right h3).mp (h5 ▸ h4)
|
/root/DuelModelResearch/mathlib4/Archive/Imo/Imo1959Q1.lean
|
{
"open": [
"Nat"
],
"variables": []
}
|
[
{
"line": "ring",
"before_state": "n k : ℕ\nh1 : k ∣ 21 * n + 4\nh2 : k ∣ 14 * n + 3\nh3 : k ∣ 2 * (21 * n + 4)\nh4 : k ∣ 3 * (14 * n + 3)\n⊢ 3 * (14 * n + 3) = 2 * (21 * n + 4) + 1",
"after_state": "No Goals!"
},
{
"line": "first\n| ring1\n|\n try_this ring_nf\"\\n\\nThe `ring` tactic failed to close the goal. Use `ring_nf` to obtain a normal form.\n \\nNote that `ring` works primarily in *commutative* rings. \\\n If you have a noncommutative ring, abelian group or module, consider using \\\n `noncomm_ring`, `abel` or `module` instead.\"",
"before_state": "n k : ℕ\nh1 : k ∣ 21 * n + 4\nh2 : k ∣ 14 * n + 3\nh3 : k ∣ 2 * (21 * n + 4)\nh4 : k ∣ 3 * (14 * n + 3)\n⊢ 3 * (14 * n + 3) = 2 * (21 * n + 4) + 1",
"after_state": "No Goals!"
},
{
"line": "ring1",
"before_state": "n k : ℕ\nh1 : k ∣ 21 * n + 4\nh2 : k ∣ 14 * n + 3\nh3 : k ∣ 2 * (21 * n + 4)\nh4 : k ∣ 3 * (14 * n + 3)\n⊢ 3 * (14 * n + 3) = 2 * (21 * n + 4) + 1",
"after_state": "No Goals!"
}
] |
theorem Imo1961Q3 {n : ℕ} {x : ℝ} (h₀ : n ≠ 0) :
(cos x) ^ n - (sin x) ^ n = 1 ↔
(∃ k : ℤ, k * π = x) ∧ Even n ∨ (∃ k : ℤ, k * (2 * π) = x) ∧ Odd n ∨
(∃ k : ℤ, -(π / 2) + k * (2 * π) = x) ∧ Odd n := by
constructor
· intro h
rcases eq_or_ne (sin x) 0 with hsinx | hsinx
· rw [hsinx, zero_pow h₀, sub_zero, pow_eq_one_iff_of_ne_zero h₀, cos_eq_one_iff,
cos_eq_neg_one_iff] at h
rcases h with ⟨k, rfl⟩ | ⟨⟨k, rfl⟩, hn⟩
· cases n.even_or_odd with
| inl hn => refine .inl ⟨⟨k * 2, ?_⟩, hn⟩; simp [mul_assoc]
| inr hn => exact .inr <| .inl ⟨⟨_, rfl⟩, hn⟩
· exact .inl ⟨⟨2 * k + 1, by push_cast; ring⟩, hn⟩
· rcases eq_or_ne (cos x) 0 with hcosx | hcosx
· right; right
rw [hcosx] at h
rw [zero_pow h₀] at h
rw [zero_sub] at h
rw [← neg_inj] at h
rw [neg_neg] at h
rw [pow_eq_neg_one_iff] at h
rw [sin_eq_neg_one_iff] at h
simpa only [eq_comm] using h
· have hcos1 : |cos x| < 1 := by
rw [abs_cos_eq_sqrt_one_sub_sin_sq]
rw [sqrt_lt' one_pos]
simp [sq_pos_of_ne_zero hsinx]
have hsin1 : |sin x| < 1 := by
rw [abs_sin_eq_sqrt_one_sub_cos_sq]
rw [sqrt_lt' one_pos]
simp [sq_pos_of_ne_zero hcosx]
match n with
| 1 =>
rw [pow_one] at h
rw [pow_one] at h
rw [sub_eq_iff_eq_add] at h
have : 2 * sin x * cos x = 0 := by
simpa [h, add_sq, add_assoc, ← two_mul, mul_add, mul_assoc, ← sq]
using cos_sq_add_sin_sq x
simp [hsinx, hcosx] at this
| 2 =>
rw [← cos_sq_add_sin_sq x] at h
rw [sub_eq_add_neg] at h
rw [add_right_inj] at h
rw [neg_eq_self ℝ] at h
exact absurd (pow_eq_zero h) hsinx
| (n + 1 + 2) =>
set m := n + 1
refine absurd ?_ h.not_lt
calc
(cos x) ^ (m + 2) - (sin x) ^ (m + 2) ≤ |cos x| ^ (m + 2) + |sin x| ^ (m + 2) := by
simp only [← abs_pow]
simp only [sub_eq_add_neg]
gcongr
exacts [le_abs_self _, neg_le_abs _]
_ = |cos x| ^ m * cos x ^ 2 + |sin x| ^ m * sin x ^ 2 := by simp [pow_add]
_ < 1 ^ m * cos x ^ 2 + 1 ^ m * sin x ^ 2 := by gcongr
_ = 1 := by simp
· rintro (⟨⟨k, rfl⟩, hn⟩ | ⟨⟨k, rfl⟩, -⟩ | ⟨⟨k, rfl⟩, hn⟩)
· rw [sin_int_mul_pi, zero_pow h₀, sub_zero, ← hn.pow_abs, abs_cos_int_mul_pi, one_pow]
· have : sin (k * (2 * π)) = 0 := by simpa [mul_assoc] using sin_int_mul_pi (k * 2)
simp [h₀, this]
· simp [hn.neg_pow, h₀]
|
/root/DuelModelResearch/mathlib4/Archive/Imo/Imo1961Q3.lean
|
{
"open": [
"Real"
],
"variables": []
}
|
[
{
"line": "constructor",
"before_state": "n : ℕ\nx : ℝ\nh₀ : n ≠ 0\n⊢ cos x ^ n - sin x ^ n = 1 ↔\n (∃ k, ↑k * π = x) ∧ Even n ∨ (∃ k, ↑k * (2 * π) = x) ∧ Odd n ∨ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd n",
"after_state": "case mp\nn : ℕ\nx : ℝ\nh₀ : n ≠ 0\n⊢ cos x ^ n - sin x ^ n = 1 →\n (∃ k, ↑k * π = x) ∧ Even n ∨ (∃ k, ↑k * (2 * π) = x) ∧ Odd n ∨ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd n\n---\ncase mpr\nn : ℕ\nx : ℝ\nh₀ : n ≠ 0\n⊢ (∃ k, ↑k * π = x) ∧ Even n ∨ (∃ k, ↑k * (2 * π) = x) ∧ Odd n ∨ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd n →\n cos x ^ n - sin x ^ n = 1"
},
{
"line": "intro h",
"before_state": "case mp\nn : ℕ\nx : ℝ\nh₀ : n ≠ 0\n⊢ cos x ^ n - sin x ^ n = 1 →\n (∃ k, ↑k * π = x) ∧ Even n ∨ (∃ k, ↑k * (2 * π) = x) ∧ Odd n ∨ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd n",
"after_state": "case mp\nn : ℕ\nx : ℝ\nh₀ : n ≠ 0\nh : cos x ^ n - sin x ^ n = 1\n⊢ (∃ k, ↑k * π = x) ∧ Even n ∨ (∃ k, ↑k * (2 * π) = x) ∧ Odd n ∨ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd n"
},
{
"line": "rcases eq_or_ne (sin x) 0 with hsinx | hsinx",
"before_state": "case mp\nn : ℕ\nx : ℝ\nh₀ : n ≠ 0\nh : cos x ^ n - sin x ^ n = 1\n⊢ (∃ k, ↑k * π = x) ∧ Even n ∨ (∃ k, ↑k * (2 * π) = x) ∧ Odd n ∨ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd n",
"after_state": "case mp.inl\nn : ℕ\nx : ℝ\nh₀ : n ≠ 0\nh : cos x ^ n - sin x ^ n = 1\nhsinx : sin x = 0\n⊢ (∃ k, ↑k * π = x) ∧ Even n ∨ (∃ k, ↑k * (2 * π) = x) ∧ Odd n ∨ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd n\n---\ncase mp.inr\nn : ℕ\nx : ℝ\nh₀ : n ≠ 0\nh : cos x ^ n - sin x ^ n = 1\nhsinx : sin x ≠ 0\n⊢ (∃ k, ↑k * π = x) ∧ Even n ∨ (∃ k, ↑k * (2 * π) = x) ∧ Odd n ∨ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd n"
},
{
"line": "rw [hsinx, zero_pow h₀, sub_zero, pow_eq_one_iff_of_ne_zero h₀, cos_eq_one_iff, cos_eq_neg_one_iff] at h",
"before_state": "case mp.inl\nn : ℕ\nx : ℝ\nh₀ : n ≠ 0\nh : cos x ^ n - sin x ^ n = 1\nhsinx : sin x = 0\n⊢ (∃ k, ↑k * π = x) ∧ Even n ∨ (∃ k, ↑k * (2 * π) = x) ∧ Odd n ∨ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd n",
"after_state": "case mp.inl\nn : ℕ\nx : ℝ\nh₀ : n ≠ 0\nh : (∃ n, ↑n * (2 * π) = x) ∨ (∃ k, π + ↑k * (2 * π) = x) ∧ Even n\nhsinx : sin x = 0\n⊢ (∃ k, ↑k * π = x) ∧ Even n ∨ (∃ k, ↑k * (2 * π) = x) ∧ Odd n ∨ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd n"
},
{
"line": "rewrite [hsinx, zero_pow h₀, sub_zero, pow_eq_one_iff_of_ne_zero h₀, cos_eq_one_iff, cos_eq_neg_one_iff] at h",
"before_state": "case mp.inl\nn : ℕ\nx : ℝ\nh₀ : n ≠ 0\nh : cos x ^ n - sin x ^ n = 1\nhsinx : sin x = 0\n⊢ (∃ k, ↑k * π = x) ∧ Even n ∨ (∃ k, ↑k * (2 * π) = x) ∧ Odd n ∨ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd n",
"after_state": "case mp.inl\nn : ℕ\nx : ℝ\nh₀ : n ≠ 0\nh : (∃ n, ↑n * (2 * π) = x) ∨ (∃ k, π + ↑k * (2 * π) = x) ∧ Even n\nhsinx : sin x = 0\n⊢ (∃ k, ↑k * π = x) ∧ Even n ∨ (∃ k, ↑k * (2 * π) = x) ∧ Odd n ∨ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd n"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "case mp.inl\nn : ℕ\nx : ℝ\nh₀ : n ≠ 0\nh : (∃ n, ↑n * (2 * π) = x) ∨ (∃ k, π + ↑k * (2 * π) = x) ∧ Even n\nhsinx : sin x = 0\n⊢ (∃ k, ↑k * π = x) ∧ Even n ∨ (∃ k, ↑k * (2 * π) = x) ∧ Odd n ∨ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd n",
"after_state": "case mp.inl\nn : ℕ\nx : ℝ\nh₀ : n ≠ 0\nh : (∃ n, ↑n * (2 * π) = x) ∨ (∃ k, π + ↑k * (2 * π) = x) ∧ Even n\nhsinx : sin x = 0\n⊢ (∃ k, ↑k * π = x) ∧ Even n ∨ (∃ k, ↑k * (2 * π) = x) ∧ Odd n ∨ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd n"
},
{
"line": "try (with_reducible rfl)",
"before_state": "case mp.inl\nn : ℕ\nx : ℝ\nh₀ : n ≠ 0\nh : (∃ n, ↑n * (2 * π) = x) ∨ (∃ k, π + ↑k * (2 * π) = x) ∧ Even n\nhsinx : sin x = 0\n⊢ (∃ k, ↑k * π = x) ∧ Even n ∨ (∃ k, ↑k * (2 * π) = x) ∧ Odd n ∨ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd n",
"after_state": "case mp.inl\nn : ℕ\nx : ℝ\nh₀ : n ≠ 0\nh : (∃ n, ↑n * (2 * π) = x) ∨ (∃ k, π + ↑k * (2 * π) = x) ∧ Even n\nhsinx : sin x = 0\n⊢ (∃ k, ↑k * π = x) ∧ Even n ∨ (∃ k, ↑k * (2 * π) = x) ∧ Odd n ∨ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd n"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "case mp.inl\nn : ℕ\nx : ℝ\nh₀ : n ≠ 0\nh : (∃ n, ↑n * (2 * π) = x) ∨ (∃ k, π + ↑k * (2 * π) = x) ∧ Even n\nhsinx : sin x = 0\n⊢ (∃ k, ↑k * π = x) ∧ Even n ∨ (∃ k, ↑k * (2 * π) = x) ∧ Odd n ∨ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd n",
"after_state": "case mp.inl\nn : ℕ\nx : ℝ\nh₀ : n ≠ 0\nh : (∃ n, ↑n * (2 * π) = x) ∨ (∃ k, π + ↑k * (2 * π) = x) ∧ Even n\nhsinx : sin x = 0\n⊢ (∃ k, ↑k * π = x) ∧ Even n ∨ (∃ k, ↑k * (2 * π) = x) ∧ Odd n ∨ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd n"
},
{
"line": "with_reducible rfl",
"before_state": "case mp.inl\nn : ℕ\nx : ℝ\nh₀ : n ≠ 0\nh : (∃ n, ↑n * (2 * π) = x) ∨ (∃ k, π + ↑k * (2 * π) = x) ∧ Even n\nhsinx : sin x = 0\n⊢ (∃ k, ↑k * π = x) ∧ Even n ∨ (∃ k, ↑k * (2 * π) = x) ∧ Odd n ∨ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd n",
"after_state": "case mp.inl\nn : ℕ\nx : ℝ\nh₀ : n ≠ 0\nh : (∃ n, ↑n * (2 * π) = x) ∨ (∃ k, π + ↑k * (2 * π) = x) ∧ Even n\nhsinx : sin x = 0\n⊢ (∃ k, ↑k * π = x) ∧ Even n ∨ (∃ k, ↑k * (2 * π) = x) ∧ Odd n ∨ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd n"
},
{
"line": "rfl",
"before_state": "case mp.inl\nn : ℕ\nx : ℝ\nh₀ : n ≠ 0\nh : (∃ n, ↑n * (2 * π) = x) ∨ (∃ k, π + ↑k * (2 * π) = x) ∧ Even n\nhsinx : sin x = 0\n⊢ (∃ k, ↑k * π = x) ∧ Even n ∨ (∃ k, ↑k * (2 * π) = x) ∧ Odd n ∨ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd n",
"after_state": "case mp.inl\nn : ℕ\nx : ℝ\nh₀ : n ≠ 0\nh : (∃ n, ↑n * (2 * π) = x) ∨ (∃ k, π + ↑k * (2 * π) = x) ∧ Even n\nhsinx : sin x = 0\n⊢ (∃ k, ↑k * π = x) ∧ Even n ∨ (∃ k, ↑k * (2 * π) = x) ∧ Odd n ∨ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd n"
},
{
"line": "apply_rfl",
"before_state": "case mp.inl\nn : ℕ\nx : ℝ\nh₀ : n ≠ 0\nh : (∃ n, ↑n * (2 * π) = x) ∨ (∃ k, π + ↑k * (2 * π) = x) ∧ Even n\nhsinx : sin x = 0\n⊢ (∃ k, ↑k * π = x) ∧ Even n ∨ (∃ k, ↑k * (2 * π) = x) ∧ Odd n ∨ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd n",
"after_state": "case mp.inl\nn : ℕ\nx : ℝ\nh₀ : n ≠ 0\nh : (∃ n, ↑n * (2 * π) = x) ∨ (∃ k, π + ↑k * (2 * π) = x) ∧ Even n\nhsinx : sin x = 0\n⊢ (∃ k, ↑k * π = x) ∧ Even n ∨ (∃ k, ↑k * (2 * π) = x) ∧ Odd n ∨ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd n"
},
{
"line": "skip",
"before_state": "case mp.inl\nn : ℕ\nx : ℝ\nh₀ : n ≠ 0\nh : (∃ n, ↑n * (2 * π) = x) ∨ (∃ k, π + ↑k * (2 * π) = x) ∧ Even n\nhsinx : sin x = 0\n⊢ (∃ k, ↑k * π = x) ∧ Even n ∨ (∃ k, ↑k * (2 * π) = x) ∧ Odd n ∨ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd n",
"after_state": "case mp.inl\nn : ℕ\nx : ℝ\nh₀ : n ≠ 0\nh : (∃ n, ↑n * (2 * π) = x) ∨ (∃ k, π + ↑k * (2 * π) = x) ∧ Even n\nhsinx : sin x = 0\n⊢ (∃ k, ↑k * π = x) ∧ Even n ∨ (∃ k, ↑k * (2 * π) = x) ∧ Odd n ∨ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd n"
},
{
"line": "rcases h with ⟨k, rfl⟩ | ⟨⟨k, rfl⟩, hn⟩",
"before_state": "case mp.inl\nn : ℕ\nx : ℝ\nh₀ : n ≠ 0\nh : (∃ n, ↑n * (2 * π) = x) ∨ (∃ k, π + ↑k * (2 * π) = x) ∧ Even n\nhsinx : sin x = 0\n⊢ (∃ k, ↑k * π = x) ∧ Even n ∨ (∃ k, ↑k * (2 * π) = x) ∧ Odd n ∨ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd n",
"after_state": "case mp.inl.inl.intro\nn : ℕ\nh₀ : n ≠ 0\nk : ℤ\nhsinx : sin (↑k * (2 * π)) = 0\n⊢ (∃ k_1, ↑k_1 * π = ↑k * (2 * π)) ∧ Even n ∨\n (∃ k_1, ↑k_1 * (2 * π) = ↑k * (2 * π)) ∧ Odd n ∨ (∃ k_1, -(π / 2) + ↑k_1 * (2 * π) = ↑k * (2 * π)) ∧ Odd n\n---\ncase mp.inl.inr.intro.intro\nn : ℕ\nh₀ : n ≠ 0\nhn : Even n\nk : ℤ\nhsinx : sin (π + ↑k * (2 * π)) = 0\n⊢ (∃ k_1, ↑k_1 * π = π + ↑k * (2 * π)) ∧ Even n ∨\n (∃ k_1, ↑k_1 * (2 * π) = π + ↑k * (2 * π)) ∧ Odd n ∨ (∃ k_1, -(π / 2) + ↑k_1 * (2 * π) = π + ↑k * (2 * π)) ∧ Odd n"
},
{
"line": "cases n.even_or_odd with\n| inl hn => refine .inl ⟨⟨k * 2, ?_⟩, hn⟩; simp [mul_assoc]\n| inr hn => exact .inr <| .inl ⟨⟨_, rfl⟩, hn⟩",
"before_state": "case mp.inl.inl.intro\nn : ℕ\nh₀ : n ≠ 0\nk : ℤ\nhsinx : sin (↑k * (2 * π)) = 0\n⊢ (∃ k_1, ↑k_1 * π = ↑k * (2 * π)) ∧ Even n ∨\n (∃ k_1, ↑k_1 * (2 * π) = ↑k * (2 * π)) ∧ Odd n ∨ (∃ k_1, -(π / 2) + ↑k_1 * (2 * π) = ↑k * (2 * π)) ∧ Odd n",
"after_state": "No Goals!"
},
{
"line": "cases n.even_or_odd with\n| inl hn => refine .inl ⟨⟨k * 2, ?_⟩, hn⟩; simp [mul_assoc]\n| inr hn => exact .inr <| .inl ⟨⟨_, rfl⟩, hn⟩",
"before_state": "case mp.inl.inl.intro\nn : ℕ\nh₀ : n ≠ 0\nk : ℤ\nhsinx : sin (↑k * (2 * π)) = 0\nx✝ : Even n ∨ Odd n\n⊢ (∃ k_1, ↑k_1 * π = ↑k * (2 * π)) ∧ Even n ∨\n (∃ k_1, ↑k_1 * (2 * π) = ↑k * (2 * π)) ∧ Odd n ∨ (∃ k_1, -(π / 2) + ↑k_1 * (2 * π) = ↑k * (2 * π)) ∧ Odd n",
"after_state": "case mp.inl.inl.intro\nn : ℕ\nh₀ : n ≠ 0\nk : ℤ\nhsinx : sin (↑k * (2 * π)) = 0\nx✝ : Even n ∨ Odd n\n⊢ (∃ k_1, ↑k_1 * π = ↑k * (2 * π)) ∧ Even n ∨\n (∃ k_1, ↑k_1 * (2 * π) = ↑k * (2 * π)) ∧ Odd n ∨ (∃ k_1, -(π / 2) + ↑k_1 * (2 * π) = ↑k * (2 * π)) ∧ Odd n"
},
{
"line": "refine .inl ⟨⟨k * 2, ?_⟩, hn⟩",
"before_state": "case mp.inl.inl.intro.inl\nn : ℕ\nh₀ : n ≠ 0\nk : ℤ\nhsinx : sin (↑k * (2 * π)) = 0\nhn : Even n\n⊢ (∃ k_1, ↑k_1 * π = ↑k * (2 * π)) ∧ Even n ∨\n (∃ k_1, ↑k_1 * (2 * π) = ↑k * (2 * π)) ∧ Odd n ∨ (∃ k_1, -(π / 2) + ↑k_1 * (2 * π) = ↑k * (2 * π)) ∧ Odd n",
"after_state": "case mp.inl.inl.intro.inl\nn : ℕ\nh₀ : n ≠ 0\nk : ℤ\nhsinx : sin (↑k * (2 * π)) = 0\nhn : Even n\n⊢ ↑(k * 2) * π = ↑k * (2 * π)"
},
{
"line": "simp [mul_assoc]",
"before_state": "case mp.inl.inl.intro.inl\nn : ℕ\nh₀ : n ≠ 0\nk : ℤ\nhsinx : sin (↑k * (2 * π)) = 0\nhn : Even n\n⊢ ↑(k * 2) * π = ↑k * (2 * π)",
"after_state": "No Goals!"
},
{
"line": "exact .inr <| .inl ⟨⟨_, rfl⟩, hn⟩",
"before_state": "case mp.inl.inl.intro.inr\nn : ℕ\nh₀ : n ≠ 0\nk : ℤ\nhsinx : sin (↑k * (2 * π)) = 0\nhn : Odd n\n⊢ (∃ k_1, ↑k_1 * π = ↑k * (2 * π)) ∧ Even n ∨\n (∃ k_1, ↑k_1 * (2 * π) = ↑k * (2 * π)) ∧ Odd n ∨ (∃ k_1, -(π / 2) + ↑k_1 * (2 * π) = ↑k * (2 * π)) ∧ Odd n",
"after_state": "No Goals!"
},
{
"line": "exact .inl ⟨⟨2 * k + 1, by push_cast; ring⟩, hn⟩",
"before_state": "case mp.inl.inr.intro.intro\nn : ℕ\nh₀ : n ≠ 0\nhn : Even n\nk : ℤ\nhsinx : sin (π + ↑k * (2 * π)) = 0\n⊢ (∃ k_1, ↑k_1 * π = π + ↑k * (2 * π)) ∧ Even n ∨\n (∃ k_1, ↑k_1 * (2 * π) = π + ↑k * (2 * π)) ∧ Odd n ∨ (∃ k_1, -(π / 2) + ↑k_1 * (2 * π) = π + ↑k * (2 * π)) ∧ Odd n",
"after_state": "No Goals!"
},
{
"line": "push_cast",
"before_state": "n : ℕ\nh₀ : n ≠ 0\nhn : Even n\nk : ℤ\nhsinx : sin (π + ↑k * (2 * π)) = 0\n⊢ ↑(2 * k + 1) * π = π + ↑k * (2 * π)",
"after_state": "n : ℕ\nh₀ : n ≠ 0\nhn : Even n\nk : ℤ\nhsinx : sin (π + ↑k * (2 * π)) = 0\n⊢ (2 * ↑k + 1) * π = π + ↑k * (2 * π)"
},
{
"line": "ring",
"before_state": "n : ℕ\nh₀ : n ≠ 0\nhn : Even n\nk : ℤ\nhsinx : sin (π + ↑k * (2 * π)) = 0\n⊢ (2 * ↑k + 1) * π = π + ↑k * (2 * π)",
"after_state": "No Goals!"
},
{
"line": "first\n| ring1\n|\n try_this ring_nf\"\\n\\nThe `ring` tactic failed to close the goal. Use `ring_nf` to obtain a normal form.\n \\nNote that `ring` works primarily in *commutative* rings. \\\n If you have a noncommutative ring, abelian group or module, consider using \\\n `noncomm_ring`, `abel` or `module` instead.\"",
"before_state": "n : ℕ\nh₀ : n ≠ 0\nhn : Even n\nk : ℤ\nhsinx : sin (π + ↑k * (2 * π)) = 0\n⊢ (2 * ↑k + 1) * π = π + ↑k * (2 * π)",
"after_state": "No Goals!"
},
{
"line": "ring1",
"before_state": "n : ℕ\nh₀ : n ≠ 0\nhn : Even n\nk : ℤ\nhsinx : sin (π + ↑k * (2 * π)) = 0\n⊢ (2 * ↑k + 1) * π = π + ↑k * (2 * π)",
"after_state": "No Goals!"
},
{
"line": "rcases eq_or_ne (cos x) 0 with hcosx | hcosx",
"before_state": "case mp.inr\nn : ℕ\nx : ℝ\nh₀ : n ≠ 0\nh : cos x ^ n - sin x ^ n = 1\nhsinx : sin x ≠ 0\n⊢ (∃ k, ↑k * π = x) ∧ Even n ∨ (∃ k, ↑k * (2 * π) = x) ∧ Odd n ∨ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd n",
"after_state": "case mp.inr.inl\nn : ℕ\nx : ℝ\nh₀ : n ≠ 0\nh : cos x ^ n - sin x ^ n = 1\nhsinx : sin x ≠ 0\nhcosx : cos x = 0\n⊢ (∃ k, ↑k * π = x) ∧ Even n ∨ (∃ k, ↑k * (2 * π) = x) ∧ Odd n ∨ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd n\n---\ncase mp.inr.inr\nn : ℕ\nx : ℝ\nh₀ : n ≠ 0\nh : cos x ^ n - sin x ^ n = 1\nhsinx : sin x ≠ 0\nhcosx : cos x ≠ 0\n⊢ (∃ k, ↑k * π = x) ∧ Even n ∨ (∃ k, ↑k * (2 * π) = x) ∧ Odd n ∨ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd n"
},
{
"line": "right",
"before_state": "case mp.inr.inl\nn : ℕ\nx : ℝ\nh₀ : n ≠ 0\nh : cos x ^ n - sin x ^ n = 1\nhsinx : sin x ≠ 0\nhcosx : cos x = 0\n⊢ (∃ k, ↑k * π = x) ∧ Even n ∨ (∃ k, ↑k * (2 * π) = x) ∧ Odd n ∨ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd n",
"after_state": "case mp.inr.inl.h\nn : ℕ\nx : ℝ\nh₀ : n ≠ 0\nh : cos x ^ n - sin x ^ n = 1\nhsinx : sin x ≠ 0\nhcosx : cos x = 0\n⊢ (∃ k, ↑k * (2 * π) = x) ∧ Odd n ∨ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd n"
},
{
"line": "right",
"before_state": "case mp.inr.inl.h\nn : ℕ\nx : ℝ\nh₀ : n ≠ 0\nh : cos x ^ n - sin x ^ n = 1\nhsinx : sin x ≠ 0\nhcosx : cos x = 0\n⊢ (∃ k, ↑k * (2 * π) = x) ∧ Odd n ∨ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd n",
"after_state": "case mp.inr.inl.h.h\nn : ℕ\nx : ℝ\nh₀ : n ≠ 0\nh : cos x ^ n - sin x ^ n = 1\nhsinx : sin x ≠ 0\nhcosx : cos x = 0\n⊢ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd n"
},
{
"line": "rw [hcosx] at h",
"before_state": "case mp.inr.inl.h.h\nn : ℕ\nx : ℝ\nh₀ : n ≠ 0\nh : cos x ^ n - sin x ^ n = 1\nhsinx : sin x ≠ 0\nhcosx : cos x = 0\n⊢ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd n",
"after_state": "case mp.inr.inl.h.h\nn : ℕ\nx : ℝ\nh₀ : n ≠ 0\nh : 0 ^ n - sin x ^ n = 1\nhsinx : sin x ≠ 0\nhcosx : cos x = 0\n⊢ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd n"
},
{
"line": "rewrite [hcosx] at h",
"before_state": "case mp.inr.inl.h.h\nn : ℕ\nx : ℝ\nh₀ : n ≠ 0\nh : cos x ^ n - sin x ^ n = 1\nhsinx : sin x ≠ 0\nhcosx : cos x = 0\n⊢ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd n",
"after_state": "case mp.inr.inl.h.h\nn : ℕ\nx : ℝ\nh₀ : n ≠ 0\nh : 0 ^ n - sin x ^ n = 1\nhsinx : sin x ≠ 0\nhcosx : cos x = 0\n⊢ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd n"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "case mp.inr.inl.h.h\nn : ℕ\nx : ℝ\nh₀ : n ≠ 0\nh : 0 ^ n - sin x ^ n = 1\nhsinx : sin x ≠ 0\nhcosx : cos x = 0\n⊢ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd n",
"after_state": "case mp.inr.inl.h.h\nn : ℕ\nx : ℝ\nh₀ : n ≠ 0\nh : 0 ^ n - sin x ^ n = 1\nhsinx : sin x ≠ 0\nhcosx : cos x = 0\n⊢ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd n"
},
{
"line": "try (with_reducible rfl)",
"before_state": "case mp.inr.inl.h.h\nn : ℕ\nx : ℝ\nh₀ : n ≠ 0\nh : 0 ^ n - sin x ^ n = 1\nhsinx : sin x ≠ 0\nhcosx : cos x = 0\n⊢ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd n",
"after_state": "case mp.inr.inl.h.h\nn : ℕ\nx : ℝ\nh₀ : n ≠ 0\nh : 0 ^ n - sin x ^ n = 1\nhsinx : sin x ≠ 0\nhcosx : cos x = 0\n⊢ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd n"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "case mp.inr.inl.h.h\nn : ℕ\nx : ℝ\nh₀ : n ≠ 0\nh : 0 ^ n - sin x ^ n = 1\nhsinx : sin x ≠ 0\nhcosx : cos x = 0\n⊢ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd n",
"after_state": "case mp.inr.inl.h.h\nn : ℕ\nx : ℝ\nh₀ : n ≠ 0\nh : 0 ^ n - sin x ^ n = 1\nhsinx : sin x ≠ 0\nhcosx : cos x = 0\n⊢ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd n"
},
{
"line": "with_reducible rfl",
"before_state": "case mp.inr.inl.h.h\nn : ℕ\nx : ℝ\nh₀ : n ≠ 0\nh : 0 ^ n - sin x ^ n = 1\nhsinx : sin x ≠ 0\nhcosx : cos x = 0\n⊢ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd n",
"after_state": "case mp.inr.inl.h.h\nn : ℕ\nx : ℝ\nh₀ : n ≠ 0\nh : 0 ^ n - sin x ^ n = 1\nhsinx : sin x ≠ 0\nhcosx : cos x = 0\n⊢ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd n"
},
{
"line": "rfl",
"before_state": "case mp.inr.inl.h.h\nn : ℕ\nx : ℝ\nh₀ : n ≠ 0\nh : 0 ^ n - sin x ^ n = 1\nhsinx : sin x ≠ 0\nhcosx : cos x = 0\n⊢ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd n",
"after_state": "case mp.inr.inl.h.h\nn : ℕ\nx : ℝ\nh₀ : n ≠ 0\nh : 0 ^ n - sin x ^ n = 1\nhsinx : sin x ≠ 0\nhcosx : cos x = 0\n⊢ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd n"
},
{
"line": "apply_rfl",
"before_state": "case mp.inr.inl.h.h\nn : ℕ\nx : ℝ\nh₀ : n ≠ 0\nh : 0 ^ n - sin x ^ n = 1\nhsinx : sin x ≠ 0\nhcosx : cos x = 0\n⊢ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd n",
"after_state": "case mp.inr.inl.h.h\nn : ℕ\nx : ℝ\nh₀ : n ≠ 0\nh : 0 ^ n - sin x ^ n = 1\nhsinx : sin x ≠ 0\nhcosx : cos x = 0\n⊢ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd n"
},
{
"line": "skip",
"before_state": "case mp.inr.inl.h.h\nn : ℕ\nx : ℝ\nh₀ : n ≠ 0\nh : 0 ^ n - sin x ^ n = 1\nhsinx : sin x ≠ 0\nhcosx : cos x = 0\n⊢ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd n",
"after_state": "case mp.inr.inl.h.h\nn : ℕ\nx : ℝ\nh₀ : n ≠ 0\nh : 0 ^ n - sin x ^ n = 1\nhsinx : sin x ≠ 0\nhcosx : cos x = 0\n⊢ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd n"
},
{
"line": "rw [zero_pow h₀] at h",
"before_state": "case mp.inr.inl.h.h\nn : ℕ\nx : ℝ\nh₀ : n ≠ 0\nh : 0 ^ n - sin x ^ n = 1\nhsinx : sin x ≠ 0\nhcosx : cos x = 0\n⊢ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd n",
"after_state": "case mp.inr.inl.h.h\nn : ℕ\nx : ℝ\nh₀ : n ≠ 0\nh : 0 - sin x ^ n = 1\nhsinx : sin x ≠ 0\nhcosx : cos x = 0\n⊢ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd n"
},
{
"line": "rewrite [zero_pow h₀] at h",
"before_state": "case mp.inr.inl.h.h\nn : ℕ\nx : ℝ\nh₀ : n ≠ 0\nh : 0 ^ n - sin x ^ n = 1\nhsinx : sin x ≠ 0\nhcosx : cos x = 0\n⊢ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd n",
"after_state": "case mp.inr.inl.h.h\nn : ℕ\nx : ℝ\nh₀ : n ≠ 0\nh : 0 - sin x ^ n = 1\nhsinx : sin x ≠ 0\nhcosx : cos x = 0\n⊢ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd n"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "case mp.inr.inl.h.h\nn : ℕ\nx : ℝ\nh₀ : n ≠ 0\nh : 0 - sin x ^ n = 1\nhsinx : sin x ≠ 0\nhcosx : cos x = 0\n⊢ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd n",
"after_state": "case mp.inr.inl.h.h\nn : ℕ\nx : ℝ\nh₀ : n ≠ 0\nh : 0 - sin x ^ n = 1\nhsinx : sin x ≠ 0\nhcosx : cos x = 0\n⊢ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd n"
},
{
"line": "try (with_reducible rfl)",
"before_state": "case mp.inr.inl.h.h\nn : ℕ\nx : ℝ\nh₀ : n ≠ 0\nh : 0 - sin x ^ n = 1\nhsinx : sin x ≠ 0\nhcosx : cos x = 0\n⊢ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd n",
"after_state": "case mp.inr.inl.h.h\nn : ℕ\nx : ℝ\nh₀ : n ≠ 0\nh : 0 - sin x ^ n = 1\nhsinx : sin x ≠ 0\nhcosx : cos x = 0\n⊢ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd n"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "case mp.inr.inl.h.h\nn : ℕ\nx : ℝ\nh₀ : n ≠ 0\nh : 0 - sin x ^ n = 1\nhsinx : sin x ≠ 0\nhcosx : cos x = 0\n⊢ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd n",
"after_state": "case mp.inr.inl.h.h\nn : ℕ\nx : ℝ\nh₀ : n ≠ 0\nh : 0 - sin x ^ n = 1\nhsinx : sin x ≠ 0\nhcosx : cos x = 0\n⊢ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd n"
},
{
"line": "with_reducible rfl",
"before_state": "case mp.inr.inl.h.h\nn : ℕ\nx : ℝ\nh₀ : n ≠ 0\nh : 0 - sin x ^ n = 1\nhsinx : sin x ≠ 0\nhcosx : cos x = 0\n⊢ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd n",
"after_state": "case mp.inr.inl.h.h\nn : ℕ\nx : ℝ\nh₀ : n ≠ 0\nh : 0 - sin x ^ n = 1\nhsinx : sin x ≠ 0\nhcosx : cos x = 0\n⊢ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd n"
},
{
"line": "rfl",
"before_state": "case mp.inr.inl.h.h\nn : ℕ\nx : ℝ\nh₀ : n ≠ 0\nh : 0 - sin x ^ n = 1\nhsinx : sin x ≠ 0\nhcosx : cos x = 0\n⊢ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd n",
"after_state": "case mp.inr.inl.h.h\nn : ℕ\nx : ℝ\nh₀ : n ≠ 0\nh : 0 - sin x ^ n = 1\nhsinx : sin x ≠ 0\nhcosx : cos x = 0\n⊢ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd n"
},
{
"line": "apply_rfl",
"before_state": "case mp.inr.inl.h.h\nn : ℕ\nx : ℝ\nh₀ : n ≠ 0\nh : 0 - sin x ^ n = 1\nhsinx : sin x ≠ 0\nhcosx : cos x = 0\n⊢ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd n",
"after_state": "case mp.inr.inl.h.h\nn : ℕ\nx : ℝ\nh₀ : n ≠ 0\nh : 0 - sin x ^ n = 1\nhsinx : sin x ≠ 0\nhcosx : cos x = 0\n⊢ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd n"
},
{
"line": "skip",
"before_state": "case mp.inr.inl.h.h\nn : ℕ\nx : ℝ\nh₀ : n ≠ 0\nh : 0 - sin x ^ n = 1\nhsinx : sin x ≠ 0\nhcosx : cos x = 0\n⊢ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd n",
"after_state": "case mp.inr.inl.h.h\nn : ℕ\nx : ℝ\nh₀ : n ≠ 0\nh : 0 - sin x ^ n = 1\nhsinx : sin x ≠ 0\nhcosx : cos x = 0\n⊢ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd n"
},
{
"line": "rw [zero_sub] at h",
"before_state": "case mp.inr.inl.h.h\nn : ℕ\nx : ℝ\nh₀ : n ≠ 0\nh : 0 - sin x ^ n = 1\nhsinx : sin x ≠ 0\nhcosx : cos x = 0\n⊢ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd n",
"after_state": "case mp.inr.inl.h.h\nn : ℕ\nx : ℝ\nh₀ : n ≠ 0\nh : -sin x ^ n = 1\nhsinx : sin x ≠ 0\nhcosx : cos x = 0\n⊢ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd n"
},
{
"line": "rewrite [zero_sub] at h",
"before_state": "case mp.inr.inl.h.h\nn : ℕ\nx : ℝ\nh₀ : n ≠ 0\nh : 0 - sin x ^ n = 1\nhsinx : sin x ≠ 0\nhcosx : cos x = 0\n⊢ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd n",
"after_state": "case mp.inr.inl.h.h\nn : ℕ\nx : ℝ\nh₀ : n ≠ 0\nh : -sin x ^ n = 1\nhsinx : sin x ≠ 0\nhcosx : cos x = 0\n⊢ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd n"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "case mp.inr.inl.h.h\nn : ℕ\nx : ℝ\nh₀ : n ≠ 0\nh : -sin x ^ n = 1\nhsinx : sin x ≠ 0\nhcosx : cos x = 0\n⊢ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd n",
"after_state": "case mp.inr.inl.h.h\nn : ℕ\nx : ℝ\nh₀ : n ≠ 0\nh : -sin x ^ n = 1\nhsinx : sin x ≠ 0\nhcosx : cos x = 0\n⊢ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd n"
},
{
"line": "try (with_reducible rfl)",
"before_state": "case mp.inr.inl.h.h\nn : ℕ\nx : ℝ\nh₀ : n ≠ 0\nh : -sin x ^ n = 1\nhsinx : sin x ≠ 0\nhcosx : cos x = 0\n⊢ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd n",
"after_state": "case mp.inr.inl.h.h\nn : ℕ\nx : ℝ\nh₀ : n ≠ 0\nh : -sin x ^ n = 1\nhsinx : sin x ≠ 0\nhcosx : cos x = 0\n⊢ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd n"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "case mp.inr.inl.h.h\nn : ℕ\nx : ℝ\nh₀ : n ≠ 0\nh : -sin x ^ n = 1\nhsinx : sin x ≠ 0\nhcosx : cos x = 0\n⊢ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd n",
"after_state": "case mp.inr.inl.h.h\nn : ℕ\nx : ℝ\nh₀ : n ≠ 0\nh : -sin x ^ n = 1\nhsinx : sin x ≠ 0\nhcosx : cos x = 0\n⊢ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd n"
},
{
"line": "with_reducible rfl",
"before_state": "case mp.inr.inl.h.h\nn : ℕ\nx : ℝ\nh₀ : n ≠ 0\nh : -sin x ^ n = 1\nhsinx : sin x ≠ 0\nhcosx : cos x = 0\n⊢ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd n",
"after_state": "case mp.inr.inl.h.h\nn : ℕ\nx : ℝ\nh₀ : n ≠ 0\nh : -sin x ^ n = 1\nhsinx : sin x ≠ 0\nhcosx : cos x = 0\n⊢ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd n"
},
{
"line": "rfl",
"before_state": "case mp.inr.inl.h.h\nn : ℕ\nx : ℝ\nh₀ : n ≠ 0\nh : -sin x ^ n = 1\nhsinx : sin x ≠ 0\nhcosx : cos x = 0\n⊢ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd n",
"after_state": "case mp.inr.inl.h.h\nn : ℕ\nx : ℝ\nh₀ : n ≠ 0\nh : -sin x ^ n = 1\nhsinx : sin x ≠ 0\nhcosx : cos x = 0\n⊢ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd n"
},
{
"line": "apply_rfl",
"before_state": "case mp.inr.inl.h.h\nn : ℕ\nx : ℝ\nh₀ : n ≠ 0\nh : -sin x ^ n = 1\nhsinx : sin x ≠ 0\nhcosx : cos x = 0\n⊢ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd n",
"after_state": "case mp.inr.inl.h.h\nn : ℕ\nx : ℝ\nh₀ : n ≠ 0\nh : -sin x ^ n = 1\nhsinx : sin x ≠ 0\nhcosx : cos x = 0\n⊢ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd n"
},
{
"line": "skip",
"before_state": "case mp.inr.inl.h.h\nn : ℕ\nx : ℝ\nh₀ : n ≠ 0\nh : -sin x ^ n = 1\nhsinx : sin x ≠ 0\nhcosx : cos x = 0\n⊢ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd n",
"after_state": "case mp.inr.inl.h.h\nn : ℕ\nx : ℝ\nh₀ : n ≠ 0\nh : -sin x ^ n = 1\nhsinx : sin x ≠ 0\nhcosx : cos x = 0\n⊢ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd n"
},
{
"line": "rw [← neg_inj] at h",
"before_state": "case mp.inr.inl.h.h\nn : ℕ\nx : ℝ\nh₀ : n ≠ 0\nh : -sin x ^ n = 1\nhsinx : sin x ≠ 0\nhcosx : cos x = 0\n⊢ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd n",
"after_state": "case mp.inr.inl.h.h\nn : ℕ\nx : ℝ\nh₀ : n ≠ 0\nh : - -sin x ^ n = -1\nhsinx : sin x ≠ 0\nhcosx : cos x = 0\n⊢ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd n"
},
{
"line": "rewrite [← neg_inj] at h",
"before_state": "case mp.inr.inl.h.h\nn : ℕ\nx : ℝ\nh₀ : n ≠ 0\nh : -sin x ^ n = 1\nhsinx : sin x ≠ 0\nhcosx : cos x = 0\n⊢ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd n",
"after_state": "case mp.inr.inl.h.h\nn : ℕ\nx : ℝ\nh₀ : n ≠ 0\nh : - -sin x ^ n = -1\nhsinx : sin x ≠ 0\nhcosx : cos x = 0\n⊢ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd n"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "case mp.inr.inl.h.h\nn : ℕ\nx : ℝ\nh₀ : n ≠ 0\nh : - -sin x ^ n = -1\nhsinx : sin x ≠ 0\nhcosx : cos x = 0\n⊢ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd n",
"after_state": "case mp.inr.inl.h.h\nn : ℕ\nx : ℝ\nh₀ : n ≠ 0\nh : - -sin x ^ n = -1\nhsinx : sin x ≠ 0\nhcosx : cos x = 0\n⊢ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd n"
},
{
"line": "try (with_reducible rfl)",
"before_state": "case mp.inr.inl.h.h\nn : ℕ\nx : ℝ\nh₀ : n ≠ 0\nh : - -sin x ^ n = -1\nhsinx : sin x ≠ 0\nhcosx : cos x = 0\n⊢ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd n",
"after_state": "case mp.inr.inl.h.h\nn : ℕ\nx : ℝ\nh₀ : n ≠ 0\nh : - -sin x ^ n = -1\nhsinx : sin x ≠ 0\nhcosx : cos x = 0\n⊢ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd n"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "case mp.inr.inl.h.h\nn : ℕ\nx : ℝ\nh₀ : n ≠ 0\nh : - -sin x ^ n = -1\nhsinx : sin x ≠ 0\nhcosx : cos x = 0\n⊢ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd n",
"after_state": "case mp.inr.inl.h.h\nn : ℕ\nx : ℝ\nh₀ : n ≠ 0\nh : - -sin x ^ n = -1\nhsinx : sin x ≠ 0\nhcosx : cos x = 0\n⊢ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd n"
},
{
"line": "with_reducible rfl",
"before_state": "case mp.inr.inl.h.h\nn : ℕ\nx : ℝ\nh₀ : n ≠ 0\nh : - -sin x ^ n = -1\nhsinx : sin x ≠ 0\nhcosx : cos x = 0\n⊢ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd n",
"after_state": "case mp.inr.inl.h.h\nn : ℕ\nx : ℝ\nh₀ : n ≠ 0\nh : - -sin x ^ n = -1\nhsinx : sin x ≠ 0\nhcosx : cos x = 0\n⊢ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd n"
},
{
"line": "rfl",
"before_state": "case mp.inr.inl.h.h\nn : ℕ\nx : ℝ\nh₀ : n ≠ 0\nh : - -sin x ^ n = -1\nhsinx : sin x ≠ 0\nhcosx : cos x = 0\n⊢ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd n",
"after_state": "case mp.inr.inl.h.h\nn : ℕ\nx : ℝ\nh₀ : n ≠ 0\nh : - -sin x ^ n = -1\nhsinx : sin x ≠ 0\nhcosx : cos x = 0\n⊢ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd n"
},
{
"line": "apply_rfl",
"before_state": "case mp.inr.inl.h.h\nn : ℕ\nx : ℝ\nh₀ : n ≠ 0\nh : - -sin x ^ n = -1\nhsinx : sin x ≠ 0\nhcosx : cos x = 0\n⊢ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd n",
"after_state": "case mp.inr.inl.h.h\nn : ℕ\nx : ℝ\nh₀ : n ≠ 0\nh : - -sin x ^ n = -1\nhsinx : sin x ≠ 0\nhcosx : cos x = 0\n⊢ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd n"
},
{
"line": "skip",
"before_state": "case mp.inr.inl.h.h\nn : ℕ\nx : ℝ\nh₀ : n ≠ 0\nh : - -sin x ^ n = -1\nhsinx : sin x ≠ 0\nhcosx : cos x = 0\n⊢ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd n",
"after_state": "case mp.inr.inl.h.h\nn : ℕ\nx : ℝ\nh₀ : n ≠ 0\nh : - -sin x ^ n = -1\nhsinx : sin x ≠ 0\nhcosx : cos x = 0\n⊢ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd n"
},
{
"line": "rw [neg_neg] at h",
"before_state": "case mp.inr.inl.h.h\nn : ℕ\nx : ℝ\nh₀ : n ≠ 0\nh : - -sin x ^ n = -1\nhsinx : sin x ≠ 0\nhcosx : cos x = 0\n⊢ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd n",
"after_state": "case mp.inr.inl.h.h\nn : ℕ\nx : ℝ\nh₀ : n ≠ 0\nh : sin x ^ n = -1\nhsinx : sin x ≠ 0\nhcosx : cos x = 0\n⊢ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd n"
},
{
"line": "rewrite [neg_neg] at h",
"before_state": "case mp.inr.inl.h.h\nn : ℕ\nx : ℝ\nh₀ : n ≠ 0\nh : - -sin x ^ n = -1\nhsinx : sin x ≠ 0\nhcosx : cos x = 0\n⊢ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd n",
"after_state": "case mp.inr.inl.h.h\nn : ℕ\nx : ℝ\nh₀ : n ≠ 0\nh : sin x ^ n = -1\nhsinx : sin x ≠ 0\nhcosx : cos x = 0\n⊢ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd n"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "case mp.inr.inl.h.h\nn : ℕ\nx : ℝ\nh₀ : n ≠ 0\nh : sin x ^ n = -1\nhsinx : sin x ≠ 0\nhcosx : cos x = 0\n⊢ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd n",
"after_state": "case mp.inr.inl.h.h\nn : ℕ\nx : ℝ\nh₀ : n ≠ 0\nh : sin x ^ n = -1\nhsinx : sin x ≠ 0\nhcosx : cos x = 0\n⊢ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd n"
},
{
"line": "try (with_reducible rfl)",
"before_state": "case mp.inr.inl.h.h\nn : ℕ\nx : ℝ\nh₀ : n ≠ 0\nh : sin x ^ n = -1\nhsinx : sin x ≠ 0\nhcosx : cos x = 0\n⊢ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd n",
"after_state": "case mp.inr.inl.h.h\nn : ℕ\nx : ℝ\nh₀ : n ≠ 0\nh : sin x ^ n = -1\nhsinx : sin x ≠ 0\nhcosx : cos x = 0\n⊢ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd n"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "case mp.inr.inl.h.h\nn : ℕ\nx : ℝ\nh₀ : n ≠ 0\nh : sin x ^ n = -1\nhsinx : sin x ≠ 0\nhcosx : cos x = 0\n⊢ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd n",
"after_state": "case mp.inr.inl.h.h\nn : ℕ\nx : ℝ\nh₀ : n ≠ 0\nh : sin x ^ n = -1\nhsinx : sin x ≠ 0\nhcosx : cos x = 0\n⊢ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd n"
},
{
"line": "with_reducible rfl",
"before_state": "case mp.inr.inl.h.h\nn : ℕ\nx : ℝ\nh₀ : n ≠ 0\nh : sin x ^ n = -1\nhsinx : sin x ≠ 0\nhcosx : cos x = 0\n⊢ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd n",
"after_state": "case mp.inr.inl.h.h\nn : ℕ\nx : ℝ\nh₀ : n ≠ 0\nh : sin x ^ n = -1\nhsinx : sin x ≠ 0\nhcosx : cos x = 0\n⊢ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd n"
},
{
"line": "rfl",
"before_state": "case mp.inr.inl.h.h\nn : ℕ\nx : ℝ\nh₀ : n ≠ 0\nh : sin x ^ n = -1\nhsinx : sin x ≠ 0\nhcosx : cos x = 0\n⊢ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd n",
"after_state": "case mp.inr.inl.h.h\nn : ℕ\nx : ℝ\nh₀ : n ≠ 0\nh : sin x ^ n = -1\nhsinx : sin x ≠ 0\nhcosx : cos x = 0\n⊢ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd n"
},
{
"line": "apply_rfl",
"before_state": "case mp.inr.inl.h.h\nn : ℕ\nx : ℝ\nh₀ : n ≠ 0\nh : sin x ^ n = -1\nhsinx : sin x ≠ 0\nhcosx : cos x = 0\n⊢ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd n",
"after_state": "case mp.inr.inl.h.h\nn : ℕ\nx : ℝ\nh₀ : n ≠ 0\nh : sin x ^ n = -1\nhsinx : sin x ≠ 0\nhcosx : cos x = 0\n⊢ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd n"
},
{
"line": "skip",
"before_state": "case mp.inr.inl.h.h\nn : ℕ\nx : ℝ\nh₀ : n ≠ 0\nh : sin x ^ n = -1\nhsinx : sin x ≠ 0\nhcosx : cos x = 0\n⊢ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd n",
"after_state": "case mp.inr.inl.h.h\nn : ℕ\nx : ℝ\nh₀ : n ≠ 0\nh : sin x ^ n = -1\nhsinx : sin x ≠ 0\nhcosx : cos x = 0\n⊢ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd n"
},
{
"line": "rw [pow_eq_neg_one_iff] at h",
"before_state": "case mp.inr.inl.h.h\nn : ℕ\nx : ℝ\nh₀ : n ≠ 0\nh : sin x ^ n = -1\nhsinx : sin x ≠ 0\nhcosx : cos x = 0\n⊢ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd n",
"after_state": "case mp.inr.inl.h.h\nn : ℕ\nx : ℝ\nh₀ : n ≠ 0\nh : sin x = -1 ∧ Odd n\nhsinx : sin x ≠ 0\nhcosx : cos x = 0\n⊢ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd n"
},
{
"line": "rewrite [pow_eq_neg_one_iff] at h",
"before_state": "case mp.inr.inl.h.h\nn : ℕ\nx : ℝ\nh₀ : n ≠ 0\nh : sin x ^ n = -1\nhsinx : sin x ≠ 0\nhcosx : cos x = 0\n⊢ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd n",
"after_state": "case mp.inr.inl.h.h\nn : ℕ\nx : ℝ\nh₀ : n ≠ 0\nh : sin x = -1 ∧ Odd n\nhsinx : sin x ≠ 0\nhcosx : cos x = 0\n⊢ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd n"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "case mp.inr.inl.h.h\nn : ℕ\nx : ℝ\nh₀ : n ≠ 0\nh : sin x = -1 ∧ Odd n\nhsinx : sin x ≠ 0\nhcosx : cos x = 0\n⊢ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd n",
"after_state": "case mp.inr.inl.h.h\nn : ℕ\nx : ℝ\nh₀ : n ≠ 0\nh : sin x = -1 ∧ Odd n\nhsinx : sin x ≠ 0\nhcosx : cos x = 0\n⊢ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd n"
},
{
"line": "try (with_reducible rfl)",
"before_state": "case mp.inr.inl.h.h\nn : ℕ\nx : ℝ\nh₀ : n ≠ 0\nh : sin x = -1 ∧ Odd n\nhsinx : sin x ≠ 0\nhcosx : cos x = 0\n⊢ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd n",
"after_state": "case mp.inr.inl.h.h\nn : ℕ\nx : ℝ\nh₀ : n ≠ 0\nh : sin x = -1 ∧ Odd n\nhsinx : sin x ≠ 0\nhcosx : cos x = 0\n⊢ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd n"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "case mp.inr.inl.h.h\nn : ℕ\nx : ℝ\nh₀ : n ≠ 0\nh : sin x = -1 ∧ Odd n\nhsinx : sin x ≠ 0\nhcosx : cos x = 0\n⊢ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd n",
"after_state": "case mp.inr.inl.h.h\nn : ℕ\nx : ℝ\nh₀ : n ≠ 0\nh : sin x = -1 ∧ Odd n\nhsinx : sin x ≠ 0\nhcosx : cos x = 0\n⊢ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd n"
},
{
"line": "with_reducible rfl",
"before_state": "case mp.inr.inl.h.h\nn : ℕ\nx : ℝ\nh₀ : n ≠ 0\nh : sin x = -1 ∧ Odd n\nhsinx : sin x ≠ 0\nhcosx : cos x = 0\n⊢ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd n",
"after_state": "case mp.inr.inl.h.h\nn : ℕ\nx : ℝ\nh₀ : n ≠ 0\nh : sin x = -1 ∧ Odd n\nhsinx : sin x ≠ 0\nhcosx : cos x = 0\n⊢ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd n"
},
{
"line": "rfl",
"before_state": "case mp.inr.inl.h.h\nn : ℕ\nx : ℝ\nh₀ : n ≠ 0\nh : sin x = -1 ∧ Odd n\nhsinx : sin x ≠ 0\nhcosx : cos x = 0\n⊢ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd n",
"after_state": "case mp.inr.inl.h.h\nn : ℕ\nx : ℝ\nh₀ : n ≠ 0\nh : sin x = -1 ∧ Odd n\nhsinx : sin x ≠ 0\nhcosx : cos x = 0\n⊢ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd n"
},
{
"line": "apply_rfl",
"before_state": "case mp.inr.inl.h.h\nn : ℕ\nx : ℝ\nh₀ : n ≠ 0\nh : sin x = -1 ∧ Odd n\nhsinx : sin x ≠ 0\nhcosx : cos x = 0\n⊢ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd n",
"after_state": "case mp.inr.inl.h.h\nn : ℕ\nx : ℝ\nh₀ : n ≠ 0\nh : sin x = -1 ∧ Odd n\nhsinx : sin x ≠ 0\nhcosx : cos x = 0\n⊢ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd n"
},
{
"line": "skip",
"before_state": "case mp.inr.inl.h.h\nn : ℕ\nx : ℝ\nh₀ : n ≠ 0\nh : sin x = -1 ∧ Odd n\nhsinx : sin x ≠ 0\nhcosx : cos x = 0\n⊢ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd n",
"after_state": "case mp.inr.inl.h.h\nn : ℕ\nx : ℝ\nh₀ : n ≠ 0\nh : sin x = -1 ∧ Odd n\nhsinx : sin x ≠ 0\nhcosx : cos x = 0\n⊢ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd n"
},
{
"line": "rw [sin_eq_neg_one_iff] at h",
"before_state": "case mp.inr.inl.h.h\nn : ℕ\nx : ℝ\nh₀ : n ≠ 0\nh : sin x = -1 ∧ Odd n\nhsinx : sin x ≠ 0\nhcosx : cos x = 0\n⊢ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd n",
"after_state": "case mp.inr.inl.h.h\nn : ℕ\nx : ℝ\nh₀ : n ≠ 0\nh : (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd n\nhsinx : sin x ≠ 0\nhcosx : cos x = 0\n⊢ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd n"
},
{
"line": "rewrite [sin_eq_neg_one_iff] at h",
"before_state": "case mp.inr.inl.h.h\nn : ℕ\nx : ℝ\nh₀ : n ≠ 0\nh : sin x = -1 ∧ Odd n\nhsinx : sin x ≠ 0\nhcosx : cos x = 0\n⊢ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd n",
"after_state": "case mp.inr.inl.h.h\nn : ℕ\nx : ℝ\nh₀ : n ≠ 0\nh : (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd n\nhsinx : sin x ≠ 0\nhcosx : cos x = 0\n⊢ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd n"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "case mp.inr.inl.h.h\nn : ℕ\nx : ℝ\nh₀ : n ≠ 0\nh : (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd n\nhsinx : sin x ≠ 0\nhcosx : cos x = 0\n⊢ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd n",
"after_state": "case mp.inr.inl.h.h\nn : ℕ\nx : ℝ\nh₀ : n ≠ 0\nh : (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd n\nhsinx : sin x ≠ 0\nhcosx : cos x = 0\n⊢ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd n"
},
{
"line": "try (with_reducible rfl)",
"before_state": "case mp.inr.inl.h.h\nn : ℕ\nx : ℝ\nh₀ : n ≠ 0\nh : (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd n\nhsinx : sin x ≠ 0\nhcosx : cos x = 0\n⊢ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd n",
"after_state": "case mp.inr.inl.h.h\nn : ℕ\nx : ℝ\nh₀ : n ≠ 0\nh : (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd n\nhsinx : sin x ≠ 0\nhcosx : cos x = 0\n⊢ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd n"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "case mp.inr.inl.h.h\nn : ℕ\nx : ℝ\nh₀ : n ≠ 0\nh : (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd n\nhsinx : sin x ≠ 0\nhcosx : cos x = 0\n⊢ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd n",
"after_state": "case mp.inr.inl.h.h\nn : ℕ\nx : ℝ\nh₀ : n ≠ 0\nh : (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd n\nhsinx : sin x ≠ 0\nhcosx : cos x = 0\n⊢ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd n"
},
{
"line": "with_reducible rfl",
"before_state": "case mp.inr.inl.h.h\nn : ℕ\nx : ℝ\nh₀ : n ≠ 0\nh : (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd n\nhsinx : sin x ≠ 0\nhcosx : cos x = 0\n⊢ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd n",
"after_state": "case mp.inr.inl.h.h\nn : ℕ\nx : ℝ\nh₀ : n ≠ 0\nh : (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd n\nhsinx : sin x ≠ 0\nhcosx : cos x = 0\n⊢ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd n"
},
{
"line": "rfl",
"before_state": "case mp.inr.inl.h.h\nn : ℕ\nx : ℝ\nh₀ : n ≠ 0\nh : (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd n\nhsinx : sin x ≠ 0\nhcosx : cos x = 0\n⊢ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd n",
"after_state": "case mp.inr.inl.h.h\nn : ℕ\nx : ℝ\nh₀ : n ≠ 0\nh : (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd n\nhsinx : sin x ≠ 0\nhcosx : cos x = 0\n⊢ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd n"
},
{
"line": "apply_rfl",
"before_state": "case mp.inr.inl.h.h\nn : ℕ\nx : ℝ\nh₀ : n ≠ 0\nh : (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd n\nhsinx : sin x ≠ 0\nhcosx : cos x = 0\n⊢ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd n",
"after_state": "case mp.inr.inl.h.h\nn : ℕ\nx : ℝ\nh₀ : n ≠ 0\nh : (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd n\nhsinx : sin x ≠ 0\nhcosx : cos x = 0\n⊢ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd n"
},
{
"line": "skip",
"before_state": "case mp.inr.inl.h.h\nn : ℕ\nx : ℝ\nh₀ : n ≠ 0\nh : (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd n\nhsinx : sin x ≠ 0\nhcosx : cos x = 0\n⊢ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd n",
"after_state": "case mp.inr.inl.h.h\nn : ℕ\nx : ℝ\nh₀ : n ≠ 0\nh : (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd n\nhsinx : sin x ≠ 0\nhcosx : cos x = 0\n⊢ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd n"
},
{
"line": "simpa only [eq_comm] using h",
"before_state": "case mp.inr.inl.h.h\nn : ℕ\nx : ℝ\nh₀ : n ≠ 0\nh : (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd n\nhsinx : sin x ≠ 0\nhcosx : cos x = 0\n⊢ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd n",
"after_state": "No Goals!"
},
{
"line": "have hcos1 : |cos x| < 1 := by\n rw [abs_cos_eq_sqrt_one_sub_sin_sq]\n rw [sqrt_lt' one_pos]\n simp [sq_pos_of_ne_zero hsinx]",
"before_state": "case mp.inr.inr\nn : ℕ\nx : ℝ\nh₀ : n ≠ 0\nh : cos x ^ n - sin x ^ n = 1\nhsinx : sin x ≠ 0\nhcosx : cos x ≠ 0\n⊢ (∃ k, ↑k * π = x) ∧ Even n ∨ (∃ k, ↑k * (2 * π) = x) ∧ Odd n ∨ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd n",
"after_state": "case mp.inr.inr\nn : ℕ\nx : ℝ\nh₀ : n ≠ 0\nh : cos x ^ n - sin x ^ n = 1\nhsinx : sin x ≠ 0\nhcosx : cos x ≠ 0\nhcos1 : |cos x| < 1\n⊢ (∃ k, ↑k * π = x) ∧ Even n ∨ (∃ k, ↑k * (2 * π) = x) ∧ Odd n ∨ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd n"
},
{
"line": "focus\n refine\n no_implicit_lambda%\n (have hcos1 : |cos x| < 1 := ?body✝;\n ?_)\n case body✝ =>\n with_annotate_state\"by\"\n ( rw [abs_cos_eq_sqrt_one_sub_sin_sq]\n rw [sqrt_lt' one_pos]\n simp [sq_pos_of_ne_zero hsinx])",
"before_state": "case mp.inr.inr\nn : ℕ\nx : ℝ\nh₀ : n ≠ 0\nh : cos x ^ n - sin x ^ n = 1\nhsinx : sin x ≠ 0\nhcosx : cos x ≠ 0\n⊢ (∃ k, ↑k * π = x) ∧ Even n ∨ (∃ k, ↑k * (2 * π) = x) ∧ Odd n ∨ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd n",
"after_state": "case mp.inr.inr\nn : ℕ\nx : ℝ\nh₀ : n ≠ 0\nh : cos x ^ n - sin x ^ n = 1\nhsinx : sin x ≠ 0\nhcosx : cos x ≠ 0\nhcos1 : |cos x| < 1\n⊢ (∃ k, ↑k * π = x) ∧ Even n ∨ (∃ k, ↑k * (2 * π) = x) ∧ Odd n ∨ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd n"
},
{
"line": "refine\n no_implicit_lambda%\n (have hcos1 : |cos x| < 1 := ?body✝;\n ?_)",
"before_state": "case mp.inr.inr\nn : ℕ\nx : ℝ\nh₀ : n ≠ 0\nh : cos x ^ n - sin x ^ n = 1\nhsinx : sin x ≠ 0\nhcosx : cos x ≠ 0\n⊢ (∃ k, ↑k * π = x) ∧ Even n ∨ (∃ k, ↑k * (2 * π) = x) ∧ Odd n ∨ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd n",
"after_state": "case body\nn : ℕ\nx : ℝ\nh₀ : n ≠ 0\nh : cos x ^ n - sin x ^ n = 1\nhsinx : sin x ≠ 0\nhcosx : cos x ≠ 0\n⊢ |cos x| < 1\n---\ncase mp.inr.inr\nn : ℕ\nx : ℝ\nh₀ : n ≠ 0\nh : cos x ^ n - sin x ^ n = 1\nhsinx : sin x ≠ 0\nhcosx : cos x ≠ 0\nhcos1 : |cos x| < 1\n⊢ (∃ k, ↑k * π = x) ∧ Even n ∨ (∃ k, ↑k * (2 * π) = x) ∧ Odd n ∨ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd n"
},
{
"line": "case body✝ =>\n with_annotate_state\"by\"\n ( rw [abs_cos_eq_sqrt_one_sub_sin_sq]\n rw [sqrt_lt' one_pos]\n simp [sq_pos_of_ne_zero hsinx])",
"before_state": "case body\nn : ℕ\nx : ℝ\nh₀ : n ≠ 0\nh : cos x ^ n - sin x ^ n = 1\nhsinx : sin x ≠ 0\nhcosx : cos x ≠ 0\n⊢ |cos x| < 1\n---\ncase mp.inr.inr\nn : ℕ\nx : ℝ\nh₀ : n ≠ 0\nh : cos x ^ n - sin x ^ n = 1\nhsinx : sin x ≠ 0\nhcosx : cos x ≠ 0\nhcos1 : |cos x| < 1\n⊢ (∃ k, ↑k * π = x) ∧ Even n ∨ (∃ k, ↑k * (2 * π) = x) ∧ Odd n ∨ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd n",
"after_state": "case mp.inr.inr\nn : ℕ\nx : ℝ\nh₀ : n ≠ 0\nh : cos x ^ n - sin x ^ n = 1\nhsinx : sin x ≠ 0\nhcosx : cos x ≠ 0\nhcos1 : |cos x| < 1\n⊢ (∃ k, ↑k * π = x) ∧ Even n ∨ (∃ k, ↑k * (2 * π) = x) ∧ Odd n ∨ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd n"
},
{
"line": "with_annotate_state\"by\"\n ( rw [abs_cos_eq_sqrt_one_sub_sin_sq]\n rw [sqrt_lt' one_pos]\n simp [sq_pos_of_ne_zero hsinx])",
"before_state": "n : ℕ\nx : ℝ\nh₀ : n ≠ 0\nh : cos x ^ n - sin x ^ n = 1\nhsinx : sin x ≠ 0\nhcosx : cos x ≠ 0\n⊢ |cos x| < 1",
"after_state": "No Goals!"
},
{
"line": "rw [abs_cos_eq_sqrt_one_sub_sin_sq]",
"before_state": "n : ℕ\nx : ℝ\nh₀ : n ≠ 0\nh : cos x ^ n - sin x ^ n = 1\nhsinx : sin x ≠ 0\nhcosx : cos x ≠ 0\n⊢ |cos x| < 1",
"after_state": "n : ℕ\nx : ℝ\nh₀ : n ≠ 0\nh : cos x ^ n - sin x ^ n = 1\nhsinx : sin x ≠ 0\nhcosx : cos x ≠ 0\n⊢ √(1 - sin x ^ 2) < 1"
},
{
"line": "rewrite [abs_cos_eq_sqrt_one_sub_sin_sq]",
"before_state": "n : ℕ\nx : ℝ\nh₀ : n ≠ 0\nh : cos x ^ n - sin x ^ n = 1\nhsinx : sin x ≠ 0\nhcosx : cos x ≠ 0\n⊢ |cos x| < 1",
"after_state": "n : ℕ\nx : ℝ\nh₀ : n ≠ 0\nh : cos x ^ n - sin x ^ n = 1\nhsinx : sin x ≠ 0\nhcosx : cos x ≠ 0\n⊢ √(1 - sin x ^ 2) < 1"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "n : ℕ\nx : ℝ\nh₀ : n ≠ 0\nh : cos x ^ n - sin x ^ n = 1\nhsinx : sin x ≠ 0\nhcosx : cos x ≠ 0\n⊢ √(1 - sin x ^ 2) < 1",
"after_state": "n : ℕ\nx : ℝ\nh₀ : n ≠ 0\nh : cos x ^ n - sin x ^ n = 1\nhsinx : sin x ≠ 0\nhcosx : cos x ≠ 0\n⊢ √(1 - sin x ^ 2) < 1"
},
{
"line": "try (with_reducible rfl)",
"before_state": "n : ℕ\nx : ℝ\nh₀ : n ≠ 0\nh : cos x ^ n - sin x ^ n = 1\nhsinx : sin x ≠ 0\nhcosx : cos x ≠ 0\n⊢ √(1 - sin x ^ 2) < 1",
"after_state": "n : ℕ\nx : ℝ\nh₀ : n ≠ 0\nh : cos x ^ n - sin x ^ n = 1\nhsinx : sin x ≠ 0\nhcosx : cos x ≠ 0\n⊢ √(1 - sin x ^ 2) < 1"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "n : ℕ\nx : ℝ\nh₀ : n ≠ 0\nh : cos x ^ n - sin x ^ n = 1\nhsinx : sin x ≠ 0\nhcosx : cos x ≠ 0\n⊢ √(1 - sin x ^ 2) < 1",
"after_state": "n : ℕ\nx : ℝ\nh₀ : n ≠ 0\nh : cos x ^ n - sin x ^ n = 1\nhsinx : sin x ≠ 0\nhcosx : cos x ≠ 0\n⊢ √(1 - sin x ^ 2) < 1"
},
{
"line": "with_reducible rfl",
"before_state": "n : ℕ\nx : ℝ\nh₀ : n ≠ 0\nh : cos x ^ n - sin x ^ n = 1\nhsinx : sin x ≠ 0\nhcosx : cos x ≠ 0\n⊢ √(1 - sin x ^ 2) < 1",
"after_state": "n : ℕ\nx : ℝ\nh₀ : n ≠ 0\nh : cos x ^ n - sin x ^ n = 1\nhsinx : sin x ≠ 0\nhcosx : cos x ≠ 0\n⊢ √(1 - sin x ^ 2) < 1"
},
{
"line": "rfl",
"before_state": "n : ℕ\nx : ℝ\nh₀ : n ≠ 0\nh : cos x ^ n - sin x ^ n = 1\nhsinx : sin x ≠ 0\nhcosx : cos x ≠ 0\n⊢ √(1 - sin x ^ 2) < 1",
"after_state": "n : ℕ\nx : ℝ\nh₀ : n ≠ 0\nh : cos x ^ n - sin x ^ n = 1\nhsinx : sin x ≠ 0\nhcosx : cos x ≠ 0\n⊢ √(1 - sin x ^ 2) < 1"
},
{
"line": "apply_rfl",
"before_state": "n : ℕ\nx : ℝ\nh₀ : n ≠ 0\nh : cos x ^ n - sin x ^ n = 1\nhsinx : sin x ≠ 0\nhcosx : cos x ≠ 0\n⊢ √(1 - sin x ^ 2) < 1",
"after_state": "n : ℕ\nx : ℝ\nh₀ : n ≠ 0\nh : cos x ^ n - sin x ^ n = 1\nhsinx : sin x ≠ 0\nhcosx : cos x ≠ 0\n⊢ √(1 - sin x ^ 2) < 1"
},
{
"line": "skip",
"before_state": "n : ℕ\nx : ℝ\nh₀ : n ≠ 0\nh : cos x ^ n - sin x ^ n = 1\nhsinx : sin x ≠ 0\nhcosx : cos x ≠ 0\n⊢ √(1 - sin x ^ 2) < 1",
"after_state": "n : ℕ\nx : ℝ\nh₀ : n ≠ 0\nh : cos x ^ n - sin x ^ n = 1\nhsinx : sin x ≠ 0\nhcosx : cos x ≠ 0\n⊢ √(1 - sin x ^ 2) < 1"
},
{
"line": "rw [sqrt_lt' one_pos]",
"before_state": "n : ℕ\nx : ℝ\nh₀ : n ≠ 0\nh : cos x ^ n - sin x ^ n = 1\nhsinx : sin x ≠ 0\nhcosx : cos x ≠ 0\n⊢ √(1 - sin x ^ 2) < 1",
"after_state": "n : ℕ\nx : ℝ\nh₀ : n ≠ 0\nh : cos x ^ n - sin x ^ n = 1\nhsinx : sin x ≠ 0\nhcosx : cos x ≠ 0\n⊢ 1 - sin x ^ 2 < 1 ^ 2"
},
{
"line": "rewrite [sqrt_lt' one_pos]",
"before_state": "n : ℕ\nx : ℝ\nh₀ : n ≠ 0\nh : cos x ^ n - sin x ^ n = 1\nhsinx : sin x ≠ 0\nhcosx : cos x ≠ 0\n⊢ √(1 - sin x ^ 2) < 1",
"after_state": "n : ℕ\nx : ℝ\nh₀ : n ≠ 0\nh : cos x ^ n - sin x ^ n = 1\nhsinx : sin x ≠ 0\nhcosx : cos x ≠ 0\n⊢ 1 - sin x ^ 2 < 1 ^ 2"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "n : ℕ\nx : ℝ\nh₀ : n ≠ 0\nh : cos x ^ n - sin x ^ n = 1\nhsinx : sin x ≠ 0\nhcosx : cos x ≠ 0\n⊢ 1 - sin x ^ 2 < 1 ^ 2",
"after_state": "n : ℕ\nx : ℝ\nh₀ : n ≠ 0\nh : cos x ^ n - sin x ^ n = 1\nhsinx : sin x ≠ 0\nhcosx : cos x ≠ 0\n⊢ 1 - sin x ^ 2 < 1 ^ 2"
},
{
"line": "try (with_reducible rfl)",
"before_state": "n : ℕ\nx : ℝ\nh₀ : n ≠ 0\nh : cos x ^ n - sin x ^ n = 1\nhsinx : sin x ≠ 0\nhcosx : cos x ≠ 0\n⊢ 1 - sin x ^ 2 < 1 ^ 2",
"after_state": "n : ℕ\nx : ℝ\nh₀ : n ≠ 0\nh : cos x ^ n - sin x ^ n = 1\nhsinx : sin x ≠ 0\nhcosx : cos x ≠ 0\n⊢ 1 - sin x ^ 2 < 1 ^ 2"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "n : ℕ\nx : ℝ\nh₀ : n ≠ 0\nh : cos x ^ n - sin x ^ n = 1\nhsinx : sin x ≠ 0\nhcosx : cos x ≠ 0\n⊢ 1 - sin x ^ 2 < 1 ^ 2",
"after_state": "n : ℕ\nx : ℝ\nh₀ : n ≠ 0\nh : cos x ^ n - sin x ^ n = 1\nhsinx : sin x ≠ 0\nhcosx : cos x ≠ 0\n⊢ 1 - sin x ^ 2 < 1 ^ 2"
},
{
"line": "with_reducible rfl",
"before_state": "n : ℕ\nx : ℝ\nh₀ : n ≠ 0\nh : cos x ^ n - sin x ^ n = 1\nhsinx : sin x ≠ 0\nhcosx : cos x ≠ 0\n⊢ 1 - sin x ^ 2 < 1 ^ 2",
"after_state": "n : ℕ\nx : ℝ\nh₀ : n ≠ 0\nh : cos x ^ n - sin x ^ n = 1\nhsinx : sin x ≠ 0\nhcosx : cos x ≠ 0\n⊢ 1 - sin x ^ 2 < 1 ^ 2"
},
{
"line": "rfl",
"before_state": "n : ℕ\nx : ℝ\nh₀ : n ≠ 0\nh : cos x ^ n - sin x ^ n = 1\nhsinx : sin x ≠ 0\nhcosx : cos x ≠ 0\n⊢ 1 - sin x ^ 2 < 1 ^ 2",
"after_state": "n : ℕ\nx : ℝ\nh₀ : n ≠ 0\nh : cos x ^ n - sin x ^ n = 1\nhsinx : sin x ≠ 0\nhcosx : cos x ≠ 0\n⊢ 1 - sin x ^ 2 < 1 ^ 2"
},
{
"line": "apply_rfl",
"before_state": "n : ℕ\nx : ℝ\nh₀ : n ≠ 0\nh : cos x ^ n - sin x ^ n = 1\nhsinx : sin x ≠ 0\nhcosx : cos x ≠ 0\n⊢ 1 - sin x ^ 2 < 1 ^ 2",
"after_state": "n : ℕ\nx : ℝ\nh₀ : n ≠ 0\nh : cos x ^ n - sin x ^ n = 1\nhsinx : sin x ≠ 0\nhcosx : cos x ≠ 0\n⊢ 1 - sin x ^ 2 < 1 ^ 2"
},
{
"line": "skip",
"before_state": "n : ℕ\nx : ℝ\nh₀ : n ≠ 0\nh : cos x ^ n - sin x ^ n = 1\nhsinx : sin x ≠ 0\nhcosx : cos x ≠ 0\n⊢ 1 - sin x ^ 2 < 1 ^ 2",
"after_state": "n : ℕ\nx : ℝ\nh₀ : n ≠ 0\nh : cos x ^ n - sin x ^ n = 1\nhsinx : sin x ≠ 0\nhcosx : cos x ≠ 0\n⊢ 1 - sin x ^ 2 < 1 ^ 2"
},
{
"line": "simp [sq_pos_of_ne_zero hsinx]",
"before_state": "n : ℕ\nx : ℝ\nh₀ : n ≠ 0\nh : cos x ^ n - sin x ^ n = 1\nhsinx : sin x ≠ 0\nhcosx : cos x ≠ 0\n⊢ 1 - sin x ^ 2 < 1 ^ 2",
"after_state": "No Goals!"
},
{
"line": "have hsin1 : |sin x| < 1 := by\n rw [abs_sin_eq_sqrt_one_sub_cos_sq]\n rw [sqrt_lt' one_pos]\n simp [sq_pos_of_ne_zero hcosx]",
"before_state": "case mp.inr.inr\nn : ℕ\nx : ℝ\nh₀ : n ≠ 0\nh : cos x ^ n - sin x ^ n = 1\nhsinx : sin x ≠ 0\nhcosx : cos x ≠ 0\nhcos1 : |cos x| < 1\n⊢ (∃ k, ↑k * π = x) ∧ Even n ∨ (∃ k, ↑k * (2 * π) = x) ∧ Odd n ∨ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd n",
"after_state": "case mp.inr.inr\nn : ℕ\nx : ℝ\nh₀ : n ≠ 0\nh : cos x ^ n - sin x ^ n = 1\nhsinx : sin x ≠ 0\nhcosx : cos x ≠ 0\nhcos1 : |cos x| < 1\nhsin1 : |sin x| < 1\n⊢ (∃ k, ↑k * π = x) ∧ Even n ∨ (∃ k, ↑k * (2 * π) = x) ∧ Odd n ∨ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd n"
},
{
"line": "focus\n refine\n no_implicit_lambda%\n (have hsin1 : |sin x| < 1 := ?body✝;\n ?_)\n case body✝ =>\n with_annotate_state\"by\"\n ( rw [abs_sin_eq_sqrt_one_sub_cos_sq]\n rw [sqrt_lt' one_pos]\n simp [sq_pos_of_ne_zero hcosx])",
"before_state": "case mp.inr.inr\nn : ℕ\nx : ℝ\nh₀ : n ≠ 0\nh : cos x ^ n - sin x ^ n = 1\nhsinx : sin x ≠ 0\nhcosx : cos x ≠ 0\nhcos1 : |cos x| < 1\n⊢ (∃ k, ↑k * π = x) ∧ Even n ∨ (∃ k, ↑k * (2 * π) = x) ∧ Odd n ∨ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd n",
"after_state": "case mp.inr.inr\nn : ℕ\nx : ℝ\nh₀ : n ≠ 0\nh : cos x ^ n - sin x ^ n = 1\nhsinx : sin x ≠ 0\nhcosx : cos x ≠ 0\nhcos1 : |cos x| < 1\nhsin1 : |sin x| < 1\n⊢ (∃ k, ↑k * π = x) ∧ Even n ∨ (∃ k, ↑k * (2 * π) = x) ∧ Odd n ∨ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd n"
},
{
"line": "refine\n no_implicit_lambda%\n (have hsin1 : |sin x| < 1 := ?body✝;\n ?_)",
"before_state": "case mp.inr.inr\nn : ℕ\nx : ℝ\nh₀ : n ≠ 0\nh : cos x ^ n - sin x ^ n = 1\nhsinx : sin x ≠ 0\nhcosx : cos x ≠ 0\nhcos1 : |cos x| < 1\n⊢ (∃ k, ↑k * π = x) ∧ Even n ∨ (∃ k, ↑k * (2 * π) = x) ∧ Odd n ∨ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd n",
"after_state": "case body\nn : ℕ\nx : ℝ\nh₀ : n ≠ 0\nh : cos x ^ n - sin x ^ n = 1\nhsinx : sin x ≠ 0\nhcosx : cos x ≠ 0\nhcos1 : |cos x| < 1\n⊢ |sin x| < 1\n---\ncase mp.inr.inr\nn : ℕ\nx : ℝ\nh₀ : n ≠ 0\nh : cos x ^ n - sin x ^ n = 1\nhsinx : sin x ≠ 0\nhcosx : cos x ≠ 0\nhcos1 : |cos x| < 1\nhsin1 : |sin x| < 1\n⊢ (∃ k, ↑k * π = x) ∧ Even n ∨ (∃ k, ↑k * (2 * π) = x) ∧ Odd n ∨ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd n"
},
{
"line": "case body✝ =>\n with_annotate_state\"by\"\n ( rw [abs_sin_eq_sqrt_one_sub_cos_sq]\n rw [sqrt_lt' one_pos]\n simp [sq_pos_of_ne_zero hcosx])",
"before_state": "case body\nn : ℕ\nx : ℝ\nh₀ : n ≠ 0\nh : cos x ^ n - sin x ^ n = 1\nhsinx : sin x ≠ 0\nhcosx : cos x ≠ 0\nhcos1 : |cos x| < 1\n⊢ |sin x| < 1\n---\ncase mp.inr.inr\nn : ℕ\nx : ℝ\nh₀ : n ≠ 0\nh : cos x ^ n - sin x ^ n = 1\nhsinx : sin x ≠ 0\nhcosx : cos x ≠ 0\nhcos1 : |cos x| < 1\nhsin1 : |sin x| < 1\n⊢ (∃ k, ↑k * π = x) ∧ Even n ∨ (∃ k, ↑k * (2 * π) = x) ∧ Odd n ∨ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd n",
"after_state": "case mp.inr.inr\nn : ℕ\nx : ℝ\nh₀ : n ≠ 0\nh : cos x ^ n - sin x ^ n = 1\nhsinx : sin x ≠ 0\nhcosx : cos x ≠ 0\nhcos1 : |cos x| < 1\nhsin1 : |sin x| < 1\n⊢ (∃ k, ↑k * π = x) ∧ Even n ∨ (∃ k, ↑k * (2 * π) = x) ∧ Odd n ∨ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd n"
},
{
"line": "with_annotate_state\"by\"\n ( rw [abs_sin_eq_sqrt_one_sub_cos_sq]\n rw [sqrt_lt' one_pos]\n simp [sq_pos_of_ne_zero hcosx])",
"before_state": "n : ℕ\nx : ℝ\nh₀ : n ≠ 0\nh : cos x ^ n - sin x ^ n = 1\nhsinx : sin x ≠ 0\nhcosx : cos x ≠ 0\nhcos1 : |cos x| < 1\n⊢ |sin x| < 1",
"after_state": "No Goals!"
},
{
"line": "rw [abs_sin_eq_sqrt_one_sub_cos_sq]",
"before_state": "n : ℕ\nx : ℝ\nh₀ : n ≠ 0\nh : cos x ^ n - sin x ^ n = 1\nhsinx : sin x ≠ 0\nhcosx : cos x ≠ 0\nhcos1 : |cos x| < 1\n⊢ |sin x| < 1",
"after_state": "n : ℕ\nx : ℝ\nh₀ : n ≠ 0\nh : cos x ^ n - sin x ^ n = 1\nhsinx : sin x ≠ 0\nhcosx : cos x ≠ 0\nhcos1 : |cos x| < 1\n⊢ √(1 - cos x ^ 2) < 1"
},
{
"line": "rewrite [abs_sin_eq_sqrt_one_sub_cos_sq]",
"before_state": "n : ℕ\nx : ℝ\nh₀ : n ≠ 0\nh : cos x ^ n - sin x ^ n = 1\nhsinx : sin x ≠ 0\nhcosx : cos x ≠ 0\nhcos1 : |cos x| < 1\n⊢ |sin x| < 1",
"after_state": "n : ℕ\nx : ℝ\nh₀ : n ≠ 0\nh : cos x ^ n - sin x ^ n = 1\nhsinx : sin x ≠ 0\nhcosx : cos x ≠ 0\nhcos1 : |cos x| < 1\n⊢ √(1 - cos x ^ 2) < 1"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "n : ℕ\nx : ℝ\nh₀ : n ≠ 0\nh : cos x ^ n - sin x ^ n = 1\nhsinx : sin x ≠ 0\nhcosx : cos x ≠ 0\nhcos1 : |cos x| < 1\n⊢ √(1 - cos x ^ 2) < 1",
"after_state": "n : ℕ\nx : ℝ\nh₀ : n ≠ 0\nh : cos x ^ n - sin x ^ n = 1\nhsinx : sin x ≠ 0\nhcosx : cos x ≠ 0\nhcos1 : |cos x| < 1\n⊢ √(1 - cos x ^ 2) < 1"
},
{
"line": "try (with_reducible rfl)",
"before_state": "n : ℕ\nx : ℝ\nh₀ : n ≠ 0\nh : cos x ^ n - sin x ^ n = 1\nhsinx : sin x ≠ 0\nhcosx : cos x ≠ 0\nhcos1 : |cos x| < 1\n⊢ √(1 - cos x ^ 2) < 1",
"after_state": "n : ℕ\nx : ℝ\nh₀ : n ≠ 0\nh : cos x ^ n - sin x ^ n = 1\nhsinx : sin x ≠ 0\nhcosx : cos x ≠ 0\nhcos1 : |cos x| < 1\n⊢ √(1 - cos x ^ 2) < 1"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "n : ℕ\nx : ℝ\nh₀ : n ≠ 0\nh : cos x ^ n - sin x ^ n = 1\nhsinx : sin x ≠ 0\nhcosx : cos x ≠ 0\nhcos1 : |cos x| < 1\n⊢ √(1 - cos x ^ 2) < 1",
"after_state": "n : ℕ\nx : ℝ\nh₀ : n ≠ 0\nh : cos x ^ n - sin x ^ n = 1\nhsinx : sin x ≠ 0\nhcosx : cos x ≠ 0\nhcos1 : |cos x| < 1\n⊢ √(1 - cos x ^ 2) < 1"
},
{
"line": "with_reducible rfl",
"before_state": "n : ℕ\nx : ℝ\nh₀ : n ≠ 0\nh : cos x ^ n - sin x ^ n = 1\nhsinx : sin x ≠ 0\nhcosx : cos x ≠ 0\nhcos1 : |cos x| < 1\n⊢ √(1 - cos x ^ 2) < 1",
"after_state": "n : ℕ\nx : ℝ\nh₀ : n ≠ 0\nh : cos x ^ n - sin x ^ n = 1\nhsinx : sin x ≠ 0\nhcosx : cos x ≠ 0\nhcos1 : |cos x| < 1\n⊢ √(1 - cos x ^ 2) < 1"
},
{
"line": "rfl",
"before_state": "n : ℕ\nx : ℝ\nh₀ : n ≠ 0\nh : cos x ^ n - sin x ^ n = 1\nhsinx : sin x ≠ 0\nhcosx : cos x ≠ 0\nhcos1 : |cos x| < 1\n⊢ √(1 - cos x ^ 2) < 1",
"after_state": "n : ℕ\nx : ℝ\nh₀ : n ≠ 0\nh : cos x ^ n - sin x ^ n = 1\nhsinx : sin x ≠ 0\nhcosx : cos x ≠ 0\nhcos1 : |cos x| < 1\n⊢ √(1 - cos x ^ 2) < 1"
},
{
"line": "apply_rfl",
"before_state": "n : ℕ\nx : ℝ\nh₀ : n ≠ 0\nh : cos x ^ n - sin x ^ n = 1\nhsinx : sin x ≠ 0\nhcosx : cos x ≠ 0\nhcos1 : |cos x| < 1\n⊢ √(1 - cos x ^ 2) < 1",
"after_state": "n : ℕ\nx : ℝ\nh₀ : n ≠ 0\nh : cos x ^ n - sin x ^ n = 1\nhsinx : sin x ≠ 0\nhcosx : cos x ≠ 0\nhcos1 : |cos x| < 1\n⊢ √(1 - cos x ^ 2) < 1"
},
{
"line": "skip",
"before_state": "n : ℕ\nx : ℝ\nh₀ : n ≠ 0\nh : cos x ^ n - sin x ^ n = 1\nhsinx : sin x ≠ 0\nhcosx : cos x ≠ 0\nhcos1 : |cos x| < 1\n⊢ √(1 - cos x ^ 2) < 1",
"after_state": "n : ℕ\nx : ℝ\nh₀ : n ≠ 0\nh : cos x ^ n - sin x ^ n = 1\nhsinx : sin x ≠ 0\nhcosx : cos x ≠ 0\nhcos1 : |cos x| < 1\n⊢ √(1 - cos x ^ 2) < 1"
},
{
"line": "rw [sqrt_lt' one_pos]",
"before_state": "n : ℕ\nx : ℝ\nh₀ : n ≠ 0\nh : cos x ^ n - sin x ^ n = 1\nhsinx : sin x ≠ 0\nhcosx : cos x ≠ 0\nhcos1 : |cos x| < 1\n⊢ √(1 - cos x ^ 2) < 1",
"after_state": "n : ℕ\nx : ℝ\nh₀ : n ≠ 0\nh : cos x ^ n - sin x ^ n = 1\nhsinx : sin x ≠ 0\nhcosx : cos x ≠ 0\nhcos1 : |cos x| < 1\n⊢ 1 - cos x ^ 2 < 1 ^ 2"
},
{
"line": "rewrite [sqrt_lt' one_pos]",
"before_state": "n : ℕ\nx : ℝ\nh₀ : n ≠ 0\nh : cos x ^ n - sin x ^ n = 1\nhsinx : sin x ≠ 0\nhcosx : cos x ≠ 0\nhcos1 : |cos x| < 1\n⊢ √(1 - cos x ^ 2) < 1",
"after_state": "n : ℕ\nx : ℝ\nh₀ : n ≠ 0\nh : cos x ^ n - sin x ^ n = 1\nhsinx : sin x ≠ 0\nhcosx : cos x ≠ 0\nhcos1 : |cos x| < 1\n⊢ 1 - cos x ^ 2 < 1 ^ 2"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "n : ℕ\nx : ℝ\nh₀ : n ≠ 0\nh : cos x ^ n - sin x ^ n = 1\nhsinx : sin x ≠ 0\nhcosx : cos x ≠ 0\nhcos1 : |cos x| < 1\n⊢ 1 - cos x ^ 2 < 1 ^ 2",
"after_state": "n : ℕ\nx : ℝ\nh₀ : n ≠ 0\nh : cos x ^ n - sin x ^ n = 1\nhsinx : sin x ≠ 0\nhcosx : cos x ≠ 0\nhcos1 : |cos x| < 1\n⊢ 1 - cos x ^ 2 < 1 ^ 2"
},
{
"line": "try (with_reducible rfl)",
"before_state": "n : ℕ\nx : ℝ\nh₀ : n ≠ 0\nh : cos x ^ n - sin x ^ n = 1\nhsinx : sin x ≠ 0\nhcosx : cos x ≠ 0\nhcos1 : |cos x| < 1\n⊢ 1 - cos x ^ 2 < 1 ^ 2",
"after_state": "n : ℕ\nx : ℝ\nh₀ : n ≠ 0\nh : cos x ^ n - sin x ^ n = 1\nhsinx : sin x ≠ 0\nhcosx : cos x ≠ 0\nhcos1 : |cos x| < 1\n⊢ 1 - cos x ^ 2 < 1 ^ 2"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "n : ℕ\nx : ℝ\nh₀ : n ≠ 0\nh : cos x ^ n - sin x ^ n = 1\nhsinx : sin x ≠ 0\nhcosx : cos x ≠ 0\nhcos1 : |cos x| < 1\n⊢ 1 - cos x ^ 2 < 1 ^ 2",
"after_state": "n : ℕ\nx : ℝ\nh₀ : n ≠ 0\nh : cos x ^ n - sin x ^ n = 1\nhsinx : sin x ≠ 0\nhcosx : cos x ≠ 0\nhcos1 : |cos x| < 1\n⊢ 1 - cos x ^ 2 < 1 ^ 2"
},
{
"line": "with_reducible rfl",
"before_state": "n : ℕ\nx : ℝ\nh₀ : n ≠ 0\nh : cos x ^ n - sin x ^ n = 1\nhsinx : sin x ≠ 0\nhcosx : cos x ≠ 0\nhcos1 : |cos x| < 1\n⊢ 1 - cos x ^ 2 < 1 ^ 2",
"after_state": "n : ℕ\nx : ℝ\nh₀ : n ≠ 0\nh : cos x ^ n - sin x ^ n = 1\nhsinx : sin x ≠ 0\nhcosx : cos x ≠ 0\nhcos1 : |cos x| < 1\n⊢ 1 - cos x ^ 2 < 1 ^ 2"
},
{
"line": "rfl",
"before_state": "n : ℕ\nx : ℝ\nh₀ : n ≠ 0\nh : cos x ^ n - sin x ^ n = 1\nhsinx : sin x ≠ 0\nhcosx : cos x ≠ 0\nhcos1 : |cos x| < 1\n⊢ 1 - cos x ^ 2 < 1 ^ 2",
"after_state": "n : ℕ\nx : ℝ\nh₀ : n ≠ 0\nh : cos x ^ n - sin x ^ n = 1\nhsinx : sin x ≠ 0\nhcosx : cos x ≠ 0\nhcos1 : |cos x| < 1\n⊢ 1 - cos x ^ 2 < 1 ^ 2"
},
{
"line": "apply_rfl",
"before_state": "n : ℕ\nx : ℝ\nh₀ : n ≠ 0\nh : cos x ^ n - sin x ^ n = 1\nhsinx : sin x ≠ 0\nhcosx : cos x ≠ 0\nhcos1 : |cos x| < 1\n⊢ 1 - cos x ^ 2 < 1 ^ 2",
"after_state": "n : ℕ\nx : ℝ\nh₀ : n ≠ 0\nh : cos x ^ n - sin x ^ n = 1\nhsinx : sin x ≠ 0\nhcosx : cos x ≠ 0\nhcos1 : |cos x| < 1\n⊢ 1 - cos x ^ 2 < 1 ^ 2"
},
{
"line": "skip",
"before_state": "n : ℕ\nx : ℝ\nh₀ : n ≠ 0\nh : cos x ^ n - sin x ^ n = 1\nhsinx : sin x ≠ 0\nhcosx : cos x ≠ 0\nhcos1 : |cos x| < 1\n⊢ 1 - cos x ^ 2 < 1 ^ 2",
"after_state": "n : ℕ\nx : ℝ\nh₀ : n ≠ 0\nh : cos x ^ n - sin x ^ n = 1\nhsinx : sin x ≠ 0\nhcosx : cos x ≠ 0\nhcos1 : |cos x| < 1\n⊢ 1 - cos x ^ 2 < 1 ^ 2"
},
{
"line": "simp [sq_pos_of_ne_zero hcosx]",
"before_state": "n : ℕ\nx : ℝ\nh₀ : n ≠ 0\nh : cos x ^ n - sin x ^ n = 1\nhsinx : sin x ≠ 0\nhcosx : cos x ≠ 0\nhcos1 : |cos x| < 1\n⊢ 1 - cos x ^ 2 < 1 ^ 2",
"after_state": "No Goals!"
},
{
"line": "match n with\n| 1 =>\n rw [pow_one] at h\n rw [pow_one] at h\n rw [sub_eq_iff_eq_add] at h\n have : 2 * sin x * cos x = 0 := by\n simpa [h, add_sq, add_assoc, ← two_mul, mul_add, mul_assoc, ← sq] using cos_sq_add_sin_sq x\n simp [hsinx, hcosx] at this\n| 2 =>\n rw [← cos_sq_add_sin_sq x] at h\n rw [sub_eq_add_neg] at h\n rw [add_right_inj] at h\n rw [neg_eq_self ℝ] at h\n exact absurd (pow_eq_zero h) hsinx\n| (n + 1 + 2) =>\n set m := n + 1\n refine absurd ?_ h.not_lt\n calc\n (cos x) ^ (m + 2) - (sin x) ^ (m + 2) ≤ |cos x| ^ (m + 2) + |sin x| ^ (m + 2) :=\n by\n simp only [← abs_pow]\n simp only [sub_eq_add_neg]\n gcongr\n exacts [le_abs_self _, neg_le_abs _]\n _ = |cos x| ^ m * cos x ^ 2 + |sin x| ^ m * sin x ^ 2 := by simp [pow_add]\n _ < 1 ^ m * cos x ^ 2 + 1 ^ m * sin x ^ 2 := by gcongr\n _ = 1 := by simp",
"before_state": "case mp.inr.inr\nn : ℕ\nx : ℝ\nh₀ : n ≠ 0\nh : cos x ^ n - sin x ^ n = 1\nhsinx : sin x ≠ 0\nhcosx : cos x ≠ 0\nhcos1 : |cos x| < 1\nhsin1 : |sin x| < 1\n⊢ (∃ k, ↑k * π = x) ∧ Even n ∨ (∃ k, ↑k * (2 * π) = x) ∧ Odd n ∨ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd n",
"after_state": "No Goals!"
},
{
"line": "refine\n no_implicit_lambda%\n (match n with\n | 1 => ?rhs✝\n | 2 => ?rhs✝¹\n | (n + 1 + 2) => ?rhs✝²)",
"before_state": "case mp.inr.inr\nn : ℕ\nx : ℝ\nh₀ : n ≠ 0\nh : cos x ^ n - sin x ^ n = 1\nhsinx : sin x ≠ 0\nhcosx : cos x ≠ 0\nhcos1 : |cos x| < 1\nhsin1 : |sin x| < 1\n⊢ (∃ k, ↑k * π = x) ∧ Even n ∨ (∃ k, ↑k * (2 * π) = x) ∧ Odd n ∨ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd n",
"after_state": "case rhs\nn : ℕ\nx : ℝ\nhsinx : sin x ≠ 0\nhcosx : cos x ≠ 0\nhcos1 : |cos x| < 1\nhsin1 : |sin x| < 1\nh₀ : 1 ≠ 0\nh : cos x ^ 1 - sin x ^ 1 = 1\n⊢ (∃ k, ↑k * π = x) ∧ Even 1 ∨ (∃ k, ↑k * (2 * π) = x) ∧ Odd 1 ∨ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd 1\n---\ncase rhs\nn : ℕ\nx : ℝ\nhsinx : sin x ≠ 0\nhcosx : cos x ≠ 0\nhcos1 : |cos x| < 1\nhsin1 : |sin x| < 1\nh₀ : 2 ≠ 0\nh : cos x ^ 2 - sin x ^ 2 = 1\n⊢ (∃ k, ↑k * π = x) ∧ Even 2 ∨ (∃ k, ↑k * (2 * π) = x) ∧ Odd 2 ∨ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd 2\n---\ncase rhs\nn✝ : ℕ\nx : ℝ\nhsinx : sin x ≠ 0\nhcosx : cos x ≠ 0\nhcos1 : |cos x| < 1\nhsin1 : |sin x| < 1\nn : ℕ\nh₀ : n + 1 + 2 ≠ 0\nh : cos x ^ (n + 1 + 2) - sin x ^ (n + 1 + 2) = 1\n⊢ (∃ k, ↑k * π = x) ∧ Even (n + 1 + 2) ∨\n (∃ k, ↑k * (2 * π) = x) ∧ Odd (n + 1 + 2) ∨ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd (n + 1 + 2)"
},
{
"line": "case rhs✝ =>\n with_annotate_state[\"|\" \"=>\"] skip\n rw [pow_one] at h\n rw [pow_one] at h\n rw [sub_eq_iff_eq_add] at h\n have : 2 * sin x * cos x = 0 := by\n simpa [h, add_sq, add_assoc, ← two_mul, mul_add, mul_assoc, ← sq] using cos_sq_add_sin_sq x\n simp [hsinx, hcosx] at this",
"before_state": "case rhs\nn : ℕ\nx : ℝ\nhsinx : sin x ≠ 0\nhcosx : cos x ≠ 0\nhcos1 : |cos x| < 1\nhsin1 : |sin x| < 1\nh₀ : 1 ≠ 0\nh : cos x ^ 1 - sin x ^ 1 = 1\n⊢ (∃ k, ↑k * π = x) ∧ Even 1 ∨ (∃ k, ↑k * (2 * π) = x) ∧ Odd 1 ∨ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd 1\n---\ncase rhs\nn : ℕ\nx : ℝ\nhsinx : sin x ≠ 0\nhcosx : cos x ≠ 0\nhcos1 : |cos x| < 1\nhsin1 : |sin x| < 1\nh₀ : 2 ≠ 0\nh : cos x ^ 2 - sin x ^ 2 = 1\n⊢ (∃ k, ↑k * π = x) ∧ Even 2 ∨ (∃ k, ↑k * (2 * π) = x) ∧ Odd 2 ∨ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd 2\n---\ncase rhs\nn✝ : ℕ\nx : ℝ\nhsinx : sin x ≠ 0\nhcosx : cos x ≠ 0\nhcos1 : |cos x| < 1\nhsin1 : |sin x| < 1\nn : ℕ\nh₀ : n + 1 + 2 ≠ 0\nh : cos x ^ (n + 1 + 2) - sin x ^ (n + 1 + 2) = 1\n⊢ (∃ k, ↑k * π = x) ∧ Even (n + 1 + 2) ∨\n (∃ k, ↑k * (2 * π) = x) ∧ Odd (n + 1 + 2) ∨ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd (n + 1 + 2)",
"after_state": "case rhs\nn : ℕ\nx : ℝ\nhsinx : sin x ≠ 0\nhcosx : cos x ≠ 0\nhcos1 : |cos x| < 1\nhsin1 : |sin x| < 1\nh₀ : 2 ≠ 0\nh : cos x ^ 2 - sin x ^ 2 = 1\n⊢ (∃ k, ↑k * π = x) ∧ Even 2 ∨ (∃ k, ↑k * (2 * π) = x) ∧ Odd 2 ∨ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd 2\n---\ncase rhs\nn✝ : ℕ\nx : ℝ\nhsinx : sin x ≠ 0\nhcosx : cos x ≠ 0\nhcos1 : |cos x| < 1\nhsin1 : |sin x| < 1\nn : ℕ\nh₀ : n + 1 + 2 ≠ 0\nh : cos x ^ (n + 1 + 2) - sin x ^ (n + 1 + 2) = 1\n⊢ (∃ k, ↑k * π = x) ∧ Even (n + 1 + 2) ∨\n (∃ k, ↑k * (2 * π) = x) ∧ Odd (n + 1 + 2) ∨ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd (n + 1 + 2)"
},
{
"line": "with_annotate_state[\"|\" \"=>\"] skip",
"before_state": "n : ℕ\nx : ℝ\nhsinx : sin x ≠ 0\nhcosx : cos x ≠ 0\nhcos1 : |cos x| < 1\nhsin1 : |sin x| < 1\nh₀ : 1 ≠ 0\nh : cos x ^ 1 - sin x ^ 1 = 1\n⊢ (∃ k, ↑k * π = x) ∧ Even 1 ∨ (∃ k, ↑k * (2 * π) = x) ∧ Odd 1 ∨ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd 1",
"after_state": "n : ℕ\nx : ℝ\nhsinx : sin x ≠ 0\nhcosx : cos x ≠ 0\nhcos1 : |cos x| < 1\nhsin1 : |sin x| < 1\nh₀ : 1 ≠ 0\nh : cos x ^ 1 - sin x ^ 1 = 1\n⊢ (∃ k, ↑k * π = x) ∧ Even 1 ∨ (∃ k, ↑k * (2 * π) = x) ∧ Odd 1 ∨ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd 1"
},
{
"line": "skip",
"before_state": "n : ℕ\nx : ℝ\nhsinx : sin x ≠ 0\nhcosx : cos x ≠ 0\nhcos1 : |cos x| < 1\nhsin1 : |sin x| < 1\nh₀ : 1 ≠ 0\nh : cos x ^ 1 - sin x ^ 1 = 1\n⊢ (∃ k, ↑k * π = x) ∧ Even 1 ∨ (∃ k, ↑k * (2 * π) = x) ∧ Odd 1 ∨ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd 1",
"after_state": "n : ℕ\nx : ℝ\nhsinx : sin x ≠ 0\nhcosx : cos x ≠ 0\nhcos1 : |cos x| < 1\nhsin1 : |sin x| < 1\nh₀ : 1 ≠ 0\nh : cos x ^ 1 - sin x ^ 1 = 1\n⊢ (∃ k, ↑k * π = x) ∧ Even 1 ∨ (∃ k, ↑k * (2 * π) = x) ∧ Odd 1 ∨ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd 1"
},
{
"line": "rw [pow_one] at h",
"before_state": "n : ℕ\nx : ℝ\nhsinx : sin x ≠ 0\nhcosx : cos x ≠ 0\nhcos1 : |cos x| < 1\nhsin1 : |sin x| < 1\nh₀ : 1 ≠ 0\nh : cos x ^ 1 - sin x ^ 1 = 1\n⊢ (∃ k, ↑k * π = x) ∧ Even 1 ∨ (∃ k, ↑k * (2 * π) = x) ∧ Odd 1 ∨ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd 1",
"after_state": "n : ℕ\nx : ℝ\nhsinx : sin x ≠ 0\nhcosx : cos x ≠ 0\nhcos1 : |cos x| < 1\nhsin1 : |sin x| < 1\nh₀ : 1 ≠ 0\nh : cos x - sin x ^ 1 = 1\n⊢ (∃ k, ↑k * π = x) ∧ Even 1 ∨ (∃ k, ↑k * (2 * π) = x) ∧ Odd 1 ∨ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd 1"
},
{
"line": "rewrite [pow_one] at h",
"before_state": "n : ℕ\nx : ℝ\nhsinx : sin x ≠ 0\nhcosx : cos x ≠ 0\nhcos1 : |cos x| < 1\nhsin1 : |sin x| < 1\nh₀ : 1 ≠ 0\nh : cos x ^ 1 - sin x ^ 1 = 1\n⊢ (∃ k, ↑k * π = x) ∧ Even 1 ∨ (∃ k, ↑k * (2 * π) = x) ∧ Odd 1 ∨ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd 1",
"after_state": "n : ℕ\nx : ℝ\nhsinx : sin x ≠ 0\nhcosx : cos x ≠ 0\nhcos1 : |cos x| < 1\nhsin1 : |sin x| < 1\nh₀ : 1 ≠ 0\nh : cos x - sin x ^ 1 = 1\n⊢ (∃ k, ↑k * π = x) ∧ Even 1 ∨ (∃ k, ↑k * (2 * π) = x) ∧ Odd 1 ∨ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd 1"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "n : ℕ\nx : ℝ\nhsinx : sin x ≠ 0\nhcosx : cos x ≠ 0\nhcos1 : |cos x| < 1\nhsin1 : |sin x| < 1\nh₀ : 1 ≠ 0\nh : cos x - sin x ^ 1 = 1\n⊢ (∃ k, ↑k * π = x) ∧ Even 1 ∨ (∃ k, ↑k * (2 * π) = x) ∧ Odd 1 ∨ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd 1",
"after_state": "n : ℕ\nx : ℝ\nhsinx : sin x ≠ 0\nhcosx : cos x ≠ 0\nhcos1 : |cos x| < 1\nhsin1 : |sin x| < 1\nh₀ : 1 ≠ 0\nh : cos x - sin x ^ 1 = 1\n⊢ (∃ k, ↑k * π = x) ∧ Even 1 ∨ (∃ k, ↑k * (2 * π) = x) ∧ Odd 1 ∨ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd 1"
},
{
"line": "try (with_reducible rfl)",
"before_state": "n : ℕ\nx : ℝ\nhsinx : sin x ≠ 0\nhcosx : cos x ≠ 0\nhcos1 : |cos x| < 1\nhsin1 : |sin x| < 1\nh₀ : 1 ≠ 0\nh : cos x - sin x ^ 1 = 1\n⊢ (∃ k, ↑k * π = x) ∧ Even 1 ∨ (∃ k, ↑k * (2 * π) = x) ∧ Odd 1 ∨ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd 1",
"after_state": "n : ℕ\nx : ℝ\nhsinx : sin x ≠ 0\nhcosx : cos x ≠ 0\nhcos1 : |cos x| < 1\nhsin1 : |sin x| < 1\nh₀ : 1 ≠ 0\nh : cos x - sin x ^ 1 = 1\n⊢ (∃ k, ↑k * π = x) ∧ Even 1 ∨ (∃ k, ↑k * (2 * π) = x) ∧ Odd 1 ∨ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd 1"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "n : ℕ\nx : ℝ\nhsinx : sin x ≠ 0\nhcosx : cos x ≠ 0\nhcos1 : |cos x| < 1\nhsin1 : |sin x| < 1\nh₀ : 1 ≠ 0\nh : cos x - sin x ^ 1 = 1\n⊢ (∃ k, ↑k * π = x) ∧ Even 1 ∨ (∃ k, ↑k * (2 * π) = x) ∧ Odd 1 ∨ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd 1",
"after_state": "n : ℕ\nx : ℝ\nhsinx : sin x ≠ 0\nhcosx : cos x ≠ 0\nhcos1 : |cos x| < 1\nhsin1 : |sin x| < 1\nh₀ : 1 ≠ 0\nh : cos x - sin x ^ 1 = 1\n⊢ (∃ k, ↑k * π = x) ∧ Even 1 ∨ (∃ k, ↑k * (2 * π) = x) ∧ Odd 1 ∨ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd 1"
},
{
"line": "with_reducible rfl",
"before_state": "n : ℕ\nx : ℝ\nhsinx : sin x ≠ 0\nhcosx : cos x ≠ 0\nhcos1 : |cos x| < 1\nhsin1 : |sin x| < 1\nh₀ : 1 ≠ 0\nh : cos x - sin x ^ 1 = 1\n⊢ (∃ k, ↑k * π = x) ∧ Even 1 ∨ (∃ k, ↑k * (2 * π) = x) ∧ Odd 1 ∨ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd 1",
"after_state": "n : ℕ\nx : ℝ\nhsinx : sin x ≠ 0\nhcosx : cos x ≠ 0\nhcos1 : |cos x| < 1\nhsin1 : |sin x| < 1\nh₀ : 1 ≠ 0\nh : cos x - sin x ^ 1 = 1\n⊢ (∃ k, ↑k * π = x) ∧ Even 1 ∨ (∃ k, ↑k * (2 * π) = x) ∧ Odd 1 ∨ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd 1"
},
{
"line": "rfl",
"before_state": "n : ℕ\nx : ℝ\nhsinx : sin x ≠ 0\nhcosx : cos x ≠ 0\nhcos1 : |cos x| < 1\nhsin1 : |sin x| < 1\nh₀ : 1 ≠ 0\nh : cos x - sin x ^ 1 = 1\n⊢ (∃ k, ↑k * π = x) ∧ Even 1 ∨ (∃ k, ↑k * (2 * π) = x) ∧ Odd 1 ∨ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd 1",
"after_state": "n : ℕ\nx : ℝ\nhsinx : sin x ≠ 0\nhcosx : cos x ≠ 0\nhcos1 : |cos x| < 1\nhsin1 : |sin x| < 1\nh₀ : 1 ≠ 0\nh : cos x - sin x ^ 1 = 1\n⊢ (∃ k, ↑k * π = x) ∧ Even 1 ∨ (∃ k, ↑k * (2 * π) = x) ∧ Odd 1 ∨ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd 1"
},
{
"line": "apply_rfl",
"before_state": "n : ℕ\nx : ℝ\nhsinx : sin x ≠ 0\nhcosx : cos x ≠ 0\nhcos1 : |cos x| < 1\nhsin1 : |sin x| < 1\nh₀ : 1 ≠ 0\nh : cos x - sin x ^ 1 = 1\n⊢ (∃ k, ↑k * π = x) ∧ Even 1 ∨ (∃ k, ↑k * (2 * π) = x) ∧ Odd 1 ∨ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd 1",
"after_state": "n : ℕ\nx : ℝ\nhsinx : sin x ≠ 0\nhcosx : cos x ≠ 0\nhcos1 : |cos x| < 1\nhsin1 : |sin x| < 1\nh₀ : 1 ≠ 0\nh : cos x - sin x ^ 1 = 1\n⊢ (∃ k, ↑k * π = x) ∧ Even 1 ∨ (∃ k, ↑k * (2 * π) = x) ∧ Odd 1 ∨ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd 1"
},
{
"line": "skip",
"before_state": "n : ℕ\nx : ℝ\nhsinx : sin x ≠ 0\nhcosx : cos x ≠ 0\nhcos1 : |cos x| < 1\nhsin1 : |sin x| < 1\nh₀ : 1 ≠ 0\nh : cos x - sin x ^ 1 = 1\n⊢ (∃ k, ↑k * π = x) ∧ Even 1 ∨ (∃ k, ↑k * (2 * π) = x) ∧ Odd 1 ∨ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd 1",
"after_state": "n : ℕ\nx : ℝ\nhsinx : sin x ≠ 0\nhcosx : cos x ≠ 0\nhcos1 : |cos x| < 1\nhsin1 : |sin x| < 1\nh₀ : 1 ≠ 0\nh : cos x - sin x ^ 1 = 1\n⊢ (∃ k, ↑k * π = x) ∧ Even 1 ∨ (∃ k, ↑k * (2 * π) = x) ∧ Odd 1 ∨ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd 1"
},
{
"line": "rw [pow_one] at h",
"before_state": "n : ℕ\nx : ℝ\nhsinx : sin x ≠ 0\nhcosx : cos x ≠ 0\nhcos1 : |cos x| < 1\nhsin1 : |sin x| < 1\nh₀ : 1 ≠ 0\nh : cos x - sin x ^ 1 = 1\n⊢ (∃ k, ↑k * π = x) ∧ Even 1 ∨ (∃ k, ↑k * (2 * π) = x) ∧ Odd 1 ∨ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd 1",
"after_state": "n : ℕ\nx : ℝ\nhsinx : sin x ≠ 0\nhcosx : cos x ≠ 0\nhcos1 : |cos x| < 1\nhsin1 : |sin x| < 1\nh₀ : 1 ≠ 0\nh : cos x - sin x = 1\n⊢ (∃ k, ↑k * π = x) ∧ Even 1 ∨ (∃ k, ↑k * (2 * π) = x) ∧ Odd 1 ∨ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd 1"
},
{
"line": "rewrite [pow_one] at h",
"before_state": "n : ℕ\nx : ℝ\nhsinx : sin x ≠ 0\nhcosx : cos x ≠ 0\nhcos1 : |cos x| < 1\nhsin1 : |sin x| < 1\nh₀ : 1 ≠ 0\nh : cos x - sin x ^ 1 = 1\n⊢ (∃ k, ↑k * π = x) ∧ Even 1 ∨ (∃ k, ↑k * (2 * π) = x) ∧ Odd 1 ∨ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd 1",
"after_state": "n : ℕ\nx : ℝ\nhsinx : sin x ≠ 0\nhcosx : cos x ≠ 0\nhcos1 : |cos x| < 1\nhsin1 : |sin x| < 1\nh₀ : 1 ≠ 0\nh : cos x - sin x = 1\n⊢ (∃ k, ↑k * π = x) ∧ Even 1 ∨ (∃ k, ↑k * (2 * π) = x) ∧ Odd 1 ∨ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd 1"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "n : ℕ\nx : ℝ\nhsinx : sin x ≠ 0\nhcosx : cos x ≠ 0\nhcos1 : |cos x| < 1\nhsin1 : |sin x| < 1\nh₀ : 1 ≠ 0\nh : cos x - sin x = 1\n⊢ (∃ k, ↑k * π = x) ∧ Even 1 ∨ (∃ k, ↑k * (2 * π) = x) ∧ Odd 1 ∨ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd 1",
"after_state": "n : ℕ\nx : ℝ\nhsinx : sin x ≠ 0\nhcosx : cos x ≠ 0\nhcos1 : |cos x| < 1\nhsin1 : |sin x| < 1\nh₀ : 1 ≠ 0\nh : cos x - sin x = 1\n⊢ (∃ k, ↑k * π = x) ∧ Even 1 ∨ (∃ k, ↑k * (2 * π) = x) ∧ Odd 1 ∨ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd 1"
},
{
"line": "try (with_reducible rfl)",
"before_state": "n : ℕ\nx : ℝ\nhsinx : sin x ≠ 0\nhcosx : cos x ≠ 0\nhcos1 : |cos x| < 1\nhsin1 : |sin x| < 1\nh₀ : 1 ≠ 0\nh : cos x - sin x = 1\n⊢ (∃ k, ↑k * π = x) ∧ Even 1 ∨ (∃ k, ↑k * (2 * π) = x) ∧ Odd 1 ∨ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd 1",
"after_state": "n : ℕ\nx : ℝ\nhsinx : sin x ≠ 0\nhcosx : cos x ≠ 0\nhcos1 : |cos x| < 1\nhsin1 : |sin x| < 1\nh₀ : 1 ≠ 0\nh : cos x - sin x = 1\n⊢ (∃ k, ↑k * π = x) ∧ Even 1 ∨ (∃ k, ↑k * (2 * π) = x) ∧ Odd 1 ∨ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd 1"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "n : ℕ\nx : ℝ\nhsinx : sin x ≠ 0\nhcosx : cos x ≠ 0\nhcos1 : |cos x| < 1\nhsin1 : |sin x| < 1\nh₀ : 1 ≠ 0\nh : cos x - sin x = 1\n⊢ (∃ k, ↑k * π = x) ∧ Even 1 ∨ (∃ k, ↑k * (2 * π) = x) ∧ Odd 1 ∨ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd 1",
"after_state": "n : ℕ\nx : ℝ\nhsinx : sin x ≠ 0\nhcosx : cos x ≠ 0\nhcos1 : |cos x| < 1\nhsin1 : |sin x| < 1\nh₀ : 1 ≠ 0\nh : cos x - sin x = 1\n⊢ (∃ k, ↑k * π = x) ∧ Even 1 ∨ (∃ k, ↑k * (2 * π) = x) ∧ Odd 1 ∨ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd 1"
},
{
"line": "with_reducible rfl",
"before_state": "n : ℕ\nx : ℝ\nhsinx : sin x ≠ 0\nhcosx : cos x ≠ 0\nhcos1 : |cos x| < 1\nhsin1 : |sin x| < 1\nh₀ : 1 ≠ 0\nh : cos x - sin x = 1\n⊢ (∃ k, ↑k * π = x) ∧ Even 1 ∨ (∃ k, ↑k * (2 * π) = x) ∧ Odd 1 ∨ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd 1",
"after_state": "n : ℕ\nx : ℝ\nhsinx : sin x ≠ 0\nhcosx : cos x ≠ 0\nhcos1 : |cos x| < 1\nhsin1 : |sin x| < 1\nh₀ : 1 ≠ 0\nh : cos x - sin x = 1\n⊢ (∃ k, ↑k * π = x) ∧ Even 1 ∨ (∃ k, ↑k * (2 * π) = x) ∧ Odd 1 ∨ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd 1"
},
{
"line": "rfl",
"before_state": "n : ℕ\nx : ℝ\nhsinx : sin x ≠ 0\nhcosx : cos x ≠ 0\nhcos1 : |cos x| < 1\nhsin1 : |sin x| < 1\nh₀ : 1 ≠ 0\nh : cos x - sin x = 1\n⊢ (∃ k, ↑k * π = x) ∧ Even 1 ∨ (∃ k, ↑k * (2 * π) = x) ∧ Odd 1 ∨ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd 1",
"after_state": "n : ℕ\nx : ℝ\nhsinx : sin x ≠ 0\nhcosx : cos x ≠ 0\nhcos1 : |cos x| < 1\nhsin1 : |sin x| < 1\nh₀ : 1 ≠ 0\nh : cos x - sin x = 1\n⊢ (∃ k, ↑k * π = x) ∧ Even 1 ∨ (∃ k, ↑k * (2 * π) = x) ∧ Odd 1 ∨ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd 1"
},
{
"line": "apply_rfl",
"before_state": "n : ℕ\nx : ℝ\nhsinx : sin x ≠ 0\nhcosx : cos x ≠ 0\nhcos1 : |cos x| < 1\nhsin1 : |sin x| < 1\nh₀ : 1 ≠ 0\nh : cos x - sin x = 1\n⊢ (∃ k, ↑k * π = x) ∧ Even 1 ∨ (∃ k, ↑k * (2 * π) = x) ∧ Odd 1 ∨ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd 1",
"after_state": "n : ℕ\nx : ℝ\nhsinx : sin x ≠ 0\nhcosx : cos x ≠ 0\nhcos1 : |cos x| < 1\nhsin1 : |sin x| < 1\nh₀ : 1 ≠ 0\nh : cos x - sin x = 1\n⊢ (∃ k, ↑k * π = x) ∧ Even 1 ∨ (∃ k, ↑k * (2 * π) = x) ∧ Odd 1 ∨ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd 1"
},
{
"line": "skip",
"before_state": "n : ℕ\nx : ℝ\nhsinx : sin x ≠ 0\nhcosx : cos x ≠ 0\nhcos1 : |cos x| < 1\nhsin1 : |sin x| < 1\nh₀ : 1 ≠ 0\nh : cos x - sin x = 1\n⊢ (∃ k, ↑k * π = x) ∧ Even 1 ∨ (∃ k, ↑k * (2 * π) = x) ∧ Odd 1 ∨ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd 1",
"after_state": "n : ℕ\nx : ℝ\nhsinx : sin x ≠ 0\nhcosx : cos x ≠ 0\nhcos1 : |cos x| < 1\nhsin1 : |sin x| < 1\nh₀ : 1 ≠ 0\nh : cos x - sin x = 1\n⊢ (∃ k, ↑k * π = x) ∧ Even 1 ∨ (∃ k, ↑k * (2 * π) = x) ∧ Odd 1 ∨ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd 1"
},
{
"line": "rw [sub_eq_iff_eq_add] at h",
"before_state": "n : ℕ\nx : ℝ\nhsinx : sin x ≠ 0\nhcosx : cos x ≠ 0\nhcos1 : |cos x| < 1\nhsin1 : |sin x| < 1\nh₀ : 1 ≠ 0\nh : cos x - sin x = 1\n⊢ (∃ k, ↑k * π = x) ∧ Even 1 ∨ (∃ k, ↑k * (2 * π) = x) ∧ Odd 1 ∨ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd 1",
"after_state": "n : ℕ\nx : ℝ\nhsinx : sin x ≠ 0\nhcosx : cos x ≠ 0\nhcos1 : |cos x| < 1\nhsin1 : |sin x| < 1\nh₀ : 1 ≠ 0\nh : cos x = 1 + sin x\n⊢ (∃ k, ↑k * π = x) ∧ Even 1 ∨ (∃ k, ↑k * (2 * π) = x) ∧ Odd 1 ∨ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd 1"
},
{
"line": "rewrite [sub_eq_iff_eq_add] at h",
"before_state": "n : ℕ\nx : ℝ\nhsinx : sin x ≠ 0\nhcosx : cos x ≠ 0\nhcos1 : |cos x| < 1\nhsin1 : |sin x| < 1\nh₀ : 1 ≠ 0\nh : cos x - sin x = 1\n⊢ (∃ k, ↑k * π = x) ∧ Even 1 ∨ (∃ k, ↑k * (2 * π) = x) ∧ Odd 1 ∨ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd 1",
"after_state": "n : ℕ\nx : ℝ\nhsinx : sin x ≠ 0\nhcosx : cos x ≠ 0\nhcos1 : |cos x| < 1\nhsin1 : |sin x| < 1\nh₀ : 1 ≠ 0\nh : cos x = 1 + sin x\n⊢ (∃ k, ↑k * π = x) ∧ Even 1 ∨ (∃ k, ↑k * (2 * π) = x) ∧ Odd 1 ∨ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd 1"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "n : ℕ\nx : ℝ\nhsinx : sin x ≠ 0\nhcosx : cos x ≠ 0\nhcos1 : |cos x| < 1\nhsin1 : |sin x| < 1\nh₀ : 1 ≠ 0\nh : cos x = 1 + sin x\n⊢ (∃ k, ↑k * π = x) ∧ Even 1 ∨ (∃ k, ↑k * (2 * π) = x) ∧ Odd 1 ∨ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd 1",
"after_state": "n : ℕ\nx : ℝ\nhsinx : sin x ≠ 0\nhcosx : cos x ≠ 0\nhcos1 : |cos x| < 1\nhsin1 : |sin x| < 1\nh₀ : 1 ≠ 0\nh : cos x = 1 + sin x\n⊢ (∃ k, ↑k * π = x) ∧ Even 1 ∨ (∃ k, ↑k * (2 * π) = x) ∧ Odd 1 ∨ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd 1"
},
{
"line": "try (with_reducible rfl)",
"before_state": "n : ℕ\nx : ℝ\nhsinx : sin x ≠ 0\nhcosx : cos x ≠ 0\nhcos1 : |cos x| < 1\nhsin1 : |sin x| < 1\nh₀ : 1 ≠ 0\nh : cos x = 1 + sin x\n⊢ (∃ k, ↑k * π = x) ∧ Even 1 ∨ (∃ k, ↑k * (2 * π) = x) ∧ Odd 1 ∨ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd 1",
"after_state": "n : ℕ\nx : ℝ\nhsinx : sin x ≠ 0\nhcosx : cos x ≠ 0\nhcos1 : |cos x| < 1\nhsin1 : |sin x| < 1\nh₀ : 1 ≠ 0\nh : cos x = 1 + sin x\n⊢ (∃ k, ↑k * π = x) ∧ Even 1 ∨ (∃ k, ↑k * (2 * π) = x) ∧ Odd 1 ∨ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd 1"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "n : ℕ\nx : ℝ\nhsinx : sin x ≠ 0\nhcosx : cos x ≠ 0\nhcos1 : |cos x| < 1\nhsin1 : |sin x| < 1\nh₀ : 1 ≠ 0\nh : cos x = 1 + sin x\n⊢ (∃ k, ↑k * π = x) ∧ Even 1 ∨ (∃ k, ↑k * (2 * π) = x) ∧ Odd 1 ∨ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd 1",
"after_state": "n : ℕ\nx : ℝ\nhsinx : sin x ≠ 0\nhcosx : cos x ≠ 0\nhcos1 : |cos x| < 1\nhsin1 : |sin x| < 1\nh₀ : 1 ≠ 0\nh : cos x = 1 + sin x\n⊢ (∃ k, ↑k * π = x) ∧ Even 1 ∨ (∃ k, ↑k * (2 * π) = x) ∧ Odd 1 ∨ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd 1"
},
{
"line": "with_reducible rfl",
"before_state": "n : ℕ\nx : ℝ\nhsinx : sin x ≠ 0\nhcosx : cos x ≠ 0\nhcos1 : |cos x| < 1\nhsin1 : |sin x| < 1\nh₀ : 1 ≠ 0\nh : cos x = 1 + sin x\n⊢ (∃ k, ↑k * π = x) ∧ Even 1 ∨ (∃ k, ↑k * (2 * π) = x) ∧ Odd 1 ∨ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd 1",
"after_state": "n : ℕ\nx : ℝ\nhsinx : sin x ≠ 0\nhcosx : cos x ≠ 0\nhcos1 : |cos x| < 1\nhsin1 : |sin x| < 1\nh₀ : 1 ≠ 0\nh : cos x = 1 + sin x\n⊢ (∃ k, ↑k * π = x) ∧ Even 1 ∨ (∃ k, ↑k * (2 * π) = x) ∧ Odd 1 ∨ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd 1"
},
{
"line": "rfl",
"before_state": "n : ℕ\nx : ℝ\nhsinx : sin x ≠ 0\nhcosx : cos x ≠ 0\nhcos1 : |cos x| < 1\nhsin1 : |sin x| < 1\nh₀ : 1 ≠ 0\nh : cos x = 1 + sin x\n⊢ (∃ k, ↑k * π = x) ∧ Even 1 ∨ (∃ k, ↑k * (2 * π) = x) ∧ Odd 1 ∨ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd 1",
"after_state": "n : ℕ\nx : ℝ\nhsinx : sin x ≠ 0\nhcosx : cos x ≠ 0\nhcos1 : |cos x| < 1\nhsin1 : |sin x| < 1\nh₀ : 1 ≠ 0\nh : cos x = 1 + sin x\n⊢ (∃ k, ↑k * π = x) ∧ Even 1 ∨ (∃ k, ↑k * (2 * π) = x) ∧ Odd 1 ∨ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd 1"
},
{
"line": "apply_rfl",
"before_state": "n : ℕ\nx : ℝ\nhsinx : sin x ≠ 0\nhcosx : cos x ≠ 0\nhcos1 : |cos x| < 1\nhsin1 : |sin x| < 1\nh₀ : 1 ≠ 0\nh : cos x = 1 + sin x\n⊢ (∃ k, ↑k * π = x) ∧ Even 1 ∨ (∃ k, ↑k * (2 * π) = x) ∧ Odd 1 ∨ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd 1",
"after_state": "n : ℕ\nx : ℝ\nhsinx : sin x ≠ 0\nhcosx : cos x ≠ 0\nhcos1 : |cos x| < 1\nhsin1 : |sin x| < 1\nh₀ : 1 ≠ 0\nh : cos x = 1 + sin x\n⊢ (∃ k, ↑k * π = x) ∧ Even 1 ∨ (∃ k, ↑k * (2 * π) = x) ∧ Odd 1 ∨ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd 1"
},
{
"line": "skip",
"before_state": "n : ℕ\nx : ℝ\nhsinx : sin x ≠ 0\nhcosx : cos x ≠ 0\nhcos1 : |cos x| < 1\nhsin1 : |sin x| < 1\nh₀ : 1 ≠ 0\nh : cos x = 1 + sin x\n⊢ (∃ k, ↑k * π = x) ∧ Even 1 ∨ (∃ k, ↑k * (2 * π) = x) ∧ Odd 1 ∨ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd 1",
"after_state": "n : ℕ\nx : ℝ\nhsinx : sin x ≠ 0\nhcosx : cos x ≠ 0\nhcos1 : |cos x| < 1\nhsin1 : |sin x| < 1\nh₀ : 1 ≠ 0\nh : cos x = 1 + sin x\n⊢ (∃ k, ↑k * π = x) ∧ Even 1 ∨ (∃ k, ↑k * (2 * π) = x) ∧ Odd 1 ∨ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd 1"
},
{
"line": "have : 2 * sin x * cos x = 0 := by\n simpa [h, add_sq, add_assoc, ← two_mul, mul_add, mul_assoc, ← sq] using cos_sq_add_sin_sq x",
"before_state": "n : ℕ\nx : ℝ\nhsinx : sin x ≠ 0\nhcosx : cos x ≠ 0\nhcos1 : |cos x| < 1\nhsin1 : |sin x| < 1\nh₀ : 1 ≠ 0\nh : cos x = 1 + sin x\n⊢ (∃ k, ↑k * π = x) ∧ Even 1 ∨ (∃ k, ↑k * (2 * π) = x) ∧ Odd 1 ∨ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd 1",
"after_state": "n : ℕ\nx : ℝ\nhsinx : sin x ≠ 0\nhcosx : cos x ≠ 0\nhcos1 : |cos x| < 1\nhsin1 : |sin x| < 1\nh₀ : 1 ≠ 0\nh : cos x = 1 + sin x\nthis : 2 * sin x * cos x = 0\n⊢ (∃ k, ↑k * π = x) ∧ Even 1 ∨ (∃ k, ↑k * (2 * π) = x) ∧ Odd 1 ∨ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd 1"
},
{
"line": "focus\n refine\n no_implicit_lambda%\n (have : 2 * sin x * cos x = 0 := ?body✝;\n ?_)\n case body✝ =>\n with_annotate_state\"by\"\n (simpa [h, add_sq, add_assoc, ← two_mul, mul_add, mul_assoc, ← sq] using cos_sq_add_sin_sq x)",
"before_state": "n : ℕ\nx : ℝ\nhsinx : sin x ≠ 0\nhcosx : cos x ≠ 0\nhcos1 : |cos x| < 1\nhsin1 : |sin x| < 1\nh₀ : 1 ≠ 0\nh : cos x = 1 + sin x\n⊢ (∃ k, ↑k * π = x) ∧ Even 1 ∨ (∃ k, ↑k * (2 * π) = x) ∧ Odd 1 ∨ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd 1",
"after_state": "n : ℕ\nx : ℝ\nhsinx : sin x ≠ 0\nhcosx : cos x ≠ 0\nhcos1 : |cos x| < 1\nhsin1 : |sin x| < 1\nh₀ : 1 ≠ 0\nh : cos x = 1 + sin x\nthis : 2 * sin x * cos x = 0\n⊢ (∃ k, ↑k * π = x) ∧ Even 1 ∨ (∃ k, ↑k * (2 * π) = x) ∧ Odd 1 ∨ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd 1"
},
{
"line": "refine\n no_implicit_lambda%\n (have : 2 * sin x * cos x = 0 := ?body✝;\n ?_)",
"before_state": "n : ℕ\nx : ℝ\nhsinx : sin x ≠ 0\nhcosx : cos x ≠ 0\nhcos1 : |cos x| < 1\nhsin1 : |sin x| < 1\nh₀ : 1 ≠ 0\nh : cos x = 1 + sin x\n⊢ (∃ k, ↑k * π = x) ∧ Even 1 ∨ (∃ k, ↑k * (2 * π) = x) ∧ Odd 1 ∨ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd 1",
"after_state": "case body\nn : ℕ\nx : ℝ\nhsinx : sin x ≠ 0\nhcosx : cos x ≠ 0\nhcos1 : |cos x| < 1\nhsin1 : |sin x| < 1\nh₀ : 1 ≠ 0\nh : cos x = 1 + sin x\n⊢ 2 * sin x * cos x = 0\n---\nn : ℕ\nx : ℝ\nhsinx : sin x ≠ 0\nhcosx : cos x ≠ 0\nhcos1 : |cos x| < 1\nhsin1 : |sin x| < 1\nh₀ : 1 ≠ 0\nh : cos x = 1 + sin x\nthis : 2 * sin x * cos x = 0\n⊢ (∃ k, ↑k * π = x) ∧ Even 1 ∨ (∃ k, ↑k * (2 * π) = x) ∧ Odd 1 ∨ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd 1"
},
{
"line": "case body✝ =>\n with_annotate_state\"by\" (simpa [h, add_sq, add_assoc, ← two_mul, mul_add, mul_assoc, ← sq] using cos_sq_add_sin_sq x)",
"before_state": "case body\nn : ℕ\nx : ℝ\nhsinx : sin x ≠ 0\nhcosx : cos x ≠ 0\nhcos1 : |cos x| < 1\nhsin1 : |sin x| < 1\nh₀ : 1 ≠ 0\nh : cos x = 1 + sin x\n⊢ 2 * sin x * cos x = 0\n---\nn : ℕ\nx : ℝ\nhsinx : sin x ≠ 0\nhcosx : cos x ≠ 0\nhcos1 : |cos x| < 1\nhsin1 : |sin x| < 1\nh₀ : 1 ≠ 0\nh : cos x = 1 + sin x\nthis : 2 * sin x * cos x = 0\n⊢ (∃ k, ↑k * π = x) ∧ Even 1 ∨ (∃ k, ↑k * (2 * π) = x) ∧ Odd 1 ∨ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd 1",
"after_state": "n : ℕ\nx : ℝ\nhsinx : sin x ≠ 0\nhcosx : cos x ≠ 0\nhcos1 : |cos x| < 1\nhsin1 : |sin x| < 1\nh₀ : 1 ≠ 0\nh : cos x = 1 + sin x\nthis : 2 * sin x * cos x = 0\n⊢ (∃ k, ↑k * π = x) ∧ Even 1 ∨ (∃ k, ↑k * (2 * π) = x) ∧ Odd 1 ∨ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd 1"
},
{
"line": "with_annotate_state\"by\" (simpa [h, add_sq, add_assoc, ← two_mul, mul_add, mul_assoc, ← sq] using cos_sq_add_sin_sq x)",
"before_state": "n : ℕ\nx : ℝ\nhsinx : sin x ≠ 0\nhcosx : cos x ≠ 0\nhcos1 : |cos x| < 1\nhsin1 : |sin x| < 1\nh₀ : 1 ≠ 0\nh : cos x = 1 + sin x\n⊢ 2 * sin x * cos x = 0",
"after_state": "No Goals!"
},
{
"line": "simpa [h, add_sq, add_assoc, ← two_mul, mul_add, mul_assoc, ← sq] using cos_sq_add_sin_sq x",
"before_state": "n : ℕ\nx : ℝ\nhsinx : sin x ≠ 0\nhcosx : cos x ≠ 0\nhcos1 : |cos x| < 1\nhsin1 : |sin x| < 1\nh₀ : 1 ≠ 0\nh : cos x = 1 + sin x\n⊢ 2 * sin x * cos x = 0",
"after_state": "No Goals!"
},
{
"line": "simp [hsinx, hcosx] at this",
"before_state": "n : ℕ\nx : ℝ\nhsinx : sin x ≠ 0\nhcosx : cos x ≠ 0\nhcos1 : |cos x| < 1\nhsin1 : |sin x| < 1\nh₀ : 1 ≠ 0\nh : cos x = 1 + sin x\nthis : 2 * sin x * cos x = 0\n⊢ (∃ k, ↑k * π = x) ∧ Even 1 ∨ (∃ k, ↑k * (2 * π) = x) ∧ Odd 1 ∨ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd 1",
"after_state": "No Goals!"
},
{
"line": "case rhs✝ =>\n with_annotate_state[\"|\" \"=>\"] skip\n rw [← cos_sq_add_sin_sq x] at h\n rw [sub_eq_add_neg] at h\n rw [add_right_inj] at h\n rw [neg_eq_self ℝ] at h\n exact absurd (pow_eq_zero h) hsinx",
"before_state": "case rhs\nn : ℕ\nx : ℝ\nhsinx : sin x ≠ 0\nhcosx : cos x ≠ 0\nhcos1 : |cos x| < 1\nhsin1 : |sin x| < 1\nh₀ : 2 ≠ 0\nh : cos x ^ 2 - sin x ^ 2 = 1\n⊢ (∃ k, ↑k * π = x) ∧ Even 2 ∨ (∃ k, ↑k * (2 * π) = x) ∧ Odd 2 ∨ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd 2\n---\ncase rhs\nn✝ : ℕ\nx : ℝ\nhsinx : sin x ≠ 0\nhcosx : cos x ≠ 0\nhcos1 : |cos x| < 1\nhsin1 : |sin x| < 1\nn : ℕ\nh₀ : n + 1 + 2 ≠ 0\nh : cos x ^ (n + 1 + 2) - sin x ^ (n + 1 + 2) = 1\n⊢ (∃ k, ↑k * π = x) ∧ Even (n + 1 + 2) ∨\n (∃ k, ↑k * (2 * π) = x) ∧ Odd (n + 1 + 2) ∨ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd (n + 1 + 2)",
"after_state": "case rhs\nn✝ : ℕ\nx : ℝ\nhsinx : sin x ≠ 0\nhcosx : cos x ≠ 0\nhcos1 : |cos x| < 1\nhsin1 : |sin x| < 1\nn : ℕ\nh₀ : n + 1 + 2 ≠ 0\nh : cos x ^ (n + 1 + 2) - sin x ^ (n + 1 + 2) = 1\n⊢ (∃ k, ↑k * π = x) ∧ Even (n + 1 + 2) ∨\n (∃ k, ↑k * (2 * π) = x) ∧ Odd (n + 1 + 2) ∨ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd (n + 1 + 2)"
},
{
"line": "with_annotate_state[\"|\" \"=>\"] skip",
"before_state": "n : ℕ\nx : ℝ\nhsinx : sin x ≠ 0\nhcosx : cos x ≠ 0\nhcos1 : |cos x| < 1\nhsin1 : |sin x| < 1\nh₀ : 2 ≠ 0\nh : cos x ^ 2 - sin x ^ 2 = 1\n⊢ (∃ k, ↑k * π = x) ∧ Even 2 ∨ (∃ k, ↑k * (2 * π) = x) ∧ Odd 2 ∨ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd 2",
"after_state": "n : ℕ\nx : ℝ\nhsinx : sin x ≠ 0\nhcosx : cos x ≠ 0\nhcos1 : |cos x| < 1\nhsin1 : |sin x| < 1\nh₀ : 2 ≠ 0\nh : cos x ^ 2 - sin x ^ 2 = 1\n⊢ (∃ k, ↑k * π = x) ∧ Even 2 ∨ (∃ k, ↑k * (2 * π) = x) ∧ Odd 2 ∨ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd 2"
},
{
"line": "skip",
"before_state": "n : ℕ\nx : ℝ\nhsinx : sin x ≠ 0\nhcosx : cos x ≠ 0\nhcos1 : |cos x| < 1\nhsin1 : |sin x| < 1\nh₀ : 2 ≠ 0\nh : cos x ^ 2 - sin x ^ 2 = 1\n⊢ (∃ k, ↑k * π = x) ∧ Even 2 ∨ (∃ k, ↑k * (2 * π) = x) ∧ Odd 2 ∨ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd 2",
"after_state": "n : ℕ\nx : ℝ\nhsinx : sin x ≠ 0\nhcosx : cos x ≠ 0\nhcos1 : |cos x| < 1\nhsin1 : |sin x| < 1\nh₀ : 2 ≠ 0\nh : cos x ^ 2 - sin x ^ 2 = 1\n⊢ (∃ k, ↑k * π = x) ∧ Even 2 ∨ (∃ k, ↑k * (2 * π) = x) ∧ Odd 2 ∨ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd 2"
},
{
"line": "rw [← cos_sq_add_sin_sq x] at h",
"before_state": "n : ℕ\nx : ℝ\nhsinx : sin x ≠ 0\nhcosx : cos x ≠ 0\nhcos1 : |cos x| < 1\nhsin1 : |sin x| < 1\nh₀ : 2 ≠ 0\nh : cos x ^ 2 - sin x ^ 2 = 1\n⊢ (∃ k, ↑k * π = x) ∧ Even 2 ∨ (∃ k, ↑k * (2 * π) = x) ∧ Odd 2 ∨ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd 2",
"after_state": "n : ℕ\nx : ℝ\nhsinx : sin x ≠ 0\nhcosx : cos x ≠ 0\nhcos1 : |cos x| < 1\nhsin1 : |sin x| < 1\nh₀ : 2 ≠ 0\nh : cos x ^ 2 - sin x ^ 2 = cos x ^ 2 + sin x ^ 2\n⊢ (∃ k, ↑k * π = x) ∧ Even 2 ∨ (∃ k, ↑k * (2 * π) = x) ∧ Odd 2 ∨ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd 2"
},
{
"line": "rewrite [← cos_sq_add_sin_sq x] at h",
"before_state": "n : ℕ\nx : ℝ\nhsinx : sin x ≠ 0\nhcosx : cos x ≠ 0\nhcos1 : |cos x| < 1\nhsin1 : |sin x| < 1\nh₀ : 2 ≠ 0\nh : cos x ^ 2 - sin x ^ 2 = 1\n⊢ (∃ k, ↑k * π = x) ∧ Even 2 ∨ (∃ k, ↑k * (2 * π) = x) ∧ Odd 2 ∨ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd 2",
"after_state": "n : ℕ\nx : ℝ\nhsinx : sin x ≠ 0\nhcosx : cos x ≠ 0\nhcos1 : |cos x| < 1\nhsin1 : |sin x| < 1\nh₀ : 2 ≠ 0\nh : cos x ^ 2 - sin x ^ 2 = cos x ^ 2 + sin x ^ 2\n⊢ (∃ k, ↑k * π = x) ∧ Even 2 ∨ (∃ k, ↑k * (2 * π) = x) ∧ Odd 2 ∨ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd 2"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "n : ℕ\nx : ℝ\nhsinx : sin x ≠ 0\nhcosx : cos x ≠ 0\nhcos1 : |cos x| < 1\nhsin1 : |sin x| < 1\nh₀ : 2 ≠ 0\nh : cos x ^ 2 - sin x ^ 2 = cos x ^ 2 + sin x ^ 2\n⊢ (∃ k, ↑k * π = x) ∧ Even 2 ∨ (∃ k, ↑k * (2 * π) = x) ∧ Odd 2 ∨ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd 2",
"after_state": "n : ℕ\nx : ℝ\nhsinx : sin x ≠ 0\nhcosx : cos x ≠ 0\nhcos1 : |cos x| < 1\nhsin1 : |sin x| < 1\nh₀ : 2 ≠ 0\nh : cos x ^ 2 - sin x ^ 2 = cos x ^ 2 + sin x ^ 2\n⊢ (∃ k, ↑k * π = x) ∧ Even 2 ∨ (∃ k, ↑k * (2 * π) = x) ∧ Odd 2 ∨ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd 2"
},
{
"line": "try (with_reducible rfl)",
"before_state": "n : ℕ\nx : ℝ\nhsinx : sin x ≠ 0\nhcosx : cos x ≠ 0\nhcos1 : |cos x| < 1\nhsin1 : |sin x| < 1\nh₀ : 2 ≠ 0\nh : cos x ^ 2 - sin x ^ 2 = cos x ^ 2 + sin x ^ 2\n⊢ (∃ k, ↑k * π = x) ∧ Even 2 ∨ (∃ k, ↑k * (2 * π) = x) ∧ Odd 2 ∨ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd 2",
"after_state": "n : ℕ\nx : ℝ\nhsinx : sin x ≠ 0\nhcosx : cos x ≠ 0\nhcos1 : |cos x| < 1\nhsin1 : |sin x| < 1\nh₀ : 2 ≠ 0\nh : cos x ^ 2 - sin x ^ 2 = cos x ^ 2 + sin x ^ 2\n⊢ (∃ k, ↑k * π = x) ∧ Even 2 ∨ (∃ k, ↑k * (2 * π) = x) ∧ Odd 2 ∨ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd 2"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "n : ℕ\nx : ℝ\nhsinx : sin x ≠ 0\nhcosx : cos x ≠ 0\nhcos1 : |cos x| < 1\nhsin1 : |sin x| < 1\nh₀ : 2 ≠ 0\nh : cos x ^ 2 - sin x ^ 2 = cos x ^ 2 + sin x ^ 2\n⊢ (∃ k, ↑k * π = x) ∧ Even 2 ∨ (∃ k, ↑k * (2 * π) = x) ∧ Odd 2 ∨ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd 2",
"after_state": "n : ℕ\nx : ℝ\nhsinx : sin x ≠ 0\nhcosx : cos x ≠ 0\nhcos1 : |cos x| < 1\nhsin1 : |sin x| < 1\nh₀ : 2 ≠ 0\nh : cos x ^ 2 - sin x ^ 2 = cos x ^ 2 + sin x ^ 2\n⊢ (∃ k, ↑k * π = x) ∧ Even 2 ∨ (∃ k, ↑k * (2 * π) = x) ∧ Odd 2 ∨ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd 2"
},
{
"line": "with_reducible rfl",
"before_state": "n : ℕ\nx : ℝ\nhsinx : sin x ≠ 0\nhcosx : cos x ≠ 0\nhcos1 : |cos x| < 1\nhsin1 : |sin x| < 1\nh₀ : 2 ≠ 0\nh : cos x ^ 2 - sin x ^ 2 = cos x ^ 2 + sin x ^ 2\n⊢ (∃ k, ↑k * π = x) ∧ Even 2 ∨ (∃ k, ↑k * (2 * π) = x) ∧ Odd 2 ∨ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd 2",
"after_state": "n : ℕ\nx : ℝ\nhsinx : sin x ≠ 0\nhcosx : cos x ≠ 0\nhcos1 : |cos x| < 1\nhsin1 : |sin x| < 1\nh₀ : 2 ≠ 0\nh : cos x ^ 2 - sin x ^ 2 = cos x ^ 2 + sin x ^ 2\n⊢ (∃ k, ↑k * π = x) ∧ Even 2 ∨ (∃ k, ↑k * (2 * π) = x) ∧ Odd 2 ∨ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd 2"
},
{
"line": "rfl",
"before_state": "n : ℕ\nx : ℝ\nhsinx : sin x ≠ 0\nhcosx : cos x ≠ 0\nhcos1 : |cos x| < 1\nhsin1 : |sin x| < 1\nh₀ : 2 ≠ 0\nh : cos x ^ 2 - sin x ^ 2 = cos x ^ 2 + sin x ^ 2\n⊢ (∃ k, ↑k * π = x) ∧ Even 2 ∨ (∃ k, ↑k * (2 * π) = x) ∧ Odd 2 ∨ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd 2",
"after_state": "n : ℕ\nx : ℝ\nhsinx : sin x ≠ 0\nhcosx : cos x ≠ 0\nhcos1 : |cos x| < 1\nhsin1 : |sin x| < 1\nh₀ : 2 ≠ 0\nh : cos x ^ 2 - sin x ^ 2 = cos x ^ 2 + sin x ^ 2\n⊢ (∃ k, ↑k * π = x) ∧ Even 2 ∨ (∃ k, ↑k * (2 * π) = x) ∧ Odd 2 ∨ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd 2"
},
{
"line": "apply_rfl",
"before_state": "n : ℕ\nx : ℝ\nhsinx : sin x ≠ 0\nhcosx : cos x ≠ 0\nhcos1 : |cos x| < 1\nhsin1 : |sin x| < 1\nh₀ : 2 ≠ 0\nh : cos x ^ 2 - sin x ^ 2 = cos x ^ 2 + sin x ^ 2\n⊢ (∃ k, ↑k * π = x) ∧ Even 2 ∨ (∃ k, ↑k * (2 * π) = x) ∧ Odd 2 ∨ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd 2",
"after_state": "n : ℕ\nx : ℝ\nhsinx : sin x ≠ 0\nhcosx : cos x ≠ 0\nhcos1 : |cos x| < 1\nhsin1 : |sin x| < 1\nh₀ : 2 ≠ 0\nh : cos x ^ 2 - sin x ^ 2 = cos x ^ 2 + sin x ^ 2\n⊢ (∃ k, ↑k * π = x) ∧ Even 2 ∨ (∃ k, ↑k * (2 * π) = x) ∧ Odd 2 ∨ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd 2"
},
{
"line": "skip",
"before_state": "n : ℕ\nx : ℝ\nhsinx : sin x ≠ 0\nhcosx : cos x ≠ 0\nhcos1 : |cos x| < 1\nhsin1 : |sin x| < 1\nh₀ : 2 ≠ 0\nh : cos x ^ 2 - sin x ^ 2 = cos x ^ 2 + sin x ^ 2\n⊢ (∃ k, ↑k * π = x) ∧ Even 2 ∨ (∃ k, ↑k * (2 * π) = x) ∧ Odd 2 ∨ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd 2",
"after_state": "n : ℕ\nx : ℝ\nhsinx : sin x ≠ 0\nhcosx : cos x ≠ 0\nhcos1 : |cos x| < 1\nhsin1 : |sin x| < 1\nh₀ : 2 ≠ 0\nh : cos x ^ 2 - sin x ^ 2 = cos x ^ 2 + sin x ^ 2\n⊢ (∃ k, ↑k * π = x) ∧ Even 2 ∨ (∃ k, ↑k * (2 * π) = x) ∧ Odd 2 ∨ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd 2"
},
{
"line": "rw [sub_eq_add_neg] at h",
"before_state": "n : ℕ\nx : ℝ\nhsinx : sin x ≠ 0\nhcosx : cos x ≠ 0\nhcos1 : |cos x| < 1\nhsin1 : |sin x| < 1\nh₀ : 2 ≠ 0\nh : cos x ^ 2 - sin x ^ 2 = cos x ^ 2 + sin x ^ 2\n⊢ (∃ k, ↑k * π = x) ∧ Even 2 ∨ (∃ k, ↑k * (2 * π) = x) ∧ Odd 2 ∨ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd 2",
"after_state": "n : ℕ\nx : ℝ\nhsinx : sin x ≠ 0\nhcosx : cos x ≠ 0\nhcos1 : |cos x| < 1\nhsin1 : |sin x| < 1\nh₀ : 2 ≠ 0\nh : cos x ^ 2 + -sin x ^ 2 = cos x ^ 2 + sin x ^ 2\n⊢ (∃ k, ↑k * π = x) ∧ Even 2 ∨ (∃ k, ↑k * (2 * π) = x) ∧ Odd 2 ∨ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd 2"
},
{
"line": "rewrite [sub_eq_add_neg] at h",
"before_state": "n : ℕ\nx : ℝ\nhsinx : sin x ≠ 0\nhcosx : cos x ≠ 0\nhcos1 : |cos x| < 1\nhsin1 : |sin x| < 1\nh₀ : 2 ≠ 0\nh : cos x ^ 2 - sin x ^ 2 = cos x ^ 2 + sin x ^ 2\n⊢ (∃ k, ↑k * π = x) ∧ Even 2 ∨ (∃ k, ↑k * (2 * π) = x) ∧ Odd 2 ∨ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd 2",
"after_state": "n : ℕ\nx : ℝ\nhsinx : sin x ≠ 0\nhcosx : cos x ≠ 0\nhcos1 : |cos x| < 1\nhsin1 : |sin x| < 1\nh₀ : 2 ≠ 0\nh : cos x ^ 2 + -sin x ^ 2 = cos x ^ 2 + sin x ^ 2\n⊢ (∃ k, ↑k * π = x) ∧ Even 2 ∨ (∃ k, ↑k * (2 * π) = x) ∧ Odd 2 ∨ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd 2"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "n : ℕ\nx : ℝ\nhsinx : sin x ≠ 0\nhcosx : cos x ≠ 0\nhcos1 : |cos x| < 1\nhsin1 : |sin x| < 1\nh₀ : 2 ≠ 0\nh : cos x ^ 2 + -sin x ^ 2 = cos x ^ 2 + sin x ^ 2\n⊢ (∃ k, ↑k * π = x) ∧ Even 2 ∨ (∃ k, ↑k * (2 * π) = x) ∧ Odd 2 ∨ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd 2",
"after_state": "n : ℕ\nx : ℝ\nhsinx : sin x ≠ 0\nhcosx : cos x ≠ 0\nhcos1 : |cos x| < 1\nhsin1 : |sin x| < 1\nh₀ : 2 ≠ 0\nh : cos x ^ 2 + -sin x ^ 2 = cos x ^ 2 + sin x ^ 2\n⊢ (∃ k, ↑k * π = x) ∧ Even 2 ∨ (∃ k, ↑k * (2 * π) = x) ∧ Odd 2 ∨ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd 2"
},
{
"line": "try (with_reducible rfl)",
"before_state": "n : ℕ\nx : ℝ\nhsinx : sin x ≠ 0\nhcosx : cos x ≠ 0\nhcos1 : |cos x| < 1\nhsin1 : |sin x| < 1\nh₀ : 2 ≠ 0\nh : cos x ^ 2 + -sin x ^ 2 = cos x ^ 2 + sin x ^ 2\n⊢ (∃ k, ↑k * π = x) ∧ Even 2 ∨ (∃ k, ↑k * (2 * π) = x) ∧ Odd 2 ∨ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd 2",
"after_state": "n : ℕ\nx : ℝ\nhsinx : sin x ≠ 0\nhcosx : cos x ≠ 0\nhcos1 : |cos x| < 1\nhsin1 : |sin x| < 1\nh₀ : 2 ≠ 0\nh : cos x ^ 2 + -sin x ^ 2 = cos x ^ 2 + sin x ^ 2\n⊢ (∃ k, ↑k * π = x) ∧ Even 2 ∨ (∃ k, ↑k * (2 * π) = x) ∧ Odd 2 ∨ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd 2"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "n : ℕ\nx : ℝ\nhsinx : sin x ≠ 0\nhcosx : cos x ≠ 0\nhcos1 : |cos x| < 1\nhsin1 : |sin x| < 1\nh₀ : 2 ≠ 0\nh : cos x ^ 2 + -sin x ^ 2 = cos x ^ 2 + sin x ^ 2\n⊢ (∃ k, ↑k * π = x) ∧ Even 2 ∨ (∃ k, ↑k * (2 * π) = x) ∧ Odd 2 ∨ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd 2",
"after_state": "n : ℕ\nx : ℝ\nhsinx : sin x ≠ 0\nhcosx : cos x ≠ 0\nhcos1 : |cos x| < 1\nhsin1 : |sin x| < 1\nh₀ : 2 ≠ 0\nh : cos x ^ 2 + -sin x ^ 2 = cos x ^ 2 + sin x ^ 2\n⊢ (∃ k, ↑k * π = x) ∧ Even 2 ∨ (∃ k, ↑k * (2 * π) = x) ∧ Odd 2 ∨ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd 2"
},
{
"line": "with_reducible rfl",
"before_state": "n : ℕ\nx : ℝ\nhsinx : sin x ≠ 0\nhcosx : cos x ≠ 0\nhcos1 : |cos x| < 1\nhsin1 : |sin x| < 1\nh₀ : 2 ≠ 0\nh : cos x ^ 2 + -sin x ^ 2 = cos x ^ 2 + sin x ^ 2\n⊢ (∃ k, ↑k * π = x) ∧ Even 2 ∨ (∃ k, ↑k * (2 * π) = x) ∧ Odd 2 ∨ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd 2",
"after_state": "n : ℕ\nx : ℝ\nhsinx : sin x ≠ 0\nhcosx : cos x ≠ 0\nhcos1 : |cos x| < 1\nhsin1 : |sin x| < 1\nh₀ : 2 ≠ 0\nh : cos x ^ 2 + -sin x ^ 2 = cos x ^ 2 + sin x ^ 2\n⊢ (∃ k, ↑k * π = x) ∧ Even 2 ∨ (∃ k, ↑k * (2 * π) = x) ∧ Odd 2 ∨ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd 2"
},
{
"line": "rfl",
"before_state": "n : ℕ\nx : ℝ\nhsinx : sin x ≠ 0\nhcosx : cos x ≠ 0\nhcos1 : |cos x| < 1\nhsin1 : |sin x| < 1\nh₀ : 2 ≠ 0\nh : cos x ^ 2 + -sin x ^ 2 = cos x ^ 2 + sin x ^ 2\n⊢ (∃ k, ↑k * π = x) ∧ Even 2 ∨ (∃ k, ↑k * (2 * π) = x) ∧ Odd 2 ∨ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd 2",
"after_state": "n : ℕ\nx : ℝ\nhsinx : sin x ≠ 0\nhcosx : cos x ≠ 0\nhcos1 : |cos x| < 1\nhsin1 : |sin x| < 1\nh₀ : 2 ≠ 0\nh : cos x ^ 2 + -sin x ^ 2 = cos x ^ 2 + sin x ^ 2\n⊢ (∃ k, ↑k * π = x) ∧ Even 2 ∨ (∃ k, ↑k * (2 * π) = x) ∧ Odd 2 ∨ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd 2"
},
{
"line": "apply_rfl",
"before_state": "n : ℕ\nx : ℝ\nhsinx : sin x ≠ 0\nhcosx : cos x ≠ 0\nhcos1 : |cos x| < 1\nhsin1 : |sin x| < 1\nh₀ : 2 ≠ 0\nh : cos x ^ 2 + -sin x ^ 2 = cos x ^ 2 + sin x ^ 2\n⊢ (∃ k, ↑k * π = x) ∧ Even 2 ∨ (∃ k, ↑k * (2 * π) = x) ∧ Odd 2 ∨ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd 2",
"after_state": "n : ℕ\nx : ℝ\nhsinx : sin x ≠ 0\nhcosx : cos x ≠ 0\nhcos1 : |cos x| < 1\nhsin1 : |sin x| < 1\nh₀ : 2 ≠ 0\nh : cos x ^ 2 + -sin x ^ 2 = cos x ^ 2 + sin x ^ 2\n⊢ (∃ k, ↑k * π = x) ∧ Even 2 ∨ (∃ k, ↑k * (2 * π) = x) ∧ Odd 2 ∨ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd 2"
},
{
"line": "skip",
"before_state": "n : ℕ\nx : ℝ\nhsinx : sin x ≠ 0\nhcosx : cos x ≠ 0\nhcos1 : |cos x| < 1\nhsin1 : |sin x| < 1\nh₀ : 2 ≠ 0\nh : cos x ^ 2 + -sin x ^ 2 = cos x ^ 2 + sin x ^ 2\n⊢ (∃ k, ↑k * π = x) ∧ Even 2 ∨ (∃ k, ↑k * (2 * π) = x) ∧ Odd 2 ∨ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd 2",
"after_state": "n : ℕ\nx : ℝ\nhsinx : sin x ≠ 0\nhcosx : cos x ≠ 0\nhcos1 : |cos x| < 1\nhsin1 : |sin x| < 1\nh₀ : 2 ≠ 0\nh : cos x ^ 2 + -sin x ^ 2 = cos x ^ 2 + sin x ^ 2\n⊢ (∃ k, ↑k * π = x) ∧ Even 2 ∨ (∃ k, ↑k * (2 * π) = x) ∧ Odd 2 ∨ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd 2"
},
{
"line": "rw [add_right_inj] at h",
"before_state": "n : ℕ\nx : ℝ\nhsinx : sin x ≠ 0\nhcosx : cos x ≠ 0\nhcos1 : |cos x| < 1\nhsin1 : |sin x| < 1\nh₀ : 2 ≠ 0\nh : cos x ^ 2 + -sin x ^ 2 = cos x ^ 2 + sin x ^ 2\n⊢ (∃ k, ↑k * π = x) ∧ Even 2 ∨ (∃ k, ↑k * (2 * π) = x) ∧ Odd 2 ∨ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd 2",
"after_state": "n : ℕ\nx : ℝ\nhsinx : sin x ≠ 0\nhcosx : cos x ≠ 0\nhcos1 : |cos x| < 1\nhsin1 : |sin x| < 1\nh₀ : 2 ≠ 0\nh : -sin x ^ 2 = sin x ^ 2\n⊢ (∃ k, ↑k * π = x) ∧ Even 2 ∨ (∃ k, ↑k * (2 * π) = x) ∧ Odd 2 ∨ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd 2"
},
{
"line": "rewrite [add_right_inj] at h",
"before_state": "n : ℕ\nx : ℝ\nhsinx : sin x ≠ 0\nhcosx : cos x ≠ 0\nhcos1 : |cos x| < 1\nhsin1 : |sin x| < 1\nh₀ : 2 ≠ 0\nh : cos x ^ 2 + -sin x ^ 2 = cos x ^ 2 + sin x ^ 2\n⊢ (∃ k, ↑k * π = x) ∧ Even 2 ∨ (∃ k, ↑k * (2 * π) = x) ∧ Odd 2 ∨ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd 2",
"after_state": "n : ℕ\nx : ℝ\nhsinx : sin x ≠ 0\nhcosx : cos x ≠ 0\nhcos1 : |cos x| < 1\nhsin1 : |sin x| < 1\nh₀ : 2 ≠ 0\nh : -sin x ^ 2 = sin x ^ 2\n⊢ (∃ k, ↑k * π = x) ∧ Even 2 ∨ (∃ k, ↑k * (2 * π) = x) ∧ Odd 2 ∨ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd 2"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "n : ℕ\nx : ℝ\nhsinx : sin x ≠ 0\nhcosx : cos x ≠ 0\nhcos1 : |cos x| < 1\nhsin1 : |sin x| < 1\nh₀ : 2 ≠ 0\nh : -sin x ^ 2 = sin x ^ 2\n⊢ (∃ k, ↑k * π = x) ∧ Even 2 ∨ (∃ k, ↑k * (2 * π) = x) ∧ Odd 2 ∨ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd 2",
"after_state": "n : ℕ\nx : ℝ\nhsinx : sin x ≠ 0\nhcosx : cos x ≠ 0\nhcos1 : |cos x| < 1\nhsin1 : |sin x| < 1\nh₀ : 2 ≠ 0\nh : -sin x ^ 2 = sin x ^ 2\n⊢ (∃ k, ↑k * π = x) ∧ Even 2 ∨ (∃ k, ↑k * (2 * π) = x) ∧ Odd 2 ∨ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd 2"
},
{
"line": "try (with_reducible rfl)",
"before_state": "n : ℕ\nx : ℝ\nhsinx : sin x ≠ 0\nhcosx : cos x ≠ 0\nhcos1 : |cos x| < 1\nhsin1 : |sin x| < 1\nh₀ : 2 ≠ 0\nh : -sin x ^ 2 = sin x ^ 2\n⊢ (∃ k, ↑k * π = x) ∧ Even 2 ∨ (∃ k, ↑k * (2 * π) = x) ∧ Odd 2 ∨ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd 2",
"after_state": "n : ℕ\nx : ℝ\nhsinx : sin x ≠ 0\nhcosx : cos x ≠ 0\nhcos1 : |cos x| < 1\nhsin1 : |sin x| < 1\nh₀ : 2 ≠ 0\nh : -sin x ^ 2 = sin x ^ 2\n⊢ (∃ k, ↑k * π = x) ∧ Even 2 ∨ (∃ k, ↑k * (2 * π) = x) ∧ Odd 2 ∨ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd 2"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "n : ℕ\nx : ℝ\nhsinx : sin x ≠ 0\nhcosx : cos x ≠ 0\nhcos1 : |cos x| < 1\nhsin1 : |sin x| < 1\nh₀ : 2 ≠ 0\nh : -sin x ^ 2 = sin x ^ 2\n⊢ (∃ k, ↑k * π = x) ∧ Even 2 ∨ (∃ k, ↑k * (2 * π) = x) ∧ Odd 2 ∨ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd 2",
"after_state": "n : ℕ\nx : ℝ\nhsinx : sin x ≠ 0\nhcosx : cos x ≠ 0\nhcos1 : |cos x| < 1\nhsin1 : |sin x| < 1\nh₀ : 2 ≠ 0\nh : -sin x ^ 2 = sin x ^ 2\n⊢ (∃ k, ↑k * π = x) ∧ Even 2 ∨ (∃ k, ↑k * (2 * π) = x) ∧ Odd 2 ∨ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd 2"
},
{
"line": "with_reducible rfl",
"before_state": "n : ℕ\nx : ℝ\nhsinx : sin x ≠ 0\nhcosx : cos x ≠ 0\nhcos1 : |cos x| < 1\nhsin1 : |sin x| < 1\nh₀ : 2 ≠ 0\nh : -sin x ^ 2 = sin x ^ 2\n⊢ (∃ k, ↑k * π = x) ∧ Even 2 ∨ (∃ k, ↑k * (2 * π) = x) ∧ Odd 2 ∨ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd 2",
"after_state": "n : ℕ\nx : ℝ\nhsinx : sin x ≠ 0\nhcosx : cos x ≠ 0\nhcos1 : |cos x| < 1\nhsin1 : |sin x| < 1\nh₀ : 2 ≠ 0\nh : -sin x ^ 2 = sin x ^ 2\n⊢ (∃ k, ↑k * π = x) ∧ Even 2 ∨ (∃ k, ↑k * (2 * π) = x) ∧ Odd 2 ∨ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd 2"
},
{
"line": "rfl",
"before_state": "n : ℕ\nx : ℝ\nhsinx : sin x ≠ 0\nhcosx : cos x ≠ 0\nhcos1 : |cos x| < 1\nhsin1 : |sin x| < 1\nh₀ : 2 ≠ 0\nh : -sin x ^ 2 = sin x ^ 2\n⊢ (∃ k, ↑k * π = x) ∧ Even 2 ∨ (∃ k, ↑k * (2 * π) = x) ∧ Odd 2 ∨ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd 2",
"after_state": "n : ℕ\nx : ℝ\nhsinx : sin x ≠ 0\nhcosx : cos x ≠ 0\nhcos1 : |cos x| < 1\nhsin1 : |sin x| < 1\nh₀ : 2 ≠ 0\nh : -sin x ^ 2 = sin x ^ 2\n⊢ (∃ k, ↑k * π = x) ∧ Even 2 ∨ (∃ k, ↑k * (2 * π) = x) ∧ Odd 2 ∨ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd 2"
},
{
"line": "apply_rfl",
"before_state": "n : ℕ\nx : ℝ\nhsinx : sin x ≠ 0\nhcosx : cos x ≠ 0\nhcos1 : |cos x| < 1\nhsin1 : |sin x| < 1\nh₀ : 2 ≠ 0\nh : -sin x ^ 2 = sin x ^ 2\n⊢ (∃ k, ↑k * π = x) ∧ Even 2 ∨ (∃ k, ↑k * (2 * π) = x) ∧ Odd 2 ∨ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd 2",
"after_state": "n : ℕ\nx : ℝ\nhsinx : sin x ≠ 0\nhcosx : cos x ≠ 0\nhcos1 : |cos x| < 1\nhsin1 : |sin x| < 1\nh₀ : 2 ≠ 0\nh : -sin x ^ 2 = sin x ^ 2\n⊢ (∃ k, ↑k * π = x) ∧ Even 2 ∨ (∃ k, ↑k * (2 * π) = x) ∧ Odd 2 ∨ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd 2"
},
{
"line": "skip",
"before_state": "n : ℕ\nx : ℝ\nhsinx : sin x ≠ 0\nhcosx : cos x ≠ 0\nhcos1 : |cos x| < 1\nhsin1 : |sin x| < 1\nh₀ : 2 ≠ 0\nh : -sin x ^ 2 = sin x ^ 2\n⊢ (∃ k, ↑k * π = x) ∧ Even 2 ∨ (∃ k, ↑k * (2 * π) = x) ∧ Odd 2 ∨ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd 2",
"after_state": "n : ℕ\nx : ℝ\nhsinx : sin x ≠ 0\nhcosx : cos x ≠ 0\nhcos1 : |cos x| < 1\nhsin1 : |sin x| < 1\nh₀ : 2 ≠ 0\nh : -sin x ^ 2 = sin x ^ 2\n⊢ (∃ k, ↑k * π = x) ∧ Even 2 ∨ (∃ k, ↑k * (2 * π) = x) ∧ Odd 2 ∨ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd 2"
},
{
"line": "rw [neg_eq_self ℝ] at h",
"before_state": "n : ℕ\nx : ℝ\nhsinx : sin x ≠ 0\nhcosx : cos x ≠ 0\nhcos1 : |cos x| < 1\nhsin1 : |sin x| < 1\nh₀ : 2 ≠ 0\nh : -sin x ^ 2 = sin x ^ 2\n⊢ (∃ k, ↑k * π = x) ∧ Even 2 ∨ (∃ k, ↑k * (2 * π) = x) ∧ Odd 2 ∨ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd 2",
"after_state": "n : ℕ\nx : ℝ\nhsinx : sin x ≠ 0\nhcosx : cos x ≠ 0\nhcos1 : |cos x| < 1\nhsin1 : |sin x| < 1\nh₀ : 2 ≠ 0\nh : sin x ^ 2 = 0\n⊢ (∃ k, ↑k * π = x) ∧ Even 2 ∨ (∃ k, ↑k * (2 * π) = x) ∧ Odd 2 ∨ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd 2"
},
{
"line": "rewrite [neg_eq_self ℝ] at h",
"before_state": "n : ℕ\nx : ℝ\nhsinx : sin x ≠ 0\nhcosx : cos x ≠ 0\nhcos1 : |cos x| < 1\nhsin1 : |sin x| < 1\nh₀ : 2 ≠ 0\nh : -sin x ^ 2 = sin x ^ 2\n⊢ (∃ k, ↑k * π = x) ∧ Even 2 ∨ (∃ k, ↑k * (2 * π) = x) ∧ Odd 2 ∨ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd 2",
"after_state": "n : ℕ\nx : ℝ\nhsinx : sin x ≠ 0\nhcosx : cos x ≠ 0\nhcos1 : |cos x| < 1\nhsin1 : |sin x| < 1\nh₀ : 2 ≠ 0\nh : sin x ^ 2 = 0\n⊢ (∃ k, ↑k * π = x) ∧ Even 2 ∨ (∃ k, ↑k * (2 * π) = x) ∧ Odd 2 ∨ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd 2"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "n : ℕ\nx : ℝ\nhsinx : sin x ≠ 0\nhcosx : cos x ≠ 0\nhcos1 : |cos x| < 1\nhsin1 : |sin x| < 1\nh₀ : 2 ≠ 0\nh : sin x ^ 2 = 0\n⊢ (∃ k, ↑k * π = x) ∧ Even 2 ∨ (∃ k, ↑k * (2 * π) = x) ∧ Odd 2 ∨ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd 2",
"after_state": "n : ℕ\nx : ℝ\nhsinx : sin x ≠ 0\nhcosx : cos x ≠ 0\nhcos1 : |cos x| < 1\nhsin1 : |sin x| < 1\nh₀ : 2 ≠ 0\nh : sin x ^ 2 = 0\n⊢ (∃ k, ↑k * π = x) ∧ Even 2 ∨ (∃ k, ↑k * (2 * π) = x) ∧ Odd 2 ∨ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd 2"
},
{
"line": "try (with_reducible rfl)",
"before_state": "n : ℕ\nx : ℝ\nhsinx : sin x ≠ 0\nhcosx : cos x ≠ 0\nhcos1 : |cos x| < 1\nhsin1 : |sin x| < 1\nh₀ : 2 ≠ 0\nh : sin x ^ 2 = 0\n⊢ (∃ k, ↑k * π = x) ∧ Even 2 ∨ (∃ k, ↑k * (2 * π) = x) ∧ Odd 2 ∨ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd 2",
"after_state": "n : ℕ\nx : ℝ\nhsinx : sin x ≠ 0\nhcosx : cos x ≠ 0\nhcos1 : |cos x| < 1\nhsin1 : |sin x| < 1\nh₀ : 2 ≠ 0\nh : sin x ^ 2 = 0\n⊢ (∃ k, ↑k * π = x) ∧ Even 2 ∨ (∃ k, ↑k * (2 * π) = x) ∧ Odd 2 ∨ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd 2"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "n : ℕ\nx : ℝ\nhsinx : sin x ≠ 0\nhcosx : cos x ≠ 0\nhcos1 : |cos x| < 1\nhsin1 : |sin x| < 1\nh₀ : 2 ≠ 0\nh : sin x ^ 2 = 0\n⊢ (∃ k, ↑k * π = x) ∧ Even 2 ∨ (∃ k, ↑k * (2 * π) = x) ∧ Odd 2 ∨ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd 2",
"after_state": "n : ℕ\nx : ℝ\nhsinx : sin x ≠ 0\nhcosx : cos x ≠ 0\nhcos1 : |cos x| < 1\nhsin1 : |sin x| < 1\nh₀ : 2 ≠ 0\nh : sin x ^ 2 = 0\n⊢ (∃ k, ↑k * π = x) ∧ Even 2 ∨ (∃ k, ↑k * (2 * π) = x) ∧ Odd 2 ∨ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd 2"
},
{
"line": "with_reducible rfl",
"before_state": "n : ℕ\nx : ℝ\nhsinx : sin x ≠ 0\nhcosx : cos x ≠ 0\nhcos1 : |cos x| < 1\nhsin1 : |sin x| < 1\nh₀ : 2 ≠ 0\nh : sin x ^ 2 = 0\n⊢ (∃ k, ↑k * π = x) ∧ Even 2 ∨ (∃ k, ↑k * (2 * π) = x) ∧ Odd 2 ∨ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd 2",
"after_state": "n : ℕ\nx : ℝ\nhsinx : sin x ≠ 0\nhcosx : cos x ≠ 0\nhcos1 : |cos x| < 1\nhsin1 : |sin x| < 1\nh₀ : 2 ≠ 0\nh : sin x ^ 2 = 0\n⊢ (∃ k, ↑k * π = x) ∧ Even 2 ∨ (∃ k, ↑k * (2 * π) = x) ∧ Odd 2 ∨ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd 2"
},
{
"line": "rfl",
"before_state": "n : ℕ\nx : ℝ\nhsinx : sin x ≠ 0\nhcosx : cos x ≠ 0\nhcos1 : |cos x| < 1\nhsin1 : |sin x| < 1\nh₀ : 2 ≠ 0\nh : sin x ^ 2 = 0\n⊢ (∃ k, ↑k * π = x) ∧ Even 2 ∨ (∃ k, ↑k * (2 * π) = x) ∧ Odd 2 ∨ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd 2",
"after_state": "n : ℕ\nx : ℝ\nhsinx : sin x ≠ 0\nhcosx : cos x ≠ 0\nhcos1 : |cos x| < 1\nhsin1 : |sin x| < 1\nh₀ : 2 ≠ 0\nh : sin x ^ 2 = 0\n⊢ (∃ k, ↑k * π = x) ∧ Even 2 ∨ (∃ k, ↑k * (2 * π) = x) ∧ Odd 2 ∨ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd 2"
},
{
"line": "apply_rfl",
"before_state": "n : ℕ\nx : ℝ\nhsinx : sin x ≠ 0\nhcosx : cos x ≠ 0\nhcos1 : |cos x| < 1\nhsin1 : |sin x| < 1\nh₀ : 2 ≠ 0\nh : sin x ^ 2 = 0\n⊢ (∃ k, ↑k * π = x) ∧ Even 2 ∨ (∃ k, ↑k * (2 * π) = x) ∧ Odd 2 ∨ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd 2",
"after_state": "n : ℕ\nx : ℝ\nhsinx : sin x ≠ 0\nhcosx : cos x ≠ 0\nhcos1 : |cos x| < 1\nhsin1 : |sin x| < 1\nh₀ : 2 ≠ 0\nh : sin x ^ 2 = 0\n⊢ (∃ k, ↑k * π = x) ∧ Even 2 ∨ (∃ k, ↑k * (2 * π) = x) ∧ Odd 2 ∨ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd 2"
},
{
"line": "skip",
"before_state": "n : ℕ\nx : ℝ\nhsinx : sin x ≠ 0\nhcosx : cos x ≠ 0\nhcos1 : |cos x| < 1\nhsin1 : |sin x| < 1\nh₀ : 2 ≠ 0\nh : sin x ^ 2 = 0\n⊢ (∃ k, ↑k * π = x) ∧ Even 2 ∨ (∃ k, ↑k * (2 * π) = x) ∧ Odd 2 ∨ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd 2",
"after_state": "n : ℕ\nx : ℝ\nhsinx : sin x ≠ 0\nhcosx : cos x ≠ 0\nhcos1 : |cos x| < 1\nhsin1 : |sin x| < 1\nh₀ : 2 ≠ 0\nh : sin x ^ 2 = 0\n⊢ (∃ k, ↑k * π = x) ∧ Even 2 ∨ (∃ k, ↑k * (2 * π) = x) ∧ Odd 2 ∨ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd 2"
},
{
"line": "exact absurd (pow_eq_zero h) hsinx",
"before_state": "n : ℕ\nx : ℝ\nhsinx : sin x ≠ 0\nhcosx : cos x ≠ 0\nhcos1 : |cos x| < 1\nhsin1 : |sin x| < 1\nh₀ : 2 ≠ 0\nh : sin x ^ 2 = 0\n⊢ (∃ k, ↑k * π = x) ∧ Even 2 ∨ (∃ k, ↑k * (2 * π) = x) ∧ Odd 2 ∨ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd 2",
"after_state": "No Goals!"
},
{
"line": "case rhs✝ =>\n with_annotate_state[\"|\" \"=>\"] skip\n set m := n + 1\n refine absurd ?_ h.not_lt\n calc\n (cos x) ^ (m + 2) - (sin x) ^ (m + 2) ≤ |cos x| ^ (m + 2) + |sin x| ^ (m + 2) :=\n by\n simp only [← abs_pow]\n simp only [sub_eq_add_neg]\n gcongr\n exacts [le_abs_self _, neg_le_abs _]\n _ = |cos x| ^ m * cos x ^ 2 + |sin x| ^ m * sin x ^ 2 := by simp [pow_add]\n _ < 1 ^ m * cos x ^ 2 + 1 ^ m * sin x ^ 2 := by gcongr\n _ = 1 := by simp",
"before_state": "case rhs\nn✝ : ℕ\nx : ℝ\nhsinx : sin x ≠ 0\nhcosx : cos x ≠ 0\nhcos1 : |cos x| < 1\nhsin1 : |sin x| < 1\nn : ℕ\nh₀ : n + 1 + 2 ≠ 0\nh : cos x ^ (n + 1 + 2) - sin x ^ (n + 1 + 2) = 1\n⊢ (∃ k, ↑k * π = x) ∧ Even (n + 1 + 2) ∨\n (∃ k, ↑k * (2 * π) = x) ∧ Odd (n + 1 + 2) ∨ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd (n + 1 + 2)",
"after_state": "No Goals!"
},
{
"line": "with_annotate_state[\"|\" \"=>\"] skip",
"before_state": "n✝ : ℕ\nx : ℝ\nhsinx : sin x ≠ 0\nhcosx : cos x ≠ 0\nhcos1 : |cos x| < 1\nhsin1 : |sin x| < 1\nn : ℕ\nh₀ : n + 1 + 2 ≠ 0\nh : cos x ^ (n + 1 + 2) - sin x ^ (n + 1 + 2) = 1\n⊢ (∃ k, ↑k * π = x) ∧ Even (n + 1 + 2) ∨\n (∃ k, ↑k * (2 * π) = x) ∧ Odd (n + 1 + 2) ∨ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd (n + 1 + 2)",
"after_state": "n✝ : ℕ\nx : ℝ\nhsinx : sin x ≠ 0\nhcosx : cos x ≠ 0\nhcos1 : |cos x| < 1\nhsin1 : |sin x| < 1\nn : ℕ\nh₀ : n + 1 + 2 ≠ 0\nh : cos x ^ (n + 1 + 2) - sin x ^ (n + 1 + 2) = 1\n⊢ (∃ k, ↑k * π = x) ∧ Even (n + 1 + 2) ∨\n (∃ k, ↑k * (2 * π) = x) ∧ Odd (n + 1 + 2) ∨ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd (n + 1 + 2)"
},
{
"line": "skip",
"before_state": "n✝ : ℕ\nx : ℝ\nhsinx : sin x ≠ 0\nhcosx : cos x ≠ 0\nhcos1 : |cos x| < 1\nhsin1 : |sin x| < 1\nn : ℕ\nh₀ : n + 1 + 2 ≠ 0\nh : cos x ^ (n + 1 + 2) - sin x ^ (n + 1 + 2) = 1\n⊢ (∃ k, ↑k * π = x) ∧ Even (n + 1 + 2) ∨\n (∃ k, ↑k * (2 * π) = x) ∧ Odd (n + 1 + 2) ∨ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd (n + 1 + 2)",
"after_state": "n✝ : ℕ\nx : ℝ\nhsinx : sin x ≠ 0\nhcosx : cos x ≠ 0\nhcos1 : |cos x| < 1\nhsin1 : |sin x| < 1\nn : ℕ\nh₀ : n + 1 + 2 ≠ 0\nh : cos x ^ (n + 1 + 2) - sin x ^ (n + 1 + 2) = 1\n⊢ (∃ k, ↑k * π = x) ∧ Even (n + 1 + 2) ∨\n (∃ k, ↑k * (2 * π) = x) ∧ Odd (n + 1 + 2) ∨ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd (n + 1 + 2)"
},
{
"line": "set m := n + 1",
"before_state": "n✝ : ℕ\nx : ℝ\nhsinx : sin x ≠ 0\nhcosx : cos x ≠ 0\nhcos1 : |cos x| < 1\nhsin1 : |sin x| < 1\nn : ℕ\nh₀ : n + 1 + 2 ≠ 0\nh : cos x ^ (n + 1 + 2) - sin x ^ (n + 1 + 2) = 1\n⊢ (∃ k, ↑k * π = x) ∧ Even (n + 1 + 2) ∨\n (∃ k, ↑k * (2 * π) = x) ∧ Odd (n + 1 + 2) ∨ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd (n + 1 + 2)",
"after_state": "n✝ : ℕ\nx : ℝ\nhsinx : sin x ≠ 0\nhcosx : cos x ≠ 0\nhcos1 : |cos x| < 1\nhsin1 : |sin x| < 1\nn : ℕ\nm : ℕ := n + 1\nh₀ : m + 2 ≠ 0\nh : cos x ^ (m + 2) - sin x ^ (m + 2) = 1\n⊢ (∃ k, ↑k * π = x) ∧ Even (m + 2) ∨\n (∃ k, ↑k * (2 * π) = x) ∧ Odd (m + 2) ∨ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd (m + 2)"
},
{
"line": "try rewrite [show ?m✝ = m from rfl✝] at *",
"before_state": "n✝ : ℕ\nx : ℝ\nhsinx : sin x ≠ 0\nhcosx : cos x ≠ 0\nhcos1 : |cos x| < 1\nhsin1 : |sin x| < 1\nn : ℕ\nh₀ : n + 1 + 2 ≠ 0\nh : cos x ^ (n + 1 + 2) - sin x ^ (n + 1 + 2) = 1\nm : ℕ := n + 1\n⊢ (∃ k, ↑k * π = x) ∧ Even (n + 1 + 2) ∨\n (∃ k, ↑k * (2 * π) = x) ∧ Odd (n + 1 + 2) ∨ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd (n + 1 + 2)",
"after_state": "n✝ : ℕ\nx : ℝ\nhsinx : sin x ≠ 0\nhcosx : cos x ≠ 0\nhcos1 : |cos x| < 1\nhsin1 : |sin x| < 1\nn : ℕ\nm : ℕ := n + 1\nh₀ : m + 2 ≠ 0\nh : cos x ^ (m + 2) - sin x ^ (m + 2) = 1\n⊢ (∃ k, ↑k * π = x) ∧ Even (m + 2) ∨\n (∃ k, ↑k * (2 * π) = x) ∧ Odd (m + 2) ∨ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd (m + 2)"
},
{
"line": "first\n| rewrite [show ?m✝ = m from rfl✝] at *\n| skip",
"before_state": "n✝ : ℕ\nx : ℝ\nhsinx : sin x ≠ 0\nhcosx : cos x ≠ 0\nhcos1 : |cos x| < 1\nhsin1 : |sin x| < 1\nn : ℕ\nh₀ : n + 1 + 2 ≠ 0\nh : cos x ^ (n + 1 + 2) - sin x ^ (n + 1 + 2) = 1\nm : ℕ := n + 1\n⊢ (∃ k, ↑k * π = x) ∧ Even (n + 1 + 2) ∨\n (∃ k, ↑k * (2 * π) = x) ∧ Odd (n + 1 + 2) ∨ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd (n + 1 + 2)",
"after_state": "n✝ : ℕ\nx : ℝ\nhsinx : sin x ≠ 0\nhcosx : cos x ≠ 0\nhcos1 : |cos x| < 1\nhsin1 : |sin x| < 1\nn : ℕ\nm : ℕ := n + 1\nh₀ : m + 2 ≠ 0\nh : cos x ^ (m + 2) - sin x ^ (m + 2) = 1\n⊢ (∃ k, ↑k * π = x) ∧ Even (m + 2) ∨\n (∃ k, ↑k * (2 * π) = x) ∧ Odd (m + 2) ∨ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd (m + 2)"
},
{
"line": "rewrite [show ?m✝ = m from rfl✝] at *",
"before_state": "n✝ : ℕ\nx : ℝ\nhsinx : sin x ≠ 0\nhcosx : cos x ≠ 0\nhcos1 : |cos x| < 1\nhsin1 : |sin x| < 1\nn : ℕ\nh₀ : n + 1 + 2 ≠ 0\nh : cos x ^ (n + 1 + 2) - sin x ^ (n + 1 + 2) = 1\nm : ℕ := n + 1\n⊢ (∃ k, ↑k * π = x) ∧ Even (n + 1 + 2) ∨\n (∃ k, ↑k * (2 * π) = x) ∧ Odd (n + 1 + 2) ∨ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd (n + 1 + 2)",
"after_state": "n✝ : ℕ\nx : ℝ\nhsinx : sin x ≠ 0\nhcosx : cos x ≠ 0\nhcos1 : |cos x| < 1\nhsin1 : |sin x| < 1\nn : ℕ\nm : ℕ := n + 1\nh₀ : m + 2 ≠ 0\nh : cos x ^ (m + 2) - sin x ^ (m + 2) = 1\n⊢ (∃ k, ↑k * π = x) ∧ Even (m + 2) ∨\n (∃ k, ↑k * (2 * π) = x) ∧ Odd (m + 2) ∨ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd (m + 2)"
},
{
"line": "refine absurd ?_ h.not_lt",
"before_state": "n✝ : ℕ\nx : ℝ\nhsinx : sin x ≠ 0\nhcosx : cos x ≠ 0\nhcos1 : |cos x| < 1\nhsin1 : |sin x| < 1\nn : ℕ\nm : ℕ := n + 1\nh₀ : m + 2 ≠ 0\nh : cos x ^ (m + 2) - sin x ^ (m + 2) = 1\n⊢ (∃ k, ↑k * π = x) ∧ Even (m + 2) ∨\n (∃ k, ↑k * (2 * π) = x) ∧ Odd (m + 2) ∨ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd (m + 2)",
"after_state": "n✝ : ℕ\nx : ℝ\nhsinx : sin x ≠ 0\nhcosx : cos x ≠ 0\nhcos1 : |cos x| < 1\nhsin1 : |sin x| < 1\nn : ℕ\nm : ℕ := n + 1\nh₀ : m + 2 ≠ 0\nh : cos x ^ (m + 2) - sin x ^ (m + 2) = 1\n⊢ cos x ^ (m + 2) - sin x ^ (m + 2) < 1"
},
{
"line": "calc\n (cos x) ^ (m + 2) - (sin x) ^ (m + 2) ≤ |cos x| ^ (m + 2) + |sin x| ^ (m + 2) :=\n by\n simp only [← abs_pow]\n simp only [sub_eq_add_neg]\n gcongr\n exacts [le_abs_self _, neg_le_abs _]\n _ = |cos x| ^ m * cos x ^ 2 + |sin x| ^ m * sin x ^ 2 := by simp [pow_add]\n _ < 1 ^ m * cos x ^ 2 + 1 ^ m * sin x ^ 2 := by gcongr\n _ = 1 := by simp",
"before_state": "n✝ : ℕ\nx : ℝ\nhsinx : sin x ≠ 0\nhcosx : cos x ≠ 0\nhcos1 : |cos x| < 1\nhsin1 : |sin x| < 1\nn : ℕ\nm : ℕ := n + 1\nh₀ : m + 2 ≠ 0\nh : cos x ^ (m + 2) - sin x ^ (m + 2) = 1\n⊢ cos x ^ (m + 2) - sin x ^ (m + 2) < 1",
"after_state": "No Goals!"
},
{
"line": "simp only [← abs_pow]",
"before_state": "n✝ : ℕ\nx : ℝ\nhsinx : sin x ≠ 0\nhcosx : cos x ≠ 0\nhcos1 : |cos x| < 1\nhsin1 : |sin x| < 1\nn : ℕ\nm : ℕ := n + 1\nh₀ : m + 2 ≠ 0\nh : cos x ^ (m + 2) - sin x ^ (m + 2) = 1\n⊢ cos x ^ (m + 2) - sin x ^ (m + 2) ≤ |cos x| ^ (m + 2) + |sin x| ^ (m + 2)",
"after_state": "n✝ : ℕ\nx : ℝ\nhsinx : sin x ≠ 0\nhcosx : cos x ≠ 0\nhcos1 : |cos x| < 1\nhsin1 : |sin x| < 1\nn : ℕ\nm : ℕ := n + 1\nh₀ : m + 2 ≠ 0\nh : cos x ^ (m + 2) - sin x ^ (m + 2) = 1\n⊢ cos x ^ (m + 2) - sin x ^ (m + 2) ≤ |cos x ^ (m + 2)| + |sin x ^ (m + 2)|"
},
{
"line": "simp only [sub_eq_add_neg]",
"before_state": "n✝ : ℕ\nx : ℝ\nhsinx : sin x ≠ 0\nhcosx : cos x ≠ 0\nhcos1 : |cos x| < 1\nhsin1 : |sin x| < 1\nn : ℕ\nm : ℕ := n + 1\nh₀ : m + 2 ≠ 0\nh : cos x ^ (m + 2) - sin x ^ (m + 2) = 1\n⊢ cos x ^ (m + 2) - sin x ^ (m + 2) ≤ |cos x ^ (m + 2)| + |sin x ^ (m + 2)|",
"after_state": "n✝ : ℕ\nx : ℝ\nhsinx : sin x ≠ 0\nhcosx : cos x ≠ 0\nhcos1 : |cos x| < 1\nhsin1 : |sin x| < 1\nn : ℕ\nm : ℕ := n + 1\nh₀ : m + 2 ≠ 0\nh : cos x ^ (m + 2) - sin x ^ (m + 2) = 1\n⊢ cos x ^ (m + 2) + -sin x ^ (m + 2) ≤ |cos x ^ (m + 2)| + |sin x ^ (m + 2)|"
},
{
"line": "gcongr",
"before_state": "n✝ : ℕ\nx : ℝ\nhsinx : sin x ≠ 0\nhcosx : cos x ≠ 0\nhcos1 : |cos x| < 1\nhsin1 : |sin x| < 1\nn : ℕ\nm : ℕ := n + 1\nh₀ : m + 2 ≠ 0\nh : cos x ^ (m + 2) - sin x ^ (m + 2) = 1\n⊢ cos x ^ (m + 2) + -sin x ^ (m + 2) ≤ |cos x ^ (m + 2)| + |sin x ^ (m + 2)|",
"after_state": "case h₁\nn✝ : ℕ\nx : ℝ\nhsinx : sin x ≠ 0\nhcosx : cos x ≠ 0\nhcos1 : |cos x| < 1\nhsin1 : |sin x| < 1\nn : ℕ\nm : ℕ := n + 1\nh₀ : m + 2 ≠ 0\nh : cos x ^ (m + 2) - sin x ^ (m + 2) = 1\n⊢ cos x ^ (m + 2) ≤ |cos x ^ (m + 2)|\n---\ncase h₂\nn✝ : ℕ\nx : ℝ\nhsinx : sin x ≠ 0\nhcosx : cos x ≠ 0\nhcos1 : |cos x| < 1\nhsin1 : |sin x| < 1\nn : ℕ\nm : ℕ := n + 1\nh₀ : m + 2 ≠ 0\nh : cos x ^ (m + 2) - sin x ^ (m + 2) = 1\n⊢ -sin x ^ (m + 2) ≤ |sin x ^ (m + 2)|"
},
{
"line": "exacts [le_abs_self _, neg_le_abs _]",
"before_state": "case h₁\nn✝ : ℕ\nx : ℝ\nhsinx : sin x ≠ 0\nhcosx : cos x ≠ 0\nhcos1 : |cos x| < 1\nhsin1 : |sin x| < 1\nn : ℕ\nm : ℕ := n + 1\nh₀ : m + 2 ≠ 0\nh : cos x ^ (m + 2) - sin x ^ (m + 2) = 1\n⊢ cos x ^ (m + 2) ≤ |cos x ^ (m + 2)|\n---\ncase h₂\nn✝ : ℕ\nx : ℝ\nhsinx : sin x ≠ 0\nhcosx : cos x ≠ 0\nhcos1 : |cos x| < 1\nhsin1 : |sin x| < 1\nn : ℕ\nm : ℕ := n + 1\nh₀ : m + 2 ≠ 0\nh : cos x ^ (m + 2) - sin x ^ (m + 2) = 1\n⊢ -sin x ^ (m + 2) ≤ |sin x ^ (m + 2)|",
"after_state": "No Goals!"
},
{
"line": "exact le_abs_self _",
"before_state": "case h₁\nn✝ : ℕ\nx : ℝ\nhsinx : sin x ≠ 0\nhcosx : cos x ≠ 0\nhcos1 : |cos x| < 1\nhsin1 : |sin x| < 1\nn : ℕ\nm : ℕ := n + 1\nh₀ : m + 2 ≠ 0\nh : cos x ^ (m + 2) - sin x ^ (m + 2) = 1\n⊢ cos x ^ (m + 2) ≤ |cos x ^ (m + 2)|\n---\ncase h₂\nn✝ : ℕ\nx : ℝ\nhsinx : sin x ≠ 0\nhcosx : cos x ≠ 0\nhcos1 : |cos x| < 1\nhsin1 : |sin x| < 1\nn : ℕ\nm : ℕ := n + 1\nh₀ : m + 2 ≠ 0\nh : cos x ^ (m + 2) - sin x ^ (m + 2) = 1\n⊢ -sin x ^ (m + 2) ≤ |sin x ^ (m + 2)|",
"after_state": "case h₂\nn✝ : ℕ\nx : ℝ\nhsinx : sin x ≠ 0\nhcosx : cos x ≠ 0\nhcos1 : |cos x| < 1\nhsin1 : |sin x| < 1\nn : ℕ\nm : ℕ := n + 1\nh₀ : m + 2 ≠ 0\nh : cos x ^ (m + 2) - sin x ^ (m + 2) = 1\n⊢ -sin x ^ (m + 2) ≤ |sin x ^ (m + 2)|"
},
{
"line": "exact neg_le_abs _",
"before_state": "case h₂\nn✝ : ℕ\nx : ℝ\nhsinx : sin x ≠ 0\nhcosx : cos x ≠ 0\nhcos1 : |cos x| < 1\nhsin1 : |sin x| < 1\nn : ℕ\nm : ℕ := n + 1\nh₀ : m + 2 ≠ 0\nh : cos x ^ (m + 2) - sin x ^ (m + 2) = 1\n⊢ -sin x ^ (m + 2) ≤ |sin x ^ (m + 2)|",
"after_state": "No Goals!"
},
{
"line": "simp [pow_add]",
"before_state": "n✝ : ℕ\nx : ℝ\nhsinx : sin x ≠ 0\nhcosx : cos x ≠ 0\nhcos1 : |cos x| < 1\nhsin1 : |sin x| < 1\nn : ℕ\nm : ℕ := n + 1\nh₀ : m + 2 ≠ 0\nh : cos x ^ (m + 2) - sin x ^ (m + 2) = 1\n⊢ |cos x| ^ (m + 2) + |sin x| ^ (m + 2) = |cos x| ^ m * cos x ^ 2 + |sin x| ^ m * sin x ^ 2",
"after_state": "No Goals!"
},
{
"line": "gcongr",
"before_state": "n✝ : ℕ\nx : ℝ\nhsinx : sin x ≠ 0\nhcosx : cos x ≠ 0\nhcos1 : |cos x| < 1\nhsin1 : |sin x| < 1\nn : ℕ\nm : ℕ := n + 1\nh₀ : m + 2 ≠ 0\nh : cos x ^ (m + 2) - sin x ^ (m + 2) = 1\n⊢ |cos x| ^ m * cos x ^ 2 + |sin x| ^ m * sin x ^ 2 < 1 ^ m * cos x ^ 2 + 1 ^ m * sin x ^ 2",
"after_state": "No Goals!"
},
{
"line": "gcongr_discharger",
"before_state": "case h₁.bc.ha\nn✝ : ℕ\nx : ℝ\nhsinx : sin x ≠ 0\nhcosx : cos x ≠ 0\nhcos1 : |cos x| < 1\nhsin1 : |sin x| < 1\nn : ℕ\nm : ℕ := n + 1\nh₀ : m + 2 ≠ 0\nh : cos x ^ (m + 2) - sin x ^ (m + 2) = 1\n⊢ 0 ≤ |cos x|",
"after_state": "No Goals!"
},
{
"line": "positivity",
"before_state": "case h₁.bc.ha\nn✝ : ℕ\nx : ℝ\nhsinx : sin x ≠ 0\nhcosx : cos x ≠ 0\nhcos1 : |cos x| < 1\nhsin1 : |sin x| < 1\nn : ℕ\nm : ℕ := n + 1\nh₀ : m + 2 ≠ 0\nh : cos x ^ (m + 2) - sin x ^ (m + 2) = 1\n⊢ 0 ≤ |cos x|",
"after_state": "No Goals!"
},
{
"line": "gcongr_discharger",
"before_state": "case h₁.bc.a\nn✝ : ℕ\nx : ℝ\nhsinx : sin x ≠ 0\nhcosx : cos x ≠ 0\nhcos1 : |cos x| < 1\nhsin1 : |sin x| < 1\nn : ℕ\nm : ℕ := n + 1\nh₀ : m + 2 ≠ 0\nh : cos x ^ (m + 2) - sin x ^ (m + 2) = 1\n⊢ m ≠ 0",
"after_state": "No Goals!"
},
{
"line": "positivity",
"before_state": "case h₁.bc.a\nn✝ : ℕ\nx : ℝ\nhsinx : sin x ≠ 0\nhcosx : cos x ≠ 0\nhcos1 : |cos x| < 1\nhsin1 : |sin x| < 1\nn : ℕ\nm : ℕ := n + 1\nh₀ : m + 2 ≠ 0\nh : cos x ^ (m + 2) - sin x ^ (m + 2) = 1\n⊢ m ≠ 0",
"after_state": "No Goals!"
},
{
"line": "gcongr_discharger",
"before_state": "case h₁.a0\nn✝ : ℕ\nx : ℝ\nhsinx : sin x ≠ 0\nhcosx : cos x ≠ 0\nhcos1 : |cos x| < 1\nhsin1 : |sin x| < 1\nn : ℕ\nm : ℕ := n + 1\nh₀ : m + 2 ≠ 0\nh : cos x ^ (m + 2) - sin x ^ (m + 2) = 1\n⊢ 0 < cos x ^ 2",
"after_state": "No Goals!"
},
{
"line": "positivity",
"before_state": "case h₁.a0\nn✝ : ℕ\nx : ℝ\nhsinx : sin x ≠ 0\nhcosx : cos x ≠ 0\nhcos1 : |cos x| < 1\nhsin1 : |sin x| < 1\nn : ℕ\nm : ℕ := n + 1\nh₀ : m + 2 ≠ 0\nh : cos x ^ (m + 2) - sin x ^ (m + 2) = 1\n⊢ 0 < cos x ^ 2",
"after_state": "No Goals!"
},
{
"line": "gcongr_discharger",
"before_state": "case h₂.bc.ha\nn✝ : ℕ\nx : ℝ\nhsinx : sin x ≠ 0\nhcosx : cos x ≠ 0\nhcos1 : |cos x| < 1\nhsin1 : |sin x| < 1\nn : ℕ\nm : ℕ := n + 1\nh₀ : m + 2 ≠ 0\nh : cos x ^ (m + 2) - sin x ^ (m + 2) = 1\n⊢ 0 ≤ |sin x|",
"after_state": "No Goals!"
},
{
"line": "positivity",
"before_state": "case h₂.bc.ha\nn✝ : ℕ\nx : ℝ\nhsinx : sin x ≠ 0\nhcosx : cos x ≠ 0\nhcos1 : |cos x| < 1\nhsin1 : |sin x| < 1\nn : ℕ\nm : ℕ := n + 1\nh₀ : m + 2 ≠ 0\nh : cos x ^ (m + 2) - sin x ^ (m + 2) = 1\n⊢ 0 ≤ |sin x|",
"after_state": "No Goals!"
},
{
"line": "gcongr_discharger",
"before_state": "case h₂.bc.a\nn✝ : ℕ\nx : ℝ\nhsinx : sin x ≠ 0\nhcosx : cos x ≠ 0\nhcos1 : |cos x| < 1\nhsin1 : |sin x| < 1\nn : ℕ\nm : ℕ := n + 1\nh₀ : m + 2 ≠ 0\nh : cos x ^ (m + 2) - sin x ^ (m + 2) = 1\n⊢ m ≠ 0",
"after_state": "No Goals!"
},
{
"line": "positivity",
"before_state": "case h₂.bc.a\nn✝ : ℕ\nx : ℝ\nhsinx : sin x ≠ 0\nhcosx : cos x ≠ 0\nhcos1 : |cos x| < 1\nhsin1 : |sin x| < 1\nn : ℕ\nm : ℕ := n + 1\nh₀ : m + 2 ≠ 0\nh : cos x ^ (m + 2) - sin x ^ (m + 2) = 1\n⊢ m ≠ 0",
"after_state": "No Goals!"
},
{
"line": "gcongr_discharger",
"before_state": "case h₂.a0\nn✝ : ℕ\nx : ℝ\nhsinx : sin x ≠ 0\nhcosx : cos x ≠ 0\nhcos1 : |cos x| < 1\nhsin1 : |sin x| < 1\nn : ℕ\nm : ℕ := n + 1\nh₀ : m + 2 ≠ 0\nh : cos x ^ (m + 2) - sin x ^ (m + 2) = 1\n⊢ 0 < sin x ^ 2",
"after_state": "No Goals!"
},
{
"line": "positivity",
"before_state": "case h₂.a0\nn✝ : ℕ\nx : ℝ\nhsinx : sin x ≠ 0\nhcosx : cos x ≠ 0\nhcos1 : |cos x| < 1\nhsin1 : |sin x| < 1\nn : ℕ\nm : ℕ := n + 1\nh₀ : m + 2 ≠ 0\nh : cos x ^ (m + 2) - sin x ^ (m + 2) = 1\n⊢ 0 < sin x ^ 2",
"after_state": "No Goals!"
},
{
"line": "simp",
"before_state": "n✝ : ℕ\nx : ℝ\nhsinx : sin x ≠ 0\nhcosx : cos x ≠ 0\nhcos1 : |cos x| < 1\nhsin1 : |sin x| < 1\nn : ℕ\nm : ℕ := n + 1\nh₀ : m + 2 ≠ 0\nh : cos x ^ (m + 2) - sin x ^ (m + 2) = 1\n⊢ 1 ^ m * cos x ^ 2 + 1 ^ m * sin x ^ 2 = 1",
"after_state": "No Goals!"
},
{
"line": "rintro (⟨⟨k, rfl⟩, hn⟩ | ⟨⟨k, rfl⟩, -⟩ | ⟨⟨k, rfl⟩, hn⟩)",
"before_state": "case mpr\nn : ℕ\nx : ℝ\nh₀ : n ≠ 0\n⊢ (∃ k, ↑k * π = x) ∧ Even n ∨ (∃ k, ↑k * (2 * π) = x) ∧ Odd n ∨ (∃ k, -(π / 2) + ↑k * (2 * π) = x) ∧ Odd n →\n cos x ^ n - sin x ^ n = 1",
"after_state": "case mpr.inl.intro.intro\nn : ℕ\nh₀ : n ≠ 0\nhn : Even n\nk : ℤ\n⊢ cos (↑k * π) ^ n - sin (↑k * π) ^ n = 1\n---\ncase mpr.inr.inl.intro.intro\nn : ℕ\nh₀ : n ≠ 0\nk : ℤ\n⊢ cos (↑k * (2 * π)) ^ n - sin (↑k * (2 * π)) ^ n = 1\n---\ncase mpr.inr.inr.intro.intro\nn : ℕ\nh₀ : n ≠ 0\nhn : Odd n\nk : ℤ\n⊢ cos (-(π / 2) + ↑k * (2 * π)) ^ n - sin (-(π / 2) + ↑k * (2 * π)) ^ n = 1"
},
{
"line": "rw [sin_int_mul_pi, zero_pow h₀, sub_zero, ← hn.pow_abs, abs_cos_int_mul_pi, one_pow]",
"before_state": "case mpr.inl.intro.intro\nn : ℕ\nh₀ : n ≠ 0\nhn : Even n\nk : ℤ\n⊢ cos (↑k * π) ^ n - sin (↑k * π) ^ n = 1",
"after_state": "No Goals!"
},
{
"line": "rewrite [sin_int_mul_pi, zero_pow h₀, sub_zero, ← hn.pow_abs, abs_cos_int_mul_pi, one_pow]",
"before_state": "case mpr.inl.intro.intro\nn : ℕ\nh₀ : n ≠ 0\nhn : Even n\nk : ℤ\n⊢ cos (↑k * π) ^ n - sin (↑k * π) ^ n = 1",
"after_state": "case mpr.inl.intro.intro\nn : ℕ\nh₀ : n ≠ 0\nhn : Even n\nk : ℤ\n⊢ 1 = 1"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "case mpr.inl.intro.intro\nn : ℕ\nh₀ : n ≠ 0\nhn : Even n\nk : ℤ\n⊢ 1 = 1",
"after_state": "No Goals!"
},
{
"line": "try (with_reducible rfl)",
"before_state": "case mpr.inl.intro.intro\nn : ℕ\nh₀ : n ≠ 0\nhn : Even n\nk : ℤ\n⊢ 1 = 1",
"after_state": "No Goals!"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "case mpr.inl.intro.intro\nn : ℕ\nh₀ : n ≠ 0\nhn : Even n\nk : ℤ\n⊢ 1 = 1",
"after_state": "No Goals!"
},
{
"line": "with_reducible rfl",
"before_state": "case mpr.inl.intro.intro\nn : ℕ\nh₀ : n ≠ 0\nhn : Even n\nk : ℤ\n⊢ 1 = 1",
"after_state": "No Goals!"
},
{
"line": "rfl",
"before_state": "case mpr.inl.intro.intro\nn : ℕ\nh₀ : n ≠ 0\nhn : Even n\nk : ℤ\n⊢ 1 = 1",
"after_state": "No Goals!"
},
{
"line": "eq_refl",
"before_state": "case mpr.inl.intro.intro\nn : ℕ\nh₀ : n ≠ 0\nhn : Even n\nk : ℤ\n⊢ 1 = 1",
"after_state": "No Goals!"
},
{
"line": "have : sin (k * (2 * π)) = 0 := by simpa [mul_assoc] using sin_int_mul_pi (k * 2)",
"before_state": "case mpr.inr.inl.intro.intro\nn : ℕ\nh₀ : n ≠ 0\nk : ℤ\n⊢ cos (↑k * (2 * π)) ^ n - sin (↑k * (2 * π)) ^ n = 1",
"after_state": "case mpr.inr.inl.intro.intro\nn : ℕ\nh₀ : n ≠ 0\nk : ℤ\nthis : sin (↑k * (2 * π)) = 0\n⊢ cos (↑k * (2 * π)) ^ n - sin (↑k * (2 * π)) ^ n = 1"
},
{
"line": "focus\n refine\n no_implicit_lambda%\n (have : sin (k * (2 * π)) = 0 := ?body✝;\n ?_)\n case body✝ => with_annotate_state\"by\" (simpa [mul_assoc] using sin_int_mul_pi (k * 2))",
"before_state": "case mpr.inr.inl.intro.intro\nn : ℕ\nh₀ : n ≠ 0\nk : ℤ\n⊢ cos (↑k * (2 * π)) ^ n - sin (↑k * (2 * π)) ^ n = 1",
"after_state": "case mpr.inr.inl.intro.intro\nn : ℕ\nh₀ : n ≠ 0\nk : ℤ\nthis : sin (↑k * (2 * π)) = 0\n⊢ cos (↑k * (2 * π)) ^ n - sin (↑k * (2 * π)) ^ n = 1"
},
{
"line": "refine\n no_implicit_lambda%\n (have : sin (k * (2 * π)) = 0 := ?body✝;\n ?_)",
"before_state": "case mpr.inr.inl.intro.intro\nn : ℕ\nh₀ : n ≠ 0\nk : ℤ\n⊢ cos (↑k * (2 * π)) ^ n - sin (↑k * (2 * π)) ^ n = 1",
"after_state": "case body\nn : ℕ\nh₀ : n ≠ 0\nk : ℤ\n⊢ sin (↑k * (2 * π)) = 0\n---\ncase mpr.inr.inl.intro.intro\nn : ℕ\nh₀ : n ≠ 0\nk : ℤ\nthis : sin (↑k * (2 * π)) = 0\n⊢ cos (↑k * (2 * π)) ^ n - sin (↑k * (2 * π)) ^ n = 1"
},
{
"line": "case body✝ => with_annotate_state\"by\" (simpa [mul_assoc] using sin_int_mul_pi (k * 2))",
"before_state": "case body\nn : ℕ\nh₀ : n ≠ 0\nk : ℤ\n⊢ sin (↑k * (2 * π)) = 0\n---\ncase mpr.inr.inl.intro.intro\nn : ℕ\nh₀ : n ≠ 0\nk : ℤ\nthis : sin (↑k * (2 * π)) = 0\n⊢ cos (↑k * (2 * π)) ^ n - sin (↑k * (2 * π)) ^ n = 1",
"after_state": "case mpr.inr.inl.intro.intro\nn : ℕ\nh₀ : n ≠ 0\nk : ℤ\nthis : sin (↑k * (2 * π)) = 0\n⊢ cos (↑k * (2 * π)) ^ n - sin (↑k * (2 * π)) ^ n = 1"
},
{
"line": "with_annotate_state\"by\" (simpa [mul_assoc] using sin_int_mul_pi (k * 2))",
"before_state": "n : ℕ\nh₀ : n ≠ 0\nk : ℤ\n⊢ sin (↑k * (2 * π)) = 0",
"after_state": "No Goals!"
},
{
"line": "simpa [mul_assoc] using sin_int_mul_pi (k * 2)",
"before_state": "n : ℕ\nh₀ : n ≠ 0\nk : ℤ\n⊢ sin (↑k * (2 * π)) = 0",
"after_state": "No Goals!"
},
{
"line": "simp [h₀, this]",
"before_state": "case mpr.inr.inl.intro.intro\nn : ℕ\nh₀ : n ≠ 0\nk : ℤ\nthis : sin (↑k * (2 * π)) = 0\n⊢ cos (↑k * (2 * π)) ^ n - sin (↑k * (2 * π)) ^ n = 1",
"after_state": "No Goals!"
},
{
"line": "simp [hn.neg_pow, h₀]",
"before_state": "case mpr.inr.inr.intro.intro\nn : ℕ\nh₀ : n ≠ 0\nhn : Odd n\nk : ℤ\n⊢ cos (-(π / 2) + ↑k * (2 * π)) ^ n - sin (-(π / 2) + ↑k * (2 * π)) ^ n = 1",
"after_state": "No Goals!"
}
] |
theorem solve_cos2_half {x : ℝ} : cos x ^ 2 = 1 / 2 ↔ ∃ k : ℤ, x = (2 * ↑k + 1) * π / 4 := by
rw [cos_sq]
simp only [add_eq_left]
simp only [div_eq_zero_iff]
norm_num
rw [cos_eq_zero_iff]
constructor <;>
· rintro ⟨k, h⟩
use k
linarith
|
/root/DuelModelResearch/mathlib4/Archive/Imo/Imo1962Q4.lean
|
{
"open": [
"Real",
"scoped Real"
],
"variables": []
}
|
[
{
"line": "rw [cos_sq]",
"before_state": "x : ℝ\n⊢ cos x ^ 2 = 1 / 2 ↔ ∃ k, x = (2 * ↑k + 1) * π / 4",
"after_state": "x : ℝ\n⊢ 1 / 2 + cos (2 * x) / 2 = 1 / 2 ↔ ∃ k, x = (2 * ↑k + 1) * π / 4"
},
{
"line": "rewrite [cos_sq]",
"before_state": "x : ℝ\n⊢ cos x ^ 2 = 1 / 2 ↔ ∃ k, x = (2 * ↑k + 1) * π / 4",
"after_state": "x : ℝ\n⊢ 1 / 2 + cos (2 * x) / 2 = 1 / 2 ↔ ∃ k, x = (2 * ↑k + 1) * π / 4"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "x : ℝ\n⊢ 1 / 2 + cos (2 * x) / 2 = 1 / 2 ↔ ∃ k, x = (2 * ↑k + 1) * π / 4",
"after_state": "x : ℝ\n⊢ 1 / 2 + cos (2 * x) / 2 = 1 / 2 ↔ ∃ k, x = (2 * ↑k + 1) * π / 4"
},
{
"line": "try (with_reducible rfl)",
"before_state": "x : ℝ\n⊢ 1 / 2 + cos (2 * x) / 2 = 1 / 2 ↔ ∃ k, x = (2 * ↑k + 1) * π / 4",
"after_state": "x : ℝ\n⊢ 1 / 2 + cos (2 * x) / 2 = 1 / 2 ↔ ∃ k, x = (2 * ↑k + 1) * π / 4"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "x : ℝ\n⊢ 1 / 2 + cos (2 * x) / 2 = 1 / 2 ↔ ∃ k, x = (2 * ↑k + 1) * π / 4",
"after_state": "x : ℝ\n⊢ 1 / 2 + cos (2 * x) / 2 = 1 / 2 ↔ ∃ k, x = (2 * ↑k + 1) * π / 4"
},
{
"line": "with_reducible rfl",
"before_state": "x : ℝ\n⊢ 1 / 2 + cos (2 * x) / 2 = 1 / 2 ↔ ∃ k, x = (2 * ↑k + 1) * π / 4",
"after_state": "x : ℝ\n⊢ 1 / 2 + cos (2 * x) / 2 = 1 / 2 ↔ ∃ k, x = (2 * ↑k + 1) * π / 4"
},
{
"line": "rfl",
"before_state": "x : ℝ\n⊢ 1 / 2 + cos (2 * x) / 2 = 1 / 2 ↔ ∃ k, x = (2 * ↑k + 1) * π / 4",
"after_state": "x : ℝ\n⊢ 1 / 2 + cos (2 * x) / 2 = 1 / 2 ↔ ∃ k, x = (2 * ↑k + 1) * π / 4"
},
{
"line": "apply_rfl",
"before_state": "x : ℝ\n⊢ 1 / 2 + cos (2 * x) / 2 = 1 / 2 ↔ ∃ k, x = (2 * ↑k + 1) * π / 4",
"after_state": "x : ℝ\n⊢ 1 / 2 + cos (2 * x) / 2 = 1 / 2 ↔ ∃ k, x = (2 * ↑k + 1) * π / 4"
},
{
"line": "skip",
"before_state": "x : ℝ\n⊢ 1 / 2 + cos (2 * x) / 2 = 1 / 2 ↔ ∃ k, x = (2 * ↑k + 1) * π / 4",
"after_state": "x : ℝ\n⊢ 1 / 2 + cos (2 * x) / 2 = 1 / 2 ↔ ∃ k, x = (2 * ↑k + 1) * π / 4"
},
{
"line": "simp only [add_eq_left]",
"before_state": "x : ℝ\n⊢ 1 / 2 + cos (2 * x) / 2 = 1 / 2 ↔ ∃ k, x = (2 * ↑k + 1) * π / 4",
"after_state": "x : ℝ\n⊢ cos (2 * x) / 2 = 0 ↔ ∃ k, x = (2 * ↑k + 1) * π / 4"
},
{
"line": "simp only [div_eq_zero_iff]",
"before_state": "x : ℝ\n⊢ cos (2 * x) / 2 = 0 ↔ ∃ k, x = (2 * ↑k + 1) * π / 4",
"after_state": "x : ℝ\n⊢ cos (2 * x) = 0 ∨ 2 = 0 ↔ ∃ k, x = (2 * ↑k + 1) * π / 4"
},
{
"line": "norm_num",
"before_state": "x : ℝ\n⊢ cos (2 * x) = 0 ∨ 2 = 0 ↔ ∃ k, x = (2 * ↑k + 1) * π / 4",
"after_state": "x : ℝ\n⊢ cos (2 * x) = 0 ↔ ∃ k, x = (2 * ↑k + 1) * π / 4"
},
{
"line": "rw [cos_eq_zero_iff]",
"before_state": "x : ℝ\n⊢ cos (2 * x) = 0 ↔ ∃ k, x = (2 * ↑k + 1) * π / 4",
"after_state": "x : ℝ\n⊢ (∃ k, 2 * x = (2 * ↑k + 1) * π / 2) ↔ ∃ k, x = (2 * ↑k + 1) * π / 4"
},
{
"line": "rewrite [cos_eq_zero_iff]",
"before_state": "x : ℝ\n⊢ cos (2 * x) = 0 ↔ ∃ k, x = (2 * ↑k + 1) * π / 4",
"after_state": "x : ℝ\n⊢ (∃ k, 2 * x = (2 * ↑k + 1) * π / 2) ↔ ∃ k, x = (2 * ↑k + 1) * π / 4"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "x : ℝ\n⊢ (∃ k, 2 * x = (2 * ↑k + 1) * π / 2) ↔ ∃ k, x = (2 * ↑k + 1) * π / 4",
"after_state": "x : ℝ\n⊢ (∃ k, 2 * x = (2 * ↑k + 1) * π / 2) ↔ ∃ k, x = (2 * ↑k + 1) * π / 4"
},
{
"line": "try (with_reducible rfl)",
"before_state": "x : ℝ\n⊢ (∃ k, 2 * x = (2 * ↑k + 1) * π / 2) ↔ ∃ k, x = (2 * ↑k + 1) * π / 4",
"after_state": "x : ℝ\n⊢ (∃ k, 2 * x = (2 * ↑k + 1) * π / 2) ↔ ∃ k, x = (2 * ↑k + 1) * π / 4"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "x : ℝ\n⊢ (∃ k, 2 * x = (2 * ↑k + 1) * π / 2) ↔ ∃ k, x = (2 * ↑k + 1) * π / 4",
"after_state": "x : ℝ\n⊢ (∃ k, 2 * x = (2 * ↑k + 1) * π / 2) ↔ ∃ k, x = (2 * ↑k + 1) * π / 4"
},
{
"line": "with_reducible rfl",
"before_state": "x : ℝ\n⊢ (∃ k, 2 * x = (2 * ↑k + 1) * π / 2) ↔ ∃ k, x = (2 * ↑k + 1) * π / 4",
"after_state": "x : ℝ\n⊢ (∃ k, 2 * x = (2 * ↑k + 1) * π / 2) ↔ ∃ k, x = (2 * ↑k + 1) * π / 4"
},
{
"line": "rfl",
"before_state": "x : ℝ\n⊢ (∃ k, 2 * x = (2 * ↑k + 1) * π / 2) ↔ ∃ k, x = (2 * ↑k + 1) * π / 4",
"after_state": "x : ℝ\n⊢ (∃ k, 2 * x = (2 * ↑k + 1) * π / 2) ↔ ∃ k, x = (2 * ↑k + 1) * π / 4"
},
{
"line": "apply_rfl",
"before_state": "x : ℝ\n⊢ (∃ k, 2 * x = (2 * ↑k + 1) * π / 2) ↔ ∃ k, x = (2 * ↑k + 1) * π / 4",
"after_state": "x : ℝ\n⊢ (∃ k, 2 * x = (2 * ↑k + 1) * π / 2) ↔ ∃ k, x = (2 * ↑k + 1) * π / 4"
},
{
"line": "skip",
"before_state": "x : ℝ\n⊢ (∃ k, 2 * x = (2 * ↑k + 1) * π / 2) ↔ ∃ k, x = (2 * ↑k + 1) * π / 4",
"after_state": "x : ℝ\n⊢ (∃ k, 2 * x = (2 * ↑k + 1) * π / 2) ↔ ∃ k, x = (2 * ↑k + 1) * π / 4"
},
{
"line": "focus\n constructor\n with_annotate_state\"<;>\" skip\n all_goals\n · rintro ⟨k, h⟩\n use k\n linarith",
"before_state": "x : ℝ\n⊢ (∃ k, 2 * x = (2 * ↑k + 1) * π / 2) ↔ ∃ k, x = (2 * ↑k + 1) * π / 4",
"after_state": "No Goals!"
},
{
"line": "constructor",
"before_state": "x : ℝ\n⊢ (∃ k, 2 * x = (2 * ↑k + 1) * π / 2) ↔ ∃ k, x = (2 * ↑k + 1) * π / 4",
"after_state": "case mp\nx : ℝ\n⊢ (∃ k, 2 * x = (2 * ↑k + 1) * π / 2) → ∃ k, x = (2 * ↑k + 1) * π / 4\n---\ncase mpr\nx : ℝ\n⊢ (∃ k, x = (2 * ↑k + 1) * π / 4) → ∃ k, 2 * x = (2 * ↑k + 1) * π / 2"
},
{
"line": "with_annotate_state\"<;>\" skip",
"before_state": "case mp\nx : ℝ\n⊢ (∃ k, 2 * x = (2 * ↑k + 1) * π / 2) → ∃ k, x = (2 * ↑k + 1) * π / 4\n---\ncase mpr\nx : ℝ\n⊢ (∃ k, x = (2 * ↑k + 1) * π / 4) → ∃ k, 2 * x = (2 * ↑k + 1) * π / 2",
"after_state": "case mp\nx : ℝ\n⊢ (∃ k, 2 * x = (2 * ↑k + 1) * π / 2) → ∃ k, x = (2 * ↑k + 1) * π / 4\n---\ncase mpr\nx : ℝ\n⊢ (∃ k, x = (2 * ↑k + 1) * π / 4) → ∃ k, 2 * x = (2 * ↑k + 1) * π / 2"
},
{
"line": "skip",
"before_state": "case mp\nx : ℝ\n⊢ (∃ k, 2 * x = (2 * ↑k + 1) * π / 2) → ∃ k, x = (2 * ↑k + 1) * π / 4\n---\ncase mpr\nx : ℝ\n⊢ (∃ k, x = (2 * ↑k + 1) * π / 4) → ∃ k, 2 * x = (2 * ↑k + 1) * π / 2",
"after_state": "case mp\nx : ℝ\n⊢ (∃ k, 2 * x = (2 * ↑k + 1) * π / 2) → ∃ k, x = (2 * ↑k + 1) * π / 4\n---\ncase mpr\nx : ℝ\n⊢ (∃ k, x = (2 * ↑k + 1) * π / 4) → ∃ k, 2 * x = (2 * ↑k + 1) * π / 2"
},
{
"line": "all_goals\n · rintro ⟨k, h⟩\n use k\n linarith",
"before_state": "case mp\nx : ℝ\n⊢ (∃ k, 2 * x = (2 * ↑k + 1) * π / 2) → ∃ k, x = (2 * ↑k + 1) * π / 4\n---\ncase mpr\nx : ℝ\n⊢ (∃ k, x = (2 * ↑k + 1) * π / 4) → ∃ k, 2 * x = (2 * ↑k + 1) * π / 2",
"after_state": "No Goals!"
},
{
"line": "rintro ⟨k, h⟩",
"before_state": "case mp\nx : ℝ\n⊢ (∃ k, 2 * x = (2 * ↑k + 1) * π / 2) → ∃ k, x = (2 * ↑k + 1) * π / 4",
"after_state": "case mp.intro\nx : ℝ\nk : ℤ\nh : 2 * x = (2 * ↑k + 1) * π / 2\n⊢ ∃ k, x = (2 * ↑k + 1) * π / 4"
},
{
"line": "use k",
"before_state": "case mp.intro\nx : ℝ\nk : ℤ\nh : 2 * x = (2 * ↑k + 1) * π / 2\n⊢ ∃ k, x = (2 * ↑k + 1) * π / 4",
"after_state": "case h\nx : ℝ\nk : ℤ\nh : 2 * x = (2 * ↑k + 1) * π / 2\n⊢ x = (2 * ↑k + 1) * π / 4"
},
{
"line": "refine without_cdot(k : ?m✝)",
"before_state": "case w\nx : ℝ\nk : ℤ\nh : 2 * x = (2 * ↑k + 1) * π / 2\n⊢ ℤ",
"after_state": "No Goals!"
},
{
"line": "try with_reducible use_discharger",
"before_state": "case h\nx : ℝ\nk : ℤ\nh : 2 * x = (2 * ↑k + 1) * π / 2\n⊢ x = (2 * ↑k + 1) * π / 4",
"after_state": "case h\nx : ℝ\nk : ℤ\nh : 2 * x = (2 * ↑k + 1) * π / 2\n⊢ x = (2 * ↑k + 1) * π / 4"
},
{
"line": "first\n| with_reducible use_discharger\n| skip",
"before_state": "case h\nx : ℝ\nk : ℤ\nh : 2 * x = (2 * ↑k + 1) * π / 2\n⊢ x = (2 * ↑k + 1) * π / 4",
"after_state": "case h\nx : ℝ\nk : ℤ\nh : 2 * x = (2 * ↑k + 1) * π / 2\n⊢ x = (2 * ↑k + 1) * π / 4"
},
{
"line": "with_reducible use_discharger",
"before_state": "case h\nx : ℝ\nk : ℤ\nh : 2 * x = (2 * ↑k + 1) * π / 2\n⊢ x = (2 * ↑k + 1) * π / 4",
"after_state": "case h\nx : ℝ\nk : ℤ\nh : 2 * x = (2 * ↑k + 1) * π / 2\n⊢ x = (2 * ↑k + 1) * π / 4"
},
{
"line": "use_discharger",
"before_state": "case h\nx : ℝ\nk : ℤ\nh : 2 * x = (2 * ↑k + 1) * π / 2\n⊢ x = (2 * ↑k + 1) * π / 4",
"after_state": "case h\nx : ℝ\nk : ℤ\nh : 2 * x = (2 * ↑k + 1) * π / 2\n⊢ x = (2 * ↑k + 1) * π / 4"
},
{
"line": "focus\n apply exists_prop.mpr✝\n with_annotate_state\"<;>\" skip\n all_goals use_discharger",
"before_state": "case h\nx : ℝ\nk : ℤ\nh : 2 * x = (2 * ↑k + 1) * π / 2\n⊢ x = (2 * ↑k + 1) * π / 4",
"after_state": "case h\nx : ℝ\nk : ℤ\nh : 2 * x = (2 * ↑k + 1) * π / 2\n⊢ x = (2 * ↑k + 1) * π / 4"
},
{
"line": "apply exists_prop.mpr✝",
"before_state": "case h\nx : ℝ\nk : ℤ\nh : 2 * x = (2 * ↑k + 1) * π / 2\n⊢ x = (2 * ↑k + 1) * π / 4",
"after_state": "case h\nx : ℝ\nk : ℤ\nh : 2 * x = (2 * ↑k + 1) * π / 2\n⊢ x = (2 * ↑k + 1) * π / 4"
},
{
"line": "skip",
"before_state": "case h\nx : ℝ\nk : ℤ\nh : 2 * x = (2 * ↑k + 1) * π / 2\n⊢ x = (2 * ↑k + 1) * π / 4",
"after_state": "case h\nx : ℝ\nk : ℤ\nh : 2 * x = (2 * ↑k + 1) * π / 2\n⊢ x = (2 * ↑k + 1) * π / 4"
},
{
"line": "linarith",
"before_state": "case h\nx : ℝ\nk : ℤ\nh : 2 * x = (2 * ↑k + 1) * π / 2\n⊢ x = (2 * ↑k + 1) * π / 4",
"after_state": "No Goals!"
},
{
"line": "norm_num",
"before_state": "x : ℝ\nk : ℤ\nh : 2 * x = (2 * ↑k + 1) * π / 2\na✝ : x < (2 * ↑k + 1) * π / 4\n⊢ 1 * 1 = 1",
"after_state": "No Goals!"
},
{
"line": "norm_num",
"before_state": "x : ℝ\nk : ℤ\nh : 2 * x = (2 * ↑k + 1) * π / 2\na✝ : x < (2 * ↑k + 1) * π / 4\n⊢ 1 * 1 = 1",
"after_state": "No Goals!"
},
{
"line": "norm_num",
"before_state": "x : ℝ\nk : ℤ\nh : 2 * x = (2 * ↑k + 1) * π / 2\na✝ : x < (2 * ↑k + 1) * π / 4\n⊢ 4 / 4 = 1",
"after_state": "No Goals!"
},
{
"line": "norm_num",
"before_state": "x : ℝ\nk : ℤ\nh : 2 * x = (2 * ↑k + 1) * π / 2\na✝ : x < (2 * ↑k + 1) * π / 4\n⊢ 1 * 4 = 4",
"after_state": "No Goals!"
},
{
"line": "norm_num",
"before_state": "x : ℝ\nk : ℤ\nh : 2 * x = (2 * ↑k + 1) * π / 2\na✝ : x < (2 * ↑k + 1) * π / 4\n⊢ 4 > 0",
"after_state": "No Goals!"
},
{
"line": "norm_num",
"before_state": "x : ℝ\nk : ℤ\nh : 2 * x = (2 * ↑k + 1) * π / 2\na✝ : x < (2 * ↑k + 1) * π / 4\n⊢ 1 * 2 = 2",
"after_state": "No Goals!"
},
{
"line": "norm_num",
"before_state": "x : ℝ\nk : ℤ\nh : 2 * x = (2 * ↑k + 1) * π / 2\na✝ : x < (2 * ↑k + 1) * π / 4\n⊢ 1 * 1 = 1",
"after_state": "No Goals!"
},
{
"line": "norm_num",
"before_state": "x : ℝ\nk : ℤ\nh : 2 * x = (2 * ↑k + 1) * π / 2\na✝ : x < (2 * ↑k + 1) * π / 4\n⊢ 1 * 1 = 1",
"after_state": "No Goals!"
},
{
"line": "norm_num",
"before_state": "x : ℝ\nk : ℤ\nh : 2 * x = (2 * ↑k + 1) * π / 2\na✝ : x < (2 * ↑k + 1) * π / 4\n⊢ 2 / 2 = 1",
"after_state": "No Goals!"
},
{
"line": "norm_num",
"before_state": "x : ℝ\nk : ℤ\nh : 2 * x = (2 * ↑k + 1) * π / 2\na✝ : x < (2 * ↑k + 1) * π / 4\n⊢ 1 * 2 = 2",
"after_state": "No Goals!"
},
{
"line": "norm_num",
"before_state": "x : ℝ\nk : ℤ\nh : 2 * x = (2 * ↑k + 1) * π / 2\na✝ : x < (2 * ↑k + 1) * π / 4\n⊢ 2 > 0",
"after_state": "No Goals!"
},
{
"line": "ring1",
"before_state": "x : ℝ\nk : ℤ\nh : 2 * x = (2 * ↑k + 1) * π / 2\na✝ : x < (2 * ↑k + 1) * π / 4\n⊢ -(1 * 2 * (2 * x) - (1 * 2 * (1 * ↑k) + 1 * 1) * (1 * π)) + (4 * x - (1 * 2 * (1 * ↑k) + 1 * 1) * (1 * π)) = 0",
"after_state": "No Goals!"
},
{
"line": "norm_num",
"before_state": "x : ℝ\nk : ℤ\nh : 2 * x = (2 * ↑k + 1) * π / 2\na✝ : (2 * ↑k + 1) * π / 4 < x\n⊢ 1 * 1 = 1",
"after_state": "No Goals!"
},
{
"line": "norm_num",
"before_state": "x : ℝ\nk : ℤ\nh : 2 * x = (2 * ↑k + 1) * π / 2\na✝ : (2 * ↑k + 1) * π / 4 < x\n⊢ 1 * 1 = 1",
"after_state": "No Goals!"
},
{
"line": "norm_num",
"before_state": "x : ℝ\nk : ℤ\nh : 2 * x = (2 * ↑k + 1) * π / 2\na✝ : (2 * ↑k + 1) * π / 4 < x\n⊢ 4 / 4 = 1",
"after_state": "No Goals!"
},
{
"line": "norm_num",
"before_state": "x : ℝ\nk : ℤ\nh : 2 * x = (2 * ↑k + 1) * π / 2\na✝ : (2 * ↑k + 1) * π / 4 < x\n⊢ 1 * 4 = 4",
"after_state": "No Goals!"
},
{
"line": "norm_num",
"before_state": "x : ℝ\nk : ℤ\nh : 2 * x = (2 * ↑k + 1) * π / 2\na✝ : (2 * ↑k + 1) * π / 4 < x\n⊢ 4 > 0",
"after_state": "No Goals!"
},
{
"line": "norm_num",
"before_state": "x : ℝ\nk : ℤ\nh : 2 * x = (2 * ↑k + 1) * π / 2\na✝ : (2 * ↑k + 1) * π / 4 < x\n⊢ 1 * 2 = 2",
"after_state": "No Goals!"
},
{
"line": "norm_num",
"before_state": "x : ℝ\nk : ℤ\nh : 2 * x = (2 * ↑k + 1) * π / 2\na✝ : (2 * ↑k + 1) * π / 4 < x\n⊢ 1 * 1 = 1",
"after_state": "No Goals!"
},
{
"line": "norm_num",
"before_state": "x : ℝ\nk : ℤ\nh : 2 * x = (2 * ↑k + 1) * π / 2\na✝ : (2 * ↑k + 1) * π / 4 < x\n⊢ 1 * 1 = 1",
"after_state": "No Goals!"
},
{
"line": "norm_num",
"before_state": "x : ℝ\nk : ℤ\nh : 2 * x = (2 * ↑k + 1) * π / 2\na✝ : (2 * ↑k + 1) * π / 4 < x\n⊢ 2 / 2 = 1",
"after_state": "No Goals!"
},
{
"line": "norm_num",
"before_state": "x : ℝ\nk : ℤ\nh : 2 * x = (2 * ↑k + 1) * π / 2\na✝ : (2 * ↑k + 1) * π / 4 < x\n⊢ 1 * 2 = 2",
"after_state": "No Goals!"
},
{
"line": "norm_num",
"before_state": "x : ℝ\nk : ℤ\nh : 2 * x = (2 * ↑k + 1) * π / 2\na✝ : (2 * ↑k + 1) * π / 4 < x\n⊢ 2 > 0",
"after_state": "No Goals!"
},
{
"line": "ring1",
"before_state": "x : ℝ\nk : ℤ\nh : 2 * x = (2 * ↑k + 1) * π / 2\na✝ : (2 * ↑k + 1) * π / 4 < x\n⊢ 1 * 2 * (2 * x) - (1 * 2 * (1 * ↑k) + 1 * 1) * (1 * π) + ((1 * 2 * (1 * ↑k) + 1 * 1) * (1 * π) - 4 * x) = 0",
"after_state": "No Goals!"
},
{
"line": "rintro ⟨k, h⟩",
"before_state": "case mpr\nx : ℝ\n⊢ (∃ k, x = (2 * ↑k + 1) * π / 4) → ∃ k, 2 * x = (2 * ↑k + 1) * π / 2",
"after_state": "case mpr.intro\nx : ℝ\nk : ℤ\nh : x = (2 * ↑k + 1) * π / 4\n⊢ ∃ k, 2 * x = (2 * ↑k + 1) * π / 2"
},
{
"line": "use k",
"before_state": "case mpr.intro\nx : ℝ\nk : ℤ\nh : x = (2 * ↑k + 1) * π / 4\n⊢ ∃ k, 2 * x = (2 * ↑k + 1) * π / 2",
"after_state": "case h\nx : ℝ\nk : ℤ\nh : x = (2 * ↑k + 1) * π / 4\n⊢ 2 * x = (2 * ↑k + 1) * π / 2"
},
{
"line": "refine without_cdot(k : ?m✝)",
"before_state": "case w\nx : ℝ\nk : ℤ\nh : x = (2 * ↑k + 1) * π / 4\n⊢ ℤ",
"after_state": "No Goals!"
},
{
"line": "try with_reducible use_discharger",
"before_state": "case h\nx : ℝ\nk : ℤ\nh : x = (2 * ↑k + 1) * π / 4\n⊢ 2 * x = (2 * ↑k + 1) * π / 2",
"after_state": "case h\nx : ℝ\nk : ℤ\nh : x = (2 * ↑k + 1) * π / 4\n⊢ 2 * x = (2 * ↑k + 1) * π / 2"
},
{
"line": "first\n| with_reducible use_discharger\n| skip",
"before_state": "case h\nx : ℝ\nk : ℤ\nh : x = (2 * ↑k + 1) * π / 4\n⊢ 2 * x = (2 * ↑k + 1) * π / 2",
"after_state": "case h\nx : ℝ\nk : ℤ\nh : x = (2 * ↑k + 1) * π / 4\n⊢ 2 * x = (2 * ↑k + 1) * π / 2"
},
{
"line": "with_reducible use_discharger",
"before_state": "case h\nx : ℝ\nk : ℤ\nh : x = (2 * ↑k + 1) * π / 4\n⊢ 2 * x = (2 * ↑k + 1) * π / 2",
"after_state": "case h\nx : ℝ\nk : ℤ\nh : x = (2 * ↑k + 1) * π / 4\n⊢ 2 * x = (2 * ↑k + 1) * π / 2"
},
{
"line": "use_discharger",
"before_state": "case h\nx : ℝ\nk : ℤ\nh : x = (2 * ↑k + 1) * π / 4\n⊢ 2 * x = (2 * ↑k + 1) * π / 2",
"after_state": "case h\nx : ℝ\nk : ℤ\nh : x = (2 * ↑k + 1) * π / 4\n⊢ 2 * x = (2 * ↑k + 1) * π / 2"
},
{
"line": "focus\n apply exists_prop.mpr✝\n with_annotate_state\"<;>\" skip\n all_goals use_discharger",
"before_state": "case h\nx : ℝ\nk : ℤ\nh : x = (2 * ↑k + 1) * π / 4\n⊢ 2 * x = (2 * ↑k + 1) * π / 2",
"after_state": "case h\nx : ℝ\nk : ℤ\nh : x = (2 * ↑k + 1) * π / 4\n⊢ 2 * x = (2 * ↑k + 1) * π / 2"
},
{
"line": "apply exists_prop.mpr✝",
"before_state": "case h\nx : ℝ\nk : ℤ\nh : x = (2 * ↑k + 1) * π / 4\n⊢ 2 * x = (2 * ↑k + 1) * π / 2",
"after_state": "case h\nx : ℝ\nk : ℤ\nh : x = (2 * ↑k + 1) * π / 4\n⊢ 2 * x = (2 * ↑k + 1) * π / 2"
},
{
"line": "skip",
"before_state": "case h\nx : ℝ\nk : ℤ\nh : x = (2 * ↑k + 1) * π / 4\n⊢ 2 * x = (2 * ↑k + 1) * π / 2",
"after_state": "case h\nx : ℝ\nk : ℤ\nh : x = (2 * ↑k + 1) * π / 4\n⊢ 2 * x = (2 * ↑k + 1) * π / 2"
},
{
"line": "linarith",
"before_state": "case h\nx : ℝ\nk : ℤ\nh : x = (2 * ↑k + 1) * π / 4\n⊢ 2 * x = (2 * ↑k + 1) * π / 2",
"after_state": "No Goals!"
},
{
"line": "norm_num",
"before_state": "x : ℝ\nk : ℤ\nh : x = (2 * ↑k + 1) * π / 4\na✝ : 2 * x < (2 * ↑k + 1) * π / 2\n⊢ 1 * 2 = 2",
"after_state": "No Goals!"
},
{
"line": "norm_num",
"before_state": "x : ℝ\nk : ℤ\nh : x = (2 * ↑k + 1) * π / 4\na✝ : 2 * x < (2 * ↑k + 1) * π / 2\n⊢ 1 * 1 = 1",
"after_state": "No Goals!"
},
{
"line": "norm_num",
"before_state": "x : ℝ\nk : ℤ\nh : x = (2 * ↑k + 1) * π / 4\na✝ : 2 * x < (2 * ↑k + 1) * π / 2\n⊢ 1 * 1 = 1",
"after_state": "No Goals!"
},
{
"line": "norm_num",
"before_state": "x : ℝ\nk : ℤ\nh : x = (2 * ↑k + 1) * π / 4\na✝ : 2 * x < (2 * ↑k + 1) * π / 2\n⊢ 2 / 2 = 1",
"after_state": "No Goals!"
},
{
"line": "norm_num",
"before_state": "x : ℝ\nk : ℤ\nh : x = (2 * ↑k + 1) * π / 4\na✝ : 2 * x < (2 * ↑k + 1) * π / 2\n⊢ 1 * 2 = 2",
"after_state": "No Goals!"
},
{
"line": "norm_num",
"before_state": "x : ℝ\nk : ℤ\nh : x = (2 * ↑k + 1) * π / 4\na✝ : 2 * x < (2 * ↑k + 1) * π / 2\n⊢ 2 > 0",
"after_state": "No Goals!"
},
{
"line": "norm_num",
"before_state": "x : ℝ\nk : ℤ\nh : x = (2 * ↑k + 1) * π / 4\na✝ : 2 * x < (2 * ↑k + 1) * π / 2\n⊢ 1 * 1 = 1",
"after_state": "No Goals!"
},
{
"line": "norm_num",
"before_state": "x : ℝ\nk : ℤ\nh : x = (2 * ↑k + 1) * π / 4\na✝ : 2 * x < (2 * ↑k + 1) * π / 2\n⊢ 1 * 1 = 1",
"after_state": "No Goals!"
},
{
"line": "norm_num",
"before_state": "x : ℝ\nk : ℤ\nh : x = (2 * ↑k + 1) * π / 4\na✝ : 2 * x < (2 * ↑k + 1) * π / 2\n⊢ 4 / 4 = 1",
"after_state": "No Goals!"
},
{
"line": "norm_num",
"before_state": "x : ℝ\nk : ℤ\nh : x = (2 * ↑k + 1) * π / 4\na✝ : 2 * x < (2 * ↑k + 1) * π / 2\n⊢ 1 * 4 = 4",
"after_state": "No Goals!"
},
{
"line": "norm_num",
"before_state": "x : ℝ\nk : ℤ\nh : x = (2 * ↑k + 1) * π / 4\na✝ : 2 * x < (2 * ↑k + 1) * π / 2\n⊢ 4 > 0",
"after_state": "No Goals!"
},
{
"line": "ring1",
"before_state": "x : ℝ\nk : ℤ\nh : x = (2 * ↑k + 1) * π / 4\na✝ : 2 * x < (2 * ↑k + 1) * π / 2\n⊢ -(4 * x - (1 * 2 * (1 * ↑k) + 1 * 1) * (1 * π)) + (1 * 2 * (2 * x) - (1 * 2 * (1 * ↑k) + 1 * 1) * (1 * π)) = 0",
"after_state": "No Goals!"
},
{
"line": "norm_num",
"before_state": "x : ℝ\nk : ℤ\nh : x = (2 * ↑k + 1) * π / 4\na✝ : (2 * ↑k + 1) * π / 2 < 2 * x\n⊢ 1 * 1 = 1",
"after_state": "No Goals!"
},
{
"line": "norm_num",
"before_state": "x : ℝ\nk : ℤ\nh : x = (2 * ↑k + 1) * π / 4\na✝ : (2 * ↑k + 1) * π / 2 < 2 * x\n⊢ 1 * 1 = 1",
"after_state": "No Goals!"
},
{
"line": "norm_num",
"before_state": "x : ℝ\nk : ℤ\nh : x = (2 * ↑k + 1) * π / 4\na✝ : (2 * ↑k + 1) * π / 2 < 2 * x\n⊢ 2 / 2 = 1",
"after_state": "No Goals!"
},
{
"line": "norm_num",
"before_state": "x : ℝ\nk : ℤ\nh : x = (2 * ↑k + 1) * π / 4\na✝ : (2 * ↑k + 1) * π / 2 < 2 * x\n⊢ 1 * 2 = 2",
"after_state": "No Goals!"
},
{
"line": "norm_num",
"before_state": "x : ℝ\nk : ℤ\nh : x = (2 * ↑k + 1) * π / 4\na✝ : (2 * ↑k + 1) * π / 2 < 2 * x\n⊢ 1 * 2 = 2",
"after_state": "No Goals!"
},
{
"line": "norm_num",
"before_state": "x : ℝ\nk : ℤ\nh : x = (2 * ↑k + 1) * π / 4\na✝ : (2 * ↑k + 1) * π / 2 < 2 * x\n⊢ 2 > 0",
"after_state": "No Goals!"
},
{
"line": "norm_num",
"before_state": "x : ℝ\nk : ℤ\nh : x = (2 * ↑k + 1) * π / 4\na✝ : (2 * ↑k + 1) * π / 2 < 2 * x\n⊢ 1 * 1 = 1",
"after_state": "No Goals!"
},
{
"line": "norm_num",
"before_state": "x : ℝ\nk : ℤ\nh : x = (2 * ↑k + 1) * π / 4\na✝ : (2 * ↑k + 1) * π / 2 < 2 * x\n⊢ 1 * 1 = 1",
"after_state": "No Goals!"
},
{
"line": "norm_num",
"before_state": "x : ℝ\nk : ℤ\nh : x = (2 * ↑k + 1) * π / 4\na✝ : (2 * ↑k + 1) * π / 2 < 2 * x\n⊢ 4 / 4 = 1",
"after_state": "No Goals!"
},
{
"line": "norm_num",
"before_state": "x : ℝ\nk : ℤ\nh : x = (2 * ↑k + 1) * π / 4\na✝ : (2 * ↑k + 1) * π / 2 < 2 * x\n⊢ 1 * 4 = 4",
"after_state": "No Goals!"
},
{
"line": "norm_num",
"before_state": "x : ℝ\nk : ℤ\nh : x = (2 * ↑k + 1) * π / 4\na✝ : (2 * ↑k + 1) * π / 2 < 2 * x\n⊢ 4 > 0",
"after_state": "No Goals!"
},
{
"line": "ring1",
"before_state": "x : ℝ\nk : ℤ\nh : x = (2 * ↑k + 1) * π / 4\na✝ : (2 * ↑k + 1) * π / 2 < 2 * x\n⊢ 4 * x - (1 * 2 * (1 * ↑k) + 1 * 1) * (1 * π) + ((1 * 2 * (1 * ↑k) + 1 * 1) * (1 * π) - 1 * 2 * (2 * x)) = 0",
"after_state": "No Goals!"
}
] |
theorem solve_cos3x_0 {x : ℝ} : cos (3 * x) = 0 ↔ ∃ k : ℤ, x = (2 * ↑k + 1) * π / 6 := by
rw [cos_eq_zero_iff]
refine exists_congr fun k => ?_
constructor <;> intro <;> linarith
|
/root/DuelModelResearch/mathlib4/Archive/Imo/Imo1962Q4.lean
|
{
"open": [
"Real",
"scoped Real"
],
"variables": []
}
|
[
{
"line": "rw [cos_eq_zero_iff]",
"before_state": "x : ℝ\n⊢ cos (3 * x) = 0 ↔ ∃ k, x = (2 * ↑k + 1) * π / 6",
"after_state": "x : ℝ\n⊢ (∃ k, 3 * x = (2 * ↑k + 1) * π / 2) ↔ ∃ k, x = (2 * ↑k + 1) * π / 6"
},
{
"line": "rewrite [cos_eq_zero_iff]",
"before_state": "x : ℝ\n⊢ cos (3 * x) = 0 ↔ ∃ k, x = (2 * ↑k + 1) * π / 6",
"after_state": "x : ℝ\n⊢ (∃ k, 3 * x = (2 * ↑k + 1) * π / 2) ↔ ∃ k, x = (2 * ↑k + 1) * π / 6"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "x : ℝ\n⊢ (∃ k, 3 * x = (2 * ↑k + 1) * π / 2) ↔ ∃ k, x = (2 * ↑k + 1) * π / 6",
"after_state": "x : ℝ\n⊢ (∃ k, 3 * x = (2 * ↑k + 1) * π / 2) ↔ ∃ k, x = (2 * ↑k + 1) * π / 6"
},
{
"line": "try (with_reducible rfl)",
"before_state": "x : ℝ\n⊢ (∃ k, 3 * x = (2 * ↑k + 1) * π / 2) ↔ ∃ k, x = (2 * ↑k + 1) * π / 6",
"after_state": "x : ℝ\n⊢ (∃ k, 3 * x = (2 * ↑k + 1) * π / 2) ↔ ∃ k, x = (2 * ↑k + 1) * π / 6"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "x : ℝ\n⊢ (∃ k, 3 * x = (2 * ↑k + 1) * π / 2) ↔ ∃ k, x = (2 * ↑k + 1) * π / 6",
"after_state": "x : ℝ\n⊢ (∃ k, 3 * x = (2 * ↑k + 1) * π / 2) ↔ ∃ k, x = (2 * ↑k + 1) * π / 6"
},
{
"line": "with_reducible rfl",
"before_state": "x : ℝ\n⊢ (∃ k, 3 * x = (2 * ↑k + 1) * π / 2) ↔ ∃ k, x = (2 * ↑k + 1) * π / 6",
"after_state": "x : ℝ\n⊢ (∃ k, 3 * x = (2 * ↑k + 1) * π / 2) ↔ ∃ k, x = (2 * ↑k + 1) * π / 6"
},
{
"line": "rfl",
"before_state": "x : ℝ\n⊢ (∃ k, 3 * x = (2 * ↑k + 1) * π / 2) ↔ ∃ k, x = (2 * ↑k + 1) * π / 6",
"after_state": "x : ℝ\n⊢ (∃ k, 3 * x = (2 * ↑k + 1) * π / 2) ↔ ∃ k, x = (2 * ↑k + 1) * π / 6"
},
{
"line": "apply_rfl",
"before_state": "x : ℝ\n⊢ (∃ k, 3 * x = (2 * ↑k + 1) * π / 2) ↔ ∃ k, x = (2 * ↑k + 1) * π / 6",
"after_state": "x : ℝ\n⊢ (∃ k, 3 * x = (2 * ↑k + 1) * π / 2) ↔ ∃ k, x = (2 * ↑k + 1) * π / 6"
},
{
"line": "skip",
"before_state": "x : ℝ\n⊢ (∃ k, 3 * x = (2 * ↑k + 1) * π / 2) ↔ ∃ k, x = (2 * ↑k + 1) * π / 6",
"after_state": "x : ℝ\n⊢ (∃ k, 3 * x = (2 * ↑k + 1) * π / 2) ↔ ∃ k, x = (2 * ↑k + 1) * π / 6"
},
{
"line": "refine exists_congr fun k => ?_",
"before_state": "x : ℝ\n⊢ (∃ k, 3 * x = (2 * ↑k + 1) * π / 2) ↔ ∃ k, x = (2 * ↑k + 1) * π / 6",
"after_state": "x : ℝ\nk : ℤ\n⊢ 3 * x = (2 * ↑k + 1) * π / 2 ↔ x = (2 * ↑k + 1) * π / 6"
},
{
"line": "focus\n constructor <;> intro\n with_annotate_state\"<;>\" skip\n all_goals linarith",
"before_state": "x : ℝ\nk : ℤ\n⊢ 3 * x = (2 * ↑k + 1) * π / 2 ↔ x = (2 * ↑k + 1) * π / 6",
"after_state": "No Goals!"
},
{
"line": "focus\n constructor\n with_annotate_state\"<;>\" skip\n all_goals intro",
"before_state": "x : ℝ\nk : ℤ\n⊢ 3 * x = (2 * ↑k + 1) * π / 2 ↔ x = (2 * ↑k + 1) * π / 6",
"after_state": "case mp\nx : ℝ\nk : ℤ\na✝ : 3 * x = (2 * ↑k + 1) * π / 2\n⊢ x = (2 * ↑k + 1) * π / 6\n---\ncase mpr\nx : ℝ\nk : ℤ\na✝ : x = (2 * ↑k + 1) * π / 6\n⊢ 3 * x = (2 * ↑k + 1) * π / 2"
},
{
"line": "constructor",
"before_state": "x : ℝ\nk : ℤ\n⊢ 3 * x = (2 * ↑k + 1) * π / 2 ↔ x = (2 * ↑k + 1) * π / 6",
"after_state": "case mp\nx : ℝ\nk : ℤ\n⊢ 3 * x = (2 * ↑k + 1) * π / 2 → x = (2 * ↑k + 1) * π / 6\n---\ncase mpr\nx : ℝ\nk : ℤ\n⊢ x = (2 * ↑k + 1) * π / 6 → 3 * x = (2 * ↑k + 1) * π / 2"
},
{
"line": "with_annotate_state\"<;>\" skip",
"before_state": "case mp\nx : ℝ\nk : ℤ\n⊢ 3 * x = (2 * ↑k + 1) * π / 2 → x = (2 * ↑k + 1) * π / 6\n---\ncase mpr\nx : ℝ\nk : ℤ\n⊢ x = (2 * ↑k + 1) * π / 6 → 3 * x = (2 * ↑k + 1) * π / 2",
"after_state": "case mp\nx : ℝ\nk : ℤ\n⊢ 3 * x = (2 * ↑k + 1) * π / 2 → x = (2 * ↑k + 1) * π / 6\n---\ncase mpr\nx : ℝ\nk : ℤ\n⊢ x = (2 * ↑k + 1) * π / 6 → 3 * x = (2 * ↑k + 1) * π / 2"
},
{
"line": "skip",
"before_state": "case mp\nx : ℝ\nk : ℤ\n⊢ 3 * x = (2 * ↑k + 1) * π / 2 → x = (2 * ↑k + 1) * π / 6\n---\ncase mpr\nx : ℝ\nk : ℤ\n⊢ x = (2 * ↑k + 1) * π / 6 → 3 * x = (2 * ↑k + 1) * π / 2",
"after_state": "case mp\nx : ℝ\nk : ℤ\n⊢ 3 * x = (2 * ↑k + 1) * π / 2 → x = (2 * ↑k + 1) * π / 6\n---\ncase mpr\nx : ℝ\nk : ℤ\n⊢ x = (2 * ↑k + 1) * π / 6 → 3 * x = (2 * ↑k + 1) * π / 2"
},
{
"line": "all_goals intro",
"before_state": "case mp\nx : ℝ\nk : ℤ\n⊢ 3 * x = (2 * ↑k + 1) * π / 2 → x = (2 * ↑k + 1) * π / 6\n---\ncase mpr\nx : ℝ\nk : ℤ\n⊢ x = (2 * ↑k + 1) * π / 6 → 3 * x = (2 * ↑k + 1) * π / 2",
"after_state": "case mp\nx : ℝ\nk : ℤ\na✝ : 3 * x = (2 * ↑k + 1) * π / 2\n⊢ x = (2 * ↑k + 1) * π / 6\n---\ncase mpr\nx : ℝ\nk : ℤ\na✝ : x = (2 * ↑k + 1) * π / 6\n⊢ 3 * x = (2 * ↑k + 1) * π / 2"
},
{
"line": "intro",
"before_state": "case mp\nx : ℝ\nk : ℤ\n⊢ 3 * x = (2 * ↑k + 1) * π / 2 → x = (2 * ↑k + 1) * π / 6",
"after_state": "case mp\nx : ℝ\nk : ℤ\na✝ : 3 * x = (2 * ↑k + 1) * π / 2\n⊢ x = (2 * ↑k + 1) * π / 6"
},
{
"line": "intro",
"before_state": "case mpr\nx : ℝ\nk : ℤ\n⊢ x = (2 * ↑k + 1) * π / 6 → 3 * x = (2 * ↑k + 1) * π / 2",
"after_state": "case mpr\nx : ℝ\nk : ℤ\na✝ : x = (2 * ↑k + 1) * π / 6\n⊢ 3 * x = (2 * ↑k + 1) * π / 2"
},
{
"line": "with_annotate_state\"<;>\" skip",
"before_state": "case mp\nx : ℝ\nk : ℤ\na✝ : 3 * x = (2 * ↑k + 1) * π / 2\n⊢ x = (2 * ↑k + 1) * π / 6\n---\ncase mpr\nx : ℝ\nk : ℤ\na✝ : x = (2 * ↑k + 1) * π / 6\n⊢ 3 * x = (2 * ↑k + 1) * π / 2",
"after_state": "case mp\nx : ℝ\nk : ℤ\na✝ : 3 * x = (2 * ↑k + 1) * π / 2\n⊢ x = (2 * ↑k + 1) * π / 6\n---\ncase mpr\nx : ℝ\nk : ℤ\na✝ : x = (2 * ↑k + 1) * π / 6\n⊢ 3 * x = (2 * ↑k + 1) * π / 2"
},
{
"line": "skip",
"before_state": "case mp\nx : ℝ\nk : ℤ\na✝ : 3 * x = (2 * ↑k + 1) * π / 2\n⊢ x = (2 * ↑k + 1) * π / 6\n---\ncase mpr\nx : ℝ\nk : ℤ\na✝ : x = (2 * ↑k + 1) * π / 6\n⊢ 3 * x = (2 * ↑k + 1) * π / 2",
"after_state": "case mp\nx : ℝ\nk : ℤ\na✝ : 3 * x = (2 * ↑k + 1) * π / 2\n⊢ x = (2 * ↑k + 1) * π / 6\n---\ncase mpr\nx : ℝ\nk : ℤ\na✝ : x = (2 * ↑k + 1) * π / 6\n⊢ 3 * x = (2 * ↑k + 1) * π / 2"
},
{
"line": "all_goals linarith",
"before_state": "case mp\nx : ℝ\nk : ℤ\na✝ : 3 * x = (2 * ↑k + 1) * π / 2\n⊢ x = (2 * ↑k + 1) * π / 6\n---\ncase mpr\nx : ℝ\nk : ℤ\na✝ : x = (2 * ↑k + 1) * π / 6\n⊢ 3 * x = (2 * ↑k + 1) * π / 2",
"after_state": "No Goals!"
},
{
"line": "linarith",
"before_state": "case mp\nx : ℝ\nk : ℤ\na✝ : 3 * x = (2 * ↑k + 1) * π / 2\n⊢ x = (2 * ↑k + 1) * π / 6",
"after_state": "No Goals!"
},
{
"line": "norm_num",
"before_state": "x : ℝ\nk : ℤ\na✝¹ : 3 * x = (2 * ↑k + 1) * π / 2\na✝ : x < (2 * ↑k + 1) * π / 6\n⊢ 1 * 1 = 1",
"after_state": "No Goals!"
},
{
"line": "norm_num",
"before_state": "x : ℝ\nk : ℤ\na✝¹ : 3 * x = (2 * ↑k + 1) * π / 2\na✝ : x < (2 * ↑k + 1) * π / 6\n⊢ 1 * 1 = 1",
"after_state": "No Goals!"
},
{
"line": "norm_num",
"before_state": "x : ℝ\nk : ℤ\na✝¹ : 3 * x = (2 * ↑k + 1) * π / 2\na✝ : x < (2 * ↑k + 1) * π / 6\n⊢ 6 / 6 = 1",
"after_state": "No Goals!"
},
{
"line": "norm_num",
"before_state": "x : ℝ\nk : ℤ\na✝¹ : 3 * x = (2 * ↑k + 1) * π / 2\na✝ : x < (2 * ↑k + 1) * π / 6\n⊢ 1 * 6 = 6",
"after_state": "No Goals!"
},
{
"line": "norm_num",
"before_state": "x : ℝ\nk : ℤ\na✝¹ : 3 * x = (2 * ↑k + 1) * π / 2\na✝ : x < (2 * ↑k + 1) * π / 6\n⊢ 6 > 0",
"after_state": "No Goals!"
},
{
"line": "norm_num",
"before_state": "x : ℝ\nk : ℤ\na✝¹ : 3 * x = (2 * ↑k + 1) * π / 2\na✝ : x < (2 * ↑k + 1) * π / 6\n⊢ 1 * 2 = 2",
"after_state": "No Goals!"
},
{
"line": "norm_num",
"before_state": "x : ℝ\nk : ℤ\na✝¹ : 3 * x = (2 * ↑k + 1) * π / 2\na✝ : x < (2 * ↑k + 1) * π / 6\n⊢ 1 * 1 = 1",
"after_state": "No Goals!"
},
{
"line": "norm_num",
"before_state": "x : ℝ\nk : ℤ\na✝¹ : 3 * x = (2 * ↑k + 1) * π / 2\na✝ : x < (2 * ↑k + 1) * π / 6\n⊢ 1 * 1 = 1",
"after_state": "No Goals!"
},
{
"line": "norm_num",
"before_state": "x : ℝ\nk : ℤ\na✝¹ : 3 * x = (2 * ↑k + 1) * π / 2\na✝ : x < (2 * ↑k + 1) * π / 6\n⊢ 2 / 2 = 1",
"after_state": "No Goals!"
},
{
"line": "norm_num",
"before_state": "x : ℝ\nk : ℤ\na✝¹ : 3 * x = (2 * ↑k + 1) * π / 2\na✝ : x < (2 * ↑k + 1) * π / 6\n⊢ 1 * 2 = 2",
"after_state": "No Goals!"
},
{
"line": "norm_num",
"before_state": "x : ℝ\nk : ℤ\na✝¹ : 3 * x = (2 * ↑k + 1) * π / 2\na✝ : x < (2 * ↑k + 1) * π / 6\n⊢ 2 > 0",
"after_state": "No Goals!"
},
{
"line": "ring1",
"before_state": "x : ℝ\nk : ℤ\na✝¹ : 3 * x = (2 * ↑k + 1) * π / 2\na✝ : x < (2 * ↑k + 1) * π / 6\n⊢ -(1 * 3 * (2 * x) - (1 * 2 * (1 * ↑k) + 1 * 1) * (1 * π)) + (6 * x - (1 * 2 * (1 * ↑k) + 1 * 1) * (1 * π)) = 0",
"after_state": "No Goals!"
},
{
"line": "norm_num",
"before_state": "x : ℝ\nk : ℤ\na✝¹ : 3 * x = (2 * ↑k + 1) * π / 2\na✝ : (2 * ↑k + 1) * π / 6 < x\n⊢ 1 * 1 = 1",
"after_state": "No Goals!"
},
{
"line": "norm_num",
"before_state": "x : ℝ\nk : ℤ\na✝¹ : 3 * x = (2 * ↑k + 1) * π / 2\na✝ : (2 * ↑k + 1) * π / 6 < x\n⊢ 1 * 1 = 1",
"after_state": "No Goals!"
},
{
"line": "norm_num",
"before_state": "x : ℝ\nk : ℤ\na✝¹ : 3 * x = (2 * ↑k + 1) * π / 2\na✝ : (2 * ↑k + 1) * π / 6 < x\n⊢ 6 / 6 = 1",
"after_state": "No Goals!"
},
{
"line": "norm_num",
"before_state": "x : ℝ\nk : ℤ\na✝¹ : 3 * x = (2 * ↑k + 1) * π / 2\na✝ : (2 * ↑k + 1) * π / 6 < x\n⊢ 1 * 6 = 6",
"after_state": "No Goals!"
},
{
"line": "norm_num",
"before_state": "x : ℝ\nk : ℤ\na✝¹ : 3 * x = (2 * ↑k + 1) * π / 2\na✝ : (2 * ↑k + 1) * π / 6 < x\n⊢ 6 > 0",
"after_state": "No Goals!"
},
{
"line": "norm_num",
"before_state": "x : ℝ\nk : ℤ\na✝¹ : 3 * x = (2 * ↑k + 1) * π / 2\na✝ : (2 * ↑k + 1) * π / 6 < x\n⊢ 1 * 2 = 2",
"after_state": "No Goals!"
},
{
"line": "norm_num",
"before_state": "x : ℝ\nk : ℤ\na✝¹ : 3 * x = (2 * ↑k + 1) * π / 2\na✝ : (2 * ↑k + 1) * π / 6 < x\n⊢ 1 * 1 = 1",
"after_state": "No Goals!"
},
{
"line": "norm_num",
"before_state": "x : ℝ\nk : ℤ\na✝¹ : 3 * x = (2 * ↑k + 1) * π / 2\na✝ : (2 * ↑k + 1) * π / 6 < x\n⊢ 1 * 1 = 1",
"after_state": "No Goals!"
},
{
"line": "norm_num",
"before_state": "x : ℝ\nk : ℤ\na✝¹ : 3 * x = (2 * ↑k + 1) * π / 2\na✝ : (2 * ↑k + 1) * π / 6 < x\n⊢ 2 / 2 = 1",
"after_state": "No Goals!"
},
{
"line": "norm_num",
"before_state": "x : ℝ\nk : ℤ\na✝¹ : 3 * x = (2 * ↑k + 1) * π / 2\na✝ : (2 * ↑k + 1) * π / 6 < x\n⊢ 1 * 2 = 2",
"after_state": "No Goals!"
},
{
"line": "norm_num",
"before_state": "x : ℝ\nk : ℤ\na✝¹ : 3 * x = (2 * ↑k + 1) * π / 2\na✝ : (2 * ↑k + 1) * π / 6 < x\n⊢ 2 > 0",
"after_state": "No Goals!"
},
{
"line": "ring1",
"before_state": "x : ℝ\nk : ℤ\na✝¹ : 3 * x = (2 * ↑k + 1) * π / 2\na✝ : (2 * ↑k + 1) * π / 6 < x\n⊢ 1 * 3 * (2 * x) - (1 * 2 * (1 * ↑k) + 1 * 1) * (1 * π) + ((1 * 2 * (1 * ↑k) + 1 * 1) * (1 * π) - 6 * x) = 0",
"after_state": "No Goals!"
},
{
"line": "linarith",
"before_state": "case mpr\nx : ℝ\nk : ℤ\na✝ : x = (2 * ↑k + 1) * π / 6\n⊢ 3 * x = (2 * ↑k + 1) * π / 2",
"after_state": "No Goals!"
},
{
"line": "norm_num",
"before_state": "x : ℝ\nk : ℤ\na✝¹ : x = (2 * ↑k + 1) * π / 6\na✝ : 3 * x < (2 * ↑k + 1) * π / 2\n⊢ 1 * 2 = 2",
"after_state": "No Goals!"
},
{
"line": "norm_num",
"before_state": "x : ℝ\nk : ℤ\na✝¹ : x = (2 * ↑k + 1) * π / 6\na✝ : 3 * x < (2 * ↑k + 1) * π / 2\n⊢ 1 * 1 = 1",
"after_state": "No Goals!"
},
{
"line": "norm_num",
"before_state": "x : ℝ\nk : ℤ\na✝¹ : x = (2 * ↑k + 1) * π / 6\na✝ : 3 * x < (2 * ↑k + 1) * π / 2\n⊢ 1 * 1 = 1",
"after_state": "No Goals!"
},
{
"line": "norm_num",
"before_state": "x : ℝ\nk : ℤ\na✝¹ : x = (2 * ↑k + 1) * π / 6\na✝ : 3 * x < (2 * ↑k + 1) * π / 2\n⊢ 2 / 2 = 1",
"after_state": "No Goals!"
},
{
"line": "norm_num",
"before_state": "x : ℝ\nk : ℤ\na✝¹ : x = (2 * ↑k + 1) * π / 6\na✝ : 3 * x < (2 * ↑k + 1) * π / 2\n⊢ 1 * 2 = 2",
"after_state": "No Goals!"
},
{
"line": "norm_num",
"before_state": "x : ℝ\nk : ℤ\na✝¹ : x = (2 * ↑k + 1) * π / 6\na✝ : 3 * x < (2 * ↑k + 1) * π / 2\n⊢ 2 > 0",
"after_state": "No Goals!"
},
{
"line": "norm_num",
"before_state": "x : ℝ\nk : ℤ\na✝¹ : x = (2 * ↑k + 1) * π / 6\na✝ : 3 * x < (2 * ↑k + 1) * π / 2\n⊢ 1 * 1 = 1",
"after_state": "No Goals!"
},
{
"line": "norm_num",
"before_state": "x : ℝ\nk : ℤ\na✝¹ : x = (2 * ↑k + 1) * π / 6\na✝ : 3 * x < (2 * ↑k + 1) * π / 2\n⊢ 1 * 1 = 1",
"after_state": "No Goals!"
},
{
"line": "norm_num",
"before_state": "x : ℝ\nk : ℤ\na✝¹ : x = (2 * ↑k + 1) * π / 6\na✝ : 3 * x < (2 * ↑k + 1) * π / 2\n⊢ 6 / 6 = 1",
"after_state": "No Goals!"
},
{
"line": "norm_num",
"before_state": "x : ℝ\nk : ℤ\na✝¹ : x = (2 * ↑k + 1) * π / 6\na✝ : 3 * x < (2 * ↑k + 1) * π / 2\n⊢ 1 * 6 = 6",
"after_state": "No Goals!"
},
{
"line": "norm_num",
"before_state": "x : ℝ\nk : ℤ\na✝¹ : x = (2 * ↑k + 1) * π / 6\na✝ : 3 * x < (2 * ↑k + 1) * π / 2\n⊢ 6 > 0",
"after_state": "No Goals!"
},
{
"line": "ring1",
"before_state": "x : ℝ\nk : ℤ\na✝¹ : x = (2 * ↑k + 1) * π / 6\na✝ : 3 * x < (2 * ↑k + 1) * π / 2\n⊢ -(6 * x - (1 * 2 * (1 * ↑k) + 1 * 1) * (1 * π)) + (1 * 3 * (2 * x) - (1 * 2 * (1 * ↑k) + 1 * 1) * (1 * π)) = 0",
"after_state": "No Goals!"
},
{
"line": "norm_num",
"before_state": "x : ℝ\nk : ℤ\na✝¹ : x = (2 * ↑k + 1) * π / 6\na✝ : (2 * ↑k + 1) * π / 2 < 3 * x\n⊢ 1 * 1 = 1",
"after_state": "No Goals!"
},
{
"line": "norm_num",
"before_state": "x : ℝ\nk : ℤ\na✝¹ : x = (2 * ↑k + 1) * π / 6\na✝ : (2 * ↑k + 1) * π / 2 < 3 * x\n⊢ 1 * 1 = 1",
"after_state": "No Goals!"
},
{
"line": "norm_num",
"before_state": "x : ℝ\nk : ℤ\na✝¹ : x = (2 * ↑k + 1) * π / 6\na✝ : (2 * ↑k + 1) * π / 2 < 3 * x\n⊢ 2 / 2 = 1",
"after_state": "No Goals!"
},
{
"line": "norm_num",
"before_state": "x : ℝ\nk : ℤ\na✝¹ : x = (2 * ↑k + 1) * π / 6\na✝ : (2 * ↑k + 1) * π / 2 < 3 * x\n⊢ 1 * 2 = 2",
"after_state": "No Goals!"
},
{
"line": "norm_num",
"before_state": "x : ℝ\nk : ℤ\na✝¹ : x = (2 * ↑k + 1) * π / 6\na✝ : (2 * ↑k + 1) * π / 2 < 3 * x\n⊢ 1 * 2 = 2",
"after_state": "No Goals!"
},
{
"line": "norm_num",
"before_state": "x : ℝ\nk : ℤ\na✝¹ : x = (2 * ↑k + 1) * π / 6\na✝ : (2 * ↑k + 1) * π / 2 < 3 * x\n⊢ 2 > 0",
"after_state": "No Goals!"
},
{
"line": "norm_num",
"before_state": "x : ℝ\nk : ℤ\na✝¹ : x = (2 * ↑k + 1) * π / 6\na✝ : (2 * ↑k + 1) * π / 2 < 3 * x\n⊢ 1 * 1 = 1",
"after_state": "No Goals!"
},
{
"line": "norm_num",
"before_state": "x : ℝ\nk : ℤ\na✝¹ : x = (2 * ↑k + 1) * π / 6\na✝ : (2 * ↑k + 1) * π / 2 < 3 * x\n⊢ 1 * 1 = 1",
"after_state": "No Goals!"
},
{
"line": "norm_num",
"before_state": "x : ℝ\nk : ℤ\na✝¹ : x = (2 * ↑k + 1) * π / 6\na✝ : (2 * ↑k + 1) * π / 2 < 3 * x\n⊢ 6 / 6 = 1",
"after_state": "No Goals!"
},
{
"line": "norm_num",
"before_state": "x : ℝ\nk : ℤ\na✝¹ : x = (2 * ↑k + 1) * π / 6\na✝ : (2 * ↑k + 1) * π / 2 < 3 * x\n⊢ 1 * 6 = 6",
"after_state": "No Goals!"
},
{
"line": "norm_num",
"before_state": "x : ℝ\nk : ℤ\na✝¹ : x = (2 * ↑k + 1) * π / 6\na✝ : (2 * ↑k + 1) * π / 2 < 3 * x\n⊢ 6 > 0",
"after_state": "No Goals!"
},
{
"line": "ring1",
"before_state": "x : ℝ\nk : ℤ\na✝¹ : x = (2 * ↑k + 1) * π / 6\na✝ : (2 * ↑k + 1) * π / 2 < 3 * x\n⊢ 6 * x - (1 * 2 * (1 * ↑k) + 1 * 1) * (1 * π) + ((1 * 2 * (1 * ↑k) + 1 * 1) * (1 * π) - 1 * 3 * (2 * x)) = 0",
"after_state": "No Goals!"
}
] |
theorem formula {R : Type*} [CommRing R] [IsDomain R] [CharZero R] (a : R) :
a ^ 2 + ((2 : R) * a ^ 2 - (1 : R)) ^ 2 + ((4 : R) * a ^ 3 - 3 * a) ^ 2 = 1 ↔
((2 : R) * a ^ 2 - (1 : R)) * ((4 : R) * a ^ 3 - 3 * a) = 0 := by
constructor <;> intro h
· apply pow_eq_zero (n := 2)
apply mul_left_injective₀ (b := 2) (by norm_num)
linear_combination (8 * a ^ 4 - 10 * a ^ 2 + 3) * h
· linear_combination 2 * a * h
|
/root/DuelModelResearch/mathlib4/Archive/Imo/Imo1962Q4.lean
|
{
"open": [
"Real",
"scoped Real",
"Imo1962Q4"
],
"variables": []
}
|
[
{
"line": "focus\n constructor\n with_annotate_state\"<;>\" skip\n all_goals intro h",
"before_state": "R : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : CharZero R\na : R\n⊢ a ^ 2 + (2 * a ^ 2 - 1) ^ 2 + (4 * a ^ 3 - 3 * a) ^ 2 = 1 ↔ (2 * a ^ 2 - 1) * (4 * a ^ 3 - 3 * a) = 0",
"after_state": "case mp\nR : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : CharZero R\na : R\nh : a ^ 2 + (2 * a ^ 2 - 1) ^ 2 + (4 * a ^ 3 - 3 * a) ^ 2 = 1\n⊢ (2 * a ^ 2 - 1) * (4 * a ^ 3 - 3 * a) = 0\n---\ncase mpr\nR : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : CharZero R\na : R\nh : (2 * a ^ 2 - 1) * (4 * a ^ 3 - 3 * a) = 0\n⊢ a ^ 2 + (2 * a ^ 2 - 1) ^ 2 + (4 * a ^ 3 - 3 * a) ^ 2 = 1"
},
{
"line": "constructor",
"before_state": "R : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : CharZero R\na : R\n⊢ a ^ 2 + (2 * a ^ 2 - 1) ^ 2 + (4 * a ^ 3 - 3 * a) ^ 2 = 1 ↔ (2 * a ^ 2 - 1) * (4 * a ^ 3 - 3 * a) = 0",
"after_state": "case mp\nR : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : CharZero R\na : R\n⊢ a ^ 2 + (2 * a ^ 2 - 1) ^ 2 + (4 * a ^ 3 - 3 * a) ^ 2 = 1 → (2 * a ^ 2 - 1) * (4 * a ^ 3 - 3 * a) = 0\n---\ncase mpr\nR : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : CharZero R\na : R\n⊢ (2 * a ^ 2 - 1) * (4 * a ^ 3 - 3 * a) = 0 → a ^ 2 + (2 * a ^ 2 - 1) ^ 2 + (4 * a ^ 3 - 3 * a) ^ 2 = 1"
},
{
"line": "with_annotate_state\"<;>\" skip",
"before_state": "case mp\nR : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : CharZero R\na : R\n⊢ a ^ 2 + (2 * a ^ 2 - 1) ^ 2 + (4 * a ^ 3 - 3 * a) ^ 2 = 1 → (2 * a ^ 2 - 1) * (4 * a ^ 3 - 3 * a) = 0\n---\ncase mpr\nR : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : CharZero R\na : R\n⊢ (2 * a ^ 2 - 1) * (4 * a ^ 3 - 3 * a) = 0 → a ^ 2 + (2 * a ^ 2 - 1) ^ 2 + (4 * a ^ 3 - 3 * a) ^ 2 = 1",
"after_state": "case mp\nR : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : CharZero R\na : R\n⊢ a ^ 2 + (2 * a ^ 2 - 1) ^ 2 + (4 * a ^ 3 - 3 * a) ^ 2 = 1 → (2 * a ^ 2 - 1) * (4 * a ^ 3 - 3 * a) = 0\n---\ncase mpr\nR : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : CharZero R\na : R\n⊢ (2 * a ^ 2 - 1) * (4 * a ^ 3 - 3 * a) = 0 → a ^ 2 + (2 * a ^ 2 - 1) ^ 2 + (4 * a ^ 3 - 3 * a) ^ 2 = 1"
},
{
"line": "skip",
"before_state": "case mp\nR : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : CharZero R\na : R\n⊢ a ^ 2 + (2 * a ^ 2 - 1) ^ 2 + (4 * a ^ 3 - 3 * a) ^ 2 = 1 → (2 * a ^ 2 - 1) * (4 * a ^ 3 - 3 * a) = 0\n---\ncase mpr\nR : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : CharZero R\na : R\n⊢ (2 * a ^ 2 - 1) * (4 * a ^ 3 - 3 * a) = 0 → a ^ 2 + (2 * a ^ 2 - 1) ^ 2 + (4 * a ^ 3 - 3 * a) ^ 2 = 1",
"after_state": "case mp\nR : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : CharZero R\na : R\n⊢ a ^ 2 + (2 * a ^ 2 - 1) ^ 2 + (4 * a ^ 3 - 3 * a) ^ 2 = 1 → (2 * a ^ 2 - 1) * (4 * a ^ 3 - 3 * a) = 0\n---\ncase mpr\nR : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : CharZero R\na : R\n⊢ (2 * a ^ 2 - 1) * (4 * a ^ 3 - 3 * a) = 0 → a ^ 2 + (2 * a ^ 2 - 1) ^ 2 + (4 * a ^ 3 - 3 * a) ^ 2 = 1"
},
{
"line": "all_goals intro h",
"before_state": "case mp\nR : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : CharZero R\na : R\n⊢ a ^ 2 + (2 * a ^ 2 - 1) ^ 2 + (4 * a ^ 3 - 3 * a) ^ 2 = 1 → (2 * a ^ 2 - 1) * (4 * a ^ 3 - 3 * a) = 0\n---\ncase mpr\nR : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : CharZero R\na : R\n⊢ (2 * a ^ 2 - 1) * (4 * a ^ 3 - 3 * a) = 0 → a ^ 2 + (2 * a ^ 2 - 1) ^ 2 + (4 * a ^ 3 - 3 * a) ^ 2 = 1",
"after_state": "case mp\nR : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : CharZero R\na : R\nh : a ^ 2 + (2 * a ^ 2 - 1) ^ 2 + (4 * a ^ 3 - 3 * a) ^ 2 = 1\n⊢ (2 * a ^ 2 - 1) * (4 * a ^ 3 - 3 * a) = 0\n---\ncase mpr\nR : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : CharZero R\na : R\nh : (2 * a ^ 2 - 1) * (4 * a ^ 3 - 3 * a) = 0\n⊢ a ^ 2 + (2 * a ^ 2 - 1) ^ 2 + (4 * a ^ 3 - 3 * a) ^ 2 = 1"
},
{
"line": "intro h",
"before_state": "case mp\nR : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : CharZero R\na : R\n⊢ a ^ 2 + (2 * a ^ 2 - 1) ^ 2 + (4 * a ^ 3 - 3 * a) ^ 2 = 1 → (2 * a ^ 2 - 1) * (4 * a ^ 3 - 3 * a) = 0",
"after_state": "case mp\nR : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : CharZero R\na : R\nh : a ^ 2 + (2 * a ^ 2 - 1) ^ 2 + (4 * a ^ 3 - 3 * a) ^ 2 = 1\n⊢ (2 * a ^ 2 - 1) * (4 * a ^ 3 - 3 * a) = 0"
},
{
"line": "intro h",
"before_state": "case mpr\nR : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : CharZero R\na : R\n⊢ (2 * a ^ 2 - 1) * (4 * a ^ 3 - 3 * a) = 0 → a ^ 2 + (2 * a ^ 2 - 1) ^ 2 + (4 * a ^ 3 - 3 * a) ^ 2 = 1",
"after_state": "case mpr\nR : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : CharZero R\na : R\nh : (2 * a ^ 2 - 1) * (4 * a ^ 3 - 3 * a) = 0\n⊢ a ^ 2 + (2 * a ^ 2 - 1) ^ 2 + (4 * a ^ 3 - 3 * a) ^ 2 = 1"
},
{
"line": "apply pow_eq_zero (n := 2)",
"before_state": "case mp\nR : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : CharZero R\na : R\nh : a ^ 2 + (2 * a ^ 2 - 1) ^ 2 + (4 * a ^ 3 - 3 * a) ^ 2 = 1\n⊢ (2 * a ^ 2 - 1) * (4 * a ^ 3 - 3 * a) = 0",
"after_state": "case mp\nR : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : CharZero R\na : R\nh : a ^ 2 + (2 * a ^ 2 - 1) ^ 2 + (4 * a ^ 3 - 3 * a) ^ 2 = 1\n⊢ ((2 * a ^ 2 - 1) * (4 * a ^ 3 - 3 * a)) ^ 2 = 0"
},
{
"line": "apply mul_left_injective₀ (b := 2) (by norm_num)",
"before_state": "case mp\nR : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : CharZero R\na : R\nh : a ^ 2 + (2 * a ^ 2 - 1) ^ 2 + (4 * a ^ 3 - 3 * a) ^ 2 = 1\n⊢ ((2 * a ^ 2 - 1) * (4 * a ^ 3 - 3 * a)) ^ 2 = 0",
"after_state": "case mp.a\nR : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : CharZero R\na : R\nh : a ^ 2 + (2 * a ^ 2 - 1) ^ 2 + (4 * a ^ 3 - 3 * a) ^ 2 = 1\n⊢ (fun a => a * 2) (((2 * a ^ 2 - 1) * (4 * a ^ 3 - 3 * a)) ^ 2) = (fun a => a * 2) 0"
},
{
"line": "norm_num",
"before_state": "R : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : CharZero R\na : R\nh : a ^ 2 + (2 * a ^ 2 - 1) ^ 2 + (4 * a ^ 3 - 3 * a) ^ 2 = 1\n⊢ 2 ≠ 0",
"after_state": "No Goals!"
},
{
"line": "linear_combination (8 * a ^ 4 - 10 * a ^ 2 + 3) * h",
"before_state": "case mp.a\nR : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : CharZero R\na : R\nh : a ^ 2 + (2 * a ^ 2 - 1) ^ 2 + (4 * a ^ 3 - 3 * a) ^ 2 = 1\n⊢ (fun a => a * 2) (((2 * a ^ 2 - 1) * (4 * a ^ 3 - 3 * a)) ^ 2) = (fun a => a * 2) 0",
"after_state": "No Goals!"
},
{
"line": "linear_combination 2 * a * h",
"before_state": "case mpr\nR : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : CharZero R\na : R\nh : (2 * a ^ 2 - 1) * (4 * a ^ 3 - 3 * a) = 0\n⊢ a ^ 2 + (2 * a ^ 2 - 1) ^ 2 + (4 * a ^ 3 - 3 * a) ^ 2 = 1",
"after_state": "No Goals!"
}
] |
theorem solve_cos2x_0 {x : ℝ} : cos (2 * x) = 0 ↔ ∃ k : ℤ, x = (2 * ↑k + 1) * π / 4 := by
rw [cos_eq_zero_iff]
refine exists_congr fun k => ?_
constructor <;> intro <;> linarith
|
/root/DuelModelResearch/mathlib4/Archive/Imo/Imo1962Q4.lean
|
{
"open": [
"Real",
"scoped Real",
"Imo1962Q4"
],
"variables": []
}
|
[
{
"line": "rw [cos_eq_zero_iff]",
"before_state": "x : ℝ\n⊢ cos (2 * x) = 0 ↔ ∃ k, x = (2 * ↑k + 1) * π / 4",
"after_state": "x : ℝ\n⊢ (∃ k, 2 * x = (2 * ↑k + 1) * π / 2) ↔ ∃ k, x = (2 * ↑k + 1) * π / 4"
},
{
"line": "rewrite [cos_eq_zero_iff]",
"before_state": "x : ℝ\n⊢ cos (2 * x) = 0 ↔ ∃ k, x = (2 * ↑k + 1) * π / 4",
"after_state": "x : ℝ\n⊢ (∃ k, 2 * x = (2 * ↑k + 1) * π / 2) ↔ ∃ k, x = (2 * ↑k + 1) * π / 4"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "x : ℝ\n⊢ (∃ k, 2 * x = (2 * ↑k + 1) * π / 2) ↔ ∃ k, x = (2 * ↑k + 1) * π / 4",
"after_state": "x : ℝ\n⊢ (∃ k, 2 * x = (2 * ↑k + 1) * π / 2) ↔ ∃ k, x = (2 * ↑k + 1) * π / 4"
},
{
"line": "try (with_reducible rfl)",
"before_state": "x : ℝ\n⊢ (∃ k, 2 * x = (2 * ↑k + 1) * π / 2) ↔ ∃ k, x = (2 * ↑k + 1) * π / 4",
"after_state": "x : ℝ\n⊢ (∃ k, 2 * x = (2 * ↑k + 1) * π / 2) ↔ ∃ k, x = (2 * ↑k + 1) * π / 4"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "x : ℝ\n⊢ (∃ k, 2 * x = (2 * ↑k + 1) * π / 2) ↔ ∃ k, x = (2 * ↑k + 1) * π / 4",
"after_state": "x : ℝ\n⊢ (∃ k, 2 * x = (2 * ↑k + 1) * π / 2) ↔ ∃ k, x = (2 * ↑k + 1) * π / 4"
},
{
"line": "with_reducible rfl",
"before_state": "x : ℝ\n⊢ (∃ k, 2 * x = (2 * ↑k + 1) * π / 2) ↔ ∃ k, x = (2 * ↑k + 1) * π / 4",
"after_state": "x : ℝ\n⊢ (∃ k, 2 * x = (2 * ↑k + 1) * π / 2) ↔ ∃ k, x = (2 * ↑k + 1) * π / 4"
},
{
"line": "rfl",
"before_state": "x : ℝ\n⊢ (∃ k, 2 * x = (2 * ↑k + 1) * π / 2) ↔ ∃ k, x = (2 * ↑k + 1) * π / 4",
"after_state": "x : ℝ\n⊢ (∃ k, 2 * x = (2 * ↑k + 1) * π / 2) ↔ ∃ k, x = (2 * ↑k + 1) * π / 4"
},
{
"line": "apply_rfl",
"before_state": "x : ℝ\n⊢ (∃ k, 2 * x = (2 * ↑k + 1) * π / 2) ↔ ∃ k, x = (2 * ↑k + 1) * π / 4",
"after_state": "x : ℝ\n⊢ (∃ k, 2 * x = (2 * ↑k + 1) * π / 2) ↔ ∃ k, x = (2 * ↑k + 1) * π / 4"
},
{
"line": "skip",
"before_state": "x : ℝ\n⊢ (∃ k, 2 * x = (2 * ↑k + 1) * π / 2) ↔ ∃ k, x = (2 * ↑k + 1) * π / 4",
"after_state": "x : ℝ\n⊢ (∃ k, 2 * x = (2 * ↑k + 1) * π / 2) ↔ ∃ k, x = (2 * ↑k + 1) * π / 4"
},
{
"line": "refine exists_congr fun k => ?_",
"before_state": "x : ℝ\n⊢ (∃ k, 2 * x = (2 * ↑k + 1) * π / 2) ↔ ∃ k, x = (2 * ↑k + 1) * π / 4",
"after_state": "x : ℝ\nk : ℤ\n⊢ 2 * x = (2 * ↑k + 1) * π / 2 ↔ x = (2 * ↑k + 1) * π / 4"
},
{
"line": "focus\n constructor <;> intro\n with_annotate_state\"<;>\" skip\n all_goals linarith",
"before_state": "x : ℝ\nk : ℤ\n⊢ 2 * x = (2 * ↑k + 1) * π / 2 ↔ x = (2 * ↑k + 1) * π / 4",
"after_state": "No Goals!"
},
{
"line": "focus\n constructor\n with_annotate_state\"<;>\" skip\n all_goals intro",
"before_state": "x : ℝ\nk : ℤ\n⊢ 2 * x = (2 * ↑k + 1) * π / 2 ↔ x = (2 * ↑k + 1) * π / 4",
"after_state": "case mp\nx : ℝ\nk : ℤ\na✝ : 2 * x = (2 * ↑k + 1) * π / 2\n⊢ x = (2 * ↑k + 1) * π / 4\n---\ncase mpr\nx : ℝ\nk : ℤ\na✝ : x = (2 * ↑k + 1) * π / 4\n⊢ 2 * x = (2 * ↑k + 1) * π / 2"
},
{
"line": "constructor",
"before_state": "x : ℝ\nk : ℤ\n⊢ 2 * x = (2 * ↑k + 1) * π / 2 ↔ x = (2 * ↑k + 1) * π / 4",
"after_state": "case mp\nx : ℝ\nk : ℤ\n⊢ 2 * x = (2 * ↑k + 1) * π / 2 → x = (2 * ↑k + 1) * π / 4\n---\ncase mpr\nx : ℝ\nk : ℤ\n⊢ x = (2 * ↑k + 1) * π / 4 → 2 * x = (2 * ↑k + 1) * π / 2"
},
{
"line": "with_annotate_state\"<;>\" skip",
"before_state": "case mp\nx : ℝ\nk : ℤ\n⊢ 2 * x = (2 * ↑k + 1) * π / 2 → x = (2 * ↑k + 1) * π / 4\n---\ncase mpr\nx : ℝ\nk : ℤ\n⊢ x = (2 * ↑k + 1) * π / 4 → 2 * x = (2 * ↑k + 1) * π / 2",
"after_state": "case mp\nx : ℝ\nk : ℤ\n⊢ 2 * x = (2 * ↑k + 1) * π / 2 → x = (2 * ↑k + 1) * π / 4\n---\ncase mpr\nx : ℝ\nk : ℤ\n⊢ x = (2 * ↑k + 1) * π / 4 → 2 * x = (2 * ↑k + 1) * π / 2"
},
{
"line": "skip",
"before_state": "case mp\nx : ℝ\nk : ℤ\n⊢ 2 * x = (2 * ↑k + 1) * π / 2 → x = (2 * ↑k + 1) * π / 4\n---\ncase mpr\nx : ℝ\nk : ℤ\n⊢ x = (2 * ↑k + 1) * π / 4 → 2 * x = (2 * ↑k + 1) * π / 2",
"after_state": "case mp\nx : ℝ\nk : ℤ\n⊢ 2 * x = (2 * ↑k + 1) * π / 2 → x = (2 * ↑k + 1) * π / 4\n---\ncase mpr\nx : ℝ\nk : ℤ\n⊢ x = (2 * ↑k + 1) * π / 4 → 2 * x = (2 * ↑k + 1) * π / 2"
},
{
"line": "all_goals intro",
"before_state": "case mp\nx : ℝ\nk : ℤ\n⊢ 2 * x = (2 * ↑k + 1) * π / 2 → x = (2 * ↑k + 1) * π / 4\n---\ncase mpr\nx : ℝ\nk : ℤ\n⊢ x = (2 * ↑k + 1) * π / 4 → 2 * x = (2 * ↑k + 1) * π / 2",
"after_state": "case mp\nx : ℝ\nk : ℤ\na✝ : 2 * x = (2 * ↑k + 1) * π / 2\n⊢ x = (2 * ↑k + 1) * π / 4\n---\ncase mpr\nx : ℝ\nk : ℤ\na✝ : x = (2 * ↑k + 1) * π / 4\n⊢ 2 * x = (2 * ↑k + 1) * π / 2"
},
{
"line": "intro",
"before_state": "case mp\nx : ℝ\nk : ℤ\n⊢ 2 * x = (2 * ↑k + 1) * π / 2 → x = (2 * ↑k + 1) * π / 4",
"after_state": "case mp\nx : ℝ\nk : ℤ\na✝ : 2 * x = (2 * ↑k + 1) * π / 2\n⊢ x = (2 * ↑k + 1) * π / 4"
},
{
"line": "intro",
"before_state": "case mpr\nx : ℝ\nk : ℤ\n⊢ x = (2 * ↑k + 1) * π / 4 → 2 * x = (2 * ↑k + 1) * π / 2",
"after_state": "case mpr\nx : ℝ\nk : ℤ\na✝ : x = (2 * ↑k + 1) * π / 4\n⊢ 2 * x = (2 * ↑k + 1) * π / 2"
},
{
"line": "with_annotate_state\"<;>\" skip",
"before_state": "case mp\nx : ℝ\nk : ℤ\na✝ : 2 * x = (2 * ↑k + 1) * π / 2\n⊢ x = (2 * ↑k + 1) * π / 4\n---\ncase mpr\nx : ℝ\nk : ℤ\na✝ : x = (2 * ↑k + 1) * π / 4\n⊢ 2 * x = (2 * ↑k + 1) * π / 2",
"after_state": "case mp\nx : ℝ\nk : ℤ\na✝ : 2 * x = (2 * ↑k + 1) * π / 2\n⊢ x = (2 * ↑k + 1) * π / 4\n---\ncase mpr\nx : ℝ\nk : ℤ\na✝ : x = (2 * ↑k + 1) * π / 4\n⊢ 2 * x = (2 * ↑k + 1) * π / 2"
},
{
"line": "skip",
"before_state": "case mp\nx : ℝ\nk : ℤ\na✝ : 2 * x = (2 * ↑k + 1) * π / 2\n⊢ x = (2 * ↑k + 1) * π / 4\n---\ncase mpr\nx : ℝ\nk : ℤ\na✝ : x = (2 * ↑k + 1) * π / 4\n⊢ 2 * x = (2 * ↑k + 1) * π / 2",
"after_state": "case mp\nx : ℝ\nk : ℤ\na✝ : 2 * x = (2 * ↑k + 1) * π / 2\n⊢ x = (2 * ↑k + 1) * π / 4\n---\ncase mpr\nx : ℝ\nk : ℤ\na✝ : x = (2 * ↑k + 1) * π / 4\n⊢ 2 * x = (2 * ↑k + 1) * π / 2"
},
{
"line": "all_goals linarith",
"before_state": "case mp\nx : ℝ\nk : ℤ\na✝ : 2 * x = (2 * ↑k + 1) * π / 2\n⊢ x = (2 * ↑k + 1) * π / 4\n---\ncase mpr\nx : ℝ\nk : ℤ\na✝ : x = (2 * ↑k + 1) * π / 4\n⊢ 2 * x = (2 * ↑k + 1) * π / 2",
"after_state": "No Goals!"
},
{
"line": "linarith",
"before_state": "case mp\nx : ℝ\nk : ℤ\na✝ : 2 * x = (2 * ↑k + 1) * π / 2\n⊢ x = (2 * ↑k + 1) * π / 4",
"after_state": "No Goals!"
},
{
"line": "norm_num",
"before_state": "x : ℝ\nk : ℤ\na✝¹ : 2 * x = (2 * ↑k + 1) * π / 2\na✝ : x < (2 * ↑k + 1) * π / 4\n⊢ 1 * 1 = 1",
"after_state": "No Goals!"
},
{
"line": "norm_num",
"before_state": "x : ℝ\nk : ℤ\na✝¹ : 2 * x = (2 * ↑k + 1) * π / 2\na✝ : x < (2 * ↑k + 1) * π / 4\n⊢ 1 * 1 = 1",
"after_state": "No Goals!"
},
{
"line": "norm_num",
"before_state": "x : ℝ\nk : ℤ\na✝¹ : 2 * x = (2 * ↑k + 1) * π / 2\na✝ : x < (2 * ↑k + 1) * π / 4\n⊢ 4 / 4 = 1",
"after_state": "No Goals!"
},
{
"line": "norm_num",
"before_state": "x : ℝ\nk : ℤ\na✝¹ : 2 * x = (2 * ↑k + 1) * π / 2\na✝ : x < (2 * ↑k + 1) * π / 4\n⊢ 1 * 4 = 4",
"after_state": "No Goals!"
},
{
"line": "norm_num",
"before_state": "x : ℝ\nk : ℤ\na✝¹ : 2 * x = (2 * ↑k + 1) * π / 2\na✝ : x < (2 * ↑k + 1) * π / 4\n⊢ 4 > 0",
"after_state": "No Goals!"
},
{
"line": "norm_num",
"before_state": "x : ℝ\nk : ℤ\na✝¹ : 2 * x = (2 * ↑k + 1) * π / 2\na✝ : x < (2 * ↑k + 1) * π / 4\n⊢ 1 * 2 = 2",
"after_state": "No Goals!"
},
{
"line": "norm_num",
"before_state": "x : ℝ\nk : ℤ\na✝¹ : 2 * x = (2 * ↑k + 1) * π / 2\na✝ : x < (2 * ↑k + 1) * π / 4\n⊢ 1 * 1 = 1",
"after_state": "No Goals!"
},
{
"line": "norm_num",
"before_state": "x : ℝ\nk : ℤ\na✝¹ : 2 * x = (2 * ↑k + 1) * π / 2\na✝ : x < (2 * ↑k + 1) * π / 4\n⊢ 1 * 1 = 1",
"after_state": "No Goals!"
},
{
"line": "norm_num",
"before_state": "x : ℝ\nk : ℤ\na✝¹ : 2 * x = (2 * ↑k + 1) * π / 2\na✝ : x < (2 * ↑k + 1) * π / 4\n⊢ 2 / 2 = 1",
"after_state": "No Goals!"
},
{
"line": "norm_num",
"before_state": "x : ℝ\nk : ℤ\na✝¹ : 2 * x = (2 * ↑k + 1) * π / 2\na✝ : x < (2 * ↑k + 1) * π / 4\n⊢ 1 * 2 = 2",
"after_state": "No Goals!"
},
{
"line": "norm_num",
"before_state": "x : ℝ\nk : ℤ\na✝¹ : 2 * x = (2 * ↑k + 1) * π / 2\na✝ : x < (2 * ↑k + 1) * π / 4\n⊢ 2 > 0",
"after_state": "No Goals!"
},
{
"line": "ring1",
"before_state": "x : ℝ\nk : ℤ\na✝¹ : 2 * x = (2 * ↑k + 1) * π / 2\na✝ : x < (2 * ↑k + 1) * π / 4\n⊢ -(1 * 2 * (2 * x) - (1 * 2 * (1 * ↑k) + 1 * 1) * (1 * π)) + (4 * x - (1 * 2 * (1 * ↑k) + 1 * 1) * (1 * π)) = 0",
"after_state": "No Goals!"
},
{
"line": "norm_num",
"before_state": "x : ℝ\nk : ℤ\na✝¹ : 2 * x = (2 * ↑k + 1) * π / 2\na✝ : (2 * ↑k + 1) * π / 4 < x\n⊢ 1 * 1 = 1",
"after_state": "No Goals!"
},
{
"line": "norm_num",
"before_state": "x : ℝ\nk : ℤ\na✝¹ : 2 * x = (2 * ↑k + 1) * π / 2\na✝ : (2 * ↑k + 1) * π / 4 < x\n⊢ 1 * 1 = 1",
"after_state": "No Goals!"
},
{
"line": "norm_num",
"before_state": "x : ℝ\nk : ℤ\na✝¹ : 2 * x = (2 * ↑k + 1) * π / 2\na✝ : (2 * ↑k + 1) * π / 4 < x\n⊢ 4 / 4 = 1",
"after_state": "No Goals!"
},
{
"line": "norm_num",
"before_state": "x : ℝ\nk : ℤ\na✝¹ : 2 * x = (2 * ↑k + 1) * π / 2\na✝ : (2 * ↑k + 1) * π / 4 < x\n⊢ 1 * 4 = 4",
"after_state": "No Goals!"
},
{
"line": "norm_num",
"before_state": "x : ℝ\nk : ℤ\na✝¹ : 2 * x = (2 * ↑k + 1) * π / 2\na✝ : (2 * ↑k + 1) * π / 4 < x\n⊢ 4 > 0",
"after_state": "No Goals!"
},
{
"line": "norm_num",
"before_state": "x : ℝ\nk : ℤ\na✝¹ : 2 * x = (2 * ↑k + 1) * π / 2\na✝ : (2 * ↑k + 1) * π / 4 < x\n⊢ 1 * 2 = 2",
"after_state": "No Goals!"
},
{
"line": "norm_num",
"before_state": "x : ℝ\nk : ℤ\na✝¹ : 2 * x = (2 * ↑k + 1) * π / 2\na✝ : (2 * ↑k + 1) * π / 4 < x\n⊢ 1 * 1 = 1",
"after_state": "No Goals!"
},
{
"line": "norm_num",
"before_state": "x : ℝ\nk : ℤ\na✝¹ : 2 * x = (2 * ↑k + 1) * π / 2\na✝ : (2 * ↑k + 1) * π / 4 < x\n⊢ 1 * 1 = 1",
"after_state": "No Goals!"
},
{
"line": "norm_num",
"before_state": "x : ℝ\nk : ℤ\na✝¹ : 2 * x = (2 * ↑k + 1) * π / 2\na✝ : (2 * ↑k + 1) * π / 4 < x\n⊢ 2 / 2 = 1",
"after_state": "No Goals!"
},
{
"line": "norm_num",
"before_state": "x : ℝ\nk : ℤ\na✝¹ : 2 * x = (2 * ↑k + 1) * π / 2\na✝ : (2 * ↑k + 1) * π / 4 < x\n⊢ 1 * 2 = 2",
"after_state": "No Goals!"
},
{
"line": "norm_num",
"before_state": "x : ℝ\nk : ℤ\na✝¹ : 2 * x = (2 * ↑k + 1) * π / 2\na✝ : (2 * ↑k + 1) * π / 4 < x\n⊢ 2 > 0",
"after_state": "No Goals!"
},
{
"line": "ring1",
"before_state": "x : ℝ\nk : ℤ\na✝¹ : 2 * x = (2 * ↑k + 1) * π / 2\na✝ : (2 * ↑k + 1) * π / 4 < x\n⊢ 1 * 2 * (2 * x) - (1 * 2 * (1 * ↑k) + 1 * 1) * (1 * π) + ((1 * 2 * (1 * ↑k) + 1 * 1) * (1 * π) - 4 * x) = 0",
"after_state": "No Goals!"
},
{
"line": "linarith",
"before_state": "case mpr\nx : ℝ\nk : ℤ\na✝ : x = (2 * ↑k + 1) * π / 4\n⊢ 2 * x = (2 * ↑k + 1) * π / 2",
"after_state": "No Goals!"
},
{
"line": "norm_num",
"before_state": "x : ℝ\nk : ℤ\na✝¹ : x = (2 * ↑k + 1) * π / 4\na✝ : 2 * x < (2 * ↑k + 1) * π / 2\n⊢ 1 * 2 = 2",
"after_state": "No Goals!"
},
{
"line": "norm_num",
"before_state": "x : ℝ\nk : ℤ\na✝¹ : x = (2 * ↑k + 1) * π / 4\na✝ : 2 * x < (2 * ↑k + 1) * π / 2\n⊢ 1 * 1 = 1",
"after_state": "No Goals!"
},
{
"line": "norm_num",
"before_state": "x : ℝ\nk : ℤ\na✝¹ : x = (2 * ↑k + 1) * π / 4\na✝ : 2 * x < (2 * ↑k + 1) * π / 2\n⊢ 1 * 1 = 1",
"after_state": "No Goals!"
},
{
"line": "norm_num",
"before_state": "x : ℝ\nk : ℤ\na✝¹ : x = (2 * ↑k + 1) * π / 4\na✝ : 2 * x < (2 * ↑k + 1) * π / 2\n⊢ 2 / 2 = 1",
"after_state": "No Goals!"
},
{
"line": "norm_num",
"before_state": "x : ℝ\nk : ℤ\na✝¹ : x = (2 * ↑k + 1) * π / 4\na✝ : 2 * x < (2 * ↑k + 1) * π / 2\n⊢ 1 * 2 = 2",
"after_state": "No Goals!"
},
{
"line": "norm_num",
"before_state": "x : ℝ\nk : ℤ\na✝¹ : x = (2 * ↑k + 1) * π / 4\na✝ : 2 * x < (2 * ↑k + 1) * π / 2\n⊢ 2 > 0",
"after_state": "No Goals!"
},
{
"line": "norm_num",
"before_state": "x : ℝ\nk : ℤ\na✝¹ : x = (2 * ↑k + 1) * π / 4\na✝ : 2 * x < (2 * ↑k + 1) * π / 2\n⊢ 1 * 1 = 1",
"after_state": "No Goals!"
},
{
"line": "norm_num",
"before_state": "x : ℝ\nk : ℤ\na✝¹ : x = (2 * ↑k + 1) * π / 4\na✝ : 2 * x < (2 * ↑k + 1) * π / 2\n⊢ 1 * 1 = 1",
"after_state": "No Goals!"
},
{
"line": "norm_num",
"before_state": "x : ℝ\nk : ℤ\na✝¹ : x = (2 * ↑k + 1) * π / 4\na✝ : 2 * x < (2 * ↑k + 1) * π / 2\n⊢ 4 / 4 = 1",
"after_state": "No Goals!"
},
{
"line": "norm_num",
"before_state": "x : ℝ\nk : ℤ\na✝¹ : x = (2 * ↑k + 1) * π / 4\na✝ : 2 * x < (2 * ↑k + 1) * π / 2\n⊢ 1 * 4 = 4",
"after_state": "No Goals!"
},
{
"line": "norm_num",
"before_state": "x : ℝ\nk : ℤ\na✝¹ : x = (2 * ↑k + 1) * π / 4\na✝ : 2 * x < (2 * ↑k + 1) * π / 2\n⊢ 4 > 0",
"after_state": "No Goals!"
},
{
"line": "ring1",
"before_state": "x : ℝ\nk : ℤ\na✝¹ : x = (2 * ↑k + 1) * π / 4\na✝ : 2 * x < (2 * ↑k + 1) * π / 2\n⊢ -(4 * x - (1 * 2 * (1 * ↑k) + 1 * 1) * (1 * π)) + (1 * 2 * (2 * x) - (1 * 2 * (1 * ↑k) + 1 * 1) * (1 * π)) = 0",
"after_state": "No Goals!"
},
{
"line": "norm_num",
"before_state": "x : ℝ\nk : ℤ\na✝¹ : x = (2 * ↑k + 1) * π / 4\na✝ : (2 * ↑k + 1) * π / 2 < 2 * x\n⊢ 1 * 1 = 1",
"after_state": "No Goals!"
},
{
"line": "norm_num",
"before_state": "x : ℝ\nk : ℤ\na✝¹ : x = (2 * ↑k + 1) * π / 4\na✝ : (2 * ↑k + 1) * π / 2 < 2 * x\n⊢ 1 * 1 = 1",
"after_state": "No Goals!"
},
{
"line": "norm_num",
"before_state": "x : ℝ\nk : ℤ\na✝¹ : x = (2 * ↑k + 1) * π / 4\na✝ : (2 * ↑k + 1) * π / 2 < 2 * x\n⊢ 2 / 2 = 1",
"after_state": "No Goals!"
},
{
"line": "norm_num",
"before_state": "x : ℝ\nk : ℤ\na✝¹ : x = (2 * ↑k + 1) * π / 4\na✝ : (2 * ↑k + 1) * π / 2 < 2 * x\n⊢ 1 * 2 = 2",
"after_state": "No Goals!"
},
{
"line": "norm_num",
"before_state": "x : ℝ\nk : ℤ\na✝¹ : x = (2 * ↑k + 1) * π / 4\na✝ : (2 * ↑k + 1) * π / 2 < 2 * x\n⊢ 1 * 2 = 2",
"after_state": "No Goals!"
},
{
"line": "norm_num",
"before_state": "x : ℝ\nk : ℤ\na✝¹ : x = (2 * ↑k + 1) * π / 4\na✝ : (2 * ↑k + 1) * π / 2 < 2 * x\n⊢ 2 > 0",
"after_state": "No Goals!"
},
{
"line": "norm_num",
"before_state": "x : ℝ\nk : ℤ\na✝¹ : x = (2 * ↑k + 1) * π / 4\na✝ : (2 * ↑k + 1) * π / 2 < 2 * x\n⊢ 1 * 1 = 1",
"after_state": "No Goals!"
},
{
"line": "norm_num",
"before_state": "x : ℝ\nk : ℤ\na✝¹ : x = (2 * ↑k + 1) * π / 4\na✝ : (2 * ↑k + 1) * π / 2 < 2 * x\n⊢ 1 * 1 = 1",
"after_state": "No Goals!"
},
{
"line": "norm_num",
"before_state": "x : ℝ\nk : ℤ\na✝¹ : x = (2 * ↑k + 1) * π / 4\na✝ : (2 * ↑k + 1) * π / 2 < 2 * x\n⊢ 4 / 4 = 1",
"after_state": "No Goals!"
},
{
"line": "norm_num",
"before_state": "x : ℝ\nk : ℤ\na✝¹ : x = (2 * ↑k + 1) * π / 4\na✝ : (2 * ↑k + 1) * π / 2 < 2 * x\n⊢ 1 * 4 = 4",
"after_state": "No Goals!"
},
{
"line": "norm_num",
"before_state": "x : ℝ\nk : ℤ\na✝¹ : x = (2 * ↑k + 1) * π / 4\na✝ : (2 * ↑k + 1) * π / 2 < 2 * x\n⊢ 4 > 0",
"after_state": "No Goals!"
},
{
"line": "ring1",
"before_state": "x : ℝ\nk : ℤ\na✝¹ : x = (2 * ↑k + 1) * π / 4\na✝ : (2 * ↑k + 1) * π / 2 < 2 * x\n⊢ 4 * x - (1 * 2 * (1 * ↑k) + 1 * 1) * (1 * π) + ((1 * 2 * (1 * ↑k) + 1 * 1) * (1 * π) - 1 * 2 * (2 * x)) = 0",
"after_state": "No Goals!"
}
] |
lemma two_sin_pi_div_seven_ne_zero : 2 * sin (π / 7) ≠ 0 := by
apply mul_ne_zero two_ne_zero (Real.sin_pos_of_pos_of_lt_pi _ _).ne' <;> linarith [pi_pos]
|
/root/DuelModelResearch/mathlib4/Archive/Imo/Imo1963Q5.lean
|
{
"open": [
"Real"
],
"variables": []
}
|
[
{
"line": "focus\n apply mul_ne_zero two_ne_zero (Real.sin_pos_of_pos_of_lt_pi _ _).ne'\n with_annotate_state\"<;>\" skip\n all_goals linarith [pi_pos]",
"before_state": "⊢ 2 * sin (π / 7) ≠ 0",
"after_state": "No Goals!"
},
{
"line": "apply mul_ne_zero two_ne_zero (Real.sin_pos_of_pos_of_lt_pi _ _).ne'",
"before_state": "⊢ 2 * sin (π / 7) ≠ 0",
"after_state": "⊢ 0 < π / 7\n---\n⊢ π / 7 < π"
},
{
"line": "with_annotate_state\"<;>\" skip",
"before_state": "⊢ 0 < π / 7\n---\n⊢ π / 7 < π",
"after_state": "⊢ 0 < π / 7\n---\n⊢ π / 7 < π"
},
{
"line": "skip",
"before_state": "⊢ 0 < π / 7\n---\n⊢ π / 7 < π",
"after_state": "⊢ 0 < π / 7\n---\n⊢ π / 7 < π"
},
{
"line": "all_goals linarith [pi_pos]",
"before_state": "⊢ 0 < π / 7\n---\n⊢ π / 7 < π",
"after_state": "No Goals!"
},
{
"line": "linarith [pi_pos]",
"before_state": "⊢ 0 < π / 7",
"after_state": "No Goals!"
},
{
"line": "norm_num",
"before_state": "a✝ : 0 ≥ π / 7\n⊢ 7 / 7 = 1",
"after_state": "No Goals!"
},
{
"line": "norm_num",
"before_state": "a✝ : 0 ≥ π / 7\n⊢ 1 * 7 = 7",
"after_state": "No Goals!"
},
{
"line": "norm_num",
"before_state": "a✝ : 0 ≥ π / 7\n⊢ 7 > 0",
"after_state": "No Goals!"
},
{
"line": "ring1",
"before_state": "a✝ : 0 ≥ π / 7\n⊢ 1 * π - 7 * 0 + (0 - π) = 0",
"after_state": "No Goals!"
},
{
"line": "linarith [pi_pos]",
"before_state": "⊢ π / 7 < π",
"after_state": "No Goals!"
},
{
"line": "norm_num",
"before_state": "a✝ : π / 7 ≥ π\n⊢ 7 / 7 = 1",
"after_state": "No Goals!"
},
{
"line": "norm_num",
"before_state": "a✝ : π / 7 ≥ π\n⊢ 1 * 7 = 7",
"after_state": "No Goals!"
},
{
"line": "norm_num",
"before_state": "a✝ : π / 7 ≥ π\n⊢ 7 > 0",
"after_state": "No Goals!"
},
{
"line": "ring1",
"before_state": "a✝ : π / 7 ≥ π\n⊢ 7 * π - 1 * π + 6 * (0 - π) = 0",
"after_state": "No Goals!"
},
{
"line": "norm_num",
"before_state": "a✝ : π / 7 ≥ π\n⊢ 6 > 0",
"after_state": "No Goals!"
}
] |
lemma sin_pi_mul_neg_div (a b : ℝ) : sin (π * (- a / b)) = - sin (π * (a / b)) := by
ring_nf
exact sin_neg _
|
/root/DuelModelResearch/mathlib4/Archive/Imo/Imo1963Q5.lean
|
{
"open": [
"Real"
],
"variables": []
}
|
[
{
"line": "ring_nf",
"before_state": "a b : ℝ\n⊢ sin (π * (-a / b)) = -sin (π * (a / b))",
"after_state": "a b : ℝ\n⊢ sin (-(π * a * b⁻¹)) = -sin (π * a * b⁻¹)"
},
{
"line": "exact sin_neg _",
"before_state": "a b : ℝ\n⊢ sin (-(π * a * b⁻¹)) = -sin (π * a * b⁻¹)",
"after_state": "No Goals!"
}
] |
theorem two_pow_mod_seven (n : ℕ) : 2 ^ n ≡ 2 ^ (n % 3) [MOD 7] :=
let t := n % 3
calc 2 ^ n = 2 ^ (3 * (n / 3) + t) := by rw [Nat.div_add_mod]
_ = (2 ^ 3) ^ (n / 3) * 2 ^ t := by rw [pow_add, pow_mul]
_ ≡ 1 ^ (n / 3) * 2 ^ t [MOD 7] := by gcongr; decide
_ = 2 ^ t := by ring
|
/root/DuelModelResearch/mathlib4/Archive/Imo/Imo1964Q1.lean
|
{
"open": [
"Nat"
],
"variables": []
}
|
[
{
"line": "rw [Nat.div_add_mod]",
"before_state": "n : ℕ\nt : ℕ := n % 3\n⊢ 2 ^ n = 2 ^ (3 * (n / 3) + t)",
"after_state": "No Goals!"
},
{
"line": "rewrite [Nat.div_add_mod]",
"before_state": "n : ℕ\nt : ℕ := n % 3\n⊢ 2 ^ n = 2 ^ (3 * (n / 3) + t)",
"after_state": "n : ℕ\nt : ℕ := n % 3\n⊢ 2 ^ n = 2 ^ n"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "n : ℕ\nt : ℕ := n % 3\n⊢ 2 ^ n = 2 ^ n",
"after_state": "No Goals!"
},
{
"line": "try (with_reducible rfl)",
"before_state": "n : ℕ\nt : ℕ := n % 3\n⊢ 2 ^ n = 2 ^ n",
"after_state": "No Goals!"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "n : ℕ\nt : ℕ := n % 3\n⊢ 2 ^ n = 2 ^ n",
"after_state": "No Goals!"
},
{
"line": "with_reducible rfl",
"before_state": "n : ℕ\nt : ℕ := n % 3\n⊢ 2 ^ n = 2 ^ n",
"after_state": "No Goals!"
},
{
"line": "rfl",
"before_state": "n : ℕ\nt : ℕ := n % 3\n⊢ 2 ^ n = 2 ^ n",
"after_state": "No Goals!"
},
{
"line": "eq_refl",
"before_state": "n : ℕ\nt : ℕ := n % 3\n⊢ 2 ^ n = 2 ^ n",
"after_state": "No Goals!"
},
{
"line": "rw [pow_add, pow_mul]",
"before_state": "n : ℕ\nt : ℕ := n % 3\n⊢ 2 ^ (3 * (n / 3) + t) = (2 ^ 3) ^ (n / 3) * 2 ^ t",
"after_state": "No Goals!"
},
{
"line": "rewrite [pow_add, pow_mul]",
"before_state": "n : ℕ\nt : ℕ := n % 3\n⊢ 2 ^ (3 * (n / 3) + t) = (2 ^ 3) ^ (n / 3) * 2 ^ t",
"after_state": "n : ℕ\nt : ℕ := n % 3\n⊢ (2 ^ 3) ^ (n / 3) * 2 ^ t = (2 ^ 3) ^ (n / 3) * 2 ^ t"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "n : ℕ\nt : ℕ := n % 3\n⊢ (2 ^ 3) ^ (n / 3) * 2 ^ t = (2 ^ 3) ^ (n / 3) * 2 ^ t",
"after_state": "No Goals!"
},
{
"line": "try (with_reducible rfl)",
"before_state": "n : ℕ\nt : ℕ := n % 3\n⊢ (2 ^ 3) ^ (n / 3) * 2 ^ t = (2 ^ 3) ^ (n / 3) * 2 ^ t",
"after_state": "No Goals!"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "n : ℕ\nt : ℕ := n % 3\n⊢ (2 ^ 3) ^ (n / 3) * 2 ^ t = (2 ^ 3) ^ (n / 3) * 2 ^ t",
"after_state": "No Goals!"
},
{
"line": "with_reducible rfl",
"before_state": "n : ℕ\nt : ℕ := n % 3\n⊢ (2 ^ 3) ^ (n / 3) * 2 ^ t = (2 ^ 3) ^ (n / 3) * 2 ^ t",
"after_state": "No Goals!"
},
{
"line": "rfl",
"before_state": "n : ℕ\nt : ℕ := n % 3\n⊢ (2 ^ 3) ^ (n / 3) * 2 ^ t = (2 ^ 3) ^ (n / 3) * 2 ^ t",
"after_state": "No Goals!"
},
{
"line": "eq_refl",
"before_state": "n : ℕ\nt : ℕ := n % 3\n⊢ (2 ^ 3) ^ (n / 3) * 2 ^ t = (2 ^ 3) ^ (n / 3) * 2 ^ t",
"after_state": "No Goals!"
},
{
"line": "gcongr",
"before_state": "n : ℕ\nt : ℕ := n % 3\n⊢ (2 ^ 3) ^ (n / 3) * 2 ^ t ≡ 1 ^ (n / 3) * 2 ^ t [MOD 7]",
"after_state": "case h.h\nn : ℕ\nt : ℕ := n % 3\n⊢ 2 ^ 3 ≡ 1 [MOD 7]"
},
{
"line": "decide",
"before_state": "case h.h\nn : ℕ\nt : ℕ := n % 3\n⊢ 2 ^ 3 ≡ 1 [MOD 7]",
"after_state": "No Goals!"
},
{
"line": "ring",
"before_state": "n : ℕ\nt : ℕ := n % 3\n⊢ 1 ^ (n / 3) * 2 ^ t = 2 ^ t",
"after_state": "No Goals!"
},
{
"line": "first\n| ring1\n|\n try_this ring_nf\"\\n\\nThe `ring` tactic failed to close the goal. Use `ring_nf` to obtain a normal form.\n \\nNote that `ring` works primarily in *commutative* rings. \\\n If you have a noncommutative ring, abelian group or module, consider using \\\n `noncomm_ring`, `abel` or `module` instead.\"",
"before_state": "n : ℕ\nt : ℕ := n % 3\n⊢ 1 ^ (n / 3) * 2 ^ t = 2 ^ t",
"after_state": "No Goals!"
},
{
"line": "ring1",
"before_state": "n : ℕ\nt : ℕ := n % 3\n⊢ 1 ^ (n / 3) * 2 ^ t = 2 ^ t",
"after_state": "No Goals!"
}
] |
theorem imo1964_q1a (n : ℕ) (_ : 0 < n) : 7 ∣ 2 ^ n - 1 ↔ 3 ∣ n := by
let t := n % 3
have : t < 3 := Nat.mod_lt _ (by decide)
calc 7 ∣ 2 ^ n - 1 ↔ 2 ^ n ≡ 1 [MOD 7] := by
rw [Nat.ModEq.comm]
rw [Nat.modEq_iff_dvd']
apply Nat.one_le_pow'
_ ↔ 2 ^ t ≡ 1 [MOD 7] := ⟨(two_pow_mod_seven n).symm.trans, (two_pow_mod_seven n).trans⟩
_ ↔ t = 0 := by interval_cases t <;> decide
_ ↔ 3 ∣ n := by rw [dvd_iff_mod_eq_zero]
|
/root/DuelModelResearch/mathlib4/Archive/Imo/Imo1964Q1.lean
|
{
"open": [
"Nat",
"Imo1964Q1"
],
"variables": []
}
|
[
{
"line": "let t := n % 3",
"before_state": "n : ℕ\nx✝ : 0 < n\n⊢ 7 ∣ 2 ^ n - 1 ↔ 3 ∣ n",
"after_state": "n : ℕ\nx✝ : 0 < n\nt : ℕ := n % 3\n⊢ 7 ∣ 2 ^ n - 1 ↔ 3 ∣ n"
},
{
"line": "refine_lift\n let t := n % 3;\n ?_",
"before_state": "n : ℕ\nx✝ : 0 < n\n⊢ 7 ∣ 2 ^ n - 1 ↔ 3 ∣ n",
"after_state": "n : ℕ\nx✝ : 0 < n\nt : ℕ := n % 3\n⊢ 7 ∣ 2 ^ n - 1 ↔ 3 ∣ n"
},
{
"line": "focus\n (refine\n no_implicit_lambda%\n (let t := n % 3;\n ?_);\n rotate_right)",
"before_state": "n : ℕ\nx✝ : 0 < n\n⊢ 7 ∣ 2 ^ n - 1 ↔ 3 ∣ n",
"after_state": "n : ℕ\nx✝ : 0 < n\nt : ℕ := n % 3\n⊢ 7 ∣ 2 ^ n - 1 ↔ 3 ∣ n"
},
{
"line": "refine\n no_implicit_lambda%\n (let t := n % 3;\n ?_)",
"before_state": "n : ℕ\nx✝ : 0 < n\n⊢ 7 ∣ 2 ^ n - 1 ↔ 3 ∣ n",
"after_state": "n : ℕ\nx✝ : 0 < n\nt : ℕ := n % 3\n⊢ 7 ∣ 2 ^ n - 1 ↔ 3 ∣ n"
},
{
"line": "rotate_right",
"before_state": "n : ℕ\nx✝ : 0 < n\nt : ℕ := n % 3\n⊢ 7 ∣ 2 ^ n - 1 ↔ 3 ∣ n",
"after_state": "n : ℕ\nx✝ : 0 < n\nt : ℕ := n % 3\n⊢ 7 ∣ 2 ^ n - 1 ↔ 3 ∣ n"
},
{
"line": "have : t < 3 := Nat.mod_lt _ (by decide)",
"before_state": "n : ℕ\nx✝ : 0 < n\nt : ℕ := n % 3\n⊢ 7 ∣ 2 ^ n - 1 ↔ 3 ∣ n",
"after_state": "n : ℕ\nx✝ : 0 < n\nt : ℕ := n % 3\nthis : t < 3\n⊢ 7 ∣ 2 ^ n - 1 ↔ 3 ∣ n"
},
{
"line": "refine_lift\n have : t < 3 := Nat.mod_lt _ (by decide);\n ?_",
"before_state": "n : ℕ\nx✝ : 0 < n\nt : ℕ := n % 3\n⊢ 7 ∣ 2 ^ n - 1 ↔ 3 ∣ n",
"after_state": "n : ℕ\nx✝ : 0 < n\nt : ℕ := n % 3\nthis : t < 3\n⊢ 7 ∣ 2 ^ n - 1 ↔ 3 ∣ n"
},
{
"line": "focus\n (refine\n no_implicit_lambda%\n (have : t < 3 := Nat.mod_lt _ (by decide);\n ?_);\n rotate_right)",
"before_state": "n : ℕ\nx✝ : 0 < n\nt : ℕ := n % 3\n⊢ 7 ∣ 2 ^ n - 1 ↔ 3 ∣ n",
"after_state": "n : ℕ\nx✝ : 0 < n\nt : ℕ := n % 3\nthis : t < 3\n⊢ 7 ∣ 2 ^ n - 1 ↔ 3 ∣ n"
},
{
"line": "refine\n no_implicit_lambda%\n (have : t < 3 := Nat.mod_lt _ (by decide);\n ?_)",
"before_state": "n : ℕ\nx✝ : 0 < n\nt : ℕ := n % 3\n⊢ 7 ∣ 2 ^ n - 1 ↔ 3 ∣ n",
"after_state": "n : ℕ\nx✝ : 0 < n\nt : ℕ := n % 3\nthis : t < 3\n⊢ 7 ∣ 2 ^ n - 1 ↔ 3 ∣ n"
},
{
"line": "decide",
"before_state": "n : ℕ\nx✝ : 0 < n\nt : ℕ := n % 3\n⊢ 3 > 0",
"after_state": "No Goals!"
},
{
"line": "rotate_right",
"before_state": "n : ℕ\nx✝ : 0 < n\nt : ℕ := n % 3\nthis : t < 3\n⊢ 7 ∣ 2 ^ n - 1 ↔ 3 ∣ n",
"after_state": "n : ℕ\nx✝ : 0 < n\nt : ℕ := n % 3\nthis : t < 3\n⊢ 7 ∣ 2 ^ n - 1 ↔ 3 ∣ n"
},
{
"line": "calc\n 7 ∣ 2 ^ n - 1 ↔ 2 ^ n ≡ 1 [MOD 7] := by\n rw [Nat.ModEq.comm]\n rw [Nat.modEq_iff_dvd']\n apply Nat.one_le_pow'\n _ ↔ 2 ^ t ≡ 1 [MOD 7] := ⟨(two_pow_mod_seven n).symm.trans, (two_pow_mod_seven n).trans⟩\n _ ↔ t = 0 := by interval_cases t <;> decide\n _ ↔ 3 ∣ n := by rw [dvd_iff_mod_eq_zero]",
"before_state": "n : ℕ\nx✝ : 0 < n\nt : ℕ := n % 3\nthis : t < 3\n⊢ 7 ∣ 2 ^ n - 1 ↔ 3 ∣ n",
"after_state": "No Goals!"
},
{
"line": "rw [Nat.ModEq.comm]",
"before_state": "n : ℕ\nx✝ : 0 < n\nt : ℕ := n % 3\nthis : t < 3\n⊢ 7 ∣ 2 ^ n - 1 ↔ 2 ^ n ≡ 1 [MOD 7]",
"after_state": "n : ℕ\nx✝ : 0 < n\nt : ℕ := n % 3\nthis : t < 3\n⊢ 7 ∣ 2 ^ n - 1 ↔ 1 ≡ 2 ^ n [MOD 7]"
},
{
"line": "rewrite [Nat.ModEq.comm]",
"before_state": "n : ℕ\nx✝ : 0 < n\nt : ℕ := n % 3\nthis : t < 3\n⊢ 7 ∣ 2 ^ n - 1 ↔ 2 ^ n ≡ 1 [MOD 7]",
"after_state": "n : ℕ\nx✝ : 0 < n\nt : ℕ := n % 3\nthis : t < 3\n⊢ 7 ∣ 2 ^ n - 1 ↔ 1 ≡ 2 ^ n [MOD 7]"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "n : ℕ\nx✝ : 0 < n\nt : ℕ := n % 3\nthis : t < 3\n⊢ 7 ∣ 2 ^ n - 1 ↔ 1 ≡ 2 ^ n [MOD 7]",
"after_state": "n : ℕ\nx✝ : 0 < n\nt : ℕ := n % 3\nthis : t < 3\n⊢ 7 ∣ 2 ^ n - 1 ↔ 1 ≡ 2 ^ n [MOD 7]"
},
{
"line": "try (with_reducible rfl)",
"before_state": "n : ℕ\nx✝ : 0 < n\nt : ℕ := n % 3\nthis : t < 3\n⊢ 7 ∣ 2 ^ n - 1 ↔ 1 ≡ 2 ^ n [MOD 7]",
"after_state": "n : ℕ\nx✝ : 0 < n\nt : ℕ := n % 3\nthis : t < 3\n⊢ 7 ∣ 2 ^ n - 1 ↔ 1 ≡ 2 ^ n [MOD 7]"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "n : ℕ\nx✝ : 0 < n\nt : ℕ := n % 3\nthis : t < 3\n⊢ 7 ∣ 2 ^ n - 1 ↔ 1 ≡ 2 ^ n [MOD 7]",
"after_state": "n : ℕ\nx✝ : 0 < n\nt : ℕ := n % 3\nthis : t < 3\n⊢ 7 ∣ 2 ^ n - 1 ↔ 1 ≡ 2 ^ n [MOD 7]"
},
{
"line": "with_reducible rfl",
"before_state": "n : ℕ\nx✝ : 0 < n\nt : ℕ := n % 3\nthis : t < 3\n⊢ 7 ∣ 2 ^ n - 1 ↔ 1 ≡ 2 ^ n [MOD 7]",
"after_state": "n : ℕ\nx✝ : 0 < n\nt : ℕ := n % 3\nthis : t < 3\n⊢ 7 ∣ 2 ^ n - 1 ↔ 1 ≡ 2 ^ n [MOD 7]"
},
{
"line": "rfl",
"before_state": "n : ℕ\nx✝ : 0 < n\nt : ℕ := n % 3\nthis : t < 3\n⊢ 7 ∣ 2 ^ n - 1 ↔ 1 ≡ 2 ^ n [MOD 7]",
"after_state": "n : ℕ\nx✝ : 0 < n\nt : ℕ := n % 3\nthis : t < 3\n⊢ 7 ∣ 2 ^ n - 1 ↔ 1 ≡ 2 ^ n [MOD 7]"
},
{
"line": "apply_rfl",
"before_state": "n : ℕ\nx✝ : 0 < n\nt : ℕ := n % 3\nthis : t < 3\n⊢ 7 ∣ 2 ^ n - 1 ↔ 1 ≡ 2 ^ n [MOD 7]",
"after_state": "n : ℕ\nx✝ : 0 < n\nt : ℕ := n % 3\nthis : t < 3\n⊢ 7 ∣ 2 ^ n - 1 ↔ 1 ≡ 2 ^ n [MOD 7]"
},
{
"line": "skip",
"before_state": "n : ℕ\nx✝ : 0 < n\nt : ℕ := n % 3\nthis : t < 3\n⊢ 7 ∣ 2 ^ n - 1 ↔ 1 ≡ 2 ^ n [MOD 7]",
"after_state": "n : ℕ\nx✝ : 0 < n\nt : ℕ := n % 3\nthis : t < 3\n⊢ 7 ∣ 2 ^ n - 1 ↔ 1 ≡ 2 ^ n [MOD 7]"
},
{
"line": "rw [Nat.modEq_iff_dvd']",
"before_state": "n : ℕ\nx✝ : 0 < n\nt : ℕ := n % 3\nthis : t < 3\n⊢ 7 ∣ 2 ^ n - 1 ↔ 1 ≡ 2 ^ n [MOD 7]",
"after_state": "n : ℕ\nx✝ : 0 < n\nt : ℕ := n % 3\nthis : t < 3\n⊢ 1 ≤ 2 ^ n"
},
{
"line": "rewrite [Nat.modEq_iff_dvd']",
"before_state": "n : ℕ\nx✝ : 0 < n\nt : ℕ := n % 3\nthis : t < 3\n⊢ 7 ∣ 2 ^ n - 1 ↔ 1 ≡ 2 ^ n [MOD 7]",
"after_state": "n : ℕ\nx✝ : 0 < n\nt : ℕ := n % 3\nthis : t < 3\n⊢ 7 ∣ 2 ^ n - 1 ↔ 7 ∣ 2 ^ n - 1\n---\nn : ℕ\nx✝ : 0 < n\nt : ℕ := n % 3\nthis : t < 3\n⊢ 1 ≤ 2 ^ n"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "n : ℕ\nx✝ : 0 < n\nt : ℕ := n % 3\nthis : t < 3\n⊢ 7 ∣ 2 ^ n - 1 ↔ 7 ∣ 2 ^ n - 1\n---\nn : ℕ\nx✝ : 0 < n\nt : ℕ := n % 3\nthis : t < 3\n⊢ 1 ≤ 2 ^ n",
"after_state": "n : ℕ\nx✝ : 0 < n\nt : ℕ := n % 3\nthis : t < 3\n⊢ 1 ≤ 2 ^ n"
},
{
"line": "try (with_reducible rfl)",
"before_state": "n : ℕ\nx✝ : 0 < n\nt : ℕ := n % 3\nthis : t < 3\n⊢ 7 ∣ 2 ^ n - 1 ↔ 7 ∣ 2 ^ n - 1\n---\nn : ℕ\nx✝ : 0 < n\nt : ℕ := n % 3\nthis : t < 3\n⊢ 1 ≤ 2 ^ n",
"after_state": "n : ℕ\nx✝ : 0 < n\nt : ℕ := n % 3\nthis : t < 3\n⊢ 1 ≤ 2 ^ n"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "n : ℕ\nx✝ : 0 < n\nt : ℕ := n % 3\nthis : t < 3\n⊢ 7 ∣ 2 ^ n - 1 ↔ 7 ∣ 2 ^ n - 1\n---\nn : ℕ\nx✝ : 0 < n\nt : ℕ := n % 3\nthis : t < 3\n⊢ 1 ≤ 2 ^ n",
"after_state": "n : ℕ\nx✝ : 0 < n\nt : ℕ := n % 3\nthis : t < 3\n⊢ 1 ≤ 2 ^ n"
},
{
"line": "with_reducible rfl",
"before_state": "n : ℕ\nx✝ : 0 < n\nt : ℕ := n % 3\nthis : t < 3\n⊢ 7 ∣ 2 ^ n - 1 ↔ 7 ∣ 2 ^ n - 1\n---\nn : ℕ\nx✝ : 0 < n\nt : ℕ := n % 3\nthis : t < 3\n⊢ 1 ≤ 2 ^ n",
"after_state": "n : ℕ\nx✝ : 0 < n\nt : ℕ := n % 3\nthis : t < 3\n⊢ 1 ≤ 2 ^ n"
},
{
"line": "rfl",
"before_state": "n : ℕ\nx✝ : 0 < n\nt : ℕ := n % 3\nthis : t < 3\n⊢ 7 ∣ 2 ^ n - 1 ↔ 7 ∣ 2 ^ n - 1\n---\nn : ℕ\nx✝ : 0 < n\nt : ℕ := n % 3\nthis : t < 3\n⊢ 1 ≤ 2 ^ n",
"after_state": "n : ℕ\nx✝ : 0 < n\nt : ℕ := n % 3\nthis : t < 3\n⊢ 1 ≤ 2 ^ n"
},
{
"line": "exact Iff.rfl✝",
"before_state": "n : ℕ\nx✝ : 0 < n\nt : ℕ := n % 3\nthis : t < 3\n⊢ 7 ∣ 2 ^ n - 1 ↔ 7 ∣ 2 ^ n - 1\n---\nn : ℕ\nx✝ : 0 < n\nt : ℕ := n % 3\nthis : t < 3\n⊢ 1 ≤ 2 ^ n",
"after_state": "n : ℕ\nx✝ : 0 < n\nt : ℕ := n % 3\nthis : t < 3\n⊢ 1 ≤ 2 ^ n"
},
{
"line": "apply Nat.one_le_pow'",
"before_state": "n : ℕ\nx✝ : 0 < n\nt : ℕ := n % 3\nthis : t < 3\n⊢ 1 ≤ 2 ^ n",
"after_state": "No Goals!"
},
{
"line": "focus\n interval_cases t\n with_annotate_state\"<;>\" skip\n all_goals decide",
"before_state": "n : ℕ\nx✝ : 0 < n\nt : ℕ := n % 3\nthis : t < 3\n⊢ 2 ^ t ≡ 1 [MOD 7] ↔ t = 0",
"after_state": "No Goals!"
},
{
"line": "interval_cases t",
"before_state": "n : ℕ\nx✝ : 0 < n\nt : ℕ := n % 3\nthis : t < 3\n⊢ 2 ^ t ≡ 1 [MOD 7] ↔ t = 0",
"after_state": "case «0»\nn : ℕ\nx✝ : 0 < n\nt : ℕ := n % 3\nthis : 0 < 3\n⊢ 2 ^ 0 ≡ 1 [MOD 7] ↔ 0 = 0\n---\ncase «1»\nn : ℕ\nx✝ : 0 < n\nt : ℕ := n % 3\nthis : 1 < 3\n⊢ 2 ^ 1 ≡ 1 [MOD 7] ↔ 1 = 0\n---\ncase «2»\nn : ℕ\nx✝ : 0 < n\nt : ℕ := n % 3\nthis : 2 < 3\n⊢ 2 ^ 2 ≡ 1 [MOD 7] ↔ 2 = 0"
},
{
"line": "with_annotate_state\"<;>\" skip",
"before_state": "case «0»\nn : ℕ\nx✝ : 0 < n\nt : ℕ := n % 3\nthis : 0 < 3\n⊢ 2 ^ 0 ≡ 1 [MOD 7] ↔ 0 = 0\n---\ncase «1»\nn : ℕ\nx✝ : 0 < n\nt : ℕ := n % 3\nthis : 1 < 3\n⊢ 2 ^ 1 ≡ 1 [MOD 7] ↔ 1 = 0\n---\ncase «2»\nn : ℕ\nx✝ : 0 < n\nt : ℕ := n % 3\nthis : 2 < 3\n⊢ 2 ^ 2 ≡ 1 [MOD 7] ↔ 2 = 0",
"after_state": "case «0»\nn : ℕ\nx✝ : 0 < n\nt : ℕ := n % 3\nthis : 0 < 3\n⊢ 2 ^ 0 ≡ 1 [MOD 7] ↔ 0 = 0\n---\ncase «1»\nn : ℕ\nx✝ : 0 < n\nt : ℕ := n % 3\nthis : 1 < 3\n⊢ 2 ^ 1 ≡ 1 [MOD 7] ↔ 1 = 0\n---\ncase «2»\nn : ℕ\nx✝ : 0 < n\nt : ℕ := n % 3\nthis : 2 < 3\n⊢ 2 ^ 2 ≡ 1 [MOD 7] ↔ 2 = 0"
},
{
"line": "skip",
"before_state": "case «0»\nn : ℕ\nx✝ : 0 < n\nt : ℕ := n % 3\nthis : 0 < 3\n⊢ 2 ^ 0 ≡ 1 [MOD 7] ↔ 0 = 0\n---\ncase «1»\nn : ℕ\nx✝ : 0 < n\nt : ℕ := n % 3\nthis : 1 < 3\n⊢ 2 ^ 1 ≡ 1 [MOD 7] ↔ 1 = 0\n---\ncase «2»\nn : ℕ\nx✝ : 0 < n\nt : ℕ := n % 3\nthis : 2 < 3\n⊢ 2 ^ 2 ≡ 1 [MOD 7] ↔ 2 = 0",
"after_state": "case «0»\nn : ℕ\nx✝ : 0 < n\nt : ℕ := n % 3\nthis : 0 < 3\n⊢ 2 ^ 0 ≡ 1 [MOD 7] ↔ 0 = 0\n---\ncase «1»\nn : ℕ\nx✝ : 0 < n\nt : ℕ := n % 3\nthis : 1 < 3\n⊢ 2 ^ 1 ≡ 1 [MOD 7] ↔ 1 = 0\n---\ncase «2»\nn : ℕ\nx✝ : 0 < n\nt : ℕ := n % 3\nthis : 2 < 3\n⊢ 2 ^ 2 ≡ 1 [MOD 7] ↔ 2 = 0"
},
{
"line": "all_goals decide",
"before_state": "case «0»\nn : ℕ\nx✝ : 0 < n\nt : ℕ := n % 3\nthis : 0 < 3\n⊢ 2 ^ 0 ≡ 1 [MOD 7] ↔ 0 = 0\n---\ncase «1»\nn : ℕ\nx✝ : 0 < n\nt : ℕ := n % 3\nthis : 1 < 3\n⊢ 2 ^ 1 ≡ 1 [MOD 7] ↔ 1 = 0\n---\ncase «2»\nn : ℕ\nx✝ : 0 < n\nt : ℕ := n % 3\nthis : 2 < 3\n⊢ 2 ^ 2 ≡ 1 [MOD 7] ↔ 2 = 0",
"after_state": "No Goals!"
},
{
"line": "decide",
"before_state": "case «0»\nn : ℕ\nx✝ : 0 < n\nt : ℕ := n % 3\nthis : 0 < 3\n⊢ 2 ^ 0 ≡ 1 [MOD 7] ↔ 0 = 0",
"after_state": "No Goals!"
},
{
"line": "decide",
"before_state": "case «1»\nn : ℕ\nx✝ : 0 < n\nt : ℕ := n % 3\nthis : 1 < 3\n⊢ 2 ^ 1 ≡ 1 [MOD 7] ↔ 1 = 0",
"after_state": "No Goals!"
},
{
"line": "decide",
"before_state": "case «2»\nn : ℕ\nx✝ : 0 < n\nt : ℕ := n % 3\nthis : 2 < 3\n⊢ 2 ^ 2 ≡ 1 [MOD 7] ↔ 2 = 0",
"after_state": "No Goals!"
},
{
"line": "rw [dvd_iff_mod_eq_zero]",
"before_state": "n : ℕ\nx✝ : 0 < n\nt : ℕ := n % 3\nthis : t < 3\n⊢ t = 0 ↔ 3 ∣ n",
"after_state": "No Goals!"
},
{
"line": "rewrite [dvd_iff_mod_eq_zero]",
"before_state": "n : ℕ\nx✝ : 0 < n\nt : ℕ := n % 3\nthis : t < 3\n⊢ t = 0 ↔ 3 ∣ n",
"after_state": "n : ℕ\nx✝ : 0 < n\nt : ℕ := n % 3\nthis : t < 3\n⊢ t = 0 ↔ n % 3 = 0"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "n : ℕ\nx✝ : 0 < n\nt : ℕ := n % 3\nthis : t < 3\n⊢ t = 0 ↔ n % 3 = 0",
"after_state": "No Goals!"
},
{
"line": "try (with_reducible rfl)",
"before_state": "n : ℕ\nx✝ : 0 < n\nt : ℕ := n % 3\nthis : t < 3\n⊢ t = 0 ↔ n % 3 = 0",
"after_state": "No Goals!"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "n : ℕ\nx✝ : 0 < n\nt : ℕ := n % 3\nthis : t < 3\n⊢ t = 0 ↔ n % 3 = 0",
"after_state": "No Goals!"
},
{
"line": "with_reducible rfl",
"before_state": "n : ℕ\nx✝ : 0 < n\nt : ℕ := n % 3\nthis : t < 3\n⊢ t = 0 ↔ n % 3 = 0",
"after_state": "No Goals!"
},
{
"line": "rfl",
"before_state": "n : ℕ\nx✝ : 0 < n\nt : ℕ := n % 3\nthis : t < 3\n⊢ t = 0 ↔ n % 3 = 0",
"after_state": "No Goals!"
},
{
"line": "exact Iff.rfl✝",
"before_state": "n : ℕ\nx✝ : 0 < n\nt : ℕ := n % 3\nthis : t < 3\n⊢ t = 0 ↔ n % 3 = 0",
"after_state": "No Goals!"
}
] |
theorem imo1964_q1b (n : ℕ) : ¬7 ∣ 2 ^ n + 1 := by
intro h
let t := n % 3
have : t < 3 := Nat.mod_lt _ (by decide)
have H : 2 ^ t + 1 ≡ 0 [MOD 7] := calc
2 ^ t + 1 ≡ 2 ^ n + 1 [MOD 7] := by gcongr ?_ + 1; exact (two_pow_mod_seven n).symm
_ ≡ 0 [MOD 7] := h.modEq_zero_nat
interval_cases t <;> contradiction
|
/root/DuelModelResearch/mathlib4/Archive/Imo/Imo1964Q1.lean
|
{
"open": [
"Nat",
"Imo1964Q1"
],
"variables": []
}
|
[
{
"line": "intro h",
"before_state": "n : ℕ\n⊢ ¬7 ∣ 2 ^ n + 1",
"after_state": "n : ℕ\nh : 7 ∣ 2 ^ n + 1\n⊢ False"
},
{
"line": "let t := n % 3",
"before_state": "n : ℕ\nh : 7 ∣ 2 ^ n + 1\n⊢ False",
"after_state": "n : ℕ\nh : 7 ∣ 2 ^ n + 1\nt : ℕ := n % 3\n⊢ False"
},
{
"line": "refine_lift\n let t := n % 3;\n ?_",
"before_state": "n : ℕ\nh : 7 ∣ 2 ^ n + 1\n⊢ False",
"after_state": "n : ℕ\nh : 7 ∣ 2 ^ n + 1\nt : ℕ := n % 3\n⊢ False"
},
{
"line": "focus\n (refine\n no_implicit_lambda%\n (let t := n % 3;\n ?_);\n rotate_right)",
"before_state": "n : ℕ\nh : 7 ∣ 2 ^ n + 1\n⊢ False",
"after_state": "n : ℕ\nh : 7 ∣ 2 ^ n + 1\nt : ℕ := n % 3\n⊢ False"
},
{
"line": "refine\n no_implicit_lambda%\n (let t := n % 3;\n ?_)",
"before_state": "n : ℕ\nh : 7 ∣ 2 ^ n + 1\n⊢ False",
"after_state": "n : ℕ\nh : 7 ∣ 2 ^ n + 1\nt : ℕ := n % 3\n⊢ False"
},
{
"line": "rotate_right",
"before_state": "n : ℕ\nh : 7 ∣ 2 ^ n + 1\nt : ℕ := n % 3\n⊢ False",
"after_state": "n : ℕ\nh : 7 ∣ 2 ^ n + 1\nt : ℕ := n % 3\n⊢ False"
},
{
"line": "have : t < 3 := Nat.mod_lt _ (by decide)",
"before_state": "n : ℕ\nh : 7 ∣ 2 ^ n + 1\nt : ℕ := n % 3\n⊢ False",
"after_state": "n : ℕ\nh : 7 ∣ 2 ^ n + 1\nt : ℕ := n % 3\nthis : t < 3\n⊢ False"
},
{
"line": "refine_lift\n have : t < 3 := Nat.mod_lt _ (by decide);\n ?_",
"before_state": "n : ℕ\nh : 7 ∣ 2 ^ n + 1\nt : ℕ := n % 3\n⊢ False",
"after_state": "n : ℕ\nh : 7 ∣ 2 ^ n + 1\nt : ℕ := n % 3\nthis : t < 3\n⊢ False"
},
{
"line": "focus\n (refine\n no_implicit_lambda%\n (have : t < 3 := Nat.mod_lt _ (by decide);\n ?_);\n rotate_right)",
"before_state": "n : ℕ\nh : 7 ∣ 2 ^ n + 1\nt : ℕ := n % 3\n⊢ False",
"after_state": "n : ℕ\nh : 7 ∣ 2 ^ n + 1\nt : ℕ := n % 3\nthis : t < 3\n⊢ False"
},
{
"line": "refine\n no_implicit_lambda%\n (have : t < 3 := Nat.mod_lt _ (by decide);\n ?_)",
"before_state": "n : ℕ\nh : 7 ∣ 2 ^ n + 1\nt : ℕ := n % 3\n⊢ False",
"after_state": "n : ℕ\nh : 7 ∣ 2 ^ n + 1\nt : ℕ := n % 3\nthis : t < 3\n⊢ False"
},
{
"line": "decide",
"before_state": "n : ℕ\nh : 7 ∣ 2 ^ n + 1\nt : ℕ := n % 3\n⊢ 3 > 0",
"after_state": "No Goals!"
},
{
"line": "rotate_right",
"before_state": "n : ℕ\nh : 7 ∣ 2 ^ n + 1\nt : ℕ := n % 3\nthis : t < 3\n⊢ False",
"after_state": "n : ℕ\nh : 7 ∣ 2 ^ n + 1\nt : ℕ := n % 3\nthis : t < 3\n⊢ False"
},
{
"line": "have H : 2 ^ t + 1 ≡ 0 [MOD 7] :=\n calc\n 2 ^ t + 1 ≡ 2 ^ n + 1 [MOD 7] := by gcongr ?_ + 1; exact (two_pow_mod_seven n).symm\n _ ≡ 0 [MOD 7] := h.modEq_zero_nat",
"before_state": "n : ℕ\nh : 7 ∣ 2 ^ n + 1\nt : ℕ := n % 3\nthis : t < 3\n⊢ False",
"after_state": "n : ℕ\nh : 7 ∣ 2 ^ n + 1\nt : ℕ := n % 3\nthis : t < 3\nH : 2 ^ t + 1 ≡ 0 [MOD 7]\n⊢ False"
},
{
"line": "refine_lift\n have H : 2 ^ t + 1 ≡ 0 [MOD 7] :=\n calc\n 2 ^ t + 1 ≡ 2 ^ n + 1 [MOD 7] := by gcongr ?_ + 1; exact (two_pow_mod_seven n).symm\n _ ≡ 0 [MOD 7] := h.modEq_zero_nat;\n ?_",
"before_state": "n : ℕ\nh : 7 ∣ 2 ^ n + 1\nt : ℕ := n % 3\nthis : t < 3\n⊢ False",
"after_state": "n : ℕ\nh : 7 ∣ 2 ^ n + 1\nt : ℕ := n % 3\nthis : t < 3\nH : 2 ^ t + 1 ≡ 0 [MOD 7]\n⊢ False"
},
{
"line": "focus\n (refine\n no_implicit_lambda%\n (have H : 2 ^ t + 1 ≡ 0 [MOD 7] :=\n calc\n 2 ^ t + 1 ≡ 2 ^ n + 1 [MOD 7] := by gcongr ?_ + 1; exact (two_pow_mod_seven n).symm\n _ ≡ 0 [MOD 7] := h.modEq_zero_nat;\n ?_);\n rotate_right)",
"before_state": "n : ℕ\nh : 7 ∣ 2 ^ n + 1\nt : ℕ := n % 3\nthis : t < 3\n⊢ False",
"after_state": "n : ℕ\nh : 7 ∣ 2 ^ n + 1\nt : ℕ := n % 3\nthis : t < 3\nH : 2 ^ t + 1 ≡ 0 [MOD 7]\n⊢ False"
},
{
"line": "refine\n no_implicit_lambda%\n (have H : 2 ^ t + 1 ≡ 0 [MOD 7] :=\n calc\n 2 ^ t + 1 ≡ 2 ^ n + 1 [MOD 7] := by gcongr ?_ + 1; exact (two_pow_mod_seven n).symm\n _ ≡ 0 [MOD 7] := h.modEq_zero_nat;\n ?_)",
"before_state": "n : ℕ\nh : 7 ∣ 2 ^ n + 1\nt : ℕ := n % 3\nthis : t < 3\n⊢ False",
"after_state": "n : ℕ\nh : 7 ∣ 2 ^ n + 1\nt : ℕ := n % 3\nthis : t < 3\nH : 2 ^ t + 1 ≡ 0 [MOD 7]\n⊢ False"
},
{
"line": "gcongr ?_ + 1",
"before_state": "n : ℕ\nh : 7 ∣ 2 ^ n + 1\nt : ℕ := n % 3\nthis : t < 3\n⊢ 2 ^ t + 1 ≡ 2 ^ n + 1 [MOD 7]",
"after_state": "case h\nn : ℕ\nh : 7 ∣ 2 ^ n + 1\nt : ℕ := n % 3\nthis : t < 3\n⊢ 2 ^ t ≡ 2 ^ n [MOD 7]"
},
{
"line": "exact (two_pow_mod_seven n).symm",
"before_state": "case h\nn : ℕ\nh : 7 ∣ 2 ^ n + 1\nt : ℕ := n % 3\nthis : t < 3\n⊢ 2 ^ t ≡ 2 ^ n [MOD 7]",
"after_state": "No Goals!"
},
{
"line": "rotate_right",
"before_state": "n : ℕ\nh : 7 ∣ 2 ^ n + 1\nt : ℕ := n % 3\nthis : t < 3\nH : 2 ^ t + 1 ≡ 0 [MOD 7]\n⊢ False",
"after_state": "n : ℕ\nh : 7 ∣ 2 ^ n + 1\nt : ℕ := n % 3\nthis : t < 3\nH : 2 ^ t + 1 ≡ 0 [MOD 7]\n⊢ False"
},
{
"line": "focus\n interval_cases t\n with_annotate_state\"<;>\" skip\n all_goals contradiction",
"before_state": "n : ℕ\nh : 7 ∣ 2 ^ n + 1\nt : ℕ := n % 3\nthis : t < 3\nH : 2 ^ t + 1 ≡ 0 [MOD 7]\n⊢ False",
"after_state": "No Goals!"
},
{
"line": "interval_cases t",
"before_state": "n : ℕ\nh : 7 ∣ 2 ^ n + 1\nt : ℕ := n % 3\nthis : t < 3\nH : 2 ^ t + 1 ≡ 0 [MOD 7]\n⊢ False",
"after_state": "case «0»\nn : ℕ\nh : 7 ∣ 2 ^ n + 1\nt : ℕ := n % 3\nthis : 0 < 3\nH : 2 ^ 0 + 1 ≡ 0 [MOD 7]\n⊢ False\n---\ncase «1»\nn : ℕ\nh : 7 ∣ 2 ^ n + 1\nt : ℕ := n % 3\nthis : 1 < 3\nH : 2 ^ 1 + 1 ≡ 0 [MOD 7]\n⊢ False\n---\ncase «2»\nn : ℕ\nh : 7 ∣ 2 ^ n + 1\nt : ℕ := n % 3\nthis : 2 < 3\nH : 2 ^ 2 + 1 ≡ 0 [MOD 7]\n⊢ False"
},
{
"line": "with_annotate_state\"<;>\" skip",
"before_state": "case «0»\nn : ℕ\nh : 7 ∣ 2 ^ n + 1\nt : ℕ := n % 3\nthis : 0 < 3\nH : 2 ^ 0 + 1 ≡ 0 [MOD 7]\n⊢ False\n---\ncase «1»\nn : ℕ\nh : 7 ∣ 2 ^ n + 1\nt : ℕ := n % 3\nthis : 1 < 3\nH : 2 ^ 1 + 1 ≡ 0 [MOD 7]\n⊢ False\n---\ncase «2»\nn : ℕ\nh : 7 ∣ 2 ^ n + 1\nt : ℕ := n % 3\nthis : 2 < 3\nH : 2 ^ 2 + 1 ≡ 0 [MOD 7]\n⊢ False",
"after_state": "case «0»\nn : ℕ\nh : 7 ∣ 2 ^ n + 1\nt : ℕ := n % 3\nthis : 0 < 3\nH : 2 ^ 0 + 1 ≡ 0 [MOD 7]\n⊢ False\n---\ncase «1»\nn : ℕ\nh : 7 ∣ 2 ^ n + 1\nt : ℕ := n % 3\nthis : 1 < 3\nH : 2 ^ 1 + 1 ≡ 0 [MOD 7]\n⊢ False\n---\ncase «2»\nn : ℕ\nh : 7 ∣ 2 ^ n + 1\nt : ℕ := n % 3\nthis : 2 < 3\nH : 2 ^ 2 + 1 ≡ 0 [MOD 7]\n⊢ False"
},
{
"line": "skip",
"before_state": "case «0»\nn : ℕ\nh : 7 ∣ 2 ^ n + 1\nt : ℕ := n % 3\nthis : 0 < 3\nH : 2 ^ 0 + 1 ≡ 0 [MOD 7]\n⊢ False\n---\ncase «1»\nn : ℕ\nh : 7 ∣ 2 ^ n + 1\nt : ℕ := n % 3\nthis : 1 < 3\nH : 2 ^ 1 + 1 ≡ 0 [MOD 7]\n⊢ False\n---\ncase «2»\nn : ℕ\nh : 7 ∣ 2 ^ n + 1\nt : ℕ := n % 3\nthis : 2 < 3\nH : 2 ^ 2 + 1 ≡ 0 [MOD 7]\n⊢ False",
"after_state": "case «0»\nn : ℕ\nh : 7 ∣ 2 ^ n + 1\nt : ℕ := n % 3\nthis : 0 < 3\nH : 2 ^ 0 + 1 ≡ 0 [MOD 7]\n⊢ False\n---\ncase «1»\nn : ℕ\nh : 7 ∣ 2 ^ n + 1\nt : ℕ := n % 3\nthis : 1 < 3\nH : 2 ^ 1 + 1 ≡ 0 [MOD 7]\n⊢ False\n---\ncase «2»\nn : ℕ\nh : 7 ∣ 2 ^ n + 1\nt : ℕ := n % 3\nthis : 2 < 3\nH : 2 ^ 2 + 1 ≡ 0 [MOD 7]\n⊢ False"
},
{
"line": "all_goals contradiction",
"before_state": "case «0»\nn : ℕ\nh : 7 ∣ 2 ^ n + 1\nt : ℕ := n % 3\nthis : 0 < 3\nH : 2 ^ 0 + 1 ≡ 0 [MOD 7]\n⊢ False\n---\ncase «1»\nn : ℕ\nh : 7 ∣ 2 ^ n + 1\nt : ℕ := n % 3\nthis : 1 < 3\nH : 2 ^ 1 + 1 ≡ 0 [MOD 7]\n⊢ False\n---\ncase «2»\nn : ℕ\nh : 7 ∣ 2 ^ n + 1\nt : ℕ := n % 3\nthis : 2 < 3\nH : 2 ^ 2 + 1 ≡ 0 [MOD 7]\n⊢ False",
"after_state": "No Goals!"
},
{
"line": "contradiction",
"before_state": "case «0»\nn : ℕ\nh : 7 ∣ 2 ^ n + 1\nt : ℕ := n % 3\nthis : 0 < 3\nH : 2 ^ 0 + 1 ≡ 0 [MOD 7]\n⊢ False",
"after_state": "No Goals!"
},
{
"line": "contradiction",
"before_state": "case «1»\nn : ℕ\nh : 7 ∣ 2 ^ n + 1\nt : ℕ := n % 3\nthis : 1 < 3\nH : 2 ^ 1 + 1 ≡ 0 [MOD 7]\n⊢ False",
"after_state": "No Goals!"
},
{
"line": "contradiction",
"before_state": "case «2»\nn : ℕ\nh : 7 ∣ 2 ^ n + 1\nt : ℕ := n % 3\nthis : 2 < 3\nH : 2 ^ 2 + 1 ≡ 0 [MOD 7]\n⊢ False",
"after_state": "No Goals!"
}
] |
theorem left_factor_large {m : ℤ} (n : ℤ) (h : 1 < m) : 1 < (n - m) ^ 2 + m ^ 2 := by nlinarith
|
/root/DuelModelResearch/mathlib4/Archive/Imo/Imo1969Q1.lean
|
{
"open": [
"Int Nat"
],
"variables": []
}
|
[
{
"line": "nlinarith",
"before_state": "m n : ℤ\nh : 1 < m\n⊢ 1 < (n - m) ^ 2 + m ^ 2",
"after_state": "No Goals!"
},
{
"line": "ring1",
"before_state": "m n : ℤ\nh : 1 < m\na✝ : 1 ≥ (n - m) ^ 2 + m ^ 2\n⊢ 3 * -1 + (0 - (n - m) ^ 2) + 4 * (1 + 1 - m) + ((n - m) ^ 2 + m ^ 2 - 1) + (0 - (1 + 1 - m) * (1 + 1 - m)) = 0",
"after_state": "No Goals!"
},
{
"line": "norm_num",
"before_state": "m n : ℤ\nh : 1 < m\na✝ : 1 ≥ (n - m) ^ 2 + m ^ 2\n⊢ 3 > 0",
"after_state": "No Goals!"
},
{
"line": "norm_num",
"before_state": "m n : ℤ\nh : 1 < m\na✝ : 1 ≥ (n - m) ^ 2 + m ^ 2\n⊢ 4 > 0",
"after_state": "No Goals!"
}
] |
theorem right_factor_large {m : ℤ} (n : ℤ) (h : 1 < m) : 1 < (n + m) ^ 2 + m ^ 2 := by nlinarith
|
/root/DuelModelResearch/mathlib4/Archive/Imo/Imo1969Q1.lean
|
{
"open": [
"Int Nat"
],
"variables": []
}
|
[
{
"line": "nlinarith",
"before_state": "m n : ℤ\nh : 1 < m\n⊢ 1 < (n + m) ^ 2 + m ^ 2",
"after_state": "No Goals!"
},
{
"line": "ring1",
"before_state": "m n : ℤ\nh : 1 < m\na✝ : 1 ≥ (n + m) ^ 2 + m ^ 2\n⊢ 3 * -1 + (0 - (n + m) ^ 2) + 4 * (1 + 1 - m) + ((n + m) ^ 2 + m ^ 2 - 1) + (0 - (1 + 1 - m) * (1 + 1 - m)) = 0",
"after_state": "No Goals!"
},
{
"line": "norm_num",
"before_state": "m n : ℤ\nh : 1 < m\na✝ : 1 ≥ (n + m) ^ 2 + m ^ 2\n⊢ 3 > 0",
"after_state": "No Goals!"
},
{
"line": "norm_num",
"before_state": "m n : ℤ\nh : 1 < m\na✝ : 1 ≥ (n + m) ^ 2 + m ^ 2\n⊢ 4 > 0",
"after_state": "No Goals!"
}
] |
theorem int_large {m : ℤ} (h : 1 < m) : 1 < m.natAbs := by
exact_mod_cast lt_of_lt_of_le h le_natAbs
|
/root/DuelModelResearch/mathlib4/Archive/Imo/Imo1969Q1.lean
|
{
"open": [
"Int Nat"
],
"variables": []
}
|
[
{
"line": "exact_mod_cast lt_of_lt_of_le h le_natAbs",
"before_state": "m : ℤ\nh : 1 < m\n⊢ 1 < m.natAbs",
"after_state": "No Goals!"
},
{
"line": "exact mod_cast (lt_of_lt_of_le h le_natAbs : _)",
"before_state": "m : ℤ\nh : 1 < m\n⊢ 1 < m.natAbs",
"after_state": "No Goals!"
}
] |
theorem polynomial_not_prime {m : ℕ} (h1 : 1 < m) (n : ℕ) : ¬Nat.Prime (n ^ 4 + 4 * m ^ 4) := by
have h2 : 1 < (m : ℤ) := Int.ofNat_lt.mpr h1
refine not_prime_of_int_mul' (left_factor_large (n : ℤ) h2) (right_factor_large (n : ℤ) h2) ?_
apply factorization
|
/root/DuelModelResearch/mathlib4/Archive/Imo/Imo1969Q1.lean
|
{
"open": [
"Int Nat"
],
"variables": []
}
|
[
{
"line": "have h2 : 1 < (m : ℤ) := Int.ofNat_lt.mpr h1",
"before_state": "m : ℕ\nh1 : 1 < m\nn : ℕ\n⊢ ¬Nat.Prime (n ^ 4 + 4 * m ^ 4)",
"after_state": "m : ℕ\nh1 : 1 < m\nn : ℕ\nh2 : 1 < ↑m\n⊢ ¬Nat.Prime (n ^ 4 + 4 * m ^ 4)"
},
{
"line": "refine_lift\n have h2 : 1 < (m : ℤ) := Int.ofNat_lt.mpr h1;\n ?_",
"before_state": "m : ℕ\nh1 : 1 < m\nn : ℕ\n⊢ ¬Nat.Prime (n ^ 4 + 4 * m ^ 4)",
"after_state": "m : ℕ\nh1 : 1 < m\nn : ℕ\nh2 : 1 < ↑m\n⊢ ¬Nat.Prime (n ^ 4 + 4 * m ^ 4)"
},
{
"line": "focus\n (refine\n no_implicit_lambda%\n (have h2 : 1 < (m : ℤ) := Int.ofNat_lt.mpr h1;\n ?_);\n rotate_right)",
"before_state": "m : ℕ\nh1 : 1 < m\nn : ℕ\n⊢ ¬Nat.Prime (n ^ 4 + 4 * m ^ 4)",
"after_state": "m : ℕ\nh1 : 1 < m\nn : ℕ\nh2 : 1 < ↑m\n⊢ ¬Nat.Prime (n ^ 4 + 4 * m ^ 4)"
},
{
"line": "refine\n no_implicit_lambda%\n (have h2 : 1 < (m : ℤ) := Int.ofNat_lt.mpr h1;\n ?_)",
"before_state": "m : ℕ\nh1 : 1 < m\nn : ℕ\n⊢ ¬Nat.Prime (n ^ 4 + 4 * m ^ 4)",
"after_state": "m : ℕ\nh1 : 1 < m\nn : ℕ\nh2 : 1 < ↑m\n⊢ ¬Nat.Prime (n ^ 4 + 4 * m ^ 4)"
},
{
"line": "rotate_right",
"before_state": "m : ℕ\nh1 : 1 < m\nn : ℕ\nh2 : 1 < ↑m\n⊢ ¬Nat.Prime (n ^ 4 + 4 * m ^ 4)",
"after_state": "m : ℕ\nh1 : 1 < m\nn : ℕ\nh2 : 1 < ↑m\n⊢ ¬Nat.Prime (n ^ 4 + 4 * m ^ 4)"
},
{
"line": "refine not_prime_of_int_mul' (left_factor_large (n : ℤ) h2) (right_factor_large (n : ℤ) h2) ?_",
"before_state": "m : ℕ\nh1 : 1 < m\nn : ℕ\nh2 : 1 < ↑m\n⊢ ¬Nat.Prime (n ^ 4 + 4 * m ^ 4)",
"after_state": "No Goals!"
}
] |
theorem bound (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) :
a ^ 4 / (a ^ 4 + b ^ 4 + c ^ 4) ≤ a ^ 3 / sqrt ((a ^ 3) ^ 2 + ↑8 * b ^ 3 * c ^ 3) := by
rw [div_le_div_iff₀ (by positivity) (by positivity)]
calc a ^ 4 * sqrt ((a ^ 3) ^ 2 + (8:ℝ) * b ^ 3 * c ^ 3)
= a ^ 3 * (a * sqrt ((a ^ 3) ^ 2 + (8:ℝ) * b ^ 3 * c ^ 3)) := by ring
_ ≤ a ^ 3 * (a ^ 4 + b ^ 4 + c ^ 4) := ?_
gcongr
apply le_of_pow_le_pow_left₀ two_ne_zero (by positivity)
rw [mul_pow]
rw [sq_sqrt (by positivity)]
rw [← sub_nonneg]
calc
(a ^ 4 + b ^ 4 + c ^ 4) ^ 2 - a ^ 2 * ((a ^ 3) ^ 2 + 8 * b ^ 3 * c ^ 3)
= 2 * (a ^ 2 * (b ^ 2 - c ^ 2)) ^ 2 + (b ^ 4 - c ^ 4) ^ 2 +
(2 * (a ^ 2 * b * c - b ^ 2 * c ^ 2)) ^ 2 := by ring
_ ≥ 0 := by positivity
|
/root/DuelModelResearch/mathlib4/Archive/Imo/Imo2001Q2.lean
|
{
"open": [
"Real"
],
"variables": [
"{a b c : ℝ}"
]
}
|
[
{
"line": "rw [div_le_div_iff₀ (by positivity) (by positivity)]",
"before_state": "a b c : ℝ\nha : 0 < a\nhb : 0 < b\nhc : 0 < c\n⊢ a ^ 4 / (a ^ 4 + b ^ 4 + c ^ 4) ≤ a ^ 3 / √((a ^ 3) ^ 2 + 8 * b ^ 3 * c ^ 3)",
"after_state": "a b c : ℝ\nha : 0 < a\nhb : 0 < b\nhc : 0 < c\n⊢ a ^ 4 * √((a ^ 3) ^ 2 + 8 * b ^ 3 * c ^ 3) ≤ a ^ 3 * (a ^ 4 + b ^ 4 + c ^ 4)"
},
{
"line": "rewrite [div_le_div_iff₀ (by positivity) (by positivity)]",
"before_state": "a b c : ℝ\nha : 0 < a\nhb : 0 < b\nhc : 0 < c\n⊢ a ^ 4 / (a ^ 4 + b ^ 4 + c ^ 4) ≤ a ^ 3 / √((a ^ 3) ^ 2 + 8 * b ^ 3 * c ^ 3)",
"after_state": "a b c : ℝ\nha : 0 < a\nhb : 0 < b\nhc : 0 < c\n⊢ a ^ 4 * √((a ^ 3) ^ 2 + 8 * b ^ 3 * c ^ 3) ≤ a ^ 3 * (a ^ 4 + b ^ 4 + c ^ 4)"
},
{
"line": "positivity",
"before_state": "a b c : ℝ\nha : 0 < a\nhb : 0 < b\nhc : 0 < c\n⊢ 0 < a ^ 4 + b ^ 4 + c ^ 4",
"after_state": "No Goals!"
},
{
"line": "positivity",
"before_state": "a b c : ℝ\nha : 0 < a\nhb : 0 < b\nhc : 0 < c\n⊢ 0 < √((a ^ 3) ^ 2 + 8 * b ^ 3 * c ^ 3)",
"after_state": "No Goals!"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "a b c : ℝ\nha : 0 < a\nhb : 0 < b\nhc : 0 < c\n⊢ a ^ 4 * √((a ^ 3) ^ 2 + 8 * b ^ 3 * c ^ 3) ≤ a ^ 3 * (a ^ 4 + b ^ 4 + c ^ 4)",
"after_state": "a b c : ℝ\nha : 0 < a\nhb : 0 < b\nhc : 0 < c\n⊢ a ^ 4 * √((a ^ 3) ^ 2 + 8 * b ^ 3 * c ^ 3) ≤ a ^ 3 * (a ^ 4 + b ^ 4 + c ^ 4)"
},
{
"line": "try (with_reducible rfl)",
"before_state": "a b c : ℝ\nha : 0 < a\nhb : 0 < b\nhc : 0 < c\n⊢ a ^ 4 * √((a ^ 3) ^ 2 + 8 * b ^ 3 * c ^ 3) ≤ a ^ 3 * (a ^ 4 + b ^ 4 + c ^ 4)",
"after_state": "a b c : ℝ\nha : 0 < a\nhb : 0 < b\nhc : 0 < c\n⊢ a ^ 4 * √((a ^ 3) ^ 2 + 8 * b ^ 3 * c ^ 3) ≤ a ^ 3 * (a ^ 4 + b ^ 4 + c ^ 4)"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "a b c : ℝ\nha : 0 < a\nhb : 0 < b\nhc : 0 < c\n⊢ a ^ 4 * √((a ^ 3) ^ 2 + 8 * b ^ 3 * c ^ 3) ≤ a ^ 3 * (a ^ 4 + b ^ 4 + c ^ 4)",
"after_state": "a b c : ℝ\nha : 0 < a\nhb : 0 < b\nhc : 0 < c\n⊢ a ^ 4 * √((a ^ 3) ^ 2 + 8 * b ^ 3 * c ^ 3) ≤ a ^ 3 * (a ^ 4 + b ^ 4 + c ^ 4)"
},
{
"line": "with_reducible rfl",
"before_state": "a b c : ℝ\nha : 0 < a\nhb : 0 < b\nhc : 0 < c\n⊢ a ^ 4 * √((a ^ 3) ^ 2 + 8 * b ^ 3 * c ^ 3) ≤ a ^ 3 * (a ^ 4 + b ^ 4 + c ^ 4)",
"after_state": "a b c : ℝ\nha : 0 < a\nhb : 0 < b\nhc : 0 < c\n⊢ a ^ 4 * √((a ^ 3) ^ 2 + 8 * b ^ 3 * c ^ 3) ≤ a ^ 3 * (a ^ 4 + b ^ 4 + c ^ 4)"
},
{
"line": "rfl",
"before_state": "a b c : ℝ\nha : 0 < a\nhb : 0 < b\nhc : 0 < c\n⊢ a ^ 4 * √((a ^ 3) ^ 2 + 8 * b ^ 3 * c ^ 3) ≤ a ^ 3 * (a ^ 4 + b ^ 4 + c ^ 4)",
"after_state": "a b c : ℝ\nha : 0 < a\nhb : 0 < b\nhc : 0 < c\n⊢ a ^ 4 * √((a ^ 3) ^ 2 + 8 * b ^ 3 * c ^ 3) ≤ a ^ 3 * (a ^ 4 + b ^ 4 + c ^ 4)"
},
{
"line": "apply_rfl",
"before_state": "a b c : ℝ\nha : 0 < a\nhb : 0 < b\nhc : 0 < c\n⊢ a ^ 4 * √((a ^ 3) ^ 2 + 8 * b ^ 3 * c ^ 3) ≤ a ^ 3 * (a ^ 4 + b ^ 4 + c ^ 4)",
"after_state": "a b c : ℝ\nha : 0 < a\nhb : 0 < b\nhc : 0 < c\n⊢ a ^ 4 * √((a ^ 3) ^ 2 + 8 * b ^ 3 * c ^ 3) ≤ a ^ 3 * (a ^ 4 + b ^ 4 + c ^ 4)"
},
{
"line": "skip",
"before_state": "a b c : ℝ\nha : 0 < a\nhb : 0 < b\nhc : 0 < c\n⊢ a ^ 4 * √((a ^ 3) ^ 2 + 8 * b ^ 3 * c ^ 3) ≤ a ^ 3 * (a ^ 4 + b ^ 4 + c ^ 4)",
"after_state": "a b c : ℝ\nha : 0 < a\nhb : 0 < b\nhc : 0 < c\n⊢ a ^ 4 * √((a ^ 3) ^ 2 + 8 * b ^ 3 * c ^ 3) ≤ a ^ 3 * (a ^ 4 + b ^ 4 + c ^ 4)"
},
{
"line": "calc\n a ^ 4 * sqrt ((a ^ 3) ^ 2 + (8 : ℝ) * b ^ 3 * c ^ 3) = a ^ 3 * (a * sqrt ((a ^ 3) ^ 2 + (8 : ℝ) * b ^ 3 * c ^ 3)) :=\n by ring\n _ ≤ a ^ 3 * (a ^ 4 + b ^ 4 + c ^ 4) := ?_",
"before_state": "a b c : ℝ\nha : 0 < a\nhb : 0 < b\nhc : 0 < c\n⊢ a ^ 4 * √((a ^ 3) ^ 2 + 8 * b ^ 3 * c ^ 3) ≤ a ^ 3 * (a ^ 4 + b ^ 4 + c ^ 4)",
"after_state": "a b c : ℝ\nha : 0 < a\nhb : 0 < b\nhc : 0 < c\n⊢ a ^ 3 * (a * √((a ^ 3) ^ 2 + 8 * b ^ 3 * c ^ 3)) ≤ a ^ 3 * (a ^ 4 + b ^ 4 + c ^ 4)"
},
{
"line": "ring",
"before_state": "a b c : ℝ\nha : 0 < a\nhb : 0 < b\nhc : 0 < c\n⊢ a ^ 4 * √((a ^ 3) ^ 2 + 8 * b ^ 3 * c ^ 3) = a ^ 3 * (a * √((a ^ 3) ^ 2 + 8 * b ^ 3 * c ^ 3))",
"after_state": "No Goals!"
},
{
"line": "first\n| ring1\n|\n try_this ring_nf\"\\n\\nThe `ring` tactic failed to close the goal. Use `ring_nf` to obtain a normal form.\n \\nNote that `ring` works primarily in *commutative* rings. \\\n If you have a noncommutative ring, abelian group or module, consider using \\\n `noncomm_ring`, `abel` or `module` instead.\"",
"before_state": "a b c : ℝ\nha : 0 < a\nhb : 0 < b\nhc : 0 < c\n⊢ a ^ 4 * √((a ^ 3) ^ 2 + 8 * b ^ 3 * c ^ 3) = a ^ 3 * (a * √((a ^ 3) ^ 2 + 8 * b ^ 3 * c ^ 3))",
"after_state": "No Goals!"
},
{
"line": "ring1",
"before_state": "a b c : ℝ\nha : 0 < a\nhb : 0 < b\nhc : 0 < c\n⊢ a ^ 4 * √((a ^ 3) ^ 2 + 8 * b ^ 3 * c ^ 3) = a ^ 3 * (a * √((a ^ 3) ^ 2 + 8 * b ^ 3 * c ^ 3))",
"after_state": "No Goals!"
},
{
"line": "gcongr",
"before_state": "a b c : ℝ\nha : 0 < a\nhb : 0 < b\nhc : 0 < c\n⊢ a ^ 3 * (a * √((a ^ 3) ^ 2 + 8 * b ^ 3 * c ^ 3)) ≤ a ^ 3 * (a ^ 4 + b ^ 4 + c ^ 4)",
"after_state": "case h\na b c : ℝ\nha : 0 < a\nhb : 0 < b\nhc : 0 < c\n⊢ a * √((a ^ 3) ^ 2 + 8 * b ^ 3 * c ^ 3) ≤ a ^ 4 + b ^ 4 + c ^ 4"
},
{
"line": "gcongr_discharger",
"before_state": "case a0\na b c : ℝ\nha : 0 < a\nhb : 0 < b\nhc : 0 < c\n⊢ 0 ≤ a ^ 3",
"after_state": "No Goals!"
},
{
"line": "positivity",
"before_state": "case a0\na b c : ℝ\nha : 0 < a\nhb : 0 < b\nhc : 0 < c\n⊢ 0 ≤ a ^ 3",
"after_state": "No Goals!"
},
{
"line": "apply le_of_pow_le_pow_left₀ two_ne_zero (by positivity)",
"before_state": "case h\na b c : ℝ\nha : 0 < a\nhb : 0 < b\nhc : 0 < c\n⊢ a * √((a ^ 3) ^ 2 + 8 * b ^ 3 * c ^ 3) ≤ a ^ 4 + b ^ 4 + c ^ 4",
"after_state": "case h\na b c : ℝ\nha : 0 < a\nhb : 0 < b\nhc : 0 < c\n⊢ (a * √((a ^ 3) ^ 2 + 8 * b ^ 3 * c ^ 3)) ^ 2 ≤ (a ^ 4 + b ^ 4 + c ^ 4) ^ 2"
},
{
"line": "positivity",
"before_state": "a b c : ℝ\nha : 0 < a\nhb : 0 < b\nhc : 0 < c\n⊢ 0 ≤ a ^ 4 + b ^ 4 + c ^ 4",
"after_state": "No Goals!"
},
{
"line": "rw [mul_pow]",
"before_state": "case h\na b c : ℝ\nha : 0 < a\nhb : 0 < b\nhc : 0 < c\n⊢ (a * √((a ^ 3) ^ 2 + 8 * b ^ 3 * c ^ 3)) ^ 2 ≤ (a ^ 4 + b ^ 4 + c ^ 4) ^ 2",
"after_state": "case h\na b c : ℝ\nha : 0 < a\nhb : 0 < b\nhc : 0 < c\n⊢ a ^ 2 * √((a ^ 3) ^ 2 + 8 * b ^ 3 * c ^ 3) ^ 2 ≤ (a ^ 4 + b ^ 4 + c ^ 4) ^ 2"
},
{
"line": "rewrite [mul_pow]",
"before_state": "case h\na b c : ℝ\nha : 0 < a\nhb : 0 < b\nhc : 0 < c\n⊢ (a * √((a ^ 3) ^ 2 + 8 * b ^ 3 * c ^ 3)) ^ 2 ≤ (a ^ 4 + b ^ 4 + c ^ 4) ^ 2",
"after_state": "case h\na b c : ℝ\nha : 0 < a\nhb : 0 < b\nhc : 0 < c\n⊢ a ^ 2 * √((a ^ 3) ^ 2 + 8 * b ^ 3 * c ^ 3) ^ 2 ≤ (a ^ 4 + b ^ 4 + c ^ 4) ^ 2"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "case h\na b c : ℝ\nha : 0 < a\nhb : 0 < b\nhc : 0 < c\n⊢ a ^ 2 * √((a ^ 3) ^ 2 + 8 * b ^ 3 * c ^ 3) ^ 2 ≤ (a ^ 4 + b ^ 4 + c ^ 4) ^ 2",
"after_state": "case h\na b c : ℝ\nha : 0 < a\nhb : 0 < b\nhc : 0 < c\n⊢ a ^ 2 * √((a ^ 3) ^ 2 + 8 * b ^ 3 * c ^ 3) ^ 2 ≤ (a ^ 4 + b ^ 4 + c ^ 4) ^ 2"
},
{
"line": "try (with_reducible rfl)",
"before_state": "case h\na b c : ℝ\nha : 0 < a\nhb : 0 < b\nhc : 0 < c\n⊢ a ^ 2 * √((a ^ 3) ^ 2 + 8 * b ^ 3 * c ^ 3) ^ 2 ≤ (a ^ 4 + b ^ 4 + c ^ 4) ^ 2",
"after_state": "case h\na b c : ℝ\nha : 0 < a\nhb : 0 < b\nhc : 0 < c\n⊢ a ^ 2 * √((a ^ 3) ^ 2 + 8 * b ^ 3 * c ^ 3) ^ 2 ≤ (a ^ 4 + b ^ 4 + c ^ 4) ^ 2"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "case h\na b c : ℝ\nha : 0 < a\nhb : 0 < b\nhc : 0 < c\n⊢ a ^ 2 * √((a ^ 3) ^ 2 + 8 * b ^ 3 * c ^ 3) ^ 2 ≤ (a ^ 4 + b ^ 4 + c ^ 4) ^ 2",
"after_state": "case h\na b c : ℝ\nha : 0 < a\nhb : 0 < b\nhc : 0 < c\n⊢ a ^ 2 * √((a ^ 3) ^ 2 + 8 * b ^ 3 * c ^ 3) ^ 2 ≤ (a ^ 4 + b ^ 4 + c ^ 4) ^ 2"
},
{
"line": "with_reducible rfl",
"before_state": "case h\na b c : ℝ\nha : 0 < a\nhb : 0 < b\nhc : 0 < c\n⊢ a ^ 2 * √((a ^ 3) ^ 2 + 8 * b ^ 3 * c ^ 3) ^ 2 ≤ (a ^ 4 + b ^ 4 + c ^ 4) ^ 2",
"after_state": "case h\na b c : ℝ\nha : 0 < a\nhb : 0 < b\nhc : 0 < c\n⊢ a ^ 2 * √((a ^ 3) ^ 2 + 8 * b ^ 3 * c ^ 3) ^ 2 ≤ (a ^ 4 + b ^ 4 + c ^ 4) ^ 2"
},
{
"line": "rfl",
"before_state": "case h\na b c : ℝ\nha : 0 < a\nhb : 0 < b\nhc : 0 < c\n⊢ a ^ 2 * √((a ^ 3) ^ 2 + 8 * b ^ 3 * c ^ 3) ^ 2 ≤ (a ^ 4 + b ^ 4 + c ^ 4) ^ 2",
"after_state": "case h\na b c : ℝ\nha : 0 < a\nhb : 0 < b\nhc : 0 < c\n⊢ a ^ 2 * √((a ^ 3) ^ 2 + 8 * b ^ 3 * c ^ 3) ^ 2 ≤ (a ^ 4 + b ^ 4 + c ^ 4) ^ 2"
},
{
"line": "apply_rfl",
"before_state": "case h\na b c : ℝ\nha : 0 < a\nhb : 0 < b\nhc : 0 < c\n⊢ a ^ 2 * √((a ^ 3) ^ 2 + 8 * b ^ 3 * c ^ 3) ^ 2 ≤ (a ^ 4 + b ^ 4 + c ^ 4) ^ 2",
"after_state": "case h\na b c : ℝ\nha : 0 < a\nhb : 0 < b\nhc : 0 < c\n⊢ a ^ 2 * √((a ^ 3) ^ 2 + 8 * b ^ 3 * c ^ 3) ^ 2 ≤ (a ^ 4 + b ^ 4 + c ^ 4) ^ 2"
},
{
"line": "skip",
"before_state": "case h\na b c : ℝ\nha : 0 < a\nhb : 0 < b\nhc : 0 < c\n⊢ a ^ 2 * √((a ^ 3) ^ 2 + 8 * b ^ 3 * c ^ 3) ^ 2 ≤ (a ^ 4 + b ^ 4 + c ^ 4) ^ 2",
"after_state": "case h\na b c : ℝ\nha : 0 < a\nhb : 0 < b\nhc : 0 < c\n⊢ a ^ 2 * √((a ^ 3) ^ 2 + 8 * b ^ 3 * c ^ 3) ^ 2 ≤ (a ^ 4 + b ^ 4 + c ^ 4) ^ 2"
},
{
"line": "rw [sq_sqrt (by positivity)]",
"before_state": "case h\na b c : ℝ\nha : 0 < a\nhb : 0 < b\nhc : 0 < c\n⊢ a ^ 2 * √((a ^ 3) ^ 2 + 8 * b ^ 3 * c ^ 3) ^ 2 ≤ (a ^ 4 + b ^ 4 + c ^ 4) ^ 2",
"after_state": "case h\na b c : ℝ\nha : 0 < a\nhb : 0 < b\nhc : 0 < c\n⊢ a ^ 2 * ((a ^ 3) ^ 2 + 8 * b ^ 3 * c ^ 3) ≤ (a ^ 4 + b ^ 4 + c ^ 4) ^ 2"
},
{
"line": "rewrite [sq_sqrt (by positivity)]",
"before_state": "case h\na b c : ℝ\nha : 0 < a\nhb : 0 < b\nhc : 0 < c\n⊢ a ^ 2 * √((a ^ 3) ^ 2 + 8 * b ^ 3 * c ^ 3) ^ 2 ≤ (a ^ 4 + b ^ 4 + c ^ 4) ^ 2",
"after_state": "case h\na b c : ℝ\nha : 0 < a\nhb : 0 < b\nhc : 0 < c\n⊢ a ^ 2 * ((a ^ 3) ^ 2 + 8 * b ^ 3 * c ^ 3) ≤ (a ^ 4 + b ^ 4 + c ^ 4) ^ 2"
},
{
"line": "positivity",
"before_state": "a b c : ℝ\nha : 0 < a\nhb : 0 < b\nhc : 0 < c\n⊢ 0 ≤ (a ^ 3) ^ 2 + 8 * b ^ 3 * c ^ 3",
"after_state": "No Goals!"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "case h\na b c : ℝ\nha : 0 < a\nhb : 0 < b\nhc : 0 < c\n⊢ a ^ 2 * ((a ^ 3) ^ 2 + 8 * b ^ 3 * c ^ 3) ≤ (a ^ 4 + b ^ 4 + c ^ 4) ^ 2",
"after_state": "case h\na b c : ℝ\nha : 0 < a\nhb : 0 < b\nhc : 0 < c\n⊢ a ^ 2 * ((a ^ 3) ^ 2 + 8 * b ^ 3 * c ^ 3) ≤ (a ^ 4 + b ^ 4 + c ^ 4) ^ 2"
},
{
"line": "try (with_reducible rfl)",
"before_state": "case h\na b c : ℝ\nha : 0 < a\nhb : 0 < b\nhc : 0 < c\n⊢ a ^ 2 * ((a ^ 3) ^ 2 + 8 * b ^ 3 * c ^ 3) ≤ (a ^ 4 + b ^ 4 + c ^ 4) ^ 2",
"after_state": "case h\na b c : ℝ\nha : 0 < a\nhb : 0 < b\nhc : 0 < c\n⊢ a ^ 2 * ((a ^ 3) ^ 2 + 8 * b ^ 3 * c ^ 3) ≤ (a ^ 4 + b ^ 4 + c ^ 4) ^ 2"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "case h\na b c : ℝ\nha : 0 < a\nhb : 0 < b\nhc : 0 < c\n⊢ a ^ 2 * ((a ^ 3) ^ 2 + 8 * b ^ 3 * c ^ 3) ≤ (a ^ 4 + b ^ 4 + c ^ 4) ^ 2",
"after_state": "case h\na b c : ℝ\nha : 0 < a\nhb : 0 < b\nhc : 0 < c\n⊢ a ^ 2 * ((a ^ 3) ^ 2 + 8 * b ^ 3 * c ^ 3) ≤ (a ^ 4 + b ^ 4 + c ^ 4) ^ 2"
},
{
"line": "with_reducible rfl",
"before_state": "case h\na b c : ℝ\nha : 0 < a\nhb : 0 < b\nhc : 0 < c\n⊢ a ^ 2 * ((a ^ 3) ^ 2 + 8 * b ^ 3 * c ^ 3) ≤ (a ^ 4 + b ^ 4 + c ^ 4) ^ 2",
"after_state": "case h\na b c : ℝ\nha : 0 < a\nhb : 0 < b\nhc : 0 < c\n⊢ a ^ 2 * ((a ^ 3) ^ 2 + 8 * b ^ 3 * c ^ 3) ≤ (a ^ 4 + b ^ 4 + c ^ 4) ^ 2"
},
{
"line": "rfl",
"before_state": "case h\na b c : ℝ\nha : 0 < a\nhb : 0 < b\nhc : 0 < c\n⊢ a ^ 2 * ((a ^ 3) ^ 2 + 8 * b ^ 3 * c ^ 3) ≤ (a ^ 4 + b ^ 4 + c ^ 4) ^ 2",
"after_state": "case h\na b c : ℝ\nha : 0 < a\nhb : 0 < b\nhc : 0 < c\n⊢ a ^ 2 * ((a ^ 3) ^ 2 + 8 * b ^ 3 * c ^ 3) ≤ (a ^ 4 + b ^ 4 + c ^ 4) ^ 2"
},
{
"line": "apply_rfl",
"before_state": "case h\na b c : ℝ\nha : 0 < a\nhb : 0 < b\nhc : 0 < c\n⊢ a ^ 2 * ((a ^ 3) ^ 2 + 8 * b ^ 3 * c ^ 3) ≤ (a ^ 4 + b ^ 4 + c ^ 4) ^ 2",
"after_state": "case h\na b c : ℝ\nha : 0 < a\nhb : 0 < b\nhc : 0 < c\n⊢ a ^ 2 * ((a ^ 3) ^ 2 + 8 * b ^ 3 * c ^ 3) ≤ (a ^ 4 + b ^ 4 + c ^ 4) ^ 2"
},
{
"line": "skip",
"before_state": "case h\na b c : ℝ\nha : 0 < a\nhb : 0 < b\nhc : 0 < c\n⊢ a ^ 2 * ((a ^ 3) ^ 2 + 8 * b ^ 3 * c ^ 3) ≤ (a ^ 4 + b ^ 4 + c ^ 4) ^ 2",
"after_state": "case h\na b c : ℝ\nha : 0 < a\nhb : 0 < b\nhc : 0 < c\n⊢ a ^ 2 * ((a ^ 3) ^ 2 + 8 * b ^ 3 * c ^ 3) ≤ (a ^ 4 + b ^ 4 + c ^ 4) ^ 2"
},
{
"line": "rw [← sub_nonneg]",
"before_state": "case h\na b c : ℝ\nha : 0 < a\nhb : 0 < b\nhc : 0 < c\n⊢ a ^ 2 * ((a ^ 3) ^ 2 + 8 * b ^ 3 * c ^ 3) ≤ (a ^ 4 + b ^ 4 + c ^ 4) ^ 2",
"after_state": "case h\na b c : ℝ\nha : 0 < a\nhb : 0 < b\nhc : 0 < c\n⊢ 0 ≤ (a ^ 4 + b ^ 4 + c ^ 4) ^ 2 - a ^ 2 * ((a ^ 3) ^ 2 + 8 * b ^ 3 * c ^ 3)"
},
{
"line": "rewrite [← sub_nonneg]",
"before_state": "case h\na b c : ℝ\nha : 0 < a\nhb : 0 < b\nhc : 0 < c\n⊢ a ^ 2 * ((a ^ 3) ^ 2 + 8 * b ^ 3 * c ^ 3) ≤ (a ^ 4 + b ^ 4 + c ^ 4) ^ 2",
"after_state": "case h\na b c : ℝ\nha : 0 < a\nhb : 0 < b\nhc : 0 < c\n⊢ 0 ≤ (a ^ 4 + b ^ 4 + c ^ 4) ^ 2 - a ^ 2 * ((a ^ 3) ^ 2 + 8 * b ^ 3 * c ^ 3)"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "case h\na b c : ℝ\nha : 0 < a\nhb : 0 < b\nhc : 0 < c\n⊢ 0 ≤ (a ^ 4 + b ^ 4 + c ^ 4) ^ 2 - a ^ 2 * ((a ^ 3) ^ 2 + 8 * b ^ 3 * c ^ 3)",
"after_state": "case h\na b c : ℝ\nha : 0 < a\nhb : 0 < b\nhc : 0 < c\n⊢ 0 ≤ (a ^ 4 + b ^ 4 + c ^ 4) ^ 2 - a ^ 2 * ((a ^ 3) ^ 2 + 8 * b ^ 3 * c ^ 3)"
},
{
"line": "try (with_reducible rfl)",
"before_state": "case h\na b c : ℝ\nha : 0 < a\nhb : 0 < b\nhc : 0 < c\n⊢ 0 ≤ (a ^ 4 + b ^ 4 + c ^ 4) ^ 2 - a ^ 2 * ((a ^ 3) ^ 2 + 8 * b ^ 3 * c ^ 3)",
"after_state": "case h\na b c : ℝ\nha : 0 < a\nhb : 0 < b\nhc : 0 < c\n⊢ 0 ≤ (a ^ 4 + b ^ 4 + c ^ 4) ^ 2 - a ^ 2 * ((a ^ 3) ^ 2 + 8 * b ^ 3 * c ^ 3)"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "case h\na b c : ℝ\nha : 0 < a\nhb : 0 < b\nhc : 0 < c\n⊢ 0 ≤ (a ^ 4 + b ^ 4 + c ^ 4) ^ 2 - a ^ 2 * ((a ^ 3) ^ 2 + 8 * b ^ 3 * c ^ 3)",
"after_state": "case h\na b c : ℝ\nha : 0 < a\nhb : 0 < b\nhc : 0 < c\n⊢ 0 ≤ (a ^ 4 + b ^ 4 + c ^ 4) ^ 2 - a ^ 2 * ((a ^ 3) ^ 2 + 8 * b ^ 3 * c ^ 3)"
},
{
"line": "with_reducible rfl",
"before_state": "case h\na b c : ℝ\nha : 0 < a\nhb : 0 < b\nhc : 0 < c\n⊢ 0 ≤ (a ^ 4 + b ^ 4 + c ^ 4) ^ 2 - a ^ 2 * ((a ^ 3) ^ 2 + 8 * b ^ 3 * c ^ 3)",
"after_state": "case h\na b c : ℝ\nha : 0 < a\nhb : 0 < b\nhc : 0 < c\n⊢ 0 ≤ (a ^ 4 + b ^ 4 + c ^ 4) ^ 2 - a ^ 2 * ((a ^ 3) ^ 2 + 8 * b ^ 3 * c ^ 3)"
},
{
"line": "rfl",
"before_state": "case h\na b c : ℝ\nha : 0 < a\nhb : 0 < b\nhc : 0 < c\n⊢ 0 ≤ (a ^ 4 + b ^ 4 + c ^ 4) ^ 2 - a ^ 2 * ((a ^ 3) ^ 2 + 8 * b ^ 3 * c ^ 3)",
"after_state": "case h\na b c : ℝ\nha : 0 < a\nhb : 0 < b\nhc : 0 < c\n⊢ 0 ≤ (a ^ 4 + b ^ 4 + c ^ 4) ^ 2 - a ^ 2 * ((a ^ 3) ^ 2 + 8 * b ^ 3 * c ^ 3)"
},
{
"line": "apply_rfl",
"before_state": "case h\na b c : ℝ\nha : 0 < a\nhb : 0 < b\nhc : 0 < c\n⊢ 0 ≤ (a ^ 4 + b ^ 4 + c ^ 4) ^ 2 - a ^ 2 * ((a ^ 3) ^ 2 + 8 * b ^ 3 * c ^ 3)",
"after_state": "case h\na b c : ℝ\nha : 0 < a\nhb : 0 < b\nhc : 0 < c\n⊢ 0 ≤ (a ^ 4 + b ^ 4 + c ^ 4) ^ 2 - a ^ 2 * ((a ^ 3) ^ 2 + 8 * b ^ 3 * c ^ 3)"
},
{
"line": "skip",
"before_state": "case h\na b c : ℝ\nha : 0 < a\nhb : 0 < b\nhc : 0 < c\n⊢ 0 ≤ (a ^ 4 + b ^ 4 + c ^ 4) ^ 2 - a ^ 2 * ((a ^ 3) ^ 2 + 8 * b ^ 3 * c ^ 3)",
"after_state": "case h\na b c : ℝ\nha : 0 < a\nhb : 0 < b\nhc : 0 < c\n⊢ 0 ≤ (a ^ 4 + b ^ 4 + c ^ 4) ^ 2 - a ^ 2 * ((a ^ 3) ^ 2 + 8 * b ^ 3 * c ^ 3)"
},
{
"line": "calc\n (a ^ 4 + b ^ 4 + c ^ 4) ^ 2 - a ^ 2 * ((a ^ 3) ^ 2 + 8 * b ^ 3 * c ^ 3) =\n 2 * (a ^ 2 * (b ^ 2 - c ^ 2)) ^ 2 + (b ^ 4 - c ^ 4) ^ 2 + (2 * (a ^ 2 * b * c - b ^ 2 * c ^ 2)) ^ 2 :=\n by ring\n _ ≥ 0 := by positivity",
"before_state": "case h\na b c : ℝ\nha : 0 < a\nhb : 0 < b\nhc : 0 < c\n⊢ 0 ≤ (a ^ 4 + b ^ 4 + c ^ 4) ^ 2 - a ^ 2 * ((a ^ 3) ^ 2 + 8 * b ^ 3 * c ^ 3)",
"after_state": "No Goals!"
},
{
"line": "ring",
"before_state": "a b c : ℝ\nha : 0 < a\nhb : 0 < b\nhc : 0 < c\n⊢ (a ^ 4 + b ^ 4 + c ^ 4) ^ 2 - a ^ 2 * ((a ^ 3) ^ 2 + 8 * b ^ 3 * c ^ 3) =\n 2 * (a ^ 2 * (b ^ 2 - c ^ 2)) ^ 2 + (b ^ 4 - c ^ 4) ^ 2 + (2 * (a ^ 2 * b * c - b ^ 2 * c ^ 2)) ^ 2",
"after_state": "No Goals!"
},
{
"line": "first\n| ring1\n|\n try_this ring_nf\"\\n\\nThe `ring` tactic failed to close the goal. Use `ring_nf` to obtain a normal form.\n \\nNote that `ring` works primarily in *commutative* rings. \\\n If you have a noncommutative ring, abelian group or module, consider using \\\n `noncomm_ring`, `abel` or `module` instead.\"",
"before_state": "a b c : ℝ\nha : 0 < a\nhb : 0 < b\nhc : 0 < c\n⊢ (a ^ 4 + b ^ 4 + c ^ 4) ^ 2 - a ^ 2 * ((a ^ 3) ^ 2 + 8 * b ^ 3 * c ^ 3) =\n 2 * (a ^ 2 * (b ^ 2 - c ^ 2)) ^ 2 + (b ^ 4 - c ^ 4) ^ 2 + (2 * (a ^ 2 * b * c - b ^ 2 * c ^ 2)) ^ 2",
"after_state": "No Goals!"
},
{
"line": "ring1",
"before_state": "a b c : ℝ\nha : 0 < a\nhb : 0 < b\nhc : 0 < c\n⊢ (a ^ 4 + b ^ 4 + c ^ 4) ^ 2 - a ^ 2 * ((a ^ 3) ^ 2 + 8 * b ^ 3 * c ^ 3) =\n 2 * (a ^ 2 * (b ^ 2 - c ^ 2)) ^ 2 + (b ^ 4 - c ^ 4) ^ 2 + (2 * (a ^ 2 * b * c - b ^ 2 * c ^ 2)) ^ 2",
"after_state": "No Goals!"
},
{
"line": "positivity",
"before_state": "a b c : ℝ\nha : 0 < a\nhb : 0 < b\nhc : 0 < c\n⊢ 2 * (a ^ 2 * (b ^ 2 - c ^ 2)) ^ 2 + (b ^ 4 - c ^ 4) ^ 2 + (2 * (a ^ 2 * b * c - b ^ 2 * c ^ 2)) ^ 2 ≥ 0",
"after_state": "No Goals!"
}
] |
theorem imo2001_q6 (hd : 0 < d) (hdc : d < c) (hcb : c < b) (hba : b < a)
(h : a * c + b * d = (a + b - c + d) * (-a + b + c + d)) : ¬Prime (a * b + c * d) := by
intro (h0 : Prime (a * b + c * d))
have ha : 0 < a := by omega
have hb : 0 < b := by omega
have hc : 0 < c := by omega
-- the key step is to show that `a*c + b*d` divides the product `(a*b + c*d) * (a*d + b*c)`
have dvd_mul : a * c + b * d ∣ (a * b + c * d) * (a * d + b * c) := by
use b ^ 2 + b * d + d ^ 2
linear_combination b * d * h
-- since `a*b + c*d` is prime (by assumption), it must divide `a*c + b*d` or `a*d + b*c`
obtain (h1 : a * b + c * d ∣ a * c + b * d) | (h2 : a * c + b * d ∣ a * d + b * c) :=
h0.left_dvd_or_dvd_right_of_dvd_mul dvd_mul
-- in both cases, we derive a contradiction
· have aux : 0 < a * c + b * d := by nlinarith only [ha, hb, hc, hd]
have : a * b + c * d ≤ a * c + b * d := Int.le_of_dvd aux h1
nlinarith only [hba, hcb, hdc, h, this]
· have aux : 0 < a * d + b * c := by nlinarith only [ha, hb, hc, hd]
have : a * c + b * d ≤ a * d + b * c := Int.le_of_dvd aux h2
nlinarith only [hba, hdc, h, this]
|
/root/DuelModelResearch/mathlib4/Archive/Imo/Imo2001Q6.lean
|
{
"open": [],
"variables": [
"{a b c d : ℤ}"
]
}
|
[
{
"line": "intro (h0 : Prime (a * b + c * d))",
"before_state": "a b c d : ℤ\nhd : 0 < d\nhdc : d < c\nhcb : c < b\nhba : b < a\nh : a * c + b * d = (a + b - c + d) * (-a + b + c + d)\n⊢ ¬Prime (a * b + c * d)",
"after_state": "a b c d : ℤ\nhd : 0 < d\nhdc : d < c\nhcb : c < b\nhba : b < a\nh : a * c + b * d = (a + b - c + d) * (-a + b + c + d)\nh0 : Prime (a * b + c * d)\n⊢ False"
},
{
"line": "have ha : 0 < a := by omega",
"before_state": "a b c d : ℤ\nhd : 0 < d\nhdc : d < c\nhcb : c < b\nhba : b < a\nh : a * c + b * d = (a + b - c + d) * (-a + b + c + d)\nh0 : Prime (a * b + c * d)\n⊢ False",
"after_state": "a b c d : ℤ\nhd : 0 < d\nhdc : d < c\nhcb : c < b\nhba : b < a\nh : a * c + b * d = (a + b - c + d) * (-a + b + c + d)\nh0 : Prime (a * b + c * d)\nha : 0 < a\n⊢ False"
},
{
"line": "focus\n refine\n no_implicit_lambda%\n (have ha : 0 < a := ?body✝;\n ?_)\n case body✝ => with_annotate_state\"by\" (omega)",
"before_state": "a b c d : ℤ\nhd : 0 < d\nhdc : d < c\nhcb : c < b\nhba : b < a\nh : a * c + b * d = (a + b - c + d) * (-a + b + c + d)\nh0 : Prime (a * b + c * d)\n⊢ False",
"after_state": "a b c d : ℤ\nhd : 0 < d\nhdc : d < c\nhcb : c < b\nhba : b < a\nh : a * c + b * d = (a + b - c + d) * (-a + b + c + d)\nh0 : Prime (a * b + c * d)\nha : 0 < a\n⊢ False"
},
{
"line": "refine\n no_implicit_lambda%\n (have ha : 0 < a := ?body✝;\n ?_)",
"before_state": "a b c d : ℤ\nhd : 0 < d\nhdc : d < c\nhcb : c < b\nhba : b < a\nh : a * c + b * d = (a + b - c + d) * (-a + b + c + d)\nh0 : Prime (a * b + c * d)\n⊢ False",
"after_state": "case body\na b c d : ℤ\nhd : 0 < d\nhdc : d < c\nhcb : c < b\nhba : b < a\nh : a * c + b * d = (a + b - c + d) * (-a + b + c + d)\nh0 : Prime (a * b + c * d)\n⊢ 0 < a\n---\na b c d : ℤ\nhd : 0 < d\nhdc : d < c\nhcb : c < b\nhba : b < a\nh : a * c + b * d = (a + b - c + d) * (-a + b + c + d)\nh0 : Prime (a * b + c * d)\nha : 0 < a\n⊢ False"
},
{
"line": "case body✝ => with_annotate_state\"by\" (omega)",
"before_state": "case body\na b c d : ℤ\nhd : 0 < d\nhdc : d < c\nhcb : c < b\nhba : b < a\nh : a * c + b * d = (a + b - c + d) * (-a + b + c + d)\nh0 : Prime (a * b + c * d)\n⊢ 0 < a\n---\na b c d : ℤ\nhd : 0 < d\nhdc : d < c\nhcb : c < b\nhba : b < a\nh : a * c + b * d = (a + b - c + d) * (-a + b + c + d)\nh0 : Prime (a * b + c * d)\nha : 0 < a\n⊢ False",
"after_state": "a b c d : ℤ\nhd : 0 < d\nhdc : d < c\nhcb : c < b\nhba : b < a\nh : a * c + b * d = (a + b - c + d) * (-a + b + c + d)\nh0 : Prime (a * b + c * d)\nha : 0 < a\n⊢ False"
},
{
"line": "with_annotate_state\"by\" (omega)",
"before_state": "a b c d : ℤ\nhd : 0 < d\nhdc : d < c\nhcb : c < b\nhba : b < a\nh : a * c + b * d = (a + b - c + d) * (-a + b + c + d)\nh0 : Prime (a * b + c * d)\n⊢ 0 < a",
"after_state": "No Goals!"
},
{
"line": "omega",
"before_state": "a b c d : ℤ\nhd : 0 < d\nhdc : d < c\nhcb : c < b\nhba : b < a\nh : a * c + b * d = (a + b - c + d) * (-a + b + c + d)\nh0 : Prime (a * b + c * d)\n⊢ 0 < a",
"after_state": "No Goals!"
},
{
"line": "have hb : 0 < b := by omega",
"before_state": "a b c d : ℤ\nhd : 0 < d\nhdc : d < c\nhcb : c < b\nhba : b < a\nh : a * c + b * d = (a + b - c + d) * (-a + b + c + d)\nh0 : Prime (a * b + c * d)\nha : 0 < a\n⊢ False",
"after_state": "a b c d : ℤ\nhd : 0 < d\nhdc : d < c\nhcb : c < b\nhba : b < a\nh : a * c + b * d = (a + b - c + d) * (-a + b + c + d)\nh0 : Prime (a * b + c * d)\nha : 0 < a\nhb : 0 < b\n⊢ False"
},
{
"line": "focus\n refine\n no_implicit_lambda%\n (have hb : 0 < b := ?body✝;\n ?_)\n case body✝ => with_annotate_state\"by\" (omega)",
"before_state": "a b c d : ℤ\nhd : 0 < d\nhdc : d < c\nhcb : c < b\nhba : b < a\nh : a * c + b * d = (a + b - c + d) * (-a + b + c + d)\nh0 : Prime (a * b + c * d)\nha : 0 < a\n⊢ False",
"after_state": "a b c d : ℤ\nhd : 0 < d\nhdc : d < c\nhcb : c < b\nhba : b < a\nh : a * c + b * d = (a + b - c + d) * (-a + b + c + d)\nh0 : Prime (a * b + c * d)\nha : 0 < a\nhb : 0 < b\n⊢ False"
},
{
"line": "refine\n no_implicit_lambda%\n (have hb : 0 < b := ?body✝;\n ?_)",
"before_state": "a b c d : ℤ\nhd : 0 < d\nhdc : d < c\nhcb : c < b\nhba : b < a\nh : a * c + b * d = (a + b - c + d) * (-a + b + c + d)\nh0 : Prime (a * b + c * d)\nha : 0 < a\n⊢ False",
"after_state": "case body\na b c d : ℤ\nhd : 0 < d\nhdc : d < c\nhcb : c < b\nhba : b < a\nh : a * c + b * d = (a + b - c + d) * (-a + b + c + d)\nh0 : Prime (a * b + c * d)\nha : 0 < a\n⊢ 0 < b\n---\na b c d : ℤ\nhd : 0 < d\nhdc : d < c\nhcb : c < b\nhba : b < a\nh : a * c + b * d = (a + b - c + d) * (-a + b + c + d)\nh0 : Prime (a * b + c * d)\nha : 0 < a\nhb : 0 < b\n⊢ False"
},
{
"line": "case body✝ => with_annotate_state\"by\" (omega)",
"before_state": "case body\na b c d : ℤ\nhd : 0 < d\nhdc : d < c\nhcb : c < b\nhba : b < a\nh : a * c + b * d = (a + b - c + d) * (-a + b + c + d)\nh0 : Prime (a * b + c * d)\nha : 0 < a\n⊢ 0 < b\n---\na b c d : ℤ\nhd : 0 < d\nhdc : d < c\nhcb : c < b\nhba : b < a\nh : a * c + b * d = (a + b - c + d) * (-a + b + c + d)\nh0 : Prime (a * b + c * d)\nha : 0 < a\nhb : 0 < b\n⊢ False",
"after_state": "a b c d : ℤ\nhd : 0 < d\nhdc : d < c\nhcb : c < b\nhba : b < a\nh : a * c + b * d = (a + b - c + d) * (-a + b + c + d)\nh0 : Prime (a * b + c * d)\nha : 0 < a\nhb : 0 < b\n⊢ False"
},
{
"line": "with_annotate_state\"by\" (omega)",
"before_state": "a b c d : ℤ\nhd : 0 < d\nhdc : d < c\nhcb : c < b\nhba : b < a\nh : a * c + b * d = (a + b - c + d) * (-a + b + c + d)\nh0 : Prime (a * b + c * d)\nha : 0 < a\n⊢ 0 < b",
"after_state": "No Goals!"
},
{
"line": "omega",
"before_state": "a b c d : ℤ\nhd : 0 < d\nhdc : d < c\nhcb : c < b\nhba : b < a\nh : a * c + b * d = (a + b - c + d) * (-a + b + c + d)\nh0 : Prime (a * b + c * d)\nha : 0 < a\n⊢ 0 < b",
"after_state": "No Goals!"
},
{
"line": "have hc : 0 < c := by\n omega\n -- the key step is to show that `a*c + b*d` divides the product `(a*b + c*d) * (a*d + b*c)`",
"before_state": "a b c d : ℤ\nhd : 0 < d\nhdc : d < c\nhcb : c < b\nhba : b < a\nh : a * c + b * d = (a + b - c + d) * (-a + b + c + d)\nh0 : Prime (a * b + c * d)\nha : 0 < a\nhb : 0 < b\n⊢ False",
"after_state": "a b c d : ℤ\nhd : 0 < d\nhdc : d < c\nhcb : c < b\nhba : b < a\nh : a * c + b * d = (a + b - c + d) * (-a + b + c + d)\nh0 : Prime (a * b + c * d)\nha : 0 < a\nhb : 0 < b\nhc : 0 < c\n⊢ False"
},
{
"line": "focus\n refine\n no_implicit_lambda%\n (have hc : 0 < c := ?body✝;\n ?_)\n case body✝ =>\n with_annotate_state\"by\"\n (omega\n -- the key step is to show that `a*c + b*d` divides the product `(a*b + c*d) * (a*d + b*c)`)",
"before_state": "a b c d : ℤ\nhd : 0 < d\nhdc : d < c\nhcb : c < b\nhba : b < a\nh : a * c + b * d = (a + b - c + d) * (-a + b + c + d)\nh0 : Prime (a * b + c * d)\nha : 0 < a\nhb : 0 < b\n⊢ False",
"after_state": "a b c d : ℤ\nhd : 0 < d\nhdc : d < c\nhcb : c < b\nhba : b < a\nh : a * c + b * d = (a + b - c + d) * (-a + b + c + d)\nh0 : Prime (a * b + c * d)\nha : 0 < a\nhb : 0 < b\nhc : 0 < c\n⊢ False"
},
{
"line": "refine\n no_implicit_lambda%\n (have hc : 0 < c := ?body✝;\n ?_)",
"before_state": "a b c d : ℤ\nhd : 0 < d\nhdc : d < c\nhcb : c < b\nhba : b < a\nh : a * c + b * d = (a + b - c + d) * (-a + b + c + d)\nh0 : Prime (a * b + c * d)\nha : 0 < a\nhb : 0 < b\n⊢ False",
"after_state": "case body\na b c d : ℤ\nhd : 0 < d\nhdc : d < c\nhcb : c < b\nhba : b < a\nh : a * c + b * d = (a + b - c + d) * (-a + b + c + d)\nh0 : Prime (a * b + c * d)\nha : 0 < a\nhb : 0 < b\n⊢ 0 < c\n---\na b c d : ℤ\nhd : 0 < d\nhdc : d < c\nhcb : c < b\nhba : b < a\nh : a * c + b * d = (a + b - c + d) * (-a + b + c + d)\nh0 : Prime (a * b + c * d)\nha : 0 < a\nhb : 0 < b\nhc : 0 < c\n⊢ False"
},
{
"line": "case body✝ =>\n with_annotate_state\"by\"\n (omega\n -- the key step is to show that `a*c + b*d` divides the product `(a*b + c*d) * (a*d + b*c)`)",
"before_state": "case body\na b c d : ℤ\nhd : 0 < d\nhdc : d < c\nhcb : c < b\nhba : b < a\nh : a * c + b * d = (a + b - c + d) * (-a + b + c + d)\nh0 : Prime (a * b + c * d)\nha : 0 < a\nhb : 0 < b\n⊢ 0 < c\n---\na b c d : ℤ\nhd : 0 < d\nhdc : d < c\nhcb : c < b\nhba : b < a\nh : a * c + b * d = (a + b - c + d) * (-a + b + c + d)\nh0 : Prime (a * b + c * d)\nha : 0 < a\nhb : 0 < b\nhc : 0 < c\n⊢ False",
"after_state": "a b c d : ℤ\nhd : 0 < d\nhdc : d < c\nhcb : c < b\nhba : b < a\nh : a * c + b * d = (a + b - c + d) * (-a + b + c + d)\nh0 : Prime (a * b + c * d)\nha : 0 < a\nhb : 0 < b\nhc : 0 < c\n⊢ False"
},
{
"line": "with_annotate_state\"by\"\n (omega\n -- the key step is to show that `a*c + b*d` divides the product `(a*b + c*d) * (a*d + b*c)`)",
"before_state": "a b c d : ℤ\nhd : 0 < d\nhdc : d < c\nhcb : c < b\nhba : b < a\nh : a * c + b * d = (a + b - c + d) * (-a + b + c + d)\nh0 : Prime (a * b + c * d)\nha : 0 < a\nhb : 0 < b\n⊢ 0 < c",
"after_state": "No Goals!"
},
{
"line": "omega\n -- the key step is to show that `a*c + b*d` divides the product `(a*b + c*d) * (a*d + b*c)`",
"before_state": "a b c d : ℤ\nhd : 0 < d\nhdc : d < c\nhcb : c < b\nhba : b < a\nh : a * c + b * d = (a + b - c + d) * (-a + b + c + d)\nh0 : Prime (a * b + c * d)\nha : 0 < a\nhb : 0 < b\n⊢ 0 < c",
"after_state": "No Goals!"
},
{
"line": "have dvd_mul : a * c + b * d ∣ (a * b + c * d) * (a * d + b * c) :=\n by\n use b ^ 2 + b * d + d ^ 2\n linear_combination b * d * h",
"before_state": "a b c d : ℤ\nhd : 0 < d\nhdc : d < c\nhcb : c < b\nhba : b < a\nh : a * c + b * d = (a + b - c + d) * (-a + b + c + d)\nh0 : Prime (a * b + c * d)\nha : 0 < a\nhb : 0 < b\nhc : 0 < c\n⊢ False",
"after_state": "a b c d : ℤ\nhd : 0 < d\nhdc : d < c\nhcb : c < b\nhba : b < a\nh : a * c + b * d = (a + b - c + d) * (-a + b + c + d)\nh0 : Prime (a * b + c * d)\nha : 0 < a\nhb : 0 < b\nhc : 0 < c\ndvd_mul : a * c + b * d ∣ (a * b + c * d) * (a * d + b * c)\n⊢ False"
},
{
"line": "focus\n refine\n no_implicit_lambda%\n (have dvd_mul : a * c + b * d ∣ (a * b + c * d) * (a * d + b * c) := ?body✝;\n ?_)\n case body✝ =>\n with_annotate_state\"by\"\n ( use b ^ 2 + b * d + d ^ 2\n linear_combination b * d * h)",
"before_state": "a b c d : ℤ\nhd : 0 < d\nhdc : d < c\nhcb : c < b\nhba : b < a\nh : a * c + b * d = (a + b - c + d) * (-a + b + c + d)\nh0 : Prime (a * b + c * d)\nha : 0 < a\nhb : 0 < b\nhc : 0 < c\n⊢ False",
"after_state": "a b c d : ℤ\nhd : 0 < d\nhdc : d < c\nhcb : c < b\nhba : b < a\nh : a * c + b * d = (a + b - c + d) * (-a + b + c + d)\nh0 : Prime (a * b + c * d)\nha : 0 < a\nhb : 0 < b\nhc : 0 < c\ndvd_mul : a * c + b * d ∣ (a * b + c * d) * (a * d + b * c)\n⊢ False"
},
{
"line": "refine\n no_implicit_lambda%\n (have dvd_mul : a * c + b * d ∣ (a * b + c * d) * (a * d + b * c) := ?body✝;\n ?_)",
"before_state": "a b c d : ℤ\nhd : 0 < d\nhdc : d < c\nhcb : c < b\nhba : b < a\nh : a * c + b * d = (a + b - c + d) * (-a + b + c + d)\nh0 : Prime (a * b + c * d)\nha : 0 < a\nhb : 0 < b\nhc : 0 < c\n⊢ False",
"after_state": "case body\na b c d : ℤ\nhd : 0 < d\nhdc : d < c\nhcb : c < b\nhba : b < a\nh : a * c + b * d = (a + b - c + d) * (-a + b + c + d)\nh0 : Prime (a * b + c * d)\nha : 0 < a\nhb : 0 < b\nhc : 0 < c\n⊢ a * c + b * d ∣ (a * b + c * d) * (a * d + b * c)\n---\na b c d : ℤ\nhd : 0 < d\nhdc : d < c\nhcb : c < b\nhba : b < a\nh : a * c + b * d = (a + b - c + d) * (-a + b + c + d)\nh0 : Prime (a * b + c * d)\nha : 0 < a\nhb : 0 < b\nhc : 0 < c\ndvd_mul : a * c + b * d ∣ (a * b + c * d) * (a * d + b * c)\n⊢ False"
},
{
"line": "case body✝ =>\n with_annotate_state\"by\"\n ( use b ^ 2 + b * d + d ^ 2\n linear_combination b * d * h)",
"before_state": "case body\na b c d : ℤ\nhd : 0 < d\nhdc : d < c\nhcb : c < b\nhba : b < a\nh : a * c + b * d = (a + b - c + d) * (-a + b + c + d)\nh0 : Prime (a * b + c * d)\nha : 0 < a\nhb : 0 < b\nhc : 0 < c\n⊢ a * c + b * d ∣ (a * b + c * d) * (a * d + b * c)\n---\na b c d : ℤ\nhd : 0 < d\nhdc : d < c\nhcb : c < b\nhba : b < a\nh : a * c + b * d = (a + b - c + d) * (-a + b + c + d)\nh0 : Prime (a * b + c * d)\nha : 0 < a\nhb : 0 < b\nhc : 0 < c\ndvd_mul : a * c + b * d ∣ (a * b + c * d) * (a * d + b * c)\n⊢ False",
"after_state": "a b c d : ℤ\nhd : 0 < d\nhdc : d < c\nhcb : c < b\nhba : b < a\nh : a * c + b * d = (a + b - c + d) * (-a + b + c + d)\nh0 : Prime (a * b + c * d)\nha : 0 < a\nhb : 0 < b\nhc : 0 < c\ndvd_mul : a * c + b * d ∣ (a * b + c * d) * (a * d + b * c)\n⊢ False"
},
{
"line": "with_annotate_state\"by\"\n ( use b ^ 2 + b * d + d ^ 2\n linear_combination b * d * h)",
"before_state": "a b c d : ℤ\nhd : 0 < d\nhdc : d < c\nhcb : c < b\nhba : b < a\nh : a * c + b * d = (a + b - c + d) * (-a + b + c + d)\nh0 : Prime (a * b + c * d)\nha : 0 < a\nhb : 0 < b\nhc : 0 < c\n⊢ a * c + b * d ∣ (a * b + c * d) * (a * d + b * c)",
"after_state": "No Goals!"
},
{
"line": "use b ^ 2 + b * d + d ^ 2",
"before_state": "a b c d : ℤ\nhd : 0 < d\nhdc : d < c\nhcb : c < b\nhba : b < a\nh : a * c + b * d = (a + b - c + d) * (-a + b + c + d)\nh0 : Prime (a * b + c * d)\nha : 0 < a\nhb : 0 < b\nhc : 0 < c\n⊢ a * c + b * d ∣ (a * b + c * d) * (a * d + b * c)",
"after_state": "case h\na b c d : ℤ\nhd : 0 < d\nhdc : d < c\nhcb : c < b\nhba : b < a\nh : a * c + b * d = (a + b - c + d) * (-a + b + c + d)\nh0 : Prime (a * b + c * d)\nha : 0 < a\nhb : 0 < b\nhc : 0 < c\n⊢ (a * b + c * d) * (a * d + b * c) = (a * c + b * d) * (b ^ 2 + b * d + d ^ 2)"
},
{
"line": "refine without_cdot(b ^ 2 + b * d + d ^ 2 : ?m✝)",
"before_state": "case w\na b c d : ℤ\nhd : 0 < d\nhdc : d < c\nhcb : c < b\nhba : b < a\nh : a * c + b * d = (a + b - c + d) * (-a + b + c + d)\nh0 : Prime (a * b + c * d)\nha : 0 < a\nhb : 0 < b\nhc : 0 < c\n⊢ ℤ",
"after_state": "No Goals!"
},
{
"line": "try with_reducible use_discharger",
"before_state": "case h\na b c d : ℤ\nhd : 0 < d\nhdc : d < c\nhcb : c < b\nhba : b < a\nh : a * c + b * d = (a + b - c + d) * (-a + b + c + d)\nh0 : Prime (a * b + c * d)\nha : 0 < a\nhb : 0 < b\nhc : 0 < c\n⊢ (a * b + c * d) * (a * d + b * c) = (a * c + b * d) * (b ^ 2 + b * d + d ^ 2)",
"after_state": "case h\na b c d : ℤ\nhd : 0 < d\nhdc : d < c\nhcb : c < b\nhba : b < a\nh : a * c + b * d = (a + b - c + d) * (-a + b + c + d)\nh0 : Prime (a * b + c * d)\nha : 0 < a\nhb : 0 < b\nhc : 0 < c\n⊢ (a * b + c * d) * (a * d + b * c) = (a * c + b * d) * (b ^ 2 + b * d + d ^ 2)"
},
{
"line": "first\n| with_reducible use_discharger\n| skip",
"before_state": "case h\na b c d : ℤ\nhd : 0 < d\nhdc : d < c\nhcb : c < b\nhba : b < a\nh : a * c + b * d = (a + b - c + d) * (-a + b + c + d)\nh0 : Prime (a * b + c * d)\nha : 0 < a\nhb : 0 < b\nhc : 0 < c\n⊢ (a * b + c * d) * (a * d + b * c) = (a * c + b * d) * (b ^ 2 + b * d + d ^ 2)",
"after_state": "case h\na b c d : ℤ\nhd : 0 < d\nhdc : d < c\nhcb : c < b\nhba : b < a\nh : a * c + b * d = (a + b - c + d) * (-a + b + c + d)\nh0 : Prime (a * b + c * d)\nha : 0 < a\nhb : 0 < b\nhc : 0 < c\n⊢ (a * b + c * d) * (a * d + b * c) = (a * c + b * d) * (b ^ 2 + b * d + d ^ 2)"
},
{
"line": "with_reducible use_discharger",
"before_state": "case h\na b c d : ℤ\nhd : 0 < d\nhdc : d < c\nhcb : c < b\nhba : b < a\nh : a * c + b * d = (a + b - c + d) * (-a + b + c + d)\nh0 : Prime (a * b + c * d)\nha : 0 < a\nhb : 0 < b\nhc : 0 < c\n⊢ (a * b + c * d) * (a * d + b * c) = (a * c + b * d) * (b ^ 2 + b * d + d ^ 2)",
"after_state": "case h\na b c d : ℤ\nhd : 0 < d\nhdc : d < c\nhcb : c < b\nhba : b < a\nh : a * c + b * d = (a + b - c + d) * (-a + b + c + d)\nh0 : Prime (a * b + c * d)\nha : 0 < a\nhb : 0 < b\nhc : 0 < c\n⊢ (a * b + c * d) * (a * d + b * c) = (a * c + b * d) * (b ^ 2 + b * d + d ^ 2)"
},
{
"line": "use_discharger",
"before_state": "case h\na b c d : ℤ\nhd : 0 < d\nhdc : d < c\nhcb : c < b\nhba : b < a\nh : a * c + b * d = (a + b - c + d) * (-a + b + c + d)\nh0 : Prime (a * b + c * d)\nha : 0 < a\nhb : 0 < b\nhc : 0 < c\n⊢ (a * b + c * d) * (a * d + b * c) = (a * c + b * d) * (b ^ 2 + b * d + d ^ 2)",
"after_state": "case h\na b c d : ℤ\nhd : 0 < d\nhdc : d < c\nhcb : c < b\nhba : b < a\nh : a * c + b * d = (a + b - c + d) * (-a + b + c + d)\nh0 : Prime (a * b + c * d)\nha : 0 < a\nhb : 0 < b\nhc : 0 < c\n⊢ (a * b + c * d) * (a * d + b * c) = (a * c + b * d) * (b ^ 2 + b * d + d ^ 2)"
},
{
"line": "focus\n apply exists_prop.mpr✝\n with_annotate_state\"<;>\" skip\n all_goals use_discharger",
"before_state": "case h\na b c d : ℤ\nhd : 0 < d\nhdc : d < c\nhcb : c < b\nhba : b < a\nh : a * c + b * d = (a + b - c + d) * (-a + b + c + d)\nh0 : Prime (a * b + c * d)\nha : 0 < a\nhb : 0 < b\nhc : 0 < c\n⊢ (a * b + c * d) * (a * d + b * c) = (a * c + b * d) * (b ^ 2 + b * d + d ^ 2)",
"after_state": "case h\na b c d : ℤ\nhd : 0 < d\nhdc : d < c\nhcb : c < b\nhba : b < a\nh : a * c + b * d = (a + b - c + d) * (-a + b + c + d)\nh0 : Prime (a * b + c * d)\nha : 0 < a\nhb : 0 < b\nhc : 0 < c\n⊢ (a * b + c * d) * (a * d + b * c) = (a * c + b * d) * (b ^ 2 + b * d + d ^ 2)"
},
{
"line": "apply exists_prop.mpr✝",
"before_state": "case h\na b c d : ℤ\nhd : 0 < d\nhdc : d < c\nhcb : c < b\nhba : b < a\nh : a * c + b * d = (a + b - c + d) * (-a + b + c + d)\nh0 : Prime (a * b + c * d)\nha : 0 < a\nhb : 0 < b\nhc : 0 < c\n⊢ (a * b + c * d) * (a * d + b * c) = (a * c + b * d) * (b ^ 2 + b * d + d ^ 2)",
"after_state": "case h\na b c d : ℤ\nhd : 0 < d\nhdc : d < c\nhcb : c < b\nhba : b < a\nh : a * c + b * d = (a + b - c + d) * (-a + b + c + d)\nh0 : Prime (a * b + c * d)\nha : 0 < a\nhb : 0 < b\nhc : 0 < c\n⊢ (a * b + c * d) * (a * d + b * c) = (a * c + b * d) * (b ^ 2 + b * d + d ^ 2)"
},
{
"line": "skip",
"before_state": "case h\na b c d : ℤ\nhd : 0 < d\nhdc : d < c\nhcb : c < b\nhba : b < a\nh : a * c + b * d = (a + b - c + d) * (-a + b + c + d)\nh0 : Prime (a * b + c * d)\nha : 0 < a\nhb : 0 < b\nhc : 0 < c\n⊢ (a * b + c * d) * (a * d + b * c) = (a * c + b * d) * (b ^ 2 + b * d + d ^ 2)",
"after_state": "case h\na b c d : ℤ\nhd : 0 < d\nhdc : d < c\nhcb : c < b\nhba : b < a\nh : a * c + b * d = (a + b - c + d) * (-a + b + c + d)\nh0 : Prime (a * b + c * d)\nha : 0 < a\nhb : 0 < b\nhc : 0 < c\n⊢ (a * b + c * d) * (a * d + b * c) = (a * c + b * d) * (b ^ 2 + b * d + d ^ 2)"
},
{
"line": "linear_combination b * d * h",
"before_state": "case h\na b c d : ℤ\nhd : 0 < d\nhdc : d < c\nhcb : c < b\nhba : b < a\nh : a * c + b * d = (a + b - c + d) * (-a + b + c + d)\nh0 : Prime (a * b + c * d)\nha : 0 < a\nhb : 0 < b\nhc : 0 < c\n⊢ (a * b + c * d) * (a * d + b * c) = (a * c + b * d) * (b ^ 2 + b * d + d ^ 2)",
"after_state": "No Goals!"
},
{
"line": "obtain (h1 : a * b + c * d ∣ a * c + b * d) | (h2 : a * c + b * d ∣ a * d + b * c) :=\n h0.left_dvd_or_dvd_right_of_dvd_mul dvd_mul",
"before_state": "a b c d : ℤ\nhd : 0 < d\nhdc : d < c\nhcb : c < b\nhba : b < a\nh : a * c + b * d = (a + b - c + d) * (-a + b + c + d)\nh0 : Prime (a * b + c * d)\nha : 0 < a\nhb : 0 < b\nhc : 0 < c\ndvd_mul : a * c + b * d ∣ (a * b + c * d) * (a * d + b * c)\n⊢ False",
"after_state": "case inl\na b c d : ℤ\nhd : 0 < d\nhdc : d < c\nhcb : c < b\nhba : b < a\nh : a * c + b * d = (a + b - c + d) * (-a + b + c + d)\nh0 : Prime (a * b + c * d)\nha : 0 < a\nhb : 0 < b\nhc : 0 < c\ndvd_mul : a * c + b * d ∣ (a * b + c * d) * (a * d + b * c)\nh1 : a * b + c * d ∣ a * c + b * d\n⊢ False\n---\ncase inr\na b c d : ℤ\nhd : 0 < d\nhdc : d < c\nhcb : c < b\nhba : b < a\nh : a * c + b * d = (a + b - c + d) * (-a + b + c + d)\nh0 : Prime (a * b + c * d)\nha : 0 < a\nhb : 0 < b\nhc : 0 < c\ndvd_mul : a * c + b * d ∣ (a * b + c * d) * (a * d + b * c)\nh2 : a * c + b * d ∣ a * d + b * c\n⊢ False"
},
{
"line": "have aux : 0 < a * c + b * d := by nlinarith only [ha, hb, hc, hd]",
"before_state": "case inl\na b c d : ℤ\nhd : 0 < d\nhdc : d < c\nhcb : c < b\nhba : b < a\nh : a * c + b * d = (a + b - c + d) * (-a + b + c + d)\nh0 : Prime (a * b + c * d)\nha : 0 < a\nhb : 0 < b\nhc : 0 < c\ndvd_mul : a * c + b * d ∣ (a * b + c * d) * (a * d + b * c)\nh1 : a * b + c * d ∣ a * c + b * d\n⊢ False",
"after_state": "case inl\na b c d : ℤ\nhd : 0 < d\nhdc : d < c\nhcb : c < b\nhba : b < a\nh : a * c + b * d = (a + b - c + d) * (-a + b + c + d)\nh0 : Prime (a * b + c * d)\nha : 0 < a\nhb : 0 < b\nhc : 0 < c\ndvd_mul : a * c + b * d ∣ (a * b + c * d) * (a * d + b * c)\nh1 : a * b + c * d ∣ a * c + b * d\naux : 0 < a * c + b * d\n⊢ False"
},
{
"line": "focus\n refine\n no_implicit_lambda%\n (have aux : 0 < a * c + b * d := ?body✝;\n ?_)\n case body✝ => with_annotate_state\"by\" (nlinarith only [ha, hb, hc, hd])",
"before_state": "case inl\na b c d : ℤ\nhd : 0 < d\nhdc : d < c\nhcb : c < b\nhba : b < a\nh : a * c + b * d = (a + b - c + d) * (-a + b + c + d)\nh0 : Prime (a * b + c * d)\nha : 0 < a\nhb : 0 < b\nhc : 0 < c\ndvd_mul : a * c + b * d ∣ (a * b + c * d) * (a * d + b * c)\nh1 : a * b + c * d ∣ a * c + b * d\n⊢ False",
"after_state": "case inl\na b c d : ℤ\nhd : 0 < d\nhdc : d < c\nhcb : c < b\nhba : b < a\nh : a * c + b * d = (a + b - c + d) * (-a + b + c + d)\nh0 : Prime (a * b + c * d)\nha : 0 < a\nhb : 0 < b\nhc : 0 < c\ndvd_mul : a * c + b * d ∣ (a * b + c * d) * (a * d + b * c)\nh1 : a * b + c * d ∣ a * c + b * d\naux : 0 < a * c + b * d\n⊢ False"
},
{
"line": "refine\n no_implicit_lambda%\n (have aux : 0 < a * c + b * d := ?body✝;\n ?_)",
"before_state": "case inl\na b c d : ℤ\nhd : 0 < d\nhdc : d < c\nhcb : c < b\nhba : b < a\nh : a * c + b * d = (a + b - c + d) * (-a + b + c + d)\nh0 : Prime (a * b + c * d)\nha : 0 < a\nhb : 0 < b\nhc : 0 < c\ndvd_mul : a * c + b * d ∣ (a * b + c * d) * (a * d + b * c)\nh1 : a * b + c * d ∣ a * c + b * d\n⊢ False",
"after_state": "case body\na b c d : ℤ\nhd : 0 < d\nhdc : d < c\nhcb : c < b\nhba : b < a\nh : a * c + b * d = (a + b - c + d) * (-a + b + c + d)\nh0 : Prime (a * b + c * d)\nha : 0 < a\nhb : 0 < b\nhc : 0 < c\ndvd_mul : a * c + b * d ∣ (a * b + c * d) * (a * d + b * c)\nh1 : a * b + c * d ∣ a * c + b * d\n⊢ 0 < a * c + b * d\n---\ncase inl\na b c d : ℤ\nhd : 0 < d\nhdc : d < c\nhcb : c < b\nhba : b < a\nh : a * c + b * d = (a + b - c + d) * (-a + b + c + d)\nh0 : Prime (a * b + c * d)\nha : 0 < a\nhb : 0 < b\nhc : 0 < c\ndvd_mul : a * c + b * d ∣ (a * b + c * d) * (a * d + b * c)\nh1 : a * b + c * d ∣ a * c + b * d\naux : 0 < a * c + b * d\n⊢ False"
},
{
"line": "case body✝ => with_annotate_state\"by\" (nlinarith only [ha, hb, hc, hd])",
"before_state": "case body\na b c d : ℤ\nhd : 0 < d\nhdc : d < c\nhcb : c < b\nhba : b < a\nh : a * c + b * d = (a + b - c + d) * (-a + b + c + d)\nh0 : Prime (a * b + c * d)\nha : 0 < a\nhb : 0 < b\nhc : 0 < c\ndvd_mul : a * c + b * d ∣ (a * b + c * d) * (a * d + b * c)\nh1 : a * b + c * d ∣ a * c + b * d\n⊢ 0 < a * c + b * d\n---\ncase inl\na b c d : ℤ\nhd : 0 < d\nhdc : d < c\nhcb : c < b\nhba : b < a\nh : a * c + b * d = (a + b - c + d) * (-a + b + c + d)\nh0 : Prime (a * b + c * d)\nha : 0 < a\nhb : 0 < b\nhc : 0 < c\ndvd_mul : a * c + b * d ∣ (a * b + c * d) * (a * d + b * c)\nh1 : a * b + c * d ∣ a * c + b * d\naux : 0 < a * c + b * d\n⊢ False",
"after_state": "case inl\na b c d : ℤ\nhd : 0 < d\nhdc : d < c\nhcb : c < b\nhba : b < a\nh : a * c + b * d = (a + b - c + d) * (-a + b + c + d)\nh0 : Prime (a * b + c * d)\nha : 0 < a\nhb : 0 < b\nhc : 0 < c\ndvd_mul : a * c + b * d ∣ (a * b + c * d) * (a * d + b * c)\nh1 : a * b + c * d ∣ a * c + b * d\naux : 0 < a * c + b * d\n⊢ False"
},
{
"line": "with_annotate_state\"by\" (nlinarith only [ha, hb, hc, hd])",
"before_state": "a b c d : ℤ\nhd : 0 < d\nhdc : d < c\nhcb : c < b\nhba : b < a\nh : a * c + b * d = (a + b - c + d) * (-a + b + c + d)\nh0 : Prime (a * b + c * d)\nha : 0 < a\nhb : 0 < b\nhc : 0 < c\ndvd_mul : a * c + b * d ∣ (a * b + c * d) * (a * d + b * c)\nh1 : a * b + c * d ∣ a * c + b * d\n⊢ 0 < a * c + b * d",
"after_state": "No Goals!"
},
{
"line": "nlinarith only [ha, hb, hc, hd]",
"before_state": "a b c d : ℤ\nhd : 0 < d\nhdc : d < c\nhcb : c < b\nhba : b < a\nh : a * c + b * d = (a + b - c + d) * (-a + b + c + d)\nh0 : Prime (a * b + c * d)\nha : 0 < a\nhb : 0 < b\nhc : 0 < c\ndvd_mul : a * c + b * d ∣ (a * b + c * d) * (a * d + b * c)\nh1 : a * b + c * d ∣ a * c + b * d\n⊢ 0 < a * c + b * d",
"after_state": "No Goals!"
},
{
"line": "ring1",
"before_state": "a b c d : ℤ\nhd : 0 < d\nhdc : d < c\nhcb : c < b\nhba : b < a\nh : a * c + b * d = (a + b - c + d) * (-a + b + c + d)\nh0 : Prime (a * b + c * d)\nha : 0 < a\nhb : 0 < b\nhc : 0 < c\ndvd_mul : a * c + b * d ∣ (a * b + c * d) * (a * d + b * c)\nh1 : a * b + c * d ∣ a * c + b * d\na✝ : 0 ≥ a * c + b * d\n⊢ 2 * -1 + (a * c + b * d - 0) + (0 + 1 - a) + (0 + 1 - b) + (0 + 1 - c) + (0 + 1 - d) +\n (0 - (0 + 1 - a) * (0 + 1 - c)) +\n (0 - (0 + 1 - b) * (0 + 1 - d)) =\n 0",
"after_state": "No Goals!"
},
{
"line": "norm_num",
"before_state": "a b c d : ℤ\nhd : 0 < d\nhdc : d < c\nhcb : c < b\nhba : b < a\nh : a * c + b * d = (a + b - c + d) * (-a + b + c + d)\nh0 : Prime (a * b + c * d)\nha : 0 < a\nhb : 0 < b\nhc : 0 < c\ndvd_mul : a * c + b * d ∣ (a * b + c * d) * (a * d + b * c)\nh1 : a * b + c * d ∣ a * c + b * d\na✝ : 0 ≥ a * c + b * d\n⊢ 2 > 0",
"after_state": "No Goals!"
},
{
"line": "have : a * b + c * d ≤ a * c + b * d := Int.le_of_dvd aux h1",
"before_state": "case inl\na b c d : ℤ\nhd : 0 < d\nhdc : d < c\nhcb : c < b\nhba : b < a\nh : a * c + b * d = (a + b - c + d) * (-a + b + c + d)\nh0 : Prime (a * b + c * d)\nha : 0 < a\nhb : 0 < b\nhc : 0 < c\ndvd_mul : a * c + b * d ∣ (a * b + c * d) * (a * d + b * c)\nh1 : a * b + c * d ∣ a * c + b * d\naux : 0 < a * c + b * d\n⊢ False",
"after_state": "case inl\na b c d : ℤ\nhd : 0 < d\nhdc : d < c\nhcb : c < b\nhba : b < a\nh : a * c + b * d = (a + b - c + d) * (-a + b + c + d)\nh0 : Prime (a * b + c * d)\nha : 0 < a\nhb : 0 < b\nhc : 0 < c\ndvd_mul : a * c + b * d ∣ (a * b + c * d) * (a * d + b * c)\nh1 : a * b + c * d ∣ a * c + b * d\naux : 0 < a * c + b * d\nthis : a * b + c * d ≤ a * c + b * d\n⊢ False"
},
{
"line": "refine_lift\n have : a * b + c * d ≤ a * c + b * d := Int.le_of_dvd aux h1;\n ?_",
"before_state": "case inl\na b c d : ℤ\nhd : 0 < d\nhdc : d < c\nhcb : c < b\nhba : b < a\nh : a * c + b * d = (a + b - c + d) * (-a + b + c + d)\nh0 : Prime (a * b + c * d)\nha : 0 < a\nhb : 0 < b\nhc : 0 < c\ndvd_mul : a * c + b * d ∣ (a * b + c * d) * (a * d + b * c)\nh1 : a * b + c * d ∣ a * c + b * d\naux : 0 < a * c + b * d\n⊢ False",
"after_state": "case inl\na b c d : ℤ\nhd : 0 < d\nhdc : d < c\nhcb : c < b\nhba : b < a\nh : a * c + b * d = (a + b - c + d) * (-a + b + c + d)\nh0 : Prime (a * b + c * d)\nha : 0 < a\nhb : 0 < b\nhc : 0 < c\ndvd_mul : a * c + b * d ∣ (a * b + c * d) * (a * d + b * c)\nh1 : a * b + c * d ∣ a * c + b * d\naux : 0 < a * c + b * d\nthis : a * b + c * d ≤ a * c + b * d\n⊢ False"
},
{
"line": "focus\n (refine\n no_implicit_lambda%\n (have : a * b + c * d ≤ a * c + b * d := Int.le_of_dvd aux h1;\n ?_);\n rotate_right)",
"before_state": "case inl\na b c d : ℤ\nhd : 0 < d\nhdc : d < c\nhcb : c < b\nhba : b < a\nh : a * c + b * d = (a + b - c + d) * (-a + b + c + d)\nh0 : Prime (a * b + c * d)\nha : 0 < a\nhb : 0 < b\nhc : 0 < c\ndvd_mul : a * c + b * d ∣ (a * b + c * d) * (a * d + b * c)\nh1 : a * b + c * d ∣ a * c + b * d\naux : 0 < a * c + b * d\n⊢ False",
"after_state": "case inl\na b c d : ℤ\nhd : 0 < d\nhdc : d < c\nhcb : c < b\nhba : b < a\nh : a * c + b * d = (a + b - c + d) * (-a + b + c + d)\nh0 : Prime (a * b + c * d)\nha : 0 < a\nhb : 0 < b\nhc : 0 < c\ndvd_mul : a * c + b * d ∣ (a * b + c * d) * (a * d + b * c)\nh1 : a * b + c * d ∣ a * c + b * d\naux : 0 < a * c + b * d\nthis : a * b + c * d ≤ a * c + b * d\n⊢ False"
},
{
"line": "refine\n no_implicit_lambda%\n (have : a * b + c * d ≤ a * c + b * d := Int.le_of_dvd aux h1;\n ?_)",
"before_state": "case inl\na b c d : ℤ\nhd : 0 < d\nhdc : d < c\nhcb : c < b\nhba : b < a\nh : a * c + b * d = (a + b - c + d) * (-a + b + c + d)\nh0 : Prime (a * b + c * d)\nha : 0 < a\nhb : 0 < b\nhc : 0 < c\ndvd_mul : a * c + b * d ∣ (a * b + c * d) * (a * d + b * c)\nh1 : a * b + c * d ∣ a * c + b * d\naux : 0 < a * c + b * d\n⊢ False",
"after_state": "case inl\na b c d : ℤ\nhd : 0 < d\nhdc : d < c\nhcb : c < b\nhba : b < a\nh : a * c + b * d = (a + b - c + d) * (-a + b + c + d)\nh0 : Prime (a * b + c * d)\nha : 0 < a\nhb : 0 < b\nhc : 0 < c\ndvd_mul : a * c + b * d ∣ (a * b + c * d) * (a * d + b * c)\nh1 : a * b + c * d ∣ a * c + b * d\naux : 0 < a * c + b * d\nthis : a * b + c * d ≤ a * c + b * d\n⊢ False"
},
{
"line": "rotate_right",
"before_state": "case inl\na b c d : ℤ\nhd : 0 < d\nhdc : d < c\nhcb : c < b\nhba : b < a\nh : a * c + b * d = (a + b - c + d) * (-a + b + c + d)\nh0 : Prime (a * b + c * d)\nha : 0 < a\nhb : 0 < b\nhc : 0 < c\ndvd_mul : a * c + b * d ∣ (a * b + c * d) * (a * d + b * c)\nh1 : a * b + c * d ∣ a * c + b * d\naux : 0 < a * c + b * d\nthis : a * b + c * d ≤ a * c + b * d\n⊢ False",
"after_state": "case inl\na b c d : ℤ\nhd : 0 < d\nhdc : d < c\nhcb : c < b\nhba : b < a\nh : a * c + b * d = (a + b - c + d) * (-a + b + c + d)\nh0 : Prime (a * b + c * d)\nha : 0 < a\nhb : 0 < b\nhc : 0 < c\ndvd_mul : a * c + b * d ∣ (a * b + c * d) * (a * d + b * c)\nh1 : a * b + c * d ∣ a * c + b * d\naux : 0 < a * c + b * d\nthis : a * b + c * d ≤ a * c + b * d\n⊢ False"
},
{
"line": "nlinarith only [hba, hcb, hdc, h, this]",
"before_state": "case inl\na b c d : ℤ\nhd : 0 < d\nhdc : d < c\nhcb : c < b\nhba : b < a\nh : a * c + b * d = (a + b - c + d) * (-a + b + c + d)\nh0 : Prime (a * b + c * d)\nha : 0 < a\nhb : 0 < b\nhc : 0 < c\ndvd_mul : a * c + b * d ∣ (a * b + c * d) * (a * d + b * c)\nh1 : a * b + c * d ∣ a * c + b * d\naux : 0 < a * c + b * d\nthis : a * b + c * d ≤ a * c + b * d\n⊢ False",
"after_state": "No Goals!"
},
{
"line": "ring1",
"before_state": "a b c d : ℤ\nhd : 0 < d\nhdc : d < c\nhcb : c < b\nhba : b < a\nh : a * c + b * d = (a + b - c + d) * (-a + b + c + d)\nh0 : Prime (a * b + c * d)\nha : 0 < a\nhb : 0 < b\nhc : 0 < c\ndvd_mul : a * c + b * d ∣ (a * b + c * d) * (a * d + b * c)\nh1 : a * b + c * d ∣ a * c + b * d\naux : 0 < a * c + b * d\nthis : a * b + c * d ≤ a * c + b * d\n⊢ 3 * -1 + (b + 1 - a) + 4 * (c + 1 - b) + (d + 1 - c) + (a * b + c * d - (a * c + b * d)) +\n (0 - (b + 1 - a) * (c + 1 - b)) +\n (0 - (c + 1 - b) * (c + 1 - b)) +\n (0 - (c + 1 - b) * (d + 1 - c)) =\n 0",
"after_state": "No Goals!"
},
{
"line": "norm_num",
"before_state": "a b c d : ℤ\nhd : 0 < d\nhdc : d < c\nhcb : c < b\nhba : b < a\nh : a * c + b * d = (a + b - c + d) * (-a + b + c + d)\nh0 : Prime (a * b + c * d)\nha : 0 < a\nhb : 0 < b\nhc : 0 < c\ndvd_mul : a * c + b * d ∣ (a * b + c * d) * (a * d + b * c)\nh1 : a * b + c * d ∣ a * c + b * d\naux : 0 < a * c + b * d\nthis : a * b + c * d ≤ a * c + b * d\n⊢ 3 > 0",
"after_state": "No Goals!"
},
{
"line": "norm_num",
"before_state": "a b c d : ℤ\nhd : 0 < d\nhdc : d < c\nhcb : c < b\nhba : b < a\nh : a * c + b * d = (a + b - c + d) * (-a + b + c + d)\nh0 : Prime (a * b + c * d)\nha : 0 < a\nhb : 0 < b\nhc : 0 < c\ndvd_mul : a * c + b * d ∣ (a * b + c * d) * (a * d + b * c)\nh1 : a * b + c * d ∣ a * c + b * d\naux : 0 < a * c + b * d\nthis : a * b + c * d ≤ a * c + b * d\n⊢ 4 > 0",
"after_state": "No Goals!"
},
{
"line": "have aux : 0 < a * d + b * c := by nlinarith only [ha, hb, hc, hd]",
"before_state": "case inr\na b c d : ℤ\nhd : 0 < d\nhdc : d < c\nhcb : c < b\nhba : b < a\nh : a * c + b * d = (a + b - c + d) * (-a + b + c + d)\nh0 : Prime (a * b + c * d)\nha : 0 < a\nhb : 0 < b\nhc : 0 < c\ndvd_mul : a * c + b * d ∣ (a * b + c * d) * (a * d + b * c)\nh2 : a * c + b * d ∣ a * d + b * c\n⊢ False",
"after_state": "case inr\na b c d : ℤ\nhd : 0 < d\nhdc : d < c\nhcb : c < b\nhba : b < a\nh : a * c + b * d = (a + b - c + d) * (-a + b + c + d)\nh0 : Prime (a * b + c * d)\nha : 0 < a\nhb : 0 < b\nhc : 0 < c\ndvd_mul : a * c + b * d ∣ (a * b + c * d) * (a * d + b * c)\nh2 : a * c + b * d ∣ a * d + b * c\naux : 0 < a * d + b * c\n⊢ False"
},
{
"line": "focus\n refine\n no_implicit_lambda%\n (have aux : 0 < a * d + b * c := ?body✝;\n ?_)\n case body✝ => with_annotate_state\"by\" (nlinarith only [ha, hb, hc, hd])",
"before_state": "case inr\na b c d : ℤ\nhd : 0 < d\nhdc : d < c\nhcb : c < b\nhba : b < a\nh : a * c + b * d = (a + b - c + d) * (-a + b + c + d)\nh0 : Prime (a * b + c * d)\nha : 0 < a\nhb : 0 < b\nhc : 0 < c\ndvd_mul : a * c + b * d ∣ (a * b + c * d) * (a * d + b * c)\nh2 : a * c + b * d ∣ a * d + b * c\n⊢ False",
"after_state": "case inr\na b c d : ℤ\nhd : 0 < d\nhdc : d < c\nhcb : c < b\nhba : b < a\nh : a * c + b * d = (a + b - c + d) * (-a + b + c + d)\nh0 : Prime (a * b + c * d)\nha : 0 < a\nhb : 0 < b\nhc : 0 < c\ndvd_mul : a * c + b * d ∣ (a * b + c * d) * (a * d + b * c)\nh2 : a * c + b * d ∣ a * d + b * c\naux : 0 < a * d + b * c\n⊢ False"
},
{
"line": "refine\n no_implicit_lambda%\n (have aux : 0 < a * d + b * c := ?body✝;\n ?_)",
"before_state": "case inr\na b c d : ℤ\nhd : 0 < d\nhdc : d < c\nhcb : c < b\nhba : b < a\nh : a * c + b * d = (a + b - c + d) * (-a + b + c + d)\nh0 : Prime (a * b + c * d)\nha : 0 < a\nhb : 0 < b\nhc : 0 < c\ndvd_mul : a * c + b * d ∣ (a * b + c * d) * (a * d + b * c)\nh2 : a * c + b * d ∣ a * d + b * c\n⊢ False",
"after_state": "case body\na b c d : ℤ\nhd : 0 < d\nhdc : d < c\nhcb : c < b\nhba : b < a\nh : a * c + b * d = (a + b - c + d) * (-a + b + c + d)\nh0 : Prime (a * b + c * d)\nha : 0 < a\nhb : 0 < b\nhc : 0 < c\ndvd_mul : a * c + b * d ∣ (a * b + c * d) * (a * d + b * c)\nh2 : a * c + b * d ∣ a * d + b * c\n⊢ 0 < a * d + b * c\n---\ncase inr\na b c d : ℤ\nhd : 0 < d\nhdc : d < c\nhcb : c < b\nhba : b < a\nh : a * c + b * d = (a + b - c + d) * (-a + b + c + d)\nh0 : Prime (a * b + c * d)\nha : 0 < a\nhb : 0 < b\nhc : 0 < c\ndvd_mul : a * c + b * d ∣ (a * b + c * d) * (a * d + b * c)\nh2 : a * c + b * d ∣ a * d + b * c\naux : 0 < a * d + b * c\n⊢ False"
},
{
"line": "case body✝ => with_annotate_state\"by\" (nlinarith only [ha, hb, hc, hd])",
"before_state": "case body\na b c d : ℤ\nhd : 0 < d\nhdc : d < c\nhcb : c < b\nhba : b < a\nh : a * c + b * d = (a + b - c + d) * (-a + b + c + d)\nh0 : Prime (a * b + c * d)\nha : 0 < a\nhb : 0 < b\nhc : 0 < c\ndvd_mul : a * c + b * d ∣ (a * b + c * d) * (a * d + b * c)\nh2 : a * c + b * d ∣ a * d + b * c\n⊢ 0 < a * d + b * c\n---\ncase inr\na b c d : ℤ\nhd : 0 < d\nhdc : d < c\nhcb : c < b\nhba : b < a\nh : a * c + b * d = (a + b - c + d) * (-a + b + c + d)\nh0 : Prime (a * b + c * d)\nha : 0 < a\nhb : 0 < b\nhc : 0 < c\ndvd_mul : a * c + b * d ∣ (a * b + c * d) * (a * d + b * c)\nh2 : a * c + b * d ∣ a * d + b * c\naux : 0 < a * d + b * c\n⊢ False",
"after_state": "case inr\na b c d : ℤ\nhd : 0 < d\nhdc : d < c\nhcb : c < b\nhba : b < a\nh : a * c + b * d = (a + b - c + d) * (-a + b + c + d)\nh0 : Prime (a * b + c * d)\nha : 0 < a\nhb : 0 < b\nhc : 0 < c\ndvd_mul : a * c + b * d ∣ (a * b + c * d) * (a * d + b * c)\nh2 : a * c + b * d ∣ a * d + b * c\naux : 0 < a * d + b * c\n⊢ False"
},
{
"line": "with_annotate_state\"by\" (nlinarith only [ha, hb, hc, hd])",
"before_state": "a b c d : ℤ\nhd : 0 < d\nhdc : d < c\nhcb : c < b\nhba : b < a\nh : a * c + b * d = (a + b - c + d) * (-a + b + c + d)\nh0 : Prime (a * b + c * d)\nha : 0 < a\nhb : 0 < b\nhc : 0 < c\ndvd_mul : a * c + b * d ∣ (a * b + c * d) * (a * d + b * c)\nh2 : a * c + b * d ∣ a * d + b * c\n⊢ 0 < a * d + b * c",
"after_state": "No Goals!"
},
{
"line": "nlinarith only [ha, hb, hc, hd]",
"before_state": "a b c d : ℤ\nhd : 0 < d\nhdc : d < c\nhcb : c < b\nhba : b < a\nh : a * c + b * d = (a + b - c + d) * (-a + b + c + d)\nh0 : Prime (a * b + c * d)\nha : 0 < a\nhb : 0 < b\nhc : 0 < c\ndvd_mul : a * c + b * d ∣ (a * b + c * d) * (a * d + b * c)\nh2 : a * c + b * d ∣ a * d + b * c\n⊢ 0 < a * d + b * c",
"after_state": "No Goals!"
},
{
"line": "ring1",
"before_state": "a b c d : ℤ\nhd : 0 < d\nhdc : d < c\nhcb : c < b\nhba : b < a\nh : a * c + b * d = (a + b - c + d) * (-a + b + c + d)\nh0 : Prime (a * b + c * d)\nha : 0 < a\nhb : 0 < b\nhc : 0 < c\ndvd_mul : a * c + b * d ∣ (a * b + c * d) * (a * d + b * c)\nh2 : a * c + b * d ∣ a * d + b * c\na✝ : 0 ≥ a * d + b * c\n⊢ 2 * -1 + (a * d + b * c - 0) + (0 + 1 - a) + (0 + 1 - b) + (0 + 1 - c) + (0 + 1 - d) +\n (0 - (0 + 1 - a) * (0 + 1 - d)) +\n (0 - (0 + 1 - b) * (0 + 1 - c)) =\n 0",
"after_state": "No Goals!"
},
{
"line": "norm_num",
"before_state": "a b c d : ℤ\nhd : 0 < d\nhdc : d < c\nhcb : c < b\nhba : b < a\nh : a * c + b * d = (a + b - c + d) * (-a + b + c + d)\nh0 : Prime (a * b + c * d)\nha : 0 < a\nhb : 0 < b\nhc : 0 < c\ndvd_mul : a * c + b * d ∣ (a * b + c * d) * (a * d + b * c)\nh2 : a * c + b * d ∣ a * d + b * c\na✝ : 0 ≥ a * d + b * c\n⊢ 2 > 0",
"after_state": "No Goals!"
},
{
"line": "have : a * c + b * d ≤ a * d + b * c := Int.le_of_dvd aux h2",
"before_state": "case inr\na b c d : ℤ\nhd : 0 < d\nhdc : d < c\nhcb : c < b\nhba : b < a\nh : a * c + b * d = (a + b - c + d) * (-a + b + c + d)\nh0 : Prime (a * b + c * d)\nha : 0 < a\nhb : 0 < b\nhc : 0 < c\ndvd_mul : a * c + b * d ∣ (a * b + c * d) * (a * d + b * c)\nh2 : a * c + b * d ∣ a * d + b * c\naux : 0 < a * d + b * c\n⊢ False",
"after_state": "case inr\na b c d : ℤ\nhd : 0 < d\nhdc : d < c\nhcb : c < b\nhba : b < a\nh : a * c + b * d = (a + b - c + d) * (-a + b + c + d)\nh0 : Prime (a * b + c * d)\nha : 0 < a\nhb : 0 < b\nhc : 0 < c\ndvd_mul : a * c + b * d ∣ (a * b + c * d) * (a * d + b * c)\nh2 : a * c + b * d ∣ a * d + b * c\naux : 0 < a * d + b * c\nthis : a * c + b * d ≤ a * d + b * c\n⊢ False"
},
{
"line": "refine_lift\n have : a * c + b * d ≤ a * d + b * c := Int.le_of_dvd aux h2;\n ?_",
"before_state": "case inr\na b c d : ℤ\nhd : 0 < d\nhdc : d < c\nhcb : c < b\nhba : b < a\nh : a * c + b * d = (a + b - c + d) * (-a + b + c + d)\nh0 : Prime (a * b + c * d)\nha : 0 < a\nhb : 0 < b\nhc : 0 < c\ndvd_mul : a * c + b * d ∣ (a * b + c * d) * (a * d + b * c)\nh2 : a * c + b * d ∣ a * d + b * c\naux : 0 < a * d + b * c\n⊢ False",
"after_state": "case inr\na b c d : ℤ\nhd : 0 < d\nhdc : d < c\nhcb : c < b\nhba : b < a\nh : a * c + b * d = (a + b - c + d) * (-a + b + c + d)\nh0 : Prime (a * b + c * d)\nha : 0 < a\nhb : 0 < b\nhc : 0 < c\ndvd_mul : a * c + b * d ∣ (a * b + c * d) * (a * d + b * c)\nh2 : a * c + b * d ∣ a * d + b * c\naux : 0 < a * d + b * c\nthis : a * c + b * d ≤ a * d + b * c\n⊢ False"
},
{
"line": "focus\n (refine\n no_implicit_lambda%\n (have : a * c + b * d ≤ a * d + b * c := Int.le_of_dvd aux h2;\n ?_);\n rotate_right)",
"before_state": "case inr\na b c d : ℤ\nhd : 0 < d\nhdc : d < c\nhcb : c < b\nhba : b < a\nh : a * c + b * d = (a + b - c + d) * (-a + b + c + d)\nh0 : Prime (a * b + c * d)\nha : 0 < a\nhb : 0 < b\nhc : 0 < c\ndvd_mul : a * c + b * d ∣ (a * b + c * d) * (a * d + b * c)\nh2 : a * c + b * d ∣ a * d + b * c\naux : 0 < a * d + b * c\n⊢ False",
"after_state": "case inr\na b c d : ℤ\nhd : 0 < d\nhdc : d < c\nhcb : c < b\nhba : b < a\nh : a * c + b * d = (a + b - c + d) * (-a + b + c + d)\nh0 : Prime (a * b + c * d)\nha : 0 < a\nhb : 0 < b\nhc : 0 < c\ndvd_mul : a * c + b * d ∣ (a * b + c * d) * (a * d + b * c)\nh2 : a * c + b * d ∣ a * d + b * c\naux : 0 < a * d + b * c\nthis : a * c + b * d ≤ a * d + b * c\n⊢ False"
},
{
"line": "refine\n no_implicit_lambda%\n (have : a * c + b * d ≤ a * d + b * c := Int.le_of_dvd aux h2;\n ?_)",
"before_state": "case inr\na b c d : ℤ\nhd : 0 < d\nhdc : d < c\nhcb : c < b\nhba : b < a\nh : a * c + b * d = (a + b - c + d) * (-a + b + c + d)\nh0 : Prime (a * b + c * d)\nha : 0 < a\nhb : 0 < b\nhc : 0 < c\ndvd_mul : a * c + b * d ∣ (a * b + c * d) * (a * d + b * c)\nh2 : a * c + b * d ∣ a * d + b * c\naux : 0 < a * d + b * c\n⊢ False",
"after_state": "case inr\na b c d : ℤ\nhd : 0 < d\nhdc : d < c\nhcb : c < b\nhba : b < a\nh : a * c + b * d = (a + b - c + d) * (-a + b + c + d)\nh0 : Prime (a * b + c * d)\nha : 0 < a\nhb : 0 < b\nhc : 0 < c\ndvd_mul : a * c + b * d ∣ (a * b + c * d) * (a * d + b * c)\nh2 : a * c + b * d ∣ a * d + b * c\naux : 0 < a * d + b * c\nthis : a * c + b * d ≤ a * d + b * c\n⊢ False"
},
{
"line": "rotate_right",
"before_state": "case inr\na b c d : ℤ\nhd : 0 < d\nhdc : d < c\nhcb : c < b\nhba : b < a\nh : a * c + b * d = (a + b - c + d) * (-a + b + c + d)\nh0 : Prime (a * b + c * d)\nha : 0 < a\nhb : 0 < b\nhc : 0 < c\ndvd_mul : a * c + b * d ∣ (a * b + c * d) * (a * d + b * c)\nh2 : a * c + b * d ∣ a * d + b * c\naux : 0 < a * d + b * c\nthis : a * c + b * d ≤ a * d + b * c\n⊢ False",
"after_state": "case inr\na b c d : ℤ\nhd : 0 < d\nhdc : d < c\nhcb : c < b\nhba : b < a\nh : a * c + b * d = (a + b - c + d) * (-a + b + c + d)\nh0 : Prime (a * b + c * d)\nha : 0 < a\nhb : 0 < b\nhc : 0 < c\ndvd_mul : a * c + b * d ∣ (a * b + c * d) * (a * d + b * c)\nh2 : a * c + b * d ∣ a * d + b * c\naux : 0 < a * d + b * c\nthis : a * c + b * d ≤ a * d + b * c\n⊢ False"
},
{
"line": "nlinarith only [hba, hdc, h, this]",
"before_state": "case inr\na b c d : ℤ\nhd : 0 < d\nhdc : d < c\nhcb : c < b\nhba : b < a\nh : a * c + b * d = (a + b - c + d) * (-a + b + c + d)\nh0 : Prime (a * b + c * d)\nha : 0 < a\nhb : 0 < b\nhc : 0 < c\ndvd_mul : a * c + b * d ∣ (a * b + c * d) * (a * d + b * c)\nh2 : a * c + b * d ∣ a * d + b * c\naux : 0 < a * d + b * c\nthis : a * c + b * d ≤ a * d + b * c\n⊢ False",
"after_state": "No Goals!"
},
{
"line": "ring1",
"before_state": "a b c d : ℤ\nhd : 0 < d\nhdc : d < c\nhcb : c < b\nhba : b < a\nh : a * c + b * d = (a + b - c + d) * (-a + b + c + d)\nh0 : Prime (a * b + c * d)\nha : 0 < a\nhb : 0 < b\nhc : 0 < c\ndvd_mul : a * c + b * d ∣ (a * b + c * d) * (a * d + b * c)\nh2 : a * c + b * d ∣ a * d + b * c\naux : 0 < a * d + b * c\nthis : a * c + b * d ≤ a * d + b * c\n⊢ -1 + (b + 1 - a) + (d + 1 - c) + (a * c + b * d - (a * d + b * c)) + (0 - (b + 1 - a) * (d + 1 - c)) = 0",
"after_state": "No Goals!"
}
] |
theorem key_insight (x y z : ℝ) (hx : x > 0) (hy : y > 0) (hz : z > 0) (h : x * y * z ≥ 1) :
(x ^ 5 - x ^ 2) / (x ^ 5 + y ^ 2 + z ^ 2) ≥ (x ^ 2 - y * z) / (x ^ 2 + y ^ 2 + z ^ 2) := by
have key :
(x ^ 5 - x ^ 2) / (x ^ 5 + y ^ 2 + z ^ 2) -
(x ^ 5 - x ^ 2 * 1) / (x ^ 3 * (x ^ 2 + y ^ 2 + z ^ 2)) =
(x ^ 3 - 1) ^ 2 * x ^ 2 * (y ^ 2 + z ^ 2) /
((x ^ 5 + y ^ 2 + z ^ 2) * (x ^ 3 * (x ^ 2 + y ^ 2 + z ^ 2))) := by
field_simp
ring
have h₅ :
(x ^ 3 - 1) ^ 2 * x ^ 2 * (y ^ 2 + z ^ 2) /
((x ^ 5 + y ^ 2 + z ^ 2) * (x ^ 3 * (x ^ 2 + y ^ 2 + z ^ 2))) ≥ 0 := by positivity
calc
(x ^ 5 - x ^ 2) / (x ^ 5 + y ^ 2 + z ^ 2)
≥ (x ^ 5 - x ^ 2 * 1) / (x ^ 3 * (x ^ 2 + y ^ 2 + z ^ 2)) := by linarith only [key, h₅]
_ ≥ (x ^ 5 - x ^ 2 * (x * y * z)) / (x ^ 3 * (x ^ 2 + y ^ 2 + z ^ 2)) := by gcongr
_ = (x ^ 2 - y * z) / (x ^ 2 + y ^ 2 + z ^ 2) := by field_simp; ring
|
/root/DuelModelResearch/mathlib4/Archive/Imo/Imo2005Q3.lean
|
{
"open": [],
"variables": []
}
|
[
{
"line": "have key :\n (x ^ 5 - x ^ 2) / (x ^ 5 + y ^ 2 + z ^ 2) - (x ^ 5 - x ^ 2 * 1) / (x ^ 3 * (x ^ 2 + y ^ 2 + z ^ 2)) =\n (x ^ 3 - 1) ^ 2 * x ^ 2 * (y ^ 2 + z ^ 2) / ((x ^ 5 + y ^ 2 + z ^ 2) * (x ^ 3 * (x ^ 2 + y ^ 2 + z ^ 2))) :=\n by\n field_simp\n ring",
"before_state": "x y z : ℝ\nhx : x > 0\nhy : y > 0\nhz : z > 0\nh : x * y * z ≥ 1\n⊢ (x ^ 5 - x ^ 2) / (x ^ 5 + y ^ 2 + z ^ 2) ≥ (x ^ 2 - y * z) / (x ^ 2 + y ^ 2 + z ^ 2)",
"after_state": "x y z : ℝ\nhx : x > 0\nhy : y > 0\nhz : z > 0\nh : x * y * z ≥ 1\nkey :\n (x ^ 5 - x ^ 2) / (x ^ 5 + y ^ 2 + z ^ 2) - (x ^ 5 - x ^ 2 * 1) / (x ^ 3 * (x ^ 2 + y ^ 2 + z ^ 2)) =\n (x ^ 3 - 1) ^ 2 * x ^ 2 * (y ^ 2 + z ^ 2) / ((x ^ 5 + y ^ 2 + z ^ 2) * (x ^ 3 * (x ^ 2 + y ^ 2 + z ^ 2)))\n⊢ (x ^ 5 - x ^ 2) / (x ^ 5 + y ^ 2 + z ^ 2) ≥ (x ^ 2 - y * z) / (x ^ 2 + y ^ 2 + z ^ 2)"
},
{
"line": "focus\n refine\n no_implicit_lambda%\n (have key :\n (x ^ 5 - x ^ 2) / (x ^ 5 + y ^ 2 + z ^ 2) - (x ^ 5 - x ^ 2 * 1) / (x ^ 3 * (x ^ 2 + y ^ 2 + z ^ 2)) =\n (x ^ 3 - 1) ^ 2 * x ^ 2 * (y ^ 2 + z ^ 2) / ((x ^ 5 + y ^ 2 + z ^ 2) * (x ^ 3 * (x ^ 2 + y ^ 2 + z ^ 2))) :=\n ?body✝;\n ?_)\n case body✝ =>\n with_annotate_state\"by\"\n ( field_simp\n ring)",
"before_state": "x y z : ℝ\nhx : x > 0\nhy : y > 0\nhz : z > 0\nh : x * y * z ≥ 1\n⊢ (x ^ 5 - x ^ 2) / (x ^ 5 + y ^ 2 + z ^ 2) ≥ (x ^ 2 - y * z) / (x ^ 2 + y ^ 2 + z ^ 2)",
"after_state": "x y z : ℝ\nhx : x > 0\nhy : y > 0\nhz : z > 0\nh : x * y * z ≥ 1\nkey :\n (x ^ 5 - x ^ 2) / (x ^ 5 + y ^ 2 + z ^ 2) - (x ^ 5 - x ^ 2 * 1) / (x ^ 3 * (x ^ 2 + y ^ 2 + z ^ 2)) =\n (x ^ 3 - 1) ^ 2 * x ^ 2 * (y ^ 2 + z ^ 2) / ((x ^ 5 + y ^ 2 + z ^ 2) * (x ^ 3 * (x ^ 2 + y ^ 2 + z ^ 2)))\n⊢ (x ^ 5 - x ^ 2) / (x ^ 5 + y ^ 2 + z ^ 2) ≥ (x ^ 2 - y * z) / (x ^ 2 + y ^ 2 + z ^ 2)"
},
{
"line": "refine\n no_implicit_lambda%\n (have key :\n (x ^ 5 - x ^ 2) / (x ^ 5 + y ^ 2 + z ^ 2) - (x ^ 5 - x ^ 2 * 1) / (x ^ 3 * (x ^ 2 + y ^ 2 + z ^ 2)) =\n (x ^ 3 - 1) ^ 2 * x ^ 2 * (y ^ 2 + z ^ 2) / ((x ^ 5 + y ^ 2 + z ^ 2) * (x ^ 3 * (x ^ 2 + y ^ 2 + z ^ 2))) :=\n ?body✝;\n ?_)",
"before_state": "x y z : ℝ\nhx : x > 0\nhy : y > 0\nhz : z > 0\nh : x * y * z ≥ 1\n⊢ (x ^ 5 - x ^ 2) / (x ^ 5 + y ^ 2 + z ^ 2) ≥ (x ^ 2 - y * z) / (x ^ 2 + y ^ 2 + z ^ 2)",
"after_state": "case body\nx y z : ℝ\nhx : x > 0\nhy : y > 0\nhz : z > 0\nh : x * y * z ≥ 1\n⊢ (x ^ 5 - x ^ 2) / (x ^ 5 + y ^ 2 + z ^ 2) - (x ^ 5 - x ^ 2 * 1) / (x ^ 3 * (x ^ 2 + y ^ 2 + z ^ 2)) =\n (x ^ 3 - 1) ^ 2 * x ^ 2 * (y ^ 2 + z ^ 2) / ((x ^ 5 + y ^ 2 + z ^ 2) * (x ^ 3 * (x ^ 2 + y ^ 2 + z ^ 2)))\n---\nx y z : ℝ\nhx : x > 0\nhy : y > 0\nhz : z > 0\nh : x * y * z ≥ 1\nkey :\n (x ^ 5 - x ^ 2) / (x ^ 5 + y ^ 2 + z ^ 2) - (x ^ 5 - x ^ 2 * 1) / (x ^ 3 * (x ^ 2 + y ^ 2 + z ^ 2)) =\n (x ^ 3 - 1) ^ 2 * x ^ 2 * (y ^ 2 + z ^ 2) / ((x ^ 5 + y ^ 2 + z ^ 2) * (x ^ 3 * (x ^ 2 + y ^ 2 + z ^ 2)))\n⊢ (x ^ 5 - x ^ 2) / (x ^ 5 + y ^ 2 + z ^ 2) ≥ (x ^ 2 - y * z) / (x ^ 2 + y ^ 2 + z ^ 2)"
},
{
"line": "case body✝ =>\n with_annotate_state\"by\"\n ( field_simp\n ring)",
"before_state": "case body\nx y z : ℝ\nhx : x > 0\nhy : y > 0\nhz : z > 0\nh : x * y * z ≥ 1\n⊢ (x ^ 5 - x ^ 2) / (x ^ 5 + y ^ 2 + z ^ 2) - (x ^ 5 - x ^ 2 * 1) / (x ^ 3 * (x ^ 2 + y ^ 2 + z ^ 2)) =\n (x ^ 3 - 1) ^ 2 * x ^ 2 * (y ^ 2 + z ^ 2) / ((x ^ 5 + y ^ 2 + z ^ 2) * (x ^ 3 * (x ^ 2 + y ^ 2 + z ^ 2)))\n---\nx y z : ℝ\nhx : x > 0\nhy : y > 0\nhz : z > 0\nh : x * y * z ≥ 1\nkey :\n (x ^ 5 - x ^ 2) / (x ^ 5 + y ^ 2 + z ^ 2) - (x ^ 5 - x ^ 2 * 1) / (x ^ 3 * (x ^ 2 + y ^ 2 + z ^ 2)) =\n (x ^ 3 - 1) ^ 2 * x ^ 2 * (y ^ 2 + z ^ 2) / ((x ^ 5 + y ^ 2 + z ^ 2) * (x ^ 3 * (x ^ 2 + y ^ 2 + z ^ 2)))\n⊢ (x ^ 5 - x ^ 2) / (x ^ 5 + y ^ 2 + z ^ 2) ≥ (x ^ 2 - y * z) / (x ^ 2 + y ^ 2 + z ^ 2)",
"after_state": "x y z : ℝ\nhx : x > 0\nhy : y > 0\nhz : z > 0\nh : x * y * z ≥ 1\nkey :\n (x ^ 5 - x ^ 2) / (x ^ 5 + y ^ 2 + z ^ 2) - (x ^ 5 - x ^ 2 * 1) / (x ^ 3 * (x ^ 2 + y ^ 2 + z ^ 2)) =\n (x ^ 3 - 1) ^ 2 * x ^ 2 * (y ^ 2 + z ^ 2) / ((x ^ 5 + y ^ 2 + z ^ 2) * (x ^ 3 * (x ^ 2 + y ^ 2 + z ^ 2)))\n⊢ (x ^ 5 - x ^ 2) / (x ^ 5 + y ^ 2 + z ^ 2) ≥ (x ^ 2 - y * z) / (x ^ 2 + y ^ 2 + z ^ 2)"
},
{
"line": "with_annotate_state\"by\"\n ( field_simp\n ring)",
"before_state": "x y z : ℝ\nhx : x > 0\nhy : y > 0\nhz : z > 0\nh : x * y * z ≥ 1\n⊢ (x ^ 5 - x ^ 2) / (x ^ 5 + y ^ 2 + z ^ 2) - (x ^ 5 - x ^ 2 * 1) / (x ^ 3 * (x ^ 2 + y ^ 2 + z ^ 2)) =\n (x ^ 3 - 1) ^ 2 * x ^ 2 * (y ^ 2 + z ^ 2) / ((x ^ 5 + y ^ 2 + z ^ 2) * (x ^ 3 * (x ^ 2 + y ^ 2 + z ^ 2)))",
"after_state": "No Goals!"
},
{
"line": "field_simp",
"before_state": "x y z : ℝ\nhx : x > 0\nhy : y > 0\nhz : z > 0\nh : x * y * z ≥ 1\n⊢ (x ^ 5 - x ^ 2) / (x ^ 5 + y ^ 2 + z ^ 2) - (x ^ 5 - x ^ 2 * 1) / (x ^ 3 * (x ^ 2 + y ^ 2 + z ^ 2)) =\n (x ^ 3 - 1) ^ 2 * x ^ 2 * (y ^ 2 + z ^ 2) / ((x ^ 5 + y ^ 2 + z ^ 2) * (x ^ 3 * (x ^ 2 + y ^ 2 + z ^ 2)))",
"after_state": "x y z : ℝ\nhx : x > 0\nhy : y > 0\nhz : z > 0\nh : x * y * z ≥ 1\n⊢ (x ^ 5 - x ^ 2) * (x ^ 3 * (x ^ 2 + y ^ 2 + z ^ 2)) - (x ^ 5 + y ^ 2 + z ^ 2) * (x ^ 5 - x ^ 2) =\n (x ^ 3 - 1) ^ 2 * x ^ 2 * (y ^ 2 + z ^ 2)"
},
{
"line": "ring",
"before_state": "x y z : ℝ\nhx : x > 0\nhy : y > 0\nhz : z > 0\nh : x * y * z ≥ 1\n⊢ (x ^ 5 - x ^ 2) * (x ^ 3 * (x ^ 2 + y ^ 2 + z ^ 2)) - (x ^ 5 + y ^ 2 + z ^ 2) * (x ^ 5 - x ^ 2) =\n (x ^ 3 - 1) ^ 2 * x ^ 2 * (y ^ 2 + z ^ 2)",
"after_state": "No Goals!"
},
{
"line": "first\n| ring1\n|\n try_this ring_nf\"\\n\\nThe `ring` tactic failed to close the goal. Use `ring_nf` to obtain a normal form.\n \\nNote that `ring` works primarily in *commutative* rings. \\\n If you have a noncommutative ring, abelian group or module, consider using \\\n `noncomm_ring`, `abel` or `module` instead.\"",
"before_state": "x y z : ℝ\nhx : x > 0\nhy : y > 0\nhz : z > 0\nh : x * y * z ≥ 1\n⊢ (x ^ 5 - x ^ 2) * (x ^ 3 * (x ^ 2 + y ^ 2 + z ^ 2)) - (x ^ 5 + y ^ 2 + z ^ 2) * (x ^ 5 - x ^ 2) =\n (x ^ 3 - 1) ^ 2 * x ^ 2 * (y ^ 2 + z ^ 2)",
"after_state": "No Goals!"
},
{
"line": "ring1",
"before_state": "x y z : ℝ\nhx : x > 0\nhy : y > 0\nhz : z > 0\nh : x * y * z ≥ 1\n⊢ (x ^ 5 - x ^ 2) * (x ^ 3 * (x ^ 2 + y ^ 2 + z ^ 2)) - (x ^ 5 + y ^ 2 + z ^ 2) * (x ^ 5 - x ^ 2) =\n (x ^ 3 - 1) ^ 2 * x ^ 2 * (y ^ 2 + z ^ 2)",
"after_state": "No Goals!"
},
{
"line": "have h₅ :\n (x ^ 3 - 1) ^ 2 * x ^ 2 * (y ^ 2 + z ^ 2) / ((x ^ 5 + y ^ 2 + z ^ 2) * (x ^ 3 * (x ^ 2 + y ^ 2 + z ^ 2))) ≥ 0 := by\n positivity",
"before_state": "x y z : ℝ\nhx : x > 0\nhy : y > 0\nhz : z > 0\nh : x * y * z ≥ 1\nkey :\n (x ^ 5 - x ^ 2) / (x ^ 5 + y ^ 2 + z ^ 2) - (x ^ 5 - x ^ 2 * 1) / (x ^ 3 * (x ^ 2 + y ^ 2 + z ^ 2)) =\n (x ^ 3 - 1) ^ 2 * x ^ 2 * (y ^ 2 + z ^ 2) / ((x ^ 5 + y ^ 2 + z ^ 2) * (x ^ 3 * (x ^ 2 + y ^ 2 + z ^ 2)))\n⊢ (x ^ 5 - x ^ 2) / (x ^ 5 + y ^ 2 + z ^ 2) ≥ (x ^ 2 - y * z) / (x ^ 2 + y ^ 2 + z ^ 2)",
"after_state": "x y z : ℝ\nhx : x > 0\nhy : y > 0\nhz : z > 0\nh : x * y * z ≥ 1\nkey :\n (x ^ 5 - x ^ 2) / (x ^ 5 + y ^ 2 + z ^ 2) - (x ^ 5 - x ^ 2 * 1) / (x ^ 3 * (x ^ 2 + y ^ 2 + z ^ 2)) =\n (x ^ 3 - 1) ^ 2 * x ^ 2 * (y ^ 2 + z ^ 2) / ((x ^ 5 + y ^ 2 + z ^ 2) * (x ^ 3 * (x ^ 2 + y ^ 2 + z ^ 2)))\nh₅ : (x ^ 3 - 1) ^ 2 * x ^ 2 * (y ^ 2 + z ^ 2) / ((x ^ 5 + y ^ 2 + z ^ 2) * (x ^ 3 * (x ^ 2 + y ^ 2 + z ^ 2))) ≥ 0\n⊢ (x ^ 5 - x ^ 2) / (x ^ 5 + y ^ 2 + z ^ 2) ≥ (x ^ 2 - y * z) / (x ^ 2 + y ^ 2 + z ^ 2)"
},
{
"line": "focus\n refine\n no_implicit_lambda%\n (have h₅ :\n (x ^ 3 - 1) ^ 2 * x ^ 2 * (y ^ 2 + z ^ 2) / ((x ^ 5 + y ^ 2 + z ^ 2) * (x ^ 3 * (x ^ 2 + y ^ 2 + z ^ 2))) ≥ 0 :=\n ?body✝;\n ?_)\n case body✝ => with_annotate_state\"by\" (positivity)",
"before_state": "x y z : ℝ\nhx : x > 0\nhy : y > 0\nhz : z > 0\nh : x * y * z ≥ 1\nkey :\n (x ^ 5 - x ^ 2) / (x ^ 5 + y ^ 2 + z ^ 2) - (x ^ 5 - x ^ 2 * 1) / (x ^ 3 * (x ^ 2 + y ^ 2 + z ^ 2)) =\n (x ^ 3 - 1) ^ 2 * x ^ 2 * (y ^ 2 + z ^ 2) / ((x ^ 5 + y ^ 2 + z ^ 2) * (x ^ 3 * (x ^ 2 + y ^ 2 + z ^ 2)))\n⊢ (x ^ 5 - x ^ 2) / (x ^ 5 + y ^ 2 + z ^ 2) ≥ (x ^ 2 - y * z) / (x ^ 2 + y ^ 2 + z ^ 2)",
"after_state": "x y z : ℝ\nhx : x > 0\nhy : y > 0\nhz : z > 0\nh : x * y * z ≥ 1\nkey :\n (x ^ 5 - x ^ 2) / (x ^ 5 + y ^ 2 + z ^ 2) - (x ^ 5 - x ^ 2 * 1) / (x ^ 3 * (x ^ 2 + y ^ 2 + z ^ 2)) =\n (x ^ 3 - 1) ^ 2 * x ^ 2 * (y ^ 2 + z ^ 2) / ((x ^ 5 + y ^ 2 + z ^ 2) * (x ^ 3 * (x ^ 2 + y ^ 2 + z ^ 2)))\nh₅ : (x ^ 3 - 1) ^ 2 * x ^ 2 * (y ^ 2 + z ^ 2) / ((x ^ 5 + y ^ 2 + z ^ 2) * (x ^ 3 * (x ^ 2 + y ^ 2 + z ^ 2))) ≥ 0\n⊢ (x ^ 5 - x ^ 2) / (x ^ 5 + y ^ 2 + z ^ 2) ≥ (x ^ 2 - y * z) / (x ^ 2 + y ^ 2 + z ^ 2)"
},
{
"line": "refine\n no_implicit_lambda%\n (have h₅ :\n (x ^ 3 - 1) ^ 2 * x ^ 2 * (y ^ 2 + z ^ 2) / ((x ^ 5 + y ^ 2 + z ^ 2) * (x ^ 3 * (x ^ 2 + y ^ 2 + z ^ 2))) ≥ 0 :=\n ?body✝;\n ?_)",
"before_state": "x y z : ℝ\nhx : x > 0\nhy : y > 0\nhz : z > 0\nh : x * y * z ≥ 1\nkey :\n (x ^ 5 - x ^ 2) / (x ^ 5 + y ^ 2 + z ^ 2) - (x ^ 5 - x ^ 2 * 1) / (x ^ 3 * (x ^ 2 + y ^ 2 + z ^ 2)) =\n (x ^ 3 - 1) ^ 2 * x ^ 2 * (y ^ 2 + z ^ 2) / ((x ^ 5 + y ^ 2 + z ^ 2) * (x ^ 3 * (x ^ 2 + y ^ 2 + z ^ 2)))\n⊢ (x ^ 5 - x ^ 2) / (x ^ 5 + y ^ 2 + z ^ 2) ≥ (x ^ 2 - y * z) / (x ^ 2 + y ^ 2 + z ^ 2)",
"after_state": "case body\nx y z : ℝ\nhx : x > 0\nhy : y > 0\nhz : z > 0\nh : x * y * z ≥ 1\nkey :\n (x ^ 5 - x ^ 2) / (x ^ 5 + y ^ 2 + z ^ 2) - (x ^ 5 - x ^ 2 * 1) / (x ^ 3 * (x ^ 2 + y ^ 2 + z ^ 2)) =\n (x ^ 3 - 1) ^ 2 * x ^ 2 * (y ^ 2 + z ^ 2) / ((x ^ 5 + y ^ 2 + z ^ 2) * (x ^ 3 * (x ^ 2 + y ^ 2 + z ^ 2)))\n⊢ (x ^ 3 - 1) ^ 2 * x ^ 2 * (y ^ 2 + z ^ 2) / ((x ^ 5 + y ^ 2 + z ^ 2) * (x ^ 3 * (x ^ 2 + y ^ 2 + z ^ 2))) ≥ 0\n---\nx y z : ℝ\nhx : x > 0\nhy : y > 0\nhz : z > 0\nh : x * y * z ≥ 1\nkey :\n (x ^ 5 - x ^ 2) / (x ^ 5 + y ^ 2 + z ^ 2) - (x ^ 5 - x ^ 2 * 1) / (x ^ 3 * (x ^ 2 + y ^ 2 + z ^ 2)) =\n (x ^ 3 - 1) ^ 2 * x ^ 2 * (y ^ 2 + z ^ 2) / ((x ^ 5 + y ^ 2 + z ^ 2) * (x ^ 3 * (x ^ 2 + y ^ 2 + z ^ 2)))\nh₅ : (x ^ 3 - 1) ^ 2 * x ^ 2 * (y ^ 2 + z ^ 2) / ((x ^ 5 + y ^ 2 + z ^ 2) * (x ^ 3 * (x ^ 2 + y ^ 2 + z ^ 2))) ≥ 0\n⊢ (x ^ 5 - x ^ 2) / (x ^ 5 + y ^ 2 + z ^ 2) ≥ (x ^ 2 - y * z) / (x ^ 2 + y ^ 2 + z ^ 2)"
},
{
"line": "case body✝ => with_annotate_state\"by\" (positivity)",
"before_state": "case body\nx y z : ℝ\nhx : x > 0\nhy : y > 0\nhz : z > 0\nh : x * y * z ≥ 1\nkey :\n (x ^ 5 - x ^ 2) / (x ^ 5 + y ^ 2 + z ^ 2) - (x ^ 5 - x ^ 2 * 1) / (x ^ 3 * (x ^ 2 + y ^ 2 + z ^ 2)) =\n (x ^ 3 - 1) ^ 2 * x ^ 2 * (y ^ 2 + z ^ 2) / ((x ^ 5 + y ^ 2 + z ^ 2) * (x ^ 3 * (x ^ 2 + y ^ 2 + z ^ 2)))\n⊢ (x ^ 3 - 1) ^ 2 * x ^ 2 * (y ^ 2 + z ^ 2) / ((x ^ 5 + y ^ 2 + z ^ 2) * (x ^ 3 * (x ^ 2 + y ^ 2 + z ^ 2))) ≥ 0\n---\nx y z : ℝ\nhx : x > 0\nhy : y > 0\nhz : z > 0\nh : x * y * z ≥ 1\nkey :\n (x ^ 5 - x ^ 2) / (x ^ 5 + y ^ 2 + z ^ 2) - (x ^ 5 - x ^ 2 * 1) / (x ^ 3 * (x ^ 2 + y ^ 2 + z ^ 2)) =\n (x ^ 3 - 1) ^ 2 * x ^ 2 * (y ^ 2 + z ^ 2) / ((x ^ 5 + y ^ 2 + z ^ 2) * (x ^ 3 * (x ^ 2 + y ^ 2 + z ^ 2)))\nh₅ : (x ^ 3 - 1) ^ 2 * x ^ 2 * (y ^ 2 + z ^ 2) / ((x ^ 5 + y ^ 2 + z ^ 2) * (x ^ 3 * (x ^ 2 + y ^ 2 + z ^ 2))) ≥ 0\n⊢ (x ^ 5 - x ^ 2) / (x ^ 5 + y ^ 2 + z ^ 2) ≥ (x ^ 2 - y * z) / (x ^ 2 + y ^ 2 + z ^ 2)",
"after_state": "x y z : ℝ\nhx : x > 0\nhy : y > 0\nhz : z > 0\nh : x * y * z ≥ 1\nkey :\n (x ^ 5 - x ^ 2) / (x ^ 5 + y ^ 2 + z ^ 2) - (x ^ 5 - x ^ 2 * 1) / (x ^ 3 * (x ^ 2 + y ^ 2 + z ^ 2)) =\n (x ^ 3 - 1) ^ 2 * x ^ 2 * (y ^ 2 + z ^ 2) / ((x ^ 5 + y ^ 2 + z ^ 2) * (x ^ 3 * (x ^ 2 + y ^ 2 + z ^ 2)))\nh₅ : (x ^ 3 - 1) ^ 2 * x ^ 2 * (y ^ 2 + z ^ 2) / ((x ^ 5 + y ^ 2 + z ^ 2) * (x ^ 3 * (x ^ 2 + y ^ 2 + z ^ 2))) ≥ 0\n⊢ (x ^ 5 - x ^ 2) / (x ^ 5 + y ^ 2 + z ^ 2) ≥ (x ^ 2 - y * z) / (x ^ 2 + y ^ 2 + z ^ 2)"
},
{
"line": "with_annotate_state\"by\" (positivity)",
"before_state": "x y z : ℝ\nhx : x > 0\nhy : y > 0\nhz : z > 0\nh : x * y * z ≥ 1\nkey :\n (x ^ 5 - x ^ 2) / (x ^ 5 + y ^ 2 + z ^ 2) - (x ^ 5 - x ^ 2 * 1) / (x ^ 3 * (x ^ 2 + y ^ 2 + z ^ 2)) =\n (x ^ 3 - 1) ^ 2 * x ^ 2 * (y ^ 2 + z ^ 2) / ((x ^ 5 + y ^ 2 + z ^ 2) * (x ^ 3 * (x ^ 2 + y ^ 2 + z ^ 2)))\n⊢ (x ^ 3 - 1) ^ 2 * x ^ 2 * (y ^ 2 + z ^ 2) / ((x ^ 5 + y ^ 2 + z ^ 2) * (x ^ 3 * (x ^ 2 + y ^ 2 + z ^ 2))) ≥ 0",
"after_state": "No Goals!"
},
{
"line": "positivity",
"before_state": "x y z : ℝ\nhx : x > 0\nhy : y > 0\nhz : z > 0\nh : x * y * z ≥ 1\nkey :\n (x ^ 5 - x ^ 2) / (x ^ 5 + y ^ 2 + z ^ 2) - (x ^ 5 - x ^ 2 * 1) / (x ^ 3 * (x ^ 2 + y ^ 2 + z ^ 2)) =\n (x ^ 3 - 1) ^ 2 * x ^ 2 * (y ^ 2 + z ^ 2) / ((x ^ 5 + y ^ 2 + z ^ 2) * (x ^ 3 * (x ^ 2 + y ^ 2 + z ^ 2)))\n⊢ (x ^ 3 - 1) ^ 2 * x ^ 2 * (y ^ 2 + z ^ 2) / ((x ^ 5 + y ^ 2 + z ^ 2) * (x ^ 3 * (x ^ 2 + y ^ 2 + z ^ 2))) ≥ 0",
"after_state": "No Goals!"
},
{
"line": "calc\n (x ^ 5 - x ^ 2) / (x ^ 5 + y ^ 2 + z ^ 2) ≥ (x ^ 5 - x ^ 2 * 1) / (x ^ 3 * (x ^ 2 + y ^ 2 + z ^ 2)) := by\n linarith only [key, h₅]\n _ ≥ (x ^ 5 - x ^ 2 * (x * y * z)) / (x ^ 3 * (x ^ 2 + y ^ 2 + z ^ 2)) := by gcongr\n _ = (x ^ 2 - y * z) / (x ^ 2 + y ^ 2 + z ^ 2) := by field_simp; ring",
"before_state": "x y z : ℝ\nhx : x > 0\nhy : y > 0\nhz : z > 0\nh : x * y * z ≥ 1\nkey :\n (x ^ 5 - x ^ 2) / (x ^ 5 + y ^ 2 + z ^ 2) - (x ^ 5 - x ^ 2 * 1) / (x ^ 3 * (x ^ 2 + y ^ 2 + z ^ 2)) =\n (x ^ 3 - 1) ^ 2 * x ^ 2 * (y ^ 2 + z ^ 2) / ((x ^ 5 + y ^ 2 + z ^ 2) * (x ^ 3 * (x ^ 2 + y ^ 2 + z ^ 2)))\nh₅ : (x ^ 3 - 1) ^ 2 * x ^ 2 * (y ^ 2 + z ^ 2) / ((x ^ 5 + y ^ 2 + z ^ 2) * (x ^ 3 * (x ^ 2 + y ^ 2 + z ^ 2))) ≥ 0\n⊢ (x ^ 5 - x ^ 2) / (x ^ 5 + y ^ 2 + z ^ 2) ≥ (x ^ 2 - y * z) / (x ^ 2 + y ^ 2 + z ^ 2)",
"after_state": "No Goals!"
},
{
"line": "linarith only [key, h₅]",
"before_state": "x y z : ℝ\nhx : x > 0\nhy : y > 0\nhz : z > 0\nh : x * y * z ≥ 1\nkey :\n (x ^ 5 - x ^ 2) / (x ^ 5 + y ^ 2 + z ^ 2) - (x ^ 5 - x ^ 2 * 1) / (x ^ 3 * (x ^ 2 + y ^ 2 + z ^ 2)) =\n (x ^ 3 - 1) ^ 2 * x ^ 2 * (y ^ 2 + z ^ 2) / ((x ^ 5 + y ^ 2 + z ^ 2) * (x ^ 3 * (x ^ 2 + y ^ 2 + z ^ 2)))\nh₅ : (x ^ 3 - 1) ^ 2 * x ^ 2 * (y ^ 2 + z ^ 2) / ((x ^ 5 + y ^ 2 + z ^ 2) * (x ^ 3 * (x ^ 2 + y ^ 2 + z ^ 2))) ≥ 0\n⊢ (x ^ 5 - x ^ 2) / (x ^ 5 + y ^ 2 + z ^ 2) ≥ (x ^ 5 - x ^ 2 * 1) / (x ^ 3 * (x ^ 2 + y ^ 2 + z ^ 2))",
"after_state": "No Goals!"
},
{
"line": "ring1",
"before_state": "x y z : ℝ\nhx : x > 0\nhy : y > 0\nhz : z > 0\nh : x * y * z ≥ 1\nkey :\n (x ^ 5 - x ^ 2) / (x ^ 5 + y ^ 2 + z ^ 2) - (x ^ 5 - x ^ 2 * 1) / (x ^ 3 * (x ^ 2 + y ^ 2 + z ^ 2)) =\n (x ^ 3 - 1) ^ 2 * x ^ 2 * (y ^ 2 + z ^ 2) / ((x ^ 5 + y ^ 2 + z ^ 2) * (x ^ 3 * (x ^ 2 + y ^ 2 + z ^ 2)))\nh₅ : (x ^ 3 - 1) ^ 2 * x ^ 2 * (y ^ 2 + z ^ 2) / ((x ^ 5 + y ^ 2 + z ^ 2) * (x ^ 3 * (x ^ 2 + y ^ 2 + z ^ 2))) ≥ 0\na✝ : (x ^ 5 - x ^ 2 * 1) / (x ^ 3 * (x ^ 2 + y ^ 2 + z ^ 2)) > (x ^ 5 - x ^ 2) / (x ^ 5 + y ^ 2 + z ^ 2)\n⊢ 1 * ((x ^ 5 - x ^ 2) / (x ^ 5 + y ^ 2 + z ^ 2)) - 1 * ((x ^ 5 - x ^ 2 * 1) / (x ^ 3 * (x ^ 2 + y ^ 2 + z ^ 2))) +\n -(1 * ((x ^ 5 - x ^ 2) / (x ^ 5 + y ^ 2 + z ^ 2)) -\n 1 * ((x ^ 5 - x ^ 2 * 1) / (x ^ 3 * (x ^ 2 + y ^ 2 + z ^ 2))) -\n 1 *\n ((x ^ 3 - 1) ^ 2 * x ^ 2 * (y ^ 2 + z ^ 2) /\n ((x ^ 5 + y ^ 2 + z ^ 2) * (x ^ 3 * (x ^ 2 + y ^ 2 + z ^ 2))))) +\n (1 * 0 -\n 1 *\n ((x ^ 3 - 1) ^ 2 * x ^ 2 * (y ^ 2 + z ^ 2) / ((x ^ 5 + y ^ 2 + z ^ 2) * (x ^ 3 * (x ^ 2 + y ^ 2 + z ^ 2))))) =\n 0",
"after_state": "No Goals!"
},
{
"line": "gcongr",
"before_state": "x y z : ℝ\nhx : x > 0\nhy : y > 0\nhz : z > 0\nh : x * y * z ≥ 1\nkey :\n (x ^ 5 - x ^ 2) / (x ^ 5 + y ^ 2 + z ^ 2) - (x ^ 5 - x ^ 2 * 1) / (x ^ 3 * (x ^ 2 + y ^ 2 + z ^ 2)) =\n (x ^ 3 - 1) ^ 2 * x ^ 2 * (y ^ 2 + z ^ 2) / ((x ^ 5 + y ^ 2 + z ^ 2) * (x ^ 3 * (x ^ 2 + y ^ 2 + z ^ 2)))\nh₅ : (x ^ 3 - 1) ^ 2 * x ^ 2 * (y ^ 2 + z ^ 2) / ((x ^ 5 + y ^ 2 + z ^ 2) * (x ^ 3 * (x ^ 2 + y ^ 2 + z ^ 2))) ≥ 0\n⊢ (x ^ 5 - x ^ 2 * 1) / (x ^ 3 * (x ^ 2 + y ^ 2 + z ^ 2)) ≥\n (x ^ 5 - x ^ 2 * (x * y * z)) / (x ^ 3 * (x ^ 2 + y ^ 2 + z ^ 2))",
"after_state": "No Goals!"
},
{
"line": "gcongr_discharger",
"before_state": "case hab.h.a0\nx y z : ℝ\nhx : x > 0\nhy : y > 0\nhz : z > 0\nh : x * y * z ≥ 1\nkey :\n (x ^ 5 - x ^ 2) / (x ^ 5 + y ^ 2 + z ^ 2) - (x ^ 5 - x ^ 2 * 1) / (x ^ 3 * (x ^ 2 + y ^ 2 + z ^ 2)) =\n (x ^ 3 - 1) ^ 2 * x ^ 2 * (y ^ 2 + z ^ 2) / ((x ^ 5 + y ^ 2 + z ^ 2) * (x ^ 3 * (x ^ 2 + y ^ 2 + z ^ 2)))\nh₅ : (x ^ 3 - 1) ^ 2 * x ^ 2 * (y ^ 2 + z ^ 2) / ((x ^ 5 + y ^ 2 + z ^ 2) * (x ^ 3 * (x ^ 2 + y ^ 2 + z ^ 2))) ≥ 0\n⊢ 0 ≤ x ^ 2",
"after_state": "No Goals!"
},
{
"line": "positivity",
"before_state": "case hab.h.a0\nx y z : ℝ\nhx : x > 0\nhy : y > 0\nhz : z > 0\nh : x * y * z ≥ 1\nkey :\n (x ^ 5 - x ^ 2) / (x ^ 5 + y ^ 2 + z ^ 2) - (x ^ 5 - x ^ 2 * 1) / (x ^ 3 * (x ^ 2 + y ^ 2 + z ^ 2)) =\n (x ^ 3 - 1) ^ 2 * x ^ 2 * (y ^ 2 + z ^ 2) / ((x ^ 5 + y ^ 2 + z ^ 2) * (x ^ 3 * (x ^ 2 + y ^ 2 + z ^ 2)))\nh₅ : (x ^ 3 - 1) ^ 2 * x ^ 2 * (y ^ 2 + z ^ 2) / ((x ^ 5 + y ^ 2 + z ^ 2) * (x ^ 3 * (x ^ 2 + y ^ 2 + z ^ 2))) ≥ 0\n⊢ 0 ≤ x ^ 2",
"after_state": "No Goals!"
},
{
"line": "gcongr_discharger",
"before_state": "case hc\nx y z : ℝ\nhx : x > 0\nhy : y > 0\nhz : z > 0\nh : x * y * z ≥ 1\nkey :\n (x ^ 5 - x ^ 2) / (x ^ 5 + y ^ 2 + z ^ 2) - (x ^ 5 - x ^ 2 * 1) / (x ^ 3 * (x ^ 2 + y ^ 2 + z ^ 2)) =\n (x ^ 3 - 1) ^ 2 * x ^ 2 * (y ^ 2 + z ^ 2) / ((x ^ 5 + y ^ 2 + z ^ 2) * (x ^ 3 * (x ^ 2 + y ^ 2 + z ^ 2)))\nh₅ : (x ^ 3 - 1) ^ 2 * x ^ 2 * (y ^ 2 + z ^ 2) / ((x ^ 5 + y ^ 2 + z ^ 2) * (x ^ 3 * (x ^ 2 + y ^ 2 + z ^ 2))) ≥ 0\n⊢ 0 ≤ x ^ 3 * (x ^ 2 + y ^ 2 + z ^ 2)",
"after_state": "No Goals!"
},
{
"line": "positivity",
"before_state": "case hc\nx y z : ℝ\nhx : x > 0\nhy : y > 0\nhz : z > 0\nh : x * y * z ≥ 1\nkey :\n (x ^ 5 - x ^ 2) / (x ^ 5 + y ^ 2 + z ^ 2) - (x ^ 5 - x ^ 2 * 1) / (x ^ 3 * (x ^ 2 + y ^ 2 + z ^ 2)) =\n (x ^ 3 - 1) ^ 2 * x ^ 2 * (y ^ 2 + z ^ 2) / ((x ^ 5 + y ^ 2 + z ^ 2) * (x ^ 3 * (x ^ 2 + y ^ 2 + z ^ 2)))\nh₅ : (x ^ 3 - 1) ^ 2 * x ^ 2 * (y ^ 2 + z ^ 2) / ((x ^ 5 + y ^ 2 + z ^ 2) * (x ^ 3 * (x ^ 2 + y ^ 2 + z ^ 2))) ≥ 0\n⊢ 0 ≤ x ^ 3 * (x ^ 2 + y ^ 2 + z ^ 2)",
"after_state": "No Goals!"
},
{
"line": "field_simp",
"before_state": "x y z : ℝ\nhx : x > 0\nhy : y > 0\nhz : z > 0\nh : x * y * z ≥ 1\nkey :\n (x ^ 5 - x ^ 2) / (x ^ 5 + y ^ 2 + z ^ 2) - (x ^ 5 - x ^ 2 * 1) / (x ^ 3 * (x ^ 2 + y ^ 2 + z ^ 2)) =\n (x ^ 3 - 1) ^ 2 * x ^ 2 * (y ^ 2 + z ^ 2) / ((x ^ 5 + y ^ 2 + z ^ 2) * (x ^ 3 * (x ^ 2 + y ^ 2 + z ^ 2)))\nh₅ : (x ^ 3 - 1) ^ 2 * x ^ 2 * (y ^ 2 + z ^ 2) / ((x ^ 5 + y ^ 2 + z ^ 2) * (x ^ 3 * (x ^ 2 + y ^ 2 + z ^ 2))) ≥ 0\n⊢ (x ^ 5 - x ^ 2 * (x * y * z)) / (x ^ 3 * (x ^ 2 + y ^ 2 + z ^ 2)) = (x ^ 2 - y * z) / (x ^ 2 + y ^ 2 + z ^ 2)",
"after_state": "x y z : ℝ\nhx : x > 0\nhy : y > 0\nhz : z > 0\nh : x * y * z ≥ 1\nkey :\n (x ^ 5 - x ^ 2) / (x ^ 5 + y ^ 2 + z ^ 2) - (x ^ 5 - x ^ 2 * 1) / (x ^ 3 * (x ^ 2 + y ^ 2 + z ^ 2)) =\n (x ^ 3 - 1) ^ 2 * x ^ 2 * (y ^ 2 + z ^ 2) / ((x ^ 5 + y ^ 2 + z ^ 2) * (x ^ 3 * (x ^ 2 + y ^ 2 + z ^ 2)))\nh₅ : (x ^ 3 - 1) ^ 2 * x ^ 2 * (y ^ 2 + z ^ 2) / ((x ^ 5 + y ^ 2 + z ^ 2) * (x ^ 3 * (x ^ 2 + y ^ 2 + z ^ 2))) ≥ 0\n⊢ (x ^ 5 - x ^ 2 * (x * y * z)) * (x ^ 2 + y ^ 2 + z ^ 2) = (x ^ 2 - y * z) * (x ^ 3 * (x ^ 2 + y ^ 2 + z ^ 2))"
},
{
"line": "ring",
"before_state": "x y z : ℝ\nhx : x > 0\nhy : y > 0\nhz : z > 0\nh : x * y * z ≥ 1\nkey :\n (x ^ 5 - x ^ 2) / (x ^ 5 + y ^ 2 + z ^ 2) - (x ^ 5 - x ^ 2 * 1) / (x ^ 3 * (x ^ 2 + y ^ 2 + z ^ 2)) =\n (x ^ 3 - 1) ^ 2 * x ^ 2 * (y ^ 2 + z ^ 2) / ((x ^ 5 + y ^ 2 + z ^ 2) * (x ^ 3 * (x ^ 2 + y ^ 2 + z ^ 2)))\nh₅ : (x ^ 3 - 1) ^ 2 * x ^ 2 * (y ^ 2 + z ^ 2) / ((x ^ 5 + y ^ 2 + z ^ 2) * (x ^ 3 * (x ^ 2 + y ^ 2 + z ^ 2))) ≥ 0\n⊢ (x ^ 5 - x ^ 2 * (x * y * z)) * (x ^ 2 + y ^ 2 + z ^ 2) = (x ^ 2 - y * z) * (x ^ 3 * (x ^ 2 + y ^ 2 + z ^ 2))",
"after_state": "No Goals!"
},
{
"line": "first\n| ring1\n|\n try_this ring_nf\"\\n\\nThe `ring` tactic failed to close the goal. Use `ring_nf` to obtain a normal form.\n \\nNote that `ring` works primarily in *commutative* rings. \\\n If you have a noncommutative ring, abelian group or module, consider using \\\n `noncomm_ring`, `abel` or `module` instead.\"",
"before_state": "x y z : ℝ\nhx : x > 0\nhy : y > 0\nhz : z > 0\nh : x * y * z ≥ 1\nkey :\n (x ^ 5 - x ^ 2) / (x ^ 5 + y ^ 2 + z ^ 2) - (x ^ 5 - x ^ 2 * 1) / (x ^ 3 * (x ^ 2 + y ^ 2 + z ^ 2)) =\n (x ^ 3 - 1) ^ 2 * x ^ 2 * (y ^ 2 + z ^ 2) / ((x ^ 5 + y ^ 2 + z ^ 2) * (x ^ 3 * (x ^ 2 + y ^ 2 + z ^ 2)))\nh₅ : (x ^ 3 - 1) ^ 2 * x ^ 2 * (y ^ 2 + z ^ 2) / ((x ^ 5 + y ^ 2 + z ^ 2) * (x ^ 3 * (x ^ 2 + y ^ 2 + z ^ 2))) ≥ 0\n⊢ (x ^ 5 - x ^ 2 * (x * y * z)) * (x ^ 2 + y ^ 2 + z ^ 2) = (x ^ 2 - y * z) * (x ^ 3 * (x ^ 2 + y ^ 2 + z ^ 2))",
"after_state": "No Goals!"
},
{
"line": "ring1",
"before_state": "x y z : ℝ\nhx : x > 0\nhy : y > 0\nhz : z > 0\nh : x * y * z ≥ 1\nkey :\n (x ^ 5 - x ^ 2) / (x ^ 5 + y ^ 2 + z ^ 2) - (x ^ 5 - x ^ 2 * 1) / (x ^ 3 * (x ^ 2 + y ^ 2 + z ^ 2)) =\n (x ^ 3 - 1) ^ 2 * x ^ 2 * (y ^ 2 + z ^ 2) / ((x ^ 5 + y ^ 2 + z ^ 2) * (x ^ 3 * (x ^ 2 + y ^ 2 + z ^ 2)))\nh₅ : (x ^ 3 - 1) ^ 2 * x ^ 2 * (y ^ 2 + z ^ 2) / ((x ^ 5 + y ^ 2 + z ^ 2) * (x ^ 3 * (x ^ 2 + y ^ 2 + z ^ 2))) ≥ 0\n⊢ (x ^ 5 - x ^ 2 * (x * y * z)) * (x ^ 2 + y ^ 2 + z ^ 2) = (x ^ 2 - y * z) * (x ^ 3 * (x ^ 2 + y ^ 2 + z ^ 2))",
"after_state": "No Goals!"
}
] |
theorem imo2005_q3 (x y z : ℝ) (hx : x > 0) (hy : y > 0) (hz : z > 0) (h : x * y * z ≥ 1) :
(x ^ 5 - x ^ 2) / (x ^ 5 + y ^ 2 + z ^ 2) + (y ^ 5 - y ^ 2) / (y ^ 5 + z ^ 2 + x ^ 2) +
(z ^ 5 - z ^ 2) / (z ^ 5 + x ^ 2 + y ^ 2) ≥
0 := by
calc
(x ^ 5 - x ^ 2) / (x ^ 5 + y ^ 2 + z ^ 2) + (y ^ 5 - y ^ 2) / (y ^ 5 + z ^ 2 + x ^ 2) +
(z ^ 5 - z ^ 2) / (z ^ 5 + x ^ 2 + y ^ 2) ≥
(x ^ 2 - y * z) / (x ^ 2 + y ^ 2 + z ^ 2) + (y ^ 2 - z * x) / (y ^ 2 + z ^ 2 + x ^ 2) +
(z ^ 2 - x * y) / (z ^ 2 + x ^ 2 + y ^ 2) := by
gcongr ?_ + ?_ + ?_ <;> apply key_insight <;> linarith
_ = 1 / 2 * ((x - y) ^ 2 + (y - z) ^ 2 + (z - x) ^ 2) / (x ^ 2 + y ^ 2 + z ^ 2) := by ring
_ ≥ 0 := by positivity
|
/root/DuelModelResearch/mathlib4/Archive/Imo/Imo2005Q3.lean
|
{
"open": [
"Imo2005Q3"
],
"variables": []
}
|
[
{
"line": "calc\n (x ^ 5 - x ^ 2) / (x ^ 5 + y ^ 2 + z ^ 2) + (y ^ 5 - y ^ 2) / (y ^ 5 + z ^ 2 + x ^ 2) +\n (z ^ 5 - z ^ 2) / (z ^ 5 + x ^ 2 + y ^ 2) ≥\n (x ^ 2 - y * z) / (x ^ 2 + y ^ 2 + z ^ 2) + (y ^ 2 - z * x) / (y ^ 2 + z ^ 2 + x ^ 2) +\n (z ^ 2 - x * y) / (z ^ 2 + x ^ 2 + y ^ 2) :=\n by gcongr ?_ + ?_ + ?_ <;> apply key_insight <;> linarith\n _ = 1 / 2 * ((x - y) ^ 2 + (y - z) ^ 2 + (z - x) ^ 2) / (x ^ 2 + y ^ 2 + z ^ 2) := by ring\n _ ≥ 0 := by positivity",
"before_state": "x y z : ℝ\nhx : x > 0\nhy : y > 0\nhz : z > 0\nh : x * y * z ≥ 1\n⊢ (x ^ 5 - x ^ 2) / (x ^ 5 + y ^ 2 + z ^ 2) + (y ^ 5 - y ^ 2) / (y ^ 5 + z ^ 2 + x ^ 2) +\n (z ^ 5 - z ^ 2) / (z ^ 5 + x ^ 2 + y ^ 2) ≥\n 0",
"after_state": "No Goals!"
},
{
"line": "focus\n gcongr ?_ + ?_ + ?_ <;> apply key_insight\n with_annotate_state\"<;>\" skip\n all_goals linarith",
"before_state": "x y z : ℝ\nhx : x > 0\nhy : y > 0\nhz : z > 0\nh : x * y * z ≥ 1\n⊢ (x ^ 5 - x ^ 2) / (x ^ 5 + y ^ 2 + z ^ 2) + (y ^ 5 - y ^ 2) / (y ^ 5 + z ^ 2 + x ^ 2) +\n (z ^ 5 - z ^ 2) / (z ^ 5 + x ^ 2 + y ^ 2) ≥\n (x ^ 2 - y * z) / (x ^ 2 + y ^ 2 + z ^ 2) + (y ^ 2 - z * x) / (y ^ 2 + z ^ 2 + x ^ 2) +\n (z ^ 2 - x * y) / (z ^ 2 + x ^ 2 + y ^ 2)",
"after_state": "No Goals!"
},
{
"line": "focus\n gcongr ?_ + ?_ + ?_\n with_annotate_state\"<;>\" skip\n all_goals apply key_insight",
"before_state": "x y z : ℝ\nhx : x > 0\nhy : y > 0\nhz : z > 0\nh : x * y * z ≥ 1\n⊢ (x ^ 5 - x ^ 2) / (x ^ 5 + y ^ 2 + z ^ 2) + (y ^ 5 - y ^ 2) / (y ^ 5 + z ^ 2 + x ^ 2) +\n (z ^ 5 - z ^ 2) / (z ^ 5 + x ^ 2 + y ^ 2) ≥\n (x ^ 2 - y * z) / (x ^ 2 + y ^ 2 + z ^ 2) + (y ^ 2 - z * x) / (y ^ 2 + z ^ 2 + x ^ 2) +\n (z ^ 2 - x * y) / (z ^ 2 + x ^ 2 + y ^ 2)",
"after_state": "No Goals!"
},
{
"line": "gcongr ?_ + ?_ + ?_",
"before_state": "x y z : ℝ\nhx : x > 0\nhy : y > 0\nhz : z > 0\nh : x * y * z ≥ 1\n⊢ (x ^ 5 - x ^ 2) / (x ^ 5 + y ^ 2 + z ^ 2) + (y ^ 5 - y ^ 2) / (y ^ 5 + z ^ 2 + x ^ 2) +\n (z ^ 5 - z ^ 2) / (z ^ 5 + x ^ 2 + y ^ 2) ≥\n (x ^ 2 - y * z) / (x ^ 2 + y ^ 2 + z ^ 2) + (y ^ 2 - z * x) / (y ^ 2 + z ^ 2 + x ^ 2) +\n (z ^ 2 - x * y) / (z ^ 2 + x ^ 2 + y ^ 2)",
"after_state": "case h₁.h₁\nx y z : ℝ\nhx : x > 0\nhy : y > 0\nhz : z > 0\nh : x * y * z ≥ 1\n⊢ (x ^ 2 - y * z) / (x ^ 2 + y ^ 2 + z ^ 2) ≤ (x ^ 5 - x ^ 2) / (x ^ 5 + y ^ 2 + z ^ 2)\n---\ncase h₁.h₂\nx y z : ℝ\nhx : x > 0\nhy : y > 0\nhz : z > 0\nh : x * y * z ≥ 1\n⊢ (y ^ 2 - z * x) / (y ^ 2 + z ^ 2 + x ^ 2) ≤ (y ^ 5 - y ^ 2) / (y ^ 5 + z ^ 2 + x ^ 2)\n---\ncase h₂\nx y z : ℝ\nhx : x > 0\nhy : y > 0\nhz : z > 0\nh : x * y * z ≥ 1\n⊢ (z ^ 2 - x * y) / (z ^ 2 + x ^ 2 + y ^ 2) ≤ (z ^ 5 - z ^ 2) / (z ^ 5 + x ^ 2 + y ^ 2)"
},
{
"line": "with_annotate_state\"<;>\" skip",
"before_state": "case h₁.h₁\nx y z : ℝ\nhx : x > 0\nhy : y > 0\nhz : z > 0\nh : x * y * z ≥ 1\n⊢ (x ^ 2 - y * z) / (x ^ 2 + y ^ 2 + z ^ 2) ≤ (x ^ 5 - x ^ 2) / (x ^ 5 + y ^ 2 + z ^ 2)\n---\ncase h₁.h₂\nx y z : ℝ\nhx : x > 0\nhy : y > 0\nhz : z > 0\nh : x * y * z ≥ 1\n⊢ (y ^ 2 - z * x) / (y ^ 2 + z ^ 2 + x ^ 2) ≤ (y ^ 5 - y ^ 2) / (y ^ 5 + z ^ 2 + x ^ 2)\n---\ncase h₂\nx y z : ℝ\nhx : x > 0\nhy : y > 0\nhz : z > 0\nh : x * y * z ≥ 1\n⊢ (z ^ 2 - x * y) / (z ^ 2 + x ^ 2 + y ^ 2) ≤ (z ^ 5 - z ^ 2) / (z ^ 5 + x ^ 2 + y ^ 2)",
"after_state": "case h₁.h₁\nx y z : ℝ\nhx : x > 0\nhy : y > 0\nhz : z > 0\nh : x * y * z ≥ 1\n⊢ (x ^ 2 - y * z) / (x ^ 2 + y ^ 2 + z ^ 2) ≤ (x ^ 5 - x ^ 2) / (x ^ 5 + y ^ 2 + z ^ 2)\n---\ncase h₁.h₂\nx y z : ℝ\nhx : x > 0\nhy : y > 0\nhz : z > 0\nh : x * y * z ≥ 1\n⊢ (y ^ 2 - z * x) / (y ^ 2 + z ^ 2 + x ^ 2) ≤ (y ^ 5 - y ^ 2) / (y ^ 5 + z ^ 2 + x ^ 2)\n---\ncase h₂\nx y z : ℝ\nhx : x > 0\nhy : y > 0\nhz : z > 0\nh : x * y * z ≥ 1\n⊢ (z ^ 2 - x * y) / (z ^ 2 + x ^ 2 + y ^ 2) ≤ (z ^ 5 - z ^ 2) / (z ^ 5 + x ^ 2 + y ^ 2)"
},
{
"line": "skip",
"before_state": "case h₁.h₁\nx y z : ℝ\nhx : x > 0\nhy : y > 0\nhz : z > 0\nh : x * y * z ≥ 1\n⊢ (x ^ 2 - y * z) / (x ^ 2 + y ^ 2 + z ^ 2) ≤ (x ^ 5 - x ^ 2) / (x ^ 5 + y ^ 2 + z ^ 2)\n---\ncase h₁.h₂\nx y z : ℝ\nhx : x > 0\nhy : y > 0\nhz : z > 0\nh : x * y * z ≥ 1\n⊢ (y ^ 2 - z * x) / (y ^ 2 + z ^ 2 + x ^ 2) ≤ (y ^ 5 - y ^ 2) / (y ^ 5 + z ^ 2 + x ^ 2)\n---\ncase h₂\nx y z : ℝ\nhx : x > 0\nhy : y > 0\nhz : z > 0\nh : x * y * z ≥ 1\n⊢ (z ^ 2 - x * y) / (z ^ 2 + x ^ 2 + y ^ 2) ≤ (z ^ 5 - z ^ 2) / (z ^ 5 + x ^ 2 + y ^ 2)",
"after_state": "case h₁.h₁\nx y z : ℝ\nhx : x > 0\nhy : y > 0\nhz : z > 0\nh : x * y * z ≥ 1\n⊢ (x ^ 2 - y * z) / (x ^ 2 + y ^ 2 + z ^ 2) ≤ (x ^ 5 - x ^ 2) / (x ^ 5 + y ^ 2 + z ^ 2)\n---\ncase h₁.h₂\nx y z : ℝ\nhx : x > 0\nhy : y > 0\nhz : z > 0\nh : x * y * z ≥ 1\n⊢ (y ^ 2 - z * x) / (y ^ 2 + z ^ 2 + x ^ 2) ≤ (y ^ 5 - y ^ 2) / (y ^ 5 + z ^ 2 + x ^ 2)\n---\ncase h₂\nx y z : ℝ\nhx : x > 0\nhy : y > 0\nhz : z > 0\nh : x * y * z ≥ 1\n⊢ (z ^ 2 - x * y) / (z ^ 2 + x ^ 2 + y ^ 2) ≤ (z ^ 5 - z ^ 2) / (z ^ 5 + x ^ 2 + y ^ 2)"
},
{
"line": "all_goals apply key_insight",
"before_state": "case h₁.h₁\nx y z : ℝ\nhx : x > 0\nhy : y > 0\nhz : z > 0\nh : x * y * z ≥ 1\n⊢ (x ^ 2 - y * z) / (x ^ 2 + y ^ 2 + z ^ 2) ≤ (x ^ 5 - x ^ 2) / (x ^ 5 + y ^ 2 + z ^ 2)\n---\ncase h₁.h₂\nx y z : ℝ\nhx : x > 0\nhy : y > 0\nhz : z > 0\nh : x * y * z ≥ 1\n⊢ (y ^ 2 - z * x) / (y ^ 2 + z ^ 2 + x ^ 2) ≤ (y ^ 5 - y ^ 2) / (y ^ 5 + z ^ 2 + x ^ 2)\n---\ncase h₂\nx y z : ℝ\nhx : x > 0\nhy : y > 0\nhz : z > 0\nh : x * y * z ≥ 1\n⊢ (z ^ 2 - x * y) / (z ^ 2 + x ^ 2 + y ^ 2) ≤ (z ^ 5 - z ^ 2) / (z ^ 5 + x ^ 2 + y ^ 2)",
"after_state": "No Goals!"
},
{
"line": "apply key_insight",
"before_state": "case h₁.h₁\nx y z : ℝ\nhx : x > 0\nhy : y > 0\nhz : z > 0\nh : x * y * z ≥ 1\n⊢ (x ^ 2 - y * z) / (x ^ 2 + y ^ 2 + z ^ 2) ≤ (x ^ 5 - x ^ 2) / (x ^ 5 + y ^ 2 + z ^ 2)",
"after_state": "No Goals!"
},
{
"line": "apply key_insight",
"before_state": "case h₁.h₂\nx y z : ℝ\nhx : x > 0\nhy : y > 0\nhz : z > 0\nh : x * y * z ≥ 1\n⊢ (y ^ 2 - z * x) / (y ^ 2 + z ^ 2 + x ^ 2) ≤ (y ^ 5 - y ^ 2) / (y ^ 5 + z ^ 2 + x ^ 2)",
"after_state": "No Goals!"
},
{
"line": "apply key_insight",
"before_state": "case h₂\nx y z : ℝ\nhx : x > 0\nhy : y > 0\nhz : z > 0\nh : x * y * z ≥ 1\n⊢ (z ^ 2 - x * y) / (z ^ 2 + x ^ 2 + y ^ 2) ≤ (z ^ 5 - z ^ 2) / (z ^ 5 + x ^ 2 + y ^ 2)",
"after_state": "No Goals!"
},
{
"line": "ring",
"before_state": "x y z : ℝ\nhx : x > 0\nhy : y > 0\nhz : z > 0\nh : x * y * z ≥ 1\n⊢ (x ^ 2 - y * z) / (x ^ 2 + y ^ 2 + z ^ 2) + (y ^ 2 - z * x) / (y ^ 2 + z ^ 2 + x ^ 2) +\n (z ^ 2 - x * y) / (z ^ 2 + x ^ 2 + y ^ 2) =\n 1 / 2 * ((x - y) ^ 2 + (y - z) ^ 2 + (z - x) ^ 2) / (x ^ 2 + y ^ 2 + z ^ 2)",
"after_state": "No Goals!"
},
{
"line": "first\n| ring1\n|\n try_this ring_nf\"\\n\\nThe `ring` tactic failed to close the goal. Use `ring_nf` to obtain a normal form.\n \\nNote that `ring` works primarily in *commutative* rings. \\\n If you have a noncommutative ring, abelian group or module, consider using \\\n `noncomm_ring`, `abel` or `module` instead.\"",
"before_state": "x y z : ℝ\nhx : x > 0\nhy : y > 0\nhz : z > 0\nh : x * y * z ≥ 1\n⊢ (x ^ 2 - y * z) / (x ^ 2 + y ^ 2 + z ^ 2) + (y ^ 2 - z * x) / (y ^ 2 + z ^ 2 + x ^ 2) +\n (z ^ 2 - x * y) / (z ^ 2 + x ^ 2 + y ^ 2) =\n 1 / 2 * ((x - y) ^ 2 + (y - z) ^ 2 + (z - x) ^ 2) / (x ^ 2 + y ^ 2 + z ^ 2)",
"after_state": "No Goals!"
},
{
"line": "ring1",
"before_state": "x y z : ℝ\nhx : x > 0\nhy : y > 0\nhz : z > 0\nh : x * y * z ≥ 1\n⊢ (x ^ 2 - y * z) / (x ^ 2 + y ^ 2 + z ^ 2) + (y ^ 2 - z * x) / (y ^ 2 + z ^ 2 + x ^ 2) +\n (z ^ 2 - x * y) / (z ^ 2 + x ^ 2 + y ^ 2) =\n 1 / 2 * ((x - y) ^ 2 + (y - z) ^ 2 + (z - x) ^ 2) / (x ^ 2 + y ^ 2 + z ^ 2)",
"after_state": "No Goals!"
},
{
"line": "positivity",
"before_state": "x y z : ℝ\nhx : x > 0\nhy : y > 0\nhz : z > 0\nh : x * y * z ≥ 1\n⊢ 1 / 2 * ((x - y) ^ 2 + (y - z) ^ 2 + (z - x) ^ 2) / (x ^ 2 + y ^ 2 + z ^ 2) ≥ 0",
"after_state": "No Goals!"
}
] |
theorem lhs_ineq {x y : ℝ} (hxy : 0 ≤ x * y) :
16 * x ^ 2 * y ^ 2 * (x + y) ^ 2 ≤ ((x + y) ^ 2) ^ 3 := by
have : (x - y) ^ 2 * ((x + y) ^ 2 + 4 * (x * y)) ≥ 0 := by positivity
calc 16 * x ^ 2 * y ^ 2 * (x + y) ^ 2 ≤ ((x + y) ^ 2) ^ 2 * (x + y) ^ 2 := by gcongr; linarith
_ = ((x + y) ^ 2) ^ 3 := by ring
|
/root/DuelModelResearch/mathlib4/Archive/Imo/Imo2006Q3.lean
|
{
"open": [
"Real"
],
"variables": []
}
|
[
{
"line": "have : (x - y) ^ 2 * ((x + y) ^ 2 + 4 * (x * y)) ≥ 0 := by positivity",
"before_state": "x y : ℝ\nhxy : 0 ≤ x * y\n⊢ 16 * x ^ 2 * y ^ 2 * (x + y) ^ 2 ≤ ((x + y) ^ 2) ^ 3",
"after_state": "x y : ℝ\nhxy : 0 ≤ x * y\nthis : (x - y) ^ 2 * ((x + y) ^ 2 + 4 * (x * y)) ≥ 0\n⊢ 16 * x ^ 2 * y ^ 2 * (x + y) ^ 2 ≤ ((x + y) ^ 2) ^ 3"
},
{
"line": "focus\n refine\n no_implicit_lambda%\n (have : (x - y) ^ 2 * ((x + y) ^ 2 + 4 * (x * y)) ≥ 0 := ?body✝;\n ?_)\n case body✝ => with_annotate_state\"by\" (positivity)",
"before_state": "x y : ℝ\nhxy : 0 ≤ x * y\n⊢ 16 * x ^ 2 * y ^ 2 * (x + y) ^ 2 ≤ ((x + y) ^ 2) ^ 3",
"after_state": "x y : ℝ\nhxy : 0 ≤ x * y\nthis : (x - y) ^ 2 * ((x + y) ^ 2 + 4 * (x * y)) ≥ 0\n⊢ 16 * x ^ 2 * y ^ 2 * (x + y) ^ 2 ≤ ((x + y) ^ 2) ^ 3"
},
{
"line": "refine\n no_implicit_lambda%\n (have : (x - y) ^ 2 * ((x + y) ^ 2 + 4 * (x * y)) ≥ 0 := ?body✝;\n ?_)",
"before_state": "x y : ℝ\nhxy : 0 ≤ x * y\n⊢ 16 * x ^ 2 * y ^ 2 * (x + y) ^ 2 ≤ ((x + y) ^ 2) ^ 3",
"after_state": "case body\nx y : ℝ\nhxy : 0 ≤ x * y\n⊢ (x - y) ^ 2 * ((x + y) ^ 2 + 4 * (x * y)) ≥ 0\n---\nx y : ℝ\nhxy : 0 ≤ x * y\nthis : (x - y) ^ 2 * ((x + y) ^ 2 + 4 * (x * y)) ≥ 0\n⊢ 16 * x ^ 2 * y ^ 2 * (x + y) ^ 2 ≤ ((x + y) ^ 2) ^ 3"
},
{
"line": "case body✝ => with_annotate_state\"by\" (positivity)",
"before_state": "case body\nx y : ℝ\nhxy : 0 ≤ x * y\n⊢ (x - y) ^ 2 * ((x + y) ^ 2 + 4 * (x * y)) ≥ 0\n---\nx y : ℝ\nhxy : 0 ≤ x * y\nthis : (x - y) ^ 2 * ((x + y) ^ 2 + 4 * (x * y)) ≥ 0\n⊢ 16 * x ^ 2 * y ^ 2 * (x + y) ^ 2 ≤ ((x + y) ^ 2) ^ 3",
"after_state": "x y : ℝ\nhxy : 0 ≤ x * y\nthis : (x - y) ^ 2 * ((x + y) ^ 2 + 4 * (x * y)) ≥ 0\n⊢ 16 * x ^ 2 * y ^ 2 * (x + y) ^ 2 ≤ ((x + y) ^ 2) ^ 3"
},
{
"line": "with_annotate_state\"by\" (positivity)",
"before_state": "x y : ℝ\nhxy : 0 ≤ x * y\n⊢ (x - y) ^ 2 * ((x + y) ^ 2 + 4 * (x * y)) ≥ 0",
"after_state": "No Goals!"
},
{
"line": "positivity",
"before_state": "x y : ℝ\nhxy : 0 ≤ x * y\n⊢ (x - y) ^ 2 * ((x + y) ^ 2 + 4 * (x * y)) ≥ 0",
"after_state": "No Goals!"
},
{
"line": "calc\n 16 * x ^ 2 * y ^ 2 * (x + y) ^ 2 ≤ ((x + y) ^ 2) ^ 2 * (x + y) ^ 2 := by gcongr; linarith\n _ = ((x + y) ^ 2) ^ 3 := by ring",
"before_state": "x y : ℝ\nhxy : 0 ≤ x * y\nthis : (x - y) ^ 2 * ((x + y) ^ 2 + 4 * (x * y)) ≥ 0\n⊢ 16 * x ^ 2 * y ^ 2 * (x + y) ^ 2 ≤ ((x + y) ^ 2) ^ 3",
"after_state": "No Goals!"
},
{
"line": "gcongr",
"before_state": "x y : ℝ\nhxy : 0 ≤ x * y\nthis : (x - y) ^ 2 * ((x + y) ^ 2 + 4 * (x * y)) ≥ 0\n⊢ 16 * x ^ 2 * y ^ 2 * (x + y) ^ 2 ≤ ((x + y) ^ 2) ^ 2 * (x + y) ^ 2",
"after_state": "case h\nx y : ℝ\nhxy : 0 ≤ x * y\nthis : (x - y) ^ 2 * ((x + y) ^ 2 + 4 * (x * y)) ≥ 0\n⊢ 16 * x ^ 2 * y ^ 2 ≤ ((x + y) ^ 2) ^ 2"
},
{
"line": "gcongr_discharger",
"before_state": "case a0\nx y : ℝ\nhxy : 0 ≤ x * y\nthis : (x - y) ^ 2 * ((x + y) ^ 2 + 4 * (x * y)) ≥ 0\n⊢ 0 ≤ (x + y) ^ 2",
"after_state": "No Goals!"
},
{
"line": "positivity",
"before_state": "case a0\nx y : ℝ\nhxy : 0 ≤ x * y\nthis : (x - y) ^ 2 * ((x + y) ^ 2 + 4 * (x * y)) ≥ 0\n⊢ 0 ≤ (x + y) ^ 2",
"after_state": "No Goals!"
},
{
"line": "linarith",
"before_state": "case h\nx y : ℝ\nhxy : 0 ≤ x * y\nthis : (x - y) ^ 2 * ((x + y) ^ 2 + 4 * (x * y)) ≥ 0\n⊢ 16 * x ^ 2 * y ^ 2 ≤ ((x + y) ^ 2) ^ 2",
"after_state": "No Goals!"
},
{
"line": "ring1",
"before_state": "x y : ℝ\nhxy : 0 ≤ x * y\nthis : (x - y) ^ 2 * ((x + y) ^ 2 + 4 * (x * y)) ≥ 0\na✝ : 16 * x ^ 2 * y ^ 2 > ((x + y) ^ 2) ^ 2\n⊢ 0 - (x - y) ^ 2 * ((x + y) ^ 2 + 4 * (x * y)) + (((x + y) ^ 2) ^ 2 - 16 * x ^ 2 * y ^ 2) = 0",
"after_state": "No Goals!"
},
{
"line": "ring",
"before_state": "x y : ℝ\nhxy : 0 ≤ x * y\nthis : (x - y) ^ 2 * ((x + y) ^ 2 + 4 * (x * y)) ≥ 0\n⊢ ((x + y) ^ 2) ^ 2 * (x + y) ^ 2 = ((x + y) ^ 2) ^ 3",
"after_state": "No Goals!"
},
{
"line": "first\n| ring1\n|\n try_this ring_nf\"\\n\\nThe `ring` tactic failed to close the goal. Use `ring_nf` to obtain a normal form.\n \\nNote that `ring` works primarily in *commutative* rings. \\\n If you have a noncommutative ring, abelian group or module, consider using \\\n `noncomm_ring`, `abel` or `module` instead.\"",
"before_state": "x y : ℝ\nhxy : 0 ≤ x * y\nthis : (x - y) ^ 2 * ((x + y) ^ 2 + 4 * (x * y)) ≥ 0\n⊢ ((x + y) ^ 2) ^ 2 * (x + y) ^ 2 = ((x + y) ^ 2) ^ 3",
"after_state": "No Goals!"
},
{
"line": "ring1",
"before_state": "x y : ℝ\nhxy : 0 ≤ x * y\nthis : (x - y) ^ 2 * ((x + y) ^ 2 + 4 * (x * y)) ≥ 0\n⊢ ((x + y) ^ 2) ^ 2 * (x + y) ^ 2 = ((x + y) ^ 2) ^ 3",
"after_state": "No Goals!"
}
] |
theorem four_pow_four_pos : (0 : ℝ) < 4 ^ 4 := by norm_num
|
/root/DuelModelResearch/mathlib4/Archive/Imo/Imo2006Q3.lean
|
{
"open": [
"Real"
],
"variables": []
}
|
[
{
"line": "norm_num",
"before_state": "⊢ 0 < 4 ^ 4",
"after_state": "No Goals!"
}
] |
theorem rhs_ineq {x y : ℝ} : 3 * (x + y) ^ 2 ≤ 2 * (x ^ 2 + y ^ 2 + (x + y) ^ 2) := by
have : 0 ≤ (x - y) ^ 2 := by positivity
linarith
|
/root/DuelModelResearch/mathlib4/Archive/Imo/Imo2006Q3.lean
|
{
"open": [
"Real"
],
"variables": []
}
|
[
{
"line": "have : 0 ≤ (x - y) ^ 2 := by positivity",
"before_state": "x y : ℝ\n⊢ 3 * (x + y) ^ 2 ≤ 2 * (x ^ 2 + y ^ 2 + (x + y) ^ 2)",
"after_state": "x y : ℝ\nthis : 0 ≤ (x - y) ^ 2\n⊢ 3 * (x + y) ^ 2 ≤ 2 * (x ^ 2 + y ^ 2 + (x + y) ^ 2)"
},
{
"line": "focus\n refine\n no_implicit_lambda%\n (have : 0 ≤ (x - y) ^ 2 := ?body✝;\n ?_)\n case body✝ => with_annotate_state\"by\" (positivity)",
"before_state": "x y : ℝ\n⊢ 3 * (x + y) ^ 2 ≤ 2 * (x ^ 2 + y ^ 2 + (x + y) ^ 2)",
"after_state": "x y : ℝ\nthis : 0 ≤ (x - y) ^ 2\n⊢ 3 * (x + y) ^ 2 ≤ 2 * (x ^ 2 + y ^ 2 + (x + y) ^ 2)"
},
{
"line": "refine\n no_implicit_lambda%\n (have : 0 ≤ (x - y) ^ 2 := ?body✝;\n ?_)",
"before_state": "x y : ℝ\n⊢ 3 * (x + y) ^ 2 ≤ 2 * (x ^ 2 + y ^ 2 + (x + y) ^ 2)",
"after_state": "case body\nx y : ℝ\n⊢ 0 ≤ (x - y) ^ 2\n---\nx y : ℝ\nthis : 0 ≤ (x - y) ^ 2\n⊢ 3 * (x + y) ^ 2 ≤ 2 * (x ^ 2 + y ^ 2 + (x + y) ^ 2)"
},
{
"line": "case body✝ => with_annotate_state\"by\" (positivity)",
"before_state": "case body\nx y : ℝ\n⊢ 0 ≤ (x - y) ^ 2\n---\nx y : ℝ\nthis : 0 ≤ (x - y) ^ 2\n⊢ 3 * (x + y) ^ 2 ≤ 2 * (x ^ 2 + y ^ 2 + (x + y) ^ 2)",
"after_state": "x y : ℝ\nthis : 0 ≤ (x - y) ^ 2\n⊢ 3 * (x + y) ^ 2 ≤ 2 * (x ^ 2 + y ^ 2 + (x + y) ^ 2)"
},
{
"line": "with_annotate_state\"by\" (positivity)",
"before_state": "x y : ℝ\n⊢ 0 ≤ (x - y) ^ 2",
"after_state": "No Goals!"
},
{
"line": "positivity",
"before_state": "x y : ℝ\n⊢ 0 ≤ (x - y) ^ 2",
"after_state": "No Goals!"
},
{
"line": "linarith",
"before_state": "x y : ℝ\nthis : 0 ≤ (x - y) ^ 2\n⊢ 3 * (x + y) ^ 2 ≤ 2 * (x ^ 2 + y ^ 2 + (x + y) ^ 2)",
"after_state": "No Goals!"
},
{
"line": "ring1",
"before_state": "x y : ℝ\nthis : 0 ≤ (x - y) ^ 2\na✝ : 3 * (x + y) ^ 2 > 2 * (x ^ 2 + y ^ 2 + (x + y) ^ 2)\n⊢ 0 - (x - y) ^ 2 + (2 * (x ^ 2 + y ^ 2 + (x + y) ^ 2) - 3 * (x + y) ^ 2) = 0",
"after_state": "No Goals!"
}
] |
theorem zero_lt_32 : (0 : ℝ) < 32 := by norm_num
|
/root/DuelModelResearch/mathlib4/Archive/Imo/Imo2006Q3.lean
|
{
"open": [
"Real"
],
"variables": []
}
|
[
{
"line": "norm_num",
"before_state": "⊢ 0 < 32",
"after_state": "No Goals!"
}
] |
theorem subst_wlog {x y z s : ℝ} (hxy : 0 ≤ x * y) (hxyz : x + y + z = 0) :
32 * |x * y * z * s| ≤ sqrt 2 * (x ^ 2 + y ^ 2 + z ^ 2 + s ^ 2) ^ 2 := by
have hz : (x + y) ^ 2 = z ^ 2 := by linear_combination (x + y - z) * hxyz
have this :=
calc
2 * s ^ 2 * (16 * x ^ 2 * y ^ 2 * (x + y) ^ 2)
≤ _ * _ ^ 3 := by gcongr; exact lhs_ineq hxy
_ ≤ (3 * (x + y) ^ 2 + 2 * s ^ 2) ^ 4 / 4 ^ 4 := mid_ineq
_ ≤ (2 * (x ^ 2 + y ^ 2 + (x + y) ^ 2) + 2 * s ^ 2) ^ 4 / 4 ^ 4 := by
gcongr (?_ + _) ^ 4 / _
apply rhs_ineq
refine le_of_pow_le_pow_left₀ two_ne_zero (by positivity) ?_
calc
(32 * |x * y * z * s|) ^ 2 = 32 * (2 * s ^ 2 * (16 * x ^ 2 * y ^ 2 * (x + y) ^ 2)) := by
rw [mul_pow]; ring
rw [sq_abs]; ring
rw [hz]; ring
_ ≤ 32 * ((2 * (x ^ 2 + y ^ 2 + (x + y) ^ 2) + 2 * s ^ 2) ^ 4 / 4 ^ 4) := by gcongr
_ = (sqrt 2 * (x ^ 2 + y ^ 2 + z ^ 2 + s ^ 2) ^ 2) ^ 2 := by
field_simp
rw [mul_pow]
rw [sq_sqrt zero_le_two]
rw [hz]
ring
|
/root/DuelModelResearch/mathlib4/Archive/Imo/Imo2006Q3.lean
|
{
"open": [
"Real"
],
"variables": []
}
|
[
{
"line": "have hz : (x + y) ^ 2 = z ^ 2 := by linear_combination (x + y - z) * hxyz",
"before_state": "x y z s : ℝ\nhxy : 0 ≤ x * y\nhxyz : x + y + z = 0\n⊢ 32 * |x * y * z * s| ≤ √2 * (x ^ 2 + y ^ 2 + z ^ 2 + s ^ 2) ^ 2",
"after_state": "x y z s : ℝ\nhxy : 0 ≤ x * y\nhxyz : x + y + z = 0\nhz : (x + y) ^ 2 = z ^ 2\n⊢ 32 * |x * y * z * s| ≤ √2 * (x ^ 2 + y ^ 2 + z ^ 2 + s ^ 2) ^ 2"
},
{
"line": "focus\n refine\n no_implicit_lambda%\n (have hz : (x + y) ^ 2 = z ^ 2 := ?body✝;\n ?_)\n case body✝ => with_annotate_state\"by\" (linear_combination (x + y - z) * hxyz)",
"before_state": "x y z s : ℝ\nhxy : 0 ≤ x * y\nhxyz : x + y + z = 0\n⊢ 32 * |x * y * z * s| ≤ √2 * (x ^ 2 + y ^ 2 + z ^ 2 + s ^ 2) ^ 2",
"after_state": "x y z s : ℝ\nhxy : 0 ≤ x * y\nhxyz : x + y + z = 0\nhz : (x + y) ^ 2 = z ^ 2\n⊢ 32 * |x * y * z * s| ≤ √2 * (x ^ 2 + y ^ 2 + z ^ 2 + s ^ 2) ^ 2"
},
{
"line": "refine\n no_implicit_lambda%\n (have hz : (x + y) ^ 2 = z ^ 2 := ?body✝;\n ?_)",
"before_state": "x y z s : ℝ\nhxy : 0 ≤ x * y\nhxyz : x + y + z = 0\n⊢ 32 * |x * y * z * s| ≤ √2 * (x ^ 2 + y ^ 2 + z ^ 2 + s ^ 2) ^ 2",
"after_state": "case body\nx y z s : ℝ\nhxy : 0 ≤ x * y\nhxyz : x + y + z = 0\n⊢ (x + y) ^ 2 = z ^ 2\n---\nx y z s : ℝ\nhxy : 0 ≤ x * y\nhxyz : x + y + z = 0\nhz : (x + y) ^ 2 = z ^ 2\n⊢ 32 * |x * y * z * s| ≤ √2 * (x ^ 2 + y ^ 2 + z ^ 2 + s ^ 2) ^ 2"
},
{
"line": "case body✝ => with_annotate_state\"by\" (linear_combination (x + y - z) * hxyz)",
"before_state": "case body\nx y z s : ℝ\nhxy : 0 ≤ x * y\nhxyz : x + y + z = 0\n⊢ (x + y) ^ 2 = z ^ 2\n---\nx y z s : ℝ\nhxy : 0 ≤ x * y\nhxyz : x + y + z = 0\nhz : (x + y) ^ 2 = z ^ 2\n⊢ 32 * |x * y * z * s| ≤ √2 * (x ^ 2 + y ^ 2 + z ^ 2 + s ^ 2) ^ 2",
"after_state": "x y z s : ℝ\nhxy : 0 ≤ x * y\nhxyz : x + y + z = 0\nhz : (x + y) ^ 2 = z ^ 2\n⊢ 32 * |x * y * z * s| ≤ √2 * (x ^ 2 + y ^ 2 + z ^ 2 + s ^ 2) ^ 2"
},
{
"line": "with_annotate_state\"by\" (linear_combination (x + y - z) * hxyz)",
"before_state": "x y z s : ℝ\nhxy : 0 ≤ x * y\nhxyz : x + y + z = 0\n⊢ (x + y) ^ 2 = z ^ 2",
"after_state": "No Goals!"
},
{
"line": "linear_combination (x + y - z) * hxyz",
"before_state": "x y z s : ℝ\nhxy : 0 ≤ x * y\nhxyz : x + y + z = 0\n⊢ (x + y) ^ 2 = z ^ 2",
"after_state": "No Goals!"
},
{
"line": "have this :=\n calc\n 2 * s ^ 2 * (16 * x ^ 2 * y ^ 2 * (x + y) ^ 2) ≤ _ * _ ^ 3 := by gcongr; exact lhs_ineq hxy\n _ ≤ (3 * (x + y) ^ 2 + 2 * s ^ 2) ^ 4 / 4 ^ 4 := mid_ineq\n _ ≤ (2 * (x ^ 2 + y ^ 2 + (x + y) ^ 2) + 2 * s ^ 2) ^ 4 / 4 ^ 4 :=\n by\n gcongr(?_ + _) ^ 4 / _\n apply rhs_ineq",
"before_state": "x y z s : ℝ\nhxy : 0 ≤ x * y\nhxyz : x + y + z = 0\nhz : (x + y) ^ 2 = z ^ 2\n⊢ 32 * |x * y * z * s| ≤ √2 * (x ^ 2 + y ^ 2 + z ^ 2 + s ^ 2) ^ 2",
"after_state": "x y z s : ℝ\nhxy : 0 ≤ x * y\nhxyz : x + y + z = 0\nhz : (x + y) ^ 2 = z ^ 2\n⊢ 32 * |x * y * z * s| ≤ √2 * (x ^ 2 + y ^ 2 + z ^ 2 + s ^ 2) ^ 2"
},
{
"line": "refine_lift\n have this :=\n calc\n 2 * s ^ 2 * (16 * x ^ 2 * y ^ 2 * (x + y) ^ 2) ≤ _ * _ ^ 3 := by gcongr; exact lhs_ineq hxy\n _ ≤ (3 * (x + y) ^ 2 + 2 * s ^ 2) ^ 4 / 4 ^ 4 := mid_ineq\n _ ≤ (2 * (x ^ 2 + y ^ 2 + (x + y) ^ 2) + 2 * s ^ 2) ^ 4 / 4 ^ 4 :=\n by\n gcongr(?_ + _) ^ 4 / _\n apply rhs_ineq;\n ?_",
"before_state": "x y z s : ℝ\nhxy : 0 ≤ x * y\nhxyz : x + y + z = 0\nhz : (x + y) ^ 2 = z ^ 2\n⊢ 32 * |x * y * z * s| ≤ √2 * (x ^ 2 + y ^ 2 + z ^ 2 + s ^ 2) ^ 2",
"after_state": "x y z s : ℝ\nhxy : 0 ≤ x * y\nhxyz : x + y + z = 0\nhz : (x + y) ^ 2 = z ^ 2\n⊢ 32 * |x * y * z * s| ≤ √2 * (x ^ 2 + y ^ 2 + z ^ 2 + s ^ 2) ^ 2"
},
{
"line": "focus\n (refine\n no_implicit_lambda%\n (have this :=\n calc\n 2 * s ^ 2 * (16 * x ^ 2 * y ^ 2 * (x + y) ^ 2) ≤ _ * _ ^ 3 := by gcongr; exact lhs_ineq hxy\n _ ≤ (3 * (x + y) ^ 2 + 2 * s ^ 2) ^ 4 / 4 ^ 4 := mid_ineq\n _ ≤ (2 * (x ^ 2 + y ^ 2 + (x + y) ^ 2) + 2 * s ^ 2) ^ 4 / 4 ^ 4 :=\n by\n gcongr(?_ + _) ^ 4 / _\n apply rhs_ineq;\n ?_);\n rotate_right)",
"before_state": "x y z s : ℝ\nhxy : 0 ≤ x * y\nhxyz : x + y + z = 0\nhz : (x + y) ^ 2 = z ^ 2\n⊢ 32 * |x * y * z * s| ≤ √2 * (x ^ 2 + y ^ 2 + z ^ 2 + s ^ 2) ^ 2",
"after_state": "x y z s : ℝ\nhxy : 0 ≤ x * y\nhxyz : x + y + z = 0\nhz : (x + y) ^ 2 = z ^ 2\n⊢ 32 * |x * y * z * s| ≤ √2 * (x ^ 2 + y ^ 2 + z ^ 2 + s ^ 2) ^ 2"
},
{
"line": "refine\n no_implicit_lambda%\n (have this :=\n calc\n 2 * s ^ 2 * (16 * x ^ 2 * y ^ 2 * (x + y) ^ 2) ≤ _ * _ ^ 3 := by gcongr; exact lhs_ineq hxy\n _ ≤ (3 * (x + y) ^ 2 + 2 * s ^ 2) ^ 4 / 4 ^ 4 := mid_ineq\n _ ≤ (2 * (x ^ 2 + y ^ 2 + (x + y) ^ 2) + 2 * s ^ 2) ^ 4 / 4 ^ 4 :=\n by\n gcongr(?_ + _) ^ 4 / _\n apply rhs_ineq;\n ?_)",
"before_state": "x y z s : ℝ\nhxy : 0 ≤ x * y\nhxyz : x + y + z = 0\nhz : (x + y) ^ 2 = z ^ 2\n⊢ 32 * |x * y * z * s| ≤ √2 * (x ^ 2 + y ^ 2 + z ^ 2 + s ^ 2) ^ 2",
"after_state": "x y z s : ℝ\nhxy : 0 ≤ x * y\nhxyz : x + y + z = 0\nhz : (x + y) ^ 2 = z ^ 2\n⊢ 32 * |x * y * z * s| ≤ √2 * (x ^ 2 + y ^ 2 + z ^ 2 + s ^ 2) ^ 2"
},
{
"line": "gcongr",
"before_state": "x y z s : ℝ\nhxy : 0 ≤ x * y\nhxyz : x + y + z = 0\nhz : (x + y) ^ 2 = z ^ 2\n⊢ 2 * s ^ 2 * (16 * x ^ 2 * y ^ 2 * (x + y) ^ 2) ≤ ?m.3750 * ?m.3756 ^ 3",
"after_state": "case h\nx y z s : ℝ\nhxy : 0 ≤ x * y\nhxyz : x + y + z = 0\nhz : (x + y) ^ 2 = z ^ 2\n⊢ 16 * x ^ 2 * y ^ 2 * (x + y) ^ 2 ≤ ?m.3756 ^ 3"
},
{
"line": "gcongr_discharger",
"before_state": "case a0\nx y z s : ℝ\nhxy : 0 ≤ x * y\nhxyz : x + y + z = 0\nhz : (x + y) ^ 2 = z ^ 2\n⊢ 0 ≤ 2 * s ^ 2",
"after_state": "No Goals!"
},
{
"line": "positivity",
"before_state": "case a0\nx y z s : ℝ\nhxy : 0 ≤ x * y\nhxyz : x + y + z = 0\nhz : (x + y) ^ 2 = z ^ 2\n⊢ 0 ≤ 2 * s ^ 2",
"after_state": "No Goals!"
},
{
"line": "exact lhs_ineq hxy",
"before_state": "case h\nx y z s : ℝ\nhxy : 0 ≤ x * y\nhxyz : x + y + z = 0\nhz : (x + y) ^ 2 = z ^ 2\n⊢ 16 * x ^ 2 * y ^ 2 * (x + y) ^ 2 ≤ ?m.3756 ^ 3",
"after_state": "No Goals!"
},
{
"line": "gcongr(?_ + _) ^ 4 / _",
"before_state": "x y z s : ℝ\nhxy : 0 ≤ x * y\nhxyz : x + y + z = 0\nhz : (x + y) ^ 2 = z ^ 2\n⊢ (3 * (x + y) ^ 2 + 2 * s ^ 2) ^ 4 / 4 ^ 4 ≤ (2 * (x ^ 2 + y ^ 2 + (x + y) ^ 2) + 2 * s ^ 2) ^ 4 / 4 ^ 4",
"after_state": "case hab.hab.bc\nx y z s : ℝ\nhxy : 0 ≤ x * y\nhxyz : x + y + z = 0\nhz : (x + y) ^ 2 = z ^ 2\n⊢ 3 * (x + y) ^ 2 ≤ 2 * (x ^ 2 + y ^ 2 + (x + y) ^ 2)"
},
{
"line": "gcongr_discharger",
"before_state": "case hab.ha\nx y z s : ℝ\nhxy : 0 ≤ x * y\nhxyz : x + y + z = 0\nhz : (x + y) ^ 2 = z ^ 2\n⊢ 0 ≤ 3 * (x + y) ^ 2 + 2 * s ^ 2",
"after_state": "No Goals!"
},
{
"line": "positivity",
"before_state": "case hab.ha\nx y z s : ℝ\nhxy : 0 ≤ x * y\nhxyz : x + y + z = 0\nhz : (x + y) ^ 2 = z ^ 2\n⊢ 0 ≤ 3 * (x + y) ^ 2 + 2 * s ^ 2",
"after_state": "No Goals!"
},
{
"line": "gcongr_discharger",
"before_state": "case hc\nx y z s : ℝ\nhxy : 0 ≤ x * y\nhxyz : x + y + z = 0\nhz : (x + y) ^ 2 = z ^ 2\n⊢ 0 ≤ 4 ^ 4",
"after_state": "No Goals!"
},
{
"line": "positivity",
"before_state": "case hc\nx y z s : ℝ\nhxy : 0 ≤ x * y\nhxyz : x + y + z = 0\nhz : (x + y) ^ 2 = z ^ 2\n⊢ 0 ≤ 4 ^ 4",
"after_state": "No Goals!"
},
{
"line": "apply rhs_ineq",
"before_state": "case hab.hab.bc\nx y z s : ℝ\nhxy : 0 ≤ x * y\nhxyz : x + y + z = 0\nhz : (x + y) ^ 2 = z ^ 2\n⊢ 3 * (x + y) ^ 2 ≤ 2 * (x ^ 2 + y ^ 2 + (x + y) ^ 2)",
"after_state": "No Goals!"
}
] |
theorem subst_proof₁ (x y z s : ℝ) (hxyz : x + y + z = 0) :
|x * y * z * s| ≤ sqrt 2 / 32 * (x ^ 2 + y ^ 2 + z ^ 2 + s ^ 2) ^ 2 := by
wlog h' : 0 ≤ x * y generalizing x y z; swap
· rw [div_mul_eq_mul_div, le_div_iff₀' zero_lt_32]
exact subst_wlog h' hxyz
rcases (mul_nonneg_of_three x y z).resolve_left h' with h | h
· convert this y z x _ h using 2 <;> linarith
· convert this z x y _ h using 2 <;> linarith
|
/root/DuelModelResearch/mathlib4/Archive/Imo/Imo2006Q3.lean
|
{
"open": [
"Real"
],
"variables": []
}
|
[
{
"line": "wlog h' : 0 ≤ x * y generalizing x y z",
"before_state": "x y z s : ℝ\nhxyz : x + y + z = 0\n⊢ |x * y * z * s| ≤ √2 / 32 * (x ^ 2 + y ^ 2 + z ^ 2 + s ^ 2) ^ 2",
"after_state": "case inr\nx y z s : ℝ\nhxyz : x + y + z = 0\nthis : ∀ (x y z : ℝ), x + y + z = 0 → 0 ≤ x * y → |x * y * z * s| ≤ √2 / 32 * (x ^ 2 + y ^ 2 + z ^ 2 + s ^ 2) ^ 2\nh' : ¬0 ≤ x * y\n⊢ |x * y * z * s| ≤ √2 / 32 * (x ^ 2 + y ^ 2 + z ^ 2 + s ^ 2) ^ 2\n---\ns x y z : ℝ\nhxyz : x + y + z = 0\nh' : 0 ≤ x * y\n⊢ |x * y * z * s| ≤ √2 / 32 * (x ^ 2 + y ^ 2 + z ^ 2 + s ^ 2) ^ 2"
},
{
"line": "swap",
"before_state": "case inr\nx y z s : ℝ\nhxyz : x + y + z = 0\nthis : ∀ (x y z : ℝ), x + y + z = 0 → 0 ≤ x * y → |x * y * z * s| ≤ √2 / 32 * (x ^ 2 + y ^ 2 + z ^ 2 + s ^ 2) ^ 2\nh' : ¬0 ≤ x * y\n⊢ |x * y * z * s| ≤ √2 / 32 * (x ^ 2 + y ^ 2 + z ^ 2 + s ^ 2) ^ 2\n---\ns x y z : ℝ\nhxyz : x + y + z = 0\nh' : 0 ≤ x * y\n⊢ |x * y * z * s| ≤ √2 / 32 * (x ^ 2 + y ^ 2 + z ^ 2 + s ^ 2) ^ 2",
"after_state": "s x y z : ℝ\nhxyz : x + y + z = 0\nh' : 0 ≤ x * y\n⊢ |x * y * z * s| ≤ √2 / 32 * (x ^ 2 + y ^ 2 + z ^ 2 + s ^ 2) ^ 2\n---\ncase inr\nx y z s : ℝ\nhxyz : x + y + z = 0\nthis : ∀ (x y z : ℝ), x + y + z = 0 → 0 ≤ x * y → |x * y * z * s| ≤ √2 / 32 * (x ^ 2 + y ^ 2 + z ^ 2 + s ^ 2) ^ 2\nh' : ¬0 ≤ x * y\n⊢ |x * y * z * s| ≤ √2 / 32 * (x ^ 2 + y ^ 2 + z ^ 2 + s ^ 2) ^ 2"
},
{
"line": "pick_goal 2",
"before_state": "case inr\nx y z s : ℝ\nhxyz : x + y + z = 0\nthis : ∀ (x y z : ℝ), x + y + z = 0 → 0 ≤ x * y → |x * y * z * s| ≤ √2 / 32 * (x ^ 2 + y ^ 2 + z ^ 2 + s ^ 2) ^ 2\nh' : ¬0 ≤ x * y\n⊢ |x * y * z * s| ≤ √2 / 32 * (x ^ 2 + y ^ 2 + z ^ 2 + s ^ 2) ^ 2\n---\ns x y z : ℝ\nhxyz : x + y + z = 0\nh' : 0 ≤ x * y\n⊢ |x * y * z * s| ≤ √2 / 32 * (x ^ 2 + y ^ 2 + z ^ 2 + s ^ 2) ^ 2",
"after_state": "s x y z : ℝ\nhxyz : x + y + z = 0\nh' : 0 ≤ x * y\n⊢ |x * y * z * s| ≤ √2 / 32 * (x ^ 2 + y ^ 2 + z ^ 2 + s ^ 2) ^ 2\n---\ncase inr\nx y z s : ℝ\nhxyz : x + y + z = 0\nthis : ∀ (x y z : ℝ), x + y + z = 0 → 0 ≤ x * y → |x * y * z * s| ≤ √2 / 32 * (x ^ 2 + y ^ 2 + z ^ 2 + s ^ 2) ^ 2\nh' : ¬0 ≤ x * y\n⊢ |x * y * z * s| ≤ √2 / 32 * (x ^ 2 + y ^ 2 + z ^ 2 + s ^ 2) ^ 2"
},
{
"line": "rw [div_mul_eq_mul_div, le_div_iff₀' zero_lt_32]",
"before_state": "s x y z : ℝ\nhxyz : x + y + z = 0\nh' : 0 ≤ x * y\n⊢ |x * y * z * s| ≤ √2 / 32 * (x ^ 2 + y ^ 2 + z ^ 2 + s ^ 2) ^ 2",
"after_state": "s x y z : ℝ\nhxyz : x + y + z = 0\nh' : 0 ≤ x * y\n⊢ |x * y * z * s| ≤ √2 * (x ^ 2 + y ^ 2 + z ^ 2 + s ^ 2) ^ 2 / 32"
},
{
"line": "rewrite [div_mul_eq_mul_div, le_div_iff₀' zero_lt_32]",
"before_state": "s x y z : ℝ\nhxyz : x + y + z = 0\nh' : 0 ≤ x * y\n⊢ |x * y * z * s| ≤ √2 / 32 * (x ^ 2 + y ^ 2 + z ^ 2 + s ^ 2) ^ 2",
"after_state": "s x y z : ℝ\nhxyz : x + y + z = 0\nh' : 0 ≤ x * y\n⊢ |x * y * z * s| ≤ √2 * (x ^ 2 + y ^ 2 + z ^ 2 + s ^ 2) ^ 2 / 32"
},
{
"line": "rcases(mul_nonneg_of_three x y z).resolve_left h' with h | h",
"before_state": "case inr\nx y z s : ℝ\nhxyz : x + y + z = 0\nthis : ∀ (x y z : ℝ), x + y + z = 0 → 0 ≤ x * y → |x * y * z * s| ≤ √2 / 32 * (x ^ 2 + y ^ 2 + z ^ 2 + s ^ 2) ^ 2\nh' : ¬0 ≤ x * y\n⊢ |x * y * z * s| ≤ √2 / 32 * (x ^ 2 + y ^ 2 + z ^ 2 + s ^ 2) ^ 2",
"after_state": "case inr.inl\nx y z s : ℝ\nhxyz : x + y + z = 0\nthis : ∀ (x y z : ℝ), x + y + z = 0 → 0 ≤ x * y → |x * y * z * s| ≤ √2 / 32 * (x ^ 2 + y ^ 2 + z ^ 2 + s ^ 2) ^ 2\nh' : ¬0 ≤ x * y\nh : 0 ≤ y * z\n⊢ |x * y * z * s| ≤ √2 / 32 * (x ^ 2 + y ^ 2 + z ^ 2 + s ^ 2) ^ 2\n---\ncase inr.inr\nx y z s : ℝ\nhxyz : x + y + z = 0\nthis : ∀ (x y z : ℝ), x + y + z = 0 → 0 ≤ x * y → |x * y * z * s| ≤ √2 / 32 * (x ^ 2 + y ^ 2 + z ^ 2 + s ^ 2) ^ 2\nh' : ¬0 ≤ x * y\nh : 0 ≤ z * x\n⊢ |x * y * z * s| ≤ √2 / 32 * (x ^ 2 + y ^ 2 + z ^ 2 + s ^ 2) ^ 2"
},
{
"line": "focus\n convert this y z x _ h using 2\n with_annotate_state\"<;>\" skip\n all_goals linarith",
"before_state": "case inr.inl\nx y z s : ℝ\nhxyz : x + y + z = 0\nthis : ∀ (x y z : ℝ), x + y + z = 0 → 0 ≤ x * y → |x * y * z * s| ≤ √2 / 32 * (x ^ 2 + y ^ 2 + z ^ 2 + s ^ 2) ^ 2\nh' : ¬0 ≤ x * y\nh : 0 ≤ y * z\n⊢ |x * y * z * s| ≤ √2 / 32 * (x ^ 2 + y ^ 2 + z ^ 2 + s ^ 2) ^ 2",
"after_state": "No Goals!"
},
{
"line": "convert this y z x _ h using 2",
"before_state": "case inr.inl\nx y z s : ℝ\nhxyz : x + y + z = 0\nthis : ∀ (x y z : ℝ), x + y + z = 0 → 0 ≤ x * y → |x * y * z * s| ≤ √2 / 32 * (x ^ 2 + y ^ 2 + z ^ 2 + s ^ 2) ^ 2\nh' : ¬0 ≤ x * y\nh : 0 ≤ y * z\n⊢ |x * y * z * s| ≤ √2 / 32 * (x ^ 2 + y ^ 2 + z ^ 2 + s ^ 2) ^ 2",
"after_state": "case h.e'_3.h.e'_4\nx y z s : ℝ\nhxyz : x + y + z = 0\nthis : ∀ (x y z : ℝ), x + y + z = 0 → 0 ≤ x * y → |x * y * z * s| ≤ √2 / 32 * (x ^ 2 + y ^ 2 + z ^ 2 + s ^ 2) ^ 2\nh' : ¬0 ≤ x * y\nh : 0 ≤ y * z\n⊢ x * y * z * s = y * z * x * s\n---\ncase h.e'_4.h.e'_6\nx y z s : ℝ\nhxyz : x + y + z = 0\nthis : ∀ (x y z : ℝ), x + y + z = 0 → 0 ≤ x * y → |x * y * z * s| ≤ √2 / 32 * (x ^ 2 + y ^ 2 + z ^ 2 + s ^ 2) ^ 2\nh' : ¬0 ≤ x * y\nh : 0 ≤ y * z\n⊢ (x ^ 2 + y ^ 2 + z ^ 2 + s ^ 2) ^ 2 = (y ^ 2 + z ^ 2 + x ^ 2 + s ^ 2) ^ 2\n---\ncase inr.inl\nx y z s : ℝ\nhxyz : x + y + z = 0\nthis : ∀ (x y z : ℝ), x + y + z = 0 → 0 ≤ x * y → |x * y * z * s| ≤ √2 / 32 * (x ^ 2 + y ^ 2 + z ^ 2 + s ^ 2) ^ 2\nh' : ¬0 ≤ x * y\nh : 0 ≤ y * z\n⊢ y + z + x = 0"
},
{
"line": "with_annotate_state\"<;>\" skip",
"before_state": "case h.e'_3.h.e'_4\nx y z s : ℝ\nhxyz : x + y + z = 0\nthis : ∀ (x y z : ℝ), x + y + z = 0 → 0 ≤ x * y → |x * y * z * s| ≤ √2 / 32 * (x ^ 2 + y ^ 2 + z ^ 2 + s ^ 2) ^ 2\nh' : ¬0 ≤ x * y\nh : 0 ≤ y * z\n⊢ x * y * z * s = y * z * x * s\n---\ncase h.e'_4.h.e'_6\nx y z s : ℝ\nhxyz : x + y + z = 0\nthis : ∀ (x y z : ℝ), x + y + z = 0 → 0 ≤ x * y → |x * y * z * s| ≤ √2 / 32 * (x ^ 2 + y ^ 2 + z ^ 2 + s ^ 2) ^ 2\nh' : ¬0 ≤ x * y\nh : 0 ≤ y * z\n⊢ (x ^ 2 + y ^ 2 + z ^ 2 + s ^ 2) ^ 2 = (y ^ 2 + z ^ 2 + x ^ 2 + s ^ 2) ^ 2\n---\ncase inr.inl\nx y z s : ℝ\nhxyz : x + y + z = 0\nthis : ∀ (x y z : ℝ), x + y + z = 0 → 0 ≤ x * y → |x * y * z * s| ≤ √2 / 32 * (x ^ 2 + y ^ 2 + z ^ 2 + s ^ 2) ^ 2\nh' : ¬0 ≤ x * y\nh : 0 ≤ y * z\n⊢ y + z + x = 0",
"after_state": "case h.e'_3.h.e'_4\nx y z s : ℝ\nhxyz : x + y + z = 0\nthis : ∀ (x y z : ℝ), x + y + z = 0 → 0 ≤ x * y → |x * y * z * s| ≤ √2 / 32 * (x ^ 2 + y ^ 2 + z ^ 2 + s ^ 2) ^ 2\nh' : ¬0 ≤ x * y\nh : 0 ≤ y * z\n⊢ x * y * z * s = y * z * x * s\n---\ncase h.e'_4.h.e'_6\nx y z s : ℝ\nhxyz : x + y + z = 0\nthis : ∀ (x y z : ℝ), x + y + z = 0 → 0 ≤ x * y → |x * y * z * s| ≤ √2 / 32 * (x ^ 2 + y ^ 2 + z ^ 2 + s ^ 2) ^ 2\nh' : ¬0 ≤ x * y\nh : 0 ≤ y * z\n⊢ (x ^ 2 + y ^ 2 + z ^ 2 + s ^ 2) ^ 2 = (y ^ 2 + z ^ 2 + x ^ 2 + s ^ 2) ^ 2\n---\ncase inr.inl\nx y z s : ℝ\nhxyz : x + y + z = 0\nthis : ∀ (x y z : ℝ), x + y + z = 0 → 0 ≤ x * y → |x * y * z * s| ≤ √2 / 32 * (x ^ 2 + y ^ 2 + z ^ 2 + s ^ 2) ^ 2\nh' : ¬0 ≤ x * y\nh : 0 ≤ y * z\n⊢ y + z + x = 0"
},
{
"line": "skip",
"before_state": "case h.e'_3.h.e'_4\nx y z s : ℝ\nhxyz : x + y + z = 0\nthis : ∀ (x y z : ℝ), x + y + z = 0 → 0 ≤ x * y → |x * y * z * s| ≤ √2 / 32 * (x ^ 2 + y ^ 2 + z ^ 2 + s ^ 2) ^ 2\nh' : ¬0 ≤ x * y\nh : 0 ≤ y * z\n⊢ x * y * z * s = y * z * x * s\n---\ncase h.e'_4.h.e'_6\nx y z s : ℝ\nhxyz : x + y + z = 0\nthis : ∀ (x y z : ℝ), x + y + z = 0 → 0 ≤ x * y → |x * y * z * s| ≤ √2 / 32 * (x ^ 2 + y ^ 2 + z ^ 2 + s ^ 2) ^ 2\nh' : ¬0 ≤ x * y\nh : 0 ≤ y * z\n⊢ (x ^ 2 + y ^ 2 + z ^ 2 + s ^ 2) ^ 2 = (y ^ 2 + z ^ 2 + x ^ 2 + s ^ 2) ^ 2\n---\ncase inr.inl\nx y z s : ℝ\nhxyz : x + y + z = 0\nthis : ∀ (x y z : ℝ), x + y + z = 0 → 0 ≤ x * y → |x * y * z * s| ≤ √2 / 32 * (x ^ 2 + y ^ 2 + z ^ 2 + s ^ 2) ^ 2\nh' : ¬0 ≤ x * y\nh : 0 ≤ y * z\n⊢ y + z + x = 0",
"after_state": "case h.e'_3.h.e'_4\nx y z s : ℝ\nhxyz : x + y + z = 0\nthis : ∀ (x y z : ℝ), x + y + z = 0 → 0 ≤ x * y → |x * y * z * s| ≤ √2 / 32 * (x ^ 2 + y ^ 2 + z ^ 2 + s ^ 2) ^ 2\nh' : ¬0 ≤ x * y\nh : 0 ≤ y * z\n⊢ x * y * z * s = y * z * x * s\n---\ncase h.e'_4.h.e'_6\nx y z s : ℝ\nhxyz : x + y + z = 0\nthis : ∀ (x y z : ℝ), x + y + z = 0 → 0 ≤ x * y → |x * y * z * s| ≤ √2 / 32 * (x ^ 2 + y ^ 2 + z ^ 2 + s ^ 2) ^ 2\nh' : ¬0 ≤ x * y\nh : 0 ≤ y * z\n⊢ (x ^ 2 + y ^ 2 + z ^ 2 + s ^ 2) ^ 2 = (y ^ 2 + z ^ 2 + x ^ 2 + s ^ 2) ^ 2\n---\ncase inr.inl\nx y z s : ℝ\nhxyz : x + y + z = 0\nthis : ∀ (x y z : ℝ), x + y + z = 0 → 0 ≤ x * y → |x * y * z * s| ≤ √2 / 32 * (x ^ 2 + y ^ 2 + z ^ 2 + s ^ 2) ^ 2\nh' : ¬0 ≤ x * y\nh : 0 ≤ y * z\n⊢ y + z + x = 0"
},
{
"line": "all_goals linarith",
"before_state": "case h.e'_3.h.e'_4\nx y z s : ℝ\nhxyz : x + y + z = 0\nthis : ∀ (x y z : ℝ), x + y + z = 0 → 0 ≤ x * y → |x * y * z * s| ≤ √2 / 32 * (x ^ 2 + y ^ 2 + z ^ 2 + s ^ 2) ^ 2\nh' : ¬0 ≤ x * y\nh : 0 ≤ y * z\n⊢ x * y * z * s = y * z * x * s\n---\ncase h.e'_4.h.e'_6\nx y z s : ℝ\nhxyz : x + y + z = 0\nthis : ∀ (x y z : ℝ), x + y + z = 0 → 0 ≤ x * y → |x * y * z * s| ≤ √2 / 32 * (x ^ 2 + y ^ 2 + z ^ 2 + s ^ 2) ^ 2\nh' : ¬0 ≤ x * y\nh : 0 ≤ y * z\n⊢ (x ^ 2 + y ^ 2 + z ^ 2 + s ^ 2) ^ 2 = (y ^ 2 + z ^ 2 + x ^ 2 + s ^ 2) ^ 2\n---\ncase inr.inl\nx y z s : ℝ\nhxyz : x + y + z = 0\nthis : ∀ (x y z : ℝ), x + y + z = 0 → 0 ≤ x * y → |x * y * z * s| ≤ √2 / 32 * (x ^ 2 + y ^ 2 + z ^ 2 + s ^ 2) ^ 2\nh' : ¬0 ≤ x * y\nh : 0 ≤ y * z\n⊢ y + z + x = 0",
"after_state": "No Goals!"
},
{
"line": "linarith",
"before_state": "case h.e'_3.h.e'_4\nx y z s : ℝ\nhxyz : x + y + z = 0\nthis : ∀ (x y z : ℝ), x + y + z = 0 → 0 ≤ x * y → |x * y * z * s| ≤ √2 / 32 * (x ^ 2 + y ^ 2 + z ^ 2 + s ^ 2) ^ 2\nh' : ¬0 ≤ x * y\nh : 0 ≤ y * z\n⊢ x * y * z * s = y * z * x * s",
"after_state": "No Goals!"
},
{
"line": "ring1",
"before_state": "x y z s : ℝ\nhxyz : x + y + z = 0\nthis : ∀ (x y z : ℝ), x + y + z = 0 → 0 ≤ x * y → |x * y * z * s| ≤ √2 / 32 * (x ^ 2 + y ^ 2 + z ^ 2 + s ^ 2) ^ 2\nh' : ¬0 ≤ x * y\nh : 0 ≤ y * z\na✝ : x * y * z * s < y * z * x * s\n⊢ x * y * z * s - y * z * x * s = 0",
"after_state": "No Goals!"
},
{
"line": "ring1",
"before_state": "x y z s : ℝ\nhxyz : x + y + z = 0\nthis : ∀ (x y z : ℝ), x + y + z = 0 → 0 ≤ x * y → |x * y * z * s| ≤ √2 / 32 * (x ^ 2 + y ^ 2 + z ^ 2 + s ^ 2) ^ 2\nh' : ¬0 ≤ x * y\nh : 0 ≤ y * z\na✝ : y * z * x * s < x * y * z * s\n⊢ y * z * x * s - x * y * z * s = 0",
"after_state": "No Goals!"
},
{
"line": "linarith",
"before_state": "case h.e'_4.h.e'_6\nx y z s : ℝ\nhxyz : x + y + z = 0\nthis : ∀ (x y z : ℝ), x + y + z = 0 → 0 ≤ x * y → |x * y * z * s| ≤ √2 / 32 * (x ^ 2 + y ^ 2 + z ^ 2 + s ^ 2) ^ 2\nh' : ¬0 ≤ x * y\nh : 0 ≤ y * z\n⊢ (x ^ 2 + y ^ 2 + z ^ 2 + s ^ 2) ^ 2 = (y ^ 2 + z ^ 2 + x ^ 2 + s ^ 2) ^ 2",
"after_state": "No Goals!"
},
{
"line": "ring1",
"before_state": "x y z s : ℝ\nhxyz : x + y + z = 0\nthis : ∀ (x y z : ℝ), x + y + z = 0 → 0 ≤ x * y → |x * y * z * s| ≤ √2 / 32 * (x ^ 2 + y ^ 2 + z ^ 2 + s ^ 2) ^ 2\nh' : ¬0 ≤ x * y\nh : 0 ≤ y * z\na✝ : (x ^ 2 + y ^ 2 + z ^ 2 + s ^ 2) ^ 2 < (y ^ 2 + z ^ 2 + x ^ 2 + s ^ 2) ^ 2\n⊢ (x ^ 2 + y ^ 2 + z ^ 2 + s ^ 2) ^ 2 - (y ^ 2 + z ^ 2 + x ^ 2 + s ^ 2) ^ 2 = 0",
"after_state": "No Goals!"
},
{
"line": "ring1",
"before_state": "x y z s : ℝ\nhxyz : x + y + z = 0\nthis : ∀ (x y z : ℝ), x + y + z = 0 → 0 ≤ x * y → |x * y * z * s| ≤ √2 / 32 * (x ^ 2 + y ^ 2 + z ^ 2 + s ^ 2) ^ 2\nh' : ¬0 ≤ x * y\nh : 0 ≤ y * z\na✝ : (y ^ 2 + z ^ 2 + x ^ 2 + s ^ 2) ^ 2 < (x ^ 2 + y ^ 2 + z ^ 2 + s ^ 2) ^ 2\n⊢ (y ^ 2 + z ^ 2 + x ^ 2 + s ^ 2) ^ 2 - (x ^ 2 + y ^ 2 + z ^ 2 + s ^ 2) ^ 2 = 0",
"after_state": "No Goals!"
},
{
"line": "linarith",
"before_state": "case inr.inl\nx y z s : ℝ\nhxyz : x + y + z = 0\nthis : ∀ (x y z : ℝ), x + y + z = 0 → 0 ≤ x * y → |x * y * z * s| ≤ √2 / 32 * (x ^ 2 + y ^ 2 + z ^ 2 + s ^ 2) ^ 2\nh' : ¬0 ≤ x * y\nh : 0 ≤ y * z\n⊢ y + z + x = 0",
"after_state": "No Goals!"
},
{
"line": "ring1",
"before_state": "x y z s : ℝ\nhxyz : x + y + z = 0\nthis : ∀ (x y z : ℝ), x + y + z = 0 → 0 ≤ x * y → |x * y * z * s| ≤ √2 / 32 * (x ^ 2 + y ^ 2 + z ^ 2 + s ^ 2) ^ 2\nh' : ¬0 ≤ x * y\nh : 0 ≤ y * z\na✝ : y + z + x < 0\n⊢ -(x + y + z - 0) + (y + z + x - 0) = 0",
"after_state": "No Goals!"
},
{
"line": "ring1",
"before_state": "x y z s : ℝ\nhxyz : x + y + z = 0\nthis : ∀ (x y z : ℝ), x + y + z = 0 → 0 ≤ x * y → |x * y * z * s| ≤ √2 / 32 * (x ^ 2 + y ^ 2 + z ^ 2 + s ^ 2) ^ 2\nh' : ¬0 ≤ x * y\nh : 0 ≤ y * z\na✝ : 0 < y + z + x\n⊢ x + y + z - 0 + (0 - (y + z + x)) = 0",
"after_state": "No Goals!"
},
{
"line": "focus\n convert this z x y _ h using 2\n with_annotate_state\"<;>\" skip\n all_goals linarith",
"before_state": "case inr.inr\nx y z s : ℝ\nhxyz : x + y + z = 0\nthis : ∀ (x y z : ℝ), x + y + z = 0 → 0 ≤ x * y → |x * y * z * s| ≤ √2 / 32 * (x ^ 2 + y ^ 2 + z ^ 2 + s ^ 2) ^ 2\nh' : ¬0 ≤ x * y\nh : 0 ≤ z * x\n⊢ |x * y * z * s| ≤ √2 / 32 * (x ^ 2 + y ^ 2 + z ^ 2 + s ^ 2) ^ 2",
"after_state": "No Goals!"
},
{
"line": "convert this z x y _ h using 2",
"before_state": "case inr.inr\nx y z s : ℝ\nhxyz : x + y + z = 0\nthis : ∀ (x y z : ℝ), x + y + z = 0 → 0 ≤ x * y → |x * y * z * s| ≤ √2 / 32 * (x ^ 2 + y ^ 2 + z ^ 2 + s ^ 2) ^ 2\nh' : ¬0 ≤ x * y\nh : 0 ≤ z * x\n⊢ |x * y * z * s| ≤ √2 / 32 * (x ^ 2 + y ^ 2 + z ^ 2 + s ^ 2) ^ 2",
"after_state": "case h.e'_3.h.e'_4\nx y z s : ℝ\nhxyz : x + y + z = 0\nthis : ∀ (x y z : ℝ), x + y + z = 0 → 0 ≤ x * y → |x * y * z * s| ≤ √2 / 32 * (x ^ 2 + y ^ 2 + z ^ 2 + s ^ 2) ^ 2\nh' : ¬0 ≤ x * y\nh : 0 ≤ z * x\n⊢ x * y * z * s = z * x * y * s\n---\ncase h.e'_4.h.e'_6\nx y z s : ℝ\nhxyz : x + y + z = 0\nthis : ∀ (x y z : ℝ), x + y + z = 0 → 0 ≤ x * y → |x * y * z * s| ≤ √2 / 32 * (x ^ 2 + y ^ 2 + z ^ 2 + s ^ 2) ^ 2\nh' : ¬0 ≤ x * y\nh : 0 ≤ z * x\n⊢ (x ^ 2 + y ^ 2 + z ^ 2 + s ^ 2) ^ 2 = (z ^ 2 + x ^ 2 + y ^ 2 + s ^ 2) ^ 2\n---\ncase inr.inr\nx y z s : ℝ\nhxyz : x + y + z = 0\nthis : ∀ (x y z : ℝ), x + y + z = 0 → 0 ≤ x * y → |x * y * z * s| ≤ √2 / 32 * (x ^ 2 + y ^ 2 + z ^ 2 + s ^ 2) ^ 2\nh' : ¬0 ≤ x * y\nh : 0 ≤ z * x\n⊢ z + x + y = 0"
},
{
"line": "with_annotate_state\"<;>\" skip",
"before_state": "case h.e'_3.h.e'_4\nx y z s : ℝ\nhxyz : x + y + z = 0\nthis : ∀ (x y z : ℝ), x + y + z = 0 → 0 ≤ x * y → |x * y * z * s| ≤ √2 / 32 * (x ^ 2 + y ^ 2 + z ^ 2 + s ^ 2) ^ 2\nh' : ¬0 ≤ x * y\nh : 0 ≤ z * x\n⊢ x * y * z * s = z * x * y * s\n---\ncase h.e'_4.h.e'_6\nx y z s : ℝ\nhxyz : x + y + z = 0\nthis : ∀ (x y z : ℝ), x + y + z = 0 → 0 ≤ x * y → |x * y * z * s| ≤ √2 / 32 * (x ^ 2 + y ^ 2 + z ^ 2 + s ^ 2) ^ 2\nh' : ¬0 ≤ x * y\nh : 0 ≤ z * x\n⊢ (x ^ 2 + y ^ 2 + z ^ 2 + s ^ 2) ^ 2 = (z ^ 2 + x ^ 2 + y ^ 2 + s ^ 2) ^ 2\n---\ncase inr.inr\nx y z s : ℝ\nhxyz : x + y + z = 0\nthis : ∀ (x y z : ℝ), x + y + z = 0 → 0 ≤ x * y → |x * y * z * s| ≤ √2 / 32 * (x ^ 2 + y ^ 2 + z ^ 2 + s ^ 2) ^ 2\nh' : ¬0 ≤ x * y\nh : 0 ≤ z * x\n⊢ z + x + y = 0",
"after_state": "case h.e'_3.h.e'_4\nx y z s : ℝ\nhxyz : x + y + z = 0\nthis : ∀ (x y z : ℝ), x + y + z = 0 → 0 ≤ x * y → |x * y * z * s| ≤ √2 / 32 * (x ^ 2 + y ^ 2 + z ^ 2 + s ^ 2) ^ 2\nh' : ¬0 ≤ x * y\nh : 0 ≤ z * x\n⊢ x * y * z * s = z * x * y * s\n---\ncase h.e'_4.h.e'_6\nx y z s : ℝ\nhxyz : x + y + z = 0\nthis : ∀ (x y z : ℝ), x + y + z = 0 → 0 ≤ x * y → |x * y * z * s| ≤ √2 / 32 * (x ^ 2 + y ^ 2 + z ^ 2 + s ^ 2) ^ 2\nh' : ¬0 ≤ x * y\nh : 0 ≤ z * x\n⊢ (x ^ 2 + y ^ 2 + z ^ 2 + s ^ 2) ^ 2 = (z ^ 2 + x ^ 2 + y ^ 2 + s ^ 2) ^ 2\n---\ncase inr.inr\nx y z s : ℝ\nhxyz : x + y + z = 0\nthis : ∀ (x y z : ℝ), x + y + z = 0 → 0 ≤ x * y → |x * y * z * s| ≤ √2 / 32 * (x ^ 2 + y ^ 2 + z ^ 2 + s ^ 2) ^ 2\nh' : ¬0 ≤ x * y\nh : 0 ≤ z * x\n⊢ z + x + y = 0"
},
{
"line": "skip",
"before_state": "case h.e'_3.h.e'_4\nx y z s : ℝ\nhxyz : x + y + z = 0\nthis : ∀ (x y z : ℝ), x + y + z = 0 → 0 ≤ x * y → |x * y * z * s| ≤ √2 / 32 * (x ^ 2 + y ^ 2 + z ^ 2 + s ^ 2) ^ 2\nh' : ¬0 ≤ x * y\nh : 0 ≤ z * x\n⊢ x * y * z * s = z * x * y * s\n---\ncase h.e'_4.h.e'_6\nx y z s : ℝ\nhxyz : x + y + z = 0\nthis : ∀ (x y z : ℝ), x + y + z = 0 → 0 ≤ x * y → |x * y * z * s| ≤ √2 / 32 * (x ^ 2 + y ^ 2 + z ^ 2 + s ^ 2) ^ 2\nh' : ¬0 ≤ x * y\nh : 0 ≤ z * x\n⊢ (x ^ 2 + y ^ 2 + z ^ 2 + s ^ 2) ^ 2 = (z ^ 2 + x ^ 2 + y ^ 2 + s ^ 2) ^ 2\n---\ncase inr.inr\nx y z s : ℝ\nhxyz : x + y + z = 0\nthis : ∀ (x y z : ℝ), x + y + z = 0 → 0 ≤ x * y → |x * y * z * s| ≤ √2 / 32 * (x ^ 2 + y ^ 2 + z ^ 2 + s ^ 2) ^ 2\nh' : ¬0 ≤ x * y\nh : 0 ≤ z * x\n⊢ z + x + y = 0",
"after_state": "case h.e'_3.h.e'_4\nx y z s : ℝ\nhxyz : x + y + z = 0\nthis : ∀ (x y z : ℝ), x + y + z = 0 → 0 ≤ x * y → |x * y * z * s| ≤ √2 / 32 * (x ^ 2 + y ^ 2 + z ^ 2 + s ^ 2) ^ 2\nh' : ¬0 ≤ x * y\nh : 0 ≤ z * x\n⊢ x * y * z * s = z * x * y * s\n---\ncase h.e'_4.h.e'_6\nx y z s : ℝ\nhxyz : x + y + z = 0\nthis : ∀ (x y z : ℝ), x + y + z = 0 → 0 ≤ x * y → |x * y * z * s| ≤ √2 / 32 * (x ^ 2 + y ^ 2 + z ^ 2 + s ^ 2) ^ 2\nh' : ¬0 ≤ x * y\nh : 0 ≤ z * x\n⊢ (x ^ 2 + y ^ 2 + z ^ 2 + s ^ 2) ^ 2 = (z ^ 2 + x ^ 2 + y ^ 2 + s ^ 2) ^ 2\n---\ncase inr.inr\nx y z s : ℝ\nhxyz : x + y + z = 0\nthis : ∀ (x y z : ℝ), x + y + z = 0 → 0 ≤ x * y → |x * y * z * s| ≤ √2 / 32 * (x ^ 2 + y ^ 2 + z ^ 2 + s ^ 2) ^ 2\nh' : ¬0 ≤ x * y\nh : 0 ≤ z * x\n⊢ z + x + y = 0"
},
{
"line": "all_goals linarith",
"before_state": "case h.e'_3.h.e'_4\nx y z s : ℝ\nhxyz : x + y + z = 0\nthis : ∀ (x y z : ℝ), x + y + z = 0 → 0 ≤ x * y → |x * y * z * s| ≤ √2 / 32 * (x ^ 2 + y ^ 2 + z ^ 2 + s ^ 2) ^ 2\nh' : ¬0 ≤ x * y\nh : 0 ≤ z * x\n⊢ x * y * z * s = z * x * y * s\n---\ncase h.e'_4.h.e'_6\nx y z s : ℝ\nhxyz : x + y + z = 0\nthis : ∀ (x y z : ℝ), x + y + z = 0 → 0 ≤ x * y → |x * y * z * s| ≤ √2 / 32 * (x ^ 2 + y ^ 2 + z ^ 2 + s ^ 2) ^ 2\nh' : ¬0 ≤ x * y\nh : 0 ≤ z * x\n⊢ (x ^ 2 + y ^ 2 + z ^ 2 + s ^ 2) ^ 2 = (z ^ 2 + x ^ 2 + y ^ 2 + s ^ 2) ^ 2\n---\ncase inr.inr\nx y z s : ℝ\nhxyz : x + y + z = 0\nthis : ∀ (x y z : ℝ), x + y + z = 0 → 0 ≤ x * y → |x * y * z * s| ≤ √2 / 32 * (x ^ 2 + y ^ 2 + z ^ 2 + s ^ 2) ^ 2\nh' : ¬0 ≤ x * y\nh : 0 ≤ z * x\n⊢ z + x + y = 0",
"after_state": "No Goals!"
},
{
"line": "linarith",
"before_state": "case h.e'_3.h.e'_4\nx y z s : ℝ\nhxyz : x + y + z = 0\nthis : ∀ (x y z : ℝ), x + y + z = 0 → 0 ≤ x * y → |x * y * z * s| ≤ √2 / 32 * (x ^ 2 + y ^ 2 + z ^ 2 + s ^ 2) ^ 2\nh' : ¬0 ≤ x * y\nh : 0 ≤ z * x\n⊢ x * y * z * s = z * x * y * s",
"after_state": "No Goals!"
},
{
"line": "ring1",
"before_state": "x y z s : ℝ\nhxyz : x + y + z = 0\nthis : ∀ (x y z : ℝ), x + y + z = 0 → 0 ≤ x * y → |x * y * z * s| ≤ √2 / 32 * (x ^ 2 + y ^ 2 + z ^ 2 + s ^ 2) ^ 2\nh' : ¬0 ≤ x * y\nh : 0 ≤ z * x\na✝ : x * y * z * s < z * x * y * s\n⊢ x * y * z * s - z * x * y * s = 0",
"after_state": "No Goals!"
},
{
"line": "ring1",
"before_state": "x y z s : ℝ\nhxyz : x + y + z = 0\nthis : ∀ (x y z : ℝ), x + y + z = 0 → 0 ≤ x * y → |x * y * z * s| ≤ √2 / 32 * (x ^ 2 + y ^ 2 + z ^ 2 + s ^ 2) ^ 2\nh' : ¬0 ≤ x * y\nh : 0 ≤ z * x\na✝ : z * x * y * s < x * y * z * s\n⊢ z * x * y * s - x * y * z * s = 0",
"after_state": "No Goals!"
},
{
"line": "linarith",
"before_state": "case h.e'_4.h.e'_6\nx y z s : ℝ\nhxyz : x + y + z = 0\nthis : ∀ (x y z : ℝ), x + y + z = 0 → 0 ≤ x * y → |x * y * z * s| ≤ √2 / 32 * (x ^ 2 + y ^ 2 + z ^ 2 + s ^ 2) ^ 2\nh' : ¬0 ≤ x * y\nh : 0 ≤ z * x\n⊢ (x ^ 2 + y ^ 2 + z ^ 2 + s ^ 2) ^ 2 = (z ^ 2 + x ^ 2 + y ^ 2 + s ^ 2) ^ 2",
"after_state": "No Goals!"
},
{
"line": "ring1",
"before_state": "x y z s : ℝ\nhxyz : x + y + z = 0\nthis : ∀ (x y z : ℝ), x + y + z = 0 → 0 ≤ x * y → |x * y * z * s| ≤ √2 / 32 * (x ^ 2 + y ^ 2 + z ^ 2 + s ^ 2) ^ 2\nh' : ¬0 ≤ x * y\nh : 0 ≤ z * x\na✝ : (x ^ 2 + y ^ 2 + z ^ 2 + s ^ 2) ^ 2 < (z ^ 2 + x ^ 2 + y ^ 2 + s ^ 2) ^ 2\n⊢ (x ^ 2 + y ^ 2 + z ^ 2 + s ^ 2) ^ 2 - (z ^ 2 + x ^ 2 + y ^ 2 + s ^ 2) ^ 2 = 0",
"after_state": "No Goals!"
},
{
"line": "ring1",
"before_state": "x y z s : ℝ\nhxyz : x + y + z = 0\nthis : ∀ (x y z : ℝ), x + y + z = 0 → 0 ≤ x * y → |x * y * z * s| ≤ √2 / 32 * (x ^ 2 + y ^ 2 + z ^ 2 + s ^ 2) ^ 2\nh' : ¬0 ≤ x * y\nh : 0 ≤ z * x\na✝ : (z ^ 2 + x ^ 2 + y ^ 2 + s ^ 2) ^ 2 < (x ^ 2 + y ^ 2 + z ^ 2 + s ^ 2) ^ 2\n⊢ (z ^ 2 + x ^ 2 + y ^ 2 + s ^ 2) ^ 2 - (x ^ 2 + y ^ 2 + z ^ 2 + s ^ 2) ^ 2 = 0",
"after_state": "No Goals!"
},
{
"line": "linarith",
"before_state": "case inr.inr\nx y z s : ℝ\nhxyz : x + y + z = 0\nthis : ∀ (x y z : ℝ), x + y + z = 0 → 0 ≤ x * y → |x * y * z * s| ≤ √2 / 32 * (x ^ 2 + y ^ 2 + z ^ 2 + s ^ 2) ^ 2\nh' : ¬0 ≤ x * y\nh : 0 ≤ z * x\n⊢ z + x + y = 0",
"after_state": "No Goals!"
},
{
"line": "ring1",
"before_state": "x y z s : ℝ\nhxyz : x + y + z = 0\nthis : ∀ (x y z : ℝ), x + y + z = 0 → 0 ≤ x * y → |x * y * z * s| ≤ √2 / 32 * (x ^ 2 + y ^ 2 + z ^ 2 + s ^ 2) ^ 2\nh' : ¬0 ≤ x * y\nh : 0 ≤ z * x\na✝ : z + x + y < 0\n⊢ -(x + y + z - 0) + (z + x + y - 0) = 0",
"after_state": "No Goals!"
},
{
"line": "ring1",
"before_state": "x y z s : ℝ\nhxyz : x + y + z = 0\nthis : ∀ (x y z : ℝ), x + y + z = 0 → 0 ≤ x * y → |x * y * z * s| ≤ √2 / 32 * (x ^ 2 + y ^ 2 + z ^ 2 + s ^ 2) ^ 2\nh' : ¬0 ≤ x * y\nh : 0 ≤ z * x\na✝ : 0 < z + x + y\n⊢ x + y + z - 0 + (0 - (z + x + y)) = 0",
"after_state": "No Goals!"
}
] |
theorem proof₂ (M : ℝ)
(h : ∀ a b c : ℝ,
|a * b * (a ^ 2 - b ^ 2) + b * c * (b ^ 2 - c ^ 2) + c * a * (c ^ 2 - a ^ 2)| ≤
M * (a ^ 2 + b ^ 2 + c ^ 2) ^ 2) :
9 * sqrt 2 / 32 ≤ M := by
set α := sqrt (2:ℝ)
have hα : α ^ 2 = 2 := sq_sqrt (by norm_num)
let a := 2 - 3 * α
let c := 2 + 3 * α
calc _ = 18 ^ 2 * 2 * α / 48 ^ 2 := by ring
_ ≤ M := ?_
rw [div_le_iff₀ (by positivity)]
calc 18 ^ 2 * 2 * α
= 18 ^ 2 * α ^ 2 * α := by linear_combination -324 * α * hα
_ = abs (-(18 ^ 2 * α ^ 2 * α)) := by rw [abs_neg, abs_of_nonneg]; positivity
_ = |a * 2 * (a ^ 2 - 2 ^ 2) + 2 * c * (2 ^ 2 - c ^ 2) + c * a * (c ^ 2 - a ^ 2)| := by ring_nf!
_ ≤ M * (a ^ 2 + 2 ^ 2 + c ^ 2) ^ 2 := by apply h
_ = M * 48 ^ 2 := by linear_combination (324 * α ^ 2 + 1080) * M * hα
|
/root/DuelModelResearch/mathlib4/Archive/Imo/Imo2006Q3.lean
|
{
"open": [
"Real"
],
"variables": []
}
|
[
{
"line": "set α := sqrt (2 : ℝ)",
"before_state": "M : ℝ\nh :\n ∀ (a b c : ℝ),\n |a * b * (a ^ 2 - b ^ 2) + b * c * (b ^ 2 - c ^ 2) + c * a * (c ^ 2 - a ^ 2)| ≤ M * (a ^ 2 + b ^ 2 + c ^ 2) ^ 2\n⊢ 9 * √2 / 32 ≤ M",
"after_state": "M : ℝ\nh :\n ∀ (a b c : ℝ),\n |a * b * (a ^ 2 - b ^ 2) + b * c * (b ^ 2 - c ^ 2) + c * a * (c ^ 2 - a ^ 2)| ≤ M * (a ^ 2 + b ^ 2 + c ^ 2) ^ 2\nα : ℝ := √2\n⊢ 9 * α / 32 ≤ M"
},
{
"line": "try rewrite [show ?m✝ = α from rfl✝] at *",
"before_state": "M : ℝ\nh :\n ∀ (a b c : ℝ),\n |a * b * (a ^ 2 - b ^ 2) + b * c * (b ^ 2 - c ^ 2) + c * a * (c ^ 2 - a ^ 2)| ≤ M * (a ^ 2 + b ^ 2 + c ^ 2) ^ 2\nα : ℝ := √2\n⊢ 9 * √2 / 32 ≤ M",
"after_state": "M : ℝ\nh :\n ∀ (a b c : ℝ),\n |a * b * (a ^ 2 - b ^ 2) + b * c * (b ^ 2 - c ^ 2) + c * a * (c ^ 2 - a ^ 2)| ≤ M * (a ^ 2 + b ^ 2 + c ^ 2) ^ 2\nα : ℝ := √2\n⊢ 9 * α / 32 ≤ M"
},
{
"line": "first\n| rewrite [show ?m✝ = α from rfl✝] at *\n| skip",
"before_state": "M : ℝ\nh :\n ∀ (a b c : ℝ),\n |a * b * (a ^ 2 - b ^ 2) + b * c * (b ^ 2 - c ^ 2) + c * a * (c ^ 2 - a ^ 2)| ≤ M * (a ^ 2 + b ^ 2 + c ^ 2) ^ 2\nα : ℝ := √2\n⊢ 9 * √2 / 32 ≤ M",
"after_state": "M : ℝ\nh :\n ∀ (a b c : ℝ),\n |a * b * (a ^ 2 - b ^ 2) + b * c * (b ^ 2 - c ^ 2) + c * a * (c ^ 2 - a ^ 2)| ≤ M * (a ^ 2 + b ^ 2 + c ^ 2) ^ 2\nα : ℝ := √2\n⊢ 9 * α / 32 ≤ M"
},
{
"line": "rewrite [show ?m✝ = α from rfl✝] at *",
"before_state": "M : ℝ\nh :\n ∀ (a b c : ℝ),\n |a * b * (a ^ 2 - b ^ 2) + b * c * (b ^ 2 - c ^ 2) + c * a * (c ^ 2 - a ^ 2)| ≤ M * (a ^ 2 + b ^ 2 + c ^ 2) ^ 2\nα : ℝ := √2\n⊢ 9 * √2 / 32 ≤ M",
"after_state": "M : ℝ\nh :\n ∀ (a b c : ℝ),\n |a * b * (a ^ 2 - b ^ 2) + b * c * (b ^ 2 - c ^ 2) + c * a * (c ^ 2 - a ^ 2)| ≤ M * (a ^ 2 + b ^ 2 + c ^ 2) ^ 2\nα : ℝ := √2\n⊢ 9 * α / 32 ≤ M"
},
{
"line": "have hα : α ^ 2 = 2 := sq_sqrt (by norm_num)",
"before_state": "M : ℝ\nh :\n ∀ (a b c : ℝ),\n |a * b * (a ^ 2 - b ^ 2) + b * c * (b ^ 2 - c ^ 2) + c * a * (c ^ 2 - a ^ 2)| ≤ M * (a ^ 2 + b ^ 2 + c ^ 2) ^ 2\nα : ℝ := √2\n⊢ 9 * α / 32 ≤ M",
"after_state": "M : ℝ\nh :\n ∀ (a b c : ℝ),\n |a * b * (a ^ 2 - b ^ 2) + b * c * (b ^ 2 - c ^ 2) + c * a * (c ^ 2 - a ^ 2)| ≤ M * (a ^ 2 + b ^ 2 + c ^ 2) ^ 2\nα : ℝ := √2\nhα : α ^ 2 = 2\n⊢ 9 * α / 32 ≤ M"
},
{
"line": "refine_lift\n have hα : α ^ 2 = 2 := sq_sqrt (by norm_num);\n ?_",
"before_state": "M : ℝ\nh :\n ∀ (a b c : ℝ),\n |a * b * (a ^ 2 - b ^ 2) + b * c * (b ^ 2 - c ^ 2) + c * a * (c ^ 2 - a ^ 2)| ≤ M * (a ^ 2 + b ^ 2 + c ^ 2) ^ 2\nα : ℝ := √2\n⊢ 9 * α / 32 ≤ M",
"after_state": "M : ℝ\nh :\n ∀ (a b c : ℝ),\n |a * b * (a ^ 2 - b ^ 2) + b * c * (b ^ 2 - c ^ 2) + c * a * (c ^ 2 - a ^ 2)| ≤ M * (a ^ 2 + b ^ 2 + c ^ 2) ^ 2\nα : ℝ := √2\nhα : α ^ 2 = 2\n⊢ 9 * α / 32 ≤ M"
},
{
"line": "focus\n (refine\n no_implicit_lambda%\n (have hα : α ^ 2 = 2 := sq_sqrt (by norm_num);\n ?_);\n rotate_right)",
"before_state": "M : ℝ\nh :\n ∀ (a b c : ℝ),\n |a * b * (a ^ 2 - b ^ 2) + b * c * (b ^ 2 - c ^ 2) + c * a * (c ^ 2 - a ^ 2)| ≤ M * (a ^ 2 + b ^ 2 + c ^ 2) ^ 2\nα : ℝ := √2\n⊢ 9 * α / 32 ≤ M",
"after_state": "M : ℝ\nh :\n ∀ (a b c : ℝ),\n |a * b * (a ^ 2 - b ^ 2) + b * c * (b ^ 2 - c ^ 2) + c * a * (c ^ 2 - a ^ 2)| ≤ M * (a ^ 2 + b ^ 2 + c ^ 2) ^ 2\nα : ℝ := √2\nhα : α ^ 2 = 2\n⊢ 9 * α / 32 ≤ M"
},
{
"line": "refine\n no_implicit_lambda%\n (have hα : α ^ 2 = 2 := sq_sqrt (by norm_num);\n ?_)",
"before_state": "M : ℝ\nh :\n ∀ (a b c : ℝ),\n |a * b * (a ^ 2 - b ^ 2) + b * c * (b ^ 2 - c ^ 2) + c * a * (c ^ 2 - a ^ 2)| ≤ M * (a ^ 2 + b ^ 2 + c ^ 2) ^ 2\nα : ℝ := √2\n⊢ 9 * α / 32 ≤ M",
"after_state": "M : ℝ\nh :\n ∀ (a b c : ℝ),\n |a * b * (a ^ 2 - b ^ 2) + b * c * (b ^ 2 - c ^ 2) + c * a * (c ^ 2 - a ^ 2)| ≤ M * (a ^ 2 + b ^ 2 + c ^ 2) ^ 2\nα : ℝ := √2\nhα : α ^ 2 = 2\n⊢ 9 * α / 32 ≤ M"
},
{
"line": "norm_num",
"before_state": "M : ℝ\nh :\n ∀ (a b c : ℝ),\n |a * b * (a ^ 2 - b ^ 2) + b * c * (b ^ 2 - c ^ 2) + c * a * (c ^ 2 - a ^ 2)| ≤ M * (a ^ 2 + b ^ 2 + c ^ 2) ^ 2\nα : ℝ := √2\n⊢ 0 ≤ 2",
"after_state": "No Goals!"
},
{
"line": "rotate_right",
"before_state": "M : ℝ\nh :\n ∀ (a b c : ℝ),\n |a * b * (a ^ 2 - b ^ 2) + b * c * (b ^ 2 - c ^ 2) + c * a * (c ^ 2 - a ^ 2)| ≤ M * (a ^ 2 + b ^ 2 + c ^ 2) ^ 2\nα : ℝ := √2\nhα : α ^ 2 = 2\n⊢ 9 * α / 32 ≤ M",
"after_state": "M : ℝ\nh :\n ∀ (a b c : ℝ),\n |a * b * (a ^ 2 - b ^ 2) + b * c * (b ^ 2 - c ^ 2) + c * a * (c ^ 2 - a ^ 2)| ≤ M * (a ^ 2 + b ^ 2 + c ^ 2) ^ 2\nα : ℝ := √2\nhα : α ^ 2 = 2\n⊢ 9 * α / 32 ≤ M"
},
{
"line": "let a := 2 - 3 * α",
"before_state": "M : ℝ\nh :\n ∀ (a b c : ℝ),\n |a * b * (a ^ 2 - b ^ 2) + b * c * (b ^ 2 - c ^ 2) + c * a * (c ^ 2 - a ^ 2)| ≤ M * (a ^ 2 + b ^ 2 + c ^ 2) ^ 2\nα : ℝ := √2\nhα : α ^ 2 = 2\n⊢ 9 * α / 32 ≤ M",
"after_state": "M : ℝ\nh :\n ∀ (a b c : ℝ),\n |a * b * (a ^ 2 - b ^ 2) + b * c * (b ^ 2 - c ^ 2) + c * a * (c ^ 2 - a ^ 2)| ≤ M * (a ^ 2 + b ^ 2 + c ^ 2) ^ 2\nα : ℝ := √2\nhα : α ^ 2 = 2\na : ℝ := 2 - 3 * α\n⊢ 9 * α / 32 ≤ M"
},
{
"line": "refine_lift\n let a := 2 - 3 * α;\n ?_",
"before_state": "M : ℝ\nh :\n ∀ (a b c : ℝ),\n |a * b * (a ^ 2 - b ^ 2) + b * c * (b ^ 2 - c ^ 2) + c * a * (c ^ 2 - a ^ 2)| ≤ M * (a ^ 2 + b ^ 2 + c ^ 2) ^ 2\nα : ℝ := √2\nhα : α ^ 2 = 2\n⊢ 9 * α / 32 ≤ M",
"after_state": "M : ℝ\nh :\n ∀ (a b c : ℝ),\n |a * b * (a ^ 2 - b ^ 2) + b * c * (b ^ 2 - c ^ 2) + c * a * (c ^ 2 - a ^ 2)| ≤ M * (a ^ 2 + b ^ 2 + c ^ 2) ^ 2\nα : ℝ := √2\nhα : α ^ 2 = 2\na : ℝ := 2 - 3 * α\n⊢ 9 * α / 32 ≤ M"
},
{
"line": "focus\n (refine\n no_implicit_lambda%\n (let a := 2 - 3 * α;\n ?_);\n rotate_right)",
"before_state": "M : ℝ\nh :\n ∀ (a b c : ℝ),\n |a * b * (a ^ 2 - b ^ 2) + b * c * (b ^ 2 - c ^ 2) + c * a * (c ^ 2 - a ^ 2)| ≤ M * (a ^ 2 + b ^ 2 + c ^ 2) ^ 2\nα : ℝ := √2\nhα : α ^ 2 = 2\n⊢ 9 * α / 32 ≤ M",
"after_state": "M : ℝ\nh :\n ∀ (a b c : ℝ),\n |a * b * (a ^ 2 - b ^ 2) + b * c * (b ^ 2 - c ^ 2) + c * a * (c ^ 2 - a ^ 2)| ≤ M * (a ^ 2 + b ^ 2 + c ^ 2) ^ 2\nα : ℝ := √2\nhα : α ^ 2 = 2\na : ℝ := 2 - 3 * α\n⊢ 9 * α / 32 ≤ M"
},
{
"line": "refine\n no_implicit_lambda%\n (let a := 2 - 3 * α;\n ?_)",
"before_state": "M : ℝ\nh :\n ∀ (a b c : ℝ),\n |a * b * (a ^ 2 - b ^ 2) + b * c * (b ^ 2 - c ^ 2) + c * a * (c ^ 2 - a ^ 2)| ≤ M * (a ^ 2 + b ^ 2 + c ^ 2) ^ 2\nα : ℝ := √2\nhα : α ^ 2 = 2\n⊢ 9 * α / 32 ≤ M",
"after_state": "M : ℝ\nh :\n ∀ (a b c : ℝ),\n |a * b * (a ^ 2 - b ^ 2) + b * c * (b ^ 2 - c ^ 2) + c * a * (c ^ 2 - a ^ 2)| ≤ M * (a ^ 2 + b ^ 2 + c ^ 2) ^ 2\nα : ℝ := √2\nhα : α ^ 2 = 2\na : ℝ := 2 - 3 * α\n⊢ 9 * α / 32 ≤ M"
},
{
"line": "rotate_right",
"before_state": "M : ℝ\nh :\n ∀ (a b c : ℝ),\n |a * b * (a ^ 2 - b ^ 2) + b * c * (b ^ 2 - c ^ 2) + c * a * (c ^ 2 - a ^ 2)| ≤ M * (a ^ 2 + b ^ 2 + c ^ 2) ^ 2\nα : ℝ := √2\nhα : α ^ 2 = 2\na : ℝ := 2 - 3 * α\n⊢ 9 * α / 32 ≤ M",
"after_state": "M : ℝ\nh :\n ∀ (a b c : ℝ),\n |a * b * (a ^ 2 - b ^ 2) + b * c * (b ^ 2 - c ^ 2) + c * a * (c ^ 2 - a ^ 2)| ≤ M * (a ^ 2 + b ^ 2 + c ^ 2) ^ 2\nα : ℝ := √2\nhα : α ^ 2 = 2\na : ℝ := 2 - 3 * α\n⊢ 9 * α / 32 ≤ M"
},
{
"line": "let c := 2 + 3 * α",
"before_state": "M : ℝ\nh :\n ∀ (a b c : ℝ),\n |a * b * (a ^ 2 - b ^ 2) + b * c * (b ^ 2 - c ^ 2) + c * a * (c ^ 2 - a ^ 2)| ≤ M * (a ^ 2 + b ^ 2 + c ^ 2) ^ 2\nα : ℝ := √2\nhα : α ^ 2 = 2\na : ℝ := 2 - 3 * α\n⊢ 9 * α / 32 ≤ M",
"after_state": "M : ℝ\nh :\n ∀ (a b c : ℝ),\n |a * b * (a ^ 2 - b ^ 2) + b * c * (b ^ 2 - c ^ 2) + c * a * (c ^ 2 - a ^ 2)| ≤ M * (a ^ 2 + b ^ 2 + c ^ 2) ^ 2\nα : ℝ := √2\nhα : α ^ 2 = 2\na : ℝ := 2 - 3 * α\nc : ℝ := 2 + 3 * α\n⊢ 9 * α / 32 ≤ M"
},
{
"line": "refine_lift\n let c := 2 + 3 * α;\n ?_",
"before_state": "M : ℝ\nh :\n ∀ (a b c : ℝ),\n |a * b * (a ^ 2 - b ^ 2) + b * c * (b ^ 2 - c ^ 2) + c * a * (c ^ 2 - a ^ 2)| ≤ M * (a ^ 2 + b ^ 2 + c ^ 2) ^ 2\nα : ℝ := √2\nhα : α ^ 2 = 2\na : ℝ := 2 - 3 * α\n⊢ 9 * α / 32 ≤ M",
"after_state": "M : ℝ\nh :\n ∀ (a b c : ℝ),\n |a * b * (a ^ 2 - b ^ 2) + b * c * (b ^ 2 - c ^ 2) + c * a * (c ^ 2 - a ^ 2)| ≤ M * (a ^ 2 + b ^ 2 + c ^ 2) ^ 2\nα : ℝ := √2\nhα : α ^ 2 = 2\na : ℝ := 2 - 3 * α\nc : ℝ := 2 + 3 * α\n⊢ 9 * α / 32 ≤ M"
},
{
"line": "focus\n (refine\n no_implicit_lambda%\n (let c := 2 + 3 * α;\n ?_);\n rotate_right)",
"before_state": "M : ℝ\nh :\n ∀ (a b c : ℝ),\n |a * b * (a ^ 2 - b ^ 2) + b * c * (b ^ 2 - c ^ 2) + c * a * (c ^ 2 - a ^ 2)| ≤ M * (a ^ 2 + b ^ 2 + c ^ 2) ^ 2\nα : ℝ := √2\nhα : α ^ 2 = 2\na : ℝ := 2 - 3 * α\n⊢ 9 * α / 32 ≤ M",
"after_state": "M : ℝ\nh :\n ∀ (a b c : ℝ),\n |a * b * (a ^ 2 - b ^ 2) + b * c * (b ^ 2 - c ^ 2) + c * a * (c ^ 2 - a ^ 2)| ≤ M * (a ^ 2 + b ^ 2 + c ^ 2) ^ 2\nα : ℝ := √2\nhα : α ^ 2 = 2\na : ℝ := 2 - 3 * α\nc : ℝ := 2 + 3 * α\n⊢ 9 * α / 32 ≤ M"
},
{
"line": "refine\n no_implicit_lambda%\n (let c := 2 + 3 * α;\n ?_)",
"before_state": "M : ℝ\nh :\n ∀ (a b c : ℝ),\n |a * b * (a ^ 2 - b ^ 2) + b * c * (b ^ 2 - c ^ 2) + c * a * (c ^ 2 - a ^ 2)| ≤ M * (a ^ 2 + b ^ 2 + c ^ 2) ^ 2\nα : ℝ := √2\nhα : α ^ 2 = 2\na : ℝ := 2 - 3 * α\n⊢ 9 * α / 32 ≤ M",
"after_state": "M : ℝ\nh :\n ∀ (a b c : ℝ),\n |a * b * (a ^ 2 - b ^ 2) + b * c * (b ^ 2 - c ^ 2) + c * a * (c ^ 2 - a ^ 2)| ≤ M * (a ^ 2 + b ^ 2 + c ^ 2) ^ 2\nα : ℝ := √2\nhα : α ^ 2 = 2\na : ℝ := 2 - 3 * α\nc : ℝ := 2 + 3 * α\n⊢ 9 * α / 32 ≤ M"
},
{
"line": "rotate_right",
"before_state": "M : ℝ\nh :\n ∀ (a b c : ℝ),\n |a * b * (a ^ 2 - b ^ 2) + b * c * (b ^ 2 - c ^ 2) + c * a * (c ^ 2 - a ^ 2)| ≤ M * (a ^ 2 + b ^ 2 + c ^ 2) ^ 2\nα : ℝ := √2\nhα : α ^ 2 = 2\na : ℝ := 2 - 3 * α\nc : ℝ := 2 + 3 * α\n⊢ 9 * α / 32 ≤ M",
"after_state": "M : ℝ\nh :\n ∀ (a b c : ℝ),\n |a * b * (a ^ 2 - b ^ 2) + b * c * (b ^ 2 - c ^ 2) + c * a * (c ^ 2 - a ^ 2)| ≤ M * (a ^ 2 + b ^ 2 + c ^ 2) ^ 2\nα : ℝ := √2\nhα : α ^ 2 = 2\na : ℝ := 2 - 3 * α\nc : ℝ := 2 + 3 * α\n⊢ 9 * α / 32 ≤ M"
},
{
"line": "calc\n _ = 18 ^ 2 * 2 * α / 48 ^ 2 := by ring\n _ ≤ M := ?_",
"before_state": "M : ℝ\nh :\n ∀ (a b c : ℝ),\n |a * b * (a ^ 2 - b ^ 2) + b * c * (b ^ 2 - c ^ 2) + c * a * (c ^ 2 - a ^ 2)| ≤ M * (a ^ 2 + b ^ 2 + c ^ 2) ^ 2\nα : ℝ := √2\nhα : α ^ 2 = 2\na : ℝ := 2 - 3 * α\nc : ℝ := 2 + 3 * α\n⊢ 9 * α / 32 ≤ M",
"after_state": "M : ℝ\nh :\n ∀ (a b c : ℝ),\n |a * b * (a ^ 2 - b ^ 2) + b * c * (b ^ 2 - c ^ 2) + c * a * (c ^ 2 - a ^ 2)| ≤ M * (a ^ 2 + b ^ 2 + c ^ 2) ^ 2\nα : ℝ := √2\nhα : α ^ 2 = 2\na : ℝ := 2 - 3 * α\nc : ℝ := 2 + 3 * α\n⊢ 18 ^ 2 * 2 * α / 48 ^ 2 ≤ M"
},
{
"line": "ring",
"before_state": "M : ℝ\nh :\n ∀ (a b c : ℝ),\n |a * b * (a ^ 2 - b ^ 2) + b * c * (b ^ 2 - c ^ 2) + c * a * (c ^ 2 - a ^ 2)| ≤ M * (a ^ 2 + b ^ 2 + c ^ 2) ^ 2\nα : ℝ := √2\nhα : α ^ 2 = 2\na : ℝ := 2 - 3 * α\nc : ℝ := 2 + 3 * α\n⊢ 9 * α / 32 = 18 ^ 2 * 2 * α / 48 ^ 2",
"after_state": "No Goals!"
},
{
"line": "first\n| ring1\n|\n try_this ring_nf\"\\n\\nThe `ring` tactic failed to close the goal. Use `ring_nf` to obtain a normal form.\n \\nNote that `ring` works primarily in *commutative* rings. \\\n If you have a noncommutative ring, abelian group or module, consider using \\\n `noncomm_ring`, `abel` or `module` instead.\"",
"before_state": "M : ℝ\nh :\n ∀ (a b c : ℝ),\n |a * b * (a ^ 2 - b ^ 2) + b * c * (b ^ 2 - c ^ 2) + c * a * (c ^ 2 - a ^ 2)| ≤ M * (a ^ 2 + b ^ 2 + c ^ 2) ^ 2\nα : ℝ := √2\nhα : α ^ 2 = 2\na : ℝ := 2 - 3 * α\nc : ℝ := 2 + 3 * α\n⊢ 9 * α / 32 = 18 ^ 2 * 2 * α / 48 ^ 2",
"after_state": "No Goals!"
},
{
"line": "ring1",
"before_state": "M : ℝ\nh :\n ∀ (a b c : ℝ),\n |a * b * (a ^ 2 - b ^ 2) + b * c * (b ^ 2 - c ^ 2) + c * a * (c ^ 2 - a ^ 2)| ≤ M * (a ^ 2 + b ^ 2 + c ^ 2) ^ 2\nα : ℝ := √2\nhα : α ^ 2 = 2\na : ℝ := 2 - 3 * α\nc : ℝ := 2 + 3 * α\n⊢ 9 * α / 32 = 18 ^ 2 * 2 * α / 48 ^ 2",
"after_state": "No Goals!"
},
{
"line": "rw [div_le_iff₀ (by positivity)]",
"before_state": "M : ℝ\nh :\n ∀ (a b c : ℝ),\n |a * b * (a ^ 2 - b ^ 2) + b * c * (b ^ 2 - c ^ 2) + c * a * (c ^ 2 - a ^ 2)| ≤ M * (a ^ 2 + b ^ 2 + c ^ 2) ^ 2\nα : ℝ := √2\nhα : α ^ 2 = 2\na : ℝ := 2 - 3 * α\nc : ℝ := 2 + 3 * α\n⊢ 18 ^ 2 * 2 * α / 48 ^ 2 ≤ M",
"after_state": "M : ℝ\nh :\n ∀ (a b c : ℝ),\n |a * b * (a ^ 2 - b ^ 2) + b * c * (b ^ 2 - c ^ 2) + c * a * (c ^ 2 - a ^ 2)| ≤ M * (a ^ 2 + b ^ 2 + c ^ 2) ^ 2\nα : ℝ := √2\nhα : α ^ 2 = 2\na : ℝ := 2 - 3 * α\nc : ℝ := 2 + 3 * α\n⊢ 18 ^ 2 * 2 * α ≤ M * 48 ^ 2"
},
{
"line": "rewrite [div_le_iff₀ (by positivity)]",
"before_state": "M : ℝ\nh :\n ∀ (a b c : ℝ),\n |a * b * (a ^ 2 - b ^ 2) + b * c * (b ^ 2 - c ^ 2) + c * a * (c ^ 2 - a ^ 2)| ≤ M * (a ^ 2 + b ^ 2 + c ^ 2) ^ 2\nα : ℝ := √2\nhα : α ^ 2 = 2\na : ℝ := 2 - 3 * α\nc : ℝ := 2 + 3 * α\n⊢ 18 ^ 2 * 2 * α / 48 ^ 2 ≤ M",
"after_state": "M : ℝ\nh :\n ∀ (a b c : ℝ),\n |a * b * (a ^ 2 - b ^ 2) + b * c * (b ^ 2 - c ^ 2) + c * a * (c ^ 2 - a ^ 2)| ≤ M * (a ^ 2 + b ^ 2 + c ^ 2) ^ 2\nα : ℝ := √2\nhα : α ^ 2 = 2\na : ℝ := 2 - 3 * α\nc : ℝ := 2 + 3 * α\n⊢ 18 ^ 2 * 2 * α ≤ M * 48 ^ 2"
},
{
"line": "positivity",
"before_state": "M : ℝ\nh :\n ∀ (a b c : ℝ),\n |a * b * (a ^ 2 - b ^ 2) + b * c * (b ^ 2 - c ^ 2) + c * a * (c ^ 2 - a ^ 2)| ≤ M * (a ^ 2 + b ^ 2 + c ^ 2) ^ 2\nα : ℝ := √2\nhα : α ^ 2 = 2\na : ℝ := 2 - 3 * α\nc : ℝ := 2 + 3 * α\n⊢ 0 < 48 ^ 2",
"after_state": "No Goals!"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "M : ℝ\nh :\n ∀ (a b c : ℝ),\n |a * b * (a ^ 2 - b ^ 2) + b * c * (b ^ 2 - c ^ 2) + c * a * (c ^ 2 - a ^ 2)| ≤ M * (a ^ 2 + b ^ 2 + c ^ 2) ^ 2\nα : ℝ := √2\nhα : α ^ 2 = 2\na : ℝ := 2 - 3 * α\nc : ℝ := 2 + 3 * α\n⊢ 18 ^ 2 * 2 * α ≤ M * 48 ^ 2",
"after_state": "M : ℝ\nh :\n ∀ (a b c : ℝ),\n |a * b * (a ^ 2 - b ^ 2) + b * c * (b ^ 2 - c ^ 2) + c * a * (c ^ 2 - a ^ 2)| ≤ M * (a ^ 2 + b ^ 2 + c ^ 2) ^ 2\nα : ℝ := √2\nhα : α ^ 2 = 2\na : ℝ := 2 - 3 * α\nc : ℝ := 2 + 3 * α\n⊢ 18 ^ 2 * 2 * α ≤ M * 48 ^ 2"
},
{
"line": "try (with_reducible rfl)",
"before_state": "M : ℝ\nh :\n ∀ (a b c : ℝ),\n |a * b * (a ^ 2 - b ^ 2) + b * c * (b ^ 2 - c ^ 2) + c * a * (c ^ 2 - a ^ 2)| ≤ M * (a ^ 2 + b ^ 2 + c ^ 2) ^ 2\nα : ℝ := √2\nhα : α ^ 2 = 2\na : ℝ := 2 - 3 * α\nc : ℝ := 2 + 3 * α\n⊢ 18 ^ 2 * 2 * α ≤ M * 48 ^ 2",
"after_state": "M : ℝ\nh :\n ∀ (a b c : ℝ),\n |a * b * (a ^ 2 - b ^ 2) + b * c * (b ^ 2 - c ^ 2) + c * a * (c ^ 2 - a ^ 2)| ≤ M * (a ^ 2 + b ^ 2 + c ^ 2) ^ 2\nα : ℝ := √2\nhα : α ^ 2 = 2\na : ℝ := 2 - 3 * α\nc : ℝ := 2 + 3 * α\n⊢ 18 ^ 2 * 2 * α ≤ M * 48 ^ 2"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "M : ℝ\nh :\n ∀ (a b c : ℝ),\n |a * b * (a ^ 2 - b ^ 2) + b * c * (b ^ 2 - c ^ 2) + c * a * (c ^ 2 - a ^ 2)| ≤ M * (a ^ 2 + b ^ 2 + c ^ 2) ^ 2\nα : ℝ := √2\nhα : α ^ 2 = 2\na : ℝ := 2 - 3 * α\nc : ℝ := 2 + 3 * α\n⊢ 18 ^ 2 * 2 * α ≤ M * 48 ^ 2",
"after_state": "M : ℝ\nh :\n ∀ (a b c : ℝ),\n |a * b * (a ^ 2 - b ^ 2) + b * c * (b ^ 2 - c ^ 2) + c * a * (c ^ 2 - a ^ 2)| ≤ M * (a ^ 2 + b ^ 2 + c ^ 2) ^ 2\nα : ℝ := √2\nhα : α ^ 2 = 2\na : ℝ := 2 - 3 * α\nc : ℝ := 2 + 3 * α\n⊢ 18 ^ 2 * 2 * α ≤ M * 48 ^ 2"
},
{
"line": "with_reducible rfl",
"before_state": "M : ℝ\nh :\n ∀ (a b c : ℝ),\n |a * b * (a ^ 2 - b ^ 2) + b * c * (b ^ 2 - c ^ 2) + c * a * (c ^ 2 - a ^ 2)| ≤ M * (a ^ 2 + b ^ 2 + c ^ 2) ^ 2\nα : ℝ := √2\nhα : α ^ 2 = 2\na : ℝ := 2 - 3 * α\nc : ℝ := 2 + 3 * α\n⊢ 18 ^ 2 * 2 * α ≤ M * 48 ^ 2",
"after_state": "M : ℝ\nh :\n ∀ (a b c : ℝ),\n |a * b * (a ^ 2 - b ^ 2) + b * c * (b ^ 2 - c ^ 2) + c * a * (c ^ 2 - a ^ 2)| ≤ M * (a ^ 2 + b ^ 2 + c ^ 2) ^ 2\nα : ℝ := √2\nhα : α ^ 2 = 2\na : ℝ := 2 - 3 * α\nc : ℝ := 2 + 3 * α\n⊢ 18 ^ 2 * 2 * α ≤ M * 48 ^ 2"
},
{
"line": "rfl",
"before_state": "M : ℝ\nh :\n ∀ (a b c : ℝ),\n |a * b * (a ^ 2 - b ^ 2) + b * c * (b ^ 2 - c ^ 2) + c * a * (c ^ 2 - a ^ 2)| ≤ M * (a ^ 2 + b ^ 2 + c ^ 2) ^ 2\nα : ℝ := √2\nhα : α ^ 2 = 2\na : ℝ := 2 - 3 * α\nc : ℝ := 2 + 3 * α\n⊢ 18 ^ 2 * 2 * α ≤ M * 48 ^ 2",
"after_state": "M : ℝ\nh :\n ∀ (a b c : ℝ),\n |a * b * (a ^ 2 - b ^ 2) + b * c * (b ^ 2 - c ^ 2) + c * a * (c ^ 2 - a ^ 2)| ≤ M * (a ^ 2 + b ^ 2 + c ^ 2) ^ 2\nα : ℝ := √2\nhα : α ^ 2 = 2\na : ℝ := 2 - 3 * α\nc : ℝ := 2 + 3 * α\n⊢ 18 ^ 2 * 2 * α ≤ M * 48 ^ 2"
},
{
"line": "apply_rfl",
"before_state": "M : ℝ\nh :\n ∀ (a b c : ℝ),\n |a * b * (a ^ 2 - b ^ 2) + b * c * (b ^ 2 - c ^ 2) + c * a * (c ^ 2 - a ^ 2)| ≤ M * (a ^ 2 + b ^ 2 + c ^ 2) ^ 2\nα : ℝ := √2\nhα : α ^ 2 = 2\na : ℝ := 2 - 3 * α\nc : ℝ := 2 + 3 * α\n⊢ 18 ^ 2 * 2 * α ≤ M * 48 ^ 2",
"after_state": "M : ℝ\nh :\n ∀ (a b c : ℝ),\n |a * b * (a ^ 2 - b ^ 2) + b * c * (b ^ 2 - c ^ 2) + c * a * (c ^ 2 - a ^ 2)| ≤ M * (a ^ 2 + b ^ 2 + c ^ 2) ^ 2\nα : ℝ := √2\nhα : α ^ 2 = 2\na : ℝ := 2 - 3 * α\nc : ℝ := 2 + 3 * α\n⊢ 18 ^ 2 * 2 * α ≤ M * 48 ^ 2"
},
{
"line": "skip",
"before_state": "M : ℝ\nh :\n ∀ (a b c : ℝ),\n |a * b * (a ^ 2 - b ^ 2) + b * c * (b ^ 2 - c ^ 2) + c * a * (c ^ 2 - a ^ 2)| ≤ M * (a ^ 2 + b ^ 2 + c ^ 2) ^ 2\nα : ℝ := √2\nhα : α ^ 2 = 2\na : ℝ := 2 - 3 * α\nc : ℝ := 2 + 3 * α\n⊢ 18 ^ 2 * 2 * α ≤ M * 48 ^ 2",
"after_state": "M : ℝ\nh :\n ∀ (a b c : ℝ),\n |a * b * (a ^ 2 - b ^ 2) + b * c * (b ^ 2 - c ^ 2) + c * a * (c ^ 2 - a ^ 2)| ≤ M * (a ^ 2 + b ^ 2 + c ^ 2) ^ 2\nα : ℝ := √2\nhα : α ^ 2 = 2\na : ℝ := 2 - 3 * α\nc : ℝ := 2 + 3 * α\n⊢ 18 ^ 2 * 2 * α ≤ M * 48 ^ 2"
},
{
"line": "calc\n 18 ^ 2 * 2 * α = 18 ^ 2 * α ^ 2 * α := by linear_combination -324 * α * hα\n _ = abs (-(18 ^ 2 * α ^ 2 * α)) := by rw [abs_neg, abs_of_nonneg]; positivity\n _ = |a * 2 * (a ^ 2 - 2 ^ 2) + 2 * c * (2 ^ 2 - c ^ 2) + c * a * (c ^ 2 - a ^ 2)| := by ring_nf!\n _ ≤ M * (a ^ 2 + 2 ^ 2 + c ^ 2) ^ 2 := by apply h\n _ = M * 48 ^ 2 := by linear_combination (324 * α ^ 2 + 1080) * M * hα",
"before_state": "M : ℝ\nh :\n ∀ (a b c : ℝ),\n |a * b * (a ^ 2 - b ^ 2) + b * c * (b ^ 2 - c ^ 2) + c * a * (c ^ 2 - a ^ 2)| ≤ M * (a ^ 2 + b ^ 2 + c ^ 2) ^ 2\nα : ℝ := √2\nhα : α ^ 2 = 2\na : ℝ := 2 - 3 * α\nc : ℝ := 2 + 3 * α\n⊢ 18 ^ 2 * 2 * α ≤ M * 48 ^ 2",
"after_state": "No Goals!"
},
{
"line": "linear_combination -324 * α * hα",
"before_state": "M : ℝ\nh :\n ∀ (a b c : ℝ),\n |a * b * (a ^ 2 - b ^ 2) + b * c * (b ^ 2 - c ^ 2) + c * a * (c ^ 2 - a ^ 2)| ≤ M * (a ^ 2 + b ^ 2 + c ^ 2) ^ 2\nα : ℝ := √2\nhα : α ^ 2 = 2\na : ℝ := 2 - 3 * α\nc : ℝ := 2 + 3 * α\n⊢ 18 ^ 2 * 2 * α = 18 ^ 2 * α ^ 2 * α",
"after_state": "No Goals!"
},
{
"line": "rw [abs_neg, abs_of_nonneg]",
"before_state": "M : ℝ\nh :\n ∀ (a b c : ℝ),\n |a * b * (a ^ 2 - b ^ 2) + b * c * (b ^ 2 - c ^ 2) + c * a * (c ^ 2 - a ^ 2)| ≤ M * (a ^ 2 + b ^ 2 + c ^ 2) ^ 2\nα : ℝ := √2\nhα : α ^ 2 = 2\na : ℝ := 2 - 3 * α\nc : ℝ := 2 + 3 * α\n⊢ 18 ^ 2 * α ^ 2 * α = |(-(18 ^ 2 * α ^ 2 * α))|",
"after_state": "M : ℝ\nh :\n ∀ (a b c : ℝ),\n |a * b * (a ^ 2 - b ^ 2) + b * c * (b ^ 2 - c ^ 2) + c * a * (c ^ 2 - a ^ 2)| ≤ M * (a ^ 2 + b ^ 2 + c ^ 2) ^ 2\nα : ℝ := √2\nhα : α ^ 2 = 2\na : ℝ := 2 - 3 * α\nc : ℝ := 2 + 3 * α\n⊢ 0 ≤ 18 ^ 2 * α ^ 2 * α"
},
{
"line": "rewrite [abs_neg, abs_of_nonneg]",
"before_state": "M : ℝ\nh :\n ∀ (a b c : ℝ),\n |a * b * (a ^ 2 - b ^ 2) + b * c * (b ^ 2 - c ^ 2) + c * a * (c ^ 2 - a ^ 2)| ≤ M * (a ^ 2 + b ^ 2 + c ^ 2) ^ 2\nα : ℝ := √2\nhα : α ^ 2 = 2\na : ℝ := 2 - 3 * α\nc : ℝ := 2 + 3 * α\n⊢ 18 ^ 2 * α ^ 2 * α = |(-(18 ^ 2 * α ^ 2 * α))|",
"after_state": "M : ℝ\nh :\n ∀ (a b c : ℝ),\n |a * b * (a ^ 2 - b ^ 2) + b * c * (b ^ 2 - c ^ 2) + c * a * (c ^ 2 - a ^ 2)| ≤ M * (a ^ 2 + b ^ 2 + c ^ 2) ^ 2\nα : ℝ := √2\nhα : α ^ 2 = 2\na : ℝ := 2 - 3 * α\nc : ℝ := 2 + 3 * α\n⊢ 18 ^ 2 * α ^ 2 * α = 18 ^ 2 * α ^ 2 * α\n---\nM : ℝ\nh :\n ∀ (a b c : ℝ),\n |a * b * (a ^ 2 - b ^ 2) + b * c * (b ^ 2 - c ^ 2) + c * a * (c ^ 2 - a ^ 2)| ≤ M * (a ^ 2 + b ^ 2 + c ^ 2) ^ 2\nα : ℝ := √2\nhα : α ^ 2 = 2\na : ℝ := 2 - 3 * α\nc : ℝ := 2 + 3 * α\n⊢ 0 ≤ 18 ^ 2 * α ^ 2 * α"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "M : ℝ\nh :\n ∀ (a b c : ℝ),\n |a * b * (a ^ 2 - b ^ 2) + b * c * (b ^ 2 - c ^ 2) + c * a * (c ^ 2 - a ^ 2)| ≤ M * (a ^ 2 + b ^ 2 + c ^ 2) ^ 2\nα : ℝ := √2\nhα : α ^ 2 = 2\na : ℝ := 2 - 3 * α\nc : ℝ := 2 + 3 * α\n⊢ 18 ^ 2 * α ^ 2 * α = 18 ^ 2 * α ^ 2 * α\n---\nM : ℝ\nh :\n ∀ (a b c : ℝ),\n |a * b * (a ^ 2 - b ^ 2) + b * c * (b ^ 2 - c ^ 2) + c * a * (c ^ 2 - a ^ 2)| ≤ M * (a ^ 2 + b ^ 2 + c ^ 2) ^ 2\nα : ℝ := √2\nhα : α ^ 2 = 2\na : ℝ := 2 - 3 * α\nc : ℝ := 2 + 3 * α\n⊢ 0 ≤ 18 ^ 2 * α ^ 2 * α",
"after_state": "M : ℝ\nh :\n ∀ (a b c : ℝ),\n |a * b * (a ^ 2 - b ^ 2) + b * c * (b ^ 2 - c ^ 2) + c * a * (c ^ 2 - a ^ 2)| ≤ M * (a ^ 2 + b ^ 2 + c ^ 2) ^ 2\nα : ℝ := √2\nhα : α ^ 2 = 2\na : ℝ := 2 - 3 * α\nc : ℝ := 2 + 3 * α\n⊢ 0 ≤ 18 ^ 2 * α ^ 2 * α"
},
{
"line": "try (with_reducible rfl)",
"before_state": "M : ℝ\nh :\n ∀ (a b c : ℝ),\n |a * b * (a ^ 2 - b ^ 2) + b * c * (b ^ 2 - c ^ 2) + c * a * (c ^ 2 - a ^ 2)| ≤ M * (a ^ 2 + b ^ 2 + c ^ 2) ^ 2\nα : ℝ := √2\nhα : α ^ 2 = 2\na : ℝ := 2 - 3 * α\nc : ℝ := 2 + 3 * α\n⊢ 18 ^ 2 * α ^ 2 * α = 18 ^ 2 * α ^ 2 * α\n---\nM : ℝ\nh :\n ∀ (a b c : ℝ),\n |a * b * (a ^ 2 - b ^ 2) + b * c * (b ^ 2 - c ^ 2) + c * a * (c ^ 2 - a ^ 2)| ≤ M * (a ^ 2 + b ^ 2 + c ^ 2) ^ 2\nα : ℝ := √2\nhα : α ^ 2 = 2\na : ℝ := 2 - 3 * α\nc : ℝ := 2 + 3 * α\n⊢ 0 ≤ 18 ^ 2 * α ^ 2 * α",
"after_state": "M : ℝ\nh :\n ∀ (a b c : ℝ),\n |a * b * (a ^ 2 - b ^ 2) + b * c * (b ^ 2 - c ^ 2) + c * a * (c ^ 2 - a ^ 2)| ≤ M * (a ^ 2 + b ^ 2 + c ^ 2) ^ 2\nα : ℝ := √2\nhα : α ^ 2 = 2\na : ℝ := 2 - 3 * α\nc : ℝ := 2 + 3 * α\n⊢ 0 ≤ 18 ^ 2 * α ^ 2 * α"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "M : ℝ\nh :\n ∀ (a b c : ℝ),\n |a * b * (a ^ 2 - b ^ 2) + b * c * (b ^ 2 - c ^ 2) + c * a * (c ^ 2 - a ^ 2)| ≤ M * (a ^ 2 + b ^ 2 + c ^ 2) ^ 2\nα : ℝ := √2\nhα : α ^ 2 = 2\na : ℝ := 2 - 3 * α\nc : ℝ := 2 + 3 * α\n⊢ 18 ^ 2 * α ^ 2 * α = 18 ^ 2 * α ^ 2 * α\n---\nM : ℝ\nh :\n ∀ (a b c : ℝ),\n |a * b * (a ^ 2 - b ^ 2) + b * c * (b ^ 2 - c ^ 2) + c * a * (c ^ 2 - a ^ 2)| ≤ M * (a ^ 2 + b ^ 2 + c ^ 2) ^ 2\nα : ℝ := √2\nhα : α ^ 2 = 2\na : ℝ := 2 - 3 * α\nc : ℝ := 2 + 3 * α\n⊢ 0 ≤ 18 ^ 2 * α ^ 2 * α",
"after_state": "M : ℝ\nh :\n ∀ (a b c : ℝ),\n |a * b * (a ^ 2 - b ^ 2) + b * c * (b ^ 2 - c ^ 2) + c * a * (c ^ 2 - a ^ 2)| ≤ M * (a ^ 2 + b ^ 2 + c ^ 2) ^ 2\nα : ℝ := √2\nhα : α ^ 2 = 2\na : ℝ := 2 - 3 * α\nc : ℝ := 2 + 3 * α\n⊢ 0 ≤ 18 ^ 2 * α ^ 2 * α"
},
{
"line": "with_reducible rfl",
"before_state": "M : ℝ\nh :\n ∀ (a b c : ℝ),\n |a * b * (a ^ 2 - b ^ 2) + b * c * (b ^ 2 - c ^ 2) + c * a * (c ^ 2 - a ^ 2)| ≤ M * (a ^ 2 + b ^ 2 + c ^ 2) ^ 2\nα : ℝ := √2\nhα : α ^ 2 = 2\na : ℝ := 2 - 3 * α\nc : ℝ := 2 + 3 * α\n⊢ 18 ^ 2 * α ^ 2 * α = 18 ^ 2 * α ^ 2 * α\n---\nM : ℝ\nh :\n ∀ (a b c : ℝ),\n |a * b * (a ^ 2 - b ^ 2) + b * c * (b ^ 2 - c ^ 2) + c * a * (c ^ 2 - a ^ 2)| ≤ M * (a ^ 2 + b ^ 2 + c ^ 2) ^ 2\nα : ℝ := √2\nhα : α ^ 2 = 2\na : ℝ := 2 - 3 * α\nc : ℝ := 2 + 3 * α\n⊢ 0 ≤ 18 ^ 2 * α ^ 2 * α",
"after_state": "M : ℝ\nh :\n ∀ (a b c : ℝ),\n |a * b * (a ^ 2 - b ^ 2) + b * c * (b ^ 2 - c ^ 2) + c * a * (c ^ 2 - a ^ 2)| ≤ M * (a ^ 2 + b ^ 2 + c ^ 2) ^ 2\nα : ℝ := √2\nhα : α ^ 2 = 2\na : ℝ := 2 - 3 * α\nc : ℝ := 2 + 3 * α\n⊢ 0 ≤ 18 ^ 2 * α ^ 2 * α"
},
{
"line": "rfl",
"before_state": "M : ℝ\nh :\n ∀ (a b c : ℝ),\n |a * b * (a ^ 2 - b ^ 2) + b * c * (b ^ 2 - c ^ 2) + c * a * (c ^ 2 - a ^ 2)| ≤ M * (a ^ 2 + b ^ 2 + c ^ 2) ^ 2\nα : ℝ := √2\nhα : α ^ 2 = 2\na : ℝ := 2 - 3 * α\nc : ℝ := 2 + 3 * α\n⊢ 18 ^ 2 * α ^ 2 * α = 18 ^ 2 * α ^ 2 * α\n---\nM : ℝ\nh :\n ∀ (a b c : ℝ),\n |a * b * (a ^ 2 - b ^ 2) + b * c * (b ^ 2 - c ^ 2) + c * a * (c ^ 2 - a ^ 2)| ≤ M * (a ^ 2 + b ^ 2 + c ^ 2) ^ 2\nα : ℝ := √2\nhα : α ^ 2 = 2\na : ℝ := 2 - 3 * α\nc : ℝ := 2 + 3 * α\n⊢ 0 ≤ 18 ^ 2 * α ^ 2 * α",
"after_state": "M : ℝ\nh :\n ∀ (a b c : ℝ),\n |a * b * (a ^ 2 - b ^ 2) + b * c * (b ^ 2 - c ^ 2) + c * a * (c ^ 2 - a ^ 2)| ≤ M * (a ^ 2 + b ^ 2 + c ^ 2) ^ 2\nα : ℝ := √2\nhα : α ^ 2 = 2\na : ℝ := 2 - 3 * α\nc : ℝ := 2 + 3 * α\n⊢ 0 ≤ 18 ^ 2 * α ^ 2 * α"
},
{
"line": "eq_refl",
"before_state": "M : ℝ\nh :\n ∀ (a b c : ℝ),\n |a * b * (a ^ 2 - b ^ 2) + b * c * (b ^ 2 - c ^ 2) + c * a * (c ^ 2 - a ^ 2)| ≤ M * (a ^ 2 + b ^ 2 + c ^ 2) ^ 2\nα : ℝ := √2\nhα : α ^ 2 = 2\na : ℝ := 2 - 3 * α\nc : ℝ := 2 + 3 * α\n⊢ 18 ^ 2 * α ^ 2 * α = 18 ^ 2 * α ^ 2 * α\n---\nM : ℝ\nh :\n ∀ (a b c : ℝ),\n |a * b * (a ^ 2 - b ^ 2) + b * c * (b ^ 2 - c ^ 2) + c * a * (c ^ 2 - a ^ 2)| ≤ M * (a ^ 2 + b ^ 2 + c ^ 2) ^ 2\nα : ℝ := √2\nhα : α ^ 2 = 2\na : ℝ := 2 - 3 * α\nc : ℝ := 2 + 3 * α\n⊢ 0 ≤ 18 ^ 2 * α ^ 2 * α",
"after_state": "M : ℝ\nh :\n ∀ (a b c : ℝ),\n |a * b * (a ^ 2 - b ^ 2) + b * c * (b ^ 2 - c ^ 2) + c * a * (c ^ 2 - a ^ 2)| ≤ M * (a ^ 2 + b ^ 2 + c ^ 2) ^ 2\nα : ℝ := √2\nhα : α ^ 2 = 2\na : ℝ := 2 - 3 * α\nc : ℝ := 2 + 3 * α\n⊢ 0 ≤ 18 ^ 2 * α ^ 2 * α"
},
{
"line": "positivity",
"before_state": "M : ℝ\nh :\n ∀ (a b c : ℝ),\n |a * b * (a ^ 2 - b ^ 2) + b * c * (b ^ 2 - c ^ 2) + c * a * (c ^ 2 - a ^ 2)| ≤ M * (a ^ 2 + b ^ 2 + c ^ 2) ^ 2\nα : ℝ := √2\nhα : α ^ 2 = 2\na : ℝ := 2 - 3 * α\nc : ℝ := 2 + 3 * α\n⊢ 0 ≤ 18 ^ 2 * α ^ 2 * α",
"after_state": "No Goals!"
},
{
"line": "ring_nf!",
"before_state": "M : ℝ\nh :\n ∀ (a b c : ℝ),\n |a * b * (a ^ 2 - b ^ 2) + b * c * (b ^ 2 - c ^ 2) + c * a * (c ^ 2 - a ^ 2)| ≤ M * (a ^ 2 + b ^ 2 + c ^ 2) ^ 2\nα : ℝ := √2\nhα : α ^ 2 = 2\na : ℝ := 2 - 3 * α\nc : ℝ := 2 + 3 * α\n⊢ |(-(18 ^ 2 * α ^ 2 * α))| = |a * 2 * (a ^ 2 - 2 ^ 2) + 2 * c * (2 ^ 2 - c ^ 2) + c * a * (c ^ 2 - a ^ 2)|",
"after_state": "No Goals!"
},
{
"line": "ring_nf !",
"before_state": "M : ℝ\nh :\n ∀ (a b c : ℝ),\n |a * b * (a ^ 2 - b ^ 2) + b * c * (b ^ 2 - c ^ 2) + c * a * (c ^ 2 - a ^ 2)| ≤ M * (a ^ 2 + b ^ 2 + c ^ 2) ^ 2\nα : ℝ := √2\nhα : α ^ 2 = 2\na : ℝ := 2 - 3 * α\nc : ℝ := 2 + 3 * α\n⊢ |(-(18 ^ 2 * α ^ 2 * α))| = |a * 2 * (a ^ 2 - 2 ^ 2) + 2 * c * (2 ^ 2 - c ^ 2) + c * a * (c ^ 2 - a ^ 2)|",
"after_state": "No Goals!"
},
{
"line": "apply h",
"before_state": "M : ℝ\nh :\n ∀ (a b c : ℝ),\n |a * b * (a ^ 2 - b ^ 2) + b * c * (b ^ 2 - c ^ 2) + c * a * (c ^ 2 - a ^ 2)| ≤ M * (a ^ 2 + b ^ 2 + c ^ 2) ^ 2\nα : ℝ := √2\nhα : α ^ 2 = 2\na : ℝ := 2 - 3 * α\nc : ℝ := 2 + 3 * α\n⊢ |a * 2 * (a ^ 2 - 2 ^ 2) + 2 * c * (2 ^ 2 - c ^ 2) + c * a * (c ^ 2 - a ^ 2)| ≤ M * (a ^ 2 + 2 ^ 2 + c ^ 2) ^ 2",
"after_state": "No Goals!"
},
{
"line": "linear_combination (324 * α ^ 2 + 1080) * M * hα",
"before_state": "M : ℝ\nh :\n ∀ (a b c : ℝ),\n |a * b * (a ^ 2 - b ^ 2) + b * c * (b ^ 2 - c ^ 2) + c * a * (c ^ 2 - a ^ 2)| ≤ M * (a ^ 2 + b ^ 2 + c ^ 2) ^ 2\nα : ℝ := √2\nhα : α ^ 2 = 2\na : ℝ := 2 - 3 * α\nc : ℝ := 2 + 3 * α\n⊢ M * (a ^ 2 + 2 ^ 2 + c ^ 2) ^ 2 = M * 48 ^ 2",
"after_state": "No Goals!"
}
] |
theorem subst_abc {x y z : ℝ} (h : x * y * z = 1) :
∃ a b c : ℝ, a ≠ 0 ∧ b ≠ 0 ∧ c ≠ 0 ∧ x = a / b ∧ y = b / c ∧ z = c / a := by
use x, 1, 1 / y
obtain ⟨⟨hx, hy⟩, _⟩ : (x ≠ 0 ∧ y ≠ 0) ∧ z ≠ 0 := by
have := h.symm ▸ one_ne_zero
simpa [not_or] using this
have : z * (y * x) = 1 := by rw [← h]; ac_rfl
field_simp [*]
|
/root/DuelModelResearch/mathlib4/Archive/Imo/Imo2008Q2.lean
|
{
"open": [],
"variables": []
}
|
[
{
"line": "use x, 1, 1 / y",
"before_state": "x y z : ℝ\nh : x * y * z = 1\n⊢ ∃ a b c, a ≠ 0 ∧ b ≠ 0 ∧ c ≠ 0 ∧ x = a / b ∧ y = b / c ∧ z = c / a",
"after_state": "case h\nx y z : ℝ\nh : x * y * z = 1\n⊢ x ≠ 0 ∧ 1 ≠ 0 ∧ 1 / y ≠ 0 ∧ x = x / 1 ∧ y = 1 / (1 / y) ∧ z = 1 / y / x"
},
{
"line": "refine without_cdot(x : ?m✝)",
"before_state": "case w\nx y z : ℝ\nh : x * y * z = 1\n⊢ ℝ",
"after_state": "No Goals!"
},
{
"line": "refine without_cdot(1 : ?m✝)",
"before_state": "case w\nx y z : ℝ\nh : x * y * z = 1\n⊢ ℝ",
"after_state": "No Goals!"
},
{
"line": "refine without_cdot(1 / y : ?m✝)",
"before_state": "case w\nx y z : ℝ\nh : x * y * z = 1\n⊢ ℝ",
"after_state": "No Goals!"
},
{
"line": "try with_reducible use_discharger",
"before_state": "case h\nx y z : ℝ\nh : x * y * z = 1\n⊢ x ≠ 0 ∧ 1 ≠ 0 ∧ 1 / y ≠ 0 ∧ x = x / 1 ∧ y = 1 / (1 / y) ∧ z = 1 / y / x",
"after_state": "case h\nx y z : ℝ\nh : x * y * z = 1\n⊢ x ≠ 0 ∧ 1 ≠ 0 ∧ 1 / y ≠ 0 ∧ x = x / 1 ∧ y = 1 / (1 / y) ∧ z = 1 / y / x"
},
{
"line": "first\n| with_reducible use_discharger\n| skip",
"before_state": "case h\nx y z : ℝ\nh : x * y * z = 1\n⊢ x ≠ 0 ∧ 1 ≠ 0 ∧ 1 / y ≠ 0 ∧ x = x / 1 ∧ y = 1 / (1 / y) ∧ z = 1 / y / x",
"after_state": "case h\nx y z : ℝ\nh : x * y * z = 1\n⊢ x ≠ 0 ∧ 1 ≠ 0 ∧ 1 / y ≠ 0 ∧ x = x / 1 ∧ y = 1 / (1 / y) ∧ z = 1 / y / x"
},
{
"line": "with_reducible use_discharger",
"before_state": "case h\nx y z : ℝ\nh : x * y * z = 1\n⊢ x ≠ 0 ∧ 1 ≠ 0 ∧ 1 / y ≠ 0 ∧ x = x / 1 ∧ y = 1 / (1 / y) ∧ z = 1 / y / x",
"after_state": "case h\nx y z : ℝ\nh : x * y * z = 1\n⊢ x ≠ 0 ∧ 1 ≠ 0 ∧ 1 / y ≠ 0 ∧ x = x / 1 ∧ y = 1 / (1 / y) ∧ z = 1 / y / x"
},
{
"line": "use_discharger",
"before_state": "case h\nx y z : ℝ\nh : x * y * z = 1\n⊢ x ≠ 0 ∧ 1 ≠ 0 ∧ 1 / y ≠ 0 ∧ x = x / 1 ∧ y = 1 / (1 / y) ∧ z = 1 / y / x",
"after_state": "case h\nx y z : ℝ\nh : x * y * z = 1\n⊢ x ≠ 0 ∧ 1 ≠ 0 ∧ 1 / y ≠ 0 ∧ x = x / 1 ∧ y = 1 / (1 / y) ∧ z = 1 / y / x"
},
{
"line": "focus\n apply exists_prop.mpr✝\n with_annotate_state\"<;>\" skip\n all_goals use_discharger",
"before_state": "case h\nx y z : ℝ\nh : x * y * z = 1\n⊢ x ≠ 0 ∧ 1 ≠ 0 ∧ 1 / y ≠ 0 ∧ x = x / 1 ∧ y = 1 / (1 / y) ∧ z = 1 / y / x",
"after_state": "case h\nx y z : ℝ\nh : x * y * z = 1\n⊢ x ≠ 0 ∧ 1 ≠ 0 ∧ 1 / y ≠ 0 ∧ x = x / 1 ∧ y = 1 / (1 / y) ∧ z = 1 / y / x"
},
{
"line": "apply exists_prop.mpr✝",
"before_state": "case h\nx y z : ℝ\nh : x * y * z = 1\n⊢ x ≠ 0 ∧ 1 ≠ 0 ∧ 1 / y ≠ 0 ∧ x = x / 1 ∧ y = 1 / (1 / y) ∧ z = 1 / y / x",
"after_state": "case h\nx y z : ℝ\nh : x * y * z = 1\n⊢ x ≠ 0 ∧ 1 ≠ 0 ∧ 1 / y ≠ 0 ∧ x = x / 1 ∧ y = 1 / (1 / y) ∧ z = 1 / y / x"
},
{
"line": "skip",
"before_state": "case h\nx y z : ℝ\nh : x * y * z = 1\n⊢ x ≠ 0 ∧ 1 ≠ 0 ∧ 1 / y ≠ 0 ∧ x = x / 1 ∧ y = 1 / (1 / y) ∧ z = 1 / y / x",
"after_state": "case h\nx y z : ℝ\nh : x * y * z = 1\n⊢ x ≠ 0 ∧ 1 ≠ 0 ∧ 1 / y ≠ 0 ∧ x = x / 1 ∧ y = 1 / (1 / y) ∧ z = 1 / y / x"
},
{
"line": "obtain ⟨⟨hx, hy⟩, _⟩ : (x ≠ 0 ∧ y ≠ 0) ∧ z ≠ 0 :=\n by\n have := h.symm ▸ one_ne_zero\n simpa [not_or] using this",
"before_state": "case h\nx y z : ℝ\nh : x * y * z = 1\n⊢ x ≠ 0 ∧ 1 ≠ 0 ∧ 1 / y ≠ 0 ∧ x = x / 1 ∧ y = 1 / (1 / y) ∧ z = 1 / y / x",
"after_state": "case h.intro.intro\nx y z : ℝ\nh : x * y * z = 1\nright✝ : z ≠ 0\nhx : x ≠ 0\nhy : y ≠ 0\n⊢ x ≠ 0 ∧ 1 ≠ 0 ∧ 1 / y ≠ 0 ∧ x = x / 1 ∧ y = 1 / (1 / y) ∧ z = 1 / y / x"
},
{
"line": "have := h.symm ▸ one_ne_zero",
"before_state": "x y z : ℝ\nh : x * y * z = 1\n⊢ (x ≠ 0 ∧ y ≠ 0) ∧ z ≠ 0",
"after_state": "x y z : ℝ\nh : x * y * z = 1\nthis : x * y * z ≠ 0\n⊢ (x ≠ 0 ∧ y ≠ 0) ∧ z ≠ 0"
},
{
"line": "refine_lift\n have := h.symm ▸ one_ne_zero;\n ?_",
"before_state": "x y z : ℝ\nh : x * y * z = 1\n⊢ (x ≠ 0 ∧ y ≠ 0) ∧ z ≠ 0",
"after_state": "x y z : ℝ\nh : x * y * z = 1\nthis : x * y * z ≠ 0\n⊢ (x ≠ 0 ∧ y ≠ 0) ∧ z ≠ 0"
},
{
"line": "focus\n (refine\n no_implicit_lambda%\n (have := h.symm ▸ one_ne_zero;\n ?_);\n rotate_right)",
"before_state": "x y z : ℝ\nh : x * y * z = 1\n⊢ (x ≠ 0 ∧ y ≠ 0) ∧ z ≠ 0",
"after_state": "x y z : ℝ\nh : x * y * z = 1\nthis : x * y * z ≠ 0\n⊢ (x ≠ 0 ∧ y ≠ 0) ∧ z ≠ 0"
},
{
"line": "refine\n no_implicit_lambda%\n (have := h.symm ▸ one_ne_zero;\n ?_)",
"before_state": "x y z : ℝ\nh : x * y * z = 1\n⊢ (x ≠ 0 ∧ y ≠ 0) ∧ z ≠ 0",
"after_state": "x y z : ℝ\nh : x * y * z = 1\nthis : x * y * z ≠ 0\n⊢ (x ≠ 0 ∧ y ≠ 0) ∧ z ≠ 0"
},
{
"line": "rotate_right",
"before_state": "x y z : ℝ\nh : x * y * z = 1\nthis : x * y * z ≠ 0\n⊢ (x ≠ 0 ∧ y ≠ 0) ∧ z ≠ 0",
"after_state": "x y z : ℝ\nh : x * y * z = 1\nthis : x * y * z ≠ 0\n⊢ (x ≠ 0 ∧ y ≠ 0) ∧ z ≠ 0"
},
{
"line": "simpa [not_or] using this",
"before_state": "x y z : ℝ\nh : x * y * z = 1\nthis : x * y * z ≠ 0\n⊢ (x ≠ 0 ∧ y ≠ 0) ∧ z ≠ 0",
"after_state": "No Goals!"
},
{
"line": "have : z * (y * x) = 1 := by rw [← h]; ac_rfl",
"before_state": "case h.intro.intro\nx y z : ℝ\nh : x * y * z = 1\nright✝ : z ≠ 0\nhx : x ≠ 0\nhy : y ≠ 0\n⊢ x ≠ 0 ∧ 1 ≠ 0 ∧ 1 / y ≠ 0 ∧ x = x / 1 ∧ y = 1 / (1 / y) ∧ z = 1 / y / x",
"after_state": "case h.intro.intro\nx y z : ℝ\nh : x * y * z = 1\nright✝ : z ≠ 0\nhx : x ≠ 0\nhy : y ≠ 0\nthis : z * (y * x) = 1\n⊢ x ≠ 0 ∧ 1 ≠ 0 ∧ 1 / y ≠ 0 ∧ x = x / 1 ∧ y = 1 / (1 / y) ∧ z = 1 / y / x"
},
{
"line": "focus\n refine\n no_implicit_lambda%\n (have : z * (y * x) = 1 := ?body✝;\n ?_)\n case body✝ => with_annotate_state\"by\" (rw [← h]; ac_rfl)",
"before_state": "case h.intro.intro\nx y z : ℝ\nh : x * y * z = 1\nright✝ : z ≠ 0\nhx : x ≠ 0\nhy : y ≠ 0\n⊢ x ≠ 0 ∧ 1 ≠ 0 ∧ 1 / y ≠ 0 ∧ x = x / 1 ∧ y = 1 / (1 / y) ∧ z = 1 / y / x",
"after_state": "case h.intro.intro\nx y z : ℝ\nh : x * y * z = 1\nright✝ : z ≠ 0\nhx : x ≠ 0\nhy : y ≠ 0\nthis : z * (y * x) = 1\n⊢ x ≠ 0 ∧ 1 ≠ 0 ∧ 1 / y ≠ 0 ∧ x = x / 1 ∧ y = 1 / (1 / y) ∧ z = 1 / y / x"
},
{
"line": "refine\n no_implicit_lambda%\n (have : z * (y * x) = 1 := ?body✝;\n ?_)",
"before_state": "case h.intro.intro\nx y z : ℝ\nh : x * y * z = 1\nright✝ : z ≠ 0\nhx : x ≠ 0\nhy : y ≠ 0\n⊢ x ≠ 0 ∧ 1 ≠ 0 ∧ 1 / y ≠ 0 ∧ x = x / 1 ∧ y = 1 / (1 / y) ∧ z = 1 / y / x",
"after_state": "case body\nx y z : ℝ\nh : x * y * z = 1\nright✝ : z ≠ 0\nhx : x ≠ 0\nhy : y ≠ 0\n⊢ z * (y * x) = 1\n---\ncase h.intro.intro\nx y z : ℝ\nh : x * y * z = 1\nright✝ : z ≠ 0\nhx : x ≠ 0\nhy : y ≠ 0\nthis : z * (y * x) = 1\n⊢ x ≠ 0 ∧ 1 ≠ 0 ∧ 1 / y ≠ 0 ∧ x = x / 1 ∧ y = 1 / (1 / y) ∧ z = 1 / y / x"
},
{
"line": "case body✝ => with_annotate_state\"by\" (rw [← h]; ac_rfl)",
"before_state": "case body\nx y z : ℝ\nh : x * y * z = 1\nright✝ : z ≠ 0\nhx : x ≠ 0\nhy : y ≠ 0\n⊢ z * (y * x) = 1\n---\ncase h.intro.intro\nx y z : ℝ\nh : x * y * z = 1\nright✝ : z ≠ 0\nhx : x ≠ 0\nhy : y ≠ 0\nthis : z * (y * x) = 1\n⊢ x ≠ 0 ∧ 1 ≠ 0 ∧ 1 / y ≠ 0 ∧ x = x / 1 ∧ y = 1 / (1 / y) ∧ z = 1 / y / x",
"after_state": "case h.intro.intro\nx y z : ℝ\nh : x * y * z = 1\nright✝ : z ≠ 0\nhx : x ≠ 0\nhy : y ≠ 0\nthis : z * (y * x) = 1\n⊢ x ≠ 0 ∧ 1 ≠ 0 ∧ 1 / y ≠ 0 ∧ x = x / 1 ∧ y = 1 / (1 / y) ∧ z = 1 / y / x"
},
{
"line": "with_annotate_state\"by\" (rw [← h]; ac_rfl)",
"before_state": "x y z : ℝ\nh : x * y * z = 1\nright✝ : z ≠ 0\nhx : x ≠ 0\nhy : y ≠ 0\n⊢ z * (y * x) = 1",
"after_state": "No Goals!"
},
{
"line": "rw [← h]",
"before_state": "x y z : ℝ\nh : x * y * z = 1\nright✝ : z ≠ 0\nhx : x ≠ 0\nhy : y ≠ 0\n⊢ z * (y * x) = 1",
"after_state": "x y z : ℝ\nh : x * y * z = 1\nright✝ : z ≠ 0\nhx : x ≠ 0\nhy : y ≠ 0\n⊢ z * (y * x) = x * y * z"
},
{
"line": "rewrite [← h]",
"before_state": "x y z : ℝ\nh : x * y * z = 1\nright✝ : z ≠ 0\nhx : x ≠ 0\nhy : y ≠ 0\n⊢ z * (y * x) = 1",
"after_state": "x y z : ℝ\nh : x * y * z = 1\nright✝ : z ≠ 0\nhx : x ≠ 0\nhy : y ≠ 0\n⊢ z * (y * x) = x * y * z"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "x y z : ℝ\nh : x * y * z = 1\nright✝ : z ≠ 0\nhx : x ≠ 0\nhy : y ≠ 0\n⊢ z * (y * x) = x * y * z",
"after_state": "x y z : ℝ\nh : x * y * z = 1\nright✝ : z ≠ 0\nhx : x ≠ 0\nhy : y ≠ 0\n⊢ z * (y * x) = x * y * z"
},
{
"line": "try (with_reducible rfl)",
"before_state": "x y z : ℝ\nh : x * y * z = 1\nright✝ : z ≠ 0\nhx : x ≠ 0\nhy : y ≠ 0\n⊢ z * (y * x) = x * y * z",
"after_state": "x y z : ℝ\nh : x * y * z = 1\nright✝ : z ≠ 0\nhx : x ≠ 0\nhy : y ≠ 0\n⊢ z * (y * x) = x * y * z"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "x y z : ℝ\nh : x * y * z = 1\nright✝ : z ≠ 0\nhx : x ≠ 0\nhy : y ≠ 0\n⊢ z * (y * x) = x * y * z",
"after_state": "x y z : ℝ\nh : x * y * z = 1\nright✝ : z ≠ 0\nhx : x ≠ 0\nhy : y ≠ 0\n⊢ z * (y * x) = x * y * z"
},
{
"line": "with_reducible rfl",
"before_state": "x y z : ℝ\nh : x * y * z = 1\nright✝ : z ≠ 0\nhx : x ≠ 0\nhy : y ≠ 0\n⊢ z * (y * x) = x * y * z",
"after_state": "x y z : ℝ\nh : x * y * z = 1\nright✝ : z ≠ 0\nhx : x ≠ 0\nhy : y ≠ 0\n⊢ z * (y * x) = x * y * z"
},
{
"line": "rfl",
"before_state": "x y z : ℝ\nh : x * y * z = 1\nright✝ : z ≠ 0\nhx : x ≠ 0\nhy : y ≠ 0\n⊢ z * (y * x) = x * y * z",
"after_state": "x y z : ℝ\nh : x * y * z = 1\nright✝ : z ≠ 0\nhx : x ≠ 0\nhy : y ≠ 0\n⊢ z * (y * x) = x * y * z"
},
{
"line": "apply_rfl",
"before_state": "x y z : ℝ\nh : x * y * z = 1\nright✝ : z ≠ 0\nhx : x ≠ 0\nhy : y ≠ 0\n⊢ z * (y * x) = x * y * z",
"after_state": "x y z : ℝ\nh : x * y * z = 1\nright✝ : z ≠ 0\nhx : x ≠ 0\nhy : y ≠ 0\n⊢ z * (y * x) = x * y * z"
},
{
"line": "skip",
"before_state": "x y z : ℝ\nh : x * y * z = 1\nright✝ : z ≠ 0\nhx : x ≠ 0\nhy : y ≠ 0\n⊢ z * (y * x) = x * y * z",
"after_state": "x y z : ℝ\nh : x * y * z = 1\nright✝ : z ≠ 0\nhx : x ≠ 0\nhy : y ≠ 0\n⊢ z * (y * x) = x * y * z"
},
{
"line": "ac_rfl",
"before_state": "x y z : ℝ\nh : x * y * z = 1\nright✝ : z ≠ 0\nhx : x ≠ 0\nhy : y ≠ 0\n⊢ z * (y * x) = x * y * z",
"after_state": "No Goals!"
},
{
"line": "field_simp [*]",
"before_state": "case h.intro.intro\nx y z : ℝ\nh : x * y * z = 1\nright✝ : z ≠ 0\nhx : x ≠ 0\nhy : y ≠ 0\nthis : z * (y * x) = 1\n⊢ x ≠ 0 ∧ 1 ≠ 0 ∧ 1 / y ≠ 0 ∧ x = x / 1 ∧ y = 1 / (1 / y) ∧ z = 1 / y / x",
"after_state": "No Goals!"
}
] |
theorem imo2008_q2a (x y z : ℝ) (h : x * y * z = 1) (hx : x ≠ 1) (hy : y ≠ 1) (hz : z ≠ 1) :
x ^ 2 / (x - 1) ^ 2 + y ^ 2 / (y - 1) ^ 2 + z ^ 2 / (z - 1) ^ 2 ≥ 1 := by
obtain ⟨a, b, c, ha, hb, hc, rfl, rfl, rfl⟩ := subst_abc h
obtain ⟨m, n, rfl, rfl⟩ : ∃ m n, b = c - m ∧ a = c - m - n := by use c - b, b - a; simp
have hm_ne_zero : m ≠ 0 := by contrapose! hy; field_simp; assumption
have hn_ne_zero : n ≠ 0 := by contrapose! hx; field_simp; assumption
have hmn_ne_zero : m + n ≠ 0 := by contrapose! hz; field_simp; linarith
have hc_sub_sub : c - (c - m - n) = m + n := by abel
rw [ge_iff_le]
rw [← sub_nonneg]
convert sq_nonneg ((c * (m ^ 2 + n ^ 2 + m * n) - m * (m + n) ^ 2) / (m * n * (m + n)))
field_simp [hc_sub_sub]; ring
|
/root/DuelModelResearch/mathlib4/Archive/Imo/Imo2008Q2.lean
|
{
"open": [],
"variables": []
}
|
[
{
"line": "obtain ⟨a, b, c, ha, hb, hc, rfl, rfl, rfl⟩ := subst_abc h",
"before_state": "x y z : ℝ\nh : x * y * z = 1\nhx : x ≠ 1\nhy : y ≠ 1\nhz : z ≠ 1\n⊢ x ^ 2 / (x - 1) ^ 2 + y ^ 2 / (y - 1) ^ 2 + z ^ 2 / (z - 1) ^ 2 ≥ 1",
"after_state": "No Goals!"
}
] |
theorem abs_eq_one_of_pow_eq_one (x : ℝ) (n : ℕ) (hn : n ≠ 0) (h : x ^ n = 1) : |x| = 1 := by
rw [← pow_left_inj₀ (abs_nonneg x) zero_le_one hn]
rw [one_pow]
rw [pow_abs]
rw [h]
rw [abs_one]
|
/root/DuelModelResearch/mathlib4/Archive/Imo/Imo2008Q4.lean
|
{
"open": [
"Real"
],
"variables": []
}
|
[
{
"line": "rw [← pow_left_inj₀ (abs_nonneg x) zero_le_one hn]",
"before_state": "x : ℝ\nn : ℕ\nhn : n ≠ 0\nh : x ^ n = 1\n⊢ |x| = 1",
"after_state": "x : ℝ\nn : ℕ\nhn : n ≠ 0\nh : x ^ n = 1\n⊢ |x| ^ n = 1 ^ n"
},
{
"line": "rewrite [← pow_left_inj₀ (abs_nonneg x) zero_le_one hn]",
"before_state": "x : ℝ\nn : ℕ\nhn : n ≠ 0\nh : x ^ n = 1\n⊢ |x| = 1",
"after_state": "x : ℝ\nn : ℕ\nhn : n ≠ 0\nh : x ^ n = 1\n⊢ |x| ^ n = 1 ^ n"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "x : ℝ\nn : ℕ\nhn : n ≠ 0\nh : x ^ n = 1\n⊢ |x| ^ n = 1 ^ n",
"after_state": "x : ℝ\nn : ℕ\nhn : n ≠ 0\nh : x ^ n = 1\n⊢ |x| ^ n = 1 ^ n"
},
{
"line": "try (with_reducible rfl)",
"before_state": "x : ℝ\nn : ℕ\nhn : n ≠ 0\nh : x ^ n = 1\n⊢ |x| ^ n = 1 ^ n",
"after_state": "x : ℝ\nn : ℕ\nhn : n ≠ 0\nh : x ^ n = 1\n⊢ |x| ^ n = 1 ^ n"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "x : ℝ\nn : ℕ\nhn : n ≠ 0\nh : x ^ n = 1\n⊢ |x| ^ n = 1 ^ n",
"after_state": "x : ℝ\nn : ℕ\nhn : n ≠ 0\nh : x ^ n = 1\n⊢ |x| ^ n = 1 ^ n"
},
{
"line": "with_reducible rfl",
"before_state": "x : ℝ\nn : ℕ\nhn : n ≠ 0\nh : x ^ n = 1\n⊢ |x| ^ n = 1 ^ n",
"after_state": "x : ℝ\nn : ℕ\nhn : n ≠ 0\nh : x ^ n = 1\n⊢ |x| ^ n = 1 ^ n"
},
{
"line": "rfl",
"before_state": "x : ℝ\nn : ℕ\nhn : n ≠ 0\nh : x ^ n = 1\n⊢ |x| ^ n = 1 ^ n",
"after_state": "x : ℝ\nn : ℕ\nhn : n ≠ 0\nh : x ^ n = 1\n⊢ |x| ^ n = 1 ^ n"
},
{
"line": "apply_rfl",
"before_state": "x : ℝ\nn : ℕ\nhn : n ≠ 0\nh : x ^ n = 1\n⊢ |x| ^ n = 1 ^ n",
"after_state": "x : ℝ\nn : ℕ\nhn : n ≠ 0\nh : x ^ n = 1\n⊢ |x| ^ n = 1 ^ n"
},
{
"line": "skip",
"before_state": "x : ℝ\nn : ℕ\nhn : n ≠ 0\nh : x ^ n = 1\n⊢ |x| ^ n = 1 ^ n",
"after_state": "x : ℝ\nn : ℕ\nhn : n ≠ 0\nh : x ^ n = 1\n⊢ |x| ^ n = 1 ^ n"
},
{
"line": "rw [one_pow]",
"before_state": "x : ℝ\nn : ℕ\nhn : n ≠ 0\nh : x ^ n = 1\n⊢ |x| ^ n = 1 ^ n",
"after_state": "x : ℝ\nn : ℕ\nhn : n ≠ 0\nh : x ^ n = 1\n⊢ |x| ^ n = 1"
},
{
"line": "rewrite [one_pow]",
"before_state": "x : ℝ\nn : ℕ\nhn : n ≠ 0\nh : x ^ n = 1\n⊢ |x| ^ n = 1 ^ n",
"after_state": "x : ℝ\nn : ℕ\nhn : n ≠ 0\nh : x ^ n = 1\n⊢ |x| ^ n = 1"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "x : ℝ\nn : ℕ\nhn : n ≠ 0\nh : x ^ n = 1\n⊢ |x| ^ n = 1",
"after_state": "x : ℝ\nn : ℕ\nhn : n ≠ 0\nh : x ^ n = 1\n⊢ |x| ^ n = 1"
},
{
"line": "try (with_reducible rfl)",
"before_state": "x : ℝ\nn : ℕ\nhn : n ≠ 0\nh : x ^ n = 1\n⊢ |x| ^ n = 1",
"after_state": "x : ℝ\nn : ℕ\nhn : n ≠ 0\nh : x ^ n = 1\n⊢ |x| ^ n = 1"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "x : ℝ\nn : ℕ\nhn : n ≠ 0\nh : x ^ n = 1\n⊢ |x| ^ n = 1",
"after_state": "x : ℝ\nn : ℕ\nhn : n ≠ 0\nh : x ^ n = 1\n⊢ |x| ^ n = 1"
},
{
"line": "with_reducible rfl",
"before_state": "x : ℝ\nn : ℕ\nhn : n ≠ 0\nh : x ^ n = 1\n⊢ |x| ^ n = 1",
"after_state": "x : ℝ\nn : ℕ\nhn : n ≠ 0\nh : x ^ n = 1\n⊢ |x| ^ n = 1"
},
{
"line": "rfl",
"before_state": "x : ℝ\nn : ℕ\nhn : n ≠ 0\nh : x ^ n = 1\n⊢ |x| ^ n = 1",
"after_state": "x : ℝ\nn : ℕ\nhn : n ≠ 0\nh : x ^ n = 1\n⊢ |x| ^ n = 1"
},
{
"line": "apply_rfl",
"before_state": "x : ℝ\nn : ℕ\nhn : n ≠ 0\nh : x ^ n = 1\n⊢ |x| ^ n = 1",
"after_state": "x : ℝ\nn : ℕ\nhn : n ≠ 0\nh : x ^ n = 1\n⊢ |x| ^ n = 1"
},
{
"line": "skip",
"before_state": "x : ℝ\nn : ℕ\nhn : n ≠ 0\nh : x ^ n = 1\n⊢ |x| ^ n = 1",
"after_state": "x : ℝ\nn : ℕ\nhn : n ≠ 0\nh : x ^ n = 1\n⊢ |x| ^ n = 1"
},
{
"line": "rw [pow_abs]",
"before_state": "x : ℝ\nn : ℕ\nhn : n ≠ 0\nh : x ^ n = 1\n⊢ |x| ^ n = 1",
"after_state": "x : ℝ\nn : ℕ\nhn : n ≠ 0\nh : x ^ n = 1\n⊢ |x ^ n| = 1"
},
{
"line": "rewrite [pow_abs]",
"before_state": "x : ℝ\nn : ℕ\nhn : n ≠ 0\nh : x ^ n = 1\n⊢ |x| ^ n = 1",
"after_state": "x : ℝ\nn : ℕ\nhn : n ≠ 0\nh : x ^ n = 1\n⊢ |x ^ n| = 1"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "x : ℝ\nn : ℕ\nhn : n ≠ 0\nh : x ^ n = 1\n⊢ |x ^ n| = 1",
"after_state": "x : ℝ\nn : ℕ\nhn : n ≠ 0\nh : x ^ n = 1\n⊢ |x ^ n| = 1"
},
{
"line": "try (with_reducible rfl)",
"before_state": "x : ℝ\nn : ℕ\nhn : n ≠ 0\nh : x ^ n = 1\n⊢ |x ^ n| = 1",
"after_state": "x : ℝ\nn : ℕ\nhn : n ≠ 0\nh : x ^ n = 1\n⊢ |x ^ n| = 1"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "x : ℝ\nn : ℕ\nhn : n ≠ 0\nh : x ^ n = 1\n⊢ |x ^ n| = 1",
"after_state": "x : ℝ\nn : ℕ\nhn : n ≠ 0\nh : x ^ n = 1\n⊢ |x ^ n| = 1"
},
{
"line": "with_reducible rfl",
"before_state": "x : ℝ\nn : ℕ\nhn : n ≠ 0\nh : x ^ n = 1\n⊢ |x ^ n| = 1",
"after_state": "x : ℝ\nn : ℕ\nhn : n ≠ 0\nh : x ^ n = 1\n⊢ |x ^ n| = 1"
},
{
"line": "rfl",
"before_state": "x : ℝ\nn : ℕ\nhn : n ≠ 0\nh : x ^ n = 1\n⊢ |x ^ n| = 1",
"after_state": "x : ℝ\nn : ℕ\nhn : n ≠ 0\nh : x ^ n = 1\n⊢ |x ^ n| = 1"
},
{
"line": "apply_rfl",
"before_state": "x : ℝ\nn : ℕ\nhn : n ≠ 0\nh : x ^ n = 1\n⊢ |x ^ n| = 1",
"after_state": "x : ℝ\nn : ℕ\nhn : n ≠ 0\nh : x ^ n = 1\n⊢ |x ^ n| = 1"
},
{
"line": "skip",
"before_state": "x : ℝ\nn : ℕ\nhn : n ≠ 0\nh : x ^ n = 1\n⊢ |x ^ n| = 1",
"after_state": "x : ℝ\nn : ℕ\nhn : n ≠ 0\nh : x ^ n = 1\n⊢ |x ^ n| = 1"
},
{
"line": "rw [h]",
"before_state": "x : ℝ\nn : ℕ\nhn : n ≠ 0\nh : x ^ n = 1\n⊢ |x ^ n| = 1",
"after_state": "x : ℝ\nn : ℕ\nhn : n ≠ 0\nh : x ^ n = 1\n⊢ |1| = 1"
},
{
"line": "rewrite [h]",
"before_state": "x : ℝ\nn : ℕ\nhn : n ≠ 0\nh : x ^ n = 1\n⊢ |x ^ n| = 1",
"after_state": "x : ℝ\nn : ℕ\nhn : n ≠ 0\nh : x ^ n = 1\n⊢ |1| = 1"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "x : ℝ\nn : ℕ\nhn : n ≠ 0\nh : x ^ n = 1\n⊢ |1| = 1",
"after_state": "x : ℝ\nn : ℕ\nhn : n ≠ 0\nh : x ^ n = 1\n⊢ |1| = 1"
},
{
"line": "try (with_reducible rfl)",
"before_state": "x : ℝ\nn : ℕ\nhn : n ≠ 0\nh : x ^ n = 1\n⊢ |1| = 1",
"after_state": "x : ℝ\nn : ℕ\nhn : n ≠ 0\nh : x ^ n = 1\n⊢ |1| = 1"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "x : ℝ\nn : ℕ\nhn : n ≠ 0\nh : x ^ n = 1\n⊢ |1| = 1",
"after_state": "x : ℝ\nn : ℕ\nhn : n ≠ 0\nh : x ^ n = 1\n⊢ |1| = 1"
},
{
"line": "with_reducible rfl",
"before_state": "x : ℝ\nn : ℕ\nhn : n ≠ 0\nh : x ^ n = 1\n⊢ |1| = 1",
"after_state": "x : ℝ\nn : ℕ\nhn : n ≠ 0\nh : x ^ n = 1\n⊢ |1| = 1"
},
{
"line": "rfl",
"before_state": "x : ℝ\nn : ℕ\nhn : n ≠ 0\nh : x ^ n = 1\n⊢ |1| = 1",
"after_state": "x : ℝ\nn : ℕ\nhn : n ≠ 0\nh : x ^ n = 1\n⊢ |1| = 1"
},
{
"line": "apply_rfl",
"before_state": "x : ℝ\nn : ℕ\nhn : n ≠ 0\nh : x ^ n = 1\n⊢ |1| = 1",
"after_state": "x : ℝ\nn : ℕ\nhn : n ≠ 0\nh : x ^ n = 1\n⊢ |1| = 1"
},
{
"line": "skip",
"before_state": "x : ℝ\nn : ℕ\nhn : n ≠ 0\nh : x ^ n = 1\n⊢ |1| = 1",
"after_state": "x : ℝ\nn : ℕ\nhn : n ≠ 0\nh : x ^ n = 1\n⊢ |1| = 1"
},
{
"line": "rw [abs_one]",
"before_state": "x : ℝ\nn : ℕ\nhn : n ≠ 0\nh : x ^ n = 1\n⊢ |1| = 1",
"after_state": "No Goals!"
},
{
"line": "rewrite [abs_one]",
"before_state": "x : ℝ\nn : ℕ\nhn : n ≠ 0\nh : x ^ n = 1\n⊢ |1| = 1",
"after_state": "x : ℝ\nn : ℕ\nhn : n ≠ 0\nh : x ^ n = 1\n⊢ 1 = 1"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "x : ℝ\nn : ℕ\nhn : n ≠ 0\nh : x ^ n = 1\n⊢ 1 = 1",
"after_state": "No Goals!"
},
{
"line": "try (with_reducible rfl)",
"before_state": "x : ℝ\nn : ℕ\nhn : n ≠ 0\nh : x ^ n = 1\n⊢ 1 = 1",
"after_state": "No Goals!"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "x : ℝ\nn : ℕ\nhn : n ≠ 0\nh : x ^ n = 1\n⊢ 1 = 1",
"after_state": "No Goals!"
},
{
"line": "with_reducible rfl",
"before_state": "x : ℝ\nn : ℕ\nhn : n ≠ 0\nh : x ^ n = 1\n⊢ 1 = 1",
"after_state": "No Goals!"
},
{
"line": "rfl",
"before_state": "x : ℝ\nn : ℕ\nhn : n ≠ 0\nh : x ^ n = 1\n⊢ 1 = 1",
"after_state": "No Goals!"
},
{
"line": "eq_refl",
"before_state": "x : ℝ\nn : ℕ\nhn : n ≠ 0\nh : x ^ n = 1\n⊢ 1 = 1",
"after_state": "No Goals!"
}
] |
theorem imo2011_q3 (f : ℝ → ℝ) (hf : ∀ x y, f (x + y) ≤ y * f x + f (f x)) : ∀ x ≤ 0, f x = 0 := by
-- reparameterize
have hxt : ∀ x t, f t ≤ t * f x - x * f x + f (f x) := fun x t =>
calc
f t = f (x + (t - x)) := by rw [add_eq_of_eq_sub' rfl]
_ ≤ (t - x) * f x + f (f x) := hf x (t - x)
_ = t * f x - x * f x + f (f x) := by rw [sub_mul]
have h_ab_combined : ∀ a b, a * f a + b * f b ≤ 2 * f a * f b := fun a b => by
linarith [hxt b (f a), hxt a (f b)]
have h_f_nonneg_of_pos : ∀ a < 0, 0 ≤ f a := fun a han =>
suffices a * f a ≤ 0 from nonneg_of_mul_nonpos_right this han
add_le_iff_nonpos_left.mp (h_ab_combined a (2 * f a))
have h_f_nonpos : ∀ x, f x ≤ 0 := fun x => by
by_contra h_suppose_not
-- If we choose a small enough argument for f, then we get a contradiction.
let s := (x * f x - f (f x)) / f x
have hm : min 0 s - 1 < s := (sub_one_lt _).trans_le (min_le_right 0 s)
have hml : min 0 s - 1 < 0 := (sub_one_lt _).trans_le (min_le_left 0 s)
suffices f (min 0 s - 1) < 0 from not_le.mpr this (h_f_nonneg_of_pos (min 0 s - 1) hml)
have hp : 0 < f x := not_le.mp h_suppose_not
calc
f (min 0 s - 1) ≤ (min 0 s - 1) * f x - x * f x + f (f x) := hxt x (min 0 s - 1)
_ < s * f x - x * f x + f (f x) := by linarith [(mul_lt_mul_right hp).mpr hm]
_ = 0 := by rw [(eq_div_iff hp.ne.symm).mp rfl]; linarith
have h_fx_zero_of_neg : ∀ x < 0, f x = 0 := fun x hxz =>
(h_f_nonpos x).antisymm (h_f_nonneg_of_pos x hxz)
intro x hx
obtain (h_x_neg : x < 0) | (rfl : x = 0) := hx.lt_or_eq
· exact h_fx_zero_of_neg _ h_x_neg
· suffices 0 ≤ f 0 from le_antisymm (h_f_nonpos 0) this
have hno : f (-1) = 0 := h_fx_zero_of_neg (-1) neg_one_lt_zero
have hp := hxt (-1) (-1)
rw [hno] at hp
linarith
|
/root/DuelModelResearch/mathlib4/Archive/Imo/Imo2011Q3.lean
|
{
"open": [],
"variables": []
}
|
[
{
"line": "have hxt : ∀ x t, f t ≤ t * f x - x * f x + f (f x) := fun x t =>\n calc\n f t = f (x + (t - x)) := by rw [add_eq_of_eq_sub' rfl]\n _ ≤ (t - x) * f x + f (f x) := (hf x (t - x))\n _ = t * f x - x * f x + f (f x) := by rw [sub_mul]",
"before_state": "f : ℝ → ℝ\nhf : ∀ (x y : ℝ), f (x + y) ≤ y * f x + f (f x)\n⊢ ∀ x ≤ 0, f x = 0",
"after_state": "f : ℝ → ℝ\nhf : ∀ (x y : ℝ), f (x + y) ≤ y * f x + f (f x)\nhxt : ∀ (x t : ℝ), f t ≤ t * f x - x * f x + f (f x)\n⊢ ∀ x ≤ 0, f x = 0"
},
{
"line": "refine_lift\n have hxt : ∀ x t, f t ≤ t * f x - x * f x + f (f x) := fun x t =>\n calc\n f t = f (x + (t - x)) := by rw [add_eq_of_eq_sub' rfl]\n _ ≤ (t - x) * f x + f (f x) := (hf x (t - x))\n _ = t * f x - x * f x + f (f x) := by rw [sub_mul];\n ?_",
"before_state": "f : ℝ → ℝ\nhf : ∀ (x y : ℝ), f (x + y) ≤ y * f x + f (f x)\n⊢ ∀ x ≤ 0, f x = 0",
"after_state": "f : ℝ → ℝ\nhf : ∀ (x y : ℝ), f (x + y) ≤ y * f x + f (f x)\nhxt : ∀ (x t : ℝ), f t ≤ t * f x - x * f x + f (f x)\n⊢ ∀ x ≤ 0, f x = 0"
},
{
"line": "focus\n (refine\n no_implicit_lambda%\n (have hxt : ∀ x t, f t ≤ t * f x - x * f x + f (f x) := fun x t =>\n calc\n f t = f (x + (t - x)) := by rw [add_eq_of_eq_sub' rfl]\n _ ≤ (t - x) * f x + f (f x) := (hf x (t - x))\n _ = t * f x - x * f x + f (f x) := by rw [sub_mul];\n ?_);\n rotate_right)",
"before_state": "f : ℝ → ℝ\nhf : ∀ (x y : ℝ), f (x + y) ≤ y * f x + f (f x)\n⊢ ∀ x ≤ 0, f x = 0",
"after_state": "f : ℝ → ℝ\nhf : ∀ (x y : ℝ), f (x + y) ≤ y * f x + f (f x)\nhxt : ∀ (x t : ℝ), f t ≤ t * f x - x * f x + f (f x)\n⊢ ∀ x ≤ 0, f x = 0"
},
{
"line": "refine\n no_implicit_lambda%\n (have hxt : ∀ x t, f t ≤ t * f x - x * f x + f (f x) := fun x t =>\n calc\n f t = f (x + (t - x)) := by rw [add_eq_of_eq_sub' rfl]\n _ ≤ (t - x) * f x + f (f x) := (hf x (t - x))\n _ = t * f x - x * f x + f (f x) := by rw [sub_mul];\n ?_)",
"before_state": "f : ℝ → ℝ\nhf : ∀ (x y : ℝ), f (x + y) ≤ y * f x + f (f x)\n⊢ ∀ x ≤ 0, f x = 0",
"after_state": "f : ℝ → ℝ\nhf : ∀ (x y : ℝ), f (x + y) ≤ y * f x + f (f x)\nhxt : ∀ (x t : ℝ), f t ≤ t * f x - x * f x + f (f x)\n⊢ ∀ x ≤ 0, f x = 0"
},
{
"line": "rw [add_eq_of_eq_sub' rfl]",
"before_state": "f : ℝ → ℝ\nhf : ∀ (x y : ℝ), f (x + y) ≤ y * f x + f (f x)\nx t : ℝ\n⊢ f t = f (x + (t - x))",
"after_state": "No Goals!"
},
{
"line": "rewrite [add_eq_of_eq_sub' rfl]",
"before_state": "f : ℝ → ℝ\nhf : ∀ (x y : ℝ), f (x + y) ≤ y * f x + f (f x)\nx t : ℝ\n⊢ f t = f (x + (t - x))",
"after_state": "f : ℝ → ℝ\nhf : ∀ (x y : ℝ), f (x + y) ≤ y * f x + f (f x)\nx t : ℝ\n⊢ f t = f t"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "f : ℝ → ℝ\nhf : ∀ (x y : ℝ), f (x + y) ≤ y * f x + f (f x)\nx t : ℝ\n⊢ f t = f t",
"after_state": "No Goals!"
},
{
"line": "try (with_reducible rfl)",
"before_state": "f : ℝ → ℝ\nhf : ∀ (x y : ℝ), f (x + y) ≤ y * f x + f (f x)\nx t : ℝ\n⊢ f t = f t",
"after_state": "No Goals!"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "f : ℝ → ℝ\nhf : ∀ (x y : ℝ), f (x + y) ≤ y * f x + f (f x)\nx t : ℝ\n⊢ f t = f t",
"after_state": "No Goals!"
},
{
"line": "with_reducible rfl",
"before_state": "f : ℝ → ℝ\nhf : ∀ (x y : ℝ), f (x + y) ≤ y * f x + f (f x)\nx t : ℝ\n⊢ f t = f t",
"after_state": "No Goals!"
},
{
"line": "rfl",
"before_state": "f : ℝ → ℝ\nhf : ∀ (x y : ℝ), f (x + y) ≤ y * f x + f (f x)\nx t : ℝ\n⊢ f t = f t",
"after_state": "No Goals!"
},
{
"line": "eq_refl",
"before_state": "f : ℝ → ℝ\nhf : ∀ (x y : ℝ), f (x + y) ≤ y * f x + f (f x)\nx t : ℝ\n⊢ f t = f t",
"after_state": "No Goals!"
},
{
"line": "rw [sub_mul]",
"before_state": "f : ℝ → ℝ\nhf : ∀ (x y : ℝ), f (x + y) ≤ y * f x + f (f x)\nx t : ℝ\n⊢ (t - x) * f x + f (f x) = t * f x - x * f x + f (f x)",
"after_state": "No Goals!"
},
{
"line": "rewrite [sub_mul]",
"before_state": "f : ℝ → ℝ\nhf : ∀ (x y : ℝ), f (x + y) ≤ y * f x + f (f x)\nx t : ℝ\n⊢ (t - x) * f x + f (f x) = t * f x - x * f x + f (f x)",
"after_state": "f : ℝ → ℝ\nhf : ∀ (x y : ℝ), f (x + y) ≤ y * f x + f (f x)\nx t : ℝ\n⊢ t * f x - x * f x + f (f x) = t * f x - x * f x + f (f x)"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "f : ℝ → ℝ\nhf : ∀ (x y : ℝ), f (x + y) ≤ y * f x + f (f x)\nx t : ℝ\n⊢ t * f x - x * f x + f (f x) = t * f x - x * f x + f (f x)",
"after_state": "No Goals!"
},
{
"line": "try (with_reducible rfl)",
"before_state": "f : ℝ → ℝ\nhf : ∀ (x y : ℝ), f (x + y) ≤ y * f x + f (f x)\nx t : ℝ\n⊢ t * f x - x * f x + f (f x) = t * f x - x * f x + f (f x)",
"after_state": "No Goals!"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "f : ℝ → ℝ\nhf : ∀ (x y : ℝ), f (x + y) ≤ y * f x + f (f x)\nx t : ℝ\n⊢ t * f x - x * f x + f (f x) = t * f x - x * f x + f (f x)",
"after_state": "No Goals!"
},
{
"line": "with_reducible rfl",
"before_state": "f : ℝ → ℝ\nhf : ∀ (x y : ℝ), f (x + y) ≤ y * f x + f (f x)\nx t : ℝ\n⊢ t * f x - x * f x + f (f x) = t * f x - x * f x + f (f x)",
"after_state": "No Goals!"
},
{
"line": "rfl",
"before_state": "f : ℝ → ℝ\nhf : ∀ (x y : ℝ), f (x + y) ≤ y * f x + f (f x)\nx t : ℝ\n⊢ t * f x - x * f x + f (f x) = t * f x - x * f x + f (f x)",
"after_state": "No Goals!"
},
{
"line": "eq_refl",
"before_state": "f : ℝ → ℝ\nhf : ∀ (x y : ℝ), f (x + y) ≤ y * f x + f (f x)\nx t : ℝ\n⊢ t * f x - x * f x + f (f x) = t * f x - x * f x + f (f x)",
"after_state": "No Goals!"
},
{
"line": "rotate_right",
"before_state": "f : ℝ → ℝ\nhf : ∀ (x y : ℝ), f (x + y) ≤ y * f x + f (f x)\nhxt : ∀ (x t : ℝ), f t ≤ t * f x - x * f x + f (f x)\n⊢ ∀ x ≤ 0, f x = 0",
"after_state": "f : ℝ → ℝ\nhf : ∀ (x y : ℝ), f (x + y) ≤ y * f x + f (f x)\nhxt : ∀ (x t : ℝ), f t ≤ t * f x - x * f x + f (f x)\n⊢ ∀ x ≤ 0, f x = 0"
},
{
"line": "have h_ab_combined : ∀ a b, a * f a + b * f b ≤ 2 * f a * f b := fun a b => by linarith [hxt b (f a), hxt a (f b)]",
"before_state": "f : ℝ → ℝ\nhf : ∀ (x y : ℝ), f (x + y) ≤ y * f x + f (f x)\nhxt : ∀ (x t : ℝ), f t ≤ t * f x - x * f x + f (f x)\n⊢ ∀ x ≤ 0, f x = 0",
"after_state": "f : ℝ → ℝ\nhf : ∀ (x y : ℝ), f (x + y) ≤ y * f x + f (f x)\nhxt : ∀ (x t : ℝ), f t ≤ t * f x - x * f x + f (f x)\nh_ab_combined : ∀ (a b : ℝ), a * f a + b * f b ≤ 2 * f a * f b\n⊢ ∀ x ≤ 0, f x = 0"
},
{
"line": "refine_lift\n have h_ab_combined : ∀ a b, a * f a + b * f b ≤ 2 * f a * f b := fun a b => by linarith [hxt b (f a), hxt a (f b)];\n ?_",
"before_state": "f : ℝ → ℝ\nhf : ∀ (x y : ℝ), f (x + y) ≤ y * f x + f (f x)\nhxt : ∀ (x t : ℝ), f t ≤ t * f x - x * f x + f (f x)\n⊢ ∀ x ≤ 0, f x = 0",
"after_state": "f : ℝ → ℝ\nhf : ∀ (x y : ℝ), f (x + y) ≤ y * f x + f (f x)\nhxt : ∀ (x t : ℝ), f t ≤ t * f x - x * f x + f (f x)\nh_ab_combined : ∀ (a b : ℝ), a * f a + b * f b ≤ 2 * f a * f b\n⊢ ∀ x ≤ 0, f x = 0"
},
{
"line": "focus\n (refine\n no_implicit_lambda%\n (have h_ab_combined : ∀ a b, a * f a + b * f b ≤ 2 * f a * f b := fun a b => by\n linarith [hxt b (f a), hxt a (f b)];\n ?_);\n rotate_right)",
"before_state": "f : ℝ → ℝ\nhf : ∀ (x y : ℝ), f (x + y) ≤ y * f x + f (f x)\nhxt : ∀ (x t : ℝ), f t ≤ t * f x - x * f x + f (f x)\n⊢ ∀ x ≤ 0, f x = 0",
"after_state": "f : ℝ → ℝ\nhf : ∀ (x y : ℝ), f (x + y) ≤ y * f x + f (f x)\nhxt : ∀ (x t : ℝ), f t ≤ t * f x - x * f x + f (f x)\nh_ab_combined : ∀ (a b : ℝ), a * f a + b * f b ≤ 2 * f a * f b\n⊢ ∀ x ≤ 0, f x = 0"
},
{
"line": "refine\n no_implicit_lambda%\n (have h_ab_combined : ∀ a b, a * f a + b * f b ≤ 2 * f a * f b := fun a b => by linarith [hxt b (f a), hxt a (f b)];\n ?_)",
"before_state": "f : ℝ → ℝ\nhf : ∀ (x y : ℝ), f (x + y) ≤ y * f x + f (f x)\nhxt : ∀ (x t : ℝ), f t ≤ t * f x - x * f x + f (f x)\n⊢ ∀ x ≤ 0, f x = 0",
"after_state": "f : ℝ → ℝ\nhf : ∀ (x y : ℝ), f (x + y) ≤ y * f x + f (f x)\nhxt : ∀ (x t : ℝ), f t ≤ t * f x - x * f x + f (f x)\nh_ab_combined : ∀ (a b : ℝ), a * f a + b * f b ≤ 2 * f a * f b\n⊢ ∀ x ≤ 0, f x = 0"
},
{
"line": "linarith [hxt b (f a), hxt a (f b)]",
"before_state": "f : ℝ → ℝ\nhf : ∀ (x y : ℝ), f (x + y) ≤ y * f x + f (f x)\nhxt : ∀ (x t : ℝ), f t ≤ t * f x - x * f x + f (f x)\na b : ℝ\n⊢ a * f a + b * f b ≤ 2 * f a * f b",
"after_state": "No Goals!"
},
{
"line": "ring1",
"before_state": "f : ℝ → ℝ\nhf : ∀ (x y : ℝ), f (x + y) ≤ y * f x + f (f x)\nhxt : ∀ (x t : ℝ), f t ≤ t * f x - x * f x + f (f x)\na b : ℝ\na✝ : a * f a + b * f b > 2 * f a * f b\n⊢ 2 * f a * f b - (a * f a + b * f b) + (f (f a) - (f a * f b - b * f b + f (f b))) +\n (f (f b) - (f b * f a - a * f a + f (f a))) =\n 0",
"after_state": "No Goals!"
},
{
"line": "rotate_right",
"before_state": "f : ℝ → ℝ\nhf : ∀ (x y : ℝ), f (x + y) ≤ y * f x + f (f x)\nhxt : ∀ (x t : ℝ), f t ≤ t * f x - x * f x + f (f x)\nh_ab_combined : ∀ (a b : ℝ), a * f a + b * f b ≤ 2 * f a * f b\n⊢ ∀ x ≤ 0, f x = 0",
"after_state": "f : ℝ → ℝ\nhf : ∀ (x y : ℝ), f (x + y) ≤ y * f x + f (f x)\nhxt : ∀ (x t : ℝ), f t ≤ t * f x - x * f x + f (f x)\nh_ab_combined : ∀ (a b : ℝ), a * f a + b * f b ≤ 2 * f a * f b\n⊢ ∀ x ≤ 0, f x = 0"
},
{
"line": "have h_f_nonneg_of_pos : ∀ a < 0, 0 ≤ f a := fun a han =>\n suffices a * f a ≤ 0 from nonneg_of_mul_nonpos_right this han\n add_le_iff_nonpos_left.mp (h_ab_combined a (2 * f a))",
"before_state": "f : ℝ → ℝ\nhf : ∀ (x y : ℝ), f (x + y) ≤ y * f x + f (f x)\nhxt : ∀ (x t : ℝ), f t ≤ t * f x - x * f x + f (f x)\nh_ab_combined : ∀ (a b : ℝ), a * f a + b * f b ≤ 2 * f a * f b\n⊢ ∀ x ≤ 0, f x = 0",
"after_state": "f : ℝ → ℝ\nhf : ∀ (x y : ℝ), f (x + y) ≤ y * f x + f (f x)\nhxt : ∀ (x t : ℝ), f t ≤ t * f x - x * f x + f (f x)\nh_ab_combined : ∀ (a b : ℝ), a * f a + b * f b ≤ 2 * f a * f b\nh_f_nonneg_of_pos : ∀ a < 0, 0 ≤ f a\n⊢ ∀ x ≤ 0, f x = 0"
},
{
"line": "refine_lift\n have h_f_nonneg_of_pos : ∀ a < 0, 0 ≤ f a := fun a han =>\n suffices a * f a ≤ 0 from nonneg_of_mul_nonpos_right this han\n add_le_iff_nonpos_left.mp (h_ab_combined a (2 * f a));\n ?_",
"before_state": "f : ℝ → ℝ\nhf : ∀ (x y : ℝ), f (x + y) ≤ y * f x + f (f x)\nhxt : ∀ (x t : ℝ), f t ≤ t * f x - x * f x + f (f x)\nh_ab_combined : ∀ (a b : ℝ), a * f a + b * f b ≤ 2 * f a * f b\n⊢ ∀ x ≤ 0, f x = 0",
"after_state": "f : ℝ → ℝ\nhf : ∀ (x y : ℝ), f (x + y) ≤ y * f x + f (f x)\nhxt : ∀ (x t : ℝ), f t ≤ t * f x - x * f x + f (f x)\nh_ab_combined : ∀ (a b : ℝ), a * f a + b * f b ≤ 2 * f a * f b\nh_f_nonneg_of_pos : ∀ a < 0, 0 ≤ f a\n⊢ ∀ x ≤ 0, f x = 0"
},
{
"line": "focus\n (refine\n no_implicit_lambda%\n (have h_f_nonneg_of_pos : ∀ a < 0, 0 ≤ f a := fun a han =>\n suffices a * f a ≤ 0 from nonneg_of_mul_nonpos_right this han\n add_le_iff_nonpos_left.mp (h_ab_combined a (2 * f a));\n ?_);\n rotate_right)",
"before_state": "f : ℝ → ℝ\nhf : ∀ (x y : ℝ), f (x + y) ≤ y * f x + f (f x)\nhxt : ∀ (x t : ℝ), f t ≤ t * f x - x * f x + f (f x)\nh_ab_combined : ∀ (a b : ℝ), a * f a + b * f b ≤ 2 * f a * f b\n⊢ ∀ x ≤ 0, f x = 0",
"after_state": "f : ℝ → ℝ\nhf : ∀ (x y : ℝ), f (x + y) ≤ y * f x + f (f x)\nhxt : ∀ (x t : ℝ), f t ≤ t * f x - x * f x + f (f x)\nh_ab_combined : ∀ (a b : ℝ), a * f a + b * f b ≤ 2 * f a * f b\nh_f_nonneg_of_pos : ∀ a < 0, 0 ≤ f a\n⊢ ∀ x ≤ 0, f x = 0"
},
{
"line": "refine\n no_implicit_lambda%\n (have h_f_nonneg_of_pos : ∀ a < 0, 0 ≤ f a := fun a han =>\n suffices a * f a ≤ 0 from nonneg_of_mul_nonpos_right this han\n add_le_iff_nonpos_left.mp (h_ab_combined a (2 * f a));\n ?_)",
"before_state": "f : ℝ → ℝ\nhf : ∀ (x y : ℝ), f (x + y) ≤ y * f x + f (f x)\nhxt : ∀ (x t : ℝ), f t ≤ t * f x - x * f x + f (f x)\nh_ab_combined : ∀ (a b : ℝ), a * f a + b * f b ≤ 2 * f a * f b\n⊢ ∀ x ≤ 0, f x = 0",
"after_state": "f : ℝ → ℝ\nhf : ∀ (x y : ℝ), f (x + y) ≤ y * f x + f (f x)\nhxt : ∀ (x t : ℝ), f t ≤ t * f x - x * f x + f (f x)\nh_ab_combined : ∀ (a b : ℝ), a * f a + b * f b ≤ 2 * f a * f b\nh_f_nonneg_of_pos : ∀ a < 0, 0 ≤ f a\n⊢ ∀ x ≤ 0, f x = 0"
},
{
"line": "rotate_right",
"before_state": "f : ℝ → ℝ\nhf : ∀ (x y : ℝ), f (x + y) ≤ y * f x + f (f x)\nhxt : ∀ (x t : ℝ), f t ≤ t * f x - x * f x + f (f x)\nh_ab_combined : ∀ (a b : ℝ), a * f a + b * f b ≤ 2 * f a * f b\nh_f_nonneg_of_pos : ∀ a < 0, 0 ≤ f a\n⊢ ∀ x ≤ 0, f x = 0",
"after_state": "f : ℝ → ℝ\nhf : ∀ (x y : ℝ), f (x + y) ≤ y * f x + f (f x)\nhxt : ∀ (x t : ℝ), f t ≤ t * f x - x * f x + f (f x)\nh_ab_combined : ∀ (a b : ℝ), a * f a + b * f b ≤ 2 * f a * f b\nh_f_nonneg_of_pos : ∀ a < 0, 0 ≤ f a\n⊢ ∀ x ≤ 0, f x = 0"
},
{
"line": "have h_f_nonpos : ∀ x, f x ≤ 0 := fun x => by\n by_contra h_suppose_not\n let s := (x * f x - f (f x)) / f x\n have hm : min 0 s - 1 < s := (sub_one_lt _).trans_le (min_le_right 0 s)\n have hml : min 0 s - 1 < 0 := (sub_one_lt _).trans_le (min_le_left 0 s)\n suffices f (min 0 s - 1) < 0 from not_le.mpr this (h_f_nonneg_of_pos (min 0 s - 1) hml)\n have hp : 0 < f x := not_le.mp h_suppose_not\n calc\n f (min 0 s - 1) ≤ (min 0 s - 1) * f x - x * f x + f (f x) := hxt x (min 0 s - 1)\n _ < s * f x - x * f x + f (f x) := by linarith [(mul_lt_mul_right hp).mpr hm]\n _ = 0 := by rw [(eq_div_iff hp.ne.symm).mp rfl]; linarith",
"before_state": "f : ℝ → ℝ\nhf : ∀ (x y : ℝ), f (x + y) ≤ y * f x + f (f x)\nhxt : ∀ (x t : ℝ), f t ≤ t * f x - x * f x + f (f x)\nh_ab_combined : ∀ (a b : ℝ), a * f a + b * f b ≤ 2 * f a * f b\nh_f_nonneg_of_pos : ∀ a < 0, 0 ≤ f a\n⊢ ∀ x ≤ 0, f x = 0",
"after_state": "f : ℝ → ℝ\nhf : ∀ (x y : ℝ), f (x + y) ≤ y * f x + f (f x)\nhxt : ∀ (x t : ℝ), f t ≤ t * f x - x * f x + f (f x)\nh_ab_combined : ∀ (a b : ℝ), a * f a + b * f b ≤ 2 * f a * f b\nh_f_nonneg_of_pos : ∀ a < 0, 0 ≤ f a\nh_f_nonpos : ∀ (x : ℝ), f x ≤ 0\n⊢ ∀ x ≤ 0, f x = 0"
},
{
"line": "refine_lift\n have h_f_nonpos : ∀ x, f x ≤ 0 := fun x => by\n by_contra h_suppose_not\n let s := (x * f x - f (f x)) / f x\n have hm : min 0 s - 1 < s := (sub_one_lt _).trans_le (min_le_right 0 s)\n have hml : min 0 s - 1 < 0 := (sub_one_lt _).trans_le (min_le_left 0 s)\n suffices f (min 0 s - 1) < 0 from not_le.mpr this (h_f_nonneg_of_pos (min 0 s - 1) hml)\n have hp : 0 < f x := not_le.mp h_suppose_not\n calc\n f (min 0 s - 1) ≤ (min 0 s - 1) * f x - x * f x + f (f x) := hxt x (min 0 s - 1)\n _ < s * f x - x * f x + f (f x) := by linarith [(mul_lt_mul_right hp).mpr hm]\n _ = 0 := by rw [(eq_div_iff hp.ne.symm).mp rfl]; linarith;\n ?_",
"before_state": "f : ℝ → ℝ\nhf : ∀ (x y : ℝ), f (x + y) ≤ y * f x + f (f x)\nhxt : ∀ (x t : ℝ), f t ≤ t * f x - x * f x + f (f x)\nh_ab_combined : ∀ (a b : ℝ), a * f a + b * f b ≤ 2 * f a * f b\nh_f_nonneg_of_pos : ∀ a < 0, 0 ≤ f a\n⊢ ∀ x ≤ 0, f x = 0",
"after_state": "f : ℝ → ℝ\nhf : ∀ (x y : ℝ), f (x + y) ≤ y * f x + f (f x)\nhxt : ∀ (x t : ℝ), f t ≤ t * f x - x * f x + f (f x)\nh_ab_combined : ∀ (a b : ℝ), a * f a + b * f b ≤ 2 * f a * f b\nh_f_nonneg_of_pos : ∀ a < 0, 0 ≤ f a\nh_f_nonpos : ∀ (x : ℝ), f x ≤ 0\n⊢ ∀ x ≤ 0, f x = 0"
},
{
"line": "focus\n (refine\n no_implicit_lambda%\n (have h_f_nonpos : ∀ x, f x ≤ 0 := fun x => by\n by_contra h_suppose_not\n let s := (x * f x - f (f x)) / f x\n have hm : min 0 s - 1 < s := (sub_one_lt _).trans_le (min_le_right 0 s)\n have hml : min 0 s - 1 < 0 := (sub_one_lt _).trans_le (min_le_left 0 s)\n suffices f (min 0 s - 1) < 0 from not_le.mpr this (h_f_nonneg_of_pos (min 0 s - 1) hml)\n have hp : 0 < f x := not_le.mp h_suppose_not\n calc\n f (min 0 s - 1) ≤ (min 0 s - 1) * f x - x * f x + f (f x) := hxt x (min 0 s - 1)\n _ < s * f x - x * f x + f (f x) := by linarith [(mul_lt_mul_right hp).mpr hm]\n _ = 0 := by rw [(eq_div_iff hp.ne.symm).mp rfl]; linarith;\n ?_);\n rotate_right)",
"before_state": "f : ℝ → ℝ\nhf : ∀ (x y : ℝ), f (x + y) ≤ y * f x + f (f x)\nhxt : ∀ (x t : ℝ), f t ≤ t * f x - x * f x + f (f x)\nh_ab_combined : ∀ (a b : ℝ), a * f a + b * f b ≤ 2 * f a * f b\nh_f_nonneg_of_pos : ∀ a < 0, 0 ≤ f a\n⊢ ∀ x ≤ 0, f x = 0",
"after_state": "f : ℝ → ℝ\nhf : ∀ (x y : ℝ), f (x + y) ≤ y * f x + f (f x)\nhxt : ∀ (x t : ℝ), f t ≤ t * f x - x * f x + f (f x)\nh_ab_combined : ∀ (a b : ℝ), a * f a + b * f b ≤ 2 * f a * f b\nh_f_nonneg_of_pos : ∀ a < 0, 0 ≤ f a\nh_f_nonpos : ∀ (x : ℝ), f x ≤ 0\n⊢ ∀ x ≤ 0, f x = 0"
},
{
"line": "refine\n no_implicit_lambda%\n (have h_f_nonpos : ∀ x, f x ≤ 0 := fun x => by\n by_contra h_suppose_not\n let s := (x * f x - f (f x)) / f x\n have hm : min 0 s - 1 < s := (sub_one_lt _).trans_le (min_le_right 0 s)\n have hml : min 0 s - 1 < 0 := (sub_one_lt _).trans_le (min_le_left 0 s)\n suffices f (min 0 s - 1) < 0 from not_le.mpr this (h_f_nonneg_of_pos (min 0 s - 1) hml)\n have hp : 0 < f x := not_le.mp h_suppose_not\n calc\n f (min 0 s - 1) ≤ (min 0 s - 1) * f x - x * f x + f (f x) := hxt x (min 0 s - 1)\n _ < s * f x - x * f x + f (f x) := by linarith [(mul_lt_mul_right hp).mpr hm]\n _ = 0 := by rw [(eq_div_iff hp.ne.symm).mp rfl]; linarith;\n ?_)",
"before_state": "f : ℝ → ℝ\nhf : ∀ (x y : ℝ), f (x + y) ≤ y * f x + f (f x)\nhxt : ∀ (x t : ℝ), f t ≤ t * f x - x * f x + f (f x)\nh_ab_combined : ∀ (a b : ℝ), a * f a + b * f b ≤ 2 * f a * f b\nh_f_nonneg_of_pos : ∀ a < 0, 0 ≤ f a\n⊢ ∀ x ≤ 0, f x = 0",
"after_state": "f : ℝ → ℝ\nhf : ∀ (x y : ℝ), f (x + y) ≤ y * f x + f (f x)\nhxt : ∀ (x t : ℝ), f t ≤ t * f x - x * f x + f (f x)\nh_ab_combined : ∀ (a b : ℝ), a * f a + b * f b ≤ 2 * f a * f b\nh_f_nonneg_of_pos : ∀ a < 0, 0 ≤ f a\nh_f_nonpos : ∀ (x : ℝ), f x ≤ 0\n⊢ ∀ x ≤ 0, f x = 0"
},
{
"line": "by_contra h_suppose_not",
"before_state": "f : ℝ → ℝ\nhf : ∀ (x y : ℝ), f (x + y) ≤ y * f x + f (f x)\nhxt : ∀ (x t : ℝ), f t ≤ t * f x - x * f x + f (f x)\nh_ab_combined : ∀ (a b : ℝ), a * f a + b * f b ≤ 2 * f a * f b\nh_f_nonneg_of_pos : ∀ a < 0, 0 ≤ f a\nx : ℝ\n⊢ f x ≤ 0",
"after_state": "f : ℝ → ℝ\nhf : ∀ (x y : ℝ), f (x + y) ≤ y * f x + f (f x)\nhxt : ∀ (x t : ℝ), f t ≤ t * f x - x * f x + f (f x)\nh_ab_combined : ∀ (a b : ℝ), a * f a + b * f b ≤ 2 * f a * f b\nh_f_nonneg_of_pos : ∀ a < 0, 0 ≤ f a\nx : ℝ\nh_suppose_not : ¬f x ≤ 0\n⊢ False"
},
{
"line": "first\n| guard_target = Not✝ _; intro h_suppose_not\n| refine (Decidable.byContradiction✝ fun h_suppose_not => ?_ :)\n| refine (Classical.byContradiction✝ fun h_suppose_not => ?_ :)",
"before_state": "f : ℝ → ℝ\nhf : ∀ (x y : ℝ), f (x + y) ≤ y * f x + f (f x)\nhxt : ∀ (x t : ℝ), f t ≤ t * f x - x * f x + f (f x)\nh_ab_combined : ∀ (a b : ℝ), a * f a + b * f b ≤ 2 * f a * f b\nh_f_nonneg_of_pos : ∀ a < 0, 0 ≤ f a\nx : ℝ\n⊢ f x ≤ 0",
"after_state": "f : ℝ → ℝ\nhf : ∀ (x y : ℝ), f (x + y) ≤ y * f x + f (f x)\nhxt : ∀ (x t : ℝ), f t ≤ t * f x - x * f x + f (f x)\nh_ab_combined : ∀ (a b : ℝ), a * f a + b * f b ≤ 2 * f a * f b\nh_f_nonneg_of_pos : ∀ a < 0, 0 ≤ f a\nx : ℝ\nh_suppose_not : ¬f x ≤ 0\n⊢ False"
},
{
"line": "guard_target = Not✝ _",
"before_state": "f : ℝ → ℝ\nhf : ∀ (x y : ℝ), f (x + y) ≤ y * f x + f (f x)\nhxt : ∀ (x t : ℝ), f t ≤ t * f x - x * f x + f (f x)\nh_ab_combined : ∀ (a b : ℝ), a * f a + b * f b ≤ 2 * f a * f b\nh_f_nonneg_of_pos : ∀ a < 0, 0 ≤ f a\nx : ℝ\n⊢ f x ≤ 0",
"after_state": "f : ℝ → ℝ\nhf : ∀ (x y : ℝ), f (x + y) ≤ y * f x + f (f x)\nhxt : ∀ (x t : ℝ), f t ≤ t * f x - x * f x + f (f x)\nh_ab_combined : ∀ (a b : ℝ), a * f a + b * f b ≤ 2 * f a * f b\nh_f_nonneg_of_pos : ∀ a < 0, 0 ≤ f a\nx : ℝ\n⊢ f x ≤ 0"
},
{
"line": "refine (Decidable.byContradiction✝ fun h_suppose_not => ?_ :)",
"before_state": "f : ℝ → ℝ\nhf : ∀ (x y : ℝ), f (x + y) ≤ y * f x + f (f x)\nhxt : ∀ (x t : ℝ), f t ≤ t * f x - x * f x + f (f x)\nh_ab_combined : ∀ (a b : ℝ), a * f a + b * f b ≤ 2 * f a * f b\nh_f_nonneg_of_pos : ∀ a < 0, 0 ≤ f a\nx : ℝ\n⊢ f x ≤ 0",
"after_state": "f : ℝ → ℝ\nhf : ∀ (x y : ℝ), f (x + y) ≤ y * f x + f (f x)\nhxt : ∀ (x t : ℝ), f t ≤ t * f x - x * f x + f (f x)\nh_ab_combined : ∀ (a b : ℝ), a * f a + b * f b ≤ 2 * f a * f b\nh_f_nonneg_of_pos : ∀ a < 0, 0 ≤ f a\nx : ℝ\n⊢ f x ≤ 0"
},
{
"line": "refine (Classical.byContradiction✝ fun h_suppose_not => ?_ :)",
"before_state": "f : ℝ → ℝ\nhf : ∀ (x y : ℝ), f (x + y) ≤ y * f x + f (f x)\nhxt : ∀ (x t : ℝ), f t ≤ t * f x - x * f x + f (f x)\nh_ab_combined : ∀ (a b : ℝ), a * f a + b * f b ≤ 2 * f a * f b\nh_f_nonneg_of_pos : ∀ a < 0, 0 ≤ f a\nx : ℝ\n⊢ f x ≤ 0",
"after_state": "f : ℝ → ℝ\nhf : ∀ (x y : ℝ), f (x + y) ≤ y * f x + f (f x)\nhxt : ∀ (x t : ℝ), f t ≤ t * f x - x * f x + f (f x)\nh_ab_combined : ∀ (a b : ℝ), a * f a + b * f b ≤ 2 * f a * f b\nh_f_nonneg_of_pos : ∀ a < 0, 0 ≤ f a\nx : ℝ\nh_suppose_not : ¬f x ≤ 0\n⊢ False"
},
{
"line": "let s := (x * f x - f (f x)) / f x",
"before_state": "f : ℝ → ℝ\nhf : ∀ (x y : ℝ), f (x + y) ≤ y * f x + f (f x)\nhxt : ∀ (x t : ℝ), f t ≤ t * f x - x * f x + f (f x)\nh_ab_combined : ∀ (a b : ℝ), a * f a + b * f b ≤ 2 * f a * f b\nh_f_nonneg_of_pos : ∀ a < 0, 0 ≤ f a\nx : ℝ\nh_suppose_not : ¬f x ≤ 0\n⊢ False",
"after_state": "f : ℝ → ℝ\nhf : ∀ (x y : ℝ), f (x + y) ≤ y * f x + f (f x)\nhxt : ∀ (x t : ℝ), f t ≤ t * f x - x * f x + f (f x)\nh_ab_combined : ∀ (a b : ℝ), a * f a + b * f b ≤ 2 * f a * f b\nh_f_nonneg_of_pos : ∀ a < 0, 0 ≤ f a\nx : ℝ\nh_suppose_not : ¬f x ≤ 0\ns : ℝ := (x * f x - f (f x)) / f x\n⊢ False"
},
{
"line": "refine_lift\n let s := (x * f x - f (f x)) / f x;\n ?_",
"before_state": "f : ℝ → ℝ\nhf : ∀ (x y : ℝ), f (x + y) ≤ y * f x + f (f x)\nhxt : ∀ (x t : ℝ), f t ≤ t * f x - x * f x + f (f x)\nh_ab_combined : ∀ (a b : ℝ), a * f a + b * f b ≤ 2 * f a * f b\nh_f_nonneg_of_pos : ∀ a < 0, 0 ≤ f a\nx : ℝ\nh_suppose_not : ¬f x ≤ 0\n⊢ False",
"after_state": "f : ℝ → ℝ\nhf : ∀ (x y : ℝ), f (x + y) ≤ y * f x + f (f x)\nhxt : ∀ (x t : ℝ), f t ≤ t * f x - x * f x + f (f x)\nh_ab_combined : ∀ (a b : ℝ), a * f a + b * f b ≤ 2 * f a * f b\nh_f_nonneg_of_pos : ∀ a < 0, 0 ≤ f a\nx : ℝ\nh_suppose_not : ¬f x ≤ 0\ns : ℝ := (x * f x - f (f x)) / f x\n⊢ False"
},
{
"line": "focus\n (refine\n no_implicit_lambda%\n (let s := (x * f x - f (f x)) / f x;\n ?_);\n rotate_right)",
"before_state": "f : ℝ → ℝ\nhf : ∀ (x y : ℝ), f (x + y) ≤ y * f x + f (f x)\nhxt : ∀ (x t : ℝ), f t ≤ t * f x - x * f x + f (f x)\nh_ab_combined : ∀ (a b : ℝ), a * f a + b * f b ≤ 2 * f a * f b\nh_f_nonneg_of_pos : ∀ a < 0, 0 ≤ f a\nx : ℝ\nh_suppose_not : ¬f x ≤ 0\n⊢ False",
"after_state": "f : ℝ → ℝ\nhf : ∀ (x y : ℝ), f (x + y) ≤ y * f x + f (f x)\nhxt : ∀ (x t : ℝ), f t ≤ t * f x - x * f x + f (f x)\nh_ab_combined : ∀ (a b : ℝ), a * f a + b * f b ≤ 2 * f a * f b\nh_f_nonneg_of_pos : ∀ a < 0, 0 ≤ f a\nx : ℝ\nh_suppose_not : ¬f x ≤ 0\ns : ℝ := (x * f x - f (f x)) / f x\n⊢ False"
},
{
"line": "refine\n no_implicit_lambda%\n (let s := (x * f x - f (f x)) / f x;\n ?_)",
"before_state": "f : ℝ → ℝ\nhf : ∀ (x y : ℝ), f (x + y) ≤ y * f x + f (f x)\nhxt : ∀ (x t : ℝ), f t ≤ t * f x - x * f x + f (f x)\nh_ab_combined : ∀ (a b : ℝ), a * f a + b * f b ≤ 2 * f a * f b\nh_f_nonneg_of_pos : ∀ a < 0, 0 ≤ f a\nx : ℝ\nh_suppose_not : ¬f x ≤ 0\n⊢ False",
"after_state": "f : ℝ → ℝ\nhf : ∀ (x y : ℝ), f (x + y) ≤ y * f x + f (f x)\nhxt : ∀ (x t : ℝ), f t ≤ t * f x - x * f x + f (f x)\nh_ab_combined : ∀ (a b : ℝ), a * f a + b * f b ≤ 2 * f a * f b\nh_f_nonneg_of_pos : ∀ a < 0, 0 ≤ f a\nx : ℝ\nh_suppose_not : ¬f x ≤ 0\ns : ℝ := (x * f x - f (f x)) / f x\n⊢ False"
},
{
"line": "rotate_right",
"before_state": "f : ℝ → ℝ\nhf : ∀ (x y : ℝ), f (x + y) ≤ y * f x + f (f x)\nhxt : ∀ (x t : ℝ), f t ≤ t * f x - x * f x + f (f x)\nh_ab_combined : ∀ (a b : ℝ), a * f a + b * f b ≤ 2 * f a * f b\nh_f_nonneg_of_pos : ∀ a < 0, 0 ≤ f a\nx : ℝ\nh_suppose_not : ¬f x ≤ 0\ns : ℝ := (x * f x - f (f x)) / f x\n⊢ False",
"after_state": "f : ℝ → ℝ\nhf : ∀ (x y : ℝ), f (x + y) ≤ y * f x + f (f x)\nhxt : ∀ (x t : ℝ), f t ≤ t * f x - x * f x + f (f x)\nh_ab_combined : ∀ (a b : ℝ), a * f a + b * f b ≤ 2 * f a * f b\nh_f_nonneg_of_pos : ∀ a < 0, 0 ≤ f a\nx : ℝ\nh_suppose_not : ¬f x ≤ 0\ns : ℝ := (x * f x - f (f x)) / f x\n⊢ False"
},
{
"line": "have hm : min 0 s - 1 < s := (sub_one_lt _).trans_le (min_le_right 0 s)",
"before_state": "f : ℝ → ℝ\nhf : ∀ (x y : ℝ), f (x + y) ≤ y * f x + f (f x)\nhxt : ∀ (x t : ℝ), f t ≤ t * f x - x * f x + f (f x)\nh_ab_combined : ∀ (a b : ℝ), a * f a + b * f b ≤ 2 * f a * f b\nh_f_nonneg_of_pos : ∀ a < 0, 0 ≤ f a\nx : ℝ\nh_suppose_not : ¬f x ≤ 0\ns : ℝ := (x * f x - f (f x)) / f x\n⊢ False",
"after_state": "f : ℝ → ℝ\nhf : ∀ (x y : ℝ), f (x + y) ≤ y * f x + f (f x)\nhxt : ∀ (x t : ℝ), f t ≤ t * f x - x * f x + f (f x)\nh_ab_combined : ∀ (a b : ℝ), a * f a + b * f b ≤ 2 * f a * f b\nh_f_nonneg_of_pos : ∀ a < 0, 0 ≤ f a\nx : ℝ\nh_suppose_not : ¬f x ≤ 0\ns : ℝ := (x * f x - f (f x)) / f x\nhm : min 0 s - 1 < s\n⊢ False"
},
{
"line": "refine_lift\n have hm : min 0 s - 1 < s := (sub_one_lt _).trans_le (min_le_right 0 s);\n ?_",
"before_state": "f : ℝ → ℝ\nhf : ∀ (x y : ℝ), f (x + y) ≤ y * f x + f (f x)\nhxt : ∀ (x t : ℝ), f t ≤ t * f x - x * f x + f (f x)\nh_ab_combined : ∀ (a b : ℝ), a * f a + b * f b ≤ 2 * f a * f b\nh_f_nonneg_of_pos : ∀ a < 0, 0 ≤ f a\nx : ℝ\nh_suppose_not : ¬f x ≤ 0\ns : ℝ := (x * f x - f (f x)) / f x\n⊢ False",
"after_state": "f : ℝ → ℝ\nhf : ∀ (x y : ℝ), f (x + y) ≤ y * f x + f (f x)\nhxt : ∀ (x t : ℝ), f t ≤ t * f x - x * f x + f (f x)\nh_ab_combined : ∀ (a b : ℝ), a * f a + b * f b ≤ 2 * f a * f b\nh_f_nonneg_of_pos : ∀ a < 0, 0 ≤ f a\nx : ℝ\nh_suppose_not : ¬f x ≤ 0\ns : ℝ := (x * f x - f (f x)) / f x\nhm : min 0 s - 1 < s\n⊢ False"
},
{
"line": "focus\n (refine\n no_implicit_lambda%\n (have hm : min 0 s - 1 < s := (sub_one_lt _).trans_le (min_le_right 0 s);\n ?_);\n rotate_right)",
"before_state": "f : ℝ → ℝ\nhf : ∀ (x y : ℝ), f (x + y) ≤ y * f x + f (f x)\nhxt : ∀ (x t : ℝ), f t ≤ t * f x - x * f x + f (f x)\nh_ab_combined : ∀ (a b : ℝ), a * f a + b * f b ≤ 2 * f a * f b\nh_f_nonneg_of_pos : ∀ a < 0, 0 ≤ f a\nx : ℝ\nh_suppose_not : ¬f x ≤ 0\ns : ℝ := (x * f x - f (f x)) / f x\n⊢ False",
"after_state": "f : ℝ → ℝ\nhf : ∀ (x y : ℝ), f (x + y) ≤ y * f x + f (f x)\nhxt : ∀ (x t : ℝ), f t ≤ t * f x - x * f x + f (f x)\nh_ab_combined : ∀ (a b : ℝ), a * f a + b * f b ≤ 2 * f a * f b\nh_f_nonneg_of_pos : ∀ a < 0, 0 ≤ f a\nx : ℝ\nh_suppose_not : ¬f x ≤ 0\ns : ℝ := (x * f x - f (f x)) / f x\nhm : min 0 s - 1 < s\n⊢ False"
},
{
"line": "refine\n no_implicit_lambda%\n (have hm : min 0 s - 1 < s := (sub_one_lt _).trans_le (min_le_right 0 s);\n ?_)",
"before_state": "f : ℝ → ℝ\nhf : ∀ (x y : ℝ), f (x + y) ≤ y * f x + f (f x)\nhxt : ∀ (x t : ℝ), f t ≤ t * f x - x * f x + f (f x)\nh_ab_combined : ∀ (a b : ℝ), a * f a + b * f b ≤ 2 * f a * f b\nh_f_nonneg_of_pos : ∀ a < 0, 0 ≤ f a\nx : ℝ\nh_suppose_not : ¬f x ≤ 0\ns : ℝ := (x * f x - f (f x)) / f x\n⊢ False",
"after_state": "f : ℝ → ℝ\nhf : ∀ (x y : ℝ), f (x + y) ≤ y * f x + f (f x)\nhxt : ∀ (x t : ℝ), f t ≤ t * f x - x * f x + f (f x)\nh_ab_combined : ∀ (a b : ℝ), a * f a + b * f b ≤ 2 * f a * f b\nh_f_nonneg_of_pos : ∀ a < 0, 0 ≤ f a\nx : ℝ\nh_suppose_not : ¬f x ≤ 0\ns : ℝ := (x * f x - f (f x)) / f x\nhm : min 0 s - 1 < s\n⊢ False"
},
{
"line": "rotate_right",
"before_state": "f : ℝ → ℝ\nhf : ∀ (x y : ℝ), f (x + y) ≤ y * f x + f (f x)\nhxt : ∀ (x t : ℝ), f t ≤ t * f x - x * f x + f (f x)\nh_ab_combined : ∀ (a b : ℝ), a * f a + b * f b ≤ 2 * f a * f b\nh_f_nonneg_of_pos : ∀ a < 0, 0 ≤ f a\nx : ℝ\nh_suppose_not : ¬f x ≤ 0\ns : ℝ := (x * f x - f (f x)) / f x\nhm : min 0 s - 1 < s\n⊢ False",
"after_state": "f : ℝ → ℝ\nhf : ∀ (x y : ℝ), f (x + y) ≤ y * f x + f (f x)\nhxt : ∀ (x t : ℝ), f t ≤ t * f x - x * f x + f (f x)\nh_ab_combined : ∀ (a b : ℝ), a * f a + b * f b ≤ 2 * f a * f b\nh_f_nonneg_of_pos : ∀ a < 0, 0 ≤ f a\nx : ℝ\nh_suppose_not : ¬f x ≤ 0\ns : ℝ := (x * f x - f (f x)) / f x\nhm : min 0 s - 1 < s\n⊢ False"
},
{
"line": "have hml : min 0 s - 1 < 0 := (sub_one_lt _).trans_le (min_le_left 0 s)",
"before_state": "f : ℝ → ℝ\nhf : ∀ (x y : ℝ), f (x + y) ≤ y * f x + f (f x)\nhxt : ∀ (x t : ℝ), f t ≤ t * f x - x * f x + f (f x)\nh_ab_combined : ∀ (a b : ℝ), a * f a + b * f b ≤ 2 * f a * f b\nh_f_nonneg_of_pos : ∀ a < 0, 0 ≤ f a\nx : ℝ\nh_suppose_not : ¬f x ≤ 0\ns : ℝ := (x * f x - f (f x)) / f x\nhm : min 0 s - 1 < s\n⊢ False",
"after_state": "f : ℝ → ℝ\nhf : ∀ (x y : ℝ), f (x + y) ≤ y * f x + f (f x)\nhxt : ∀ (x t : ℝ), f t ≤ t * f x - x * f x + f (f x)\nh_ab_combined : ∀ (a b : ℝ), a * f a + b * f b ≤ 2 * f a * f b\nh_f_nonneg_of_pos : ∀ a < 0, 0 ≤ f a\nx : ℝ\nh_suppose_not : ¬f x ≤ 0\ns : ℝ := (x * f x - f (f x)) / f x\nhm : min 0 s - 1 < s\nhml : min 0 s - 1 < 0\n⊢ False"
},
{
"line": "refine_lift\n have hml : min 0 s - 1 < 0 := (sub_one_lt _).trans_le (min_le_left 0 s);\n ?_",
"before_state": "f : ℝ → ℝ\nhf : ∀ (x y : ℝ), f (x + y) ≤ y * f x + f (f x)\nhxt : ∀ (x t : ℝ), f t ≤ t * f x - x * f x + f (f x)\nh_ab_combined : ∀ (a b : ℝ), a * f a + b * f b ≤ 2 * f a * f b\nh_f_nonneg_of_pos : ∀ a < 0, 0 ≤ f a\nx : ℝ\nh_suppose_not : ¬f x ≤ 0\ns : ℝ := (x * f x - f (f x)) / f x\nhm : min 0 s - 1 < s\n⊢ False",
"after_state": "f : ℝ → ℝ\nhf : ∀ (x y : ℝ), f (x + y) ≤ y * f x + f (f x)\nhxt : ∀ (x t : ℝ), f t ≤ t * f x - x * f x + f (f x)\nh_ab_combined : ∀ (a b : ℝ), a * f a + b * f b ≤ 2 * f a * f b\nh_f_nonneg_of_pos : ∀ a < 0, 0 ≤ f a\nx : ℝ\nh_suppose_not : ¬f x ≤ 0\ns : ℝ := (x * f x - f (f x)) / f x\nhm : min 0 s - 1 < s\nhml : min 0 s - 1 < 0\n⊢ False"
},
{
"line": "focus\n (refine\n no_implicit_lambda%\n (have hml : min 0 s - 1 < 0 := (sub_one_lt _).trans_le (min_le_left 0 s);\n ?_);\n rotate_right)",
"before_state": "f : ℝ → ℝ\nhf : ∀ (x y : ℝ), f (x + y) ≤ y * f x + f (f x)\nhxt : ∀ (x t : ℝ), f t ≤ t * f x - x * f x + f (f x)\nh_ab_combined : ∀ (a b : ℝ), a * f a + b * f b ≤ 2 * f a * f b\nh_f_nonneg_of_pos : ∀ a < 0, 0 ≤ f a\nx : ℝ\nh_suppose_not : ¬f x ≤ 0\ns : ℝ := (x * f x - f (f x)) / f x\nhm : min 0 s - 1 < s\n⊢ False",
"after_state": "f : ℝ → ℝ\nhf : ∀ (x y : ℝ), f (x + y) ≤ y * f x + f (f x)\nhxt : ∀ (x t : ℝ), f t ≤ t * f x - x * f x + f (f x)\nh_ab_combined : ∀ (a b : ℝ), a * f a + b * f b ≤ 2 * f a * f b\nh_f_nonneg_of_pos : ∀ a < 0, 0 ≤ f a\nx : ℝ\nh_suppose_not : ¬f x ≤ 0\ns : ℝ := (x * f x - f (f x)) / f x\nhm : min 0 s - 1 < s\nhml : min 0 s - 1 < 0\n⊢ False"
},
{
"line": "refine\n no_implicit_lambda%\n (have hml : min 0 s - 1 < 0 := (sub_one_lt _).trans_le (min_le_left 0 s);\n ?_)",
"before_state": "f : ℝ → ℝ\nhf : ∀ (x y : ℝ), f (x + y) ≤ y * f x + f (f x)\nhxt : ∀ (x t : ℝ), f t ≤ t * f x - x * f x + f (f x)\nh_ab_combined : ∀ (a b : ℝ), a * f a + b * f b ≤ 2 * f a * f b\nh_f_nonneg_of_pos : ∀ a < 0, 0 ≤ f a\nx : ℝ\nh_suppose_not : ¬f x ≤ 0\ns : ℝ := (x * f x - f (f x)) / f x\nhm : min 0 s - 1 < s\n⊢ False",
"after_state": "f : ℝ → ℝ\nhf : ∀ (x y : ℝ), f (x + y) ≤ y * f x + f (f x)\nhxt : ∀ (x t : ℝ), f t ≤ t * f x - x * f x + f (f x)\nh_ab_combined : ∀ (a b : ℝ), a * f a + b * f b ≤ 2 * f a * f b\nh_f_nonneg_of_pos : ∀ a < 0, 0 ≤ f a\nx : ℝ\nh_suppose_not : ¬f x ≤ 0\ns : ℝ := (x * f x - f (f x)) / f x\nhm : min 0 s - 1 < s\nhml : min 0 s - 1 < 0\n⊢ False"
},
{
"line": "rotate_right",
"before_state": "f : ℝ → ℝ\nhf : ∀ (x y : ℝ), f (x + y) ≤ y * f x + f (f x)\nhxt : ∀ (x t : ℝ), f t ≤ t * f x - x * f x + f (f x)\nh_ab_combined : ∀ (a b : ℝ), a * f a + b * f b ≤ 2 * f a * f b\nh_f_nonneg_of_pos : ∀ a < 0, 0 ≤ f a\nx : ℝ\nh_suppose_not : ¬f x ≤ 0\ns : ℝ := (x * f x - f (f x)) / f x\nhm : min 0 s - 1 < s\nhml : min 0 s - 1 < 0\n⊢ False",
"after_state": "f : ℝ → ℝ\nhf : ∀ (x y : ℝ), f (x + y) ≤ y * f x + f (f x)\nhxt : ∀ (x t : ℝ), f t ≤ t * f x - x * f x + f (f x)\nh_ab_combined : ∀ (a b : ℝ), a * f a + b * f b ≤ 2 * f a * f b\nh_f_nonneg_of_pos : ∀ a < 0, 0 ≤ f a\nx : ℝ\nh_suppose_not : ¬f x ≤ 0\ns : ℝ := (x * f x - f (f x)) / f x\nhm : min 0 s - 1 < s\nhml : min 0 s - 1 < 0\n⊢ False"
},
{
"line": "suffices f (min 0 s - 1) < 0 from not_le.mpr this (h_f_nonneg_of_pos (min 0 s - 1) hml)",
"before_state": "f : ℝ → ℝ\nhf : ∀ (x y : ℝ), f (x + y) ≤ y * f x + f (f x)\nhxt : ∀ (x t : ℝ), f t ≤ t * f x - x * f x + f (f x)\nh_ab_combined : ∀ (a b : ℝ), a * f a + b * f b ≤ 2 * f a * f b\nh_f_nonneg_of_pos : ∀ a < 0, 0 ≤ f a\nx : ℝ\nh_suppose_not : ¬f x ≤ 0\ns : ℝ := (x * f x - f (f x)) / f x\nhm : min 0 s - 1 < s\nhml : min 0 s - 1 < 0\n⊢ False",
"after_state": "f : ℝ → ℝ\nhf : ∀ (x y : ℝ), f (x + y) ≤ y * f x + f (f x)\nhxt : ∀ (x t : ℝ), f t ≤ t * f x - x * f x + f (f x)\nh_ab_combined : ∀ (a b : ℝ), a * f a + b * f b ≤ 2 * f a * f b\nh_f_nonneg_of_pos : ∀ a < 0, 0 ≤ f a\nx : ℝ\nh_suppose_not : ¬f x ≤ 0\ns : ℝ := (x * f x - f (f x)) / f x\nhm : min 0 s - 1 < s\nhml : min 0 s - 1 < 0\n⊢ f (min 0 s - 1) < 0"
},
{
"line": "refine_lift\n suffices f (min 0 s - 1) < 0 from not_le.mpr this (h_f_nonneg_of_pos (min 0 s - 1) hml);\n ?_",
"before_state": "f : ℝ → ℝ\nhf : ∀ (x y : ℝ), f (x + y) ≤ y * f x + f (f x)\nhxt : ∀ (x t : ℝ), f t ≤ t * f x - x * f x + f (f x)\nh_ab_combined : ∀ (a b : ℝ), a * f a + b * f b ≤ 2 * f a * f b\nh_f_nonneg_of_pos : ∀ a < 0, 0 ≤ f a\nx : ℝ\nh_suppose_not : ¬f x ≤ 0\ns : ℝ := (x * f x - f (f x)) / f x\nhm : min 0 s - 1 < s\nhml : min 0 s - 1 < 0\n⊢ False",
"after_state": "f : ℝ → ℝ\nhf : ∀ (x y : ℝ), f (x + y) ≤ y * f x + f (f x)\nhxt : ∀ (x t : ℝ), f t ≤ t * f x - x * f x + f (f x)\nh_ab_combined : ∀ (a b : ℝ), a * f a + b * f b ≤ 2 * f a * f b\nh_f_nonneg_of_pos : ∀ a < 0, 0 ≤ f a\nx : ℝ\nh_suppose_not : ¬f x ≤ 0\ns : ℝ := (x * f x - f (f x)) / f x\nhm : min 0 s - 1 < s\nhml : min 0 s - 1 < 0\n⊢ f (min 0 s - 1) < 0"
},
{
"line": "focus\n (refine\n no_implicit_lambda%\n (suffices f (min 0 s - 1) < 0 from not_le.mpr this (h_f_nonneg_of_pos (min 0 s - 1) hml);\n ?_);\n rotate_right)",
"before_state": "f : ℝ → ℝ\nhf : ∀ (x y : ℝ), f (x + y) ≤ y * f x + f (f x)\nhxt : ∀ (x t : ℝ), f t ≤ t * f x - x * f x + f (f x)\nh_ab_combined : ∀ (a b : ℝ), a * f a + b * f b ≤ 2 * f a * f b\nh_f_nonneg_of_pos : ∀ a < 0, 0 ≤ f a\nx : ℝ\nh_suppose_not : ¬f x ≤ 0\ns : ℝ := (x * f x - f (f x)) / f x\nhm : min 0 s - 1 < s\nhml : min 0 s - 1 < 0\n⊢ False",
"after_state": "f : ℝ → ℝ\nhf : ∀ (x y : ℝ), f (x + y) ≤ y * f x + f (f x)\nhxt : ∀ (x t : ℝ), f t ≤ t * f x - x * f x + f (f x)\nh_ab_combined : ∀ (a b : ℝ), a * f a + b * f b ≤ 2 * f a * f b\nh_f_nonneg_of_pos : ∀ a < 0, 0 ≤ f a\nx : ℝ\nh_suppose_not : ¬f x ≤ 0\ns : ℝ := (x * f x - f (f x)) / f x\nhm : min 0 s - 1 < s\nhml : min 0 s - 1 < 0\n⊢ f (min 0 s - 1) < 0"
},
{
"line": "refine\n no_implicit_lambda%\n (suffices f (min 0 s - 1) < 0 from not_le.mpr this (h_f_nonneg_of_pos (min 0 s - 1) hml);\n ?_)",
"before_state": "f : ℝ → ℝ\nhf : ∀ (x y : ℝ), f (x + y) ≤ y * f x + f (f x)\nhxt : ∀ (x t : ℝ), f t ≤ t * f x - x * f x + f (f x)\nh_ab_combined : ∀ (a b : ℝ), a * f a + b * f b ≤ 2 * f a * f b\nh_f_nonneg_of_pos : ∀ a < 0, 0 ≤ f a\nx : ℝ\nh_suppose_not : ¬f x ≤ 0\ns : ℝ := (x * f x - f (f x)) / f x\nhm : min 0 s - 1 < s\nhml : min 0 s - 1 < 0\n⊢ False",
"after_state": "f : ℝ → ℝ\nhf : ∀ (x y : ℝ), f (x + y) ≤ y * f x + f (f x)\nhxt : ∀ (x t : ℝ), f t ≤ t * f x - x * f x + f (f x)\nh_ab_combined : ∀ (a b : ℝ), a * f a + b * f b ≤ 2 * f a * f b\nh_f_nonneg_of_pos : ∀ a < 0, 0 ≤ f a\nx : ℝ\nh_suppose_not : ¬f x ≤ 0\ns : ℝ := (x * f x - f (f x)) / f x\nhm : min 0 s - 1 < s\nhml : min 0 s - 1 < 0\n⊢ f (min 0 s - 1) < 0"
},
{
"line": "rotate_right",
"before_state": "f : ℝ → ℝ\nhf : ∀ (x y : ℝ), f (x + y) ≤ y * f x + f (f x)\nhxt : ∀ (x t : ℝ), f t ≤ t * f x - x * f x + f (f x)\nh_ab_combined : ∀ (a b : ℝ), a * f a + b * f b ≤ 2 * f a * f b\nh_f_nonneg_of_pos : ∀ a < 0, 0 ≤ f a\nx : ℝ\nh_suppose_not : ¬f x ≤ 0\ns : ℝ := (x * f x - f (f x)) / f x\nhm : min 0 s - 1 < s\nhml : min 0 s - 1 < 0\n⊢ f (min 0 s - 1) < 0",
"after_state": "f : ℝ → ℝ\nhf : ∀ (x y : ℝ), f (x + y) ≤ y * f x + f (f x)\nhxt : ∀ (x t : ℝ), f t ≤ t * f x - x * f x + f (f x)\nh_ab_combined : ∀ (a b : ℝ), a * f a + b * f b ≤ 2 * f a * f b\nh_f_nonneg_of_pos : ∀ a < 0, 0 ≤ f a\nx : ℝ\nh_suppose_not : ¬f x ≤ 0\ns : ℝ := (x * f x - f (f x)) / f x\nhm : min 0 s - 1 < s\nhml : min 0 s - 1 < 0\n⊢ f (min 0 s - 1) < 0"
},
{
"line": "have hp : 0 < f x := not_le.mp h_suppose_not",
"before_state": "f : ℝ → ℝ\nhf : ∀ (x y : ℝ), f (x + y) ≤ y * f x + f (f x)\nhxt : ∀ (x t : ℝ), f t ≤ t * f x - x * f x + f (f x)\nh_ab_combined : ∀ (a b : ℝ), a * f a + b * f b ≤ 2 * f a * f b\nh_f_nonneg_of_pos : ∀ a < 0, 0 ≤ f a\nx : ℝ\nh_suppose_not : ¬f x ≤ 0\ns : ℝ := (x * f x - f (f x)) / f x\nhm : min 0 s - 1 < s\nhml : min 0 s - 1 < 0\n⊢ f (min 0 s - 1) < 0",
"after_state": "f : ℝ → ℝ\nhf : ∀ (x y : ℝ), f (x + y) ≤ y * f x + f (f x)\nhxt : ∀ (x t : ℝ), f t ≤ t * f x - x * f x + f (f x)\nh_ab_combined : ∀ (a b : ℝ), a * f a + b * f b ≤ 2 * f a * f b\nh_f_nonneg_of_pos : ∀ a < 0, 0 ≤ f a\nx : ℝ\nh_suppose_not : ¬f x ≤ 0\ns : ℝ := (x * f x - f (f x)) / f x\nhm : min 0 s - 1 < s\nhml : min 0 s - 1 < 0\nhp : 0 < f x\n⊢ f (min 0 s - 1) < 0"
},
{
"line": "refine_lift\n have hp : 0 < f x := not_le.mp h_suppose_not;\n ?_",
"before_state": "f : ℝ → ℝ\nhf : ∀ (x y : ℝ), f (x + y) ≤ y * f x + f (f x)\nhxt : ∀ (x t : ℝ), f t ≤ t * f x - x * f x + f (f x)\nh_ab_combined : ∀ (a b : ℝ), a * f a + b * f b ≤ 2 * f a * f b\nh_f_nonneg_of_pos : ∀ a < 0, 0 ≤ f a\nx : ℝ\nh_suppose_not : ¬f x ≤ 0\ns : ℝ := (x * f x - f (f x)) / f x\nhm : min 0 s - 1 < s\nhml : min 0 s - 1 < 0\n⊢ f (min 0 s - 1) < 0",
"after_state": "f : ℝ → ℝ\nhf : ∀ (x y : ℝ), f (x + y) ≤ y * f x + f (f x)\nhxt : ∀ (x t : ℝ), f t ≤ t * f x - x * f x + f (f x)\nh_ab_combined : ∀ (a b : ℝ), a * f a + b * f b ≤ 2 * f a * f b\nh_f_nonneg_of_pos : ∀ a < 0, 0 ≤ f a\nx : ℝ\nh_suppose_not : ¬f x ≤ 0\ns : ℝ := (x * f x - f (f x)) / f x\nhm : min 0 s - 1 < s\nhml : min 0 s - 1 < 0\nhp : 0 < f x\n⊢ f (min 0 s - 1) < 0"
},
{
"line": "focus\n (refine\n no_implicit_lambda%\n (have hp : 0 < f x := not_le.mp h_suppose_not;\n ?_);\n rotate_right)",
"before_state": "f : ℝ → ℝ\nhf : ∀ (x y : ℝ), f (x + y) ≤ y * f x + f (f x)\nhxt : ∀ (x t : ℝ), f t ≤ t * f x - x * f x + f (f x)\nh_ab_combined : ∀ (a b : ℝ), a * f a + b * f b ≤ 2 * f a * f b\nh_f_nonneg_of_pos : ∀ a < 0, 0 ≤ f a\nx : ℝ\nh_suppose_not : ¬f x ≤ 0\ns : ℝ := (x * f x - f (f x)) / f x\nhm : min 0 s - 1 < s\nhml : min 0 s - 1 < 0\n⊢ f (min 0 s - 1) < 0",
"after_state": "f : ℝ → ℝ\nhf : ∀ (x y : ℝ), f (x + y) ≤ y * f x + f (f x)\nhxt : ∀ (x t : ℝ), f t ≤ t * f x - x * f x + f (f x)\nh_ab_combined : ∀ (a b : ℝ), a * f a + b * f b ≤ 2 * f a * f b\nh_f_nonneg_of_pos : ∀ a < 0, 0 ≤ f a\nx : ℝ\nh_suppose_not : ¬f x ≤ 0\ns : ℝ := (x * f x - f (f x)) / f x\nhm : min 0 s - 1 < s\nhml : min 0 s - 1 < 0\nhp : 0 < f x\n⊢ f (min 0 s - 1) < 0"
},
{
"line": "refine\n no_implicit_lambda%\n (have hp : 0 < f x := not_le.mp h_suppose_not;\n ?_)",
"before_state": "f : ℝ → ℝ\nhf : ∀ (x y : ℝ), f (x + y) ≤ y * f x + f (f x)\nhxt : ∀ (x t : ℝ), f t ≤ t * f x - x * f x + f (f x)\nh_ab_combined : ∀ (a b : ℝ), a * f a + b * f b ≤ 2 * f a * f b\nh_f_nonneg_of_pos : ∀ a < 0, 0 ≤ f a\nx : ℝ\nh_suppose_not : ¬f x ≤ 0\ns : ℝ := (x * f x - f (f x)) / f x\nhm : min 0 s - 1 < s\nhml : min 0 s - 1 < 0\n⊢ f (min 0 s - 1) < 0",
"after_state": "f : ℝ → ℝ\nhf : ∀ (x y : ℝ), f (x + y) ≤ y * f x + f (f x)\nhxt : ∀ (x t : ℝ), f t ≤ t * f x - x * f x + f (f x)\nh_ab_combined : ∀ (a b : ℝ), a * f a + b * f b ≤ 2 * f a * f b\nh_f_nonneg_of_pos : ∀ a < 0, 0 ≤ f a\nx : ℝ\nh_suppose_not : ¬f x ≤ 0\ns : ℝ := (x * f x - f (f x)) / f x\nhm : min 0 s - 1 < s\nhml : min 0 s - 1 < 0\nhp : 0 < f x\n⊢ f (min 0 s - 1) < 0"
},
{
"line": "rotate_right",
"before_state": "f : ℝ → ℝ\nhf : ∀ (x y : ℝ), f (x + y) ≤ y * f x + f (f x)\nhxt : ∀ (x t : ℝ), f t ≤ t * f x - x * f x + f (f x)\nh_ab_combined : ∀ (a b : ℝ), a * f a + b * f b ≤ 2 * f a * f b\nh_f_nonneg_of_pos : ∀ a < 0, 0 ≤ f a\nx : ℝ\nh_suppose_not : ¬f x ≤ 0\ns : ℝ := (x * f x - f (f x)) / f x\nhm : min 0 s - 1 < s\nhml : min 0 s - 1 < 0\nhp : 0 < f x\n⊢ f (min 0 s - 1) < 0",
"after_state": "f : ℝ → ℝ\nhf : ∀ (x y : ℝ), f (x + y) ≤ y * f x + f (f x)\nhxt : ∀ (x t : ℝ), f t ≤ t * f x - x * f x + f (f x)\nh_ab_combined : ∀ (a b : ℝ), a * f a + b * f b ≤ 2 * f a * f b\nh_f_nonneg_of_pos : ∀ a < 0, 0 ≤ f a\nx : ℝ\nh_suppose_not : ¬f x ≤ 0\ns : ℝ := (x * f x - f (f x)) / f x\nhm : min 0 s - 1 < s\nhml : min 0 s - 1 < 0\nhp : 0 < f x\n⊢ f (min 0 s - 1) < 0"
},
{
"line": "calc\n f (min 0 s - 1) ≤ (min 0 s - 1) * f x - x * f x + f (f x) := hxt x (min 0 s - 1)\n _ < s * f x - x * f x + f (f x) := by linarith [(mul_lt_mul_right hp).mpr hm]\n _ = 0 := by rw [(eq_div_iff hp.ne.symm).mp rfl]; linarith",
"before_state": "f : ℝ → ℝ\nhf : ∀ (x y : ℝ), f (x + y) ≤ y * f x + f (f x)\nhxt : ∀ (x t : ℝ), f t ≤ t * f x - x * f x + f (f x)\nh_ab_combined : ∀ (a b : ℝ), a * f a + b * f b ≤ 2 * f a * f b\nh_f_nonneg_of_pos : ∀ a < 0, 0 ≤ f a\nx : ℝ\nh_suppose_not : ¬f x ≤ 0\ns : ℝ := (x * f x - f (f x)) / f x\nhm : min 0 s - 1 < s\nhml : min 0 s - 1 < 0\nhp : 0 < f x\n⊢ f (min 0 s - 1) < 0",
"after_state": "No Goals!"
},
{
"line": "linarith [(mul_lt_mul_right hp).mpr hm]",
"before_state": "f : ℝ → ℝ\nhf : ∀ (x y : ℝ), f (x + y) ≤ y * f x + f (f x)\nhxt : ∀ (x t : ℝ), f t ≤ t * f x - x * f x + f (f x)\nh_ab_combined : ∀ (a b : ℝ), a * f a + b * f b ≤ 2 * f a * f b\nh_f_nonneg_of_pos : ∀ a < 0, 0 ≤ f a\nx : ℝ\nh_suppose_not : ¬f x ≤ 0\ns : ℝ := (x * f x - f (f x)) / f x\nhm : min 0 s - 1 < s\nhml : min 0 s - 1 < 0\nhp : 0 < f x\n⊢ (min 0 s - 1) * f x - x * f x + f (f x) < s * f x - x * f x + f (f x)",
"after_state": "No Goals!"
},
{
"line": "ring1",
"before_state": "f : ℝ → ℝ\nhf : ∀ (x y : ℝ), f (x + y) ≤ y * f x + f (f x)\nhxt : ∀ (x t : ℝ), f t ≤ t * f x - x * f x + f (f x)\nh_ab_combined : ∀ (a b : ℝ), a * f a + b * f b ≤ 2 * f a * f b\nh_f_nonneg_of_pos : ∀ a < 0, 0 ≤ f a\nx : ℝ\nh_suppose_not : ¬f x ≤ 0\ns : ℝ := (x * f x - f (f x)) / f x\nhm : min 0 s - 1 < s\nhml : min 0 s - 1 < 0\nhp : 0 < f x\na✝ : (min 0 s - 1) * f x - x * f x + f (f x) ≥ s * f x - x * f x + f (f x)\n⊢ s * f x - x * f x + f (f x) - ((min 0 s - 1) * f x - x * f x + f (f x)) + ((min 0 s - 1) * f x - s * f x) = 0",
"after_state": "No Goals!"
},
{
"line": "rw [(eq_div_iff hp.ne.symm).mp rfl]",
"before_state": "f : ℝ → ℝ\nhf : ∀ (x y : ℝ), f (x + y) ≤ y * f x + f (f x)\nhxt : ∀ (x t : ℝ), f t ≤ t * f x - x * f x + f (f x)\nh_ab_combined : ∀ (a b : ℝ), a * f a + b * f b ≤ 2 * f a * f b\nh_f_nonneg_of_pos : ∀ a < 0, 0 ≤ f a\nx : ℝ\nh_suppose_not : ¬f x ≤ 0\ns : ℝ := (x * f x - f (f x)) / f x\nhm : min 0 s - 1 < s\nhml : min 0 s - 1 < 0\nhp : 0 < f x\n⊢ s * f x - x * f x + f (f x) = 0",
"after_state": "f : ℝ → ℝ\nhf : ∀ (x y : ℝ), f (x + y) ≤ y * f x + f (f x)\nhxt : ∀ (x t : ℝ), f t ≤ t * f x - x * f x + f (f x)\nh_ab_combined : ∀ (a b : ℝ), a * f a + b * f b ≤ 2 * f a * f b\nh_f_nonneg_of_pos : ∀ a < 0, 0 ≤ f a\nx : ℝ\nh_suppose_not : ¬f x ≤ 0\ns : ℝ := (x * f x - f (f x)) / f x\nhm : min 0 s - 1 < s\nhml : min 0 s - 1 < 0\nhp : 0 < f x\n⊢ x * f x - f (f x) - x * f x + f (f x) = 0"
},
{
"line": "rewrite [(eq_div_iff hp.ne.symm).mp rfl]",
"before_state": "f : ℝ → ℝ\nhf : ∀ (x y : ℝ), f (x + y) ≤ y * f x + f (f x)\nhxt : ∀ (x t : ℝ), f t ≤ t * f x - x * f x + f (f x)\nh_ab_combined : ∀ (a b : ℝ), a * f a + b * f b ≤ 2 * f a * f b\nh_f_nonneg_of_pos : ∀ a < 0, 0 ≤ f a\nx : ℝ\nh_suppose_not : ¬f x ≤ 0\ns : ℝ := (x * f x - f (f x)) / f x\nhm : min 0 s - 1 < s\nhml : min 0 s - 1 < 0\nhp : 0 < f x\n⊢ s * f x - x * f x + f (f x) = 0",
"after_state": "f : ℝ → ℝ\nhf : ∀ (x y : ℝ), f (x + y) ≤ y * f x + f (f x)\nhxt : ∀ (x t : ℝ), f t ≤ t * f x - x * f x + f (f x)\nh_ab_combined : ∀ (a b : ℝ), a * f a + b * f b ≤ 2 * f a * f b\nh_f_nonneg_of_pos : ∀ a < 0, 0 ≤ f a\nx : ℝ\nh_suppose_not : ¬f x ≤ 0\ns : ℝ := (x * f x - f (f x)) / f x\nhm : min 0 s - 1 < s\nhml : min 0 s - 1 < 0\nhp : 0 < f x\n⊢ x * f x - f (f x) - x * f x + f (f x) = 0"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "f : ℝ → ℝ\nhf : ∀ (x y : ℝ), f (x + y) ≤ y * f x + f (f x)\nhxt : ∀ (x t : ℝ), f t ≤ t * f x - x * f x + f (f x)\nh_ab_combined : ∀ (a b : ℝ), a * f a + b * f b ≤ 2 * f a * f b\nh_f_nonneg_of_pos : ∀ a < 0, 0 ≤ f a\nx : ℝ\nh_suppose_not : ¬f x ≤ 0\ns : ℝ := (x * f x - f (f x)) / f x\nhm : min 0 s - 1 < s\nhml : min 0 s - 1 < 0\nhp : 0 < f x\n⊢ x * f x - f (f x) - x * f x + f (f x) = 0",
"after_state": "f : ℝ → ℝ\nhf : ∀ (x y : ℝ), f (x + y) ≤ y * f x + f (f x)\nhxt : ∀ (x t : ℝ), f t ≤ t * f x - x * f x + f (f x)\nh_ab_combined : ∀ (a b : ℝ), a * f a + b * f b ≤ 2 * f a * f b\nh_f_nonneg_of_pos : ∀ a < 0, 0 ≤ f a\nx : ℝ\nh_suppose_not : ¬f x ≤ 0\ns : ℝ := (x * f x - f (f x)) / f x\nhm : min 0 s - 1 < s\nhml : min 0 s - 1 < 0\nhp : 0 < f x\n⊢ x * f x - f (f x) - x * f x + f (f x) = 0"
},
{
"line": "try (with_reducible rfl)",
"before_state": "f : ℝ → ℝ\nhf : ∀ (x y : ℝ), f (x + y) ≤ y * f x + f (f x)\nhxt : ∀ (x t : ℝ), f t ≤ t * f x - x * f x + f (f x)\nh_ab_combined : ∀ (a b : ℝ), a * f a + b * f b ≤ 2 * f a * f b\nh_f_nonneg_of_pos : ∀ a < 0, 0 ≤ f a\nx : ℝ\nh_suppose_not : ¬f x ≤ 0\ns : ℝ := (x * f x - f (f x)) / f x\nhm : min 0 s - 1 < s\nhml : min 0 s - 1 < 0\nhp : 0 < f x\n⊢ x * f x - f (f x) - x * f x + f (f x) = 0",
"after_state": "f : ℝ → ℝ\nhf : ∀ (x y : ℝ), f (x + y) ≤ y * f x + f (f x)\nhxt : ∀ (x t : ℝ), f t ≤ t * f x - x * f x + f (f x)\nh_ab_combined : ∀ (a b : ℝ), a * f a + b * f b ≤ 2 * f a * f b\nh_f_nonneg_of_pos : ∀ a < 0, 0 ≤ f a\nx : ℝ\nh_suppose_not : ¬f x ≤ 0\ns : ℝ := (x * f x - f (f x)) / f x\nhm : min 0 s - 1 < s\nhml : min 0 s - 1 < 0\nhp : 0 < f x\n⊢ x * f x - f (f x) - x * f x + f (f x) = 0"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "f : ℝ → ℝ\nhf : ∀ (x y : ℝ), f (x + y) ≤ y * f x + f (f x)\nhxt : ∀ (x t : ℝ), f t ≤ t * f x - x * f x + f (f x)\nh_ab_combined : ∀ (a b : ℝ), a * f a + b * f b ≤ 2 * f a * f b\nh_f_nonneg_of_pos : ∀ a < 0, 0 ≤ f a\nx : ℝ\nh_suppose_not : ¬f x ≤ 0\ns : ℝ := (x * f x - f (f x)) / f x\nhm : min 0 s - 1 < s\nhml : min 0 s - 1 < 0\nhp : 0 < f x\n⊢ x * f x - f (f x) - x * f x + f (f x) = 0",
"after_state": "f : ℝ → ℝ\nhf : ∀ (x y : ℝ), f (x + y) ≤ y * f x + f (f x)\nhxt : ∀ (x t : ℝ), f t ≤ t * f x - x * f x + f (f x)\nh_ab_combined : ∀ (a b : ℝ), a * f a + b * f b ≤ 2 * f a * f b\nh_f_nonneg_of_pos : ∀ a < 0, 0 ≤ f a\nx : ℝ\nh_suppose_not : ¬f x ≤ 0\ns : ℝ := (x * f x - f (f x)) / f x\nhm : min 0 s - 1 < s\nhml : min 0 s - 1 < 0\nhp : 0 < f x\n⊢ x * f x - f (f x) - x * f x + f (f x) = 0"
},
{
"line": "with_reducible rfl",
"before_state": "f : ℝ → ℝ\nhf : ∀ (x y : ℝ), f (x + y) ≤ y * f x + f (f x)\nhxt : ∀ (x t : ℝ), f t ≤ t * f x - x * f x + f (f x)\nh_ab_combined : ∀ (a b : ℝ), a * f a + b * f b ≤ 2 * f a * f b\nh_f_nonneg_of_pos : ∀ a < 0, 0 ≤ f a\nx : ℝ\nh_suppose_not : ¬f x ≤ 0\ns : ℝ := (x * f x - f (f x)) / f x\nhm : min 0 s - 1 < s\nhml : min 0 s - 1 < 0\nhp : 0 < f x\n⊢ x * f x - f (f x) - x * f x + f (f x) = 0",
"after_state": "f : ℝ → ℝ\nhf : ∀ (x y : ℝ), f (x + y) ≤ y * f x + f (f x)\nhxt : ∀ (x t : ℝ), f t ≤ t * f x - x * f x + f (f x)\nh_ab_combined : ∀ (a b : ℝ), a * f a + b * f b ≤ 2 * f a * f b\nh_f_nonneg_of_pos : ∀ a < 0, 0 ≤ f a\nx : ℝ\nh_suppose_not : ¬f x ≤ 0\ns : ℝ := (x * f x - f (f x)) / f x\nhm : min 0 s - 1 < s\nhml : min 0 s - 1 < 0\nhp : 0 < f x\n⊢ x * f x - f (f x) - x * f x + f (f x) = 0"
},
{
"line": "rfl",
"before_state": "f : ℝ → ℝ\nhf : ∀ (x y : ℝ), f (x + y) ≤ y * f x + f (f x)\nhxt : ∀ (x t : ℝ), f t ≤ t * f x - x * f x + f (f x)\nh_ab_combined : ∀ (a b : ℝ), a * f a + b * f b ≤ 2 * f a * f b\nh_f_nonneg_of_pos : ∀ a < 0, 0 ≤ f a\nx : ℝ\nh_suppose_not : ¬f x ≤ 0\ns : ℝ := (x * f x - f (f x)) / f x\nhm : min 0 s - 1 < s\nhml : min 0 s - 1 < 0\nhp : 0 < f x\n⊢ x * f x - f (f x) - x * f x + f (f x) = 0",
"after_state": "f : ℝ → ℝ\nhf : ∀ (x y : ℝ), f (x + y) ≤ y * f x + f (f x)\nhxt : ∀ (x t : ℝ), f t ≤ t * f x - x * f x + f (f x)\nh_ab_combined : ∀ (a b : ℝ), a * f a + b * f b ≤ 2 * f a * f b\nh_f_nonneg_of_pos : ∀ a < 0, 0 ≤ f a\nx : ℝ\nh_suppose_not : ¬f x ≤ 0\ns : ℝ := (x * f x - f (f x)) / f x\nhm : min 0 s - 1 < s\nhml : min 0 s - 1 < 0\nhp : 0 < f x\n⊢ x * f x - f (f x) - x * f x + f (f x) = 0"
},
{
"line": "apply_rfl",
"before_state": "f : ℝ → ℝ\nhf : ∀ (x y : ℝ), f (x + y) ≤ y * f x + f (f x)\nhxt : ∀ (x t : ℝ), f t ≤ t * f x - x * f x + f (f x)\nh_ab_combined : ∀ (a b : ℝ), a * f a + b * f b ≤ 2 * f a * f b\nh_f_nonneg_of_pos : ∀ a < 0, 0 ≤ f a\nx : ℝ\nh_suppose_not : ¬f x ≤ 0\ns : ℝ := (x * f x - f (f x)) / f x\nhm : min 0 s - 1 < s\nhml : min 0 s - 1 < 0\nhp : 0 < f x\n⊢ x * f x - f (f x) - x * f x + f (f x) = 0",
"after_state": "f : ℝ → ℝ\nhf : ∀ (x y : ℝ), f (x + y) ≤ y * f x + f (f x)\nhxt : ∀ (x t : ℝ), f t ≤ t * f x - x * f x + f (f x)\nh_ab_combined : ∀ (a b : ℝ), a * f a + b * f b ≤ 2 * f a * f b\nh_f_nonneg_of_pos : ∀ a < 0, 0 ≤ f a\nx : ℝ\nh_suppose_not : ¬f x ≤ 0\ns : ℝ := (x * f x - f (f x)) / f x\nhm : min 0 s - 1 < s\nhml : min 0 s - 1 < 0\nhp : 0 < f x\n⊢ x * f x - f (f x) - x * f x + f (f x) = 0"
},
{
"line": "skip",
"before_state": "f : ℝ → ℝ\nhf : ∀ (x y : ℝ), f (x + y) ≤ y * f x + f (f x)\nhxt : ∀ (x t : ℝ), f t ≤ t * f x - x * f x + f (f x)\nh_ab_combined : ∀ (a b : ℝ), a * f a + b * f b ≤ 2 * f a * f b\nh_f_nonneg_of_pos : ∀ a < 0, 0 ≤ f a\nx : ℝ\nh_suppose_not : ¬f x ≤ 0\ns : ℝ := (x * f x - f (f x)) / f x\nhm : min 0 s - 1 < s\nhml : min 0 s - 1 < 0\nhp : 0 < f x\n⊢ x * f x - f (f x) - x * f x + f (f x) = 0",
"after_state": "f : ℝ → ℝ\nhf : ∀ (x y : ℝ), f (x + y) ≤ y * f x + f (f x)\nhxt : ∀ (x t : ℝ), f t ≤ t * f x - x * f x + f (f x)\nh_ab_combined : ∀ (a b : ℝ), a * f a + b * f b ≤ 2 * f a * f b\nh_f_nonneg_of_pos : ∀ a < 0, 0 ≤ f a\nx : ℝ\nh_suppose_not : ¬f x ≤ 0\ns : ℝ := (x * f x - f (f x)) / f x\nhm : min 0 s - 1 < s\nhml : min 0 s - 1 < 0\nhp : 0 < f x\n⊢ x * f x - f (f x) - x * f x + f (f x) = 0"
},
{
"line": "linarith",
"before_state": "f : ℝ → ℝ\nhf : ∀ (x y : ℝ), f (x + y) ≤ y * f x + f (f x)\nhxt : ∀ (x t : ℝ), f t ≤ t * f x - x * f x + f (f x)\nh_ab_combined : ∀ (a b : ℝ), a * f a + b * f b ≤ 2 * f a * f b\nh_f_nonneg_of_pos : ∀ a < 0, 0 ≤ f a\nx : ℝ\nh_suppose_not : ¬f x ≤ 0\ns : ℝ := (x * f x - f (f x)) / f x\nhm : min 0 s - 1 < s\nhml : min 0 s - 1 < 0\nhp : 0 < f x\n⊢ x * f x - f (f x) - x * f x + f (f x) = 0",
"after_state": "No Goals!"
},
{
"line": "ring1",
"before_state": "f : ℝ → ℝ\nhf : ∀ (x y : ℝ), f (x + y) ≤ y * f x + f (f x)\nhxt : ∀ (x t : ℝ), f t ≤ t * f x - x * f x + f (f x)\nh_ab_combined : ∀ (a b : ℝ), a * f a + b * f b ≤ 2 * f a * f b\nh_f_nonneg_of_pos : ∀ a < 0, 0 ≤ f a\nx : ℝ\nh_suppose_not : ¬f x ≤ 0\ns : ℝ := (x * f x - f (f x)) / f x\nhm : min 0 s - 1 < s\nhml : min 0 s - 1 < 0\nhp : 0 < f x\na✝ : x * f x - f (f x) - x * f x + f (f x) < 0\n⊢ x * f x - f (f x) - x * f x + f (f x) - 0 = 0",
"after_state": "No Goals!"
},
{
"line": "ring1",
"before_state": "f : ℝ → ℝ\nhf : ∀ (x y : ℝ), f (x + y) ≤ y * f x + f (f x)\nhxt : ∀ (x t : ℝ), f t ≤ t * f x - x * f x + f (f x)\nh_ab_combined : ∀ (a b : ℝ), a * f a + b * f b ≤ 2 * f a * f b\nh_f_nonneg_of_pos : ∀ a < 0, 0 ≤ f a\nx : ℝ\nh_suppose_not : ¬f x ≤ 0\ns : ℝ := (x * f x - f (f x)) / f x\nhm : min 0 s - 1 < s\nhml : min 0 s - 1 < 0\nhp : 0 < f x\na✝ : 0 < x * f x - f (f x) - x * f x + f (f x)\n⊢ 0 - (x * f x - f (f x) - x * f x + f (f x)) = 0",
"after_state": "No Goals!"
},
{
"line": "rotate_right",
"before_state": "f : ℝ → ℝ\nhf : ∀ (x y : ℝ), f (x + y) ≤ y * f x + f (f x)\nhxt : ∀ (x t : ℝ), f t ≤ t * f x - x * f x + f (f x)\nh_ab_combined : ∀ (a b : ℝ), a * f a + b * f b ≤ 2 * f a * f b\nh_f_nonneg_of_pos : ∀ a < 0, 0 ≤ f a\nh_f_nonpos : ∀ (x : ℝ), f x ≤ 0\n⊢ ∀ x ≤ 0, f x = 0",
"after_state": "f : ℝ → ℝ\nhf : ∀ (x y : ℝ), f (x + y) ≤ y * f x + f (f x)\nhxt : ∀ (x t : ℝ), f t ≤ t * f x - x * f x + f (f x)\nh_ab_combined : ∀ (a b : ℝ), a * f a + b * f b ≤ 2 * f a * f b\nh_f_nonneg_of_pos : ∀ a < 0, 0 ≤ f a\nh_f_nonpos : ∀ (x : ℝ), f x ≤ 0\n⊢ ∀ x ≤ 0, f x = 0"
},
{
"line": "have h_fx_zero_of_neg : ∀ x < 0, f x = 0 := fun x hxz => (h_f_nonpos x).antisymm (h_f_nonneg_of_pos x hxz)",
"before_state": "f : ℝ → ℝ\nhf : ∀ (x y : ℝ), f (x + y) ≤ y * f x + f (f x)\nhxt : ∀ (x t : ℝ), f t ≤ t * f x - x * f x + f (f x)\nh_ab_combined : ∀ (a b : ℝ), a * f a + b * f b ≤ 2 * f a * f b\nh_f_nonneg_of_pos : ∀ a < 0, 0 ≤ f a\nh_f_nonpos : ∀ (x : ℝ), f x ≤ 0\n⊢ ∀ x ≤ 0, f x = 0",
"after_state": "f : ℝ → ℝ\nhf : ∀ (x y : ℝ), f (x + y) ≤ y * f x + f (f x)\nhxt : ∀ (x t : ℝ), f t ≤ t * f x - x * f x + f (f x)\nh_ab_combined : ∀ (a b : ℝ), a * f a + b * f b ≤ 2 * f a * f b\nh_f_nonneg_of_pos : ∀ a < 0, 0 ≤ f a\nh_f_nonpos : ∀ (x : ℝ), f x ≤ 0\nh_fx_zero_of_neg : ∀ x < 0, f x = 0\n⊢ ∀ x ≤ 0, f x = 0"
},
{
"line": "refine_lift\n have h_fx_zero_of_neg : ∀ x < 0, f x = 0 := fun x hxz => (h_f_nonpos x).antisymm (h_f_nonneg_of_pos x hxz);\n ?_",
"before_state": "f : ℝ → ℝ\nhf : ∀ (x y : ℝ), f (x + y) ≤ y * f x + f (f x)\nhxt : ∀ (x t : ℝ), f t ≤ t * f x - x * f x + f (f x)\nh_ab_combined : ∀ (a b : ℝ), a * f a + b * f b ≤ 2 * f a * f b\nh_f_nonneg_of_pos : ∀ a < 0, 0 ≤ f a\nh_f_nonpos : ∀ (x : ℝ), f x ≤ 0\n⊢ ∀ x ≤ 0, f x = 0",
"after_state": "f : ℝ → ℝ\nhf : ∀ (x y : ℝ), f (x + y) ≤ y * f x + f (f x)\nhxt : ∀ (x t : ℝ), f t ≤ t * f x - x * f x + f (f x)\nh_ab_combined : ∀ (a b : ℝ), a * f a + b * f b ≤ 2 * f a * f b\nh_f_nonneg_of_pos : ∀ a < 0, 0 ≤ f a\nh_f_nonpos : ∀ (x : ℝ), f x ≤ 0\nh_fx_zero_of_neg : ∀ x < 0, f x = 0\n⊢ ∀ x ≤ 0, f x = 0"
},
{
"line": "focus\n (refine\n no_implicit_lambda%\n (have h_fx_zero_of_neg : ∀ x < 0, f x = 0 := fun x hxz => (h_f_nonpos x).antisymm (h_f_nonneg_of_pos x hxz);\n ?_);\n rotate_right)",
"before_state": "f : ℝ → ℝ\nhf : ∀ (x y : ℝ), f (x + y) ≤ y * f x + f (f x)\nhxt : ∀ (x t : ℝ), f t ≤ t * f x - x * f x + f (f x)\nh_ab_combined : ∀ (a b : ℝ), a * f a + b * f b ≤ 2 * f a * f b\nh_f_nonneg_of_pos : ∀ a < 0, 0 ≤ f a\nh_f_nonpos : ∀ (x : ℝ), f x ≤ 0\n⊢ ∀ x ≤ 0, f x = 0",
"after_state": "f : ℝ → ℝ\nhf : ∀ (x y : ℝ), f (x + y) ≤ y * f x + f (f x)\nhxt : ∀ (x t : ℝ), f t ≤ t * f x - x * f x + f (f x)\nh_ab_combined : ∀ (a b : ℝ), a * f a + b * f b ≤ 2 * f a * f b\nh_f_nonneg_of_pos : ∀ a < 0, 0 ≤ f a\nh_f_nonpos : ∀ (x : ℝ), f x ≤ 0\nh_fx_zero_of_neg : ∀ x < 0, f x = 0\n⊢ ∀ x ≤ 0, f x = 0"
},
{
"line": "refine\n no_implicit_lambda%\n (have h_fx_zero_of_neg : ∀ x < 0, f x = 0 := fun x hxz => (h_f_nonpos x).antisymm (h_f_nonneg_of_pos x hxz);\n ?_)",
"before_state": "f : ℝ → ℝ\nhf : ∀ (x y : ℝ), f (x + y) ≤ y * f x + f (f x)\nhxt : ∀ (x t : ℝ), f t ≤ t * f x - x * f x + f (f x)\nh_ab_combined : ∀ (a b : ℝ), a * f a + b * f b ≤ 2 * f a * f b\nh_f_nonneg_of_pos : ∀ a < 0, 0 ≤ f a\nh_f_nonpos : ∀ (x : ℝ), f x ≤ 0\n⊢ ∀ x ≤ 0, f x = 0",
"after_state": "f : ℝ → ℝ\nhf : ∀ (x y : ℝ), f (x + y) ≤ y * f x + f (f x)\nhxt : ∀ (x t : ℝ), f t ≤ t * f x - x * f x + f (f x)\nh_ab_combined : ∀ (a b : ℝ), a * f a + b * f b ≤ 2 * f a * f b\nh_f_nonneg_of_pos : ∀ a < 0, 0 ≤ f a\nh_f_nonpos : ∀ (x : ℝ), f x ≤ 0\nh_fx_zero_of_neg : ∀ x < 0, f x = 0\n⊢ ∀ x ≤ 0, f x = 0"
},
{
"line": "rotate_right",
"before_state": "f : ℝ → ℝ\nhf : ∀ (x y : ℝ), f (x + y) ≤ y * f x + f (f x)\nhxt : ∀ (x t : ℝ), f t ≤ t * f x - x * f x + f (f x)\nh_ab_combined : ∀ (a b : ℝ), a * f a + b * f b ≤ 2 * f a * f b\nh_f_nonneg_of_pos : ∀ a < 0, 0 ≤ f a\nh_f_nonpos : ∀ (x : ℝ), f x ≤ 0\nh_fx_zero_of_neg : ∀ x < 0, f x = 0\n⊢ ∀ x ≤ 0, f x = 0",
"after_state": "f : ℝ → ℝ\nhf : ∀ (x y : ℝ), f (x + y) ≤ y * f x + f (f x)\nhxt : ∀ (x t : ℝ), f t ≤ t * f x - x * f x + f (f x)\nh_ab_combined : ∀ (a b : ℝ), a * f a + b * f b ≤ 2 * f a * f b\nh_f_nonneg_of_pos : ∀ a < 0, 0 ≤ f a\nh_f_nonpos : ∀ (x : ℝ), f x ≤ 0\nh_fx_zero_of_neg : ∀ x < 0, f x = 0\n⊢ ∀ x ≤ 0, f x = 0"
},
{
"line": "intro x hx",
"before_state": "f : ℝ → ℝ\nhf : ∀ (x y : ℝ), f (x + y) ≤ y * f x + f (f x)\nhxt : ∀ (x t : ℝ), f t ≤ t * f x - x * f x + f (f x)\nh_ab_combined : ∀ (a b : ℝ), a * f a + b * f b ≤ 2 * f a * f b\nh_f_nonneg_of_pos : ∀ a < 0, 0 ≤ f a\nh_f_nonpos : ∀ (x : ℝ), f x ≤ 0\nh_fx_zero_of_neg : ∀ x < 0, f x = 0\n⊢ ∀ x ≤ 0, f x = 0",
"after_state": "f : ℝ → ℝ\nhf : ∀ (x y : ℝ), f (x + y) ≤ y * f x + f (f x)\nhxt : ∀ (x t : ℝ), f t ≤ t * f x - x * f x + f (f x)\nh_ab_combined : ∀ (a b : ℝ), a * f a + b * f b ≤ 2 * f a * f b\nh_f_nonneg_of_pos : ∀ a < 0, 0 ≤ f a\nh_f_nonpos : ∀ (x : ℝ), f x ≤ 0\nh_fx_zero_of_neg : ∀ x < 0, f x = 0\nx : ℝ\nhx : x ≤ 0\n⊢ f x = 0"
},
{
"line": "intro x;\n intro hx",
"before_state": "f : ℝ → ℝ\nhf : ∀ (x y : ℝ), f (x + y) ≤ y * f x + f (f x)\nhxt : ∀ (x t : ℝ), f t ≤ t * f x - x * f x + f (f x)\nh_ab_combined : ∀ (a b : ℝ), a * f a + b * f b ≤ 2 * f a * f b\nh_f_nonneg_of_pos : ∀ a < 0, 0 ≤ f a\nh_f_nonpos : ∀ (x : ℝ), f x ≤ 0\nh_fx_zero_of_neg : ∀ x < 0, f x = 0\n⊢ ∀ x ≤ 0, f x = 0",
"after_state": "f : ℝ → ℝ\nhf : ∀ (x y : ℝ), f (x + y) ≤ y * f x + f (f x)\nhxt : ∀ (x t : ℝ), f t ≤ t * f x - x * f x + f (f x)\nh_ab_combined : ∀ (a b : ℝ), a * f a + b * f b ≤ 2 * f a * f b\nh_f_nonneg_of_pos : ∀ a < 0, 0 ≤ f a\nh_f_nonpos : ∀ (x : ℝ), f x ≤ 0\nh_fx_zero_of_neg : ∀ x < 0, f x = 0\nx : ℝ\nhx : x ≤ 0\n⊢ f x = 0"
},
{
"line": "intro x",
"before_state": "f : ℝ → ℝ\nhf : ∀ (x y : ℝ), f (x + y) ≤ y * f x + f (f x)\nhxt : ∀ (x t : ℝ), f t ≤ t * f x - x * f x + f (f x)\nh_ab_combined : ∀ (a b : ℝ), a * f a + b * f b ≤ 2 * f a * f b\nh_f_nonneg_of_pos : ∀ a < 0, 0 ≤ f a\nh_f_nonpos : ∀ (x : ℝ), f x ≤ 0\nh_fx_zero_of_neg : ∀ x < 0, f x = 0\n⊢ ∀ x ≤ 0, f x = 0",
"after_state": "f : ℝ → ℝ\nhf : ∀ (x y : ℝ), f (x + y) ≤ y * f x + f (f x)\nhxt : ∀ (x t : ℝ), f t ≤ t * f x - x * f x + f (f x)\nh_ab_combined : ∀ (a b : ℝ), a * f a + b * f b ≤ 2 * f a * f b\nh_f_nonneg_of_pos : ∀ a < 0, 0 ≤ f a\nh_f_nonpos : ∀ (x : ℝ), f x ≤ 0\nh_fx_zero_of_neg : ∀ x < 0, f x = 0\nx : ℝ\n⊢ x ≤ 0 → f x = 0"
},
{
"line": "intro hx",
"before_state": "f : ℝ → ℝ\nhf : ∀ (x y : ℝ), f (x + y) ≤ y * f x + f (f x)\nhxt : ∀ (x t : ℝ), f t ≤ t * f x - x * f x + f (f x)\nh_ab_combined : ∀ (a b : ℝ), a * f a + b * f b ≤ 2 * f a * f b\nh_f_nonneg_of_pos : ∀ a < 0, 0 ≤ f a\nh_f_nonpos : ∀ (x : ℝ), f x ≤ 0\nh_fx_zero_of_neg : ∀ x < 0, f x = 0\nx : ℝ\n⊢ x ≤ 0 → f x = 0",
"after_state": "f : ℝ → ℝ\nhf : ∀ (x y : ℝ), f (x + y) ≤ y * f x + f (f x)\nhxt : ∀ (x t : ℝ), f t ≤ t * f x - x * f x + f (f x)\nh_ab_combined : ∀ (a b : ℝ), a * f a + b * f b ≤ 2 * f a * f b\nh_f_nonneg_of_pos : ∀ a < 0, 0 ≤ f a\nh_f_nonpos : ∀ (x : ℝ), f x ≤ 0\nh_fx_zero_of_neg : ∀ x < 0, f x = 0\nx : ℝ\nhx : x ≤ 0\n⊢ f x = 0"
},
{
"line": "obtain (h_x_neg : x < 0) | (rfl : x = 0) := hx.lt_or_eq",
"before_state": "f : ℝ → ℝ\nhf : ∀ (x y : ℝ), f (x + y) ≤ y * f x + f (f x)\nhxt : ∀ (x t : ℝ), f t ≤ t * f x - x * f x + f (f x)\nh_ab_combined : ∀ (a b : ℝ), a * f a + b * f b ≤ 2 * f a * f b\nh_f_nonneg_of_pos : ∀ a < 0, 0 ≤ f a\nh_f_nonpos : ∀ (x : ℝ), f x ≤ 0\nh_fx_zero_of_neg : ∀ x < 0, f x = 0\nx : ℝ\nhx : x ≤ 0\n⊢ f x = 0",
"after_state": "case inl\nf : ℝ → ℝ\nhf : ∀ (x y : ℝ), f (x + y) ≤ y * f x + f (f x)\nhxt : ∀ (x t : ℝ), f t ≤ t * f x - x * f x + f (f x)\nh_ab_combined : ∀ (a b : ℝ), a * f a + b * f b ≤ 2 * f a * f b\nh_f_nonneg_of_pos : ∀ a < 0, 0 ≤ f a\nh_f_nonpos : ∀ (x : ℝ), f x ≤ 0\nh_fx_zero_of_neg : ∀ x < 0, f x = 0\nx : ℝ\nhx : x ≤ 0\nh_x_neg : x < 0\n⊢ f x = 0\n---\ncase inr\nf : ℝ → ℝ\nhf : ∀ (x y : ℝ), f (x + y) ≤ y * f x + f (f x)\nhxt : ∀ (x t : ℝ), f t ≤ t * f x - x * f x + f (f x)\nh_ab_combined : ∀ (a b : ℝ), a * f a + b * f b ≤ 2 * f a * f b\nh_f_nonneg_of_pos : ∀ a < 0, 0 ≤ f a\nh_f_nonpos : ∀ (x : ℝ), f x ≤ 0\nh_fx_zero_of_neg : ∀ x < 0, f x = 0\nhx : 0 ≤ 0\n⊢ f 0 = 0"
},
{
"line": "exact h_fx_zero_of_neg _ h_x_neg",
"before_state": "case inl\nf : ℝ → ℝ\nhf : ∀ (x y : ℝ), f (x + y) ≤ y * f x + f (f x)\nhxt : ∀ (x t : ℝ), f t ≤ t * f x - x * f x + f (f x)\nh_ab_combined : ∀ (a b : ℝ), a * f a + b * f b ≤ 2 * f a * f b\nh_f_nonneg_of_pos : ∀ a < 0, 0 ≤ f a\nh_f_nonpos : ∀ (x : ℝ), f x ≤ 0\nh_fx_zero_of_neg : ∀ x < 0, f x = 0\nx : ℝ\nhx : x ≤ 0\nh_x_neg : x < 0\n⊢ f x = 0",
"after_state": "No Goals!"
},
{
"line": "suffices 0 ≤ f 0 from le_antisymm (h_f_nonpos 0) this",
"before_state": "case inr\nf : ℝ → ℝ\nhf : ∀ (x y : ℝ), f (x + y) ≤ y * f x + f (f x)\nhxt : ∀ (x t : ℝ), f t ≤ t * f x - x * f x + f (f x)\nh_ab_combined : ∀ (a b : ℝ), a * f a + b * f b ≤ 2 * f a * f b\nh_f_nonneg_of_pos : ∀ a < 0, 0 ≤ f a\nh_f_nonpos : ∀ (x : ℝ), f x ≤ 0\nh_fx_zero_of_neg : ∀ x < 0, f x = 0\nhx : 0 ≤ 0\n⊢ f 0 = 0",
"after_state": "case inr\nf : ℝ → ℝ\nhf : ∀ (x y : ℝ), f (x + y) ≤ y * f x + f (f x)\nhxt : ∀ (x t : ℝ), f t ≤ t * f x - x * f x + f (f x)\nh_ab_combined : ∀ (a b : ℝ), a * f a + b * f b ≤ 2 * f a * f b\nh_f_nonneg_of_pos : ∀ a < 0, 0 ≤ f a\nh_f_nonpos : ∀ (x : ℝ), f x ≤ 0\nh_fx_zero_of_neg : ∀ x < 0, f x = 0\nhx : 0 ≤ 0\n⊢ 0 ≤ f 0"
},
{
"line": "refine_lift\n suffices 0 ≤ f 0 from le_antisymm (h_f_nonpos 0) this;\n ?_",
"before_state": "case inr\nf : ℝ → ℝ\nhf : ∀ (x y : ℝ), f (x + y) ≤ y * f x + f (f x)\nhxt : ∀ (x t : ℝ), f t ≤ t * f x - x * f x + f (f x)\nh_ab_combined : ∀ (a b : ℝ), a * f a + b * f b ≤ 2 * f a * f b\nh_f_nonneg_of_pos : ∀ a < 0, 0 ≤ f a\nh_f_nonpos : ∀ (x : ℝ), f x ≤ 0\nh_fx_zero_of_neg : ∀ x < 0, f x = 0\nhx : 0 ≤ 0\n⊢ f 0 = 0",
"after_state": "case inr\nf : ℝ → ℝ\nhf : ∀ (x y : ℝ), f (x + y) ≤ y * f x + f (f x)\nhxt : ∀ (x t : ℝ), f t ≤ t * f x - x * f x + f (f x)\nh_ab_combined : ∀ (a b : ℝ), a * f a + b * f b ≤ 2 * f a * f b\nh_f_nonneg_of_pos : ∀ a < 0, 0 ≤ f a\nh_f_nonpos : ∀ (x : ℝ), f x ≤ 0\nh_fx_zero_of_neg : ∀ x < 0, f x = 0\nhx : 0 ≤ 0\n⊢ 0 ≤ f 0"
},
{
"line": "focus\n (refine\n no_implicit_lambda%\n (suffices 0 ≤ f 0 from le_antisymm (h_f_nonpos 0) this;\n ?_);\n rotate_right)",
"before_state": "case inr\nf : ℝ → ℝ\nhf : ∀ (x y : ℝ), f (x + y) ≤ y * f x + f (f x)\nhxt : ∀ (x t : ℝ), f t ≤ t * f x - x * f x + f (f x)\nh_ab_combined : ∀ (a b : ℝ), a * f a + b * f b ≤ 2 * f a * f b\nh_f_nonneg_of_pos : ∀ a < 0, 0 ≤ f a\nh_f_nonpos : ∀ (x : ℝ), f x ≤ 0\nh_fx_zero_of_neg : ∀ x < 0, f x = 0\nhx : 0 ≤ 0\n⊢ f 0 = 0",
"after_state": "case inr\nf : ℝ → ℝ\nhf : ∀ (x y : ℝ), f (x + y) ≤ y * f x + f (f x)\nhxt : ∀ (x t : ℝ), f t ≤ t * f x - x * f x + f (f x)\nh_ab_combined : ∀ (a b : ℝ), a * f a + b * f b ≤ 2 * f a * f b\nh_f_nonneg_of_pos : ∀ a < 0, 0 ≤ f a\nh_f_nonpos : ∀ (x : ℝ), f x ≤ 0\nh_fx_zero_of_neg : ∀ x < 0, f x = 0\nhx : 0 ≤ 0\n⊢ 0 ≤ f 0"
},
{
"line": "refine\n no_implicit_lambda%\n (suffices 0 ≤ f 0 from le_antisymm (h_f_nonpos 0) this;\n ?_)",
"before_state": "case inr\nf : ℝ → ℝ\nhf : ∀ (x y : ℝ), f (x + y) ≤ y * f x + f (f x)\nhxt : ∀ (x t : ℝ), f t ≤ t * f x - x * f x + f (f x)\nh_ab_combined : ∀ (a b : ℝ), a * f a + b * f b ≤ 2 * f a * f b\nh_f_nonneg_of_pos : ∀ a < 0, 0 ≤ f a\nh_f_nonpos : ∀ (x : ℝ), f x ≤ 0\nh_fx_zero_of_neg : ∀ x < 0, f x = 0\nhx : 0 ≤ 0\n⊢ f 0 = 0",
"after_state": "case inr\nf : ℝ → ℝ\nhf : ∀ (x y : ℝ), f (x + y) ≤ y * f x + f (f x)\nhxt : ∀ (x t : ℝ), f t ≤ t * f x - x * f x + f (f x)\nh_ab_combined : ∀ (a b : ℝ), a * f a + b * f b ≤ 2 * f a * f b\nh_f_nonneg_of_pos : ∀ a < 0, 0 ≤ f a\nh_f_nonpos : ∀ (x : ℝ), f x ≤ 0\nh_fx_zero_of_neg : ∀ x < 0, f x = 0\nhx : 0 ≤ 0\n⊢ 0 ≤ f 0"
},
{
"line": "rotate_right",
"before_state": "case inr\nf : ℝ → ℝ\nhf : ∀ (x y : ℝ), f (x + y) ≤ y * f x + f (f x)\nhxt : ∀ (x t : ℝ), f t ≤ t * f x - x * f x + f (f x)\nh_ab_combined : ∀ (a b : ℝ), a * f a + b * f b ≤ 2 * f a * f b\nh_f_nonneg_of_pos : ∀ a < 0, 0 ≤ f a\nh_f_nonpos : ∀ (x : ℝ), f x ≤ 0\nh_fx_zero_of_neg : ∀ x < 0, f x = 0\nhx : 0 ≤ 0\n⊢ 0 ≤ f 0",
"after_state": "case inr\nf : ℝ → ℝ\nhf : ∀ (x y : ℝ), f (x + y) ≤ y * f x + f (f x)\nhxt : ∀ (x t : ℝ), f t ≤ t * f x - x * f x + f (f x)\nh_ab_combined : ∀ (a b : ℝ), a * f a + b * f b ≤ 2 * f a * f b\nh_f_nonneg_of_pos : ∀ a < 0, 0 ≤ f a\nh_f_nonpos : ∀ (x : ℝ), f x ≤ 0\nh_fx_zero_of_neg : ∀ x < 0, f x = 0\nhx : 0 ≤ 0\n⊢ 0 ≤ f 0"
},
{
"line": "have hno : f (-1) = 0 := h_fx_zero_of_neg (-1) neg_one_lt_zero",
"before_state": "case inr\nf : ℝ → ℝ\nhf : ∀ (x y : ℝ), f (x + y) ≤ y * f x + f (f x)\nhxt : ∀ (x t : ℝ), f t ≤ t * f x - x * f x + f (f x)\nh_ab_combined : ∀ (a b : ℝ), a * f a + b * f b ≤ 2 * f a * f b\nh_f_nonneg_of_pos : ∀ a < 0, 0 ≤ f a\nh_f_nonpos : ∀ (x : ℝ), f x ≤ 0\nh_fx_zero_of_neg : ∀ x < 0, f x = 0\nhx : 0 ≤ 0\n⊢ 0 ≤ f 0",
"after_state": "case inr\nf : ℝ → ℝ\nhf : ∀ (x y : ℝ), f (x + y) ≤ y * f x + f (f x)\nhxt : ∀ (x t : ℝ), f t ≤ t * f x - x * f x + f (f x)\nh_ab_combined : ∀ (a b : ℝ), a * f a + b * f b ≤ 2 * f a * f b\nh_f_nonneg_of_pos : ∀ a < 0, 0 ≤ f a\nh_f_nonpos : ∀ (x : ℝ), f x ≤ 0\nh_fx_zero_of_neg : ∀ x < 0, f x = 0\nhx : 0 ≤ 0\nhno : f (-1) = 0\n⊢ 0 ≤ f 0"
},
{
"line": "refine_lift\n have hno : f (-1) = 0 := h_fx_zero_of_neg (-1) neg_one_lt_zero;\n ?_",
"before_state": "case inr\nf : ℝ → ℝ\nhf : ∀ (x y : ℝ), f (x + y) ≤ y * f x + f (f x)\nhxt : ∀ (x t : ℝ), f t ≤ t * f x - x * f x + f (f x)\nh_ab_combined : ∀ (a b : ℝ), a * f a + b * f b ≤ 2 * f a * f b\nh_f_nonneg_of_pos : ∀ a < 0, 0 ≤ f a\nh_f_nonpos : ∀ (x : ℝ), f x ≤ 0\nh_fx_zero_of_neg : ∀ x < 0, f x = 0\nhx : 0 ≤ 0\n⊢ 0 ≤ f 0",
"after_state": "case inr\nf : ℝ → ℝ\nhf : ∀ (x y : ℝ), f (x + y) ≤ y * f x + f (f x)\nhxt : ∀ (x t : ℝ), f t ≤ t * f x - x * f x + f (f x)\nh_ab_combined : ∀ (a b : ℝ), a * f a + b * f b ≤ 2 * f a * f b\nh_f_nonneg_of_pos : ∀ a < 0, 0 ≤ f a\nh_f_nonpos : ∀ (x : ℝ), f x ≤ 0\nh_fx_zero_of_neg : ∀ x < 0, f x = 0\nhx : 0 ≤ 0\nhno : f (-1) = 0\n⊢ 0 ≤ f 0"
},
{
"line": "focus\n (refine\n no_implicit_lambda%\n (have hno : f (-1) = 0 := h_fx_zero_of_neg (-1) neg_one_lt_zero;\n ?_);\n rotate_right)",
"before_state": "case inr\nf : ℝ → ℝ\nhf : ∀ (x y : ℝ), f (x + y) ≤ y * f x + f (f x)\nhxt : ∀ (x t : ℝ), f t ≤ t * f x - x * f x + f (f x)\nh_ab_combined : ∀ (a b : ℝ), a * f a + b * f b ≤ 2 * f a * f b\nh_f_nonneg_of_pos : ∀ a < 0, 0 ≤ f a\nh_f_nonpos : ∀ (x : ℝ), f x ≤ 0\nh_fx_zero_of_neg : ∀ x < 0, f x = 0\nhx : 0 ≤ 0\n⊢ 0 ≤ f 0",
"after_state": "case inr\nf : ℝ → ℝ\nhf : ∀ (x y : ℝ), f (x + y) ≤ y * f x + f (f x)\nhxt : ∀ (x t : ℝ), f t ≤ t * f x - x * f x + f (f x)\nh_ab_combined : ∀ (a b : ℝ), a * f a + b * f b ≤ 2 * f a * f b\nh_f_nonneg_of_pos : ∀ a < 0, 0 ≤ f a\nh_f_nonpos : ∀ (x : ℝ), f x ≤ 0\nh_fx_zero_of_neg : ∀ x < 0, f x = 0\nhx : 0 ≤ 0\nhno : f (-1) = 0\n⊢ 0 ≤ f 0"
},
{
"line": "refine\n no_implicit_lambda%\n (have hno : f (-1) = 0 := h_fx_zero_of_neg (-1) neg_one_lt_zero;\n ?_)",
"before_state": "case inr\nf : ℝ → ℝ\nhf : ∀ (x y : ℝ), f (x + y) ≤ y * f x + f (f x)\nhxt : ∀ (x t : ℝ), f t ≤ t * f x - x * f x + f (f x)\nh_ab_combined : ∀ (a b : ℝ), a * f a + b * f b ≤ 2 * f a * f b\nh_f_nonneg_of_pos : ∀ a < 0, 0 ≤ f a\nh_f_nonpos : ∀ (x : ℝ), f x ≤ 0\nh_fx_zero_of_neg : ∀ x < 0, f x = 0\nhx : 0 ≤ 0\n⊢ 0 ≤ f 0",
"after_state": "case inr\nf : ℝ → ℝ\nhf : ∀ (x y : ℝ), f (x + y) ≤ y * f x + f (f x)\nhxt : ∀ (x t : ℝ), f t ≤ t * f x - x * f x + f (f x)\nh_ab_combined : ∀ (a b : ℝ), a * f a + b * f b ≤ 2 * f a * f b\nh_f_nonneg_of_pos : ∀ a < 0, 0 ≤ f a\nh_f_nonpos : ∀ (x : ℝ), f x ≤ 0\nh_fx_zero_of_neg : ∀ x < 0, f x = 0\nhx : 0 ≤ 0\nhno : f (-1) = 0\n⊢ 0 ≤ f 0"
},
{
"line": "rotate_right",
"before_state": "case inr\nf : ℝ → ℝ\nhf : ∀ (x y : ℝ), f (x + y) ≤ y * f x + f (f x)\nhxt : ∀ (x t : ℝ), f t ≤ t * f x - x * f x + f (f x)\nh_ab_combined : ∀ (a b : ℝ), a * f a + b * f b ≤ 2 * f a * f b\nh_f_nonneg_of_pos : ∀ a < 0, 0 ≤ f a\nh_f_nonpos : ∀ (x : ℝ), f x ≤ 0\nh_fx_zero_of_neg : ∀ x < 0, f x = 0\nhx : 0 ≤ 0\nhno : f (-1) = 0\n⊢ 0 ≤ f 0",
"after_state": "case inr\nf : ℝ → ℝ\nhf : ∀ (x y : ℝ), f (x + y) ≤ y * f x + f (f x)\nhxt : ∀ (x t : ℝ), f t ≤ t * f x - x * f x + f (f x)\nh_ab_combined : ∀ (a b : ℝ), a * f a + b * f b ≤ 2 * f a * f b\nh_f_nonneg_of_pos : ∀ a < 0, 0 ≤ f a\nh_f_nonpos : ∀ (x : ℝ), f x ≤ 0\nh_fx_zero_of_neg : ∀ x < 0, f x = 0\nhx : 0 ≤ 0\nhno : f (-1) = 0\n⊢ 0 ≤ f 0"
},
{
"line": "have hp := hxt (-1) (-1)",
"before_state": "case inr\nf : ℝ → ℝ\nhf : ∀ (x y : ℝ), f (x + y) ≤ y * f x + f (f x)\nhxt : ∀ (x t : ℝ), f t ≤ t * f x - x * f x + f (f x)\nh_ab_combined : ∀ (a b : ℝ), a * f a + b * f b ≤ 2 * f a * f b\nh_f_nonneg_of_pos : ∀ a < 0, 0 ≤ f a\nh_f_nonpos : ∀ (x : ℝ), f x ≤ 0\nh_fx_zero_of_neg : ∀ x < 0, f x = 0\nhx : 0 ≤ 0\nhno : f (-1) = 0\n⊢ 0 ≤ f 0",
"after_state": "case inr\nf : ℝ → ℝ\nhf : ∀ (x y : ℝ), f (x + y) ≤ y * f x + f (f x)\nhxt : ∀ (x t : ℝ), f t ≤ t * f x - x * f x + f (f x)\nh_ab_combined : ∀ (a b : ℝ), a * f a + b * f b ≤ 2 * f a * f b\nh_f_nonneg_of_pos : ∀ a < 0, 0 ≤ f a\nh_f_nonpos : ∀ (x : ℝ), f x ≤ 0\nh_fx_zero_of_neg : ∀ x < 0, f x = 0\nhx : 0 ≤ 0\nhno : f (-1) = 0\nhp : f (-1) ≤ -1 * f (-1) - -1 * f (-1) + f (f (-1))\n⊢ 0 ≤ f 0"
},
{
"line": "refine_lift\n have hp := hxt (-1) (-1);\n ?_",
"before_state": "case inr\nf : ℝ → ℝ\nhf : ∀ (x y : ℝ), f (x + y) ≤ y * f x + f (f x)\nhxt : ∀ (x t : ℝ), f t ≤ t * f x - x * f x + f (f x)\nh_ab_combined : ∀ (a b : ℝ), a * f a + b * f b ≤ 2 * f a * f b\nh_f_nonneg_of_pos : ∀ a < 0, 0 ≤ f a\nh_f_nonpos : ∀ (x : ℝ), f x ≤ 0\nh_fx_zero_of_neg : ∀ x < 0, f x = 0\nhx : 0 ≤ 0\nhno : f (-1) = 0\n⊢ 0 ≤ f 0",
"after_state": "case inr\nf : ℝ → ℝ\nhf : ∀ (x y : ℝ), f (x + y) ≤ y * f x + f (f x)\nhxt : ∀ (x t : ℝ), f t ≤ t * f x - x * f x + f (f x)\nh_ab_combined : ∀ (a b : ℝ), a * f a + b * f b ≤ 2 * f a * f b\nh_f_nonneg_of_pos : ∀ a < 0, 0 ≤ f a\nh_f_nonpos : ∀ (x : ℝ), f x ≤ 0\nh_fx_zero_of_neg : ∀ x < 0, f x = 0\nhx : 0 ≤ 0\nhno : f (-1) = 0\nhp : f (-1) ≤ -1 * f (-1) - -1 * f (-1) + f (f (-1))\n⊢ 0 ≤ f 0"
},
{
"line": "focus\n (refine\n no_implicit_lambda%\n (have hp := hxt (-1) (-1);\n ?_);\n rotate_right)",
"before_state": "case inr\nf : ℝ → ℝ\nhf : ∀ (x y : ℝ), f (x + y) ≤ y * f x + f (f x)\nhxt : ∀ (x t : ℝ), f t ≤ t * f x - x * f x + f (f x)\nh_ab_combined : ∀ (a b : ℝ), a * f a + b * f b ≤ 2 * f a * f b\nh_f_nonneg_of_pos : ∀ a < 0, 0 ≤ f a\nh_f_nonpos : ∀ (x : ℝ), f x ≤ 0\nh_fx_zero_of_neg : ∀ x < 0, f x = 0\nhx : 0 ≤ 0\nhno : f (-1) = 0\n⊢ 0 ≤ f 0",
"after_state": "case inr\nf : ℝ → ℝ\nhf : ∀ (x y : ℝ), f (x + y) ≤ y * f x + f (f x)\nhxt : ∀ (x t : ℝ), f t ≤ t * f x - x * f x + f (f x)\nh_ab_combined : ∀ (a b : ℝ), a * f a + b * f b ≤ 2 * f a * f b\nh_f_nonneg_of_pos : ∀ a < 0, 0 ≤ f a\nh_f_nonpos : ∀ (x : ℝ), f x ≤ 0\nh_fx_zero_of_neg : ∀ x < 0, f x = 0\nhx : 0 ≤ 0\nhno : f (-1) = 0\nhp : f (-1) ≤ -1 * f (-1) - -1 * f (-1) + f (f (-1))\n⊢ 0 ≤ f 0"
},
{
"line": "refine\n no_implicit_lambda%\n (have hp := hxt (-1) (-1);\n ?_)",
"before_state": "case inr\nf : ℝ → ℝ\nhf : ∀ (x y : ℝ), f (x + y) ≤ y * f x + f (f x)\nhxt : ∀ (x t : ℝ), f t ≤ t * f x - x * f x + f (f x)\nh_ab_combined : ∀ (a b : ℝ), a * f a + b * f b ≤ 2 * f a * f b\nh_f_nonneg_of_pos : ∀ a < 0, 0 ≤ f a\nh_f_nonpos : ∀ (x : ℝ), f x ≤ 0\nh_fx_zero_of_neg : ∀ x < 0, f x = 0\nhx : 0 ≤ 0\nhno : f (-1) = 0\n⊢ 0 ≤ f 0",
"after_state": "case inr\nf : ℝ → ℝ\nhf : ∀ (x y : ℝ), f (x + y) ≤ y * f x + f (f x)\nhxt : ∀ (x t : ℝ), f t ≤ t * f x - x * f x + f (f x)\nh_ab_combined : ∀ (a b : ℝ), a * f a + b * f b ≤ 2 * f a * f b\nh_f_nonneg_of_pos : ∀ a < 0, 0 ≤ f a\nh_f_nonpos : ∀ (x : ℝ), f x ≤ 0\nh_fx_zero_of_neg : ∀ x < 0, f x = 0\nhx : 0 ≤ 0\nhno : f (-1) = 0\nhp : f (-1) ≤ -1 * f (-1) - -1 * f (-1) + f (f (-1))\n⊢ 0 ≤ f 0"
},
{
"line": "rotate_right",
"before_state": "case inr\nf : ℝ → ℝ\nhf : ∀ (x y : ℝ), f (x + y) ≤ y * f x + f (f x)\nhxt : ∀ (x t : ℝ), f t ≤ t * f x - x * f x + f (f x)\nh_ab_combined : ∀ (a b : ℝ), a * f a + b * f b ≤ 2 * f a * f b\nh_f_nonneg_of_pos : ∀ a < 0, 0 ≤ f a\nh_f_nonpos : ∀ (x : ℝ), f x ≤ 0\nh_fx_zero_of_neg : ∀ x < 0, f x = 0\nhx : 0 ≤ 0\nhno : f (-1) = 0\nhp : f (-1) ≤ -1 * f (-1) - -1 * f (-1) + f (f (-1))\n⊢ 0 ≤ f 0",
"after_state": "case inr\nf : ℝ → ℝ\nhf : ∀ (x y : ℝ), f (x + y) ≤ y * f x + f (f x)\nhxt : ∀ (x t : ℝ), f t ≤ t * f x - x * f x + f (f x)\nh_ab_combined : ∀ (a b : ℝ), a * f a + b * f b ≤ 2 * f a * f b\nh_f_nonneg_of_pos : ∀ a < 0, 0 ≤ f a\nh_f_nonpos : ∀ (x : ℝ), f x ≤ 0\nh_fx_zero_of_neg : ∀ x < 0, f x = 0\nhx : 0 ≤ 0\nhno : f (-1) = 0\nhp : f (-1) ≤ -1 * f (-1) - -1 * f (-1) + f (f (-1))\n⊢ 0 ≤ f 0"
},
{
"line": "rw [hno] at hp",
"before_state": "case inr\nf : ℝ → ℝ\nhf : ∀ (x y : ℝ), f (x + y) ≤ y * f x + f (f x)\nhxt : ∀ (x t : ℝ), f t ≤ t * f x - x * f x + f (f x)\nh_ab_combined : ∀ (a b : ℝ), a * f a + b * f b ≤ 2 * f a * f b\nh_f_nonneg_of_pos : ∀ a < 0, 0 ≤ f a\nh_f_nonpos : ∀ (x : ℝ), f x ≤ 0\nh_fx_zero_of_neg : ∀ x < 0, f x = 0\nhx : 0 ≤ 0\nhno : f (-1) = 0\nhp : f (-1) ≤ -1 * f (-1) - -1 * f (-1) + f (f (-1))\n⊢ 0 ≤ f 0",
"after_state": "case inr\nf : ℝ → ℝ\nhf : ∀ (x y : ℝ), f (x + y) ≤ y * f x + f (f x)\nhxt : ∀ (x t : ℝ), f t ≤ t * f x - x * f x + f (f x)\nh_ab_combined : ∀ (a b : ℝ), a * f a + b * f b ≤ 2 * f a * f b\nh_f_nonneg_of_pos : ∀ a < 0, 0 ≤ f a\nh_f_nonpos : ∀ (x : ℝ), f x ≤ 0\nh_fx_zero_of_neg : ∀ x < 0, f x = 0\nhx : 0 ≤ 0\nhno : f (-1) = 0\nhp : 0 ≤ -1 * 0 - -1 * 0 + f 0\n⊢ 0 ≤ f 0"
},
{
"line": "rewrite [hno] at hp",
"before_state": "case inr\nf : ℝ → ℝ\nhf : ∀ (x y : ℝ), f (x + y) ≤ y * f x + f (f x)\nhxt : ∀ (x t : ℝ), f t ≤ t * f x - x * f x + f (f x)\nh_ab_combined : ∀ (a b : ℝ), a * f a + b * f b ≤ 2 * f a * f b\nh_f_nonneg_of_pos : ∀ a < 0, 0 ≤ f a\nh_f_nonpos : ∀ (x : ℝ), f x ≤ 0\nh_fx_zero_of_neg : ∀ x < 0, f x = 0\nhx : 0 ≤ 0\nhno : f (-1) = 0\nhp : f (-1) ≤ -1 * f (-1) - -1 * f (-1) + f (f (-1))\n⊢ 0 ≤ f 0",
"after_state": "case inr\nf : ℝ → ℝ\nhf : ∀ (x y : ℝ), f (x + y) ≤ y * f x + f (f x)\nhxt : ∀ (x t : ℝ), f t ≤ t * f x - x * f x + f (f x)\nh_ab_combined : ∀ (a b : ℝ), a * f a + b * f b ≤ 2 * f a * f b\nh_f_nonneg_of_pos : ∀ a < 0, 0 ≤ f a\nh_f_nonpos : ∀ (x : ℝ), f x ≤ 0\nh_fx_zero_of_neg : ∀ x < 0, f x = 0\nhx : 0 ≤ 0\nhno : f (-1) = 0\nhp : 0 ≤ -1 * 0 - -1 * 0 + f 0\n⊢ 0 ≤ f 0"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "case inr\nf : ℝ → ℝ\nhf : ∀ (x y : ℝ), f (x + y) ≤ y * f x + f (f x)\nhxt : ∀ (x t : ℝ), f t ≤ t * f x - x * f x + f (f x)\nh_ab_combined : ∀ (a b : ℝ), a * f a + b * f b ≤ 2 * f a * f b\nh_f_nonneg_of_pos : ∀ a < 0, 0 ≤ f a\nh_f_nonpos : ∀ (x : ℝ), f x ≤ 0\nh_fx_zero_of_neg : ∀ x < 0, f x = 0\nhx : 0 ≤ 0\nhno : f (-1) = 0\nhp : 0 ≤ -1 * 0 - -1 * 0 + f 0\n⊢ 0 ≤ f 0",
"after_state": "case inr\nf : ℝ → ℝ\nhf : ∀ (x y : ℝ), f (x + y) ≤ y * f x + f (f x)\nhxt : ∀ (x t : ℝ), f t ≤ t * f x - x * f x + f (f x)\nh_ab_combined : ∀ (a b : ℝ), a * f a + b * f b ≤ 2 * f a * f b\nh_f_nonneg_of_pos : ∀ a < 0, 0 ≤ f a\nh_f_nonpos : ∀ (x : ℝ), f x ≤ 0\nh_fx_zero_of_neg : ∀ x < 0, f x = 0\nhx : 0 ≤ 0\nhno : f (-1) = 0\nhp : 0 ≤ -1 * 0 - -1 * 0 + f 0\n⊢ 0 ≤ f 0"
},
{
"line": "try (with_reducible rfl)",
"before_state": "case inr\nf : ℝ → ℝ\nhf : ∀ (x y : ℝ), f (x + y) ≤ y * f x + f (f x)\nhxt : ∀ (x t : ℝ), f t ≤ t * f x - x * f x + f (f x)\nh_ab_combined : ∀ (a b : ℝ), a * f a + b * f b ≤ 2 * f a * f b\nh_f_nonneg_of_pos : ∀ a < 0, 0 ≤ f a\nh_f_nonpos : ∀ (x : ℝ), f x ≤ 0\nh_fx_zero_of_neg : ∀ x < 0, f x = 0\nhx : 0 ≤ 0\nhno : f (-1) = 0\nhp : 0 ≤ -1 * 0 - -1 * 0 + f 0\n⊢ 0 ≤ f 0",
"after_state": "case inr\nf : ℝ → ℝ\nhf : ∀ (x y : ℝ), f (x + y) ≤ y * f x + f (f x)\nhxt : ∀ (x t : ℝ), f t ≤ t * f x - x * f x + f (f x)\nh_ab_combined : ∀ (a b : ℝ), a * f a + b * f b ≤ 2 * f a * f b\nh_f_nonneg_of_pos : ∀ a < 0, 0 ≤ f a\nh_f_nonpos : ∀ (x : ℝ), f x ≤ 0\nh_fx_zero_of_neg : ∀ x < 0, f x = 0\nhx : 0 ≤ 0\nhno : f (-1) = 0\nhp : 0 ≤ -1 * 0 - -1 * 0 + f 0\n⊢ 0 ≤ f 0"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "case inr\nf : ℝ → ℝ\nhf : ∀ (x y : ℝ), f (x + y) ≤ y * f x + f (f x)\nhxt : ∀ (x t : ℝ), f t ≤ t * f x - x * f x + f (f x)\nh_ab_combined : ∀ (a b : ℝ), a * f a + b * f b ≤ 2 * f a * f b\nh_f_nonneg_of_pos : ∀ a < 0, 0 ≤ f a\nh_f_nonpos : ∀ (x : ℝ), f x ≤ 0\nh_fx_zero_of_neg : ∀ x < 0, f x = 0\nhx : 0 ≤ 0\nhno : f (-1) = 0\nhp : 0 ≤ -1 * 0 - -1 * 0 + f 0\n⊢ 0 ≤ f 0",
"after_state": "case inr\nf : ℝ → ℝ\nhf : ∀ (x y : ℝ), f (x + y) ≤ y * f x + f (f x)\nhxt : ∀ (x t : ℝ), f t ≤ t * f x - x * f x + f (f x)\nh_ab_combined : ∀ (a b : ℝ), a * f a + b * f b ≤ 2 * f a * f b\nh_f_nonneg_of_pos : ∀ a < 0, 0 ≤ f a\nh_f_nonpos : ∀ (x : ℝ), f x ≤ 0\nh_fx_zero_of_neg : ∀ x < 0, f x = 0\nhx : 0 ≤ 0\nhno : f (-1) = 0\nhp : 0 ≤ -1 * 0 - -1 * 0 + f 0\n⊢ 0 ≤ f 0"
},
{
"line": "with_reducible rfl",
"before_state": "case inr\nf : ℝ → ℝ\nhf : ∀ (x y : ℝ), f (x + y) ≤ y * f x + f (f x)\nhxt : ∀ (x t : ℝ), f t ≤ t * f x - x * f x + f (f x)\nh_ab_combined : ∀ (a b : ℝ), a * f a + b * f b ≤ 2 * f a * f b\nh_f_nonneg_of_pos : ∀ a < 0, 0 ≤ f a\nh_f_nonpos : ∀ (x : ℝ), f x ≤ 0\nh_fx_zero_of_neg : ∀ x < 0, f x = 0\nhx : 0 ≤ 0\nhno : f (-1) = 0\nhp : 0 ≤ -1 * 0 - -1 * 0 + f 0\n⊢ 0 ≤ f 0",
"after_state": "case inr\nf : ℝ → ℝ\nhf : ∀ (x y : ℝ), f (x + y) ≤ y * f x + f (f x)\nhxt : ∀ (x t : ℝ), f t ≤ t * f x - x * f x + f (f x)\nh_ab_combined : ∀ (a b : ℝ), a * f a + b * f b ≤ 2 * f a * f b\nh_f_nonneg_of_pos : ∀ a < 0, 0 ≤ f a\nh_f_nonpos : ∀ (x : ℝ), f x ≤ 0\nh_fx_zero_of_neg : ∀ x < 0, f x = 0\nhx : 0 ≤ 0\nhno : f (-1) = 0\nhp : 0 ≤ -1 * 0 - -1 * 0 + f 0\n⊢ 0 ≤ f 0"
},
{
"line": "rfl",
"before_state": "case inr\nf : ℝ → ℝ\nhf : ∀ (x y : ℝ), f (x + y) ≤ y * f x + f (f x)\nhxt : ∀ (x t : ℝ), f t ≤ t * f x - x * f x + f (f x)\nh_ab_combined : ∀ (a b : ℝ), a * f a + b * f b ≤ 2 * f a * f b\nh_f_nonneg_of_pos : ∀ a < 0, 0 ≤ f a\nh_f_nonpos : ∀ (x : ℝ), f x ≤ 0\nh_fx_zero_of_neg : ∀ x < 0, f x = 0\nhx : 0 ≤ 0\nhno : f (-1) = 0\nhp : 0 ≤ -1 * 0 - -1 * 0 + f 0\n⊢ 0 ≤ f 0",
"after_state": "case inr\nf : ℝ → ℝ\nhf : ∀ (x y : ℝ), f (x + y) ≤ y * f x + f (f x)\nhxt : ∀ (x t : ℝ), f t ≤ t * f x - x * f x + f (f x)\nh_ab_combined : ∀ (a b : ℝ), a * f a + b * f b ≤ 2 * f a * f b\nh_f_nonneg_of_pos : ∀ a < 0, 0 ≤ f a\nh_f_nonpos : ∀ (x : ℝ), f x ≤ 0\nh_fx_zero_of_neg : ∀ x < 0, f x = 0\nhx : 0 ≤ 0\nhno : f (-1) = 0\nhp : 0 ≤ -1 * 0 - -1 * 0 + f 0\n⊢ 0 ≤ f 0"
},
{
"line": "apply_rfl",
"before_state": "case inr\nf : ℝ → ℝ\nhf : ∀ (x y : ℝ), f (x + y) ≤ y * f x + f (f x)\nhxt : ∀ (x t : ℝ), f t ≤ t * f x - x * f x + f (f x)\nh_ab_combined : ∀ (a b : ℝ), a * f a + b * f b ≤ 2 * f a * f b\nh_f_nonneg_of_pos : ∀ a < 0, 0 ≤ f a\nh_f_nonpos : ∀ (x : ℝ), f x ≤ 0\nh_fx_zero_of_neg : ∀ x < 0, f x = 0\nhx : 0 ≤ 0\nhno : f (-1) = 0\nhp : 0 ≤ -1 * 0 - -1 * 0 + f 0\n⊢ 0 ≤ f 0",
"after_state": "case inr\nf : ℝ → ℝ\nhf : ∀ (x y : ℝ), f (x + y) ≤ y * f x + f (f x)\nhxt : ∀ (x t : ℝ), f t ≤ t * f x - x * f x + f (f x)\nh_ab_combined : ∀ (a b : ℝ), a * f a + b * f b ≤ 2 * f a * f b\nh_f_nonneg_of_pos : ∀ a < 0, 0 ≤ f a\nh_f_nonpos : ∀ (x : ℝ), f x ≤ 0\nh_fx_zero_of_neg : ∀ x < 0, f x = 0\nhx : 0 ≤ 0\nhno : f (-1) = 0\nhp : 0 ≤ -1 * 0 - -1 * 0 + f 0\n⊢ 0 ≤ f 0"
},
{
"line": "skip",
"before_state": "case inr\nf : ℝ → ℝ\nhf : ∀ (x y : ℝ), f (x + y) ≤ y * f x + f (f x)\nhxt : ∀ (x t : ℝ), f t ≤ t * f x - x * f x + f (f x)\nh_ab_combined : ∀ (a b : ℝ), a * f a + b * f b ≤ 2 * f a * f b\nh_f_nonneg_of_pos : ∀ a < 0, 0 ≤ f a\nh_f_nonpos : ∀ (x : ℝ), f x ≤ 0\nh_fx_zero_of_neg : ∀ x < 0, f x = 0\nhx : 0 ≤ 0\nhno : f (-1) = 0\nhp : 0 ≤ -1 * 0 - -1 * 0 + f 0\n⊢ 0 ≤ f 0",
"after_state": "case inr\nf : ℝ → ℝ\nhf : ∀ (x y : ℝ), f (x + y) ≤ y * f x + f (f x)\nhxt : ∀ (x t : ℝ), f t ≤ t * f x - x * f x + f (f x)\nh_ab_combined : ∀ (a b : ℝ), a * f a + b * f b ≤ 2 * f a * f b\nh_f_nonneg_of_pos : ∀ a < 0, 0 ≤ f a\nh_f_nonpos : ∀ (x : ℝ), f x ≤ 0\nh_fx_zero_of_neg : ∀ x < 0, f x = 0\nhx : 0 ≤ 0\nhno : f (-1) = 0\nhp : 0 ≤ -1 * 0 - -1 * 0 + f 0\n⊢ 0 ≤ f 0"
},
{
"line": "linarith",
"before_state": "case inr\nf : ℝ → ℝ\nhf : ∀ (x y : ℝ), f (x + y) ≤ y * f x + f (f x)\nhxt : ∀ (x t : ℝ), f t ≤ t * f x - x * f x + f (f x)\nh_ab_combined : ∀ (a b : ℝ), a * f a + b * f b ≤ 2 * f a * f b\nh_f_nonneg_of_pos : ∀ a < 0, 0 ≤ f a\nh_f_nonpos : ∀ (x : ℝ), f x ≤ 0\nh_fx_zero_of_neg : ∀ x < 0, f x = 0\nhx : 0 ≤ 0\nhno : f (-1) = 0\nhp : 0 ≤ -1 * 0 - -1 * 0 + f 0\n⊢ 0 ≤ f 0",
"after_state": "No Goals!"
},
{
"line": "ring1",
"before_state": "f : ℝ → ℝ\nhf : ∀ (x y : ℝ), f (x + y) ≤ y * f x + f (f x)\nhxt : ∀ (x t : ℝ), f t ≤ t * f x - x * f x + f (f x)\nh_ab_combined : ∀ (a b : ℝ), a * f a + b * f b ≤ 2 * f a * f b\nh_f_nonneg_of_pos : ∀ a < 0, 0 ≤ f a\nh_f_nonpos : ∀ (x : ℝ), f x ≤ 0\nh_fx_zero_of_neg : ∀ x < 0, f x = 0\nhx : 0 ≤ 0\nhno : f (-1) = 0\nhp : 0 ≤ -1 * 0 - -1 * 0 + f 0\na✝ : 0 > f 0\n⊢ 0 - (-1 * 0 - -1 * 0 + f 0) + (f 0 - 0) = 0",
"after_state": "No Goals!"
}
] |
theorem imo2011_q5 (f : ℤ → ℤ) (hpos : ∀ n : ℤ, 0 < f n) (hdvd : ∀ m n : ℤ, f (m - n) ∣ f m - f n) :
∀ m n : ℤ, f m ≤ f n → f m ∣ f n := by
intro m n h_fm_le_fn
rcases lt_or_eq_of_le h_fm_le_fn with h_fm_lt_fn | h_fm_eq_fn
· -- m < n
let d := f m - f (m - n)
have h_fn_dvd_d : f n ∣ d := by
rw [← sub_sub_self m n]
exact hdvd m (m - n)
have h_d_lt_fn : d < f n := calc
d < f m := sub_lt_self _ (hpos (m - n))
_ < f n := h_fm_lt_fn
have h_neg_d_lt_fn : -d < f n := by
calc
-d = f (m - n) - f m := neg_sub _ _
_ < f (m - n) := sub_lt_self _ (hpos m)
_ ≤ f n - f m := le_of_dvd (sub_pos.mpr h_fm_lt_fn) ?_
_ < f n := sub_lt_self _ (hpos m)
-- ⊢ f (m - n) ∣ f n - f m
rw [← Int.dvd_neg]
rw [neg_sub]
exact hdvd m n
have h_d_eq_zero : d = 0 := by
obtain hd | hd | hd : d > 0 ∨ d = 0 ∨ d < 0 := trichotomous d 0
· -- d > 0
have h₁ : f n ≤ d := le_of_dvd hd h_fn_dvd_d
have h₂ : ¬f n ≤ d := not_le.mpr h_d_lt_fn
contradiction
· -- d = 0
exact hd
· -- d < 0
have h₁ : f n ≤ -d := le_of_dvd (neg_pos.mpr hd) h_fn_dvd_d.neg_right
have h₂ : ¬f n ≤ -d := not_le.mpr h_neg_d_lt_fn
contradiction
have h₁ : f m = f (m - n) := sub_eq_zero.mp h_d_eq_zero
have h₂ : f (m - n) ∣ f m - f n := hdvd m n
rw [← h₁] at h₂
exact (dvd_iff_dvd_of_dvd_sub h₂).mp dvd_rfl
· -- m = n
rw [h_fm_eq_fn]
|
/root/DuelModelResearch/mathlib4/Archive/Imo/Imo2011Q5.lean
|
{
"open": [
"Int"
],
"variables": []
}
|
[
{
"line": "intro m n h_fm_le_fn",
"before_state": "f : ℤ → ℤ\nhpos : ∀ (n : ℤ), 0 < f n\nhdvd : ∀ (m n : ℤ), f (m - n) ∣ f m - f n\n⊢ ∀ (m n : ℤ), f m ≤ f n → f m ∣ f n",
"after_state": "f : ℤ → ℤ\nhpos : ∀ (n : ℤ), 0 < f n\nhdvd : ∀ (m n : ℤ), f (m - n) ∣ f m - f n\nm n : ℤ\nh_fm_le_fn : f m ≤ f n\n⊢ f m ∣ f n"
},
{
"line": "intro m;\n intro n h_fm_le_fn",
"before_state": "f : ℤ → ℤ\nhpos : ∀ (n : ℤ), 0 < f n\nhdvd : ∀ (m n : ℤ), f (m - n) ∣ f m - f n\n⊢ ∀ (m n : ℤ), f m ≤ f n → f m ∣ f n",
"after_state": "f : ℤ → ℤ\nhpos : ∀ (n : ℤ), 0 < f n\nhdvd : ∀ (m n : ℤ), f (m - n) ∣ f m - f n\nm n : ℤ\nh_fm_le_fn : f m ≤ f n\n⊢ f m ∣ f n"
},
{
"line": "intro m",
"before_state": "f : ℤ → ℤ\nhpos : ∀ (n : ℤ), 0 < f n\nhdvd : ∀ (m n : ℤ), f (m - n) ∣ f m - f n\n⊢ ∀ (m n : ℤ), f m ≤ f n → f m ∣ f n",
"after_state": "f : ℤ → ℤ\nhpos : ∀ (n : ℤ), 0 < f n\nhdvd : ∀ (m n : ℤ), f (m - n) ∣ f m - f n\nm : ℤ\n⊢ ∀ (n : ℤ), f m ≤ f n → f m ∣ f n"
},
{
"line": "intro n h_fm_le_fn",
"before_state": "f : ℤ → ℤ\nhpos : ∀ (n : ℤ), 0 < f n\nhdvd : ∀ (m n : ℤ), f (m - n) ∣ f m - f n\nm : ℤ\n⊢ ∀ (n : ℤ), f m ≤ f n → f m ∣ f n",
"after_state": "f : ℤ → ℤ\nhpos : ∀ (n : ℤ), 0 < f n\nhdvd : ∀ (m n : ℤ), f (m - n) ∣ f m - f n\nm n : ℤ\nh_fm_le_fn : f m ≤ f n\n⊢ f m ∣ f n"
},
{
"line": "intro n;\n intro h_fm_le_fn",
"before_state": "f : ℤ → ℤ\nhpos : ∀ (n : ℤ), 0 < f n\nhdvd : ∀ (m n : ℤ), f (m - n) ∣ f m - f n\nm : ℤ\n⊢ ∀ (n : ℤ), f m ≤ f n → f m ∣ f n",
"after_state": "f : ℤ → ℤ\nhpos : ∀ (n : ℤ), 0 < f n\nhdvd : ∀ (m n : ℤ), f (m - n) ∣ f m - f n\nm n : ℤ\nh_fm_le_fn : f m ≤ f n\n⊢ f m ∣ f n"
},
{
"line": "intro n",
"before_state": "f : ℤ → ℤ\nhpos : ∀ (n : ℤ), 0 < f n\nhdvd : ∀ (m n : ℤ), f (m - n) ∣ f m - f n\nm : ℤ\n⊢ ∀ (n : ℤ), f m ≤ f n → f m ∣ f n",
"after_state": "f : ℤ → ℤ\nhpos : ∀ (n : ℤ), 0 < f n\nhdvd : ∀ (m n : ℤ), f (m - n) ∣ f m - f n\nm n : ℤ\n⊢ f m ≤ f n → f m ∣ f n"
},
{
"line": "intro h_fm_le_fn",
"before_state": "f : ℤ → ℤ\nhpos : ∀ (n : ℤ), 0 < f n\nhdvd : ∀ (m n : ℤ), f (m - n) ∣ f m - f n\nm n : ℤ\n⊢ f m ≤ f n → f m ∣ f n",
"after_state": "f : ℤ → ℤ\nhpos : ∀ (n : ℤ), 0 < f n\nhdvd : ∀ (m n : ℤ), f (m - n) ∣ f m - f n\nm n : ℤ\nh_fm_le_fn : f m ≤ f n\n⊢ f m ∣ f n"
},
{
"line": "rcases lt_or_eq_of_le h_fm_le_fn with h_fm_lt_fn | h_fm_eq_fn",
"before_state": "f : ℤ → ℤ\nhpos : ∀ (n : ℤ), 0 < f n\nhdvd : ∀ (m n : ℤ), f (m - n) ∣ f m - f n\nm n : ℤ\nh_fm_le_fn : f m ≤ f n\n⊢ f m ∣ f n",
"after_state": "case inl\nf : ℤ → ℤ\nhpos : ∀ (n : ℤ), 0 < f n\nhdvd : ∀ (m n : ℤ), f (m - n) ∣ f m - f n\nm n : ℤ\nh_fm_le_fn : f m ≤ f n\nh_fm_lt_fn : f m < f n\n⊢ f m ∣ f n\n---\ncase inr\nf : ℤ → ℤ\nhpos : ∀ (n : ℤ), 0 < f n\nhdvd : ∀ (m n : ℤ), f (m - n) ∣ f m - f n\nm n : ℤ\nh_fm_le_fn : f m ≤ f n\nh_fm_eq_fn : f m = f n\n⊢ f m ∣ f n"
},
{
"line": "let d := f m - f (m - n)",
"before_state": "case inl\nf : ℤ → ℤ\nhpos : ∀ (n : ℤ), 0 < f n\nhdvd : ∀ (m n : ℤ), f (m - n) ∣ f m - f n\nm n : ℤ\nh_fm_le_fn : f m ≤ f n\nh_fm_lt_fn : f m < f n\n⊢ f m ∣ f n",
"after_state": "case inl\nf : ℤ → ℤ\nhpos : ∀ (n : ℤ), 0 < f n\nhdvd : ∀ (m n : ℤ), f (m - n) ∣ f m - f n\nm n : ℤ\nh_fm_le_fn : f m ≤ f n\nh_fm_lt_fn : f m < f n\nd : ℤ := f m - f (m - n)\n⊢ f m ∣ f n"
},
{
"line": "refine_lift\n let d := f m - f (m - n);\n ?_",
"before_state": "case inl\nf : ℤ → ℤ\nhpos : ∀ (n : ℤ), 0 < f n\nhdvd : ∀ (m n : ℤ), f (m - n) ∣ f m - f n\nm n : ℤ\nh_fm_le_fn : f m ≤ f n\nh_fm_lt_fn : f m < f n\n⊢ f m ∣ f n",
"after_state": "case inl\nf : ℤ → ℤ\nhpos : ∀ (n : ℤ), 0 < f n\nhdvd : ∀ (m n : ℤ), f (m - n) ∣ f m - f n\nm n : ℤ\nh_fm_le_fn : f m ≤ f n\nh_fm_lt_fn : f m < f n\nd : ℤ := f m - f (m - n)\n⊢ f m ∣ f n"
},
{
"line": "focus\n (refine\n no_implicit_lambda%\n (let d := f m - f (m - n);\n ?_);\n rotate_right)",
"before_state": "case inl\nf : ℤ → ℤ\nhpos : ∀ (n : ℤ), 0 < f n\nhdvd : ∀ (m n : ℤ), f (m - n) ∣ f m - f n\nm n : ℤ\nh_fm_le_fn : f m ≤ f n\nh_fm_lt_fn : f m < f n\n⊢ f m ∣ f n",
"after_state": "case inl\nf : ℤ → ℤ\nhpos : ∀ (n : ℤ), 0 < f n\nhdvd : ∀ (m n : ℤ), f (m - n) ∣ f m - f n\nm n : ℤ\nh_fm_le_fn : f m ≤ f n\nh_fm_lt_fn : f m < f n\nd : ℤ := f m - f (m - n)\n⊢ f m ∣ f n"
},
{
"line": "refine\n no_implicit_lambda%\n (let d := f m - f (m - n);\n ?_)",
"before_state": "case inl\nf : ℤ → ℤ\nhpos : ∀ (n : ℤ), 0 < f n\nhdvd : ∀ (m n : ℤ), f (m - n) ∣ f m - f n\nm n : ℤ\nh_fm_le_fn : f m ≤ f n\nh_fm_lt_fn : f m < f n\n⊢ f m ∣ f n",
"after_state": "case inl\nf : ℤ → ℤ\nhpos : ∀ (n : ℤ), 0 < f n\nhdvd : ∀ (m n : ℤ), f (m - n) ∣ f m - f n\nm n : ℤ\nh_fm_le_fn : f m ≤ f n\nh_fm_lt_fn : f m < f n\nd : ℤ := f m - f (m - n)\n⊢ f m ∣ f n"
},
{
"line": "rotate_right",
"before_state": "case inl\nf : ℤ → ℤ\nhpos : ∀ (n : ℤ), 0 < f n\nhdvd : ∀ (m n : ℤ), f (m - n) ∣ f m - f n\nm n : ℤ\nh_fm_le_fn : f m ≤ f n\nh_fm_lt_fn : f m < f n\nd : ℤ := f m - f (m - n)\n⊢ f m ∣ f n",
"after_state": "case inl\nf : ℤ → ℤ\nhpos : ∀ (n : ℤ), 0 < f n\nhdvd : ∀ (m n : ℤ), f (m - n) ∣ f m - f n\nm n : ℤ\nh_fm_le_fn : f m ≤ f n\nh_fm_lt_fn : f m < f n\nd : ℤ := f m - f (m - n)\n⊢ f m ∣ f n"
},
{
"line": "have h_fn_dvd_d : f n ∣ d := by\n rw [← sub_sub_self m n]\n exact hdvd m (m - n)",
"before_state": "case inl\nf : ℤ → ℤ\nhpos : ∀ (n : ℤ), 0 < f n\nhdvd : ∀ (m n : ℤ), f (m - n) ∣ f m - f n\nm n : ℤ\nh_fm_le_fn : f m ≤ f n\nh_fm_lt_fn : f m < f n\nd : ℤ := f m - f (m - n)\n⊢ f m ∣ f n",
"after_state": "case inl\nf : ℤ → ℤ\nhpos : ∀ (n : ℤ), 0 < f n\nhdvd : ∀ (m n : ℤ), f (m - n) ∣ f m - f n\nm n : ℤ\nh_fm_le_fn : f m ≤ f n\nh_fm_lt_fn : f m < f n\nd : ℤ := f m - f (m - n)\nh_fn_dvd_d : f n ∣ d\n⊢ f m ∣ f n"
},
{
"line": "focus\n refine\n no_implicit_lambda%\n (have h_fn_dvd_d : f n ∣ d := ?body✝;\n ?_)\n case body✝ =>\n with_annotate_state\"by\"\n ( rw [← sub_sub_self m n]\n exact hdvd m (m - n))",
"before_state": "case inl\nf : ℤ → ℤ\nhpos : ∀ (n : ℤ), 0 < f n\nhdvd : ∀ (m n : ℤ), f (m - n) ∣ f m - f n\nm n : ℤ\nh_fm_le_fn : f m ≤ f n\nh_fm_lt_fn : f m < f n\nd : ℤ := f m - f (m - n)\n⊢ f m ∣ f n",
"after_state": "case inl\nf : ℤ → ℤ\nhpos : ∀ (n : ℤ), 0 < f n\nhdvd : ∀ (m n : ℤ), f (m - n) ∣ f m - f n\nm n : ℤ\nh_fm_le_fn : f m ≤ f n\nh_fm_lt_fn : f m < f n\nd : ℤ := f m - f (m - n)\nh_fn_dvd_d : f n ∣ d\n⊢ f m ∣ f n"
},
{
"line": "refine\n no_implicit_lambda%\n (have h_fn_dvd_d : f n ∣ d := ?body✝;\n ?_)",
"before_state": "case inl\nf : ℤ → ℤ\nhpos : ∀ (n : ℤ), 0 < f n\nhdvd : ∀ (m n : ℤ), f (m - n) ∣ f m - f n\nm n : ℤ\nh_fm_le_fn : f m ≤ f n\nh_fm_lt_fn : f m < f n\nd : ℤ := f m - f (m - n)\n⊢ f m ∣ f n",
"after_state": "case body\nf : ℤ → ℤ\nhpos : ∀ (n : ℤ), 0 < f n\nhdvd : ∀ (m n : ℤ), f (m - n) ∣ f m - f n\nm n : ℤ\nh_fm_le_fn : f m ≤ f n\nh_fm_lt_fn : f m < f n\nd : ℤ := f m - f (m - n)\n⊢ f n ∣ d\n---\ncase inl\nf : ℤ → ℤ\nhpos : ∀ (n : ℤ), 0 < f n\nhdvd : ∀ (m n : ℤ), f (m - n) ∣ f m - f n\nm n : ℤ\nh_fm_le_fn : f m ≤ f n\nh_fm_lt_fn : f m < f n\nd : ℤ := f m - f (m - n)\nh_fn_dvd_d : f n ∣ d\n⊢ f m ∣ f n"
},
{
"line": "case body✝ =>\n with_annotate_state\"by\"\n ( rw [← sub_sub_self m n]\n exact hdvd m (m - n))",
"before_state": "case body\nf : ℤ → ℤ\nhpos : ∀ (n : ℤ), 0 < f n\nhdvd : ∀ (m n : ℤ), f (m - n) ∣ f m - f n\nm n : ℤ\nh_fm_le_fn : f m ≤ f n\nh_fm_lt_fn : f m < f n\nd : ℤ := f m - f (m - n)\n⊢ f n ∣ d\n---\ncase inl\nf : ℤ → ℤ\nhpos : ∀ (n : ℤ), 0 < f n\nhdvd : ∀ (m n : ℤ), f (m - n) ∣ f m - f n\nm n : ℤ\nh_fm_le_fn : f m ≤ f n\nh_fm_lt_fn : f m < f n\nd : ℤ := f m - f (m - n)\nh_fn_dvd_d : f n ∣ d\n⊢ f m ∣ f n",
"after_state": "case inl\nf : ℤ → ℤ\nhpos : ∀ (n : ℤ), 0 < f n\nhdvd : ∀ (m n : ℤ), f (m - n) ∣ f m - f n\nm n : ℤ\nh_fm_le_fn : f m ≤ f n\nh_fm_lt_fn : f m < f n\nd : ℤ := f m - f (m - n)\nh_fn_dvd_d : f n ∣ d\n⊢ f m ∣ f n"
},
{
"line": "with_annotate_state\"by\"\n ( rw [← sub_sub_self m n]\n exact hdvd m (m - n))",
"before_state": "f : ℤ → ℤ\nhpos : ∀ (n : ℤ), 0 < f n\nhdvd : ∀ (m n : ℤ), f (m - n) ∣ f m - f n\nm n : ℤ\nh_fm_le_fn : f m ≤ f n\nh_fm_lt_fn : f m < f n\nd : ℤ := f m - f (m - n)\n⊢ f n ∣ d",
"after_state": "No Goals!"
},
{
"line": "rw [← sub_sub_self m n]",
"before_state": "f : ℤ → ℤ\nhpos : ∀ (n : ℤ), 0 < f n\nhdvd : ∀ (m n : ℤ), f (m - n) ∣ f m - f n\nm n : ℤ\nh_fm_le_fn : f m ≤ f n\nh_fm_lt_fn : f m < f n\nd : ℤ := f m - f (m - n)\n⊢ f n ∣ d",
"after_state": "f : ℤ → ℤ\nhpos : ∀ (n : ℤ), 0 < f n\nhdvd : ∀ (m n : ℤ), f (m - n) ∣ f m - f n\nm n : ℤ\nh_fm_le_fn : f m ≤ f n\nh_fm_lt_fn : f m < f n\nd : ℤ := f m - f (m - n)\n⊢ f (m - (m - n)) ∣ d"
},
{
"line": "rewrite [← sub_sub_self m n]",
"before_state": "f : ℤ → ℤ\nhpos : ∀ (n : ℤ), 0 < f n\nhdvd : ∀ (m n : ℤ), f (m - n) ∣ f m - f n\nm n : ℤ\nh_fm_le_fn : f m ≤ f n\nh_fm_lt_fn : f m < f n\nd : ℤ := f m - f (m - n)\n⊢ f n ∣ d",
"after_state": "f : ℤ → ℤ\nhpos : ∀ (n : ℤ), 0 < f n\nhdvd : ∀ (m n : ℤ), f (m - n) ∣ f m - f n\nm n : ℤ\nh_fm_le_fn : f m ≤ f n\nh_fm_lt_fn : f m < f n\nd : ℤ := f m - f (m - n)\n⊢ f (m - (m - n)) ∣ d"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "f : ℤ → ℤ\nhpos : ∀ (n : ℤ), 0 < f n\nhdvd : ∀ (m n : ℤ), f (m - n) ∣ f m - f n\nm n : ℤ\nh_fm_le_fn : f m ≤ f n\nh_fm_lt_fn : f m < f n\nd : ℤ := f m - f (m - n)\n⊢ f (m - (m - n)) ∣ d",
"after_state": "f : ℤ → ℤ\nhpos : ∀ (n : ℤ), 0 < f n\nhdvd : ∀ (m n : ℤ), f (m - n) ∣ f m - f n\nm n : ℤ\nh_fm_le_fn : f m ≤ f n\nh_fm_lt_fn : f m < f n\nd : ℤ := f m - f (m - n)\n⊢ f (m - (m - n)) ∣ d"
},
{
"line": "try (with_reducible rfl)",
"before_state": "f : ℤ → ℤ\nhpos : ∀ (n : ℤ), 0 < f n\nhdvd : ∀ (m n : ℤ), f (m - n) ∣ f m - f n\nm n : ℤ\nh_fm_le_fn : f m ≤ f n\nh_fm_lt_fn : f m < f n\nd : ℤ := f m - f (m - n)\n⊢ f (m - (m - n)) ∣ d",
"after_state": "f : ℤ → ℤ\nhpos : ∀ (n : ℤ), 0 < f n\nhdvd : ∀ (m n : ℤ), f (m - n) ∣ f m - f n\nm n : ℤ\nh_fm_le_fn : f m ≤ f n\nh_fm_lt_fn : f m < f n\nd : ℤ := f m - f (m - n)\n⊢ f (m - (m - n)) ∣ d"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "f : ℤ → ℤ\nhpos : ∀ (n : ℤ), 0 < f n\nhdvd : ∀ (m n : ℤ), f (m - n) ∣ f m - f n\nm n : ℤ\nh_fm_le_fn : f m ≤ f n\nh_fm_lt_fn : f m < f n\nd : ℤ := f m - f (m - n)\n⊢ f (m - (m - n)) ∣ d",
"after_state": "f : ℤ → ℤ\nhpos : ∀ (n : ℤ), 0 < f n\nhdvd : ∀ (m n : ℤ), f (m - n) ∣ f m - f n\nm n : ℤ\nh_fm_le_fn : f m ≤ f n\nh_fm_lt_fn : f m < f n\nd : ℤ := f m - f (m - n)\n⊢ f (m - (m - n)) ∣ d"
},
{
"line": "with_reducible rfl",
"before_state": "f : ℤ → ℤ\nhpos : ∀ (n : ℤ), 0 < f n\nhdvd : ∀ (m n : ℤ), f (m - n) ∣ f m - f n\nm n : ℤ\nh_fm_le_fn : f m ≤ f n\nh_fm_lt_fn : f m < f n\nd : ℤ := f m - f (m - n)\n⊢ f (m - (m - n)) ∣ d",
"after_state": "f : ℤ → ℤ\nhpos : ∀ (n : ℤ), 0 < f n\nhdvd : ∀ (m n : ℤ), f (m - n) ∣ f m - f n\nm n : ℤ\nh_fm_le_fn : f m ≤ f n\nh_fm_lt_fn : f m < f n\nd : ℤ := f m - f (m - n)\n⊢ f (m - (m - n)) ∣ d"
},
{
"line": "rfl",
"before_state": "f : ℤ → ℤ\nhpos : ∀ (n : ℤ), 0 < f n\nhdvd : ∀ (m n : ℤ), f (m - n) ∣ f m - f n\nm n : ℤ\nh_fm_le_fn : f m ≤ f n\nh_fm_lt_fn : f m < f n\nd : ℤ := f m - f (m - n)\n⊢ f (m - (m - n)) ∣ d",
"after_state": "f : ℤ → ℤ\nhpos : ∀ (n : ℤ), 0 < f n\nhdvd : ∀ (m n : ℤ), f (m - n) ∣ f m - f n\nm n : ℤ\nh_fm_le_fn : f m ≤ f n\nh_fm_lt_fn : f m < f n\nd : ℤ := f m - f (m - n)\n⊢ f (m - (m - n)) ∣ d"
},
{
"line": "apply_rfl",
"before_state": "f : ℤ → ℤ\nhpos : ∀ (n : ℤ), 0 < f n\nhdvd : ∀ (m n : ℤ), f (m - n) ∣ f m - f n\nm n : ℤ\nh_fm_le_fn : f m ≤ f n\nh_fm_lt_fn : f m < f n\nd : ℤ := f m - f (m - n)\n⊢ f (m - (m - n)) ∣ d",
"after_state": "f : ℤ → ℤ\nhpos : ∀ (n : ℤ), 0 < f n\nhdvd : ∀ (m n : ℤ), f (m - n) ∣ f m - f n\nm n : ℤ\nh_fm_le_fn : f m ≤ f n\nh_fm_lt_fn : f m < f n\nd : ℤ := f m - f (m - n)\n⊢ f (m - (m - n)) ∣ d"
},
{
"line": "skip",
"before_state": "f : ℤ → ℤ\nhpos : ∀ (n : ℤ), 0 < f n\nhdvd : ∀ (m n : ℤ), f (m - n) ∣ f m - f n\nm n : ℤ\nh_fm_le_fn : f m ≤ f n\nh_fm_lt_fn : f m < f n\nd : ℤ := f m - f (m - n)\n⊢ f (m - (m - n)) ∣ d",
"after_state": "f : ℤ → ℤ\nhpos : ∀ (n : ℤ), 0 < f n\nhdvd : ∀ (m n : ℤ), f (m - n) ∣ f m - f n\nm n : ℤ\nh_fm_le_fn : f m ≤ f n\nh_fm_lt_fn : f m < f n\nd : ℤ := f m - f (m - n)\n⊢ f (m - (m - n)) ∣ d"
},
{
"line": "exact hdvd m (m - n)",
"before_state": "f : ℤ → ℤ\nhpos : ∀ (n : ℤ), 0 < f n\nhdvd : ∀ (m n : ℤ), f (m - n) ∣ f m - f n\nm n : ℤ\nh_fm_le_fn : f m ≤ f n\nh_fm_lt_fn : f m < f n\nd : ℤ := f m - f (m - n)\n⊢ f (m - (m - n)) ∣ d",
"after_state": "No Goals!"
},
{
"line": "have h_d_lt_fn : d < f n :=\n calc\n d < f m := sub_lt_self _ (hpos (m - n))\n _ < f n := h_fm_lt_fn",
"before_state": "case inl\nf : ℤ → ℤ\nhpos : ∀ (n : ℤ), 0 < f n\nhdvd : ∀ (m n : ℤ), f (m - n) ∣ f m - f n\nm n : ℤ\nh_fm_le_fn : f m ≤ f n\nh_fm_lt_fn : f m < f n\nd : ℤ := f m - f (m - n)\nh_fn_dvd_d : f n ∣ d\n⊢ f m ∣ f n",
"after_state": "case inl\nf : ℤ → ℤ\nhpos : ∀ (n : ℤ), 0 < f n\nhdvd : ∀ (m n : ℤ), f (m - n) ∣ f m - f n\nm n : ℤ\nh_fm_le_fn : f m ≤ f n\nh_fm_lt_fn : f m < f n\nd : ℤ := f m - f (m - n)\nh_fn_dvd_d : f n ∣ d\nh_d_lt_fn : d < f n\n⊢ f m ∣ f n"
},
{
"line": "refine_lift\n have h_d_lt_fn : d < f n :=\n calc\n d < f m := sub_lt_self _ (hpos (m - n))\n _ < f n := h_fm_lt_fn;\n ?_",
"before_state": "case inl\nf : ℤ → ℤ\nhpos : ∀ (n : ℤ), 0 < f n\nhdvd : ∀ (m n : ℤ), f (m - n) ∣ f m - f n\nm n : ℤ\nh_fm_le_fn : f m ≤ f n\nh_fm_lt_fn : f m < f n\nd : ℤ := f m - f (m - n)\nh_fn_dvd_d : f n ∣ d\n⊢ f m ∣ f n",
"after_state": "case inl\nf : ℤ → ℤ\nhpos : ∀ (n : ℤ), 0 < f n\nhdvd : ∀ (m n : ℤ), f (m - n) ∣ f m - f n\nm n : ℤ\nh_fm_le_fn : f m ≤ f n\nh_fm_lt_fn : f m < f n\nd : ℤ := f m - f (m - n)\nh_fn_dvd_d : f n ∣ d\nh_d_lt_fn : d < f n\n⊢ f m ∣ f n"
},
{
"line": "focus\n (refine\n no_implicit_lambda%\n (have h_d_lt_fn : d < f n :=\n calc\n d < f m := sub_lt_self _ (hpos (m - n))\n _ < f n := h_fm_lt_fn;\n ?_);\n rotate_right)",
"before_state": "case inl\nf : ℤ → ℤ\nhpos : ∀ (n : ℤ), 0 < f n\nhdvd : ∀ (m n : ℤ), f (m - n) ∣ f m - f n\nm n : ℤ\nh_fm_le_fn : f m ≤ f n\nh_fm_lt_fn : f m < f n\nd : ℤ := f m - f (m - n)\nh_fn_dvd_d : f n ∣ d\n⊢ f m ∣ f n",
"after_state": "case inl\nf : ℤ → ℤ\nhpos : ∀ (n : ℤ), 0 < f n\nhdvd : ∀ (m n : ℤ), f (m - n) ∣ f m - f n\nm n : ℤ\nh_fm_le_fn : f m ≤ f n\nh_fm_lt_fn : f m < f n\nd : ℤ := f m - f (m - n)\nh_fn_dvd_d : f n ∣ d\nh_d_lt_fn : d < f n\n⊢ f m ∣ f n"
},
{
"line": "refine\n no_implicit_lambda%\n (have h_d_lt_fn : d < f n :=\n calc\n d < f m := sub_lt_self _ (hpos (m - n))\n _ < f n := h_fm_lt_fn;\n ?_)",
"before_state": "case inl\nf : ℤ → ℤ\nhpos : ∀ (n : ℤ), 0 < f n\nhdvd : ∀ (m n : ℤ), f (m - n) ∣ f m - f n\nm n : ℤ\nh_fm_le_fn : f m ≤ f n\nh_fm_lt_fn : f m < f n\nd : ℤ := f m - f (m - n)\nh_fn_dvd_d : f n ∣ d\n⊢ f m ∣ f n",
"after_state": "case inl\nf : ℤ → ℤ\nhpos : ∀ (n : ℤ), 0 < f n\nhdvd : ∀ (m n : ℤ), f (m - n) ∣ f m - f n\nm n : ℤ\nh_fm_le_fn : f m ≤ f n\nh_fm_lt_fn : f m < f n\nd : ℤ := f m - f (m - n)\nh_fn_dvd_d : f n ∣ d\nh_d_lt_fn : d < f n\n⊢ f m ∣ f n"
},
{
"line": "rotate_right",
"before_state": "case inl\nf : ℤ → ℤ\nhpos : ∀ (n : ℤ), 0 < f n\nhdvd : ∀ (m n : ℤ), f (m - n) ∣ f m - f n\nm n : ℤ\nh_fm_le_fn : f m ≤ f n\nh_fm_lt_fn : f m < f n\nd : ℤ := f m - f (m - n)\nh_fn_dvd_d : f n ∣ d\nh_d_lt_fn : d < f n\n⊢ f m ∣ f n",
"after_state": "case inl\nf : ℤ → ℤ\nhpos : ∀ (n : ℤ), 0 < f n\nhdvd : ∀ (m n : ℤ), f (m - n) ∣ f m - f n\nm n : ℤ\nh_fm_le_fn : f m ≤ f n\nh_fm_lt_fn : f m < f n\nd : ℤ := f m - f (m - n)\nh_fn_dvd_d : f n ∣ d\nh_d_lt_fn : d < f n\n⊢ f m ∣ f n"
},
{
"line": "have h_neg_d_lt_fn : -d < f n :=\n by\n calc\n -d = f (m - n) - f m := neg_sub _ _\n _ < f (m - n) := (sub_lt_self _ (hpos m))\n _ ≤ f n - f m := (le_of_dvd (sub_pos.mpr h_fm_lt_fn) ?_)\n _ < f n :=\n sub_lt_self _\n (hpos m)\n -- ⊢ f (m - n) ∣ f n - f m\n rw [← Int.dvd_neg]\n rw [neg_sub]\n exact hdvd m n",
"before_state": "case inl\nf : ℤ → ℤ\nhpos : ∀ (n : ℤ), 0 < f n\nhdvd : ∀ (m n : ℤ), f (m - n) ∣ f m - f n\nm n : ℤ\nh_fm_le_fn : f m ≤ f n\nh_fm_lt_fn : f m < f n\nd : ℤ := f m - f (m - n)\nh_fn_dvd_d : f n ∣ d\nh_d_lt_fn : d < f n\n⊢ f m ∣ f n",
"after_state": "case inl\nf : ℤ → ℤ\nhpos : ∀ (n : ℤ), 0 < f n\nhdvd : ∀ (m n : ℤ), f (m - n) ∣ f m - f n\nm n : ℤ\nh_fm_le_fn : f m ≤ f n\nh_fm_lt_fn : f m < f n\nd : ℤ := f m - f (m - n)\nh_fn_dvd_d : f n ∣ d\nh_d_lt_fn : d < f n\nh_neg_d_lt_fn : -d < f n\n⊢ f m ∣ f n"
},
{
"line": "focus\n refine\n no_implicit_lambda%\n (have h_neg_d_lt_fn : -d < f n := ?body✝;\n ?_)\n case body✝ =>\n with_annotate_state\"by\"\n ( calc\n -d = f (m - n) - f m := neg_sub _ _\n _ < f (m - n) := (sub_lt_self _ (hpos m))\n _ ≤ f n - f m := (le_of_dvd (sub_pos.mpr h_fm_lt_fn) ?_)\n _ < f n :=\n sub_lt_self _\n (hpos m)\n -- ⊢ f (m - n) ∣ f n - f m\n rw [← Int.dvd_neg]\n rw [neg_sub]\n exact hdvd m n)",
"before_state": "case inl\nf : ℤ → ℤ\nhpos : ∀ (n : ℤ), 0 < f n\nhdvd : ∀ (m n : ℤ), f (m - n) ∣ f m - f n\nm n : ℤ\nh_fm_le_fn : f m ≤ f n\nh_fm_lt_fn : f m < f n\nd : ℤ := f m - f (m - n)\nh_fn_dvd_d : f n ∣ d\nh_d_lt_fn : d < f n\n⊢ f m ∣ f n",
"after_state": "case inl\nf : ℤ → ℤ\nhpos : ∀ (n : ℤ), 0 < f n\nhdvd : ∀ (m n : ℤ), f (m - n) ∣ f m - f n\nm n : ℤ\nh_fm_le_fn : f m ≤ f n\nh_fm_lt_fn : f m < f n\nd : ℤ := f m - f (m - n)\nh_fn_dvd_d : f n ∣ d\nh_d_lt_fn : d < f n\nh_neg_d_lt_fn : -d < f n\n⊢ f m ∣ f n"
},
{
"line": "refine\n no_implicit_lambda%\n (have h_neg_d_lt_fn : -d < f n := ?body✝;\n ?_)",
"before_state": "case inl\nf : ℤ → ℤ\nhpos : ∀ (n : ℤ), 0 < f n\nhdvd : ∀ (m n : ℤ), f (m - n) ∣ f m - f n\nm n : ℤ\nh_fm_le_fn : f m ≤ f n\nh_fm_lt_fn : f m < f n\nd : ℤ := f m - f (m - n)\nh_fn_dvd_d : f n ∣ d\nh_d_lt_fn : d < f n\n⊢ f m ∣ f n",
"after_state": "case body\nf : ℤ → ℤ\nhpos : ∀ (n : ℤ), 0 < f n\nhdvd : ∀ (m n : ℤ), f (m - n) ∣ f m - f n\nm n : ℤ\nh_fm_le_fn : f m ≤ f n\nh_fm_lt_fn : f m < f n\nd : ℤ := f m - f (m - n)\nh_fn_dvd_d : f n ∣ d\nh_d_lt_fn : d < f n\n⊢ -d < f n\n---\ncase inl\nf : ℤ → ℤ\nhpos : ∀ (n : ℤ), 0 < f n\nhdvd : ∀ (m n : ℤ), f (m - n) ∣ f m - f n\nm n : ℤ\nh_fm_le_fn : f m ≤ f n\nh_fm_lt_fn : f m < f n\nd : ℤ := f m - f (m - n)\nh_fn_dvd_d : f n ∣ d\nh_d_lt_fn : d < f n\nh_neg_d_lt_fn : -d < f n\n⊢ f m ∣ f n"
},
{
"line": "case body✝ =>\n with_annotate_state\"by\"\n ( calc\n -d = f (m - n) - f m := neg_sub _ _\n _ < f (m - n) := (sub_lt_self _ (hpos m))\n _ ≤ f n - f m := (le_of_dvd (sub_pos.mpr h_fm_lt_fn) ?_)\n _ < f n :=\n sub_lt_self _\n (hpos m)\n -- ⊢ f (m - n) ∣ f n - f m\n rw [← Int.dvd_neg]\n rw [neg_sub]\n exact hdvd m n)",
"before_state": "case body\nf : ℤ → ℤ\nhpos : ∀ (n : ℤ), 0 < f n\nhdvd : ∀ (m n : ℤ), f (m - n) ∣ f m - f n\nm n : ℤ\nh_fm_le_fn : f m ≤ f n\nh_fm_lt_fn : f m < f n\nd : ℤ := f m - f (m - n)\nh_fn_dvd_d : f n ∣ d\nh_d_lt_fn : d < f n\n⊢ -d < f n\n---\ncase inl\nf : ℤ → ℤ\nhpos : ∀ (n : ℤ), 0 < f n\nhdvd : ∀ (m n : ℤ), f (m - n) ∣ f m - f n\nm n : ℤ\nh_fm_le_fn : f m ≤ f n\nh_fm_lt_fn : f m < f n\nd : ℤ := f m - f (m - n)\nh_fn_dvd_d : f n ∣ d\nh_d_lt_fn : d < f n\nh_neg_d_lt_fn : -d < f n\n⊢ f m ∣ f n",
"after_state": "case inl\nf : ℤ → ℤ\nhpos : ∀ (n : ℤ), 0 < f n\nhdvd : ∀ (m n : ℤ), f (m - n) ∣ f m - f n\nm n : ℤ\nh_fm_le_fn : f m ≤ f n\nh_fm_lt_fn : f m < f n\nd : ℤ := f m - f (m - n)\nh_fn_dvd_d : f n ∣ d\nh_d_lt_fn : d < f n\nh_neg_d_lt_fn : -d < f n\n⊢ f m ∣ f n"
},
{
"line": "with_annotate_state\"by\"\n ( calc\n -d = f (m - n) - f m := neg_sub _ _\n _ < f (m - n) := (sub_lt_self _ (hpos m))\n _ ≤ f n - f m := (le_of_dvd (sub_pos.mpr h_fm_lt_fn) ?_)\n _ < f n :=\n sub_lt_self _\n (hpos m)\n -- ⊢ f (m - n) ∣ f n - f m\n rw [← Int.dvd_neg]\n rw [neg_sub]\n exact hdvd m n)",
"before_state": "f : ℤ → ℤ\nhpos : ∀ (n : ℤ), 0 < f n\nhdvd : ∀ (m n : ℤ), f (m - n) ∣ f m - f n\nm n : ℤ\nh_fm_le_fn : f m ≤ f n\nh_fm_lt_fn : f m < f n\nd : ℤ := f m - f (m - n)\nh_fn_dvd_d : f n ∣ d\nh_d_lt_fn : d < f n\n⊢ -d < f n",
"after_state": "No Goals!"
},
{
"line": "calc\n -d = f (m - n) - f m := neg_sub _ _\n _ < f (m - n) := (sub_lt_self _ (hpos m))\n _ ≤ f n - f m := (le_of_dvd (sub_pos.mpr h_fm_lt_fn) ?_)\n _ < f n :=\n sub_lt_self _\n (hpos m)\n -- ⊢ f (m - n) ∣ f n - f m",
"before_state": "f : ℤ → ℤ\nhpos : ∀ (n : ℤ), 0 < f n\nhdvd : ∀ (m n : ℤ), f (m - n) ∣ f m - f n\nm n : ℤ\nh_fm_le_fn : f m ≤ f n\nh_fm_lt_fn : f m < f n\nd : ℤ := f m - f (m - n)\nh_fn_dvd_d : f n ∣ d\nh_d_lt_fn : d < f n\n⊢ -d < f n",
"after_state": "f : ℤ → ℤ\nhpos : ∀ (n : ℤ), 0 < f n\nhdvd : ∀ (m n : ℤ), f (m - n) ∣ f m - f n\nm n : ℤ\nh_fm_le_fn : f m ≤ f n\nh_fm_lt_fn : f m < f n\nd : ℤ := f m - f (m - n)\nh_fn_dvd_d : f n ∣ d\nh_d_lt_fn : d < f n\n⊢ f (m - n) ∣ f n - f m"
},
{
"line": "rw [← Int.dvd_neg]",
"before_state": "f : ℤ → ℤ\nhpos : ∀ (n : ℤ), 0 < f n\nhdvd : ∀ (m n : ℤ), f (m - n) ∣ f m - f n\nm n : ℤ\nh_fm_le_fn : f m ≤ f n\nh_fm_lt_fn : f m < f n\nd : ℤ := f m - f (m - n)\nh_fn_dvd_d : f n ∣ d\nh_d_lt_fn : d < f n\n⊢ f (m - n) ∣ f n - f m",
"after_state": "f : ℤ → ℤ\nhpos : ∀ (n : ℤ), 0 < f n\nhdvd : ∀ (m n : ℤ), f (m - n) ∣ f m - f n\nm n : ℤ\nh_fm_le_fn : f m ≤ f n\nh_fm_lt_fn : f m < f n\nd : ℤ := f m - f (m - n)\nh_fn_dvd_d : f n ∣ d\nh_d_lt_fn : d < f n\n⊢ f (m - n) ∣ -(f n - f m)"
},
{
"line": "rewrite [← Int.dvd_neg]",
"before_state": "f : ℤ → ℤ\nhpos : ∀ (n : ℤ), 0 < f n\nhdvd : ∀ (m n : ℤ), f (m - n) ∣ f m - f n\nm n : ℤ\nh_fm_le_fn : f m ≤ f n\nh_fm_lt_fn : f m < f n\nd : ℤ := f m - f (m - n)\nh_fn_dvd_d : f n ∣ d\nh_d_lt_fn : d < f n\n⊢ f (m - n) ∣ f n - f m",
"after_state": "f : ℤ → ℤ\nhpos : ∀ (n : ℤ), 0 < f n\nhdvd : ∀ (m n : ℤ), f (m - n) ∣ f m - f n\nm n : ℤ\nh_fm_le_fn : f m ≤ f n\nh_fm_lt_fn : f m < f n\nd : ℤ := f m - f (m - n)\nh_fn_dvd_d : f n ∣ d\nh_d_lt_fn : d < f n\n⊢ f (m - n) ∣ -(f n - f m)"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "f : ℤ → ℤ\nhpos : ∀ (n : ℤ), 0 < f n\nhdvd : ∀ (m n : ℤ), f (m - n) ∣ f m - f n\nm n : ℤ\nh_fm_le_fn : f m ≤ f n\nh_fm_lt_fn : f m < f n\nd : ℤ := f m - f (m - n)\nh_fn_dvd_d : f n ∣ d\nh_d_lt_fn : d < f n\n⊢ f (m - n) ∣ -(f n - f m)",
"after_state": "f : ℤ → ℤ\nhpos : ∀ (n : ℤ), 0 < f n\nhdvd : ∀ (m n : ℤ), f (m - n) ∣ f m - f n\nm n : ℤ\nh_fm_le_fn : f m ≤ f n\nh_fm_lt_fn : f m < f n\nd : ℤ := f m - f (m - n)\nh_fn_dvd_d : f n ∣ d\nh_d_lt_fn : d < f n\n⊢ f (m - n) ∣ -(f n - f m)"
},
{
"line": "try (with_reducible rfl)",
"before_state": "f : ℤ → ℤ\nhpos : ∀ (n : ℤ), 0 < f n\nhdvd : ∀ (m n : ℤ), f (m - n) ∣ f m - f n\nm n : ℤ\nh_fm_le_fn : f m ≤ f n\nh_fm_lt_fn : f m < f n\nd : ℤ := f m - f (m - n)\nh_fn_dvd_d : f n ∣ d\nh_d_lt_fn : d < f n\n⊢ f (m - n) ∣ -(f n - f m)",
"after_state": "f : ℤ → ℤ\nhpos : ∀ (n : ℤ), 0 < f n\nhdvd : ∀ (m n : ℤ), f (m - n) ∣ f m - f n\nm n : ℤ\nh_fm_le_fn : f m ≤ f n\nh_fm_lt_fn : f m < f n\nd : ℤ := f m - f (m - n)\nh_fn_dvd_d : f n ∣ d\nh_d_lt_fn : d < f n\n⊢ f (m - n) ∣ -(f n - f m)"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "f : ℤ → ℤ\nhpos : ∀ (n : ℤ), 0 < f n\nhdvd : ∀ (m n : ℤ), f (m - n) ∣ f m - f n\nm n : ℤ\nh_fm_le_fn : f m ≤ f n\nh_fm_lt_fn : f m < f n\nd : ℤ := f m - f (m - n)\nh_fn_dvd_d : f n ∣ d\nh_d_lt_fn : d < f n\n⊢ f (m - n) ∣ -(f n - f m)",
"after_state": "f : ℤ → ℤ\nhpos : ∀ (n : ℤ), 0 < f n\nhdvd : ∀ (m n : ℤ), f (m - n) ∣ f m - f n\nm n : ℤ\nh_fm_le_fn : f m ≤ f n\nh_fm_lt_fn : f m < f n\nd : ℤ := f m - f (m - n)\nh_fn_dvd_d : f n ∣ d\nh_d_lt_fn : d < f n\n⊢ f (m - n) ∣ -(f n - f m)"
},
{
"line": "with_reducible rfl",
"before_state": "f : ℤ → ℤ\nhpos : ∀ (n : ℤ), 0 < f n\nhdvd : ∀ (m n : ℤ), f (m - n) ∣ f m - f n\nm n : ℤ\nh_fm_le_fn : f m ≤ f n\nh_fm_lt_fn : f m < f n\nd : ℤ := f m - f (m - n)\nh_fn_dvd_d : f n ∣ d\nh_d_lt_fn : d < f n\n⊢ f (m - n) ∣ -(f n - f m)",
"after_state": "f : ℤ → ℤ\nhpos : ∀ (n : ℤ), 0 < f n\nhdvd : ∀ (m n : ℤ), f (m - n) ∣ f m - f n\nm n : ℤ\nh_fm_le_fn : f m ≤ f n\nh_fm_lt_fn : f m < f n\nd : ℤ := f m - f (m - n)\nh_fn_dvd_d : f n ∣ d\nh_d_lt_fn : d < f n\n⊢ f (m - n) ∣ -(f n - f m)"
},
{
"line": "rfl",
"before_state": "f : ℤ → ℤ\nhpos : ∀ (n : ℤ), 0 < f n\nhdvd : ∀ (m n : ℤ), f (m - n) ∣ f m - f n\nm n : ℤ\nh_fm_le_fn : f m ≤ f n\nh_fm_lt_fn : f m < f n\nd : ℤ := f m - f (m - n)\nh_fn_dvd_d : f n ∣ d\nh_d_lt_fn : d < f n\n⊢ f (m - n) ∣ -(f n - f m)",
"after_state": "f : ℤ → ℤ\nhpos : ∀ (n : ℤ), 0 < f n\nhdvd : ∀ (m n : ℤ), f (m - n) ∣ f m - f n\nm n : ℤ\nh_fm_le_fn : f m ≤ f n\nh_fm_lt_fn : f m < f n\nd : ℤ := f m - f (m - n)\nh_fn_dvd_d : f n ∣ d\nh_d_lt_fn : d < f n\n⊢ f (m - n) ∣ -(f n - f m)"
},
{
"line": "apply_rfl",
"before_state": "f : ℤ → ℤ\nhpos : ∀ (n : ℤ), 0 < f n\nhdvd : ∀ (m n : ℤ), f (m - n) ∣ f m - f n\nm n : ℤ\nh_fm_le_fn : f m ≤ f n\nh_fm_lt_fn : f m < f n\nd : ℤ := f m - f (m - n)\nh_fn_dvd_d : f n ∣ d\nh_d_lt_fn : d < f n\n⊢ f (m - n) ∣ -(f n - f m)",
"after_state": "f : ℤ → ℤ\nhpos : ∀ (n : ℤ), 0 < f n\nhdvd : ∀ (m n : ℤ), f (m - n) ∣ f m - f n\nm n : ℤ\nh_fm_le_fn : f m ≤ f n\nh_fm_lt_fn : f m < f n\nd : ℤ := f m - f (m - n)\nh_fn_dvd_d : f n ∣ d\nh_d_lt_fn : d < f n\n⊢ f (m - n) ∣ -(f n - f m)"
},
{
"line": "skip",
"before_state": "f : ℤ → ℤ\nhpos : ∀ (n : ℤ), 0 < f n\nhdvd : ∀ (m n : ℤ), f (m - n) ∣ f m - f n\nm n : ℤ\nh_fm_le_fn : f m ≤ f n\nh_fm_lt_fn : f m < f n\nd : ℤ := f m - f (m - n)\nh_fn_dvd_d : f n ∣ d\nh_d_lt_fn : d < f n\n⊢ f (m - n) ∣ -(f n - f m)",
"after_state": "f : ℤ → ℤ\nhpos : ∀ (n : ℤ), 0 < f n\nhdvd : ∀ (m n : ℤ), f (m - n) ∣ f m - f n\nm n : ℤ\nh_fm_le_fn : f m ≤ f n\nh_fm_lt_fn : f m < f n\nd : ℤ := f m - f (m - n)\nh_fn_dvd_d : f n ∣ d\nh_d_lt_fn : d < f n\n⊢ f (m - n) ∣ -(f n - f m)"
},
{
"line": "rw [neg_sub]",
"before_state": "f : ℤ → ℤ\nhpos : ∀ (n : ℤ), 0 < f n\nhdvd : ∀ (m n : ℤ), f (m - n) ∣ f m - f n\nm n : ℤ\nh_fm_le_fn : f m ≤ f n\nh_fm_lt_fn : f m < f n\nd : ℤ := f m - f (m - n)\nh_fn_dvd_d : f n ∣ d\nh_d_lt_fn : d < f n\n⊢ f (m - n) ∣ -(f n - f m)",
"after_state": "f : ℤ → ℤ\nhpos : ∀ (n : ℤ), 0 < f n\nhdvd : ∀ (m n : ℤ), f (m - n) ∣ f m - f n\nm n : ℤ\nh_fm_le_fn : f m ≤ f n\nh_fm_lt_fn : f m < f n\nd : ℤ := f m - f (m - n)\nh_fn_dvd_d : f n ∣ d\nh_d_lt_fn : d < f n\n⊢ f (m - n) ∣ f m - f n"
},
{
"line": "rewrite [neg_sub]",
"before_state": "f : ℤ → ℤ\nhpos : ∀ (n : ℤ), 0 < f n\nhdvd : ∀ (m n : ℤ), f (m - n) ∣ f m - f n\nm n : ℤ\nh_fm_le_fn : f m ≤ f n\nh_fm_lt_fn : f m < f n\nd : ℤ := f m - f (m - n)\nh_fn_dvd_d : f n ∣ d\nh_d_lt_fn : d < f n\n⊢ f (m - n) ∣ -(f n - f m)",
"after_state": "f : ℤ → ℤ\nhpos : ∀ (n : ℤ), 0 < f n\nhdvd : ∀ (m n : ℤ), f (m - n) ∣ f m - f n\nm n : ℤ\nh_fm_le_fn : f m ≤ f n\nh_fm_lt_fn : f m < f n\nd : ℤ := f m - f (m - n)\nh_fn_dvd_d : f n ∣ d\nh_d_lt_fn : d < f n\n⊢ f (m - n) ∣ f m - f n"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "f : ℤ → ℤ\nhpos : ∀ (n : ℤ), 0 < f n\nhdvd : ∀ (m n : ℤ), f (m - n) ∣ f m - f n\nm n : ℤ\nh_fm_le_fn : f m ≤ f n\nh_fm_lt_fn : f m < f n\nd : ℤ := f m - f (m - n)\nh_fn_dvd_d : f n ∣ d\nh_d_lt_fn : d < f n\n⊢ f (m - n) ∣ f m - f n",
"after_state": "f : ℤ → ℤ\nhpos : ∀ (n : ℤ), 0 < f n\nhdvd : ∀ (m n : ℤ), f (m - n) ∣ f m - f n\nm n : ℤ\nh_fm_le_fn : f m ≤ f n\nh_fm_lt_fn : f m < f n\nd : ℤ := f m - f (m - n)\nh_fn_dvd_d : f n ∣ d\nh_d_lt_fn : d < f n\n⊢ f (m - n) ∣ f m - f n"
},
{
"line": "try (with_reducible rfl)",
"before_state": "f : ℤ → ℤ\nhpos : ∀ (n : ℤ), 0 < f n\nhdvd : ∀ (m n : ℤ), f (m - n) ∣ f m - f n\nm n : ℤ\nh_fm_le_fn : f m ≤ f n\nh_fm_lt_fn : f m < f n\nd : ℤ := f m - f (m - n)\nh_fn_dvd_d : f n ∣ d\nh_d_lt_fn : d < f n\n⊢ f (m - n) ∣ f m - f n",
"after_state": "f : ℤ → ℤ\nhpos : ∀ (n : ℤ), 0 < f n\nhdvd : ∀ (m n : ℤ), f (m - n) ∣ f m - f n\nm n : ℤ\nh_fm_le_fn : f m ≤ f n\nh_fm_lt_fn : f m < f n\nd : ℤ := f m - f (m - n)\nh_fn_dvd_d : f n ∣ d\nh_d_lt_fn : d < f n\n⊢ f (m - n) ∣ f m - f n"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "f : ℤ → ℤ\nhpos : ∀ (n : ℤ), 0 < f n\nhdvd : ∀ (m n : ℤ), f (m - n) ∣ f m - f n\nm n : ℤ\nh_fm_le_fn : f m ≤ f n\nh_fm_lt_fn : f m < f n\nd : ℤ := f m - f (m - n)\nh_fn_dvd_d : f n ∣ d\nh_d_lt_fn : d < f n\n⊢ f (m - n) ∣ f m - f n",
"after_state": "f : ℤ → ℤ\nhpos : ∀ (n : ℤ), 0 < f n\nhdvd : ∀ (m n : ℤ), f (m - n) ∣ f m - f n\nm n : ℤ\nh_fm_le_fn : f m ≤ f n\nh_fm_lt_fn : f m < f n\nd : ℤ := f m - f (m - n)\nh_fn_dvd_d : f n ∣ d\nh_d_lt_fn : d < f n\n⊢ f (m - n) ∣ f m - f n"
},
{
"line": "with_reducible rfl",
"before_state": "f : ℤ → ℤ\nhpos : ∀ (n : ℤ), 0 < f n\nhdvd : ∀ (m n : ℤ), f (m - n) ∣ f m - f n\nm n : ℤ\nh_fm_le_fn : f m ≤ f n\nh_fm_lt_fn : f m < f n\nd : ℤ := f m - f (m - n)\nh_fn_dvd_d : f n ∣ d\nh_d_lt_fn : d < f n\n⊢ f (m - n) ∣ f m - f n",
"after_state": "f : ℤ → ℤ\nhpos : ∀ (n : ℤ), 0 < f n\nhdvd : ∀ (m n : ℤ), f (m - n) ∣ f m - f n\nm n : ℤ\nh_fm_le_fn : f m ≤ f n\nh_fm_lt_fn : f m < f n\nd : ℤ := f m - f (m - n)\nh_fn_dvd_d : f n ∣ d\nh_d_lt_fn : d < f n\n⊢ f (m - n) ∣ f m - f n"
},
{
"line": "rfl",
"before_state": "f : ℤ → ℤ\nhpos : ∀ (n : ℤ), 0 < f n\nhdvd : ∀ (m n : ℤ), f (m - n) ∣ f m - f n\nm n : ℤ\nh_fm_le_fn : f m ≤ f n\nh_fm_lt_fn : f m < f n\nd : ℤ := f m - f (m - n)\nh_fn_dvd_d : f n ∣ d\nh_d_lt_fn : d < f n\n⊢ f (m - n) ∣ f m - f n",
"after_state": "f : ℤ → ℤ\nhpos : ∀ (n : ℤ), 0 < f n\nhdvd : ∀ (m n : ℤ), f (m - n) ∣ f m - f n\nm n : ℤ\nh_fm_le_fn : f m ≤ f n\nh_fm_lt_fn : f m < f n\nd : ℤ := f m - f (m - n)\nh_fn_dvd_d : f n ∣ d\nh_d_lt_fn : d < f n\n⊢ f (m - n) ∣ f m - f n"
},
{
"line": "apply_rfl",
"before_state": "f : ℤ → ℤ\nhpos : ∀ (n : ℤ), 0 < f n\nhdvd : ∀ (m n : ℤ), f (m - n) ∣ f m - f n\nm n : ℤ\nh_fm_le_fn : f m ≤ f n\nh_fm_lt_fn : f m < f n\nd : ℤ := f m - f (m - n)\nh_fn_dvd_d : f n ∣ d\nh_d_lt_fn : d < f n\n⊢ f (m - n) ∣ f m - f n",
"after_state": "f : ℤ → ℤ\nhpos : ∀ (n : ℤ), 0 < f n\nhdvd : ∀ (m n : ℤ), f (m - n) ∣ f m - f n\nm n : ℤ\nh_fm_le_fn : f m ≤ f n\nh_fm_lt_fn : f m < f n\nd : ℤ := f m - f (m - n)\nh_fn_dvd_d : f n ∣ d\nh_d_lt_fn : d < f n\n⊢ f (m - n) ∣ f m - f n"
},
{
"line": "skip",
"before_state": "f : ℤ → ℤ\nhpos : ∀ (n : ℤ), 0 < f n\nhdvd : ∀ (m n : ℤ), f (m - n) ∣ f m - f n\nm n : ℤ\nh_fm_le_fn : f m ≤ f n\nh_fm_lt_fn : f m < f n\nd : ℤ := f m - f (m - n)\nh_fn_dvd_d : f n ∣ d\nh_d_lt_fn : d < f n\n⊢ f (m - n) ∣ f m - f n",
"after_state": "f : ℤ → ℤ\nhpos : ∀ (n : ℤ), 0 < f n\nhdvd : ∀ (m n : ℤ), f (m - n) ∣ f m - f n\nm n : ℤ\nh_fm_le_fn : f m ≤ f n\nh_fm_lt_fn : f m < f n\nd : ℤ := f m - f (m - n)\nh_fn_dvd_d : f n ∣ d\nh_d_lt_fn : d < f n\n⊢ f (m - n) ∣ f m - f n"
},
{
"line": "exact hdvd m n",
"before_state": "f : ℤ → ℤ\nhpos : ∀ (n : ℤ), 0 < f n\nhdvd : ∀ (m n : ℤ), f (m - n) ∣ f m - f n\nm n : ℤ\nh_fm_le_fn : f m ≤ f n\nh_fm_lt_fn : f m < f n\nd : ℤ := f m - f (m - n)\nh_fn_dvd_d : f n ∣ d\nh_d_lt_fn : d < f n\n⊢ f (m - n) ∣ f m - f n",
"after_state": "No Goals!"
},
{
"line": "have h_d_eq_zero : d = 0 := by\n obtain hd | hd | hd : d > 0 ∨ d = 0 ∨ d < 0 := trichotomous d 0\n ·\n -- d > 0\n have h₁ : f n ≤ d := le_of_dvd hd h_fn_dvd_d\n have h₂ : ¬f n ≤ d := not_le.mpr h_d_lt_fn\n contradiction\n ·\n -- d = 0exact hd\n ·\n -- d < 0\n have h₁ : f n ≤ -d := le_of_dvd (neg_pos.mpr hd) h_fn_dvd_d.neg_right\n have h₂ : ¬f n ≤ -d := not_le.mpr h_neg_d_lt_fn\n contradiction",
"before_state": "case inl\nf : ℤ → ℤ\nhpos : ∀ (n : ℤ), 0 < f n\nhdvd : ∀ (m n : ℤ), f (m - n) ∣ f m - f n\nm n : ℤ\nh_fm_le_fn : f m ≤ f n\nh_fm_lt_fn : f m < f n\nd : ℤ := f m - f (m - n)\nh_fn_dvd_d : f n ∣ d\nh_d_lt_fn : d < f n\nh_neg_d_lt_fn : -d < f n\n⊢ f m ∣ f n",
"after_state": "case inl\nf : ℤ → ℤ\nhpos : ∀ (n : ℤ), 0 < f n\nhdvd : ∀ (m n : ℤ), f (m - n) ∣ f m - f n\nm n : ℤ\nh_fm_le_fn : f m ≤ f n\nh_fm_lt_fn : f m < f n\nd : ℤ := f m - f (m - n)\nh_fn_dvd_d : f n ∣ d\nh_d_lt_fn : d < f n\nh_neg_d_lt_fn : -d < f n\nh_d_eq_zero : d = 0\n⊢ f m ∣ f n"
},
{
"line": "focus\n refine\n no_implicit_lambda%\n (have h_d_eq_zero : d = 0 := ?body✝;\n ?_)\n case body✝ =>\n with_annotate_state\"by\"\n ( obtain hd | hd | hd : d > 0 ∨ d = 0 ∨ d < 0 := trichotomous d 0\n ·\n -- d > 0\n have h₁ : f n ≤ d := le_of_dvd hd h_fn_dvd_d\n have h₂ : ¬f n ≤ d := not_le.mpr h_d_lt_fn\n contradiction\n ·\n -- d = 0exact hd\n ·\n -- d < 0\n have h₁ : f n ≤ -d := le_of_dvd (neg_pos.mpr hd) h_fn_dvd_d.neg_right\n have h₂ : ¬f n ≤ -d := not_le.mpr h_neg_d_lt_fn\n contradiction)",
"before_state": "case inl\nf : ℤ → ℤ\nhpos : ∀ (n : ℤ), 0 < f n\nhdvd : ∀ (m n : ℤ), f (m - n) ∣ f m - f n\nm n : ℤ\nh_fm_le_fn : f m ≤ f n\nh_fm_lt_fn : f m < f n\nd : ℤ := f m - f (m - n)\nh_fn_dvd_d : f n ∣ d\nh_d_lt_fn : d < f n\nh_neg_d_lt_fn : -d < f n\n⊢ f m ∣ f n",
"after_state": "case inl\nf : ℤ → ℤ\nhpos : ∀ (n : ℤ), 0 < f n\nhdvd : ∀ (m n : ℤ), f (m - n) ∣ f m - f n\nm n : ℤ\nh_fm_le_fn : f m ≤ f n\nh_fm_lt_fn : f m < f n\nd : ℤ := f m - f (m - n)\nh_fn_dvd_d : f n ∣ d\nh_d_lt_fn : d < f n\nh_neg_d_lt_fn : -d < f n\nh_d_eq_zero : d = 0\n⊢ f m ∣ f n"
},
{
"line": "refine\n no_implicit_lambda%\n (have h_d_eq_zero : d = 0 := ?body✝;\n ?_)",
"before_state": "case inl\nf : ℤ → ℤ\nhpos : ∀ (n : ℤ), 0 < f n\nhdvd : ∀ (m n : ℤ), f (m - n) ∣ f m - f n\nm n : ℤ\nh_fm_le_fn : f m ≤ f n\nh_fm_lt_fn : f m < f n\nd : ℤ := f m - f (m - n)\nh_fn_dvd_d : f n ∣ d\nh_d_lt_fn : d < f n\nh_neg_d_lt_fn : -d < f n\n⊢ f m ∣ f n",
"after_state": "case body\nf : ℤ → ℤ\nhpos : ∀ (n : ℤ), 0 < f n\nhdvd : ∀ (m n : ℤ), f (m - n) ∣ f m - f n\nm n : ℤ\nh_fm_le_fn : f m ≤ f n\nh_fm_lt_fn : f m < f n\nd : ℤ := f m - f (m - n)\nh_fn_dvd_d : f n ∣ d\nh_d_lt_fn : d < f n\nh_neg_d_lt_fn : -d < f n\n⊢ d = 0\n---\ncase inl\nf : ℤ → ℤ\nhpos : ∀ (n : ℤ), 0 < f n\nhdvd : ∀ (m n : ℤ), f (m - n) ∣ f m - f n\nm n : ℤ\nh_fm_le_fn : f m ≤ f n\nh_fm_lt_fn : f m < f n\nd : ℤ := f m - f (m - n)\nh_fn_dvd_d : f n ∣ d\nh_d_lt_fn : d < f n\nh_neg_d_lt_fn : -d < f n\nh_d_eq_zero : d = 0\n⊢ f m ∣ f n"
},
{
"line": "case body✝ =>\n with_annotate_state\"by\"\n ( obtain hd | hd | hd : d > 0 ∨ d = 0 ∨ d < 0 := trichotomous d 0\n ·\n -- d > 0\n have h₁ : f n ≤ d := le_of_dvd hd h_fn_dvd_d\n have h₂ : ¬f n ≤ d := not_le.mpr h_d_lt_fn\n contradiction\n ·\n -- d = 0exact hd\n ·\n -- d < 0\n have h₁ : f n ≤ -d := le_of_dvd (neg_pos.mpr hd) h_fn_dvd_d.neg_right\n have h₂ : ¬f n ≤ -d := not_le.mpr h_neg_d_lt_fn\n contradiction)",
"before_state": "case body\nf : ℤ → ℤ\nhpos : ∀ (n : ℤ), 0 < f n\nhdvd : ∀ (m n : ℤ), f (m - n) ∣ f m - f n\nm n : ℤ\nh_fm_le_fn : f m ≤ f n\nh_fm_lt_fn : f m < f n\nd : ℤ := f m - f (m - n)\nh_fn_dvd_d : f n ∣ d\nh_d_lt_fn : d < f n\nh_neg_d_lt_fn : -d < f n\n⊢ d = 0\n---\ncase inl\nf : ℤ → ℤ\nhpos : ∀ (n : ℤ), 0 < f n\nhdvd : ∀ (m n : ℤ), f (m - n) ∣ f m - f n\nm n : ℤ\nh_fm_le_fn : f m ≤ f n\nh_fm_lt_fn : f m < f n\nd : ℤ := f m - f (m - n)\nh_fn_dvd_d : f n ∣ d\nh_d_lt_fn : d < f n\nh_neg_d_lt_fn : -d < f n\nh_d_eq_zero : d = 0\n⊢ f m ∣ f n",
"after_state": "case inl\nf : ℤ → ℤ\nhpos : ∀ (n : ℤ), 0 < f n\nhdvd : ∀ (m n : ℤ), f (m - n) ∣ f m - f n\nm n : ℤ\nh_fm_le_fn : f m ≤ f n\nh_fm_lt_fn : f m < f n\nd : ℤ := f m - f (m - n)\nh_fn_dvd_d : f n ∣ d\nh_d_lt_fn : d < f n\nh_neg_d_lt_fn : -d < f n\nh_d_eq_zero : d = 0\n⊢ f m ∣ f n"
},
{
"line": "with_annotate_state\"by\"\n ( obtain hd | hd | hd : d > 0 ∨ d = 0 ∨ d < 0 := trichotomous d 0\n ·\n -- d > 0\n have h₁ : f n ≤ d := le_of_dvd hd h_fn_dvd_d\n have h₂ : ¬f n ≤ d := not_le.mpr h_d_lt_fn\n contradiction\n ·\n -- d = 0exact hd\n ·\n -- d < 0\n have h₁ : f n ≤ -d := le_of_dvd (neg_pos.mpr hd) h_fn_dvd_d.neg_right\n have h₂ : ¬f n ≤ -d := not_le.mpr h_neg_d_lt_fn\n contradiction)",
"before_state": "f : ℤ → ℤ\nhpos : ∀ (n : ℤ), 0 < f n\nhdvd : ∀ (m n : ℤ), f (m - n) ∣ f m - f n\nm n : ℤ\nh_fm_le_fn : f m ≤ f n\nh_fm_lt_fn : f m < f n\nd : ℤ := f m - f (m - n)\nh_fn_dvd_d : f n ∣ d\nh_d_lt_fn : d < f n\nh_neg_d_lt_fn : -d < f n\n⊢ d = 0",
"after_state": "No Goals!"
},
{
"line": "obtain hd | hd | hd : d > 0 ∨ d = 0 ∨ d < 0 := trichotomous d 0",
"before_state": "f : ℤ → ℤ\nhpos : ∀ (n : ℤ), 0 < f n\nhdvd : ∀ (m n : ℤ), f (m - n) ∣ f m - f n\nm n : ℤ\nh_fm_le_fn : f m ≤ f n\nh_fm_lt_fn : f m < f n\nd : ℤ := f m - f (m - n)\nh_fn_dvd_d : f n ∣ d\nh_d_lt_fn : d < f n\nh_neg_d_lt_fn : -d < f n\n⊢ d = 0",
"after_state": "case inl\nf : ℤ → ℤ\nhpos : ∀ (n : ℤ), 0 < f n\nhdvd : ∀ (m n : ℤ), f (m - n) ∣ f m - f n\nm n : ℤ\nh_fm_le_fn : f m ≤ f n\nh_fm_lt_fn : f m < f n\nd : ℤ := f m - f (m - n)\nh_fn_dvd_d : f n ∣ d\nh_d_lt_fn : d < f n\nh_neg_d_lt_fn : -d < f n\nhd : d > 0\n⊢ d = 0\n---\ncase inr.inl\nf : ℤ → ℤ\nhpos : ∀ (n : ℤ), 0 < f n\nhdvd : ∀ (m n : ℤ), f (m - n) ∣ f m - f n\nm n : ℤ\nh_fm_le_fn : f m ≤ f n\nh_fm_lt_fn : f m < f n\nd : ℤ := f m - f (m - n)\nh_fn_dvd_d : f n ∣ d\nh_d_lt_fn : d < f n\nh_neg_d_lt_fn : -d < f n\nhd : d = 0\n⊢ d = 0\n---\ncase inr.inr\nf : ℤ → ℤ\nhpos : ∀ (n : ℤ), 0 < f n\nhdvd : ∀ (m n : ℤ), f (m - n) ∣ f m - f n\nm n : ℤ\nh_fm_le_fn : f m ≤ f n\nh_fm_lt_fn : f m < f n\nd : ℤ := f m - f (m - n)\nh_fn_dvd_d : f n ∣ d\nh_d_lt_fn : d < f n\nh_neg_d_lt_fn : -d < f n\nhd : d < 0\n⊢ d = 0"
},
{
"line": "have h₁ : f n ≤ d := le_of_dvd hd h_fn_dvd_d",
"before_state": "case inl\nf : ℤ → ℤ\nhpos : ∀ (n : ℤ), 0 < f n\nhdvd : ∀ (m n : ℤ), f (m - n) ∣ f m - f n\nm n : ℤ\nh_fm_le_fn : f m ≤ f n\nh_fm_lt_fn : f m < f n\nd : ℤ := f m - f (m - n)\nh_fn_dvd_d : f n ∣ d\nh_d_lt_fn : d < f n\nh_neg_d_lt_fn : -d < f n\nhd : d > 0\n⊢ d = 0",
"after_state": "case inl\nf : ℤ → ℤ\nhpos : ∀ (n : ℤ), 0 < f n\nhdvd : ∀ (m n : ℤ), f (m - n) ∣ f m - f n\nm n : ℤ\nh_fm_le_fn : f m ≤ f n\nh_fm_lt_fn : f m < f n\nd : ℤ := f m - f (m - n)\nh_fn_dvd_d : f n ∣ d\nh_d_lt_fn : d < f n\nh_neg_d_lt_fn : -d < f n\nhd : d > 0\nh₁ : f n ≤ d\n⊢ d = 0"
},
{
"line": "refine_lift\n have h₁ : f n ≤ d := le_of_dvd hd h_fn_dvd_d;\n ?_",
"before_state": "case inl\nf : ℤ → ℤ\nhpos : ∀ (n : ℤ), 0 < f n\nhdvd : ∀ (m n : ℤ), f (m - n) ∣ f m - f n\nm n : ℤ\nh_fm_le_fn : f m ≤ f n\nh_fm_lt_fn : f m < f n\nd : ℤ := f m - f (m - n)\nh_fn_dvd_d : f n ∣ d\nh_d_lt_fn : d < f n\nh_neg_d_lt_fn : -d < f n\nhd : d > 0\n⊢ d = 0",
"after_state": "case inl\nf : ℤ → ℤ\nhpos : ∀ (n : ℤ), 0 < f n\nhdvd : ∀ (m n : ℤ), f (m - n) ∣ f m - f n\nm n : ℤ\nh_fm_le_fn : f m ≤ f n\nh_fm_lt_fn : f m < f n\nd : ℤ := f m - f (m - n)\nh_fn_dvd_d : f n ∣ d\nh_d_lt_fn : d < f n\nh_neg_d_lt_fn : -d < f n\nhd : d > 0\nh₁ : f n ≤ d\n⊢ d = 0"
},
{
"line": "focus\n (refine\n no_implicit_lambda%\n (have h₁ : f n ≤ d := le_of_dvd hd h_fn_dvd_d;\n ?_);\n rotate_right)",
"before_state": "case inl\nf : ℤ → ℤ\nhpos : ∀ (n : ℤ), 0 < f n\nhdvd : ∀ (m n : ℤ), f (m - n) ∣ f m - f n\nm n : ℤ\nh_fm_le_fn : f m ≤ f n\nh_fm_lt_fn : f m < f n\nd : ℤ := f m - f (m - n)\nh_fn_dvd_d : f n ∣ d\nh_d_lt_fn : d < f n\nh_neg_d_lt_fn : -d < f n\nhd : d > 0\n⊢ d = 0",
"after_state": "case inl\nf : ℤ → ℤ\nhpos : ∀ (n : ℤ), 0 < f n\nhdvd : ∀ (m n : ℤ), f (m - n) ∣ f m - f n\nm n : ℤ\nh_fm_le_fn : f m ≤ f n\nh_fm_lt_fn : f m < f n\nd : ℤ := f m - f (m - n)\nh_fn_dvd_d : f n ∣ d\nh_d_lt_fn : d < f n\nh_neg_d_lt_fn : -d < f n\nhd : d > 0\nh₁ : f n ≤ d\n⊢ d = 0"
},
{
"line": "refine\n no_implicit_lambda%\n (have h₁ : f n ≤ d := le_of_dvd hd h_fn_dvd_d;\n ?_)",
"before_state": "case inl\nf : ℤ → ℤ\nhpos : ∀ (n : ℤ), 0 < f n\nhdvd : ∀ (m n : ℤ), f (m - n) ∣ f m - f n\nm n : ℤ\nh_fm_le_fn : f m ≤ f n\nh_fm_lt_fn : f m < f n\nd : ℤ := f m - f (m - n)\nh_fn_dvd_d : f n ∣ d\nh_d_lt_fn : d < f n\nh_neg_d_lt_fn : -d < f n\nhd : d > 0\n⊢ d = 0",
"after_state": "case inl\nf : ℤ → ℤ\nhpos : ∀ (n : ℤ), 0 < f n\nhdvd : ∀ (m n : ℤ), f (m - n) ∣ f m - f n\nm n : ℤ\nh_fm_le_fn : f m ≤ f n\nh_fm_lt_fn : f m < f n\nd : ℤ := f m - f (m - n)\nh_fn_dvd_d : f n ∣ d\nh_d_lt_fn : d < f n\nh_neg_d_lt_fn : -d < f n\nhd : d > 0\nh₁ : f n ≤ d\n⊢ d = 0"
},
{
"line": "rotate_right",
"before_state": "case inl\nf : ℤ → ℤ\nhpos : ∀ (n : ℤ), 0 < f n\nhdvd : ∀ (m n : ℤ), f (m - n) ∣ f m - f n\nm n : ℤ\nh_fm_le_fn : f m ≤ f n\nh_fm_lt_fn : f m < f n\nd : ℤ := f m - f (m - n)\nh_fn_dvd_d : f n ∣ d\nh_d_lt_fn : d < f n\nh_neg_d_lt_fn : -d < f n\nhd : d > 0\nh₁ : f n ≤ d\n⊢ d = 0",
"after_state": "case inl\nf : ℤ → ℤ\nhpos : ∀ (n : ℤ), 0 < f n\nhdvd : ∀ (m n : ℤ), f (m - n) ∣ f m - f n\nm n : ℤ\nh_fm_le_fn : f m ≤ f n\nh_fm_lt_fn : f m < f n\nd : ℤ := f m - f (m - n)\nh_fn_dvd_d : f n ∣ d\nh_d_lt_fn : d < f n\nh_neg_d_lt_fn : -d < f n\nhd : d > 0\nh₁ : f n ≤ d\n⊢ d = 0"
},
{
"line": "have h₂ : ¬f n ≤ d := not_le.mpr h_d_lt_fn",
"before_state": "case inl\nf : ℤ → ℤ\nhpos : ∀ (n : ℤ), 0 < f n\nhdvd : ∀ (m n : ℤ), f (m - n) ∣ f m - f n\nm n : ℤ\nh_fm_le_fn : f m ≤ f n\nh_fm_lt_fn : f m < f n\nd : ℤ := f m - f (m - n)\nh_fn_dvd_d : f n ∣ d\nh_d_lt_fn : d < f n\nh_neg_d_lt_fn : -d < f n\nhd : d > 0\nh₁ : f n ≤ d\n⊢ d = 0",
"after_state": "case inl\nf : ℤ → ℤ\nhpos : ∀ (n : ℤ), 0 < f n\nhdvd : ∀ (m n : ℤ), f (m - n) ∣ f m - f n\nm n : ℤ\nh_fm_le_fn : f m ≤ f n\nh_fm_lt_fn : f m < f n\nd : ℤ := f m - f (m - n)\nh_fn_dvd_d : f n ∣ d\nh_d_lt_fn : d < f n\nh_neg_d_lt_fn : -d < f n\nhd : d > 0\nh₁ : f n ≤ d\nh₂ : ¬f n ≤ d\n⊢ d = 0"
},
{
"line": "refine_lift\n have h₂ : ¬f n ≤ d := not_le.mpr h_d_lt_fn;\n ?_",
"before_state": "case inl\nf : ℤ → ℤ\nhpos : ∀ (n : ℤ), 0 < f n\nhdvd : ∀ (m n : ℤ), f (m - n) ∣ f m - f n\nm n : ℤ\nh_fm_le_fn : f m ≤ f n\nh_fm_lt_fn : f m < f n\nd : ℤ := f m - f (m - n)\nh_fn_dvd_d : f n ∣ d\nh_d_lt_fn : d < f n\nh_neg_d_lt_fn : -d < f n\nhd : d > 0\nh₁ : f n ≤ d\n⊢ d = 0",
"after_state": "case inl\nf : ℤ → ℤ\nhpos : ∀ (n : ℤ), 0 < f n\nhdvd : ∀ (m n : ℤ), f (m - n) ∣ f m - f n\nm n : ℤ\nh_fm_le_fn : f m ≤ f n\nh_fm_lt_fn : f m < f n\nd : ℤ := f m - f (m - n)\nh_fn_dvd_d : f n ∣ d\nh_d_lt_fn : d < f n\nh_neg_d_lt_fn : -d < f n\nhd : d > 0\nh₁ : f n ≤ d\nh₂ : ¬f n ≤ d\n⊢ d = 0"
},
{
"line": "focus\n (refine\n no_implicit_lambda%\n (have h₂ : ¬f n ≤ d := not_le.mpr h_d_lt_fn;\n ?_);\n rotate_right)",
"before_state": "case inl\nf : ℤ → ℤ\nhpos : ∀ (n : ℤ), 0 < f n\nhdvd : ∀ (m n : ℤ), f (m - n) ∣ f m - f n\nm n : ℤ\nh_fm_le_fn : f m ≤ f n\nh_fm_lt_fn : f m < f n\nd : ℤ := f m - f (m - n)\nh_fn_dvd_d : f n ∣ d\nh_d_lt_fn : d < f n\nh_neg_d_lt_fn : -d < f n\nhd : d > 0\nh₁ : f n ≤ d\n⊢ d = 0",
"after_state": "case inl\nf : ℤ → ℤ\nhpos : ∀ (n : ℤ), 0 < f n\nhdvd : ∀ (m n : ℤ), f (m - n) ∣ f m - f n\nm n : ℤ\nh_fm_le_fn : f m ≤ f n\nh_fm_lt_fn : f m < f n\nd : ℤ := f m - f (m - n)\nh_fn_dvd_d : f n ∣ d\nh_d_lt_fn : d < f n\nh_neg_d_lt_fn : -d < f n\nhd : d > 0\nh₁ : f n ≤ d\nh₂ : ¬f n ≤ d\n⊢ d = 0"
},
{
"line": "refine\n no_implicit_lambda%\n (have h₂ : ¬f n ≤ d := not_le.mpr h_d_lt_fn;\n ?_)",
"before_state": "case inl\nf : ℤ → ℤ\nhpos : ∀ (n : ℤ), 0 < f n\nhdvd : ∀ (m n : ℤ), f (m - n) ∣ f m - f n\nm n : ℤ\nh_fm_le_fn : f m ≤ f n\nh_fm_lt_fn : f m < f n\nd : ℤ := f m - f (m - n)\nh_fn_dvd_d : f n ∣ d\nh_d_lt_fn : d < f n\nh_neg_d_lt_fn : -d < f n\nhd : d > 0\nh₁ : f n ≤ d\n⊢ d = 0",
"after_state": "case inl\nf : ℤ → ℤ\nhpos : ∀ (n : ℤ), 0 < f n\nhdvd : ∀ (m n : ℤ), f (m - n) ∣ f m - f n\nm n : ℤ\nh_fm_le_fn : f m ≤ f n\nh_fm_lt_fn : f m < f n\nd : ℤ := f m - f (m - n)\nh_fn_dvd_d : f n ∣ d\nh_d_lt_fn : d < f n\nh_neg_d_lt_fn : -d < f n\nhd : d > 0\nh₁ : f n ≤ d\nh₂ : ¬f n ≤ d\n⊢ d = 0"
},
{
"line": "rotate_right",
"before_state": "case inl\nf : ℤ → ℤ\nhpos : ∀ (n : ℤ), 0 < f n\nhdvd : ∀ (m n : ℤ), f (m - n) ∣ f m - f n\nm n : ℤ\nh_fm_le_fn : f m ≤ f n\nh_fm_lt_fn : f m < f n\nd : ℤ := f m - f (m - n)\nh_fn_dvd_d : f n ∣ d\nh_d_lt_fn : d < f n\nh_neg_d_lt_fn : -d < f n\nhd : d > 0\nh₁ : f n ≤ d\nh₂ : ¬f n ≤ d\n⊢ d = 0",
"after_state": "case inl\nf : ℤ → ℤ\nhpos : ∀ (n : ℤ), 0 < f n\nhdvd : ∀ (m n : ℤ), f (m - n) ∣ f m - f n\nm n : ℤ\nh_fm_le_fn : f m ≤ f n\nh_fm_lt_fn : f m < f n\nd : ℤ := f m - f (m - n)\nh_fn_dvd_d : f n ∣ d\nh_d_lt_fn : d < f n\nh_neg_d_lt_fn : -d < f n\nhd : d > 0\nh₁ : f n ≤ d\nh₂ : ¬f n ≤ d\n⊢ d = 0"
},
{
"line": "contradiction",
"before_state": "case inl\nf : ℤ → ℤ\nhpos : ∀ (n : ℤ), 0 < f n\nhdvd : ∀ (m n : ℤ), f (m - n) ∣ f m - f n\nm n : ℤ\nh_fm_le_fn : f m ≤ f n\nh_fm_lt_fn : f m < f n\nd : ℤ := f m - f (m - n)\nh_fn_dvd_d : f n ∣ d\nh_d_lt_fn : d < f n\nh_neg_d_lt_fn : -d < f n\nhd : d > 0\nh₁ : f n ≤ d\nh₂ : ¬f n ≤ d\n⊢ d = 0",
"after_state": "No Goals!"
},
{
"line": "exact hd",
"before_state": "case inr.inl\nf : ℤ → ℤ\nhpos : ∀ (n : ℤ), 0 < f n\nhdvd : ∀ (m n : ℤ), f (m - n) ∣ f m - f n\nm n : ℤ\nh_fm_le_fn : f m ≤ f n\nh_fm_lt_fn : f m < f n\nd : ℤ := f m - f (m - n)\nh_fn_dvd_d : f n ∣ d\nh_d_lt_fn : d < f n\nh_neg_d_lt_fn : -d < f n\nhd : d = 0\n⊢ d = 0",
"after_state": "No Goals!"
},
{
"line": "have h₁ : f n ≤ -d := le_of_dvd (neg_pos.mpr hd) h_fn_dvd_d.neg_right",
"before_state": "case inr.inr\nf : ℤ → ℤ\nhpos : ∀ (n : ℤ), 0 < f n\nhdvd : ∀ (m n : ℤ), f (m - n) ∣ f m - f n\nm n : ℤ\nh_fm_le_fn : f m ≤ f n\nh_fm_lt_fn : f m < f n\nd : ℤ := f m - f (m - n)\nh_fn_dvd_d : f n ∣ d\nh_d_lt_fn : d < f n\nh_neg_d_lt_fn : -d < f n\nhd : d < 0\n⊢ d = 0",
"after_state": "case inr.inr\nf : ℤ → ℤ\nhpos : ∀ (n : ℤ), 0 < f n\nhdvd : ∀ (m n : ℤ), f (m - n) ∣ f m - f n\nm n : ℤ\nh_fm_le_fn : f m ≤ f n\nh_fm_lt_fn : f m < f n\nd : ℤ := f m - f (m - n)\nh_fn_dvd_d : f n ∣ d\nh_d_lt_fn : d < f n\nh_neg_d_lt_fn : -d < f n\nhd : d < 0\nh₁ : f n ≤ -d\n⊢ d = 0"
},
{
"line": "refine_lift\n have h₁ : f n ≤ -d := le_of_dvd (neg_pos.mpr hd) h_fn_dvd_d.neg_right;\n ?_",
"before_state": "case inr.inr\nf : ℤ → ℤ\nhpos : ∀ (n : ℤ), 0 < f n\nhdvd : ∀ (m n : ℤ), f (m - n) ∣ f m - f n\nm n : ℤ\nh_fm_le_fn : f m ≤ f n\nh_fm_lt_fn : f m < f n\nd : ℤ := f m - f (m - n)\nh_fn_dvd_d : f n ∣ d\nh_d_lt_fn : d < f n\nh_neg_d_lt_fn : -d < f n\nhd : d < 0\n⊢ d = 0",
"after_state": "case inr.inr\nf : ℤ → ℤ\nhpos : ∀ (n : ℤ), 0 < f n\nhdvd : ∀ (m n : ℤ), f (m - n) ∣ f m - f n\nm n : ℤ\nh_fm_le_fn : f m ≤ f n\nh_fm_lt_fn : f m < f n\nd : ℤ := f m - f (m - n)\nh_fn_dvd_d : f n ∣ d\nh_d_lt_fn : d < f n\nh_neg_d_lt_fn : -d < f n\nhd : d < 0\nh₁ : f n ≤ -d\n⊢ d = 0"
},
{
"line": "focus\n (refine\n no_implicit_lambda%\n (have h₁ : f n ≤ -d := le_of_dvd (neg_pos.mpr hd) h_fn_dvd_d.neg_right;\n ?_);\n rotate_right)",
"before_state": "case inr.inr\nf : ℤ → ℤ\nhpos : ∀ (n : ℤ), 0 < f n\nhdvd : ∀ (m n : ℤ), f (m - n) ∣ f m - f n\nm n : ℤ\nh_fm_le_fn : f m ≤ f n\nh_fm_lt_fn : f m < f n\nd : ℤ := f m - f (m - n)\nh_fn_dvd_d : f n ∣ d\nh_d_lt_fn : d < f n\nh_neg_d_lt_fn : -d < f n\nhd : d < 0\n⊢ d = 0",
"after_state": "case inr.inr\nf : ℤ → ℤ\nhpos : ∀ (n : ℤ), 0 < f n\nhdvd : ∀ (m n : ℤ), f (m - n) ∣ f m - f n\nm n : ℤ\nh_fm_le_fn : f m ≤ f n\nh_fm_lt_fn : f m < f n\nd : ℤ := f m - f (m - n)\nh_fn_dvd_d : f n ∣ d\nh_d_lt_fn : d < f n\nh_neg_d_lt_fn : -d < f n\nhd : d < 0\nh₁ : f n ≤ -d\n⊢ d = 0"
},
{
"line": "refine\n no_implicit_lambda%\n (have h₁ : f n ≤ -d := le_of_dvd (neg_pos.mpr hd) h_fn_dvd_d.neg_right;\n ?_)",
"before_state": "case inr.inr\nf : ℤ → ℤ\nhpos : ∀ (n : ℤ), 0 < f n\nhdvd : ∀ (m n : ℤ), f (m - n) ∣ f m - f n\nm n : ℤ\nh_fm_le_fn : f m ≤ f n\nh_fm_lt_fn : f m < f n\nd : ℤ := f m - f (m - n)\nh_fn_dvd_d : f n ∣ d\nh_d_lt_fn : d < f n\nh_neg_d_lt_fn : -d < f n\nhd : d < 0\n⊢ d = 0",
"after_state": "case inr.inr\nf : ℤ → ℤ\nhpos : ∀ (n : ℤ), 0 < f n\nhdvd : ∀ (m n : ℤ), f (m - n) ∣ f m - f n\nm n : ℤ\nh_fm_le_fn : f m ≤ f n\nh_fm_lt_fn : f m < f n\nd : ℤ := f m - f (m - n)\nh_fn_dvd_d : f n ∣ d\nh_d_lt_fn : d < f n\nh_neg_d_lt_fn : -d < f n\nhd : d < 0\nh₁ : f n ≤ -d\n⊢ d = 0"
},
{
"line": "rotate_right",
"before_state": "case inr.inr\nf : ℤ → ℤ\nhpos : ∀ (n : ℤ), 0 < f n\nhdvd : ∀ (m n : ℤ), f (m - n) ∣ f m - f n\nm n : ℤ\nh_fm_le_fn : f m ≤ f n\nh_fm_lt_fn : f m < f n\nd : ℤ := f m - f (m - n)\nh_fn_dvd_d : f n ∣ d\nh_d_lt_fn : d < f n\nh_neg_d_lt_fn : -d < f n\nhd : d < 0\nh₁ : f n ≤ -d\n⊢ d = 0",
"after_state": "case inr.inr\nf : ℤ → ℤ\nhpos : ∀ (n : ℤ), 0 < f n\nhdvd : ∀ (m n : ℤ), f (m - n) ∣ f m - f n\nm n : ℤ\nh_fm_le_fn : f m ≤ f n\nh_fm_lt_fn : f m < f n\nd : ℤ := f m - f (m - n)\nh_fn_dvd_d : f n ∣ d\nh_d_lt_fn : d < f n\nh_neg_d_lt_fn : -d < f n\nhd : d < 0\nh₁ : f n ≤ -d\n⊢ d = 0"
},
{
"line": "have h₂ : ¬f n ≤ -d := not_le.mpr h_neg_d_lt_fn",
"before_state": "case inr.inr\nf : ℤ → ℤ\nhpos : ∀ (n : ℤ), 0 < f n\nhdvd : ∀ (m n : ℤ), f (m - n) ∣ f m - f n\nm n : ℤ\nh_fm_le_fn : f m ≤ f n\nh_fm_lt_fn : f m < f n\nd : ℤ := f m - f (m - n)\nh_fn_dvd_d : f n ∣ d\nh_d_lt_fn : d < f n\nh_neg_d_lt_fn : -d < f n\nhd : d < 0\nh₁ : f n ≤ -d\n⊢ d = 0",
"after_state": "case inr.inr\nf : ℤ → ℤ\nhpos : ∀ (n : ℤ), 0 < f n\nhdvd : ∀ (m n : ℤ), f (m - n) ∣ f m - f n\nm n : ℤ\nh_fm_le_fn : f m ≤ f n\nh_fm_lt_fn : f m < f n\nd : ℤ := f m - f (m - n)\nh_fn_dvd_d : f n ∣ d\nh_d_lt_fn : d < f n\nh_neg_d_lt_fn : -d < f n\nhd : d < 0\nh₁ : f n ≤ -d\nh₂ : ¬f n ≤ -d\n⊢ d = 0"
},
{
"line": "refine_lift\n have h₂ : ¬f n ≤ -d := not_le.mpr h_neg_d_lt_fn;\n ?_",
"before_state": "case inr.inr\nf : ℤ → ℤ\nhpos : ∀ (n : ℤ), 0 < f n\nhdvd : ∀ (m n : ℤ), f (m - n) ∣ f m - f n\nm n : ℤ\nh_fm_le_fn : f m ≤ f n\nh_fm_lt_fn : f m < f n\nd : ℤ := f m - f (m - n)\nh_fn_dvd_d : f n ∣ d\nh_d_lt_fn : d < f n\nh_neg_d_lt_fn : -d < f n\nhd : d < 0\nh₁ : f n ≤ -d\n⊢ d = 0",
"after_state": "case inr.inr\nf : ℤ → ℤ\nhpos : ∀ (n : ℤ), 0 < f n\nhdvd : ∀ (m n : ℤ), f (m - n) ∣ f m - f n\nm n : ℤ\nh_fm_le_fn : f m ≤ f n\nh_fm_lt_fn : f m < f n\nd : ℤ := f m - f (m - n)\nh_fn_dvd_d : f n ∣ d\nh_d_lt_fn : d < f n\nh_neg_d_lt_fn : -d < f n\nhd : d < 0\nh₁ : f n ≤ -d\nh₂ : ¬f n ≤ -d\n⊢ d = 0"
},
{
"line": "focus\n (refine\n no_implicit_lambda%\n (have h₂ : ¬f n ≤ -d := not_le.mpr h_neg_d_lt_fn;\n ?_);\n rotate_right)",
"before_state": "case inr.inr\nf : ℤ → ℤ\nhpos : ∀ (n : ℤ), 0 < f n\nhdvd : ∀ (m n : ℤ), f (m - n) ∣ f m - f n\nm n : ℤ\nh_fm_le_fn : f m ≤ f n\nh_fm_lt_fn : f m < f n\nd : ℤ := f m - f (m - n)\nh_fn_dvd_d : f n ∣ d\nh_d_lt_fn : d < f n\nh_neg_d_lt_fn : -d < f n\nhd : d < 0\nh₁ : f n ≤ -d\n⊢ d = 0",
"after_state": "case inr.inr\nf : ℤ → ℤ\nhpos : ∀ (n : ℤ), 0 < f n\nhdvd : ∀ (m n : ℤ), f (m - n) ∣ f m - f n\nm n : ℤ\nh_fm_le_fn : f m ≤ f n\nh_fm_lt_fn : f m < f n\nd : ℤ := f m - f (m - n)\nh_fn_dvd_d : f n ∣ d\nh_d_lt_fn : d < f n\nh_neg_d_lt_fn : -d < f n\nhd : d < 0\nh₁ : f n ≤ -d\nh₂ : ¬f n ≤ -d\n⊢ d = 0"
},
{
"line": "refine\n no_implicit_lambda%\n (have h₂ : ¬f n ≤ -d := not_le.mpr h_neg_d_lt_fn;\n ?_)",
"before_state": "case inr.inr\nf : ℤ → ℤ\nhpos : ∀ (n : ℤ), 0 < f n\nhdvd : ∀ (m n : ℤ), f (m - n) ∣ f m - f n\nm n : ℤ\nh_fm_le_fn : f m ≤ f n\nh_fm_lt_fn : f m < f n\nd : ℤ := f m - f (m - n)\nh_fn_dvd_d : f n ∣ d\nh_d_lt_fn : d < f n\nh_neg_d_lt_fn : -d < f n\nhd : d < 0\nh₁ : f n ≤ -d\n⊢ d = 0",
"after_state": "case inr.inr\nf : ℤ → ℤ\nhpos : ∀ (n : ℤ), 0 < f n\nhdvd : ∀ (m n : ℤ), f (m - n) ∣ f m - f n\nm n : ℤ\nh_fm_le_fn : f m ≤ f n\nh_fm_lt_fn : f m < f n\nd : ℤ := f m - f (m - n)\nh_fn_dvd_d : f n ∣ d\nh_d_lt_fn : d < f n\nh_neg_d_lt_fn : -d < f n\nhd : d < 0\nh₁ : f n ≤ -d\nh₂ : ¬f n ≤ -d\n⊢ d = 0"
},
{
"line": "rotate_right",
"before_state": "case inr.inr\nf : ℤ → ℤ\nhpos : ∀ (n : ℤ), 0 < f n\nhdvd : ∀ (m n : ℤ), f (m - n) ∣ f m - f n\nm n : ℤ\nh_fm_le_fn : f m ≤ f n\nh_fm_lt_fn : f m < f n\nd : ℤ := f m - f (m - n)\nh_fn_dvd_d : f n ∣ d\nh_d_lt_fn : d < f n\nh_neg_d_lt_fn : -d < f n\nhd : d < 0\nh₁ : f n ≤ -d\nh₂ : ¬f n ≤ -d\n⊢ d = 0",
"after_state": "case inr.inr\nf : ℤ → ℤ\nhpos : ∀ (n : ℤ), 0 < f n\nhdvd : ∀ (m n : ℤ), f (m - n) ∣ f m - f n\nm n : ℤ\nh_fm_le_fn : f m ≤ f n\nh_fm_lt_fn : f m < f n\nd : ℤ := f m - f (m - n)\nh_fn_dvd_d : f n ∣ d\nh_d_lt_fn : d < f n\nh_neg_d_lt_fn : -d < f n\nhd : d < 0\nh₁ : f n ≤ -d\nh₂ : ¬f n ≤ -d\n⊢ d = 0"
},
{
"line": "contradiction",
"before_state": "case inr.inr\nf : ℤ → ℤ\nhpos : ∀ (n : ℤ), 0 < f n\nhdvd : ∀ (m n : ℤ), f (m - n) ∣ f m - f n\nm n : ℤ\nh_fm_le_fn : f m ≤ f n\nh_fm_lt_fn : f m < f n\nd : ℤ := f m - f (m - n)\nh_fn_dvd_d : f n ∣ d\nh_d_lt_fn : d < f n\nh_neg_d_lt_fn : -d < f n\nhd : d < 0\nh₁ : f n ≤ -d\nh₂ : ¬f n ≤ -d\n⊢ d = 0",
"after_state": "No Goals!"
},
{
"line": "have h₁ : f m = f (m - n) := sub_eq_zero.mp h_d_eq_zero",
"before_state": "case inl\nf : ℤ → ℤ\nhpos : ∀ (n : ℤ), 0 < f n\nhdvd : ∀ (m n : ℤ), f (m - n) ∣ f m - f n\nm n : ℤ\nh_fm_le_fn : f m ≤ f n\nh_fm_lt_fn : f m < f n\nd : ℤ := f m - f (m - n)\nh_fn_dvd_d : f n ∣ d\nh_d_lt_fn : d < f n\nh_neg_d_lt_fn : -d < f n\nh_d_eq_zero : d = 0\n⊢ f m ∣ f n",
"after_state": "case inl\nf : ℤ → ℤ\nhpos : ∀ (n : ℤ), 0 < f n\nhdvd : ∀ (m n : ℤ), f (m - n) ∣ f m - f n\nm n : ℤ\nh_fm_le_fn : f m ≤ f n\nh_fm_lt_fn : f m < f n\nd : ℤ := f m - f (m - n)\nh_fn_dvd_d : f n ∣ d\nh_d_lt_fn : d < f n\nh_neg_d_lt_fn : -d < f n\nh_d_eq_zero : d = 0\nh₁ : f m = f (m - n)\n⊢ f m ∣ f n"
},
{
"line": "refine_lift\n have h₁ : f m = f (m - n) := sub_eq_zero.mp h_d_eq_zero;\n ?_",
"before_state": "case inl\nf : ℤ → ℤ\nhpos : ∀ (n : ℤ), 0 < f n\nhdvd : ∀ (m n : ℤ), f (m - n) ∣ f m - f n\nm n : ℤ\nh_fm_le_fn : f m ≤ f n\nh_fm_lt_fn : f m < f n\nd : ℤ := f m - f (m - n)\nh_fn_dvd_d : f n ∣ d\nh_d_lt_fn : d < f n\nh_neg_d_lt_fn : -d < f n\nh_d_eq_zero : d = 0\n⊢ f m ∣ f n",
"after_state": "case inl\nf : ℤ → ℤ\nhpos : ∀ (n : ℤ), 0 < f n\nhdvd : ∀ (m n : ℤ), f (m - n) ∣ f m - f n\nm n : ℤ\nh_fm_le_fn : f m ≤ f n\nh_fm_lt_fn : f m < f n\nd : ℤ := f m - f (m - n)\nh_fn_dvd_d : f n ∣ d\nh_d_lt_fn : d < f n\nh_neg_d_lt_fn : -d < f n\nh_d_eq_zero : d = 0\nh₁ : f m = f (m - n)\n⊢ f m ∣ f n"
},
{
"line": "focus\n (refine\n no_implicit_lambda%\n (have h₁ : f m = f (m - n) := sub_eq_zero.mp h_d_eq_zero;\n ?_);\n rotate_right)",
"before_state": "case inl\nf : ℤ → ℤ\nhpos : ∀ (n : ℤ), 0 < f n\nhdvd : ∀ (m n : ℤ), f (m - n) ∣ f m - f n\nm n : ℤ\nh_fm_le_fn : f m ≤ f n\nh_fm_lt_fn : f m < f n\nd : ℤ := f m - f (m - n)\nh_fn_dvd_d : f n ∣ d\nh_d_lt_fn : d < f n\nh_neg_d_lt_fn : -d < f n\nh_d_eq_zero : d = 0\n⊢ f m ∣ f n",
"after_state": "case inl\nf : ℤ → ℤ\nhpos : ∀ (n : ℤ), 0 < f n\nhdvd : ∀ (m n : ℤ), f (m - n) ∣ f m - f n\nm n : ℤ\nh_fm_le_fn : f m ≤ f n\nh_fm_lt_fn : f m < f n\nd : ℤ := f m - f (m - n)\nh_fn_dvd_d : f n ∣ d\nh_d_lt_fn : d < f n\nh_neg_d_lt_fn : -d < f n\nh_d_eq_zero : d = 0\nh₁ : f m = f (m - n)\n⊢ f m ∣ f n"
},
{
"line": "refine\n no_implicit_lambda%\n (have h₁ : f m = f (m - n) := sub_eq_zero.mp h_d_eq_zero;\n ?_)",
"before_state": "case inl\nf : ℤ → ℤ\nhpos : ∀ (n : ℤ), 0 < f n\nhdvd : ∀ (m n : ℤ), f (m - n) ∣ f m - f n\nm n : ℤ\nh_fm_le_fn : f m ≤ f n\nh_fm_lt_fn : f m < f n\nd : ℤ := f m - f (m - n)\nh_fn_dvd_d : f n ∣ d\nh_d_lt_fn : d < f n\nh_neg_d_lt_fn : -d < f n\nh_d_eq_zero : d = 0\n⊢ f m ∣ f n",
"after_state": "case inl\nf : ℤ → ℤ\nhpos : ∀ (n : ℤ), 0 < f n\nhdvd : ∀ (m n : ℤ), f (m - n) ∣ f m - f n\nm n : ℤ\nh_fm_le_fn : f m ≤ f n\nh_fm_lt_fn : f m < f n\nd : ℤ := f m - f (m - n)\nh_fn_dvd_d : f n ∣ d\nh_d_lt_fn : d < f n\nh_neg_d_lt_fn : -d < f n\nh_d_eq_zero : d = 0\nh₁ : f m = f (m - n)\n⊢ f m ∣ f n"
},
{
"line": "rotate_right",
"before_state": "case inl\nf : ℤ → ℤ\nhpos : ∀ (n : ℤ), 0 < f n\nhdvd : ∀ (m n : ℤ), f (m - n) ∣ f m - f n\nm n : ℤ\nh_fm_le_fn : f m ≤ f n\nh_fm_lt_fn : f m < f n\nd : ℤ := f m - f (m - n)\nh_fn_dvd_d : f n ∣ d\nh_d_lt_fn : d < f n\nh_neg_d_lt_fn : -d < f n\nh_d_eq_zero : d = 0\nh₁ : f m = f (m - n)\n⊢ f m ∣ f n",
"after_state": "case inl\nf : ℤ → ℤ\nhpos : ∀ (n : ℤ), 0 < f n\nhdvd : ∀ (m n : ℤ), f (m - n) ∣ f m - f n\nm n : ℤ\nh_fm_le_fn : f m ≤ f n\nh_fm_lt_fn : f m < f n\nd : ℤ := f m - f (m - n)\nh_fn_dvd_d : f n ∣ d\nh_d_lt_fn : d < f n\nh_neg_d_lt_fn : -d < f n\nh_d_eq_zero : d = 0\nh₁ : f m = f (m - n)\n⊢ f m ∣ f n"
},
{
"line": "have h₂ : f (m - n) ∣ f m - f n := hdvd m n",
"before_state": "case inl\nf : ℤ → ℤ\nhpos : ∀ (n : ℤ), 0 < f n\nhdvd : ∀ (m n : ℤ), f (m - n) ∣ f m - f n\nm n : ℤ\nh_fm_le_fn : f m ≤ f n\nh_fm_lt_fn : f m < f n\nd : ℤ := f m - f (m - n)\nh_fn_dvd_d : f n ∣ d\nh_d_lt_fn : d < f n\nh_neg_d_lt_fn : -d < f n\nh_d_eq_zero : d = 0\nh₁ : f m = f (m - n)\n⊢ f m ∣ f n",
"after_state": "case inl\nf : ℤ → ℤ\nhpos : ∀ (n : ℤ), 0 < f n\nhdvd : ∀ (m n : ℤ), f (m - n) ∣ f m - f n\nm n : ℤ\nh_fm_le_fn : f m ≤ f n\nh_fm_lt_fn : f m < f n\nd : ℤ := f m - f (m - n)\nh_fn_dvd_d : f n ∣ d\nh_d_lt_fn : d < f n\nh_neg_d_lt_fn : -d < f n\nh_d_eq_zero : d = 0\nh₁ : f m = f (m - n)\nh₂ : f (m - n) ∣ f m - f n\n⊢ f m ∣ f n"
},
{
"line": "refine_lift\n have h₂ : f (m - n) ∣ f m - f n := hdvd m n;\n ?_",
"before_state": "case inl\nf : ℤ → ℤ\nhpos : ∀ (n : ℤ), 0 < f n\nhdvd : ∀ (m n : ℤ), f (m - n) ∣ f m - f n\nm n : ℤ\nh_fm_le_fn : f m ≤ f n\nh_fm_lt_fn : f m < f n\nd : ℤ := f m - f (m - n)\nh_fn_dvd_d : f n ∣ d\nh_d_lt_fn : d < f n\nh_neg_d_lt_fn : -d < f n\nh_d_eq_zero : d = 0\nh₁ : f m = f (m - n)\n⊢ f m ∣ f n",
"after_state": "case inl\nf : ℤ → ℤ\nhpos : ∀ (n : ℤ), 0 < f n\nhdvd : ∀ (m n : ℤ), f (m - n) ∣ f m - f n\nm n : ℤ\nh_fm_le_fn : f m ≤ f n\nh_fm_lt_fn : f m < f n\nd : ℤ := f m - f (m - n)\nh_fn_dvd_d : f n ∣ d\nh_d_lt_fn : d < f n\nh_neg_d_lt_fn : -d < f n\nh_d_eq_zero : d = 0\nh₁ : f m = f (m - n)\nh₂ : f (m - n) ∣ f m - f n\n⊢ f m ∣ f n"
},
{
"line": "focus\n (refine\n no_implicit_lambda%\n (have h₂ : f (m - n) ∣ f m - f n := hdvd m n;\n ?_);\n rotate_right)",
"before_state": "case inl\nf : ℤ → ℤ\nhpos : ∀ (n : ℤ), 0 < f n\nhdvd : ∀ (m n : ℤ), f (m - n) ∣ f m - f n\nm n : ℤ\nh_fm_le_fn : f m ≤ f n\nh_fm_lt_fn : f m < f n\nd : ℤ := f m - f (m - n)\nh_fn_dvd_d : f n ∣ d\nh_d_lt_fn : d < f n\nh_neg_d_lt_fn : -d < f n\nh_d_eq_zero : d = 0\nh₁ : f m = f (m - n)\n⊢ f m ∣ f n",
"after_state": "case inl\nf : ℤ → ℤ\nhpos : ∀ (n : ℤ), 0 < f n\nhdvd : ∀ (m n : ℤ), f (m - n) ∣ f m - f n\nm n : ℤ\nh_fm_le_fn : f m ≤ f n\nh_fm_lt_fn : f m < f n\nd : ℤ := f m - f (m - n)\nh_fn_dvd_d : f n ∣ d\nh_d_lt_fn : d < f n\nh_neg_d_lt_fn : -d < f n\nh_d_eq_zero : d = 0\nh₁ : f m = f (m - n)\nh₂ : f (m - n) ∣ f m - f n\n⊢ f m ∣ f n"
},
{
"line": "refine\n no_implicit_lambda%\n (have h₂ : f (m - n) ∣ f m - f n := hdvd m n;\n ?_)",
"before_state": "case inl\nf : ℤ → ℤ\nhpos : ∀ (n : ℤ), 0 < f n\nhdvd : ∀ (m n : ℤ), f (m - n) ∣ f m - f n\nm n : ℤ\nh_fm_le_fn : f m ≤ f n\nh_fm_lt_fn : f m < f n\nd : ℤ := f m - f (m - n)\nh_fn_dvd_d : f n ∣ d\nh_d_lt_fn : d < f n\nh_neg_d_lt_fn : -d < f n\nh_d_eq_zero : d = 0\nh₁ : f m = f (m - n)\n⊢ f m ∣ f n",
"after_state": "case inl\nf : ℤ → ℤ\nhpos : ∀ (n : ℤ), 0 < f n\nhdvd : ∀ (m n : ℤ), f (m - n) ∣ f m - f n\nm n : ℤ\nh_fm_le_fn : f m ≤ f n\nh_fm_lt_fn : f m < f n\nd : ℤ := f m - f (m - n)\nh_fn_dvd_d : f n ∣ d\nh_d_lt_fn : d < f n\nh_neg_d_lt_fn : -d < f n\nh_d_eq_zero : d = 0\nh₁ : f m = f (m - n)\nh₂ : f (m - n) ∣ f m - f n\n⊢ f m ∣ f n"
},
{
"line": "rotate_right",
"before_state": "case inl\nf : ℤ → ℤ\nhpos : ∀ (n : ℤ), 0 < f n\nhdvd : ∀ (m n : ℤ), f (m - n) ∣ f m - f n\nm n : ℤ\nh_fm_le_fn : f m ≤ f n\nh_fm_lt_fn : f m < f n\nd : ℤ := f m - f (m - n)\nh_fn_dvd_d : f n ∣ d\nh_d_lt_fn : d < f n\nh_neg_d_lt_fn : -d < f n\nh_d_eq_zero : d = 0\nh₁ : f m = f (m - n)\nh₂ : f (m - n) ∣ f m - f n\n⊢ f m ∣ f n",
"after_state": "case inl\nf : ℤ → ℤ\nhpos : ∀ (n : ℤ), 0 < f n\nhdvd : ∀ (m n : ℤ), f (m - n) ∣ f m - f n\nm n : ℤ\nh_fm_le_fn : f m ≤ f n\nh_fm_lt_fn : f m < f n\nd : ℤ := f m - f (m - n)\nh_fn_dvd_d : f n ∣ d\nh_d_lt_fn : d < f n\nh_neg_d_lt_fn : -d < f n\nh_d_eq_zero : d = 0\nh₁ : f m = f (m - n)\nh₂ : f (m - n) ∣ f m - f n\n⊢ f m ∣ f n"
},
{
"line": "rw [← h₁] at h₂",
"before_state": "case inl\nf : ℤ → ℤ\nhpos : ∀ (n : ℤ), 0 < f n\nhdvd : ∀ (m n : ℤ), f (m - n) ∣ f m - f n\nm n : ℤ\nh_fm_le_fn : f m ≤ f n\nh_fm_lt_fn : f m < f n\nd : ℤ := f m - f (m - n)\nh_fn_dvd_d : f n ∣ d\nh_d_lt_fn : d < f n\nh_neg_d_lt_fn : -d < f n\nh_d_eq_zero : d = 0\nh₁ : f m = f (m - n)\nh₂ : f (m - n) ∣ f m - f n\n⊢ f m ∣ f n",
"after_state": "case inl\nf : ℤ → ℤ\nhpos : ∀ (n : ℤ), 0 < f n\nhdvd : ∀ (m n : ℤ), f (m - n) ∣ f m - f n\nm n : ℤ\nh_fm_le_fn : f m ≤ f n\nh_fm_lt_fn : f m < f n\nd : ℤ := f m - f (m - n)\nh_fn_dvd_d : f n ∣ d\nh_d_lt_fn : d < f n\nh_neg_d_lt_fn : -d < f n\nh_d_eq_zero : d = 0\nh₁ : f m = f (m - n)\nh₂ : f m ∣ f m - f n\n⊢ f m ∣ f n"
},
{
"line": "rewrite [← h₁] at h₂",
"before_state": "case inl\nf : ℤ → ℤ\nhpos : ∀ (n : ℤ), 0 < f n\nhdvd : ∀ (m n : ℤ), f (m - n) ∣ f m - f n\nm n : ℤ\nh_fm_le_fn : f m ≤ f n\nh_fm_lt_fn : f m < f n\nd : ℤ := f m - f (m - n)\nh_fn_dvd_d : f n ∣ d\nh_d_lt_fn : d < f n\nh_neg_d_lt_fn : -d < f n\nh_d_eq_zero : d = 0\nh₁ : f m = f (m - n)\nh₂ : f (m - n) ∣ f m - f n\n⊢ f m ∣ f n",
"after_state": "case inl\nf : ℤ → ℤ\nhpos : ∀ (n : ℤ), 0 < f n\nhdvd : ∀ (m n : ℤ), f (m - n) ∣ f m - f n\nm n : ℤ\nh_fm_le_fn : f m ≤ f n\nh_fm_lt_fn : f m < f n\nd : ℤ := f m - f (m - n)\nh_fn_dvd_d : f n ∣ d\nh_d_lt_fn : d < f n\nh_neg_d_lt_fn : -d < f n\nh_d_eq_zero : d = 0\nh₁ : f m = f (m - n)\nh₂ : f m ∣ f m - f n\n⊢ f m ∣ f n"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "case inl\nf : ℤ → ℤ\nhpos : ∀ (n : ℤ), 0 < f n\nhdvd : ∀ (m n : ℤ), f (m - n) ∣ f m - f n\nm n : ℤ\nh_fm_le_fn : f m ≤ f n\nh_fm_lt_fn : f m < f n\nd : ℤ := f m - f (m - n)\nh_fn_dvd_d : f n ∣ d\nh_d_lt_fn : d < f n\nh_neg_d_lt_fn : -d < f n\nh_d_eq_zero : d = 0\nh₁ : f m = f (m - n)\nh₂ : f m ∣ f m - f n\n⊢ f m ∣ f n",
"after_state": "case inl\nf : ℤ → ℤ\nhpos : ∀ (n : ℤ), 0 < f n\nhdvd : ∀ (m n : ℤ), f (m - n) ∣ f m - f n\nm n : ℤ\nh_fm_le_fn : f m ≤ f n\nh_fm_lt_fn : f m < f n\nd : ℤ := f m - f (m - n)\nh_fn_dvd_d : f n ∣ d\nh_d_lt_fn : d < f n\nh_neg_d_lt_fn : -d < f n\nh_d_eq_zero : d = 0\nh₁ : f m = f (m - n)\nh₂ : f m ∣ f m - f n\n⊢ f m ∣ f n"
},
{
"line": "try (with_reducible rfl)",
"before_state": "case inl\nf : ℤ → ℤ\nhpos : ∀ (n : ℤ), 0 < f n\nhdvd : ∀ (m n : ℤ), f (m - n) ∣ f m - f n\nm n : ℤ\nh_fm_le_fn : f m ≤ f n\nh_fm_lt_fn : f m < f n\nd : ℤ := f m - f (m - n)\nh_fn_dvd_d : f n ∣ d\nh_d_lt_fn : d < f n\nh_neg_d_lt_fn : -d < f n\nh_d_eq_zero : d = 0\nh₁ : f m = f (m - n)\nh₂ : f m ∣ f m - f n\n⊢ f m ∣ f n",
"after_state": "case inl\nf : ℤ → ℤ\nhpos : ∀ (n : ℤ), 0 < f n\nhdvd : ∀ (m n : ℤ), f (m - n) ∣ f m - f n\nm n : ℤ\nh_fm_le_fn : f m ≤ f n\nh_fm_lt_fn : f m < f n\nd : ℤ := f m - f (m - n)\nh_fn_dvd_d : f n ∣ d\nh_d_lt_fn : d < f n\nh_neg_d_lt_fn : -d < f n\nh_d_eq_zero : d = 0\nh₁ : f m = f (m - n)\nh₂ : f m ∣ f m - f n\n⊢ f m ∣ f n"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "case inl\nf : ℤ → ℤ\nhpos : ∀ (n : ℤ), 0 < f n\nhdvd : ∀ (m n : ℤ), f (m - n) ∣ f m - f n\nm n : ℤ\nh_fm_le_fn : f m ≤ f n\nh_fm_lt_fn : f m < f n\nd : ℤ := f m - f (m - n)\nh_fn_dvd_d : f n ∣ d\nh_d_lt_fn : d < f n\nh_neg_d_lt_fn : -d < f n\nh_d_eq_zero : d = 0\nh₁ : f m = f (m - n)\nh₂ : f m ∣ f m - f n\n⊢ f m ∣ f n",
"after_state": "case inl\nf : ℤ → ℤ\nhpos : ∀ (n : ℤ), 0 < f n\nhdvd : ∀ (m n : ℤ), f (m - n) ∣ f m - f n\nm n : ℤ\nh_fm_le_fn : f m ≤ f n\nh_fm_lt_fn : f m < f n\nd : ℤ := f m - f (m - n)\nh_fn_dvd_d : f n ∣ d\nh_d_lt_fn : d < f n\nh_neg_d_lt_fn : -d < f n\nh_d_eq_zero : d = 0\nh₁ : f m = f (m - n)\nh₂ : f m ∣ f m - f n\n⊢ f m ∣ f n"
},
{
"line": "with_reducible rfl",
"before_state": "case inl\nf : ℤ → ℤ\nhpos : ∀ (n : ℤ), 0 < f n\nhdvd : ∀ (m n : ℤ), f (m - n) ∣ f m - f n\nm n : ℤ\nh_fm_le_fn : f m ≤ f n\nh_fm_lt_fn : f m < f n\nd : ℤ := f m - f (m - n)\nh_fn_dvd_d : f n ∣ d\nh_d_lt_fn : d < f n\nh_neg_d_lt_fn : -d < f n\nh_d_eq_zero : d = 0\nh₁ : f m = f (m - n)\nh₂ : f m ∣ f m - f n\n⊢ f m ∣ f n",
"after_state": "case inl\nf : ℤ → ℤ\nhpos : ∀ (n : ℤ), 0 < f n\nhdvd : ∀ (m n : ℤ), f (m - n) ∣ f m - f n\nm n : ℤ\nh_fm_le_fn : f m ≤ f n\nh_fm_lt_fn : f m < f n\nd : ℤ := f m - f (m - n)\nh_fn_dvd_d : f n ∣ d\nh_d_lt_fn : d < f n\nh_neg_d_lt_fn : -d < f n\nh_d_eq_zero : d = 0\nh₁ : f m = f (m - n)\nh₂ : f m ∣ f m - f n\n⊢ f m ∣ f n"
},
{
"line": "rfl",
"before_state": "case inl\nf : ℤ → ℤ\nhpos : ∀ (n : ℤ), 0 < f n\nhdvd : ∀ (m n : ℤ), f (m - n) ∣ f m - f n\nm n : ℤ\nh_fm_le_fn : f m ≤ f n\nh_fm_lt_fn : f m < f n\nd : ℤ := f m - f (m - n)\nh_fn_dvd_d : f n ∣ d\nh_d_lt_fn : d < f n\nh_neg_d_lt_fn : -d < f n\nh_d_eq_zero : d = 0\nh₁ : f m = f (m - n)\nh₂ : f m ∣ f m - f n\n⊢ f m ∣ f n",
"after_state": "case inl\nf : ℤ → ℤ\nhpos : ∀ (n : ℤ), 0 < f n\nhdvd : ∀ (m n : ℤ), f (m - n) ∣ f m - f n\nm n : ℤ\nh_fm_le_fn : f m ≤ f n\nh_fm_lt_fn : f m < f n\nd : ℤ := f m - f (m - n)\nh_fn_dvd_d : f n ∣ d\nh_d_lt_fn : d < f n\nh_neg_d_lt_fn : -d < f n\nh_d_eq_zero : d = 0\nh₁ : f m = f (m - n)\nh₂ : f m ∣ f m - f n\n⊢ f m ∣ f n"
},
{
"line": "apply_rfl",
"before_state": "case inl\nf : ℤ → ℤ\nhpos : ∀ (n : ℤ), 0 < f n\nhdvd : ∀ (m n : ℤ), f (m - n) ∣ f m - f n\nm n : ℤ\nh_fm_le_fn : f m ≤ f n\nh_fm_lt_fn : f m < f n\nd : ℤ := f m - f (m - n)\nh_fn_dvd_d : f n ∣ d\nh_d_lt_fn : d < f n\nh_neg_d_lt_fn : -d < f n\nh_d_eq_zero : d = 0\nh₁ : f m = f (m - n)\nh₂ : f m ∣ f m - f n\n⊢ f m ∣ f n",
"after_state": "case inl\nf : ℤ → ℤ\nhpos : ∀ (n : ℤ), 0 < f n\nhdvd : ∀ (m n : ℤ), f (m - n) ∣ f m - f n\nm n : ℤ\nh_fm_le_fn : f m ≤ f n\nh_fm_lt_fn : f m < f n\nd : ℤ := f m - f (m - n)\nh_fn_dvd_d : f n ∣ d\nh_d_lt_fn : d < f n\nh_neg_d_lt_fn : -d < f n\nh_d_eq_zero : d = 0\nh₁ : f m = f (m - n)\nh₂ : f m ∣ f m - f n\n⊢ f m ∣ f n"
},
{
"line": "skip",
"before_state": "case inl\nf : ℤ → ℤ\nhpos : ∀ (n : ℤ), 0 < f n\nhdvd : ∀ (m n : ℤ), f (m - n) ∣ f m - f n\nm n : ℤ\nh_fm_le_fn : f m ≤ f n\nh_fm_lt_fn : f m < f n\nd : ℤ := f m - f (m - n)\nh_fn_dvd_d : f n ∣ d\nh_d_lt_fn : d < f n\nh_neg_d_lt_fn : -d < f n\nh_d_eq_zero : d = 0\nh₁ : f m = f (m - n)\nh₂ : f m ∣ f m - f n\n⊢ f m ∣ f n",
"after_state": "case inl\nf : ℤ → ℤ\nhpos : ∀ (n : ℤ), 0 < f n\nhdvd : ∀ (m n : ℤ), f (m - n) ∣ f m - f n\nm n : ℤ\nh_fm_le_fn : f m ≤ f n\nh_fm_lt_fn : f m < f n\nd : ℤ := f m - f (m - n)\nh_fn_dvd_d : f n ∣ d\nh_d_lt_fn : d < f n\nh_neg_d_lt_fn : -d < f n\nh_d_eq_zero : d = 0\nh₁ : f m = f (m - n)\nh₂ : f m ∣ f m - f n\n⊢ f m ∣ f n"
},
{
"line": "exact (dvd_iff_dvd_of_dvd_sub h₂).mp dvd_rfl",
"before_state": "case inl\nf : ℤ → ℤ\nhpos : ∀ (n : ℤ), 0 < f n\nhdvd : ∀ (m n : ℤ), f (m - n) ∣ f m - f n\nm n : ℤ\nh_fm_le_fn : f m ≤ f n\nh_fm_lt_fn : f m < f n\nd : ℤ := f m - f (m - n)\nh_fn_dvd_d : f n ∣ d\nh_d_lt_fn : d < f n\nh_neg_d_lt_fn : -d < f n\nh_d_eq_zero : d = 0\nh₁ : f m = f (m - n)\nh₂ : f m ∣ f m - f n\n⊢ f m ∣ f n",
"after_state": "No Goals!"
},
{
"line": "rw [h_fm_eq_fn]",
"before_state": "case inr\nf : ℤ → ℤ\nhpos : ∀ (n : ℤ), 0 < f n\nhdvd : ∀ (m n : ℤ), f (m - n) ∣ f m - f n\nm n : ℤ\nh_fm_le_fn : f m ≤ f n\nh_fm_eq_fn : f m = f n\n⊢ f m ∣ f n",
"after_state": "No Goals!"
},
{
"line": "rewrite [h_fm_eq_fn]",
"before_state": "case inr\nf : ℤ → ℤ\nhpos : ∀ (n : ℤ), 0 < f n\nhdvd : ∀ (m n : ℤ), f (m - n) ∣ f m - f n\nm n : ℤ\nh_fm_le_fn : f m ≤ f n\nh_fm_eq_fn : f m = f n\n⊢ f m ∣ f n",
"after_state": "case inr\nf : ℤ → ℤ\nhpos : ∀ (n : ℤ), 0 < f n\nhdvd : ∀ (m n : ℤ), f (m - n) ∣ f m - f n\nm n : ℤ\nh_fm_le_fn : f m ≤ f n\nh_fm_eq_fn : f m = f n\n⊢ f n ∣ f n"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "case inr\nf : ℤ → ℤ\nhpos : ∀ (n : ℤ), 0 < f n\nhdvd : ∀ (m n : ℤ), f (m - n) ∣ f m - f n\nm n : ℤ\nh_fm_le_fn : f m ≤ f n\nh_fm_eq_fn : f m = f n\n⊢ f n ∣ f n",
"after_state": "No Goals!"
},
{
"line": "try (with_reducible rfl)",
"before_state": "case inr\nf : ℤ → ℤ\nhpos : ∀ (n : ℤ), 0 < f n\nhdvd : ∀ (m n : ℤ), f (m - n) ∣ f m - f n\nm n : ℤ\nh_fm_le_fn : f m ≤ f n\nh_fm_eq_fn : f m = f n\n⊢ f n ∣ f n",
"after_state": "No Goals!"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "case inr\nf : ℤ → ℤ\nhpos : ∀ (n : ℤ), 0 < f n\nhdvd : ∀ (m n : ℤ), f (m - n) ∣ f m - f n\nm n : ℤ\nh_fm_le_fn : f m ≤ f n\nh_fm_eq_fn : f m = f n\n⊢ f n ∣ f n",
"after_state": "No Goals!"
},
{
"line": "with_reducible rfl",
"before_state": "case inr\nf : ℤ → ℤ\nhpos : ∀ (n : ℤ), 0 < f n\nhdvd : ∀ (m n : ℤ), f (m - n) ∣ f m - f n\nm n : ℤ\nh_fm_le_fn : f m ≤ f n\nh_fm_eq_fn : f m = f n\n⊢ f n ∣ f n",
"after_state": "No Goals!"
},
{
"line": "rfl",
"before_state": "case inr\nf : ℤ → ℤ\nhpos : ∀ (n : ℤ), 0 < f n\nhdvd : ∀ (m n : ℤ), f (m - n) ∣ f m - f n\nm n : ℤ\nh_fm_le_fn : f m ≤ f n\nh_fm_eq_fn : f m = f n\n⊢ f n ∣ f n",
"after_state": "No Goals!"
},
{
"line": "apply_rfl",
"before_state": "case inr\nf : ℤ → ℤ\nhpos : ∀ (n : ℤ), 0 < f n\nhdvd : ∀ (m n : ℤ), f (m - n) ∣ f m - f n\nm n : ℤ\nh_fm_le_fn : f m ≤ f n\nh_fm_eq_fn : f m = f n\n⊢ f n ∣ f n",
"after_state": "No Goals!"
}
] |
theorem imo2020_q2 (a b c d : ℝ) (hd0 : 0 < d) (hdc : d ≤ c) (hcb : c ≤ b) (hba : b ≤ a)
(h1 : a + b + c + d = 1) : (a + 2 * b + 3 * c + 4 * d) * a ^ a * b ^ b * c ^ c * d ^ d < 1 := by
have hp : a ^ a * b ^ b * c ^ c * d ^ d ≤ a * a + b * b + c * c + d * d := by
refine geom_mean_le_arith_mean4_weighted ?_ ?_ ?_ ?_ ?_ ?_ ?_ ?_ h1 <;> linarith
calc
(a + 2 * b + 3 * c + 4 * d) * a ^ a * b ^ b * c ^ c * d ^ d =
(a + 2 * b + 3 * c + 4 * d) * (a ^ a * b ^ b * c ^ c * d ^ d) := by ac_rfl
_ ≤ (a + 2 * b + 3 * c + 4 * d) * (a * a + b * b + c * c + d * d) := by gcongr; linarith
_ = (a + 2 * b + 3 * c + 4 * d) * a ^ 2 + (a + 2 * b + 3 * c + 4 * d) * b ^ 2
+ (a + 2 * b + 3 * c + 4 * d) * c ^ 2 + (a + 2 * b + 3 * c + 4 * d) * d ^ 2 := by ring
_ ≤ (a + 3 * b + 3 * c + 3 * d) * a ^ 2 + (3 * a + b + 3 * c + 3 * d) * b ^ 2
+ (3 * a + 3 * b + c + 3 * d) * c ^ 2 + (3 * a + 3 * b + 3 * c + d) * d ^ 2 := by
gcongr ?_ * _ + ?_ * _ + ?_ * _ + ?_ * _ <;> linarith
_ < (a + 3 * b + 3 * c + 3 * d) * a ^ 2 + (3 * a + b + 3 * c + 3 * d) * b ^ 2
+ (3 * a + 3 * b + c + 3 * d) * c ^ 2 + (3 * a + 3 * b + 3 * c + d) * d ^ 2
+ (6 * a * b * c + 6 * a * b * d + 6 * a * c * d + 6 * b * c * d) :=
(lt_add_of_pos_right _ (by apply_rules [add_pos, mul_pos, zero_lt_one] <;> linarith))
_ = (a + b + c + d) ^ 3 := by ring
_ = 1 := by simp [h1]
|
/root/DuelModelResearch/mathlib4/Archive/Imo/Imo2020Q2.lean
|
{
"open": [
"Real"
],
"variables": []
}
|
[
{
"line": "have hp : a ^ a * b ^ b * c ^ c * d ^ d ≤ a * a + b * b + c * c + d * d := by\n refine geom_mean_le_arith_mean4_weighted ?_ ?_ ?_ ?_ ?_ ?_ ?_ ?_ h1 <;> linarith",
"before_state": "a b c d : ℝ\nhd0 : 0 < d\nhdc : d ≤ c\nhcb : c ≤ b\nhba : b ≤ a\nh1 : a + b + c + d = 1\n⊢ (a + 2 * b + 3 * c + 4 * d) * a ^ a * b ^ b * c ^ c * d ^ d < 1",
"after_state": "a b c d : ℝ\nhd0 : 0 < d\nhdc : d ≤ c\nhcb : c ≤ b\nhba : b ≤ a\nh1 : a + b + c + d = 1\nhp : a ^ a * b ^ b * c ^ c * d ^ d ≤ a * a + b * b + c * c + d * d\n⊢ (a + 2 * b + 3 * c + 4 * d) * a ^ a * b ^ b * c ^ c * d ^ d < 1"
},
{
"line": "focus\n refine\n no_implicit_lambda%\n (have hp : a ^ a * b ^ b * c ^ c * d ^ d ≤ a * a + b * b + c * c + d * d := ?body✝;\n ?_)\n case body✝ =>\n with_annotate_state\"by\" (refine geom_mean_le_arith_mean4_weighted ?_ ?_ ?_ ?_ ?_ ?_ ?_ ?_ h1 <;> linarith)",
"before_state": "a b c d : ℝ\nhd0 : 0 < d\nhdc : d ≤ c\nhcb : c ≤ b\nhba : b ≤ a\nh1 : a + b + c + d = 1\n⊢ (a + 2 * b + 3 * c + 4 * d) * a ^ a * b ^ b * c ^ c * d ^ d < 1",
"after_state": "a b c d : ℝ\nhd0 : 0 < d\nhdc : d ≤ c\nhcb : c ≤ b\nhba : b ≤ a\nh1 : a + b + c + d = 1\nhp : a ^ a * b ^ b * c ^ c * d ^ d ≤ a * a + b * b + c * c + d * d\n⊢ (a + 2 * b + 3 * c + 4 * d) * a ^ a * b ^ b * c ^ c * d ^ d < 1"
},
{
"line": "refine\n no_implicit_lambda%\n (have hp : a ^ a * b ^ b * c ^ c * d ^ d ≤ a * a + b * b + c * c + d * d := ?body✝;\n ?_)",
"before_state": "a b c d : ℝ\nhd0 : 0 < d\nhdc : d ≤ c\nhcb : c ≤ b\nhba : b ≤ a\nh1 : a + b + c + d = 1\n⊢ (a + 2 * b + 3 * c + 4 * d) * a ^ a * b ^ b * c ^ c * d ^ d < 1",
"after_state": "case body\na b c d : ℝ\nhd0 : 0 < d\nhdc : d ≤ c\nhcb : c ≤ b\nhba : b ≤ a\nh1 : a + b + c + d = 1\n⊢ a ^ a * b ^ b * c ^ c * d ^ d ≤ a * a + b * b + c * c + d * d\n---\na b c d : ℝ\nhd0 : 0 < d\nhdc : d ≤ c\nhcb : c ≤ b\nhba : b ≤ a\nh1 : a + b + c + d = 1\nhp : a ^ a * b ^ b * c ^ c * d ^ d ≤ a * a + b * b + c * c + d * d\n⊢ (a + 2 * b + 3 * c + 4 * d) * a ^ a * b ^ b * c ^ c * d ^ d < 1"
},
{
"line": "case body✝ => with_annotate_state\"by\" (refine geom_mean_le_arith_mean4_weighted ?_ ?_ ?_ ?_ ?_ ?_ ?_ ?_ h1 <;> linarith)",
"before_state": "case body\na b c d : ℝ\nhd0 : 0 < d\nhdc : d ≤ c\nhcb : c ≤ b\nhba : b ≤ a\nh1 : a + b + c + d = 1\n⊢ a ^ a * b ^ b * c ^ c * d ^ d ≤ a * a + b * b + c * c + d * d\n---\na b c d : ℝ\nhd0 : 0 < d\nhdc : d ≤ c\nhcb : c ≤ b\nhba : b ≤ a\nh1 : a + b + c + d = 1\nhp : a ^ a * b ^ b * c ^ c * d ^ d ≤ a * a + b * b + c * c + d * d\n⊢ (a + 2 * b + 3 * c + 4 * d) * a ^ a * b ^ b * c ^ c * d ^ d < 1",
"after_state": "a b c d : ℝ\nhd0 : 0 < d\nhdc : d ≤ c\nhcb : c ≤ b\nhba : b ≤ a\nh1 : a + b + c + d = 1\nhp : a ^ a * b ^ b * c ^ c * d ^ d ≤ a * a + b * b + c * c + d * d\n⊢ (a + 2 * b + 3 * c + 4 * d) * a ^ a * b ^ b * c ^ c * d ^ d < 1"
},
{
"line": "with_annotate_state\"by\" (refine geom_mean_le_arith_mean4_weighted ?_ ?_ ?_ ?_ ?_ ?_ ?_ ?_ h1 <;> linarith)",
"before_state": "a b c d : ℝ\nhd0 : 0 < d\nhdc : d ≤ c\nhcb : c ≤ b\nhba : b ≤ a\nh1 : a + b + c + d = 1\n⊢ a ^ a * b ^ b * c ^ c * d ^ d ≤ a * a + b * b + c * c + d * d",
"after_state": "No Goals!"
},
{
"line": "focus\n refine geom_mean_le_arith_mean4_weighted ?_ ?_ ?_ ?_ ?_ ?_ ?_ ?_ h1\n with_annotate_state\"<;>\" skip\n all_goals linarith",
"before_state": "a b c d : ℝ\nhd0 : 0 < d\nhdc : d ≤ c\nhcb : c ≤ b\nhba : b ≤ a\nh1 : a + b + c + d = 1\n⊢ a ^ a * b ^ b * c ^ c * d ^ d ≤ a * a + b * b + c * c + d * d",
"after_state": "No Goals!"
},
{
"line": "refine geom_mean_le_arith_mean4_weighted ?_ ?_ ?_ ?_ ?_ ?_ ?_ ?_ h1",
"before_state": "a b c d : ℝ\nhd0 : 0 < d\nhdc : d ≤ c\nhcb : c ≤ b\nhba : b ≤ a\nh1 : a + b + c + d = 1\n⊢ a ^ a * b ^ b * c ^ c * d ^ d ≤ a * a + b * b + c * c + d * d",
"after_state": "case refine_1\na b c d : ℝ\nhd0 : 0 < d\nhdc : d ≤ c\nhcb : c ≤ b\nhba : b ≤ a\nh1 : a + b + c + d = 1\n⊢ 0 ≤ a\n---\ncase refine_2\na b c d : ℝ\nhd0 : 0 < d\nhdc : d ≤ c\nhcb : c ≤ b\nhba : b ≤ a\nh1 : a + b + c + d = 1\n⊢ 0 ≤ b\n---\ncase refine_3\na b c d : ℝ\nhd0 : 0 < d\nhdc : d ≤ c\nhcb : c ≤ b\nhba : b ≤ a\nh1 : a + b + c + d = 1\n⊢ 0 ≤ c\n---\ncase refine_4\na b c d : ℝ\nhd0 : 0 < d\nhdc : d ≤ c\nhcb : c ≤ b\nhba : b ≤ a\nh1 : a + b + c + d = 1\n⊢ 0 ≤ d\n---\ncase refine_5\na b c d : ℝ\nhd0 : 0 < d\nhdc : d ≤ c\nhcb : c ≤ b\nhba : b ≤ a\nh1 : a + b + c + d = 1\n⊢ 0 ≤ a\n---\ncase refine_6\na b c d : ℝ\nhd0 : 0 < d\nhdc : d ≤ c\nhcb : c ≤ b\nhba : b ≤ a\nh1 : a + b + c + d = 1\n⊢ 0 ≤ b\n---\ncase refine_7\na b c d : ℝ\nhd0 : 0 < d\nhdc : d ≤ c\nhcb : c ≤ b\nhba : b ≤ a\nh1 : a + b + c + d = 1\n⊢ 0 ≤ c\n---\ncase refine_8\na b c d : ℝ\nhd0 : 0 < d\nhdc : d ≤ c\nhcb : c ≤ b\nhba : b ≤ a\nh1 : a + b + c + d = 1\n⊢ 0 ≤ d"
},
{
"line": "with_annotate_state\"<;>\" skip",
"before_state": "case refine_1\na b c d : ℝ\nhd0 : 0 < d\nhdc : d ≤ c\nhcb : c ≤ b\nhba : b ≤ a\nh1 : a + b + c + d = 1\n⊢ 0 ≤ a\n---\ncase refine_2\na b c d : ℝ\nhd0 : 0 < d\nhdc : d ≤ c\nhcb : c ≤ b\nhba : b ≤ a\nh1 : a + b + c + d = 1\n⊢ 0 ≤ b\n---\ncase refine_3\na b c d : ℝ\nhd0 : 0 < d\nhdc : d ≤ c\nhcb : c ≤ b\nhba : b ≤ a\nh1 : a + b + c + d = 1\n⊢ 0 ≤ c\n---\ncase refine_4\na b c d : ℝ\nhd0 : 0 < d\nhdc : d ≤ c\nhcb : c ≤ b\nhba : b ≤ a\nh1 : a + b + c + d = 1\n⊢ 0 ≤ d\n---\ncase refine_5\na b c d : ℝ\nhd0 : 0 < d\nhdc : d ≤ c\nhcb : c ≤ b\nhba : b ≤ a\nh1 : a + b + c + d = 1\n⊢ 0 ≤ a\n---\ncase refine_6\na b c d : ℝ\nhd0 : 0 < d\nhdc : d ≤ c\nhcb : c ≤ b\nhba : b ≤ a\nh1 : a + b + c + d = 1\n⊢ 0 ≤ b\n---\ncase refine_7\na b c d : ℝ\nhd0 : 0 < d\nhdc : d ≤ c\nhcb : c ≤ b\nhba : b ≤ a\nh1 : a + b + c + d = 1\n⊢ 0 ≤ c\n---\ncase refine_8\na b c d : ℝ\nhd0 : 0 < d\nhdc : d ≤ c\nhcb : c ≤ b\nhba : b ≤ a\nh1 : a + b + c + d = 1\n⊢ 0 ≤ d",
"after_state": "case refine_1\na b c d : ℝ\nhd0 : 0 < d\nhdc : d ≤ c\nhcb : c ≤ b\nhba : b ≤ a\nh1 : a + b + c + d = 1\n⊢ 0 ≤ a\n---\ncase refine_2\na b c d : ℝ\nhd0 : 0 < d\nhdc : d ≤ c\nhcb : c ≤ b\nhba : b ≤ a\nh1 : a + b + c + d = 1\n⊢ 0 ≤ b\n---\ncase refine_3\na b c d : ℝ\nhd0 : 0 < d\nhdc : d ≤ c\nhcb : c ≤ b\nhba : b ≤ a\nh1 : a + b + c + d = 1\n⊢ 0 ≤ c\n---\ncase refine_4\na b c d : ℝ\nhd0 : 0 < d\nhdc : d ≤ c\nhcb : c ≤ b\nhba : b ≤ a\nh1 : a + b + c + d = 1\n⊢ 0 ≤ d\n---\ncase refine_5\na b c d : ℝ\nhd0 : 0 < d\nhdc : d ≤ c\nhcb : c ≤ b\nhba : b ≤ a\nh1 : a + b + c + d = 1\n⊢ 0 ≤ a\n---\ncase refine_6\na b c d : ℝ\nhd0 : 0 < d\nhdc : d ≤ c\nhcb : c ≤ b\nhba : b ≤ a\nh1 : a + b + c + d = 1\n⊢ 0 ≤ b\n---\ncase refine_7\na b c d : ℝ\nhd0 : 0 < d\nhdc : d ≤ c\nhcb : c ≤ b\nhba : b ≤ a\nh1 : a + b + c + d = 1\n⊢ 0 ≤ c\n---\ncase refine_8\na b c d : ℝ\nhd0 : 0 < d\nhdc : d ≤ c\nhcb : c ≤ b\nhba : b ≤ a\nh1 : a + b + c + d = 1\n⊢ 0 ≤ d"
},
{
"line": "skip",
"before_state": "case refine_1\na b c d : ℝ\nhd0 : 0 < d\nhdc : d ≤ c\nhcb : c ≤ b\nhba : b ≤ a\nh1 : a + b + c + d = 1\n⊢ 0 ≤ a\n---\ncase refine_2\na b c d : ℝ\nhd0 : 0 < d\nhdc : d ≤ c\nhcb : c ≤ b\nhba : b ≤ a\nh1 : a + b + c + d = 1\n⊢ 0 ≤ b\n---\ncase refine_3\na b c d : ℝ\nhd0 : 0 < d\nhdc : d ≤ c\nhcb : c ≤ b\nhba : b ≤ a\nh1 : a + b + c + d = 1\n⊢ 0 ≤ c\n---\ncase refine_4\na b c d : ℝ\nhd0 : 0 < d\nhdc : d ≤ c\nhcb : c ≤ b\nhba : b ≤ a\nh1 : a + b + c + d = 1\n⊢ 0 ≤ d\n---\ncase refine_5\na b c d : ℝ\nhd0 : 0 < d\nhdc : d ≤ c\nhcb : c ≤ b\nhba : b ≤ a\nh1 : a + b + c + d = 1\n⊢ 0 ≤ a\n---\ncase refine_6\na b c d : ℝ\nhd0 : 0 < d\nhdc : d ≤ c\nhcb : c ≤ b\nhba : b ≤ a\nh1 : a + b + c + d = 1\n⊢ 0 ≤ b\n---\ncase refine_7\na b c d : ℝ\nhd0 : 0 < d\nhdc : d ≤ c\nhcb : c ≤ b\nhba : b ≤ a\nh1 : a + b + c + d = 1\n⊢ 0 ≤ c\n---\ncase refine_8\na b c d : ℝ\nhd0 : 0 < d\nhdc : d ≤ c\nhcb : c ≤ b\nhba : b ≤ a\nh1 : a + b + c + d = 1\n⊢ 0 ≤ d",
"after_state": "case refine_1\na b c d : ℝ\nhd0 : 0 < d\nhdc : d ≤ c\nhcb : c ≤ b\nhba : b ≤ a\nh1 : a + b + c + d = 1\n⊢ 0 ≤ a\n---\ncase refine_2\na b c d : ℝ\nhd0 : 0 < d\nhdc : d ≤ c\nhcb : c ≤ b\nhba : b ≤ a\nh1 : a + b + c + d = 1\n⊢ 0 ≤ b\n---\ncase refine_3\na b c d : ℝ\nhd0 : 0 < d\nhdc : d ≤ c\nhcb : c ≤ b\nhba : b ≤ a\nh1 : a + b + c + d = 1\n⊢ 0 ≤ c\n---\ncase refine_4\na b c d : ℝ\nhd0 : 0 < d\nhdc : d ≤ c\nhcb : c ≤ b\nhba : b ≤ a\nh1 : a + b + c + d = 1\n⊢ 0 ≤ d\n---\ncase refine_5\na b c d : ℝ\nhd0 : 0 < d\nhdc : d ≤ c\nhcb : c ≤ b\nhba : b ≤ a\nh1 : a + b + c + d = 1\n⊢ 0 ≤ a\n---\ncase refine_6\na b c d : ℝ\nhd0 : 0 < d\nhdc : d ≤ c\nhcb : c ≤ b\nhba : b ≤ a\nh1 : a + b + c + d = 1\n⊢ 0 ≤ b\n---\ncase refine_7\na b c d : ℝ\nhd0 : 0 < d\nhdc : d ≤ c\nhcb : c ≤ b\nhba : b ≤ a\nh1 : a + b + c + d = 1\n⊢ 0 ≤ c\n---\ncase refine_8\na b c d : ℝ\nhd0 : 0 < d\nhdc : d ≤ c\nhcb : c ≤ b\nhba : b ≤ a\nh1 : a + b + c + d = 1\n⊢ 0 ≤ d"
},
{
"line": "all_goals linarith",
"before_state": "case refine_1\na b c d : ℝ\nhd0 : 0 < d\nhdc : d ≤ c\nhcb : c ≤ b\nhba : b ≤ a\nh1 : a + b + c + d = 1\n⊢ 0 ≤ a\n---\ncase refine_2\na b c d : ℝ\nhd0 : 0 < d\nhdc : d ≤ c\nhcb : c ≤ b\nhba : b ≤ a\nh1 : a + b + c + d = 1\n⊢ 0 ≤ b\n---\ncase refine_3\na b c d : ℝ\nhd0 : 0 < d\nhdc : d ≤ c\nhcb : c ≤ b\nhba : b ≤ a\nh1 : a + b + c + d = 1\n⊢ 0 ≤ c\n---\ncase refine_4\na b c d : ℝ\nhd0 : 0 < d\nhdc : d ≤ c\nhcb : c ≤ b\nhba : b ≤ a\nh1 : a + b + c + d = 1\n⊢ 0 ≤ d\n---\ncase refine_5\na b c d : ℝ\nhd0 : 0 < d\nhdc : d ≤ c\nhcb : c ≤ b\nhba : b ≤ a\nh1 : a + b + c + d = 1\n⊢ 0 ≤ a\n---\ncase refine_6\na b c d : ℝ\nhd0 : 0 < d\nhdc : d ≤ c\nhcb : c ≤ b\nhba : b ≤ a\nh1 : a + b + c + d = 1\n⊢ 0 ≤ b\n---\ncase refine_7\na b c d : ℝ\nhd0 : 0 < d\nhdc : d ≤ c\nhcb : c ≤ b\nhba : b ≤ a\nh1 : a + b + c + d = 1\n⊢ 0 ≤ c\n---\ncase refine_8\na b c d : ℝ\nhd0 : 0 < d\nhdc : d ≤ c\nhcb : c ≤ b\nhba : b ≤ a\nh1 : a + b + c + d = 1\n⊢ 0 ≤ d",
"after_state": "No Goals!"
},
{
"line": "linarith",
"before_state": "case refine_1\na b c d : ℝ\nhd0 : 0 < d\nhdc : d ≤ c\nhcb : c ≤ b\nhba : b ≤ a\nh1 : a + b + c + d = 1\n⊢ 0 ≤ a",
"after_state": "No Goals!"
},
{
"line": "ring1",
"before_state": "a b c d : ℝ\nhd0 : 0 < d\nhdc : d ≤ c\nhcb : c ≤ b\nhba : b ≤ a\nh1 : a + b + c + d = 1\na✝ : 0 > a\n⊢ 0 - d + (d - c) + (c - b) + (b - a) + (a - 0) = 0",
"after_state": "No Goals!"
},
{
"line": "linarith",
"before_state": "case refine_2\na b c d : ℝ\nhd0 : 0 < d\nhdc : d ≤ c\nhcb : c ≤ b\nhba : b ≤ a\nh1 : a + b + c + d = 1\n⊢ 0 ≤ b",
"after_state": "No Goals!"
},
{
"line": "ring1",
"before_state": "a b c d : ℝ\nhd0 : 0 < d\nhdc : d ≤ c\nhcb : c ≤ b\nhba : b ≤ a\nh1 : a + b + c + d = 1\na✝ : 0 > b\n⊢ 0 - d + (d - c) + (c - b) + (b - 0) = 0",
"after_state": "No Goals!"
},
{
"line": "linarith",
"before_state": "case refine_3\na b c d : ℝ\nhd0 : 0 < d\nhdc : d ≤ c\nhcb : c ≤ b\nhba : b ≤ a\nh1 : a + b + c + d = 1\n⊢ 0 ≤ c",
"after_state": "No Goals!"
},
{
"line": "ring1",
"before_state": "a b c d : ℝ\nhd0 : 0 < d\nhdc : d ≤ c\nhcb : c ≤ b\nhba : b ≤ a\nh1 : a + b + c + d = 1\na✝ : 0 > c\n⊢ 0 - d + (d - c) + (c - 0) = 0",
"after_state": "No Goals!"
},
{
"line": "linarith",
"before_state": "case refine_4\na b c d : ℝ\nhd0 : 0 < d\nhdc : d ≤ c\nhcb : c ≤ b\nhba : b ≤ a\nh1 : a + b + c + d = 1\n⊢ 0 ≤ d",
"after_state": "No Goals!"
},
{
"line": "ring1",
"before_state": "a b c d : ℝ\nhd0 : 0 < d\nhdc : d ≤ c\nhcb : c ≤ b\nhba : b ≤ a\nh1 : a + b + c + d = 1\na✝ : 0 > d\n⊢ 0 - d + (d - 0) = 0",
"after_state": "No Goals!"
},
{
"line": "linarith",
"before_state": "case refine_5\na b c d : ℝ\nhd0 : 0 < d\nhdc : d ≤ c\nhcb : c ≤ b\nhba : b ≤ a\nh1 : a + b + c + d = 1\n⊢ 0 ≤ a",
"after_state": "No Goals!"
},
{
"line": "ring1",
"before_state": "a b c d : ℝ\nhd0 : 0 < d\nhdc : d ≤ c\nhcb : c ≤ b\nhba : b ≤ a\nh1 : a + b + c + d = 1\na✝ : 0 > a\n⊢ 0 - d + (d - c) + (c - b) + (b - a) + (a - 0) = 0",
"after_state": "No Goals!"
},
{
"line": "linarith",
"before_state": "case refine_6\na b c d : ℝ\nhd0 : 0 < d\nhdc : d ≤ c\nhcb : c ≤ b\nhba : b ≤ a\nh1 : a + b + c + d = 1\n⊢ 0 ≤ b",
"after_state": "No Goals!"
},
{
"line": "ring1",
"before_state": "a b c d : ℝ\nhd0 : 0 < d\nhdc : d ≤ c\nhcb : c ≤ b\nhba : b ≤ a\nh1 : a + b + c + d = 1\na✝ : 0 > b\n⊢ 0 - d + (d - c) + (c - b) + (b - 0) = 0",
"after_state": "No Goals!"
},
{
"line": "linarith",
"before_state": "case refine_7\na b c d : ℝ\nhd0 : 0 < d\nhdc : d ≤ c\nhcb : c ≤ b\nhba : b ≤ a\nh1 : a + b + c + d = 1\n⊢ 0 ≤ c",
"after_state": "No Goals!"
},
{
"line": "ring1",
"before_state": "a b c d : ℝ\nhd0 : 0 < d\nhdc : d ≤ c\nhcb : c ≤ b\nhba : b ≤ a\nh1 : a + b + c + d = 1\na✝ : 0 > c\n⊢ 0 - d + (d - c) + (c - 0) = 0",
"after_state": "No Goals!"
},
{
"line": "linarith",
"before_state": "case refine_8\na b c d : ℝ\nhd0 : 0 < d\nhdc : d ≤ c\nhcb : c ≤ b\nhba : b ≤ a\nh1 : a + b + c + d = 1\n⊢ 0 ≤ d",
"after_state": "No Goals!"
},
{
"line": "ring1",
"before_state": "a b c d : ℝ\nhd0 : 0 < d\nhdc : d ≤ c\nhcb : c ≤ b\nhba : b ≤ a\nh1 : a + b + c + d = 1\na✝ : 0 > d\n⊢ 0 - d + (d - 0) = 0",
"after_state": "No Goals!"
},
{
"line": "calc\n (a + 2 * b + 3 * c + 4 * d) * a ^ a * b ^ b * c ^ c * d ^ d =\n (a + 2 * b + 3 * c + 4 * d) * (a ^ a * b ^ b * c ^ c * d ^ d) :=\n by ac_rfl\n _ ≤ (a + 2 * b + 3 * c + 4 * d) * (a * a + b * b + c * c + d * d) := by gcongr; linarith\n _ =\n (a + 2 * b + 3 * c + 4 * d) * a ^ 2 + (a + 2 * b + 3 * c + 4 * d) * b ^ 2 + (a + 2 * b + 3 * c + 4 * d) * c ^ 2 +\n (a + 2 * b + 3 * c + 4 * d) * d ^ 2 :=\n by ring\n _ ≤\n (a + 3 * b + 3 * c + 3 * d) * a ^ 2 + (3 * a + b + 3 * c + 3 * d) * b ^ 2 + (3 * a + 3 * b + c + 3 * d) * c ^ 2 +\n (3 * a + 3 * b + 3 * c + d) * d ^ 2 :=\n by gcongr ?_ * _ + ?_ * _ + ?_ * _ + ?_ * _ <;> linarith\n _ <\n (a + 3 * b + 3 * c + 3 * d) * a ^ 2 + (3 * a + b + 3 * c + 3 * d) * b ^ 2 + (3 * a + 3 * b + c + 3 * d) * c ^ 2 +\n (3 * a + 3 * b + 3 * c + d) * d ^ 2 +\n (6 * a * b * c + 6 * a * b * d + 6 * a * c * d + 6 * b * c * d) :=\n (lt_add_of_pos_right _ (by apply_rules [add_pos, mul_pos, zero_lt_one] <;> linarith))\n _ = (a + b + c + d) ^ 3 := by ring\n _ = 1 := by simp [h1]",
"before_state": "a b c d : ℝ\nhd0 : 0 < d\nhdc : d ≤ c\nhcb : c ≤ b\nhba : b ≤ a\nh1 : a + b + c + d = 1\nhp : a ^ a * b ^ b * c ^ c * d ^ d ≤ a * a + b * b + c * c + d * d\n⊢ (a + 2 * b + 3 * c + 4 * d) * a ^ a * b ^ b * c ^ c * d ^ d < 1",
"after_state": "No Goals!"
},
{
"line": "ac_rfl",
"before_state": "a b c d : ℝ\nhd0 : 0 < d\nhdc : d ≤ c\nhcb : c ≤ b\nhba : b ≤ a\nh1 : a + b + c + d = 1\nhp : a ^ a * b ^ b * c ^ c * d ^ d ≤ a * a + b * b + c * c + d * d\n⊢ (a + 2 * b + 3 * c + 4 * d) * a ^ a * b ^ b * c ^ c * d ^ d =\n (a + 2 * b + 3 * c + 4 * d) * (a ^ a * b ^ b * c ^ c * d ^ d)",
"after_state": "No Goals!"
},
{
"line": "gcongr",
"before_state": "a b c d : ℝ\nhd0 : 0 < d\nhdc : d ≤ c\nhcb : c ≤ b\nhba : b ≤ a\nh1 : a + b + c + d = 1\nhp : a ^ a * b ^ b * c ^ c * d ^ d ≤ a * a + b * b + c * c + d * d\n⊢ (a + 2 * b + 3 * c + 4 * d) * (a ^ a * b ^ b * c ^ c * d ^ d) ≤\n (a + 2 * b + 3 * c + 4 * d) * (a * a + b * b + c * c + d * d)",
"after_state": "case a0\na b c d : ℝ\nhd0 : 0 < d\nhdc : d ≤ c\nhcb : c ≤ b\nhba : b ≤ a\nh1 : a + b + c + d = 1\nhp : a ^ a * b ^ b * c ^ c * d ^ d ≤ a * a + b * b + c * c + d * d\n⊢ 0 ≤ a + 2 * b + 3 * c + 4 * d"
},
{
"line": "linarith",
"before_state": "case a0\na b c d : ℝ\nhd0 : 0 < d\nhdc : d ≤ c\nhcb : c ≤ b\nhba : b ≤ a\nh1 : a + b + c + d = 1\nhp : a ^ a * b ^ b * c ^ c * d ^ d ≤ a * a + b * b + c * c + d * d\n⊢ 0 ≤ a + 2 * b + 3 * c + 4 * d",
"after_state": "No Goals!"
},
{
"line": "ring1",
"before_state": "a b c d : ℝ\nhd0 : 0 < d\nhdc : d ≤ c\nhcb : c ≤ b\nhba : b ≤ a\nh1 : a + b + c + d = 1\nhp : a ^ a * b ^ b * c ^ c * d ^ d ≤ a * a + b * b + c * c + d * d\na✝ : 0 > a + 2 * b + 3 * c + 4 * d\n⊢ 10 * (0 - d) + 6 * (d - c) + 3 * (c - b) + (b - a) + (a + 2 * b + 3 * c + 4 * d - 0) = 0",
"after_state": "No Goals!"
},
{
"line": "norm_num",
"before_state": "a b c d : ℝ\nhd0 : 0 < d\nhdc : d ≤ c\nhcb : c ≤ b\nhba : b ≤ a\nh1 : a + b + c + d = 1\nhp : a ^ a * b ^ b * c ^ c * d ^ d ≤ a * a + b * b + c * c + d * d\na✝ : 0 > a + 2 * b + 3 * c + 4 * d\n⊢ 10 > 0",
"after_state": "No Goals!"
},
{
"line": "norm_num",
"before_state": "a b c d : ℝ\nhd0 : 0 < d\nhdc : d ≤ c\nhcb : c ≤ b\nhba : b ≤ a\nh1 : a + b + c + d = 1\nhp : a ^ a * b ^ b * c ^ c * d ^ d ≤ a * a + b * b + c * c + d * d\na✝ : 0 > a + 2 * b + 3 * c + 4 * d\n⊢ 6 > 0",
"after_state": "No Goals!"
},
{
"line": "norm_num",
"before_state": "a b c d : ℝ\nhd0 : 0 < d\nhdc : d ≤ c\nhcb : c ≤ b\nhba : b ≤ a\nh1 : a + b + c + d = 1\nhp : a ^ a * b ^ b * c ^ c * d ^ d ≤ a * a + b * b + c * c + d * d\na✝ : 0 > a + 2 * b + 3 * c + 4 * d\n⊢ 3 > 0",
"after_state": "No Goals!"
},
{
"line": "ring",
"before_state": "a b c d : ℝ\nhd0 : 0 < d\nhdc : d ≤ c\nhcb : c ≤ b\nhba : b ≤ a\nh1 : a + b + c + d = 1\nhp : a ^ a * b ^ b * c ^ c * d ^ d ≤ a * a + b * b + c * c + d * d\n⊢ (a + 2 * b + 3 * c + 4 * d) * (a * a + b * b + c * c + d * d) =\n (a + 2 * b + 3 * c + 4 * d) * a ^ 2 + (a + 2 * b + 3 * c + 4 * d) * b ^ 2 + (a + 2 * b + 3 * c + 4 * d) * c ^ 2 +\n (a + 2 * b + 3 * c + 4 * d) * d ^ 2",
"after_state": "No Goals!"
},
{
"line": "first\n| ring1\n|\n try_this ring_nf\"\\n\\nThe `ring` tactic failed to close the goal. Use `ring_nf` to obtain a normal form.\n \\nNote that `ring` works primarily in *commutative* rings. \\\n If you have a noncommutative ring, abelian group or module, consider using \\\n `noncomm_ring`, `abel` or `module` instead.\"",
"before_state": "a b c d : ℝ\nhd0 : 0 < d\nhdc : d ≤ c\nhcb : c ≤ b\nhba : b ≤ a\nh1 : a + b + c + d = 1\nhp : a ^ a * b ^ b * c ^ c * d ^ d ≤ a * a + b * b + c * c + d * d\n⊢ (a + 2 * b + 3 * c + 4 * d) * (a * a + b * b + c * c + d * d) =\n (a + 2 * b + 3 * c + 4 * d) * a ^ 2 + (a + 2 * b + 3 * c + 4 * d) * b ^ 2 + (a + 2 * b + 3 * c + 4 * d) * c ^ 2 +\n (a + 2 * b + 3 * c + 4 * d) * d ^ 2",
"after_state": "No Goals!"
},
{
"line": "ring1",
"before_state": "a b c d : ℝ\nhd0 : 0 < d\nhdc : d ≤ c\nhcb : c ≤ b\nhba : b ≤ a\nh1 : a + b + c + d = 1\nhp : a ^ a * b ^ b * c ^ c * d ^ d ≤ a * a + b * b + c * c + d * d\n⊢ (a + 2 * b + 3 * c + 4 * d) * (a * a + b * b + c * c + d * d) =\n (a + 2 * b + 3 * c + 4 * d) * a ^ 2 + (a + 2 * b + 3 * c + 4 * d) * b ^ 2 + (a + 2 * b + 3 * c + 4 * d) * c ^ 2 +\n (a + 2 * b + 3 * c + 4 * d) * d ^ 2",
"after_state": "No Goals!"
},
{
"line": "focus\n gcongr ?_ * _ + ?_ * _ + ?_ * _ + ?_ * _\n with_annotate_state\"<;>\" skip\n all_goals linarith",
"before_state": "a b c d : ℝ\nhd0 : 0 < d\nhdc : d ≤ c\nhcb : c ≤ b\nhba : b ≤ a\nh1 : a + b + c + d = 1\nhp : a ^ a * b ^ b * c ^ c * d ^ d ≤ a * a + b * b + c * c + d * d\n⊢ (a + 2 * b + 3 * c + 4 * d) * a ^ 2 + (a + 2 * b + 3 * c + 4 * d) * b ^ 2 + (a + 2 * b + 3 * c + 4 * d) * c ^ 2 +\n (a + 2 * b + 3 * c + 4 * d) * d ^ 2 ≤\n (a + 3 * b + 3 * c + 3 * d) * a ^ 2 + (3 * a + b + 3 * c + 3 * d) * b ^ 2 + (3 * a + 3 * b + c + 3 * d) * c ^ 2 +\n (3 * a + 3 * b + 3 * c + d) * d ^ 2",
"after_state": "No Goals!"
},
{
"line": "gcongr ?_ * _ + ?_ * _ + ?_ * _ + ?_ * _",
"before_state": "a b c d : ℝ\nhd0 : 0 < d\nhdc : d ≤ c\nhcb : c ≤ b\nhba : b ≤ a\nh1 : a + b + c + d = 1\nhp : a ^ a * b ^ b * c ^ c * d ^ d ≤ a * a + b * b + c * c + d * d\n⊢ (a + 2 * b + 3 * c + 4 * d) * a ^ 2 + (a + 2 * b + 3 * c + 4 * d) * b ^ 2 + (a + 2 * b + 3 * c + 4 * d) * c ^ 2 +\n (a + 2 * b + 3 * c + 4 * d) * d ^ 2 ≤\n (a + 3 * b + 3 * c + 3 * d) * a ^ 2 + (3 * a + b + 3 * c + 3 * d) * b ^ 2 + (3 * a + 3 * b + c + 3 * d) * c ^ 2 +\n (3 * a + 3 * b + 3 * c + d) * d ^ 2",
"after_state": "case h₁.h₁.h₁.h\na b c d : ℝ\nhd0 : 0 < d\nhdc : d ≤ c\nhcb : c ≤ b\nhba : b ≤ a\nh1 : a + b + c + d = 1\nhp : a ^ a * b ^ b * c ^ c * d ^ d ≤ a * a + b * b + c * c + d * d\n⊢ a + 2 * b + 3 * c + 4 * d ≤ a + 3 * b + 3 * c + 3 * d\n---\ncase h₁.h₁.h₂.h\na b c d : ℝ\nhd0 : 0 < d\nhdc : d ≤ c\nhcb : c ≤ b\nhba : b ≤ a\nh1 : a + b + c + d = 1\nhp : a ^ a * b ^ b * c ^ c * d ^ d ≤ a * a + b * b + c * c + d * d\n⊢ a + 2 * b + 3 * c + 4 * d ≤ 3 * a + b + 3 * c + 3 * d\n---\ncase h₁.h₂.h\na b c d : ℝ\nhd0 : 0 < d\nhdc : d ≤ c\nhcb : c ≤ b\nhba : b ≤ a\nh1 : a + b + c + d = 1\nhp : a ^ a * b ^ b * c ^ c * d ^ d ≤ a * a + b * b + c * c + d * d\n⊢ a + 2 * b + 3 * c + 4 * d ≤ 3 * a + 3 * b + c + 3 * d\n---\ncase h₂.h\na b c d : ℝ\nhd0 : 0 < d\nhdc : d ≤ c\nhcb : c ≤ b\nhba : b ≤ a\nh1 : a + b + c + d = 1\nhp : a ^ a * b ^ b * c ^ c * d ^ d ≤ a * a + b * b + c * c + d * d\n⊢ a + 2 * b + 3 * c + 4 * d ≤ 3 * a + 3 * b + 3 * c + d"
},
{
"line": "gcongr_discharger",
"before_state": "case h₁.h₁.h₁.a0\na b c d : ℝ\nhd0 : 0 < d\nhdc : d ≤ c\nhcb : c ≤ b\nhba : b ≤ a\nh1 : a + b + c + d = 1\nhp : a ^ a * b ^ b * c ^ c * d ^ d ≤ a * a + b * b + c * c + d * d\n⊢ 0 ≤ a ^ 2",
"after_state": "No Goals!"
},
{
"line": "positivity",
"before_state": "case h₁.h₁.h₁.a0\na b c d : ℝ\nhd0 : 0 < d\nhdc : d ≤ c\nhcb : c ≤ b\nhba : b ≤ a\nh1 : a + b + c + d = 1\nhp : a ^ a * b ^ b * c ^ c * d ^ d ≤ a * a + b * b + c * c + d * d\n⊢ 0 ≤ a ^ 2",
"after_state": "No Goals!"
},
{
"line": "gcongr_discharger",
"before_state": "case h₁.h₁.h₂.a0\na b c d : ℝ\nhd0 : 0 < d\nhdc : d ≤ c\nhcb : c ≤ b\nhba : b ≤ a\nh1 : a + b + c + d = 1\nhp : a ^ a * b ^ b * c ^ c * d ^ d ≤ a * a + b * b + c * c + d * d\n⊢ 0 ≤ b ^ 2",
"after_state": "No Goals!"
},
{
"line": "positivity",
"before_state": "case h₁.h₁.h₂.a0\na b c d : ℝ\nhd0 : 0 < d\nhdc : d ≤ c\nhcb : c ≤ b\nhba : b ≤ a\nh1 : a + b + c + d = 1\nhp : a ^ a * b ^ b * c ^ c * d ^ d ≤ a * a + b * b + c * c + d * d\n⊢ 0 ≤ b ^ 2",
"after_state": "No Goals!"
},
{
"line": "gcongr_discharger",
"before_state": "case h₁.h₂.a0\na b c d : ℝ\nhd0 : 0 < d\nhdc : d ≤ c\nhcb : c ≤ b\nhba : b ≤ a\nh1 : a + b + c + d = 1\nhp : a ^ a * b ^ b * c ^ c * d ^ d ≤ a * a + b * b + c * c + d * d\n⊢ 0 ≤ c ^ 2",
"after_state": "No Goals!"
},
{
"line": "positivity",
"before_state": "case h₁.h₂.a0\na b c d : ℝ\nhd0 : 0 < d\nhdc : d ≤ c\nhcb : c ≤ b\nhba : b ≤ a\nh1 : a + b + c + d = 1\nhp : a ^ a * b ^ b * c ^ c * d ^ d ≤ a * a + b * b + c * c + d * d\n⊢ 0 ≤ c ^ 2",
"after_state": "No Goals!"
},
{
"line": "gcongr_discharger",
"before_state": "case h₂.a0\na b c d : ℝ\nhd0 : 0 < d\nhdc : d ≤ c\nhcb : c ≤ b\nhba : b ≤ a\nh1 : a + b + c + d = 1\nhp : a ^ a * b ^ b * c ^ c * d ^ d ≤ a * a + b * b + c * c + d * d\n⊢ 0 ≤ d ^ 2",
"after_state": "No Goals!"
},
{
"line": "positivity",
"before_state": "case h₂.a0\na b c d : ℝ\nhd0 : 0 < d\nhdc : d ≤ c\nhcb : c ≤ b\nhba : b ≤ a\nh1 : a + b + c + d = 1\nhp : a ^ a * b ^ b * c ^ c * d ^ d ≤ a * a + b * b + c * c + d * d\n⊢ 0 ≤ d ^ 2",
"after_state": "No Goals!"
},
{
"line": "with_annotate_state\"<;>\" skip",
"before_state": "case h₁.h₁.h₁.h\na b c d : ℝ\nhd0 : 0 < d\nhdc : d ≤ c\nhcb : c ≤ b\nhba : b ≤ a\nh1 : a + b + c + d = 1\nhp : a ^ a * b ^ b * c ^ c * d ^ d ≤ a * a + b * b + c * c + d * d\n⊢ a + 2 * b + 3 * c + 4 * d ≤ a + 3 * b + 3 * c + 3 * d\n---\ncase h₁.h₁.h₂.h\na b c d : ℝ\nhd0 : 0 < d\nhdc : d ≤ c\nhcb : c ≤ b\nhba : b ≤ a\nh1 : a + b + c + d = 1\nhp : a ^ a * b ^ b * c ^ c * d ^ d ≤ a * a + b * b + c * c + d * d\n⊢ a + 2 * b + 3 * c + 4 * d ≤ 3 * a + b + 3 * c + 3 * d\n---\ncase h₁.h₂.h\na b c d : ℝ\nhd0 : 0 < d\nhdc : d ≤ c\nhcb : c ≤ b\nhba : b ≤ a\nh1 : a + b + c + d = 1\nhp : a ^ a * b ^ b * c ^ c * d ^ d ≤ a * a + b * b + c * c + d * d\n⊢ a + 2 * b + 3 * c + 4 * d ≤ 3 * a + 3 * b + c + 3 * d\n---\ncase h₂.h\na b c d : ℝ\nhd0 : 0 < d\nhdc : d ≤ c\nhcb : c ≤ b\nhba : b ≤ a\nh1 : a + b + c + d = 1\nhp : a ^ a * b ^ b * c ^ c * d ^ d ≤ a * a + b * b + c * c + d * d\n⊢ a + 2 * b + 3 * c + 4 * d ≤ 3 * a + 3 * b + 3 * c + d",
"after_state": "case h₁.h₁.h₁.h\na b c d : ℝ\nhd0 : 0 < d\nhdc : d ≤ c\nhcb : c ≤ b\nhba : b ≤ a\nh1 : a + b + c + d = 1\nhp : a ^ a * b ^ b * c ^ c * d ^ d ≤ a * a + b * b + c * c + d * d\n⊢ a + 2 * b + 3 * c + 4 * d ≤ a + 3 * b + 3 * c + 3 * d\n---\ncase h₁.h₁.h₂.h\na b c d : ℝ\nhd0 : 0 < d\nhdc : d ≤ c\nhcb : c ≤ b\nhba : b ≤ a\nh1 : a + b + c + d = 1\nhp : a ^ a * b ^ b * c ^ c * d ^ d ≤ a * a + b * b + c * c + d * d\n⊢ a + 2 * b + 3 * c + 4 * d ≤ 3 * a + b + 3 * c + 3 * d\n---\ncase h₁.h₂.h\na b c d : ℝ\nhd0 : 0 < d\nhdc : d ≤ c\nhcb : c ≤ b\nhba : b ≤ a\nh1 : a + b + c + d = 1\nhp : a ^ a * b ^ b * c ^ c * d ^ d ≤ a * a + b * b + c * c + d * d\n⊢ a + 2 * b + 3 * c + 4 * d ≤ 3 * a + 3 * b + c + 3 * d\n---\ncase h₂.h\na b c d : ℝ\nhd0 : 0 < d\nhdc : d ≤ c\nhcb : c ≤ b\nhba : b ≤ a\nh1 : a + b + c + d = 1\nhp : a ^ a * b ^ b * c ^ c * d ^ d ≤ a * a + b * b + c * c + d * d\n⊢ a + 2 * b + 3 * c + 4 * d ≤ 3 * a + 3 * b + 3 * c + d"
},
{
"line": "skip",
"before_state": "case h₁.h₁.h₁.h\na b c d : ℝ\nhd0 : 0 < d\nhdc : d ≤ c\nhcb : c ≤ b\nhba : b ≤ a\nh1 : a + b + c + d = 1\nhp : a ^ a * b ^ b * c ^ c * d ^ d ≤ a * a + b * b + c * c + d * d\n⊢ a + 2 * b + 3 * c + 4 * d ≤ a + 3 * b + 3 * c + 3 * d\n---\ncase h₁.h₁.h₂.h\na b c d : ℝ\nhd0 : 0 < d\nhdc : d ≤ c\nhcb : c ≤ b\nhba : b ≤ a\nh1 : a + b + c + d = 1\nhp : a ^ a * b ^ b * c ^ c * d ^ d ≤ a * a + b * b + c * c + d * d\n⊢ a + 2 * b + 3 * c + 4 * d ≤ 3 * a + b + 3 * c + 3 * d\n---\ncase h₁.h₂.h\na b c d : ℝ\nhd0 : 0 < d\nhdc : d ≤ c\nhcb : c ≤ b\nhba : b ≤ a\nh1 : a + b + c + d = 1\nhp : a ^ a * b ^ b * c ^ c * d ^ d ≤ a * a + b * b + c * c + d * d\n⊢ a + 2 * b + 3 * c + 4 * d ≤ 3 * a + 3 * b + c + 3 * d\n---\ncase h₂.h\na b c d : ℝ\nhd0 : 0 < d\nhdc : d ≤ c\nhcb : c ≤ b\nhba : b ≤ a\nh1 : a + b + c + d = 1\nhp : a ^ a * b ^ b * c ^ c * d ^ d ≤ a * a + b * b + c * c + d * d\n⊢ a + 2 * b + 3 * c + 4 * d ≤ 3 * a + 3 * b + 3 * c + d",
"after_state": "case h₁.h₁.h₁.h\na b c d : ℝ\nhd0 : 0 < d\nhdc : d ≤ c\nhcb : c ≤ b\nhba : b ≤ a\nh1 : a + b + c + d = 1\nhp : a ^ a * b ^ b * c ^ c * d ^ d ≤ a * a + b * b + c * c + d * d\n⊢ a + 2 * b + 3 * c + 4 * d ≤ a + 3 * b + 3 * c + 3 * d\n---\ncase h₁.h₁.h₂.h\na b c d : ℝ\nhd0 : 0 < d\nhdc : d ≤ c\nhcb : c ≤ b\nhba : b ≤ a\nh1 : a + b + c + d = 1\nhp : a ^ a * b ^ b * c ^ c * d ^ d ≤ a * a + b * b + c * c + d * d\n⊢ a + 2 * b + 3 * c + 4 * d ≤ 3 * a + b + 3 * c + 3 * d\n---\ncase h₁.h₂.h\na b c d : ℝ\nhd0 : 0 < d\nhdc : d ≤ c\nhcb : c ≤ b\nhba : b ≤ a\nh1 : a + b + c + d = 1\nhp : a ^ a * b ^ b * c ^ c * d ^ d ≤ a * a + b * b + c * c + d * d\n⊢ a + 2 * b + 3 * c + 4 * d ≤ 3 * a + 3 * b + c + 3 * d\n---\ncase h₂.h\na b c d : ℝ\nhd0 : 0 < d\nhdc : d ≤ c\nhcb : c ≤ b\nhba : b ≤ a\nh1 : a + b + c + d = 1\nhp : a ^ a * b ^ b * c ^ c * d ^ d ≤ a * a + b * b + c * c + d * d\n⊢ a + 2 * b + 3 * c + 4 * d ≤ 3 * a + 3 * b + 3 * c + d"
},
{
"line": "all_goals linarith",
"before_state": "case h₁.h₁.h₁.h\na b c d : ℝ\nhd0 : 0 < d\nhdc : d ≤ c\nhcb : c ≤ b\nhba : b ≤ a\nh1 : a + b + c + d = 1\nhp : a ^ a * b ^ b * c ^ c * d ^ d ≤ a * a + b * b + c * c + d * d\n⊢ a + 2 * b + 3 * c + 4 * d ≤ a + 3 * b + 3 * c + 3 * d\n---\ncase h₁.h₁.h₂.h\na b c d : ℝ\nhd0 : 0 < d\nhdc : d ≤ c\nhcb : c ≤ b\nhba : b ≤ a\nh1 : a + b + c + d = 1\nhp : a ^ a * b ^ b * c ^ c * d ^ d ≤ a * a + b * b + c * c + d * d\n⊢ a + 2 * b + 3 * c + 4 * d ≤ 3 * a + b + 3 * c + 3 * d\n---\ncase h₁.h₂.h\na b c d : ℝ\nhd0 : 0 < d\nhdc : d ≤ c\nhcb : c ≤ b\nhba : b ≤ a\nh1 : a + b + c + d = 1\nhp : a ^ a * b ^ b * c ^ c * d ^ d ≤ a * a + b * b + c * c + d * d\n⊢ a + 2 * b + 3 * c + 4 * d ≤ 3 * a + 3 * b + c + 3 * d\n---\ncase h₂.h\na b c d : ℝ\nhd0 : 0 < d\nhdc : d ≤ c\nhcb : c ≤ b\nhba : b ≤ a\nh1 : a + b + c + d = 1\nhp : a ^ a * b ^ b * c ^ c * d ^ d ≤ a * a + b * b + c * c + d * d\n⊢ a + 2 * b + 3 * c + 4 * d ≤ 3 * a + 3 * b + 3 * c + d",
"after_state": "No Goals!"
},
{
"line": "linarith",
"before_state": "case h₁.h₁.h₁.h\na b c d : ℝ\nhd0 : 0 < d\nhdc : d ≤ c\nhcb : c ≤ b\nhba : b ≤ a\nh1 : a + b + c + d = 1\nhp : a ^ a * b ^ b * c ^ c * d ^ d ≤ a * a + b * b + c * c + d * d\n⊢ a + 2 * b + 3 * c + 4 * d ≤ a + 3 * b + 3 * c + 3 * d",
"after_state": "No Goals!"
},
{
"line": "ring1",
"before_state": "a b c d : ℝ\nhd0 : 0 < d\nhdc : d ≤ c\nhcb : c ≤ b\nhba : b ≤ a\nh1 : a + b + c + d = 1\nhp : a ^ a * b ^ b * c ^ c * d ^ d ≤ a * a + b * b + c * c + d * d\na✝ : a + 2 * b + 3 * c + 4 * d > a + 3 * b + 3 * c + 3 * d\n⊢ d - c + (c - b) + (a + 3 * b + 3 * c + 3 * d - (a + 2 * b + 3 * c + 4 * d)) = 0",
"after_state": "No Goals!"
},
{
"line": "linarith",
"before_state": "case h₁.h₁.h₂.h\na b c d : ℝ\nhd0 : 0 < d\nhdc : d ≤ c\nhcb : c ≤ b\nhba : b ≤ a\nh1 : a + b + c + d = 1\nhp : a ^ a * b ^ b * c ^ c * d ^ d ≤ a * a + b * b + c * c + d * d\n⊢ a + 2 * b + 3 * c + 4 * d ≤ 3 * a + b + 3 * c + 3 * d",
"after_state": "No Goals!"
},
{
"line": "ring1",
"before_state": "a b c d : ℝ\nhd0 : 0 < d\nhdc : d ≤ c\nhcb : c ≤ b\nhba : b ≤ a\nh1 : a + b + c + d = 1\nhp : a ^ a * b ^ b * c ^ c * d ^ d ≤ a * a + b * b + c * c + d * d\na✝ : a + 2 * b + 3 * c + 4 * d > 3 * a + b + 3 * c + 3 * d\n⊢ d - c + (c - b) + 2 * (b - a) + (3 * a + b + 3 * c + 3 * d - (a + 2 * b + 3 * c + 4 * d)) = 0",
"after_state": "No Goals!"
},
{
"line": "norm_num",
"before_state": "a b c d : ℝ\nhd0 : 0 < d\nhdc : d ≤ c\nhcb : c ≤ b\nhba : b ≤ a\nh1 : a + b + c + d = 1\nhp : a ^ a * b ^ b * c ^ c * d ^ d ≤ a * a + b * b + c * c + d * d\na✝ : a + 2 * b + 3 * c + 4 * d > 3 * a + b + 3 * c + 3 * d\n⊢ 2 > 0",
"after_state": "No Goals!"
},
{
"line": "linarith",
"before_state": "case h₁.h₂.h\na b c d : ℝ\nhd0 : 0 < d\nhdc : d ≤ c\nhcb : c ≤ b\nhba : b ≤ a\nh1 : a + b + c + d = 1\nhp : a ^ a * b ^ b * c ^ c * d ^ d ≤ a * a + b * b + c * c + d * d\n⊢ a + 2 * b + 3 * c + 4 * d ≤ 3 * a + 3 * b + c + 3 * d",
"after_state": "No Goals!"
},
{
"line": "ring1",
"before_state": "a b c d : ℝ\nhd0 : 0 < d\nhdc : d ≤ c\nhcb : c ≤ b\nhba : b ≤ a\nh1 : a + b + c + d = 1\nhp : a ^ a * b ^ b * c ^ c * d ^ d ≤ a * a + b * b + c * c + d * d\na✝ : a + 2 * b + 3 * c + 4 * d > 3 * a + 3 * b + c + 3 * d\n⊢ d - c + 3 * (c - b) + 2 * (b - a) + (3 * a + 3 * b + c + 3 * d - (a + 2 * b + 3 * c + 4 * d)) = 0",
"after_state": "No Goals!"
},
{
"line": "norm_num",
"before_state": "a b c d : ℝ\nhd0 : 0 < d\nhdc : d ≤ c\nhcb : c ≤ b\nhba : b ≤ a\nh1 : a + b + c + d = 1\nhp : a ^ a * b ^ b * c ^ c * d ^ d ≤ a * a + b * b + c * c + d * d\na✝ : a + 2 * b + 3 * c + 4 * d > 3 * a + 3 * b + c + 3 * d\n⊢ 3 > 0",
"after_state": "No Goals!"
},
{
"line": "norm_num",
"before_state": "a b c d : ℝ\nhd0 : 0 < d\nhdc : d ≤ c\nhcb : c ≤ b\nhba : b ≤ a\nh1 : a + b + c + d = 1\nhp : a ^ a * b ^ b * c ^ c * d ^ d ≤ a * a + b * b + c * c + d * d\na✝ : a + 2 * b + 3 * c + 4 * d > 3 * a + 3 * b + c + 3 * d\n⊢ 2 > 0",
"after_state": "No Goals!"
},
{
"line": "linarith",
"before_state": "case h₂.h\na b c d : ℝ\nhd0 : 0 < d\nhdc : d ≤ c\nhcb : c ≤ b\nhba : b ≤ a\nh1 : a + b + c + d = 1\nhp : a ^ a * b ^ b * c ^ c * d ^ d ≤ a * a + b * b + c * c + d * d\n⊢ a + 2 * b + 3 * c + 4 * d ≤ 3 * a + 3 * b + 3 * c + d",
"after_state": "No Goals!"
},
{
"line": "ring1",
"before_state": "a b c d : ℝ\nhd0 : 0 < d\nhdc : d ≤ c\nhcb : c ≤ b\nhba : b ≤ a\nh1 : a + b + c + d = 1\nhp : a ^ a * b ^ b * c ^ c * d ^ d ≤ a * a + b * b + c * c + d * d\na✝ : a + 2 * b + 3 * c + 4 * d > 3 * a + 3 * b + 3 * c + d\n⊢ 3 * (d - c) + 3 * (c - b) + 2 * (b - a) + (3 * a + 3 * b + 3 * c + d - (a + 2 * b + 3 * c + 4 * d)) = 0",
"after_state": "No Goals!"
},
{
"line": "norm_num",
"before_state": "a b c d : ℝ\nhd0 : 0 < d\nhdc : d ≤ c\nhcb : c ≤ b\nhba : b ≤ a\nh1 : a + b + c + d = 1\nhp : a ^ a * b ^ b * c ^ c * d ^ d ≤ a * a + b * b + c * c + d * d\na✝ : a + 2 * b + 3 * c + 4 * d > 3 * a + 3 * b + 3 * c + d\n⊢ 3 > 0",
"after_state": "No Goals!"
},
{
"line": "norm_num",
"before_state": "a b c d : ℝ\nhd0 : 0 < d\nhdc : d ≤ c\nhcb : c ≤ b\nhba : b ≤ a\nh1 : a + b + c + d = 1\nhp : a ^ a * b ^ b * c ^ c * d ^ d ≤ a * a + b * b + c * c + d * d\na✝ : a + 2 * b + 3 * c + 4 * d > 3 * a + 3 * b + 3 * c + d\n⊢ 3 > 0",
"after_state": "No Goals!"
},
{
"line": "norm_num",
"before_state": "a b c d : ℝ\nhd0 : 0 < d\nhdc : d ≤ c\nhcb : c ≤ b\nhba : b ≤ a\nh1 : a + b + c + d = 1\nhp : a ^ a * b ^ b * c ^ c * d ^ d ≤ a * a + b * b + c * c + d * d\na✝ : a + 2 * b + 3 * c + 4 * d > 3 * a + 3 * b + 3 * c + d\n⊢ 2 > 0",
"after_state": "No Goals!"
},
{
"line": "focus\n apply_rules [add_pos, mul_pos, zero_lt_one]\n with_annotate_state\"<;>\" skip\n all_goals linarith",
"before_state": "a b c d : ℝ\nhd0 : 0 < d\nhdc : d ≤ c\nhcb : c ≤ b\nhba : b ≤ a\nh1 : a + b + c + d = 1\nhp : a ^ a * b ^ b * c ^ c * d ^ d ≤ a * a + b * b + c * c + d * d\n⊢ 0 < 6 * a * b * c + 6 * a * b * d + 6 * a * c * d + 6 * b * c * d",
"after_state": "No Goals!"
},
{
"line": "apply_rules [add_pos, mul_pos, zero_lt_one]",
"before_state": "a b c d : ℝ\nhd0 : 0 < d\nhdc : d ≤ c\nhcb : c ≤ b\nhba : b ≤ a\nh1 : a + b + c + d = 1\nhp : a ^ a * b ^ b * c ^ c * d ^ d ≤ a * a + b * b + c * c + d * d\n⊢ 0 < 6 * a * b * c + 6 * a * b * d + 6 * a * c * d + 6 * b * c * d",
"after_state": "case ha.ha.ha.ha.ha.ha\na b c d : ℝ\nhd0 : 0 < d\nhdc : d ≤ c\nhcb : c ≤ b\nhba : b ≤ a\nh1 : a + b + c + d = 1\nhp : a ^ a * b ^ b * c ^ c * d ^ d ≤ a * a + b * b + c * c + d * d\n⊢ 0 < 6\n---\ncase ha.ha.ha.ha.ha.hb\na b c d : ℝ\nhd0 : 0 < d\nhdc : d ≤ c\nhcb : c ≤ b\nhba : b ≤ a\nh1 : a + b + c + d = 1\nhp : a ^ a * b ^ b * c ^ c * d ^ d ≤ a * a + b * b + c * c + d * d\n⊢ 0 < a\n---\ncase ha.ha.ha.ha.hb\na b c d : ℝ\nhd0 : 0 < d\nhdc : d ≤ c\nhcb : c ≤ b\nhba : b ≤ a\nh1 : a + b + c + d = 1\nhp : a ^ a * b ^ b * c ^ c * d ^ d ≤ a * a + b * b + c * c + d * d\n⊢ 0 < b\n---\ncase ha.ha.ha.hb\na b c d : ℝ\nhd0 : 0 < d\nhdc : d ≤ c\nhcb : c ≤ b\nhba : b ≤ a\nh1 : a + b + c + d = 1\nhp : a ^ a * b ^ b * c ^ c * d ^ d ≤ a * a + b * b + c * c + d * d\n⊢ 0 < c\n---\ncase ha.ha.hb.ha.ha.ha\na b c d : ℝ\nhd0 : 0 < d\nhdc : d ≤ c\nhcb : c ≤ b\nhba : b ≤ a\nh1 : a + b + c + d = 1\nhp : a ^ a * b ^ b * c ^ c * d ^ d ≤ a * a + b * b + c * c + d * d\n⊢ 0 < 6\n---\ncase ha.ha.hb.ha.ha.hb\na b c d : ℝ\nhd0 : 0 < d\nhdc : d ≤ c\nhcb : c ≤ b\nhba : b ≤ a\nh1 : a + b + c + d = 1\nhp : a ^ a * b ^ b * c ^ c * d ^ d ≤ a * a + b * b + c * c + d * d\n⊢ 0 < a\n---\ncase ha.ha.hb.ha.hb\na b c d : ℝ\nhd0 : 0 < d\nhdc : d ≤ c\nhcb : c ≤ b\nhba : b ≤ a\nh1 : a + b + c + d = 1\nhp : a ^ a * b ^ b * c ^ c * d ^ d ≤ a * a + b * b + c * c + d * d\n⊢ 0 < b\n---\ncase ha.hb.ha.ha.ha\na b c d : ℝ\nhd0 : 0 < d\nhdc : d ≤ c\nhcb : c ≤ b\nhba : b ≤ a\nh1 : a + b + c + d = 1\nhp : a ^ a * b ^ b * c ^ c * d ^ d ≤ a * a + b * b + c * c + d * d\n⊢ 0 < 6\n---\ncase ha.hb.ha.ha.hb\na b c d : ℝ\nhd0 : 0 < d\nhdc : d ≤ c\nhcb : c ≤ b\nhba : b ≤ a\nh1 : a + b + c + d = 1\nhp : a ^ a * b ^ b * c ^ c * d ^ d ≤ a * a + b * b + c * c + d * d\n⊢ 0 < a\n---\ncase ha.hb.ha.hb\na b c d : ℝ\nhd0 : 0 < d\nhdc : d ≤ c\nhcb : c ≤ b\nhba : b ≤ a\nh1 : a + b + c + d = 1\nhp : a ^ a * b ^ b * c ^ c * d ^ d ≤ a * a + b * b + c * c + d * d\n⊢ 0 < c\n---\ncase hb.ha.ha.ha\na b c d : ℝ\nhd0 : 0 < d\nhdc : d ≤ c\nhcb : c ≤ b\nhba : b ≤ a\nh1 : a + b + c + d = 1\nhp : a ^ a * b ^ b * c ^ c * d ^ d ≤ a * a + b * b + c * c + d * d\n⊢ 0 < 6\n---\ncase hb.ha.ha.hb\na b c d : ℝ\nhd0 : 0 < d\nhdc : d ≤ c\nhcb : c ≤ b\nhba : b ≤ a\nh1 : a + b + c + d = 1\nhp : a ^ a * b ^ b * c ^ c * d ^ d ≤ a * a + b * b + c * c + d * d\n⊢ 0 < b\n---\ncase hb.ha.hb\na b c d : ℝ\nhd0 : 0 < d\nhdc : d ≤ c\nhcb : c ≤ b\nhba : b ≤ a\nh1 : a + b + c + d = 1\nhp : a ^ a * b ^ b * c ^ c * d ^ d ≤ a * a + b * b + c * c + d * d\n⊢ 0 < c"
},
{
"line": "with_annotate_state\"<;>\" skip",
"before_state": "case ha.ha.ha.ha.ha.ha\na b c d : ℝ\nhd0 : 0 < d\nhdc : d ≤ c\nhcb : c ≤ b\nhba : b ≤ a\nh1 : a + b + c + d = 1\nhp : a ^ a * b ^ b * c ^ c * d ^ d ≤ a * a + b * b + c * c + d * d\n⊢ 0 < 6\n---\ncase ha.ha.ha.ha.ha.hb\na b c d : ℝ\nhd0 : 0 < d\nhdc : d ≤ c\nhcb : c ≤ b\nhba : b ≤ a\nh1 : a + b + c + d = 1\nhp : a ^ a * b ^ b * c ^ c * d ^ d ≤ a * a + b * b + c * c + d * d\n⊢ 0 < a\n---\ncase ha.ha.ha.ha.hb\na b c d : ℝ\nhd0 : 0 < d\nhdc : d ≤ c\nhcb : c ≤ b\nhba : b ≤ a\nh1 : a + b + c + d = 1\nhp : a ^ a * b ^ b * c ^ c * d ^ d ≤ a * a + b * b + c * c + d * d\n⊢ 0 < b\n---\ncase ha.ha.ha.hb\na b c d : ℝ\nhd0 : 0 < d\nhdc : d ≤ c\nhcb : c ≤ b\nhba : b ≤ a\nh1 : a + b + c + d = 1\nhp : a ^ a * b ^ b * c ^ c * d ^ d ≤ a * a + b * b + c * c + d * d\n⊢ 0 < c\n---\ncase ha.ha.hb.ha.ha.ha\na b c d : ℝ\nhd0 : 0 < d\nhdc : d ≤ c\nhcb : c ≤ b\nhba : b ≤ a\nh1 : a + b + c + d = 1\nhp : a ^ a * b ^ b * c ^ c * d ^ d ≤ a * a + b * b + c * c + d * d\n⊢ 0 < 6\n---\ncase ha.ha.hb.ha.ha.hb\na b c d : ℝ\nhd0 : 0 < d\nhdc : d ≤ c\nhcb : c ≤ b\nhba : b ≤ a\nh1 : a + b + c + d = 1\nhp : a ^ a * b ^ b * c ^ c * d ^ d ≤ a * a + b * b + c * c + d * d\n⊢ 0 < a\n---\ncase ha.ha.hb.ha.hb\na b c d : ℝ\nhd0 : 0 < d\nhdc : d ≤ c\nhcb : c ≤ b\nhba : b ≤ a\nh1 : a + b + c + d = 1\nhp : a ^ a * b ^ b * c ^ c * d ^ d ≤ a * a + b * b + c * c + d * d\n⊢ 0 < b\n---\ncase ha.hb.ha.ha.ha\na b c d : ℝ\nhd0 : 0 < d\nhdc : d ≤ c\nhcb : c ≤ b\nhba : b ≤ a\nh1 : a + b + c + d = 1\nhp : a ^ a * b ^ b * c ^ c * d ^ d ≤ a * a + b * b + c * c + d * d\n⊢ 0 < 6\n---\ncase ha.hb.ha.ha.hb\na b c d : ℝ\nhd0 : 0 < d\nhdc : d ≤ c\nhcb : c ≤ b\nhba : b ≤ a\nh1 : a + b + c + d = 1\nhp : a ^ a * b ^ b * c ^ c * d ^ d ≤ a * a + b * b + c * c + d * d\n⊢ 0 < a\n---\ncase ha.hb.ha.hb\na b c d : ℝ\nhd0 : 0 < d\nhdc : d ≤ c\nhcb : c ≤ b\nhba : b ≤ a\nh1 : a + b + c + d = 1\nhp : a ^ a * b ^ b * c ^ c * d ^ d ≤ a * a + b * b + c * c + d * d\n⊢ 0 < c\n---\ncase hb.ha.ha.ha\na b c d : ℝ\nhd0 : 0 < d\nhdc : d ≤ c\nhcb : c ≤ b\nhba : b ≤ a\nh1 : a + b + c + d = 1\nhp : a ^ a * b ^ b * c ^ c * d ^ d ≤ a * a + b * b + c * c + d * d\n⊢ 0 < 6\n---\ncase hb.ha.ha.hb\na b c d : ℝ\nhd0 : 0 < d\nhdc : d ≤ c\nhcb : c ≤ b\nhba : b ≤ a\nh1 : a + b + c + d = 1\nhp : a ^ a * b ^ b * c ^ c * d ^ d ≤ a * a + b * b + c * c + d * d\n⊢ 0 < b\n---\ncase hb.ha.hb\na b c d : ℝ\nhd0 : 0 < d\nhdc : d ≤ c\nhcb : c ≤ b\nhba : b ≤ a\nh1 : a + b + c + d = 1\nhp : a ^ a * b ^ b * c ^ c * d ^ d ≤ a * a + b * b + c * c + d * d\n⊢ 0 < c",
"after_state": "case ha.ha.ha.ha.ha.ha\na b c d : ℝ\nhd0 : 0 < d\nhdc : d ≤ c\nhcb : c ≤ b\nhba : b ≤ a\nh1 : a + b + c + d = 1\nhp : a ^ a * b ^ b * c ^ c * d ^ d ≤ a * a + b * b + c * c + d * d\n⊢ 0 < 6\n---\ncase ha.ha.ha.ha.ha.hb\na b c d : ℝ\nhd0 : 0 < d\nhdc : d ≤ c\nhcb : c ≤ b\nhba : b ≤ a\nh1 : a + b + c + d = 1\nhp : a ^ a * b ^ b * c ^ c * d ^ d ≤ a * a + b * b + c * c + d * d\n⊢ 0 < a\n---\ncase ha.ha.ha.ha.hb\na b c d : ℝ\nhd0 : 0 < d\nhdc : d ≤ c\nhcb : c ≤ b\nhba : b ≤ a\nh1 : a + b + c + d = 1\nhp : a ^ a * b ^ b * c ^ c * d ^ d ≤ a * a + b * b + c * c + d * d\n⊢ 0 < b\n---\ncase ha.ha.ha.hb\na b c d : ℝ\nhd0 : 0 < d\nhdc : d ≤ c\nhcb : c ≤ b\nhba : b ≤ a\nh1 : a + b + c + d = 1\nhp : a ^ a * b ^ b * c ^ c * d ^ d ≤ a * a + b * b + c * c + d * d\n⊢ 0 < c\n---\ncase ha.ha.hb.ha.ha.ha\na b c d : ℝ\nhd0 : 0 < d\nhdc : d ≤ c\nhcb : c ≤ b\nhba : b ≤ a\nh1 : a + b + c + d = 1\nhp : a ^ a * b ^ b * c ^ c * d ^ d ≤ a * a + b * b + c * c + d * d\n⊢ 0 < 6\n---\ncase ha.ha.hb.ha.ha.hb\na b c d : ℝ\nhd0 : 0 < d\nhdc : d ≤ c\nhcb : c ≤ b\nhba : b ≤ a\nh1 : a + b + c + d = 1\nhp : a ^ a * b ^ b * c ^ c * d ^ d ≤ a * a + b * b + c * c + d * d\n⊢ 0 < a\n---\ncase ha.ha.hb.ha.hb\na b c d : ℝ\nhd0 : 0 < d\nhdc : d ≤ c\nhcb : c ≤ b\nhba : b ≤ a\nh1 : a + b + c + d = 1\nhp : a ^ a * b ^ b * c ^ c * d ^ d ≤ a * a + b * b + c * c + d * d\n⊢ 0 < b\n---\ncase ha.hb.ha.ha.ha\na b c d : ℝ\nhd0 : 0 < d\nhdc : d ≤ c\nhcb : c ≤ b\nhba : b ≤ a\nh1 : a + b + c + d = 1\nhp : a ^ a * b ^ b * c ^ c * d ^ d ≤ a * a + b * b + c * c + d * d\n⊢ 0 < 6\n---\ncase ha.hb.ha.ha.hb\na b c d : ℝ\nhd0 : 0 < d\nhdc : d ≤ c\nhcb : c ≤ b\nhba : b ≤ a\nh1 : a + b + c + d = 1\nhp : a ^ a * b ^ b * c ^ c * d ^ d ≤ a * a + b * b + c * c + d * d\n⊢ 0 < a\n---\ncase ha.hb.ha.hb\na b c d : ℝ\nhd0 : 0 < d\nhdc : d ≤ c\nhcb : c ≤ b\nhba : b ≤ a\nh1 : a + b + c + d = 1\nhp : a ^ a * b ^ b * c ^ c * d ^ d ≤ a * a + b * b + c * c + d * d\n⊢ 0 < c\n---\ncase hb.ha.ha.ha\na b c d : ℝ\nhd0 : 0 < d\nhdc : d ≤ c\nhcb : c ≤ b\nhba : b ≤ a\nh1 : a + b + c + d = 1\nhp : a ^ a * b ^ b * c ^ c * d ^ d ≤ a * a + b * b + c * c + d * d\n⊢ 0 < 6\n---\ncase hb.ha.ha.hb\na b c d : ℝ\nhd0 : 0 < d\nhdc : d ≤ c\nhcb : c ≤ b\nhba : b ≤ a\nh1 : a + b + c + d = 1\nhp : a ^ a * b ^ b * c ^ c * d ^ d ≤ a * a + b * b + c * c + d * d\n⊢ 0 < b\n---\ncase hb.ha.hb\na b c d : ℝ\nhd0 : 0 < d\nhdc : d ≤ c\nhcb : c ≤ b\nhba : b ≤ a\nh1 : a + b + c + d = 1\nhp : a ^ a * b ^ b * c ^ c * d ^ d ≤ a * a + b * b + c * c + d * d\n⊢ 0 < c"
},
{
"line": "skip",
"before_state": "case ha.ha.ha.ha.ha.ha\na b c d : ℝ\nhd0 : 0 < d\nhdc : d ≤ c\nhcb : c ≤ b\nhba : b ≤ a\nh1 : a + b + c + d = 1\nhp : a ^ a * b ^ b * c ^ c * d ^ d ≤ a * a + b * b + c * c + d * d\n⊢ 0 < 6\n---\ncase ha.ha.ha.ha.ha.hb\na b c d : ℝ\nhd0 : 0 < d\nhdc : d ≤ c\nhcb : c ≤ b\nhba : b ≤ a\nh1 : a + b + c + d = 1\nhp : a ^ a * b ^ b * c ^ c * d ^ d ≤ a * a + b * b + c * c + d * d\n⊢ 0 < a\n---\ncase ha.ha.ha.ha.hb\na b c d : ℝ\nhd0 : 0 < d\nhdc : d ≤ c\nhcb : c ≤ b\nhba : b ≤ a\nh1 : a + b + c + d = 1\nhp : a ^ a * b ^ b * c ^ c * d ^ d ≤ a * a + b * b + c * c + d * d\n⊢ 0 < b\n---\ncase ha.ha.ha.hb\na b c d : ℝ\nhd0 : 0 < d\nhdc : d ≤ c\nhcb : c ≤ b\nhba : b ≤ a\nh1 : a + b + c + d = 1\nhp : a ^ a * b ^ b * c ^ c * d ^ d ≤ a * a + b * b + c * c + d * d\n⊢ 0 < c\n---\ncase ha.ha.hb.ha.ha.ha\na b c d : ℝ\nhd0 : 0 < d\nhdc : d ≤ c\nhcb : c ≤ b\nhba : b ≤ a\nh1 : a + b + c + d = 1\nhp : a ^ a * b ^ b * c ^ c * d ^ d ≤ a * a + b * b + c * c + d * d\n⊢ 0 < 6\n---\ncase ha.ha.hb.ha.ha.hb\na b c d : ℝ\nhd0 : 0 < d\nhdc : d ≤ c\nhcb : c ≤ b\nhba : b ≤ a\nh1 : a + b + c + d = 1\nhp : a ^ a * b ^ b * c ^ c * d ^ d ≤ a * a + b * b + c * c + d * d\n⊢ 0 < a\n---\ncase ha.ha.hb.ha.hb\na b c d : ℝ\nhd0 : 0 < d\nhdc : d ≤ c\nhcb : c ≤ b\nhba : b ≤ a\nh1 : a + b + c + d = 1\nhp : a ^ a * b ^ b * c ^ c * d ^ d ≤ a * a + b * b + c * c + d * d\n⊢ 0 < b\n---\ncase ha.hb.ha.ha.ha\na b c d : ℝ\nhd0 : 0 < d\nhdc : d ≤ c\nhcb : c ≤ b\nhba : b ≤ a\nh1 : a + b + c + d = 1\nhp : a ^ a * b ^ b * c ^ c * d ^ d ≤ a * a + b * b + c * c + d * d\n⊢ 0 < 6\n---\ncase ha.hb.ha.ha.hb\na b c d : ℝ\nhd0 : 0 < d\nhdc : d ≤ c\nhcb : c ≤ b\nhba : b ≤ a\nh1 : a + b + c + d = 1\nhp : a ^ a * b ^ b * c ^ c * d ^ d ≤ a * a + b * b + c * c + d * d\n⊢ 0 < a\n---\ncase ha.hb.ha.hb\na b c d : ℝ\nhd0 : 0 < d\nhdc : d ≤ c\nhcb : c ≤ b\nhba : b ≤ a\nh1 : a + b + c + d = 1\nhp : a ^ a * b ^ b * c ^ c * d ^ d ≤ a * a + b * b + c * c + d * d\n⊢ 0 < c\n---\ncase hb.ha.ha.ha\na b c d : ℝ\nhd0 : 0 < d\nhdc : d ≤ c\nhcb : c ≤ b\nhba : b ≤ a\nh1 : a + b + c + d = 1\nhp : a ^ a * b ^ b * c ^ c * d ^ d ≤ a * a + b * b + c * c + d * d\n⊢ 0 < 6\n---\ncase hb.ha.ha.hb\na b c d : ℝ\nhd0 : 0 < d\nhdc : d ≤ c\nhcb : c ≤ b\nhba : b ≤ a\nh1 : a + b + c + d = 1\nhp : a ^ a * b ^ b * c ^ c * d ^ d ≤ a * a + b * b + c * c + d * d\n⊢ 0 < b\n---\ncase hb.ha.hb\na b c d : ℝ\nhd0 : 0 < d\nhdc : d ≤ c\nhcb : c ≤ b\nhba : b ≤ a\nh1 : a + b + c + d = 1\nhp : a ^ a * b ^ b * c ^ c * d ^ d ≤ a * a + b * b + c * c + d * d\n⊢ 0 < c",
"after_state": "case ha.ha.ha.ha.ha.ha\na b c d : ℝ\nhd0 : 0 < d\nhdc : d ≤ c\nhcb : c ≤ b\nhba : b ≤ a\nh1 : a + b + c + d = 1\nhp : a ^ a * b ^ b * c ^ c * d ^ d ≤ a * a + b * b + c * c + d * d\n⊢ 0 < 6\n---\ncase ha.ha.ha.ha.ha.hb\na b c d : ℝ\nhd0 : 0 < d\nhdc : d ≤ c\nhcb : c ≤ b\nhba : b ≤ a\nh1 : a + b + c + d = 1\nhp : a ^ a * b ^ b * c ^ c * d ^ d ≤ a * a + b * b + c * c + d * d\n⊢ 0 < a\n---\ncase ha.ha.ha.ha.hb\na b c d : ℝ\nhd0 : 0 < d\nhdc : d ≤ c\nhcb : c ≤ b\nhba : b ≤ a\nh1 : a + b + c + d = 1\nhp : a ^ a * b ^ b * c ^ c * d ^ d ≤ a * a + b * b + c * c + d * d\n⊢ 0 < b\n---\ncase ha.ha.ha.hb\na b c d : ℝ\nhd0 : 0 < d\nhdc : d ≤ c\nhcb : c ≤ b\nhba : b ≤ a\nh1 : a + b + c + d = 1\nhp : a ^ a * b ^ b * c ^ c * d ^ d ≤ a * a + b * b + c * c + d * d\n⊢ 0 < c\n---\ncase ha.ha.hb.ha.ha.ha\na b c d : ℝ\nhd0 : 0 < d\nhdc : d ≤ c\nhcb : c ≤ b\nhba : b ≤ a\nh1 : a + b + c + d = 1\nhp : a ^ a * b ^ b * c ^ c * d ^ d ≤ a * a + b * b + c * c + d * d\n⊢ 0 < 6\n---\ncase ha.ha.hb.ha.ha.hb\na b c d : ℝ\nhd0 : 0 < d\nhdc : d ≤ c\nhcb : c ≤ b\nhba : b ≤ a\nh1 : a + b + c + d = 1\nhp : a ^ a * b ^ b * c ^ c * d ^ d ≤ a * a + b * b + c * c + d * d\n⊢ 0 < a\n---\ncase ha.ha.hb.ha.hb\na b c d : ℝ\nhd0 : 0 < d\nhdc : d ≤ c\nhcb : c ≤ b\nhba : b ≤ a\nh1 : a + b + c + d = 1\nhp : a ^ a * b ^ b * c ^ c * d ^ d ≤ a * a + b * b + c * c + d * d\n⊢ 0 < b\n---\ncase ha.hb.ha.ha.ha\na b c d : ℝ\nhd0 : 0 < d\nhdc : d ≤ c\nhcb : c ≤ b\nhba : b ≤ a\nh1 : a + b + c + d = 1\nhp : a ^ a * b ^ b * c ^ c * d ^ d ≤ a * a + b * b + c * c + d * d\n⊢ 0 < 6\n---\ncase ha.hb.ha.ha.hb\na b c d : ℝ\nhd0 : 0 < d\nhdc : d ≤ c\nhcb : c ≤ b\nhba : b ≤ a\nh1 : a + b + c + d = 1\nhp : a ^ a * b ^ b * c ^ c * d ^ d ≤ a * a + b * b + c * c + d * d\n⊢ 0 < a\n---\ncase ha.hb.ha.hb\na b c d : ℝ\nhd0 : 0 < d\nhdc : d ≤ c\nhcb : c ≤ b\nhba : b ≤ a\nh1 : a + b + c + d = 1\nhp : a ^ a * b ^ b * c ^ c * d ^ d ≤ a * a + b * b + c * c + d * d\n⊢ 0 < c\n---\ncase hb.ha.ha.ha\na b c d : ℝ\nhd0 : 0 < d\nhdc : d ≤ c\nhcb : c ≤ b\nhba : b ≤ a\nh1 : a + b + c + d = 1\nhp : a ^ a * b ^ b * c ^ c * d ^ d ≤ a * a + b * b + c * c + d * d\n⊢ 0 < 6\n---\ncase hb.ha.ha.hb\na b c d : ℝ\nhd0 : 0 < d\nhdc : d ≤ c\nhcb : c ≤ b\nhba : b ≤ a\nh1 : a + b + c + d = 1\nhp : a ^ a * b ^ b * c ^ c * d ^ d ≤ a * a + b * b + c * c + d * d\n⊢ 0 < b\n---\ncase hb.ha.hb\na b c d : ℝ\nhd0 : 0 < d\nhdc : d ≤ c\nhcb : c ≤ b\nhba : b ≤ a\nh1 : a + b + c + d = 1\nhp : a ^ a * b ^ b * c ^ c * d ^ d ≤ a * a + b * b + c * c + d * d\n⊢ 0 < c"
},
{
"line": "all_goals linarith",
"before_state": "case ha.ha.ha.ha.ha.ha\na b c d : ℝ\nhd0 : 0 < d\nhdc : d ≤ c\nhcb : c ≤ b\nhba : b ≤ a\nh1 : a + b + c + d = 1\nhp : a ^ a * b ^ b * c ^ c * d ^ d ≤ a * a + b * b + c * c + d * d\n⊢ 0 < 6\n---\ncase ha.ha.ha.ha.ha.hb\na b c d : ℝ\nhd0 : 0 < d\nhdc : d ≤ c\nhcb : c ≤ b\nhba : b ≤ a\nh1 : a + b + c + d = 1\nhp : a ^ a * b ^ b * c ^ c * d ^ d ≤ a * a + b * b + c * c + d * d\n⊢ 0 < a\n---\ncase ha.ha.ha.ha.hb\na b c d : ℝ\nhd0 : 0 < d\nhdc : d ≤ c\nhcb : c ≤ b\nhba : b ≤ a\nh1 : a + b + c + d = 1\nhp : a ^ a * b ^ b * c ^ c * d ^ d ≤ a * a + b * b + c * c + d * d\n⊢ 0 < b\n---\ncase ha.ha.ha.hb\na b c d : ℝ\nhd0 : 0 < d\nhdc : d ≤ c\nhcb : c ≤ b\nhba : b ≤ a\nh1 : a + b + c + d = 1\nhp : a ^ a * b ^ b * c ^ c * d ^ d ≤ a * a + b * b + c * c + d * d\n⊢ 0 < c\n---\ncase ha.ha.hb.ha.ha.ha\na b c d : ℝ\nhd0 : 0 < d\nhdc : d ≤ c\nhcb : c ≤ b\nhba : b ≤ a\nh1 : a + b + c + d = 1\nhp : a ^ a * b ^ b * c ^ c * d ^ d ≤ a * a + b * b + c * c + d * d\n⊢ 0 < 6\n---\ncase ha.ha.hb.ha.ha.hb\na b c d : ℝ\nhd0 : 0 < d\nhdc : d ≤ c\nhcb : c ≤ b\nhba : b ≤ a\nh1 : a + b + c + d = 1\nhp : a ^ a * b ^ b * c ^ c * d ^ d ≤ a * a + b * b + c * c + d * d\n⊢ 0 < a\n---\ncase ha.ha.hb.ha.hb\na b c d : ℝ\nhd0 : 0 < d\nhdc : d ≤ c\nhcb : c ≤ b\nhba : b ≤ a\nh1 : a + b + c + d = 1\nhp : a ^ a * b ^ b * c ^ c * d ^ d ≤ a * a + b * b + c * c + d * d\n⊢ 0 < b\n---\ncase ha.hb.ha.ha.ha\na b c d : ℝ\nhd0 : 0 < d\nhdc : d ≤ c\nhcb : c ≤ b\nhba : b ≤ a\nh1 : a + b + c + d = 1\nhp : a ^ a * b ^ b * c ^ c * d ^ d ≤ a * a + b * b + c * c + d * d\n⊢ 0 < 6\n---\ncase ha.hb.ha.ha.hb\na b c d : ℝ\nhd0 : 0 < d\nhdc : d ≤ c\nhcb : c ≤ b\nhba : b ≤ a\nh1 : a + b + c + d = 1\nhp : a ^ a * b ^ b * c ^ c * d ^ d ≤ a * a + b * b + c * c + d * d\n⊢ 0 < a\n---\ncase ha.hb.ha.hb\na b c d : ℝ\nhd0 : 0 < d\nhdc : d ≤ c\nhcb : c ≤ b\nhba : b ≤ a\nh1 : a + b + c + d = 1\nhp : a ^ a * b ^ b * c ^ c * d ^ d ≤ a * a + b * b + c * c + d * d\n⊢ 0 < c\n---\ncase hb.ha.ha.ha\na b c d : ℝ\nhd0 : 0 < d\nhdc : d ≤ c\nhcb : c ≤ b\nhba : b ≤ a\nh1 : a + b + c + d = 1\nhp : a ^ a * b ^ b * c ^ c * d ^ d ≤ a * a + b * b + c * c + d * d\n⊢ 0 < 6\n---\ncase hb.ha.ha.hb\na b c d : ℝ\nhd0 : 0 < d\nhdc : d ≤ c\nhcb : c ≤ b\nhba : b ≤ a\nh1 : a + b + c + d = 1\nhp : a ^ a * b ^ b * c ^ c * d ^ d ≤ a * a + b * b + c * c + d * d\n⊢ 0 < b\n---\ncase hb.ha.hb\na b c d : ℝ\nhd0 : 0 < d\nhdc : d ≤ c\nhcb : c ≤ b\nhba : b ≤ a\nh1 : a + b + c + d = 1\nhp : a ^ a * b ^ b * c ^ c * d ^ d ≤ a * a + b * b + c * c + d * d\n⊢ 0 < c",
"after_state": "No Goals!"
},
{
"line": "linarith",
"before_state": "case ha.ha.ha.ha.ha.ha\na b c d : ℝ\nhd0 : 0 < d\nhdc : d ≤ c\nhcb : c ≤ b\nhba : b ≤ a\nh1 : a + b + c + d = 1\nhp : a ^ a * b ^ b * c ^ c * d ^ d ≤ a * a + b * b + c * c + d * d\n⊢ 0 < 6",
"after_state": "No Goals!"
},
{
"line": "ring1",
"before_state": "a b c d : ℝ\nhd0 : 0 < d\nhdc : d ≤ c\nhcb : c ≤ b\nhba : b ≤ a\nh1 : a + b + c + d = 1\nhp : a ^ a * b ^ b * c ^ c * d ^ d ≤ a * a + b * b + c * c + d * d\na✝ : 0 ≥ 6\n⊢ 24 * (0 - d) + 18 * (d - c) + 12 * (c - b) + 6 * (b - a) + 6 * (a + b + c + d - 1) + (6 - 0) = 0",
"after_state": "No Goals!"
},
{
"line": "norm_num",
"before_state": "a b c d : ℝ\nhd0 : 0 < d\nhdc : d ≤ c\nhcb : c ≤ b\nhba : b ≤ a\nh1 : a + b + c + d = 1\nhp : a ^ a * b ^ b * c ^ c * d ^ d ≤ a * a + b * b + c * c + d * d\na✝ : 0 ≥ 6\n⊢ 24 > 0",
"after_state": "No Goals!"
},
{
"line": "norm_num",
"before_state": "a b c d : ℝ\nhd0 : 0 < d\nhdc : d ≤ c\nhcb : c ≤ b\nhba : b ≤ a\nh1 : a + b + c + d = 1\nhp : a ^ a * b ^ b * c ^ c * d ^ d ≤ a * a + b * b + c * c + d * d\na✝ : 0 ≥ 6\n⊢ 18 > 0",
"after_state": "No Goals!"
},
{
"line": "norm_num",
"before_state": "a b c d : ℝ\nhd0 : 0 < d\nhdc : d ≤ c\nhcb : c ≤ b\nhba : b ≤ a\nh1 : a + b + c + d = 1\nhp : a ^ a * b ^ b * c ^ c * d ^ d ≤ a * a + b * b + c * c + d * d\na✝ : 0 ≥ 6\n⊢ 12 > 0",
"after_state": "No Goals!"
},
{
"line": "norm_num",
"before_state": "a b c d : ℝ\nhd0 : 0 < d\nhdc : d ≤ c\nhcb : c ≤ b\nhba : b ≤ a\nh1 : a + b + c + d = 1\nhp : a ^ a * b ^ b * c ^ c * d ^ d ≤ a * a + b * b + c * c + d * d\na✝ : 0 ≥ 6\n⊢ 6 > 0",
"after_state": "No Goals!"
},
{
"line": "norm_num",
"before_state": "a b c d : ℝ\nhd0 : 0 < d\nhdc : d ≤ c\nhcb : c ≤ b\nhba : b ≤ a\nh1 : a + b + c + d = 1\nhp : a ^ a * b ^ b * c ^ c * d ^ d ≤ a * a + b * b + c * c + d * d\na✝ : 0 ≥ 6\n⊢ 6 > 0",
"after_state": "No Goals!"
},
{
"line": "linarith",
"before_state": "case ha.ha.ha.ha.ha.hb\na b c d : ℝ\nhd0 : 0 < d\nhdc : d ≤ c\nhcb : c ≤ b\nhba : b ≤ a\nh1 : a + b + c + d = 1\nhp : a ^ a * b ^ b * c ^ c * d ^ d ≤ a * a + b * b + c * c + d * d\n⊢ 0 < a",
"after_state": "No Goals!"
},
{
"line": "ring1",
"before_state": "a b c d : ℝ\nhd0 : 0 < d\nhdc : d ≤ c\nhcb : c ≤ b\nhba : b ≤ a\nh1 : a + b + c + d = 1\nhp : a ^ a * b ^ b * c ^ c * d ^ d ≤ a * a + b * b + c * c + d * d\na✝ : 0 ≥ a\n⊢ 0 - d + (d - c) + (c - b) + (b - a) + (a - 0) = 0",
"after_state": "No Goals!"
},
{
"line": "linarith",
"before_state": "case ha.ha.ha.ha.hb\na b c d : ℝ\nhd0 : 0 < d\nhdc : d ≤ c\nhcb : c ≤ b\nhba : b ≤ a\nh1 : a + b + c + d = 1\nhp : a ^ a * b ^ b * c ^ c * d ^ d ≤ a * a + b * b + c * c + d * d\n⊢ 0 < b",
"after_state": "No Goals!"
},
{
"line": "ring1",
"before_state": "a b c d : ℝ\nhd0 : 0 < d\nhdc : d ≤ c\nhcb : c ≤ b\nhba : b ≤ a\nh1 : a + b + c + d = 1\nhp : a ^ a * b ^ b * c ^ c * d ^ d ≤ a * a + b * b + c * c + d * d\na✝ : 0 ≥ b\n⊢ 0 - d + (d - c) + (c - b) + (b - 0) = 0",
"after_state": "No Goals!"
},
{
"line": "linarith",
"before_state": "case ha.ha.ha.hb\na b c d : ℝ\nhd0 : 0 < d\nhdc : d ≤ c\nhcb : c ≤ b\nhba : b ≤ a\nh1 : a + b + c + d = 1\nhp : a ^ a * b ^ b * c ^ c * d ^ d ≤ a * a + b * b + c * c + d * d\n⊢ 0 < c",
"after_state": "No Goals!"
},
{
"line": "ring1",
"before_state": "a b c d : ℝ\nhd0 : 0 < d\nhdc : d ≤ c\nhcb : c ≤ b\nhba : b ≤ a\nh1 : a + b + c + d = 1\nhp : a ^ a * b ^ b * c ^ c * d ^ d ≤ a * a + b * b + c * c + d * d\na✝ : 0 ≥ c\n⊢ 0 - d + (d - c) + (c - 0) = 0",
"after_state": "No Goals!"
},
{
"line": "linarith",
"before_state": "case ha.ha.hb.ha.ha.ha\na b c d : ℝ\nhd0 : 0 < d\nhdc : d ≤ c\nhcb : c ≤ b\nhba : b ≤ a\nh1 : a + b + c + d = 1\nhp : a ^ a * b ^ b * c ^ c * d ^ d ≤ a * a + b * b + c * c + d * d\n⊢ 0 < 6",
"after_state": "No Goals!"
},
{
"line": "ring1",
"before_state": "a b c d : ℝ\nhd0 : 0 < d\nhdc : d ≤ c\nhcb : c ≤ b\nhba : b ≤ a\nh1 : a + b + c + d = 1\nhp : a ^ a * b ^ b * c ^ c * d ^ d ≤ a * a + b * b + c * c + d * d\na✝ : 0 ≥ 6\n⊢ 24 * (0 - d) + 18 * (d - c) + 12 * (c - b) + 6 * (b - a) + 6 * (a + b + c + d - 1) + (6 - 0) = 0",
"after_state": "No Goals!"
},
{
"line": "norm_num",
"before_state": "a b c d : ℝ\nhd0 : 0 < d\nhdc : d ≤ c\nhcb : c ≤ b\nhba : b ≤ a\nh1 : a + b + c + d = 1\nhp : a ^ a * b ^ b * c ^ c * d ^ d ≤ a * a + b * b + c * c + d * d\na✝ : 0 ≥ 6\n⊢ 24 > 0",
"after_state": "No Goals!"
},
{
"line": "norm_num",
"before_state": "a b c d : ℝ\nhd0 : 0 < d\nhdc : d ≤ c\nhcb : c ≤ b\nhba : b ≤ a\nh1 : a + b + c + d = 1\nhp : a ^ a * b ^ b * c ^ c * d ^ d ≤ a * a + b * b + c * c + d * d\na✝ : 0 ≥ 6\n⊢ 18 > 0",
"after_state": "No Goals!"
},
{
"line": "norm_num",
"before_state": "a b c d : ℝ\nhd0 : 0 < d\nhdc : d ≤ c\nhcb : c ≤ b\nhba : b ≤ a\nh1 : a + b + c + d = 1\nhp : a ^ a * b ^ b * c ^ c * d ^ d ≤ a * a + b * b + c * c + d * d\na✝ : 0 ≥ 6\n⊢ 12 > 0",
"after_state": "No Goals!"
},
{
"line": "norm_num",
"before_state": "a b c d : ℝ\nhd0 : 0 < d\nhdc : d ≤ c\nhcb : c ≤ b\nhba : b ≤ a\nh1 : a + b + c + d = 1\nhp : a ^ a * b ^ b * c ^ c * d ^ d ≤ a * a + b * b + c * c + d * d\na✝ : 0 ≥ 6\n⊢ 6 > 0",
"after_state": "No Goals!"
},
{
"line": "norm_num",
"before_state": "a b c d : ℝ\nhd0 : 0 < d\nhdc : d ≤ c\nhcb : c ≤ b\nhba : b ≤ a\nh1 : a + b + c + d = 1\nhp : a ^ a * b ^ b * c ^ c * d ^ d ≤ a * a + b * b + c * c + d * d\na✝ : 0 ≥ 6\n⊢ 6 > 0",
"after_state": "No Goals!"
},
{
"line": "linarith",
"before_state": "case ha.ha.hb.ha.ha.hb\na b c d : ℝ\nhd0 : 0 < d\nhdc : d ≤ c\nhcb : c ≤ b\nhba : b ≤ a\nh1 : a + b + c + d = 1\nhp : a ^ a * b ^ b * c ^ c * d ^ d ≤ a * a + b * b + c * c + d * d\n⊢ 0 < a",
"after_state": "No Goals!"
},
{
"line": "ring1",
"before_state": "a b c d : ℝ\nhd0 : 0 < d\nhdc : d ≤ c\nhcb : c ≤ b\nhba : b ≤ a\nh1 : a + b + c + d = 1\nhp : a ^ a * b ^ b * c ^ c * d ^ d ≤ a * a + b * b + c * c + d * d\na✝ : 0 ≥ a\n⊢ 0 - d + (d - c) + (c - b) + (b - a) + (a - 0) = 0",
"after_state": "No Goals!"
},
{
"line": "linarith",
"before_state": "case ha.ha.hb.ha.hb\na b c d : ℝ\nhd0 : 0 < d\nhdc : d ≤ c\nhcb : c ≤ b\nhba : b ≤ a\nh1 : a + b + c + d = 1\nhp : a ^ a * b ^ b * c ^ c * d ^ d ≤ a * a + b * b + c * c + d * d\n⊢ 0 < b",
"after_state": "No Goals!"
},
{
"line": "ring1",
"before_state": "a b c d : ℝ\nhd0 : 0 < d\nhdc : d ≤ c\nhcb : c ≤ b\nhba : b ≤ a\nh1 : a + b + c + d = 1\nhp : a ^ a * b ^ b * c ^ c * d ^ d ≤ a * a + b * b + c * c + d * d\na✝ : 0 ≥ b\n⊢ 0 - d + (d - c) + (c - b) + (b - 0) = 0",
"after_state": "No Goals!"
},
{
"line": "linarith",
"before_state": "case ha.hb.ha.ha.ha\na b c d : ℝ\nhd0 : 0 < d\nhdc : d ≤ c\nhcb : c ≤ b\nhba : b ≤ a\nh1 : a + b + c + d = 1\nhp : a ^ a * b ^ b * c ^ c * d ^ d ≤ a * a + b * b + c * c + d * d\n⊢ 0 < 6",
"after_state": "No Goals!"
},
{
"line": "ring1",
"before_state": "a b c d : ℝ\nhd0 : 0 < d\nhdc : d ≤ c\nhcb : c ≤ b\nhba : b ≤ a\nh1 : a + b + c + d = 1\nhp : a ^ a * b ^ b * c ^ c * d ^ d ≤ a * a + b * b + c * c + d * d\na✝ : 0 ≥ 6\n⊢ 24 * (0 - d) + 18 * (d - c) + 12 * (c - b) + 6 * (b - a) + 6 * (a + b + c + d - 1) + (6 - 0) = 0",
"after_state": "No Goals!"
},
{
"line": "norm_num",
"before_state": "a b c d : ℝ\nhd0 : 0 < d\nhdc : d ≤ c\nhcb : c ≤ b\nhba : b ≤ a\nh1 : a + b + c + d = 1\nhp : a ^ a * b ^ b * c ^ c * d ^ d ≤ a * a + b * b + c * c + d * d\na✝ : 0 ≥ 6\n⊢ 24 > 0",
"after_state": "No Goals!"
},
{
"line": "norm_num",
"before_state": "a b c d : ℝ\nhd0 : 0 < d\nhdc : d ≤ c\nhcb : c ≤ b\nhba : b ≤ a\nh1 : a + b + c + d = 1\nhp : a ^ a * b ^ b * c ^ c * d ^ d ≤ a * a + b * b + c * c + d * d\na✝ : 0 ≥ 6\n⊢ 18 > 0",
"after_state": "No Goals!"
},
{
"line": "norm_num",
"before_state": "a b c d : ℝ\nhd0 : 0 < d\nhdc : d ≤ c\nhcb : c ≤ b\nhba : b ≤ a\nh1 : a + b + c + d = 1\nhp : a ^ a * b ^ b * c ^ c * d ^ d ≤ a * a + b * b + c * c + d * d\na✝ : 0 ≥ 6\n⊢ 12 > 0",
"after_state": "No Goals!"
},
{
"line": "norm_num",
"before_state": "a b c d : ℝ\nhd0 : 0 < d\nhdc : d ≤ c\nhcb : c ≤ b\nhba : b ≤ a\nh1 : a + b + c + d = 1\nhp : a ^ a * b ^ b * c ^ c * d ^ d ≤ a * a + b * b + c * c + d * d\na✝ : 0 ≥ 6\n⊢ 6 > 0",
"after_state": "No Goals!"
},
{
"line": "norm_num",
"before_state": "a b c d : ℝ\nhd0 : 0 < d\nhdc : d ≤ c\nhcb : c ≤ b\nhba : b ≤ a\nh1 : a + b + c + d = 1\nhp : a ^ a * b ^ b * c ^ c * d ^ d ≤ a * a + b * b + c * c + d * d\na✝ : 0 ≥ 6\n⊢ 6 > 0",
"after_state": "No Goals!"
},
{
"line": "linarith",
"before_state": "case ha.hb.ha.ha.hb\na b c d : ℝ\nhd0 : 0 < d\nhdc : d ≤ c\nhcb : c ≤ b\nhba : b ≤ a\nh1 : a + b + c + d = 1\nhp : a ^ a * b ^ b * c ^ c * d ^ d ≤ a * a + b * b + c * c + d * d\n⊢ 0 < a",
"after_state": "No Goals!"
},
{
"line": "ring1",
"before_state": "a b c d : ℝ\nhd0 : 0 < d\nhdc : d ≤ c\nhcb : c ≤ b\nhba : b ≤ a\nh1 : a + b + c + d = 1\nhp : a ^ a * b ^ b * c ^ c * d ^ d ≤ a * a + b * b + c * c + d * d\na✝ : 0 ≥ a\n⊢ 0 - d + (d - c) + (c - b) + (b - a) + (a - 0) = 0",
"after_state": "No Goals!"
},
{
"line": "linarith",
"before_state": "case ha.hb.ha.hb\na b c d : ℝ\nhd0 : 0 < d\nhdc : d ≤ c\nhcb : c ≤ b\nhba : b ≤ a\nh1 : a + b + c + d = 1\nhp : a ^ a * b ^ b * c ^ c * d ^ d ≤ a * a + b * b + c * c + d * d\n⊢ 0 < c",
"after_state": "No Goals!"
},
{
"line": "ring1",
"before_state": "a b c d : ℝ\nhd0 : 0 < d\nhdc : d ≤ c\nhcb : c ≤ b\nhba : b ≤ a\nh1 : a + b + c + d = 1\nhp : a ^ a * b ^ b * c ^ c * d ^ d ≤ a * a + b * b + c * c + d * d\na✝ : 0 ≥ c\n⊢ 0 - d + (d - c) + (c - 0) = 0",
"after_state": "No Goals!"
},
{
"line": "linarith",
"before_state": "case hb.ha.ha.ha\na b c d : ℝ\nhd0 : 0 < d\nhdc : d ≤ c\nhcb : c ≤ b\nhba : b ≤ a\nh1 : a + b + c + d = 1\nhp : a ^ a * b ^ b * c ^ c * d ^ d ≤ a * a + b * b + c * c + d * d\n⊢ 0 < 6",
"after_state": "No Goals!"
},
{
"line": "ring1",
"before_state": "a b c d : ℝ\nhd0 : 0 < d\nhdc : d ≤ c\nhcb : c ≤ b\nhba : b ≤ a\nh1 : a + b + c + d = 1\nhp : a ^ a * b ^ b * c ^ c * d ^ d ≤ a * a + b * b + c * c + d * d\na✝ : 0 ≥ 6\n⊢ 24 * (0 - d) + 18 * (d - c) + 12 * (c - b) + 6 * (b - a) + 6 * (a + b + c + d - 1) + (6 - 0) = 0",
"after_state": "No Goals!"
},
{
"line": "norm_num",
"before_state": "a b c d : ℝ\nhd0 : 0 < d\nhdc : d ≤ c\nhcb : c ≤ b\nhba : b ≤ a\nh1 : a + b + c + d = 1\nhp : a ^ a * b ^ b * c ^ c * d ^ d ≤ a * a + b * b + c * c + d * d\na✝ : 0 ≥ 6\n⊢ 24 > 0",
"after_state": "No Goals!"
},
{
"line": "norm_num",
"before_state": "a b c d : ℝ\nhd0 : 0 < d\nhdc : d ≤ c\nhcb : c ≤ b\nhba : b ≤ a\nh1 : a + b + c + d = 1\nhp : a ^ a * b ^ b * c ^ c * d ^ d ≤ a * a + b * b + c * c + d * d\na✝ : 0 ≥ 6\n⊢ 18 > 0",
"after_state": "No Goals!"
},
{
"line": "norm_num",
"before_state": "a b c d : ℝ\nhd0 : 0 < d\nhdc : d ≤ c\nhcb : c ≤ b\nhba : b ≤ a\nh1 : a + b + c + d = 1\nhp : a ^ a * b ^ b * c ^ c * d ^ d ≤ a * a + b * b + c * c + d * d\na✝ : 0 ≥ 6\n⊢ 12 > 0",
"after_state": "No Goals!"
},
{
"line": "norm_num",
"before_state": "a b c d : ℝ\nhd0 : 0 < d\nhdc : d ≤ c\nhcb : c ≤ b\nhba : b ≤ a\nh1 : a + b + c + d = 1\nhp : a ^ a * b ^ b * c ^ c * d ^ d ≤ a * a + b * b + c * c + d * d\na✝ : 0 ≥ 6\n⊢ 6 > 0",
"after_state": "No Goals!"
},
{
"line": "norm_num",
"before_state": "a b c d : ℝ\nhd0 : 0 < d\nhdc : d ≤ c\nhcb : c ≤ b\nhba : b ≤ a\nh1 : a + b + c + d = 1\nhp : a ^ a * b ^ b * c ^ c * d ^ d ≤ a * a + b * b + c * c + d * d\na✝ : 0 ≥ 6\n⊢ 6 > 0",
"after_state": "No Goals!"
},
{
"line": "linarith",
"before_state": "case hb.ha.ha.hb\na b c d : ℝ\nhd0 : 0 < d\nhdc : d ≤ c\nhcb : c ≤ b\nhba : b ≤ a\nh1 : a + b + c + d = 1\nhp : a ^ a * b ^ b * c ^ c * d ^ d ≤ a * a + b * b + c * c + d * d\n⊢ 0 < b",
"after_state": "No Goals!"
},
{
"line": "ring1",
"before_state": "a b c d : ℝ\nhd0 : 0 < d\nhdc : d ≤ c\nhcb : c ≤ b\nhba : b ≤ a\nh1 : a + b + c + d = 1\nhp : a ^ a * b ^ b * c ^ c * d ^ d ≤ a * a + b * b + c * c + d * d\na✝ : 0 ≥ b\n⊢ 0 - d + (d - c) + (c - b) + (b - 0) = 0",
"after_state": "No Goals!"
},
{
"line": "linarith",
"before_state": "case hb.ha.hb\na b c d : ℝ\nhd0 : 0 < d\nhdc : d ≤ c\nhcb : c ≤ b\nhba : b ≤ a\nh1 : a + b + c + d = 1\nhp : a ^ a * b ^ b * c ^ c * d ^ d ≤ a * a + b * b + c * c + d * d\n⊢ 0 < c",
"after_state": "No Goals!"
},
{
"line": "ring1",
"before_state": "a b c d : ℝ\nhd0 : 0 < d\nhdc : d ≤ c\nhcb : c ≤ b\nhba : b ≤ a\nh1 : a + b + c + d = 1\nhp : a ^ a * b ^ b * c ^ c * d ^ d ≤ a * a + b * b + c * c + d * d\na✝ : 0 ≥ c\n⊢ 0 - d + (d - c) + (c - 0) = 0",
"after_state": "No Goals!"
},
{
"line": "ring",
"before_state": "a b c d : ℝ\nhd0 : 0 < d\nhdc : d ≤ c\nhcb : c ≤ b\nhba : b ≤ a\nh1 : a + b + c + d = 1\nhp : a ^ a * b ^ b * c ^ c * d ^ d ≤ a * a + b * b + c * c + d * d\n⊢ (a + 3 * b + 3 * c + 3 * d) * a ^ 2 + (3 * a + b + 3 * c + 3 * d) * b ^ 2 + (3 * a + 3 * b + c + 3 * d) * c ^ 2 +\n (3 * a + 3 * b + 3 * c + d) * d ^ 2 +\n (6 * a * b * c + 6 * a * b * d + 6 * a * c * d + 6 * b * c * d) =\n (a + b + c + d) ^ 3",
"after_state": "No Goals!"
},
{
"line": "first\n| ring1\n|\n try_this ring_nf\"\\n\\nThe `ring` tactic failed to close the goal. Use `ring_nf` to obtain a normal form.\n \\nNote that `ring` works primarily in *commutative* rings. \\\n If you have a noncommutative ring, abelian group or module, consider using \\\n `noncomm_ring`, `abel` or `module` instead.\"",
"before_state": "a b c d : ℝ\nhd0 : 0 < d\nhdc : d ≤ c\nhcb : c ≤ b\nhba : b ≤ a\nh1 : a + b + c + d = 1\nhp : a ^ a * b ^ b * c ^ c * d ^ d ≤ a * a + b * b + c * c + d * d\n⊢ (a + 3 * b + 3 * c + 3 * d) * a ^ 2 + (3 * a + b + 3 * c + 3 * d) * b ^ 2 + (3 * a + 3 * b + c + 3 * d) * c ^ 2 +\n (3 * a + 3 * b + 3 * c + d) * d ^ 2 +\n (6 * a * b * c + 6 * a * b * d + 6 * a * c * d + 6 * b * c * d) =\n (a + b + c + d) ^ 3",
"after_state": "No Goals!"
},
{
"line": "ring1",
"before_state": "a b c d : ℝ\nhd0 : 0 < d\nhdc : d ≤ c\nhcb : c ≤ b\nhba : b ≤ a\nh1 : a + b + c + d = 1\nhp : a ^ a * b ^ b * c ^ c * d ^ d ≤ a * a + b * b + c * c + d * d\n⊢ (a + 3 * b + 3 * c + 3 * d) * a ^ 2 + (3 * a + b + 3 * c + 3 * d) * b ^ 2 + (3 * a + 3 * b + c + 3 * d) * c ^ 2 +\n (3 * a + 3 * b + 3 * c + d) * d ^ 2 +\n (6 * a * b * c + 6 * a * b * d + 6 * a * c * d + 6 * b * c * d) =\n (a + b + c + d) ^ 3",
"after_state": "No Goals!"
},
{
"line": "simp [h1]",
"before_state": "a b c d : ℝ\nhd0 : 0 < d\nhdc : d ≤ c\nhcb : c ≤ b\nhba : b ≤ a\nh1 : a + b + c + d = 1\nhp : a ^ a * b ^ b * c ^ c * d ^ d ≤ a * a + b * b + c * c + d * d\n⊢ (a + b + c + d) ^ 3 = 1",
"after_state": "No Goals!"
}
] |
theorem sqrt_two_mul_sub_one_le_one : sqrt (2 * x - 1) ≤ 1 ↔ x ≤ 1 := by
simp [sqrt_le_iff, ← two_mul]
|
/root/DuelModelResearch/mathlib4/Archive/Imo/Imo1959Q2.lean
|
{
"open": [
"Set Real"
],
"variables": [
"{x A : ℝ}"
]
}
|
[
{
"line": "simp [sqrt_le_iff, ← two_mul]",
"before_state": "x : ℝ\n⊢ √(2 * x - 1) ≤ 1 ↔ x ≤ 1",
"after_state": "No Goals!"
}
] |
private lemma helper_5_digits {c : ℤ} (hc : 6 * 10 ^ 5 + c = 4 * (10 * c + 6)) : c = 15384 := by
omega
|
/root/DuelModelResearch/mathlib4/Archive/Imo/Imo1962Q1.lean
|
{
"open": [
"Nat"
],
"variables": []
}
|
[
{
"line": "omega",
"before_state": "c : ℤ\nhc : 6 * 10 ^ 5 + c = 4 * (10 * c + 6)\n⊢ c = 15384",
"after_state": "No Goals!"
}
] |
theorem imo1972_q5 (f g : ℝ → ℝ) (hf1 : ∀ x, ∀ y, f (x + y) + f (x - y) = 2 * f x * g y)
(hf2 : ∀ y, ‖f y‖ ≤ 1) (hf3 : ∃ x, f x ≠ 0) (y : ℝ) : ‖g y‖ ≤ 1 := by
-- Suppose the conclusion does not hold.
by_contra! hneg
set S := Set.range fun x => ‖f x‖
-- Introduce `k`, the supremum of `f`.
let k : ℝ := sSup S
-- Show that `‖f x‖ ≤ k`.
have hk₁ : ∀ x, ‖f x‖ ≤ k := by
have h : BddAbove S := ⟨1, Set.forall_mem_range.mpr hf2⟩
intro x
exact le_csSup h (Set.mem_range_self x)
-- Show that `2 * (‖f x‖ * ‖g y‖) ≤ 2 * k`.
have hk₂ : ∀ x, 2 * (‖f x‖ * ‖g y‖) ≤ 2 * k := fun x ↦
calc
2 * (‖f x‖ * ‖g y‖) = ‖2 * f x * g y‖ := by simp [abs_mul, mul_assoc]
_ = ‖f (x + y) + f (x - y)‖ := by rw [hf1]
_ ≤ ‖f (x + y)‖ + ‖f (x - y)‖ := norm_add_le _ _
_ ≤ k + k := add_le_add (hk₁ _) (hk₁ _)
_ = 2 * k := (two_mul _).symm
set k' := k / ‖g y‖
-- Demonstrate that `k' < k` using `hneg`.
have H₁ : k' < k := by
have h₁ : 0 < k := by
obtain ⟨x, hx⟩ := hf3
calc
0 < ‖f x‖ := norm_pos_iff.mpr hx
_ ≤ k := hk₁ x
rw [div_lt_iff₀]
· apply lt_mul_of_one_lt_right h₁ hneg
· exact zero_lt_one.trans hneg
-- Demonstrate that `k ≤ k'` using `hk₂`.
have H₂ : k ≤ k' := by
have h₁ : ∃ x : ℝ, x ∈ S := by use ‖f 0‖; exact Set.mem_range_self 0
have h₂ : ∀ x, ‖f x‖ ≤ k' := by
intro x
rw [le_div_iff₀]
· apply (mul_le_mul_left zero_lt_two).mp (hk₂ x)
· exact zero_lt_one.trans hneg
apply csSup_le h₁
rintro y' ⟨yy, rfl⟩
exact h₂ yy
-- Conclude by obtaining a contradiction, `k' < k'`.
apply lt_irrefl k'
calc
k' < k := H₁
_ ≤ k' := H₂
|
/root/DuelModelResearch/mathlib4/Archive/Imo/Imo1972Q5.lean
|
{
"open": [],
"variables": []
}
|
[
{
"line": "by_contra! hneg",
"before_state": "f g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : ∀ (y : ℝ), ‖f y‖ ≤ 1\nhf3 : ∃ x, f x ≠ 0\ny : ℝ\n⊢ ‖g y‖ ≤ 1",
"after_state": "f g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : ∀ (y : ℝ), ‖f y‖ ≤ 1\nhf3 : ∃ x, f x ≠ 0\ny : ℝ\nhneg : 1 < ‖g y‖\n⊢ False"
},
{
"line": "by_contra hneg",
"before_state": "f g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : ∀ (y : ℝ), ‖f y‖ ≤ 1\nhf3 : ∃ x, f x ≠ 0\ny : ℝ\n⊢ ‖g y‖ ≤ 1",
"after_state": "f g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : ∀ (y : ℝ), ‖f y‖ ≤ 1\nhf3 : ∃ x, f x ≠ 0\ny : ℝ\nhneg : ¬‖g y‖ ≤ 1\n⊢ False"
},
{
"line": "first\n| guard_target = Not✝ _; intro hneg\n| refine (Decidable.byContradiction✝ fun hneg => ?_ :)\n| refine (Classical.byContradiction✝ fun hneg => ?_ :)",
"before_state": "f g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : ∀ (y : ℝ), ‖f y‖ ≤ 1\nhf3 : ∃ x, f x ≠ 0\ny : ℝ\n⊢ ‖g y‖ ≤ 1",
"after_state": "f g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : ∀ (y : ℝ), ‖f y‖ ≤ 1\nhf3 : ∃ x, f x ≠ 0\ny : ℝ\nhneg : ¬‖g y‖ ≤ 1\n⊢ False"
},
{
"line": "guard_target = Not✝ _",
"before_state": "f g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : ∀ (y : ℝ), ‖f y‖ ≤ 1\nhf3 : ∃ x, f x ≠ 0\ny : ℝ\n⊢ ‖g y‖ ≤ 1",
"after_state": "f g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : ∀ (y : ℝ), ‖f y‖ ≤ 1\nhf3 : ∃ x, f x ≠ 0\ny : ℝ\n⊢ ‖g y‖ ≤ 1"
},
{
"line": "refine (Decidable.byContradiction✝ fun hneg => ?_ :)",
"before_state": "f g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : ∀ (y : ℝ), ‖f y‖ ≤ 1\nhf3 : ∃ x, f x ≠ 0\ny : ℝ\n⊢ ‖g y‖ ≤ 1",
"after_state": "f g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : ∀ (y : ℝ), ‖f y‖ ≤ 1\nhf3 : ∃ x, f x ≠ 0\ny : ℝ\n⊢ ‖g y‖ ≤ 1"
},
{
"line": "refine (Classical.byContradiction✝ fun hneg => ?_ :)",
"before_state": "f g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : ∀ (y : ℝ), ‖f y‖ ≤ 1\nhf3 : ∃ x, f x ≠ 0\ny : ℝ\n⊢ ‖g y‖ ≤ 1",
"after_state": "f g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : ∀ (y : ℝ), ‖f y‖ ≤ 1\nhf3 : ∃ x, f x ≠ 0\ny : ℝ\nhneg : ¬‖g y‖ ≤ 1\n⊢ False"
},
{
"line": "try push_neg at hneg",
"before_state": "f g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : ∀ (y : ℝ), ‖f y‖ ≤ 1\nhf3 : ∃ x, f x ≠ 0\ny : ℝ\nhneg : ¬‖g y‖ ≤ 1\n⊢ False",
"after_state": "f g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : ∀ (y : ℝ), ‖f y‖ ≤ 1\nhf3 : ∃ x, f x ≠ 0\ny : ℝ\nhneg : 1 < ‖g y‖\n⊢ False"
},
{
"line": "first\n| push_neg at hneg\n| skip",
"before_state": "f g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : ∀ (y : ℝ), ‖f y‖ ≤ 1\nhf3 : ∃ x, f x ≠ 0\ny : ℝ\nhneg : ¬‖g y‖ ≤ 1\n⊢ False",
"after_state": "f g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : ∀ (y : ℝ), ‖f y‖ ≤ 1\nhf3 : ∃ x, f x ≠ 0\ny : ℝ\nhneg : 1 < ‖g y‖\n⊢ False"
},
{
"line": "push_neg at hneg",
"before_state": "f g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : ∀ (y : ℝ), ‖f y‖ ≤ 1\nhf3 : ∃ x, f x ≠ 0\ny : ℝ\nhneg : ¬‖g y‖ ≤ 1\n⊢ False",
"after_state": "f g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : ∀ (y : ℝ), ‖f y‖ ≤ 1\nhf3 : ∃ x, f x ≠ 0\ny : ℝ\nhneg : 1 < ‖g y‖\n⊢ False"
},
{
"line": "set S :=\n Set.range fun x =>\n ‖f x‖\n -- Introduce `k`, the supremum of `f`.",
"before_state": "f g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : ∀ (y : ℝ), ‖f y‖ ≤ 1\nhf3 : ∃ x, f x ≠ 0\ny : ℝ\nhneg : 1 < ‖g y‖\n⊢ False",
"after_state": "f g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : ∀ (y : ℝ), ‖f y‖ ≤ 1\nhf3 : ∃ x, f x ≠ 0\ny : ℝ\nhneg : 1 < ‖g y‖\nS : Set ℝ := Set.range fun x => ‖f x‖\n⊢ False"
},
{
"line": "try rewrite [show ?m✝ = S from rfl✝] at *",
"before_state": "f g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : ∀ (y : ℝ), ‖f y‖ ≤ 1\nhf3 : ∃ x, f x ≠ 0\ny : ℝ\nhneg : 1 < ‖g y‖\nS : Set ℝ := Set.range fun x => ‖f x‖\n⊢ False",
"after_state": "f g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : ∀ (y : ℝ), ‖f y‖ ≤ 1\nhf3 : ∃ x, f x ≠ 0\ny : ℝ\nhneg : 1 < ‖g y‖\nS : Set ℝ := Set.range fun x => ‖f x‖\n⊢ False"
},
{
"line": "first\n| rewrite [show ?m✝ = S from rfl✝] at *\n| skip",
"before_state": "f g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : ∀ (y : ℝ), ‖f y‖ ≤ 1\nhf3 : ∃ x, f x ≠ 0\ny : ℝ\nhneg : 1 < ‖g y‖\nS : Set ℝ := Set.range fun x => ‖f x‖\n⊢ False",
"after_state": "f g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : ∀ (y : ℝ), ‖f y‖ ≤ 1\nhf3 : ∃ x, f x ≠ 0\ny : ℝ\nhneg : 1 < ‖g y‖\nS : Set ℝ := Set.range fun x => ‖f x‖\n⊢ False"
},
{
"line": "rewrite [show ?m✝ = S from rfl✝] at *",
"before_state": "f g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : ∀ (y : ℝ), ‖f y‖ ≤ 1\nhf3 : ∃ x, f x ≠ 0\ny : ℝ\nhneg : 1 < ‖g y‖\nS : Set ℝ := Set.range fun x => ‖f x‖\n⊢ False",
"after_state": "f g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : ∀ (y : ℝ), ‖f y‖ ≤ 1\nhf3 : ∃ x, f x ≠ 0\ny : ℝ\nhneg : 1 < ‖g y‖\nS : Set ℝ := Set.range fun x => ‖f x‖\n⊢ False"
},
{
"line": "skip",
"before_state": "f g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : ∀ (y : ℝ), ‖f y‖ ≤ 1\nhf3 : ∃ x, f x ≠ 0\ny : ℝ\nhneg : 1 < ‖g y‖\nS : Set ℝ := Set.range fun x => ‖f x‖\n⊢ False",
"after_state": "f g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : ∀ (y : ℝ), ‖f y‖ ≤ 1\nhf3 : ∃ x, f x ≠ 0\ny : ℝ\nhneg : 1 < ‖g y‖\nS : Set ℝ := Set.range fun x => ‖f x‖\n⊢ False"
},
{
"line": "let k : ℝ := sSup S",
"before_state": "f g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : ∀ (y : ℝ), ‖f y‖ ≤ 1\nhf3 : ∃ x, f x ≠ 0\ny : ℝ\nhneg : 1 < ‖g y‖\nS : Set ℝ := Set.range fun x => ‖f x‖\n⊢ False",
"after_state": "f g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : ∀ (y : ℝ), ‖f y‖ ≤ 1\nhf3 : ∃ x, f x ≠ 0\ny : ℝ\nhneg : 1 < ‖g y‖\nS : Set ℝ := Set.range fun x => ‖f x‖\nk : ℝ := sSup S\n⊢ False"
},
{
"line": "refine_lift\n let k : ℝ := sSup S;\n ?_",
"before_state": "f g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : ∀ (y : ℝ), ‖f y‖ ≤ 1\nhf3 : ∃ x, f x ≠ 0\ny : ℝ\nhneg : 1 < ‖g y‖\nS : Set ℝ := Set.range fun x => ‖f x‖\n⊢ False",
"after_state": "f g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : ∀ (y : ℝ), ‖f y‖ ≤ 1\nhf3 : ∃ x, f x ≠ 0\ny : ℝ\nhneg : 1 < ‖g y‖\nS : Set ℝ := Set.range fun x => ‖f x‖\nk : ℝ := sSup S\n⊢ False"
},
{
"line": "focus\n (refine\n no_implicit_lambda%\n (let k : ℝ := sSup S;\n ?_);\n rotate_right)",
"before_state": "f g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : ∀ (y : ℝ), ‖f y‖ ≤ 1\nhf3 : ∃ x, f x ≠ 0\ny : ℝ\nhneg : 1 < ‖g y‖\nS : Set ℝ := Set.range fun x => ‖f x‖\n⊢ False",
"after_state": "f g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : ∀ (y : ℝ), ‖f y‖ ≤ 1\nhf3 : ∃ x, f x ≠ 0\ny : ℝ\nhneg : 1 < ‖g y‖\nS : Set ℝ := Set.range fun x => ‖f x‖\nk : ℝ := sSup S\n⊢ False"
},
{
"line": "refine\n no_implicit_lambda%\n (let k : ℝ := sSup S;\n ?_)",
"before_state": "f g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : ∀ (y : ℝ), ‖f y‖ ≤ 1\nhf3 : ∃ x, f x ≠ 0\ny : ℝ\nhneg : 1 < ‖g y‖\nS : Set ℝ := Set.range fun x => ‖f x‖\n⊢ False",
"after_state": "f g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : ∀ (y : ℝ), ‖f y‖ ≤ 1\nhf3 : ∃ x, f x ≠ 0\ny : ℝ\nhneg : 1 < ‖g y‖\nS : Set ℝ := Set.range fun x => ‖f x‖\nk : ℝ := sSup S\n⊢ False"
},
{
"line": "rotate_right",
"before_state": "f g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : ∀ (y : ℝ), ‖f y‖ ≤ 1\nhf3 : ∃ x, f x ≠ 0\ny : ℝ\nhneg : 1 < ‖g y‖\nS : Set ℝ := Set.range fun x => ‖f x‖\nk : ℝ := sSup S\n⊢ False",
"after_state": "f g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : ∀ (y : ℝ), ‖f y‖ ≤ 1\nhf3 : ∃ x, f x ≠ 0\ny : ℝ\nhneg : 1 < ‖g y‖\nS : Set ℝ := Set.range fun x => ‖f x‖\nk : ℝ := sSup S\n⊢ False"
},
{
"line": "have hk₁ : ∀ x, ‖f x‖ ≤ k := by\n have h : BddAbove S := ⟨1, Set.forall_mem_range.mpr hf2⟩\n intro x\n exact\n le_csSup h\n (Set.mem_range_self x)\n -- Show that `2 * (‖f x‖ * ‖g y‖) ≤ 2 * k`.",
"before_state": "f g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : ∀ (y : ℝ), ‖f y‖ ≤ 1\nhf3 : ∃ x, f x ≠ 0\ny : ℝ\nhneg : 1 < ‖g y‖\nS : Set ℝ := Set.range fun x => ‖f x‖\nk : ℝ := sSup S\n⊢ False",
"after_state": "f g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : ∀ (y : ℝ), ‖f y‖ ≤ 1\nhf3 : ∃ x, f x ≠ 0\ny : ℝ\nhneg : 1 < ‖g y‖\nS : Set ℝ := Set.range fun x => ‖f x‖\nk : ℝ := sSup S\nhk₁ : ∀ (x : ℝ), ‖f x‖ ≤ k\n⊢ False"
},
{
"line": "focus\n refine\n no_implicit_lambda%\n (have hk₁ : ∀ x, ‖f x‖ ≤ k := ?body✝;\n ?_)\n case body✝ =>\n with_annotate_state\"by\"\n ( have h : BddAbove S := ⟨1, Set.forall_mem_range.mpr hf2⟩\n intro x\n exact\n le_csSup h\n (Set.mem_range_self x)\n -- Show that `2 * (‖f x‖ * ‖g y‖) ≤ 2 * k`.)",
"before_state": "f g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : ∀ (y : ℝ), ‖f y‖ ≤ 1\nhf3 : ∃ x, f x ≠ 0\ny : ℝ\nhneg : 1 < ‖g y‖\nS : Set ℝ := Set.range fun x => ‖f x‖\nk : ℝ := sSup S\n⊢ False",
"after_state": "f g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : ∀ (y : ℝ), ‖f y‖ ≤ 1\nhf3 : ∃ x, f x ≠ 0\ny : ℝ\nhneg : 1 < ‖g y‖\nS : Set ℝ := Set.range fun x => ‖f x‖\nk : ℝ := sSup S\nhk₁ : ∀ (x : ℝ), ‖f x‖ ≤ k\n⊢ False"
},
{
"line": "refine\n no_implicit_lambda%\n (have hk₁ : ∀ x, ‖f x‖ ≤ k := ?body✝;\n ?_)",
"before_state": "f g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : ∀ (y : ℝ), ‖f y‖ ≤ 1\nhf3 : ∃ x, f x ≠ 0\ny : ℝ\nhneg : 1 < ‖g y‖\nS : Set ℝ := Set.range fun x => ‖f x‖\nk : ℝ := sSup S\n⊢ False",
"after_state": "case body\nf g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : ∀ (y : ℝ), ‖f y‖ ≤ 1\nhf3 : ∃ x, f x ≠ 0\ny : ℝ\nhneg : 1 < ‖g y‖\nS : Set ℝ := Set.range fun x => ‖f x‖\nk : ℝ := sSup S\n⊢ ∀ (x : ℝ), ‖f x‖ ≤ k\n---\nf g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : ∀ (y : ℝ), ‖f y‖ ≤ 1\nhf3 : ∃ x, f x ≠ 0\ny : ℝ\nhneg : 1 < ‖g y‖\nS : Set ℝ := Set.range fun x => ‖f x‖\nk : ℝ := sSup S\nhk₁ : ∀ (x : ℝ), ‖f x‖ ≤ k\n⊢ False"
},
{
"line": "case body✝ =>\n with_annotate_state\"by\"\n ( have h : BddAbove S := ⟨1, Set.forall_mem_range.mpr hf2⟩\n intro x\n exact\n le_csSup h\n (Set.mem_range_self x)\n -- Show that `2 * (‖f x‖ * ‖g y‖) ≤ 2 * k`.)",
"before_state": "case body\nf g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : ∀ (y : ℝ), ‖f y‖ ≤ 1\nhf3 : ∃ x, f x ≠ 0\ny : ℝ\nhneg : 1 < ‖g y‖\nS : Set ℝ := Set.range fun x => ‖f x‖\nk : ℝ := sSup S\n⊢ ∀ (x : ℝ), ‖f x‖ ≤ k\n---\nf g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : ∀ (y : ℝ), ‖f y‖ ≤ 1\nhf3 : ∃ x, f x ≠ 0\ny : ℝ\nhneg : 1 < ‖g y‖\nS : Set ℝ := Set.range fun x => ‖f x‖\nk : ℝ := sSup S\nhk₁ : ∀ (x : ℝ), ‖f x‖ ≤ k\n⊢ False",
"after_state": "f g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : ∀ (y : ℝ), ‖f y‖ ≤ 1\nhf3 : ∃ x, f x ≠ 0\ny : ℝ\nhneg : 1 < ‖g y‖\nS : Set ℝ := Set.range fun x => ‖f x‖\nk : ℝ := sSup S\nhk₁ : ∀ (x : ℝ), ‖f x‖ ≤ k\n⊢ False"
},
{
"line": "with_annotate_state\"by\"\n ( have h : BddAbove S := ⟨1, Set.forall_mem_range.mpr hf2⟩\n intro x\n exact\n le_csSup h\n (Set.mem_range_self x)\n -- Show that `2 * (‖f x‖ * ‖g y‖) ≤ 2 * k`.)",
"before_state": "f g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : ∀ (y : ℝ), ‖f y‖ ≤ 1\nhf3 : ∃ x, f x ≠ 0\ny : ℝ\nhneg : 1 < ‖g y‖\nS : Set ℝ := Set.range fun x => ‖f x‖\nk : ℝ := sSup S\n⊢ ∀ (x : ℝ), ‖f x‖ ≤ k",
"after_state": "No Goals!"
},
{
"line": "have h : BddAbove S := ⟨1, Set.forall_mem_range.mpr hf2⟩",
"before_state": "f g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : ∀ (y : ℝ), ‖f y‖ ≤ 1\nhf3 : ∃ x, f x ≠ 0\ny : ℝ\nhneg : 1 < ‖g y‖\nS : Set ℝ := Set.range fun x => ‖f x‖\nk : ℝ := sSup S\n⊢ ∀ (x : ℝ), ‖f x‖ ≤ k",
"after_state": "f g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : ∀ (y : ℝ), ‖f y‖ ≤ 1\nhf3 : ∃ x, f x ≠ 0\ny : ℝ\nhneg : 1 < ‖g y‖\nS : Set ℝ := Set.range fun x => ‖f x‖\nk : ℝ := sSup S\nh : BddAbove S\n⊢ ∀ (x : ℝ), ‖f x‖ ≤ k"
},
{
"line": "refine_lift\n have h : BddAbove S := ⟨1, Set.forall_mem_range.mpr hf2⟩;\n ?_",
"before_state": "f g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : ∀ (y : ℝ), ‖f y‖ ≤ 1\nhf3 : ∃ x, f x ≠ 0\ny : ℝ\nhneg : 1 < ‖g y‖\nS : Set ℝ := Set.range fun x => ‖f x‖\nk : ℝ := sSup S\n⊢ ∀ (x : ℝ), ‖f x‖ ≤ k",
"after_state": "f g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : ∀ (y : ℝ), ‖f y‖ ≤ 1\nhf3 : ∃ x, f x ≠ 0\ny : ℝ\nhneg : 1 < ‖g y‖\nS : Set ℝ := Set.range fun x => ‖f x‖\nk : ℝ := sSup S\nh : BddAbove S\n⊢ ∀ (x : ℝ), ‖f x‖ ≤ k"
},
{
"line": "focus\n (refine\n no_implicit_lambda%\n (have h : BddAbove S := ⟨1, Set.forall_mem_range.mpr hf2⟩;\n ?_);\n rotate_right)",
"before_state": "f g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : ∀ (y : ℝ), ‖f y‖ ≤ 1\nhf3 : ∃ x, f x ≠ 0\ny : ℝ\nhneg : 1 < ‖g y‖\nS : Set ℝ := Set.range fun x => ‖f x‖\nk : ℝ := sSup S\n⊢ ∀ (x : ℝ), ‖f x‖ ≤ k",
"after_state": "f g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : ∀ (y : ℝ), ‖f y‖ ≤ 1\nhf3 : ∃ x, f x ≠ 0\ny : ℝ\nhneg : 1 < ‖g y‖\nS : Set ℝ := Set.range fun x => ‖f x‖\nk : ℝ := sSup S\nh : BddAbove S\n⊢ ∀ (x : ℝ), ‖f x‖ ≤ k"
},
{
"line": "refine\n no_implicit_lambda%\n (have h : BddAbove S := ⟨1, Set.forall_mem_range.mpr hf2⟩;\n ?_)",
"before_state": "f g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : ∀ (y : ℝ), ‖f y‖ ≤ 1\nhf3 : ∃ x, f x ≠ 0\ny : ℝ\nhneg : 1 < ‖g y‖\nS : Set ℝ := Set.range fun x => ‖f x‖\nk : ℝ := sSup S\n⊢ ∀ (x : ℝ), ‖f x‖ ≤ k",
"after_state": "f g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : ∀ (y : ℝ), ‖f y‖ ≤ 1\nhf3 : ∃ x, f x ≠ 0\ny : ℝ\nhneg : 1 < ‖g y‖\nS : Set ℝ := Set.range fun x => ‖f x‖\nk : ℝ := sSup S\nh : BddAbove S\n⊢ ∀ (x : ℝ), ‖f x‖ ≤ k"
},
{
"line": "rotate_right",
"before_state": "f g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : ∀ (y : ℝ), ‖f y‖ ≤ 1\nhf3 : ∃ x, f x ≠ 0\ny : ℝ\nhneg : 1 < ‖g y‖\nS : Set ℝ := Set.range fun x => ‖f x‖\nk : ℝ := sSup S\nh : BddAbove S\n⊢ ∀ (x : ℝ), ‖f x‖ ≤ k",
"after_state": "f g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : ∀ (y : ℝ), ‖f y‖ ≤ 1\nhf3 : ∃ x, f x ≠ 0\ny : ℝ\nhneg : 1 < ‖g y‖\nS : Set ℝ := Set.range fun x => ‖f x‖\nk : ℝ := sSup S\nh : BddAbove S\n⊢ ∀ (x : ℝ), ‖f x‖ ≤ k"
},
{
"line": "intro x",
"before_state": "f g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : ∀ (y : ℝ), ‖f y‖ ≤ 1\nhf3 : ∃ x, f x ≠ 0\ny : ℝ\nhneg : 1 < ‖g y‖\nS : Set ℝ := Set.range fun x => ‖f x‖\nk : ℝ := sSup S\nh : BddAbove S\n⊢ ∀ (x : ℝ), ‖f x‖ ≤ k",
"after_state": "f g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : ∀ (y : ℝ), ‖f y‖ ≤ 1\nhf3 : ∃ x, f x ≠ 0\ny : ℝ\nhneg : 1 < ‖g y‖\nS : Set ℝ := Set.range fun x => ‖f x‖\nk : ℝ := sSup S\nh : BddAbove S\nx : ℝ\n⊢ ‖f x‖ ≤ k"
},
{
"line": "exact\n le_csSup h\n (Set.mem_range_self x)\n -- Show that `2 * (‖f x‖ * ‖g y‖) ≤ 2 * k`.",
"before_state": "f g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : ∀ (y : ℝ), ‖f y‖ ≤ 1\nhf3 : ∃ x, f x ≠ 0\ny : ℝ\nhneg : 1 < ‖g y‖\nS : Set ℝ := Set.range fun x => ‖f x‖\nk : ℝ := sSup S\nh : BddAbove S\nx : ℝ\n⊢ ‖f x‖ ≤ k",
"after_state": "No Goals!"
},
{
"line": "have hk₂ : ∀ x, 2 * (‖f x‖ * ‖g y‖) ≤ 2 * k := fun x ↦\n calc\n 2 * (‖f x‖ * ‖g y‖) = ‖2 * f x * g y‖ := by simp [abs_mul, mul_assoc]\n _ = ‖f (x + y) + f (x - y)‖ := by rw [hf1]\n _ ≤ ‖f (x + y)‖ + ‖f (x - y)‖ := (norm_add_le _ _)\n _ ≤ k + k := (add_le_add (hk₁ _) (hk₁ _))\n _ = 2 * k := (two_mul _).symm",
"before_state": "f g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : ∀ (y : ℝ), ‖f y‖ ≤ 1\nhf3 : ∃ x, f x ≠ 0\ny : ℝ\nhneg : 1 < ‖g y‖\nS : Set ℝ := Set.range fun x => ‖f x‖\nk : ℝ := sSup S\nhk₁ : ∀ (x : ℝ), ‖f x‖ ≤ k\n⊢ False",
"after_state": "f g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : ∀ (y : ℝ), ‖f y‖ ≤ 1\nhf3 : ∃ x, f x ≠ 0\ny : ℝ\nhneg : 1 < ‖g y‖\nS : Set ℝ := Set.range fun x => ‖f x‖\nk : ℝ := sSup S\nhk₁ : ∀ (x : ℝ), ‖f x‖ ≤ k\nhk₂ : ∀ (x : ℝ), 2 * (‖f x‖ * ‖g y‖) ≤ 2 * k\n⊢ False"
},
{
"line": "refine_lift\n have hk₂ : ∀ x, 2 * (‖f x‖ * ‖g y‖) ≤ 2 * k := fun x ↦\n calc\n 2 * (‖f x‖ * ‖g y‖) = ‖2 * f x * g y‖ := by simp [abs_mul, mul_assoc]\n _ = ‖f (x + y) + f (x - y)‖ := by rw [hf1]\n _ ≤ ‖f (x + y)‖ + ‖f (x - y)‖ := (norm_add_le _ _)\n _ ≤ k + k := (add_le_add (hk₁ _) (hk₁ _))\n _ = 2 * k := (two_mul _).symm;\n ?_",
"before_state": "f g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : ∀ (y : ℝ), ‖f y‖ ≤ 1\nhf3 : ∃ x, f x ≠ 0\ny : ℝ\nhneg : 1 < ‖g y‖\nS : Set ℝ := Set.range fun x => ‖f x‖\nk : ℝ := sSup S\nhk₁ : ∀ (x : ℝ), ‖f x‖ ≤ k\n⊢ False",
"after_state": "f g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : ∀ (y : ℝ), ‖f y‖ ≤ 1\nhf3 : ∃ x, f x ≠ 0\ny : ℝ\nhneg : 1 < ‖g y‖\nS : Set ℝ := Set.range fun x => ‖f x‖\nk : ℝ := sSup S\nhk₁ : ∀ (x : ℝ), ‖f x‖ ≤ k\nhk₂ : ∀ (x : ℝ), 2 * (‖f x‖ * ‖g y‖) ≤ 2 * k\n⊢ False"
},
{
"line": "focus\n (refine\n no_implicit_lambda%\n (have hk₂ : ∀ x, 2 * (‖f x‖ * ‖g y‖) ≤ 2 * k := fun x ↦\n calc\n 2 * (‖f x‖ * ‖g y‖) = ‖2 * f x * g y‖ := by simp [abs_mul, mul_assoc]\n _ = ‖f (x + y) + f (x - y)‖ := by rw [hf1]\n _ ≤ ‖f (x + y)‖ + ‖f (x - y)‖ := (norm_add_le _ _)\n _ ≤ k + k := (add_le_add (hk₁ _) (hk₁ _))\n _ = 2 * k := (two_mul _).symm;\n ?_);\n rotate_right)",
"before_state": "f g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : ∀ (y : ℝ), ‖f y‖ ≤ 1\nhf3 : ∃ x, f x ≠ 0\ny : ℝ\nhneg : 1 < ‖g y‖\nS : Set ℝ := Set.range fun x => ‖f x‖\nk : ℝ := sSup S\nhk₁ : ∀ (x : ℝ), ‖f x‖ ≤ k\n⊢ False",
"after_state": "f g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : ∀ (y : ℝ), ‖f y‖ ≤ 1\nhf3 : ∃ x, f x ≠ 0\ny : ℝ\nhneg : 1 < ‖g y‖\nS : Set ℝ := Set.range fun x => ‖f x‖\nk : ℝ := sSup S\nhk₁ : ∀ (x : ℝ), ‖f x‖ ≤ k\nhk₂ : ∀ (x : ℝ), 2 * (‖f x‖ * ‖g y‖) ≤ 2 * k\n⊢ False"
},
{
"line": "refine\n no_implicit_lambda%\n (have hk₂ : ∀ x, 2 * (‖f x‖ * ‖g y‖) ≤ 2 * k := fun x ↦\n calc\n 2 * (‖f x‖ * ‖g y‖) = ‖2 * f x * g y‖ := by simp [abs_mul, mul_assoc]\n _ = ‖f (x + y) + f (x - y)‖ := by rw [hf1]\n _ ≤ ‖f (x + y)‖ + ‖f (x - y)‖ := (norm_add_le _ _)\n _ ≤ k + k := (add_le_add (hk₁ _) (hk₁ _))\n _ = 2 * k := (two_mul _).symm;\n ?_)",
"before_state": "f g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : ∀ (y : ℝ), ‖f y‖ ≤ 1\nhf3 : ∃ x, f x ≠ 0\ny : ℝ\nhneg : 1 < ‖g y‖\nS : Set ℝ := Set.range fun x => ‖f x‖\nk : ℝ := sSup S\nhk₁ : ∀ (x : ℝ), ‖f x‖ ≤ k\n⊢ False",
"after_state": "f g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : ∀ (y : ℝ), ‖f y‖ ≤ 1\nhf3 : ∃ x, f x ≠ 0\ny : ℝ\nhneg : 1 < ‖g y‖\nS : Set ℝ := Set.range fun x => ‖f x‖\nk : ℝ := sSup S\nhk₁ : ∀ (x : ℝ), ‖f x‖ ≤ k\nhk₂ : ∀ (x : ℝ), 2 * (‖f x‖ * ‖g y‖) ≤ 2 * k\n⊢ False"
},
{
"line": "simp [abs_mul, mul_assoc]",
"before_state": "f g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : ∀ (y : ℝ), ‖f y‖ ≤ 1\nhf3 : ∃ x, f x ≠ 0\ny : ℝ\nhneg : 1 < ‖g y‖\nS : Set ℝ := Set.range fun x => ‖f x‖\nk : ℝ := sSup S\nhk₁ : ∀ (x : ℝ), ‖f x‖ ≤ k\nx : ℝ\n⊢ 2 * (‖f x‖ * ‖g y‖) = ‖2 * f x * g y‖",
"after_state": "No Goals!"
},
{
"line": "rw [hf1]",
"before_state": "f g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : ∀ (y : ℝ), ‖f y‖ ≤ 1\nhf3 : ∃ x, f x ≠ 0\ny : ℝ\nhneg : 1 < ‖g y‖\nS : Set ℝ := Set.range fun x => ‖f x‖\nk : ℝ := sSup S\nhk₁ : ∀ (x : ℝ), ‖f x‖ ≤ k\nx : ℝ\n⊢ ‖2 * f x * g y‖ = ‖f (x + y) + f (x - y)‖",
"after_state": "No Goals!"
},
{
"line": "rewrite [hf1]",
"before_state": "f g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : ∀ (y : ℝ), ‖f y‖ ≤ 1\nhf3 : ∃ x, f x ≠ 0\ny : ℝ\nhneg : 1 < ‖g y‖\nS : Set ℝ := Set.range fun x => ‖f x‖\nk : ℝ := sSup S\nhk₁ : ∀ (x : ℝ), ‖f x‖ ≤ k\nx : ℝ\n⊢ ‖2 * f x * g y‖ = ‖f (x + y) + f (x - y)‖",
"after_state": "f g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : ∀ (y : ℝ), ‖f y‖ ≤ 1\nhf3 : ∃ x, f x ≠ 0\ny : ℝ\nhneg : 1 < ‖g y‖\nS : Set ℝ := Set.range fun x => ‖f x‖\nk : ℝ := sSup S\nhk₁ : ∀ (x : ℝ), ‖f x‖ ≤ k\nx : ℝ\n⊢ ‖2 * f x * g y‖ = ‖2 * f x * g y‖"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "f g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : ∀ (y : ℝ), ‖f y‖ ≤ 1\nhf3 : ∃ x, f x ≠ 0\ny : ℝ\nhneg : 1 < ‖g y‖\nS : Set ℝ := Set.range fun x => ‖f x‖\nk : ℝ := sSup S\nhk₁ : ∀ (x : ℝ), ‖f x‖ ≤ k\nx : ℝ\n⊢ ‖2 * f x * g y‖ = ‖2 * f x * g y‖",
"after_state": "No Goals!"
},
{
"line": "try (with_reducible rfl)",
"before_state": "f g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : ∀ (y : ℝ), ‖f y‖ ≤ 1\nhf3 : ∃ x, f x ≠ 0\ny : ℝ\nhneg : 1 < ‖g y‖\nS : Set ℝ := Set.range fun x => ‖f x‖\nk : ℝ := sSup S\nhk₁ : ∀ (x : ℝ), ‖f x‖ ≤ k\nx : ℝ\n⊢ ‖2 * f x * g y‖ = ‖2 * f x * g y‖",
"after_state": "No Goals!"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "f g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : ∀ (y : ℝ), ‖f y‖ ≤ 1\nhf3 : ∃ x, f x ≠ 0\ny : ℝ\nhneg : 1 < ‖g y‖\nS : Set ℝ := Set.range fun x => ‖f x‖\nk : ℝ := sSup S\nhk₁ : ∀ (x : ℝ), ‖f x‖ ≤ k\nx : ℝ\n⊢ ‖2 * f x * g y‖ = ‖2 * f x * g y‖",
"after_state": "No Goals!"
},
{
"line": "with_reducible rfl",
"before_state": "f g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : ∀ (y : ℝ), ‖f y‖ ≤ 1\nhf3 : ∃ x, f x ≠ 0\ny : ℝ\nhneg : 1 < ‖g y‖\nS : Set ℝ := Set.range fun x => ‖f x‖\nk : ℝ := sSup S\nhk₁ : ∀ (x : ℝ), ‖f x‖ ≤ k\nx : ℝ\n⊢ ‖2 * f x * g y‖ = ‖2 * f x * g y‖",
"after_state": "No Goals!"
},
{
"line": "rfl",
"before_state": "f g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : ∀ (y : ℝ), ‖f y‖ ≤ 1\nhf3 : ∃ x, f x ≠ 0\ny : ℝ\nhneg : 1 < ‖g y‖\nS : Set ℝ := Set.range fun x => ‖f x‖\nk : ℝ := sSup S\nhk₁ : ∀ (x : ℝ), ‖f x‖ ≤ k\nx : ℝ\n⊢ ‖2 * f x * g y‖ = ‖2 * f x * g y‖",
"after_state": "No Goals!"
},
{
"line": "eq_refl",
"before_state": "f g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : ∀ (y : ℝ), ‖f y‖ ≤ 1\nhf3 : ∃ x, f x ≠ 0\ny : ℝ\nhneg : 1 < ‖g y‖\nS : Set ℝ := Set.range fun x => ‖f x‖\nk : ℝ := sSup S\nhk₁ : ∀ (x : ℝ), ‖f x‖ ≤ k\nx : ℝ\n⊢ ‖2 * f x * g y‖ = ‖2 * f x * g y‖",
"after_state": "No Goals!"
},
{
"line": "rotate_right",
"before_state": "f g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : ∀ (y : ℝ), ‖f y‖ ≤ 1\nhf3 : ∃ x, f x ≠ 0\ny : ℝ\nhneg : 1 < ‖g y‖\nS : Set ℝ := Set.range fun x => ‖f x‖\nk : ℝ := sSup S\nhk₁ : ∀ (x : ℝ), ‖f x‖ ≤ k\nhk₂ : ∀ (x : ℝ), 2 * (‖f x‖ * ‖g y‖) ≤ 2 * k\n⊢ False",
"after_state": "f g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : ∀ (y : ℝ), ‖f y‖ ≤ 1\nhf3 : ∃ x, f x ≠ 0\ny : ℝ\nhneg : 1 < ‖g y‖\nS : Set ℝ := Set.range fun x => ‖f x‖\nk : ℝ := sSup S\nhk₁ : ∀ (x : ℝ), ‖f x‖ ≤ k\nhk₂ : ∀ (x : ℝ), 2 * (‖f x‖ * ‖g y‖) ≤ 2 * k\n⊢ False"
},
{
"line": "set k' :=\n k /\n ‖g y‖\n -- Demonstrate that `k' < k` using `hneg`.",
"before_state": "f g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : ∀ (y : ℝ), ‖f y‖ ≤ 1\nhf3 : ∃ x, f x ≠ 0\ny : ℝ\nhneg : 1 < ‖g y‖\nS : Set ℝ := Set.range fun x => ‖f x‖\nk : ℝ := sSup S\nhk₁ : ∀ (x : ℝ), ‖f x‖ ≤ k\nhk₂ : ∀ (x : ℝ), 2 * (‖f x‖ * ‖g y‖) ≤ 2 * k\n⊢ False",
"after_state": "f g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : ∀ (y : ℝ), ‖f y‖ ≤ 1\nhf3 : ∃ x, f x ≠ 0\ny : ℝ\nhneg : 1 < ‖g y‖\nS : Set ℝ := Set.range fun x => ‖f x‖\nk : ℝ := sSup S\nhk₁ : ∀ (x : ℝ), ‖f x‖ ≤ k\nhk₂ : ∀ (x : ℝ), 2 * (‖f x‖ * ‖g y‖) ≤ 2 * k\nk' : ℝ := k / ‖g y‖\n⊢ False"
},
{
"line": "try rewrite [show ?m✝ = k' from rfl✝] at *",
"before_state": "f g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : ∀ (y : ℝ), ‖f y‖ ≤ 1\nhf3 : ∃ x, f x ≠ 0\ny : ℝ\nhneg : 1 < ‖g y‖\nS : Set ℝ := Set.range fun x => ‖f x‖\nk : ℝ := sSup S\nhk₁ : ∀ (x : ℝ), ‖f x‖ ≤ k\nhk₂ : ∀ (x : ℝ), 2 * (‖f x‖ * ‖g y‖) ≤ 2 * k\nk' : ℝ := k / ‖g y‖\n⊢ False",
"after_state": "f g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : ∀ (y : ℝ), ‖f y‖ ≤ 1\nhf3 : ∃ x, f x ≠ 0\ny : ℝ\nhneg : 1 < ‖g y‖\nS : Set ℝ := Set.range fun x => ‖f x‖\nk : ℝ := sSup S\nhk₁ : ∀ (x : ℝ), ‖f x‖ ≤ k\nhk₂ : ∀ (x : ℝ), 2 * (‖f x‖ * ‖g y‖) ≤ 2 * k\nk' : ℝ := k / ‖g y‖\n⊢ False"
},
{
"line": "first\n| rewrite [show ?m✝ = k' from rfl✝] at *\n| skip",
"before_state": "f g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : ∀ (y : ℝ), ‖f y‖ ≤ 1\nhf3 : ∃ x, f x ≠ 0\ny : ℝ\nhneg : 1 < ‖g y‖\nS : Set ℝ := Set.range fun x => ‖f x‖\nk : ℝ := sSup S\nhk₁ : ∀ (x : ℝ), ‖f x‖ ≤ k\nhk₂ : ∀ (x : ℝ), 2 * (‖f x‖ * ‖g y‖) ≤ 2 * k\nk' : ℝ := k / ‖g y‖\n⊢ False",
"after_state": "f g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : ∀ (y : ℝ), ‖f y‖ ≤ 1\nhf3 : ∃ x, f x ≠ 0\ny : ℝ\nhneg : 1 < ‖g y‖\nS : Set ℝ := Set.range fun x => ‖f x‖\nk : ℝ := sSup S\nhk₁ : ∀ (x : ℝ), ‖f x‖ ≤ k\nhk₂ : ∀ (x : ℝ), 2 * (‖f x‖ * ‖g y‖) ≤ 2 * k\nk' : ℝ := k / ‖g y‖\n⊢ False"
},
{
"line": "rewrite [show ?m✝ = k' from rfl✝] at *",
"before_state": "f g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : ∀ (y : ℝ), ‖f y‖ ≤ 1\nhf3 : ∃ x, f x ≠ 0\ny : ℝ\nhneg : 1 < ‖g y‖\nS : Set ℝ := Set.range fun x => ‖f x‖\nk : ℝ := sSup S\nhk₁ : ∀ (x : ℝ), ‖f x‖ ≤ k\nhk₂ : ∀ (x : ℝ), 2 * (‖f x‖ * ‖g y‖) ≤ 2 * k\nk' : ℝ := k / ‖g y‖\n⊢ False",
"after_state": "f g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : ∀ (y : ℝ), ‖f y‖ ≤ 1\nhf3 : ∃ x, f x ≠ 0\ny : ℝ\nhneg : 1 < ‖g y‖\nS : Set ℝ := Set.range fun x => ‖f x‖\nk : ℝ := sSup S\nhk₁ : ∀ (x : ℝ), ‖f x‖ ≤ k\nhk₂ : ∀ (x : ℝ), 2 * (‖f x‖ * ‖g y‖) ≤ 2 * k\nk' : ℝ := k / ‖g y‖\n⊢ False"
},
{
"line": "skip",
"before_state": "f g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : ∀ (y : ℝ), ‖f y‖ ≤ 1\nhf3 : ∃ x, f x ≠ 0\ny : ℝ\nhneg : 1 < ‖g y‖\nS : Set ℝ := Set.range fun x => ‖f x‖\nk : ℝ := sSup S\nhk₁ : ∀ (x : ℝ), ‖f x‖ ≤ k\nhk₂ : ∀ (x : ℝ), 2 * (‖f x‖ * ‖g y‖) ≤ 2 * k\nk' : ℝ := k / ‖g y‖\n⊢ False",
"after_state": "f g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : ∀ (y : ℝ), ‖f y‖ ≤ 1\nhf3 : ∃ x, f x ≠ 0\ny : ℝ\nhneg : 1 < ‖g y‖\nS : Set ℝ := Set.range fun x => ‖f x‖\nk : ℝ := sSup S\nhk₁ : ∀ (x : ℝ), ‖f x‖ ≤ k\nhk₂ : ∀ (x : ℝ), 2 * (‖f x‖ * ‖g y‖) ≤ 2 * k\nk' : ℝ := k / ‖g y‖\n⊢ False"
},
{
"line": "have H₁ : k' < k :=\n by\n have h₁ : 0 < k := by\n obtain ⟨x, hx⟩ := hf3\n calc\n 0 < ‖f x‖ := norm_pos_iff.mpr hx\n _ ≤ k := hk₁ x\n rw [div_lt_iff₀]\n · apply lt_mul_of_one_lt_right h₁ hneg\n · exact zero_lt_one.trans hneg",
"before_state": "f g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : ∀ (y : ℝ), ‖f y‖ ≤ 1\nhf3 : ∃ x, f x ≠ 0\ny : ℝ\nhneg : 1 < ‖g y‖\nS : Set ℝ := Set.range fun x => ‖f x‖\nk : ℝ := sSup S\nhk₁ : ∀ (x : ℝ), ‖f x‖ ≤ k\nhk₂ : ∀ (x : ℝ), 2 * (‖f x‖ * ‖g y‖) ≤ 2 * k\nk' : ℝ := k / ‖g y‖\n⊢ False",
"after_state": "f g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : ∀ (y : ℝ), ‖f y‖ ≤ 1\nhf3 : ∃ x, f x ≠ 0\ny : ℝ\nhneg : 1 < ‖g y‖\nS : Set ℝ := Set.range fun x => ‖f x‖\nk : ℝ := sSup S\nhk₁ : ∀ (x : ℝ), ‖f x‖ ≤ k\nhk₂ : ∀ (x : ℝ), 2 * (‖f x‖ * ‖g y‖) ≤ 2 * k\nk' : ℝ := k / ‖g y‖\nH₁ : k' < k\n⊢ False"
},
{
"line": "focus\n refine\n no_implicit_lambda%\n (have H₁ : k' < k := ?body✝;\n ?_)\n case body✝ =>\n with_annotate_state\"by\"\n ( have h₁ : 0 < k := by\n obtain ⟨x, hx⟩ := hf3\n calc\n 0 < ‖f x‖ := norm_pos_iff.mpr hx\n _ ≤ k := hk₁ x\n rw [div_lt_iff₀]\n · apply lt_mul_of_one_lt_right h₁ hneg\n · exact zero_lt_one.trans hneg)",
"before_state": "f g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : ∀ (y : ℝ), ‖f y‖ ≤ 1\nhf3 : ∃ x, f x ≠ 0\ny : ℝ\nhneg : 1 < ‖g y‖\nS : Set ℝ := Set.range fun x => ‖f x‖\nk : ℝ := sSup S\nhk₁ : ∀ (x : ℝ), ‖f x‖ ≤ k\nhk₂ : ∀ (x : ℝ), 2 * (‖f x‖ * ‖g y‖) ≤ 2 * k\nk' : ℝ := k / ‖g y‖\n⊢ False",
"after_state": "f g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : ∀ (y : ℝ), ‖f y‖ ≤ 1\nhf3 : ∃ x, f x ≠ 0\ny : ℝ\nhneg : 1 < ‖g y‖\nS : Set ℝ := Set.range fun x => ‖f x‖\nk : ℝ := sSup S\nhk₁ : ∀ (x : ℝ), ‖f x‖ ≤ k\nhk₂ : ∀ (x : ℝ), 2 * (‖f x‖ * ‖g y‖) ≤ 2 * k\nk' : ℝ := k / ‖g y‖\nH₁ : k' < k\n⊢ False"
},
{
"line": "refine\n no_implicit_lambda%\n (have H₁ : k' < k := ?body✝;\n ?_)",
"before_state": "f g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : ∀ (y : ℝ), ‖f y‖ ≤ 1\nhf3 : ∃ x, f x ≠ 0\ny : ℝ\nhneg : 1 < ‖g y‖\nS : Set ℝ := Set.range fun x => ‖f x‖\nk : ℝ := sSup S\nhk₁ : ∀ (x : ℝ), ‖f x‖ ≤ k\nhk₂ : ∀ (x : ℝ), 2 * (‖f x‖ * ‖g y‖) ≤ 2 * k\nk' : ℝ := k / ‖g y‖\n⊢ False",
"after_state": "case body\nf g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : ∀ (y : ℝ), ‖f y‖ ≤ 1\nhf3 : ∃ x, f x ≠ 0\ny : ℝ\nhneg : 1 < ‖g y‖\nS : Set ℝ := Set.range fun x => ‖f x‖\nk : ℝ := sSup S\nhk₁ : ∀ (x : ℝ), ‖f x‖ ≤ k\nhk₂ : ∀ (x : ℝ), 2 * (‖f x‖ * ‖g y‖) ≤ 2 * k\nk' : ℝ := k / ‖g y‖\n⊢ k' < k\n---\nf g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : ∀ (y : ℝ), ‖f y‖ ≤ 1\nhf3 : ∃ x, f x ≠ 0\ny : ℝ\nhneg : 1 < ‖g y‖\nS : Set ℝ := Set.range fun x => ‖f x‖\nk : ℝ := sSup S\nhk₁ : ∀ (x : ℝ), ‖f x‖ ≤ k\nhk₂ : ∀ (x : ℝ), 2 * (‖f x‖ * ‖g y‖) ≤ 2 * k\nk' : ℝ := k / ‖g y‖\nH₁ : k' < k\n⊢ False"
},
{
"line": "case body✝ =>\n with_annotate_state\"by\"\n ( have h₁ : 0 < k := by\n obtain ⟨x, hx⟩ := hf3\n calc\n 0 < ‖f x‖ := norm_pos_iff.mpr hx\n _ ≤ k := hk₁ x\n rw [div_lt_iff₀]\n · apply lt_mul_of_one_lt_right h₁ hneg\n · exact zero_lt_one.trans hneg)",
"before_state": "case body\nf g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : ∀ (y : ℝ), ‖f y‖ ≤ 1\nhf3 : ∃ x, f x ≠ 0\ny : ℝ\nhneg : 1 < ‖g y‖\nS : Set ℝ := Set.range fun x => ‖f x‖\nk : ℝ := sSup S\nhk₁ : ∀ (x : ℝ), ‖f x‖ ≤ k\nhk₂ : ∀ (x : ℝ), 2 * (‖f x‖ * ‖g y‖) ≤ 2 * k\nk' : ℝ := k / ‖g y‖\n⊢ k' < k\n---\nf g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : ∀ (y : ℝ), ‖f y‖ ≤ 1\nhf3 : ∃ x, f x ≠ 0\ny : ℝ\nhneg : 1 < ‖g y‖\nS : Set ℝ := Set.range fun x => ‖f x‖\nk : ℝ := sSup S\nhk₁ : ∀ (x : ℝ), ‖f x‖ ≤ k\nhk₂ : ∀ (x : ℝ), 2 * (‖f x‖ * ‖g y‖) ≤ 2 * k\nk' : ℝ := k / ‖g y‖\nH₁ : k' < k\n⊢ False",
"after_state": "f g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : ∀ (y : ℝ), ‖f y‖ ≤ 1\nhf3 : ∃ x, f x ≠ 0\ny : ℝ\nhneg : 1 < ‖g y‖\nS : Set ℝ := Set.range fun x => ‖f x‖\nk : ℝ := sSup S\nhk₁ : ∀ (x : ℝ), ‖f x‖ ≤ k\nhk₂ : ∀ (x : ℝ), 2 * (‖f x‖ * ‖g y‖) ≤ 2 * k\nk' : ℝ := k / ‖g y‖\nH₁ : k' < k\n⊢ False"
},
{
"line": "with_annotate_state\"by\"\n ( have h₁ : 0 < k := by\n obtain ⟨x, hx⟩ := hf3\n calc\n 0 < ‖f x‖ := norm_pos_iff.mpr hx\n _ ≤ k := hk₁ x\n rw [div_lt_iff₀]\n · apply lt_mul_of_one_lt_right h₁ hneg\n · exact zero_lt_one.trans hneg)",
"before_state": "f g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : ∀ (y : ℝ), ‖f y‖ ≤ 1\nhf3 : ∃ x, f x ≠ 0\ny : ℝ\nhneg : 1 < ‖g y‖\nS : Set ℝ := Set.range fun x => ‖f x‖\nk : ℝ := sSup S\nhk₁ : ∀ (x : ℝ), ‖f x‖ ≤ k\nhk₂ : ∀ (x : ℝ), 2 * (‖f x‖ * ‖g y‖) ≤ 2 * k\nk' : ℝ := k / ‖g y‖\n⊢ k' < k",
"after_state": "No Goals!"
},
{
"line": "have h₁ : 0 < k := by\n obtain ⟨x, hx⟩ := hf3\n calc\n 0 < ‖f x‖ := norm_pos_iff.mpr hx\n _ ≤ k := hk₁ x",
"before_state": "f g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : ∀ (y : ℝ), ‖f y‖ ≤ 1\nhf3 : ∃ x, f x ≠ 0\ny : ℝ\nhneg : 1 < ‖g y‖\nS : Set ℝ := Set.range fun x => ‖f x‖\nk : ℝ := sSup S\nhk₁ : ∀ (x : ℝ), ‖f x‖ ≤ k\nhk₂ : ∀ (x : ℝ), 2 * (‖f x‖ * ‖g y‖) ≤ 2 * k\nk' : ℝ := k / ‖g y‖\n⊢ k' < k",
"after_state": "f g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : ∀ (y : ℝ), ‖f y‖ ≤ 1\nhf3 : ∃ x, f x ≠ 0\ny : ℝ\nhneg : 1 < ‖g y‖\nS : Set ℝ := Set.range fun x => ‖f x‖\nk : ℝ := sSup S\nhk₁ : ∀ (x : ℝ), ‖f x‖ ≤ k\nhk₂ : ∀ (x : ℝ), 2 * (‖f x‖ * ‖g y‖) ≤ 2 * k\nk' : ℝ := k / ‖g y‖\nh₁ : 0 < k\n⊢ k' < k"
},
{
"line": "focus\n refine\n no_implicit_lambda%\n (have h₁ : 0 < k := ?body✝;\n ?_)\n case body✝ =>\n with_annotate_state\"by\"\n ( obtain ⟨x, hx⟩ := hf3\n calc\n 0 < ‖f x‖ := norm_pos_iff.mpr hx\n _ ≤ k := hk₁ x)",
"before_state": "f g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : ∀ (y : ℝ), ‖f y‖ ≤ 1\nhf3 : ∃ x, f x ≠ 0\ny : ℝ\nhneg : 1 < ‖g y‖\nS : Set ℝ := Set.range fun x => ‖f x‖\nk : ℝ := sSup S\nhk₁ : ∀ (x : ℝ), ‖f x‖ ≤ k\nhk₂ : ∀ (x : ℝ), 2 * (‖f x‖ * ‖g y‖) ≤ 2 * k\nk' : ℝ := k / ‖g y‖\n⊢ k' < k",
"after_state": "f g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : ∀ (y : ℝ), ‖f y‖ ≤ 1\nhf3 : ∃ x, f x ≠ 0\ny : ℝ\nhneg : 1 < ‖g y‖\nS : Set ℝ := Set.range fun x => ‖f x‖\nk : ℝ := sSup S\nhk₁ : ∀ (x : ℝ), ‖f x‖ ≤ k\nhk₂ : ∀ (x : ℝ), 2 * (‖f x‖ * ‖g y‖) ≤ 2 * k\nk' : ℝ := k / ‖g y‖\nh₁ : 0 < k\n⊢ k' < k"
},
{
"line": "refine\n no_implicit_lambda%\n (have h₁ : 0 < k := ?body✝;\n ?_)",
"before_state": "f g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : ∀ (y : ℝ), ‖f y‖ ≤ 1\nhf3 : ∃ x, f x ≠ 0\ny : ℝ\nhneg : 1 < ‖g y‖\nS : Set ℝ := Set.range fun x => ‖f x‖\nk : ℝ := sSup S\nhk₁ : ∀ (x : ℝ), ‖f x‖ ≤ k\nhk₂ : ∀ (x : ℝ), 2 * (‖f x‖ * ‖g y‖) ≤ 2 * k\nk' : ℝ := k / ‖g y‖\n⊢ k' < k",
"after_state": "case body\nf g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : ∀ (y : ℝ), ‖f y‖ ≤ 1\nhf3 : ∃ x, f x ≠ 0\ny : ℝ\nhneg : 1 < ‖g y‖\nS : Set ℝ := Set.range fun x => ‖f x‖\nk : ℝ := sSup S\nhk₁ : ∀ (x : ℝ), ‖f x‖ ≤ k\nhk₂ : ∀ (x : ℝ), 2 * (‖f x‖ * ‖g y‖) ≤ 2 * k\nk' : ℝ := k / ‖g y‖\n⊢ 0 < k\n---\nf g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : ∀ (y : ℝ), ‖f y‖ ≤ 1\nhf3 : ∃ x, f x ≠ 0\ny : ℝ\nhneg : 1 < ‖g y‖\nS : Set ℝ := Set.range fun x => ‖f x‖\nk : ℝ := sSup S\nhk₁ : ∀ (x : ℝ), ‖f x‖ ≤ k\nhk₂ : ∀ (x : ℝ), 2 * (‖f x‖ * ‖g y‖) ≤ 2 * k\nk' : ℝ := k / ‖g y‖\nh₁ : 0 < k\n⊢ k' < k"
},
{
"line": "case body✝ =>\n with_annotate_state\"by\"\n ( obtain ⟨x, hx⟩ := hf3\n calc\n 0 < ‖f x‖ := norm_pos_iff.mpr hx\n _ ≤ k := hk₁ x)",
"before_state": "case body\nf g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : ∀ (y : ℝ), ‖f y‖ ≤ 1\nhf3 : ∃ x, f x ≠ 0\ny : ℝ\nhneg : 1 < ‖g y‖\nS : Set ℝ := Set.range fun x => ‖f x‖\nk : ℝ := sSup S\nhk₁ : ∀ (x : ℝ), ‖f x‖ ≤ k\nhk₂ : ∀ (x : ℝ), 2 * (‖f x‖ * ‖g y‖) ≤ 2 * k\nk' : ℝ := k / ‖g y‖\n⊢ 0 < k\n---\nf g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : ∀ (y : ℝ), ‖f y‖ ≤ 1\nhf3 : ∃ x, f x ≠ 0\ny : ℝ\nhneg : 1 < ‖g y‖\nS : Set ℝ := Set.range fun x => ‖f x‖\nk : ℝ := sSup S\nhk₁ : ∀ (x : ℝ), ‖f x‖ ≤ k\nhk₂ : ∀ (x : ℝ), 2 * (‖f x‖ * ‖g y‖) ≤ 2 * k\nk' : ℝ := k / ‖g y‖\nh₁ : 0 < k\n⊢ k' < k",
"after_state": "f g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : ∀ (y : ℝ), ‖f y‖ ≤ 1\nhf3 : ∃ x, f x ≠ 0\ny : ℝ\nhneg : 1 < ‖g y‖\nS : Set ℝ := Set.range fun x => ‖f x‖\nk : ℝ := sSup S\nhk₁ : ∀ (x : ℝ), ‖f x‖ ≤ k\nhk₂ : ∀ (x : ℝ), 2 * (‖f x‖ * ‖g y‖) ≤ 2 * k\nk' : ℝ := k / ‖g y‖\nh₁ : 0 < k\n⊢ k' < k"
},
{
"line": "with_annotate_state\"by\"\n ( obtain ⟨x, hx⟩ := hf3\n calc\n 0 < ‖f x‖ := norm_pos_iff.mpr hx\n _ ≤ k := hk₁ x)",
"before_state": "f g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : ∀ (y : ℝ), ‖f y‖ ≤ 1\nhf3 : ∃ x, f x ≠ 0\ny : ℝ\nhneg : 1 < ‖g y‖\nS : Set ℝ := Set.range fun x => ‖f x‖\nk : ℝ := sSup S\nhk₁ : ∀ (x : ℝ), ‖f x‖ ≤ k\nhk₂ : ∀ (x : ℝ), 2 * (‖f x‖ * ‖g y‖) ≤ 2 * k\nk' : ℝ := k / ‖g y‖\n⊢ 0 < k",
"after_state": "No Goals!"
},
{
"line": "obtain ⟨x, hx⟩ := hf3",
"before_state": "f g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : ∀ (y : ℝ), ‖f y‖ ≤ 1\nhf3 : ∃ x, f x ≠ 0\ny : ℝ\nhneg : 1 < ‖g y‖\nS : Set ℝ := Set.range fun x => ‖f x‖\nk : ℝ := sSup S\nhk₁ : ∀ (x : ℝ), ‖f x‖ ≤ k\nhk₂ : ∀ (x : ℝ), 2 * (‖f x‖ * ‖g y‖) ≤ 2 * k\nk' : ℝ := k / ‖g y‖\n⊢ 0 < k",
"after_state": "case intro\nf g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : ∀ (y : ℝ), ‖f y‖ ≤ 1\ny : ℝ\nhneg : 1 < ‖g y‖\nS : Set ℝ := Set.range fun x => ‖f x‖\nk : ℝ := sSup S\nhk₁ : ∀ (x : ℝ), ‖f x‖ ≤ k\nhk₂ : ∀ (x : ℝ), 2 * (‖f x‖ * ‖g y‖) ≤ 2 * k\nk' : ℝ := k / ‖g y‖\nx : ℝ\nhx : f x ≠ 0\n⊢ 0 < k"
},
{
"line": "calc\n 0 < ‖f x‖ := norm_pos_iff.mpr hx\n _ ≤ k := hk₁ x",
"before_state": "case intro\nf g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : ∀ (y : ℝ), ‖f y‖ ≤ 1\ny : ℝ\nhneg : 1 < ‖g y‖\nS : Set ℝ := Set.range fun x => ‖f x‖\nk : ℝ := sSup S\nhk₁ : ∀ (x : ℝ), ‖f x‖ ≤ k\nhk₂ : ∀ (x : ℝ), 2 * (‖f x‖ * ‖g y‖) ≤ 2 * k\nk' : ℝ := k / ‖g y‖\nx : ℝ\nhx : f x ≠ 0\n⊢ 0 < k",
"after_state": "No Goals!"
},
{
"line": "rw [div_lt_iff₀]",
"before_state": "f g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : ∀ (y : ℝ), ‖f y‖ ≤ 1\nhf3 : ∃ x, f x ≠ 0\ny : ℝ\nhneg : 1 < ‖g y‖\nS : Set ℝ := Set.range fun x => ‖f x‖\nk : ℝ := sSup S\nhk₁ : ∀ (x : ℝ), ‖f x‖ ≤ k\nhk₂ : ∀ (x : ℝ), 2 * (‖f x‖ * ‖g y‖) ≤ 2 * k\nk' : ℝ := k / ‖g y‖\nh₁ : 0 < k\n⊢ k' < k",
"after_state": "f g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : ∀ (y : ℝ), ‖f y‖ ≤ 1\nhf3 : ∃ x, f x ≠ 0\ny : ℝ\nhneg : 1 < ‖g y‖\nS : Set ℝ := Set.range fun x => ‖f x‖\nk : ℝ := sSup S\nhk₁ : ∀ (x : ℝ), ‖f x‖ ≤ k\nhk₂ : ∀ (x : ℝ), 2 * (‖f x‖ * ‖g y‖) ≤ 2 * k\nk' : ℝ := k / ‖g y‖\nh₁ : 0 < k\n⊢ k < k * ‖g y‖\n---\nf g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : ∀ (y : ℝ), ‖f y‖ ≤ 1\nhf3 : ∃ x, f x ≠ 0\ny : ℝ\nhneg : 1 < ‖g y‖\nS : Set ℝ := Set.range fun x => ‖f x‖\nk : ℝ := sSup S\nhk₁ : ∀ (x : ℝ), ‖f x‖ ≤ k\nhk₂ : ∀ (x : ℝ), 2 * (‖f x‖ * ‖g y‖) ≤ 2 * k\nk' : ℝ := k / ‖g y‖\nh₁ : 0 < k\n⊢ 0 < ‖g y‖"
},
{
"line": "rewrite [div_lt_iff₀]",
"before_state": "f g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : ∀ (y : ℝ), ‖f y‖ ≤ 1\nhf3 : ∃ x, f x ≠ 0\ny : ℝ\nhneg : 1 < ‖g y‖\nS : Set ℝ := Set.range fun x => ‖f x‖\nk : ℝ := sSup S\nhk₁ : ∀ (x : ℝ), ‖f x‖ ≤ k\nhk₂ : ∀ (x : ℝ), 2 * (‖f x‖ * ‖g y‖) ≤ 2 * k\nk' : ℝ := k / ‖g y‖\nh₁ : 0 < k\n⊢ k' < k",
"after_state": "f g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : ∀ (y : ℝ), ‖f y‖ ≤ 1\nhf3 : ∃ x, f x ≠ 0\ny : ℝ\nhneg : 1 < ‖g y‖\nS : Set ℝ := Set.range fun x => ‖f x‖\nk : ℝ := sSup S\nhk₁ : ∀ (x : ℝ), ‖f x‖ ≤ k\nhk₂ : ∀ (x : ℝ), 2 * (‖f x‖ * ‖g y‖) ≤ 2 * k\nk' : ℝ := k / ‖g y‖\nh₁ : 0 < k\n⊢ k < k * ‖g y‖\n---\nf g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : ∀ (y : ℝ), ‖f y‖ ≤ 1\nhf3 : ∃ x, f x ≠ 0\ny : ℝ\nhneg : 1 < ‖g y‖\nS : Set ℝ := Set.range fun x => ‖f x‖\nk : ℝ := sSup S\nhk₁ : ∀ (x : ℝ), ‖f x‖ ≤ k\nhk₂ : ∀ (x : ℝ), 2 * (‖f x‖ * ‖g y‖) ≤ 2 * k\nk' : ℝ := k / ‖g y‖\nh₁ : 0 < k\n⊢ 0 < ‖g y‖"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "f g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : ∀ (y : ℝ), ‖f y‖ ≤ 1\nhf3 : ∃ x, f x ≠ 0\ny : ℝ\nhneg : 1 < ‖g y‖\nS : Set ℝ := Set.range fun x => ‖f x‖\nk : ℝ := sSup S\nhk₁ : ∀ (x : ℝ), ‖f x‖ ≤ k\nhk₂ : ∀ (x : ℝ), 2 * (‖f x‖ * ‖g y‖) ≤ 2 * k\nk' : ℝ := k / ‖g y‖\nh₁ : 0 < k\n⊢ k < k * ‖g y‖\n---\nf g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : ∀ (y : ℝ), ‖f y‖ ≤ 1\nhf3 : ∃ x, f x ≠ 0\ny : ℝ\nhneg : 1 < ‖g y‖\nS : Set ℝ := Set.range fun x => ‖f x‖\nk : ℝ := sSup S\nhk₁ : ∀ (x : ℝ), ‖f x‖ ≤ k\nhk₂ : ∀ (x : ℝ), 2 * (‖f x‖ * ‖g y‖) ≤ 2 * k\nk' : ℝ := k / ‖g y‖\nh₁ : 0 < k\n⊢ 0 < ‖g y‖",
"after_state": "f g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : ∀ (y : ℝ), ‖f y‖ ≤ 1\nhf3 : ∃ x, f x ≠ 0\ny : ℝ\nhneg : 1 < ‖g y‖\nS : Set ℝ := Set.range fun x => ‖f x‖\nk : ℝ := sSup S\nhk₁ : ∀ (x : ℝ), ‖f x‖ ≤ k\nhk₂ : ∀ (x : ℝ), 2 * (‖f x‖ * ‖g y‖) ≤ 2 * k\nk' : ℝ := k / ‖g y‖\nh₁ : 0 < k\n⊢ k < k * ‖g y‖\n---\nf g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : ∀ (y : ℝ), ‖f y‖ ≤ 1\nhf3 : ∃ x, f x ≠ 0\ny : ℝ\nhneg : 1 < ‖g y‖\nS : Set ℝ := Set.range fun x => ‖f x‖\nk : ℝ := sSup S\nhk₁ : ∀ (x : ℝ), ‖f x‖ ≤ k\nhk₂ : ∀ (x : ℝ), 2 * (‖f x‖ * ‖g y‖) ≤ 2 * k\nk' : ℝ := k / ‖g y‖\nh₁ : 0 < k\n⊢ 0 < ‖g y‖"
},
{
"line": "try (with_reducible rfl)",
"before_state": "f g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : ∀ (y : ℝ), ‖f y‖ ≤ 1\nhf3 : ∃ x, f x ≠ 0\ny : ℝ\nhneg : 1 < ‖g y‖\nS : Set ℝ := Set.range fun x => ‖f x‖\nk : ℝ := sSup S\nhk₁ : ∀ (x : ℝ), ‖f x‖ ≤ k\nhk₂ : ∀ (x : ℝ), 2 * (‖f x‖ * ‖g y‖) ≤ 2 * k\nk' : ℝ := k / ‖g y‖\nh₁ : 0 < k\n⊢ k < k * ‖g y‖\n---\nf g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : ∀ (y : ℝ), ‖f y‖ ≤ 1\nhf3 : ∃ x, f x ≠ 0\ny : ℝ\nhneg : 1 < ‖g y‖\nS : Set ℝ := Set.range fun x => ‖f x‖\nk : ℝ := sSup S\nhk₁ : ∀ (x : ℝ), ‖f x‖ ≤ k\nhk₂ : ∀ (x : ℝ), 2 * (‖f x‖ * ‖g y‖) ≤ 2 * k\nk' : ℝ := k / ‖g y‖\nh₁ : 0 < k\n⊢ 0 < ‖g y‖",
"after_state": "f g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : ∀ (y : ℝ), ‖f y‖ ≤ 1\nhf3 : ∃ x, f x ≠ 0\ny : ℝ\nhneg : 1 < ‖g y‖\nS : Set ℝ := Set.range fun x => ‖f x‖\nk : ℝ := sSup S\nhk₁ : ∀ (x : ℝ), ‖f x‖ ≤ k\nhk₂ : ∀ (x : ℝ), 2 * (‖f x‖ * ‖g y‖) ≤ 2 * k\nk' : ℝ := k / ‖g y‖\nh₁ : 0 < k\n⊢ k < k * ‖g y‖\n---\nf g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : ∀ (y : ℝ), ‖f y‖ ≤ 1\nhf3 : ∃ x, f x ≠ 0\ny : ℝ\nhneg : 1 < ‖g y‖\nS : Set ℝ := Set.range fun x => ‖f x‖\nk : ℝ := sSup S\nhk₁ : ∀ (x : ℝ), ‖f x‖ ≤ k\nhk₂ : ∀ (x : ℝ), 2 * (‖f x‖ * ‖g y‖) ≤ 2 * k\nk' : ℝ := k / ‖g y‖\nh₁ : 0 < k\n⊢ 0 < ‖g y‖"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "f g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : ∀ (y : ℝ), ‖f y‖ ≤ 1\nhf3 : ∃ x, f x ≠ 0\ny : ℝ\nhneg : 1 < ‖g y‖\nS : Set ℝ := Set.range fun x => ‖f x‖\nk : ℝ := sSup S\nhk₁ : ∀ (x : ℝ), ‖f x‖ ≤ k\nhk₂ : ∀ (x : ℝ), 2 * (‖f x‖ * ‖g y‖) ≤ 2 * k\nk' : ℝ := k / ‖g y‖\nh₁ : 0 < k\n⊢ k < k * ‖g y‖\n---\nf g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : ∀ (y : ℝ), ‖f y‖ ≤ 1\nhf3 : ∃ x, f x ≠ 0\ny : ℝ\nhneg : 1 < ‖g y‖\nS : Set ℝ := Set.range fun x => ‖f x‖\nk : ℝ := sSup S\nhk₁ : ∀ (x : ℝ), ‖f x‖ ≤ k\nhk₂ : ∀ (x : ℝ), 2 * (‖f x‖ * ‖g y‖) ≤ 2 * k\nk' : ℝ := k / ‖g y‖\nh₁ : 0 < k\n⊢ 0 < ‖g y‖",
"after_state": "f g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : ∀ (y : ℝ), ‖f y‖ ≤ 1\nhf3 : ∃ x, f x ≠ 0\ny : ℝ\nhneg : 1 < ‖g y‖\nS : Set ℝ := Set.range fun x => ‖f x‖\nk : ℝ := sSup S\nhk₁ : ∀ (x : ℝ), ‖f x‖ ≤ k\nhk₂ : ∀ (x : ℝ), 2 * (‖f x‖ * ‖g y‖) ≤ 2 * k\nk' : ℝ := k / ‖g y‖\nh₁ : 0 < k\n⊢ k < k * ‖g y‖\n---\nf g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : ∀ (y : ℝ), ‖f y‖ ≤ 1\nhf3 : ∃ x, f x ≠ 0\ny : ℝ\nhneg : 1 < ‖g y‖\nS : Set ℝ := Set.range fun x => ‖f x‖\nk : ℝ := sSup S\nhk₁ : ∀ (x : ℝ), ‖f x‖ ≤ k\nhk₂ : ∀ (x : ℝ), 2 * (‖f x‖ * ‖g y‖) ≤ 2 * k\nk' : ℝ := k / ‖g y‖\nh₁ : 0 < k\n⊢ 0 < ‖g y‖"
},
{
"line": "with_reducible rfl",
"before_state": "f g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : ∀ (y : ℝ), ‖f y‖ ≤ 1\nhf3 : ∃ x, f x ≠ 0\ny : ℝ\nhneg : 1 < ‖g y‖\nS : Set ℝ := Set.range fun x => ‖f x‖\nk : ℝ := sSup S\nhk₁ : ∀ (x : ℝ), ‖f x‖ ≤ k\nhk₂ : ∀ (x : ℝ), 2 * (‖f x‖ * ‖g y‖) ≤ 2 * k\nk' : ℝ := k / ‖g y‖\nh₁ : 0 < k\n⊢ k < k * ‖g y‖\n---\nf g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : ∀ (y : ℝ), ‖f y‖ ≤ 1\nhf3 : ∃ x, f x ≠ 0\ny : ℝ\nhneg : 1 < ‖g y‖\nS : Set ℝ := Set.range fun x => ‖f x‖\nk : ℝ := sSup S\nhk₁ : ∀ (x : ℝ), ‖f x‖ ≤ k\nhk₂ : ∀ (x : ℝ), 2 * (‖f x‖ * ‖g y‖) ≤ 2 * k\nk' : ℝ := k / ‖g y‖\nh₁ : 0 < k\n⊢ 0 < ‖g y‖",
"after_state": "f g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : ∀ (y : ℝ), ‖f y‖ ≤ 1\nhf3 : ∃ x, f x ≠ 0\ny : ℝ\nhneg : 1 < ‖g y‖\nS : Set ℝ := Set.range fun x => ‖f x‖\nk : ℝ := sSup S\nhk₁ : ∀ (x : ℝ), ‖f x‖ ≤ k\nhk₂ : ∀ (x : ℝ), 2 * (‖f x‖ * ‖g y‖) ≤ 2 * k\nk' : ℝ := k / ‖g y‖\nh₁ : 0 < k\n⊢ k < k * ‖g y‖\n---\nf g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : ∀ (y : ℝ), ‖f y‖ ≤ 1\nhf3 : ∃ x, f x ≠ 0\ny : ℝ\nhneg : 1 < ‖g y‖\nS : Set ℝ := Set.range fun x => ‖f x‖\nk : ℝ := sSup S\nhk₁ : ∀ (x : ℝ), ‖f x‖ ≤ k\nhk₂ : ∀ (x : ℝ), 2 * (‖f x‖ * ‖g y‖) ≤ 2 * k\nk' : ℝ := k / ‖g y‖\nh₁ : 0 < k\n⊢ 0 < ‖g y‖"
},
{
"line": "rfl",
"before_state": "f g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : ∀ (y : ℝ), ‖f y‖ ≤ 1\nhf3 : ∃ x, f x ≠ 0\ny : ℝ\nhneg : 1 < ‖g y‖\nS : Set ℝ := Set.range fun x => ‖f x‖\nk : ℝ := sSup S\nhk₁ : ∀ (x : ℝ), ‖f x‖ ≤ k\nhk₂ : ∀ (x : ℝ), 2 * (‖f x‖ * ‖g y‖) ≤ 2 * k\nk' : ℝ := k / ‖g y‖\nh₁ : 0 < k\n⊢ k < k * ‖g y‖\n---\nf g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : ∀ (y : ℝ), ‖f y‖ ≤ 1\nhf3 : ∃ x, f x ≠ 0\ny : ℝ\nhneg : 1 < ‖g y‖\nS : Set ℝ := Set.range fun x => ‖f x‖\nk : ℝ := sSup S\nhk₁ : ∀ (x : ℝ), ‖f x‖ ≤ k\nhk₂ : ∀ (x : ℝ), 2 * (‖f x‖ * ‖g y‖) ≤ 2 * k\nk' : ℝ := k / ‖g y‖\nh₁ : 0 < k\n⊢ 0 < ‖g y‖",
"after_state": "f g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : ∀ (y : ℝ), ‖f y‖ ≤ 1\nhf3 : ∃ x, f x ≠ 0\ny : ℝ\nhneg : 1 < ‖g y‖\nS : Set ℝ := Set.range fun x => ‖f x‖\nk : ℝ := sSup S\nhk₁ : ∀ (x : ℝ), ‖f x‖ ≤ k\nhk₂ : ∀ (x : ℝ), 2 * (‖f x‖ * ‖g y‖) ≤ 2 * k\nk' : ℝ := k / ‖g y‖\nh₁ : 0 < k\n⊢ k < k * ‖g y‖\n---\nf g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : ∀ (y : ℝ), ‖f y‖ ≤ 1\nhf3 : ∃ x, f x ≠ 0\ny : ℝ\nhneg : 1 < ‖g y‖\nS : Set ℝ := Set.range fun x => ‖f x‖\nk : ℝ := sSup S\nhk₁ : ∀ (x : ℝ), ‖f x‖ ≤ k\nhk₂ : ∀ (x : ℝ), 2 * (‖f x‖ * ‖g y‖) ≤ 2 * k\nk' : ℝ := k / ‖g y‖\nh₁ : 0 < k\n⊢ 0 < ‖g y‖"
},
{
"line": "apply_rfl",
"before_state": "f g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : ∀ (y : ℝ), ‖f y‖ ≤ 1\nhf3 : ∃ x, f x ≠ 0\ny : ℝ\nhneg : 1 < ‖g y‖\nS : Set ℝ := Set.range fun x => ‖f x‖\nk : ℝ := sSup S\nhk₁ : ∀ (x : ℝ), ‖f x‖ ≤ k\nhk₂ : ∀ (x : ℝ), 2 * (‖f x‖ * ‖g y‖) ≤ 2 * k\nk' : ℝ := k / ‖g y‖\nh₁ : 0 < k\n⊢ k < k * ‖g y‖\n---\nf g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : ∀ (y : ℝ), ‖f y‖ ≤ 1\nhf3 : ∃ x, f x ≠ 0\ny : ℝ\nhneg : 1 < ‖g y‖\nS : Set ℝ := Set.range fun x => ‖f x‖\nk : ℝ := sSup S\nhk₁ : ∀ (x : ℝ), ‖f x‖ ≤ k\nhk₂ : ∀ (x : ℝ), 2 * (‖f x‖ * ‖g y‖) ≤ 2 * k\nk' : ℝ := k / ‖g y‖\nh₁ : 0 < k\n⊢ 0 < ‖g y‖",
"after_state": "f g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : ∀ (y : ℝ), ‖f y‖ ≤ 1\nhf3 : ∃ x, f x ≠ 0\ny : ℝ\nhneg : 1 < ‖g y‖\nS : Set ℝ := Set.range fun x => ‖f x‖\nk : ℝ := sSup S\nhk₁ : ∀ (x : ℝ), ‖f x‖ ≤ k\nhk₂ : ∀ (x : ℝ), 2 * (‖f x‖ * ‖g y‖) ≤ 2 * k\nk' : ℝ := k / ‖g y‖\nh₁ : 0 < k\n⊢ k < k * ‖g y‖\n---\nf g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : ∀ (y : ℝ), ‖f y‖ ≤ 1\nhf3 : ∃ x, f x ≠ 0\ny : ℝ\nhneg : 1 < ‖g y‖\nS : Set ℝ := Set.range fun x => ‖f x‖\nk : ℝ := sSup S\nhk₁ : ∀ (x : ℝ), ‖f x‖ ≤ k\nhk₂ : ∀ (x : ℝ), 2 * (‖f x‖ * ‖g y‖) ≤ 2 * k\nk' : ℝ := k / ‖g y‖\nh₁ : 0 < k\n⊢ 0 < ‖g y‖"
},
{
"line": "skip",
"before_state": "f g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : ∀ (y : ℝ), ‖f y‖ ≤ 1\nhf3 : ∃ x, f x ≠ 0\ny : ℝ\nhneg : 1 < ‖g y‖\nS : Set ℝ := Set.range fun x => ‖f x‖\nk : ℝ := sSup S\nhk₁ : ∀ (x : ℝ), ‖f x‖ ≤ k\nhk₂ : ∀ (x : ℝ), 2 * (‖f x‖ * ‖g y‖) ≤ 2 * k\nk' : ℝ := k / ‖g y‖\nh₁ : 0 < k\n⊢ k < k * ‖g y‖\n---\nf g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : ∀ (y : ℝ), ‖f y‖ ≤ 1\nhf3 : ∃ x, f x ≠ 0\ny : ℝ\nhneg : 1 < ‖g y‖\nS : Set ℝ := Set.range fun x => ‖f x‖\nk : ℝ := sSup S\nhk₁ : ∀ (x : ℝ), ‖f x‖ ≤ k\nhk₂ : ∀ (x : ℝ), 2 * (‖f x‖ * ‖g y‖) ≤ 2 * k\nk' : ℝ := k / ‖g y‖\nh₁ : 0 < k\n⊢ 0 < ‖g y‖",
"after_state": "f g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : ∀ (y : ℝ), ‖f y‖ ≤ 1\nhf3 : ∃ x, f x ≠ 0\ny : ℝ\nhneg : 1 < ‖g y‖\nS : Set ℝ := Set.range fun x => ‖f x‖\nk : ℝ := sSup S\nhk₁ : ∀ (x : ℝ), ‖f x‖ ≤ k\nhk₂ : ∀ (x : ℝ), 2 * (‖f x‖ * ‖g y‖) ≤ 2 * k\nk' : ℝ := k / ‖g y‖\nh₁ : 0 < k\n⊢ k < k * ‖g y‖\n---\nf g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : ∀ (y : ℝ), ‖f y‖ ≤ 1\nhf3 : ∃ x, f x ≠ 0\ny : ℝ\nhneg : 1 < ‖g y‖\nS : Set ℝ := Set.range fun x => ‖f x‖\nk : ℝ := sSup S\nhk₁ : ∀ (x : ℝ), ‖f x‖ ≤ k\nhk₂ : ∀ (x : ℝ), 2 * (‖f x‖ * ‖g y‖) ≤ 2 * k\nk' : ℝ := k / ‖g y‖\nh₁ : 0 < k\n⊢ 0 < ‖g y‖"
},
{
"line": "apply lt_mul_of_one_lt_right h₁ hneg",
"before_state": "f g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : ∀ (y : ℝ), ‖f y‖ ≤ 1\nhf3 : ∃ x, f x ≠ 0\ny : ℝ\nhneg : 1 < ‖g y‖\nS : Set ℝ := Set.range fun x => ‖f x‖\nk : ℝ := sSup S\nhk₁ : ∀ (x : ℝ), ‖f x‖ ≤ k\nhk₂ : ∀ (x : ℝ), 2 * (‖f x‖ * ‖g y‖) ≤ 2 * k\nk' : ℝ := k / ‖g y‖\nh₁ : 0 < k\n⊢ k < k * ‖g y‖",
"after_state": "No Goals!"
},
{
"line": "exact zero_lt_one.trans hneg",
"before_state": "f g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : ∀ (y : ℝ), ‖f y‖ ≤ 1\nhf3 : ∃ x, f x ≠ 0\ny : ℝ\nhneg : 1 < ‖g y‖\nS : Set ℝ := Set.range fun x => ‖f x‖\nk : ℝ := sSup S\nhk₁ : ∀ (x : ℝ), ‖f x‖ ≤ k\nhk₂ : ∀ (x : ℝ), 2 * (‖f x‖ * ‖g y‖) ≤ 2 * k\nk' : ℝ := k / ‖g y‖\nh₁ : 0 < k\n⊢ 0 < ‖g y‖",
"after_state": "No Goals!"
},
{
"line": "have H₂ : k ≤ k' := by\n have h₁ : ∃ x : ℝ, x ∈ S := by use ‖f 0‖; exact Set.mem_range_self 0\n have h₂ : ∀ x, ‖f x‖ ≤ k' := by\n intro x\n rw [le_div_iff₀]\n · apply (mul_le_mul_left zero_lt_two).mp (hk₂ x)\n · exact zero_lt_one.trans hneg\n apply csSup_le h₁\n rintro y' ⟨yy, rfl⟩\n exact h₂ yy",
"before_state": "f g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : ∀ (y : ℝ), ‖f y‖ ≤ 1\nhf3 : ∃ x, f x ≠ 0\ny : ℝ\nhneg : 1 < ‖g y‖\nS : Set ℝ := Set.range fun x => ‖f x‖\nk : ℝ := sSup S\nhk₁ : ∀ (x : ℝ), ‖f x‖ ≤ k\nhk₂ : ∀ (x : ℝ), 2 * (‖f x‖ * ‖g y‖) ≤ 2 * k\nk' : ℝ := k / ‖g y‖\nH₁ : k' < k\n⊢ False",
"after_state": "f g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : ∀ (y : ℝ), ‖f y‖ ≤ 1\nhf3 : ∃ x, f x ≠ 0\ny : ℝ\nhneg : 1 < ‖g y‖\nS : Set ℝ := Set.range fun x => ‖f x‖\nk : ℝ := sSup S\nhk₁ : ∀ (x : ℝ), ‖f x‖ ≤ k\nhk₂ : ∀ (x : ℝ), 2 * (‖f x‖ * ‖g y‖) ≤ 2 * k\nk' : ℝ := k / ‖g y‖\nH₁ : k' < k\nH₂ : k ≤ k'\n⊢ False"
},
{
"line": "focus\n refine\n no_implicit_lambda%\n (have H₂ : k ≤ k' := ?body✝;\n ?_)\n case body✝ =>\n with_annotate_state\"by\"\n ( have h₁ : ∃ x : ℝ, x ∈ S := by use ‖f 0‖; exact Set.mem_range_self 0\n have h₂ : ∀ x, ‖f x‖ ≤ k' := by\n intro x\n rw [le_div_iff₀]\n · apply (mul_le_mul_left zero_lt_two).mp (hk₂ x)\n · exact zero_lt_one.trans hneg\n apply csSup_le h₁\n rintro y' ⟨yy, rfl⟩\n exact h₂ yy)",
"before_state": "f g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : ∀ (y : ℝ), ‖f y‖ ≤ 1\nhf3 : ∃ x, f x ≠ 0\ny : ℝ\nhneg : 1 < ‖g y‖\nS : Set ℝ := Set.range fun x => ‖f x‖\nk : ℝ := sSup S\nhk₁ : ∀ (x : ℝ), ‖f x‖ ≤ k\nhk₂ : ∀ (x : ℝ), 2 * (‖f x‖ * ‖g y‖) ≤ 2 * k\nk' : ℝ := k / ‖g y‖\nH₁ : k' < k\n⊢ False",
"after_state": "f g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : ∀ (y : ℝ), ‖f y‖ ≤ 1\nhf3 : ∃ x, f x ≠ 0\ny : ℝ\nhneg : 1 < ‖g y‖\nS : Set ℝ := Set.range fun x => ‖f x‖\nk : ℝ := sSup S\nhk₁ : ∀ (x : ℝ), ‖f x‖ ≤ k\nhk₂ : ∀ (x : ℝ), 2 * (‖f x‖ * ‖g y‖) ≤ 2 * k\nk' : ℝ := k / ‖g y‖\nH₁ : k' < k\nH₂ : k ≤ k'\n⊢ False"
},
{
"line": "refine\n no_implicit_lambda%\n (have H₂ : k ≤ k' := ?body✝;\n ?_)",
"before_state": "f g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : ∀ (y : ℝ), ‖f y‖ ≤ 1\nhf3 : ∃ x, f x ≠ 0\ny : ℝ\nhneg : 1 < ‖g y‖\nS : Set ℝ := Set.range fun x => ‖f x‖\nk : ℝ := sSup S\nhk₁ : ∀ (x : ℝ), ‖f x‖ ≤ k\nhk₂ : ∀ (x : ℝ), 2 * (‖f x‖ * ‖g y‖) ≤ 2 * k\nk' : ℝ := k / ‖g y‖\nH₁ : k' < k\n⊢ False",
"after_state": "case body\nf g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : ∀ (y : ℝ), ‖f y‖ ≤ 1\nhf3 : ∃ x, f x ≠ 0\ny : ℝ\nhneg : 1 < ‖g y‖\nS : Set ℝ := Set.range fun x => ‖f x‖\nk : ℝ := sSup S\nhk₁ : ∀ (x : ℝ), ‖f x‖ ≤ k\nhk₂ : ∀ (x : ℝ), 2 * (‖f x‖ * ‖g y‖) ≤ 2 * k\nk' : ℝ := k / ‖g y‖\nH₁ : k' < k\n⊢ k ≤ k'\n---\nf g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : ∀ (y : ℝ), ‖f y‖ ≤ 1\nhf3 : ∃ x, f x ≠ 0\ny : ℝ\nhneg : 1 < ‖g y‖\nS : Set ℝ := Set.range fun x => ‖f x‖\nk : ℝ := sSup S\nhk₁ : ∀ (x : ℝ), ‖f x‖ ≤ k\nhk₂ : ∀ (x : ℝ), 2 * (‖f x‖ * ‖g y‖) ≤ 2 * k\nk' : ℝ := k / ‖g y‖\nH₁ : k' < k\nH₂ : k ≤ k'\n⊢ False"
},
{
"line": "case body✝ =>\n with_annotate_state\"by\"\n ( have h₁ : ∃ x : ℝ, x ∈ S := by use ‖f 0‖; exact Set.mem_range_self 0\n have h₂ : ∀ x, ‖f x‖ ≤ k' := by\n intro x\n rw [le_div_iff₀]\n · apply (mul_le_mul_left zero_lt_two).mp (hk₂ x)\n · exact zero_lt_one.trans hneg\n apply csSup_le h₁\n rintro y' ⟨yy, rfl⟩\n exact h₂ yy)",
"before_state": "case body\nf g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : ∀ (y : ℝ), ‖f y‖ ≤ 1\nhf3 : ∃ x, f x ≠ 0\ny : ℝ\nhneg : 1 < ‖g y‖\nS : Set ℝ := Set.range fun x => ‖f x‖\nk : ℝ := sSup S\nhk₁ : ∀ (x : ℝ), ‖f x‖ ≤ k\nhk₂ : ∀ (x : ℝ), 2 * (‖f x‖ * ‖g y‖) ≤ 2 * k\nk' : ℝ := k / ‖g y‖\nH₁ : k' < k\n⊢ k ≤ k'\n---\nf g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : ∀ (y : ℝ), ‖f y‖ ≤ 1\nhf3 : ∃ x, f x ≠ 0\ny : ℝ\nhneg : 1 < ‖g y‖\nS : Set ℝ := Set.range fun x => ‖f x‖\nk : ℝ := sSup S\nhk₁ : ∀ (x : ℝ), ‖f x‖ ≤ k\nhk₂ : ∀ (x : ℝ), 2 * (‖f x‖ * ‖g y‖) ≤ 2 * k\nk' : ℝ := k / ‖g y‖\nH₁ : k' < k\nH₂ : k ≤ k'\n⊢ False",
"after_state": "f g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : ∀ (y : ℝ), ‖f y‖ ≤ 1\nhf3 : ∃ x, f x ≠ 0\ny : ℝ\nhneg : 1 < ‖g y‖\nS : Set ℝ := Set.range fun x => ‖f x‖\nk : ℝ := sSup S\nhk₁ : ∀ (x : ℝ), ‖f x‖ ≤ k\nhk₂ : ∀ (x : ℝ), 2 * (‖f x‖ * ‖g y‖) ≤ 2 * k\nk' : ℝ := k / ‖g y‖\nH₁ : k' < k\nH₂ : k ≤ k'\n⊢ False"
},
{
"line": "with_annotate_state\"by\"\n ( have h₁ : ∃ x : ℝ, x ∈ S := by use ‖f 0‖; exact Set.mem_range_self 0\n have h₂ : ∀ x, ‖f x‖ ≤ k' := by\n intro x\n rw [le_div_iff₀]\n · apply (mul_le_mul_left zero_lt_two).mp (hk₂ x)\n · exact zero_lt_one.trans hneg\n apply csSup_le h₁\n rintro y' ⟨yy, rfl⟩\n exact h₂ yy)",
"before_state": "f g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : ∀ (y : ℝ), ‖f y‖ ≤ 1\nhf3 : ∃ x, f x ≠ 0\ny : ℝ\nhneg : 1 < ‖g y‖\nS : Set ℝ := Set.range fun x => ‖f x‖\nk : ℝ := sSup S\nhk₁ : ∀ (x : ℝ), ‖f x‖ ≤ k\nhk₂ : ∀ (x : ℝ), 2 * (‖f x‖ * ‖g y‖) ≤ 2 * k\nk' : ℝ := k / ‖g y‖\nH₁ : k' < k\n⊢ k ≤ k'",
"after_state": "No Goals!"
},
{
"line": "have h₁ : ∃ x : ℝ, x ∈ S := by use ‖f 0‖; exact Set.mem_range_self 0",
"before_state": "f g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : ∀ (y : ℝ), ‖f y‖ ≤ 1\nhf3 : ∃ x, f x ≠ 0\ny : ℝ\nhneg : 1 < ‖g y‖\nS : Set ℝ := Set.range fun x => ‖f x‖\nk : ℝ := sSup S\nhk₁ : ∀ (x : ℝ), ‖f x‖ ≤ k\nhk₂ : ∀ (x : ℝ), 2 * (‖f x‖ * ‖g y‖) ≤ 2 * k\nk' : ℝ := k / ‖g y‖\nH₁ : k' < k\n⊢ k ≤ k'",
"after_state": "f g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : ∀ (y : ℝ), ‖f y‖ ≤ 1\nhf3 : ∃ x, f x ≠ 0\ny : ℝ\nhneg : 1 < ‖g y‖\nS : Set ℝ := Set.range fun x => ‖f x‖\nk : ℝ := sSup S\nhk₁ : ∀ (x : ℝ), ‖f x‖ ≤ k\nhk₂ : ∀ (x : ℝ), 2 * (‖f x‖ * ‖g y‖) ≤ 2 * k\nk' : ℝ := k / ‖g y‖\nH₁ : k' < k\nh₁ : ∃ x, x ∈ S\n⊢ k ≤ k'"
},
{
"line": "focus\n refine\n no_implicit_lambda%\n (have h₁ : ∃ x : ℝ, x ∈ S := ?body✝;\n ?_)\n case body✝ => with_annotate_state\"by\" (use ‖f 0‖; exact Set.mem_range_self 0)",
"before_state": "f g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : ∀ (y : ℝ), ‖f y‖ ≤ 1\nhf3 : ∃ x, f x ≠ 0\ny : ℝ\nhneg : 1 < ‖g y‖\nS : Set ℝ := Set.range fun x => ‖f x‖\nk : ℝ := sSup S\nhk₁ : ∀ (x : ℝ), ‖f x‖ ≤ k\nhk₂ : ∀ (x : ℝ), 2 * (‖f x‖ * ‖g y‖) ≤ 2 * k\nk' : ℝ := k / ‖g y‖\nH₁ : k' < k\n⊢ k ≤ k'",
"after_state": "f g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : ∀ (y : ℝ), ‖f y‖ ≤ 1\nhf3 : ∃ x, f x ≠ 0\ny : ℝ\nhneg : 1 < ‖g y‖\nS : Set ℝ := Set.range fun x => ‖f x‖\nk : ℝ := sSup S\nhk₁ : ∀ (x : ℝ), ‖f x‖ ≤ k\nhk₂ : ∀ (x : ℝ), 2 * (‖f x‖ * ‖g y‖) ≤ 2 * k\nk' : ℝ := k / ‖g y‖\nH₁ : k' < k\nh₁ : ∃ x, x ∈ S\n⊢ k ≤ k'"
},
{
"line": "refine\n no_implicit_lambda%\n (have h₁ : ∃ x : ℝ, x ∈ S := ?body✝;\n ?_)",
"before_state": "f g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : ∀ (y : ℝ), ‖f y‖ ≤ 1\nhf3 : ∃ x, f x ≠ 0\ny : ℝ\nhneg : 1 < ‖g y‖\nS : Set ℝ := Set.range fun x => ‖f x‖\nk : ℝ := sSup S\nhk₁ : ∀ (x : ℝ), ‖f x‖ ≤ k\nhk₂ : ∀ (x : ℝ), 2 * (‖f x‖ * ‖g y‖) ≤ 2 * k\nk' : ℝ := k / ‖g y‖\nH₁ : k' < k\n⊢ k ≤ k'",
"after_state": "case body\nf g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : ∀ (y : ℝ), ‖f y‖ ≤ 1\nhf3 : ∃ x, f x ≠ 0\ny : ℝ\nhneg : 1 < ‖g y‖\nS : Set ℝ := Set.range fun x => ‖f x‖\nk : ℝ := sSup S\nhk₁ : ∀ (x : ℝ), ‖f x‖ ≤ k\nhk₂ : ∀ (x : ℝ), 2 * (‖f x‖ * ‖g y‖) ≤ 2 * k\nk' : ℝ := k / ‖g y‖\nH₁ : k' < k\n⊢ ∃ x, x ∈ S\n---\nf g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : ∀ (y : ℝ), ‖f y‖ ≤ 1\nhf3 : ∃ x, f x ≠ 0\ny : ℝ\nhneg : 1 < ‖g y‖\nS : Set ℝ := Set.range fun x => ‖f x‖\nk : ℝ := sSup S\nhk₁ : ∀ (x : ℝ), ‖f x‖ ≤ k\nhk₂ : ∀ (x : ℝ), 2 * (‖f x‖ * ‖g y‖) ≤ 2 * k\nk' : ℝ := k / ‖g y‖\nH₁ : k' < k\nh₁ : ∃ x, x ∈ S\n⊢ k ≤ k'"
},
{
"line": "case body✝ => with_annotate_state\"by\" (use ‖f 0‖; exact Set.mem_range_self 0)",
"before_state": "case body\nf g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : ∀ (y : ℝ), ‖f y‖ ≤ 1\nhf3 : ∃ x, f x ≠ 0\ny : ℝ\nhneg : 1 < ‖g y‖\nS : Set ℝ := Set.range fun x => ‖f x‖\nk : ℝ := sSup S\nhk₁ : ∀ (x : ℝ), ‖f x‖ ≤ k\nhk₂ : ∀ (x : ℝ), 2 * (‖f x‖ * ‖g y‖) ≤ 2 * k\nk' : ℝ := k / ‖g y‖\nH₁ : k' < k\n⊢ ∃ x, x ∈ S\n---\nf g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : ∀ (y : ℝ), ‖f y‖ ≤ 1\nhf3 : ∃ x, f x ≠ 0\ny : ℝ\nhneg : 1 < ‖g y‖\nS : Set ℝ := Set.range fun x => ‖f x‖\nk : ℝ := sSup S\nhk₁ : ∀ (x : ℝ), ‖f x‖ ≤ k\nhk₂ : ∀ (x : ℝ), 2 * (‖f x‖ * ‖g y‖) ≤ 2 * k\nk' : ℝ := k / ‖g y‖\nH₁ : k' < k\nh₁ : ∃ x, x ∈ S\n⊢ k ≤ k'",
"after_state": "f g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : ∀ (y : ℝ), ‖f y‖ ≤ 1\nhf3 : ∃ x, f x ≠ 0\ny : ℝ\nhneg : 1 < ‖g y‖\nS : Set ℝ := Set.range fun x => ‖f x‖\nk : ℝ := sSup S\nhk₁ : ∀ (x : ℝ), ‖f x‖ ≤ k\nhk₂ : ∀ (x : ℝ), 2 * (‖f x‖ * ‖g y‖) ≤ 2 * k\nk' : ℝ := k / ‖g y‖\nH₁ : k' < k\nh₁ : ∃ x, x ∈ S\n⊢ k ≤ k'"
},
{
"line": "with_annotate_state\"by\" (use ‖f 0‖; exact Set.mem_range_self 0)",
"before_state": "f g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : ∀ (y : ℝ), ‖f y‖ ≤ 1\nhf3 : ∃ x, f x ≠ 0\ny : ℝ\nhneg : 1 < ‖g y‖\nS : Set ℝ := Set.range fun x => ‖f x‖\nk : ℝ := sSup S\nhk₁ : ∀ (x : ℝ), ‖f x‖ ≤ k\nhk₂ : ∀ (x : ℝ), 2 * (‖f x‖ * ‖g y‖) ≤ 2 * k\nk' : ℝ := k / ‖g y‖\nH₁ : k' < k\n⊢ ∃ x, x ∈ S",
"after_state": "No Goals!"
},
{
"line": "use ‖f 0‖",
"before_state": "f g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : ∀ (y : ℝ), ‖f y‖ ≤ 1\nhf3 : ∃ x, f x ≠ 0\ny : ℝ\nhneg : 1 < ‖g y‖\nS : Set ℝ := Set.range fun x => ‖f x‖\nk : ℝ := sSup S\nhk₁ : ∀ (x : ℝ), ‖f x‖ ≤ k\nhk₂ : ∀ (x : ℝ), 2 * (‖f x‖ * ‖g y‖) ≤ 2 * k\nk' : ℝ := k / ‖g y‖\nH₁ : k' < k\n⊢ ∃ x, x ∈ S",
"after_state": "case h\nf g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : ∀ (y : ℝ), ‖f y‖ ≤ 1\nhf3 : ∃ x, f x ≠ 0\ny : ℝ\nhneg : 1 < ‖g y‖\nS : Set ℝ := Set.range fun x => ‖f x‖\nk : ℝ := sSup S\nhk₁ : ∀ (x : ℝ), ‖f x‖ ≤ k\nhk₂ : ∀ (x : ℝ), 2 * (‖f x‖ * ‖g y‖) ≤ 2 * k\nk' : ℝ := k / ‖g y‖\nH₁ : k' < k\n⊢ ‖f 0‖ ∈ S"
},
{
"line": "refine without_cdot(‖f 0‖ : ?m✝)",
"before_state": "case w\nf g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : ∀ (y : ℝ), ‖f y‖ ≤ 1\nhf3 : ∃ x, f x ≠ 0\ny : ℝ\nhneg : 1 < ‖g y‖\nS : Set ℝ := Set.range fun x => ‖f x‖\nk : ℝ := sSup S\nhk₁ : ∀ (x : ℝ), ‖f x‖ ≤ k\nhk₂ : ∀ (x : ℝ), 2 * (‖f x‖ * ‖g y‖) ≤ 2 * k\nk' : ℝ := k / ‖g y‖\nH₁ : k' < k\n⊢ ℝ",
"after_state": "No Goals!"
},
{
"line": "try with_reducible use_discharger",
"before_state": "case h\nf g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : ∀ (y : ℝ), ‖f y‖ ≤ 1\nhf3 : ∃ x, f x ≠ 0\ny : ℝ\nhneg : 1 < ‖g y‖\nS : Set ℝ := Set.range fun x => ‖f x‖\nk : ℝ := sSup S\nhk₁ : ∀ (x : ℝ), ‖f x‖ ≤ k\nhk₂ : ∀ (x : ℝ), 2 * (‖f x‖ * ‖g y‖) ≤ 2 * k\nk' : ℝ := k / ‖g y‖\nH₁ : k' < k\n⊢ ‖f 0‖ ∈ S",
"after_state": "case h\nf g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : ∀ (y : ℝ), ‖f y‖ ≤ 1\nhf3 : ∃ x, f x ≠ 0\ny : ℝ\nhneg : 1 < ‖g y‖\nS : Set ℝ := Set.range fun x => ‖f x‖\nk : ℝ := sSup S\nhk₁ : ∀ (x : ℝ), ‖f x‖ ≤ k\nhk₂ : ∀ (x : ℝ), 2 * (‖f x‖ * ‖g y‖) ≤ 2 * k\nk' : ℝ := k / ‖g y‖\nH₁ : k' < k\n⊢ ‖f 0‖ ∈ S"
},
{
"line": "first\n| with_reducible use_discharger\n| skip",
"before_state": "case h\nf g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : ∀ (y : ℝ), ‖f y‖ ≤ 1\nhf3 : ∃ x, f x ≠ 0\ny : ℝ\nhneg : 1 < ‖g y‖\nS : Set ℝ := Set.range fun x => ‖f x‖\nk : ℝ := sSup S\nhk₁ : ∀ (x : ℝ), ‖f x‖ ≤ k\nhk₂ : ∀ (x : ℝ), 2 * (‖f x‖ * ‖g y‖) ≤ 2 * k\nk' : ℝ := k / ‖g y‖\nH₁ : k' < k\n⊢ ‖f 0‖ ∈ S",
"after_state": "case h\nf g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : ∀ (y : ℝ), ‖f y‖ ≤ 1\nhf3 : ∃ x, f x ≠ 0\ny : ℝ\nhneg : 1 < ‖g y‖\nS : Set ℝ := Set.range fun x => ‖f x‖\nk : ℝ := sSup S\nhk₁ : ∀ (x : ℝ), ‖f x‖ ≤ k\nhk₂ : ∀ (x : ℝ), 2 * (‖f x‖ * ‖g y‖) ≤ 2 * k\nk' : ℝ := k / ‖g y‖\nH₁ : k' < k\n⊢ ‖f 0‖ ∈ S"
},
{
"line": "with_reducible use_discharger",
"before_state": "case h\nf g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : ∀ (y : ℝ), ‖f y‖ ≤ 1\nhf3 : ∃ x, f x ≠ 0\ny : ℝ\nhneg : 1 < ‖g y‖\nS : Set ℝ := Set.range fun x => ‖f x‖\nk : ℝ := sSup S\nhk₁ : ∀ (x : ℝ), ‖f x‖ ≤ k\nhk₂ : ∀ (x : ℝ), 2 * (‖f x‖ * ‖g y‖) ≤ 2 * k\nk' : ℝ := k / ‖g y‖\nH₁ : k' < k\n⊢ ‖f 0‖ ∈ S",
"after_state": "case h\nf g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : ∀ (y : ℝ), ‖f y‖ ≤ 1\nhf3 : ∃ x, f x ≠ 0\ny : ℝ\nhneg : 1 < ‖g y‖\nS : Set ℝ := Set.range fun x => ‖f x‖\nk : ℝ := sSup S\nhk₁ : ∀ (x : ℝ), ‖f x‖ ≤ k\nhk₂ : ∀ (x : ℝ), 2 * (‖f x‖ * ‖g y‖) ≤ 2 * k\nk' : ℝ := k / ‖g y‖\nH₁ : k' < k\n⊢ ‖f 0‖ ∈ S"
},
{
"line": "use_discharger",
"before_state": "case h\nf g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : ∀ (y : ℝ), ‖f y‖ ≤ 1\nhf3 : ∃ x, f x ≠ 0\ny : ℝ\nhneg : 1 < ‖g y‖\nS : Set ℝ := Set.range fun x => ‖f x‖\nk : ℝ := sSup S\nhk₁ : ∀ (x : ℝ), ‖f x‖ ≤ k\nhk₂ : ∀ (x : ℝ), 2 * (‖f x‖ * ‖g y‖) ≤ 2 * k\nk' : ℝ := k / ‖g y‖\nH₁ : k' < k\n⊢ ‖f 0‖ ∈ S",
"after_state": "case h\nf g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : ∀ (y : ℝ), ‖f y‖ ≤ 1\nhf3 : ∃ x, f x ≠ 0\ny : ℝ\nhneg : 1 < ‖g y‖\nS : Set ℝ := Set.range fun x => ‖f x‖\nk : ℝ := sSup S\nhk₁ : ∀ (x : ℝ), ‖f x‖ ≤ k\nhk₂ : ∀ (x : ℝ), 2 * (‖f x‖ * ‖g y‖) ≤ 2 * k\nk' : ℝ := k / ‖g y‖\nH₁ : k' < k\n⊢ ‖f 0‖ ∈ S"
},
{
"line": "focus\n apply exists_prop.mpr✝\n with_annotate_state\"<;>\" skip\n all_goals use_discharger",
"before_state": "case h\nf g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : ∀ (y : ℝ), ‖f y‖ ≤ 1\nhf3 : ∃ x, f x ≠ 0\ny : ℝ\nhneg : 1 < ‖g y‖\nS : Set ℝ := Set.range fun x => ‖f x‖\nk : ℝ := sSup S\nhk₁ : ∀ (x : ℝ), ‖f x‖ ≤ k\nhk₂ : ∀ (x : ℝ), 2 * (‖f x‖ * ‖g y‖) ≤ 2 * k\nk' : ℝ := k / ‖g y‖\nH₁ : k' < k\n⊢ ‖f 0‖ ∈ S",
"after_state": "case h\nf g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : ∀ (y : ℝ), ‖f y‖ ≤ 1\nhf3 : ∃ x, f x ≠ 0\ny : ℝ\nhneg : 1 < ‖g y‖\nS : Set ℝ := Set.range fun x => ‖f x‖\nk : ℝ := sSup S\nhk₁ : ∀ (x : ℝ), ‖f x‖ ≤ k\nhk₂ : ∀ (x : ℝ), 2 * (‖f x‖ * ‖g y‖) ≤ 2 * k\nk' : ℝ := k / ‖g y‖\nH₁ : k' < k\n⊢ ‖f 0‖ ∈ S"
},
{
"line": "apply exists_prop.mpr✝",
"before_state": "case h\nf g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : ∀ (y : ℝ), ‖f y‖ ≤ 1\nhf3 : ∃ x, f x ≠ 0\ny : ℝ\nhneg : 1 < ‖g y‖\nS : Set ℝ := Set.range fun x => ‖f x‖\nk : ℝ := sSup S\nhk₁ : ∀ (x : ℝ), ‖f x‖ ≤ k\nhk₂ : ∀ (x : ℝ), 2 * (‖f x‖ * ‖g y‖) ≤ 2 * k\nk' : ℝ := k / ‖g y‖\nH₁ : k' < k\n⊢ ‖f 0‖ ∈ S",
"after_state": "case h\nf g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : ∀ (y : ℝ), ‖f y‖ ≤ 1\nhf3 : ∃ x, f x ≠ 0\ny : ℝ\nhneg : 1 < ‖g y‖\nS : Set ℝ := Set.range fun x => ‖f x‖\nk : ℝ := sSup S\nhk₁ : ∀ (x : ℝ), ‖f x‖ ≤ k\nhk₂ : ∀ (x : ℝ), 2 * (‖f x‖ * ‖g y‖) ≤ 2 * k\nk' : ℝ := k / ‖g y‖\nH₁ : k' < k\n⊢ ‖f 0‖ ∈ S"
},
{
"line": "skip",
"before_state": "case h\nf g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : ∀ (y : ℝ), ‖f y‖ ≤ 1\nhf3 : ∃ x, f x ≠ 0\ny : ℝ\nhneg : 1 < ‖g y‖\nS : Set ℝ := Set.range fun x => ‖f x‖\nk : ℝ := sSup S\nhk₁ : ∀ (x : ℝ), ‖f x‖ ≤ k\nhk₂ : ∀ (x : ℝ), 2 * (‖f x‖ * ‖g y‖) ≤ 2 * k\nk' : ℝ := k / ‖g y‖\nH₁ : k' < k\n⊢ ‖f 0‖ ∈ S",
"after_state": "case h\nf g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : ∀ (y : ℝ), ‖f y‖ ≤ 1\nhf3 : ∃ x, f x ≠ 0\ny : ℝ\nhneg : 1 < ‖g y‖\nS : Set ℝ := Set.range fun x => ‖f x‖\nk : ℝ := sSup S\nhk₁ : ∀ (x : ℝ), ‖f x‖ ≤ k\nhk₂ : ∀ (x : ℝ), 2 * (‖f x‖ * ‖g y‖) ≤ 2 * k\nk' : ℝ := k / ‖g y‖\nH₁ : k' < k\n⊢ ‖f 0‖ ∈ S"
},
{
"line": "exact Set.mem_range_self 0",
"before_state": "case h\nf g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : ∀ (y : ℝ), ‖f y‖ ≤ 1\nhf3 : ∃ x, f x ≠ 0\ny : ℝ\nhneg : 1 < ‖g y‖\nS : Set ℝ := Set.range fun x => ‖f x‖\nk : ℝ := sSup S\nhk₁ : ∀ (x : ℝ), ‖f x‖ ≤ k\nhk₂ : ∀ (x : ℝ), 2 * (‖f x‖ * ‖g y‖) ≤ 2 * k\nk' : ℝ := k / ‖g y‖\nH₁ : k' < k\n⊢ ‖f 0‖ ∈ S",
"after_state": "No Goals!"
},
{
"line": "have h₂ : ∀ x, ‖f x‖ ≤ k' := by\n intro x\n rw [le_div_iff₀]\n · apply (mul_le_mul_left zero_lt_two).mp (hk₂ x)\n · exact zero_lt_one.trans hneg",
"before_state": "f g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : ∀ (y : ℝ), ‖f y‖ ≤ 1\nhf3 : ∃ x, f x ≠ 0\ny : ℝ\nhneg : 1 < ‖g y‖\nS : Set ℝ := Set.range fun x => ‖f x‖\nk : ℝ := sSup S\nhk₁ : ∀ (x : ℝ), ‖f x‖ ≤ k\nhk₂ : ∀ (x : ℝ), 2 * (‖f x‖ * ‖g y‖) ≤ 2 * k\nk' : ℝ := k / ‖g y‖\nH₁ : k' < k\nh₁ : ∃ x, x ∈ S\n⊢ k ≤ k'",
"after_state": "f g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : ∀ (y : ℝ), ‖f y‖ ≤ 1\nhf3 : ∃ x, f x ≠ 0\ny : ℝ\nhneg : 1 < ‖g y‖\nS : Set ℝ := Set.range fun x => ‖f x‖\nk : ℝ := sSup S\nhk₁ : ∀ (x : ℝ), ‖f x‖ ≤ k\nhk₂ : ∀ (x : ℝ), 2 * (‖f x‖ * ‖g y‖) ≤ 2 * k\nk' : ℝ := k / ‖g y‖\nH₁ : k' < k\nh₁ : ∃ x, x ∈ S\nh₂ : ∀ (x : ℝ), ‖f x‖ ≤ k'\n⊢ k ≤ k'"
},
{
"line": "focus\n refine\n no_implicit_lambda%\n (have h₂ : ∀ x, ‖f x‖ ≤ k' := ?body✝;\n ?_)\n case body✝ =>\n with_annotate_state\"by\"\n ( intro x\n rw [le_div_iff₀]\n · apply (mul_le_mul_left zero_lt_two).mp (hk₂ x)\n · exact zero_lt_one.trans hneg)",
"before_state": "f g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : ∀ (y : ℝ), ‖f y‖ ≤ 1\nhf3 : ∃ x, f x ≠ 0\ny : ℝ\nhneg : 1 < ‖g y‖\nS : Set ℝ := Set.range fun x => ‖f x‖\nk : ℝ := sSup S\nhk₁ : ∀ (x : ℝ), ‖f x‖ ≤ k\nhk₂ : ∀ (x : ℝ), 2 * (‖f x‖ * ‖g y‖) ≤ 2 * k\nk' : ℝ := k / ‖g y‖\nH₁ : k' < k\nh₁ : ∃ x, x ∈ S\n⊢ k ≤ k'",
"after_state": "f g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : ∀ (y : ℝ), ‖f y‖ ≤ 1\nhf3 : ∃ x, f x ≠ 0\ny : ℝ\nhneg : 1 < ‖g y‖\nS : Set ℝ := Set.range fun x => ‖f x‖\nk : ℝ := sSup S\nhk₁ : ∀ (x : ℝ), ‖f x‖ ≤ k\nhk₂ : ∀ (x : ℝ), 2 * (‖f x‖ * ‖g y‖) ≤ 2 * k\nk' : ℝ := k / ‖g y‖\nH₁ : k' < k\nh₁ : ∃ x, x ∈ S\nh₂ : ∀ (x : ℝ), ‖f x‖ ≤ k'\n⊢ k ≤ k'"
},
{
"line": "refine\n no_implicit_lambda%\n (have h₂ : ∀ x, ‖f x‖ ≤ k' := ?body✝;\n ?_)",
"before_state": "f g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : ∀ (y : ℝ), ‖f y‖ ≤ 1\nhf3 : ∃ x, f x ≠ 0\ny : ℝ\nhneg : 1 < ‖g y‖\nS : Set ℝ := Set.range fun x => ‖f x‖\nk : ℝ := sSup S\nhk₁ : ∀ (x : ℝ), ‖f x‖ ≤ k\nhk₂ : ∀ (x : ℝ), 2 * (‖f x‖ * ‖g y‖) ≤ 2 * k\nk' : ℝ := k / ‖g y‖\nH₁ : k' < k\nh₁ : ∃ x, x ∈ S\n⊢ k ≤ k'",
"after_state": "case body\nf g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : ∀ (y : ℝ), ‖f y‖ ≤ 1\nhf3 : ∃ x, f x ≠ 0\ny : ℝ\nhneg : 1 < ‖g y‖\nS : Set ℝ := Set.range fun x => ‖f x‖\nk : ℝ := sSup S\nhk₁ : ∀ (x : ℝ), ‖f x‖ ≤ k\nhk₂ : ∀ (x : ℝ), 2 * (‖f x‖ * ‖g y‖) ≤ 2 * k\nk' : ℝ := k / ‖g y‖\nH₁ : k' < k\nh₁ : ∃ x, x ∈ S\n⊢ ∀ (x : ℝ), ‖f x‖ ≤ k'\n---\nf g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : ∀ (y : ℝ), ‖f y‖ ≤ 1\nhf3 : ∃ x, f x ≠ 0\ny : ℝ\nhneg : 1 < ‖g y‖\nS : Set ℝ := Set.range fun x => ‖f x‖\nk : ℝ := sSup S\nhk₁ : ∀ (x : ℝ), ‖f x‖ ≤ k\nhk₂ : ∀ (x : ℝ), 2 * (‖f x‖ * ‖g y‖) ≤ 2 * k\nk' : ℝ := k / ‖g y‖\nH₁ : k' < k\nh₁ : ∃ x, x ∈ S\nh₂ : ∀ (x : ℝ), ‖f x‖ ≤ k'\n⊢ k ≤ k'"
},
{
"line": "case body✝ =>\n with_annotate_state\"by\"\n ( intro x\n rw [le_div_iff₀]\n · apply (mul_le_mul_left zero_lt_two).mp (hk₂ x)\n · exact zero_lt_one.trans hneg)",
"before_state": "case body\nf g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : ∀ (y : ℝ), ‖f y‖ ≤ 1\nhf3 : ∃ x, f x ≠ 0\ny : ℝ\nhneg : 1 < ‖g y‖\nS : Set ℝ := Set.range fun x => ‖f x‖\nk : ℝ := sSup S\nhk₁ : ∀ (x : ℝ), ‖f x‖ ≤ k\nhk₂ : ∀ (x : ℝ), 2 * (‖f x‖ * ‖g y‖) ≤ 2 * k\nk' : ℝ := k / ‖g y‖\nH₁ : k' < k\nh₁ : ∃ x, x ∈ S\n⊢ ∀ (x : ℝ), ‖f x‖ ≤ k'\n---\nf g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : ∀ (y : ℝ), ‖f y‖ ≤ 1\nhf3 : ∃ x, f x ≠ 0\ny : ℝ\nhneg : 1 < ‖g y‖\nS : Set ℝ := Set.range fun x => ‖f x‖\nk : ℝ := sSup S\nhk₁ : ∀ (x : ℝ), ‖f x‖ ≤ k\nhk₂ : ∀ (x : ℝ), 2 * (‖f x‖ * ‖g y‖) ≤ 2 * k\nk' : ℝ := k / ‖g y‖\nH₁ : k' < k\nh₁ : ∃ x, x ∈ S\nh₂ : ∀ (x : ℝ), ‖f x‖ ≤ k'\n⊢ k ≤ k'",
"after_state": "f g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : ∀ (y : ℝ), ‖f y‖ ≤ 1\nhf3 : ∃ x, f x ≠ 0\ny : ℝ\nhneg : 1 < ‖g y‖\nS : Set ℝ := Set.range fun x => ‖f x‖\nk : ℝ := sSup S\nhk₁ : ∀ (x : ℝ), ‖f x‖ ≤ k\nhk₂ : ∀ (x : ℝ), 2 * (‖f x‖ * ‖g y‖) ≤ 2 * k\nk' : ℝ := k / ‖g y‖\nH₁ : k' < k\nh₁ : ∃ x, x ∈ S\nh₂ : ∀ (x : ℝ), ‖f x‖ ≤ k'\n⊢ k ≤ k'"
},
{
"line": "with_annotate_state\"by\"\n ( intro x\n rw [le_div_iff₀]\n · apply (mul_le_mul_left zero_lt_two).mp (hk₂ x)\n · exact zero_lt_one.trans hneg)",
"before_state": "f g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : ∀ (y : ℝ), ‖f y‖ ≤ 1\nhf3 : ∃ x, f x ≠ 0\ny : ℝ\nhneg : 1 < ‖g y‖\nS : Set ℝ := Set.range fun x => ‖f x‖\nk : ℝ := sSup S\nhk₁ : ∀ (x : ℝ), ‖f x‖ ≤ k\nhk₂ : ∀ (x : ℝ), 2 * (‖f x‖ * ‖g y‖) ≤ 2 * k\nk' : ℝ := k / ‖g y‖\nH₁ : k' < k\nh₁ : ∃ x, x ∈ S\n⊢ ∀ (x : ℝ), ‖f x‖ ≤ k'",
"after_state": "No Goals!"
},
{
"line": "intro x",
"before_state": "f g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : ∀ (y : ℝ), ‖f y‖ ≤ 1\nhf3 : ∃ x, f x ≠ 0\ny : ℝ\nhneg : 1 < ‖g y‖\nS : Set ℝ := Set.range fun x => ‖f x‖\nk : ℝ := sSup S\nhk₁ : ∀ (x : ℝ), ‖f x‖ ≤ k\nhk₂ : ∀ (x : ℝ), 2 * (‖f x‖ * ‖g y‖) ≤ 2 * k\nk' : ℝ := k / ‖g y‖\nH₁ : k' < k\nh₁ : ∃ x, x ∈ S\n⊢ ∀ (x : ℝ), ‖f x‖ ≤ k'",
"after_state": "f g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : ∀ (y : ℝ), ‖f y‖ ≤ 1\nhf3 : ∃ x, f x ≠ 0\ny : ℝ\nhneg : 1 < ‖g y‖\nS : Set ℝ := Set.range fun x => ‖f x‖\nk : ℝ := sSup S\nhk₁ : ∀ (x : ℝ), ‖f x‖ ≤ k\nhk₂ : ∀ (x : ℝ), 2 * (‖f x‖ * ‖g y‖) ≤ 2 * k\nk' : ℝ := k / ‖g y‖\nH₁ : k' < k\nh₁ : ∃ x, x ∈ S\nx : ℝ\n⊢ ‖f x‖ ≤ k'"
},
{
"line": "rw [le_div_iff₀]",
"before_state": "f g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : ∀ (y : ℝ), ‖f y‖ ≤ 1\nhf3 : ∃ x, f x ≠ 0\ny : ℝ\nhneg : 1 < ‖g y‖\nS : Set ℝ := Set.range fun x => ‖f x‖\nk : ℝ := sSup S\nhk₁ : ∀ (x : ℝ), ‖f x‖ ≤ k\nhk₂ : ∀ (x : ℝ), 2 * (‖f x‖ * ‖g y‖) ≤ 2 * k\nk' : ℝ := k / ‖g y‖\nH₁ : k' < k\nh₁ : ∃ x, x ∈ S\nx : ℝ\n⊢ ‖f x‖ ≤ k'",
"after_state": "f g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : ∀ (y : ℝ), ‖f y‖ ≤ 1\nhf3 : ∃ x, f x ≠ 0\ny : ℝ\nhneg : 1 < ‖g y‖\nS : Set ℝ := Set.range fun x => ‖f x‖\nk : ℝ := sSup S\nhk₁ : ∀ (x : ℝ), ‖f x‖ ≤ k\nhk₂ : ∀ (x : ℝ), 2 * (‖f x‖ * ‖g y‖) ≤ 2 * k\nk' : ℝ := k / ‖g y‖\nH₁ : k' < k\nh₁ : ∃ x, x ∈ S\nx : ℝ\n⊢ ‖f x‖ * ‖g y‖ ≤ k\n---\nf g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : ∀ (y : ℝ), ‖f y‖ ≤ 1\nhf3 : ∃ x, f x ≠ 0\ny : ℝ\nhneg : 1 < ‖g y‖\nS : Set ℝ := Set.range fun x => ‖f x‖\nk : ℝ := sSup S\nhk₁ : ∀ (x : ℝ), ‖f x‖ ≤ k\nhk₂ : ∀ (x : ℝ), 2 * (‖f x‖ * ‖g y‖) ≤ 2 * k\nk' : ℝ := k / ‖g y‖\nH₁ : k' < k\nh₁ : ∃ x, x ∈ S\nx : ℝ\n⊢ 0 < ‖g y‖"
},
{
"line": "rewrite [le_div_iff₀]",
"before_state": "f g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : ∀ (y : ℝ), ‖f y‖ ≤ 1\nhf3 : ∃ x, f x ≠ 0\ny : ℝ\nhneg : 1 < ‖g y‖\nS : Set ℝ := Set.range fun x => ‖f x‖\nk : ℝ := sSup S\nhk₁ : ∀ (x : ℝ), ‖f x‖ ≤ k\nhk₂ : ∀ (x : ℝ), 2 * (‖f x‖ * ‖g y‖) ≤ 2 * k\nk' : ℝ := k / ‖g y‖\nH₁ : k' < k\nh₁ : ∃ x, x ∈ S\nx : ℝ\n⊢ ‖f x‖ ≤ k'",
"after_state": "f g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : ∀ (y : ℝ), ‖f y‖ ≤ 1\nhf3 : ∃ x, f x ≠ 0\ny : ℝ\nhneg : 1 < ‖g y‖\nS : Set ℝ := Set.range fun x => ‖f x‖\nk : ℝ := sSup S\nhk₁ : ∀ (x : ℝ), ‖f x‖ ≤ k\nhk₂ : ∀ (x : ℝ), 2 * (‖f x‖ * ‖g y‖) ≤ 2 * k\nk' : ℝ := k / ‖g y‖\nH₁ : k' < k\nh₁ : ∃ x, x ∈ S\nx : ℝ\n⊢ ‖f x‖ * ‖g y‖ ≤ k\n---\nf g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : ∀ (y : ℝ), ‖f y‖ ≤ 1\nhf3 : ∃ x, f x ≠ 0\ny : ℝ\nhneg : 1 < ‖g y‖\nS : Set ℝ := Set.range fun x => ‖f x‖\nk : ℝ := sSup S\nhk₁ : ∀ (x : ℝ), ‖f x‖ ≤ k\nhk₂ : ∀ (x : ℝ), 2 * (‖f x‖ * ‖g y‖) ≤ 2 * k\nk' : ℝ := k / ‖g y‖\nH₁ : k' < k\nh₁ : ∃ x, x ∈ S\nx : ℝ\n⊢ 0 < ‖g y‖"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "f g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : ∀ (y : ℝ), ‖f y‖ ≤ 1\nhf3 : ∃ x, f x ≠ 0\ny : ℝ\nhneg : 1 < ‖g y‖\nS : Set ℝ := Set.range fun x => ‖f x‖\nk : ℝ := sSup S\nhk₁ : ∀ (x : ℝ), ‖f x‖ ≤ k\nhk₂ : ∀ (x : ℝ), 2 * (‖f x‖ * ‖g y‖) ≤ 2 * k\nk' : ℝ := k / ‖g y‖\nH₁ : k' < k\nh₁ : ∃ x, x ∈ S\nx : ℝ\n⊢ ‖f x‖ * ‖g y‖ ≤ k\n---\nf g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : ∀ (y : ℝ), ‖f y‖ ≤ 1\nhf3 : ∃ x, f x ≠ 0\ny : ℝ\nhneg : 1 < ‖g y‖\nS : Set ℝ := Set.range fun x => ‖f x‖\nk : ℝ := sSup S\nhk₁ : ∀ (x : ℝ), ‖f x‖ ≤ k\nhk₂ : ∀ (x : ℝ), 2 * (‖f x‖ * ‖g y‖) ≤ 2 * k\nk' : ℝ := k / ‖g y‖\nH₁ : k' < k\nh₁ : ∃ x, x ∈ S\nx : ℝ\n⊢ 0 < ‖g y‖",
"after_state": "f g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : ∀ (y : ℝ), ‖f y‖ ≤ 1\nhf3 : ∃ x, f x ≠ 0\ny : ℝ\nhneg : 1 < ‖g y‖\nS : Set ℝ := Set.range fun x => ‖f x‖\nk : ℝ := sSup S\nhk₁ : ∀ (x : ℝ), ‖f x‖ ≤ k\nhk₂ : ∀ (x : ℝ), 2 * (‖f x‖ * ‖g y‖) ≤ 2 * k\nk' : ℝ := k / ‖g y‖\nH₁ : k' < k\nh₁ : ∃ x, x ∈ S\nx : ℝ\n⊢ ‖f x‖ * ‖g y‖ ≤ k\n---\nf g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : ∀ (y : ℝ), ‖f y‖ ≤ 1\nhf3 : ∃ x, f x ≠ 0\ny : ℝ\nhneg : 1 < ‖g y‖\nS : Set ℝ := Set.range fun x => ‖f x‖\nk : ℝ := sSup S\nhk₁ : ∀ (x : ℝ), ‖f x‖ ≤ k\nhk₂ : ∀ (x : ℝ), 2 * (‖f x‖ * ‖g y‖) ≤ 2 * k\nk' : ℝ := k / ‖g y‖\nH₁ : k' < k\nh₁ : ∃ x, x ∈ S\nx : ℝ\n⊢ 0 < ‖g y‖"
},
{
"line": "try (with_reducible rfl)",
"before_state": "f g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : ∀ (y : ℝ), ‖f y‖ ≤ 1\nhf3 : ∃ x, f x ≠ 0\ny : ℝ\nhneg : 1 < ‖g y‖\nS : Set ℝ := Set.range fun x => ‖f x‖\nk : ℝ := sSup S\nhk₁ : ∀ (x : ℝ), ‖f x‖ ≤ k\nhk₂ : ∀ (x : ℝ), 2 * (‖f x‖ * ‖g y‖) ≤ 2 * k\nk' : ℝ := k / ‖g y‖\nH₁ : k' < k\nh₁ : ∃ x, x ∈ S\nx : ℝ\n⊢ ‖f x‖ * ‖g y‖ ≤ k\n---\nf g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : ∀ (y : ℝ), ‖f y‖ ≤ 1\nhf3 : ∃ x, f x ≠ 0\ny : ℝ\nhneg : 1 < ‖g y‖\nS : Set ℝ := Set.range fun x => ‖f x‖\nk : ℝ := sSup S\nhk₁ : ∀ (x : ℝ), ‖f x‖ ≤ k\nhk₂ : ∀ (x : ℝ), 2 * (‖f x‖ * ‖g y‖) ≤ 2 * k\nk' : ℝ := k / ‖g y‖\nH₁ : k' < k\nh₁ : ∃ x, x ∈ S\nx : ℝ\n⊢ 0 < ‖g y‖",
"after_state": "f g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : ∀ (y : ℝ), ‖f y‖ ≤ 1\nhf3 : ∃ x, f x ≠ 0\ny : ℝ\nhneg : 1 < ‖g y‖\nS : Set ℝ := Set.range fun x => ‖f x‖\nk : ℝ := sSup S\nhk₁ : ∀ (x : ℝ), ‖f x‖ ≤ k\nhk₂ : ∀ (x : ℝ), 2 * (‖f x‖ * ‖g y‖) ≤ 2 * k\nk' : ℝ := k / ‖g y‖\nH₁ : k' < k\nh₁ : ∃ x, x ∈ S\nx : ℝ\n⊢ ‖f x‖ * ‖g y‖ ≤ k\n---\nf g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : ∀ (y : ℝ), ‖f y‖ ≤ 1\nhf3 : ∃ x, f x ≠ 0\ny : ℝ\nhneg : 1 < ‖g y‖\nS : Set ℝ := Set.range fun x => ‖f x‖\nk : ℝ := sSup S\nhk₁ : ∀ (x : ℝ), ‖f x‖ ≤ k\nhk₂ : ∀ (x : ℝ), 2 * (‖f x‖ * ‖g y‖) ≤ 2 * k\nk' : ℝ := k / ‖g y‖\nH₁ : k' < k\nh₁ : ∃ x, x ∈ S\nx : ℝ\n⊢ 0 < ‖g y‖"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "f g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : ∀ (y : ℝ), ‖f y‖ ≤ 1\nhf3 : ∃ x, f x ≠ 0\ny : ℝ\nhneg : 1 < ‖g y‖\nS : Set ℝ := Set.range fun x => ‖f x‖\nk : ℝ := sSup S\nhk₁ : ∀ (x : ℝ), ‖f x‖ ≤ k\nhk₂ : ∀ (x : ℝ), 2 * (‖f x‖ * ‖g y‖) ≤ 2 * k\nk' : ℝ := k / ‖g y‖\nH₁ : k' < k\nh₁ : ∃ x, x ∈ S\nx : ℝ\n⊢ ‖f x‖ * ‖g y‖ ≤ k\n---\nf g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : ∀ (y : ℝ), ‖f y‖ ≤ 1\nhf3 : ∃ x, f x ≠ 0\ny : ℝ\nhneg : 1 < ‖g y‖\nS : Set ℝ := Set.range fun x => ‖f x‖\nk : ℝ := sSup S\nhk₁ : ∀ (x : ℝ), ‖f x‖ ≤ k\nhk₂ : ∀ (x : ℝ), 2 * (‖f x‖ * ‖g y‖) ≤ 2 * k\nk' : ℝ := k / ‖g y‖\nH₁ : k' < k\nh₁ : ∃ x, x ∈ S\nx : ℝ\n⊢ 0 < ‖g y‖",
"after_state": "f g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : ∀ (y : ℝ), ‖f y‖ ≤ 1\nhf3 : ∃ x, f x ≠ 0\ny : ℝ\nhneg : 1 < ‖g y‖\nS : Set ℝ := Set.range fun x => ‖f x‖\nk : ℝ := sSup S\nhk₁ : ∀ (x : ℝ), ‖f x‖ ≤ k\nhk₂ : ∀ (x : ℝ), 2 * (‖f x‖ * ‖g y‖) ≤ 2 * k\nk' : ℝ := k / ‖g y‖\nH₁ : k' < k\nh₁ : ∃ x, x ∈ S\nx : ℝ\n⊢ ‖f x‖ * ‖g y‖ ≤ k\n---\nf g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : ∀ (y : ℝ), ‖f y‖ ≤ 1\nhf3 : ∃ x, f x ≠ 0\ny : ℝ\nhneg : 1 < ‖g y‖\nS : Set ℝ := Set.range fun x => ‖f x‖\nk : ℝ := sSup S\nhk₁ : ∀ (x : ℝ), ‖f x‖ ≤ k\nhk₂ : ∀ (x : ℝ), 2 * (‖f x‖ * ‖g y‖) ≤ 2 * k\nk' : ℝ := k / ‖g y‖\nH₁ : k' < k\nh₁ : ∃ x, x ∈ S\nx : ℝ\n⊢ 0 < ‖g y‖"
},
{
"line": "with_reducible rfl",
"before_state": "f g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : ∀ (y : ℝ), ‖f y‖ ≤ 1\nhf3 : ∃ x, f x ≠ 0\ny : ℝ\nhneg : 1 < ‖g y‖\nS : Set ℝ := Set.range fun x => ‖f x‖\nk : ℝ := sSup S\nhk₁ : ∀ (x : ℝ), ‖f x‖ ≤ k\nhk₂ : ∀ (x : ℝ), 2 * (‖f x‖ * ‖g y‖) ≤ 2 * k\nk' : ℝ := k / ‖g y‖\nH₁ : k' < k\nh₁ : ∃ x, x ∈ S\nx : ℝ\n⊢ ‖f x‖ * ‖g y‖ ≤ k\n---\nf g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : ∀ (y : ℝ), ‖f y‖ ≤ 1\nhf3 : ∃ x, f x ≠ 0\ny : ℝ\nhneg : 1 < ‖g y‖\nS : Set ℝ := Set.range fun x => ‖f x‖\nk : ℝ := sSup S\nhk₁ : ∀ (x : ℝ), ‖f x‖ ≤ k\nhk₂ : ∀ (x : ℝ), 2 * (‖f x‖ * ‖g y‖) ≤ 2 * k\nk' : ℝ := k / ‖g y‖\nH₁ : k' < k\nh₁ : ∃ x, x ∈ S\nx : ℝ\n⊢ 0 < ‖g y‖",
"after_state": "f g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : ∀ (y : ℝ), ‖f y‖ ≤ 1\nhf3 : ∃ x, f x ≠ 0\ny : ℝ\nhneg : 1 < ‖g y‖\nS : Set ℝ := Set.range fun x => ‖f x‖\nk : ℝ := sSup S\nhk₁ : ∀ (x : ℝ), ‖f x‖ ≤ k\nhk₂ : ∀ (x : ℝ), 2 * (‖f x‖ * ‖g y‖) ≤ 2 * k\nk' : ℝ := k / ‖g y‖\nH₁ : k' < k\nh₁ : ∃ x, x ∈ S\nx : ℝ\n⊢ ‖f x‖ * ‖g y‖ ≤ k\n---\nf g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : ∀ (y : ℝ), ‖f y‖ ≤ 1\nhf3 : ∃ x, f x ≠ 0\ny : ℝ\nhneg : 1 < ‖g y‖\nS : Set ℝ := Set.range fun x => ‖f x‖\nk : ℝ := sSup S\nhk₁ : ∀ (x : ℝ), ‖f x‖ ≤ k\nhk₂ : ∀ (x : ℝ), 2 * (‖f x‖ * ‖g y‖) ≤ 2 * k\nk' : ℝ := k / ‖g y‖\nH₁ : k' < k\nh₁ : ∃ x, x ∈ S\nx : ℝ\n⊢ 0 < ‖g y‖"
},
{
"line": "rfl",
"before_state": "f g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : ∀ (y : ℝ), ‖f y‖ ≤ 1\nhf3 : ∃ x, f x ≠ 0\ny : ℝ\nhneg : 1 < ‖g y‖\nS : Set ℝ := Set.range fun x => ‖f x‖\nk : ℝ := sSup S\nhk₁ : ∀ (x : ℝ), ‖f x‖ ≤ k\nhk₂ : ∀ (x : ℝ), 2 * (‖f x‖ * ‖g y‖) ≤ 2 * k\nk' : ℝ := k / ‖g y‖\nH₁ : k' < k\nh₁ : ∃ x, x ∈ S\nx : ℝ\n⊢ ‖f x‖ * ‖g y‖ ≤ k\n---\nf g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : ∀ (y : ℝ), ‖f y‖ ≤ 1\nhf3 : ∃ x, f x ≠ 0\ny : ℝ\nhneg : 1 < ‖g y‖\nS : Set ℝ := Set.range fun x => ‖f x‖\nk : ℝ := sSup S\nhk₁ : ∀ (x : ℝ), ‖f x‖ ≤ k\nhk₂ : ∀ (x : ℝ), 2 * (‖f x‖ * ‖g y‖) ≤ 2 * k\nk' : ℝ := k / ‖g y‖\nH₁ : k' < k\nh₁ : ∃ x, x ∈ S\nx : ℝ\n⊢ 0 < ‖g y‖",
"after_state": "f g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : ∀ (y : ℝ), ‖f y‖ ≤ 1\nhf3 : ∃ x, f x ≠ 0\ny : ℝ\nhneg : 1 < ‖g y‖\nS : Set ℝ := Set.range fun x => ‖f x‖\nk : ℝ := sSup S\nhk₁ : ∀ (x : ℝ), ‖f x‖ ≤ k\nhk₂ : ∀ (x : ℝ), 2 * (‖f x‖ * ‖g y‖) ≤ 2 * k\nk' : ℝ := k / ‖g y‖\nH₁ : k' < k\nh₁ : ∃ x, x ∈ S\nx : ℝ\n⊢ ‖f x‖ * ‖g y‖ ≤ k\n---\nf g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : ∀ (y : ℝ), ‖f y‖ ≤ 1\nhf3 : ∃ x, f x ≠ 0\ny : ℝ\nhneg : 1 < ‖g y‖\nS : Set ℝ := Set.range fun x => ‖f x‖\nk : ℝ := sSup S\nhk₁ : ∀ (x : ℝ), ‖f x‖ ≤ k\nhk₂ : ∀ (x : ℝ), 2 * (‖f x‖ * ‖g y‖) ≤ 2 * k\nk' : ℝ := k / ‖g y‖\nH₁ : k' < k\nh₁ : ∃ x, x ∈ S\nx : ℝ\n⊢ 0 < ‖g y‖"
},
{
"line": "apply_rfl",
"before_state": "f g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : ∀ (y : ℝ), ‖f y‖ ≤ 1\nhf3 : ∃ x, f x ≠ 0\ny : ℝ\nhneg : 1 < ‖g y‖\nS : Set ℝ := Set.range fun x => ‖f x‖\nk : ℝ := sSup S\nhk₁ : ∀ (x : ℝ), ‖f x‖ ≤ k\nhk₂ : ∀ (x : ℝ), 2 * (‖f x‖ * ‖g y‖) ≤ 2 * k\nk' : ℝ := k / ‖g y‖\nH₁ : k' < k\nh₁ : ∃ x, x ∈ S\nx : ℝ\n⊢ ‖f x‖ * ‖g y‖ ≤ k\n---\nf g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : ∀ (y : ℝ), ‖f y‖ ≤ 1\nhf3 : ∃ x, f x ≠ 0\ny : ℝ\nhneg : 1 < ‖g y‖\nS : Set ℝ := Set.range fun x => ‖f x‖\nk : ℝ := sSup S\nhk₁ : ∀ (x : ℝ), ‖f x‖ ≤ k\nhk₂ : ∀ (x : ℝ), 2 * (‖f x‖ * ‖g y‖) ≤ 2 * k\nk' : ℝ := k / ‖g y‖\nH₁ : k' < k\nh₁ : ∃ x, x ∈ S\nx : ℝ\n⊢ 0 < ‖g y‖",
"after_state": "f g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : ∀ (y : ℝ), ‖f y‖ ≤ 1\nhf3 : ∃ x, f x ≠ 0\ny : ℝ\nhneg : 1 < ‖g y‖\nS : Set ℝ := Set.range fun x => ‖f x‖\nk : ℝ := sSup S\nhk₁ : ∀ (x : ℝ), ‖f x‖ ≤ k\nhk₂ : ∀ (x : ℝ), 2 * (‖f x‖ * ‖g y‖) ≤ 2 * k\nk' : ℝ := k / ‖g y‖\nH₁ : k' < k\nh₁ : ∃ x, x ∈ S\nx : ℝ\n⊢ ‖f x‖ * ‖g y‖ ≤ k\n---\nf g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : ∀ (y : ℝ), ‖f y‖ ≤ 1\nhf3 : ∃ x, f x ≠ 0\ny : ℝ\nhneg : 1 < ‖g y‖\nS : Set ℝ := Set.range fun x => ‖f x‖\nk : ℝ := sSup S\nhk₁ : ∀ (x : ℝ), ‖f x‖ ≤ k\nhk₂ : ∀ (x : ℝ), 2 * (‖f x‖ * ‖g y‖) ≤ 2 * k\nk' : ℝ := k / ‖g y‖\nH₁ : k' < k\nh₁ : ∃ x, x ∈ S\nx : ℝ\n⊢ 0 < ‖g y‖"
},
{
"line": "skip",
"before_state": "f g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : ∀ (y : ℝ), ‖f y‖ ≤ 1\nhf3 : ∃ x, f x ≠ 0\ny : ℝ\nhneg : 1 < ‖g y‖\nS : Set ℝ := Set.range fun x => ‖f x‖\nk : ℝ := sSup S\nhk₁ : ∀ (x : ℝ), ‖f x‖ ≤ k\nhk₂ : ∀ (x : ℝ), 2 * (‖f x‖ * ‖g y‖) ≤ 2 * k\nk' : ℝ := k / ‖g y‖\nH₁ : k' < k\nh₁ : ∃ x, x ∈ S\nx : ℝ\n⊢ ‖f x‖ * ‖g y‖ ≤ k\n---\nf g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : ∀ (y : ℝ), ‖f y‖ ≤ 1\nhf3 : ∃ x, f x ≠ 0\ny : ℝ\nhneg : 1 < ‖g y‖\nS : Set ℝ := Set.range fun x => ‖f x‖\nk : ℝ := sSup S\nhk₁ : ∀ (x : ℝ), ‖f x‖ ≤ k\nhk₂ : ∀ (x : ℝ), 2 * (‖f x‖ * ‖g y‖) ≤ 2 * k\nk' : ℝ := k / ‖g y‖\nH₁ : k' < k\nh₁ : ∃ x, x ∈ S\nx : ℝ\n⊢ 0 < ‖g y‖",
"after_state": "f g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : ∀ (y : ℝ), ‖f y‖ ≤ 1\nhf3 : ∃ x, f x ≠ 0\ny : ℝ\nhneg : 1 < ‖g y‖\nS : Set ℝ := Set.range fun x => ‖f x‖\nk : ℝ := sSup S\nhk₁ : ∀ (x : ℝ), ‖f x‖ ≤ k\nhk₂ : ∀ (x : ℝ), 2 * (‖f x‖ * ‖g y‖) ≤ 2 * k\nk' : ℝ := k / ‖g y‖\nH₁ : k' < k\nh₁ : ∃ x, x ∈ S\nx : ℝ\n⊢ ‖f x‖ * ‖g y‖ ≤ k\n---\nf g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : ∀ (y : ℝ), ‖f y‖ ≤ 1\nhf3 : ∃ x, f x ≠ 0\ny : ℝ\nhneg : 1 < ‖g y‖\nS : Set ℝ := Set.range fun x => ‖f x‖\nk : ℝ := sSup S\nhk₁ : ∀ (x : ℝ), ‖f x‖ ≤ k\nhk₂ : ∀ (x : ℝ), 2 * (‖f x‖ * ‖g y‖) ≤ 2 * k\nk' : ℝ := k / ‖g y‖\nH₁ : k' < k\nh₁ : ∃ x, x ∈ S\nx : ℝ\n⊢ 0 < ‖g y‖"
},
{
"line": "apply (mul_le_mul_left zero_lt_two).mp (hk₂ x)",
"before_state": "f g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : ∀ (y : ℝ), ‖f y‖ ≤ 1\nhf3 : ∃ x, f x ≠ 0\ny : ℝ\nhneg : 1 < ‖g y‖\nS : Set ℝ := Set.range fun x => ‖f x‖\nk : ℝ := sSup S\nhk₁ : ∀ (x : ℝ), ‖f x‖ ≤ k\nhk₂ : ∀ (x : ℝ), 2 * (‖f x‖ * ‖g y‖) ≤ 2 * k\nk' : ℝ := k / ‖g y‖\nH₁ : k' < k\nh₁ : ∃ x, x ∈ S\nx : ℝ\n⊢ ‖f x‖ * ‖g y‖ ≤ k",
"after_state": "No Goals!"
},
{
"line": "exact zero_lt_one.trans hneg",
"before_state": "f g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : ∀ (y : ℝ), ‖f y‖ ≤ 1\nhf3 : ∃ x, f x ≠ 0\ny : ℝ\nhneg : 1 < ‖g y‖\nS : Set ℝ := Set.range fun x => ‖f x‖\nk : ℝ := sSup S\nhk₁ : ∀ (x : ℝ), ‖f x‖ ≤ k\nhk₂ : ∀ (x : ℝ), 2 * (‖f x‖ * ‖g y‖) ≤ 2 * k\nk' : ℝ := k / ‖g y‖\nH₁ : k' < k\nh₁ : ∃ x, x ∈ S\nx : ℝ\n⊢ 0 < ‖g y‖",
"after_state": "No Goals!"
},
{
"line": "apply csSup_le h₁",
"before_state": "f g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : ∀ (y : ℝ), ‖f y‖ ≤ 1\nhf3 : ∃ x, f x ≠ 0\ny : ℝ\nhneg : 1 < ‖g y‖\nS : Set ℝ := Set.range fun x => ‖f x‖\nk : ℝ := sSup S\nhk₁ : ∀ (x : ℝ), ‖f x‖ ≤ k\nhk₂ : ∀ (x : ℝ), 2 * (‖f x‖ * ‖g y‖) ≤ 2 * k\nk' : ℝ := k / ‖g y‖\nH₁ : k' < k\nh₁ : ∃ x, x ∈ S\nh₂ : ∀ (x : ℝ), ‖f x‖ ≤ k'\n⊢ k ≤ k'",
"after_state": "f g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : ∀ (y : ℝ), ‖f y‖ ≤ 1\nhf3 : ∃ x, f x ≠ 0\ny : ℝ\nhneg : 1 < ‖g y‖\nS : Set ℝ := Set.range fun x => ‖f x‖\nk : ℝ := sSup S\nhk₁ : ∀ (x : ℝ), ‖f x‖ ≤ k\nhk₂ : ∀ (x : ℝ), 2 * (‖f x‖ * ‖g y‖) ≤ 2 * k\nk' : ℝ := k / ‖g y‖\nH₁ : k' < k\nh₁ : ∃ x, x ∈ S\nh₂ : ∀ (x : ℝ), ‖f x‖ ≤ k'\n⊢ ∀ b ∈ S, b ≤ k'"
},
{
"line": "rintro y' ⟨yy, rfl⟩",
"before_state": "f g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : ∀ (y : ℝ), ‖f y‖ ≤ 1\nhf3 : ∃ x, f x ≠ 0\ny : ℝ\nhneg : 1 < ‖g y‖\nS : Set ℝ := Set.range fun x => ‖f x‖\nk : ℝ := sSup S\nhk₁ : ∀ (x : ℝ), ‖f x‖ ≤ k\nhk₂ : ∀ (x : ℝ), 2 * (‖f x‖ * ‖g y‖) ≤ 2 * k\nk' : ℝ := k / ‖g y‖\nH₁ : k' < k\nh₁ : ∃ x, x ∈ S\nh₂ : ∀ (x : ℝ), ‖f x‖ ≤ k'\n⊢ ∀ b ∈ S, b ≤ k'",
"after_state": "case intro\nf g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : ∀ (y : ℝ), ‖f y‖ ≤ 1\nhf3 : ∃ x, f x ≠ 0\ny : ℝ\nhneg : 1 < ‖g y‖\nS : Set ℝ := Set.range fun x => ‖f x‖\nk : ℝ := sSup S\nhk₁ : ∀ (x : ℝ), ‖f x‖ ≤ k\nhk₂ : ∀ (x : ℝ), 2 * (‖f x‖ * ‖g y‖) ≤ 2 * k\nk' : ℝ := k / ‖g y‖\nH₁ : k' < k\nh₁ : ∃ x, x ∈ S\nh₂ : ∀ (x : ℝ), ‖f x‖ ≤ k'\nyy : ℝ\n⊢ (fun x => ‖f x‖) yy ≤ k'"
},
{
"line": "exact h₂ yy",
"before_state": "case intro\nf g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : ∀ (y : ℝ), ‖f y‖ ≤ 1\nhf3 : ∃ x, f x ≠ 0\ny : ℝ\nhneg : 1 < ‖g y‖\nS : Set ℝ := Set.range fun x => ‖f x‖\nk : ℝ := sSup S\nhk₁ : ∀ (x : ℝ), ‖f x‖ ≤ k\nhk₂ : ∀ (x : ℝ), 2 * (‖f x‖ * ‖g y‖) ≤ 2 * k\nk' : ℝ := k / ‖g y‖\nH₁ : k' < k\nh₁ : ∃ x, x ∈ S\nh₂ : ∀ (x : ℝ), ‖f x‖ ≤ k'\nyy : ℝ\n⊢ (fun x => ‖f x‖) yy ≤ k'",
"after_state": "No Goals!"
},
{
"line": "apply lt_irrefl k'",
"before_state": "f g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : ∀ (y : ℝ), ‖f y‖ ≤ 1\nhf3 : ∃ x, f x ≠ 0\ny : ℝ\nhneg : 1 < ‖g y‖\nS : Set ℝ := Set.range fun x => ‖f x‖\nk : ℝ := sSup S\nhk₁ : ∀ (x : ℝ), ‖f x‖ ≤ k\nhk₂ : ∀ (x : ℝ), 2 * (‖f x‖ * ‖g y‖) ≤ 2 * k\nk' : ℝ := k / ‖g y‖\nH₁ : k' < k\nH₂ : k ≤ k'\n⊢ False",
"after_state": "f g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : ∀ (y : ℝ), ‖f y‖ ≤ 1\nhf3 : ∃ x, f x ≠ 0\ny : ℝ\nhneg : 1 < ‖g y‖\nS : Set ℝ := Set.range fun x => ‖f x‖\nk : ℝ := sSup S\nhk₁ : ∀ (x : ℝ), ‖f x‖ ≤ k\nhk₂ : ∀ (x : ℝ), 2 * (‖f x‖ * ‖g y‖) ≤ 2 * k\nk' : ℝ := k / ‖g y‖\nH₁ : k' < k\nH₂ : k ≤ k'\n⊢ k' < k'"
},
{
"line": "calc\n k' < k := H₁\n _ ≤ k' := H₂",
"before_state": "f g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : ∀ (y : ℝ), ‖f y‖ ≤ 1\nhf3 : ∃ x, f x ≠ 0\ny : ℝ\nhneg : 1 < ‖g y‖\nS : Set ℝ := Set.range fun x => ‖f x‖\nk : ℝ := sSup S\nhk₁ : ∀ (x : ℝ), ‖f x‖ ≤ k\nhk₂ : ∀ (x : ℝ), 2 * (‖f x‖ * ‖g y‖) ≤ 2 * k\nk' : ℝ := k / ‖g y‖\nH₁ : k' < k\nH₂ : k ≤ k'\n⊢ k' < k'",
"after_state": "No Goals!"
}
] |
theorem imo1972_q5' (f g : ℝ → ℝ) (hf1 : ∀ x, ∀ y, f (x + y) + f (x - y) = 2 * f x * g y)
(hf2 : BddAbove (Set.range fun x => ‖f x‖)) (hf3 : ∃ x, f x ≠ 0) (y : ℝ) : ‖g y‖ ≤ 1 := by
obtain ⟨x, hx⟩ := hf3
set k := ⨆ x, ‖f x‖
have h : ∀ x, ‖f x‖ ≤ k := le_ciSup hf2
by_contra! H
have hgy : 0 < ‖g y‖ := by linarith
have k_pos : 0 < k := lt_of_lt_of_le (norm_pos_iff.mpr hx) (h x)
have : k / ‖g y‖ < k := (div_lt_iff₀ hgy).mpr (lt_mul_of_one_lt_right k_pos H)
have : k ≤ k / ‖g y‖ := by
suffices ∀ x, ‖f x‖ ≤ k / ‖g y‖ from ciSup_le this
intro x
suffices 2 * (‖f x‖ * ‖g y‖) ≤ 2 * k by
rwa [le_div_iff₀ hgy, ← mul_le_mul_left (zero_lt_two : (0 : ℝ) < 2)]
calc
2 * (‖f x‖ * ‖g y‖) = ‖2 * f x * g y‖ := by simp [abs_mul, mul_assoc]
_ = ‖f (x + y) + f (x - y)‖ := by rw [hf1]
_ ≤ ‖f (x + y)‖ + ‖f (x - y)‖ := abs_add _ _
_ ≤ 2 * k := by linarith [h (x + y), h (x - y)]
linarith
|
/root/DuelModelResearch/mathlib4/Archive/Imo/Imo1972Q5.lean
|
{
"open": [],
"variables": []
}
|
[
{
"line": "obtain ⟨x, hx⟩ := hf3",
"before_state": "f g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : BddAbove (Set.range fun x => ‖f x‖)\nhf3 : ∃ x, f x ≠ 0\ny : ℝ\n⊢ ‖g y‖ ≤ 1",
"after_state": "case intro\nf g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : BddAbove (Set.range fun x => ‖f x‖)\ny x : ℝ\nhx : f x ≠ 0\n⊢ ‖g y‖ ≤ 1"
},
{
"line": "set k := ⨆ x, ‖f x‖",
"before_state": "case intro\nf g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : BddAbove (Set.range fun x => ‖f x‖)\ny x : ℝ\nhx : f x ≠ 0\n⊢ ‖g y‖ ≤ 1",
"after_state": "case intro\nf g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : BddAbove (Set.range fun x => ‖f x‖)\ny x : ℝ\nhx : f x ≠ 0\nk : ℝ := ⨆ x, ‖f x‖\n⊢ ‖g y‖ ≤ 1"
},
{
"line": "try rewrite [show ?m✝ = k from rfl✝] at *",
"before_state": "case intro\nf g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : BddAbove (Set.range fun x => ‖f x‖)\ny x : ℝ\nhx : f x ≠ 0\nk : ℝ := ⨆ x, ‖f x‖\n⊢ ‖g y‖ ≤ 1",
"after_state": "case intro\nf g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : BddAbove (Set.range fun x => ‖f x‖)\ny x : ℝ\nhx : f x ≠ 0\nk : ℝ := ⨆ x, ‖f x‖\n⊢ ‖g y‖ ≤ 1"
},
{
"line": "first\n| rewrite [show ?m✝ = k from rfl✝] at *\n| skip",
"before_state": "case intro\nf g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : BddAbove (Set.range fun x => ‖f x‖)\ny x : ℝ\nhx : f x ≠ 0\nk : ℝ := ⨆ x, ‖f x‖\n⊢ ‖g y‖ ≤ 1",
"after_state": "case intro\nf g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : BddAbove (Set.range fun x => ‖f x‖)\ny x : ℝ\nhx : f x ≠ 0\nk : ℝ := ⨆ x, ‖f x‖\n⊢ ‖g y‖ ≤ 1"
},
{
"line": "rewrite [show ?m✝ = k from rfl✝] at *",
"before_state": "case intro\nf g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : BddAbove (Set.range fun x => ‖f x‖)\ny x : ℝ\nhx : f x ≠ 0\nk : ℝ := ⨆ x, ‖f x‖\n⊢ ‖g y‖ ≤ 1",
"after_state": "case intro\nf g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : BddAbove (Set.range fun x => ‖f x‖)\ny x : ℝ\nhx : f x ≠ 0\nk : ℝ := ⨆ x, ‖f x‖\n⊢ ‖g y‖ ≤ 1"
},
{
"line": "skip",
"before_state": "case intro\nf g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : BddAbove (Set.range fun x => ‖f x‖)\ny x : ℝ\nhx : f x ≠ 0\nk : ℝ := ⨆ x, ‖f x‖\n⊢ ‖g y‖ ≤ 1",
"after_state": "case intro\nf g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : BddAbove (Set.range fun x => ‖f x‖)\ny x : ℝ\nhx : f x ≠ 0\nk : ℝ := ⨆ x, ‖f x‖\n⊢ ‖g y‖ ≤ 1"
},
{
"line": "have h : ∀ x, ‖f x‖ ≤ k := le_ciSup hf2",
"before_state": "case intro\nf g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : BddAbove (Set.range fun x => ‖f x‖)\ny x : ℝ\nhx : f x ≠ 0\nk : ℝ := ⨆ x, ‖f x‖\n⊢ ‖g y‖ ≤ 1",
"after_state": "case intro\nf g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : BddAbove (Set.range fun x => ‖f x‖)\ny x : ℝ\nhx : f x ≠ 0\nk : ℝ := ⨆ x, ‖f x‖\nh : ∀ (x : ℝ), ‖f x‖ ≤ k\n⊢ ‖g y‖ ≤ 1"
},
{
"line": "refine_lift\n have h : ∀ x, ‖f x‖ ≤ k := le_ciSup hf2;\n ?_",
"before_state": "case intro\nf g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : BddAbove (Set.range fun x => ‖f x‖)\ny x : ℝ\nhx : f x ≠ 0\nk : ℝ := ⨆ x, ‖f x‖\n⊢ ‖g y‖ ≤ 1",
"after_state": "case intro\nf g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : BddAbove (Set.range fun x => ‖f x‖)\ny x : ℝ\nhx : f x ≠ 0\nk : ℝ := ⨆ x, ‖f x‖\nh : ∀ (x : ℝ), ‖f x‖ ≤ k\n⊢ ‖g y‖ ≤ 1"
},
{
"line": "focus\n (refine\n no_implicit_lambda%\n (have h : ∀ x, ‖f x‖ ≤ k := le_ciSup hf2;\n ?_);\n rotate_right)",
"before_state": "case intro\nf g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : BddAbove (Set.range fun x => ‖f x‖)\ny x : ℝ\nhx : f x ≠ 0\nk : ℝ := ⨆ x, ‖f x‖\n⊢ ‖g y‖ ≤ 1",
"after_state": "case intro\nf g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : BddAbove (Set.range fun x => ‖f x‖)\ny x : ℝ\nhx : f x ≠ 0\nk : ℝ := ⨆ x, ‖f x‖\nh : ∀ (x : ℝ), ‖f x‖ ≤ k\n⊢ ‖g y‖ ≤ 1"
},
{
"line": "refine\n no_implicit_lambda%\n (have h : ∀ x, ‖f x‖ ≤ k := le_ciSup hf2;\n ?_)",
"before_state": "case intro\nf g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : BddAbove (Set.range fun x => ‖f x‖)\ny x : ℝ\nhx : f x ≠ 0\nk : ℝ := ⨆ x, ‖f x‖\n⊢ ‖g y‖ ≤ 1",
"after_state": "case intro\nf g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : BddAbove (Set.range fun x => ‖f x‖)\ny x : ℝ\nhx : f x ≠ 0\nk : ℝ := ⨆ x, ‖f x‖\nh : ∀ (x : ℝ), ‖f x‖ ≤ k\n⊢ ‖g y‖ ≤ 1"
},
{
"line": "rotate_right",
"before_state": "case intro\nf g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : BddAbove (Set.range fun x => ‖f x‖)\ny x : ℝ\nhx : f x ≠ 0\nk : ℝ := ⨆ x, ‖f x‖\nh : ∀ (x : ℝ), ‖f x‖ ≤ k\n⊢ ‖g y‖ ≤ 1",
"after_state": "case intro\nf g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : BddAbove (Set.range fun x => ‖f x‖)\ny x : ℝ\nhx : f x ≠ 0\nk : ℝ := ⨆ x, ‖f x‖\nh : ∀ (x : ℝ), ‖f x‖ ≤ k\n⊢ ‖g y‖ ≤ 1"
},
{
"line": "by_contra! H",
"before_state": "case intro\nf g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : BddAbove (Set.range fun x => ‖f x‖)\ny x : ℝ\nhx : f x ≠ 0\nk : ℝ := ⨆ x, ‖f x‖\nh : ∀ (x : ℝ), ‖f x‖ ≤ k\n⊢ ‖g y‖ ≤ 1",
"after_state": "case intro\nf g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : BddAbove (Set.range fun x => ‖f x‖)\ny x : ℝ\nhx : f x ≠ 0\nk : ℝ := ⨆ x, ‖f x‖\nh : ∀ (x : ℝ), ‖f x‖ ≤ k\nH : 1 < ‖g y‖\n⊢ False"
},
{
"line": "by_contra H",
"before_state": "case intro\nf g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : BddAbove (Set.range fun x => ‖f x‖)\ny x : ℝ\nhx : f x ≠ 0\nk : ℝ := ⨆ x, ‖f x‖\nh : ∀ (x : ℝ), ‖f x‖ ≤ k\n⊢ ‖g y‖ ≤ 1",
"after_state": "case intro\nf g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : BddAbove (Set.range fun x => ‖f x‖)\ny x : ℝ\nhx : f x ≠ 0\nk : ℝ := ⨆ x, ‖f x‖\nh : ∀ (x : ℝ), ‖f x‖ ≤ k\nH : ¬‖g y‖ ≤ 1\n⊢ False"
},
{
"line": "first\n| guard_target = Not✝ _; intro H\n| refine (Decidable.byContradiction✝ fun H => ?_ :)\n| refine (Classical.byContradiction✝ fun H => ?_ :)",
"before_state": "case intro\nf g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : BddAbove (Set.range fun x => ‖f x‖)\ny x : ℝ\nhx : f x ≠ 0\nk : ℝ := ⨆ x, ‖f x‖\nh : ∀ (x : ℝ), ‖f x‖ ≤ k\n⊢ ‖g y‖ ≤ 1",
"after_state": "case intro\nf g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : BddAbove (Set.range fun x => ‖f x‖)\ny x : ℝ\nhx : f x ≠ 0\nk : ℝ := ⨆ x, ‖f x‖\nh : ∀ (x : ℝ), ‖f x‖ ≤ k\nH : ¬‖g y‖ ≤ 1\n⊢ False"
},
{
"line": "guard_target = Not✝ _",
"before_state": "case intro\nf g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : BddAbove (Set.range fun x => ‖f x‖)\ny x : ℝ\nhx : f x ≠ 0\nk : ℝ := ⨆ x, ‖f x‖\nh : ∀ (x : ℝ), ‖f x‖ ≤ k\n⊢ ‖g y‖ ≤ 1",
"after_state": "case intro\nf g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : BddAbove (Set.range fun x => ‖f x‖)\ny x : ℝ\nhx : f x ≠ 0\nk : ℝ := ⨆ x, ‖f x‖\nh : ∀ (x : ℝ), ‖f x‖ ≤ k\n⊢ ‖g y‖ ≤ 1"
},
{
"line": "refine (Decidable.byContradiction✝ fun H => ?_ :)",
"before_state": "case intro\nf g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : BddAbove (Set.range fun x => ‖f x‖)\ny x : ℝ\nhx : f x ≠ 0\nk : ℝ := ⨆ x, ‖f x‖\nh : ∀ (x : ℝ), ‖f x‖ ≤ k\n⊢ ‖g y‖ ≤ 1",
"after_state": "case intro\nf g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : BddAbove (Set.range fun x => ‖f x‖)\ny x : ℝ\nhx : f x ≠ 0\nk : ℝ := ⨆ x, ‖f x‖\nh : ∀ (x : ℝ), ‖f x‖ ≤ k\n⊢ ‖g y‖ ≤ 1"
},
{
"line": "refine (Classical.byContradiction✝ fun H => ?_ :)",
"before_state": "case intro\nf g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : BddAbove (Set.range fun x => ‖f x‖)\ny x : ℝ\nhx : f x ≠ 0\nk : ℝ := ⨆ x, ‖f x‖\nh : ∀ (x : ℝ), ‖f x‖ ≤ k\n⊢ ‖g y‖ ≤ 1",
"after_state": "case intro\nf g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : BddAbove (Set.range fun x => ‖f x‖)\ny x : ℝ\nhx : f x ≠ 0\nk : ℝ := ⨆ x, ‖f x‖\nh : ∀ (x : ℝ), ‖f x‖ ≤ k\nH : ¬‖g y‖ ≤ 1\n⊢ False"
},
{
"line": "try push_neg at H",
"before_state": "case intro\nf g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : BddAbove (Set.range fun x => ‖f x‖)\ny x : ℝ\nhx : f x ≠ 0\nk : ℝ := ⨆ x, ‖f x‖\nh : ∀ (x : ℝ), ‖f x‖ ≤ k\nH : ¬‖g y‖ ≤ 1\n⊢ False",
"after_state": "case intro\nf g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : BddAbove (Set.range fun x => ‖f x‖)\ny x : ℝ\nhx : f x ≠ 0\nk : ℝ := ⨆ x, ‖f x‖\nh : ∀ (x : ℝ), ‖f x‖ ≤ k\nH : 1 < ‖g y‖\n⊢ False"
},
{
"line": "first\n| push_neg at H\n| skip",
"before_state": "case intro\nf g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : BddAbove (Set.range fun x => ‖f x‖)\ny x : ℝ\nhx : f x ≠ 0\nk : ℝ := ⨆ x, ‖f x‖\nh : ∀ (x : ℝ), ‖f x‖ ≤ k\nH : ¬‖g y‖ ≤ 1\n⊢ False",
"after_state": "case intro\nf g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : BddAbove (Set.range fun x => ‖f x‖)\ny x : ℝ\nhx : f x ≠ 0\nk : ℝ := ⨆ x, ‖f x‖\nh : ∀ (x : ℝ), ‖f x‖ ≤ k\nH : 1 < ‖g y‖\n⊢ False"
},
{
"line": "push_neg at H",
"before_state": "case intro\nf g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : BddAbove (Set.range fun x => ‖f x‖)\ny x : ℝ\nhx : f x ≠ 0\nk : ℝ := ⨆ x, ‖f x‖\nh : ∀ (x : ℝ), ‖f x‖ ≤ k\nH : ¬‖g y‖ ≤ 1\n⊢ False",
"after_state": "case intro\nf g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : BddAbove (Set.range fun x => ‖f x‖)\ny x : ℝ\nhx : f x ≠ 0\nk : ℝ := ⨆ x, ‖f x‖\nh : ∀ (x : ℝ), ‖f x‖ ≤ k\nH : 1 < ‖g y‖\n⊢ False"
},
{
"line": "have hgy : 0 < ‖g y‖ := by linarith",
"before_state": "case intro\nf g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : BddAbove (Set.range fun x => ‖f x‖)\ny x : ℝ\nhx : f x ≠ 0\nk : ℝ := ⨆ x, ‖f x‖\nh : ∀ (x : ℝ), ‖f x‖ ≤ k\nH : 1 < ‖g y‖\n⊢ False",
"after_state": "case intro\nf g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : BddAbove (Set.range fun x => ‖f x‖)\ny x : ℝ\nhx : f x ≠ 0\nk : ℝ := ⨆ x, ‖f x‖\nh : ∀ (x : ℝ), ‖f x‖ ≤ k\nH : 1 < ‖g y‖\nhgy : 0 < ‖g y‖\n⊢ False"
},
{
"line": "focus\n refine\n no_implicit_lambda%\n (have hgy : 0 < ‖g y‖ := ?body✝;\n ?_)\n case body✝ => with_annotate_state\"by\" (linarith)",
"before_state": "case intro\nf g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : BddAbove (Set.range fun x => ‖f x‖)\ny x : ℝ\nhx : f x ≠ 0\nk : ℝ := ⨆ x, ‖f x‖\nh : ∀ (x : ℝ), ‖f x‖ ≤ k\nH : 1 < ‖g y‖\n⊢ False",
"after_state": "case intro\nf g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : BddAbove (Set.range fun x => ‖f x‖)\ny x : ℝ\nhx : f x ≠ 0\nk : ℝ := ⨆ x, ‖f x‖\nh : ∀ (x : ℝ), ‖f x‖ ≤ k\nH : 1 < ‖g y‖\nhgy : 0 < ‖g y‖\n⊢ False"
},
{
"line": "refine\n no_implicit_lambda%\n (have hgy : 0 < ‖g y‖ := ?body✝;\n ?_)",
"before_state": "case intro\nf g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : BddAbove (Set.range fun x => ‖f x‖)\ny x : ℝ\nhx : f x ≠ 0\nk : ℝ := ⨆ x, ‖f x‖\nh : ∀ (x : ℝ), ‖f x‖ ≤ k\nH : 1 < ‖g y‖\n⊢ False",
"after_state": "case body\nf g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : BddAbove (Set.range fun x => ‖f x‖)\ny x : ℝ\nhx : f x ≠ 0\nk : ℝ := ⨆ x, ‖f x‖\nh : ∀ (x : ℝ), ‖f x‖ ≤ k\nH : 1 < ‖g y‖\n⊢ 0 < ‖g y‖\n---\ncase intro\nf g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : BddAbove (Set.range fun x => ‖f x‖)\ny x : ℝ\nhx : f x ≠ 0\nk : ℝ := ⨆ x, ‖f x‖\nh : ∀ (x : ℝ), ‖f x‖ ≤ k\nH : 1 < ‖g y‖\nhgy : 0 < ‖g y‖\n⊢ False"
},
{
"line": "case body✝ => with_annotate_state\"by\" (linarith)",
"before_state": "case body\nf g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : BddAbove (Set.range fun x => ‖f x‖)\ny x : ℝ\nhx : f x ≠ 0\nk : ℝ := ⨆ x, ‖f x‖\nh : ∀ (x : ℝ), ‖f x‖ ≤ k\nH : 1 < ‖g y‖\n⊢ 0 < ‖g y‖\n---\ncase intro\nf g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : BddAbove (Set.range fun x => ‖f x‖)\ny x : ℝ\nhx : f x ≠ 0\nk : ℝ := ⨆ x, ‖f x‖\nh : ∀ (x : ℝ), ‖f x‖ ≤ k\nH : 1 < ‖g y‖\nhgy : 0 < ‖g y‖\n⊢ False",
"after_state": "case intro\nf g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : BddAbove (Set.range fun x => ‖f x‖)\ny x : ℝ\nhx : f x ≠ 0\nk : ℝ := ⨆ x, ‖f x‖\nh : ∀ (x : ℝ), ‖f x‖ ≤ k\nH : 1 < ‖g y‖\nhgy : 0 < ‖g y‖\n⊢ False"
},
{
"line": "with_annotate_state\"by\" (linarith)",
"before_state": "f g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : BddAbove (Set.range fun x => ‖f x‖)\ny x : ℝ\nhx : f x ≠ 0\nk : ℝ := ⨆ x, ‖f x‖\nh : ∀ (x : ℝ), ‖f x‖ ≤ k\nH : 1 < ‖g y‖\n⊢ 0 < ‖g y‖",
"after_state": "No Goals!"
},
{
"line": "linarith",
"before_state": "f g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : BddAbove (Set.range fun x => ‖f x‖)\ny x : ℝ\nhx : f x ≠ 0\nk : ℝ := ⨆ x, ‖f x‖\nh : ∀ (x : ℝ), ‖f x‖ ≤ k\nH : 1 < ‖g y‖\n⊢ 0 < ‖g y‖",
"after_state": "No Goals!"
},
{
"line": "ring1",
"before_state": "f g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : BddAbove (Set.range fun x => ‖f x‖)\ny x : ℝ\nhx : f x ≠ 0\nk : ℝ := ⨆ x, ‖f x‖\nh : ∀ (x : ℝ), ‖f x‖ ≤ k\nH : 1 < ‖g y‖\na✝ : 0 ≥ ‖g y‖\n⊢ -1 + (1 - ‖g y‖) + (‖g y‖ - 0) = 0",
"after_state": "No Goals!"
},
{
"line": "have k_pos : 0 < k := lt_of_lt_of_le (norm_pos_iff.mpr hx) (h x)",
"before_state": "case intro\nf g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : BddAbove (Set.range fun x => ‖f x‖)\ny x : ℝ\nhx : f x ≠ 0\nk : ℝ := ⨆ x, ‖f x‖\nh : ∀ (x : ℝ), ‖f x‖ ≤ k\nH : 1 < ‖g y‖\nhgy : 0 < ‖g y‖\n⊢ False",
"after_state": "case intro\nf g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : BddAbove (Set.range fun x => ‖f x‖)\ny x : ℝ\nhx : f x ≠ 0\nk : ℝ := ⨆ x, ‖f x‖\nh : ∀ (x : ℝ), ‖f x‖ ≤ k\nH : 1 < ‖g y‖\nhgy : 0 < ‖g y‖\nk_pos : 0 < k\n⊢ False"
},
{
"line": "refine_lift\n have k_pos : 0 < k := lt_of_lt_of_le (norm_pos_iff.mpr hx) (h x);\n ?_",
"before_state": "case intro\nf g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : BddAbove (Set.range fun x => ‖f x‖)\ny x : ℝ\nhx : f x ≠ 0\nk : ℝ := ⨆ x, ‖f x‖\nh : ∀ (x : ℝ), ‖f x‖ ≤ k\nH : 1 < ‖g y‖\nhgy : 0 < ‖g y‖\n⊢ False",
"after_state": "case intro\nf g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : BddAbove (Set.range fun x => ‖f x‖)\ny x : ℝ\nhx : f x ≠ 0\nk : ℝ := ⨆ x, ‖f x‖\nh : ∀ (x : ℝ), ‖f x‖ ≤ k\nH : 1 < ‖g y‖\nhgy : 0 < ‖g y‖\nk_pos : 0 < k\n⊢ False"
},
{
"line": "focus\n (refine\n no_implicit_lambda%\n (have k_pos : 0 < k := lt_of_lt_of_le (norm_pos_iff.mpr hx) (h x);\n ?_);\n rotate_right)",
"before_state": "case intro\nf g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : BddAbove (Set.range fun x => ‖f x‖)\ny x : ℝ\nhx : f x ≠ 0\nk : ℝ := ⨆ x, ‖f x‖\nh : ∀ (x : ℝ), ‖f x‖ ≤ k\nH : 1 < ‖g y‖\nhgy : 0 < ‖g y‖\n⊢ False",
"after_state": "case intro\nf g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : BddAbove (Set.range fun x => ‖f x‖)\ny x : ℝ\nhx : f x ≠ 0\nk : ℝ := ⨆ x, ‖f x‖\nh : ∀ (x : ℝ), ‖f x‖ ≤ k\nH : 1 < ‖g y‖\nhgy : 0 < ‖g y‖\nk_pos : 0 < k\n⊢ False"
},
{
"line": "refine\n no_implicit_lambda%\n (have k_pos : 0 < k := lt_of_lt_of_le (norm_pos_iff.mpr hx) (h x);\n ?_)",
"before_state": "case intro\nf g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : BddAbove (Set.range fun x => ‖f x‖)\ny x : ℝ\nhx : f x ≠ 0\nk : ℝ := ⨆ x, ‖f x‖\nh : ∀ (x : ℝ), ‖f x‖ ≤ k\nH : 1 < ‖g y‖\nhgy : 0 < ‖g y‖\n⊢ False",
"after_state": "case intro\nf g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : BddAbove (Set.range fun x => ‖f x‖)\ny x : ℝ\nhx : f x ≠ 0\nk : ℝ := ⨆ x, ‖f x‖\nh : ∀ (x : ℝ), ‖f x‖ ≤ k\nH : 1 < ‖g y‖\nhgy : 0 < ‖g y‖\nk_pos : 0 < k\n⊢ False"
},
{
"line": "rotate_right",
"before_state": "case intro\nf g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : BddAbove (Set.range fun x => ‖f x‖)\ny x : ℝ\nhx : f x ≠ 0\nk : ℝ := ⨆ x, ‖f x‖\nh : ∀ (x : ℝ), ‖f x‖ ≤ k\nH : 1 < ‖g y‖\nhgy : 0 < ‖g y‖\nk_pos : 0 < k\n⊢ False",
"after_state": "case intro\nf g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : BddAbove (Set.range fun x => ‖f x‖)\ny x : ℝ\nhx : f x ≠ 0\nk : ℝ := ⨆ x, ‖f x‖\nh : ∀ (x : ℝ), ‖f x‖ ≤ k\nH : 1 < ‖g y‖\nhgy : 0 < ‖g y‖\nk_pos : 0 < k\n⊢ False"
},
{
"line": "have : k / ‖g y‖ < k := (div_lt_iff₀ hgy).mpr (lt_mul_of_one_lt_right k_pos H)",
"before_state": "case intro\nf g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : BddAbove (Set.range fun x => ‖f x‖)\ny x : ℝ\nhx : f x ≠ 0\nk : ℝ := ⨆ x, ‖f x‖\nh : ∀ (x : ℝ), ‖f x‖ ≤ k\nH : 1 < ‖g y‖\nhgy : 0 < ‖g y‖\nk_pos : 0 < k\n⊢ False",
"after_state": "case intro\nf g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : BddAbove (Set.range fun x => ‖f x‖)\ny x : ℝ\nhx : f x ≠ 0\nk : ℝ := ⨆ x, ‖f x‖\nh : ∀ (x : ℝ), ‖f x‖ ≤ k\nH : 1 < ‖g y‖\nhgy : 0 < ‖g y‖\nk_pos : 0 < k\nthis : k / ‖g y‖ < k\n⊢ False"
},
{
"line": "refine_lift\n have : k / ‖g y‖ < k := (div_lt_iff₀ hgy).mpr (lt_mul_of_one_lt_right k_pos H);\n ?_",
"before_state": "case intro\nf g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : BddAbove (Set.range fun x => ‖f x‖)\ny x : ℝ\nhx : f x ≠ 0\nk : ℝ := ⨆ x, ‖f x‖\nh : ∀ (x : ℝ), ‖f x‖ ≤ k\nH : 1 < ‖g y‖\nhgy : 0 < ‖g y‖\nk_pos : 0 < k\n⊢ False",
"after_state": "case intro\nf g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : BddAbove (Set.range fun x => ‖f x‖)\ny x : ℝ\nhx : f x ≠ 0\nk : ℝ := ⨆ x, ‖f x‖\nh : ∀ (x : ℝ), ‖f x‖ ≤ k\nH : 1 < ‖g y‖\nhgy : 0 < ‖g y‖\nk_pos : 0 < k\nthis : k / ‖g y‖ < k\n⊢ False"
},
{
"line": "focus\n (refine\n no_implicit_lambda%\n (have : k / ‖g y‖ < k := (div_lt_iff₀ hgy).mpr (lt_mul_of_one_lt_right k_pos H);\n ?_);\n rotate_right)",
"before_state": "case intro\nf g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : BddAbove (Set.range fun x => ‖f x‖)\ny x : ℝ\nhx : f x ≠ 0\nk : ℝ := ⨆ x, ‖f x‖\nh : ∀ (x : ℝ), ‖f x‖ ≤ k\nH : 1 < ‖g y‖\nhgy : 0 < ‖g y‖\nk_pos : 0 < k\n⊢ False",
"after_state": "case intro\nf g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : BddAbove (Set.range fun x => ‖f x‖)\ny x : ℝ\nhx : f x ≠ 0\nk : ℝ := ⨆ x, ‖f x‖\nh : ∀ (x : ℝ), ‖f x‖ ≤ k\nH : 1 < ‖g y‖\nhgy : 0 < ‖g y‖\nk_pos : 0 < k\nthis : k / ‖g y‖ < k\n⊢ False"
},
{
"line": "refine\n no_implicit_lambda%\n (have : k / ‖g y‖ < k := (div_lt_iff₀ hgy).mpr (lt_mul_of_one_lt_right k_pos H);\n ?_)",
"before_state": "case intro\nf g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : BddAbove (Set.range fun x => ‖f x‖)\ny x : ℝ\nhx : f x ≠ 0\nk : ℝ := ⨆ x, ‖f x‖\nh : ∀ (x : ℝ), ‖f x‖ ≤ k\nH : 1 < ‖g y‖\nhgy : 0 < ‖g y‖\nk_pos : 0 < k\n⊢ False",
"after_state": "case intro\nf g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : BddAbove (Set.range fun x => ‖f x‖)\ny x : ℝ\nhx : f x ≠ 0\nk : ℝ := ⨆ x, ‖f x‖\nh : ∀ (x : ℝ), ‖f x‖ ≤ k\nH : 1 < ‖g y‖\nhgy : 0 < ‖g y‖\nk_pos : 0 < k\nthis : k / ‖g y‖ < k\n⊢ False"
},
{
"line": "rotate_right",
"before_state": "case intro\nf g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : BddAbove (Set.range fun x => ‖f x‖)\ny x : ℝ\nhx : f x ≠ 0\nk : ℝ := ⨆ x, ‖f x‖\nh : ∀ (x : ℝ), ‖f x‖ ≤ k\nH : 1 < ‖g y‖\nhgy : 0 < ‖g y‖\nk_pos : 0 < k\nthis : k / ‖g y‖ < k\n⊢ False",
"after_state": "case intro\nf g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : BddAbove (Set.range fun x => ‖f x‖)\ny x : ℝ\nhx : f x ≠ 0\nk : ℝ := ⨆ x, ‖f x‖\nh : ∀ (x : ℝ), ‖f x‖ ≤ k\nH : 1 < ‖g y‖\nhgy : 0 < ‖g y‖\nk_pos : 0 < k\nthis : k / ‖g y‖ < k\n⊢ False"
},
{
"line": "have : k ≤ k / ‖g y‖ := by\n suffices ∀ x, ‖f x‖ ≤ k / ‖g y‖ from ciSup_le this\n intro x\n suffices 2 * (‖f x‖ * ‖g y‖) ≤ 2 * k by rwa [le_div_iff₀ hgy, ← mul_le_mul_left (zero_lt_two : (0 : ℝ) < 2)]\n calc\n 2 * (‖f x‖ * ‖g y‖) = ‖2 * f x * g y‖ := by simp [abs_mul, mul_assoc]\n _ = ‖f (x + y) + f (x - y)‖ := by rw [hf1]\n _ ≤ ‖f (x + y)‖ + ‖f (x - y)‖ := (abs_add _ _)\n _ ≤ 2 * k := by linarith [h (x + y), h (x - y)]",
"before_state": "case intro\nf g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : BddAbove (Set.range fun x => ‖f x‖)\ny x : ℝ\nhx : f x ≠ 0\nk : ℝ := ⨆ x, ‖f x‖\nh : ∀ (x : ℝ), ‖f x‖ ≤ k\nH : 1 < ‖g y‖\nhgy : 0 < ‖g y‖\nk_pos : 0 < k\nthis : k / ‖g y‖ < k\n⊢ False",
"after_state": "case intro\nf g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : BddAbove (Set.range fun x => ‖f x‖)\ny x : ℝ\nhx : f x ≠ 0\nk : ℝ := ⨆ x, ‖f x‖\nh : ∀ (x : ℝ), ‖f x‖ ≤ k\nH : 1 < ‖g y‖\nhgy : 0 < ‖g y‖\nk_pos : 0 < k\nthis✝ : k / ‖g y‖ < k\nthis : k ≤ k / ‖g y‖\n⊢ False"
},
{
"line": "focus\n refine\n no_implicit_lambda%\n (have : k ≤ k / ‖g y‖ := ?body✝;\n ?_)\n case body✝ =>\n with_annotate_state\"by\"\n ( suffices ∀ x, ‖f x‖ ≤ k / ‖g y‖ from ciSup_le this\n intro x\n suffices 2 * (‖f x‖ * ‖g y‖) ≤ 2 * k by rwa [le_div_iff₀ hgy, ← mul_le_mul_left (zero_lt_two : (0 : ℝ) < 2)]\n calc\n 2 * (‖f x‖ * ‖g y‖) = ‖2 * f x * g y‖ := by simp [abs_mul, mul_assoc]\n _ = ‖f (x + y) + f (x - y)‖ := by rw [hf1]\n _ ≤ ‖f (x + y)‖ + ‖f (x - y)‖ := (abs_add _ _)\n _ ≤ 2 * k := by linarith [h (x + y), h (x - y)])",
"before_state": "case intro\nf g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : BddAbove (Set.range fun x => ‖f x‖)\ny x : ℝ\nhx : f x ≠ 0\nk : ℝ := ⨆ x, ‖f x‖\nh : ∀ (x : ℝ), ‖f x‖ ≤ k\nH : 1 < ‖g y‖\nhgy : 0 < ‖g y‖\nk_pos : 0 < k\nthis : k / ‖g y‖ < k\n⊢ False",
"after_state": "case intro\nf g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : BddAbove (Set.range fun x => ‖f x‖)\ny x : ℝ\nhx : f x ≠ 0\nk : ℝ := ⨆ x, ‖f x‖\nh : ∀ (x : ℝ), ‖f x‖ ≤ k\nH : 1 < ‖g y‖\nhgy : 0 < ‖g y‖\nk_pos : 0 < k\nthis✝ : k / ‖g y‖ < k\nthis : k ≤ k / ‖g y‖\n⊢ False"
},
{
"line": "refine\n no_implicit_lambda%\n (have : k ≤ k / ‖g y‖ := ?body✝;\n ?_)",
"before_state": "case intro\nf g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : BddAbove (Set.range fun x => ‖f x‖)\ny x : ℝ\nhx : f x ≠ 0\nk : ℝ := ⨆ x, ‖f x‖\nh : ∀ (x : ℝ), ‖f x‖ ≤ k\nH : 1 < ‖g y‖\nhgy : 0 < ‖g y‖\nk_pos : 0 < k\nthis : k / ‖g y‖ < k\n⊢ False",
"after_state": "case body\nf g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : BddAbove (Set.range fun x => ‖f x‖)\ny x : ℝ\nhx : f x ≠ 0\nk : ℝ := ⨆ x, ‖f x‖\nh : ∀ (x : ℝ), ‖f x‖ ≤ k\nH : 1 < ‖g y‖\nhgy : 0 < ‖g y‖\nk_pos : 0 < k\nthis : k / ‖g y‖ < k\n⊢ k ≤ k / ‖g y‖\n---\ncase intro\nf g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : BddAbove (Set.range fun x => ‖f x‖)\ny x : ℝ\nhx : f x ≠ 0\nk : ℝ := ⨆ x, ‖f x‖\nh : ∀ (x : ℝ), ‖f x‖ ≤ k\nH : 1 < ‖g y‖\nhgy : 0 < ‖g y‖\nk_pos : 0 < k\nthis✝ : k / ‖g y‖ < k\nthis : k ≤ k / ‖g y‖\n⊢ False"
},
{
"line": "case body✝ =>\n with_annotate_state\"by\"\n ( suffices ∀ x, ‖f x‖ ≤ k / ‖g y‖ from ciSup_le this\n intro x\n suffices 2 * (‖f x‖ * ‖g y‖) ≤ 2 * k by rwa [le_div_iff₀ hgy, ← mul_le_mul_left (zero_lt_two : (0 : ℝ) < 2)]\n calc\n 2 * (‖f x‖ * ‖g y‖) = ‖2 * f x * g y‖ := by simp [abs_mul, mul_assoc]\n _ = ‖f (x + y) + f (x - y)‖ := by rw [hf1]\n _ ≤ ‖f (x + y)‖ + ‖f (x - y)‖ := (abs_add _ _)\n _ ≤ 2 * k := by linarith [h (x + y), h (x - y)])",
"before_state": "case body\nf g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : BddAbove (Set.range fun x => ‖f x‖)\ny x : ℝ\nhx : f x ≠ 0\nk : ℝ := ⨆ x, ‖f x‖\nh : ∀ (x : ℝ), ‖f x‖ ≤ k\nH : 1 < ‖g y‖\nhgy : 0 < ‖g y‖\nk_pos : 0 < k\nthis : k / ‖g y‖ < k\n⊢ k ≤ k / ‖g y‖\n---\ncase intro\nf g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : BddAbove (Set.range fun x => ‖f x‖)\ny x : ℝ\nhx : f x ≠ 0\nk : ℝ := ⨆ x, ‖f x‖\nh : ∀ (x : ℝ), ‖f x‖ ≤ k\nH : 1 < ‖g y‖\nhgy : 0 < ‖g y‖\nk_pos : 0 < k\nthis✝ : k / ‖g y‖ < k\nthis : k ≤ k / ‖g y‖\n⊢ False",
"after_state": "case intro\nf g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : BddAbove (Set.range fun x => ‖f x‖)\ny x : ℝ\nhx : f x ≠ 0\nk : ℝ := ⨆ x, ‖f x‖\nh : ∀ (x : ℝ), ‖f x‖ ≤ k\nH : 1 < ‖g y‖\nhgy : 0 < ‖g y‖\nk_pos : 0 < k\nthis✝ : k / ‖g y‖ < k\nthis : k ≤ k / ‖g y‖\n⊢ False"
},
{
"line": "with_annotate_state\"by\"\n ( suffices ∀ x, ‖f x‖ ≤ k / ‖g y‖ from ciSup_le this\n intro x\n suffices 2 * (‖f x‖ * ‖g y‖) ≤ 2 * k by rwa [le_div_iff₀ hgy, ← mul_le_mul_left (zero_lt_two : (0 : ℝ) < 2)]\n calc\n 2 * (‖f x‖ * ‖g y‖) = ‖2 * f x * g y‖ := by simp [abs_mul, mul_assoc]\n _ = ‖f (x + y) + f (x - y)‖ := by rw [hf1]\n _ ≤ ‖f (x + y)‖ + ‖f (x - y)‖ := (abs_add _ _)\n _ ≤ 2 * k := by linarith [h (x + y), h (x - y)])",
"before_state": "f g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : BddAbove (Set.range fun x => ‖f x‖)\ny x : ℝ\nhx : f x ≠ 0\nk : ℝ := ⨆ x, ‖f x‖\nh : ∀ (x : ℝ), ‖f x‖ ≤ k\nH : 1 < ‖g y‖\nhgy : 0 < ‖g y‖\nk_pos : 0 < k\nthis : k / ‖g y‖ < k\n⊢ k ≤ k / ‖g y‖",
"after_state": "No Goals!"
},
{
"line": "suffices ∀ x, ‖f x‖ ≤ k / ‖g y‖ from ciSup_le this",
"before_state": "f g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : BddAbove (Set.range fun x => ‖f x‖)\ny x : ℝ\nhx : f x ≠ 0\nk : ℝ := ⨆ x, ‖f x‖\nh : ∀ (x : ℝ), ‖f x‖ ≤ k\nH : 1 < ‖g y‖\nhgy : 0 < ‖g y‖\nk_pos : 0 < k\nthis : k / ‖g y‖ < k\n⊢ k ≤ k / ‖g y‖",
"after_state": "f g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : BddAbove (Set.range fun x => ‖f x‖)\ny x : ℝ\nhx : f x ≠ 0\nk : ℝ := ⨆ x, ‖f x‖\nh : ∀ (x : ℝ), ‖f x‖ ≤ k\nH : 1 < ‖g y‖\nhgy : 0 < ‖g y‖\nk_pos : 0 < k\nthis : k / ‖g y‖ < k\n⊢ ∀ (x : ℝ), ‖f x‖ ≤ k / ‖g y‖"
},
{
"line": "refine_lift\n suffices ∀ x, ‖f x‖ ≤ k / ‖g y‖ from ciSup_le this;\n ?_",
"before_state": "f g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : BddAbove (Set.range fun x => ‖f x‖)\ny x : ℝ\nhx : f x ≠ 0\nk : ℝ := ⨆ x, ‖f x‖\nh : ∀ (x : ℝ), ‖f x‖ ≤ k\nH : 1 < ‖g y‖\nhgy : 0 < ‖g y‖\nk_pos : 0 < k\nthis : k / ‖g y‖ < k\n⊢ k ≤ k / ‖g y‖",
"after_state": "f g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : BddAbove (Set.range fun x => ‖f x‖)\ny x : ℝ\nhx : f x ≠ 0\nk : ℝ := ⨆ x, ‖f x‖\nh : ∀ (x : ℝ), ‖f x‖ ≤ k\nH : 1 < ‖g y‖\nhgy : 0 < ‖g y‖\nk_pos : 0 < k\nthis : k / ‖g y‖ < k\n⊢ ∀ (x : ℝ), ‖f x‖ ≤ k / ‖g y‖"
},
{
"line": "focus\n (refine\n no_implicit_lambda%\n (suffices ∀ x, ‖f x‖ ≤ k / ‖g y‖ from ciSup_le this;\n ?_);\n rotate_right)",
"before_state": "f g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : BddAbove (Set.range fun x => ‖f x‖)\ny x : ℝ\nhx : f x ≠ 0\nk : ℝ := ⨆ x, ‖f x‖\nh : ∀ (x : ℝ), ‖f x‖ ≤ k\nH : 1 < ‖g y‖\nhgy : 0 < ‖g y‖\nk_pos : 0 < k\nthis : k / ‖g y‖ < k\n⊢ k ≤ k / ‖g y‖",
"after_state": "f g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : BddAbove (Set.range fun x => ‖f x‖)\ny x : ℝ\nhx : f x ≠ 0\nk : ℝ := ⨆ x, ‖f x‖\nh : ∀ (x : ℝ), ‖f x‖ ≤ k\nH : 1 < ‖g y‖\nhgy : 0 < ‖g y‖\nk_pos : 0 < k\nthis : k / ‖g y‖ < k\n⊢ ∀ (x : ℝ), ‖f x‖ ≤ k / ‖g y‖"
},
{
"line": "refine\n no_implicit_lambda%\n (suffices ∀ x, ‖f x‖ ≤ k / ‖g y‖ from ciSup_le this;\n ?_)",
"before_state": "f g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : BddAbove (Set.range fun x => ‖f x‖)\ny x : ℝ\nhx : f x ≠ 0\nk : ℝ := ⨆ x, ‖f x‖\nh : ∀ (x : ℝ), ‖f x‖ ≤ k\nH : 1 < ‖g y‖\nhgy : 0 < ‖g y‖\nk_pos : 0 < k\nthis : k / ‖g y‖ < k\n⊢ k ≤ k / ‖g y‖",
"after_state": "f g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : BddAbove (Set.range fun x => ‖f x‖)\ny x : ℝ\nhx : f x ≠ 0\nk : ℝ := ⨆ x, ‖f x‖\nh : ∀ (x : ℝ), ‖f x‖ ≤ k\nH : 1 < ‖g y‖\nhgy : 0 < ‖g y‖\nk_pos : 0 < k\nthis : k / ‖g y‖ < k\n⊢ ∀ (x : ℝ), ‖f x‖ ≤ k / ‖g y‖"
},
{
"line": "rotate_right",
"before_state": "f g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : BddAbove (Set.range fun x => ‖f x‖)\ny x : ℝ\nhx : f x ≠ 0\nk : ℝ := ⨆ x, ‖f x‖\nh : ∀ (x : ℝ), ‖f x‖ ≤ k\nH : 1 < ‖g y‖\nhgy : 0 < ‖g y‖\nk_pos : 0 < k\nthis : k / ‖g y‖ < k\n⊢ ∀ (x : ℝ), ‖f x‖ ≤ k / ‖g y‖",
"after_state": "f g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : BddAbove (Set.range fun x => ‖f x‖)\ny x : ℝ\nhx : f x ≠ 0\nk : ℝ := ⨆ x, ‖f x‖\nh : ∀ (x : ℝ), ‖f x‖ ≤ k\nH : 1 < ‖g y‖\nhgy : 0 < ‖g y‖\nk_pos : 0 < k\nthis : k / ‖g y‖ < k\n⊢ ∀ (x : ℝ), ‖f x‖ ≤ k / ‖g y‖"
},
{
"line": "intro x",
"before_state": "f g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : BddAbove (Set.range fun x => ‖f x‖)\ny x : ℝ\nhx : f x ≠ 0\nk : ℝ := ⨆ x, ‖f x‖\nh : ∀ (x : ℝ), ‖f x‖ ≤ k\nH : 1 < ‖g y‖\nhgy : 0 < ‖g y‖\nk_pos : 0 < k\nthis : k / ‖g y‖ < k\n⊢ ∀ (x : ℝ), ‖f x‖ ≤ k / ‖g y‖",
"after_state": "f g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : BddAbove (Set.range fun x => ‖f x‖)\ny x✝ : ℝ\nhx : f x✝ ≠ 0\nk : ℝ := ⨆ x, ‖f x‖\nh : ∀ (x : ℝ), ‖f x‖ ≤ k\nH : 1 < ‖g y‖\nhgy : 0 < ‖g y‖\nk_pos : 0 < k\nthis : k / ‖g y‖ < k\nx : ℝ\n⊢ ‖f x‖ ≤ k / ‖g y‖"
},
{
"line": "suffices 2 * (‖f x‖ * ‖g y‖) ≤ 2 * k by rwa [le_div_iff₀ hgy, ← mul_le_mul_left (zero_lt_two : (0 : ℝ) < 2)]",
"before_state": "f g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : BddAbove (Set.range fun x => ‖f x‖)\ny x✝ : ℝ\nhx : f x✝ ≠ 0\nk : ℝ := ⨆ x, ‖f x‖\nh : ∀ (x : ℝ), ‖f x‖ ≤ k\nH : 1 < ‖g y‖\nhgy : 0 < ‖g y‖\nk_pos : 0 < k\nthis : k / ‖g y‖ < k\nx : ℝ\n⊢ ‖f x‖ ≤ k / ‖g y‖",
"after_state": "f g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : BddAbove (Set.range fun x => ‖f x‖)\ny x✝ : ℝ\nhx : f x✝ ≠ 0\nk : ℝ := ⨆ x, ‖f x‖\nh : ∀ (x : ℝ), ‖f x‖ ≤ k\nH : 1 < ‖g y‖\nhgy : 0 < ‖g y‖\nk_pos : 0 < k\nthis : k / ‖g y‖ < k\nx : ℝ\n⊢ 2 * (‖f x‖ * ‖g y‖) ≤ 2 * k"
},
{
"line": "refine_lift\n suffices 2 * (‖f x‖ * ‖g y‖) ≤ 2 * k by rwa [le_div_iff₀ hgy, ← mul_le_mul_left (zero_lt_two : (0 : ℝ) < 2)];\n ?_",
"before_state": "f g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : BddAbove (Set.range fun x => ‖f x‖)\ny x✝ : ℝ\nhx : f x✝ ≠ 0\nk : ℝ := ⨆ x, ‖f x‖\nh : ∀ (x : ℝ), ‖f x‖ ≤ k\nH : 1 < ‖g y‖\nhgy : 0 < ‖g y‖\nk_pos : 0 < k\nthis : k / ‖g y‖ < k\nx : ℝ\n⊢ ‖f x‖ ≤ k / ‖g y‖",
"after_state": "f g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : BddAbove (Set.range fun x => ‖f x‖)\ny x✝ : ℝ\nhx : f x✝ ≠ 0\nk : ℝ := ⨆ x, ‖f x‖\nh : ∀ (x : ℝ), ‖f x‖ ≤ k\nH : 1 < ‖g y‖\nhgy : 0 < ‖g y‖\nk_pos : 0 < k\nthis : k / ‖g y‖ < k\nx : ℝ\n⊢ 2 * (‖f x‖ * ‖g y‖) ≤ 2 * k"
},
{
"line": "focus\n (refine\n no_implicit_lambda%\n (suffices 2 * (‖f x‖ * ‖g y‖) ≤ 2 * k by rwa [le_div_iff₀ hgy, ← mul_le_mul_left (zero_lt_two : (0 : ℝ) < 2)];\n ?_);\n rotate_right)",
"before_state": "f g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : BddAbove (Set.range fun x => ‖f x‖)\ny x✝ : ℝ\nhx : f x✝ ≠ 0\nk : ℝ := ⨆ x, ‖f x‖\nh : ∀ (x : ℝ), ‖f x‖ ≤ k\nH : 1 < ‖g y‖\nhgy : 0 < ‖g y‖\nk_pos : 0 < k\nthis : k / ‖g y‖ < k\nx : ℝ\n⊢ ‖f x‖ ≤ k / ‖g y‖",
"after_state": "f g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : BddAbove (Set.range fun x => ‖f x‖)\ny x✝ : ℝ\nhx : f x✝ ≠ 0\nk : ℝ := ⨆ x, ‖f x‖\nh : ∀ (x : ℝ), ‖f x‖ ≤ k\nH : 1 < ‖g y‖\nhgy : 0 < ‖g y‖\nk_pos : 0 < k\nthis : k / ‖g y‖ < k\nx : ℝ\n⊢ 2 * (‖f x‖ * ‖g y‖) ≤ 2 * k"
},
{
"line": "refine\n no_implicit_lambda%\n (suffices 2 * (‖f x‖ * ‖g y‖) ≤ 2 * k by rwa [le_div_iff₀ hgy, ← mul_le_mul_left (zero_lt_two : (0 : ℝ) < 2)];\n ?_)",
"before_state": "f g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : BddAbove (Set.range fun x => ‖f x‖)\ny x✝ : ℝ\nhx : f x✝ ≠ 0\nk : ℝ := ⨆ x, ‖f x‖\nh : ∀ (x : ℝ), ‖f x‖ ≤ k\nH : 1 < ‖g y‖\nhgy : 0 < ‖g y‖\nk_pos : 0 < k\nthis : k / ‖g y‖ < k\nx : ℝ\n⊢ ‖f x‖ ≤ k / ‖g y‖",
"after_state": "f g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : BddAbove (Set.range fun x => ‖f x‖)\ny x✝ : ℝ\nhx : f x✝ ≠ 0\nk : ℝ := ⨆ x, ‖f x‖\nh : ∀ (x : ℝ), ‖f x‖ ≤ k\nH : 1 < ‖g y‖\nhgy : 0 < ‖g y‖\nk_pos : 0 < k\nthis : k / ‖g y‖ < k\nx : ℝ\n⊢ 2 * (‖f x‖ * ‖g y‖) ≤ 2 * k"
},
{
"line": "rwa [le_div_iff₀ hgy, ← mul_le_mul_left (zero_lt_two : (0 : ℝ) < 2)]",
"before_state": "f g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : BddAbove (Set.range fun x => ‖f x‖)\ny x✝ : ℝ\nhx : f x✝ ≠ 0\nk : ℝ := ⨆ x, ‖f x‖\nh : ∀ (x : ℝ), ‖f x‖ ≤ k\nH : 1 < ‖g y‖\nhgy : 0 < ‖g y‖\nk_pos : 0 < k\nthis✝ : k / ‖g y‖ < k\nx : ℝ\nthis : 2 * (‖f x‖ * ‖g y‖) ≤ 2 * k\n⊢ ‖f x‖ ≤ k / ‖g y‖",
"after_state": "No Goals!"
},
{
"line": "rw [le_div_iff₀ hgy, ← mul_le_mul_left (zero_lt_two : (0 : ℝ) < 2)]",
"before_state": "f g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : BddAbove (Set.range fun x => ‖f x‖)\ny x✝ : ℝ\nhx : f x✝ ≠ 0\nk : ℝ := ⨆ x, ‖f x‖\nh : ∀ (x : ℝ), ‖f x‖ ≤ k\nH : 1 < ‖g y‖\nhgy : 0 < ‖g y‖\nk_pos : 0 < k\nthis✝ : k / ‖g y‖ < k\nx : ℝ\nthis : 2 * (‖f x‖ * ‖g y‖) ≤ 2 * k\n⊢ ‖f x‖ ≤ k / ‖g y‖",
"after_state": "f g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : BddAbove (Set.range fun x => ‖f x‖)\ny x✝ : ℝ\nhx : f x✝ ≠ 0\nk : ℝ := ⨆ x, ‖f x‖\nh : ∀ (x : ℝ), ‖f x‖ ≤ k\nH : 1 < ‖g y‖\nhgy : 0 < ‖g y‖\nk_pos : 0 < k\nthis✝ : k / ‖g y‖ < k\nx : ℝ\nthis : 2 * (‖f x‖ * ‖g y‖) ≤ 2 * k\n⊢ 2 * (‖f x‖ * ‖g y‖) ≤ 2 * k"
},
{
"line": "rewrite [le_div_iff₀ hgy, ← mul_le_mul_left (zero_lt_two : (0 : ℝ) < 2)]",
"before_state": "f g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : BddAbove (Set.range fun x => ‖f x‖)\ny x✝ : ℝ\nhx : f x✝ ≠ 0\nk : ℝ := ⨆ x, ‖f x‖\nh : ∀ (x : ℝ), ‖f x‖ ≤ k\nH : 1 < ‖g y‖\nhgy : 0 < ‖g y‖\nk_pos : 0 < k\nthis✝ : k / ‖g y‖ < k\nx : ℝ\nthis : 2 * (‖f x‖ * ‖g y‖) ≤ 2 * k\n⊢ ‖f x‖ ≤ k / ‖g y‖",
"after_state": "f g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : BddAbove (Set.range fun x => ‖f x‖)\ny x✝ : ℝ\nhx : f x✝ ≠ 0\nk : ℝ := ⨆ x, ‖f x‖\nh : ∀ (x : ℝ), ‖f x‖ ≤ k\nH : 1 < ‖g y‖\nhgy : 0 < ‖g y‖\nk_pos : 0 < k\nthis✝ : k / ‖g y‖ < k\nx : ℝ\nthis : 2 * (‖f x‖ * ‖g y‖) ≤ 2 * k\n⊢ 2 * (‖f x‖ * ‖g y‖) ≤ 2 * k"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "f g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : BddAbove (Set.range fun x => ‖f x‖)\ny x✝ : ℝ\nhx : f x✝ ≠ 0\nk : ℝ := ⨆ x, ‖f x‖\nh : ∀ (x : ℝ), ‖f x‖ ≤ k\nH : 1 < ‖g y‖\nhgy : 0 < ‖g y‖\nk_pos : 0 < k\nthis✝ : k / ‖g y‖ < k\nx : ℝ\nthis : 2 * (‖f x‖ * ‖g y‖) ≤ 2 * k\n⊢ 2 * (‖f x‖ * ‖g y‖) ≤ 2 * k",
"after_state": "f g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : BddAbove (Set.range fun x => ‖f x‖)\ny x✝ : ℝ\nhx : f x✝ ≠ 0\nk : ℝ := ⨆ x, ‖f x‖\nh : ∀ (x : ℝ), ‖f x‖ ≤ k\nH : 1 < ‖g y‖\nhgy : 0 < ‖g y‖\nk_pos : 0 < k\nthis✝ : k / ‖g y‖ < k\nx : ℝ\nthis : 2 * (‖f x‖ * ‖g y‖) ≤ 2 * k\n⊢ 2 * (‖f x‖ * ‖g y‖) ≤ 2 * k"
},
{
"line": "try (with_reducible rfl)",
"before_state": "f g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : BddAbove (Set.range fun x => ‖f x‖)\ny x✝ : ℝ\nhx : f x✝ ≠ 0\nk : ℝ := ⨆ x, ‖f x‖\nh : ∀ (x : ℝ), ‖f x‖ ≤ k\nH : 1 < ‖g y‖\nhgy : 0 < ‖g y‖\nk_pos : 0 < k\nthis✝ : k / ‖g y‖ < k\nx : ℝ\nthis : 2 * (‖f x‖ * ‖g y‖) ≤ 2 * k\n⊢ 2 * (‖f x‖ * ‖g y‖) ≤ 2 * k",
"after_state": "f g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : BddAbove (Set.range fun x => ‖f x‖)\ny x✝ : ℝ\nhx : f x✝ ≠ 0\nk : ℝ := ⨆ x, ‖f x‖\nh : ∀ (x : ℝ), ‖f x‖ ≤ k\nH : 1 < ‖g y‖\nhgy : 0 < ‖g y‖\nk_pos : 0 < k\nthis✝ : k / ‖g y‖ < k\nx : ℝ\nthis : 2 * (‖f x‖ * ‖g y‖) ≤ 2 * k\n⊢ 2 * (‖f x‖ * ‖g y‖) ≤ 2 * k"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "f g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : BddAbove (Set.range fun x => ‖f x‖)\ny x✝ : ℝ\nhx : f x✝ ≠ 0\nk : ℝ := ⨆ x, ‖f x‖\nh : ∀ (x : ℝ), ‖f x‖ ≤ k\nH : 1 < ‖g y‖\nhgy : 0 < ‖g y‖\nk_pos : 0 < k\nthis✝ : k / ‖g y‖ < k\nx : ℝ\nthis : 2 * (‖f x‖ * ‖g y‖) ≤ 2 * k\n⊢ 2 * (‖f x‖ * ‖g y‖) ≤ 2 * k",
"after_state": "f g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : BddAbove (Set.range fun x => ‖f x‖)\ny x✝ : ℝ\nhx : f x✝ ≠ 0\nk : ℝ := ⨆ x, ‖f x‖\nh : ∀ (x : ℝ), ‖f x‖ ≤ k\nH : 1 < ‖g y‖\nhgy : 0 < ‖g y‖\nk_pos : 0 < k\nthis✝ : k / ‖g y‖ < k\nx : ℝ\nthis : 2 * (‖f x‖ * ‖g y‖) ≤ 2 * k\n⊢ 2 * (‖f x‖ * ‖g y‖) ≤ 2 * k"
},
{
"line": "with_reducible rfl",
"before_state": "f g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : BddAbove (Set.range fun x => ‖f x‖)\ny x✝ : ℝ\nhx : f x✝ ≠ 0\nk : ℝ := ⨆ x, ‖f x‖\nh : ∀ (x : ℝ), ‖f x‖ ≤ k\nH : 1 < ‖g y‖\nhgy : 0 < ‖g y‖\nk_pos : 0 < k\nthis✝ : k / ‖g y‖ < k\nx : ℝ\nthis : 2 * (‖f x‖ * ‖g y‖) ≤ 2 * k\n⊢ 2 * (‖f x‖ * ‖g y‖) ≤ 2 * k",
"after_state": "f g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : BddAbove (Set.range fun x => ‖f x‖)\ny x✝ : ℝ\nhx : f x✝ ≠ 0\nk : ℝ := ⨆ x, ‖f x‖\nh : ∀ (x : ℝ), ‖f x‖ ≤ k\nH : 1 < ‖g y‖\nhgy : 0 < ‖g y‖\nk_pos : 0 < k\nthis✝ : k / ‖g y‖ < k\nx : ℝ\nthis : 2 * (‖f x‖ * ‖g y‖) ≤ 2 * k\n⊢ 2 * (‖f x‖ * ‖g y‖) ≤ 2 * k"
},
{
"line": "rfl",
"before_state": "f g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : BddAbove (Set.range fun x => ‖f x‖)\ny x✝ : ℝ\nhx : f x✝ ≠ 0\nk : ℝ := ⨆ x, ‖f x‖\nh : ∀ (x : ℝ), ‖f x‖ ≤ k\nH : 1 < ‖g y‖\nhgy : 0 < ‖g y‖\nk_pos : 0 < k\nthis✝ : k / ‖g y‖ < k\nx : ℝ\nthis : 2 * (‖f x‖ * ‖g y‖) ≤ 2 * k\n⊢ 2 * (‖f x‖ * ‖g y‖) ≤ 2 * k",
"after_state": "f g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : BddAbove (Set.range fun x => ‖f x‖)\ny x✝ : ℝ\nhx : f x✝ ≠ 0\nk : ℝ := ⨆ x, ‖f x‖\nh : ∀ (x : ℝ), ‖f x‖ ≤ k\nH : 1 < ‖g y‖\nhgy : 0 < ‖g y‖\nk_pos : 0 < k\nthis✝ : k / ‖g y‖ < k\nx : ℝ\nthis : 2 * (‖f x‖ * ‖g y‖) ≤ 2 * k\n⊢ 2 * (‖f x‖ * ‖g y‖) ≤ 2 * k"
},
{
"line": "apply_rfl",
"before_state": "f g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : BddAbove (Set.range fun x => ‖f x‖)\ny x✝ : ℝ\nhx : f x✝ ≠ 0\nk : ℝ := ⨆ x, ‖f x‖\nh : ∀ (x : ℝ), ‖f x‖ ≤ k\nH : 1 < ‖g y‖\nhgy : 0 < ‖g y‖\nk_pos : 0 < k\nthis✝ : k / ‖g y‖ < k\nx : ℝ\nthis : 2 * (‖f x‖ * ‖g y‖) ≤ 2 * k\n⊢ 2 * (‖f x‖ * ‖g y‖) ≤ 2 * k",
"after_state": "f g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : BddAbove (Set.range fun x => ‖f x‖)\ny x✝ : ℝ\nhx : f x✝ ≠ 0\nk : ℝ := ⨆ x, ‖f x‖\nh : ∀ (x : ℝ), ‖f x‖ ≤ k\nH : 1 < ‖g y‖\nhgy : 0 < ‖g y‖\nk_pos : 0 < k\nthis✝ : k / ‖g y‖ < k\nx : ℝ\nthis : 2 * (‖f x‖ * ‖g y‖) ≤ 2 * k\n⊢ 2 * (‖f x‖ * ‖g y‖) ≤ 2 * k"
},
{
"line": "skip",
"before_state": "f g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : BddAbove (Set.range fun x => ‖f x‖)\ny x✝ : ℝ\nhx : f x✝ ≠ 0\nk : ℝ := ⨆ x, ‖f x‖\nh : ∀ (x : ℝ), ‖f x‖ ≤ k\nH : 1 < ‖g y‖\nhgy : 0 < ‖g y‖\nk_pos : 0 < k\nthis✝ : k / ‖g y‖ < k\nx : ℝ\nthis : 2 * (‖f x‖ * ‖g y‖) ≤ 2 * k\n⊢ 2 * (‖f x‖ * ‖g y‖) ≤ 2 * k",
"after_state": "f g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : BddAbove (Set.range fun x => ‖f x‖)\ny x✝ : ℝ\nhx : f x✝ ≠ 0\nk : ℝ := ⨆ x, ‖f x‖\nh : ∀ (x : ℝ), ‖f x‖ ≤ k\nH : 1 < ‖g y‖\nhgy : 0 < ‖g y‖\nk_pos : 0 < k\nthis✝ : k / ‖g y‖ < k\nx : ℝ\nthis : 2 * (‖f x‖ * ‖g y‖) ≤ 2 * k\n⊢ 2 * (‖f x‖ * ‖g y‖) ≤ 2 * k"
},
{
"line": "assumption",
"before_state": "f g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : BddAbove (Set.range fun x => ‖f x‖)\ny x✝ : ℝ\nhx : f x✝ ≠ 0\nk : ℝ := ⨆ x, ‖f x‖\nh : ∀ (x : ℝ), ‖f x‖ ≤ k\nH : 1 < ‖g y‖\nhgy : 0 < ‖g y‖\nk_pos : 0 < k\nthis✝ : k / ‖g y‖ < k\nx : ℝ\nthis : 2 * (‖f x‖ * ‖g y‖) ≤ 2 * k\n⊢ 2 * (‖f x‖ * ‖g y‖) ≤ 2 * k",
"after_state": "No Goals!"
},
{
"line": "rotate_right",
"before_state": "f g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : BddAbove (Set.range fun x => ‖f x‖)\ny x✝ : ℝ\nhx : f x✝ ≠ 0\nk : ℝ := ⨆ x, ‖f x‖\nh : ∀ (x : ℝ), ‖f x‖ ≤ k\nH : 1 < ‖g y‖\nhgy : 0 < ‖g y‖\nk_pos : 0 < k\nthis : k / ‖g y‖ < k\nx : ℝ\n⊢ 2 * (‖f x‖ * ‖g y‖) ≤ 2 * k",
"after_state": "f g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : BddAbove (Set.range fun x => ‖f x‖)\ny x✝ : ℝ\nhx : f x✝ ≠ 0\nk : ℝ := ⨆ x, ‖f x‖\nh : ∀ (x : ℝ), ‖f x‖ ≤ k\nH : 1 < ‖g y‖\nhgy : 0 < ‖g y‖\nk_pos : 0 < k\nthis : k / ‖g y‖ < k\nx : ℝ\n⊢ 2 * (‖f x‖ * ‖g y‖) ≤ 2 * k"
},
{
"line": "calc\n 2 * (‖f x‖ * ‖g y‖) = ‖2 * f x * g y‖ := by simp [abs_mul, mul_assoc]\n _ = ‖f (x + y) + f (x - y)‖ := by rw [hf1]\n _ ≤ ‖f (x + y)‖ + ‖f (x - y)‖ := (abs_add _ _)\n _ ≤ 2 * k := by linarith [h (x + y), h (x - y)]",
"before_state": "f g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : BddAbove (Set.range fun x => ‖f x‖)\ny x✝ : ℝ\nhx : f x✝ ≠ 0\nk : ℝ := ⨆ x, ‖f x‖\nh : ∀ (x : ℝ), ‖f x‖ ≤ k\nH : 1 < ‖g y‖\nhgy : 0 < ‖g y‖\nk_pos : 0 < k\nthis : k / ‖g y‖ < k\nx : ℝ\n⊢ 2 * (‖f x‖ * ‖g y‖) ≤ 2 * k",
"after_state": "No Goals!"
},
{
"line": "simp [abs_mul, mul_assoc]",
"before_state": "f g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : BddAbove (Set.range fun x => ‖f x‖)\ny x✝ : ℝ\nhx : f x✝ ≠ 0\nk : ℝ := ⨆ x, ‖f x‖\nh : ∀ (x : ℝ), ‖f x‖ ≤ k\nH : 1 < ‖g y‖\nhgy : 0 < ‖g y‖\nk_pos : 0 < k\nthis : k / ‖g y‖ < k\nx : ℝ\n⊢ 2 * (‖f x‖ * ‖g y‖) = ‖2 * f x * g y‖",
"after_state": "No Goals!"
},
{
"line": "rw [hf1]",
"before_state": "f g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : BddAbove (Set.range fun x => ‖f x‖)\ny x✝ : ℝ\nhx : f x✝ ≠ 0\nk : ℝ := ⨆ x, ‖f x‖\nh : ∀ (x : ℝ), ‖f x‖ ≤ k\nH : 1 < ‖g y‖\nhgy : 0 < ‖g y‖\nk_pos : 0 < k\nthis : k / ‖g y‖ < k\nx : ℝ\n⊢ ‖2 * f x * g y‖ = ‖f (x + y) + f (x - y)‖",
"after_state": "No Goals!"
},
{
"line": "rewrite [hf1]",
"before_state": "f g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : BddAbove (Set.range fun x => ‖f x‖)\ny x✝ : ℝ\nhx : f x✝ ≠ 0\nk : ℝ := ⨆ x, ‖f x‖\nh : ∀ (x : ℝ), ‖f x‖ ≤ k\nH : 1 < ‖g y‖\nhgy : 0 < ‖g y‖\nk_pos : 0 < k\nthis : k / ‖g y‖ < k\nx : ℝ\n⊢ ‖2 * f x * g y‖ = ‖f (x + y) + f (x - y)‖",
"after_state": "f g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : BddAbove (Set.range fun x => ‖f x‖)\ny x✝ : ℝ\nhx : f x✝ ≠ 0\nk : ℝ := ⨆ x, ‖f x‖\nh : ∀ (x : ℝ), ‖f x‖ ≤ k\nH : 1 < ‖g y‖\nhgy : 0 < ‖g y‖\nk_pos : 0 < k\nthis : k / ‖g y‖ < k\nx : ℝ\n⊢ ‖2 * f x * g y‖ = ‖2 * f x * g y‖"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "f g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : BddAbove (Set.range fun x => ‖f x‖)\ny x✝ : ℝ\nhx : f x✝ ≠ 0\nk : ℝ := ⨆ x, ‖f x‖\nh : ∀ (x : ℝ), ‖f x‖ ≤ k\nH : 1 < ‖g y‖\nhgy : 0 < ‖g y‖\nk_pos : 0 < k\nthis : k / ‖g y‖ < k\nx : ℝ\n⊢ ‖2 * f x * g y‖ = ‖2 * f x * g y‖",
"after_state": "No Goals!"
},
{
"line": "try (with_reducible rfl)",
"before_state": "f g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : BddAbove (Set.range fun x => ‖f x‖)\ny x✝ : ℝ\nhx : f x✝ ≠ 0\nk : ℝ := ⨆ x, ‖f x‖\nh : ∀ (x : ℝ), ‖f x‖ ≤ k\nH : 1 < ‖g y‖\nhgy : 0 < ‖g y‖\nk_pos : 0 < k\nthis : k / ‖g y‖ < k\nx : ℝ\n⊢ ‖2 * f x * g y‖ = ‖2 * f x * g y‖",
"after_state": "No Goals!"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "f g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : BddAbove (Set.range fun x => ‖f x‖)\ny x✝ : ℝ\nhx : f x✝ ≠ 0\nk : ℝ := ⨆ x, ‖f x‖\nh : ∀ (x : ℝ), ‖f x‖ ≤ k\nH : 1 < ‖g y‖\nhgy : 0 < ‖g y‖\nk_pos : 0 < k\nthis : k / ‖g y‖ < k\nx : ℝ\n⊢ ‖2 * f x * g y‖ = ‖2 * f x * g y‖",
"after_state": "No Goals!"
},
{
"line": "with_reducible rfl",
"before_state": "f g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : BddAbove (Set.range fun x => ‖f x‖)\ny x✝ : ℝ\nhx : f x✝ ≠ 0\nk : ℝ := ⨆ x, ‖f x‖\nh : ∀ (x : ℝ), ‖f x‖ ≤ k\nH : 1 < ‖g y‖\nhgy : 0 < ‖g y‖\nk_pos : 0 < k\nthis : k / ‖g y‖ < k\nx : ℝ\n⊢ ‖2 * f x * g y‖ = ‖2 * f x * g y‖",
"after_state": "No Goals!"
},
{
"line": "rfl",
"before_state": "f g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : BddAbove (Set.range fun x => ‖f x‖)\ny x✝ : ℝ\nhx : f x✝ ≠ 0\nk : ℝ := ⨆ x, ‖f x‖\nh : ∀ (x : ℝ), ‖f x‖ ≤ k\nH : 1 < ‖g y‖\nhgy : 0 < ‖g y‖\nk_pos : 0 < k\nthis : k / ‖g y‖ < k\nx : ℝ\n⊢ ‖2 * f x * g y‖ = ‖2 * f x * g y‖",
"after_state": "No Goals!"
},
{
"line": "eq_refl",
"before_state": "f g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : BddAbove (Set.range fun x => ‖f x‖)\ny x✝ : ℝ\nhx : f x✝ ≠ 0\nk : ℝ := ⨆ x, ‖f x‖\nh : ∀ (x : ℝ), ‖f x‖ ≤ k\nH : 1 < ‖g y‖\nhgy : 0 < ‖g y‖\nk_pos : 0 < k\nthis : k / ‖g y‖ < k\nx : ℝ\n⊢ ‖2 * f x * g y‖ = ‖2 * f x * g y‖",
"after_state": "No Goals!"
},
{
"line": "linarith [h (x + y), h (x - y)]",
"before_state": "f g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : BddAbove (Set.range fun x => ‖f x‖)\ny x✝ : ℝ\nhx : f x✝ ≠ 0\nk : ℝ := ⨆ x, ‖f x‖\nh : ∀ (x : ℝ), ‖f x‖ ≤ k\nH : 1 < ‖g y‖\nhgy : 0 < ‖g y‖\nk_pos : 0 < k\nthis : k / ‖g y‖ < k\nx : ℝ\n⊢ ‖f (x + y)‖ + ‖f (x - y)‖ ≤ 2 * k",
"after_state": "No Goals!"
},
{
"line": "ring1",
"before_state": "f g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : BddAbove (Set.range fun x => ‖f x‖)\ny x✝ : ℝ\nhx : f x✝ ≠ 0\nk : ℝ := ⨆ x, ‖f x‖\nh : ∀ (x : ℝ), ‖f x‖ ≤ k\nH : 1 < ‖g y‖\nhgy : 0 < ‖g y‖\nk_pos : 0 < k\nthis : k / ‖g y‖ < k\nx : ℝ\na✝ : ‖f (x + y)‖ + ‖f (x - y)‖ > 2 * k\n⊢ 2 * k - (‖f (x + y)‖ + ‖f (x - y)‖) + (‖f (x + y)‖ - k) + (‖f (x - y)‖ - k) = 0",
"after_state": "No Goals!"
},
{
"line": "linarith",
"before_state": "case intro\nf g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : BddAbove (Set.range fun x => ‖f x‖)\ny x : ℝ\nhx : f x ≠ 0\nk : ℝ := ⨆ x, ‖f x‖\nh : ∀ (x : ℝ), ‖f x‖ ≤ k\nH : 1 < ‖g y‖\nhgy : 0 < ‖g y‖\nk_pos : 0 < k\nthis✝ : k / ‖g y‖ < k\nthis : k ≤ k / ‖g y‖\n⊢ False",
"after_state": "No Goals!"
},
{
"line": "ring1",
"before_state": "f g : ℝ → ℝ\nhf1 : ∀ (x y : ℝ), f (x + y) + f (x - y) = 2 * f x * g y\nhf2 : BddAbove (Set.range fun x => ‖f x‖)\ny x : ℝ\nhx : f x ≠ 0\nk : ℝ := ⨆ x, ‖f x‖\nh : ∀ (x : ℝ), ‖f x‖ ≤ k\nH : 1 < ‖g y‖\nhgy : 0 < ‖g y‖\nk_pos : 0 < k\nthis✝ : k / ‖g y‖ < k\nthis : k ≤ k / ‖g y‖\n⊢ 1 * (k / ‖g y‖) - 1 * k + (1 * k - 1 * (k / ‖g y‖)) = 0",
"after_state": "No Goals!"
}
] |
theorem imo1977_q6_nat (f : ℕ → ℕ) (h : ∀ n, f (f n) < f (n + 1)) : ∀ n, f n = n := by
have h' : ∀ k n : ℕ, k ≤ n → k ≤ f n := by
intro k
induction' k with k h_ind
· intros; exact Nat.zero_le _
· intro n hk
apply Nat.succ_le_of_lt
calc
k ≤ f (f (n - 1)) := h_ind _ (h_ind (n - 1) (le_tsub_of_add_le_right hk))
_ < f n := tsub_add_cancel_of_le (le_trans (Nat.succ_le_succ (Nat.zero_le _)) hk) ▸ h _
have hf : ∀ n, n ≤ f n := fun n => h' n n rfl.le
have hf_mono : StrictMono f := strictMono_nat_of_lt_succ fun _ => lt_of_le_of_lt (hf _) (h _)
intro
exact Nat.eq_of_le_of_lt_succ (hf _) (hf_mono.lt_iff_lt.mp (h _))
|
/root/DuelModelResearch/mathlib4/Archive/Imo/Imo1977Q6.lean
|
{
"open": [],
"variables": []
}
|
[
{
"line": "have h' : ∀ k n : ℕ, k ≤ n → k ≤ f n := by\n intro k\n induction' k with k h_ind\n · intros; exact Nat.zero_le _\n · intro n hk\n apply Nat.succ_le_of_lt\n calc\n k ≤ f (f (n - 1)) := h_ind _ (h_ind (n - 1) (le_tsub_of_add_le_right hk))\n _ < f n := tsub_add_cancel_of_le (le_trans (Nat.succ_le_succ (Nat.zero_le _)) hk) ▸ h _",
"before_state": "f : ℕ → ℕ\nh : ∀ (n : ℕ), f (f n) < f (n + 1)\n⊢ ∀ (n : ℕ), f n = n",
"after_state": "f : ℕ → ℕ\nh : ∀ (n : ℕ), f (f n) < f (n + 1)\nh' : ∀ (k n : ℕ), k ≤ n → k ≤ f n\n⊢ ∀ (n : ℕ), f n = n"
},
{
"line": "focus\n refine\n no_implicit_lambda%\n (have h' : ∀ k n : ℕ, k ≤ n → k ≤ f n := ?body✝;\n ?_)\n case body✝ =>\n with_annotate_state\"by\"\n ( intro k\n induction' k with k h_ind\n · intros; exact Nat.zero_le _\n · intro n hk\n apply Nat.succ_le_of_lt\n calc\n k ≤ f (f (n - 1)) := h_ind _ (h_ind (n - 1) (le_tsub_of_add_le_right hk))\n _ < f n := tsub_add_cancel_of_le (le_trans (Nat.succ_le_succ (Nat.zero_le _)) hk) ▸ h _)",
"before_state": "f : ℕ → ℕ\nh : ∀ (n : ℕ), f (f n) < f (n + 1)\n⊢ ∀ (n : ℕ), f n = n",
"after_state": "f : ℕ → ℕ\nh : ∀ (n : ℕ), f (f n) < f (n + 1)\nh' : ∀ (k n : ℕ), k ≤ n → k ≤ f n\n⊢ ∀ (n : ℕ), f n = n"
},
{
"line": "refine\n no_implicit_lambda%\n (have h' : ∀ k n : ℕ, k ≤ n → k ≤ f n := ?body✝;\n ?_)",
"before_state": "f : ℕ → ℕ\nh : ∀ (n : ℕ), f (f n) < f (n + 1)\n⊢ ∀ (n : ℕ), f n = n",
"after_state": "case body\nf : ℕ → ℕ\nh : ∀ (n : ℕ), f (f n) < f (n + 1)\n⊢ ∀ (k n : ℕ), k ≤ n → k ≤ f n\n---\nf : ℕ → ℕ\nh : ∀ (n : ℕ), f (f n) < f (n + 1)\nh' : ∀ (k n : ℕ), k ≤ n → k ≤ f n\n⊢ ∀ (n : ℕ), f n = n"
},
{
"line": "case body✝ =>\n with_annotate_state\"by\"\n ( intro k\n induction' k with k h_ind\n · intros; exact Nat.zero_le _\n · intro n hk\n apply Nat.succ_le_of_lt\n calc\n k ≤ f (f (n - 1)) := h_ind _ (h_ind (n - 1) (le_tsub_of_add_le_right hk))\n _ < f n := tsub_add_cancel_of_le (le_trans (Nat.succ_le_succ (Nat.zero_le _)) hk) ▸ h _)",
"before_state": "case body\nf : ℕ → ℕ\nh : ∀ (n : ℕ), f (f n) < f (n + 1)\n⊢ ∀ (k n : ℕ), k ≤ n → k ≤ f n\n---\nf : ℕ → ℕ\nh : ∀ (n : ℕ), f (f n) < f (n + 1)\nh' : ∀ (k n : ℕ), k ≤ n → k ≤ f n\n⊢ ∀ (n : ℕ), f n = n",
"after_state": "f : ℕ → ℕ\nh : ∀ (n : ℕ), f (f n) < f (n + 1)\nh' : ∀ (k n : ℕ), k ≤ n → k ≤ f n\n⊢ ∀ (n : ℕ), f n = n"
},
{
"line": "with_annotate_state\"by\"\n ( intro k\n induction' k with k h_ind\n · intros; exact Nat.zero_le _\n · intro n hk\n apply Nat.succ_le_of_lt\n calc\n k ≤ f (f (n - 1)) := h_ind _ (h_ind (n - 1) (le_tsub_of_add_le_right hk))\n _ < f n := tsub_add_cancel_of_le (le_trans (Nat.succ_le_succ (Nat.zero_le _)) hk) ▸ h _)",
"before_state": "f : ℕ → ℕ\nh : ∀ (n : ℕ), f (f n) < f (n + 1)\n⊢ ∀ (k n : ℕ), k ≤ n → k ≤ f n",
"after_state": "No Goals!"
},
{
"line": "intro k",
"before_state": "f : ℕ → ℕ\nh : ∀ (n : ℕ), f (f n) < f (n + 1)\n⊢ ∀ (k n : ℕ), k ≤ n → k ≤ f n",
"after_state": "f : ℕ → ℕ\nh : ∀ (n : ℕ), f (f n) < f (n + 1)\nk : ℕ\n⊢ ∀ (n : ℕ), k ≤ n → k ≤ f n"
},
{
"line": "induction' k with k h_ind",
"before_state": "f : ℕ → ℕ\nh : ∀ (n : ℕ), f (f n) < f (n + 1)\nk : ℕ\n⊢ ∀ (n : ℕ), k ≤ n → k ≤ f n",
"after_state": "case zero\nf : ℕ → ℕ\nh : ∀ (n : ℕ), f (f n) < f (n + 1)\n⊢ ∀ (n : ℕ), 0 ≤ n → 0 ≤ f n\n---\ncase succ\nf : ℕ → ℕ\nh : ∀ (n : ℕ), f (f n) < f (n + 1)\nk : ℕ\nh_ind : ∀ (n : ℕ), k ≤ n → k ≤ f n\n⊢ ∀ (n : ℕ), k + 1 ≤ n → k + 1 ≤ f n"
},
{
"line": "intros",
"before_state": "case zero\nf : ℕ → ℕ\nh : ∀ (n : ℕ), f (f n) < f (n + 1)\n⊢ ∀ (n : ℕ), 0 ≤ n → 0 ≤ f n",
"after_state": "case zero\nf : ℕ → ℕ\nh : ∀ (n : ℕ), f (f n) < f (n + 1)\nn✝ : ℕ\na✝ : 0 ≤ n✝\n⊢ 0 ≤ f n✝"
},
{
"line": "exact Nat.zero_le _",
"before_state": "case zero\nf : ℕ → ℕ\nh : ∀ (n : ℕ), f (f n) < f (n + 1)\nn✝ : ℕ\na✝ : 0 ≤ n✝\n⊢ 0 ≤ f n✝",
"after_state": "No Goals!"
},
{
"line": "intro n hk",
"before_state": "case succ\nf : ℕ → ℕ\nh : ∀ (n : ℕ), f (f n) < f (n + 1)\nk : ℕ\nh_ind : ∀ (n : ℕ), k ≤ n → k ≤ f n\n⊢ ∀ (n : ℕ), k + 1 ≤ n → k + 1 ≤ f n",
"after_state": "case succ\nf : ℕ → ℕ\nh : ∀ (n : ℕ), f (f n) < f (n + 1)\nk : ℕ\nh_ind : ∀ (n : ℕ), k ≤ n → k ≤ f n\nn : ℕ\nhk : k + 1 ≤ n\n⊢ k + 1 ≤ f n"
},
{
"line": "intro n;\n intro hk",
"before_state": "case succ\nf : ℕ → ℕ\nh : ∀ (n : ℕ), f (f n) < f (n + 1)\nk : ℕ\nh_ind : ∀ (n : ℕ), k ≤ n → k ≤ f n\n⊢ ∀ (n : ℕ), k + 1 ≤ n → k + 1 ≤ f n",
"after_state": "case succ\nf : ℕ → ℕ\nh : ∀ (n : ℕ), f (f n) < f (n + 1)\nk : ℕ\nh_ind : ∀ (n : ℕ), k ≤ n → k ≤ f n\nn : ℕ\nhk : k + 1 ≤ n\n⊢ k + 1 ≤ f n"
},
{
"line": "intro n",
"before_state": "case succ\nf : ℕ → ℕ\nh : ∀ (n : ℕ), f (f n) < f (n + 1)\nk : ℕ\nh_ind : ∀ (n : ℕ), k ≤ n → k ≤ f n\n⊢ ∀ (n : ℕ), k + 1 ≤ n → k + 1 ≤ f n",
"after_state": "case succ\nf : ℕ → ℕ\nh : ∀ (n : ℕ), f (f n) < f (n + 1)\nk : ℕ\nh_ind : ∀ (n : ℕ), k ≤ n → k ≤ f n\nn : ℕ\n⊢ k + 1 ≤ n → k + 1 ≤ f n"
},
{
"line": "intro hk",
"before_state": "case succ\nf : ℕ → ℕ\nh : ∀ (n : ℕ), f (f n) < f (n + 1)\nk : ℕ\nh_ind : ∀ (n : ℕ), k ≤ n → k ≤ f n\nn : ℕ\n⊢ k + 1 ≤ n → k + 1 ≤ f n",
"after_state": "case succ\nf : ℕ → ℕ\nh : ∀ (n : ℕ), f (f n) < f (n + 1)\nk : ℕ\nh_ind : ∀ (n : ℕ), k ≤ n → k ≤ f n\nn : ℕ\nhk : k + 1 ≤ n\n⊢ k + 1 ≤ f n"
},
{
"line": "apply Nat.succ_le_of_lt",
"before_state": "case succ\nf : ℕ → ℕ\nh : ∀ (n : ℕ), f (f n) < f (n + 1)\nk : ℕ\nh_ind : ∀ (n : ℕ), k ≤ n → k ≤ f n\nn : ℕ\nhk : k + 1 ≤ n\n⊢ k + 1 ≤ f n",
"after_state": "case succ.h\nf : ℕ → ℕ\nh : ∀ (n : ℕ), f (f n) < f (n + 1)\nk : ℕ\nh_ind : ∀ (n : ℕ), k ≤ n → k ≤ f n\nn : ℕ\nhk : k + 1 ≤ n\n⊢ k < f n"
},
{
"line": "calc\n k ≤ f (f (n - 1)) := h_ind _ (h_ind (n - 1) (le_tsub_of_add_le_right hk))\n _ < f n := tsub_add_cancel_of_le (le_trans (Nat.succ_le_succ (Nat.zero_le _)) hk) ▸ h _",
"before_state": "case succ.h\nf : ℕ → ℕ\nh : ∀ (n : ℕ), f (f n) < f (n + 1)\nk : ℕ\nh_ind : ∀ (n : ℕ), k ≤ n → k ≤ f n\nn : ℕ\nhk : k + 1 ≤ n\n⊢ k < f n",
"after_state": "No Goals!"
},
{
"line": "have hf : ∀ n, n ≤ f n := fun n => h' n n rfl.le",
"before_state": "f : ℕ → ℕ\nh : ∀ (n : ℕ), f (f n) < f (n + 1)\nh' : ∀ (k n : ℕ), k ≤ n → k ≤ f n\n⊢ ∀ (n : ℕ), f n = n",
"after_state": "f : ℕ → ℕ\nh : ∀ (n : ℕ), f (f n) < f (n + 1)\nh' : ∀ (k n : ℕ), k ≤ n → k ≤ f n\nhf : ∀ (n : ℕ), n ≤ f n\n⊢ ∀ (n : ℕ), f n = n"
},
{
"line": "refine_lift\n have hf : ∀ n, n ≤ f n := fun n => h' n n rfl.le;\n ?_",
"before_state": "f : ℕ → ℕ\nh : ∀ (n : ℕ), f (f n) < f (n + 1)\nh' : ∀ (k n : ℕ), k ≤ n → k ≤ f n\n⊢ ∀ (n : ℕ), f n = n",
"after_state": "f : ℕ → ℕ\nh : ∀ (n : ℕ), f (f n) < f (n + 1)\nh' : ∀ (k n : ℕ), k ≤ n → k ≤ f n\nhf : ∀ (n : ℕ), n ≤ f n\n⊢ ∀ (n : ℕ), f n = n"
},
{
"line": "focus\n (refine\n no_implicit_lambda%\n (have hf : ∀ n, n ≤ f n := fun n => h' n n rfl.le;\n ?_);\n rotate_right)",
"before_state": "f : ℕ → ℕ\nh : ∀ (n : ℕ), f (f n) < f (n + 1)\nh' : ∀ (k n : ℕ), k ≤ n → k ≤ f n\n⊢ ∀ (n : ℕ), f n = n",
"after_state": "f : ℕ → ℕ\nh : ∀ (n : ℕ), f (f n) < f (n + 1)\nh' : ∀ (k n : ℕ), k ≤ n → k ≤ f n\nhf : ∀ (n : ℕ), n ≤ f n\n⊢ ∀ (n : ℕ), f n = n"
},
{
"line": "refine\n no_implicit_lambda%\n (have hf : ∀ n, n ≤ f n := fun n => h' n n rfl.le;\n ?_)",
"before_state": "f : ℕ → ℕ\nh : ∀ (n : ℕ), f (f n) < f (n + 1)\nh' : ∀ (k n : ℕ), k ≤ n → k ≤ f n\n⊢ ∀ (n : ℕ), f n = n",
"after_state": "f : ℕ → ℕ\nh : ∀ (n : ℕ), f (f n) < f (n + 1)\nh' : ∀ (k n : ℕ), k ≤ n → k ≤ f n\nhf : ∀ (n : ℕ), n ≤ f n\n⊢ ∀ (n : ℕ), f n = n"
},
{
"line": "rotate_right",
"before_state": "f : ℕ → ℕ\nh : ∀ (n : ℕ), f (f n) < f (n + 1)\nh' : ∀ (k n : ℕ), k ≤ n → k ≤ f n\nhf : ∀ (n : ℕ), n ≤ f n\n⊢ ∀ (n : ℕ), f n = n",
"after_state": "f : ℕ → ℕ\nh : ∀ (n : ℕ), f (f n) < f (n + 1)\nh' : ∀ (k n : ℕ), k ≤ n → k ≤ f n\nhf : ∀ (n : ℕ), n ≤ f n\n⊢ ∀ (n : ℕ), f n = n"
},
{
"line": "have hf_mono : StrictMono f := strictMono_nat_of_lt_succ fun _ => lt_of_le_of_lt (hf _) (h _)",
"before_state": "f : ℕ → ℕ\nh : ∀ (n : ℕ), f (f n) < f (n + 1)\nh' : ∀ (k n : ℕ), k ≤ n → k ≤ f n\nhf : ∀ (n : ℕ), n ≤ f n\n⊢ ∀ (n : ℕ), f n = n",
"after_state": "f : ℕ → ℕ\nh : ∀ (n : ℕ), f (f n) < f (n + 1)\nh' : ∀ (k n : ℕ), k ≤ n → k ≤ f n\nhf : ∀ (n : ℕ), n ≤ f n\nhf_mono : StrictMono f\n⊢ ∀ (n : ℕ), f n = n"
},
{
"line": "refine_lift\n have hf_mono : StrictMono f := strictMono_nat_of_lt_succ fun _ => lt_of_le_of_lt (hf _) (h _);\n ?_",
"before_state": "f : ℕ → ℕ\nh : ∀ (n : ℕ), f (f n) < f (n + 1)\nh' : ∀ (k n : ℕ), k ≤ n → k ≤ f n\nhf : ∀ (n : ℕ), n ≤ f n\n⊢ ∀ (n : ℕ), f n = n",
"after_state": "f : ℕ → ℕ\nh : ∀ (n : ℕ), f (f n) < f (n + 1)\nh' : ∀ (k n : ℕ), k ≤ n → k ≤ f n\nhf : ∀ (n : ℕ), n ≤ f n\nhf_mono : StrictMono f\n⊢ ∀ (n : ℕ), f n = n"
},
{
"line": "focus\n (refine\n no_implicit_lambda%\n (have hf_mono : StrictMono f := strictMono_nat_of_lt_succ fun _ => lt_of_le_of_lt (hf _) (h _);\n ?_);\n rotate_right)",
"before_state": "f : ℕ → ℕ\nh : ∀ (n : ℕ), f (f n) < f (n + 1)\nh' : ∀ (k n : ℕ), k ≤ n → k ≤ f n\nhf : ∀ (n : ℕ), n ≤ f n\n⊢ ∀ (n : ℕ), f n = n",
"after_state": "f : ℕ → ℕ\nh : ∀ (n : ℕ), f (f n) < f (n + 1)\nh' : ∀ (k n : ℕ), k ≤ n → k ≤ f n\nhf : ∀ (n : ℕ), n ≤ f n\nhf_mono : StrictMono f\n⊢ ∀ (n : ℕ), f n = n"
},
{
"line": "refine\n no_implicit_lambda%\n (have hf_mono : StrictMono f := strictMono_nat_of_lt_succ fun _ => lt_of_le_of_lt (hf _) (h _);\n ?_)",
"before_state": "f : ℕ → ℕ\nh : ∀ (n : ℕ), f (f n) < f (n + 1)\nh' : ∀ (k n : ℕ), k ≤ n → k ≤ f n\nhf : ∀ (n : ℕ), n ≤ f n\n⊢ ∀ (n : ℕ), f n = n",
"after_state": "f : ℕ → ℕ\nh : ∀ (n : ℕ), f (f n) < f (n + 1)\nh' : ∀ (k n : ℕ), k ≤ n → k ≤ f n\nhf : ∀ (n : ℕ), n ≤ f n\nhf_mono : StrictMono f\n⊢ ∀ (n : ℕ), f n = n"
},
{
"line": "rotate_right",
"before_state": "f : ℕ → ℕ\nh : ∀ (n : ℕ), f (f n) < f (n + 1)\nh' : ∀ (k n : ℕ), k ≤ n → k ≤ f n\nhf : ∀ (n : ℕ), n ≤ f n\nhf_mono : StrictMono f\n⊢ ∀ (n : ℕ), f n = n",
"after_state": "f : ℕ → ℕ\nh : ∀ (n : ℕ), f (f n) < f (n + 1)\nh' : ∀ (k n : ℕ), k ≤ n → k ≤ f n\nhf : ∀ (n : ℕ), n ≤ f n\nhf_mono : StrictMono f\n⊢ ∀ (n : ℕ), f n = n"
},
{
"line": "intro",
"before_state": "f : ℕ → ℕ\nh : ∀ (n : ℕ), f (f n) < f (n + 1)\nh' : ∀ (k n : ℕ), k ≤ n → k ≤ f n\nhf : ∀ (n : ℕ), n ≤ f n\nhf_mono : StrictMono f\n⊢ ∀ (n : ℕ), f n = n",
"after_state": "f : ℕ → ℕ\nh : ∀ (n : ℕ), f (f n) < f (n + 1)\nh' : ∀ (k n : ℕ), k ≤ n → k ≤ f n\nhf : ∀ (n : ℕ), n ≤ f n\nhf_mono : StrictMono f\nn✝ : ℕ\n⊢ f n✝ = n✝"
},
{
"line": "exact Nat.eq_of_le_of_lt_succ (hf _) (hf_mono.lt_iff_lt.mp (h _))",
"before_state": "f : ℕ → ℕ\nh : ∀ (n : ℕ), f (f n) < f (n + 1)\nh' : ∀ (k n : ℕ), k ≤ n → k ≤ f n\nhf : ∀ (n : ℕ), n ≤ f n\nhf_mono : StrictMono f\nn✝ : ℕ\n⊢ f n✝ = n✝",
"after_state": "No Goals!"
}
] |
lemma le_avg : ∑ k ∈ range (n + 1), x k ≤ (∑ k ∈ range n, x k) * (1 + 1 / n) := by
rw [sum_range_succ]
rw [mul_one_add]
rw [add_le_add_iff_left]
rw [mul_one_div]
rw [le_div_iff₀ (mod_cast hn.bot_lt)]
rw [mul_comm]
rw [← nsmul_eq_mul]
conv_lhs => rw [← card_range n, ← sum_const]
refine sum_le_sum fun k hk ↦ hx (le_of_lt ?_)
simpa using hk
|
/root/DuelModelResearch/mathlib4/Archive/Imo/Imo1982Q3.lean
|
{
"open": [
"Finset NNReal"
],
"variables": [
"{x : ℕ → ℝ} {n : ℕ} (hn : n ≠ 0) (hx : Antitone x)"
]
}
|
[
{
"line": "rw [sum_range_succ]",
"before_state": "x : ℕ → ℝ\nn : ℕ\n⊢ ∑ k ∈ range (n + 1), x k ≤ (∑ k ∈ range n, x k) * (1 + 1 / ↑n)",
"after_state": "x : ℕ → ℝ\nn : ℕ\n⊢ ∑ x_1 ∈ range n, x x_1 + x n ≤ (∑ k ∈ range n, x k) * (1 + 1 / ↑n)"
},
{
"line": "rewrite [sum_range_succ]",
"before_state": "x : ℕ → ℝ\nn : ℕ\n⊢ ∑ k ∈ range (n + 1), x k ≤ (∑ k ∈ range n, x k) * (1 + 1 / ↑n)",
"after_state": "x : ℕ → ℝ\nn : ℕ\n⊢ ∑ x_1 ∈ range n, x x_1 + x n ≤ (∑ k ∈ range n, x k) * (1 + 1 / ↑n)"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "x : ℕ → ℝ\nn : ℕ\n⊢ ∑ x_1 ∈ range n, x x_1 + x n ≤ (∑ k ∈ range n, x k) * (1 + 1 / ↑n)",
"after_state": "x : ℕ → ℝ\nn : ℕ\n⊢ ∑ x_1 ∈ range n, x x_1 + x n ≤ (∑ k ∈ range n, x k) * (1 + 1 / ↑n)"
},
{
"line": "try (with_reducible rfl)",
"before_state": "x : ℕ → ℝ\nn : ℕ\n⊢ ∑ x_1 ∈ range n, x x_1 + x n ≤ (∑ k ∈ range n, x k) * (1 + 1 / ↑n)",
"after_state": "x : ℕ → ℝ\nn : ℕ\n⊢ ∑ x_1 ∈ range n, x x_1 + x n ≤ (∑ k ∈ range n, x k) * (1 + 1 / ↑n)"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "x : ℕ → ℝ\nn : ℕ\n⊢ ∑ x_1 ∈ range n, x x_1 + x n ≤ (∑ k ∈ range n, x k) * (1 + 1 / ↑n)",
"after_state": "x : ℕ → ℝ\nn : ℕ\n⊢ ∑ x_1 ∈ range n, x x_1 + x n ≤ (∑ k ∈ range n, x k) * (1 + 1 / ↑n)"
},
{
"line": "with_reducible rfl",
"before_state": "x : ℕ → ℝ\nn : ℕ\n⊢ ∑ x_1 ∈ range n, x x_1 + x n ≤ (∑ k ∈ range n, x k) * (1 + 1 / ↑n)",
"after_state": "x : ℕ → ℝ\nn : ℕ\n⊢ ∑ x_1 ∈ range n, x x_1 + x n ≤ (∑ k ∈ range n, x k) * (1 + 1 / ↑n)"
},
{
"line": "rfl",
"before_state": "x : ℕ → ℝ\nn : ℕ\n⊢ ∑ x_1 ∈ range n, x x_1 + x n ≤ (∑ k ∈ range n, x k) * (1 + 1 / ↑n)",
"after_state": "x : ℕ → ℝ\nn : ℕ\n⊢ ∑ x_1 ∈ range n, x x_1 + x n ≤ (∑ k ∈ range n, x k) * (1 + 1 / ↑n)"
},
{
"line": "apply_rfl",
"before_state": "x : ℕ → ℝ\nn : ℕ\n⊢ ∑ x_1 ∈ range n, x x_1 + x n ≤ (∑ k ∈ range n, x k) * (1 + 1 / ↑n)",
"after_state": "x : ℕ → ℝ\nn : ℕ\n⊢ ∑ x_1 ∈ range n, x x_1 + x n ≤ (∑ k ∈ range n, x k) * (1 + 1 / ↑n)"
},
{
"line": "skip",
"before_state": "x : ℕ → ℝ\nn : ℕ\n⊢ ∑ x_1 ∈ range n, x x_1 + x n ≤ (∑ k ∈ range n, x k) * (1 + 1 / ↑n)",
"after_state": "x : ℕ → ℝ\nn : ℕ\n⊢ ∑ x_1 ∈ range n, x x_1 + x n ≤ (∑ k ∈ range n, x k) * (1 + 1 / ↑n)"
},
{
"line": "rw [mul_one_add]",
"before_state": "x : ℕ → ℝ\nn : ℕ\n⊢ ∑ x_1 ∈ range n, x x_1 + x n ≤ (∑ k ∈ range n, x k) * (1 + 1 / ↑n)",
"after_state": "x : ℕ → ℝ\nn : ℕ\n⊢ ∑ x_1 ∈ range n, x x_1 + x n ≤ ∑ k ∈ range n, x k + (∑ k ∈ range n, x k) * (1 / ↑n)"
},
{
"line": "rewrite [mul_one_add]",
"before_state": "x : ℕ → ℝ\nn : ℕ\n⊢ ∑ x_1 ∈ range n, x x_1 + x n ≤ (∑ k ∈ range n, x k) * (1 + 1 / ↑n)",
"after_state": "x : ℕ → ℝ\nn : ℕ\n⊢ ∑ x_1 ∈ range n, x x_1 + x n ≤ ∑ k ∈ range n, x k + (∑ k ∈ range n, x k) * (1 / ↑n)"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "x : ℕ → ℝ\nn : ℕ\n⊢ ∑ x_1 ∈ range n, x x_1 + x n ≤ ∑ k ∈ range n, x k + (∑ k ∈ range n, x k) * (1 / ↑n)",
"after_state": "x : ℕ → ℝ\nn : ℕ\n⊢ ∑ x_1 ∈ range n, x x_1 + x n ≤ ∑ k ∈ range n, x k + (∑ k ∈ range n, x k) * (1 / ↑n)"
},
{
"line": "try (with_reducible rfl)",
"before_state": "x : ℕ → ℝ\nn : ℕ\n⊢ ∑ x_1 ∈ range n, x x_1 + x n ≤ ∑ k ∈ range n, x k + (∑ k ∈ range n, x k) * (1 / ↑n)",
"after_state": "x : ℕ → ℝ\nn : ℕ\n⊢ ∑ x_1 ∈ range n, x x_1 + x n ≤ ∑ k ∈ range n, x k + (∑ k ∈ range n, x k) * (1 / ↑n)"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "x : ℕ → ℝ\nn : ℕ\n⊢ ∑ x_1 ∈ range n, x x_1 + x n ≤ ∑ k ∈ range n, x k + (∑ k ∈ range n, x k) * (1 / ↑n)",
"after_state": "x : ℕ → ℝ\nn : ℕ\n⊢ ∑ x_1 ∈ range n, x x_1 + x n ≤ ∑ k ∈ range n, x k + (∑ k ∈ range n, x k) * (1 / ↑n)"
},
{
"line": "with_reducible rfl",
"before_state": "x : ℕ → ℝ\nn : ℕ\n⊢ ∑ x_1 ∈ range n, x x_1 + x n ≤ ∑ k ∈ range n, x k + (∑ k ∈ range n, x k) * (1 / ↑n)",
"after_state": "x : ℕ → ℝ\nn : ℕ\n⊢ ∑ x_1 ∈ range n, x x_1 + x n ≤ ∑ k ∈ range n, x k + (∑ k ∈ range n, x k) * (1 / ↑n)"
},
{
"line": "rfl",
"before_state": "x : ℕ → ℝ\nn : ℕ\n⊢ ∑ x_1 ∈ range n, x x_1 + x n ≤ ∑ k ∈ range n, x k + (∑ k ∈ range n, x k) * (1 / ↑n)",
"after_state": "x : ℕ → ℝ\nn : ℕ\n⊢ ∑ x_1 ∈ range n, x x_1 + x n ≤ ∑ k ∈ range n, x k + (∑ k ∈ range n, x k) * (1 / ↑n)"
},
{
"line": "apply_rfl",
"before_state": "x : ℕ → ℝ\nn : ℕ\n⊢ ∑ x_1 ∈ range n, x x_1 + x n ≤ ∑ k ∈ range n, x k + (∑ k ∈ range n, x k) * (1 / ↑n)",
"after_state": "x : ℕ → ℝ\nn : ℕ\n⊢ ∑ x_1 ∈ range n, x x_1 + x n ≤ ∑ k ∈ range n, x k + (∑ k ∈ range n, x k) * (1 / ↑n)"
},
{
"line": "skip",
"before_state": "x : ℕ → ℝ\nn : ℕ\n⊢ ∑ x_1 ∈ range n, x x_1 + x n ≤ ∑ k ∈ range n, x k + (∑ k ∈ range n, x k) * (1 / ↑n)",
"after_state": "x : ℕ → ℝ\nn : ℕ\n⊢ ∑ x_1 ∈ range n, x x_1 + x n ≤ ∑ k ∈ range n, x k + (∑ k ∈ range n, x k) * (1 / ↑n)"
},
{
"line": "rw [add_le_add_iff_left]",
"before_state": "x : ℕ → ℝ\nn : ℕ\n⊢ ∑ x_1 ∈ range n, x x_1 + x n ≤ ∑ k ∈ range n, x k + (∑ k ∈ range n, x k) * (1 / ↑n)",
"after_state": "x : ℕ → ℝ\nn : ℕ\n⊢ x n ≤ (∑ k ∈ range n, x k) * (1 / ↑n)"
},
{
"line": "rewrite [add_le_add_iff_left]",
"before_state": "x : ℕ → ℝ\nn : ℕ\n⊢ ∑ x_1 ∈ range n, x x_1 + x n ≤ ∑ k ∈ range n, x k + (∑ k ∈ range n, x k) * (1 / ↑n)",
"after_state": "x : ℕ → ℝ\nn : ℕ\n⊢ x n ≤ (∑ k ∈ range n, x k) * (1 / ↑n)"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "x : ℕ → ℝ\nn : ℕ\n⊢ x n ≤ (∑ k ∈ range n, x k) * (1 / ↑n)",
"after_state": "x : ℕ → ℝ\nn : ℕ\n⊢ x n ≤ (∑ k ∈ range n, x k) * (1 / ↑n)"
},
{
"line": "try (with_reducible rfl)",
"before_state": "x : ℕ → ℝ\nn : ℕ\n⊢ x n ≤ (∑ k ∈ range n, x k) * (1 / ↑n)",
"after_state": "x : ℕ → ℝ\nn : ℕ\n⊢ x n ≤ (∑ k ∈ range n, x k) * (1 / ↑n)"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "x : ℕ → ℝ\nn : ℕ\n⊢ x n ≤ (∑ k ∈ range n, x k) * (1 / ↑n)",
"after_state": "x : ℕ → ℝ\nn : ℕ\n⊢ x n ≤ (∑ k ∈ range n, x k) * (1 / ↑n)"
},
{
"line": "with_reducible rfl",
"before_state": "x : ℕ → ℝ\nn : ℕ\n⊢ x n ≤ (∑ k ∈ range n, x k) * (1 / ↑n)",
"after_state": "x : ℕ → ℝ\nn : ℕ\n⊢ x n ≤ (∑ k ∈ range n, x k) * (1 / ↑n)"
},
{
"line": "rfl",
"before_state": "x : ℕ → ℝ\nn : ℕ\n⊢ x n ≤ (∑ k ∈ range n, x k) * (1 / ↑n)",
"after_state": "x : ℕ → ℝ\nn : ℕ\n⊢ x n ≤ (∑ k ∈ range n, x k) * (1 / ↑n)"
},
{
"line": "apply_rfl",
"before_state": "x : ℕ → ℝ\nn : ℕ\n⊢ x n ≤ (∑ k ∈ range n, x k) * (1 / ↑n)",
"after_state": "x : ℕ → ℝ\nn : ℕ\n⊢ x n ≤ (∑ k ∈ range n, x k) * (1 / ↑n)"
},
{
"line": "skip",
"before_state": "x : ℕ → ℝ\nn : ℕ\n⊢ x n ≤ (∑ k ∈ range n, x k) * (1 / ↑n)",
"after_state": "x : ℕ → ℝ\nn : ℕ\n⊢ x n ≤ (∑ k ∈ range n, x k) * (1 / ↑n)"
},
{
"line": "rw [mul_one_div]",
"before_state": "x : ℕ → ℝ\nn : ℕ\n⊢ x n ≤ (∑ k ∈ range n, x k) * (1 / ↑n)",
"after_state": "x : ℕ → ℝ\nn : ℕ\n⊢ x n ≤ (∑ k ∈ range n, x k) / ↑n"
},
{
"line": "rewrite [mul_one_div]",
"before_state": "x : ℕ → ℝ\nn : ℕ\n⊢ x n ≤ (∑ k ∈ range n, x k) * (1 / ↑n)",
"after_state": "x : ℕ → ℝ\nn : ℕ\n⊢ x n ≤ (∑ k ∈ range n, x k) / ↑n"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "x : ℕ → ℝ\nn : ℕ\n⊢ x n ≤ (∑ k ∈ range n, x k) / ↑n",
"after_state": "x : ℕ → ℝ\nn : ℕ\n⊢ x n ≤ (∑ k ∈ range n, x k) / ↑n"
},
{
"line": "try (with_reducible rfl)",
"before_state": "x : ℕ → ℝ\nn : ℕ\n⊢ x n ≤ (∑ k ∈ range n, x k) / ↑n",
"after_state": "x : ℕ → ℝ\nn : ℕ\n⊢ x n ≤ (∑ k ∈ range n, x k) / ↑n"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "x : ℕ → ℝ\nn : ℕ\n⊢ x n ≤ (∑ k ∈ range n, x k) / ↑n",
"after_state": "x : ℕ → ℝ\nn : ℕ\n⊢ x n ≤ (∑ k ∈ range n, x k) / ↑n"
},
{
"line": "with_reducible rfl",
"before_state": "x : ℕ → ℝ\nn : ℕ\n⊢ x n ≤ (∑ k ∈ range n, x k) / ↑n",
"after_state": "x : ℕ → ℝ\nn : ℕ\n⊢ x n ≤ (∑ k ∈ range n, x k) / ↑n"
},
{
"line": "rfl",
"before_state": "x : ℕ → ℝ\nn : ℕ\n⊢ x n ≤ (∑ k ∈ range n, x k) / ↑n",
"after_state": "x : ℕ → ℝ\nn : ℕ\n⊢ x n ≤ (∑ k ∈ range n, x k) / ↑n"
},
{
"line": "apply_rfl",
"before_state": "x : ℕ → ℝ\nn : ℕ\n⊢ x n ≤ (∑ k ∈ range n, x k) / ↑n",
"after_state": "x : ℕ → ℝ\nn : ℕ\n⊢ x n ≤ (∑ k ∈ range n, x k) / ↑n"
},
{
"line": "skip",
"before_state": "x : ℕ → ℝ\nn : ℕ\n⊢ x n ≤ (∑ k ∈ range n, x k) / ↑n",
"after_state": "x : ℕ → ℝ\nn : ℕ\n⊢ x n ≤ (∑ k ∈ range n, x k) / ↑n"
},
{
"line": "rw [le_div_iff₀ (mod_cast hn.bot_lt)]",
"before_state": "x : ℕ → ℝ\nn : ℕ\n⊢ x n ≤ (∑ k ∈ range n, x k) / ↑n",
"after_state": "x : ℕ → ℝ\nn : ℕ\n⊢ x n * ↑n ≤ ∑ k ∈ range n, x k"
},
{
"line": "rewrite [le_div_iff₀ (mod_cast hn.bot_lt)]",
"before_state": "x : ℕ → ℝ\nn : ℕ\n⊢ x n ≤ (∑ k ∈ range n, x k) / ↑n",
"after_state": "x : ℕ → ℝ\nn : ℕ\n⊢ x n * ↑n ≤ ∑ k ∈ range n, x k"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "x : ℕ → ℝ\nn : ℕ\n⊢ x n * ↑n ≤ ∑ k ∈ range n, x k",
"after_state": "x : ℕ → ℝ\nn : ℕ\n⊢ x n * ↑n ≤ ∑ k ∈ range n, x k"
},
{
"line": "try (with_reducible rfl)",
"before_state": "x : ℕ → ℝ\nn : ℕ\n⊢ x n * ↑n ≤ ∑ k ∈ range n, x k",
"after_state": "x : ℕ → ℝ\nn : ℕ\n⊢ x n * ↑n ≤ ∑ k ∈ range n, x k"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "x : ℕ → ℝ\nn : ℕ\n⊢ x n * ↑n ≤ ∑ k ∈ range n, x k",
"after_state": "x : ℕ → ℝ\nn : ℕ\n⊢ x n * ↑n ≤ ∑ k ∈ range n, x k"
},
{
"line": "with_reducible rfl",
"before_state": "x : ℕ → ℝ\nn : ℕ\n⊢ x n * ↑n ≤ ∑ k ∈ range n, x k",
"after_state": "x : ℕ → ℝ\nn : ℕ\n⊢ x n * ↑n ≤ ∑ k ∈ range n, x k"
},
{
"line": "rfl",
"before_state": "x : ℕ → ℝ\nn : ℕ\n⊢ x n * ↑n ≤ ∑ k ∈ range n, x k",
"after_state": "x : ℕ → ℝ\nn : ℕ\n⊢ x n * ↑n ≤ ∑ k ∈ range n, x k"
},
{
"line": "apply_rfl",
"before_state": "x : ℕ → ℝ\nn : ℕ\n⊢ x n * ↑n ≤ ∑ k ∈ range n, x k",
"after_state": "x : ℕ → ℝ\nn : ℕ\n⊢ x n * ↑n ≤ ∑ k ∈ range n, x k"
},
{
"line": "skip",
"before_state": "x : ℕ → ℝ\nn : ℕ\n⊢ x n * ↑n ≤ ∑ k ∈ range n, x k",
"after_state": "x : ℕ → ℝ\nn : ℕ\n⊢ x n * ↑n ≤ ∑ k ∈ range n, x k"
},
{
"line": "rw [mul_comm]",
"before_state": "x : ℕ → ℝ\nn : ℕ\n⊢ x n * ↑n ≤ ∑ k ∈ range n, x k",
"after_state": "x : ℕ → ℝ\nn : ℕ\n⊢ ↑n * x n ≤ ∑ k ∈ range n, x k"
},
{
"line": "rewrite [mul_comm]",
"before_state": "x : ℕ → ℝ\nn : ℕ\n⊢ x n * ↑n ≤ ∑ k ∈ range n, x k",
"after_state": "x : ℕ → ℝ\nn : ℕ\n⊢ ↑n * x n ≤ ∑ k ∈ range n, x k"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "x : ℕ → ℝ\nn : ℕ\n⊢ ↑n * x n ≤ ∑ k ∈ range n, x k",
"after_state": "x : ℕ → ℝ\nn : ℕ\n⊢ ↑n * x n ≤ ∑ k ∈ range n, x k"
},
{
"line": "try (with_reducible rfl)",
"before_state": "x : ℕ → ℝ\nn : ℕ\n⊢ ↑n * x n ≤ ∑ k ∈ range n, x k",
"after_state": "x : ℕ → ℝ\nn : ℕ\n⊢ ↑n * x n ≤ ∑ k ∈ range n, x k"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "x : ℕ → ℝ\nn : ℕ\n⊢ ↑n * x n ≤ ∑ k ∈ range n, x k",
"after_state": "x : ℕ → ℝ\nn : ℕ\n⊢ ↑n * x n ≤ ∑ k ∈ range n, x k"
},
{
"line": "with_reducible rfl",
"before_state": "x : ℕ → ℝ\nn : ℕ\n⊢ ↑n * x n ≤ ∑ k ∈ range n, x k",
"after_state": "x : ℕ → ℝ\nn : ℕ\n⊢ ↑n * x n ≤ ∑ k ∈ range n, x k"
},
{
"line": "rfl",
"before_state": "x : ℕ → ℝ\nn : ℕ\n⊢ ↑n * x n ≤ ∑ k ∈ range n, x k",
"after_state": "x : ℕ → ℝ\nn : ℕ\n⊢ ↑n * x n ≤ ∑ k ∈ range n, x k"
},
{
"line": "apply_rfl",
"before_state": "x : ℕ → ℝ\nn : ℕ\n⊢ ↑n * x n ≤ ∑ k ∈ range n, x k",
"after_state": "x : ℕ → ℝ\nn : ℕ\n⊢ ↑n * x n ≤ ∑ k ∈ range n, x k"
},
{
"line": "skip",
"before_state": "x : ℕ → ℝ\nn : ℕ\n⊢ ↑n * x n ≤ ∑ k ∈ range n, x k",
"after_state": "x : ℕ → ℝ\nn : ℕ\n⊢ ↑n * x n ≤ ∑ k ∈ range n, x k"
},
{
"line": "rw [← nsmul_eq_mul]",
"before_state": "x : ℕ → ℝ\nn : ℕ\n⊢ ↑n * x n ≤ ∑ k ∈ range n, x k",
"after_state": "x : ℕ → ℝ\nn : ℕ\n⊢ n • x n ≤ ∑ k ∈ range n, x k"
},
{
"line": "rewrite [← nsmul_eq_mul]",
"before_state": "x : ℕ → ℝ\nn : ℕ\n⊢ ↑n * x n ≤ ∑ k ∈ range n, x k",
"after_state": "x : ℕ → ℝ\nn : ℕ\n⊢ n • x n ≤ ∑ k ∈ range n, x k"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "x : ℕ → ℝ\nn : ℕ\n⊢ n • x n ≤ ∑ k ∈ range n, x k",
"after_state": "x : ℕ → ℝ\nn : ℕ\n⊢ n • x n ≤ ∑ k ∈ range n, x k"
},
{
"line": "try (with_reducible rfl)",
"before_state": "x : ℕ → ℝ\nn : ℕ\n⊢ n • x n ≤ ∑ k ∈ range n, x k",
"after_state": "x : ℕ → ℝ\nn : ℕ\n⊢ n • x n ≤ ∑ k ∈ range n, x k"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "x : ℕ → ℝ\nn : ℕ\n⊢ n • x n ≤ ∑ k ∈ range n, x k",
"after_state": "x : ℕ → ℝ\nn : ℕ\n⊢ n • x n ≤ ∑ k ∈ range n, x k"
},
{
"line": "with_reducible rfl",
"before_state": "x : ℕ → ℝ\nn : ℕ\n⊢ n • x n ≤ ∑ k ∈ range n, x k",
"after_state": "x : ℕ → ℝ\nn : ℕ\n⊢ n • x n ≤ ∑ k ∈ range n, x k"
},
{
"line": "rfl",
"before_state": "x : ℕ → ℝ\nn : ℕ\n⊢ n • x n ≤ ∑ k ∈ range n, x k",
"after_state": "x : ℕ → ℝ\nn : ℕ\n⊢ n • x n ≤ ∑ k ∈ range n, x k"
},
{
"line": "apply_rfl",
"before_state": "x : ℕ → ℝ\nn : ℕ\n⊢ n • x n ≤ ∑ k ∈ range n, x k",
"after_state": "x : ℕ → ℝ\nn : ℕ\n⊢ n • x n ≤ ∑ k ∈ range n, x k"
},
{
"line": "skip",
"before_state": "x : ℕ → ℝ\nn : ℕ\n⊢ n • x n ≤ ∑ k ∈ range n, x k",
"after_state": "x : ℕ → ℝ\nn : ℕ\n⊢ n • x n ≤ ∑ k ∈ range n, x k"
},
{
"line": "conv_lhs => rw [← card_range n, ← sum_const]",
"before_state": "x : ℕ → ℝ\nn : ℕ\n⊢ n • x n ≤ ∑ k ∈ range n, x k",
"after_state": "x : ℕ → ℝ\nn : ℕ\n⊢ ∑ _x ∈ range n, x #(range n) ≤ ∑ k ∈ range n, x k"
},
{
"line": "conv => lhs; (rw [← card_range n, ← sum_const])",
"before_state": "x : ℕ → ℝ\nn : ℕ\n⊢ n • x n ≤ ∑ k ∈ range n, x k",
"after_state": "x : ℕ → ℝ\nn : ℕ\n⊢ ∑ _x ∈ range n, x #(range n) ≤ ∑ k ∈ range n, x k"
},
{
"line": "lhs; (rw [← card_range n, ← sum_const])",
"before_state": "x : ℕ → ℝ\nn : ℕ\n| n • x n ≤ ∑ k ∈ range n, x k",
"after_state": "x : ℕ → ℝ\nn : ℕ\n| ∑ _x ∈ range n, x #(range n)"
},
{
"line": "lhs; (rw [← card_range n, ← sum_const])",
"before_state": "x : ℕ → ℝ\nn : ℕ\n| n • x n ≤ ∑ k ∈ range n, x k",
"after_state": "x : ℕ → ℝ\nn : ℕ\n| ∑ _x ∈ range n, x #(range n)"
},
{
"line": "lhs",
"before_state": "x : ℕ → ℝ\nn : ℕ\n| n • x n ≤ ∑ k ∈ range n, x k",
"after_state": "x : ℕ → ℝ\nn : ℕ\n| n • x n"
},
{
"line": "(rw [← card_range n, ← sum_const])",
"before_state": "x : ℕ → ℝ\nn : ℕ\n| n • x n",
"after_state": "x : ℕ → ℝ\nn : ℕ\n| ∑ _x ∈ range n, x #(range n)"
},
{
"line": "rw [← card_range n, ← sum_const]",
"before_state": "x : ℕ → ℝ\nn : ℕ\n| n • x n",
"after_state": "x : ℕ → ℝ\nn : ℕ\n| ∑ _x ∈ range n, x #(range n)"
},
{
"line": "rw [← card_range n, ← sum_const]",
"before_state": "x : ℕ → ℝ\nn : ℕ\n| n • x n",
"after_state": "x : ℕ → ℝ\nn : ℕ\n| ∑ _x ∈ range n, x #(range n)"
},
{
"line": "rw [← card_range n, ← sum_const]",
"before_state": "x : ℕ → ℝ\nn : ℕ\n| n • x n",
"after_state": "x : ℕ → ℝ\nn : ℕ\n| ∑ _x ∈ range n, x #(range n)"
},
{
"line": "rewrite [← card_range n, ← sum_const]",
"before_state": "x : ℕ → ℝ\nn : ℕ\n| n • x n",
"after_state": "x : ℕ → ℝ\nn : ℕ\n| ∑ _x ∈ range n, x #(range n)"
},
{
"line": "try with_reducible rfl",
"before_state": "x : ℕ → ℝ\nn : ℕ\n⊢ ∑ _x ∈ range n, x #(range n) ≤ ∑ k ∈ range n, x k",
"after_state": "x : ℕ → ℝ\nn : ℕ\n⊢ ∑ _x ∈ range n, x #(range n) ≤ ∑ k ∈ range n, x k"
},
{
"line": "first\n| with_reducible rfl\n| skip",
"before_state": "x : ℕ → ℝ\nn : ℕ\n⊢ ∑ _x ∈ range n, x #(range n) ≤ ∑ k ∈ range n, x k",
"after_state": "x : ℕ → ℝ\nn : ℕ\n⊢ ∑ _x ∈ range n, x #(range n) ≤ ∑ k ∈ range n, x k"
},
{
"line": "with_reducible rfl",
"before_state": "x : ℕ → ℝ\nn : ℕ\n⊢ ∑ _x ∈ range n, x #(range n) ≤ ∑ k ∈ range n, x k",
"after_state": "x : ℕ → ℝ\nn : ℕ\n⊢ ∑ _x ∈ range n, x #(range n) ≤ ∑ k ∈ range n, x k"
},
{
"line": "rfl",
"before_state": "x : ℕ → ℝ\nn : ℕ\n⊢ ∑ _x ∈ range n, x #(range n) ≤ ∑ k ∈ range n, x k",
"after_state": "x : ℕ → ℝ\nn : ℕ\n⊢ ∑ _x ∈ range n, x #(range n) ≤ ∑ k ∈ range n, x k"
},
{
"line": "apply_rfl",
"before_state": "x : ℕ → ℝ\nn : ℕ\n⊢ ∑ _x ∈ range n, x #(range n) ≤ ∑ k ∈ range n, x k",
"after_state": "x : ℕ → ℝ\nn : ℕ\n⊢ ∑ _x ∈ range n, x #(range n) ≤ ∑ k ∈ range n, x k"
},
{
"line": "skip",
"before_state": "x : ℕ → ℝ\nn : ℕ\n⊢ ∑ _x ∈ range n, x #(range n) ≤ ∑ k ∈ range n, x k",
"after_state": "x : ℕ → ℝ\nn : ℕ\n⊢ ∑ _x ∈ range n, x #(range n) ≤ ∑ k ∈ range n, x k"
},
{
"line": "refine sum_le_sum fun k hk ↦ hx (le_of_lt ?_)",
"before_state": "x : ℕ → ℝ\nn : ℕ\n⊢ ∑ _x ∈ range n, x #(range n) ≤ ∑ k ∈ range n, x k",
"after_state": "No Goals!"
}
] |
lemma ineq (h0 : x 0 = 1) (hp : ∀ k, 0 < x k) :
4 * n / (n + 1) ≤ ∑ k ∈ range (n + 1), x k ^ 2 / x (k + 1) := by
calc
-- We first use AM-GM.
_ ≤ (∑ k ∈ range n, x (k + 1) + 1) ^ 2 / (∑ k ∈ range n, x (k + 1)) * n / (n + 1) := by
gcongr
rw [le_div_iff₀]
· simpa using four_mul_le_sq_add (∑ k ∈ range n, x (k + 1)) 1
· exact sum_pos (fun k _ ↦ hp _) (nonempty_range_iff.2 hn)
-- We move the fraction into the denominator.
_ = (∑ k ∈ range n, x (k + 1) + 1) ^ 2 / ((∑ k ∈ range n, x (k + 1)) * (1 + 1 / n)) := by
field_simp
-- We make use of the `le_avg` lemma.
_ ≤ (∑ k ∈ range (n + 1), x k) ^ 2 / ∑ k ∈ range (n + 1), x (k + 1) := by
gcongr
· exact sum_pos (fun k _ ↦ hp _) nonempty_range_succ
· exact add_nonneg (sum_nonneg fun k _ ↦ (hp _).le) zero_le_one
· rw [sum_range_succ', h0]
· exact le_avg hn (hx.comp_monotone @Nat.succ_le_succ)
-- We conclude by Sedrakyan.
_ ≤ _ := sq_sum_div_le_sum_sq_div _ x fun k _ ↦ hp (k + 1)
|
/root/DuelModelResearch/mathlib4/Archive/Imo/Imo1982Q3.lean
|
{
"open": [
"Finset NNReal"
],
"variables": [
"{x : ℕ → ℝ} {n : ℕ} (hn : n ≠ 0) (hx : Antitone x)"
]
}
|
[
{
"line": "calc\n -- We first use AM-GM.\n _ ≤ (∑ k ∈ range n, x (k + 1) + 1) ^ 2 / (∑ k ∈ range n, x (k + 1)) * n / (n + 1) :=\n by\n gcongr\n rw [le_div_iff₀]\n · simpa using four_mul_le_sq_add (∑ k ∈ range n, x (k + 1)) 1\n ·\n exact\n sum_pos (fun k _ ↦ hp _)\n (nonempty_range_iff.2 hn)\n -- We move the fraction into the denominator.\n _ = (∑ k ∈ range n, x (k + 1) + 1) ^ 2 / ((∑ k ∈ range n, x (k + 1)) * (1 + 1 / n)) := by\n field_simp\n -- We make use of the `le_avg` lemma.\n _ ≤ (∑ k ∈ range (n + 1), x k) ^ 2 / ∑ k ∈ range (n + 1), x (k + 1) :=\n by\n gcongr\n · exact sum_pos (fun k _ ↦ hp _) nonempty_range_succ\n · exact add_nonneg (sum_nonneg fun k _ ↦ (hp _).le) zero_le_one\n · rw [sum_range_succ', h0]\n ·\n exact\n le_avg hn\n (hx.comp_monotone @Nat.succ_le_succ)\n -- We conclude by Sedrakyan.\n _ ≤ _ := sq_sum_div_le_sum_sq_div _ x fun k _ ↦ hp (k + 1)",
"before_state": "x : ℕ → ℝ\nn : ℕ\nh0 : x 0 = 1\nhp : ∀ (k : ℕ), 0 < x k\n⊢ 4 * ↑n / (↑n + 1) ≤ ∑ k ∈ range (n + 1), x k ^ 2 / x (k + 1)",
"after_state": "No Goals!"
},
{
"line": "gcongr",
"before_state": "x : ℕ → ℝ\nn : ℕ\nh0 : x 0 = 1\nhp : ∀ (k : ℕ), 0 < x k\n⊢ 4 * ↑n / (↑n + 1) ≤ ((∑ k ∈ range n, x (k + 1) + 1) ^ 2 / ∑ k ∈ range n, x (k + 1)) * ↑n / (↑n + 1)",
"after_state": "case hab.h\nx : ℕ → ℝ\nn : ℕ\nh0 : x 0 = 1\nhp : ∀ (k : ℕ), 0 < x k\n⊢ 4 ≤ (∑ k ∈ range n, x (k + 1) + 1) ^ 2 / ∑ k ∈ range n, x (k + 1)"
},
{
"line": "gcongr_discharger",
"before_state": "case hab.a0\nx : ℕ → ℝ\nn : ℕ\nh0 : x 0 = 1\nhp : ∀ (k : ℕ), 0 < x k\n⊢ 0 ≤ ↑n",
"after_state": "No Goals!"
},
{
"line": "positivity",
"before_state": "case hab.a0\nx : ℕ → ℝ\nn : ℕ\nh0 : x 0 = 1\nhp : ∀ (k : ℕ), 0 < x k\n⊢ 0 ≤ ↑n",
"after_state": "No Goals!"
},
{
"line": "gcongr_discharger",
"before_state": "case hc\nx : ℕ → ℝ\nn : ℕ\nh0 : x 0 = 1\nhp : ∀ (k : ℕ), 0 < x k\n⊢ 0 ≤ ↑n + 1",
"after_state": "No Goals!"
},
{
"line": "positivity",
"before_state": "case hc\nx : ℕ → ℝ\nn : ℕ\nh0 : x 0 = 1\nhp : ∀ (k : ℕ), 0 < x k\n⊢ 0 ≤ ↑n + 1",
"after_state": "No Goals!"
},
{
"line": "rw [le_div_iff₀]",
"before_state": "case hab.h\nx : ℕ → ℝ\nn : ℕ\nh0 : x 0 = 1\nhp : ∀ (k : ℕ), 0 < x k\n⊢ 4 ≤ (∑ k ∈ range n, x (k + 1) + 1) ^ 2 / ∑ k ∈ range n, x (k + 1)",
"after_state": "case hab.h\nx : ℕ → ℝ\nn : ℕ\nh0 : x 0 = 1\nhp : ∀ (k : ℕ), 0 < x k\n⊢ 4 * ∑ k ∈ range n, x (k + 1) ≤ (∑ k ∈ range n, x (k + 1) + 1) ^ 2\n---\ncase hab.h\nx : ℕ → ℝ\nn : ℕ\nh0 : x 0 = 1\nhp : ∀ (k : ℕ), 0 < x k\n⊢ 0 < ∑ k ∈ range n, x (k + 1)"
},
{
"line": "rewrite [le_div_iff₀]",
"before_state": "case hab.h\nx : ℕ → ℝ\nn : ℕ\nh0 : x 0 = 1\nhp : ∀ (k : ℕ), 0 < x k\n⊢ 4 ≤ (∑ k ∈ range n, x (k + 1) + 1) ^ 2 / ∑ k ∈ range n, x (k + 1)",
"after_state": "case hab.h\nx : ℕ → ℝ\nn : ℕ\nh0 : x 0 = 1\nhp : ∀ (k : ℕ), 0 < x k\n⊢ 4 * ∑ k ∈ range n, x (k + 1) ≤ (∑ k ∈ range n, x (k + 1) + 1) ^ 2\n---\ncase hab.h\nx : ℕ → ℝ\nn : ℕ\nh0 : x 0 = 1\nhp : ∀ (k : ℕ), 0 < x k\n⊢ 0 < ∑ k ∈ range n, x (k + 1)"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "case hab.h\nx : ℕ → ℝ\nn : ℕ\nh0 : x 0 = 1\nhp : ∀ (k : ℕ), 0 < x k\n⊢ 4 * ∑ k ∈ range n, x (k + 1) ≤ (∑ k ∈ range n, x (k + 1) + 1) ^ 2\n---\ncase hab.h\nx : ℕ → ℝ\nn : ℕ\nh0 : x 0 = 1\nhp : ∀ (k : ℕ), 0 < x k\n⊢ 0 < ∑ k ∈ range n, x (k + 1)",
"after_state": "case hab.h\nx : ℕ → ℝ\nn : ℕ\nh0 : x 0 = 1\nhp : ∀ (k : ℕ), 0 < x k\n⊢ 4 * ∑ k ∈ range n, x (k + 1) ≤ (∑ k ∈ range n, x (k + 1) + 1) ^ 2\n---\ncase hab.h\nx : ℕ → ℝ\nn : ℕ\nh0 : x 0 = 1\nhp : ∀ (k : ℕ), 0 < x k\n⊢ 0 < ∑ k ∈ range n, x (k + 1)"
},
{
"line": "try (with_reducible rfl)",
"before_state": "case hab.h\nx : ℕ → ℝ\nn : ℕ\nh0 : x 0 = 1\nhp : ∀ (k : ℕ), 0 < x k\n⊢ 4 * ∑ k ∈ range n, x (k + 1) ≤ (∑ k ∈ range n, x (k + 1) + 1) ^ 2\n---\ncase hab.h\nx : ℕ → ℝ\nn : ℕ\nh0 : x 0 = 1\nhp : ∀ (k : ℕ), 0 < x k\n⊢ 0 < ∑ k ∈ range n, x (k + 1)",
"after_state": "case hab.h\nx : ℕ → ℝ\nn : ℕ\nh0 : x 0 = 1\nhp : ∀ (k : ℕ), 0 < x k\n⊢ 4 * ∑ k ∈ range n, x (k + 1) ≤ (∑ k ∈ range n, x (k + 1) + 1) ^ 2\n---\ncase hab.h\nx : ℕ → ℝ\nn : ℕ\nh0 : x 0 = 1\nhp : ∀ (k : ℕ), 0 < x k\n⊢ 0 < ∑ k ∈ range n, x (k + 1)"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "case hab.h\nx : ℕ → ℝ\nn : ℕ\nh0 : x 0 = 1\nhp : ∀ (k : ℕ), 0 < x k\n⊢ 4 * ∑ k ∈ range n, x (k + 1) ≤ (∑ k ∈ range n, x (k + 1) + 1) ^ 2\n---\ncase hab.h\nx : ℕ → ℝ\nn : ℕ\nh0 : x 0 = 1\nhp : ∀ (k : ℕ), 0 < x k\n⊢ 0 < ∑ k ∈ range n, x (k + 1)",
"after_state": "case hab.h\nx : ℕ → ℝ\nn : ℕ\nh0 : x 0 = 1\nhp : ∀ (k : ℕ), 0 < x k\n⊢ 4 * ∑ k ∈ range n, x (k + 1) ≤ (∑ k ∈ range n, x (k + 1) + 1) ^ 2\n---\ncase hab.h\nx : ℕ → ℝ\nn : ℕ\nh0 : x 0 = 1\nhp : ∀ (k : ℕ), 0 < x k\n⊢ 0 < ∑ k ∈ range n, x (k + 1)"
},
{
"line": "with_reducible rfl",
"before_state": "case hab.h\nx : ℕ → ℝ\nn : ℕ\nh0 : x 0 = 1\nhp : ∀ (k : ℕ), 0 < x k\n⊢ 4 * ∑ k ∈ range n, x (k + 1) ≤ (∑ k ∈ range n, x (k + 1) + 1) ^ 2\n---\ncase hab.h\nx : ℕ → ℝ\nn : ℕ\nh0 : x 0 = 1\nhp : ∀ (k : ℕ), 0 < x k\n⊢ 0 < ∑ k ∈ range n, x (k + 1)",
"after_state": "case hab.h\nx : ℕ → ℝ\nn : ℕ\nh0 : x 0 = 1\nhp : ∀ (k : ℕ), 0 < x k\n⊢ 4 * ∑ k ∈ range n, x (k + 1) ≤ (∑ k ∈ range n, x (k + 1) + 1) ^ 2\n---\ncase hab.h\nx : ℕ → ℝ\nn : ℕ\nh0 : x 0 = 1\nhp : ∀ (k : ℕ), 0 < x k\n⊢ 0 < ∑ k ∈ range n, x (k + 1)"
},
{
"line": "rfl",
"before_state": "case hab.h\nx : ℕ → ℝ\nn : ℕ\nh0 : x 0 = 1\nhp : ∀ (k : ℕ), 0 < x k\n⊢ 4 * ∑ k ∈ range n, x (k + 1) ≤ (∑ k ∈ range n, x (k + 1) + 1) ^ 2\n---\ncase hab.h\nx : ℕ → ℝ\nn : ℕ\nh0 : x 0 = 1\nhp : ∀ (k : ℕ), 0 < x k\n⊢ 0 < ∑ k ∈ range n, x (k + 1)",
"after_state": "case hab.h\nx : ℕ → ℝ\nn : ℕ\nh0 : x 0 = 1\nhp : ∀ (k : ℕ), 0 < x k\n⊢ 4 * ∑ k ∈ range n, x (k + 1) ≤ (∑ k ∈ range n, x (k + 1) + 1) ^ 2\n---\ncase hab.h\nx : ℕ → ℝ\nn : ℕ\nh0 : x 0 = 1\nhp : ∀ (k : ℕ), 0 < x k\n⊢ 0 < ∑ k ∈ range n, x (k + 1)"
},
{
"line": "apply_rfl",
"before_state": "case hab.h\nx : ℕ → ℝ\nn : ℕ\nh0 : x 0 = 1\nhp : ∀ (k : ℕ), 0 < x k\n⊢ 4 * ∑ k ∈ range n, x (k + 1) ≤ (∑ k ∈ range n, x (k + 1) + 1) ^ 2\n---\ncase hab.h\nx : ℕ → ℝ\nn : ℕ\nh0 : x 0 = 1\nhp : ∀ (k : ℕ), 0 < x k\n⊢ 0 < ∑ k ∈ range n, x (k + 1)",
"after_state": "case hab.h\nx : ℕ → ℝ\nn : ℕ\nh0 : x 0 = 1\nhp : ∀ (k : ℕ), 0 < x k\n⊢ 4 * ∑ k ∈ range n, x (k + 1) ≤ (∑ k ∈ range n, x (k + 1) + 1) ^ 2\n---\ncase hab.h\nx : ℕ → ℝ\nn : ℕ\nh0 : x 0 = 1\nhp : ∀ (k : ℕ), 0 < x k\n⊢ 0 < ∑ k ∈ range n, x (k + 1)"
},
{
"line": "skip",
"before_state": "case hab.h\nx : ℕ → ℝ\nn : ℕ\nh0 : x 0 = 1\nhp : ∀ (k : ℕ), 0 < x k\n⊢ 4 * ∑ k ∈ range n, x (k + 1) ≤ (∑ k ∈ range n, x (k + 1) + 1) ^ 2\n---\ncase hab.h\nx : ℕ → ℝ\nn : ℕ\nh0 : x 0 = 1\nhp : ∀ (k : ℕ), 0 < x k\n⊢ 0 < ∑ k ∈ range n, x (k + 1)",
"after_state": "case hab.h\nx : ℕ → ℝ\nn : ℕ\nh0 : x 0 = 1\nhp : ∀ (k : ℕ), 0 < x k\n⊢ 4 * ∑ k ∈ range n, x (k + 1) ≤ (∑ k ∈ range n, x (k + 1) + 1) ^ 2\n---\ncase hab.h\nx : ℕ → ℝ\nn : ℕ\nh0 : x 0 = 1\nhp : ∀ (k : ℕ), 0 < x k\n⊢ 0 < ∑ k ∈ range n, x (k + 1)"
},
{
"line": "simpa using four_mul_le_sq_add (∑ k ∈ range n, x (k + 1)) 1",
"before_state": "case hab.h\nx : ℕ → ℝ\nn : ℕ\nh0 : x 0 = 1\nhp : ∀ (k : ℕ), 0 < x k\n⊢ 4 * ∑ k ∈ range n, x (k + 1) ≤ (∑ k ∈ range n, x (k + 1) + 1) ^ 2",
"after_state": "No Goals!"
},
{
"line": "exact\n sum_pos (fun k _ ↦ hp _)\n (nonempty_range_iff.2 hn)\n -- We move the fraction into the denominator.",
"before_state": "case hab.h\nx : ℕ → ℝ\nn : ℕ\nh0 : x 0 = 1\nhp : ∀ (k : ℕ), 0 < x k\n⊢ 0 < ∑ k ∈ range n, x (k + 1)",
"after_state": "No Goals!"
},
{
"line": "field_simp\n -- We make use of the `le_avg` lemma.",
"before_state": "x : ℕ → ℝ\nn : ℕ\nh0 : x 0 = 1\nhp : ∀ (k : ℕ), 0 < x k\n⊢ ((∑ k ∈ range n, x (k + 1) + 1) ^ 2 / ∑ k ∈ range n, x (k + 1)) * ↑n / (↑n + 1) =\n (∑ k ∈ range n, x (k + 1) + 1) ^ 2 / ((∑ k ∈ range n, x (k + 1)) * (1 + 1 / ↑n))",
"after_state": "x : ℕ → ℝ\nn : ℕ\nh0 : x 0 = 1\nhp : ∀ (k : ℕ), 0 < x k\n⊢ (∑ k ∈ range n, x (k + 1) + 1) ^ 2 * ↑n / ((∑ k ∈ range n, x (k + 1)) * (↑n + 1)) =\n (∑ k ∈ range n, x (k + 1) + 1) ^ 2 / ((∑ k ∈ range n, x (k + 1)) * (1 + 1 / ↑n))"
},
{
"line": "gcongr",
"before_state": "x : ℕ → ℝ\nn : ℕ\nh0 : x 0 = 1\nhp : ∀ (k : ℕ), 0 < x k\n⊢ (∑ k ∈ range n, x (k + 1) + 1) ^ 2 / ((∑ k ∈ range n, x (k + 1)) * (1 + 1 / ↑n)) ≤\n (∑ k ∈ range (n + 1), x k) ^ 2 / ∑ k ∈ range (n + 1), x (k + 1)",
"after_state": "case hd\nx : ℕ → ℝ\nn : ℕ\nh0 : x 0 = 1\nhp : ∀ (k : ℕ), 0 < x k\n⊢ 0 < ∑ k ∈ range (n + 1), x (k + 1)\n---\ncase hac.ha\nx : ℕ → ℝ\nn : ℕ\nh0 : x 0 = 1\nhp : ∀ (k : ℕ), 0 < x k\n⊢ 0 ≤ ∑ k ∈ range n, x (k + 1) + 1\n---\ncase hac.hab\nx : ℕ → ℝ\nn : ℕ\nh0 : x 0 = 1\nhp : ∀ (k : ℕ), 0 < x k\n⊢ ∑ k ∈ range n, x (k + 1) + 1 ≤ ∑ k ∈ range (n + 1), x k\n---\ncase hdb\nx : ℕ → ℝ\nn : ℕ\nh0 : x 0 = 1\nhp : ∀ (k : ℕ), 0 < x k\n⊢ ∑ k ∈ range (n + 1), x (k + 1) ≤ (∑ k ∈ range n, x (k + 1)) * (1 + 1 / ↑n)"
},
{
"line": "gcongr_discharger",
"before_state": "case hc\nx : ℕ → ℝ\nn : ℕ\nh0 : x 0 = 1\nhp : ∀ (k : ℕ), 0 < x k\n⊢ 0 ≤ (∑ k ∈ range (n + 1), x k) ^ 2",
"after_state": "No Goals!"
},
{
"line": "positivity",
"before_state": "case hc\nx : ℕ → ℝ\nn : ℕ\nh0 : x 0 = 1\nhp : ∀ (k : ℕ), 0 < x k\n⊢ 0 ≤ (∑ k ∈ range (n + 1), x k) ^ 2",
"after_state": "No Goals!"
},
{
"line": "exact sum_pos (fun k _ ↦ hp _) nonempty_range_succ",
"before_state": "case hd\nx : ℕ → ℝ\nn : ℕ\nh0 : x 0 = 1\nhp : ∀ (k : ℕ), 0 < x k\n⊢ 0 < ∑ k ∈ range (n + 1), x (k + 1)",
"after_state": "No Goals!"
},
{
"line": "exact add_nonneg (sum_nonneg fun k _ ↦ (hp _).le) zero_le_one",
"before_state": "case hac.ha\nx : ℕ → ℝ\nn : ℕ\nh0 : x 0 = 1\nhp : ∀ (k : ℕ), 0 < x k\n⊢ 0 ≤ ∑ k ∈ range n, x (k + 1) + 1",
"after_state": "No Goals!"
},
{
"line": "rw [sum_range_succ', h0]",
"before_state": "case hac.hab\nx : ℕ → ℝ\nn : ℕ\nh0 : x 0 = 1\nhp : ∀ (k : ℕ), 0 < x k\n⊢ ∑ k ∈ range n, x (k + 1) + 1 ≤ ∑ k ∈ range (n + 1), x k",
"after_state": "No Goals!"
},
{
"line": "rewrite [sum_range_succ', h0]",
"before_state": "case hac.hab\nx : ℕ → ℝ\nn : ℕ\nh0 : x 0 = 1\nhp : ∀ (k : ℕ), 0 < x k\n⊢ ∑ k ∈ range n, x (k + 1) + 1 ≤ ∑ k ∈ range (n + 1), x k",
"after_state": "case hac.hab\nx : ℕ → ℝ\nn : ℕ\nh0 : x 0 = 1\nhp : ∀ (k : ℕ), 0 < x k\n⊢ ∑ k ∈ range n, x (k + 1) + 1 ≤ ∑ k ∈ range n, x (k + 1) + 1"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "case hac.hab\nx : ℕ → ℝ\nn : ℕ\nh0 : x 0 = 1\nhp : ∀ (k : ℕ), 0 < x k\n⊢ ∑ k ∈ range n, x (k + 1) + 1 ≤ ∑ k ∈ range n, x (k + 1) + 1",
"after_state": "No Goals!"
},
{
"line": "try (with_reducible rfl)",
"before_state": "case hac.hab\nx : ℕ → ℝ\nn : ℕ\nh0 : x 0 = 1\nhp : ∀ (k : ℕ), 0 < x k\n⊢ ∑ k ∈ range n, x (k + 1) + 1 ≤ ∑ k ∈ range n, x (k + 1) + 1",
"after_state": "No Goals!"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "case hac.hab\nx : ℕ → ℝ\nn : ℕ\nh0 : x 0 = 1\nhp : ∀ (k : ℕ), 0 < x k\n⊢ ∑ k ∈ range n, x (k + 1) + 1 ≤ ∑ k ∈ range n, x (k + 1) + 1",
"after_state": "No Goals!"
},
{
"line": "with_reducible rfl",
"before_state": "case hac.hab\nx : ℕ → ℝ\nn : ℕ\nh0 : x 0 = 1\nhp : ∀ (k : ℕ), 0 < x k\n⊢ ∑ k ∈ range n, x (k + 1) + 1 ≤ ∑ k ∈ range n, x (k + 1) + 1",
"after_state": "No Goals!"
},
{
"line": "rfl",
"before_state": "case hac.hab\nx : ℕ → ℝ\nn : ℕ\nh0 : x 0 = 1\nhp : ∀ (k : ℕ), 0 < x k\n⊢ ∑ k ∈ range n, x (k + 1) + 1 ≤ ∑ k ∈ range n, x (k + 1) + 1",
"after_state": "No Goals!"
},
{
"line": "apply_rfl",
"before_state": "case hac.hab\nx : ℕ → ℝ\nn : ℕ\nh0 : x 0 = 1\nhp : ∀ (k : ℕ), 0 < x k\n⊢ ∑ k ∈ range n, x (k + 1) + 1 ≤ ∑ k ∈ range n, x (k + 1) + 1",
"after_state": "No Goals!"
},
{
"line": "exact\n le_avg hn\n (hx.comp_monotone @Nat.succ_le_succ)\n -- We conclude by Sedrakyan.",
"before_state": "case hdb\nx : ℕ → ℝ\nn : ℕ\nh0 : x 0 = 1\nhp : ∀ (k : ℕ), 0 < x k\n⊢ ∑ k ∈ range (n + 1), x (k + 1) ≤ (∑ k ∈ range n, x (k + 1)) * (1 + 1 / ↑n)",
"after_state": "No Goals!"
}
] |
theorem imo1982_q3a (hx : Antitone x) (h0 : x 0 = 1) (hp : ∀ k, 0 < x k) :
∃ n : ℕ, 3.999 ≤ ∑ k ∈ range n, (x k) ^ 2 / x (k + 1) := by
use 4000
convert Imo1982Q3.ineq (Nat.succ_ne_zero 3998) hx h0 hp
norm_num
|
/root/DuelModelResearch/mathlib4/Archive/Imo/Imo1982Q3.lean
|
{
"open": [
"Finset NNReal"
],
"variables": [
"{x : ℕ → ℝ} {n : ℕ} (hn : n ≠ 0) (hx : Antitone x)"
]
}
|
[
{
"line": "use 4000",
"before_state": "x : ℕ → ℝ\nhx : Antitone x\nh0 : x 0 = 1\nhp : ∀ (k : ℕ), 0 < x k\n⊢ ∃ n, 3.999 ≤ ∑ k ∈ range n, x k ^ 2 / x (k + 1)",
"after_state": "case h\nx : ℕ → ℝ\nhx : Antitone x\nh0 : x 0 = 1\nhp : ∀ (k : ℕ), 0 < x k\n⊢ 3.999 ≤ ∑ k ∈ range 4000, x k ^ 2 / x (k + 1)"
},
{
"line": "refine without_cdot(4000 : ?m✝)",
"before_state": "case w\nx : ℕ → ℝ\nhx : Antitone x\nh0 : x 0 = 1\nhp : ∀ (k : ℕ), 0 < x k\n⊢ ℕ",
"after_state": "No Goals!"
},
{
"line": "try with_reducible use_discharger",
"before_state": "case h\nx : ℕ → ℝ\nhx : Antitone x\nh0 : x 0 = 1\nhp : ∀ (k : ℕ), 0 < x k\n⊢ 3.999 ≤ ∑ k ∈ range 4000, x k ^ 2 / x (k + 1)",
"after_state": "case h\nx : ℕ → ℝ\nhx : Antitone x\nh0 : x 0 = 1\nhp : ∀ (k : ℕ), 0 < x k\n⊢ 3.999 ≤ ∑ k ∈ range 4000, x k ^ 2 / x (k + 1)"
},
{
"line": "first\n| with_reducible use_discharger\n| skip",
"before_state": "case h\nx : ℕ → ℝ\nhx : Antitone x\nh0 : x 0 = 1\nhp : ∀ (k : ℕ), 0 < x k\n⊢ 3.999 ≤ ∑ k ∈ range 4000, x k ^ 2 / x (k + 1)",
"after_state": "case h\nx : ℕ → ℝ\nhx : Antitone x\nh0 : x 0 = 1\nhp : ∀ (k : ℕ), 0 < x k\n⊢ 3.999 ≤ ∑ k ∈ range 4000, x k ^ 2 / x (k + 1)"
},
{
"line": "with_reducible use_discharger",
"before_state": "case h\nx : ℕ → ℝ\nhx : Antitone x\nh0 : x 0 = 1\nhp : ∀ (k : ℕ), 0 < x k\n⊢ 3.999 ≤ ∑ k ∈ range 4000, x k ^ 2 / x (k + 1)",
"after_state": "case h\nx : ℕ → ℝ\nhx : Antitone x\nh0 : x 0 = 1\nhp : ∀ (k : ℕ), 0 < x k\n⊢ 3.999 ≤ ∑ k ∈ range 4000, x k ^ 2 / x (k + 1)"
},
{
"line": "use_discharger",
"before_state": "case h\nx : ℕ → ℝ\nhx : Antitone x\nh0 : x 0 = 1\nhp : ∀ (k : ℕ), 0 < x k\n⊢ 3.999 ≤ ∑ k ∈ range 4000, x k ^ 2 / x (k + 1)",
"after_state": "case h\nx : ℕ → ℝ\nhx : Antitone x\nh0 : x 0 = 1\nhp : ∀ (k : ℕ), 0 < x k\n⊢ 3.999 ≤ ∑ k ∈ range 4000, x k ^ 2 / x (k + 1)"
},
{
"line": "focus\n apply exists_prop.mpr✝\n with_annotate_state\"<;>\" skip\n all_goals use_discharger",
"before_state": "case h\nx : ℕ → ℝ\nhx : Antitone x\nh0 : x 0 = 1\nhp : ∀ (k : ℕ), 0 < x k\n⊢ 3.999 ≤ ∑ k ∈ range 4000, x k ^ 2 / x (k + 1)",
"after_state": "case h\nx : ℕ → ℝ\nhx : Antitone x\nh0 : x 0 = 1\nhp : ∀ (k : ℕ), 0 < x k\n⊢ 3.999 ≤ ∑ k ∈ range 4000, x k ^ 2 / x (k + 1)"
},
{
"line": "apply exists_prop.mpr✝",
"before_state": "case h\nx : ℕ → ℝ\nhx : Antitone x\nh0 : x 0 = 1\nhp : ∀ (k : ℕ), 0 < x k\n⊢ 3.999 ≤ ∑ k ∈ range 4000, x k ^ 2 / x (k + 1)",
"after_state": "case h\nx : ℕ → ℝ\nhx : Antitone x\nh0 : x 0 = 1\nhp : ∀ (k : ℕ), 0 < x k\n⊢ 3.999 ≤ ∑ k ∈ range 4000, x k ^ 2 / x (k + 1)"
},
{
"line": "skip",
"before_state": "case h\nx : ℕ → ℝ\nhx : Antitone x\nh0 : x 0 = 1\nhp : ∀ (k : ℕ), 0 < x k\n⊢ 3.999 ≤ ∑ k ∈ range 4000, x k ^ 2 / x (k + 1)",
"after_state": "case h\nx : ℕ → ℝ\nhx : Antitone x\nh0 : x 0 = 1\nhp : ∀ (k : ℕ), 0 < x k\n⊢ 3.999 ≤ ∑ k ∈ range 4000, x k ^ 2 / x (k + 1)"
},
{
"line": "convert Imo1982Q3.ineq (Nat.succ_ne_zero 3998) hx h0 hp",
"before_state": "case h\nx : ℕ → ℝ\nhx : Antitone x\nh0 : x 0 = 1\nhp : ∀ (k : ℕ), 0 < x k\n⊢ 3.999 ≤ ∑ k ∈ range 4000, x k ^ 2 / x (k + 1)",
"after_state": "No Goals!"
}
] |
theorem imo1982_q3b : ∃ x : ℕ → ℝ, Antitone x ∧ x 0 = 1 ∧ (∀ k, 0 < x k)
∧ ∀ n, ∑ k ∈ range n, x k ^ 2 / x (k + 1) < 4 := by
refine ⟨fun k ↦ 2⁻¹ ^ k, ?_, pow_zero _, ?_, fun n ↦ ?_⟩
· apply (pow_right_strictAnti₀ _ _).antitone <;> norm_num
· simp
· have {k : ℕ} : (2 : ℝ)⁻¹ ^ (k * 2) * ((2 : ℝ)⁻¹ ^ k)⁻¹ = (2 : ℝ)⁻¹ ^ k := by
rw [← pow_sub₀] <;> simp [mul_two]
simp_rw [← pow_mul, pow_succ, ← div_eq_mul_inv, div_div_eq_mul_div, mul_comm, mul_div_assoc,
← mul_sum, div_eq_mul_inv, this, ← two_add_two_eq_four, ← mul_two,
mul_lt_mul_iff_of_pos_left two_pos]
convert NNReal.coe_lt_coe.2 <| geom_sum_lt (inv_ne_zero two_ne_zero) two_inv_lt_one n
· simp
· norm_num
|
/root/DuelModelResearch/mathlib4/Archive/Imo/Imo1982Q3.lean
|
{
"open": [
"Finset NNReal"
],
"variables": [
"{x : ℕ → ℝ} {n : ℕ} (hn : n ≠ 0) (hx : Antitone x)"
]
}
|
[
{
"line": "refine ⟨fun k ↦ 2⁻¹ ^ k, ?_, pow_zero _, ?_, fun n ↦ ?_⟩",
"before_state": "⊢ ∃ x, Antitone x ∧ x 0 = 1 ∧ (∀ (k : ℕ), 0 < x k) ∧ ∀ (n : ℕ), ∑ k ∈ range n, x k ^ 2 / x (k + 1) < 4",
"after_state": "case refine_1\n⊢ Antitone fun k => 2⁻¹ ^ k\n---\ncase refine_2\n⊢ ∀ (k : ℕ), 0 < (fun k => 2⁻¹ ^ k) k\n---\ncase refine_3\nn : ℕ\n⊢ ∑ k ∈ range n, (fun k => 2⁻¹ ^ k) k ^ 2 / (fun k => 2⁻¹ ^ k) (k + 1) < 4"
},
{
"line": "focus\n apply (pow_right_strictAnti₀ _ _).antitone\n with_annotate_state\"<;>\" skip\n all_goals norm_num",
"before_state": "case refine_1\n⊢ Antitone fun k => 2⁻¹ ^ k",
"after_state": "No Goals!"
},
{
"line": "apply (pow_right_strictAnti₀ _ _).antitone",
"before_state": "case refine_1\n⊢ Antitone fun k => 2⁻¹ ^ k",
"after_state": "⊢ 0 < 2⁻¹\n---\n⊢ 2⁻¹ < 1"
},
{
"line": "with_annotate_state\"<;>\" skip",
"before_state": "⊢ 0 < 2⁻¹\n---\n⊢ 2⁻¹ < 1",
"after_state": "⊢ 0 < 2⁻¹\n---\n⊢ 2⁻¹ < 1"
},
{
"line": "skip",
"before_state": "⊢ 0 < 2⁻¹\n---\n⊢ 2⁻¹ < 1",
"after_state": "⊢ 0 < 2⁻¹\n---\n⊢ 2⁻¹ < 1"
},
{
"line": "all_goals norm_num",
"before_state": "⊢ 0 < 2⁻¹\n---\n⊢ 2⁻¹ < 1",
"after_state": "No Goals!"
},
{
"line": "norm_num",
"before_state": "⊢ 0 < 2⁻¹",
"after_state": "No Goals!"
},
{
"line": "norm_num",
"before_state": "⊢ 2⁻¹ < 1",
"after_state": "No Goals!"
},
{
"line": "simp",
"before_state": "case refine_2\n⊢ ∀ (k : ℕ), 0 < (fun k => 2⁻¹ ^ k) k",
"after_state": "No Goals!"
},
{
"line": "have {k : ℕ} : (2 : ℝ)⁻¹ ^ (k * 2) * ((2 : ℝ)⁻¹ ^ k)⁻¹ = (2 : ℝ)⁻¹ ^ k := by rw [← pow_sub₀] <;> simp [mul_two]",
"before_state": "case refine_3\nn : ℕ\n⊢ ∑ k ∈ range n, (fun k => 2⁻¹ ^ k) k ^ 2 / (fun k => 2⁻¹ ^ k) (k + 1) < 4",
"after_state": "case refine_3\nn : ℕ\nthis : ∀ {k : ℕ}, 2⁻¹ ^ (k * 2) * (2⁻¹ ^ k)⁻¹ = 2⁻¹ ^ k\n⊢ ∑ k ∈ range n, (fun k => 2⁻¹ ^ k) k ^ 2 / (fun k => 2⁻¹ ^ k) (k + 1) < 4"
},
{
"line": "focus\n refine\n no_implicit_lambda%\n (have {k : ℕ} : (2 : ℝ)⁻¹ ^ (k * 2) * ((2 : ℝ)⁻¹ ^ k)⁻¹ = (2 : ℝ)⁻¹ ^ k := ?body✝;\n ?_)\n case body✝ => with_annotate_state\"by\" (rw [← pow_sub₀] <;> simp [mul_two])",
"before_state": "case refine_3\nn : ℕ\n⊢ ∑ k ∈ range n, (fun k => 2⁻¹ ^ k) k ^ 2 / (fun k => 2⁻¹ ^ k) (k + 1) < 4",
"after_state": "case refine_3\nn : ℕ\nthis : ∀ {k : ℕ}, 2⁻¹ ^ (k * 2) * (2⁻¹ ^ k)⁻¹ = 2⁻¹ ^ k\n⊢ ∑ k ∈ range n, (fun k => 2⁻¹ ^ k) k ^ 2 / (fun k => 2⁻¹ ^ k) (k + 1) < 4"
},
{
"line": "refine\n no_implicit_lambda%\n (have {k : ℕ} : (2 : ℝ)⁻¹ ^ (k * 2) * ((2 : ℝ)⁻¹ ^ k)⁻¹ = (2 : ℝ)⁻¹ ^ k := ?body✝;\n ?_)",
"before_state": "case refine_3\nn : ℕ\n⊢ ∑ k ∈ range n, (fun k => 2⁻¹ ^ k) k ^ 2 / (fun k => 2⁻¹ ^ k) (k + 1) < 4",
"after_state": "case body\nn k : ℕ\n⊢ 2⁻¹ ^ (k * 2) * (2⁻¹ ^ k)⁻¹ = 2⁻¹ ^ k\n---\ncase refine_3\nn : ℕ\nthis : ∀ {k : ℕ}, 2⁻¹ ^ (k * 2) * (2⁻¹ ^ k)⁻¹ = 2⁻¹ ^ k\n⊢ ∑ k ∈ range n, (fun k => 2⁻¹ ^ k) k ^ 2 / (fun k => 2⁻¹ ^ k) (k + 1) < 4"
},
{
"line": "case body✝ => with_annotate_state\"by\" (rw [← pow_sub₀] <;> simp [mul_two])",
"before_state": "case body\nn k : ℕ\n⊢ 2⁻¹ ^ (k * 2) * (2⁻¹ ^ k)⁻¹ = 2⁻¹ ^ k\n---\ncase refine_3\nn : ℕ\nthis : ∀ {k : ℕ}, 2⁻¹ ^ (k * 2) * (2⁻¹ ^ k)⁻¹ = 2⁻¹ ^ k\n⊢ ∑ k ∈ range n, (fun k => 2⁻¹ ^ k) k ^ 2 / (fun k => 2⁻¹ ^ k) (k + 1) < 4",
"after_state": "case refine_3\nn : ℕ\nthis : ∀ {k : ℕ}, 2⁻¹ ^ (k * 2) * (2⁻¹ ^ k)⁻¹ = 2⁻¹ ^ k\n⊢ ∑ k ∈ range n, (fun k => 2⁻¹ ^ k) k ^ 2 / (fun k => 2⁻¹ ^ k) (k + 1) < 4"
},
{
"line": "with_annotate_state\"by\" (rw [← pow_sub₀] <;> simp [mul_two])",
"before_state": "n k : ℕ\n⊢ 2⁻¹ ^ (k * 2) * (2⁻¹ ^ k)⁻¹ = 2⁻¹ ^ k",
"after_state": "No Goals!"
},
{
"line": "focus\n rw [← pow_sub₀]\n with_annotate_state\"<;>\" skip\n all_goals simp [mul_two]",
"before_state": "n k : ℕ\n⊢ 2⁻¹ ^ (k * 2) * (2⁻¹ ^ k)⁻¹ = 2⁻¹ ^ k",
"after_state": "No Goals!"
},
{
"line": "rw [← pow_sub₀]",
"before_state": "n k : ℕ\n⊢ 2⁻¹ ^ (k * 2) * (2⁻¹ ^ k)⁻¹ = 2⁻¹ ^ k",
"after_state": "n k : ℕ\n⊢ 2⁻¹ ^ (k * 2 - k) = 2⁻¹ ^ k\n---\ncase ha\nn k : ℕ\n⊢ 2⁻¹ ≠ 0\n---\ncase h\nn k : ℕ\n⊢ k ≤ k * 2"
},
{
"line": "rewrite [← pow_sub₀]",
"before_state": "n k : ℕ\n⊢ 2⁻¹ ^ (k * 2) * (2⁻¹ ^ k)⁻¹ = 2⁻¹ ^ k",
"after_state": "n k : ℕ\n⊢ 2⁻¹ ^ (k * 2 - k) = 2⁻¹ ^ k\n---\ncase ha\nn k : ℕ\n⊢ 2⁻¹ ≠ 0\n---\ncase h\nn k : ℕ\n⊢ k ≤ k * 2"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "n k : ℕ\n⊢ 2⁻¹ ^ (k * 2 - k) = 2⁻¹ ^ k\n---\ncase ha\nn k : ℕ\n⊢ 2⁻¹ ≠ 0\n---\ncase h\nn k : ℕ\n⊢ k ≤ k * 2",
"after_state": "n k : ℕ\n⊢ 2⁻¹ ^ (k * 2 - k) = 2⁻¹ ^ k\n---\ncase ha\nn k : ℕ\n⊢ 2⁻¹ ≠ 0\n---\ncase h\nn k : ℕ\n⊢ k ≤ k * 2"
},
{
"line": "try (with_reducible rfl)",
"before_state": "n k : ℕ\n⊢ 2⁻¹ ^ (k * 2 - k) = 2⁻¹ ^ k\n---\ncase ha\nn k : ℕ\n⊢ 2⁻¹ ≠ 0\n---\ncase h\nn k : ℕ\n⊢ k ≤ k * 2",
"after_state": "n k : ℕ\n⊢ 2⁻¹ ^ (k * 2 - k) = 2⁻¹ ^ k\n---\ncase ha\nn k : ℕ\n⊢ 2⁻¹ ≠ 0\n---\ncase h\nn k : ℕ\n⊢ k ≤ k * 2"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "n k : ℕ\n⊢ 2⁻¹ ^ (k * 2 - k) = 2⁻¹ ^ k\n---\ncase ha\nn k : ℕ\n⊢ 2⁻¹ ≠ 0\n---\ncase h\nn k : ℕ\n⊢ k ≤ k * 2",
"after_state": "n k : ℕ\n⊢ 2⁻¹ ^ (k * 2 - k) = 2⁻¹ ^ k\n---\ncase ha\nn k : ℕ\n⊢ 2⁻¹ ≠ 0\n---\ncase h\nn k : ℕ\n⊢ k ≤ k * 2"
},
{
"line": "with_reducible rfl",
"before_state": "n k : ℕ\n⊢ 2⁻¹ ^ (k * 2 - k) = 2⁻¹ ^ k\n---\ncase ha\nn k : ℕ\n⊢ 2⁻¹ ≠ 0\n---\ncase h\nn k : ℕ\n⊢ k ≤ k * 2",
"after_state": "n k : ℕ\n⊢ 2⁻¹ ^ (k * 2 - k) = 2⁻¹ ^ k\n---\ncase ha\nn k : ℕ\n⊢ 2⁻¹ ≠ 0\n---\ncase h\nn k : ℕ\n⊢ k ≤ k * 2"
},
{
"line": "rfl",
"before_state": "n k : ℕ\n⊢ 2⁻¹ ^ (k * 2 - k) = 2⁻¹ ^ k\n---\ncase ha\nn k : ℕ\n⊢ 2⁻¹ ≠ 0\n---\ncase h\nn k : ℕ\n⊢ k ≤ k * 2",
"after_state": "n k : ℕ\n⊢ 2⁻¹ ^ (k * 2 - k) = 2⁻¹ ^ k\n---\ncase ha\nn k : ℕ\n⊢ 2⁻¹ ≠ 0\n---\ncase h\nn k : ℕ\n⊢ k ≤ k * 2"
},
{
"line": "apply_rfl",
"before_state": "n k : ℕ\n⊢ 2⁻¹ ^ (k * 2 - k) = 2⁻¹ ^ k\n---\ncase ha\nn k : ℕ\n⊢ 2⁻¹ ≠ 0\n---\ncase h\nn k : ℕ\n⊢ k ≤ k * 2",
"after_state": "n k : ℕ\n⊢ 2⁻¹ ^ (k * 2 - k) = 2⁻¹ ^ k\n---\ncase ha\nn k : ℕ\n⊢ 2⁻¹ ≠ 0\n---\ncase h\nn k : ℕ\n⊢ k ≤ k * 2"
},
{
"line": "skip",
"before_state": "n k : ℕ\n⊢ 2⁻¹ ^ (k * 2 - k) = 2⁻¹ ^ k\n---\ncase ha\nn k : ℕ\n⊢ 2⁻¹ ≠ 0\n---\ncase h\nn k : ℕ\n⊢ k ≤ k * 2",
"after_state": "n k : ℕ\n⊢ 2⁻¹ ^ (k * 2 - k) = 2⁻¹ ^ k\n---\ncase ha\nn k : ℕ\n⊢ 2⁻¹ ≠ 0\n---\ncase h\nn k : ℕ\n⊢ k ≤ k * 2"
},
{
"line": "with_annotate_state\"<;>\" skip",
"before_state": "n k : ℕ\n⊢ 2⁻¹ ^ (k * 2 - k) = 2⁻¹ ^ k\n---\ncase ha\nn k : ℕ\n⊢ 2⁻¹ ≠ 0\n---\ncase h\nn k : ℕ\n⊢ k ≤ k * 2",
"after_state": "n k : ℕ\n⊢ 2⁻¹ ^ (k * 2 - k) = 2⁻¹ ^ k\n---\ncase ha\nn k : ℕ\n⊢ 2⁻¹ ≠ 0\n---\ncase h\nn k : ℕ\n⊢ k ≤ k * 2"
},
{
"line": "skip",
"before_state": "n k : ℕ\n⊢ 2⁻¹ ^ (k * 2 - k) = 2⁻¹ ^ k\n---\ncase ha\nn k : ℕ\n⊢ 2⁻¹ ≠ 0\n---\ncase h\nn k : ℕ\n⊢ k ≤ k * 2",
"after_state": "n k : ℕ\n⊢ 2⁻¹ ^ (k * 2 - k) = 2⁻¹ ^ k\n---\ncase ha\nn k : ℕ\n⊢ 2⁻¹ ≠ 0\n---\ncase h\nn k : ℕ\n⊢ k ≤ k * 2"
},
{
"line": "all_goals simp [mul_two]",
"before_state": "n k : ℕ\n⊢ 2⁻¹ ^ (k * 2 - k) = 2⁻¹ ^ k\n---\ncase ha\nn k : ℕ\n⊢ 2⁻¹ ≠ 0\n---\ncase h\nn k : ℕ\n⊢ k ≤ k * 2",
"after_state": "No Goals!"
},
{
"line": "simp [mul_two]",
"before_state": "n k : ℕ\n⊢ 2⁻¹ ^ (k * 2 - k) = 2⁻¹ ^ k",
"after_state": "No Goals!"
},
{
"line": "simp [mul_two]",
"before_state": "case ha\nn k : ℕ\n⊢ 2⁻¹ ≠ 0",
"after_state": "No Goals!"
},
{
"line": "simp [mul_two]",
"before_state": "case h\nn k : ℕ\n⊢ k ≤ k * 2",
"after_state": "No Goals!"
},
{
"line": "simp_rw [← pow_mul, pow_succ, ← div_eq_mul_inv, div_div_eq_mul_div, mul_comm, mul_div_assoc, ← mul_sum, div_eq_mul_inv,\n this, ← two_add_two_eq_four, ← mul_two, mul_lt_mul_iff_of_pos_left two_pos]",
"before_state": "case refine_3\nn : ℕ\nthis : ∀ {k : ℕ}, 2⁻¹ ^ (k * 2) * (2⁻¹ ^ k)⁻¹ = 2⁻¹ ^ k\n⊢ ∑ k ∈ range n, (fun k => 2⁻¹ ^ k) k ^ 2 / (fun k => 2⁻¹ ^ k) (k + 1) < 4",
"after_state": "case refine_3\nn : ℕ\nthis : ∀ {k : ℕ}, 2⁻¹ ^ (k * 2) * (2⁻¹ ^ k)⁻¹ = 2⁻¹ ^ k\n⊢ ∑ x ∈ range n, 2⁻¹ ^ x < 2"
},
{
"line": "simp (failIfUnchanged✝ := false✝) only",
"before_state": "case refine_3\nn : ℕ\nthis : ∀ {k : ℕ}, 2⁻¹ ^ (k * 2) * (2⁻¹ ^ k)⁻¹ = 2⁻¹ ^ k\n⊢ ∑ k ∈ range n, (fun k => 2⁻¹ ^ k) k ^ 2 / (fun k => 2⁻¹ ^ k) (k + 1) < 4",
"after_state": "case refine_3\nn : ℕ\nthis : ∀ {k : ℕ}, 2⁻¹ ^ (k * 2) * (2⁻¹ ^ k)⁻¹ = 2⁻¹ ^ k\n⊢ ∑ x ∈ range n, (2⁻¹ ^ x) ^ 2 / 2⁻¹ ^ (x + 1) < 4"
},
{
"line": "simp only [← pow_mul]",
"before_state": "case refine_3\nn : ℕ\nthis : ∀ {k : ℕ}, 2⁻¹ ^ (k * 2) * (2⁻¹ ^ k)⁻¹ = 2⁻¹ ^ k\n⊢ ∑ x ∈ range n, (2⁻¹ ^ x) ^ 2 / 2⁻¹ ^ (x + 1) < 4",
"after_state": "case refine_3\nn : ℕ\nthis : ∀ {k : ℕ}, 2⁻¹ ^ (k * 2) * (2⁻¹ ^ k)⁻¹ = 2⁻¹ ^ k\n⊢ ∑ x ∈ range n, 2⁻¹ ^ (x * 2) / 2⁻¹ ^ (x + 1) < 4"
},
{
"line": "simp only [pow_succ]",
"before_state": "case refine_3\nn : ℕ\nthis : ∀ {k : ℕ}, 2⁻¹ ^ (k * 2) * (2⁻¹ ^ k)⁻¹ = 2⁻¹ ^ k\n⊢ ∑ x ∈ range n, 2⁻¹ ^ (x * 2) / 2⁻¹ ^ (x + 1) < 4",
"after_state": "case refine_3\nn : ℕ\nthis : ∀ {k : ℕ}, 2⁻¹ ^ (k * 2) * (2⁻¹ ^ k)⁻¹ = 2⁻¹ ^ k\n⊢ ∑ x ∈ range n, 2⁻¹ ^ (x * 2) / (2⁻¹ ^ x * 2⁻¹) < 4"
},
{
"line": "simp only [← div_eq_mul_inv]",
"before_state": "case refine_3\nn : ℕ\nthis : ∀ {k : ℕ}, 2⁻¹ ^ (k * 2) * (2⁻¹ ^ k)⁻¹ = 2⁻¹ ^ k\n⊢ ∑ x ∈ range n, 2⁻¹ ^ (x * 2) / (2⁻¹ ^ x * 2⁻¹) < 4",
"after_state": "case refine_3\nn : ℕ\nthis : ∀ {k : ℕ}, 2⁻¹ ^ (k * 2) * (2⁻¹ ^ k)⁻¹ = 2⁻¹ ^ k\n⊢ ∑ x ∈ range n, 2⁻¹ ^ (x * 2) / (2⁻¹ ^ x / 2) < 4"
},
{
"line": "simp only [div_div_eq_mul_div]",
"before_state": "case refine_3\nn : ℕ\nthis : ∀ {k : ℕ}, 2⁻¹ ^ (k * 2) * (2⁻¹ ^ k)⁻¹ = 2⁻¹ ^ k\n⊢ ∑ x ∈ range n, 2⁻¹ ^ (x * 2) / (2⁻¹ ^ x / 2) < 4",
"after_state": "case refine_3\nn : ℕ\nthis : ∀ {k : ℕ}, 2⁻¹ ^ (k * 2) * (2⁻¹ ^ k)⁻¹ = 2⁻¹ ^ k\n⊢ ∑ x ∈ range n, 2⁻¹ ^ (x * 2) * 2 / 2⁻¹ ^ x < 4"
},
{
"line": "simp only [mul_comm]",
"before_state": "case refine_3\nn : ℕ\nthis : ∀ {k : ℕ}, 2⁻¹ ^ (k * 2) * (2⁻¹ ^ k)⁻¹ = 2⁻¹ ^ k\n⊢ ∑ x ∈ range n, 2⁻¹ ^ (x * 2) * 2 / 2⁻¹ ^ x < 4",
"after_state": "case refine_3\nn : ℕ\nthis : ∀ {k : ℕ}, 2⁻¹ ^ (k * 2) * (2⁻¹ ^ k)⁻¹ = 2⁻¹ ^ k\n⊢ ∑ x ∈ range n, 2 * 2⁻¹ ^ (x * 2) / 2⁻¹ ^ x < 4"
},
{
"line": "simp only [mul_div_assoc]",
"before_state": "case refine_3\nn : ℕ\nthis : ∀ {k : ℕ}, 2⁻¹ ^ (k * 2) * (2⁻¹ ^ k)⁻¹ = 2⁻¹ ^ k\n⊢ ∑ x ∈ range n, 2 * 2⁻¹ ^ (x * 2) / 2⁻¹ ^ x < 4",
"after_state": "case refine_3\nn : ℕ\nthis : ∀ {k : ℕ}, 2⁻¹ ^ (k * 2) * (2⁻¹ ^ k)⁻¹ = 2⁻¹ ^ k\n⊢ ∑ x ∈ range n, 2 * (2⁻¹ ^ (x * 2) / 2⁻¹ ^ x) < 4"
},
{
"line": "simp only [← mul_sum]",
"before_state": "case refine_3\nn : ℕ\nthis : ∀ {k : ℕ}, 2⁻¹ ^ (k * 2) * (2⁻¹ ^ k)⁻¹ = 2⁻¹ ^ k\n⊢ ∑ x ∈ range n, 2 * (2⁻¹ ^ (x * 2) / 2⁻¹ ^ x) < 4",
"after_state": "case refine_3\nn : ℕ\nthis : ∀ {k : ℕ}, 2⁻¹ ^ (k * 2) * (2⁻¹ ^ k)⁻¹ = 2⁻¹ ^ k\n⊢ 2 * ∑ i ∈ range n, 2⁻¹ ^ (i * 2) / 2⁻¹ ^ i < 4"
},
{
"line": "simp only [div_eq_mul_inv]",
"before_state": "case refine_3\nn : ℕ\nthis : ∀ {k : ℕ}, 2⁻¹ ^ (k * 2) * (2⁻¹ ^ k)⁻¹ = 2⁻¹ ^ k\n⊢ 2 * ∑ i ∈ range n, 2⁻¹ ^ (i * 2) / 2⁻¹ ^ i < 4",
"after_state": "case refine_3\nn : ℕ\nthis : ∀ {k : ℕ}, 2⁻¹ ^ (k * 2) * (2⁻¹ ^ k)⁻¹ = 2⁻¹ ^ k\n⊢ 2 * ∑ x ∈ range n, 2⁻¹ ^ (x * 2) * (2⁻¹ ^ x)⁻¹ < 4"
},
{
"line": "simp only [this]",
"before_state": "case refine_3\nn : ℕ\nthis : ∀ {k : ℕ}, 2⁻¹ ^ (k * 2) * (2⁻¹ ^ k)⁻¹ = 2⁻¹ ^ k\n⊢ 2 * ∑ x ∈ range n, 2⁻¹ ^ (x * 2) * (2⁻¹ ^ x)⁻¹ < 4",
"after_state": "case refine_3\nn : ℕ\nthis : ∀ {k : ℕ}, 2⁻¹ ^ (k * 2) * (2⁻¹ ^ k)⁻¹ = 2⁻¹ ^ k\n⊢ 2 * ∑ x ∈ range n, 2⁻¹ ^ x < 4"
},
{
"line": "simp only [← two_add_two_eq_four]",
"before_state": "case refine_3\nn : ℕ\nthis : ∀ {k : ℕ}, 2⁻¹ ^ (k * 2) * (2⁻¹ ^ k)⁻¹ = 2⁻¹ ^ k\n⊢ 2 * ∑ x ∈ range n, 2⁻¹ ^ x < 4",
"after_state": "case refine_3\nn : ℕ\nthis : ∀ {k : ℕ}, 2⁻¹ ^ (k * 2) * (2⁻¹ ^ k)⁻¹ = 2⁻¹ ^ k\n⊢ 2 * ∑ x ∈ range n, 2⁻¹ ^ x < 2 + 2"
},
{
"line": "simp only [← mul_two]",
"before_state": "case refine_3\nn : ℕ\nthis : ∀ {k : ℕ}, 2⁻¹ ^ (k * 2) * (2⁻¹ ^ k)⁻¹ = 2⁻¹ ^ k\n⊢ 2 * ∑ x ∈ range n, 2⁻¹ ^ x < 2 + 2",
"after_state": "case refine_3\nn : ℕ\nthis : ∀ {k : ℕ}, 2⁻¹ ^ (k * 2) * (2⁻¹ ^ k)⁻¹ = 2⁻¹ ^ k\n⊢ 2 * ∑ x ∈ range n, 2⁻¹ ^ x < 2 * 2"
},
{
"line": "simp only [mul_lt_mul_iff_of_pos_left two_pos]",
"before_state": "case refine_3\nn : ℕ\nthis : ∀ {k : ℕ}, 2⁻¹ ^ (k * 2) * (2⁻¹ ^ k)⁻¹ = 2⁻¹ ^ k\n⊢ 2 * ∑ x ∈ range n, 2⁻¹ ^ x < 2 * 2",
"after_state": "case refine_3\nn : ℕ\nthis : ∀ {k : ℕ}, 2⁻¹ ^ (k * 2) * (2⁻¹ ^ k)⁻¹ = 2⁻¹ ^ k\n⊢ ∑ x ∈ range n, 2⁻¹ ^ x < 2"
},
{
"line": "convert NNReal.coe_lt_coe.2 <| geom_sum_lt (inv_ne_zero two_ne_zero) two_inv_lt_one n",
"before_state": "case refine_3\nn : ℕ\nthis : ∀ {k : ℕ}, 2⁻¹ ^ (k * 2) * (2⁻¹ ^ k)⁻¹ = 2⁻¹ ^ k\n⊢ ∑ x ∈ range n, 2⁻¹ ^ x < 2",
"after_state": "case h.e'_3\nn : ℕ\nthis : ∀ {k : ℕ}, 2⁻¹ ^ (k * 2) * (2⁻¹ ^ k)⁻¹ = 2⁻¹ ^ k\n⊢ ∑ x ∈ range n, 2⁻¹ ^ x = ↑(∑ i ∈ range n, 2⁻¹ ^ i)\n---\ncase h.e'_4\nn : ℕ\nthis : ∀ {k : ℕ}, 2⁻¹ ^ (k * 2) * (2⁻¹ ^ k)⁻¹ = 2⁻¹ ^ k\n⊢ 2 = ↑(1 - 2⁻¹)⁻¹"
},
{
"line": "simp",
"before_state": "case h.e'_3\nn : ℕ\nthis : ∀ {k : ℕ}, 2⁻¹ ^ (k * 2) * (2⁻¹ ^ k)⁻¹ = 2⁻¹ ^ k\n⊢ ∑ x ∈ range n, 2⁻¹ ^ x = ↑(∑ i ∈ range n, 2⁻¹ ^ i)",
"after_state": "No Goals!"
},
{
"line": "norm_num",
"before_state": "case h.e'_4\nn : ℕ\nthis : ∀ {k : ℕ}, 2⁻¹ ^ (k * 2) * (2⁻¹ ^ k)⁻¹ = 2⁻¹ ^ k\n⊢ 2 = ↑(1 - 2⁻¹)⁻¹",
"after_state": "No Goals!"
}
] |
theorem imo1988_q6 {a b : ℕ} (h : a * b + 1 ∣ a ^ 2 + b ^ 2) :
∃ d, d ^ 2 = (a ^ 2 + b ^ 2) / (a * b + 1) := by
rcases h with ⟨k, hk⟩
rw [hk]
rw [Nat.mul_div_cancel_left _ (Nat.succ_pos (a * b))]
simp only [sq] at hk
apply constant_descent_vieta_jumping a b (H := fun a b => a * a + b * b = (a * b + 1) * k)
hk (fun x => k * x) (fun x => x * x - k) fun _ _ => False <;>
clear hk a b
· -- We will now show that the fibers of the solution set are described by a quadratic equation.
intro x y
rw [← Int.natCast_inj]
rw [← sub_eq_zero]
apply eq_iff_eq_cancel_right.2
simp; ring
· -- Show that the solution set is symmetric in a and b.
intro x y
simp [add_comm (x * x), mul_comm x]
· -- Show that the claim is true if b = 0.
suffices ∀ a, a * a = k → ∃ d, d * d = k by simpa
rintro x rfl; use x
· -- Show that the claim is true if a = b.
intro x hx
suffices k ≤ 1 by
rw [Nat.le_add_one_iff] at this
rw [Nat.le_zero] at this
rcases this with (rfl | rfl)
· use 0; simp
· use 1; simp
contrapose! hx with k_lt_one
apply ne_of_lt
calc
x * x + x * x = x * x * 2 := by rw [mul_two]
_ ≤ x * x * k := Nat.mul_le_mul_left (x * x) k_lt_one
_ < (x * x + 1) * k := by linarith
· -- Show the descent step.
intro x y hx x_lt_y _ _ z h_root _ hV₀
constructor
· have hpos : z * z + x * x > 0 := by
apply add_pos_of_nonneg_of_pos
· apply mul_self_nonneg
· apply mul_pos <;> exact mod_cast hx
have hzx : z * z + x * x = (z * x + 1) * k := by
rw [← sub_eq_zero]
rw [← h_root]
ring
rw [hzx] at hpos
replace hpos : z * x + 1 > 0 := pos_of_mul_pos_left hpos (Int.ofNat_zero_le k)
replace hpos : z * x ≥ 0 := Int.le_of_lt_add_one hpos
apply nonneg_of_mul_nonneg_left hpos (mod_cast hx)
· contrapose! hV₀ with x_lt_z
apply ne_of_gt
calc
z * y > x * x := by apply mul_lt_mul' <;> omega
_ ≥ x * x - k := sub_le_self _ (Int.ofNat_zero_le k)
· -- There is no base case in this application of Vieta jumping.
simp
|
/root/DuelModelResearch/mathlib4/Archive/Imo/Imo1988Q6.lean
|
{
"open": [
"Imo1988Q6"
],
"variables": []
}
|
[
{
"line": "rcases h with ⟨k, hk⟩",
"before_state": "a b : ℕ\nh : a * b + 1 ∣ a ^ 2 + b ^ 2\n⊢ ∃ d, d ^ 2 = (a ^ 2 + b ^ 2) / (a * b + 1)",
"after_state": "case intro\na b k : ℕ\nhk : a ^ 2 + b ^ 2 = (a * b + 1) * k\n⊢ ∃ d, d ^ 2 = (a ^ 2 + b ^ 2) / (a * b + 1)"
},
{
"line": "rw [hk]",
"before_state": "case intro\na b k : ℕ\nhk : a ^ 2 + b ^ 2 = (a * b + 1) * k\n⊢ ∃ d, d ^ 2 = (a ^ 2 + b ^ 2) / (a * b + 1)",
"after_state": "case intro\na b k : ℕ\nhk : a ^ 2 + b ^ 2 = (a * b + 1) * k\n⊢ ∃ d, d ^ 2 = (a * b + 1) * k / (a * b + 1)"
},
{
"line": "rewrite [hk]",
"before_state": "case intro\na b k : ℕ\nhk : a ^ 2 + b ^ 2 = (a * b + 1) * k\n⊢ ∃ d, d ^ 2 = (a ^ 2 + b ^ 2) / (a * b + 1)",
"after_state": "case intro\na b k : ℕ\nhk : a ^ 2 + b ^ 2 = (a * b + 1) * k\n⊢ ∃ d, d ^ 2 = (a * b + 1) * k / (a * b + 1)"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "case intro\na b k : ℕ\nhk : a ^ 2 + b ^ 2 = (a * b + 1) * k\n⊢ ∃ d, d ^ 2 = (a * b + 1) * k / (a * b + 1)",
"after_state": "case intro\na b k : ℕ\nhk : a ^ 2 + b ^ 2 = (a * b + 1) * k\n⊢ ∃ d, d ^ 2 = (a * b + 1) * k / (a * b + 1)"
},
{
"line": "try (with_reducible rfl)",
"before_state": "case intro\na b k : ℕ\nhk : a ^ 2 + b ^ 2 = (a * b + 1) * k\n⊢ ∃ d, d ^ 2 = (a * b + 1) * k / (a * b + 1)",
"after_state": "case intro\na b k : ℕ\nhk : a ^ 2 + b ^ 2 = (a * b + 1) * k\n⊢ ∃ d, d ^ 2 = (a * b + 1) * k / (a * b + 1)"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "case intro\na b k : ℕ\nhk : a ^ 2 + b ^ 2 = (a * b + 1) * k\n⊢ ∃ d, d ^ 2 = (a * b + 1) * k / (a * b + 1)",
"after_state": "case intro\na b k : ℕ\nhk : a ^ 2 + b ^ 2 = (a * b + 1) * k\n⊢ ∃ d, d ^ 2 = (a * b + 1) * k / (a * b + 1)"
},
{
"line": "with_reducible rfl",
"before_state": "case intro\na b k : ℕ\nhk : a ^ 2 + b ^ 2 = (a * b + 1) * k\n⊢ ∃ d, d ^ 2 = (a * b + 1) * k / (a * b + 1)",
"after_state": "case intro\na b k : ℕ\nhk : a ^ 2 + b ^ 2 = (a * b + 1) * k\n⊢ ∃ d, d ^ 2 = (a * b + 1) * k / (a * b + 1)"
},
{
"line": "rfl",
"before_state": "case intro\na b k : ℕ\nhk : a ^ 2 + b ^ 2 = (a * b + 1) * k\n⊢ ∃ d, d ^ 2 = (a * b + 1) * k / (a * b + 1)",
"after_state": "case intro\na b k : ℕ\nhk : a ^ 2 + b ^ 2 = (a * b + 1) * k\n⊢ ∃ d, d ^ 2 = (a * b + 1) * k / (a * b + 1)"
},
{
"line": "apply_rfl",
"before_state": "case intro\na b k : ℕ\nhk : a ^ 2 + b ^ 2 = (a * b + 1) * k\n⊢ ∃ d, d ^ 2 = (a * b + 1) * k / (a * b + 1)",
"after_state": "case intro\na b k : ℕ\nhk : a ^ 2 + b ^ 2 = (a * b + 1) * k\n⊢ ∃ d, d ^ 2 = (a * b + 1) * k / (a * b + 1)"
},
{
"line": "skip",
"before_state": "case intro\na b k : ℕ\nhk : a ^ 2 + b ^ 2 = (a * b + 1) * k\n⊢ ∃ d, d ^ 2 = (a * b + 1) * k / (a * b + 1)",
"after_state": "case intro\na b k : ℕ\nhk : a ^ 2 + b ^ 2 = (a * b + 1) * k\n⊢ ∃ d, d ^ 2 = (a * b + 1) * k / (a * b + 1)"
},
{
"line": "rw [Nat.mul_div_cancel_left _ (Nat.succ_pos (a * b))]",
"before_state": "case intro\na b k : ℕ\nhk : a ^ 2 + b ^ 2 = (a * b + 1) * k\n⊢ ∃ d, d ^ 2 = (a * b + 1) * k / (a * b + 1)",
"after_state": "case intro\na b k : ℕ\nhk : a ^ 2 + b ^ 2 = (a * b + 1) * k\n⊢ ∃ d, d ^ 2 = k"
},
{
"line": "rewrite [Nat.mul_div_cancel_left _ (Nat.succ_pos (a * b))]",
"before_state": "case intro\na b k : ℕ\nhk : a ^ 2 + b ^ 2 = (a * b + 1) * k\n⊢ ∃ d, d ^ 2 = (a * b + 1) * k / (a * b + 1)",
"after_state": "case intro\na b k : ℕ\nhk : a ^ 2 + b ^ 2 = (a * b + 1) * k\n⊢ ∃ d, d ^ 2 = k"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "case intro\na b k : ℕ\nhk : a ^ 2 + b ^ 2 = (a * b + 1) * k\n⊢ ∃ d, d ^ 2 = k",
"after_state": "case intro\na b k : ℕ\nhk : a ^ 2 + b ^ 2 = (a * b + 1) * k\n⊢ ∃ d, d ^ 2 = k"
},
{
"line": "try (with_reducible rfl)",
"before_state": "case intro\na b k : ℕ\nhk : a ^ 2 + b ^ 2 = (a * b + 1) * k\n⊢ ∃ d, d ^ 2 = k",
"after_state": "case intro\na b k : ℕ\nhk : a ^ 2 + b ^ 2 = (a * b + 1) * k\n⊢ ∃ d, d ^ 2 = k"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "case intro\na b k : ℕ\nhk : a ^ 2 + b ^ 2 = (a * b + 1) * k\n⊢ ∃ d, d ^ 2 = k",
"after_state": "case intro\na b k : ℕ\nhk : a ^ 2 + b ^ 2 = (a * b + 1) * k\n⊢ ∃ d, d ^ 2 = k"
},
{
"line": "with_reducible rfl",
"before_state": "case intro\na b k : ℕ\nhk : a ^ 2 + b ^ 2 = (a * b + 1) * k\n⊢ ∃ d, d ^ 2 = k",
"after_state": "case intro\na b k : ℕ\nhk : a ^ 2 + b ^ 2 = (a * b + 1) * k\n⊢ ∃ d, d ^ 2 = k"
},
{
"line": "rfl",
"before_state": "case intro\na b k : ℕ\nhk : a ^ 2 + b ^ 2 = (a * b + 1) * k\n⊢ ∃ d, d ^ 2 = k",
"after_state": "case intro\na b k : ℕ\nhk : a ^ 2 + b ^ 2 = (a * b + 1) * k\n⊢ ∃ d, d ^ 2 = k"
},
{
"line": "apply_rfl",
"before_state": "case intro\na b k : ℕ\nhk : a ^ 2 + b ^ 2 = (a * b + 1) * k\n⊢ ∃ d, d ^ 2 = k",
"after_state": "case intro\na b k : ℕ\nhk : a ^ 2 + b ^ 2 = (a * b + 1) * k\n⊢ ∃ d, d ^ 2 = k"
},
{
"line": "skip",
"before_state": "case intro\na b k : ℕ\nhk : a ^ 2 + b ^ 2 = (a * b + 1) * k\n⊢ ∃ d, d ^ 2 = k",
"after_state": "case intro\na b k : ℕ\nhk : a ^ 2 + b ^ 2 = (a * b + 1) * k\n⊢ ∃ d, d ^ 2 = k"
},
{
"line": "simp only [sq] at hk",
"before_state": "case intro\na b k : ℕ\nhk : a ^ 2 + b ^ 2 = (a * b + 1) * k\n⊢ ∃ d, d ^ 2 = k",
"after_state": "case intro\na b k : ℕ\nhk : a * a + b * b = (a * b + 1) * k\n⊢ ∃ d, d ^ 2 = k"
},
{
"line": "focus\n apply\n constant_descent_vieta_jumping a b (H := fun a b => a * a + b * b = (a * b + 1) * k) hk (fun x => k * x)\n (fun x => x * x - k) fun _ _ => False\n with_annotate_state\"<;>\" skip\n all_goals clear hk a b",
"before_state": "case intro\na b k : ℕ\nhk : a * a + b * b = (a * b + 1) * k\n⊢ ∃ d, d ^ 2 = k",
"after_state": "No Goals!"
},
{
"line": "apply\n constant_descent_vieta_jumping a b (H := fun a b => a * a + b * b = (a * b + 1) * k) hk (fun x => k * x)\n (fun x => x * x - k) fun _ _ => False",
"before_state": "case intro\na b k : ℕ\nhk : a * a + b * b = (a * b + 1) * k\n⊢ ∃ d, d ^ 2 = k",
"after_state": "No Goals!"
}
] |
example {a b : ℕ} (h : a * b ∣ a ^ 2 + b ^ 2 + 1) : 3 * a * b = a ^ 2 + b ^ 2 + 1 := by
rcases h with ⟨k, hk⟩
suffices k = 3 by simp_all; ring
simp only [sq] at hk
apply constant_descent_vieta_jumping a b (H := fun a b => a * a + b * b + 1 = a * b * k)
hk (fun x => k * x) (fun x => x * x + 1) fun x _ => x ≤ 1 <;>
clear hk a b
· -- We will now show that the fibers of the solution set are described by a quadratic equation.
intro x y
rw [← Int.natCast_inj]
rw [← sub_eq_zero]
apply eq_iff_eq_cancel_right.2
simp; ring
· -- Show that the solution set is symmetric in a and b.
intro x y; ring_nf
· -- Show that the claim is true if b = 0.
simp
· -- Show that the claim is true if a = b.
intro x hx
have x_sq_dvd : x * x ∣ x * x * k := dvd_mul_right (x * x) k
rw [← hx] at x_sq_dvd
obtain ⟨y, hy⟩ : x * x ∣ 1 := by simpa only [Nat.dvd_add_self_left, add_assoc] using x_sq_dvd
obtain ⟨rfl, rfl⟩ : x = 1 ∧ y = 1 := by simpa [mul_eq_one] using hy.symm
simpa using hx.symm
· -- Show the descent step.
intro x y _ hx h_base _ z _ _ hV₀
constructor
· have zy_pos : z * y ≥ 0 := by rw [hV₀]; exact mod_cast Nat.zero_le _
apply nonneg_of_mul_nonneg_left zy_pos
omega
· contrapose! hV₀ with x_lt_z
apply ne_of_gt
push_neg at h_base
calc
z * y > x * y := by apply mul_lt_mul_of_pos_right <;> omega
_ ≥ x * (x + 1) := by apply mul_le_mul <;> omega
_ > x * x + 1 := by
rw [mul_add]
omega
· -- Show the base case.
intro x y h h_base
obtain rfl | rfl : x = 0 ∨ x = 1 := by rwa [Nat.le_add_one_iff, Nat.le_zero] at h_base
· simp at h
· rw [mul_one, one_mul, add_right_comm] at h
have y_dvd : y ∣ y * k := dvd_mul_right y k
rw [← h] at y_dvd
rw [Nat.dvd_add_left (dvd_mul_left y y)] at y_dvd
obtain rfl | rfl := (Nat.dvd_prime Nat.prime_two).mp y_dvd <;> apply mul_left_cancel₀
exacts [one_ne_zero, h.symm, two_ne_zero, h.symm]
|
/root/DuelModelResearch/mathlib4/Archive/Imo/Imo1988Q6.lean
|
{
"open": [
"Imo1988Q6"
],
"variables": []
}
|
[
{
"line": "rcases h with ⟨k, hk⟩",
"before_state": "a b : ℕ\nh : a * b ∣ a ^ 2 + b ^ 2 + 1\n⊢ 3 * a * b = a ^ 2 + b ^ 2 + 1",
"after_state": "case intro\na b k : ℕ\nhk : a ^ 2 + b ^ 2 + 1 = a * b * k\n⊢ 3 * a * b = a ^ 2 + b ^ 2 + 1"
},
{
"line": "suffices k = 3 by simp_all; ring",
"before_state": "case intro\na b k : ℕ\nhk : a ^ 2 + b ^ 2 + 1 = a * b * k\n⊢ 3 * a * b = a ^ 2 + b ^ 2 + 1",
"after_state": "case intro\na b k : ℕ\nhk : a ^ 2 + b ^ 2 + 1 = a * b * k\n⊢ k = 3"
},
{
"line": "refine_lift\n suffices k = 3 by simp_all; ring;\n ?_",
"before_state": "case intro\na b k : ℕ\nhk : a ^ 2 + b ^ 2 + 1 = a * b * k\n⊢ 3 * a * b = a ^ 2 + b ^ 2 + 1",
"after_state": "case intro\na b k : ℕ\nhk : a ^ 2 + b ^ 2 + 1 = a * b * k\n⊢ k = 3"
},
{
"line": "focus\n (refine\n no_implicit_lambda%\n (suffices k = 3 by simp_all; ring;\n ?_);\n rotate_right)",
"before_state": "case intro\na b k : ℕ\nhk : a ^ 2 + b ^ 2 + 1 = a * b * k\n⊢ 3 * a * b = a ^ 2 + b ^ 2 + 1",
"after_state": "case intro\na b k : ℕ\nhk : a ^ 2 + b ^ 2 + 1 = a * b * k\n⊢ k = 3"
},
{
"line": "refine\n no_implicit_lambda%\n (suffices k = 3 by simp_all; ring;\n ?_)",
"before_state": "case intro\na b k : ℕ\nhk : a ^ 2 + b ^ 2 + 1 = a * b * k\n⊢ 3 * a * b = a ^ 2 + b ^ 2 + 1",
"after_state": "case intro\na b k : ℕ\nhk : a ^ 2 + b ^ 2 + 1 = a * b * k\n⊢ k = 3"
},
{
"line": "simp_all",
"before_state": "a b k : ℕ\nhk : a ^ 2 + b ^ 2 + 1 = a * b * k\nthis : k = 3\n⊢ 3 * a * b = a ^ 2 + b ^ 2 + 1",
"after_state": "a b k : ℕ\nhk : a ^ 2 + b ^ 2 + 1 = a * b * 3\nthis : k = 3\n⊢ 3 * a * b = a * b * 3"
},
{
"line": "ring",
"before_state": "a b k : ℕ\nhk : a ^ 2 + b ^ 2 + 1 = a * b * 3\nthis : k = 3\n⊢ 3 * a * b = a * b * 3",
"after_state": "No Goals!"
},
{
"line": "first\n| ring1\n|\n try_this ring_nf\"\\n\\nThe `ring` tactic failed to close the goal. Use `ring_nf` to obtain a normal form.\n \\nNote that `ring` works primarily in *commutative* rings. \\\n If you have a noncommutative ring, abelian group or module, consider using \\\n `noncomm_ring`, `abel` or `module` instead.\"",
"before_state": "a b k : ℕ\nhk : a ^ 2 + b ^ 2 + 1 = a * b * 3\nthis : k = 3\n⊢ 3 * a * b = a * b * 3",
"after_state": "No Goals!"
},
{
"line": "ring1",
"before_state": "a b k : ℕ\nhk : a ^ 2 + b ^ 2 + 1 = a * b * 3\nthis : k = 3\n⊢ 3 * a * b = a * b * 3",
"after_state": "No Goals!"
},
{
"line": "rotate_right",
"before_state": "case intro\na b k : ℕ\nhk : a ^ 2 + b ^ 2 + 1 = a * b * k\n⊢ k = 3",
"after_state": "case intro\na b k : ℕ\nhk : a ^ 2 + b ^ 2 + 1 = a * b * k\n⊢ k = 3"
},
{
"line": "simp only [sq] at hk",
"before_state": "case intro\na b k : ℕ\nhk : a ^ 2 + b ^ 2 + 1 = a * b * k\n⊢ k = 3",
"after_state": "case intro\na b k : ℕ\nhk : a * a + b * b + 1 = a * b * k\n⊢ k = 3"
},
{
"line": "focus\n apply\n constant_descent_vieta_jumping a b (H := fun a b => a * a + b * b + 1 = a * b * k) hk (fun x => k * x)\n (fun x => x * x + 1) fun x _ => x ≤ 1\n with_annotate_state\"<;>\" skip\n all_goals clear hk a b",
"before_state": "case intro\na b k : ℕ\nhk : a * a + b * b + 1 = a * b * k\n⊢ k = 3",
"after_state": "No Goals!"
},
{
"line": "apply\n constant_descent_vieta_jumping a b (H := fun a b => a * a + b * b + 1 = a * b * k) hk (fun x => k * x)\n (fun x => x * x + 1) fun x _ => x ≤ 1",
"before_state": "case intro\na b k : ℕ\nhk : a * a + b * b + 1 = a * b * k\n⊢ k = 3",
"after_state": "No Goals!"
}
] |
theorem tedious (m : ℕ) (k : Fin (m + 1)) : m - ((m + 1 - ↑k) + m) % (m + 1) = ↑k := by
obtain ⟨k, hk⟩ := k
rw [Nat.lt_succ_iff] at hk
rw [le_iff_exists_add] at hk
rcases hk with ⟨c, rfl⟩
have : (k + c + 1 - k) + (k + c) = c + (k + c + 1) := by omega
rw [Fin.val_mk]
rw [this]
rw [Nat.add_mod_right]
rw [Nat.mod_eq_of_lt]
rw [Nat.add_sub_cancel]
omega
|
/root/DuelModelResearch/mathlib4/Archive/Imo/Imo1994Q1.lean
|
{
"open": [
"Finset"
],
"variables": []
}
|
[
{
"line": "obtain ⟨k, hk⟩ := k",
"before_state": "m : ℕ\nk : Fin (m + 1)\n⊢ m - (m + 1 - ↑k + m) % (m + 1) = ↑k",
"after_state": "case mk\nm k : ℕ\nhk : k < m + 1\n⊢ m - (m + 1 - ↑⟨k, hk⟩ + m) % (m + 1) = ↑⟨k, hk⟩"
},
{
"line": "rw [Nat.lt_succ_iff] at hk",
"before_state": "case mk\nm k : ℕ\nhk : k < m + 1\n⊢ m - (m + 1 - ↑⟨k, hk⟩ + m) % (m + 1) = ↑⟨k, hk⟩",
"after_state": "case mk\nm k : ℕ\nhk✝ : k < m + 1\nhk : k ≤ m\n⊢ m - (m + 1 - ↑⟨k, hk✝⟩ + m) % (m + 1) = ↑⟨k, hk✝⟩"
},
{
"line": "rewrite [Nat.lt_succ_iff] at hk",
"before_state": "case mk\nm k : ℕ\nhk : k < m + 1\n⊢ m - (m + 1 - ↑⟨k, hk⟩ + m) % (m + 1) = ↑⟨k, hk⟩",
"after_state": "case mk\nm k : ℕ\nhk✝ : k < m + 1\nhk : k ≤ m\n⊢ m - (m + 1 - ↑⟨k, hk✝⟩ + m) % (m + 1) = ↑⟨k, hk✝⟩"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "case mk\nm k : ℕ\nhk✝ : k < m + 1\nhk : k ≤ m\n⊢ m - (m + 1 - ↑⟨k, hk✝⟩ + m) % (m + 1) = ↑⟨k, hk✝⟩",
"after_state": "case mk\nm k : ℕ\nhk✝ : k < m + 1\nhk : k ≤ m\n⊢ m - (m + 1 - ↑⟨k, hk✝⟩ + m) % (m + 1) = ↑⟨k, hk✝⟩"
},
{
"line": "try (with_reducible rfl)",
"before_state": "case mk\nm k : ℕ\nhk✝ : k < m + 1\nhk : k ≤ m\n⊢ m - (m + 1 - ↑⟨k, hk✝⟩ + m) % (m + 1) = ↑⟨k, hk✝⟩",
"after_state": "case mk\nm k : ℕ\nhk✝ : k < m + 1\nhk : k ≤ m\n⊢ m - (m + 1 - ↑⟨k, hk✝⟩ + m) % (m + 1) = ↑⟨k, hk✝⟩"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "case mk\nm k : ℕ\nhk✝ : k < m + 1\nhk : k ≤ m\n⊢ m - (m + 1 - ↑⟨k, hk✝⟩ + m) % (m + 1) = ↑⟨k, hk✝⟩",
"after_state": "case mk\nm k : ℕ\nhk✝ : k < m + 1\nhk : k ≤ m\n⊢ m - (m + 1 - ↑⟨k, hk✝⟩ + m) % (m + 1) = ↑⟨k, hk✝⟩"
},
{
"line": "with_reducible rfl",
"before_state": "case mk\nm k : ℕ\nhk✝ : k < m + 1\nhk : k ≤ m\n⊢ m - (m + 1 - ↑⟨k, hk✝⟩ + m) % (m + 1) = ↑⟨k, hk✝⟩",
"after_state": "case mk\nm k : ℕ\nhk✝ : k < m + 1\nhk : k ≤ m\n⊢ m - (m + 1 - ↑⟨k, hk✝⟩ + m) % (m + 1) = ↑⟨k, hk✝⟩"
},
{
"line": "rfl",
"before_state": "case mk\nm k : ℕ\nhk✝ : k < m + 1\nhk : k ≤ m\n⊢ m - (m + 1 - ↑⟨k, hk✝⟩ + m) % (m + 1) = ↑⟨k, hk✝⟩",
"after_state": "case mk\nm k : ℕ\nhk✝ : k < m + 1\nhk : k ≤ m\n⊢ m - (m + 1 - ↑⟨k, hk✝⟩ + m) % (m + 1) = ↑⟨k, hk✝⟩"
},
{
"line": "apply_rfl",
"before_state": "case mk\nm k : ℕ\nhk✝ : k < m + 1\nhk : k ≤ m\n⊢ m - (m + 1 - ↑⟨k, hk✝⟩ + m) % (m + 1) = ↑⟨k, hk✝⟩",
"after_state": "case mk\nm k : ℕ\nhk✝ : k < m + 1\nhk : k ≤ m\n⊢ m - (m + 1 - ↑⟨k, hk✝⟩ + m) % (m + 1) = ↑⟨k, hk✝⟩"
},
{
"line": "skip",
"before_state": "case mk\nm k : ℕ\nhk✝ : k < m + 1\nhk : k ≤ m\n⊢ m - (m + 1 - ↑⟨k, hk✝⟩ + m) % (m + 1) = ↑⟨k, hk✝⟩",
"after_state": "case mk\nm k : ℕ\nhk✝ : k < m + 1\nhk : k ≤ m\n⊢ m - (m + 1 - ↑⟨k, hk✝⟩ + m) % (m + 1) = ↑⟨k, hk✝⟩"
},
{
"line": "rw [le_iff_exists_add] at hk",
"before_state": "case mk\nm k : ℕ\nhk✝ : k < m + 1\nhk : k ≤ m\n⊢ m - (m + 1 - ↑⟨k, hk✝⟩ + m) % (m + 1) = ↑⟨k, hk✝⟩",
"after_state": "case mk\nm k : ℕ\nhk✝ : k < m + 1\nhk : ∃ c, m = k + c\n⊢ m - (m + 1 - ↑⟨k, hk✝⟩ + m) % (m + 1) = ↑⟨k, hk✝⟩"
},
{
"line": "rewrite [le_iff_exists_add] at hk",
"before_state": "case mk\nm k : ℕ\nhk✝ : k < m + 1\nhk : k ≤ m\n⊢ m - (m + 1 - ↑⟨k, hk✝⟩ + m) % (m + 1) = ↑⟨k, hk✝⟩",
"after_state": "case mk\nm k : ℕ\nhk✝ : k < m + 1\nhk : ∃ c, m = k + c\n⊢ m - (m + 1 - ↑⟨k, hk✝⟩ + m) % (m + 1) = ↑⟨k, hk✝⟩"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "case mk\nm k : ℕ\nhk✝ : k < m + 1\nhk : ∃ c, m = k + c\n⊢ m - (m + 1 - ↑⟨k, hk✝⟩ + m) % (m + 1) = ↑⟨k, hk✝⟩",
"after_state": "case mk\nm k : ℕ\nhk✝ : k < m + 1\nhk : ∃ c, m = k + c\n⊢ m - (m + 1 - ↑⟨k, hk✝⟩ + m) % (m + 1) = ↑⟨k, hk✝⟩"
},
{
"line": "try (with_reducible rfl)",
"before_state": "case mk\nm k : ℕ\nhk✝ : k < m + 1\nhk : ∃ c, m = k + c\n⊢ m - (m + 1 - ↑⟨k, hk✝⟩ + m) % (m + 1) = ↑⟨k, hk✝⟩",
"after_state": "case mk\nm k : ℕ\nhk✝ : k < m + 1\nhk : ∃ c, m = k + c\n⊢ m - (m + 1 - ↑⟨k, hk✝⟩ + m) % (m + 1) = ↑⟨k, hk✝⟩"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "case mk\nm k : ℕ\nhk✝ : k < m + 1\nhk : ∃ c, m = k + c\n⊢ m - (m + 1 - ↑⟨k, hk✝⟩ + m) % (m + 1) = ↑⟨k, hk✝⟩",
"after_state": "case mk\nm k : ℕ\nhk✝ : k < m + 1\nhk : ∃ c, m = k + c\n⊢ m - (m + 1 - ↑⟨k, hk✝⟩ + m) % (m + 1) = ↑⟨k, hk✝⟩"
},
{
"line": "with_reducible rfl",
"before_state": "case mk\nm k : ℕ\nhk✝ : k < m + 1\nhk : ∃ c, m = k + c\n⊢ m - (m + 1 - ↑⟨k, hk✝⟩ + m) % (m + 1) = ↑⟨k, hk✝⟩",
"after_state": "case mk\nm k : ℕ\nhk✝ : k < m + 1\nhk : ∃ c, m = k + c\n⊢ m - (m + 1 - ↑⟨k, hk✝⟩ + m) % (m + 1) = ↑⟨k, hk✝⟩"
},
{
"line": "rfl",
"before_state": "case mk\nm k : ℕ\nhk✝ : k < m + 1\nhk : ∃ c, m = k + c\n⊢ m - (m + 1 - ↑⟨k, hk✝⟩ + m) % (m + 1) = ↑⟨k, hk✝⟩",
"after_state": "case mk\nm k : ℕ\nhk✝ : k < m + 1\nhk : ∃ c, m = k + c\n⊢ m - (m + 1 - ↑⟨k, hk✝⟩ + m) % (m + 1) = ↑⟨k, hk✝⟩"
},
{
"line": "apply_rfl",
"before_state": "case mk\nm k : ℕ\nhk✝ : k < m + 1\nhk : ∃ c, m = k + c\n⊢ m - (m + 1 - ↑⟨k, hk✝⟩ + m) % (m + 1) = ↑⟨k, hk✝⟩",
"after_state": "case mk\nm k : ℕ\nhk✝ : k < m + 1\nhk : ∃ c, m = k + c\n⊢ m - (m + 1 - ↑⟨k, hk✝⟩ + m) % (m + 1) = ↑⟨k, hk✝⟩"
},
{
"line": "skip",
"before_state": "case mk\nm k : ℕ\nhk✝ : k < m + 1\nhk : ∃ c, m = k + c\n⊢ m - (m + 1 - ↑⟨k, hk✝⟩ + m) % (m + 1) = ↑⟨k, hk✝⟩",
"after_state": "case mk\nm k : ℕ\nhk✝ : k < m + 1\nhk : ∃ c, m = k + c\n⊢ m - (m + 1 - ↑⟨k, hk✝⟩ + m) % (m + 1) = ↑⟨k, hk✝⟩"
},
{
"line": "rcases hk with ⟨c, rfl⟩",
"before_state": "case mk\nm k : ℕ\nhk✝ : k < m + 1\nhk : ∃ c, m = k + c\n⊢ m - (m + 1 - ↑⟨k, hk✝⟩ + m) % (m + 1) = ↑⟨k, hk✝⟩",
"after_state": "case mk.intro\nk c : ℕ\nhk : k < k + c + 1\n⊢ k + c - (k + c + 1 - ↑⟨k, hk⟩ + (k + c)) % (k + c + 1) = ↑⟨k, hk⟩"
},
{
"line": "have : (k + c + 1 - k) + (k + c) = c + (k + c + 1) := by omega",
"before_state": "case mk.intro\nk c : ℕ\nhk : k < k + c + 1\n⊢ k + c - (k + c + 1 - ↑⟨k, hk⟩ + (k + c)) % (k + c + 1) = ↑⟨k, hk⟩",
"after_state": "case mk.intro\nk c : ℕ\nhk : k < k + c + 1\nthis : k + c + 1 - k + (k + c) = c + (k + c + 1)\n⊢ k + c - (k + c + 1 - ↑⟨k, hk⟩ + (k + c)) % (k + c + 1) = ↑⟨k, hk⟩"
},
{
"line": "focus\n refine\n no_implicit_lambda%\n (have : (k + c + 1 - k) + (k + c) = c + (k + c + 1) := ?body✝;\n ?_)\n case body✝ => with_annotate_state\"by\" (omega)",
"before_state": "case mk.intro\nk c : ℕ\nhk : k < k + c + 1\n⊢ k + c - (k + c + 1 - ↑⟨k, hk⟩ + (k + c)) % (k + c + 1) = ↑⟨k, hk⟩",
"after_state": "case mk.intro\nk c : ℕ\nhk : k < k + c + 1\nthis : k + c + 1 - k + (k + c) = c + (k + c + 1)\n⊢ k + c - (k + c + 1 - ↑⟨k, hk⟩ + (k + c)) % (k + c + 1) = ↑⟨k, hk⟩"
},
{
"line": "refine\n no_implicit_lambda%\n (have : (k + c + 1 - k) + (k + c) = c + (k + c + 1) := ?body✝;\n ?_)",
"before_state": "case mk.intro\nk c : ℕ\nhk : k < k + c + 1\n⊢ k + c - (k + c + 1 - ↑⟨k, hk⟩ + (k + c)) % (k + c + 1) = ↑⟨k, hk⟩",
"after_state": "case body\nk c : ℕ\nhk : k < k + c + 1\n⊢ k + c + 1 - k + (k + c) = c + (k + c + 1)\n---\ncase mk.intro\nk c : ℕ\nhk : k < k + c + 1\nthis : k + c + 1 - k + (k + c) = c + (k + c + 1)\n⊢ k + c - (k + c + 1 - ↑⟨k, hk⟩ + (k + c)) % (k + c + 1) = ↑⟨k, hk⟩"
},
{
"line": "case body✝ => with_annotate_state\"by\" (omega)",
"before_state": "case body\nk c : ℕ\nhk : k < k + c + 1\n⊢ k + c + 1 - k + (k + c) = c + (k + c + 1)\n---\ncase mk.intro\nk c : ℕ\nhk : k < k + c + 1\nthis : k + c + 1 - k + (k + c) = c + (k + c + 1)\n⊢ k + c - (k + c + 1 - ↑⟨k, hk⟩ + (k + c)) % (k + c + 1) = ↑⟨k, hk⟩",
"after_state": "case mk.intro\nk c : ℕ\nhk : k < k + c + 1\nthis : k + c + 1 - k + (k + c) = c + (k + c + 1)\n⊢ k + c - (k + c + 1 - ↑⟨k, hk⟩ + (k + c)) % (k + c + 1) = ↑⟨k, hk⟩"
},
{
"line": "with_annotate_state\"by\" (omega)",
"before_state": "k c : ℕ\nhk : k < k + c + 1\n⊢ k + c + 1 - k + (k + c) = c + (k + c + 1)",
"after_state": "No Goals!"
},
{
"line": "omega",
"before_state": "k c : ℕ\nhk : k < k + c + 1\n⊢ k + c + 1 - k + (k + c) = c + (k + c + 1)",
"after_state": "No Goals!"
},
{
"line": "rw [Fin.val_mk]",
"before_state": "case mk.intro\nk c : ℕ\nhk : k < k + c + 1\nthis : k + c + 1 - k + (k + c) = c + (k + c + 1)\n⊢ k + c - (k + c + 1 - ↑⟨k, hk⟩ + (k + c)) % (k + c + 1) = ↑⟨k, hk⟩",
"after_state": "case mk.intro\nk c : ℕ\nhk : k < k + c + 1\nthis : k + c + 1 - k + (k + c) = c + (k + c + 1)\n⊢ k + c - (k + c + 1 - k + (k + c)) % (k + c + 1) = k"
},
{
"line": "rewrite [Fin.val_mk]",
"before_state": "case mk.intro\nk c : ℕ\nhk : k < k + c + 1\nthis : k + c + 1 - k + (k + c) = c + (k + c + 1)\n⊢ k + c - (k + c + 1 - ↑⟨k, hk⟩ + (k + c)) % (k + c + 1) = ↑⟨k, hk⟩",
"after_state": "case mk.intro\nk c : ℕ\nhk : k < k + c + 1\nthis : k + c + 1 - k + (k + c) = c + (k + c + 1)\n⊢ k + c - (k + c + 1 - k + (k + c)) % (k + c + 1) = k"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "case mk.intro\nk c : ℕ\nhk : k < k + c + 1\nthis : k + c + 1 - k + (k + c) = c + (k + c + 1)\n⊢ k + c - (k + c + 1 - k + (k + c)) % (k + c + 1) = k",
"after_state": "case mk.intro\nk c : ℕ\nhk : k < k + c + 1\nthis : k + c + 1 - k + (k + c) = c + (k + c + 1)\n⊢ k + c - (k + c + 1 - k + (k + c)) % (k + c + 1) = k"
},
{
"line": "try (with_reducible rfl)",
"before_state": "case mk.intro\nk c : ℕ\nhk : k < k + c + 1\nthis : k + c + 1 - k + (k + c) = c + (k + c + 1)\n⊢ k + c - (k + c + 1 - k + (k + c)) % (k + c + 1) = k",
"after_state": "case mk.intro\nk c : ℕ\nhk : k < k + c + 1\nthis : k + c + 1 - k + (k + c) = c + (k + c + 1)\n⊢ k + c - (k + c + 1 - k + (k + c)) % (k + c + 1) = k"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "case mk.intro\nk c : ℕ\nhk : k < k + c + 1\nthis : k + c + 1 - k + (k + c) = c + (k + c + 1)\n⊢ k + c - (k + c + 1 - k + (k + c)) % (k + c + 1) = k",
"after_state": "case mk.intro\nk c : ℕ\nhk : k < k + c + 1\nthis : k + c + 1 - k + (k + c) = c + (k + c + 1)\n⊢ k + c - (k + c + 1 - k + (k + c)) % (k + c + 1) = k"
},
{
"line": "with_reducible rfl",
"before_state": "case mk.intro\nk c : ℕ\nhk : k < k + c + 1\nthis : k + c + 1 - k + (k + c) = c + (k + c + 1)\n⊢ k + c - (k + c + 1 - k + (k + c)) % (k + c + 1) = k",
"after_state": "case mk.intro\nk c : ℕ\nhk : k < k + c + 1\nthis : k + c + 1 - k + (k + c) = c + (k + c + 1)\n⊢ k + c - (k + c + 1 - k + (k + c)) % (k + c + 1) = k"
},
{
"line": "rfl",
"before_state": "case mk.intro\nk c : ℕ\nhk : k < k + c + 1\nthis : k + c + 1 - k + (k + c) = c + (k + c + 1)\n⊢ k + c - (k + c + 1 - k + (k + c)) % (k + c + 1) = k",
"after_state": "case mk.intro\nk c : ℕ\nhk : k < k + c + 1\nthis : k + c + 1 - k + (k + c) = c + (k + c + 1)\n⊢ k + c - (k + c + 1 - k + (k + c)) % (k + c + 1) = k"
},
{
"line": "apply_rfl",
"before_state": "case mk.intro\nk c : ℕ\nhk : k < k + c + 1\nthis : k + c + 1 - k + (k + c) = c + (k + c + 1)\n⊢ k + c - (k + c + 1 - k + (k + c)) % (k + c + 1) = k",
"after_state": "case mk.intro\nk c : ℕ\nhk : k < k + c + 1\nthis : k + c + 1 - k + (k + c) = c + (k + c + 1)\n⊢ k + c - (k + c + 1 - k + (k + c)) % (k + c + 1) = k"
},
{
"line": "skip",
"before_state": "case mk.intro\nk c : ℕ\nhk : k < k + c + 1\nthis : k + c + 1 - k + (k + c) = c + (k + c + 1)\n⊢ k + c - (k + c + 1 - k + (k + c)) % (k + c + 1) = k",
"after_state": "case mk.intro\nk c : ℕ\nhk : k < k + c + 1\nthis : k + c + 1 - k + (k + c) = c + (k + c + 1)\n⊢ k + c - (k + c + 1 - k + (k + c)) % (k + c + 1) = k"
},
{
"line": "rw [this]",
"before_state": "case mk.intro\nk c : ℕ\nhk : k < k + c + 1\nthis : k + c + 1 - k + (k + c) = c + (k + c + 1)\n⊢ k + c - (k + c + 1 - k + (k + c)) % (k + c + 1) = k",
"after_state": "case mk.intro\nk c : ℕ\nhk : k < k + c + 1\nthis : k + c + 1 - k + (k + c) = c + (k + c + 1)\n⊢ k + c - (c + (k + c + 1)) % (k + c + 1) = k"
},
{
"line": "rewrite [this]",
"before_state": "case mk.intro\nk c : ℕ\nhk : k < k + c + 1\nthis : k + c + 1 - k + (k + c) = c + (k + c + 1)\n⊢ k + c - (k + c + 1 - k + (k + c)) % (k + c + 1) = k",
"after_state": "case mk.intro\nk c : ℕ\nhk : k < k + c + 1\nthis : k + c + 1 - k + (k + c) = c + (k + c + 1)\n⊢ k + c - (c + (k + c + 1)) % (k + c + 1) = k"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "case mk.intro\nk c : ℕ\nhk : k < k + c + 1\nthis : k + c + 1 - k + (k + c) = c + (k + c + 1)\n⊢ k + c - (c + (k + c + 1)) % (k + c + 1) = k",
"after_state": "case mk.intro\nk c : ℕ\nhk : k < k + c + 1\nthis : k + c + 1 - k + (k + c) = c + (k + c + 1)\n⊢ k + c - (c + (k + c + 1)) % (k + c + 1) = k"
},
{
"line": "try (with_reducible rfl)",
"before_state": "case mk.intro\nk c : ℕ\nhk : k < k + c + 1\nthis : k + c + 1 - k + (k + c) = c + (k + c + 1)\n⊢ k + c - (c + (k + c + 1)) % (k + c + 1) = k",
"after_state": "case mk.intro\nk c : ℕ\nhk : k < k + c + 1\nthis : k + c + 1 - k + (k + c) = c + (k + c + 1)\n⊢ k + c - (c + (k + c + 1)) % (k + c + 1) = k"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "case mk.intro\nk c : ℕ\nhk : k < k + c + 1\nthis : k + c + 1 - k + (k + c) = c + (k + c + 1)\n⊢ k + c - (c + (k + c + 1)) % (k + c + 1) = k",
"after_state": "case mk.intro\nk c : ℕ\nhk : k < k + c + 1\nthis : k + c + 1 - k + (k + c) = c + (k + c + 1)\n⊢ k + c - (c + (k + c + 1)) % (k + c + 1) = k"
},
{
"line": "with_reducible rfl",
"before_state": "case mk.intro\nk c : ℕ\nhk : k < k + c + 1\nthis : k + c + 1 - k + (k + c) = c + (k + c + 1)\n⊢ k + c - (c + (k + c + 1)) % (k + c + 1) = k",
"after_state": "case mk.intro\nk c : ℕ\nhk : k < k + c + 1\nthis : k + c + 1 - k + (k + c) = c + (k + c + 1)\n⊢ k + c - (c + (k + c + 1)) % (k + c + 1) = k"
},
{
"line": "rfl",
"before_state": "case mk.intro\nk c : ℕ\nhk : k < k + c + 1\nthis : k + c + 1 - k + (k + c) = c + (k + c + 1)\n⊢ k + c - (c + (k + c + 1)) % (k + c + 1) = k",
"after_state": "case mk.intro\nk c : ℕ\nhk : k < k + c + 1\nthis : k + c + 1 - k + (k + c) = c + (k + c + 1)\n⊢ k + c - (c + (k + c + 1)) % (k + c + 1) = k"
},
{
"line": "apply_rfl",
"before_state": "case mk.intro\nk c : ℕ\nhk : k < k + c + 1\nthis : k + c + 1 - k + (k + c) = c + (k + c + 1)\n⊢ k + c - (c + (k + c + 1)) % (k + c + 1) = k",
"after_state": "case mk.intro\nk c : ℕ\nhk : k < k + c + 1\nthis : k + c + 1 - k + (k + c) = c + (k + c + 1)\n⊢ k + c - (c + (k + c + 1)) % (k + c + 1) = k"
},
{
"line": "skip",
"before_state": "case mk.intro\nk c : ℕ\nhk : k < k + c + 1\nthis : k + c + 1 - k + (k + c) = c + (k + c + 1)\n⊢ k + c - (c + (k + c + 1)) % (k + c + 1) = k",
"after_state": "case mk.intro\nk c : ℕ\nhk : k < k + c + 1\nthis : k + c + 1 - k + (k + c) = c + (k + c + 1)\n⊢ k + c - (c + (k + c + 1)) % (k + c + 1) = k"
},
{
"line": "rw [Nat.add_mod_right]",
"before_state": "case mk.intro\nk c : ℕ\nhk : k < k + c + 1\nthis : k + c + 1 - k + (k + c) = c + (k + c + 1)\n⊢ k + c - (c + (k + c + 1)) % (k + c + 1) = k",
"after_state": "case mk.intro\nk c : ℕ\nhk : k < k + c + 1\nthis : k + c + 1 - k + (k + c) = c + (k + c + 1)\n⊢ k + c - c % (k + c + 1) = k"
},
{
"line": "rewrite [Nat.add_mod_right]",
"before_state": "case mk.intro\nk c : ℕ\nhk : k < k + c + 1\nthis : k + c + 1 - k + (k + c) = c + (k + c + 1)\n⊢ k + c - (c + (k + c + 1)) % (k + c + 1) = k",
"after_state": "case mk.intro\nk c : ℕ\nhk : k < k + c + 1\nthis : k + c + 1 - k + (k + c) = c + (k + c + 1)\n⊢ k + c - c % (k + c + 1) = k"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "case mk.intro\nk c : ℕ\nhk : k < k + c + 1\nthis : k + c + 1 - k + (k + c) = c + (k + c + 1)\n⊢ k + c - c % (k + c + 1) = k",
"after_state": "case mk.intro\nk c : ℕ\nhk : k < k + c + 1\nthis : k + c + 1 - k + (k + c) = c + (k + c + 1)\n⊢ k + c - c % (k + c + 1) = k"
},
{
"line": "try (with_reducible rfl)",
"before_state": "case mk.intro\nk c : ℕ\nhk : k < k + c + 1\nthis : k + c + 1 - k + (k + c) = c + (k + c + 1)\n⊢ k + c - c % (k + c + 1) = k",
"after_state": "case mk.intro\nk c : ℕ\nhk : k < k + c + 1\nthis : k + c + 1 - k + (k + c) = c + (k + c + 1)\n⊢ k + c - c % (k + c + 1) = k"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "case mk.intro\nk c : ℕ\nhk : k < k + c + 1\nthis : k + c + 1 - k + (k + c) = c + (k + c + 1)\n⊢ k + c - c % (k + c + 1) = k",
"after_state": "case mk.intro\nk c : ℕ\nhk : k < k + c + 1\nthis : k + c + 1 - k + (k + c) = c + (k + c + 1)\n⊢ k + c - c % (k + c + 1) = k"
},
{
"line": "with_reducible rfl",
"before_state": "case mk.intro\nk c : ℕ\nhk : k < k + c + 1\nthis : k + c + 1 - k + (k + c) = c + (k + c + 1)\n⊢ k + c - c % (k + c + 1) = k",
"after_state": "case mk.intro\nk c : ℕ\nhk : k < k + c + 1\nthis : k + c + 1 - k + (k + c) = c + (k + c + 1)\n⊢ k + c - c % (k + c + 1) = k"
},
{
"line": "rfl",
"before_state": "case mk.intro\nk c : ℕ\nhk : k < k + c + 1\nthis : k + c + 1 - k + (k + c) = c + (k + c + 1)\n⊢ k + c - c % (k + c + 1) = k",
"after_state": "case mk.intro\nk c : ℕ\nhk : k < k + c + 1\nthis : k + c + 1 - k + (k + c) = c + (k + c + 1)\n⊢ k + c - c % (k + c + 1) = k"
},
{
"line": "apply_rfl",
"before_state": "case mk.intro\nk c : ℕ\nhk : k < k + c + 1\nthis : k + c + 1 - k + (k + c) = c + (k + c + 1)\n⊢ k + c - c % (k + c + 1) = k",
"after_state": "case mk.intro\nk c : ℕ\nhk : k < k + c + 1\nthis : k + c + 1 - k + (k + c) = c + (k + c + 1)\n⊢ k + c - c % (k + c + 1) = k"
},
{
"line": "skip",
"before_state": "case mk.intro\nk c : ℕ\nhk : k < k + c + 1\nthis : k + c + 1 - k + (k + c) = c + (k + c + 1)\n⊢ k + c - c % (k + c + 1) = k",
"after_state": "case mk.intro\nk c : ℕ\nhk : k < k + c + 1\nthis : k + c + 1 - k + (k + c) = c + (k + c + 1)\n⊢ k + c - c % (k + c + 1) = k"
},
{
"line": "rw [Nat.mod_eq_of_lt]",
"before_state": "case mk.intro\nk c : ℕ\nhk : k < k + c + 1\nthis : k + c + 1 - k + (k + c) = c + (k + c + 1)\n⊢ k + c - c % (k + c + 1) = k",
"after_state": "case mk.intro\nk c : ℕ\nhk : k < k + c + 1\nthis : k + c + 1 - k + (k + c) = c + (k + c + 1)\n⊢ k + c - c = k\n---\ncase mk.intro\nk c : ℕ\nhk : k < k + c + 1\nthis : k + c + 1 - k + (k + c) = c + (k + c + 1)\n⊢ c < k + c + 1"
},
{
"line": "rewrite [Nat.mod_eq_of_lt]",
"before_state": "case mk.intro\nk c : ℕ\nhk : k < k + c + 1\nthis : k + c + 1 - k + (k + c) = c + (k + c + 1)\n⊢ k + c - c % (k + c + 1) = k",
"after_state": "case mk.intro\nk c : ℕ\nhk : k < k + c + 1\nthis : k + c + 1 - k + (k + c) = c + (k + c + 1)\n⊢ k + c - c = k\n---\ncase mk.intro\nk c : ℕ\nhk : k < k + c + 1\nthis : k + c + 1 - k + (k + c) = c + (k + c + 1)\n⊢ c < k + c + 1"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "case mk.intro\nk c : ℕ\nhk : k < k + c + 1\nthis : k + c + 1 - k + (k + c) = c + (k + c + 1)\n⊢ k + c - c = k\n---\ncase mk.intro\nk c : ℕ\nhk : k < k + c + 1\nthis : k + c + 1 - k + (k + c) = c + (k + c + 1)\n⊢ c < k + c + 1",
"after_state": "case mk.intro\nk c : ℕ\nhk : k < k + c + 1\nthis : k + c + 1 - k + (k + c) = c + (k + c + 1)\n⊢ k + c - c = k\n---\ncase mk.intro\nk c : ℕ\nhk : k < k + c + 1\nthis : k + c + 1 - k + (k + c) = c + (k + c + 1)\n⊢ c < k + c + 1"
},
{
"line": "try (with_reducible rfl)",
"before_state": "case mk.intro\nk c : ℕ\nhk : k < k + c + 1\nthis : k + c + 1 - k + (k + c) = c + (k + c + 1)\n⊢ k + c - c = k\n---\ncase mk.intro\nk c : ℕ\nhk : k < k + c + 1\nthis : k + c + 1 - k + (k + c) = c + (k + c + 1)\n⊢ c < k + c + 1",
"after_state": "case mk.intro\nk c : ℕ\nhk : k < k + c + 1\nthis : k + c + 1 - k + (k + c) = c + (k + c + 1)\n⊢ k + c - c = k\n---\ncase mk.intro\nk c : ℕ\nhk : k < k + c + 1\nthis : k + c + 1 - k + (k + c) = c + (k + c + 1)\n⊢ c < k + c + 1"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "case mk.intro\nk c : ℕ\nhk : k < k + c + 1\nthis : k + c + 1 - k + (k + c) = c + (k + c + 1)\n⊢ k + c - c = k\n---\ncase mk.intro\nk c : ℕ\nhk : k < k + c + 1\nthis : k + c + 1 - k + (k + c) = c + (k + c + 1)\n⊢ c < k + c + 1",
"after_state": "case mk.intro\nk c : ℕ\nhk : k < k + c + 1\nthis : k + c + 1 - k + (k + c) = c + (k + c + 1)\n⊢ k + c - c = k\n---\ncase mk.intro\nk c : ℕ\nhk : k < k + c + 1\nthis : k + c + 1 - k + (k + c) = c + (k + c + 1)\n⊢ c < k + c + 1"
},
{
"line": "with_reducible rfl",
"before_state": "case mk.intro\nk c : ℕ\nhk : k < k + c + 1\nthis : k + c + 1 - k + (k + c) = c + (k + c + 1)\n⊢ k + c - c = k\n---\ncase mk.intro\nk c : ℕ\nhk : k < k + c + 1\nthis : k + c + 1 - k + (k + c) = c + (k + c + 1)\n⊢ c < k + c + 1",
"after_state": "case mk.intro\nk c : ℕ\nhk : k < k + c + 1\nthis : k + c + 1 - k + (k + c) = c + (k + c + 1)\n⊢ k + c - c = k\n---\ncase mk.intro\nk c : ℕ\nhk : k < k + c + 1\nthis : k + c + 1 - k + (k + c) = c + (k + c + 1)\n⊢ c < k + c + 1"
},
{
"line": "rfl",
"before_state": "case mk.intro\nk c : ℕ\nhk : k < k + c + 1\nthis : k + c + 1 - k + (k + c) = c + (k + c + 1)\n⊢ k + c - c = k\n---\ncase mk.intro\nk c : ℕ\nhk : k < k + c + 1\nthis : k + c + 1 - k + (k + c) = c + (k + c + 1)\n⊢ c < k + c + 1",
"after_state": "case mk.intro\nk c : ℕ\nhk : k < k + c + 1\nthis : k + c + 1 - k + (k + c) = c + (k + c + 1)\n⊢ k + c - c = k\n---\ncase mk.intro\nk c : ℕ\nhk : k < k + c + 1\nthis : k + c + 1 - k + (k + c) = c + (k + c + 1)\n⊢ c < k + c + 1"
},
{
"line": "apply_rfl",
"before_state": "case mk.intro\nk c : ℕ\nhk : k < k + c + 1\nthis : k + c + 1 - k + (k + c) = c + (k + c + 1)\n⊢ k + c - c = k\n---\ncase mk.intro\nk c : ℕ\nhk : k < k + c + 1\nthis : k + c + 1 - k + (k + c) = c + (k + c + 1)\n⊢ c < k + c + 1",
"after_state": "case mk.intro\nk c : ℕ\nhk : k < k + c + 1\nthis : k + c + 1 - k + (k + c) = c + (k + c + 1)\n⊢ k + c - c = k\n---\ncase mk.intro\nk c : ℕ\nhk : k < k + c + 1\nthis : k + c + 1 - k + (k + c) = c + (k + c + 1)\n⊢ c < k + c + 1"
},
{
"line": "skip",
"before_state": "case mk.intro\nk c : ℕ\nhk : k < k + c + 1\nthis : k + c + 1 - k + (k + c) = c + (k + c + 1)\n⊢ k + c - c = k\n---\ncase mk.intro\nk c : ℕ\nhk : k < k + c + 1\nthis : k + c + 1 - k + (k + c) = c + (k + c + 1)\n⊢ c < k + c + 1",
"after_state": "case mk.intro\nk c : ℕ\nhk : k < k + c + 1\nthis : k + c + 1 - k + (k + c) = c + (k + c + 1)\n⊢ k + c - c = k\n---\ncase mk.intro\nk c : ℕ\nhk : k < k + c + 1\nthis : k + c + 1 - k + (k + c) = c + (k + c + 1)\n⊢ c < k + c + 1"
},
{
"line": "rw [Nat.add_sub_cancel]",
"before_state": "case mk.intro\nk c : ℕ\nhk : k < k + c + 1\nthis : k + c + 1 - k + (k + c) = c + (k + c + 1)\n⊢ k + c - c = k\n---\ncase mk.intro\nk c : ℕ\nhk : k < k + c + 1\nthis : k + c + 1 - k + (k + c) = c + (k + c + 1)\n⊢ c < k + c + 1",
"after_state": "case mk.intro\nk c : ℕ\nhk : k < k + c + 1\nthis : k + c + 1 - k + (k + c) = c + (k + c + 1)\n⊢ c < k + c + 1"
},
{
"line": "rewrite [Nat.add_sub_cancel]",
"before_state": "case mk.intro\nk c : ℕ\nhk : k < k + c + 1\nthis : k + c + 1 - k + (k + c) = c + (k + c + 1)\n⊢ k + c - c = k\n---\ncase mk.intro\nk c : ℕ\nhk : k < k + c + 1\nthis : k + c + 1 - k + (k + c) = c + (k + c + 1)\n⊢ c < k + c + 1",
"after_state": "case mk.intro\nk c : ℕ\nhk : k < k + c + 1\nthis : k + c + 1 - k + (k + c) = c + (k + c + 1)\n⊢ k = k\n---\ncase mk.intro\nk c : ℕ\nhk : k < k + c + 1\nthis : k + c + 1 - k + (k + c) = c + (k + c + 1)\n⊢ c < k + c + 1"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "case mk.intro\nk c : ℕ\nhk : k < k + c + 1\nthis : k + c + 1 - k + (k + c) = c + (k + c + 1)\n⊢ k = k\n---\ncase mk.intro\nk c : ℕ\nhk : k < k + c + 1\nthis : k + c + 1 - k + (k + c) = c + (k + c + 1)\n⊢ c < k + c + 1",
"after_state": "case mk.intro\nk c : ℕ\nhk : k < k + c + 1\nthis : k + c + 1 - k + (k + c) = c + (k + c + 1)\n⊢ c < k + c + 1"
},
{
"line": "try (with_reducible rfl)",
"before_state": "case mk.intro\nk c : ℕ\nhk : k < k + c + 1\nthis : k + c + 1 - k + (k + c) = c + (k + c + 1)\n⊢ k = k\n---\ncase mk.intro\nk c : ℕ\nhk : k < k + c + 1\nthis : k + c + 1 - k + (k + c) = c + (k + c + 1)\n⊢ c < k + c + 1",
"after_state": "case mk.intro\nk c : ℕ\nhk : k < k + c + 1\nthis : k + c + 1 - k + (k + c) = c + (k + c + 1)\n⊢ c < k + c + 1"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "case mk.intro\nk c : ℕ\nhk : k < k + c + 1\nthis : k + c + 1 - k + (k + c) = c + (k + c + 1)\n⊢ k = k\n---\ncase mk.intro\nk c : ℕ\nhk : k < k + c + 1\nthis : k + c + 1 - k + (k + c) = c + (k + c + 1)\n⊢ c < k + c + 1",
"after_state": "case mk.intro\nk c : ℕ\nhk : k < k + c + 1\nthis : k + c + 1 - k + (k + c) = c + (k + c + 1)\n⊢ c < k + c + 1"
},
{
"line": "with_reducible rfl",
"before_state": "case mk.intro\nk c : ℕ\nhk : k < k + c + 1\nthis : k + c + 1 - k + (k + c) = c + (k + c + 1)\n⊢ k = k\n---\ncase mk.intro\nk c : ℕ\nhk : k < k + c + 1\nthis : k + c + 1 - k + (k + c) = c + (k + c + 1)\n⊢ c < k + c + 1",
"after_state": "case mk.intro\nk c : ℕ\nhk : k < k + c + 1\nthis : k + c + 1 - k + (k + c) = c + (k + c + 1)\n⊢ c < k + c + 1"
},
{
"line": "rfl",
"before_state": "case mk.intro\nk c : ℕ\nhk : k < k + c + 1\nthis : k + c + 1 - k + (k + c) = c + (k + c + 1)\n⊢ k = k\n---\ncase mk.intro\nk c : ℕ\nhk : k < k + c + 1\nthis : k + c + 1 - k + (k + c) = c + (k + c + 1)\n⊢ c < k + c + 1",
"after_state": "case mk.intro\nk c : ℕ\nhk : k < k + c + 1\nthis : k + c + 1 - k + (k + c) = c + (k + c + 1)\n⊢ c < k + c + 1"
},
{
"line": "eq_refl",
"before_state": "case mk.intro\nk c : ℕ\nhk : k < k + c + 1\nthis : k + c + 1 - k + (k + c) = c + (k + c + 1)\n⊢ k = k\n---\ncase mk.intro\nk c : ℕ\nhk : k < k + c + 1\nthis : k + c + 1 - k + (k + c) = c + (k + c + 1)\n⊢ c < k + c + 1",
"after_state": "case mk.intro\nk c : ℕ\nhk : k < k + c + 1\nthis : k + c + 1 - k + (k + c) = c + (k + c + 1)\n⊢ c < k + c + 1"
},
{
"line": "omega",
"before_state": "case mk.intro\nk c : ℕ\nhk : k < k + c + 1\nthis : k + c + 1 - k + (k + c) = c + (k + c + 1)\n⊢ c < k + c + 1",
"after_state": "No Goals!"
}
] |
theorem add_sq_add_sq_sub {α : Type*} [Ring α] (x y : α) :
(x + y) * (x + y) + (x - y) * (x - y) = 2 * x * x + 2 * y * y := by noncomm_ring
|
/root/DuelModelResearch/mathlib4/Archive/Imo/Imo1998Q2.lean
|
{
"open": [
"scoped Classical in",
"scoped Classical in",
"scoped Classical in",
"scoped Classical in",
"scoped Classical in",
"scoped Classical in",
"scoped Classical in"
],
"variables": [
"{C J : Type*} (r : C → J → Prop)",
"[Fintype J] [Fintype C]"
]
}
|
[
{
"line": "noncomm_ring",
"before_state": "α : Type u_1\ninst✝ : Ring α\nx y : α\n⊢ (x + y) * (x + y) + (x - y) * (x - y) = 2 * x * x + 2 * y * y",
"after_state": "No Goals!"
},
{
"line": "focus\n (first\n |\n simp only [add_mul✝, mul_add✝, sub_eq_add_neg✝, mul_assoc✝, pow_one✝, pow_zero✝, pow_succ✝, one_mul✝, mul_one✝,\n zero_mul✝, mul_zero✝, nat_lit_mul_eq_nsmul✝, mul_nat_lit_eq_nsmul✝, mul_smul_comm✝, smul_mul_assoc✝, neg_mul✝,\n mul_neg✝, ]\n | fail \"`noncomm_ring` simp lemmas don't apply; try `abel` instead\")\n with_annotate_state\"<;>\" skip\n all_goals\n first\n | abel1\n | abel_nf",
"before_state": "α : Type u_1\ninst✝ : Ring α\nx y : α\n⊢ (x + y) * (x + y) + (x - y) * (x - y) = 2 * x * x + 2 * y * y",
"after_state": "No Goals!"
},
{
"line": "first\n|\n simp only [add_mul✝, mul_add✝, sub_eq_add_neg✝, mul_assoc✝, pow_one✝, pow_zero✝, pow_succ✝, one_mul✝, mul_one✝,\n zero_mul✝, mul_zero✝, nat_lit_mul_eq_nsmul✝, mul_nat_lit_eq_nsmul✝, mul_smul_comm✝, smul_mul_assoc✝, neg_mul✝,\n mul_neg✝, ]\n| fail \"`noncomm_ring` simp lemmas don't apply; try `abel` instead\"",
"before_state": "α : Type u_1\ninst✝ : Ring α\nx y : α\n⊢ (x + y) * (x + y) + (x - y) * (x - y) = 2 * x * x + 2 * y * y",
"after_state": "α : Type u_1\ninst✝ : Ring α\nx y : α\n⊢ x * x + y * x + (x * y + y * y) + (x * x + -(y * x) + -(x * y + -(y * y))) = 2 • (x * x) + 2 • (y * y)"
},
{
"line": "simp only [add_mul✝, mul_add✝, sub_eq_add_neg✝, mul_assoc✝, pow_one✝, pow_zero✝, pow_succ✝, one_mul✝, mul_one✝,\n zero_mul✝, mul_zero✝, nat_lit_mul_eq_nsmul✝, mul_nat_lit_eq_nsmul✝, mul_smul_comm✝, smul_mul_assoc✝, neg_mul✝,\n mul_neg✝, ]",
"before_state": "α : Type u_1\ninst✝ : Ring α\nx y : α\n⊢ (x + y) * (x + y) + (x - y) * (x - y) = 2 * x * x + 2 * y * y",
"after_state": "α : Type u_1\ninst✝ : Ring α\nx y : α\n⊢ x * x + y * x + (x * y + y * y) + (x * x + -(y * x) + -(x * y + -(y * y))) = 2 • (x * x) + 2 • (y * y)"
},
{
"line": "with_annotate_state\"<;>\" skip",
"before_state": "α : Type u_1\ninst✝ : Ring α\nx y : α\n⊢ x * x + y * x + (x * y + y * y) + (x * x + -(y * x) + -(x * y + -(y * y))) = 2 • (x * x) + 2 • (y * y)",
"after_state": "α : Type u_1\ninst✝ : Ring α\nx y : α\n⊢ x * x + y * x + (x * y + y * y) + (x * x + -(y * x) + -(x * y + -(y * y))) = 2 • (x * x) + 2 • (y * y)"
},
{
"line": "skip",
"before_state": "α : Type u_1\ninst✝ : Ring α\nx y : α\n⊢ x * x + y * x + (x * y + y * y) + (x * x + -(y * x) + -(x * y + -(y * y))) = 2 • (x * x) + 2 • (y * y)",
"after_state": "α : Type u_1\ninst✝ : Ring α\nx y : α\n⊢ x * x + y * x + (x * y + y * y) + (x * x + -(y * x) + -(x * y + -(y * y))) = 2 • (x * x) + 2 • (y * y)"
},
{
"line": "all_goals\n first\n | abel1\n | abel_nf",
"before_state": "α : Type u_1\ninst✝ : Ring α\nx y : α\n⊢ x * x + y * x + (x * y + y * y) + (x * x + -(y * x) + -(x * y + -(y * y))) = 2 • (x * x) + 2 • (y * y)",
"after_state": "No Goals!"
},
{
"line": "first\n| abel1\n| abel_nf",
"before_state": "α : Type u_1\ninst✝ : Ring α\nx y : α\n⊢ x * x + y * x + (x * y + y * y) + (x * x + -(y * x) + -(x * y + -(y * y))) = 2 • (x * x) + 2 • (y * y)",
"after_state": "No Goals!"
},
{
"line": "abel1",
"before_state": "α : Type u_1\ninst✝ : Ring α\nx y : α\n⊢ x * x + y * x + (x * y + y * y) + (x * x + -(y * x) + -(x * y + -(y * y))) = 2 • (x * x) + 2 • (y * y)",
"after_state": "No Goals!"
}
] |
theorem clear_denominators {a b k : ℕ} (ha : 0 < a) (hb : 0 < b) :
(b - 1 : ℚ) / (2 * b) ≤ k / a ↔ ((b : ℕ) - 1) * a ≤ k * (2 * b) := by
rw [div_le_div_iff₀]
on_goal 1 => convert Nat.cast_le (α := ℚ)
all_goals simp [ha, hb]
|
/root/DuelModelResearch/mathlib4/Archive/Imo/Imo1998Q2.lean
|
{
"open": [
"scoped Classical in",
"scoped Classical in",
"scoped Classical in",
"scoped Classical in",
"scoped Classical in",
"scoped Classical in",
"scoped Classical in",
"scoped Classical in",
"scoped Classical in"
],
"variables": [
"{C J : Type*} (r : C → J → Prop)",
"[Fintype J] [Fintype C]",
"[Fintype J]"
]
}
|
[
{
"line": "rw [div_le_div_iff₀]",
"before_state": "a b k : ℕ\nha : 0 < a\nhb : 0 < b\n⊢ (↑b - 1) / (2 * ↑b) ≤ ↑k / ↑a ↔ (b - 1) * a ≤ k * (2 * b)",
"after_state": "a b k : ℕ\nha : 0 < a\nhb : 0 < b\n⊢ (↑b - 1) * ↑a ≤ ↑k * (2 * ↑b) ↔ (b - 1) * a ≤ k * (2 * b)\n---\ncase hb\na b k : ℕ\nha : 0 < a\nhb : 0 < b\n⊢ 0 < 2 * ↑b\n---\ncase hd\na b k : ℕ\nha : 0 < a\nhb : 0 < b\n⊢ 0 < ↑a"
},
{
"line": "rewrite [div_le_div_iff₀]",
"before_state": "a b k : ℕ\nha : 0 < a\nhb : 0 < b\n⊢ (↑b - 1) / (2 * ↑b) ≤ ↑k / ↑a ↔ (b - 1) * a ≤ k * (2 * b)",
"after_state": "a b k : ℕ\nha : 0 < a\nhb : 0 < b\n⊢ (↑b - 1) * ↑a ≤ ↑k * (2 * ↑b) ↔ (b - 1) * a ≤ k * (2 * b)\n---\ncase hb\na b k : ℕ\nha : 0 < a\nhb : 0 < b\n⊢ 0 < 2 * ↑b\n---\ncase hd\na b k : ℕ\nha : 0 < a\nhb : 0 < b\n⊢ 0 < ↑a"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "a b k : ℕ\nha : 0 < a\nhb : 0 < b\n⊢ (↑b - 1) * ↑a ≤ ↑k * (2 * ↑b) ↔ (b - 1) * a ≤ k * (2 * b)\n---\ncase hb\na b k : ℕ\nha : 0 < a\nhb : 0 < b\n⊢ 0 < 2 * ↑b\n---\ncase hd\na b k : ℕ\nha : 0 < a\nhb : 0 < b\n⊢ 0 < ↑a",
"after_state": "a b k : ℕ\nha : 0 < a\nhb : 0 < b\n⊢ (↑b - 1) * ↑a ≤ ↑k * (2 * ↑b) ↔ (b - 1) * a ≤ k * (2 * b)\n---\ncase hb\na b k : ℕ\nha : 0 < a\nhb : 0 < b\n⊢ 0 < 2 * ↑b\n---\ncase hd\na b k : ℕ\nha : 0 < a\nhb : 0 < b\n⊢ 0 < ↑a"
},
{
"line": "try (with_reducible rfl)",
"before_state": "a b k : ℕ\nha : 0 < a\nhb : 0 < b\n⊢ (↑b - 1) * ↑a ≤ ↑k * (2 * ↑b) ↔ (b - 1) * a ≤ k * (2 * b)\n---\ncase hb\na b k : ℕ\nha : 0 < a\nhb : 0 < b\n⊢ 0 < 2 * ↑b\n---\ncase hd\na b k : ℕ\nha : 0 < a\nhb : 0 < b\n⊢ 0 < ↑a",
"after_state": "a b k : ℕ\nha : 0 < a\nhb : 0 < b\n⊢ (↑b - 1) * ↑a ≤ ↑k * (2 * ↑b) ↔ (b - 1) * a ≤ k * (2 * b)\n---\ncase hb\na b k : ℕ\nha : 0 < a\nhb : 0 < b\n⊢ 0 < 2 * ↑b\n---\ncase hd\na b k : ℕ\nha : 0 < a\nhb : 0 < b\n⊢ 0 < ↑a"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "a b k : ℕ\nha : 0 < a\nhb : 0 < b\n⊢ (↑b - 1) * ↑a ≤ ↑k * (2 * ↑b) ↔ (b - 1) * a ≤ k * (2 * b)\n---\ncase hb\na b k : ℕ\nha : 0 < a\nhb : 0 < b\n⊢ 0 < 2 * ↑b\n---\ncase hd\na b k : ℕ\nha : 0 < a\nhb : 0 < b\n⊢ 0 < ↑a",
"after_state": "a b k : ℕ\nha : 0 < a\nhb : 0 < b\n⊢ (↑b - 1) * ↑a ≤ ↑k * (2 * ↑b) ↔ (b - 1) * a ≤ k * (2 * b)\n---\ncase hb\na b k : ℕ\nha : 0 < a\nhb : 0 < b\n⊢ 0 < 2 * ↑b\n---\ncase hd\na b k : ℕ\nha : 0 < a\nhb : 0 < b\n⊢ 0 < ↑a"
},
{
"line": "with_reducible rfl",
"before_state": "a b k : ℕ\nha : 0 < a\nhb : 0 < b\n⊢ (↑b - 1) * ↑a ≤ ↑k * (2 * ↑b) ↔ (b - 1) * a ≤ k * (2 * b)\n---\ncase hb\na b k : ℕ\nha : 0 < a\nhb : 0 < b\n⊢ 0 < 2 * ↑b\n---\ncase hd\na b k : ℕ\nha : 0 < a\nhb : 0 < b\n⊢ 0 < ↑a",
"after_state": "a b k : ℕ\nha : 0 < a\nhb : 0 < b\n⊢ (↑b - 1) * ↑a ≤ ↑k * (2 * ↑b) ↔ (b - 1) * a ≤ k * (2 * b)\n---\ncase hb\na b k : ℕ\nha : 0 < a\nhb : 0 < b\n⊢ 0 < 2 * ↑b\n---\ncase hd\na b k : ℕ\nha : 0 < a\nhb : 0 < b\n⊢ 0 < ↑a"
},
{
"line": "rfl",
"before_state": "a b k : ℕ\nha : 0 < a\nhb : 0 < b\n⊢ (↑b - 1) * ↑a ≤ ↑k * (2 * ↑b) ↔ (b - 1) * a ≤ k * (2 * b)\n---\ncase hb\na b k : ℕ\nha : 0 < a\nhb : 0 < b\n⊢ 0 < 2 * ↑b\n---\ncase hd\na b k : ℕ\nha : 0 < a\nhb : 0 < b\n⊢ 0 < ↑a",
"after_state": "a b k : ℕ\nha : 0 < a\nhb : 0 < b\n⊢ (↑b - 1) * ↑a ≤ ↑k * (2 * ↑b) ↔ (b - 1) * a ≤ k * (2 * b)\n---\ncase hb\na b k : ℕ\nha : 0 < a\nhb : 0 < b\n⊢ 0 < 2 * ↑b\n---\ncase hd\na b k : ℕ\nha : 0 < a\nhb : 0 < b\n⊢ 0 < ↑a"
},
{
"line": "apply_rfl",
"before_state": "a b k : ℕ\nha : 0 < a\nhb : 0 < b\n⊢ (↑b - 1) * ↑a ≤ ↑k * (2 * ↑b) ↔ (b - 1) * a ≤ k * (2 * b)\n---\ncase hb\na b k : ℕ\nha : 0 < a\nhb : 0 < b\n⊢ 0 < 2 * ↑b\n---\ncase hd\na b k : ℕ\nha : 0 < a\nhb : 0 < b\n⊢ 0 < ↑a",
"after_state": "a b k : ℕ\nha : 0 < a\nhb : 0 < b\n⊢ (↑b - 1) * ↑a ≤ ↑k * (2 * ↑b) ↔ (b - 1) * a ≤ k * (2 * b)\n---\ncase hb\na b k : ℕ\nha : 0 < a\nhb : 0 < b\n⊢ 0 < 2 * ↑b\n---\ncase hd\na b k : ℕ\nha : 0 < a\nhb : 0 < b\n⊢ 0 < ↑a"
},
{
"line": "skip",
"before_state": "a b k : ℕ\nha : 0 < a\nhb : 0 < b\n⊢ (↑b - 1) * ↑a ≤ ↑k * (2 * ↑b) ↔ (b - 1) * a ≤ k * (2 * b)\n---\ncase hb\na b k : ℕ\nha : 0 < a\nhb : 0 < b\n⊢ 0 < 2 * ↑b\n---\ncase hd\na b k : ℕ\nha : 0 < a\nhb : 0 < b\n⊢ 0 < ↑a",
"after_state": "a b k : ℕ\nha : 0 < a\nhb : 0 < b\n⊢ (↑b - 1) * ↑a ≤ ↑k * (2 * ↑b) ↔ (b - 1) * a ≤ k * (2 * b)\n---\ncase hb\na b k : ℕ\nha : 0 < a\nhb : 0 < b\n⊢ 0 < 2 * ↑b\n---\ncase hd\na b k : ℕ\nha : 0 < a\nhb : 0 < b\n⊢ 0 < ↑a"
},
{
"line": "on_goal 1 => convert Nat.cast_le (α := ℚ)",
"before_state": "a b k : ℕ\nha : 0 < a\nhb : 0 < b\n⊢ (↑b - 1) * ↑a ≤ ↑k * (2 * ↑b) ↔ (b - 1) * a ≤ k * (2 * b)\n---\ncase hb\na b k : ℕ\nha : 0 < a\nhb : 0 < b\n⊢ 0 < 2 * ↑b\n---\ncase hd\na b k : ℕ\nha : 0 < a\nhb : 0 < b\n⊢ 0 < ↑a",
"after_state": "case h.e'_1.h.e'_3\na b k : ℕ\nha : 0 < a\nhb : 0 < b\n⊢ (↑b - 1) * ↑a = ↑((b - 1) * a)\n---\ncase h.e'_1.h.e'_4\na b k : ℕ\nha : 0 < a\nhb : 0 < b\n⊢ ↑k * (2 * ↑b) = ↑(k * (2 * b))\n---\ncase hb\na b k : ℕ\nha : 0 < a\nhb : 0 < b\n⊢ 0 < 2 * ↑b\n---\ncase hd\na b k : ℕ\nha : 0 < a\nhb : 0 < b\n⊢ 0 < ↑a"
},
{
"line": "convert Nat.cast_le (α := ℚ)",
"before_state": "a b k : ℕ\nha : 0 < a\nhb : 0 < b\n⊢ (↑b - 1) * ↑a ≤ ↑k * (2 * ↑b) ↔ (b - 1) * a ≤ k * (2 * b)",
"after_state": "case h.e'_1.h.e'_3\na b k : ℕ\nha : 0 < a\nhb : 0 < b\n⊢ (↑b - 1) * ↑a = ↑((b - 1) * a)\n---\ncase h.e'_1.h.e'_4\na b k : ℕ\nha : 0 < a\nhb : 0 < b\n⊢ ↑k * (2 * ↑b) = ↑(k * (2 * b))"
},
{
"line": "all_goals simp [ha, hb]",
"before_state": "case h.e'_1.h.e'_3\na b k : ℕ\nha : 0 < a\nhb : 0 < b\n⊢ (↑b - 1) * ↑a = ↑((b - 1) * a)\n---\ncase h.e'_1.h.e'_4\na b k : ℕ\nha : 0 < a\nhb : 0 < b\n⊢ ↑k * (2 * ↑b) = ↑(k * (2 * b))\n---\ncase hb\na b k : ℕ\nha : 0 < a\nhb : 0 < b\n⊢ 0 < 2 * ↑b\n---\ncase hd\na b k : ℕ\nha : 0 < a\nhb : 0 < b\n⊢ 0 < ↑a",
"after_state": "No Goals!"
},
{
"line": "simp [ha, hb]",
"before_state": "case h.e'_1.h.e'_3\na b k : ℕ\nha : 0 < a\nhb : 0 < b\n⊢ (↑b - 1) * ↑a = ↑((b - 1) * a)",
"after_state": "No Goals!"
},
{
"line": "simp [ha, hb]",
"before_state": "case h.e'_1.h.e'_4\na b k : ℕ\nha : 0 < a\nhb : 0 < b\n⊢ ↑k * (2 * ↑b) = ↑(k * (2 * b))",
"after_state": "No Goals!"
},
{
"line": "simp [ha, hb]",
"before_state": "case hb\na b k : ℕ\nha : 0 < a\nhb : 0 < b\n⊢ 0 < 2 * ↑b",
"after_state": "No Goals!"
},
{
"line": "simp [ha, hb]",
"before_state": "case hd\na b k : ℕ\nha : 0 < a\nhb : 0 < b\n⊢ 0 < ↑a",
"after_state": "No Goals!"
}
] |
theorem Int.natAbs_eq_of_chain_dvd {l : Cycle ℤ} {x y : ℤ} (hl : l.Chain (· ∣ ·)) (hx : x ∈ l)
(hy : y ∈ l) : x.natAbs = y.natAbs := by
rw [Cycle.chain_iff_pairwise] at hl
exact Int.natAbs_eq_of_dvd_dvd (hl x hx y hy) (hl y hy x hx)
|
/root/DuelModelResearch/mathlib4/Archive/Imo/Imo2006Q5.lean
|
{
"open": [
"Function Polynomial"
],
"variables": []
}
|
[
{
"line": "rw [Cycle.chain_iff_pairwise] at hl",
"before_state": "l : Cycle ℤ\nx y : ℤ\nhl : Cycle.Chain (fun x1 x2 => x1 ∣ x2) l\nhx : x ∈ l\nhy : y ∈ l\n⊢ x.natAbs = y.natAbs",
"after_state": "l : Cycle ℤ\nx y : ℤ\nhl : ∀ a ∈ l, ∀ b ∈ l, a ∣ b\nhx : x ∈ l\nhy : y ∈ l\n⊢ x.natAbs = y.natAbs"
},
{
"line": "rewrite [Cycle.chain_iff_pairwise] at hl",
"before_state": "l : Cycle ℤ\nx y : ℤ\nhl : Cycle.Chain (fun x1 x2 => x1 ∣ x2) l\nhx : x ∈ l\nhy : y ∈ l\n⊢ x.natAbs = y.natAbs",
"after_state": "l : Cycle ℤ\nx y : ℤ\nhl : ∀ a ∈ l, ∀ b ∈ l, a ∣ b\nhx : x ∈ l\nhy : y ∈ l\n⊢ x.natAbs = y.natAbs"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "l : Cycle ℤ\nx y : ℤ\nhl : ∀ a ∈ l, ∀ b ∈ l, a ∣ b\nhx : x ∈ l\nhy : y ∈ l\n⊢ x.natAbs = y.natAbs",
"after_state": "l : Cycle ℤ\nx y : ℤ\nhl : ∀ a ∈ l, ∀ b ∈ l, a ∣ b\nhx : x ∈ l\nhy : y ∈ l\n⊢ x.natAbs = y.natAbs"
},
{
"line": "try (with_reducible rfl)",
"before_state": "l : Cycle ℤ\nx y : ℤ\nhl : ∀ a ∈ l, ∀ b ∈ l, a ∣ b\nhx : x ∈ l\nhy : y ∈ l\n⊢ x.natAbs = y.natAbs",
"after_state": "l : Cycle ℤ\nx y : ℤ\nhl : ∀ a ∈ l, ∀ b ∈ l, a ∣ b\nhx : x ∈ l\nhy : y ∈ l\n⊢ x.natAbs = y.natAbs"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "l : Cycle ℤ\nx y : ℤ\nhl : ∀ a ∈ l, ∀ b ∈ l, a ∣ b\nhx : x ∈ l\nhy : y ∈ l\n⊢ x.natAbs = y.natAbs",
"after_state": "l : Cycle ℤ\nx y : ℤ\nhl : ∀ a ∈ l, ∀ b ∈ l, a ∣ b\nhx : x ∈ l\nhy : y ∈ l\n⊢ x.natAbs = y.natAbs"
},
{
"line": "with_reducible rfl",
"before_state": "l : Cycle ℤ\nx y : ℤ\nhl : ∀ a ∈ l, ∀ b ∈ l, a ∣ b\nhx : x ∈ l\nhy : y ∈ l\n⊢ x.natAbs = y.natAbs",
"after_state": "l : Cycle ℤ\nx y : ℤ\nhl : ∀ a ∈ l, ∀ b ∈ l, a ∣ b\nhx : x ∈ l\nhy : y ∈ l\n⊢ x.natAbs = y.natAbs"
},
{
"line": "rfl",
"before_state": "l : Cycle ℤ\nx y : ℤ\nhl : ∀ a ∈ l, ∀ b ∈ l, a ∣ b\nhx : x ∈ l\nhy : y ∈ l\n⊢ x.natAbs = y.natAbs",
"after_state": "l : Cycle ℤ\nx y : ℤ\nhl : ∀ a ∈ l, ∀ b ∈ l, a ∣ b\nhx : x ∈ l\nhy : y ∈ l\n⊢ x.natAbs = y.natAbs"
},
{
"line": "apply_rfl",
"before_state": "l : Cycle ℤ\nx y : ℤ\nhl : ∀ a ∈ l, ∀ b ∈ l, a ∣ b\nhx : x ∈ l\nhy : y ∈ l\n⊢ x.natAbs = y.natAbs",
"after_state": "l : Cycle ℤ\nx y : ℤ\nhl : ∀ a ∈ l, ∀ b ∈ l, a ∣ b\nhx : x ∈ l\nhy : y ∈ l\n⊢ x.natAbs = y.natAbs"
},
{
"line": "skip",
"before_state": "l : Cycle ℤ\nx y : ℤ\nhl : ∀ a ∈ l, ∀ b ∈ l, a ∣ b\nhx : x ∈ l\nhy : y ∈ l\n⊢ x.natAbs = y.natAbs",
"after_state": "l : Cycle ℤ\nx y : ℤ\nhl : ∀ a ∈ l, ∀ b ∈ l, a ∣ b\nhx : x ∈ l\nhy : y ∈ l\n⊢ x.natAbs = y.natAbs"
},
{
"line": "exact Int.natAbs_eq_of_dvd_dvd (hl x hx y hy) (hl y hy x hx)",
"before_state": "l : Cycle ℤ\nx y : ℤ\nhl : ∀ a ∈ l, ∀ b ∈ l, a ∣ b\nhx : x ∈ l\nhy : y ∈ l\n⊢ x.natAbs = y.natAbs",
"after_state": "No Goals!"
}
] |
theorem Int.add_eq_add_of_natAbs_eq_of_natAbs_eq {a b c d : ℤ} (hne : a ≠ b)
(h₁ : (c - a).natAbs = (d - b).natAbs) (h₂ : (c - b).natAbs = (d - a).natAbs) :
a + b = c + d := by
rcases Int.natAbs_eq_natAbs_iff.1 h₁ with h₁ | h₁
· rcases Int.natAbs_eq_natAbs_iff.1 h₂ with h₂ | h₂
· exact (hne <| by linarith).elim
· linarith
· linarith
|
/root/DuelModelResearch/mathlib4/Archive/Imo/Imo2006Q5.lean
|
{
"open": [
"Function Polynomial"
],
"variables": []
}
|
[
{
"line": "rcases Int.natAbs_eq_natAbs_iff.1 h₁ with h₁ | h₁",
"before_state": "a b c d : ℤ\nhne : a ≠ b\nh₁ : (c - a).natAbs = (d - b).natAbs\nh₂ : (c - b).natAbs = (d - a).natAbs\n⊢ a + b = c + d",
"after_state": "case inl\na b c d : ℤ\nhne : a ≠ b\nh₁✝ : (c - a).natAbs = (d - b).natAbs\nh₂ : (c - b).natAbs = (d - a).natAbs\nh₁ : c - a = d - b\n⊢ a + b = c + d\n---\ncase inr\na b c d : ℤ\nhne : a ≠ b\nh₁✝ : (c - a).natAbs = (d - b).natAbs\nh₂ : (c - b).natAbs = (d - a).natAbs\nh₁ : c - a = -(d - b)\n⊢ a + b = c + d"
},
{
"line": "rcases Int.natAbs_eq_natAbs_iff.1 h₂ with h₂ | h₂",
"before_state": "case inl\na b c d : ℤ\nhne : a ≠ b\nh₁✝ : (c - a).natAbs = (d - b).natAbs\nh₂ : (c - b).natAbs = (d - a).natAbs\nh₁ : c - a = d - b\n⊢ a + b = c + d",
"after_state": "case inl.inl\na b c d : ℤ\nhne : a ≠ b\nh₁✝ : (c - a).natAbs = (d - b).natAbs\nh₂✝ : (c - b).natAbs = (d - a).natAbs\nh₁ : c - a = d - b\nh₂ : c - b = d - a\n⊢ a + b = c + d\n---\ncase inl.inr\na b c d : ℤ\nhne : a ≠ b\nh₁✝ : (c - a).natAbs = (d - b).natAbs\nh₂✝ : (c - b).natAbs = (d - a).natAbs\nh₁ : c - a = d - b\nh₂ : c - b = -(d - a)\n⊢ a + b = c + d"
},
{
"line": "exact (hne <| by linarith).elim",
"before_state": "case inl.inl\na b c d : ℤ\nhne : a ≠ b\nh₁✝ : (c - a).natAbs = (d - b).natAbs\nh₂✝ : (c - b).natAbs = (d - a).natAbs\nh₁ : c - a = d - b\nh₂ : c - b = d - a\n⊢ a + b = c + d",
"after_state": "No Goals!"
},
{
"line": "linarith",
"before_state": "a b c d : ℤ\nhne : a ≠ b\nh₁✝ : (c - a).natAbs = (d - b).natAbs\nh₂✝ : (c - b).natAbs = (d - a).natAbs\nh₁ : c - a = d - b\nh₂ : c - b = d - a\n⊢ a = b",
"after_state": "No Goals!"
},
{
"line": "ring1",
"before_state": "a b c d : ℤ\nhne : a ≠ b\nh₁✝ : (c - a).natAbs = (d - b).natAbs\nh₂✝ : (c - b).natAbs = (d - a).natAbs\nh₁ : c - a = d - b\nh₂ : c - b = d - a\na✝ : a < b\n⊢ 2 * -1 + (c - a - (d - b)) + -(c - b - (d - a)) + 2 * (a + 1 - b) = 0",
"after_state": "No Goals!"
},
{
"line": "norm_num",
"before_state": "a b c d : ℤ\nhne : a ≠ b\nh₁✝ : (c - a).natAbs = (d - b).natAbs\nh₂✝ : (c - b).natAbs = (d - a).natAbs\nh₁ : c - a = d - b\nh₂ : c - b = d - a\na✝ : a < b\n⊢ 2 > 0",
"after_state": "No Goals!"
},
{
"line": "norm_num",
"before_state": "a b c d : ℤ\nhne : a ≠ b\nh₁✝ : (c - a).natAbs = (d - b).natAbs\nh₂✝ : (c - b).natAbs = (d - a).natAbs\nh₁ : c - a = d - b\nh₂ : c - b = d - a\na✝ : a < b\n⊢ 2 > 0",
"after_state": "No Goals!"
},
{
"line": "ring1",
"before_state": "a b c d : ℤ\nhne : a ≠ b\nh₁✝ : (c - a).natAbs = (d - b).natAbs\nh₂✝ : (c - b).natAbs = (d - a).natAbs\nh₁ : c - a = d - b\nh₂ : c - b = d - a\na✝ : b < a\n⊢ 2 * -1 + -(c - a - (d - b)) + (c - b - (d - a)) + 2 * (b + 1 - a) = 0",
"after_state": "No Goals!"
},
{
"line": "norm_num",
"before_state": "a b c d : ℤ\nhne : a ≠ b\nh₁✝ : (c - a).natAbs = (d - b).natAbs\nh₂✝ : (c - b).natAbs = (d - a).natAbs\nh₁ : c - a = d - b\nh₂ : c - b = d - a\na✝ : b < a\n⊢ 2 > 0",
"after_state": "No Goals!"
},
{
"line": "norm_num",
"before_state": "a b c d : ℤ\nhne : a ≠ b\nh₁✝ : (c - a).natAbs = (d - b).natAbs\nh₂✝ : (c - b).natAbs = (d - a).natAbs\nh₁ : c - a = d - b\nh₂ : c - b = d - a\na✝ : b < a\n⊢ 2 > 0",
"after_state": "No Goals!"
},
{
"line": "linarith",
"before_state": "case inl.inr\na b c d : ℤ\nhne : a ≠ b\nh₁✝ : (c - a).natAbs = (d - b).natAbs\nh₂✝ : (c - b).natAbs = (d - a).natAbs\nh₁ : c - a = d - b\nh₂ : c - b = -(d - a)\n⊢ a + b = c + d",
"after_state": "No Goals!"
},
{
"line": "ring1",
"before_state": "a b c d : ℤ\nhne : a ≠ b\nh₁✝ : (c - a).natAbs = (d - b).natAbs\nh₂✝ : (c - b).natAbs = (d - a).natAbs\nh₁ : c - a = d - b\nh₂ : c - b = -(d - a)\na✝ : a + b < c + d\n⊢ -1 + (c - b - -(d - a)) + (a + b + 1 - (c + d)) = 0",
"after_state": "No Goals!"
},
{
"line": "ring1",
"before_state": "a b c d : ℤ\nhne : a ≠ b\nh₁✝ : (c - a).natAbs = (d - b).natAbs\nh₂✝ : (c - b).natAbs = (d - a).natAbs\nh₁ : c - a = d - b\nh₂ : c - b = -(d - a)\na✝ : c + d < a + b\n⊢ -1 + -(c - b - -(d - a)) + (c + d + 1 - (a + b)) = 0",
"after_state": "No Goals!"
},
{
"line": "linarith",
"before_state": "case inr\na b c d : ℤ\nhne : a ≠ b\nh₁✝ : (c - a).natAbs = (d - b).natAbs\nh₂ : (c - b).natAbs = (d - a).natAbs\nh₁ : c - a = -(d - b)\n⊢ a + b = c + d",
"after_state": "No Goals!"
},
{
"line": "ring1",
"before_state": "a b c d : ℤ\nhne : a ≠ b\nh₁✝ : (c - a).natAbs = (d - b).natAbs\nh₂ : (c - b).natAbs = (d - a).natAbs\nh₁ : c - a = -(d - b)\na✝ : a + b < c + d\n⊢ -1 + (c - a - -(d - b)) + (a + b + 1 - (c + d)) = 0",
"after_state": "No Goals!"
},
{
"line": "ring1",
"before_state": "a b c d : ℤ\nhne : a ≠ b\nh₁✝ : (c - a).natAbs = (d - b).natAbs\nh₂ : (c - b).natAbs = (d - a).natAbs\nh₁ : c - a = -(d - b)\na✝ : c + d < a + b\n⊢ -1 + -(c - a - -(d - b)) + (c + d + 1 - (a + b)) = 0",
"after_state": "No Goals!"
}
] |
theorem Polynomial.isPeriodicPt_eval_two {P : Polynomial ℤ} {t : ℤ}
(ht : t ∈ periodicPts fun x => P.eval x) : IsPeriodicPt (fun x => P.eval x) 2 t := by
-- The cycle [P(t) - t, P(P(t)) - P(t), ...]
let C : Cycle ℤ := (periodicOrbit (fun x => P.eval x) t).map fun x => P.eval x - x
have HC : ∀ {n : ℕ}, (fun x => P.eval x)^[n + 1] t - (fun x => P.eval x)^[n] t ∈ C := by
intro n
rw [Cycle.mem_map]
rw [Function.iterate_succ_apply']
exact ⟨_, iterate_mem_periodicOrbit ht n, rfl⟩
-- Elements in C are all divisible by one another.
have Hdvd : C.Chain (· ∣ ·) := by
rw [Cycle.chain_map]
rw [periodicOrbit_chain' _ ht]
intro n
convert sub_dvd_eval_sub ((fun x => P.eval x)^[n + 1] t) ((fun x => P.eval x)^[n] t) P <;>
rw [Function.iterate_succ_apply']
-- Any two entries in C have the same absolute value.
have Habs :
∀ m n : ℕ,
((fun x => P.eval x)^[m + 1] t - (fun x => P.eval x)^[m] t).natAbs =
((fun x => P.eval x)^[n + 1] t - (fun x => P.eval x)^[n] t).natAbs :=
fun m n => Int.natAbs_eq_of_chain_dvd Hdvd HC HC
-- We case on whether the elements on C are pairwise equal.
by_cases HC' : C.Chain (· = ·)
· -- Any two entries in C are equal.
have Heq :
∀ m n : ℕ,
(fun x => P.eval x)^[m + 1] t - (fun x => P.eval x)^[m] t =
(fun x => P.eval x)^[n + 1] t - (fun x => P.eval x)^[n] t :=
fun m n => Cycle.chain_iff_pairwise.1 HC' _ HC _ HC
-- The sign of P^n(t) - t is the same as P(t) - t for positive n. Proven by induction on n.
have IH : ∀ n : ℕ, ((fun x => P.eval x)^[n + 1] t - t).sign = (P.eval t - t).sign := by
intro n
induction' n with n IH
· rfl
· apply Eq.trans _ (Int.sign_add_eq_of_sign_eq IH)
have H := Heq n.succ 0
dsimp at H ⊢
rw [← H]
rw [sub_add_sub_cancel']
-- This implies that the sign of P(t) - t is the same as the sign of P^k(t) - t, which is 0.
-- Hence P(t) = t and P(P(t)) = P(t).
rcases ht with ⟨_ | k, hk, hk'⟩
· exact (irrefl 0 hk).elim
· have H := IH k
rw [hk'.isFixedPt.eq] at H
rw [sub_self] at H
rw [Int.sign_zero] at H
rw [eq_comm] at H
rw [Int.sign_eq_zero_iff_zero] at H
rw [sub_eq_zero] at H
simp [IsPeriodicPt, IsFixedPt, H]
· -- We take two nonequal consecutive entries.
rw [Cycle.chain_map] at HC'
rw [periodicOrbit_chain' _ ht] at HC'
push_neg at HC'
obtain ⟨n, hn⟩ := HC'
-- They must have opposite sign, so that P^{k + 1}(t) - P^k(t) = P^{k + 2}(t) - P^{k + 1}(t).
rcases Int.natAbs_eq_natAbs_iff.1 (Habs n n.succ) with hn' | hn'
· apply (hn _).elim
convert hn' <;> simp only [Function.iterate_succ_apply']
-- We deduce P^{k + 2}(t) = P^k(t) and hence P(P(t)) = t.
· rw [neg_sub, sub_right_inj] at hn'
simp only [Function.iterate_succ_apply'] at hn'
exact isPeriodicPt_of_mem_periodicPts_of_isPeriodicPt_iterate ht hn'.symm
|
/root/DuelModelResearch/mathlib4/Archive/Imo/Imo2006Q5.lean
|
{
"open": [
"Function Polynomial"
],
"variables": []
}
|
[
{
"line": "let C : Cycle ℤ := (periodicOrbit (fun x => P.eval x) t).map fun x => P.eval x - x",
"before_state": "P : ℤ[X]\nt : ℤ\nht : t ∈ periodicPts fun x => eval x P\n⊢ IsPeriodicPt (fun x => eval x P) 2 t",
"after_state": "P : ℤ[X]\nt : ℤ\nht : t ∈ periodicPts fun x => eval x P\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\n⊢ IsPeriodicPt (fun x => eval x P) 2 t"
},
{
"line": "refine_lift\n let C : Cycle ℤ := (periodicOrbit (fun x => P.eval x) t).map fun x => P.eval x - x;\n ?_",
"before_state": "P : ℤ[X]\nt : ℤ\nht : t ∈ periodicPts fun x => eval x P\n⊢ IsPeriodicPt (fun x => eval x P) 2 t",
"after_state": "P : ℤ[X]\nt : ℤ\nht : t ∈ periodicPts fun x => eval x P\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\n⊢ IsPeriodicPt (fun x => eval x P) 2 t"
},
{
"line": "focus\n (refine\n no_implicit_lambda%\n (let C : Cycle ℤ := (periodicOrbit (fun x => P.eval x) t).map fun x => P.eval x - x;\n ?_);\n rotate_right)",
"before_state": "P : ℤ[X]\nt : ℤ\nht : t ∈ periodicPts fun x => eval x P\n⊢ IsPeriodicPt (fun x => eval x P) 2 t",
"after_state": "P : ℤ[X]\nt : ℤ\nht : t ∈ periodicPts fun x => eval x P\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\n⊢ IsPeriodicPt (fun x => eval x P) 2 t"
},
{
"line": "refine\n no_implicit_lambda%\n (let C : Cycle ℤ := (periodicOrbit (fun x => P.eval x) t).map fun x => P.eval x - x;\n ?_)",
"before_state": "P : ℤ[X]\nt : ℤ\nht : t ∈ periodicPts fun x => eval x P\n⊢ IsPeriodicPt (fun x => eval x P) 2 t",
"after_state": "P : ℤ[X]\nt : ℤ\nht : t ∈ periodicPts fun x => eval x P\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\n⊢ IsPeriodicPt (fun x => eval x P) 2 t"
},
{
"line": "rotate_right",
"before_state": "P : ℤ[X]\nt : ℤ\nht : t ∈ periodicPts fun x => eval x P\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\n⊢ IsPeriodicPt (fun x => eval x P) 2 t",
"after_state": "P : ℤ[X]\nt : ℤ\nht : t ∈ periodicPts fun x => eval x P\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\n⊢ IsPeriodicPt (fun x => eval x P) 2 t"
},
{
"line": "have HC : ∀ {n : ℕ}, (fun x => P.eval x)^[n + 1] t - (fun x => P.eval x)^[n] t ∈ C :=\n by\n intro n\n rw [Cycle.mem_map]\n rw [Function.iterate_succ_apply']\n exact\n ⟨_, iterate_mem_periodicOrbit ht n, rfl⟩\n -- Elements in C are all divisible by one another.",
"before_state": "P : ℤ[X]\nt : ℤ\nht : t ∈ periodicPts fun x => eval x P\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\n⊢ IsPeriodicPt (fun x => eval x P) 2 t",
"after_state": "P : ℤ[X]\nt : ℤ\nht : t ∈ periodicPts fun x => eval x P\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\n⊢ IsPeriodicPt (fun x => eval x P) 2 t"
},
{
"line": "focus\n refine\n no_implicit_lambda%\n (have HC : ∀ {n : ℕ}, (fun x => P.eval x)^[n + 1] t - (fun x => P.eval x)^[n] t ∈ C := ?body✝;\n ?_)\n case body✝ =>\n with_annotate_state\"by\"\n ( intro n\n rw [Cycle.mem_map]\n rw [Function.iterate_succ_apply']\n exact\n ⟨_, iterate_mem_periodicOrbit ht n, rfl⟩\n -- Elements in C are all divisible by one another.)",
"before_state": "P : ℤ[X]\nt : ℤ\nht : t ∈ periodicPts fun x => eval x P\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\n⊢ IsPeriodicPt (fun x => eval x P) 2 t",
"after_state": "P : ℤ[X]\nt : ℤ\nht : t ∈ periodicPts fun x => eval x P\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\n⊢ IsPeriodicPt (fun x => eval x P) 2 t"
},
{
"line": "refine\n no_implicit_lambda%\n (have HC : ∀ {n : ℕ}, (fun x => P.eval x)^[n + 1] t - (fun x => P.eval x)^[n] t ∈ C := ?body✝;\n ?_)",
"before_state": "P : ℤ[X]\nt : ℤ\nht : t ∈ periodicPts fun x => eval x P\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\n⊢ IsPeriodicPt (fun x => eval x P) 2 t",
"after_state": "case body\nP : ℤ[X]\nt : ℤ\nht : t ∈ periodicPts fun x => eval x P\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\n⊢ ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\n---\nP : ℤ[X]\nt : ℤ\nht : t ∈ periodicPts fun x => eval x P\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\n⊢ IsPeriodicPt (fun x => eval x P) 2 t"
},
{
"line": "case body✝ =>\n with_annotate_state\"by\"\n ( intro n\n rw [Cycle.mem_map]\n rw [Function.iterate_succ_apply']\n exact\n ⟨_, iterate_mem_periodicOrbit ht n, rfl⟩\n -- Elements in C are all divisible by one another.)",
"before_state": "case body\nP : ℤ[X]\nt : ℤ\nht : t ∈ periodicPts fun x => eval x P\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\n⊢ ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\n---\nP : ℤ[X]\nt : ℤ\nht : t ∈ periodicPts fun x => eval x P\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\n⊢ IsPeriodicPt (fun x => eval x P) 2 t",
"after_state": "P : ℤ[X]\nt : ℤ\nht : t ∈ periodicPts fun x => eval x P\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\n⊢ IsPeriodicPt (fun x => eval x P) 2 t"
},
{
"line": "with_annotate_state\"by\"\n ( intro n\n rw [Cycle.mem_map]\n rw [Function.iterate_succ_apply']\n exact\n ⟨_, iterate_mem_periodicOrbit ht n, rfl⟩\n -- Elements in C are all divisible by one another.)",
"before_state": "P : ℤ[X]\nt : ℤ\nht : t ∈ periodicPts fun x => eval x P\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\n⊢ ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C",
"after_state": "No Goals!"
},
{
"line": "intro n",
"before_state": "P : ℤ[X]\nt : ℤ\nht : t ∈ periodicPts fun x => eval x P\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\n⊢ ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C",
"after_state": "P : ℤ[X]\nt : ℤ\nht : t ∈ periodicPts fun x => eval x P\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nn : ℕ\n⊢ (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C"
},
{
"line": "rw [Cycle.mem_map]",
"before_state": "P : ℤ[X]\nt : ℤ\nht : t ∈ periodicPts fun x => eval x P\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nn : ℕ\n⊢ (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C",
"after_state": "P : ℤ[X]\nt : ℤ\nht : t ∈ periodicPts fun x => eval x P\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nn : ℕ\n⊢ ∃ a ∈ periodicOrbit (fun x => eval x P) t, eval a P - a = (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t"
},
{
"line": "rewrite [Cycle.mem_map]",
"before_state": "P : ℤ[X]\nt : ℤ\nht : t ∈ periodicPts fun x => eval x P\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nn : ℕ\n⊢ (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C",
"after_state": "P : ℤ[X]\nt : ℤ\nht : t ∈ periodicPts fun x => eval x P\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nn : ℕ\n⊢ ∃ a ∈ periodicOrbit (fun x => eval x P) t, eval a P - a = (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "P : ℤ[X]\nt : ℤ\nht : t ∈ periodicPts fun x => eval x P\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nn : ℕ\n⊢ ∃ a ∈ periodicOrbit (fun x => eval x P) t, eval a P - a = (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t",
"after_state": "P : ℤ[X]\nt : ℤ\nht : t ∈ periodicPts fun x => eval x P\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nn : ℕ\n⊢ ∃ a ∈ periodicOrbit (fun x => eval x P) t, eval a P - a = (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t"
},
{
"line": "try (with_reducible rfl)",
"before_state": "P : ℤ[X]\nt : ℤ\nht : t ∈ periodicPts fun x => eval x P\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nn : ℕ\n⊢ ∃ a ∈ periodicOrbit (fun x => eval x P) t, eval a P - a = (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t",
"after_state": "P : ℤ[X]\nt : ℤ\nht : t ∈ periodicPts fun x => eval x P\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nn : ℕ\n⊢ ∃ a ∈ periodicOrbit (fun x => eval x P) t, eval a P - a = (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "P : ℤ[X]\nt : ℤ\nht : t ∈ periodicPts fun x => eval x P\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nn : ℕ\n⊢ ∃ a ∈ periodicOrbit (fun x => eval x P) t, eval a P - a = (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t",
"after_state": "P : ℤ[X]\nt : ℤ\nht : t ∈ periodicPts fun x => eval x P\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nn : ℕ\n⊢ ∃ a ∈ periodicOrbit (fun x => eval x P) t, eval a P - a = (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t"
},
{
"line": "with_reducible rfl",
"before_state": "P : ℤ[X]\nt : ℤ\nht : t ∈ periodicPts fun x => eval x P\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nn : ℕ\n⊢ ∃ a ∈ periodicOrbit (fun x => eval x P) t, eval a P - a = (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t",
"after_state": "P : ℤ[X]\nt : ℤ\nht : t ∈ periodicPts fun x => eval x P\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nn : ℕ\n⊢ ∃ a ∈ periodicOrbit (fun x => eval x P) t, eval a P - a = (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t"
},
{
"line": "rfl",
"before_state": "P : ℤ[X]\nt : ℤ\nht : t ∈ periodicPts fun x => eval x P\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nn : ℕ\n⊢ ∃ a ∈ periodicOrbit (fun x => eval x P) t, eval a P - a = (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t",
"after_state": "P : ℤ[X]\nt : ℤ\nht : t ∈ periodicPts fun x => eval x P\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nn : ℕ\n⊢ ∃ a ∈ periodicOrbit (fun x => eval x P) t, eval a P - a = (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t"
},
{
"line": "apply_rfl",
"before_state": "P : ℤ[X]\nt : ℤ\nht : t ∈ periodicPts fun x => eval x P\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nn : ℕ\n⊢ ∃ a ∈ periodicOrbit (fun x => eval x P) t, eval a P - a = (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t",
"after_state": "P : ℤ[X]\nt : ℤ\nht : t ∈ periodicPts fun x => eval x P\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nn : ℕ\n⊢ ∃ a ∈ periodicOrbit (fun x => eval x P) t, eval a P - a = (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t"
},
{
"line": "skip",
"before_state": "P : ℤ[X]\nt : ℤ\nht : t ∈ periodicPts fun x => eval x P\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nn : ℕ\n⊢ ∃ a ∈ periodicOrbit (fun x => eval x P) t, eval a P - a = (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t",
"after_state": "P : ℤ[X]\nt : ℤ\nht : t ∈ periodicPts fun x => eval x P\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nn : ℕ\n⊢ ∃ a ∈ periodicOrbit (fun x => eval x P) t, eval a P - a = (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t"
},
{
"line": "rw [Function.iterate_succ_apply']",
"before_state": "P : ℤ[X]\nt : ℤ\nht : t ∈ periodicPts fun x => eval x P\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nn : ℕ\n⊢ ∃ a ∈ periodicOrbit (fun x => eval x P) t, eval a P - a = (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t",
"after_state": "P : ℤ[X]\nt : ℤ\nht : t ∈ periodicPts fun x => eval x P\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nn : ℕ\n⊢ ∃ a ∈ periodicOrbit (fun x => eval x P) t,\n eval a P - a = eval ((fun x => eval x P)^[n] t) P - (fun x => eval x P)^[n] t"
},
{
"line": "rewrite [Function.iterate_succ_apply']",
"before_state": "P : ℤ[X]\nt : ℤ\nht : t ∈ periodicPts fun x => eval x P\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nn : ℕ\n⊢ ∃ a ∈ periodicOrbit (fun x => eval x P) t, eval a P - a = (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t",
"after_state": "P : ℤ[X]\nt : ℤ\nht : t ∈ periodicPts fun x => eval x P\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nn : ℕ\n⊢ ∃ a ∈ periodicOrbit (fun x => eval x P) t,\n eval a P - a = eval ((fun x => eval x P)^[n] t) P - (fun x => eval x P)^[n] t"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "P : ℤ[X]\nt : ℤ\nht : t ∈ periodicPts fun x => eval x P\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nn : ℕ\n⊢ ∃ a ∈ periodicOrbit (fun x => eval x P) t,\n eval a P - a = eval ((fun x => eval x P)^[n] t) P - (fun x => eval x P)^[n] t",
"after_state": "P : ℤ[X]\nt : ℤ\nht : t ∈ periodicPts fun x => eval x P\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nn : ℕ\n⊢ ∃ a ∈ periodicOrbit (fun x => eval x P) t,\n eval a P - a = eval ((fun x => eval x P)^[n] t) P - (fun x => eval x P)^[n] t"
},
{
"line": "try (with_reducible rfl)",
"before_state": "P : ℤ[X]\nt : ℤ\nht : t ∈ periodicPts fun x => eval x P\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nn : ℕ\n⊢ ∃ a ∈ periodicOrbit (fun x => eval x P) t,\n eval a P - a = eval ((fun x => eval x P)^[n] t) P - (fun x => eval x P)^[n] t",
"after_state": "P : ℤ[X]\nt : ℤ\nht : t ∈ periodicPts fun x => eval x P\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nn : ℕ\n⊢ ∃ a ∈ periodicOrbit (fun x => eval x P) t,\n eval a P - a = eval ((fun x => eval x P)^[n] t) P - (fun x => eval x P)^[n] t"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "P : ℤ[X]\nt : ℤ\nht : t ∈ periodicPts fun x => eval x P\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nn : ℕ\n⊢ ∃ a ∈ periodicOrbit (fun x => eval x P) t,\n eval a P - a = eval ((fun x => eval x P)^[n] t) P - (fun x => eval x P)^[n] t",
"after_state": "P : ℤ[X]\nt : ℤ\nht : t ∈ periodicPts fun x => eval x P\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nn : ℕ\n⊢ ∃ a ∈ periodicOrbit (fun x => eval x P) t,\n eval a P - a = eval ((fun x => eval x P)^[n] t) P - (fun x => eval x P)^[n] t"
},
{
"line": "with_reducible rfl",
"before_state": "P : ℤ[X]\nt : ℤ\nht : t ∈ periodicPts fun x => eval x P\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nn : ℕ\n⊢ ∃ a ∈ periodicOrbit (fun x => eval x P) t,\n eval a P - a = eval ((fun x => eval x P)^[n] t) P - (fun x => eval x P)^[n] t",
"after_state": "P : ℤ[X]\nt : ℤ\nht : t ∈ periodicPts fun x => eval x P\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nn : ℕ\n⊢ ∃ a ∈ periodicOrbit (fun x => eval x P) t,\n eval a P - a = eval ((fun x => eval x P)^[n] t) P - (fun x => eval x P)^[n] t"
},
{
"line": "rfl",
"before_state": "P : ℤ[X]\nt : ℤ\nht : t ∈ periodicPts fun x => eval x P\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nn : ℕ\n⊢ ∃ a ∈ periodicOrbit (fun x => eval x P) t,\n eval a P - a = eval ((fun x => eval x P)^[n] t) P - (fun x => eval x P)^[n] t",
"after_state": "P : ℤ[X]\nt : ℤ\nht : t ∈ periodicPts fun x => eval x P\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nn : ℕ\n⊢ ∃ a ∈ periodicOrbit (fun x => eval x P) t,\n eval a P - a = eval ((fun x => eval x P)^[n] t) P - (fun x => eval x P)^[n] t"
},
{
"line": "apply_rfl",
"before_state": "P : ℤ[X]\nt : ℤ\nht : t ∈ periodicPts fun x => eval x P\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nn : ℕ\n⊢ ∃ a ∈ periodicOrbit (fun x => eval x P) t,\n eval a P - a = eval ((fun x => eval x P)^[n] t) P - (fun x => eval x P)^[n] t",
"after_state": "P : ℤ[X]\nt : ℤ\nht : t ∈ periodicPts fun x => eval x P\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nn : ℕ\n⊢ ∃ a ∈ periodicOrbit (fun x => eval x P) t,\n eval a P - a = eval ((fun x => eval x P)^[n] t) P - (fun x => eval x P)^[n] t"
},
{
"line": "skip",
"before_state": "P : ℤ[X]\nt : ℤ\nht : t ∈ periodicPts fun x => eval x P\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nn : ℕ\n⊢ ∃ a ∈ periodicOrbit (fun x => eval x P) t,\n eval a P - a = eval ((fun x => eval x P)^[n] t) P - (fun x => eval x P)^[n] t",
"after_state": "P : ℤ[X]\nt : ℤ\nht : t ∈ periodicPts fun x => eval x P\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nn : ℕ\n⊢ ∃ a ∈ periodicOrbit (fun x => eval x P) t,\n eval a P - a = eval ((fun x => eval x P)^[n] t) P - (fun x => eval x P)^[n] t"
},
{
"line": "exact\n ⟨_, iterate_mem_periodicOrbit ht n, rfl⟩\n -- Elements in C are all divisible by one another.",
"before_state": "P : ℤ[X]\nt : ℤ\nht : t ∈ periodicPts fun x => eval x P\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nn : ℕ\n⊢ ∃ a ∈ periodicOrbit (fun x => eval x P) t,\n eval a P - a = eval ((fun x => eval x P)^[n] t) P - (fun x => eval x P)^[n] t",
"after_state": "No Goals!"
},
{
"line": "have Hdvd : C.Chain (· ∣ ·) := by\n rw [Cycle.chain_map]\n rw [periodicOrbit_chain' _ ht]\n intro n\n convert sub_dvd_eval_sub ((fun x => P.eval x)^[n + 1] t) ((fun x => P.eval x)^[n] t) P <;>\n rw [Function.iterate_succ_apply']\n -- Any two entries in C have the same absolute value.",
"before_state": "P : ℤ[X]\nt : ℤ\nht : t ∈ periodicPts fun x => eval x P\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\n⊢ IsPeriodicPt (fun x => eval x P) 2 t",
"after_state": "P : ℤ[X]\nt : ℤ\nht : t ∈ periodicPts fun x => eval x P\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\nHdvd : Cycle.Chain (fun x1 x2 => x1 ∣ x2) C\n⊢ IsPeriodicPt (fun x => eval x P) 2 t"
},
{
"line": "focus\n refine\n no_implicit_lambda%\n (have Hdvd : C.Chain (· ∣ ·) := ?body✝;\n ?_)\n case body✝ =>\n with_annotate_state\"by\"\n ( rw [Cycle.chain_map]\n rw [periodicOrbit_chain' _ ht]\n intro n\n convert sub_dvd_eval_sub ((fun x => P.eval x)^[n + 1] t) ((fun x => P.eval x)^[n] t) P <;>\n rw [Function.iterate_succ_apply']\n -- Any two entries in C have the same absolute value.)",
"before_state": "P : ℤ[X]\nt : ℤ\nht : t ∈ periodicPts fun x => eval x P\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\n⊢ IsPeriodicPt (fun x => eval x P) 2 t",
"after_state": "P : ℤ[X]\nt : ℤ\nht : t ∈ periodicPts fun x => eval x P\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\nHdvd : Cycle.Chain (fun x1 x2 => x1 ∣ x2) C\n⊢ IsPeriodicPt (fun x => eval x P) 2 t"
},
{
"line": "refine\n no_implicit_lambda%\n (have Hdvd : C.Chain (· ∣ ·) := ?body✝;\n ?_)",
"before_state": "P : ℤ[X]\nt : ℤ\nht : t ∈ periodicPts fun x => eval x P\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\n⊢ IsPeriodicPt (fun x => eval x P) 2 t",
"after_state": "case body\nP : ℤ[X]\nt : ℤ\nht : t ∈ periodicPts fun x => eval x P\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\n⊢ Cycle.Chain (fun x1 x2 => x1 ∣ x2) C\n---\nP : ℤ[X]\nt : ℤ\nht : t ∈ periodicPts fun x => eval x P\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\nHdvd : Cycle.Chain (fun x1 x2 => x1 ∣ x2) C\n⊢ IsPeriodicPt (fun x => eval x P) 2 t"
},
{
"line": "case body✝ =>\n with_annotate_state\"by\"\n ( rw [Cycle.chain_map]\n rw [periodicOrbit_chain' _ ht]\n intro n\n convert sub_dvd_eval_sub ((fun x => P.eval x)^[n + 1] t) ((fun x => P.eval x)^[n] t) P <;>\n rw [Function.iterate_succ_apply']\n -- Any two entries in C have the same absolute value.)",
"before_state": "case body\nP : ℤ[X]\nt : ℤ\nht : t ∈ periodicPts fun x => eval x P\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\n⊢ Cycle.Chain (fun x1 x2 => x1 ∣ x2) C\n---\nP : ℤ[X]\nt : ℤ\nht : t ∈ periodicPts fun x => eval x P\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\nHdvd : Cycle.Chain (fun x1 x2 => x1 ∣ x2) C\n⊢ IsPeriodicPt (fun x => eval x P) 2 t",
"after_state": "P : ℤ[X]\nt : ℤ\nht : t ∈ periodicPts fun x => eval x P\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\nHdvd : Cycle.Chain (fun x1 x2 => x1 ∣ x2) C\n⊢ IsPeriodicPt (fun x => eval x P) 2 t"
},
{
"line": "with_annotate_state\"by\"\n ( rw [Cycle.chain_map]\n rw [periodicOrbit_chain' _ ht]\n intro n\n convert sub_dvd_eval_sub ((fun x => P.eval x)^[n + 1] t) ((fun x => P.eval x)^[n] t) P <;>\n rw [Function.iterate_succ_apply']\n -- Any two entries in C have the same absolute value.)",
"before_state": "P : ℤ[X]\nt : ℤ\nht : t ∈ periodicPts fun x => eval x P\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\n⊢ Cycle.Chain (fun x1 x2 => x1 ∣ x2) C",
"after_state": "No Goals!"
},
{
"line": "rw [Cycle.chain_map]",
"before_state": "P : ℤ[X]\nt : ℤ\nht : t ∈ periodicPts fun x => eval x P\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\n⊢ Cycle.Chain (fun x1 x2 => x1 ∣ x2) C",
"after_state": "P : ℤ[X]\nt : ℤ\nht : t ∈ periodicPts fun x => eval x P\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\n⊢ Cycle.Chain (fun a b => eval a P - a ∣ eval b P - b) (periodicOrbit (fun x => eval x P) t)"
},
{
"line": "rewrite [Cycle.chain_map]",
"before_state": "P : ℤ[X]\nt : ℤ\nht : t ∈ periodicPts fun x => eval x P\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\n⊢ Cycle.Chain (fun x1 x2 => x1 ∣ x2) C",
"after_state": "P : ℤ[X]\nt : ℤ\nht : t ∈ periodicPts fun x => eval x P\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\n⊢ Cycle.Chain (fun a b => eval a P - a ∣ eval b P - b) (periodicOrbit (fun x => eval x P) t)"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "P : ℤ[X]\nt : ℤ\nht : t ∈ periodicPts fun x => eval x P\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\n⊢ Cycle.Chain (fun a b => eval a P - a ∣ eval b P - b) (periodicOrbit (fun x => eval x P) t)",
"after_state": "P : ℤ[X]\nt : ℤ\nht : t ∈ periodicPts fun x => eval x P\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\n⊢ Cycle.Chain (fun a b => eval a P - a ∣ eval b P - b) (periodicOrbit (fun x => eval x P) t)"
},
{
"line": "try (with_reducible rfl)",
"before_state": "P : ℤ[X]\nt : ℤ\nht : t ∈ periodicPts fun x => eval x P\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\n⊢ Cycle.Chain (fun a b => eval a P - a ∣ eval b P - b) (periodicOrbit (fun x => eval x P) t)",
"after_state": "P : ℤ[X]\nt : ℤ\nht : t ∈ periodicPts fun x => eval x P\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\n⊢ Cycle.Chain (fun a b => eval a P - a ∣ eval b P - b) (periodicOrbit (fun x => eval x P) t)"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "P : ℤ[X]\nt : ℤ\nht : t ∈ periodicPts fun x => eval x P\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\n⊢ Cycle.Chain (fun a b => eval a P - a ∣ eval b P - b) (periodicOrbit (fun x => eval x P) t)",
"after_state": "P : ℤ[X]\nt : ℤ\nht : t ∈ periodicPts fun x => eval x P\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\n⊢ Cycle.Chain (fun a b => eval a P - a ∣ eval b P - b) (periodicOrbit (fun x => eval x P) t)"
},
{
"line": "with_reducible rfl",
"before_state": "P : ℤ[X]\nt : ℤ\nht : t ∈ periodicPts fun x => eval x P\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\n⊢ Cycle.Chain (fun a b => eval a P - a ∣ eval b P - b) (periodicOrbit (fun x => eval x P) t)",
"after_state": "P : ℤ[X]\nt : ℤ\nht : t ∈ periodicPts fun x => eval x P\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\n⊢ Cycle.Chain (fun a b => eval a P - a ∣ eval b P - b) (periodicOrbit (fun x => eval x P) t)"
},
{
"line": "rfl",
"before_state": "P : ℤ[X]\nt : ℤ\nht : t ∈ periodicPts fun x => eval x P\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\n⊢ Cycle.Chain (fun a b => eval a P - a ∣ eval b P - b) (periodicOrbit (fun x => eval x P) t)",
"after_state": "P : ℤ[X]\nt : ℤ\nht : t ∈ periodicPts fun x => eval x P\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\n⊢ Cycle.Chain (fun a b => eval a P - a ∣ eval b P - b) (periodicOrbit (fun x => eval x P) t)"
},
{
"line": "apply_rfl",
"before_state": "P : ℤ[X]\nt : ℤ\nht : t ∈ periodicPts fun x => eval x P\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\n⊢ Cycle.Chain (fun a b => eval a P - a ∣ eval b P - b) (periodicOrbit (fun x => eval x P) t)",
"after_state": "P : ℤ[X]\nt : ℤ\nht : t ∈ periodicPts fun x => eval x P\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\n⊢ Cycle.Chain (fun a b => eval a P - a ∣ eval b P - b) (periodicOrbit (fun x => eval x P) t)"
},
{
"line": "skip",
"before_state": "P : ℤ[X]\nt : ℤ\nht : t ∈ periodicPts fun x => eval x P\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\n⊢ Cycle.Chain (fun a b => eval a P - a ∣ eval b P - b) (periodicOrbit (fun x => eval x P) t)",
"after_state": "P : ℤ[X]\nt : ℤ\nht : t ∈ periodicPts fun x => eval x P\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\n⊢ Cycle.Chain (fun a b => eval a P - a ∣ eval b P - b) (periodicOrbit (fun x => eval x P) t)"
},
{
"line": "rw [periodicOrbit_chain' _ ht]",
"before_state": "P : ℤ[X]\nt : ℤ\nht : t ∈ periodicPts fun x => eval x P\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\n⊢ Cycle.Chain (fun a b => eval a P - a ∣ eval b P - b) (periodicOrbit (fun x => eval x P) t)",
"after_state": "P : ℤ[X]\nt : ℤ\nht : t ∈ periodicPts fun x => eval x P\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\n⊢ ∀ (n : ℕ),\n eval ((fun x => eval x P)^[n] t) P - (fun x => eval x P)^[n] t ∣\n eval ((fun x => eval x P)^[n + 1] t) P - (fun x => eval x P)^[n + 1] t"
},
{
"line": "rewrite [periodicOrbit_chain' _ ht]",
"before_state": "P : ℤ[X]\nt : ℤ\nht : t ∈ periodicPts fun x => eval x P\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\n⊢ Cycle.Chain (fun a b => eval a P - a ∣ eval b P - b) (periodicOrbit (fun x => eval x P) t)",
"after_state": "P : ℤ[X]\nt : ℤ\nht : t ∈ periodicPts fun x => eval x P\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\n⊢ ∀ (n : ℕ),\n eval ((fun x => eval x P)^[n] t) P - (fun x => eval x P)^[n] t ∣\n eval ((fun x => eval x P)^[n + 1] t) P - (fun x => eval x P)^[n + 1] t"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "P : ℤ[X]\nt : ℤ\nht : t ∈ periodicPts fun x => eval x P\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\n⊢ ∀ (n : ℕ),\n eval ((fun x => eval x P)^[n] t) P - (fun x => eval x P)^[n] t ∣\n eval ((fun x => eval x P)^[n + 1] t) P - (fun x => eval x P)^[n + 1] t",
"after_state": "P : ℤ[X]\nt : ℤ\nht : t ∈ periodicPts fun x => eval x P\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\n⊢ ∀ (n : ℕ),\n eval ((fun x => eval x P)^[n] t) P - (fun x => eval x P)^[n] t ∣\n eval ((fun x => eval x P)^[n + 1] t) P - (fun x => eval x P)^[n + 1] t"
},
{
"line": "try (with_reducible rfl)",
"before_state": "P : ℤ[X]\nt : ℤ\nht : t ∈ periodicPts fun x => eval x P\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\n⊢ ∀ (n : ℕ),\n eval ((fun x => eval x P)^[n] t) P - (fun x => eval x P)^[n] t ∣\n eval ((fun x => eval x P)^[n + 1] t) P - (fun x => eval x P)^[n + 1] t",
"after_state": "P : ℤ[X]\nt : ℤ\nht : t ∈ periodicPts fun x => eval x P\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\n⊢ ∀ (n : ℕ),\n eval ((fun x => eval x P)^[n] t) P - (fun x => eval x P)^[n] t ∣\n eval ((fun x => eval x P)^[n + 1] t) P - (fun x => eval x P)^[n + 1] t"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "P : ℤ[X]\nt : ℤ\nht : t ∈ periodicPts fun x => eval x P\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\n⊢ ∀ (n : ℕ),\n eval ((fun x => eval x P)^[n] t) P - (fun x => eval x P)^[n] t ∣\n eval ((fun x => eval x P)^[n + 1] t) P - (fun x => eval x P)^[n + 1] t",
"after_state": "P : ℤ[X]\nt : ℤ\nht : t ∈ periodicPts fun x => eval x P\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\n⊢ ∀ (n : ℕ),\n eval ((fun x => eval x P)^[n] t) P - (fun x => eval x P)^[n] t ∣\n eval ((fun x => eval x P)^[n + 1] t) P - (fun x => eval x P)^[n + 1] t"
},
{
"line": "with_reducible rfl",
"before_state": "P : ℤ[X]\nt : ℤ\nht : t ∈ periodicPts fun x => eval x P\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\n⊢ ∀ (n : ℕ),\n eval ((fun x => eval x P)^[n] t) P - (fun x => eval x P)^[n] t ∣\n eval ((fun x => eval x P)^[n + 1] t) P - (fun x => eval x P)^[n + 1] t",
"after_state": "P : ℤ[X]\nt : ℤ\nht : t ∈ periodicPts fun x => eval x P\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\n⊢ ∀ (n : ℕ),\n eval ((fun x => eval x P)^[n] t) P - (fun x => eval x P)^[n] t ∣\n eval ((fun x => eval x P)^[n + 1] t) P - (fun x => eval x P)^[n + 1] t"
},
{
"line": "rfl",
"before_state": "P : ℤ[X]\nt : ℤ\nht : t ∈ periodicPts fun x => eval x P\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\n⊢ ∀ (n : ℕ),\n eval ((fun x => eval x P)^[n] t) P - (fun x => eval x P)^[n] t ∣\n eval ((fun x => eval x P)^[n + 1] t) P - (fun x => eval x P)^[n + 1] t",
"after_state": "P : ℤ[X]\nt : ℤ\nht : t ∈ periodicPts fun x => eval x P\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\n⊢ ∀ (n : ℕ),\n eval ((fun x => eval x P)^[n] t) P - (fun x => eval x P)^[n] t ∣\n eval ((fun x => eval x P)^[n + 1] t) P - (fun x => eval x P)^[n + 1] t"
},
{
"line": "apply_rfl",
"before_state": "P : ℤ[X]\nt : ℤ\nht : t ∈ periodicPts fun x => eval x P\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\n⊢ ∀ (n : ℕ),\n eval ((fun x => eval x P)^[n] t) P - (fun x => eval x P)^[n] t ∣\n eval ((fun x => eval x P)^[n + 1] t) P - (fun x => eval x P)^[n + 1] t",
"after_state": "P : ℤ[X]\nt : ℤ\nht : t ∈ periodicPts fun x => eval x P\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\n⊢ ∀ (n : ℕ),\n eval ((fun x => eval x P)^[n] t) P - (fun x => eval x P)^[n] t ∣\n eval ((fun x => eval x P)^[n + 1] t) P - (fun x => eval x P)^[n + 1] t"
},
{
"line": "skip",
"before_state": "P : ℤ[X]\nt : ℤ\nht : t ∈ periodicPts fun x => eval x P\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\n⊢ ∀ (n : ℕ),\n eval ((fun x => eval x P)^[n] t) P - (fun x => eval x P)^[n] t ∣\n eval ((fun x => eval x P)^[n + 1] t) P - (fun x => eval x P)^[n + 1] t",
"after_state": "P : ℤ[X]\nt : ℤ\nht : t ∈ periodicPts fun x => eval x P\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\n⊢ ∀ (n : ℕ),\n eval ((fun x => eval x P)^[n] t) P - (fun x => eval x P)^[n] t ∣\n eval ((fun x => eval x P)^[n + 1] t) P - (fun x => eval x P)^[n + 1] t"
},
{
"line": "intro n",
"before_state": "P : ℤ[X]\nt : ℤ\nht : t ∈ periodicPts fun x => eval x P\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\n⊢ ∀ (n : ℕ),\n eval ((fun x => eval x P)^[n] t) P - (fun x => eval x P)^[n] t ∣\n eval ((fun x => eval x P)^[n + 1] t) P - (fun x => eval x P)^[n + 1] t",
"after_state": "P : ℤ[X]\nt : ℤ\nht : t ∈ periodicPts fun x => eval x P\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\nn : ℕ\n⊢ eval ((fun x => eval x P)^[n] t) P - (fun x => eval x P)^[n] t ∣\n eval ((fun x => eval x P)^[n + 1] t) P - (fun x => eval x P)^[n + 1] t"
},
{
"line": "focus\n convert sub_dvd_eval_sub ((fun x => P.eval x)^[n + 1] t) ((fun x => P.eval x)^[n] t) P\n with_annotate_state\"<;>\" skip\n all_goals\n rw [Function.iterate_succ_apply']\n -- Any two entries in C have the same absolute value.",
"before_state": "P : ℤ[X]\nt : ℤ\nht : t ∈ periodicPts fun x => eval x P\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\nn : ℕ\n⊢ eval ((fun x => eval x P)^[n] t) P - (fun x => eval x P)^[n] t ∣\n eval ((fun x => eval x P)^[n + 1] t) P - (fun x => eval x P)^[n + 1] t",
"after_state": "No Goals!"
},
{
"line": "convert sub_dvd_eval_sub ((fun x => P.eval x)^[n + 1] t) ((fun x => P.eval x)^[n] t) P",
"before_state": "P : ℤ[X]\nt : ℤ\nht : t ∈ periodicPts fun x => eval x P\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\nn : ℕ\n⊢ eval ((fun x => eval x P)^[n] t) P - (fun x => eval x P)^[n] t ∣\n eval ((fun x => eval x P)^[n + 1] t) P - (fun x => eval x P)^[n + 1] t",
"after_state": "case h.e'_3.h.e'_5\nP : ℤ[X]\nt : ℤ\nht : t ∈ periodicPts fun x => eval x P\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\nn : ℕ\n⊢ eval ((fun x => eval x P)^[n] t) P = (fun x => eval x P)^[n + 1] t\n---\ncase h.e'_4.h.e'_6\nP : ℤ[X]\nt : ℤ\nht : t ∈ periodicPts fun x => eval x P\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\nn : ℕ\n⊢ (fun x => eval x P)^[n + 1] t = eval ((fun x => eval x P)^[n] t) P"
},
{
"line": "with_annotate_state\"<;>\" skip",
"before_state": "case h.e'_3.h.e'_5\nP : ℤ[X]\nt : ℤ\nht : t ∈ periodicPts fun x => eval x P\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\nn : ℕ\n⊢ eval ((fun x => eval x P)^[n] t) P = (fun x => eval x P)^[n + 1] t\n---\ncase h.e'_4.h.e'_6\nP : ℤ[X]\nt : ℤ\nht : t ∈ periodicPts fun x => eval x P\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\nn : ℕ\n⊢ (fun x => eval x P)^[n + 1] t = eval ((fun x => eval x P)^[n] t) P",
"after_state": "case h.e'_3.h.e'_5\nP : ℤ[X]\nt : ℤ\nht : t ∈ periodicPts fun x => eval x P\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\nn : ℕ\n⊢ eval ((fun x => eval x P)^[n] t) P = (fun x => eval x P)^[n + 1] t\n---\ncase h.e'_4.h.e'_6\nP : ℤ[X]\nt : ℤ\nht : t ∈ periodicPts fun x => eval x P\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\nn : ℕ\n⊢ (fun x => eval x P)^[n + 1] t = eval ((fun x => eval x P)^[n] t) P"
},
{
"line": "skip",
"before_state": "case h.e'_3.h.e'_5\nP : ℤ[X]\nt : ℤ\nht : t ∈ periodicPts fun x => eval x P\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\nn : ℕ\n⊢ eval ((fun x => eval x P)^[n] t) P = (fun x => eval x P)^[n + 1] t\n---\ncase h.e'_4.h.e'_6\nP : ℤ[X]\nt : ℤ\nht : t ∈ periodicPts fun x => eval x P\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\nn : ℕ\n⊢ (fun x => eval x P)^[n + 1] t = eval ((fun x => eval x P)^[n] t) P",
"after_state": "case h.e'_3.h.e'_5\nP : ℤ[X]\nt : ℤ\nht : t ∈ periodicPts fun x => eval x P\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\nn : ℕ\n⊢ eval ((fun x => eval x P)^[n] t) P = (fun x => eval x P)^[n + 1] t\n---\ncase h.e'_4.h.e'_6\nP : ℤ[X]\nt : ℤ\nht : t ∈ periodicPts fun x => eval x P\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\nn : ℕ\n⊢ (fun x => eval x P)^[n + 1] t = eval ((fun x => eval x P)^[n] t) P"
},
{
"line": "all_goals\n rw [Function.iterate_succ_apply']\n -- Any two entries in C have the same absolute value.",
"before_state": "case h.e'_3.h.e'_5\nP : ℤ[X]\nt : ℤ\nht : t ∈ periodicPts fun x => eval x P\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\nn : ℕ\n⊢ eval ((fun x => eval x P)^[n] t) P = (fun x => eval x P)^[n + 1] t\n---\ncase h.e'_4.h.e'_6\nP : ℤ[X]\nt : ℤ\nht : t ∈ periodicPts fun x => eval x P\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\nn : ℕ\n⊢ (fun x => eval x P)^[n + 1] t = eval ((fun x => eval x P)^[n] t) P",
"after_state": "No Goals!"
},
{
"line": "rw [Function.iterate_succ_apply']\n -- Any two entries in C have the same absolute value.",
"before_state": "case h.e'_3.h.e'_5\nP : ℤ[X]\nt : ℤ\nht : t ∈ periodicPts fun x => eval x P\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\nn : ℕ\n⊢ eval ((fun x => eval x P)^[n] t) P = (fun x => eval x P)^[n + 1] t",
"after_state": "No Goals!"
},
{
"line": "rewrite [Function.iterate_succ_apply']",
"before_state": "case h.e'_3.h.e'_5\nP : ℤ[X]\nt : ℤ\nht : t ∈ periodicPts fun x => eval x P\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\nn : ℕ\n⊢ eval ((fun x => eval x P)^[n] t) P = (fun x => eval x P)^[n + 1] t",
"after_state": "case h.e'_3.h.e'_5\nP : ℤ[X]\nt : ℤ\nht : t ∈ periodicPts fun x => eval x P\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\nn : ℕ\n⊢ eval ((fun x => eval x P)^[n] t) P = eval ((fun x => eval x P)^[n] t) P"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "case h.e'_3.h.e'_5\nP : ℤ[X]\nt : ℤ\nht : t ∈ periodicPts fun x => eval x P\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\nn : ℕ\n⊢ eval ((fun x => eval x P)^[n] t) P = eval ((fun x => eval x P)^[n] t) P",
"after_state": "No Goals!"
},
{
"line": "try (with_reducible rfl)",
"before_state": "case h.e'_3.h.e'_5\nP : ℤ[X]\nt : ℤ\nht : t ∈ periodicPts fun x => eval x P\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\nn : ℕ\n⊢ eval ((fun x => eval x P)^[n] t) P = eval ((fun x => eval x P)^[n] t) P",
"after_state": "No Goals!"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "case h.e'_3.h.e'_5\nP : ℤ[X]\nt : ℤ\nht : t ∈ periodicPts fun x => eval x P\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\nn : ℕ\n⊢ eval ((fun x => eval x P)^[n] t) P = eval ((fun x => eval x P)^[n] t) P",
"after_state": "No Goals!"
},
{
"line": "with_reducible rfl",
"before_state": "case h.e'_3.h.e'_5\nP : ℤ[X]\nt : ℤ\nht : t ∈ periodicPts fun x => eval x P\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\nn : ℕ\n⊢ eval ((fun x => eval x P)^[n] t) P = eval ((fun x => eval x P)^[n] t) P",
"after_state": "No Goals!"
},
{
"line": "rfl",
"before_state": "case h.e'_3.h.e'_5\nP : ℤ[X]\nt : ℤ\nht : t ∈ periodicPts fun x => eval x P\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\nn : ℕ\n⊢ eval ((fun x => eval x P)^[n] t) P = eval ((fun x => eval x P)^[n] t) P",
"after_state": "No Goals!"
},
{
"line": "eq_refl",
"before_state": "case h.e'_3.h.e'_5\nP : ℤ[X]\nt : ℤ\nht : t ∈ periodicPts fun x => eval x P\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\nn : ℕ\n⊢ eval ((fun x => eval x P)^[n] t) P = eval ((fun x => eval x P)^[n] t) P",
"after_state": "No Goals!"
},
{
"line": "rw [Function.iterate_succ_apply']\n -- Any two entries in C have the same absolute value.",
"before_state": "case h.e'_4.h.e'_6\nP : ℤ[X]\nt : ℤ\nht : t ∈ periodicPts fun x => eval x P\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\nn : ℕ\n⊢ (fun x => eval x P)^[n + 1] t = eval ((fun x => eval x P)^[n] t) P",
"after_state": "No Goals!"
},
{
"line": "rewrite [Function.iterate_succ_apply']",
"before_state": "case h.e'_4.h.e'_6\nP : ℤ[X]\nt : ℤ\nht : t ∈ periodicPts fun x => eval x P\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\nn : ℕ\n⊢ (fun x => eval x P)^[n + 1] t = eval ((fun x => eval x P)^[n] t) P",
"after_state": "case h.e'_4.h.e'_6\nP : ℤ[X]\nt : ℤ\nht : t ∈ periodicPts fun x => eval x P\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\nn : ℕ\n⊢ eval ((fun x => eval x P)^[n] t) P = eval ((fun x => eval x P)^[n] t) P"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "case h.e'_4.h.e'_6\nP : ℤ[X]\nt : ℤ\nht : t ∈ periodicPts fun x => eval x P\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\nn : ℕ\n⊢ eval ((fun x => eval x P)^[n] t) P = eval ((fun x => eval x P)^[n] t) P",
"after_state": "No Goals!"
},
{
"line": "try (with_reducible rfl)",
"before_state": "case h.e'_4.h.e'_6\nP : ℤ[X]\nt : ℤ\nht : t ∈ periodicPts fun x => eval x P\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\nn : ℕ\n⊢ eval ((fun x => eval x P)^[n] t) P = eval ((fun x => eval x P)^[n] t) P",
"after_state": "No Goals!"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "case h.e'_4.h.e'_6\nP : ℤ[X]\nt : ℤ\nht : t ∈ periodicPts fun x => eval x P\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\nn : ℕ\n⊢ eval ((fun x => eval x P)^[n] t) P = eval ((fun x => eval x P)^[n] t) P",
"after_state": "No Goals!"
},
{
"line": "with_reducible rfl",
"before_state": "case h.e'_4.h.e'_6\nP : ℤ[X]\nt : ℤ\nht : t ∈ periodicPts fun x => eval x P\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\nn : ℕ\n⊢ eval ((fun x => eval x P)^[n] t) P = eval ((fun x => eval x P)^[n] t) P",
"after_state": "No Goals!"
},
{
"line": "rfl",
"before_state": "case h.e'_4.h.e'_6\nP : ℤ[X]\nt : ℤ\nht : t ∈ periodicPts fun x => eval x P\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\nn : ℕ\n⊢ eval ((fun x => eval x P)^[n] t) P = eval ((fun x => eval x P)^[n] t) P",
"after_state": "No Goals!"
},
{
"line": "eq_refl",
"before_state": "case h.e'_4.h.e'_6\nP : ℤ[X]\nt : ℤ\nht : t ∈ periodicPts fun x => eval x P\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\nn : ℕ\n⊢ eval ((fun x => eval x P)^[n] t) P = eval ((fun x => eval x P)^[n] t) P",
"after_state": "No Goals!"
},
{
"line": "have Habs :\n ∀ m n : ℕ,\n ((fun x => P.eval x)^[m + 1] t - (fun x => P.eval x)^[m] t).natAbs =\n ((fun x => P.eval x)^[n + 1] t - (fun x => P.eval x)^[n] t).natAbs :=\n fun m n => Int.natAbs_eq_of_chain_dvd Hdvd HC HC",
"before_state": "P : ℤ[X]\nt : ℤ\nht : t ∈ periodicPts fun x => eval x P\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\nHdvd : Cycle.Chain (fun x1 x2 => x1 ∣ x2) C\n⊢ IsPeriodicPt (fun x => eval x P) 2 t",
"after_state": "P : ℤ[X]\nt : ℤ\nht : t ∈ periodicPts fun x => eval x P\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\nHdvd : Cycle.Chain (fun x1 x2 => x1 ∣ x2) C\nHabs :\n ∀ (m n : ℕ),\n ((fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t).natAbs =\n ((fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t).natAbs\n⊢ IsPeriodicPt (fun x => eval x P) 2 t"
},
{
"line": "refine_lift\n have Habs :\n ∀ m n : ℕ,\n ((fun x => P.eval x)^[m + 1] t - (fun x => P.eval x)^[m] t).natAbs =\n ((fun x => P.eval x)^[n + 1] t - (fun x => P.eval x)^[n] t).natAbs :=\n fun m n => Int.natAbs_eq_of_chain_dvd Hdvd HC HC;\n ?_",
"before_state": "P : ℤ[X]\nt : ℤ\nht : t ∈ periodicPts fun x => eval x P\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\nHdvd : Cycle.Chain (fun x1 x2 => x1 ∣ x2) C\n⊢ IsPeriodicPt (fun x => eval x P) 2 t",
"after_state": "P : ℤ[X]\nt : ℤ\nht : t ∈ periodicPts fun x => eval x P\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\nHdvd : Cycle.Chain (fun x1 x2 => x1 ∣ x2) C\nHabs :\n ∀ (m n : ℕ),\n ((fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t).natAbs =\n ((fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t).natAbs\n⊢ IsPeriodicPt (fun x => eval x P) 2 t"
},
{
"line": "focus\n (refine\n no_implicit_lambda%\n (have Habs :\n ∀ m n : ℕ,\n ((fun x => P.eval x)^[m + 1] t - (fun x => P.eval x)^[m] t).natAbs =\n ((fun x => P.eval x)^[n + 1] t - (fun x => P.eval x)^[n] t).natAbs :=\n fun m n => Int.natAbs_eq_of_chain_dvd Hdvd HC HC;\n ?_);\n rotate_right)",
"before_state": "P : ℤ[X]\nt : ℤ\nht : t ∈ periodicPts fun x => eval x P\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\nHdvd : Cycle.Chain (fun x1 x2 => x1 ∣ x2) C\n⊢ IsPeriodicPt (fun x => eval x P) 2 t",
"after_state": "P : ℤ[X]\nt : ℤ\nht : t ∈ periodicPts fun x => eval x P\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\nHdvd : Cycle.Chain (fun x1 x2 => x1 ∣ x2) C\nHabs :\n ∀ (m n : ℕ),\n ((fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t).natAbs =\n ((fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t).natAbs\n⊢ IsPeriodicPt (fun x => eval x P) 2 t"
},
{
"line": "refine\n no_implicit_lambda%\n (have Habs :\n ∀ m n : ℕ,\n ((fun x => P.eval x)^[m + 1] t - (fun x => P.eval x)^[m] t).natAbs =\n ((fun x => P.eval x)^[n + 1] t - (fun x => P.eval x)^[n] t).natAbs :=\n fun m n => Int.natAbs_eq_of_chain_dvd Hdvd HC HC;\n ?_)",
"before_state": "P : ℤ[X]\nt : ℤ\nht : t ∈ periodicPts fun x => eval x P\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\nHdvd : Cycle.Chain (fun x1 x2 => x1 ∣ x2) C\n⊢ IsPeriodicPt (fun x => eval x P) 2 t",
"after_state": "P : ℤ[X]\nt : ℤ\nht : t ∈ periodicPts fun x => eval x P\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\nHdvd : Cycle.Chain (fun x1 x2 => x1 ∣ x2) C\nHabs :\n ∀ (m n : ℕ),\n ((fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t).natAbs =\n ((fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t).natAbs\n⊢ IsPeriodicPt (fun x => eval x P) 2 t"
},
{
"line": "rotate_right",
"before_state": "P : ℤ[X]\nt : ℤ\nht : t ∈ periodicPts fun x => eval x P\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\nHdvd : Cycle.Chain (fun x1 x2 => x1 ∣ x2) C\nHabs :\n ∀ (m n : ℕ),\n ((fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t).natAbs =\n ((fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t).natAbs\n⊢ IsPeriodicPt (fun x => eval x P) 2 t",
"after_state": "P : ℤ[X]\nt : ℤ\nht : t ∈ periodicPts fun x => eval x P\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\nHdvd : Cycle.Chain (fun x1 x2 => x1 ∣ x2) C\nHabs :\n ∀ (m n : ℕ),\n ((fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t).natAbs =\n ((fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t).natAbs\n⊢ IsPeriodicPt (fun x => eval x P) 2 t"
},
{
"line": "by_cases HC' : C.Chain (· = ·)",
"before_state": "P : ℤ[X]\nt : ℤ\nht : t ∈ periodicPts fun x => eval x P\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\nHdvd : Cycle.Chain (fun x1 x2 => x1 ∣ x2) C\nHabs :\n ∀ (m n : ℕ),\n ((fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t).natAbs =\n ((fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t).natAbs\n⊢ IsPeriodicPt (fun x => eval x P) 2 t",
"after_state": "case pos\nP : ℤ[X]\nt : ℤ\nht : t ∈ periodicPts fun x => eval x P\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\nHdvd : Cycle.Chain (fun x1 x2 => x1 ∣ x2) C\nHabs :\n ∀ (m n : ℕ),\n ((fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t).natAbs =\n ((fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t).natAbs\nHC' : Cycle.Chain (fun x1 x2 => x1 = x2) C\n⊢ IsPeriodicPt (fun x => eval x P) 2 t\n---\ncase neg\nP : ℤ[X]\nt : ℤ\nht : t ∈ periodicPts fun x => eval x P\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\nHdvd : Cycle.Chain (fun x1 x2 => x1 ∣ x2) C\nHabs :\n ∀ (m n : ℕ),\n ((fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t).natAbs =\n ((fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t).natAbs\nHC' : ¬Cycle.Chain (fun x1 x2 => x1 = x2) C\n⊢ IsPeriodicPt (fun x => eval x P) 2 t"
},
{
"line": "open Classical✝ in refine if HC' : C.Chain (· = ·) then ?pos✝ else ?neg✝",
"before_state": "P : ℤ[X]\nt : ℤ\nht : t ∈ periodicPts fun x => eval x P\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\nHdvd : Cycle.Chain (fun x1 x2 => x1 ∣ x2) C\nHabs :\n ∀ (m n : ℕ),\n ((fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t).natAbs =\n ((fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t).natAbs\n⊢ IsPeriodicPt (fun x => eval x P) 2 t",
"after_state": "case pos\nP : ℤ[X]\nt : ℤ\nht : t ∈ periodicPts fun x => eval x P\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\nHdvd : Cycle.Chain (fun x1 x2 => x1 ∣ x2) C\nHabs :\n ∀ (m n : ℕ),\n ((fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t).natAbs =\n ((fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t).natAbs\nHC' : Cycle.Chain (fun x1 x2 => x1 = x2) C\n⊢ IsPeriodicPt (fun x => eval x P) 2 t\n---\ncase neg\nP : ℤ[X]\nt : ℤ\nht : t ∈ periodicPts fun x => eval x P\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\nHdvd : Cycle.Chain (fun x1 x2 => x1 ∣ x2) C\nHabs :\n ∀ (m n : ℕ),\n ((fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t).natAbs =\n ((fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t).natAbs\nHC' : ¬Cycle.Chain (fun x1 x2 => x1 = x2) C\n⊢ IsPeriodicPt (fun x => eval x P) 2 t"
},
{
"line": "refine if HC' : C.Chain (· = ·) then ?pos✝ else ?neg✝",
"before_state": "P : ℤ[X]\nt : ℤ\nht : t ∈ periodicPts fun x => eval x P\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\nHdvd : Cycle.Chain (fun x1 x2 => x1 ∣ x2) C\nHabs :\n ∀ (m n : ℕ),\n ((fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t).natAbs =\n ((fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t).natAbs\n⊢ IsPeriodicPt (fun x => eval x P) 2 t",
"after_state": "case pos\nP : ℤ[X]\nt : ℤ\nht : t ∈ periodicPts fun x => eval x P\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\nHdvd : Cycle.Chain (fun x1 x2 => x1 ∣ x2) C\nHabs :\n ∀ (m n : ℕ),\n ((fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t).natAbs =\n ((fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t).natAbs\nHC' : Cycle.Chain (fun x1 x2 => x1 = x2) C\n⊢ IsPeriodicPt (fun x => eval x P) 2 t\n---\ncase neg\nP : ℤ[X]\nt : ℤ\nht : t ∈ periodicPts fun x => eval x P\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\nHdvd : Cycle.Chain (fun x1 x2 => x1 ∣ x2) C\nHabs :\n ∀ (m n : ℕ),\n ((fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t).natAbs =\n ((fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t).natAbs\nHC' : ¬Cycle.Chain (fun x1 x2 => x1 = x2) C\n⊢ IsPeriodicPt (fun x => eval x P) 2 t"
},
{
"line": "have Heq :\n ∀ m n : ℕ,\n (fun x => P.eval x)^[m + 1] t - (fun x => P.eval x)^[m] t =\n (fun x => P.eval x)^[n + 1] t - (fun x => P.eval x)^[n] t :=\n fun m n => Cycle.chain_iff_pairwise.1 HC' _ HC _ HC",
"before_state": "case pos\nP : ℤ[X]\nt : ℤ\nht : t ∈ periodicPts fun x => eval x P\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\nHdvd : Cycle.Chain (fun x1 x2 => x1 ∣ x2) C\nHabs :\n ∀ (m n : ℕ),\n ((fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t).natAbs =\n ((fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t).natAbs\nHC' : Cycle.Chain (fun x1 x2 => x1 = x2) C\n⊢ IsPeriodicPt (fun x => eval x P) 2 t",
"after_state": "case pos\nP : ℤ[X]\nt : ℤ\nht : t ∈ periodicPts fun x => eval x P\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\nHdvd : Cycle.Chain (fun x1 x2 => x1 ∣ x2) C\nHabs :\n ∀ (m n : ℕ),\n ((fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t).natAbs =\n ((fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t).natAbs\nHC' : Cycle.Chain (fun x1 x2 => x1 = x2) C\nHeq :\n ∀ (m n : ℕ),\n (fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t =\n (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t\n⊢ IsPeriodicPt (fun x => eval x P) 2 t"
},
{
"line": "refine_lift\n have Heq :\n ∀ m n : ℕ,\n (fun x => P.eval x)^[m + 1] t - (fun x => P.eval x)^[m] t =\n (fun x => P.eval x)^[n + 1] t - (fun x => P.eval x)^[n] t :=\n fun m n => Cycle.chain_iff_pairwise.1 HC' _ HC _ HC;\n ?_",
"before_state": "case pos\nP : ℤ[X]\nt : ℤ\nht : t ∈ periodicPts fun x => eval x P\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\nHdvd : Cycle.Chain (fun x1 x2 => x1 ∣ x2) C\nHabs :\n ∀ (m n : ℕ),\n ((fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t).natAbs =\n ((fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t).natAbs\nHC' : Cycle.Chain (fun x1 x2 => x1 = x2) C\n⊢ IsPeriodicPt (fun x => eval x P) 2 t",
"after_state": "case pos\nP : ℤ[X]\nt : ℤ\nht : t ∈ periodicPts fun x => eval x P\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\nHdvd : Cycle.Chain (fun x1 x2 => x1 ∣ x2) C\nHabs :\n ∀ (m n : ℕ),\n ((fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t).natAbs =\n ((fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t).natAbs\nHC' : Cycle.Chain (fun x1 x2 => x1 = x2) C\nHeq :\n ∀ (m n : ℕ),\n (fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t =\n (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t\n⊢ IsPeriodicPt (fun x => eval x P) 2 t"
},
{
"line": "focus\n (refine\n no_implicit_lambda%\n (have Heq :\n ∀ m n : ℕ,\n (fun x => P.eval x)^[m + 1] t - (fun x => P.eval x)^[m] t =\n (fun x => P.eval x)^[n + 1] t - (fun x => P.eval x)^[n] t :=\n fun m n => Cycle.chain_iff_pairwise.1 HC' _ HC _ HC;\n ?_);\n rotate_right)",
"before_state": "case pos\nP : ℤ[X]\nt : ℤ\nht : t ∈ periodicPts fun x => eval x P\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\nHdvd : Cycle.Chain (fun x1 x2 => x1 ∣ x2) C\nHabs :\n ∀ (m n : ℕ),\n ((fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t).natAbs =\n ((fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t).natAbs\nHC' : Cycle.Chain (fun x1 x2 => x1 = x2) C\n⊢ IsPeriodicPt (fun x => eval x P) 2 t",
"after_state": "case pos\nP : ℤ[X]\nt : ℤ\nht : t ∈ periodicPts fun x => eval x P\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\nHdvd : Cycle.Chain (fun x1 x2 => x1 ∣ x2) C\nHabs :\n ∀ (m n : ℕ),\n ((fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t).natAbs =\n ((fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t).natAbs\nHC' : Cycle.Chain (fun x1 x2 => x1 = x2) C\nHeq :\n ∀ (m n : ℕ),\n (fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t =\n (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t\n⊢ IsPeriodicPt (fun x => eval x P) 2 t"
},
{
"line": "refine\n no_implicit_lambda%\n (have Heq :\n ∀ m n : ℕ,\n (fun x => P.eval x)^[m + 1] t - (fun x => P.eval x)^[m] t =\n (fun x => P.eval x)^[n + 1] t - (fun x => P.eval x)^[n] t :=\n fun m n => Cycle.chain_iff_pairwise.1 HC' _ HC _ HC;\n ?_)",
"before_state": "case pos\nP : ℤ[X]\nt : ℤ\nht : t ∈ periodicPts fun x => eval x P\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\nHdvd : Cycle.Chain (fun x1 x2 => x1 ∣ x2) C\nHabs :\n ∀ (m n : ℕ),\n ((fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t).natAbs =\n ((fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t).natAbs\nHC' : Cycle.Chain (fun x1 x2 => x1 = x2) C\n⊢ IsPeriodicPt (fun x => eval x P) 2 t",
"after_state": "case pos\nP : ℤ[X]\nt : ℤ\nht : t ∈ periodicPts fun x => eval x P\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\nHdvd : Cycle.Chain (fun x1 x2 => x1 ∣ x2) C\nHabs :\n ∀ (m n : ℕ),\n ((fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t).natAbs =\n ((fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t).natAbs\nHC' : Cycle.Chain (fun x1 x2 => x1 = x2) C\nHeq :\n ∀ (m n : ℕ),\n (fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t =\n (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t\n⊢ IsPeriodicPt (fun x => eval x P) 2 t"
},
{
"line": "rotate_right",
"before_state": "case pos\nP : ℤ[X]\nt : ℤ\nht : t ∈ periodicPts fun x => eval x P\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\nHdvd : Cycle.Chain (fun x1 x2 => x1 ∣ x2) C\nHabs :\n ∀ (m n : ℕ),\n ((fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t).natAbs =\n ((fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t).natAbs\nHC' : Cycle.Chain (fun x1 x2 => x1 = x2) C\nHeq :\n ∀ (m n : ℕ),\n (fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t =\n (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t\n⊢ IsPeriodicPt (fun x => eval x P) 2 t",
"after_state": "case pos\nP : ℤ[X]\nt : ℤ\nht : t ∈ periodicPts fun x => eval x P\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\nHdvd : Cycle.Chain (fun x1 x2 => x1 ∣ x2) C\nHabs :\n ∀ (m n : ℕ),\n ((fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t).natAbs =\n ((fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t).natAbs\nHC' : Cycle.Chain (fun x1 x2 => x1 = x2) C\nHeq :\n ∀ (m n : ℕ),\n (fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t =\n (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t\n⊢ IsPeriodicPt (fun x => eval x P) 2 t"
},
{
"line": "have IH : ∀ n : ℕ, ((fun x => P.eval x)^[n + 1] t - t).sign = (P.eval t - t).sign :=\n by\n intro n\n induction' n with n IH\n · rfl\n · apply Eq.trans _ (Int.sign_add_eq_of_sign_eq IH)\n have H := Heq n.succ 0\n dsimp at H ⊢\n rw [← H]\n rw [sub_add_sub_cancel']\n -- This implies that the sign of P(t) - t is the same as the sign of P^k(t) - t, which is 0.\n -- Hence P(t) = t and P(P(t)) = P(t).",
"before_state": "case pos\nP : ℤ[X]\nt : ℤ\nht : t ∈ periodicPts fun x => eval x P\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\nHdvd : Cycle.Chain (fun x1 x2 => x1 ∣ x2) C\nHabs :\n ∀ (m n : ℕ),\n ((fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t).natAbs =\n ((fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t).natAbs\nHC' : Cycle.Chain (fun x1 x2 => x1 = x2) C\nHeq :\n ∀ (m n : ℕ),\n (fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t =\n (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t\n⊢ IsPeriodicPt (fun x => eval x P) 2 t",
"after_state": "case pos\nP : ℤ[X]\nt : ℤ\nht : t ∈ periodicPts fun x => eval x P\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\nHdvd : Cycle.Chain (fun x1 x2 => x1 ∣ x2) C\nHabs :\n ∀ (m n : ℕ),\n ((fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t).natAbs =\n ((fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t).natAbs\nHC' : Cycle.Chain (fun x1 x2 => x1 = x2) C\nHeq :\n ∀ (m n : ℕ),\n (fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t =\n (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t\nIH : ∀ (n : ℕ), ((fun x => eval x P)^[n + 1] t - t).sign = (eval t P - t).sign\n⊢ IsPeriodicPt (fun x => eval x P) 2 t"
},
{
"line": "focus\n refine\n no_implicit_lambda%\n (have IH : ∀ n : ℕ, ((fun x => P.eval x)^[n + 1] t - t).sign = (P.eval t - t).sign := ?body✝;\n ?_)\n case body✝ =>\n with_annotate_state\"by\"\n ( intro n\n induction' n with n IH\n · rfl\n · apply Eq.trans _ (Int.sign_add_eq_of_sign_eq IH)\n have H := Heq n.succ 0\n dsimp at H ⊢\n rw [← H]\n rw [sub_add_sub_cancel']\n -- This implies that the sign of P(t) - t is the same as the sign of P^k(t) - t, which is 0.\n -- Hence P(t) = t and P(P(t)) = P(t).)",
"before_state": "case pos\nP : ℤ[X]\nt : ℤ\nht : t ∈ periodicPts fun x => eval x P\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\nHdvd : Cycle.Chain (fun x1 x2 => x1 ∣ x2) C\nHabs :\n ∀ (m n : ℕ),\n ((fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t).natAbs =\n ((fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t).natAbs\nHC' : Cycle.Chain (fun x1 x2 => x1 = x2) C\nHeq :\n ∀ (m n : ℕ),\n (fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t =\n (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t\n⊢ IsPeriodicPt (fun x => eval x P) 2 t",
"after_state": "case pos\nP : ℤ[X]\nt : ℤ\nht : t ∈ periodicPts fun x => eval x P\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\nHdvd : Cycle.Chain (fun x1 x2 => x1 ∣ x2) C\nHabs :\n ∀ (m n : ℕ),\n ((fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t).natAbs =\n ((fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t).natAbs\nHC' : Cycle.Chain (fun x1 x2 => x1 = x2) C\nHeq :\n ∀ (m n : ℕ),\n (fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t =\n (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t\nIH : ∀ (n : ℕ), ((fun x => eval x P)^[n + 1] t - t).sign = (eval t P - t).sign\n⊢ IsPeriodicPt (fun x => eval x P) 2 t"
},
{
"line": "refine\n no_implicit_lambda%\n (have IH : ∀ n : ℕ, ((fun x => P.eval x)^[n + 1] t - t).sign = (P.eval t - t).sign := ?body✝;\n ?_)",
"before_state": "case pos\nP : ℤ[X]\nt : ℤ\nht : t ∈ periodicPts fun x => eval x P\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\nHdvd : Cycle.Chain (fun x1 x2 => x1 ∣ x2) C\nHabs :\n ∀ (m n : ℕ),\n ((fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t).natAbs =\n ((fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t).natAbs\nHC' : Cycle.Chain (fun x1 x2 => x1 = x2) C\nHeq :\n ∀ (m n : ℕ),\n (fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t =\n (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t\n⊢ IsPeriodicPt (fun x => eval x P) 2 t",
"after_state": "case body\nP : ℤ[X]\nt : ℤ\nht : t ∈ periodicPts fun x => eval x P\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\nHdvd : Cycle.Chain (fun x1 x2 => x1 ∣ x2) C\nHabs :\n ∀ (m n : ℕ),\n ((fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t).natAbs =\n ((fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t).natAbs\nHC' : Cycle.Chain (fun x1 x2 => x1 = x2) C\nHeq :\n ∀ (m n : ℕ),\n (fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t =\n (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t\n⊢ ∀ (n : ℕ), ((fun x => eval x P)^[n + 1] t - t).sign = (eval t P - t).sign\n---\ncase pos\nP : ℤ[X]\nt : ℤ\nht : t ∈ periodicPts fun x => eval x P\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\nHdvd : Cycle.Chain (fun x1 x2 => x1 ∣ x2) C\nHabs :\n ∀ (m n : ℕ),\n ((fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t).natAbs =\n ((fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t).natAbs\nHC' : Cycle.Chain (fun x1 x2 => x1 = x2) C\nHeq :\n ∀ (m n : ℕ),\n (fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t =\n (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t\nIH : ∀ (n : ℕ), ((fun x => eval x P)^[n + 1] t - t).sign = (eval t P - t).sign\n⊢ IsPeriodicPt (fun x => eval x P) 2 t"
},
{
"line": "case body✝ =>\n with_annotate_state\"by\"\n ( intro n\n induction' n with n IH\n · rfl\n · apply Eq.trans _ (Int.sign_add_eq_of_sign_eq IH)\n have H := Heq n.succ 0\n dsimp at H ⊢\n rw [← H]\n rw [sub_add_sub_cancel']\n -- This implies that the sign of P(t) - t is the same as the sign of P^k(t) - t, which is 0.\n -- Hence P(t) = t and P(P(t)) = P(t).)",
"before_state": "case body\nP : ℤ[X]\nt : ℤ\nht : t ∈ periodicPts fun x => eval x P\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\nHdvd : Cycle.Chain (fun x1 x2 => x1 ∣ x2) C\nHabs :\n ∀ (m n : ℕ),\n ((fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t).natAbs =\n ((fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t).natAbs\nHC' : Cycle.Chain (fun x1 x2 => x1 = x2) C\nHeq :\n ∀ (m n : ℕ),\n (fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t =\n (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t\n⊢ ∀ (n : ℕ), ((fun x => eval x P)^[n + 1] t - t).sign = (eval t P - t).sign\n---\ncase pos\nP : ℤ[X]\nt : ℤ\nht : t ∈ periodicPts fun x => eval x P\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\nHdvd : Cycle.Chain (fun x1 x2 => x1 ∣ x2) C\nHabs :\n ∀ (m n : ℕ),\n ((fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t).natAbs =\n ((fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t).natAbs\nHC' : Cycle.Chain (fun x1 x2 => x1 = x2) C\nHeq :\n ∀ (m n : ℕ),\n (fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t =\n (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t\nIH : ∀ (n : ℕ), ((fun x => eval x P)^[n + 1] t - t).sign = (eval t P - t).sign\n⊢ IsPeriodicPt (fun x => eval x P) 2 t",
"after_state": "case pos\nP : ℤ[X]\nt : ℤ\nht : t ∈ periodicPts fun x => eval x P\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\nHdvd : Cycle.Chain (fun x1 x2 => x1 ∣ x2) C\nHabs :\n ∀ (m n : ℕ),\n ((fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t).natAbs =\n ((fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t).natAbs\nHC' : Cycle.Chain (fun x1 x2 => x1 = x2) C\nHeq :\n ∀ (m n : ℕ),\n (fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t =\n (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t\nIH : ∀ (n : ℕ), ((fun x => eval x P)^[n + 1] t - t).sign = (eval t P - t).sign\n⊢ IsPeriodicPt (fun x => eval x P) 2 t"
},
{
"line": "with_annotate_state\"by\"\n ( intro n\n induction' n with n IH\n · rfl\n · apply Eq.trans _ (Int.sign_add_eq_of_sign_eq IH)\n have H := Heq n.succ 0\n dsimp at H ⊢\n rw [← H]\n rw [sub_add_sub_cancel']\n -- This implies that the sign of P(t) - t is the same as the sign of P^k(t) - t, which is 0.\n -- Hence P(t) = t and P(P(t)) = P(t).)",
"before_state": "P : ℤ[X]\nt : ℤ\nht : t ∈ periodicPts fun x => eval x P\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\nHdvd : Cycle.Chain (fun x1 x2 => x1 ∣ x2) C\nHabs :\n ∀ (m n : ℕ),\n ((fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t).natAbs =\n ((fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t).natAbs\nHC' : Cycle.Chain (fun x1 x2 => x1 = x2) C\nHeq :\n ∀ (m n : ℕ),\n (fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t =\n (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t\n⊢ ∀ (n : ℕ), ((fun x => eval x P)^[n + 1] t - t).sign = (eval t P - t).sign",
"after_state": "No Goals!"
},
{
"line": "intro n",
"before_state": "P : ℤ[X]\nt : ℤ\nht : t ∈ periodicPts fun x => eval x P\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\nHdvd : Cycle.Chain (fun x1 x2 => x1 ∣ x2) C\nHabs :\n ∀ (m n : ℕ),\n ((fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t).natAbs =\n ((fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t).natAbs\nHC' : Cycle.Chain (fun x1 x2 => x1 = x2) C\nHeq :\n ∀ (m n : ℕ),\n (fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t =\n (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t\n⊢ ∀ (n : ℕ), ((fun x => eval x P)^[n + 1] t - t).sign = (eval t P - t).sign",
"after_state": "P : ℤ[X]\nt : ℤ\nht : t ∈ periodicPts fun x => eval x P\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\nHdvd : Cycle.Chain (fun x1 x2 => x1 ∣ x2) C\nHabs :\n ∀ (m n : ℕ),\n ((fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t).natAbs =\n ((fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t).natAbs\nHC' : Cycle.Chain (fun x1 x2 => x1 = x2) C\nHeq :\n ∀ (m n : ℕ),\n (fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t =\n (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t\nn : ℕ\n⊢ ((fun x => eval x P)^[n + 1] t - t).sign = (eval t P - t).sign"
},
{
"line": "induction' n with n IH",
"before_state": "P : ℤ[X]\nt : ℤ\nht : t ∈ periodicPts fun x => eval x P\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\nHdvd : Cycle.Chain (fun x1 x2 => x1 ∣ x2) C\nHabs :\n ∀ (m n : ℕ),\n ((fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t).natAbs =\n ((fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t).natAbs\nHC' : Cycle.Chain (fun x1 x2 => x1 = x2) C\nHeq :\n ∀ (m n : ℕ),\n (fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t =\n (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t\nn : ℕ\n⊢ ((fun x => eval x P)^[n + 1] t - t).sign = (eval t P - t).sign",
"after_state": "case zero\nP : ℤ[X]\nt : ℤ\nht : t ∈ periodicPts fun x => eval x P\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\nHdvd : Cycle.Chain (fun x1 x2 => x1 ∣ x2) C\nHabs :\n ∀ (m n : ℕ),\n ((fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t).natAbs =\n ((fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t).natAbs\nHC' : Cycle.Chain (fun x1 x2 => x1 = x2) C\nHeq :\n ∀ (m n : ℕ),\n (fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t =\n (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t\n⊢ ((fun x => eval x P)^[0 + 1] t - t).sign = (eval t P - t).sign\n---\ncase succ\nP : ℤ[X]\nt : ℤ\nht : t ∈ periodicPts fun x => eval x P\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\nHdvd : Cycle.Chain (fun x1 x2 => x1 ∣ x2) C\nHabs :\n ∀ (m n : ℕ),\n ((fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t).natAbs =\n ((fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t).natAbs\nHC' : Cycle.Chain (fun x1 x2 => x1 = x2) C\nHeq :\n ∀ (m n : ℕ),\n (fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t =\n (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t\nn : ℕ\nIH : ((fun x => eval x P)^[n + 1] t - t).sign = (eval t P - t).sign\n⊢ ((fun x => eval x P)^[n + 1 + 1] t - t).sign = (eval t P - t).sign"
},
{
"line": "rfl",
"before_state": "case zero\nP : ℤ[X]\nt : ℤ\nht : t ∈ periodicPts fun x => eval x P\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\nHdvd : Cycle.Chain (fun x1 x2 => x1 ∣ x2) C\nHabs :\n ∀ (m n : ℕ),\n ((fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t).natAbs =\n ((fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t).natAbs\nHC' : Cycle.Chain (fun x1 x2 => x1 = x2) C\nHeq :\n ∀ (m n : ℕ),\n (fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t =\n (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t\n⊢ ((fun x => eval x P)^[0 + 1] t - t).sign = (eval t P - t).sign",
"after_state": "No Goals!"
},
{
"line": "eq_refl",
"before_state": "case zero\nP : ℤ[X]\nt : ℤ\nht : t ∈ periodicPts fun x => eval x P\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\nHdvd : Cycle.Chain (fun x1 x2 => x1 ∣ x2) C\nHabs :\n ∀ (m n : ℕ),\n ((fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t).natAbs =\n ((fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t).natAbs\nHC' : Cycle.Chain (fun x1 x2 => x1 = x2) C\nHeq :\n ∀ (m n : ℕ),\n (fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t =\n (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t\n⊢ ((fun x => eval x P)^[0 + 1] t - t).sign = (eval t P - t).sign",
"after_state": "No Goals!"
},
{
"line": "apply Eq.trans _ (Int.sign_add_eq_of_sign_eq IH)",
"before_state": "case succ\nP : ℤ[X]\nt : ℤ\nht : t ∈ periodicPts fun x => eval x P\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\nHdvd : Cycle.Chain (fun x1 x2 => x1 ∣ x2) C\nHabs :\n ∀ (m n : ℕ),\n ((fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t).natAbs =\n ((fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t).natAbs\nHC' : Cycle.Chain (fun x1 x2 => x1 = x2) C\nHeq :\n ∀ (m n : ℕ),\n (fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t =\n (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t\nn : ℕ\nIH : ((fun x => eval x P)^[n + 1] t - t).sign = (eval t P - t).sign\n⊢ ((fun x => eval x P)^[n + 1 + 1] t - t).sign = (eval t P - t).sign",
"after_state": "P : ℤ[X]\nt : ℤ\nht : t ∈ periodicPts fun x => eval x P\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\nHdvd : Cycle.Chain (fun x1 x2 => x1 ∣ x2) C\nHabs :\n ∀ (m n : ℕ),\n ((fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t).natAbs =\n ((fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t).natAbs\nHC' : Cycle.Chain (fun x1 x2 => x1 = x2) C\nHeq :\n ∀ (m n : ℕ),\n (fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t =\n (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t\nn : ℕ\nIH : ((fun x => eval x P)^[n + 1] t - t).sign = (eval t P - t).sign\n⊢ ((fun x => eval x P)^[n + 1 + 1] t - t).sign = ((fun x => eval x P)^[n + 1] t - t + (eval t P - t)).sign"
},
{
"line": "have H := Heq n.succ 0",
"before_state": "P : ℤ[X]\nt : ℤ\nht : t ∈ periodicPts fun x => eval x P\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\nHdvd : Cycle.Chain (fun x1 x2 => x1 ∣ x2) C\nHabs :\n ∀ (m n : ℕ),\n ((fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t).natAbs =\n ((fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t).natAbs\nHC' : Cycle.Chain (fun x1 x2 => x1 = x2) C\nHeq :\n ∀ (m n : ℕ),\n (fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t =\n (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t\nn : ℕ\nIH : ((fun x => eval x P)^[n + 1] t - t).sign = (eval t P - t).sign\n⊢ ((fun x => eval x P)^[n + 1 + 1] t - t).sign = ((fun x => eval x P)^[n + 1] t - t + (eval t P - t)).sign",
"after_state": "P : ℤ[X]\nt : ℤ\nht : t ∈ periodicPts fun x => eval x P\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\nHdvd : Cycle.Chain (fun x1 x2 => x1 ∣ x2) C\nHabs :\n ∀ (m n : ℕ),\n ((fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t).natAbs =\n ((fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t).natAbs\nHC' : Cycle.Chain (fun x1 x2 => x1 = x2) C\nHeq :\n ∀ (m n : ℕ),\n (fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t =\n (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t\nn : ℕ\nIH : ((fun x => eval x P)^[n + 1] t - t).sign = (eval t P - t).sign\nH :\n (fun x => eval x P)^[n.succ + 1] t - (fun x => eval x P)^[n.succ] t =\n (fun x => eval x P)^[0 + 1] t - (fun x => eval x P)^[0] t\n⊢ ((fun x => eval x P)^[n + 1 + 1] t - t).sign = ((fun x => eval x P)^[n + 1] t - t + (eval t P - t)).sign"
},
{
"line": "refine_lift\n have H := Heq n.succ 0;\n ?_",
"before_state": "P : ℤ[X]\nt : ℤ\nht : t ∈ periodicPts fun x => eval x P\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\nHdvd : Cycle.Chain (fun x1 x2 => x1 ∣ x2) C\nHabs :\n ∀ (m n : ℕ),\n ((fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t).natAbs =\n ((fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t).natAbs\nHC' : Cycle.Chain (fun x1 x2 => x1 = x2) C\nHeq :\n ∀ (m n : ℕ),\n (fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t =\n (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t\nn : ℕ\nIH : ((fun x => eval x P)^[n + 1] t - t).sign = (eval t P - t).sign\n⊢ ((fun x => eval x P)^[n + 1 + 1] t - t).sign = ((fun x => eval x P)^[n + 1] t - t + (eval t P - t)).sign",
"after_state": "P : ℤ[X]\nt : ℤ\nht : t ∈ periodicPts fun x => eval x P\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\nHdvd : Cycle.Chain (fun x1 x2 => x1 ∣ x2) C\nHabs :\n ∀ (m n : ℕ),\n ((fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t).natAbs =\n ((fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t).natAbs\nHC' : Cycle.Chain (fun x1 x2 => x1 = x2) C\nHeq :\n ∀ (m n : ℕ),\n (fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t =\n (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t\nn : ℕ\nIH : ((fun x => eval x P)^[n + 1] t - t).sign = (eval t P - t).sign\nH :\n (fun x => eval x P)^[n.succ + 1] t - (fun x => eval x P)^[n.succ] t =\n (fun x => eval x P)^[0 + 1] t - (fun x => eval x P)^[0] t\n⊢ ((fun x => eval x P)^[n + 1 + 1] t - t).sign = ((fun x => eval x P)^[n + 1] t - t + (eval t P - t)).sign"
},
{
"line": "focus\n (refine\n no_implicit_lambda%\n (have H := Heq n.succ 0;\n ?_);\n rotate_right)",
"before_state": "P : ℤ[X]\nt : ℤ\nht : t ∈ periodicPts fun x => eval x P\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\nHdvd : Cycle.Chain (fun x1 x2 => x1 ∣ x2) C\nHabs :\n ∀ (m n : ℕ),\n ((fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t).natAbs =\n ((fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t).natAbs\nHC' : Cycle.Chain (fun x1 x2 => x1 = x2) C\nHeq :\n ∀ (m n : ℕ),\n (fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t =\n (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t\nn : ℕ\nIH : ((fun x => eval x P)^[n + 1] t - t).sign = (eval t P - t).sign\n⊢ ((fun x => eval x P)^[n + 1 + 1] t - t).sign = ((fun x => eval x P)^[n + 1] t - t + (eval t P - t)).sign",
"after_state": "P : ℤ[X]\nt : ℤ\nht : t ∈ periodicPts fun x => eval x P\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\nHdvd : Cycle.Chain (fun x1 x2 => x1 ∣ x2) C\nHabs :\n ∀ (m n : ℕ),\n ((fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t).natAbs =\n ((fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t).natAbs\nHC' : Cycle.Chain (fun x1 x2 => x1 = x2) C\nHeq :\n ∀ (m n : ℕ),\n (fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t =\n (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t\nn : ℕ\nIH : ((fun x => eval x P)^[n + 1] t - t).sign = (eval t P - t).sign\nH :\n (fun x => eval x P)^[n.succ + 1] t - (fun x => eval x P)^[n.succ] t =\n (fun x => eval x P)^[0 + 1] t - (fun x => eval x P)^[0] t\n⊢ ((fun x => eval x P)^[n + 1 + 1] t - t).sign = ((fun x => eval x P)^[n + 1] t - t + (eval t P - t)).sign"
},
{
"line": "refine\n no_implicit_lambda%\n (have H := Heq n.succ 0;\n ?_)",
"before_state": "P : ℤ[X]\nt : ℤ\nht : t ∈ periodicPts fun x => eval x P\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\nHdvd : Cycle.Chain (fun x1 x2 => x1 ∣ x2) C\nHabs :\n ∀ (m n : ℕ),\n ((fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t).natAbs =\n ((fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t).natAbs\nHC' : Cycle.Chain (fun x1 x2 => x1 = x2) C\nHeq :\n ∀ (m n : ℕ),\n (fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t =\n (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t\nn : ℕ\nIH : ((fun x => eval x P)^[n + 1] t - t).sign = (eval t P - t).sign\n⊢ ((fun x => eval x P)^[n + 1 + 1] t - t).sign = ((fun x => eval x P)^[n + 1] t - t + (eval t P - t)).sign",
"after_state": "P : ℤ[X]\nt : ℤ\nht : t ∈ periodicPts fun x => eval x P\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\nHdvd : Cycle.Chain (fun x1 x2 => x1 ∣ x2) C\nHabs :\n ∀ (m n : ℕ),\n ((fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t).natAbs =\n ((fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t).natAbs\nHC' : Cycle.Chain (fun x1 x2 => x1 = x2) C\nHeq :\n ∀ (m n : ℕ),\n (fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t =\n (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t\nn : ℕ\nIH : ((fun x => eval x P)^[n + 1] t - t).sign = (eval t P - t).sign\nH :\n (fun x => eval x P)^[n.succ + 1] t - (fun x => eval x P)^[n.succ] t =\n (fun x => eval x P)^[0 + 1] t - (fun x => eval x P)^[0] t\n⊢ ((fun x => eval x P)^[n + 1 + 1] t - t).sign = ((fun x => eval x P)^[n + 1] t - t + (eval t P - t)).sign"
},
{
"line": "rotate_right",
"before_state": "P : ℤ[X]\nt : ℤ\nht : t ∈ periodicPts fun x => eval x P\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\nHdvd : Cycle.Chain (fun x1 x2 => x1 ∣ x2) C\nHabs :\n ∀ (m n : ℕ),\n ((fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t).natAbs =\n ((fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t).natAbs\nHC' : Cycle.Chain (fun x1 x2 => x1 = x2) C\nHeq :\n ∀ (m n : ℕ),\n (fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t =\n (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t\nn : ℕ\nIH : ((fun x => eval x P)^[n + 1] t - t).sign = (eval t P - t).sign\nH :\n (fun x => eval x P)^[n.succ + 1] t - (fun x => eval x P)^[n.succ] t =\n (fun x => eval x P)^[0 + 1] t - (fun x => eval x P)^[0] t\n⊢ ((fun x => eval x P)^[n + 1 + 1] t - t).sign = ((fun x => eval x P)^[n + 1] t - t + (eval t P - t)).sign",
"after_state": "P : ℤ[X]\nt : ℤ\nht : t ∈ periodicPts fun x => eval x P\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\nHdvd : Cycle.Chain (fun x1 x2 => x1 ∣ x2) C\nHabs :\n ∀ (m n : ℕ),\n ((fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t).natAbs =\n ((fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t).natAbs\nHC' : Cycle.Chain (fun x1 x2 => x1 = x2) C\nHeq :\n ∀ (m n : ℕ),\n (fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t =\n (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t\nn : ℕ\nIH : ((fun x => eval x P)^[n + 1] t - t).sign = (eval t P - t).sign\nH :\n (fun x => eval x P)^[n.succ + 1] t - (fun x => eval x P)^[n.succ] t =\n (fun x => eval x P)^[0 + 1] t - (fun x => eval x P)^[0] t\n⊢ ((fun x => eval x P)^[n + 1 + 1] t - t).sign = ((fun x => eval x P)^[n + 1] t - t + (eval t P - t)).sign"
},
{
"line": "dsimp at H ⊢",
"before_state": "P : ℤ[X]\nt : ℤ\nht : t ∈ periodicPts fun x => eval x P\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\nHdvd : Cycle.Chain (fun x1 x2 => x1 ∣ x2) C\nHabs :\n ∀ (m n : ℕ),\n ((fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t).natAbs =\n ((fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t).natAbs\nHC' : Cycle.Chain (fun x1 x2 => x1 = x2) C\nHeq :\n ∀ (m n : ℕ),\n (fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t =\n (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t\nn : ℕ\nIH : ((fun x => eval x P)^[n + 1] t - t).sign = (eval t P - t).sign\nH :\n (fun x => eval x P)^[n.succ + 1] t - (fun x => eval x P)^[n.succ] t =\n (fun x => eval x P)^[0 + 1] t - (fun x => eval x P)^[0] t\n⊢ ((fun x => eval x P)^[n + 1 + 1] t - t).sign = ((fun x => eval x P)^[n + 1] t - t + (eval t P - t)).sign",
"after_state": "P : ℤ[X]\nt : ℤ\nht : t ∈ periodicPts fun x => eval x P\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\nHdvd : Cycle.Chain (fun x1 x2 => x1 ∣ x2) C\nHabs :\n ∀ (m n : ℕ),\n ((fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t).natAbs =\n ((fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t).natAbs\nHC' : Cycle.Chain (fun x1 x2 => x1 = x2) C\nHeq :\n ∀ (m n : ℕ),\n (fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t =\n (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t\nn : ℕ\nIH : ((fun x => eval x P)^[n + 1] t - t).sign = (eval t P - t).sign\nH : (fun x => eval x P)^[n] (eval (eval t P) P) - (fun x => eval x P)^[n] (eval t P) = eval t P - t\n⊢ ((fun x => eval x P)^[n] (eval (eval t P) P) - t).sign =\n ((fun x => eval x P)^[n] (eval t P) - t + (eval t P - t)).sign"
},
{
"line": "rw [← H]",
"before_state": "P : ℤ[X]\nt : ℤ\nht : t ∈ periodicPts fun x => eval x P\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\nHdvd : Cycle.Chain (fun x1 x2 => x1 ∣ x2) C\nHabs :\n ∀ (m n : ℕ),\n ((fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t).natAbs =\n ((fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t).natAbs\nHC' : Cycle.Chain (fun x1 x2 => x1 = x2) C\nHeq :\n ∀ (m n : ℕ),\n (fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t =\n (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t\nn : ℕ\nIH : ((fun x => eval x P)^[n + 1] t - t).sign = (eval t P - t).sign\nH : (fun x => eval x P)^[n] (eval (eval t P) P) - (fun x => eval x P)^[n] (eval t P) = eval t P - t\n⊢ ((fun x => eval x P)^[n] (eval (eval t P) P) - t).sign =\n ((fun x => eval x P)^[n] (eval t P) - t + (eval t P - t)).sign",
"after_state": "P : ℤ[X]\nt : ℤ\nht : t ∈ periodicPts fun x => eval x P\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\nHdvd : Cycle.Chain (fun x1 x2 => x1 ∣ x2) C\nHabs :\n ∀ (m n : ℕ),\n ((fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t).natAbs =\n ((fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t).natAbs\nHC' : Cycle.Chain (fun x1 x2 => x1 = x2) C\nHeq :\n ∀ (m n : ℕ),\n (fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t =\n (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t\nn : ℕ\nIH : ((fun x => eval x P)^[n + 1] t - t).sign = (eval t P - t).sign\nH : (fun x => eval x P)^[n] (eval (eval t P) P) - (fun x => eval x P)^[n] (eval t P) = eval t P - t\n⊢ ((fun x => eval x P)^[n] (eval (eval t P) P) - t).sign =\n ((fun x => eval x P)^[n] (eval t P) - t +\n ((fun x => eval x P)^[n] (eval (eval t P) P) - (fun x => eval x P)^[n] (eval t P))).sign"
},
{
"line": "rewrite [← H]",
"before_state": "P : ℤ[X]\nt : ℤ\nht : t ∈ periodicPts fun x => eval x P\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\nHdvd : Cycle.Chain (fun x1 x2 => x1 ∣ x2) C\nHabs :\n ∀ (m n : ℕ),\n ((fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t).natAbs =\n ((fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t).natAbs\nHC' : Cycle.Chain (fun x1 x2 => x1 = x2) C\nHeq :\n ∀ (m n : ℕ),\n (fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t =\n (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t\nn : ℕ\nIH : ((fun x => eval x P)^[n + 1] t - t).sign = (eval t P - t).sign\nH : (fun x => eval x P)^[n] (eval (eval t P) P) - (fun x => eval x P)^[n] (eval t P) = eval t P - t\n⊢ ((fun x => eval x P)^[n] (eval (eval t P) P) - t).sign =\n ((fun x => eval x P)^[n] (eval t P) - t + (eval t P - t)).sign",
"after_state": "P : ℤ[X]\nt : ℤ\nht : t ∈ periodicPts fun x => eval x P\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\nHdvd : Cycle.Chain (fun x1 x2 => x1 ∣ x2) C\nHabs :\n ∀ (m n : ℕ),\n ((fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t).natAbs =\n ((fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t).natAbs\nHC' : Cycle.Chain (fun x1 x2 => x1 = x2) C\nHeq :\n ∀ (m n : ℕ),\n (fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t =\n (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t\nn : ℕ\nIH : ((fun x => eval x P)^[n + 1] t - t).sign = (eval t P - t).sign\nH : (fun x => eval x P)^[n] (eval (eval t P) P) - (fun x => eval x P)^[n] (eval t P) = eval t P - t\n⊢ ((fun x => eval x P)^[n] (eval (eval t P) P) - t).sign =\n ((fun x => eval x P)^[n] (eval t P) - t +\n ((fun x => eval x P)^[n] (eval (eval t P) P) - (fun x => eval x P)^[n] (eval t P))).sign"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "P : ℤ[X]\nt : ℤ\nht : t ∈ periodicPts fun x => eval x P\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\nHdvd : Cycle.Chain (fun x1 x2 => x1 ∣ x2) C\nHabs :\n ∀ (m n : ℕ),\n ((fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t).natAbs =\n ((fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t).natAbs\nHC' : Cycle.Chain (fun x1 x2 => x1 = x2) C\nHeq :\n ∀ (m n : ℕ),\n (fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t =\n (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t\nn : ℕ\nIH : ((fun x => eval x P)^[n + 1] t - t).sign = (eval t P - t).sign\nH : (fun x => eval x P)^[n] (eval (eval t P) P) - (fun x => eval x P)^[n] (eval t P) = eval t P - t\n⊢ ((fun x => eval x P)^[n] (eval (eval t P) P) - t).sign =\n ((fun x => eval x P)^[n] (eval t P) - t +\n ((fun x => eval x P)^[n] (eval (eval t P) P) - (fun x => eval x P)^[n] (eval t P))).sign",
"after_state": "P : ℤ[X]\nt : ℤ\nht : t ∈ periodicPts fun x => eval x P\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\nHdvd : Cycle.Chain (fun x1 x2 => x1 ∣ x2) C\nHabs :\n ∀ (m n : ℕ),\n ((fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t).natAbs =\n ((fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t).natAbs\nHC' : Cycle.Chain (fun x1 x2 => x1 = x2) C\nHeq :\n ∀ (m n : ℕ),\n (fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t =\n (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t\nn : ℕ\nIH : ((fun x => eval x P)^[n + 1] t - t).sign = (eval t P - t).sign\nH : (fun x => eval x P)^[n] (eval (eval t P) P) - (fun x => eval x P)^[n] (eval t P) = eval t P - t\n⊢ ((fun x => eval x P)^[n] (eval (eval t P) P) - t).sign =\n ((fun x => eval x P)^[n] (eval t P) - t +\n ((fun x => eval x P)^[n] (eval (eval t P) P) - (fun x => eval x P)^[n] (eval t P))).sign"
},
{
"line": "try (with_reducible rfl)",
"before_state": "P : ℤ[X]\nt : ℤ\nht : t ∈ periodicPts fun x => eval x P\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\nHdvd : Cycle.Chain (fun x1 x2 => x1 ∣ x2) C\nHabs :\n ∀ (m n : ℕ),\n ((fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t).natAbs =\n ((fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t).natAbs\nHC' : Cycle.Chain (fun x1 x2 => x1 = x2) C\nHeq :\n ∀ (m n : ℕ),\n (fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t =\n (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t\nn : ℕ\nIH : ((fun x => eval x P)^[n + 1] t - t).sign = (eval t P - t).sign\nH : (fun x => eval x P)^[n] (eval (eval t P) P) - (fun x => eval x P)^[n] (eval t P) = eval t P - t\n⊢ ((fun x => eval x P)^[n] (eval (eval t P) P) - t).sign =\n ((fun x => eval x P)^[n] (eval t P) - t +\n ((fun x => eval x P)^[n] (eval (eval t P) P) - (fun x => eval x P)^[n] (eval t P))).sign",
"after_state": "P : ℤ[X]\nt : ℤ\nht : t ∈ periodicPts fun x => eval x P\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\nHdvd : Cycle.Chain (fun x1 x2 => x1 ∣ x2) C\nHabs :\n ∀ (m n : ℕ),\n ((fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t).natAbs =\n ((fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t).natAbs\nHC' : Cycle.Chain (fun x1 x2 => x1 = x2) C\nHeq :\n ∀ (m n : ℕ),\n (fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t =\n (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t\nn : ℕ\nIH : ((fun x => eval x P)^[n + 1] t - t).sign = (eval t P - t).sign\nH : (fun x => eval x P)^[n] (eval (eval t P) P) - (fun x => eval x P)^[n] (eval t P) = eval t P - t\n⊢ ((fun x => eval x P)^[n] (eval (eval t P) P) - t).sign =\n ((fun x => eval x P)^[n] (eval t P) - t +\n ((fun x => eval x P)^[n] (eval (eval t P) P) - (fun x => eval x P)^[n] (eval t P))).sign"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "P : ℤ[X]\nt : ℤ\nht : t ∈ periodicPts fun x => eval x P\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\nHdvd : Cycle.Chain (fun x1 x2 => x1 ∣ x2) C\nHabs :\n ∀ (m n : ℕ),\n ((fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t).natAbs =\n ((fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t).natAbs\nHC' : Cycle.Chain (fun x1 x2 => x1 = x2) C\nHeq :\n ∀ (m n : ℕ),\n (fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t =\n (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t\nn : ℕ\nIH : ((fun x => eval x P)^[n + 1] t - t).sign = (eval t P - t).sign\nH : (fun x => eval x P)^[n] (eval (eval t P) P) - (fun x => eval x P)^[n] (eval t P) = eval t P - t\n⊢ ((fun x => eval x P)^[n] (eval (eval t P) P) - t).sign =\n ((fun x => eval x P)^[n] (eval t P) - t +\n ((fun x => eval x P)^[n] (eval (eval t P) P) - (fun x => eval x P)^[n] (eval t P))).sign",
"after_state": "P : ℤ[X]\nt : ℤ\nht : t ∈ periodicPts fun x => eval x P\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\nHdvd : Cycle.Chain (fun x1 x2 => x1 ∣ x2) C\nHabs :\n ∀ (m n : ℕ),\n ((fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t).natAbs =\n ((fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t).natAbs\nHC' : Cycle.Chain (fun x1 x2 => x1 = x2) C\nHeq :\n ∀ (m n : ℕ),\n (fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t =\n (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t\nn : ℕ\nIH : ((fun x => eval x P)^[n + 1] t - t).sign = (eval t P - t).sign\nH : (fun x => eval x P)^[n] (eval (eval t P) P) - (fun x => eval x P)^[n] (eval t P) = eval t P - t\n⊢ ((fun x => eval x P)^[n] (eval (eval t P) P) - t).sign =\n ((fun x => eval x P)^[n] (eval t P) - t +\n ((fun x => eval x P)^[n] (eval (eval t P) P) - (fun x => eval x P)^[n] (eval t P))).sign"
},
{
"line": "with_reducible rfl",
"before_state": "P : ℤ[X]\nt : ℤ\nht : t ∈ periodicPts fun x => eval x P\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\nHdvd : Cycle.Chain (fun x1 x2 => x1 ∣ x2) C\nHabs :\n ∀ (m n : ℕ),\n ((fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t).natAbs =\n ((fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t).natAbs\nHC' : Cycle.Chain (fun x1 x2 => x1 = x2) C\nHeq :\n ∀ (m n : ℕ),\n (fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t =\n (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t\nn : ℕ\nIH : ((fun x => eval x P)^[n + 1] t - t).sign = (eval t P - t).sign\nH : (fun x => eval x P)^[n] (eval (eval t P) P) - (fun x => eval x P)^[n] (eval t P) = eval t P - t\n⊢ ((fun x => eval x P)^[n] (eval (eval t P) P) - t).sign =\n ((fun x => eval x P)^[n] (eval t P) - t +\n ((fun x => eval x P)^[n] (eval (eval t P) P) - (fun x => eval x P)^[n] (eval t P))).sign",
"after_state": "P : ℤ[X]\nt : ℤ\nht : t ∈ periodicPts fun x => eval x P\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\nHdvd : Cycle.Chain (fun x1 x2 => x1 ∣ x2) C\nHabs :\n ∀ (m n : ℕ),\n ((fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t).natAbs =\n ((fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t).natAbs\nHC' : Cycle.Chain (fun x1 x2 => x1 = x2) C\nHeq :\n ∀ (m n : ℕ),\n (fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t =\n (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t\nn : ℕ\nIH : ((fun x => eval x P)^[n + 1] t - t).sign = (eval t P - t).sign\nH : (fun x => eval x P)^[n] (eval (eval t P) P) - (fun x => eval x P)^[n] (eval t P) = eval t P - t\n⊢ ((fun x => eval x P)^[n] (eval (eval t P) P) - t).sign =\n ((fun x => eval x P)^[n] (eval t P) - t +\n ((fun x => eval x P)^[n] (eval (eval t P) P) - (fun x => eval x P)^[n] (eval t P))).sign"
},
{
"line": "rfl",
"before_state": "P : ℤ[X]\nt : ℤ\nht : t ∈ periodicPts fun x => eval x P\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\nHdvd : Cycle.Chain (fun x1 x2 => x1 ∣ x2) C\nHabs :\n ∀ (m n : ℕ),\n ((fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t).natAbs =\n ((fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t).natAbs\nHC' : Cycle.Chain (fun x1 x2 => x1 = x2) C\nHeq :\n ∀ (m n : ℕ),\n (fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t =\n (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t\nn : ℕ\nIH : ((fun x => eval x P)^[n + 1] t - t).sign = (eval t P - t).sign\nH : (fun x => eval x P)^[n] (eval (eval t P) P) - (fun x => eval x P)^[n] (eval t P) = eval t P - t\n⊢ ((fun x => eval x P)^[n] (eval (eval t P) P) - t).sign =\n ((fun x => eval x P)^[n] (eval t P) - t +\n ((fun x => eval x P)^[n] (eval (eval t P) P) - (fun x => eval x P)^[n] (eval t P))).sign",
"after_state": "P : ℤ[X]\nt : ℤ\nht : t ∈ periodicPts fun x => eval x P\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\nHdvd : Cycle.Chain (fun x1 x2 => x1 ∣ x2) C\nHabs :\n ∀ (m n : ℕ),\n ((fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t).natAbs =\n ((fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t).natAbs\nHC' : Cycle.Chain (fun x1 x2 => x1 = x2) C\nHeq :\n ∀ (m n : ℕ),\n (fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t =\n (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t\nn : ℕ\nIH : ((fun x => eval x P)^[n + 1] t - t).sign = (eval t P - t).sign\nH : (fun x => eval x P)^[n] (eval (eval t P) P) - (fun x => eval x P)^[n] (eval t P) = eval t P - t\n⊢ ((fun x => eval x P)^[n] (eval (eval t P) P) - t).sign =\n ((fun x => eval x P)^[n] (eval t P) - t +\n ((fun x => eval x P)^[n] (eval (eval t P) P) - (fun x => eval x P)^[n] (eval t P))).sign"
},
{
"line": "apply_rfl",
"before_state": "P : ℤ[X]\nt : ℤ\nht : t ∈ periodicPts fun x => eval x P\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\nHdvd : Cycle.Chain (fun x1 x2 => x1 ∣ x2) C\nHabs :\n ∀ (m n : ℕ),\n ((fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t).natAbs =\n ((fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t).natAbs\nHC' : Cycle.Chain (fun x1 x2 => x1 = x2) C\nHeq :\n ∀ (m n : ℕ),\n (fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t =\n (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t\nn : ℕ\nIH : ((fun x => eval x P)^[n + 1] t - t).sign = (eval t P - t).sign\nH : (fun x => eval x P)^[n] (eval (eval t P) P) - (fun x => eval x P)^[n] (eval t P) = eval t P - t\n⊢ ((fun x => eval x P)^[n] (eval (eval t P) P) - t).sign =\n ((fun x => eval x P)^[n] (eval t P) - t +\n ((fun x => eval x P)^[n] (eval (eval t P) P) - (fun x => eval x P)^[n] (eval t P))).sign",
"after_state": "P : ℤ[X]\nt : ℤ\nht : t ∈ periodicPts fun x => eval x P\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\nHdvd : Cycle.Chain (fun x1 x2 => x1 ∣ x2) C\nHabs :\n ∀ (m n : ℕ),\n ((fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t).natAbs =\n ((fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t).natAbs\nHC' : Cycle.Chain (fun x1 x2 => x1 = x2) C\nHeq :\n ∀ (m n : ℕ),\n (fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t =\n (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t\nn : ℕ\nIH : ((fun x => eval x P)^[n + 1] t - t).sign = (eval t P - t).sign\nH : (fun x => eval x P)^[n] (eval (eval t P) P) - (fun x => eval x P)^[n] (eval t P) = eval t P - t\n⊢ ((fun x => eval x P)^[n] (eval (eval t P) P) - t).sign =\n ((fun x => eval x P)^[n] (eval t P) - t +\n ((fun x => eval x P)^[n] (eval (eval t P) P) - (fun x => eval x P)^[n] (eval t P))).sign"
},
{
"line": "skip",
"before_state": "P : ℤ[X]\nt : ℤ\nht : t ∈ periodicPts fun x => eval x P\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\nHdvd : Cycle.Chain (fun x1 x2 => x1 ∣ x2) C\nHabs :\n ∀ (m n : ℕ),\n ((fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t).natAbs =\n ((fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t).natAbs\nHC' : Cycle.Chain (fun x1 x2 => x1 = x2) C\nHeq :\n ∀ (m n : ℕ),\n (fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t =\n (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t\nn : ℕ\nIH : ((fun x => eval x P)^[n + 1] t - t).sign = (eval t P - t).sign\nH : (fun x => eval x P)^[n] (eval (eval t P) P) - (fun x => eval x P)^[n] (eval t P) = eval t P - t\n⊢ ((fun x => eval x P)^[n] (eval (eval t P) P) - t).sign =\n ((fun x => eval x P)^[n] (eval t P) - t +\n ((fun x => eval x P)^[n] (eval (eval t P) P) - (fun x => eval x P)^[n] (eval t P))).sign",
"after_state": "P : ℤ[X]\nt : ℤ\nht : t ∈ periodicPts fun x => eval x P\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\nHdvd : Cycle.Chain (fun x1 x2 => x1 ∣ x2) C\nHabs :\n ∀ (m n : ℕ),\n ((fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t).natAbs =\n ((fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t).natAbs\nHC' : Cycle.Chain (fun x1 x2 => x1 = x2) C\nHeq :\n ∀ (m n : ℕ),\n (fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t =\n (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t\nn : ℕ\nIH : ((fun x => eval x P)^[n + 1] t - t).sign = (eval t P - t).sign\nH : (fun x => eval x P)^[n] (eval (eval t P) P) - (fun x => eval x P)^[n] (eval t P) = eval t P - t\n⊢ ((fun x => eval x P)^[n] (eval (eval t P) P) - t).sign =\n ((fun x => eval x P)^[n] (eval t P) - t +\n ((fun x => eval x P)^[n] (eval (eval t P) P) - (fun x => eval x P)^[n] (eval t P))).sign"
},
{
"line": "rw [sub_add_sub_cancel']\n -- This implies that the sign of P(t) - t is the same as the sign of P^k(t) - t, which is 0.\n -- Hence P(t) = t and P(P(t)) = P(t).",
"before_state": "P : ℤ[X]\nt : ℤ\nht : t ∈ periodicPts fun x => eval x P\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\nHdvd : Cycle.Chain (fun x1 x2 => x1 ∣ x2) C\nHabs :\n ∀ (m n : ℕ),\n ((fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t).natAbs =\n ((fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t).natAbs\nHC' : Cycle.Chain (fun x1 x2 => x1 = x2) C\nHeq :\n ∀ (m n : ℕ),\n (fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t =\n (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t\nn : ℕ\nIH : ((fun x => eval x P)^[n + 1] t - t).sign = (eval t P - t).sign\nH : (fun x => eval x P)^[n] (eval (eval t P) P) - (fun x => eval x P)^[n] (eval t P) = eval t P - t\n⊢ ((fun x => eval x P)^[n] (eval (eval t P) P) - t).sign =\n ((fun x => eval x P)^[n] (eval t P) - t +\n ((fun x => eval x P)^[n] (eval (eval t P) P) - (fun x => eval x P)^[n] (eval t P))).sign",
"after_state": "No Goals!"
},
{
"line": "rewrite [sub_add_sub_cancel']",
"before_state": "P : ℤ[X]\nt : ℤ\nht : t ∈ periodicPts fun x => eval x P\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\nHdvd : Cycle.Chain (fun x1 x2 => x1 ∣ x2) C\nHabs :\n ∀ (m n : ℕ),\n ((fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t).natAbs =\n ((fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t).natAbs\nHC' : Cycle.Chain (fun x1 x2 => x1 = x2) C\nHeq :\n ∀ (m n : ℕ),\n (fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t =\n (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t\nn : ℕ\nIH : ((fun x => eval x P)^[n + 1] t - t).sign = (eval t P - t).sign\nH : (fun x => eval x P)^[n] (eval (eval t P) P) - (fun x => eval x P)^[n] (eval t P) = eval t P - t\n⊢ ((fun x => eval x P)^[n] (eval (eval t P) P) - t).sign =\n ((fun x => eval x P)^[n] (eval t P) - t +\n ((fun x => eval x P)^[n] (eval (eval t P) P) - (fun x => eval x P)^[n] (eval t P))).sign",
"after_state": "P : ℤ[X]\nt : ℤ\nht : t ∈ periodicPts fun x => eval x P\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\nHdvd : Cycle.Chain (fun x1 x2 => x1 ∣ x2) C\nHabs :\n ∀ (m n : ℕ),\n ((fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t).natAbs =\n ((fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t).natAbs\nHC' : Cycle.Chain (fun x1 x2 => x1 = x2) C\nHeq :\n ∀ (m n : ℕ),\n (fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t =\n (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t\nn : ℕ\nIH : ((fun x => eval x P)^[n + 1] t - t).sign = (eval t P - t).sign\nH : (fun x => eval x P)^[n] (eval (eval t P) P) - (fun x => eval x P)^[n] (eval t P) = eval t P - t\n⊢ ((fun x => eval x P)^[n] (eval (eval t P) P) - t).sign = ((fun x => eval x P)^[n] (eval (eval t P) P) - t).sign"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "P : ℤ[X]\nt : ℤ\nht : t ∈ periodicPts fun x => eval x P\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\nHdvd : Cycle.Chain (fun x1 x2 => x1 ∣ x2) C\nHabs :\n ∀ (m n : ℕ),\n ((fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t).natAbs =\n ((fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t).natAbs\nHC' : Cycle.Chain (fun x1 x2 => x1 = x2) C\nHeq :\n ∀ (m n : ℕ),\n (fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t =\n (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t\nn : ℕ\nIH : ((fun x => eval x P)^[n + 1] t - t).sign = (eval t P - t).sign\nH : (fun x => eval x P)^[n] (eval (eval t P) P) - (fun x => eval x P)^[n] (eval t P) = eval t P - t\n⊢ ((fun x => eval x P)^[n] (eval (eval t P) P) - t).sign = ((fun x => eval x P)^[n] (eval (eval t P) P) - t).sign",
"after_state": "No Goals!"
},
{
"line": "try (with_reducible rfl)",
"before_state": "P : ℤ[X]\nt : ℤ\nht : t ∈ periodicPts fun x => eval x P\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\nHdvd : Cycle.Chain (fun x1 x2 => x1 ∣ x2) C\nHabs :\n ∀ (m n : ℕ),\n ((fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t).natAbs =\n ((fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t).natAbs\nHC' : Cycle.Chain (fun x1 x2 => x1 = x2) C\nHeq :\n ∀ (m n : ℕ),\n (fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t =\n (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t\nn : ℕ\nIH : ((fun x => eval x P)^[n + 1] t - t).sign = (eval t P - t).sign\nH : (fun x => eval x P)^[n] (eval (eval t P) P) - (fun x => eval x P)^[n] (eval t P) = eval t P - t\n⊢ ((fun x => eval x P)^[n] (eval (eval t P) P) - t).sign = ((fun x => eval x P)^[n] (eval (eval t P) P) - t).sign",
"after_state": "No Goals!"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "P : ℤ[X]\nt : ℤ\nht : t ∈ periodicPts fun x => eval x P\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\nHdvd : Cycle.Chain (fun x1 x2 => x1 ∣ x2) C\nHabs :\n ∀ (m n : ℕ),\n ((fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t).natAbs =\n ((fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t).natAbs\nHC' : Cycle.Chain (fun x1 x2 => x1 = x2) C\nHeq :\n ∀ (m n : ℕ),\n (fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t =\n (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t\nn : ℕ\nIH : ((fun x => eval x P)^[n + 1] t - t).sign = (eval t P - t).sign\nH : (fun x => eval x P)^[n] (eval (eval t P) P) - (fun x => eval x P)^[n] (eval t P) = eval t P - t\n⊢ ((fun x => eval x P)^[n] (eval (eval t P) P) - t).sign = ((fun x => eval x P)^[n] (eval (eval t P) P) - t).sign",
"after_state": "No Goals!"
},
{
"line": "with_reducible rfl",
"before_state": "P : ℤ[X]\nt : ℤ\nht : t ∈ periodicPts fun x => eval x P\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\nHdvd : Cycle.Chain (fun x1 x2 => x1 ∣ x2) C\nHabs :\n ∀ (m n : ℕ),\n ((fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t).natAbs =\n ((fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t).natAbs\nHC' : Cycle.Chain (fun x1 x2 => x1 = x2) C\nHeq :\n ∀ (m n : ℕ),\n (fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t =\n (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t\nn : ℕ\nIH : ((fun x => eval x P)^[n + 1] t - t).sign = (eval t P - t).sign\nH : (fun x => eval x P)^[n] (eval (eval t P) P) - (fun x => eval x P)^[n] (eval t P) = eval t P - t\n⊢ ((fun x => eval x P)^[n] (eval (eval t P) P) - t).sign = ((fun x => eval x P)^[n] (eval (eval t P) P) - t).sign",
"after_state": "No Goals!"
},
{
"line": "rfl",
"before_state": "P : ℤ[X]\nt : ℤ\nht : t ∈ periodicPts fun x => eval x P\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\nHdvd : Cycle.Chain (fun x1 x2 => x1 ∣ x2) C\nHabs :\n ∀ (m n : ℕ),\n ((fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t).natAbs =\n ((fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t).natAbs\nHC' : Cycle.Chain (fun x1 x2 => x1 = x2) C\nHeq :\n ∀ (m n : ℕ),\n (fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t =\n (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t\nn : ℕ\nIH : ((fun x => eval x P)^[n + 1] t - t).sign = (eval t P - t).sign\nH : (fun x => eval x P)^[n] (eval (eval t P) P) - (fun x => eval x P)^[n] (eval t P) = eval t P - t\n⊢ ((fun x => eval x P)^[n] (eval (eval t P) P) - t).sign = ((fun x => eval x P)^[n] (eval (eval t P) P) - t).sign",
"after_state": "No Goals!"
},
{
"line": "eq_refl",
"before_state": "P : ℤ[X]\nt : ℤ\nht : t ∈ periodicPts fun x => eval x P\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\nHdvd : Cycle.Chain (fun x1 x2 => x1 ∣ x2) C\nHabs :\n ∀ (m n : ℕ),\n ((fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t).natAbs =\n ((fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t).natAbs\nHC' : Cycle.Chain (fun x1 x2 => x1 = x2) C\nHeq :\n ∀ (m n : ℕ),\n (fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t =\n (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t\nn : ℕ\nIH : ((fun x => eval x P)^[n + 1] t - t).sign = (eval t P - t).sign\nH : (fun x => eval x P)^[n] (eval (eval t P) P) - (fun x => eval x P)^[n] (eval t P) = eval t P - t\n⊢ ((fun x => eval x P)^[n] (eval (eval t P) P) - t).sign = ((fun x => eval x P)^[n] (eval (eval t P) P) - t).sign",
"after_state": "No Goals!"
},
{
"line": "rcases ht with ⟨_ | k, hk, hk'⟩",
"before_state": "case pos\nP : ℤ[X]\nt : ℤ\nht : t ∈ periodicPts fun x => eval x P\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\nHdvd : Cycle.Chain (fun x1 x2 => x1 ∣ x2) C\nHabs :\n ∀ (m n : ℕ),\n ((fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t).natAbs =\n ((fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t).natAbs\nHC' : Cycle.Chain (fun x1 x2 => x1 = x2) C\nHeq :\n ∀ (m n : ℕ),\n (fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t =\n (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t\nIH : ∀ (n : ℕ), ((fun x => eval x P)^[n + 1] t - t).sign = (eval t P - t).sign\n⊢ IsPeriodicPt (fun x => eval x P) 2 t",
"after_state": "case pos.intro.zero.intro\nP : ℤ[X]\nt : ℤ\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\nHdvd : Cycle.Chain (fun x1 x2 => x1 ∣ x2) C\nHabs :\n ∀ (m n : ℕ),\n ((fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t).natAbs =\n ((fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t).natAbs\nHC' : Cycle.Chain (fun x1 x2 => x1 = x2) C\nHeq :\n ∀ (m n : ℕ),\n (fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t =\n (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t\nIH : ∀ (n : ℕ), ((fun x => eval x P)^[n + 1] t - t).sign = (eval t P - t).sign\nhk : 0 > 0\nhk' : IsPeriodicPt (fun x => eval x P) 0 t\n⊢ IsPeriodicPt (fun x => eval x P) 2 t\n---\ncase pos.intro.succ.intro\nP : ℤ[X]\nt : ℤ\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\nHdvd : Cycle.Chain (fun x1 x2 => x1 ∣ x2) C\nHabs :\n ∀ (m n : ℕ),\n ((fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t).natAbs =\n ((fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t).natAbs\nHC' : Cycle.Chain (fun x1 x2 => x1 = x2) C\nHeq :\n ∀ (m n : ℕ),\n (fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t =\n (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t\nIH : ∀ (n : ℕ), ((fun x => eval x P)^[n + 1] t - t).sign = (eval t P - t).sign\nk : ℕ\nhk : k + 1 > 0\nhk' : IsPeriodicPt (fun x => eval x P) (k + 1) t\n⊢ IsPeriodicPt (fun x => eval x P) 2 t"
},
{
"line": "exact (irrefl 0 hk).elim",
"before_state": "case pos.intro.zero.intro\nP : ℤ[X]\nt : ℤ\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\nHdvd : Cycle.Chain (fun x1 x2 => x1 ∣ x2) C\nHabs :\n ∀ (m n : ℕ),\n ((fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t).natAbs =\n ((fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t).natAbs\nHC' : Cycle.Chain (fun x1 x2 => x1 = x2) C\nHeq :\n ∀ (m n : ℕ),\n (fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t =\n (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t\nIH : ∀ (n : ℕ), ((fun x => eval x P)^[n + 1] t - t).sign = (eval t P - t).sign\nhk : 0 > 0\nhk' : IsPeriodicPt (fun x => eval x P) 0 t\n⊢ IsPeriodicPt (fun x => eval x P) 2 t",
"after_state": "No Goals!"
},
{
"line": "have H := IH k",
"before_state": "case pos.intro.succ.intro\nP : ℤ[X]\nt : ℤ\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\nHdvd : Cycle.Chain (fun x1 x2 => x1 ∣ x2) C\nHabs :\n ∀ (m n : ℕ),\n ((fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t).natAbs =\n ((fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t).natAbs\nHC' : Cycle.Chain (fun x1 x2 => x1 = x2) C\nHeq :\n ∀ (m n : ℕ),\n (fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t =\n (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t\nIH : ∀ (n : ℕ), ((fun x => eval x P)^[n + 1] t - t).sign = (eval t P - t).sign\nk : ℕ\nhk : k + 1 > 0\nhk' : IsPeriodicPt (fun x => eval x P) (k + 1) t\n⊢ IsPeriodicPt (fun x => eval x P) 2 t",
"after_state": "case pos.intro.succ.intro\nP : ℤ[X]\nt : ℤ\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\nHdvd : Cycle.Chain (fun x1 x2 => x1 ∣ x2) C\nHabs :\n ∀ (m n : ℕ),\n ((fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t).natAbs =\n ((fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t).natAbs\nHC' : Cycle.Chain (fun x1 x2 => x1 = x2) C\nHeq :\n ∀ (m n : ℕ),\n (fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t =\n (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t\nIH : ∀ (n : ℕ), ((fun x => eval x P)^[n + 1] t - t).sign = (eval t P - t).sign\nk : ℕ\nhk : k + 1 > 0\nhk' : IsPeriodicPt (fun x => eval x P) (k + 1) t\nH : ((fun x => eval x P)^[k + 1] t - t).sign = (eval t P - t).sign\n⊢ IsPeriodicPt (fun x => eval x P) 2 t"
},
{
"line": "refine_lift\n have H := IH k;\n ?_",
"before_state": "case pos.intro.succ.intro\nP : ℤ[X]\nt : ℤ\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\nHdvd : Cycle.Chain (fun x1 x2 => x1 ∣ x2) C\nHabs :\n ∀ (m n : ℕ),\n ((fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t).natAbs =\n ((fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t).natAbs\nHC' : Cycle.Chain (fun x1 x2 => x1 = x2) C\nHeq :\n ∀ (m n : ℕ),\n (fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t =\n (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t\nIH : ∀ (n : ℕ), ((fun x => eval x P)^[n + 1] t - t).sign = (eval t P - t).sign\nk : ℕ\nhk : k + 1 > 0\nhk' : IsPeriodicPt (fun x => eval x P) (k + 1) t\n⊢ IsPeriodicPt (fun x => eval x P) 2 t",
"after_state": "case pos.intro.succ.intro\nP : ℤ[X]\nt : ℤ\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\nHdvd : Cycle.Chain (fun x1 x2 => x1 ∣ x2) C\nHabs :\n ∀ (m n : ℕ),\n ((fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t).natAbs =\n ((fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t).natAbs\nHC' : Cycle.Chain (fun x1 x2 => x1 = x2) C\nHeq :\n ∀ (m n : ℕ),\n (fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t =\n (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t\nIH : ∀ (n : ℕ), ((fun x => eval x P)^[n + 1] t - t).sign = (eval t P - t).sign\nk : ℕ\nhk : k + 1 > 0\nhk' : IsPeriodicPt (fun x => eval x P) (k + 1) t\nH : ((fun x => eval x P)^[k + 1] t - t).sign = (eval t P - t).sign\n⊢ IsPeriodicPt (fun x => eval x P) 2 t"
},
{
"line": "focus\n (refine\n no_implicit_lambda%\n (have H := IH k;\n ?_);\n rotate_right)",
"before_state": "case pos.intro.succ.intro\nP : ℤ[X]\nt : ℤ\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\nHdvd : Cycle.Chain (fun x1 x2 => x1 ∣ x2) C\nHabs :\n ∀ (m n : ℕ),\n ((fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t).natAbs =\n ((fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t).natAbs\nHC' : Cycle.Chain (fun x1 x2 => x1 = x2) C\nHeq :\n ∀ (m n : ℕ),\n (fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t =\n (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t\nIH : ∀ (n : ℕ), ((fun x => eval x P)^[n + 1] t - t).sign = (eval t P - t).sign\nk : ℕ\nhk : k + 1 > 0\nhk' : IsPeriodicPt (fun x => eval x P) (k + 1) t\n⊢ IsPeriodicPt (fun x => eval x P) 2 t",
"after_state": "case pos.intro.succ.intro\nP : ℤ[X]\nt : ℤ\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\nHdvd : Cycle.Chain (fun x1 x2 => x1 ∣ x2) C\nHabs :\n ∀ (m n : ℕ),\n ((fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t).natAbs =\n ((fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t).natAbs\nHC' : Cycle.Chain (fun x1 x2 => x1 = x2) C\nHeq :\n ∀ (m n : ℕ),\n (fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t =\n (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t\nIH : ∀ (n : ℕ), ((fun x => eval x P)^[n + 1] t - t).sign = (eval t P - t).sign\nk : ℕ\nhk : k + 1 > 0\nhk' : IsPeriodicPt (fun x => eval x P) (k + 1) t\nH : ((fun x => eval x P)^[k + 1] t - t).sign = (eval t P - t).sign\n⊢ IsPeriodicPt (fun x => eval x P) 2 t"
},
{
"line": "refine\n no_implicit_lambda%\n (have H := IH k;\n ?_)",
"before_state": "case pos.intro.succ.intro\nP : ℤ[X]\nt : ℤ\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\nHdvd : Cycle.Chain (fun x1 x2 => x1 ∣ x2) C\nHabs :\n ∀ (m n : ℕ),\n ((fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t).natAbs =\n ((fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t).natAbs\nHC' : Cycle.Chain (fun x1 x2 => x1 = x2) C\nHeq :\n ∀ (m n : ℕ),\n (fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t =\n (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t\nIH : ∀ (n : ℕ), ((fun x => eval x P)^[n + 1] t - t).sign = (eval t P - t).sign\nk : ℕ\nhk : k + 1 > 0\nhk' : IsPeriodicPt (fun x => eval x P) (k + 1) t\n⊢ IsPeriodicPt (fun x => eval x P) 2 t",
"after_state": "case pos.intro.succ.intro\nP : ℤ[X]\nt : ℤ\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\nHdvd : Cycle.Chain (fun x1 x2 => x1 ∣ x2) C\nHabs :\n ∀ (m n : ℕ),\n ((fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t).natAbs =\n ((fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t).natAbs\nHC' : Cycle.Chain (fun x1 x2 => x1 = x2) C\nHeq :\n ∀ (m n : ℕ),\n (fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t =\n (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t\nIH : ∀ (n : ℕ), ((fun x => eval x P)^[n + 1] t - t).sign = (eval t P - t).sign\nk : ℕ\nhk : k + 1 > 0\nhk' : IsPeriodicPt (fun x => eval x P) (k + 1) t\nH : ((fun x => eval x P)^[k + 1] t - t).sign = (eval t P - t).sign\n⊢ IsPeriodicPt (fun x => eval x P) 2 t"
},
{
"line": "rotate_right",
"before_state": "case pos.intro.succ.intro\nP : ℤ[X]\nt : ℤ\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\nHdvd : Cycle.Chain (fun x1 x2 => x1 ∣ x2) C\nHabs :\n ∀ (m n : ℕ),\n ((fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t).natAbs =\n ((fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t).natAbs\nHC' : Cycle.Chain (fun x1 x2 => x1 = x2) C\nHeq :\n ∀ (m n : ℕ),\n (fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t =\n (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t\nIH : ∀ (n : ℕ), ((fun x => eval x P)^[n + 1] t - t).sign = (eval t P - t).sign\nk : ℕ\nhk : k + 1 > 0\nhk' : IsPeriodicPt (fun x => eval x P) (k + 1) t\nH : ((fun x => eval x P)^[k + 1] t - t).sign = (eval t P - t).sign\n⊢ IsPeriodicPt (fun x => eval x P) 2 t",
"after_state": "case pos.intro.succ.intro\nP : ℤ[X]\nt : ℤ\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\nHdvd : Cycle.Chain (fun x1 x2 => x1 ∣ x2) C\nHabs :\n ∀ (m n : ℕ),\n ((fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t).natAbs =\n ((fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t).natAbs\nHC' : Cycle.Chain (fun x1 x2 => x1 = x2) C\nHeq :\n ∀ (m n : ℕ),\n (fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t =\n (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t\nIH : ∀ (n : ℕ), ((fun x => eval x P)^[n + 1] t - t).sign = (eval t P - t).sign\nk : ℕ\nhk : k + 1 > 0\nhk' : IsPeriodicPt (fun x => eval x P) (k + 1) t\nH : ((fun x => eval x P)^[k + 1] t - t).sign = (eval t P - t).sign\n⊢ IsPeriodicPt (fun x => eval x P) 2 t"
},
{
"line": "rw [hk'.isFixedPt.eq] at H",
"before_state": "case pos.intro.succ.intro\nP : ℤ[X]\nt : ℤ\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\nHdvd : Cycle.Chain (fun x1 x2 => x1 ∣ x2) C\nHabs :\n ∀ (m n : ℕ),\n ((fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t).natAbs =\n ((fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t).natAbs\nHC' : Cycle.Chain (fun x1 x2 => x1 = x2) C\nHeq :\n ∀ (m n : ℕ),\n (fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t =\n (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t\nIH : ∀ (n : ℕ), ((fun x => eval x P)^[n + 1] t - t).sign = (eval t P - t).sign\nk : ℕ\nhk : k + 1 > 0\nhk' : IsPeriodicPt (fun x => eval x P) (k + 1) t\nH : ((fun x => eval x P)^[k + 1] t - t).sign = (eval t P - t).sign\n⊢ IsPeriodicPt (fun x => eval x P) 2 t",
"after_state": "case pos.intro.succ.intro\nP : ℤ[X]\nt : ℤ\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\nHdvd : Cycle.Chain (fun x1 x2 => x1 ∣ x2) C\nHabs :\n ∀ (m n : ℕ),\n ((fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t).natAbs =\n ((fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t).natAbs\nHC' : Cycle.Chain (fun x1 x2 => x1 = x2) C\nHeq :\n ∀ (m n : ℕ),\n (fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t =\n (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t\nIH : ∀ (n : ℕ), ((fun x => eval x P)^[n + 1] t - t).sign = (eval t P - t).sign\nk : ℕ\nhk : k + 1 > 0\nhk' : IsPeriodicPt (fun x => eval x P) (k + 1) t\nH : (t - t).sign = (eval t P - t).sign\n⊢ IsPeriodicPt (fun x => eval x P) 2 t"
},
{
"line": "rewrite [hk'.isFixedPt.eq] at H",
"before_state": "case pos.intro.succ.intro\nP : ℤ[X]\nt : ℤ\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\nHdvd : Cycle.Chain (fun x1 x2 => x1 ∣ x2) C\nHabs :\n ∀ (m n : ℕ),\n ((fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t).natAbs =\n ((fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t).natAbs\nHC' : Cycle.Chain (fun x1 x2 => x1 = x2) C\nHeq :\n ∀ (m n : ℕ),\n (fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t =\n (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t\nIH : ∀ (n : ℕ), ((fun x => eval x P)^[n + 1] t - t).sign = (eval t P - t).sign\nk : ℕ\nhk : k + 1 > 0\nhk' : IsPeriodicPt (fun x => eval x P) (k + 1) t\nH : ((fun x => eval x P)^[k + 1] t - t).sign = (eval t P - t).sign\n⊢ IsPeriodicPt (fun x => eval x P) 2 t",
"after_state": "case pos.intro.succ.intro\nP : ℤ[X]\nt : ℤ\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\nHdvd : Cycle.Chain (fun x1 x2 => x1 ∣ x2) C\nHabs :\n ∀ (m n : ℕ),\n ((fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t).natAbs =\n ((fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t).natAbs\nHC' : Cycle.Chain (fun x1 x2 => x1 = x2) C\nHeq :\n ∀ (m n : ℕ),\n (fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t =\n (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t\nIH : ∀ (n : ℕ), ((fun x => eval x P)^[n + 1] t - t).sign = (eval t P - t).sign\nk : ℕ\nhk : k + 1 > 0\nhk' : IsPeriodicPt (fun x => eval x P) (k + 1) t\nH : (t - t).sign = (eval t P - t).sign\n⊢ IsPeriodicPt (fun x => eval x P) 2 t"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "case pos.intro.succ.intro\nP : ℤ[X]\nt : ℤ\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\nHdvd : Cycle.Chain (fun x1 x2 => x1 ∣ x2) C\nHabs :\n ∀ (m n : ℕ),\n ((fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t).natAbs =\n ((fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t).natAbs\nHC' : Cycle.Chain (fun x1 x2 => x1 = x2) C\nHeq :\n ∀ (m n : ℕ),\n (fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t =\n (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t\nIH : ∀ (n : ℕ), ((fun x => eval x P)^[n + 1] t - t).sign = (eval t P - t).sign\nk : ℕ\nhk : k + 1 > 0\nhk' : IsPeriodicPt (fun x => eval x P) (k + 1) t\nH : (t - t).sign = (eval t P - t).sign\n⊢ IsPeriodicPt (fun x => eval x P) 2 t",
"after_state": "case pos.intro.succ.intro\nP : ℤ[X]\nt : ℤ\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\nHdvd : Cycle.Chain (fun x1 x2 => x1 ∣ x2) C\nHabs :\n ∀ (m n : ℕ),\n ((fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t).natAbs =\n ((fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t).natAbs\nHC' : Cycle.Chain (fun x1 x2 => x1 = x2) C\nHeq :\n ∀ (m n : ℕ),\n (fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t =\n (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t\nIH : ∀ (n : ℕ), ((fun x => eval x P)^[n + 1] t - t).sign = (eval t P - t).sign\nk : ℕ\nhk : k + 1 > 0\nhk' : IsPeriodicPt (fun x => eval x P) (k + 1) t\nH : (t - t).sign = (eval t P - t).sign\n⊢ IsPeriodicPt (fun x => eval x P) 2 t"
},
{
"line": "try (with_reducible rfl)",
"before_state": "case pos.intro.succ.intro\nP : ℤ[X]\nt : ℤ\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\nHdvd : Cycle.Chain (fun x1 x2 => x1 ∣ x2) C\nHabs :\n ∀ (m n : ℕ),\n ((fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t).natAbs =\n ((fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t).natAbs\nHC' : Cycle.Chain (fun x1 x2 => x1 = x2) C\nHeq :\n ∀ (m n : ℕ),\n (fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t =\n (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t\nIH : ∀ (n : ℕ), ((fun x => eval x P)^[n + 1] t - t).sign = (eval t P - t).sign\nk : ℕ\nhk : k + 1 > 0\nhk' : IsPeriodicPt (fun x => eval x P) (k + 1) t\nH : (t - t).sign = (eval t P - t).sign\n⊢ IsPeriodicPt (fun x => eval x P) 2 t",
"after_state": "case pos.intro.succ.intro\nP : ℤ[X]\nt : ℤ\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\nHdvd : Cycle.Chain (fun x1 x2 => x1 ∣ x2) C\nHabs :\n ∀ (m n : ℕ),\n ((fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t).natAbs =\n ((fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t).natAbs\nHC' : Cycle.Chain (fun x1 x2 => x1 = x2) C\nHeq :\n ∀ (m n : ℕ),\n (fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t =\n (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t\nIH : ∀ (n : ℕ), ((fun x => eval x P)^[n + 1] t - t).sign = (eval t P - t).sign\nk : ℕ\nhk : k + 1 > 0\nhk' : IsPeriodicPt (fun x => eval x P) (k + 1) t\nH : (t - t).sign = (eval t P - t).sign\n⊢ IsPeriodicPt (fun x => eval x P) 2 t"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "case pos.intro.succ.intro\nP : ℤ[X]\nt : ℤ\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\nHdvd : Cycle.Chain (fun x1 x2 => x1 ∣ x2) C\nHabs :\n ∀ (m n : ℕ),\n ((fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t).natAbs =\n ((fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t).natAbs\nHC' : Cycle.Chain (fun x1 x2 => x1 = x2) C\nHeq :\n ∀ (m n : ℕ),\n (fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t =\n (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t\nIH : ∀ (n : ℕ), ((fun x => eval x P)^[n + 1] t - t).sign = (eval t P - t).sign\nk : ℕ\nhk : k + 1 > 0\nhk' : IsPeriodicPt (fun x => eval x P) (k + 1) t\nH : (t - t).sign = (eval t P - t).sign\n⊢ IsPeriodicPt (fun x => eval x P) 2 t",
"after_state": "case pos.intro.succ.intro\nP : ℤ[X]\nt : ℤ\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\nHdvd : Cycle.Chain (fun x1 x2 => x1 ∣ x2) C\nHabs :\n ∀ (m n : ℕ),\n ((fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t).natAbs =\n ((fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t).natAbs\nHC' : Cycle.Chain (fun x1 x2 => x1 = x2) C\nHeq :\n ∀ (m n : ℕ),\n (fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t =\n (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t\nIH : ∀ (n : ℕ), ((fun x => eval x P)^[n + 1] t - t).sign = (eval t P - t).sign\nk : ℕ\nhk : k + 1 > 0\nhk' : IsPeriodicPt (fun x => eval x P) (k + 1) t\nH : (t - t).sign = (eval t P - t).sign\n⊢ IsPeriodicPt (fun x => eval x P) 2 t"
},
{
"line": "with_reducible rfl",
"before_state": "case pos.intro.succ.intro\nP : ℤ[X]\nt : ℤ\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\nHdvd : Cycle.Chain (fun x1 x2 => x1 ∣ x2) C\nHabs :\n ∀ (m n : ℕ),\n ((fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t).natAbs =\n ((fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t).natAbs\nHC' : Cycle.Chain (fun x1 x2 => x1 = x2) C\nHeq :\n ∀ (m n : ℕ),\n (fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t =\n (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t\nIH : ∀ (n : ℕ), ((fun x => eval x P)^[n + 1] t - t).sign = (eval t P - t).sign\nk : ℕ\nhk : k + 1 > 0\nhk' : IsPeriodicPt (fun x => eval x P) (k + 1) t\nH : (t - t).sign = (eval t P - t).sign\n⊢ IsPeriodicPt (fun x => eval x P) 2 t",
"after_state": "case pos.intro.succ.intro\nP : ℤ[X]\nt : ℤ\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\nHdvd : Cycle.Chain (fun x1 x2 => x1 ∣ x2) C\nHabs :\n ∀ (m n : ℕ),\n ((fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t).natAbs =\n ((fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t).natAbs\nHC' : Cycle.Chain (fun x1 x2 => x1 = x2) C\nHeq :\n ∀ (m n : ℕ),\n (fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t =\n (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t\nIH : ∀ (n : ℕ), ((fun x => eval x P)^[n + 1] t - t).sign = (eval t P - t).sign\nk : ℕ\nhk : k + 1 > 0\nhk' : IsPeriodicPt (fun x => eval x P) (k + 1) t\nH : (t - t).sign = (eval t P - t).sign\n⊢ IsPeriodicPt (fun x => eval x P) 2 t"
},
{
"line": "rfl",
"before_state": "case pos.intro.succ.intro\nP : ℤ[X]\nt : ℤ\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\nHdvd : Cycle.Chain (fun x1 x2 => x1 ∣ x2) C\nHabs :\n ∀ (m n : ℕ),\n ((fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t).natAbs =\n ((fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t).natAbs\nHC' : Cycle.Chain (fun x1 x2 => x1 = x2) C\nHeq :\n ∀ (m n : ℕ),\n (fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t =\n (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t\nIH : ∀ (n : ℕ), ((fun x => eval x P)^[n + 1] t - t).sign = (eval t P - t).sign\nk : ℕ\nhk : k + 1 > 0\nhk' : IsPeriodicPt (fun x => eval x P) (k + 1) t\nH : (t - t).sign = (eval t P - t).sign\n⊢ IsPeriodicPt (fun x => eval x P) 2 t",
"after_state": "case pos.intro.succ.intro\nP : ℤ[X]\nt : ℤ\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\nHdvd : Cycle.Chain (fun x1 x2 => x1 ∣ x2) C\nHabs :\n ∀ (m n : ℕ),\n ((fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t).natAbs =\n ((fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t).natAbs\nHC' : Cycle.Chain (fun x1 x2 => x1 = x2) C\nHeq :\n ∀ (m n : ℕ),\n (fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t =\n (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t\nIH : ∀ (n : ℕ), ((fun x => eval x P)^[n + 1] t - t).sign = (eval t P - t).sign\nk : ℕ\nhk : k + 1 > 0\nhk' : IsPeriodicPt (fun x => eval x P) (k + 1) t\nH : (t - t).sign = (eval t P - t).sign\n⊢ IsPeriodicPt (fun x => eval x P) 2 t"
},
{
"line": "apply_rfl",
"before_state": "case pos.intro.succ.intro\nP : ℤ[X]\nt : ℤ\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\nHdvd : Cycle.Chain (fun x1 x2 => x1 ∣ x2) C\nHabs :\n ∀ (m n : ℕ),\n ((fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t).natAbs =\n ((fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t).natAbs\nHC' : Cycle.Chain (fun x1 x2 => x1 = x2) C\nHeq :\n ∀ (m n : ℕ),\n (fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t =\n (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t\nIH : ∀ (n : ℕ), ((fun x => eval x P)^[n + 1] t - t).sign = (eval t P - t).sign\nk : ℕ\nhk : k + 1 > 0\nhk' : IsPeriodicPt (fun x => eval x P) (k + 1) t\nH : (t - t).sign = (eval t P - t).sign\n⊢ IsPeriodicPt (fun x => eval x P) 2 t",
"after_state": "case pos.intro.succ.intro\nP : ℤ[X]\nt : ℤ\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\nHdvd : Cycle.Chain (fun x1 x2 => x1 ∣ x2) C\nHabs :\n ∀ (m n : ℕ),\n ((fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t).natAbs =\n ((fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t).natAbs\nHC' : Cycle.Chain (fun x1 x2 => x1 = x2) C\nHeq :\n ∀ (m n : ℕ),\n (fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t =\n (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t\nIH : ∀ (n : ℕ), ((fun x => eval x P)^[n + 1] t - t).sign = (eval t P - t).sign\nk : ℕ\nhk : k + 1 > 0\nhk' : IsPeriodicPt (fun x => eval x P) (k + 1) t\nH : (t - t).sign = (eval t P - t).sign\n⊢ IsPeriodicPt (fun x => eval x P) 2 t"
},
{
"line": "skip",
"before_state": "case pos.intro.succ.intro\nP : ℤ[X]\nt : ℤ\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\nHdvd : Cycle.Chain (fun x1 x2 => x1 ∣ x2) C\nHabs :\n ∀ (m n : ℕ),\n ((fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t).natAbs =\n ((fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t).natAbs\nHC' : Cycle.Chain (fun x1 x2 => x1 = x2) C\nHeq :\n ∀ (m n : ℕ),\n (fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t =\n (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t\nIH : ∀ (n : ℕ), ((fun x => eval x P)^[n + 1] t - t).sign = (eval t P - t).sign\nk : ℕ\nhk : k + 1 > 0\nhk' : IsPeriodicPt (fun x => eval x P) (k + 1) t\nH : (t - t).sign = (eval t P - t).sign\n⊢ IsPeriodicPt (fun x => eval x P) 2 t",
"after_state": "case pos.intro.succ.intro\nP : ℤ[X]\nt : ℤ\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\nHdvd : Cycle.Chain (fun x1 x2 => x1 ∣ x2) C\nHabs :\n ∀ (m n : ℕ),\n ((fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t).natAbs =\n ((fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t).natAbs\nHC' : Cycle.Chain (fun x1 x2 => x1 = x2) C\nHeq :\n ∀ (m n : ℕ),\n (fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t =\n (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t\nIH : ∀ (n : ℕ), ((fun x => eval x P)^[n + 1] t - t).sign = (eval t P - t).sign\nk : ℕ\nhk : k + 1 > 0\nhk' : IsPeriodicPt (fun x => eval x P) (k + 1) t\nH : (t - t).sign = (eval t P - t).sign\n⊢ IsPeriodicPt (fun x => eval x P) 2 t"
},
{
"line": "rw [sub_self] at H",
"before_state": "case pos.intro.succ.intro\nP : ℤ[X]\nt : ℤ\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\nHdvd : Cycle.Chain (fun x1 x2 => x1 ∣ x2) C\nHabs :\n ∀ (m n : ℕ),\n ((fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t).natAbs =\n ((fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t).natAbs\nHC' : Cycle.Chain (fun x1 x2 => x1 = x2) C\nHeq :\n ∀ (m n : ℕ),\n (fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t =\n (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t\nIH : ∀ (n : ℕ), ((fun x => eval x P)^[n + 1] t - t).sign = (eval t P - t).sign\nk : ℕ\nhk : k + 1 > 0\nhk' : IsPeriodicPt (fun x => eval x P) (k + 1) t\nH : (t - t).sign = (eval t P - t).sign\n⊢ IsPeriodicPt (fun x => eval x P) 2 t",
"after_state": "case pos.intro.succ.intro\nP : ℤ[X]\nt : ℤ\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\nHdvd : Cycle.Chain (fun x1 x2 => x1 ∣ x2) C\nHabs :\n ∀ (m n : ℕ),\n ((fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t).natAbs =\n ((fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t).natAbs\nHC' : Cycle.Chain (fun x1 x2 => x1 = x2) C\nHeq :\n ∀ (m n : ℕ),\n (fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t =\n (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t\nIH : ∀ (n : ℕ), ((fun x => eval x P)^[n + 1] t - t).sign = (eval t P - t).sign\nk : ℕ\nhk : k + 1 > 0\nhk' : IsPeriodicPt (fun x => eval x P) (k + 1) t\nH : Int.sign 0 = (eval t P - t).sign\n⊢ IsPeriodicPt (fun x => eval x P) 2 t"
},
{
"line": "rewrite [sub_self] at H",
"before_state": "case pos.intro.succ.intro\nP : ℤ[X]\nt : ℤ\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\nHdvd : Cycle.Chain (fun x1 x2 => x1 ∣ x2) C\nHabs :\n ∀ (m n : ℕ),\n ((fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t).natAbs =\n ((fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t).natAbs\nHC' : Cycle.Chain (fun x1 x2 => x1 = x2) C\nHeq :\n ∀ (m n : ℕ),\n (fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t =\n (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t\nIH : ∀ (n : ℕ), ((fun x => eval x P)^[n + 1] t - t).sign = (eval t P - t).sign\nk : ℕ\nhk : k + 1 > 0\nhk' : IsPeriodicPt (fun x => eval x P) (k + 1) t\nH : (t - t).sign = (eval t P - t).sign\n⊢ IsPeriodicPt (fun x => eval x P) 2 t",
"after_state": "case pos.intro.succ.intro\nP : ℤ[X]\nt : ℤ\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\nHdvd : Cycle.Chain (fun x1 x2 => x1 ∣ x2) C\nHabs :\n ∀ (m n : ℕ),\n ((fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t).natAbs =\n ((fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t).natAbs\nHC' : Cycle.Chain (fun x1 x2 => x1 = x2) C\nHeq :\n ∀ (m n : ℕ),\n (fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t =\n (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t\nIH : ∀ (n : ℕ), ((fun x => eval x P)^[n + 1] t - t).sign = (eval t P - t).sign\nk : ℕ\nhk : k + 1 > 0\nhk' : IsPeriodicPt (fun x => eval x P) (k + 1) t\nH : Int.sign 0 = (eval t P - t).sign\n⊢ IsPeriodicPt (fun x => eval x P) 2 t"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "case pos.intro.succ.intro\nP : ℤ[X]\nt : ℤ\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\nHdvd : Cycle.Chain (fun x1 x2 => x1 ∣ x2) C\nHabs :\n ∀ (m n : ℕ),\n ((fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t).natAbs =\n ((fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t).natAbs\nHC' : Cycle.Chain (fun x1 x2 => x1 = x2) C\nHeq :\n ∀ (m n : ℕ),\n (fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t =\n (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t\nIH : ∀ (n : ℕ), ((fun x => eval x P)^[n + 1] t - t).sign = (eval t P - t).sign\nk : ℕ\nhk : k + 1 > 0\nhk' : IsPeriodicPt (fun x => eval x P) (k + 1) t\nH : Int.sign 0 = (eval t P - t).sign\n⊢ IsPeriodicPt (fun x => eval x P) 2 t",
"after_state": "case pos.intro.succ.intro\nP : ℤ[X]\nt : ℤ\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\nHdvd : Cycle.Chain (fun x1 x2 => x1 ∣ x2) C\nHabs :\n ∀ (m n : ℕ),\n ((fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t).natAbs =\n ((fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t).natAbs\nHC' : Cycle.Chain (fun x1 x2 => x1 = x2) C\nHeq :\n ∀ (m n : ℕ),\n (fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t =\n (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t\nIH : ∀ (n : ℕ), ((fun x => eval x P)^[n + 1] t - t).sign = (eval t P - t).sign\nk : ℕ\nhk : k + 1 > 0\nhk' : IsPeriodicPt (fun x => eval x P) (k + 1) t\nH : Int.sign 0 = (eval t P - t).sign\n⊢ IsPeriodicPt (fun x => eval x P) 2 t"
},
{
"line": "try (with_reducible rfl)",
"before_state": "case pos.intro.succ.intro\nP : ℤ[X]\nt : ℤ\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\nHdvd : Cycle.Chain (fun x1 x2 => x1 ∣ x2) C\nHabs :\n ∀ (m n : ℕ),\n ((fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t).natAbs =\n ((fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t).natAbs\nHC' : Cycle.Chain (fun x1 x2 => x1 = x2) C\nHeq :\n ∀ (m n : ℕ),\n (fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t =\n (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t\nIH : ∀ (n : ℕ), ((fun x => eval x P)^[n + 1] t - t).sign = (eval t P - t).sign\nk : ℕ\nhk : k + 1 > 0\nhk' : IsPeriodicPt (fun x => eval x P) (k + 1) t\nH : Int.sign 0 = (eval t P - t).sign\n⊢ IsPeriodicPt (fun x => eval x P) 2 t",
"after_state": "case pos.intro.succ.intro\nP : ℤ[X]\nt : ℤ\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\nHdvd : Cycle.Chain (fun x1 x2 => x1 ∣ x2) C\nHabs :\n ∀ (m n : ℕ),\n ((fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t).natAbs =\n ((fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t).natAbs\nHC' : Cycle.Chain (fun x1 x2 => x1 = x2) C\nHeq :\n ∀ (m n : ℕ),\n (fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t =\n (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t\nIH : ∀ (n : ℕ), ((fun x => eval x P)^[n + 1] t - t).sign = (eval t P - t).sign\nk : ℕ\nhk : k + 1 > 0\nhk' : IsPeriodicPt (fun x => eval x P) (k + 1) t\nH : Int.sign 0 = (eval t P - t).sign\n⊢ IsPeriodicPt (fun x => eval x P) 2 t"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "case pos.intro.succ.intro\nP : ℤ[X]\nt : ℤ\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\nHdvd : Cycle.Chain (fun x1 x2 => x1 ∣ x2) C\nHabs :\n ∀ (m n : ℕ),\n ((fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t).natAbs =\n ((fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t).natAbs\nHC' : Cycle.Chain (fun x1 x2 => x1 = x2) C\nHeq :\n ∀ (m n : ℕ),\n (fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t =\n (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t\nIH : ∀ (n : ℕ), ((fun x => eval x P)^[n + 1] t - t).sign = (eval t P - t).sign\nk : ℕ\nhk : k + 1 > 0\nhk' : IsPeriodicPt (fun x => eval x P) (k + 1) t\nH : Int.sign 0 = (eval t P - t).sign\n⊢ IsPeriodicPt (fun x => eval x P) 2 t",
"after_state": "case pos.intro.succ.intro\nP : ℤ[X]\nt : ℤ\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\nHdvd : Cycle.Chain (fun x1 x2 => x1 ∣ x2) C\nHabs :\n ∀ (m n : ℕ),\n ((fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t).natAbs =\n ((fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t).natAbs\nHC' : Cycle.Chain (fun x1 x2 => x1 = x2) C\nHeq :\n ∀ (m n : ℕ),\n (fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t =\n (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t\nIH : ∀ (n : ℕ), ((fun x => eval x P)^[n + 1] t - t).sign = (eval t P - t).sign\nk : ℕ\nhk : k + 1 > 0\nhk' : IsPeriodicPt (fun x => eval x P) (k + 1) t\nH : Int.sign 0 = (eval t P - t).sign\n⊢ IsPeriodicPt (fun x => eval x P) 2 t"
},
{
"line": "with_reducible rfl",
"before_state": "case pos.intro.succ.intro\nP : ℤ[X]\nt : ℤ\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\nHdvd : Cycle.Chain (fun x1 x2 => x1 ∣ x2) C\nHabs :\n ∀ (m n : ℕ),\n ((fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t).natAbs =\n ((fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t).natAbs\nHC' : Cycle.Chain (fun x1 x2 => x1 = x2) C\nHeq :\n ∀ (m n : ℕ),\n (fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t =\n (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t\nIH : ∀ (n : ℕ), ((fun x => eval x P)^[n + 1] t - t).sign = (eval t P - t).sign\nk : ℕ\nhk : k + 1 > 0\nhk' : IsPeriodicPt (fun x => eval x P) (k + 1) t\nH : Int.sign 0 = (eval t P - t).sign\n⊢ IsPeriodicPt (fun x => eval x P) 2 t",
"after_state": "case pos.intro.succ.intro\nP : ℤ[X]\nt : ℤ\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\nHdvd : Cycle.Chain (fun x1 x2 => x1 ∣ x2) C\nHabs :\n ∀ (m n : ℕ),\n ((fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t).natAbs =\n ((fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t).natAbs\nHC' : Cycle.Chain (fun x1 x2 => x1 = x2) C\nHeq :\n ∀ (m n : ℕ),\n (fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t =\n (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t\nIH : ∀ (n : ℕ), ((fun x => eval x P)^[n + 1] t - t).sign = (eval t P - t).sign\nk : ℕ\nhk : k + 1 > 0\nhk' : IsPeriodicPt (fun x => eval x P) (k + 1) t\nH : Int.sign 0 = (eval t P - t).sign\n⊢ IsPeriodicPt (fun x => eval x P) 2 t"
},
{
"line": "rfl",
"before_state": "case pos.intro.succ.intro\nP : ℤ[X]\nt : ℤ\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\nHdvd : Cycle.Chain (fun x1 x2 => x1 ∣ x2) C\nHabs :\n ∀ (m n : ℕ),\n ((fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t).natAbs =\n ((fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t).natAbs\nHC' : Cycle.Chain (fun x1 x2 => x1 = x2) C\nHeq :\n ∀ (m n : ℕ),\n (fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t =\n (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t\nIH : ∀ (n : ℕ), ((fun x => eval x P)^[n + 1] t - t).sign = (eval t P - t).sign\nk : ℕ\nhk : k + 1 > 0\nhk' : IsPeriodicPt (fun x => eval x P) (k + 1) t\nH : Int.sign 0 = (eval t P - t).sign\n⊢ IsPeriodicPt (fun x => eval x P) 2 t",
"after_state": "case pos.intro.succ.intro\nP : ℤ[X]\nt : ℤ\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\nHdvd : Cycle.Chain (fun x1 x2 => x1 ∣ x2) C\nHabs :\n ∀ (m n : ℕ),\n ((fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t).natAbs =\n ((fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t).natAbs\nHC' : Cycle.Chain (fun x1 x2 => x1 = x2) C\nHeq :\n ∀ (m n : ℕ),\n (fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t =\n (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t\nIH : ∀ (n : ℕ), ((fun x => eval x P)^[n + 1] t - t).sign = (eval t P - t).sign\nk : ℕ\nhk : k + 1 > 0\nhk' : IsPeriodicPt (fun x => eval x P) (k + 1) t\nH : Int.sign 0 = (eval t P - t).sign\n⊢ IsPeriodicPt (fun x => eval x P) 2 t"
},
{
"line": "apply_rfl",
"before_state": "case pos.intro.succ.intro\nP : ℤ[X]\nt : ℤ\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\nHdvd : Cycle.Chain (fun x1 x2 => x1 ∣ x2) C\nHabs :\n ∀ (m n : ℕ),\n ((fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t).natAbs =\n ((fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t).natAbs\nHC' : Cycle.Chain (fun x1 x2 => x1 = x2) C\nHeq :\n ∀ (m n : ℕ),\n (fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t =\n (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t\nIH : ∀ (n : ℕ), ((fun x => eval x P)^[n + 1] t - t).sign = (eval t P - t).sign\nk : ℕ\nhk : k + 1 > 0\nhk' : IsPeriodicPt (fun x => eval x P) (k + 1) t\nH : Int.sign 0 = (eval t P - t).sign\n⊢ IsPeriodicPt (fun x => eval x P) 2 t",
"after_state": "case pos.intro.succ.intro\nP : ℤ[X]\nt : ℤ\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\nHdvd : Cycle.Chain (fun x1 x2 => x1 ∣ x2) C\nHabs :\n ∀ (m n : ℕ),\n ((fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t).natAbs =\n ((fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t).natAbs\nHC' : Cycle.Chain (fun x1 x2 => x1 = x2) C\nHeq :\n ∀ (m n : ℕ),\n (fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t =\n (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t\nIH : ∀ (n : ℕ), ((fun x => eval x P)^[n + 1] t - t).sign = (eval t P - t).sign\nk : ℕ\nhk : k + 1 > 0\nhk' : IsPeriodicPt (fun x => eval x P) (k + 1) t\nH : Int.sign 0 = (eval t P - t).sign\n⊢ IsPeriodicPt (fun x => eval x P) 2 t"
},
{
"line": "skip",
"before_state": "case pos.intro.succ.intro\nP : ℤ[X]\nt : ℤ\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\nHdvd : Cycle.Chain (fun x1 x2 => x1 ∣ x2) C\nHabs :\n ∀ (m n : ℕ),\n ((fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t).natAbs =\n ((fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t).natAbs\nHC' : Cycle.Chain (fun x1 x2 => x1 = x2) C\nHeq :\n ∀ (m n : ℕ),\n (fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t =\n (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t\nIH : ∀ (n : ℕ), ((fun x => eval x P)^[n + 1] t - t).sign = (eval t P - t).sign\nk : ℕ\nhk : k + 1 > 0\nhk' : IsPeriodicPt (fun x => eval x P) (k + 1) t\nH : Int.sign 0 = (eval t P - t).sign\n⊢ IsPeriodicPt (fun x => eval x P) 2 t",
"after_state": "case pos.intro.succ.intro\nP : ℤ[X]\nt : ℤ\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\nHdvd : Cycle.Chain (fun x1 x2 => x1 ∣ x2) C\nHabs :\n ∀ (m n : ℕ),\n ((fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t).natAbs =\n ((fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t).natAbs\nHC' : Cycle.Chain (fun x1 x2 => x1 = x2) C\nHeq :\n ∀ (m n : ℕ),\n (fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t =\n (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t\nIH : ∀ (n : ℕ), ((fun x => eval x P)^[n + 1] t - t).sign = (eval t P - t).sign\nk : ℕ\nhk : k + 1 > 0\nhk' : IsPeriodicPt (fun x => eval x P) (k + 1) t\nH : Int.sign 0 = (eval t P - t).sign\n⊢ IsPeriodicPt (fun x => eval x P) 2 t"
},
{
"line": "rw [Int.sign_zero] at H",
"before_state": "case pos.intro.succ.intro\nP : ℤ[X]\nt : ℤ\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\nHdvd : Cycle.Chain (fun x1 x2 => x1 ∣ x2) C\nHabs :\n ∀ (m n : ℕ),\n ((fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t).natAbs =\n ((fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t).natAbs\nHC' : Cycle.Chain (fun x1 x2 => x1 = x2) C\nHeq :\n ∀ (m n : ℕ),\n (fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t =\n (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t\nIH : ∀ (n : ℕ), ((fun x => eval x P)^[n + 1] t - t).sign = (eval t P - t).sign\nk : ℕ\nhk : k + 1 > 0\nhk' : IsPeriodicPt (fun x => eval x P) (k + 1) t\nH : Int.sign 0 = (eval t P - t).sign\n⊢ IsPeriodicPt (fun x => eval x P) 2 t",
"after_state": "case pos.intro.succ.intro\nP : ℤ[X]\nt : ℤ\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\nHdvd : Cycle.Chain (fun x1 x2 => x1 ∣ x2) C\nHabs :\n ∀ (m n : ℕ),\n ((fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t).natAbs =\n ((fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t).natAbs\nHC' : Cycle.Chain (fun x1 x2 => x1 = x2) C\nHeq :\n ∀ (m n : ℕ),\n (fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t =\n (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t\nIH : ∀ (n : ℕ), ((fun x => eval x P)^[n + 1] t - t).sign = (eval t P - t).sign\nk : ℕ\nhk : k + 1 > 0\nhk' : IsPeriodicPt (fun x => eval x P) (k + 1) t\nH : 0 = (eval t P - t).sign\n⊢ IsPeriodicPt (fun x => eval x P) 2 t"
},
{
"line": "rewrite [Int.sign_zero] at H",
"before_state": "case pos.intro.succ.intro\nP : ℤ[X]\nt : ℤ\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\nHdvd : Cycle.Chain (fun x1 x2 => x1 ∣ x2) C\nHabs :\n ∀ (m n : ℕ),\n ((fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t).natAbs =\n ((fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t).natAbs\nHC' : Cycle.Chain (fun x1 x2 => x1 = x2) C\nHeq :\n ∀ (m n : ℕ),\n (fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t =\n (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t\nIH : ∀ (n : ℕ), ((fun x => eval x P)^[n + 1] t - t).sign = (eval t P - t).sign\nk : ℕ\nhk : k + 1 > 0\nhk' : IsPeriodicPt (fun x => eval x P) (k + 1) t\nH : Int.sign 0 = (eval t P - t).sign\n⊢ IsPeriodicPt (fun x => eval x P) 2 t",
"after_state": "case pos.intro.succ.intro\nP : ℤ[X]\nt : ℤ\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\nHdvd : Cycle.Chain (fun x1 x2 => x1 ∣ x2) C\nHabs :\n ∀ (m n : ℕ),\n ((fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t).natAbs =\n ((fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t).natAbs\nHC' : Cycle.Chain (fun x1 x2 => x1 = x2) C\nHeq :\n ∀ (m n : ℕ),\n (fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t =\n (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t\nIH : ∀ (n : ℕ), ((fun x => eval x P)^[n + 1] t - t).sign = (eval t P - t).sign\nk : ℕ\nhk : k + 1 > 0\nhk' : IsPeriodicPt (fun x => eval x P) (k + 1) t\nH : 0 = (eval t P - t).sign\n⊢ IsPeriodicPt (fun x => eval x P) 2 t"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "case pos.intro.succ.intro\nP : ℤ[X]\nt : ℤ\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\nHdvd : Cycle.Chain (fun x1 x2 => x1 ∣ x2) C\nHabs :\n ∀ (m n : ℕ),\n ((fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t).natAbs =\n ((fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t).natAbs\nHC' : Cycle.Chain (fun x1 x2 => x1 = x2) C\nHeq :\n ∀ (m n : ℕ),\n (fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t =\n (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t\nIH : ∀ (n : ℕ), ((fun x => eval x P)^[n + 1] t - t).sign = (eval t P - t).sign\nk : ℕ\nhk : k + 1 > 0\nhk' : IsPeriodicPt (fun x => eval x P) (k + 1) t\nH : 0 = (eval t P - t).sign\n⊢ IsPeriodicPt (fun x => eval x P) 2 t",
"after_state": "case pos.intro.succ.intro\nP : ℤ[X]\nt : ℤ\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\nHdvd : Cycle.Chain (fun x1 x2 => x1 ∣ x2) C\nHabs :\n ∀ (m n : ℕ),\n ((fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t).natAbs =\n ((fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t).natAbs\nHC' : Cycle.Chain (fun x1 x2 => x1 = x2) C\nHeq :\n ∀ (m n : ℕ),\n (fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t =\n (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t\nIH : ∀ (n : ℕ), ((fun x => eval x P)^[n + 1] t - t).sign = (eval t P - t).sign\nk : ℕ\nhk : k + 1 > 0\nhk' : IsPeriodicPt (fun x => eval x P) (k + 1) t\nH : 0 = (eval t P - t).sign\n⊢ IsPeriodicPt (fun x => eval x P) 2 t"
},
{
"line": "try (with_reducible rfl)",
"before_state": "case pos.intro.succ.intro\nP : ℤ[X]\nt : ℤ\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\nHdvd : Cycle.Chain (fun x1 x2 => x1 ∣ x2) C\nHabs :\n ∀ (m n : ℕ),\n ((fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t).natAbs =\n ((fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t).natAbs\nHC' : Cycle.Chain (fun x1 x2 => x1 = x2) C\nHeq :\n ∀ (m n : ℕ),\n (fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t =\n (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t\nIH : ∀ (n : ℕ), ((fun x => eval x P)^[n + 1] t - t).sign = (eval t P - t).sign\nk : ℕ\nhk : k + 1 > 0\nhk' : IsPeriodicPt (fun x => eval x P) (k + 1) t\nH : 0 = (eval t P - t).sign\n⊢ IsPeriodicPt (fun x => eval x P) 2 t",
"after_state": "case pos.intro.succ.intro\nP : ℤ[X]\nt : ℤ\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\nHdvd : Cycle.Chain (fun x1 x2 => x1 ∣ x2) C\nHabs :\n ∀ (m n : ℕ),\n ((fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t).natAbs =\n ((fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t).natAbs\nHC' : Cycle.Chain (fun x1 x2 => x1 = x2) C\nHeq :\n ∀ (m n : ℕ),\n (fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t =\n (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t\nIH : ∀ (n : ℕ), ((fun x => eval x P)^[n + 1] t - t).sign = (eval t P - t).sign\nk : ℕ\nhk : k + 1 > 0\nhk' : IsPeriodicPt (fun x => eval x P) (k + 1) t\nH : 0 = (eval t P - t).sign\n⊢ IsPeriodicPt (fun x => eval x P) 2 t"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "case pos.intro.succ.intro\nP : ℤ[X]\nt : ℤ\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\nHdvd : Cycle.Chain (fun x1 x2 => x1 ∣ x2) C\nHabs :\n ∀ (m n : ℕ),\n ((fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t).natAbs =\n ((fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t).natAbs\nHC' : Cycle.Chain (fun x1 x2 => x1 = x2) C\nHeq :\n ∀ (m n : ℕ),\n (fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t =\n (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t\nIH : ∀ (n : ℕ), ((fun x => eval x P)^[n + 1] t - t).sign = (eval t P - t).sign\nk : ℕ\nhk : k + 1 > 0\nhk' : IsPeriodicPt (fun x => eval x P) (k + 1) t\nH : 0 = (eval t P - t).sign\n⊢ IsPeriodicPt (fun x => eval x P) 2 t",
"after_state": "case pos.intro.succ.intro\nP : ℤ[X]\nt : ℤ\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\nHdvd : Cycle.Chain (fun x1 x2 => x1 ∣ x2) C\nHabs :\n ∀ (m n : ℕ),\n ((fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t).natAbs =\n ((fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t).natAbs\nHC' : Cycle.Chain (fun x1 x2 => x1 = x2) C\nHeq :\n ∀ (m n : ℕ),\n (fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t =\n (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t\nIH : ∀ (n : ℕ), ((fun x => eval x P)^[n + 1] t - t).sign = (eval t P - t).sign\nk : ℕ\nhk : k + 1 > 0\nhk' : IsPeriodicPt (fun x => eval x P) (k + 1) t\nH : 0 = (eval t P - t).sign\n⊢ IsPeriodicPt (fun x => eval x P) 2 t"
},
{
"line": "with_reducible rfl",
"before_state": "case pos.intro.succ.intro\nP : ℤ[X]\nt : ℤ\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\nHdvd : Cycle.Chain (fun x1 x2 => x1 ∣ x2) C\nHabs :\n ∀ (m n : ℕ),\n ((fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t).natAbs =\n ((fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t).natAbs\nHC' : Cycle.Chain (fun x1 x2 => x1 = x2) C\nHeq :\n ∀ (m n : ℕ),\n (fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t =\n (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t\nIH : ∀ (n : ℕ), ((fun x => eval x P)^[n + 1] t - t).sign = (eval t P - t).sign\nk : ℕ\nhk : k + 1 > 0\nhk' : IsPeriodicPt (fun x => eval x P) (k + 1) t\nH : 0 = (eval t P - t).sign\n⊢ IsPeriodicPt (fun x => eval x P) 2 t",
"after_state": "case pos.intro.succ.intro\nP : ℤ[X]\nt : ℤ\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\nHdvd : Cycle.Chain (fun x1 x2 => x1 ∣ x2) C\nHabs :\n ∀ (m n : ℕ),\n ((fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t).natAbs =\n ((fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t).natAbs\nHC' : Cycle.Chain (fun x1 x2 => x1 = x2) C\nHeq :\n ∀ (m n : ℕ),\n (fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t =\n (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t\nIH : ∀ (n : ℕ), ((fun x => eval x P)^[n + 1] t - t).sign = (eval t P - t).sign\nk : ℕ\nhk : k + 1 > 0\nhk' : IsPeriodicPt (fun x => eval x P) (k + 1) t\nH : 0 = (eval t P - t).sign\n⊢ IsPeriodicPt (fun x => eval x P) 2 t"
},
{
"line": "rfl",
"before_state": "case pos.intro.succ.intro\nP : ℤ[X]\nt : ℤ\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\nHdvd : Cycle.Chain (fun x1 x2 => x1 ∣ x2) C\nHabs :\n ∀ (m n : ℕ),\n ((fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t).natAbs =\n ((fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t).natAbs\nHC' : Cycle.Chain (fun x1 x2 => x1 = x2) C\nHeq :\n ∀ (m n : ℕ),\n (fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t =\n (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t\nIH : ∀ (n : ℕ), ((fun x => eval x P)^[n + 1] t - t).sign = (eval t P - t).sign\nk : ℕ\nhk : k + 1 > 0\nhk' : IsPeriodicPt (fun x => eval x P) (k + 1) t\nH : 0 = (eval t P - t).sign\n⊢ IsPeriodicPt (fun x => eval x P) 2 t",
"after_state": "case pos.intro.succ.intro\nP : ℤ[X]\nt : ℤ\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\nHdvd : Cycle.Chain (fun x1 x2 => x1 ∣ x2) C\nHabs :\n ∀ (m n : ℕ),\n ((fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t).natAbs =\n ((fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t).natAbs\nHC' : Cycle.Chain (fun x1 x2 => x1 = x2) C\nHeq :\n ∀ (m n : ℕ),\n (fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t =\n (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t\nIH : ∀ (n : ℕ), ((fun x => eval x P)^[n + 1] t - t).sign = (eval t P - t).sign\nk : ℕ\nhk : k + 1 > 0\nhk' : IsPeriodicPt (fun x => eval x P) (k + 1) t\nH : 0 = (eval t P - t).sign\n⊢ IsPeriodicPt (fun x => eval x P) 2 t"
},
{
"line": "apply_rfl",
"before_state": "case pos.intro.succ.intro\nP : ℤ[X]\nt : ℤ\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\nHdvd : Cycle.Chain (fun x1 x2 => x1 ∣ x2) C\nHabs :\n ∀ (m n : ℕ),\n ((fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t).natAbs =\n ((fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t).natAbs\nHC' : Cycle.Chain (fun x1 x2 => x1 = x2) C\nHeq :\n ∀ (m n : ℕ),\n (fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t =\n (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t\nIH : ∀ (n : ℕ), ((fun x => eval x P)^[n + 1] t - t).sign = (eval t P - t).sign\nk : ℕ\nhk : k + 1 > 0\nhk' : IsPeriodicPt (fun x => eval x P) (k + 1) t\nH : 0 = (eval t P - t).sign\n⊢ IsPeriodicPt (fun x => eval x P) 2 t",
"after_state": "case pos.intro.succ.intro\nP : ℤ[X]\nt : ℤ\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\nHdvd : Cycle.Chain (fun x1 x2 => x1 ∣ x2) C\nHabs :\n ∀ (m n : ℕ),\n ((fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t).natAbs =\n ((fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t).natAbs\nHC' : Cycle.Chain (fun x1 x2 => x1 = x2) C\nHeq :\n ∀ (m n : ℕ),\n (fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t =\n (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t\nIH : ∀ (n : ℕ), ((fun x => eval x P)^[n + 1] t - t).sign = (eval t P - t).sign\nk : ℕ\nhk : k + 1 > 0\nhk' : IsPeriodicPt (fun x => eval x P) (k + 1) t\nH : 0 = (eval t P - t).sign\n⊢ IsPeriodicPt (fun x => eval x P) 2 t"
},
{
"line": "skip",
"before_state": "case pos.intro.succ.intro\nP : ℤ[X]\nt : ℤ\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\nHdvd : Cycle.Chain (fun x1 x2 => x1 ∣ x2) C\nHabs :\n ∀ (m n : ℕ),\n ((fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t).natAbs =\n ((fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t).natAbs\nHC' : Cycle.Chain (fun x1 x2 => x1 = x2) C\nHeq :\n ∀ (m n : ℕ),\n (fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t =\n (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t\nIH : ∀ (n : ℕ), ((fun x => eval x P)^[n + 1] t - t).sign = (eval t P - t).sign\nk : ℕ\nhk : k + 1 > 0\nhk' : IsPeriodicPt (fun x => eval x P) (k + 1) t\nH : 0 = (eval t P - t).sign\n⊢ IsPeriodicPt (fun x => eval x P) 2 t",
"after_state": "case pos.intro.succ.intro\nP : ℤ[X]\nt : ℤ\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\nHdvd : Cycle.Chain (fun x1 x2 => x1 ∣ x2) C\nHabs :\n ∀ (m n : ℕ),\n ((fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t).natAbs =\n ((fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t).natAbs\nHC' : Cycle.Chain (fun x1 x2 => x1 = x2) C\nHeq :\n ∀ (m n : ℕ),\n (fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t =\n (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t\nIH : ∀ (n : ℕ), ((fun x => eval x P)^[n + 1] t - t).sign = (eval t P - t).sign\nk : ℕ\nhk : k + 1 > 0\nhk' : IsPeriodicPt (fun x => eval x P) (k + 1) t\nH : 0 = (eval t P - t).sign\n⊢ IsPeriodicPt (fun x => eval x P) 2 t"
},
{
"line": "rw [eq_comm] at H",
"before_state": "case pos.intro.succ.intro\nP : ℤ[X]\nt : ℤ\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\nHdvd : Cycle.Chain (fun x1 x2 => x1 ∣ x2) C\nHabs :\n ∀ (m n : ℕ),\n ((fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t).natAbs =\n ((fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t).natAbs\nHC' : Cycle.Chain (fun x1 x2 => x1 = x2) C\nHeq :\n ∀ (m n : ℕ),\n (fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t =\n (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t\nIH : ∀ (n : ℕ), ((fun x => eval x P)^[n + 1] t - t).sign = (eval t P - t).sign\nk : ℕ\nhk : k + 1 > 0\nhk' : IsPeriodicPt (fun x => eval x P) (k + 1) t\nH : 0 = (eval t P - t).sign\n⊢ IsPeriodicPt (fun x => eval x P) 2 t",
"after_state": "case pos.intro.succ.intro\nP : ℤ[X]\nt : ℤ\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\nHdvd : Cycle.Chain (fun x1 x2 => x1 ∣ x2) C\nHabs :\n ∀ (m n : ℕ),\n ((fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t).natAbs =\n ((fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t).natAbs\nHC' : Cycle.Chain (fun x1 x2 => x1 = x2) C\nHeq :\n ∀ (m n : ℕ),\n (fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t =\n (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t\nIH : ∀ (n : ℕ), ((fun x => eval x P)^[n + 1] t - t).sign = (eval t P - t).sign\nk : ℕ\nhk : k + 1 > 0\nhk' : IsPeriodicPt (fun x => eval x P) (k + 1) t\nH : (eval t P - t).sign = 0\n⊢ IsPeriodicPt (fun x => eval x P) 2 t"
},
{
"line": "rewrite [eq_comm] at H",
"before_state": "case pos.intro.succ.intro\nP : ℤ[X]\nt : ℤ\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\nHdvd : Cycle.Chain (fun x1 x2 => x1 ∣ x2) C\nHabs :\n ∀ (m n : ℕ),\n ((fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t).natAbs =\n ((fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t).natAbs\nHC' : Cycle.Chain (fun x1 x2 => x1 = x2) C\nHeq :\n ∀ (m n : ℕ),\n (fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t =\n (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t\nIH : ∀ (n : ℕ), ((fun x => eval x P)^[n + 1] t - t).sign = (eval t P - t).sign\nk : ℕ\nhk : k + 1 > 0\nhk' : IsPeriodicPt (fun x => eval x P) (k + 1) t\nH : 0 = (eval t P - t).sign\n⊢ IsPeriodicPt (fun x => eval x P) 2 t",
"after_state": "case pos.intro.succ.intro\nP : ℤ[X]\nt : ℤ\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\nHdvd : Cycle.Chain (fun x1 x2 => x1 ∣ x2) C\nHabs :\n ∀ (m n : ℕ),\n ((fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t).natAbs =\n ((fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t).natAbs\nHC' : Cycle.Chain (fun x1 x2 => x1 = x2) C\nHeq :\n ∀ (m n : ℕ),\n (fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t =\n (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t\nIH : ∀ (n : ℕ), ((fun x => eval x P)^[n + 1] t - t).sign = (eval t P - t).sign\nk : ℕ\nhk : k + 1 > 0\nhk' : IsPeriodicPt (fun x => eval x P) (k + 1) t\nH : (eval t P - t).sign = 0\n⊢ IsPeriodicPt (fun x => eval x P) 2 t"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "case pos.intro.succ.intro\nP : ℤ[X]\nt : ℤ\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\nHdvd : Cycle.Chain (fun x1 x2 => x1 ∣ x2) C\nHabs :\n ∀ (m n : ℕ),\n ((fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t).natAbs =\n ((fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t).natAbs\nHC' : Cycle.Chain (fun x1 x2 => x1 = x2) C\nHeq :\n ∀ (m n : ℕ),\n (fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t =\n (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t\nIH : ∀ (n : ℕ), ((fun x => eval x P)^[n + 1] t - t).sign = (eval t P - t).sign\nk : ℕ\nhk : k + 1 > 0\nhk' : IsPeriodicPt (fun x => eval x P) (k + 1) t\nH : (eval t P - t).sign = 0\n⊢ IsPeriodicPt (fun x => eval x P) 2 t",
"after_state": "case pos.intro.succ.intro\nP : ℤ[X]\nt : ℤ\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\nHdvd : Cycle.Chain (fun x1 x2 => x1 ∣ x2) C\nHabs :\n ∀ (m n : ℕ),\n ((fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t).natAbs =\n ((fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t).natAbs\nHC' : Cycle.Chain (fun x1 x2 => x1 = x2) C\nHeq :\n ∀ (m n : ℕ),\n (fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t =\n (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t\nIH : ∀ (n : ℕ), ((fun x => eval x P)^[n + 1] t - t).sign = (eval t P - t).sign\nk : ℕ\nhk : k + 1 > 0\nhk' : IsPeriodicPt (fun x => eval x P) (k + 1) t\nH : (eval t P - t).sign = 0\n⊢ IsPeriodicPt (fun x => eval x P) 2 t"
},
{
"line": "try (with_reducible rfl)",
"before_state": "case pos.intro.succ.intro\nP : ℤ[X]\nt : ℤ\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\nHdvd : Cycle.Chain (fun x1 x2 => x1 ∣ x2) C\nHabs :\n ∀ (m n : ℕ),\n ((fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t).natAbs =\n ((fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t).natAbs\nHC' : Cycle.Chain (fun x1 x2 => x1 = x2) C\nHeq :\n ∀ (m n : ℕ),\n (fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t =\n (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t\nIH : ∀ (n : ℕ), ((fun x => eval x P)^[n + 1] t - t).sign = (eval t P - t).sign\nk : ℕ\nhk : k + 1 > 0\nhk' : IsPeriodicPt (fun x => eval x P) (k + 1) t\nH : (eval t P - t).sign = 0\n⊢ IsPeriodicPt (fun x => eval x P) 2 t",
"after_state": "case pos.intro.succ.intro\nP : ℤ[X]\nt : ℤ\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\nHdvd : Cycle.Chain (fun x1 x2 => x1 ∣ x2) C\nHabs :\n ∀ (m n : ℕ),\n ((fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t).natAbs =\n ((fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t).natAbs\nHC' : Cycle.Chain (fun x1 x2 => x1 = x2) C\nHeq :\n ∀ (m n : ℕ),\n (fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t =\n (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t\nIH : ∀ (n : ℕ), ((fun x => eval x P)^[n + 1] t - t).sign = (eval t P - t).sign\nk : ℕ\nhk : k + 1 > 0\nhk' : IsPeriodicPt (fun x => eval x P) (k + 1) t\nH : (eval t P - t).sign = 0\n⊢ IsPeriodicPt (fun x => eval x P) 2 t"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "case pos.intro.succ.intro\nP : ℤ[X]\nt : ℤ\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\nHdvd : Cycle.Chain (fun x1 x2 => x1 ∣ x2) C\nHabs :\n ∀ (m n : ℕ),\n ((fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t).natAbs =\n ((fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t).natAbs\nHC' : Cycle.Chain (fun x1 x2 => x1 = x2) C\nHeq :\n ∀ (m n : ℕ),\n (fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t =\n (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t\nIH : ∀ (n : ℕ), ((fun x => eval x P)^[n + 1] t - t).sign = (eval t P - t).sign\nk : ℕ\nhk : k + 1 > 0\nhk' : IsPeriodicPt (fun x => eval x P) (k + 1) t\nH : (eval t P - t).sign = 0\n⊢ IsPeriodicPt (fun x => eval x P) 2 t",
"after_state": "case pos.intro.succ.intro\nP : ℤ[X]\nt : ℤ\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\nHdvd : Cycle.Chain (fun x1 x2 => x1 ∣ x2) C\nHabs :\n ∀ (m n : ℕ),\n ((fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t).natAbs =\n ((fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t).natAbs\nHC' : Cycle.Chain (fun x1 x2 => x1 = x2) C\nHeq :\n ∀ (m n : ℕ),\n (fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t =\n (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t\nIH : ∀ (n : ℕ), ((fun x => eval x P)^[n + 1] t - t).sign = (eval t P - t).sign\nk : ℕ\nhk : k + 1 > 0\nhk' : IsPeriodicPt (fun x => eval x P) (k + 1) t\nH : (eval t P - t).sign = 0\n⊢ IsPeriodicPt (fun x => eval x P) 2 t"
},
{
"line": "with_reducible rfl",
"before_state": "case pos.intro.succ.intro\nP : ℤ[X]\nt : ℤ\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\nHdvd : Cycle.Chain (fun x1 x2 => x1 ∣ x2) C\nHabs :\n ∀ (m n : ℕ),\n ((fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t).natAbs =\n ((fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t).natAbs\nHC' : Cycle.Chain (fun x1 x2 => x1 = x2) C\nHeq :\n ∀ (m n : ℕ),\n (fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t =\n (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t\nIH : ∀ (n : ℕ), ((fun x => eval x P)^[n + 1] t - t).sign = (eval t P - t).sign\nk : ℕ\nhk : k + 1 > 0\nhk' : IsPeriodicPt (fun x => eval x P) (k + 1) t\nH : (eval t P - t).sign = 0\n⊢ IsPeriodicPt (fun x => eval x P) 2 t",
"after_state": "case pos.intro.succ.intro\nP : ℤ[X]\nt : ℤ\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\nHdvd : Cycle.Chain (fun x1 x2 => x1 ∣ x2) C\nHabs :\n ∀ (m n : ℕ),\n ((fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t).natAbs =\n ((fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t).natAbs\nHC' : Cycle.Chain (fun x1 x2 => x1 = x2) C\nHeq :\n ∀ (m n : ℕ),\n (fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t =\n (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t\nIH : ∀ (n : ℕ), ((fun x => eval x P)^[n + 1] t - t).sign = (eval t P - t).sign\nk : ℕ\nhk : k + 1 > 0\nhk' : IsPeriodicPt (fun x => eval x P) (k + 1) t\nH : (eval t P - t).sign = 0\n⊢ IsPeriodicPt (fun x => eval x P) 2 t"
},
{
"line": "rfl",
"before_state": "case pos.intro.succ.intro\nP : ℤ[X]\nt : ℤ\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\nHdvd : Cycle.Chain (fun x1 x2 => x1 ∣ x2) C\nHabs :\n ∀ (m n : ℕ),\n ((fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t).natAbs =\n ((fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t).natAbs\nHC' : Cycle.Chain (fun x1 x2 => x1 = x2) C\nHeq :\n ∀ (m n : ℕ),\n (fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t =\n (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t\nIH : ∀ (n : ℕ), ((fun x => eval x P)^[n + 1] t - t).sign = (eval t P - t).sign\nk : ℕ\nhk : k + 1 > 0\nhk' : IsPeriodicPt (fun x => eval x P) (k + 1) t\nH : (eval t P - t).sign = 0\n⊢ IsPeriodicPt (fun x => eval x P) 2 t",
"after_state": "case pos.intro.succ.intro\nP : ℤ[X]\nt : ℤ\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\nHdvd : Cycle.Chain (fun x1 x2 => x1 ∣ x2) C\nHabs :\n ∀ (m n : ℕ),\n ((fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t).natAbs =\n ((fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t).natAbs\nHC' : Cycle.Chain (fun x1 x2 => x1 = x2) C\nHeq :\n ∀ (m n : ℕ),\n (fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t =\n (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t\nIH : ∀ (n : ℕ), ((fun x => eval x P)^[n + 1] t - t).sign = (eval t P - t).sign\nk : ℕ\nhk : k + 1 > 0\nhk' : IsPeriodicPt (fun x => eval x P) (k + 1) t\nH : (eval t P - t).sign = 0\n⊢ IsPeriodicPt (fun x => eval x P) 2 t"
},
{
"line": "apply_rfl",
"before_state": "case pos.intro.succ.intro\nP : ℤ[X]\nt : ℤ\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\nHdvd : Cycle.Chain (fun x1 x2 => x1 ∣ x2) C\nHabs :\n ∀ (m n : ℕ),\n ((fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t).natAbs =\n ((fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t).natAbs\nHC' : Cycle.Chain (fun x1 x2 => x1 = x2) C\nHeq :\n ∀ (m n : ℕ),\n (fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t =\n (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t\nIH : ∀ (n : ℕ), ((fun x => eval x P)^[n + 1] t - t).sign = (eval t P - t).sign\nk : ℕ\nhk : k + 1 > 0\nhk' : IsPeriodicPt (fun x => eval x P) (k + 1) t\nH : (eval t P - t).sign = 0\n⊢ IsPeriodicPt (fun x => eval x P) 2 t",
"after_state": "case pos.intro.succ.intro\nP : ℤ[X]\nt : ℤ\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\nHdvd : Cycle.Chain (fun x1 x2 => x1 ∣ x2) C\nHabs :\n ∀ (m n : ℕ),\n ((fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t).natAbs =\n ((fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t).natAbs\nHC' : Cycle.Chain (fun x1 x2 => x1 = x2) C\nHeq :\n ∀ (m n : ℕ),\n (fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t =\n (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t\nIH : ∀ (n : ℕ), ((fun x => eval x P)^[n + 1] t - t).sign = (eval t P - t).sign\nk : ℕ\nhk : k + 1 > 0\nhk' : IsPeriodicPt (fun x => eval x P) (k + 1) t\nH : (eval t P - t).sign = 0\n⊢ IsPeriodicPt (fun x => eval x P) 2 t"
},
{
"line": "skip",
"before_state": "case pos.intro.succ.intro\nP : ℤ[X]\nt : ℤ\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\nHdvd : Cycle.Chain (fun x1 x2 => x1 ∣ x2) C\nHabs :\n ∀ (m n : ℕ),\n ((fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t).natAbs =\n ((fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t).natAbs\nHC' : Cycle.Chain (fun x1 x2 => x1 = x2) C\nHeq :\n ∀ (m n : ℕ),\n (fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t =\n (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t\nIH : ∀ (n : ℕ), ((fun x => eval x P)^[n + 1] t - t).sign = (eval t P - t).sign\nk : ℕ\nhk : k + 1 > 0\nhk' : IsPeriodicPt (fun x => eval x P) (k + 1) t\nH : (eval t P - t).sign = 0\n⊢ IsPeriodicPt (fun x => eval x P) 2 t",
"after_state": "case pos.intro.succ.intro\nP : ℤ[X]\nt : ℤ\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\nHdvd : Cycle.Chain (fun x1 x2 => x1 ∣ x2) C\nHabs :\n ∀ (m n : ℕ),\n ((fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t).natAbs =\n ((fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t).natAbs\nHC' : Cycle.Chain (fun x1 x2 => x1 = x2) C\nHeq :\n ∀ (m n : ℕ),\n (fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t =\n (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t\nIH : ∀ (n : ℕ), ((fun x => eval x P)^[n + 1] t - t).sign = (eval t P - t).sign\nk : ℕ\nhk : k + 1 > 0\nhk' : IsPeriodicPt (fun x => eval x P) (k + 1) t\nH : (eval t P - t).sign = 0\n⊢ IsPeriodicPt (fun x => eval x P) 2 t"
},
{
"line": "rw [Int.sign_eq_zero_iff_zero] at H",
"before_state": "case pos.intro.succ.intro\nP : ℤ[X]\nt : ℤ\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\nHdvd : Cycle.Chain (fun x1 x2 => x1 ∣ x2) C\nHabs :\n ∀ (m n : ℕ),\n ((fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t).natAbs =\n ((fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t).natAbs\nHC' : Cycle.Chain (fun x1 x2 => x1 = x2) C\nHeq :\n ∀ (m n : ℕ),\n (fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t =\n (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t\nIH : ∀ (n : ℕ), ((fun x => eval x P)^[n + 1] t - t).sign = (eval t P - t).sign\nk : ℕ\nhk : k + 1 > 0\nhk' : IsPeriodicPt (fun x => eval x P) (k + 1) t\nH : (eval t P - t).sign = 0\n⊢ IsPeriodicPt (fun x => eval x P) 2 t",
"after_state": "case pos.intro.succ.intro\nP : ℤ[X]\nt : ℤ\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\nHdvd : Cycle.Chain (fun x1 x2 => x1 ∣ x2) C\nHabs :\n ∀ (m n : ℕ),\n ((fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t).natAbs =\n ((fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t).natAbs\nHC' : Cycle.Chain (fun x1 x2 => x1 = x2) C\nHeq :\n ∀ (m n : ℕ),\n (fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t =\n (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t\nIH : ∀ (n : ℕ), ((fun x => eval x P)^[n + 1] t - t).sign = (eval t P - t).sign\nk : ℕ\nhk : k + 1 > 0\nhk' : IsPeriodicPt (fun x => eval x P) (k + 1) t\nH : eval t P - t = 0\n⊢ IsPeriodicPt (fun x => eval x P) 2 t"
},
{
"line": "rewrite [Int.sign_eq_zero_iff_zero] at H",
"before_state": "case pos.intro.succ.intro\nP : ℤ[X]\nt : ℤ\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\nHdvd : Cycle.Chain (fun x1 x2 => x1 ∣ x2) C\nHabs :\n ∀ (m n : ℕ),\n ((fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t).natAbs =\n ((fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t).natAbs\nHC' : Cycle.Chain (fun x1 x2 => x1 = x2) C\nHeq :\n ∀ (m n : ℕ),\n (fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t =\n (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t\nIH : ∀ (n : ℕ), ((fun x => eval x P)^[n + 1] t - t).sign = (eval t P - t).sign\nk : ℕ\nhk : k + 1 > 0\nhk' : IsPeriodicPt (fun x => eval x P) (k + 1) t\nH : (eval t P - t).sign = 0\n⊢ IsPeriodicPt (fun x => eval x P) 2 t",
"after_state": "case pos.intro.succ.intro\nP : ℤ[X]\nt : ℤ\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\nHdvd : Cycle.Chain (fun x1 x2 => x1 ∣ x2) C\nHabs :\n ∀ (m n : ℕ),\n ((fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t).natAbs =\n ((fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t).natAbs\nHC' : Cycle.Chain (fun x1 x2 => x1 = x2) C\nHeq :\n ∀ (m n : ℕ),\n (fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t =\n (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t\nIH : ∀ (n : ℕ), ((fun x => eval x P)^[n + 1] t - t).sign = (eval t P - t).sign\nk : ℕ\nhk : k + 1 > 0\nhk' : IsPeriodicPt (fun x => eval x P) (k + 1) t\nH : eval t P - t = 0\n⊢ IsPeriodicPt (fun x => eval x P) 2 t"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "case pos.intro.succ.intro\nP : ℤ[X]\nt : ℤ\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\nHdvd : Cycle.Chain (fun x1 x2 => x1 ∣ x2) C\nHabs :\n ∀ (m n : ℕ),\n ((fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t).natAbs =\n ((fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t).natAbs\nHC' : Cycle.Chain (fun x1 x2 => x1 = x2) C\nHeq :\n ∀ (m n : ℕ),\n (fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t =\n (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t\nIH : ∀ (n : ℕ), ((fun x => eval x P)^[n + 1] t - t).sign = (eval t P - t).sign\nk : ℕ\nhk : k + 1 > 0\nhk' : IsPeriodicPt (fun x => eval x P) (k + 1) t\nH : eval t P - t = 0\n⊢ IsPeriodicPt (fun x => eval x P) 2 t",
"after_state": "case pos.intro.succ.intro\nP : ℤ[X]\nt : ℤ\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\nHdvd : Cycle.Chain (fun x1 x2 => x1 ∣ x2) C\nHabs :\n ∀ (m n : ℕ),\n ((fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t).natAbs =\n ((fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t).natAbs\nHC' : Cycle.Chain (fun x1 x2 => x1 = x2) C\nHeq :\n ∀ (m n : ℕ),\n (fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t =\n (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t\nIH : ∀ (n : ℕ), ((fun x => eval x P)^[n + 1] t - t).sign = (eval t P - t).sign\nk : ℕ\nhk : k + 1 > 0\nhk' : IsPeriodicPt (fun x => eval x P) (k + 1) t\nH : eval t P - t = 0\n⊢ IsPeriodicPt (fun x => eval x P) 2 t"
},
{
"line": "try (with_reducible rfl)",
"before_state": "case pos.intro.succ.intro\nP : ℤ[X]\nt : ℤ\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\nHdvd : Cycle.Chain (fun x1 x2 => x1 ∣ x2) C\nHabs :\n ∀ (m n : ℕ),\n ((fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t).natAbs =\n ((fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t).natAbs\nHC' : Cycle.Chain (fun x1 x2 => x1 = x2) C\nHeq :\n ∀ (m n : ℕ),\n (fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t =\n (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t\nIH : ∀ (n : ℕ), ((fun x => eval x P)^[n + 1] t - t).sign = (eval t P - t).sign\nk : ℕ\nhk : k + 1 > 0\nhk' : IsPeriodicPt (fun x => eval x P) (k + 1) t\nH : eval t P - t = 0\n⊢ IsPeriodicPt (fun x => eval x P) 2 t",
"after_state": "case pos.intro.succ.intro\nP : ℤ[X]\nt : ℤ\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\nHdvd : Cycle.Chain (fun x1 x2 => x1 ∣ x2) C\nHabs :\n ∀ (m n : ℕ),\n ((fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t).natAbs =\n ((fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t).natAbs\nHC' : Cycle.Chain (fun x1 x2 => x1 = x2) C\nHeq :\n ∀ (m n : ℕ),\n (fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t =\n (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t\nIH : ∀ (n : ℕ), ((fun x => eval x P)^[n + 1] t - t).sign = (eval t P - t).sign\nk : ℕ\nhk : k + 1 > 0\nhk' : IsPeriodicPt (fun x => eval x P) (k + 1) t\nH : eval t P - t = 0\n⊢ IsPeriodicPt (fun x => eval x P) 2 t"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "case pos.intro.succ.intro\nP : ℤ[X]\nt : ℤ\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\nHdvd : Cycle.Chain (fun x1 x2 => x1 ∣ x2) C\nHabs :\n ∀ (m n : ℕ),\n ((fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t).natAbs =\n ((fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t).natAbs\nHC' : Cycle.Chain (fun x1 x2 => x1 = x2) C\nHeq :\n ∀ (m n : ℕ),\n (fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t =\n (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t\nIH : ∀ (n : ℕ), ((fun x => eval x P)^[n + 1] t - t).sign = (eval t P - t).sign\nk : ℕ\nhk : k + 1 > 0\nhk' : IsPeriodicPt (fun x => eval x P) (k + 1) t\nH : eval t P - t = 0\n⊢ IsPeriodicPt (fun x => eval x P) 2 t",
"after_state": "case pos.intro.succ.intro\nP : ℤ[X]\nt : ℤ\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\nHdvd : Cycle.Chain (fun x1 x2 => x1 ∣ x2) C\nHabs :\n ∀ (m n : ℕ),\n ((fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t).natAbs =\n ((fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t).natAbs\nHC' : Cycle.Chain (fun x1 x2 => x1 = x2) C\nHeq :\n ∀ (m n : ℕ),\n (fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t =\n (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t\nIH : ∀ (n : ℕ), ((fun x => eval x P)^[n + 1] t - t).sign = (eval t P - t).sign\nk : ℕ\nhk : k + 1 > 0\nhk' : IsPeriodicPt (fun x => eval x P) (k + 1) t\nH : eval t P - t = 0\n⊢ IsPeriodicPt (fun x => eval x P) 2 t"
},
{
"line": "with_reducible rfl",
"before_state": "case pos.intro.succ.intro\nP : ℤ[X]\nt : ℤ\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\nHdvd : Cycle.Chain (fun x1 x2 => x1 ∣ x2) C\nHabs :\n ∀ (m n : ℕ),\n ((fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t).natAbs =\n ((fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t).natAbs\nHC' : Cycle.Chain (fun x1 x2 => x1 = x2) C\nHeq :\n ∀ (m n : ℕ),\n (fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t =\n (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t\nIH : ∀ (n : ℕ), ((fun x => eval x P)^[n + 1] t - t).sign = (eval t P - t).sign\nk : ℕ\nhk : k + 1 > 0\nhk' : IsPeriodicPt (fun x => eval x P) (k + 1) t\nH : eval t P - t = 0\n⊢ IsPeriodicPt (fun x => eval x P) 2 t",
"after_state": "case pos.intro.succ.intro\nP : ℤ[X]\nt : ℤ\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\nHdvd : Cycle.Chain (fun x1 x2 => x1 ∣ x2) C\nHabs :\n ∀ (m n : ℕ),\n ((fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t).natAbs =\n ((fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t).natAbs\nHC' : Cycle.Chain (fun x1 x2 => x1 = x2) C\nHeq :\n ∀ (m n : ℕ),\n (fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t =\n (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t\nIH : ∀ (n : ℕ), ((fun x => eval x P)^[n + 1] t - t).sign = (eval t P - t).sign\nk : ℕ\nhk : k + 1 > 0\nhk' : IsPeriodicPt (fun x => eval x P) (k + 1) t\nH : eval t P - t = 0\n⊢ IsPeriodicPt (fun x => eval x P) 2 t"
},
{
"line": "rfl",
"before_state": "case pos.intro.succ.intro\nP : ℤ[X]\nt : ℤ\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\nHdvd : Cycle.Chain (fun x1 x2 => x1 ∣ x2) C\nHabs :\n ∀ (m n : ℕ),\n ((fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t).natAbs =\n ((fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t).natAbs\nHC' : Cycle.Chain (fun x1 x2 => x1 = x2) C\nHeq :\n ∀ (m n : ℕ),\n (fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t =\n (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t\nIH : ∀ (n : ℕ), ((fun x => eval x P)^[n + 1] t - t).sign = (eval t P - t).sign\nk : ℕ\nhk : k + 1 > 0\nhk' : IsPeriodicPt (fun x => eval x P) (k + 1) t\nH : eval t P - t = 0\n⊢ IsPeriodicPt (fun x => eval x P) 2 t",
"after_state": "case pos.intro.succ.intro\nP : ℤ[X]\nt : ℤ\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\nHdvd : Cycle.Chain (fun x1 x2 => x1 ∣ x2) C\nHabs :\n ∀ (m n : ℕ),\n ((fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t).natAbs =\n ((fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t).natAbs\nHC' : Cycle.Chain (fun x1 x2 => x1 = x2) C\nHeq :\n ∀ (m n : ℕ),\n (fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t =\n (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t\nIH : ∀ (n : ℕ), ((fun x => eval x P)^[n + 1] t - t).sign = (eval t P - t).sign\nk : ℕ\nhk : k + 1 > 0\nhk' : IsPeriodicPt (fun x => eval x P) (k + 1) t\nH : eval t P - t = 0\n⊢ IsPeriodicPt (fun x => eval x P) 2 t"
},
{
"line": "apply_rfl",
"before_state": "case pos.intro.succ.intro\nP : ℤ[X]\nt : ℤ\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\nHdvd : Cycle.Chain (fun x1 x2 => x1 ∣ x2) C\nHabs :\n ∀ (m n : ℕ),\n ((fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t).natAbs =\n ((fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t).natAbs\nHC' : Cycle.Chain (fun x1 x2 => x1 = x2) C\nHeq :\n ∀ (m n : ℕ),\n (fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t =\n (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t\nIH : ∀ (n : ℕ), ((fun x => eval x P)^[n + 1] t - t).sign = (eval t P - t).sign\nk : ℕ\nhk : k + 1 > 0\nhk' : IsPeriodicPt (fun x => eval x P) (k + 1) t\nH : eval t P - t = 0\n⊢ IsPeriodicPt (fun x => eval x P) 2 t",
"after_state": "case pos.intro.succ.intro\nP : ℤ[X]\nt : ℤ\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\nHdvd : Cycle.Chain (fun x1 x2 => x1 ∣ x2) C\nHabs :\n ∀ (m n : ℕ),\n ((fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t).natAbs =\n ((fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t).natAbs\nHC' : Cycle.Chain (fun x1 x2 => x1 = x2) C\nHeq :\n ∀ (m n : ℕ),\n (fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t =\n (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t\nIH : ∀ (n : ℕ), ((fun x => eval x P)^[n + 1] t - t).sign = (eval t P - t).sign\nk : ℕ\nhk : k + 1 > 0\nhk' : IsPeriodicPt (fun x => eval x P) (k + 1) t\nH : eval t P - t = 0\n⊢ IsPeriodicPt (fun x => eval x P) 2 t"
},
{
"line": "skip",
"before_state": "case pos.intro.succ.intro\nP : ℤ[X]\nt : ℤ\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\nHdvd : Cycle.Chain (fun x1 x2 => x1 ∣ x2) C\nHabs :\n ∀ (m n : ℕ),\n ((fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t).natAbs =\n ((fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t).natAbs\nHC' : Cycle.Chain (fun x1 x2 => x1 = x2) C\nHeq :\n ∀ (m n : ℕ),\n (fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t =\n (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t\nIH : ∀ (n : ℕ), ((fun x => eval x P)^[n + 1] t - t).sign = (eval t P - t).sign\nk : ℕ\nhk : k + 1 > 0\nhk' : IsPeriodicPt (fun x => eval x P) (k + 1) t\nH : eval t P - t = 0\n⊢ IsPeriodicPt (fun x => eval x P) 2 t",
"after_state": "case pos.intro.succ.intro\nP : ℤ[X]\nt : ℤ\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\nHdvd : Cycle.Chain (fun x1 x2 => x1 ∣ x2) C\nHabs :\n ∀ (m n : ℕ),\n ((fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t).natAbs =\n ((fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t).natAbs\nHC' : Cycle.Chain (fun x1 x2 => x1 = x2) C\nHeq :\n ∀ (m n : ℕ),\n (fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t =\n (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t\nIH : ∀ (n : ℕ), ((fun x => eval x P)^[n + 1] t - t).sign = (eval t P - t).sign\nk : ℕ\nhk : k + 1 > 0\nhk' : IsPeriodicPt (fun x => eval x P) (k + 1) t\nH : eval t P - t = 0\n⊢ IsPeriodicPt (fun x => eval x P) 2 t"
},
{
"line": "rw [sub_eq_zero] at H",
"before_state": "case pos.intro.succ.intro\nP : ℤ[X]\nt : ℤ\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\nHdvd : Cycle.Chain (fun x1 x2 => x1 ∣ x2) C\nHabs :\n ∀ (m n : ℕ),\n ((fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t).natAbs =\n ((fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t).natAbs\nHC' : Cycle.Chain (fun x1 x2 => x1 = x2) C\nHeq :\n ∀ (m n : ℕ),\n (fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t =\n (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t\nIH : ∀ (n : ℕ), ((fun x => eval x P)^[n + 1] t - t).sign = (eval t P - t).sign\nk : ℕ\nhk : k + 1 > 0\nhk' : IsPeriodicPt (fun x => eval x P) (k + 1) t\nH : eval t P - t = 0\n⊢ IsPeriodicPt (fun x => eval x P) 2 t",
"after_state": "case pos.intro.succ.intro\nP : ℤ[X]\nt : ℤ\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\nHdvd : Cycle.Chain (fun x1 x2 => x1 ∣ x2) C\nHabs :\n ∀ (m n : ℕ),\n ((fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t).natAbs =\n ((fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t).natAbs\nHC' : Cycle.Chain (fun x1 x2 => x1 = x2) C\nHeq :\n ∀ (m n : ℕ),\n (fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t =\n (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t\nIH : ∀ (n : ℕ), ((fun x => eval x P)^[n + 1] t - t).sign = (eval t P - t).sign\nk : ℕ\nhk : k + 1 > 0\nhk' : IsPeriodicPt (fun x => eval x P) (k + 1) t\nH : eval t P = t\n⊢ IsPeriodicPt (fun x => eval x P) 2 t"
},
{
"line": "rewrite [sub_eq_zero] at H",
"before_state": "case pos.intro.succ.intro\nP : ℤ[X]\nt : ℤ\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\nHdvd : Cycle.Chain (fun x1 x2 => x1 ∣ x2) C\nHabs :\n ∀ (m n : ℕ),\n ((fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t).natAbs =\n ((fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t).natAbs\nHC' : Cycle.Chain (fun x1 x2 => x1 = x2) C\nHeq :\n ∀ (m n : ℕ),\n (fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t =\n (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t\nIH : ∀ (n : ℕ), ((fun x => eval x P)^[n + 1] t - t).sign = (eval t P - t).sign\nk : ℕ\nhk : k + 1 > 0\nhk' : IsPeriodicPt (fun x => eval x P) (k + 1) t\nH : eval t P - t = 0\n⊢ IsPeriodicPt (fun x => eval x P) 2 t",
"after_state": "case pos.intro.succ.intro\nP : ℤ[X]\nt : ℤ\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\nHdvd : Cycle.Chain (fun x1 x2 => x1 ∣ x2) C\nHabs :\n ∀ (m n : ℕ),\n ((fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t).natAbs =\n ((fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t).natAbs\nHC' : Cycle.Chain (fun x1 x2 => x1 = x2) C\nHeq :\n ∀ (m n : ℕ),\n (fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t =\n (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t\nIH : ∀ (n : ℕ), ((fun x => eval x P)^[n + 1] t - t).sign = (eval t P - t).sign\nk : ℕ\nhk : k + 1 > 0\nhk' : IsPeriodicPt (fun x => eval x P) (k + 1) t\nH : eval t P = t\n⊢ IsPeriodicPt (fun x => eval x P) 2 t"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "case pos.intro.succ.intro\nP : ℤ[X]\nt : ℤ\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\nHdvd : Cycle.Chain (fun x1 x2 => x1 ∣ x2) C\nHabs :\n ∀ (m n : ℕ),\n ((fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t).natAbs =\n ((fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t).natAbs\nHC' : Cycle.Chain (fun x1 x2 => x1 = x2) C\nHeq :\n ∀ (m n : ℕ),\n (fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t =\n (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t\nIH : ∀ (n : ℕ), ((fun x => eval x P)^[n + 1] t - t).sign = (eval t P - t).sign\nk : ℕ\nhk : k + 1 > 0\nhk' : IsPeriodicPt (fun x => eval x P) (k + 1) t\nH : eval t P = t\n⊢ IsPeriodicPt (fun x => eval x P) 2 t",
"after_state": "case pos.intro.succ.intro\nP : ℤ[X]\nt : ℤ\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\nHdvd : Cycle.Chain (fun x1 x2 => x1 ∣ x2) C\nHabs :\n ∀ (m n : ℕ),\n ((fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t).natAbs =\n ((fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t).natAbs\nHC' : Cycle.Chain (fun x1 x2 => x1 = x2) C\nHeq :\n ∀ (m n : ℕ),\n (fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t =\n (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t\nIH : ∀ (n : ℕ), ((fun x => eval x P)^[n + 1] t - t).sign = (eval t P - t).sign\nk : ℕ\nhk : k + 1 > 0\nhk' : IsPeriodicPt (fun x => eval x P) (k + 1) t\nH : eval t P = t\n⊢ IsPeriodicPt (fun x => eval x P) 2 t"
},
{
"line": "try (with_reducible rfl)",
"before_state": "case pos.intro.succ.intro\nP : ℤ[X]\nt : ℤ\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\nHdvd : Cycle.Chain (fun x1 x2 => x1 ∣ x2) C\nHabs :\n ∀ (m n : ℕ),\n ((fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t).natAbs =\n ((fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t).natAbs\nHC' : Cycle.Chain (fun x1 x2 => x1 = x2) C\nHeq :\n ∀ (m n : ℕ),\n (fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t =\n (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t\nIH : ∀ (n : ℕ), ((fun x => eval x P)^[n + 1] t - t).sign = (eval t P - t).sign\nk : ℕ\nhk : k + 1 > 0\nhk' : IsPeriodicPt (fun x => eval x P) (k + 1) t\nH : eval t P = t\n⊢ IsPeriodicPt (fun x => eval x P) 2 t",
"after_state": "case pos.intro.succ.intro\nP : ℤ[X]\nt : ℤ\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\nHdvd : Cycle.Chain (fun x1 x2 => x1 ∣ x2) C\nHabs :\n ∀ (m n : ℕ),\n ((fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t).natAbs =\n ((fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t).natAbs\nHC' : Cycle.Chain (fun x1 x2 => x1 = x2) C\nHeq :\n ∀ (m n : ℕ),\n (fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t =\n (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t\nIH : ∀ (n : ℕ), ((fun x => eval x P)^[n + 1] t - t).sign = (eval t P - t).sign\nk : ℕ\nhk : k + 1 > 0\nhk' : IsPeriodicPt (fun x => eval x P) (k + 1) t\nH : eval t P = t\n⊢ IsPeriodicPt (fun x => eval x P) 2 t"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "case pos.intro.succ.intro\nP : ℤ[X]\nt : ℤ\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\nHdvd : Cycle.Chain (fun x1 x2 => x1 ∣ x2) C\nHabs :\n ∀ (m n : ℕ),\n ((fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t).natAbs =\n ((fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t).natAbs\nHC' : Cycle.Chain (fun x1 x2 => x1 = x2) C\nHeq :\n ∀ (m n : ℕ),\n (fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t =\n (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t\nIH : ∀ (n : ℕ), ((fun x => eval x P)^[n + 1] t - t).sign = (eval t P - t).sign\nk : ℕ\nhk : k + 1 > 0\nhk' : IsPeriodicPt (fun x => eval x P) (k + 1) t\nH : eval t P = t\n⊢ IsPeriodicPt (fun x => eval x P) 2 t",
"after_state": "case pos.intro.succ.intro\nP : ℤ[X]\nt : ℤ\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\nHdvd : Cycle.Chain (fun x1 x2 => x1 ∣ x2) C\nHabs :\n ∀ (m n : ℕ),\n ((fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t).natAbs =\n ((fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t).natAbs\nHC' : Cycle.Chain (fun x1 x2 => x1 = x2) C\nHeq :\n ∀ (m n : ℕ),\n (fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t =\n (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t\nIH : ∀ (n : ℕ), ((fun x => eval x P)^[n + 1] t - t).sign = (eval t P - t).sign\nk : ℕ\nhk : k + 1 > 0\nhk' : IsPeriodicPt (fun x => eval x P) (k + 1) t\nH : eval t P = t\n⊢ IsPeriodicPt (fun x => eval x P) 2 t"
},
{
"line": "with_reducible rfl",
"before_state": "case pos.intro.succ.intro\nP : ℤ[X]\nt : ℤ\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\nHdvd : Cycle.Chain (fun x1 x2 => x1 ∣ x2) C\nHabs :\n ∀ (m n : ℕ),\n ((fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t).natAbs =\n ((fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t).natAbs\nHC' : Cycle.Chain (fun x1 x2 => x1 = x2) C\nHeq :\n ∀ (m n : ℕ),\n (fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t =\n (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t\nIH : ∀ (n : ℕ), ((fun x => eval x P)^[n + 1] t - t).sign = (eval t P - t).sign\nk : ℕ\nhk : k + 1 > 0\nhk' : IsPeriodicPt (fun x => eval x P) (k + 1) t\nH : eval t P = t\n⊢ IsPeriodicPt (fun x => eval x P) 2 t",
"after_state": "case pos.intro.succ.intro\nP : ℤ[X]\nt : ℤ\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\nHdvd : Cycle.Chain (fun x1 x2 => x1 ∣ x2) C\nHabs :\n ∀ (m n : ℕ),\n ((fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t).natAbs =\n ((fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t).natAbs\nHC' : Cycle.Chain (fun x1 x2 => x1 = x2) C\nHeq :\n ∀ (m n : ℕ),\n (fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t =\n (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t\nIH : ∀ (n : ℕ), ((fun x => eval x P)^[n + 1] t - t).sign = (eval t P - t).sign\nk : ℕ\nhk : k + 1 > 0\nhk' : IsPeriodicPt (fun x => eval x P) (k + 1) t\nH : eval t P = t\n⊢ IsPeriodicPt (fun x => eval x P) 2 t"
},
{
"line": "rfl",
"before_state": "case pos.intro.succ.intro\nP : ℤ[X]\nt : ℤ\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\nHdvd : Cycle.Chain (fun x1 x2 => x1 ∣ x2) C\nHabs :\n ∀ (m n : ℕ),\n ((fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t).natAbs =\n ((fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t).natAbs\nHC' : Cycle.Chain (fun x1 x2 => x1 = x2) C\nHeq :\n ∀ (m n : ℕ),\n (fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t =\n (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t\nIH : ∀ (n : ℕ), ((fun x => eval x P)^[n + 1] t - t).sign = (eval t P - t).sign\nk : ℕ\nhk : k + 1 > 0\nhk' : IsPeriodicPt (fun x => eval x P) (k + 1) t\nH : eval t P = t\n⊢ IsPeriodicPt (fun x => eval x P) 2 t",
"after_state": "case pos.intro.succ.intro\nP : ℤ[X]\nt : ℤ\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\nHdvd : Cycle.Chain (fun x1 x2 => x1 ∣ x2) C\nHabs :\n ∀ (m n : ℕ),\n ((fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t).natAbs =\n ((fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t).natAbs\nHC' : Cycle.Chain (fun x1 x2 => x1 = x2) C\nHeq :\n ∀ (m n : ℕ),\n (fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t =\n (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t\nIH : ∀ (n : ℕ), ((fun x => eval x P)^[n + 1] t - t).sign = (eval t P - t).sign\nk : ℕ\nhk : k + 1 > 0\nhk' : IsPeriodicPt (fun x => eval x P) (k + 1) t\nH : eval t P = t\n⊢ IsPeriodicPt (fun x => eval x P) 2 t"
},
{
"line": "apply_rfl",
"before_state": "case pos.intro.succ.intro\nP : ℤ[X]\nt : ℤ\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\nHdvd : Cycle.Chain (fun x1 x2 => x1 ∣ x2) C\nHabs :\n ∀ (m n : ℕ),\n ((fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t).natAbs =\n ((fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t).natAbs\nHC' : Cycle.Chain (fun x1 x2 => x1 = x2) C\nHeq :\n ∀ (m n : ℕ),\n (fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t =\n (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t\nIH : ∀ (n : ℕ), ((fun x => eval x P)^[n + 1] t - t).sign = (eval t P - t).sign\nk : ℕ\nhk : k + 1 > 0\nhk' : IsPeriodicPt (fun x => eval x P) (k + 1) t\nH : eval t P = t\n⊢ IsPeriodicPt (fun x => eval x P) 2 t",
"after_state": "case pos.intro.succ.intro\nP : ℤ[X]\nt : ℤ\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\nHdvd : Cycle.Chain (fun x1 x2 => x1 ∣ x2) C\nHabs :\n ∀ (m n : ℕ),\n ((fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t).natAbs =\n ((fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t).natAbs\nHC' : Cycle.Chain (fun x1 x2 => x1 = x2) C\nHeq :\n ∀ (m n : ℕ),\n (fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t =\n (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t\nIH : ∀ (n : ℕ), ((fun x => eval x P)^[n + 1] t - t).sign = (eval t P - t).sign\nk : ℕ\nhk : k + 1 > 0\nhk' : IsPeriodicPt (fun x => eval x P) (k + 1) t\nH : eval t P = t\n⊢ IsPeriodicPt (fun x => eval x P) 2 t"
},
{
"line": "skip",
"before_state": "case pos.intro.succ.intro\nP : ℤ[X]\nt : ℤ\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\nHdvd : Cycle.Chain (fun x1 x2 => x1 ∣ x2) C\nHabs :\n ∀ (m n : ℕ),\n ((fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t).natAbs =\n ((fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t).natAbs\nHC' : Cycle.Chain (fun x1 x2 => x1 = x2) C\nHeq :\n ∀ (m n : ℕ),\n (fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t =\n (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t\nIH : ∀ (n : ℕ), ((fun x => eval x P)^[n + 1] t - t).sign = (eval t P - t).sign\nk : ℕ\nhk : k + 1 > 0\nhk' : IsPeriodicPt (fun x => eval x P) (k + 1) t\nH : eval t P = t\n⊢ IsPeriodicPt (fun x => eval x P) 2 t",
"after_state": "case pos.intro.succ.intro\nP : ℤ[X]\nt : ℤ\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\nHdvd : Cycle.Chain (fun x1 x2 => x1 ∣ x2) C\nHabs :\n ∀ (m n : ℕ),\n ((fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t).natAbs =\n ((fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t).natAbs\nHC' : Cycle.Chain (fun x1 x2 => x1 = x2) C\nHeq :\n ∀ (m n : ℕ),\n (fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t =\n (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t\nIH : ∀ (n : ℕ), ((fun x => eval x P)^[n + 1] t - t).sign = (eval t P - t).sign\nk : ℕ\nhk : k + 1 > 0\nhk' : IsPeriodicPt (fun x => eval x P) (k + 1) t\nH : eval t P = t\n⊢ IsPeriodicPt (fun x => eval x P) 2 t"
},
{
"line": "simp [IsPeriodicPt, IsFixedPt, H]",
"before_state": "case pos.intro.succ.intro\nP : ℤ[X]\nt : ℤ\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\nHdvd : Cycle.Chain (fun x1 x2 => x1 ∣ x2) C\nHabs :\n ∀ (m n : ℕ),\n ((fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t).natAbs =\n ((fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t).natAbs\nHC' : Cycle.Chain (fun x1 x2 => x1 = x2) C\nHeq :\n ∀ (m n : ℕ),\n (fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t =\n (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t\nIH : ∀ (n : ℕ), ((fun x => eval x P)^[n + 1] t - t).sign = (eval t P - t).sign\nk : ℕ\nhk : k + 1 > 0\nhk' : IsPeriodicPt (fun x => eval x P) (k + 1) t\nH : eval t P = t\n⊢ IsPeriodicPt (fun x => eval x P) 2 t",
"after_state": "No Goals!"
},
{
"line": "rw [Cycle.chain_map] at HC'",
"before_state": "case neg\nP : ℤ[X]\nt : ℤ\nht : t ∈ periodicPts fun x => eval x P\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\nHdvd : Cycle.Chain (fun x1 x2 => x1 ∣ x2) C\nHabs :\n ∀ (m n : ℕ),\n ((fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t).natAbs =\n ((fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t).natAbs\nHC' : ¬Cycle.Chain (fun x1 x2 => x1 = x2) C\n⊢ IsPeriodicPt (fun x => eval x P) 2 t",
"after_state": "case neg\nP : ℤ[X]\nt : ℤ\nht : t ∈ periodicPts fun x => eval x P\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\nHdvd : Cycle.Chain (fun x1 x2 => x1 ∣ x2) C\nHabs :\n ∀ (m n : ℕ),\n ((fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t).natAbs =\n ((fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t).natAbs\nHC' : ¬Cycle.Chain (fun a b => eval a P - a = eval b P - b) (periodicOrbit (fun x => eval x P) t)\n⊢ IsPeriodicPt (fun x => eval x P) 2 t"
},
{
"line": "rewrite [Cycle.chain_map] at HC'",
"before_state": "case neg\nP : ℤ[X]\nt : ℤ\nht : t ∈ periodicPts fun x => eval x P\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\nHdvd : Cycle.Chain (fun x1 x2 => x1 ∣ x2) C\nHabs :\n ∀ (m n : ℕ),\n ((fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t).natAbs =\n ((fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t).natAbs\nHC' : ¬Cycle.Chain (fun x1 x2 => x1 = x2) C\n⊢ IsPeriodicPt (fun x => eval x P) 2 t",
"after_state": "case neg\nP : ℤ[X]\nt : ℤ\nht : t ∈ periodicPts fun x => eval x P\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\nHdvd : Cycle.Chain (fun x1 x2 => x1 ∣ x2) C\nHabs :\n ∀ (m n : ℕ),\n ((fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t).natAbs =\n ((fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t).natAbs\nHC' : ¬Cycle.Chain (fun a b => eval a P - a = eval b P - b) (periodicOrbit (fun x => eval x P) t)\n⊢ IsPeriodicPt (fun x => eval x P) 2 t"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "case neg\nP : ℤ[X]\nt : ℤ\nht : t ∈ periodicPts fun x => eval x P\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\nHdvd : Cycle.Chain (fun x1 x2 => x1 ∣ x2) C\nHabs :\n ∀ (m n : ℕ),\n ((fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t).natAbs =\n ((fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t).natAbs\nHC' : ¬Cycle.Chain (fun a b => eval a P - a = eval b P - b) (periodicOrbit (fun x => eval x P) t)\n⊢ IsPeriodicPt (fun x => eval x P) 2 t",
"after_state": "case neg\nP : ℤ[X]\nt : ℤ\nht : t ∈ periodicPts fun x => eval x P\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\nHdvd : Cycle.Chain (fun x1 x2 => x1 ∣ x2) C\nHabs :\n ∀ (m n : ℕ),\n ((fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t).natAbs =\n ((fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t).natAbs\nHC' : ¬Cycle.Chain (fun a b => eval a P - a = eval b P - b) (periodicOrbit (fun x => eval x P) t)\n⊢ IsPeriodicPt (fun x => eval x P) 2 t"
},
{
"line": "try (with_reducible rfl)",
"before_state": "case neg\nP : ℤ[X]\nt : ℤ\nht : t ∈ periodicPts fun x => eval x P\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\nHdvd : Cycle.Chain (fun x1 x2 => x1 ∣ x2) C\nHabs :\n ∀ (m n : ℕ),\n ((fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t).natAbs =\n ((fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t).natAbs\nHC' : ¬Cycle.Chain (fun a b => eval a P - a = eval b P - b) (periodicOrbit (fun x => eval x P) t)\n⊢ IsPeriodicPt (fun x => eval x P) 2 t",
"after_state": "case neg\nP : ℤ[X]\nt : ℤ\nht : t ∈ periodicPts fun x => eval x P\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\nHdvd : Cycle.Chain (fun x1 x2 => x1 ∣ x2) C\nHabs :\n ∀ (m n : ℕ),\n ((fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t).natAbs =\n ((fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t).natAbs\nHC' : ¬Cycle.Chain (fun a b => eval a P - a = eval b P - b) (periodicOrbit (fun x => eval x P) t)\n⊢ IsPeriodicPt (fun x => eval x P) 2 t"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "case neg\nP : ℤ[X]\nt : ℤ\nht : t ∈ periodicPts fun x => eval x P\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\nHdvd : Cycle.Chain (fun x1 x2 => x1 ∣ x2) C\nHabs :\n ∀ (m n : ℕ),\n ((fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t).natAbs =\n ((fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t).natAbs\nHC' : ¬Cycle.Chain (fun a b => eval a P - a = eval b P - b) (periodicOrbit (fun x => eval x P) t)\n⊢ IsPeriodicPt (fun x => eval x P) 2 t",
"after_state": "case neg\nP : ℤ[X]\nt : ℤ\nht : t ∈ periodicPts fun x => eval x P\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\nHdvd : Cycle.Chain (fun x1 x2 => x1 ∣ x2) C\nHabs :\n ∀ (m n : ℕ),\n ((fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t).natAbs =\n ((fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t).natAbs\nHC' : ¬Cycle.Chain (fun a b => eval a P - a = eval b P - b) (periodicOrbit (fun x => eval x P) t)\n⊢ IsPeriodicPt (fun x => eval x P) 2 t"
},
{
"line": "with_reducible rfl",
"before_state": "case neg\nP : ℤ[X]\nt : ℤ\nht : t ∈ periodicPts fun x => eval x P\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\nHdvd : Cycle.Chain (fun x1 x2 => x1 ∣ x2) C\nHabs :\n ∀ (m n : ℕ),\n ((fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t).natAbs =\n ((fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t).natAbs\nHC' : ¬Cycle.Chain (fun a b => eval a P - a = eval b P - b) (periodicOrbit (fun x => eval x P) t)\n⊢ IsPeriodicPt (fun x => eval x P) 2 t",
"after_state": "case neg\nP : ℤ[X]\nt : ℤ\nht : t ∈ periodicPts fun x => eval x P\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\nHdvd : Cycle.Chain (fun x1 x2 => x1 ∣ x2) C\nHabs :\n ∀ (m n : ℕ),\n ((fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t).natAbs =\n ((fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t).natAbs\nHC' : ¬Cycle.Chain (fun a b => eval a P - a = eval b P - b) (periodicOrbit (fun x => eval x P) t)\n⊢ IsPeriodicPt (fun x => eval x P) 2 t"
},
{
"line": "rfl",
"before_state": "case neg\nP : ℤ[X]\nt : ℤ\nht : t ∈ periodicPts fun x => eval x P\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\nHdvd : Cycle.Chain (fun x1 x2 => x1 ∣ x2) C\nHabs :\n ∀ (m n : ℕ),\n ((fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t).natAbs =\n ((fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t).natAbs\nHC' : ¬Cycle.Chain (fun a b => eval a P - a = eval b P - b) (periodicOrbit (fun x => eval x P) t)\n⊢ IsPeriodicPt (fun x => eval x P) 2 t",
"after_state": "case neg\nP : ℤ[X]\nt : ℤ\nht : t ∈ periodicPts fun x => eval x P\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\nHdvd : Cycle.Chain (fun x1 x2 => x1 ∣ x2) C\nHabs :\n ∀ (m n : ℕ),\n ((fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t).natAbs =\n ((fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t).natAbs\nHC' : ¬Cycle.Chain (fun a b => eval a P - a = eval b P - b) (periodicOrbit (fun x => eval x P) t)\n⊢ IsPeriodicPt (fun x => eval x P) 2 t"
},
{
"line": "apply_rfl",
"before_state": "case neg\nP : ℤ[X]\nt : ℤ\nht : t ∈ periodicPts fun x => eval x P\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\nHdvd : Cycle.Chain (fun x1 x2 => x1 ∣ x2) C\nHabs :\n ∀ (m n : ℕ),\n ((fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t).natAbs =\n ((fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t).natAbs\nHC' : ¬Cycle.Chain (fun a b => eval a P - a = eval b P - b) (periodicOrbit (fun x => eval x P) t)\n⊢ IsPeriodicPt (fun x => eval x P) 2 t",
"after_state": "case neg\nP : ℤ[X]\nt : ℤ\nht : t ∈ periodicPts fun x => eval x P\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\nHdvd : Cycle.Chain (fun x1 x2 => x1 ∣ x2) C\nHabs :\n ∀ (m n : ℕ),\n ((fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t).natAbs =\n ((fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t).natAbs\nHC' : ¬Cycle.Chain (fun a b => eval a P - a = eval b P - b) (periodicOrbit (fun x => eval x P) t)\n⊢ IsPeriodicPt (fun x => eval x P) 2 t"
},
{
"line": "skip",
"before_state": "case neg\nP : ℤ[X]\nt : ℤ\nht : t ∈ periodicPts fun x => eval x P\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\nHdvd : Cycle.Chain (fun x1 x2 => x1 ∣ x2) C\nHabs :\n ∀ (m n : ℕ),\n ((fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t).natAbs =\n ((fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t).natAbs\nHC' : ¬Cycle.Chain (fun a b => eval a P - a = eval b P - b) (periodicOrbit (fun x => eval x P) t)\n⊢ IsPeriodicPt (fun x => eval x P) 2 t",
"after_state": "case neg\nP : ℤ[X]\nt : ℤ\nht : t ∈ periodicPts fun x => eval x P\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\nHdvd : Cycle.Chain (fun x1 x2 => x1 ∣ x2) C\nHabs :\n ∀ (m n : ℕ),\n ((fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t).natAbs =\n ((fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t).natAbs\nHC' : ¬Cycle.Chain (fun a b => eval a P - a = eval b P - b) (periodicOrbit (fun x => eval x P) t)\n⊢ IsPeriodicPt (fun x => eval x P) 2 t"
},
{
"line": "rw [periodicOrbit_chain' _ ht] at HC'",
"before_state": "case neg\nP : ℤ[X]\nt : ℤ\nht : t ∈ periodicPts fun x => eval x P\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\nHdvd : Cycle.Chain (fun x1 x2 => x1 ∣ x2) C\nHabs :\n ∀ (m n : ℕ),\n ((fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t).natAbs =\n ((fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t).natAbs\nHC' : ¬Cycle.Chain (fun a b => eval a P - a = eval b P - b) (periodicOrbit (fun x => eval x P) t)\n⊢ IsPeriodicPt (fun x => eval x P) 2 t",
"after_state": "case neg\nP : ℤ[X]\nt : ℤ\nht : t ∈ periodicPts fun x => eval x P\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\nHdvd : Cycle.Chain (fun x1 x2 => x1 ∣ x2) C\nHabs :\n ∀ (m n : ℕ),\n ((fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t).natAbs =\n ((fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t).natAbs\nHC' :\n ¬∀ (n : ℕ),\n eval ((fun x => eval x P)^[n] t) P - (fun x => eval x P)^[n] t =\n eval ((fun x => eval x P)^[n + 1] t) P - (fun x => eval x P)^[n + 1] t\n⊢ IsPeriodicPt (fun x => eval x P) 2 t"
},
{
"line": "rewrite [periodicOrbit_chain' _ ht] at HC'",
"before_state": "case neg\nP : ℤ[X]\nt : ℤ\nht : t ∈ periodicPts fun x => eval x P\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\nHdvd : Cycle.Chain (fun x1 x2 => x1 ∣ x2) C\nHabs :\n ∀ (m n : ℕ),\n ((fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t).natAbs =\n ((fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t).natAbs\nHC' : ¬Cycle.Chain (fun a b => eval a P - a = eval b P - b) (periodicOrbit (fun x => eval x P) t)\n⊢ IsPeriodicPt (fun x => eval x P) 2 t",
"after_state": "case neg\nP : ℤ[X]\nt : ℤ\nht : t ∈ periodicPts fun x => eval x P\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\nHdvd : Cycle.Chain (fun x1 x2 => x1 ∣ x2) C\nHabs :\n ∀ (m n : ℕ),\n ((fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t).natAbs =\n ((fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t).natAbs\nHC' :\n ¬∀ (n : ℕ),\n eval ((fun x => eval x P)^[n] t) P - (fun x => eval x P)^[n] t =\n eval ((fun x => eval x P)^[n + 1] t) P - (fun x => eval x P)^[n + 1] t\n⊢ IsPeriodicPt (fun x => eval x P) 2 t"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "case neg\nP : ℤ[X]\nt : ℤ\nht : t ∈ periodicPts fun x => eval x P\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\nHdvd : Cycle.Chain (fun x1 x2 => x1 ∣ x2) C\nHabs :\n ∀ (m n : ℕ),\n ((fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t).natAbs =\n ((fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t).natAbs\nHC' :\n ¬∀ (n : ℕ),\n eval ((fun x => eval x P)^[n] t) P - (fun x => eval x P)^[n] t =\n eval ((fun x => eval x P)^[n + 1] t) P - (fun x => eval x P)^[n + 1] t\n⊢ IsPeriodicPt (fun x => eval x P) 2 t",
"after_state": "case neg\nP : ℤ[X]\nt : ℤ\nht : t ∈ periodicPts fun x => eval x P\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\nHdvd : Cycle.Chain (fun x1 x2 => x1 ∣ x2) C\nHabs :\n ∀ (m n : ℕ),\n ((fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t).natAbs =\n ((fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t).natAbs\nHC' :\n ¬∀ (n : ℕ),\n eval ((fun x => eval x P)^[n] t) P - (fun x => eval x P)^[n] t =\n eval ((fun x => eval x P)^[n + 1] t) P - (fun x => eval x P)^[n + 1] t\n⊢ IsPeriodicPt (fun x => eval x P) 2 t"
},
{
"line": "try (with_reducible rfl)",
"before_state": "case neg\nP : ℤ[X]\nt : ℤ\nht : t ∈ periodicPts fun x => eval x P\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\nHdvd : Cycle.Chain (fun x1 x2 => x1 ∣ x2) C\nHabs :\n ∀ (m n : ℕ),\n ((fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t).natAbs =\n ((fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t).natAbs\nHC' :\n ¬∀ (n : ℕ),\n eval ((fun x => eval x P)^[n] t) P - (fun x => eval x P)^[n] t =\n eval ((fun x => eval x P)^[n + 1] t) P - (fun x => eval x P)^[n + 1] t\n⊢ IsPeriodicPt (fun x => eval x P) 2 t",
"after_state": "case neg\nP : ℤ[X]\nt : ℤ\nht : t ∈ periodicPts fun x => eval x P\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\nHdvd : Cycle.Chain (fun x1 x2 => x1 ∣ x2) C\nHabs :\n ∀ (m n : ℕ),\n ((fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t).natAbs =\n ((fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t).natAbs\nHC' :\n ¬∀ (n : ℕ),\n eval ((fun x => eval x P)^[n] t) P - (fun x => eval x P)^[n] t =\n eval ((fun x => eval x P)^[n + 1] t) P - (fun x => eval x P)^[n + 1] t\n⊢ IsPeriodicPt (fun x => eval x P) 2 t"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "case neg\nP : ℤ[X]\nt : ℤ\nht : t ∈ periodicPts fun x => eval x P\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\nHdvd : Cycle.Chain (fun x1 x2 => x1 ∣ x2) C\nHabs :\n ∀ (m n : ℕ),\n ((fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t).natAbs =\n ((fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t).natAbs\nHC' :\n ¬∀ (n : ℕ),\n eval ((fun x => eval x P)^[n] t) P - (fun x => eval x P)^[n] t =\n eval ((fun x => eval x P)^[n + 1] t) P - (fun x => eval x P)^[n + 1] t\n⊢ IsPeriodicPt (fun x => eval x P) 2 t",
"after_state": "case neg\nP : ℤ[X]\nt : ℤ\nht : t ∈ periodicPts fun x => eval x P\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\nHdvd : Cycle.Chain (fun x1 x2 => x1 ∣ x2) C\nHabs :\n ∀ (m n : ℕ),\n ((fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t).natAbs =\n ((fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t).natAbs\nHC' :\n ¬∀ (n : ℕ),\n eval ((fun x => eval x P)^[n] t) P - (fun x => eval x P)^[n] t =\n eval ((fun x => eval x P)^[n + 1] t) P - (fun x => eval x P)^[n + 1] t\n⊢ IsPeriodicPt (fun x => eval x P) 2 t"
},
{
"line": "with_reducible rfl",
"before_state": "case neg\nP : ℤ[X]\nt : ℤ\nht : t ∈ periodicPts fun x => eval x P\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\nHdvd : Cycle.Chain (fun x1 x2 => x1 ∣ x2) C\nHabs :\n ∀ (m n : ℕ),\n ((fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t).natAbs =\n ((fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t).natAbs\nHC' :\n ¬∀ (n : ℕ),\n eval ((fun x => eval x P)^[n] t) P - (fun x => eval x P)^[n] t =\n eval ((fun x => eval x P)^[n + 1] t) P - (fun x => eval x P)^[n + 1] t\n⊢ IsPeriodicPt (fun x => eval x P) 2 t",
"after_state": "case neg\nP : ℤ[X]\nt : ℤ\nht : t ∈ periodicPts fun x => eval x P\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\nHdvd : Cycle.Chain (fun x1 x2 => x1 ∣ x2) C\nHabs :\n ∀ (m n : ℕ),\n ((fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t).natAbs =\n ((fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t).natAbs\nHC' :\n ¬∀ (n : ℕ),\n eval ((fun x => eval x P)^[n] t) P - (fun x => eval x P)^[n] t =\n eval ((fun x => eval x P)^[n + 1] t) P - (fun x => eval x P)^[n + 1] t\n⊢ IsPeriodicPt (fun x => eval x P) 2 t"
},
{
"line": "rfl",
"before_state": "case neg\nP : ℤ[X]\nt : ℤ\nht : t ∈ periodicPts fun x => eval x P\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\nHdvd : Cycle.Chain (fun x1 x2 => x1 ∣ x2) C\nHabs :\n ∀ (m n : ℕ),\n ((fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t).natAbs =\n ((fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t).natAbs\nHC' :\n ¬∀ (n : ℕ),\n eval ((fun x => eval x P)^[n] t) P - (fun x => eval x P)^[n] t =\n eval ((fun x => eval x P)^[n + 1] t) P - (fun x => eval x P)^[n + 1] t\n⊢ IsPeriodicPt (fun x => eval x P) 2 t",
"after_state": "case neg\nP : ℤ[X]\nt : ℤ\nht : t ∈ periodicPts fun x => eval x P\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\nHdvd : Cycle.Chain (fun x1 x2 => x1 ∣ x2) C\nHabs :\n ∀ (m n : ℕ),\n ((fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t).natAbs =\n ((fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t).natAbs\nHC' :\n ¬∀ (n : ℕ),\n eval ((fun x => eval x P)^[n] t) P - (fun x => eval x P)^[n] t =\n eval ((fun x => eval x P)^[n + 1] t) P - (fun x => eval x P)^[n + 1] t\n⊢ IsPeriodicPt (fun x => eval x P) 2 t"
},
{
"line": "apply_rfl",
"before_state": "case neg\nP : ℤ[X]\nt : ℤ\nht : t ∈ periodicPts fun x => eval x P\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\nHdvd : Cycle.Chain (fun x1 x2 => x1 ∣ x2) C\nHabs :\n ∀ (m n : ℕ),\n ((fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t).natAbs =\n ((fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t).natAbs\nHC' :\n ¬∀ (n : ℕ),\n eval ((fun x => eval x P)^[n] t) P - (fun x => eval x P)^[n] t =\n eval ((fun x => eval x P)^[n + 1] t) P - (fun x => eval x P)^[n + 1] t\n⊢ IsPeriodicPt (fun x => eval x P) 2 t",
"after_state": "case neg\nP : ℤ[X]\nt : ℤ\nht : t ∈ periodicPts fun x => eval x P\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\nHdvd : Cycle.Chain (fun x1 x2 => x1 ∣ x2) C\nHabs :\n ∀ (m n : ℕ),\n ((fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t).natAbs =\n ((fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t).natAbs\nHC' :\n ¬∀ (n : ℕ),\n eval ((fun x => eval x P)^[n] t) P - (fun x => eval x P)^[n] t =\n eval ((fun x => eval x P)^[n + 1] t) P - (fun x => eval x P)^[n + 1] t\n⊢ IsPeriodicPt (fun x => eval x P) 2 t"
},
{
"line": "skip",
"before_state": "case neg\nP : ℤ[X]\nt : ℤ\nht : t ∈ periodicPts fun x => eval x P\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\nHdvd : Cycle.Chain (fun x1 x2 => x1 ∣ x2) C\nHabs :\n ∀ (m n : ℕ),\n ((fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t).natAbs =\n ((fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t).natAbs\nHC' :\n ¬∀ (n : ℕ),\n eval ((fun x => eval x P)^[n] t) P - (fun x => eval x P)^[n] t =\n eval ((fun x => eval x P)^[n + 1] t) P - (fun x => eval x P)^[n + 1] t\n⊢ IsPeriodicPt (fun x => eval x P) 2 t",
"after_state": "case neg\nP : ℤ[X]\nt : ℤ\nht : t ∈ periodicPts fun x => eval x P\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\nHdvd : Cycle.Chain (fun x1 x2 => x1 ∣ x2) C\nHabs :\n ∀ (m n : ℕ),\n ((fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t).natAbs =\n ((fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t).natAbs\nHC' :\n ¬∀ (n : ℕ),\n eval ((fun x => eval x P)^[n] t) P - (fun x => eval x P)^[n] t =\n eval ((fun x => eval x P)^[n + 1] t) P - (fun x => eval x P)^[n + 1] t\n⊢ IsPeriodicPt (fun x => eval x P) 2 t"
},
{
"line": "push_neg at HC'",
"before_state": "case neg\nP : ℤ[X]\nt : ℤ\nht : t ∈ periodicPts fun x => eval x P\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\nHdvd : Cycle.Chain (fun x1 x2 => x1 ∣ x2) C\nHabs :\n ∀ (m n : ℕ),\n ((fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t).natAbs =\n ((fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t).natAbs\nHC' :\n ¬∀ (n : ℕ),\n eval ((fun x => eval x P)^[n] t) P - (fun x => eval x P)^[n] t =\n eval ((fun x => eval x P)^[n + 1] t) P - (fun x => eval x P)^[n + 1] t\n⊢ IsPeriodicPt (fun x => eval x P) 2 t",
"after_state": "case neg\nP : ℤ[X]\nt : ℤ\nht : t ∈ periodicPts fun x => eval x P\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\nHdvd : Cycle.Chain (fun x1 x2 => x1 ∣ x2) C\nHabs :\n ∀ (m n : ℕ),\n ((fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t).natAbs =\n ((fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t).natAbs\nHC' :\n ∃ n,\n eval ((fun x => eval x P)^[n] t) P - (fun x => eval x P)^[n] t ≠\n eval ((fun x => eval x P)^[n + 1] t) P - (fun x => eval x P)^[n + 1] t\n⊢ IsPeriodicPt (fun x => eval x P) 2 t"
},
{
"line": "obtain ⟨n, hn⟩ := HC'",
"before_state": "case neg\nP : ℤ[X]\nt : ℤ\nht : t ∈ periodicPts fun x => eval x P\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\nHdvd : Cycle.Chain (fun x1 x2 => x1 ∣ x2) C\nHabs :\n ∀ (m n : ℕ),\n ((fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t).natAbs =\n ((fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t).natAbs\nHC' :\n ∃ n,\n eval ((fun x => eval x P)^[n] t) P - (fun x => eval x P)^[n] t ≠\n eval ((fun x => eval x P)^[n + 1] t) P - (fun x => eval x P)^[n + 1] t\n⊢ IsPeriodicPt (fun x => eval x P) 2 t",
"after_state": "case neg.intro\nP : ℤ[X]\nt : ℤ\nht : t ∈ periodicPts fun x => eval x P\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\nHdvd : Cycle.Chain (fun x1 x2 => x1 ∣ x2) C\nHabs :\n ∀ (m n : ℕ),\n ((fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t).natAbs =\n ((fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t).natAbs\nn : ℕ\nhn :\n eval ((fun x => eval x P)^[n] t) P - (fun x => eval x P)^[n] t ≠\n eval ((fun x => eval x P)^[n + 1] t) P - (fun x => eval x P)^[n + 1] t\n⊢ IsPeriodicPt (fun x => eval x P) 2 t"
},
{
"line": "rcases Int.natAbs_eq_natAbs_iff.1 (Habs n n.succ) with hn' | hn'",
"before_state": "case neg.intro\nP : ℤ[X]\nt : ℤ\nht : t ∈ periodicPts fun x => eval x P\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\nHdvd : Cycle.Chain (fun x1 x2 => x1 ∣ x2) C\nHabs :\n ∀ (m n : ℕ),\n ((fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t).natAbs =\n ((fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t).natAbs\nn : ℕ\nhn :\n eval ((fun x => eval x P)^[n] t) P - (fun x => eval x P)^[n] t ≠\n eval ((fun x => eval x P)^[n + 1] t) P - (fun x => eval x P)^[n + 1] t\n⊢ IsPeriodicPt (fun x => eval x P) 2 t",
"after_state": "case neg.intro.inl\nP : ℤ[X]\nt : ℤ\nht : t ∈ periodicPts fun x => eval x P\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\nHdvd : Cycle.Chain (fun x1 x2 => x1 ∣ x2) C\nHabs :\n ∀ (m n : ℕ),\n ((fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t).natAbs =\n ((fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t).natAbs\nn : ℕ\nhn :\n eval ((fun x => eval x P)^[n] t) P - (fun x => eval x P)^[n] t ≠\n eval ((fun x => eval x P)^[n + 1] t) P - (fun x => eval x P)^[n + 1] t\nhn' :\n (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t =\n (fun x => eval x P)^[n.succ + 1] t - (fun x => eval x P)^[n.succ] t\n⊢ IsPeriodicPt (fun x => eval x P) 2 t\n---\ncase neg.intro.inr\nP : ℤ[X]\nt : ℤ\nht : t ∈ periodicPts fun x => eval x P\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\nHdvd : Cycle.Chain (fun x1 x2 => x1 ∣ x2) C\nHabs :\n ∀ (m n : ℕ),\n ((fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t).natAbs =\n ((fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t).natAbs\nn : ℕ\nhn :\n eval ((fun x => eval x P)^[n] t) P - (fun x => eval x P)^[n] t ≠\n eval ((fun x => eval x P)^[n + 1] t) P - (fun x => eval x P)^[n + 1] t\nhn' :\n (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t =\n -((fun x => eval x P)^[n.succ + 1] t - (fun x => eval x P)^[n.succ] t)\n⊢ IsPeriodicPt (fun x => eval x P) 2 t"
},
{
"line": "apply (hn _).elim",
"before_state": "case neg.intro.inl\nP : ℤ[X]\nt : ℤ\nht : t ∈ periodicPts fun x => eval x P\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\nHdvd : Cycle.Chain (fun x1 x2 => x1 ∣ x2) C\nHabs :\n ∀ (m n : ℕ),\n ((fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t).natAbs =\n ((fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t).natAbs\nn : ℕ\nhn :\n eval ((fun x => eval x P)^[n] t) P - (fun x => eval x P)^[n] t ≠\n eval ((fun x => eval x P)^[n + 1] t) P - (fun x => eval x P)^[n + 1] t\nhn' :\n (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t =\n (fun x => eval x P)^[n.succ + 1] t - (fun x => eval x P)^[n.succ] t\n⊢ IsPeriodicPt (fun x => eval x P) 2 t",
"after_state": "P : ℤ[X]\nt : ℤ\nht : t ∈ periodicPts fun x => eval x P\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\nHdvd : Cycle.Chain (fun x1 x2 => x1 ∣ x2) C\nHabs :\n ∀ (m n : ℕ),\n ((fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t).natAbs =\n ((fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t).natAbs\nn : ℕ\nhn :\n eval ((fun x => eval x P)^[n] t) P - (fun x => eval x P)^[n] t ≠\n eval ((fun x => eval x P)^[n + 1] t) P - (fun x => eval x P)^[n + 1] t\nhn' :\n (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t =\n (fun x => eval x P)^[n.succ + 1] t - (fun x => eval x P)^[n.succ] t\n⊢ eval ((fun x => eval x P)^[n] t) P - (fun x => eval x P)^[n] t =\n eval ((fun x => eval x P)^[n + 1] t) P - (fun x => eval x P)^[n + 1] t"
},
{
"line": "focus\n convert hn'\n with_annotate_state\"<;>\" skip\n all_goals\n simp only [Function.iterate_succ_apply']\n -- We deduce P^{k + 2}(t) = P^k(t) and hence P(P(t)) = t.",
"before_state": "P : ℤ[X]\nt : ℤ\nht : t ∈ periodicPts fun x => eval x P\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\nHdvd : Cycle.Chain (fun x1 x2 => x1 ∣ x2) C\nHabs :\n ∀ (m n : ℕ),\n ((fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t).natAbs =\n ((fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t).natAbs\nn : ℕ\nhn :\n eval ((fun x => eval x P)^[n] t) P - (fun x => eval x P)^[n] t ≠\n eval ((fun x => eval x P)^[n + 1] t) P - (fun x => eval x P)^[n + 1] t\nhn' :\n (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t =\n (fun x => eval x P)^[n.succ + 1] t - (fun x => eval x P)^[n.succ] t\n⊢ eval ((fun x => eval x P)^[n] t) P - (fun x => eval x P)^[n] t =\n eval ((fun x => eval x P)^[n + 1] t) P - (fun x => eval x P)^[n + 1] t",
"after_state": "No Goals!"
},
{
"line": "convert hn'",
"before_state": "P : ℤ[X]\nt : ℤ\nht : t ∈ periodicPts fun x => eval x P\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\nHdvd : Cycle.Chain (fun x1 x2 => x1 ∣ x2) C\nHabs :\n ∀ (m n : ℕ),\n ((fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t).natAbs =\n ((fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t).natAbs\nn : ℕ\nhn :\n eval ((fun x => eval x P)^[n] t) P - (fun x => eval x P)^[n] t ≠\n eval ((fun x => eval x P)^[n + 1] t) P - (fun x => eval x P)^[n + 1] t\nhn' :\n (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t =\n (fun x => eval x P)^[n.succ + 1] t - (fun x => eval x P)^[n.succ] t\n⊢ eval ((fun x => eval x P)^[n] t) P - (fun x => eval x P)^[n] t =\n eval ((fun x => eval x P)^[n + 1] t) P - (fun x => eval x P)^[n + 1] t",
"after_state": "case h.e'_2.h.e'_5\nP : ℤ[X]\nt : ℤ\nht : t ∈ periodicPts fun x => eval x P\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\nHdvd : Cycle.Chain (fun x1 x2 => x1 ∣ x2) C\nHabs :\n ∀ (m n : ℕ),\n ((fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t).natAbs =\n ((fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t).natAbs\nn : ℕ\nhn :\n eval ((fun x => eval x P)^[n] t) P - (fun x => eval x P)^[n] t ≠\n eval ((fun x => eval x P)^[n + 1] t) P - (fun x => eval x P)^[n + 1] t\nhn' :\n (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t =\n (fun x => eval x P)^[n.succ + 1] t - (fun x => eval x P)^[n.succ] t\n⊢ eval ((fun x => eval x P)^[n] t) P = (fun x => eval x P)^[n + 1] t\n---\ncase h.e'_3.h.e'_5\nP : ℤ[X]\nt : ℤ\nht : t ∈ periodicPts fun x => eval x P\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\nHdvd : Cycle.Chain (fun x1 x2 => x1 ∣ x2) C\nHabs :\n ∀ (m n : ℕ),\n ((fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t).natAbs =\n ((fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t).natAbs\nn : ℕ\nhn :\n eval ((fun x => eval x P)^[n] t) P - (fun x => eval x P)^[n] t ≠\n eval ((fun x => eval x P)^[n + 1] t) P - (fun x => eval x P)^[n + 1] t\nhn' :\n (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t =\n (fun x => eval x P)^[n.succ + 1] t - (fun x => eval x P)^[n.succ] t\n⊢ eval ((fun x => eval x P)^[n + 1] t) P = (fun x => eval x P)^[n.succ + 1] t"
},
{
"line": "with_annotate_state\"<;>\" skip",
"before_state": "case h.e'_2.h.e'_5\nP : ℤ[X]\nt : ℤ\nht : t ∈ periodicPts fun x => eval x P\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\nHdvd : Cycle.Chain (fun x1 x2 => x1 ∣ x2) C\nHabs :\n ∀ (m n : ℕ),\n ((fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t).natAbs =\n ((fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t).natAbs\nn : ℕ\nhn :\n eval ((fun x => eval x P)^[n] t) P - (fun x => eval x P)^[n] t ≠\n eval ((fun x => eval x P)^[n + 1] t) P - (fun x => eval x P)^[n + 1] t\nhn' :\n (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t =\n (fun x => eval x P)^[n.succ + 1] t - (fun x => eval x P)^[n.succ] t\n⊢ eval ((fun x => eval x P)^[n] t) P = (fun x => eval x P)^[n + 1] t\n---\ncase h.e'_3.h.e'_5\nP : ℤ[X]\nt : ℤ\nht : t ∈ periodicPts fun x => eval x P\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\nHdvd : Cycle.Chain (fun x1 x2 => x1 ∣ x2) C\nHabs :\n ∀ (m n : ℕ),\n ((fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t).natAbs =\n ((fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t).natAbs\nn : ℕ\nhn :\n eval ((fun x => eval x P)^[n] t) P - (fun x => eval x P)^[n] t ≠\n eval ((fun x => eval x P)^[n + 1] t) P - (fun x => eval x P)^[n + 1] t\nhn' :\n (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t =\n (fun x => eval x P)^[n.succ + 1] t - (fun x => eval x P)^[n.succ] t\n⊢ eval ((fun x => eval x P)^[n + 1] t) P = (fun x => eval x P)^[n.succ + 1] t",
"after_state": "case h.e'_2.h.e'_5\nP : ℤ[X]\nt : ℤ\nht : t ∈ periodicPts fun x => eval x P\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\nHdvd : Cycle.Chain (fun x1 x2 => x1 ∣ x2) C\nHabs :\n ∀ (m n : ℕ),\n ((fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t).natAbs =\n ((fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t).natAbs\nn : ℕ\nhn :\n eval ((fun x => eval x P)^[n] t) P - (fun x => eval x P)^[n] t ≠\n eval ((fun x => eval x P)^[n + 1] t) P - (fun x => eval x P)^[n + 1] t\nhn' :\n (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t =\n (fun x => eval x P)^[n.succ + 1] t - (fun x => eval x P)^[n.succ] t\n⊢ eval ((fun x => eval x P)^[n] t) P = (fun x => eval x P)^[n + 1] t\n---\ncase h.e'_3.h.e'_5\nP : ℤ[X]\nt : ℤ\nht : t ∈ periodicPts fun x => eval x P\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\nHdvd : Cycle.Chain (fun x1 x2 => x1 ∣ x2) C\nHabs :\n ∀ (m n : ℕ),\n ((fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t).natAbs =\n ((fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t).natAbs\nn : ℕ\nhn :\n eval ((fun x => eval x P)^[n] t) P - (fun x => eval x P)^[n] t ≠\n eval ((fun x => eval x P)^[n + 1] t) P - (fun x => eval x P)^[n + 1] t\nhn' :\n (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t =\n (fun x => eval x P)^[n.succ + 1] t - (fun x => eval x P)^[n.succ] t\n⊢ eval ((fun x => eval x P)^[n + 1] t) P = (fun x => eval x P)^[n.succ + 1] t"
},
{
"line": "skip",
"before_state": "case h.e'_2.h.e'_5\nP : ℤ[X]\nt : ℤ\nht : t ∈ periodicPts fun x => eval x P\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\nHdvd : Cycle.Chain (fun x1 x2 => x1 ∣ x2) C\nHabs :\n ∀ (m n : ℕ),\n ((fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t).natAbs =\n ((fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t).natAbs\nn : ℕ\nhn :\n eval ((fun x => eval x P)^[n] t) P - (fun x => eval x P)^[n] t ≠\n eval ((fun x => eval x P)^[n + 1] t) P - (fun x => eval x P)^[n + 1] t\nhn' :\n (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t =\n (fun x => eval x P)^[n.succ + 1] t - (fun x => eval x P)^[n.succ] t\n⊢ eval ((fun x => eval x P)^[n] t) P = (fun x => eval x P)^[n + 1] t\n---\ncase h.e'_3.h.e'_5\nP : ℤ[X]\nt : ℤ\nht : t ∈ periodicPts fun x => eval x P\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\nHdvd : Cycle.Chain (fun x1 x2 => x1 ∣ x2) C\nHabs :\n ∀ (m n : ℕ),\n ((fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t).natAbs =\n ((fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t).natAbs\nn : ℕ\nhn :\n eval ((fun x => eval x P)^[n] t) P - (fun x => eval x P)^[n] t ≠\n eval ((fun x => eval x P)^[n + 1] t) P - (fun x => eval x P)^[n + 1] t\nhn' :\n (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t =\n (fun x => eval x P)^[n.succ + 1] t - (fun x => eval x P)^[n.succ] t\n⊢ eval ((fun x => eval x P)^[n + 1] t) P = (fun x => eval x P)^[n.succ + 1] t",
"after_state": "case h.e'_2.h.e'_5\nP : ℤ[X]\nt : ℤ\nht : t ∈ periodicPts fun x => eval x P\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\nHdvd : Cycle.Chain (fun x1 x2 => x1 ∣ x2) C\nHabs :\n ∀ (m n : ℕ),\n ((fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t).natAbs =\n ((fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t).natAbs\nn : ℕ\nhn :\n eval ((fun x => eval x P)^[n] t) P - (fun x => eval x P)^[n] t ≠\n eval ((fun x => eval x P)^[n + 1] t) P - (fun x => eval x P)^[n + 1] t\nhn' :\n (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t =\n (fun x => eval x P)^[n.succ + 1] t - (fun x => eval x P)^[n.succ] t\n⊢ eval ((fun x => eval x P)^[n] t) P = (fun x => eval x P)^[n + 1] t\n---\ncase h.e'_3.h.e'_5\nP : ℤ[X]\nt : ℤ\nht : t ∈ periodicPts fun x => eval x P\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\nHdvd : Cycle.Chain (fun x1 x2 => x1 ∣ x2) C\nHabs :\n ∀ (m n : ℕ),\n ((fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t).natAbs =\n ((fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t).natAbs\nn : ℕ\nhn :\n eval ((fun x => eval x P)^[n] t) P - (fun x => eval x P)^[n] t ≠\n eval ((fun x => eval x P)^[n + 1] t) P - (fun x => eval x P)^[n + 1] t\nhn' :\n (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t =\n (fun x => eval x P)^[n.succ + 1] t - (fun x => eval x P)^[n.succ] t\n⊢ eval ((fun x => eval x P)^[n + 1] t) P = (fun x => eval x P)^[n.succ + 1] t"
},
{
"line": "all_goals\n simp only [Function.iterate_succ_apply']\n -- We deduce P^{k + 2}(t) = P^k(t) and hence P(P(t)) = t.",
"before_state": "case h.e'_2.h.e'_5\nP : ℤ[X]\nt : ℤ\nht : t ∈ periodicPts fun x => eval x P\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\nHdvd : Cycle.Chain (fun x1 x2 => x1 ∣ x2) C\nHabs :\n ∀ (m n : ℕ),\n ((fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t).natAbs =\n ((fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t).natAbs\nn : ℕ\nhn :\n eval ((fun x => eval x P)^[n] t) P - (fun x => eval x P)^[n] t ≠\n eval ((fun x => eval x P)^[n + 1] t) P - (fun x => eval x P)^[n + 1] t\nhn' :\n (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t =\n (fun x => eval x P)^[n.succ + 1] t - (fun x => eval x P)^[n.succ] t\n⊢ eval ((fun x => eval x P)^[n] t) P = (fun x => eval x P)^[n + 1] t\n---\ncase h.e'_3.h.e'_5\nP : ℤ[X]\nt : ℤ\nht : t ∈ periodicPts fun x => eval x P\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\nHdvd : Cycle.Chain (fun x1 x2 => x1 ∣ x2) C\nHabs :\n ∀ (m n : ℕ),\n ((fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t).natAbs =\n ((fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t).natAbs\nn : ℕ\nhn :\n eval ((fun x => eval x P)^[n] t) P - (fun x => eval x P)^[n] t ≠\n eval ((fun x => eval x P)^[n + 1] t) P - (fun x => eval x P)^[n + 1] t\nhn' :\n (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t =\n (fun x => eval x P)^[n.succ + 1] t - (fun x => eval x P)^[n.succ] t\n⊢ eval ((fun x => eval x P)^[n + 1] t) P = (fun x => eval x P)^[n.succ + 1] t",
"after_state": "No Goals!"
},
{
"line": "simp only [Function.iterate_succ_apply']\n -- We deduce P^{k + 2}(t) = P^k(t) and hence P(P(t)) = t.",
"before_state": "case h.e'_2.h.e'_5\nP : ℤ[X]\nt : ℤ\nht : t ∈ periodicPts fun x => eval x P\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\nHdvd : Cycle.Chain (fun x1 x2 => x1 ∣ x2) C\nHabs :\n ∀ (m n : ℕ),\n ((fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t).natAbs =\n ((fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t).natAbs\nn : ℕ\nhn :\n eval ((fun x => eval x P)^[n] t) P - (fun x => eval x P)^[n] t ≠\n eval ((fun x => eval x P)^[n + 1] t) P - (fun x => eval x P)^[n + 1] t\nhn' :\n (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t =\n (fun x => eval x P)^[n.succ + 1] t - (fun x => eval x P)^[n.succ] t\n⊢ eval ((fun x => eval x P)^[n] t) P = (fun x => eval x P)^[n + 1] t",
"after_state": "No Goals!"
},
{
"line": "simp only [Function.iterate_succ_apply']\n -- We deduce P^{k + 2}(t) = P^k(t) and hence P(P(t)) = t.",
"before_state": "case h.e'_3.h.e'_5\nP : ℤ[X]\nt : ℤ\nht : t ∈ periodicPts fun x => eval x P\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\nHdvd : Cycle.Chain (fun x1 x2 => x1 ∣ x2) C\nHabs :\n ∀ (m n : ℕ),\n ((fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t).natAbs =\n ((fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t).natAbs\nn : ℕ\nhn :\n eval ((fun x => eval x P)^[n] t) P - (fun x => eval x P)^[n] t ≠\n eval ((fun x => eval x P)^[n + 1] t) P - (fun x => eval x P)^[n + 1] t\nhn' :\n (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t =\n (fun x => eval x P)^[n.succ + 1] t - (fun x => eval x P)^[n.succ] t\n⊢ eval ((fun x => eval x P)^[n + 1] t) P = (fun x => eval x P)^[n.succ + 1] t",
"after_state": "No Goals!"
},
{
"line": "rw [neg_sub, sub_right_inj] at hn'",
"before_state": "case neg.intro.inr\nP : ℤ[X]\nt : ℤ\nht : t ∈ periodicPts fun x => eval x P\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\nHdvd : Cycle.Chain (fun x1 x2 => x1 ∣ x2) C\nHabs :\n ∀ (m n : ℕ),\n ((fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t).natAbs =\n ((fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t).natAbs\nn : ℕ\nhn :\n eval ((fun x => eval x P)^[n] t) P - (fun x => eval x P)^[n] t ≠\n eval ((fun x => eval x P)^[n + 1] t) P - (fun x => eval x P)^[n + 1] t\nhn' :\n (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t =\n -((fun x => eval x P)^[n.succ + 1] t - (fun x => eval x P)^[n.succ] t)\n⊢ IsPeriodicPt (fun x => eval x P) 2 t",
"after_state": "case neg.intro.inr\nP : ℤ[X]\nt : ℤ\nht : t ∈ periodicPts fun x => eval x P\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\nHdvd : Cycle.Chain (fun x1 x2 => x1 ∣ x2) C\nHabs :\n ∀ (m n : ℕ),\n ((fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t).natAbs =\n ((fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t).natAbs\nn : ℕ\nhn :\n eval ((fun x => eval x P)^[n] t) P - (fun x => eval x P)^[n] t ≠\n eval ((fun x => eval x P)^[n + 1] t) P - (fun x => eval x P)^[n + 1] t\nhn' : (fun x => eval x P)^[n] t = (fun x => eval x P)^[n.succ + 1] t\n⊢ IsPeriodicPt (fun x => eval x P) 2 t"
},
{
"line": "rewrite [neg_sub, sub_right_inj] at hn'",
"before_state": "case neg.intro.inr\nP : ℤ[X]\nt : ℤ\nht : t ∈ periodicPts fun x => eval x P\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\nHdvd : Cycle.Chain (fun x1 x2 => x1 ∣ x2) C\nHabs :\n ∀ (m n : ℕ),\n ((fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t).natAbs =\n ((fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t).natAbs\nn : ℕ\nhn :\n eval ((fun x => eval x P)^[n] t) P - (fun x => eval x P)^[n] t ≠\n eval ((fun x => eval x P)^[n + 1] t) P - (fun x => eval x P)^[n + 1] t\nhn' :\n (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t =\n -((fun x => eval x P)^[n.succ + 1] t - (fun x => eval x P)^[n.succ] t)\n⊢ IsPeriodicPt (fun x => eval x P) 2 t",
"after_state": "case neg.intro.inr\nP : ℤ[X]\nt : ℤ\nht : t ∈ periodicPts fun x => eval x P\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\nHdvd : Cycle.Chain (fun x1 x2 => x1 ∣ x2) C\nHabs :\n ∀ (m n : ℕ),\n ((fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t).natAbs =\n ((fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t).natAbs\nn : ℕ\nhn :\n eval ((fun x => eval x P)^[n] t) P - (fun x => eval x P)^[n] t ≠\n eval ((fun x => eval x P)^[n + 1] t) P - (fun x => eval x P)^[n + 1] t\nhn' : (fun x => eval x P)^[n] t = (fun x => eval x P)^[n.succ + 1] t\n⊢ IsPeriodicPt (fun x => eval x P) 2 t"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "case neg.intro.inr\nP : ℤ[X]\nt : ℤ\nht : t ∈ periodicPts fun x => eval x P\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\nHdvd : Cycle.Chain (fun x1 x2 => x1 ∣ x2) C\nHabs :\n ∀ (m n : ℕ),\n ((fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t).natAbs =\n ((fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t).natAbs\nn : ℕ\nhn :\n eval ((fun x => eval x P)^[n] t) P - (fun x => eval x P)^[n] t ≠\n eval ((fun x => eval x P)^[n + 1] t) P - (fun x => eval x P)^[n + 1] t\nhn' : (fun x => eval x P)^[n] t = (fun x => eval x P)^[n.succ + 1] t\n⊢ IsPeriodicPt (fun x => eval x P) 2 t",
"after_state": "case neg.intro.inr\nP : ℤ[X]\nt : ℤ\nht : t ∈ periodicPts fun x => eval x P\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\nHdvd : Cycle.Chain (fun x1 x2 => x1 ∣ x2) C\nHabs :\n ∀ (m n : ℕ),\n ((fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t).natAbs =\n ((fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t).natAbs\nn : ℕ\nhn :\n eval ((fun x => eval x P)^[n] t) P - (fun x => eval x P)^[n] t ≠\n eval ((fun x => eval x P)^[n + 1] t) P - (fun x => eval x P)^[n + 1] t\nhn' : (fun x => eval x P)^[n] t = (fun x => eval x P)^[n.succ + 1] t\n⊢ IsPeriodicPt (fun x => eval x P) 2 t"
},
{
"line": "try (with_reducible rfl)",
"before_state": "case neg.intro.inr\nP : ℤ[X]\nt : ℤ\nht : t ∈ periodicPts fun x => eval x P\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\nHdvd : Cycle.Chain (fun x1 x2 => x1 ∣ x2) C\nHabs :\n ∀ (m n : ℕ),\n ((fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t).natAbs =\n ((fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t).natAbs\nn : ℕ\nhn :\n eval ((fun x => eval x P)^[n] t) P - (fun x => eval x P)^[n] t ≠\n eval ((fun x => eval x P)^[n + 1] t) P - (fun x => eval x P)^[n + 1] t\nhn' : (fun x => eval x P)^[n] t = (fun x => eval x P)^[n.succ + 1] t\n⊢ IsPeriodicPt (fun x => eval x P) 2 t",
"after_state": "case neg.intro.inr\nP : ℤ[X]\nt : ℤ\nht : t ∈ periodicPts fun x => eval x P\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\nHdvd : Cycle.Chain (fun x1 x2 => x1 ∣ x2) C\nHabs :\n ∀ (m n : ℕ),\n ((fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t).natAbs =\n ((fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t).natAbs\nn : ℕ\nhn :\n eval ((fun x => eval x P)^[n] t) P - (fun x => eval x P)^[n] t ≠\n eval ((fun x => eval x P)^[n + 1] t) P - (fun x => eval x P)^[n + 1] t\nhn' : (fun x => eval x P)^[n] t = (fun x => eval x P)^[n.succ + 1] t\n⊢ IsPeriodicPt (fun x => eval x P) 2 t"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "case neg.intro.inr\nP : ℤ[X]\nt : ℤ\nht : t ∈ periodicPts fun x => eval x P\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\nHdvd : Cycle.Chain (fun x1 x2 => x1 ∣ x2) C\nHabs :\n ∀ (m n : ℕ),\n ((fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t).natAbs =\n ((fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t).natAbs\nn : ℕ\nhn :\n eval ((fun x => eval x P)^[n] t) P - (fun x => eval x P)^[n] t ≠\n eval ((fun x => eval x P)^[n + 1] t) P - (fun x => eval x P)^[n + 1] t\nhn' : (fun x => eval x P)^[n] t = (fun x => eval x P)^[n.succ + 1] t\n⊢ IsPeriodicPt (fun x => eval x P) 2 t",
"after_state": "case neg.intro.inr\nP : ℤ[X]\nt : ℤ\nht : t ∈ periodicPts fun x => eval x P\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\nHdvd : Cycle.Chain (fun x1 x2 => x1 ∣ x2) C\nHabs :\n ∀ (m n : ℕ),\n ((fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t).natAbs =\n ((fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t).natAbs\nn : ℕ\nhn :\n eval ((fun x => eval x P)^[n] t) P - (fun x => eval x P)^[n] t ≠\n eval ((fun x => eval x P)^[n + 1] t) P - (fun x => eval x P)^[n + 1] t\nhn' : (fun x => eval x P)^[n] t = (fun x => eval x P)^[n.succ + 1] t\n⊢ IsPeriodicPt (fun x => eval x P) 2 t"
},
{
"line": "with_reducible rfl",
"before_state": "case neg.intro.inr\nP : ℤ[X]\nt : ℤ\nht : t ∈ periodicPts fun x => eval x P\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\nHdvd : Cycle.Chain (fun x1 x2 => x1 ∣ x2) C\nHabs :\n ∀ (m n : ℕ),\n ((fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t).natAbs =\n ((fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t).natAbs\nn : ℕ\nhn :\n eval ((fun x => eval x P)^[n] t) P - (fun x => eval x P)^[n] t ≠\n eval ((fun x => eval x P)^[n + 1] t) P - (fun x => eval x P)^[n + 1] t\nhn' : (fun x => eval x P)^[n] t = (fun x => eval x P)^[n.succ + 1] t\n⊢ IsPeriodicPt (fun x => eval x P) 2 t",
"after_state": "case neg.intro.inr\nP : ℤ[X]\nt : ℤ\nht : t ∈ periodicPts fun x => eval x P\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\nHdvd : Cycle.Chain (fun x1 x2 => x1 ∣ x2) C\nHabs :\n ∀ (m n : ℕ),\n ((fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t).natAbs =\n ((fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t).natAbs\nn : ℕ\nhn :\n eval ((fun x => eval x P)^[n] t) P - (fun x => eval x P)^[n] t ≠\n eval ((fun x => eval x P)^[n + 1] t) P - (fun x => eval x P)^[n + 1] t\nhn' : (fun x => eval x P)^[n] t = (fun x => eval x P)^[n.succ + 1] t\n⊢ IsPeriodicPt (fun x => eval x P) 2 t"
},
{
"line": "rfl",
"before_state": "case neg.intro.inr\nP : ℤ[X]\nt : ℤ\nht : t ∈ periodicPts fun x => eval x P\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\nHdvd : Cycle.Chain (fun x1 x2 => x1 ∣ x2) C\nHabs :\n ∀ (m n : ℕ),\n ((fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t).natAbs =\n ((fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t).natAbs\nn : ℕ\nhn :\n eval ((fun x => eval x P)^[n] t) P - (fun x => eval x P)^[n] t ≠\n eval ((fun x => eval x P)^[n + 1] t) P - (fun x => eval x P)^[n + 1] t\nhn' : (fun x => eval x P)^[n] t = (fun x => eval x P)^[n.succ + 1] t\n⊢ IsPeriodicPt (fun x => eval x P) 2 t",
"after_state": "case neg.intro.inr\nP : ℤ[X]\nt : ℤ\nht : t ∈ periodicPts fun x => eval x P\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\nHdvd : Cycle.Chain (fun x1 x2 => x1 ∣ x2) C\nHabs :\n ∀ (m n : ℕ),\n ((fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t).natAbs =\n ((fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t).natAbs\nn : ℕ\nhn :\n eval ((fun x => eval x P)^[n] t) P - (fun x => eval x P)^[n] t ≠\n eval ((fun x => eval x P)^[n + 1] t) P - (fun x => eval x P)^[n + 1] t\nhn' : (fun x => eval x P)^[n] t = (fun x => eval x P)^[n.succ + 1] t\n⊢ IsPeriodicPt (fun x => eval x P) 2 t"
},
{
"line": "apply_rfl",
"before_state": "case neg.intro.inr\nP : ℤ[X]\nt : ℤ\nht : t ∈ periodicPts fun x => eval x P\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\nHdvd : Cycle.Chain (fun x1 x2 => x1 ∣ x2) C\nHabs :\n ∀ (m n : ℕ),\n ((fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t).natAbs =\n ((fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t).natAbs\nn : ℕ\nhn :\n eval ((fun x => eval x P)^[n] t) P - (fun x => eval x P)^[n] t ≠\n eval ((fun x => eval x P)^[n + 1] t) P - (fun x => eval x P)^[n + 1] t\nhn' : (fun x => eval x P)^[n] t = (fun x => eval x P)^[n.succ + 1] t\n⊢ IsPeriodicPt (fun x => eval x P) 2 t",
"after_state": "case neg.intro.inr\nP : ℤ[X]\nt : ℤ\nht : t ∈ periodicPts fun x => eval x P\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\nHdvd : Cycle.Chain (fun x1 x2 => x1 ∣ x2) C\nHabs :\n ∀ (m n : ℕ),\n ((fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t).natAbs =\n ((fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t).natAbs\nn : ℕ\nhn :\n eval ((fun x => eval x P)^[n] t) P - (fun x => eval x P)^[n] t ≠\n eval ((fun x => eval x P)^[n + 1] t) P - (fun x => eval x P)^[n + 1] t\nhn' : (fun x => eval x P)^[n] t = (fun x => eval x P)^[n.succ + 1] t\n⊢ IsPeriodicPt (fun x => eval x P) 2 t"
},
{
"line": "skip",
"before_state": "case neg.intro.inr\nP : ℤ[X]\nt : ℤ\nht : t ∈ periodicPts fun x => eval x P\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\nHdvd : Cycle.Chain (fun x1 x2 => x1 ∣ x2) C\nHabs :\n ∀ (m n : ℕ),\n ((fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t).natAbs =\n ((fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t).natAbs\nn : ℕ\nhn :\n eval ((fun x => eval x P)^[n] t) P - (fun x => eval x P)^[n] t ≠\n eval ((fun x => eval x P)^[n + 1] t) P - (fun x => eval x P)^[n + 1] t\nhn' : (fun x => eval x P)^[n] t = (fun x => eval x P)^[n.succ + 1] t\n⊢ IsPeriodicPt (fun x => eval x P) 2 t",
"after_state": "case neg.intro.inr\nP : ℤ[X]\nt : ℤ\nht : t ∈ periodicPts fun x => eval x P\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\nHdvd : Cycle.Chain (fun x1 x2 => x1 ∣ x2) C\nHabs :\n ∀ (m n : ℕ),\n ((fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t).natAbs =\n ((fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t).natAbs\nn : ℕ\nhn :\n eval ((fun x => eval x P)^[n] t) P - (fun x => eval x P)^[n] t ≠\n eval ((fun x => eval x P)^[n + 1] t) P - (fun x => eval x P)^[n + 1] t\nhn' : (fun x => eval x P)^[n] t = (fun x => eval x P)^[n.succ + 1] t\n⊢ IsPeriodicPt (fun x => eval x P) 2 t"
},
{
"line": "simp only [Function.iterate_succ_apply'] at hn'",
"before_state": "case neg.intro.inr\nP : ℤ[X]\nt : ℤ\nht : t ∈ periodicPts fun x => eval x P\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\nHdvd : Cycle.Chain (fun x1 x2 => x1 ∣ x2) C\nHabs :\n ∀ (m n : ℕ),\n ((fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t).natAbs =\n ((fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t).natAbs\nn : ℕ\nhn :\n eval ((fun x => eval x P)^[n] t) P - (fun x => eval x P)^[n] t ≠\n eval ((fun x => eval x P)^[n + 1] t) P - (fun x => eval x P)^[n + 1] t\nhn' : (fun x => eval x P)^[n] t = (fun x => eval x P)^[n.succ + 1] t\n⊢ IsPeriodicPt (fun x => eval x P) 2 t",
"after_state": "case neg.intro.inr\nP : ℤ[X]\nt : ℤ\nht : t ∈ periodicPts fun x => eval x P\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\nHdvd : Cycle.Chain (fun x1 x2 => x1 ∣ x2) C\nHabs :\n ∀ (m n : ℕ),\n ((fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t).natAbs =\n ((fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t).natAbs\nn : ℕ\nhn :\n eval ((fun x => eval x P)^[n] t) P - (fun x => eval x P)^[n] t ≠\n eval ((fun x => eval x P)^[n + 1] t) P - (fun x => eval x P)^[n + 1] t\nhn' : (fun x => eval x P)^[n] t = eval (eval ((fun x => eval x P)^[n] t) P) P\n⊢ IsPeriodicPt (fun x => eval x P) 2 t"
},
{
"line": "exact isPeriodicPt_of_mem_periodicPts_of_isPeriodicPt_iterate ht hn'.symm",
"before_state": "case neg.intro.inr\nP : ℤ[X]\nt : ℤ\nht : t ∈ periodicPts fun x => eval x P\nC : Cycle ℤ := Cycle.map (fun x => eval x P - x) (periodicOrbit (fun x => eval x P) t)\nHC : ∀ {n : ℕ}, (fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t ∈ C\nHdvd : Cycle.Chain (fun x1 x2 => x1 ∣ x2) C\nHabs :\n ∀ (m n : ℕ),\n ((fun x => eval x P)^[m + 1] t - (fun x => eval x P)^[m] t).natAbs =\n ((fun x => eval x P)^[n + 1] t - (fun x => eval x P)^[n] t).natAbs\nn : ℕ\nhn :\n eval ((fun x => eval x P)^[n] t) P - (fun x => eval x P)^[n] t ≠\n eval ((fun x => eval x P)^[n + 1] t) P - (fun x => eval x P)^[n + 1] t\nhn' : (fun x => eval x P)^[n] t = eval (eval ((fun x => eval x P)^[n] t) P) P\n⊢ IsPeriodicPt (fun x => eval x P) 2 t",
"after_state": "No Goals!"
}
] |
theorem Polynomial.iterate_comp_sub_X_ne {P : Polynomial ℤ} (hP : 1 < P.natDegree) {k : ℕ}
(hk : 0 < k) : P.comp^[k] X - X ≠ 0 := by
rw [sub_ne_zero]
apply_fun natDegree
simpa using (one_lt_pow₀ hP hk.ne').ne'
|
/root/DuelModelResearch/mathlib4/Archive/Imo/Imo2006Q5.lean
|
{
"open": [
"Function Polynomial"
],
"variables": []
}
|
[
{
"line": "rw [sub_ne_zero]",
"before_state": "P : ℤ[X]\nhP : 1 < P.natDegree\nk : ℕ\nhk : 0 < k\n⊢ P.comp^[k] X - X ≠ 0",
"after_state": "P : ℤ[X]\nhP : 1 < P.natDegree\nk : ℕ\nhk : 0 < k\n⊢ P.comp^[k] X ≠ X"
},
{
"line": "rewrite [sub_ne_zero]",
"before_state": "P : ℤ[X]\nhP : 1 < P.natDegree\nk : ℕ\nhk : 0 < k\n⊢ P.comp^[k] X - X ≠ 0",
"after_state": "P : ℤ[X]\nhP : 1 < P.natDegree\nk : ℕ\nhk : 0 < k\n⊢ P.comp^[k] X ≠ X"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "P : ℤ[X]\nhP : 1 < P.natDegree\nk : ℕ\nhk : 0 < k\n⊢ P.comp^[k] X ≠ X",
"after_state": "P : ℤ[X]\nhP : 1 < P.natDegree\nk : ℕ\nhk : 0 < k\n⊢ P.comp^[k] X ≠ X"
},
{
"line": "try (with_reducible rfl)",
"before_state": "P : ℤ[X]\nhP : 1 < P.natDegree\nk : ℕ\nhk : 0 < k\n⊢ P.comp^[k] X ≠ X",
"after_state": "P : ℤ[X]\nhP : 1 < P.natDegree\nk : ℕ\nhk : 0 < k\n⊢ P.comp^[k] X ≠ X"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "P : ℤ[X]\nhP : 1 < P.natDegree\nk : ℕ\nhk : 0 < k\n⊢ P.comp^[k] X ≠ X",
"after_state": "P : ℤ[X]\nhP : 1 < P.natDegree\nk : ℕ\nhk : 0 < k\n⊢ P.comp^[k] X ≠ X"
},
{
"line": "with_reducible rfl",
"before_state": "P : ℤ[X]\nhP : 1 < P.natDegree\nk : ℕ\nhk : 0 < k\n⊢ P.comp^[k] X ≠ X",
"after_state": "P : ℤ[X]\nhP : 1 < P.natDegree\nk : ℕ\nhk : 0 < k\n⊢ P.comp^[k] X ≠ X"
},
{
"line": "rfl",
"before_state": "P : ℤ[X]\nhP : 1 < P.natDegree\nk : ℕ\nhk : 0 < k\n⊢ P.comp^[k] X ≠ X",
"after_state": "P : ℤ[X]\nhP : 1 < P.natDegree\nk : ℕ\nhk : 0 < k\n⊢ P.comp^[k] X ≠ X"
},
{
"line": "apply_rfl",
"before_state": "P : ℤ[X]\nhP : 1 < P.natDegree\nk : ℕ\nhk : 0 < k\n⊢ P.comp^[k] X ≠ X",
"after_state": "P : ℤ[X]\nhP : 1 < P.natDegree\nk : ℕ\nhk : 0 < k\n⊢ P.comp^[k] X ≠ X"
},
{
"line": "skip",
"before_state": "P : ℤ[X]\nhP : 1 < P.natDegree\nk : ℕ\nhk : 0 < k\n⊢ P.comp^[k] X ≠ X",
"after_state": "P : ℤ[X]\nhP : 1 < P.natDegree\nk : ℕ\nhk : 0 < k\n⊢ P.comp^[k] X ≠ X"
},
{
"line": "apply_fun natDegree",
"before_state": "P : ℤ[X]\nhP : 1 < P.natDegree\nk : ℕ\nhk : 0 < k\n⊢ P.comp^[k] X ≠ X",
"after_state": "P : ℤ[X]\nhP : 1 < P.natDegree\nk : ℕ\nhk : 0 < k\n⊢ (P.comp^[k] X).natDegree ≠ X.natDegree"
},
{
"line": "simpa using (one_lt_pow₀ hP hk.ne').ne'",
"before_state": "P : ℤ[X]\nhP : 1 < P.natDegree\nk : ℕ\nhk : 0 < k\n⊢ (P.comp^[k] X).natDegree ≠ X.natDegree",
"after_state": "No Goals!"
}
] |
theorem p_lemma (p : ℕ) (hpp : Nat.Prime p) (hp_mod_4_eq_1 : p ≡ 1 [MOD 4]) (hp_gt_20 : p > 20) :
∃ n : ℕ, p ∣ n ^ 2 + 1 ∧ (p : ℝ) > 2 * n + sqrt (2 * n) := by
haveI := Fact.mk hpp
have hp_mod_4_ne_3 : p % 4 ≠ 3 := by linarith [show p % 4 = 1 from hp_mod_4_eq_1]
obtain ⟨y, hy⟩ := ZMod.exists_sq_eq_neg_one_iff.mpr hp_mod_4_ne_3
let m := ZMod.valMinAbs y
let n := Int.natAbs m
have hnat₁ : p ∣ n ^ 2 + 1 := by
refine Int.natCast_dvd_natCast.mp ?_
simp only [n]
simp only [Int.natAbs_sq]
simp only [Int.natCast_pow]
simp only [Int.natCast_succ]
simp only [Int.natCast_dvd_natCast.mp]
refine (ZMod.intCast_zmod_eq_zero_iff_dvd (m ^ 2 + 1) p).mp ?_
simp only [m]
simp only [Int.cast_pow]
simp only [Int.cast_add]
simp only [Int.cast_one]
simp only [ZMod.coe_valMinAbs]
rw [pow_two]; exact neg_add_cancel 1
rw [← hy]; exact neg_add_cancel 1
have hnat₂ : n ≤ p / 2 := ZMod.natAbs_valMinAbs_le y
have hnat₃ : p ≥ 2 * n := by omega
set k : ℕ := p - 2 * n with hnat₄
have hnat₅ : p ∣ k ^ 2 + 4 := by
obtain ⟨x, hx⟩ := hnat₁
have : (p : ℤ) ∣ (k : ℤ) ^ 2 + 4 := by
use (p : ℤ) - 4 * n + 4 * x
have hcast₁ : (k : ℤ) = p - 2 * n := by assumption_mod_cast
have hcast₂ : (n : ℤ) ^ 2 + 1 = p * x := by assumption_mod_cast
linear_combination ((k : ℤ) + p - 2 * n) * hcast₁ + 4 * hcast₂
assumption_mod_cast
have hnat₆ : k ^ 2 + 4 ≥ p := Nat.le_of_dvd (k ^ 2 + 3).succ_pos hnat₅
have hreal₁ : (k : ℝ) = p - 2 * n := by assumption_mod_cast
have hreal₂ : (p : ℝ) > 20 := by assumption_mod_cast
have hreal₃ : (k : ℝ) ^ 2 + 4 ≥ p := by assumption_mod_cast
have hreal₅ : (k : ℝ) > 4 := by
refine lt_of_pow_lt_pow_left₀ 2 k.cast_nonneg ?_
linarith only [hreal₂, hreal₃]
have hreal₆ : (k : ℝ) > sqrt (2 * n) := by
refine lt_of_pow_lt_pow_left₀ 2 k.cast_nonneg ?_
rw [sq_sqrt (mul_nonneg zero_le_two n.cast_nonneg)]
linarith only [hreal₁, hreal₃, hreal₅]
exact ⟨n, hnat₁, by linarith only [hreal₆, hreal₁]⟩
|
/root/DuelModelResearch/mathlib4/Archive/Imo/Imo2008Q3.lean
|
{
"open": [
"Real"
],
"variables": []
}
|
[
{
"line": "haveI := Fact.mk hpp",
"before_state": "p : ℕ\nhpp : Nat.Prime p\nhp_mod_4_eq_1 : p ≡ 1 [MOD 4]\nhp_gt_20 : p > 20\n⊢ ∃ n, p ∣ n ^ 2 + 1 ∧ ↑p > 2 * ↑n + √(2 * ↑n)",
"after_state": "p : ℕ\nhpp : Nat.Prime p\nhp_mod_4_eq_1 : p ≡ 1 [MOD 4]\nhp_gt_20 : p > 20\nthis : Fact (Nat.Prime p)\n⊢ ∃ n, p ∣ n ^ 2 + 1 ∧ ↑p > 2 * ↑n + √(2 * ↑n)"
},
{
"line": "refine_lift\n haveI := Fact.mk hpp;\n ?_",
"before_state": "p : ℕ\nhpp : Nat.Prime p\nhp_mod_4_eq_1 : p ≡ 1 [MOD 4]\nhp_gt_20 : p > 20\n⊢ ∃ n, p ∣ n ^ 2 + 1 ∧ ↑p > 2 * ↑n + √(2 * ↑n)",
"after_state": "p : ℕ\nhpp : Nat.Prime p\nhp_mod_4_eq_1 : p ≡ 1 [MOD 4]\nhp_gt_20 : p > 20\nthis : Fact (Nat.Prime p)\n⊢ ∃ n, p ∣ n ^ 2 + 1 ∧ ↑p > 2 * ↑n + √(2 * ↑n)"
},
{
"line": "focus\n (refine\n no_implicit_lambda%\n haveI := Fact.mk hpp;\n ?_;\n rotate_right)",
"before_state": "p : ℕ\nhpp : Nat.Prime p\nhp_mod_4_eq_1 : p ≡ 1 [MOD 4]\nhp_gt_20 : p > 20\n⊢ ∃ n, p ∣ n ^ 2 + 1 ∧ ↑p > 2 * ↑n + √(2 * ↑n)",
"after_state": "p : ℕ\nhpp : Nat.Prime p\nhp_mod_4_eq_1 : p ≡ 1 [MOD 4]\nhp_gt_20 : p > 20\nthis : Fact (Nat.Prime p)\n⊢ ∃ n, p ∣ n ^ 2 + 1 ∧ ↑p > 2 * ↑n + √(2 * ↑n)"
},
{
"line": "refine\n no_implicit_lambda%\n haveI := Fact.mk hpp;\n ?_",
"before_state": "p : ℕ\nhpp : Nat.Prime p\nhp_mod_4_eq_1 : p ≡ 1 [MOD 4]\nhp_gt_20 : p > 20\n⊢ ∃ n, p ∣ n ^ 2 + 1 ∧ ↑p > 2 * ↑n + √(2 * ↑n)",
"after_state": "p : ℕ\nhpp : Nat.Prime p\nhp_mod_4_eq_1 : p ≡ 1 [MOD 4]\nhp_gt_20 : p > 20\nthis : Fact (Nat.Prime p)\n⊢ ∃ n, p ∣ n ^ 2 + 1 ∧ ↑p > 2 * ↑n + √(2 * ↑n)"
},
{
"line": "rotate_right",
"before_state": "p : ℕ\nhpp : Nat.Prime p\nhp_mod_4_eq_1 : p ≡ 1 [MOD 4]\nhp_gt_20 : p > 20\nthis : Fact (Nat.Prime p)\n⊢ ∃ n, p ∣ n ^ 2 + 1 ∧ ↑p > 2 * ↑n + √(2 * ↑n)",
"after_state": "p : ℕ\nhpp : Nat.Prime p\nhp_mod_4_eq_1 : p ≡ 1 [MOD 4]\nhp_gt_20 : p > 20\nthis : Fact (Nat.Prime p)\n⊢ ∃ n, p ∣ n ^ 2 + 1 ∧ ↑p > 2 * ↑n + √(2 * ↑n)"
},
{
"line": "have hp_mod_4_ne_3 : p % 4 ≠ 3 := by linarith [show p % 4 = 1 from hp_mod_4_eq_1]",
"before_state": "p : ℕ\nhpp : Nat.Prime p\nhp_mod_4_eq_1 : p ≡ 1 [MOD 4]\nhp_gt_20 : p > 20\nthis : Fact (Nat.Prime p)\n⊢ ∃ n, p ∣ n ^ 2 + 1 ∧ ↑p > 2 * ↑n + √(2 * ↑n)",
"after_state": "p : ℕ\nhpp : Nat.Prime p\nhp_mod_4_eq_1 : p ≡ 1 [MOD 4]\nhp_gt_20 : p > 20\nthis : Fact (Nat.Prime p)\nhp_mod_4_ne_3 : p % 4 ≠ 3\n⊢ ∃ n, p ∣ n ^ 2 + 1 ∧ ↑p > 2 * ↑n + √(2 * ↑n)"
},
{
"line": "focus\n refine\n no_implicit_lambda%\n (have hp_mod_4_ne_3 : p % 4 ≠ 3 := ?body✝;\n ?_)\n case body✝ => with_annotate_state\"by\" (linarith [show p % 4 = 1 from hp_mod_4_eq_1])",
"before_state": "p : ℕ\nhpp : Nat.Prime p\nhp_mod_4_eq_1 : p ≡ 1 [MOD 4]\nhp_gt_20 : p > 20\nthis : Fact (Nat.Prime p)\n⊢ ∃ n, p ∣ n ^ 2 + 1 ∧ ↑p > 2 * ↑n + √(2 * ↑n)",
"after_state": "p : ℕ\nhpp : Nat.Prime p\nhp_mod_4_eq_1 : p ≡ 1 [MOD 4]\nhp_gt_20 : p > 20\nthis : Fact (Nat.Prime p)\nhp_mod_4_ne_3 : p % 4 ≠ 3\n⊢ ∃ n, p ∣ n ^ 2 + 1 ∧ ↑p > 2 * ↑n + √(2 * ↑n)"
},
{
"line": "refine\n no_implicit_lambda%\n (have hp_mod_4_ne_3 : p % 4 ≠ 3 := ?body✝;\n ?_)",
"before_state": "p : ℕ\nhpp : Nat.Prime p\nhp_mod_4_eq_1 : p ≡ 1 [MOD 4]\nhp_gt_20 : p > 20\nthis : Fact (Nat.Prime p)\n⊢ ∃ n, p ∣ n ^ 2 + 1 ∧ ↑p > 2 * ↑n + √(2 * ↑n)",
"after_state": "case body\np : ℕ\nhpp : Nat.Prime p\nhp_mod_4_eq_1 : p ≡ 1 [MOD 4]\nhp_gt_20 : p > 20\nthis : Fact (Nat.Prime p)\n⊢ p % 4 ≠ 3\n---\np : ℕ\nhpp : Nat.Prime p\nhp_mod_4_eq_1 : p ≡ 1 [MOD 4]\nhp_gt_20 : p > 20\nthis : Fact (Nat.Prime p)\nhp_mod_4_ne_3 : p % 4 ≠ 3\n⊢ ∃ n, p ∣ n ^ 2 + 1 ∧ ↑p > 2 * ↑n + √(2 * ↑n)"
},
{
"line": "case body✝ => with_annotate_state\"by\" (linarith [show p % 4 = 1 from hp_mod_4_eq_1])",
"before_state": "case body\np : ℕ\nhpp : Nat.Prime p\nhp_mod_4_eq_1 : p ≡ 1 [MOD 4]\nhp_gt_20 : p > 20\nthis : Fact (Nat.Prime p)\n⊢ p % 4 ≠ 3\n---\np : ℕ\nhpp : Nat.Prime p\nhp_mod_4_eq_1 : p ≡ 1 [MOD 4]\nhp_gt_20 : p > 20\nthis : Fact (Nat.Prime p)\nhp_mod_4_ne_3 : p % 4 ≠ 3\n⊢ ∃ n, p ∣ n ^ 2 + 1 ∧ ↑p > 2 * ↑n + √(2 * ↑n)",
"after_state": "p : ℕ\nhpp : Nat.Prime p\nhp_mod_4_eq_1 : p ≡ 1 [MOD 4]\nhp_gt_20 : p > 20\nthis : Fact (Nat.Prime p)\nhp_mod_4_ne_3 : p % 4 ≠ 3\n⊢ ∃ n, p ∣ n ^ 2 + 1 ∧ ↑p > 2 * ↑n + √(2 * ↑n)"
},
{
"line": "with_annotate_state\"by\" (linarith [show p % 4 = 1 from hp_mod_4_eq_1])",
"before_state": "p : ℕ\nhpp : Nat.Prime p\nhp_mod_4_eq_1 : p ≡ 1 [MOD 4]\nhp_gt_20 : p > 20\nthis : Fact (Nat.Prime p)\n⊢ p % 4 ≠ 3",
"after_state": "No Goals!"
},
{
"line": "linarith [show p % 4 = 1 from hp_mod_4_eq_1]",
"before_state": "p : ℕ\nhpp : Nat.Prime p\nhp_mod_4_eq_1 : p ≡ 1 [MOD 4]\nhp_gt_20 : p > 20\nthis : Fact (Nat.Prime p)\n⊢ p % 4 ≠ 3",
"after_state": "No Goals!"
},
{
"line": "ring1",
"before_state": "p : ℕ\nhpp : Nat.Prime p\nhp_mod_4_eq_1 : p ≡ 1 [MOD 4]\nhp_gt_20 : p > 20\nthis : Fact (Nat.Prime p)\na✝ : p % 4 = 3\n⊢ 2 * -1 + -(↑p % 4 - 3) + (↑p % 4 - 1) = 0",
"after_state": "No Goals!"
},
{
"line": "norm_num",
"before_state": "p : ℕ\nhpp : Nat.Prime p\nhp_mod_4_eq_1 : p ≡ 1 [MOD 4]\nhp_gt_20 : p > 20\nthis : Fact (Nat.Prime p)\na✝ : p % 4 = 3\n⊢ 2 > 0",
"after_state": "No Goals!"
},
{
"line": "obtain ⟨y, hy⟩ := ZMod.exists_sq_eq_neg_one_iff.mpr hp_mod_4_ne_3",
"before_state": "p : ℕ\nhpp : Nat.Prime p\nhp_mod_4_eq_1 : p ≡ 1 [MOD 4]\nhp_gt_20 : p > 20\nthis : Fact (Nat.Prime p)\nhp_mod_4_ne_3 : p % 4 ≠ 3\n⊢ ∃ n, p ∣ n ^ 2 + 1 ∧ ↑p > 2 * ↑n + √(2 * ↑n)",
"after_state": "case intro\np : ℕ\nhpp : Nat.Prime p\nhp_mod_4_eq_1 : p ≡ 1 [MOD 4]\nhp_gt_20 : p > 20\nthis : Fact (Nat.Prime p)\nhp_mod_4_ne_3 : p % 4 ≠ 3\ny : ZMod p\nhy : -1 = y * y\n⊢ ∃ n, p ∣ n ^ 2 + 1 ∧ ↑p > 2 * ↑n + √(2 * ↑n)"
},
{
"line": "let m := ZMod.valMinAbs y",
"before_state": "case intro\np : ℕ\nhpp : Nat.Prime p\nhp_mod_4_eq_1 : p ≡ 1 [MOD 4]\nhp_gt_20 : p > 20\nthis : Fact (Nat.Prime p)\nhp_mod_4_ne_3 : p % 4 ≠ 3\ny : ZMod p\nhy : -1 = y * y\n⊢ ∃ n, p ∣ n ^ 2 + 1 ∧ ↑p > 2 * ↑n + √(2 * ↑n)",
"after_state": "case intro\np : ℕ\nhpp : Nat.Prime p\nhp_mod_4_eq_1 : p ≡ 1 [MOD 4]\nhp_gt_20 : p > 20\nthis : Fact (Nat.Prime p)\nhp_mod_4_ne_3 : p % 4 ≠ 3\ny : ZMod p\nhy : -1 = y * y\nm : ℤ := y.valMinAbs\n⊢ ∃ n, p ∣ n ^ 2 + 1 ∧ ↑p > 2 * ↑n + √(2 * ↑n)"
},
{
"line": "refine_lift\n let m := ZMod.valMinAbs y;\n ?_",
"before_state": "case intro\np : ℕ\nhpp : Nat.Prime p\nhp_mod_4_eq_1 : p ≡ 1 [MOD 4]\nhp_gt_20 : p > 20\nthis : Fact (Nat.Prime p)\nhp_mod_4_ne_3 : p % 4 ≠ 3\ny : ZMod p\nhy : -1 = y * y\n⊢ ∃ n, p ∣ n ^ 2 + 1 ∧ ↑p > 2 * ↑n + √(2 * ↑n)",
"after_state": "case intro\np : ℕ\nhpp : Nat.Prime p\nhp_mod_4_eq_1 : p ≡ 1 [MOD 4]\nhp_gt_20 : p > 20\nthis : Fact (Nat.Prime p)\nhp_mod_4_ne_3 : p % 4 ≠ 3\ny : ZMod p\nhy : -1 = y * y\nm : ℤ := y.valMinAbs\n⊢ ∃ n, p ∣ n ^ 2 + 1 ∧ ↑p > 2 * ↑n + √(2 * ↑n)"
},
{
"line": "focus\n (refine\n no_implicit_lambda%\n (let m := ZMod.valMinAbs y;\n ?_);\n rotate_right)",
"before_state": "case intro\np : ℕ\nhpp : Nat.Prime p\nhp_mod_4_eq_1 : p ≡ 1 [MOD 4]\nhp_gt_20 : p > 20\nthis : Fact (Nat.Prime p)\nhp_mod_4_ne_3 : p % 4 ≠ 3\ny : ZMod p\nhy : -1 = y * y\n⊢ ∃ n, p ∣ n ^ 2 + 1 ∧ ↑p > 2 * ↑n + √(2 * ↑n)",
"after_state": "case intro\np : ℕ\nhpp : Nat.Prime p\nhp_mod_4_eq_1 : p ≡ 1 [MOD 4]\nhp_gt_20 : p > 20\nthis : Fact (Nat.Prime p)\nhp_mod_4_ne_3 : p % 4 ≠ 3\ny : ZMod p\nhy : -1 = y * y\nm : ℤ := y.valMinAbs\n⊢ ∃ n, p ∣ n ^ 2 + 1 ∧ ↑p > 2 * ↑n + √(2 * ↑n)"
},
{
"line": "refine\n no_implicit_lambda%\n (let m := ZMod.valMinAbs y;\n ?_)",
"before_state": "case intro\np : ℕ\nhpp : Nat.Prime p\nhp_mod_4_eq_1 : p ≡ 1 [MOD 4]\nhp_gt_20 : p > 20\nthis : Fact (Nat.Prime p)\nhp_mod_4_ne_3 : p % 4 ≠ 3\ny : ZMod p\nhy : -1 = y * y\n⊢ ∃ n, p ∣ n ^ 2 + 1 ∧ ↑p > 2 * ↑n + √(2 * ↑n)",
"after_state": "case intro\np : ℕ\nhpp : Nat.Prime p\nhp_mod_4_eq_1 : p ≡ 1 [MOD 4]\nhp_gt_20 : p > 20\nthis : Fact (Nat.Prime p)\nhp_mod_4_ne_3 : p % 4 ≠ 3\ny : ZMod p\nhy : -1 = y * y\nm : ℤ := y.valMinAbs\n⊢ ∃ n, p ∣ n ^ 2 + 1 ∧ ↑p > 2 * ↑n + √(2 * ↑n)"
},
{
"line": "rotate_right",
"before_state": "case intro\np : ℕ\nhpp : Nat.Prime p\nhp_mod_4_eq_1 : p ≡ 1 [MOD 4]\nhp_gt_20 : p > 20\nthis : Fact (Nat.Prime p)\nhp_mod_4_ne_3 : p % 4 ≠ 3\ny : ZMod p\nhy : -1 = y * y\nm : ℤ := y.valMinAbs\n⊢ ∃ n, p ∣ n ^ 2 + 1 ∧ ↑p > 2 * ↑n + √(2 * ↑n)",
"after_state": "case intro\np : ℕ\nhpp : Nat.Prime p\nhp_mod_4_eq_1 : p ≡ 1 [MOD 4]\nhp_gt_20 : p > 20\nthis : Fact (Nat.Prime p)\nhp_mod_4_ne_3 : p % 4 ≠ 3\ny : ZMod p\nhy : -1 = y * y\nm : ℤ := y.valMinAbs\n⊢ ∃ n, p ∣ n ^ 2 + 1 ∧ ↑p > 2 * ↑n + √(2 * ↑n)"
},
{
"line": "let n := Int.natAbs m",
"before_state": "case intro\np : ℕ\nhpp : Nat.Prime p\nhp_mod_4_eq_1 : p ≡ 1 [MOD 4]\nhp_gt_20 : p > 20\nthis : Fact (Nat.Prime p)\nhp_mod_4_ne_3 : p % 4 ≠ 3\ny : ZMod p\nhy : -1 = y * y\nm : ℤ := y.valMinAbs\n⊢ ∃ n, p ∣ n ^ 2 + 1 ∧ ↑p > 2 * ↑n + √(2 * ↑n)",
"after_state": "case intro\np : ℕ\nhpp : Nat.Prime p\nhp_mod_4_eq_1 : p ≡ 1 [MOD 4]\nhp_gt_20 : p > 20\nthis : Fact (Nat.Prime p)\nhp_mod_4_ne_3 : p % 4 ≠ 3\ny : ZMod p\nhy : -1 = y * y\nm : ℤ := y.valMinAbs\nn : ℕ := m.natAbs\n⊢ ∃ n, p ∣ n ^ 2 + 1 ∧ ↑p > 2 * ↑n + √(2 * ↑n)"
},
{
"line": "refine_lift\n let n := Int.natAbs m;\n ?_",
"before_state": "case intro\np : ℕ\nhpp : Nat.Prime p\nhp_mod_4_eq_1 : p ≡ 1 [MOD 4]\nhp_gt_20 : p > 20\nthis : Fact (Nat.Prime p)\nhp_mod_4_ne_3 : p % 4 ≠ 3\ny : ZMod p\nhy : -1 = y * y\nm : ℤ := y.valMinAbs\n⊢ ∃ n, p ∣ n ^ 2 + 1 ∧ ↑p > 2 * ↑n + √(2 * ↑n)",
"after_state": "case intro\np : ℕ\nhpp : Nat.Prime p\nhp_mod_4_eq_1 : p ≡ 1 [MOD 4]\nhp_gt_20 : p > 20\nthis : Fact (Nat.Prime p)\nhp_mod_4_ne_3 : p % 4 ≠ 3\ny : ZMod p\nhy : -1 = y * y\nm : ℤ := y.valMinAbs\nn : ℕ := m.natAbs\n⊢ ∃ n, p ∣ n ^ 2 + 1 ∧ ↑p > 2 * ↑n + √(2 * ↑n)"
},
{
"line": "focus\n (refine\n no_implicit_lambda%\n (let n := Int.natAbs m;\n ?_);\n rotate_right)",
"before_state": "case intro\np : ℕ\nhpp : Nat.Prime p\nhp_mod_4_eq_1 : p ≡ 1 [MOD 4]\nhp_gt_20 : p > 20\nthis : Fact (Nat.Prime p)\nhp_mod_4_ne_3 : p % 4 ≠ 3\ny : ZMod p\nhy : -1 = y * y\nm : ℤ := y.valMinAbs\n⊢ ∃ n, p ∣ n ^ 2 + 1 ∧ ↑p > 2 * ↑n + √(2 * ↑n)",
"after_state": "case intro\np : ℕ\nhpp : Nat.Prime p\nhp_mod_4_eq_1 : p ≡ 1 [MOD 4]\nhp_gt_20 : p > 20\nthis : Fact (Nat.Prime p)\nhp_mod_4_ne_3 : p % 4 ≠ 3\ny : ZMod p\nhy : -1 = y * y\nm : ℤ := y.valMinAbs\nn : ℕ := m.natAbs\n⊢ ∃ n, p ∣ n ^ 2 + 1 ∧ ↑p > 2 * ↑n + √(2 * ↑n)"
},
{
"line": "refine\n no_implicit_lambda%\n (let n := Int.natAbs m;\n ?_)",
"before_state": "case intro\np : ℕ\nhpp : Nat.Prime p\nhp_mod_4_eq_1 : p ≡ 1 [MOD 4]\nhp_gt_20 : p > 20\nthis : Fact (Nat.Prime p)\nhp_mod_4_ne_3 : p % 4 ≠ 3\ny : ZMod p\nhy : -1 = y * y\nm : ℤ := y.valMinAbs\n⊢ ∃ n, p ∣ n ^ 2 + 1 ∧ ↑p > 2 * ↑n + √(2 * ↑n)",
"after_state": "case intro\np : ℕ\nhpp : Nat.Prime p\nhp_mod_4_eq_1 : p ≡ 1 [MOD 4]\nhp_gt_20 : p > 20\nthis : Fact (Nat.Prime p)\nhp_mod_4_ne_3 : p % 4 ≠ 3\ny : ZMod p\nhy : -1 = y * y\nm : ℤ := y.valMinAbs\nn : ℕ := m.natAbs\n⊢ ∃ n, p ∣ n ^ 2 + 1 ∧ ↑p > 2 * ↑n + √(2 * ↑n)"
},
{
"line": "rotate_right",
"before_state": "case intro\np : ℕ\nhpp : Nat.Prime p\nhp_mod_4_eq_1 : p ≡ 1 [MOD 4]\nhp_gt_20 : p > 20\nthis : Fact (Nat.Prime p)\nhp_mod_4_ne_3 : p % 4 ≠ 3\ny : ZMod p\nhy : -1 = y * y\nm : ℤ := y.valMinAbs\nn : ℕ := m.natAbs\n⊢ ∃ n, p ∣ n ^ 2 + 1 ∧ ↑p > 2 * ↑n + √(2 * ↑n)",
"after_state": "case intro\np : ℕ\nhpp : Nat.Prime p\nhp_mod_4_eq_1 : p ≡ 1 [MOD 4]\nhp_gt_20 : p > 20\nthis : Fact (Nat.Prime p)\nhp_mod_4_ne_3 : p % 4 ≠ 3\ny : ZMod p\nhy : -1 = y * y\nm : ℤ := y.valMinAbs\nn : ℕ := m.natAbs\n⊢ ∃ n, p ∣ n ^ 2 + 1 ∧ ↑p > 2 * ↑n + √(2 * ↑n)"
},
{
"line": "have hnat₁ : p ∣ n ^ 2 + 1 := by\n refine Int.natCast_dvd_natCast.mp ?_\n simp only [n]\n simp only [Int.natAbs_sq]\n simp only [Int.natCast_pow]\n simp only [Int.natCast_succ]\n simp only [Int.natCast_dvd_natCast.mp]\n refine (ZMod.intCast_zmod_eq_zero_iff_dvd (m ^ 2 + 1) p).mp ?_\n simp only [m]\n simp only [Int.cast_pow]\n simp only [Int.cast_add]\n simp only [Int.cast_one]\n simp only [ZMod.coe_valMinAbs]\n rw [pow_two]; exact neg_add_cancel 1\n rw [← hy]; exact neg_add_cancel 1",
"before_state": "case intro\np : ℕ\nhpp : Nat.Prime p\nhp_mod_4_eq_1 : p ≡ 1 [MOD 4]\nhp_gt_20 : p > 20\nthis : Fact (Nat.Prime p)\nhp_mod_4_ne_3 : p % 4 ≠ 3\ny : ZMod p\nhy : -1 = y * y\nm : ℤ := y.valMinAbs\nn : ℕ := m.natAbs\n⊢ ∃ n, p ∣ n ^ 2 + 1 ∧ ↑p > 2 * ↑n + √(2 * ↑n)",
"after_state": "case intro\np : ℕ\nhpp : Nat.Prime p\nhp_mod_4_eq_1 : p ≡ 1 [MOD 4]\nhp_gt_20 : p > 20\nthis : Fact (Nat.Prime p)\nhp_mod_4_ne_3 : p % 4 ≠ 3\ny : ZMod p\nhy : -1 = y * y\nm : ℤ := y.valMinAbs\nn : ℕ := m.natAbs\nhnat₁ : p ∣ n ^ 2 + 1\n⊢ ∃ n, p ∣ n ^ 2 + 1 ∧ ↑p > 2 * ↑n + √(2 * ↑n)"
},
{
"line": "focus\n refine\n no_implicit_lambda%\n (have hnat₁ : p ∣ n ^ 2 + 1 := ?body✝;\n ?_)\n case body✝ =>\n with_annotate_state\"by\"\n ( refine Int.natCast_dvd_natCast.mp ?_\n simp only [n]\n simp only [Int.natAbs_sq]\n simp only [Int.natCast_pow]\n simp only [Int.natCast_succ]\n simp only [Int.natCast_dvd_natCast.mp]\n refine (ZMod.intCast_zmod_eq_zero_iff_dvd (m ^ 2 + 1) p).mp ?_\n simp only [m]\n simp only [Int.cast_pow]\n simp only [Int.cast_add]\n simp only [Int.cast_one]\n simp only [ZMod.coe_valMinAbs]\n rw [pow_two]; exact neg_add_cancel 1\n rw [← hy]; exact neg_add_cancel 1)",
"before_state": "case intro\np : ℕ\nhpp : Nat.Prime p\nhp_mod_4_eq_1 : p ≡ 1 [MOD 4]\nhp_gt_20 : p > 20\nthis : Fact (Nat.Prime p)\nhp_mod_4_ne_3 : p % 4 ≠ 3\ny : ZMod p\nhy : -1 = y * y\nm : ℤ := y.valMinAbs\nn : ℕ := m.natAbs\n⊢ ∃ n, p ∣ n ^ 2 + 1 ∧ ↑p > 2 * ↑n + √(2 * ↑n)",
"after_state": "case intro\np : ℕ\nhpp : Nat.Prime p\nhp_mod_4_eq_1 : p ≡ 1 [MOD 4]\nhp_gt_20 : p > 20\nthis : Fact (Nat.Prime p)\nhp_mod_4_ne_3 : p % 4 ≠ 3\ny : ZMod p\nhy : -1 = y * y\nm : ℤ := y.valMinAbs\nn : ℕ := m.natAbs\nhnat₁ : p ∣ n ^ 2 + 1\n⊢ ∃ n, p ∣ n ^ 2 + 1 ∧ ↑p > 2 * ↑n + √(2 * ↑n)"
},
{
"line": "refine\n no_implicit_lambda%\n (have hnat₁ : p ∣ n ^ 2 + 1 := ?body✝;\n ?_)",
"before_state": "case intro\np : ℕ\nhpp : Nat.Prime p\nhp_mod_4_eq_1 : p ≡ 1 [MOD 4]\nhp_gt_20 : p > 20\nthis : Fact (Nat.Prime p)\nhp_mod_4_ne_3 : p % 4 ≠ 3\ny : ZMod p\nhy : -1 = y * y\nm : ℤ := y.valMinAbs\nn : ℕ := m.natAbs\n⊢ ∃ n, p ∣ n ^ 2 + 1 ∧ ↑p > 2 * ↑n + √(2 * ↑n)",
"after_state": "case body\np : ℕ\nhpp : Nat.Prime p\nhp_mod_4_eq_1 : p ≡ 1 [MOD 4]\nhp_gt_20 : p > 20\nthis : Fact (Nat.Prime p)\nhp_mod_4_ne_3 : p % 4 ≠ 3\ny : ZMod p\nhy : -1 = y * y\nm : ℤ := y.valMinAbs\nn : ℕ := m.natAbs\n⊢ p ∣ n ^ 2 + 1\n---\ncase intro\np : ℕ\nhpp : Nat.Prime p\nhp_mod_4_eq_1 : p ≡ 1 [MOD 4]\nhp_gt_20 : p > 20\nthis : Fact (Nat.Prime p)\nhp_mod_4_ne_3 : p % 4 ≠ 3\ny : ZMod p\nhy : -1 = y * y\nm : ℤ := y.valMinAbs\nn : ℕ := m.natAbs\nhnat₁ : p ∣ n ^ 2 + 1\n⊢ ∃ n, p ∣ n ^ 2 + 1 ∧ ↑p > 2 * ↑n + √(2 * ↑n)"
},
{
"line": "case body✝ =>\n with_annotate_state\"by\"\n ( refine Int.natCast_dvd_natCast.mp ?_\n simp only [n]\n simp only [Int.natAbs_sq]\n simp only [Int.natCast_pow]\n simp only [Int.natCast_succ]\n simp only [Int.natCast_dvd_natCast.mp]\n refine (ZMod.intCast_zmod_eq_zero_iff_dvd (m ^ 2 + 1) p).mp ?_\n simp only [m]\n simp only [Int.cast_pow]\n simp only [Int.cast_add]\n simp only [Int.cast_one]\n simp only [ZMod.coe_valMinAbs]\n rw [pow_two]; exact neg_add_cancel 1\n rw [← hy]; exact neg_add_cancel 1)",
"before_state": "case body\np : ℕ\nhpp : Nat.Prime p\nhp_mod_4_eq_1 : p ≡ 1 [MOD 4]\nhp_gt_20 : p > 20\nthis : Fact (Nat.Prime p)\nhp_mod_4_ne_3 : p % 4 ≠ 3\ny : ZMod p\nhy : -1 = y * y\nm : ℤ := y.valMinAbs\nn : ℕ := m.natAbs\n⊢ p ∣ n ^ 2 + 1\n---\ncase intro\np : ℕ\nhpp : Nat.Prime p\nhp_mod_4_eq_1 : p ≡ 1 [MOD 4]\nhp_gt_20 : p > 20\nthis : Fact (Nat.Prime p)\nhp_mod_4_ne_3 : p % 4 ≠ 3\ny : ZMod p\nhy : -1 = y * y\nm : ℤ := y.valMinAbs\nn : ℕ := m.natAbs\nhnat₁ : p ∣ n ^ 2 + 1\n⊢ ∃ n, p ∣ n ^ 2 + 1 ∧ ↑p > 2 * ↑n + √(2 * ↑n)",
"after_state": "case intro\np : ℕ\nhpp : Nat.Prime p\nhp_mod_4_eq_1 : p ≡ 1 [MOD 4]\nhp_gt_20 : p > 20\nthis : Fact (Nat.Prime p)\nhp_mod_4_ne_3 : p % 4 ≠ 3\ny : ZMod p\nhy : -1 = y * y\nm : ℤ := y.valMinAbs\nn : ℕ := m.natAbs\nhnat₁ : p ∣ n ^ 2 + 1\n⊢ ∃ n, p ∣ n ^ 2 + 1 ∧ ↑p > 2 * ↑n + √(2 * ↑n)"
},
{
"line": "with_annotate_state\"by\"\n ( refine Int.natCast_dvd_natCast.mp ?_\n simp only [n]\n simp only [Int.natAbs_sq]\n simp only [Int.natCast_pow]\n simp only [Int.natCast_succ]\n simp only [Int.natCast_dvd_natCast.mp]\n refine (ZMod.intCast_zmod_eq_zero_iff_dvd (m ^ 2 + 1) p).mp ?_\n simp only [m]\n simp only [Int.cast_pow]\n simp only [Int.cast_add]\n simp only [Int.cast_one]\n simp only [ZMod.coe_valMinAbs]\n rw [pow_two]; exact neg_add_cancel 1\n rw [← hy]; exact neg_add_cancel 1)",
"before_state": "p : ℕ\nhpp : Nat.Prime p\nhp_mod_4_eq_1 : p ≡ 1 [MOD 4]\nhp_gt_20 : p > 20\nthis : Fact (Nat.Prime p)\nhp_mod_4_ne_3 : p % 4 ≠ 3\ny : ZMod p\nhy : -1 = y * y\nm : ℤ := y.valMinAbs\nn : ℕ := m.natAbs\n⊢ p ∣ n ^ 2 + 1",
"after_state": "p : ℕ\nhpp : Nat.Prime p\nhp_mod_4_eq_1 : p ≡ 1 [MOD 4]\nhp_gt_20 : p > 20\nthis : Fact (Nat.Prime p)\nhp_mod_4_ne_3 : p % 4 ≠ 3\ny : ZMod p\nhy : -1 = y * y\nm : ℤ := y.valMinAbs\nn : ℕ := m.natAbs\n⊢ ↑p ∣ ↑(m.natAbs ^ 2 + 1)"
},
{
"line": "refine Int.natCast_dvd_natCast.mp ?_",
"before_state": "p : ℕ\nhpp : Nat.Prime p\nhp_mod_4_eq_1 : p ≡ 1 [MOD 4]\nhp_gt_20 : p > 20\nthis : Fact (Nat.Prime p)\nhp_mod_4_ne_3 : p % 4 ≠ 3\ny : ZMod p\nhy : -1 = y * y\nm : ℤ := y.valMinAbs\nn : ℕ := m.natAbs\n⊢ p ∣ n ^ 2 + 1",
"after_state": "p : ℕ\nhpp : Nat.Prime p\nhp_mod_4_eq_1 : p ≡ 1 [MOD 4]\nhp_gt_20 : p > 20\nthis : Fact (Nat.Prime p)\nhp_mod_4_ne_3 : p % 4 ≠ 3\ny : ZMod p\nhy : -1 = y * y\nm : ℤ := y.valMinAbs\nn : ℕ := m.natAbs\n⊢ ↑p ∣ ↑(n ^ 2 + 1)"
},
{
"line": "simp only [n]",
"before_state": "p : ℕ\nhpp : Nat.Prime p\nhp_mod_4_eq_1 : p ≡ 1 [MOD 4]\nhp_gt_20 : p > 20\nthis : Fact (Nat.Prime p)\nhp_mod_4_ne_3 : p % 4 ≠ 3\ny : ZMod p\nhy : -1 = y * y\nm : ℤ := y.valMinAbs\nn : ℕ := m.natAbs\n⊢ ↑p ∣ ↑(n ^ 2 + 1)",
"after_state": "p : ℕ\nhpp : Nat.Prime p\nhp_mod_4_eq_1 : p ≡ 1 [MOD 4]\nhp_gt_20 : p > 20\nthis : Fact (Nat.Prime p)\nhp_mod_4_ne_3 : p % 4 ≠ 3\ny : ZMod p\nhy : -1 = y * y\nm : ℤ := y.valMinAbs\nn : ℕ := m.natAbs\n⊢ ↑p ∣ ↑(m.natAbs ^ 2 + 1)"
},
{
"line": "simp only [Int.natAbs_sq]",
"before_state": "p : ℕ\nhpp : Nat.Prime p\nhp_mod_4_eq_1 : p ≡ 1 [MOD 4]\nhp_gt_20 : p > 20\nthis : Fact (Nat.Prime p)\nhp_mod_4_ne_3 : p % 4 ≠ 3\ny : ZMod p\nhy : -1 = y * y\nm : ℤ := y.valMinAbs\nn : ℕ := m.natAbs\n⊢ ↑p ∣ ↑(m.natAbs ^ 2 + 1)",
"after_state": "p : ℕ\nhpp : Nat.Prime p\nhp_mod_4_eq_1 : p ≡ 1 [MOD 4]\nhp_gt_20 : p > 20\nthis : Fact (Nat.Prime p)\nhp_mod_4_ne_3 : p % 4 ≠ 3\ny : ZMod p\nhy : -1 = y * y\nm : ℤ := y.valMinAbs\nn : ℕ := m.natAbs\n⊢ ↑p ∣ ↑(m.natAbs ^ 2 + 1)"
},
{
"line": "have hnat₂ : n ≤ p / 2 := ZMod.natAbs_valMinAbs_le y",
"before_state": "case intro\np : ℕ\nhpp : Nat.Prime p\nhp_mod_4_eq_1 : p ≡ 1 [MOD 4]\nhp_gt_20 : p > 20\nthis : Fact (Nat.Prime p)\nhp_mod_4_ne_3 : p % 4 ≠ 3\ny : ZMod p\nhy : -1 = y * y\nm : ℤ := y.valMinAbs\nn : ℕ := m.natAbs\nhnat₁ : p ∣ n ^ 2 + 1\n⊢ ∃ n, p ∣ n ^ 2 + 1 ∧ ↑p > 2 * ↑n + √(2 * ↑n)",
"after_state": "case intro\np : ℕ\nhpp : Nat.Prime p\nhp_mod_4_eq_1 : p ≡ 1 [MOD 4]\nhp_gt_20 : p > 20\nthis : Fact (Nat.Prime p)\nhp_mod_4_ne_3 : p % 4 ≠ 3\ny : ZMod p\nhy : -1 = y * y\nm : ℤ := y.valMinAbs\nn : ℕ := m.natAbs\nhnat₁ : p ∣ n ^ 2 + 1\nhnat₂ : n ≤ p / 2\n⊢ ∃ n, p ∣ n ^ 2 + 1 ∧ ↑p > 2 * ↑n + √(2 * ↑n)"
},
{
"line": "refine_lift\n have hnat₂ : n ≤ p / 2 := ZMod.natAbs_valMinAbs_le y;\n ?_",
"before_state": "case intro\np : ℕ\nhpp : Nat.Prime p\nhp_mod_4_eq_1 : p ≡ 1 [MOD 4]\nhp_gt_20 : p > 20\nthis : Fact (Nat.Prime p)\nhp_mod_4_ne_3 : p % 4 ≠ 3\ny : ZMod p\nhy : -1 = y * y\nm : ℤ := y.valMinAbs\nn : ℕ := m.natAbs\nhnat₁ : p ∣ n ^ 2 + 1\n⊢ ∃ n, p ∣ n ^ 2 + 1 ∧ ↑p > 2 * ↑n + √(2 * ↑n)",
"after_state": "case intro\np : ℕ\nhpp : Nat.Prime p\nhp_mod_4_eq_1 : p ≡ 1 [MOD 4]\nhp_gt_20 : p > 20\nthis : Fact (Nat.Prime p)\nhp_mod_4_ne_3 : p % 4 ≠ 3\ny : ZMod p\nhy : -1 = y * y\nm : ℤ := y.valMinAbs\nn : ℕ := m.natAbs\nhnat₁ : p ∣ n ^ 2 + 1\nhnat₂ : n ≤ p / 2\n⊢ ∃ n, p ∣ n ^ 2 + 1 ∧ ↑p > 2 * ↑n + √(2 * ↑n)"
},
{
"line": "focus\n (refine\n no_implicit_lambda%\n (have hnat₂ : n ≤ p / 2 := ZMod.natAbs_valMinAbs_le y;\n ?_);\n rotate_right)",
"before_state": "case intro\np : ℕ\nhpp : Nat.Prime p\nhp_mod_4_eq_1 : p ≡ 1 [MOD 4]\nhp_gt_20 : p > 20\nthis : Fact (Nat.Prime p)\nhp_mod_4_ne_3 : p % 4 ≠ 3\ny : ZMod p\nhy : -1 = y * y\nm : ℤ := y.valMinAbs\nn : ℕ := m.natAbs\nhnat₁ : p ∣ n ^ 2 + 1\n⊢ ∃ n, p ∣ n ^ 2 + 1 ∧ ↑p > 2 * ↑n + √(2 * ↑n)",
"after_state": "case intro\np : ℕ\nhpp : Nat.Prime p\nhp_mod_4_eq_1 : p ≡ 1 [MOD 4]\nhp_gt_20 : p > 20\nthis : Fact (Nat.Prime p)\nhp_mod_4_ne_3 : p % 4 ≠ 3\ny : ZMod p\nhy : -1 = y * y\nm : ℤ := y.valMinAbs\nn : ℕ := m.natAbs\nhnat₁ : p ∣ n ^ 2 + 1\nhnat₂ : n ≤ p / 2\n⊢ ∃ n, p ∣ n ^ 2 + 1 ∧ ↑p > 2 * ↑n + √(2 * ↑n)"
},
{
"line": "refine\n no_implicit_lambda%\n (have hnat₂ : n ≤ p / 2 := ZMod.natAbs_valMinAbs_le y;\n ?_)",
"before_state": "case intro\np : ℕ\nhpp : Nat.Prime p\nhp_mod_4_eq_1 : p ≡ 1 [MOD 4]\nhp_gt_20 : p > 20\nthis : Fact (Nat.Prime p)\nhp_mod_4_ne_3 : p % 4 ≠ 3\ny : ZMod p\nhy : -1 = y * y\nm : ℤ := y.valMinAbs\nn : ℕ := m.natAbs\nhnat₁ : p ∣ n ^ 2 + 1\n⊢ ∃ n, p ∣ n ^ 2 + 1 ∧ ↑p > 2 * ↑n + √(2 * ↑n)",
"after_state": "case intro\np : ℕ\nhpp : Nat.Prime p\nhp_mod_4_eq_1 : p ≡ 1 [MOD 4]\nhp_gt_20 : p > 20\nthis : Fact (Nat.Prime p)\nhp_mod_4_ne_3 : p % 4 ≠ 3\ny : ZMod p\nhy : -1 = y * y\nm : ℤ := y.valMinAbs\nn : ℕ := m.natAbs\nhnat₁ : p ∣ n ^ 2 + 1\nhnat₂ : n ≤ p / 2\n⊢ ∃ n, p ∣ n ^ 2 + 1 ∧ ↑p > 2 * ↑n + √(2 * ↑n)"
},
{
"line": "rotate_right",
"before_state": "case intro\np : ℕ\nhpp : Nat.Prime p\nhp_mod_4_eq_1 : p ≡ 1 [MOD 4]\nhp_gt_20 : p > 20\nthis : Fact (Nat.Prime p)\nhp_mod_4_ne_3 : p % 4 ≠ 3\ny : ZMod p\nhy : -1 = y * y\nm : ℤ := y.valMinAbs\nn : ℕ := m.natAbs\nhnat₁ : p ∣ n ^ 2 + 1\nhnat₂ : n ≤ p / 2\n⊢ ∃ n, p ∣ n ^ 2 + 1 ∧ ↑p > 2 * ↑n + √(2 * ↑n)",
"after_state": "case intro\np : ℕ\nhpp : Nat.Prime p\nhp_mod_4_eq_1 : p ≡ 1 [MOD 4]\nhp_gt_20 : p > 20\nthis : Fact (Nat.Prime p)\nhp_mod_4_ne_3 : p % 4 ≠ 3\ny : ZMod p\nhy : -1 = y * y\nm : ℤ := y.valMinAbs\nn : ℕ := m.natAbs\nhnat₁ : p ∣ n ^ 2 + 1\nhnat₂ : n ≤ p / 2\n⊢ ∃ n, p ∣ n ^ 2 + 1 ∧ ↑p > 2 * ↑n + √(2 * ↑n)"
},
{
"line": "have hnat₃ : p ≥ 2 * n := by omega",
"before_state": "case intro\np : ℕ\nhpp : Nat.Prime p\nhp_mod_4_eq_1 : p ≡ 1 [MOD 4]\nhp_gt_20 : p > 20\nthis : Fact (Nat.Prime p)\nhp_mod_4_ne_3 : p % 4 ≠ 3\ny : ZMod p\nhy : -1 = y * y\nm : ℤ := y.valMinAbs\nn : ℕ := m.natAbs\nhnat₁ : p ∣ n ^ 2 + 1\nhnat₂ : n ≤ p / 2\n⊢ ∃ n, p ∣ n ^ 2 + 1 ∧ ↑p > 2 * ↑n + √(2 * ↑n)",
"after_state": "case intro\np : ℕ\nhpp : Nat.Prime p\nhp_mod_4_eq_1 : p ≡ 1 [MOD 4]\nhp_gt_20 : p > 20\nthis : Fact (Nat.Prime p)\nhp_mod_4_ne_3 : p % 4 ≠ 3\ny : ZMod p\nhy : -1 = y * y\nm : ℤ := y.valMinAbs\nn : ℕ := m.natAbs\nhnat₁ : p ∣ n ^ 2 + 1\nhnat₂ : n ≤ p / 2\nhnat₃ : p ≥ 2 * n\n⊢ ∃ n, p ∣ n ^ 2 + 1 ∧ ↑p > 2 * ↑n + √(2 * ↑n)"
},
{
"line": "focus\n refine\n no_implicit_lambda%\n (have hnat₃ : p ≥ 2 * n := ?body✝;\n ?_)\n case body✝ => with_annotate_state\"by\" (omega)",
"before_state": "case intro\np : ℕ\nhpp : Nat.Prime p\nhp_mod_4_eq_1 : p ≡ 1 [MOD 4]\nhp_gt_20 : p > 20\nthis : Fact (Nat.Prime p)\nhp_mod_4_ne_3 : p % 4 ≠ 3\ny : ZMod p\nhy : -1 = y * y\nm : ℤ := y.valMinAbs\nn : ℕ := m.natAbs\nhnat₁ : p ∣ n ^ 2 + 1\nhnat₂ : n ≤ p / 2\n⊢ ∃ n, p ∣ n ^ 2 + 1 ∧ ↑p > 2 * ↑n + √(2 * ↑n)",
"after_state": "case intro\np : ℕ\nhpp : Nat.Prime p\nhp_mod_4_eq_1 : p ≡ 1 [MOD 4]\nhp_gt_20 : p > 20\nthis : Fact (Nat.Prime p)\nhp_mod_4_ne_3 : p % 4 ≠ 3\ny : ZMod p\nhy : -1 = y * y\nm : ℤ := y.valMinAbs\nn : ℕ := m.natAbs\nhnat₁ : p ∣ n ^ 2 + 1\nhnat₂ : n ≤ p / 2\nhnat₃ : p ≥ 2 * n\n⊢ ∃ n, p ∣ n ^ 2 + 1 ∧ ↑p > 2 * ↑n + √(2 * ↑n)"
},
{
"line": "refine\n no_implicit_lambda%\n (have hnat₃ : p ≥ 2 * n := ?body✝;\n ?_)",
"before_state": "case intro\np : ℕ\nhpp : Nat.Prime p\nhp_mod_4_eq_1 : p ≡ 1 [MOD 4]\nhp_gt_20 : p > 20\nthis : Fact (Nat.Prime p)\nhp_mod_4_ne_3 : p % 4 ≠ 3\ny : ZMod p\nhy : -1 = y * y\nm : ℤ := y.valMinAbs\nn : ℕ := m.natAbs\nhnat₁ : p ∣ n ^ 2 + 1\nhnat₂ : n ≤ p / 2\n⊢ ∃ n, p ∣ n ^ 2 + 1 ∧ ↑p > 2 * ↑n + √(2 * ↑n)",
"after_state": "case body\np : ℕ\nhpp : Nat.Prime p\nhp_mod_4_eq_1 : p ≡ 1 [MOD 4]\nhp_gt_20 : p > 20\nthis : Fact (Nat.Prime p)\nhp_mod_4_ne_3 : p % 4 ≠ 3\ny : ZMod p\nhy : -1 = y * y\nm : ℤ := y.valMinAbs\nn : ℕ := m.natAbs\nhnat₁ : p ∣ n ^ 2 + 1\nhnat₂ : n ≤ p / 2\n⊢ p ≥ 2 * n\n---\ncase intro\np : ℕ\nhpp : Nat.Prime p\nhp_mod_4_eq_1 : p ≡ 1 [MOD 4]\nhp_gt_20 : p > 20\nthis : Fact (Nat.Prime p)\nhp_mod_4_ne_3 : p % 4 ≠ 3\ny : ZMod p\nhy : -1 = y * y\nm : ℤ := y.valMinAbs\nn : ℕ := m.natAbs\nhnat₁ : p ∣ n ^ 2 + 1\nhnat₂ : n ≤ p / 2\nhnat₃ : p ≥ 2 * n\n⊢ ∃ n, p ∣ n ^ 2 + 1 ∧ ↑p > 2 * ↑n + √(2 * ↑n)"
},
{
"line": "case body✝ => with_annotate_state\"by\" (omega)",
"before_state": "case body\np : ℕ\nhpp : Nat.Prime p\nhp_mod_4_eq_1 : p ≡ 1 [MOD 4]\nhp_gt_20 : p > 20\nthis : Fact (Nat.Prime p)\nhp_mod_4_ne_3 : p % 4 ≠ 3\ny : ZMod p\nhy : -1 = y * y\nm : ℤ := y.valMinAbs\nn : ℕ := m.natAbs\nhnat₁ : p ∣ n ^ 2 + 1\nhnat₂ : n ≤ p / 2\n⊢ p ≥ 2 * n\n---\ncase intro\np : ℕ\nhpp : Nat.Prime p\nhp_mod_4_eq_1 : p ≡ 1 [MOD 4]\nhp_gt_20 : p > 20\nthis : Fact (Nat.Prime p)\nhp_mod_4_ne_3 : p % 4 ≠ 3\ny : ZMod p\nhy : -1 = y * y\nm : ℤ := y.valMinAbs\nn : ℕ := m.natAbs\nhnat₁ : p ∣ n ^ 2 + 1\nhnat₂ : n ≤ p / 2\nhnat₃ : p ≥ 2 * n\n⊢ ∃ n, p ∣ n ^ 2 + 1 ∧ ↑p > 2 * ↑n + √(2 * ↑n)",
"after_state": "case intro\np : ℕ\nhpp : Nat.Prime p\nhp_mod_4_eq_1 : p ≡ 1 [MOD 4]\nhp_gt_20 : p > 20\nthis : Fact (Nat.Prime p)\nhp_mod_4_ne_3 : p % 4 ≠ 3\ny : ZMod p\nhy : -1 = y * y\nm : ℤ := y.valMinAbs\nn : ℕ := m.natAbs\nhnat₁ : p ∣ n ^ 2 + 1\nhnat₂ : n ≤ p / 2\nhnat₃ : p ≥ 2 * n\n⊢ ∃ n, p ∣ n ^ 2 + 1 ∧ ↑p > 2 * ↑n + √(2 * ↑n)"
},
{
"line": "with_annotate_state\"by\" (omega)",
"before_state": "p : ℕ\nhpp : Nat.Prime p\nhp_mod_4_eq_1 : p ≡ 1 [MOD 4]\nhp_gt_20 : p > 20\nthis : Fact (Nat.Prime p)\nhp_mod_4_ne_3 : p % 4 ≠ 3\ny : ZMod p\nhy : -1 = y * y\nm : ℤ := y.valMinAbs\nn : ℕ := m.natAbs\nhnat₁ : p ∣ n ^ 2 + 1\nhnat₂ : n ≤ p / 2\n⊢ p ≥ 2 * n",
"after_state": "No Goals!"
},
{
"line": "omega",
"before_state": "p : ℕ\nhpp : Nat.Prime p\nhp_mod_4_eq_1 : p ≡ 1 [MOD 4]\nhp_gt_20 : p > 20\nthis : Fact (Nat.Prime p)\nhp_mod_4_ne_3 : p % 4 ≠ 3\ny : ZMod p\nhy : -1 = y * y\nm : ℤ := y.valMinAbs\nn : ℕ := m.natAbs\nhnat₁ : p ∣ n ^ 2 + 1\nhnat₂ : n ≤ p / 2\n⊢ p ≥ 2 * n",
"after_state": "No Goals!"
},
{
"line": "set k : ℕ := p - 2 * n with hnat₄",
"before_state": "case intro\np : ℕ\nhpp : Nat.Prime p\nhp_mod_4_eq_1 : p ≡ 1 [MOD 4]\nhp_gt_20 : p > 20\nthis : Fact (Nat.Prime p)\nhp_mod_4_ne_3 : p % 4 ≠ 3\ny : ZMod p\nhy : -1 = y * y\nm : ℤ := y.valMinAbs\nn : ℕ := m.natAbs\nhnat₁ : p ∣ n ^ 2 + 1\nhnat₂ : n ≤ p / 2\nhnat₃ : p ≥ 2 * n\n⊢ ∃ n, p ∣ n ^ 2 + 1 ∧ ↑p > 2 * ↑n + √(2 * ↑n)",
"after_state": "case intro\np : ℕ\nhpp : Nat.Prime p\nhp_mod_4_eq_1 : p ≡ 1 [MOD 4]\nhp_gt_20 : p > 20\nthis : Fact (Nat.Prime p)\nhp_mod_4_ne_3 : p % 4 ≠ 3\ny : ZMod p\nhy : -1 = y * y\nm : ℤ := y.valMinAbs\nn : ℕ := m.natAbs\nhnat₁ : p ∣ n ^ 2 + 1\nhnat₂ : n ≤ p / 2\nhnat₃ : p ≥ 2 * n\nk : ℕ := p - 2 * n\nhnat₄ : k = p - 2 * n\n⊢ ∃ n, p ∣ n ^ 2 + 1 ∧ ↑p > 2 * ↑n + √(2 * ↑n)"
},
{
"line": "try rewrite [show ?m✝ = k from rfl✝] at *",
"before_state": "case intro\np : ℕ\nhpp : Nat.Prime p\nhp_mod_4_eq_1 : p ≡ 1 [MOD 4]\nhp_gt_20 : p > 20\nthis : Fact (Nat.Prime p)\nhp_mod_4_ne_3 : p % 4 ≠ 3\ny : ZMod p\nhy : -1 = y * y\nm : ℤ := y.valMinAbs\nn : ℕ := m.natAbs\nhnat₁ : p ∣ n ^ 2 + 1\nhnat₂ : n ≤ p / 2\nhnat₃ : p ≥ 2 * n\nk : ℕ := p - 2 * n\n⊢ ∃ n, p ∣ n ^ 2 + 1 ∧ ↑p > 2 * ↑n + √(2 * ↑n)",
"after_state": "case intro\np : ℕ\nhpp : Nat.Prime p\nhp_mod_4_eq_1 : p ≡ 1 [MOD 4]\nhp_gt_20 : p > 20\nthis : Fact (Nat.Prime p)\nhp_mod_4_ne_3 : p % 4 ≠ 3\ny : ZMod p\nhy : -1 = y * y\nm : ℤ := y.valMinAbs\nn : ℕ := m.natAbs\nhnat₁ : p ∣ n ^ 2 + 1\nhnat₂ : n ≤ p / 2\nhnat₃ : p ≥ 2 * n\nk : ℕ := p - 2 * n\n⊢ ∃ n, p ∣ n ^ 2 + 1 ∧ ↑p > 2 * ↑n + √(2 * ↑n)"
},
{
"line": "first\n| rewrite [show ?m✝ = k from rfl✝] at *\n| skip",
"before_state": "case intro\np : ℕ\nhpp : Nat.Prime p\nhp_mod_4_eq_1 : p ≡ 1 [MOD 4]\nhp_gt_20 : p > 20\nthis : Fact (Nat.Prime p)\nhp_mod_4_ne_3 : p % 4 ≠ 3\ny : ZMod p\nhy : -1 = y * y\nm : ℤ := y.valMinAbs\nn : ℕ := m.natAbs\nhnat₁ : p ∣ n ^ 2 + 1\nhnat₂ : n ≤ p / 2\nhnat₃ : p ≥ 2 * n\nk : ℕ := p - 2 * n\n⊢ ∃ n, p ∣ n ^ 2 + 1 ∧ ↑p > 2 * ↑n + √(2 * ↑n)",
"after_state": "case intro\np : ℕ\nhpp : Nat.Prime p\nhp_mod_4_eq_1 : p ≡ 1 [MOD 4]\nhp_gt_20 : p > 20\nthis : Fact (Nat.Prime p)\nhp_mod_4_ne_3 : p % 4 ≠ 3\ny : ZMod p\nhy : -1 = y * y\nm : ℤ := y.valMinAbs\nn : ℕ := m.natAbs\nhnat₁ : p ∣ n ^ 2 + 1\nhnat₂ : n ≤ p / 2\nhnat₃ : p ≥ 2 * n\nk : ℕ := p - 2 * n\n⊢ ∃ n, p ∣ n ^ 2 + 1 ∧ ↑p > 2 * ↑n + √(2 * ↑n)"
},
{
"line": "rewrite [show ?m✝ = k from rfl✝] at *",
"before_state": "case intro\np : ℕ\nhpp : Nat.Prime p\nhp_mod_4_eq_1 : p ≡ 1 [MOD 4]\nhp_gt_20 : p > 20\nthis : Fact (Nat.Prime p)\nhp_mod_4_ne_3 : p % 4 ≠ 3\ny : ZMod p\nhy : -1 = y * y\nm : ℤ := y.valMinAbs\nn : ℕ := m.natAbs\nhnat₁ : p ∣ n ^ 2 + 1\nhnat₂ : n ≤ p / 2\nhnat₃ : p ≥ 2 * n\nk : ℕ := p - 2 * n\n⊢ ∃ n, p ∣ n ^ 2 + 1 ∧ ↑p > 2 * ↑n + √(2 * ↑n)",
"after_state": "case intro\np : ℕ\nhpp : Nat.Prime p\nhp_mod_4_eq_1 : p ≡ 1 [MOD 4]\nhp_gt_20 : p > 20\nthis : Fact (Nat.Prime p)\nhp_mod_4_ne_3 : p % 4 ≠ 3\ny : ZMod p\nhy : -1 = y * y\nm : ℤ := y.valMinAbs\nn : ℕ := m.natAbs\nhnat₁ : p ∣ n ^ 2 + 1\nhnat₂ : n ≤ p / 2\nhnat₃ : p ≥ 2 * n\nk : ℕ := p - 2 * n\n⊢ ∃ n, p ∣ n ^ 2 + 1 ∧ ↑p > 2 * ↑n + √(2 * ↑n)"
},
{
"line": "skip",
"before_state": "case intro\np : ℕ\nhpp : Nat.Prime p\nhp_mod_4_eq_1 : p ≡ 1 [MOD 4]\nhp_gt_20 : p > 20\nthis : Fact (Nat.Prime p)\nhp_mod_4_ne_3 : p % 4 ≠ 3\ny : ZMod p\nhy : -1 = y * y\nm : ℤ := y.valMinAbs\nn : ℕ := m.natAbs\nhnat₁ : p ∣ n ^ 2 + 1\nhnat₂ : n ≤ p / 2\nhnat₃ : p ≥ 2 * n\nk : ℕ := p - 2 * n\n⊢ ∃ n, p ∣ n ^ 2 + 1 ∧ ↑p > 2 * ↑n + √(2 * ↑n)",
"after_state": "case intro\np : ℕ\nhpp : Nat.Prime p\nhp_mod_4_eq_1 : p ≡ 1 [MOD 4]\nhp_gt_20 : p > 20\nthis : Fact (Nat.Prime p)\nhp_mod_4_ne_3 : p % 4 ≠ 3\ny : ZMod p\nhy : -1 = y * y\nm : ℤ := y.valMinAbs\nn : ℕ := m.natAbs\nhnat₁ : p ∣ n ^ 2 + 1\nhnat₂ : n ≤ p / 2\nhnat₃ : p ≥ 2 * n\nk : ℕ := p - 2 * n\n⊢ ∃ n, p ∣ n ^ 2 + 1 ∧ ↑p > 2 * ↑n + √(2 * ↑n)"
},
{
"line": "have hnat₄ : k = (?m✝ : ?m✝¹) := rfl✝",
"before_state": "case intro\np : ℕ\nhpp : Nat.Prime p\nhp_mod_4_eq_1 : p ≡ 1 [MOD 4]\nhp_gt_20 : p > 20\nthis : Fact (Nat.Prime p)\nhp_mod_4_ne_3 : p % 4 ≠ 3\ny : ZMod p\nhy : -1 = y * y\nm : ℤ := y.valMinAbs\nn : ℕ := m.natAbs\nhnat₁ : p ∣ n ^ 2 + 1\nhnat₂ : n ≤ p / 2\nhnat₃ : p ≥ 2 * n\nk : ℕ := p - 2 * n\n⊢ ∃ n, p ∣ n ^ 2 + 1 ∧ ↑p > 2 * ↑n + √(2 * ↑n)",
"after_state": "case intro\np : ℕ\nhpp : Nat.Prime p\nhp_mod_4_eq_1 : p ≡ 1 [MOD 4]\nhp_gt_20 : p > 20\nthis : Fact (Nat.Prime p)\nhp_mod_4_ne_3 : p % 4 ≠ 3\ny : ZMod p\nhy : -1 = y * y\nm : ℤ := y.valMinAbs\nn : ℕ := m.natAbs\nhnat₁ : p ∣ n ^ 2 + 1\nhnat₂ : n ≤ p / 2\nhnat₃ : p ≥ 2 * n\nk : ℕ := p - 2 * n\nhnat₄ : k = p - 2 * n\n⊢ ∃ n, p ∣ n ^ 2 + 1 ∧ ↑p > 2 * ↑n + √(2 * ↑n)"
},
{
"line": "refine_lift\n have hnat₄ : k = (?m✝ : ?m✝¹) := rfl✝;\n ?_",
"before_state": "case intro\np : ℕ\nhpp : Nat.Prime p\nhp_mod_4_eq_1 : p ≡ 1 [MOD 4]\nhp_gt_20 : p > 20\nthis : Fact (Nat.Prime p)\nhp_mod_4_ne_3 : p % 4 ≠ 3\ny : ZMod p\nhy : -1 = y * y\nm : ℤ := y.valMinAbs\nn : ℕ := m.natAbs\nhnat₁ : p ∣ n ^ 2 + 1\nhnat₂ : n ≤ p / 2\nhnat₃ : p ≥ 2 * n\nk : ℕ := p - 2 * n\n⊢ ∃ n, p ∣ n ^ 2 + 1 ∧ ↑p > 2 * ↑n + √(2 * ↑n)",
"after_state": "case intro\np : ℕ\nhpp : Nat.Prime p\nhp_mod_4_eq_1 : p ≡ 1 [MOD 4]\nhp_gt_20 : p > 20\nthis : Fact (Nat.Prime p)\nhp_mod_4_ne_3 : p % 4 ≠ 3\ny : ZMod p\nhy : -1 = y * y\nm : ℤ := y.valMinAbs\nn : ℕ := m.natAbs\nhnat₁ : p ∣ n ^ 2 + 1\nhnat₂ : n ≤ p / 2\nhnat₃ : p ≥ 2 * n\nk : ℕ := p - 2 * n\nhnat₄ : k = p - 2 * n\n⊢ ∃ n, p ∣ n ^ 2 + 1 ∧ ↑p > 2 * ↑n + √(2 * ↑n)"
},
{
"line": "focus\n (refine\n no_implicit_lambda%\n (have hnat₄ : k = (?m✝ : ?m✝¹) := rfl✝;\n ?_);\n rotate_right)",
"before_state": "case intro\np : ℕ\nhpp : Nat.Prime p\nhp_mod_4_eq_1 : p ≡ 1 [MOD 4]\nhp_gt_20 : p > 20\nthis : Fact (Nat.Prime p)\nhp_mod_4_ne_3 : p % 4 ≠ 3\ny : ZMod p\nhy : -1 = y * y\nm : ℤ := y.valMinAbs\nn : ℕ := m.natAbs\nhnat₁ : p ∣ n ^ 2 + 1\nhnat₂ : n ≤ p / 2\nhnat₃ : p ≥ 2 * n\nk : ℕ := p - 2 * n\n⊢ ∃ n, p ∣ n ^ 2 + 1 ∧ ↑p > 2 * ↑n + √(2 * ↑n)",
"after_state": "case intro\np : ℕ\nhpp : Nat.Prime p\nhp_mod_4_eq_1 : p ≡ 1 [MOD 4]\nhp_gt_20 : p > 20\nthis : Fact (Nat.Prime p)\nhp_mod_4_ne_3 : p % 4 ≠ 3\ny : ZMod p\nhy : -1 = y * y\nm : ℤ := y.valMinAbs\nn : ℕ := m.natAbs\nhnat₁ : p ∣ n ^ 2 + 1\nhnat₂ : n ≤ p / 2\nhnat₃ : p ≥ 2 * n\nk : ℕ := p - 2 * n\nhnat₄ : k = p - 2 * n\n⊢ ∃ n, p ∣ n ^ 2 + 1 ∧ ↑p > 2 * ↑n + √(2 * ↑n)"
},
{
"line": "refine\n no_implicit_lambda%\n (have hnat₄ : k = (?m✝ : ?m✝¹) := rfl✝;\n ?_)",
"before_state": "case intro\np : ℕ\nhpp : Nat.Prime p\nhp_mod_4_eq_1 : p ≡ 1 [MOD 4]\nhp_gt_20 : p > 20\nthis : Fact (Nat.Prime p)\nhp_mod_4_ne_3 : p % 4 ≠ 3\ny : ZMod p\nhy : -1 = y * y\nm : ℤ := y.valMinAbs\nn : ℕ := m.natAbs\nhnat₁ : p ∣ n ^ 2 + 1\nhnat₂ : n ≤ p / 2\nhnat₃ : p ≥ 2 * n\nk : ℕ := p - 2 * n\n⊢ ∃ n, p ∣ n ^ 2 + 1 ∧ ↑p > 2 * ↑n + √(2 * ↑n)",
"after_state": "case intro\np : ℕ\nhpp : Nat.Prime p\nhp_mod_4_eq_1 : p ≡ 1 [MOD 4]\nhp_gt_20 : p > 20\nthis : Fact (Nat.Prime p)\nhp_mod_4_ne_3 : p % 4 ≠ 3\ny : ZMod p\nhy : -1 = y * y\nm : ℤ := y.valMinAbs\nn : ℕ := m.natAbs\nhnat₁ : p ∣ n ^ 2 + 1\nhnat₂ : n ≤ p / 2\nhnat₃ : p ≥ 2 * n\nk : ℕ := p - 2 * n\nhnat₄ : k = p - 2 * n\n⊢ ∃ n, p ∣ n ^ 2 + 1 ∧ ↑p > 2 * ↑n + √(2 * ↑n)"
},
{
"line": "rotate_right",
"before_state": "case intro\np : ℕ\nhpp : Nat.Prime p\nhp_mod_4_eq_1 : p ≡ 1 [MOD 4]\nhp_gt_20 : p > 20\nthis : Fact (Nat.Prime p)\nhp_mod_4_ne_3 : p % 4 ≠ 3\ny : ZMod p\nhy : -1 = y * y\nm : ℤ := y.valMinAbs\nn : ℕ := m.natAbs\nhnat₁ : p ∣ n ^ 2 + 1\nhnat₂ : n ≤ p / 2\nhnat₃ : p ≥ 2 * n\nk : ℕ := p - 2 * n\nhnat₄ : k = p - 2 * n\n⊢ ∃ n, p ∣ n ^ 2 + 1 ∧ ↑p > 2 * ↑n + √(2 * ↑n)",
"after_state": "case intro\np : ℕ\nhpp : Nat.Prime p\nhp_mod_4_eq_1 : p ≡ 1 [MOD 4]\nhp_gt_20 : p > 20\nthis : Fact (Nat.Prime p)\nhp_mod_4_ne_3 : p % 4 ≠ 3\ny : ZMod p\nhy : -1 = y * y\nm : ℤ := y.valMinAbs\nn : ℕ := m.natAbs\nhnat₁ : p ∣ n ^ 2 + 1\nhnat₂ : n ≤ p / 2\nhnat₃ : p ≥ 2 * n\nk : ℕ := p - 2 * n\nhnat₄ : k = p - 2 * n\n⊢ ∃ n, p ∣ n ^ 2 + 1 ∧ ↑p > 2 * ↑n + √(2 * ↑n)"
},
{
"line": "have hnat₅ : p ∣ k ^ 2 + 4 := by\n obtain ⟨x, hx⟩ := hnat₁\n have : (p : ℤ) ∣ (k : ℤ) ^ 2 + 4 := by\n use (p : ℤ) - 4 * n + 4 * x\n have hcast₁ : (k : ℤ) = p - 2 * n := by assumption_mod_cast\n have hcast₂ : (n : ℤ) ^ 2 + 1 = p * x := by assumption_mod_cast\n linear_combination ((k : ℤ) + p - 2 * n) * hcast₁ + 4 * hcast₂\n assumption_mod_cast",
"before_state": "case intro\np : ℕ\nhpp : Nat.Prime p\nhp_mod_4_eq_1 : p ≡ 1 [MOD 4]\nhp_gt_20 : p > 20\nthis : Fact (Nat.Prime p)\nhp_mod_4_ne_3 : p % 4 ≠ 3\ny : ZMod p\nhy : -1 = y * y\nm : ℤ := y.valMinAbs\nn : ℕ := m.natAbs\nhnat₁ : p ∣ n ^ 2 + 1\nhnat₂ : n ≤ p / 2\nhnat₃ : p ≥ 2 * n\nk : ℕ := p - 2 * n\nhnat₄ : k = p - 2 * n\n⊢ ∃ n, p ∣ n ^ 2 + 1 ∧ ↑p > 2 * ↑n + √(2 * ↑n)",
"after_state": "case intro\np : ℕ\nhpp : Nat.Prime p\nhp_mod_4_eq_1 : p ≡ 1 [MOD 4]\nhp_gt_20 : p > 20\nthis : Fact (Nat.Prime p)\nhp_mod_4_ne_3 : p % 4 ≠ 3\ny : ZMod p\nhy : -1 = y * y\nm : ℤ := y.valMinAbs\nn : ℕ := m.natAbs\nhnat₁ : p ∣ n ^ 2 + 1\nhnat₂ : n ≤ p / 2\nhnat₃ : p ≥ 2 * n\nk : ℕ := p - 2 * n\nhnat₄ : k = p - 2 * n\nhnat₅ : p ∣ k ^ 2 + 4\n⊢ ∃ n, p ∣ n ^ 2 + 1 ∧ ↑p > 2 * ↑n + √(2 * ↑n)"
},
{
"line": "focus\n refine\n no_implicit_lambda%\n (have hnat₅ : p ∣ k ^ 2 + 4 := ?body✝;\n ?_)\n case body✝ =>\n with_annotate_state\"by\"\n ( obtain ⟨x, hx⟩ := hnat₁\n have : (p : ℤ) ∣ (k : ℤ) ^ 2 + 4 := by\n use (p : ℤ) - 4 * n + 4 * x\n have hcast₁ : (k : ℤ) = p - 2 * n := by assumption_mod_cast\n have hcast₂ : (n : ℤ) ^ 2 + 1 = p * x := by assumption_mod_cast\n linear_combination ((k : ℤ) + p - 2 * n) * hcast₁ + 4 * hcast₂\n assumption_mod_cast)",
"before_state": "case intro\np : ℕ\nhpp : Nat.Prime p\nhp_mod_4_eq_1 : p ≡ 1 [MOD 4]\nhp_gt_20 : p > 20\nthis : Fact (Nat.Prime p)\nhp_mod_4_ne_3 : p % 4 ≠ 3\ny : ZMod p\nhy : -1 = y * y\nm : ℤ := y.valMinAbs\nn : ℕ := m.natAbs\nhnat₁ : p ∣ n ^ 2 + 1\nhnat₂ : n ≤ p / 2\nhnat₃ : p ≥ 2 * n\nk : ℕ := p - 2 * n\nhnat₄ : k = p - 2 * n\n⊢ ∃ n, p ∣ n ^ 2 + 1 ∧ ↑p > 2 * ↑n + √(2 * ↑n)",
"after_state": "case intro\np : ℕ\nhpp : Nat.Prime p\nhp_mod_4_eq_1 : p ≡ 1 [MOD 4]\nhp_gt_20 : p > 20\nthis : Fact (Nat.Prime p)\nhp_mod_4_ne_3 : p % 4 ≠ 3\ny : ZMod p\nhy : -1 = y * y\nm : ℤ := y.valMinAbs\nn : ℕ := m.natAbs\nhnat₁ : p ∣ n ^ 2 + 1\nhnat₂ : n ≤ p / 2\nhnat₃ : p ≥ 2 * n\nk : ℕ := p - 2 * n\nhnat₄ : k = p - 2 * n\nhnat₅ : p ∣ k ^ 2 + 4\n⊢ ∃ n, p ∣ n ^ 2 + 1 ∧ ↑p > 2 * ↑n + √(2 * ↑n)"
},
{
"line": "refine\n no_implicit_lambda%\n (have hnat₅ : p ∣ k ^ 2 + 4 := ?body✝;\n ?_)",
"before_state": "case intro\np : ℕ\nhpp : Nat.Prime p\nhp_mod_4_eq_1 : p ≡ 1 [MOD 4]\nhp_gt_20 : p > 20\nthis : Fact (Nat.Prime p)\nhp_mod_4_ne_3 : p % 4 ≠ 3\ny : ZMod p\nhy : -1 = y * y\nm : ℤ := y.valMinAbs\nn : ℕ := m.natAbs\nhnat₁ : p ∣ n ^ 2 + 1\nhnat₂ : n ≤ p / 2\nhnat₃ : p ≥ 2 * n\nk : ℕ := p - 2 * n\nhnat₄ : k = p - 2 * n\n⊢ ∃ n, p ∣ n ^ 2 + 1 ∧ ↑p > 2 * ↑n + √(2 * ↑n)",
"after_state": "case body\np : ℕ\nhpp : Nat.Prime p\nhp_mod_4_eq_1 : p ≡ 1 [MOD 4]\nhp_gt_20 : p > 20\nthis : Fact (Nat.Prime p)\nhp_mod_4_ne_3 : p % 4 ≠ 3\ny : ZMod p\nhy : -1 = y * y\nm : ℤ := y.valMinAbs\nn : ℕ := m.natAbs\nhnat₁ : p ∣ n ^ 2 + 1\nhnat₂ : n ≤ p / 2\nhnat₃ : p ≥ 2 * n\nk : ℕ := p - 2 * n\nhnat₄ : k = p - 2 * n\n⊢ p ∣ k ^ 2 + 4\n---\ncase intro\np : ℕ\nhpp : Nat.Prime p\nhp_mod_4_eq_1 : p ≡ 1 [MOD 4]\nhp_gt_20 : p > 20\nthis : Fact (Nat.Prime p)\nhp_mod_4_ne_3 : p % 4 ≠ 3\ny : ZMod p\nhy : -1 = y * y\nm : ℤ := y.valMinAbs\nn : ℕ := m.natAbs\nhnat₁ : p ∣ n ^ 2 + 1\nhnat₂ : n ≤ p / 2\nhnat₃ : p ≥ 2 * n\nk : ℕ := p - 2 * n\nhnat₄ : k = p - 2 * n\nhnat₅ : p ∣ k ^ 2 + 4\n⊢ ∃ n, p ∣ n ^ 2 + 1 ∧ ↑p > 2 * ↑n + √(2 * ↑n)"
},
{
"line": "case body✝ =>\n with_annotate_state\"by\"\n ( obtain ⟨x, hx⟩ := hnat₁\n have : (p : ℤ) ∣ (k : ℤ) ^ 2 + 4 := by\n use (p : ℤ) - 4 * n + 4 * x\n have hcast₁ : (k : ℤ) = p - 2 * n := by assumption_mod_cast\n have hcast₂ : (n : ℤ) ^ 2 + 1 = p * x := by assumption_mod_cast\n linear_combination ((k : ℤ) + p - 2 * n) * hcast₁ + 4 * hcast₂\n assumption_mod_cast)",
"before_state": "case body\np : ℕ\nhpp : Nat.Prime p\nhp_mod_4_eq_1 : p ≡ 1 [MOD 4]\nhp_gt_20 : p > 20\nthis : Fact (Nat.Prime p)\nhp_mod_4_ne_3 : p % 4 ≠ 3\ny : ZMod p\nhy : -1 = y * y\nm : ℤ := y.valMinAbs\nn : ℕ := m.natAbs\nhnat₁ : p ∣ n ^ 2 + 1\nhnat₂ : n ≤ p / 2\nhnat₃ : p ≥ 2 * n\nk : ℕ := p - 2 * n\nhnat₄ : k = p - 2 * n\n⊢ p ∣ k ^ 2 + 4\n---\ncase intro\np : ℕ\nhpp : Nat.Prime p\nhp_mod_4_eq_1 : p ≡ 1 [MOD 4]\nhp_gt_20 : p > 20\nthis : Fact (Nat.Prime p)\nhp_mod_4_ne_3 : p % 4 ≠ 3\ny : ZMod p\nhy : -1 = y * y\nm : ℤ := y.valMinAbs\nn : ℕ := m.natAbs\nhnat₁ : p ∣ n ^ 2 + 1\nhnat₂ : n ≤ p / 2\nhnat₃ : p ≥ 2 * n\nk : ℕ := p - 2 * n\nhnat₄ : k = p - 2 * n\nhnat₅ : p ∣ k ^ 2 + 4\n⊢ ∃ n, p ∣ n ^ 2 + 1 ∧ ↑p > 2 * ↑n + √(2 * ↑n)",
"after_state": "case intro\np : ℕ\nhpp : Nat.Prime p\nhp_mod_4_eq_1 : p ≡ 1 [MOD 4]\nhp_gt_20 : p > 20\nthis : Fact (Nat.Prime p)\nhp_mod_4_ne_3 : p % 4 ≠ 3\ny : ZMod p\nhy : -1 = y * y\nm : ℤ := y.valMinAbs\nn : ℕ := m.natAbs\nhnat₁ : p ∣ n ^ 2 + 1\nhnat₂ : n ≤ p / 2\nhnat₃ : p ≥ 2 * n\nk : ℕ := p - 2 * n\nhnat₄ : k = p - 2 * n\nhnat₅ : p ∣ k ^ 2 + 4\n⊢ ∃ n, p ∣ n ^ 2 + 1 ∧ ↑p > 2 * ↑n + √(2 * ↑n)"
},
{
"line": "with_annotate_state\"by\"\n ( obtain ⟨x, hx⟩ := hnat₁\n have : (p : ℤ) ∣ (k : ℤ) ^ 2 + 4 := by\n use (p : ℤ) - 4 * n + 4 * x\n have hcast₁ : (k : ℤ) = p - 2 * n := by assumption_mod_cast\n have hcast₂ : (n : ℤ) ^ 2 + 1 = p * x := by assumption_mod_cast\n linear_combination ((k : ℤ) + p - 2 * n) * hcast₁ + 4 * hcast₂\n assumption_mod_cast)",
"before_state": "p : ℕ\nhpp : Nat.Prime p\nhp_mod_4_eq_1 : p ≡ 1 [MOD 4]\nhp_gt_20 : p > 20\nthis : Fact (Nat.Prime p)\nhp_mod_4_ne_3 : p % 4 ≠ 3\ny : ZMod p\nhy : -1 = y * y\nm : ℤ := y.valMinAbs\nn : ℕ := m.natAbs\nhnat₁ : p ∣ n ^ 2 + 1\nhnat₂ : n ≤ p / 2\nhnat₃ : p ≥ 2 * n\nk : ℕ := p - 2 * n\nhnat₄ : k = p - 2 * n\n⊢ p ∣ k ^ 2 + 4",
"after_state": "No Goals!"
},
{
"line": "obtain ⟨x, hx⟩ := hnat₁",
"before_state": "p : ℕ\nhpp : Nat.Prime p\nhp_mod_4_eq_1 : p ≡ 1 [MOD 4]\nhp_gt_20 : p > 20\nthis : Fact (Nat.Prime p)\nhp_mod_4_ne_3 : p % 4 ≠ 3\ny : ZMod p\nhy : -1 = y * y\nm : ℤ := y.valMinAbs\nn : ℕ := m.natAbs\nhnat₁ : p ∣ n ^ 2 + 1\nhnat₂ : n ≤ p / 2\nhnat₃ : p ≥ 2 * n\nk : ℕ := p - 2 * n\nhnat₄ : k = p - 2 * n\n⊢ p ∣ k ^ 2 + 4",
"after_state": "case intro\np : ℕ\nhpp : Nat.Prime p\nhp_mod_4_eq_1 : p ≡ 1 [MOD 4]\nhp_gt_20 : p > 20\nthis : Fact (Nat.Prime p)\nhp_mod_4_ne_3 : p % 4 ≠ 3\ny : ZMod p\nhy : -1 = y * y\nm : ℤ := y.valMinAbs\nn : ℕ := m.natAbs\nhnat₂ : n ≤ p / 2\nhnat₃ : p ≥ 2 * n\nk : ℕ := p - 2 * n\nhnat₄ : k = p - 2 * n\nx : ℕ\nhx : n ^ 2 + 1 = p * x\n⊢ p ∣ k ^ 2 + 4"
},
{
"line": "have : (p : ℤ) ∣ (k : ℤ) ^ 2 + 4 := by\n use (p : ℤ) - 4 * n + 4 * x\n have hcast₁ : (k : ℤ) = p - 2 * n := by assumption_mod_cast\n have hcast₂ : (n : ℤ) ^ 2 + 1 = p * x := by assumption_mod_cast\n linear_combination ((k : ℤ) + p - 2 * n) * hcast₁ + 4 * hcast₂",
"before_state": "case intro\np : ℕ\nhpp : Nat.Prime p\nhp_mod_4_eq_1 : p ≡ 1 [MOD 4]\nhp_gt_20 : p > 20\nthis : Fact (Nat.Prime p)\nhp_mod_4_ne_3 : p % 4 ≠ 3\ny : ZMod p\nhy : -1 = y * y\nm : ℤ := y.valMinAbs\nn : ℕ := m.natAbs\nhnat₂ : n ≤ p / 2\nhnat₃ : p ≥ 2 * n\nk : ℕ := p - 2 * n\nhnat₄ : k = p - 2 * n\nx : ℕ\nhx : n ^ 2 + 1 = p * x\n⊢ p ∣ k ^ 2 + 4",
"after_state": "case intro\np : ℕ\nhpp : Nat.Prime p\nhp_mod_4_eq_1 : p ≡ 1 [MOD 4]\nhp_gt_20 : p > 20\nthis✝ : Fact (Nat.Prime p)\nhp_mod_4_ne_3 : p % 4 ≠ 3\ny : ZMod p\nhy : -1 = y * y\nm : ℤ := y.valMinAbs\nn : ℕ := m.natAbs\nhnat₂ : n ≤ p / 2\nhnat₃ : p ≥ 2 * n\nk : ℕ := p - 2 * n\nhnat₄ : k = p - 2 * n\nx : ℕ\nhx : n ^ 2 + 1 = p * x\nthis : ↑p ∣ ↑k ^ 2 + 4\n⊢ p ∣ k ^ 2 + 4"
},
{
"line": "focus\n refine\n no_implicit_lambda%\n (have : (p : ℤ) ∣ (k : ℤ) ^ 2 + 4 := ?body✝;\n ?_)\n case body✝ =>\n with_annotate_state\"by\"\n ( use (p : ℤ) - 4 * n + 4 * x\n have hcast₁ : (k : ℤ) = p - 2 * n := by assumption_mod_cast\n have hcast₂ : (n : ℤ) ^ 2 + 1 = p * x := by assumption_mod_cast\n linear_combination ((k : ℤ) + p - 2 * n) * hcast₁ + 4 * hcast₂)",
"before_state": "case intro\np : ℕ\nhpp : Nat.Prime p\nhp_mod_4_eq_1 : p ≡ 1 [MOD 4]\nhp_gt_20 : p > 20\nthis : Fact (Nat.Prime p)\nhp_mod_4_ne_3 : p % 4 ≠ 3\ny : ZMod p\nhy : -1 = y * y\nm : ℤ := y.valMinAbs\nn : ℕ := m.natAbs\nhnat₂ : n ≤ p / 2\nhnat₃ : p ≥ 2 * n\nk : ℕ := p - 2 * n\nhnat₄ : k = p - 2 * n\nx : ℕ\nhx : n ^ 2 + 1 = p * x\n⊢ p ∣ k ^ 2 + 4",
"after_state": "case intro\np : ℕ\nhpp : Nat.Prime p\nhp_mod_4_eq_1 : p ≡ 1 [MOD 4]\nhp_gt_20 : p > 20\nthis✝ : Fact (Nat.Prime p)\nhp_mod_4_ne_3 : p % 4 ≠ 3\ny : ZMod p\nhy : -1 = y * y\nm : ℤ := y.valMinAbs\nn : ℕ := m.natAbs\nhnat₂ : n ≤ p / 2\nhnat₃ : p ≥ 2 * n\nk : ℕ := p - 2 * n\nhnat₄ : k = p - 2 * n\nx : ℕ\nhx : n ^ 2 + 1 = p * x\nthis : ↑p ∣ ↑k ^ 2 + 4\n⊢ p ∣ k ^ 2 + 4"
},
{
"line": "refine\n no_implicit_lambda%\n (have : (p : ℤ) ∣ (k : ℤ) ^ 2 + 4 := ?body✝;\n ?_)",
"before_state": "case intro\np : ℕ\nhpp : Nat.Prime p\nhp_mod_4_eq_1 : p ≡ 1 [MOD 4]\nhp_gt_20 : p > 20\nthis : Fact (Nat.Prime p)\nhp_mod_4_ne_3 : p % 4 ≠ 3\ny : ZMod p\nhy : -1 = y * y\nm : ℤ := y.valMinAbs\nn : ℕ := m.natAbs\nhnat₂ : n ≤ p / 2\nhnat₃ : p ≥ 2 * n\nk : ℕ := p - 2 * n\nhnat₄ : k = p - 2 * n\nx : ℕ\nhx : n ^ 2 + 1 = p * x\n⊢ p ∣ k ^ 2 + 4",
"after_state": "case body\np : ℕ\nhpp : Nat.Prime p\nhp_mod_4_eq_1 : p ≡ 1 [MOD 4]\nhp_gt_20 : p > 20\nthis : Fact (Nat.Prime p)\nhp_mod_4_ne_3 : p % 4 ≠ 3\ny : ZMod p\nhy : -1 = y * y\nm : ℤ := y.valMinAbs\nn : ℕ := m.natAbs\nhnat₂ : n ≤ p / 2\nhnat₃ : p ≥ 2 * n\nk : ℕ := p - 2 * n\nhnat₄ : k = p - 2 * n\nx : ℕ\nhx : n ^ 2 + 1 = p * x\n⊢ ↑p ∣ ↑k ^ 2 + 4\n---\ncase intro\np : ℕ\nhpp : Nat.Prime p\nhp_mod_4_eq_1 : p ≡ 1 [MOD 4]\nhp_gt_20 : p > 20\nthis✝ : Fact (Nat.Prime p)\nhp_mod_4_ne_3 : p % 4 ≠ 3\ny : ZMod p\nhy : -1 = y * y\nm : ℤ := y.valMinAbs\nn : ℕ := m.natAbs\nhnat₂ : n ≤ p / 2\nhnat₃ : p ≥ 2 * n\nk : ℕ := p - 2 * n\nhnat₄ : k = p - 2 * n\nx : ℕ\nhx : n ^ 2 + 1 = p * x\nthis : ↑p ∣ ↑k ^ 2 + 4\n⊢ p ∣ k ^ 2 + 4"
},
{
"line": "case body✝ =>\n with_annotate_state\"by\"\n ( use (p : ℤ) - 4 * n + 4 * x\n have hcast₁ : (k : ℤ) = p - 2 * n := by assumption_mod_cast\n have hcast₂ : (n : ℤ) ^ 2 + 1 = p * x := by assumption_mod_cast\n linear_combination ((k : ℤ) + p - 2 * n) * hcast₁ + 4 * hcast₂)",
"before_state": "case body\np : ℕ\nhpp : Nat.Prime p\nhp_mod_4_eq_1 : p ≡ 1 [MOD 4]\nhp_gt_20 : p > 20\nthis : Fact (Nat.Prime p)\nhp_mod_4_ne_3 : p % 4 ≠ 3\ny : ZMod p\nhy : -1 = y * y\nm : ℤ := y.valMinAbs\nn : ℕ := m.natAbs\nhnat₂ : n ≤ p / 2\nhnat₃ : p ≥ 2 * n\nk : ℕ := p - 2 * n\nhnat₄ : k = p - 2 * n\nx : ℕ\nhx : n ^ 2 + 1 = p * x\n⊢ ↑p ∣ ↑k ^ 2 + 4\n---\ncase intro\np : ℕ\nhpp : Nat.Prime p\nhp_mod_4_eq_1 : p ≡ 1 [MOD 4]\nhp_gt_20 : p > 20\nthis✝ : Fact (Nat.Prime p)\nhp_mod_4_ne_3 : p % 4 ≠ 3\ny : ZMod p\nhy : -1 = y * y\nm : ℤ := y.valMinAbs\nn : ℕ := m.natAbs\nhnat₂ : n ≤ p / 2\nhnat₃ : p ≥ 2 * n\nk : ℕ := p - 2 * n\nhnat₄ : k = p - 2 * n\nx : ℕ\nhx : n ^ 2 + 1 = p * x\nthis : ↑p ∣ ↑k ^ 2 + 4\n⊢ p ∣ k ^ 2 + 4",
"after_state": "case intro\np : ℕ\nhpp : Nat.Prime p\nhp_mod_4_eq_1 : p ≡ 1 [MOD 4]\nhp_gt_20 : p > 20\nthis✝ : Fact (Nat.Prime p)\nhp_mod_4_ne_3 : p % 4 ≠ 3\ny : ZMod p\nhy : -1 = y * y\nm : ℤ := y.valMinAbs\nn : ℕ := m.natAbs\nhnat₂ : n ≤ p / 2\nhnat₃ : p ≥ 2 * n\nk : ℕ := p - 2 * n\nhnat₄ : k = p - 2 * n\nx : ℕ\nhx : n ^ 2 + 1 = p * x\nthis : ↑p ∣ ↑k ^ 2 + 4\n⊢ p ∣ k ^ 2 + 4"
},
{
"line": "with_annotate_state\"by\"\n ( use (p : ℤ) - 4 * n + 4 * x\n have hcast₁ : (k : ℤ) = p - 2 * n := by assumption_mod_cast\n have hcast₂ : (n : ℤ) ^ 2 + 1 = p * x := by assumption_mod_cast\n linear_combination ((k : ℤ) + p - 2 * n) * hcast₁ + 4 * hcast₂)",
"before_state": "p : ℕ\nhpp : Nat.Prime p\nhp_mod_4_eq_1 : p ≡ 1 [MOD 4]\nhp_gt_20 : p > 20\nthis : Fact (Nat.Prime p)\nhp_mod_4_ne_3 : p % 4 ≠ 3\ny : ZMod p\nhy : -1 = y * y\nm : ℤ := y.valMinAbs\nn : ℕ := m.natAbs\nhnat₂ : n ≤ p / 2\nhnat₃ : p ≥ 2 * n\nk : ℕ := p - 2 * n\nhnat₄ : k = p - 2 * n\nx : ℕ\nhx : n ^ 2 + 1 = p * x\n⊢ ↑p ∣ ↑k ^ 2 + 4",
"after_state": "No Goals!"
},
{
"line": "use (p : ℤ) - 4 * n + 4 * x",
"before_state": "p : ℕ\nhpp : Nat.Prime p\nhp_mod_4_eq_1 : p ≡ 1 [MOD 4]\nhp_gt_20 : p > 20\nthis : Fact (Nat.Prime p)\nhp_mod_4_ne_3 : p % 4 ≠ 3\ny : ZMod p\nhy : -1 = y * y\nm : ℤ := y.valMinAbs\nn : ℕ := m.natAbs\nhnat₂ : n ≤ p / 2\nhnat₃ : p ≥ 2 * n\nk : ℕ := p - 2 * n\nhnat₄ : k = p - 2 * n\nx : ℕ\nhx : n ^ 2 + 1 = p * x\n⊢ ↑p ∣ ↑k ^ 2 + 4",
"after_state": "case h\np : ℕ\nhpp : Nat.Prime p\nhp_mod_4_eq_1 : p ≡ 1 [MOD 4]\nhp_gt_20 : p > 20\nthis : Fact (Nat.Prime p)\nhp_mod_4_ne_3 : p % 4 ≠ 3\ny : ZMod p\nhy : -1 = y * y\nm : ℤ := y.valMinAbs\nn : ℕ := m.natAbs\nhnat₂ : n ≤ p / 2\nhnat₃ : p ≥ 2 * n\nk : ℕ := p - 2 * n\nhnat₄ : k = p - 2 * n\nx : ℕ\nhx : n ^ 2 + 1 = p * x\n⊢ ↑k ^ 2 + 4 = ↑p * (↑p - 4 * ↑n + 4 * ↑x)"
},
{
"line": "refine without_cdot((p : ℤ) - 4 * n + 4 * x : ?m✝)",
"before_state": "case w\np : ℕ\nhpp : Nat.Prime p\nhp_mod_4_eq_1 : p ≡ 1 [MOD 4]\nhp_gt_20 : p > 20\nthis : Fact (Nat.Prime p)\nhp_mod_4_ne_3 : p % 4 ≠ 3\ny : ZMod p\nhy : -1 = y * y\nm : ℤ := y.valMinAbs\nn : ℕ := m.natAbs\nhnat₂ : n ≤ p / 2\nhnat₃ : p ≥ 2 * n\nk : ℕ := p - 2 * n\nhnat₄ : k = p - 2 * n\nx : ℕ\nhx : n ^ 2 + 1 = p * x\n⊢ ℤ",
"after_state": "No Goals!"
},
{
"line": "try with_reducible use_discharger",
"before_state": "case h\np : ℕ\nhpp : Nat.Prime p\nhp_mod_4_eq_1 : p ≡ 1 [MOD 4]\nhp_gt_20 : p > 20\nthis : Fact (Nat.Prime p)\nhp_mod_4_ne_3 : p % 4 ≠ 3\ny : ZMod p\nhy : -1 = y * y\nm : ℤ := y.valMinAbs\nn : ℕ := m.natAbs\nhnat₂ : n ≤ p / 2\nhnat₃ : p ≥ 2 * n\nk : ℕ := p - 2 * n\nhnat₄ : k = p - 2 * n\nx : ℕ\nhx : n ^ 2 + 1 = p * x\n⊢ ↑k ^ 2 + 4 = ↑p * (↑p - 4 * ↑n + 4 * ↑x)",
"after_state": "case h\np : ℕ\nhpp : Nat.Prime p\nhp_mod_4_eq_1 : p ≡ 1 [MOD 4]\nhp_gt_20 : p > 20\nthis : Fact (Nat.Prime p)\nhp_mod_4_ne_3 : p % 4 ≠ 3\ny : ZMod p\nhy : -1 = y * y\nm : ℤ := y.valMinAbs\nn : ℕ := m.natAbs\nhnat₂ : n ≤ p / 2\nhnat₃ : p ≥ 2 * n\nk : ℕ := p - 2 * n\nhnat₄ : k = p - 2 * n\nx : ℕ\nhx : n ^ 2 + 1 = p * x\n⊢ ↑k ^ 2 + 4 = ↑p * (↑p - 4 * ↑n + 4 * ↑x)"
},
{
"line": "first\n| with_reducible use_discharger\n| skip",
"before_state": "case h\np : ℕ\nhpp : Nat.Prime p\nhp_mod_4_eq_1 : p ≡ 1 [MOD 4]\nhp_gt_20 : p > 20\nthis : Fact (Nat.Prime p)\nhp_mod_4_ne_3 : p % 4 ≠ 3\ny : ZMod p\nhy : -1 = y * y\nm : ℤ := y.valMinAbs\nn : ℕ := m.natAbs\nhnat₂ : n ≤ p / 2\nhnat₃ : p ≥ 2 * n\nk : ℕ := p - 2 * n\nhnat₄ : k = p - 2 * n\nx : ℕ\nhx : n ^ 2 + 1 = p * x\n⊢ ↑k ^ 2 + 4 = ↑p * (↑p - 4 * ↑n + 4 * ↑x)",
"after_state": "case h\np : ℕ\nhpp : Nat.Prime p\nhp_mod_4_eq_1 : p ≡ 1 [MOD 4]\nhp_gt_20 : p > 20\nthis : Fact (Nat.Prime p)\nhp_mod_4_ne_3 : p % 4 ≠ 3\ny : ZMod p\nhy : -1 = y * y\nm : ℤ := y.valMinAbs\nn : ℕ := m.natAbs\nhnat₂ : n ≤ p / 2\nhnat₃ : p ≥ 2 * n\nk : ℕ := p - 2 * n\nhnat₄ : k = p - 2 * n\nx : ℕ\nhx : n ^ 2 + 1 = p * x\n⊢ ↑k ^ 2 + 4 = ↑p * (↑p - 4 * ↑n + 4 * ↑x)"
},
{
"line": "with_reducible use_discharger",
"before_state": "case h\np : ℕ\nhpp : Nat.Prime p\nhp_mod_4_eq_1 : p ≡ 1 [MOD 4]\nhp_gt_20 : p > 20\nthis : Fact (Nat.Prime p)\nhp_mod_4_ne_3 : p % 4 ≠ 3\ny : ZMod p\nhy : -1 = y * y\nm : ℤ := y.valMinAbs\nn : ℕ := m.natAbs\nhnat₂ : n ≤ p / 2\nhnat₃ : p ≥ 2 * n\nk : ℕ := p - 2 * n\nhnat₄ : k = p - 2 * n\nx : ℕ\nhx : n ^ 2 + 1 = p * x\n⊢ ↑k ^ 2 + 4 = ↑p * (↑p - 4 * ↑n + 4 * ↑x)",
"after_state": "case h\np : ℕ\nhpp : Nat.Prime p\nhp_mod_4_eq_1 : p ≡ 1 [MOD 4]\nhp_gt_20 : p > 20\nthis : Fact (Nat.Prime p)\nhp_mod_4_ne_3 : p % 4 ≠ 3\ny : ZMod p\nhy : -1 = y * y\nm : ℤ := y.valMinAbs\nn : ℕ := m.natAbs\nhnat₂ : n ≤ p / 2\nhnat₃ : p ≥ 2 * n\nk : ℕ := p - 2 * n\nhnat₄ : k = p - 2 * n\nx : ℕ\nhx : n ^ 2 + 1 = p * x\n⊢ ↑k ^ 2 + 4 = ↑p * (↑p - 4 * ↑n + 4 * ↑x)"
},
{
"line": "use_discharger",
"before_state": "case h\np : ℕ\nhpp : Nat.Prime p\nhp_mod_4_eq_1 : p ≡ 1 [MOD 4]\nhp_gt_20 : p > 20\nthis : Fact (Nat.Prime p)\nhp_mod_4_ne_3 : p % 4 ≠ 3\ny : ZMod p\nhy : -1 = y * y\nm : ℤ := y.valMinAbs\nn : ℕ := m.natAbs\nhnat₂ : n ≤ p / 2\nhnat₃ : p ≥ 2 * n\nk : ℕ := p - 2 * n\nhnat₄ : k = p - 2 * n\nx : ℕ\nhx : n ^ 2 + 1 = p * x\n⊢ ↑k ^ 2 + 4 = ↑p * (↑p - 4 * ↑n + 4 * ↑x)",
"after_state": "case h\np : ℕ\nhpp : Nat.Prime p\nhp_mod_4_eq_1 : p ≡ 1 [MOD 4]\nhp_gt_20 : p > 20\nthis : Fact (Nat.Prime p)\nhp_mod_4_ne_3 : p % 4 ≠ 3\ny : ZMod p\nhy : -1 = y * y\nm : ℤ := y.valMinAbs\nn : ℕ := m.natAbs\nhnat₂ : n ≤ p / 2\nhnat₃ : p ≥ 2 * n\nk : ℕ := p - 2 * n\nhnat₄ : k = p - 2 * n\nx : ℕ\nhx : n ^ 2 + 1 = p * x\n⊢ ↑k ^ 2 + 4 = ↑p * (↑p - 4 * ↑n + 4 * ↑x)"
},
{
"line": "focus\n apply exists_prop.mpr✝\n with_annotate_state\"<;>\" skip\n all_goals use_discharger",
"before_state": "case h\np : ℕ\nhpp : Nat.Prime p\nhp_mod_4_eq_1 : p ≡ 1 [MOD 4]\nhp_gt_20 : p > 20\nthis : Fact (Nat.Prime p)\nhp_mod_4_ne_3 : p % 4 ≠ 3\ny : ZMod p\nhy : -1 = y * y\nm : ℤ := y.valMinAbs\nn : ℕ := m.natAbs\nhnat₂ : n ≤ p / 2\nhnat₃ : p ≥ 2 * n\nk : ℕ := p - 2 * n\nhnat₄ : k = p - 2 * n\nx : ℕ\nhx : n ^ 2 + 1 = p * x\n⊢ ↑k ^ 2 + 4 = ↑p * (↑p - 4 * ↑n + 4 * ↑x)",
"after_state": "case h\np : ℕ\nhpp : Nat.Prime p\nhp_mod_4_eq_1 : p ≡ 1 [MOD 4]\nhp_gt_20 : p > 20\nthis : Fact (Nat.Prime p)\nhp_mod_4_ne_3 : p % 4 ≠ 3\ny : ZMod p\nhy : -1 = y * y\nm : ℤ := y.valMinAbs\nn : ℕ := m.natAbs\nhnat₂ : n ≤ p / 2\nhnat₃ : p ≥ 2 * n\nk : ℕ := p - 2 * n\nhnat₄ : k = p - 2 * n\nx : ℕ\nhx : n ^ 2 + 1 = p * x\n⊢ ↑k ^ 2 + 4 = ↑p * (↑p - 4 * ↑n + 4 * ↑x)"
},
{
"line": "apply exists_prop.mpr✝",
"before_state": "case h\np : ℕ\nhpp : Nat.Prime p\nhp_mod_4_eq_1 : p ≡ 1 [MOD 4]\nhp_gt_20 : p > 20\nthis : Fact (Nat.Prime p)\nhp_mod_4_ne_3 : p % 4 ≠ 3\ny : ZMod p\nhy : -1 = y * y\nm : ℤ := y.valMinAbs\nn : ℕ := m.natAbs\nhnat₂ : n ≤ p / 2\nhnat₃ : p ≥ 2 * n\nk : ℕ := p - 2 * n\nhnat₄ : k = p - 2 * n\nx : ℕ\nhx : n ^ 2 + 1 = p * x\n⊢ ↑k ^ 2 + 4 = ↑p * (↑p - 4 * ↑n + 4 * ↑x)",
"after_state": "case h\np : ℕ\nhpp : Nat.Prime p\nhp_mod_4_eq_1 : p ≡ 1 [MOD 4]\nhp_gt_20 : p > 20\nthis : Fact (Nat.Prime p)\nhp_mod_4_ne_3 : p % 4 ≠ 3\ny : ZMod p\nhy : -1 = y * y\nm : ℤ := y.valMinAbs\nn : ℕ := m.natAbs\nhnat₂ : n ≤ p / 2\nhnat₃ : p ≥ 2 * n\nk : ℕ := p - 2 * n\nhnat₄ : k = p - 2 * n\nx : ℕ\nhx : n ^ 2 + 1 = p * x\n⊢ ↑k ^ 2 + 4 = ↑p * (↑p - 4 * ↑n + 4 * ↑x)"
},
{
"line": "skip",
"before_state": "case h\np : ℕ\nhpp : Nat.Prime p\nhp_mod_4_eq_1 : p ≡ 1 [MOD 4]\nhp_gt_20 : p > 20\nthis : Fact (Nat.Prime p)\nhp_mod_4_ne_3 : p % 4 ≠ 3\ny : ZMod p\nhy : -1 = y * y\nm : ℤ := y.valMinAbs\nn : ℕ := m.natAbs\nhnat₂ : n ≤ p / 2\nhnat₃ : p ≥ 2 * n\nk : ℕ := p - 2 * n\nhnat₄ : k = p - 2 * n\nx : ℕ\nhx : n ^ 2 + 1 = p * x\n⊢ ↑k ^ 2 + 4 = ↑p * (↑p - 4 * ↑n + 4 * ↑x)",
"after_state": "case h\np : ℕ\nhpp : Nat.Prime p\nhp_mod_4_eq_1 : p ≡ 1 [MOD 4]\nhp_gt_20 : p > 20\nthis : Fact (Nat.Prime p)\nhp_mod_4_ne_3 : p % 4 ≠ 3\ny : ZMod p\nhy : -1 = y * y\nm : ℤ := y.valMinAbs\nn : ℕ := m.natAbs\nhnat₂ : n ≤ p / 2\nhnat₃ : p ≥ 2 * n\nk : ℕ := p - 2 * n\nhnat₄ : k = p - 2 * n\nx : ℕ\nhx : n ^ 2 + 1 = p * x\n⊢ ↑k ^ 2 + 4 = ↑p * (↑p - 4 * ↑n + 4 * ↑x)"
},
{
"line": "have hcast₁ : (k : ℤ) = p - 2 * n := by assumption_mod_cast",
"before_state": "case h\np : ℕ\nhpp : Nat.Prime p\nhp_mod_4_eq_1 : p ≡ 1 [MOD 4]\nhp_gt_20 : p > 20\nthis : Fact (Nat.Prime p)\nhp_mod_4_ne_3 : p % 4 ≠ 3\ny : ZMod p\nhy : -1 = y * y\nm : ℤ := y.valMinAbs\nn : ℕ := m.natAbs\nhnat₂ : n ≤ p / 2\nhnat₃ : p ≥ 2 * n\nk : ℕ := p - 2 * n\nhnat₄ : k = p - 2 * n\nx : ℕ\nhx : n ^ 2 + 1 = p * x\n⊢ ↑k ^ 2 + 4 = ↑p * (↑p - 4 * ↑n + 4 * ↑x)",
"after_state": "case h\np : ℕ\nhpp : Nat.Prime p\nhp_mod_4_eq_1 : p ≡ 1 [MOD 4]\nhp_gt_20 : p > 20\nthis : Fact (Nat.Prime p)\nhp_mod_4_ne_3 : p % 4 ≠ 3\ny : ZMod p\nhy : -1 = y * y\nm : ℤ := y.valMinAbs\nn : ℕ := m.natAbs\nhnat₂ : n ≤ p / 2\nhnat₃ : p ≥ 2 * n\nk : ℕ := p - 2 * n\nhnat₄ : k = p - 2 * n\nx : ℕ\nhx : n ^ 2 + 1 = p * x\nhcast₁ : ↑k = ↑p - 2 * ↑n\n⊢ ↑k ^ 2 + 4 = ↑p * (↑p - 4 * ↑n + 4 * ↑x)"
},
{
"line": "focus\n refine\n no_implicit_lambda%\n (have hcast₁ : (k : ℤ) = p - 2 * n := ?body✝;\n ?_)\n case body✝ => with_annotate_state\"by\" (assumption_mod_cast)",
"before_state": "case h\np : ℕ\nhpp : Nat.Prime p\nhp_mod_4_eq_1 : p ≡ 1 [MOD 4]\nhp_gt_20 : p > 20\nthis : Fact (Nat.Prime p)\nhp_mod_4_ne_3 : p % 4 ≠ 3\ny : ZMod p\nhy : -1 = y * y\nm : ℤ := y.valMinAbs\nn : ℕ := m.natAbs\nhnat₂ : n ≤ p / 2\nhnat₃ : p ≥ 2 * n\nk : ℕ := p - 2 * n\nhnat₄ : k = p - 2 * n\nx : ℕ\nhx : n ^ 2 + 1 = p * x\n⊢ ↑k ^ 2 + 4 = ↑p * (↑p - 4 * ↑n + 4 * ↑x)",
"after_state": "case h\np : ℕ\nhpp : Nat.Prime p\nhp_mod_4_eq_1 : p ≡ 1 [MOD 4]\nhp_gt_20 : p > 20\nthis : Fact (Nat.Prime p)\nhp_mod_4_ne_3 : p % 4 ≠ 3\ny : ZMod p\nhy : -1 = y * y\nm : ℤ := y.valMinAbs\nn : ℕ := m.natAbs\nhnat₂ : n ≤ p / 2\nhnat₃ : p ≥ 2 * n\nk : ℕ := p - 2 * n\nhnat₄ : k = p - 2 * n\nx : ℕ\nhx : n ^ 2 + 1 = p * x\nhcast₁ : ↑k = ↑p - 2 * ↑n\n⊢ ↑k ^ 2 + 4 = ↑p * (↑p - 4 * ↑n + 4 * ↑x)"
},
{
"line": "refine\n no_implicit_lambda%\n (have hcast₁ : (k : ℤ) = p - 2 * n := ?body✝;\n ?_)",
"before_state": "case h\np : ℕ\nhpp : Nat.Prime p\nhp_mod_4_eq_1 : p ≡ 1 [MOD 4]\nhp_gt_20 : p > 20\nthis : Fact (Nat.Prime p)\nhp_mod_4_ne_3 : p % 4 ≠ 3\ny : ZMod p\nhy : -1 = y * y\nm : ℤ := y.valMinAbs\nn : ℕ := m.natAbs\nhnat₂ : n ≤ p / 2\nhnat₃ : p ≥ 2 * n\nk : ℕ := p - 2 * n\nhnat₄ : k = p - 2 * n\nx : ℕ\nhx : n ^ 2 + 1 = p * x\n⊢ ↑k ^ 2 + 4 = ↑p * (↑p - 4 * ↑n + 4 * ↑x)",
"after_state": "case body\np : ℕ\nhpp : Nat.Prime p\nhp_mod_4_eq_1 : p ≡ 1 [MOD 4]\nhp_gt_20 : p > 20\nthis : Fact (Nat.Prime p)\nhp_mod_4_ne_3 : p % 4 ≠ 3\ny : ZMod p\nhy : -1 = y * y\nm : ℤ := y.valMinAbs\nn : ℕ := m.natAbs\nhnat₂ : n ≤ p / 2\nhnat₃ : p ≥ 2 * n\nk : ℕ := p - 2 * n\nhnat₄ : k = p - 2 * n\nx : ℕ\nhx : n ^ 2 + 1 = p * x\n⊢ ↑k = ↑p - 2 * ↑n\n---\ncase h\np : ℕ\nhpp : Nat.Prime p\nhp_mod_4_eq_1 : p ≡ 1 [MOD 4]\nhp_gt_20 : p > 20\nthis : Fact (Nat.Prime p)\nhp_mod_4_ne_3 : p % 4 ≠ 3\ny : ZMod p\nhy : -1 = y * y\nm : ℤ := y.valMinAbs\nn : ℕ := m.natAbs\nhnat₂ : n ≤ p / 2\nhnat₃ : p ≥ 2 * n\nk : ℕ := p - 2 * n\nhnat₄ : k = p - 2 * n\nx : ℕ\nhx : n ^ 2 + 1 = p * x\nhcast₁ : ↑k = ↑p - 2 * ↑n\n⊢ ↑k ^ 2 + 4 = ↑p * (↑p - 4 * ↑n + 4 * ↑x)"
},
{
"line": "case body✝ => with_annotate_state\"by\" (assumption_mod_cast)",
"before_state": "case body\np : ℕ\nhpp : Nat.Prime p\nhp_mod_4_eq_1 : p ≡ 1 [MOD 4]\nhp_gt_20 : p > 20\nthis : Fact (Nat.Prime p)\nhp_mod_4_ne_3 : p % 4 ≠ 3\ny : ZMod p\nhy : -1 = y * y\nm : ℤ := y.valMinAbs\nn : ℕ := m.natAbs\nhnat₂ : n ≤ p / 2\nhnat₃ : p ≥ 2 * n\nk : ℕ := p - 2 * n\nhnat₄ : k = p - 2 * n\nx : ℕ\nhx : n ^ 2 + 1 = p * x\n⊢ ↑k = ↑p - 2 * ↑n\n---\ncase h\np : ℕ\nhpp : Nat.Prime p\nhp_mod_4_eq_1 : p ≡ 1 [MOD 4]\nhp_gt_20 : p > 20\nthis : Fact (Nat.Prime p)\nhp_mod_4_ne_3 : p % 4 ≠ 3\ny : ZMod p\nhy : -1 = y * y\nm : ℤ := y.valMinAbs\nn : ℕ := m.natAbs\nhnat₂ : n ≤ p / 2\nhnat₃ : p ≥ 2 * n\nk : ℕ := p - 2 * n\nhnat₄ : k = p - 2 * n\nx : ℕ\nhx : n ^ 2 + 1 = p * x\nhcast₁ : ↑k = ↑p - 2 * ↑n\n⊢ ↑k ^ 2 + 4 = ↑p * (↑p - 4 * ↑n + 4 * ↑x)",
"after_state": "case h\np : ℕ\nhpp : Nat.Prime p\nhp_mod_4_eq_1 : p ≡ 1 [MOD 4]\nhp_gt_20 : p > 20\nthis : Fact (Nat.Prime p)\nhp_mod_4_ne_3 : p % 4 ≠ 3\ny : ZMod p\nhy : -1 = y * y\nm : ℤ := y.valMinAbs\nn : ℕ := m.natAbs\nhnat₂ : n ≤ p / 2\nhnat₃ : p ≥ 2 * n\nk : ℕ := p - 2 * n\nhnat₄ : k = p - 2 * n\nx : ℕ\nhx : n ^ 2 + 1 = p * x\nhcast₁ : ↑k = ↑p - 2 * ↑n\n⊢ ↑k ^ 2 + 4 = ↑p * (↑p - 4 * ↑n + 4 * ↑x)"
},
{
"line": "with_annotate_state\"by\" (assumption_mod_cast)",
"before_state": "p : ℕ\nhpp : Nat.Prime p\nhp_mod_4_eq_1 : p ≡ 1 [MOD 4]\nhp_gt_20 : p > 20\nthis : Fact (Nat.Prime p)\nhp_mod_4_ne_3 : p % 4 ≠ 3\ny : ZMod p\nhy : -1 = y * y\nm : ℤ := y.valMinAbs\nn : ℕ := m.natAbs\nhnat₂ : n ≤ p / 2\nhnat₃ : p ≥ 2 * n\nk : ℕ := p - 2 * n\nhnat₄ : k = p - 2 * n\nx : ℕ\nhx : n ^ 2 + 1 = p * x\n⊢ ↑k = ↑p - 2 * ↑n",
"after_state": "No Goals!"
},
{
"line": "assumption_mod_cast",
"before_state": "p : ℕ\nhpp : Nat.Prime p\nhp_mod_4_eq_1 : p ≡ 1 [MOD 4]\nhp_gt_20 : p > 20\nthis : Fact (Nat.Prime p)\nhp_mod_4_ne_3 : p % 4 ≠ 3\ny : ZMod p\nhy : -1 = y * y\nm : ℤ := y.valMinAbs\nn : ℕ := m.natAbs\nhnat₂ : n ≤ p / 2\nhnat₃ : p ≥ 2 * n\nk : ℕ := p - 2 * n\nhnat₄ : k = p - 2 * n\nx : ℕ\nhx : n ^ 2 + 1 = p * x\n⊢ ↑k = ↑p - 2 * ↑n",
"after_state": "No Goals!"
},
{
"line": "focus\n norm_cast0 at *\n with_annotate_state\"<;>\" skip\n all_goals assumption",
"before_state": "p : ℕ\nhpp : Nat.Prime p\nhp_mod_4_eq_1 : p ≡ 1 [MOD 4]\nhp_gt_20 : p > 20\nthis : Fact (Nat.Prime p)\nhp_mod_4_ne_3 : p % 4 ≠ 3\ny : ZMod p\nhy : -1 = y * y\nm : ℤ := y.valMinAbs\nn : ℕ := m.natAbs\nhnat₂ : n ≤ p / 2\nhnat₃ : p ≥ 2 * n\nk : ℕ := p - 2 * n\nhnat₄ : k = p - 2 * n\nx : ℕ\nhx : n ^ 2 + 1 = p * x\n⊢ ↑k = ↑p - 2 * ↑n",
"after_state": "No Goals!"
},
{
"line": "norm_cast0 at *",
"before_state": "p : ℕ\nhpp : Nat.Prime p\nhp_mod_4_eq_1 : p ≡ 1 [MOD 4]\nhp_gt_20 : p > 20\nthis : Fact (Nat.Prime p)\nhp_mod_4_ne_3 : p % 4 ≠ 3\ny : ZMod p\nhy : -1 = y * y\nm : ℤ := y.valMinAbs\nn : ℕ := m.natAbs\nhnat₂ : n ≤ p / 2\nhnat₃ : p ≥ 2 * n\nk : ℕ := p - 2 * n\nhnat₄ : k = p - 2 * n\nx : ℕ\nhx : n ^ 2 + 1 = p * x\n⊢ ↑k = ↑p - 2 * ↑n",
"after_state": "p : ℕ\nhpp : Nat.Prime p\nhp_mod_4_eq_1 : p ≡ 1 [MOD 4]\nhp_gt_20 : p > 20\nthis : Fact (Nat.Prime p)\nhp_mod_4_ne_3 : ¬p % 4 = 3\ny : ZMod p\nm : ℤ := y.valMinAbs\nn : ℕ := m.natAbs\nhnat₂ : n ≤ p / 2\nhnat₃ : p ≥ 2 * n\nk : ℕ := p - 2 * n\nhnat₄ : k = p - 2 * n\nx : ℕ\nhx : n ^ 2 + 1 = p * x\nhy : ↑(Int.negSucc 0) = y * y\n⊢ k = p - 2 * n"
},
{
"line": "with_annotate_state\"<;>\" skip",
"before_state": "p : ℕ\nhpp : Nat.Prime p\nhp_mod_4_eq_1 : p ≡ 1 [MOD 4]\nhp_gt_20 : p > 20\nthis : Fact (Nat.Prime p)\nhp_mod_4_ne_3 : ¬p % 4 = 3\ny : ZMod p\nm : ℤ := y.valMinAbs\nn : ℕ := m.natAbs\nhnat₂ : n ≤ p / 2\nhnat₃ : p ≥ 2 * n\nk : ℕ := p - 2 * n\nhnat₄ : k = p - 2 * n\nx : ℕ\nhx : n ^ 2 + 1 = p * x\nhy : ↑(Int.negSucc 0) = y * y\n⊢ k = p - 2 * n",
"after_state": "p : ℕ\nhpp : Nat.Prime p\nhp_mod_4_eq_1 : p ≡ 1 [MOD 4]\nhp_gt_20 : p > 20\nthis : Fact (Nat.Prime p)\nhp_mod_4_ne_3 : ¬p % 4 = 3\ny : ZMod p\nm : ℤ := y.valMinAbs\nn : ℕ := m.natAbs\nhnat₂ : n ≤ p / 2\nhnat₃ : p ≥ 2 * n\nk : ℕ := p - 2 * n\nhnat₄ : k = p - 2 * n\nx : ℕ\nhx : n ^ 2 + 1 = p * x\nhy : ↑(Int.negSucc 0) = y * y\n⊢ k = p - 2 * n"
},
{
"line": "skip",
"before_state": "p : ℕ\nhpp : Nat.Prime p\nhp_mod_4_eq_1 : p ≡ 1 [MOD 4]\nhp_gt_20 : p > 20\nthis : Fact (Nat.Prime p)\nhp_mod_4_ne_3 : ¬p % 4 = 3\ny : ZMod p\nm : ℤ := y.valMinAbs\nn : ℕ := m.natAbs\nhnat₂ : n ≤ p / 2\nhnat₃ : p ≥ 2 * n\nk : ℕ := p - 2 * n\nhnat₄ : k = p - 2 * n\nx : ℕ\nhx : n ^ 2 + 1 = p * x\nhy : ↑(Int.negSucc 0) = y * y\n⊢ k = p - 2 * n",
"after_state": "p : ℕ\nhpp : Nat.Prime p\nhp_mod_4_eq_1 : p ≡ 1 [MOD 4]\nhp_gt_20 : p > 20\nthis : Fact (Nat.Prime p)\nhp_mod_4_ne_3 : ¬p % 4 = 3\ny : ZMod p\nm : ℤ := y.valMinAbs\nn : ℕ := m.natAbs\nhnat₂ : n ≤ p / 2\nhnat₃ : p ≥ 2 * n\nk : ℕ := p - 2 * n\nhnat₄ : k = p - 2 * n\nx : ℕ\nhx : n ^ 2 + 1 = p * x\nhy : ↑(Int.negSucc 0) = y * y\n⊢ k = p - 2 * n"
},
{
"line": "all_goals assumption",
"before_state": "p : ℕ\nhpp : Nat.Prime p\nhp_mod_4_eq_1 : p ≡ 1 [MOD 4]\nhp_gt_20 : p > 20\nthis : Fact (Nat.Prime p)\nhp_mod_4_ne_3 : ¬p % 4 = 3\ny : ZMod p\nm : ℤ := y.valMinAbs\nn : ℕ := m.natAbs\nhnat₂ : n ≤ p / 2\nhnat₃ : p ≥ 2 * n\nk : ℕ := p - 2 * n\nhnat₄ : k = p - 2 * n\nx : ℕ\nhx : n ^ 2 + 1 = p * x\nhy : ↑(Int.negSucc 0) = y * y\n⊢ k = p - 2 * n",
"after_state": "No Goals!"
},
{
"line": "assumption",
"before_state": "p : ℕ\nhpp : Nat.Prime p\nhp_mod_4_eq_1 : p ≡ 1 [MOD 4]\nhp_gt_20 : p > 20\nthis : Fact (Nat.Prime p)\nhp_mod_4_ne_3 : ¬p % 4 = 3\ny : ZMod p\nm : ℤ := y.valMinAbs\nn : ℕ := m.natAbs\nhnat₂ : n ≤ p / 2\nhnat₃ : p ≥ 2 * n\nk : ℕ := p - 2 * n\nhnat₄ : k = p - 2 * n\nx : ℕ\nhx : n ^ 2 + 1 = p * x\nhy : ↑(Int.negSucc 0) = y * y\n⊢ k = p - 2 * n",
"after_state": "No Goals!"
},
{
"line": "have hcast₂ : (n : ℤ) ^ 2 + 1 = p * x := by assumption_mod_cast",
"before_state": "case h\np : ℕ\nhpp : Nat.Prime p\nhp_mod_4_eq_1 : p ≡ 1 [MOD 4]\nhp_gt_20 : p > 20\nthis : Fact (Nat.Prime p)\nhp_mod_4_ne_3 : p % 4 ≠ 3\ny : ZMod p\nhy : -1 = y * y\nm : ℤ := y.valMinAbs\nn : ℕ := m.natAbs\nhnat₂ : n ≤ p / 2\nhnat₃ : p ≥ 2 * n\nk : ℕ := p - 2 * n\nhnat₄ : k = p - 2 * n\nx : ℕ\nhx : n ^ 2 + 1 = p * x\nhcast₁ : ↑k = ↑p - 2 * ↑n\n⊢ ↑k ^ 2 + 4 = ↑p * (↑p - 4 * ↑n + 4 * ↑x)",
"after_state": "case h\np : ℕ\nhpp : Nat.Prime p\nhp_mod_4_eq_1 : p ≡ 1 [MOD 4]\nhp_gt_20 : p > 20\nthis : Fact (Nat.Prime p)\nhp_mod_4_ne_3 : p % 4 ≠ 3\ny : ZMod p\nhy : -1 = y * y\nm : ℤ := y.valMinAbs\nn : ℕ := m.natAbs\nhnat₂ : n ≤ p / 2\nhnat₃ : p ≥ 2 * n\nk : ℕ := p - 2 * n\nhnat₄ : k = p - 2 * n\nx : ℕ\nhx : n ^ 2 + 1 = p * x\nhcast₁ : ↑k = ↑p - 2 * ↑n\nhcast₂ : ↑n ^ 2 + 1 = ↑p * ↑x\n⊢ ↑k ^ 2 + 4 = ↑p * (↑p - 4 * ↑n + 4 * ↑x)"
},
{
"line": "focus\n refine\n no_implicit_lambda%\n (have hcast₂ : (n : ℤ) ^ 2 + 1 = p * x := ?body✝;\n ?_)\n case body✝ => with_annotate_state\"by\" (assumption_mod_cast)",
"before_state": "case h\np : ℕ\nhpp : Nat.Prime p\nhp_mod_4_eq_1 : p ≡ 1 [MOD 4]\nhp_gt_20 : p > 20\nthis : Fact (Nat.Prime p)\nhp_mod_4_ne_3 : p % 4 ≠ 3\ny : ZMod p\nhy : -1 = y * y\nm : ℤ := y.valMinAbs\nn : ℕ := m.natAbs\nhnat₂ : n ≤ p / 2\nhnat₃ : p ≥ 2 * n\nk : ℕ := p - 2 * n\nhnat₄ : k = p - 2 * n\nx : ℕ\nhx : n ^ 2 + 1 = p * x\nhcast₁ : ↑k = ↑p - 2 * ↑n\n⊢ ↑k ^ 2 + 4 = ↑p * (↑p - 4 * ↑n + 4 * ↑x)",
"after_state": "case h\np : ℕ\nhpp : Nat.Prime p\nhp_mod_4_eq_1 : p ≡ 1 [MOD 4]\nhp_gt_20 : p > 20\nthis : Fact (Nat.Prime p)\nhp_mod_4_ne_3 : p % 4 ≠ 3\ny : ZMod p\nhy : -1 = y * y\nm : ℤ := y.valMinAbs\nn : ℕ := m.natAbs\nhnat₂ : n ≤ p / 2\nhnat₃ : p ≥ 2 * n\nk : ℕ := p - 2 * n\nhnat₄ : k = p - 2 * n\nx : ℕ\nhx : n ^ 2 + 1 = p * x\nhcast₁ : ↑k = ↑p - 2 * ↑n\nhcast₂ : ↑n ^ 2 + 1 = ↑p * ↑x\n⊢ ↑k ^ 2 + 4 = ↑p * (↑p - 4 * ↑n + 4 * ↑x)"
},
{
"line": "refine\n no_implicit_lambda%\n (have hcast₂ : (n : ℤ) ^ 2 + 1 = p * x := ?body✝;\n ?_)",
"before_state": "case h\np : ℕ\nhpp : Nat.Prime p\nhp_mod_4_eq_1 : p ≡ 1 [MOD 4]\nhp_gt_20 : p > 20\nthis : Fact (Nat.Prime p)\nhp_mod_4_ne_3 : p % 4 ≠ 3\ny : ZMod p\nhy : -1 = y * y\nm : ℤ := y.valMinAbs\nn : ℕ := m.natAbs\nhnat₂ : n ≤ p / 2\nhnat₃ : p ≥ 2 * n\nk : ℕ := p - 2 * n\nhnat₄ : k = p - 2 * n\nx : ℕ\nhx : n ^ 2 + 1 = p * x\nhcast₁ : ↑k = ↑p - 2 * ↑n\n⊢ ↑k ^ 2 + 4 = ↑p * (↑p - 4 * ↑n + 4 * ↑x)",
"after_state": "case body\np : ℕ\nhpp : Nat.Prime p\nhp_mod_4_eq_1 : p ≡ 1 [MOD 4]\nhp_gt_20 : p > 20\nthis : Fact (Nat.Prime p)\nhp_mod_4_ne_3 : p % 4 ≠ 3\ny : ZMod p\nhy : -1 = y * y\nm : ℤ := y.valMinAbs\nn : ℕ := m.natAbs\nhnat₂ : n ≤ p / 2\nhnat₃ : p ≥ 2 * n\nk : ℕ := p - 2 * n\nhnat₄ : k = p - 2 * n\nx : ℕ\nhx : n ^ 2 + 1 = p * x\nhcast₁ : ↑k = ↑p - 2 * ↑n\n⊢ ↑n ^ 2 + 1 = ↑p * ↑x\n---\ncase h\np : ℕ\nhpp : Nat.Prime p\nhp_mod_4_eq_1 : p ≡ 1 [MOD 4]\nhp_gt_20 : p > 20\nthis : Fact (Nat.Prime p)\nhp_mod_4_ne_3 : p % 4 ≠ 3\ny : ZMod p\nhy : -1 = y * y\nm : ℤ := y.valMinAbs\nn : ℕ := m.natAbs\nhnat₂ : n ≤ p / 2\nhnat₃ : p ≥ 2 * n\nk : ℕ := p - 2 * n\nhnat₄ : k = p - 2 * n\nx : ℕ\nhx : n ^ 2 + 1 = p * x\nhcast₁ : ↑k = ↑p - 2 * ↑n\nhcast₂ : ↑n ^ 2 + 1 = ↑p * ↑x\n⊢ ↑k ^ 2 + 4 = ↑p * (↑p - 4 * ↑n + 4 * ↑x)"
},
{
"line": "case body✝ => with_annotate_state\"by\" (assumption_mod_cast)",
"before_state": "case body\np : ℕ\nhpp : Nat.Prime p\nhp_mod_4_eq_1 : p ≡ 1 [MOD 4]\nhp_gt_20 : p > 20\nthis : Fact (Nat.Prime p)\nhp_mod_4_ne_3 : p % 4 ≠ 3\ny : ZMod p\nhy : -1 = y * y\nm : ℤ := y.valMinAbs\nn : ℕ := m.natAbs\nhnat₂ : n ≤ p / 2\nhnat₃ : p ≥ 2 * n\nk : ℕ := p - 2 * n\nhnat₄ : k = p - 2 * n\nx : ℕ\nhx : n ^ 2 + 1 = p * x\nhcast₁ : ↑k = ↑p - 2 * ↑n\n⊢ ↑n ^ 2 + 1 = ↑p * ↑x\n---\ncase h\np : ℕ\nhpp : Nat.Prime p\nhp_mod_4_eq_1 : p ≡ 1 [MOD 4]\nhp_gt_20 : p > 20\nthis : Fact (Nat.Prime p)\nhp_mod_4_ne_3 : p % 4 ≠ 3\ny : ZMod p\nhy : -1 = y * y\nm : ℤ := y.valMinAbs\nn : ℕ := m.natAbs\nhnat₂ : n ≤ p / 2\nhnat₃ : p ≥ 2 * n\nk : ℕ := p - 2 * n\nhnat₄ : k = p - 2 * n\nx : ℕ\nhx : n ^ 2 + 1 = p * x\nhcast₁ : ↑k = ↑p - 2 * ↑n\nhcast₂ : ↑n ^ 2 + 1 = ↑p * ↑x\n⊢ ↑k ^ 2 + 4 = ↑p * (↑p - 4 * ↑n + 4 * ↑x)",
"after_state": "case h\np : ℕ\nhpp : Nat.Prime p\nhp_mod_4_eq_1 : p ≡ 1 [MOD 4]\nhp_gt_20 : p > 20\nthis : Fact (Nat.Prime p)\nhp_mod_4_ne_3 : p % 4 ≠ 3\ny : ZMod p\nhy : -1 = y * y\nm : ℤ := y.valMinAbs\nn : ℕ := m.natAbs\nhnat₂ : n ≤ p / 2\nhnat₃ : p ≥ 2 * n\nk : ℕ := p - 2 * n\nhnat₄ : k = p - 2 * n\nx : ℕ\nhx : n ^ 2 + 1 = p * x\nhcast₁ : ↑k = ↑p - 2 * ↑n\nhcast₂ : ↑n ^ 2 + 1 = ↑p * ↑x\n⊢ ↑k ^ 2 + 4 = ↑p * (↑p - 4 * ↑n + 4 * ↑x)"
},
{
"line": "with_annotate_state\"by\" (assumption_mod_cast)",
"before_state": "p : ℕ\nhpp : Nat.Prime p\nhp_mod_4_eq_1 : p ≡ 1 [MOD 4]\nhp_gt_20 : p > 20\nthis : Fact (Nat.Prime p)\nhp_mod_4_ne_3 : p % 4 ≠ 3\ny : ZMod p\nhy : -1 = y * y\nm : ℤ := y.valMinAbs\nn : ℕ := m.natAbs\nhnat₂ : n ≤ p / 2\nhnat₃ : p ≥ 2 * n\nk : ℕ := p - 2 * n\nhnat₄ : k = p - 2 * n\nx : ℕ\nhx : n ^ 2 + 1 = p * x\nhcast₁ : ↑k = ↑p - 2 * ↑n\n⊢ ↑n ^ 2 + 1 = ↑p * ↑x",
"after_state": "No Goals!"
},
{
"line": "assumption_mod_cast",
"before_state": "p : ℕ\nhpp : Nat.Prime p\nhp_mod_4_eq_1 : p ≡ 1 [MOD 4]\nhp_gt_20 : p > 20\nthis : Fact (Nat.Prime p)\nhp_mod_4_ne_3 : p % 4 ≠ 3\ny : ZMod p\nhy : -1 = y * y\nm : ℤ := y.valMinAbs\nn : ℕ := m.natAbs\nhnat₂ : n ≤ p / 2\nhnat₃ : p ≥ 2 * n\nk : ℕ := p - 2 * n\nhnat₄ : k = p - 2 * n\nx : ℕ\nhx : n ^ 2 + 1 = p * x\nhcast₁ : ↑k = ↑p - 2 * ↑n\n⊢ ↑n ^ 2 + 1 = ↑p * ↑x",
"after_state": "No Goals!"
},
{
"line": "focus\n norm_cast0 at *\n with_annotate_state\"<;>\" skip\n all_goals assumption",
"before_state": "p : ℕ\nhpp : Nat.Prime p\nhp_mod_4_eq_1 : p ≡ 1 [MOD 4]\nhp_gt_20 : p > 20\nthis : Fact (Nat.Prime p)\nhp_mod_4_ne_3 : p % 4 ≠ 3\ny : ZMod p\nhy : -1 = y * y\nm : ℤ := y.valMinAbs\nn : ℕ := m.natAbs\nhnat₂ : n ≤ p / 2\nhnat₃ : p ≥ 2 * n\nk : ℕ := p - 2 * n\nhnat₄ : k = p - 2 * n\nx : ℕ\nhx : n ^ 2 + 1 = p * x\nhcast₁ : ↑k = ↑p - 2 * ↑n\n⊢ ↑n ^ 2 + 1 = ↑p * ↑x",
"after_state": "No Goals!"
},
{
"line": "norm_cast0 at *",
"before_state": "p : ℕ\nhpp : Nat.Prime p\nhp_mod_4_eq_1 : p ≡ 1 [MOD 4]\nhp_gt_20 : p > 20\nthis : Fact (Nat.Prime p)\nhp_mod_4_ne_3 : p % 4 ≠ 3\ny : ZMod p\nhy : -1 = y * y\nm : ℤ := y.valMinAbs\nn : ℕ := m.natAbs\nhnat₂ : n ≤ p / 2\nhnat₃ : p ≥ 2 * n\nk : ℕ := p - 2 * n\nhnat₄ : k = p - 2 * n\nx : ℕ\nhx : n ^ 2 + 1 = p * x\nhcast₁ : ↑k = ↑p - 2 * ↑n\n⊢ ↑n ^ 2 + 1 = ↑p * ↑x",
"after_state": "p : ℕ\nhpp : Nat.Prime p\nhp_mod_4_eq_1 : p ≡ 1 [MOD 4]\nhp_gt_20 : p > 20\nthis : Fact (Nat.Prime p)\nhp_mod_4_ne_3 : ¬p % 4 = 3\ny : ZMod p\nm : ℤ := y.valMinAbs\nn : ℕ := m.natAbs\nhnat₂ : n ≤ p / 2\nhnat₃ : p ≥ 2 * n\nk : ℕ := p - 2 * n\nhnat₄ : k = p - 2 * n\nx : ℕ\nhx : n ^ 2 + 1 = p * x\nhy : ↑(Int.negSucc 0) = y * y\nhcast₁ : k = p - 2 * n\n⊢ n ^ 2 + 1 = p * x"
},
{
"line": "with_annotate_state\"<;>\" skip",
"before_state": "p : ℕ\nhpp : Nat.Prime p\nhp_mod_4_eq_1 : p ≡ 1 [MOD 4]\nhp_gt_20 : p > 20\nthis : Fact (Nat.Prime p)\nhp_mod_4_ne_3 : ¬p % 4 = 3\ny : ZMod p\nm : ℤ := y.valMinAbs\nn : ℕ := m.natAbs\nhnat₂ : n ≤ p / 2\nhnat₃ : p ≥ 2 * n\nk : ℕ := p - 2 * n\nhnat₄ : k = p - 2 * n\nx : ℕ\nhx : n ^ 2 + 1 = p * x\nhy : ↑(Int.negSucc 0) = y * y\nhcast₁ : k = p - 2 * n\n⊢ n ^ 2 + 1 = p * x",
"after_state": "p : ℕ\nhpp : Nat.Prime p\nhp_mod_4_eq_1 : p ≡ 1 [MOD 4]\nhp_gt_20 : p > 20\nthis : Fact (Nat.Prime p)\nhp_mod_4_ne_3 : ¬p % 4 = 3\ny : ZMod p\nm : ℤ := y.valMinAbs\nn : ℕ := m.natAbs\nhnat₂ : n ≤ p / 2\nhnat₃ : p ≥ 2 * n\nk : ℕ := p - 2 * n\nhnat₄ : k = p - 2 * n\nx : ℕ\nhx : n ^ 2 + 1 = p * x\nhy : ↑(Int.negSucc 0) = y * y\nhcast₁ : k = p - 2 * n\n⊢ n ^ 2 + 1 = p * x"
},
{
"line": "skip",
"before_state": "p : ℕ\nhpp : Nat.Prime p\nhp_mod_4_eq_1 : p ≡ 1 [MOD 4]\nhp_gt_20 : p > 20\nthis : Fact (Nat.Prime p)\nhp_mod_4_ne_3 : ¬p % 4 = 3\ny : ZMod p\nm : ℤ := y.valMinAbs\nn : ℕ := m.natAbs\nhnat₂ : n ≤ p / 2\nhnat₃ : p ≥ 2 * n\nk : ℕ := p - 2 * n\nhnat₄ : k = p - 2 * n\nx : ℕ\nhx : n ^ 2 + 1 = p * x\nhy : ↑(Int.negSucc 0) = y * y\nhcast₁ : k = p - 2 * n\n⊢ n ^ 2 + 1 = p * x",
"after_state": "p : ℕ\nhpp : Nat.Prime p\nhp_mod_4_eq_1 : p ≡ 1 [MOD 4]\nhp_gt_20 : p > 20\nthis : Fact (Nat.Prime p)\nhp_mod_4_ne_3 : ¬p % 4 = 3\ny : ZMod p\nm : ℤ := y.valMinAbs\nn : ℕ := m.natAbs\nhnat₂ : n ≤ p / 2\nhnat₃ : p ≥ 2 * n\nk : ℕ := p - 2 * n\nhnat₄ : k = p - 2 * n\nx : ℕ\nhx : n ^ 2 + 1 = p * x\nhy : ↑(Int.negSucc 0) = y * y\nhcast₁ : k = p - 2 * n\n⊢ n ^ 2 + 1 = p * x"
},
{
"line": "all_goals assumption",
"before_state": "p : ℕ\nhpp : Nat.Prime p\nhp_mod_4_eq_1 : p ≡ 1 [MOD 4]\nhp_gt_20 : p > 20\nthis : Fact (Nat.Prime p)\nhp_mod_4_ne_3 : ¬p % 4 = 3\ny : ZMod p\nm : ℤ := y.valMinAbs\nn : ℕ := m.natAbs\nhnat₂ : n ≤ p / 2\nhnat₃ : p ≥ 2 * n\nk : ℕ := p - 2 * n\nhnat₄ : k = p - 2 * n\nx : ℕ\nhx : n ^ 2 + 1 = p * x\nhy : ↑(Int.negSucc 0) = y * y\nhcast₁ : k = p - 2 * n\n⊢ n ^ 2 + 1 = p * x",
"after_state": "No Goals!"
},
{
"line": "assumption",
"before_state": "p : ℕ\nhpp : Nat.Prime p\nhp_mod_4_eq_1 : p ≡ 1 [MOD 4]\nhp_gt_20 : p > 20\nthis : Fact (Nat.Prime p)\nhp_mod_4_ne_3 : ¬p % 4 = 3\ny : ZMod p\nm : ℤ := y.valMinAbs\nn : ℕ := m.natAbs\nhnat₂ : n ≤ p / 2\nhnat₃ : p ≥ 2 * n\nk : ℕ := p - 2 * n\nhnat₄ : k = p - 2 * n\nx : ℕ\nhx : n ^ 2 + 1 = p * x\nhy : ↑(Int.negSucc 0) = y * y\nhcast₁ : k = p - 2 * n\n⊢ n ^ 2 + 1 = p * x",
"after_state": "No Goals!"
},
{
"line": "linear_combination ((k : ℤ) + p - 2 * n) * hcast₁ + 4 * hcast₂",
"before_state": "case h\np : ℕ\nhpp : Nat.Prime p\nhp_mod_4_eq_1 : p ≡ 1 [MOD 4]\nhp_gt_20 : p > 20\nthis : Fact (Nat.Prime p)\nhp_mod_4_ne_3 : p % 4 ≠ 3\ny : ZMod p\nhy : -1 = y * y\nm : ℤ := y.valMinAbs\nn : ℕ := m.natAbs\nhnat₂ : n ≤ p / 2\nhnat₃ : p ≥ 2 * n\nk : ℕ := p - 2 * n\nhnat₄ : k = p - 2 * n\nx : ℕ\nhx : n ^ 2 + 1 = p * x\nhcast₁ : ↑k = ↑p - 2 * ↑n\nhcast₂ : ↑n ^ 2 + 1 = ↑p * ↑x\n⊢ ↑k ^ 2 + 4 = ↑p * (↑p - 4 * ↑n + 4 * ↑x)",
"after_state": "No Goals!"
},
{
"line": "assumption_mod_cast",
"before_state": "case intro\np : ℕ\nhpp : Nat.Prime p\nhp_mod_4_eq_1 : p ≡ 1 [MOD 4]\nhp_gt_20 : p > 20\nthis✝ : Fact (Nat.Prime p)\nhp_mod_4_ne_3 : p % 4 ≠ 3\ny : ZMod p\nhy : -1 = y * y\nm : ℤ := y.valMinAbs\nn : ℕ := m.natAbs\nhnat₂ : n ≤ p / 2\nhnat₃ : p ≥ 2 * n\nk : ℕ := p - 2 * n\nhnat₄ : k = p - 2 * n\nx : ℕ\nhx : n ^ 2 + 1 = p * x\nthis : ↑p ∣ ↑k ^ 2 + 4\n⊢ p ∣ k ^ 2 + 4",
"after_state": "No Goals!"
},
{
"line": "focus\n norm_cast0 at *\n with_annotate_state\"<;>\" skip\n all_goals assumption",
"before_state": "case intro\np : ℕ\nhpp : Nat.Prime p\nhp_mod_4_eq_1 : p ≡ 1 [MOD 4]\nhp_gt_20 : p > 20\nthis✝ : Fact (Nat.Prime p)\nhp_mod_4_ne_3 : p % 4 ≠ 3\ny : ZMod p\nhy : -1 = y * y\nm : ℤ := y.valMinAbs\nn : ℕ := m.natAbs\nhnat₂ : n ≤ p / 2\nhnat₃ : p ≥ 2 * n\nk : ℕ := p - 2 * n\nhnat₄ : k = p - 2 * n\nx : ℕ\nhx : n ^ 2 + 1 = p * x\nthis : ↑p ∣ ↑k ^ 2 + 4\n⊢ p ∣ k ^ 2 + 4",
"after_state": "No Goals!"
},
{
"line": "norm_cast0 at *",
"before_state": "case intro\np : ℕ\nhpp : Nat.Prime p\nhp_mod_4_eq_1 : p ≡ 1 [MOD 4]\nhp_gt_20 : p > 20\nthis✝ : Fact (Nat.Prime p)\nhp_mod_4_ne_3 : p % 4 ≠ 3\ny : ZMod p\nhy : -1 = y * y\nm : ℤ := y.valMinAbs\nn : ℕ := m.natAbs\nhnat₂ : n ≤ p / 2\nhnat₃ : p ≥ 2 * n\nk : ℕ := p - 2 * n\nhnat₄ : k = p - 2 * n\nx : ℕ\nhx : n ^ 2 + 1 = p * x\nthis : ↑p ∣ ↑k ^ 2 + 4\n⊢ p ∣ k ^ 2 + 4",
"after_state": "case intro\np : ℕ\nhpp : Nat.Prime p\nhp_mod_4_eq_1 : p ≡ 1 [MOD 4]\nhp_gt_20 : p > 20\nthis✝ : Fact (Nat.Prime p)\nhp_mod_4_ne_3 : ¬p % 4 = 3\ny : ZMod p\nm : ℤ := y.valMinAbs\nn : ℕ := m.natAbs\nhnat₂ : n ≤ p / 2\nhnat₃ : p ≥ 2 * n\nk : ℕ := p - 2 * n\nhnat₄ : k = p - 2 * n\nx : ℕ\nhx : n ^ 2 + 1 = p * x\nhy : ↑(Int.negSucc 0) = y * y\nthis : p ∣ k ^ 2 + 4\n⊢ p ∣ k ^ 2 + 4"
},
{
"line": "with_annotate_state\"<;>\" skip",
"before_state": "case intro\np : ℕ\nhpp : Nat.Prime p\nhp_mod_4_eq_1 : p ≡ 1 [MOD 4]\nhp_gt_20 : p > 20\nthis✝ : Fact (Nat.Prime p)\nhp_mod_4_ne_3 : ¬p % 4 = 3\ny : ZMod p\nm : ℤ := y.valMinAbs\nn : ℕ := m.natAbs\nhnat₂ : n ≤ p / 2\nhnat₃ : p ≥ 2 * n\nk : ℕ := p - 2 * n\nhnat₄ : k = p - 2 * n\nx : ℕ\nhx : n ^ 2 + 1 = p * x\nhy : ↑(Int.negSucc 0) = y * y\nthis : p ∣ k ^ 2 + 4\n⊢ p ∣ k ^ 2 + 4",
"after_state": "case intro\np : ℕ\nhpp : Nat.Prime p\nhp_mod_4_eq_1 : p ≡ 1 [MOD 4]\nhp_gt_20 : p > 20\nthis✝ : Fact (Nat.Prime p)\nhp_mod_4_ne_3 : ¬p % 4 = 3\ny : ZMod p\nm : ℤ := y.valMinAbs\nn : ℕ := m.natAbs\nhnat₂ : n ≤ p / 2\nhnat₃ : p ≥ 2 * n\nk : ℕ := p - 2 * n\nhnat₄ : k = p - 2 * n\nx : ℕ\nhx : n ^ 2 + 1 = p * x\nhy : ↑(Int.negSucc 0) = y * y\nthis : p ∣ k ^ 2 + 4\n⊢ p ∣ k ^ 2 + 4"
},
{
"line": "skip",
"before_state": "case intro\np : ℕ\nhpp : Nat.Prime p\nhp_mod_4_eq_1 : p ≡ 1 [MOD 4]\nhp_gt_20 : p > 20\nthis✝ : Fact (Nat.Prime p)\nhp_mod_4_ne_3 : ¬p % 4 = 3\ny : ZMod p\nm : ℤ := y.valMinAbs\nn : ℕ := m.natAbs\nhnat₂ : n ≤ p / 2\nhnat₃ : p ≥ 2 * n\nk : ℕ := p - 2 * n\nhnat₄ : k = p - 2 * n\nx : ℕ\nhx : n ^ 2 + 1 = p * x\nhy : ↑(Int.negSucc 0) = y * y\nthis : p ∣ k ^ 2 + 4\n⊢ p ∣ k ^ 2 + 4",
"after_state": "case intro\np : ℕ\nhpp : Nat.Prime p\nhp_mod_4_eq_1 : p ≡ 1 [MOD 4]\nhp_gt_20 : p > 20\nthis✝ : Fact (Nat.Prime p)\nhp_mod_4_ne_3 : ¬p % 4 = 3\ny : ZMod p\nm : ℤ := y.valMinAbs\nn : ℕ := m.natAbs\nhnat₂ : n ≤ p / 2\nhnat₃ : p ≥ 2 * n\nk : ℕ := p - 2 * n\nhnat₄ : k = p - 2 * n\nx : ℕ\nhx : n ^ 2 + 1 = p * x\nhy : ↑(Int.negSucc 0) = y * y\nthis : p ∣ k ^ 2 + 4\n⊢ p ∣ k ^ 2 + 4"
},
{
"line": "all_goals assumption",
"before_state": "case intro\np : ℕ\nhpp : Nat.Prime p\nhp_mod_4_eq_1 : p ≡ 1 [MOD 4]\nhp_gt_20 : p > 20\nthis✝ : Fact (Nat.Prime p)\nhp_mod_4_ne_3 : ¬p % 4 = 3\ny : ZMod p\nm : ℤ := y.valMinAbs\nn : ℕ := m.natAbs\nhnat₂ : n ≤ p / 2\nhnat₃ : p ≥ 2 * n\nk : ℕ := p - 2 * n\nhnat₄ : k = p - 2 * n\nx : ℕ\nhx : n ^ 2 + 1 = p * x\nhy : ↑(Int.negSucc 0) = y * y\nthis : p ∣ k ^ 2 + 4\n⊢ p ∣ k ^ 2 + 4",
"after_state": "No Goals!"
},
{
"line": "assumption",
"before_state": "case intro\np : ℕ\nhpp : Nat.Prime p\nhp_mod_4_eq_1 : p ≡ 1 [MOD 4]\nhp_gt_20 : p > 20\nthis✝ : Fact (Nat.Prime p)\nhp_mod_4_ne_3 : ¬p % 4 = 3\ny : ZMod p\nm : ℤ := y.valMinAbs\nn : ℕ := m.natAbs\nhnat₂ : n ≤ p / 2\nhnat₃ : p ≥ 2 * n\nk : ℕ := p - 2 * n\nhnat₄ : k = p - 2 * n\nx : ℕ\nhx : n ^ 2 + 1 = p * x\nhy : ↑(Int.negSucc 0) = y * y\nthis : p ∣ k ^ 2 + 4\n⊢ p ∣ k ^ 2 + 4",
"after_state": "No Goals!"
},
{
"line": "have hnat₆ : k ^ 2 + 4 ≥ p := Nat.le_of_dvd (k ^ 2 + 3).succ_pos hnat₅",
"before_state": "case intro\np : ℕ\nhpp : Nat.Prime p\nhp_mod_4_eq_1 : p ≡ 1 [MOD 4]\nhp_gt_20 : p > 20\nthis : Fact (Nat.Prime p)\nhp_mod_4_ne_3 : p % 4 ≠ 3\ny : ZMod p\nhy : -1 = y * y\nm : ℤ := y.valMinAbs\nn : ℕ := m.natAbs\nhnat₁ : p ∣ n ^ 2 + 1\nhnat₂ : n ≤ p / 2\nhnat₃ : p ≥ 2 * n\nk : ℕ := p - 2 * n\nhnat₄ : k = p - 2 * n\nhnat₅ : p ∣ k ^ 2 + 4\n⊢ ∃ n, p ∣ n ^ 2 + 1 ∧ ↑p > 2 * ↑n + √(2 * ↑n)",
"after_state": "case intro\np : ℕ\nhpp : Nat.Prime p\nhp_mod_4_eq_1 : p ≡ 1 [MOD 4]\nhp_gt_20 : p > 20\nthis : Fact (Nat.Prime p)\nhp_mod_4_ne_3 : p % 4 ≠ 3\ny : ZMod p\nhy : -1 = y * y\nm : ℤ := y.valMinAbs\nn : ℕ := m.natAbs\nhnat₁ : p ∣ n ^ 2 + 1\nhnat₂ : n ≤ p / 2\nhnat₃ : p ≥ 2 * n\nk : ℕ := p - 2 * n\nhnat₄ : k = p - 2 * n\nhnat₅ : p ∣ k ^ 2 + 4\nhnat₆ : k ^ 2 + 4 ≥ p\n⊢ ∃ n, p ∣ n ^ 2 + 1 ∧ ↑p > 2 * ↑n + √(2 * ↑n)"
},
{
"line": "refine_lift\n have hnat₆ : k ^ 2 + 4 ≥ p := Nat.le_of_dvd (k ^ 2 + 3).succ_pos hnat₅;\n ?_",
"before_state": "case intro\np : ℕ\nhpp : Nat.Prime p\nhp_mod_4_eq_1 : p ≡ 1 [MOD 4]\nhp_gt_20 : p > 20\nthis : Fact (Nat.Prime p)\nhp_mod_4_ne_3 : p % 4 ≠ 3\ny : ZMod p\nhy : -1 = y * y\nm : ℤ := y.valMinAbs\nn : ℕ := m.natAbs\nhnat₁ : p ∣ n ^ 2 + 1\nhnat₂ : n ≤ p / 2\nhnat₃ : p ≥ 2 * n\nk : ℕ := p - 2 * n\nhnat₄ : k = p - 2 * n\nhnat₅ : p ∣ k ^ 2 + 4\n⊢ ∃ n, p ∣ n ^ 2 + 1 ∧ ↑p > 2 * ↑n + √(2 * ↑n)",
"after_state": "case intro\np : ℕ\nhpp : Nat.Prime p\nhp_mod_4_eq_1 : p ≡ 1 [MOD 4]\nhp_gt_20 : p > 20\nthis : Fact (Nat.Prime p)\nhp_mod_4_ne_3 : p % 4 ≠ 3\ny : ZMod p\nhy : -1 = y * y\nm : ℤ := y.valMinAbs\nn : ℕ := m.natAbs\nhnat₁ : p ∣ n ^ 2 + 1\nhnat₂ : n ≤ p / 2\nhnat₃ : p ≥ 2 * n\nk : ℕ := p - 2 * n\nhnat₄ : k = p - 2 * n\nhnat₅ : p ∣ k ^ 2 + 4\nhnat₆ : k ^ 2 + 4 ≥ p\n⊢ ∃ n, p ∣ n ^ 2 + 1 ∧ ↑p > 2 * ↑n + √(2 * ↑n)"
},
{
"line": "focus\n (refine\n no_implicit_lambda%\n (have hnat₆ : k ^ 2 + 4 ≥ p := Nat.le_of_dvd (k ^ 2 + 3).succ_pos hnat₅;\n ?_);\n rotate_right)",
"before_state": "case intro\np : ℕ\nhpp : Nat.Prime p\nhp_mod_4_eq_1 : p ≡ 1 [MOD 4]\nhp_gt_20 : p > 20\nthis : Fact (Nat.Prime p)\nhp_mod_4_ne_3 : p % 4 ≠ 3\ny : ZMod p\nhy : -1 = y * y\nm : ℤ := y.valMinAbs\nn : ℕ := m.natAbs\nhnat₁ : p ∣ n ^ 2 + 1\nhnat₂ : n ≤ p / 2\nhnat₃ : p ≥ 2 * n\nk : ℕ := p - 2 * n\nhnat₄ : k = p - 2 * n\nhnat₅ : p ∣ k ^ 2 + 4\n⊢ ∃ n, p ∣ n ^ 2 + 1 ∧ ↑p > 2 * ↑n + √(2 * ↑n)",
"after_state": "case intro\np : ℕ\nhpp : Nat.Prime p\nhp_mod_4_eq_1 : p ≡ 1 [MOD 4]\nhp_gt_20 : p > 20\nthis : Fact (Nat.Prime p)\nhp_mod_4_ne_3 : p % 4 ≠ 3\ny : ZMod p\nhy : -1 = y * y\nm : ℤ := y.valMinAbs\nn : ℕ := m.natAbs\nhnat₁ : p ∣ n ^ 2 + 1\nhnat₂ : n ≤ p / 2\nhnat₃ : p ≥ 2 * n\nk : ℕ := p - 2 * n\nhnat₄ : k = p - 2 * n\nhnat₅ : p ∣ k ^ 2 + 4\nhnat₆ : k ^ 2 + 4 ≥ p\n⊢ ∃ n, p ∣ n ^ 2 + 1 ∧ ↑p > 2 * ↑n + √(2 * ↑n)"
},
{
"line": "refine\n no_implicit_lambda%\n (have hnat₆ : k ^ 2 + 4 ≥ p := Nat.le_of_dvd (k ^ 2 + 3).succ_pos hnat₅;\n ?_)",
"before_state": "case intro\np : ℕ\nhpp : Nat.Prime p\nhp_mod_4_eq_1 : p ≡ 1 [MOD 4]\nhp_gt_20 : p > 20\nthis : Fact (Nat.Prime p)\nhp_mod_4_ne_3 : p % 4 ≠ 3\ny : ZMod p\nhy : -1 = y * y\nm : ℤ := y.valMinAbs\nn : ℕ := m.natAbs\nhnat₁ : p ∣ n ^ 2 + 1\nhnat₂ : n ≤ p / 2\nhnat₃ : p ≥ 2 * n\nk : ℕ := p - 2 * n\nhnat₄ : k = p - 2 * n\nhnat₅ : p ∣ k ^ 2 + 4\n⊢ ∃ n, p ∣ n ^ 2 + 1 ∧ ↑p > 2 * ↑n + √(2 * ↑n)",
"after_state": "case intro\np : ℕ\nhpp : Nat.Prime p\nhp_mod_4_eq_1 : p ≡ 1 [MOD 4]\nhp_gt_20 : p > 20\nthis : Fact (Nat.Prime p)\nhp_mod_4_ne_3 : p % 4 ≠ 3\ny : ZMod p\nhy : -1 = y * y\nm : ℤ := y.valMinAbs\nn : ℕ := m.natAbs\nhnat₁ : p ∣ n ^ 2 + 1\nhnat₂ : n ≤ p / 2\nhnat₃ : p ≥ 2 * n\nk : ℕ := p - 2 * n\nhnat₄ : k = p - 2 * n\nhnat₅ : p ∣ k ^ 2 + 4\nhnat₆ : k ^ 2 + 4 ≥ p\n⊢ ∃ n, p ∣ n ^ 2 + 1 ∧ ↑p > 2 * ↑n + √(2 * ↑n)"
},
{
"line": "rotate_right",
"before_state": "case intro\np : ℕ\nhpp : Nat.Prime p\nhp_mod_4_eq_1 : p ≡ 1 [MOD 4]\nhp_gt_20 : p > 20\nthis : Fact (Nat.Prime p)\nhp_mod_4_ne_3 : p % 4 ≠ 3\ny : ZMod p\nhy : -1 = y * y\nm : ℤ := y.valMinAbs\nn : ℕ := m.natAbs\nhnat₁ : p ∣ n ^ 2 + 1\nhnat₂ : n ≤ p / 2\nhnat₃ : p ≥ 2 * n\nk : ℕ := p - 2 * n\nhnat₄ : k = p - 2 * n\nhnat₅ : p ∣ k ^ 2 + 4\nhnat₆ : k ^ 2 + 4 ≥ p\n⊢ ∃ n, p ∣ n ^ 2 + 1 ∧ ↑p > 2 * ↑n + √(2 * ↑n)",
"after_state": "case intro\np : ℕ\nhpp : Nat.Prime p\nhp_mod_4_eq_1 : p ≡ 1 [MOD 4]\nhp_gt_20 : p > 20\nthis : Fact (Nat.Prime p)\nhp_mod_4_ne_3 : p % 4 ≠ 3\ny : ZMod p\nhy : -1 = y * y\nm : ℤ := y.valMinAbs\nn : ℕ := m.natAbs\nhnat₁ : p ∣ n ^ 2 + 1\nhnat₂ : n ≤ p / 2\nhnat₃ : p ≥ 2 * n\nk : ℕ := p - 2 * n\nhnat₄ : k = p - 2 * n\nhnat₅ : p ∣ k ^ 2 + 4\nhnat₆ : k ^ 2 + 4 ≥ p\n⊢ ∃ n, p ∣ n ^ 2 + 1 ∧ ↑p > 2 * ↑n + √(2 * ↑n)"
},
{
"line": "have hreal₁ : (k : ℝ) = p - 2 * n := by assumption_mod_cast",
"before_state": "case intro\np : ℕ\nhpp : Nat.Prime p\nhp_mod_4_eq_1 : p ≡ 1 [MOD 4]\nhp_gt_20 : p > 20\nthis : Fact (Nat.Prime p)\nhp_mod_4_ne_3 : p % 4 ≠ 3\ny : ZMod p\nhy : -1 = y * y\nm : ℤ := y.valMinAbs\nn : ℕ := m.natAbs\nhnat₁ : p ∣ n ^ 2 + 1\nhnat₂ : n ≤ p / 2\nhnat₃ : p ≥ 2 * n\nk : ℕ := p - 2 * n\nhnat₄ : k = p - 2 * n\nhnat₅ : p ∣ k ^ 2 + 4\nhnat₆ : k ^ 2 + 4 ≥ p\n⊢ ∃ n, p ∣ n ^ 2 + 1 ∧ ↑p > 2 * ↑n + √(2 * ↑n)",
"after_state": "case intro\np : ℕ\nhpp : Nat.Prime p\nhp_mod_4_eq_1 : p ≡ 1 [MOD 4]\nhp_gt_20 : p > 20\nthis : Fact (Nat.Prime p)\nhp_mod_4_ne_3 : p % 4 ≠ 3\ny : ZMod p\nhy : -1 = y * y\nm : ℤ := y.valMinAbs\nn : ℕ := m.natAbs\nhnat₁ : p ∣ n ^ 2 + 1\nhnat₂ : n ≤ p / 2\nhnat₃ : p ≥ 2 * n\nk : ℕ := p - 2 * n\nhnat₄ : k = p - 2 * n\nhnat₅ : p ∣ k ^ 2 + 4\nhnat₆ : k ^ 2 + 4 ≥ p\nhreal₁ : ↑k = ↑p - 2 * ↑n\n⊢ ∃ n, p ∣ n ^ 2 + 1 ∧ ↑p > 2 * ↑n + √(2 * ↑n)"
},
{
"line": "focus\n refine\n no_implicit_lambda%\n (have hreal₁ : (k : ℝ) = p - 2 * n := ?body✝;\n ?_)\n case body✝ => with_annotate_state\"by\" (assumption_mod_cast)",
"before_state": "case intro\np : ℕ\nhpp : Nat.Prime p\nhp_mod_4_eq_1 : p ≡ 1 [MOD 4]\nhp_gt_20 : p > 20\nthis : Fact (Nat.Prime p)\nhp_mod_4_ne_3 : p % 4 ≠ 3\ny : ZMod p\nhy : -1 = y * y\nm : ℤ := y.valMinAbs\nn : ℕ := m.natAbs\nhnat₁ : p ∣ n ^ 2 + 1\nhnat₂ : n ≤ p / 2\nhnat₃ : p ≥ 2 * n\nk : ℕ := p - 2 * n\nhnat₄ : k = p - 2 * n\nhnat₅ : p ∣ k ^ 2 + 4\nhnat₆ : k ^ 2 + 4 ≥ p\n⊢ ∃ n, p ∣ n ^ 2 + 1 ∧ ↑p > 2 * ↑n + √(2 * ↑n)",
"after_state": "case intro\np : ℕ\nhpp : Nat.Prime p\nhp_mod_4_eq_1 : p ≡ 1 [MOD 4]\nhp_gt_20 : p > 20\nthis : Fact (Nat.Prime p)\nhp_mod_4_ne_3 : p % 4 ≠ 3\ny : ZMod p\nhy : -1 = y * y\nm : ℤ := y.valMinAbs\nn : ℕ := m.natAbs\nhnat₁ : p ∣ n ^ 2 + 1\nhnat₂ : n ≤ p / 2\nhnat₃ : p ≥ 2 * n\nk : ℕ := p - 2 * n\nhnat₄ : k = p - 2 * n\nhnat₅ : p ∣ k ^ 2 + 4\nhnat₆ : k ^ 2 + 4 ≥ p\nhreal₁ : ↑k = ↑p - 2 * ↑n\n⊢ ∃ n, p ∣ n ^ 2 + 1 ∧ ↑p > 2 * ↑n + √(2 * ↑n)"
},
{
"line": "refine\n no_implicit_lambda%\n (have hreal₁ : (k : ℝ) = p - 2 * n := ?body✝;\n ?_)",
"before_state": "case intro\np : ℕ\nhpp : Nat.Prime p\nhp_mod_4_eq_1 : p ≡ 1 [MOD 4]\nhp_gt_20 : p > 20\nthis : Fact (Nat.Prime p)\nhp_mod_4_ne_3 : p % 4 ≠ 3\ny : ZMod p\nhy : -1 = y * y\nm : ℤ := y.valMinAbs\nn : ℕ := m.natAbs\nhnat₁ : p ∣ n ^ 2 + 1\nhnat₂ : n ≤ p / 2\nhnat₃ : p ≥ 2 * n\nk : ℕ := p - 2 * n\nhnat₄ : k = p - 2 * n\nhnat₅ : p ∣ k ^ 2 + 4\nhnat₆ : k ^ 2 + 4 ≥ p\n⊢ ∃ n, p ∣ n ^ 2 + 1 ∧ ↑p > 2 * ↑n + √(2 * ↑n)",
"after_state": "case body\np : ℕ\nhpp : Nat.Prime p\nhp_mod_4_eq_1 : p ≡ 1 [MOD 4]\nhp_gt_20 : p > 20\nthis : Fact (Nat.Prime p)\nhp_mod_4_ne_3 : p % 4 ≠ 3\ny : ZMod p\nhy : -1 = y * y\nm : ℤ := y.valMinAbs\nn : ℕ := m.natAbs\nhnat₁ : p ∣ n ^ 2 + 1\nhnat₂ : n ≤ p / 2\nhnat₃ : p ≥ 2 * n\nk : ℕ := p - 2 * n\nhnat₄ : k = p - 2 * n\nhnat₅ : p ∣ k ^ 2 + 4\nhnat₆ : k ^ 2 + 4 ≥ p\n⊢ ↑k = ↑p - 2 * ↑n\n---\ncase intro\np : ℕ\nhpp : Nat.Prime p\nhp_mod_4_eq_1 : p ≡ 1 [MOD 4]\nhp_gt_20 : p > 20\nthis : Fact (Nat.Prime p)\nhp_mod_4_ne_3 : p % 4 ≠ 3\ny : ZMod p\nhy : -1 = y * y\nm : ℤ := y.valMinAbs\nn : ℕ := m.natAbs\nhnat₁ : p ∣ n ^ 2 + 1\nhnat₂ : n ≤ p / 2\nhnat₃ : p ≥ 2 * n\nk : ℕ := p - 2 * n\nhnat₄ : k = p - 2 * n\nhnat₅ : p ∣ k ^ 2 + 4\nhnat₆ : k ^ 2 + 4 ≥ p\nhreal₁ : ↑k = ↑p - 2 * ↑n\n⊢ ∃ n, p ∣ n ^ 2 + 1 ∧ ↑p > 2 * ↑n + √(2 * ↑n)"
},
{
"line": "case body✝ => with_annotate_state\"by\" (assumption_mod_cast)",
"before_state": "case body\np : ℕ\nhpp : Nat.Prime p\nhp_mod_4_eq_1 : p ≡ 1 [MOD 4]\nhp_gt_20 : p > 20\nthis : Fact (Nat.Prime p)\nhp_mod_4_ne_3 : p % 4 ≠ 3\ny : ZMod p\nhy : -1 = y * y\nm : ℤ := y.valMinAbs\nn : ℕ := m.natAbs\nhnat₁ : p ∣ n ^ 2 + 1\nhnat₂ : n ≤ p / 2\nhnat₃ : p ≥ 2 * n\nk : ℕ := p - 2 * n\nhnat₄ : k = p - 2 * n\nhnat₅ : p ∣ k ^ 2 + 4\nhnat₆ : k ^ 2 + 4 ≥ p\n⊢ ↑k = ↑p - 2 * ↑n\n---\ncase intro\np : ℕ\nhpp : Nat.Prime p\nhp_mod_4_eq_1 : p ≡ 1 [MOD 4]\nhp_gt_20 : p > 20\nthis : Fact (Nat.Prime p)\nhp_mod_4_ne_3 : p % 4 ≠ 3\ny : ZMod p\nhy : -1 = y * y\nm : ℤ := y.valMinAbs\nn : ℕ := m.natAbs\nhnat₁ : p ∣ n ^ 2 + 1\nhnat₂ : n ≤ p / 2\nhnat₃ : p ≥ 2 * n\nk : ℕ := p - 2 * n\nhnat₄ : k = p - 2 * n\nhnat₅ : p ∣ k ^ 2 + 4\nhnat₆ : k ^ 2 + 4 ≥ p\nhreal₁ : ↑k = ↑p - 2 * ↑n\n⊢ ∃ n, p ∣ n ^ 2 + 1 ∧ ↑p > 2 * ↑n + √(2 * ↑n)",
"after_state": "case intro\np : ℕ\nhpp : Nat.Prime p\nhp_mod_4_eq_1 : p ≡ 1 [MOD 4]\nhp_gt_20 : p > 20\nthis : Fact (Nat.Prime p)\nhp_mod_4_ne_3 : p % 4 ≠ 3\ny : ZMod p\nhy : -1 = y * y\nm : ℤ := y.valMinAbs\nn : ℕ := m.natAbs\nhnat₁ : p ∣ n ^ 2 + 1\nhnat₂ : n ≤ p / 2\nhnat₃ : p ≥ 2 * n\nk : ℕ := p - 2 * n\nhnat₄ : k = p - 2 * n\nhnat₅ : p ∣ k ^ 2 + 4\nhnat₆ : k ^ 2 + 4 ≥ p\nhreal₁ : ↑k = ↑p - 2 * ↑n\n⊢ ∃ n, p ∣ n ^ 2 + 1 ∧ ↑p > 2 * ↑n + √(2 * ↑n)"
},
{
"line": "with_annotate_state\"by\" (assumption_mod_cast)",
"before_state": "p : ℕ\nhpp : Nat.Prime p\nhp_mod_4_eq_1 : p ≡ 1 [MOD 4]\nhp_gt_20 : p > 20\nthis : Fact (Nat.Prime p)\nhp_mod_4_ne_3 : p % 4 ≠ 3\ny : ZMod p\nhy : -1 = y * y\nm : ℤ := y.valMinAbs\nn : ℕ := m.natAbs\nhnat₁ : p ∣ n ^ 2 + 1\nhnat₂ : n ≤ p / 2\nhnat₃ : p ≥ 2 * n\nk : ℕ := p - 2 * n\nhnat₄ : k = p - 2 * n\nhnat₅ : p ∣ k ^ 2 + 4\nhnat₆ : k ^ 2 + 4 ≥ p\n⊢ ↑k = ↑p - 2 * ↑n",
"after_state": "No Goals!"
},
{
"line": "assumption_mod_cast",
"before_state": "p : ℕ\nhpp : Nat.Prime p\nhp_mod_4_eq_1 : p ≡ 1 [MOD 4]\nhp_gt_20 : p > 20\nthis : Fact (Nat.Prime p)\nhp_mod_4_ne_3 : p % 4 ≠ 3\ny : ZMod p\nhy : -1 = y * y\nm : ℤ := y.valMinAbs\nn : ℕ := m.natAbs\nhnat₁ : p ∣ n ^ 2 + 1\nhnat₂ : n ≤ p / 2\nhnat₃ : p ≥ 2 * n\nk : ℕ := p - 2 * n\nhnat₄ : k = p - 2 * n\nhnat₅ : p ∣ k ^ 2 + 4\nhnat₆ : k ^ 2 + 4 ≥ p\n⊢ ↑k = ↑p - 2 * ↑n",
"after_state": "No Goals!"
},
{
"line": "focus\n norm_cast0 at *\n with_annotate_state\"<;>\" skip\n all_goals assumption",
"before_state": "p : ℕ\nhpp : Nat.Prime p\nhp_mod_4_eq_1 : p ≡ 1 [MOD 4]\nhp_gt_20 : p > 20\nthis : Fact (Nat.Prime p)\nhp_mod_4_ne_3 : p % 4 ≠ 3\ny : ZMod p\nhy : -1 = y * y\nm : ℤ := y.valMinAbs\nn : ℕ := m.natAbs\nhnat₁ : p ∣ n ^ 2 + 1\nhnat₂ : n ≤ p / 2\nhnat₃ : p ≥ 2 * n\nk : ℕ := p - 2 * n\nhnat₄ : k = p - 2 * n\nhnat₅ : p ∣ k ^ 2 + 4\nhnat₆ : k ^ 2 + 4 ≥ p\n⊢ ↑k = ↑p - 2 * ↑n",
"after_state": "No Goals!"
},
{
"line": "norm_cast0 at *",
"before_state": "p : ℕ\nhpp : Nat.Prime p\nhp_mod_4_eq_1 : p ≡ 1 [MOD 4]\nhp_gt_20 : p > 20\nthis : Fact (Nat.Prime p)\nhp_mod_4_ne_3 : p % 4 ≠ 3\ny : ZMod p\nhy : -1 = y * y\nm : ℤ := y.valMinAbs\nn : ℕ := m.natAbs\nhnat₁ : p ∣ n ^ 2 + 1\nhnat₂ : n ≤ p / 2\nhnat₃ : p ≥ 2 * n\nk : ℕ := p - 2 * n\nhnat₄ : k = p - 2 * n\nhnat₅ : p ∣ k ^ 2 + 4\nhnat₆ : k ^ 2 + 4 ≥ p\n⊢ ↑k = ↑p - 2 * ↑n",
"after_state": "p : ℕ\nhpp : Nat.Prime p\nhp_mod_4_eq_1 : p ≡ 1 [MOD 4]\nhp_gt_20 : p > 20\nthis : Fact (Nat.Prime p)\nhp_mod_4_ne_3 : ¬p % 4 = 3\ny : ZMod p\nm : ℤ := y.valMinAbs\nn : ℕ := m.natAbs\nhnat₁ : p ∣ n ^ 2 + 1\nhnat₂ : n ≤ p / 2\nhnat₃ : p ≥ 2 * n\nk : ℕ := p - 2 * n\nhnat₄ : k = p - 2 * n\nhnat₅ : p ∣ k ^ 2 + 4\nhnat₆ : k ^ 2 + 4 ≥ p\nhy : ↑(Int.negSucc 0) = y * y\n⊢ k = p - 2 * n"
},
{
"line": "with_annotate_state\"<;>\" skip",
"before_state": "p : ℕ\nhpp : Nat.Prime p\nhp_mod_4_eq_1 : p ≡ 1 [MOD 4]\nhp_gt_20 : p > 20\nthis : Fact (Nat.Prime p)\nhp_mod_4_ne_3 : ¬p % 4 = 3\ny : ZMod p\nm : ℤ := y.valMinAbs\nn : ℕ := m.natAbs\nhnat₁ : p ∣ n ^ 2 + 1\nhnat₂ : n ≤ p / 2\nhnat₃ : p ≥ 2 * n\nk : ℕ := p - 2 * n\nhnat₄ : k = p - 2 * n\nhnat₅ : p ∣ k ^ 2 + 4\nhnat₆ : k ^ 2 + 4 ≥ p\nhy : ↑(Int.negSucc 0) = y * y\n⊢ k = p - 2 * n",
"after_state": "p : ℕ\nhpp : Nat.Prime p\nhp_mod_4_eq_1 : p ≡ 1 [MOD 4]\nhp_gt_20 : p > 20\nthis : Fact (Nat.Prime p)\nhp_mod_4_ne_3 : ¬p % 4 = 3\ny : ZMod p\nm : ℤ := y.valMinAbs\nn : ℕ := m.natAbs\nhnat₁ : p ∣ n ^ 2 + 1\nhnat₂ : n ≤ p / 2\nhnat₃ : p ≥ 2 * n\nk : ℕ := p - 2 * n\nhnat₄ : k = p - 2 * n\nhnat₅ : p ∣ k ^ 2 + 4\nhnat₆ : k ^ 2 + 4 ≥ p\nhy : ↑(Int.negSucc 0) = y * y\n⊢ k = p - 2 * n"
},
{
"line": "skip",
"before_state": "p : ℕ\nhpp : Nat.Prime p\nhp_mod_4_eq_1 : p ≡ 1 [MOD 4]\nhp_gt_20 : p > 20\nthis : Fact (Nat.Prime p)\nhp_mod_4_ne_3 : ¬p % 4 = 3\ny : ZMod p\nm : ℤ := y.valMinAbs\nn : ℕ := m.natAbs\nhnat₁ : p ∣ n ^ 2 + 1\nhnat₂ : n ≤ p / 2\nhnat₃ : p ≥ 2 * n\nk : ℕ := p - 2 * n\nhnat₄ : k = p - 2 * n\nhnat₅ : p ∣ k ^ 2 + 4\nhnat₆ : k ^ 2 + 4 ≥ p\nhy : ↑(Int.negSucc 0) = y * y\n⊢ k = p - 2 * n",
"after_state": "p : ℕ\nhpp : Nat.Prime p\nhp_mod_4_eq_1 : p ≡ 1 [MOD 4]\nhp_gt_20 : p > 20\nthis : Fact (Nat.Prime p)\nhp_mod_4_ne_3 : ¬p % 4 = 3\ny : ZMod p\nm : ℤ := y.valMinAbs\nn : ℕ := m.natAbs\nhnat₁ : p ∣ n ^ 2 + 1\nhnat₂ : n ≤ p / 2\nhnat₃ : p ≥ 2 * n\nk : ℕ := p - 2 * n\nhnat₄ : k = p - 2 * n\nhnat₅ : p ∣ k ^ 2 + 4\nhnat₆ : k ^ 2 + 4 ≥ p\nhy : ↑(Int.negSucc 0) = y * y\n⊢ k = p - 2 * n"
},
{
"line": "all_goals assumption",
"before_state": "p : ℕ\nhpp : Nat.Prime p\nhp_mod_4_eq_1 : p ≡ 1 [MOD 4]\nhp_gt_20 : p > 20\nthis : Fact (Nat.Prime p)\nhp_mod_4_ne_3 : ¬p % 4 = 3\ny : ZMod p\nm : ℤ := y.valMinAbs\nn : ℕ := m.natAbs\nhnat₁ : p ∣ n ^ 2 + 1\nhnat₂ : n ≤ p / 2\nhnat₃ : p ≥ 2 * n\nk : ℕ := p - 2 * n\nhnat₄ : k = p - 2 * n\nhnat₅ : p ∣ k ^ 2 + 4\nhnat₆ : k ^ 2 + 4 ≥ p\nhy : ↑(Int.negSucc 0) = y * y\n⊢ k = p - 2 * n",
"after_state": "No Goals!"
},
{
"line": "assumption",
"before_state": "p : ℕ\nhpp : Nat.Prime p\nhp_mod_4_eq_1 : p ≡ 1 [MOD 4]\nhp_gt_20 : p > 20\nthis : Fact (Nat.Prime p)\nhp_mod_4_ne_3 : ¬p % 4 = 3\ny : ZMod p\nm : ℤ := y.valMinAbs\nn : ℕ := m.natAbs\nhnat₁ : p ∣ n ^ 2 + 1\nhnat₂ : n ≤ p / 2\nhnat₃ : p ≥ 2 * n\nk : ℕ := p - 2 * n\nhnat₄ : k = p - 2 * n\nhnat₅ : p ∣ k ^ 2 + 4\nhnat₆ : k ^ 2 + 4 ≥ p\nhy : ↑(Int.negSucc 0) = y * y\n⊢ k = p - 2 * n",
"after_state": "No Goals!"
},
{
"line": "have hreal₂ : (p : ℝ) > 20 := by assumption_mod_cast",
"before_state": "case intro\np : ℕ\nhpp : Nat.Prime p\nhp_mod_4_eq_1 : p ≡ 1 [MOD 4]\nhp_gt_20 : p > 20\nthis : Fact (Nat.Prime p)\nhp_mod_4_ne_3 : p % 4 ≠ 3\ny : ZMod p\nhy : -1 = y * y\nm : ℤ := y.valMinAbs\nn : ℕ := m.natAbs\nhnat₁ : p ∣ n ^ 2 + 1\nhnat₂ : n ≤ p / 2\nhnat₃ : p ≥ 2 * n\nk : ℕ := p - 2 * n\nhnat₄ : k = p - 2 * n\nhnat₅ : p ∣ k ^ 2 + 4\nhnat₆ : k ^ 2 + 4 ≥ p\nhreal₁ : ↑k = ↑p - 2 * ↑n\n⊢ ∃ n, p ∣ n ^ 2 + 1 ∧ ↑p > 2 * ↑n + √(2 * ↑n)",
"after_state": "case intro\np : ℕ\nhpp : Nat.Prime p\nhp_mod_4_eq_1 : p ≡ 1 [MOD 4]\nhp_gt_20 : p > 20\nthis : Fact (Nat.Prime p)\nhp_mod_4_ne_3 : p % 4 ≠ 3\ny : ZMod p\nhy : -1 = y * y\nm : ℤ := y.valMinAbs\nn : ℕ := m.natAbs\nhnat₁ : p ∣ n ^ 2 + 1\nhnat₂ : n ≤ p / 2\nhnat₃ : p ≥ 2 * n\nk : ℕ := p - 2 * n\nhnat₄ : k = p - 2 * n\nhnat₅ : p ∣ k ^ 2 + 4\nhnat₆ : k ^ 2 + 4 ≥ p\nhreal₁ : ↑k = ↑p - 2 * ↑n\nhreal₂ : ↑p > 20\n⊢ ∃ n, p ∣ n ^ 2 + 1 ∧ ↑p > 2 * ↑n + √(2 * ↑n)"
},
{
"line": "focus\n refine\n no_implicit_lambda%\n (have hreal₂ : (p : ℝ) > 20 := ?body✝;\n ?_)\n case body✝ => with_annotate_state\"by\" (assumption_mod_cast)",
"before_state": "case intro\np : ℕ\nhpp : Nat.Prime p\nhp_mod_4_eq_1 : p ≡ 1 [MOD 4]\nhp_gt_20 : p > 20\nthis : Fact (Nat.Prime p)\nhp_mod_4_ne_3 : p % 4 ≠ 3\ny : ZMod p\nhy : -1 = y * y\nm : ℤ := y.valMinAbs\nn : ℕ := m.natAbs\nhnat₁ : p ∣ n ^ 2 + 1\nhnat₂ : n ≤ p / 2\nhnat₃ : p ≥ 2 * n\nk : ℕ := p - 2 * n\nhnat₄ : k = p - 2 * n\nhnat₅ : p ∣ k ^ 2 + 4\nhnat₆ : k ^ 2 + 4 ≥ p\nhreal₁ : ↑k = ↑p - 2 * ↑n\n⊢ ∃ n, p ∣ n ^ 2 + 1 ∧ ↑p > 2 * ↑n + √(2 * ↑n)",
"after_state": "case intro\np : ℕ\nhpp : Nat.Prime p\nhp_mod_4_eq_1 : p ≡ 1 [MOD 4]\nhp_gt_20 : p > 20\nthis : Fact (Nat.Prime p)\nhp_mod_4_ne_3 : p % 4 ≠ 3\ny : ZMod p\nhy : -1 = y * y\nm : ℤ := y.valMinAbs\nn : ℕ := m.natAbs\nhnat₁ : p ∣ n ^ 2 + 1\nhnat₂ : n ≤ p / 2\nhnat₃ : p ≥ 2 * n\nk : ℕ := p - 2 * n\nhnat₄ : k = p - 2 * n\nhnat₅ : p ∣ k ^ 2 + 4\nhnat₆ : k ^ 2 + 4 ≥ p\nhreal₁ : ↑k = ↑p - 2 * ↑n\nhreal₂ : ↑p > 20\n⊢ ∃ n, p ∣ n ^ 2 + 1 ∧ ↑p > 2 * ↑n + √(2 * ↑n)"
},
{
"line": "refine\n no_implicit_lambda%\n (have hreal₂ : (p : ℝ) > 20 := ?body✝;\n ?_)",
"before_state": "case intro\np : ℕ\nhpp : Nat.Prime p\nhp_mod_4_eq_1 : p ≡ 1 [MOD 4]\nhp_gt_20 : p > 20\nthis : Fact (Nat.Prime p)\nhp_mod_4_ne_3 : p % 4 ≠ 3\ny : ZMod p\nhy : -1 = y * y\nm : ℤ := y.valMinAbs\nn : ℕ := m.natAbs\nhnat₁ : p ∣ n ^ 2 + 1\nhnat₂ : n ≤ p / 2\nhnat₃ : p ≥ 2 * n\nk : ℕ := p - 2 * n\nhnat₄ : k = p - 2 * n\nhnat₅ : p ∣ k ^ 2 + 4\nhnat₆ : k ^ 2 + 4 ≥ p\nhreal₁ : ↑k = ↑p - 2 * ↑n\n⊢ ∃ n, p ∣ n ^ 2 + 1 ∧ ↑p > 2 * ↑n + √(2 * ↑n)",
"after_state": "case body\np : ℕ\nhpp : Nat.Prime p\nhp_mod_4_eq_1 : p ≡ 1 [MOD 4]\nhp_gt_20 : p > 20\nthis : Fact (Nat.Prime p)\nhp_mod_4_ne_3 : p % 4 ≠ 3\ny : ZMod p\nhy : -1 = y * y\nm : ℤ := y.valMinAbs\nn : ℕ := m.natAbs\nhnat₁ : p ∣ n ^ 2 + 1\nhnat₂ : n ≤ p / 2\nhnat₃ : p ≥ 2 * n\nk : ℕ := p - 2 * n\nhnat₄ : k = p - 2 * n\nhnat₅ : p ∣ k ^ 2 + 4\nhnat₆ : k ^ 2 + 4 ≥ p\nhreal₁ : ↑k = ↑p - 2 * ↑n\n⊢ ↑p > 20\n---\ncase intro\np : ℕ\nhpp : Nat.Prime p\nhp_mod_4_eq_1 : p ≡ 1 [MOD 4]\nhp_gt_20 : p > 20\nthis : Fact (Nat.Prime p)\nhp_mod_4_ne_3 : p % 4 ≠ 3\ny : ZMod p\nhy : -1 = y * y\nm : ℤ := y.valMinAbs\nn : ℕ := m.natAbs\nhnat₁ : p ∣ n ^ 2 + 1\nhnat₂ : n ≤ p / 2\nhnat₃ : p ≥ 2 * n\nk : ℕ := p - 2 * n\nhnat₄ : k = p - 2 * n\nhnat₅ : p ∣ k ^ 2 + 4\nhnat₆ : k ^ 2 + 4 ≥ p\nhreal₁ : ↑k = ↑p - 2 * ↑n\nhreal₂ : ↑p > 20\n⊢ ∃ n, p ∣ n ^ 2 + 1 ∧ ↑p > 2 * ↑n + √(2 * ↑n)"
},
{
"line": "case body✝ => with_annotate_state\"by\" (assumption_mod_cast)",
"before_state": "case body\np : ℕ\nhpp : Nat.Prime p\nhp_mod_4_eq_1 : p ≡ 1 [MOD 4]\nhp_gt_20 : p > 20\nthis : Fact (Nat.Prime p)\nhp_mod_4_ne_3 : p % 4 ≠ 3\ny : ZMod p\nhy : -1 = y * y\nm : ℤ := y.valMinAbs\nn : ℕ := m.natAbs\nhnat₁ : p ∣ n ^ 2 + 1\nhnat₂ : n ≤ p / 2\nhnat₃ : p ≥ 2 * n\nk : ℕ := p - 2 * n\nhnat₄ : k = p - 2 * n\nhnat₅ : p ∣ k ^ 2 + 4\nhnat₆ : k ^ 2 + 4 ≥ p\nhreal₁ : ↑k = ↑p - 2 * ↑n\n⊢ ↑p > 20\n---\ncase intro\np : ℕ\nhpp : Nat.Prime p\nhp_mod_4_eq_1 : p ≡ 1 [MOD 4]\nhp_gt_20 : p > 20\nthis : Fact (Nat.Prime p)\nhp_mod_4_ne_3 : p % 4 ≠ 3\ny : ZMod p\nhy : -1 = y * y\nm : ℤ := y.valMinAbs\nn : ℕ := m.natAbs\nhnat₁ : p ∣ n ^ 2 + 1\nhnat₂ : n ≤ p / 2\nhnat₃ : p ≥ 2 * n\nk : ℕ := p - 2 * n\nhnat₄ : k = p - 2 * n\nhnat₅ : p ∣ k ^ 2 + 4\nhnat₆ : k ^ 2 + 4 ≥ p\nhreal₁ : ↑k = ↑p - 2 * ↑n\nhreal₂ : ↑p > 20\n⊢ ∃ n, p ∣ n ^ 2 + 1 ∧ ↑p > 2 * ↑n + √(2 * ↑n)",
"after_state": "case intro\np : ℕ\nhpp : Nat.Prime p\nhp_mod_4_eq_1 : p ≡ 1 [MOD 4]\nhp_gt_20 : p > 20\nthis : Fact (Nat.Prime p)\nhp_mod_4_ne_3 : p % 4 ≠ 3\ny : ZMod p\nhy : -1 = y * y\nm : ℤ := y.valMinAbs\nn : ℕ := m.natAbs\nhnat₁ : p ∣ n ^ 2 + 1\nhnat₂ : n ≤ p / 2\nhnat₃ : p ≥ 2 * n\nk : ℕ := p - 2 * n\nhnat₄ : k = p - 2 * n\nhnat₅ : p ∣ k ^ 2 + 4\nhnat₆ : k ^ 2 + 4 ≥ p\nhreal₁ : ↑k = ↑p - 2 * ↑n\nhreal₂ : ↑p > 20\n⊢ ∃ n, p ∣ n ^ 2 + 1 ∧ ↑p > 2 * ↑n + √(2 * ↑n)"
},
{
"line": "with_annotate_state\"by\" (assumption_mod_cast)",
"before_state": "p : ℕ\nhpp : Nat.Prime p\nhp_mod_4_eq_1 : p ≡ 1 [MOD 4]\nhp_gt_20 : p > 20\nthis : Fact (Nat.Prime p)\nhp_mod_4_ne_3 : p % 4 ≠ 3\ny : ZMod p\nhy : -1 = y * y\nm : ℤ := y.valMinAbs\nn : ℕ := m.natAbs\nhnat₁ : p ∣ n ^ 2 + 1\nhnat₂ : n ≤ p / 2\nhnat₃ : p ≥ 2 * n\nk : ℕ := p - 2 * n\nhnat₄ : k = p - 2 * n\nhnat₅ : p ∣ k ^ 2 + 4\nhnat₆ : k ^ 2 + 4 ≥ p\nhreal₁ : ↑k = ↑p - 2 * ↑n\n⊢ ↑p > 20",
"after_state": "No Goals!"
},
{
"line": "assumption_mod_cast",
"before_state": "p : ℕ\nhpp : Nat.Prime p\nhp_mod_4_eq_1 : p ≡ 1 [MOD 4]\nhp_gt_20 : p > 20\nthis : Fact (Nat.Prime p)\nhp_mod_4_ne_3 : p % 4 ≠ 3\ny : ZMod p\nhy : -1 = y * y\nm : ℤ := y.valMinAbs\nn : ℕ := m.natAbs\nhnat₁ : p ∣ n ^ 2 + 1\nhnat₂ : n ≤ p / 2\nhnat₃ : p ≥ 2 * n\nk : ℕ := p - 2 * n\nhnat₄ : k = p - 2 * n\nhnat₅ : p ∣ k ^ 2 + 4\nhnat₆ : k ^ 2 + 4 ≥ p\nhreal₁ : ↑k = ↑p - 2 * ↑n\n⊢ ↑p > 20",
"after_state": "No Goals!"
},
{
"line": "focus\n norm_cast0 at *\n with_annotate_state\"<;>\" skip\n all_goals assumption",
"before_state": "p : ℕ\nhpp : Nat.Prime p\nhp_mod_4_eq_1 : p ≡ 1 [MOD 4]\nhp_gt_20 : p > 20\nthis : Fact (Nat.Prime p)\nhp_mod_4_ne_3 : p % 4 ≠ 3\ny : ZMod p\nhy : -1 = y * y\nm : ℤ := y.valMinAbs\nn : ℕ := m.natAbs\nhnat₁ : p ∣ n ^ 2 + 1\nhnat₂ : n ≤ p / 2\nhnat₃ : p ≥ 2 * n\nk : ℕ := p - 2 * n\nhnat₄ : k = p - 2 * n\nhnat₅ : p ∣ k ^ 2 + 4\nhnat₆ : k ^ 2 + 4 ≥ p\nhreal₁ : ↑k = ↑p - 2 * ↑n\n⊢ ↑p > 20",
"after_state": "No Goals!"
},
{
"line": "norm_cast0 at *",
"before_state": "p : ℕ\nhpp : Nat.Prime p\nhp_mod_4_eq_1 : p ≡ 1 [MOD 4]\nhp_gt_20 : p > 20\nthis : Fact (Nat.Prime p)\nhp_mod_4_ne_3 : p % 4 ≠ 3\ny : ZMod p\nhy : -1 = y * y\nm : ℤ := y.valMinAbs\nn : ℕ := m.natAbs\nhnat₁ : p ∣ n ^ 2 + 1\nhnat₂ : n ≤ p / 2\nhnat₃ : p ≥ 2 * n\nk : ℕ := p - 2 * n\nhnat₄ : k = p - 2 * n\nhnat₅ : p ∣ k ^ 2 + 4\nhnat₆ : k ^ 2 + 4 ≥ p\nhreal₁ : ↑k = ↑p - 2 * ↑n\n⊢ ↑p > 20",
"after_state": "p : ℕ\nhpp : Nat.Prime p\nhp_mod_4_eq_1 : p ≡ 1 [MOD 4]\nhp_gt_20 : p > 20\nthis : Fact (Nat.Prime p)\nhp_mod_4_ne_3 : ¬p % 4 = 3\ny : ZMod p\nm : ℤ := y.valMinAbs\nn : ℕ := m.natAbs\nhnat₁ : p ∣ n ^ 2 + 1\nhnat₂ : n ≤ p / 2\nhnat₃ : p ≥ 2 * n\nk : ℕ := p - 2 * n\nhnat₄ : k = p - 2 * n\nhnat₅ : p ∣ k ^ 2 + 4\nhnat₆ : k ^ 2 + 4 ≥ p\nhy : ↑(Int.negSucc 0) = y * y\nhreal₁ : k = p - 2 * n\n⊢ 20 < p"
},
{
"line": "with_annotate_state\"<;>\" skip",
"before_state": "p : ℕ\nhpp : Nat.Prime p\nhp_mod_4_eq_1 : p ≡ 1 [MOD 4]\nhp_gt_20 : p > 20\nthis : Fact (Nat.Prime p)\nhp_mod_4_ne_3 : ¬p % 4 = 3\ny : ZMod p\nm : ℤ := y.valMinAbs\nn : ℕ := m.natAbs\nhnat₁ : p ∣ n ^ 2 + 1\nhnat₂ : n ≤ p / 2\nhnat₃ : p ≥ 2 * n\nk : ℕ := p - 2 * n\nhnat₄ : k = p - 2 * n\nhnat₅ : p ∣ k ^ 2 + 4\nhnat₆ : k ^ 2 + 4 ≥ p\nhy : ↑(Int.negSucc 0) = y * y\nhreal₁ : k = p - 2 * n\n⊢ 20 < p",
"after_state": "p : ℕ\nhpp : Nat.Prime p\nhp_mod_4_eq_1 : p ≡ 1 [MOD 4]\nhp_gt_20 : p > 20\nthis : Fact (Nat.Prime p)\nhp_mod_4_ne_3 : ¬p % 4 = 3\ny : ZMod p\nm : ℤ := y.valMinAbs\nn : ℕ := m.natAbs\nhnat₁ : p ∣ n ^ 2 + 1\nhnat₂ : n ≤ p / 2\nhnat₃ : p ≥ 2 * n\nk : ℕ := p - 2 * n\nhnat₄ : k = p - 2 * n\nhnat₅ : p ∣ k ^ 2 + 4\nhnat₆ : k ^ 2 + 4 ≥ p\nhy : ↑(Int.negSucc 0) = y * y\nhreal₁ : k = p - 2 * n\n⊢ 20 < p"
},
{
"line": "skip",
"before_state": "p : ℕ\nhpp : Nat.Prime p\nhp_mod_4_eq_1 : p ≡ 1 [MOD 4]\nhp_gt_20 : p > 20\nthis : Fact (Nat.Prime p)\nhp_mod_4_ne_3 : ¬p % 4 = 3\ny : ZMod p\nm : ℤ := y.valMinAbs\nn : ℕ := m.natAbs\nhnat₁ : p ∣ n ^ 2 + 1\nhnat₂ : n ≤ p / 2\nhnat₃ : p ≥ 2 * n\nk : ℕ := p - 2 * n\nhnat₄ : k = p - 2 * n\nhnat₅ : p ∣ k ^ 2 + 4\nhnat₆ : k ^ 2 + 4 ≥ p\nhy : ↑(Int.negSucc 0) = y * y\nhreal₁ : k = p - 2 * n\n⊢ 20 < p",
"after_state": "p : ℕ\nhpp : Nat.Prime p\nhp_mod_4_eq_1 : p ≡ 1 [MOD 4]\nhp_gt_20 : p > 20\nthis : Fact (Nat.Prime p)\nhp_mod_4_ne_3 : ¬p % 4 = 3\ny : ZMod p\nm : ℤ := y.valMinAbs\nn : ℕ := m.natAbs\nhnat₁ : p ∣ n ^ 2 + 1\nhnat₂ : n ≤ p / 2\nhnat₃ : p ≥ 2 * n\nk : ℕ := p - 2 * n\nhnat₄ : k = p - 2 * n\nhnat₅ : p ∣ k ^ 2 + 4\nhnat₆ : k ^ 2 + 4 ≥ p\nhy : ↑(Int.negSucc 0) = y * y\nhreal₁ : k = p - 2 * n\n⊢ 20 < p"
},
{
"line": "all_goals assumption",
"before_state": "p : ℕ\nhpp : Nat.Prime p\nhp_mod_4_eq_1 : p ≡ 1 [MOD 4]\nhp_gt_20 : p > 20\nthis : Fact (Nat.Prime p)\nhp_mod_4_ne_3 : ¬p % 4 = 3\ny : ZMod p\nm : ℤ := y.valMinAbs\nn : ℕ := m.natAbs\nhnat₁ : p ∣ n ^ 2 + 1\nhnat₂ : n ≤ p / 2\nhnat₃ : p ≥ 2 * n\nk : ℕ := p - 2 * n\nhnat₄ : k = p - 2 * n\nhnat₅ : p ∣ k ^ 2 + 4\nhnat₆ : k ^ 2 + 4 ≥ p\nhy : ↑(Int.negSucc 0) = y * y\nhreal₁ : k = p - 2 * n\n⊢ 20 < p",
"after_state": "No Goals!"
},
{
"line": "assumption",
"before_state": "p : ℕ\nhpp : Nat.Prime p\nhp_mod_4_eq_1 : p ≡ 1 [MOD 4]\nhp_gt_20 : p > 20\nthis : Fact (Nat.Prime p)\nhp_mod_4_ne_3 : ¬p % 4 = 3\ny : ZMod p\nm : ℤ := y.valMinAbs\nn : ℕ := m.natAbs\nhnat₁ : p ∣ n ^ 2 + 1\nhnat₂ : n ≤ p / 2\nhnat₃ : p ≥ 2 * n\nk : ℕ := p - 2 * n\nhnat₄ : k = p - 2 * n\nhnat₅ : p ∣ k ^ 2 + 4\nhnat₆ : k ^ 2 + 4 ≥ p\nhy : ↑(Int.negSucc 0) = y * y\nhreal₁ : k = p - 2 * n\n⊢ 20 < p",
"after_state": "No Goals!"
},
{
"line": "have hreal₃ : (k : ℝ) ^ 2 + 4 ≥ p := by assumption_mod_cast",
"before_state": "case intro\np : ℕ\nhpp : Nat.Prime p\nhp_mod_4_eq_1 : p ≡ 1 [MOD 4]\nhp_gt_20 : p > 20\nthis : Fact (Nat.Prime p)\nhp_mod_4_ne_3 : p % 4 ≠ 3\ny : ZMod p\nhy : -1 = y * y\nm : ℤ := y.valMinAbs\nn : ℕ := m.natAbs\nhnat₁ : p ∣ n ^ 2 + 1\nhnat₂ : n ≤ p / 2\nhnat₃ : p ≥ 2 * n\nk : ℕ := p - 2 * n\nhnat₄ : k = p - 2 * n\nhnat₅ : p ∣ k ^ 2 + 4\nhnat₆ : k ^ 2 + 4 ≥ p\nhreal₁ : ↑k = ↑p - 2 * ↑n\nhreal₂ : ↑p > 20\n⊢ ∃ n, p ∣ n ^ 2 + 1 ∧ ↑p > 2 * ↑n + √(2 * ↑n)",
"after_state": "case intro\np : ℕ\nhpp : Nat.Prime p\nhp_mod_4_eq_1 : p ≡ 1 [MOD 4]\nhp_gt_20 : p > 20\nthis : Fact (Nat.Prime p)\nhp_mod_4_ne_3 : p % 4 ≠ 3\ny : ZMod p\nhy : -1 = y * y\nm : ℤ := y.valMinAbs\nn : ℕ := m.natAbs\nhnat₁ : p ∣ n ^ 2 + 1\nhnat₂ : n ≤ p / 2\nhnat₃ : p ≥ 2 * n\nk : ℕ := p - 2 * n\nhnat₄ : k = p - 2 * n\nhnat₅ : p ∣ k ^ 2 + 4\nhnat₆ : k ^ 2 + 4 ≥ p\nhreal₁ : ↑k = ↑p - 2 * ↑n\nhreal₂ : ↑p > 20\nhreal₃ : ↑k ^ 2 + 4 ≥ ↑p\n⊢ ∃ n, p ∣ n ^ 2 + 1 ∧ ↑p > 2 * ↑n + √(2 * ↑n)"
},
{
"line": "focus\n refine\n no_implicit_lambda%\n (have hreal₃ : (k : ℝ) ^ 2 + 4 ≥ p := ?body✝;\n ?_)\n case body✝ => with_annotate_state\"by\" (assumption_mod_cast)",
"before_state": "case intro\np : ℕ\nhpp : Nat.Prime p\nhp_mod_4_eq_1 : p ≡ 1 [MOD 4]\nhp_gt_20 : p > 20\nthis : Fact (Nat.Prime p)\nhp_mod_4_ne_3 : p % 4 ≠ 3\ny : ZMod p\nhy : -1 = y * y\nm : ℤ := y.valMinAbs\nn : ℕ := m.natAbs\nhnat₁ : p ∣ n ^ 2 + 1\nhnat₂ : n ≤ p / 2\nhnat₃ : p ≥ 2 * n\nk : ℕ := p - 2 * n\nhnat₄ : k = p - 2 * n\nhnat₅ : p ∣ k ^ 2 + 4\nhnat₆ : k ^ 2 + 4 ≥ p\nhreal₁ : ↑k = ↑p - 2 * ↑n\nhreal₂ : ↑p > 20\n⊢ ∃ n, p ∣ n ^ 2 + 1 ∧ ↑p > 2 * ↑n + √(2 * ↑n)",
"after_state": "case intro\np : ℕ\nhpp : Nat.Prime p\nhp_mod_4_eq_1 : p ≡ 1 [MOD 4]\nhp_gt_20 : p > 20\nthis : Fact (Nat.Prime p)\nhp_mod_4_ne_3 : p % 4 ≠ 3\ny : ZMod p\nhy : -1 = y * y\nm : ℤ := y.valMinAbs\nn : ℕ := m.natAbs\nhnat₁ : p ∣ n ^ 2 + 1\nhnat₂ : n ≤ p / 2\nhnat₃ : p ≥ 2 * n\nk : ℕ := p - 2 * n\nhnat₄ : k = p - 2 * n\nhnat₅ : p ∣ k ^ 2 + 4\nhnat₆ : k ^ 2 + 4 ≥ p\nhreal₁ : ↑k = ↑p - 2 * ↑n\nhreal₂ : ↑p > 20\nhreal₃ : ↑k ^ 2 + 4 ≥ ↑p\n⊢ ∃ n, p ∣ n ^ 2 + 1 ∧ ↑p > 2 * ↑n + √(2 * ↑n)"
},
{
"line": "refine\n no_implicit_lambda%\n (have hreal₃ : (k : ℝ) ^ 2 + 4 ≥ p := ?body✝;\n ?_)",
"before_state": "case intro\np : ℕ\nhpp : Nat.Prime p\nhp_mod_4_eq_1 : p ≡ 1 [MOD 4]\nhp_gt_20 : p > 20\nthis : Fact (Nat.Prime p)\nhp_mod_4_ne_3 : p % 4 ≠ 3\ny : ZMod p\nhy : -1 = y * y\nm : ℤ := y.valMinAbs\nn : ℕ := m.natAbs\nhnat₁ : p ∣ n ^ 2 + 1\nhnat₂ : n ≤ p / 2\nhnat₃ : p ≥ 2 * n\nk : ℕ := p - 2 * n\nhnat₄ : k = p - 2 * n\nhnat₅ : p ∣ k ^ 2 + 4\nhnat₆ : k ^ 2 + 4 ≥ p\nhreal₁ : ↑k = ↑p - 2 * ↑n\nhreal₂ : ↑p > 20\n⊢ ∃ n, p ∣ n ^ 2 + 1 ∧ ↑p > 2 * ↑n + √(2 * ↑n)",
"after_state": "case body\np : ℕ\nhpp : Nat.Prime p\nhp_mod_4_eq_1 : p ≡ 1 [MOD 4]\nhp_gt_20 : p > 20\nthis : Fact (Nat.Prime p)\nhp_mod_4_ne_3 : p % 4 ≠ 3\ny : ZMod p\nhy : -1 = y * y\nm : ℤ := y.valMinAbs\nn : ℕ := m.natAbs\nhnat₁ : p ∣ n ^ 2 + 1\nhnat₂ : n ≤ p / 2\nhnat₃ : p ≥ 2 * n\nk : ℕ := p - 2 * n\nhnat₄ : k = p - 2 * n\nhnat₅ : p ∣ k ^ 2 + 4\nhnat₆ : k ^ 2 + 4 ≥ p\nhreal₁ : ↑k = ↑p - 2 * ↑n\nhreal₂ : ↑p > 20\n⊢ ↑k ^ 2 + 4 ≥ ↑p\n---\ncase intro\np : ℕ\nhpp : Nat.Prime p\nhp_mod_4_eq_1 : p ≡ 1 [MOD 4]\nhp_gt_20 : p > 20\nthis : Fact (Nat.Prime p)\nhp_mod_4_ne_3 : p % 4 ≠ 3\ny : ZMod p\nhy : -1 = y * y\nm : ℤ := y.valMinAbs\nn : ℕ := m.natAbs\nhnat₁ : p ∣ n ^ 2 + 1\nhnat₂ : n ≤ p / 2\nhnat₃ : p ≥ 2 * n\nk : ℕ := p - 2 * n\nhnat₄ : k = p - 2 * n\nhnat₅ : p ∣ k ^ 2 + 4\nhnat₆ : k ^ 2 + 4 ≥ p\nhreal₁ : ↑k = ↑p - 2 * ↑n\nhreal₂ : ↑p > 20\nhreal₃ : ↑k ^ 2 + 4 ≥ ↑p\n⊢ ∃ n, p ∣ n ^ 2 + 1 ∧ ↑p > 2 * ↑n + √(2 * ↑n)"
},
{
"line": "case body✝ => with_annotate_state\"by\" (assumption_mod_cast)",
"before_state": "case body\np : ℕ\nhpp : Nat.Prime p\nhp_mod_4_eq_1 : p ≡ 1 [MOD 4]\nhp_gt_20 : p > 20\nthis : Fact (Nat.Prime p)\nhp_mod_4_ne_3 : p % 4 ≠ 3\ny : ZMod p\nhy : -1 = y * y\nm : ℤ := y.valMinAbs\nn : ℕ := m.natAbs\nhnat₁ : p ∣ n ^ 2 + 1\nhnat₂ : n ≤ p / 2\nhnat₃ : p ≥ 2 * n\nk : ℕ := p - 2 * n\nhnat₄ : k = p - 2 * n\nhnat₅ : p ∣ k ^ 2 + 4\nhnat₆ : k ^ 2 + 4 ≥ p\nhreal₁ : ↑k = ↑p - 2 * ↑n\nhreal₂ : ↑p > 20\n⊢ ↑k ^ 2 + 4 ≥ ↑p\n---\ncase intro\np : ℕ\nhpp : Nat.Prime p\nhp_mod_4_eq_1 : p ≡ 1 [MOD 4]\nhp_gt_20 : p > 20\nthis : Fact (Nat.Prime p)\nhp_mod_4_ne_3 : p % 4 ≠ 3\ny : ZMod p\nhy : -1 = y * y\nm : ℤ := y.valMinAbs\nn : ℕ := m.natAbs\nhnat₁ : p ∣ n ^ 2 + 1\nhnat₂ : n ≤ p / 2\nhnat₃ : p ≥ 2 * n\nk : ℕ := p - 2 * n\nhnat₄ : k = p - 2 * n\nhnat₅ : p ∣ k ^ 2 + 4\nhnat₆ : k ^ 2 + 4 ≥ p\nhreal₁ : ↑k = ↑p - 2 * ↑n\nhreal₂ : ↑p > 20\nhreal₃ : ↑k ^ 2 + 4 ≥ ↑p\n⊢ ∃ n, p ∣ n ^ 2 + 1 ∧ ↑p > 2 * ↑n + √(2 * ↑n)",
"after_state": "case intro\np : ℕ\nhpp : Nat.Prime p\nhp_mod_4_eq_1 : p ≡ 1 [MOD 4]\nhp_gt_20 : p > 20\nthis : Fact (Nat.Prime p)\nhp_mod_4_ne_3 : p % 4 ≠ 3\ny : ZMod p\nhy : -1 = y * y\nm : ℤ := y.valMinAbs\nn : ℕ := m.natAbs\nhnat₁ : p ∣ n ^ 2 + 1\nhnat₂ : n ≤ p / 2\nhnat₃ : p ≥ 2 * n\nk : ℕ := p - 2 * n\nhnat₄ : k = p - 2 * n\nhnat₅ : p ∣ k ^ 2 + 4\nhnat₆ : k ^ 2 + 4 ≥ p\nhreal₁ : ↑k = ↑p - 2 * ↑n\nhreal₂ : ↑p > 20\nhreal₃ : ↑k ^ 2 + 4 ≥ ↑p\n⊢ ∃ n, p ∣ n ^ 2 + 1 ∧ ↑p > 2 * ↑n + √(2 * ↑n)"
},
{
"line": "with_annotate_state\"by\" (assumption_mod_cast)",
"before_state": "p : ℕ\nhpp : Nat.Prime p\nhp_mod_4_eq_1 : p ≡ 1 [MOD 4]\nhp_gt_20 : p > 20\nthis : Fact (Nat.Prime p)\nhp_mod_4_ne_3 : p % 4 ≠ 3\ny : ZMod p\nhy : -1 = y * y\nm : ℤ := y.valMinAbs\nn : ℕ := m.natAbs\nhnat₁ : p ∣ n ^ 2 + 1\nhnat₂ : n ≤ p / 2\nhnat₃ : p ≥ 2 * n\nk : ℕ := p - 2 * n\nhnat₄ : k = p - 2 * n\nhnat₅ : p ∣ k ^ 2 + 4\nhnat₆ : k ^ 2 + 4 ≥ p\nhreal₁ : ↑k = ↑p - 2 * ↑n\nhreal₂ : ↑p > 20\n⊢ ↑k ^ 2 + 4 ≥ ↑p",
"after_state": "No Goals!"
},
{
"line": "assumption_mod_cast",
"before_state": "p : ℕ\nhpp : Nat.Prime p\nhp_mod_4_eq_1 : p ≡ 1 [MOD 4]\nhp_gt_20 : p > 20\nthis : Fact (Nat.Prime p)\nhp_mod_4_ne_3 : p % 4 ≠ 3\ny : ZMod p\nhy : -1 = y * y\nm : ℤ := y.valMinAbs\nn : ℕ := m.natAbs\nhnat₁ : p ∣ n ^ 2 + 1\nhnat₂ : n ≤ p / 2\nhnat₃ : p ≥ 2 * n\nk : ℕ := p - 2 * n\nhnat₄ : k = p - 2 * n\nhnat₅ : p ∣ k ^ 2 + 4\nhnat₆ : k ^ 2 + 4 ≥ p\nhreal₁ : ↑k = ↑p - 2 * ↑n\nhreal₂ : ↑p > 20\n⊢ ↑k ^ 2 + 4 ≥ ↑p",
"after_state": "No Goals!"
},
{
"line": "focus\n norm_cast0 at *\n with_annotate_state\"<;>\" skip\n all_goals assumption",
"before_state": "p : ℕ\nhpp : Nat.Prime p\nhp_mod_4_eq_1 : p ≡ 1 [MOD 4]\nhp_gt_20 : p > 20\nthis : Fact (Nat.Prime p)\nhp_mod_4_ne_3 : p % 4 ≠ 3\ny : ZMod p\nhy : -1 = y * y\nm : ℤ := y.valMinAbs\nn : ℕ := m.natAbs\nhnat₁ : p ∣ n ^ 2 + 1\nhnat₂ : n ≤ p / 2\nhnat₃ : p ≥ 2 * n\nk : ℕ := p - 2 * n\nhnat₄ : k = p - 2 * n\nhnat₅ : p ∣ k ^ 2 + 4\nhnat₆ : k ^ 2 + 4 ≥ p\nhreal₁ : ↑k = ↑p - 2 * ↑n\nhreal₂ : ↑p > 20\n⊢ ↑k ^ 2 + 4 ≥ ↑p",
"after_state": "No Goals!"
},
{
"line": "norm_cast0 at *",
"before_state": "p : ℕ\nhpp : Nat.Prime p\nhp_mod_4_eq_1 : p ≡ 1 [MOD 4]\nhp_gt_20 : p > 20\nthis : Fact (Nat.Prime p)\nhp_mod_4_ne_3 : p % 4 ≠ 3\ny : ZMod p\nhy : -1 = y * y\nm : ℤ := y.valMinAbs\nn : ℕ := m.natAbs\nhnat₁ : p ∣ n ^ 2 + 1\nhnat₂ : n ≤ p / 2\nhnat₃ : p ≥ 2 * n\nk : ℕ := p - 2 * n\nhnat₄ : k = p - 2 * n\nhnat₅ : p ∣ k ^ 2 + 4\nhnat₆ : k ^ 2 + 4 ≥ p\nhreal₁ : ↑k = ↑p - 2 * ↑n\nhreal₂ : ↑p > 20\n⊢ ↑k ^ 2 + 4 ≥ ↑p",
"after_state": "p : ℕ\nhpp : Nat.Prime p\nhp_mod_4_eq_1 : p ≡ 1 [MOD 4]\nhp_gt_20 : p > 20\nthis : Fact (Nat.Prime p)\nhp_mod_4_ne_3 : ¬p % 4 = 3\ny : ZMod p\nm : ℤ := y.valMinAbs\nn : ℕ := m.natAbs\nhnat₁ : p ∣ n ^ 2 + 1\nhnat₂ : n ≤ p / 2\nhnat₃ : p ≥ 2 * n\nk : ℕ := p - 2 * n\nhnat₄ : k = p - 2 * n\nhnat₅ : p ∣ k ^ 2 + 4\nhnat₆ : k ^ 2 + 4 ≥ p\nhy : ↑(Int.negSucc 0) = y * y\nhreal₁ : k = p - 2 * n\nhreal₂ : 20 < p\n⊢ p ≤ k ^ 2 + 4"
},
{
"line": "with_annotate_state\"<;>\" skip",
"before_state": "p : ℕ\nhpp : Nat.Prime p\nhp_mod_4_eq_1 : p ≡ 1 [MOD 4]\nhp_gt_20 : p > 20\nthis : Fact (Nat.Prime p)\nhp_mod_4_ne_3 : ¬p % 4 = 3\ny : ZMod p\nm : ℤ := y.valMinAbs\nn : ℕ := m.natAbs\nhnat₁ : p ∣ n ^ 2 + 1\nhnat₂ : n ≤ p / 2\nhnat₃ : p ≥ 2 * n\nk : ℕ := p - 2 * n\nhnat₄ : k = p - 2 * n\nhnat₅ : p ∣ k ^ 2 + 4\nhnat₆ : k ^ 2 + 4 ≥ p\nhy : ↑(Int.negSucc 0) = y * y\nhreal₁ : k = p - 2 * n\nhreal₂ : 20 < p\n⊢ p ≤ k ^ 2 + 4",
"after_state": "p : ℕ\nhpp : Nat.Prime p\nhp_mod_4_eq_1 : p ≡ 1 [MOD 4]\nhp_gt_20 : p > 20\nthis : Fact (Nat.Prime p)\nhp_mod_4_ne_3 : ¬p % 4 = 3\ny : ZMod p\nm : ℤ := y.valMinAbs\nn : ℕ := m.natAbs\nhnat₁ : p ∣ n ^ 2 + 1\nhnat₂ : n ≤ p / 2\nhnat₃ : p ≥ 2 * n\nk : ℕ := p - 2 * n\nhnat₄ : k = p - 2 * n\nhnat₅ : p ∣ k ^ 2 + 4\nhnat₆ : k ^ 2 + 4 ≥ p\nhy : ↑(Int.negSucc 0) = y * y\nhreal₁ : k = p - 2 * n\nhreal₂ : 20 < p\n⊢ p ≤ k ^ 2 + 4"
},
{
"line": "skip",
"before_state": "p : ℕ\nhpp : Nat.Prime p\nhp_mod_4_eq_1 : p ≡ 1 [MOD 4]\nhp_gt_20 : p > 20\nthis : Fact (Nat.Prime p)\nhp_mod_4_ne_3 : ¬p % 4 = 3\ny : ZMod p\nm : ℤ := y.valMinAbs\nn : ℕ := m.natAbs\nhnat₁ : p ∣ n ^ 2 + 1\nhnat₂ : n ≤ p / 2\nhnat₃ : p ≥ 2 * n\nk : ℕ := p - 2 * n\nhnat₄ : k = p - 2 * n\nhnat₅ : p ∣ k ^ 2 + 4\nhnat₆ : k ^ 2 + 4 ≥ p\nhy : ↑(Int.negSucc 0) = y * y\nhreal₁ : k = p - 2 * n\nhreal₂ : 20 < p\n⊢ p ≤ k ^ 2 + 4",
"after_state": "p : ℕ\nhpp : Nat.Prime p\nhp_mod_4_eq_1 : p ≡ 1 [MOD 4]\nhp_gt_20 : p > 20\nthis : Fact (Nat.Prime p)\nhp_mod_4_ne_3 : ¬p % 4 = 3\ny : ZMod p\nm : ℤ := y.valMinAbs\nn : ℕ := m.natAbs\nhnat₁ : p ∣ n ^ 2 + 1\nhnat₂ : n ≤ p / 2\nhnat₃ : p ≥ 2 * n\nk : ℕ := p - 2 * n\nhnat₄ : k = p - 2 * n\nhnat₅ : p ∣ k ^ 2 + 4\nhnat₆ : k ^ 2 + 4 ≥ p\nhy : ↑(Int.negSucc 0) = y * y\nhreal₁ : k = p - 2 * n\nhreal₂ : 20 < p\n⊢ p ≤ k ^ 2 + 4"
},
{
"line": "all_goals assumption",
"before_state": "p : ℕ\nhpp : Nat.Prime p\nhp_mod_4_eq_1 : p ≡ 1 [MOD 4]\nhp_gt_20 : p > 20\nthis : Fact (Nat.Prime p)\nhp_mod_4_ne_3 : ¬p % 4 = 3\ny : ZMod p\nm : ℤ := y.valMinAbs\nn : ℕ := m.natAbs\nhnat₁ : p ∣ n ^ 2 + 1\nhnat₂ : n ≤ p / 2\nhnat₃ : p ≥ 2 * n\nk : ℕ := p - 2 * n\nhnat₄ : k = p - 2 * n\nhnat₅ : p ∣ k ^ 2 + 4\nhnat₆ : k ^ 2 + 4 ≥ p\nhy : ↑(Int.negSucc 0) = y * y\nhreal₁ : k = p - 2 * n\nhreal₂ : 20 < p\n⊢ p ≤ k ^ 2 + 4",
"after_state": "No Goals!"
},
{
"line": "assumption",
"before_state": "p : ℕ\nhpp : Nat.Prime p\nhp_mod_4_eq_1 : p ≡ 1 [MOD 4]\nhp_gt_20 : p > 20\nthis : Fact (Nat.Prime p)\nhp_mod_4_ne_3 : ¬p % 4 = 3\ny : ZMod p\nm : ℤ := y.valMinAbs\nn : ℕ := m.natAbs\nhnat₁ : p ∣ n ^ 2 + 1\nhnat₂ : n ≤ p / 2\nhnat₃ : p ≥ 2 * n\nk : ℕ := p - 2 * n\nhnat₄ : k = p - 2 * n\nhnat₅ : p ∣ k ^ 2 + 4\nhnat₆ : k ^ 2 + 4 ≥ p\nhy : ↑(Int.negSucc 0) = y * y\nhreal₁ : k = p - 2 * n\nhreal₂ : 20 < p\n⊢ p ≤ k ^ 2 + 4",
"after_state": "No Goals!"
},
{
"line": "have hreal₅ : (k : ℝ) > 4 := by\n refine lt_of_pow_lt_pow_left₀ 2 k.cast_nonneg ?_\n linarith only [hreal₂, hreal₃]",
"before_state": "case intro\np : ℕ\nhpp : Nat.Prime p\nhp_mod_4_eq_1 : p ≡ 1 [MOD 4]\nhp_gt_20 : p > 20\nthis : Fact (Nat.Prime p)\nhp_mod_4_ne_3 : p % 4 ≠ 3\ny : ZMod p\nhy : -1 = y * y\nm : ℤ := y.valMinAbs\nn : ℕ := m.natAbs\nhnat₁ : p ∣ n ^ 2 + 1\nhnat₂ : n ≤ p / 2\nhnat₃ : p ≥ 2 * n\nk : ℕ := p - 2 * n\nhnat₄ : k = p - 2 * n\nhnat₅ : p ∣ k ^ 2 + 4\nhnat₆ : k ^ 2 + 4 ≥ p\nhreal₁ : ↑k = ↑p - 2 * ↑n\nhreal₂ : ↑p > 20\nhreal₃ : ↑k ^ 2 + 4 ≥ ↑p\n⊢ ∃ n, p ∣ n ^ 2 + 1 ∧ ↑p > 2 * ↑n + √(2 * ↑n)",
"after_state": "case intro\np : ℕ\nhpp : Nat.Prime p\nhp_mod_4_eq_1 : p ≡ 1 [MOD 4]\nhp_gt_20 : p > 20\nthis : Fact (Nat.Prime p)\nhp_mod_4_ne_3 : p % 4 ≠ 3\ny : ZMod p\nhy : -1 = y * y\nm : ℤ := y.valMinAbs\nn : ℕ := m.natAbs\nhnat₁ : p ∣ n ^ 2 + 1\nhnat₂ : n ≤ p / 2\nhnat₃ : p ≥ 2 * n\nk : ℕ := p - 2 * n\nhnat₄ : k = p - 2 * n\nhnat₅ : p ∣ k ^ 2 + 4\nhnat₆ : k ^ 2 + 4 ≥ p\nhreal₁ : ↑k = ↑p - 2 * ↑n\nhreal₂ : ↑p > 20\nhreal₃ : ↑k ^ 2 + 4 ≥ ↑p\nhreal₅ : ↑k > 4\n⊢ ∃ n, p ∣ n ^ 2 + 1 ∧ ↑p > 2 * ↑n + √(2 * ↑n)"
},
{
"line": "focus\n refine\n no_implicit_lambda%\n (have hreal₅ : (k : ℝ) > 4 := ?body✝;\n ?_)\n case body✝ =>\n with_annotate_state\"by\"\n ( refine lt_of_pow_lt_pow_left₀ 2 k.cast_nonneg ?_\n linarith only [hreal₂, hreal₃])",
"before_state": "case intro\np : ℕ\nhpp : Nat.Prime p\nhp_mod_4_eq_1 : p ≡ 1 [MOD 4]\nhp_gt_20 : p > 20\nthis : Fact (Nat.Prime p)\nhp_mod_4_ne_3 : p % 4 ≠ 3\ny : ZMod p\nhy : -1 = y * y\nm : ℤ := y.valMinAbs\nn : ℕ := m.natAbs\nhnat₁ : p ∣ n ^ 2 + 1\nhnat₂ : n ≤ p / 2\nhnat₃ : p ≥ 2 * n\nk : ℕ := p - 2 * n\nhnat₄ : k = p - 2 * n\nhnat₅ : p ∣ k ^ 2 + 4\nhnat₆ : k ^ 2 + 4 ≥ p\nhreal₁ : ↑k = ↑p - 2 * ↑n\nhreal₂ : ↑p > 20\nhreal₃ : ↑k ^ 2 + 4 ≥ ↑p\n⊢ ∃ n, p ∣ n ^ 2 + 1 ∧ ↑p > 2 * ↑n + √(2 * ↑n)",
"after_state": "case intro\np : ℕ\nhpp : Nat.Prime p\nhp_mod_4_eq_1 : p ≡ 1 [MOD 4]\nhp_gt_20 : p > 20\nthis : Fact (Nat.Prime p)\nhp_mod_4_ne_3 : p % 4 ≠ 3\ny : ZMod p\nhy : -1 = y * y\nm : ℤ := y.valMinAbs\nn : ℕ := m.natAbs\nhnat₁ : p ∣ n ^ 2 + 1\nhnat₂ : n ≤ p / 2\nhnat₃ : p ≥ 2 * n\nk : ℕ := p - 2 * n\nhnat₄ : k = p - 2 * n\nhnat₅ : p ∣ k ^ 2 + 4\nhnat₆ : k ^ 2 + 4 ≥ p\nhreal₁ : ↑k = ↑p - 2 * ↑n\nhreal₂ : ↑p > 20\nhreal₃ : ↑k ^ 2 + 4 ≥ ↑p\nhreal₅ : ↑k > 4\n⊢ ∃ n, p ∣ n ^ 2 + 1 ∧ ↑p > 2 * ↑n + √(2 * ↑n)"
},
{
"line": "refine\n no_implicit_lambda%\n (have hreal₅ : (k : ℝ) > 4 := ?body✝;\n ?_)",
"before_state": "case intro\np : ℕ\nhpp : Nat.Prime p\nhp_mod_4_eq_1 : p ≡ 1 [MOD 4]\nhp_gt_20 : p > 20\nthis : Fact (Nat.Prime p)\nhp_mod_4_ne_3 : p % 4 ≠ 3\ny : ZMod p\nhy : -1 = y * y\nm : ℤ := y.valMinAbs\nn : ℕ := m.natAbs\nhnat₁ : p ∣ n ^ 2 + 1\nhnat₂ : n ≤ p / 2\nhnat₃ : p ≥ 2 * n\nk : ℕ := p - 2 * n\nhnat₄ : k = p - 2 * n\nhnat₅ : p ∣ k ^ 2 + 4\nhnat₆ : k ^ 2 + 4 ≥ p\nhreal₁ : ↑k = ↑p - 2 * ↑n\nhreal₂ : ↑p > 20\nhreal₃ : ↑k ^ 2 + 4 ≥ ↑p\n⊢ ∃ n, p ∣ n ^ 2 + 1 ∧ ↑p > 2 * ↑n + √(2 * ↑n)",
"after_state": "case body\np : ℕ\nhpp : Nat.Prime p\nhp_mod_4_eq_1 : p ≡ 1 [MOD 4]\nhp_gt_20 : p > 20\nthis : Fact (Nat.Prime p)\nhp_mod_4_ne_3 : p % 4 ≠ 3\ny : ZMod p\nhy : -1 = y * y\nm : ℤ := y.valMinAbs\nn : ℕ := m.natAbs\nhnat₁ : p ∣ n ^ 2 + 1\nhnat₂ : n ≤ p / 2\nhnat₃ : p ≥ 2 * n\nk : ℕ := p - 2 * n\nhnat₄ : k = p - 2 * n\nhnat₅ : p ∣ k ^ 2 + 4\nhnat₆ : k ^ 2 + 4 ≥ p\nhreal₁ : ↑k = ↑p - 2 * ↑n\nhreal₂ : ↑p > 20\nhreal₃ : ↑k ^ 2 + 4 ≥ ↑p\n⊢ ↑k > 4\n---\ncase intro\np : ℕ\nhpp : Nat.Prime p\nhp_mod_4_eq_1 : p ≡ 1 [MOD 4]\nhp_gt_20 : p > 20\nthis : Fact (Nat.Prime p)\nhp_mod_4_ne_3 : p % 4 ≠ 3\ny : ZMod p\nhy : -1 = y * y\nm : ℤ := y.valMinAbs\nn : ℕ := m.natAbs\nhnat₁ : p ∣ n ^ 2 + 1\nhnat₂ : n ≤ p / 2\nhnat₃ : p ≥ 2 * n\nk : ℕ := p - 2 * n\nhnat₄ : k = p - 2 * n\nhnat₅ : p ∣ k ^ 2 + 4\nhnat₆ : k ^ 2 + 4 ≥ p\nhreal₁ : ↑k = ↑p - 2 * ↑n\nhreal₂ : ↑p > 20\nhreal₃ : ↑k ^ 2 + 4 ≥ ↑p\nhreal₅ : ↑k > 4\n⊢ ∃ n, p ∣ n ^ 2 + 1 ∧ ↑p > 2 * ↑n + √(2 * ↑n)"
},
{
"line": "case body✝ =>\n with_annotate_state\"by\"\n ( refine lt_of_pow_lt_pow_left₀ 2 k.cast_nonneg ?_\n linarith only [hreal₂, hreal₃])",
"before_state": "case body\np : ℕ\nhpp : Nat.Prime p\nhp_mod_4_eq_1 : p ≡ 1 [MOD 4]\nhp_gt_20 : p > 20\nthis : Fact (Nat.Prime p)\nhp_mod_4_ne_3 : p % 4 ≠ 3\ny : ZMod p\nhy : -1 = y * y\nm : ℤ := y.valMinAbs\nn : ℕ := m.natAbs\nhnat₁ : p ∣ n ^ 2 + 1\nhnat₂ : n ≤ p / 2\nhnat₃ : p ≥ 2 * n\nk : ℕ := p - 2 * n\nhnat₄ : k = p - 2 * n\nhnat₅ : p ∣ k ^ 2 + 4\nhnat₆ : k ^ 2 + 4 ≥ p\nhreal₁ : ↑k = ↑p - 2 * ↑n\nhreal₂ : ↑p > 20\nhreal₃ : ↑k ^ 2 + 4 ≥ ↑p\n⊢ ↑k > 4\n---\ncase intro\np : ℕ\nhpp : Nat.Prime p\nhp_mod_4_eq_1 : p ≡ 1 [MOD 4]\nhp_gt_20 : p > 20\nthis : Fact (Nat.Prime p)\nhp_mod_4_ne_3 : p % 4 ≠ 3\ny : ZMod p\nhy : -1 = y * y\nm : ℤ := y.valMinAbs\nn : ℕ := m.natAbs\nhnat₁ : p ∣ n ^ 2 + 1\nhnat₂ : n ≤ p / 2\nhnat₃ : p ≥ 2 * n\nk : ℕ := p - 2 * n\nhnat₄ : k = p - 2 * n\nhnat₅ : p ∣ k ^ 2 + 4\nhnat₆ : k ^ 2 + 4 ≥ p\nhreal₁ : ↑k = ↑p - 2 * ↑n\nhreal₂ : ↑p > 20\nhreal₃ : ↑k ^ 2 + 4 ≥ ↑p\nhreal₅ : ↑k > 4\n⊢ ∃ n, p ∣ n ^ 2 + 1 ∧ ↑p > 2 * ↑n + √(2 * ↑n)",
"after_state": "case intro\np : ℕ\nhpp : Nat.Prime p\nhp_mod_4_eq_1 : p ≡ 1 [MOD 4]\nhp_gt_20 : p > 20\nthis : Fact (Nat.Prime p)\nhp_mod_4_ne_3 : p % 4 ≠ 3\ny : ZMod p\nhy : -1 = y * y\nm : ℤ := y.valMinAbs\nn : ℕ := m.natAbs\nhnat₁ : p ∣ n ^ 2 + 1\nhnat₂ : n ≤ p / 2\nhnat₃ : p ≥ 2 * n\nk : ℕ := p - 2 * n\nhnat₄ : k = p - 2 * n\nhnat₅ : p ∣ k ^ 2 + 4\nhnat₆ : k ^ 2 + 4 ≥ p\nhreal₁ : ↑k = ↑p - 2 * ↑n\nhreal₂ : ↑p > 20\nhreal₃ : ↑k ^ 2 + 4 ≥ ↑p\nhreal₅ : ↑k > 4\n⊢ ∃ n, p ∣ n ^ 2 + 1 ∧ ↑p > 2 * ↑n + √(2 * ↑n)"
},
{
"line": "with_annotate_state\"by\"\n ( refine lt_of_pow_lt_pow_left₀ 2 k.cast_nonneg ?_\n linarith only [hreal₂, hreal₃])",
"before_state": "p : ℕ\nhpp : Nat.Prime p\nhp_mod_4_eq_1 : p ≡ 1 [MOD 4]\nhp_gt_20 : p > 20\nthis : Fact (Nat.Prime p)\nhp_mod_4_ne_3 : p % 4 ≠ 3\ny : ZMod p\nhy : -1 = y * y\nm : ℤ := y.valMinAbs\nn : ℕ := m.natAbs\nhnat₁ : p ∣ n ^ 2 + 1\nhnat₂ : n ≤ p / 2\nhnat₃ : p ≥ 2 * n\nk : ℕ := p - 2 * n\nhnat₄ : k = p - 2 * n\nhnat₅ : p ∣ k ^ 2 + 4\nhnat₆ : k ^ 2 + 4 ≥ p\nhreal₁ : ↑k = ↑p - 2 * ↑n\nhreal₂ : ↑p > 20\nhreal₃ : ↑k ^ 2 + 4 ≥ ↑p\n⊢ ↑k > 4",
"after_state": "No Goals!"
},
{
"line": "refine lt_of_pow_lt_pow_left₀ 2 k.cast_nonneg ?_",
"before_state": "p : ℕ\nhpp : Nat.Prime p\nhp_mod_4_eq_1 : p ≡ 1 [MOD 4]\nhp_gt_20 : p > 20\nthis : Fact (Nat.Prime p)\nhp_mod_4_ne_3 : p % 4 ≠ 3\ny : ZMod p\nhy : -1 = y * y\nm : ℤ := y.valMinAbs\nn : ℕ := m.natAbs\nhnat₁ : p ∣ n ^ 2 + 1\nhnat₂ : n ≤ p / 2\nhnat₃ : p ≥ 2 * n\nk : ℕ := p - 2 * n\nhnat₄ : k = p - 2 * n\nhnat₅ : p ∣ k ^ 2 + 4\nhnat₆ : k ^ 2 + 4 ≥ p\nhreal₁ : ↑k = ↑p - 2 * ↑n\nhreal₂ : ↑p > 20\nhreal₃ : ↑k ^ 2 + 4 ≥ ↑p\n⊢ ↑k > 4",
"after_state": "p : ℕ\nhpp : Nat.Prime p\nhp_mod_4_eq_1 : p ≡ 1 [MOD 4]\nhp_gt_20 : p > 20\nthis : Fact (Nat.Prime p)\nhp_mod_4_ne_3 : p % 4 ≠ 3\ny : ZMod p\nhy : -1 = y * y\nm : ℤ := y.valMinAbs\nn : ℕ := m.natAbs\nhnat₁ : p ∣ n ^ 2 + 1\nhnat₂ : n ≤ p / 2\nhnat₃ : p ≥ 2 * n\nk : ℕ := p - 2 * n\nhnat₄ : k = p - 2 * n\nhnat₅ : p ∣ k ^ 2 + 4\nhnat₆ : k ^ 2 + 4 ≥ p\nhreal₁ : ↑k = ↑p - 2 * ↑n\nhreal₂ : ↑p > 20\nhreal₃ : ↑k ^ 2 + 4 ≥ ↑p\n⊢ 4 ^ 2 < ↑k ^ 2"
},
{
"line": "linarith only [hreal₂, hreal₃]",
"before_state": "p : ℕ\nhpp : Nat.Prime p\nhp_mod_4_eq_1 : p ≡ 1 [MOD 4]\nhp_gt_20 : p > 20\nthis : Fact (Nat.Prime p)\nhp_mod_4_ne_3 : p % 4 ≠ 3\ny : ZMod p\nhy : -1 = y * y\nm : ℤ := y.valMinAbs\nn : ℕ := m.natAbs\nhnat₁ : p ∣ n ^ 2 + 1\nhnat₂ : n ≤ p / 2\nhnat₃ : p ≥ 2 * n\nk : ℕ := p - 2 * n\nhnat₄ : k = p - 2 * n\nhnat₅ : p ∣ k ^ 2 + 4\nhnat₆ : k ^ 2 + 4 ≥ p\nhreal₁ : ↑k = ↑p - 2 * ↑n\nhreal₂ : ↑p > 20\nhreal₃ : ↑k ^ 2 + 4 ≥ ↑p\n⊢ 4 ^ 2 < ↑k ^ 2",
"after_state": "No Goals!"
},
{
"line": "ring1",
"before_state": "p : ℕ\nhpp : Nat.Prime p\nhp_mod_4_eq_1 : p ≡ 1 [MOD 4]\nhp_gt_20 : p > 20\nthis : Fact (Nat.Prime p)\nhp_mod_4_ne_3 : p % 4 ≠ 3\ny : ZMod p\nhy : -1 = y * y\nm : ℤ := y.valMinAbs\nn : ℕ := m.natAbs\nhnat₁ : p ∣ n ^ 2 + 1\nhnat₂ : n ≤ p / 2\nhnat₃ : p ≥ 2 * n\nk : ℕ := p - 2 * n\nhnat₄ : k = p - 2 * n\nhnat₅ : p ∣ k ^ 2 + 4\nhnat₆ : k ^ 2 + 4 ≥ p\nhreal₁ : ↑k = ↑p - 2 * ↑n\nhreal₂ : ↑p > 20\nhreal₃ : ↑k ^ 2 + 4 ≥ ↑p\na✝ : 4 ^ 2 ≥ ↑k ^ 2\n⊢ ↑k ^ 2 - 4 ^ 2 + (20 - ↑p) + (↑p - (↑k ^ 2 + 4)) = 0",
"after_state": "No Goals!"
},
{
"line": "have hreal₆ : (k : ℝ) > sqrt (2 * n) :=\n by\n refine lt_of_pow_lt_pow_left₀ 2 k.cast_nonneg ?_\n rw [sq_sqrt (mul_nonneg zero_le_two n.cast_nonneg)]\n linarith only [hreal₁, hreal₃, hreal₅]",
"before_state": "case intro\np : ℕ\nhpp : Nat.Prime p\nhp_mod_4_eq_1 : p ≡ 1 [MOD 4]\nhp_gt_20 : p > 20\nthis : Fact (Nat.Prime p)\nhp_mod_4_ne_3 : p % 4 ≠ 3\ny : ZMod p\nhy : -1 = y * y\nm : ℤ := y.valMinAbs\nn : ℕ := m.natAbs\nhnat₁ : p ∣ n ^ 2 + 1\nhnat₂ : n ≤ p / 2\nhnat₃ : p ≥ 2 * n\nk : ℕ := p - 2 * n\nhnat₄ : k = p - 2 * n\nhnat₅ : p ∣ k ^ 2 + 4\nhnat₆ : k ^ 2 + 4 ≥ p\nhreal₁ : ↑k = ↑p - 2 * ↑n\nhreal₂ : ↑p > 20\nhreal₃ : ↑k ^ 2 + 4 ≥ ↑p\nhreal₅ : ↑k > 4\n⊢ ∃ n, p ∣ n ^ 2 + 1 ∧ ↑p > 2 * ↑n + √(2 * ↑n)",
"after_state": "case intro\np : ℕ\nhpp : Nat.Prime p\nhp_mod_4_eq_1 : p ≡ 1 [MOD 4]\nhp_gt_20 : p > 20\nthis : Fact (Nat.Prime p)\nhp_mod_4_ne_3 : p % 4 ≠ 3\ny : ZMod p\nhy : -1 = y * y\nm : ℤ := y.valMinAbs\nn : ℕ := m.natAbs\nhnat₁ : p ∣ n ^ 2 + 1\nhnat₂ : n ≤ p / 2\nhnat₃ : p ≥ 2 * n\nk : ℕ := p - 2 * n\nhnat₄ : k = p - 2 * n\nhnat₅ : p ∣ k ^ 2 + 4\nhnat₆ : k ^ 2 + 4 ≥ p\nhreal₁ : ↑k = ↑p - 2 * ↑n\nhreal₂ : ↑p > 20\nhreal₃ : ↑k ^ 2 + 4 ≥ ↑p\nhreal₅ : ↑k > 4\nhreal₆ : ↑k > √(2 * ↑n)\n⊢ ∃ n, p ∣ n ^ 2 + 1 ∧ ↑p > 2 * ↑n + √(2 * ↑n)"
},
{
"line": "focus\n refine\n no_implicit_lambda%\n (have hreal₆ : (k : ℝ) > sqrt (2 * n) := ?body✝;\n ?_)\n case body✝ =>\n with_annotate_state\"by\"\n ( refine lt_of_pow_lt_pow_left₀ 2 k.cast_nonneg ?_\n rw [sq_sqrt (mul_nonneg zero_le_two n.cast_nonneg)]\n linarith only [hreal₁, hreal₃, hreal₅])",
"before_state": "case intro\np : ℕ\nhpp : Nat.Prime p\nhp_mod_4_eq_1 : p ≡ 1 [MOD 4]\nhp_gt_20 : p > 20\nthis : Fact (Nat.Prime p)\nhp_mod_4_ne_3 : p % 4 ≠ 3\ny : ZMod p\nhy : -1 = y * y\nm : ℤ := y.valMinAbs\nn : ℕ := m.natAbs\nhnat₁ : p ∣ n ^ 2 + 1\nhnat₂ : n ≤ p / 2\nhnat₃ : p ≥ 2 * n\nk : ℕ := p - 2 * n\nhnat₄ : k = p - 2 * n\nhnat₅ : p ∣ k ^ 2 + 4\nhnat₆ : k ^ 2 + 4 ≥ p\nhreal₁ : ↑k = ↑p - 2 * ↑n\nhreal₂ : ↑p > 20\nhreal₃ : ↑k ^ 2 + 4 ≥ ↑p\nhreal₅ : ↑k > 4\n⊢ ∃ n, p ∣ n ^ 2 + 1 ∧ ↑p > 2 * ↑n + √(2 * ↑n)",
"after_state": "case intro\np : ℕ\nhpp : Nat.Prime p\nhp_mod_4_eq_1 : p ≡ 1 [MOD 4]\nhp_gt_20 : p > 20\nthis : Fact (Nat.Prime p)\nhp_mod_4_ne_3 : p % 4 ≠ 3\ny : ZMod p\nhy : -1 = y * y\nm : ℤ := y.valMinAbs\nn : ℕ := m.natAbs\nhnat₁ : p ∣ n ^ 2 + 1\nhnat₂ : n ≤ p / 2\nhnat₃ : p ≥ 2 * n\nk : ℕ := p - 2 * n\nhnat₄ : k = p - 2 * n\nhnat₅ : p ∣ k ^ 2 + 4\nhnat₆ : k ^ 2 + 4 ≥ p\nhreal₁ : ↑k = ↑p - 2 * ↑n\nhreal₂ : ↑p > 20\nhreal₃ : ↑k ^ 2 + 4 ≥ ↑p\nhreal₅ : ↑k > 4\nhreal₆ : ↑k > √(2 * ↑n)\n⊢ ∃ n, p ∣ n ^ 2 + 1 ∧ ↑p > 2 * ↑n + √(2 * ↑n)"
},
{
"line": "refine\n no_implicit_lambda%\n (have hreal₆ : (k : ℝ) > sqrt (2 * n) := ?body✝;\n ?_)",
"before_state": "case intro\np : ℕ\nhpp : Nat.Prime p\nhp_mod_4_eq_1 : p ≡ 1 [MOD 4]\nhp_gt_20 : p > 20\nthis : Fact (Nat.Prime p)\nhp_mod_4_ne_3 : p % 4 ≠ 3\ny : ZMod p\nhy : -1 = y * y\nm : ℤ := y.valMinAbs\nn : ℕ := m.natAbs\nhnat₁ : p ∣ n ^ 2 + 1\nhnat₂ : n ≤ p / 2\nhnat₃ : p ≥ 2 * n\nk : ℕ := p - 2 * n\nhnat₄ : k = p - 2 * n\nhnat₅ : p ∣ k ^ 2 + 4\nhnat₆ : k ^ 2 + 4 ≥ p\nhreal₁ : ↑k = ↑p - 2 * ↑n\nhreal₂ : ↑p > 20\nhreal₃ : ↑k ^ 2 + 4 ≥ ↑p\nhreal₅ : ↑k > 4\n⊢ ∃ n, p ∣ n ^ 2 + 1 ∧ ↑p > 2 * ↑n + √(2 * ↑n)",
"after_state": "case body\np : ℕ\nhpp : Nat.Prime p\nhp_mod_4_eq_1 : p ≡ 1 [MOD 4]\nhp_gt_20 : p > 20\nthis : Fact (Nat.Prime p)\nhp_mod_4_ne_3 : p % 4 ≠ 3\ny : ZMod p\nhy : -1 = y * y\nm : ℤ := y.valMinAbs\nn : ℕ := m.natAbs\nhnat₁ : p ∣ n ^ 2 + 1\nhnat₂ : n ≤ p / 2\nhnat₃ : p ≥ 2 * n\nk : ℕ := p - 2 * n\nhnat₄ : k = p - 2 * n\nhnat₅ : p ∣ k ^ 2 + 4\nhnat₆ : k ^ 2 + 4 ≥ p\nhreal₁ : ↑k = ↑p - 2 * ↑n\nhreal₂ : ↑p > 20\nhreal₃ : ↑k ^ 2 + 4 ≥ ↑p\nhreal₅ : ↑k > 4\n⊢ ↑k > √(2 * ↑n)\n---\ncase intro\np : ℕ\nhpp : Nat.Prime p\nhp_mod_4_eq_1 : p ≡ 1 [MOD 4]\nhp_gt_20 : p > 20\nthis : Fact (Nat.Prime p)\nhp_mod_4_ne_3 : p % 4 ≠ 3\ny : ZMod p\nhy : -1 = y * y\nm : ℤ := y.valMinAbs\nn : ℕ := m.natAbs\nhnat₁ : p ∣ n ^ 2 + 1\nhnat₂ : n ≤ p / 2\nhnat₃ : p ≥ 2 * n\nk : ℕ := p - 2 * n\nhnat₄ : k = p - 2 * n\nhnat₅ : p ∣ k ^ 2 + 4\nhnat₆ : k ^ 2 + 4 ≥ p\nhreal₁ : ↑k = ↑p - 2 * ↑n\nhreal₂ : ↑p > 20\nhreal₃ : ↑k ^ 2 + 4 ≥ ↑p\nhreal₅ : ↑k > 4\nhreal₆ : ↑k > √(2 * ↑n)\n⊢ ∃ n, p ∣ n ^ 2 + 1 ∧ ↑p > 2 * ↑n + √(2 * ↑n)"
},
{
"line": "case body✝ =>\n with_annotate_state\"by\"\n ( refine lt_of_pow_lt_pow_left₀ 2 k.cast_nonneg ?_\n rw [sq_sqrt (mul_nonneg zero_le_two n.cast_nonneg)]\n linarith only [hreal₁, hreal₃, hreal₅])",
"before_state": "case body\np : ℕ\nhpp : Nat.Prime p\nhp_mod_4_eq_1 : p ≡ 1 [MOD 4]\nhp_gt_20 : p > 20\nthis : Fact (Nat.Prime p)\nhp_mod_4_ne_3 : p % 4 ≠ 3\ny : ZMod p\nhy : -1 = y * y\nm : ℤ := y.valMinAbs\nn : ℕ := m.natAbs\nhnat₁ : p ∣ n ^ 2 + 1\nhnat₂ : n ≤ p / 2\nhnat₃ : p ≥ 2 * n\nk : ℕ := p - 2 * n\nhnat₄ : k = p - 2 * n\nhnat₅ : p ∣ k ^ 2 + 4\nhnat₆ : k ^ 2 + 4 ≥ p\nhreal₁ : ↑k = ↑p - 2 * ↑n\nhreal₂ : ↑p > 20\nhreal₃ : ↑k ^ 2 + 4 ≥ ↑p\nhreal₅ : ↑k > 4\n⊢ ↑k > √(2 * ↑n)\n---\ncase intro\np : ℕ\nhpp : Nat.Prime p\nhp_mod_4_eq_1 : p ≡ 1 [MOD 4]\nhp_gt_20 : p > 20\nthis : Fact (Nat.Prime p)\nhp_mod_4_ne_3 : p % 4 ≠ 3\ny : ZMod p\nhy : -1 = y * y\nm : ℤ := y.valMinAbs\nn : ℕ := m.natAbs\nhnat₁ : p ∣ n ^ 2 + 1\nhnat₂ : n ≤ p / 2\nhnat₃ : p ≥ 2 * n\nk : ℕ := p - 2 * n\nhnat₄ : k = p - 2 * n\nhnat₅ : p ∣ k ^ 2 + 4\nhnat₆ : k ^ 2 + 4 ≥ p\nhreal₁ : ↑k = ↑p - 2 * ↑n\nhreal₂ : ↑p > 20\nhreal₃ : ↑k ^ 2 + 4 ≥ ↑p\nhreal₅ : ↑k > 4\nhreal₆ : ↑k > √(2 * ↑n)\n⊢ ∃ n, p ∣ n ^ 2 + 1 ∧ ↑p > 2 * ↑n + √(2 * ↑n)",
"after_state": "case intro\np : ℕ\nhpp : Nat.Prime p\nhp_mod_4_eq_1 : p ≡ 1 [MOD 4]\nhp_gt_20 : p > 20\nthis : Fact (Nat.Prime p)\nhp_mod_4_ne_3 : p % 4 ≠ 3\ny : ZMod p\nhy : -1 = y * y\nm : ℤ := y.valMinAbs\nn : ℕ := m.natAbs\nhnat₁ : p ∣ n ^ 2 + 1\nhnat₂ : n ≤ p / 2\nhnat₃ : p ≥ 2 * n\nk : ℕ := p - 2 * n\nhnat₄ : k = p - 2 * n\nhnat₅ : p ∣ k ^ 2 + 4\nhnat₆ : k ^ 2 + 4 ≥ p\nhreal₁ : ↑k = ↑p - 2 * ↑n\nhreal₂ : ↑p > 20\nhreal₃ : ↑k ^ 2 + 4 ≥ ↑p\nhreal₅ : ↑k > 4\nhreal₆ : ↑k > √(2 * ↑n)\n⊢ ∃ n, p ∣ n ^ 2 + 1 ∧ ↑p > 2 * ↑n + √(2 * ↑n)"
},
{
"line": "with_annotate_state\"by\"\n ( refine lt_of_pow_lt_pow_left₀ 2 k.cast_nonneg ?_\n rw [sq_sqrt (mul_nonneg zero_le_two n.cast_nonneg)]\n linarith only [hreal₁, hreal₃, hreal₅])",
"before_state": "p : ℕ\nhpp : Nat.Prime p\nhp_mod_4_eq_1 : p ≡ 1 [MOD 4]\nhp_gt_20 : p > 20\nthis : Fact (Nat.Prime p)\nhp_mod_4_ne_3 : p % 4 ≠ 3\ny : ZMod p\nhy : -1 = y * y\nm : ℤ := y.valMinAbs\nn : ℕ := m.natAbs\nhnat₁ : p ∣ n ^ 2 + 1\nhnat₂ : n ≤ p / 2\nhnat₃ : p ≥ 2 * n\nk : ℕ := p - 2 * n\nhnat₄ : k = p - 2 * n\nhnat₅ : p ∣ k ^ 2 + 4\nhnat₆ : k ^ 2 + 4 ≥ p\nhreal₁ : ↑k = ↑p - 2 * ↑n\nhreal₂ : ↑p > 20\nhreal₃ : ↑k ^ 2 + 4 ≥ ↑p\nhreal₅ : ↑k > 4\n⊢ ↑k > √(2 * ↑n)",
"after_state": "No Goals!"
},
{
"line": "refine lt_of_pow_lt_pow_left₀ 2 k.cast_nonneg ?_",
"before_state": "p : ℕ\nhpp : Nat.Prime p\nhp_mod_4_eq_1 : p ≡ 1 [MOD 4]\nhp_gt_20 : p > 20\nthis : Fact (Nat.Prime p)\nhp_mod_4_ne_3 : p % 4 ≠ 3\ny : ZMod p\nhy : -1 = y * y\nm : ℤ := y.valMinAbs\nn : ℕ := m.natAbs\nhnat₁ : p ∣ n ^ 2 + 1\nhnat₂ : n ≤ p / 2\nhnat₃ : p ≥ 2 * n\nk : ℕ := p - 2 * n\nhnat₄ : k = p - 2 * n\nhnat₅ : p ∣ k ^ 2 + 4\nhnat₆ : k ^ 2 + 4 ≥ p\nhreal₁ : ↑k = ↑p - 2 * ↑n\nhreal₂ : ↑p > 20\nhreal₃ : ↑k ^ 2 + 4 ≥ ↑p\nhreal₅ : ↑k > 4\n⊢ ↑k > √(2 * ↑n)",
"after_state": "p : ℕ\nhpp : Nat.Prime p\nhp_mod_4_eq_1 : p ≡ 1 [MOD 4]\nhp_gt_20 : p > 20\nthis : Fact (Nat.Prime p)\nhp_mod_4_ne_3 : p % 4 ≠ 3\ny : ZMod p\nhy : -1 = y * y\nm : ℤ := y.valMinAbs\nn : ℕ := m.natAbs\nhnat₁ : p ∣ n ^ 2 + 1\nhnat₂ : n ≤ p / 2\nhnat₃ : p ≥ 2 * n\nk : ℕ := p - 2 * n\nhnat₄ : k = p - 2 * n\nhnat₅ : p ∣ k ^ 2 + 4\nhnat₆ : k ^ 2 + 4 ≥ p\nhreal₁ : ↑k = ↑p - 2 * ↑n\nhreal₂ : ↑p > 20\nhreal₃ : ↑k ^ 2 + 4 ≥ ↑p\nhreal₅ : ↑k > 4\n⊢ √(2 * ↑n) ^ 2 < ↑k ^ 2"
},
{
"line": "rw [sq_sqrt (mul_nonneg zero_le_two n.cast_nonneg)]",
"before_state": "p : ℕ\nhpp : Nat.Prime p\nhp_mod_4_eq_1 : p ≡ 1 [MOD 4]\nhp_gt_20 : p > 20\nthis : Fact (Nat.Prime p)\nhp_mod_4_ne_3 : p % 4 ≠ 3\ny : ZMod p\nhy : -1 = y * y\nm : ℤ := y.valMinAbs\nn : ℕ := m.natAbs\nhnat₁ : p ∣ n ^ 2 + 1\nhnat₂ : n ≤ p / 2\nhnat₃ : p ≥ 2 * n\nk : ℕ := p - 2 * n\nhnat₄ : k = p - 2 * n\nhnat₅ : p ∣ k ^ 2 + 4\nhnat₆ : k ^ 2 + 4 ≥ p\nhreal₁ : ↑k = ↑p - 2 * ↑n\nhreal₂ : ↑p > 20\nhreal₃ : ↑k ^ 2 + 4 ≥ ↑p\nhreal₅ : ↑k > 4\n⊢ √(2 * ↑n) ^ 2 < ↑k ^ 2",
"after_state": "p : ℕ\nhpp : Nat.Prime p\nhp_mod_4_eq_1 : p ≡ 1 [MOD 4]\nhp_gt_20 : p > 20\nthis : Fact (Nat.Prime p)\nhp_mod_4_ne_3 : p % 4 ≠ 3\ny : ZMod p\nhy : -1 = y * y\nm : ℤ := y.valMinAbs\nn : ℕ := m.natAbs\nhnat₁ : p ∣ n ^ 2 + 1\nhnat₂ : n ≤ p / 2\nhnat₃ : p ≥ 2 * n\nk : ℕ := p - 2 * n\nhnat₄ : k = p - 2 * n\nhnat₅ : p ∣ k ^ 2 + 4\nhnat₆ : k ^ 2 + 4 ≥ p\nhreal₁ : ↑k = ↑p - 2 * ↑n\nhreal₂ : ↑p > 20\nhreal₃ : ↑k ^ 2 + 4 ≥ ↑p\nhreal₅ : ↑k > 4\n⊢ 2 * ↑n < ↑k ^ 2"
},
{
"line": "rewrite [sq_sqrt (mul_nonneg zero_le_two n.cast_nonneg)]",
"before_state": "p : ℕ\nhpp : Nat.Prime p\nhp_mod_4_eq_1 : p ≡ 1 [MOD 4]\nhp_gt_20 : p > 20\nthis : Fact (Nat.Prime p)\nhp_mod_4_ne_3 : p % 4 ≠ 3\ny : ZMod p\nhy : -1 = y * y\nm : ℤ := y.valMinAbs\nn : ℕ := m.natAbs\nhnat₁ : p ∣ n ^ 2 + 1\nhnat₂ : n ≤ p / 2\nhnat₃ : p ≥ 2 * n\nk : ℕ := p - 2 * n\nhnat₄ : k = p - 2 * n\nhnat₅ : p ∣ k ^ 2 + 4\nhnat₆ : k ^ 2 + 4 ≥ p\nhreal₁ : ↑k = ↑p - 2 * ↑n\nhreal₂ : ↑p > 20\nhreal₃ : ↑k ^ 2 + 4 ≥ ↑p\nhreal₅ : ↑k > 4\n⊢ √(2 * ↑n) ^ 2 < ↑k ^ 2",
"after_state": "p : ℕ\nhpp : Nat.Prime p\nhp_mod_4_eq_1 : p ≡ 1 [MOD 4]\nhp_gt_20 : p > 20\nthis : Fact (Nat.Prime p)\nhp_mod_4_ne_3 : p % 4 ≠ 3\ny : ZMod p\nhy : -1 = y * y\nm : ℤ := y.valMinAbs\nn : ℕ := m.natAbs\nhnat₁ : p ∣ n ^ 2 + 1\nhnat₂ : n ≤ p / 2\nhnat₃ : p ≥ 2 * n\nk : ℕ := p - 2 * n\nhnat₄ : k = p - 2 * n\nhnat₅ : p ∣ k ^ 2 + 4\nhnat₆ : k ^ 2 + 4 ≥ p\nhreal₁ : ↑k = ↑p - 2 * ↑n\nhreal₂ : ↑p > 20\nhreal₃ : ↑k ^ 2 + 4 ≥ ↑p\nhreal₅ : ↑k > 4\n⊢ 2 * ↑n < ↑k ^ 2"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "p : ℕ\nhpp : Nat.Prime p\nhp_mod_4_eq_1 : p ≡ 1 [MOD 4]\nhp_gt_20 : p > 20\nthis : Fact (Nat.Prime p)\nhp_mod_4_ne_3 : p % 4 ≠ 3\ny : ZMod p\nhy : -1 = y * y\nm : ℤ := y.valMinAbs\nn : ℕ := m.natAbs\nhnat₁ : p ∣ n ^ 2 + 1\nhnat₂ : n ≤ p / 2\nhnat₃ : p ≥ 2 * n\nk : ℕ := p - 2 * n\nhnat₄ : k = p - 2 * n\nhnat₅ : p ∣ k ^ 2 + 4\nhnat₆ : k ^ 2 + 4 ≥ p\nhreal₁ : ↑k = ↑p - 2 * ↑n\nhreal₂ : ↑p > 20\nhreal₃ : ↑k ^ 2 + 4 ≥ ↑p\nhreal₅ : ↑k > 4\n⊢ 2 * ↑n < ↑k ^ 2",
"after_state": "p : ℕ\nhpp : Nat.Prime p\nhp_mod_4_eq_1 : p ≡ 1 [MOD 4]\nhp_gt_20 : p > 20\nthis : Fact (Nat.Prime p)\nhp_mod_4_ne_3 : p % 4 ≠ 3\ny : ZMod p\nhy : -1 = y * y\nm : ℤ := y.valMinAbs\nn : ℕ := m.natAbs\nhnat₁ : p ∣ n ^ 2 + 1\nhnat₂ : n ≤ p / 2\nhnat₃ : p ≥ 2 * n\nk : ℕ := p - 2 * n\nhnat₄ : k = p - 2 * n\nhnat₅ : p ∣ k ^ 2 + 4\nhnat₆ : k ^ 2 + 4 ≥ p\nhreal₁ : ↑k = ↑p - 2 * ↑n\nhreal₂ : ↑p > 20\nhreal₃ : ↑k ^ 2 + 4 ≥ ↑p\nhreal₅ : ↑k > 4\n⊢ 2 * ↑n < ↑k ^ 2"
},
{
"line": "try (with_reducible rfl)",
"before_state": "p : ℕ\nhpp : Nat.Prime p\nhp_mod_4_eq_1 : p ≡ 1 [MOD 4]\nhp_gt_20 : p > 20\nthis : Fact (Nat.Prime p)\nhp_mod_4_ne_3 : p % 4 ≠ 3\ny : ZMod p\nhy : -1 = y * y\nm : ℤ := y.valMinAbs\nn : ℕ := m.natAbs\nhnat₁ : p ∣ n ^ 2 + 1\nhnat₂ : n ≤ p / 2\nhnat₃ : p ≥ 2 * n\nk : ℕ := p - 2 * n\nhnat₄ : k = p - 2 * n\nhnat₅ : p ∣ k ^ 2 + 4\nhnat₆ : k ^ 2 + 4 ≥ p\nhreal₁ : ↑k = ↑p - 2 * ↑n\nhreal₂ : ↑p > 20\nhreal₃ : ↑k ^ 2 + 4 ≥ ↑p\nhreal₅ : ↑k > 4\n⊢ 2 * ↑n < ↑k ^ 2",
"after_state": "p : ℕ\nhpp : Nat.Prime p\nhp_mod_4_eq_1 : p ≡ 1 [MOD 4]\nhp_gt_20 : p > 20\nthis : Fact (Nat.Prime p)\nhp_mod_4_ne_3 : p % 4 ≠ 3\ny : ZMod p\nhy : -1 = y * y\nm : ℤ := y.valMinAbs\nn : ℕ := m.natAbs\nhnat₁ : p ∣ n ^ 2 + 1\nhnat₂ : n ≤ p / 2\nhnat₃ : p ≥ 2 * n\nk : ℕ := p - 2 * n\nhnat₄ : k = p - 2 * n\nhnat₅ : p ∣ k ^ 2 + 4\nhnat₆ : k ^ 2 + 4 ≥ p\nhreal₁ : ↑k = ↑p - 2 * ↑n\nhreal₂ : ↑p > 20\nhreal₃ : ↑k ^ 2 + 4 ≥ ↑p\nhreal₅ : ↑k > 4\n⊢ 2 * ↑n < ↑k ^ 2"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "p : ℕ\nhpp : Nat.Prime p\nhp_mod_4_eq_1 : p ≡ 1 [MOD 4]\nhp_gt_20 : p > 20\nthis : Fact (Nat.Prime p)\nhp_mod_4_ne_3 : p % 4 ≠ 3\ny : ZMod p\nhy : -1 = y * y\nm : ℤ := y.valMinAbs\nn : ℕ := m.natAbs\nhnat₁ : p ∣ n ^ 2 + 1\nhnat₂ : n ≤ p / 2\nhnat₃ : p ≥ 2 * n\nk : ℕ := p - 2 * n\nhnat₄ : k = p - 2 * n\nhnat₅ : p ∣ k ^ 2 + 4\nhnat₆ : k ^ 2 + 4 ≥ p\nhreal₁ : ↑k = ↑p - 2 * ↑n\nhreal₂ : ↑p > 20\nhreal₃ : ↑k ^ 2 + 4 ≥ ↑p\nhreal₅ : ↑k > 4\n⊢ 2 * ↑n < ↑k ^ 2",
"after_state": "p : ℕ\nhpp : Nat.Prime p\nhp_mod_4_eq_1 : p ≡ 1 [MOD 4]\nhp_gt_20 : p > 20\nthis : Fact (Nat.Prime p)\nhp_mod_4_ne_3 : p % 4 ≠ 3\ny : ZMod p\nhy : -1 = y * y\nm : ℤ := y.valMinAbs\nn : ℕ := m.natAbs\nhnat₁ : p ∣ n ^ 2 + 1\nhnat₂ : n ≤ p / 2\nhnat₃ : p ≥ 2 * n\nk : ℕ := p - 2 * n\nhnat₄ : k = p - 2 * n\nhnat₅ : p ∣ k ^ 2 + 4\nhnat₆ : k ^ 2 + 4 ≥ p\nhreal₁ : ↑k = ↑p - 2 * ↑n\nhreal₂ : ↑p > 20\nhreal₃ : ↑k ^ 2 + 4 ≥ ↑p\nhreal₅ : ↑k > 4\n⊢ 2 * ↑n < ↑k ^ 2"
},
{
"line": "with_reducible rfl",
"before_state": "p : ℕ\nhpp : Nat.Prime p\nhp_mod_4_eq_1 : p ≡ 1 [MOD 4]\nhp_gt_20 : p > 20\nthis : Fact (Nat.Prime p)\nhp_mod_4_ne_3 : p % 4 ≠ 3\ny : ZMod p\nhy : -1 = y * y\nm : ℤ := y.valMinAbs\nn : ℕ := m.natAbs\nhnat₁ : p ∣ n ^ 2 + 1\nhnat₂ : n ≤ p / 2\nhnat₃ : p ≥ 2 * n\nk : ℕ := p - 2 * n\nhnat₄ : k = p - 2 * n\nhnat₅ : p ∣ k ^ 2 + 4\nhnat₆ : k ^ 2 + 4 ≥ p\nhreal₁ : ↑k = ↑p - 2 * ↑n\nhreal₂ : ↑p > 20\nhreal₃ : ↑k ^ 2 + 4 ≥ ↑p\nhreal₅ : ↑k > 4\n⊢ 2 * ↑n < ↑k ^ 2",
"after_state": "p : ℕ\nhpp : Nat.Prime p\nhp_mod_4_eq_1 : p ≡ 1 [MOD 4]\nhp_gt_20 : p > 20\nthis : Fact (Nat.Prime p)\nhp_mod_4_ne_3 : p % 4 ≠ 3\ny : ZMod p\nhy : -1 = y * y\nm : ℤ := y.valMinAbs\nn : ℕ := m.natAbs\nhnat₁ : p ∣ n ^ 2 + 1\nhnat₂ : n ≤ p / 2\nhnat₃ : p ≥ 2 * n\nk : ℕ := p - 2 * n\nhnat₄ : k = p - 2 * n\nhnat₅ : p ∣ k ^ 2 + 4\nhnat₆ : k ^ 2 + 4 ≥ p\nhreal₁ : ↑k = ↑p - 2 * ↑n\nhreal₂ : ↑p > 20\nhreal₃ : ↑k ^ 2 + 4 ≥ ↑p\nhreal₅ : ↑k > 4\n⊢ 2 * ↑n < ↑k ^ 2"
},
{
"line": "rfl",
"before_state": "p : ℕ\nhpp : Nat.Prime p\nhp_mod_4_eq_1 : p ≡ 1 [MOD 4]\nhp_gt_20 : p > 20\nthis : Fact (Nat.Prime p)\nhp_mod_4_ne_3 : p % 4 ≠ 3\ny : ZMod p\nhy : -1 = y * y\nm : ℤ := y.valMinAbs\nn : ℕ := m.natAbs\nhnat₁ : p ∣ n ^ 2 + 1\nhnat₂ : n ≤ p / 2\nhnat₃ : p ≥ 2 * n\nk : ℕ := p - 2 * n\nhnat₄ : k = p - 2 * n\nhnat₅ : p ∣ k ^ 2 + 4\nhnat₆ : k ^ 2 + 4 ≥ p\nhreal₁ : ↑k = ↑p - 2 * ↑n\nhreal₂ : ↑p > 20\nhreal₃ : ↑k ^ 2 + 4 ≥ ↑p\nhreal₅ : ↑k > 4\n⊢ 2 * ↑n < ↑k ^ 2",
"after_state": "p : ℕ\nhpp : Nat.Prime p\nhp_mod_4_eq_1 : p ≡ 1 [MOD 4]\nhp_gt_20 : p > 20\nthis : Fact (Nat.Prime p)\nhp_mod_4_ne_3 : p % 4 ≠ 3\ny : ZMod p\nhy : -1 = y * y\nm : ℤ := y.valMinAbs\nn : ℕ := m.natAbs\nhnat₁ : p ∣ n ^ 2 + 1\nhnat₂ : n ≤ p / 2\nhnat₃ : p ≥ 2 * n\nk : ℕ := p - 2 * n\nhnat₄ : k = p - 2 * n\nhnat₅ : p ∣ k ^ 2 + 4\nhnat₆ : k ^ 2 + 4 ≥ p\nhreal₁ : ↑k = ↑p - 2 * ↑n\nhreal₂ : ↑p > 20\nhreal₃ : ↑k ^ 2 + 4 ≥ ↑p\nhreal₅ : ↑k > 4\n⊢ 2 * ↑n < ↑k ^ 2"
},
{
"line": "apply_rfl",
"before_state": "p : ℕ\nhpp : Nat.Prime p\nhp_mod_4_eq_1 : p ≡ 1 [MOD 4]\nhp_gt_20 : p > 20\nthis : Fact (Nat.Prime p)\nhp_mod_4_ne_3 : p % 4 ≠ 3\ny : ZMod p\nhy : -1 = y * y\nm : ℤ := y.valMinAbs\nn : ℕ := m.natAbs\nhnat₁ : p ∣ n ^ 2 + 1\nhnat₂ : n ≤ p / 2\nhnat₃ : p ≥ 2 * n\nk : ℕ := p - 2 * n\nhnat₄ : k = p - 2 * n\nhnat₅ : p ∣ k ^ 2 + 4\nhnat₆ : k ^ 2 + 4 ≥ p\nhreal₁ : ↑k = ↑p - 2 * ↑n\nhreal₂ : ↑p > 20\nhreal₃ : ↑k ^ 2 + 4 ≥ ↑p\nhreal₅ : ↑k > 4\n⊢ 2 * ↑n < ↑k ^ 2",
"after_state": "p : ℕ\nhpp : Nat.Prime p\nhp_mod_4_eq_1 : p ≡ 1 [MOD 4]\nhp_gt_20 : p > 20\nthis : Fact (Nat.Prime p)\nhp_mod_4_ne_3 : p % 4 ≠ 3\ny : ZMod p\nhy : -1 = y * y\nm : ℤ := y.valMinAbs\nn : ℕ := m.natAbs\nhnat₁ : p ∣ n ^ 2 + 1\nhnat₂ : n ≤ p / 2\nhnat₃ : p ≥ 2 * n\nk : ℕ := p - 2 * n\nhnat₄ : k = p - 2 * n\nhnat₅ : p ∣ k ^ 2 + 4\nhnat₆ : k ^ 2 + 4 ≥ p\nhreal₁ : ↑k = ↑p - 2 * ↑n\nhreal₂ : ↑p > 20\nhreal₃ : ↑k ^ 2 + 4 ≥ ↑p\nhreal₅ : ↑k > 4\n⊢ 2 * ↑n < ↑k ^ 2"
},
{
"line": "skip",
"before_state": "p : ℕ\nhpp : Nat.Prime p\nhp_mod_4_eq_1 : p ≡ 1 [MOD 4]\nhp_gt_20 : p > 20\nthis : Fact (Nat.Prime p)\nhp_mod_4_ne_3 : p % 4 ≠ 3\ny : ZMod p\nhy : -1 = y * y\nm : ℤ := y.valMinAbs\nn : ℕ := m.natAbs\nhnat₁ : p ∣ n ^ 2 + 1\nhnat₂ : n ≤ p / 2\nhnat₃ : p ≥ 2 * n\nk : ℕ := p - 2 * n\nhnat₄ : k = p - 2 * n\nhnat₅ : p ∣ k ^ 2 + 4\nhnat₆ : k ^ 2 + 4 ≥ p\nhreal₁ : ↑k = ↑p - 2 * ↑n\nhreal₂ : ↑p > 20\nhreal₃ : ↑k ^ 2 + 4 ≥ ↑p\nhreal₅ : ↑k > 4\n⊢ 2 * ↑n < ↑k ^ 2",
"after_state": "p : ℕ\nhpp : Nat.Prime p\nhp_mod_4_eq_1 : p ≡ 1 [MOD 4]\nhp_gt_20 : p > 20\nthis : Fact (Nat.Prime p)\nhp_mod_4_ne_3 : p % 4 ≠ 3\ny : ZMod p\nhy : -1 = y * y\nm : ℤ := y.valMinAbs\nn : ℕ := m.natAbs\nhnat₁ : p ∣ n ^ 2 + 1\nhnat₂ : n ≤ p / 2\nhnat₃ : p ≥ 2 * n\nk : ℕ := p - 2 * n\nhnat₄ : k = p - 2 * n\nhnat₅ : p ∣ k ^ 2 + 4\nhnat₆ : k ^ 2 + 4 ≥ p\nhreal₁ : ↑k = ↑p - 2 * ↑n\nhreal₂ : ↑p > 20\nhreal₃ : ↑k ^ 2 + 4 ≥ ↑p\nhreal₅ : ↑k > 4\n⊢ 2 * ↑n < ↑k ^ 2"
},
{
"line": "linarith only [hreal₁, hreal₃, hreal₅]",
"before_state": "p : ℕ\nhpp : Nat.Prime p\nhp_mod_4_eq_1 : p ≡ 1 [MOD 4]\nhp_gt_20 : p > 20\nthis : Fact (Nat.Prime p)\nhp_mod_4_ne_3 : p % 4 ≠ 3\ny : ZMod p\nhy : -1 = y * y\nm : ℤ := y.valMinAbs\nn : ℕ := m.natAbs\nhnat₁ : p ∣ n ^ 2 + 1\nhnat₂ : n ≤ p / 2\nhnat₃ : p ≥ 2 * n\nk : ℕ := p - 2 * n\nhnat₄ : k = p - 2 * n\nhnat₅ : p ∣ k ^ 2 + 4\nhnat₆ : k ^ 2 + 4 ≥ p\nhreal₁ : ↑k = ↑p - 2 * ↑n\nhreal₂ : ↑p > 20\nhreal₃ : ↑k ^ 2 + 4 ≥ ↑p\nhreal₅ : ↑k > 4\n⊢ 2 * ↑n < ↑k ^ 2",
"after_state": "No Goals!"
},
{
"line": "ring1",
"before_state": "p : ℕ\nhpp : Nat.Prime p\nhp_mod_4_eq_1 : p ≡ 1 [MOD 4]\nhp_gt_20 : p > 20\nthis : Fact (Nat.Prime p)\nhp_mod_4_ne_3 : p % 4 ≠ 3\ny : ZMod p\nhy : -1 = y * y\nm : ℤ := y.valMinAbs\nn : ℕ := m.natAbs\nhnat₁ : p ∣ n ^ 2 + 1\nhnat₂ : n ≤ p / 2\nhnat₃ : p ≥ 2 * n\nk : ℕ := p - 2 * n\nhnat₄ : k = p - 2 * n\nhnat₅ : p ∣ k ^ 2 + 4\nhnat₆ : k ^ 2 + 4 ≥ p\nhreal₁ : ↑k = ↑p - 2 * ↑n\nhreal₂ : ↑p > 20\nhreal₃ : ↑k ^ 2 + 4 ≥ ↑p\nhreal₅ : ↑k > 4\na✝ : 2 * ↑n ≥ ↑k ^ 2\n⊢ ↑k ^ 2 - 2 * ↑n + (↑k - (↑p - 2 * ↑n)) + (↑p - (↑k ^ 2 + 4)) + (4 - ↑k) = 0",
"after_state": "No Goals!"
},
{
"line": "exact ⟨n, hnat₁, by linarith only [hreal₆, hreal₁]⟩",
"before_state": "case intro\np : ℕ\nhpp : Nat.Prime p\nhp_mod_4_eq_1 : p ≡ 1 [MOD 4]\nhp_gt_20 : p > 20\nthis : Fact (Nat.Prime p)\nhp_mod_4_ne_3 : p % 4 ≠ 3\ny : ZMod p\nhy : -1 = y * y\nm : ℤ := y.valMinAbs\nn : ℕ := m.natAbs\nhnat₁ : p ∣ n ^ 2 + 1\nhnat₂ : n ≤ p / 2\nhnat₃ : p ≥ 2 * n\nk : ℕ := p - 2 * n\nhnat₄ : k = p - 2 * n\nhnat₅ : p ∣ k ^ 2 + 4\nhnat₆ : k ^ 2 + 4 ≥ p\nhreal₁ : ↑k = ↑p - 2 * ↑n\nhreal₂ : ↑p > 20\nhreal₃ : ↑k ^ 2 + 4 ≥ ↑p\nhreal₅ : ↑k > 4\nhreal₆ : ↑k > √(2 * ↑n)\n⊢ ∃ n, p ∣ n ^ 2 + 1 ∧ ↑p > 2 * ↑n + √(2 * ↑n)",
"after_state": "No Goals!"
},
{
"line": "linarith only [hreal₆, hreal₁]",
"before_state": "p : ℕ\nhpp : Nat.Prime p\nhp_mod_4_eq_1 : p ≡ 1 [MOD 4]\nhp_gt_20 : p > 20\nthis : Fact (Nat.Prime p)\nhp_mod_4_ne_3 : p % 4 ≠ 3\ny : ZMod p\nhy : -1 = y * y\nm : ℤ := y.valMinAbs\nn : ℕ := m.natAbs\nhnat₁ : p ∣ n ^ 2 + 1\nhnat₂ : n ≤ p / 2\nhnat₃ : p ≥ 2 * n\nk : ℕ := p - 2 * n\nhnat₄ : k = p - 2 * n\nhnat₅ : p ∣ k ^ 2 + 4\nhnat₆ : k ^ 2 + 4 ≥ p\nhreal₁ : ↑k = ↑p - 2 * ↑n\nhreal₂ : ↑p > 20\nhreal₃ : ↑k ^ 2 + 4 ≥ ↑p\nhreal₅ : ↑k > 4\nhreal₆ : ↑k > √(2 * ↑n)\n⊢ ↑p > 2 * ↑n + √(2 * ↑n)",
"after_state": "No Goals!"
},
{
"line": "ring1",
"before_state": "p : ℕ\nhpp : Nat.Prime p\nhp_mod_4_eq_1 : p ≡ 1 [MOD 4]\nhp_gt_20 : p > 20\nthis : Fact (Nat.Prime p)\nhp_mod_4_ne_3 : p % 4 ≠ 3\ny : ZMod p\nhy : -1 = y * y\nm : ℤ := y.valMinAbs\nn : ℕ := m.natAbs\nhnat₁ : p ∣ n ^ 2 + 1\nhnat₂ : n ≤ p / 2\nhnat₃ : p ≥ 2 * n\nk : ℕ := p - 2 * n\nhnat₄ : k = p - 2 * n\nhnat₅ : p ∣ k ^ 2 + 4\nhnat₆ : k ^ 2 + 4 ≥ p\nhreal₁ : ↑k = ↑p - 2 * ↑n\nhreal₂ : ↑p > 20\nhreal₃ : ↑k ^ 2 + 4 ≥ ↑p\nhreal₅ : ↑k > 4\nhreal₆ : ↑k > √(2 * ↑n)\na✝ : 2 * ↑n + √(2 * ↑n) ≥ ↑p\n⊢ ↑p - (2 * ↑n + √(2 * ↑n)) + (√(2 * ↑n) - ↑k) + (↑k - (↑p - 2 * ↑n)) = 0",
"after_state": "No Goals!"
}
] |
theorem imo2008_q3 : ∀ N : ℕ, ∃ n : ℕ, n ≥ N ∧
∃ p : ℕ, Nat.Prime p ∧ p ∣ n ^ 2 + 1 ∧ (p : ℝ) > 2 * n + sqrt (2 * n) := by
intro N
obtain ⟨p, hpp, hineq₁, hpmod4⟩ := Nat.exists_prime_gt_modEq_one (N ^ 2 + 20) four_ne_zero
obtain ⟨n, hnat, hreal⟩ := p_lemma p hpp hpmod4 (by linarith [hineq₁, Nat.zero_le (N ^ 2)])
have hineq₂ : n ^ 2 + 1 ≥ p := Nat.le_of_dvd (n ^ 2).succ_pos hnat
have hineq₃ : n * n ≥ N * N := by linarith [hineq₁, hineq₂]
have hn_ge_N : n ≥ N := Nat.mul_self_le_mul_self_iff.1 hineq₃
exact ⟨n, hn_ge_N, p, hpp, hnat, hreal⟩
|
/root/DuelModelResearch/mathlib4/Archive/Imo/Imo2008Q3.lean
|
{
"open": [
"Real",
"Imo2008Q3"
],
"variables": []
}
|
[
{
"line": "intro N",
"before_state": "⊢ ∀ (N : ℕ), ∃ n ≥ N, ∃ p, Nat.Prime p ∧ p ∣ n ^ 2 + 1 ∧ ↑p > 2 * ↑n + √(2 * ↑n)",
"after_state": "N : ℕ\n⊢ ∃ n ≥ N, ∃ p, Nat.Prime p ∧ p ∣ n ^ 2 + 1 ∧ ↑p > 2 * ↑n + √(2 * ↑n)"
},
{
"line": "obtain ⟨p, hpp, hineq₁, hpmod4⟩ := Nat.exists_prime_gt_modEq_one (N ^ 2 + 20) four_ne_zero",
"before_state": "N : ℕ\n⊢ ∃ n ≥ N, ∃ p, Nat.Prime p ∧ p ∣ n ^ 2 + 1 ∧ ↑p > 2 * ↑n + √(2 * ↑n)",
"after_state": "case intro.intro.intro\nN p : ℕ\nhpp : Nat.Prime p\nhineq₁ : N ^ 2 + 20 < p\nhpmod4 : p ≡ 1 [MOD 4]\n⊢ ∃ n ≥ N, ∃ p, Nat.Prime p ∧ p ∣ n ^ 2 + 1 ∧ ↑p > 2 * ↑n + √(2 * ↑n)"
},
{
"line": "obtain ⟨n, hnat, hreal⟩ := p_lemma p hpp hpmod4 (by linarith [hineq₁, Nat.zero_le (N ^ 2)])",
"before_state": "case intro.intro.intro\nN p : ℕ\nhpp : Nat.Prime p\nhineq₁ : N ^ 2 + 20 < p\nhpmod4 : p ≡ 1 [MOD 4]\n⊢ ∃ n ≥ N, ∃ p, Nat.Prime p ∧ p ∣ n ^ 2 + 1 ∧ ↑p > 2 * ↑n + √(2 * ↑n)",
"after_state": "No Goals!"
}
] |
theorem arith_lemma (k n : ℕ) : 0 < 2 * n + 2 ^ k.succ := by positivity
|
/root/DuelModelResearch/mathlib4/Archive/Imo/Imo2013Q1.lean
|
{
"open": [],
"variables": []
}
|
[
{
"line": "positivity",
"before_state": "k n : ℕ\n⊢ 0 < 2 * n + 2 ^ k.succ",
"after_state": "No Goals!"
}
] |
theorem prod_lemma (m : ℕ → ℕ+) (k : ℕ) (nm : ℕ+) :
∏ i ∈ Finset.range k, ((1 : ℚ) + 1 / ↑(if i < k then m i else nm)) =
∏ i ∈ Finset.range k, (1 + 1 / (m i : ℚ)) := by
suffices ∀ i, i ∈ Finset.range k → (1 : ℚ) + 1 / ↑(if i < k then m i else nm) = 1 + 1 / m i from
Finset.prod_congr rfl this
intro i hi
simp [Finset.mem_range.mp hi]
|
/root/DuelModelResearch/mathlib4/Archive/Imo/Imo2013Q1.lean
|
{
"open": [],
"variables": []
}
|
[
{
"line": "suffices ∀ i, i ∈ Finset.range k → (1 : ℚ) + 1 / ↑(if i < k then m i else nm) = 1 + 1 / m i from\n Finset.prod_congr rfl this",
"before_state": "m : ℕ → ℕ+\nk : ℕ\nnm : ℕ+\n⊢ ∏ i ∈ Finset.range k, (1 + 1 / ↑↑(if i < k then m i else nm)) = ∏ i ∈ Finset.range k, (1 + 1 / ↑↑(m i))",
"after_state": "m : ℕ → ℕ+\nk : ℕ\nnm : ℕ+\n⊢ ∀ i ∈ Finset.range k, 1 + 1 / ↑↑(if i < k then m i else nm) = 1 + 1 / ↑↑(m i)"
},
{
"line": "refine_lift\n suffices ∀ i, i ∈ Finset.range k → (1 : ℚ) + 1 / ↑(if i < k then m i else nm) = 1 + 1 / m i from\n Finset.prod_congr rfl this;\n ?_",
"before_state": "m : ℕ → ℕ+\nk : ℕ\nnm : ℕ+\n⊢ ∏ i ∈ Finset.range k, (1 + 1 / ↑↑(if i < k then m i else nm)) = ∏ i ∈ Finset.range k, (1 + 1 / ↑↑(m i))",
"after_state": "m : ℕ → ℕ+\nk : ℕ\nnm : ℕ+\n⊢ ∀ i ∈ Finset.range k, 1 + 1 / ↑↑(if i < k then m i else nm) = 1 + 1 / ↑↑(m i)"
},
{
"line": "focus\n (refine\n no_implicit_lambda%\n (suffices ∀ i, i ∈ Finset.range k → (1 : ℚ) + 1 / ↑(if i < k then m i else nm) = 1 + 1 / m i from\n Finset.prod_congr rfl this;\n ?_);\n rotate_right)",
"before_state": "m : ℕ → ℕ+\nk : ℕ\nnm : ℕ+\n⊢ ∏ i ∈ Finset.range k, (1 + 1 / ↑↑(if i < k then m i else nm)) = ∏ i ∈ Finset.range k, (1 + 1 / ↑↑(m i))",
"after_state": "m : ℕ → ℕ+\nk : ℕ\nnm : ℕ+\n⊢ ∀ i ∈ Finset.range k, 1 + 1 / ↑↑(if i < k then m i else nm) = 1 + 1 / ↑↑(m i)"
},
{
"line": "refine\n no_implicit_lambda%\n (suffices ∀ i, i ∈ Finset.range k → (1 : ℚ) + 1 / ↑(if i < k then m i else nm) = 1 + 1 / m i from\n Finset.prod_congr rfl this;\n ?_)",
"before_state": "m : ℕ → ℕ+\nk : ℕ\nnm : ℕ+\n⊢ ∏ i ∈ Finset.range k, (1 + 1 / ↑↑(if i < k then m i else nm)) = ∏ i ∈ Finset.range k, (1 + 1 / ↑↑(m i))",
"after_state": "m : ℕ → ℕ+\nk : ℕ\nnm : ℕ+\n⊢ ∀ i ∈ Finset.range k, 1 + 1 / ↑↑(if i < k then m i else nm) = 1 + 1 / ↑↑(m i)"
},
{
"line": "rotate_right",
"before_state": "m : ℕ → ℕ+\nk : ℕ\nnm : ℕ+\n⊢ ∀ i ∈ Finset.range k, 1 + 1 / ↑↑(if i < k then m i else nm) = 1 + 1 / ↑↑(m i)",
"after_state": "m : ℕ → ℕ+\nk : ℕ\nnm : ℕ+\n⊢ ∀ i ∈ Finset.range k, 1 + 1 / ↑↑(if i < k then m i else nm) = 1 + 1 / ↑↑(m i)"
},
{
"line": "intro i hi",
"before_state": "m : ℕ → ℕ+\nk : ℕ\nnm : ℕ+\n⊢ ∀ i ∈ Finset.range k, 1 + 1 / ↑↑(if i < k then m i else nm) = 1 + 1 / ↑↑(m i)",
"after_state": "m : ℕ → ℕ+\nk : ℕ\nnm : ℕ+\ni : ℕ\nhi : i ∈ Finset.range k\n⊢ 1 + 1 / ↑↑(if i < k then m i else nm) = 1 + 1 / ↑↑(m i)"
},
{
"line": "intro i;\n intro hi",
"before_state": "m : ℕ → ℕ+\nk : ℕ\nnm : ℕ+\n⊢ ∀ i ∈ Finset.range k, 1 + 1 / ↑↑(if i < k then m i else nm) = 1 + 1 / ↑↑(m i)",
"after_state": "m : ℕ → ℕ+\nk : ℕ\nnm : ℕ+\ni : ℕ\nhi : i ∈ Finset.range k\n⊢ 1 + 1 / ↑↑(if i < k then m i else nm) = 1 + 1 / ↑↑(m i)"
},
{
"line": "intro i",
"before_state": "m : ℕ → ℕ+\nk : ℕ\nnm : ℕ+\n⊢ ∀ i ∈ Finset.range k, 1 + 1 / ↑↑(if i < k then m i else nm) = 1 + 1 / ↑↑(m i)",
"after_state": "m : ℕ → ℕ+\nk : ℕ\nnm : ℕ+\ni : ℕ\n⊢ i ∈ Finset.range k → 1 + 1 / ↑↑(if i < k then m i else nm) = 1 + 1 / ↑↑(m i)"
},
{
"line": "intro hi",
"before_state": "m : ℕ → ℕ+\nk : ℕ\nnm : ℕ+\ni : ℕ\n⊢ i ∈ Finset.range k → 1 + 1 / ↑↑(if i < k then m i else nm) = 1 + 1 / ↑↑(m i)",
"after_state": "m : ℕ → ℕ+\nk : ℕ\nnm : ℕ+\ni : ℕ\nhi : i ∈ Finset.range k\n⊢ 1 + 1 / ↑↑(if i < k then m i else nm) = 1 + 1 / ↑↑(m i)"
},
{
"line": "simp [Finset.mem_range.mp hi]",
"before_state": "m : ℕ → ℕ+\nk : ℕ\nnm : ℕ+\ni : ℕ\nhi : i ∈ Finset.range k\n⊢ 1 + 1 / ↑↑(if i < k then m i else nm) = 1 + 1 / ↑↑(m i)",
"after_state": "No Goals!"
}
] |
theorem imo2013_q1 (n : ℕ+) (k : ℕ) :
∃ m : ℕ → ℕ+, (1 : ℚ) + (2 ^ k - 1) / n = ∏ i ∈ Finset.range k, (1 + 1 / (m i : ℚ)) := by
revert n
induction' k with pk hpk
· intro n; use fun (_ : ℕ) => (1 : ℕ+); simp
-- For the base case, any m works.
intro n
obtain ⟨t, ht : ↑n = t + t⟩ | ⟨t, ht : ↑n = 2 * t + 1⟩ := (n : ℕ).even_or_odd
· -- even case
rw [← two_mul] at ht
rcases t with - | t
-- Eliminate the zero case to simplify later calculations.
· exfalso; rw [Nat.mul_zero] at ht; exact PNat.ne_zero n ht
-- Now we have ht : ↑n = 2 * (t + 1).
let t_succ : ℕ+ := ⟨t + 1, t.succ_pos⟩
obtain ⟨pm, hpm⟩ := hpk t_succ
let m i := if i < pk then pm i else ⟨2 * t + 2 ^ pk.succ, arith_lemma pk t⟩
use m
have hmpk : (m pk : ℚ) = 2 * t + 2 ^ pk.succ := by
have : m pk = ⟨2 * t + 2 ^ pk.succ, _⟩ := if_neg (irrefl pk); simp [this]
calc
((1 : ℚ) + (2 ^ pk.succ - 1) / (n : ℚ) : ℚ)= 1 + (2 * 2 ^ pk - 1) / (2 * (t + 1) : ℕ) := by
rw [ht]
rw [pow_succ']
_ = (1 + 1 / (2 * t + 2 * 2 ^ pk)) * (1 + (2 ^ pk - 1) / (↑t + 1)) := by
field_simp
ring
_ = (1 + 1 / (2 * t + 2 ^ pk.succ)) * (1 + (2 ^ pk - 1) / t_succ) := by
simp [pow_succ', PNat.mk_coe, t_succ]
_ = (∏ i ∈ Finset.range pk, (1 + 1 / (m i : ℚ))) * (1 + 1 / m pk) := by
rw [prod_lemma]
rw [hpm]
rw [← hmpk]
rw [mul_comm]
_ = ∏ i ∈ Finset.range pk.succ, (1 + 1 / (m i : ℚ)) := by rw [← Finset.prod_range_succ _ pk]
· -- odd case
let t_succ : ℕ+ := ⟨t + 1, t.succ_pos⟩
obtain ⟨pm, hpm⟩ := hpk t_succ
let m i := if i < pk then pm i else ⟨2 * t + 1, Nat.succ_pos _⟩
use m
have hmpk : (m pk : ℚ) = 2 * t + 1 := by
have : m pk = ⟨2 * t + 1, _⟩ := if_neg (irrefl pk)
simp [this]
calc
((1 : ℚ) + (2 ^ pk.succ - 1) / ↑n : ℚ) = 1 + (2 * 2 ^ pk - 1) / (2 * t + 1 : ℕ) := by
rw [ht]
rw [pow_succ']
_ = (1 + 1 / (2 * t + 1)) * (1 + (2 ^ pk - 1) / (t + 1)) := by
field_simp
ring
_ = (1 + 1 / (2 * t + 1)) * (1 + (2 ^ pk - 1) / t_succ) := by norm_cast
_ = (∏ i ∈ Finset.range pk, (1 + 1 / (m i : ℚ))) * (1 + 1 / ↑(m pk)) := by
rw [prod_lemma]
rw [hpm]
rw [← hmpk]
rw [mul_comm]
_ = ∏ i ∈ Finset.range pk.succ, (1 + 1 / (m i : ℚ)) := by rw [← Finset.prod_range_succ _ pk]
|
/root/DuelModelResearch/mathlib4/Archive/Imo/Imo2013Q1.lean
|
{
"open": [
"Imo2013Q1"
],
"variables": []
}
|
[
{
"line": "revert n",
"before_state": "n : ℕ+\nk : ℕ\n⊢ ∃ m, 1 + (2 ^ k - 1) / ↑↑n = ∏ i ∈ Finset.range k, (1 + 1 / ↑↑(m i))",
"after_state": "k : ℕ\n⊢ ∀ (n : ℕ+), ∃ m, 1 + (2 ^ k - 1) / ↑↑n = ∏ i ∈ Finset.range k, (1 + 1 / ↑↑(m i))"
},
{
"line": "induction' k with pk hpk",
"before_state": "k : ℕ\n⊢ ∀ (n : ℕ+), ∃ m, 1 + (2 ^ k - 1) / ↑↑n = ∏ i ∈ Finset.range k, (1 + 1 / ↑↑(m i))",
"after_state": "case zero\n⊢ ∀ (n : ℕ+), ∃ m, 1 + (2 ^ 0 - 1) / ↑↑n = ∏ i ∈ Finset.range 0, (1 + 1 / ↑↑(m i))\n---\ncase succ\npk : ℕ\nhpk : ∀ (n : ℕ+), ∃ m, 1 + (2 ^ pk - 1) / ↑↑n = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(m i))\n⊢ ∀ (n : ℕ+), ∃ m, 1 + (2 ^ (pk + 1) - 1) / ↑↑n = ∏ i ∈ Finset.range (pk + 1), (1 + 1 / ↑↑(m i))"
},
{
"line": "intro n",
"before_state": "case zero\n⊢ ∀ (n : ℕ+), ∃ m, 1 + (2 ^ 0 - 1) / ↑↑n = ∏ i ∈ Finset.range 0, (1 + 1 / ↑↑(m i))",
"after_state": "case zero\nn : ℕ+\n⊢ ∃ m, 1 + (2 ^ 0 - 1) / ↑↑n = ∏ i ∈ Finset.range 0, (1 + 1 / ↑↑(m i))"
},
{
"line": "use fun (_ : ℕ) => (1 : ℕ+)",
"before_state": "case zero\nn : ℕ+\n⊢ ∃ m, 1 + (2 ^ 0 - 1) / ↑↑n = ∏ i ∈ Finset.range 0, (1 + 1 / ↑↑(m i))",
"after_state": "case h\nn : ℕ+\n⊢ 1 + (2 ^ 0 - 1) / ↑↑n = ∏ i ∈ Finset.range 0, (1 + 1 / ↑↑((fun x => 1) i))"
},
{
"line": "refine without_cdot(fun (_ : ℕ) => (1 : ℕ+) : ?m✝)",
"before_state": "case w\nn : ℕ+\n⊢ ℕ → ℕ+",
"after_state": "No Goals!"
},
{
"line": "try with_reducible use_discharger",
"before_state": "case h\nn : ℕ+\n⊢ 1 + (2 ^ 0 - 1) / ↑↑n = ∏ i ∈ Finset.range 0, (1 + 1 / ↑↑((fun x => 1) i))",
"after_state": "case h\nn : ℕ+\n⊢ 1 + (2 ^ 0 - 1) / ↑↑n = ∏ i ∈ Finset.range 0, (1 + 1 / ↑↑((fun x => 1) i))"
},
{
"line": "first\n| with_reducible use_discharger\n| skip",
"before_state": "case h\nn : ℕ+\n⊢ 1 + (2 ^ 0 - 1) / ↑↑n = ∏ i ∈ Finset.range 0, (1 + 1 / ↑↑((fun x => 1) i))",
"after_state": "case h\nn : ℕ+\n⊢ 1 + (2 ^ 0 - 1) / ↑↑n = ∏ i ∈ Finset.range 0, (1 + 1 / ↑↑((fun x => 1) i))"
},
{
"line": "with_reducible use_discharger",
"before_state": "case h\nn : ℕ+\n⊢ 1 + (2 ^ 0 - 1) / ↑↑n = ∏ i ∈ Finset.range 0, (1 + 1 / ↑↑((fun x => 1) i))",
"after_state": "case h\nn : ℕ+\n⊢ 1 + (2 ^ 0 - 1) / ↑↑n = ∏ i ∈ Finset.range 0, (1 + 1 / ↑↑((fun x => 1) i))"
},
{
"line": "use_discharger",
"before_state": "case h\nn : ℕ+\n⊢ 1 + (2 ^ 0 - 1) / ↑↑n = ∏ i ∈ Finset.range 0, (1 + 1 / ↑↑((fun x => 1) i))",
"after_state": "case h\nn : ℕ+\n⊢ 1 + (2 ^ 0 - 1) / ↑↑n = ∏ i ∈ Finset.range 0, (1 + 1 / ↑↑((fun x => 1) i))"
},
{
"line": "focus\n apply exists_prop.mpr✝\n with_annotate_state\"<;>\" skip\n all_goals use_discharger",
"before_state": "case h\nn : ℕ+\n⊢ 1 + (2 ^ 0 - 1) / ↑↑n = ∏ i ∈ Finset.range 0, (1 + 1 / ↑↑((fun x => 1) i))",
"after_state": "case h\nn : ℕ+\n⊢ 1 + (2 ^ 0 - 1) / ↑↑n = ∏ i ∈ Finset.range 0, (1 + 1 / ↑↑((fun x => 1) i))"
},
{
"line": "apply exists_prop.mpr✝",
"before_state": "case h\nn : ℕ+\n⊢ 1 + (2 ^ 0 - 1) / ↑↑n = ∏ i ∈ Finset.range 0, (1 + 1 / ↑↑((fun x => 1) i))",
"after_state": "case h\nn : ℕ+\n⊢ 1 + (2 ^ 0 - 1) / ↑↑n = ∏ i ∈ Finset.range 0, (1 + 1 / ↑↑((fun x => 1) i))"
},
{
"line": "skip",
"before_state": "case h\nn : ℕ+\n⊢ 1 + (2 ^ 0 - 1) / ↑↑n = ∏ i ∈ Finset.range 0, (1 + 1 / ↑↑((fun x => 1) i))",
"after_state": "case h\nn : ℕ+\n⊢ 1 + (2 ^ 0 - 1) / ↑↑n = ∏ i ∈ Finset.range 0, (1 + 1 / ↑↑((fun x => 1) i))"
},
{
"line": "simp\n -- For the base case, any m works.",
"before_state": "case h\nn : ℕ+\n⊢ 1 + (2 ^ 0 - 1) / ↑↑n = ∏ i ∈ Finset.range 0, (1 + 1 / ↑↑((fun x => 1) i))",
"after_state": "No Goals!"
},
{
"line": "intro n",
"before_state": "case succ\npk : ℕ\nhpk : ∀ (n : ℕ+), ∃ m, 1 + (2 ^ pk - 1) / ↑↑n = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(m i))\n⊢ ∀ (n : ℕ+), ∃ m, 1 + (2 ^ (pk + 1) - 1) / ↑↑n = ∏ i ∈ Finset.range (pk + 1), (1 + 1 / ↑↑(m i))",
"after_state": "case succ\npk : ℕ\nhpk : ∀ (n : ℕ+), ∃ m, 1 + (2 ^ pk - 1) / ↑↑n = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(m i))\nn : ℕ+\n⊢ ∃ m, 1 + (2 ^ (pk + 1) - 1) / ↑↑n = ∏ i ∈ Finset.range (pk + 1), (1 + 1 / ↑↑(m i))"
},
{
"line": "obtain ⟨t, ht : ↑n = t + t⟩ | ⟨t, ht : ↑n = 2 * t + 1⟩ := (n : ℕ).even_or_odd",
"before_state": "case succ\npk : ℕ\nhpk : ∀ (n : ℕ+), ∃ m, 1 + (2 ^ pk - 1) / ↑↑n = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(m i))\nn : ℕ+\n⊢ ∃ m, 1 + (2 ^ (pk + 1) - 1) / ↑↑n = ∏ i ∈ Finset.range (pk + 1), (1 + 1 / ↑↑(m i))",
"after_state": "case succ.inl.intro\npk : ℕ\nhpk : ∀ (n : ℕ+), ∃ m, 1 + (2 ^ pk - 1) / ↑↑n = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(m i))\nn : ℕ+\nt : ℕ\nht : ↑n = t + t\n⊢ ∃ m, 1 + (2 ^ (pk + 1) - 1) / ↑↑n = ∏ i ∈ Finset.range (pk + 1), (1 + 1 / ↑↑(m i))\n---\ncase succ.inr.intro\npk : ℕ\nhpk : ∀ (n : ℕ+), ∃ m, 1 + (2 ^ pk - 1) / ↑↑n = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(m i))\nn : ℕ+\nt : ℕ\nht : ↑n = 2 * t + 1\n⊢ ∃ m, 1 + (2 ^ (pk + 1) - 1) / ↑↑n = ∏ i ∈ Finset.range (pk + 1), (1 + 1 / ↑↑(m i))"
},
{
"line": "rw [← two_mul] at ht",
"before_state": "case succ.inl.intro\npk : ℕ\nhpk : ∀ (n : ℕ+), ∃ m, 1 + (2 ^ pk - 1) / ↑↑n = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(m i))\nn : ℕ+\nt : ℕ\nht : ↑n = t + t\n⊢ ∃ m, 1 + (2 ^ (pk + 1) - 1) / ↑↑n = ∏ i ∈ Finset.range (pk + 1), (1 + 1 / ↑↑(m i))",
"after_state": "case succ.inl.intro\npk : ℕ\nhpk : ∀ (n : ℕ+), ∃ m, 1 + (2 ^ pk - 1) / ↑↑n = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(m i))\nn : ℕ+\nt : ℕ\nht : ↑n = 2 * t\n⊢ ∃ m, 1 + (2 ^ (pk + 1) - 1) / ↑↑n = ∏ i ∈ Finset.range (pk + 1), (1 + 1 / ↑↑(m i))"
},
{
"line": "rewrite [← two_mul] at ht",
"before_state": "case succ.inl.intro\npk : ℕ\nhpk : ∀ (n : ℕ+), ∃ m, 1 + (2 ^ pk - 1) / ↑↑n = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(m i))\nn : ℕ+\nt : ℕ\nht : ↑n = t + t\n⊢ ∃ m, 1 + (2 ^ (pk + 1) - 1) / ↑↑n = ∏ i ∈ Finset.range (pk + 1), (1 + 1 / ↑↑(m i))",
"after_state": "case succ.inl.intro\npk : ℕ\nhpk : ∀ (n : ℕ+), ∃ m, 1 + (2 ^ pk - 1) / ↑↑n = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(m i))\nn : ℕ+\nt : ℕ\nht : ↑n = 2 * t\n⊢ ∃ m, 1 + (2 ^ (pk + 1) - 1) / ↑↑n = ∏ i ∈ Finset.range (pk + 1), (1 + 1 / ↑↑(m i))"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "case succ.inl.intro\npk : ℕ\nhpk : ∀ (n : ℕ+), ∃ m, 1 + (2 ^ pk - 1) / ↑↑n = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(m i))\nn : ℕ+\nt : ℕ\nht : ↑n = 2 * t\n⊢ ∃ m, 1 + (2 ^ (pk + 1) - 1) / ↑↑n = ∏ i ∈ Finset.range (pk + 1), (1 + 1 / ↑↑(m i))",
"after_state": "case succ.inl.intro\npk : ℕ\nhpk : ∀ (n : ℕ+), ∃ m, 1 + (2 ^ pk - 1) / ↑↑n = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(m i))\nn : ℕ+\nt : ℕ\nht : ↑n = 2 * t\n⊢ ∃ m, 1 + (2 ^ (pk + 1) - 1) / ↑↑n = ∏ i ∈ Finset.range (pk + 1), (1 + 1 / ↑↑(m i))"
},
{
"line": "try (with_reducible rfl)",
"before_state": "case succ.inl.intro\npk : ℕ\nhpk : ∀ (n : ℕ+), ∃ m, 1 + (2 ^ pk - 1) / ↑↑n = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(m i))\nn : ℕ+\nt : ℕ\nht : ↑n = 2 * t\n⊢ ∃ m, 1 + (2 ^ (pk + 1) - 1) / ↑↑n = ∏ i ∈ Finset.range (pk + 1), (1 + 1 / ↑↑(m i))",
"after_state": "case succ.inl.intro\npk : ℕ\nhpk : ∀ (n : ℕ+), ∃ m, 1 + (2 ^ pk - 1) / ↑↑n = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(m i))\nn : ℕ+\nt : ℕ\nht : ↑n = 2 * t\n⊢ ∃ m, 1 + (2 ^ (pk + 1) - 1) / ↑↑n = ∏ i ∈ Finset.range (pk + 1), (1 + 1 / ↑↑(m i))"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "case succ.inl.intro\npk : ℕ\nhpk : ∀ (n : ℕ+), ∃ m, 1 + (2 ^ pk - 1) / ↑↑n = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(m i))\nn : ℕ+\nt : ℕ\nht : ↑n = 2 * t\n⊢ ∃ m, 1 + (2 ^ (pk + 1) - 1) / ↑↑n = ∏ i ∈ Finset.range (pk + 1), (1 + 1 / ↑↑(m i))",
"after_state": "case succ.inl.intro\npk : ℕ\nhpk : ∀ (n : ℕ+), ∃ m, 1 + (2 ^ pk - 1) / ↑↑n = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(m i))\nn : ℕ+\nt : ℕ\nht : ↑n = 2 * t\n⊢ ∃ m, 1 + (2 ^ (pk + 1) - 1) / ↑↑n = ∏ i ∈ Finset.range (pk + 1), (1 + 1 / ↑↑(m i))"
},
{
"line": "with_reducible rfl",
"before_state": "case succ.inl.intro\npk : ℕ\nhpk : ∀ (n : ℕ+), ∃ m, 1 + (2 ^ pk - 1) / ↑↑n = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(m i))\nn : ℕ+\nt : ℕ\nht : ↑n = 2 * t\n⊢ ∃ m, 1 + (2 ^ (pk + 1) - 1) / ↑↑n = ∏ i ∈ Finset.range (pk + 1), (1 + 1 / ↑↑(m i))",
"after_state": "case succ.inl.intro\npk : ℕ\nhpk : ∀ (n : ℕ+), ∃ m, 1 + (2 ^ pk - 1) / ↑↑n = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(m i))\nn : ℕ+\nt : ℕ\nht : ↑n = 2 * t\n⊢ ∃ m, 1 + (2 ^ (pk + 1) - 1) / ↑↑n = ∏ i ∈ Finset.range (pk + 1), (1 + 1 / ↑↑(m i))"
},
{
"line": "rfl",
"before_state": "case succ.inl.intro\npk : ℕ\nhpk : ∀ (n : ℕ+), ∃ m, 1 + (2 ^ pk - 1) / ↑↑n = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(m i))\nn : ℕ+\nt : ℕ\nht : ↑n = 2 * t\n⊢ ∃ m, 1 + (2 ^ (pk + 1) - 1) / ↑↑n = ∏ i ∈ Finset.range (pk + 1), (1 + 1 / ↑↑(m i))",
"after_state": "case succ.inl.intro\npk : ℕ\nhpk : ∀ (n : ℕ+), ∃ m, 1 + (2 ^ pk - 1) / ↑↑n = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(m i))\nn : ℕ+\nt : ℕ\nht : ↑n = 2 * t\n⊢ ∃ m, 1 + (2 ^ (pk + 1) - 1) / ↑↑n = ∏ i ∈ Finset.range (pk + 1), (1 + 1 / ↑↑(m i))"
},
{
"line": "apply_rfl",
"before_state": "case succ.inl.intro\npk : ℕ\nhpk : ∀ (n : ℕ+), ∃ m, 1 + (2 ^ pk - 1) / ↑↑n = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(m i))\nn : ℕ+\nt : ℕ\nht : ↑n = 2 * t\n⊢ ∃ m, 1 + (2 ^ (pk + 1) - 1) / ↑↑n = ∏ i ∈ Finset.range (pk + 1), (1 + 1 / ↑↑(m i))",
"after_state": "case succ.inl.intro\npk : ℕ\nhpk : ∀ (n : ℕ+), ∃ m, 1 + (2 ^ pk - 1) / ↑↑n = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(m i))\nn : ℕ+\nt : ℕ\nht : ↑n = 2 * t\n⊢ ∃ m, 1 + (2 ^ (pk + 1) - 1) / ↑↑n = ∏ i ∈ Finset.range (pk + 1), (1 + 1 / ↑↑(m i))"
},
{
"line": "skip",
"before_state": "case succ.inl.intro\npk : ℕ\nhpk : ∀ (n : ℕ+), ∃ m, 1 + (2 ^ pk - 1) / ↑↑n = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(m i))\nn : ℕ+\nt : ℕ\nht : ↑n = 2 * t\n⊢ ∃ m, 1 + (2 ^ (pk + 1) - 1) / ↑↑n = ∏ i ∈ Finset.range (pk + 1), (1 + 1 / ↑↑(m i))",
"after_state": "case succ.inl.intro\npk : ℕ\nhpk : ∀ (n : ℕ+), ∃ m, 1 + (2 ^ pk - 1) / ↑↑n = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(m i))\nn : ℕ+\nt : ℕ\nht : ↑n = 2 * t\n⊢ ∃ m, 1 + (2 ^ (pk + 1) - 1) / ↑↑n = ∏ i ∈ Finset.range (pk + 1), (1 + 1 / ↑↑(m i))"
},
{
"line": "rcases t with - | t",
"before_state": "case succ.inl.intro\npk : ℕ\nhpk : ∀ (n : ℕ+), ∃ m, 1 + (2 ^ pk - 1) / ↑↑n = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(m i))\nn : ℕ+\nt : ℕ\nht : ↑n = 2 * t\n⊢ ∃ m, 1 + (2 ^ (pk + 1) - 1) / ↑↑n = ∏ i ∈ Finset.range (pk + 1), (1 + 1 / ↑↑(m i))",
"after_state": "case succ.inl.intro.zero\npk : ℕ\nhpk : ∀ (n : ℕ+), ∃ m, 1 + (2 ^ pk - 1) / ↑↑n = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(m i))\nn : ℕ+\nht : ↑n = 2 * 0\n⊢ ∃ m, 1 + (2 ^ (pk + 1) - 1) / ↑↑n = ∏ i ∈ Finset.range (pk + 1), (1 + 1 / ↑↑(m i))\n---\ncase succ.inl.intro.succ\npk : ℕ\nhpk : ∀ (n : ℕ+), ∃ m, 1 + (2 ^ pk - 1) / ↑↑n = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(m i))\nn : ℕ+\nt : ℕ\nht : ↑n = 2 * (t + 1)\n⊢ ∃ m, 1 + (2 ^ (pk + 1) - 1) / ↑↑n = ∏ i ∈ Finset.range (pk + 1), (1 + 1 / ↑↑(m i))"
},
{
"line": "exfalso",
"before_state": "case succ.inl.intro.zero\npk : ℕ\nhpk : ∀ (n : ℕ+), ∃ m, 1 + (2 ^ pk - 1) / ↑↑n = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(m i))\nn : ℕ+\nht : ↑n = 2 * 0\n⊢ ∃ m, 1 + (2 ^ (pk + 1) - 1) / ↑↑n = ∏ i ∈ Finset.range (pk + 1), (1 + 1 / ↑↑(m i))",
"after_state": "case succ.inl.intro.zero\npk : ℕ\nhpk : ∀ (n : ℕ+), ∃ m, 1 + (2 ^ pk - 1) / ↑↑n = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(m i))\nn : ℕ+\nht : ↑n = 2 * 0\n⊢ False"
},
{
"line": "refine False.elim✝ ?_",
"before_state": "case succ.inl.intro.zero\npk : ℕ\nhpk : ∀ (n : ℕ+), ∃ m, 1 + (2 ^ pk - 1) / ↑↑n = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(m i))\nn : ℕ+\nht : ↑n = 2 * 0\n⊢ ∃ m, 1 + (2 ^ (pk + 1) - 1) / ↑↑n = ∏ i ∈ Finset.range (pk + 1), (1 + 1 / ↑↑(m i))",
"after_state": "case succ.inl.intro.zero\npk : ℕ\nhpk : ∀ (n : ℕ+), ∃ m, 1 + (2 ^ pk - 1) / ↑↑n = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(m i))\nn : ℕ+\nht : ↑n = 2 * 0\n⊢ False"
},
{
"line": "rw [Nat.mul_zero] at ht",
"before_state": "case succ.inl.intro.zero\npk : ℕ\nhpk : ∀ (n : ℕ+), ∃ m, 1 + (2 ^ pk - 1) / ↑↑n = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(m i))\nn : ℕ+\nht : ↑n = 2 * 0\n⊢ False",
"after_state": "case succ.inl.intro.zero\npk : ℕ\nhpk : ∀ (n : ℕ+), ∃ m, 1 + (2 ^ pk - 1) / ↑↑n = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(m i))\nn : ℕ+\nht : ↑n = 0\n⊢ False"
},
{
"line": "rewrite [Nat.mul_zero] at ht",
"before_state": "case succ.inl.intro.zero\npk : ℕ\nhpk : ∀ (n : ℕ+), ∃ m, 1 + (2 ^ pk - 1) / ↑↑n = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(m i))\nn : ℕ+\nht : ↑n = 2 * 0\n⊢ False",
"after_state": "case succ.inl.intro.zero\npk : ℕ\nhpk : ∀ (n : ℕ+), ∃ m, 1 + (2 ^ pk - 1) / ↑↑n = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(m i))\nn : ℕ+\nht : ↑n = 0\n⊢ False"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "case succ.inl.intro.zero\npk : ℕ\nhpk : ∀ (n : ℕ+), ∃ m, 1 + (2 ^ pk - 1) / ↑↑n = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(m i))\nn : ℕ+\nht : ↑n = 0\n⊢ False",
"after_state": "case succ.inl.intro.zero\npk : ℕ\nhpk : ∀ (n : ℕ+), ∃ m, 1 + (2 ^ pk - 1) / ↑↑n = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(m i))\nn : ℕ+\nht : ↑n = 0\n⊢ False"
},
{
"line": "try (with_reducible rfl)",
"before_state": "case succ.inl.intro.zero\npk : ℕ\nhpk : ∀ (n : ℕ+), ∃ m, 1 + (2 ^ pk - 1) / ↑↑n = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(m i))\nn : ℕ+\nht : ↑n = 0\n⊢ False",
"after_state": "case succ.inl.intro.zero\npk : ℕ\nhpk : ∀ (n : ℕ+), ∃ m, 1 + (2 ^ pk - 1) / ↑↑n = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(m i))\nn : ℕ+\nht : ↑n = 0\n⊢ False"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "case succ.inl.intro.zero\npk : ℕ\nhpk : ∀ (n : ℕ+), ∃ m, 1 + (2 ^ pk - 1) / ↑↑n = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(m i))\nn : ℕ+\nht : ↑n = 0\n⊢ False",
"after_state": "case succ.inl.intro.zero\npk : ℕ\nhpk : ∀ (n : ℕ+), ∃ m, 1 + (2 ^ pk - 1) / ↑↑n = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(m i))\nn : ℕ+\nht : ↑n = 0\n⊢ False"
},
{
"line": "with_reducible rfl",
"before_state": "case succ.inl.intro.zero\npk : ℕ\nhpk : ∀ (n : ℕ+), ∃ m, 1 + (2 ^ pk - 1) / ↑↑n = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(m i))\nn : ℕ+\nht : ↑n = 0\n⊢ False",
"after_state": "case succ.inl.intro.zero\npk : ℕ\nhpk : ∀ (n : ℕ+), ∃ m, 1 + (2 ^ pk - 1) / ↑↑n = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(m i))\nn : ℕ+\nht : ↑n = 0\n⊢ False"
},
{
"line": "rfl",
"before_state": "case succ.inl.intro.zero\npk : ℕ\nhpk : ∀ (n : ℕ+), ∃ m, 1 + (2 ^ pk - 1) / ↑↑n = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(m i))\nn : ℕ+\nht : ↑n = 0\n⊢ False",
"after_state": "case succ.inl.intro.zero\npk : ℕ\nhpk : ∀ (n : ℕ+), ∃ m, 1 + (2 ^ pk - 1) / ↑↑n = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(m i))\nn : ℕ+\nht : ↑n = 0\n⊢ False"
},
{
"line": "apply_rfl",
"before_state": "case succ.inl.intro.zero\npk : ℕ\nhpk : ∀ (n : ℕ+), ∃ m, 1 + (2 ^ pk - 1) / ↑↑n = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(m i))\nn : ℕ+\nht : ↑n = 0\n⊢ False",
"after_state": "case succ.inl.intro.zero\npk : ℕ\nhpk : ∀ (n : ℕ+), ∃ m, 1 + (2 ^ pk - 1) / ↑↑n = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(m i))\nn : ℕ+\nht : ↑n = 0\n⊢ False"
},
{
"line": "skip",
"before_state": "case succ.inl.intro.zero\npk : ℕ\nhpk : ∀ (n : ℕ+), ∃ m, 1 + (2 ^ pk - 1) / ↑↑n = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(m i))\nn : ℕ+\nht : ↑n = 0\n⊢ False",
"after_state": "case succ.inl.intro.zero\npk : ℕ\nhpk : ∀ (n : ℕ+), ∃ m, 1 + (2 ^ pk - 1) / ↑↑n = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(m i))\nn : ℕ+\nht : ↑n = 0\n⊢ False"
},
{
"line": "exact PNat.ne_zero n ht",
"before_state": "case succ.inl.intro.zero\npk : ℕ\nhpk : ∀ (n : ℕ+), ∃ m, 1 + (2 ^ pk - 1) / ↑↑n = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(m i))\nn : ℕ+\nht : ↑n = 0\n⊢ False",
"after_state": "No Goals!"
},
{
"line": "let t_succ : ℕ+ := ⟨t + 1, t.succ_pos⟩",
"before_state": "case succ.inl.intro.succ\npk : ℕ\nhpk : ∀ (n : ℕ+), ∃ m, 1 + (2 ^ pk - 1) / ↑↑n = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(m i))\nn : ℕ+\nt : ℕ\nht : ↑n = 2 * (t + 1)\n⊢ ∃ m, 1 + (2 ^ (pk + 1) - 1) / ↑↑n = ∏ i ∈ Finset.range (pk + 1), (1 + 1 / ↑↑(m i))",
"after_state": "case succ.inl.intro.succ\npk : ℕ\nhpk : ∀ (n : ℕ+), ∃ m, 1 + (2 ^ pk - 1) / ↑↑n = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(m i))\nn : ℕ+\nt : ℕ\nht : ↑n = 2 * (t + 1)\nt_succ : ℕ+ := ⟨t + 1, ⋯⟩\n⊢ ∃ m, 1 + (2 ^ (pk + 1) - 1) / ↑↑n = ∏ i ∈ Finset.range (pk + 1), (1 + 1 / ↑↑(m i))"
},
{
"line": "refine_lift\n let t_succ : ℕ+ := ⟨t + 1, t.succ_pos⟩;\n ?_",
"before_state": "case succ.inl.intro.succ\npk : ℕ\nhpk : ∀ (n : ℕ+), ∃ m, 1 + (2 ^ pk - 1) / ↑↑n = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(m i))\nn : ℕ+\nt : ℕ\nht : ↑n = 2 * (t + 1)\n⊢ ∃ m, 1 + (2 ^ (pk + 1) - 1) / ↑↑n = ∏ i ∈ Finset.range (pk + 1), (1 + 1 / ↑↑(m i))",
"after_state": "case succ.inl.intro.succ\npk : ℕ\nhpk : ∀ (n : ℕ+), ∃ m, 1 + (2 ^ pk - 1) / ↑↑n = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(m i))\nn : ℕ+\nt : ℕ\nht : ↑n = 2 * (t + 1)\nt_succ : ℕ+ := ⟨t + 1, ⋯⟩\n⊢ ∃ m, 1 + (2 ^ (pk + 1) - 1) / ↑↑n = ∏ i ∈ Finset.range (pk + 1), (1 + 1 / ↑↑(m i))"
},
{
"line": "focus\n (refine\n no_implicit_lambda%\n (let t_succ : ℕ+ := ⟨t + 1, t.succ_pos⟩;\n ?_);\n rotate_right)",
"before_state": "case succ.inl.intro.succ\npk : ℕ\nhpk : ∀ (n : ℕ+), ∃ m, 1 + (2 ^ pk - 1) / ↑↑n = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(m i))\nn : ℕ+\nt : ℕ\nht : ↑n = 2 * (t + 1)\n⊢ ∃ m, 1 + (2 ^ (pk + 1) - 1) / ↑↑n = ∏ i ∈ Finset.range (pk + 1), (1 + 1 / ↑↑(m i))",
"after_state": "case succ.inl.intro.succ\npk : ℕ\nhpk : ∀ (n : ℕ+), ∃ m, 1 + (2 ^ pk - 1) / ↑↑n = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(m i))\nn : ℕ+\nt : ℕ\nht : ↑n = 2 * (t + 1)\nt_succ : ℕ+ := ⟨t + 1, ⋯⟩\n⊢ ∃ m, 1 + (2 ^ (pk + 1) - 1) / ↑↑n = ∏ i ∈ Finset.range (pk + 1), (1 + 1 / ↑↑(m i))"
},
{
"line": "refine\n no_implicit_lambda%\n (let t_succ : ℕ+ := ⟨t + 1, t.succ_pos⟩;\n ?_)",
"before_state": "case succ.inl.intro.succ\npk : ℕ\nhpk : ∀ (n : ℕ+), ∃ m, 1 + (2 ^ pk - 1) / ↑↑n = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(m i))\nn : ℕ+\nt : ℕ\nht : ↑n = 2 * (t + 1)\n⊢ ∃ m, 1 + (2 ^ (pk + 1) - 1) / ↑↑n = ∏ i ∈ Finset.range (pk + 1), (1 + 1 / ↑↑(m i))",
"after_state": "case succ.inl.intro.succ\npk : ℕ\nhpk : ∀ (n : ℕ+), ∃ m, 1 + (2 ^ pk - 1) / ↑↑n = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(m i))\nn : ℕ+\nt : ℕ\nht : ↑n = 2 * (t + 1)\nt_succ : ℕ+ := ⟨t + 1, ⋯⟩\n⊢ ∃ m, 1 + (2 ^ (pk + 1) - 1) / ↑↑n = ∏ i ∈ Finset.range (pk + 1), (1 + 1 / ↑↑(m i))"
},
{
"line": "rotate_right",
"before_state": "case succ.inl.intro.succ\npk : ℕ\nhpk : ∀ (n : ℕ+), ∃ m, 1 + (2 ^ pk - 1) / ↑↑n = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(m i))\nn : ℕ+\nt : ℕ\nht : ↑n = 2 * (t + 1)\nt_succ : ℕ+ := ⟨t + 1, ⋯⟩\n⊢ ∃ m, 1 + (2 ^ (pk + 1) - 1) / ↑↑n = ∏ i ∈ Finset.range (pk + 1), (1 + 1 / ↑↑(m i))",
"after_state": "case succ.inl.intro.succ\npk : ℕ\nhpk : ∀ (n : ℕ+), ∃ m, 1 + (2 ^ pk - 1) / ↑↑n = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(m i))\nn : ℕ+\nt : ℕ\nht : ↑n = 2 * (t + 1)\nt_succ : ℕ+ := ⟨t + 1, ⋯⟩\n⊢ ∃ m, 1 + (2 ^ (pk + 1) - 1) / ↑↑n = ∏ i ∈ Finset.range (pk + 1), (1 + 1 / ↑↑(m i))"
},
{
"line": "obtain ⟨pm, hpm⟩ := hpk t_succ",
"before_state": "case succ.inl.intro.succ\npk : ℕ\nhpk : ∀ (n : ℕ+), ∃ m, 1 + (2 ^ pk - 1) / ↑↑n = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(m i))\nn : ℕ+\nt : ℕ\nht : ↑n = 2 * (t + 1)\nt_succ : ℕ+ := ⟨t + 1, ⋯⟩\n⊢ ∃ m, 1 + (2 ^ (pk + 1) - 1) / ↑↑n = ∏ i ∈ Finset.range (pk + 1), (1 + 1 / ↑↑(m i))",
"after_state": "case succ.inl.intro.succ.intro\npk : ℕ\nhpk : ∀ (n : ℕ+), ∃ m, 1 + (2 ^ pk - 1) / ↑↑n = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(m i))\nn : ℕ+\nt : ℕ\nht : ↑n = 2 * (t + 1)\nt_succ : ℕ+ := ⟨t + 1, ⋯⟩\npm : ℕ → ℕ+\nhpm : 1 + (2 ^ pk - 1) / ↑↑t_succ = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(pm i))\n⊢ ∃ m, 1 + (2 ^ (pk + 1) - 1) / ↑↑n = ∏ i ∈ Finset.range (pk + 1), (1 + 1 / ↑↑(m i))"
},
{
"line": "let m i := if i < pk then pm i else ⟨2 * t + 2 ^ pk.succ, arith_lemma pk t⟩",
"before_state": "case succ.inl.intro.succ.intro\npk : ℕ\nhpk : ∀ (n : ℕ+), ∃ m, 1 + (2 ^ pk - 1) / ↑↑n = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(m i))\nn : ℕ+\nt : ℕ\nht : ↑n = 2 * (t + 1)\nt_succ : ℕ+ := ⟨t + 1, ⋯⟩\npm : ℕ → ℕ+\nhpm : 1 + (2 ^ pk - 1) / ↑↑t_succ = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(pm i))\n⊢ ∃ m, 1 + (2 ^ (pk + 1) - 1) / ↑↑n = ∏ i ∈ Finset.range (pk + 1), (1 + 1 / ↑↑(m i))",
"after_state": "case succ.inl.intro.succ.intro\npk : ℕ\nhpk : ∀ (n : ℕ+), ∃ m, 1 + (2 ^ pk - 1) / ↑↑n = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(m i))\nn : ℕ+\nt : ℕ\nht : ↑n = 2 * (t + 1)\nt_succ : ℕ+ := ⟨t + 1, ⋯⟩\npm : ℕ → ℕ+\nhpm : 1 + (2 ^ pk - 1) / ↑↑t_succ = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(pm i))\nm : ℕ → ℕ+ := fun i => if i < pk then pm i else ⟨2 * t + 2 ^ pk.succ, ⋯⟩\n⊢ ∃ m, 1 + (2 ^ (pk + 1) - 1) / ↑↑n = ∏ i ∈ Finset.range (pk + 1), (1 + 1 / ↑↑(m i))"
},
{
"line": "refine_lift\n let m i := if i < pk then pm i else ⟨2 * t + 2 ^ pk.succ, arith_lemma pk t⟩;\n ?_",
"before_state": "case succ.inl.intro.succ.intro\npk : ℕ\nhpk : ∀ (n : ℕ+), ∃ m, 1 + (2 ^ pk - 1) / ↑↑n = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(m i))\nn : ℕ+\nt : ℕ\nht : ↑n = 2 * (t + 1)\nt_succ : ℕ+ := ⟨t + 1, ⋯⟩\npm : ℕ → ℕ+\nhpm : 1 + (2 ^ pk - 1) / ↑↑t_succ = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(pm i))\n⊢ ∃ m, 1 + (2 ^ (pk + 1) - 1) / ↑↑n = ∏ i ∈ Finset.range (pk + 1), (1 + 1 / ↑↑(m i))",
"after_state": "case succ.inl.intro.succ.intro\npk : ℕ\nhpk : ∀ (n : ℕ+), ∃ m, 1 + (2 ^ pk - 1) / ↑↑n = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(m i))\nn : ℕ+\nt : ℕ\nht : ↑n = 2 * (t + 1)\nt_succ : ℕ+ := ⟨t + 1, ⋯⟩\npm : ℕ → ℕ+\nhpm : 1 + (2 ^ pk - 1) / ↑↑t_succ = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(pm i))\nm : ℕ → ℕ+ := fun i => if i < pk then pm i else ⟨2 * t + 2 ^ pk.succ, ⋯⟩\n⊢ ∃ m, 1 + (2 ^ (pk + 1) - 1) / ↑↑n = ∏ i ∈ Finset.range (pk + 1), (1 + 1 / ↑↑(m i))"
},
{
"line": "focus\n (refine\n no_implicit_lambda%\n (let m i := if i < pk then pm i else ⟨2 * t + 2 ^ pk.succ, arith_lemma pk t⟩;\n ?_);\n rotate_right)",
"before_state": "case succ.inl.intro.succ.intro\npk : ℕ\nhpk : ∀ (n : ℕ+), ∃ m, 1 + (2 ^ pk - 1) / ↑↑n = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(m i))\nn : ℕ+\nt : ℕ\nht : ↑n = 2 * (t + 1)\nt_succ : ℕ+ := ⟨t + 1, ⋯⟩\npm : ℕ → ℕ+\nhpm : 1 + (2 ^ pk - 1) / ↑↑t_succ = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(pm i))\n⊢ ∃ m, 1 + (2 ^ (pk + 1) - 1) / ↑↑n = ∏ i ∈ Finset.range (pk + 1), (1 + 1 / ↑↑(m i))",
"after_state": "case succ.inl.intro.succ.intro\npk : ℕ\nhpk : ∀ (n : ℕ+), ∃ m, 1 + (2 ^ pk - 1) / ↑↑n = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(m i))\nn : ℕ+\nt : ℕ\nht : ↑n = 2 * (t + 1)\nt_succ : ℕ+ := ⟨t + 1, ⋯⟩\npm : ℕ → ℕ+\nhpm : 1 + (2 ^ pk - 1) / ↑↑t_succ = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(pm i))\nm : ℕ → ℕ+ := fun i => if i < pk then pm i else ⟨2 * t + 2 ^ pk.succ, ⋯⟩\n⊢ ∃ m, 1 + (2 ^ (pk + 1) - 1) / ↑↑n = ∏ i ∈ Finset.range (pk + 1), (1 + 1 / ↑↑(m i))"
},
{
"line": "refine\n no_implicit_lambda%\n (let m i := if i < pk then pm i else ⟨2 * t + 2 ^ pk.succ, arith_lemma pk t⟩;\n ?_)",
"before_state": "case succ.inl.intro.succ.intro\npk : ℕ\nhpk : ∀ (n : ℕ+), ∃ m, 1 + (2 ^ pk - 1) / ↑↑n = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(m i))\nn : ℕ+\nt : ℕ\nht : ↑n = 2 * (t + 1)\nt_succ : ℕ+ := ⟨t + 1, ⋯⟩\npm : ℕ → ℕ+\nhpm : 1 + (2 ^ pk - 1) / ↑↑t_succ = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(pm i))\n⊢ ∃ m, 1 + (2 ^ (pk + 1) - 1) / ↑↑n = ∏ i ∈ Finset.range (pk + 1), (1 + 1 / ↑↑(m i))",
"after_state": "case succ.inl.intro.succ.intro\npk : ℕ\nhpk : ∀ (n : ℕ+), ∃ m, 1 + (2 ^ pk - 1) / ↑↑n = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(m i))\nn : ℕ+\nt : ℕ\nht : ↑n = 2 * (t + 1)\nt_succ : ℕ+ := ⟨t + 1, ⋯⟩\npm : ℕ → ℕ+\nhpm : 1 + (2 ^ pk - 1) / ↑↑t_succ = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(pm i))\nm : ℕ → ℕ+ := fun i => if i < pk then pm i else ⟨2 * t + 2 ^ pk.succ, ⋯⟩\n⊢ ∃ m, 1 + (2 ^ (pk + 1) - 1) / ↑↑n = ∏ i ∈ Finset.range (pk + 1), (1 + 1 / ↑↑(m i))"
},
{
"line": "rotate_right",
"before_state": "case succ.inl.intro.succ.intro\npk : ℕ\nhpk : ∀ (n : ℕ+), ∃ m, 1 + (2 ^ pk - 1) / ↑↑n = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(m i))\nn : ℕ+\nt : ℕ\nht : ↑n = 2 * (t + 1)\nt_succ : ℕ+ := ⟨t + 1, ⋯⟩\npm : ℕ → ℕ+\nhpm : 1 + (2 ^ pk - 1) / ↑↑t_succ = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(pm i))\nm : ℕ → ℕ+ := fun i => if i < pk then pm i else ⟨2 * t + 2 ^ pk.succ, ⋯⟩\n⊢ ∃ m, 1 + (2 ^ (pk + 1) - 1) / ↑↑n = ∏ i ∈ Finset.range (pk + 1), (1 + 1 / ↑↑(m i))",
"after_state": "case succ.inl.intro.succ.intro\npk : ℕ\nhpk : ∀ (n : ℕ+), ∃ m, 1 + (2 ^ pk - 1) / ↑↑n = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(m i))\nn : ℕ+\nt : ℕ\nht : ↑n = 2 * (t + 1)\nt_succ : ℕ+ := ⟨t + 1, ⋯⟩\npm : ℕ → ℕ+\nhpm : 1 + (2 ^ pk - 1) / ↑↑t_succ = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(pm i))\nm : ℕ → ℕ+ := fun i => if i < pk then pm i else ⟨2 * t + 2 ^ pk.succ, ⋯⟩\n⊢ ∃ m, 1 + (2 ^ (pk + 1) - 1) / ↑↑n = ∏ i ∈ Finset.range (pk + 1), (1 + 1 / ↑↑(m i))"
},
{
"line": "use m",
"before_state": "case succ.inl.intro.succ.intro\npk : ℕ\nhpk : ∀ (n : ℕ+), ∃ m, 1 + (2 ^ pk - 1) / ↑↑n = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(m i))\nn : ℕ+\nt : ℕ\nht : ↑n = 2 * (t + 1)\nt_succ : ℕ+ := ⟨t + 1, ⋯⟩\npm : ℕ → ℕ+\nhpm : 1 + (2 ^ pk - 1) / ↑↑t_succ = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(pm i))\nm : ℕ → ℕ+ := fun i => if i < pk then pm i else ⟨2 * t + 2 ^ pk.succ, ⋯⟩\n⊢ ∃ m, 1 + (2 ^ (pk + 1) - 1) / ↑↑n = ∏ i ∈ Finset.range (pk + 1), (1 + 1 / ↑↑(m i))",
"after_state": "case h\npk : ℕ\nhpk : ∀ (n : ℕ+), ∃ m, 1 + (2 ^ pk - 1) / ↑↑n = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(m i))\nn : ℕ+\nt : ℕ\nht : ↑n = 2 * (t + 1)\nt_succ : ℕ+ := ⟨t + 1, ⋯⟩\npm : ℕ → ℕ+\nhpm : 1 + (2 ^ pk - 1) / ↑↑t_succ = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(pm i))\nm : ℕ → ℕ+ := fun i => if i < pk then pm i else ⟨2 * t + 2 ^ pk.succ, ⋯⟩\n⊢ 1 + (2 ^ (pk + 1) - 1) / ↑↑n = ∏ i ∈ Finset.range (pk + 1), (1 + 1 / ↑↑(m i))"
},
{
"line": "refine without_cdot(m : ?m✝)",
"before_state": "case w\npk : ℕ\nhpk : ∀ (n : ℕ+), ∃ m, 1 + (2 ^ pk - 1) / ↑↑n = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(m i))\nn : ℕ+\nt : ℕ\nht : ↑n = 2 * (t + 1)\nt_succ : ℕ+ := ⟨t + 1, ⋯⟩\npm : ℕ → ℕ+\nhpm : 1 + (2 ^ pk - 1) / ↑↑t_succ = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(pm i))\nm : ℕ → ℕ+ := fun i => if i < pk then pm i else ⟨2 * t + 2 ^ pk.succ, ⋯⟩\n⊢ ℕ → ℕ+",
"after_state": "No Goals!"
},
{
"line": "try with_reducible use_discharger",
"before_state": "case h\npk : ℕ\nhpk : ∀ (n : ℕ+), ∃ m, 1 + (2 ^ pk - 1) / ↑↑n = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(m i))\nn : ℕ+\nt : ℕ\nht : ↑n = 2 * (t + 1)\nt_succ : ℕ+ := ⟨t + 1, ⋯⟩\npm : ℕ → ℕ+\nhpm : 1 + (2 ^ pk - 1) / ↑↑t_succ = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(pm i))\nm : ℕ → ℕ+ := fun i => if i < pk then pm i else ⟨2 * t + 2 ^ pk.succ, ⋯⟩\n⊢ 1 + (2 ^ (pk + 1) - 1) / ↑↑n = ∏ i ∈ Finset.range (pk + 1), (1 + 1 / ↑↑(m i))",
"after_state": "case h\npk : ℕ\nhpk : ∀ (n : ℕ+), ∃ m, 1 + (2 ^ pk - 1) / ↑↑n = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(m i))\nn : ℕ+\nt : ℕ\nht : ↑n = 2 * (t + 1)\nt_succ : ℕ+ := ⟨t + 1, ⋯⟩\npm : ℕ → ℕ+\nhpm : 1 + (2 ^ pk - 1) / ↑↑t_succ = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(pm i))\nm : ℕ → ℕ+ := fun i => if i < pk then pm i else ⟨2 * t + 2 ^ pk.succ, ⋯⟩\n⊢ 1 + (2 ^ (pk + 1) - 1) / ↑↑n = ∏ i ∈ Finset.range (pk + 1), (1 + 1 / ↑↑(m i))"
},
{
"line": "first\n| with_reducible use_discharger\n| skip",
"before_state": "case h\npk : ℕ\nhpk : ∀ (n : ℕ+), ∃ m, 1 + (2 ^ pk - 1) / ↑↑n = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(m i))\nn : ℕ+\nt : ℕ\nht : ↑n = 2 * (t + 1)\nt_succ : ℕ+ := ⟨t + 1, ⋯⟩\npm : ℕ → ℕ+\nhpm : 1 + (2 ^ pk - 1) / ↑↑t_succ = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(pm i))\nm : ℕ → ℕ+ := fun i => if i < pk then pm i else ⟨2 * t + 2 ^ pk.succ, ⋯⟩\n⊢ 1 + (2 ^ (pk + 1) - 1) / ↑↑n = ∏ i ∈ Finset.range (pk + 1), (1 + 1 / ↑↑(m i))",
"after_state": "case h\npk : ℕ\nhpk : ∀ (n : ℕ+), ∃ m, 1 + (2 ^ pk - 1) / ↑↑n = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(m i))\nn : ℕ+\nt : ℕ\nht : ↑n = 2 * (t + 1)\nt_succ : ℕ+ := ⟨t + 1, ⋯⟩\npm : ℕ → ℕ+\nhpm : 1 + (2 ^ pk - 1) / ↑↑t_succ = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(pm i))\nm : ℕ → ℕ+ := fun i => if i < pk then pm i else ⟨2 * t + 2 ^ pk.succ, ⋯⟩\n⊢ 1 + (2 ^ (pk + 1) - 1) / ↑↑n = ∏ i ∈ Finset.range (pk + 1), (1 + 1 / ↑↑(m i))"
},
{
"line": "with_reducible use_discharger",
"before_state": "case h\npk : ℕ\nhpk : ∀ (n : ℕ+), ∃ m, 1 + (2 ^ pk - 1) / ↑↑n = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(m i))\nn : ℕ+\nt : ℕ\nht : ↑n = 2 * (t + 1)\nt_succ : ℕ+ := ⟨t + 1, ⋯⟩\npm : ℕ → ℕ+\nhpm : 1 + (2 ^ pk - 1) / ↑↑t_succ = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(pm i))\nm : ℕ → ℕ+ := fun i => if i < pk then pm i else ⟨2 * t + 2 ^ pk.succ, ⋯⟩\n⊢ 1 + (2 ^ (pk + 1) - 1) / ↑↑n = ∏ i ∈ Finset.range (pk + 1), (1 + 1 / ↑↑(m i))",
"after_state": "case h\npk : ℕ\nhpk : ∀ (n : ℕ+), ∃ m, 1 + (2 ^ pk - 1) / ↑↑n = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(m i))\nn : ℕ+\nt : ℕ\nht : ↑n = 2 * (t + 1)\nt_succ : ℕ+ := ⟨t + 1, ⋯⟩\npm : ℕ → ℕ+\nhpm : 1 + (2 ^ pk - 1) / ↑↑t_succ = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(pm i))\nm : ℕ → ℕ+ := fun i => if i < pk then pm i else ⟨2 * t + 2 ^ pk.succ, ⋯⟩\n⊢ 1 + (2 ^ (pk + 1) - 1) / ↑↑n = ∏ i ∈ Finset.range (pk + 1), (1 + 1 / ↑↑(m i))"
},
{
"line": "use_discharger",
"before_state": "case h\npk : ℕ\nhpk : ∀ (n : ℕ+), ∃ m, 1 + (2 ^ pk - 1) / ↑↑n = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(m i))\nn : ℕ+\nt : ℕ\nht : ↑n = 2 * (t + 1)\nt_succ : ℕ+ := ⟨t + 1, ⋯⟩\npm : ℕ → ℕ+\nhpm : 1 + (2 ^ pk - 1) / ↑↑t_succ = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(pm i))\nm : ℕ → ℕ+ := fun i => if i < pk then pm i else ⟨2 * t + 2 ^ pk.succ, ⋯⟩\n⊢ 1 + (2 ^ (pk + 1) - 1) / ↑↑n = ∏ i ∈ Finset.range (pk + 1), (1 + 1 / ↑↑(m i))",
"after_state": "case h\npk : ℕ\nhpk : ∀ (n : ℕ+), ∃ m, 1 + (2 ^ pk - 1) / ↑↑n = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(m i))\nn : ℕ+\nt : ℕ\nht : ↑n = 2 * (t + 1)\nt_succ : ℕ+ := ⟨t + 1, ⋯⟩\npm : ℕ → ℕ+\nhpm : 1 + (2 ^ pk - 1) / ↑↑t_succ = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(pm i))\nm : ℕ → ℕ+ := fun i => if i < pk then pm i else ⟨2 * t + 2 ^ pk.succ, ⋯⟩\n⊢ 1 + (2 ^ (pk + 1) - 1) / ↑↑n = ∏ i ∈ Finset.range (pk + 1), (1 + 1 / ↑↑(m i))"
},
{
"line": "focus\n apply exists_prop.mpr✝\n with_annotate_state\"<;>\" skip\n all_goals use_discharger",
"before_state": "case h\npk : ℕ\nhpk : ∀ (n : ℕ+), ∃ m, 1 + (2 ^ pk - 1) / ↑↑n = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(m i))\nn : ℕ+\nt : ℕ\nht : ↑n = 2 * (t + 1)\nt_succ : ℕ+ := ⟨t + 1, ⋯⟩\npm : ℕ → ℕ+\nhpm : 1 + (2 ^ pk - 1) / ↑↑t_succ = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(pm i))\nm : ℕ → ℕ+ := fun i => if i < pk then pm i else ⟨2 * t + 2 ^ pk.succ, ⋯⟩\n⊢ 1 + (2 ^ (pk + 1) - 1) / ↑↑n = ∏ i ∈ Finset.range (pk + 1), (1 + 1 / ↑↑(m i))",
"after_state": "case h\npk : ℕ\nhpk : ∀ (n : ℕ+), ∃ m, 1 + (2 ^ pk - 1) / ↑↑n = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(m i))\nn : ℕ+\nt : ℕ\nht : ↑n = 2 * (t + 1)\nt_succ : ℕ+ := ⟨t + 1, ⋯⟩\npm : ℕ → ℕ+\nhpm : 1 + (2 ^ pk - 1) / ↑↑t_succ = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(pm i))\nm : ℕ → ℕ+ := fun i => if i < pk then pm i else ⟨2 * t + 2 ^ pk.succ, ⋯⟩\n⊢ 1 + (2 ^ (pk + 1) - 1) / ↑↑n = ∏ i ∈ Finset.range (pk + 1), (1 + 1 / ↑↑(m i))"
},
{
"line": "apply exists_prop.mpr✝",
"before_state": "case h\npk : ℕ\nhpk : ∀ (n : ℕ+), ∃ m, 1 + (2 ^ pk - 1) / ↑↑n = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(m i))\nn : ℕ+\nt : ℕ\nht : ↑n = 2 * (t + 1)\nt_succ : ℕ+ := ⟨t + 1, ⋯⟩\npm : ℕ → ℕ+\nhpm : 1 + (2 ^ pk - 1) / ↑↑t_succ = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(pm i))\nm : ℕ → ℕ+ := fun i => if i < pk then pm i else ⟨2 * t + 2 ^ pk.succ, ⋯⟩\n⊢ 1 + (2 ^ (pk + 1) - 1) / ↑↑n = ∏ i ∈ Finset.range (pk + 1), (1 + 1 / ↑↑(m i))",
"after_state": "case h\npk : ℕ\nhpk : ∀ (n : ℕ+), ∃ m, 1 + (2 ^ pk - 1) / ↑↑n = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(m i))\nn : ℕ+\nt : ℕ\nht : ↑n = 2 * (t + 1)\nt_succ : ℕ+ := ⟨t + 1, ⋯⟩\npm : ℕ → ℕ+\nhpm : 1 + (2 ^ pk - 1) / ↑↑t_succ = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(pm i))\nm : ℕ → ℕ+ := fun i => if i < pk then pm i else ⟨2 * t + 2 ^ pk.succ, ⋯⟩\n⊢ 1 + (2 ^ (pk + 1) - 1) / ↑↑n = ∏ i ∈ Finset.range (pk + 1), (1 + 1 / ↑↑(m i))"
},
{
"line": "skip",
"before_state": "case h\npk : ℕ\nhpk : ∀ (n : ℕ+), ∃ m, 1 + (2 ^ pk - 1) / ↑↑n = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(m i))\nn : ℕ+\nt : ℕ\nht : ↑n = 2 * (t + 1)\nt_succ : ℕ+ := ⟨t + 1, ⋯⟩\npm : ℕ → ℕ+\nhpm : 1 + (2 ^ pk - 1) / ↑↑t_succ = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(pm i))\nm : ℕ → ℕ+ := fun i => if i < pk then pm i else ⟨2 * t + 2 ^ pk.succ, ⋯⟩\n⊢ 1 + (2 ^ (pk + 1) - 1) / ↑↑n = ∏ i ∈ Finset.range (pk + 1), (1 + 1 / ↑↑(m i))",
"after_state": "case h\npk : ℕ\nhpk : ∀ (n : ℕ+), ∃ m, 1 + (2 ^ pk - 1) / ↑↑n = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(m i))\nn : ℕ+\nt : ℕ\nht : ↑n = 2 * (t + 1)\nt_succ : ℕ+ := ⟨t + 1, ⋯⟩\npm : ℕ → ℕ+\nhpm : 1 + (2 ^ pk - 1) / ↑↑t_succ = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(pm i))\nm : ℕ → ℕ+ := fun i => if i < pk then pm i else ⟨2 * t + 2 ^ pk.succ, ⋯⟩\n⊢ 1 + (2 ^ (pk + 1) - 1) / ↑↑n = ∏ i ∈ Finset.range (pk + 1), (1 + 1 / ↑↑(m i))"
},
{
"line": "have hmpk : (m pk : ℚ) = 2 * t + 2 ^ pk.succ := by have : m pk = ⟨2 * t + 2 ^ pk.succ, _⟩ := if_neg (irrefl pk);\n simp [this]",
"before_state": "case h\npk : ℕ\nhpk : ∀ (n : ℕ+), ∃ m, 1 + (2 ^ pk - 1) / ↑↑n = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(m i))\nn : ℕ+\nt : ℕ\nht : ↑n = 2 * (t + 1)\nt_succ : ℕ+ := ⟨t + 1, ⋯⟩\npm : ℕ → ℕ+\nhpm : 1 + (2 ^ pk - 1) / ↑↑t_succ = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(pm i))\nm : ℕ → ℕ+ := fun i => if i < pk then pm i else ⟨2 * t + 2 ^ pk.succ, ⋯⟩\n⊢ 1 + (2 ^ (pk + 1) - 1) / ↑↑n = ∏ i ∈ Finset.range (pk + 1), (1 + 1 / ↑↑(m i))",
"after_state": "case h\npk : ℕ\nhpk : ∀ (n : ℕ+), ∃ m, 1 + (2 ^ pk - 1) / ↑↑n = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(m i))\nn : ℕ+\nt : ℕ\nht : ↑n = 2 * (t + 1)\nt_succ : ℕ+ := ⟨t + 1, ⋯⟩\npm : ℕ → ℕ+\nhpm : 1 + (2 ^ pk - 1) / ↑↑t_succ = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(pm i))\nm : ℕ → ℕ+ := fun i => if i < pk then pm i else ⟨2 * t + 2 ^ pk.succ, ⋯⟩\nhmpk : ↑↑(m pk) = 2 * ↑t + 2 ^ pk.succ\n⊢ 1 + (2 ^ (pk + 1) - 1) / ↑↑n = ∏ i ∈ Finset.range (pk + 1), (1 + 1 / ↑↑(m i))"
},
{
"line": "focus\n refine\n no_implicit_lambda%\n (have hmpk : (m pk : ℚ) = 2 * t + 2 ^ pk.succ := ?body✝;\n ?_)\n case body✝ => with_annotate_state\"by\" (have : m pk = ⟨2 * t + 2 ^ pk.succ, _⟩ := if_neg (irrefl pk); simp [this])",
"before_state": "case h\npk : ℕ\nhpk : ∀ (n : ℕ+), ∃ m, 1 + (2 ^ pk - 1) / ↑↑n = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(m i))\nn : ℕ+\nt : ℕ\nht : ↑n = 2 * (t + 1)\nt_succ : ℕ+ := ⟨t + 1, ⋯⟩\npm : ℕ → ℕ+\nhpm : 1 + (2 ^ pk - 1) / ↑↑t_succ = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(pm i))\nm : ℕ → ℕ+ := fun i => if i < pk then pm i else ⟨2 * t + 2 ^ pk.succ, ⋯⟩\n⊢ 1 + (2 ^ (pk + 1) - 1) / ↑↑n = ∏ i ∈ Finset.range (pk + 1), (1 + 1 / ↑↑(m i))",
"after_state": "case h\npk : ℕ\nhpk : ∀ (n : ℕ+), ∃ m, 1 + (2 ^ pk - 1) / ↑↑n = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(m i))\nn : ℕ+\nt : ℕ\nht : ↑n = 2 * (t + 1)\nt_succ : ℕ+ := ⟨t + 1, ⋯⟩\npm : ℕ → ℕ+\nhpm : 1 + (2 ^ pk - 1) / ↑↑t_succ = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(pm i))\nm : ℕ → ℕ+ := fun i => if i < pk then pm i else ⟨2 * t + 2 ^ pk.succ, ⋯⟩\nhmpk : ↑↑(m pk) = 2 * ↑t + 2 ^ pk.succ\n⊢ 1 + (2 ^ (pk + 1) - 1) / ↑↑n = ∏ i ∈ Finset.range (pk + 1), (1 + 1 / ↑↑(m i))"
},
{
"line": "refine\n no_implicit_lambda%\n (have hmpk : (m pk : ℚ) = 2 * t + 2 ^ pk.succ := ?body✝;\n ?_)",
"before_state": "case h\npk : ℕ\nhpk : ∀ (n : ℕ+), ∃ m, 1 + (2 ^ pk - 1) / ↑↑n = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(m i))\nn : ℕ+\nt : ℕ\nht : ↑n = 2 * (t + 1)\nt_succ : ℕ+ := ⟨t + 1, ⋯⟩\npm : ℕ → ℕ+\nhpm : 1 + (2 ^ pk - 1) / ↑↑t_succ = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(pm i))\nm : ℕ → ℕ+ := fun i => if i < pk then pm i else ⟨2 * t + 2 ^ pk.succ, ⋯⟩\n⊢ 1 + (2 ^ (pk + 1) - 1) / ↑↑n = ∏ i ∈ Finset.range (pk + 1), (1 + 1 / ↑↑(m i))",
"after_state": "case body\npk : ℕ\nhpk : ∀ (n : ℕ+), ∃ m, 1 + (2 ^ pk - 1) / ↑↑n = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(m i))\nn : ℕ+\nt : ℕ\nht : ↑n = 2 * (t + 1)\nt_succ : ℕ+ := ⟨t + 1, ⋯⟩\npm : ℕ → ℕ+\nhpm : 1 + (2 ^ pk - 1) / ↑↑t_succ = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(pm i))\nm : ℕ → ℕ+ := fun i => if i < pk then pm i else ⟨2 * t + 2 ^ pk.succ, ⋯⟩\n⊢ ↑↑(m pk) = 2 * ↑t + 2 ^ pk.succ\n---\ncase h\npk : ℕ\nhpk : ∀ (n : ℕ+), ∃ m, 1 + (2 ^ pk - 1) / ↑↑n = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(m i))\nn : ℕ+\nt : ℕ\nht : ↑n = 2 * (t + 1)\nt_succ : ℕ+ := ⟨t + 1, ⋯⟩\npm : ℕ → ℕ+\nhpm : 1 + (2 ^ pk - 1) / ↑↑t_succ = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(pm i))\nm : ℕ → ℕ+ := fun i => if i < pk then pm i else ⟨2 * t + 2 ^ pk.succ, ⋯⟩\nhmpk : ↑↑(m pk) = 2 * ↑t + 2 ^ pk.succ\n⊢ 1 + (2 ^ (pk + 1) - 1) / ↑↑n = ∏ i ∈ Finset.range (pk + 1), (1 + 1 / ↑↑(m i))"
},
{
"line": "case body✝ => with_annotate_state\"by\" (have : m pk = ⟨2 * t + 2 ^ pk.succ, _⟩ := if_neg (irrefl pk); simp [this])",
"before_state": "case body\npk : ℕ\nhpk : ∀ (n : ℕ+), ∃ m, 1 + (2 ^ pk - 1) / ↑↑n = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(m i))\nn : ℕ+\nt : ℕ\nht : ↑n = 2 * (t + 1)\nt_succ : ℕ+ := ⟨t + 1, ⋯⟩\npm : ℕ → ℕ+\nhpm : 1 + (2 ^ pk - 1) / ↑↑t_succ = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(pm i))\nm : ℕ → ℕ+ := fun i => if i < pk then pm i else ⟨2 * t + 2 ^ pk.succ, ⋯⟩\n⊢ ↑↑(m pk) = 2 * ↑t + 2 ^ pk.succ\n---\ncase h\npk : ℕ\nhpk : ∀ (n : ℕ+), ∃ m, 1 + (2 ^ pk - 1) / ↑↑n = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(m i))\nn : ℕ+\nt : ℕ\nht : ↑n = 2 * (t + 1)\nt_succ : ℕ+ := ⟨t + 1, ⋯⟩\npm : ℕ → ℕ+\nhpm : 1 + (2 ^ pk - 1) / ↑↑t_succ = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(pm i))\nm : ℕ → ℕ+ := fun i => if i < pk then pm i else ⟨2 * t + 2 ^ pk.succ, ⋯⟩\nhmpk : ↑↑(m pk) = 2 * ↑t + 2 ^ pk.succ\n⊢ 1 + (2 ^ (pk + 1) - 1) / ↑↑n = ∏ i ∈ Finset.range (pk + 1), (1 + 1 / ↑↑(m i))",
"after_state": "case h\npk : ℕ\nhpk : ∀ (n : ℕ+), ∃ m, 1 + (2 ^ pk - 1) / ↑↑n = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(m i))\nn : ℕ+\nt : ℕ\nht : ↑n = 2 * (t + 1)\nt_succ : ℕ+ := ⟨t + 1, ⋯⟩\npm : ℕ → ℕ+\nhpm : 1 + (2 ^ pk - 1) / ↑↑t_succ = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(pm i))\nm : ℕ → ℕ+ := fun i => if i < pk then pm i else ⟨2 * t + 2 ^ pk.succ, ⋯⟩\nhmpk : ↑↑(m pk) = 2 * ↑t + 2 ^ pk.succ\n⊢ 1 + (2 ^ (pk + 1) - 1) / ↑↑n = ∏ i ∈ Finset.range (pk + 1), (1 + 1 / ↑↑(m i))"
},
{
"line": "with_annotate_state\"by\" (have : m pk = ⟨2 * t + 2 ^ pk.succ, _⟩ := if_neg (irrefl pk); simp [this])",
"before_state": "pk : ℕ\nhpk : ∀ (n : ℕ+), ∃ m, 1 + (2 ^ pk - 1) / ↑↑n = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(m i))\nn : ℕ+\nt : ℕ\nht : ↑n = 2 * (t + 1)\nt_succ : ℕ+ := ⟨t + 1, ⋯⟩\npm : ℕ → ℕ+\nhpm : 1 + (2 ^ pk - 1) / ↑↑t_succ = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(pm i))\nm : ℕ → ℕ+ := fun i => if i < pk then pm i else ⟨2 * t + 2 ^ pk.succ, ⋯⟩\n⊢ ↑↑(m pk) = 2 * ↑t + 2 ^ pk.succ",
"after_state": "No Goals!"
},
{
"line": "have : m pk = ⟨2 * t + 2 ^ pk.succ, _⟩ := if_neg (irrefl pk)",
"before_state": "pk : ℕ\nhpk : ∀ (n : ℕ+), ∃ m, 1 + (2 ^ pk - 1) / ↑↑n = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(m i))\nn : ℕ+\nt : ℕ\nht : ↑n = 2 * (t + 1)\nt_succ : ℕ+ := ⟨t + 1, ⋯⟩\npm : ℕ → ℕ+\nhpm : 1 + (2 ^ pk - 1) / ↑↑t_succ = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(pm i))\nm : ℕ → ℕ+ := fun i => if i < pk then pm i else ⟨2 * t + 2 ^ pk.succ, ⋯⟩\n⊢ ↑↑(m pk) = 2 * ↑t + 2 ^ pk.succ",
"after_state": "pk : ℕ\nhpk : ∀ (n : ℕ+), ∃ m, 1 + (2 ^ pk - 1) / ↑↑n = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(m i))\nn : ℕ+\nt : ℕ\nht : ↑n = 2 * (t + 1)\nt_succ : ℕ+ := ⟨t + 1, ⋯⟩\npm : ℕ → ℕ+\nhpm : 1 + (2 ^ pk - 1) / ↑↑t_succ = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(pm i))\nm : ℕ → ℕ+ := fun i => if i < pk then pm i else ⟨2 * t + 2 ^ pk.succ, ⋯⟩\nthis : m pk = ⟨2 * t + 2 ^ pk.succ, ⋯⟩\n⊢ ↑↑(m pk) = 2 * ↑t + 2 ^ pk.succ"
},
{
"line": "refine_lift\n have : m pk = ⟨2 * t + 2 ^ pk.succ, _⟩ := if_neg (irrefl pk);\n ?_",
"before_state": "pk : ℕ\nhpk : ∀ (n : ℕ+), ∃ m, 1 + (2 ^ pk - 1) / ↑↑n = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(m i))\nn : ℕ+\nt : ℕ\nht : ↑n = 2 * (t + 1)\nt_succ : ℕ+ := ⟨t + 1, ⋯⟩\npm : ℕ → ℕ+\nhpm : 1 + (2 ^ pk - 1) / ↑↑t_succ = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(pm i))\nm : ℕ → ℕ+ := fun i => if i < pk then pm i else ⟨2 * t + 2 ^ pk.succ, ⋯⟩\n⊢ ↑↑(m pk) = 2 * ↑t + 2 ^ pk.succ",
"after_state": "pk : ℕ\nhpk : ∀ (n : ℕ+), ∃ m, 1 + (2 ^ pk - 1) / ↑↑n = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(m i))\nn : ℕ+\nt : ℕ\nht : ↑n = 2 * (t + 1)\nt_succ : ℕ+ := ⟨t + 1, ⋯⟩\npm : ℕ → ℕ+\nhpm : 1 + (2 ^ pk - 1) / ↑↑t_succ = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(pm i))\nm : ℕ → ℕ+ := fun i => if i < pk then pm i else ⟨2 * t + 2 ^ pk.succ, ⋯⟩\nthis : m pk = ⟨2 * t + 2 ^ pk.succ, ⋯⟩\n⊢ ↑↑(m pk) = 2 * ↑t + 2 ^ pk.succ"
},
{
"line": "focus\n (refine\n no_implicit_lambda%\n (have : m pk = ⟨2 * t + 2 ^ pk.succ, _⟩ := if_neg (irrefl pk);\n ?_);\n rotate_right)",
"before_state": "pk : ℕ\nhpk : ∀ (n : ℕ+), ∃ m, 1 + (2 ^ pk - 1) / ↑↑n = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(m i))\nn : ℕ+\nt : ℕ\nht : ↑n = 2 * (t + 1)\nt_succ : ℕ+ := ⟨t + 1, ⋯⟩\npm : ℕ → ℕ+\nhpm : 1 + (2 ^ pk - 1) / ↑↑t_succ = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(pm i))\nm : ℕ → ℕ+ := fun i => if i < pk then pm i else ⟨2 * t + 2 ^ pk.succ, ⋯⟩\n⊢ ↑↑(m pk) = 2 * ↑t + 2 ^ pk.succ",
"after_state": "pk : ℕ\nhpk : ∀ (n : ℕ+), ∃ m, 1 + (2 ^ pk - 1) / ↑↑n = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(m i))\nn : ℕ+\nt : ℕ\nht : ↑n = 2 * (t + 1)\nt_succ : ℕ+ := ⟨t + 1, ⋯⟩\npm : ℕ → ℕ+\nhpm : 1 + (2 ^ pk - 1) / ↑↑t_succ = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(pm i))\nm : ℕ → ℕ+ := fun i => if i < pk then pm i else ⟨2 * t + 2 ^ pk.succ, ⋯⟩\nthis : m pk = ⟨2 * t + 2 ^ pk.succ, ⋯⟩\n⊢ ↑↑(m pk) = 2 * ↑t + 2 ^ pk.succ"
},
{
"line": "refine\n no_implicit_lambda%\n (have : m pk = ⟨2 * t + 2 ^ pk.succ, _⟩ := if_neg (irrefl pk);\n ?_)",
"before_state": "pk : ℕ\nhpk : ∀ (n : ℕ+), ∃ m, 1 + (2 ^ pk - 1) / ↑↑n = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(m i))\nn : ℕ+\nt : ℕ\nht : ↑n = 2 * (t + 1)\nt_succ : ℕ+ := ⟨t + 1, ⋯⟩\npm : ℕ → ℕ+\nhpm : 1 + (2 ^ pk - 1) / ↑↑t_succ = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(pm i))\nm : ℕ → ℕ+ := fun i => if i < pk then pm i else ⟨2 * t + 2 ^ pk.succ, ⋯⟩\n⊢ ↑↑(m pk) = 2 * ↑t + 2 ^ pk.succ",
"after_state": "pk : ℕ\nhpk : ∀ (n : ℕ+), ∃ m, 1 + (2 ^ pk - 1) / ↑↑n = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(m i))\nn : ℕ+\nt : ℕ\nht : ↑n = 2 * (t + 1)\nt_succ : ℕ+ := ⟨t + 1, ⋯⟩\npm : ℕ → ℕ+\nhpm : 1 + (2 ^ pk - 1) / ↑↑t_succ = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(pm i))\nm : ℕ → ℕ+ := fun i => if i < pk then pm i else ⟨2 * t + 2 ^ pk.succ, ⋯⟩\nthis : m pk = ⟨2 * t + 2 ^ pk.succ, ⋯⟩\n⊢ ↑↑(m pk) = 2 * ↑t + 2 ^ pk.succ"
},
{
"line": "rotate_right",
"before_state": "pk : ℕ\nhpk : ∀ (n : ℕ+), ∃ m, 1 + (2 ^ pk - 1) / ↑↑n = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(m i))\nn : ℕ+\nt : ℕ\nht : ↑n = 2 * (t + 1)\nt_succ : ℕ+ := ⟨t + 1, ⋯⟩\npm : ℕ → ℕ+\nhpm : 1 + (2 ^ pk - 1) / ↑↑t_succ = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(pm i))\nm : ℕ → ℕ+ := fun i => if i < pk then pm i else ⟨2 * t + 2 ^ pk.succ, ⋯⟩\nthis : m pk = ⟨2 * t + 2 ^ pk.succ, ⋯⟩\n⊢ ↑↑(m pk) = 2 * ↑t + 2 ^ pk.succ",
"after_state": "pk : ℕ\nhpk : ∀ (n : ℕ+), ∃ m, 1 + (2 ^ pk - 1) / ↑↑n = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(m i))\nn : ℕ+\nt : ℕ\nht : ↑n = 2 * (t + 1)\nt_succ : ℕ+ := ⟨t + 1, ⋯⟩\npm : ℕ → ℕ+\nhpm : 1 + (2 ^ pk - 1) / ↑↑t_succ = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(pm i))\nm : ℕ → ℕ+ := fun i => if i < pk then pm i else ⟨2 * t + 2 ^ pk.succ, ⋯⟩\nthis : m pk = ⟨2 * t + 2 ^ pk.succ, ⋯⟩\n⊢ ↑↑(m pk) = 2 * ↑t + 2 ^ pk.succ"
},
{
"line": "simp [this]",
"before_state": "pk : ℕ\nhpk : ∀ (n : ℕ+), ∃ m, 1 + (2 ^ pk - 1) / ↑↑n = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(m i))\nn : ℕ+\nt : ℕ\nht : ↑n = 2 * (t + 1)\nt_succ : ℕ+ := ⟨t + 1, ⋯⟩\npm : ℕ → ℕ+\nhpm : 1 + (2 ^ pk - 1) / ↑↑t_succ = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(pm i))\nm : ℕ → ℕ+ := fun i => if i < pk then pm i else ⟨2 * t + 2 ^ pk.succ, ⋯⟩\nthis : m pk = ⟨2 * t + 2 ^ pk.succ, ⋯⟩\n⊢ ↑↑(m pk) = 2 * ↑t + 2 ^ pk.succ",
"after_state": "No Goals!"
},
{
"line": "calc\n ((1 : ℚ) + (2 ^ pk.succ - 1) / (n : ℚ) : ℚ) = 1 + (2 * 2 ^ pk - 1) / (2 * (t + 1) : ℕ) :=\n by\n rw [ht]\n rw [pow_succ']\n _ = (1 + 1 / (2 * t + 2 * 2 ^ pk)) * (1 + (2 ^ pk - 1) / (↑t + 1)) :=\n by\n field_simp\n ring\n _ = (1 + 1 / (2 * t + 2 ^ pk.succ)) * (1 + (2 ^ pk - 1) / t_succ) := by simp [pow_succ', PNat.mk_coe, t_succ]\n _ = (∏ i ∈ Finset.range pk, (1 + 1 / (m i : ℚ))) * (1 + 1 / m pk) :=\n by\n rw [prod_lemma]\n rw [hpm]\n rw [← hmpk]\n rw [mul_comm]\n _ = ∏ i ∈ Finset.range pk.succ, (1 + 1 / (m i : ℚ)) := by rw [← Finset.prod_range_succ _ pk]",
"before_state": "case h\npk : ℕ\nhpk : ∀ (n : ℕ+), ∃ m, 1 + (2 ^ pk - 1) / ↑↑n = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(m i))\nn : ℕ+\nt : ℕ\nht : ↑n = 2 * (t + 1)\nt_succ : ℕ+ := ⟨t + 1, ⋯⟩\npm : ℕ → ℕ+\nhpm : 1 + (2 ^ pk - 1) / ↑↑t_succ = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(pm i))\nm : ℕ → ℕ+ := fun i => if i < pk then pm i else ⟨2 * t + 2 ^ pk.succ, ⋯⟩\nhmpk : ↑↑(m pk) = 2 * ↑t + 2 ^ pk.succ\n⊢ 1 + (2 ^ (pk + 1) - 1) / ↑↑n = ∏ i ∈ Finset.range (pk + 1), (1 + 1 / ↑↑(m i))",
"after_state": "No Goals!"
},
{
"line": "rw [ht]",
"before_state": "pk : ℕ\nhpk : ∀ (n : ℕ+), ∃ m, 1 + (2 ^ pk - 1) / ↑↑n = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(m i))\nn : ℕ+\nt : ℕ\nht : ↑n = 2 * (t + 1)\nt_succ : ℕ+ := ⟨t + 1, ⋯⟩\npm : ℕ → ℕ+\nhpm : 1 + (2 ^ pk - 1) / ↑↑t_succ = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(pm i))\nm : ℕ → ℕ+ := fun i => if i < pk then pm i else ⟨2 * t + 2 ^ pk.succ, ⋯⟩\nhmpk : ↑↑(m pk) = 2 * ↑t + 2 ^ pk.succ\n⊢ 1 + (2 ^ pk.succ - 1) / ↑↑n = 1 + (2 * 2 ^ pk - 1) / ↑(2 * (t + 1))",
"after_state": "pk : ℕ\nhpk : ∀ (n : ℕ+), ∃ m, 1 + (2 ^ pk - 1) / ↑↑n = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(m i))\nn : ℕ+\nt : ℕ\nht : ↑n = 2 * (t + 1)\nt_succ : ℕ+ := ⟨t + 1, ⋯⟩\npm : ℕ → ℕ+\nhpm : 1 + (2 ^ pk - 1) / ↑↑t_succ = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(pm i))\nm : ℕ → ℕ+ := fun i => if i < pk then pm i else ⟨2 * t + 2 ^ pk.succ, ⋯⟩\nhmpk : ↑↑(m pk) = 2 * ↑t + 2 ^ pk.succ\n⊢ 1 + (2 ^ pk.succ - 1) / ↑(2 * (t + 1)) = 1 + (2 * 2 ^ pk - 1) / ↑(2 * (t + 1))"
},
{
"line": "rewrite [ht]",
"before_state": "pk : ℕ\nhpk : ∀ (n : ℕ+), ∃ m, 1 + (2 ^ pk - 1) / ↑↑n = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(m i))\nn : ℕ+\nt : ℕ\nht : ↑n = 2 * (t + 1)\nt_succ : ℕ+ := ⟨t + 1, ⋯⟩\npm : ℕ → ℕ+\nhpm : 1 + (2 ^ pk - 1) / ↑↑t_succ = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(pm i))\nm : ℕ → ℕ+ := fun i => if i < pk then pm i else ⟨2 * t + 2 ^ pk.succ, ⋯⟩\nhmpk : ↑↑(m pk) = 2 * ↑t + 2 ^ pk.succ\n⊢ 1 + (2 ^ pk.succ - 1) / ↑↑n = 1 + (2 * 2 ^ pk - 1) / ↑(2 * (t + 1))",
"after_state": "pk : ℕ\nhpk : ∀ (n : ℕ+), ∃ m, 1 + (2 ^ pk - 1) / ↑↑n = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(m i))\nn : ℕ+\nt : ℕ\nht : ↑n = 2 * (t + 1)\nt_succ : ℕ+ := ⟨t + 1, ⋯⟩\npm : ℕ → ℕ+\nhpm : 1 + (2 ^ pk - 1) / ↑↑t_succ = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(pm i))\nm : ℕ → ℕ+ := fun i => if i < pk then pm i else ⟨2 * t + 2 ^ pk.succ, ⋯⟩\nhmpk : ↑↑(m pk) = 2 * ↑t + 2 ^ pk.succ\n⊢ 1 + (2 ^ pk.succ - 1) / ↑(2 * (t + 1)) = 1 + (2 * 2 ^ pk - 1) / ↑(2 * (t + 1))"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "pk : ℕ\nhpk : ∀ (n : ℕ+), ∃ m, 1 + (2 ^ pk - 1) / ↑↑n = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(m i))\nn : ℕ+\nt : ℕ\nht : ↑n = 2 * (t + 1)\nt_succ : ℕ+ := ⟨t + 1, ⋯⟩\npm : ℕ → ℕ+\nhpm : 1 + (2 ^ pk - 1) / ↑↑t_succ = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(pm i))\nm : ℕ → ℕ+ := fun i => if i < pk then pm i else ⟨2 * t + 2 ^ pk.succ, ⋯⟩\nhmpk : ↑↑(m pk) = 2 * ↑t + 2 ^ pk.succ\n⊢ 1 + (2 ^ pk.succ - 1) / ↑(2 * (t + 1)) = 1 + (2 * 2 ^ pk - 1) / ↑(2 * (t + 1))",
"after_state": "pk : ℕ\nhpk : ∀ (n : ℕ+), ∃ m, 1 + (2 ^ pk - 1) / ↑↑n = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(m i))\nn : ℕ+\nt : ℕ\nht : ↑n = 2 * (t + 1)\nt_succ : ℕ+ := ⟨t + 1, ⋯⟩\npm : ℕ → ℕ+\nhpm : 1 + (2 ^ pk - 1) / ↑↑t_succ = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(pm i))\nm : ℕ → ℕ+ := fun i => if i < pk then pm i else ⟨2 * t + 2 ^ pk.succ, ⋯⟩\nhmpk : ↑↑(m pk) = 2 * ↑t + 2 ^ pk.succ\n⊢ 1 + (2 ^ pk.succ - 1) / ↑(2 * (t + 1)) = 1 + (2 * 2 ^ pk - 1) / ↑(2 * (t + 1))"
},
{
"line": "try (with_reducible rfl)",
"before_state": "pk : ℕ\nhpk : ∀ (n : ℕ+), ∃ m, 1 + (2 ^ pk - 1) / ↑↑n = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(m i))\nn : ℕ+\nt : ℕ\nht : ↑n = 2 * (t + 1)\nt_succ : ℕ+ := ⟨t + 1, ⋯⟩\npm : ℕ → ℕ+\nhpm : 1 + (2 ^ pk - 1) / ↑↑t_succ = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(pm i))\nm : ℕ → ℕ+ := fun i => if i < pk then pm i else ⟨2 * t + 2 ^ pk.succ, ⋯⟩\nhmpk : ↑↑(m pk) = 2 * ↑t + 2 ^ pk.succ\n⊢ 1 + (2 ^ pk.succ - 1) / ↑(2 * (t + 1)) = 1 + (2 * 2 ^ pk - 1) / ↑(2 * (t + 1))",
"after_state": "pk : ℕ\nhpk : ∀ (n : ℕ+), ∃ m, 1 + (2 ^ pk - 1) / ↑↑n = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(m i))\nn : ℕ+\nt : ℕ\nht : ↑n = 2 * (t + 1)\nt_succ : ℕ+ := ⟨t + 1, ⋯⟩\npm : ℕ → ℕ+\nhpm : 1 + (2 ^ pk - 1) / ↑↑t_succ = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(pm i))\nm : ℕ → ℕ+ := fun i => if i < pk then pm i else ⟨2 * t + 2 ^ pk.succ, ⋯⟩\nhmpk : ↑↑(m pk) = 2 * ↑t + 2 ^ pk.succ\n⊢ 1 + (2 ^ pk.succ - 1) / ↑(2 * (t + 1)) = 1 + (2 * 2 ^ pk - 1) / ↑(2 * (t + 1))"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "pk : ℕ\nhpk : ∀ (n : ℕ+), ∃ m, 1 + (2 ^ pk - 1) / ↑↑n = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(m i))\nn : ℕ+\nt : ℕ\nht : ↑n = 2 * (t + 1)\nt_succ : ℕ+ := ⟨t + 1, ⋯⟩\npm : ℕ → ℕ+\nhpm : 1 + (2 ^ pk - 1) / ↑↑t_succ = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(pm i))\nm : ℕ → ℕ+ := fun i => if i < pk then pm i else ⟨2 * t + 2 ^ pk.succ, ⋯⟩\nhmpk : ↑↑(m pk) = 2 * ↑t + 2 ^ pk.succ\n⊢ 1 + (2 ^ pk.succ - 1) / ↑(2 * (t + 1)) = 1 + (2 * 2 ^ pk - 1) / ↑(2 * (t + 1))",
"after_state": "pk : ℕ\nhpk : ∀ (n : ℕ+), ∃ m, 1 + (2 ^ pk - 1) / ↑↑n = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(m i))\nn : ℕ+\nt : ℕ\nht : ↑n = 2 * (t + 1)\nt_succ : ℕ+ := ⟨t + 1, ⋯⟩\npm : ℕ → ℕ+\nhpm : 1 + (2 ^ pk - 1) / ↑↑t_succ = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(pm i))\nm : ℕ → ℕ+ := fun i => if i < pk then pm i else ⟨2 * t + 2 ^ pk.succ, ⋯⟩\nhmpk : ↑↑(m pk) = 2 * ↑t + 2 ^ pk.succ\n⊢ 1 + (2 ^ pk.succ - 1) / ↑(2 * (t + 1)) = 1 + (2 * 2 ^ pk - 1) / ↑(2 * (t + 1))"
},
{
"line": "with_reducible rfl",
"before_state": "pk : ℕ\nhpk : ∀ (n : ℕ+), ∃ m, 1 + (2 ^ pk - 1) / ↑↑n = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(m i))\nn : ℕ+\nt : ℕ\nht : ↑n = 2 * (t + 1)\nt_succ : ℕ+ := ⟨t + 1, ⋯⟩\npm : ℕ → ℕ+\nhpm : 1 + (2 ^ pk - 1) / ↑↑t_succ = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(pm i))\nm : ℕ → ℕ+ := fun i => if i < pk then pm i else ⟨2 * t + 2 ^ pk.succ, ⋯⟩\nhmpk : ↑↑(m pk) = 2 * ↑t + 2 ^ pk.succ\n⊢ 1 + (2 ^ pk.succ - 1) / ↑(2 * (t + 1)) = 1 + (2 * 2 ^ pk - 1) / ↑(2 * (t + 1))",
"after_state": "pk : ℕ\nhpk : ∀ (n : ℕ+), ∃ m, 1 + (2 ^ pk - 1) / ↑↑n = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(m i))\nn : ℕ+\nt : ℕ\nht : ↑n = 2 * (t + 1)\nt_succ : ℕ+ := ⟨t + 1, ⋯⟩\npm : ℕ → ℕ+\nhpm : 1 + (2 ^ pk - 1) / ↑↑t_succ = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(pm i))\nm : ℕ → ℕ+ := fun i => if i < pk then pm i else ⟨2 * t + 2 ^ pk.succ, ⋯⟩\nhmpk : ↑↑(m pk) = 2 * ↑t + 2 ^ pk.succ\n⊢ 1 + (2 ^ pk.succ - 1) / ↑(2 * (t + 1)) = 1 + (2 * 2 ^ pk - 1) / ↑(2 * (t + 1))"
},
{
"line": "rfl",
"before_state": "pk : ℕ\nhpk : ∀ (n : ℕ+), ∃ m, 1 + (2 ^ pk - 1) / ↑↑n = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(m i))\nn : ℕ+\nt : ℕ\nht : ↑n = 2 * (t + 1)\nt_succ : ℕ+ := ⟨t + 1, ⋯⟩\npm : ℕ → ℕ+\nhpm : 1 + (2 ^ pk - 1) / ↑↑t_succ = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(pm i))\nm : ℕ → ℕ+ := fun i => if i < pk then pm i else ⟨2 * t + 2 ^ pk.succ, ⋯⟩\nhmpk : ↑↑(m pk) = 2 * ↑t + 2 ^ pk.succ\n⊢ 1 + (2 ^ pk.succ - 1) / ↑(2 * (t + 1)) = 1 + (2 * 2 ^ pk - 1) / ↑(2 * (t + 1))",
"after_state": "pk : ℕ\nhpk : ∀ (n : ℕ+), ∃ m, 1 + (2 ^ pk - 1) / ↑↑n = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(m i))\nn : ℕ+\nt : ℕ\nht : ↑n = 2 * (t + 1)\nt_succ : ℕ+ := ⟨t + 1, ⋯⟩\npm : ℕ → ℕ+\nhpm : 1 + (2 ^ pk - 1) / ↑↑t_succ = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(pm i))\nm : ℕ → ℕ+ := fun i => if i < pk then pm i else ⟨2 * t + 2 ^ pk.succ, ⋯⟩\nhmpk : ↑↑(m pk) = 2 * ↑t + 2 ^ pk.succ\n⊢ 1 + (2 ^ pk.succ - 1) / ↑(2 * (t + 1)) = 1 + (2 * 2 ^ pk - 1) / ↑(2 * (t + 1))"
},
{
"line": "apply_rfl",
"before_state": "pk : ℕ\nhpk : ∀ (n : ℕ+), ∃ m, 1 + (2 ^ pk - 1) / ↑↑n = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(m i))\nn : ℕ+\nt : ℕ\nht : ↑n = 2 * (t + 1)\nt_succ : ℕ+ := ⟨t + 1, ⋯⟩\npm : ℕ → ℕ+\nhpm : 1 + (2 ^ pk - 1) / ↑↑t_succ = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(pm i))\nm : ℕ → ℕ+ := fun i => if i < pk then pm i else ⟨2 * t + 2 ^ pk.succ, ⋯⟩\nhmpk : ↑↑(m pk) = 2 * ↑t + 2 ^ pk.succ\n⊢ 1 + (2 ^ pk.succ - 1) / ↑(2 * (t + 1)) = 1 + (2 * 2 ^ pk - 1) / ↑(2 * (t + 1))",
"after_state": "pk : ℕ\nhpk : ∀ (n : ℕ+), ∃ m, 1 + (2 ^ pk - 1) / ↑↑n = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(m i))\nn : ℕ+\nt : ℕ\nht : ↑n = 2 * (t + 1)\nt_succ : ℕ+ := ⟨t + 1, ⋯⟩\npm : ℕ → ℕ+\nhpm : 1 + (2 ^ pk - 1) / ↑↑t_succ = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(pm i))\nm : ℕ → ℕ+ := fun i => if i < pk then pm i else ⟨2 * t + 2 ^ pk.succ, ⋯⟩\nhmpk : ↑↑(m pk) = 2 * ↑t + 2 ^ pk.succ\n⊢ 1 + (2 ^ pk.succ - 1) / ↑(2 * (t + 1)) = 1 + (2 * 2 ^ pk - 1) / ↑(2 * (t + 1))"
},
{
"line": "skip",
"before_state": "pk : ℕ\nhpk : ∀ (n : ℕ+), ∃ m, 1 + (2 ^ pk - 1) / ↑↑n = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(m i))\nn : ℕ+\nt : ℕ\nht : ↑n = 2 * (t + 1)\nt_succ : ℕ+ := ⟨t + 1, ⋯⟩\npm : ℕ → ℕ+\nhpm : 1 + (2 ^ pk - 1) / ↑↑t_succ = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(pm i))\nm : ℕ → ℕ+ := fun i => if i < pk then pm i else ⟨2 * t + 2 ^ pk.succ, ⋯⟩\nhmpk : ↑↑(m pk) = 2 * ↑t + 2 ^ pk.succ\n⊢ 1 + (2 ^ pk.succ - 1) / ↑(2 * (t + 1)) = 1 + (2 * 2 ^ pk - 1) / ↑(2 * (t + 1))",
"after_state": "pk : ℕ\nhpk : ∀ (n : ℕ+), ∃ m, 1 + (2 ^ pk - 1) / ↑↑n = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(m i))\nn : ℕ+\nt : ℕ\nht : ↑n = 2 * (t + 1)\nt_succ : ℕ+ := ⟨t + 1, ⋯⟩\npm : ℕ → ℕ+\nhpm : 1 + (2 ^ pk - 1) / ↑↑t_succ = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(pm i))\nm : ℕ → ℕ+ := fun i => if i < pk then pm i else ⟨2 * t + 2 ^ pk.succ, ⋯⟩\nhmpk : ↑↑(m pk) = 2 * ↑t + 2 ^ pk.succ\n⊢ 1 + (2 ^ pk.succ - 1) / ↑(2 * (t + 1)) = 1 + (2 * 2 ^ pk - 1) / ↑(2 * (t + 1))"
},
{
"line": "rw [pow_succ']",
"before_state": "pk : ℕ\nhpk : ∀ (n : ℕ+), ∃ m, 1 + (2 ^ pk - 1) / ↑↑n = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(m i))\nn : ℕ+\nt : ℕ\nht : ↑n = 2 * (t + 1)\nt_succ : ℕ+ := ⟨t + 1, ⋯⟩\npm : ℕ → ℕ+\nhpm : 1 + (2 ^ pk - 1) / ↑↑t_succ = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(pm i))\nm : ℕ → ℕ+ := fun i => if i < pk then pm i else ⟨2 * t + 2 ^ pk.succ, ⋯⟩\nhmpk : ↑↑(m pk) = 2 * ↑t + 2 ^ pk.succ\n⊢ 1 + (2 ^ pk.succ - 1) / ↑(2 * (t + 1)) = 1 + (2 * 2 ^ pk - 1) / ↑(2 * (t + 1))",
"after_state": "No Goals!"
},
{
"line": "rewrite [pow_succ']",
"before_state": "pk : ℕ\nhpk : ∀ (n : ℕ+), ∃ m, 1 + (2 ^ pk - 1) / ↑↑n = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(m i))\nn : ℕ+\nt : ℕ\nht : ↑n = 2 * (t + 1)\nt_succ : ℕ+ := ⟨t + 1, ⋯⟩\npm : ℕ → ℕ+\nhpm : 1 + (2 ^ pk - 1) / ↑↑t_succ = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(pm i))\nm : ℕ → ℕ+ := fun i => if i < pk then pm i else ⟨2 * t + 2 ^ pk.succ, ⋯⟩\nhmpk : ↑↑(m pk) = 2 * ↑t + 2 ^ pk.succ\n⊢ 1 + (2 ^ pk.succ - 1) / ↑(2 * (t + 1)) = 1 + (2 * 2 ^ pk - 1) / ↑(2 * (t + 1))",
"after_state": "pk : ℕ\nhpk : ∀ (n : ℕ+), ∃ m, 1 + (2 ^ pk - 1) / ↑↑n = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(m i))\nn : ℕ+\nt : ℕ\nht : ↑n = 2 * (t + 1)\nt_succ : ℕ+ := ⟨t + 1, ⋯⟩\npm : ℕ → ℕ+\nhpm : 1 + (2 ^ pk - 1) / ↑↑t_succ = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(pm i))\nm : ℕ → ℕ+ := fun i => if i < pk then pm i else ⟨2 * t + 2 ^ pk.succ, ⋯⟩\nhmpk : ↑↑(m pk) = 2 * ↑t + 2 ^ pk.succ\n⊢ 1 + (2 * 2 ^ pk - 1) / ↑(2 * (t + 1)) = 1 + (2 * 2 ^ pk - 1) / ↑(2 * (t + 1))"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "pk : ℕ\nhpk : ∀ (n : ℕ+), ∃ m, 1 + (2 ^ pk - 1) / ↑↑n = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(m i))\nn : ℕ+\nt : ℕ\nht : ↑n = 2 * (t + 1)\nt_succ : ℕ+ := ⟨t + 1, ⋯⟩\npm : ℕ → ℕ+\nhpm : 1 + (2 ^ pk - 1) / ↑↑t_succ = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(pm i))\nm : ℕ → ℕ+ := fun i => if i < pk then pm i else ⟨2 * t + 2 ^ pk.succ, ⋯⟩\nhmpk : ↑↑(m pk) = 2 * ↑t + 2 ^ pk.succ\n⊢ 1 + (2 * 2 ^ pk - 1) / ↑(2 * (t + 1)) = 1 + (2 * 2 ^ pk - 1) / ↑(2 * (t + 1))",
"after_state": "No Goals!"
},
{
"line": "try (with_reducible rfl)",
"before_state": "pk : ℕ\nhpk : ∀ (n : ℕ+), ∃ m, 1 + (2 ^ pk - 1) / ↑↑n = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(m i))\nn : ℕ+\nt : ℕ\nht : ↑n = 2 * (t + 1)\nt_succ : ℕ+ := ⟨t + 1, ⋯⟩\npm : ℕ → ℕ+\nhpm : 1 + (2 ^ pk - 1) / ↑↑t_succ = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(pm i))\nm : ℕ → ℕ+ := fun i => if i < pk then pm i else ⟨2 * t + 2 ^ pk.succ, ⋯⟩\nhmpk : ↑↑(m pk) = 2 * ↑t + 2 ^ pk.succ\n⊢ 1 + (2 * 2 ^ pk - 1) / ↑(2 * (t + 1)) = 1 + (2 * 2 ^ pk - 1) / ↑(2 * (t + 1))",
"after_state": "No Goals!"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "pk : ℕ\nhpk : ∀ (n : ℕ+), ∃ m, 1 + (2 ^ pk - 1) / ↑↑n = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(m i))\nn : ℕ+\nt : ℕ\nht : ↑n = 2 * (t + 1)\nt_succ : ℕ+ := ⟨t + 1, ⋯⟩\npm : ℕ → ℕ+\nhpm : 1 + (2 ^ pk - 1) / ↑↑t_succ = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(pm i))\nm : ℕ → ℕ+ := fun i => if i < pk then pm i else ⟨2 * t + 2 ^ pk.succ, ⋯⟩\nhmpk : ↑↑(m pk) = 2 * ↑t + 2 ^ pk.succ\n⊢ 1 + (2 * 2 ^ pk - 1) / ↑(2 * (t + 1)) = 1 + (2 * 2 ^ pk - 1) / ↑(2 * (t + 1))",
"after_state": "No Goals!"
},
{
"line": "with_reducible rfl",
"before_state": "pk : ℕ\nhpk : ∀ (n : ℕ+), ∃ m, 1 + (2 ^ pk - 1) / ↑↑n = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(m i))\nn : ℕ+\nt : ℕ\nht : ↑n = 2 * (t + 1)\nt_succ : ℕ+ := ⟨t + 1, ⋯⟩\npm : ℕ → ℕ+\nhpm : 1 + (2 ^ pk - 1) / ↑↑t_succ = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(pm i))\nm : ℕ → ℕ+ := fun i => if i < pk then pm i else ⟨2 * t + 2 ^ pk.succ, ⋯⟩\nhmpk : ↑↑(m pk) = 2 * ↑t + 2 ^ pk.succ\n⊢ 1 + (2 * 2 ^ pk - 1) / ↑(2 * (t + 1)) = 1 + (2 * 2 ^ pk - 1) / ↑(2 * (t + 1))",
"after_state": "No Goals!"
},
{
"line": "rfl",
"before_state": "pk : ℕ\nhpk : ∀ (n : ℕ+), ∃ m, 1 + (2 ^ pk - 1) / ↑↑n = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(m i))\nn : ℕ+\nt : ℕ\nht : ↑n = 2 * (t + 1)\nt_succ : ℕ+ := ⟨t + 1, ⋯⟩\npm : ℕ → ℕ+\nhpm : 1 + (2 ^ pk - 1) / ↑↑t_succ = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(pm i))\nm : ℕ → ℕ+ := fun i => if i < pk then pm i else ⟨2 * t + 2 ^ pk.succ, ⋯⟩\nhmpk : ↑↑(m pk) = 2 * ↑t + 2 ^ pk.succ\n⊢ 1 + (2 * 2 ^ pk - 1) / ↑(2 * (t + 1)) = 1 + (2 * 2 ^ pk - 1) / ↑(2 * (t + 1))",
"after_state": "No Goals!"
},
{
"line": "eq_refl",
"before_state": "pk : ℕ\nhpk : ∀ (n : ℕ+), ∃ m, 1 + (2 ^ pk - 1) / ↑↑n = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(m i))\nn : ℕ+\nt : ℕ\nht : ↑n = 2 * (t + 1)\nt_succ : ℕ+ := ⟨t + 1, ⋯⟩\npm : ℕ → ℕ+\nhpm : 1 + (2 ^ pk - 1) / ↑↑t_succ = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(pm i))\nm : ℕ → ℕ+ := fun i => if i < pk then pm i else ⟨2 * t + 2 ^ pk.succ, ⋯⟩\nhmpk : ↑↑(m pk) = 2 * ↑t + 2 ^ pk.succ\n⊢ 1 + (2 * 2 ^ pk - 1) / ↑(2 * (t + 1)) = 1 + (2 * 2 ^ pk - 1) / ↑(2 * (t + 1))",
"after_state": "No Goals!"
},
{
"line": "field_simp",
"before_state": "pk : ℕ\nhpk : ∀ (n : ℕ+), ∃ m, 1 + (2 ^ pk - 1) / ↑↑n = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(m i))\nn : ℕ+\nt : ℕ\nht : ↑n = 2 * (t + 1)\nt_succ : ℕ+ := ⟨t + 1, ⋯⟩\npm : ℕ → ℕ+\nhpm : 1 + (2 ^ pk - 1) / ↑↑t_succ = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(pm i))\nm : ℕ → ℕ+ := fun i => if i < pk then pm i else ⟨2 * t + 2 ^ pk.succ, ⋯⟩\nhmpk : ↑↑(m pk) = 2 * ↑t + 2 ^ pk.succ\n⊢ 1 + (2 * 2 ^ pk - 1) / ↑(2 * (t + 1)) = (1 + 1 / (2 * ↑t + 2 * 2 ^ pk)) * (1 + (2 ^ pk - 1) / (↑t + 1))",
"after_state": "pk : ℕ\nhpk : ∀ (n : ℕ+), ∃ m, 1 + (2 ^ pk - 1) / ↑↑n = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(m i))\nn : ℕ+\nt : ℕ\nht : ↑n = 2 * (t + 1)\nt_succ : ℕ+ := ⟨t + 1, ⋯⟩\npm : ℕ → ℕ+\nhpm : 1 + (2 ^ pk - 1) / ↑↑t_succ = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(pm i))\nm : ℕ → ℕ+ := fun i => if i < pk then pm i else ⟨2 * t + 2 ^ pk.succ, ⋯⟩\nhmpk : ↑↑(m pk) = 2 * ↑t + 2 ^ pk.succ\n⊢ (2 * (↑t + 1) + (2 * 2 ^ pk - 1)) * ((2 * ↑t + 2 * 2 ^ pk) * (↑t + 1)) =\n (2 * ↑t + 2 * 2 ^ pk + 1) * (↑t + 2 ^ pk) * (2 * (↑t + 1))"
},
{
"line": "ring",
"before_state": "pk : ℕ\nhpk : ∀ (n : ℕ+), ∃ m, 1 + (2 ^ pk - 1) / ↑↑n = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(m i))\nn : ℕ+\nt : ℕ\nht : ↑n = 2 * (t + 1)\nt_succ : ℕ+ := ⟨t + 1, ⋯⟩\npm : ℕ → ℕ+\nhpm : 1 + (2 ^ pk - 1) / ↑↑t_succ = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(pm i))\nm : ℕ → ℕ+ := fun i => if i < pk then pm i else ⟨2 * t + 2 ^ pk.succ, ⋯⟩\nhmpk : ↑↑(m pk) = 2 * ↑t + 2 ^ pk.succ\n⊢ (2 * (↑t + 1) + (2 * 2 ^ pk - 1)) * ((2 * ↑t + 2 * 2 ^ pk) * (↑t + 1)) =\n (2 * ↑t + 2 * 2 ^ pk + 1) * (↑t + 2 ^ pk) * (2 * (↑t + 1))",
"after_state": "No Goals!"
},
{
"line": "first\n| ring1\n|\n try_this ring_nf\"\\n\\nThe `ring` tactic failed to close the goal. Use `ring_nf` to obtain a normal form.\n \\nNote that `ring` works primarily in *commutative* rings. \\\n If you have a noncommutative ring, abelian group or module, consider using \\\n `noncomm_ring`, `abel` or `module` instead.\"",
"before_state": "pk : ℕ\nhpk : ∀ (n : ℕ+), ∃ m, 1 + (2 ^ pk - 1) / ↑↑n = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(m i))\nn : ℕ+\nt : ℕ\nht : ↑n = 2 * (t + 1)\nt_succ : ℕ+ := ⟨t + 1, ⋯⟩\npm : ℕ → ℕ+\nhpm : 1 + (2 ^ pk - 1) / ↑↑t_succ = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(pm i))\nm : ℕ → ℕ+ := fun i => if i < pk then pm i else ⟨2 * t + 2 ^ pk.succ, ⋯⟩\nhmpk : ↑↑(m pk) = 2 * ↑t + 2 ^ pk.succ\n⊢ (2 * (↑t + 1) + (2 * 2 ^ pk - 1)) * ((2 * ↑t + 2 * 2 ^ pk) * (↑t + 1)) =\n (2 * ↑t + 2 * 2 ^ pk + 1) * (↑t + 2 ^ pk) * (2 * (↑t + 1))",
"after_state": "No Goals!"
},
{
"line": "ring1",
"before_state": "pk : ℕ\nhpk : ∀ (n : ℕ+), ∃ m, 1 + (2 ^ pk - 1) / ↑↑n = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(m i))\nn : ℕ+\nt : ℕ\nht : ↑n = 2 * (t + 1)\nt_succ : ℕ+ := ⟨t + 1, ⋯⟩\npm : ℕ → ℕ+\nhpm : 1 + (2 ^ pk - 1) / ↑↑t_succ = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(pm i))\nm : ℕ → ℕ+ := fun i => if i < pk then pm i else ⟨2 * t + 2 ^ pk.succ, ⋯⟩\nhmpk : ↑↑(m pk) = 2 * ↑t + 2 ^ pk.succ\n⊢ (2 * (↑t + 1) + (2 * 2 ^ pk - 1)) * ((2 * ↑t + 2 * 2 ^ pk) * (↑t + 1)) =\n (2 * ↑t + 2 * 2 ^ pk + 1) * (↑t + 2 ^ pk) * (2 * (↑t + 1))",
"after_state": "No Goals!"
},
{
"line": "simp [pow_succ', PNat.mk_coe, t_succ]",
"before_state": "pk : ℕ\nhpk : ∀ (n : ℕ+), ∃ m, 1 + (2 ^ pk - 1) / ↑↑n = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(m i))\nn : ℕ+\nt : ℕ\nht : ↑n = 2 * (t + 1)\nt_succ : ℕ+ := ⟨t + 1, ⋯⟩\npm : ℕ → ℕ+\nhpm : 1 + (2 ^ pk - 1) / ↑↑t_succ = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(pm i))\nm : ℕ → ℕ+ := fun i => if i < pk then pm i else ⟨2 * t + 2 ^ pk.succ, ⋯⟩\nhmpk : ↑↑(m pk) = 2 * ↑t + 2 ^ pk.succ\n⊢ (1 + 1 / (2 * ↑t + 2 * 2 ^ pk)) * (1 + (2 ^ pk - 1) / (↑t + 1)) =\n (1 + 1 / (2 * ↑t + 2 ^ pk.succ)) * (1 + (2 ^ pk - 1) / ↑↑t_succ)",
"after_state": "No Goals!"
},
{
"line": "rw [prod_lemma]",
"before_state": "pk : ℕ\nhpk : ∀ (n : ℕ+), ∃ m, 1 + (2 ^ pk - 1) / ↑↑n = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(m i))\nn : ℕ+\nt : ℕ\nht : ↑n = 2 * (t + 1)\nt_succ : ℕ+ := ⟨t + 1, ⋯⟩\npm : ℕ → ℕ+\nhpm : 1 + (2 ^ pk - 1) / ↑↑t_succ = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(pm i))\nm : ℕ → ℕ+ := fun i => if i < pk then pm i else ⟨2 * t + 2 ^ pk.succ, ⋯⟩\nhmpk : ↑↑(m pk) = 2 * ↑t + 2 ^ pk.succ\n⊢ (1 + 1 / (2 * ↑t + 2 ^ pk.succ)) * (1 + (2 ^ pk - 1) / ↑↑t_succ) =\n (∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(m i))) * (1 + 1 / ↑↑(m pk))",
"after_state": "pk : ℕ\nhpk : ∀ (n : ℕ+), ∃ m, 1 + (2 ^ pk - 1) / ↑↑n = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(m i))\nn : ℕ+\nt : ℕ\nht : ↑n = 2 * (t + 1)\nt_succ : ℕ+ := ⟨t + 1, ⋯⟩\npm : ℕ → ℕ+\nhpm : 1 + (2 ^ pk - 1) / ↑↑t_succ = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(pm i))\nm : ℕ → ℕ+ := fun i => if i < pk then pm i else ⟨2 * t + 2 ^ pk.succ, ⋯⟩\nhmpk : ↑↑(m pk) = 2 * ↑t + 2 ^ pk.succ\n⊢ (1 + 1 / (2 * ↑t + 2 ^ pk.succ)) * (1 + (2 ^ pk - 1) / ↑↑t_succ) =\n (∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(m i))) * (1 + 1 / ↑↑(m pk))"
},
{
"line": "rewrite [prod_lemma]",
"before_state": "pk : ℕ\nhpk : ∀ (n : ℕ+), ∃ m, 1 + (2 ^ pk - 1) / ↑↑n = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(m i))\nn : ℕ+\nt : ℕ\nht : ↑n = 2 * (t + 1)\nt_succ : ℕ+ := ⟨t + 1, ⋯⟩\npm : ℕ → ℕ+\nhpm : 1 + (2 ^ pk - 1) / ↑↑t_succ = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(pm i))\nm : ℕ → ℕ+ := fun i => if i < pk then pm i else ⟨2 * t + 2 ^ pk.succ, ⋯⟩\nhmpk : ↑↑(m pk) = 2 * ↑t + 2 ^ pk.succ\n⊢ (1 + 1 / (2 * ↑t + 2 ^ pk.succ)) * (1 + (2 ^ pk - 1) / ↑↑t_succ) =\n (∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(m i))) * (1 + 1 / ↑↑(m pk))",
"after_state": "pk : ℕ\nhpk : ∀ (n : ℕ+), ∃ m, 1 + (2 ^ pk - 1) / ↑↑n = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(m i))\nn : ℕ+\nt : ℕ\nht : ↑n = 2 * (t + 1)\nt_succ : ℕ+ := ⟨t + 1, ⋯⟩\npm : ℕ → ℕ+\nhpm : 1 + (2 ^ pk - 1) / ↑↑t_succ = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(pm i))\nm : ℕ → ℕ+ := fun i => if i < pk then pm i else ⟨2 * t + 2 ^ pk.succ, ⋯⟩\nhmpk : ↑↑(m pk) = 2 * ↑t + 2 ^ pk.succ\n⊢ (1 + 1 / (2 * ↑t + 2 ^ pk.succ)) * (1 + (2 ^ pk - 1) / ↑↑t_succ) =\n (∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(m i))) * (1 + 1 / ↑↑(m pk))"
},
{
"line": "rw [← Finset.prod_range_succ _ pk]",
"before_state": "pk : ℕ\nhpk : ∀ (n : ℕ+), ∃ m, 1 + (2 ^ pk - 1) / ↑↑n = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(m i))\nn : ℕ+\nt : ℕ\nht : ↑n = 2 * (t + 1)\nt_succ : ℕ+ := ⟨t + 1, ⋯⟩\npm : ℕ → ℕ+\nhpm : 1 + (2 ^ pk - 1) / ↑↑t_succ = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(pm i))\nm : ℕ → ℕ+ := fun i => if i < pk then pm i else ⟨2 * t + 2 ^ pk.succ, ⋯⟩\nhmpk : ↑↑(m pk) = 2 * ↑t + 2 ^ pk.succ\n⊢ (∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(m i))) * (1 + 1 / ↑↑(m pk)) = ∏ i ∈ Finset.range pk.succ, (1 + 1 / ↑↑(m i))",
"after_state": "No Goals!"
},
{
"line": "rewrite [← Finset.prod_range_succ _ pk]",
"before_state": "pk : ℕ\nhpk : ∀ (n : ℕ+), ∃ m, 1 + (2 ^ pk - 1) / ↑↑n = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(m i))\nn : ℕ+\nt : ℕ\nht : ↑n = 2 * (t + 1)\nt_succ : ℕ+ := ⟨t + 1, ⋯⟩\npm : ℕ → ℕ+\nhpm : 1 + (2 ^ pk - 1) / ↑↑t_succ = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(pm i))\nm : ℕ → ℕ+ := fun i => if i < pk then pm i else ⟨2 * t + 2 ^ pk.succ, ⋯⟩\nhmpk : ↑↑(m pk) = 2 * ↑t + 2 ^ pk.succ\n⊢ (∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(m i))) * (1 + 1 / ↑↑(m pk)) = ∏ i ∈ Finset.range pk.succ, (1 + 1 / ↑↑(m i))",
"after_state": "pk : ℕ\nhpk : ∀ (n : ℕ+), ∃ m, 1 + (2 ^ pk - 1) / ↑↑n = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(m i))\nn : ℕ+\nt : ℕ\nht : ↑n = 2 * (t + 1)\nt_succ : ℕ+ := ⟨t + 1, ⋯⟩\npm : ℕ → ℕ+\nhpm : 1 + (2 ^ pk - 1) / ↑↑t_succ = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(pm i))\nm : ℕ → ℕ+ := fun i => if i < pk then pm i else ⟨2 * t + 2 ^ pk.succ, ⋯⟩\nhmpk : ↑↑(m pk) = 2 * ↑t + 2 ^ pk.succ\n⊢ ∏ x ∈ Finset.range (pk + 1), (1 + 1 / ↑↑(m x)) = ∏ i ∈ Finset.range pk.succ, (1 + 1 / ↑↑(m i))"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "pk : ℕ\nhpk : ∀ (n : ℕ+), ∃ m, 1 + (2 ^ pk - 1) / ↑↑n = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(m i))\nn : ℕ+\nt : ℕ\nht : ↑n = 2 * (t + 1)\nt_succ : ℕ+ := ⟨t + 1, ⋯⟩\npm : ℕ → ℕ+\nhpm : 1 + (2 ^ pk - 1) / ↑↑t_succ = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(pm i))\nm : ℕ → ℕ+ := fun i => if i < pk then pm i else ⟨2 * t + 2 ^ pk.succ, ⋯⟩\nhmpk : ↑↑(m pk) = 2 * ↑t + 2 ^ pk.succ\n⊢ ∏ x ∈ Finset.range (pk + 1), (1 + 1 / ↑↑(m x)) = ∏ i ∈ Finset.range pk.succ, (1 + 1 / ↑↑(m i))",
"after_state": "No Goals!"
},
{
"line": "try (with_reducible rfl)",
"before_state": "pk : ℕ\nhpk : ∀ (n : ℕ+), ∃ m, 1 + (2 ^ pk - 1) / ↑↑n = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(m i))\nn : ℕ+\nt : ℕ\nht : ↑n = 2 * (t + 1)\nt_succ : ℕ+ := ⟨t + 1, ⋯⟩\npm : ℕ → ℕ+\nhpm : 1 + (2 ^ pk - 1) / ↑↑t_succ = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(pm i))\nm : ℕ → ℕ+ := fun i => if i < pk then pm i else ⟨2 * t + 2 ^ pk.succ, ⋯⟩\nhmpk : ↑↑(m pk) = 2 * ↑t + 2 ^ pk.succ\n⊢ ∏ x ∈ Finset.range (pk + 1), (1 + 1 / ↑↑(m x)) = ∏ i ∈ Finset.range pk.succ, (1 + 1 / ↑↑(m i))",
"after_state": "No Goals!"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "pk : ℕ\nhpk : ∀ (n : ℕ+), ∃ m, 1 + (2 ^ pk - 1) / ↑↑n = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(m i))\nn : ℕ+\nt : ℕ\nht : ↑n = 2 * (t + 1)\nt_succ : ℕ+ := ⟨t + 1, ⋯⟩\npm : ℕ → ℕ+\nhpm : 1 + (2 ^ pk - 1) / ↑↑t_succ = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(pm i))\nm : ℕ → ℕ+ := fun i => if i < pk then pm i else ⟨2 * t + 2 ^ pk.succ, ⋯⟩\nhmpk : ↑↑(m pk) = 2 * ↑t + 2 ^ pk.succ\n⊢ ∏ x ∈ Finset.range (pk + 1), (1 + 1 / ↑↑(m x)) = ∏ i ∈ Finset.range pk.succ, (1 + 1 / ↑↑(m i))",
"after_state": "No Goals!"
},
{
"line": "with_reducible rfl",
"before_state": "pk : ℕ\nhpk : ∀ (n : ℕ+), ∃ m, 1 + (2 ^ pk - 1) / ↑↑n = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(m i))\nn : ℕ+\nt : ℕ\nht : ↑n = 2 * (t + 1)\nt_succ : ℕ+ := ⟨t + 1, ⋯⟩\npm : ℕ → ℕ+\nhpm : 1 + (2 ^ pk - 1) / ↑↑t_succ = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(pm i))\nm : ℕ → ℕ+ := fun i => if i < pk then pm i else ⟨2 * t + 2 ^ pk.succ, ⋯⟩\nhmpk : ↑↑(m pk) = 2 * ↑t + 2 ^ pk.succ\n⊢ ∏ x ∈ Finset.range (pk + 1), (1 + 1 / ↑↑(m x)) = ∏ i ∈ Finset.range pk.succ, (1 + 1 / ↑↑(m i))",
"after_state": "No Goals!"
},
{
"line": "rfl",
"before_state": "pk : ℕ\nhpk : ∀ (n : ℕ+), ∃ m, 1 + (2 ^ pk - 1) / ↑↑n = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(m i))\nn : ℕ+\nt : ℕ\nht : ↑n = 2 * (t + 1)\nt_succ : ℕ+ := ⟨t + 1, ⋯⟩\npm : ℕ → ℕ+\nhpm : 1 + (2 ^ pk - 1) / ↑↑t_succ = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(pm i))\nm : ℕ → ℕ+ := fun i => if i < pk then pm i else ⟨2 * t + 2 ^ pk.succ, ⋯⟩\nhmpk : ↑↑(m pk) = 2 * ↑t + 2 ^ pk.succ\n⊢ ∏ x ∈ Finset.range (pk + 1), (1 + 1 / ↑↑(m x)) = ∏ i ∈ Finset.range pk.succ, (1 + 1 / ↑↑(m i))",
"after_state": "No Goals!"
},
{
"line": "eq_refl",
"before_state": "pk : ℕ\nhpk : ∀ (n : ℕ+), ∃ m, 1 + (2 ^ pk - 1) / ↑↑n = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(m i))\nn : ℕ+\nt : ℕ\nht : ↑n = 2 * (t + 1)\nt_succ : ℕ+ := ⟨t + 1, ⋯⟩\npm : ℕ → ℕ+\nhpm : 1 + (2 ^ pk - 1) / ↑↑t_succ = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(pm i))\nm : ℕ → ℕ+ := fun i => if i < pk then pm i else ⟨2 * t + 2 ^ pk.succ, ⋯⟩\nhmpk : ↑↑(m pk) = 2 * ↑t + 2 ^ pk.succ\n⊢ ∏ x ∈ Finset.range (pk + 1), (1 + 1 / ↑↑(m x)) = ∏ i ∈ Finset.range pk.succ, (1 + 1 / ↑↑(m i))",
"after_state": "No Goals!"
},
{
"line": "let t_succ : ℕ+ := ⟨t + 1, t.succ_pos⟩",
"before_state": "case succ.inr.intro\npk : ℕ\nhpk : ∀ (n : ℕ+), ∃ m, 1 + (2 ^ pk - 1) / ↑↑n = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(m i))\nn : ℕ+\nt : ℕ\nht : ↑n = 2 * t + 1\n⊢ ∃ m, 1 + (2 ^ (pk + 1) - 1) / ↑↑n = ∏ i ∈ Finset.range (pk + 1), (1 + 1 / ↑↑(m i))",
"after_state": "case succ.inr.intro\npk : ℕ\nhpk : ∀ (n : ℕ+), ∃ m, 1 + (2 ^ pk - 1) / ↑↑n = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(m i))\nn : ℕ+\nt : ℕ\nht : ↑n = 2 * t + 1\nt_succ : ℕ+ := ⟨t + 1, ⋯⟩\n⊢ ∃ m, 1 + (2 ^ (pk + 1) - 1) / ↑↑n = ∏ i ∈ Finset.range (pk + 1), (1 + 1 / ↑↑(m i))"
},
{
"line": "refine_lift\n let t_succ : ℕ+ := ⟨t + 1, t.succ_pos⟩;\n ?_",
"before_state": "case succ.inr.intro\npk : ℕ\nhpk : ∀ (n : ℕ+), ∃ m, 1 + (2 ^ pk - 1) / ↑↑n = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(m i))\nn : ℕ+\nt : ℕ\nht : ↑n = 2 * t + 1\n⊢ ∃ m, 1 + (2 ^ (pk + 1) - 1) / ↑↑n = ∏ i ∈ Finset.range (pk + 1), (1 + 1 / ↑↑(m i))",
"after_state": "case succ.inr.intro\npk : ℕ\nhpk : ∀ (n : ℕ+), ∃ m, 1 + (2 ^ pk - 1) / ↑↑n = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(m i))\nn : ℕ+\nt : ℕ\nht : ↑n = 2 * t + 1\nt_succ : ℕ+ := ⟨t + 1, ⋯⟩\n⊢ ∃ m, 1 + (2 ^ (pk + 1) - 1) / ↑↑n = ∏ i ∈ Finset.range (pk + 1), (1 + 1 / ↑↑(m i))"
},
{
"line": "focus\n (refine\n no_implicit_lambda%\n (let t_succ : ℕ+ := ⟨t + 1, t.succ_pos⟩;\n ?_);\n rotate_right)",
"before_state": "case succ.inr.intro\npk : ℕ\nhpk : ∀ (n : ℕ+), ∃ m, 1 + (2 ^ pk - 1) / ↑↑n = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(m i))\nn : ℕ+\nt : ℕ\nht : ↑n = 2 * t + 1\n⊢ ∃ m, 1 + (2 ^ (pk + 1) - 1) / ↑↑n = ∏ i ∈ Finset.range (pk + 1), (1 + 1 / ↑↑(m i))",
"after_state": "case succ.inr.intro\npk : ℕ\nhpk : ∀ (n : ℕ+), ∃ m, 1 + (2 ^ pk - 1) / ↑↑n = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(m i))\nn : ℕ+\nt : ℕ\nht : ↑n = 2 * t + 1\nt_succ : ℕ+ := ⟨t + 1, ⋯⟩\n⊢ ∃ m, 1 + (2 ^ (pk + 1) - 1) / ↑↑n = ∏ i ∈ Finset.range (pk + 1), (1 + 1 / ↑↑(m i))"
},
{
"line": "refine\n no_implicit_lambda%\n (let t_succ : ℕ+ := ⟨t + 1, t.succ_pos⟩;\n ?_)",
"before_state": "case succ.inr.intro\npk : ℕ\nhpk : ∀ (n : ℕ+), ∃ m, 1 + (2 ^ pk - 1) / ↑↑n = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(m i))\nn : ℕ+\nt : ℕ\nht : ↑n = 2 * t + 1\n⊢ ∃ m, 1 + (2 ^ (pk + 1) - 1) / ↑↑n = ∏ i ∈ Finset.range (pk + 1), (1 + 1 / ↑↑(m i))",
"after_state": "case succ.inr.intro\npk : ℕ\nhpk : ∀ (n : ℕ+), ∃ m, 1 + (2 ^ pk - 1) / ↑↑n = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(m i))\nn : ℕ+\nt : ℕ\nht : ↑n = 2 * t + 1\nt_succ : ℕ+ := ⟨t + 1, ⋯⟩\n⊢ ∃ m, 1 + (2 ^ (pk + 1) - 1) / ↑↑n = ∏ i ∈ Finset.range (pk + 1), (1 + 1 / ↑↑(m i))"
},
{
"line": "rotate_right",
"before_state": "case succ.inr.intro\npk : ℕ\nhpk : ∀ (n : ℕ+), ∃ m, 1 + (2 ^ pk - 1) / ↑↑n = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(m i))\nn : ℕ+\nt : ℕ\nht : ↑n = 2 * t + 1\nt_succ : ℕ+ := ⟨t + 1, ⋯⟩\n⊢ ∃ m, 1 + (2 ^ (pk + 1) - 1) / ↑↑n = ∏ i ∈ Finset.range (pk + 1), (1 + 1 / ↑↑(m i))",
"after_state": "case succ.inr.intro\npk : ℕ\nhpk : ∀ (n : ℕ+), ∃ m, 1 + (2 ^ pk - 1) / ↑↑n = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(m i))\nn : ℕ+\nt : ℕ\nht : ↑n = 2 * t + 1\nt_succ : ℕ+ := ⟨t + 1, ⋯⟩\n⊢ ∃ m, 1 + (2 ^ (pk + 1) - 1) / ↑↑n = ∏ i ∈ Finset.range (pk + 1), (1 + 1 / ↑↑(m i))"
},
{
"line": "obtain ⟨pm, hpm⟩ := hpk t_succ",
"before_state": "case succ.inr.intro\npk : ℕ\nhpk : ∀ (n : ℕ+), ∃ m, 1 + (2 ^ pk - 1) / ↑↑n = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(m i))\nn : ℕ+\nt : ℕ\nht : ↑n = 2 * t + 1\nt_succ : ℕ+ := ⟨t + 1, ⋯⟩\n⊢ ∃ m, 1 + (2 ^ (pk + 1) - 1) / ↑↑n = ∏ i ∈ Finset.range (pk + 1), (1 + 1 / ↑↑(m i))",
"after_state": "case succ.inr.intro.intro\npk : ℕ\nhpk : ∀ (n : ℕ+), ∃ m, 1 + (2 ^ pk - 1) / ↑↑n = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(m i))\nn : ℕ+\nt : ℕ\nht : ↑n = 2 * t + 1\nt_succ : ℕ+ := ⟨t + 1, ⋯⟩\npm : ℕ → ℕ+\nhpm : 1 + (2 ^ pk - 1) / ↑↑t_succ = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(pm i))\n⊢ ∃ m, 1 + (2 ^ (pk + 1) - 1) / ↑↑n = ∏ i ∈ Finset.range (pk + 1), (1 + 1 / ↑↑(m i))"
},
{
"line": "let m i := if i < pk then pm i else ⟨2 * t + 1, Nat.succ_pos _⟩",
"before_state": "case succ.inr.intro.intro\npk : ℕ\nhpk : ∀ (n : ℕ+), ∃ m, 1 + (2 ^ pk - 1) / ↑↑n = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(m i))\nn : ℕ+\nt : ℕ\nht : ↑n = 2 * t + 1\nt_succ : ℕ+ := ⟨t + 1, ⋯⟩\npm : ℕ → ℕ+\nhpm : 1 + (2 ^ pk - 1) / ↑↑t_succ = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(pm i))\n⊢ ∃ m, 1 + (2 ^ (pk + 1) - 1) / ↑↑n = ∏ i ∈ Finset.range (pk + 1), (1 + 1 / ↑↑(m i))",
"after_state": "case succ.inr.intro.intro\npk : ℕ\nhpk : ∀ (n : ℕ+), ∃ m, 1 + (2 ^ pk - 1) / ↑↑n = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(m i))\nn : ℕ+\nt : ℕ\nht : ↑n = 2 * t + 1\nt_succ : ℕ+ := ⟨t + 1, ⋯⟩\npm : ℕ → ℕ+\nhpm : 1 + (2 ^ pk - 1) / ↑↑t_succ = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(pm i))\nm : ℕ → ℕ+ := fun i => if i < pk then pm i else ⟨2 * t + 1, ⋯⟩\n⊢ ∃ m, 1 + (2 ^ (pk + 1) - 1) / ↑↑n = ∏ i ∈ Finset.range (pk + 1), (1 + 1 / ↑↑(m i))"
},
{
"line": "refine_lift\n let m i := if i < pk then pm i else ⟨2 * t + 1, Nat.succ_pos _⟩;\n ?_",
"before_state": "case succ.inr.intro.intro\npk : ℕ\nhpk : ∀ (n : ℕ+), ∃ m, 1 + (2 ^ pk - 1) / ↑↑n = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(m i))\nn : ℕ+\nt : ℕ\nht : ↑n = 2 * t + 1\nt_succ : ℕ+ := ⟨t + 1, ⋯⟩\npm : ℕ → ℕ+\nhpm : 1 + (2 ^ pk - 1) / ↑↑t_succ = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(pm i))\n⊢ ∃ m, 1 + (2 ^ (pk + 1) - 1) / ↑↑n = ∏ i ∈ Finset.range (pk + 1), (1 + 1 / ↑↑(m i))",
"after_state": "case succ.inr.intro.intro\npk : ℕ\nhpk : ∀ (n : ℕ+), ∃ m, 1 + (2 ^ pk - 1) / ↑↑n = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(m i))\nn : ℕ+\nt : ℕ\nht : ↑n = 2 * t + 1\nt_succ : ℕ+ := ⟨t + 1, ⋯⟩\npm : ℕ → ℕ+\nhpm : 1 + (2 ^ pk - 1) / ↑↑t_succ = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(pm i))\nm : ℕ → ℕ+ := fun i => if i < pk then pm i else ⟨2 * t + 1, ⋯⟩\n⊢ ∃ m, 1 + (2 ^ (pk + 1) - 1) / ↑↑n = ∏ i ∈ Finset.range (pk + 1), (1 + 1 / ↑↑(m i))"
},
{
"line": "focus\n (refine\n no_implicit_lambda%\n (let m i := if i < pk then pm i else ⟨2 * t + 1, Nat.succ_pos _⟩;\n ?_);\n rotate_right)",
"before_state": "case succ.inr.intro.intro\npk : ℕ\nhpk : ∀ (n : ℕ+), ∃ m, 1 + (2 ^ pk - 1) / ↑↑n = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(m i))\nn : ℕ+\nt : ℕ\nht : ↑n = 2 * t + 1\nt_succ : ℕ+ := ⟨t + 1, ⋯⟩\npm : ℕ → ℕ+\nhpm : 1 + (2 ^ pk - 1) / ↑↑t_succ = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(pm i))\n⊢ ∃ m, 1 + (2 ^ (pk + 1) - 1) / ↑↑n = ∏ i ∈ Finset.range (pk + 1), (1 + 1 / ↑↑(m i))",
"after_state": "case succ.inr.intro.intro\npk : ℕ\nhpk : ∀ (n : ℕ+), ∃ m, 1 + (2 ^ pk - 1) / ↑↑n = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(m i))\nn : ℕ+\nt : ℕ\nht : ↑n = 2 * t + 1\nt_succ : ℕ+ := ⟨t + 1, ⋯⟩\npm : ℕ → ℕ+\nhpm : 1 + (2 ^ pk - 1) / ↑↑t_succ = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(pm i))\nm : ℕ → ℕ+ := fun i => if i < pk then pm i else ⟨2 * t + 1, ⋯⟩\n⊢ ∃ m, 1 + (2 ^ (pk + 1) - 1) / ↑↑n = ∏ i ∈ Finset.range (pk + 1), (1 + 1 / ↑↑(m i))"
},
{
"line": "refine\n no_implicit_lambda%\n (let m i := if i < pk then pm i else ⟨2 * t + 1, Nat.succ_pos _⟩;\n ?_)",
"before_state": "case succ.inr.intro.intro\npk : ℕ\nhpk : ∀ (n : ℕ+), ∃ m, 1 + (2 ^ pk - 1) / ↑↑n = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(m i))\nn : ℕ+\nt : ℕ\nht : ↑n = 2 * t + 1\nt_succ : ℕ+ := ⟨t + 1, ⋯⟩\npm : ℕ → ℕ+\nhpm : 1 + (2 ^ pk - 1) / ↑↑t_succ = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(pm i))\n⊢ ∃ m, 1 + (2 ^ (pk + 1) - 1) / ↑↑n = ∏ i ∈ Finset.range (pk + 1), (1 + 1 / ↑↑(m i))",
"after_state": "case succ.inr.intro.intro\npk : ℕ\nhpk : ∀ (n : ℕ+), ∃ m, 1 + (2 ^ pk - 1) / ↑↑n = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(m i))\nn : ℕ+\nt : ℕ\nht : ↑n = 2 * t + 1\nt_succ : ℕ+ := ⟨t + 1, ⋯⟩\npm : ℕ → ℕ+\nhpm : 1 + (2 ^ pk - 1) / ↑↑t_succ = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(pm i))\nm : ℕ → ℕ+ := fun i => if i < pk then pm i else ⟨2 * t + 1, ⋯⟩\n⊢ ∃ m, 1 + (2 ^ (pk + 1) - 1) / ↑↑n = ∏ i ∈ Finset.range (pk + 1), (1 + 1 / ↑↑(m i))"
},
{
"line": "rotate_right",
"before_state": "case succ.inr.intro.intro\npk : ℕ\nhpk : ∀ (n : ℕ+), ∃ m, 1 + (2 ^ pk - 1) / ↑↑n = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(m i))\nn : ℕ+\nt : ℕ\nht : ↑n = 2 * t + 1\nt_succ : ℕ+ := ⟨t + 1, ⋯⟩\npm : ℕ → ℕ+\nhpm : 1 + (2 ^ pk - 1) / ↑↑t_succ = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(pm i))\nm : ℕ → ℕ+ := fun i => if i < pk then pm i else ⟨2 * t + 1, ⋯⟩\n⊢ ∃ m, 1 + (2 ^ (pk + 1) - 1) / ↑↑n = ∏ i ∈ Finset.range (pk + 1), (1 + 1 / ↑↑(m i))",
"after_state": "case succ.inr.intro.intro\npk : ℕ\nhpk : ∀ (n : ℕ+), ∃ m, 1 + (2 ^ pk - 1) / ↑↑n = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(m i))\nn : ℕ+\nt : ℕ\nht : ↑n = 2 * t + 1\nt_succ : ℕ+ := ⟨t + 1, ⋯⟩\npm : ℕ → ℕ+\nhpm : 1 + (2 ^ pk - 1) / ↑↑t_succ = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(pm i))\nm : ℕ → ℕ+ := fun i => if i < pk then pm i else ⟨2 * t + 1, ⋯⟩\n⊢ ∃ m, 1 + (2 ^ (pk + 1) - 1) / ↑↑n = ∏ i ∈ Finset.range (pk + 1), (1 + 1 / ↑↑(m i))"
},
{
"line": "use m",
"before_state": "case succ.inr.intro.intro\npk : ℕ\nhpk : ∀ (n : ℕ+), ∃ m, 1 + (2 ^ pk - 1) / ↑↑n = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(m i))\nn : ℕ+\nt : ℕ\nht : ↑n = 2 * t + 1\nt_succ : ℕ+ := ⟨t + 1, ⋯⟩\npm : ℕ → ℕ+\nhpm : 1 + (2 ^ pk - 1) / ↑↑t_succ = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(pm i))\nm : ℕ → ℕ+ := fun i => if i < pk then pm i else ⟨2 * t + 1, ⋯⟩\n⊢ ∃ m, 1 + (2 ^ (pk + 1) - 1) / ↑↑n = ∏ i ∈ Finset.range (pk + 1), (1 + 1 / ↑↑(m i))",
"after_state": "case h\npk : ℕ\nhpk : ∀ (n : ℕ+), ∃ m, 1 + (2 ^ pk - 1) / ↑↑n = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(m i))\nn : ℕ+\nt : ℕ\nht : ↑n = 2 * t + 1\nt_succ : ℕ+ := ⟨t + 1, ⋯⟩\npm : ℕ → ℕ+\nhpm : 1 + (2 ^ pk - 1) / ↑↑t_succ = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(pm i))\nm : ℕ → ℕ+ := fun i => if i < pk then pm i else ⟨2 * t + 1, ⋯⟩\n⊢ 1 + (2 ^ (pk + 1) - 1) / ↑↑n = ∏ i ∈ Finset.range (pk + 1), (1 + 1 / ↑↑(m i))"
},
{
"line": "refine without_cdot(m : ?m✝)",
"before_state": "case w\npk : ℕ\nhpk : ∀ (n : ℕ+), ∃ m, 1 + (2 ^ pk - 1) / ↑↑n = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(m i))\nn : ℕ+\nt : ℕ\nht : ↑n = 2 * t + 1\nt_succ : ℕ+ := ⟨t + 1, ⋯⟩\npm : ℕ → ℕ+\nhpm : 1 + (2 ^ pk - 1) / ↑↑t_succ = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(pm i))\nm : ℕ → ℕ+ := fun i => if i < pk then pm i else ⟨2 * t + 1, ⋯⟩\n⊢ ℕ → ℕ+",
"after_state": "No Goals!"
},
{
"line": "try with_reducible use_discharger",
"before_state": "case h\npk : ℕ\nhpk : ∀ (n : ℕ+), ∃ m, 1 + (2 ^ pk - 1) / ↑↑n = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(m i))\nn : ℕ+\nt : ℕ\nht : ↑n = 2 * t + 1\nt_succ : ℕ+ := ⟨t + 1, ⋯⟩\npm : ℕ → ℕ+\nhpm : 1 + (2 ^ pk - 1) / ↑↑t_succ = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(pm i))\nm : ℕ → ℕ+ := fun i => if i < pk then pm i else ⟨2 * t + 1, ⋯⟩\n⊢ 1 + (2 ^ (pk + 1) - 1) / ↑↑n = ∏ i ∈ Finset.range (pk + 1), (1 + 1 / ↑↑(m i))",
"after_state": "case h\npk : ℕ\nhpk : ∀ (n : ℕ+), ∃ m, 1 + (2 ^ pk - 1) / ↑↑n = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(m i))\nn : ℕ+\nt : ℕ\nht : ↑n = 2 * t + 1\nt_succ : ℕ+ := ⟨t + 1, ⋯⟩\npm : ℕ → ℕ+\nhpm : 1 + (2 ^ pk - 1) / ↑↑t_succ = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(pm i))\nm : ℕ → ℕ+ := fun i => if i < pk then pm i else ⟨2 * t + 1, ⋯⟩\n⊢ 1 + (2 ^ (pk + 1) - 1) / ↑↑n = ∏ i ∈ Finset.range (pk + 1), (1 + 1 / ↑↑(m i))"
},
{
"line": "first\n| with_reducible use_discharger\n| skip",
"before_state": "case h\npk : ℕ\nhpk : ∀ (n : ℕ+), ∃ m, 1 + (2 ^ pk - 1) / ↑↑n = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(m i))\nn : ℕ+\nt : ℕ\nht : ↑n = 2 * t + 1\nt_succ : ℕ+ := ⟨t + 1, ⋯⟩\npm : ℕ → ℕ+\nhpm : 1 + (2 ^ pk - 1) / ↑↑t_succ = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(pm i))\nm : ℕ → ℕ+ := fun i => if i < pk then pm i else ⟨2 * t + 1, ⋯⟩\n⊢ 1 + (2 ^ (pk + 1) - 1) / ↑↑n = ∏ i ∈ Finset.range (pk + 1), (1 + 1 / ↑↑(m i))",
"after_state": "case h\npk : ℕ\nhpk : ∀ (n : ℕ+), ∃ m, 1 + (2 ^ pk - 1) / ↑↑n = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(m i))\nn : ℕ+\nt : ℕ\nht : ↑n = 2 * t + 1\nt_succ : ℕ+ := ⟨t + 1, ⋯⟩\npm : ℕ → ℕ+\nhpm : 1 + (2 ^ pk - 1) / ↑↑t_succ = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(pm i))\nm : ℕ → ℕ+ := fun i => if i < pk then pm i else ⟨2 * t + 1, ⋯⟩\n⊢ 1 + (2 ^ (pk + 1) - 1) / ↑↑n = ∏ i ∈ Finset.range (pk + 1), (1 + 1 / ↑↑(m i))"
},
{
"line": "with_reducible use_discharger",
"before_state": "case h\npk : ℕ\nhpk : ∀ (n : ℕ+), ∃ m, 1 + (2 ^ pk - 1) / ↑↑n = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(m i))\nn : ℕ+\nt : ℕ\nht : ↑n = 2 * t + 1\nt_succ : ℕ+ := ⟨t + 1, ⋯⟩\npm : ℕ → ℕ+\nhpm : 1 + (2 ^ pk - 1) / ↑↑t_succ = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(pm i))\nm : ℕ → ℕ+ := fun i => if i < pk then pm i else ⟨2 * t + 1, ⋯⟩\n⊢ 1 + (2 ^ (pk + 1) - 1) / ↑↑n = ∏ i ∈ Finset.range (pk + 1), (1 + 1 / ↑↑(m i))",
"after_state": "case h\npk : ℕ\nhpk : ∀ (n : ℕ+), ∃ m, 1 + (2 ^ pk - 1) / ↑↑n = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(m i))\nn : ℕ+\nt : ℕ\nht : ↑n = 2 * t + 1\nt_succ : ℕ+ := ⟨t + 1, ⋯⟩\npm : ℕ → ℕ+\nhpm : 1 + (2 ^ pk - 1) / ↑↑t_succ = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(pm i))\nm : ℕ → ℕ+ := fun i => if i < pk then pm i else ⟨2 * t + 1, ⋯⟩\n⊢ 1 + (2 ^ (pk + 1) - 1) / ↑↑n = ∏ i ∈ Finset.range (pk + 1), (1 + 1 / ↑↑(m i))"
},
{
"line": "use_discharger",
"before_state": "case h\npk : ℕ\nhpk : ∀ (n : ℕ+), ∃ m, 1 + (2 ^ pk - 1) / ↑↑n = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(m i))\nn : ℕ+\nt : ℕ\nht : ↑n = 2 * t + 1\nt_succ : ℕ+ := ⟨t + 1, ⋯⟩\npm : ℕ → ℕ+\nhpm : 1 + (2 ^ pk - 1) / ↑↑t_succ = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(pm i))\nm : ℕ → ℕ+ := fun i => if i < pk then pm i else ⟨2 * t + 1, ⋯⟩\n⊢ 1 + (2 ^ (pk + 1) - 1) / ↑↑n = ∏ i ∈ Finset.range (pk + 1), (1 + 1 / ↑↑(m i))",
"after_state": "case h\npk : ℕ\nhpk : ∀ (n : ℕ+), ∃ m, 1 + (2 ^ pk - 1) / ↑↑n = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(m i))\nn : ℕ+\nt : ℕ\nht : ↑n = 2 * t + 1\nt_succ : ℕ+ := ⟨t + 1, ⋯⟩\npm : ℕ → ℕ+\nhpm : 1 + (2 ^ pk - 1) / ↑↑t_succ = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(pm i))\nm : ℕ → ℕ+ := fun i => if i < pk then pm i else ⟨2 * t + 1, ⋯⟩\n⊢ 1 + (2 ^ (pk + 1) - 1) / ↑↑n = ∏ i ∈ Finset.range (pk + 1), (1 + 1 / ↑↑(m i))"
},
{
"line": "focus\n apply exists_prop.mpr✝\n with_annotate_state\"<;>\" skip\n all_goals use_discharger",
"before_state": "case h\npk : ℕ\nhpk : ∀ (n : ℕ+), ∃ m, 1 + (2 ^ pk - 1) / ↑↑n = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(m i))\nn : ℕ+\nt : ℕ\nht : ↑n = 2 * t + 1\nt_succ : ℕ+ := ⟨t + 1, ⋯⟩\npm : ℕ → ℕ+\nhpm : 1 + (2 ^ pk - 1) / ↑↑t_succ = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(pm i))\nm : ℕ → ℕ+ := fun i => if i < pk then pm i else ⟨2 * t + 1, ⋯⟩\n⊢ 1 + (2 ^ (pk + 1) - 1) / ↑↑n = ∏ i ∈ Finset.range (pk + 1), (1 + 1 / ↑↑(m i))",
"after_state": "case h\npk : ℕ\nhpk : ∀ (n : ℕ+), ∃ m, 1 + (2 ^ pk - 1) / ↑↑n = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(m i))\nn : ℕ+\nt : ℕ\nht : ↑n = 2 * t + 1\nt_succ : ℕ+ := ⟨t + 1, ⋯⟩\npm : ℕ → ℕ+\nhpm : 1 + (2 ^ pk - 1) / ↑↑t_succ = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(pm i))\nm : ℕ → ℕ+ := fun i => if i < pk then pm i else ⟨2 * t + 1, ⋯⟩\n⊢ 1 + (2 ^ (pk + 1) - 1) / ↑↑n = ∏ i ∈ Finset.range (pk + 1), (1 + 1 / ↑↑(m i))"
},
{
"line": "apply exists_prop.mpr✝",
"before_state": "case h\npk : ℕ\nhpk : ∀ (n : ℕ+), ∃ m, 1 + (2 ^ pk - 1) / ↑↑n = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(m i))\nn : ℕ+\nt : ℕ\nht : ↑n = 2 * t + 1\nt_succ : ℕ+ := ⟨t + 1, ⋯⟩\npm : ℕ → ℕ+\nhpm : 1 + (2 ^ pk - 1) / ↑↑t_succ = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(pm i))\nm : ℕ → ℕ+ := fun i => if i < pk then pm i else ⟨2 * t + 1, ⋯⟩\n⊢ 1 + (2 ^ (pk + 1) - 1) / ↑↑n = ∏ i ∈ Finset.range (pk + 1), (1 + 1 / ↑↑(m i))",
"after_state": "case h\npk : ℕ\nhpk : ∀ (n : ℕ+), ∃ m, 1 + (2 ^ pk - 1) / ↑↑n = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(m i))\nn : ℕ+\nt : ℕ\nht : ↑n = 2 * t + 1\nt_succ : ℕ+ := ⟨t + 1, ⋯⟩\npm : ℕ → ℕ+\nhpm : 1 + (2 ^ pk - 1) / ↑↑t_succ = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(pm i))\nm : ℕ → ℕ+ := fun i => if i < pk then pm i else ⟨2 * t + 1, ⋯⟩\n⊢ 1 + (2 ^ (pk + 1) - 1) / ↑↑n = ∏ i ∈ Finset.range (pk + 1), (1 + 1 / ↑↑(m i))"
},
{
"line": "skip",
"before_state": "case h\npk : ℕ\nhpk : ∀ (n : ℕ+), ∃ m, 1 + (2 ^ pk - 1) / ↑↑n = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(m i))\nn : ℕ+\nt : ℕ\nht : ↑n = 2 * t + 1\nt_succ : ℕ+ := ⟨t + 1, ⋯⟩\npm : ℕ → ℕ+\nhpm : 1 + (2 ^ pk - 1) / ↑↑t_succ = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(pm i))\nm : ℕ → ℕ+ := fun i => if i < pk then pm i else ⟨2 * t + 1, ⋯⟩\n⊢ 1 + (2 ^ (pk + 1) - 1) / ↑↑n = ∏ i ∈ Finset.range (pk + 1), (1 + 1 / ↑↑(m i))",
"after_state": "case h\npk : ℕ\nhpk : ∀ (n : ℕ+), ∃ m, 1 + (2 ^ pk - 1) / ↑↑n = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(m i))\nn : ℕ+\nt : ℕ\nht : ↑n = 2 * t + 1\nt_succ : ℕ+ := ⟨t + 1, ⋯⟩\npm : ℕ → ℕ+\nhpm : 1 + (2 ^ pk - 1) / ↑↑t_succ = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(pm i))\nm : ℕ → ℕ+ := fun i => if i < pk then pm i else ⟨2 * t + 1, ⋯⟩\n⊢ 1 + (2 ^ (pk + 1) - 1) / ↑↑n = ∏ i ∈ Finset.range (pk + 1), (1 + 1 / ↑↑(m i))"
},
{
"line": "have hmpk : (m pk : ℚ) = 2 * t + 1 :=\n by\n have : m pk = ⟨2 * t + 1, _⟩ := if_neg (irrefl pk)\n simp [this]",
"before_state": "case h\npk : ℕ\nhpk : ∀ (n : ℕ+), ∃ m, 1 + (2 ^ pk - 1) / ↑↑n = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(m i))\nn : ℕ+\nt : ℕ\nht : ↑n = 2 * t + 1\nt_succ : ℕ+ := ⟨t + 1, ⋯⟩\npm : ℕ → ℕ+\nhpm : 1 + (2 ^ pk - 1) / ↑↑t_succ = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(pm i))\nm : ℕ → ℕ+ := fun i => if i < pk then pm i else ⟨2 * t + 1, ⋯⟩\n⊢ 1 + (2 ^ (pk + 1) - 1) / ↑↑n = ∏ i ∈ Finset.range (pk + 1), (1 + 1 / ↑↑(m i))",
"after_state": "case h\npk : ℕ\nhpk : ∀ (n : ℕ+), ∃ m, 1 + (2 ^ pk - 1) / ↑↑n = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(m i))\nn : ℕ+\nt : ℕ\nht : ↑n = 2 * t + 1\nt_succ : ℕ+ := ⟨t + 1, ⋯⟩\npm : ℕ → ℕ+\nhpm : 1 + (2 ^ pk - 1) / ↑↑t_succ = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(pm i))\nm : ℕ → ℕ+ := fun i => if i < pk then pm i else ⟨2 * t + 1, ⋯⟩\nhmpk : ↑↑(m pk) = 2 * ↑t + 1\n⊢ 1 + (2 ^ (pk + 1) - 1) / ↑↑n = ∏ i ∈ Finset.range (pk + 1), (1 + 1 / ↑↑(m i))"
},
{
"line": "focus\n refine\n no_implicit_lambda%\n (have hmpk : (m pk : ℚ) = 2 * t + 1 := ?body✝;\n ?_)\n case body✝ =>\n with_annotate_state\"by\"\n ( have : m pk = ⟨2 * t + 1, _⟩ := if_neg (irrefl pk)\n simp [this])",
"before_state": "case h\npk : ℕ\nhpk : ∀ (n : ℕ+), ∃ m, 1 + (2 ^ pk - 1) / ↑↑n = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(m i))\nn : ℕ+\nt : ℕ\nht : ↑n = 2 * t + 1\nt_succ : ℕ+ := ⟨t + 1, ⋯⟩\npm : ℕ → ℕ+\nhpm : 1 + (2 ^ pk - 1) / ↑↑t_succ = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(pm i))\nm : ℕ → ℕ+ := fun i => if i < pk then pm i else ⟨2 * t + 1, ⋯⟩\n⊢ 1 + (2 ^ (pk + 1) - 1) / ↑↑n = ∏ i ∈ Finset.range (pk + 1), (1 + 1 / ↑↑(m i))",
"after_state": "case h\npk : ℕ\nhpk : ∀ (n : ℕ+), ∃ m, 1 + (2 ^ pk - 1) / ↑↑n = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(m i))\nn : ℕ+\nt : ℕ\nht : ↑n = 2 * t + 1\nt_succ : ℕ+ := ⟨t + 1, ⋯⟩\npm : ℕ → ℕ+\nhpm : 1 + (2 ^ pk - 1) / ↑↑t_succ = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(pm i))\nm : ℕ → ℕ+ := fun i => if i < pk then pm i else ⟨2 * t + 1, ⋯⟩\nhmpk : ↑↑(m pk) = 2 * ↑t + 1\n⊢ 1 + (2 ^ (pk + 1) - 1) / ↑↑n = ∏ i ∈ Finset.range (pk + 1), (1 + 1 / ↑↑(m i))"
},
{
"line": "refine\n no_implicit_lambda%\n (have hmpk : (m pk : ℚ) = 2 * t + 1 := ?body✝;\n ?_)",
"before_state": "case h\npk : ℕ\nhpk : ∀ (n : ℕ+), ∃ m, 1 + (2 ^ pk - 1) / ↑↑n = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(m i))\nn : ℕ+\nt : ℕ\nht : ↑n = 2 * t + 1\nt_succ : ℕ+ := ⟨t + 1, ⋯⟩\npm : ℕ → ℕ+\nhpm : 1 + (2 ^ pk - 1) / ↑↑t_succ = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(pm i))\nm : ℕ → ℕ+ := fun i => if i < pk then pm i else ⟨2 * t + 1, ⋯⟩\n⊢ 1 + (2 ^ (pk + 1) - 1) / ↑↑n = ∏ i ∈ Finset.range (pk + 1), (1 + 1 / ↑↑(m i))",
"after_state": "case body\npk : ℕ\nhpk : ∀ (n : ℕ+), ∃ m, 1 + (2 ^ pk - 1) / ↑↑n = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(m i))\nn : ℕ+\nt : ℕ\nht : ↑n = 2 * t + 1\nt_succ : ℕ+ := ⟨t + 1, ⋯⟩\npm : ℕ → ℕ+\nhpm : 1 + (2 ^ pk - 1) / ↑↑t_succ = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(pm i))\nm : ℕ → ℕ+ := fun i => if i < pk then pm i else ⟨2 * t + 1, ⋯⟩\n⊢ ↑↑(m pk) = 2 * ↑t + 1\n---\ncase h\npk : ℕ\nhpk : ∀ (n : ℕ+), ∃ m, 1 + (2 ^ pk - 1) / ↑↑n = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(m i))\nn : ℕ+\nt : ℕ\nht : ↑n = 2 * t + 1\nt_succ : ℕ+ := ⟨t + 1, ⋯⟩\npm : ℕ → ℕ+\nhpm : 1 + (2 ^ pk - 1) / ↑↑t_succ = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(pm i))\nm : ℕ → ℕ+ := fun i => if i < pk then pm i else ⟨2 * t + 1, ⋯⟩\nhmpk : ↑↑(m pk) = 2 * ↑t + 1\n⊢ 1 + (2 ^ (pk + 1) - 1) / ↑↑n = ∏ i ∈ Finset.range (pk + 1), (1 + 1 / ↑↑(m i))"
},
{
"line": "case body✝ =>\n with_annotate_state\"by\"\n ( have : m pk = ⟨2 * t + 1, _⟩ := if_neg (irrefl pk)\n simp [this])",
"before_state": "case body\npk : ℕ\nhpk : ∀ (n : ℕ+), ∃ m, 1 + (2 ^ pk - 1) / ↑↑n = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(m i))\nn : ℕ+\nt : ℕ\nht : ↑n = 2 * t + 1\nt_succ : ℕ+ := ⟨t + 1, ⋯⟩\npm : ℕ → ℕ+\nhpm : 1 + (2 ^ pk - 1) / ↑↑t_succ = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(pm i))\nm : ℕ → ℕ+ := fun i => if i < pk then pm i else ⟨2 * t + 1, ⋯⟩\n⊢ ↑↑(m pk) = 2 * ↑t + 1\n---\ncase h\npk : ℕ\nhpk : ∀ (n : ℕ+), ∃ m, 1 + (2 ^ pk - 1) / ↑↑n = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(m i))\nn : ℕ+\nt : ℕ\nht : ↑n = 2 * t + 1\nt_succ : ℕ+ := ⟨t + 1, ⋯⟩\npm : ℕ → ℕ+\nhpm : 1 + (2 ^ pk - 1) / ↑↑t_succ = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(pm i))\nm : ℕ → ℕ+ := fun i => if i < pk then pm i else ⟨2 * t + 1, ⋯⟩\nhmpk : ↑↑(m pk) = 2 * ↑t + 1\n⊢ 1 + (2 ^ (pk + 1) - 1) / ↑↑n = ∏ i ∈ Finset.range (pk + 1), (1 + 1 / ↑↑(m i))",
"after_state": "case h\npk : ℕ\nhpk : ∀ (n : ℕ+), ∃ m, 1 + (2 ^ pk - 1) / ↑↑n = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(m i))\nn : ℕ+\nt : ℕ\nht : ↑n = 2 * t + 1\nt_succ : ℕ+ := ⟨t + 1, ⋯⟩\npm : ℕ → ℕ+\nhpm : 1 + (2 ^ pk - 1) / ↑↑t_succ = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(pm i))\nm : ℕ → ℕ+ := fun i => if i < pk then pm i else ⟨2 * t + 1, ⋯⟩\nhmpk : ↑↑(m pk) = 2 * ↑t + 1\n⊢ 1 + (2 ^ (pk + 1) - 1) / ↑↑n = ∏ i ∈ Finset.range (pk + 1), (1 + 1 / ↑↑(m i))"
},
{
"line": "with_annotate_state\"by\"\n ( have : m pk = ⟨2 * t + 1, _⟩ := if_neg (irrefl pk)\n simp [this])",
"before_state": "pk : ℕ\nhpk : ∀ (n : ℕ+), ∃ m, 1 + (2 ^ pk - 1) / ↑↑n = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(m i))\nn : ℕ+\nt : ℕ\nht : ↑n = 2 * t + 1\nt_succ : ℕ+ := ⟨t + 1, ⋯⟩\npm : ℕ → ℕ+\nhpm : 1 + (2 ^ pk - 1) / ↑↑t_succ = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(pm i))\nm : ℕ → ℕ+ := fun i => if i < pk then pm i else ⟨2 * t + 1, ⋯⟩\n⊢ ↑↑(m pk) = 2 * ↑t + 1",
"after_state": "No Goals!"
},
{
"line": "have : m pk = ⟨2 * t + 1, _⟩ := if_neg (irrefl pk)",
"before_state": "pk : ℕ\nhpk : ∀ (n : ℕ+), ∃ m, 1 + (2 ^ pk - 1) / ↑↑n = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(m i))\nn : ℕ+\nt : ℕ\nht : ↑n = 2 * t + 1\nt_succ : ℕ+ := ⟨t + 1, ⋯⟩\npm : ℕ → ℕ+\nhpm : 1 + (2 ^ pk - 1) / ↑↑t_succ = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(pm i))\nm : ℕ → ℕ+ := fun i => if i < pk then pm i else ⟨2 * t + 1, ⋯⟩\n⊢ ↑↑(m pk) = 2 * ↑t + 1",
"after_state": "pk : ℕ\nhpk : ∀ (n : ℕ+), ∃ m, 1 + (2 ^ pk - 1) / ↑↑n = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(m i))\nn : ℕ+\nt : ℕ\nht : ↑n = 2 * t + 1\nt_succ : ℕ+ := ⟨t + 1, ⋯⟩\npm : ℕ → ℕ+\nhpm : 1 + (2 ^ pk - 1) / ↑↑t_succ = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(pm i))\nm : ℕ → ℕ+ := fun i => if i < pk then pm i else ⟨2 * t + 1, ⋯⟩\nthis : m pk = ⟨2 * t + 1, ⋯⟩\n⊢ ↑↑(m pk) = 2 * ↑t + 1"
},
{
"line": "refine_lift\n have : m pk = ⟨2 * t + 1, _⟩ := if_neg (irrefl pk);\n ?_",
"before_state": "pk : ℕ\nhpk : ∀ (n : ℕ+), ∃ m, 1 + (2 ^ pk - 1) / ↑↑n = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(m i))\nn : ℕ+\nt : ℕ\nht : ↑n = 2 * t + 1\nt_succ : ℕ+ := ⟨t + 1, ⋯⟩\npm : ℕ → ℕ+\nhpm : 1 + (2 ^ pk - 1) / ↑↑t_succ = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(pm i))\nm : ℕ → ℕ+ := fun i => if i < pk then pm i else ⟨2 * t + 1, ⋯⟩\n⊢ ↑↑(m pk) = 2 * ↑t + 1",
"after_state": "pk : ℕ\nhpk : ∀ (n : ℕ+), ∃ m, 1 + (2 ^ pk - 1) / ↑↑n = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(m i))\nn : ℕ+\nt : ℕ\nht : ↑n = 2 * t + 1\nt_succ : ℕ+ := ⟨t + 1, ⋯⟩\npm : ℕ → ℕ+\nhpm : 1 + (2 ^ pk - 1) / ↑↑t_succ = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(pm i))\nm : ℕ → ℕ+ := fun i => if i < pk then pm i else ⟨2 * t + 1, ⋯⟩\nthis : m pk = ⟨2 * t + 1, ⋯⟩\n⊢ ↑↑(m pk) = 2 * ↑t + 1"
},
{
"line": "focus\n (refine\n no_implicit_lambda%\n (have : m pk = ⟨2 * t + 1, _⟩ := if_neg (irrefl pk);\n ?_);\n rotate_right)",
"before_state": "pk : ℕ\nhpk : ∀ (n : ℕ+), ∃ m, 1 + (2 ^ pk - 1) / ↑↑n = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(m i))\nn : ℕ+\nt : ℕ\nht : ↑n = 2 * t + 1\nt_succ : ℕ+ := ⟨t + 1, ⋯⟩\npm : ℕ → ℕ+\nhpm : 1 + (2 ^ pk - 1) / ↑↑t_succ = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(pm i))\nm : ℕ → ℕ+ := fun i => if i < pk then pm i else ⟨2 * t + 1, ⋯⟩\n⊢ ↑↑(m pk) = 2 * ↑t + 1",
"after_state": "pk : ℕ\nhpk : ∀ (n : ℕ+), ∃ m, 1 + (2 ^ pk - 1) / ↑↑n = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(m i))\nn : ℕ+\nt : ℕ\nht : ↑n = 2 * t + 1\nt_succ : ℕ+ := ⟨t + 1, ⋯⟩\npm : ℕ → ℕ+\nhpm : 1 + (2 ^ pk - 1) / ↑↑t_succ = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(pm i))\nm : ℕ → ℕ+ := fun i => if i < pk then pm i else ⟨2 * t + 1, ⋯⟩\nthis : m pk = ⟨2 * t + 1, ⋯⟩\n⊢ ↑↑(m pk) = 2 * ↑t + 1"
},
{
"line": "refine\n no_implicit_lambda%\n (have : m pk = ⟨2 * t + 1, _⟩ := if_neg (irrefl pk);\n ?_)",
"before_state": "pk : ℕ\nhpk : ∀ (n : ℕ+), ∃ m, 1 + (2 ^ pk - 1) / ↑↑n = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(m i))\nn : ℕ+\nt : ℕ\nht : ↑n = 2 * t + 1\nt_succ : ℕ+ := ⟨t + 1, ⋯⟩\npm : ℕ → ℕ+\nhpm : 1 + (2 ^ pk - 1) / ↑↑t_succ = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(pm i))\nm : ℕ → ℕ+ := fun i => if i < pk then pm i else ⟨2 * t + 1, ⋯⟩\n⊢ ↑↑(m pk) = 2 * ↑t + 1",
"after_state": "pk : ℕ\nhpk : ∀ (n : ℕ+), ∃ m, 1 + (2 ^ pk - 1) / ↑↑n = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(m i))\nn : ℕ+\nt : ℕ\nht : ↑n = 2 * t + 1\nt_succ : ℕ+ := ⟨t + 1, ⋯⟩\npm : ℕ → ℕ+\nhpm : 1 + (2 ^ pk - 1) / ↑↑t_succ = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(pm i))\nm : ℕ → ℕ+ := fun i => if i < pk then pm i else ⟨2 * t + 1, ⋯⟩\nthis : m pk = ⟨2 * t + 1, ⋯⟩\n⊢ ↑↑(m pk) = 2 * ↑t + 1"
},
{
"line": "rotate_right",
"before_state": "pk : ℕ\nhpk : ∀ (n : ℕ+), ∃ m, 1 + (2 ^ pk - 1) / ↑↑n = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(m i))\nn : ℕ+\nt : ℕ\nht : ↑n = 2 * t + 1\nt_succ : ℕ+ := ⟨t + 1, ⋯⟩\npm : ℕ → ℕ+\nhpm : 1 + (2 ^ pk - 1) / ↑↑t_succ = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(pm i))\nm : ℕ → ℕ+ := fun i => if i < pk then pm i else ⟨2 * t + 1, ⋯⟩\nthis : m pk = ⟨2 * t + 1, ⋯⟩\n⊢ ↑↑(m pk) = 2 * ↑t + 1",
"after_state": "pk : ℕ\nhpk : ∀ (n : ℕ+), ∃ m, 1 + (2 ^ pk - 1) / ↑↑n = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(m i))\nn : ℕ+\nt : ℕ\nht : ↑n = 2 * t + 1\nt_succ : ℕ+ := ⟨t + 1, ⋯⟩\npm : ℕ → ℕ+\nhpm : 1 + (2 ^ pk - 1) / ↑↑t_succ = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(pm i))\nm : ℕ → ℕ+ := fun i => if i < pk then pm i else ⟨2 * t + 1, ⋯⟩\nthis : m pk = ⟨2 * t + 1, ⋯⟩\n⊢ ↑↑(m pk) = 2 * ↑t + 1"
},
{
"line": "simp [this]",
"before_state": "pk : ℕ\nhpk : ∀ (n : ℕ+), ∃ m, 1 + (2 ^ pk - 1) / ↑↑n = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(m i))\nn : ℕ+\nt : ℕ\nht : ↑n = 2 * t + 1\nt_succ : ℕ+ := ⟨t + 1, ⋯⟩\npm : ℕ → ℕ+\nhpm : 1 + (2 ^ pk - 1) / ↑↑t_succ = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(pm i))\nm : ℕ → ℕ+ := fun i => if i < pk then pm i else ⟨2 * t + 1, ⋯⟩\nthis : m pk = ⟨2 * t + 1, ⋯⟩\n⊢ ↑↑(m pk) = 2 * ↑t + 1",
"after_state": "No Goals!"
},
{
"line": "calc\n ((1 : ℚ) + (2 ^ pk.succ - 1) / ↑n : ℚ) = 1 + (2 * 2 ^ pk - 1) / (2 * t + 1 : ℕ) :=\n by\n rw [ht]\n rw [pow_succ']\n _ = (1 + 1 / (2 * t + 1)) * (1 + (2 ^ pk - 1) / (t + 1)) :=\n by\n field_simp\n ring\n _ = (1 + 1 / (2 * t + 1)) * (1 + (2 ^ pk - 1) / t_succ) := by norm_cast\n _ = (∏ i ∈ Finset.range pk, (1 + 1 / (m i : ℚ))) * (1 + 1 / ↑(m pk)) :=\n by\n rw [prod_lemma]\n rw [hpm]\n rw [← hmpk]\n rw [mul_comm]\n _ = ∏ i ∈ Finset.range pk.succ, (1 + 1 / (m i : ℚ)) := by rw [← Finset.prod_range_succ _ pk]",
"before_state": "case h\npk : ℕ\nhpk : ∀ (n : ℕ+), ∃ m, 1 + (2 ^ pk - 1) / ↑↑n = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(m i))\nn : ℕ+\nt : ℕ\nht : ↑n = 2 * t + 1\nt_succ : ℕ+ := ⟨t + 1, ⋯⟩\npm : ℕ → ℕ+\nhpm : 1 + (2 ^ pk - 1) / ↑↑t_succ = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(pm i))\nm : ℕ → ℕ+ := fun i => if i < pk then pm i else ⟨2 * t + 1, ⋯⟩\nhmpk : ↑↑(m pk) = 2 * ↑t + 1\n⊢ 1 + (2 ^ (pk + 1) - 1) / ↑↑n = ∏ i ∈ Finset.range (pk + 1), (1 + 1 / ↑↑(m i))",
"after_state": "No Goals!"
},
{
"line": "rw [ht]",
"before_state": "pk : ℕ\nhpk : ∀ (n : ℕ+), ∃ m, 1 + (2 ^ pk - 1) / ↑↑n = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(m i))\nn : ℕ+\nt : ℕ\nht : ↑n = 2 * t + 1\nt_succ : ℕ+ := ⟨t + 1, ⋯⟩\npm : ℕ → ℕ+\nhpm : 1 + (2 ^ pk - 1) / ↑↑t_succ = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(pm i))\nm : ℕ → ℕ+ := fun i => if i < pk then pm i else ⟨2 * t + 1, ⋯⟩\nhmpk : ↑↑(m pk) = 2 * ↑t + 1\n⊢ 1 + (2 ^ pk.succ - 1) / ↑↑n = 1 + (2 * 2 ^ pk - 1) / ↑(2 * t + 1)",
"after_state": "pk : ℕ\nhpk : ∀ (n : ℕ+), ∃ m, 1 + (2 ^ pk - 1) / ↑↑n = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(m i))\nn : ℕ+\nt : ℕ\nht : ↑n = 2 * t + 1\nt_succ : ℕ+ := ⟨t + 1, ⋯⟩\npm : ℕ → ℕ+\nhpm : 1 + (2 ^ pk - 1) / ↑↑t_succ = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(pm i))\nm : ℕ → ℕ+ := fun i => if i < pk then pm i else ⟨2 * t + 1, ⋯⟩\nhmpk : ↑↑(m pk) = 2 * ↑t + 1\n⊢ 1 + (2 ^ pk.succ - 1) / ↑(2 * t + 1) = 1 + (2 * 2 ^ pk - 1) / ↑(2 * t + 1)"
},
{
"line": "rewrite [ht]",
"before_state": "pk : ℕ\nhpk : ∀ (n : ℕ+), ∃ m, 1 + (2 ^ pk - 1) / ↑↑n = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(m i))\nn : ℕ+\nt : ℕ\nht : ↑n = 2 * t + 1\nt_succ : ℕ+ := ⟨t + 1, ⋯⟩\npm : ℕ → ℕ+\nhpm : 1 + (2 ^ pk - 1) / ↑↑t_succ = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(pm i))\nm : ℕ → ℕ+ := fun i => if i < pk then pm i else ⟨2 * t + 1, ⋯⟩\nhmpk : ↑↑(m pk) = 2 * ↑t + 1\n⊢ 1 + (2 ^ pk.succ - 1) / ↑↑n = 1 + (2 * 2 ^ pk - 1) / ↑(2 * t + 1)",
"after_state": "pk : ℕ\nhpk : ∀ (n : ℕ+), ∃ m, 1 + (2 ^ pk - 1) / ↑↑n = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(m i))\nn : ℕ+\nt : ℕ\nht : ↑n = 2 * t + 1\nt_succ : ℕ+ := ⟨t + 1, ⋯⟩\npm : ℕ → ℕ+\nhpm : 1 + (2 ^ pk - 1) / ↑↑t_succ = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(pm i))\nm : ℕ → ℕ+ := fun i => if i < pk then pm i else ⟨2 * t + 1, ⋯⟩\nhmpk : ↑↑(m pk) = 2 * ↑t + 1\n⊢ 1 + (2 ^ pk.succ - 1) / ↑(2 * t + 1) = 1 + (2 * 2 ^ pk - 1) / ↑(2 * t + 1)"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "pk : ℕ\nhpk : ∀ (n : ℕ+), ∃ m, 1 + (2 ^ pk - 1) / ↑↑n = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(m i))\nn : ℕ+\nt : ℕ\nht : ↑n = 2 * t + 1\nt_succ : ℕ+ := ⟨t + 1, ⋯⟩\npm : ℕ → ℕ+\nhpm : 1 + (2 ^ pk - 1) / ↑↑t_succ = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(pm i))\nm : ℕ → ℕ+ := fun i => if i < pk then pm i else ⟨2 * t + 1, ⋯⟩\nhmpk : ↑↑(m pk) = 2 * ↑t + 1\n⊢ 1 + (2 ^ pk.succ - 1) / ↑(2 * t + 1) = 1 + (2 * 2 ^ pk - 1) / ↑(2 * t + 1)",
"after_state": "pk : ℕ\nhpk : ∀ (n : ℕ+), ∃ m, 1 + (2 ^ pk - 1) / ↑↑n = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(m i))\nn : ℕ+\nt : ℕ\nht : ↑n = 2 * t + 1\nt_succ : ℕ+ := ⟨t + 1, ⋯⟩\npm : ℕ → ℕ+\nhpm : 1 + (2 ^ pk - 1) / ↑↑t_succ = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(pm i))\nm : ℕ → ℕ+ := fun i => if i < pk then pm i else ⟨2 * t + 1, ⋯⟩\nhmpk : ↑↑(m pk) = 2 * ↑t + 1\n⊢ 1 + (2 ^ pk.succ - 1) / ↑(2 * t + 1) = 1 + (2 * 2 ^ pk - 1) / ↑(2 * t + 1)"
},
{
"line": "try (with_reducible rfl)",
"before_state": "pk : ℕ\nhpk : ∀ (n : ℕ+), ∃ m, 1 + (2 ^ pk - 1) / ↑↑n = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(m i))\nn : ℕ+\nt : ℕ\nht : ↑n = 2 * t + 1\nt_succ : ℕ+ := ⟨t + 1, ⋯⟩\npm : ℕ → ℕ+\nhpm : 1 + (2 ^ pk - 1) / ↑↑t_succ = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(pm i))\nm : ℕ → ℕ+ := fun i => if i < pk then pm i else ⟨2 * t + 1, ⋯⟩\nhmpk : ↑↑(m pk) = 2 * ↑t + 1\n⊢ 1 + (2 ^ pk.succ - 1) / ↑(2 * t + 1) = 1 + (2 * 2 ^ pk - 1) / ↑(2 * t + 1)",
"after_state": "pk : ℕ\nhpk : ∀ (n : ℕ+), ∃ m, 1 + (2 ^ pk - 1) / ↑↑n = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(m i))\nn : ℕ+\nt : ℕ\nht : ↑n = 2 * t + 1\nt_succ : ℕ+ := ⟨t + 1, ⋯⟩\npm : ℕ → ℕ+\nhpm : 1 + (2 ^ pk - 1) / ↑↑t_succ = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(pm i))\nm : ℕ → ℕ+ := fun i => if i < pk then pm i else ⟨2 * t + 1, ⋯⟩\nhmpk : ↑↑(m pk) = 2 * ↑t + 1\n⊢ 1 + (2 ^ pk.succ - 1) / ↑(2 * t + 1) = 1 + (2 * 2 ^ pk - 1) / ↑(2 * t + 1)"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "pk : ℕ\nhpk : ∀ (n : ℕ+), ∃ m, 1 + (2 ^ pk - 1) / ↑↑n = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(m i))\nn : ℕ+\nt : ℕ\nht : ↑n = 2 * t + 1\nt_succ : ℕ+ := ⟨t + 1, ⋯⟩\npm : ℕ → ℕ+\nhpm : 1 + (2 ^ pk - 1) / ↑↑t_succ = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(pm i))\nm : ℕ → ℕ+ := fun i => if i < pk then pm i else ⟨2 * t + 1, ⋯⟩\nhmpk : ↑↑(m pk) = 2 * ↑t + 1\n⊢ 1 + (2 ^ pk.succ - 1) / ↑(2 * t + 1) = 1 + (2 * 2 ^ pk - 1) / ↑(2 * t + 1)",
"after_state": "pk : ℕ\nhpk : ∀ (n : ℕ+), ∃ m, 1 + (2 ^ pk - 1) / ↑↑n = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(m i))\nn : ℕ+\nt : ℕ\nht : ↑n = 2 * t + 1\nt_succ : ℕ+ := ⟨t + 1, ⋯⟩\npm : ℕ → ℕ+\nhpm : 1 + (2 ^ pk - 1) / ↑↑t_succ = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(pm i))\nm : ℕ → ℕ+ := fun i => if i < pk then pm i else ⟨2 * t + 1, ⋯⟩\nhmpk : ↑↑(m pk) = 2 * ↑t + 1\n⊢ 1 + (2 ^ pk.succ - 1) / ↑(2 * t + 1) = 1 + (2 * 2 ^ pk - 1) / ↑(2 * t + 1)"
},
{
"line": "with_reducible rfl",
"before_state": "pk : ℕ\nhpk : ∀ (n : ℕ+), ∃ m, 1 + (2 ^ pk - 1) / ↑↑n = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(m i))\nn : ℕ+\nt : ℕ\nht : ↑n = 2 * t + 1\nt_succ : ℕ+ := ⟨t + 1, ⋯⟩\npm : ℕ → ℕ+\nhpm : 1 + (2 ^ pk - 1) / ↑↑t_succ = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(pm i))\nm : ℕ → ℕ+ := fun i => if i < pk then pm i else ⟨2 * t + 1, ⋯⟩\nhmpk : ↑↑(m pk) = 2 * ↑t + 1\n⊢ 1 + (2 ^ pk.succ - 1) / ↑(2 * t + 1) = 1 + (2 * 2 ^ pk - 1) / ↑(2 * t + 1)",
"after_state": "pk : ℕ\nhpk : ∀ (n : ℕ+), ∃ m, 1 + (2 ^ pk - 1) / ↑↑n = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(m i))\nn : ℕ+\nt : ℕ\nht : ↑n = 2 * t + 1\nt_succ : ℕ+ := ⟨t + 1, ⋯⟩\npm : ℕ → ℕ+\nhpm : 1 + (2 ^ pk - 1) / ↑↑t_succ = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(pm i))\nm : ℕ → ℕ+ := fun i => if i < pk then pm i else ⟨2 * t + 1, ⋯⟩\nhmpk : ↑↑(m pk) = 2 * ↑t + 1\n⊢ 1 + (2 ^ pk.succ - 1) / ↑(2 * t + 1) = 1 + (2 * 2 ^ pk - 1) / ↑(2 * t + 1)"
},
{
"line": "rfl",
"before_state": "pk : ℕ\nhpk : ∀ (n : ℕ+), ∃ m, 1 + (2 ^ pk - 1) / ↑↑n = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(m i))\nn : ℕ+\nt : ℕ\nht : ↑n = 2 * t + 1\nt_succ : ℕ+ := ⟨t + 1, ⋯⟩\npm : ℕ → ℕ+\nhpm : 1 + (2 ^ pk - 1) / ↑↑t_succ = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(pm i))\nm : ℕ → ℕ+ := fun i => if i < pk then pm i else ⟨2 * t + 1, ⋯⟩\nhmpk : ↑↑(m pk) = 2 * ↑t + 1\n⊢ 1 + (2 ^ pk.succ - 1) / ↑(2 * t + 1) = 1 + (2 * 2 ^ pk - 1) / ↑(2 * t + 1)",
"after_state": "pk : ℕ\nhpk : ∀ (n : ℕ+), ∃ m, 1 + (2 ^ pk - 1) / ↑↑n = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(m i))\nn : ℕ+\nt : ℕ\nht : ↑n = 2 * t + 1\nt_succ : ℕ+ := ⟨t + 1, ⋯⟩\npm : ℕ → ℕ+\nhpm : 1 + (2 ^ pk - 1) / ↑↑t_succ = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(pm i))\nm : ℕ → ℕ+ := fun i => if i < pk then pm i else ⟨2 * t + 1, ⋯⟩\nhmpk : ↑↑(m pk) = 2 * ↑t + 1\n⊢ 1 + (2 ^ pk.succ - 1) / ↑(2 * t + 1) = 1 + (2 * 2 ^ pk - 1) / ↑(2 * t + 1)"
},
{
"line": "apply_rfl",
"before_state": "pk : ℕ\nhpk : ∀ (n : ℕ+), ∃ m, 1 + (2 ^ pk - 1) / ↑↑n = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(m i))\nn : ℕ+\nt : ℕ\nht : ↑n = 2 * t + 1\nt_succ : ℕ+ := ⟨t + 1, ⋯⟩\npm : ℕ → ℕ+\nhpm : 1 + (2 ^ pk - 1) / ↑↑t_succ = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(pm i))\nm : ℕ → ℕ+ := fun i => if i < pk then pm i else ⟨2 * t + 1, ⋯⟩\nhmpk : ↑↑(m pk) = 2 * ↑t + 1\n⊢ 1 + (2 ^ pk.succ - 1) / ↑(2 * t + 1) = 1 + (2 * 2 ^ pk - 1) / ↑(2 * t + 1)",
"after_state": "pk : ℕ\nhpk : ∀ (n : ℕ+), ∃ m, 1 + (2 ^ pk - 1) / ↑↑n = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(m i))\nn : ℕ+\nt : ℕ\nht : ↑n = 2 * t + 1\nt_succ : ℕ+ := ⟨t + 1, ⋯⟩\npm : ℕ → ℕ+\nhpm : 1 + (2 ^ pk - 1) / ↑↑t_succ = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(pm i))\nm : ℕ → ℕ+ := fun i => if i < pk then pm i else ⟨2 * t + 1, ⋯⟩\nhmpk : ↑↑(m pk) = 2 * ↑t + 1\n⊢ 1 + (2 ^ pk.succ - 1) / ↑(2 * t + 1) = 1 + (2 * 2 ^ pk - 1) / ↑(2 * t + 1)"
},
{
"line": "skip",
"before_state": "pk : ℕ\nhpk : ∀ (n : ℕ+), ∃ m, 1 + (2 ^ pk - 1) / ↑↑n = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(m i))\nn : ℕ+\nt : ℕ\nht : ↑n = 2 * t + 1\nt_succ : ℕ+ := ⟨t + 1, ⋯⟩\npm : ℕ → ℕ+\nhpm : 1 + (2 ^ pk - 1) / ↑↑t_succ = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(pm i))\nm : ℕ → ℕ+ := fun i => if i < pk then pm i else ⟨2 * t + 1, ⋯⟩\nhmpk : ↑↑(m pk) = 2 * ↑t + 1\n⊢ 1 + (2 ^ pk.succ - 1) / ↑(2 * t + 1) = 1 + (2 * 2 ^ pk - 1) / ↑(2 * t + 1)",
"after_state": "pk : ℕ\nhpk : ∀ (n : ℕ+), ∃ m, 1 + (2 ^ pk - 1) / ↑↑n = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(m i))\nn : ℕ+\nt : ℕ\nht : ↑n = 2 * t + 1\nt_succ : ℕ+ := ⟨t + 1, ⋯⟩\npm : ℕ → ℕ+\nhpm : 1 + (2 ^ pk - 1) / ↑↑t_succ = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(pm i))\nm : ℕ → ℕ+ := fun i => if i < pk then pm i else ⟨2 * t + 1, ⋯⟩\nhmpk : ↑↑(m pk) = 2 * ↑t + 1\n⊢ 1 + (2 ^ pk.succ - 1) / ↑(2 * t + 1) = 1 + (2 * 2 ^ pk - 1) / ↑(2 * t + 1)"
},
{
"line": "rw [pow_succ']",
"before_state": "pk : ℕ\nhpk : ∀ (n : ℕ+), ∃ m, 1 + (2 ^ pk - 1) / ↑↑n = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(m i))\nn : ℕ+\nt : ℕ\nht : ↑n = 2 * t + 1\nt_succ : ℕ+ := ⟨t + 1, ⋯⟩\npm : ℕ → ℕ+\nhpm : 1 + (2 ^ pk - 1) / ↑↑t_succ = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(pm i))\nm : ℕ → ℕ+ := fun i => if i < pk then pm i else ⟨2 * t + 1, ⋯⟩\nhmpk : ↑↑(m pk) = 2 * ↑t + 1\n⊢ 1 + (2 ^ pk.succ - 1) / ↑(2 * t + 1) = 1 + (2 * 2 ^ pk - 1) / ↑(2 * t + 1)",
"after_state": "No Goals!"
},
{
"line": "rewrite [pow_succ']",
"before_state": "pk : ℕ\nhpk : ∀ (n : ℕ+), ∃ m, 1 + (2 ^ pk - 1) / ↑↑n = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(m i))\nn : ℕ+\nt : ℕ\nht : ↑n = 2 * t + 1\nt_succ : ℕ+ := ⟨t + 1, ⋯⟩\npm : ℕ → ℕ+\nhpm : 1 + (2 ^ pk - 1) / ↑↑t_succ = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(pm i))\nm : ℕ → ℕ+ := fun i => if i < pk then pm i else ⟨2 * t + 1, ⋯⟩\nhmpk : ↑↑(m pk) = 2 * ↑t + 1\n⊢ 1 + (2 ^ pk.succ - 1) / ↑(2 * t + 1) = 1 + (2 * 2 ^ pk - 1) / ↑(2 * t + 1)",
"after_state": "pk : ℕ\nhpk : ∀ (n : ℕ+), ∃ m, 1 + (2 ^ pk - 1) / ↑↑n = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(m i))\nn : ℕ+\nt : ℕ\nht : ↑n = 2 * t + 1\nt_succ : ℕ+ := ⟨t + 1, ⋯⟩\npm : ℕ → ℕ+\nhpm : 1 + (2 ^ pk - 1) / ↑↑t_succ = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(pm i))\nm : ℕ → ℕ+ := fun i => if i < pk then pm i else ⟨2 * t + 1, ⋯⟩\nhmpk : ↑↑(m pk) = 2 * ↑t + 1\n⊢ 1 + (2 * 2 ^ pk - 1) / ↑(2 * t + 1) = 1 + (2 * 2 ^ pk - 1) / ↑(2 * t + 1)"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "pk : ℕ\nhpk : ∀ (n : ℕ+), ∃ m, 1 + (2 ^ pk - 1) / ↑↑n = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(m i))\nn : ℕ+\nt : ℕ\nht : ↑n = 2 * t + 1\nt_succ : ℕ+ := ⟨t + 1, ⋯⟩\npm : ℕ → ℕ+\nhpm : 1 + (2 ^ pk - 1) / ↑↑t_succ = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(pm i))\nm : ℕ → ℕ+ := fun i => if i < pk then pm i else ⟨2 * t + 1, ⋯⟩\nhmpk : ↑↑(m pk) = 2 * ↑t + 1\n⊢ 1 + (2 * 2 ^ pk - 1) / ↑(2 * t + 1) = 1 + (2 * 2 ^ pk - 1) / ↑(2 * t + 1)",
"after_state": "No Goals!"
},
{
"line": "try (with_reducible rfl)",
"before_state": "pk : ℕ\nhpk : ∀ (n : ℕ+), ∃ m, 1 + (2 ^ pk - 1) / ↑↑n = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(m i))\nn : ℕ+\nt : ℕ\nht : ↑n = 2 * t + 1\nt_succ : ℕ+ := ⟨t + 1, ⋯⟩\npm : ℕ → ℕ+\nhpm : 1 + (2 ^ pk - 1) / ↑↑t_succ = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(pm i))\nm : ℕ → ℕ+ := fun i => if i < pk then pm i else ⟨2 * t + 1, ⋯⟩\nhmpk : ↑↑(m pk) = 2 * ↑t + 1\n⊢ 1 + (2 * 2 ^ pk - 1) / ↑(2 * t + 1) = 1 + (2 * 2 ^ pk - 1) / ↑(2 * t + 1)",
"after_state": "No Goals!"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "pk : ℕ\nhpk : ∀ (n : ℕ+), ∃ m, 1 + (2 ^ pk - 1) / ↑↑n = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(m i))\nn : ℕ+\nt : ℕ\nht : ↑n = 2 * t + 1\nt_succ : ℕ+ := ⟨t + 1, ⋯⟩\npm : ℕ → ℕ+\nhpm : 1 + (2 ^ pk - 1) / ↑↑t_succ = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(pm i))\nm : ℕ → ℕ+ := fun i => if i < pk then pm i else ⟨2 * t + 1, ⋯⟩\nhmpk : ↑↑(m pk) = 2 * ↑t + 1\n⊢ 1 + (2 * 2 ^ pk - 1) / ↑(2 * t + 1) = 1 + (2 * 2 ^ pk - 1) / ↑(2 * t + 1)",
"after_state": "No Goals!"
},
{
"line": "with_reducible rfl",
"before_state": "pk : ℕ\nhpk : ∀ (n : ℕ+), ∃ m, 1 + (2 ^ pk - 1) / ↑↑n = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(m i))\nn : ℕ+\nt : ℕ\nht : ↑n = 2 * t + 1\nt_succ : ℕ+ := ⟨t + 1, ⋯⟩\npm : ℕ → ℕ+\nhpm : 1 + (2 ^ pk - 1) / ↑↑t_succ = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(pm i))\nm : ℕ → ℕ+ := fun i => if i < pk then pm i else ⟨2 * t + 1, ⋯⟩\nhmpk : ↑↑(m pk) = 2 * ↑t + 1\n⊢ 1 + (2 * 2 ^ pk - 1) / ↑(2 * t + 1) = 1 + (2 * 2 ^ pk - 1) / ↑(2 * t + 1)",
"after_state": "No Goals!"
},
{
"line": "rfl",
"before_state": "pk : ℕ\nhpk : ∀ (n : ℕ+), ∃ m, 1 + (2 ^ pk - 1) / ↑↑n = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(m i))\nn : ℕ+\nt : ℕ\nht : ↑n = 2 * t + 1\nt_succ : ℕ+ := ⟨t + 1, ⋯⟩\npm : ℕ → ℕ+\nhpm : 1 + (2 ^ pk - 1) / ↑↑t_succ = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(pm i))\nm : ℕ → ℕ+ := fun i => if i < pk then pm i else ⟨2 * t + 1, ⋯⟩\nhmpk : ↑↑(m pk) = 2 * ↑t + 1\n⊢ 1 + (2 * 2 ^ pk - 1) / ↑(2 * t + 1) = 1 + (2 * 2 ^ pk - 1) / ↑(2 * t + 1)",
"after_state": "No Goals!"
},
{
"line": "eq_refl",
"before_state": "pk : ℕ\nhpk : ∀ (n : ℕ+), ∃ m, 1 + (2 ^ pk - 1) / ↑↑n = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(m i))\nn : ℕ+\nt : ℕ\nht : ↑n = 2 * t + 1\nt_succ : ℕ+ := ⟨t + 1, ⋯⟩\npm : ℕ → ℕ+\nhpm : 1 + (2 ^ pk - 1) / ↑↑t_succ = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(pm i))\nm : ℕ → ℕ+ := fun i => if i < pk then pm i else ⟨2 * t + 1, ⋯⟩\nhmpk : ↑↑(m pk) = 2 * ↑t + 1\n⊢ 1 + (2 * 2 ^ pk - 1) / ↑(2 * t + 1) = 1 + (2 * 2 ^ pk - 1) / ↑(2 * t + 1)",
"after_state": "No Goals!"
},
{
"line": "field_simp",
"before_state": "pk : ℕ\nhpk : ∀ (n : ℕ+), ∃ m, 1 + (2 ^ pk - 1) / ↑↑n = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(m i))\nn : ℕ+\nt : ℕ\nht : ↑n = 2 * t + 1\nt_succ : ℕ+ := ⟨t + 1, ⋯⟩\npm : ℕ → ℕ+\nhpm : 1 + (2 ^ pk - 1) / ↑↑t_succ = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(pm i))\nm : ℕ → ℕ+ := fun i => if i < pk then pm i else ⟨2 * t + 1, ⋯⟩\nhmpk : ↑↑(m pk) = 2 * ↑t + 1\n⊢ 1 + (2 * 2 ^ pk - 1) / ↑(2 * t + 1) = (1 + 1 / (2 * ↑t + 1)) * (1 + (2 ^ pk - 1) / (↑t + 1))",
"after_state": "pk : ℕ\nhpk : ∀ (n : ℕ+), ∃ m, 1 + (2 ^ pk - 1) / ↑↑n = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(m i))\nn : ℕ+\nt : ℕ\nht : ↑n = 2 * t + 1\nt_succ : ℕ+ := ⟨t + 1, ⋯⟩\npm : ℕ → ℕ+\nhpm : 1 + (2 ^ pk - 1) / ↑↑t_succ = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(pm i))\nm : ℕ → ℕ+ := fun i => if i < pk then pm i else ⟨2 * t + 1, ⋯⟩\nhmpk : ↑↑(m pk) = 2 * ↑t + 1\n⊢ (2 * ↑t + 2 * 2 ^ pk) * ((2 * ↑t + 1) * (↑t + 1)) = (2 * ↑t + 1 + 1) * (↑t + 2 ^ pk) * (2 * ↑t + 1)"
},
{
"line": "ring",
"before_state": "pk : ℕ\nhpk : ∀ (n : ℕ+), ∃ m, 1 + (2 ^ pk - 1) / ↑↑n = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(m i))\nn : ℕ+\nt : ℕ\nht : ↑n = 2 * t + 1\nt_succ : ℕ+ := ⟨t + 1, ⋯⟩\npm : ℕ → ℕ+\nhpm : 1 + (2 ^ pk - 1) / ↑↑t_succ = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(pm i))\nm : ℕ → ℕ+ := fun i => if i < pk then pm i else ⟨2 * t + 1, ⋯⟩\nhmpk : ↑↑(m pk) = 2 * ↑t + 1\n⊢ (2 * ↑t + 2 * 2 ^ pk) * ((2 * ↑t + 1) * (↑t + 1)) = (2 * ↑t + 1 + 1) * (↑t + 2 ^ pk) * (2 * ↑t + 1)",
"after_state": "No Goals!"
},
{
"line": "first\n| ring1\n|\n try_this ring_nf\"\\n\\nThe `ring` tactic failed to close the goal. Use `ring_nf` to obtain a normal form.\n \\nNote that `ring` works primarily in *commutative* rings. \\\n If you have a noncommutative ring, abelian group or module, consider using \\\n `noncomm_ring`, `abel` or `module` instead.\"",
"before_state": "pk : ℕ\nhpk : ∀ (n : ℕ+), ∃ m, 1 + (2 ^ pk - 1) / ↑↑n = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(m i))\nn : ℕ+\nt : ℕ\nht : ↑n = 2 * t + 1\nt_succ : ℕ+ := ⟨t + 1, ⋯⟩\npm : ℕ → ℕ+\nhpm : 1 + (2 ^ pk - 1) / ↑↑t_succ = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(pm i))\nm : ℕ → ℕ+ := fun i => if i < pk then pm i else ⟨2 * t + 1, ⋯⟩\nhmpk : ↑↑(m pk) = 2 * ↑t + 1\n⊢ (2 * ↑t + 2 * 2 ^ pk) * ((2 * ↑t + 1) * (↑t + 1)) = (2 * ↑t + 1 + 1) * (↑t + 2 ^ pk) * (2 * ↑t + 1)",
"after_state": "No Goals!"
},
{
"line": "ring1",
"before_state": "pk : ℕ\nhpk : ∀ (n : ℕ+), ∃ m, 1 + (2 ^ pk - 1) / ↑↑n = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(m i))\nn : ℕ+\nt : ℕ\nht : ↑n = 2 * t + 1\nt_succ : ℕ+ := ⟨t + 1, ⋯⟩\npm : ℕ → ℕ+\nhpm : 1 + (2 ^ pk - 1) / ↑↑t_succ = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(pm i))\nm : ℕ → ℕ+ := fun i => if i < pk then pm i else ⟨2 * t + 1, ⋯⟩\nhmpk : ↑↑(m pk) = 2 * ↑t + 1\n⊢ (2 * ↑t + 2 * 2 ^ pk) * ((2 * ↑t + 1) * (↑t + 1)) = (2 * ↑t + 1 + 1) * (↑t + 2 ^ pk) * (2 * ↑t + 1)",
"after_state": "No Goals!"
},
{
"line": "norm_cast",
"before_state": "pk : ℕ\nhpk : ∀ (n : ℕ+), ∃ m, 1 + (2 ^ pk - 1) / ↑↑n = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(m i))\nn : ℕ+\nt : ℕ\nht : ↑n = 2 * t + 1\nt_succ : ℕ+ := ⟨t + 1, ⋯⟩\npm : ℕ → ℕ+\nhpm : 1 + (2 ^ pk - 1) / ↑↑t_succ = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(pm i))\nm : ℕ → ℕ+ := fun i => if i < pk then pm i else ⟨2 * t + 1, ⋯⟩\nhmpk : ↑↑(m pk) = 2 * ↑t + 1\n⊢ (1 + 1 / (2 * ↑t + 1)) * (1 + (2 ^ pk - 1) / (↑t + 1)) = (1 + 1 / (2 * ↑t + 1)) * (1 + (2 ^ pk - 1) / ↑↑t_succ)",
"after_state": "No Goals!"
},
{
"line": "focus\n norm_cast0\n with_annotate_state\"<;>\" skip\n all_goals try trivial",
"before_state": "pk : ℕ\nhpk : ∀ (n : ℕ+), ∃ m, 1 + (2 ^ pk - 1) / ↑↑n = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(m i))\nn : ℕ+\nt : ℕ\nht : ↑n = 2 * t + 1\nt_succ : ℕ+ := ⟨t + 1, ⋯⟩\npm : ℕ → ℕ+\nhpm : 1 + (2 ^ pk - 1) / ↑↑t_succ = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(pm i))\nm : ℕ → ℕ+ := fun i => if i < pk then pm i else ⟨2 * t + 1, ⋯⟩\nhmpk : ↑↑(m pk) = 2 * ↑t + 1\n⊢ (1 + 1 / (2 * ↑t + 1)) * (1 + (2 ^ pk - 1) / (↑t + 1)) = (1 + 1 / (2 * ↑t + 1)) * (1 + (2 ^ pk - 1) / ↑↑t_succ)",
"after_state": "No Goals!"
},
{
"line": "norm_cast0",
"before_state": "pk : ℕ\nhpk : ∀ (n : ℕ+), ∃ m, 1 + (2 ^ pk - 1) / ↑↑n = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(m i))\nn : ℕ+\nt : ℕ\nht : ↑n = 2 * t + 1\nt_succ : ℕ+ := ⟨t + 1, ⋯⟩\npm : ℕ → ℕ+\nhpm : 1 + (2 ^ pk - 1) / ↑↑t_succ = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(pm i))\nm : ℕ → ℕ+ := fun i => if i < pk then pm i else ⟨2 * t + 1, ⋯⟩\nhmpk : ↑↑(m pk) = 2 * ↑t + 1\n⊢ (1 + 1 / (2 * ↑t + 1)) * (1 + (2 ^ pk - 1) / (↑t + 1)) = (1 + 1 / (2 * ↑t + 1)) * (1 + (2 ^ pk - 1) / ↑↑t_succ)",
"after_state": "pk : ℕ\nhpk : ∀ (n : ℕ+), ∃ m, 1 + (2 ^ pk - 1) / ↑↑n = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(m i))\nn : ℕ+\nt : ℕ\nht : ↑n = 2 * t + 1\nt_succ : ℕ+ := ⟨t + 1, ⋯⟩\npm : ℕ → ℕ+\nhpm : 1 + (2 ^ pk - 1) / ↑↑t_succ = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(pm i))\nm : ℕ → ℕ+ := fun i => if i < pk then pm i else ⟨2 * t + 1, ⋯⟩\nhmpk : ↑↑(m pk) = 2 * ↑t + 1\n⊢ (1 + 1 / ↑(2 * t + 1)) * (1 + Rat.divInt (Int.subNatNat (2 ^ pk) 1) ↑(t + 1)) =\n (1 + 1 / ↑(2 * t + 1)) * (1 + Rat.divInt (Int.subNatNat (2 ^ pk) 1) ↑↑t_succ)"
},
{
"line": "with_annotate_state\"<;>\" skip",
"before_state": "pk : ℕ\nhpk : ∀ (n : ℕ+), ∃ m, 1 + (2 ^ pk - 1) / ↑↑n = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(m i))\nn : ℕ+\nt : ℕ\nht : ↑n = 2 * t + 1\nt_succ : ℕ+ := ⟨t + 1, ⋯⟩\npm : ℕ → ℕ+\nhpm : 1 + (2 ^ pk - 1) / ↑↑t_succ = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(pm i))\nm : ℕ → ℕ+ := fun i => if i < pk then pm i else ⟨2 * t + 1, ⋯⟩\nhmpk : ↑↑(m pk) = 2 * ↑t + 1\n⊢ (1 + 1 / ↑(2 * t + 1)) * (1 + Rat.divInt (Int.subNatNat (2 ^ pk) 1) ↑(t + 1)) =\n (1 + 1 / ↑(2 * t + 1)) * (1 + Rat.divInt (Int.subNatNat (2 ^ pk) 1) ↑↑t_succ)",
"after_state": "pk : ℕ\nhpk : ∀ (n : ℕ+), ∃ m, 1 + (2 ^ pk - 1) / ↑↑n = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(m i))\nn : ℕ+\nt : ℕ\nht : ↑n = 2 * t + 1\nt_succ : ℕ+ := ⟨t + 1, ⋯⟩\npm : ℕ → ℕ+\nhpm : 1 + (2 ^ pk - 1) / ↑↑t_succ = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(pm i))\nm : ℕ → ℕ+ := fun i => if i < pk then pm i else ⟨2 * t + 1, ⋯⟩\nhmpk : ↑↑(m pk) = 2 * ↑t + 1\n⊢ (1 + 1 / ↑(2 * t + 1)) * (1 + Rat.divInt (Int.subNatNat (2 ^ pk) 1) ↑(t + 1)) =\n (1 + 1 / ↑(2 * t + 1)) * (1 + Rat.divInt (Int.subNatNat (2 ^ pk) 1) ↑↑t_succ)"
},
{
"line": "skip",
"before_state": "pk : ℕ\nhpk : ∀ (n : ℕ+), ∃ m, 1 + (2 ^ pk - 1) / ↑↑n = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(m i))\nn : ℕ+\nt : ℕ\nht : ↑n = 2 * t + 1\nt_succ : ℕ+ := ⟨t + 1, ⋯⟩\npm : ℕ → ℕ+\nhpm : 1 + (2 ^ pk - 1) / ↑↑t_succ = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(pm i))\nm : ℕ → ℕ+ := fun i => if i < pk then pm i else ⟨2 * t + 1, ⋯⟩\nhmpk : ↑↑(m pk) = 2 * ↑t + 1\n⊢ (1 + 1 / ↑(2 * t + 1)) * (1 + Rat.divInt (Int.subNatNat (2 ^ pk) 1) ↑(t + 1)) =\n (1 + 1 / ↑(2 * t + 1)) * (1 + Rat.divInt (Int.subNatNat (2 ^ pk) 1) ↑↑t_succ)",
"after_state": "pk : ℕ\nhpk : ∀ (n : ℕ+), ∃ m, 1 + (2 ^ pk - 1) / ↑↑n = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(m i))\nn : ℕ+\nt : ℕ\nht : ↑n = 2 * t + 1\nt_succ : ℕ+ := ⟨t + 1, ⋯⟩\npm : ℕ → ℕ+\nhpm : 1 + (2 ^ pk - 1) / ↑↑t_succ = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(pm i))\nm : ℕ → ℕ+ := fun i => if i < pk then pm i else ⟨2 * t + 1, ⋯⟩\nhmpk : ↑↑(m pk) = 2 * ↑t + 1\n⊢ (1 + 1 / ↑(2 * t + 1)) * (1 + Rat.divInt (Int.subNatNat (2 ^ pk) 1) ↑(t + 1)) =\n (1 + 1 / ↑(2 * t + 1)) * (1 + Rat.divInt (Int.subNatNat (2 ^ pk) 1) ↑↑t_succ)"
},
{
"line": "all_goals try trivial",
"before_state": "pk : ℕ\nhpk : ∀ (n : ℕ+), ∃ m, 1 + (2 ^ pk - 1) / ↑↑n = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(m i))\nn : ℕ+\nt : ℕ\nht : ↑n = 2 * t + 1\nt_succ : ℕ+ := ⟨t + 1, ⋯⟩\npm : ℕ → ℕ+\nhpm : 1 + (2 ^ pk - 1) / ↑↑t_succ = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(pm i))\nm : ℕ → ℕ+ := fun i => if i < pk then pm i else ⟨2 * t + 1, ⋯⟩\nhmpk : ↑↑(m pk) = 2 * ↑t + 1\n⊢ (1 + 1 / ↑(2 * t + 1)) * (1 + Rat.divInt (Int.subNatNat (2 ^ pk) 1) ↑(t + 1)) =\n (1 + 1 / ↑(2 * t + 1)) * (1 + Rat.divInt (Int.subNatNat (2 ^ pk) 1) ↑↑t_succ)",
"after_state": "No Goals!"
},
{
"line": "try trivial",
"before_state": "pk : ℕ\nhpk : ∀ (n : ℕ+), ∃ m, 1 + (2 ^ pk - 1) / ↑↑n = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(m i))\nn : ℕ+\nt : ℕ\nht : ↑n = 2 * t + 1\nt_succ : ℕ+ := ⟨t + 1, ⋯⟩\npm : ℕ → ℕ+\nhpm : 1 + (2 ^ pk - 1) / ↑↑t_succ = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(pm i))\nm : ℕ → ℕ+ := fun i => if i < pk then pm i else ⟨2 * t + 1, ⋯⟩\nhmpk : ↑↑(m pk) = 2 * ↑t + 1\n⊢ (1 + 1 / ↑(2 * t + 1)) * (1 + Rat.divInt (Int.subNatNat (2 ^ pk) 1) ↑(t + 1)) =\n (1 + 1 / ↑(2 * t + 1)) * (1 + Rat.divInt (Int.subNatNat (2 ^ pk) 1) ↑↑t_succ)",
"after_state": "No Goals!"
},
{
"line": "first\n| trivial\n| skip",
"before_state": "pk : ℕ\nhpk : ∀ (n : ℕ+), ∃ m, 1 + (2 ^ pk - 1) / ↑↑n = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(m i))\nn : ℕ+\nt : ℕ\nht : ↑n = 2 * t + 1\nt_succ : ℕ+ := ⟨t + 1, ⋯⟩\npm : ℕ → ℕ+\nhpm : 1 + (2 ^ pk - 1) / ↑↑t_succ = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(pm i))\nm : ℕ → ℕ+ := fun i => if i < pk then pm i else ⟨2 * t + 1, ⋯⟩\nhmpk : ↑↑(m pk) = 2 * ↑t + 1\n⊢ (1 + 1 / ↑(2 * t + 1)) * (1 + Rat.divInt (Int.subNatNat (2 ^ pk) 1) ↑(t + 1)) =\n (1 + 1 / ↑(2 * t + 1)) * (1 + Rat.divInt (Int.subNatNat (2 ^ pk) 1) ↑↑t_succ)",
"after_state": "No Goals!"
},
{
"line": "trivial",
"before_state": "pk : ℕ\nhpk : ∀ (n : ℕ+), ∃ m, 1 + (2 ^ pk - 1) / ↑↑n = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(m i))\nn : ℕ+\nt : ℕ\nht : ↑n = 2 * t + 1\nt_succ : ℕ+ := ⟨t + 1, ⋯⟩\npm : ℕ → ℕ+\nhpm : 1 + (2 ^ pk - 1) / ↑↑t_succ = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(pm i))\nm : ℕ → ℕ+ := fun i => if i < pk then pm i else ⟨2 * t + 1, ⋯⟩\nhmpk : ↑↑(m pk) = 2 * ↑t + 1\n⊢ (1 + 1 / ↑(2 * t + 1)) * (1 + Rat.divInt (Int.subNatNat (2 ^ pk) 1) ↑(t + 1)) =\n (1 + 1 / ↑(2 * t + 1)) * (1 + Rat.divInt (Int.subNatNat (2 ^ pk) 1) ↑↑t_succ)",
"after_state": "No Goals!"
},
{
"line": "rfl",
"before_state": "pk : ℕ\nhpk : ∀ (n : ℕ+), ∃ m, 1 + (2 ^ pk - 1) / ↑↑n = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(m i))\nn : ℕ+\nt : ℕ\nht : ↑n = 2 * t + 1\nt_succ : ℕ+ := ⟨t + 1, ⋯⟩\npm : ℕ → ℕ+\nhpm : 1 + (2 ^ pk - 1) / ↑↑t_succ = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(pm i))\nm : ℕ → ℕ+ := fun i => if i < pk then pm i else ⟨2 * t + 1, ⋯⟩\nhmpk : ↑↑(m pk) = 2 * ↑t + 1\n⊢ (1 + 1 / ↑(2 * t + 1)) * (1 + Rat.divInt (Int.subNatNat (2 ^ pk) 1) ↑(t + 1)) =\n (1 + 1 / ↑(2 * t + 1)) * (1 + Rat.divInt (Int.subNatNat (2 ^ pk) 1) ↑↑t_succ)",
"after_state": "No Goals!"
},
{
"line": "eq_refl",
"before_state": "pk : ℕ\nhpk : ∀ (n : ℕ+), ∃ m, 1 + (2 ^ pk - 1) / ↑↑n = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(m i))\nn : ℕ+\nt : ℕ\nht : ↑n = 2 * t + 1\nt_succ : ℕ+ := ⟨t + 1, ⋯⟩\npm : ℕ → ℕ+\nhpm : 1 + (2 ^ pk - 1) / ↑↑t_succ = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(pm i))\nm : ℕ → ℕ+ := fun i => if i < pk then pm i else ⟨2 * t + 1, ⋯⟩\nhmpk : ↑↑(m pk) = 2 * ↑t + 1\n⊢ (1 + 1 / ↑(2 * t + 1)) * (1 + Rat.divInt (Int.subNatNat (2 ^ pk) 1) ↑(t + 1)) =\n (1 + 1 / ↑(2 * t + 1)) * (1 + Rat.divInt (Int.subNatNat (2 ^ pk) 1) ↑↑t_succ)",
"after_state": "No Goals!"
},
{
"line": "rw [prod_lemma]",
"before_state": "pk : ℕ\nhpk : ∀ (n : ℕ+), ∃ m, 1 + (2 ^ pk - 1) / ↑↑n = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(m i))\nn : ℕ+\nt : ℕ\nht : ↑n = 2 * t + 1\nt_succ : ℕ+ := ⟨t + 1, ⋯⟩\npm : ℕ → ℕ+\nhpm : 1 + (2 ^ pk - 1) / ↑↑t_succ = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(pm i))\nm : ℕ → ℕ+ := fun i => if i < pk then pm i else ⟨2 * t + 1, ⋯⟩\nhmpk : ↑↑(m pk) = 2 * ↑t + 1\n⊢ (1 + 1 / (2 * ↑t + 1)) * (1 + (2 ^ pk - 1) / ↑↑t_succ) =\n (∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(m i))) * (1 + 1 / ↑↑(m pk))",
"after_state": "pk : ℕ\nhpk : ∀ (n : ℕ+), ∃ m, 1 + (2 ^ pk - 1) / ↑↑n = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(m i))\nn : ℕ+\nt : ℕ\nht : ↑n = 2 * t + 1\nt_succ : ℕ+ := ⟨t + 1, ⋯⟩\npm : ℕ → ℕ+\nhpm : 1 + (2 ^ pk - 1) / ↑↑t_succ = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(pm i))\nm : ℕ → ℕ+ := fun i => if i < pk then pm i else ⟨2 * t + 1, ⋯⟩\nhmpk : ↑↑(m pk) = 2 * ↑t + 1\n⊢ (1 + 1 / (2 * ↑t + 1)) * (1 + (2 ^ pk - 1) / ↑↑t_succ) =\n (∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(m i))) * (1 + 1 / ↑↑(m pk))"
},
{
"line": "rewrite [prod_lemma]",
"before_state": "pk : ℕ\nhpk : ∀ (n : ℕ+), ∃ m, 1 + (2 ^ pk - 1) / ↑↑n = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(m i))\nn : ℕ+\nt : ℕ\nht : ↑n = 2 * t + 1\nt_succ : ℕ+ := ⟨t + 1, ⋯⟩\npm : ℕ → ℕ+\nhpm : 1 + (2 ^ pk - 1) / ↑↑t_succ = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(pm i))\nm : ℕ → ℕ+ := fun i => if i < pk then pm i else ⟨2 * t + 1, ⋯⟩\nhmpk : ↑↑(m pk) = 2 * ↑t + 1\n⊢ (1 + 1 / (2 * ↑t + 1)) * (1 + (2 ^ pk - 1) / ↑↑t_succ) =\n (∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(m i))) * (1 + 1 / ↑↑(m pk))",
"after_state": "pk : ℕ\nhpk : ∀ (n : ℕ+), ∃ m, 1 + (2 ^ pk - 1) / ↑↑n = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(m i))\nn : ℕ+\nt : ℕ\nht : ↑n = 2 * t + 1\nt_succ : ℕ+ := ⟨t + 1, ⋯⟩\npm : ℕ → ℕ+\nhpm : 1 + (2 ^ pk - 1) / ↑↑t_succ = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(pm i))\nm : ℕ → ℕ+ := fun i => if i < pk then pm i else ⟨2 * t + 1, ⋯⟩\nhmpk : ↑↑(m pk) = 2 * ↑t + 1\n⊢ (1 + 1 / (2 * ↑t + 1)) * (1 + (2 ^ pk - 1) / ↑↑t_succ) =\n (∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(m i))) * (1 + 1 / ↑↑(m pk))"
},
{
"line": "rw [← Finset.prod_range_succ _ pk]",
"before_state": "pk : ℕ\nhpk : ∀ (n : ℕ+), ∃ m, 1 + (2 ^ pk - 1) / ↑↑n = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(m i))\nn : ℕ+\nt : ℕ\nht : ↑n = 2 * t + 1\nt_succ : ℕ+ := ⟨t + 1, ⋯⟩\npm : ℕ → ℕ+\nhpm : 1 + (2 ^ pk - 1) / ↑↑t_succ = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(pm i))\nm : ℕ → ℕ+ := fun i => if i < pk then pm i else ⟨2 * t + 1, ⋯⟩\nhmpk : ↑↑(m pk) = 2 * ↑t + 1\n⊢ (∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(m i))) * (1 + 1 / ↑↑(m pk)) = ∏ i ∈ Finset.range pk.succ, (1 + 1 / ↑↑(m i))",
"after_state": "No Goals!"
},
{
"line": "rewrite [← Finset.prod_range_succ _ pk]",
"before_state": "pk : ℕ\nhpk : ∀ (n : ℕ+), ∃ m, 1 + (2 ^ pk - 1) / ↑↑n = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(m i))\nn : ℕ+\nt : ℕ\nht : ↑n = 2 * t + 1\nt_succ : ℕ+ := ⟨t + 1, ⋯⟩\npm : ℕ → ℕ+\nhpm : 1 + (2 ^ pk - 1) / ↑↑t_succ = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(pm i))\nm : ℕ → ℕ+ := fun i => if i < pk then pm i else ⟨2 * t + 1, ⋯⟩\nhmpk : ↑↑(m pk) = 2 * ↑t + 1\n⊢ (∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(m i))) * (1 + 1 / ↑↑(m pk)) = ∏ i ∈ Finset.range pk.succ, (1 + 1 / ↑↑(m i))",
"after_state": "pk : ℕ\nhpk : ∀ (n : ℕ+), ∃ m, 1 + (2 ^ pk - 1) / ↑↑n = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(m i))\nn : ℕ+\nt : ℕ\nht : ↑n = 2 * t + 1\nt_succ : ℕ+ := ⟨t + 1, ⋯⟩\npm : ℕ → ℕ+\nhpm : 1 + (2 ^ pk - 1) / ↑↑t_succ = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(pm i))\nm : ℕ → ℕ+ := fun i => if i < pk then pm i else ⟨2 * t + 1, ⋯⟩\nhmpk : ↑↑(m pk) = 2 * ↑t + 1\n⊢ ∏ x ∈ Finset.range (pk + 1), (1 + 1 / ↑↑(m x)) = ∏ i ∈ Finset.range pk.succ, (1 + 1 / ↑↑(m i))"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "pk : ℕ\nhpk : ∀ (n : ℕ+), ∃ m, 1 + (2 ^ pk - 1) / ↑↑n = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(m i))\nn : ℕ+\nt : ℕ\nht : ↑n = 2 * t + 1\nt_succ : ℕ+ := ⟨t + 1, ⋯⟩\npm : ℕ → ℕ+\nhpm : 1 + (2 ^ pk - 1) / ↑↑t_succ = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(pm i))\nm : ℕ → ℕ+ := fun i => if i < pk then pm i else ⟨2 * t + 1, ⋯⟩\nhmpk : ↑↑(m pk) = 2 * ↑t + 1\n⊢ ∏ x ∈ Finset.range (pk + 1), (1 + 1 / ↑↑(m x)) = ∏ i ∈ Finset.range pk.succ, (1 + 1 / ↑↑(m i))",
"after_state": "No Goals!"
},
{
"line": "try (with_reducible rfl)",
"before_state": "pk : ℕ\nhpk : ∀ (n : ℕ+), ∃ m, 1 + (2 ^ pk - 1) / ↑↑n = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(m i))\nn : ℕ+\nt : ℕ\nht : ↑n = 2 * t + 1\nt_succ : ℕ+ := ⟨t + 1, ⋯⟩\npm : ℕ → ℕ+\nhpm : 1 + (2 ^ pk - 1) / ↑↑t_succ = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(pm i))\nm : ℕ → ℕ+ := fun i => if i < pk then pm i else ⟨2 * t + 1, ⋯⟩\nhmpk : ↑↑(m pk) = 2 * ↑t + 1\n⊢ ∏ x ∈ Finset.range (pk + 1), (1 + 1 / ↑↑(m x)) = ∏ i ∈ Finset.range pk.succ, (1 + 1 / ↑↑(m i))",
"after_state": "No Goals!"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "pk : ℕ\nhpk : ∀ (n : ℕ+), ∃ m, 1 + (2 ^ pk - 1) / ↑↑n = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(m i))\nn : ℕ+\nt : ℕ\nht : ↑n = 2 * t + 1\nt_succ : ℕ+ := ⟨t + 1, ⋯⟩\npm : ℕ → ℕ+\nhpm : 1 + (2 ^ pk - 1) / ↑↑t_succ = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(pm i))\nm : ℕ → ℕ+ := fun i => if i < pk then pm i else ⟨2 * t + 1, ⋯⟩\nhmpk : ↑↑(m pk) = 2 * ↑t + 1\n⊢ ∏ x ∈ Finset.range (pk + 1), (1 + 1 / ↑↑(m x)) = ∏ i ∈ Finset.range pk.succ, (1 + 1 / ↑↑(m i))",
"after_state": "No Goals!"
},
{
"line": "with_reducible rfl",
"before_state": "pk : ℕ\nhpk : ∀ (n : ℕ+), ∃ m, 1 + (2 ^ pk - 1) / ↑↑n = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(m i))\nn : ℕ+\nt : ℕ\nht : ↑n = 2 * t + 1\nt_succ : ℕ+ := ⟨t + 1, ⋯⟩\npm : ℕ → ℕ+\nhpm : 1 + (2 ^ pk - 1) / ↑↑t_succ = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(pm i))\nm : ℕ → ℕ+ := fun i => if i < pk then pm i else ⟨2 * t + 1, ⋯⟩\nhmpk : ↑↑(m pk) = 2 * ↑t + 1\n⊢ ∏ x ∈ Finset.range (pk + 1), (1 + 1 / ↑↑(m x)) = ∏ i ∈ Finset.range pk.succ, (1 + 1 / ↑↑(m i))",
"after_state": "No Goals!"
},
{
"line": "rfl",
"before_state": "pk : ℕ\nhpk : ∀ (n : ℕ+), ∃ m, 1 + (2 ^ pk - 1) / ↑↑n = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(m i))\nn : ℕ+\nt : ℕ\nht : ↑n = 2 * t + 1\nt_succ : ℕ+ := ⟨t + 1, ⋯⟩\npm : ℕ → ℕ+\nhpm : 1 + (2 ^ pk - 1) / ↑↑t_succ = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(pm i))\nm : ℕ → ℕ+ := fun i => if i < pk then pm i else ⟨2 * t + 1, ⋯⟩\nhmpk : ↑↑(m pk) = 2 * ↑t + 1\n⊢ ∏ x ∈ Finset.range (pk + 1), (1 + 1 / ↑↑(m x)) = ∏ i ∈ Finset.range pk.succ, (1 + 1 / ↑↑(m i))",
"after_state": "No Goals!"
},
{
"line": "eq_refl",
"before_state": "pk : ℕ\nhpk : ∀ (n : ℕ+), ∃ m, 1 + (2 ^ pk - 1) / ↑↑n = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(m i))\nn : ℕ+\nt : ℕ\nht : ↑n = 2 * t + 1\nt_succ : ℕ+ := ⟨t + 1, ⋯⟩\npm : ℕ → ℕ+\nhpm : 1 + (2 ^ pk - 1) / ↑↑t_succ = ∏ i ∈ Finset.range pk, (1 + 1 / ↑↑(pm i))\nm : ℕ → ℕ+ := fun i => if i < pk then pm i else ⟨2 * t + 1, ⋯⟩\nhmpk : ↑↑(m pk) = 2 * ↑t + 1\n⊢ ∏ x ∈ Finset.range (pk + 1), (1 + 1 / ↑↑(m x)) = ∏ i ∈ Finset.range pk.succ, (1 + 1 / ↑↑(m i))",
"after_state": "No Goals!"
}
] |
theorem le_of_all_pow_lt_succ {x y : ℝ} (hx : 1 < x) (hy : 1 < y)
(h : ∀ n : ℕ, 0 < n → x ^ n - 1 < y ^ n) : x ≤ y := by
by_contra! hxy
have hxmy : 0 < x - y := sub_pos.mpr hxy
have hn : ∀ n : ℕ, 0 < n → (x - y) * (n : ℝ) ≤ x ^ n - y ^ n := by
intro n _
have hterm : ∀ i : ℕ, i ∈ Finset.range n → 1 ≤ x ^ i * y ^ (n - 1 - i) := by
intro i _
calc
1 ≤ x ^ i := one_le_pow₀ hx.le
_ = x ^ i * 1 := by ring
_ ≤ x ^ i * y ^ (n - 1 - i) := by gcongr; apply one_le_pow₀ hy.le
calc
(x - y) * (n : ℝ) = (n : ℝ) * (x - y) := by ring
_ = (∑ _i ∈ Finset.range n, (1 : ℝ)) * (x - y) := by
simp only [mul_one]
simp only [Finset.sum_const]
simp only [nsmul_eq_mul]
simp only [Finset.card_range]
_ ≤ (∑ i ∈ Finset.range n, x ^ i * y ^ (n - 1 - i)) * (x - y) := by
gcongr with i hi; apply hterm i hi
_ = x ^ n - y ^ n := geom_sum₂_mul x y n
-- Choose n larger than 1 / (x - y).
obtain ⟨N, hN⟩ := exists_nat_gt (1 / (x - y))
have hNp : 0 < N := mod_cast (one_div_pos.mpr hxmy).trans hN
have :=
calc
1 = (x - y) * (1 / (x - y)) := by field_simp
_ < (x - y) * N := by gcongr
_ ≤ x ^ N - y ^ N := hn N hNp
linarith [h N hNp]
|
/root/DuelModelResearch/mathlib4/Archive/Imo/Imo2013Q5.lean
|
{
"open": [],
"variables": []
}
|
[
{
"line": "by_contra! hxy",
"before_state": "x y : ℝ\nhx : 1 < x\nhy : 1 < y\nh : ∀ (n : ℕ), 0 < n → x ^ n - 1 < y ^ n\n⊢ x ≤ y",
"after_state": "x y : ℝ\nhx : 1 < x\nhy : 1 < y\nh : ∀ (n : ℕ), 0 < n → x ^ n - 1 < y ^ n\nhxy : y < x\n⊢ False"
},
{
"line": "by_contra hxy",
"before_state": "x y : ℝ\nhx : 1 < x\nhy : 1 < y\nh : ∀ (n : ℕ), 0 < n → x ^ n - 1 < y ^ n\n⊢ x ≤ y",
"after_state": "x y : ℝ\nhx : 1 < x\nhy : 1 < y\nh : ∀ (n : ℕ), 0 < n → x ^ n - 1 < y ^ n\nhxy : ¬x ≤ y\n⊢ False"
},
{
"line": "first\n| guard_target = Not✝ _; intro hxy\n| refine (Decidable.byContradiction✝ fun hxy => ?_ :)\n| refine (Classical.byContradiction✝ fun hxy => ?_ :)",
"before_state": "x y : ℝ\nhx : 1 < x\nhy : 1 < y\nh : ∀ (n : ℕ), 0 < n → x ^ n - 1 < y ^ n\n⊢ x ≤ y",
"after_state": "x y : ℝ\nhx : 1 < x\nhy : 1 < y\nh : ∀ (n : ℕ), 0 < n → x ^ n - 1 < y ^ n\nhxy : ¬x ≤ y\n⊢ False"
},
{
"line": "guard_target = Not✝ _",
"before_state": "x y : ℝ\nhx : 1 < x\nhy : 1 < y\nh : ∀ (n : ℕ), 0 < n → x ^ n - 1 < y ^ n\n⊢ x ≤ y",
"after_state": "x y : ℝ\nhx : 1 < x\nhy : 1 < y\nh : ∀ (n : ℕ), 0 < n → x ^ n - 1 < y ^ n\n⊢ x ≤ y"
},
{
"line": "refine (Decidable.byContradiction✝ fun hxy => ?_ :)",
"before_state": "x y : ℝ\nhx : 1 < x\nhy : 1 < y\nh : ∀ (n : ℕ), 0 < n → x ^ n - 1 < y ^ n\n⊢ x ≤ y",
"after_state": "x y : ℝ\nhx : 1 < x\nhy : 1 < y\nh : ∀ (n : ℕ), 0 < n → x ^ n - 1 < y ^ n\n⊢ x ≤ y"
},
{
"line": "refine (Classical.byContradiction✝ fun hxy => ?_ :)",
"before_state": "x y : ℝ\nhx : 1 < x\nhy : 1 < y\nh : ∀ (n : ℕ), 0 < n → x ^ n - 1 < y ^ n\n⊢ x ≤ y",
"after_state": "x y : ℝ\nhx : 1 < x\nhy : 1 < y\nh : ∀ (n : ℕ), 0 < n → x ^ n - 1 < y ^ n\nhxy : ¬x ≤ y\n⊢ False"
},
{
"line": "try push_neg at hxy",
"before_state": "x y : ℝ\nhx : 1 < x\nhy : 1 < y\nh : ∀ (n : ℕ), 0 < n → x ^ n - 1 < y ^ n\nhxy : ¬x ≤ y\n⊢ False",
"after_state": "x y : ℝ\nhx : 1 < x\nhy : 1 < y\nh : ∀ (n : ℕ), 0 < n → x ^ n - 1 < y ^ n\nhxy : y < x\n⊢ False"
},
{
"line": "first\n| push_neg at hxy\n| skip",
"before_state": "x y : ℝ\nhx : 1 < x\nhy : 1 < y\nh : ∀ (n : ℕ), 0 < n → x ^ n - 1 < y ^ n\nhxy : ¬x ≤ y\n⊢ False",
"after_state": "x y : ℝ\nhx : 1 < x\nhy : 1 < y\nh : ∀ (n : ℕ), 0 < n → x ^ n - 1 < y ^ n\nhxy : y < x\n⊢ False"
},
{
"line": "push_neg at hxy",
"before_state": "x y : ℝ\nhx : 1 < x\nhy : 1 < y\nh : ∀ (n : ℕ), 0 < n → x ^ n - 1 < y ^ n\nhxy : ¬x ≤ y\n⊢ False",
"after_state": "x y : ℝ\nhx : 1 < x\nhy : 1 < y\nh : ∀ (n : ℕ), 0 < n → x ^ n - 1 < y ^ n\nhxy : y < x\n⊢ False"
},
{
"line": "have hxmy : 0 < x - y := sub_pos.mpr hxy",
"before_state": "x y : ℝ\nhx : 1 < x\nhy : 1 < y\nh : ∀ (n : ℕ), 0 < n → x ^ n - 1 < y ^ n\nhxy : y < x\n⊢ False",
"after_state": "x y : ℝ\nhx : 1 < x\nhy : 1 < y\nh : ∀ (n : ℕ), 0 < n → x ^ n - 1 < y ^ n\nhxy : y < x\nhxmy : 0 < x - y\n⊢ False"
},
{
"line": "refine_lift\n have hxmy : 0 < x - y := sub_pos.mpr hxy;\n ?_",
"before_state": "x y : ℝ\nhx : 1 < x\nhy : 1 < y\nh : ∀ (n : ℕ), 0 < n → x ^ n - 1 < y ^ n\nhxy : y < x\n⊢ False",
"after_state": "x y : ℝ\nhx : 1 < x\nhy : 1 < y\nh : ∀ (n : ℕ), 0 < n → x ^ n - 1 < y ^ n\nhxy : y < x\nhxmy : 0 < x - y\n⊢ False"
},
{
"line": "focus\n (refine\n no_implicit_lambda%\n (have hxmy : 0 < x - y := sub_pos.mpr hxy;\n ?_);\n rotate_right)",
"before_state": "x y : ℝ\nhx : 1 < x\nhy : 1 < y\nh : ∀ (n : ℕ), 0 < n → x ^ n - 1 < y ^ n\nhxy : y < x\n⊢ False",
"after_state": "x y : ℝ\nhx : 1 < x\nhy : 1 < y\nh : ∀ (n : ℕ), 0 < n → x ^ n - 1 < y ^ n\nhxy : y < x\nhxmy : 0 < x - y\n⊢ False"
},
{
"line": "refine\n no_implicit_lambda%\n (have hxmy : 0 < x - y := sub_pos.mpr hxy;\n ?_)",
"before_state": "x y : ℝ\nhx : 1 < x\nhy : 1 < y\nh : ∀ (n : ℕ), 0 < n → x ^ n - 1 < y ^ n\nhxy : y < x\n⊢ False",
"after_state": "x y : ℝ\nhx : 1 < x\nhy : 1 < y\nh : ∀ (n : ℕ), 0 < n → x ^ n - 1 < y ^ n\nhxy : y < x\nhxmy : 0 < x - y\n⊢ False"
},
{
"line": "rotate_right",
"before_state": "x y : ℝ\nhx : 1 < x\nhy : 1 < y\nh : ∀ (n : ℕ), 0 < n → x ^ n - 1 < y ^ n\nhxy : y < x\nhxmy : 0 < x - y\n⊢ False",
"after_state": "x y : ℝ\nhx : 1 < x\nhy : 1 < y\nh : ∀ (n : ℕ), 0 < n → x ^ n - 1 < y ^ n\nhxy : y < x\nhxmy : 0 < x - y\n⊢ False"
},
{
"line": "have hn : ∀ n : ℕ, 0 < n → (x - y) * (n : ℝ) ≤ x ^ n - y ^ n :=\n by\n intro n _\n have hterm : ∀ i : ℕ, i ∈ Finset.range n → 1 ≤ x ^ i * y ^ (n - 1 - i) :=\n by\n intro i _\n calc\n 1 ≤ x ^ i := one_le_pow₀ hx.le\n _ = x ^ i * 1 := by ring\n _ ≤ x ^ i * y ^ (n - 1 - i) := by gcongr; apply one_le_pow₀ hy.le\n calc\n (x - y) * (n : ℝ) = (n : ℝ) * (x - y) := by ring\n _ = (∑ _i ∈ Finset.range n, (1 : ℝ)) * (x - y) :=\n by\n simp only [mul_one]\n simp only [Finset.sum_const]\n simp only [nsmul_eq_mul]\n simp only [Finset.card_range]\n _ ≤ (∑ i ∈ Finset.range n, x ^ i * y ^ (n - 1 - i)) * (x - y) := by gcongr with i hi; apply hterm i hi\n _ = x ^ n - y ^ n := geom_sum₂_mul x y n",
"before_state": "x y : ℝ\nhx : 1 < x\nhy : 1 < y\nh : ∀ (n : ℕ), 0 < n → x ^ n - 1 < y ^ n\nhxy : y < x\nhxmy : 0 < x - y\n⊢ False",
"after_state": "x y : ℝ\nhx : 1 < x\nhy : 1 < y\nh : ∀ (n : ℕ), 0 < n → x ^ n - 1 < y ^ n\nhxy : y < x\nhxmy : 0 < x - y\nhn : ∀ (n : ℕ), 0 < n → (x - y) * ↑n ≤ x ^ n - y ^ n\n⊢ False"
},
{
"line": "focus\n refine\n no_implicit_lambda%\n (have hn : ∀ n : ℕ, 0 < n → (x - y) * (n : ℝ) ≤ x ^ n - y ^ n := ?body✝;\n ?_)\n case body✝ =>\n with_annotate_state\"by\"\n ( intro n _\n have hterm : ∀ i : ℕ, i ∈ Finset.range n → 1 ≤ x ^ i * y ^ (n - 1 - i) :=\n by\n intro i _\n calc\n 1 ≤ x ^ i := one_le_pow₀ hx.le\n _ = x ^ i * 1 := by ring\n _ ≤ x ^ i * y ^ (n - 1 - i) := by gcongr; apply one_le_pow₀ hy.le\n calc\n (x - y) * (n : ℝ) = (n : ℝ) * (x - y) := by ring\n _ = (∑ _i ∈ Finset.range n, (1 : ℝ)) * (x - y) :=\n by\n simp only [mul_one]\n simp only [Finset.sum_const]\n simp only [nsmul_eq_mul]\n simp only [Finset.card_range]\n _ ≤ (∑ i ∈ Finset.range n, x ^ i * y ^ (n - 1 - i)) * (x - y) := by gcongr with i hi; apply hterm i hi\n _ = x ^ n - y ^ n := geom_sum₂_mul x y n)",
"before_state": "x y : ℝ\nhx : 1 < x\nhy : 1 < y\nh : ∀ (n : ℕ), 0 < n → x ^ n - 1 < y ^ n\nhxy : y < x\nhxmy : 0 < x - y\n⊢ False",
"after_state": "x y : ℝ\nhx : 1 < x\nhy : 1 < y\nh : ∀ (n : ℕ), 0 < n → x ^ n - 1 < y ^ n\nhxy : y < x\nhxmy : 0 < x - y\nhn : ∀ (n : ℕ), 0 < n → (x - y) * ↑n ≤ x ^ n - y ^ n\n⊢ False"
},
{
"line": "refine\n no_implicit_lambda%\n (have hn : ∀ n : ℕ, 0 < n → (x - y) * (n : ℝ) ≤ x ^ n - y ^ n := ?body✝;\n ?_)",
"before_state": "x y : ℝ\nhx : 1 < x\nhy : 1 < y\nh : ∀ (n : ℕ), 0 < n → x ^ n - 1 < y ^ n\nhxy : y < x\nhxmy : 0 < x - y\n⊢ False",
"after_state": "case body\nx y : ℝ\nhx : 1 < x\nhy : 1 < y\nh : ∀ (n : ℕ), 0 < n → x ^ n - 1 < y ^ n\nhxy : y < x\nhxmy : 0 < x - y\n⊢ ∀ (n : ℕ), 0 < n → (x - y) * ↑n ≤ x ^ n - y ^ n\n---\nx y : ℝ\nhx : 1 < x\nhy : 1 < y\nh : ∀ (n : ℕ), 0 < n → x ^ n - 1 < y ^ n\nhxy : y < x\nhxmy : 0 < x - y\nhn : ∀ (n : ℕ), 0 < n → (x - y) * ↑n ≤ x ^ n - y ^ n\n⊢ False"
},
{
"line": "case body✝ =>\n with_annotate_state\"by\"\n ( intro n _\n have hterm : ∀ i : ℕ, i ∈ Finset.range n → 1 ≤ x ^ i * y ^ (n - 1 - i) :=\n by\n intro i _\n calc\n 1 ≤ x ^ i := one_le_pow₀ hx.le\n _ = x ^ i * 1 := by ring\n _ ≤ x ^ i * y ^ (n - 1 - i) := by gcongr; apply one_le_pow₀ hy.le\n calc\n (x - y) * (n : ℝ) = (n : ℝ) * (x - y) := by ring\n _ = (∑ _i ∈ Finset.range n, (1 : ℝ)) * (x - y) :=\n by\n simp only [mul_one]\n simp only [Finset.sum_const]\n simp only [nsmul_eq_mul]\n simp only [Finset.card_range]\n _ ≤ (∑ i ∈ Finset.range n, x ^ i * y ^ (n - 1 - i)) * (x - y) := by gcongr with i hi; apply hterm i hi\n _ = x ^ n - y ^ n := geom_sum₂_mul x y n)",
"before_state": "case body\nx y : ℝ\nhx : 1 < x\nhy : 1 < y\nh : ∀ (n : ℕ), 0 < n → x ^ n - 1 < y ^ n\nhxy : y < x\nhxmy : 0 < x - y\n⊢ ∀ (n : ℕ), 0 < n → (x - y) * ↑n ≤ x ^ n - y ^ n\n---\nx y : ℝ\nhx : 1 < x\nhy : 1 < y\nh : ∀ (n : ℕ), 0 < n → x ^ n - 1 < y ^ n\nhxy : y < x\nhxmy : 0 < x - y\nhn : ∀ (n : ℕ), 0 < n → (x - y) * ↑n ≤ x ^ n - y ^ n\n⊢ False",
"after_state": "x y : ℝ\nhx : 1 < x\nhy : 1 < y\nh : ∀ (n : ℕ), 0 < n → x ^ n - 1 < y ^ n\nhxy : y < x\nhxmy : 0 < x - y\nhn : ∀ (n : ℕ), 0 < n → (x - y) * ↑n ≤ x ^ n - y ^ n\n⊢ False"
},
{
"line": "with_annotate_state\"by\"\n ( intro n _\n have hterm : ∀ i : ℕ, i ∈ Finset.range n → 1 ≤ x ^ i * y ^ (n - 1 - i) :=\n by\n intro i _\n calc\n 1 ≤ x ^ i := one_le_pow₀ hx.le\n _ = x ^ i * 1 := by ring\n _ ≤ x ^ i * y ^ (n - 1 - i) := by gcongr; apply one_le_pow₀ hy.le\n calc\n (x - y) * (n : ℝ) = (n : ℝ) * (x - y) := by ring\n _ = (∑ _i ∈ Finset.range n, (1 : ℝ)) * (x - y) :=\n by\n simp only [mul_one]\n simp only [Finset.sum_const]\n simp only [nsmul_eq_mul]\n simp only [Finset.card_range]\n _ ≤ (∑ i ∈ Finset.range n, x ^ i * y ^ (n - 1 - i)) * (x - y) := by gcongr with i hi; apply hterm i hi\n _ = x ^ n - y ^ n := geom_sum₂_mul x y n)",
"before_state": "x y : ℝ\nhx : 1 < x\nhy : 1 < y\nh : ∀ (n : ℕ), 0 < n → x ^ n - 1 < y ^ n\nhxy : y < x\nhxmy : 0 < x - y\n⊢ ∀ (n : ℕ), 0 < n → (x - y) * ↑n ≤ x ^ n - y ^ n",
"after_state": "No Goals!"
},
{
"line": "intro n _",
"before_state": "x y : ℝ\nhx : 1 < x\nhy : 1 < y\nh : ∀ (n : ℕ), 0 < n → x ^ n - 1 < y ^ n\nhxy : y < x\nhxmy : 0 < x - y\n⊢ ∀ (n : ℕ), 0 < n → (x - y) * ↑n ≤ x ^ n - y ^ n",
"after_state": "x y : ℝ\nhx : 1 < x\nhy : 1 < y\nh : ∀ (n : ℕ), 0 < n → x ^ n - 1 < y ^ n\nhxy : y < x\nhxmy : 0 < x - y\nn : ℕ\na✝ : 0 < n\n⊢ (x - y) * ↑n ≤ x ^ n - y ^ n"
},
{
"line": "intro n;\n intro _",
"before_state": "x y : ℝ\nhx : 1 < x\nhy : 1 < y\nh : ∀ (n : ℕ), 0 < n → x ^ n - 1 < y ^ n\nhxy : y < x\nhxmy : 0 < x - y\n⊢ ∀ (n : ℕ), 0 < n → (x - y) * ↑n ≤ x ^ n - y ^ n",
"after_state": "x y : ℝ\nhx : 1 < x\nhy : 1 < y\nh : ∀ (n : ℕ), 0 < n → x ^ n - 1 < y ^ n\nhxy : y < x\nhxmy : 0 < x - y\nn : ℕ\na✝ : 0 < n\n⊢ (x - y) * ↑n ≤ x ^ n - y ^ n"
},
{
"line": "intro n",
"before_state": "x y : ℝ\nhx : 1 < x\nhy : 1 < y\nh : ∀ (n : ℕ), 0 < n → x ^ n - 1 < y ^ n\nhxy : y < x\nhxmy : 0 < x - y\n⊢ ∀ (n : ℕ), 0 < n → (x - y) * ↑n ≤ x ^ n - y ^ n",
"after_state": "x y : ℝ\nhx : 1 < x\nhy : 1 < y\nh : ∀ (n : ℕ), 0 < n → x ^ n - 1 < y ^ n\nhxy : y < x\nhxmy : 0 < x - y\nn : ℕ\n⊢ 0 < n → (x - y) * ↑n ≤ x ^ n - y ^ n"
},
{
"line": "intro _",
"before_state": "x y : ℝ\nhx : 1 < x\nhy : 1 < y\nh : ∀ (n : ℕ), 0 < n → x ^ n - 1 < y ^ n\nhxy : y < x\nhxmy : 0 < x - y\nn : ℕ\n⊢ 0 < n → (x - y) * ↑n ≤ x ^ n - y ^ n",
"after_state": "x y : ℝ\nhx : 1 < x\nhy : 1 < y\nh : ∀ (n : ℕ), 0 < n → x ^ n - 1 < y ^ n\nhxy : y < x\nhxmy : 0 < x - y\nn : ℕ\na✝ : 0 < n\n⊢ (x - y) * ↑n ≤ x ^ n - y ^ n"
},
{
"line": "have hterm : ∀ i : ℕ, i ∈ Finset.range n → 1 ≤ x ^ i * y ^ (n - 1 - i) :=\n by\n intro i _\n calc\n 1 ≤ x ^ i := one_le_pow₀ hx.le\n _ = x ^ i * 1 := by ring\n _ ≤ x ^ i * y ^ (n - 1 - i) := by gcongr; apply one_le_pow₀ hy.le",
"before_state": "x y : ℝ\nhx : 1 < x\nhy : 1 < y\nh : ∀ (n : ℕ), 0 < n → x ^ n - 1 < y ^ n\nhxy : y < x\nhxmy : 0 < x - y\nn : ℕ\na✝ : 0 < n\n⊢ (x - y) * ↑n ≤ x ^ n - y ^ n",
"after_state": "x y : ℝ\nhx : 1 < x\nhy : 1 < y\nh : ∀ (n : ℕ), 0 < n → x ^ n - 1 < y ^ n\nhxy : y < x\nhxmy : 0 < x - y\nn : ℕ\na✝ : 0 < n\nhterm : ∀ i ∈ Finset.range n, 1 ≤ x ^ i * y ^ (n - 1 - i)\n⊢ (x - y) * ↑n ≤ x ^ n - y ^ n"
},
{
"line": "focus\n refine\n no_implicit_lambda%\n (have hterm : ∀ i : ℕ, i ∈ Finset.range n → 1 ≤ x ^ i * y ^ (n - 1 - i) := ?body✝;\n ?_)\n case body✝ =>\n with_annotate_state\"by\"\n ( intro i _\n calc\n 1 ≤ x ^ i := one_le_pow₀ hx.le\n _ = x ^ i * 1 := by ring\n _ ≤ x ^ i * y ^ (n - 1 - i) := by gcongr; apply one_le_pow₀ hy.le)",
"before_state": "x y : ℝ\nhx : 1 < x\nhy : 1 < y\nh : ∀ (n : ℕ), 0 < n → x ^ n - 1 < y ^ n\nhxy : y < x\nhxmy : 0 < x - y\nn : ℕ\na✝ : 0 < n\n⊢ (x - y) * ↑n ≤ x ^ n - y ^ n",
"after_state": "x y : ℝ\nhx : 1 < x\nhy : 1 < y\nh : ∀ (n : ℕ), 0 < n → x ^ n - 1 < y ^ n\nhxy : y < x\nhxmy : 0 < x - y\nn : ℕ\na✝ : 0 < n\nhterm : ∀ i ∈ Finset.range n, 1 ≤ x ^ i * y ^ (n - 1 - i)\n⊢ (x - y) * ↑n ≤ x ^ n - y ^ n"
},
{
"line": "refine\n no_implicit_lambda%\n (have hterm : ∀ i : ℕ, i ∈ Finset.range n → 1 ≤ x ^ i * y ^ (n - 1 - i) := ?body✝;\n ?_)",
"before_state": "x y : ℝ\nhx : 1 < x\nhy : 1 < y\nh : ∀ (n : ℕ), 0 < n → x ^ n - 1 < y ^ n\nhxy : y < x\nhxmy : 0 < x - y\nn : ℕ\na✝ : 0 < n\n⊢ (x - y) * ↑n ≤ x ^ n - y ^ n",
"after_state": "case body\nx y : ℝ\nhx : 1 < x\nhy : 1 < y\nh : ∀ (n : ℕ), 0 < n → x ^ n - 1 < y ^ n\nhxy : y < x\nhxmy : 0 < x - y\nn : ℕ\na✝ : 0 < n\n⊢ ∀ i ∈ Finset.range n, 1 ≤ x ^ i * y ^ (n - 1 - i)\n---\nx y : ℝ\nhx : 1 < x\nhy : 1 < y\nh : ∀ (n : ℕ), 0 < n → x ^ n - 1 < y ^ n\nhxy : y < x\nhxmy : 0 < x - y\nn : ℕ\na✝ : 0 < n\nhterm : ∀ i ∈ Finset.range n, 1 ≤ x ^ i * y ^ (n - 1 - i)\n⊢ (x - y) * ↑n ≤ x ^ n - y ^ n"
},
{
"line": "case body✝ =>\n with_annotate_state\"by\"\n ( intro i _\n calc\n 1 ≤ x ^ i := one_le_pow₀ hx.le\n _ = x ^ i * 1 := by ring\n _ ≤ x ^ i * y ^ (n - 1 - i) := by gcongr; apply one_le_pow₀ hy.le)",
"before_state": "case body\nx y : ℝ\nhx : 1 < x\nhy : 1 < y\nh : ∀ (n : ℕ), 0 < n → x ^ n - 1 < y ^ n\nhxy : y < x\nhxmy : 0 < x - y\nn : ℕ\na✝ : 0 < n\n⊢ ∀ i ∈ Finset.range n, 1 ≤ x ^ i * y ^ (n - 1 - i)\n---\nx y : ℝ\nhx : 1 < x\nhy : 1 < y\nh : ∀ (n : ℕ), 0 < n → x ^ n - 1 < y ^ n\nhxy : y < x\nhxmy : 0 < x - y\nn : ℕ\na✝ : 0 < n\nhterm : ∀ i ∈ Finset.range n, 1 ≤ x ^ i * y ^ (n - 1 - i)\n⊢ (x - y) * ↑n ≤ x ^ n - y ^ n",
"after_state": "x y : ℝ\nhx : 1 < x\nhy : 1 < y\nh : ∀ (n : ℕ), 0 < n → x ^ n - 1 < y ^ n\nhxy : y < x\nhxmy : 0 < x - y\nn : ℕ\na✝ : 0 < n\nhterm : ∀ i ∈ Finset.range n, 1 ≤ x ^ i * y ^ (n - 1 - i)\n⊢ (x - y) * ↑n ≤ x ^ n - y ^ n"
},
{
"line": "with_annotate_state\"by\"\n ( intro i _\n calc\n 1 ≤ x ^ i := one_le_pow₀ hx.le\n _ = x ^ i * 1 := by ring\n _ ≤ x ^ i * y ^ (n - 1 - i) := by gcongr; apply one_le_pow₀ hy.le)",
"before_state": "x y : ℝ\nhx : 1 < x\nhy : 1 < y\nh : ∀ (n : ℕ), 0 < n → x ^ n - 1 < y ^ n\nhxy : y < x\nhxmy : 0 < x - y\nn : ℕ\na✝ : 0 < n\n⊢ ∀ i ∈ Finset.range n, 1 ≤ x ^ i * y ^ (n - 1 - i)",
"after_state": "No Goals!"
},
{
"line": "intro i _",
"before_state": "x y : ℝ\nhx : 1 < x\nhy : 1 < y\nh : ∀ (n : ℕ), 0 < n → x ^ n - 1 < y ^ n\nhxy : y < x\nhxmy : 0 < x - y\nn : ℕ\na✝ : 0 < n\n⊢ ∀ i ∈ Finset.range n, 1 ≤ x ^ i * y ^ (n - 1 - i)",
"after_state": "x y : ℝ\nhx : 1 < x\nhy : 1 < y\nh : ∀ (n : ℕ), 0 < n → x ^ n - 1 < y ^ n\nhxy : y < x\nhxmy : 0 < x - y\nn : ℕ\na✝¹ : 0 < n\ni : ℕ\na✝ : i ∈ Finset.range n\n⊢ 1 ≤ x ^ i * y ^ (n - 1 - i)"
},
{
"line": "intro i;\n intro _",
"before_state": "x y : ℝ\nhx : 1 < x\nhy : 1 < y\nh : ∀ (n : ℕ), 0 < n → x ^ n - 1 < y ^ n\nhxy : y < x\nhxmy : 0 < x - y\nn : ℕ\na✝ : 0 < n\n⊢ ∀ i ∈ Finset.range n, 1 ≤ x ^ i * y ^ (n - 1 - i)",
"after_state": "x y : ℝ\nhx : 1 < x\nhy : 1 < y\nh : ∀ (n : ℕ), 0 < n → x ^ n - 1 < y ^ n\nhxy : y < x\nhxmy : 0 < x - y\nn : ℕ\na✝¹ : 0 < n\ni : ℕ\na✝ : i ∈ Finset.range n\n⊢ 1 ≤ x ^ i * y ^ (n - 1 - i)"
},
{
"line": "intro i",
"before_state": "x y : ℝ\nhx : 1 < x\nhy : 1 < y\nh : ∀ (n : ℕ), 0 < n → x ^ n - 1 < y ^ n\nhxy : y < x\nhxmy : 0 < x - y\nn : ℕ\na✝ : 0 < n\n⊢ ∀ i ∈ Finset.range n, 1 ≤ x ^ i * y ^ (n - 1 - i)",
"after_state": "x y : ℝ\nhx : 1 < x\nhy : 1 < y\nh : ∀ (n : ℕ), 0 < n → x ^ n - 1 < y ^ n\nhxy : y < x\nhxmy : 0 < x - y\nn : ℕ\na✝ : 0 < n\ni : ℕ\n⊢ i ∈ Finset.range n → 1 ≤ x ^ i * y ^ (n - 1 - i)"
},
{
"line": "intro _",
"before_state": "x y : ℝ\nhx : 1 < x\nhy : 1 < y\nh : ∀ (n : ℕ), 0 < n → x ^ n - 1 < y ^ n\nhxy : y < x\nhxmy : 0 < x - y\nn : ℕ\na✝ : 0 < n\ni : ℕ\n⊢ i ∈ Finset.range n → 1 ≤ x ^ i * y ^ (n - 1 - i)",
"after_state": "x y : ℝ\nhx : 1 < x\nhy : 1 < y\nh : ∀ (n : ℕ), 0 < n → x ^ n - 1 < y ^ n\nhxy : y < x\nhxmy : 0 < x - y\nn : ℕ\na✝¹ : 0 < n\ni : ℕ\na✝ : i ∈ Finset.range n\n⊢ 1 ≤ x ^ i * y ^ (n - 1 - i)"
},
{
"line": "calc\n 1 ≤ x ^ i := one_le_pow₀ hx.le\n _ = x ^ i * 1 := by ring\n _ ≤ x ^ i * y ^ (n - 1 - i) := by gcongr; apply one_le_pow₀ hy.le",
"before_state": "x y : ℝ\nhx : 1 < x\nhy : 1 < y\nh : ∀ (n : ℕ), 0 < n → x ^ n - 1 < y ^ n\nhxy : y < x\nhxmy : 0 < x - y\nn : ℕ\na✝¹ : 0 < n\ni : ℕ\na✝ : i ∈ Finset.range n\n⊢ 1 ≤ x ^ i * y ^ (n - 1 - i)",
"after_state": "No Goals!"
},
{
"line": "ring",
"before_state": "x y : ℝ\nhx : 1 < x\nhy : 1 < y\nh : ∀ (n : ℕ), 0 < n → x ^ n - 1 < y ^ n\nhxy : y < x\nhxmy : 0 < x - y\nn : ℕ\na✝¹ : 0 < n\ni : ℕ\na✝ : i ∈ Finset.range n\n⊢ x ^ i = x ^ i * 1",
"after_state": "No Goals!"
},
{
"line": "first\n| ring1\n|\n try_this ring_nf\"\\n\\nThe `ring` tactic failed to close the goal. Use `ring_nf` to obtain a normal form.\n \\nNote that `ring` works primarily in *commutative* rings. \\\n If you have a noncommutative ring, abelian group or module, consider using \\\n `noncomm_ring`, `abel` or `module` instead.\"",
"before_state": "x y : ℝ\nhx : 1 < x\nhy : 1 < y\nh : ∀ (n : ℕ), 0 < n → x ^ n - 1 < y ^ n\nhxy : y < x\nhxmy : 0 < x - y\nn : ℕ\na✝¹ : 0 < n\ni : ℕ\na✝ : i ∈ Finset.range n\n⊢ x ^ i = x ^ i * 1",
"after_state": "No Goals!"
},
{
"line": "ring1",
"before_state": "x y : ℝ\nhx : 1 < x\nhy : 1 < y\nh : ∀ (n : ℕ), 0 < n → x ^ n - 1 < y ^ n\nhxy : y < x\nhxmy : 0 < x - y\nn : ℕ\na✝¹ : 0 < n\ni : ℕ\na✝ : i ∈ Finset.range n\n⊢ x ^ i = x ^ i * 1",
"after_state": "No Goals!"
},
{
"line": "gcongr",
"before_state": "x y : ℝ\nhx : 1 < x\nhy : 1 < y\nh : ∀ (n : ℕ), 0 < n → x ^ n - 1 < y ^ n\nhxy : y < x\nhxmy : 0 < x - y\nn : ℕ\na✝¹ : 0 < n\ni : ℕ\na✝ : i ∈ Finset.range n\n⊢ x ^ i * 1 ≤ x ^ i * y ^ (n - 1 - i)",
"after_state": "case h\nx y : ℝ\nhx : 1 < x\nhy : 1 < y\nh : ∀ (n : ℕ), 0 < n → x ^ n - 1 < y ^ n\nhxy : y < x\nhxmy : 0 < x - y\nn : ℕ\na✝¹ : 0 < n\ni : ℕ\na✝ : i ∈ Finset.range n\n⊢ 1 ≤ y ^ (n - 1 - i)"
},
{
"line": "gcongr_discharger",
"before_state": "case a0\nx y : ℝ\nhx : 1 < x\nhy : 1 < y\nh : ∀ (n : ℕ), 0 < n → x ^ n - 1 < y ^ n\nhxy : y < x\nhxmy : 0 < x - y\nn : ℕ\na✝¹ : 0 < n\ni : ℕ\na✝ : i ∈ Finset.range n\n⊢ 0 ≤ x ^ i",
"after_state": "No Goals!"
},
{
"line": "positivity",
"before_state": "case a0\nx y : ℝ\nhx : 1 < x\nhy : 1 < y\nh : ∀ (n : ℕ), 0 < n → x ^ n - 1 < y ^ n\nhxy : y < x\nhxmy : 0 < x - y\nn : ℕ\na✝¹ : 0 < n\ni : ℕ\na✝ : i ∈ Finset.range n\n⊢ 0 ≤ x ^ i",
"after_state": "No Goals!"
},
{
"line": "apply one_le_pow₀ hy.le",
"before_state": "case h\nx y : ℝ\nhx : 1 < x\nhy : 1 < y\nh : ∀ (n : ℕ), 0 < n → x ^ n - 1 < y ^ n\nhxy : y < x\nhxmy : 0 < x - y\nn : ℕ\na✝¹ : 0 < n\ni : ℕ\na✝ : i ∈ Finset.range n\n⊢ 1 ≤ y ^ (n - 1 - i)",
"after_state": "No Goals!"
},
{
"line": "calc\n (x - y) * (n : ℝ) = (n : ℝ) * (x - y) := by ring\n _ = (∑ _i ∈ Finset.range n, (1 : ℝ)) * (x - y) :=\n by\n simp only [mul_one]\n simp only [Finset.sum_const]\n simp only [nsmul_eq_mul]\n simp only [Finset.card_range]\n _ ≤ (∑ i ∈ Finset.range n, x ^ i * y ^ (n - 1 - i)) * (x - y) := by gcongr with i hi; apply hterm i hi\n _ = x ^ n - y ^ n := geom_sum₂_mul x y n",
"before_state": "x y : ℝ\nhx : 1 < x\nhy : 1 < y\nh : ∀ (n : ℕ), 0 < n → x ^ n - 1 < y ^ n\nhxy : y < x\nhxmy : 0 < x - y\nn : ℕ\na✝ : 0 < n\nhterm : ∀ i ∈ Finset.range n, 1 ≤ x ^ i * y ^ (n - 1 - i)\n⊢ (x - y) * ↑n ≤ x ^ n - y ^ n",
"after_state": "No Goals!"
},
{
"line": "ring",
"before_state": "x y : ℝ\nhx : 1 < x\nhy : 1 < y\nh : ∀ (n : ℕ), 0 < n → x ^ n - 1 < y ^ n\nhxy : y < x\nhxmy : 0 < x - y\nn : ℕ\na✝ : 0 < n\nhterm : ∀ i ∈ Finset.range n, 1 ≤ x ^ i * y ^ (n - 1 - i)\n⊢ (x - y) * ↑n = ↑n * (x - y)",
"after_state": "No Goals!"
},
{
"line": "first\n| ring1\n|\n try_this ring_nf\"\\n\\nThe `ring` tactic failed to close the goal. Use `ring_nf` to obtain a normal form.\n \\nNote that `ring` works primarily in *commutative* rings. \\\n If you have a noncommutative ring, abelian group or module, consider using \\\n `noncomm_ring`, `abel` or `module` instead.\"",
"before_state": "x y : ℝ\nhx : 1 < x\nhy : 1 < y\nh : ∀ (n : ℕ), 0 < n → x ^ n - 1 < y ^ n\nhxy : y < x\nhxmy : 0 < x - y\nn : ℕ\na✝ : 0 < n\nhterm : ∀ i ∈ Finset.range n, 1 ≤ x ^ i * y ^ (n - 1 - i)\n⊢ (x - y) * ↑n = ↑n * (x - y)",
"after_state": "No Goals!"
},
{
"line": "ring1",
"before_state": "x y : ℝ\nhx : 1 < x\nhy : 1 < y\nh : ∀ (n : ℕ), 0 < n → x ^ n - 1 < y ^ n\nhxy : y < x\nhxmy : 0 < x - y\nn : ℕ\na✝ : 0 < n\nhterm : ∀ i ∈ Finset.range n, 1 ≤ x ^ i * y ^ (n - 1 - i)\n⊢ (x - y) * ↑n = ↑n * (x - y)",
"after_state": "No Goals!"
},
{
"line": "simp only [mul_one]",
"before_state": "x y : ℝ\nhx : 1 < x\nhy : 1 < y\nh : ∀ (n : ℕ), 0 < n → x ^ n - 1 < y ^ n\nhxy : y < x\nhxmy : 0 < x - y\nn : ℕ\na✝ : 0 < n\nhterm : ∀ i ∈ Finset.range n, 1 ≤ x ^ i * y ^ (n - 1 - i)\n⊢ ↑n * (x - y) = (∑ _i ∈ Finset.range n, 1) * (x - y)",
"after_state": "x y : ℝ\nhx : 1 < x\nhy : 1 < y\nh : ∀ (n : ℕ), 0 < n → x ^ n - 1 < y ^ n\nhxy : y < x\nhxmy : 0 < x - y\nn : ℕ\na✝ : 0 < n\nhterm : ∀ i ∈ Finset.range n, 1 ≤ x ^ i * y ^ (n - 1 - i)\n⊢ ↑n * (x - y) = (∑ _i ∈ Finset.range n, 1) * (x - y)"
},
{
"line": "gcongr with i hi",
"before_state": "x y : ℝ\nhx : 1 < x\nhy : 1 < y\nh : ∀ (n : ℕ), 0 < n → x ^ n - 1 < y ^ n\nhxy : y < x\nhxmy : 0 < x - y\nn : ℕ\na✝ : 0 < n\nhterm : ∀ i ∈ Finset.range n, 1 ≤ x ^ i * y ^ (n - 1 - i)\n⊢ (∑ _i ∈ Finset.range n, 1) * (x - y) ≤ (∑ i ∈ Finset.range n, x ^ i * y ^ (n - 1 - i)) * (x - y)",
"after_state": "case h.h\nx y : ℝ\nhx : 1 < x\nhy : 1 < y\nh : ∀ (n : ℕ), 0 < n → x ^ n - 1 < y ^ n\nhxy : y < x\nhxmy : 0 < x - y\nn : ℕ\na✝ : 0 < n\nhterm : ∀ i ∈ Finset.range n, 1 ≤ x ^ i * y ^ (n - 1 - i)\ni : ℕ\nhi : i ∈ Finset.range n\n⊢ 1 ≤ x ^ i * y ^ (n - 1 - i)"
},
{
"line": "gcongr_discharger",
"before_state": "case a0\nx y : ℝ\nhx : 1 < x\nhy : 1 < y\nh : ∀ (n : ℕ), 0 < n → x ^ n - 1 < y ^ n\nhxy : y < x\nhxmy : 0 < x - y\nn : ℕ\na✝ : 0 < n\nhterm : ∀ i ∈ Finset.range n, 1 ≤ x ^ i * y ^ (n - 1 - i)\n⊢ 0 ≤ x - y",
"after_state": "No Goals!"
},
{
"line": "positivity",
"before_state": "case a0\nx y : ℝ\nhx : 1 < x\nhy : 1 < y\nh : ∀ (n : ℕ), 0 < n → x ^ n - 1 < y ^ n\nhxy : y < x\nhxmy : 0 < x - y\nn : ℕ\na✝ : 0 < n\nhterm : ∀ i ∈ Finset.range n, 1 ≤ x ^ i * y ^ (n - 1 - i)\n⊢ 0 ≤ x - y",
"after_state": "No Goals!"
},
{
"line": "apply hterm i hi",
"before_state": "case h.h\nx y : ℝ\nhx : 1 < x\nhy : 1 < y\nh : ∀ (n : ℕ), 0 < n → x ^ n - 1 < y ^ n\nhxy : y < x\nhxmy : 0 < x - y\nn : ℕ\na✝ : 0 < n\nhterm : ∀ i ∈ Finset.range n, 1 ≤ x ^ i * y ^ (n - 1 - i)\ni : ℕ\nhi : i ∈ Finset.range n\n⊢ 1 ≤ x ^ i * y ^ (n - 1 - i)",
"after_state": "No Goals!"
},
{
"line": "obtain ⟨N, hN⟩ := exists_nat_gt (1 / (x - y))",
"before_state": "x y : ℝ\nhx : 1 < x\nhy : 1 < y\nh : ∀ (n : ℕ), 0 < n → x ^ n - 1 < y ^ n\nhxy : y < x\nhxmy : 0 < x - y\nhn : ∀ (n : ℕ), 0 < n → (x - y) * ↑n ≤ x ^ n - y ^ n\n⊢ False",
"after_state": "case intro\nx y : ℝ\nhx : 1 < x\nhy : 1 < y\nh : ∀ (n : ℕ), 0 < n → x ^ n - 1 < y ^ n\nhxy : y < x\nhxmy : 0 < x - y\nhn : ∀ (n : ℕ), 0 < n → (x - y) * ↑n ≤ x ^ n - y ^ n\nN : ℕ\nhN : 1 / (x - y) < ↑N\n⊢ False"
},
{
"line": "have hNp : 0 < N := mod_cast (one_div_pos.mpr hxmy).trans hN",
"before_state": "case intro\nx y : ℝ\nhx : 1 < x\nhy : 1 < y\nh : ∀ (n : ℕ), 0 < n → x ^ n - 1 < y ^ n\nhxy : y < x\nhxmy : 0 < x - y\nhn : ∀ (n : ℕ), 0 < n → (x - y) * ↑n ≤ x ^ n - y ^ n\nN : ℕ\nhN : 1 / (x - y) < ↑N\n⊢ False",
"after_state": "case intro\nx y : ℝ\nhx : 1 < x\nhy : 1 < y\nh : ∀ (n : ℕ), 0 < n → x ^ n - 1 < y ^ n\nhxy : y < x\nhxmy : 0 < x - y\nhn : ∀ (n : ℕ), 0 < n → (x - y) * ↑n ≤ x ^ n - y ^ n\nN : ℕ\nhN : 1 / (x - y) < ↑N\nhNp : 0 < N\n⊢ False"
},
{
"line": "refine_lift\n have hNp : 0 < N := mod_cast (one_div_pos.mpr hxmy).trans hN;\n ?_",
"before_state": "case intro\nx y : ℝ\nhx : 1 < x\nhy : 1 < y\nh : ∀ (n : ℕ), 0 < n → x ^ n - 1 < y ^ n\nhxy : y < x\nhxmy : 0 < x - y\nhn : ∀ (n : ℕ), 0 < n → (x - y) * ↑n ≤ x ^ n - y ^ n\nN : ℕ\nhN : 1 / (x - y) < ↑N\n⊢ False",
"after_state": "case intro\nx y : ℝ\nhx : 1 < x\nhy : 1 < y\nh : ∀ (n : ℕ), 0 < n → x ^ n - 1 < y ^ n\nhxy : y < x\nhxmy : 0 < x - y\nhn : ∀ (n : ℕ), 0 < n → (x - y) * ↑n ≤ x ^ n - y ^ n\nN : ℕ\nhN : 1 / (x - y) < ↑N\nhNp : 0 < N\n⊢ False"
},
{
"line": "focus\n (refine\n no_implicit_lambda%\n (have hNp : 0 < N := mod_cast (one_div_pos.mpr hxmy).trans hN;\n ?_);\n rotate_right)",
"before_state": "case intro\nx y : ℝ\nhx : 1 < x\nhy : 1 < y\nh : ∀ (n : ℕ), 0 < n → x ^ n - 1 < y ^ n\nhxy : y < x\nhxmy : 0 < x - y\nhn : ∀ (n : ℕ), 0 < n → (x - y) * ↑n ≤ x ^ n - y ^ n\nN : ℕ\nhN : 1 / (x - y) < ↑N\n⊢ False",
"after_state": "case intro\nx y : ℝ\nhx : 1 < x\nhy : 1 < y\nh : ∀ (n : ℕ), 0 < n → x ^ n - 1 < y ^ n\nhxy : y < x\nhxmy : 0 < x - y\nhn : ∀ (n : ℕ), 0 < n → (x - y) * ↑n ≤ x ^ n - y ^ n\nN : ℕ\nhN : 1 / (x - y) < ↑N\nhNp : 0 < N\n⊢ False"
},
{
"line": "refine\n no_implicit_lambda%\n (have hNp : 0 < N := mod_cast (one_div_pos.mpr hxmy).trans hN;\n ?_)",
"before_state": "case intro\nx y : ℝ\nhx : 1 < x\nhy : 1 < y\nh : ∀ (n : ℕ), 0 < n → x ^ n - 1 < y ^ n\nhxy : y < x\nhxmy : 0 < x - y\nhn : ∀ (n : ℕ), 0 < n → (x - y) * ↑n ≤ x ^ n - y ^ n\nN : ℕ\nhN : 1 / (x - y) < ↑N\n⊢ False",
"after_state": "case intro\nx y : ℝ\nhx : 1 < x\nhy : 1 < y\nh : ∀ (n : ℕ), 0 < n → x ^ n - 1 < y ^ n\nhxy : y < x\nhxmy : 0 < x - y\nhn : ∀ (n : ℕ), 0 < n → (x - y) * ↑n ≤ x ^ n - y ^ n\nN : ℕ\nhN : 1 / (x - y) < ↑N\nhNp : 0 < N\n⊢ False"
},
{
"line": "rotate_right",
"before_state": "case intro\nx y : ℝ\nhx : 1 < x\nhy : 1 < y\nh : ∀ (n : ℕ), 0 < n → x ^ n - 1 < y ^ n\nhxy : y < x\nhxmy : 0 < x - y\nhn : ∀ (n : ℕ), 0 < n → (x - y) * ↑n ≤ x ^ n - y ^ n\nN : ℕ\nhN : 1 / (x - y) < ↑N\nhNp : 0 < N\n⊢ False",
"after_state": "case intro\nx y : ℝ\nhx : 1 < x\nhy : 1 < y\nh : ∀ (n : ℕ), 0 < n → x ^ n - 1 < y ^ n\nhxy : y < x\nhxmy : 0 < x - y\nhn : ∀ (n : ℕ), 0 < n → (x - y) * ↑n ≤ x ^ n - y ^ n\nN : ℕ\nhN : 1 / (x - y) < ↑N\nhNp : 0 < N\n⊢ False"
},
{
"line": "have :=\n calc\n 1 = (x - y) * (1 / (x - y)) := by field_simp\n _ < (x - y) * N := by gcongr\n _ ≤ x ^ N - y ^ N := hn N hNp",
"before_state": "case intro\nx y : ℝ\nhx : 1 < x\nhy : 1 < y\nh : ∀ (n : ℕ), 0 < n → x ^ n - 1 < y ^ n\nhxy : y < x\nhxmy : 0 < x - y\nhn : ∀ (n : ℕ), 0 < n → (x - y) * ↑n ≤ x ^ n - y ^ n\nN : ℕ\nhN : 1 / (x - y) < ↑N\nhNp : 0 < N\n⊢ False",
"after_state": "case intro\nx y : ℝ\nhx : 1 < x\nhy : 1 < y\nh : ∀ (n : ℕ), 0 < n → x ^ n - 1 < y ^ n\nhxy : y < x\nhxmy : 0 < x - y\nhn : ∀ (n : ℕ), 0 < n → (x - y) * ↑n ≤ x ^ n - y ^ n\nN : ℕ\nhN : 1 / (x - y) < ↑N\nhNp : 0 < N\nthis : 1 < x ^ N - y ^ N\n⊢ False"
},
{
"line": "refine_lift\n have :=\n calc\n 1 = (x - y) * (1 / (x - y)) := by field_simp\n _ < (x - y) * N := by gcongr\n _ ≤ x ^ N - y ^ N := hn N hNp;\n ?_",
"before_state": "case intro\nx y : ℝ\nhx : 1 < x\nhy : 1 < y\nh : ∀ (n : ℕ), 0 < n → x ^ n - 1 < y ^ n\nhxy : y < x\nhxmy : 0 < x - y\nhn : ∀ (n : ℕ), 0 < n → (x - y) * ↑n ≤ x ^ n - y ^ n\nN : ℕ\nhN : 1 / (x - y) < ↑N\nhNp : 0 < N\n⊢ False",
"after_state": "case intro\nx y : ℝ\nhx : 1 < x\nhy : 1 < y\nh : ∀ (n : ℕ), 0 < n → x ^ n - 1 < y ^ n\nhxy : y < x\nhxmy : 0 < x - y\nhn : ∀ (n : ℕ), 0 < n → (x - y) * ↑n ≤ x ^ n - y ^ n\nN : ℕ\nhN : 1 / (x - y) < ↑N\nhNp : 0 < N\nthis : 1 < x ^ N - y ^ N\n⊢ False"
},
{
"line": "focus\n (refine\n no_implicit_lambda%\n (have :=\n calc\n 1 = (x - y) * (1 / (x - y)) := by field_simp\n _ < (x - y) * N := by gcongr\n _ ≤ x ^ N - y ^ N := hn N hNp;\n ?_);\n rotate_right)",
"before_state": "case intro\nx y : ℝ\nhx : 1 < x\nhy : 1 < y\nh : ∀ (n : ℕ), 0 < n → x ^ n - 1 < y ^ n\nhxy : y < x\nhxmy : 0 < x - y\nhn : ∀ (n : ℕ), 0 < n → (x - y) * ↑n ≤ x ^ n - y ^ n\nN : ℕ\nhN : 1 / (x - y) < ↑N\nhNp : 0 < N\n⊢ False",
"after_state": "case intro\nx y : ℝ\nhx : 1 < x\nhy : 1 < y\nh : ∀ (n : ℕ), 0 < n → x ^ n - 1 < y ^ n\nhxy : y < x\nhxmy : 0 < x - y\nhn : ∀ (n : ℕ), 0 < n → (x - y) * ↑n ≤ x ^ n - y ^ n\nN : ℕ\nhN : 1 / (x - y) < ↑N\nhNp : 0 < N\nthis : 1 < x ^ N - y ^ N\n⊢ False"
},
{
"line": "refine\n no_implicit_lambda%\n (have :=\n calc\n 1 = (x - y) * (1 / (x - y)) := by field_simp\n _ < (x - y) * N := by gcongr\n _ ≤ x ^ N - y ^ N := hn N hNp;\n ?_)",
"before_state": "case intro\nx y : ℝ\nhx : 1 < x\nhy : 1 < y\nh : ∀ (n : ℕ), 0 < n → x ^ n - 1 < y ^ n\nhxy : y < x\nhxmy : 0 < x - y\nhn : ∀ (n : ℕ), 0 < n → (x - y) * ↑n ≤ x ^ n - y ^ n\nN : ℕ\nhN : 1 / (x - y) < ↑N\nhNp : 0 < N\n⊢ False",
"after_state": "case intro\nx y : ℝ\nhx : 1 < x\nhy : 1 < y\nh : ∀ (n : ℕ), 0 < n → x ^ n - 1 < y ^ n\nhxy : y < x\nhxmy : 0 < x - y\nhn : ∀ (n : ℕ), 0 < n → (x - y) * ↑n ≤ x ^ n - y ^ n\nN : ℕ\nhN : 1 / (x - y) < ↑N\nhNp : 0 < N\nthis : 1 < x ^ N - y ^ N\n⊢ False"
},
{
"line": "field_simp",
"before_state": "x y : ℝ\nhx : 1 < x\nhy : 1 < y\nh : ∀ (n : ℕ), 0 < n → x ^ n - 1 < y ^ n\nhxy : y < x\nhxmy : 0 < x - y\nhn : ∀ (n : ℕ), 0 < n → (x - y) * ↑n ≤ x ^ n - y ^ n\nN : ℕ\nhN : 1 / (x - y) < ↑N\nhNp : 0 < N\n⊢ 1 = (x - y) * (1 / (x - y))",
"after_state": "No Goals!"
},
{
"line": "gcongr",
"before_state": "x y : ℝ\nhx : 1 < x\nhy : 1 < y\nh : ∀ (n : ℕ), 0 < n → x ^ n - 1 < y ^ n\nhxy : y < x\nhxmy : 0 < x - y\nhn : ∀ (n : ℕ), 0 < n → (x - y) * ↑n ≤ x ^ n - y ^ n\nN : ℕ\nhN : 1 / (x - y) < ↑N\nhNp : 0 < N\n⊢ (x - y) * (1 / (x - y)) < (x - y) * ↑N",
"after_state": "No Goals!"
},
{
"line": "gcongr_discharger",
"before_state": "case a0\nx y : ℝ\nhx : 1 < x\nhy : 1 < y\nh : ∀ (n : ℕ), 0 < n → x ^ n - 1 < y ^ n\nhxy : y < x\nhxmy : 0 < x - y\nhn : ∀ (n : ℕ), 0 < n → (x - y) * ↑n ≤ x ^ n - y ^ n\nN : ℕ\nhN : 1 / (x - y) < ↑N\nhNp : 0 < N\n⊢ 0 < x - y",
"after_state": "No Goals!"
},
{
"line": "positivity",
"before_state": "case a0\nx y : ℝ\nhx : 1 < x\nhy : 1 < y\nh : ∀ (n : ℕ), 0 < n → x ^ n - 1 < y ^ n\nhxy : y < x\nhxmy : 0 < x - y\nhn : ∀ (n : ℕ), 0 < n → (x - y) * ↑n ≤ x ^ n - y ^ n\nN : ℕ\nhN : 1 / (x - y) < ↑N\nhNp : 0 < N\n⊢ 0 < x - y",
"after_state": "No Goals!"
},
{
"line": "rotate_right",
"before_state": "case intro\nx y : ℝ\nhx : 1 < x\nhy : 1 < y\nh : ∀ (n : ℕ), 0 < n → x ^ n - 1 < y ^ n\nhxy : y < x\nhxmy : 0 < x - y\nhn : ∀ (n : ℕ), 0 < n → (x - y) * ↑n ≤ x ^ n - y ^ n\nN : ℕ\nhN : 1 / (x - y) < ↑N\nhNp : 0 < N\nthis : 1 < x ^ N - y ^ N\n⊢ False",
"after_state": "case intro\nx y : ℝ\nhx : 1 < x\nhy : 1 < y\nh : ∀ (n : ℕ), 0 < n → x ^ n - 1 < y ^ n\nhxy : y < x\nhxmy : 0 < x - y\nhn : ∀ (n : ℕ), 0 < n → (x - y) * ↑n ≤ x ^ n - y ^ n\nN : ℕ\nhN : 1 / (x - y) < ↑N\nhNp : 0 < N\nthis : 1 < x ^ N - y ^ N\n⊢ False"
},
{
"line": "linarith [h N hNp]",
"before_state": "case intro\nx y : ℝ\nhx : 1 < x\nhy : 1 < y\nh : ∀ (n : ℕ), 0 < n → x ^ n - 1 < y ^ n\nhxy : y < x\nhxmy : 0 < x - y\nhn : ∀ (n : ℕ), 0 < n → (x - y) * ↑n ≤ x ^ n - y ^ n\nN : ℕ\nhN : 1 / (x - y) < ↑N\nhNp : 0 < N\nthis : 1 < x ^ N - y ^ N\n⊢ False",
"after_state": "No Goals!"
},
{
"line": "ring1",
"before_state": "x y : ℝ\nhx : 1 < x\nhy : 1 < y\nh : ∀ (n : ℕ), 0 < n → x ^ n - 1 < y ^ n\nhxy : y < x\nhxmy : 0 < x - y\nhn : ∀ (n : ℕ), 0 < n → (x - y) * ↑n ≤ x ^ n - y ^ n\nN : ℕ\nhN : 1 / (x - y) < ↑N\nhNp : 0 < N\nthis : 1 < x ^ N - y ^ N\n⊢ 1 - (x ^ N - y ^ N) + (x ^ N - 1 - y ^ N) = 0",
"after_state": "No Goals!"
}
] |
theorem le_of_all_pow_lt_succ' {x y : ℝ} (hx : 1 < x) (hy : 0 < y)
(h : ∀ n : ℕ, 0 < n → x ^ n - 1 < y ^ n) : x ≤ y := by
refine le_of_all_pow_lt_succ hx ?_ h
by_contra! hy'' : y ≤ 1
-- Then there exists y' such that 0 < y ≤ 1 < y' < x.
have h_y'_lt_x : (x + 1) / 2 < x := by linarith
have h1_lt_y' : 1 < (x + 1) / 2 := by linarith
set y' := (x + 1) / 2
have h_y_lt_y' : y < y' := by linarith
have hh : ∀ n, 0 < n → x ^ n - 1 < y' ^ n := by
intro n hn
calc
x ^ n - 1 < y ^ n := h n hn
_ ≤ y' ^ n := by gcongr
exact h_y'_lt_x.not_le (le_of_all_pow_lt_succ hx h1_lt_y' hh)
|
/root/DuelModelResearch/mathlib4/Archive/Imo/Imo2013Q5.lean
|
{
"open": [],
"variables": []
}
|
[
{
"line": "refine le_of_all_pow_lt_succ hx ?_ h",
"before_state": "x y : ℝ\nhx : 1 < x\nhy : 0 < y\nh : ∀ (n : ℕ), 0 < n → x ^ n - 1 < y ^ n\n⊢ x ≤ y",
"after_state": "No Goals!"
}
] |
theorem f_pos_of_pos {f : ℚ → ℝ} {q : ℚ} (hq : 0 < q)
(H1 : ∀ x y, 0 < x → 0 < y → f (x * y) ≤ f x * f y) (H4 : ∀ n : ℕ, 0 < n → (n : ℝ) ≤ f n) :
0 < f q := by
have num_pos : 0 < q.num := Rat.num_pos.mpr hq
have hmul_pos :=
calc
(0 : ℝ) < q.num := Int.cast_pos.mpr num_pos
_ = ((q.num.natAbs : ℤ) : ℝ) := congr_arg Int.cast (Int.natAbs_of_nonneg num_pos.le).symm
_ ≤ f q.num.natAbs := (H4 q.num.natAbs ((@Int.natAbs_pos q.num).mpr num_pos.ne.symm))
_ = f q.num := by rw [Nat.cast_natAbs, abs_of_nonneg num_pos.le]
_ = f (q * q.den) := by rw [← Rat.mul_den_eq_num]
_ ≤ f q * f q.den := H1 q q.den hq (Nat.cast_pos.mpr q.pos)
have h_f_denom_pos :=
calc
(0 : ℝ) < q.den := Nat.cast_pos.mpr q.pos
_ ≤ f q.den := H4 q.den q.pos
exact pos_of_mul_pos_left hmul_pos h_f_denom_pos.le
|
/root/DuelModelResearch/mathlib4/Archive/Imo/Imo2013Q5.lean
|
{
"open": [],
"variables": []
}
|
[
{
"line": "have num_pos : 0 < q.num := Rat.num_pos.mpr hq",
"before_state": "f : ℚ → ℝ\nq : ℚ\nhq : 0 < q\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\n⊢ 0 < f q",
"after_state": "f : ℚ → ℝ\nq : ℚ\nhq : 0 < q\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nnum_pos : 0 < q.num\n⊢ 0 < f q"
},
{
"line": "refine_lift\n have num_pos : 0 < q.num := Rat.num_pos.mpr hq;\n ?_",
"before_state": "f : ℚ → ℝ\nq : ℚ\nhq : 0 < q\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\n⊢ 0 < f q",
"after_state": "f : ℚ → ℝ\nq : ℚ\nhq : 0 < q\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nnum_pos : 0 < q.num\n⊢ 0 < f q"
},
{
"line": "focus\n (refine\n no_implicit_lambda%\n (have num_pos : 0 < q.num := Rat.num_pos.mpr hq;\n ?_);\n rotate_right)",
"before_state": "f : ℚ → ℝ\nq : ℚ\nhq : 0 < q\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\n⊢ 0 < f q",
"after_state": "f : ℚ → ℝ\nq : ℚ\nhq : 0 < q\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nnum_pos : 0 < q.num\n⊢ 0 < f q"
},
{
"line": "refine\n no_implicit_lambda%\n (have num_pos : 0 < q.num := Rat.num_pos.mpr hq;\n ?_)",
"before_state": "f : ℚ → ℝ\nq : ℚ\nhq : 0 < q\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\n⊢ 0 < f q",
"after_state": "f : ℚ → ℝ\nq : ℚ\nhq : 0 < q\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nnum_pos : 0 < q.num\n⊢ 0 < f q"
},
{
"line": "rotate_right",
"before_state": "f : ℚ → ℝ\nq : ℚ\nhq : 0 < q\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nnum_pos : 0 < q.num\n⊢ 0 < f q",
"after_state": "f : ℚ → ℝ\nq : ℚ\nhq : 0 < q\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nnum_pos : 0 < q.num\n⊢ 0 < f q"
},
{
"line": "have hmul_pos :=\n calc\n (0 : ℝ) < q.num := Int.cast_pos.mpr num_pos\n _ = ((q.num.natAbs : ℤ) : ℝ) := (congr_arg Int.cast (Int.natAbs_of_nonneg num_pos.le).symm)\n _ ≤ f q.num.natAbs := (H4 q.num.natAbs ((@Int.natAbs_pos q.num).mpr num_pos.ne.symm))\n _ = f q.num := by rw [Nat.cast_natAbs, abs_of_nonneg num_pos.le]\n _ = f (q * q.den) := by rw [← Rat.mul_den_eq_num]\n _ ≤ f q * f q.den := H1 q q.den hq (Nat.cast_pos.mpr q.pos)",
"before_state": "f : ℚ → ℝ\nq : ℚ\nhq : 0 < q\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nnum_pos : 0 < q.num\n⊢ 0 < f q",
"after_state": "f : ℚ → ℝ\nq : ℚ\nhq : 0 < q\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nnum_pos : 0 < q.num\nhmul_pos : 0 < f q * f ↑q.den\n⊢ 0 < f q"
},
{
"line": "refine_lift\n have hmul_pos :=\n calc\n (0 : ℝ) < q.num := Int.cast_pos.mpr num_pos\n _ = ((q.num.natAbs : ℤ) : ℝ) := (congr_arg Int.cast (Int.natAbs_of_nonneg num_pos.le).symm)\n _ ≤ f q.num.natAbs := (H4 q.num.natAbs ((@Int.natAbs_pos q.num).mpr num_pos.ne.symm))\n _ = f q.num := by rw [Nat.cast_natAbs, abs_of_nonneg num_pos.le]\n _ = f (q * q.den) := by rw [← Rat.mul_den_eq_num]\n _ ≤ f q * f q.den := H1 q q.den hq (Nat.cast_pos.mpr q.pos);\n ?_",
"before_state": "f : ℚ → ℝ\nq : ℚ\nhq : 0 < q\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nnum_pos : 0 < q.num\n⊢ 0 < f q",
"after_state": "f : ℚ → ℝ\nq : ℚ\nhq : 0 < q\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nnum_pos : 0 < q.num\nhmul_pos : 0 < f q * f ↑q.den\n⊢ 0 < f q"
},
{
"line": "focus\n (refine\n no_implicit_lambda%\n (have hmul_pos :=\n calc\n (0 : ℝ) < q.num := Int.cast_pos.mpr num_pos\n _ = ((q.num.natAbs : ℤ) : ℝ) := (congr_arg Int.cast (Int.natAbs_of_nonneg num_pos.le).symm)\n _ ≤ f q.num.natAbs := (H4 q.num.natAbs ((@Int.natAbs_pos q.num).mpr num_pos.ne.symm))\n _ = f q.num := by rw [Nat.cast_natAbs, abs_of_nonneg num_pos.le]\n _ = f (q * q.den) := by rw [← Rat.mul_den_eq_num]\n _ ≤ f q * f q.den := H1 q q.den hq (Nat.cast_pos.mpr q.pos);\n ?_);\n rotate_right)",
"before_state": "f : ℚ → ℝ\nq : ℚ\nhq : 0 < q\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nnum_pos : 0 < q.num\n⊢ 0 < f q",
"after_state": "f : ℚ → ℝ\nq : ℚ\nhq : 0 < q\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nnum_pos : 0 < q.num\nhmul_pos : 0 < f q * f ↑q.den\n⊢ 0 < f q"
},
{
"line": "refine\n no_implicit_lambda%\n (have hmul_pos :=\n calc\n (0 : ℝ) < q.num := Int.cast_pos.mpr num_pos\n _ = ((q.num.natAbs : ℤ) : ℝ) := (congr_arg Int.cast (Int.natAbs_of_nonneg num_pos.le).symm)\n _ ≤ f q.num.natAbs := (H4 q.num.natAbs ((@Int.natAbs_pos q.num).mpr num_pos.ne.symm))\n _ = f q.num := by rw [Nat.cast_natAbs, abs_of_nonneg num_pos.le]\n _ = f (q * q.den) := by rw [← Rat.mul_den_eq_num]\n _ ≤ f q * f q.den := H1 q q.den hq (Nat.cast_pos.mpr q.pos);\n ?_)",
"before_state": "f : ℚ → ℝ\nq : ℚ\nhq : 0 < q\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nnum_pos : 0 < q.num\n⊢ 0 < f q",
"after_state": "f : ℚ → ℝ\nq : ℚ\nhq : 0 < q\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nnum_pos : 0 < q.num\nhmul_pos : 0 < f q * f ↑q.den\n⊢ 0 < f q"
},
{
"line": "rw [Nat.cast_natAbs, abs_of_nonneg num_pos.le]",
"before_state": "f : ℚ → ℝ\nq : ℚ\nhq : 0 < q\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nnum_pos : 0 < q.num\n⊢ f ↑q.num.natAbs = f ↑q.num",
"after_state": "No Goals!"
},
{
"line": "rewrite [Nat.cast_natAbs, abs_of_nonneg num_pos.le]",
"before_state": "f : ℚ → ℝ\nq : ℚ\nhq : 0 < q\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nnum_pos : 0 < q.num\n⊢ f ↑q.num.natAbs = f ↑q.num",
"after_state": "f : ℚ → ℝ\nq : ℚ\nhq : 0 < q\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nnum_pos : 0 < q.num\n⊢ f ↑q.num = f ↑q.num"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "f : ℚ → ℝ\nq : ℚ\nhq : 0 < q\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nnum_pos : 0 < q.num\n⊢ f ↑q.num = f ↑q.num",
"after_state": "No Goals!"
},
{
"line": "try (with_reducible rfl)",
"before_state": "f : ℚ → ℝ\nq : ℚ\nhq : 0 < q\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nnum_pos : 0 < q.num\n⊢ f ↑q.num = f ↑q.num",
"after_state": "No Goals!"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "f : ℚ → ℝ\nq : ℚ\nhq : 0 < q\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nnum_pos : 0 < q.num\n⊢ f ↑q.num = f ↑q.num",
"after_state": "No Goals!"
},
{
"line": "with_reducible rfl",
"before_state": "f : ℚ → ℝ\nq : ℚ\nhq : 0 < q\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nnum_pos : 0 < q.num\n⊢ f ↑q.num = f ↑q.num",
"after_state": "No Goals!"
},
{
"line": "rfl",
"before_state": "f : ℚ → ℝ\nq : ℚ\nhq : 0 < q\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nnum_pos : 0 < q.num\n⊢ f ↑q.num = f ↑q.num",
"after_state": "No Goals!"
},
{
"line": "eq_refl",
"before_state": "f : ℚ → ℝ\nq : ℚ\nhq : 0 < q\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nnum_pos : 0 < q.num\n⊢ f ↑q.num = f ↑q.num",
"after_state": "No Goals!"
},
{
"line": "rw [← Rat.mul_den_eq_num]",
"before_state": "f : ℚ → ℝ\nq : ℚ\nhq : 0 < q\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nnum_pos : 0 < q.num\n⊢ f ↑q.num = f (q * ↑q.den)",
"after_state": "No Goals!"
},
{
"line": "rewrite [← Rat.mul_den_eq_num]",
"before_state": "f : ℚ → ℝ\nq : ℚ\nhq : 0 < q\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nnum_pos : 0 < q.num\n⊢ f ↑q.num = f (q * ↑q.den)",
"after_state": "f : ℚ → ℝ\nq : ℚ\nhq : 0 < q\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nnum_pos : 0 < q.num\n⊢ f (q * ↑q.den) = f (q * ↑q.den)"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "f : ℚ → ℝ\nq : ℚ\nhq : 0 < q\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nnum_pos : 0 < q.num\n⊢ f (q * ↑q.den) = f (q * ↑q.den)",
"after_state": "No Goals!"
},
{
"line": "try (with_reducible rfl)",
"before_state": "f : ℚ → ℝ\nq : ℚ\nhq : 0 < q\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nnum_pos : 0 < q.num\n⊢ f (q * ↑q.den) = f (q * ↑q.den)",
"after_state": "No Goals!"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "f : ℚ → ℝ\nq : ℚ\nhq : 0 < q\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nnum_pos : 0 < q.num\n⊢ f (q * ↑q.den) = f (q * ↑q.den)",
"after_state": "No Goals!"
},
{
"line": "with_reducible rfl",
"before_state": "f : ℚ → ℝ\nq : ℚ\nhq : 0 < q\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nnum_pos : 0 < q.num\n⊢ f (q * ↑q.den) = f (q * ↑q.den)",
"after_state": "No Goals!"
},
{
"line": "rfl",
"before_state": "f : ℚ → ℝ\nq : ℚ\nhq : 0 < q\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nnum_pos : 0 < q.num\n⊢ f (q * ↑q.den) = f (q * ↑q.den)",
"after_state": "No Goals!"
},
{
"line": "eq_refl",
"before_state": "f : ℚ → ℝ\nq : ℚ\nhq : 0 < q\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nnum_pos : 0 < q.num\n⊢ f (q * ↑q.den) = f (q * ↑q.den)",
"after_state": "No Goals!"
},
{
"line": "rotate_right",
"before_state": "f : ℚ → ℝ\nq : ℚ\nhq : 0 < q\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nnum_pos : 0 < q.num\nhmul_pos : 0 < f q * f ↑q.den\n⊢ 0 < f q",
"after_state": "f : ℚ → ℝ\nq : ℚ\nhq : 0 < q\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nnum_pos : 0 < q.num\nhmul_pos : 0 < f q * f ↑q.den\n⊢ 0 < f q"
},
{
"line": "have h_f_denom_pos :=\n calc\n (0 : ℝ) < q.den := Nat.cast_pos.mpr q.pos\n _ ≤ f q.den := H4 q.den q.pos",
"before_state": "f : ℚ → ℝ\nq : ℚ\nhq : 0 < q\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nnum_pos : 0 < q.num\nhmul_pos : 0 < f q * f ↑q.den\n⊢ 0 < f q",
"after_state": "f : ℚ → ℝ\nq : ℚ\nhq : 0 < q\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nnum_pos : 0 < q.num\nhmul_pos : 0 < f q * f ↑q.den\nh_f_denom_pos : 0 < f ↑q.den\n⊢ 0 < f q"
},
{
"line": "refine_lift\n have h_f_denom_pos :=\n calc\n (0 : ℝ) < q.den := Nat.cast_pos.mpr q.pos\n _ ≤ f q.den := H4 q.den q.pos;\n ?_",
"before_state": "f : ℚ → ℝ\nq : ℚ\nhq : 0 < q\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nnum_pos : 0 < q.num\nhmul_pos : 0 < f q * f ↑q.den\n⊢ 0 < f q",
"after_state": "f : ℚ → ℝ\nq : ℚ\nhq : 0 < q\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nnum_pos : 0 < q.num\nhmul_pos : 0 < f q * f ↑q.den\nh_f_denom_pos : 0 < f ↑q.den\n⊢ 0 < f q"
},
{
"line": "focus\n (refine\n no_implicit_lambda%\n (have h_f_denom_pos :=\n calc\n (0 : ℝ) < q.den := Nat.cast_pos.mpr q.pos\n _ ≤ f q.den := H4 q.den q.pos;\n ?_);\n rotate_right)",
"before_state": "f : ℚ → ℝ\nq : ℚ\nhq : 0 < q\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nnum_pos : 0 < q.num\nhmul_pos : 0 < f q * f ↑q.den\n⊢ 0 < f q",
"after_state": "f : ℚ → ℝ\nq : ℚ\nhq : 0 < q\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nnum_pos : 0 < q.num\nhmul_pos : 0 < f q * f ↑q.den\nh_f_denom_pos : 0 < f ↑q.den\n⊢ 0 < f q"
},
{
"line": "refine\n no_implicit_lambda%\n (have h_f_denom_pos :=\n calc\n (0 : ℝ) < q.den := Nat.cast_pos.mpr q.pos\n _ ≤ f q.den := H4 q.den q.pos;\n ?_)",
"before_state": "f : ℚ → ℝ\nq : ℚ\nhq : 0 < q\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nnum_pos : 0 < q.num\nhmul_pos : 0 < f q * f ↑q.den\n⊢ 0 < f q",
"after_state": "f : ℚ → ℝ\nq : ℚ\nhq : 0 < q\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nnum_pos : 0 < q.num\nhmul_pos : 0 < f q * f ↑q.den\nh_f_denom_pos : 0 < f ↑q.den\n⊢ 0 < f q"
},
{
"line": "rotate_right",
"before_state": "f : ℚ → ℝ\nq : ℚ\nhq : 0 < q\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nnum_pos : 0 < q.num\nhmul_pos : 0 < f q * f ↑q.den\nh_f_denom_pos : 0 < f ↑q.den\n⊢ 0 < f q",
"after_state": "f : ℚ → ℝ\nq : ℚ\nhq : 0 < q\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nnum_pos : 0 < q.num\nhmul_pos : 0 < f q * f ↑q.den\nh_f_denom_pos : 0 < f ↑q.den\n⊢ 0 < f q"
},
{
"line": "exact pos_of_mul_pos_left hmul_pos h_f_denom_pos.le",
"before_state": "f : ℚ → ℝ\nq : ℚ\nhq : 0 < q\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nnum_pos : 0 < q.num\nhmul_pos : 0 < f q * f ↑q.den\nh_f_denom_pos : 0 < f ↑q.den\n⊢ 0 < f q",
"after_state": "No Goals!"
}
] |
theorem fx_gt_xm1 {f : ℚ → ℝ} {x : ℚ} (hx : 1 ≤ x)
(H1 : ∀ x y, 0 < x → 0 < y → f (x * y) ≤ f x * f y)
(H2 : ∀ x y, 0 < x → 0 < y → f x + f y ≤ f (x + y)) (H4 : ∀ n : ℕ, 0 < n → (n : ℝ) ≤ f n) :
(x - 1 : ℝ) < f x := by
have hx0 :=
calc
(x - 1 : ℝ) < ⌊x⌋₊ := mod_cast Nat.sub_one_lt_floor x
_ ≤ f ⌊x⌋₊ := H4 _ (Nat.floor_pos.2 hx)
obtain h_eq | h_lt := (Nat.floor_le <| zero_le_one.trans hx).eq_or_lt
· rwa [h_eq] at hx0
calc
(x - 1 : ℝ) < f ⌊x⌋₊ := hx0
_ < f (x - ⌊x⌋₊) + f ⌊x⌋₊ := (lt_add_of_pos_left _ (f_pos_of_pos (sub_pos.mpr h_lt) H1 H4))
_ ≤ f (x - ⌊x⌋₊ + ⌊x⌋₊) := (H2 _ _ (sub_pos.mpr h_lt) (Nat.cast_pos.2 (Nat.floor_pos.2 hx)))
_ = f x := by ring_nf
|
/root/DuelModelResearch/mathlib4/Archive/Imo/Imo2013Q5.lean
|
{
"open": [],
"variables": []
}
|
[
{
"line": "have hx0 :=\n calc\n (x - 1 : ℝ) < ⌊x⌋₊ := mod_cast Nat.sub_one_lt_floor x\n _ ≤ f ⌊x⌋₊ := H4 _ (Nat.floor_pos.2 hx)",
"before_state": "f : ℚ → ℝ\nx : ℚ\nhx : 1 ≤ x\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\n⊢ ↑x - 1 < f x",
"after_state": "f : ℚ → ℝ\nx : ℚ\nhx : 1 ≤ x\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nhx0 : ↑x - 1 < f ↑⌊x⌋₊\n⊢ ↑x - 1 < f x"
},
{
"line": "refine_lift\n have hx0 :=\n calc\n (x - 1 : ℝ) < ⌊x⌋₊ := mod_cast Nat.sub_one_lt_floor x\n _ ≤ f ⌊x⌋₊ := H4 _ (Nat.floor_pos.2 hx);\n ?_",
"before_state": "f : ℚ → ℝ\nx : ℚ\nhx : 1 ≤ x\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\n⊢ ↑x - 1 < f x",
"after_state": "f : ℚ → ℝ\nx : ℚ\nhx : 1 ≤ x\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nhx0 : ↑x - 1 < f ↑⌊x⌋₊\n⊢ ↑x - 1 < f x"
},
{
"line": "focus\n (refine\n no_implicit_lambda%\n (have hx0 :=\n calc\n (x - 1 : ℝ) < ⌊x⌋₊ := mod_cast Nat.sub_one_lt_floor x\n _ ≤ f ⌊x⌋₊ := H4 _ (Nat.floor_pos.2 hx);\n ?_);\n rotate_right)",
"before_state": "f : ℚ → ℝ\nx : ℚ\nhx : 1 ≤ x\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\n⊢ ↑x - 1 < f x",
"after_state": "f : ℚ → ℝ\nx : ℚ\nhx : 1 ≤ x\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nhx0 : ↑x - 1 < f ↑⌊x⌋₊\n⊢ ↑x - 1 < f x"
},
{
"line": "refine\n no_implicit_lambda%\n (have hx0 :=\n calc\n (x - 1 : ℝ) < ⌊x⌋₊ := mod_cast Nat.sub_one_lt_floor x\n _ ≤ f ⌊x⌋₊ := H4 _ (Nat.floor_pos.2 hx);\n ?_)",
"before_state": "f : ℚ → ℝ\nx : ℚ\nhx : 1 ≤ x\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\n⊢ ↑x - 1 < f x",
"after_state": "f : ℚ → ℝ\nx : ℚ\nhx : 1 ≤ x\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nhx0 : ↑x - 1 < f ↑⌊x⌋₊\n⊢ ↑x - 1 < f x"
},
{
"line": "rotate_right",
"before_state": "f : ℚ → ℝ\nx : ℚ\nhx : 1 ≤ x\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nhx0 : ↑x - 1 < f ↑⌊x⌋₊\n⊢ ↑x - 1 < f x",
"after_state": "f : ℚ → ℝ\nx : ℚ\nhx : 1 ≤ x\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nhx0 : ↑x - 1 < f ↑⌊x⌋₊\n⊢ ↑x - 1 < f x"
},
{
"line": "obtain h_eq | h_lt := (Nat.floor_le <| zero_le_one.trans hx).eq_or_lt",
"before_state": "f : ℚ → ℝ\nx : ℚ\nhx : 1 ≤ x\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nhx0 : ↑x - 1 < f ↑⌊x⌋₊\n⊢ ↑x - 1 < f x",
"after_state": "case inl\nf : ℚ → ℝ\nx : ℚ\nhx : 1 ≤ x\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nhx0 : ↑x - 1 < f ↑⌊x⌋₊\nh_eq : ↑⌊x⌋₊ = x\n⊢ ↑x - 1 < f x\n---\ncase inr\nf : ℚ → ℝ\nx : ℚ\nhx : 1 ≤ x\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nhx0 : ↑x - 1 < f ↑⌊x⌋₊\nh_lt : ↑⌊x⌋₊ < x\n⊢ ↑x - 1 < f x"
},
{
"line": "rwa [h_eq] at hx0",
"before_state": "case inl\nf : ℚ → ℝ\nx : ℚ\nhx : 1 ≤ x\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nhx0 : ↑x - 1 < f ↑⌊x⌋₊\nh_eq : ↑⌊x⌋₊ = x\n⊢ ↑x - 1 < f x",
"after_state": "No Goals!"
},
{
"line": "rw [h_eq] at hx0",
"before_state": "case inl\nf : ℚ → ℝ\nx : ℚ\nhx : 1 ≤ x\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nhx0 : ↑x - 1 < f ↑⌊x⌋₊\nh_eq : ↑⌊x⌋₊ = x\n⊢ ↑x - 1 < f x",
"after_state": "case inl\nf : ℚ → ℝ\nx : ℚ\nhx : 1 ≤ x\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nhx0 : ↑x - 1 < f x\nh_eq : ↑⌊x⌋₊ = x\n⊢ ↑x - 1 < f x"
},
{
"line": "rewrite [h_eq] at hx0",
"before_state": "case inl\nf : ℚ → ℝ\nx : ℚ\nhx : 1 ≤ x\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nhx0 : ↑x - 1 < f ↑⌊x⌋₊\nh_eq : ↑⌊x⌋₊ = x\n⊢ ↑x - 1 < f x",
"after_state": "case inl\nf : ℚ → ℝ\nx : ℚ\nhx : 1 ≤ x\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nhx0 : ↑x - 1 < f x\nh_eq : ↑⌊x⌋₊ = x\n⊢ ↑x - 1 < f x"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "case inl\nf : ℚ → ℝ\nx : ℚ\nhx : 1 ≤ x\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nhx0 : ↑x - 1 < f x\nh_eq : ↑⌊x⌋₊ = x\n⊢ ↑x - 1 < f x",
"after_state": "case inl\nf : ℚ → ℝ\nx : ℚ\nhx : 1 ≤ x\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nhx0 : ↑x - 1 < f x\nh_eq : ↑⌊x⌋₊ = x\n⊢ ↑x - 1 < f x"
},
{
"line": "try (with_reducible rfl)",
"before_state": "case inl\nf : ℚ → ℝ\nx : ℚ\nhx : 1 ≤ x\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nhx0 : ↑x - 1 < f x\nh_eq : ↑⌊x⌋₊ = x\n⊢ ↑x - 1 < f x",
"after_state": "case inl\nf : ℚ → ℝ\nx : ℚ\nhx : 1 ≤ x\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nhx0 : ↑x - 1 < f x\nh_eq : ↑⌊x⌋₊ = x\n⊢ ↑x - 1 < f x"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "case inl\nf : ℚ → ℝ\nx : ℚ\nhx : 1 ≤ x\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nhx0 : ↑x - 1 < f x\nh_eq : ↑⌊x⌋₊ = x\n⊢ ↑x - 1 < f x",
"after_state": "case inl\nf : ℚ → ℝ\nx : ℚ\nhx : 1 ≤ x\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nhx0 : ↑x - 1 < f x\nh_eq : ↑⌊x⌋₊ = x\n⊢ ↑x - 1 < f x"
},
{
"line": "with_reducible rfl",
"before_state": "case inl\nf : ℚ → ℝ\nx : ℚ\nhx : 1 ≤ x\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nhx0 : ↑x - 1 < f x\nh_eq : ↑⌊x⌋₊ = x\n⊢ ↑x - 1 < f x",
"after_state": "case inl\nf : ℚ → ℝ\nx : ℚ\nhx : 1 ≤ x\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nhx0 : ↑x - 1 < f x\nh_eq : ↑⌊x⌋₊ = x\n⊢ ↑x - 1 < f x"
},
{
"line": "rfl",
"before_state": "case inl\nf : ℚ → ℝ\nx : ℚ\nhx : 1 ≤ x\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nhx0 : ↑x - 1 < f x\nh_eq : ↑⌊x⌋₊ = x\n⊢ ↑x - 1 < f x",
"after_state": "case inl\nf : ℚ → ℝ\nx : ℚ\nhx : 1 ≤ x\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nhx0 : ↑x - 1 < f x\nh_eq : ↑⌊x⌋₊ = x\n⊢ ↑x - 1 < f x"
},
{
"line": "apply_rfl",
"before_state": "case inl\nf : ℚ → ℝ\nx : ℚ\nhx : 1 ≤ x\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nhx0 : ↑x - 1 < f x\nh_eq : ↑⌊x⌋₊ = x\n⊢ ↑x - 1 < f x",
"after_state": "case inl\nf : ℚ → ℝ\nx : ℚ\nhx : 1 ≤ x\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nhx0 : ↑x - 1 < f x\nh_eq : ↑⌊x⌋₊ = x\n⊢ ↑x - 1 < f x"
},
{
"line": "skip",
"before_state": "case inl\nf : ℚ → ℝ\nx : ℚ\nhx : 1 ≤ x\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nhx0 : ↑x - 1 < f x\nh_eq : ↑⌊x⌋₊ = x\n⊢ ↑x - 1 < f x",
"after_state": "case inl\nf : ℚ → ℝ\nx : ℚ\nhx : 1 ≤ x\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nhx0 : ↑x - 1 < f x\nh_eq : ↑⌊x⌋₊ = x\n⊢ ↑x - 1 < f x"
},
{
"line": "assumption",
"before_state": "case inl\nf : ℚ → ℝ\nx : ℚ\nhx : 1 ≤ x\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nhx0 : ↑x - 1 < f x\nh_eq : ↑⌊x⌋₊ = x\n⊢ ↑x - 1 < f x",
"after_state": "No Goals!"
},
{
"line": "calc\n (x - 1 : ℝ) < f ⌊x⌋₊ := hx0\n _ < f (x - ⌊x⌋₊) + f ⌊x⌋₊ := (lt_add_of_pos_left _ (f_pos_of_pos (sub_pos.mpr h_lt) H1 H4))\n _ ≤ f (x - ⌊x⌋₊ + ⌊x⌋₊) := (H2 _ _ (sub_pos.mpr h_lt) (Nat.cast_pos.2 (Nat.floor_pos.2 hx)))\n _ = f x := by ring_nf",
"before_state": "case inr\nf : ℚ → ℝ\nx : ℚ\nhx : 1 ≤ x\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nhx0 : ↑x - 1 < f ↑⌊x⌋₊\nh_lt : ↑⌊x⌋₊ < x\n⊢ ↑x - 1 < f x",
"after_state": "No Goals!"
},
{
"line": "ring_nf",
"before_state": "f : ℚ → ℝ\nx : ℚ\nhx : 1 ≤ x\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nhx0 : ↑x - 1 < f ↑⌊x⌋₊\nh_lt : ↑⌊x⌋₊ < x\n⊢ f (x - ↑⌊x⌋₊ + ↑⌊x⌋₊) = f x",
"after_state": "No Goals!"
}
] |
theorem pow_f_le_f_pow {f : ℚ → ℝ} {n : ℕ} (hn : 0 < n) {x : ℚ} (hx : 1 < x)
(H1 : ∀ x y, 0 < x → 0 < y → f (x * y) ≤ f x * f y) (H4 : ∀ n : ℕ, 0 < n → (n : ℝ) ≤ f n) :
f (x ^ n) ≤ f x ^ n := by
induction' n with pn hpn
· exfalso; exact Nat.lt_asymm hn hn
rcases pn with - | pn
· norm_num
have hpn' := hpn pn.succ_pos
rw [pow_succ x (pn + 1)]
rw [pow_succ (f x) (pn + 1)]
have hxp : 0 < x := by positivity
calc
f (x ^ (pn + 1) * x) ≤ f (x ^ (pn + 1)) * f x := H1 (x ^ (pn + 1)) x (pow_pos hxp (pn + 1)) hxp
_ ≤ f x ^ (pn + 1) * f x := by gcongr; exact (f_pos_of_pos hxp H1 H4).le
|
/root/DuelModelResearch/mathlib4/Archive/Imo/Imo2013Q5.lean
|
{
"open": [],
"variables": []
}
|
[
{
"line": "induction' n with pn hpn",
"before_state": "f : ℚ → ℝ\nn : ℕ\nhn : 0 < n\nx : ℚ\nhx : 1 < x\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\n⊢ f (x ^ n) ≤ f x ^ n",
"after_state": "case zero\nf : ℚ → ℝ\nx : ℚ\nhx : 1 < x\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nhn : 0 < 0\n⊢ f (x ^ 0) ≤ f x ^ 0\n---\ncase succ\nf : ℚ → ℝ\nx : ℚ\nhx : 1 < x\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\npn : ℕ\nhpn : 0 < pn → f (x ^ pn) ≤ f x ^ pn\nhn : 0 < pn + 1\n⊢ f (x ^ (pn + 1)) ≤ f x ^ (pn + 1)"
},
{
"line": "exfalso",
"before_state": "case zero\nf : ℚ → ℝ\nx : ℚ\nhx : 1 < x\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nhn : 0 < 0\n⊢ f (x ^ 0) ≤ f x ^ 0",
"after_state": "case zero\nf : ℚ → ℝ\nx : ℚ\nhx : 1 < x\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nhn : 0 < 0\n⊢ False"
},
{
"line": "refine False.elim✝ ?_",
"before_state": "case zero\nf : ℚ → ℝ\nx : ℚ\nhx : 1 < x\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nhn : 0 < 0\n⊢ f (x ^ 0) ≤ f x ^ 0",
"after_state": "case zero\nf : ℚ → ℝ\nx : ℚ\nhx : 1 < x\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nhn : 0 < 0\n⊢ False"
},
{
"line": "exact Nat.lt_asymm hn hn",
"before_state": "case zero\nf : ℚ → ℝ\nx : ℚ\nhx : 1 < x\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nhn : 0 < 0\n⊢ False",
"after_state": "No Goals!"
},
{
"line": "rcases pn with - | pn",
"before_state": "case succ\nf : ℚ → ℝ\nx : ℚ\nhx : 1 < x\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\npn : ℕ\nhpn : 0 < pn → f (x ^ pn) ≤ f x ^ pn\nhn : 0 < pn + 1\n⊢ f (x ^ (pn + 1)) ≤ f x ^ (pn + 1)",
"after_state": "case succ.zero\nf : ℚ → ℝ\nx : ℚ\nhx : 1 < x\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nhpn : 0 < 0 → f (x ^ 0) ≤ f x ^ 0\nhn : 0 < 0 + 1\n⊢ f (x ^ (0 + 1)) ≤ f x ^ (0 + 1)\n---\ncase succ.succ\nf : ℚ → ℝ\nx : ℚ\nhx : 1 < x\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\npn : ℕ\nhpn : 0 < pn + 1 → f (x ^ (pn + 1)) ≤ f x ^ (pn + 1)\nhn : 0 < pn + 1 + 1\n⊢ f (x ^ (pn + 1 + 1)) ≤ f x ^ (pn + 1 + 1)"
},
{
"line": "norm_num",
"before_state": "case succ.zero\nf : ℚ → ℝ\nx : ℚ\nhx : 1 < x\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nhpn : 0 < 0 → f (x ^ 0) ≤ f x ^ 0\nhn : 0 < 0 + 1\n⊢ f (x ^ (0 + 1)) ≤ f x ^ (0 + 1)",
"after_state": "No Goals!"
},
{
"line": "have hpn' := hpn pn.succ_pos",
"before_state": "case succ.succ\nf : ℚ → ℝ\nx : ℚ\nhx : 1 < x\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\npn : ℕ\nhpn : 0 < pn + 1 → f (x ^ (pn + 1)) ≤ f x ^ (pn + 1)\nhn : 0 < pn + 1 + 1\n⊢ f (x ^ (pn + 1 + 1)) ≤ f x ^ (pn + 1 + 1)",
"after_state": "case succ.succ\nf : ℚ → ℝ\nx : ℚ\nhx : 1 < x\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\npn : ℕ\nhpn : 0 < pn + 1 → f (x ^ (pn + 1)) ≤ f x ^ (pn + 1)\nhn : 0 < pn + 1 + 1\nhpn' : f (x ^ (pn + 1)) ≤ f x ^ (pn + 1)\n⊢ f (x ^ (pn + 1 + 1)) ≤ f x ^ (pn + 1 + 1)"
},
{
"line": "refine_lift\n have hpn' := hpn pn.succ_pos;\n ?_",
"before_state": "case succ.succ\nf : ℚ → ℝ\nx : ℚ\nhx : 1 < x\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\npn : ℕ\nhpn : 0 < pn + 1 → f (x ^ (pn + 1)) ≤ f x ^ (pn + 1)\nhn : 0 < pn + 1 + 1\n⊢ f (x ^ (pn + 1 + 1)) ≤ f x ^ (pn + 1 + 1)",
"after_state": "case succ.succ\nf : ℚ → ℝ\nx : ℚ\nhx : 1 < x\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\npn : ℕ\nhpn : 0 < pn + 1 → f (x ^ (pn + 1)) ≤ f x ^ (pn + 1)\nhn : 0 < pn + 1 + 1\nhpn' : f (x ^ (pn + 1)) ≤ f x ^ (pn + 1)\n⊢ f (x ^ (pn + 1 + 1)) ≤ f x ^ (pn + 1 + 1)"
},
{
"line": "focus\n (refine\n no_implicit_lambda%\n (have hpn' := hpn pn.succ_pos;\n ?_);\n rotate_right)",
"before_state": "case succ.succ\nf : ℚ → ℝ\nx : ℚ\nhx : 1 < x\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\npn : ℕ\nhpn : 0 < pn + 1 → f (x ^ (pn + 1)) ≤ f x ^ (pn + 1)\nhn : 0 < pn + 1 + 1\n⊢ f (x ^ (pn + 1 + 1)) ≤ f x ^ (pn + 1 + 1)",
"after_state": "case succ.succ\nf : ℚ → ℝ\nx : ℚ\nhx : 1 < x\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\npn : ℕ\nhpn : 0 < pn + 1 → f (x ^ (pn + 1)) ≤ f x ^ (pn + 1)\nhn : 0 < pn + 1 + 1\nhpn' : f (x ^ (pn + 1)) ≤ f x ^ (pn + 1)\n⊢ f (x ^ (pn + 1 + 1)) ≤ f x ^ (pn + 1 + 1)"
},
{
"line": "refine\n no_implicit_lambda%\n (have hpn' := hpn pn.succ_pos;\n ?_)",
"before_state": "case succ.succ\nf : ℚ → ℝ\nx : ℚ\nhx : 1 < x\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\npn : ℕ\nhpn : 0 < pn + 1 → f (x ^ (pn + 1)) ≤ f x ^ (pn + 1)\nhn : 0 < pn + 1 + 1\n⊢ f (x ^ (pn + 1 + 1)) ≤ f x ^ (pn + 1 + 1)",
"after_state": "case succ.succ\nf : ℚ → ℝ\nx : ℚ\nhx : 1 < x\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\npn : ℕ\nhpn : 0 < pn + 1 → f (x ^ (pn + 1)) ≤ f x ^ (pn + 1)\nhn : 0 < pn + 1 + 1\nhpn' : f (x ^ (pn + 1)) ≤ f x ^ (pn + 1)\n⊢ f (x ^ (pn + 1 + 1)) ≤ f x ^ (pn + 1 + 1)"
},
{
"line": "rotate_right",
"before_state": "case succ.succ\nf : ℚ → ℝ\nx : ℚ\nhx : 1 < x\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\npn : ℕ\nhpn : 0 < pn + 1 → f (x ^ (pn + 1)) ≤ f x ^ (pn + 1)\nhn : 0 < pn + 1 + 1\nhpn' : f (x ^ (pn + 1)) ≤ f x ^ (pn + 1)\n⊢ f (x ^ (pn + 1 + 1)) ≤ f x ^ (pn + 1 + 1)",
"after_state": "case succ.succ\nf : ℚ → ℝ\nx : ℚ\nhx : 1 < x\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\npn : ℕ\nhpn : 0 < pn + 1 → f (x ^ (pn + 1)) ≤ f x ^ (pn + 1)\nhn : 0 < pn + 1 + 1\nhpn' : f (x ^ (pn + 1)) ≤ f x ^ (pn + 1)\n⊢ f (x ^ (pn + 1 + 1)) ≤ f x ^ (pn + 1 + 1)"
},
{
"line": "rw [pow_succ x (pn + 1)]",
"before_state": "case succ.succ\nf : ℚ → ℝ\nx : ℚ\nhx : 1 < x\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\npn : ℕ\nhpn : 0 < pn + 1 → f (x ^ (pn + 1)) ≤ f x ^ (pn + 1)\nhn : 0 < pn + 1 + 1\nhpn' : f (x ^ (pn + 1)) ≤ f x ^ (pn + 1)\n⊢ f (x ^ (pn + 1 + 1)) ≤ f x ^ (pn + 1 + 1)",
"after_state": "case succ.succ\nf : ℚ → ℝ\nx : ℚ\nhx : 1 < x\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\npn : ℕ\nhpn : 0 < pn + 1 → f (x ^ (pn + 1)) ≤ f x ^ (pn + 1)\nhn : 0 < pn + 1 + 1\nhpn' : f (x ^ (pn + 1)) ≤ f x ^ (pn + 1)\n⊢ f (x ^ (pn + 1) * x) ≤ f x ^ (pn + 1 + 1)"
},
{
"line": "rewrite [pow_succ x (pn + 1)]",
"before_state": "case succ.succ\nf : ℚ → ℝ\nx : ℚ\nhx : 1 < x\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\npn : ℕ\nhpn : 0 < pn + 1 → f (x ^ (pn + 1)) ≤ f x ^ (pn + 1)\nhn : 0 < pn + 1 + 1\nhpn' : f (x ^ (pn + 1)) ≤ f x ^ (pn + 1)\n⊢ f (x ^ (pn + 1 + 1)) ≤ f x ^ (pn + 1 + 1)",
"after_state": "case succ.succ\nf : ℚ → ℝ\nx : ℚ\nhx : 1 < x\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\npn : ℕ\nhpn : 0 < pn + 1 → f (x ^ (pn + 1)) ≤ f x ^ (pn + 1)\nhn : 0 < pn + 1 + 1\nhpn' : f (x ^ (pn + 1)) ≤ f x ^ (pn + 1)\n⊢ f (x ^ (pn + 1) * x) ≤ f x ^ (pn + 1 + 1)"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "case succ.succ\nf : ℚ → ℝ\nx : ℚ\nhx : 1 < x\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\npn : ℕ\nhpn : 0 < pn + 1 → f (x ^ (pn + 1)) ≤ f x ^ (pn + 1)\nhn : 0 < pn + 1 + 1\nhpn' : f (x ^ (pn + 1)) ≤ f x ^ (pn + 1)\n⊢ f (x ^ (pn + 1) * x) ≤ f x ^ (pn + 1 + 1)",
"after_state": "case succ.succ\nf : ℚ → ℝ\nx : ℚ\nhx : 1 < x\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\npn : ℕ\nhpn : 0 < pn + 1 → f (x ^ (pn + 1)) ≤ f x ^ (pn + 1)\nhn : 0 < pn + 1 + 1\nhpn' : f (x ^ (pn + 1)) ≤ f x ^ (pn + 1)\n⊢ f (x ^ (pn + 1) * x) ≤ f x ^ (pn + 1 + 1)"
},
{
"line": "try (with_reducible rfl)",
"before_state": "case succ.succ\nf : ℚ → ℝ\nx : ℚ\nhx : 1 < x\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\npn : ℕ\nhpn : 0 < pn + 1 → f (x ^ (pn + 1)) ≤ f x ^ (pn + 1)\nhn : 0 < pn + 1 + 1\nhpn' : f (x ^ (pn + 1)) ≤ f x ^ (pn + 1)\n⊢ f (x ^ (pn + 1) * x) ≤ f x ^ (pn + 1 + 1)",
"after_state": "case succ.succ\nf : ℚ → ℝ\nx : ℚ\nhx : 1 < x\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\npn : ℕ\nhpn : 0 < pn + 1 → f (x ^ (pn + 1)) ≤ f x ^ (pn + 1)\nhn : 0 < pn + 1 + 1\nhpn' : f (x ^ (pn + 1)) ≤ f x ^ (pn + 1)\n⊢ f (x ^ (pn + 1) * x) ≤ f x ^ (pn + 1 + 1)"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "case succ.succ\nf : ℚ → ℝ\nx : ℚ\nhx : 1 < x\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\npn : ℕ\nhpn : 0 < pn + 1 → f (x ^ (pn + 1)) ≤ f x ^ (pn + 1)\nhn : 0 < pn + 1 + 1\nhpn' : f (x ^ (pn + 1)) ≤ f x ^ (pn + 1)\n⊢ f (x ^ (pn + 1) * x) ≤ f x ^ (pn + 1 + 1)",
"after_state": "case succ.succ\nf : ℚ → ℝ\nx : ℚ\nhx : 1 < x\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\npn : ℕ\nhpn : 0 < pn + 1 → f (x ^ (pn + 1)) ≤ f x ^ (pn + 1)\nhn : 0 < pn + 1 + 1\nhpn' : f (x ^ (pn + 1)) ≤ f x ^ (pn + 1)\n⊢ f (x ^ (pn + 1) * x) ≤ f x ^ (pn + 1 + 1)"
},
{
"line": "with_reducible rfl",
"before_state": "case succ.succ\nf : ℚ → ℝ\nx : ℚ\nhx : 1 < x\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\npn : ℕ\nhpn : 0 < pn + 1 → f (x ^ (pn + 1)) ≤ f x ^ (pn + 1)\nhn : 0 < pn + 1 + 1\nhpn' : f (x ^ (pn + 1)) ≤ f x ^ (pn + 1)\n⊢ f (x ^ (pn + 1) * x) ≤ f x ^ (pn + 1 + 1)",
"after_state": "case succ.succ\nf : ℚ → ℝ\nx : ℚ\nhx : 1 < x\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\npn : ℕ\nhpn : 0 < pn + 1 → f (x ^ (pn + 1)) ≤ f x ^ (pn + 1)\nhn : 0 < pn + 1 + 1\nhpn' : f (x ^ (pn + 1)) ≤ f x ^ (pn + 1)\n⊢ f (x ^ (pn + 1) * x) ≤ f x ^ (pn + 1 + 1)"
},
{
"line": "rfl",
"before_state": "case succ.succ\nf : ℚ → ℝ\nx : ℚ\nhx : 1 < x\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\npn : ℕ\nhpn : 0 < pn + 1 → f (x ^ (pn + 1)) ≤ f x ^ (pn + 1)\nhn : 0 < pn + 1 + 1\nhpn' : f (x ^ (pn + 1)) ≤ f x ^ (pn + 1)\n⊢ f (x ^ (pn + 1) * x) ≤ f x ^ (pn + 1 + 1)",
"after_state": "case succ.succ\nf : ℚ → ℝ\nx : ℚ\nhx : 1 < x\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\npn : ℕ\nhpn : 0 < pn + 1 → f (x ^ (pn + 1)) ≤ f x ^ (pn + 1)\nhn : 0 < pn + 1 + 1\nhpn' : f (x ^ (pn + 1)) ≤ f x ^ (pn + 1)\n⊢ f (x ^ (pn + 1) * x) ≤ f x ^ (pn + 1 + 1)"
},
{
"line": "apply_rfl",
"before_state": "case succ.succ\nf : ℚ → ℝ\nx : ℚ\nhx : 1 < x\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\npn : ℕ\nhpn : 0 < pn + 1 → f (x ^ (pn + 1)) ≤ f x ^ (pn + 1)\nhn : 0 < pn + 1 + 1\nhpn' : f (x ^ (pn + 1)) ≤ f x ^ (pn + 1)\n⊢ f (x ^ (pn + 1) * x) ≤ f x ^ (pn + 1 + 1)",
"after_state": "case succ.succ\nf : ℚ → ℝ\nx : ℚ\nhx : 1 < x\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\npn : ℕ\nhpn : 0 < pn + 1 → f (x ^ (pn + 1)) ≤ f x ^ (pn + 1)\nhn : 0 < pn + 1 + 1\nhpn' : f (x ^ (pn + 1)) ≤ f x ^ (pn + 1)\n⊢ f (x ^ (pn + 1) * x) ≤ f x ^ (pn + 1 + 1)"
},
{
"line": "skip",
"before_state": "case succ.succ\nf : ℚ → ℝ\nx : ℚ\nhx : 1 < x\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\npn : ℕ\nhpn : 0 < pn + 1 → f (x ^ (pn + 1)) ≤ f x ^ (pn + 1)\nhn : 0 < pn + 1 + 1\nhpn' : f (x ^ (pn + 1)) ≤ f x ^ (pn + 1)\n⊢ f (x ^ (pn + 1) * x) ≤ f x ^ (pn + 1 + 1)",
"after_state": "case succ.succ\nf : ℚ → ℝ\nx : ℚ\nhx : 1 < x\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\npn : ℕ\nhpn : 0 < pn + 1 → f (x ^ (pn + 1)) ≤ f x ^ (pn + 1)\nhn : 0 < pn + 1 + 1\nhpn' : f (x ^ (pn + 1)) ≤ f x ^ (pn + 1)\n⊢ f (x ^ (pn + 1) * x) ≤ f x ^ (pn + 1 + 1)"
},
{
"line": "rw [pow_succ (f x) (pn + 1)]",
"before_state": "case succ.succ\nf : ℚ → ℝ\nx : ℚ\nhx : 1 < x\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\npn : ℕ\nhpn : 0 < pn + 1 → f (x ^ (pn + 1)) ≤ f x ^ (pn + 1)\nhn : 0 < pn + 1 + 1\nhpn' : f (x ^ (pn + 1)) ≤ f x ^ (pn + 1)\n⊢ f (x ^ (pn + 1) * x) ≤ f x ^ (pn + 1 + 1)",
"after_state": "case succ.succ\nf : ℚ → ℝ\nx : ℚ\nhx : 1 < x\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\npn : ℕ\nhpn : 0 < pn + 1 → f (x ^ (pn + 1)) ≤ f x ^ (pn + 1)\nhn : 0 < pn + 1 + 1\nhpn' : f (x ^ (pn + 1)) ≤ f x ^ (pn + 1)\n⊢ f (x ^ (pn + 1) * x) ≤ f x ^ (pn + 1) * f x"
},
{
"line": "rewrite [pow_succ (f x) (pn + 1)]",
"before_state": "case succ.succ\nf : ℚ → ℝ\nx : ℚ\nhx : 1 < x\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\npn : ℕ\nhpn : 0 < pn + 1 → f (x ^ (pn + 1)) ≤ f x ^ (pn + 1)\nhn : 0 < pn + 1 + 1\nhpn' : f (x ^ (pn + 1)) ≤ f x ^ (pn + 1)\n⊢ f (x ^ (pn + 1) * x) ≤ f x ^ (pn + 1 + 1)",
"after_state": "case succ.succ\nf : ℚ → ℝ\nx : ℚ\nhx : 1 < x\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\npn : ℕ\nhpn : 0 < pn + 1 → f (x ^ (pn + 1)) ≤ f x ^ (pn + 1)\nhn : 0 < pn + 1 + 1\nhpn' : f (x ^ (pn + 1)) ≤ f x ^ (pn + 1)\n⊢ f (x ^ (pn + 1) * x) ≤ f x ^ (pn + 1) * f x"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "case succ.succ\nf : ℚ → ℝ\nx : ℚ\nhx : 1 < x\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\npn : ℕ\nhpn : 0 < pn + 1 → f (x ^ (pn + 1)) ≤ f x ^ (pn + 1)\nhn : 0 < pn + 1 + 1\nhpn' : f (x ^ (pn + 1)) ≤ f x ^ (pn + 1)\n⊢ f (x ^ (pn + 1) * x) ≤ f x ^ (pn + 1) * f x",
"after_state": "case succ.succ\nf : ℚ → ℝ\nx : ℚ\nhx : 1 < x\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\npn : ℕ\nhpn : 0 < pn + 1 → f (x ^ (pn + 1)) ≤ f x ^ (pn + 1)\nhn : 0 < pn + 1 + 1\nhpn' : f (x ^ (pn + 1)) ≤ f x ^ (pn + 1)\n⊢ f (x ^ (pn + 1) * x) ≤ f x ^ (pn + 1) * f x"
},
{
"line": "try (with_reducible rfl)",
"before_state": "case succ.succ\nf : ℚ → ℝ\nx : ℚ\nhx : 1 < x\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\npn : ℕ\nhpn : 0 < pn + 1 → f (x ^ (pn + 1)) ≤ f x ^ (pn + 1)\nhn : 0 < pn + 1 + 1\nhpn' : f (x ^ (pn + 1)) ≤ f x ^ (pn + 1)\n⊢ f (x ^ (pn + 1) * x) ≤ f x ^ (pn + 1) * f x",
"after_state": "case succ.succ\nf : ℚ → ℝ\nx : ℚ\nhx : 1 < x\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\npn : ℕ\nhpn : 0 < pn + 1 → f (x ^ (pn + 1)) ≤ f x ^ (pn + 1)\nhn : 0 < pn + 1 + 1\nhpn' : f (x ^ (pn + 1)) ≤ f x ^ (pn + 1)\n⊢ f (x ^ (pn + 1) * x) ≤ f x ^ (pn + 1) * f x"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "case succ.succ\nf : ℚ → ℝ\nx : ℚ\nhx : 1 < x\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\npn : ℕ\nhpn : 0 < pn + 1 → f (x ^ (pn + 1)) ≤ f x ^ (pn + 1)\nhn : 0 < pn + 1 + 1\nhpn' : f (x ^ (pn + 1)) ≤ f x ^ (pn + 1)\n⊢ f (x ^ (pn + 1) * x) ≤ f x ^ (pn + 1) * f x",
"after_state": "case succ.succ\nf : ℚ → ℝ\nx : ℚ\nhx : 1 < x\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\npn : ℕ\nhpn : 0 < pn + 1 → f (x ^ (pn + 1)) ≤ f x ^ (pn + 1)\nhn : 0 < pn + 1 + 1\nhpn' : f (x ^ (pn + 1)) ≤ f x ^ (pn + 1)\n⊢ f (x ^ (pn + 1) * x) ≤ f x ^ (pn + 1) * f x"
},
{
"line": "with_reducible rfl",
"before_state": "case succ.succ\nf : ℚ → ℝ\nx : ℚ\nhx : 1 < x\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\npn : ℕ\nhpn : 0 < pn + 1 → f (x ^ (pn + 1)) ≤ f x ^ (pn + 1)\nhn : 0 < pn + 1 + 1\nhpn' : f (x ^ (pn + 1)) ≤ f x ^ (pn + 1)\n⊢ f (x ^ (pn + 1) * x) ≤ f x ^ (pn + 1) * f x",
"after_state": "case succ.succ\nf : ℚ → ℝ\nx : ℚ\nhx : 1 < x\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\npn : ℕ\nhpn : 0 < pn + 1 → f (x ^ (pn + 1)) ≤ f x ^ (pn + 1)\nhn : 0 < pn + 1 + 1\nhpn' : f (x ^ (pn + 1)) ≤ f x ^ (pn + 1)\n⊢ f (x ^ (pn + 1) * x) ≤ f x ^ (pn + 1) * f x"
},
{
"line": "rfl",
"before_state": "case succ.succ\nf : ℚ → ℝ\nx : ℚ\nhx : 1 < x\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\npn : ℕ\nhpn : 0 < pn + 1 → f (x ^ (pn + 1)) ≤ f x ^ (pn + 1)\nhn : 0 < pn + 1 + 1\nhpn' : f (x ^ (pn + 1)) ≤ f x ^ (pn + 1)\n⊢ f (x ^ (pn + 1) * x) ≤ f x ^ (pn + 1) * f x",
"after_state": "case succ.succ\nf : ℚ → ℝ\nx : ℚ\nhx : 1 < x\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\npn : ℕ\nhpn : 0 < pn + 1 → f (x ^ (pn + 1)) ≤ f x ^ (pn + 1)\nhn : 0 < pn + 1 + 1\nhpn' : f (x ^ (pn + 1)) ≤ f x ^ (pn + 1)\n⊢ f (x ^ (pn + 1) * x) ≤ f x ^ (pn + 1) * f x"
},
{
"line": "apply_rfl",
"before_state": "case succ.succ\nf : ℚ → ℝ\nx : ℚ\nhx : 1 < x\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\npn : ℕ\nhpn : 0 < pn + 1 → f (x ^ (pn + 1)) ≤ f x ^ (pn + 1)\nhn : 0 < pn + 1 + 1\nhpn' : f (x ^ (pn + 1)) ≤ f x ^ (pn + 1)\n⊢ f (x ^ (pn + 1) * x) ≤ f x ^ (pn + 1) * f x",
"after_state": "case succ.succ\nf : ℚ → ℝ\nx : ℚ\nhx : 1 < x\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\npn : ℕ\nhpn : 0 < pn + 1 → f (x ^ (pn + 1)) ≤ f x ^ (pn + 1)\nhn : 0 < pn + 1 + 1\nhpn' : f (x ^ (pn + 1)) ≤ f x ^ (pn + 1)\n⊢ f (x ^ (pn + 1) * x) ≤ f x ^ (pn + 1) * f x"
},
{
"line": "skip",
"before_state": "case succ.succ\nf : ℚ → ℝ\nx : ℚ\nhx : 1 < x\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\npn : ℕ\nhpn : 0 < pn + 1 → f (x ^ (pn + 1)) ≤ f x ^ (pn + 1)\nhn : 0 < pn + 1 + 1\nhpn' : f (x ^ (pn + 1)) ≤ f x ^ (pn + 1)\n⊢ f (x ^ (pn + 1) * x) ≤ f x ^ (pn + 1) * f x",
"after_state": "case succ.succ\nf : ℚ → ℝ\nx : ℚ\nhx : 1 < x\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\npn : ℕ\nhpn : 0 < pn + 1 → f (x ^ (pn + 1)) ≤ f x ^ (pn + 1)\nhn : 0 < pn + 1 + 1\nhpn' : f (x ^ (pn + 1)) ≤ f x ^ (pn + 1)\n⊢ f (x ^ (pn + 1) * x) ≤ f x ^ (pn + 1) * f x"
},
{
"line": "have hxp : 0 < x := by positivity",
"before_state": "case succ.succ\nf : ℚ → ℝ\nx : ℚ\nhx : 1 < x\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\npn : ℕ\nhpn : 0 < pn + 1 → f (x ^ (pn + 1)) ≤ f x ^ (pn + 1)\nhn : 0 < pn + 1 + 1\nhpn' : f (x ^ (pn + 1)) ≤ f x ^ (pn + 1)\n⊢ f (x ^ (pn + 1) * x) ≤ f x ^ (pn + 1) * f x",
"after_state": "case succ.succ\nf : ℚ → ℝ\nx : ℚ\nhx : 1 < x\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\npn : ℕ\nhpn : 0 < pn + 1 → f (x ^ (pn + 1)) ≤ f x ^ (pn + 1)\nhn : 0 < pn + 1 + 1\nhpn' : f (x ^ (pn + 1)) ≤ f x ^ (pn + 1)\nhxp : 0 < x\n⊢ f (x ^ (pn + 1) * x) ≤ f x ^ (pn + 1) * f x"
},
{
"line": "focus\n refine\n no_implicit_lambda%\n (have hxp : 0 < x := ?body✝;\n ?_)\n case body✝ => with_annotate_state\"by\" (positivity)",
"before_state": "case succ.succ\nf : ℚ → ℝ\nx : ℚ\nhx : 1 < x\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\npn : ℕ\nhpn : 0 < pn + 1 → f (x ^ (pn + 1)) ≤ f x ^ (pn + 1)\nhn : 0 < pn + 1 + 1\nhpn' : f (x ^ (pn + 1)) ≤ f x ^ (pn + 1)\n⊢ f (x ^ (pn + 1) * x) ≤ f x ^ (pn + 1) * f x",
"after_state": "case succ.succ\nf : ℚ → ℝ\nx : ℚ\nhx : 1 < x\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\npn : ℕ\nhpn : 0 < pn + 1 → f (x ^ (pn + 1)) ≤ f x ^ (pn + 1)\nhn : 0 < pn + 1 + 1\nhpn' : f (x ^ (pn + 1)) ≤ f x ^ (pn + 1)\nhxp : 0 < x\n⊢ f (x ^ (pn + 1) * x) ≤ f x ^ (pn + 1) * f x"
},
{
"line": "refine\n no_implicit_lambda%\n (have hxp : 0 < x := ?body✝;\n ?_)",
"before_state": "case succ.succ\nf : ℚ → ℝ\nx : ℚ\nhx : 1 < x\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\npn : ℕ\nhpn : 0 < pn + 1 → f (x ^ (pn + 1)) ≤ f x ^ (pn + 1)\nhn : 0 < pn + 1 + 1\nhpn' : f (x ^ (pn + 1)) ≤ f x ^ (pn + 1)\n⊢ f (x ^ (pn + 1) * x) ≤ f x ^ (pn + 1) * f x",
"after_state": "case body\nf : ℚ → ℝ\nx : ℚ\nhx : 1 < x\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\npn : ℕ\nhpn : 0 < pn + 1 → f (x ^ (pn + 1)) ≤ f x ^ (pn + 1)\nhn : 0 < pn + 1 + 1\nhpn' : f (x ^ (pn + 1)) ≤ f x ^ (pn + 1)\n⊢ 0 < x\n---\ncase succ.succ\nf : ℚ → ℝ\nx : ℚ\nhx : 1 < x\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\npn : ℕ\nhpn : 0 < pn + 1 → f (x ^ (pn + 1)) ≤ f x ^ (pn + 1)\nhn : 0 < pn + 1 + 1\nhpn' : f (x ^ (pn + 1)) ≤ f x ^ (pn + 1)\nhxp : 0 < x\n⊢ f (x ^ (pn + 1) * x) ≤ f x ^ (pn + 1) * f x"
},
{
"line": "case body✝ => with_annotate_state\"by\" (positivity)",
"before_state": "case body\nf : ℚ → ℝ\nx : ℚ\nhx : 1 < x\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\npn : ℕ\nhpn : 0 < pn + 1 → f (x ^ (pn + 1)) ≤ f x ^ (pn + 1)\nhn : 0 < pn + 1 + 1\nhpn' : f (x ^ (pn + 1)) ≤ f x ^ (pn + 1)\n⊢ 0 < x\n---\ncase succ.succ\nf : ℚ → ℝ\nx : ℚ\nhx : 1 < x\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\npn : ℕ\nhpn : 0 < pn + 1 → f (x ^ (pn + 1)) ≤ f x ^ (pn + 1)\nhn : 0 < pn + 1 + 1\nhpn' : f (x ^ (pn + 1)) ≤ f x ^ (pn + 1)\nhxp : 0 < x\n⊢ f (x ^ (pn + 1) * x) ≤ f x ^ (pn + 1) * f x",
"after_state": "case succ.succ\nf : ℚ → ℝ\nx : ℚ\nhx : 1 < x\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\npn : ℕ\nhpn : 0 < pn + 1 → f (x ^ (pn + 1)) ≤ f x ^ (pn + 1)\nhn : 0 < pn + 1 + 1\nhpn' : f (x ^ (pn + 1)) ≤ f x ^ (pn + 1)\nhxp : 0 < x\n⊢ f (x ^ (pn + 1) * x) ≤ f x ^ (pn + 1) * f x"
},
{
"line": "with_annotate_state\"by\" (positivity)",
"before_state": "f : ℚ → ℝ\nx : ℚ\nhx : 1 < x\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\npn : ℕ\nhpn : 0 < pn + 1 → f (x ^ (pn + 1)) ≤ f x ^ (pn + 1)\nhn : 0 < pn + 1 + 1\nhpn' : f (x ^ (pn + 1)) ≤ f x ^ (pn + 1)\n⊢ 0 < x",
"after_state": "No Goals!"
},
{
"line": "positivity",
"before_state": "f : ℚ → ℝ\nx : ℚ\nhx : 1 < x\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\npn : ℕ\nhpn : 0 < pn + 1 → f (x ^ (pn + 1)) ≤ f x ^ (pn + 1)\nhn : 0 < pn + 1 + 1\nhpn' : f (x ^ (pn + 1)) ≤ f x ^ (pn + 1)\n⊢ 0 < x",
"after_state": "No Goals!"
},
{
"line": "calc\n f (x ^ (pn + 1) * x) ≤ f (x ^ (pn + 1)) * f x := H1 (x ^ (pn + 1)) x (pow_pos hxp (pn + 1)) hxp\n _ ≤ f x ^ (pn + 1) * f x := by gcongr; exact (f_pos_of_pos hxp H1 H4).le",
"before_state": "case succ.succ\nf : ℚ → ℝ\nx : ℚ\nhx : 1 < x\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\npn : ℕ\nhpn : 0 < pn + 1 → f (x ^ (pn + 1)) ≤ f x ^ (pn + 1)\nhn : 0 < pn + 1 + 1\nhpn' : f (x ^ (pn + 1)) ≤ f x ^ (pn + 1)\nhxp : 0 < x\n⊢ f (x ^ (pn + 1) * x) ≤ f x ^ (pn + 1) * f x",
"after_state": "No Goals!"
},
{
"line": "gcongr",
"before_state": "f : ℚ → ℝ\nx : ℚ\nhx : 1 < x\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\npn : ℕ\nhpn : 0 < pn + 1 → f (x ^ (pn + 1)) ≤ f x ^ (pn + 1)\nhn : 0 < pn + 1 + 1\nhpn' : f (x ^ (pn + 1)) ≤ f x ^ (pn + 1)\nhxp : 0 < x\n⊢ f (x ^ (pn + 1)) * f x ≤ f x ^ (pn + 1) * f x",
"after_state": "case a0\nf : ℚ → ℝ\nx : ℚ\nhx : 1 < x\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\npn : ℕ\nhpn : 0 < pn + 1 → f (x ^ (pn + 1)) ≤ f x ^ (pn + 1)\nhn : 0 < pn + 1 + 1\nhpn' : f (x ^ (pn + 1)) ≤ f x ^ (pn + 1)\nhxp : 0 < x\n⊢ 0 ≤ f x"
},
{
"line": "exact (f_pos_of_pos hxp H1 H4).le",
"before_state": "case a0\nf : ℚ → ℝ\nx : ℚ\nhx : 1 < x\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\npn : ℕ\nhpn : 0 < pn + 1 → f (x ^ (pn + 1)) ≤ f x ^ (pn + 1)\nhn : 0 < pn + 1 + 1\nhpn' : f (x ^ (pn + 1)) ≤ f x ^ (pn + 1)\nhxp : 0 < x\n⊢ 0 ≤ f x",
"after_state": "No Goals!"
}
] |
theorem fixed_point_of_pos_nat_pow {f : ℚ → ℝ} {n : ℕ} (hn : 0 < n)
(H1 : ∀ x y, 0 < x → 0 < y → f (x * y) ≤ f x * f y) (H4 : ∀ n : ℕ, 0 < n → (n : ℝ) ≤ f n)
(H5 : ∀ x : ℚ, 1 < x → (x : ℝ) ≤ f x) {a : ℚ} (ha1 : 1 < a) (hae : f a = a) :
f (a ^ n) = a ^ n := by
have hh0 : (a : ℝ) ^ n ≤ f (a ^ n) := mod_cast H5 (a ^ n) (one_lt_pow₀ ha1 hn.ne')
have hh1 :=
calc
f (a ^ n) ≤ f a ^ n := pow_f_le_f_pow hn ha1 H1 H4
_ = (a : ℝ) ^ n := by rw [← hae]
exact mod_cast hh1.antisymm hh0
|
/root/DuelModelResearch/mathlib4/Archive/Imo/Imo2013Q5.lean
|
{
"open": [],
"variables": []
}
|
[
{
"line": "have hh0 : (a : ℝ) ^ n ≤ f (a ^ n) := mod_cast H5 (a ^ n) (one_lt_pow₀ ha1 hn.ne')",
"before_state": "f : ℚ → ℝ\nn : ℕ\nhn : 0 < n\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\n⊢ f (a ^ n) = ↑a ^ n",
"after_state": "f : ℚ → ℝ\nn : ℕ\nhn : 0 < n\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nhh0 : ↑a ^ n ≤ f (a ^ n)\n⊢ f (a ^ n) = ↑a ^ n"
},
{
"line": "refine_lift\n have hh0 : (a : ℝ) ^ n ≤ f (a ^ n) := mod_cast H5 (a ^ n) (one_lt_pow₀ ha1 hn.ne');\n ?_",
"before_state": "f : ℚ → ℝ\nn : ℕ\nhn : 0 < n\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\n⊢ f (a ^ n) = ↑a ^ n",
"after_state": "f : ℚ → ℝ\nn : ℕ\nhn : 0 < n\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nhh0 : ↑a ^ n ≤ f (a ^ n)\n⊢ f (a ^ n) = ↑a ^ n"
},
{
"line": "focus\n (refine\n no_implicit_lambda%\n (have hh0 : (a : ℝ) ^ n ≤ f (a ^ n) := mod_cast H5 (a ^ n) (one_lt_pow₀ ha1 hn.ne');\n ?_);\n rotate_right)",
"before_state": "f : ℚ → ℝ\nn : ℕ\nhn : 0 < n\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\n⊢ f (a ^ n) = ↑a ^ n",
"after_state": "f : ℚ → ℝ\nn : ℕ\nhn : 0 < n\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nhh0 : ↑a ^ n ≤ f (a ^ n)\n⊢ f (a ^ n) = ↑a ^ n"
},
{
"line": "refine\n no_implicit_lambda%\n (have hh0 : (a : ℝ) ^ n ≤ f (a ^ n) := mod_cast H5 (a ^ n) (one_lt_pow₀ ha1 hn.ne');\n ?_)",
"before_state": "f : ℚ → ℝ\nn : ℕ\nhn : 0 < n\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\n⊢ f (a ^ n) = ↑a ^ n",
"after_state": "f : ℚ → ℝ\nn : ℕ\nhn : 0 < n\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nhh0 : ↑a ^ n ≤ f (a ^ n)\n⊢ f (a ^ n) = ↑a ^ n"
},
{
"line": "rotate_right",
"before_state": "f : ℚ → ℝ\nn : ℕ\nhn : 0 < n\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nhh0 : ↑a ^ n ≤ f (a ^ n)\n⊢ f (a ^ n) = ↑a ^ n",
"after_state": "f : ℚ → ℝ\nn : ℕ\nhn : 0 < n\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nhh0 : ↑a ^ n ≤ f (a ^ n)\n⊢ f (a ^ n) = ↑a ^ n"
},
{
"line": "have hh1 :=\n calc\n f (a ^ n) ≤ f a ^ n := pow_f_le_f_pow hn ha1 H1 H4\n _ = (a : ℝ) ^ n := by rw [← hae]",
"before_state": "f : ℚ → ℝ\nn : ℕ\nhn : 0 < n\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nhh0 : ↑a ^ n ≤ f (a ^ n)\n⊢ f (a ^ n) = ↑a ^ n",
"after_state": "f : ℚ → ℝ\nn : ℕ\nhn : 0 < n\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nhh0 : ↑a ^ n ≤ f (a ^ n)\nhh1 : f (a ^ n) ≤ ↑a ^ n\n⊢ f (a ^ n) = ↑a ^ n"
},
{
"line": "refine_lift\n have hh1 :=\n calc\n f (a ^ n) ≤ f a ^ n := pow_f_le_f_pow hn ha1 H1 H4\n _ = (a : ℝ) ^ n := by rw [← hae];\n ?_",
"before_state": "f : ℚ → ℝ\nn : ℕ\nhn : 0 < n\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nhh0 : ↑a ^ n ≤ f (a ^ n)\n⊢ f (a ^ n) = ↑a ^ n",
"after_state": "f : ℚ → ℝ\nn : ℕ\nhn : 0 < n\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nhh0 : ↑a ^ n ≤ f (a ^ n)\nhh1 : f (a ^ n) ≤ ↑a ^ n\n⊢ f (a ^ n) = ↑a ^ n"
},
{
"line": "focus\n (refine\n no_implicit_lambda%\n (have hh1 :=\n calc\n f (a ^ n) ≤ f a ^ n := pow_f_le_f_pow hn ha1 H1 H4\n _ = (a : ℝ) ^ n := by rw [← hae];\n ?_);\n rotate_right)",
"before_state": "f : ℚ → ℝ\nn : ℕ\nhn : 0 < n\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nhh0 : ↑a ^ n ≤ f (a ^ n)\n⊢ f (a ^ n) = ↑a ^ n",
"after_state": "f : ℚ → ℝ\nn : ℕ\nhn : 0 < n\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nhh0 : ↑a ^ n ≤ f (a ^ n)\nhh1 : f (a ^ n) ≤ ↑a ^ n\n⊢ f (a ^ n) = ↑a ^ n"
},
{
"line": "refine\n no_implicit_lambda%\n (have hh1 :=\n calc\n f (a ^ n) ≤ f a ^ n := pow_f_le_f_pow hn ha1 H1 H4\n _ = (a : ℝ) ^ n := by rw [← hae];\n ?_)",
"before_state": "f : ℚ → ℝ\nn : ℕ\nhn : 0 < n\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nhh0 : ↑a ^ n ≤ f (a ^ n)\n⊢ f (a ^ n) = ↑a ^ n",
"after_state": "f : ℚ → ℝ\nn : ℕ\nhn : 0 < n\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nhh0 : ↑a ^ n ≤ f (a ^ n)\nhh1 : f (a ^ n) ≤ ↑a ^ n\n⊢ f (a ^ n) = ↑a ^ n"
},
{
"line": "rw [← hae]",
"before_state": "f : ℚ → ℝ\nn : ℕ\nhn : 0 < n\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nhh0 : ↑a ^ n ≤ f (a ^ n)\n⊢ f a ^ n = ↑a ^ n",
"after_state": "No Goals!"
},
{
"line": "rewrite [← hae]",
"before_state": "f : ℚ → ℝ\nn : ℕ\nhn : 0 < n\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nhh0 : ↑a ^ n ≤ f (a ^ n)\n⊢ f a ^ n = ↑a ^ n",
"after_state": "f : ℚ → ℝ\nn : ℕ\nhn : 0 < n\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nhh0 : ↑a ^ n ≤ f (a ^ n)\n⊢ f a ^ n = f a ^ n"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "f : ℚ → ℝ\nn : ℕ\nhn : 0 < n\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nhh0 : ↑a ^ n ≤ f (a ^ n)\n⊢ f a ^ n = f a ^ n",
"after_state": "No Goals!"
},
{
"line": "try (with_reducible rfl)",
"before_state": "f : ℚ → ℝ\nn : ℕ\nhn : 0 < n\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nhh0 : ↑a ^ n ≤ f (a ^ n)\n⊢ f a ^ n = f a ^ n",
"after_state": "No Goals!"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "f : ℚ → ℝ\nn : ℕ\nhn : 0 < n\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nhh0 : ↑a ^ n ≤ f (a ^ n)\n⊢ f a ^ n = f a ^ n",
"after_state": "No Goals!"
},
{
"line": "with_reducible rfl",
"before_state": "f : ℚ → ℝ\nn : ℕ\nhn : 0 < n\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nhh0 : ↑a ^ n ≤ f (a ^ n)\n⊢ f a ^ n = f a ^ n",
"after_state": "No Goals!"
},
{
"line": "rfl",
"before_state": "f : ℚ → ℝ\nn : ℕ\nhn : 0 < n\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nhh0 : ↑a ^ n ≤ f (a ^ n)\n⊢ f a ^ n = f a ^ n",
"after_state": "No Goals!"
},
{
"line": "eq_refl",
"before_state": "f : ℚ → ℝ\nn : ℕ\nhn : 0 < n\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nhh0 : ↑a ^ n ≤ f (a ^ n)\n⊢ f a ^ n = f a ^ n",
"after_state": "No Goals!"
},
{
"line": "rotate_right",
"before_state": "f : ℚ → ℝ\nn : ℕ\nhn : 0 < n\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nhh0 : ↑a ^ n ≤ f (a ^ n)\nhh1 : f (a ^ n) ≤ ↑a ^ n\n⊢ f (a ^ n) = ↑a ^ n",
"after_state": "f : ℚ → ℝ\nn : ℕ\nhn : 0 < n\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nhh0 : ↑a ^ n ≤ f (a ^ n)\nhh1 : f (a ^ n) ≤ ↑a ^ n\n⊢ f (a ^ n) = ↑a ^ n"
},
{
"line": "exact mod_cast hh1.antisymm hh0",
"before_state": "f : ℚ → ℝ\nn : ℕ\nhn : 0 < n\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nhh0 : ↑a ^ n ≤ f (a ^ n)\nhh1 : f (a ^ n) ≤ ↑a ^ n\n⊢ f (a ^ n) = ↑a ^ n",
"after_state": "No Goals!"
}
] |
theorem fixed_point_of_gt_1 {f : ℚ → ℝ} {x : ℚ} (hx : 1 < x)
(H1 : ∀ x y, 0 < x → 0 < y → f (x * y) ≤ f x * f y)
(H2 : ∀ x y, 0 < x → 0 < y → f x + f y ≤ f (x + y)) (H4 : ∀ n : ℕ, 0 < n → (n : ℝ) ≤ f n)
(H5 : ∀ x : ℚ, 1 < x → (x : ℝ) ≤ f x) {a : ℚ} (ha1 : 1 < a) (hae : f a = a) : f x = x := by
-- Choose n such that 1 + x < a^n.
obtain ⟨N, hN⟩ := pow_unbounded_of_one_lt (1 + x) ha1
have h_big_enough : (1 : ℚ) < a ^ N - x := lt_sub_iff_add_lt.mpr hN
have h1 :=
calc
(x : ℝ) + (a ^ N - x : ℚ) ≤ f x + (a ^ N - x : ℚ) := by gcongr; exact H5 x hx
_ ≤ f x + f (a ^ N - x) := by gcongr; exact H5 _ h_big_enough
have hxp : 0 < x := by positivity
have hNp : 0 < N := by by_contra! H; rw [Nat.le_zero.mp H] at hN; linarith
have h2 :=
calc
f x + f (a ^ N - x) ≤ f (x + (a ^ N - x)) := H2 x (a ^ N - x) hxp (by positivity)
_ = f (a ^ N) := by ring_nf
_ = a ^ N := fixed_point_of_pos_nat_pow hNp H1 H4 H5 ha1 hae
_ = x + (a ^ N - x) := by ring
have heq := h1.antisymm (mod_cast h2)
linarith [H5 x hx, H5 _ h_big_enough]
|
/root/DuelModelResearch/mathlib4/Archive/Imo/Imo2013Q5.lean
|
{
"open": [],
"variables": []
}
|
[
{
"line": "obtain ⟨N, hN⟩ := pow_unbounded_of_one_lt (1 + x) ha1",
"before_state": "f : ℚ → ℝ\nx : ℚ\nhx : 1 < x\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\n⊢ f x = ↑x",
"after_state": "case intro\nf : ℚ → ℝ\nx : ℚ\nhx : 1 < x\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nN : ℕ\nhN : 1 + x < a ^ N\n⊢ f x = ↑x"
},
{
"line": "have h_big_enough : (1 : ℚ) < a ^ N - x := lt_sub_iff_add_lt.mpr hN",
"before_state": "case intro\nf : ℚ → ℝ\nx : ℚ\nhx : 1 < x\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nN : ℕ\nhN : 1 + x < a ^ N\n⊢ f x = ↑x",
"after_state": "case intro\nf : ℚ → ℝ\nx : ℚ\nhx : 1 < x\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nN : ℕ\nhN : 1 + x < a ^ N\nh_big_enough : 1 < a ^ N - x\n⊢ f x = ↑x"
},
{
"line": "refine_lift\n have h_big_enough : (1 : ℚ) < a ^ N - x := lt_sub_iff_add_lt.mpr hN;\n ?_",
"before_state": "case intro\nf : ℚ → ℝ\nx : ℚ\nhx : 1 < x\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nN : ℕ\nhN : 1 + x < a ^ N\n⊢ f x = ↑x",
"after_state": "case intro\nf : ℚ → ℝ\nx : ℚ\nhx : 1 < x\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nN : ℕ\nhN : 1 + x < a ^ N\nh_big_enough : 1 < a ^ N - x\n⊢ f x = ↑x"
},
{
"line": "focus\n (refine\n no_implicit_lambda%\n (have h_big_enough : (1 : ℚ) < a ^ N - x := lt_sub_iff_add_lt.mpr hN;\n ?_);\n rotate_right)",
"before_state": "case intro\nf : ℚ → ℝ\nx : ℚ\nhx : 1 < x\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nN : ℕ\nhN : 1 + x < a ^ N\n⊢ f x = ↑x",
"after_state": "case intro\nf : ℚ → ℝ\nx : ℚ\nhx : 1 < x\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nN : ℕ\nhN : 1 + x < a ^ N\nh_big_enough : 1 < a ^ N - x\n⊢ f x = ↑x"
},
{
"line": "refine\n no_implicit_lambda%\n (have h_big_enough : (1 : ℚ) < a ^ N - x := lt_sub_iff_add_lt.mpr hN;\n ?_)",
"before_state": "case intro\nf : ℚ → ℝ\nx : ℚ\nhx : 1 < x\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nN : ℕ\nhN : 1 + x < a ^ N\n⊢ f x = ↑x",
"after_state": "case intro\nf : ℚ → ℝ\nx : ℚ\nhx : 1 < x\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nN : ℕ\nhN : 1 + x < a ^ N\nh_big_enough : 1 < a ^ N - x\n⊢ f x = ↑x"
},
{
"line": "rotate_right",
"before_state": "case intro\nf : ℚ → ℝ\nx : ℚ\nhx : 1 < x\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nN : ℕ\nhN : 1 + x < a ^ N\nh_big_enough : 1 < a ^ N - x\n⊢ f x = ↑x",
"after_state": "case intro\nf : ℚ → ℝ\nx : ℚ\nhx : 1 < x\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nN : ℕ\nhN : 1 + x < a ^ N\nh_big_enough : 1 < a ^ N - x\n⊢ f x = ↑x"
},
{
"line": "have h1 :=\n calc\n (x : ℝ) + (a ^ N - x : ℚ) ≤ f x + (a ^ N - x : ℚ) := by gcongr; exact H5 x hx\n _ ≤ f x + f (a ^ N - x) := by gcongr; exact H5 _ h_big_enough",
"before_state": "case intro\nf : ℚ → ℝ\nx : ℚ\nhx : 1 < x\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nN : ℕ\nhN : 1 + x < a ^ N\nh_big_enough : 1 < a ^ N - x\n⊢ f x = ↑x",
"after_state": "case intro\nf : ℚ → ℝ\nx : ℚ\nhx : 1 < x\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nN : ℕ\nhN : 1 + x < a ^ N\nh_big_enough : 1 < a ^ N - x\nh1 : ↑x + ↑(a ^ N - x) ≤ f x + f (a ^ N - x)\n⊢ f x = ↑x"
},
{
"line": "refine_lift\n have h1 :=\n calc\n (x : ℝ) + (a ^ N - x : ℚ) ≤ f x + (a ^ N - x : ℚ) := by gcongr; exact H5 x hx\n _ ≤ f x + f (a ^ N - x) := by gcongr; exact H5 _ h_big_enough;\n ?_",
"before_state": "case intro\nf : ℚ → ℝ\nx : ℚ\nhx : 1 < x\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nN : ℕ\nhN : 1 + x < a ^ N\nh_big_enough : 1 < a ^ N - x\n⊢ f x = ↑x",
"after_state": "case intro\nf : ℚ → ℝ\nx : ℚ\nhx : 1 < x\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nN : ℕ\nhN : 1 + x < a ^ N\nh_big_enough : 1 < a ^ N - x\nh1 : ↑x + ↑(a ^ N - x) ≤ f x + f (a ^ N - x)\n⊢ f x = ↑x"
},
{
"line": "focus\n (refine\n no_implicit_lambda%\n (have h1 :=\n calc\n (x : ℝ) + (a ^ N - x : ℚ) ≤ f x + (a ^ N - x : ℚ) := by gcongr; exact H5 x hx\n _ ≤ f x + f (a ^ N - x) := by gcongr; exact H5 _ h_big_enough;\n ?_);\n rotate_right)",
"before_state": "case intro\nf : ℚ → ℝ\nx : ℚ\nhx : 1 < x\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nN : ℕ\nhN : 1 + x < a ^ N\nh_big_enough : 1 < a ^ N - x\n⊢ f x = ↑x",
"after_state": "case intro\nf : ℚ → ℝ\nx : ℚ\nhx : 1 < x\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nN : ℕ\nhN : 1 + x < a ^ N\nh_big_enough : 1 < a ^ N - x\nh1 : ↑x + ↑(a ^ N - x) ≤ f x + f (a ^ N - x)\n⊢ f x = ↑x"
},
{
"line": "refine\n no_implicit_lambda%\n (have h1 :=\n calc\n (x : ℝ) + (a ^ N - x : ℚ) ≤ f x + (a ^ N - x : ℚ) := by gcongr; exact H5 x hx\n _ ≤ f x + f (a ^ N - x) := by gcongr; exact H5 _ h_big_enough;\n ?_)",
"before_state": "case intro\nf : ℚ → ℝ\nx : ℚ\nhx : 1 < x\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nN : ℕ\nhN : 1 + x < a ^ N\nh_big_enough : 1 < a ^ N - x\n⊢ f x = ↑x",
"after_state": "case intro\nf : ℚ → ℝ\nx : ℚ\nhx : 1 < x\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nN : ℕ\nhN : 1 + x < a ^ N\nh_big_enough : 1 < a ^ N - x\nh1 : ↑x + ↑(a ^ N - x) ≤ f x + f (a ^ N - x)\n⊢ f x = ↑x"
},
{
"line": "gcongr",
"before_state": "f : ℚ → ℝ\nx : ℚ\nhx : 1 < x\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nN : ℕ\nhN : 1 + x < a ^ N\nh_big_enough : 1 < a ^ N - x\n⊢ ↑x + ↑(a ^ N - x) ≤ f x + ↑(a ^ N - x)",
"after_state": "case bc\nf : ℚ → ℝ\nx : ℚ\nhx : 1 < x\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nN : ℕ\nhN : 1 + x < a ^ N\nh_big_enough : 1 < a ^ N - x\n⊢ ↑x ≤ f x"
},
{
"line": "exact H5 x hx",
"before_state": "case bc\nf : ℚ → ℝ\nx : ℚ\nhx : 1 < x\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nN : ℕ\nhN : 1 + x < a ^ N\nh_big_enough : 1 < a ^ N - x\n⊢ ↑x ≤ f x",
"after_state": "No Goals!"
},
{
"line": "gcongr",
"before_state": "f : ℚ → ℝ\nx : ℚ\nhx : 1 < x\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nN : ℕ\nhN : 1 + x < a ^ N\nh_big_enough : 1 < a ^ N - x\n⊢ f x + ↑(a ^ N - x) ≤ f x + f (a ^ N - x)",
"after_state": "case bc\nf : ℚ → ℝ\nx : ℚ\nhx : 1 < x\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nN : ℕ\nhN : 1 + x < a ^ N\nh_big_enough : 1 < a ^ N - x\n⊢ ↑(a ^ N - x) ≤ f (a ^ N - x)"
},
{
"line": "exact H5 _ h_big_enough",
"before_state": "case bc\nf : ℚ → ℝ\nx : ℚ\nhx : 1 < x\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nN : ℕ\nhN : 1 + x < a ^ N\nh_big_enough : 1 < a ^ N - x\n⊢ ↑(a ^ N - x) ≤ f (a ^ N - x)",
"after_state": "No Goals!"
},
{
"line": "rotate_right",
"before_state": "case intro\nf : ℚ → ℝ\nx : ℚ\nhx : 1 < x\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nN : ℕ\nhN : 1 + x < a ^ N\nh_big_enough : 1 < a ^ N - x\nh1 : ↑x + ↑(a ^ N - x) ≤ f x + f (a ^ N - x)\n⊢ f x = ↑x",
"after_state": "case intro\nf : ℚ → ℝ\nx : ℚ\nhx : 1 < x\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nN : ℕ\nhN : 1 + x < a ^ N\nh_big_enough : 1 < a ^ N - x\nh1 : ↑x + ↑(a ^ N - x) ≤ f x + f (a ^ N - x)\n⊢ f x = ↑x"
},
{
"line": "have hxp : 0 < x := by positivity",
"before_state": "case intro\nf : ℚ → ℝ\nx : ℚ\nhx : 1 < x\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nN : ℕ\nhN : 1 + x < a ^ N\nh_big_enough : 1 < a ^ N - x\nh1 : ↑x + ↑(a ^ N - x) ≤ f x + f (a ^ N - x)\n⊢ f x = ↑x",
"after_state": "case intro\nf : ℚ → ℝ\nx : ℚ\nhx : 1 < x\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nN : ℕ\nhN : 1 + x < a ^ N\nh_big_enough : 1 < a ^ N - x\nh1 : ↑x + ↑(a ^ N - x) ≤ f x + f (a ^ N - x)\nhxp : 0 < x\n⊢ f x = ↑x"
},
{
"line": "focus\n refine\n no_implicit_lambda%\n (have hxp : 0 < x := ?body✝;\n ?_)\n case body✝ => with_annotate_state\"by\" (positivity)",
"before_state": "case intro\nf : ℚ → ℝ\nx : ℚ\nhx : 1 < x\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nN : ℕ\nhN : 1 + x < a ^ N\nh_big_enough : 1 < a ^ N - x\nh1 : ↑x + ↑(a ^ N - x) ≤ f x + f (a ^ N - x)\n⊢ f x = ↑x",
"after_state": "case intro\nf : ℚ → ℝ\nx : ℚ\nhx : 1 < x\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nN : ℕ\nhN : 1 + x < a ^ N\nh_big_enough : 1 < a ^ N - x\nh1 : ↑x + ↑(a ^ N - x) ≤ f x + f (a ^ N - x)\nhxp : 0 < x\n⊢ f x = ↑x"
},
{
"line": "refine\n no_implicit_lambda%\n (have hxp : 0 < x := ?body✝;\n ?_)",
"before_state": "case intro\nf : ℚ → ℝ\nx : ℚ\nhx : 1 < x\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nN : ℕ\nhN : 1 + x < a ^ N\nh_big_enough : 1 < a ^ N - x\nh1 : ↑x + ↑(a ^ N - x) ≤ f x + f (a ^ N - x)\n⊢ f x = ↑x",
"after_state": "case body\nf : ℚ → ℝ\nx : ℚ\nhx : 1 < x\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nN : ℕ\nhN : 1 + x < a ^ N\nh_big_enough : 1 < a ^ N - x\nh1 : ↑x + ↑(a ^ N - x) ≤ f x + f (a ^ N - x)\n⊢ 0 < x\n---\ncase intro\nf : ℚ → ℝ\nx : ℚ\nhx : 1 < x\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nN : ℕ\nhN : 1 + x < a ^ N\nh_big_enough : 1 < a ^ N - x\nh1 : ↑x + ↑(a ^ N - x) ≤ f x + f (a ^ N - x)\nhxp : 0 < x\n⊢ f x = ↑x"
},
{
"line": "case body✝ => with_annotate_state\"by\" (positivity)",
"before_state": "case body\nf : ℚ → ℝ\nx : ℚ\nhx : 1 < x\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nN : ℕ\nhN : 1 + x < a ^ N\nh_big_enough : 1 < a ^ N - x\nh1 : ↑x + ↑(a ^ N - x) ≤ f x + f (a ^ N - x)\n⊢ 0 < x\n---\ncase intro\nf : ℚ → ℝ\nx : ℚ\nhx : 1 < x\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nN : ℕ\nhN : 1 + x < a ^ N\nh_big_enough : 1 < a ^ N - x\nh1 : ↑x + ↑(a ^ N - x) ≤ f x + f (a ^ N - x)\nhxp : 0 < x\n⊢ f x = ↑x",
"after_state": "case intro\nf : ℚ → ℝ\nx : ℚ\nhx : 1 < x\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nN : ℕ\nhN : 1 + x < a ^ N\nh_big_enough : 1 < a ^ N - x\nh1 : ↑x + ↑(a ^ N - x) ≤ f x + f (a ^ N - x)\nhxp : 0 < x\n⊢ f x = ↑x"
},
{
"line": "with_annotate_state\"by\" (positivity)",
"before_state": "f : ℚ → ℝ\nx : ℚ\nhx : 1 < x\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nN : ℕ\nhN : 1 + x < a ^ N\nh_big_enough : 1 < a ^ N - x\nh1 : ↑x + ↑(a ^ N - x) ≤ f x + f (a ^ N - x)\n⊢ 0 < x",
"after_state": "No Goals!"
},
{
"line": "positivity",
"before_state": "f : ℚ → ℝ\nx : ℚ\nhx : 1 < x\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nN : ℕ\nhN : 1 + x < a ^ N\nh_big_enough : 1 < a ^ N - x\nh1 : ↑x + ↑(a ^ N - x) ≤ f x + f (a ^ N - x)\n⊢ 0 < x",
"after_state": "No Goals!"
},
{
"line": "have hNp : 0 < N := by by_contra! H; rw [Nat.le_zero.mp H] at hN; linarith",
"before_state": "case intro\nf : ℚ → ℝ\nx : ℚ\nhx : 1 < x\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nN : ℕ\nhN : 1 + x < a ^ N\nh_big_enough : 1 < a ^ N - x\nh1 : ↑x + ↑(a ^ N - x) ≤ f x + f (a ^ N - x)\nhxp : 0 < x\n⊢ f x = ↑x",
"after_state": "case intro\nf : ℚ → ℝ\nx : ℚ\nhx : 1 < x\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nN : ℕ\nhN : 1 + x < a ^ N\nh_big_enough : 1 < a ^ N - x\nh1 : ↑x + ↑(a ^ N - x) ≤ f x + f (a ^ N - x)\nhxp : 0 < x\nhNp : 0 < N\n⊢ f x = ↑x"
},
{
"line": "focus\n refine\n no_implicit_lambda%\n (have hNp : 0 < N := ?body✝;\n ?_)\n case body✝ => with_annotate_state\"by\" (by_contra! H; rw [Nat.le_zero.mp H] at hN; linarith)",
"before_state": "case intro\nf : ℚ → ℝ\nx : ℚ\nhx : 1 < x\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nN : ℕ\nhN : 1 + x < a ^ N\nh_big_enough : 1 < a ^ N - x\nh1 : ↑x + ↑(a ^ N - x) ≤ f x + f (a ^ N - x)\nhxp : 0 < x\n⊢ f x = ↑x",
"after_state": "case intro\nf : ℚ → ℝ\nx : ℚ\nhx : 1 < x\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nN : ℕ\nhN : 1 + x < a ^ N\nh_big_enough : 1 < a ^ N - x\nh1 : ↑x + ↑(a ^ N - x) ≤ f x + f (a ^ N - x)\nhxp : 0 < x\nhNp : 0 < N\n⊢ f x = ↑x"
},
{
"line": "refine\n no_implicit_lambda%\n (have hNp : 0 < N := ?body✝;\n ?_)",
"before_state": "case intro\nf : ℚ → ℝ\nx : ℚ\nhx : 1 < x\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nN : ℕ\nhN : 1 + x < a ^ N\nh_big_enough : 1 < a ^ N - x\nh1 : ↑x + ↑(a ^ N - x) ≤ f x + f (a ^ N - x)\nhxp : 0 < x\n⊢ f x = ↑x",
"after_state": "case body\nf : ℚ → ℝ\nx : ℚ\nhx : 1 < x\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nN : ℕ\nhN : 1 + x < a ^ N\nh_big_enough : 1 < a ^ N - x\nh1 : ↑x + ↑(a ^ N - x) ≤ f x + f (a ^ N - x)\nhxp : 0 < x\n⊢ 0 < N\n---\ncase intro\nf : ℚ → ℝ\nx : ℚ\nhx : 1 < x\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nN : ℕ\nhN : 1 + x < a ^ N\nh_big_enough : 1 < a ^ N - x\nh1 : ↑x + ↑(a ^ N - x) ≤ f x + f (a ^ N - x)\nhxp : 0 < x\nhNp : 0 < N\n⊢ f x = ↑x"
},
{
"line": "case body✝ => with_annotate_state\"by\" (by_contra! H; rw [Nat.le_zero.mp H] at hN; linarith)",
"before_state": "case body\nf : ℚ → ℝ\nx : ℚ\nhx : 1 < x\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nN : ℕ\nhN : 1 + x < a ^ N\nh_big_enough : 1 < a ^ N - x\nh1 : ↑x + ↑(a ^ N - x) ≤ f x + f (a ^ N - x)\nhxp : 0 < x\n⊢ 0 < N\n---\ncase intro\nf : ℚ → ℝ\nx : ℚ\nhx : 1 < x\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nN : ℕ\nhN : 1 + x < a ^ N\nh_big_enough : 1 < a ^ N - x\nh1 : ↑x + ↑(a ^ N - x) ≤ f x + f (a ^ N - x)\nhxp : 0 < x\nhNp : 0 < N\n⊢ f x = ↑x",
"after_state": "case intro\nf : ℚ → ℝ\nx : ℚ\nhx : 1 < x\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nN : ℕ\nhN : 1 + x < a ^ N\nh_big_enough : 1 < a ^ N - x\nh1 : ↑x + ↑(a ^ N - x) ≤ f x + f (a ^ N - x)\nhxp : 0 < x\nhNp : 0 < N\n⊢ f x = ↑x"
},
{
"line": "with_annotate_state\"by\" (by_contra! H; rw [Nat.le_zero.mp H] at hN; linarith)",
"before_state": "f : ℚ → ℝ\nx : ℚ\nhx : 1 < x\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nN : ℕ\nhN : 1 + x < a ^ N\nh_big_enough : 1 < a ^ N - x\nh1 : ↑x + ↑(a ^ N - x) ≤ f x + f (a ^ N - x)\nhxp : 0 < x\n⊢ 0 < N",
"after_state": "No Goals!"
},
{
"line": "by_contra! H",
"before_state": "f : ℚ → ℝ\nx : ℚ\nhx : 1 < x\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nN : ℕ\nhN : 1 + x < a ^ N\nh_big_enough : 1 < a ^ N - x\nh1 : ↑x + ↑(a ^ N - x) ≤ f x + f (a ^ N - x)\nhxp : 0 < x\n⊢ 0 < N",
"after_state": "f : ℚ → ℝ\nx : ℚ\nhx : 1 < x\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nN : ℕ\nhN : 1 + x < a ^ N\nh_big_enough : 1 < a ^ N - x\nh1 : ↑x + ↑(a ^ N - x) ≤ f x + f (a ^ N - x)\nhxp : 0 < x\nH : N ≤ 0\n⊢ False"
},
{
"line": "by_contra H",
"before_state": "f : ℚ → ℝ\nx : ℚ\nhx : 1 < x\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nN : ℕ\nhN : 1 + x < a ^ N\nh_big_enough : 1 < a ^ N - x\nh1 : ↑x + ↑(a ^ N - x) ≤ f x + f (a ^ N - x)\nhxp : 0 < x\n⊢ 0 < N",
"after_state": "f : ℚ → ℝ\nx : ℚ\nhx : 1 < x\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nN : ℕ\nhN : 1 + x < a ^ N\nh_big_enough : 1 < a ^ N - x\nh1 : ↑x + ↑(a ^ N - x) ≤ f x + f (a ^ N - x)\nhxp : 0 < x\nH : ¬0 < N\n⊢ False"
},
{
"line": "first\n| guard_target = Not✝ _; intro H\n| refine (Decidable.byContradiction✝ fun H => ?_ :)\n| refine (Classical.byContradiction✝ fun H => ?_ :)",
"before_state": "f : ℚ → ℝ\nx : ℚ\nhx : 1 < x\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nN : ℕ\nhN : 1 + x < a ^ N\nh_big_enough : 1 < a ^ N - x\nh1 : ↑x + ↑(a ^ N - x) ≤ f x + f (a ^ N - x)\nhxp : 0 < x\n⊢ 0 < N",
"after_state": "f : ℚ → ℝ\nx : ℚ\nhx : 1 < x\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nN : ℕ\nhN : 1 + x < a ^ N\nh_big_enough : 1 < a ^ N - x\nh1 : ↑x + ↑(a ^ N - x) ≤ f x + f (a ^ N - x)\nhxp : 0 < x\nH : ¬0 < N\n⊢ False"
},
{
"line": "guard_target = Not✝ _",
"before_state": "f : ℚ → ℝ\nx : ℚ\nhx : 1 < x\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nN : ℕ\nhN : 1 + x < a ^ N\nh_big_enough : 1 < a ^ N - x\nh1 : ↑x + ↑(a ^ N - x) ≤ f x + f (a ^ N - x)\nhxp : 0 < x\n⊢ 0 < N",
"after_state": "f : ℚ → ℝ\nx : ℚ\nhx : 1 < x\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nN : ℕ\nhN : 1 + x < a ^ N\nh_big_enough : 1 < a ^ N - x\nh1 : ↑x + ↑(a ^ N - x) ≤ f x + f (a ^ N - x)\nhxp : 0 < x\n⊢ 0 < N"
},
{
"line": "refine (Decidable.byContradiction✝ fun H => ?_ :)",
"before_state": "f : ℚ → ℝ\nx : ℚ\nhx : 1 < x\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nN : ℕ\nhN : 1 + x < a ^ N\nh_big_enough : 1 < a ^ N - x\nh1 : ↑x + ↑(a ^ N - x) ≤ f x + f (a ^ N - x)\nhxp : 0 < x\n⊢ 0 < N",
"after_state": "f : ℚ → ℝ\nx : ℚ\nhx : 1 < x\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nN : ℕ\nhN : 1 + x < a ^ N\nh_big_enough : 1 < a ^ N - x\nh1 : ↑x + ↑(a ^ N - x) ≤ f x + f (a ^ N - x)\nhxp : 0 < x\n⊢ 0 < N"
},
{
"line": "refine (Classical.byContradiction✝ fun H => ?_ :)",
"before_state": "f : ℚ → ℝ\nx : ℚ\nhx : 1 < x\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nN : ℕ\nhN : 1 + x < a ^ N\nh_big_enough : 1 < a ^ N - x\nh1 : ↑x + ↑(a ^ N - x) ≤ f x + f (a ^ N - x)\nhxp : 0 < x\n⊢ 0 < N",
"after_state": "f : ℚ → ℝ\nx : ℚ\nhx : 1 < x\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nN : ℕ\nhN : 1 + x < a ^ N\nh_big_enough : 1 < a ^ N - x\nh1 : ↑x + ↑(a ^ N - x) ≤ f x + f (a ^ N - x)\nhxp : 0 < x\nH : ¬0 < N\n⊢ False"
},
{
"line": "try push_neg at H",
"before_state": "f : ℚ → ℝ\nx : ℚ\nhx : 1 < x\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nN : ℕ\nhN : 1 + x < a ^ N\nh_big_enough : 1 < a ^ N - x\nh1 : ↑x + ↑(a ^ N - x) ≤ f x + f (a ^ N - x)\nhxp : 0 < x\nH : ¬0 < N\n⊢ False",
"after_state": "f : ℚ → ℝ\nx : ℚ\nhx : 1 < x\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nN : ℕ\nhN : 1 + x < a ^ N\nh_big_enough : 1 < a ^ N - x\nh1 : ↑x + ↑(a ^ N - x) ≤ f x + f (a ^ N - x)\nhxp : 0 < x\nH : N ≤ 0\n⊢ False"
},
{
"line": "first\n| push_neg at H\n| skip",
"before_state": "f : ℚ → ℝ\nx : ℚ\nhx : 1 < x\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nN : ℕ\nhN : 1 + x < a ^ N\nh_big_enough : 1 < a ^ N - x\nh1 : ↑x + ↑(a ^ N - x) ≤ f x + f (a ^ N - x)\nhxp : 0 < x\nH : ¬0 < N\n⊢ False",
"after_state": "f : ℚ → ℝ\nx : ℚ\nhx : 1 < x\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nN : ℕ\nhN : 1 + x < a ^ N\nh_big_enough : 1 < a ^ N - x\nh1 : ↑x + ↑(a ^ N - x) ≤ f x + f (a ^ N - x)\nhxp : 0 < x\nH : N ≤ 0\n⊢ False"
},
{
"line": "push_neg at H",
"before_state": "f : ℚ → ℝ\nx : ℚ\nhx : 1 < x\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nN : ℕ\nhN : 1 + x < a ^ N\nh_big_enough : 1 < a ^ N - x\nh1 : ↑x + ↑(a ^ N - x) ≤ f x + f (a ^ N - x)\nhxp : 0 < x\nH : ¬0 < N\n⊢ False",
"after_state": "f : ℚ → ℝ\nx : ℚ\nhx : 1 < x\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nN : ℕ\nhN : 1 + x < a ^ N\nh_big_enough : 1 < a ^ N - x\nh1 : ↑x + ↑(a ^ N - x) ≤ f x + f (a ^ N - x)\nhxp : 0 < x\nH : N ≤ 0\n⊢ False"
},
{
"line": "rw [Nat.le_zero.mp H] at hN",
"before_state": "f : ℚ → ℝ\nx : ℚ\nhx : 1 < x\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nN : ℕ\nhN : 1 + x < a ^ N\nh_big_enough : 1 < a ^ N - x\nh1 : ↑x + ↑(a ^ N - x) ≤ f x + f (a ^ N - x)\nhxp : 0 < x\nH : N ≤ 0\n⊢ False",
"after_state": "f : ℚ → ℝ\nx : ℚ\nhx : 1 < x\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nN : ℕ\nhN : 1 + x < a ^ 0\nh_big_enough : 1 < a ^ N - x\nh1 : ↑x + ↑(a ^ N - x) ≤ f x + f (a ^ N - x)\nhxp : 0 < x\nH : N ≤ 0\n⊢ False"
},
{
"line": "rewrite [Nat.le_zero.mp H] at hN",
"before_state": "f : ℚ → ℝ\nx : ℚ\nhx : 1 < x\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nN : ℕ\nhN : 1 + x < a ^ N\nh_big_enough : 1 < a ^ N - x\nh1 : ↑x + ↑(a ^ N - x) ≤ f x + f (a ^ N - x)\nhxp : 0 < x\nH : N ≤ 0\n⊢ False",
"after_state": "f : ℚ → ℝ\nx : ℚ\nhx : 1 < x\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nN : ℕ\nhN : 1 + x < a ^ 0\nh_big_enough : 1 < a ^ N - x\nh1 : ↑x + ↑(a ^ N - x) ≤ f x + f (a ^ N - x)\nhxp : 0 < x\nH : N ≤ 0\n⊢ False"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "f : ℚ → ℝ\nx : ℚ\nhx : 1 < x\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nN : ℕ\nhN : 1 + x < a ^ 0\nh_big_enough : 1 < a ^ N - x\nh1 : ↑x + ↑(a ^ N - x) ≤ f x + f (a ^ N - x)\nhxp : 0 < x\nH : N ≤ 0\n⊢ False",
"after_state": "f : ℚ → ℝ\nx : ℚ\nhx : 1 < x\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nN : ℕ\nhN : 1 + x < a ^ 0\nh_big_enough : 1 < a ^ N - x\nh1 : ↑x + ↑(a ^ N - x) ≤ f x + f (a ^ N - x)\nhxp : 0 < x\nH : N ≤ 0\n⊢ False"
},
{
"line": "try (with_reducible rfl)",
"before_state": "f : ℚ → ℝ\nx : ℚ\nhx : 1 < x\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nN : ℕ\nhN : 1 + x < a ^ 0\nh_big_enough : 1 < a ^ N - x\nh1 : ↑x + ↑(a ^ N - x) ≤ f x + f (a ^ N - x)\nhxp : 0 < x\nH : N ≤ 0\n⊢ False",
"after_state": "f : ℚ → ℝ\nx : ℚ\nhx : 1 < x\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nN : ℕ\nhN : 1 + x < a ^ 0\nh_big_enough : 1 < a ^ N - x\nh1 : ↑x + ↑(a ^ N - x) ≤ f x + f (a ^ N - x)\nhxp : 0 < x\nH : N ≤ 0\n⊢ False"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "f : ℚ → ℝ\nx : ℚ\nhx : 1 < x\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nN : ℕ\nhN : 1 + x < a ^ 0\nh_big_enough : 1 < a ^ N - x\nh1 : ↑x + ↑(a ^ N - x) ≤ f x + f (a ^ N - x)\nhxp : 0 < x\nH : N ≤ 0\n⊢ False",
"after_state": "f : ℚ → ℝ\nx : ℚ\nhx : 1 < x\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nN : ℕ\nhN : 1 + x < a ^ 0\nh_big_enough : 1 < a ^ N - x\nh1 : ↑x + ↑(a ^ N - x) ≤ f x + f (a ^ N - x)\nhxp : 0 < x\nH : N ≤ 0\n⊢ False"
},
{
"line": "with_reducible rfl",
"before_state": "f : ℚ → ℝ\nx : ℚ\nhx : 1 < x\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nN : ℕ\nhN : 1 + x < a ^ 0\nh_big_enough : 1 < a ^ N - x\nh1 : ↑x + ↑(a ^ N - x) ≤ f x + f (a ^ N - x)\nhxp : 0 < x\nH : N ≤ 0\n⊢ False",
"after_state": "f : ℚ → ℝ\nx : ℚ\nhx : 1 < x\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nN : ℕ\nhN : 1 + x < a ^ 0\nh_big_enough : 1 < a ^ N - x\nh1 : ↑x + ↑(a ^ N - x) ≤ f x + f (a ^ N - x)\nhxp : 0 < x\nH : N ≤ 0\n⊢ False"
},
{
"line": "rfl",
"before_state": "f : ℚ → ℝ\nx : ℚ\nhx : 1 < x\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nN : ℕ\nhN : 1 + x < a ^ 0\nh_big_enough : 1 < a ^ N - x\nh1 : ↑x + ↑(a ^ N - x) ≤ f x + f (a ^ N - x)\nhxp : 0 < x\nH : N ≤ 0\n⊢ False",
"after_state": "f : ℚ → ℝ\nx : ℚ\nhx : 1 < x\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nN : ℕ\nhN : 1 + x < a ^ 0\nh_big_enough : 1 < a ^ N - x\nh1 : ↑x + ↑(a ^ N - x) ≤ f x + f (a ^ N - x)\nhxp : 0 < x\nH : N ≤ 0\n⊢ False"
},
{
"line": "apply_rfl",
"before_state": "f : ℚ → ℝ\nx : ℚ\nhx : 1 < x\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nN : ℕ\nhN : 1 + x < a ^ 0\nh_big_enough : 1 < a ^ N - x\nh1 : ↑x + ↑(a ^ N - x) ≤ f x + f (a ^ N - x)\nhxp : 0 < x\nH : N ≤ 0\n⊢ False",
"after_state": "f : ℚ → ℝ\nx : ℚ\nhx : 1 < x\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nN : ℕ\nhN : 1 + x < a ^ 0\nh_big_enough : 1 < a ^ N - x\nh1 : ↑x + ↑(a ^ N - x) ≤ f x + f (a ^ N - x)\nhxp : 0 < x\nH : N ≤ 0\n⊢ False"
},
{
"line": "skip",
"before_state": "f : ℚ → ℝ\nx : ℚ\nhx : 1 < x\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nN : ℕ\nhN : 1 + x < a ^ 0\nh_big_enough : 1 < a ^ N - x\nh1 : ↑x + ↑(a ^ N - x) ≤ f x + f (a ^ N - x)\nhxp : 0 < x\nH : N ≤ 0\n⊢ False",
"after_state": "f : ℚ → ℝ\nx : ℚ\nhx : 1 < x\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nN : ℕ\nhN : 1 + x < a ^ 0\nh_big_enough : 1 < a ^ N - x\nh1 : ↑x + ↑(a ^ N - x) ≤ f x + f (a ^ N - x)\nhxp : 0 < x\nH : N ≤ 0\n⊢ False"
},
{
"line": "linarith",
"before_state": "f : ℚ → ℝ\nx : ℚ\nhx : 1 < x\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nN : ℕ\nhN : 1 + x < a ^ 0\nh_big_enough : 1 < a ^ N - x\nh1 : ↑x + ↑(a ^ N - x) ≤ f x + f (a ^ N - x)\nhxp : 0 < x\nH : N ≤ 0\n⊢ False",
"after_state": "No Goals!"
},
{
"line": "ring1",
"before_state": "f : ℚ → ℝ\nx : ℚ\nhx : 1 < x\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nN : ℕ\nhN : 1 + x < a ^ 0\nh_big_enough : 1 < a ^ N - x\nh1 : ↑x + ↑(a ^ N - x) ≤ f x + f (a ^ N - x)\nhxp : 0 < x\nH : N ≤ 0\n⊢ -1 + (1 - x) + (1 + x - a ^ 0) = 0",
"after_state": "No Goals!"
},
{
"line": "have h2 :=\n calc\n f x + f (a ^ N - x) ≤ f (x + (a ^ N - x)) := H2 x (a ^ N - x) hxp (by positivity)\n _ = f (a ^ N) := by ring_nf\n _ = a ^ N := (fixed_point_of_pos_nat_pow hNp H1 H4 H5 ha1 hae)\n _ = x + (a ^ N - x) := by ring",
"before_state": "case intro\nf : ℚ → ℝ\nx : ℚ\nhx : 1 < x\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nN : ℕ\nhN : 1 + x < a ^ N\nh_big_enough : 1 < a ^ N - x\nh1 : ↑x + ↑(a ^ N - x) ≤ f x + f (a ^ N - x)\nhxp : 0 < x\nhNp : 0 < N\n⊢ f x = ↑x",
"after_state": "case intro\nf : ℚ → ℝ\nx : ℚ\nhx : 1 < x\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nN : ℕ\nhN : 1 + x < a ^ N\nh_big_enough : 1 < a ^ N - x\nh1 : ↑x + ↑(a ^ N - x) ≤ f x + f (a ^ N - x)\nhxp : 0 < x\nhNp : 0 < N\nh2 : f x + f (a ^ N - x) ≤ ↑x + (↑a ^ N - ↑x)\n⊢ f x = ↑x"
},
{
"line": "refine_lift\n have h2 :=\n calc\n f x + f (a ^ N - x) ≤ f (x + (a ^ N - x)) := H2 x (a ^ N - x) hxp (by positivity)\n _ = f (a ^ N) := by ring_nf\n _ = a ^ N := (fixed_point_of_pos_nat_pow hNp H1 H4 H5 ha1 hae)\n _ = x + (a ^ N - x) := by ring;\n ?_",
"before_state": "case intro\nf : ℚ → ℝ\nx : ℚ\nhx : 1 < x\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nN : ℕ\nhN : 1 + x < a ^ N\nh_big_enough : 1 < a ^ N - x\nh1 : ↑x + ↑(a ^ N - x) ≤ f x + f (a ^ N - x)\nhxp : 0 < x\nhNp : 0 < N\n⊢ f x = ↑x",
"after_state": "case intro\nf : ℚ → ℝ\nx : ℚ\nhx : 1 < x\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nN : ℕ\nhN : 1 + x < a ^ N\nh_big_enough : 1 < a ^ N - x\nh1 : ↑x + ↑(a ^ N - x) ≤ f x + f (a ^ N - x)\nhxp : 0 < x\nhNp : 0 < N\nh2 : f x + f (a ^ N - x) ≤ ↑x + (↑a ^ N - ↑x)\n⊢ f x = ↑x"
},
{
"line": "focus\n (refine\n no_implicit_lambda%\n (have h2 :=\n calc\n f x + f (a ^ N - x) ≤ f (x + (a ^ N - x)) := H2 x (a ^ N - x) hxp (by positivity)\n _ = f (a ^ N) := by ring_nf\n _ = a ^ N := (fixed_point_of_pos_nat_pow hNp H1 H4 H5 ha1 hae)\n _ = x + (a ^ N - x) := by ring;\n ?_);\n rotate_right)",
"before_state": "case intro\nf : ℚ → ℝ\nx : ℚ\nhx : 1 < x\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nN : ℕ\nhN : 1 + x < a ^ N\nh_big_enough : 1 < a ^ N - x\nh1 : ↑x + ↑(a ^ N - x) ≤ f x + f (a ^ N - x)\nhxp : 0 < x\nhNp : 0 < N\n⊢ f x = ↑x",
"after_state": "case intro\nf : ℚ → ℝ\nx : ℚ\nhx : 1 < x\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nN : ℕ\nhN : 1 + x < a ^ N\nh_big_enough : 1 < a ^ N - x\nh1 : ↑x + ↑(a ^ N - x) ≤ f x + f (a ^ N - x)\nhxp : 0 < x\nhNp : 0 < N\nh2 : f x + f (a ^ N - x) ≤ ↑x + (↑a ^ N - ↑x)\n⊢ f x = ↑x"
},
{
"line": "refine\n no_implicit_lambda%\n (have h2 :=\n calc\n f x + f (a ^ N - x) ≤ f (x + (a ^ N - x)) := H2 x (a ^ N - x) hxp (by positivity)\n _ = f (a ^ N) := by ring_nf\n _ = a ^ N := (fixed_point_of_pos_nat_pow hNp H1 H4 H5 ha1 hae)\n _ = x + (a ^ N - x) := by ring;\n ?_)",
"before_state": "case intro\nf : ℚ → ℝ\nx : ℚ\nhx : 1 < x\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nN : ℕ\nhN : 1 + x < a ^ N\nh_big_enough : 1 < a ^ N - x\nh1 : ↑x + ↑(a ^ N - x) ≤ f x + f (a ^ N - x)\nhxp : 0 < x\nhNp : 0 < N\n⊢ f x = ↑x",
"after_state": "case intro\nf : ℚ → ℝ\nx : ℚ\nhx : 1 < x\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nN : ℕ\nhN : 1 + x < a ^ N\nh_big_enough : 1 < a ^ N - x\nh1 : ↑x + ↑(a ^ N - x) ≤ f x + f (a ^ N - x)\nhxp : 0 < x\nhNp : 0 < N\nh2 : f x + f (a ^ N - x) ≤ ↑x + (↑a ^ N - ↑x)\n⊢ f x = ↑x"
},
{
"line": "positivity",
"before_state": "f : ℚ → ℝ\nx : ℚ\nhx : 1 < x\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nN : ℕ\nhN : 1 + x < a ^ N\nh_big_enough : 1 < a ^ N - x\nh1 : ↑x + ↑(a ^ N - x) ≤ f x + f (a ^ N - x)\nhxp : 0 < x\nhNp : 0 < N\n⊢ 0 < a ^ N - x",
"after_state": "No Goals!"
},
{
"line": "ring_nf",
"before_state": "f : ℚ → ℝ\nx : ℚ\nhx : 1 < x\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nN : ℕ\nhN : 1 + x < a ^ N\nh_big_enough : 1 < a ^ N - x\nh1 : ↑x + ↑(a ^ N - x) ≤ f x + f (a ^ N - x)\nhxp : 0 < x\nhNp : 0 < N\n⊢ f (x + (a ^ N - x)) = f (a ^ N)",
"after_state": "No Goals!"
},
{
"line": "ring",
"before_state": "f : ℚ → ℝ\nx : ℚ\nhx : 1 < x\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nN : ℕ\nhN : 1 + x < a ^ N\nh_big_enough : 1 < a ^ N - x\nh1 : ↑x + ↑(a ^ N - x) ≤ f x + f (a ^ N - x)\nhxp : 0 < x\nhNp : 0 < N\n⊢ ↑a ^ N = ↑x + (↑a ^ N - ↑x)",
"after_state": "No Goals!"
},
{
"line": "first\n| ring1\n|\n try_this ring_nf\"\\n\\nThe `ring` tactic failed to close the goal. Use `ring_nf` to obtain a normal form.\n \\nNote that `ring` works primarily in *commutative* rings. \\\n If you have a noncommutative ring, abelian group or module, consider using \\\n `noncomm_ring`, `abel` or `module` instead.\"",
"before_state": "f : ℚ → ℝ\nx : ℚ\nhx : 1 < x\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nN : ℕ\nhN : 1 + x < a ^ N\nh_big_enough : 1 < a ^ N - x\nh1 : ↑x + ↑(a ^ N - x) ≤ f x + f (a ^ N - x)\nhxp : 0 < x\nhNp : 0 < N\n⊢ ↑a ^ N = ↑x + (↑a ^ N - ↑x)",
"after_state": "No Goals!"
},
{
"line": "ring1",
"before_state": "f : ℚ → ℝ\nx : ℚ\nhx : 1 < x\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nN : ℕ\nhN : 1 + x < a ^ N\nh_big_enough : 1 < a ^ N - x\nh1 : ↑x + ↑(a ^ N - x) ≤ f x + f (a ^ N - x)\nhxp : 0 < x\nhNp : 0 < N\n⊢ ↑a ^ N = ↑x + (↑a ^ N - ↑x)",
"after_state": "No Goals!"
},
{
"line": "rotate_right",
"before_state": "case intro\nf : ℚ → ℝ\nx : ℚ\nhx : 1 < x\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nN : ℕ\nhN : 1 + x < a ^ N\nh_big_enough : 1 < a ^ N - x\nh1 : ↑x + ↑(a ^ N - x) ≤ f x + f (a ^ N - x)\nhxp : 0 < x\nhNp : 0 < N\nh2 : f x + f (a ^ N - x) ≤ ↑x + (↑a ^ N - ↑x)\n⊢ f x = ↑x",
"after_state": "case intro\nf : ℚ → ℝ\nx : ℚ\nhx : 1 < x\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nN : ℕ\nhN : 1 + x < a ^ N\nh_big_enough : 1 < a ^ N - x\nh1 : ↑x + ↑(a ^ N - x) ≤ f x + f (a ^ N - x)\nhxp : 0 < x\nhNp : 0 < N\nh2 : f x + f (a ^ N - x) ≤ ↑x + (↑a ^ N - ↑x)\n⊢ f x = ↑x"
},
{
"line": "have heq := h1.antisymm (mod_cast h2)",
"before_state": "case intro\nf : ℚ → ℝ\nx : ℚ\nhx : 1 < x\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nN : ℕ\nhN : 1 + x < a ^ N\nh_big_enough : 1 < a ^ N - x\nh1 : ↑x + ↑(a ^ N - x) ≤ f x + f (a ^ N - x)\nhxp : 0 < x\nhNp : 0 < N\nh2 : f x + f (a ^ N - x) ≤ ↑x + (↑a ^ N - ↑x)\n⊢ f x = ↑x",
"after_state": "case intro\nf : ℚ → ℝ\nx : ℚ\nhx : 1 < x\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nN : ℕ\nhN : 1 + x < a ^ N\nh_big_enough : 1 < a ^ N - x\nh1 : ↑x + ↑(a ^ N - x) ≤ f x + f (a ^ N - x)\nhxp : 0 < x\nhNp : 0 < N\nh2 : f x + f (a ^ N - x) ≤ ↑x + (↑a ^ N - ↑x)\nheq : ↑x + ↑(a ^ N - x) = f x + f (a ^ N - x)\n⊢ f x = ↑x"
},
{
"line": "refine_lift\n have heq := h1.antisymm (mod_cast h2);\n ?_",
"before_state": "case intro\nf : ℚ → ℝ\nx : ℚ\nhx : 1 < x\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nN : ℕ\nhN : 1 + x < a ^ N\nh_big_enough : 1 < a ^ N - x\nh1 : ↑x + ↑(a ^ N - x) ≤ f x + f (a ^ N - x)\nhxp : 0 < x\nhNp : 0 < N\nh2 : f x + f (a ^ N - x) ≤ ↑x + (↑a ^ N - ↑x)\n⊢ f x = ↑x",
"after_state": "case intro\nf : ℚ → ℝ\nx : ℚ\nhx : 1 < x\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nN : ℕ\nhN : 1 + x < a ^ N\nh_big_enough : 1 < a ^ N - x\nh1 : ↑x + ↑(a ^ N - x) ≤ f x + f (a ^ N - x)\nhxp : 0 < x\nhNp : 0 < N\nh2 : f x + f (a ^ N - x) ≤ ↑x + (↑a ^ N - ↑x)\nheq : ↑x + ↑(a ^ N - x) = f x + f (a ^ N - x)\n⊢ f x = ↑x"
},
{
"line": "focus\n (refine\n no_implicit_lambda%\n (have heq := h1.antisymm (mod_cast h2);\n ?_);\n rotate_right)",
"before_state": "case intro\nf : ℚ → ℝ\nx : ℚ\nhx : 1 < x\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nN : ℕ\nhN : 1 + x < a ^ N\nh_big_enough : 1 < a ^ N - x\nh1 : ↑x + ↑(a ^ N - x) ≤ f x + f (a ^ N - x)\nhxp : 0 < x\nhNp : 0 < N\nh2 : f x + f (a ^ N - x) ≤ ↑x + (↑a ^ N - ↑x)\n⊢ f x = ↑x",
"after_state": "case intro\nf : ℚ → ℝ\nx : ℚ\nhx : 1 < x\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nN : ℕ\nhN : 1 + x < a ^ N\nh_big_enough : 1 < a ^ N - x\nh1 : ↑x + ↑(a ^ N - x) ≤ f x + f (a ^ N - x)\nhxp : 0 < x\nhNp : 0 < N\nh2 : f x + f (a ^ N - x) ≤ ↑x + (↑a ^ N - ↑x)\nheq : ↑x + ↑(a ^ N - x) = f x + f (a ^ N - x)\n⊢ f x = ↑x"
},
{
"line": "refine\n no_implicit_lambda%\n (have heq := h1.antisymm (mod_cast h2);\n ?_)",
"before_state": "case intro\nf : ℚ → ℝ\nx : ℚ\nhx : 1 < x\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nN : ℕ\nhN : 1 + x < a ^ N\nh_big_enough : 1 < a ^ N - x\nh1 : ↑x + ↑(a ^ N - x) ≤ f x + f (a ^ N - x)\nhxp : 0 < x\nhNp : 0 < N\nh2 : f x + f (a ^ N - x) ≤ ↑x + (↑a ^ N - ↑x)\n⊢ f x = ↑x",
"after_state": "case intro\nf : ℚ → ℝ\nx : ℚ\nhx : 1 < x\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nN : ℕ\nhN : 1 + x < a ^ N\nh_big_enough : 1 < a ^ N - x\nh1 : ↑x + ↑(a ^ N - x) ≤ f x + f (a ^ N - x)\nhxp : 0 < x\nhNp : 0 < N\nh2 : f x + f (a ^ N - x) ≤ ↑x + (↑a ^ N - ↑x)\nheq : ↑x + ↑(a ^ N - x) = f x + f (a ^ N - x)\n⊢ f x = ↑x"
},
{
"line": "rotate_right",
"before_state": "case intro\nf : ℚ → ℝ\nx : ℚ\nhx : 1 < x\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nN : ℕ\nhN : 1 + x < a ^ N\nh_big_enough : 1 < a ^ N - x\nh1 : ↑x + ↑(a ^ N - x) ≤ f x + f (a ^ N - x)\nhxp : 0 < x\nhNp : 0 < N\nh2 : f x + f (a ^ N - x) ≤ ↑x + (↑a ^ N - ↑x)\nheq : ↑x + ↑(a ^ N - x) = f x + f (a ^ N - x)\n⊢ f x = ↑x",
"after_state": "case intro\nf : ℚ → ℝ\nx : ℚ\nhx : 1 < x\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nN : ℕ\nhN : 1 + x < a ^ N\nh_big_enough : 1 < a ^ N - x\nh1 : ↑x + ↑(a ^ N - x) ≤ f x + f (a ^ N - x)\nhxp : 0 < x\nhNp : 0 < N\nh2 : f x + f (a ^ N - x) ≤ ↑x + (↑a ^ N - ↑x)\nheq : ↑x + ↑(a ^ N - x) = f x + f (a ^ N - x)\n⊢ f x = ↑x"
},
{
"line": "linarith [H5 x hx, H5 _ h_big_enough]",
"before_state": "case intro\nf : ℚ → ℝ\nx : ℚ\nhx : 1 < x\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nN : ℕ\nhN : 1 + x < a ^ N\nh_big_enough : 1 < a ^ N - x\nh1 : ↑x + ↑(a ^ N - x) ≤ f x + f (a ^ N - x)\nhxp : 0 < x\nhNp : 0 < N\nh2 : f x + f (a ^ N - x) ≤ ↑x + (↑a ^ N - ↑x)\nheq : ↑x + ↑(a ^ N - x) = f x + f (a ^ N - x)\n⊢ f x = ↑x",
"after_state": "No Goals!"
},
{
"line": "ring1",
"before_state": "f : ℚ → ℝ\nx : ℚ\nhx : 1 < x\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nN : ℕ\nhN : 1 + x < a ^ N\nh_big_enough : 1 < a ^ N - x\nh1 : ↑x + ↑(a ^ N - x) ≤ f x + f (a ^ N - x)\nhxp : 0 < x\nhNp : 0 < N\nh2 : f x + f (a ^ N - x) ≤ ↑x + (↑a ^ N - ↑x)\nheq : ↑x + ↑(a ^ N - x) = f x + f (a ^ N - x)\na✝ : f x < ↑x\n⊢ f x - ↑x + (↑x - f x) = 0",
"after_state": "No Goals!"
},
{
"line": "ring1",
"before_state": "f : ℚ → ℝ\nx : ℚ\nhx : 1 < x\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nN : ℕ\nhN : 1 + x < a ^ N\nh_big_enough : 1 < a ^ N - x\nh1 : ↑x + ↑(a ^ N - x) ≤ f x + f (a ^ N - x)\nhxp : 0 < x\nhNp : 0 < N\nh2 : f x + f (a ^ N - x) ≤ ↑x + (↑a ^ N - ↑x)\nheq : ↑x + ↑(a ^ N - x) = f x + f (a ^ N - x)\na✝ : ↑x < f x\n⊢ -(↑x + ↑(a ^ N - x) - (f x + f (a ^ N - x))) + (↑x - f x) + (↑(a ^ N - x) - f (a ^ N - x)) = 0",
"after_state": "No Goals!"
}
] |
theorem imo2013_q5 (f : ℚ → ℝ) (H1 : ∀ x y, 0 < x → 0 < y → f (x * y) ≤ f x * f y)
(H2 : ∀ x y, 0 < x → 0 < y → f x + f y ≤ f (x + y)) (H_fixed_point : ∃ a, 1 < a ∧ f a = a) :
∀ x, 0 < x → f x = x := by
obtain ⟨a, ha1, hae⟩ := H_fixed_point
have H3 : ∀ x : ℚ, 0 < x → ∀ n : ℕ, 0 < n → ↑n * f x ≤ f (n * x) := by
intro x hx n hn
rcases n with - | n
· exact (lt_irrefl 0 hn).elim
induction' n with pn hpn
· norm_num
calc
↑(pn + 2) * f x = (↑pn + 1 + 1) * f x := by norm_cast
_ = (↑pn + 1) * f x + f x := by ring
_ ≤ f (↑pn.succ * x) + f x := mod_cast add_le_add_right (hpn pn.succ_pos) (f x)
_ ≤ f ((↑pn + 1) * x + x) := by exact_mod_cast H2 _ _ (mul_pos pn.cast_add_one_pos hx) hx
_ = f ((↑pn + 1 + 1) * x) := by ring_nf
_ = f (↑(pn + 2) * x) := by norm_cast
have H4 : ∀ n : ℕ, 0 < n → (n : ℝ) ≤ f n := by
intro n hn
have hf1 : 1 ≤ f 1 := by
have a_pos : (0 : ℝ) < a := Rat.cast_pos.mpr (zero_lt_one.trans ha1)
suffices ↑a * 1 ≤ ↑a * f 1 by rwa [← mul_le_mul_left a_pos]
calc
↑a * 1 = ↑a := mul_one (a : ℝ)
_ = f a := hae.symm
_ = f (a * 1) := by rw [mul_one]
_ ≤ f a * f 1 := (H1 a 1) (zero_lt_one.trans ha1) zero_lt_one
_ = ↑a * f 1 := by rw [hae]
calc
(n : ℝ) = (n : ℝ) * 1 := (mul_one _).symm
_ ≤ (n : ℝ) * f 1 := by gcongr
_ ≤ f (n * 1) := H3 1 zero_lt_one n hn
_ = f n := by rw [mul_one]
have H5 : ∀ x : ℚ, 1 < x → (x : ℝ) ≤ f x := by
intro x hx
have hxnm1 : ∀ n : ℕ, 0 < n → (x : ℝ) ^ n - 1 < f x ^ n := by
intro n hn
calc
(x : ℝ) ^ n - 1 < f (x ^ n) :=
mod_cast fx_gt_xm1 (one_le_pow₀ hx.le) H1 H2 H4
_ ≤ f x ^ n := pow_f_le_f_pow hn hx H1 H4
have hx' : 1 < (x : ℝ) := mod_cast hx
have hxp : 0 < x := by positivity
exact le_of_all_pow_lt_succ' hx' (f_pos_of_pos hxp H1 H4) hxnm1
have h_f_commutes_with_pos_nat_mul : ∀ n : ℕ, 0 < n → ∀ x : ℚ, 0 < x → f (n * x) = n * f x := by
intro n hn x hx
have h2 : f (n * x) ≤ n * f x := by
rcases n with - | n
· exfalso; exact Nat.lt_asymm hn hn
rcases n with - | n
· norm_num
have hfneq : f n.succ.succ = n.succ.succ := by
have :=
fixed_point_of_gt_1 (Nat.one_lt_cast.mpr (Nat.succ_lt_succ n.succ_pos)) H1 H2 H4 H5 ha1
hae
rwa [Rat.cast_natCast n.succ.succ] at this
rw [← hfneq]
exact H1 (n.succ.succ : ℚ) x (Nat.cast_pos.mpr hn) hx
exact h2.antisymm (H3 x hx n hn)
-- For the final calculation, we expand x as (2 * x.num) / (2 * x.den), because
-- we need the top of the fraction to be strictly greater than 1 in order
-- to apply `fixed_point_of_gt_1`.
intro x hx
have H₀ : x * x.den = x.num := x.mul_den_eq_num
have H : x * (↑(2 * x.den) : ℚ) = (↑(2 * x.num) : ℚ) := by push_cast; linear_combination 2 * H₀
set x2denom := 2 * x.den
set x2num := 2 * x.num
have hx2pos : 0 < 2 * x.den := by positivity
have hx2cnezr : (x2denom : ℝ) ≠ (0 : ℝ) := by positivity
have : 0 < x.num := by rwa [Rat.num_pos]
have hx2num_gt_one : (1 : ℚ) < (2 * x.num : ℤ) := by norm_cast; linarith
apply mul_left_cancel₀ hx2cnezr
calc
x2denom * f x
= f (x2denom * x) := (h_f_commutes_with_pos_nat_mul x2denom hx2pos x hx).symm
_ = f x2num := by congr; linear_combination H
_ = x2num := fixed_point_of_gt_1 hx2num_gt_one H1 H2 H4 H5 ha1 hae
_ = ((x2num : ℚ) : ℝ) := by norm_cast
_ = (↑(x2denom * x) : ℝ) := by congr; linear_combination -H
_ = x2denom * x := by push_cast; rfl
|
/root/DuelModelResearch/mathlib4/Archive/Imo/Imo2013Q5.lean
|
{
"open": [
"Imo2013Q5"
],
"variables": []
}
|
[
{
"line": "obtain ⟨a, ha1, hae⟩ := H_fixed_point",
"before_state": "f : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\nH_fixed_point : ∃ a, 1 < a ∧ f a = ↑a\n⊢ ∀ (x : ℚ), 0 < x → f x = ↑x",
"after_state": "case intro.intro\nf : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\n⊢ ∀ (x : ℚ), 0 < x → f x = ↑x"
},
{
"line": "have H3 : ∀ x : ℚ, 0 < x → ∀ n : ℕ, 0 < n → ↑n * f x ≤ f (n * x) :=\n by\n intro x hx n hn\n rcases n with - | n\n · exact (lt_irrefl 0 hn).elim\n induction' n with pn hpn\n · norm_num\n calc\n ↑(pn + 2) * f x = (↑pn + 1 + 1) * f x := by norm_cast\n _ = (↑pn + 1) * f x + f x := by ring\n _ ≤ f (↑pn.succ * x) + f x := (mod_cast add_le_add_right (hpn pn.succ_pos) (f x))\n _ ≤ f ((↑pn + 1) * x + x) := by exact_mod_cast H2 _ _ (mul_pos pn.cast_add_one_pos hx) hx\n _ = f ((↑pn + 1 + 1) * x) := by ring_nf\n _ = f (↑(pn + 2) * x) := by norm_cast",
"before_state": "case intro.intro\nf : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\n⊢ ∀ (x : ℚ), 0 < x → f x = ↑x",
"after_state": "case intro.intro\nf : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\n⊢ ∀ (x : ℚ), 0 < x → f x = ↑x"
},
{
"line": "focus\n refine\n no_implicit_lambda%\n (have H3 : ∀ x : ℚ, 0 < x → ∀ n : ℕ, 0 < n → ↑n * f x ≤ f (n * x) := ?body✝;\n ?_)\n case body✝ =>\n with_annotate_state\"by\"\n ( intro x hx n hn\n rcases n with - | n\n · exact (lt_irrefl 0 hn).elim\n induction' n with pn hpn\n · norm_num\n calc\n ↑(pn + 2) * f x = (↑pn + 1 + 1) * f x := by norm_cast\n _ = (↑pn + 1) * f x + f x := by ring\n _ ≤ f (↑pn.succ * x) + f x := (mod_cast add_le_add_right (hpn pn.succ_pos) (f x))\n _ ≤ f ((↑pn + 1) * x + x) := by exact_mod_cast H2 _ _ (mul_pos pn.cast_add_one_pos hx) hx\n _ = f ((↑pn + 1 + 1) * x) := by ring_nf\n _ = f (↑(pn + 2) * x) := by norm_cast)",
"before_state": "case intro.intro\nf : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\n⊢ ∀ (x : ℚ), 0 < x → f x = ↑x",
"after_state": "case intro.intro\nf : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\n⊢ ∀ (x : ℚ), 0 < x → f x = ↑x"
},
{
"line": "refine\n no_implicit_lambda%\n (have H3 : ∀ x : ℚ, 0 < x → ∀ n : ℕ, 0 < n → ↑n * f x ≤ f (n * x) := ?body✝;\n ?_)",
"before_state": "case intro.intro\nf : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\n⊢ ∀ (x : ℚ), 0 < x → f x = ↑x",
"after_state": "case body\nf : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\n⊢ ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\n---\ncase intro.intro\nf : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\n⊢ ∀ (x : ℚ), 0 < x → f x = ↑x"
},
{
"line": "case body✝ =>\n with_annotate_state\"by\"\n ( intro x hx n hn\n rcases n with - | n\n · exact (lt_irrefl 0 hn).elim\n induction' n with pn hpn\n · norm_num\n calc\n ↑(pn + 2) * f x = (↑pn + 1 + 1) * f x := by norm_cast\n _ = (↑pn + 1) * f x + f x := by ring\n _ ≤ f (↑pn.succ * x) + f x := (mod_cast add_le_add_right (hpn pn.succ_pos) (f x))\n _ ≤ f ((↑pn + 1) * x + x) := by exact_mod_cast H2 _ _ (mul_pos pn.cast_add_one_pos hx) hx\n _ = f ((↑pn + 1 + 1) * x) := by ring_nf\n _ = f (↑(pn + 2) * x) := by norm_cast)",
"before_state": "case body\nf : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\n⊢ ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\n---\ncase intro.intro\nf : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\n⊢ ∀ (x : ℚ), 0 < x → f x = ↑x",
"after_state": "case intro.intro\nf : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\n⊢ ∀ (x : ℚ), 0 < x → f x = ↑x"
},
{
"line": "with_annotate_state\"by\"\n ( intro x hx n hn\n rcases n with - | n\n · exact (lt_irrefl 0 hn).elim\n induction' n with pn hpn\n · norm_num\n calc\n ↑(pn + 2) * f x = (↑pn + 1 + 1) * f x := by norm_cast\n _ = (↑pn + 1) * f x + f x := by ring\n _ ≤ f (↑pn.succ * x) + f x := (mod_cast add_le_add_right (hpn pn.succ_pos) (f x))\n _ ≤ f ((↑pn + 1) * x + x) := by exact_mod_cast H2 _ _ (mul_pos pn.cast_add_one_pos hx) hx\n _ = f ((↑pn + 1 + 1) * x) := by ring_nf\n _ = f (↑(pn + 2) * x) := by norm_cast)",
"before_state": "f : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\n⊢ ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)",
"after_state": "No Goals!"
},
{
"line": "intro x hx n hn",
"before_state": "f : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\n⊢ ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)",
"after_state": "f : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nx : ℚ\nhx : 0 < x\nn : ℕ\nhn : 0 < n\n⊢ ↑n * f x ≤ f (↑n * x)"
},
{
"line": "intro x;\n intro hx n hn",
"before_state": "f : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\n⊢ ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)",
"after_state": "f : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nx : ℚ\nhx : 0 < x\nn : ℕ\nhn : 0 < n\n⊢ ↑n * f x ≤ f (↑n * x)"
},
{
"line": "intro x",
"before_state": "f : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\n⊢ ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)",
"after_state": "f : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nx : ℚ\n⊢ 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)"
},
{
"line": "intro hx n hn",
"before_state": "f : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nx : ℚ\n⊢ 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)",
"after_state": "f : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nx : ℚ\nhx : 0 < x\nn : ℕ\nhn : 0 < n\n⊢ ↑n * f x ≤ f (↑n * x)"
},
{
"line": "intro hx;\n intro n hn",
"before_state": "f : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nx : ℚ\n⊢ 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)",
"after_state": "f : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nx : ℚ\nhx : 0 < x\nn : ℕ\nhn : 0 < n\n⊢ ↑n * f x ≤ f (↑n * x)"
},
{
"line": "intro hx",
"before_state": "f : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nx : ℚ\n⊢ 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)",
"after_state": "f : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nx : ℚ\nhx : 0 < x\n⊢ ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)"
},
{
"line": "intro n hn",
"before_state": "f : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nx : ℚ\nhx : 0 < x\n⊢ ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)",
"after_state": "f : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nx : ℚ\nhx : 0 < x\nn : ℕ\nhn : 0 < n\n⊢ ↑n * f x ≤ f (↑n * x)"
},
{
"line": "intro n;\n intro hn",
"before_state": "f : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nx : ℚ\nhx : 0 < x\n⊢ ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)",
"after_state": "f : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nx : ℚ\nhx : 0 < x\nn : ℕ\nhn : 0 < n\n⊢ ↑n * f x ≤ f (↑n * x)"
},
{
"line": "intro n",
"before_state": "f : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nx : ℚ\nhx : 0 < x\n⊢ ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)",
"after_state": "f : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nx : ℚ\nhx : 0 < x\nn : ℕ\n⊢ 0 < n → ↑n * f x ≤ f (↑n * x)"
},
{
"line": "intro hn",
"before_state": "f : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nx : ℚ\nhx : 0 < x\nn : ℕ\n⊢ 0 < n → ↑n * f x ≤ f (↑n * x)",
"after_state": "f : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nx : ℚ\nhx : 0 < x\nn : ℕ\nhn : 0 < n\n⊢ ↑n * f x ≤ f (↑n * x)"
},
{
"line": "rcases n with - | n",
"before_state": "f : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nx : ℚ\nhx : 0 < x\nn : ℕ\nhn : 0 < n\n⊢ ↑n * f x ≤ f (↑n * x)",
"after_state": "case zero\nf : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nx : ℚ\nhx : 0 < x\nhn : 0 < 0\n⊢ ↑0 * f x ≤ f (↑0 * x)\n---\ncase succ\nf : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nx : ℚ\nhx : 0 < x\nn : ℕ\nhn : 0 < n + 1\n⊢ ↑(n + 1) * f x ≤ f (↑(n + 1) * x)"
},
{
"line": "exact (lt_irrefl 0 hn).elim",
"before_state": "case zero\nf : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nx : ℚ\nhx : 0 < x\nhn : 0 < 0\n⊢ ↑0 * f x ≤ f (↑0 * x)",
"after_state": "No Goals!"
},
{
"line": "induction' n with pn hpn",
"before_state": "case succ\nf : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nx : ℚ\nhx : 0 < x\nn : ℕ\nhn : 0 < n + 1\n⊢ ↑(n + 1) * f x ≤ f (↑(n + 1) * x)",
"after_state": "case succ.zero\nf : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nx : ℚ\nhx : 0 < x\nhn : 0 < 0 + 1\n⊢ ↑(0 + 1) * f x ≤ f (↑(0 + 1) * x)\n---\ncase succ.succ\nf : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nx : ℚ\nhx : 0 < x\npn : ℕ\nhpn : 0 < pn + 1 → ↑(pn + 1) * f x ≤ f (↑(pn + 1) * x)\nhn : 0 < pn + 1 + 1\n⊢ ↑(pn + 1 + 1) * f x ≤ f (↑(pn + 1 + 1) * x)"
},
{
"line": "norm_num",
"before_state": "case succ.zero\nf : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nx : ℚ\nhx : 0 < x\nhn : 0 < 0 + 1\n⊢ ↑(0 + 1) * f x ≤ f (↑(0 + 1) * x)",
"after_state": "No Goals!"
},
{
"line": "calc\n ↑(pn + 2) * f x = (↑pn + 1 + 1) * f x := by norm_cast\n _ = (↑pn + 1) * f x + f x := by ring\n _ ≤ f (↑pn.succ * x) + f x := (mod_cast add_le_add_right (hpn pn.succ_pos) (f x))\n _ ≤ f ((↑pn + 1) * x + x) := by exact_mod_cast H2 _ _ (mul_pos pn.cast_add_one_pos hx) hx\n _ = f ((↑pn + 1 + 1) * x) := by ring_nf\n _ = f (↑(pn + 2) * x) := by norm_cast",
"before_state": "case succ.succ\nf : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nx : ℚ\nhx : 0 < x\npn : ℕ\nhpn : 0 < pn + 1 → ↑(pn + 1) * f x ≤ f (↑(pn + 1) * x)\nhn : 0 < pn + 1 + 1\n⊢ ↑(pn + 1 + 1) * f x ≤ f (↑(pn + 1 + 1) * x)",
"after_state": "No Goals!"
},
{
"line": "norm_cast",
"before_state": "f : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nx : ℚ\nhx : 0 < x\npn : ℕ\nhpn : 0 < pn + 1 → ↑(pn + 1) * f x ≤ f (↑(pn + 1) * x)\nhn : 0 < pn + 1 + 1\n⊢ ↑(pn + 2) * f x = (↑pn + 1 + 1) * f x",
"after_state": "No Goals!"
},
{
"line": "focus\n norm_cast0\n with_annotate_state\"<;>\" skip\n all_goals try trivial",
"before_state": "f : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nx : ℚ\nhx : 0 < x\npn : ℕ\nhpn : 0 < pn + 1 → ↑(pn + 1) * f x ≤ f (↑(pn + 1) * x)\nhn : 0 < pn + 1 + 1\n⊢ ↑(pn + 2) * f x = (↑pn + 1 + 1) * f x",
"after_state": "No Goals!"
},
{
"line": "norm_cast0",
"before_state": "f : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nx : ℚ\nhx : 0 < x\npn : ℕ\nhpn : 0 < pn + 1 → ↑(pn + 1) * f x ≤ f (↑(pn + 1) * x)\nhn : 0 < pn + 1 + 1\n⊢ ↑(pn + 2) * f x = (↑pn + 1 + 1) * f x",
"after_state": "f : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nx : ℚ\nhx : 0 < x\npn : ℕ\nhpn : 0 < pn + 1 → ↑(pn + 1) * f x ≤ f (↑(pn + 1) * x)\nhn : 0 < pn + 1 + 1\n⊢ ↑(pn + 2) * f x = ↑(pn + 1 + 1) * f x"
},
{
"line": "with_annotate_state\"<;>\" skip",
"before_state": "f : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nx : ℚ\nhx : 0 < x\npn : ℕ\nhpn : 0 < pn + 1 → ↑(pn + 1) * f x ≤ f (↑(pn + 1) * x)\nhn : 0 < pn + 1 + 1\n⊢ ↑(pn + 2) * f x = ↑(pn + 1 + 1) * f x",
"after_state": "f : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nx : ℚ\nhx : 0 < x\npn : ℕ\nhpn : 0 < pn + 1 → ↑(pn + 1) * f x ≤ f (↑(pn + 1) * x)\nhn : 0 < pn + 1 + 1\n⊢ ↑(pn + 2) * f x = ↑(pn + 1 + 1) * f x"
},
{
"line": "skip",
"before_state": "f : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nx : ℚ\nhx : 0 < x\npn : ℕ\nhpn : 0 < pn + 1 → ↑(pn + 1) * f x ≤ f (↑(pn + 1) * x)\nhn : 0 < pn + 1 + 1\n⊢ ↑(pn + 2) * f x = ↑(pn + 1 + 1) * f x",
"after_state": "f : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nx : ℚ\nhx : 0 < x\npn : ℕ\nhpn : 0 < pn + 1 → ↑(pn + 1) * f x ≤ f (↑(pn + 1) * x)\nhn : 0 < pn + 1 + 1\n⊢ ↑(pn + 2) * f x = ↑(pn + 1 + 1) * f x"
},
{
"line": "all_goals try trivial",
"before_state": "f : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nx : ℚ\nhx : 0 < x\npn : ℕ\nhpn : 0 < pn + 1 → ↑(pn + 1) * f x ≤ f (↑(pn + 1) * x)\nhn : 0 < pn + 1 + 1\n⊢ ↑(pn + 2) * f x = ↑(pn + 1 + 1) * f x",
"after_state": "No Goals!"
},
{
"line": "try trivial",
"before_state": "f : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nx : ℚ\nhx : 0 < x\npn : ℕ\nhpn : 0 < pn + 1 → ↑(pn + 1) * f x ≤ f (↑(pn + 1) * x)\nhn : 0 < pn + 1 + 1\n⊢ ↑(pn + 2) * f x = ↑(pn + 1 + 1) * f x",
"after_state": "No Goals!"
},
{
"line": "first\n| trivial\n| skip",
"before_state": "f : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nx : ℚ\nhx : 0 < x\npn : ℕ\nhpn : 0 < pn + 1 → ↑(pn + 1) * f x ≤ f (↑(pn + 1) * x)\nhn : 0 < pn + 1 + 1\n⊢ ↑(pn + 2) * f x = ↑(pn + 1 + 1) * f x",
"after_state": "No Goals!"
},
{
"line": "trivial",
"before_state": "f : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nx : ℚ\nhx : 0 < x\npn : ℕ\nhpn : 0 < pn + 1 → ↑(pn + 1) * f x ≤ f (↑(pn + 1) * x)\nhn : 0 < pn + 1 + 1\n⊢ ↑(pn + 2) * f x = ↑(pn + 1 + 1) * f x",
"after_state": "No Goals!"
},
{
"line": "rfl",
"before_state": "f : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nx : ℚ\nhx : 0 < x\npn : ℕ\nhpn : 0 < pn + 1 → ↑(pn + 1) * f x ≤ f (↑(pn + 1) * x)\nhn : 0 < pn + 1 + 1\n⊢ ↑(pn + 2) * f x = ↑(pn + 1 + 1) * f x",
"after_state": "No Goals!"
},
{
"line": "eq_refl",
"before_state": "f : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nx : ℚ\nhx : 0 < x\npn : ℕ\nhpn : 0 < pn + 1 → ↑(pn + 1) * f x ≤ f (↑(pn + 1) * x)\nhn : 0 < pn + 1 + 1\n⊢ ↑(pn + 2) * f x = ↑(pn + 1 + 1) * f x",
"after_state": "No Goals!"
},
{
"line": "ring",
"before_state": "f : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nx : ℚ\nhx : 0 < x\npn : ℕ\nhpn : 0 < pn + 1 → ↑(pn + 1) * f x ≤ f (↑(pn + 1) * x)\nhn : 0 < pn + 1 + 1\n⊢ (↑pn + 1 + 1) * f x = (↑pn + 1) * f x + f x",
"after_state": "No Goals!"
},
{
"line": "first\n| ring1\n|\n try_this ring_nf\"\\n\\nThe `ring` tactic failed to close the goal. Use `ring_nf` to obtain a normal form.\n \\nNote that `ring` works primarily in *commutative* rings. \\\n If you have a noncommutative ring, abelian group or module, consider using \\\n `noncomm_ring`, `abel` or `module` instead.\"",
"before_state": "f : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nx : ℚ\nhx : 0 < x\npn : ℕ\nhpn : 0 < pn + 1 → ↑(pn + 1) * f x ≤ f (↑(pn + 1) * x)\nhn : 0 < pn + 1 + 1\n⊢ (↑pn + 1 + 1) * f x = (↑pn + 1) * f x + f x",
"after_state": "No Goals!"
},
{
"line": "ring1",
"before_state": "f : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nx : ℚ\nhx : 0 < x\npn : ℕ\nhpn : 0 < pn + 1 → ↑(pn + 1) * f x ≤ f (↑(pn + 1) * x)\nhn : 0 < pn + 1 + 1\n⊢ (↑pn + 1 + 1) * f x = (↑pn + 1) * f x + f x",
"after_state": "No Goals!"
},
{
"line": "exact_mod_cast H2 _ _ (mul_pos pn.cast_add_one_pos hx) hx",
"before_state": "f : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nx : ℚ\nhx : 0 < x\npn : ℕ\nhpn : 0 < pn + 1 → ↑(pn + 1) * f x ≤ f (↑(pn + 1) * x)\nhn : 0 < pn + 1 + 1\n⊢ f (↑pn.succ * x) + f x ≤ f ((↑pn + 1) * x + x)",
"after_state": "No Goals!"
},
{
"line": "exact mod_cast (H2 _ _ (mul_pos pn.cast_add_one_pos hx) hx : _)",
"before_state": "f : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nx : ℚ\nhx : 0 < x\npn : ℕ\nhpn : 0 < pn + 1 → ↑(pn + 1) * f x ≤ f (↑(pn + 1) * x)\nhn : 0 < pn + 1 + 1\n⊢ f (↑pn.succ * x) + f x ≤ f ((↑pn + 1) * x + x)",
"after_state": "No Goals!"
},
{
"line": "ring_nf",
"before_state": "f : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nx : ℚ\nhx : 0 < x\npn : ℕ\nhpn : 0 < pn + 1 → ↑(pn + 1) * f x ≤ f (↑(pn + 1) * x)\nhn : 0 < pn + 1 + 1\n⊢ f ((↑pn + 1) * x + x) = f ((↑pn + 1 + 1) * x)",
"after_state": "No Goals!"
},
{
"line": "norm_cast",
"before_state": "f : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nx : ℚ\nhx : 0 < x\npn : ℕ\nhpn : 0 < pn + 1 → ↑(pn + 1) * f x ≤ f (↑(pn + 1) * x)\nhn : 0 < pn + 1 + 1\n⊢ f ((↑pn + 1 + 1) * x) = f (↑(pn + 2) * x)",
"after_state": "No Goals!"
},
{
"line": "focus\n norm_cast0\n with_annotate_state\"<;>\" skip\n all_goals try trivial",
"before_state": "f : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nx : ℚ\nhx : 0 < x\npn : ℕ\nhpn : 0 < pn + 1 → ↑(pn + 1) * f x ≤ f (↑(pn + 1) * x)\nhn : 0 < pn + 1 + 1\n⊢ f ((↑pn + 1 + 1) * x) = f (↑(pn + 2) * x)",
"after_state": "No Goals!"
},
{
"line": "norm_cast0",
"before_state": "f : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nx : ℚ\nhx : 0 < x\npn : ℕ\nhpn : 0 < pn + 1 → ↑(pn + 1) * f x ≤ f (↑(pn + 1) * x)\nhn : 0 < pn + 1 + 1\n⊢ f ((↑pn + 1 + 1) * x) = f (↑(pn + 2) * x)",
"after_state": "f : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nx : ℚ\nhx : 0 < x\npn : ℕ\nhpn : 0 < pn + 1 → ↑(pn + 1) * f x ≤ f (↑(pn + 1) * x)\nhn : 0 < pn + 1 + 1\n⊢ f (↑(pn + 1 + 1) * x) = f (↑(pn + 2) * x)"
},
{
"line": "with_annotate_state\"<;>\" skip",
"before_state": "f : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nx : ℚ\nhx : 0 < x\npn : ℕ\nhpn : 0 < pn + 1 → ↑(pn + 1) * f x ≤ f (↑(pn + 1) * x)\nhn : 0 < pn + 1 + 1\n⊢ f (↑(pn + 1 + 1) * x) = f (↑(pn + 2) * x)",
"after_state": "f : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nx : ℚ\nhx : 0 < x\npn : ℕ\nhpn : 0 < pn + 1 → ↑(pn + 1) * f x ≤ f (↑(pn + 1) * x)\nhn : 0 < pn + 1 + 1\n⊢ f (↑(pn + 1 + 1) * x) = f (↑(pn + 2) * x)"
},
{
"line": "skip",
"before_state": "f : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nx : ℚ\nhx : 0 < x\npn : ℕ\nhpn : 0 < pn + 1 → ↑(pn + 1) * f x ≤ f (↑(pn + 1) * x)\nhn : 0 < pn + 1 + 1\n⊢ f (↑(pn + 1 + 1) * x) = f (↑(pn + 2) * x)",
"after_state": "f : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nx : ℚ\nhx : 0 < x\npn : ℕ\nhpn : 0 < pn + 1 → ↑(pn + 1) * f x ≤ f (↑(pn + 1) * x)\nhn : 0 < pn + 1 + 1\n⊢ f (↑(pn + 1 + 1) * x) = f (↑(pn + 2) * x)"
},
{
"line": "all_goals try trivial",
"before_state": "f : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nx : ℚ\nhx : 0 < x\npn : ℕ\nhpn : 0 < pn + 1 → ↑(pn + 1) * f x ≤ f (↑(pn + 1) * x)\nhn : 0 < pn + 1 + 1\n⊢ f (↑(pn + 1 + 1) * x) = f (↑(pn + 2) * x)",
"after_state": "No Goals!"
},
{
"line": "try trivial",
"before_state": "f : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nx : ℚ\nhx : 0 < x\npn : ℕ\nhpn : 0 < pn + 1 → ↑(pn + 1) * f x ≤ f (↑(pn + 1) * x)\nhn : 0 < pn + 1 + 1\n⊢ f (↑(pn + 1 + 1) * x) = f (↑(pn + 2) * x)",
"after_state": "No Goals!"
},
{
"line": "first\n| trivial\n| skip",
"before_state": "f : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nx : ℚ\nhx : 0 < x\npn : ℕ\nhpn : 0 < pn + 1 → ↑(pn + 1) * f x ≤ f (↑(pn + 1) * x)\nhn : 0 < pn + 1 + 1\n⊢ f (↑(pn + 1 + 1) * x) = f (↑(pn + 2) * x)",
"after_state": "No Goals!"
},
{
"line": "trivial",
"before_state": "f : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nx : ℚ\nhx : 0 < x\npn : ℕ\nhpn : 0 < pn + 1 → ↑(pn + 1) * f x ≤ f (↑(pn + 1) * x)\nhn : 0 < pn + 1 + 1\n⊢ f (↑(pn + 1 + 1) * x) = f (↑(pn + 2) * x)",
"after_state": "No Goals!"
},
{
"line": "rfl",
"before_state": "f : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nx : ℚ\nhx : 0 < x\npn : ℕ\nhpn : 0 < pn + 1 → ↑(pn + 1) * f x ≤ f (↑(pn + 1) * x)\nhn : 0 < pn + 1 + 1\n⊢ f (↑(pn + 1 + 1) * x) = f (↑(pn + 2) * x)",
"after_state": "No Goals!"
},
{
"line": "eq_refl",
"before_state": "f : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nx : ℚ\nhx : 0 < x\npn : ℕ\nhpn : 0 < pn + 1 → ↑(pn + 1) * f x ≤ f (↑(pn + 1) * x)\nhn : 0 < pn + 1 + 1\n⊢ f (↑(pn + 1 + 1) * x) = f (↑(pn + 2) * x)",
"after_state": "No Goals!"
},
{
"line": "have H4 : ∀ n : ℕ, 0 < n → (n : ℝ) ≤ f n := by\n intro n hn\n have hf1 : 1 ≤ f 1 := by\n have a_pos : (0 : ℝ) < a := Rat.cast_pos.mpr (zero_lt_one.trans ha1)\n suffices ↑a * 1 ≤ ↑a * f 1 by rwa [← mul_le_mul_left a_pos]\n calc\n ↑a * 1 = ↑a := mul_one (a : ℝ)\n _ = f a := hae.symm\n _ = f (a * 1) := by rw [mul_one]\n _ ≤ f a * f 1 := ((H1 a 1) (zero_lt_one.trans ha1) zero_lt_one)\n _ = ↑a * f 1 := by rw [hae]\n calc\n (n : ℝ) = (n : ℝ) * 1 := (mul_one _).symm\n _ ≤ (n : ℝ) * f 1 := by gcongr\n _ ≤ f (n * 1) := (H3 1 zero_lt_one n hn)\n _ = f n := by rw [mul_one]",
"before_state": "case intro.intro\nf : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\n⊢ ∀ (x : ℚ), 0 < x → f x = ↑x",
"after_state": "case intro.intro\nf : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\n⊢ ∀ (x : ℚ), 0 < x → f x = ↑x"
},
{
"line": "focus\n refine\n no_implicit_lambda%\n (have H4 : ∀ n : ℕ, 0 < n → (n : ℝ) ≤ f n := ?body✝;\n ?_)\n case body✝ =>\n with_annotate_state\"by\"\n ( intro n hn\n have hf1 : 1 ≤ f 1 :=\n by\n have a_pos : (0 : ℝ) < a := Rat.cast_pos.mpr (zero_lt_one.trans ha1)\n suffices ↑a * 1 ≤ ↑a * f 1 by rwa [← mul_le_mul_left a_pos]\n calc\n ↑a * 1 = ↑a := mul_one (a : ℝ)\n _ = f a := hae.symm\n _ = f (a * 1) := by rw [mul_one]\n _ ≤ f a * f 1 := ((H1 a 1) (zero_lt_one.trans ha1) zero_lt_one)\n _ = ↑a * f 1 := by rw [hae]\n calc\n (n : ℝ) = (n : ℝ) * 1 := (mul_one _).symm\n _ ≤ (n : ℝ) * f 1 := by gcongr\n _ ≤ f (n * 1) := (H3 1 zero_lt_one n hn)\n _ = f n := by rw [mul_one])",
"before_state": "case intro.intro\nf : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\n⊢ ∀ (x : ℚ), 0 < x → f x = ↑x",
"after_state": "case intro.intro\nf : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\n⊢ ∀ (x : ℚ), 0 < x → f x = ↑x"
},
{
"line": "refine\n no_implicit_lambda%\n (have H4 : ∀ n : ℕ, 0 < n → (n : ℝ) ≤ f n := ?body✝;\n ?_)",
"before_state": "case intro.intro\nf : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\n⊢ ∀ (x : ℚ), 0 < x → f x = ↑x",
"after_state": "case body\nf : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\n⊢ ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\n---\ncase intro.intro\nf : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\n⊢ ∀ (x : ℚ), 0 < x → f x = ↑x"
},
{
"line": "case body✝ =>\n with_annotate_state\"by\"\n ( intro n hn\n have hf1 : 1 ≤ f 1 := by\n have a_pos : (0 : ℝ) < a := Rat.cast_pos.mpr (zero_lt_one.trans ha1)\n suffices ↑a * 1 ≤ ↑a * f 1 by rwa [← mul_le_mul_left a_pos]\n calc\n ↑a * 1 = ↑a := mul_one (a : ℝ)\n _ = f a := hae.symm\n _ = f (a * 1) := by rw [mul_one]\n _ ≤ f a * f 1 := ((H1 a 1) (zero_lt_one.trans ha1) zero_lt_one)\n _ = ↑a * f 1 := by rw [hae]\n calc\n (n : ℝ) = (n : ℝ) * 1 := (mul_one _).symm\n _ ≤ (n : ℝ) * f 1 := by gcongr\n _ ≤ f (n * 1) := (H3 1 zero_lt_one n hn)\n _ = f n := by rw [mul_one])",
"before_state": "case body\nf : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\n⊢ ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\n---\ncase intro.intro\nf : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\n⊢ ∀ (x : ℚ), 0 < x → f x = ↑x",
"after_state": "case intro.intro\nf : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\n⊢ ∀ (x : ℚ), 0 < x → f x = ↑x"
},
{
"line": "with_annotate_state\"by\"\n ( intro n hn\n have hf1 : 1 ≤ f 1 := by\n have a_pos : (0 : ℝ) < a := Rat.cast_pos.mpr (zero_lt_one.trans ha1)\n suffices ↑a * 1 ≤ ↑a * f 1 by rwa [← mul_le_mul_left a_pos]\n calc\n ↑a * 1 = ↑a := mul_one (a : ℝ)\n _ = f a := hae.symm\n _ = f (a * 1) := by rw [mul_one]\n _ ≤ f a * f 1 := ((H1 a 1) (zero_lt_one.trans ha1) zero_lt_one)\n _ = ↑a * f 1 := by rw [hae]\n calc\n (n : ℝ) = (n : ℝ) * 1 := (mul_one _).symm\n _ ≤ (n : ℝ) * f 1 := by gcongr\n _ ≤ f (n * 1) := (H3 1 zero_lt_one n hn)\n _ = f n := by rw [mul_one])",
"before_state": "f : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\n⊢ ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n",
"after_state": "No Goals!"
},
{
"line": "intro n hn",
"before_state": "f : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\n⊢ ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n",
"after_state": "f : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nn : ℕ\nhn : 0 < n\n⊢ ↑n ≤ f ↑n"
},
{
"line": "intro n;\n intro hn",
"before_state": "f : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\n⊢ ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n",
"after_state": "f : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nn : ℕ\nhn : 0 < n\n⊢ ↑n ≤ f ↑n"
},
{
"line": "intro n",
"before_state": "f : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\n⊢ ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n",
"after_state": "f : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nn : ℕ\n⊢ 0 < n → ↑n ≤ f ↑n"
},
{
"line": "intro hn",
"before_state": "f : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nn : ℕ\n⊢ 0 < n → ↑n ≤ f ↑n",
"after_state": "f : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nn : ℕ\nhn : 0 < n\n⊢ ↑n ≤ f ↑n"
},
{
"line": "have hf1 : 1 ≤ f 1 := by\n have a_pos : (0 : ℝ) < a := Rat.cast_pos.mpr (zero_lt_one.trans ha1)\n suffices ↑a * 1 ≤ ↑a * f 1 by rwa [← mul_le_mul_left a_pos]\n calc\n ↑a * 1 = ↑a := mul_one (a : ℝ)\n _ = f a := hae.symm\n _ = f (a * 1) := by rw [mul_one]\n _ ≤ f a * f 1 := ((H1 a 1) (zero_lt_one.trans ha1) zero_lt_one)\n _ = ↑a * f 1 := by rw [hae]",
"before_state": "f : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nn : ℕ\nhn : 0 < n\n⊢ ↑n ≤ f ↑n",
"after_state": "f : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nn : ℕ\nhn : 0 < n\nhf1 : 1 ≤ f 1\n⊢ ↑n ≤ f ↑n"
},
{
"line": "focus\n refine\n no_implicit_lambda%\n (have hf1 : 1 ≤ f 1 := ?body✝;\n ?_)\n case body✝ =>\n with_annotate_state\"by\"\n ( have a_pos : (0 : ℝ) < a := Rat.cast_pos.mpr (zero_lt_one.trans ha1)\n suffices ↑a * 1 ≤ ↑a * f 1 by rwa [← mul_le_mul_left a_pos]\n calc\n ↑a * 1 = ↑a := mul_one (a : ℝ)\n _ = f a := hae.symm\n _ = f (a * 1) := by rw [mul_one]\n _ ≤ f a * f 1 := ((H1 a 1) (zero_lt_one.trans ha1) zero_lt_one)\n _ = ↑a * f 1 := by rw [hae])",
"before_state": "f : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nn : ℕ\nhn : 0 < n\n⊢ ↑n ≤ f ↑n",
"after_state": "f : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nn : ℕ\nhn : 0 < n\nhf1 : 1 ≤ f 1\n⊢ ↑n ≤ f ↑n"
},
{
"line": "refine\n no_implicit_lambda%\n (have hf1 : 1 ≤ f 1 := ?body✝;\n ?_)",
"before_state": "f : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nn : ℕ\nhn : 0 < n\n⊢ ↑n ≤ f ↑n",
"after_state": "case body\nf : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nn : ℕ\nhn : 0 < n\n⊢ 1 ≤ f 1\n---\nf : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nn : ℕ\nhn : 0 < n\nhf1 : 1 ≤ f 1\n⊢ ↑n ≤ f ↑n"
},
{
"line": "case body✝ =>\n with_annotate_state\"by\"\n ( have a_pos : (0 : ℝ) < a := Rat.cast_pos.mpr (zero_lt_one.trans ha1)\n suffices ↑a * 1 ≤ ↑a * f 1 by rwa [← mul_le_mul_left a_pos]\n calc\n ↑a * 1 = ↑a := mul_one (a : ℝ)\n _ = f a := hae.symm\n _ = f (a * 1) := by rw [mul_one]\n _ ≤ f a * f 1 := ((H1 a 1) (zero_lt_one.trans ha1) zero_lt_one)\n _ = ↑a * f 1 := by rw [hae])",
"before_state": "case body\nf : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nn : ℕ\nhn : 0 < n\n⊢ 1 ≤ f 1\n---\nf : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nn : ℕ\nhn : 0 < n\nhf1 : 1 ≤ f 1\n⊢ ↑n ≤ f ↑n",
"after_state": "f : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nn : ℕ\nhn : 0 < n\nhf1 : 1 ≤ f 1\n⊢ ↑n ≤ f ↑n"
},
{
"line": "with_annotate_state\"by\"\n ( have a_pos : (0 : ℝ) < a := Rat.cast_pos.mpr (zero_lt_one.trans ha1)\n suffices ↑a * 1 ≤ ↑a * f 1 by rwa [← mul_le_mul_left a_pos]\n calc\n ↑a * 1 = ↑a := mul_one (a : ℝ)\n _ = f a := hae.symm\n _ = f (a * 1) := by rw [mul_one]\n _ ≤ f a * f 1 := ((H1 a 1) (zero_lt_one.trans ha1) zero_lt_one)\n _ = ↑a * f 1 := by rw [hae])",
"before_state": "f : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nn : ℕ\nhn : 0 < n\n⊢ 1 ≤ f 1",
"after_state": "No Goals!"
},
{
"line": "have a_pos : (0 : ℝ) < a := Rat.cast_pos.mpr (zero_lt_one.trans ha1)",
"before_state": "f : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nn : ℕ\nhn : 0 < n\n⊢ 1 ≤ f 1",
"after_state": "f : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nn : ℕ\nhn : 0 < n\na_pos : 0 < ↑a\n⊢ 1 ≤ f 1"
},
{
"line": "refine_lift\n have a_pos : (0 : ℝ) < a := Rat.cast_pos.mpr (zero_lt_one.trans ha1);\n ?_",
"before_state": "f : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nn : ℕ\nhn : 0 < n\n⊢ 1 ≤ f 1",
"after_state": "f : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nn : ℕ\nhn : 0 < n\na_pos : 0 < ↑a\n⊢ 1 ≤ f 1"
},
{
"line": "focus\n (refine\n no_implicit_lambda%\n (have a_pos : (0 : ℝ) < a := Rat.cast_pos.mpr (zero_lt_one.trans ha1);\n ?_);\n rotate_right)",
"before_state": "f : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nn : ℕ\nhn : 0 < n\n⊢ 1 ≤ f 1",
"after_state": "f : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nn : ℕ\nhn : 0 < n\na_pos : 0 < ↑a\n⊢ 1 ≤ f 1"
},
{
"line": "refine\n no_implicit_lambda%\n (have a_pos : (0 : ℝ) < a := Rat.cast_pos.mpr (zero_lt_one.trans ha1);\n ?_)",
"before_state": "f : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nn : ℕ\nhn : 0 < n\n⊢ 1 ≤ f 1",
"after_state": "f : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nn : ℕ\nhn : 0 < n\na_pos : 0 < ↑a\n⊢ 1 ≤ f 1"
},
{
"line": "rotate_right",
"before_state": "f : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nn : ℕ\nhn : 0 < n\na_pos : 0 < ↑a\n⊢ 1 ≤ f 1",
"after_state": "f : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nn : ℕ\nhn : 0 < n\na_pos : 0 < ↑a\n⊢ 1 ≤ f 1"
},
{
"line": "suffices ↑a * 1 ≤ ↑a * f 1 by rwa [← mul_le_mul_left a_pos]",
"before_state": "f : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nn : ℕ\nhn : 0 < n\na_pos : 0 < ↑a\n⊢ 1 ≤ f 1",
"after_state": "f : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nn : ℕ\nhn : 0 < n\na_pos : 0 < ↑a\n⊢ ↑a * 1 ≤ ↑a * f 1"
},
{
"line": "refine_lift\n suffices ↑a * 1 ≤ ↑a * f 1 by rwa [← mul_le_mul_left a_pos];\n ?_",
"before_state": "f : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nn : ℕ\nhn : 0 < n\na_pos : 0 < ↑a\n⊢ 1 ≤ f 1",
"after_state": "f : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nn : ℕ\nhn : 0 < n\na_pos : 0 < ↑a\n⊢ ↑a * 1 ≤ ↑a * f 1"
},
{
"line": "focus\n (refine\n no_implicit_lambda%\n (suffices ↑a * 1 ≤ ↑a * f 1 by rwa [← mul_le_mul_left a_pos];\n ?_);\n rotate_right)",
"before_state": "f : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nn : ℕ\nhn : 0 < n\na_pos : 0 < ↑a\n⊢ 1 ≤ f 1",
"after_state": "f : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nn : ℕ\nhn : 0 < n\na_pos : 0 < ↑a\n⊢ ↑a * 1 ≤ ↑a * f 1"
},
{
"line": "refine\n no_implicit_lambda%\n (suffices ↑a * 1 ≤ ↑a * f 1 by rwa [← mul_le_mul_left a_pos];\n ?_)",
"before_state": "f : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nn : ℕ\nhn : 0 < n\na_pos : 0 < ↑a\n⊢ 1 ≤ f 1",
"after_state": "f : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nn : ℕ\nhn : 0 < n\na_pos : 0 < ↑a\n⊢ ↑a * 1 ≤ ↑a * f 1"
},
{
"line": "rwa [← mul_le_mul_left a_pos]",
"before_state": "f : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nn : ℕ\nhn : 0 < n\na_pos : 0 < ↑a\nthis : ↑a * 1 ≤ ↑a * f 1\n⊢ 1 ≤ f 1",
"after_state": "No Goals!"
},
{
"line": "rw [← mul_le_mul_left a_pos]",
"before_state": "f : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nn : ℕ\nhn : 0 < n\na_pos : 0 < ↑a\nthis : ↑a * 1 ≤ ↑a * f 1\n⊢ 1 ≤ f 1",
"after_state": "f : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nn : ℕ\nhn : 0 < n\na_pos : 0 < ↑a\nthis : ↑a * 1 ≤ ↑a * f 1\n⊢ ↑a * 1 ≤ ↑a * f 1"
},
{
"line": "rewrite [← mul_le_mul_left a_pos]",
"before_state": "f : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nn : ℕ\nhn : 0 < n\na_pos : 0 < ↑a\nthis : ↑a * 1 ≤ ↑a * f 1\n⊢ 1 ≤ f 1",
"after_state": "f : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nn : ℕ\nhn : 0 < n\na_pos : 0 < ↑a\nthis : ↑a * 1 ≤ ↑a * f 1\n⊢ ↑a * 1 ≤ ↑a * f 1"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "f : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nn : ℕ\nhn : 0 < n\na_pos : 0 < ↑a\nthis : ↑a * 1 ≤ ↑a * f 1\n⊢ ↑a * 1 ≤ ↑a * f 1",
"after_state": "f : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nn : ℕ\nhn : 0 < n\na_pos : 0 < ↑a\nthis : ↑a * 1 ≤ ↑a * f 1\n⊢ ↑a * 1 ≤ ↑a * f 1"
},
{
"line": "try (with_reducible rfl)",
"before_state": "f : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nn : ℕ\nhn : 0 < n\na_pos : 0 < ↑a\nthis : ↑a * 1 ≤ ↑a * f 1\n⊢ ↑a * 1 ≤ ↑a * f 1",
"after_state": "f : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nn : ℕ\nhn : 0 < n\na_pos : 0 < ↑a\nthis : ↑a * 1 ≤ ↑a * f 1\n⊢ ↑a * 1 ≤ ↑a * f 1"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "f : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nn : ℕ\nhn : 0 < n\na_pos : 0 < ↑a\nthis : ↑a * 1 ≤ ↑a * f 1\n⊢ ↑a * 1 ≤ ↑a * f 1",
"after_state": "f : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nn : ℕ\nhn : 0 < n\na_pos : 0 < ↑a\nthis : ↑a * 1 ≤ ↑a * f 1\n⊢ ↑a * 1 ≤ ↑a * f 1"
},
{
"line": "with_reducible rfl",
"before_state": "f : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nn : ℕ\nhn : 0 < n\na_pos : 0 < ↑a\nthis : ↑a * 1 ≤ ↑a * f 1\n⊢ ↑a * 1 ≤ ↑a * f 1",
"after_state": "f : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nn : ℕ\nhn : 0 < n\na_pos : 0 < ↑a\nthis : ↑a * 1 ≤ ↑a * f 1\n⊢ ↑a * 1 ≤ ↑a * f 1"
},
{
"line": "rfl",
"before_state": "f : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nn : ℕ\nhn : 0 < n\na_pos : 0 < ↑a\nthis : ↑a * 1 ≤ ↑a * f 1\n⊢ ↑a * 1 ≤ ↑a * f 1",
"after_state": "f : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nn : ℕ\nhn : 0 < n\na_pos : 0 < ↑a\nthis : ↑a * 1 ≤ ↑a * f 1\n⊢ ↑a * 1 ≤ ↑a * f 1"
},
{
"line": "apply_rfl",
"before_state": "f : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nn : ℕ\nhn : 0 < n\na_pos : 0 < ↑a\nthis : ↑a * 1 ≤ ↑a * f 1\n⊢ ↑a * 1 ≤ ↑a * f 1",
"after_state": "f : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nn : ℕ\nhn : 0 < n\na_pos : 0 < ↑a\nthis : ↑a * 1 ≤ ↑a * f 1\n⊢ ↑a * 1 ≤ ↑a * f 1"
},
{
"line": "skip",
"before_state": "f : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nn : ℕ\nhn : 0 < n\na_pos : 0 < ↑a\nthis : ↑a * 1 ≤ ↑a * f 1\n⊢ ↑a * 1 ≤ ↑a * f 1",
"after_state": "f : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nn : ℕ\nhn : 0 < n\na_pos : 0 < ↑a\nthis : ↑a * 1 ≤ ↑a * f 1\n⊢ ↑a * 1 ≤ ↑a * f 1"
},
{
"line": "assumption",
"before_state": "f : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nn : ℕ\nhn : 0 < n\na_pos : 0 < ↑a\nthis : ↑a * 1 ≤ ↑a * f 1\n⊢ ↑a * 1 ≤ ↑a * f 1",
"after_state": "No Goals!"
},
{
"line": "rotate_right",
"before_state": "f : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nn : ℕ\nhn : 0 < n\na_pos : 0 < ↑a\n⊢ ↑a * 1 ≤ ↑a * f 1",
"after_state": "f : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nn : ℕ\nhn : 0 < n\na_pos : 0 < ↑a\n⊢ ↑a * 1 ≤ ↑a * f 1"
},
{
"line": "calc\n ↑a * 1 = ↑a := mul_one (a : ℝ)\n _ = f a := hae.symm\n _ = f (a * 1) := by rw [mul_one]\n _ ≤ f a * f 1 := ((H1 a 1) (zero_lt_one.trans ha1) zero_lt_one)\n _ = ↑a * f 1 := by rw [hae]",
"before_state": "f : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nn : ℕ\nhn : 0 < n\na_pos : 0 < ↑a\n⊢ ↑a * 1 ≤ ↑a * f 1",
"after_state": "No Goals!"
},
{
"line": "rw [mul_one]",
"before_state": "f : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nn : ℕ\nhn : 0 < n\na_pos : 0 < ↑a\n⊢ f a = f (a * 1)",
"after_state": "No Goals!"
},
{
"line": "rewrite [mul_one]",
"before_state": "f : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nn : ℕ\nhn : 0 < n\na_pos : 0 < ↑a\n⊢ f a = f (a * 1)",
"after_state": "f : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nn : ℕ\nhn : 0 < n\na_pos : 0 < ↑a\n⊢ f a = f a"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "f : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nn : ℕ\nhn : 0 < n\na_pos : 0 < ↑a\n⊢ f a = f a",
"after_state": "No Goals!"
},
{
"line": "try (with_reducible rfl)",
"before_state": "f : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nn : ℕ\nhn : 0 < n\na_pos : 0 < ↑a\n⊢ f a = f a",
"after_state": "No Goals!"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "f : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nn : ℕ\nhn : 0 < n\na_pos : 0 < ↑a\n⊢ f a = f a",
"after_state": "No Goals!"
},
{
"line": "with_reducible rfl",
"before_state": "f : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nn : ℕ\nhn : 0 < n\na_pos : 0 < ↑a\n⊢ f a = f a",
"after_state": "No Goals!"
},
{
"line": "rfl",
"before_state": "f : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nn : ℕ\nhn : 0 < n\na_pos : 0 < ↑a\n⊢ f a = f a",
"after_state": "No Goals!"
},
{
"line": "eq_refl",
"before_state": "f : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nn : ℕ\nhn : 0 < n\na_pos : 0 < ↑a\n⊢ f a = f a",
"after_state": "No Goals!"
},
{
"line": "rw [hae]",
"before_state": "f : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nn : ℕ\nhn : 0 < n\na_pos : 0 < ↑a\n⊢ f a * f 1 = ↑a * f 1",
"after_state": "No Goals!"
},
{
"line": "rewrite [hae]",
"before_state": "f : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nn : ℕ\nhn : 0 < n\na_pos : 0 < ↑a\n⊢ f a * f 1 = ↑a * f 1",
"after_state": "f : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nn : ℕ\nhn : 0 < n\na_pos : 0 < ↑a\n⊢ ↑a * f 1 = ↑a * f 1"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "f : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nn : ℕ\nhn : 0 < n\na_pos : 0 < ↑a\n⊢ ↑a * f 1 = ↑a * f 1",
"after_state": "No Goals!"
},
{
"line": "try (with_reducible rfl)",
"before_state": "f : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nn : ℕ\nhn : 0 < n\na_pos : 0 < ↑a\n⊢ ↑a * f 1 = ↑a * f 1",
"after_state": "No Goals!"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "f : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nn : ℕ\nhn : 0 < n\na_pos : 0 < ↑a\n⊢ ↑a * f 1 = ↑a * f 1",
"after_state": "No Goals!"
},
{
"line": "with_reducible rfl",
"before_state": "f : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nn : ℕ\nhn : 0 < n\na_pos : 0 < ↑a\n⊢ ↑a * f 1 = ↑a * f 1",
"after_state": "No Goals!"
},
{
"line": "rfl",
"before_state": "f : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nn : ℕ\nhn : 0 < n\na_pos : 0 < ↑a\n⊢ ↑a * f 1 = ↑a * f 1",
"after_state": "No Goals!"
},
{
"line": "eq_refl",
"before_state": "f : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nn : ℕ\nhn : 0 < n\na_pos : 0 < ↑a\n⊢ ↑a * f 1 = ↑a * f 1",
"after_state": "No Goals!"
},
{
"line": "calc\n (n : ℝ) = (n : ℝ) * 1 := (mul_one _).symm\n _ ≤ (n : ℝ) * f 1 := by gcongr\n _ ≤ f (n * 1) := (H3 1 zero_lt_one n hn)\n _ = f n := by rw [mul_one]",
"before_state": "f : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nn : ℕ\nhn : 0 < n\nhf1 : 1 ≤ f 1\n⊢ ↑n ≤ f ↑n",
"after_state": "No Goals!"
},
{
"line": "gcongr",
"before_state": "f : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nn : ℕ\nhn : 0 < n\nhf1 : 1 ≤ f 1\n⊢ ↑n * 1 ≤ ↑n * f 1",
"after_state": "No Goals!"
},
{
"line": "gcongr_discharger",
"before_state": "case a0\nf : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nn : ℕ\nhn : 0 < n\nhf1 : 1 ≤ f 1\n⊢ 0 ≤ ↑n",
"after_state": "No Goals!"
},
{
"line": "positivity",
"before_state": "case a0\nf : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nn : ℕ\nhn : 0 < n\nhf1 : 1 ≤ f 1\n⊢ 0 ≤ ↑n",
"after_state": "No Goals!"
},
{
"line": "rw [mul_one]",
"before_state": "f : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nn : ℕ\nhn : 0 < n\nhf1 : 1 ≤ f 1\n⊢ f (↑n * 1) = f ↑n",
"after_state": "No Goals!"
},
{
"line": "rewrite [mul_one]",
"before_state": "f : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nn : ℕ\nhn : 0 < n\nhf1 : 1 ≤ f 1\n⊢ f (↑n * 1) = f ↑n",
"after_state": "f : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nn : ℕ\nhn : 0 < n\nhf1 : 1 ≤ f 1\n⊢ f ↑n = f ↑n"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "f : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nn : ℕ\nhn : 0 < n\nhf1 : 1 ≤ f 1\n⊢ f ↑n = f ↑n",
"after_state": "No Goals!"
},
{
"line": "try (with_reducible rfl)",
"before_state": "f : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nn : ℕ\nhn : 0 < n\nhf1 : 1 ≤ f 1\n⊢ f ↑n = f ↑n",
"after_state": "No Goals!"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "f : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nn : ℕ\nhn : 0 < n\nhf1 : 1 ≤ f 1\n⊢ f ↑n = f ↑n",
"after_state": "No Goals!"
},
{
"line": "with_reducible rfl",
"before_state": "f : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nn : ℕ\nhn : 0 < n\nhf1 : 1 ≤ f 1\n⊢ f ↑n = f ↑n",
"after_state": "No Goals!"
},
{
"line": "rfl",
"before_state": "f : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nn : ℕ\nhn : 0 < n\nhf1 : 1 ≤ f 1\n⊢ f ↑n = f ↑n",
"after_state": "No Goals!"
},
{
"line": "eq_refl",
"before_state": "f : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nn : ℕ\nhn : 0 < n\nhf1 : 1 ≤ f 1\n⊢ f ↑n = f ↑n",
"after_state": "No Goals!"
},
{
"line": "have H5 : ∀ x : ℚ, 1 < x → (x : ℝ) ≤ f x := by\n intro x hx\n have hxnm1 : ∀ n : ℕ, 0 < n → (x : ℝ) ^ n - 1 < f x ^ n :=\n by\n intro n hn\n calc\n (x : ℝ) ^ n - 1 < f (x ^ n) := mod_cast fx_gt_xm1 (one_le_pow₀ hx.le) H1 H2 H4\n _ ≤ f x ^ n := pow_f_le_f_pow hn hx H1 H4\n have hx' : 1 < (x : ℝ) := mod_cast hx\n have hxp : 0 < x := by positivity\n exact le_of_all_pow_lt_succ' hx' (f_pos_of_pos hxp H1 H4) hxnm1",
"before_state": "case intro.intro\nf : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\n⊢ ∀ (x : ℚ), 0 < x → f x = ↑x",
"after_state": "case intro.intro\nf : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\n⊢ ∀ (x : ℚ), 0 < x → f x = ↑x"
},
{
"line": "focus\n refine\n no_implicit_lambda%\n (have H5 : ∀ x : ℚ, 1 < x → (x : ℝ) ≤ f x := ?body✝;\n ?_)\n case body✝ =>\n with_annotate_state\"by\"\n ( intro x hx\n have hxnm1 : ∀ n : ℕ, 0 < n → (x : ℝ) ^ n - 1 < f x ^ n :=\n by\n intro n hn\n calc\n (x : ℝ) ^ n - 1 < f (x ^ n) := mod_cast fx_gt_xm1 (one_le_pow₀ hx.le) H1 H2 H4\n _ ≤ f x ^ n := pow_f_le_f_pow hn hx H1 H4\n have hx' : 1 < (x : ℝ) := mod_cast hx\n have hxp : 0 < x := by positivity\n exact le_of_all_pow_lt_succ' hx' (f_pos_of_pos hxp H1 H4) hxnm1)",
"before_state": "case intro.intro\nf : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\n⊢ ∀ (x : ℚ), 0 < x → f x = ↑x",
"after_state": "case intro.intro\nf : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\n⊢ ∀ (x : ℚ), 0 < x → f x = ↑x"
},
{
"line": "refine\n no_implicit_lambda%\n (have H5 : ∀ x : ℚ, 1 < x → (x : ℝ) ≤ f x := ?body✝;\n ?_)",
"before_state": "case intro.intro\nf : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\n⊢ ∀ (x : ℚ), 0 < x → f x = ↑x",
"after_state": "case body\nf : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\n⊢ ∀ (x : ℚ), 1 < x → ↑x ≤ f x\n---\ncase intro.intro\nf : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\n⊢ ∀ (x : ℚ), 0 < x → f x = ↑x"
},
{
"line": "case body✝ =>\n with_annotate_state\"by\"\n ( intro x hx\n have hxnm1 : ∀ n : ℕ, 0 < n → (x : ℝ) ^ n - 1 < f x ^ n :=\n by\n intro n hn\n calc\n (x : ℝ) ^ n - 1 < f (x ^ n) := mod_cast fx_gt_xm1 (one_le_pow₀ hx.le) H1 H2 H4\n _ ≤ f x ^ n := pow_f_le_f_pow hn hx H1 H4\n have hx' : 1 < (x : ℝ) := mod_cast hx\n have hxp : 0 < x := by positivity\n exact le_of_all_pow_lt_succ' hx' (f_pos_of_pos hxp H1 H4) hxnm1)",
"before_state": "case body\nf : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\n⊢ ∀ (x : ℚ), 1 < x → ↑x ≤ f x\n---\ncase intro.intro\nf : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\n⊢ ∀ (x : ℚ), 0 < x → f x = ↑x",
"after_state": "case intro.intro\nf : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\n⊢ ∀ (x : ℚ), 0 < x → f x = ↑x"
},
{
"line": "with_annotate_state\"by\"\n ( intro x hx\n have hxnm1 : ∀ n : ℕ, 0 < n → (x : ℝ) ^ n - 1 < f x ^ n :=\n by\n intro n hn\n calc\n (x : ℝ) ^ n - 1 < f (x ^ n) := mod_cast fx_gt_xm1 (one_le_pow₀ hx.le) H1 H2 H4\n _ ≤ f x ^ n := pow_f_le_f_pow hn hx H1 H4\n have hx' : 1 < (x : ℝ) := mod_cast hx\n have hxp : 0 < x := by positivity\n exact le_of_all_pow_lt_succ' hx' (f_pos_of_pos hxp H1 H4) hxnm1)",
"before_state": "f : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\n⊢ ∀ (x : ℚ), 1 < x → ↑x ≤ f x",
"after_state": "No Goals!"
},
{
"line": "intro x hx",
"before_state": "f : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\n⊢ ∀ (x : ℚ), 1 < x → ↑x ≤ f x",
"after_state": "f : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nx : ℚ\nhx : 1 < x\n⊢ ↑x ≤ f x"
},
{
"line": "intro x;\n intro hx",
"before_state": "f : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\n⊢ ∀ (x : ℚ), 1 < x → ↑x ≤ f x",
"after_state": "f : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nx : ℚ\nhx : 1 < x\n⊢ ↑x ≤ f x"
},
{
"line": "intro x",
"before_state": "f : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\n⊢ ∀ (x : ℚ), 1 < x → ↑x ≤ f x",
"after_state": "f : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nx : ℚ\n⊢ 1 < x → ↑x ≤ f x"
},
{
"line": "intro hx",
"before_state": "f : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nx : ℚ\n⊢ 1 < x → ↑x ≤ f x",
"after_state": "f : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nx : ℚ\nhx : 1 < x\n⊢ ↑x ≤ f x"
},
{
"line": "have hxnm1 : ∀ n : ℕ, 0 < n → (x : ℝ) ^ n - 1 < f x ^ n :=\n by\n intro n hn\n calc\n (x : ℝ) ^ n - 1 < f (x ^ n) := mod_cast fx_gt_xm1 (one_le_pow₀ hx.le) H1 H2 H4\n _ ≤ f x ^ n := pow_f_le_f_pow hn hx H1 H4",
"before_state": "f : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nx : ℚ\nhx : 1 < x\n⊢ ↑x ≤ f x",
"after_state": "f : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nx : ℚ\nhx : 1 < x\nhxnm1 : ∀ (n : ℕ), 0 < n → ↑x ^ n - 1 < f x ^ n\n⊢ ↑x ≤ f x"
},
{
"line": "focus\n refine\n no_implicit_lambda%\n (have hxnm1 : ∀ n : ℕ, 0 < n → (x : ℝ) ^ n - 1 < f x ^ n := ?body✝;\n ?_)\n case body✝ =>\n with_annotate_state\"by\"\n ( intro n hn\n calc\n (x : ℝ) ^ n - 1 < f (x ^ n) := mod_cast fx_gt_xm1 (one_le_pow₀ hx.le) H1 H2 H4\n _ ≤ f x ^ n := pow_f_le_f_pow hn hx H1 H4)",
"before_state": "f : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nx : ℚ\nhx : 1 < x\n⊢ ↑x ≤ f x",
"after_state": "f : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nx : ℚ\nhx : 1 < x\nhxnm1 : ∀ (n : ℕ), 0 < n → ↑x ^ n - 1 < f x ^ n\n⊢ ↑x ≤ f x"
},
{
"line": "refine\n no_implicit_lambda%\n (have hxnm1 : ∀ n : ℕ, 0 < n → (x : ℝ) ^ n - 1 < f x ^ n := ?body✝;\n ?_)",
"before_state": "f : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nx : ℚ\nhx : 1 < x\n⊢ ↑x ≤ f x",
"after_state": "case body\nf : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nx : ℚ\nhx : 1 < x\n⊢ ∀ (n : ℕ), 0 < n → ↑x ^ n - 1 < f x ^ n\n---\nf : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nx : ℚ\nhx : 1 < x\nhxnm1 : ∀ (n : ℕ), 0 < n → ↑x ^ n - 1 < f x ^ n\n⊢ ↑x ≤ f x"
},
{
"line": "case body✝ =>\n with_annotate_state\"by\"\n ( intro n hn\n calc\n (x : ℝ) ^ n - 1 < f (x ^ n) := mod_cast fx_gt_xm1 (one_le_pow₀ hx.le) H1 H2 H4\n _ ≤ f x ^ n := pow_f_le_f_pow hn hx H1 H4)",
"before_state": "case body\nf : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nx : ℚ\nhx : 1 < x\n⊢ ∀ (n : ℕ), 0 < n → ↑x ^ n - 1 < f x ^ n\n---\nf : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nx : ℚ\nhx : 1 < x\nhxnm1 : ∀ (n : ℕ), 0 < n → ↑x ^ n - 1 < f x ^ n\n⊢ ↑x ≤ f x",
"after_state": "f : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nx : ℚ\nhx : 1 < x\nhxnm1 : ∀ (n : ℕ), 0 < n → ↑x ^ n - 1 < f x ^ n\n⊢ ↑x ≤ f x"
},
{
"line": "with_annotate_state\"by\"\n ( intro n hn\n calc\n (x : ℝ) ^ n - 1 < f (x ^ n) := mod_cast fx_gt_xm1 (one_le_pow₀ hx.le) H1 H2 H4\n _ ≤ f x ^ n := pow_f_le_f_pow hn hx H1 H4)",
"before_state": "f : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nx : ℚ\nhx : 1 < x\n⊢ ∀ (n : ℕ), 0 < n → ↑x ^ n - 1 < f x ^ n",
"after_state": "No Goals!"
},
{
"line": "intro n hn",
"before_state": "f : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nx : ℚ\nhx : 1 < x\n⊢ ∀ (n : ℕ), 0 < n → ↑x ^ n - 1 < f x ^ n",
"after_state": "f : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nx : ℚ\nhx : 1 < x\nn : ℕ\nhn : 0 < n\n⊢ ↑x ^ n - 1 < f x ^ n"
},
{
"line": "intro n;\n intro hn",
"before_state": "f : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nx : ℚ\nhx : 1 < x\n⊢ ∀ (n : ℕ), 0 < n → ↑x ^ n - 1 < f x ^ n",
"after_state": "f : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nx : ℚ\nhx : 1 < x\nn : ℕ\nhn : 0 < n\n⊢ ↑x ^ n - 1 < f x ^ n"
},
{
"line": "intro n",
"before_state": "f : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nx : ℚ\nhx : 1 < x\n⊢ ∀ (n : ℕ), 0 < n → ↑x ^ n - 1 < f x ^ n",
"after_state": "f : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nx : ℚ\nhx : 1 < x\nn : ℕ\n⊢ 0 < n → ↑x ^ n - 1 < f x ^ n"
},
{
"line": "intro hn",
"before_state": "f : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nx : ℚ\nhx : 1 < x\nn : ℕ\n⊢ 0 < n → ↑x ^ n - 1 < f x ^ n",
"after_state": "f : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nx : ℚ\nhx : 1 < x\nn : ℕ\nhn : 0 < n\n⊢ ↑x ^ n - 1 < f x ^ n"
},
{
"line": "calc\n (x : ℝ) ^ n - 1 < f (x ^ n) := mod_cast fx_gt_xm1 (one_le_pow₀ hx.le) H1 H2 H4\n _ ≤ f x ^ n := pow_f_le_f_pow hn hx H1 H4",
"before_state": "f : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nx : ℚ\nhx : 1 < x\nn : ℕ\nhn : 0 < n\n⊢ ↑x ^ n - 1 < f x ^ n",
"after_state": "No Goals!"
},
{
"line": "have hx' : 1 < (x : ℝ) := mod_cast hx",
"before_state": "f : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nx : ℚ\nhx : 1 < x\nhxnm1 : ∀ (n : ℕ), 0 < n → ↑x ^ n - 1 < f x ^ n\n⊢ ↑x ≤ f x",
"after_state": "f : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nx : ℚ\nhx : 1 < x\nhxnm1 : ∀ (n : ℕ), 0 < n → ↑x ^ n - 1 < f x ^ n\nhx' : 1 < ↑x\n⊢ ↑x ≤ f x"
},
{
"line": "refine_lift\n have hx' : 1 < (x : ℝ) := mod_cast hx;\n ?_",
"before_state": "f : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nx : ℚ\nhx : 1 < x\nhxnm1 : ∀ (n : ℕ), 0 < n → ↑x ^ n - 1 < f x ^ n\n⊢ ↑x ≤ f x",
"after_state": "f : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nx : ℚ\nhx : 1 < x\nhxnm1 : ∀ (n : ℕ), 0 < n → ↑x ^ n - 1 < f x ^ n\nhx' : 1 < ↑x\n⊢ ↑x ≤ f x"
},
{
"line": "focus\n (refine\n no_implicit_lambda%\n (have hx' : 1 < (x : ℝ) := mod_cast hx;\n ?_);\n rotate_right)",
"before_state": "f : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nx : ℚ\nhx : 1 < x\nhxnm1 : ∀ (n : ℕ), 0 < n → ↑x ^ n - 1 < f x ^ n\n⊢ ↑x ≤ f x",
"after_state": "f : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nx : ℚ\nhx : 1 < x\nhxnm1 : ∀ (n : ℕ), 0 < n → ↑x ^ n - 1 < f x ^ n\nhx' : 1 < ↑x\n⊢ ↑x ≤ f x"
},
{
"line": "refine\n no_implicit_lambda%\n (have hx' : 1 < (x : ℝ) := mod_cast hx;\n ?_)",
"before_state": "f : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nx : ℚ\nhx : 1 < x\nhxnm1 : ∀ (n : ℕ), 0 < n → ↑x ^ n - 1 < f x ^ n\n⊢ ↑x ≤ f x",
"after_state": "f : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nx : ℚ\nhx : 1 < x\nhxnm1 : ∀ (n : ℕ), 0 < n → ↑x ^ n - 1 < f x ^ n\nhx' : 1 < ↑x\n⊢ ↑x ≤ f x"
},
{
"line": "rotate_right",
"before_state": "f : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nx : ℚ\nhx : 1 < x\nhxnm1 : ∀ (n : ℕ), 0 < n → ↑x ^ n - 1 < f x ^ n\nhx' : 1 < ↑x\n⊢ ↑x ≤ f x",
"after_state": "f : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nx : ℚ\nhx : 1 < x\nhxnm1 : ∀ (n : ℕ), 0 < n → ↑x ^ n - 1 < f x ^ n\nhx' : 1 < ↑x\n⊢ ↑x ≤ f x"
},
{
"line": "have hxp : 0 < x := by positivity",
"before_state": "f : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nx : ℚ\nhx : 1 < x\nhxnm1 : ∀ (n : ℕ), 0 < n → ↑x ^ n - 1 < f x ^ n\nhx' : 1 < ↑x\n⊢ ↑x ≤ f x",
"after_state": "f : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nx : ℚ\nhx : 1 < x\nhxnm1 : ∀ (n : ℕ), 0 < n → ↑x ^ n - 1 < f x ^ n\nhx' : 1 < ↑x\nhxp : 0 < x\n⊢ ↑x ≤ f x"
},
{
"line": "focus\n refine\n no_implicit_lambda%\n (have hxp : 0 < x := ?body✝;\n ?_)\n case body✝ => with_annotate_state\"by\" (positivity)",
"before_state": "f : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nx : ℚ\nhx : 1 < x\nhxnm1 : ∀ (n : ℕ), 0 < n → ↑x ^ n - 1 < f x ^ n\nhx' : 1 < ↑x\n⊢ ↑x ≤ f x",
"after_state": "f : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nx : ℚ\nhx : 1 < x\nhxnm1 : ∀ (n : ℕ), 0 < n → ↑x ^ n - 1 < f x ^ n\nhx' : 1 < ↑x\nhxp : 0 < x\n⊢ ↑x ≤ f x"
},
{
"line": "refine\n no_implicit_lambda%\n (have hxp : 0 < x := ?body✝;\n ?_)",
"before_state": "f : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nx : ℚ\nhx : 1 < x\nhxnm1 : ∀ (n : ℕ), 0 < n → ↑x ^ n - 1 < f x ^ n\nhx' : 1 < ↑x\n⊢ ↑x ≤ f x",
"after_state": "case body\nf : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nx : ℚ\nhx : 1 < x\nhxnm1 : ∀ (n : ℕ), 0 < n → ↑x ^ n - 1 < f x ^ n\nhx' : 1 < ↑x\n⊢ 0 < x\n---\nf : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nx : ℚ\nhx : 1 < x\nhxnm1 : ∀ (n : ℕ), 0 < n → ↑x ^ n - 1 < f x ^ n\nhx' : 1 < ↑x\nhxp : 0 < x\n⊢ ↑x ≤ f x"
},
{
"line": "case body✝ => with_annotate_state\"by\" (positivity)",
"before_state": "case body\nf : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nx : ℚ\nhx : 1 < x\nhxnm1 : ∀ (n : ℕ), 0 < n → ↑x ^ n - 1 < f x ^ n\nhx' : 1 < ↑x\n⊢ 0 < x\n---\nf : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nx : ℚ\nhx : 1 < x\nhxnm1 : ∀ (n : ℕ), 0 < n → ↑x ^ n - 1 < f x ^ n\nhx' : 1 < ↑x\nhxp : 0 < x\n⊢ ↑x ≤ f x",
"after_state": "f : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nx : ℚ\nhx : 1 < x\nhxnm1 : ∀ (n : ℕ), 0 < n → ↑x ^ n - 1 < f x ^ n\nhx' : 1 < ↑x\nhxp : 0 < x\n⊢ ↑x ≤ f x"
},
{
"line": "with_annotate_state\"by\" (positivity)",
"before_state": "f : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nx : ℚ\nhx : 1 < x\nhxnm1 : ∀ (n : ℕ), 0 < n → ↑x ^ n - 1 < f x ^ n\nhx' : 1 < ↑x\n⊢ 0 < x",
"after_state": "No Goals!"
},
{
"line": "positivity",
"before_state": "f : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nx : ℚ\nhx : 1 < x\nhxnm1 : ∀ (n : ℕ), 0 < n → ↑x ^ n - 1 < f x ^ n\nhx' : 1 < ↑x\n⊢ 0 < x",
"after_state": "No Goals!"
},
{
"line": "exact le_of_all_pow_lt_succ' hx' (f_pos_of_pos hxp H1 H4) hxnm1",
"before_state": "f : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nx : ℚ\nhx : 1 < x\nhxnm1 : ∀ (n : ℕ), 0 < n → ↑x ^ n - 1 < f x ^ n\nhx' : 1 < ↑x\nhxp : 0 < x\n⊢ ↑x ≤ f x",
"after_state": "No Goals!"
},
{
"line": "have h_f_commutes_with_pos_nat_mul : ∀ n : ℕ, 0 < n → ∀ x : ℚ, 0 < x → f (n * x) = n * f x :=\n by\n intro n hn x hx\n have h2 : f (n * x) ≤ n * f x := by\n rcases n with - | n\n · exfalso; exact Nat.lt_asymm hn hn\n rcases n with - | n\n · norm_num\n have hfneq : f n.succ.succ = n.succ.succ :=\n by\n have := fixed_point_of_gt_1 (Nat.one_lt_cast.mpr (Nat.succ_lt_succ n.succ_pos)) H1 H2 H4 H5 ha1 hae\n rwa [Rat.cast_natCast n.succ.succ] at this\n rw [← hfneq]\n exact H1 (n.succ.succ : ℚ) x (Nat.cast_pos.mpr hn) hx\n exact\n h2.antisymm\n (H3 x hx n hn)\n -- For the final calculation, we expand x as (2 * x.num) / (2 * x.den), because\n -- we need the top of the fraction to be strictly greater than 1 in order\n -- to apply `fixed_point_of_gt_1`.",
"before_state": "case intro.intro\nf : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\n⊢ ∀ (x : ℚ), 0 < x → f x = ↑x",
"after_state": "case intro.intro\nf : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\nh_f_commutes_with_pos_nat_mul : ∀ (n : ℕ), 0 < n → ∀ (x : ℚ), 0 < x → f (↑n * x) = ↑n * f x\n⊢ ∀ (x : ℚ), 0 < x → f x = ↑x"
},
{
"line": "focus\n refine\n no_implicit_lambda%\n (have h_f_commutes_with_pos_nat_mul : ∀ n : ℕ, 0 < n → ∀ x : ℚ, 0 < x → f (n * x) = n * f x := ?body✝;\n ?_)\n case body✝ =>\n with_annotate_state\"by\"\n ( intro n hn x hx\n have h2 : f (n * x) ≤ n * f x := by\n rcases n with - | n\n · exfalso; exact Nat.lt_asymm hn hn\n rcases n with - | n\n · norm_num\n have hfneq : f n.succ.succ = n.succ.succ :=\n by\n have := fixed_point_of_gt_1 (Nat.one_lt_cast.mpr (Nat.succ_lt_succ n.succ_pos)) H1 H2 H4 H5 ha1 hae\n rwa [Rat.cast_natCast n.succ.succ] at this\n rw [← hfneq]\n exact H1 (n.succ.succ : ℚ) x (Nat.cast_pos.mpr hn) hx\n exact\n h2.antisymm\n (H3 x hx n hn)\n -- For the final calculation, we expand x as (2 * x.num) / (2 * x.den), because\n -- we need the top of the fraction to be strictly greater than 1 in order\n -- to apply `fixed_point_of_gt_1`.)",
"before_state": "case intro.intro\nf : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\n⊢ ∀ (x : ℚ), 0 < x → f x = ↑x",
"after_state": "case intro.intro\nf : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\nh_f_commutes_with_pos_nat_mul : ∀ (n : ℕ), 0 < n → ∀ (x : ℚ), 0 < x → f (↑n * x) = ↑n * f x\n⊢ ∀ (x : ℚ), 0 < x → f x = ↑x"
},
{
"line": "refine\n no_implicit_lambda%\n (have h_f_commutes_with_pos_nat_mul : ∀ n : ℕ, 0 < n → ∀ x : ℚ, 0 < x → f (n * x) = n * f x := ?body✝;\n ?_)",
"before_state": "case intro.intro\nf : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\n⊢ ∀ (x : ℚ), 0 < x → f x = ↑x",
"after_state": "case body\nf : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\n⊢ ∀ (n : ℕ), 0 < n → ∀ (x : ℚ), 0 < x → f (↑n * x) = ↑n * f x\n---\ncase intro.intro\nf : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\nh_f_commutes_with_pos_nat_mul : ∀ (n : ℕ), 0 < n → ∀ (x : ℚ), 0 < x → f (↑n * x) = ↑n * f x\n⊢ ∀ (x : ℚ), 0 < x → f x = ↑x"
},
{
"line": "case body✝ =>\n with_annotate_state\"by\"\n ( intro n hn x hx\n have h2 : f (n * x) ≤ n * f x := by\n rcases n with - | n\n · exfalso; exact Nat.lt_asymm hn hn\n rcases n with - | n\n · norm_num\n have hfneq : f n.succ.succ = n.succ.succ :=\n by\n have := fixed_point_of_gt_1 (Nat.one_lt_cast.mpr (Nat.succ_lt_succ n.succ_pos)) H1 H2 H4 H5 ha1 hae\n rwa [Rat.cast_natCast n.succ.succ] at this\n rw [← hfneq]\n exact H1 (n.succ.succ : ℚ) x (Nat.cast_pos.mpr hn) hx\n exact\n h2.antisymm\n (H3 x hx n hn)\n -- For the final calculation, we expand x as (2 * x.num) / (2 * x.den), because\n -- we need the top of the fraction to be strictly greater than 1 in order\n -- to apply `fixed_point_of_gt_1`.)",
"before_state": "case body\nf : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\n⊢ ∀ (n : ℕ), 0 < n → ∀ (x : ℚ), 0 < x → f (↑n * x) = ↑n * f x\n---\ncase intro.intro\nf : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\nh_f_commutes_with_pos_nat_mul : ∀ (n : ℕ), 0 < n → ∀ (x : ℚ), 0 < x → f (↑n * x) = ↑n * f x\n⊢ ∀ (x : ℚ), 0 < x → f x = ↑x",
"after_state": "case intro.intro\nf : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\nh_f_commutes_with_pos_nat_mul : ∀ (n : ℕ), 0 < n → ∀ (x : ℚ), 0 < x → f (↑n * x) = ↑n * f x\n⊢ ∀ (x : ℚ), 0 < x → f x = ↑x"
},
{
"line": "with_annotate_state\"by\"\n ( intro n hn x hx\n have h2 : f (n * x) ≤ n * f x := by\n rcases n with - | n\n · exfalso; exact Nat.lt_asymm hn hn\n rcases n with - | n\n · norm_num\n have hfneq : f n.succ.succ = n.succ.succ :=\n by\n have := fixed_point_of_gt_1 (Nat.one_lt_cast.mpr (Nat.succ_lt_succ n.succ_pos)) H1 H2 H4 H5 ha1 hae\n rwa [Rat.cast_natCast n.succ.succ] at this\n rw [← hfneq]\n exact H1 (n.succ.succ : ℚ) x (Nat.cast_pos.mpr hn) hx\n exact\n h2.antisymm\n (H3 x hx n hn)\n -- For the final calculation, we expand x as (2 * x.num) / (2 * x.den), because\n -- we need the top of the fraction to be strictly greater than 1 in order\n -- to apply `fixed_point_of_gt_1`.)",
"before_state": "f : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\n⊢ ∀ (n : ℕ), 0 < n → ∀ (x : ℚ), 0 < x → f (↑n * x) = ↑n * f x",
"after_state": "No Goals!"
},
{
"line": "intro n hn x hx",
"before_state": "f : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\n⊢ ∀ (n : ℕ), 0 < n → ∀ (x : ℚ), 0 < x → f (↑n * x) = ↑n * f x",
"after_state": "f : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\nn : ℕ\nhn : 0 < n\nx : ℚ\nhx : 0 < x\n⊢ f (↑n * x) = ↑n * f x"
},
{
"line": "intro n;\n intro hn x hx",
"before_state": "f : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\n⊢ ∀ (n : ℕ), 0 < n → ∀ (x : ℚ), 0 < x → f (↑n * x) = ↑n * f x",
"after_state": "f : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\nn : ℕ\nhn : 0 < n\nx : ℚ\nhx : 0 < x\n⊢ f (↑n * x) = ↑n * f x"
},
{
"line": "intro n",
"before_state": "f : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\n⊢ ∀ (n : ℕ), 0 < n → ∀ (x : ℚ), 0 < x → f (↑n * x) = ↑n * f x",
"after_state": "f : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\nn : ℕ\n⊢ 0 < n → ∀ (x : ℚ), 0 < x → f (↑n * x) = ↑n * f x"
},
{
"line": "intro hn x hx",
"before_state": "f : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\nn : ℕ\n⊢ 0 < n → ∀ (x : ℚ), 0 < x → f (↑n * x) = ↑n * f x",
"after_state": "f : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\nn : ℕ\nhn : 0 < n\nx : ℚ\nhx : 0 < x\n⊢ f (↑n * x) = ↑n * f x"
},
{
"line": "intro hn;\n intro x hx",
"before_state": "f : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\nn : ℕ\n⊢ 0 < n → ∀ (x : ℚ), 0 < x → f (↑n * x) = ↑n * f x",
"after_state": "f : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\nn : ℕ\nhn : 0 < n\nx : ℚ\nhx : 0 < x\n⊢ f (↑n * x) = ↑n * f x"
},
{
"line": "intro hn",
"before_state": "f : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\nn : ℕ\n⊢ 0 < n → ∀ (x : ℚ), 0 < x → f (↑n * x) = ↑n * f x",
"after_state": "f : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\nn : ℕ\nhn : 0 < n\n⊢ ∀ (x : ℚ), 0 < x → f (↑n * x) = ↑n * f x"
},
{
"line": "intro x hx",
"before_state": "f : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\nn : ℕ\nhn : 0 < n\n⊢ ∀ (x : ℚ), 0 < x → f (↑n * x) = ↑n * f x",
"after_state": "f : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\nn : ℕ\nhn : 0 < n\nx : ℚ\nhx : 0 < x\n⊢ f (↑n * x) = ↑n * f x"
},
{
"line": "intro x;\n intro hx",
"before_state": "f : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\nn : ℕ\nhn : 0 < n\n⊢ ∀ (x : ℚ), 0 < x → f (↑n * x) = ↑n * f x",
"after_state": "f : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\nn : ℕ\nhn : 0 < n\nx : ℚ\nhx : 0 < x\n⊢ f (↑n * x) = ↑n * f x"
},
{
"line": "intro x",
"before_state": "f : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\nn : ℕ\nhn : 0 < n\n⊢ ∀ (x : ℚ), 0 < x → f (↑n * x) = ↑n * f x",
"after_state": "f : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\nn : ℕ\nhn : 0 < n\nx : ℚ\n⊢ 0 < x → f (↑n * x) = ↑n * f x"
},
{
"line": "intro hx",
"before_state": "f : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\nn : ℕ\nhn : 0 < n\nx : ℚ\n⊢ 0 < x → f (↑n * x) = ↑n * f x",
"after_state": "f : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\nn : ℕ\nhn : 0 < n\nx : ℚ\nhx : 0 < x\n⊢ f (↑n * x) = ↑n * f x"
},
{
"line": "have h2 : f (n * x) ≤ n * f x := by\n rcases n with - | n\n · exfalso; exact Nat.lt_asymm hn hn\n rcases n with - | n\n · norm_num\n have hfneq : f n.succ.succ = n.succ.succ :=\n by\n have := fixed_point_of_gt_1 (Nat.one_lt_cast.mpr (Nat.succ_lt_succ n.succ_pos)) H1 H2 H4 H5 ha1 hae\n rwa [Rat.cast_natCast n.succ.succ] at this\n rw [← hfneq]\n exact H1 (n.succ.succ : ℚ) x (Nat.cast_pos.mpr hn) hx",
"before_state": "f : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\nn : ℕ\nhn : 0 < n\nx : ℚ\nhx : 0 < x\n⊢ f (↑n * x) = ↑n * f x",
"after_state": "f : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\nn : ℕ\nhn : 0 < n\nx : ℚ\nhx : 0 < x\nh2 : f (↑n * x) ≤ ↑n * f x\n⊢ f (↑n * x) = ↑n * f x"
},
{
"line": "focus\n refine\n no_implicit_lambda%\n (have h2 : f (n * x) ≤ n * f x := ?body✝;\n ?_)\n case body✝ =>\n with_annotate_state\"by\"\n ( rcases n with - | n\n · exfalso; exact Nat.lt_asymm hn hn\n rcases n with - | n\n · norm_num\n have hfneq : f n.succ.succ = n.succ.succ :=\n by\n have := fixed_point_of_gt_1 (Nat.one_lt_cast.mpr (Nat.succ_lt_succ n.succ_pos)) H1 H2 H4 H5 ha1 hae\n rwa [Rat.cast_natCast n.succ.succ] at this\n rw [← hfneq]\n exact H1 (n.succ.succ : ℚ) x (Nat.cast_pos.mpr hn) hx)",
"before_state": "f : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\nn : ℕ\nhn : 0 < n\nx : ℚ\nhx : 0 < x\n⊢ f (↑n * x) = ↑n * f x",
"after_state": "f : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\nn : ℕ\nhn : 0 < n\nx : ℚ\nhx : 0 < x\nh2 : f (↑n * x) ≤ ↑n * f x\n⊢ f (↑n * x) = ↑n * f x"
},
{
"line": "refine\n no_implicit_lambda%\n (have h2 : f (n * x) ≤ n * f x := ?body✝;\n ?_)",
"before_state": "f : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\nn : ℕ\nhn : 0 < n\nx : ℚ\nhx : 0 < x\n⊢ f (↑n * x) = ↑n * f x",
"after_state": "case body\nf : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\nn : ℕ\nhn : 0 < n\nx : ℚ\nhx : 0 < x\n⊢ f (↑n * x) ≤ ↑n * f x\n---\nf : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\nn : ℕ\nhn : 0 < n\nx : ℚ\nhx : 0 < x\nh2 : f (↑n * x) ≤ ↑n * f x\n⊢ f (↑n * x) = ↑n * f x"
},
{
"line": "case body✝ =>\n with_annotate_state\"by\"\n ( rcases n with - | n\n · exfalso; exact Nat.lt_asymm hn hn\n rcases n with - | n\n · norm_num\n have hfneq : f n.succ.succ = n.succ.succ :=\n by\n have := fixed_point_of_gt_1 (Nat.one_lt_cast.mpr (Nat.succ_lt_succ n.succ_pos)) H1 H2 H4 H5 ha1 hae\n rwa [Rat.cast_natCast n.succ.succ] at this\n rw [← hfneq]\n exact H1 (n.succ.succ : ℚ) x (Nat.cast_pos.mpr hn) hx)",
"before_state": "case body\nf : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\nn : ℕ\nhn : 0 < n\nx : ℚ\nhx : 0 < x\n⊢ f (↑n * x) ≤ ↑n * f x\n---\nf : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\nn : ℕ\nhn : 0 < n\nx : ℚ\nhx : 0 < x\nh2 : f (↑n * x) ≤ ↑n * f x\n⊢ f (↑n * x) = ↑n * f x",
"after_state": "f : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\nn : ℕ\nhn : 0 < n\nx : ℚ\nhx : 0 < x\nh2 : f (↑n * x) ≤ ↑n * f x\n⊢ f (↑n * x) = ↑n * f x"
},
{
"line": "with_annotate_state\"by\"\n ( rcases n with - | n\n · exfalso; exact Nat.lt_asymm hn hn\n rcases n with - | n\n · norm_num\n have hfneq : f n.succ.succ = n.succ.succ :=\n by\n have := fixed_point_of_gt_1 (Nat.one_lt_cast.mpr (Nat.succ_lt_succ n.succ_pos)) H1 H2 H4 H5 ha1 hae\n rwa [Rat.cast_natCast n.succ.succ] at this\n rw [← hfneq]\n exact H1 (n.succ.succ : ℚ) x (Nat.cast_pos.mpr hn) hx)",
"before_state": "f : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\nn : ℕ\nhn : 0 < n\nx : ℚ\nhx : 0 < x\n⊢ f (↑n * x) ≤ ↑n * f x",
"after_state": "No Goals!"
},
{
"line": "rcases n with - | n",
"before_state": "f : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\nn : ℕ\nhn : 0 < n\nx : ℚ\nhx : 0 < x\n⊢ f (↑n * x) ≤ ↑n * f x",
"after_state": "case zero\nf : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\nx : ℚ\nhx : 0 < x\nhn : 0 < 0\n⊢ f (↑0 * x) ≤ ↑0 * f x\n---\ncase succ\nf : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\nx : ℚ\nhx : 0 < x\nn : ℕ\nhn : 0 < n + 1\n⊢ f (↑(n + 1) * x) ≤ ↑(n + 1) * f x"
},
{
"line": "exfalso",
"before_state": "case zero\nf : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\nx : ℚ\nhx : 0 < x\nhn : 0 < 0\n⊢ f (↑0 * x) ≤ ↑0 * f x",
"after_state": "case zero\nf : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\nx : ℚ\nhx : 0 < x\nhn : 0 < 0\n⊢ False"
},
{
"line": "refine False.elim✝ ?_",
"before_state": "case zero\nf : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\nx : ℚ\nhx : 0 < x\nhn : 0 < 0\n⊢ f (↑0 * x) ≤ ↑0 * f x",
"after_state": "case zero\nf : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\nx : ℚ\nhx : 0 < x\nhn : 0 < 0\n⊢ False"
},
{
"line": "exact Nat.lt_asymm hn hn",
"before_state": "case zero\nf : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\nx : ℚ\nhx : 0 < x\nhn : 0 < 0\n⊢ False",
"after_state": "No Goals!"
},
{
"line": "rcases n with - | n",
"before_state": "case succ\nf : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\nx : ℚ\nhx : 0 < x\nn : ℕ\nhn : 0 < n + 1\n⊢ f (↑(n + 1) * x) ≤ ↑(n + 1) * f x",
"after_state": "case succ.zero\nf : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\nx : ℚ\nhx : 0 < x\nhn : 0 < 0 + 1\n⊢ f (↑(0 + 1) * x) ≤ ↑(0 + 1) * f x\n---\ncase succ.succ\nf : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\nx : ℚ\nhx : 0 < x\nn : ℕ\nhn : 0 < n + 1 + 1\n⊢ f (↑(n + 1 + 1) * x) ≤ ↑(n + 1 + 1) * f x"
},
{
"line": "norm_num",
"before_state": "case succ.zero\nf : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\nx : ℚ\nhx : 0 < x\nhn : 0 < 0 + 1\n⊢ f (↑(0 + 1) * x) ≤ ↑(0 + 1) * f x",
"after_state": "No Goals!"
},
{
"line": "have hfneq : f n.succ.succ = n.succ.succ :=\n by\n have := fixed_point_of_gt_1 (Nat.one_lt_cast.mpr (Nat.succ_lt_succ n.succ_pos)) H1 H2 H4 H5 ha1 hae\n rwa [Rat.cast_natCast n.succ.succ] at this",
"before_state": "case succ.succ\nf : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\nx : ℚ\nhx : 0 < x\nn : ℕ\nhn : 0 < n + 1 + 1\n⊢ f (↑(n + 1 + 1) * x) ≤ ↑(n + 1 + 1) * f x",
"after_state": "case succ.succ\nf : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\nx : ℚ\nhx : 0 < x\nn : ℕ\nhn : 0 < n + 1 + 1\nhfneq : f ↑n.succ.succ = ↑n.succ.succ\n⊢ f (↑(n + 1 + 1) * x) ≤ ↑(n + 1 + 1) * f x"
},
{
"line": "focus\n refine\n no_implicit_lambda%\n (have hfneq : f n.succ.succ = n.succ.succ := ?body✝;\n ?_)\n case body✝ =>\n with_annotate_state\"by\"\n ( have := fixed_point_of_gt_1 (Nat.one_lt_cast.mpr (Nat.succ_lt_succ n.succ_pos)) H1 H2 H4 H5 ha1 hae\n rwa [Rat.cast_natCast n.succ.succ] at this)",
"before_state": "case succ.succ\nf : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\nx : ℚ\nhx : 0 < x\nn : ℕ\nhn : 0 < n + 1 + 1\n⊢ f (↑(n + 1 + 1) * x) ≤ ↑(n + 1 + 1) * f x",
"after_state": "case succ.succ\nf : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\nx : ℚ\nhx : 0 < x\nn : ℕ\nhn : 0 < n + 1 + 1\nhfneq : f ↑n.succ.succ = ↑n.succ.succ\n⊢ f (↑(n + 1 + 1) * x) ≤ ↑(n + 1 + 1) * f x"
},
{
"line": "refine\n no_implicit_lambda%\n (have hfneq : f n.succ.succ = n.succ.succ := ?body✝;\n ?_)",
"before_state": "case succ.succ\nf : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\nx : ℚ\nhx : 0 < x\nn : ℕ\nhn : 0 < n + 1 + 1\n⊢ f (↑(n + 1 + 1) * x) ≤ ↑(n + 1 + 1) * f x",
"after_state": "case body\nf : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\nx : ℚ\nhx : 0 < x\nn : ℕ\nhn : 0 < n + 1 + 1\n⊢ f ↑n.succ.succ = ↑n.succ.succ\n---\ncase succ.succ\nf : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\nx : ℚ\nhx : 0 < x\nn : ℕ\nhn : 0 < n + 1 + 1\nhfneq : f ↑n.succ.succ = ↑n.succ.succ\n⊢ f (↑(n + 1 + 1) * x) ≤ ↑(n + 1 + 1) * f x"
},
{
"line": "case body✝ =>\n with_annotate_state\"by\"\n ( have := fixed_point_of_gt_1 (Nat.one_lt_cast.mpr (Nat.succ_lt_succ n.succ_pos)) H1 H2 H4 H5 ha1 hae\n rwa [Rat.cast_natCast n.succ.succ] at this)",
"before_state": "case body\nf : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\nx : ℚ\nhx : 0 < x\nn : ℕ\nhn : 0 < n + 1 + 1\n⊢ f ↑n.succ.succ = ↑n.succ.succ\n---\ncase succ.succ\nf : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\nx : ℚ\nhx : 0 < x\nn : ℕ\nhn : 0 < n + 1 + 1\nhfneq : f ↑n.succ.succ = ↑n.succ.succ\n⊢ f (↑(n + 1 + 1) * x) ≤ ↑(n + 1 + 1) * f x",
"after_state": "case succ.succ\nf : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\nx : ℚ\nhx : 0 < x\nn : ℕ\nhn : 0 < n + 1 + 1\nhfneq : f ↑n.succ.succ = ↑n.succ.succ\n⊢ f (↑(n + 1 + 1) * x) ≤ ↑(n + 1 + 1) * f x"
},
{
"line": "with_annotate_state\"by\"\n ( have := fixed_point_of_gt_1 (Nat.one_lt_cast.mpr (Nat.succ_lt_succ n.succ_pos)) H1 H2 H4 H5 ha1 hae\n rwa [Rat.cast_natCast n.succ.succ] at this)",
"before_state": "f : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\nx : ℚ\nhx : 0 < x\nn : ℕ\nhn : 0 < n + 1 + 1\n⊢ f ↑n.succ.succ = ↑n.succ.succ",
"after_state": "f : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\nx : ℚ\nhx : 0 < x\nn : ℕ\nhn : 0 < n + 1 + 1\n⊢ f ↑n.succ.succ = ↑n.succ.succ"
},
{
"line": "have := fixed_point_of_gt_1 (Nat.one_lt_cast.mpr (Nat.succ_lt_succ n.succ_pos)) H1 H2 H4 H5 ha1 hae",
"before_state": "f : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\nx : ℚ\nhx : 0 < x\nn : ℕ\nhn : 0 < n + 1 + 1\n⊢ f ↑n.succ.succ = ↑n.succ.succ",
"after_state": "f : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\nx : ℚ\nhx : 0 < x\nn : ℕ\nhn : 0 < n + 1 + 1\n⊢ f ↑n.succ.succ = ↑n.succ.succ"
},
{
"line": "refine_lift\n have := fixed_point_of_gt_1 (Nat.one_lt_cast.mpr (Nat.succ_lt_succ n.succ_pos)) H1 H2 H4 H5 ha1 hae;\n ?_",
"before_state": "f : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\nx : ℚ\nhx : 0 < x\nn : ℕ\nhn : 0 < n + 1 + 1\n⊢ f ↑n.succ.succ = ↑n.succ.succ",
"after_state": "f : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\nx : ℚ\nhx : 0 < x\nn : ℕ\nhn : 0 < n + 1 + 1\n⊢ f ↑n.succ.succ = ↑n.succ.succ"
},
{
"line": "focus\n (refine\n no_implicit_lambda%\n (have := fixed_point_of_gt_1 (Nat.one_lt_cast.mpr (Nat.succ_lt_succ n.succ_pos)) H1 H2 H4 H5 ha1 hae;\n ?_);\n rotate_right)",
"before_state": "f : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\nx : ℚ\nhx : 0 < x\nn : ℕ\nhn : 0 < n + 1 + 1\n⊢ f ↑n.succ.succ = ↑n.succ.succ",
"after_state": "f : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\nx : ℚ\nhx : 0 < x\nn : ℕ\nhn : 0 < n + 1 + 1\n⊢ f ↑n.succ.succ = ↑n.succ.succ"
},
{
"line": "refine\n no_implicit_lambda%\n (have := fixed_point_of_gt_1 (Nat.one_lt_cast.mpr (Nat.succ_lt_succ n.succ_pos)) H1 H2 H4 H5 ha1 hae;\n ?_)",
"before_state": "f : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\nx : ℚ\nhx : 0 < x\nn : ℕ\nhn : 0 < n + 1 + 1\n⊢ f ↑n.succ.succ = ↑n.succ.succ",
"after_state": "f : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\nx : ℚ\nhx : 0 < x\nn : ℕ\nhn : 0 < n + 1 + 1\n⊢ f ↑n.succ.succ = ↑n.succ.succ"
},
{
"line": "rw [← hfneq]",
"before_state": "case succ.succ\nf : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\nx : ℚ\nhx : 0 < x\nn : ℕ\nhn : 0 < n + 1 + 1\nhfneq : f ↑n.succ.succ = ↑n.succ.succ\n⊢ f (↑(n + 1 + 1) * x) ≤ ↑(n + 1 + 1) * f x",
"after_state": "case succ.succ\nf : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\nx : ℚ\nhx : 0 < x\nn : ℕ\nhn : 0 < n + 1 + 1\nhfneq : f ↑n.succ.succ = ↑n.succ.succ\n⊢ f (↑(n + 1 + 1) * x) ≤ f ↑n.succ.succ * f x"
},
{
"line": "rewrite [← hfneq]",
"before_state": "case succ.succ\nf : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\nx : ℚ\nhx : 0 < x\nn : ℕ\nhn : 0 < n + 1 + 1\nhfneq : f ↑n.succ.succ = ↑n.succ.succ\n⊢ f (↑(n + 1 + 1) * x) ≤ ↑(n + 1 + 1) * f x",
"after_state": "case succ.succ\nf : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\nx : ℚ\nhx : 0 < x\nn : ℕ\nhn : 0 < n + 1 + 1\nhfneq : f ↑n.succ.succ = ↑n.succ.succ\n⊢ f (↑(n + 1 + 1) * x) ≤ f ↑n.succ.succ * f x"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "case succ.succ\nf : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\nx : ℚ\nhx : 0 < x\nn : ℕ\nhn : 0 < n + 1 + 1\nhfneq : f ↑n.succ.succ = ↑n.succ.succ\n⊢ f (↑(n + 1 + 1) * x) ≤ f ↑n.succ.succ * f x",
"after_state": "case succ.succ\nf : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\nx : ℚ\nhx : 0 < x\nn : ℕ\nhn : 0 < n + 1 + 1\nhfneq : f ↑n.succ.succ = ↑n.succ.succ\n⊢ f (↑(n + 1 + 1) * x) ≤ f ↑n.succ.succ * f x"
},
{
"line": "try (with_reducible rfl)",
"before_state": "case succ.succ\nf : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\nx : ℚ\nhx : 0 < x\nn : ℕ\nhn : 0 < n + 1 + 1\nhfneq : f ↑n.succ.succ = ↑n.succ.succ\n⊢ f (↑(n + 1 + 1) * x) ≤ f ↑n.succ.succ * f x",
"after_state": "case succ.succ\nf : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\nx : ℚ\nhx : 0 < x\nn : ℕ\nhn : 0 < n + 1 + 1\nhfneq : f ↑n.succ.succ = ↑n.succ.succ\n⊢ f (↑(n + 1 + 1) * x) ≤ f ↑n.succ.succ * f x"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "case succ.succ\nf : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\nx : ℚ\nhx : 0 < x\nn : ℕ\nhn : 0 < n + 1 + 1\nhfneq : f ↑n.succ.succ = ↑n.succ.succ\n⊢ f (↑(n + 1 + 1) * x) ≤ f ↑n.succ.succ * f x",
"after_state": "case succ.succ\nf : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\nx : ℚ\nhx : 0 < x\nn : ℕ\nhn : 0 < n + 1 + 1\nhfneq : f ↑n.succ.succ = ↑n.succ.succ\n⊢ f (↑(n + 1 + 1) * x) ≤ f ↑n.succ.succ * f x"
},
{
"line": "with_reducible rfl",
"before_state": "case succ.succ\nf : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\nx : ℚ\nhx : 0 < x\nn : ℕ\nhn : 0 < n + 1 + 1\nhfneq : f ↑n.succ.succ = ↑n.succ.succ\n⊢ f (↑(n + 1 + 1) * x) ≤ f ↑n.succ.succ * f x",
"after_state": "case succ.succ\nf : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\nx : ℚ\nhx : 0 < x\nn : ℕ\nhn : 0 < n + 1 + 1\nhfneq : f ↑n.succ.succ = ↑n.succ.succ\n⊢ f (↑(n + 1 + 1) * x) ≤ f ↑n.succ.succ * f x"
},
{
"line": "rfl",
"before_state": "case succ.succ\nf : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\nx : ℚ\nhx : 0 < x\nn : ℕ\nhn : 0 < n + 1 + 1\nhfneq : f ↑n.succ.succ = ↑n.succ.succ\n⊢ f (↑(n + 1 + 1) * x) ≤ f ↑n.succ.succ * f x",
"after_state": "case succ.succ\nf : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\nx : ℚ\nhx : 0 < x\nn : ℕ\nhn : 0 < n + 1 + 1\nhfneq : f ↑n.succ.succ = ↑n.succ.succ\n⊢ f (↑(n + 1 + 1) * x) ≤ f ↑n.succ.succ * f x"
},
{
"line": "apply_rfl",
"before_state": "case succ.succ\nf : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\nx : ℚ\nhx : 0 < x\nn : ℕ\nhn : 0 < n + 1 + 1\nhfneq : f ↑n.succ.succ = ↑n.succ.succ\n⊢ f (↑(n + 1 + 1) * x) ≤ f ↑n.succ.succ * f x",
"after_state": "case succ.succ\nf : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\nx : ℚ\nhx : 0 < x\nn : ℕ\nhn : 0 < n + 1 + 1\nhfneq : f ↑n.succ.succ = ↑n.succ.succ\n⊢ f (↑(n + 1 + 1) * x) ≤ f ↑n.succ.succ * f x"
},
{
"line": "skip",
"before_state": "case succ.succ\nf : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\nx : ℚ\nhx : 0 < x\nn : ℕ\nhn : 0 < n + 1 + 1\nhfneq : f ↑n.succ.succ = ↑n.succ.succ\n⊢ f (↑(n + 1 + 1) * x) ≤ f ↑n.succ.succ * f x",
"after_state": "case succ.succ\nf : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\nx : ℚ\nhx : 0 < x\nn : ℕ\nhn : 0 < n + 1 + 1\nhfneq : f ↑n.succ.succ = ↑n.succ.succ\n⊢ f (↑(n + 1 + 1) * x) ≤ f ↑n.succ.succ * f x"
},
{
"line": "exact H1 (n.succ.succ : ℚ) x (Nat.cast_pos.mpr hn) hx",
"before_state": "case succ.succ\nf : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\nx : ℚ\nhx : 0 < x\nn : ℕ\nhn : 0 < n + 1 + 1\nhfneq : f ↑n.succ.succ = ↑n.succ.succ\n⊢ f (↑(n + 1 + 1) * x) ≤ f ↑n.succ.succ * f x",
"after_state": "No Goals!"
},
{
"line": "exact\n h2.antisymm\n (H3 x hx n hn)\n -- For the final calculation, we expand x as (2 * x.num) / (2 * x.den), because\n -- we need the top of the fraction to be strictly greater than 1 in order\n -- to apply `fixed_point_of_gt_1`.",
"before_state": "f : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\nn : ℕ\nhn : 0 < n\nx : ℚ\nhx : 0 < x\nh2 : f (↑n * x) ≤ ↑n * f x\n⊢ f (↑n * x) = ↑n * f x",
"after_state": "No Goals!"
},
{
"line": "intro x hx",
"before_state": "case intro.intro\nf : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\nh_f_commutes_with_pos_nat_mul : ∀ (n : ℕ), 0 < n → ∀ (x : ℚ), 0 < x → f (↑n * x) = ↑n * f x\n⊢ ∀ (x : ℚ), 0 < x → f x = ↑x",
"after_state": "case intro.intro\nf : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\nh_f_commutes_with_pos_nat_mul : ∀ (n : ℕ), 0 < n → ∀ (x : ℚ), 0 < x → f (↑n * x) = ↑n * f x\nx : ℚ\nhx : 0 < x\n⊢ f x = ↑x"
},
{
"line": "intro x;\n intro hx",
"before_state": "case intro.intro\nf : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\nh_f_commutes_with_pos_nat_mul : ∀ (n : ℕ), 0 < n → ∀ (x : ℚ), 0 < x → f (↑n * x) = ↑n * f x\n⊢ ∀ (x : ℚ), 0 < x → f x = ↑x",
"after_state": "case intro.intro\nf : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\nh_f_commutes_with_pos_nat_mul : ∀ (n : ℕ), 0 < n → ∀ (x : ℚ), 0 < x → f (↑n * x) = ↑n * f x\nx : ℚ\nhx : 0 < x\n⊢ f x = ↑x"
},
{
"line": "intro x",
"before_state": "case intro.intro\nf : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\nh_f_commutes_with_pos_nat_mul : ∀ (n : ℕ), 0 < n → ∀ (x : ℚ), 0 < x → f (↑n * x) = ↑n * f x\n⊢ ∀ (x : ℚ), 0 < x → f x = ↑x",
"after_state": "case intro.intro\nf : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\nh_f_commutes_with_pos_nat_mul : ∀ (n : ℕ), 0 < n → ∀ (x : ℚ), 0 < x → f (↑n * x) = ↑n * f x\nx : ℚ\n⊢ 0 < x → f x = ↑x"
},
{
"line": "intro hx",
"before_state": "case intro.intro\nf : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\nh_f_commutes_with_pos_nat_mul : ∀ (n : ℕ), 0 < n → ∀ (x : ℚ), 0 < x → f (↑n * x) = ↑n * f x\nx : ℚ\n⊢ 0 < x → f x = ↑x",
"after_state": "case intro.intro\nf : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\nh_f_commutes_with_pos_nat_mul : ∀ (n : ℕ), 0 < n → ∀ (x : ℚ), 0 < x → f (↑n * x) = ↑n * f x\nx : ℚ\nhx : 0 < x\n⊢ f x = ↑x"
},
{
"line": "have H₀ : x * x.den = x.num := x.mul_den_eq_num",
"before_state": "case intro.intro\nf : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\nh_f_commutes_with_pos_nat_mul : ∀ (n : ℕ), 0 < n → ∀ (x : ℚ), 0 < x → f (↑n * x) = ↑n * f x\nx : ℚ\nhx : 0 < x\n⊢ f x = ↑x",
"after_state": "case intro.intro\nf : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\nh_f_commutes_with_pos_nat_mul : ∀ (n : ℕ), 0 < n → ∀ (x : ℚ), 0 < x → f (↑n * x) = ↑n * f x\nx : ℚ\nhx : 0 < x\nH₀ : x * ↑x.den = ↑x.num\n⊢ f x = ↑x"
},
{
"line": "refine_lift\n have H₀ : x * x.den = x.num := x.mul_den_eq_num;\n ?_",
"before_state": "case intro.intro\nf : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\nh_f_commutes_with_pos_nat_mul : ∀ (n : ℕ), 0 < n → ∀ (x : ℚ), 0 < x → f (↑n * x) = ↑n * f x\nx : ℚ\nhx : 0 < x\n⊢ f x = ↑x",
"after_state": "case intro.intro\nf : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\nh_f_commutes_with_pos_nat_mul : ∀ (n : ℕ), 0 < n → ∀ (x : ℚ), 0 < x → f (↑n * x) = ↑n * f x\nx : ℚ\nhx : 0 < x\nH₀ : x * ↑x.den = ↑x.num\n⊢ f x = ↑x"
},
{
"line": "focus\n (refine\n no_implicit_lambda%\n (have H₀ : x * x.den = x.num := x.mul_den_eq_num;\n ?_);\n rotate_right)",
"before_state": "case intro.intro\nf : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\nh_f_commutes_with_pos_nat_mul : ∀ (n : ℕ), 0 < n → ∀ (x : ℚ), 0 < x → f (↑n * x) = ↑n * f x\nx : ℚ\nhx : 0 < x\n⊢ f x = ↑x",
"after_state": "case intro.intro\nf : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\nh_f_commutes_with_pos_nat_mul : ∀ (n : ℕ), 0 < n → ∀ (x : ℚ), 0 < x → f (↑n * x) = ↑n * f x\nx : ℚ\nhx : 0 < x\nH₀ : x * ↑x.den = ↑x.num\n⊢ f x = ↑x"
},
{
"line": "refine\n no_implicit_lambda%\n (have H₀ : x * x.den = x.num := x.mul_den_eq_num;\n ?_)",
"before_state": "case intro.intro\nf : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\nh_f_commutes_with_pos_nat_mul : ∀ (n : ℕ), 0 < n → ∀ (x : ℚ), 0 < x → f (↑n * x) = ↑n * f x\nx : ℚ\nhx : 0 < x\n⊢ f x = ↑x",
"after_state": "case intro.intro\nf : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\nh_f_commutes_with_pos_nat_mul : ∀ (n : ℕ), 0 < n → ∀ (x : ℚ), 0 < x → f (↑n * x) = ↑n * f x\nx : ℚ\nhx : 0 < x\nH₀ : x * ↑x.den = ↑x.num\n⊢ f x = ↑x"
},
{
"line": "rotate_right",
"before_state": "case intro.intro\nf : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\nh_f_commutes_with_pos_nat_mul : ∀ (n : ℕ), 0 < n → ∀ (x : ℚ), 0 < x → f (↑n * x) = ↑n * f x\nx : ℚ\nhx : 0 < x\nH₀ : x * ↑x.den = ↑x.num\n⊢ f x = ↑x",
"after_state": "case intro.intro\nf : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\nh_f_commutes_with_pos_nat_mul : ∀ (n : ℕ), 0 < n → ∀ (x : ℚ), 0 < x → f (↑n * x) = ↑n * f x\nx : ℚ\nhx : 0 < x\nH₀ : x * ↑x.den = ↑x.num\n⊢ f x = ↑x"
},
{
"line": "have H : x * (↑(2 * x.den) : ℚ) = (↑(2 * x.num) : ℚ) := by push_cast; linear_combination 2 * H₀",
"before_state": "case intro.intro\nf : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\nh_f_commutes_with_pos_nat_mul : ∀ (n : ℕ), 0 < n → ∀ (x : ℚ), 0 < x → f (↑n * x) = ↑n * f x\nx : ℚ\nhx : 0 < x\nH₀ : x * ↑x.den = ↑x.num\n⊢ f x = ↑x",
"after_state": "case intro.intro\nf : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\nh_f_commutes_with_pos_nat_mul : ∀ (n : ℕ), 0 < n → ∀ (x : ℚ), 0 < x → f (↑n * x) = ↑n * f x\nx : ℚ\nhx : 0 < x\nH₀ : x * ↑x.den = ↑x.num\nH : x * ↑(2 * x.den) = ↑(2 * x.num)\n⊢ f x = ↑x"
},
{
"line": "focus\n refine\n no_implicit_lambda%\n (have H : x * (↑(2 * x.den) : ℚ) = (↑(2 * x.num) : ℚ) := ?body✝;\n ?_)\n case body✝ => with_annotate_state\"by\" (push_cast; linear_combination 2 * H₀)",
"before_state": "case intro.intro\nf : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\nh_f_commutes_with_pos_nat_mul : ∀ (n : ℕ), 0 < n → ∀ (x : ℚ), 0 < x → f (↑n * x) = ↑n * f x\nx : ℚ\nhx : 0 < x\nH₀ : x * ↑x.den = ↑x.num\n⊢ f x = ↑x",
"after_state": "case intro.intro\nf : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\nh_f_commutes_with_pos_nat_mul : ∀ (n : ℕ), 0 < n → ∀ (x : ℚ), 0 < x → f (↑n * x) = ↑n * f x\nx : ℚ\nhx : 0 < x\nH₀ : x * ↑x.den = ↑x.num\nH : x * ↑(2 * x.den) = ↑(2 * x.num)\n⊢ f x = ↑x"
},
{
"line": "refine\n no_implicit_lambda%\n (have H : x * (↑(2 * x.den) : ℚ) = (↑(2 * x.num) : ℚ) := ?body✝;\n ?_)",
"before_state": "case intro.intro\nf : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\nh_f_commutes_with_pos_nat_mul : ∀ (n : ℕ), 0 < n → ∀ (x : ℚ), 0 < x → f (↑n * x) = ↑n * f x\nx : ℚ\nhx : 0 < x\nH₀ : x * ↑x.den = ↑x.num\n⊢ f x = ↑x",
"after_state": "case body\nf : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\nh_f_commutes_with_pos_nat_mul : ∀ (n : ℕ), 0 < n → ∀ (x : ℚ), 0 < x → f (↑n * x) = ↑n * f x\nx : ℚ\nhx : 0 < x\nH₀ : x * ↑x.den = ↑x.num\n⊢ x * ↑(2 * x.den) = ↑(2 * x.num)\n---\ncase intro.intro\nf : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\nh_f_commutes_with_pos_nat_mul : ∀ (n : ℕ), 0 < n → ∀ (x : ℚ), 0 < x → f (↑n * x) = ↑n * f x\nx : ℚ\nhx : 0 < x\nH₀ : x * ↑x.den = ↑x.num\nH : x * ↑(2 * x.den) = ↑(2 * x.num)\n⊢ f x = ↑x"
},
{
"line": "case body✝ => with_annotate_state\"by\" (push_cast; linear_combination 2 * H₀)",
"before_state": "case body\nf : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\nh_f_commutes_with_pos_nat_mul : ∀ (n : ℕ), 0 < n → ∀ (x : ℚ), 0 < x → f (↑n * x) = ↑n * f x\nx : ℚ\nhx : 0 < x\nH₀ : x * ↑x.den = ↑x.num\n⊢ x * ↑(2 * x.den) = ↑(2 * x.num)\n---\ncase intro.intro\nf : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\nh_f_commutes_with_pos_nat_mul : ∀ (n : ℕ), 0 < n → ∀ (x : ℚ), 0 < x → f (↑n * x) = ↑n * f x\nx : ℚ\nhx : 0 < x\nH₀ : x * ↑x.den = ↑x.num\nH : x * ↑(2 * x.den) = ↑(2 * x.num)\n⊢ f x = ↑x",
"after_state": "case intro.intro\nf : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\nh_f_commutes_with_pos_nat_mul : ∀ (n : ℕ), 0 < n → ∀ (x : ℚ), 0 < x → f (↑n * x) = ↑n * f x\nx : ℚ\nhx : 0 < x\nH₀ : x * ↑x.den = ↑x.num\nH : x * ↑(2 * x.den) = ↑(2 * x.num)\n⊢ f x = ↑x"
},
{
"line": "with_annotate_state\"by\" (push_cast; linear_combination 2 * H₀)",
"before_state": "f : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\nh_f_commutes_with_pos_nat_mul : ∀ (n : ℕ), 0 < n → ∀ (x : ℚ), 0 < x → f (↑n * x) = ↑n * f x\nx : ℚ\nhx : 0 < x\nH₀ : x * ↑x.den = ↑x.num\n⊢ x * ↑(2 * x.den) = ↑(2 * x.num)",
"after_state": "No Goals!"
},
{
"line": "push_cast",
"before_state": "f : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\nh_f_commutes_with_pos_nat_mul : ∀ (n : ℕ), 0 < n → ∀ (x : ℚ), 0 < x → f (↑n * x) = ↑n * f x\nx : ℚ\nhx : 0 < x\nH₀ : x * ↑x.den = ↑x.num\n⊢ x * ↑(2 * x.den) = ↑(2 * x.num)",
"after_state": "f : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\nh_f_commutes_with_pos_nat_mul : ∀ (n : ℕ), 0 < n → ∀ (x : ℚ), 0 < x → f (↑n * x) = ↑n * f x\nx : ℚ\nhx : 0 < x\nH₀ : x * ↑x.den = ↑x.num\n⊢ x * (2 * ↑x.den) = 2 * ↑x.num"
},
{
"line": "linear_combination 2 * H₀",
"before_state": "f : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\nh_f_commutes_with_pos_nat_mul : ∀ (n : ℕ), 0 < n → ∀ (x : ℚ), 0 < x → f (↑n * x) = ↑n * f x\nx : ℚ\nhx : 0 < x\nH₀ : x * ↑x.den = ↑x.num\n⊢ x * (2 * ↑x.den) = 2 * ↑x.num",
"after_state": "No Goals!"
},
{
"line": "set x2denom := 2 * x.den",
"before_state": "case intro.intro\nf : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\nh_f_commutes_with_pos_nat_mul : ∀ (n : ℕ), 0 < n → ∀ (x : ℚ), 0 < x → f (↑n * x) = ↑n * f x\nx : ℚ\nhx : 0 < x\nH₀ : x * ↑x.den = ↑x.num\nH : x * ↑(2 * x.den) = ↑(2 * x.num)\n⊢ f x = ↑x",
"after_state": "case intro.intro\nf : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\nh_f_commutes_with_pos_nat_mul : ∀ (n : ℕ), 0 < n → ∀ (x : ℚ), 0 < x → f (↑n * x) = ↑n * f x\nx : ℚ\nhx : 0 < x\nH₀ : x * ↑x.den = ↑x.num\nx2denom : ℕ := 2 * x.den\nH : x * ↑x2denom = ↑(2 * x.num)\n⊢ f x = ↑x"
},
{
"line": "try rewrite [show ?m✝ = x2denom from rfl✝] at *",
"before_state": "case intro.intro\nf : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\nh_f_commutes_with_pos_nat_mul : ∀ (n : ℕ), 0 < n → ∀ (x : ℚ), 0 < x → f (↑n * x) = ↑n * f x\nx : ℚ\nhx : 0 < x\nH₀ : x * ↑x.den = ↑x.num\nH : x * ↑(2 * x.den) = ↑(2 * x.num)\nx2denom : ℕ := 2 * x.den\n⊢ f x = ↑x",
"after_state": "case intro.intro\nf : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\nh_f_commutes_with_pos_nat_mul : ∀ (n : ℕ), 0 < n → ∀ (x : ℚ), 0 < x → f (↑n * x) = ↑n * f x\nx : ℚ\nhx : 0 < x\nH₀ : x * ↑x.den = ↑x.num\nx2denom : ℕ := 2 * x.den\nH : x * ↑x2denom = ↑(2 * x.num)\n⊢ f x = ↑x"
},
{
"line": "first\n| rewrite [show ?m✝ = x2denom from rfl✝] at *\n| skip",
"before_state": "case intro.intro\nf : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\nh_f_commutes_with_pos_nat_mul : ∀ (n : ℕ), 0 < n → ∀ (x : ℚ), 0 < x → f (↑n * x) = ↑n * f x\nx : ℚ\nhx : 0 < x\nH₀ : x * ↑x.den = ↑x.num\nH : x * ↑(2 * x.den) = ↑(2 * x.num)\nx2denom : ℕ := 2 * x.den\n⊢ f x = ↑x",
"after_state": "case intro.intro\nf : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\nh_f_commutes_with_pos_nat_mul : ∀ (n : ℕ), 0 < n → ∀ (x : ℚ), 0 < x → f (↑n * x) = ↑n * f x\nx : ℚ\nhx : 0 < x\nH₀ : x * ↑x.den = ↑x.num\nx2denom : ℕ := 2 * x.den\nH : x * ↑x2denom = ↑(2 * x.num)\n⊢ f x = ↑x"
},
{
"line": "rewrite [show ?m✝ = x2denom from rfl✝] at *",
"before_state": "case intro.intro\nf : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\nh_f_commutes_with_pos_nat_mul : ∀ (n : ℕ), 0 < n → ∀ (x : ℚ), 0 < x → f (↑n * x) = ↑n * f x\nx : ℚ\nhx : 0 < x\nH₀ : x * ↑x.den = ↑x.num\nH : x * ↑(2 * x.den) = ↑(2 * x.num)\nx2denom : ℕ := 2 * x.den\n⊢ f x = ↑x",
"after_state": "case intro.intro\nf : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\nh_f_commutes_with_pos_nat_mul : ∀ (n : ℕ), 0 < n → ∀ (x : ℚ), 0 < x → f (↑n * x) = ↑n * f x\nx : ℚ\nhx : 0 < x\nH₀ : x * ↑x.den = ↑x.num\nx2denom : ℕ := 2 * x.den\nH : x * ↑x2denom = ↑(2 * x.num)\n⊢ f x = ↑x"
},
{
"line": "set x2num := 2 * x.num",
"before_state": "case intro.intro\nf : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\nh_f_commutes_with_pos_nat_mul : ∀ (n : ℕ), 0 < n → ∀ (x : ℚ), 0 < x → f (↑n * x) = ↑n * f x\nx : ℚ\nhx : 0 < x\nH₀ : x * ↑x.den = ↑x.num\nx2denom : ℕ := 2 * x.den\nH : x * ↑x2denom = ↑(2 * x.num)\n⊢ f x = ↑x",
"after_state": "case intro.intro\nf : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\nh_f_commutes_with_pos_nat_mul : ∀ (n : ℕ), 0 < n → ∀ (x : ℚ), 0 < x → f (↑n * x) = ↑n * f x\nx : ℚ\nhx : 0 < x\nH₀ : x * ↑x.den = ↑x.num\nx2denom : ℕ := 2 * x.den\nx2num : ℤ := 2 * x.num\nH : x * ↑x2denom = ↑x2num\n⊢ f x = ↑x"
},
{
"line": "try rewrite [show ?m✝ = x2num from rfl✝] at *",
"before_state": "case intro.intro\nf : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\nh_f_commutes_with_pos_nat_mul : ∀ (n : ℕ), 0 < n → ∀ (x : ℚ), 0 < x → f (↑n * x) = ↑n * f x\nx : ℚ\nhx : 0 < x\nH₀ : x * ↑x.den = ↑x.num\nx2denom : ℕ := 2 * x.den\nH : x * ↑x2denom = ↑(2 * x.num)\nx2num : ℤ := 2 * x.num\n⊢ f x = ↑x",
"after_state": "case intro.intro\nf : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\nh_f_commutes_with_pos_nat_mul : ∀ (n : ℕ), 0 < n → ∀ (x : ℚ), 0 < x → f (↑n * x) = ↑n * f x\nx : ℚ\nhx : 0 < x\nH₀ : x * ↑x.den = ↑x.num\nx2denom : ℕ := 2 * x.den\nx2num : ℤ := 2 * x.num\nH : x * ↑x2denom = ↑x2num\n⊢ f x = ↑x"
},
{
"line": "first\n| rewrite [show ?m✝ = x2num from rfl✝] at *\n| skip",
"before_state": "case intro.intro\nf : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\nh_f_commutes_with_pos_nat_mul : ∀ (n : ℕ), 0 < n → ∀ (x : ℚ), 0 < x → f (↑n * x) = ↑n * f x\nx : ℚ\nhx : 0 < x\nH₀ : x * ↑x.den = ↑x.num\nx2denom : ℕ := 2 * x.den\nH : x * ↑x2denom = ↑(2 * x.num)\nx2num : ℤ := 2 * x.num\n⊢ f x = ↑x",
"after_state": "case intro.intro\nf : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\nh_f_commutes_with_pos_nat_mul : ∀ (n : ℕ), 0 < n → ∀ (x : ℚ), 0 < x → f (↑n * x) = ↑n * f x\nx : ℚ\nhx : 0 < x\nH₀ : x * ↑x.den = ↑x.num\nx2denom : ℕ := 2 * x.den\nx2num : ℤ := 2 * x.num\nH : x * ↑x2denom = ↑x2num\n⊢ f x = ↑x"
},
{
"line": "rewrite [show ?m✝ = x2num from rfl✝] at *",
"before_state": "case intro.intro\nf : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\nh_f_commutes_with_pos_nat_mul : ∀ (n : ℕ), 0 < n → ∀ (x : ℚ), 0 < x → f (↑n * x) = ↑n * f x\nx : ℚ\nhx : 0 < x\nH₀ : x * ↑x.den = ↑x.num\nx2denom : ℕ := 2 * x.den\nH : x * ↑x2denom = ↑(2 * x.num)\nx2num : ℤ := 2 * x.num\n⊢ f x = ↑x",
"after_state": "case intro.intro\nf : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\nh_f_commutes_with_pos_nat_mul : ∀ (n : ℕ), 0 < n → ∀ (x : ℚ), 0 < x → f (↑n * x) = ↑n * f x\nx : ℚ\nhx : 0 < x\nH₀ : x * ↑x.den = ↑x.num\nx2denom : ℕ := 2 * x.den\nx2num : ℤ := 2 * x.num\nH : x * ↑x2denom = ↑x2num\n⊢ f x = ↑x"
},
{
"line": "have hx2pos : 0 < 2 * x.den := by positivity",
"before_state": "case intro.intro\nf : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\nh_f_commutes_with_pos_nat_mul : ∀ (n : ℕ), 0 < n → ∀ (x : ℚ), 0 < x → f (↑n * x) = ↑n * f x\nx : ℚ\nhx : 0 < x\nH₀ : x * ↑x.den = ↑x.num\nx2denom : ℕ := 2 * x.den\nx2num : ℤ := 2 * x.num\nH : x * ↑x2denom = ↑x2num\n⊢ f x = ↑x",
"after_state": "case intro.intro\nf : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\nh_f_commutes_with_pos_nat_mul : ∀ (n : ℕ), 0 < n → ∀ (x : ℚ), 0 < x → f (↑n * x) = ↑n * f x\nx : ℚ\nhx : 0 < x\nH₀ : x * ↑x.den = ↑x.num\nx2denom : ℕ := 2 * x.den\nx2num : ℤ := 2 * x.num\nH : x * ↑x2denom = ↑x2num\nhx2pos : 0 < 2 * x.den\n⊢ f x = ↑x"
},
{
"line": "focus\n refine\n no_implicit_lambda%\n (have hx2pos : 0 < 2 * x.den := ?body✝;\n ?_)\n case body✝ => with_annotate_state\"by\" (positivity)",
"before_state": "case intro.intro\nf : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\nh_f_commutes_with_pos_nat_mul : ∀ (n : ℕ), 0 < n → ∀ (x : ℚ), 0 < x → f (↑n * x) = ↑n * f x\nx : ℚ\nhx : 0 < x\nH₀ : x * ↑x.den = ↑x.num\nx2denom : ℕ := 2 * x.den\nx2num : ℤ := 2 * x.num\nH : x * ↑x2denom = ↑x2num\n⊢ f x = ↑x",
"after_state": "case intro.intro\nf : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\nh_f_commutes_with_pos_nat_mul : ∀ (n : ℕ), 0 < n → ∀ (x : ℚ), 0 < x → f (↑n * x) = ↑n * f x\nx : ℚ\nhx : 0 < x\nH₀ : x * ↑x.den = ↑x.num\nx2denom : ℕ := 2 * x.den\nx2num : ℤ := 2 * x.num\nH : x * ↑x2denom = ↑x2num\nhx2pos : 0 < 2 * x.den\n⊢ f x = ↑x"
},
{
"line": "refine\n no_implicit_lambda%\n (have hx2pos : 0 < 2 * x.den := ?body✝;\n ?_)",
"before_state": "case intro.intro\nf : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\nh_f_commutes_with_pos_nat_mul : ∀ (n : ℕ), 0 < n → ∀ (x : ℚ), 0 < x → f (↑n * x) = ↑n * f x\nx : ℚ\nhx : 0 < x\nH₀ : x * ↑x.den = ↑x.num\nx2denom : ℕ := 2 * x.den\nx2num : ℤ := 2 * x.num\nH : x * ↑x2denom = ↑x2num\n⊢ f x = ↑x",
"after_state": "case body\nf : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\nh_f_commutes_with_pos_nat_mul : ∀ (n : ℕ), 0 < n → ∀ (x : ℚ), 0 < x → f (↑n * x) = ↑n * f x\nx : ℚ\nhx : 0 < x\nH₀ : x * ↑x.den = ↑x.num\nx2denom : ℕ := 2 * x.den\nx2num : ℤ := 2 * x.num\nH : x * ↑x2denom = ↑x2num\n⊢ 0 < 2 * x.den\n---\ncase intro.intro\nf : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\nh_f_commutes_with_pos_nat_mul : ∀ (n : ℕ), 0 < n → ∀ (x : ℚ), 0 < x → f (↑n * x) = ↑n * f x\nx : ℚ\nhx : 0 < x\nH₀ : x * ↑x.den = ↑x.num\nx2denom : ℕ := 2 * x.den\nx2num : ℤ := 2 * x.num\nH : x * ↑x2denom = ↑x2num\nhx2pos : 0 < 2 * x.den\n⊢ f x = ↑x"
},
{
"line": "case body✝ => with_annotate_state\"by\" (positivity)",
"before_state": "case body\nf : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\nh_f_commutes_with_pos_nat_mul : ∀ (n : ℕ), 0 < n → ∀ (x : ℚ), 0 < x → f (↑n * x) = ↑n * f x\nx : ℚ\nhx : 0 < x\nH₀ : x * ↑x.den = ↑x.num\nx2denom : ℕ := 2 * x.den\nx2num : ℤ := 2 * x.num\nH : x * ↑x2denom = ↑x2num\n⊢ 0 < 2 * x.den\n---\ncase intro.intro\nf : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\nh_f_commutes_with_pos_nat_mul : ∀ (n : ℕ), 0 < n → ∀ (x : ℚ), 0 < x → f (↑n * x) = ↑n * f x\nx : ℚ\nhx : 0 < x\nH₀ : x * ↑x.den = ↑x.num\nx2denom : ℕ := 2 * x.den\nx2num : ℤ := 2 * x.num\nH : x * ↑x2denom = ↑x2num\nhx2pos : 0 < 2 * x.den\n⊢ f x = ↑x",
"after_state": "case intro.intro\nf : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\nh_f_commutes_with_pos_nat_mul : ∀ (n : ℕ), 0 < n → ∀ (x : ℚ), 0 < x → f (↑n * x) = ↑n * f x\nx : ℚ\nhx : 0 < x\nH₀ : x * ↑x.den = ↑x.num\nx2denom : ℕ := 2 * x.den\nx2num : ℤ := 2 * x.num\nH : x * ↑x2denom = ↑x2num\nhx2pos : 0 < 2 * x.den\n⊢ f x = ↑x"
},
{
"line": "with_annotate_state\"by\" (positivity)",
"before_state": "f : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\nh_f_commutes_with_pos_nat_mul : ∀ (n : ℕ), 0 < n → ∀ (x : ℚ), 0 < x → f (↑n * x) = ↑n * f x\nx : ℚ\nhx : 0 < x\nH₀ : x * ↑x.den = ↑x.num\nx2denom : ℕ := 2 * x.den\nx2num : ℤ := 2 * x.num\nH : x * ↑x2denom = ↑x2num\n⊢ 0 < 2 * x.den",
"after_state": "No Goals!"
},
{
"line": "positivity",
"before_state": "f : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\nh_f_commutes_with_pos_nat_mul : ∀ (n : ℕ), 0 < n → ∀ (x : ℚ), 0 < x → f (↑n * x) = ↑n * f x\nx : ℚ\nhx : 0 < x\nH₀ : x * ↑x.den = ↑x.num\nx2denom : ℕ := 2 * x.den\nx2num : ℤ := 2 * x.num\nH : x * ↑x2denom = ↑x2num\n⊢ 0 < 2 * x.den",
"after_state": "No Goals!"
},
{
"line": "have hx2cnezr : (x2denom : ℝ) ≠ (0 : ℝ) := by positivity",
"before_state": "case intro.intro\nf : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\nh_f_commutes_with_pos_nat_mul : ∀ (n : ℕ), 0 < n → ∀ (x : ℚ), 0 < x → f (↑n * x) = ↑n * f x\nx : ℚ\nhx : 0 < x\nH₀ : x * ↑x.den = ↑x.num\nx2denom : ℕ := 2 * x.den\nx2num : ℤ := 2 * x.num\nH : x * ↑x2denom = ↑x2num\nhx2pos : 0 < 2 * x.den\n⊢ f x = ↑x",
"after_state": "case intro.intro\nf : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\nh_f_commutes_with_pos_nat_mul : ∀ (n : ℕ), 0 < n → ∀ (x : ℚ), 0 < x → f (↑n * x) = ↑n * f x\nx : ℚ\nhx : 0 < x\nH₀ : x * ↑x.den = ↑x.num\nx2denom : ℕ := 2 * x.den\nx2num : ℤ := 2 * x.num\nH : x * ↑x2denom = ↑x2num\nhx2pos : 0 < 2 * x.den\nhx2cnezr : ↑x2denom ≠ 0\n⊢ f x = ↑x"
},
{
"line": "focus\n refine\n no_implicit_lambda%\n (have hx2cnezr : (x2denom : ℝ) ≠ (0 : ℝ) := ?body✝;\n ?_)\n case body✝ => with_annotate_state\"by\" (positivity)",
"before_state": "case intro.intro\nf : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\nh_f_commutes_with_pos_nat_mul : ∀ (n : ℕ), 0 < n → ∀ (x : ℚ), 0 < x → f (↑n * x) = ↑n * f x\nx : ℚ\nhx : 0 < x\nH₀ : x * ↑x.den = ↑x.num\nx2denom : ℕ := 2 * x.den\nx2num : ℤ := 2 * x.num\nH : x * ↑x2denom = ↑x2num\nhx2pos : 0 < 2 * x.den\n⊢ f x = ↑x",
"after_state": "case intro.intro\nf : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\nh_f_commutes_with_pos_nat_mul : ∀ (n : ℕ), 0 < n → ∀ (x : ℚ), 0 < x → f (↑n * x) = ↑n * f x\nx : ℚ\nhx : 0 < x\nH₀ : x * ↑x.den = ↑x.num\nx2denom : ℕ := 2 * x.den\nx2num : ℤ := 2 * x.num\nH : x * ↑x2denom = ↑x2num\nhx2pos : 0 < 2 * x.den\nhx2cnezr : ↑x2denom ≠ 0\n⊢ f x = ↑x"
},
{
"line": "refine\n no_implicit_lambda%\n (have hx2cnezr : (x2denom : ℝ) ≠ (0 : ℝ) := ?body✝;\n ?_)",
"before_state": "case intro.intro\nf : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\nh_f_commutes_with_pos_nat_mul : ∀ (n : ℕ), 0 < n → ∀ (x : ℚ), 0 < x → f (↑n * x) = ↑n * f x\nx : ℚ\nhx : 0 < x\nH₀ : x * ↑x.den = ↑x.num\nx2denom : ℕ := 2 * x.den\nx2num : ℤ := 2 * x.num\nH : x * ↑x2denom = ↑x2num\nhx2pos : 0 < 2 * x.den\n⊢ f x = ↑x",
"after_state": "case body\nf : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\nh_f_commutes_with_pos_nat_mul : ∀ (n : ℕ), 0 < n → ∀ (x : ℚ), 0 < x → f (↑n * x) = ↑n * f x\nx : ℚ\nhx : 0 < x\nH₀ : x * ↑x.den = ↑x.num\nx2denom : ℕ := 2 * x.den\nx2num : ℤ := 2 * x.num\nH : x * ↑x2denom = ↑x2num\nhx2pos : 0 < 2 * x.den\n⊢ ↑x2denom ≠ 0\n---\ncase intro.intro\nf : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\nh_f_commutes_with_pos_nat_mul : ∀ (n : ℕ), 0 < n → ∀ (x : ℚ), 0 < x → f (↑n * x) = ↑n * f x\nx : ℚ\nhx : 0 < x\nH₀ : x * ↑x.den = ↑x.num\nx2denom : ℕ := 2 * x.den\nx2num : ℤ := 2 * x.num\nH : x * ↑x2denom = ↑x2num\nhx2pos : 0 < 2 * x.den\nhx2cnezr : ↑x2denom ≠ 0\n⊢ f x = ↑x"
},
{
"line": "case body✝ => with_annotate_state\"by\" (positivity)",
"before_state": "case body\nf : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\nh_f_commutes_with_pos_nat_mul : ∀ (n : ℕ), 0 < n → ∀ (x : ℚ), 0 < x → f (↑n * x) = ↑n * f x\nx : ℚ\nhx : 0 < x\nH₀ : x * ↑x.den = ↑x.num\nx2denom : ℕ := 2 * x.den\nx2num : ℤ := 2 * x.num\nH : x * ↑x2denom = ↑x2num\nhx2pos : 0 < 2 * x.den\n⊢ ↑x2denom ≠ 0\n---\ncase intro.intro\nf : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\nh_f_commutes_with_pos_nat_mul : ∀ (n : ℕ), 0 < n → ∀ (x : ℚ), 0 < x → f (↑n * x) = ↑n * f x\nx : ℚ\nhx : 0 < x\nH₀ : x * ↑x.den = ↑x.num\nx2denom : ℕ := 2 * x.den\nx2num : ℤ := 2 * x.num\nH : x * ↑x2denom = ↑x2num\nhx2pos : 0 < 2 * x.den\nhx2cnezr : ↑x2denom ≠ 0\n⊢ f x = ↑x",
"after_state": "case intro.intro\nf : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\nh_f_commutes_with_pos_nat_mul : ∀ (n : ℕ), 0 < n → ∀ (x : ℚ), 0 < x → f (↑n * x) = ↑n * f x\nx : ℚ\nhx : 0 < x\nH₀ : x * ↑x.den = ↑x.num\nx2denom : ℕ := 2 * x.den\nx2num : ℤ := 2 * x.num\nH : x * ↑x2denom = ↑x2num\nhx2pos : 0 < 2 * x.den\nhx2cnezr : ↑x2denom ≠ 0\n⊢ f x = ↑x"
},
{
"line": "with_annotate_state\"by\" (positivity)",
"before_state": "f : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\nh_f_commutes_with_pos_nat_mul : ∀ (n : ℕ), 0 < n → ∀ (x : ℚ), 0 < x → f (↑n * x) = ↑n * f x\nx : ℚ\nhx : 0 < x\nH₀ : x * ↑x.den = ↑x.num\nx2denom : ℕ := 2 * x.den\nx2num : ℤ := 2 * x.num\nH : x * ↑x2denom = ↑x2num\nhx2pos : 0 < 2 * x.den\n⊢ ↑x2denom ≠ 0",
"after_state": "No Goals!"
},
{
"line": "positivity",
"before_state": "f : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\nh_f_commutes_with_pos_nat_mul : ∀ (n : ℕ), 0 < n → ∀ (x : ℚ), 0 < x → f (↑n * x) = ↑n * f x\nx : ℚ\nhx : 0 < x\nH₀ : x * ↑x.den = ↑x.num\nx2denom : ℕ := 2 * x.den\nx2num : ℤ := 2 * x.num\nH : x * ↑x2denom = ↑x2num\nhx2pos : 0 < 2 * x.den\n⊢ ↑x2denom ≠ 0",
"after_state": "No Goals!"
},
{
"line": "have : 0 < x.num := by rwa [Rat.num_pos]",
"before_state": "case intro.intro\nf : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\nh_f_commutes_with_pos_nat_mul : ∀ (n : ℕ), 0 < n → ∀ (x : ℚ), 0 < x → f (↑n * x) = ↑n * f x\nx : ℚ\nhx : 0 < x\nH₀ : x * ↑x.den = ↑x.num\nx2denom : ℕ := 2 * x.den\nx2num : ℤ := 2 * x.num\nH : x * ↑x2denom = ↑x2num\nhx2pos : 0 < 2 * x.den\nhx2cnezr : ↑x2denom ≠ 0\n⊢ f x = ↑x",
"after_state": "case intro.intro\nf : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\nh_f_commutes_with_pos_nat_mul : ∀ (n : ℕ), 0 < n → ∀ (x : ℚ), 0 < x → f (↑n * x) = ↑n * f x\nx : ℚ\nhx : 0 < x\nH₀ : x * ↑x.den = ↑x.num\nx2denom : ℕ := 2 * x.den\nx2num : ℤ := 2 * x.num\nH : x * ↑x2denom = ↑x2num\nhx2pos : 0 < 2 * x.den\nhx2cnezr : ↑x2denom ≠ 0\nthis : 0 < x.num\n⊢ f x = ↑x"
},
{
"line": "focus\n refine\n no_implicit_lambda%\n (have : 0 < x.num := ?body✝;\n ?_)\n case body✝ => with_annotate_state\"by\" (rwa [Rat.num_pos])",
"before_state": "case intro.intro\nf : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\nh_f_commutes_with_pos_nat_mul : ∀ (n : ℕ), 0 < n → ∀ (x : ℚ), 0 < x → f (↑n * x) = ↑n * f x\nx : ℚ\nhx : 0 < x\nH₀ : x * ↑x.den = ↑x.num\nx2denom : ℕ := 2 * x.den\nx2num : ℤ := 2 * x.num\nH : x * ↑x2denom = ↑x2num\nhx2pos : 0 < 2 * x.den\nhx2cnezr : ↑x2denom ≠ 0\n⊢ f x = ↑x",
"after_state": "case intro.intro\nf : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\nh_f_commutes_with_pos_nat_mul : ∀ (n : ℕ), 0 < n → ∀ (x : ℚ), 0 < x → f (↑n * x) = ↑n * f x\nx : ℚ\nhx : 0 < x\nH₀ : x * ↑x.den = ↑x.num\nx2denom : ℕ := 2 * x.den\nx2num : ℤ := 2 * x.num\nH : x * ↑x2denom = ↑x2num\nhx2pos : 0 < 2 * x.den\nhx2cnezr : ↑x2denom ≠ 0\nthis : 0 < x.num\n⊢ f x = ↑x"
},
{
"line": "refine\n no_implicit_lambda%\n (have : 0 < x.num := ?body✝;\n ?_)",
"before_state": "case intro.intro\nf : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\nh_f_commutes_with_pos_nat_mul : ∀ (n : ℕ), 0 < n → ∀ (x : ℚ), 0 < x → f (↑n * x) = ↑n * f x\nx : ℚ\nhx : 0 < x\nH₀ : x * ↑x.den = ↑x.num\nx2denom : ℕ := 2 * x.den\nx2num : ℤ := 2 * x.num\nH : x * ↑x2denom = ↑x2num\nhx2pos : 0 < 2 * x.den\nhx2cnezr : ↑x2denom ≠ 0\n⊢ f x = ↑x",
"after_state": "case body\nf : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\nh_f_commutes_with_pos_nat_mul : ∀ (n : ℕ), 0 < n → ∀ (x : ℚ), 0 < x → f (↑n * x) = ↑n * f x\nx : ℚ\nhx : 0 < x\nH₀ : x * ↑x.den = ↑x.num\nx2denom : ℕ := 2 * x.den\nx2num : ℤ := 2 * x.num\nH : x * ↑x2denom = ↑x2num\nhx2pos : 0 < 2 * x.den\nhx2cnezr : ↑x2denom ≠ 0\n⊢ 0 < x.num\n---\ncase intro.intro\nf : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\nh_f_commutes_with_pos_nat_mul : ∀ (n : ℕ), 0 < n → ∀ (x : ℚ), 0 < x → f (↑n * x) = ↑n * f x\nx : ℚ\nhx : 0 < x\nH₀ : x * ↑x.den = ↑x.num\nx2denom : ℕ := 2 * x.den\nx2num : ℤ := 2 * x.num\nH : x * ↑x2denom = ↑x2num\nhx2pos : 0 < 2 * x.den\nhx2cnezr : ↑x2denom ≠ 0\nthis : 0 < x.num\n⊢ f x = ↑x"
},
{
"line": "case body✝ => with_annotate_state\"by\" (rwa [Rat.num_pos])",
"before_state": "case body\nf : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\nh_f_commutes_with_pos_nat_mul : ∀ (n : ℕ), 0 < n → ∀ (x : ℚ), 0 < x → f (↑n * x) = ↑n * f x\nx : ℚ\nhx : 0 < x\nH₀ : x * ↑x.den = ↑x.num\nx2denom : ℕ := 2 * x.den\nx2num : ℤ := 2 * x.num\nH : x * ↑x2denom = ↑x2num\nhx2pos : 0 < 2 * x.den\nhx2cnezr : ↑x2denom ≠ 0\n⊢ 0 < x.num\n---\ncase intro.intro\nf : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\nh_f_commutes_with_pos_nat_mul : ∀ (n : ℕ), 0 < n → ∀ (x : ℚ), 0 < x → f (↑n * x) = ↑n * f x\nx : ℚ\nhx : 0 < x\nH₀ : x * ↑x.den = ↑x.num\nx2denom : ℕ := 2 * x.den\nx2num : ℤ := 2 * x.num\nH : x * ↑x2denom = ↑x2num\nhx2pos : 0 < 2 * x.den\nhx2cnezr : ↑x2denom ≠ 0\nthis : 0 < x.num\n⊢ f x = ↑x",
"after_state": "case intro.intro\nf : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\nh_f_commutes_with_pos_nat_mul : ∀ (n : ℕ), 0 < n → ∀ (x : ℚ), 0 < x → f (↑n * x) = ↑n * f x\nx : ℚ\nhx : 0 < x\nH₀ : x * ↑x.den = ↑x.num\nx2denom : ℕ := 2 * x.den\nx2num : ℤ := 2 * x.num\nH : x * ↑x2denom = ↑x2num\nhx2pos : 0 < 2 * x.den\nhx2cnezr : ↑x2denom ≠ 0\nthis : 0 < x.num\n⊢ f x = ↑x"
},
{
"line": "with_annotate_state\"by\" (rwa [Rat.num_pos])",
"before_state": "f : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\nh_f_commutes_with_pos_nat_mul : ∀ (n : ℕ), 0 < n → ∀ (x : ℚ), 0 < x → f (↑n * x) = ↑n * f x\nx : ℚ\nhx : 0 < x\nH₀ : x * ↑x.den = ↑x.num\nx2denom : ℕ := 2 * x.den\nx2num : ℤ := 2 * x.num\nH : x * ↑x2denom = ↑x2num\nhx2pos : 0 < 2 * x.den\nhx2cnezr : ↑x2denom ≠ 0\n⊢ 0 < x.num",
"after_state": "No Goals!"
},
{
"line": "rwa [Rat.num_pos]",
"before_state": "f : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\nh_f_commutes_with_pos_nat_mul : ∀ (n : ℕ), 0 < n → ∀ (x : ℚ), 0 < x → f (↑n * x) = ↑n * f x\nx : ℚ\nhx : 0 < x\nH₀ : x * ↑x.den = ↑x.num\nx2denom : ℕ := 2 * x.den\nx2num : ℤ := 2 * x.num\nH : x * ↑x2denom = ↑x2num\nhx2pos : 0 < 2 * x.den\nhx2cnezr : ↑x2denom ≠ 0\n⊢ 0 < x.num",
"after_state": "No Goals!"
},
{
"line": "rw [Rat.num_pos]",
"before_state": "f : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\nh_f_commutes_with_pos_nat_mul : ∀ (n : ℕ), 0 < n → ∀ (x : ℚ), 0 < x → f (↑n * x) = ↑n * f x\nx : ℚ\nhx : 0 < x\nH₀ : x * ↑x.den = ↑x.num\nx2denom : ℕ := 2 * x.den\nx2num : ℤ := 2 * x.num\nH : x * ↑x2denom = ↑x2num\nhx2pos : 0 < 2 * x.den\nhx2cnezr : ↑x2denom ≠ 0\n⊢ 0 < x.num",
"after_state": "f : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\nh_f_commutes_with_pos_nat_mul : ∀ (n : ℕ), 0 < n → ∀ (x : ℚ), 0 < x → f (↑n * x) = ↑n * f x\nx : ℚ\nhx : 0 < x\nH₀ : x * ↑x.den = ↑x.num\nx2denom : ℕ := 2 * x.den\nx2num : ℤ := 2 * x.num\nH : x * ↑x2denom = ↑x2num\nhx2pos : 0 < 2 * x.den\nhx2cnezr : ↑x2denom ≠ 0\n⊢ 0 < x"
},
{
"line": "rewrite [Rat.num_pos]",
"before_state": "f : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\nh_f_commutes_with_pos_nat_mul : ∀ (n : ℕ), 0 < n → ∀ (x : ℚ), 0 < x → f (↑n * x) = ↑n * f x\nx : ℚ\nhx : 0 < x\nH₀ : x * ↑x.den = ↑x.num\nx2denom : ℕ := 2 * x.den\nx2num : ℤ := 2 * x.num\nH : x * ↑x2denom = ↑x2num\nhx2pos : 0 < 2 * x.den\nhx2cnezr : ↑x2denom ≠ 0\n⊢ 0 < x.num",
"after_state": "f : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\nh_f_commutes_with_pos_nat_mul : ∀ (n : ℕ), 0 < n → ∀ (x : ℚ), 0 < x → f (↑n * x) = ↑n * f x\nx : ℚ\nhx : 0 < x\nH₀ : x * ↑x.den = ↑x.num\nx2denom : ℕ := 2 * x.den\nx2num : ℤ := 2 * x.num\nH : x * ↑x2denom = ↑x2num\nhx2pos : 0 < 2 * x.den\nhx2cnezr : ↑x2denom ≠ 0\n⊢ 0 < x"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "f : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\nh_f_commutes_with_pos_nat_mul : ∀ (n : ℕ), 0 < n → ∀ (x : ℚ), 0 < x → f (↑n * x) = ↑n * f x\nx : ℚ\nhx : 0 < x\nH₀ : x * ↑x.den = ↑x.num\nx2denom : ℕ := 2 * x.den\nx2num : ℤ := 2 * x.num\nH : x * ↑x2denom = ↑x2num\nhx2pos : 0 < 2 * x.den\nhx2cnezr : ↑x2denom ≠ 0\n⊢ 0 < x",
"after_state": "f : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\nh_f_commutes_with_pos_nat_mul : ∀ (n : ℕ), 0 < n → ∀ (x : ℚ), 0 < x → f (↑n * x) = ↑n * f x\nx : ℚ\nhx : 0 < x\nH₀ : x * ↑x.den = ↑x.num\nx2denom : ℕ := 2 * x.den\nx2num : ℤ := 2 * x.num\nH : x * ↑x2denom = ↑x2num\nhx2pos : 0 < 2 * x.den\nhx2cnezr : ↑x2denom ≠ 0\n⊢ 0 < x"
},
{
"line": "try (with_reducible rfl)",
"before_state": "f : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\nh_f_commutes_with_pos_nat_mul : ∀ (n : ℕ), 0 < n → ∀ (x : ℚ), 0 < x → f (↑n * x) = ↑n * f x\nx : ℚ\nhx : 0 < x\nH₀ : x * ↑x.den = ↑x.num\nx2denom : ℕ := 2 * x.den\nx2num : ℤ := 2 * x.num\nH : x * ↑x2denom = ↑x2num\nhx2pos : 0 < 2 * x.den\nhx2cnezr : ↑x2denom ≠ 0\n⊢ 0 < x",
"after_state": "f : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\nh_f_commutes_with_pos_nat_mul : ∀ (n : ℕ), 0 < n → ∀ (x : ℚ), 0 < x → f (↑n * x) = ↑n * f x\nx : ℚ\nhx : 0 < x\nH₀ : x * ↑x.den = ↑x.num\nx2denom : ℕ := 2 * x.den\nx2num : ℤ := 2 * x.num\nH : x * ↑x2denom = ↑x2num\nhx2pos : 0 < 2 * x.den\nhx2cnezr : ↑x2denom ≠ 0\n⊢ 0 < x"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "f : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\nh_f_commutes_with_pos_nat_mul : ∀ (n : ℕ), 0 < n → ∀ (x : ℚ), 0 < x → f (↑n * x) = ↑n * f x\nx : ℚ\nhx : 0 < x\nH₀ : x * ↑x.den = ↑x.num\nx2denom : ℕ := 2 * x.den\nx2num : ℤ := 2 * x.num\nH : x * ↑x2denom = ↑x2num\nhx2pos : 0 < 2 * x.den\nhx2cnezr : ↑x2denom ≠ 0\n⊢ 0 < x",
"after_state": "f : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\nh_f_commutes_with_pos_nat_mul : ∀ (n : ℕ), 0 < n → ∀ (x : ℚ), 0 < x → f (↑n * x) = ↑n * f x\nx : ℚ\nhx : 0 < x\nH₀ : x * ↑x.den = ↑x.num\nx2denom : ℕ := 2 * x.den\nx2num : ℤ := 2 * x.num\nH : x * ↑x2denom = ↑x2num\nhx2pos : 0 < 2 * x.den\nhx2cnezr : ↑x2denom ≠ 0\n⊢ 0 < x"
},
{
"line": "with_reducible rfl",
"before_state": "f : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\nh_f_commutes_with_pos_nat_mul : ∀ (n : ℕ), 0 < n → ∀ (x : ℚ), 0 < x → f (↑n * x) = ↑n * f x\nx : ℚ\nhx : 0 < x\nH₀ : x * ↑x.den = ↑x.num\nx2denom : ℕ := 2 * x.den\nx2num : ℤ := 2 * x.num\nH : x * ↑x2denom = ↑x2num\nhx2pos : 0 < 2 * x.den\nhx2cnezr : ↑x2denom ≠ 0\n⊢ 0 < x",
"after_state": "f : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\nh_f_commutes_with_pos_nat_mul : ∀ (n : ℕ), 0 < n → ∀ (x : ℚ), 0 < x → f (↑n * x) = ↑n * f x\nx : ℚ\nhx : 0 < x\nH₀ : x * ↑x.den = ↑x.num\nx2denom : ℕ := 2 * x.den\nx2num : ℤ := 2 * x.num\nH : x * ↑x2denom = ↑x2num\nhx2pos : 0 < 2 * x.den\nhx2cnezr : ↑x2denom ≠ 0\n⊢ 0 < x"
},
{
"line": "rfl",
"before_state": "f : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\nh_f_commutes_with_pos_nat_mul : ∀ (n : ℕ), 0 < n → ∀ (x : ℚ), 0 < x → f (↑n * x) = ↑n * f x\nx : ℚ\nhx : 0 < x\nH₀ : x * ↑x.den = ↑x.num\nx2denom : ℕ := 2 * x.den\nx2num : ℤ := 2 * x.num\nH : x * ↑x2denom = ↑x2num\nhx2pos : 0 < 2 * x.den\nhx2cnezr : ↑x2denom ≠ 0\n⊢ 0 < x",
"after_state": "f : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\nh_f_commutes_with_pos_nat_mul : ∀ (n : ℕ), 0 < n → ∀ (x : ℚ), 0 < x → f (↑n * x) = ↑n * f x\nx : ℚ\nhx : 0 < x\nH₀ : x * ↑x.den = ↑x.num\nx2denom : ℕ := 2 * x.den\nx2num : ℤ := 2 * x.num\nH : x * ↑x2denom = ↑x2num\nhx2pos : 0 < 2 * x.den\nhx2cnezr : ↑x2denom ≠ 0\n⊢ 0 < x"
},
{
"line": "apply_rfl",
"before_state": "f : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\nh_f_commutes_with_pos_nat_mul : ∀ (n : ℕ), 0 < n → ∀ (x : ℚ), 0 < x → f (↑n * x) = ↑n * f x\nx : ℚ\nhx : 0 < x\nH₀ : x * ↑x.den = ↑x.num\nx2denom : ℕ := 2 * x.den\nx2num : ℤ := 2 * x.num\nH : x * ↑x2denom = ↑x2num\nhx2pos : 0 < 2 * x.den\nhx2cnezr : ↑x2denom ≠ 0\n⊢ 0 < x",
"after_state": "f : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\nh_f_commutes_with_pos_nat_mul : ∀ (n : ℕ), 0 < n → ∀ (x : ℚ), 0 < x → f (↑n * x) = ↑n * f x\nx : ℚ\nhx : 0 < x\nH₀ : x * ↑x.den = ↑x.num\nx2denom : ℕ := 2 * x.den\nx2num : ℤ := 2 * x.num\nH : x * ↑x2denom = ↑x2num\nhx2pos : 0 < 2 * x.den\nhx2cnezr : ↑x2denom ≠ 0\n⊢ 0 < x"
},
{
"line": "skip",
"before_state": "f : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\nh_f_commutes_with_pos_nat_mul : ∀ (n : ℕ), 0 < n → ∀ (x : ℚ), 0 < x → f (↑n * x) = ↑n * f x\nx : ℚ\nhx : 0 < x\nH₀ : x * ↑x.den = ↑x.num\nx2denom : ℕ := 2 * x.den\nx2num : ℤ := 2 * x.num\nH : x * ↑x2denom = ↑x2num\nhx2pos : 0 < 2 * x.den\nhx2cnezr : ↑x2denom ≠ 0\n⊢ 0 < x",
"after_state": "f : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\nh_f_commutes_with_pos_nat_mul : ∀ (n : ℕ), 0 < n → ∀ (x : ℚ), 0 < x → f (↑n * x) = ↑n * f x\nx : ℚ\nhx : 0 < x\nH₀ : x * ↑x.den = ↑x.num\nx2denom : ℕ := 2 * x.den\nx2num : ℤ := 2 * x.num\nH : x * ↑x2denom = ↑x2num\nhx2pos : 0 < 2 * x.den\nhx2cnezr : ↑x2denom ≠ 0\n⊢ 0 < x"
},
{
"line": "assumption",
"before_state": "f : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\nh_f_commutes_with_pos_nat_mul : ∀ (n : ℕ), 0 < n → ∀ (x : ℚ), 0 < x → f (↑n * x) = ↑n * f x\nx : ℚ\nhx : 0 < x\nH₀ : x * ↑x.den = ↑x.num\nx2denom : ℕ := 2 * x.den\nx2num : ℤ := 2 * x.num\nH : x * ↑x2denom = ↑x2num\nhx2pos : 0 < 2 * x.den\nhx2cnezr : ↑x2denom ≠ 0\n⊢ 0 < x",
"after_state": "No Goals!"
},
{
"line": "have hx2num_gt_one : (1 : ℚ) < (2 * x.num : ℤ) := by norm_cast; linarith",
"before_state": "case intro.intro\nf : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\nh_f_commutes_with_pos_nat_mul : ∀ (n : ℕ), 0 < n → ∀ (x : ℚ), 0 < x → f (↑n * x) = ↑n * f x\nx : ℚ\nhx : 0 < x\nH₀ : x * ↑x.den = ↑x.num\nx2denom : ℕ := 2 * x.den\nx2num : ℤ := 2 * x.num\nH : x * ↑x2denom = ↑x2num\nhx2pos : 0 < 2 * x.den\nhx2cnezr : ↑x2denom ≠ 0\nthis : 0 < x.num\n⊢ f x = ↑x",
"after_state": "case intro.intro\nf : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\nh_f_commutes_with_pos_nat_mul : ∀ (n : ℕ), 0 < n → ∀ (x : ℚ), 0 < x → f (↑n * x) = ↑n * f x\nx : ℚ\nhx : 0 < x\nH₀ : x * ↑x.den = ↑x.num\nx2denom : ℕ := 2 * x.den\nx2num : ℤ := 2 * x.num\nH : x * ↑x2denom = ↑x2num\nhx2pos : 0 < 2 * x.den\nhx2cnezr : ↑x2denom ≠ 0\nthis : 0 < x.num\nhx2num_gt_one : 1 < ↑(2 * x.num)\n⊢ f x = ↑x"
},
{
"line": "focus\n refine\n no_implicit_lambda%\n (have hx2num_gt_one : (1 : ℚ) < (2 * x.num : ℤ) := ?body✝;\n ?_)\n case body✝ => with_annotate_state\"by\" (norm_cast; linarith)",
"before_state": "case intro.intro\nf : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\nh_f_commutes_with_pos_nat_mul : ∀ (n : ℕ), 0 < n → ∀ (x : ℚ), 0 < x → f (↑n * x) = ↑n * f x\nx : ℚ\nhx : 0 < x\nH₀ : x * ↑x.den = ↑x.num\nx2denom : ℕ := 2 * x.den\nx2num : ℤ := 2 * x.num\nH : x * ↑x2denom = ↑x2num\nhx2pos : 0 < 2 * x.den\nhx2cnezr : ↑x2denom ≠ 0\nthis : 0 < x.num\n⊢ f x = ↑x",
"after_state": "case intro.intro\nf : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\nh_f_commutes_with_pos_nat_mul : ∀ (n : ℕ), 0 < n → ∀ (x : ℚ), 0 < x → f (↑n * x) = ↑n * f x\nx : ℚ\nhx : 0 < x\nH₀ : x * ↑x.den = ↑x.num\nx2denom : ℕ := 2 * x.den\nx2num : ℤ := 2 * x.num\nH : x * ↑x2denom = ↑x2num\nhx2pos : 0 < 2 * x.den\nhx2cnezr : ↑x2denom ≠ 0\nthis : 0 < x.num\nhx2num_gt_one : 1 < ↑(2 * x.num)\n⊢ f x = ↑x"
},
{
"line": "refine\n no_implicit_lambda%\n (have hx2num_gt_one : (1 : ℚ) < (2 * x.num : ℤ) := ?body✝;\n ?_)",
"before_state": "case intro.intro\nf : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\nh_f_commutes_with_pos_nat_mul : ∀ (n : ℕ), 0 < n → ∀ (x : ℚ), 0 < x → f (↑n * x) = ↑n * f x\nx : ℚ\nhx : 0 < x\nH₀ : x * ↑x.den = ↑x.num\nx2denom : ℕ := 2 * x.den\nx2num : ℤ := 2 * x.num\nH : x * ↑x2denom = ↑x2num\nhx2pos : 0 < 2 * x.den\nhx2cnezr : ↑x2denom ≠ 0\nthis : 0 < x.num\n⊢ f x = ↑x",
"after_state": "case body\nf : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\nh_f_commutes_with_pos_nat_mul : ∀ (n : ℕ), 0 < n → ∀ (x : ℚ), 0 < x → f (↑n * x) = ↑n * f x\nx : ℚ\nhx : 0 < x\nH₀ : x * ↑x.den = ↑x.num\nx2denom : ℕ := 2 * x.den\nx2num : ℤ := 2 * x.num\nH : x * ↑x2denom = ↑x2num\nhx2pos : 0 < 2 * x.den\nhx2cnezr : ↑x2denom ≠ 0\nthis : 0 < x.num\n⊢ 1 < ↑(2 * x.num)\n---\ncase intro.intro\nf : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\nh_f_commutes_with_pos_nat_mul : ∀ (n : ℕ), 0 < n → ∀ (x : ℚ), 0 < x → f (↑n * x) = ↑n * f x\nx : ℚ\nhx : 0 < x\nH₀ : x * ↑x.den = ↑x.num\nx2denom : ℕ := 2 * x.den\nx2num : ℤ := 2 * x.num\nH : x * ↑x2denom = ↑x2num\nhx2pos : 0 < 2 * x.den\nhx2cnezr : ↑x2denom ≠ 0\nthis : 0 < x.num\nhx2num_gt_one : 1 < ↑(2 * x.num)\n⊢ f x = ↑x"
},
{
"line": "case body✝ => with_annotate_state\"by\" (norm_cast; linarith)",
"before_state": "case body\nf : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\nh_f_commutes_with_pos_nat_mul : ∀ (n : ℕ), 0 < n → ∀ (x : ℚ), 0 < x → f (↑n * x) = ↑n * f x\nx : ℚ\nhx : 0 < x\nH₀ : x * ↑x.den = ↑x.num\nx2denom : ℕ := 2 * x.den\nx2num : ℤ := 2 * x.num\nH : x * ↑x2denom = ↑x2num\nhx2pos : 0 < 2 * x.den\nhx2cnezr : ↑x2denom ≠ 0\nthis : 0 < x.num\n⊢ 1 < ↑(2 * x.num)\n---\ncase intro.intro\nf : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\nh_f_commutes_with_pos_nat_mul : ∀ (n : ℕ), 0 < n → ∀ (x : ℚ), 0 < x → f (↑n * x) = ↑n * f x\nx : ℚ\nhx : 0 < x\nH₀ : x * ↑x.den = ↑x.num\nx2denom : ℕ := 2 * x.den\nx2num : ℤ := 2 * x.num\nH : x * ↑x2denom = ↑x2num\nhx2pos : 0 < 2 * x.den\nhx2cnezr : ↑x2denom ≠ 0\nthis : 0 < x.num\nhx2num_gt_one : 1 < ↑(2 * x.num)\n⊢ f x = ↑x",
"after_state": "case intro.intro\nf : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\nh_f_commutes_with_pos_nat_mul : ∀ (n : ℕ), 0 < n → ∀ (x : ℚ), 0 < x → f (↑n * x) = ↑n * f x\nx : ℚ\nhx : 0 < x\nH₀ : x * ↑x.den = ↑x.num\nx2denom : ℕ := 2 * x.den\nx2num : ℤ := 2 * x.num\nH : x * ↑x2denom = ↑x2num\nhx2pos : 0 < 2 * x.den\nhx2cnezr : ↑x2denom ≠ 0\nthis : 0 < x.num\nhx2num_gt_one : 1 < ↑(2 * x.num)\n⊢ f x = ↑x"
},
{
"line": "with_annotate_state\"by\" (norm_cast; linarith)",
"before_state": "f : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\nh_f_commutes_with_pos_nat_mul : ∀ (n : ℕ), 0 < n → ∀ (x : ℚ), 0 < x → f (↑n * x) = ↑n * f x\nx : ℚ\nhx : 0 < x\nH₀ : x * ↑x.den = ↑x.num\nx2denom : ℕ := 2 * x.den\nx2num : ℤ := 2 * x.num\nH : x * ↑x2denom = ↑x2num\nhx2pos : 0 < 2 * x.den\nhx2cnezr : ↑x2denom ≠ 0\nthis : 0 < x.num\n⊢ 1 < ↑(2 * x.num)",
"after_state": "No Goals!"
},
{
"line": "norm_cast",
"before_state": "f : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\nh_f_commutes_with_pos_nat_mul : ∀ (n : ℕ), 0 < n → ∀ (x : ℚ), 0 < x → f (↑n * x) = ↑n * f x\nx : ℚ\nhx : 0 < x\nH₀ : x * ↑x.den = ↑x.num\nx2denom : ℕ := 2 * x.den\nx2num : ℤ := 2 * x.num\nH : x * ↑x2denom = ↑x2num\nhx2pos : 0 < 2 * x.den\nhx2cnezr : ↑x2denom ≠ 0\nthis : 0 < x.num\n⊢ 1 < ↑(2 * x.num)",
"after_state": "f : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\nh_f_commutes_with_pos_nat_mul : ∀ (n : ℕ), 0 < n → ∀ (x : ℚ), 0 < x → f (↑n * x) = ↑n * f x\nx : ℚ\nhx : 0 < x\nH₀ : x * ↑x.den = ↑x.num\nx2denom : ℕ := 2 * x.den\nx2num : ℤ := 2 * x.num\nH : x * ↑x2denom = ↑x2num\nhx2pos : 0 < 2 * x.den\nhx2cnezr : ↑x2denom ≠ 0\nthis : 0 < x.num\n⊢ 1 < 2 * x.num"
},
{
"line": "focus\n norm_cast0\n with_annotate_state\"<;>\" skip\n all_goals try trivial",
"before_state": "f : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\nh_f_commutes_with_pos_nat_mul : ∀ (n : ℕ), 0 < n → ∀ (x : ℚ), 0 < x → f (↑n * x) = ↑n * f x\nx : ℚ\nhx : 0 < x\nH₀ : x * ↑x.den = ↑x.num\nx2denom : ℕ := 2 * x.den\nx2num : ℤ := 2 * x.num\nH : x * ↑x2denom = ↑x2num\nhx2pos : 0 < 2 * x.den\nhx2cnezr : ↑x2denom ≠ 0\nthis : 0 < x.num\n⊢ 1 < ↑(2 * x.num)",
"after_state": "f : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\nh_f_commutes_with_pos_nat_mul : ∀ (n : ℕ), 0 < n → ∀ (x : ℚ), 0 < x → f (↑n * x) = ↑n * f x\nx : ℚ\nhx : 0 < x\nH₀ : x * ↑x.den = ↑x.num\nx2denom : ℕ := 2 * x.den\nx2num : ℤ := 2 * x.num\nH : x * ↑x2denom = ↑x2num\nhx2pos : 0 < 2 * x.den\nhx2cnezr : ↑x2denom ≠ 0\nthis : 0 < x.num\n⊢ 1 < 2 * x.num"
},
{
"line": "norm_cast0",
"before_state": "f : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\nh_f_commutes_with_pos_nat_mul : ∀ (n : ℕ), 0 < n → ∀ (x : ℚ), 0 < x → f (↑n * x) = ↑n * f x\nx : ℚ\nhx : 0 < x\nH₀ : x * ↑x.den = ↑x.num\nx2denom : ℕ := 2 * x.den\nx2num : ℤ := 2 * x.num\nH : x * ↑x2denom = ↑x2num\nhx2pos : 0 < 2 * x.den\nhx2cnezr : ↑x2denom ≠ 0\nthis : 0 < x.num\n⊢ 1 < ↑(2 * x.num)",
"after_state": "f : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\nh_f_commutes_with_pos_nat_mul : ∀ (n : ℕ), 0 < n → ∀ (x : ℚ), 0 < x → f (↑n * x) = ↑n * f x\nx : ℚ\nhx : 0 < x\nH₀ : x * ↑x.den = ↑x.num\nx2denom : ℕ := 2 * x.den\nx2num : ℤ := 2 * x.num\nH : x * ↑x2denom = ↑x2num\nhx2pos : 0 < 2 * x.den\nhx2cnezr : ↑x2denom ≠ 0\nthis : 0 < x.num\n⊢ 1 < 2 * x.num"
},
{
"line": "with_annotate_state\"<;>\" skip",
"before_state": "f : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\nh_f_commutes_with_pos_nat_mul : ∀ (n : ℕ), 0 < n → ∀ (x : ℚ), 0 < x → f (↑n * x) = ↑n * f x\nx : ℚ\nhx : 0 < x\nH₀ : x * ↑x.den = ↑x.num\nx2denom : ℕ := 2 * x.den\nx2num : ℤ := 2 * x.num\nH : x * ↑x2denom = ↑x2num\nhx2pos : 0 < 2 * x.den\nhx2cnezr : ↑x2denom ≠ 0\nthis : 0 < x.num\n⊢ 1 < 2 * x.num",
"after_state": "f : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\nh_f_commutes_with_pos_nat_mul : ∀ (n : ℕ), 0 < n → ∀ (x : ℚ), 0 < x → f (↑n * x) = ↑n * f x\nx : ℚ\nhx : 0 < x\nH₀ : x * ↑x.den = ↑x.num\nx2denom : ℕ := 2 * x.den\nx2num : ℤ := 2 * x.num\nH : x * ↑x2denom = ↑x2num\nhx2pos : 0 < 2 * x.den\nhx2cnezr : ↑x2denom ≠ 0\nthis : 0 < x.num\n⊢ 1 < 2 * x.num"
},
{
"line": "skip",
"before_state": "f : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\nh_f_commutes_with_pos_nat_mul : ∀ (n : ℕ), 0 < n → ∀ (x : ℚ), 0 < x → f (↑n * x) = ↑n * f x\nx : ℚ\nhx : 0 < x\nH₀ : x * ↑x.den = ↑x.num\nx2denom : ℕ := 2 * x.den\nx2num : ℤ := 2 * x.num\nH : x * ↑x2denom = ↑x2num\nhx2pos : 0 < 2 * x.den\nhx2cnezr : ↑x2denom ≠ 0\nthis : 0 < x.num\n⊢ 1 < 2 * x.num",
"after_state": "f : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\nh_f_commutes_with_pos_nat_mul : ∀ (n : ℕ), 0 < n → ∀ (x : ℚ), 0 < x → f (↑n * x) = ↑n * f x\nx : ℚ\nhx : 0 < x\nH₀ : x * ↑x.den = ↑x.num\nx2denom : ℕ := 2 * x.den\nx2num : ℤ := 2 * x.num\nH : x * ↑x2denom = ↑x2num\nhx2pos : 0 < 2 * x.den\nhx2cnezr : ↑x2denom ≠ 0\nthis : 0 < x.num\n⊢ 1 < 2 * x.num"
},
{
"line": "all_goals try trivial",
"before_state": "f : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\nh_f_commutes_with_pos_nat_mul : ∀ (n : ℕ), 0 < n → ∀ (x : ℚ), 0 < x → f (↑n * x) = ↑n * f x\nx : ℚ\nhx : 0 < x\nH₀ : x * ↑x.den = ↑x.num\nx2denom : ℕ := 2 * x.den\nx2num : ℤ := 2 * x.num\nH : x * ↑x2denom = ↑x2num\nhx2pos : 0 < 2 * x.den\nhx2cnezr : ↑x2denom ≠ 0\nthis : 0 < x.num\n⊢ 1 < 2 * x.num",
"after_state": "f : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\nh_f_commutes_with_pos_nat_mul : ∀ (n : ℕ), 0 < n → ∀ (x : ℚ), 0 < x → f (↑n * x) = ↑n * f x\nx : ℚ\nhx : 0 < x\nH₀ : x * ↑x.den = ↑x.num\nx2denom : ℕ := 2 * x.den\nx2num : ℤ := 2 * x.num\nH : x * ↑x2denom = ↑x2num\nhx2pos : 0 < 2 * x.den\nhx2cnezr : ↑x2denom ≠ 0\nthis : 0 < x.num\n⊢ 1 < 2 * x.num"
},
{
"line": "try trivial",
"before_state": "f : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\nh_f_commutes_with_pos_nat_mul : ∀ (n : ℕ), 0 < n → ∀ (x : ℚ), 0 < x → f (↑n * x) = ↑n * f x\nx : ℚ\nhx : 0 < x\nH₀ : x * ↑x.den = ↑x.num\nx2denom : ℕ := 2 * x.den\nx2num : ℤ := 2 * x.num\nH : x * ↑x2denom = ↑x2num\nhx2pos : 0 < 2 * x.den\nhx2cnezr : ↑x2denom ≠ 0\nthis : 0 < x.num\n⊢ 1 < 2 * x.num",
"after_state": "f : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\nh_f_commutes_with_pos_nat_mul : ∀ (n : ℕ), 0 < n → ∀ (x : ℚ), 0 < x → f (↑n * x) = ↑n * f x\nx : ℚ\nhx : 0 < x\nH₀ : x * ↑x.den = ↑x.num\nx2denom : ℕ := 2 * x.den\nx2num : ℤ := 2 * x.num\nH : x * ↑x2denom = ↑x2num\nhx2pos : 0 < 2 * x.den\nhx2cnezr : ↑x2denom ≠ 0\nthis : 0 < x.num\n⊢ 1 < 2 * x.num"
},
{
"line": "first\n| trivial\n| skip",
"before_state": "f : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\nh_f_commutes_with_pos_nat_mul : ∀ (n : ℕ), 0 < n → ∀ (x : ℚ), 0 < x → f (↑n * x) = ↑n * f x\nx : ℚ\nhx : 0 < x\nH₀ : x * ↑x.den = ↑x.num\nx2denom : ℕ := 2 * x.den\nx2num : ℤ := 2 * x.num\nH : x * ↑x2denom = ↑x2num\nhx2pos : 0 < 2 * x.den\nhx2cnezr : ↑x2denom ≠ 0\nthis : 0 < x.num\n⊢ 1 < 2 * x.num",
"after_state": "f : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\nh_f_commutes_with_pos_nat_mul : ∀ (n : ℕ), 0 < n → ∀ (x : ℚ), 0 < x → f (↑n * x) = ↑n * f x\nx : ℚ\nhx : 0 < x\nH₀ : x * ↑x.den = ↑x.num\nx2denom : ℕ := 2 * x.den\nx2num : ℤ := 2 * x.num\nH : x * ↑x2denom = ↑x2num\nhx2pos : 0 < 2 * x.den\nhx2cnezr : ↑x2denom ≠ 0\nthis : 0 < x.num\n⊢ 1 < 2 * x.num"
},
{
"line": "trivial",
"before_state": "f : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\nh_f_commutes_with_pos_nat_mul : ∀ (n : ℕ), 0 < n → ∀ (x : ℚ), 0 < x → f (↑n * x) = ↑n * f x\nx : ℚ\nhx : 0 < x\nH₀ : x * ↑x.den = ↑x.num\nx2denom : ℕ := 2 * x.den\nx2num : ℤ := 2 * x.num\nH : x * ↑x2denom = ↑x2num\nhx2pos : 0 < 2 * x.den\nhx2cnezr : ↑x2denom ≠ 0\nthis : 0 < x.num\n⊢ 1 < 2 * x.num",
"after_state": "f : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\nh_f_commutes_with_pos_nat_mul : ∀ (n : ℕ), 0 < n → ∀ (x : ℚ), 0 < x → f (↑n * x) = ↑n * f x\nx : ℚ\nhx : 0 < x\nH₀ : x * ↑x.den = ↑x.num\nx2denom : ℕ := 2 * x.den\nx2num : ℤ := 2 * x.num\nH : x * ↑x2denom = ↑x2num\nhx2pos : 0 < 2 * x.den\nhx2cnezr : ↑x2denom ≠ 0\nthis : 0 < x.num\n⊢ 1 < 2 * x.num"
},
{
"line": "assumption",
"before_state": "f : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\nh_f_commutes_with_pos_nat_mul : ∀ (n : ℕ), 0 < n → ∀ (x : ℚ), 0 < x → f (↑n * x) = ↑n * f x\nx : ℚ\nhx : 0 < x\nH₀ : x * ↑x.den = ↑x.num\nx2denom : ℕ := 2 * x.den\nx2num : ℤ := 2 * x.num\nH : x * ↑x2denom = ↑x2num\nhx2pos : 0 < 2 * x.den\nhx2cnezr : ↑x2denom ≠ 0\nthis : 0 < x.num\n⊢ 1 < 2 * x.num",
"after_state": "f : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\nh_f_commutes_with_pos_nat_mul : ∀ (n : ℕ), 0 < n → ∀ (x : ℚ), 0 < x → f (↑n * x) = ↑n * f x\nx : ℚ\nhx : 0 < x\nH₀ : x * ↑x.den = ↑x.num\nx2denom : ℕ := 2 * x.den\nx2num : ℤ := 2 * x.num\nH : x * ↑x2denom = ↑x2num\nhx2pos : 0 < 2 * x.den\nhx2cnezr : ↑x2denom ≠ 0\nthis : 0 < x.num\n⊢ 1 < 2 * x.num"
},
{
"line": "skip",
"before_state": "f : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\nh_f_commutes_with_pos_nat_mul : ∀ (n : ℕ), 0 < n → ∀ (x : ℚ), 0 < x → f (↑n * x) = ↑n * f x\nx : ℚ\nhx : 0 < x\nH₀ : x * ↑x.den = ↑x.num\nx2denom : ℕ := 2 * x.den\nx2num : ℤ := 2 * x.num\nH : x * ↑x2denom = ↑x2num\nhx2pos : 0 < 2 * x.den\nhx2cnezr : ↑x2denom ≠ 0\nthis : 0 < x.num\n⊢ 1 < 2 * x.num",
"after_state": "f : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\nh_f_commutes_with_pos_nat_mul : ∀ (n : ℕ), 0 < n → ∀ (x : ℚ), 0 < x → f (↑n * x) = ↑n * f x\nx : ℚ\nhx : 0 < x\nH₀ : x * ↑x.den = ↑x.num\nx2denom : ℕ := 2 * x.den\nx2num : ℤ := 2 * x.num\nH : x * ↑x2denom = ↑x2num\nhx2pos : 0 < 2 * x.den\nhx2cnezr : ↑x2denom ≠ 0\nthis : 0 < x.num\n⊢ 1 < 2 * x.num"
},
{
"line": "linarith",
"before_state": "f : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\nh_f_commutes_with_pos_nat_mul : ∀ (n : ℕ), 0 < n → ∀ (x : ℚ), 0 < x → f (↑n * x) = ↑n * f x\nx : ℚ\nhx : 0 < x\nH₀ : x * ↑x.den = ↑x.num\nx2denom : ℕ := 2 * x.den\nx2num : ℤ := 2 * x.num\nH : x * ↑x2denom = ↑x2num\nhx2pos : 0 < 2 * x.den\nhx2cnezr : ↑x2denom ≠ 0\nthis : 0 < x.num\n⊢ 1 < 2 * x.num",
"after_state": "No Goals!"
},
{
"line": "ring1",
"before_state": "f : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\nh_f_commutes_with_pos_nat_mul : ∀ (n : ℕ), 0 < n → ∀ (x : ℚ), 0 < x → f (↑n * x) = ↑n * f x\nx : ℚ\nhx : 0 < x\nH₀ : x * ↑x.den = ↑x.num\nx2denom : ℕ := 2 * x.den\nx2num : ℤ := 2 * x.num\nH : x * ↑x2denom = ↑x2num\nhx2pos : 0 < 2 * x.den\nhx2cnezr : ↑x2denom ≠ 0\nthis : 0 < x.num\na✝ : 1 ≥ 2 * x.num\n⊢ -1 + 2 * (0 + 1 - x.num) + (2 * x.num - 1) = 0",
"after_state": "No Goals!"
},
{
"line": "norm_num",
"before_state": "f : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\nh_f_commutes_with_pos_nat_mul : ∀ (n : ℕ), 0 < n → ∀ (x : ℚ), 0 < x → f (↑n * x) = ↑n * f x\nx : ℚ\nhx : 0 < x\nH₀ : x * ↑x.den = ↑x.num\nx2denom : ℕ := 2 * x.den\nx2num : ℤ := 2 * x.num\nH : x * ↑x2denom = ↑x2num\nhx2pos : 0 < 2 * x.den\nhx2cnezr : ↑x2denom ≠ 0\nthis : 0 < x.num\na✝ : 1 ≥ 2 * x.num\n⊢ 2 > 0",
"after_state": "No Goals!"
},
{
"line": "apply mul_left_cancel₀ hx2cnezr",
"before_state": "case intro.intro\nf : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\nh_f_commutes_with_pos_nat_mul : ∀ (n : ℕ), 0 < n → ∀ (x : ℚ), 0 < x → f (↑n * x) = ↑n * f x\nx : ℚ\nhx : 0 < x\nH₀ : x * ↑x.den = ↑x.num\nx2denom : ℕ := 2 * x.den\nx2num : ℤ := 2 * x.num\nH : x * ↑x2denom = ↑x2num\nhx2pos : 0 < 2 * x.den\nhx2cnezr : ↑x2denom ≠ 0\nthis : 0 < x.num\nhx2num_gt_one : 1 < ↑(2 * x.num)\n⊢ f x = ↑x",
"after_state": "case intro.intro\nf : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\nh_f_commutes_with_pos_nat_mul : ∀ (n : ℕ), 0 < n → ∀ (x : ℚ), 0 < x → f (↑n * x) = ↑n * f x\nx : ℚ\nhx : 0 < x\nH₀ : x * ↑x.den = ↑x.num\nx2denom : ℕ := 2 * x.den\nx2num : ℤ := 2 * x.num\nH : x * ↑x2denom = ↑x2num\nhx2pos : 0 < 2 * x.den\nhx2cnezr : ↑x2denom ≠ 0\nthis : 0 < x.num\nhx2num_gt_one : 1 < ↑(2 * x.num)\n⊢ ↑x2denom * f x = ↑x2denom * ↑x"
},
{
"line": "calc\n x2denom * f x = f (x2denom * x) := (h_f_commutes_with_pos_nat_mul x2denom hx2pos x hx).symm\n _ = f x2num := by congr; linear_combination H\n _ = x2num := (fixed_point_of_gt_1 hx2num_gt_one H1 H2 H4 H5 ha1 hae)\n _ = ((x2num : ℚ) : ℝ) := by norm_cast\n _ = (↑(x2denom * x) : ℝ) := by congr; linear_combination -H\n _ = x2denom * x := by push_cast; rfl",
"before_state": "case intro.intro\nf : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\nh_f_commutes_with_pos_nat_mul : ∀ (n : ℕ), 0 < n → ∀ (x : ℚ), 0 < x → f (↑n * x) = ↑n * f x\nx : ℚ\nhx : 0 < x\nH₀ : x * ↑x.den = ↑x.num\nx2denom : ℕ := 2 * x.den\nx2num : ℤ := 2 * x.num\nH : x * ↑x2denom = ↑x2num\nhx2pos : 0 < 2 * x.den\nhx2cnezr : ↑x2denom ≠ 0\nthis : 0 < x.num\nhx2num_gt_one : 1 < ↑(2 * x.num)\n⊢ ↑x2denom * f x = ↑x2denom * ↑x",
"after_state": "No Goals!"
},
{
"line": "congr",
"before_state": "f : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\nh_f_commutes_with_pos_nat_mul : ∀ (n : ℕ), 0 < n → ∀ (x : ℚ), 0 < x → f (↑n * x) = ↑n * f x\nx : ℚ\nhx : 0 < x\nH₀ : x * ↑x.den = ↑x.num\nx2denom : ℕ := 2 * x.den\nx2num : ℤ := 2 * x.num\nH : x * ↑x2denom = ↑x2num\nhx2pos : 0 < 2 * x.den\nhx2cnezr : ↑x2denom ≠ 0\nthis : 0 < x.num\nhx2num_gt_one : 1 < ↑(2 * x.num)\n⊢ f (↑x2denom * x) = f ↑x2num",
"after_state": "case e_a\nf : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\nh_f_commutes_with_pos_nat_mul : ∀ (n : ℕ), 0 < n → ∀ (x : ℚ), 0 < x → f (↑n * x) = ↑n * f x\nx : ℚ\nhx : 0 < x\nH₀ : x * ↑x.den = ↑x.num\nx2denom : ℕ := 2 * x.den\nx2num : ℤ := 2 * x.num\nH : x * ↑x2denom = ↑x2num\nhx2pos : 0 < 2 * x.den\nhx2cnezr : ↑x2denom ≠ 0\nthis : 0 < x.num\nhx2num_gt_one : 1 < ↑(2 * x.num)\n⊢ ↑x2denom * x = ↑x2num"
},
{
"line": "linear_combination H",
"before_state": "case e_a\nf : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\nh_f_commutes_with_pos_nat_mul : ∀ (n : ℕ), 0 < n → ∀ (x : ℚ), 0 < x → f (↑n * x) = ↑n * f x\nx : ℚ\nhx : 0 < x\nH₀ : x * ↑x.den = ↑x.num\nx2denom : ℕ := 2 * x.den\nx2num : ℤ := 2 * x.num\nH : x * ↑x2denom = ↑x2num\nhx2pos : 0 < 2 * x.den\nhx2cnezr : ↑x2denom ≠ 0\nthis : 0 < x.num\nhx2num_gt_one : 1 < ↑(2 * x.num)\n⊢ ↑x2denom * x = ↑x2num",
"after_state": "No Goals!"
},
{
"line": "norm_cast",
"before_state": "f : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\nh_f_commutes_with_pos_nat_mul : ∀ (n : ℕ), 0 < n → ∀ (x : ℚ), 0 < x → f (↑n * x) = ↑n * f x\nx : ℚ\nhx : 0 < x\nH₀ : x * ↑x.den = ↑x.num\nx2denom : ℕ := 2 * x.den\nx2num : ℤ := 2 * x.num\nH : x * ↑x2denom = ↑x2num\nhx2pos : 0 < 2 * x.den\nhx2cnezr : ↑x2denom ≠ 0\nthis : 0 < x.num\nhx2num_gt_one : 1 < ↑(2 * x.num)\n⊢ ↑x2num = ↑↑x2num",
"after_state": "No Goals!"
},
{
"line": "focus\n norm_cast0\n with_annotate_state\"<;>\" skip\n all_goals try trivial",
"before_state": "f : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\nh_f_commutes_with_pos_nat_mul : ∀ (n : ℕ), 0 < n → ∀ (x : ℚ), 0 < x → f (↑n * x) = ↑n * f x\nx : ℚ\nhx : 0 < x\nH₀ : x * ↑x.den = ↑x.num\nx2denom : ℕ := 2 * x.den\nx2num : ℤ := 2 * x.num\nH : x * ↑x2denom = ↑x2num\nhx2pos : 0 < 2 * x.den\nhx2cnezr : ↑x2denom ≠ 0\nthis : 0 < x.num\nhx2num_gt_one : 1 < ↑(2 * x.num)\n⊢ ↑x2num = ↑↑x2num",
"after_state": "No Goals!"
},
{
"line": "norm_cast0",
"before_state": "f : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\nh_f_commutes_with_pos_nat_mul : ∀ (n : ℕ), 0 < n → ∀ (x : ℚ), 0 < x → f (↑n * x) = ↑n * f x\nx : ℚ\nhx : 0 < x\nH₀ : x * ↑x.den = ↑x.num\nx2denom : ℕ := 2 * x.den\nx2num : ℤ := 2 * x.num\nH : x * ↑x2denom = ↑x2num\nhx2pos : 0 < 2 * x.den\nhx2cnezr : ↑x2denom ≠ 0\nthis : 0 < x.num\nhx2num_gt_one : 1 < ↑(2 * x.num)\n⊢ ↑x2num = ↑↑x2num",
"after_state": "f : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\nh_f_commutes_with_pos_nat_mul : ∀ (n : ℕ), 0 < n → ∀ (x : ℚ), 0 < x → f (↑n * x) = ↑n * f x\nx : ℚ\nhx : 0 < x\nH₀ : x * ↑x.den = ↑x.num\nx2denom : ℕ := 2 * x.den\nx2num : ℤ := 2 * x.num\nH : x * ↑x2denom = ↑x2num\nhx2pos : 0 < 2 * x.den\nhx2cnezr : ↑x2denom ≠ 0\nthis : 0 < x.num\nhx2num_gt_one : 1 < ↑(2 * x.num)\n⊢ x2num = x2num"
},
{
"line": "with_annotate_state\"<;>\" skip",
"before_state": "f : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\nh_f_commutes_with_pos_nat_mul : ∀ (n : ℕ), 0 < n → ∀ (x : ℚ), 0 < x → f (↑n * x) = ↑n * f x\nx : ℚ\nhx : 0 < x\nH₀ : x * ↑x.den = ↑x.num\nx2denom : ℕ := 2 * x.den\nx2num : ℤ := 2 * x.num\nH : x * ↑x2denom = ↑x2num\nhx2pos : 0 < 2 * x.den\nhx2cnezr : ↑x2denom ≠ 0\nthis : 0 < x.num\nhx2num_gt_one : 1 < ↑(2 * x.num)\n⊢ x2num = x2num",
"after_state": "f : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\nh_f_commutes_with_pos_nat_mul : ∀ (n : ℕ), 0 < n → ∀ (x : ℚ), 0 < x → f (↑n * x) = ↑n * f x\nx : ℚ\nhx : 0 < x\nH₀ : x * ↑x.den = ↑x.num\nx2denom : ℕ := 2 * x.den\nx2num : ℤ := 2 * x.num\nH : x * ↑x2denom = ↑x2num\nhx2pos : 0 < 2 * x.den\nhx2cnezr : ↑x2denom ≠ 0\nthis : 0 < x.num\nhx2num_gt_one : 1 < ↑(2 * x.num)\n⊢ x2num = x2num"
},
{
"line": "skip",
"before_state": "f : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\nh_f_commutes_with_pos_nat_mul : ∀ (n : ℕ), 0 < n → ∀ (x : ℚ), 0 < x → f (↑n * x) = ↑n * f x\nx : ℚ\nhx : 0 < x\nH₀ : x * ↑x.den = ↑x.num\nx2denom : ℕ := 2 * x.den\nx2num : ℤ := 2 * x.num\nH : x * ↑x2denom = ↑x2num\nhx2pos : 0 < 2 * x.den\nhx2cnezr : ↑x2denom ≠ 0\nthis : 0 < x.num\nhx2num_gt_one : 1 < ↑(2 * x.num)\n⊢ x2num = x2num",
"after_state": "f : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\nh_f_commutes_with_pos_nat_mul : ∀ (n : ℕ), 0 < n → ∀ (x : ℚ), 0 < x → f (↑n * x) = ↑n * f x\nx : ℚ\nhx : 0 < x\nH₀ : x * ↑x.den = ↑x.num\nx2denom : ℕ := 2 * x.den\nx2num : ℤ := 2 * x.num\nH : x * ↑x2denom = ↑x2num\nhx2pos : 0 < 2 * x.den\nhx2cnezr : ↑x2denom ≠ 0\nthis : 0 < x.num\nhx2num_gt_one : 1 < ↑(2 * x.num)\n⊢ x2num = x2num"
},
{
"line": "all_goals try trivial",
"before_state": "f : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\nh_f_commutes_with_pos_nat_mul : ∀ (n : ℕ), 0 < n → ∀ (x : ℚ), 0 < x → f (↑n * x) = ↑n * f x\nx : ℚ\nhx : 0 < x\nH₀ : x * ↑x.den = ↑x.num\nx2denom : ℕ := 2 * x.den\nx2num : ℤ := 2 * x.num\nH : x * ↑x2denom = ↑x2num\nhx2pos : 0 < 2 * x.den\nhx2cnezr : ↑x2denom ≠ 0\nthis : 0 < x.num\nhx2num_gt_one : 1 < ↑(2 * x.num)\n⊢ x2num = x2num",
"after_state": "No Goals!"
},
{
"line": "try trivial",
"before_state": "f : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\nh_f_commutes_with_pos_nat_mul : ∀ (n : ℕ), 0 < n → ∀ (x : ℚ), 0 < x → f (↑n * x) = ↑n * f x\nx : ℚ\nhx : 0 < x\nH₀ : x * ↑x.den = ↑x.num\nx2denom : ℕ := 2 * x.den\nx2num : ℤ := 2 * x.num\nH : x * ↑x2denom = ↑x2num\nhx2pos : 0 < 2 * x.den\nhx2cnezr : ↑x2denom ≠ 0\nthis : 0 < x.num\nhx2num_gt_one : 1 < ↑(2 * x.num)\n⊢ x2num = x2num",
"after_state": "No Goals!"
},
{
"line": "first\n| trivial\n| skip",
"before_state": "f : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\nh_f_commutes_with_pos_nat_mul : ∀ (n : ℕ), 0 < n → ∀ (x : ℚ), 0 < x → f (↑n * x) = ↑n * f x\nx : ℚ\nhx : 0 < x\nH₀ : x * ↑x.den = ↑x.num\nx2denom : ℕ := 2 * x.den\nx2num : ℤ := 2 * x.num\nH : x * ↑x2denom = ↑x2num\nhx2pos : 0 < 2 * x.den\nhx2cnezr : ↑x2denom ≠ 0\nthis : 0 < x.num\nhx2num_gt_one : 1 < ↑(2 * x.num)\n⊢ x2num = x2num",
"after_state": "No Goals!"
},
{
"line": "trivial",
"before_state": "f : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\nh_f_commutes_with_pos_nat_mul : ∀ (n : ℕ), 0 < n → ∀ (x : ℚ), 0 < x → f (↑n * x) = ↑n * f x\nx : ℚ\nhx : 0 < x\nH₀ : x * ↑x.den = ↑x.num\nx2denom : ℕ := 2 * x.den\nx2num : ℤ := 2 * x.num\nH : x * ↑x2denom = ↑x2num\nhx2pos : 0 < 2 * x.den\nhx2cnezr : ↑x2denom ≠ 0\nthis : 0 < x.num\nhx2num_gt_one : 1 < ↑(2 * x.num)\n⊢ x2num = x2num",
"after_state": "No Goals!"
},
{
"line": "rfl",
"before_state": "f : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\nh_f_commutes_with_pos_nat_mul : ∀ (n : ℕ), 0 < n → ∀ (x : ℚ), 0 < x → f (↑n * x) = ↑n * f x\nx : ℚ\nhx : 0 < x\nH₀ : x * ↑x.den = ↑x.num\nx2denom : ℕ := 2 * x.den\nx2num : ℤ := 2 * x.num\nH : x * ↑x2denom = ↑x2num\nhx2pos : 0 < 2 * x.den\nhx2cnezr : ↑x2denom ≠ 0\nthis : 0 < x.num\nhx2num_gt_one : 1 < ↑(2 * x.num)\n⊢ x2num = x2num",
"after_state": "No Goals!"
},
{
"line": "eq_refl",
"before_state": "f : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\nh_f_commutes_with_pos_nat_mul : ∀ (n : ℕ), 0 < n → ∀ (x : ℚ), 0 < x → f (↑n * x) = ↑n * f x\nx : ℚ\nhx : 0 < x\nH₀ : x * ↑x.den = ↑x.num\nx2denom : ℕ := 2 * x.den\nx2num : ℤ := 2 * x.num\nH : x * ↑x2denom = ↑x2num\nhx2pos : 0 < 2 * x.den\nhx2cnezr : ↑x2denom ≠ 0\nthis : 0 < x.num\nhx2num_gt_one : 1 < ↑(2 * x.num)\n⊢ x2num = x2num",
"after_state": "No Goals!"
},
{
"line": "congr",
"before_state": "f : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\nh_f_commutes_with_pos_nat_mul : ∀ (n : ℕ), 0 < n → ∀ (x : ℚ), 0 < x → f (↑n * x) = ↑n * f x\nx : ℚ\nhx : 0 < x\nH₀ : x * ↑x.den = ↑x.num\nx2denom : ℕ := 2 * x.den\nx2num : ℤ := 2 * x.num\nH : x * ↑x2denom = ↑x2num\nhx2pos : 0 < 2 * x.den\nhx2cnezr : ↑x2denom ≠ 0\nthis : 0 < x.num\nhx2num_gt_one : 1 < ↑(2 * x.num)\n⊢ ↑↑x2num = ↑(↑x2denom * x)",
"after_state": "case e_a\nf : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\nh_f_commutes_with_pos_nat_mul : ∀ (n : ℕ), 0 < n → ∀ (x : ℚ), 0 < x → f (↑n * x) = ↑n * f x\nx : ℚ\nhx : 0 < x\nH₀ : x * ↑x.den = ↑x.num\nx2denom : ℕ := 2 * x.den\nx2num : ℤ := 2 * x.num\nH : x * ↑x2denom = ↑x2num\nhx2pos : 0 < 2 * x.den\nhx2cnezr : ↑x2denom ≠ 0\nthis : 0 < x.num\nhx2num_gt_one : 1 < ↑(2 * x.num)\n⊢ ↑x2num = ↑x2denom * x"
},
{
"line": "linear_combination -H",
"before_state": "case e_a\nf : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\nh_f_commutes_with_pos_nat_mul : ∀ (n : ℕ), 0 < n → ∀ (x : ℚ), 0 < x → f (↑n * x) = ↑n * f x\nx : ℚ\nhx : 0 < x\nH₀ : x * ↑x.den = ↑x.num\nx2denom : ℕ := 2 * x.den\nx2num : ℤ := 2 * x.num\nH : x * ↑x2denom = ↑x2num\nhx2pos : 0 < 2 * x.den\nhx2cnezr : ↑x2denom ≠ 0\nthis : 0 < x.num\nhx2num_gt_one : 1 < ↑(2 * x.num)\n⊢ ↑x2num = ↑x2denom * x",
"after_state": "No Goals!"
},
{
"line": "push_cast",
"before_state": "f : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\nh_f_commutes_with_pos_nat_mul : ∀ (n : ℕ), 0 < n → ∀ (x : ℚ), 0 < x → f (↑n * x) = ↑n * f x\nx : ℚ\nhx : 0 < x\nH₀ : x * ↑x.den = ↑x.num\nx2denom : ℕ := 2 * x.den\nx2num : ℤ := 2 * x.num\nH : x * ↑x2denom = ↑x2num\nhx2pos : 0 < 2 * x.den\nhx2cnezr : ↑x2denom ≠ 0\nthis : 0 < x.num\nhx2num_gt_one : 1 < ↑(2 * x.num)\n⊢ ↑(↑x2denom * x) = ↑x2denom * ↑x",
"after_state": "f : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\nh_f_commutes_with_pos_nat_mul : ∀ (n : ℕ), 0 < n → ∀ (x : ℚ), 0 < x → f (↑n * x) = ↑n * f x\nx : ℚ\nhx : 0 < x\nH₀ : x * ↑x.den = ↑x.num\nx2denom : ℕ := 2 * x.den\nx2num : ℤ := 2 * x.num\nH : x * ↑x2denom = ↑x2num\nhx2pos : 0 < 2 * x.den\nhx2cnezr : ↑x2denom ≠ 0\nthis : 0 < x.num\nhx2num_gt_one : 1 < ↑(2 * x.num)\n⊢ ↑x2denom * ↑x = ↑x2denom * ↑x"
},
{
"line": "rfl",
"before_state": "f : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\nh_f_commutes_with_pos_nat_mul : ∀ (n : ℕ), 0 < n → ∀ (x : ℚ), 0 < x → f (↑n * x) = ↑n * f x\nx : ℚ\nhx : 0 < x\nH₀ : x * ↑x.den = ↑x.num\nx2denom : ℕ := 2 * x.den\nx2num : ℤ := 2 * x.num\nH : x * ↑x2denom = ↑x2num\nhx2pos : 0 < 2 * x.den\nhx2cnezr : ↑x2denom ≠ 0\nthis : 0 < x.num\nhx2num_gt_one : 1 < ↑(2 * x.num)\n⊢ ↑x2denom * ↑x = ↑x2denom * ↑x",
"after_state": "No Goals!"
},
{
"line": "eq_refl",
"before_state": "f : ℚ → ℝ\nH1 : ∀ (x y : ℚ), 0 < x → 0 < y → f (x * y) ≤ f x * f y\nH2 : ∀ (x y : ℚ), 0 < x → 0 < y → f x + f y ≤ f (x + y)\na : ℚ\nha1 : 1 < a\nhae : f a = ↑a\nH3 : ∀ (x : ℚ), 0 < x → ∀ (n : ℕ), 0 < n → ↑n * f x ≤ f (↑n * x)\nH4 : ∀ (n : ℕ), 0 < n → ↑n ≤ f ↑n\nH5 : ∀ (x : ℚ), 1 < x → ↑x ≤ f x\nh_f_commutes_with_pos_nat_mul : ∀ (n : ℕ), 0 < n → ∀ (x : ℚ), 0 < x → f (↑n * x) = ↑n * f x\nx : ℚ\nhx : 0 < x\nH₀ : x * ↑x.den = ↑x.num\nx2denom : ℕ := 2 * x.den\nx2num : ℤ := 2 * x.num\nH : x * ↑x2denom = ↑x2num\nhx2pos : 0 < 2 * x.den\nhx2cnezr : ↑x2denom ≠ 0\nthis : 0 < x.num\nhx2num_gt_one : 1 < ↑(2 * x.num)\n⊢ ↑x2denom * ↑x = ↑x2denom * ↑x",
"after_state": "No Goals!"
}
] |
theorem imo2019_q1 (f : ℤ → ℤ) :
(∀ a b : ℤ, f (2 * a) + 2 * f b = f (f (a + b))) ↔ f = 0 ∨ ∃ c, f = fun x => 2 * x + c := by
constructor; swap
-- easy way: f(x)=0 and f(x)=2x+c work.
· rintro (rfl | ⟨c, rfl⟩) <;> intros <;> norm_num; ring
-- hard way.
intro hf
-- functional equation
-- Using `h` for `(0, b)` and `(-1, b + 1)`, we get `f (b + 1) = f b + m`
obtain ⟨m, H⟩ : ∃ m, ∀ b, f (b + 1) = f b + m := by
refine ⟨(f 0 - f (-2)) / 2, fun b => ?_⟩
refine sub_eq_iff_eq_add'.1 (Int.eq_ediv_of_mul_eq_right two_ne_zero ?_)
have h1 : f 0 + 2 * f b = f (f b) := by simpa using hf 0 b
have h2 : f (-2) + 2 * f (b + 1) = f (f b) := by simpa using hf (-1) (b + 1)
linarith
-- Hence, `f` is an affine map, `f b = f 0 + m * b`
obtain ⟨c, H⟩ : ∃ c, ∀ b, f b = c + m * b := by
refine ⟨f 0, fun b => ?_⟩
induction' b with b ihb b ihb
· simp
· simp [H, ihb, mul_add, add_assoc]
· rw [← sub_eq_of_eq_add (H _)]
simp [ihb]; ring
-- Now use `hf 0 0` and `hf 0 1` to show that `m ∈ {0, 2}`
have H3 : 2 * c = m * c := by simpa [H, mul_add] using hf 0 0
obtain rfl | rfl : 2 = m ∨ m = 0 := by simpa [H, mul_add, H3] using hf 0 1
· right; use c; ext b; simp [H, add_comm]
· left; ext b; simpa [H, two_ne_zero] using H3
|
/root/DuelModelResearch/mathlib4/Archive/Imo/Imo2019Q1.lean
|
{
"open": [],
"variables": []
}
|
[
{
"line": "constructor",
"before_state": "f : ℤ → ℤ\n⊢ (∀ (a b : ℤ), f (2 * a) + 2 * f b = f (f (a + b))) ↔ f = 0 ∨ ∃ c, f = fun x => 2 * x + c",
"after_state": "case mp\nf : ℤ → ℤ\n⊢ (∀ (a b : ℤ), f (2 * a) + 2 * f b = f (f (a + b))) → f = 0 ∨ ∃ c, f = fun x => 2 * x + c\n---\ncase mpr\nf : ℤ → ℤ\n⊢ (f = 0 ∨ ∃ c, f = fun x => 2 * x + c) → ∀ (a b : ℤ), f (2 * a) + 2 * f b = f (f (a + b))"
},
{
"line": "swap\n -- easy way: f(x)=0 and f(x)=2x+c work.",
"before_state": "case mp\nf : ℤ → ℤ\n⊢ (∀ (a b : ℤ), f (2 * a) + 2 * f b = f (f (a + b))) → f = 0 ∨ ∃ c, f = fun x => 2 * x + c\n---\ncase mpr\nf : ℤ → ℤ\n⊢ (f = 0 ∨ ∃ c, f = fun x => 2 * x + c) → ∀ (a b : ℤ), f (2 * a) + 2 * f b = f (f (a + b))",
"after_state": "case mpr\nf : ℤ → ℤ\n⊢ (f = 0 ∨ ∃ c, f = fun x => 2 * x + c) → ∀ (a b : ℤ), f (2 * a) + 2 * f b = f (f (a + b))\n---\ncase mp\nf : ℤ → ℤ\n⊢ (∀ (a b : ℤ), f (2 * a) + 2 * f b = f (f (a + b))) → f = 0 ∨ ∃ c, f = fun x => 2 * x + c"
},
{
"line": "pick_goal 2",
"before_state": "case mp\nf : ℤ → ℤ\n⊢ (∀ (a b : ℤ), f (2 * a) + 2 * f b = f (f (a + b))) → f = 0 ∨ ∃ c, f = fun x => 2 * x + c\n---\ncase mpr\nf : ℤ → ℤ\n⊢ (f = 0 ∨ ∃ c, f = fun x => 2 * x + c) → ∀ (a b : ℤ), f (2 * a) + 2 * f b = f (f (a + b))",
"after_state": "case mpr\nf : ℤ → ℤ\n⊢ (f = 0 ∨ ∃ c, f = fun x => 2 * x + c) → ∀ (a b : ℤ), f (2 * a) + 2 * f b = f (f (a + b))\n---\ncase mp\nf : ℤ → ℤ\n⊢ (∀ (a b : ℤ), f (2 * a) + 2 * f b = f (f (a + b))) → f = 0 ∨ ∃ c, f = fun x => 2 * x + c"
},
{
"line": "focus\n rintro (rfl | ⟨c, rfl⟩) <;> intros\n with_annotate_state\"<;>\" skip\n all_goals norm_num",
"before_state": "case mpr\nf : ℤ → ℤ\n⊢ (f = 0 ∨ ∃ c, f = fun x => 2 * x + c) → ∀ (a b : ℤ), f (2 * a) + 2 * f b = f (f (a + b))",
"after_state": "case mpr.inr.intro\nc a✝ b✝ : ℤ\n⊢ 2 * (2 * a✝) + c + 2 * (2 * b✝ + c) = 2 * (2 * (a✝ + b✝) + c) + c"
},
{
"line": "focus\n rintro (rfl | ⟨c, rfl⟩)\n with_annotate_state\"<;>\" skip\n all_goals intros",
"before_state": "case mpr\nf : ℤ → ℤ\n⊢ (f = 0 ∨ ∃ c, f = fun x => 2 * x + c) → ∀ (a b : ℤ), f (2 * a) + 2 * f b = f (f (a + b))",
"after_state": "case mpr.inl\na✝ b✝ : ℤ\n⊢ 0 (2 * a✝) + 2 * 0 b✝ = 0 (0 (a✝ + b✝))\n---\ncase mpr.inr.intro\nc a✝ b✝ : ℤ\n⊢ (fun x => 2 * x + c) (2 * a✝) + 2 * (fun x => 2 * x + c) b✝ = (fun x => 2 * x + c) ((fun x => 2 * x + c) (a✝ + b✝))"
},
{
"line": "rintro (rfl | ⟨c, rfl⟩)",
"before_state": "case mpr\nf : ℤ → ℤ\n⊢ (f = 0 ∨ ∃ c, f = fun x => 2 * x + c) → ∀ (a b : ℤ), f (2 * a) + 2 * f b = f (f (a + b))",
"after_state": "case mpr.inl\n⊢ ∀ (a b : ℤ), 0 (2 * a) + 2 * 0 b = 0 (0 (a + b))\n---\ncase mpr.inr.intro\nc : ℤ\n⊢ ∀ (a b : ℤ),\n (fun x => 2 * x + c) (2 * a) + 2 * (fun x => 2 * x + c) b = (fun x => 2 * x + c) ((fun x => 2 * x + c) (a + b))"
},
{
"line": "with_annotate_state\"<;>\" skip",
"before_state": "case mpr.inl\n⊢ ∀ (a b : ℤ), 0 (2 * a) + 2 * 0 b = 0 (0 (a + b))\n---\ncase mpr.inr.intro\nc : ℤ\n⊢ ∀ (a b : ℤ),\n (fun x => 2 * x + c) (2 * a) + 2 * (fun x => 2 * x + c) b = (fun x => 2 * x + c) ((fun x => 2 * x + c) (a + b))",
"after_state": "case mpr.inl\n⊢ ∀ (a b : ℤ), 0 (2 * a) + 2 * 0 b = 0 (0 (a + b))\n---\ncase mpr.inr.intro\nc : ℤ\n⊢ ∀ (a b : ℤ),\n (fun x => 2 * x + c) (2 * a) + 2 * (fun x => 2 * x + c) b = (fun x => 2 * x + c) ((fun x => 2 * x + c) (a + b))"
},
{
"line": "skip",
"before_state": "case mpr.inl\n⊢ ∀ (a b : ℤ), 0 (2 * a) + 2 * 0 b = 0 (0 (a + b))\n---\ncase mpr.inr.intro\nc : ℤ\n⊢ ∀ (a b : ℤ),\n (fun x => 2 * x + c) (2 * a) + 2 * (fun x => 2 * x + c) b = (fun x => 2 * x + c) ((fun x => 2 * x + c) (a + b))",
"after_state": "case mpr.inl\n⊢ ∀ (a b : ℤ), 0 (2 * a) + 2 * 0 b = 0 (0 (a + b))\n---\ncase mpr.inr.intro\nc : ℤ\n⊢ ∀ (a b : ℤ),\n (fun x => 2 * x + c) (2 * a) + 2 * (fun x => 2 * x + c) b = (fun x => 2 * x + c) ((fun x => 2 * x + c) (a + b))"
},
{
"line": "all_goals intros",
"before_state": "case mpr.inl\n⊢ ∀ (a b : ℤ), 0 (2 * a) + 2 * 0 b = 0 (0 (a + b))\n---\ncase mpr.inr.intro\nc : ℤ\n⊢ ∀ (a b : ℤ),\n (fun x => 2 * x + c) (2 * a) + 2 * (fun x => 2 * x + c) b = (fun x => 2 * x + c) ((fun x => 2 * x + c) (a + b))",
"after_state": "case mpr.inl\na✝ b✝ : ℤ\n⊢ 0 (2 * a✝) + 2 * 0 b✝ = 0 (0 (a✝ + b✝))\n---\ncase mpr.inr.intro\nc a✝ b✝ : ℤ\n⊢ (fun x => 2 * x + c) (2 * a✝) + 2 * (fun x => 2 * x + c) b✝ = (fun x => 2 * x + c) ((fun x => 2 * x + c) (a✝ + b✝))"
},
{
"line": "intros",
"before_state": "case mpr.inl\n⊢ ∀ (a b : ℤ), 0 (2 * a) + 2 * 0 b = 0 (0 (a + b))",
"after_state": "case mpr.inl\na✝ b✝ : ℤ\n⊢ 0 (2 * a✝) + 2 * 0 b✝ = 0 (0 (a✝ + b✝))"
},
{
"line": "intros",
"before_state": "case mpr.inr.intro\nc : ℤ\n⊢ ∀ (a b : ℤ),\n (fun x => 2 * x + c) (2 * a) + 2 * (fun x => 2 * x + c) b = (fun x => 2 * x + c) ((fun x => 2 * x + c) (a + b))",
"after_state": "case mpr.inr.intro\nc a✝ b✝ : ℤ\n⊢ (fun x => 2 * x + c) (2 * a✝) + 2 * (fun x => 2 * x + c) b✝ = (fun x => 2 * x + c) ((fun x => 2 * x + c) (a✝ + b✝))"
},
{
"line": "with_annotate_state\"<;>\" skip",
"before_state": "case mpr.inl\na✝ b✝ : ℤ\n⊢ 0 (2 * a✝) + 2 * 0 b✝ = 0 (0 (a✝ + b✝))\n---\ncase mpr.inr.intro\nc a✝ b✝ : ℤ\n⊢ (fun x => 2 * x + c) (2 * a✝) + 2 * (fun x => 2 * x + c) b✝ = (fun x => 2 * x + c) ((fun x => 2 * x + c) (a✝ + b✝))",
"after_state": "case mpr.inl\na✝ b✝ : ℤ\n⊢ 0 (2 * a✝) + 2 * 0 b✝ = 0 (0 (a✝ + b✝))\n---\ncase mpr.inr.intro\nc a✝ b✝ : ℤ\n⊢ (fun x => 2 * x + c) (2 * a✝) + 2 * (fun x => 2 * x + c) b✝ = (fun x => 2 * x + c) ((fun x => 2 * x + c) (a✝ + b✝))"
},
{
"line": "skip",
"before_state": "case mpr.inl\na✝ b✝ : ℤ\n⊢ 0 (2 * a✝) + 2 * 0 b✝ = 0 (0 (a✝ + b✝))\n---\ncase mpr.inr.intro\nc a✝ b✝ : ℤ\n⊢ (fun x => 2 * x + c) (2 * a✝) + 2 * (fun x => 2 * x + c) b✝ = (fun x => 2 * x + c) ((fun x => 2 * x + c) (a✝ + b✝))",
"after_state": "case mpr.inl\na✝ b✝ : ℤ\n⊢ 0 (2 * a✝) + 2 * 0 b✝ = 0 (0 (a✝ + b✝))\n---\ncase mpr.inr.intro\nc a✝ b✝ : ℤ\n⊢ (fun x => 2 * x + c) (2 * a✝) + 2 * (fun x => 2 * x + c) b✝ = (fun x => 2 * x + c) ((fun x => 2 * x + c) (a✝ + b✝))"
},
{
"line": "all_goals norm_num",
"before_state": "case mpr.inl\na✝ b✝ : ℤ\n⊢ 0 (2 * a✝) + 2 * 0 b✝ = 0 (0 (a✝ + b✝))\n---\ncase mpr.inr.intro\nc a✝ b✝ : ℤ\n⊢ (fun x => 2 * x + c) (2 * a✝) + 2 * (fun x => 2 * x + c) b✝ = (fun x => 2 * x + c) ((fun x => 2 * x + c) (a✝ + b✝))",
"after_state": "case mpr.inr.intro\nc a✝ b✝ : ℤ\n⊢ 2 * (2 * a✝) + c + 2 * (2 * b✝ + c) = 2 * (2 * (a✝ + b✝) + c) + c"
},
{
"line": "norm_num",
"before_state": "case mpr.inl\na✝ b✝ : ℤ\n⊢ 0 (2 * a✝) + 2 * 0 b✝ = 0 (0 (a✝ + b✝))",
"after_state": "No Goals!"
},
{
"line": "norm_num",
"before_state": "case mpr.inr.intro\nc a✝ b✝ : ℤ\n⊢ (fun x => 2 * x + c) (2 * a✝) + 2 * (fun x => 2 * x + c) b✝ = (fun x => 2 * x + c) ((fun x => 2 * x + c) (a✝ + b✝))",
"after_state": "case mpr.inr.intro\nc a✝ b✝ : ℤ\n⊢ 2 * (2 * a✝) + c + 2 * (2 * b✝ + c) = 2 * (2 * (a✝ + b✝) + c) + c"
},
{
"line": "ring\n -- hard way.",
"before_state": "case mpr.inr.intro\nc a✝ b✝ : ℤ\n⊢ 2 * (2 * a✝) + c + 2 * (2 * b✝ + c) = 2 * (2 * (a✝ + b✝) + c) + c",
"after_state": "No Goals!"
},
{
"line": "first\n| ring1\n|\n try_this ring_nf\"\\n\\nThe `ring` tactic failed to close the goal. Use `ring_nf` to obtain a normal form.\n \\nNote that `ring` works primarily in *commutative* rings. \\\n If you have a noncommutative ring, abelian group or module, consider using \\\n `noncomm_ring`, `abel` or `module` instead.\"",
"before_state": "case mpr.inr.intro\nc a✝ b✝ : ℤ\n⊢ 2 * (2 * a✝) + c + 2 * (2 * b✝ + c) = 2 * (2 * (a✝ + b✝) + c) + c",
"after_state": "No Goals!"
},
{
"line": "ring1",
"before_state": "case mpr.inr.intro\nc a✝ b✝ : ℤ\n⊢ 2 * (2 * a✝) + c + 2 * (2 * b✝ + c) = 2 * (2 * (a✝ + b✝) + c) + c",
"after_state": "No Goals!"
},
{
"line": "intro hf",
"before_state": "case mp\nf : ℤ → ℤ\n⊢ (∀ (a b : ℤ), f (2 * a) + 2 * f b = f (f (a + b))) → f = 0 ∨ ∃ c, f = fun x => 2 * x + c",
"after_state": "case mp\nf : ℤ → ℤ\nhf : ∀ (a b : ℤ), f (2 * a) + 2 * f b = f (f (a + b))\n⊢ f = 0 ∨ ∃ c, f = fun x => 2 * x + c"
},
{
"line": "obtain ⟨m, H⟩ : ∃ m, ∀ b, f (b + 1) = f b + m :=\n by\n refine ⟨(f 0 - f (-2)) / 2, fun b => ?_⟩\n refine sub_eq_iff_eq_add'.1 (Int.eq_ediv_of_mul_eq_right two_ne_zero ?_)\n have h1 : f 0 + 2 * f b = f (f b) := by simpa using hf 0 b\n have h2 : f (-2) + 2 * f (b + 1) = f (f b) := by simpa using hf (-1) (b + 1)\n linarith\n -- Hence, `f` is an affine map, `f b = f 0 + m * b`",
"before_state": "case mp\nf : ℤ → ℤ\nhf : ∀ (a b : ℤ), f (2 * a) + 2 * f b = f (f (a + b))\n⊢ f = 0 ∨ ∃ c, f = fun x => 2 * x + c",
"after_state": "case mp.intro\nf : ℤ → ℤ\nhf : ∀ (a b : ℤ), f (2 * a) + 2 * f b = f (f (a + b))\nm : ℤ\nH : ∀ (b : ℤ), f (b + 1) = f b + m\n⊢ f = 0 ∨ ∃ c, f = fun x => 2 * x + c"
},
{
"line": "refine ⟨(f 0 - f (-2)) / 2, fun b => ?_⟩",
"before_state": "f : ℤ → ℤ\nhf : ∀ (a b : ℤ), f (2 * a) + 2 * f b = f (f (a + b))\n⊢ ∃ m, ∀ (b : ℤ), f (b + 1) = f b + m",
"after_state": "f : ℤ → ℤ\nhf : ∀ (a b : ℤ), f (2 * a) + 2 * f b = f (f (a + b))\nb : ℤ\n⊢ f (b + 1) = f b + (f 0 - f (-2)) / 2"
},
{
"line": "refine sub_eq_iff_eq_add'.1 (Int.eq_ediv_of_mul_eq_right two_ne_zero ?_)",
"before_state": "f : ℤ → ℤ\nhf : ∀ (a b : ℤ), f (2 * a) + 2 * f b = f (f (a + b))\nb : ℤ\n⊢ f (b + 1) = f b + (f 0 - f (-2)) / 2",
"after_state": "f : ℤ → ℤ\nhf : ∀ (a b : ℤ), f (2 * a) + 2 * f b = f (f (a + b))\nb : ℤ\n⊢ 2 * (f (b + 1) - f b) = f 0 - f (-2)"
},
{
"line": "have h1 : f 0 + 2 * f b = f (f b) := by simpa using hf 0 b",
"before_state": "f : ℤ → ℤ\nhf : ∀ (a b : ℤ), f (2 * a) + 2 * f b = f (f (a + b))\nb : ℤ\n⊢ 2 * (f (b + 1) - f b) = f 0 - f (-2)",
"after_state": "f : ℤ → ℤ\nhf : ∀ (a b : ℤ), f (2 * a) + 2 * f b = f (f (a + b))\nb : ℤ\nh1 : f 0 + 2 * f b = f (f b)\n⊢ 2 * (f (b + 1) - f b) = f 0 - f (-2)"
},
{
"line": "focus\n refine\n no_implicit_lambda%\n (have h1 : f 0 + 2 * f b = f (f b) := ?body✝;\n ?_)\n case body✝ => with_annotate_state\"by\" (simpa using hf 0 b)",
"before_state": "f : ℤ → ℤ\nhf : ∀ (a b : ℤ), f (2 * a) + 2 * f b = f (f (a + b))\nb : ℤ\n⊢ 2 * (f (b + 1) - f b) = f 0 - f (-2)",
"after_state": "f : ℤ → ℤ\nhf : ∀ (a b : ℤ), f (2 * a) + 2 * f b = f (f (a + b))\nb : ℤ\nh1 : f 0 + 2 * f b = f (f b)\n⊢ 2 * (f (b + 1) - f b) = f 0 - f (-2)"
},
{
"line": "refine\n no_implicit_lambda%\n (have h1 : f 0 + 2 * f b = f (f b) := ?body✝;\n ?_)",
"before_state": "f : ℤ → ℤ\nhf : ∀ (a b : ℤ), f (2 * a) + 2 * f b = f (f (a + b))\nb : ℤ\n⊢ 2 * (f (b + 1) - f b) = f 0 - f (-2)",
"after_state": "case body\nf : ℤ → ℤ\nhf : ∀ (a b : ℤ), f (2 * a) + 2 * f b = f (f (a + b))\nb : ℤ\n⊢ f 0 + 2 * f b = f (f b)\n---\nf : ℤ → ℤ\nhf : ∀ (a b : ℤ), f (2 * a) + 2 * f b = f (f (a + b))\nb : ℤ\nh1 : f 0 + 2 * f b = f (f b)\n⊢ 2 * (f (b + 1) - f b) = f 0 - f (-2)"
},
{
"line": "case body✝ => with_annotate_state\"by\" (simpa using hf 0 b)",
"before_state": "case body\nf : ℤ → ℤ\nhf : ∀ (a b : ℤ), f (2 * a) + 2 * f b = f (f (a + b))\nb : ℤ\n⊢ f 0 + 2 * f b = f (f b)\n---\nf : ℤ → ℤ\nhf : ∀ (a b : ℤ), f (2 * a) + 2 * f b = f (f (a + b))\nb : ℤ\nh1 : f 0 + 2 * f b = f (f b)\n⊢ 2 * (f (b + 1) - f b) = f 0 - f (-2)",
"after_state": "f : ℤ → ℤ\nhf : ∀ (a b : ℤ), f (2 * a) + 2 * f b = f (f (a + b))\nb : ℤ\nh1 : f 0 + 2 * f b = f (f b)\n⊢ 2 * (f (b + 1) - f b) = f 0 - f (-2)"
},
{
"line": "with_annotate_state\"by\" (simpa using hf 0 b)",
"before_state": "f : ℤ → ℤ\nhf : ∀ (a b : ℤ), f (2 * a) + 2 * f b = f (f (a + b))\nb : ℤ\n⊢ f 0 + 2 * f b = f (f b)",
"after_state": "No Goals!"
},
{
"line": "simpa using hf 0 b",
"before_state": "f : ℤ → ℤ\nhf : ∀ (a b : ℤ), f (2 * a) + 2 * f b = f (f (a + b))\nb : ℤ\n⊢ f 0 + 2 * f b = f (f b)",
"after_state": "No Goals!"
},
{
"line": "have h2 : f (-2) + 2 * f (b + 1) = f (f b) := by simpa using hf (-1) (b + 1)",
"before_state": "f : ℤ → ℤ\nhf : ∀ (a b : ℤ), f (2 * a) + 2 * f b = f (f (a + b))\nb : ℤ\nh1 : f 0 + 2 * f b = f (f b)\n⊢ 2 * (f (b + 1) - f b) = f 0 - f (-2)",
"after_state": "f : ℤ → ℤ\nhf : ∀ (a b : ℤ), f (2 * a) + 2 * f b = f (f (a + b))\nb : ℤ\nh1 : f 0 + 2 * f b = f (f b)\nh2 : f (-2) + 2 * f (b + 1) = f (f b)\n⊢ 2 * (f (b + 1) - f b) = f 0 - f (-2)"
},
{
"line": "focus\n refine\n no_implicit_lambda%\n (have h2 : f (-2) + 2 * f (b + 1) = f (f b) := ?body✝;\n ?_)\n case body✝ => with_annotate_state\"by\" (simpa using hf (-1) (b + 1))",
"before_state": "f : ℤ → ℤ\nhf : ∀ (a b : ℤ), f (2 * a) + 2 * f b = f (f (a + b))\nb : ℤ\nh1 : f 0 + 2 * f b = f (f b)\n⊢ 2 * (f (b + 1) - f b) = f 0 - f (-2)",
"after_state": "f : ℤ → ℤ\nhf : ∀ (a b : ℤ), f (2 * a) + 2 * f b = f (f (a + b))\nb : ℤ\nh1 : f 0 + 2 * f b = f (f b)\nh2 : f (-2) + 2 * f (b + 1) = f (f b)\n⊢ 2 * (f (b + 1) - f b) = f 0 - f (-2)"
},
{
"line": "refine\n no_implicit_lambda%\n (have h2 : f (-2) + 2 * f (b + 1) = f (f b) := ?body✝;\n ?_)",
"before_state": "f : ℤ → ℤ\nhf : ∀ (a b : ℤ), f (2 * a) + 2 * f b = f (f (a + b))\nb : ℤ\nh1 : f 0 + 2 * f b = f (f b)\n⊢ 2 * (f (b + 1) - f b) = f 0 - f (-2)",
"after_state": "case body\nf : ℤ → ℤ\nhf : ∀ (a b : ℤ), f (2 * a) + 2 * f b = f (f (a + b))\nb : ℤ\nh1 : f 0 + 2 * f b = f (f b)\n⊢ f (-2) + 2 * f (b + 1) = f (f b)\n---\nf : ℤ → ℤ\nhf : ∀ (a b : ℤ), f (2 * a) + 2 * f b = f (f (a + b))\nb : ℤ\nh1 : f 0 + 2 * f b = f (f b)\nh2 : f (-2) + 2 * f (b + 1) = f (f b)\n⊢ 2 * (f (b + 1) - f b) = f 0 - f (-2)"
},
{
"line": "case body✝ => with_annotate_state\"by\" (simpa using hf (-1) (b + 1))",
"before_state": "case body\nf : ℤ → ℤ\nhf : ∀ (a b : ℤ), f (2 * a) + 2 * f b = f (f (a + b))\nb : ℤ\nh1 : f 0 + 2 * f b = f (f b)\n⊢ f (-2) + 2 * f (b + 1) = f (f b)\n---\nf : ℤ → ℤ\nhf : ∀ (a b : ℤ), f (2 * a) + 2 * f b = f (f (a + b))\nb : ℤ\nh1 : f 0 + 2 * f b = f (f b)\nh2 : f (-2) + 2 * f (b + 1) = f (f b)\n⊢ 2 * (f (b + 1) - f b) = f 0 - f (-2)",
"after_state": "f : ℤ → ℤ\nhf : ∀ (a b : ℤ), f (2 * a) + 2 * f b = f (f (a + b))\nb : ℤ\nh1 : f 0 + 2 * f b = f (f b)\nh2 : f (-2) + 2 * f (b + 1) = f (f b)\n⊢ 2 * (f (b + 1) - f b) = f 0 - f (-2)"
},
{
"line": "with_annotate_state\"by\" (simpa using hf (-1) (b + 1))",
"before_state": "f : ℤ → ℤ\nhf : ∀ (a b : ℤ), f (2 * a) + 2 * f b = f (f (a + b))\nb : ℤ\nh1 : f 0 + 2 * f b = f (f b)\n⊢ f (-2) + 2 * f (b + 1) = f (f b)",
"after_state": "No Goals!"
},
{
"line": "simpa using hf (-1) (b + 1)",
"before_state": "f : ℤ → ℤ\nhf : ∀ (a b : ℤ), f (2 * a) + 2 * f b = f (f (a + b))\nb : ℤ\nh1 : f 0 + 2 * f b = f (f b)\n⊢ f (-2) + 2 * f (b + 1) = f (f b)",
"after_state": "No Goals!"
},
{
"line": "linarith\n -- Hence, `f` is an affine map, `f b = f 0 + m * b`",
"before_state": "f : ℤ → ℤ\nhf : ∀ (a b : ℤ), f (2 * a) + 2 * f b = f (f (a + b))\nb : ℤ\nh1 : f 0 + 2 * f b = f (f b)\nh2 : f (-2) + 2 * f (b + 1) = f (f b)\n⊢ 2 * (f (b + 1) - f b) = f 0 - f (-2)",
"after_state": "No Goals!"
},
{
"line": "ring1",
"before_state": "f : ℤ → ℤ\nhf : ∀ (a b : ℤ), f (2 * a) + 2 * f b = f (f (a + b))\nb : ℤ\nh1 : f 0 + 2 * f b = f (f b)\nh2 : f (-2) + 2 * f (b + 1) = f (f b)\na✝ : 2 * (f (b + 1) - f b) < f 0 - f (-2)\n⊢ -1 + (f 0 + 2 * f b - f (f b)) + -(f (-2) + 2 * f (b + 1) - f (f b)) + (2 * (f (b + 1) - f b) + 1 - (f 0 - f (-2))) =\n 0",
"after_state": "No Goals!"
},
{
"line": "ring1",
"before_state": "f : ℤ → ℤ\nhf : ∀ (a b : ℤ), f (2 * a) + 2 * f b = f (f (a + b))\nb : ℤ\nh1 : f 0 + 2 * f b = f (f b)\nh2 : f (-2) + 2 * f (b + 1) = f (f b)\na✝ : f 0 - f (-2) < 2 * (f (b + 1) - f b)\n⊢ -1 + -(f 0 + 2 * f b - f (f b)) + (f (-2) + 2 * f (b + 1) - f (f b)) + (f 0 - f (-2) + 1 - 2 * (f (b + 1) - f b)) = 0",
"after_state": "No Goals!"
},
{
"line": "obtain ⟨c, H⟩ : ∃ c, ∀ b, f b = c + m * b := by\n refine ⟨f 0, fun b => ?_⟩\n induction' b with b ihb b ihb\n · simp\n · simp [H, ihb, mul_add, add_assoc]\n · rw [← sub_eq_of_eq_add (H _)]\n simp [ihb];\n ring\n -- Now use `hf 0 0` and `hf 0 1` to show that `m ∈ {0, 2}`",
"before_state": "case mp.intro\nf : ℤ → ℤ\nhf : ∀ (a b : ℤ), f (2 * a) + 2 * f b = f (f (a + b))\nm : ℤ\nH : ∀ (b : ℤ), f (b + 1) = f b + m\n⊢ f = 0 ∨ ∃ c, f = fun x => 2 * x + c",
"after_state": "case mp.intro.intro\nf : ℤ → ℤ\nhf : ∀ (a b : ℤ), f (2 * a) + 2 * f b = f (f (a + b))\nm : ℤ\nH✝ : ∀ (b : ℤ), f (b + 1) = f b + m\nc : ℤ\nH : ∀ (b : ℤ), f b = c + m * b\n⊢ f = 0 ∨ ∃ c, f = fun x => 2 * x + c"
},
{
"line": "refine ⟨f 0, fun b => ?_⟩",
"before_state": "f : ℤ → ℤ\nhf : ∀ (a b : ℤ), f (2 * a) + 2 * f b = f (f (a + b))\nm : ℤ\nH : ∀ (b : ℤ), f (b + 1) = f b + m\n⊢ ∃ c, ∀ (b : ℤ), f b = c + m * b",
"after_state": "f : ℤ → ℤ\nhf : ∀ (a b : ℤ), f (2 * a) + 2 * f b = f (f (a + b))\nm : ℤ\nH : ∀ (b : ℤ), f (b + 1) = f b + m\nb : ℤ\n⊢ f b = f 0 + m * b"
},
{
"line": "induction' b with b ihb b ihb",
"before_state": "f : ℤ → ℤ\nhf : ∀ (a b : ℤ), f (2 * a) + 2 * f b = f (f (a + b))\nm : ℤ\nH : ∀ (b : ℤ), f (b + 1) = f b + m\nb : ℤ\n⊢ f b = f 0 + m * b",
"after_state": "case hz\nf : ℤ → ℤ\nhf : ∀ (a b : ℤ), f (2 * a) + 2 * f b = f (f (a + b))\nm : ℤ\nH : ∀ (b : ℤ), f (b + 1) = f b + m\n⊢ f 0 = f 0 + m * 0\n---\ncase hp\nf : ℤ → ℤ\nhf : ∀ (a b : ℤ), f (2 * a) + 2 * f b = f (f (a + b))\nm : ℤ\nH : ∀ (b : ℤ), f (b + 1) = f b + m\nb : ℕ\nihb : f ↑b = f 0 + m * ↑b\n⊢ f (↑b + 1) = f 0 + m * (↑b + 1)\n---\ncase hn\nf : ℤ → ℤ\nhf : ∀ (a b : ℤ), f (2 * a) + 2 * f b = f (f (a + b))\nm : ℤ\nH : ∀ (b : ℤ), f (b + 1) = f b + m\nb : ℕ\nihb : f (-↑b) = f 0 + m * -↑b\n⊢ f (-↑b - 1) = f 0 + m * (-↑b - 1)"
},
{
"line": "simp",
"before_state": "case hz\nf : ℤ → ℤ\nhf : ∀ (a b : ℤ), f (2 * a) + 2 * f b = f (f (a + b))\nm : ℤ\nH : ∀ (b : ℤ), f (b + 1) = f b + m\n⊢ f 0 = f 0 + m * 0",
"after_state": "No Goals!"
},
{
"line": "simp [H, ihb, mul_add, add_assoc]",
"before_state": "case hp\nf : ℤ → ℤ\nhf : ∀ (a b : ℤ), f (2 * a) + 2 * f b = f (f (a + b))\nm : ℤ\nH : ∀ (b : ℤ), f (b + 1) = f b + m\nb : ℕ\nihb : f ↑b = f 0 + m * ↑b\n⊢ f (↑b + 1) = f 0 + m * (↑b + 1)",
"after_state": "No Goals!"
},
{
"line": "rw [← sub_eq_of_eq_add (H _)]",
"before_state": "case hn\nf : ℤ → ℤ\nhf : ∀ (a b : ℤ), f (2 * a) + 2 * f b = f (f (a + b))\nm : ℤ\nH : ∀ (b : ℤ), f (b + 1) = f b + m\nb : ℕ\nihb : f (-↑b) = f 0 + m * -↑b\n⊢ f (-↑b - 1) = f 0 + m * (-↑b - 1)",
"after_state": "case hn\nf : ℤ → ℤ\nhf : ∀ (a b : ℤ), f (2 * a) + 2 * f b = f (f (a + b))\nm : ℤ\nH : ∀ (b : ℤ), f (b + 1) = f b + m\nb : ℕ\nihb : f (-↑b) = f 0 + m * -↑b\n⊢ f (-↑b - 1 + 1) - m = f 0 + m * (-↑b - 1)"
},
{
"line": "rewrite [← sub_eq_of_eq_add (H _)]",
"before_state": "case hn\nf : ℤ → ℤ\nhf : ∀ (a b : ℤ), f (2 * a) + 2 * f b = f (f (a + b))\nm : ℤ\nH : ∀ (b : ℤ), f (b + 1) = f b + m\nb : ℕ\nihb : f (-↑b) = f 0 + m * -↑b\n⊢ f (-↑b - 1) = f 0 + m * (-↑b - 1)",
"after_state": "case hn\nf : ℤ → ℤ\nhf : ∀ (a b : ℤ), f (2 * a) + 2 * f b = f (f (a + b))\nm : ℤ\nH : ∀ (b : ℤ), f (b + 1) = f b + m\nb : ℕ\nihb : f (-↑b) = f 0 + m * -↑b\n⊢ f (-↑b - 1 + 1) - m = f 0 + m * (-↑b - 1)"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "case hn\nf : ℤ → ℤ\nhf : ∀ (a b : ℤ), f (2 * a) + 2 * f b = f (f (a + b))\nm : ℤ\nH : ∀ (b : ℤ), f (b + 1) = f b + m\nb : ℕ\nihb : f (-↑b) = f 0 + m * -↑b\n⊢ f (-↑b - 1 + 1) - m = f 0 + m * (-↑b - 1)",
"after_state": "case hn\nf : ℤ → ℤ\nhf : ∀ (a b : ℤ), f (2 * a) + 2 * f b = f (f (a + b))\nm : ℤ\nH : ∀ (b : ℤ), f (b + 1) = f b + m\nb : ℕ\nihb : f (-↑b) = f 0 + m * -↑b\n⊢ f (-↑b - 1 + 1) - m = f 0 + m * (-↑b - 1)"
},
{
"line": "try (with_reducible rfl)",
"before_state": "case hn\nf : ℤ → ℤ\nhf : ∀ (a b : ℤ), f (2 * a) + 2 * f b = f (f (a + b))\nm : ℤ\nH : ∀ (b : ℤ), f (b + 1) = f b + m\nb : ℕ\nihb : f (-↑b) = f 0 + m * -↑b\n⊢ f (-↑b - 1 + 1) - m = f 0 + m * (-↑b - 1)",
"after_state": "case hn\nf : ℤ → ℤ\nhf : ∀ (a b : ℤ), f (2 * a) + 2 * f b = f (f (a + b))\nm : ℤ\nH : ∀ (b : ℤ), f (b + 1) = f b + m\nb : ℕ\nihb : f (-↑b) = f 0 + m * -↑b\n⊢ f (-↑b - 1 + 1) - m = f 0 + m * (-↑b - 1)"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "case hn\nf : ℤ → ℤ\nhf : ∀ (a b : ℤ), f (2 * a) + 2 * f b = f (f (a + b))\nm : ℤ\nH : ∀ (b : ℤ), f (b + 1) = f b + m\nb : ℕ\nihb : f (-↑b) = f 0 + m * -↑b\n⊢ f (-↑b - 1 + 1) - m = f 0 + m * (-↑b - 1)",
"after_state": "case hn\nf : ℤ → ℤ\nhf : ∀ (a b : ℤ), f (2 * a) + 2 * f b = f (f (a + b))\nm : ℤ\nH : ∀ (b : ℤ), f (b + 1) = f b + m\nb : ℕ\nihb : f (-↑b) = f 0 + m * -↑b\n⊢ f (-↑b - 1 + 1) - m = f 0 + m * (-↑b - 1)"
},
{
"line": "with_reducible rfl",
"before_state": "case hn\nf : ℤ → ℤ\nhf : ∀ (a b : ℤ), f (2 * a) + 2 * f b = f (f (a + b))\nm : ℤ\nH : ∀ (b : ℤ), f (b + 1) = f b + m\nb : ℕ\nihb : f (-↑b) = f 0 + m * -↑b\n⊢ f (-↑b - 1 + 1) - m = f 0 + m * (-↑b - 1)",
"after_state": "case hn\nf : ℤ → ℤ\nhf : ∀ (a b : ℤ), f (2 * a) + 2 * f b = f (f (a + b))\nm : ℤ\nH : ∀ (b : ℤ), f (b + 1) = f b + m\nb : ℕ\nihb : f (-↑b) = f 0 + m * -↑b\n⊢ f (-↑b - 1 + 1) - m = f 0 + m * (-↑b - 1)"
},
{
"line": "rfl",
"before_state": "case hn\nf : ℤ → ℤ\nhf : ∀ (a b : ℤ), f (2 * a) + 2 * f b = f (f (a + b))\nm : ℤ\nH : ∀ (b : ℤ), f (b + 1) = f b + m\nb : ℕ\nihb : f (-↑b) = f 0 + m * -↑b\n⊢ f (-↑b - 1 + 1) - m = f 0 + m * (-↑b - 1)",
"after_state": "case hn\nf : ℤ → ℤ\nhf : ∀ (a b : ℤ), f (2 * a) + 2 * f b = f (f (a + b))\nm : ℤ\nH : ∀ (b : ℤ), f (b + 1) = f b + m\nb : ℕ\nihb : f (-↑b) = f 0 + m * -↑b\n⊢ f (-↑b - 1 + 1) - m = f 0 + m * (-↑b - 1)"
},
{
"line": "apply_rfl",
"before_state": "case hn\nf : ℤ → ℤ\nhf : ∀ (a b : ℤ), f (2 * a) + 2 * f b = f (f (a + b))\nm : ℤ\nH : ∀ (b : ℤ), f (b + 1) = f b + m\nb : ℕ\nihb : f (-↑b) = f 0 + m * -↑b\n⊢ f (-↑b - 1 + 1) - m = f 0 + m * (-↑b - 1)",
"after_state": "case hn\nf : ℤ → ℤ\nhf : ∀ (a b : ℤ), f (2 * a) + 2 * f b = f (f (a + b))\nm : ℤ\nH : ∀ (b : ℤ), f (b + 1) = f b + m\nb : ℕ\nihb : f (-↑b) = f 0 + m * -↑b\n⊢ f (-↑b - 1 + 1) - m = f 0 + m * (-↑b - 1)"
},
{
"line": "skip",
"before_state": "case hn\nf : ℤ → ℤ\nhf : ∀ (a b : ℤ), f (2 * a) + 2 * f b = f (f (a + b))\nm : ℤ\nH : ∀ (b : ℤ), f (b + 1) = f b + m\nb : ℕ\nihb : f (-↑b) = f 0 + m * -↑b\n⊢ f (-↑b - 1 + 1) - m = f 0 + m * (-↑b - 1)",
"after_state": "case hn\nf : ℤ → ℤ\nhf : ∀ (a b : ℤ), f (2 * a) + 2 * f b = f (f (a + b))\nm : ℤ\nH : ∀ (b : ℤ), f (b + 1) = f b + m\nb : ℕ\nihb : f (-↑b) = f 0 + m * -↑b\n⊢ f (-↑b - 1 + 1) - m = f 0 + m * (-↑b - 1)"
},
{
"line": "simp [ihb]",
"before_state": "case hn\nf : ℤ → ℤ\nhf : ∀ (a b : ℤ), f (2 * a) + 2 * f b = f (f (a + b))\nm : ℤ\nH : ∀ (b : ℤ), f (b + 1) = f b + m\nb : ℕ\nihb : f (-↑b) = f 0 + m * -↑b\n⊢ f (-↑b - 1 + 1) - m = f 0 + m * (-↑b - 1)",
"after_state": "case hn\nf : ℤ → ℤ\nhf : ∀ (a b : ℤ), f (2 * a) + 2 * f b = f (f (a + b))\nm : ℤ\nH : ∀ (b : ℤ), f (b + 1) = f b + m\nb : ℕ\nihb : f (-↑b) = f 0 + m * -↑b\n⊢ f 0 + -(m * ↑b) - m = f 0 + m * (-↑b - 1)"
},
{
"line": "ring\n -- Now use `hf 0 0` and `hf 0 1` to show that `m ∈ {0, 2}`",
"before_state": "case hn\nf : ℤ → ℤ\nhf : ∀ (a b : ℤ), f (2 * a) + 2 * f b = f (f (a + b))\nm : ℤ\nH : ∀ (b : ℤ), f (b + 1) = f b + m\nb : ℕ\nihb : f (-↑b) = f 0 + m * -↑b\n⊢ f 0 + -(m * ↑b) - m = f 0 + m * (-↑b - 1)",
"after_state": "No Goals!"
},
{
"line": "first\n| ring1\n|\n try_this ring_nf\"\\n\\nThe `ring` tactic failed to close the goal. Use `ring_nf` to obtain a normal form.\n \\nNote that `ring` works primarily in *commutative* rings. \\\n If you have a noncommutative ring, abelian group or module, consider using \\\n `noncomm_ring`, `abel` or `module` instead.\"",
"before_state": "case hn\nf : ℤ → ℤ\nhf : ∀ (a b : ℤ), f (2 * a) + 2 * f b = f (f (a + b))\nm : ℤ\nH : ∀ (b : ℤ), f (b + 1) = f b + m\nb : ℕ\nihb : f (-↑b) = f 0 + m * -↑b\n⊢ f 0 + -(m * ↑b) - m = f 0 + m * (-↑b - 1)",
"after_state": "No Goals!"
},
{
"line": "ring1",
"before_state": "case hn\nf : ℤ → ℤ\nhf : ∀ (a b : ℤ), f (2 * a) + 2 * f b = f (f (a + b))\nm : ℤ\nH : ∀ (b : ℤ), f (b + 1) = f b + m\nb : ℕ\nihb : f (-↑b) = f 0 + m * -↑b\n⊢ f 0 + -(m * ↑b) - m = f 0 + m * (-↑b - 1)",
"after_state": "No Goals!"
},
{
"line": "have H3 : 2 * c = m * c := by simpa [H, mul_add] using hf 0 0",
"before_state": "case mp.intro.intro\nf : ℤ → ℤ\nhf : ∀ (a b : ℤ), f (2 * a) + 2 * f b = f (f (a + b))\nm : ℤ\nH✝ : ∀ (b : ℤ), f (b + 1) = f b + m\nc : ℤ\nH : ∀ (b : ℤ), f b = c + m * b\n⊢ f = 0 ∨ ∃ c, f = fun x => 2 * x + c",
"after_state": "case mp.intro.intro\nf : ℤ → ℤ\nhf : ∀ (a b : ℤ), f (2 * a) + 2 * f b = f (f (a + b))\nm : ℤ\nH✝ : ∀ (b : ℤ), f (b + 1) = f b + m\nc : ℤ\nH : ∀ (b : ℤ), f b = c + m * b\nH3 : 2 * c = m * c\n⊢ f = 0 ∨ ∃ c, f = fun x => 2 * x + c"
},
{
"line": "focus\n refine\n no_implicit_lambda%\n (have H3 : 2 * c = m * c := ?body✝;\n ?_)\n case body✝ => with_annotate_state\"by\" (simpa [H, mul_add] using hf 0 0)",
"before_state": "case mp.intro.intro\nf : ℤ → ℤ\nhf : ∀ (a b : ℤ), f (2 * a) + 2 * f b = f (f (a + b))\nm : ℤ\nH✝ : ∀ (b : ℤ), f (b + 1) = f b + m\nc : ℤ\nH : ∀ (b : ℤ), f b = c + m * b\n⊢ f = 0 ∨ ∃ c, f = fun x => 2 * x + c",
"after_state": "case mp.intro.intro\nf : ℤ → ℤ\nhf : ∀ (a b : ℤ), f (2 * a) + 2 * f b = f (f (a + b))\nm : ℤ\nH✝ : ∀ (b : ℤ), f (b + 1) = f b + m\nc : ℤ\nH : ∀ (b : ℤ), f b = c + m * b\nH3 : 2 * c = m * c\n⊢ f = 0 ∨ ∃ c, f = fun x => 2 * x + c"
},
{
"line": "refine\n no_implicit_lambda%\n (have H3 : 2 * c = m * c := ?body✝;\n ?_)",
"before_state": "case mp.intro.intro\nf : ℤ → ℤ\nhf : ∀ (a b : ℤ), f (2 * a) + 2 * f b = f (f (a + b))\nm : ℤ\nH✝ : ∀ (b : ℤ), f (b + 1) = f b + m\nc : ℤ\nH : ∀ (b : ℤ), f b = c + m * b\n⊢ f = 0 ∨ ∃ c, f = fun x => 2 * x + c",
"after_state": "case body\nf : ℤ → ℤ\nhf : ∀ (a b : ℤ), f (2 * a) + 2 * f b = f (f (a + b))\nm : ℤ\nH✝ : ∀ (b : ℤ), f (b + 1) = f b + m\nc : ℤ\nH : ∀ (b : ℤ), f b = c + m * b\n⊢ 2 * c = m * c\n---\ncase mp.intro.intro\nf : ℤ → ℤ\nhf : ∀ (a b : ℤ), f (2 * a) + 2 * f b = f (f (a + b))\nm : ℤ\nH✝ : ∀ (b : ℤ), f (b + 1) = f b + m\nc : ℤ\nH : ∀ (b : ℤ), f b = c + m * b\nH3 : 2 * c = m * c\n⊢ f = 0 ∨ ∃ c, f = fun x => 2 * x + c"
},
{
"line": "case body✝ => with_annotate_state\"by\" (simpa [H, mul_add] using hf 0 0)",
"before_state": "case body\nf : ℤ → ℤ\nhf : ∀ (a b : ℤ), f (2 * a) + 2 * f b = f (f (a + b))\nm : ℤ\nH✝ : ∀ (b : ℤ), f (b + 1) = f b + m\nc : ℤ\nH : ∀ (b : ℤ), f b = c + m * b\n⊢ 2 * c = m * c\n---\ncase mp.intro.intro\nf : ℤ → ℤ\nhf : ∀ (a b : ℤ), f (2 * a) + 2 * f b = f (f (a + b))\nm : ℤ\nH✝ : ∀ (b : ℤ), f (b + 1) = f b + m\nc : ℤ\nH : ∀ (b : ℤ), f b = c + m * b\nH3 : 2 * c = m * c\n⊢ f = 0 ∨ ∃ c, f = fun x => 2 * x + c",
"after_state": "case mp.intro.intro\nf : ℤ → ℤ\nhf : ∀ (a b : ℤ), f (2 * a) + 2 * f b = f (f (a + b))\nm : ℤ\nH✝ : ∀ (b : ℤ), f (b + 1) = f b + m\nc : ℤ\nH : ∀ (b : ℤ), f b = c + m * b\nH3 : 2 * c = m * c\n⊢ f = 0 ∨ ∃ c, f = fun x => 2 * x + c"
},
{
"line": "with_annotate_state\"by\" (simpa [H, mul_add] using hf 0 0)",
"before_state": "f : ℤ → ℤ\nhf : ∀ (a b : ℤ), f (2 * a) + 2 * f b = f (f (a + b))\nm : ℤ\nH✝ : ∀ (b : ℤ), f (b + 1) = f b + m\nc : ℤ\nH : ∀ (b : ℤ), f b = c + m * b\n⊢ 2 * c = m * c",
"after_state": "No Goals!"
},
{
"line": "simpa [H, mul_add] using hf 0 0",
"before_state": "f : ℤ → ℤ\nhf : ∀ (a b : ℤ), f (2 * a) + 2 * f b = f (f (a + b))\nm : ℤ\nH✝ : ∀ (b : ℤ), f (b + 1) = f b + m\nc : ℤ\nH : ∀ (b : ℤ), f b = c + m * b\n⊢ 2 * c = m * c",
"after_state": "No Goals!"
},
{
"line": "obtain rfl | rfl : 2 = m ∨ m = 0 := by simpa [H, mul_add, H3] using hf 0 1",
"before_state": "case mp.intro.intro\nf : ℤ → ℤ\nhf : ∀ (a b : ℤ), f (2 * a) + 2 * f b = f (f (a + b))\nm : ℤ\nH✝ : ∀ (b : ℤ), f (b + 1) = f b + m\nc : ℤ\nH : ∀ (b : ℤ), f b = c + m * b\nH3 : 2 * c = m * c\n⊢ f = 0 ∨ ∃ c, f = fun x => 2 * x + c",
"after_state": "case mp.intro.intro.inl\nf : ℤ → ℤ\nhf : ∀ (a b : ℤ), f (2 * a) + 2 * f b = f (f (a + b))\nc : ℤ\nH✝ : ∀ (b : ℤ), f (b + 1) = f b + 2\nH : ∀ (b : ℤ), f b = c + 2 * b\nH3 : 2 * c = 2 * c\n⊢ f = 0 ∨ ∃ c, f = fun x => 2 * x + c\n---\ncase mp.intro.intro.inr\nf : ℤ → ℤ\nhf : ∀ (a b : ℤ), f (2 * a) + 2 * f b = f (f (a + b))\nc : ℤ\nH✝ : ∀ (b : ℤ), f (b + 1) = f b + 0\nH : ∀ (b : ℤ), f b = c + 0 * b\nH3 : 2 * c = 0 * c\n⊢ f = 0 ∨ ∃ c, f = fun x => 2 * x + c"
},
{
"line": "simpa [H, mul_add, H3] using hf 0 1",
"before_state": "f : ℤ → ℤ\nhf : ∀ (a b : ℤ), f (2 * a) + 2 * f b = f (f (a + b))\nm : ℤ\nH✝ : ∀ (b : ℤ), f (b + 1) = f b + m\nc : ℤ\nH : ∀ (b : ℤ), f b = c + m * b\nH3 : 2 * c = m * c\n⊢ 2 = m ∨ m = 0",
"after_state": "No Goals!"
},
{
"line": "right",
"before_state": "case mp.intro.intro.inl\nf : ℤ → ℤ\nhf : ∀ (a b : ℤ), f (2 * a) + 2 * f b = f (f (a + b))\nc : ℤ\nH✝ : ∀ (b : ℤ), f (b + 1) = f b + 2\nH : ∀ (b : ℤ), f b = c + 2 * b\nH3 : 2 * c = 2 * c\n⊢ f = 0 ∨ ∃ c, f = fun x => 2 * x + c",
"after_state": "case mp.intro.intro.inl.h\nf : ℤ → ℤ\nhf : ∀ (a b : ℤ), f (2 * a) + 2 * f b = f (f (a + b))\nc : ℤ\nH✝ : ∀ (b : ℤ), f (b + 1) = f b + 2\nH : ∀ (b : ℤ), f b = c + 2 * b\nH3 : 2 * c = 2 * c\n⊢ ∃ c, f = fun x => 2 * x + c"
},
{
"line": "use c",
"before_state": "case mp.intro.intro.inl.h\nf : ℤ → ℤ\nhf : ∀ (a b : ℤ), f (2 * a) + 2 * f b = f (f (a + b))\nc : ℤ\nH✝ : ∀ (b : ℤ), f (b + 1) = f b + 2\nH : ∀ (b : ℤ), f b = c + 2 * b\nH3 : 2 * c = 2 * c\n⊢ ∃ c, f = fun x => 2 * x + c",
"after_state": "case h\nf : ℤ → ℤ\nhf : ∀ (a b : ℤ), f (2 * a) + 2 * f b = f (f (a + b))\nc : ℤ\nH✝ : ∀ (b : ℤ), f (b + 1) = f b + 2\nH : ∀ (b : ℤ), f b = c + 2 * b\nH3 : 2 * c = 2 * c\n⊢ f = fun x => 2 * x + c"
},
{
"line": "refine without_cdot(c : ?m✝)",
"before_state": "case w\nf : ℤ → ℤ\nhf : ∀ (a b : ℤ), f (2 * a) + 2 * f b = f (f (a + b))\nc : ℤ\nH✝ : ∀ (b : ℤ), f (b + 1) = f b + 2\nH : ∀ (b : ℤ), f b = c + 2 * b\nH3 : 2 * c = 2 * c\n⊢ ℤ",
"after_state": "No Goals!"
},
{
"line": "try with_reducible use_discharger",
"before_state": "case h\nf : ℤ → ℤ\nhf : ∀ (a b : ℤ), f (2 * a) + 2 * f b = f (f (a + b))\nc : ℤ\nH✝ : ∀ (b : ℤ), f (b + 1) = f b + 2\nH : ∀ (b : ℤ), f b = c + 2 * b\nH3 : 2 * c = 2 * c\n⊢ f = fun x => 2 * x + c",
"after_state": "case h\nf : ℤ → ℤ\nhf : ∀ (a b : ℤ), f (2 * a) + 2 * f b = f (f (a + b))\nc : ℤ\nH✝ : ∀ (b : ℤ), f (b + 1) = f b + 2\nH : ∀ (b : ℤ), f b = c + 2 * b\nH3 : 2 * c = 2 * c\n⊢ f = fun x => 2 * x + c"
},
{
"line": "first\n| with_reducible use_discharger\n| skip",
"before_state": "case h\nf : ℤ → ℤ\nhf : ∀ (a b : ℤ), f (2 * a) + 2 * f b = f (f (a + b))\nc : ℤ\nH✝ : ∀ (b : ℤ), f (b + 1) = f b + 2\nH : ∀ (b : ℤ), f b = c + 2 * b\nH3 : 2 * c = 2 * c\n⊢ f = fun x => 2 * x + c",
"after_state": "case h\nf : ℤ → ℤ\nhf : ∀ (a b : ℤ), f (2 * a) + 2 * f b = f (f (a + b))\nc : ℤ\nH✝ : ∀ (b : ℤ), f (b + 1) = f b + 2\nH : ∀ (b : ℤ), f b = c + 2 * b\nH3 : 2 * c = 2 * c\n⊢ f = fun x => 2 * x + c"
},
{
"line": "with_reducible use_discharger",
"before_state": "case h\nf : ℤ → ℤ\nhf : ∀ (a b : ℤ), f (2 * a) + 2 * f b = f (f (a + b))\nc : ℤ\nH✝ : ∀ (b : ℤ), f (b + 1) = f b + 2\nH : ∀ (b : ℤ), f b = c + 2 * b\nH3 : 2 * c = 2 * c\n⊢ f = fun x => 2 * x + c",
"after_state": "case h\nf : ℤ → ℤ\nhf : ∀ (a b : ℤ), f (2 * a) + 2 * f b = f (f (a + b))\nc : ℤ\nH✝ : ∀ (b : ℤ), f (b + 1) = f b + 2\nH : ∀ (b : ℤ), f b = c + 2 * b\nH3 : 2 * c = 2 * c\n⊢ f = fun x => 2 * x + c"
},
{
"line": "use_discharger",
"before_state": "case h\nf : ℤ → ℤ\nhf : ∀ (a b : ℤ), f (2 * a) + 2 * f b = f (f (a + b))\nc : ℤ\nH✝ : ∀ (b : ℤ), f (b + 1) = f b + 2\nH : ∀ (b : ℤ), f b = c + 2 * b\nH3 : 2 * c = 2 * c\n⊢ f = fun x => 2 * x + c",
"after_state": "case h\nf : ℤ → ℤ\nhf : ∀ (a b : ℤ), f (2 * a) + 2 * f b = f (f (a + b))\nc : ℤ\nH✝ : ∀ (b : ℤ), f (b + 1) = f b + 2\nH : ∀ (b : ℤ), f b = c + 2 * b\nH3 : 2 * c = 2 * c\n⊢ f = fun x => 2 * x + c"
},
{
"line": "focus\n apply exists_prop.mpr✝\n with_annotate_state\"<;>\" skip\n all_goals use_discharger",
"before_state": "case h\nf : ℤ → ℤ\nhf : ∀ (a b : ℤ), f (2 * a) + 2 * f b = f (f (a + b))\nc : ℤ\nH✝ : ∀ (b : ℤ), f (b + 1) = f b + 2\nH : ∀ (b : ℤ), f b = c + 2 * b\nH3 : 2 * c = 2 * c\n⊢ f = fun x => 2 * x + c",
"after_state": "case h\nf : ℤ → ℤ\nhf : ∀ (a b : ℤ), f (2 * a) + 2 * f b = f (f (a + b))\nc : ℤ\nH✝ : ∀ (b : ℤ), f (b + 1) = f b + 2\nH : ∀ (b : ℤ), f b = c + 2 * b\nH3 : 2 * c = 2 * c\n⊢ f = fun x => 2 * x + c"
},
{
"line": "apply exists_prop.mpr✝",
"before_state": "case h\nf : ℤ → ℤ\nhf : ∀ (a b : ℤ), f (2 * a) + 2 * f b = f (f (a + b))\nc : ℤ\nH✝ : ∀ (b : ℤ), f (b + 1) = f b + 2\nH : ∀ (b : ℤ), f b = c + 2 * b\nH3 : 2 * c = 2 * c\n⊢ f = fun x => 2 * x + c",
"after_state": "case h\nf : ℤ → ℤ\nhf : ∀ (a b : ℤ), f (2 * a) + 2 * f b = f (f (a + b))\nc : ℤ\nH✝ : ∀ (b : ℤ), f (b + 1) = f b + 2\nH : ∀ (b : ℤ), f b = c + 2 * b\nH3 : 2 * c = 2 * c\n⊢ f = fun x => 2 * x + c"
},
{
"line": "skip",
"before_state": "case h\nf : ℤ → ℤ\nhf : ∀ (a b : ℤ), f (2 * a) + 2 * f b = f (f (a + b))\nc : ℤ\nH✝ : ∀ (b : ℤ), f (b + 1) = f b + 2\nH : ∀ (b : ℤ), f b = c + 2 * b\nH3 : 2 * c = 2 * c\n⊢ f = fun x => 2 * x + c",
"after_state": "case h\nf : ℤ → ℤ\nhf : ∀ (a b : ℤ), f (2 * a) + 2 * f b = f (f (a + b))\nc : ℤ\nH✝ : ∀ (b : ℤ), f (b + 1) = f b + 2\nH : ∀ (b : ℤ), f b = c + 2 * b\nH3 : 2 * c = 2 * c\n⊢ f = fun x => 2 * x + c"
},
{
"line": "ext b",
"before_state": "case h\nf : ℤ → ℤ\nhf : ∀ (a b : ℤ), f (2 * a) + 2 * f b = f (f (a + b))\nc : ℤ\nH✝ : ∀ (b : ℤ), f (b + 1) = f b + 2\nH : ∀ (b : ℤ), f b = c + 2 * b\nH3 : 2 * c = 2 * c\n⊢ f = fun x => 2 * x + c",
"after_state": "case h.h\nf : ℤ → ℤ\nhf : ∀ (a b : ℤ), f (2 * a) + 2 * f b = f (f (a + b))\nc : ℤ\nH✝ : ∀ (b : ℤ), f (b + 1) = f b + 2\nH : ∀ (b : ℤ), f b = c + 2 * b\nH3 : 2 * c = 2 * c\nb : ℤ\n⊢ f b = 2 * b + c"
},
{
"line": "simp [H, add_comm]",
"before_state": "case h.h\nf : ℤ → ℤ\nhf : ∀ (a b : ℤ), f (2 * a) + 2 * f b = f (f (a + b))\nc : ℤ\nH✝ : ∀ (b : ℤ), f (b + 1) = f b + 2\nH : ∀ (b : ℤ), f b = c + 2 * b\nH3 : 2 * c = 2 * c\nb : ℤ\n⊢ f b = 2 * b + c",
"after_state": "No Goals!"
},
{
"line": "left",
"before_state": "case mp.intro.intro.inr\nf : ℤ → ℤ\nhf : ∀ (a b : ℤ), f (2 * a) + 2 * f b = f (f (a + b))\nc : ℤ\nH✝ : ∀ (b : ℤ), f (b + 1) = f b + 0\nH : ∀ (b : ℤ), f b = c + 0 * b\nH3 : 2 * c = 0 * c\n⊢ f = 0 ∨ ∃ c, f = fun x => 2 * x + c",
"after_state": "case mp.intro.intro.inr.h\nf : ℤ → ℤ\nhf : ∀ (a b : ℤ), f (2 * a) + 2 * f b = f (f (a + b))\nc : ℤ\nH✝ : ∀ (b : ℤ), f (b + 1) = f b + 0\nH : ∀ (b : ℤ), f b = c + 0 * b\nH3 : 2 * c = 0 * c\n⊢ f = 0"
},
{
"line": "ext b",
"before_state": "case mp.intro.intro.inr.h\nf : ℤ → ℤ\nhf : ∀ (a b : ℤ), f (2 * a) + 2 * f b = f (f (a + b))\nc : ℤ\nH✝ : ∀ (b : ℤ), f (b + 1) = f b + 0\nH : ∀ (b : ℤ), f b = c + 0 * b\nH3 : 2 * c = 0 * c\n⊢ f = 0",
"after_state": "case mp.intro.intro.inr.h.h\nf : ℤ → ℤ\nhf : ∀ (a b : ℤ), f (2 * a) + 2 * f b = f (f (a + b))\nc : ℤ\nH✝ : ∀ (b : ℤ), f (b + 1) = f b + 0\nH : ∀ (b : ℤ), f b = c + 0 * b\nH3 : 2 * c = 0 * c\nb : ℤ\n⊢ f b = 0 b"
},
{
"line": "simpa [H, two_ne_zero] using H3",
"before_state": "case mp.intro.intro.inr.h.h\nf : ℤ → ℤ\nhf : ∀ (a b : ℤ), f (2 * a) + 2 * f b = f (f (a + b))\nc : ℤ\nH✝ : ∀ (b : ℤ), f (b + 1) = f b + 0\nH : ∀ (b : ℤ), f b = c + 0 * b\nH3 : 2 * c = 0 * c\nb : ℤ\n⊢ f b = 0 b",
"after_state": "No Goals!"
}
] |
theorem upper_bound {k n : ℕ} (hk : k > 0)
(h : (k ! : ℤ) = ∏ i ∈ range n, ((2 : ℤ) ^ n - (2 : ℤ) ^ i)) : n < 6 := by
have h2 : ∑ i ∈ range n, i < k := by
suffices emultiplicity 2 (k ! : ℤ) = ↑(∑ i ∈ range n, i : ℕ) by
rw [← Nat.cast_lt (α := ℕ∞)]; change emultiplicity ((2 : ℕ) : ℤ) _ < _
rw [← this]; change emultiplicity ((2 : ℕ) : ℤ) _ < _
simp_rw [Int.natCast_emultiplicity, emultiplicity_two_factorial_lt hk.lt.ne.symm]
rw [h]
rw [Finset.emultiplicity_prod Int.prime_two]
rw [Nat.cast_sum]
apply sum_congr rfl; intro i hi
rw [emultiplicity_sub_of_gt]
rw [emultiplicity_pow_self_of_prime Int.prime_two]
rwa [emultiplicity_pow_self_of_prime Int.prime_two,
emultiplicity_pow_self_of_prime Int.prime_two, Nat.cast_lt, ← mem_range]
rw [← not_le]; intro hn
apply _root_.ne_of_gt _ h
calc ∏ i ∈ range n, ((2:ℤ) ^ n - (2:ℤ) ^ i) ≤ ∏ __ ∈ range n, (2:ℤ) ^ n := ?_
_ < ↑ k ! := ?_
· gcongr
· intro i hi
simp only [mem_range] at hi
have : (2:ℤ) ^ i ≤ (2:ℤ) ^ n := by gcongr; norm_num
linarith
· apply sub_le_self
positivity
norm_cast
calc ∏ __ ∈ range n, 2 ^ n = 2 ^ (n * n) := by rw [prod_const, card_range, ← pow_mul]
_ < (∑ i ∈ range n, i)! := ?_
_ ≤ k ! := by gcongr
clear h h2
induction' n, hn using Nat.le_induction with n' hn' IH
· decide
let A := ∑ i ∈ range n', i
have le_sum : ∑ i ∈ range 6, i ≤ A := by
apply sum_le_sum_of_subset
simpa using hn'
calc 2 ^ ((n' + 1) * (n' + 1))
≤ 2 ^ (n' * n' + 4 * n') := by gcongr <;> linarith
_ = 2 ^ (n' * n') * (2 ^ 4) ^ n' := by rw [← pow_mul, ← pow_add]
_ < A ! * (2 ^ 4) ^ n' := by gcongr
_ = A ! * (15 + 1) ^ n' := rfl
_ ≤ A ! * (A + 1) ^ n' := by gcongr; exact le_sum
_ ≤ (A + n')! := factorial_mul_pow_le_factorial
_ = (∑ i ∈ range (n' + 1), i)! := by rw [sum_range_succ]
|
/root/DuelModelResearch/mathlib4/Archive/Imo/Imo2019Q4.lean
|
{
"open": [
"scoped Nat",
"Nat hiding zero_le Prime",
"Finset"
],
"variables": []
}
|
[
{
"line": "have h2 : ∑ i ∈ range n, i < k :=\n by\n suffices emultiplicity 2 (k ! : ℤ) = ↑(∑ i ∈ range n, i : ℕ)\n by\n rw [← Nat.cast_lt (α := ℕ∞)]; change emultiplicity ((2 : ℕ) : ℤ) _ < _\n rw [← this]; change emultiplicity ((2 : ℕ) : ℤ) _ < _\n simp_rw [Int.natCast_emultiplicity, emultiplicity_two_factorial_lt hk.lt.ne.symm]\n rw [h]\n rw [Finset.emultiplicity_prod Int.prime_two]\n rw [Nat.cast_sum]\n apply sum_congr rfl; intro i hi\n rw [emultiplicity_sub_of_gt]\n rw [emultiplicity_pow_self_of_prime Int.prime_two]\n rwa [emultiplicity_pow_self_of_prime Int.prime_two, emultiplicity_pow_self_of_prime Int.prime_two, Nat.cast_lt, ←\n mem_range]",
"before_state": "k n : ℕ\nhk : k > 0\nh : ↑k ! = ∏ i ∈ range n, (2 ^ n - 2 ^ i)\n⊢ n < 6",
"after_state": "k n : ℕ\nhk : k > 0\nh : ↑k ! = ∏ i ∈ range n, (2 ^ n - 2 ^ i)\nh2 : ∑ i ∈ range n, i < k\n⊢ n < 6"
},
{
"line": "focus\n refine\n no_implicit_lambda%\n (have h2 : ∑ i ∈ range n, i < k := ?body✝;\n ?_)\n case body✝ =>\n with_annotate_state\"by\"\n ( suffices emultiplicity 2 (k ! : ℤ) = ↑(∑ i ∈ range n, i : ℕ)\n by\n rw [← Nat.cast_lt (α := ℕ∞)]; change emultiplicity ((2 : ℕ) : ℤ) _ < _\n rw [← this]; change emultiplicity ((2 : ℕ) : ℤ) _ < _\n simp_rw [Int.natCast_emultiplicity, emultiplicity_two_factorial_lt hk.lt.ne.symm]\n rw [h]\n rw [Finset.emultiplicity_prod Int.prime_two]\n rw [Nat.cast_sum]\n apply sum_congr rfl; intro i hi\n rw [emultiplicity_sub_of_gt]\n rw [emultiplicity_pow_self_of_prime Int.prime_two]\n rwa [emultiplicity_pow_self_of_prime Int.prime_two, emultiplicity_pow_self_of_prime Int.prime_two, Nat.cast_lt,\n ← mem_range])",
"before_state": "k n : ℕ\nhk : k > 0\nh : ↑k ! = ∏ i ∈ range n, (2 ^ n - 2 ^ i)\n⊢ n < 6",
"after_state": "k n : ℕ\nhk : k > 0\nh : ↑k ! = ∏ i ∈ range n, (2 ^ n - 2 ^ i)\nh2 : ∑ i ∈ range n, i < k\n⊢ n < 6"
},
{
"line": "refine\n no_implicit_lambda%\n (have h2 : ∑ i ∈ range n, i < k := ?body✝;\n ?_)",
"before_state": "k n : ℕ\nhk : k > 0\nh : ↑k ! = ∏ i ∈ range n, (2 ^ n - 2 ^ i)\n⊢ n < 6",
"after_state": "case body\nk n : ℕ\nhk : k > 0\nh : ↑k ! = ∏ i ∈ range n, (2 ^ n - 2 ^ i)\n⊢ ∑ i ∈ range n, i < k\n---\nk n : ℕ\nhk : k > 0\nh : ↑k ! = ∏ i ∈ range n, (2 ^ n - 2 ^ i)\nh2 : ∑ i ∈ range n, i < k\n⊢ n < 6"
},
{
"line": "case body✝ =>\n with_annotate_state\"by\"\n ( suffices emultiplicity 2 (k ! : ℤ) = ↑(∑ i ∈ range n, i : ℕ)\n by\n rw [← Nat.cast_lt (α := ℕ∞)]; change emultiplicity ((2 : ℕ) : ℤ) _ < _\n rw [← this]; change emultiplicity ((2 : ℕ) : ℤ) _ < _\n simp_rw [Int.natCast_emultiplicity, emultiplicity_two_factorial_lt hk.lt.ne.symm]\n rw [h]\n rw [Finset.emultiplicity_prod Int.prime_two]\n rw [Nat.cast_sum]\n apply sum_congr rfl; intro i hi\n rw [emultiplicity_sub_of_gt]\n rw [emultiplicity_pow_self_of_prime Int.prime_two]\n rwa [emultiplicity_pow_self_of_prime Int.prime_two, emultiplicity_pow_self_of_prime Int.prime_two, Nat.cast_lt, ←\n mem_range])",
"before_state": "case body\nk n : ℕ\nhk : k > 0\nh : ↑k ! = ∏ i ∈ range n, (2 ^ n - 2 ^ i)\n⊢ ∑ i ∈ range n, i < k\n---\nk n : ℕ\nhk : k > 0\nh : ↑k ! = ∏ i ∈ range n, (2 ^ n - 2 ^ i)\nh2 : ∑ i ∈ range n, i < k\n⊢ n < 6",
"after_state": "k n : ℕ\nhk : k > 0\nh : ↑k ! = ∏ i ∈ range n, (2 ^ n - 2 ^ i)\nh2 : ∑ i ∈ range n, i < k\n⊢ n < 6"
},
{
"line": "with_annotate_state\"by\"\n ( suffices emultiplicity 2 (k ! : ℤ) = ↑(∑ i ∈ range n, i : ℕ)\n by\n rw [← Nat.cast_lt (α := ℕ∞)]; change emultiplicity ((2 : ℕ) : ℤ) _ < _\n rw [← this]; change emultiplicity ((2 : ℕ) : ℤ) _ < _\n simp_rw [Int.natCast_emultiplicity, emultiplicity_two_factorial_lt hk.lt.ne.symm]\n rw [h]\n rw [Finset.emultiplicity_prod Int.prime_two]\n rw [Nat.cast_sum]\n apply sum_congr rfl; intro i hi\n rw [emultiplicity_sub_of_gt]\n rw [emultiplicity_pow_self_of_prime Int.prime_two]\n rwa [emultiplicity_pow_self_of_prime Int.prime_two, emultiplicity_pow_self_of_prime Int.prime_two, Nat.cast_lt, ←\n mem_range])",
"before_state": "k n : ℕ\nhk : k > 0\nh : ↑k ! = ∏ i ∈ range n, (2 ^ n - 2 ^ i)\n⊢ ∑ i ∈ range n, i < k",
"after_state": "No Goals!"
},
{
"line": "suffices emultiplicity 2 (k ! : ℤ) = ↑(∑ i ∈ range n, i : ℕ)\n by\n rw [← Nat.cast_lt (α := ℕ∞)]; change emultiplicity ((2 : ℕ) : ℤ) _ < _\n rw [← this]; change emultiplicity ((2 : ℕ) : ℤ) _ < _\n simp_rw [Int.natCast_emultiplicity, emultiplicity_two_factorial_lt hk.lt.ne.symm]",
"before_state": "k n : ℕ\nhk : k > 0\nh : ↑k ! = ∏ i ∈ range n, (2 ^ n - 2 ^ i)\n⊢ ∑ i ∈ range n, i < k",
"after_state": "k n : ℕ\nhk : k > 0\nh : ↑k ! = ∏ i ∈ range n, (2 ^ n - 2 ^ i)\n⊢ emultiplicity 2 ↑k ! = ↑(∑ i ∈ range n, i)"
},
{
"line": "refine_lift\n suffices emultiplicity 2 (k ! : ℤ) = ↑(∑ i ∈ range n, i : ℕ)\n by\n rw [← Nat.cast_lt (α := ℕ∞)]; change emultiplicity ((2 : ℕ) : ℤ) _ < _\n rw [← this]; change emultiplicity ((2 : ℕ) : ℤ) _ < _\n simp_rw [Int.natCast_emultiplicity, emultiplicity_two_factorial_lt hk.lt.ne.symm];\n ?_",
"before_state": "k n : ℕ\nhk : k > 0\nh : ↑k ! = ∏ i ∈ range n, (2 ^ n - 2 ^ i)\n⊢ ∑ i ∈ range n, i < k",
"after_state": "k n : ℕ\nhk : k > 0\nh : ↑k ! = ∏ i ∈ range n, (2 ^ n - 2 ^ i)\n⊢ emultiplicity 2 ↑k ! = ↑(∑ i ∈ range n, i)"
},
{
"line": "focus\n (refine\n no_implicit_lambda%\n (suffices emultiplicity 2 (k ! : ℤ) = ↑(∑ i ∈ range n, i : ℕ)\n by\n rw [← Nat.cast_lt (α := ℕ∞)]; change emultiplicity ((2 : ℕ) : ℤ) _ < _\n rw [← this]; change emultiplicity ((2 : ℕ) : ℤ) _ < _\n simp_rw [Int.natCast_emultiplicity, emultiplicity_two_factorial_lt hk.lt.ne.symm];\n ?_);\n rotate_right)",
"before_state": "k n : ℕ\nhk : k > 0\nh : ↑k ! = ∏ i ∈ range n, (2 ^ n - 2 ^ i)\n⊢ ∑ i ∈ range n, i < k",
"after_state": "k n : ℕ\nhk : k > 0\nh : ↑k ! = ∏ i ∈ range n, (2 ^ n - 2 ^ i)\n⊢ emultiplicity 2 ↑k ! = ↑(∑ i ∈ range n, i)"
},
{
"line": "refine\n no_implicit_lambda%\n (suffices emultiplicity 2 (k ! : ℤ) = ↑(∑ i ∈ range n, i : ℕ)\n by\n rw [← Nat.cast_lt (α := ℕ∞)]; change emultiplicity ((2 : ℕ) : ℤ) _ < _\n rw [← this]; change emultiplicity ((2 : ℕ) : ℤ) _ < _\n simp_rw [Int.natCast_emultiplicity, emultiplicity_two_factorial_lt hk.lt.ne.symm];\n ?_)",
"before_state": "k n : ℕ\nhk : k > 0\nh : ↑k ! = ∏ i ∈ range n, (2 ^ n - 2 ^ i)\n⊢ ∑ i ∈ range n, i < k",
"after_state": "k n : ℕ\nhk : k > 0\nh : ↑k ! = ∏ i ∈ range n, (2 ^ n - 2 ^ i)\n⊢ emultiplicity 2 ↑k ! = ↑(∑ i ∈ range n, i)"
},
{
"line": "rw [← Nat.cast_lt (α := ℕ∞)]",
"before_state": "k n : ℕ\nhk : k > 0\nh : ↑k ! = ∏ i ∈ range n, (2 ^ n - 2 ^ i)\nthis : emultiplicity 2 ↑k ! = ↑(∑ i ∈ range n, i)\n⊢ ∑ i ∈ range n, i < k",
"after_state": "k n : ℕ\nhk : k > 0\nh : ↑k ! = ∏ i ∈ range n, (2 ^ n - 2 ^ i)\nthis : emultiplicity 2 ↑k ! = ↑(∑ i ∈ range n, i)\n⊢ ↑(∑ i ∈ range n, i) < ↑k"
},
{
"line": "rewrite [← Nat.cast_lt (α := ℕ∞)]",
"before_state": "k n : ℕ\nhk : k > 0\nh : ↑k ! = ∏ i ∈ range n, (2 ^ n - 2 ^ i)\nthis : emultiplicity 2 ↑k ! = ↑(∑ i ∈ range n, i)\n⊢ ∑ i ∈ range n, i < k",
"after_state": "k n : ℕ\nhk : k > 0\nh : ↑k ! = ∏ i ∈ range n, (2 ^ n - 2 ^ i)\nthis : emultiplicity 2 ↑k ! = ↑(∑ i ∈ range n, i)\n⊢ ↑(∑ i ∈ range n, i) < ↑k"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "k n : ℕ\nhk : k > 0\nh : ↑k ! = ∏ i ∈ range n, (2 ^ n - 2 ^ i)\nthis : emultiplicity 2 ↑k ! = ↑(∑ i ∈ range n, i)\n⊢ ↑(∑ i ∈ range n, i) < ↑k",
"after_state": "k n : ℕ\nhk : k > 0\nh : ↑k ! = ∏ i ∈ range n, (2 ^ n - 2 ^ i)\nthis : emultiplicity 2 ↑k ! = ↑(∑ i ∈ range n, i)\n⊢ ↑(∑ i ∈ range n, i) < ↑k"
},
{
"line": "try (with_reducible rfl)",
"before_state": "k n : ℕ\nhk : k > 0\nh : ↑k ! = ∏ i ∈ range n, (2 ^ n - 2 ^ i)\nthis : emultiplicity 2 ↑k ! = ↑(∑ i ∈ range n, i)\n⊢ ↑(∑ i ∈ range n, i) < ↑k",
"after_state": "k n : ℕ\nhk : k > 0\nh : ↑k ! = ∏ i ∈ range n, (2 ^ n - 2 ^ i)\nthis : emultiplicity 2 ↑k ! = ↑(∑ i ∈ range n, i)\n⊢ ↑(∑ i ∈ range n, i) < ↑k"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "k n : ℕ\nhk : k > 0\nh : ↑k ! = ∏ i ∈ range n, (2 ^ n - 2 ^ i)\nthis : emultiplicity 2 ↑k ! = ↑(∑ i ∈ range n, i)\n⊢ ↑(∑ i ∈ range n, i) < ↑k",
"after_state": "k n : ℕ\nhk : k > 0\nh : ↑k ! = ∏ i ∈ range n, (2 ^ n - 2 ^ i)\nthis : emultiplicity 2 ↑k ! = ↑(∑ i ∈ range n, i)\n⊢ ↑(∑ i ∈ range n, i) < ↑k"
},
{
"line": "with_reducible rfl",
"before_state": "k n : ℕ\nhk : k > 0\nh : ↑k ! = ∏ i ∈ range n, (2 ^ n - 2 ^ i)\nthis : emultiplicity 2 ↑k ! = ↑(∑ i ∈ range n, i)\n⊢ ↑(∑ i ∈ range n, i) < ↑k",
"after_state": "k n : ℕ\nhk : k > 0\nh : ↑k ! = ∏ i ∈ range n, (2 ^ n - 2 ^ i)\nthis : emultiplicity 2 ↑k ! = ↑(∑ i ∈ range n, i)\n⊢ ↑(∑ i ∈ range n, i) < ↑k"
},
{
"line": "rfl",
"before_state": "k n : ℕ\nhk : k > 0\nh : ↑k ! = ∏ i ∈ range n, (2 ^ n - 2 ^ i)\nthis : emultiplicity 2 ↑k ! = ↑(∑ i ∈ range n, i)\n⊢ ↑(∑ i ∈ range n, i) < ↑k",
"after_state": "k n : ℕ\nhk : k > 0\nh : ↑k ! = ∏ i ∈ range n, (2 ^ n - 2 ^ i)\nthis : emultiplicity 2 ↑k ! = ↑(∑ i ∈ range n, i)\n⊢ ↑(∑ i ∈ range n, i) < ↑k"
},
{
"line": "apply_rfl",
"before_state": "k n : ℕ\nhk : k > 0\nh : ↑k ! = ∏ i ∈ range n, (2 ^ n - 2 ^ i)\nthis : emultiplicity 2 ↑k ! = ↑(∑ i ∈ range n, i)\n⊢ ↑(∑ i ∈ range n, i) < ↑k",
"after_state": "k n : ℕ\nhk : k > 0\nh : ↑k ! = ∏ i ∈ range n, (2 ^ n - 2 ^ i)\nthis : emultiplicity 2 ↑k ! = ↑(∑ i ∈ range n, i)\n⊢ ↑(∑ i ∈ range n, i) < ↑k"
},
{
"line": "skip",
"before_state": "k n : ℕ\nhk : k > 0\nh : ↑k ! = ∏ i ∈ range n, (2 ^ n - 2 ^ i)\nthis : emultiplicity 2 ↑k ! = ↑(∑ i ∈ range n, i)\n⊢ ↑(∑ i ∈ range n, i) < ↑k",
"after_state": "k n : ℕ\nhk : k > 0\nh : ↑k ! = ∏ i ∈ range n, (2 ^ n - 2 ^ i)\nthis : emultiplicity 2 ↑k ! = ↑(∑ i ∈ range n, i)\n⊢ ↑(∑ i ∈ range n, i) < ↑k"
},
{
"line": "change emultiplicity ((2 : ℕ) : ℤ) _ < _",
"before_state": "k n : ℕ\nhk : k > 0\nh : ↑k ! = ∏ i ∈ range n, (2 ^ n - 2 ^ i)\nthis : emultiplicity 2 ↑k ! = ↑(∑ i ∈ range n, i)\n⊢ ↑(∑ i ∈ range n, i) < ↑k",
"after_state": "k n : ℕ\nhk : k > 0\nh : ↑k ! = ∏ i ∈ range n, (2 ^ n - 2 ^ i)\nthis : emultiplicity 2 ↑k ! = ↑(∑ i ∈ range n, i)\n⊢ ↑(∑ i ∈ range n, i) < ↑k"
},
{
"line": "rotate_right",
"before_state": "k n : ℕ\nhk : k > 0\nh : ↑k ! = ∏ i ∈ range n, (2 ^ n - 2 ^ i)\n⊢ emultiplicity 2 ↑k ! = ↑(∑ i ∈ range n, i)",
"after_state": "k n : ℕ\nhk : k > 0\nh : ↑k ! = ∏ i ∈ range n, (2 ^ n - 2 ^ i)\n⊢ emultiplicity 2 ↑k ! = ↑(∑ i ∈ range n, i)"
},
{
"line": "rw [h]",
"before_state": "k n : ℕ\nhk : k > 0\nh : ↑k ! = ∏ i ∈ range n, (2 ^ n - 2 ^ i)\n⊢ emultiplicity 2 ↑k ! = ↑(∑ i ∈ range n, i)",
"after_state": "k n : ℕ\nhk : k > 0\nh : ↑k ! = ∏ i ∈ range n, (2 ^ n - 2 ^ i)\n⊢ emultiplicity 2 (∏ i ∈ range n, (2 ^ n - 2 ^ i)) = ↑(∑ i ∈ range n, i)"
},
{
"line": "rewrite [h]",
"before_state": "k n : ℕ\nhk : k > 0\nh : ↑k ! = ∏ i ∈ range n, (2 ^ n - 2 ^ i)\n⊢ emultiplicity 2 ↑k ! = ↑(∑ i ∈ range n, i)",
"after_state": "k n : ℕ\nhk : k > 0\nh : ↑k ! = ∏ i ∈ range n, (2 ^ n - 2 ^ i)\n⊢ emultiplicity 2 (∏ i ∈ range n, (2 ^ n - 2 ^ i)) = ↑(∑ i ∈ range n, i)"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "k n : ℕ\nhk : k > 0\nh : ↑k ! = ∏ i ∈ range n, (2 ^ n - 2 ^ i)\n⊢ emultiplicity 2 (∏ i ∈ range n, (2 ^ n - 2 ^ i)) = ↑(∑ i ∈ range n, i)",
"after_state": "k n : ℕ\nhk : k > 0\nh : ↑k ! = ∏ i ∈ range n, (2 ^ n - 2 ^ i)\n⊢ emultiplicity 2 (∏ i ∈ range n, (2 ^ n - 2 ^ i)) = ↑(∑ i ∈ range n, i)"
},
{
"line": "try (with_reducible rfl)",
"before_state": "k n : ℕ\nhk : k > 0\nh : ↑k ! = ∏ i ∈ range n, (2 ^ n - 2 ^ i)\n⊢ emultiplicity 2 (∏ i ∈ range n, (2 ^ n - 2 ^ i)) = ↑(∑ i ∈ range n, i)",
"after_state": "k n : ℕ\nhk : k > 0\nh : ↑k ! = ∏ i ∈ range n, (2 ^ n - 2 ^ i)\n⊢ emultiplicity 2 (∏ i ∈ range n, (2 ^ n - 2 ^ i)) = ↑(∑ i ∈ range n, i)"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "k n : ℕ\nhk : k > 0\nh : ↑k ! = ∏ i ∈ range n, (2 ^ n - 2 ^ i)\n⊢ emultiplicity 2 (∏ i ∈ range n, (2 ^ n - 2 ^ i)) = ↑(∑ i ∈ range n, i)",
"after_state": "k n : ℕ\nhk : k > 0\nh : ↑k ! = ∏ i ∈ range n, (2 ^ n - 2 ^ i)\n⊢ emultiplicity 2 (∏ i ∈ range n, (2 ^ n - 2 ^ i)) = ↑(∑ i ∈ range n, i)"
},
{
"line": "with_reducible rfl",
"before_state": "k n : ℕ\nhk : k > 0\nh : ↑k ! = ∏ i ∈ range n, (2 ^ n - 2 ^ i)\n⊢ emultiplicity 2 (∏ i ∈ range n, (2 ^ n - 2 ^ i)) = ↑(∑ i ∈ range n, i)",
"after_state": "k n : ℕ\nhk : k > 0\nh : ↑k ! = ∏ i ∈ range n, (2 ^ n - 2 ^ i)\n⊢ emultiplicity 2 (∏ i ∈ range n, (2 ^ n - 2 ^ i)) = ↑(∑ i ∈ range n, i)"
},
{
"line": "rfl",
"before_state": "k n : ℕ\nhk : k > 0\nh : ↑k ! = ∏ i ∈ range n, (2 ^ n - 2 ^ i)\n⊢ emultiplicity 2 (∏ i ∈ range n, (2 ^ n - 2 ^ i)) = ↑(∑ i ∈ range n, i)",
"after_state": "k n : ℕ\nhk : k > 0\nh : ↑k ! = ∏ i ∈ range n, (2 ^ n - 2 ^ i)\n⊢ emultiplicity 2 (∏ i ∈ range n, (2 ^ n - 2 ^ i)) = ↑(∑ i ∈ range n, i)"
},
{
"line": "apply_rfl",
"before_state": "k n : ℕ\nhk : k > 0\nh : ↑k ! = ∏ i ∈ range n, (2 ^ n - 2 ^ i)\n⊢ emultiplicity 2 (∏ i ∈ range n, (2 ^ n - 2 ^ i)) = ↑(∑ i ∈ range n, i)",
"after_state": "k n : ℕ\nhk : k > 0\nh : ↑k ! = ∏ i ∈ range n, (2 ^ n - 2 ^ i)\n⊢ emultiplicity 2 (∏ i ∈ range n, (2 ^ n - 2 ^ i)) = ↑(∑ i ∈ range n, i)"
},
{
"line": "skip",
"before_state": "k n : ℕ\nhk : k > 0\nh : ↑k ! = ∏ i ∈ range n, (2 ^ n - 2 ^ i)\n⊢ emultiplicity 2 (∏ i ∈ range n, (2 ^ n - 2 ^ i)) = ↑(∑ i ∈ range n, i)",
"after_state": "k n : ℕ\nhk : k > 0\nh : ↑k ! = ∏ i ∈ range n, (2 ^ n - 2 ^ i)\n⊢ emultiplicity 2 (∏ i ∈ range n, (2 ^ n - 2 ^ i)) = ↑(∑ i ∈ range n, i)"
},
{
"line": "rw [Finset.emultiplicity_prod Int.prime_two]",
"before_state": "k n : ℕ\nhk : k > 0\nh : ↑k ! = ∏ i ∈ range n, (2 ^ n - 2 ^ i)\n⊢ emultiplicity 2 (∏ i ∈ range n, (2 ^ n - 2 ^ i)) = ↑(∑ i ∈ range n, i)",
"after_state": "k n : ℕ\nhk : k > 0\nh : ↑k ! = ∏ i ∈ range n, (2 ^ n - 2 ^ i)\n⊢ ∑ x ∈ range n, emultiplicity 2 (2 ^ n - 2 ^ x) = ↑(∑ i ∈ range n, i)"
},
{
"line": "rewrite [Finset.emultiplicity_prod Int.prime_two]",
"before_state": "k n : ℕ\nhk : k > 0\nh : ↑k ! = ∏ i ∈ range n, (2 ^ n - 2 ^ i)\n⊢ emultiplicity 2 (∏ i ∈ range n, (2 ^ n - 2 ^ i)) = ↑(∑ i ∈ range n, i)",
"after_state": "k n : ℕ\nhk : k > 0\nh : ↑k ! = ∏ i ∈ range n, (2 ^ n - 2 ^ i)\n⊢ ∑ x ∈ range n, emultiplicity 2 (2 ^ n - 2 ^ x) = ↑(∑ i ∈ range n, i)"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "k n : ℕ\nhk : k > 0\nh : ↑k ! = ∏ i ∈ range n, (2 ^ n - 2 ^ i)\n⊢ ∑ x ∈ range n, emultiplicity 2 (2 ^ n - 2 ^ x) = ↑(∑ i ∈ range n, i)",
"after_state": "k n : ℕ\nhk : k > 0\nh : ↑k ! = ∏ i ∈ range n, (2 ^ n - 2 ^ i)\n⊢ ∑ x ∈ range n, emultiplicity 2 (2 ^ n - 2 ^ x) = ↑(∑ i ∈ range n, i)"
},
{
"line": "try (with_reducible rfl)",
"before_state": "k n : ℕ\nhk : k > 0\nh : ↑k ! = ∏ i ∈ range n, (2 ^ n - 2 ^ i)\n⊢ ∑ x ∈ range n, emultiplicity 2 (2 ^ n - 2 ^ x) = ↑(∑ i ∈ range n, i)",
"after_state": "k n : ℕ\nhk : k > 0\nh : ↑k ! = ∏ i ∈ range n, (2 ^ n - 2 ^ i)\n⊢ ∑ x ∈ range n, emultiplicity 2 (2 ^ n - 2 ^ x) = ↑(∑ i ∈ range n, i)"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "k n : ℕ\nhk : k > 0\nh : ↑k ! = ∏ i ∈ range n, (2 ^ n - 2 ^ i)\n⊢ ∑ x ∈ range n, emultiplicity 2 (2 ^ n - 2 ^ x) = ↑(∑ i ∈ range n, i)",
"after_state": "k n : ℕ\nhk : k > 0\nh : ↑k ! = ∏ i ∈ range n, (2 ^ n - 2 ^ i)\n⊢ ∑ x ∈ range n, emultiplicity 2 (2 ^ n - 2 ^ x) = ↑(∑ i ∈ range n, i)"
},
{
"line": "with_reducible rfl",
"before_state": "k n : ℕ\nhk : k > 0\nh : ↑k ! = ∏ i ∈ range n, (2 ^ n - 2 ^ i)\n⊢ ∑ x ∈ range n, emultiplicity 2 (2 ^ n - 2 ^ x) = ↑(∑ i ∈ range n, i)",
"after_state": "k n : ℕ\nhk : k > 0\nh : ↑k ! = ∏ i ∈ range n, (2 ^ n - 2 ^ i)\n⊢ ∑ x ∈ range n, emultiplicity 2 (2 ^ n - 2 ^ x) = ↑(∑ i ∈ range n, i)"
},
{
"line": "rfl",
"before_state": "k n : ℕ\nhk : k > 0\nh : ↑k ! = ∏ i ∈ range n, (2 ^ n - 2 ^ i)\n⊢ ∑ x ∈ range n, emultiplicity 2 (2 ^ n - 2 ^ x) = ↑(∑ i ∈ range n, i)",
"after_state": "k n : ℕ\nhk : k > 0\nh : ↑k ! = ∏ i ∈ range n, (2 ^ n - 2 ^ i)\n⊢ ∑ x ∈ range n, emultiplicity 2 (2 ^ n - 2 ^ x) = ↑(∑ i ∈ range n, i)"
},
{
"line": "apply_rfl",
"before_state": "k n : ℕ\nhk : k > 0\nh : ↑k ! = ∏ i ∈ range n, (2 ^ n - 2 ^ i)\n⊢ ∑ x ∈ range n, emultiplicity 2 (2 ^ n - 2 ^ x) = ↑(∑ i ∈ range n, i)",
"after_state": "k n : ℕ\nhk : k > 0\nh : ↑k ! = ∏ i ∈ range n, (2 ^ n - 2 ^ i)\n⊢ ∑ x ∈ range n, emultiplicity 2 (2 ^ n - 2 ^ x) = ↑(∑ i ∈ range n, i)"
},
{
"line": "skip",
"before_state": "k n : ℕ\nhk : k > 0\nh : ↑k ! = ∏ i ∈ range n, (2 ^ n - 2 ^ i)\n⊢ ∑ x ∈ range n, emultiplicity 2 (2 ^ n - 2 ^ x) = ↑(∑ i ∈ range n, i)",
"after_state": "k n : ℕ\nhk : k > 0\nh : ↑k ! = ∏ i ∈ range n, (2 ^ n - 2 ^ i)\n⊢ ∑ x ∈ range n, emultiplicity 2 (2 ^ n - 2 ^ x) = ↑(∑ i ∈ range n, i)"
},
{
"line": "rw [Nat.cast_sum]",
"before_state": "k n : ℕ\nhk : k > 0\nh : ↑k ! = ∏ i ∈ range n, (2 ^ n - 2 ^ i)\n⊢ ∑ x ∈ range n, emultiplicity 2 (2 ^ n - 2 ^ x) = ↑(∑ i ∈ range n, i)",
"after_state": "k n : ℕ\nhk : k > 0\nh : ↑k ! = ∏ i ∈ range n, (2 ^ n - 2 ^ i)\n⊢ ∑ x ∈ range n, emultiplicity 2 (2 ^ n - 2 ^ x) = ∑ x ∈ range n, ↑x"
},
{
"line": "rewrite [Nat.cast_sum]",
"before_state": "k n : ℕ\nhk : k > 0\nh : ↑k ! = ∏ i ∈ range n, (2 ^ n - 2 ^ i)\n⊢ ∑ x ∈ range n, emultiplicity 2 (2 ^ n - 2 ^ x) = ↑(∑ i ∈ range n, i)",
"after_state": "k n : ℕ\nhk : k > 0\nh : ↑k ! = ∏ i ∈ range n, (2 ^ n - 2 ^ i)\n⊢ ∑ x ∈ range n, emultiplicity 2 (2 ^ n - 2 ^ x) = ∑ x ∈ range n, ↑x"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "k n : ℕ\nhk : k > 0\nh : ↑k ! = ∏ i ∈ range n, (2 ^ n - 2 ^ i)\n⊢ ∑ x ∈ range n, emultiplicity 2 (2 ^ n - 2 ^ x) = ∑ x ∈ range n, ↑x",
"after_state": "k n : ℕ\nhk : k > 0\nh : ↑k ! = ∏ i ∈ range n, (2 ^ n - 2 ^ i)\n⊢ ∑ x ∈ range n, emultiplicity 2 (2 ^ n - 2 ^ x) = ∑ x ∈ range n, ↑x"
},
{
"line": "try (with_reducible rfl)",
"before_state": "k n : ℕ\nhk : k > 0\nh : ↑k ! = ∏ i ∈ range n, (2 ^ n - 2 ^ i)\n⊢ ∑ x ∈ range n, emultiplicity 2 (2 ^ n - 2 ^ x) = ∑ x ∈ range n, ↑x",
"after_state": "k n : ℕ\nhk : k > 0\nh : ↑k ! = ∏ i ∈ range n, (2 ^ n - 2 ^ i)\n⊢ ∑ x ∈ range n, emultiplicity 2 (2 ^ n - 2 ^ x) = ∑ x ∈ range n, ↑x"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "k n : ℕ\nhk : k > 0\nh : ↑k ! = ∏ i ∈ range n, (2 ^ n - 2 ^ i)\n⊢ ∑ x ∈ range n, emultiplicity 2 (2 ^ n - 2 ^ x) = ∑ x ∈ range n, ↑x",
"after_state": "k n : ℕ\nhk : k > 0\nh : ↑k ! = ∏ i ∈ range n, (2 ^ n - 2 ^ i)\n⊢ ∑ x ∈ range n, emultiplicity 2 (2 ^ n - 2 ^ x) = ∑ x ∈ range n, ↑x"
},
{
"line": "with_reducible rfl",
"before_state": "k n : ℕ\nhk : k > 0\nh : ↑k ! = ∏ i ∈ range n, (2 ^ n - 2 ^ i)\n⊢ ∑ x ∈ range n, emultiplicity 2 (2 ^ n - 2 ^ x) = ∑ x ∈ range n, ↑x",
"after_state": "k n : ℕ\nhk : k > 0\nh : ↑k ! = ∏ i ∈ range n, (2 ^ n - 2 ^ i)\n⊢ ∑ x ∈ range n, emultiplicity 2 (2 ^ n - 2 ^ x) = ∑ x ∈ range n, ↑x"
},
{
"line": "rfl",
"before_state": "k n : ℕ\nhk : k > 0\nh : ↑k ! = ∏ i ∈ range n, (2 ^ n - 2 ^ i)\n⊢ ∑ x ∈ range n, emultiplicity 2 (2 ^ n - 2 ^ x) = ∑ x ∈ range n, ↑x",
"after_state": "k n : ℕ\nhk : k > 0\nh : ↑k ! = ∏ i ∈ range n, (2 ^ n - 2 ^ i)\n⊢ ∑ x ∈ range n, emultiplicity 2 (2 ^ n - 2 ^ x) = ∑ x ∈ range n, ↑x"
},
{
"line": "apply_rfl",
"before_state": "k n : ℕ\nhk : k > 0\nh : ↑k ! = ∏ i ∈ range n, (2 ^ n - 2 ^ i)\n⊢ ∑ x ∈ range n, emultiplicity 2 (2 ^ n - 2 ^ x) = ∑ x ∈ range n, ↑x",
"after_state": "k n : ℕ\nhk : k > 0\nh : ↑k ! = ∏ i ∈ range n, (2 ^ n - 2 ^ i)\n⊢ ∑ x ∈ range n, emultiplicity 2 (2 ^ n - 2 ^ x) = ∑ x ∈ range n, ↑x"
},
{
"line": "skip",
"before_state": "k n : ℕ\nhk : k > 0\nh : ↑k ! = ∏ i ∈ range n, (2 ^ n - 2 ^ i)\n⊢ ∑ x ∈ range n, emultiplicity 2 (2 ^ n - 2 ^ x) = ∑ x ∈ range n, ↑x",
"after_state": "k n : ℕ\nhk : k > 0\nh : ↑k ! = ∏ i ∈ range n, (2 ^ n - 2 ^ i)\n⊢ ∑ x ∈ range n, emultiplicity 2 (2 ^ n - 2 ^ x) = ∑ x ∈ range n, ↑x"
},
{
"line": "apply sum_congr rfl",
"before_state": "k n : ℕ\nhk : k > 0\nh : ↑k ! = ∏ i ∈ range n, (2 ^ n - 2 ^ i)\n⊢ ∑ x ∈ range n, emultiplicity 2 (2 ^ n - 2 ^ x) = ∑ x ∈ range n, ↑x",
"after_state": "k n : ℕ\nhk : k > 0\nh : ↑k ! = ∏ i ∈ range n, (2 ^ n - 2 ^ i)\n⊢ ∀ x ∈ range n, emultiplicity 2 (2 ^ n - 2 ^ x) = ↑x"
},
{
"line": "intro i hi",
"before_state": "k n : ℕ\nhk : k > 0\nh : ↑k ! = ∏ i ∈ range n, (2 ^ n - 2 ^ i)\n⊢ ∀ x ∈ range n, emultiplicity 2 (2 ^ n - 2 ^ x) = ↑x",
"after_state": "k n : ℕ\nhk : k > 0\nh : ↑k ! = ∏ i ∈ range n, (2 ^ n - 2 ^ i)\ni : ℕ\nhi : i ∈ range n\n⊢ emultiplicity 2 (2 ^ n - 2 ^ i) = ↑i"
},
{
"line": "intro i;\n intro hi",
"before_state": "k n : ℕ\nhk : k > 0\nh : ↑k ! = ∏ i ∈ range n, (2 ^ n - 2 ^ i)\n⊢ ∀ x ∈ range n, emultiplicity 2 (2 ^ n - 2 ^ x) = ↑x",
"after_state": "k n : ℕ\nhk : k > 0\nh : ↑k ! = ∏ i ∈ range n, (2 ^ n - 2 ^ i)\ni : ℕ\nhi : i ∈ range n\n⊢ emultiplicity 2 (2 ^ n - 2 ^ i) = ↑i"
},
{
"line": "intro i",
"before_state": "k n : ℕ\nhk : k > 0\nh : ↑k ! = ∏ i ∈ range n, (2 ^ n - 2 ^ i)\n⊢ ∀ x ∈ range n, emultiplicity 2 (2 ^ n - 2 ^ x) = ↑x",
"after_state": "k n : ℕ\nhk : k > 0\nh : ↑k ! = ∏ i ∈ range n, (2 ^ n - 2 ^ i)\ni : ℕ\n⊢ i ∈ range n → emultiplicity 2 (2 ^ n - 2 ^ i) = ↑i"
},
{
"line": "intro hi",
"before_state": "k n : ℕ\nhk : k > 0\nh : ↑k ! = ∏ i ∈ range n, (2 ^ n - 2 ^ i)\ni : ℕ\n⊢ i ∈ range n → emultiplicity 2 (2 ^ n - 2 ^ i) = ↑i",
"after_state": "k n : ℕ\nhk : k > 0\nh : ↑k ! = ∏ i ∈ range n, (2 ^ n - 2 ^ i)\ni : ℕ\nhi : i ∈ range n\n⊢ emultiplicity 2 (2 ^ n - 2 ^ i) = ↑i"
},
{
"line": "rw [emultiplicity_sub_of_gt]",
"before_state": "k n : ℕ\nhk : k > 0\nh : ↑k ! = ∏ i ∈ range n, (2 ^ n - 2 ^ i)\ni : ℕ\nhi : i ∈ range n\n⊢ emultiplicity 2 (2 ^ n - 2 ^ i) = ↑i",
"after_state": "k n : ℕ\nhk : k > 0\nh : ↑k ! = ∏ i ∈ range n, (2 ^ n - 2 ^ i)\ni : ℕ\nhi : i ∈ range n\n⊢ emultiplicity 2 (2 ^ i) = ↑i\n---\nk n : ℕ\nhk : k > 0\nh : ↑k ! = ∏ i ∈ range n, (2 ^ n - 2 ^ i)\ni : ℕ\nhi : i ∈ range n\n⊢ emultiplicity 2 (2 ^ i) < emultiplicity 2 (2 ^ n)"
},
{
"line": "rewrite [emultiplicity_sub_of_gt]",
"before_state": "k n : ℕ\nhk : k > 0\nh : ↑k ! = ∏ i ∈ range n, (2 ^ n - 2 ^ i)\ni : ℕ\nhi : i ∈ range n\n⊢ emultiplicity 2 (2 ^ n - 2 ^ i) = ↑i",
"after_state": "k n : ℕ\nhk : k > 0\nh : ↑k ! = ∏ i ∈ range n, (2 ^ n - 2 ^ i)\ni : ℕ\nhi : i ∈ range n\n⊢ emultiplicity 2 (2 ^ i) = ↑i\n---\nk n : ℕ\nhk : k > 0\nh : ↑k ! = ∏ i ∈ range n, (2 ^ n - 2 ^ i)\ni : ℕ\nhi : i ∈ range n\n⊢ emultiplicity 2 (2 ^ i) < emultiplicity 2 (2 ^ n)"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "k n : ℕ\nhk : k > 0\nh : ↑k ! = ∏ i ∈ range n, (2 ^ n - 2 ^ i)\ni : ℕ\nhi : i ∈ range n\n⊢ emultiplicity 2 (2 ^ i) = ↑i\n---\nk n : ℕ\nhk : k > 0\nh : ↑k ! = ∏ i ∈ range n, (2 ^ n - 2 ^ i)\ni : ℕ\nhi : i ∈ range n\n⊢ emultiplicity 2 (2 ^ i) < emultiplicity 2 (2 ^ n)",
"after_state": "k n : ℕ\nhk : k > 0\nh : ↑k ! = ∏ i ∈ range n, (2 ^ n - 2 ^ i)\ni : ℕ\nhi : i ∈ range n\n⊢ emultiplicity 2 (2 ^ i) = ↑i\n---\nk n : ℕ\nhk : k > 0\nh : ↑k ! = ∏ i ∈ range n, (2 ^ n - 2 ^ i)\ni : ℕ\nhi : i ∈ range n\n⊢ emultiplicity 2 (2 ^ i) < emultiplicity 2 (2 ^ n)"
},
{
"line": "try (with_reducible rfl)",
"before_state": "k n : ℕ\nhk : k > 0\nh : ↑k ! = ∏ i ∈ range n, (2 ^ n - 2 ^ i)\ni : ℕ\nhi : i ∈ range n\n⊢ emultiplicity 2 (2 ^ i) = ↑i\n---\nk n : ℕ\nhk : k > 0\nh : ↑k ! = ∏ i ∈ range n, (2 ^ n - 2 ^ i)\ni : ℕ\nhi : i ∈ range n\n⊢ emultiplicity 2 (2 ^ i) < emultiplicity 2 (2 ^ n)",
"after_state": "k n : ℕ\nhk : k > 0\nh : ↑k ! = ∏ i ∈ range n, (2 ^ n - 2 ^ i)\ni : ℕ\nhi : i ∈ range n\n⊢ emultiplicity 2 (2 ^ i) = ↑i\n---\nk n : ℕ\nhk : k > 0\nh : ↑k ! = ∏ i ∈ range n, (2 ^ n - 2 ^ i)\ni : ℕ\nhi : i ∈ range n\n⊢ emultiplicity 2 (2 ^ i) < emultiplicity 2 (2 ^ n)"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "k n : ℕ\nhk : k > 0\nh : ↑k ! = ∏ i ∈ range n, (2 ^ n - 2 ^ i)\ni : ℕ\nhi : i ∈ range n\n⊢ emultiplicity 2 (2 ^ i) = ↑i\n---\nk n : ℕ\nhk : k > 0\nh : ↑k ! = ∏ i ∈ range n, (2 ^ n - 2 ^ i)\ni : ℕ\nhi : i ∈ range n\n⊢ emultiplicity 2 (2 ^ i) < emultiplicity 2 (2 ^ n)",
"after_state": "k n : ℕ\nhk : k > 0\nh : ↑k ! = ∏ i ∈ range n, (2 ^ n - 2 ^ i)\ni : ℕ\nhi : i ∈ range n\n⊢ emultiplicity 2 (2 ^ i) = ↑i\n---\nk n : ℕ\nhk : k > 0\nh : ↑k ! = ∏ i ∈ range n, (2 ^ n - 2 ^ i)\ni : ℕ\nhi : i ∈ range n\n⊢ emultiplicity 2 (2 ^ i) < emultiplicity 2 (2 ^ n)"
},
{
"line": "with_reducible rfl",
"before_state": "k n : ℕ\nhk : k > 0\nh : ↑k ! = ∏ i ∈ range n, (2 ^ n - 2 ^ i)\ni : ℕ\nhi : i ∈ range n\n⊢ emultiplicity 2 (2 ^ i) = ↑i\n---\nk n : ℕ\nhk : k > 0\nh : ↑k ! = ∏ i ∈ range n, (2 ^ n - 2 ^ i)\ni : ℕ\nhi : i ∈ range n\n⊢ emultiplicity 2 (2 ^ i) < emultiplicity 2 (2 ^ n)",
"after_state": "k n : ℕ\nhk : k > 0\nh : ↑k ! = ∏ i ∈ range n, (2 ^ n - 2 ^ i)\ni : ℕ\nhi : i ∈ range n\n⊢ emultiplicity 2 (2 ^ i) = ↑i\n---\nk n : ℕ\nhk : k > 0\nh : ↑k ! = ∏ i ∈ range n, (2 ^ n - 2 ^ i)\ni : ℕ\nhi : i ∈ range n\n⊢ emultiplicity 2 (2 ^ i) < emultiplicity 2 (2 ^ n)"
},
{
"line": "rfl",
"before_state": "k n : ℕ\nhk : k > 0\nh : ↑k ! = ∏ i ∈ range n, (2 ^ n - 2 ^ i)\ni : ℕ\nhi : i ∈ range n\n⊢ emultiplicity 2 (2 ^ i) = ↑i\n---\nk n : ℕ\nhk : k > 0\nh : ↑k ! = ∏ i ∈ range n, (2 ^ n - 2 ^ i)\ni : ℕ\nhi : i ∈ range n\n⊢ emultiplicity 2 (2 ^ i) < emultiplicity 2 (2 ^ n)",
"after_state": "k n : ℕ\nhk : k > 0\nh : ↑k ! = ∏ i ∈ range n, (2 ^ n - 2 ^ i)\ni : ℕ\nhi : i ∈ range n\n⊢ emultiplicity 2 (2 ^ i) = ↑i\n---\nk n : ℕ\nhk : k > 0\nh : ↑k ! = ∏ i ∈ range n, (2 ^ n - 2 ^ i)\ni : ℕ\nhi : i ∈ range n\n⊢ emultiplicity 2 (2 ^ i) < emultiplicity 2 (2 ^ n)"
},
{
"line": "apply_rfl",
"before_state": "k n : ℕ\nhk : k > 0\nh : ↑k ! = ∏ i ∈ range n, (2 ^ n - 2 ^ i)\ni : ℕ\nhi : i ∈ range n\n⊢ emultiplicity 2 (2 ^ i) = ↑i\n---\nk n : ℕ\nhk : k > 0\nh : ↑k ! = ∏ i ∈ range n, (2 ^ n - 2 ^ i)\ni : ℕ\nhi : i ∈ range n\n⊢ emultiplicity 2 (2 ^ i) < emultiplicity 2 (2 ^ n)",
"after_state": "k n : ℕ\nhk : k > 0\nh : ↑k ! = ∏ i ∈ range n, (2 ^ n - 2 ^ i)\ni : ℕ\nhi : i ∈ range n\n⊢ emultiplicity 2 (2 ^ i) = ↑i\n---\nk n : ℕ\nhk : k > 0\nh : ↑k ! = ∏ i ∈ range n, (2 ^ n - 2 ^ i)\ni : ℕ\nhi : i ∈ range n\n⊢ emultiplicity 2 (2 ^ i) < emultiplicity 2 (2 ^ n)"
},
{
"line": "skip",
"before_state": "k n : ℕ\nhk : k > 0\nh : ↑k ! = ∏ i ∈ range n, (2 ^ n - 2 ^ i)\ni : ℕ\nhi : i ∈ range n\n⊢ emultiplicity 2 (2 ^ i) = ↑i\n---\nk n : ℕ\nhk : k > 0\nh : ↑k ! = ∏ i ∈ range n, (2 ^ n - 2 ^ i)\ni : ℕ\nhi : i ∈ range n\n⊢ emultiplicity 2 (2 ^ i) < emultiplicity 2 (2 ^ n)",
"after_state": "k n : ℕ\nhk : k > 0\nh : ↑k ! = ∏ i ∈ range n, (2 ^ n - 2 ^ i)\ni : ℕ\nhi : i ∈ range n\n⊢ emultiplicity 2 (2 ^ i) = ↑i\n---\nk n : ℕ\nhk : k > 0\nh : ↑k ! = ∏ i ∈ range n, (2 ^ n - 2 ^ i)\ni : ℕ\nhi : i ∈ range n\n⊢ emultiplicity 2 (2 ^ i) < emultiplicity 2 (2 ^ n)"
},
{
"line": "rw [emultiplicity_pow_self_of_prime Int.prime_two]",
"before_state": "k n : ℕ\nhk : k > 0\nh : ↑k ! = ∏ i ∈ range n, (2 ^ n - 2 ^ i)\ni : ℕ\nhi : i ∈ range n\n⊢ emultiplicity 2 (2 ^ i) = ↑i\n---\nk n : ℕ\nhk : k > 0\nh : ↑k ! = ∏ i ∈ range n, (2 ^ n - 2 ^ i)\ni : ℕ\nhi : i ∈ range n\n⊢ emultiplicity 2 (2 ^ i) < emultiplicity 2 (2 ^ n)",
"after_state": "k n : ℕ\nhk : k > 0\nh : ↑k ! = ∏ i ∈ range n, (2 ^ n - 2 ^ i)\ni : ℕ\nhi : i ∈ range n\n⊢ emultiplicity 2 (2 ^ i) < emultiplicity 2 (2 ^ n)"
},
{
"line": "rewrite [emultiplicity_pow_self_of_prime Int.prime_two]",
"before_state": "k n : ℕ\nhk : k > 0\nh : ↑k ! = ∏ i ∈ range n, (2 ^ n - 2 ^ i)\ni : ℕ\nhi : i ∈ range n\n⊢ emultiplicity 2 (2 ^ i) = ↑i\n---\nk n : ℕ\nhk : k > 0\nh : ↑k ! = ∏ i ∈ range n, (2 ^ n - 2 ^ i)\ni : ℕ\nhi : i ∈ range n\n⊢ emultiplicity 2 (2 ^ i) < emultiplicity 2 (2 ^ n)",
"after_state": "k n : ℕ\nhk : k > 0\nh : ↑k ! = ∏ i ∈ range n, (2 ^ n - 2 ^ i)\ni : ℕ\nhi : i ∈ range n\n⊢ ↑i = ↑i\n---\nk n : ℕ\nhk : k > 0\nh : ↑k ! = ∏ i ∈ range n, (2 ^ n - 2 ^ i)\ni : ℕ\nhi : i ∈ range n\n⊢ emultiplicity 2 (2 ^ i) < emultiplicity 2 (2 ^ n)"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "k n : ℕ\nhk : k > 0\nh : ↑k ! = ∏ i ∈ range n, (2 ^ n - 2 ^ i)\ni : ℕ\nhi : i ∈ range n\n⊢ ↑i = ↑i\n---\nk n : ℕ\nhk : k > 0\nh : ↑k ! = ∏ i ∈ range n, (2 ^ n - 2 ^ i)\ni : ℕ\nhi : i ∈ range n\n⊢ emultiplicity 2 (2 ^ i) < emultiplicity 2 (2 ^ n)",
"after_state": "k n : ℕ\nhk : k > 0\nh : ↑k ! = ∏ i ∈ range n, (2 ^ n - 2 ^ i)\ni : ℕ\nhi : i ∈ range n\n⊢ emultiplicity 2 (2 ^ i) < emultiplicity 2 (2 ^ n)"
},
{
"line": "try (with_reducible rfl)",
"before_state": "k n : ℕ\nhk : k > 0\nh : ↑k ! = ∏ i ∈ range n, (2 ^ n - 2 ^ i)\ni : ℕ\nhi : i ∈ range n\n⊢ ↑i = ↑i\n---\nk n : ℕ\nhk : k > 0\nh : ↑k ! = ∏ i ∈ range n, (2 ^ n - 2 ^ i)\ni : ℕ\nhi : i ∈ range n\n⊢ emultiplicity 2 (2 ^ i) < emultiplicity 2 (2 ^ n)",
"after_state": "k n : ℕ\nhk : k > 0\nh : ↑k ! = ∏ i ∈ range n, (2 ^ n - 2 ^ i)\ni : ℕ\nhi : i ∈ range n\n⊢ emultiplicity 2 (2 ^ i) < emultiplicity 2 (2 ^ n)"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "k n : ℕ\nhk : k > 0\nh : ↑k ! = ∏ i ∈ range n, (2 ^ n - 2 ^ i)\ni : ℕ\nhi : i ∈ range n\n⊢ ↑i = ↑i\n---\nk n : ℕ\nhk : k > 0\nh : ↑k ! = ∏ i ∈ range n, (2 ^ n - 2 ^ i)\ni : ℕ\nhi : i ∈ range n\n⊢ emultiplicity 2 (2 ^ i) < emultiplicity 2 (2 ^ n)",
"after_state": "k n : ℕ\nhk : k > 0\nh : ↑k ! = ∏ i ∈ range n, (2 ^ n - 2 ^ i)\ni : ℕ\nhi : i ∈ range n\n⊢ emultiplicity 2 (2 ^ i) < emultiplicity 2 (2 ^ n)"
},
{
"line": "with_reducible rfl",
"before_state": "k n : ℕ\nhk : k > 0\nh : ↑k ! = ∏ i ∈ range n, (2 ^ n - 2 ^ i)\ni : ℕ\nhi : i ∈ range n\n⊢ ↑i = ↑i\n---\nk n : ℕ\nhk : k > 0\nh : ↑k ! = ∏ i ∈ range n, (2 ^ n - 2 ^ i)\ni : ℕ\nhi : i ∈ range n\n⊢ emultiplicity 2 (2 ^ i) < emultiplicity 2 (2 ^ n)",
"after_state": "k n : ℕ\nhk : k > 0\nh : ↑k ! = ∏ i ∈ range n, (2 ^ n - 2 ^ i)\ni : ℕ\nhi : i ∈ range n\n⊢ emultiplicity 2 (2 ^ i) < emultiplicity 2 (2 ^ n)"
},
{
"line": "rfl",
"before_state": "k n : ℕ\nhk : k > 0\nh : ↑k ! = ∏ i ∈ range n, (2 ^ n - 2 ^ i)\ni : ℕ\nhi : i ∈ range n\n⊢ ↑i = ↑i\n---\nk n : ℕ\nhk : k > 0\nh : ↑k ! = ∏ i ∈ range n, (2 ^ n - 2 ^ i)\ni : ℕ\nhi : i ∈ range n\n⊢ emultiplicity 2 (2 ^ i) < emultiplicity 2 (2 ^ n)",
"after_state": "k n : ℕ\nhk : k > 0\nh : ↑k ! = ∏ i ∈ range n, (2 ^ n - 2 ^ i)\ni : ℕ\nhi : i ∈ range n\n⊢ emultiplicity 2 (2 ^ i) < emultiplicity 2 (2 ^ n)"
},
{
"line": "eq_refl",
"before_state": "k n : ℕ\nhk : k > 0\nh : ↑k ! = ∏ i ∈ range n, (2 ^ n - 2 ^ i)\ni : ℕ\nhi : i ∈ range n\n⊢ ↑i = ↑i\n---\nk n : ℕ\nhk : k > 0\nh : ↑k ! = ∏ i ∈ range n, (2 ^ n - 2 ^ i)\ni : ℕ\nhi : i ∈ range n\n⊢ emultiplicity 2 (2 ^ i) < emultiplicity 2 (2 ^ n)",
"after_state": "k n : ℕ\nhk : k > 0\nh : ↑k ! = ∏ i ∈ range n, (2 ^ n - 2 ^ i)\ni : ℕ\nhi : i ∈ range n\n⊢ emultiplicity 2 (2 ^ i) < emultiplicity 2 (2 ^ n)"
},
{
"line": "rwa [emultiplicity_pow_self_of_prime Int.prime_two, emultiplicity_pow_self_of_prime Int.prime_two, Nat.cast_lt, ←\n mem_range]",
"before_state": "k n : ℕ\nhk : k > 0\nh : ↑k ! = ∏ i ∈ range n, (2 ^ n - 2 ^ i)\ni : ℕ\nhi : i ∈ range n\n⊢ emultiplicity 2 (2 ^ i) < emultiplicity 2 (2 ^ n)",
"after_state": "No Goals!"
},
{
"line": "rw [emultiplicity_pow_self_of_prime Int.prime_two, emultiplicity_pow_self_of_prime Int.prime_two, Nat.cast_lt, ←\n mem_range]",
"before_state": "k n : ℕ\nhk : k > 0\nh : ↑k ! = ∏ i ∈ range n, (2 ^ n - 2 ^ i)\ni : ℕ\nhi : i ∈ range n\n⊢ emultiplicity 2 (2 ^ i) < emultiplicity 2 (2 ^ n)",
"after_state": "k n : ℕ\nhk : k > 0\nh : ↑k ! = ∏ i ∈ range n, (2 ^ n - 2 ^ i)\ni : ℕ\nhi : i ∈ range n\n⊢ i ∈ range n"
},
{
"line": "rewrite [emultiplicity_pow_self_of_prime Int.prime_two, emultiplicity_pow_self_of_prime Int.prime_two, Nat.cast_lt, ←\n mem_range]",
"before_state": "k n : ℕ\nhk : k > 0\nh : ↑k ! = ∏ i ∈ range n, (2 ^ n - 2 ^ i)\ni : ℕ\nhi : i ∈ range n\n⊢ emultiplicity 2 (2 ^ i) < emultiplicity 2 (2 ^ n)",
"after_state": "k n : ℕ\nhk : k > 0\nh : ↑k ! = ∏ i ∈ range n, (2 ^ n - 2 ^ i)\ni : ℕ\nhi : i ∈ range n\n⊢ i ∈ range n"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "k n : ℕ\nhk : k > 0\nh : ↑k ! = ∏ i ∈ range n, (2 ^ n - 2 ^ i)\ni : ℕ\nhi : i ∈ range n\n⊢ i ∈ range n",
"after_state": "k n : ℕ\nhk : k > 0\nh : ↑k ! = ∏ i ∈ range n, (2 ^ n - 2 ^ i)\ni : ℕ\nhi : i ∈ range n\n⊢ i ∈ range n"
},
{
"line": "try (with_reducible rfl)",
"before_state": "k n : ℕ\nhk : k > 0\nh : ↑k ! = ∏ i ∈ range n, (2 ^ n - 2 ^ i)\ni : ℕ\nhi : i ∈ range n\n⊢ i ∈ range n",
"after_state": "k n : ℕ\nhk : k > 0\nh : ↑k ! = ∏ i ∈ range n, (2 ^ n - 2 ^ i)\ni : ℕ\nhi : i ∈ range n\n⊢ i ∈ range n"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "k n : ℕ\nhk : k > 0\nh : ↑k ! = ∏ i ∈ range n, (2 ^ n - 2 ^ i)\ni : ℕ\nhi : i ∈ range n\n⊢ i ∈ range n",
"after_state": "k n : ℕ\nhk : k > 0\nh : ↑k ! = ∏ i ∈ range n, (2 ^ n - 2 ^ i)\ni : ℕ\nhi : i ∈ range n\n⊢ i ∈ range n"
},
{
"line": "with_reducible rfl",
"before_state": "k n : ℕ\nhk : k > 0\nh : ↑k ! = ∏ i ∈ range n, (2 ^ n - 2 ^ i)\ni : ℕ\nhi : i ∈ range n\n⊢ i ∈ range n",
"after_state": "k n : ℕ\nhk : k > 0\nh : ↑k ! = ∏ i ∈ range n, (2 ^ n - 2 ^ i)\ni : ℕ\nhi : i ∈ range n\n⊢ i ∈ range n"
},
{
"line": "rfl",
"before_state": "k n : ℕ\nhk : k > 0\nh : ↑k ! = ∏ i ∈ range n, (2 ^ n - 2 ^ i)\ni : ℕ\nhi : i ∈ range n\n⊢ i ∈ range n",
"after_state": "k n : ℕ\nhk : k > 0\nh : ↑k ! = ∏ i ∈ range n, (2 ^ n - 2 ^ i)\ni : ℕ\nhi : i ∈ range n\n⊢ i ∈ range n"
},
{
"line": "apply_rfl",
"before_state": "k n : ℕ\nhk : k > 0\nh : ↑k ! = ∏ i ∈ range n, (2 ^ n - 2 ^ i)\ni : ℕ\nhi : i ∈ range n\n⊢ i ∈ range n",
"after_state": "k n : ℕ\nhk : k > 0\nh : ↑k ! = ∏ i ∈ range n, (2 ^ n - 2 ^ i)\ni : ℕ\nhi : i ∈ range n\n⊢ i ∈ range n"
},
{
"line": "skip",
"before_state": "k n : ℕ\nhk : k > 0\nh : ↑k ! = ∏ i ∈ range n, (2 ^ n - 2 ^ i)\ni : ℕ\nhi : i ∈ range n\n⊢ i ∈ range n",
"after_state": "k n : ℕ\nhk : k > 0\nh : ↑k ! = ∏ i ∈ range n, (2 ^ n - 2 ^ i)\ni : ℕ\nhi : i ∈ range n\n⊢ i ∈ range n"
},
{
"line": "assumption",
"before_state": "k n : ℕ\nhk : k > 0\nh : ↑k ! = ∏ i ∈ range n, (2 ^ n - 2 ^ i)\ni : ℕ\nhi : i ∈ range n\n⊢ i ∈ range n",
"after_state": "No Goals!"
},
{
"line": "rw [← not_le]",
"before_state": "k n : ℕ\nhk : k > 0\nh : ↑k ! = ∏ i ∈ range n, (2 ^ n - 2 ^ i)\nh2 : ∑ i ∈ range n, i < k\n⊢ n < 6",
"after_state": "k n : ℕ\nhk : k > 0\nh : ↑k ! = ∏ i ∈ range n, (2 ^ n - 2 ^ i)\nh2 : ∑ i ∈ range n, i < k\n⊢ ¬6 ≤ n"
},
{
"line": "rewrite [← not_le]",
"before_state": "k n : ℕ\nhk : k > 0\nh : ↑k ! = ∏ i ∈ range n, (2 ^ n - 2 ^ i)\nh2 : ∑ i ∈ range n, i < k\n⊢ n < 6",
"after_state": "k n : ℕ\nhk : k > 0\nh : ↑k ! = ∏ i ∈ range n, (2 ^ n - 2 ^ i)\nh2 : ∑ i ∈ range n, i < k\n⊢ ¬6 ≤ n"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "k n : ℕ\nhk : k > 0\nh : ↑k ! = ∏ i ∈ range n, (2 ^ n - 2 ^ i)\nh2 : ∑ i ∈ range n, i < k\n⊢ ¬6 ≤ n",
"after_state": "k n : ℕ\nhk : k > 0\nh : ↑k ! = ∏ i ∈ range n, (2 ^ n - 2 ^ i)\nh2 : ∑ i ∈ range n, i < k\n⊢ ¬6 ≤ n"
},
{
"line": "try (with_reducible rfl)",
"before_state": "k n : ℕ\nhk : k > 0\nh : ↑k ! = ∏ i ∈ range n, (2 ^ n - 2 ^ i)\nh2 : ∑ i ∈ range n, i < k\n⊢ ¬6 ≤ n",
"after_state": "k n : ℕ\nhk : k > 0\nh : ↑k ! = ∏ i ∈ range n, (2 ^ n - 2 ^ i)\nh2 : ∑ i ∈ range n, i < k\n⊢ ¬6 ≤ n"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "k n : ℕ\nhk : k > 0\nh : ↑k ! = ∏ i ∈ range n, (2 ^ n - 2 ^ i)\nh2 : ∑ i ∈ range n, i < k\n⊢ ¬6 ≤ n",
"after_state": "k n : ℕ\nhk : k > 0\nh : ↑k ! = ∏ i ∈ range n, (2 ^ n - 2 ^ i)\nh2 : ∑ i ∈ range n, i < k\n⊢ ¬6 ≤ n"
},
{
"line": "with_reducible rfl",
"before_state": "k n : ℕ\nhk : k > 0\nh : ↑k ! = ∏ i ∈ range n, (2 ^ n - 2 ^ i)\nh2 : ∑ i ∈ range n, i < k\n⊢ ¬6 ≤ n",
"after_state": "k n : ℕ\nhk : k > 0\nh : ↑k ! = ∏ i ∈ range n, (2 ^ n - 2 ^ i)\nh2 : ∑ i ∈ range n, i < k\n⊢ ¬6 ≤ n"
},
{
"line": "rfl",
"before_state": "k n : ℕ\nhk : k > 0\nh : ↑k ! = ∏ i ∈ range n, (2 ^ n - 2 ^ i)\nh2 : ∑ i ∈ range n, i < k\n⊢ ¬6 ≤ n",
"after_state": "k n : ℕ\nhk : k > 0\nh : ↑k ! = ∏ i ∈ range n, (2 ^ n - 2 ^ i)\nh2 : ∑ i ∈ range n, i < k\n⊢ ¬6 ≤ n"
},
{
"line": "apply_rfl",
"before_state": "k n : ℕ\nhk : k > 0\nh : ↑k ! = ∏ i ∈ range n, (2 ^ n - 2 ^ i)\nh2 : ∑ i ∈ range n, i < k\n⊢ ¬6 ≤ n",
"after_state": "k n : ℕ\nhk : k > 0\nh : ↑k ! = ∏ i ∈ range n, (2 ^ n - 2 ^ i)\nh2 : ∑ i ∈ range n, i < k\n⊢ ¬6 ≤ n"
},
{
"line": "skip",
"before_state": "k n : ℕ\nhk : k > 0\nh : ↑k ! = ∏ i ∈ range n, (2 ^ n - 2 ^ i)\nh2 : ∑ i ∈ range n, i < k\n⊢ ¬6 ≤ n",
"after_state": "k n : ℕ\nhk : k > 0\nh : ↑k ! = ∏ i ∈ range n, (2 ^ n - 2 ^ i)\nh2 : ∑ i ∈ range n, i < k\n⊢ ¬6 ≤ n"
},
{
"line": "intro hn",
"before_state": "k n : ℕ\nhk : k > 0\nh : ↑k ! = ∏ i ∈ range n, (2 ^ n - 2 ^ i)\nh2 : ∑ i ∈ range n, i < k\n⊢ ¬6 ≤ n",
"after_state": "k n : ℕ\nhk : k > 0\nh : ↑k ! = ∏ i ∈ range n, (2 ^ n - 2 ^ i)\nh2 : ∑ i ∈ range n, i < k\nhn : 6 ≤ n\n⊢ False"
},
{
"line": "apply _root_.ne_of_gt _ h",
"before_state": "k n : ℕ\nhk : k > 0\nh : ↑k ! = ∏ i ∈ range n, (2 ^ n - 2 ^ i)\nh2 : ∑ i ∈ range n, i < k\nhn : 6 ≤ n\n⊢ False",
"after_state": "k n : ℕ\nhk : k > 0\nh : ↑k ! = ∏ i ∈ range n, (2 ^ n - 2 ^ i)\nh2 : ∑ i ∈ range n, i < k\nhn : 6 ≤ n\n⊢ ∏ i ∈ range n, (2 ^ n - 2 ^ i) < ↑k !"
},
{
"line": "calc\n ∏ i ∈ range n, ((2 : ℤ) ^ n - (2 : ℤ) ^ i) ≤ ∏ __ ∈ range n, (2 : ℤ) ^ n := ?_\n _ < ↑k ! := ?_",
"before_state": "k n : ℕ\nhk : k > 0\nh : ↑k ! = ∏ i ∈ range n, (2 ^ n - 2 ^ i)\nh2 : ∑ i ∈ range n, i < k\nhn : 6 ≤ n\n⊢ ∏ i ∈ range n, (2 ^ n - 2 ^ i) < ↑k !",
"after_state": "case calc_1\nk n : ℕ\nhk : k > 0\nh : ↑k ! = ∏ i ∈ range n, (2 ^ n - 2 ^ i)\nh2 : ∑ i ∈ range n, i < k\nhn : 6 ≤ n\n⊢ ∏ i ∈ range n, (2 ^ n - 2 ^ i) ≤ ∏ __ ∈ range n, 2 ^ n\n---\ncase calc_2\nk n : ℕ\nhk : k > 0\nh : ↑k ! = ∏ i ∈ range n, (2 ^ n - 2 ^ i)\nh2 : ∑ i ∈ range n, i < k\nhn : 6 ≤ n\n⊢ ∏ __ ∈ range n, 2 ^ n < ↑k !"
},
{
"line": "gcongr",
"before_state": "case calc_1\nk n : ℕ\nhk : k > 0\nh : ↑k ! = ∏ i ∈ range n, (2 ^ n - 2 ^ i)\nh2 : ∑ i ∈ range n, i < k\nhn : 6 ≤ n\n⊢ ∏ i ∈ range n, (2 ^ n - 2 ^ i) ≤ ∏ __ ∈ range n, 2 ^ n",
"after_state": "case calc_1.h0\nk n : ℕ\nhk : k > 0\nh : ↑k ! = ∏ i ∈ range n, (2 ^ n - 2 ^ i)\nh2 : ∑ i ∈ range n, i < k\nhn : 6 ≤ n\n⊢ ∀ i ∈ range n, 0 ≤ 2 ^ n - 2 ^ i\n---\ncase calc_1.h1\nk n : ℕ\nhk : k > 0\nh : ↑k ! = ∏ i ∈ range n, (2 ^ n - 2 ^ i)\nh2 : ∑ i ∈ range n, i < k\nhn : 6 ≤ n\ni✝ : ℕ\na✝ : i✝ ∈ range n\n⊢ 2 ^ n - 2 ^ i✝ ≤ 2 ^ n"
},
{
"line": "intro i hi",
"before_state": "case calc_1.h0\nk n : ℕ\nhk : k > 0\nh : ↑k ! = ∏ i ∈ range n, (2 ^ n - 2 ^ i)\nh2 : ∑ i ∈ range n, i < k\nhn : 6 ≤ n\n⊢ ∀ i ∈ range n, 0 ≤ 2 ^ n - 2 ^ i",
"after_state": "case calc_1.h0\nk n : ℕ\nhk : k > 0\nh : ↑k ! = ∏ i ∈ range n, (2 ^ n - 2 ^ i)\nh2 : ∑ i ∈ range n, i < k\nhn : 6 ≤ n\ni : ℕ\nhi : i ∈ range n\n⊢ 0 ≤ 2 ^ n - 2 ^ i"
},
{
"line": "intro i;\n intro hi",
"before_state": "case calc_1.h0\nk n : ℕ\nhk : k > 0\nh : ↑k ! = ∏ i ∈ range n, (2 ^ n - 2 ^ i)\nh2 : ∑ i ∈ range n, i < k\nhn : 6 ≤ n\n⊢ ∀ i ∈ range n, 0 ≤ 2 ^ n - 2 ^ i",
"after_state": "case calc_1.h0\nk n : ℕ\nhk : k > 0\nh : ↑k ! = ∏ i ∈ range n, (2 ^ n - 2 ^ i)\nh2 : ∑ i ∈ range n, i < k\nhn : 6 ≤ n\ni : ℕ\nhi : i ∈ range n\n⊢ 0 ≤ 2 ^ n - 2 ^ i"
},
{
"line": "intro i",
"before_state": "case calc_1.h0\nk n : ℕ\nhk : k > 0\nh : ↑k ! = ∏ i ∈ range n, (2 ^ n - 2 ^ i)\nh2 : ∑ i ∈ range n, i < k\nhn : 6 ≤ n\n⊢ ∀ i ∈ range n, 0 ≤ 2 ^ n - 2 ^ i",
"after_state": "case calc_1.h0\nk n : ℕ\nhk : k > 0\nh : ↑k ! = ∏ i ∈ range n, (2 ^ n - 2 ^ i)\nh2 : ∑ i ∈ range n, i < k\nhn : 6 ≤ n\ni : ℕ\n⊢ i ∈ range n → 0 ≤ 2 ^ n - 2 ^ i"
},
{
"line": "intro hi",
"before_state": "case calc_1.h0\nk n : ℕ\nhk : k > 0\nh : ↑k ! = ∏ i ∈ range n, (2 ^ n - 2 ^ i)\nh2 : ∑ i ∈ range n, i < k\nhn : 6 ≤ n\ni : ℕ\n⊢ i ∈ range n → 0 ≤ 2 ^ n - 2 ^ i",
"after_state": "case calc_1.h0\nk n : ℕ\nhk : k > 0\nh : ↑k ! = ∏ i ∈ range n, (2 ^ n - 2 ^ i)\nh2 : ∑ i ∈ range n, i < k\nhn : 6 ≤ n\ni : ℕ\nhi : i ∈ range n\n⊢ 0 ≤ 2 ^ n - 2 ^ i"
},
{
"line": "simp only [mem_range] at hi",
"before_state": "case calc_1.h0\nk n : ℕ\nhk : k > 0\nh : ↑k ! = ∏ i ∈ range n, (2 ^ n - 2 ^ i)\nh2 : ∑ i ∈ range n, i < k\nhn : 6 ≤ n\ni : ℕ\nhi : i ∈ range n\n⊢ 0 ≤ 2 ^ n - 2 ^ i",
"after_state": "case calc_1.h0\nk n : ℕ\nhk : k > 0\nh : ↑k ! = ∏ i ∈ range n, (2 ^ n - 2 ^ i)\nh2 : ∑ i ∈ range n, i < k\nhn : 6 ≤ n\ni : ℕ\nhi : i < n\n⊢ 0 ≤ 2 ^ n - 2 ^ i"
},
{
"line": "have : (2 : ℤ) ^ i ≤ (2 : ℤ) ^ n := by gcongr; norm_num",
"before_state": "case calc_1.h0\nk n : ℕ\nhk : k > 0\nh : ↑k ! = ∏ i ∈ range n, (2 ^ n - 2 ^ i)\nh2 : ∑ i ∈ range n, i < k\nhn : 6 ≤ n\ni : ℕ\nhi : i < n\n⊢ 0 ≤ 2 ^ n - 2 ^ i",
"after_state": "case calc_1.h0\nk n : ℕ\nhk : k > 0\nh : ↑k ! = ∏ i ∈ range n, (2 ^ n - 2 ^ i)\nh2 : ∑ i ∈ range n, i < k\nhn : 6 ≤ n\ni : ℕ\nhi : i < n\nthis : 2 ^ i ≤ 2 ^ n\n⊢ 0 ≤ 2 ^ n - 2 ^ i"
},
{
"line": "focus\n refine\n no_implicit_lambda%\n (have : (2 : ℤ) ^ i ≤ (2 : ℤ) ^ n := ?body✝;\n ?_)\n case body✝ => with_annotate_state\"by\" (gcongr; norm_num)",
"before_state": "case calc_1.h0\nk n : ℕ\nhk : k > 0\nh : ↑k ! = ∏ i ∈ range n, (2 ^ n - 2 ^ i)\nh2 : ∑ i ∈ range n, i < k\nhn : 6 ≤ n\ni : ℕ\nhi : i < n\n⊢ 0 ≤ 2 ^ n - 2 ^ i",
"after_state": "case calc_1.h0\nk n : ℕ\nhk : k > 0\nh : ↑k ! = ∏ i ∈ range n, (2 ^ n - 2 ^ i)\nh2 : ∑ i ∈ range n, i < k\nhn : 6 ≤ n\ni : ℕ\nhi : i < n\nthis : 2 ^ i ≤ 2 ^ n\n⊢ 0 ≤ 2 ^ n - 2 ^ i"
},
{
"line": "refine\n no_implicit_lambda%\n (have : (2 : ℤ) ^ i ≤ (2 : ℤ) ^ n := ?body✝;\n ?_)",
"before_state": "case calc_1.h0\nk n : ℕ\nhk : k > 0\nh : ↑k ! = ∏ i ∈ range n, (2 ^ n - 2 ^ i)\nh2 : ∑ i ∈ range n, i < k\nhn : 6 ≤ n\ni : ℕ\nhi : i < n\n⊢ 0 ≤ 2 ^ n - 2 ^ i",
"after_state": "case body\nk n : ℕ\nhk : k > 0\nh : ↑k ! = ∏ i ∈ range n, (2 ^ n - 2 ^ i)\nh2 : ∑ i ∈ range n, i < k\nhn : 6 ≤ n\ni : ℕ\nhi : i < n\n⊢ 2 ^ i ≤ 2 ^ n\n---\ncase calc_1.h0\nk n : ℕ\nhk : k > 0\nh : ↑k ! = ∏ i ∈ range n, (2 ^ n - 2 ^ i)\nh2 : ∑ i ∈ range n, i < k\nhn : 6 ≤ n\ni : ℕ\nhi : i < n\nthis : 2 ^ i ≤ 2 ^ n\n⊢ 0 ≤ 2 ^ n - 2 ^ i"
},
{
"line": "case body✝ => with_annotate_state\"by\" (gcongr; norm_num)",
"before_state": "case body\nk n : ℕ\nhk : k > 0\nh : ↑k ! = ∏ i ∈ range n, (2 ^ n - 2 ^ i)\nh2 : ∑ i ∈ range n, i < k\nhn : 6 ≤ n\ni : ℕ\nhi : i < n\n⊢ 2 ^ i ≤ 2 ^ n\n---\ncase calc_1.h0\nk n : ℕ\nhk : k > 0\nh : ↑k ! = ∏ i ∈ range n, (2 ^ n - 2 ^ i)\nh2 : ∑ i ∈ range n, i < k\nhn : 6 ≤ n\ni : ℕ\nhi : i < n\nthis : 2 ^ i ≤ 2 ^ n\n⊢ 0 ≤ 2 ^ n - 2 ^ i",
"after_state": "case calc_1.h0\nk n : ℕ\nhk : k > 0\nh : ↑k ! = ∏ i ∈ range n, (2 ^ n - 2 ^ i)\nh2 : ∑ i ∈ range n, i < k\nhn : 6 ≤ n\ni : ℕ\nhi : i < n\nthis : 2 ^ i ≤ 2 ^ n\n⊢ 0 ≤ 2 ^ n - 2 ^ i"
},
{
"line": "with_annotate_state\"by\" (gcongr; norm_num)",
"before_state": "k n : ℕ\nhk : k > 0\nh : ↑k ! = ∏ i ∈ range n, (2 ^ n - 2 ^ i)\nh2 : ∑ i ∈ range n, i < k\nhn : 6 ≤ n\ni : ℕ\nhi : i < n\n⊢ 2 ^ i ≤ 2 ^ n",
"after_state": "No Goals!"
},
{
"line": "gcongr",
"before_state": "k n : ℕ\nhk : k > 0\nh : ↑k ! = ∏ i ∈ range n, (2 ^ n - 2 ^ i)\nh2 : ∑ i ∈ range n, i < k\nhn : 6 ≤ n\ni : ℕ\nhi : i < n\n⊢ 2 ^ i ≤ 2 ^ n",
"after_state": "case ha\nk n : ℕ\nhk : k > 0\nh : ↑k ! = ∏ i ∈ range n, (2 ^ n - 2 ^ i)\nh2 : ∑ i ∈ range n, i < k\nhn : 6 ≤ n\ni : ℕ\nhi : i < n\n⊢ 1 ≤ 2"
},
{
"line": "norm_num",
"before_state": "case ha\nk n : ℕ\nhk : k > 0\nh : ↑k ! = ∏ i ∈ range n, (2 ^ n - 2 ^ i)\nh2 : ∑ i ∈ range n, i < k\nhn : 6 ≤ n\ni : ℕ\nhi : i < n\n⊢ 1 ≤ 2",
"after_state": "No Goals!"
},
{
"line": "linarith",
"before_state": "case calc_1.h0\nk n : ℕ\nhk : k > 0\nh : ↑k ! = ∏ i ∈ range n, (2 ^ n - 2 ^ i)\nh2 : ∑ i ∈ range n, i < k\nhn : 6 ≤ n\ni : ℕ\nhi : i < n\nthis : 2 ^ i ≤ 2 ^ n\n⊢ 0 ≤ 2 ^ n - 2 ^ i",
"after_state": "No Goals!"
},
{
"line": "ring1",
"before_state": "k n : ℕ\nhk : k > 0\nh : ↑k ! = ∏ i ∈ range n, (2 ^ n - 2 ^ i)\nh2 : ∑ i ∈ range n, i < k\nhn : 6 ≤ n\ni : ℕ\nhi : i < n\nthis : 2 ^ i ≤ 2 ^ n\na✝ : 0 > 2 ^ n - 2 ^ i\n⊢ -1 + (2 ^ i - 2 ^ n) + (2 ^ n - 2 ^ i + 1 - 0) = 0",
"after_state": "No Goals!"
},
{
"line": "apply sub_le_self",
"before_state": "case calc_1.h1\nk n : ℕ\nhk : k > 0\nh : ↑k ! = ∏ i ∈ range n, (2 ^ n - 2 ^ i)\nh2 : ∑ i ∈ range n, i < k\nhn : 6 ≤ n\ni✝ : ℕ\na✝ : i✝ ∈ range n\n⊢ 2 ^ n - 2 ^ i✝ ≤ 2 ^ n",
"after_state": "case calc_1.h1.a\nk n : ℕ\nhk : k > 0\nh : ↑k ! = ∏ i ∈ range n, (2 ^ n - 2 ^ i)\nh2 : ∑ i ∈ range n, i < k\nhn : 6 ≤ n\ni✝ : ℕ\na✝ : i✝ ∈ range n\n⊢ 0 ≤ 2 ^ i✝"
},
{
"line": "positivity",
"before_state": "case calc_1.h1.a\nk n : ℕ\nhk : k > 0\nh : ↑k ! = ∏ i ∈ range n, (2 ^ n - 2 ^ i)\nh2 : ∑ i ∈ range n, i < k\nhn : 6 ≤ n\ni✝ : ℕ\na✝ : i✝ ∈ range n\n⊢ 0 ≤ 2 ^ i✝",
"after_state": "No Goals!"
},
{
"line": "norm_cast",
"before_state": "case calc_2\nk n : ℕ\nhk : k > 0\nh : ↑k ! = ∏ i ∈ range n, (2 ^ n - 2 ^ i)\nh2 : ∑ i ∈ range n, i < k\nhn : 6 ≤ n\n⊢ ∏ __ ∈ range n, 2 ^ n < ↑k !",
"after_state": "case calc_2\nk n : ℕ\nhk : k > 0\nh : ↑k ! = ∏ i ∈ range n, (2 ^ n - 2 ^ i)\nh2 : ∑ i ∈ range n, i < k\nhn : 6 ≤ n\n⊢ ∏ i ∈ range n, 2 ^ n < k !"
},
{
"line": "focus\n norm_cast0\n with_annotate_state\"<;>\" skip\n all_goals try trivial",
"before_state": "case calc_2\nk n : ℕ\nhk : k > 0\nh : ↑k ! = ∏ i ∈ range n, (2 ^ n - 2 ^ i)\nh2 : ∑ i ∈ range n, i < k\nhn : 6 ≤ n\n⊢ ∏ __ ∈ range n, 2 ^ n < ↑k !",
"after_state": "case calc_2\nk n : ℕ\nhk : k > 0\nh : ↑k ! = ∏ i ∈ range n, (2 ^ n - 2 ^ i)\nh2 : ∑ i ∈ range n, i < k\nhn : 6 ≤ n\n⊢ ∏ i ∈ range n, 2 ^ n < k !"
},
{
"line": "norm_cast0",
"before_state": "case calc_2\nk n : ℕ\nhk : k > 0\nh : ↑k ! = ∏ i ∈ range n, (2 ^ n - 2 ^ i)\nh2 : ∑ i ∈ range n, i < k\nhn : 6 ≤ n\n⊢ ∏ __ ∈ range n, 2 ^ n < ↑k !",
"after_state": "case calc_2\nk n : ℕ\nhk : k > 0\nh : ↑k ! = ∏ i ∈ range n, (2 ^ n - 2 ^ i)\nh2 : ∑ i ∈ range n, i < k\nhn : 6 ≤ n\n⊢ ∏ i ∈ range n, 2 ^ n < k !"
},
{
"line": "with_annotate_state\"<;>\" skip",
"before_state": "case calc_2\nk n : ℕ\nhk : k > 0\nh : ↑k ! = ∏ i ∈ range n, (2 ^ n - 2 ^ i)\nh2 : ∑ i ∈ range n, i < k\nhn : 6 ≤ n\n⊢ ∏ i ∈ range n, 2 ^ n < k !",
"after_state": "case calc_2\nk n : ℕ\nhk : k > 0\nh : ↑k ! = ∏ i ∈ range n, (2 ^ n - 2 ^ i)\nh2 : ∑ i ∈ range n, i < k\nhn : 6 ≤ n\n⊢ ∏ i ∈ range n, 2 ^ n < k !"
},
{
"line": "skip",
"before_state": "case calc_2\nk n : ℕ\nhk : k > 0\nh : ↑k ! = ∏ i ∈ range n, (2 ^ n - 2 ^ i)\nh2 : ∑ i ∈ range n, i < k\nhn : 6 ≤ n\n⊢ ∏ i ∈ range n, 2 ^ n < k !",
"after_state": "case calc_2\nk n : ℕ\nhk : k > 0\nh : ↑k ! = ∏ i ∈ range n, (2 ^ n - 2 ^ i)\nh2 : ∑ i ∈ range n, i < k\nhn : 6 ≤ n\n⊢ ∏ i ∈ range n, 2 ^ n < k !"
},
{
"line": "all_goals try trivial",
"before_state": "case calc_2\nk n : ℕ\nhk : k > 0\nh : ↑k ! = ∏ i ∈ range n, (2 ^ n - 2 ^ i)\nh2 : ∑ i ∈ range n, i < k\nhn : 6 ≤ n\n⊢ ∏ i ∈ range n, 2 ^ n < k !",
"after_state": "case calc_2\nk n : ℕ\nhk : k > 0\nh : ↑k ! = ∏ i ∈ range n, (2 ^ n - 2 ^ i)\nh2 : ∑ i ∈ range n, i < k\nhn : 6 ≤ n\n⊢ ∏ i ∈ range n, 2 ^ n < k !"
},
{
"line": "try trivial",
"before_state": "case calc_2\nk n : ℕ\nhk : k > 0\nh : ↑k ! = ∏ i ∈ range n, (2 ^ n - 2 ^ i)\nh2 : ∑ i ∈ range n, i < k\nhn : 6 ≤ n\n⊢ ∏ i ∈ range n, 2 ^ n < k !",
"after_state": "case calc_2\nk n : ℕ\nhk : k > 0\nh : ↑k ! = ∏ i ∈ range n, (2 ^ n - 2 ^ i)\nh2 : ∑ i ∈ range n, i < k\nhn : 6 ≤ n\n⊢ ∏ i ∈ range n, 2 ^ n < k !"
},
{
"line": "first\n| trivial\n| skip",
"before_state": "case calc_2\nk n : ℕ\nhk : k > 0\nh : ↑k ! = ∏ i ∈ range n, (2 ^ n - 2 ^ i)\nh2 : ∑ i ∈ range n, i < k\nhn : 6 ≤ n\n⊢ ∏ i ∈ range n, 2 ^ n < k !",
"after_state": "case calc_2\nk n : ℕ\nhk : k > 0\nh : ↑k ! = ∏ i ∈ range n, (2 ^ n - 2 ^ i)\nh2 : ∑ i ∈ range n, i < k\nhn : 6 ≤ n\n⊢ ∏ i ∈ range n, 2 ^ n < k !"
},
{
"line": "trivial",
"before_state": "case calc_2\nk n : ℕ\nhk : k > 0\nh : ↑k ! = ∏ i ∈ range n, (2 ^ n - 2 ^ i)\nh2 : ∑ i ∈ range n, i < k\nhn : 6 ≤ n\n⊢ ∏ i ∈ range n, 2 ^ n < k !",
"after_state": "case calc_2\nk n : ℕ\nhk : k > 0\nh : ↑k ! = ∏ i ∈ range n, (2 ^ n - 2 ^ i)\nh2 : ∑ i ∈ range n, i < k\nhn : 6 ≤ n\n⊢ ∏ i ∈ range n, 2 ^ n < k !"
},
{
"line": "assumption",
"before_state": "case calc_2\nk n : ℕ\nhk : k > 0\nh : ↑k ! = ∏ i ∈ range n, (2 ^ n - 2 ^ i)\nh2 : ∑ i ∈ range n, i < k\nhn : 6 ≤ n\n⊢ ∏ i ∈ range n, 2 ^ n < k !",
"after_state": "case calc_2\nk n : ℕ\nhk : k > 0\nh : ↑k ! = ∏ i ∈ range n, (2 ^ n - 2 ^ i)\nh2 : ∑ i ∈ range n, i < k\nhn : 6 ≤ n\n⊢ ∏ i ∈ range n, 2 ^ n < k !"
},
{
"line": "skip",
"before_state": "case calc_2\nk n : ℕ\nhk : k > 0\nh : ↑k ! = ∏ i ∈ range n, (2 ^ n - 2 ^ i)\nh2 : ∑ i ∈ range n, i < k\nhn : 6 ≤ n\n⊢ ∏ i ∈ range n, 2 ^ n < k !",
"after_state": "case calc_2\nk n : ℕ\nhk : k > 0\nh : ↑k ! = ∏ i ∈ range n, (2 ^ n - 2 ^ i)\nh2 : ∑ i ∈ range n, i < k\nhn : 6 ≤ n\n⊢ ∏ i ∈ range n, 2 ^ n < k !"
},
{
"line": "calc\n ∏ __ ∈ range n, 2 ^ n = 2 ^ (n * n) := by rw [prod_const, card_range, ← pow_mul]\n _ < (∑ i ∈ range n, i)! := ?_\n _ ≤ k ! := by gcongr",
"before_state": "case calc_2\nk n : ℕ\nhk : k > 0\nh : ↑k ! = ∏ i ∈ range n, (2 ^ n - 2 ^ i)\nh2 : ∑ i ∈ range n, i < k\nhn : 6 ≤ n\n⊢ ∏ i ∈ range n, 2 ^ n < k !",
"after_state": "case calc_2\nk n : ℕ\nhk : k > 0\nh : ↑k ! = ∏ i ∈ range n, (2 ^ n - 2 ^ i)\nh2 : ∑ i ∈ range n, i < k\nhn : 6 ≤ n\n⊢ 2 ^ (n * n) < (∑ i ∈ range n, i)!"
},
{
"line": "rw [prod_const, card_range, ← pow_mul]",
"before_state": "k n : ℕ\nhk : k > 0\nh : ↑k ! = ∏ i ∈ range n, (2 ^ n - 2 ^ i)\nh2 : ∑ i ∈ range n, i < k\nhn : 6 ≤ n\n⊢ ∏ __ ∈ range n, 2 ^ n = 2 ^ (n * n)",
"after_state": "No Goals!"
},
{
"line": "rewrite [prod_const, card_range, ← pow_mul]",
"before_state": "k n : ℕ\nhk : k > 0\nh : ↑k ! = ∏ i ∈ range n, (2 ^ n - 2 ^ i)\nh2 : ∑ i ∈ range n, i < k\nhn : 6 ≤ n\n⊢ ∏ __ ∈ range n, 2 ^ n = 2 ^ (n * n)",
"after_state": "k n : ℕ\nhk : k > 0\nh : ↑k ! = ∏ i ∈ range n, (2 ^ n - 2 ^ i)\nh2 : ∑ i ∈ range n, i < k\nhn : 6 ≤ n\n⊢ 2 ^ (n * n) = 2 ^ (n * n)"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "k n : ℕ\nhk : k > 0\nh : ↑k ! = ∏ i ∈ range n, (2 ^ n - 2 ^ i)\nh2 : ∑ i ∈ range n, i < k\nhn : 6 ≤ n\n⊢ 2 ^ (n * n) = 2 ^ (n * n)",
"after_state": "No Goals!"
},
{
"line": "try (with_reducible rfl)",
"before_state": "k n : ℕ\nhk : k > 0\nh : ↑k ! = ∏ i ∈ range n, (2 ^ n - 2 ^ i)\nh2 : ∑ i ∈ range n, i < k\nhn : 6 ≤ n\n⊢ 2 ^ (n * n) = 2 ^ (n * n)",
"after_state": "No Goals!"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "k n : ℕ\nhk : k > 0\nh : ↑k ! = ∏ i ∈ range n, (2 ^ n - 2 ^ i)\nh2 : ∑ i ∈ range n, i < k\nhn : 6 ≤ n\n⊢ 2 ^ (n * n) = 2 ^ (n * n)",
"after_state": "No Goals!"
},
{
"line": "with_reducible rfl",
"before_state": "k n : ℕ\nhk : k > 0\nh : ↑k ! = ∏ i ∈ range n, (2 ^ n - 2 ^ i)\nh2 : ∑ i ∈ range n, i < k\nhn : 6 ≤ n\n⊢ 2 ^ (n * n) = 2 ^ (n * n)",
"after_state": "No Goals!"
},
{
"line": "rfl",
"before_state": "k n : ℕ\nhk : k > 0\nh : ↑k ! = ∏ i ∈ range n, (2 ^ n - 2 ^ i)\nh2 : ∑ i ∈ range n, i < k\nhn : 6 ≤ n\n⊢ 2 ^ (n * n) = 2 ^ (n * n)",
"after_state": "No Goals!"
},
{
"line": "eq_refl",
"before_state": "k n : ℕ\nhk : k > 0\nh : ↑k ! = ∏ i ∈ range n, (2 ^ n - 2 ^ i)\nh2 : ∑ i ∈ range n, i < k\nhn : 6 ≤ n\n⊢ 2 ^ (n * n) = 2 ^ (n * n)",
"after_state": "No Goals!"
},
{
"line": "gcongr",
"before_state": "k n : ℕ\nhk : k > 0\nh : ↑k ! = ∏ i ∈ range n, (2 ^ n - 2 ^ i)\nh2 : ∑ i ∈ range n, i < k\nhn : 6 ≤ n\n⊢ (∑ i ∈ range n, i)! ≤ k !",
"after_state": "No Goals!"
},
{
"line": "clear h h2",
"before_state": "case calc_2\nk n : ℕ\nhk : k > 0\nh : ↑k ! = ∏ i ∈ range n, (2 ^ n - 2 ^ i)\nh2 : ∑ i ∈ range n, i < k\nhn : 6 ≤ n\n⊢ 2 ^ (n * n) < (∑ i ∈ range n, i)!",
"after_state": "case calc_2\nk n : ℕ\nhk : k > 0\nhn : 6 ≤ n\n⊢ 2 ^ (n * n) < (∑ i ∈ range n, i)!"
},
{
"line": "induction' n, hn using Nat.le_induction with n' hn' IH",
"before_state": "case calc_2\nk n : ℕ\nhk : k > 0\nhn : 6 ≤ n\n⊢ 2 ^ (n * n) < (∑ i ∈ range n, i)!",
"after_state": "case calc_2.base\nk n : ℕ\nhk : k > 0\n⊢ 2 ^ (6 * 6) < (∑ i ∈ range 6, i)!\n---\ncase calc_2.succ\nk n : ℕ\nhk : k > 0\nn' : ℕ\nhn' : 6 ≤ n'\nIH : 2 ^ (n' * n') < (∑ i ∈ range n', i)!\n⊢ 2 ^ ((n' + 1) * (n' + 1)) < (∑ i ∈ range (n' + 1), i)!"
},
{
"line": "decide",
"before_state": "case calc_2.base\nk n : ℕ\nhk : k > 0\n⊢ 2 ^ (6 * 6) < (∑ i ∈ range 6, i)!",
"after_state": "No Goals!"
},
{
"line": "let A := ∑ i ∈ range n', i",
"before_state": "case calc_2.succ\nk n : ℕ\nhk : k > 0\nn' : ℕ\nhn' : 6 ≤ n'\nIH : 2 ^ (n' * n') < (∑ i ∈ range n', i)!\n⊢ 2 ^ ((n' + 1) * (n' + 1)) < (∑ i ∈ range (n' + 1), i)!",
"after_state": "case calc_2.succ\nk n : ℕ\nhk : k > 0\nn' : ℕ\nhn' : 6 ≤ n'\nIH : 2 ^ (n' * n') < (∑ i ∈ range n', i)!\nA : ℕ := ∑ i ∈ range n', i\n⊢ 2 ^ ((n' + 1) * (n' + 1)) < (∑ i ∈ range (n' + 1), i)!"
},
{
"line": "refine_lift\n let A := ∑ i ∈ range n', i;\n ?_",
"before_state": "case calc_2.succ\nk n : ℕ\nhk : k > 0\nn' : ℕ\nhn' : 6 ≤ n'\nIH : 2 ^ (n' * n') < (∑ i ∈ range n', i)!\n⊢ 2 ^ ((n' + 1) * (n' + 1)) < (∑ i ∈ range (n' + 1), i)!",
"after_state": "case calc_2.succ\nk n : ℕ\nhk : k > 0\nn' : ℕ\nhn' : 6 ≤ n'\nIH : 2 ^ (n' * n') < (∑ i ∈ range n', i)!\nA : ℕ := ∑ i ∈ range n', i\n⊢ 2 ^ ((n' + 1) * (n' + 1)) < (∑ i ∈ range (n' + 1), i)!"
},
{
"line": "focus\n (refine\n no_implicit_lambda%\n (let A := ∑ i ∈ range n', i;\n ?_);\n rotate_right)",
"before_state": "case calc_2.succ\nk n : ℕ\nhk : k > 0\nn' : ℕ\nhn' : 6 ≤ n'\nIH : 2 ^ (n' * n') < (∑ i ∈ range n', i)!\n⊢ 2 ^ ((n' + 1) * (n' + 1)) < (∑ i ∈ range (n' + 1), i)!",
"after_state": "case calc_2.succ\nk n : ℕ\nhk : k > 0\nn' : ℕ\nhn' : 6 ≤ n'\nIH : 2 ^ (n' * n') < (∑ i ∈ range n', i)!\nA : ℕ := ∑ i ∈ range n', i\n⊢ 2 ^ ((n' + 1) * (n' + 1)) < (∑ i ∈ range (n' + 1), i)!"
},
{
"line": "refine\n no_implicit_lambda%\n (let A := ∑ i ∈ range n', i;\n ?_)",
"before_state": "case calc_2.succ\nk n : ℕ\nhk : k > 0\nn' : ℕ\nhn' : 6 ≤ n'\nIH : 2 ^ (n' * n') < (∑ i ∈ range n', i)!\n⊢ 2 ^ ((n' + 1) * (n' + 1)) < (∑ i ∈ range (n' + 1), i)!",
"after_state": "case calc_2.succ\nk n : ℕ\nhk : k > 0\nn' : ℕ\nhn' : 6 ≤ n'\nIH : 2 ^ (n' * n') < (∑ i ∈ range n', i)!\nA : ℕ := ∑ i ∈ range n', i\n⊢ 2 ^ ((n' + 1) * (n' + 1)) < (∑ i ∈ range (n' + 1), i)!"
},
{
"line": "rotate_right",
"before_state": "case calc_2.succ\nk n : ℕ\nhk : k > 0\nn' : ℕ\nhn' : 6 ≤ n'\nIH : 2 ^ (n' * n') < (∑ i ∈ range n', i)!\nA : ℕ := ∑ i ∈ range n', i\n⊢ 2 ^ ((n' + 1) * (n' + 1)) < (∑ i ∈ range (n' + 1), i)!",
"after_state": "case calc_2.succ\nk n : ℕ\nhk : k > 0\nn' : ℕ\nhn' : 6 ≤ n'\nIH : 2 ^ (n' * n') < (∑ i ∈ range n', i)!\nA : ℕ := ∑ i ∈ range n', i\n⊢ 2 ^ ((n' + 1) * (n' + 1)) < (∑ i ∈ range (n' + 1), i)!"
},
{
"line": "have le_sum : ∑ i ∈ range 6, i ≤ A := by\n apply sum_le_sum_of_subset\n simpa using hn'",
"before_state": "case calc_2.succ\nk n : ℕ\nhk : k > 0\nn' : ℕ\nhn' : 6 ≤ n'\nIH : 2 ^ (n' * n') < (∑ i ∈ range n', i)!\nA : ℕ := ∑ i ∈ range n', i\n⊢ 2 ^ ((n' + 1) * (n' + 1)) < (∑ i ∈ range (n' + 1), i)!",
"after_state": "case calc_2.succ\nk n : ℕ\nhk : k > 0\nn' : ℕ\nhn' : 6 ≤ n'\nIH : 2 ^ (n' * n') < (∑ i ∈ range n', i)!\nA : ℕ := ∑ i ∈ range n', i\nle_sum : ∑ i ∈ range 6, i ≤ A\n⊢ 2 ^ ((n' + 1) * (n' + 1)) < (∑ i ∈ range (n' + 1), i)!"
},
{
"line": "focus\n refine\n no_implicit_lambda%\n (have le_sum : ∑ i ∈ range 6, i ≤ A := ?body✝;\n ?_)\n case body✝ =>\n with_annotate_state\"by\"\n ( apply sum_le_sum_of_subset\n simpa using hn')",
"before_state": "case calc_2.succ\nk n : ℕ\nhk : k > 0\nn' : ℕ\nhn' : 6 ≤ n'\nIH : 2 ^ (n' * n') < (∑ i ∈ range n', i)!\nA : ℕ := ∑ i ∈ range n', i\n⊢ 2 ^ ((n' + 1) * (n' + 1)) < (∑ i ∈ range (n' + 1), i)!",
"after_state": "case calc_2.succ\nk n : ℕ\nhk : k > 0\nn' : ℕ\nhn' : 6 ≤ n'\nIH : 2 ^ (n' * n') < (∑ i ∈ range n', i)!\nA : ℕ := ∑ i ∈ range n', i\nle_sum : ∑ i ∈ range 6, i ≤ A\n⊢ 2 ^ ((n' + 1) * (n' + 1)) < (∑ i ∈ range (n' + 1), i)!"
},
{
"line": "refine\n no_implicit_lambda%\n (have le_sum : ∑ i ∈ range 6, i ≤ A := ?body✝;\n ?_)",
"before_state": "case calc_2.succ\nk n : ℕ\nhk : k > 0\nn' : ℕ\nhn' : 6 ≤ n'\nIH : 2 ^ (n' * n') < (∑ i ∈ range n', i)!\nA : ℕ := ∑ i ∈ range n', i\n⊢ 2 ^ ((n' + 1) * (n' + 1)) < (∑ i ∈ range (n' + 1), i)!",
"after_state": "case body\nk n : ℕ\nhk : k > 0\nn' : ℕ\nhn' : 6 ≤ n'\nIH : 2 ^ (n' * n') < (∑ i ∈ range n', i)!\nA : ℕ := ∑ i ∈ range n', i\n⊢ ∑ i ∈ range 6, i ≤ A\n---\ncase calc_2.succ\nk n : ℕ\nhk : k > 0\nn' : ℕ\nhn' : 6 ≤ n'\nIH : 2 ^ (n' * n') < (∑ i ∈ range n', i)!\nA : ℕ := ∑ i ∈ range n', i\nle_sum : ∑ i ∈ range 6, i ≤ A\n⊢ 2 ^ ((n' + 1) * (n' + 1)) < (∑ i ∈ range (n' + 1), i)!"
},
{
"line": "case body✝ =>\n with_annotate_state\"by\"\n ( apply sum_le_sum_of_subset\n simpa using hn')",
"before_state": "case body\nk n : ℕ\nhk : k > 0\nn' : ℕ\nhn' : 6 ≤ n'\nIH : 2 ^ (n' * n') < (∑ i ∈ range n', i)!\nA : ℕ := ∑ i ∈ range n', i\n⊢ ∑ i ∈ range 6, i ≤ A\n---\ncase calc_2.succ\nk n : ℕ\nhk : k > 0\nn' : ℕ\nhn' : 6 ≤ n'\nIH : 2 ^ (n' * n') < (∑ i ∈ range n', i)!\nA : ℕ := ∑ i ∈ range n', i\nle_sum : ∑ i ∈ range 6, i ≤ A\n⊢ 2 ^ ((n' + 1) * (n' + 1)) < (∑ i ∈ range (n' + 1), i)!",
"after_state": "case calc_2.succ\nk n : ℕ\nhk : k > 0\nn' : ℕ\nhn' : 6 ≤ n'\nIH : 2 ^ (n' * n') < (∑ i ∈ range n', i)!\nA : ℕ := ∑ i ∈ range n', i\nle_sum : ∑ i ∈ range 6, i ≤ A\n⊢ 2 ^ ((n' + 1) * (n' + 1)) < (∑ i ∈ range (n' + 1), i)!"
},
{
"line": "with_annotate_state\"by\"\n ( apply sum_le_sum_of_subset\n simpa using hn')",
"before_state": "k n : ℕ\nhk : k > 0\nn' : ℕ\nhn' : 6 ≤ n'\nIH : 2 ^ (n' * n') < (∑ i ∈ range n', i)!\nA : ℕ := ∑ i ∈ range n', i\n⊢ ∑ i ∈ range 6, i ≤ A",
"after_state": "No Goals!"
},
{
"line": "apply sum_le_sum_of_subset",
"before_state": "k n : ℕ\nhk : k > 0\nn' : ℕ\nhn' : 6 ≤ n'\nIH : 2 ^ (n' * n') < (∑ i ∈ range n', i)!\nA : ℕ := ∑ i ∈ range n', i\n⊢ ∑ i ∈ range 6, i ≤ A",
"after_state": "case h\nk n : ℕ\nhk : k > 0\nn' : ℕ\nhn' : 6 ≤ n'\nIH : 2 ^ (n' * n') < (∑ i ∈ range n', i)!\nA : ℕ := ∑ i ∈ range n', i\n⊢ range 6 ⊆ range n'"
},
{
"line": "simpa using hn'",
"before_state": "case h\nk n : ℕ\nhk : k > 0\nn' : ℕ\nhn' : 6 ≤ n'\nIH : 2 ^ (n' * n') < (∑ i ∈ range n', i)!\nA : ℕ := ∑ i ∈ range n', i\n⊢ range 6 ⊆ range n'",
"after_state": "No Goals!"
},
{
"line": "calc\n 2 ^ ((n' + 1) * (n' + 1)) ≤ 2 ^ (n' * n' + 4 * n') := by gcongr <;> linarith\n _ = 2 ^ (n' * n') * (2 ^ 4) ^ n' := by rw [← pow_mul, ← pow_add]\n _ < A ! * (2 ^ 4) ^ n' := by gcongr\n _ = A ! * (15 + 1) ^ n' := rfl\n _ ≤ A ! * (A + 1) ^ n' := by gcongr; exact le_sum\n _ ≤ (A + n')! := factorial_mul_pow_le_factorial\n _ = (∑ i ∈ range (n' + 1), i)! := by rw [sum_range_succ]",
"before_state": "case calc_2.succ\nk n : ℕ\nhk : k > 0\nn' : ℕ\nhn' : 6 ≤ n'\nIH : 2 ^ (n' * n') < (∑ i ∈ range n', i)!\nA : ℕ := ∑ i ∈ range n', i\nle_sum : ∑ i ∈ range 6, i ≤ A\n⊢ 2 ^ ((n' + 1) * (n' + 1)) < (∑ i ∈ range (n' + 1), i)!",
"after_state": "No Goals!"
},
{
"line": "focus\n gcongr\n with_annotate_state\"<;>\" skip\n all_goals linarith",
"before_state": "k n : ℕ\nhk : k > 0\nn' : ℕ\nhn' : 6 ≤ n'\nIH : 2 ^ (n' * n') < (∑ i ∈ range n', i)!\nA : ℕ := ∑ i ∈ range n', i\nle_sum : ∑ i ∈ range 6, i ≤ A\n⊢ 2 ^ ((n' + 1) * (n' + 1)) ≤ 2 ^ (n' * n' + 4 * n')",
"after_state": "No Goals!"
},
{
"line": "gcongr",
"before_state": "k n : ℕ\nhk : k > 0\nn' : ℕ\nhn' : 6 ≤ n'\nIH : 2 ^ (n' * n') < (∑ i ∈ range n', i)!\nA : ℕ := ∑ i ∈ range n', i\nle_sum : ∑ i ∈ range 6, i ≤ A\n⊢ 2 ^ ((n' + 1) * (n' + 1)) ≤ 2 ^ (n' * n' + 4 * n')",
"after_state": "case ha\nk n : ℕ\nhk : k > 0\nn' : ℕ\nhn' : 6 ≤ n'\nIH : 2 ^ (n' * n') < (∑ i ∈ range n', i)!\nA : ℕ := ∑ i ∈ range n', i\nle_sum : ∑ i ∈ range 6, i ≤ A\n⊢ 1 ≤ 2\n---\ncase hmn\nk n : ℕ\nhk : k > 0\nn' : ℕ\nhn' : 6 ≤ n'\nIH : 2 ^ (n' * n') < (∑ i ∈ range n', i)!\nA : ℕ := ∑ i ∈ range n', i\nle_sum : ∑ i ∈ range 6, i ≤ A\n⊢ (n' + 1) * (n' + 1) ≤ n' * n' + 4 * n'"
},
{
"line": "with_annotate_state\"<;>\" skip",
"before_state": "case ha\nk n : ℕ\nhk : k > 0\nn' : ℕ\nhn' : 6 ≤ n'\nIH : 2 ^ (n' * n') < (∑ i ∈ range n', i)!\nA : ℕ := ∑ i ∈ range n', i\nle_sum : ∑ i ∈ range 6, i ≤ A\n⊢ 1 ≤ 2\n---\ncase hmn\nk n : ℕ\nhk : k > 0\nn' : ℕ\nhn' : 6 ≤ n'\nIH : 2 ^ (n' * n') < (∑ i ∈ range n', i)!\nA : ℕ := ∑ i ∈ range n', i\nle_sum : ∑ i ∈ range 6, i ≤ A\n⊢ (n' + 1) * (n' + 1) ≤ n' * n' + 4 * n'",
"after_state": "case ha\nk n : ℕ\nhk : k > 0\nn' : ℕ\nhn' : 6 ≤ n'\nIH : 2 ^ (n' * n') < (∑ i ∈ range n', i)!\nA : ℕ := ∑ i ∈ range n', i\nle_sum : ∑ i ∈ range 6, i ≤ A\n⊢ 1 ≤ 2\n---\ncase hmn\nk n : ℕ\nhk : k > 0\nn' : ℕ\nhn' : 6 ≤ n'\nIH : 2 ^ (n' * n') < (∑ i ∈ range n', i)!\nA : ℕ := ∑ i ∈ range n', i\nle_sum : ∑ i ∈ range 6, i ≤ A\n⊢ (n' + 1) * (n' + 1) ≤ n' * n' + 4 * n'"
},
{
"line": "skip",
"before_state": "case ha\nk n : ℕ\nhk : k > 0\nn' : ℕ\nhn' : 6 ≤ n'\nIH : 2 ^ (n' * n') < (∑ i ∈ range n', i)!\nA : ℕ := ∑ i ∈ range n', i\nle_sum : ∑ i ∈ range 6, i ≤ A\n⊢ 1 ≤ 2\n---\ncase hmn\nk n : ℕ\nhk : k > 0\nn' : ℕ\nhn' : 6 ≤ n'\nIH : 2 ^ (n' * n') < (∑ i ∈ range n', i)!\nA : ℕ := ∑ i ∈ range n', i\nle_sum : ∑ i ∈ range 6, i ≤ A\n⊢ (n' + 1) * (n' + 1) ≤ n' * n' + 4 * n'",
"after_state": "case ha\nk n : ℕ\nhk : k > 0\nn' : ℕ\nhn' : 6 ≤ n'\nIH : 2 ^ (n' * n') < (∑ i ∈ range n', i)!\nA : ℕ := ∑ i ∈ range n', i\nle_sum : ∑ i ∈ range 6, i ≤ A\n⊢ 1 ≤ 2\n---\ncase hmn\nk n : ℕ\nhk : k > 0\nn' : ℕ\nhn' : 6 ≤ n'\nIH : 2 ^ (n' * n') < (∑ i ∈ range n', i)!\nA : ℕ := ∑ i ∈ range n', i\nle_sum : ∑ i ∈ range 6, i ≤ A\n⊢ (n' + 1) * (n' + 1) ≤ n' * n' + 4 * n'"
},
{
"line": "all_goals linarith",
"before_state": "case ha\nk n : ℕ\nhk : k > 0\nn' : ℕ\nhn' : 6 ≤ n'\nIH : 2 ^ (n' * n') < (∑ i ∈ range n', i)!\nA : ℕ := ∑ i ∈ range n', i\nle_sum : ∑ i ∈ range 6, i ≤ A\n⊢ 1 ≤ 2\n---\ncase hmn\nk n : ℕ\nhk : k > 0\nn' : ℕ\nhn' : 6 ≤ n'\nIH : 2 ^ (n' * n') < (∑ i ∈ range n', i)!\nA : ℕ := ∑ i ∈ range n', i\nle_sum : ∑ i ∈ range 6, i ≤ A\n⊢ (n' + 1) * (n' + 1) ≤ n' * n' + 4 * n'",
"after_state": "No Goals!"
},
{
"line": "linarith",
"before_state": "case ha\nk n : ℕ\nhk : k > 0\nn' : ℕ\nhn' : 6 ≤ n'\nIH : 2 ^ (n' * n') < (∑ i ∈ range n', i)!\nA : ℕ := ∑ i ∈ range n', i\nle_sum : ∑ i ∈ range 6, i ≤ A\n⊢ 1 ≤ 2",
"after_state": "No Goals!"
},
{
"line": "ring1",
"before_state": "k n : ℕ\nhk : k > 0\nn' : ℕ\nhn' : 6 ≤ n'\nIH : 2 ^ (n' * n') < (∑ i ∈ range n', i)!\nA : ℕ := ∑ i ∈ range n', i\nle_sum : ∑ i ∈ range 6, i ≤ A\na✝ : 1 > 2\n⊢ 2 * -1 + (2 + 1 - 1) = 0",
"after_state": "No Goals!"
},
{
"line": "norm_num",
"before_state": "k n : ℕ\nhk : k > 0\nn' : ℕ\nhn' : 6 ≤ n'\nIH : 2 ^ (n' * n') < (∑ i ∈ range n', i)!\nA : ℕ := ∑ i ∈ range n', i\nle_sum : ∑ i ∈ range 6, i ≤ A\na✝ : 1 > 2\n⊢ 2 > 0",
"after_state": "No Goals!"
},
{
"line": "linarith",
"before_state": "case hmn\nk n : ℕ\nhk : k > 0\nn' : ℕ\nhn' : 6 ≤ n'\nIH : 2 ^ (n' * n') < (∑ i ∈ range n', i)!\nA : ℕ := ∑ i ∈ range n', i\nle_sum : ∑ i ∈ range 6, i ≤ A\n⊢ (n' + 1) * (n' + 1) ≤ n' * n' + 4 * n'",
"after_state": "No Goals!"
},
{
"line": "ring1",
"before_state": "k n : ℕ\nhk : k > 0\nn' : ℕ\nhn' : 6 ≤ n'\nIH : 2 ^ (n' * n') < (∑ i ∈ range n', i)!\nA : ℕ := ∑ i ∈ range n', i\nle_sum : ∑ i ∈ range 6, i ≤ A\na✝ : (n' + 1) * (n' + 1) > n' * n' + 4 * n'\n⊢ 12 * -1 + 2 * (6 - ↑n') + (↑n' * ↑n' + 4 * ↑n' + 1 - (↑n' + 1) * (↑n' + 1)) = 0",
"after_state": "No Goals!"
},
{
"line": "norm_num",
"before_state": "k n : ℕ\nhk : k > 0\nn' : ℕ\nhn' : 6 ≤ n'\nIH : 2 ^ (n' * n') < (∑ i ∈ range n', i)!\nA : ℕ := ∑ i ∈ range n', i\nle_sum : ∑ i ∈ range 6, i ≤ A\na✝ : (n' + 1) * (n' + 1) > n' * n' + 4 * n'\n⊢ 12 > 0",
"after_state": "No Goals!"
},
{
"line": "norm_num",
"before_state": "k n : ℕ\nhk : k > 0\nn' : ℕ\nhn' : 6 ≤ n'\nIH : 2 ^ (n' * n') < (∑ i ∈ range n', i)!\nA : ℕ := ∑ i ∈ range n', i\nle_sum : ∑ i ∈ range 6, i ≤ A\na✝ : (n' + 1) * (n' + 1) > n' * n' + 4 * n'\n⊢ 2 > 0",
"after_state": "No Goals!"
},
{
"line": "rw [← pow_mul, ← pow_add]",
"before_state": "k n : ℕ\nhk : k > 0\nn' : ℕ\nhn' : 6 ≤ n'\nIH : 2 ^ (n' * n') < (∑ i ∈ range n', i)!\nA : ℕ := ∑ i ∈ range n', i\nle_sum : ∑ i ∈ range 6, i ≤ A\n⊢ 2 ^ (n' * n' + 4 * n') = 2 ^ (n' * n') * (2 ^ 4) ^ n'",
"after_state": "No Goals!"
},
{
"line": "rewrite [← pow_mul, ← pow_add]",
"before_state": "k n : ℕ\nhk : k > 0\nn' : ℕ\nhn' : 6 ≤ n'\nIH : 2 ^ (n' * n') < (∑ i ∈ range n', i)!\nA : ℕ := ∑ i ∈ range n', i\nle_sum : ∑ i ∈ range 6, i ≤ A\n⊢ 2 ^ (n' * n' + 4 * n') = 2 ^ (n' * n') * (2 ^ 4) ^ n'",
"after_state": "k n : ℕ\nhk : k > 0\nn' : ℕ\nhn' : 6 ≤ n'\nIH : 2 ^ (n' * n') < (∑ i ∈ range n', i)!\nA : ℕ := ∑ i ∈ range n', i\nle_sum : ∑ i ∈ range 6, i ≤ A\n⊢ 2 ^ (n' * n' + 4 * n') = 2 ^ (n' * n' + 4 * n')"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "k n : ℕ\nhk : k > 0\nn' : ℕ\nhn' : 6 ≤ n'\nIH : 2 ^ (n' * n') < (∑ i ∈ range n', i)!\nA : ℕ := ∑ i ∈ range n', i\nle_sum : ∑ i ∈ range 6, i ≤ A\n⊢ 2 ^ (n' * n' + 4 * n') = 2 ^ (n' * n' + 4 * n')",
"after_state": "No Goals!"
},
{
"line": "try (with_reducible rfl)",
"before_state": "k n : ℕ\nhk : k > 0\nn' : ℕ\nhn' : 6 ≤ n'\nIH : 2 ^ (n' * n') < (∑ i ∈ range n', i)!\nA : ℕ := ∑ i ∈ range n', i\nle_sum : ∑ i ∈ range 6, i ≤ A\n⊢ 2 ^ (n' * n' + 4 * n') = 2 ^ (n' * n' + 4 * n')",
"after_state": "No Goals!"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "k n : ℕ\nhk : k > 0\nn' : ℕ\nhn' : 6 ≤ n'\nIH : 2 ^ (n' * n') < (∑ i ∈ range n', i)!\nA : ℕ := ∑ i ∈ range n', i\nle_sum : ∑ i ∈ range 6, i ≤ A\n⊢ 2 ^ (n' * n' + 4 * n') = 2 ^ (n' * n' + 4 * n')",
"after_state": "No Goals!"
},
{
"line": "with_reducible rfl",
"before_state": "k n : ℕ\nhk : k > 0\nn' : ℕ\nhn' : 6 ≤ n'\nIH : 2 ^ (n' * n') < (∑ i ∈ range n', i)!\nA : ℕ := ∑ i ∈ range n', i\nle_sum : ∑ i ∈ range 6, i ≤ A\n⊢ 2 ^ (n' * n' + 4 * n') = 2 ^ (n' * n' + 4 * n')",
"after_state": "No Goals!"
},
{
"line": "rfl",
"before_state": "k n : ℕ\nhk : k > 0\nn' : ℕ\nhn' : 6 ≤ n'\nIH : 2 ^ (n' * n') < (∑ i ∈ range n', i)!\nA : ℕ := ∑ i ∈ range n', i\nle_sum : ∑ i ∈ range 6, i ≤ A\n⊢ 2 ^ (n' * n' + 4 * n') = 2 ^ (n' * n' + 4 * n')",
"after_state": "No Goals!"
},
{
"line": "eq_refl",
"before_state": "k n : ℕ\nhk : k > 0\nn' : ℕ\nhn' : 6 ≤ n'\nIH : 2 ^ (n' * n') < (∑ i ∈ range n', i)!\nA : ℕ := ∑ i ∈ range n', i\nle_sum : ∑ i ∈ range 6, i ≤ A\n⊢ 2 ^ (n' * n' + 4 * n') = 2 ^ (n' * n' + 4 * n')",
"after_state": "No Goals!"
},
{
"line": "gcongr",
"before_state": "k n : ℕ\nhk : k > 0\nn' : ℕ\nhn' : 6 ≤ n'\nIH : 2 ^ (n' * n') < (∑ i ∈ range n', i)!\nA : ℕ := ∑ i ∈ range n', i\nle_sum : ∑ i ∈ range 6, i ≤ A\n⊢ 2 ^ (n' * n') * (2 ^ 4) ^ n' < A ! * (2 ^ 4) ^ n'",
"after_state": "No Goals!"
},
{
"line": "gcongr_discharger",
"before_state": "case a0\nk n : ℕ\nhk : k > 0\nn' : ℕ\nhn' : 6 ≤ n'\nIH : 2 ^ (n' * n') < (∑ i ∈ range n', i)!\nA : ℕ := ∑ i ∈ range n', i\nle_sum : ∑ i ∈ range 6, i ≤ A\n⊢ 0 < (2 ^ 4) ^ n'",
"after_state": "No Goals!"
},
{
"line": "positivity",
"before_state": "case a0\nk n : ℕ\nhk : k > 0\nn' : ℕ\nhn' : 6 ≤ n'\nIH : 2 ^ (n' * n') < (∑ i ∈ range n', i)!\nA : ℕ := ∑ i ∈ range n', i\nle_sum : ∑ i ∈ range 6, i ≤ A\n⊢ 0 < (2 ^ 4) ^ n'",
"after_state": "No Goals!"
},
{
"line": "gcongr",
"before_state": "k n : ℕ\nhk : k > 0\nn' : ℕ\nhn' : 6 ≤ n'\nIH : 2 ^ (n' * n') < (∑ i ∈ range n', i)!\nA : ℕ := ∑ i ∈ range n', i\nle_sum : ∑ i ∈ range 6, i ≤ A\n⊢ A ! * (15 + 1) ^ n' ≤ A ! * (A + 1) ^ n'",
"after_state": "case bc.hab.bc\nk n : ℕ\nhk : k > 0\nn' : ℕ\nhn' : 6 ≤ n'\nIH : 2 ^ (n' * n') < (∑ i ∈ range n', i)!\nA : ℕ := ∑ i ∈ range n', i\nle_sum : ∑ i ∈ range 6, i ≤ A\n⊢ 15 ≤ A"
},
{
"line": "gcongr_discharger",
"before_state": "case bc.ha\nk n : ℕ\nhk : k > 0\nn' : ℕ\nhn' : 6 ≤ n'\nIH : 2 ^ (n' * n') < (∑ i ∈ range n', i)!\nA : ℕ := ∑ i ∈ range n', i\nle_sum : ∑ i ∈ range 6, i ≤ A\n⊢ 0 ≤ 15 + 1",
"after_state": "No Goals!"
},
{
"line": "positivity",
"before_state": "case bc.ha\nk n : ℕ\nhk : k > 0\nn' : ℕ\nhn' : 6 ≤ n'\nIH : 2 ^ (n' * n') < (∑ i ∈ range n', i)!\nA : ℕ := ∑ i ∈ range n', i\nle_sum : ∑ i ∈ range 6, i ≤ A\n⊢ 0 ≤ 15 + 1",
"after_state": "No Goals!"
},
{
"line": "exact le_sum",
"before_state": "case bc.hab.bc\nk n : ℕ\nhk : k > 0\nn' : ℕ\nhn' : 6 ≤ n'\nIH : 2 ^ (n' * n') < (∑ i ∈ range n', i)!\nA : ℕ := ∑ i ∈ range n', i\nle_sum : ∑ i ∈ range 6, i ≤ A\n⊢ 15 ≤ A",
"after_state": "No Goals!"
},
{
"line": "rw [sum_range_succ]",
"before_state": "k n : ℕ\nhk : k > 0\nn' : ℕ\nhn' : 6 ≤ n'\nIH : 2 ^ (n' * n') < (∑ i ∈ range n', i)!\nA : ℕ := ∑ i ∈ range n', i\nle_sum : ∑ i ∈ range 6, i ≤ A\n⊢ (A + n')! = (∑ i ∈ range (n' + 1), i)!",
"after_state": "No Goals!"
},
{
"line": "rewrite [sum_range_succ]",
"before_state": "k n : ℕ\nhk : k > 0\nn' : ℕ\nhn' : 6 ≤ n'\nIH : 2 ^ (n' * n') < (∑ i ∈ range n', i)!\nA : ℕ := ∑ i ∈ range n', i\nle_sum : ∑ i ∈ range 6, i ≤ A\n⊢ (A + n')! = (∑ i ∈ range (n' + 1), i)!",
"after_state": "k n : ℕ\nhk : k > 0\nn' : ℕ\nhn' : 6 ≤ n'\nIH : 2 ^ (n' * n') < (∑ i ∈ range n', i)!\nA : ℕ := ∑ i ∈ range n', i\nle_sum : ∑ i ∈ range 6, i ≤ A\n⊢ (A + n')! = (∑ x ∈ range n', x + n')!"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "k n : ℕ\nhk : k > 0\nn' : ℕ\nhn' : 6 ≤ n'\nIH : 2 ^ (n' * n') < (∑ i ∈ range n', i)!\nA : ℕ := ∑ i ∈ range n', i\nle_sum : ∑ i ∈ range 6, i ≤ A\n⊢ (A + n')! = (∑ x ∈ range n', x + n')!",
"after_state": "No Goals!"
},
{
"line": "try (with_reducible rfl)",
"before_state": "k n : ℕ\nhk : k > 0\nn' : ℕ\nhn' : 6 ≤ n'\nIH : 2 ^ (n' * n') < (∑ i ∈ range n', i)!\nA : ℕ := ∑ i ∈ range n', i\nle_sum : ∑ i ∈ range 6, i ≤ A\n⊢ (A + n')! = (∑ x ∈ range n', x + n')!",
"after_state": "No Goals!"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "k n : ℕ\nhk : k > 0\nn' : ℕ\nhn' : 6 ≤ n'\nIH : 2 ^ (n' * n') < (∑ i ∈ range n', i)!\nA : ℕ := ∑ i ∈ range n', i\nle_sum : ∑ i ∈ range 6, i ≤ A\n⊢ (A + n')! = (∑ x ∈ range n', x + n')!",
"after_state": "No Goals!"
},
{
"line": "with_reducible rfl",
"before_state": "k n : ℕ\nhk : k > 0\nn' : ℕ\nhn' : 6 ≤ n'\nIH : 2 ^ (n' * n') < (∑ i ∈ range n', i)!\nA : ℕ := ∑ i ∈ range n', i\nle_sum : ∑ i ∈ range 6, i ≤ A\n⊢ (A + n')! = (∑ x ∈ range n', x + n')!",
"after_state": "No Goals!"
},
{
"line": "rfl",
"before_state": "k n : ℕ\nhk : k > 0\nn' : ℕ\nhn' : 6 ≤ n'\nIH : 2 ^ (n' * n') < (∑ i ∈ range n', i)!\nA : ℕ := ∑ i ∈ range n', i\nle_sum : ∑ i ∈ range 6, i ≤ A\n⊢ (A + n')! = (∑ x ∈ range n', x + n')!",
"after_state": "No Goals!"
},
{
"line": "eq_refl",
"before_state": "k n : ℕ\nhk : k > 0\nn' : ℕ\nhn' : 6 ≤ n'\nIH : 2 ^ (n' * n') < (∑ i ∈ range n', i)!\nA : ℕ := ∑ i ∈ range n', i\nle_sum : ∑ i ∈ range 6, i ≤ A\n⊢ (A + n')! = (∑ x ∈ range n', x + n')!",
"after_state": "No Goals!"
}
] |
lemma exists_numbers_in_interval {n : ℕ} (hn : 100 ≤ n) :
∃ l : ℕ, n + 4 * l ≤ 2 * l ^ 2 ∧ 2 * l ^ 2 + 4 * l ≤ 2 * n := by
have hn' : 1 ≤ Nat.sqrt (n + 1) := by
rw [Nat.le_sqrt]
apply Nat.le_add_left
have h₁ := Nat.sqrt_le' (n + 1)
have h₂ := Nat.succ_le_succ_sqrt' (n + 1)
have h₃ : 10 ≤ (n + 1).sqrt := by
rw [Nat.le_sqrt]
omega
rw [← Nat.sub_add_cancel hn'] at h₁ h₂ h₃
set l := (n + 1).sqrt - 1
refine ⟨l, ?_, ?_⟩
· calc n + 4 * l ≤ (l ^ 2 + 4 * l + 2) + 4 * l := by linarith only [h₂]
_ ≤ 2 * l ^ 2 := by nlinarith only [h₃]
· linarith only [h₁]
|
/root/DuelModelResearch/mathlib4/Archive/Imo/Imo2021Q1.lean
|
{
"open": [
"Finset"
],
"variables": []
}
|
[
{
"line": "have hn' : 1 ≤ Nat.sqrt (n + 1) := by\n rw [Nat.le_sqrt]\n apply Nat.le_add_left",
"before_state": "n : ℕ\nhn : 100 ≤ n\n⊢ ∃ l, n + 4 * l ≤ 2 * l ^ 2 ∧ 2 * l ^ 2 + 4 * l ≤ 2 * n",
"after_state": "n : ℕ\nhn : 100 ≤ n\nhn' : 1 ≤ (n + 1).sqrt\n⊢ ∃ l, n + 4 * l ≤ 2 * l ^ 2 ∧ 2 * l ^ 2 + 4 * l ≤ 2 * n"
},
{
"line": "focus\n refine\n no_implicit_lambda%\n (have hn' : 1 ≤ Nat.sqrt (n + 1) := ?body✝;\n ?_)\n case body✝ =>\n with_annotate_state\"by\"\n ( rw [Nat.le_sqrt]\n apply Nat.le_add_left)",
"before_state": "n : ℕ\nhn : 100 ≤ n\n⊢ ∃ l, n + 4 * l ≤ 2 * l ^ 2 ∧ 2 * l ^ 2 + 4 * l ≤ 2 * n",
"after_state": "n : ℕ\nhn : 100 ≤ n\nhn' : 1 ≤ (n + 1).sqrt\n⊢ ∃ l, n + 4 * l ≤ 2 * l ^ 2 ∧ 2 * l ^ 2 + 4 * l ≤ 2 * n"
},
{
"line": "refine\n no_implicit_lambda%\n (have hn' : 1 ≤ Nat.sqrt (n + 1) := ?body✝;\n ?_)",
"before_state": "n : ℕ\nhn : 100 ≤ n\n⊢ ∃ l, n + 4 * l ≤ 2 * l ^ 2 ∧ 2 * l ^ 2 + 4 * l ≤ 2 * n",
"after_state": "case body\nn : ℕ\nhn : 100 ≤ n\n⊢ 1 ≤ (n + 1).sqrt\n---\nn : ℕ\nhn : 100 ≤ n\nhn' : 1 ≤ (n + 1).sqrt\n⊢ ∃ l, n + 4 * l ≤ 2 * l ^ 2 ∧ 2 * l ^ 2 + 4 * l ≤ 2 * n"
},
{
"line": "case body✝ =>\n with_annotate_state\"by\"\n ( rw [Nat.le_sqrt]\n apply Nat.le_add_left)",
"before_state": "case body\nn : ℕ\nhn : 100 ≤ n\n⊢ 1 ≤ (n + 1).sqrt\n---\nn : ℕ\nhn : 100 ≤ n\nhn' : 1 ≤ (n + 1).sqrt\n⊢ ∃ l, n + 4 * l ≤ 2 * l ^ 2 ∧ 2 * l ^ 2 + 4 * l ≤ 2 * n",
"after_state": "n : ℕ\nhn : 100 ≤ n\nhn' : 1 ≤ (n + 1).sqrt\n⊢ ∃ l, n + 4 * l ≤ 2 * l ^ 2 ∧ 2 * l ^ 2 + 4 * l ≤ 2 * n"
},
{
"line": "with_annotate_state\"by\"\n ( rw [Nat.le_sqrt]\n apply Nat.le_add_left)",
"before_state": "n : ℕ\nhn : 100 ≤ n\n⊢ 1 ≤ (n + 1).sqrt",
"after_state": "No Goals!"
},
{
"line": "rw [Nat.le_sqrt]",
"before_state": "n : ℕ\nhn : 100 ≤ n\n⊢ 1 ≤ (n + 1).sqrt",
"after_state": "n : ℕ\nhn : 100 ≤ n\n⊢ 1 * 1 ≤ n + 1"
},
{
"line": "rewrite [Nat.le_sqrt]",
"before_state": "n : ℕ\nhn : 100 ≤ n\n⊢ 1 ≤ (n + 1).sqrt",
"after_state": "n : ℕ\nhn : 100 ≤ n\n⊢ 1 * 1 ≤ n + 1"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "n : ℕ\nhn : 100 ≤ n\n⊢ 1 * 1 ≤ n + 1",
"after_state": "n : ℕ\nhn : 100 ≤ n\n⊢ 1 * 1 ≤ n + 1"
},
{
"line": "try (with_reducible rfl)",
"before_state": "n : ℕ\nhn : 100 ≤ n\n⊢ 1 * 1 ≤ n + 1",
"after_state": "n : ℕ\nhn : 100 ≤ n\n⊢ 1 * 1 ≤ n + 1"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "n : ℕ\nhn : 100 ≤ n\n⊢ 1 * 1 ≤ n + 1",
"after_state": "n : ℕ\nhn : 100 ≤ n\n⊢ 1 * 1 ≤ n + 1"
},
{
"line": "with_reducible rfl",
"before_state": "n : ℕ\nhn : 100 ≤ n\n⊢ 1 * 1 ≤ n + 1",
"after_state": "n : ℕ\nhn : 100 ≤ n\n⊢ 1 * 1 ≤ n + 1"
},
{
"line": "rfl",
"before_state": "n : ℕ\nhn : 100 ≤ n\n⊢ 1 * 1 ≤ n + 1",
"after_state": "n : ℕ\nhn : 100 ≤ n\n⊢ 1 * 1 ≤ n + 1"
},
{
"line": "apply_rfl",
"before_state": "n : ℕ\nhn : 100 ≤ n\n⊢ 1 * 1 ≤ n + 1",
"after_state": "n : ℕ\nhn : 100 ≤ n\n⊢ 1 * 1 ≤ n + 1"
},
{
"line": "skip",
"before_state": "n : ℕ\nhn : 100 ≤ n\n⊢ 1 * 1 ≤ n + 1",
"after_state": "n : ℕ\nhn : 100 ≤ n\n⊢ 1 * 1 ≤ n + 1"
},
{
"line": "apply Nat.le_add_left",
"before_state": "n : ℕ\nhn : 100 ≤ n\n⊢ 1 * 1 ≤ n + 1",
"after_state": "No Goals!"
},
{
"line": "have h₁ := Nat.sqrt_le' (n + 1)",
"before_state": "n : ℕ\nhn : 100 ≤ n\nhn' : 1 ≤ (n + 1).sqrt\n⊢ ∃ l, n + 4 * l ≤ 2 * l ^ 2 ∧ 2 * l ^ 2 + 4 * l ≤ 2 * n",
"after_state": "n : ℕ\nhn : 100 ≤ n\nhn' : 1 ≤ (n + 1).sqrt\nh₁ : (n + 1).sqrt ^ 2 ≤ n + 1\n⊢ ∃ l, n + 4 * l ≤ 2 * l ^ 2 ∧ 2 * l ^ 2 + 4 * l ≤ 2 * n"
},
{
"line": "refine_lift\n have h₁ := Nat.sqrt_le' (n + 1);\n ?_",
"before_state": "n : ℕ\nhn : 100 ≤ n\nhn' : 1 ≤ (n + 1).sqrt\n⊢ ∃ l, n + 4 * l ≤ 2 * l ^ 2 ∧ 2 * l ^ 2 + 4 * l ≤ 2 * n",
"after_state": "n : ℕ\nhn : 100 ≤ n\nhn' : 1 ≤ (n + 1).sqrt\nh₁ : (n + 1).sqrt ^ 2 ≤ n + 1\n⊢ ∃ l, n + 4 * l ≤ 2 * l ^ 2 ∧ 2 * l ^ 2 + 4 * l ≤ 2 * n"
},
{
"line": "focus\n (refine\n no_implicit_lambda%\n (have h₁ := Nat.sqrt_le' (n + 1);\n ?_);\n rotate_right)",
"before_state": "n : ℕ\nhn : 100 ≤ n\nhn' : 1 ≤ (n + 1).sqrt\n⊢ ∃ l, n + 4 * l ≤ 2 * l ^ 2 ∧ 2 * l ^ 2 + 4 * l ≤ 2 * n",
"after_state": "n : ℕ\nhn : 100 ≤ n\nhn' : 1 ≤ (n + 1).sqrt\nh₁ : (n + 1).sqrt ^ 2 ≤ n + 1\n⊢ ∃ l, n + 4 * l ≤ 2 * l ^ 2 ∧ 2 * l ^ 2 + 4 * l ≤ 2 * n"
},
{
"line": "refine\n no_implicit_lambda%\n (have h₁ := Nat.sqrt_le' (n + 1);\n ?_)",
"before_state": "n : ℕ\nhn : 100 ≤ n\nhn' : 1 ≤ (n + 1).sqrt\n⊢ ∃ l, n + 4 * l ≤ 2 * l ^ 2 ∧ 2 * l ^ 2 + 4 * l ≤ 2 * n",
"after_state": "n : ℕ\nhn : 100 ≤ n\nhn' : 1 ≤ (n + 1).sqrt\nh₁ : (n + 1).sqrt ^ 2 ≤ n + 1\n⊢ ∃ l, n + 4 * l ≤ 2 * l ^ 2 ∧ 2 * l ^ 2 + 4 * l ≤ 2 * n"
},
{
"line": "rotate_right",
"before_state": "n : ℕ\nhn : 100 ≤ n\nhn' : 1 ≤ (n + 1).sqrt\nh₁ : (n + 1).sqrt ^ 2 ≤ n + 1\n⊢ ∃ l, n + 4 * l ≤ 2 * l ^ 2 ∧ 2 * l ^ 2 + 4 * l ≤ 2 * n",
"after_state": "n : ℕ\nhn : 100 ≤ n\nhn' : 1 ≤ (n + 1).sqrt\nh₁ : (n + 1).sqrt ^ 2 ≤ n + 1\n⊢ ∃ l, n + 4 * l ≤ 2 * l ^ 2 ∧ 2 * l ^ 2 + 4 * l ≤ 2 * n"
},
{
"line": "have h₂ := Nat.succ_le_succ_sqrt' (n + 1)",
"before_state": "n : ℕ\nhn : 100 ≤ n\nhn' : 1 ≤ (n + 1).sqrt\nh₁ : (n + 1).sqrt ^ 2 ≤ n + 1\n⊢ ∃ l, n + 4 * l ≤ 2 * l ^ 2 ∧ 2 * l ^ 2 + 4 * l ≤ 2 * n",
"after_state": "n : ℕ\nhn : 100 ≤ n\nhn' : 1 ≤ (n + 1).sqrt\nh₁ : (n + 1).sqrt ^ 2 ≤ n + 1\nh₂ : n + 1 + 1 ≤ ((n + 1).sqrt + 1) ^ 2\n⊢ ∃ l, n + 4 * l ≤ 2 * l ^ 2 ∧ 2 * l ^ 2 + 4 * l ≤ 2 * n"
},
{
"line": "refine_lift\n have h₂ := Nat.succ_le_succ_sqrt' (n + 1);\n ?_",
"before_state": "n : ℕ\nhn : 100 ≤ n\nhn' : 1 ≤ (n + 1).sqrt\nh₁ : (n + 1).sqrt ^ 2 ≤ n + 1\n⊢ ∃ l, n + 4 * l ≤ 2 * l ^ 2 ∧ 2 * l ^ 2 + 4 * l ≤ 2 * n",
"after_state": "n : ℕ\nhn : 100 ≤ n\nhn' : 1 ≤ (n + 1).sqrt\nh₁ : (n + 1).sqrt ^ 2 ≤ n + 1\nh₂ : n + 1 + 1 ≤ ((n + 1).sqrt + 1) ^ 2\n⊢ ∃ l, n + 4 * l ≤ 2 * l ^ 2 ∧ 2 * l ^ 2 + 4 * l ≤ 2 * n"
},
{
"line": "focus\n (refine\n no_implicit_lambda%\n (have h₂ := Nat.succ_le_succ_sqrt' (n + 1);\n ?_);\n rotate_right)",
"before_state": "n : ℕ\nhn : 100 ≤ n\nhn' : 1 ≤ (n + 1).sqrt\nh₁ : (n + 1).sqrt ^ 2 ≤ n + 1\n⊢ ∃ l, n + 4 * l ≤ 2 * l ^ 2 ∧ 2 * l ^ 2 + 4 * l ≤ 2 * n",
"after_state": "n : ℕ\nhn : 100 ≤ n\nhn' : 1 ≤ (n + 1).sqrt\nh₁ : (n + 1).sqrt ^ 2 ≤ n + 1\nh₂ : n + 1 + 1 ≤ ((n + 1).sqrt + 1) ^ 2\n⊢ ∃ l, n + 4 * l ≤ 2 * l ^ 2 ∧ 2 * l ^ 2 + 4 * l ≤ 2 * n"
},
{
"line": "refine\n no_implicit_lambda%\n (have h₂ := Nat.succ_le_succ_sqrt' (n + 1);\n ?_)",
"before_state": "n : ℕ\nhn : 100 ≤ n\nhn' : 1 ≤ (n + 1).sqrt\nh₁ : (n + 1).sqrt ^ 2 ≤ n + 1\n⊢ ∃ l, n + 4 * l ≤ 2 * l ^ 2 ∧ 2 * l ^ 2 + 4 * l ≤ 2 * n",
"after_state": "n : ℕ\nhn : 100 ≤ n\nhn' : 1 ≤ (n + 1).sqrt\nh₁ : (n + 1).sqrt ^ 2 ≤ n + 1\nh₂ : n + 1 + 1 ≤ ((n + 1).sqrt + 1) ^ 2\n⊢ ∃ l, n + 4 * l ≤ 2 * l ^ 2 ∧ 2 * l ^ 2 + 4 * l ≤ 2 * n"
},
{
"line": "rotate_right",
"before_state": "n : ℕ\nhn : 100 ≤ n\nhn' : 1 ≤ (n + 1).sqrt\nh₁ : (n + 1).sqrt ^ 2 ≤ n + 1\nh₂ : n + 1 + 1 ≤ ((n + 1).sqrt + 1) ^ 2\n⊢ ∃ l, n + 4 * l ≤ 2 * l ^ 2 ∧ 2 * l ^ 2 + 4 * l ≤ 2 * n",
"after_state": "n : ℕ\nhn : 100 ≤ n\nhn' : 1 ≤ (n + 1).sqrt\nh₁ : (n + 1).sqrt ^ 2 ≤ n + 1\nh₂ : n + 1 + 1 ≤ ((n + 1).sqrt + 1) ^ 2\n⊢ ∃ l, n + 4 * l ≤ 2 * l ^ 2 ∧ 2 * l ^ 2 + 4 * l ≤ 2 * n"
},
{
"line": "have h₃ : 10 ≤ (n + 1).sqrt := by\n rw [Nat.le_sqrt]\n omega",
"before_state": "n : ℕ\nhn : 100 ≤ n\nhn' : 1 ≤ (n + 1).sqrt\nh₁ : (n + 1).sqrt ^ 2 ≤ n + 1\nh₂ : n + 1 + 1 ≤ ((n + 1).sqrt + 1) ^ 2\n⊢ ∃ l, n + 4 * l ≤ 2 * l ^ 2 ∧ 2 * l ^ 2 + 4 * l ≤ 2 * n",
"after_state": "n : ℕ\nhn : 100 ≤ n\nhn' : 1 ≤ (n + 1).sqrt\nh₁ : (n + 1).sqrt ^ 2 ≤ n + 1\nh₂ : n + 1 + 1 ≤ ((n + 1).sqrt + 1) ^ 2\nh₃ : 10 ≤ (n + 1).sqrt\n⊢ ∃ l, n + 4 * l ≤ 2 * l ^ 2 ∧ 2 * l ^ 2 + 4 * l ≤ 2 * n"
},
{
"line": "focus\n refine\n no_implicit_lambda%\n (have h₃ : 10 ≤ (n + 1).sqrt := ?body✝;\n ?_)\n case body✝ =>\n with_annotate_state\"by\"\n ( rw [Nat.le_sqrt]\n omega)",
"before_state": "n : ℕ\nhn : 100 ≤ n\nhn' : 1 ≤ (n + 1).sqrt\nh₁ : (n + 1).sqrt ^ 2 ≤ n + 1\nh₂ : n + 1 + 1 ≤ ((n + 1).sqrt + 1) ^ 2\n⊢ ∃ l, n + 4 * l ≤ 2 * l ^ 2 ∧ 2 * l ^ 2 + 4 * l ≤ 2 * n",
"after_state": "n : ℕ\nhn : 100 ≤ n\nhn' : 1 ≤ (n + 1).sqrt\nh₁ : (n + 1).sqrt ^ 2 ≤ n + 1\nh₂ : n + 1 + 1 ≤ ((n + 1).sqrt + 1) ^ 2\nh₃ : 10 ≤ (n + 1).sqrt\n⊢ ∃ l, n + 4 * l ≤ 2 * l ^ 2 ∧ 2 * l ^ 2 + 4 * l ≤ 2 * n"
},
{
"line": "refine\n no_implicit_lambda%\n (have h₃ : 10 ≤ (n + 1).sqrt := ?body✝;\n ?_)",
"before_state": "n : ℕ\nhn : 100 ≤ n\nhn' : 1 ≤ (n + 1).sqrt\nh₁ : (n + 1).sqrt ^ 2 ≤ n + 1\nh₂ : n + 1 + 1 ≤ ((n + 1).sqrt + 1) ^ 2\n⊢ ∃ l, n + 4 * l ≤ 2 * l ^ 2 ∧ 2 * l ^ 2 + 4 * l ≤ 2 * n",
"after_state": "case body\nn : ℕ\nhn : 100 ≤ n\nhn' : 1 ≤ (n + 1).sqrt\nh₁ : (n + 1).sqrt ^ 2 ≤ n + 1\nh₂ : n + 1 + 1 ≤ ((n + 1).sqrt + 1) ^ 2\n⊢ 10 ≤ (n + 1).sqrt\n---\nn : ℕ\nhn : 100 ≤ n\nhn' : 1 ≤ (n + 1).sqrt\nh₁ : (n + 1).sqrt ^ 2 ≤ n + 1\nh₂ : n + 1 + 1 ≤ ((n + 1).sqrt + 1) ^ 2\nh₃ : 10 ≤ (n + 1).sqrt\n⊢ ∃ l, n + 4 * l ≤ 2 * l ^ 2 ∧ 2 * l ^ 2 + 4 * l ≤ 2 * n"
},
{
"line": "case body✝ =>\n with_annotate_state\"by\"\n ( rw [Nat.le_sqrt]\n omega)",
"before_state": "case body\nn : ℕ\nhn : 100 ≤ n\nhn' : 1 ≤ (n + 1).sqrt\nh₁ : (n + 1).sqrt ^ 2 ≤ n + 1\nh₂ : n + 1 + 1 ≤ ((n + 1).sqrt + 1) ^ 2\n⊢ 10 ≤ (n + 1).sqrt\n---\nn : ℕ\nhn : 100 ≤ n\nhn' : 1 ≤ (n + 1).sqrt\nh₁ : (n + 1).sqrt ^ 2 ≤ n + 1\nh₂ : n + 1 + 1 ≤ ((n + 1).sqrt + 1) ^ 2\nh₃ : 10 ≤ (n + 1).sqrt\n⊢ ∃ l, n + 4 * l ≤ 2 * l ^ 2 ∧ 2 * l ^ 2 + 4 * l ≤ 2 * n",
"after_state": "n : ℕ\nhn : 100 ≤ n\nhn' : 1 ≤ (n + 1).sqrt\nh₁ : (n + 1).sqrt ^ 2 ≤ n + 1\nh₂ : n + 1 + 1 ≤ ((n + 1).sqrt + 1) ^ 2\nh₃ : 10 ≤ (n + 1).sqrt\n⊢ ∃ l, n + 4 * l ≤ 2 * l ^ 2 ∧ 2 * l ^ 2 + 4 * l ≤ 2 * n"
},
{
"line": "with_annotate_state\"by\"\n ( rw [Nat.le_sqrt]\n omega)",
"before_state": "n : ℕ\nhn : 100 ≤ n\nhn' : 1 ≤ (n + 1).sqrt\nh₁ : (n + 1).sqrt ^ 2 ≤ n + 1\nh₂ : n + 1 + 1 ≤ ((n + 1).sqrt + 1) ^ 2\n⊢ 10 ≤ (n + 1).sqrt",
"after_state": "No Goals!"
},
{
"line": "rw [Nat.le_sqrt]",
"before_state": "n : ℕ\nhn : 100 ≤ n\nhn' : 1 ≤ (n + 1).sqrt\nh₁ : (n + 1).sqrt ^ 2 ≤ n + 1\nh₂ : n + 1 + 1 ≤ ((n + 1).sqrt + 1) ^ 2\n⊢ 10 ≤ (n + 1).sqrt",
"after_state": "n : ℕ\nhn : 100 ≤ n\nhn' : 1 ≤ (n + 1).sqrt\nh₁ : (n + 1).sqrt ^ 2 ≤ n + 1\nh₂ : n + 1 + 1 ≤ ((n + 1).sqrt + 1) ^ 2\n⊢ 10 * 10 ≤ n + 1"
},
{
"line": "rewrite [Nat.le_sqrt]",
"before_state": "n : ℕ\nhn : 100 ≤ n\nhn' : 1 ≤ (n + 1).sqrt\nh₁ : (n + 1).sqrt ^ 2 ≤ n + 1\nh₂ : n + 1 + 1 ≤ ((n + 1).sqrt + 1) ^ 2\n⊢ 10 ≤ (n + 1).sqrt",
"after_state": "n : ℕ\nhn : 100 ≤ n\nhn' : 1 ≤ (n + 1).sqrt\nh₁ : (n + 1).sqrt ^ 2 ≤ n + 1\nh₂ : n + 1 + 1 ≤ ((n + 1).sqrt + 1) ^ 2\n⊢ 10 * 10 ≤ n + 1"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "n : ℕ\nhn : 100 ≤ n\nhn' : 1 ≤ (n + 1).sqrt\nh₁ : (n + 1).sqrt ^ 2 ≤ n + 1\nh₂ : n + 1 + 1 ≤ ((n + 1).sqrt + 1) ^ 2\n⊢ 10 * 10 ≤ n + 1",
"after_state": "n : ℕ\nhn : 100 ≤ n\nhn' : 1 ≤ (n + 1).sqrt\nh₁ : (n + 1).sqrt ^ 2 ≤ n + 1\nh₂ : n + 1 + 1 ≤ ((n + 1).sqrt + 1) ^ 2\n⊢ 10 * 10 ≤ n + 1"
},
{
"line": "try (with_reducible rfl)",
"before_state": "n : ℕ\nhn : 100 ≤ n\nhn' : 1 ≤ (n + 1).sqrt\nh₁ : (n + 1).sqrt ^ 2 ≤ n + 1\nh₂ : n + 1 + 1 ≤ ((n + 1).sqrt + 1) ^ 2\n⊢ 10 * 10 ≤ n + 1",
"after_state": "n : ℕ\nhn : 100 ≤ n\nhn' : 1 ≤ (n + 1).sqrt\nh₁ : (n + 1).sqrt ^ 2 ≤ n + 1\nh₂ : n + 1 + 1 ≤ ((n + 1).sqrt + 1) ^ 2\n⊢ 10 * 10 ≤ n + 1"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "n : ℕ\nhn : 100 ≤ n\nhn' : 1 ≤ (n + 1).sqrt\nh₁ : (n + 1).sqrt ^ 2 ≤ n + 1\nh₂ : n + 1 + 1 ≤ ((n + 1).sqrt + 1) ^ 2\n⊢ 10 * 10 ≤ n + 1",
"after_state": "n : ℕ\nhn : 100 ≤ n\nhn' : 1 ≤ (n + 1).sqrt\nh₁ : (n + 1).sqrt ^ 2 ≤ n + 1\nh₂ : n + 1 + 1 ≤ ((n + 1).sqrt + 1) ^ 2\n⊢ 10 * 10 ≤ n + 1"
},
{
"line": "with_reducible rfl",
"before_state": "n : ℕ\nhn : 100 ≤ n\nhn' : 1 ≤ (n + 1).sqrt\nh₁ : (n + 1).sqrt ^ 2 ≤ n + 1\nh₂ : n + 1 + 1 ≤ ((n + 1).sqrt + 1) ^ 2\n⊢ 10 * 10 ≤ n + 1",
"after_state": "n : ℕ\nhn : 100 ≤ n\nhn' : 1 ≤ (n + 1).sqrt\nh₁ : (n + 1).sqrt ^ 2 ≤ n + 1\nh₂ : n + 1 + 1 ≤ ((n + 1).sqrt + 1) ^ 2\n⊢ 10 * 10 ≤ n + 1"
},
{
"line": "rfl",
"before_state": "n : ℕ\nhn : 100 ≤ n\nhn' : 1 ≤ (n + 1).sqrt\nh₁ : (n + 1).sqrt ^ 2 ≤ n + 1\nh₂ : n + 1 + 1 ≤ ((n + 1).sqrt + 1) ^ 2\n⊢ 10 * 10 ≤ n + 1",
"after_state": "n : ℕ\nhn : 100 ≤ n\nhn' : 1 ≤ (n + 1).sqrt\nh₁ : (n + 1).sqrt ^ 2 ≤ n + 1\nh₂ : n + 1 + 1 ≤ ((n + 1).sqrt + 1) ^ 2\n⊢ 10 * 10 ≤ n + 1"
},
{
"line": "apply_rfl",
"before_state": "n : ℕ\nhn : 100 ≤ n\nhn' : 1 ≤ (n + 1).sqrt\nh₁ : (n + 1).sqrt ^ 2 ≤ n + 1\nh₂ : n + 1 + 1 ≤ ((n + 1).sqrt + 1) ^ 2\n⊢ 10 * 10 ≤ n + 1",
"after_state": "n : ℕ\nhn : 100 ≤ n\nhn' : 1 ≤ (n + 1).sqrt\nh₁ : (n + 1).sqrt ^ 2 ≤ n + 1\nh₂ : n + 1 + 1 ≤ ((n + 1).sqrt + 1) ^ 2\n⊢ 10 * 10 ≤ n + 1"
},
{
"line": "skip",
"before_state": "n : ℕ\nhn : 100 ≤ n\nhn' : 1 ≤ (n + 1).sqrt\nh₁ : (n + 1).sqrt ^ 2 ≤ n + 1\nh₂ : n + 1 + 1 ≤ ((n + 1).sqrt + 1) ^ 2\n⊢ 10 * 10 ≤ n + 1",
"after_state": "n : ℕ\nhn : 100 ≤ n\nhn' : 1 ≤ (n + 1).sqrt\nh₁ : (n + 1).sqrt ^ 2 ≤ n + 1\nh₂ : n + 1 + 1 ≤ ((n + 1).sqrt + 1) ^ 2\n⊢ 10 * 10 ≤ n + 1"
},
{
"line": "omega",
"before_state": "n : ℕ\nhn : 100 ≤ n\nhn' : 1 ≤ (n + 1).sqrt\nh₁ : (n + 1).sqrt ^ 2 ≤ n + 1\nh₂ : n + 1 + 1 ≤ ((n + 1).sqrt + 1) ^ 2\n⊢ 10 * 10 ≤ n + 1",
"after_state": "No Goals!"
},
{
"line": "rw [← Nat.sub_add_cancel hn'] at h₁ h₂ h₃",
"before_state": "n : ℕ\nhn : 100 ≤ n\nhn' : 1 ≤ (n + 1).sqrt\nh₁ : (n + 1).sqrt ^ 2 ≤ n + 1\nh₂ : n + 1 + 1 ≤ ((n + 1).sqrt + 1) ^ 2\nh₃ : 10 ≤ (n + 1).sqrt\n⊢ ∃ l, n + 4 * l ≤ 2 * l ^ 2 ∧ 2 * l ^ 2 + 4 * l ≤ 2 * n",
"after_state": "n : ℕ\nhn : 100 ≤ n\nhn' : 1 ≤ (n + 1).sqrt\nh₁ : ((n + 1).sqrt - 1 + 1) ^ 2 ≤ n + 1\nh₂ : n + 1 + 1 ≤ ((n + 1).sqrt - 1 + 1 + 1) ^ 2\nh₃ : 10 ≤ (n + 1).sqrt - 1 + 1\n⊢ ∃ l, n + 4 * l ≤ 2 * l ^ 2 ∧ 2 * l ^ 2 + 4 * l ≤ 2 * n"
},
{
"line": "rewrite [← Nat.sub_add_cancel hn'] at h₁ h₂ h₃",
"before_state": "n : ℕ\nhn : 100 ≤ n\nhn' : 1 ≤ (n + 1).sqrt\nh₁ : (n + 1).sqrt ^ 2 ≤ n + 1\nh₂ : n + 1 + 1 ≤ ((n + 1).sqrt + 1) ^ 2\nh₃ : 10 ≤ (n + 1).sqrt\n⊢ ∃ l, n + 4 * l ≤ 2 * l ^ 2 ∧ 2 * l ^ 2 + 4 * l ≤ 2 * n",
"after_state": "n : ℕ\nhn : 100 ≤ n\nhn' : 1 ≤ (n + 1).sqrt\nh₁ : ((n + 1).sqrt - 1 + 1) ^ 2 ≤ n + 1\nh₂ : n + 1 + 1 ≤ ((n + 1).sqrt - 1 + 1 + 1) ^ 2\nh₃ : 10 ≤ (n + 1).sqrt - 1 + 1\n⊢ ∃ l, n + 4 * l ≤ 2 * l ^ 2 ∧ 2 * l ^ 2 + 4 * l ≤ 2 * n"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "n : ℕ\nhn : 100 ≤ n\nhn' : 1 ≤ (n + 1).sqrt\nh₁ : ((n + 1).sqrt - 1 + 1) ^ 2 ≤ n + 1\nh₂ : n + 1 + 1 ≤ ((n + 1).sqrt - 1 + 1 + 1) ^ 2\nh₃ : 10 ≤ (n + 1).sqrt - 1 + 1\n⊢ ∃ l, n + 4 * l ≤ 2 * l ^ 2 ∧ 2 * l ^ 2 + 4 * l ≤ 2 * n",
"after_state": "n : ℕ\nhn : 100 ≤ n\nhn' : 1 ≤ (n + 1).sqrt\nh₁ : ((n + 1).sqrt - 1 + 1) ^ 2 ≤ n + 1\nh₂ : n + 1 + 1 ≤ ((n + 1).sqrt - 1 + 1 + 1) ^ 2\nh₃ : 10 ≤ (n + 1).sqrt - 1 + 1\n⊢ ∃ l, n + 4 * l ≤ 2 * l ^ 2 ∧ 2 * l ^ 2 + 4 * l ≤ 2 * n"
},
{
"line": "try (with_reducible rfl)",
"before_state": "n : ℕ\nhn : 100 ≤ n\nhn' : 1 ≤ (n + 1).sqrt\nh₁ : ((n + 1).sqrt - 1 + 1) ^ 2 ≤ n + 1\nh₂ : n + 1 + 1 ≤ ((n + 1).sqrt - 1 + 1 + 1) ^ 2\nh₃ : 10 ≤ (n + 1).sqrt - 1 + 1\n⊢ ∃ l, n + 4 * l ≤ 2 * l ^ 2 ∧ 2 * l ^ 2 + 4 * l ≤ 2 * n",
"after_state": "n : ℕ\nhn : 100 ≤ n\nhn' : 1 ≤ (n + 1).sqrt\nh₁ : ((n + 1).sqrt - 1 + 1) ^ 2 ≤ n + 1\nh₂ : n + 1 + 1 ≤ ((n + 1).sqrt - 1 + 1 + 1) ^ 2\nh₃ : 10 ≤ (n + 1).sqrt - 1 + 1\n⊢ ∃ l, n + 4 * l ≤ 2 * l ^ 2 ∧ 2 * l ^ 2 + 4 * l ≤ 2 * n"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "n : ℕ\nhn : 100 ≤ n\nhn' : 1 ≤ (n + 1).sqrt\nh₁ : ((n + 1).sqrt - 1 + 1) ^ 2 ≤ n + 1\nh₂ : n + 1 + 1 ≤ ((n + 1).sqrt - 1 + 1 + 1) ^ 2\nh₃ : 10 ≤ (n + 1).sqrt - 1 + 1\n⊢ ∃ l, n + 4 * l ≤ 2 * l ^ 2 ∧ 2 * l ^ 2 + 4 * l ≤ 2 * n",
"after_state": "n : ℕ\nhn : 100 ≤ n\nhn' : 1 ≤ (n + 1).sqrt\nh₁ : ((n + 1).sqrt - 1 + 1) ^ 2 ≤ n + 1\nh₂ : n + 1 + 1 ≤ ((n + 1).sqrt - 1 + 1 + 1) ^ 2\nh₃ : 10 ≤ (n + 1).sqrt - 1 + 1\n⊢ ∃ l, n + 4 * l ≤ 2 * l ^ 2 ∧ 2 * l ^ 2 + 4 * l ≤ 2 * n"
},
{
"line": "with_reducible rfl",
"before_state": "n : ℕ\nhn : 100 ≤ n\nhn' : 1 ≤ (n + 1).sqrt\nh₁ : ((n + 1).sqrt - 1 + 1) ^ 2 ≤ n + 1\nh₂ : n + 1 + 1 ≤ ((n + 1).sqrt - 1 + 1 + 1) ^ 2\nh₃ : 10 ≤ (n + 1).sqrt - 1 + 1\n⊢ ∃ l, n + 4 * l ≤ 2 * l ^ 2 ∧ 2 * l ^ 2 + 4 * l ≤ 2 * n",
"after_state": "n : ℕ\nhn : 100 ≤ n\nhn' : 1 ≤ (n + 1).sqrt\nh₁ : ((n + 1).sqrt - 1 + 1) ^ 2 ≤ n + 1\nh₂ : n + 1 + 1 ≤ ((n + 1).sqrt - 1 + 1 + 1) ^ 2\nh₃ : 10 ≤ (n + 1).sqrt - 1 + 1\n⊢ ∃ l, n + 4 * l ≤ 2 * l ^ 2 ∧ 2 * l ^ 2 + 4 * l ≤ 2 * n"
},
{
"line": "rfl",
"before_state": "n : ℕ\nhn : 100 ≤ n\nhn' : 1 ≤ (n + 1).sqrt\nh₁ : ((n + 1).sqrt - 1 + 1) ^ 2 ≤ n + 1\nh₂ : n + 1 + 1 ≤ ((n + 1).sqrt - 1 + 1 + 1) ^ 2\nh₃ : 10 ≤ (n + 1).sqrt - 1 + 1\n⊢ ∃ l, n + 4 * l ≤ 2 * l ^ 2 ∧ 2 * l ^ 2 + 4 * l ≤ 2 * n",
"after_state": "n : ℕ\nhn : 100 ≤ n\nhn' : 1 ≤ (n + 1).sqrt\nh₁ : ((n + 1).sqrt - 1 + 1) ^ 2 ≤ n + 1\nh₂ : n + 1 + 1 ≤ ((n + 1).sqrt - 1 + 1 + 1) ^ 2\nh₃ : 10 ≤ (n + 1).sqrt - 1 + 1\n⊢ ∃ l, n + 4 * l ≤ 2 * l ^ 2 ∧ 2 * l ^ 2 + 4 * l ≤ 2 * n"
},
{
"line": "apply_rfl",
"before_state": "n : ℕ\nhn : 100 ≤ n\nhn' : 1 ≤ (n + 1).sqrt\nh₁ : ((n + 1).sqrt - 1 + 1) ^ 2 ≤ n + 1\nh₂ : n + 1 + 1 ≤ ((n + 1).sqrt - 1 + 1 + 1) ^ 2\nh₃ : 10 ≤ (n + 1).sqrt - 1 + 1\n⊢ ∃ l, n + 4 * l ≤ 2 * l ^ 2 ∧ 2 * l ^ 2 + 4 * l ≤ 2 * n",
"after_state": "n : ℕ\nhn : 100 ≤ n\nhn' : 1 ≤ (n + 1).sqrt\nh₁ : ((n + 1).sqrt - 1 + 1) ^ 2 ≤ n + 1\nh₂ : n + 1 + 1 ≤ ((n + 1).sqrt - 1 + 1 + 1) ^ 2\nh₃ : 10 ≤ (n + 1).sqrt - 1 + 1\n⊢ ∃ l, n + 4 * l ≤ 2 * l ^ 2 ∧ 2 * l ^ 2 + 4 * l ≤ 2 * n"
},
{
"line": "skip",
"before_state": "n : ℕ\nhn : 100 ≤ n\nhn' : 1 ≤ (n + 1).sqrt\nh₁ : ((n + 1).sqrt - 1 + 1) ^ 2 ≤ n + 1\nh₂ : n + 1 + 1 ≤ ((n + 1).sqrt - 1 + 1 + 1) ^ 2\nh₃ : 10 ≤ (n + 1).sqrt - 1 + 1\n⊢ ∃ l, n + 4 * l ≤ 2 * l ^ 2 ∧ 2 * l ^ 2 + 4 * l ≤ 2 * n",
"after_state": "n : ℕ\nhn : 100 ≤ n\nhn' : 1 ≤ (n + 1).sqrt\nh₁ : ((n + 1).sqrt - 1 + 1) ^ 2 ≤ n + 1\nh₂ : n + 1 + 1 ≤ ((n + 1).sqrt - 1 + 1 + 1) ^ 2\nh₃ : 10 ≤ (n + 1).sqrt - 1 + 1\n⊢ ∃ l, n + 4 * l ≤ 2 * l ^ 2 ∧ 2 * l ^ 2 + 4 * l ≤ 2 * n"
},
{
"line": "set l := (n + 1).sqrt - 1",
"before_state": "n : ℕ\nhn : 100 ≤ n\nhn' : 1 ≤ (n + 1).sqrt\nh₁ : ((n + 1).sqrt - 1 + 1) ^ 2 ≤ n + 1\nh₂ : n + 1 + 1 ≤ ((n + 1).sqrt - 1 + 1 + 1) ^ 2\nh₃ : 10 ≤ (n + 1).sqrt - 1 + 1\n⊢ ∃ l, n + 4 * l ≤ 2 * l ^ 2 ∧ 2 * l ^ 2 + 4 * l ≤ 2 * n",
"after_state": "n : ℕ\nhn : 100 ≤ n\nhn' : 1 ≤ (n + 1).sqrt\nl : ℕ := (n + 1).sqrt - 1\nh₁ : (l + 1) ^ 2 ≤ n + 1\nh₂ : n + 1 + 1 ≤ (l + 1 + 1) ^ 2\nh₃ : 10 ≤ l + 1\n⊢ ∃ l, n + 4 * l ≤ 2 * l ^ 2 ∧ 2 * l ^ 2 + 4 * l ≤ 2 * n"
},
{
"line": "try rewrite [show ?m✝ = l from rfl✝] at *",
"before_state": "n : ℕ\nhn : 100 ≤ n\nhn' : 1 ≤ (n + 1).sqrt\nh₁ : ((n + 1).sqrt - 1 + 1) ^ 2 ≤ n + 1\nh₂ : n + 1 + 1 ≤ ((n + 1).sqrt - 1 + 1 + 1) ^ 2\nh₃ : 10 ≤ (n + 1).sqrt - 1 + 1\nl : ℕ := (n + 1).sqrt - 1\n⊢ ∃ l, n + 4 * l ≤ 2 * l ^ 2 ∧ 2 * l ^ 2 + 4 * l ≤ 2 * n",
"after_state": "n : ℕ\nhn : 100 ≤ n\nhn' : 1 ≤ (n + 1).sqrt\nl : ℕ := (n + 1).sqrt - 1\nh₁ : (l + 1) ^ 2 ≤ n + 1\nh₂ : n + 1 + 1 ≤ (l + 1 + 1) ^ 2\nh₃ : 10 ≤ l + 1\n⊢ ∃ l, n + 4 * l ≤ 2 * l ^ 2 ∧ 2 * l ^ 2 + 4 * l ≤ 2 * n"
},
{
"line": "first\n| rewrite [show ?m✝ = l from rfl✝] at *\n| skip",
"before_state": "n : ℕ\nhn : 100 ≤ n\nhn' : 1 ≤ (n + 1).sqrt\nh₁ : ((n + 1).sqrt - 1 + 1) ^ 2 ≤ n + 1\nh₂ : n + 1 + 1 ≤ ((n + 1).sqrt - 1 + 1 + 1) ^ 2\nh₃ : 10 ≤ (n + 1).sqrt - 1 + 1\nl : ℕ := (n + 1).sqrt - 1\n⊢ ∃ l, n + 4 * l ≤ 2 * l ^ 2 ∧ 2 * l ^ 2 + 4 * l ≤ 2 * n",
"after_state": "n : ℕ\nhn : 100 ≤ n\nhn' : 1 ≤ (n + 1).sqrt\nl : ℕ := (n + 1).sqrt - 1\nh₁ : (l + 1) ^ 2 ≤ n + 1\nh₂ : n + 1 + 1 ≤ (l + 1 + 1) ^ 2\nh₃ : 10 ≤ l + 1\n⊢ ∃ l, n + 4 * l ≤ 2 * l ^ 2 ∧ 2 * l ^ 2 + 4 * l ≤ 2 * n"
},
{
"line": "rewrite [show ?m✝ = l from rfl✝] at *",
"before_state": "n : ℕ\nhn : 100 ≤ n\nhn' : 1 ≤ (n + 1).sqrt\nh₁ : ((n + 1).sqrt - 1 + 1) ^ 2 ≤ n + 1\nh₂ : n + 1 + 1 ≤ ((n + 1).sqrt - 1 + 1 + 1) ^ 2\nh₃ : 10 ≤ (n + 1).sqrt - 1 + 1\nl : ℕ := (n + 1).sqrt - 1\n⊢ ∃ l, n + 4 * l ≤ 2 * l ^ 2 ∧ 2 * l ^ 2 + 4 * l ≤ 2 * n",
"after_state": "n : ℕ\nhn : 100 ≤ n\nhn' : 1 ≤ (n + 1).sqrt\nl : ℕ := (n + 1).sqrt - 1\nh₁ : (l + 1) ^ 2 ≤ n + 1\nh₂ : n + 1 + 1 ≤ (l + 1 + 1) ^ 2\nh₃ : 10 ≤ l + 1\n⊢ ∃ l, n + 4 * l ≤ 2 * l ^ 2 ∧ 2 * l ^ 2 + 4 * l ≤ 2 * n"
},
{
"line": "refine ⟨l, ?_, ?_⟩",
"before_state": "n : ℕ\nhn : 100 ≤ n\nhn' : 1 ≤ (n + 1).sqrt\nl : ℕ := (n + 1).sqrt - 1\nh₁ : (l + 1) ^ 2 ≤ n + 1\nh₂ : n + 1 + 1 ≤ (l + 1 + 1) ^ 2\nh₃ : 10 ≤ l + 1\n⊢ ∃ l, n + 4 * l ≤ 2 * l ^ 2 ∧ 2 * l ^ 2 + 4 * l ≤ 2 * n",
"after_state": "case refine_1\nn : ℕ\nhn : 100 ≤ n\nhn' : 1 ≤ (n + 1).sqrt\nl : ℕ := (n + 1).sqrt - 1\nh₁ : (l + 1) ^ 2 ≤ n + 1\nh₂ : n + 1 + 1 ≤ (l + 1 + 1) ^ 2\nh₃ : 10 ≤ l + 1\n⊢ n + 4 * l ≤ 2 * l ^ 2\n---\ncase refine_2\nn : ℕ\nhn : 100 ≤ n\nhn' : 1 ≤ (n + 1).sqrt\nl : ℕ := (n + 1).sqrt - 1\nh₁ : (l + 1) ^ 2 ≤ n + 1\nh₂ : n + 1 + 1 ≤ (l + 1 + 1) ^ 2\nh₃ : 10 ≤ l + 1\n⊢ 2 * l ^ 2 + 4 * l ≤ 2 * n"
},
{
"line": "calc\n n + 4 * l ≤ (l ^ 2 + 4 * l + 2) + 4 * l := by linarith only [h₂]\n _ ≤ 2 * l ^ 2 := by nlinarith only [h₃]",
"before_state": "case refine_1\nn : ℕ\nhn : 100 ≤ n\nhn' : 1 ≤ (n + 1).sqrt\nl : ℕ := (n + 1).sqrt - 1\nh₁ : (l + 1) ^ 2 ≤ n + 1\nh₂ : n + 1 + 1 ≤ (l + 1 + 1) ^ 2\nh₃ : 10 ≤ l + 1\n⊢ n + 4 * l ≤ 2 * l ^ 2",
"after_state": "No Goals!"
},
{
"line": "linarith only [h₂]",
"before_state": "n : ℕ\nhn : 100 ≤ n\nhn' : 1 ≤ (n + 1).sqrt\nl : ℕ := (n + 1).sqrt - 1\nh₁ : (l + 1) ^ 2 ≤ n + 1\nh₂ : n + 1 + 1 ≤ (l + 1 + 1) ^ 2\nh₃ : 10 ≤ l + 1\n⊢ n + 4 * l ≤ l ^ 2 + 4 * l + 2 + 4 * l",
"after_state": "No Goals!"
},
{
"line": "ring1",
"before_state": "n : ℕ\nhn : 100 ≤ n\nhn' : 1 ≤ (n + 1).sqrt\nl : ℕ := (n + 1).sqrt - 1\nh₁ : (l + 1) ^ 2 ≤ n + 1\nh₂ : n + 1 + 1 ≤ (l + 1 + 1) ^ 2\nh₃ : 10 ≤ l + 1\na✝ : n + 4 * l > l ^ 2 + 4 * l + 2 + 4 * l\n⊢ -1 + (↑l ^ 2 + 4 * ↑l + 2 + 4 * ↑l + 1 - (↑n + 4 * ↑l)) + (↑n + 1 + 1 - (↑l + 1 + 1) ^ 2) = 0",
"after_state": "No Goals!"
},
{
"line": "nlinarith only [h₃]",
"before_state": "n : ℕ\nhn : 100 ≤ n\nhn' : 1 ≤ (n + 1).sqrt\nl : ℕ := (n + 1).sqrt - 1\nh₁ : (l + 1) ^ 2 ≤ n + 1\nh₂ : n + 1 + 1 ≤ (l + 1 + 1) ^ 2\nh₃ : 10 ≤ l + 1\n⊢ l ^ 2 + 4 * l + 2 + 4 * l ≤ 2 * l ^ 2",
"after_state": "No Goals!"
},
{
"line": "ring1",
"before_state": "n : ℕ\nhn : 100 ≤ n\nhn' : 1 ≤ (n + 1).sqrt\nl : ℕ := (n + 1).sqrt - 1\nh₁ : (l + 1) ^ 2 ≤ n + 1\nh₂ : n + 1 + 1 ≤ (l + 1 + 1) ^ 2\nh₃ : 10 ≤ l + 1\na✝ : l ^ 2 + 4 * l + 2 + 4 * l > 2 * l ^ 2\n⊢ 8 * -1 + (2 * ↑l ^ 2 + 1 - (↑l ^ 2 + 4 * ↑l + 2 + 4 * ↑l)) + (10 - (↑l + 1)) + (0 - (0 - ↑l) * (10 - (↑l + 1))) = 0",
"after_state": "No Goals!"
},
{
"line": "norm_num",
"before_state": "n : ℕ\nhn : 100 ≤ n\nhn' : 1 ≤ (n + 1).sqrt\nl : ℕ := (n + 1).sqrt - 1\nh₁ : (l + 1) ^ 2 ≤ n + 1\nh₂ : n + 1 + 1 ≤ (l + 1 + 1) ^ 2\nh₃ : 10 ≤ l + 1\na✝ : l ^ 2 + 4 * l + 2 + 4 * l > 2 * l ^ 2\n⊢ 8 > 0",
"after_state": "No Goals!"
},
{
"line": "linarith only [h₁]",
"before_state": "case refine_2\nn : ℕ\nhn : 100 ≤ n\nhn' : 1 ≤ (n + 1).sqrt\nl : ℕ := (n + 1).sqrt - 1\nh₁ : (l + 1) ^ 2 ≤ n + 1\nh₂ : n + 1 + 1 ≤ (l + 1 + 1) ^ 2\nh₃ : 10 ≤ l + 1\n⊢ 2 * l ^ 2 + 4 * l ≤ 2 * n",
"after_state": "No Goals!"
},
{
"line": "ring1",
"before_state": "n : ℕ\nhn : 100 ≤ n\nhn' : 1 ≤ (n + 1).sqrt\nl : ℕ := (n + 1).sqrt - 1\nh₁ : (l + 1) ^ 2 ≤ n + 1\nh₂ : n + 1 + 1 ≤ (l + 1 + 1) ^ 2\nh₃ : 10 ≤ l + 1\na✝ : 2 * l ^ 2 + 4 * l > 2 * n\n⊢ -1 + (2 * ↑n + 1 - (2 * ↑l ^ 2 + 4 * ↑l)) + 2 * ((↑l + 1) ^ 2 - (↑n + 1)) = 0",
"after_state": "No Goals!"
},
{
"line": "norm_num",
"before_state": "n : ℕ\nhn : 100 ≤ n\nhn' : 1 ≤ (n + 1).sqrt\nl : ℕ := (n + 1).sqrt - 1\nh₁ : (l + 1) ^ 2 ≤ n + 1\nh₂ : n + 1 + 1 ≤ (l + 1 + 1) ^ 2\nh₃ : 10 ≤ l + 1\na✝ : 2 * l ^ 2 + 4 * l > 2 * n\n⊢ 2 > 0",
"after_state": "No Goals!"
}
] |
lemma exists_triplet_summing_to_squares {n : ℕ} (hn : 100 ≤ n) :
∃ a b c : ℕ, n ≤ a ∧ a < b ∧ b < c ∧ c ≤ 2 * n ∧
IsSquare (a + b) ∧ IsSquare (c + a) ∧ IsSquare (b + c) := by
obtain ⟨l, hl1, hl2⟩ := exists_numbers_in_interval hn
have hl : 1 < l := by contrapose! hl1; interval_cases l <;> linarith
have h₁ : 4 * l ≤ 2 * l ^ 2 := by omega
have h₂ : 1 ≤ 2 * l := by omega
refine ⟨2 * l ^ 2 - 4 * l, 2 * l ^ 2 + 1, 2 * l ^ 2 + 4 * l, ?_, ?_, ?_,
⟨?_, ⟨2 * l - 1, ?_⟩, ⟨2 * l, ?_⟩, 2 * l + 1, ?_⟩⟩
all_goals zify [h₁, h₂]; linarith
|
/root/DuelModelResearch/mathlib4/Archive/Imo/Imo2021Q1.lean
|
{
"open": [
"Finset"
],
"variables": []
}
|
[
{
"line": "obtain ⟨l, hl1, hl2⟩ := exists_numbers_in_interval hn",
"before_state": "n : ℕ\nhn : 100 ≤ n\n⊢ ∃ a b c, n ≤ a ∧ a < b ∧ b < c ∧ c ≤ 2 * n ∧ IsSquare (a + b) ∧ IsSquare (c + a) ∧ IsSquare (b + c)",
"after_state": "No Goals!"
}
] |
lemma exists_finset_3_le_card_with_pairs_summing_to_squares {n : ℕ} (hn : 100 ≤ n) :
∃ B : Finset ℕ,
2 * 1 + 1 ≤ #B ∧
(∀ a ∈ B, ∀ b ∈ B, a ≠ b → IsSquare (a + b)) ∧
∀ c ∈ B, n ≤ c ∧ c ≤ 2 * n := by
obtain ⟨a, b, c, hna, hab, hbc, hcn, h₁, h₂, h₃⟩ := exists_triplet_summing_to_squares hn
refine ⟨{a, b, c}, ?_, ?_, ?_⟩
· suffices a ∉ {b, c} ∧ b ∉ {c} by
rw [Finset.card_insert_of_not_mem this.1]
rw [Finset.card_insert_of_not_mem this.2]
rw [Finset.card_singleton]
rw [Finset.mem_insert]
rw [Finset.mem_singleton]
rw [Finset.mem_singleton]
push_neg
exact ⟨⟨hab.ne, (hab.trans hbc).ne⟩, hbc.ne⟩
· intro x hx y hy hxy
simp only [Finset.mem_insert] at hx hy
simp only [Finset.mem_singleton] at hx hy
rcases hx with (rfl | rfl | rfl) <;> rcases hy with (rfl | rfl | rfl)
all_goals
first
| contradiction
| assumption
| simpa only [add_comm x y]
· simp only [Finset.mem_insert, Finset.mem_singleton]
rintro d (rfl | rfl | rfl) <;> constructor <;> linarith only [hna, hab, hbc, hcn]
|
/root/DuelModelResearch/mathlib4/Archive/Imo/Imo2021Q1.lean
|
{
"open": [
"Finset"
],
"variables": []
}
|
[
{
"line": "obtain ⟨a, b, c, hna, hab, hbc, hcn, h₁, h₂, h₃⟩ := exists_triplet_summing_to_squares hn",
"before_state": "n : ℕ\nhn : 100 ≤ n\n⊢ ∃ B, 2 * 1 + 1 ≤ #B ∧ (∀ a ∈ B, ∀ b ∈ B, a ≠ b → IsSquare (a + b)) ∧ ∀ c ∈ B, n ≤ c ∧ c ≤ 2 * n",
"after_state": "No Goals!"
}
] |
theorem imo2021_q1 :
∀ n : ℕ, 100 ≤ n → ∀ A ⊆ Finset.Icc n (2 * n),
(∃ a ∈ A, ∃ b ∈ A, a ≠ b ∧ IsSquare (a + b)) ∨
∃ a ∈ Finset.Icc n (2 * n) \ A, ∃ b ∈ Finset.Icc n (2 * n) \ A, a ≠ b ∧ IsSquare (a + b) := by
intro n hn A hA
-- For each n ∈ ℕ such that 100 ≤ n, there exists a pairwise unequal triplet {a, b, c} ⊆ [n, 2n]
-- such that all pairwise sums are perfect squares. In practice, it will be easier to use
-- a finite set B ⊆ [n, 2n] such that all pairwise unequal pairs of B sum to a perfect square
-- noting that B has cardinality greater or equal to 3, by the explicit construction of the
-- triplet {a, b, c} before.
obtain ⟨B, hB, h₁, h₂⟩ := exists_finset_3_le_card_with_pairs_summing_to_squares hn
have hBsub : B ⊆ Finset.Icc n (2 * n) := by
intro c hcB; simpa only [Finset.mem_Icc] using h₂ c hcB
have hB' : 2 * 1 < #(B ∩ (Icc n (2 * n) \ A) ∪ B ∩ A) := by
rwa [← inter_union_distrib_left, sdiff_union_self_eq_union, union_eq_left.2 hA,
inter_eq_left.2 hBsub, ← Nat.succ_le_iff]
-- Since B has cardinality greater or equal to 3, there must exist a subset C ⊆ B such that
-- for any A ⊆ [n, 2n], either C ⊆ A or C ⊆ [n, 2n] \ A and C has cardinality greater
-- or equal to 2.
obtain ⟨C, hC, hCA⟩ := Finset.exists_subset_or_subset_of_two_mul_lt_card hB'
rw [Finset.one_lt_card] at hC
rcases hC with ⟨a, ha, b, hb, hab⟩
simp only [Finset.subset_iff] at hCA
simp only [Finset.mem_inter] at hCA
-- Now we split into the two cases C ⊆ [n, 2n] \ A and C ⊆ A, which can be dealt with identically.
rcases hCA with hCA | hCA <;> [right; left] <;>
exact ⟨a, (hCA ha).2, b, (hCA hb).2, hab, h₁ a (hCA ha).1 b (hCA hb).1 hab⟩
|
/root/DuelModelResearch/mathlib4/Archive/Imo/Imo2021Q1.lean
|
{
"open": [
"Finset",
"Imo2021Q1"
],
"variables": []
}
|
[
{
"line": "intro n hn A hA",
"before_state": "⊢ ∀ (n : ℕ),\n 100 ≤ n →\n ∀ A ⊆ Icc n (2 * n),\n (∃ a ∈ A, ∃ b ∈ A, a ≠ b ∧ IsSquare (a + b)) ∨\n ∃ a ∈ Icc n (2 * n) \\ A, ∃ b ∈ Icc n (2 * n) \\ A, a ≠ b ∧ IsSquare (a + b)",
"after_state": "n : ℕ\nhn : 100 ≤ n\nA : Finset ℕ\nhA : A ⊆ Icc n (2 * n)\n⊢ (∃ a ∈ A, ∃ b ∈ A, a ≠ b ∧ IsSquare (a + b)) ∨\n ∃ a ∈ Icc n (2 * n) \\ A, ∃ b ∈ Icc n (2 * n) \\ A, a ≠ b ∧ IsSquare (a + b)"
},
{
"line": "intro n;\n intro hn A hA",
"before_state": "⊢ ∀ (n : ℕ),\n 100 ≤ n →\n ∀ A ⊆ Icc n (2 * n),\n (∃ a ∈ A, ∃ b ∈ A, a ≠ b ∧ IsSquare (a + b)) ∨\n ∃ a ∈ Icc n (2 * n) \\ A, ∃ b ∈ Icc n (2 * n) \\ A, a ≠ b ∧ IsSquare (a + b)",
"after_state": "n : ℕ\nhn : 100 ≤ n\nA : Finset ℕ\nhA : A ⊆ Icc n (2 * n)\n⊢ (∃ a ∈ A, ∃ b ∈ A, a ≠ b ∧ IsSquare (a + b)) ∨\n ∃ a ∈ Icc n (2 * n) \\ A, ∃ b ∈ Icc n (2 * n) \\ A, a ≠ b ∧ IsSquare (a + b)"
},
{
"line": "intro n",
"before_state": "⊢ ∀ (n : ℕ),\n 100 ≤ n →\n ∀ A ⊆ Icc n (2 * n),\n (∃ a ∈ A, ∃ b ∈ A, a ≠ b ∧ IsSquare (a + b)) ∨\n ∃ a ∈ Icc n (2 * n) \\ A, ∃ b ∈ Icc n (2 * n) \\ A, a ≠ b ∧ IsSquare (a + b)",
"after_state": "n : ℕ\n⊢ 100 ≤ n →\n ∀ A ⊆ Icc n (2 * n),\n (∃ a ∈ A, ∃ b ∈ A, a ≠ b ∧ IsSquare (a + b)) ∨\n ∃ a ∈ Icc n (2 * n) \\ A, ∃ b ∈ Icc n (2 * n) \\ A, a ≠ b ∧ IsSquare (a + b)"
},
{
"line": "intro hn A hA",
"before_state": "n : ℕ\n⊢ 100 ≤ n →\n ∀ A ⊆ Icc n (2 * n),\n (∃ a ∈ A, ∃ b ∈ A, a ≠ b ∧ IsSquare (a + b)) ∨\n ∃ a ∈ Icc n (2 * n) \\ A, ∃ b ∈ Icc n (2 * n) \\ A, a ≠ b ∧ IsSquare (a + b)",
"after_state": "n : ℕ\nhn : 100 ≤ n\nA : Finset ℕ\nhA : A ⊆ Icc n (2 * n)\n⊢ (∃ a ∈ A, ∃ b ∈ A, a ≠ b ∧ IsSquare (a + b)) ∨\n ∃ a ∈ Icc n (2 * n) \\ A, ∃ b ∈ Icc n (2 * n) \\ A, a ≠ b ∧ IsSquare (a + b)"
},
{
"line": "intro hn;\n intro A hA",
"before_state": "n : ℕ\n⊢ 100 ≤ n →\n ∀ A ⊆ Icc n (2 * n),\n (∃ a ∈ A, ∃ b ∈ A, a ≠ b ∧ IsSquare (a + b)) ∨\n ∃ a ∈ Icc n (2 * n) \\ A, ∃ b ∈ Icc n (2 * n) \\ A, a ≠ b ∧ IsSquare (a + b)",
"after_state": "n : ℕ\nhn : 100 ≤ n\nA : Finset ℕ\nhA : A ⊆ Icc n (2 * n)\n⊢ (∃ a ∈ A, ∃ b ∈ A, a ≠ b ∧ IsSquare (a + b)) ∨\n ∃ a ∈ Icc n (2 * n) \\ A, ∃ b ∈ Icc n (2 * n) \\ A, a ≠ b ∧ IsSquare (a + b)"
},
{
"line": "intro hn",
"before_state": "n : ℕ\n⊢ 100 ≤ n →\n ∀ A ⊆ Icc n (2 * n),\n (∃ a ∈ A, ∃ b ∈ A, a ≠ b ∧ IsSquare (a + b)) ∨\n ∃ a ∈ Icc n (2 * n) \\ A, ∃ b ∈ Icc n (2 * n) \\ A, a ≠ b ∧ IsSquare (a + b)",
"after_state": "n : ℕ\nhn : 100 ≤ n\n⊢ ∀ A ⊆ Icc n (2 * n),\n (∃ a ∈ A, ∃ b ∈ A, a ≠ b ∧ IsSquare (a + b)) ∨\n ∃ a ∈ Icc n (2 * n) \\ A, ∃ b ∈ Icc n (2 * n) \\ A, a ≠ b ∧ IsSquare (a + b)"
},
{
"line": "intro A hA",
"before_state": "n : ℕ\nhn : 100 ≤ n\n⊢ ∀ A ⊆ Icc n (2 * n),\n (∃ a ∈ A, ∃ b ∈ A, a ≠ b ∧ IsSquare (a + b)) ∨\n ∃ a ∈ Icc n (2 * n) \\ A, ∃ b ∈ Icc n (2 * n) \\ A, a ≠ b ∧ IsSquare (a + b)",
"after_state": "n : ℕ\nhn : 100 ≤ n\nA : Finset ℕ\nhA : A ⊆ Icc n (2 * n)\n⊢ (∃ a ∈ A, ∃ b ∈ A, a ≠ b ∧ IsSquare (a + b)) ∨\n ∃ a ∈ Icc n (2 * n) \\ A, ∃ b ∈ Icc n (2 * n) \\ A, a ≠ b ∧ IsSquare (a + b)"
},
{
"line": "intro A;\n intro hA",
"before_state": "n : ℕ\nhn : 100 ≤ n\n⊢ ∀ A ⊆ Icc n (2 * n),\n (∃ a ∈ A, ∃ b ∈ A, a ≠ b ∧ IsSquare (a + b)) ∨\n ∃ a ∈ Icc n (2 * n) \\ A, ∃ b ∈ Icc n (2 * n) \\ A, a ≠ b ∧ IsSquare (a + b)",
"after_state": "n : ℕ\nhn : 100 ≤ n\nA : Finset ℕ\nhA : A ⊆ Icc n (2 * n)\n⊢ (∃ a ∈ A, ∃ b ∈ A, a ≠ b ∧ IsSquare (a + b)) ∨\n ∃ a ∈ Icc n (2 * n) \\ A, ∃ b ∈ Icc n (2 * n) \\ A, a ≠ b ∧ IsSquare (a + b)"
},
{
"line": "intro A",
"before_state": "n : ℕ\nhn : 100 ≤ n\n⊢ ∀ A ⊆ Icc n (2 * n),\n (∃ a ∈ A, ∃ b ∈ A, a ≠ b ∧ IsSquare (a + b)) ∨\n ∃ a ∈ Icc n (2 * n) \\ A, ∃ b ∈ Icc n (2 * n) \\ A, a ≠ b ∧ IsSquare (a + b)",
"after_state": "n : ℕ\nhn : 100 ≤ n\nA : Finset ℕ\n⊢ A ⊆ Icc n (2 * n) →\n (∃ a ∈ A, ∃ b ∈ A, a ≠ b ∧ IsSquare (a + b)) ∨\n ∃ a ∈ Icc n (2 * n) \\ A, ∃ b ∈ Icc n (2 * n) \\ A, a ≠ b ∧ IsSquare (a + b)"
},
{
"line": "intro hA",
"before_state": "n : ℕ\nhn : 100 ≤ n\nA : Finset ℕ\n⊢ A ⊆ Icc n (2 * n) →\n (∃ a ∈ A, ∃ b ∈ A, a ≠ b ∧ IsSquare (a + b)) ∨\n ∃ a ∈ Icc n (2 * n) \\ A, ∃ b ∈ Icc n (2 * n) \\ A, a ≠ b ∧ IsSquare (a + b)",
"after_state": "n : ℕ\nhn : 100 ≤ n\nA : Finset ℕ\nhA : A ⊆ Icc n (2 * n)\n⊢ (∃ a ∈ A, ∃ b ∈ A, a ≠ b ∧ IsSquare (a + b)) ∨\n ∃ a ∈ Icc n (2 * n) \\ A, ∃ b ∈ Icc n (2 * n) \\ A, a ≠ b ∧ IsSquare (a + b)"
},
{
"line": "obtain ⟨B, hB, h₁, h₂⟩ := exists_finset_3_le_card_with_pairs_summing_to_squares hn",
"before_state": "n : ℕ\nhn : 100 ≤ n\nA : Finset ℕ\nhA : A ⊆ Icc n (2 * n)\n⊢ (∃ a ∈ A, ∃ b ∈ A, a ≠ b ∧ IsSquare (a + b)) ∨\n ∃ a ∈ Icc n (2 * n) \\ A, ∃ b ∈ Icc n (2 * n) \\ A, a ≠ b ∧ IsSquare (a + b)",
"after_state": "No Goals!"
}
] |
lemma dvd_pow_iff_of_dvd_sub {a b d n : ℕ} {z : ℤ} (ha : a.Coprime d)
(hd : (φ d : ℤ) ∣ (n : ℤ) - z) :
d ∣ a ^ n + b ↔ (((ZMod.unitOfCoprime _ ha) ^ z : (ZMod d)ˣ) : ZMod d) + b = 0 := by
rcases hd with ⟨k, hk⟩
rw [← ZMod.natCast_zmod_eq_zero_iff_dvd]
convert Iff.rfl
push_cast
congr
suffices (((ZMod.unitOfCoprime _ ha) ^ z : (ZMod d)ˣ) : ZMod d) =
(((ZMod.unitOfCoprime _ ha) ^ (n : ℤ) : (ZMod d)ˣ) : ZMod d) by
convert this
rw [sub_eq_iff_eq_add] at hk
rw [hk]
rw [zpow_add]
rw [zpow_mul]
norm_cast
rw [ZMod.pow_totient]
rw [one_zpow]
rw [one_mul]
|
/root/DuelModelResearch/mathlib4/Archive/Imo/Imo2024Q2.lean
|
{
"open": [
"scoped Nat"
],
"variables": []
}
|
[
{
"line": "rcases hd with ⟨k, hk⟩",
"before_state": "a b d n : ℕ\nz : ℤ\nha : a.Coprime d\nhd : ↑(φ d) ∣ ↑n - z\n⊢ d ∣ a ^ n + b ↔ ↑(ZMod.unitOfCoprime a ha ^ z) + ↑b = 0",
"after_state": "case intro\na b d n : ℕ\nz : ℤ\nha : a.Coprime d\nk : ℤ\nhk : ↑n - z = ↑(φ d) * k\n⊢ d ∣ a ^ n + b ↔ ↑(ZMod.unitOfCoprime a ha ^ z) + ↑b = 0"
},
{
"line": "rw [← ZMod.natCast_zmod_eq_zero_iff_dvd]",
"before_state": "case intro\na b d n : ℕ\nz : ℤ\nha : a.Coprime d\nk : ℤ\nhk : ↑n - z = ↑(φ d) * k\n⊢ d ∣ a ^ n + b ↔ ↑(ZMod.unitOfCoprime a ha ^ z) + ↑b = 0",
"after_state": "case intro\na b d n : ℕ\nz : ℤ\nha : a.Coprime d\nk : ℤ\nhk : ↑n - z = ↑(φ d) * k\n⊢ ↑(a ^ n + b) = 0 ↔ ↑(ZMod.unitOfCoprime a ha ^ z) + ↑b = 0"
},
{
"line": "rewrite [← ZMod.natCast_zmod_eq_zero_iff_dvd]",
"before_state": "case intro\na b d n : ℕ\nz : ℤ\nha : a.Coprime d\nk : ℤ\nhk : ↑n - z = ↑(φ d) * k\n⊢ d ∣ a ^ n + b ↔ ↑(ZMod.unitOfCoprime a ha ^ z) + ↑b = 0",
"after_state": "case intro\na b d n : ℕ\nz : ℤ\nha : a.Coprime d\nk : ℤ\nhk : ↑n - z = ↑(φ d) * k\n⊢ ↑(a ^ n + b) = 0 ↔ ↑(ZMod.unitOfCoprime a ha ^ z) + ↑b = 0"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "case intro\na b d n : ℕ\nz : ℤ\nha : a.Coprime d\nk : ℤ\nhk : ↑n - z = ↑(φ d) * k\n⊢ ↑(a ^ n + b) = 0 ↔ ↑(ZMod.unitOfCoprime a ha ^ z) + ↑b = 0",
"after_state": "case intro\na b d n : ℕ\nz : ℤ\nha : a.Coprime d\nk : ℤ\nhk : ↑n - z = ↑(φ d) * k\n⊢ ↑(a ^ n + b) = 0 ↔ ↑(ZMod.unitOfCoprime a ha ^ z) + ↑b = 0"
},
{
"line": "try (with_reducible rfl)",
"before_state": "case intro\na b d n : ℕ\nz : ℤ\nha : a.Coprime d\nk : ℤ\nhk : ↑n - z = ↑(φ d) * k\n⊢ ↑(a ^ n + b) = 0 ↔ ↑(ZMod.unitOfCoprime a ha ^ z) + ↑b = 0",
"after_state": "case intro\na b d n : ℕ\nz : ℤ\nha : a.Coprime d\nk : ℤ\nhk : ↑n - z = ↑(φ d) * k\n⊢ ↑(a ^ n + b) = 0 ↔ ↑(ZMod.unitOfCoprime a ha ^ z) + ↑b = 0"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "case intro\na b d n : ℕ\nz : ℤ\nha : a.Coprime d\nk : ℤ\nhk : ↑n - z = ↑(φ d) * k\n⊢ ↑(a ^ n + b) = 0 ↔ ↑(ZMod.unitOfCoprime a ha ^ z) + ↑b = 0",
"after_state": "case intro\na b d n : ℕ\nz : ℤ\nha : a.Coprime d\nk : ℤ\nhk : ↑n - z = ↑(φ d) * k\n⊢ ↑(a ^ n + b) = 0 ↔ ↑(ZMod.unitOfCoprime a ha ^ z) + ↑b = 0"
},
{
"line": "with_reducible rfl",
"before_state": "case intro\na b d n : ℕ\nz : ℤ\nha : a.Coprime d\nk : ℤ\nhk : ↑n - z = ↑(φ d) * k\n⊢ ↑(a ^ n + b) = 0 ↔ ↑(ZMod.unitOfCoprime a ha ^ z) + ↑b = 0",
"after_state": "case intro\na b d n : ℕ\nz : ℤ\nha : a.Coprime d\nk : ℤ\nhk : ↑n - z = ↑(φ d) * k\n⊢ ↑(a ^ n + b) = 0 ↔ ↑(ZMod.unitOfCoprime a ha ^ z) + ↑b = 0"
},
{
"line": "rfl",
"before_state": "case intro\na b d n : ℕ\nz : ℤ\nha : a.Coprime d\nk : ℤ\nhk : ↑n - z = ↑(φ d) * k\n⊢ ↑(a ^ n + b) = 0 ↔ ↑(ZMod.unitOfCoprime a ha ^ z) + ↑b = 0",
"after_state": "case intro\na b d n : ℕ\nz : ℤ\nha : a.Coprime d\nk : ℤ\nhk : ↑n - z = ↑(φ d) * k\n⊢ ↑(a ^ n + b) = 0 ↔ ↑(ZMod.unitOfCoprime a ha ^ z) + ↑b = 0"
},
{
"line": "apply_rfl",
"before_state": "case intro\na b d n : ℕ\nz : ℤ\nha : a.Coprime d\nk : ℤ\nhk : ↑n - z = ↑(φ d) * k\n⊢ ↑(a ^ n + b) = 0 ↔ ↑(ZMod.unitOfCoprime a ha ^ z) + ↑b = 0",
"after_state": "case intro\na b d n : ℕ\nz : ℤ\nha : a.Coprime d\nk : ℤ\nhk : ↑n - z = ↑(φ d) * k\n⊢ ↑(a ^ n + b) = 0 ↔ ↑(ZMod.unitOfCoprime a ha ^ z) + ↑b = 0"
},
{
"line": "skip",
"before_state": "case intro\na b d n : ℕ\nz : ℤ\nha : a.Coprime d\nk : ℤ\nhk : ↑n - z = ↑(φ d) * k\n⊢ ↑(a ^ n + b) = 0 ↔ ↑(ZMod.unitOfCoprime a ha ^ z) + ↑b = 0",
"after_state": "case intro\na b d n : ℕ\nz : ℤ\nha : a.Coprime d\nk : ℤ\nhk : ↑n - z = ↑(φ d) * k\n⊢ ↑(a ^ n + b) = 0 ↔ ↑(ZMod.unitOfCoprime a ha ^ z) + ↑b = 0"
},
{
"line": "convert Iff.rfl",
"before_state": "case intro\na b d n : ℕ\nz : ℤ\nha : a.Coprime d\nk : ℤ\nhk : ↑n - z = ↑(φ d) * k\n⊢ ↑(a ^ n + b) = 0 ↔ ↑(ZMod.unitOfCoprime a ha ^ z) + ↑b = 0",
"after_state": "case h.e'_2.h.e'_2\na b d n : ℕ\nz : ℤ\nha : a.Coprime d\nk : ℤ\nhk : ↑n - z = ↑(φ d) * k\n⊢ ↑(ZMod.unitOfCoprime a ha ^ z) + ↑b = ↑(a ^ n + b)"
},
{
"line": "push_cast",
"before_state": "case h.e'_2.h.e'_2\na b d n : ℕ\nz : ℤ\nha : a.Coprime d\nk : ℤ\nhk : ↑n - z = ↑(φ d) * k\n⊢ ↑(ZMod.unitOfCoprime a ha ^ z) + ↑b = ↑(a ^ n + b)",
"after_state": "case h.e'_2.h.e'_2\na b d n : ℕ\nz : ℤ\nha : a.Coprime d\nk : ℤ\nhk : ↑n - z = ↑(φ d) * k\n⊢ ↑(ZMod.unitOfCoprime a ha ^ z) + ↑b = ↑a ^ n + ↑b"
},
{
"line": "congr",
"before_state": "case h.e'_2.h.e'_2\na b d n : ℕ\nz : ℤ\nha : a.Coprime d\nk : ℤ\nhk : ↑n - z = ↑(φ d) * k\n⊢ ↑(ZMod.unitOfCoprime a ha ^ z) + ↑b = ↑a ^ n + ↑b",
"after_state": "case h.e'_2.h.e'_2.e_a\na b d n : ℕ\nz : ℤ\nha : a.Coprime d\nk : ℤ\nhk : ↑n - z = ↑(φ d) * k\n⊢ ↑(ZMod.unitOfCoprime a ha ^ z) = ↑a ^ n"
},
{
"line": "suffices\n (((ZMod.unitOfCoprime _ ha) ^ z : (ZMod d)ˣ) : ZMod d) = (((ZMod.unitOfCoprime _ ha) ^ (n : ℤ) : (ZMod d)ˣ) : ZMod d)\n by convert this",
"before_state": "case h.e'_2.h.e'_2.e_a\na b d n : ℕ\nz : ℤ\nha : a.Coprime d\nk : ℤ\nhk : ↑n - z = ↑(φ d) * k\n⊢ ↑(ZMod.unitOfCoprime a ha ^ z) = ↑a ^ n",
"after_state": "case h.e'_2.h.e'_2.e_a\na b d n : ℕ\nz : ℤ\nha : a.Coprime d\nk : ℤ\nhk : ↑n - z = ↑(φ d) * k\n⊢ ↑(ZMod.unitOfCoprime a ha ^ z) = ↑(ZMod.unitOfCoprime a ha ^ ↑n)"
},
{
"line": "refine_lift\n suffices\n (((ZMod.unitOfCoprime _ ha) ^ z : (ZMod d)ˣ) : ZMod d) =\n (((ZMod.unitOfCoprime _ ha) ^ (n : ℤ) : (ZMod d)ˣ) : ZMod d)\n by convert this;\n ?_",
"before_state": "case h.e'_2.h.e'_2.e_a\na b d n : ℕ\nz : ℤ\nha : a.Coprime d\nk : ℤ\nhk : ↑n - z = ↑(φ d) * k\n⊢ ↑(ZMod.unitOfCoprime a ha ^ z) = ↑a ^ n",
"after_state": "case h.e'_2.h.e'_2.e_a\na b d n : ℕ\nz : ℤ\nha : a.Coprime d\nk : ℤ\nhk : ↑n - z = ↑(φ d) * k\n⊢ ↑(ZMod.unitOfCoprime a ha ^ z) = ↑(ZMod.unitOfCoprime a ha ^ ↑n)"
},
{
"line": "focus\n (refine\n no_implicit_lambda%\n (suffices\n (((ZMod.unitOfCoprime _ ha) ^ z : (ZMod d)ˣ) : ZMod d) =\n (((ZMod.unitOfCoprime _ ha) ^ (n : ℤ) : (ZMod d)ˣ) : ZMod d)\n by convert this;\n ?_);\n rotate_right)",
"before_state": "case h.e'_2.h.e'_2.e_a\na b d n : ℕ\nz : ℤ\nha : a.Coprime d\nk : ℤ\nhk : ↑n - z = ↑(φ d) * k\n⊢ ↑(ZMod.unitOfCoprime a ha ^ z) = ↑a ^ n",
"after_state": "case h.e'_2.h.e'_2.e_a\na b d n : ℕ\nz : ℤ\nha : a.Coprime d\nk : ℤ\nhk : ↑n - z = ↑(φ d) * k\n⊢ ↑(ZMod.unitOfCoprime a ha ^ z) = ↑(ZMod.unitOfCoprime a ha ^ ↑n)"
},
{
"line": "refine\n no_implicit_lambda%\n (suffices\n (((ZMod.unitOfCoprime _ ha) ^ z : (ZMod d)ˣ) : ZMod d) =\n (((ZMod.unitOfCoprime _ ha) ^ (n : ℤ) : (ZMod d)ˣ) : ZMod d)\n by convert this;\n ?_)",
"before_state": "case h.e'_2.h.e'_2.e_a\na b d n : ℕ\nz : ℤ\nha : a.Coprime d\nk : ℤ\nhk : ↑n - z = ↑(φ d) * k\n⊢ ↑(ZMod.unitOfCoprime a ha ^ z) = ↑a ^ n",
"after_state": "case h.e'_2.h.e'_2.e_a\na b d n : ℕ\nz : ℤ\nha : a.Coprime d\nk : ℤ\nhk : ↑n - z = ↑(φ d) * k\n⊢ ↑(ZMod.unitOfCoprime a ha ^ z) = ↑(ZMod.unitOfCoprime a ha ^ ↑n)"
},
{
"line": "convert this",
"before_state": "a b d n : ℕ\nz : ℤ\nha : a.Coprime d\nk : ℤ\nhk : ↑n - z = ↑(φ d) * k\nthis : ↑(ZMod.unitOfCoprime a ha ^ z) = ↑(ZMod.unitOfCoprime a ha ^ ↑n)\n⊢ ↑(ZMod.unitOfCoprime a ha ^ z) = ↑a ^ n",
"after_state": "No Goals!"
},
{
"line": "rotate_right",
"before_state": "case h.e'_2.h.e'_2.e_a\na b d n : ℕ\nz : ℤ\nha : a.Coprime d\nk : ℤ\nhk : ↑n - z = ↑(φ d) * k\n⊢ ↑(ZMod.unitOfCoprime a ha ^ z) = ↑(ZMod.unitOfCoprime a ha ^ ↑n)",
"after_state": "case h.e'_2.h.e'_2.e_a\na b d n : ℕ\nz : ℤ\nha : a.Coprime d\nk : ℤ\nhk : ↑n - z = ↑(φ d) * k\n⊢ ↑(ZMod.unitOfCoprime a ha ^ z) = ↑(ZMod.unitOfCoprime a ha ^ ↑n)"
},
{
"line": "rw [sub_eq_iff_eq_add] at hk",
"before_state": "case h.e'_2.h.e'_2.e_a\na b d n : ℕ\nz : ℤ\nha : a.Coprime d\nk : ℤ\nhk : ↑n - z = ↑(φ d) * k\n⊢ ↑(ZMod.unitOfCoprime a ha ^ z) = ↑(ZMod.unitOfCoprime a ha ^ ↑n)",
"after_state": "case h.e'_2.h.e'_2.e_a\na b d n : ℕ\nz : ℤ\nha : a.Coprime d\nk : ℤ\nhk : ↑n = ↑(φ d) * k + z\n⊢ ↑(ZMod.unitOfCoprime a ha ^ z) = ↑(ZMod.unitOfCoprime a ha ^ ↑n)"
},
{
"line": "rewrite [sub_eq_iff_eq_add] at hk",
"before_state": "case h.e'_2.h.e'_2.e_a\na b d n : ℕ\nz : ℤ\nha : a.Coprime d\nk : ℤ\nhk : ↑n - z = ↑(φ d) * k\n⊢ ↑(ZMod.unitOfCoprime a ha ^ z) = ↑(ZMod.unitOfCoprime a ha ^ ↑n)",
"after_state": "case h.e'_2.h.e'_2.e_a\na b d n : ℕ\nz : ℤ\nha : a.Coprime d\nk : ℤ\nhk : ↑n = ↑(φ d) * k + z\n⊢ ↑(ZMod.unitOfCoprime a ha ^ z) = ↑(ZMod.unitOfCoprime a ha ^ ↑n)"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "case h.e'_2.h.e'_2.e_a\na b d n : ℕ\nz : ℤ\nha : a.Coprime d\nk : ℤ\nhk : ↑n = ↑(φ d) * k + z\n⊢ ↑(ZMod.unitOfCoprime a ha ^ z) = ↑(ZMod.unitOfCoprime a ha ^ ↑n)",
"after_state": "case h.e'_2.h.e'_2.e_a\na b d n : ℕ\nz : ℤ\nha : a.Coprime d\nk : ℤ\nhk : ↑n = ↑(φ d) * k + z\n⊢ ↑(ZMod.unitOfCoprime a ha ^ z) = ↑(ZMod.unitOfCoprime a ha ^ ↑n)"
},
{
"line": "try (with_reducible rfl)",
"before_state": "case h.e'_2.h.e'_2.e_a\na b d n : ℕ\nz : ℤ\nha : a.Coprime d\nk : ℤ\nhk : ↑n = ↑(φ d) * k + z\n⊢ ↑(ZMod.unitOfCoprime a ha ^ z) = ↑(ZMod.unitOfCoprime a ha ^ ↑n)",
"after_state": "case h.e'_2.h.e'_2.e_a\na b d n : ℕ\nz : ℤ\nha : a.Coprime d\nk : ℤ\nhk : ↑n = ↑(φ d) * k + z\n⊢ ↑(ZMod.unitOfCoprime a ha ^ z) = ↑(ZMod.unitOfCoprime a ha ^ ↑n)"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "case h.e'_2.h.e'_2.e_a\na b d n : ℕ\nz : ℤ\nha : a.Coprime d\nk : ℤ\nhk : ↑n = ↑(φ d) * k + z\n⊢ ↑(ZMod.unitOfCoprime a ha ^ z) = ↑(ZMod.unitOfCoprime a ha ^ ↑n)",
"after_state": "case h.e'_2.h.e'_2.e_a\na b d n : ℕ\nz : ℤ\nha : a.Coprime d\nk : ℤ\nhk : ↑n = ↑(φ d) * k + z\n⊢ ↑(ZMod.unitOfCoprime a ha ^ z) = ↑(ZMod.unitOfCoprime a ha ^ ↑n)"
},
{
"line": "with_reducible rfl",
"before_state": "case h.e'_2.h.e'_2.e_a\na b d n : ℕ\nz : ℤ\nha : a.Coprime d\nk : ℤ\nhk : ↑n = ↑(φ d) * k + z\n⊢ ↑(ZMod.unitOfCoprime a ha ^ z) = ↑(ZMod.unitOfCoprime a ha ^ ↑n)",
"after_state": "case h.e'_2.h.e'_2.e_a\na b d n : ℕ\nz : ℤ\nha : a.Coprime d\nk : ℤ\nhk : ↑n = ↑(φ d) * k + z\n⊢ ↑(ZMod.unitOfCoprime a ha ^ z) = ↑(ZMod.unitOfCoprime a ha ^ ↑n)"
},
{
"line": "rfl",
"before_state": "case h.e'_2.h.e'_2.e_a\na b d n : ℕ\nz : ℤ\nha : a.Coprime d\nk : ℤ\nhk : ↑n = ↑(φ d) * k + z\n⊢ ↑(ZMod.unitOfCoprime a ha ^ z) = ↑(ZMod.unitOfCoprime a ha ^ ↑n)",
"after_state": "case h.e'_2.h.e'_2.e_a\na b d n : ℕ\nz : ℤ\nha : a.Coprime d\nk : ℤ\nhk : ↑n = ↑(φ d) * k + z\n⊢ ↑(ZMod.unitOfCoprime a ha ^ z) = ↑(ZMod.unitOfCoprime a ha ^ ↑n)"
},
{
"line": "apply_rfl",
"before_state": "case h.e'_2.h.e'_2.e_a\na b d n : ℕ\nz : ℤ\nha : a.Coprime d\nk : ℤ\nhk : ↑n = ↑(φ d) * k + z\n⊢ ↑(ZMod.unitOfCoprime a ha ^ z) = ↑(ZMod.unitOfCoprime a ha ^ ↑n)",
"after_state": "case h.e'_2.h.e'_2.e_a\na b d n : ℕ\nz : ℤ\nha : a.Coprime d\nk : ℤ\nhk : ↑n = ↑(φ d) * k + z\n⊢ ↑(ZMod.unitOfCoprime a ha ^ z) = ↑(ZMod.unitOfCoprime a ha ^ ↑n)"
},
{
"line": "skip",
"before_state": "case h.e'_2.h.e'_2.e_a\na b d n : ℕ\nz : ℤ\nha : a.Coprime d\nk : ℤ\nhk : ↑n = ↑(φ d) * k + z\n⊢ ↑(ZMod.unitOfCoprime a ha ^ z) = ↑(ZMod.unitOfCoprime a ha ^ ↑n)",
"after_state": "case h.e'_2.h.e'_2.e_a\na b d n : ℕ\nz : ℤ\nha : a.Coprime d\nk : ℤ\nhk : ↑n = ↑(φ d) * k + z\n⊢ ↑(ZMod.unitOfCoprime a ha ^ z) = ↑(ZMod.unitOfCoprime a ha ^ ↑n)"
},
{
"line": "rw [hk]",
"before_state": "case h.e'_2.h.e'_2.e_a\na b d n : ℕ\nz : ℤ\nha : a.Coprime d\nk : ℤ\nhk : ↑n = ↑(φ d) * k + z\n⊢ ↑(ZMod.unitOfCoprime a ha ^ z) = ↑(ZMod.unitOfCoprime a ha ^ ↑n)",
"after_state": "case h.e'_2.h.e'_2.e_a\na b d n : ℕ\nz : ℤ\nha : a.Coprime d\nk : ℤ\nhk : ↑n = ↑(φ d) * k + z\n⊢ ↑(ZMod.unitOfCoprime a ha ^ z) = ↑(ZMod.unitOfCoprime a ha ^ (↑(φ d) * k + z))"
},
{
"line": "rewrite [hk]",
"before_state": "case h.e'_2.h.e'_2.e_a\na b d n : ℕ\nz : ℤ\nha : a.Coprime d\nk : ℤ\nhk : ↑n = ↑(φ d) * k + z\n⊢ ↑(ZMod.unitOfCoprime a ha ^ z) = ↑(ZMod.unitOfCoprime a ha ^ ↑n)",
"after_state": "case h.e'_2.h.e'_2.e_a\na b d n : ℕ\nz : ℤ\nha : a.Coprime d\nk : ℤ\nhk : ↑n = ↑(φ d) * k + z\n⊢ ↑(ZMod.unitOfCoprime a ha ^ z) = ↑(ZMod.unitOfCoprime a ha ^ (↑(φ d) * k + z))"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "case h.e'_2.h.e'_2.e_a\na b d n : ℕ\nz : ℤ\nha : a.Coprime d\nk : ℤ\nhk : ↑n = ↑(φ d) * k + z\n⊢ ↑(ZMod.unitOfCoprime a ha ^ z) = ↑(ZMod.unitOfCoprime a ha ^ (↑(φ d) * k + z))",
"after_state": "case h.e'_2.h.e'_2.e_a\na b d n : ℕ\nz : ℤ\nha : a.Coprime d\nk : ℤ\nhk : ↑n = ↑(φ d) * k + z\n⊢ ↑(ZMod.unitOfCoprime a ha ^ z) = ↑(ZMod.unitOfCoprime a ha ^ (↑(φ d) * k + z))"
},
{
"line": "try (with_reducible rfl)",
"before_state": "case h.e'_2.h.e'_2.e_a\na b d n : ℕ\nz : ℤ\nha : a.Coprime d\nk : ℤ\nhk : ↑n = ↑(φ d) * k + z\n⊢ ↑(ZMod.unitOfCoprime a ha ^ z) = ↑(ZMod.unitOfCoprime a ha ^ (↑(φ d) * k + z))",
"after_state": "case h.e'_2.h.e'_2.e_a\na b d n : ℕ\nz : ℤ\nha : a.Coprime d\nk : ℤ\nhk : ↑n = ↑(φ d) * k + z\n⊢ ↑(ZMod.unitOfCoprime a ha ^ z) = ↑(ZMod.unitOfCoprime a ha ^ (↑(φ d) * k + z))"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "case h.e'_2.h.e'_2.e_a\na b d n : ℕ\nz : ℤ\nha : a.Coprime d\nk : ℤ\nhk : ↑n = ↑(φ d) * k + z\n⊢ ↑(ZMod.unitOfCoprime a ha ^ z) = ↑(ZMod.unitOfCoprime a ha ^ (↑(φ d) * k + z))",
"after_state": "case h.e'_2.h.e'_2.e_a\na b d n : ℕ\nz : ℤ\nha : a.Coprime d\nk : ℤ\nhk : ↑n = ↑(φ d) * k + z\n⊢ ↑(ZMod.unitOfCoprime a ha ^ z) = ↑(ZMod.unitOfCoprime a ha ^ (↑(φ d) * k + z))"
},
{
"line": "with_reducible rfl",
"before_state": "case h.e'_2.h.e'_2.e_a\na b d n : ℕ\nz : ℤ\nha : a.Coprime d\nk : ℤ\nhk : ↑n = ↑(φ d) * k + z\n⊢ ↑(ZMod.unitOfCoprime a ha ^ z) = ↑(ZMod.unitOfCoprime a ha ^ (↑(φ d) * k + z))",
"after_state": "case h.e'_2.h.e'_2.e_a\na b d n : ℕ\nz : ℤ\nha : a.Coprime d\nk : ℤ\nhk : ↑n = ↑(φ d) * k + z\n⊢ ↑(ZMod.unitOfCoprime a ha ^ z) = ↑(ZMod.unitOfCoprime a ha ^ (↑(φ d) * k + z))"
},
{
"line": "rfl",
"before_state": "case h.e'_2.h.e'_2.e_a\na b d n : ℕ\nz : ℤ\nha : a.Coprime d\nk : ℤ\nhk : ↑n = ↑(φ d) * k + z\n⊢ ↑(ZMod.unitOfCoprime a ha ^ z) = ↑(ZMod.unitOfCoprime a ha ^ (↑(φ d) * k + z))",
"after_state": "case h.e'_2.h.e'_2.e_a\na b d n : ℕ\nz : ℤ\nha : a.Coprime d\nk : ℤ\nhk : ↑n = ↑(φ d) * k + z\n⊢ ↑(ZMod.unitOfCoprime a ha ^ z) = ↑(ZMod.unitOfCoprime a ha ^ (↑(φ d) * k + z))"
},
{
"line": "apply_rfl",
"before_state": "case h.e'_2.h.e'_2.e_a\na b d n : ℕ\nz : ℤ\nha : a.Coprime d\nk : ℤ\nhk : ↑n = ↑(φ d) * k + z\n⊢ ↑(ZMod.unitOfCoprime a ha ^ z) = ↑(ZMod.unitOfCoprime a ha ^ (↑(φ d) * k + z))",
"after_state": "case h.e'_2.h.e'_2.e_a\na b d n : ℕ\nz : ℤ\nha : a.Coprime d\nk : ℤ\nhk : ↑n = ↑(φ d) * k + z\n⊢ ↑(ZMod.unitOfCoprime a ha ^ z) = ↑(ZMod.unitOfCoprime a ha ^ (↑(φ d) * k + z))"
},
{
"line": "skip",
"before_state": "case h.e'_2.h.e'_2.e_a\na b d n : ℕ\nz : ℤ\nha : a.Coprime d\nk : ℤ\nhk : ↑n = ↑(φ d) * k + z\n⊢ ↑(ZMod.unitOfCoprime a ha ^ z) = ↑(ZMod.unitOfCoprime a ha ^ (↑(φ d) * k + z))",
"after_state": "case h.e'_2.h.e'_2.e_a\na b d n : ℕ\nz : ℤ\nha : a.Coprime d\nk : ℤ\nhk : ↑n = ↑(φ d) * k + z\n⊢ ↑(ZMod.unitOfCoprime a ha ^ z) = ↑(ZMod.unitOfCoprime a ha ^ (↑(φ d) * k + z))"
},
{
"line": "rw [zpow_add]",
"before_state": "case h.e'_2.h.e'_2.e_a\na b d n : ℕ\nz : ℤ\nha : a.Coprime d\nk : ℤ\nhk : ↑n = ↑(φ d) * k + z\n⊢ ↑(ZMod.unitOfCoprime a ha ^ z) = ↑(ZMod.unitOfCoprime a ha ^ (↑(φ d) * k + z))",
"after_state": "case h.e'_2.h.e'_2.e_a\na b d n : ℕ\nz : ℤ\nha : a.Coprime d\nk : ℤ\nhk : ↑n = ↑(φ d) * k + z\n⊢ ↑(ZMod.unitOfCoprime a ha ^ z) = ↑(ZMod.unitOfCoprime a ha ^ (↑(φ d) * k) * ZMod.unitOfCoprime a ha ^ z)"
},
{
"line": "rewrite [zpow_add]",
"before_state": "case h.e'_2.h.e'_2.e_a\na b d n : ℕ\nz : ℤ\nha : a.Coprime d\nk : ℤ\nhk : ↑n = ↑(φ d) * k + z\n⊢ ↑(ZMod.unitOfCoprime a ha ^ z) = ↑(ZMod.unitOfCoprime a ha ^ (↑(φ d) * k + z))",
"after_state": "case h.e'_2.h.e'_2.e_a\na b d n : ℕ\nz : ℤ\nha : a.Coprime d\nk : ℤ\nhk : ↑n = ↑(φ d) * k + z\n⊢ ↑(ZMod.unitOfCoprime a ha ^ z) = ↑(ZMod.unitOfCoprime a ha ^ (↑(φ d) * k) * ZMod.unitOfCoprime a ha ^ z)"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "case h.e'_2.h.e'_2.e_a\na b d n : ℕ\nz : ℤ\nha : a.Coprime d\nk : ℤ\nhk : ↑n = ↑(φ d) * k + z\n⊢ ↑(ZMod.unitOfCoprime a ha ^ z) = ↑(ZMod.unitOfCoprime a ha ^ (↑(φ d) * k) * ZMod.unitOfCoprime a ha ^ z)",
"after_state": "case h.e'_2.h.e'_2.e_a\na b d n : ℕ\nz : ℤ\nha : a.Coprime d\nk : ℤ\nhk : ↑n = ↑(φ d) * k + z\n⊢ ↑(ZMod.unitOfCoprime a ha ^ z) = ↑(ZMod.unitOfCoprime a ha ^ (↑(φ d) * k) * ZMod.unitOfCoprime a ha ^ z)"
},
{
"line": "try (with_reducible rfl)",
"before_state": "case h.e'_2.h.e'_2.e_a\na b d n : ℕ\nz : ℤ\nha : a.Coprime d\nk : ℤ\nhk : ↑n = ↑(φ d) * k + z\n⊢ ↑(ZMod.unitOfCoprime a ha ^ z) = ↑(ZMod.unitOfCoprime a ha ^ (↑(φ d) * k) * ZMod.unitOfCoprime a ha ^ z)",
"after_state": "case h.e'_2.h.e'_2.e_a\na b d n : ℕ\nz : ℤ\nha : a.Coprime d\nk : ℤ\nhk : ↑n = ↑(φ d) * k + z\n⊢ ↑(ZMod.unitOfCoprime a ha ^ z) = ↑(ZMod.unitOfCoprime a ha ^ (↑(φ d) * k) * ZMod.unitOfCoprime a ha ^ z)"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "case h.e'_2.h.e'_2.e_a\na b d n : ℕ\nz : ℤ\nha : a.Coprime d\nk : ℤ\nhk : ↑n = ↑(φ d) * k + z\n⊢ ↑(ZMod.unitOfCoprime a ha ^ z) = ↑(ZMod.unitOfCoprime a ha ^ (↑(φ d) * k) * ZMod.unitOfCoprime a ha ^ z)",
"after_state": "case h.e'_2.h.e'_2.e_a\na b d n : ℕ\nz : ℤ\nha : a.Coprime d\nk : ℤ\nhk : ↑n = ↑(φ d) * k + z\n⊢ ↑(ZMod.unitOfCoprime a ha ^ z) = ↑(ZMod.unitOfCoprime a ha ^ (↑(φ d) * k) * ZMod.unitOfCoprime a ha ^ z)"
},
{
"line": "with_reducible rfl",
"before_state": "case h.e'_2.h.e'_2.e_a\na b d n : ℕ\nz : ℤ\nha : a.Coprime d\nk : ℤ\nhk : ↑n = ↑(φ d) * k + z\n⊢ ↑(ZMod.unitOfCoprime a ha ^ z) = ↑(ZMod.unitOfCoprime a ha ^ (↑(φ d) * k) * ZMod.unitOfCoprime a ha ^ z)",
"after_state": "case h.e'_2.h.e'_2.e_a\na b d n : ℕ\nz : ℤ\nha : a.Coprime d\nk : ℤ\nhk : ↑n = ↑(φ d) * k + z\n⊢ ↑(ZMod.unitOfCoprime a ha ^ z) = ↑(ZMod.unitOfCoprime a ha ^ (↑(φ d) * k) * ZMod.unitOfCoprime a ha ^ z)"
},
{
"line": "rfl",
"before_state": "case h.e'_2.h.e'_2.e_a\na b d n : ℕ\nz : ℤ\nha : a.Coprime d\nk : ℤ\nhk : ↑n = ↑(φ d) * k + z\n⊢ ↑(ZMod.unitOfCoprime a ha ^ z) = ↑(ZMod.unitOfCoprime a ha ^ (↑(φ d) * k) * ZMod.unitOfCoprime a ha ^ z)",
"after_state": "case h.e'_2.h.e'_2.e_a\na b d n : ℕ\nz : ℤ\nha : a.Coprime d\nk : ℤ\nhk : ↑n = ↑(φ d) * k + z\n⊢ ↑(ZMod.unitOfCoprime a ha ^ z) = ↑(ZMod.unitOfCoprime a ha ^ (↑(φ d) * k) * ZMod.unitOfCoprime a ha ^ z)"
},
{
"line": "apply_rfl",
"before_state": "case h.e'_2.h.e'_2.e_a\na b d n : ℕ\nz : ℤ\nha : a.Coprime d\nk : ℤ\nhk : ↑n = ↑(φ d) * k + z\n⊢ ↑(ZMod.unitOfCoprime a ha ^ z) = ↑(ZMod.unitOfCoprime a ha ^ (↑(φ d) * k) * ZMod.unitOfCoprime a ha ^ z)",
"after_state": "case h.e'_2.h.e'_2.e_a\na b d n : ℕ\nz : ℤ\nha : a.Coprime d\nk : ℤ\nhk : ↑n = ↑(φ d) * k + z\n⊢ ↑(ZMod.unitOfCoprime a ha ^ z) = ↑(ZMod.unitOfCoprime a ha ^ (↑(φ d) * k) * ZMod.unitOfCoprime a ha ^ z)"
},
{
"line": "skip",
"before_state": "case h.e'_2.h.e'_2.e_a\na b d n : ℕ\nz : ℤ\nha : a.Coprime d\nk : ℤ\nhk : ↑n = ↑(φ d) * k + z\n⊢ ↑(ZMod.unitOfCoprime a ha ^ z) = ↑(ZMod.unitOfCoprime a ha ^ (↑(φ d) * k) * ZMod.unitOfCoprime a ha ^ z)",
"after_state": "case h.e'_2.h.e'_2.e_a\na b d n : ℕ\nz : ℤ\nha : a.Coprime d\nk : ℤ\nhk : ↑n = ↑(φ d) * k + z\n⊢ ↑(ZMod.unitOfCoprime a ha ^ z) = ↑(ZMod.unitOfCoprime a ha ^ (↑(φ d) * k) * ZMod.unitOfCoprime a ha ^ z)"
},
{
"line": "rw [zpow_mul]",
"before_state": "case h.e'_2.h.e'_2.e_a\na b d n : ℕ\nz : ℤ\nha : a.Coprime d\nk : ℤ\nhk : ↑n = ↑(φ d) * k + z\n⊢ ↑(ZMod.unitOfCoprime a ha ^ z) = ↑(ZMod.unitOfCoprime a ha ^ (↑(φ d) * k) * ZMod.unitOfCoprime a ha ^ z)",
"after_state": "case h.e'_2.h.e'_2.e_a\na b d n : ℕ\nz : ℤ\nha : a.Coprime d\nk : ℤ\nhk : ↑n = ↑(φ d) * k + z\n⊢ ↑(ZMod.unitOfCoprime a ha ^ z) = ↑((ZMod.unitOfCoprime a ha ^ ↑(φ d)) ^ k * ZMod.unitOfCoprime a ha ^ z)"
},
{
"line": "rewrite [zpow_mul]",
"before_state": "case h.e'_2.h.e'_2.e_a\na b d n : ℕ\nz : ℤ\nha : a.Coprime d\nk : ℤ\nhk : ↑n = ↑(φ d) * k + z\n⊢ ↑(ZMod.unitOfCoprime a ha ^ z) = ↑(ZMod.unitOfCoprime a ha ^ (↑(φ d) * k) * ZMod.unitOfCoprime a ha ^ z)",
"after_state": "case h.e'_2.h.e'_2.e_a\na b d n : ℕ\nz : ℤ\nha : a.Coprime d\nk : ℤ\nhk : ↑n = ↑(φ d) * k + z\n⊢ ↑(ZMod.unitOfCoprime a ha ^ z) = ↑((ZMod.unitOfCoprime a ha ^ ↑(φ d)) ^ k * ZMod.unitOfCoprime a ha ^ z)"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "case h.e'_2.h.e'_2.e_a\na b d n : ℕ\nz : ℤ\nha : a.Coprime d\nk : ℤ\nhk : ↑n = ↑(φ d) * k + z\n⊢ ↑(ZMod.unitOfCoprime a ha ^ z) = ↑((ZMod.unitOfCoprime a ha ^ ↑(φ d)) ^ k * ZMod.unitOfCoprime a ha ^ z)",
"after_state": "case h.e'_2.h.e'_2.e_a\na b d n : ℕ\nz : ℤ\nha : a.Coprime d\nk : ℤ\nhk : ↑n = ↑(φ d) * k + z\n⊢ ↑(ZMod.unitOfCoprime a ha ^ z) = ↑((ZMod.unitOfCoprime a ha ^ ↑(φ d)) ^ k * ZMod.unitOfCoprime a ha ^ z)"
},
{
"line": "try (with_reducible rfl)",
"before_state": "case h.e'_2.h.e'_2.e_a\na b d n : ℕ\nz : ℤ\nha : a.Coprime d\nk : ℤ\nhk : ↑n = ↑(φ d) * k + z\n⊢ ↑(ZMod.unitOfCoprime a ha ^ z) = ↑((ZMod.unitOfCoprime a ha ^ ↑(φ d)) ^ k * ZMod.unitOfCoprime a ha ^ z)",
"after_state": "case h.e'_2.h.e'_2.e_a\na b d n : ℕ\nz : ℤ\nha : a.Coprime d\nk : ℤ\nhk : ↑n = ↑(φ d) * k + z\n⊢ ↑(ZMod.unitOfCoprime a ha ^ z) = ↑((ZMod.unitOfCoprime a ha ^ ↑(φ d)) ^ k * ZMod.unitOfCoprime a ha ^ z)"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "case h.e'_2.h.e'_2.e_a\na b d n : ℕ\nz : ℤ\nha : a.Coprime d\nk : ℤ\nhk : ↑n = ↑(φ d) * k + z\n⊢ ↑(ZMod.unitOfCoprime a ha ^ z) = ↑((ZMod.unitOfCoprime a ha ^ ↑(φ d)) ^ k * ZMod.unitOfCoprime a ha ^ z)",
"after_state": "case h.e'_2.h.e'_2.e_a\na b d n : ℕ\nz : ℤ\nha : a.Coprime d\nk : ℤ\nhk : ↑n = ↑(φ d) * k + z\n⊢ ↑(ZMod.unitOfCoprime a ha ^ z) = ↑((ZMod.unitOfCoprime a ha ^ ↑(φ d)) ^ k * ZMod.unitOfCoprime a ha ^ z)"
},
{
"line": "with_reducible rfl",
"before_state": "case h.e'_2.h.e'_2.e_a\na b d n : ℕ\nz : ℤ\nha : a.Coprime d\nk : ℤ\nhk : ↑n = ↑(φ d) * k + z\n⊢ ↑(ZMod.unitOfCoprime a ha ^ z) = ↑((ZMod.unitOfCoprime a ha ^ ↑(φ d)) ^ k * ZMod.unitOfCoprime a ha ^ z)",
"after_state": "case h.e'_2.h.e'_2.e_a\na b d n : ℕ\nz : ℤ\nha : a.Coprime d\nk : ℤ\nhk : ↑n = ↑(φ d) * k + z\n⊢ ↑(ZMod.unitOfCoprime a ha ^ z) = ↑((ZMod.unitOfCoprime a ha ^ ↑(φ d)) ^ k * ZMod.unitOfCoprime a ha ^ z)"
},
{
"line": "rfl",
"before_state": "case h.e'_2.h.e'_2.e_a\na b d n : ℕ\nz : ℤ\nha : a.Coprime d\nk : ℤ\nhk : ↑n = ↑(φ d) * k + z\n⊢ ↑(ZMod.unitOfCoprime a ha ^ z) = ↑((ZMod.unitOfCoprime a ha ^ ↑(φ d)) ^ k * ZMod.unitOfCoprime a ha ^ z)",
"after_state": "case h.e'_2.h.e'_2.e_a\na b d n : ℕ\nz : ℤ\nha : a.Coprime d\nk : ℤ\nhk : ↑n = ↑(φ d) * k + z\n⊢ ↑(ZMod.unitOfCoprime a ha ^ z) = ↑((ZMod.unitOfCoprime a ha ^ ↑(φ d)) ^ k * ZMod.unitOfCoprime a ha ^ z)"
},
{
"line": "apply_rfl",
"before_state": "case h.e'_2.h.e'_2.e_a\na b d n : ℕ\nz : ℤ\nha : a.Coprime d\nk : ℤ\nhk : ↑n = ↑(φ d) * k + z\n⊢ ↑(ZMod.unitOfCoprime a ha ^ z) = ↑((ZMod.unitOfCoprime a ha ^ ↑(φ d)) ^ k * ZMod.unitOfCoprime a ha ^ z)",
"after_state": "case h.e'_2.h.e'_2.e_a\na b d n : ℕ\nz : ℤ\nha : a.Coprime d\nk : ℤ\nhk : ↑n = ↑(φ d) * k + z\n⊢ ↑(ZMod.unitOfCoprime a ha ^ z) = ↑((ZMod.unitOfCoprime a ha ^ ↑(φ d)) ^ k * ZMod.unitOfCoprime a ha ^ z)"
},
{
"line": "skip",
"before_state": "case h.e'_2.h.e'_2.e_a\na b d n : ℕ\nz : ℤ\nha : a.Coprime d\nk : ℤ\nhk : ↑n = ↑(φ d) * k + z\n⊢ ↑(ZMod.unitOfCoprime a ha ^ z) = ↑((ZMod.unitOfCoprime a ha ^ ↑(φ d)) ^ k * ZMod.unitOfCoprime a ha ^ z)",
"after_state": "case h.e'_2.h.e'_2.e_a\na b d n : ℕ\nz : ℤ\nha : a.Coprime d\nk : ℤ\nhk : ↑n = ↑(φ d) * k + z\n⊢ ↑(ZMod.unitOfCoprime a ha ^ z) = ↑((ZMod.unitOfCoprime a ha ^ ↑(φ d)) ^ k * ZMod.unitOfCoprime a ha ^ z)"
},
{
"line": "norm_cast",
"before_state": "case h.e'_2.h.e'_2.e_a\na b d n : ℕ\nz : ℤ\nha : a.Coprime d\nk : ℤ\nhk : ↑n = ↑(φ d) * k + z\n⊢ ↑(ZMod.unitOfCoprime a ha ^ z) = ↑((ZMod.unitOfCoprime a ha ^ ↑(φ d)) ^ k * ZMod.unitOfCoprime a ha ^ z)",
"after_state": "case h.e'_2.h.e'_2.e_a\na b d n : ℕ\nz : ℤ\nha : a.Coprime d\nk : ℤ\nhk : ↑n = ↑(φ d) * k + z\n⊢ ZMod.unitOfCoprime a ha ^ z = (ZMod.unitOfCoprime a ha ^ φ d) ^ k * ZMod.unitOfCoprime a ha ^ z"
},
{
"line": "focus\n norm_cast0\n with_annotate_state\"<;>\" skip\n all_goals try trivial",
"before_state": "case h.e'_2.h.e'_2.e_a\na b d n : ℕ\nz : ℤ\nha : a.Coprime d\nk : ℤ\nhk : ↑n = ↑(φ d) * k + z\n⊢ ↑(ZMod.unitOfCoprime a ha ^ z) = ↑((ZMod.unitOfCoprime a ha ^ ↑(φ d)) ^ k * ZMod.unitOfCoprime a ha ^ z)",
"after_state": "case h.e'_2.h.e'_2.e_a\na b d n : ℕ\nz : ℤ\nha : a.Coprime d\nk : ℤ\nhk : ↑n = ↑(φ d) * k + z\n⊢ ZMod.unitOfCoprime a ha ^ z = (ZMod.unitOfCoprime a ha ^ φ d) ^ k * ZMod.unitOfCoprime a ha ^ z"
},
{
"line": "norm_cast0",
"before_state": "case h.e'_2.h.e'_2.e_a\na b d n : ℕ\nz : ℤ\nha : a.Coprime d\nk : ℤ\nhk : ↑n = ↑(φ d) * k + z\n⊢ ↑(ZMod.unitOfCoprime a ha ^ z) = ↑((ZMod.unitOfCoprime a ha ^ ↑(φ d)) ^ k * ZMod.unitOfCoprime a ha ^ z)",
"after_state": "case h.e'_2.h.e'_2.e_a\na b d n : ℕ\nz : ℤ\nha : a.Coprime d\nk : ℤ\nhk : ↑n = ↑(φ d) * k + z\n⊢ ZMod.unitOfCoprime a ha ^ z = (ZMod.unitOfCoprime a ha ^ φ d) ^ k * ZMod.unitOfCoprime a ha ^ z"
},
{
"line": "with_annotate_state\"<;>\" skip",
"before_state": "case h.e'_2.h.e'_2.e_a\na b d n : ℕ\nz : ℤ\nha : a.Coprime d\nk : ℤ\nhk : ↑n = ↑(φ d) * k + z\n⊢ ZMod.unitOfCoprime a ha ^ z = (ZMod.unitOfCoprime a ha ^ φ d) ^ k * ZMod.unitOfCoprime a ha ^ z",
"after_state": "case h.e'_2.h.e'_2.e_a\na b d n : ℕ\nz : ℤ\nha : a.Coprime d\nk : ℤ\nhk : ↑n = ↑(φ d) * k + z\n⊢ ZMod.unitOfCoprime a ha ^ z = (ZMod.unitOfCoprime a ha ^ φ d) ^ k * ZMod.unitOfCoprime a ha ^ z"
},
{
"line": "skip",
"before_state": "case h.e'_2.h.e'_2.e_a\na b d n : ℕ\nz : ℤ\nha : a.Coprime d\nk : ℤ\nhk : ↑n = ↑(φ d) * k + z\n⊢ ZMod.unitOfCoprime a ha ^ z = (ZMod.unitOfCoprime a ha ^ φ d) ^ k * ZMod.unitOfCoprime a ha ^ z",
"after_state": "case h.e'_2.h.e'_2.e_a\na b d n : ℕ\nz : ℤ\nha : a.Coprime d\nk : ℤ\nhk : ↑n = ↑(φ d) * k + z\n⊢ ZMod.unitOfCoprime a ha ^ z = (ZMod.unitOfCoprime a ha ^ φ d) ^ k * ZMod.unitOfCoprime a ha ^ z"
},
{
"line": "all_goals try trivial",
"before_state": "case h.e'_2.h.e'_2.e_a\na b d n : ℕ\nz : ℤ\nha : a.Coprime d\nk : ℤ\nhk : ↑n = ↑(φ d) * k + z\n⊢ ZMod.unitOfCoprime a ha ^ z = (ZMod.unitOfCoprime a ha ^ φ d) ^ k * ZMod.unitOfCoprime a ha ^ z",
"after_state": "case h.e'_2.h.e'_2.e_a\na b d n : ℕ\nz : ℤ\nha : a.Coprime d\nk : ℤ\nhk : ↑n = ↑(φ d) * k + z\n⊢ ZMod.unitOfCoprime a ha ^ z = (ZMod.unitOfCoprime a ha ^ φ d) ^ k * ZMod.unitOfCoprime a ha ^ z"
},
{
"line": "try trivial",
"before_state": "case h.e'_2.h.e'_2.e_a\na b d n : ℕ\nz : ℤ\nha : a.Coprime d\nk : ℤ\nhk : ↑n = ↑(φ d) * k + z\n⊢ ZMod.unitOfCoprime a ha ^ z = (ZMod.unitOfCoprime a ha ^ φ d) ^ k * ZMod.unitOfCoprime a ha ^ z",
"after_state": "case h.e'_2.h.e'_2.e_a\na b d n : ℕ\nz : ℤ\nha : a.Coprime d\nk : ℤ\nhk : ↑n = ↑(φ d) * k + z\n⊢ ZMod.unitOfCoprime a ha ^ z = (ZMod.unitOfCoprime a ha ^ φ d) ^ k * ZMod.unitOfCoprime a ha ^ z"
},
{
"line": "first\n| trivial\n| skip",
"before_state": "case h.e'_2.h.e'_2.e_a\na b d n : ℕ\nz : ℤ\nha : a.Coprime d\nk : ℤ\nhk : ↑n = ↑(φ d) * k + z\n⊢ ZMod.unitOfCoprime a ha ^ z = (ZMod.unitOfCoprime a ha ^ φ d) ^ k * ZMod.unitOfCoprime a ha ^ z",
"after_state": "case h.e'_2.h.e'_2.e_a\na b d n : ℕ\nz : ℤ\nha : a.Coprime d\nk : ℤ\nhk : ↑n = ↑(φ d) * k + z\n⊢ ZMod.unitOfCoprime a ha ^ z = (ZMod.unitOfCoprime a ha ^ φ d) ^ k * ZMod.unitOfCoprime a ha ^ z"
},
{
"line": "trivial",
"before_state": "case h.e'_2.h.e'_2.e_a\na b d n : ℕ\nz : ℤ\nha : a.Coprime d\nk : ℤ\nhk : ↑n = ↑(φ d) * k + z\n⊢ ZMod.unitOfCoprime a ha ^ z = (ZMod.unitOfCoprime a ha ^ φ d) ^ k * ZMod.unitOfCoprime a ha ^ z",
"after_state": "case h.e'_2.h.e'_2.e_a\na b d n : ℕ\nz : ℤ\nha : a.Coprime d\nk : ℤ\nhk : ↑n = ↑(φ d) * k + z\n⊢ ZMod.unitOfCoprime a ha ^ z = (ZMod.unitOfCoprime a ha ^ φ d) ^ k * ZMod.unitOfCoprime a ha ^ z"
},
{
"line": "assumption",
"before_state": "case h.e'_2.h.e'_2.e_a\na b d n : ℕ\nz : ℤ\nha : a.Coprime d\nk : ℤ\nhk : ↑n = ↑(φ d) * k + z\n⊢ ZMod.unitOfCoprime a ha ^ z = (ZMod.unitOfCoprime a ha ^ φ d) ^ k * ZMod.unitOfCoprime a ha ^ z",
"after_state": "case h.e'_2.h.e'_2.e_a\na b d n : ℕ\nz : ℤ\nha : a.Coprime d\nk : ℤ\nhk : ↑n = ↑(φ d) * k + z\n⊢ ZMod.unitOfCoprime a ha ^ z = (ZMod.unitOfCoprime a ha ^ φ d) ^ k * ZMod.unitOfCoprime a ha ^ z"
},
{
"line": "skip",
"before_state": "case h.e'_2.h.e'_2.e_a\na b d n : ℕ\nz : ℤ\nha : a.Coprime d\nk : ℤ\nhk : ↑n = ↑(φ d) * k + z\n⊢ ZMod.unitOfCoprime a ha ^ z = (ZMod.unitOfCoprime a ha ^ φ d) ^ k * ZMod.unitOfCoprime a ha ^ z",
"after_state": "case h.e'_2.h.e'_2.e_a\na b d n : ℕ\nz : ℤ\nha : a.Coprime d\nk : ℤ\nhk : ↑n = ↑(φ d) * k + z\n⊢ ZMod.unitOfCoprime a ha ^ z = (ZMod.unitOfCoprime a ha ^ φ d) ^ k * ZMod.unitOfCoprime a ha ^ z"
},
{
"line": "rw [ZMod.pow_totient]",
"before_state": "case h.e'_2.h.e'_2.e_a\na b d n : ℕ\nz : ℤ\nha : a.Coprime d\nk : ℤ\nhk : ↑n = ↑(φ d) * k + z\n⊢ ZMod.unitOfCoprime a ha ^ z = (ZMod.unitOfCoprime a ha ^ φ d) ^ k * ZMod.unitOfCoprime a ha ^ z",
"after_state": "case h.e'_2.h.e'_2.e_a\na b d n : ℕ\nz : ℤ\nha : a.Coprime d\nk : ℤ\nhk : ↑n = ↑(φ d) * k + z\n⊢ ZMod.unitOfCoprime a ha ^ z = 1 ^ k * ZMod.unitOfCoprime a ha ^ z"
},
{
"line": "rewrite [ZMod.pow_totient]",
"before_state": "case h.e'_2.h.e'_2.e_a\na b d n : ℕ\nz : ℤ\nha : a.Coprime d\nk : ℤ\nhk : ↑n = ↑(φ d) * k + z\n⊢ ZMod.unitOfCoprime a ha ^ z = (ZMod.unitOfCoprime a ha ^ φ d) ^ k * ZMod.unitOfCoprime a ha ^ z",
"after_state": "case h.e'_2.h.e'_2.e_a\na b d n : ℕ\nz : ℤ\nha : a.Coprime d\nk : ℤ\nhk : ↑n = ↑(φ d) * k + z\n⊢ ZMod.unitOfCoprime a ha ^ z = 1 ^ k * ZMod.unitOfCoprime a ha ^ z"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "case h.e'_2.h.e'_2.e_a\na b d n : ℕ\nz : ℤ\nha : a.Coprime d\nk : ℤ\nhk : ↑n = ↑(φ d) * k + z\n⊢ ZMod.unitOfCoprime a ha ^ z = 1 ^ k * ZMod.unitOfCoprime a ha ^ z",
"after_state": "case h.e'_2.h.e'_2.e_a\na b d n : ℕ\nz : ℤ\nha : a.Coprime d\nk : ℤ\nhk : ↑n = ↑(φ d) * k + z\n⊢ ZMod.unitOfCoprime a ha ^ z = 1 ^ k * ZMod.unitOfCoprime a ha ^ z"
},
{
"line": "try (with_reducible rfl)",
"before_state": "case h.e'_2.h.e'_2.e_a\na b d n : ℕ\nz : ℤ\nha : a.Coprime d\nk : ℤ\nhk : ↑n = ↑(φ d) * k + z\n⊢ ZMod.unitOfCoprime a ha ^ z = 1 ^ k * ZMod.unitOfCoprime a ha ^ z",
"after_state": "case h.e'_2.h.e'_2.e_a\na b d n : ℕ\nz : ℤ\nha : a.Coprime d\nk : ℤ\nhk : ↑n = ↑(φ d) * k + z\n⊢ ZMod.unitOfCoprime a ha ^ z = 1 ^ k * ZMod.unitOfCoprime a ha ^ z"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "case h.e'_2.h.e'_2.e_a\na b d n : ℕ\nz : ℤ\nha : a.Coprime d\nk : ℤ\nhk : ↑n = ↑(φ d) * k + z\n⊢ ZMod.unitOfCoprime a ha ^ z = 1 ^ k * ZMod.unitOfCoprime a ha ^ z",
"after_state": "case h.e'_2.h.e'_2.e_a\na b d n : ℕ\nz : ℤ\nha : a.Coprime d\nk : ℤ\nhk : ↑n = ↑(φ d) * k + z\n⊢ ZMod.unitOfCoprime a ha ^ z = 1 ^ k * ZMod.unitOfCoprime a ha ^ z"
},
{
"line": "with_reducible rfl",
"before_state": "case h.e'_2.h.e'_2.e_a\na b d n : ℕ\nz : ℤ\nha : a.Coprime d\nk : ℤ\nhk : ↑n = ↑(φ d) * k + z\n⊢ ZMod.unitOfCoprime a ha ^ z = 1 ^ k * ZMod.unitOfCoprime a ha ^ z",
"after_state": "case h.e'_2.h.e'_2.e_a\na b d n : ℕ\nz : ℤ\nha : a.Coprime d\nk : ℤ\nhk : ↑n = ↑(φ d) * k + z\n⊢ ZMod.unitOfCoprime a ha ^ z = 1 ^ k * ZMod.unitOfCoprime a ha ^ z"
},
{
"line": "rfl",
"before_state": "case h.e'_2.h.e'_2.e_a\na b d n : ℕ\nz : ℤ\nha : a.Coprime d\nk : ℤ\nhk : ↑n = ↑(φ d) * k + z\n⊢ ZMod.unitOfCoprime a ha ^ z = 1 ^ k * ZMod.unitOfCoprime a ha ^ z",
"after_state": "case h.e'_2.h.e'_2.e_a\na b d n : ℕ\nz : ℤ\nha : a.Coprime d\nk : ℤ\nhk : ↑n = ↑(φ d) * k + z\n⊢ ZMod.unitOfCoprime a ha ^ z = 1 ^ k * ZMod.unitOfCoprime a ha ^ z"
},
{
"line": "apply_rfl",
"before_state": "case h.e'_2.h.e'_2.e_a\na b d n : ℕ\nz : ℤ\nha : a.Coprime d\nk : ℤ\nhk : ↑n = ↑(φ d) * k + z\n⊢ ZMod.unitOfCoprime a ha ^ z = 1 ^ k * ZMod.unitOfCoprime a ha ^ z",
"after_state": "case h.e'_2.h.e'_2.e_a\na b d n : ℕ\nz : ℤ\nha : a.Coprime d\nk : ℤ\nhk : ↑n = ↑(φ d) * k + z\n⊢ ZMod.unitOfCoprime a ha ^ z = 1 ^ k * ZMod.unitOfCoprime a ha ^ z"
},
{
"line": "skip",
"before_state": "case h.e'_2.h.e'_2.e_a\na b d n : ℕ\nz : ℤ\nha : a.Coprime d\nk : ℤ\nhk : ↑n = ↑(φ d) * k + z\n⊢ ZMod.unitOfCoprime a ha ^ z = 1 ^ k * ZMod.unitOfCoprime a ha ^ z",
"after_state": "case h.e'_2.h.e'_2.e_a\na b d n : ℕ\nz : ℤ\nha : a.Coprime d\nk : ℤ\nhk : ↑n = ↑(φ d) * k + z\n⊢ ZMod.unitOfCoprime a ha ^ z = 1 ^ k * ZMod.unitOfCoprime a ha ^ z"
},
{
"line": "rw [one_zpow]",
"before_state": "case h.e'_2.h.e'_2.e_a\na b d n : ℕ\nz : ℤ\nha : a.Coprime d\nk : ℤ\nhk : ↑n = ↑(φ d) * k + z\n⊢ ZMod.unitOfCoprime a ha ^ z = 1 ^ k * ZMod.unitOfCoprime a ha ^ z",
"after_state": "case h.e'_2.h.e'_2.e_a\na b d n : ℕ\nz : ℤ\nha : a.Coprime d\nk : ℤ\nhk : ↑n = ↑(φ d) * k + z\n⊢ ZMod.unitOfCoprime a ha ^ z = 1 * ZMod.unitOfCoprime a ha ^ z"
},
{
"line": "rewrite [one_zpow]",
"before_state": "case h.e'_2.h.e'_2.e_a\na b d n : ℕ\nz : ℤ\nha : a.Coprime d\nk : ℤ\nhk : ↑n = ↑(φ d) * k + z\n⊢ ZMod.unitOfCoprime a ha ^ z = 1 ^ k * ZMod.unitOfCoprime a ha ^ z",
"after_state": "case h.e'_2.h.e'_2.e_a\na b d n : ℕ\nz : ℤ\nha : a.Coprime d\nk : ℤ\nhk : ↑n = ↑(φ d) * k + z\n⊢ ZMod.unitOfCoprime a ha ^ z = 1 * ZMod.unitOfCoprime a ha ^ z"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "case h.e'_2.h.e'_2.e_a\na b d n : ℕ\nz : ℤ\nha : a.Coprime d\nk : ℤ\nhk : ↑n = ↑(φ d) * k + z\n⊢ ZMod.unitOfCoprime a ha ^ z = 1 * ZMod.unitOfCoprime a ha ^ z",
"after_state": "case h.e'_2.h.e'_2.e_a\na b d n : ℕ\nz : ℤ\nha : a.Coprime d\nk : ℤ\nhk : ↑n = ↑(φ d) * k + z\n⊢ ZMod.unitOfCoprime a ha ^ z = 1 * ZMod.unitOfCoprime a ha ^ z"
},
{
"line": "try (with_reducible rfl)",
"before_state": "case h.e'_2.h.e'_2.e_a\na b d n : ℕ\nz : ℤ\nha : a.Coprime d\nk : ℤ\nhk : ↑n = ↑(φ d) * k + z\n⊢ ZMod.unitOfCoprime a ha ^ z = 1 * ZMod.unitOfCoprime a ha ^ z",
"after_state": "case h.e'_2.h.e'_2.e_a\na b d n : ℕ\nz : ℤ\nha : a.Coprime d\nk : ℤ\nhk : ↑n = ↑(φ d) * k + z\n⊢ ZMod.unitOfCoprime a ha ^ z = 1 * ZMod.unitOfCoprime a ha ^ z"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "case h.e'_2.h.e'_2.e_a\na b d n : ℕ\nz : ℤ\nha : a.Coprime d\nk : ℤ\nhk : ↑n = ↑(φ d) * k + z\n⊢ ZMod.unitOfCoprime a ha ^ z = 1 * ZMod.unitOfCoprime a ha ^ z",
"after_state": "case h.e'_2.h.e'_2.e_a\na b d n : ℕ\nz : ℤ\nha : a.Coprime d\nk : ℤ\nhk : ↑n = ↑(φ d) * k + z\n⊢ ZMod.unitOfCoprime a ha ^ z = 1 * ZMod.unitOfCoprime a ha ^ z"
},
{
"line": "with_reducible rfl",
"before_state": "case h.e'_2.h.e'_2.e_a\na b d n : ℕ\nz : ℤ\nha : a.Coprime d\nk : ℤ\nhk : ↑n = ↑(φ d) * k + z\n⊢ ZMod.unitOfCoprime a ha ^ z = 1 * ZMod.unitOfCoprime a ha ^ z",
"after_state": "case h.e'_2.h.e'_2.e_a\na b d n : ℕ\nz : ℤ\nha : a.Coprime d\nk : ℤ\nhk : ↑n = ↑(φ d) * k + z\n⊢ ZMod.unitOfCoprime a ha ^ z = 1 * ZMod.unitOfCoprime a ha ^ z"
},
{
"line": "rfl",
"before_state": "case h.e'_2.h.e'_2.e_a\na b d n : ℕ\nz : ℤ\nha : a.Coprime d\nk : ℤ\nhk : ↑n = ↑(φ d) * k + z\n⊢ ZMod.unitOfCoprime a ha ^ z = 1 * ZMod.unitOfCoprime a ha ^ z",
"after_state": "case h.e'_2.h.e'_2.e_a\na b d n : ℕ\nz : ℤ\nha : a.Coprime d\nk : ℤ\nhk : ↑n = ↑(φ d) * k + z\n⊢ ZMod.unitOfCoprime a ha ^ z = 1 * ZMod.unitOfCoprime a ha ^ z"
},
{
"line": "apply_rfl",
"before_state": "case h.e'_2.h.e'_2.e_a\na b d n : ℕ\nz : ℤ\nha : a.Coprime d\nk : ℤ\nhk : ↑n = ↑(φ d) * k + z\n⊢ ZMod.unitOfCoprime a ha ^ z = 1 * ZMod.unitOfCoprime a ha ^ z",
"after_state": "case h.e'_2.h.e'_2.e_a\na b d n : ℕ\nz : ℤ\nha : a.Coprime d\nk : ℤ\nhk : ↑n = ↑(φ d) * k + z\n⊢ ZMod.unitOfCoprime a ha ^ z = 1 * ZMod.unitOfCoprime a ha ^ z"
},
{
"line": "skip",
"before_state": "case h.e'_2.h.e'_2.e_a\na b d n : ℕ\nz : ℤ\nha : a.Coprime d\nk : ℤ\nhk : ↑n = ↑(φ d) * k + z\n⊢ ZMod.unitOfCoprime a ha ^ z = 1 * ZMod.unitOfCoprime a ha ^ z",
"after_state": "case h.e'_2.h.e'_2.e_a\na b d n : ℕ\nz : ℤ\nha : a.Coprime d\nk : ℤ\nhk : ↑n = ↑(φ d) * k + z\n⊢ ZMod.unitOfCoprime a ha ^ z = 1 * ZMod.unitOfCoprime a ha ^ z"
},
{
"line": "rw [one_mul]",
"before_state": "case h.e'_2.h.e'_2.e_a\na b d n : ℕ\nz : ℤ\nha : a.Coprime d\nk : ℤ\nhk : ↑n = ↑(φ d) * k + z\n⊢ ZMod.unitOfCoprime a ha ^ z = 1 * ZMod.unitOfCoprime a ha ^ z",
"after_state": "No Goals!"
},
{
"line": "rewrite [one_mul]",
"before_state": "case h.e'_2.h.e'_2.e_a\na b d n : ℕ\nz : ℤ\nha : a.Coprime d\nk : ℤ\nhk : ↑n = ↑(φ d) * k + z\n⊢ ZMod.unitOfCoprime a ha ^ z = 1 * ZMod.unitOfCoprime a ha ^ z",
"after_state": "case h.e'_2.h.e'_2.e_a\na b d n : ℕ\nz : ℤ\nha : a.Coprime d\nk : ℤ\nhk : ↑n = ↑(φ d) * k + z\n⊢ ZMod.unitOfCoprime a ha ^ z = ZMod.unitOfCoprime a ha ^ z"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "case h.e'_2.h.e'_2.e_a\na b d n : ℕ\nz : ℤ\nha : a.Coprime d\nk : ℤ\nhk : ↑n = ↑(φ d) * k + z\n⊢ ZMod.unitOfCoprime a ha ^ z = ZMod.unitOfCoprime a ha ^ z",
"after_state": "No Goals!"
},
{
"line": "try (with_reducible rfl)",
"before_state": "case h.e'_2.h.e'_2.e_a\na b d n : ℕ\nz : ℤ\nha : a.Coprime d\nk : ℤ\nhk : ↑n = ↑(φ d) * k + z\n⊢ ZMod.unitOfCoprime a ha ^ z = ZMod.unitOfCoprime a ha ^ z",
"after_state": "No Goals!"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "case h.e'_2.h.e'_2.e_a\na b d n : ℕ\nz : ℤ\nha : a.Coprime d\nk : ℤ\nhk : ↑n = ↑(φ d) * k + z\n⊢ ZMod.unitOfCoprime a ha ^ z = ZMod.unitOfCoprime a ha ^ z",
"after_state": "No Goals!"
},
{
"line": "with_reducible rfl",
"before_state": "case h.e'_2.h.e'_2.e_a\na b d n : ℕ\nz : ℤ\nha : a.Coprime d\nk : ℤ\nhk : ↑n = ↑(φ d) * k + z\n⊢ ZMod.unitOfCoprime a ha ^ z = ZMod.unitOfCoprime a ha ^ z",
"after_state": "No Goals!"
},
{
"line": "rfl",
"before_state": "case h.e'_2.h.e'_2.e_a\na b d n : ℕ\nz : ℤ\nha : a.Coprime d\nk : ℤ\nhk : ↑n = ↑(φ d) * k + z\n⊢ ZMod.unitOfCoprime a ha ^ z = ZMod.unitOfCoprime a ha ^ z",
"after_state": "No Goals!"
},
{
"line": "eq_refl",
"before_state": "case h.e'_2.h.e'_2.e_a\na b d n : ℕ\nz : ℤ\nha : a.Coprime d\nk : ℤ\nhk : ↑n = ↑(φ d) * k + z\n⊢ ZMod.unitOfCoprime a ha ^ z = ZMod.unitOfCoprime a ha ^ z",
"after_state": "No Goals!"
}
] |
lemma map_add_one_range (p : ℕ → Prop) [DecidablePred p] (n : ℕ) (h0 : ¬ p 0) :
{x ∈ Finset.range n | p (x + 1)}.map ⟨(· + 1), add_left_injective 1⟩ =
{x ∈ Finset.range (n + 1) | p x } := by
ext x
simp only [Finset.mem_map]
constructor
· aesop
· intro hx
use x - 1
cases x <;> simp_all
|
/root/DuelModelResearch/mathlib4/Archive/Imo/Imo2024Q3.lean
|
{
"open": [
"scoped Finset"
],
"variables": []
}
|
[
{
"line": "ext x",
"before_state": "p : ℕ → Prop\ninst✝ : DecidablePred p\nn : ℕ\nh0 : ¬p 0\n⊢ Finset.map { toFun := fun x => x + 1, inj' := ⋯ } ({x ∈ Finset.range n | p (x + 1)}) =\n {x ∈ Finset.range (n + 1) | p x}",
"after_state": "case h\np : ℕ → Prop\ninst✝ : DecidablePred p\nn : ℕ\nh0 : ¬p 0\nx : ℕ\n⊢ x ∈ Finset.map { toFun := fun x => x + 1, inj' := ⋯ } ({x ∈ Finset.range n | p (x + 1)}) ↔\n x ∈ {x ∈ Finset.range (n + 1) | p x}"
},
{
"line": "simp only [Finset.mem_map]",
"before_state": "case h\np : ℕ → Prop\ninst✝ : DecidablePred p\nn : ℕ\nh0 : ¬p 0\nx : ℕ\n⊢ x ∈ Finset.map { toFun := fun x => x + 1, inj' := ⋯ } ({x ∈ Finset.range n | p (x + 1)}) ↔\n x ∈ {x ∈ Finset.range (n + 1) | p x}",
"after_state": "case h\np : ℕ → Prop\ninst✝ : DecidablePred p\nn : ℕ\nh0 : ¬p 0\nx : ℕ\n⊢ (∃ a ∈ {x ∈ Finset.range n | p (x + 1)}, { toFun := fun x => x + 1, inj' := ⋯ } a = x) ↔\n x ∈ {x ∈ Finset.range (n + 1) | p x}"
},
{
"line": "constructor",
"before_state": "case h\np : ℕ → Prop\ninst✝ : DecidablePred p\nn : ℕ\nh0 : ¬p 0\nx : ℕ\n⊢ (∃ a ∈ {x ∈ Finset.range n | p (x + 1)}, { toFun := fun x => x + 1, inj' := ⋯ } a = x) ↔\n x ∈ {x ∈ Finset.range (n + 1) | p x}",
"after_state": "case h.mp\np : ℕ → Prop\ninst✝ : DecidablePred p\nn : ℕ\nh0 : ¬p 0\nx : ℕ\n⊢ (∃ a ∈ {x ∈ Finset.range n | p (x + 1)}, { toFun := fun x => x + 1, inj' := ⋯ } a = x) →\n x ∈ {x ∈ Finset.range (n + 1) | p x}\n---\ncase h.mpr\np : ℕ → Prop\ninst✝ : DecidablePred p\nn : ℕ\nh0 : ¬p 0\nx : ℕ\n⊢ x ∈ {x ∈ Finset.range (n + 1) | p x} →\n ∃ a ∈ {x ∈ Finset.range n | p (x + 1)}, { toFun := fun x => x + 1, inj' := ⋯ } a = x"
},
{
"line": "aesop",
"before_state": "case h.mp\np : ℕ → Prop\ninst✝ : DecidablePred p\nn : ℕ\nh0 : ¬p 0\nx : ℕ\n⊢ (∃ a ∈ {x ∈ Finset.range n | p (x + 1)}, { toFun := fun x => x + 1, inj' := ⋯ } a = x) →\n x ∈ {x ∈ Finset.range (n + 1) | p x}",
"after_state": "No Goals!"
},
{
"line": "intro hx",
"before_state": "case h.mpr\np : ℕ → Prop\ninst✝ : DecidablePred p\nn : ℕ\nh0 : ¬p 0\nx : ℕ\n⊢ x ∈ {x ∈ Finset.range (n + 1) | p x} →\n ∃ a ∈ {x ∈ Finset.range n | p (x + 1)}, { toFun := fun x => x + 1, inj' := ⋯ } a = x",
"after_state": "case h.mpr\np : ℕ → Prop\ninst✝ : DecidablePred p\nn : ℕ\nh0 : ¬p 0\nx : ℕ\nhx : x ∈ {x ∈ Finset.range (n + 1) | p x}\n⊢ ∃ a ∈ {x ∈ Finset.range n | p (x + 1)}, { toFun := fun x => x + 1, inj' := ⋯ } a = x"
},
{
"line": "use x - 1",
"before_state": "case h.mpr\np : ℕ → Prop\ninst✝ : DecidablePred p\nn : ℕ\nh0 : ¬p 0\nx : ℕ\nhx : x ∈ {x ∈ Finset.range (n + 1) | p x}\n⊢ ∃ a ∈ {x ∈ Finset.range n | p (x + 1)}, { toFun := fun x => x + 1, inj' := ⋯ } a = x",
"after_state": "case h\np : ℕ → Prop\ninst✝ : DecidablePred p\nn : ℕ\nh0 : ¬p 0\nx : ℕ\nhx : x ∈ {x ∈ Finset.range (n + 1) | p x}\n⊢ x - 1 ∈ {x ∈ Finset.range n | p (x + 1)} ∧ { toFun := fun x => x + 1, inj' := ⋯ } (x - 1) = x"
},
{
"line": "refine without_cdot(x - 1 : ?m✝)",
"before_state": "case w\np : ℕ → Prop\ninst✝ : DecidablePred p\nn : ℕ\nh0 : ¬p 0\nx : ℕ\nhx : x ∈ {x ∈ Finset.range (n + 1) | p x}\n⊢ ℕ",
"after_state": "No Goals!"
},
{
"line": "try with_reducible use_discharger",
"before_state": "case h\np : ℕ → Prop\ninst✝ : DecidablePred p\nn : ℕ\nh0 : ¬p 0\nx : ℕ\nhx : x ∈ {x ∈ Finset.range (n + 1) | p x}\n⊢ x - 1 ∈ {x ∈ Finset.range n | p (x + 1)} ∧ { toFun := fun x => x + 1, inj' := ⋯ } (x - 1) = x",
"after_state": "case h\np : ℕ → Prop\ninst✝ : DecidablePred p\nn : ℕ\nh0 : ¬p 0\nx : ℕ\nhx : x ∈ {x ∈ Finset.range (n + 1) | p x}\n⊢ x - 1 ∈ {x ∈ Finset.range n | p (x + 1)} ∧ { toFun := fun x => x + 1, inj' := ⋯ } (x - 1) = x"
},
{
"line": "first\n| with_reducible use_discharger\n| skip",
"before_state": "case h\np : ℕ → Prop\ninst✝ : DecidablePred p\nn : ℕ\nh0 : ¬p 0\nx : ℕ\nhx : x ∈ {x ∈ Finset.range (n + 1) | p x}\n⊢ x - 1 ∈ {x ∈ Finset.range n | p (x + 1)} ∧ { toFun := fun x => x + 1, inj' := ⋯ } (x - 1) = x",
"after_state": "case h\np : ℕ → Prop\ninst✝ : DecidablePred p\nn : ℕ\nh0 : ¬p 0\nx : ℕ\nhx : x ∈ {x ∈ Finset.range (n + 1) | p x}\n⊢ x - 1 ∈ {x ∈ Finset.range n | p (x + 1)} ∧ { toFun := fun x => x + 1, inj' := ⋯ } (x - 1) = x"
},
{
"line": "with_reducible use_discharger",
"before_state": "case h\np : ℕ → Prop\ninst✝ : DecidablePred p\nn : ℕ\nh0 : ¬p 0\nx : ℕ\nhx : x ∈ {x ∈ Finset.range (n + 1) | p x}\n⊢ x - 1 ∈ {x ∈ Finset.range n | p (x + 1)} ∧ { toFun := fun x => x + 1, inj' := ⋯ } (x - 1) = x",
"after_state": "case h\np : ℕ → Prop\ninst✝ : DecidablePred p\nn : ℕ\nh0 : ¬p 0\nx : ℕ\nhx : x ∈ {x ∈ Finset.range (n + 1) | p x}\n⊢ x - 1 ∈ {x ∈ Finset.range n | p (x + 1)} ∧ { toFun := fun x => x + 1, inj' := ⋯ } (x - 1) = x"
},
{
"line": "use_discharger",
"before_state": "case h\np : ℕ → Prop\ninst✝ : DecidablePred p\nn : ℕ\nh0 : ¬p 0\nx : ℕ\nhx : x ∈ {x ∈ Finset.range (n + 1) | p x}\n⊢ x - 1 ∈ {x ∈ Finset.range n | p (x + 1)} ∧ { toFun := fun x => x + 1, inj' := ⋯ } (x - 1) = x",
"after_state": "case h\np : ℕ → Prop\ninst✝ : DecidablePred p\nn : ℕ\nh0 : ¬p 0\nx : ℕ\nhx : x ∈ {x ∈ Finset.range (n + 1) | p x}\n⊢ x - 1 ∈ {x ∈ Finset.range n | p (x + 1)} ∧ { toFun := fun x => x + 1, inj' := ⋯ } (x - 1) = x"
},
{
"line": "focus\n apply exists_prop.mpr✝\n with_annotate_state\"<;>\" skip\n all_goals use_discharger",
"before_state": "case h\np : ℕ → Prop\ninst✝ : DecidablePred p\nn : ℕ\nh0 : ¬p 0\nx : ℕ\nhx : x ∈ {x ∈ Finset.range (n + 1) | p x}\n⊢ x - 1 ∈ {x ∈ Finset.range n | p (x + 1)} ∧ { toFun := fun x => x + 1, inj' := ⋯ } (x - 1) = x",
"after_state": "case h\np : ℕ → Prop\ninst✝ : DecidablePred p\nn : ℕ\nh0 : ¬p 0\nx : ℕ\nhx : x ∈ {x ∈ Finset.range (n + 1) | p x}\n⊢ x - 1 ∈ {x ∈ Finset.range n | p (x + 1)} ∧ { toFun := fun x => x + 1, inj' := ⋯ } (x - 1) = x"
},
{
"line": "apply exists_prop.mpr✝",
"before_state": "case h\np : ℕ → Prop\ninst✝ : DecidablePred p\nn : ℕ\nh0 : ¬p 0\nx : ℕ\nhx : x ∈ {x ∈ Finset.range (n + 1) | p x}\n⊢ x - 1 ∈ {x ∈ Finset.range n | p (x + 1)} ∧ { toFun := fun x => x + 1, inj' := ⋯ } (x - 1) = x",
"after_state": "case h\np : ℕ → Prop\ninst✝ : DecidablePred p\nn : ℕ\nh0 : ¬p 0\nx : ℕ\nhx : x ∈ {x ∈ Finset.range (n + 1) | p x}\n⊢ x - 1 ∈ {x ∈ Finset.range n | p (x + 1)} ∧ { toFun := fun x => x + 1, inj' := ⋯ } (x - 1) = x"
},
{
"line": "skip",
"before_state": "case h\np : ℕ → Prop\ninst✝ : DecidablePred p\nn : ℕ\nh0 : ¬p 0\nx : ℕ\nhx : x ∈ {x ∈ Finset.range (n + 1) | p x}\n⊢ x - 1 ∈ {x ∈ Finset.range n | p (x + 1)} ∧ { toFun := fun x => x + 1, inj' := ⋯ } (x - 1) = x",
"after_state": "case h\np : ℕ → Prop\ninst✝ : DecidablePred p\nn : ℕ\nh0 : ¬p 0\nx : ℕ\nhx : x ∈ {x ∈ Finset.range (n + 1) | p x}\n⊢ x - 1 ∈ {x ∈ Finset.range n | p (x + 1)} ∧ { toFun := fun x => x + 1, inj' := ⋯ } (x - 1) = x"
},
{
"line": "focus\n cases x\n with_annotate_state\"<;>\" skip\n all_goals simp_all",
"before_state": "case h\np : ℕ → Prop\ninst✝ : DecidablePred p\nn : ℕ\nh0 : ¬p 0\nx : ℕ\nhx : x ∈ {x ∈ Finset.range (n + 1) | p x}\n⊢ x - 1 ∈ {x ∈ Finset.range n | p (x + 1)} ∧ { toFun := fun x => x + 1, inj' := ⋯ } (x - 1) = x",
"after_state": "No Goals!"
},
{
"line": "cases x",
"before_state": "case h\np : ℕ → Prop\ninst✝ : DecidablePred p\nn : ℕ\nh0 : ¬p 0\nx : ℕ\nhx : x ∈ {x ∈ Finset.range (n + 1) | p x}\n⊢ x - 1 ∈ {x ∈ Finset.range n | p (x + 1)} ∧ { toFun := fun x => x + 1, inj' := ⋯ } (x - 1) = x",
"after_state": "case h.zero\np : ℕ → Prop\ninst✝ : DecidablePred p\nn : ℕ\nh0 : ¬p 0\nhx : 0 ∈ {x ∈ Finset.range (n + 1) | p x}\n⊢ 0 - 1 ∈ {x ∈ Finset.range n | p (x + 1)} ∧ { toFun := fun x => x + 1, inj' := ⋯ } (0 - 1) = 0\n---\ncase h.succ\np : ℕ → Prop\ninst✝ : DecidablePred p\nn : ℕ\nh0 : ¬p 0\nn✝ : ℕ\nhx : n✝ + 1 ∈ {x ∈ Finset.range (n + 1) | p x}\n⊢ n✝ + 1 - 1 ∈ {x ∈ Finset.range n | p (x + 1)} ∧ { toFun := fun x => x + 1, inj' := ⋯ } (n✝ + 1 - 1) = n✝ + 1"
},
{
"line": "with_annotate_state\"<;>\" skip",
"before_state": "case h.zero\np : ℕ → Prop\ninst✝ : DecidablePred p\nn : ℕ\nh0 : ¬p 0\nhx : 0 ∈ {x ∈ Finset.range (n + 1) | p x}\n⊢ 0 - 1 ∈ {x ∈ Finset.range n | p (x + 1)} ∧ { toFun := fun x => x + 1, inj' := ⋯ } (0 - 1) = 0\n---\ncase h.succ\np : ℕ → Prop\ninst✝ : DecidablePred p\nn : ℕ\nh0 : ¬p 0\nn✝ : ℕ\nhx : n✝ + 1 ∈ {x ∈ Finset.range (n + 1) | p x}\n⊢ n✝ + 1 - 1 ∈ {x ∈ Finset.range n | p (x + 1)} ∧ { toFun := fun x => x + 1, inj' := ⋯ } (n✝ + 1 - 1) = n✝ + 1",
"after_state": "case h.zero\np : ℕ → Prop\ninst✝ : DecidablePred p\nn : ℕ\nh0 : ¬p 0\nhx : 0 ∈ {x ∈ Finset.range (n + 1) | p x}\n⊢ 0 - 1 ∈ {x ∈ Finset.range n | p (x + 1)} ∧ { toFun := fun x => x + 1, inj' := ⋯ } (0 - 1) = 0\n---\ncase h.succ\np : ℕ → Prop\ninst✝ : DecidablePred p\nn : ℕ\nh0 : ¬p 0\nn✝ : ℕ\nhx : n✝ + 1 ∈ {x ∈ Finset.range (n + 1) | p x}\n⊢ n✝ + 1 - 1 ∈ {x ∈ Finset.range n | p (x + 1)} ∧ { toFun := fun x => x + 1, inj' := ⋯ } (n✝ + 1 - 1) = n✝ + 1"
},
{
"line": "skip",
"before_state": "case h.zero\np : ℕ → Prop\ninst✝ : DecidablePred p\nn : ℕ\nh0 : ¬p 0\nhx : 0 ∈ {x ∈ Finset.range (n + 1) | p x}\n⊢ 0 - 1 ∈ {x ∈ Finset.range n | p (x + 1)} ∧ { toFun := fun x => x + 1, inj' := ⋯ } (0 - 1) = 0\n---\ncase h.succ\np : ℕ → Prop\ninst✝ : DecidablePred p\nn : ℕ\nh0 : ¬p 0\nn✝ : ℕ\nhx : n✝ + 1 ∈ {x ∈ Finset.range (n + 1) | p x}\n⊢ n✝ + 1 - 1 ∈ {x ∈ Finset.range n | p (x + 1)} ∧ { toFun := fun x => x + 1, inj' := ⋯ } (n✝ + 1 - 1) = n✝ + 1",
"after_state": "case h.zero\np : ℕ → Prop\ninst✝ : DecidablePred p\nn : ℕ\nh0 : ¬p 0\nhx : 0 ∈ {x ∈ Finset.range (n + 1) | p x}\n⊢ 0 - 1 ∈ {x ∈ Finset.range n | p (x + 1)} ∧ { toFun := fun x => x + 1, inj' := ⋯ } (0 - 1) = 0\n---\ncase h.succ\np : ℕ → Prop\ninst✝ : DecidablePred p\nn : ℕ\nh0 : ¬p 0\nn✝ : ℕ\nhx : n✝ + 1 ∈ {x ∈ Finset.range (n + 1) | p x}\n⊢ n✝ + 1 - 1 ∈ {x ∈ Finset.range n | p (x + 1)} ∧ { toFun := fun x => x + 1, inj' := ⋯ } (n✝ + 1 - 1) = n✝ + 1"
},
{
"line": "all_goals simp_all",
"before_state": "case h.zero\np : ℕ → Prop\ninst✝ : DecidablePred p\nn : ℕ\nh0 : ¬p 0\nhx : 0 ∈ {x ∈ Finset.range (n + 1) | p x}\n⊢ 0 - 1 ∈ {x ∈ Finset.range n | p (x + 1)} ∧ { toFun := fun x => x + 1, inj' := ⋯ } (0 - 1) = 0\n---\ncase h.succ\np : ℕ → Prop\ninst✝ : DecidablePred p\nn : ℕ\nh0 : ¬p 0\nn✝ : ℕ\nhx : n✝ + 1 ∈ {x ∈ Finset.range (n + 1) | p x}\n⊢ n✝ + 1 - 1 ∈ {x ∈ Finset.range n | p (x + 1)} ∧ { toFun := fun x => x + 1, inj' := ⋯ } (n✝ + 1 - 1) = n✝ + 1",
"after_state": "No Goals!"
},
{
"line": "simp_all",
"before_state": "case h.zero\np : ℕ → Prop\ninst✝ : DecidablePred p\nn : ℕ\nh0 : ¬p 0\nhx : 0 ∈ {x ∈ Finset.range (n + 1) | p x}\n⊢ 0 - 1 ∈ {x ∈ Finset.range n | p (x + 1)} ∧ { toFun := fun x => x + 1, inj' := ⋯ } (0 - 1) = 0",
"after_state": "No Goals!"
},
{
"line": "simp_all",
"before_state": "case h.succ\np : ℕ → Prop\ninst✝ : DecidablePred p\nn : ℕ\nh0 : ¬p 0\nn✝ : ℕ\nhx : n✝ + 1 ∈ {x ∈ Finset.range (n + 1) | p x}\n⊢ n✝ + 1 - 1 ∈ {x ∈ Finset.range n | p (x + 1)} ∧ { toFun := fun x => x + 1, inj' := ⋯ } (n✝ + 1 - 1) = n✝ + 1",
"after_state": "No Goals!"
}
] |
theorem add_mod2 (a : ℕ) : ∃ t, a + a % 2 = t * 2 := by
simp only [mul_comm _ 2] -- write `t*2` as `2*t`
apply dvd_of_mod_eq_zero -- it suffices to prove `(a + a % 2) % 2 = 0`
rw [add_mod]
rw [mod_mod]
rw [← two_mul]
rw [mul_mod_right]
|
/root/DuelModelResearch/mathlib4/Archive/MiuLanguage/DecisionSuf.lean
|
{
"open": [
"MiuAtom List Nat"
],
"variables": []
}
|
[
{
"line": "simp only [mul_comm _ 2]\n -- write `t*2` as `2*t`",
"before_state": "a : ℕ\n⊢ ∃ t, a + a % 2 = t * 2",
"after_state": "a : ℕ\n⊢ ∃ t, a + a % 2 = 2 * t"
},
{
"line": "apply dvd_of_mod_eq_zero",
"before_state": "a : ℕ\n⊢ ∃ t, a + a % 2 = 2 * t",
"after_state": "No Goals!"
}
] |
private theorem le_pow2_and_pow2_eq_mod3' (c : ℕ) (x : ℕ) (h : c = 1 ∨ c = 2) :
∃ m : ℕ, c + 3 * x ≤ 2 ^ m ∧ 2 ^ m % 3 = c % 3 := by
induction' x with k hk
· use c + 1
rcases h with hc | hc <;> · rw [hc]; norm_num
rcases hk with ⟨g, hkg, hgmod⟩
by_cases hp : c + 3 * (k + 1) ≤ 2 ^ g
· use g, hp, hgmod
refine ⟨g + 2, ?_, ?_⟩
· rw [mul_succ, ← add_assoc, pow_add]
change c + 3 * k + 3 ≤ 2 ^ g * (1 + 3); rw [mul_add (2 ^ g) 1 3, mul_one]
linarith [hkg, @Nat.one_le_two_pow g]
· rw [pow_add, ← mul_one c]
exact ModEq.mul hgmod rfl
|
/root/DuelModelResearch/mathlib4/Archive/MiuLanguage/DecisionSuf.lean
|
{
"open": [
"MiuAtom List Nat"
],
"variables": []
}
|
[
{
"line": "induction' x with k hk",
"before_state": "c x : ℕ\nh : c = 1 ∨ c = 2\n⊢ ∃ m, c + 3 * x ≤ 2 ^ m ∧ 2 ^ m % 3 = c % 3",
"after_state": "case zero\nc : ℕ\nh : c = 1 ∨ c = 2\n⊢ ∃ m, c + 3 * 0 ≤ 2 ^ m ∧ 2 ^ m % 3 = c % 3\n---\ncase succ\nc : ℕ\nh : c = 1 ∨ c = 2\nk : ℕ\nhk : ∃ m, c + 3 * k ≤ 2 ^ m ∧ 2 ^ m % 3 = c % 3\n⊢ ∃ m, c + 3 * (k + 1) ≤ 2 ^ m ∧ 2 ^ m % 3 = c % 3"
},
{
"line": "use c + 1",
"before_state": "case zero\nc : ℕ\nh : c = 1 ∨ c = 2\n⊢ ∃ m, c + 3 * 0 ≤ 2 ^ m ∧ 2 ^ m % 3 = c % 3",
"after_state": "case h\nc : ℕ\nh : c = 1 ∨ c = 2\n⊢ c + 3 * 0 ≤ 2 ^ (c + 1) ∧ 2 ^ (c + 1) % 3 = c % 3"
},
{
"line": "refine without_cdot(c + 1 : ?m✝)",
"before_state": "case w\nc : ℕ\nh : c = 1 ∨ c = 2\n⊢ ℕ",
"after_state": "No Goals!"
},
{
"line": "try with_reducible use_discharger",
"before_state": "case h\nc : ℕ\nh : c = 1 ∨ c = 2\n⊢ c + 3 * 0 ≤ 2 ^ (c + 1) ∧ 2 ^ (c + 1) % 3 = c % 3",
"after_state": "case h\nc : ℕ\nh : c = 1 ∨ c = 2\n⊢ c + 3 * 0 ≤ 2 ^ (c + 1) ∧ 2 ^ (c + 1) % 3 = c % 3"
},
{
"line": "first\n| with_reducible use_discharger\n| skip",
"before_state": "case h\nc : ℕ\nh : c = 1 ∨ c = 2\n⊢ c + 3 * 0 ≤ 2 ^ (c + 1) ∧ 2 ^ (c + 1) % 3 = c % 3",
"after_state": "case h\nc : ℕ\nh : c = 1 ∨ c = 2\n⊢ c + 3 * 0 ≤ 2 ^ (c + 1) ∧ 2 ^ (c + 1) % 3 = c % 3"
},
{
"line": "with_reducible use_discharger",
"before_state": "case h\nc : ℕ\nh : c = 1 ∨ c = 2\n⊢ c + 3 * 0 ≤ 2 ^ (c + 1) ∧ 2 ^ (c + 1) % 3 = c % 3",
"after_state": "case h\nc : ℕ\nh : c = 1 ∨ c = 2\n⊢ c + 3 * 0 ≤ 2 ^ (c + 1) ∧ 2 ^ (c + 1) % 3 = c % 3"
},
{
"line": "use_discharger",
"before_state": "case h\nc : ℕ\nh : c = 1 ∨ c = 2\n⊢ c + 3 * 0 ≤ 2 ^ (c + 1) ∧ 2 ^ (c + 1) % 3 = c % 3",
"after_state": "case h\nc : ℕ\nh : c = 1 ∨ c = 2\n⊢ c + 3 * 0 ≤ 2 ^ (c + 1) ∧ 2 ^ (c + 1) % 3 = c % 3"
},
{
"line": "focus\n apply exists_prop.mpr✝\n with_annotate_state\"<;>\" skip\n all_goals use_discharger",
"before_state": "case h\nc : ℕ\nh : c = 1 ∨ c = 2\n⊢ c + 3 * 0 ≤ 2 ^ (c + 1) ∧ 2 ^ (c + 1) % 3 = c % 3",
"after_state": "case h\nc : ℕ\nh : c = 1 ∨ c = 2\n⊢ c + 3 * 0 ≤ 2 ^ (c + 1) ∧ 2 ^ (c + 1) % 3 = c % 3"
},
{
"line": "apply exists_prop.mpr✝",
"before_state": "case h\nc : ℕ\nh : c = 1 ∨ c = 2\n⊢ c + 3 * 0 ≤ 2 ^ (c + 1) ∧ 2 ^ (c + 1) % 3 = c % 3",
"after_state": "case h\nc : ℕ\nh : c = 1 ∨ c = 2\n⊢ c + 3 * 0 ≤ 2 ^ (c + 1) ∧ 2 ^ (c + 1) % 3 = c % 3"
},
{
"line": "skip",
"before_state": "case h\nc : ℕ\nh : c = 1 ∨ c = 2\n⊢ c + 3 * 0 ≤ 2 ^ (c + 1) ∧ 2 ^ (c + 1) % 3 = c % 3",
"after_state": "case h\nc : ℕ\nh : c = 1 ∨ c = 2\n⊢ c + 3 * 0 ≤ 2 ^ (c + 1) ∧ 2 ^ (c + 1) % 3 = c % 3"
},
{
"line": "focus\n rcases h with hc | hc\n with_annotate_state\"<;>\" skip\n all_goals · rw [hc]; norm_num",
"before_state": "case h\nc : ℕ\nh : c = 1 ∨ c = 2\n⊢ c + 3 * 0 ≤ 2 ^ (c + 1) ∧ 2 ^ (c + 1) % 3 = c % 3",
"after_state": "No Goals!"
},
{
"line": "rcases h with hc | hc",
"before_state": "case h\nc : ℕ\nh : c = 1 ∨ c = 2\n⊢ c + 3 * 0 ≤ 2 ^ (c + 1) ∧ 2 ^ (c + 1) % 3 = c % 3",
"after_state": "case h.inl\nc : ℕ\nhc : c = 1\n⊢ c + 3 * 0 ≤ 2 ^ (c + 1) ∧ 2 ^ (c + 1) % 3 = c % 3\n---\ncase h.inr\nc : ℕ\nhc : c = 2\n⊢ c + 3 * 0 ≤ 2 ^ (c + 1) ∧ 2 ^ (c + 1) % 3 = c % 3"
},
{
"line": "with_annotate_state\"<;>\" skip",
"before_state": "case h.inl\nc : ℕ\nhc : c = 1\n⊢ c + 3 * 0 ≤ 2 ^ (c + 1) ∧ 2 ^ (c + 1) % 3 = c % 3\n---\ncase h.inr\nc : ℕ\nhc : c = 2\n⊢ c + 3 * 0 ≤ 2 ^ (c + 1) ∧ 2 ^ (c + 1) % 3 = c % 3",
"after_state": "case h.inl\nc : ℕ\nhc : c = 1\n⊢ c + 3 * 0 ≤ 2 ^ (c + 1) ∧ 2 ^ (c + 1) % 3 = c % 3\n---\ncase h.inr\nc : ℕ\nhc : c = 2\n⊢ c + 3 * 0 ≤ 2 ^ (c + 1) ∧ 2 ^ (c + 1) % 3 = c % 3"
},
{
"line": "skip",
"before_state": "case h.inl\nc : ℕ\nhc : c = 1\n⊢ c + 3 * 0 ≤ 2 ^ (c + 1) ∧ 2 ^ (c + 1) % 3 = c % 3\n---\ncase h.inr\nc : ℕ\nhc : c = 2\n⊢ c + 3 * 0 ≤ 2 ^ (c + 1) ∧ 2 ^ (c + 1) % 3 = c % 3",
"after_state": "case h.inl\nc : ℕ\nhc : c = 1\n⊢ c + 3 * 0 ≤ 2 ^ (c + 1) ∧ 2 ^ (c + 1) % 3 = c % 3\n---\ncase h.inr\nc : ℕ\nhc : c = 2\n⊢ c + 3 * 0 ≤ 2 ^ (c + 1) ∧ 2 ^ (c + 1) % 3 = c % 3"
},
{
"line": "all_goals · rw [hc]; norm_num",
"before_state": "case h.inl\nc : ℕ\nhc : c = 1\n⊢ c + 3 * 0 ≤ 2 ^ (c + 1) ∧ 2 ^ (c + 1) % 3 = c % 3\n---\ncase h.inr\nc : ℕ\nhc : c = 2\n⊢ c + 3 * 0 ≤ 2 ^ (c + 1) ∧ 2 ^ (c + 1) % 3 = c % 3",
"after_state": "No Goals!"
},
{
"line": "rw [hc]",
"before_state": "case h.inl\nc : ℕ\nhc : c = 1\n⊢ c + 3 * 0 ≤ 2 ^ (c + 1) ∧ 2 ^ (c + 1) % 3 = c % 3",
"after_state": "case h.inl\nc : ℕ\nhc : c = 1\n⊢ 1 + 3 * 0 ≤ 2 ^ (1 + 1) ∧ 2 ^ (1 + 1) % 3 = 1 % 3"
},
{
"line": "rewrite [hc]",
"before_state": "case h.inl\nc : ℕ\nhc : c = 1\n⊢ c + 3 * 0 ≤ 2 ^ (c + 1) ∧ 2 ^ (c + 1) % 3 = c % 3",
"after_state": "case h.inl\nc : ℕ\nhc : c = 1\n⊢ 1 + 3 * 0 ≤ 2 ^ (1 + 1) ∧ 2 ^ (1 + 1) % 3 = 1 % 3"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "case h.inl\nc : ℕ\nhc : c = 1\n⊢ 1 + 3 * 0 ≤ 2 ^ (1 + 1) ∧ 2 ^ (1 + 1) % 3 = 1 % 3",
"after_state": "case h.inl\nc : ℕ\nhc : c = 1\n⊢ 1 + 3 * 0 ≤ 2 ^ (1 + 1) ∧ 2 ^ (1 + 1) % 3 = 1 % 3"
},
{
"line": "try (with_reducible rfl)",
"before_state": "case h.inl\nc : ℕ\nhc : c = 1\n⊢ 1 + 3 * 0 ≤ 2 ^ (1 + 1) ∧ 2 ^ (1 + 1) % 3 = 1 % 3",
"after_state": "case h.inl\nc : ℕ\nhc : c = 1\n⊢ 1 + 3 * 0 ≤ 2 ^ (1 + 1) ∧ 2 ^ (1 + 1) % 3 = 1 % 3"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "case h.inl\nc : ℕ\nhc : c = 1\n⊢ 1 + 3 * 0 ≤ 2 ^ (1 + 1) ∧ 2 ^ (1 + 1) % 3 = 1 % 3",
"after_state": "case h.inl\nc : ℕ\nhc : c = 1\n⊢ 1 + 3 * 0 ≤ 2 ^ (1 + 1) ∧ 2 ^ (1 + 1) % 3 = 1 % 3"
},
{
"line": "with_reducible rfl",
"before_state": "case h.inl\nc : ℕ\nhc : c = 1\n⊢ 1 + 3 * 0 ≤ 2 ^ (1 + 1) ∧ 2 ^ (1 + 1) % 3 = 1 % 3",
"after_state": "case h.inl\nc : ℕ\nhc : c = 1\n⊢ 1 + 3 * 0 ≤ 2 ^ (1 + 1) ∧ 2 ^ (1 + 1) % 3 = 1 % 3"
},
{
"line": "rfl",
"before_state": "case h.inl\nc : ℕ\nhc : c = 1\n⊢ 1 + 3 * 0 ≤ 2 ^ (1 + 1) ∧ 2 ^ (1 + 1) % 3 = 1 % 3",
"after_state": "case h.inl\nc : ℕ\nhc : c = 1\n⊢ 1 + 3 * 0 ≤ 2 ^ (1 + 1) ∧ 2 ^ (1 + 1) % 3 = 1 % 3"
},
{
"line": "apply_rfl",
"before_state": "case h.inl\nc : ℕ\nhc : c = 1\n⊢ 1 + 3 * 0 ≤ 2 ^ (1 + 1) ∧ 2 ^ (1 + 1) % 3 = 1 % 3",
"after_state": "case h.inl\nc : ℕ\nhc : c = 1\n⊢ 1 + 3 * 0 ≤ 2 ^ (1 + 1) ∧ 2 ^ (1 + 1) % 3 = 1 % 3"
},
{
"line": "skip",
"before_state": "case h.inl\nc : ℕ\nhc : c = 1\n⊢ 1 + 3 * 0 ≤ 2 ^ (1 + 1) ∧ 2 ^ (1 + 1) % 3 = 1 % 3",
"after_state": "case h.inl\nc : ℕ\nhc : c = 1\n⊢ 1 + 3 * 0 ≤ 2 ^ (1 + 1) ∧ 2 ^ (1 + 1) % 3 = 1 % 3"
},
{
"line": "norm_num",
"before_state": "case h.inl\nc : ℕ\nhc : c = 1\n⊢ 1 + 3 * 0 ≤ 2 ^ (1 + 1) ∧ 2 ^ (1 + 1) % 3 = 1 % 3",
"after_state": "No Goals!"
},
{
"line": "rw [hc]",
"before_state": "case h.inr\nc : ℕ\nhc : c = 2\n⊢ c + 3 * 0 ≤ 2 ^ (c + 1) ∧ 2 ^ (c + 1) % 3 = c % 3",
"after_state": "case h.inr\nc : ℕ\nhc : c = 2\n⊢ 2 + 3 * 0 ≤ 2 ^ (2 + 1) ∧ 2 ^ (2 + 1) % 3 = 2 % 3"
},
{
"line": "rewrite [hc]",
"before_state": "case h.inr\nc : ℕ\nhc : c = 2\n⊢ c + 3 * 0 ≤ 2 ^ (c + 1) ∧ 2 ^ (c + 1) % 3 = c % 3",
"after_state": "case h.inr\nc : ℕ\nhc : c = 2\n⊢ 2 + 3 * 0 ≤ 2 ^ (2 + 1) ∧ 2 ^ (2 + 1) % 3 = 2 % 3"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "case h.inr\nc : ℕ\nhc : c = 2\n⊢ 2 + 3 * 0 ≤ 2 ^ (2 + 1) ∧ 2 ^ (2 + 1) % 3 = 2 % 3",
"after_state": "case h.inr\nc : ℕ\nhc : c = 2\n⊢ 2 + 3 * 0 ≤ 2 ^ (2 + 1) ∧ 2 ^ (2 + 1) % 3 = 2 % 3"
},
{
"line": "try (with_reducible rfl)",
"before_state": "case h.inr\nc : ℕ\nhc : c = 2\n⊢ 2 + 3 * 0 ≤ 2 ^ (2 + 1) ∧ 2 ^ (2 + 1) % 3 = 2 % 3",
"after_state": "case h.inr\nc : ℕ\nhc : c = 2\n⊢ 2 + 3 * 0 ≤ 2 ^ (2 + 1) ∧ 2 ^ (2 + 1) % 3 = 2 % 3"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "case h.inr\nc : ℕ\nhc : c = 2\n⊢ 2 + 3 * 0 ≤ 2 ^ (2 + 1) ∧ 2 ^ (2 + 1) % 3 = 2 % 3",
"after_state": "case h.inr\nc : ℕ\nhc : c = 2\n⊢ 2 + 3 * 0 ≤ 2 ^ (2 + 1) ∧ 2 ^ (2 + 1) % 3 = 2 % 3"
},
{
"line": "with_reducible rfl",
"before_state": "case h.inr\nc : ℕ\nhc : c = 2\n⊢ 2 + 3 * 0 ≤ 2 ^ (2 + 1) ∧ 2 ^ (2 + 1) % 3 = 2 % 3",
"after_state": "case h.inr\nc : ℕ\nhc : c = 2\n⊢ 2 + 3 * 0 ≤ 2 ^ (2 + 1) ∧ 2 ^ (2 + 1) % 3 = 2 % 3"
},
{
"line": "rfl",
"before_state": "case h.inr\nc : ℕ\nhc : c = 2\n⊢ 2 + 3 * 0 ≤ 2 ^ (2 + 1) ∧ 2 ^ (2 + 1) % 3 = 2 % 3",
"after_state": "case h.inr\nc : ℕ\nhc : c = 2\n⊢ 2 + 3 * 0 ≤ 2 ^ (2 + 1) ∧ 2 ^ (2 + 1) % 3 = 2 % 3"
},
{
"line": "apply_rfl",
"before_state": "case h.inr\nc : ℕ\nhc : c = 2\n⊢ 2 + 3 * 0 ≤ 2 ^ (2 + 1) ∧ 2 ^ (2 + 1) % 3 = 2 % 3",
"after_state": "case h.inr\nc : ℕ\nhc : c = 2\n⊢ 2 + 3 * 0 ≤ 2 ^ (2 + 1) ∧ 2 ^ (2 + 1) % 3 = 2 % 3"
},
{
"line": "skip",
"before_state": "case h.inr\nc : ℕ\nhc : c = 2\n⊢ 2 + 3 * 0 ≤ 2 ^ (2 + 1) ∧ 2 ^ (2 + 1) % 3 = 2 % 3",
"after_state": "case h.inr\nc : ℕ\nhc : c = 2\n⊢ 2 + 3 * 0 ≤ 2 ^ (2 + 1) ∧ 2 ^ (2 + 1) % 3 = 2 % 3"
},
{
"line": "norm_num",
"before_state": "case h.inr\nc : ℕ\nhc : c = 2\n⊢ 2 + 3 * 0 ≤ 2 ^ (2 + 1) ∧ 2 ^ (2 + 1) % 3 = 2 % 3",
"after_state": "No Goals!"
},
{
"line": "rcases hk with ⟨g, hkg, hgmod⟩",
"before_state": "case succ\nc : ℕ\nh : c = 1 ∨ c = 2\nk : ℕ\nhk : ∃ m, c + 3 * k ≤ 2 ^ m ∧ 2 ^ m % 3 = c % 3\n⊢ ∃ m, c + 3 * (k + 1) ≤ 2 ^ m ∧ 2 ^ m % 3 = c % 3",
"after_state": "case succ.intro.intro\nc : ℕ\nh : c = 1 ∨ c = 2\nk g : ℕ\nhkg : c + 3 * k ≤ 2 ^ g\nhgmod : 2 ^ g % 3 = c % 3\n⊢ ∃ m, c + 3 * (k + 1) ≤ 2 ^ m ∧ 2 ^ m % 3 = c % 3"
},
{
"line": "by_cases hp : c + 3 * (k + 1) ≤ 2 ^ g",
"before_state": "case succ.intro.intro\nc : ℕ\nh : c = 1 ∨ c = 2\nk g : ℕ\nhkg : c + 3 * k ≤ 2 ^ g\nhgmod : 2 ^ g % 3 = c % 3\n⊢ ∃ m, c + 3 * (k + 1) ≤ 2 ^ m ∧ 2 ^ m % 3 = c % 3",
"after_state": "case pos\nc : ℕ\nh : c = 1 ∨ c = 2\nk g : ℕ\nhkg : c + 3 * k ≤ 2 ^ g\nhgmod : 2 ^ g % 3 = c % 3\nhp : c + 3 * (k + 1) ≤ 2 ^ g\n⊢ ∃ m, c + 3 * (k + 1) ≤ 2 ^ m ∧ 2 ^ m % 3 = c % 3\n---\ncase neg\nc : ℕ\nh : c = 1 ∨ c = 2\nk g : ℕ\nhkg : c + 3 * k ≤ 2 ^ g\nhgmod : 2 ^ g % 3 = c % 3\nhp : ¬c + 3 * (k + 1) ≤ 2 ^ g\n⊢ ∃ m, c + 3 * (k + 1) ≤ 2 ^ m ∧ 2 ^ m % 3 = c % 3"
},
{
"line": "open Classical✝ in refine if hp : c + 3 * (k + 1) ≤ 2 ^ g then ?pos✝ else ?neg✝",
"before_state": "case succ.intro.intro\nc : ℕ\nh : c = 1 ∨ c = 2\nk g : ℕ\nhkg : c + 3 * k ≤ 2 ^ g\nhgmod : 2 ^ g % 3 = c % 3\n⊢ ∃ m, c + 3 * (k + 1) ≤ 2 ^ m ∧ 2 ^ m % 3 = c % 3",
"after_state": "case pos\nc : ℕ\nh : c = 1 ∨ c = 2\nk g : ℕ\nhkg : c + 3 * k ≤ 2 ^ g\nhgmod : 2 ^ g % 3 = c % 3\nhp : c + 3 * (k + 1) ≤ 2 ^ g\n⊢ ∃ m, c + 3 * (k + 1) ≤ 2 ^ m ∧ 2 ^ m % 3 = c % 3\n---\ncase neg\nc : ℕ\nh : c = 1 ∨ c = 2\nk g : ℕ\nhkg : c + 3 * k ≤ 2 ^ g\nhgmod : 2 ^ g % 3 = c % 3\nhp : ¬c + 3 * (k + 1) ≤ 2 ^ g\n⊢ ∃ m, c + 3 * (k + 1) ≤ 2 ^ m ∧ 2 ^ m % 3 = c % 3"
},
{
"line": "refine if hp : c + 3 * (k + 1) ≤ 2 ^ g then ?pos✝ else ?neg✝",
"before_state": "case succ.intro.intro\nc : ℕ\nh : c = 1 ∨ c = 2\nk g : ℕ\nhkg : c + 3 * k ≤ 2 ^ g\nhgmod : 2 ^ g % 3 = c % 3\n⊢ ∃ m, c + 3 * (k + 1) ≤ 2 ^ m ∧ 2 ^ m % 3 = c % 3",
"after_state": "case pos\nc : ℕ\nh : c = 1 ∨ c = 2\nk g : ℕ\nhkg : c + 3 * k ≤ 2 ^ g\nhgmod : 2 ^ g % 3 = c % 3\nhp : c + 3 * (k + 1) ≤ 2 ^ g\n⊢ ∃ m, c + 3 * (k + 1) ≤ 2 ^ m ∧ 2 ^ m % 3 = c % 3\n---\ncase neg\nc : ℕ\nh : c = 1 ∨ c = 2\nk g : ℕ\nhkg : c + 3 * k ≤ 2 ^ g\nhgmod : 2 ^ g % 3 = c % 3\nhp : ¬c + 3 * (k + 1) ≤ 2 ^ g\n⊢ ∃ m, c + 3 * (k + 1) ≤ 2 ^ m ∧ 2 ^ m % 3 = c % 3"
},
{
"line": "use g, hp, hgmod",
"before_state": "case pos\nc : ℕ\nh : c = 1 ∨ c = 2\nk g : ℕ\nhkg : c + 3 * k ≤ 2 ^ g\nhgmod : 2 ^ g % 3 = c % 3\nhp : c + 3 * (k + 1) ≤ 2 ^ g\n⊢ ∃ m, c + 3 * (k + 1) ≤ 2 ^ m ∧ 2 ^ m % 3 = c % 3",
"after_state": "No Goals!"
},
{
"line": "refine without_cdot(g : ?m✝)",
"before_state": "case w\nc : ℕ\nh : c = 1 ∨ c = 2\nk g : ℕ\nhkg : c + 3 * k ≤ 2 ^ g\nhgmod : 2 ^ g % 3 = c % 3\nhp : c + 3 * (k + 1) ≤ 2 ^ g\n⊢ ℕ",
"after_state": "No Goals!"
},
{
"line": "refine without_cdot(hp : ?m✝)",
"before_state": "case left\nc : ℕ\nh : c = 1 ∨ c = 2\nk g : ℕ\nhkg : c + 3 * k ≤ 2 ^ g\nhgmod : 2 ^ g % 3 = c % 3\nhp : c + 3 * (k + 1) ≤ 2 ^ g\n⊢ c + 3 * (k + 1) ≤ 2 ^ g",
"after_state": "No Goals!"
},
{
"line": "refine without_cdot(hgmod : ?m✝)",
"before_state": "case right\nc : ℕ\nh : c = 1 ∨ c = 2\nk g : ℕ\nhkg : c + 3 * k ≤ 2 ^ g\nhgmod : 2 ^ g % 3 = c % 3\nhp : c + 3 * (k + 1) ≤ 2 ^ g\n⊢ 2 ^ g % 3 = c % 3",
"after_state": "No Goals!"
},
{
"line": "refine ⟨g + 2, ?_, ?_⟩",
"before_state": "case neg\nc : ℕ\nh : c = 1 ∨ c = 2\nk g : ℕ\nhkg : c + 3 * k ≤ 2 ^ g\nhgmod : 2 ^ g % 3 = c % 3\nhp : ¬c + 3 * (k + 1) ≤ 2 ^ g\n⊢ ∃ m, c + 3 * (k + 1) ≤ 2 ^ m ∧ 2 ^ m % 3 = c % 3",
"after_state": "case neg.refine_1\nc : ℕ\nh : c = 1 ∨ c = 2\nk g : ℕ\nhkg : c + 3 * k ≤ 2 ^ g\nhgmod : 2 ^ g % 3 = c % 3\nhp : ¬c + 3 * (k + 1) ≤ 2 ^ g\n⊢ c + 3 * (k + 1) ≤ 2 ^ (g + 2)\n---\ncase neg.refine_2\nc : ℕ\nh : c = 1 ∨ c = 2\nk g : ℕ\nhkg : c + 3 * k ≤ 2 ^ g\nhgmod : 2 ^ g % 3 = c % 3\nhp : ¬c + 3 * (k + 1) ≤ 2 ^ g\n⊢ 2 ^ (g + 2) % 3 = c % 3"
},
{
"line": "rw [mul_succ, ← add_assoc, pow_add]",
"before_state": "case neg.refine_1\nc : ℕ\nh : c = 1 ∨ c = 2\nk g : ℕ\nhkg : c + 3 * k ≤ 2 ^ g\nhgmod : 2 ^ g % 3 = c % 3\nhp : ¬c + 3 * (k + 1) ≤ 2 ^ g\n⊢ c + 3 * (k + 1) ≤ 2 ^ (g + 2)",
"after_state": "case neg.refine_1\nc : ℕ\nh : c = 1 ∨ c = 2\nk g : ℕ\nhkg : c + 3 * k ≤ 2 ^ g\nhgmod : 2 ^ g % 3 = c % 3\nhp : ¬c + 3 * (k + 1) ≤ 2 ^ g\n⊢ c + 3 * (k + 1) ≤ 2 ^ (g + 2)"
},
{
"line": "rewrite [mul_succ, ← add_assoc, pow_add]",
"before_state": "case neg.refine_1\nc : ℕ\nh : c = 1 ∨ c = 2\nk g : ℕ\nhkg : c + 3 * k ≤ 2 ^ g\nhgmod : 2 ^ g % 3 = c % 3\nhp : ¬c + 3 * (k + 1) ≤ 2 ^ g\n⊢ c + 3 * (k + 1) ≤ 2 ^ (g + 2)",
"after_state": "case neg.refine_1\nc : ℕ\nh : c = 1 ∨ c = 2\nk g : ℕ\nhkg : c + 3 * k ≤ 2 ^ g\nhgmod : 2 ^ g % 3 = c % 3\nhp : ¬c + 3 * (k + 1) ≤ 2 ^ g\n⊢ c + 3 * (k + 1) ≤ 2 ^ (g + 2)"
},
{
"line": "rw [pow_add, ← mul_one c]",
"before_state": "case neg.refine_2\nc : ℕ\nh : c = 1 ∨ c = 2\nk g : ℕ\nhkg : c + 3 * k ≤ 2 ^ g\nhgmod : 2 ^ g % 3 = c % 3\nhp : ¬c + 3 * (k + 1) ≤ 2 ^ g\n⊢ 2 ^ (g + 2) % 3 = c % 3",
"after_state": "case neg.refine_2\nc : ℕ\nh : c = 1 ∨ c = 2\nk g : ℕ\nhkg : c + 3 * k ≤ 2 ^ g\nhgmod : 2 ^ g % 3 = c % 3\nhp : ¬c + 3 * (k + 1) ≤ 2 ^ g\n⊢ 2 ^ g * 2 ^ 2 % 3 = c * 1 % 3"
},
{
"line": "rewrite [pow_add, ← mul_one c]",
"before_state": "case neg.refine_2\nc : ℕ\nh : c = 1 ∨ c = 2\nk g : ℕ\nhkg : c + 3 * k ≤ 2 ^ g\nhgmod : 2 ^ g % 3 = c % 3\nhp : ¬c + 3 * (k + 1) ≤ 2 ^ g\n⊢ 2 ^ (g + 2) % 3 = c % 3",
"after_state": "case neg.refine_2\nc : ℕ\nh : c = 1 ∨ c = 2\nk g : ℕ\nhkg : c + 3 * k ≤ 2 ^ g\nhgmod : 2 ^ g % 3 = c % 3\nhp : ¬c + 3 * (k + 1) ≤ 2 ^ g\n⊢ 2 ^ g * 2 ^ 2 % 3 = c * 1 % 3"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "case neg.refine_2\nc : ℕ\nh : c = 1 ∨ c = 2\nk g : ℕ\nhkg : c + 3 * k ≤ 2 ^ g\nhgmod : 2 ^ g % 3 = c % 3\nhp : ¬c + 3 * (k + 1) ≤ 2 ^ g\n⊢ 2 ^ g * 2 ^ 2 % 3 = c * 1 % 3",
"after_state": "case neg.refine_2\nc : ℕ\nh : c = 1 ∨ c = 2\nk g : ℕ\nhkg : c + 3 * k ≤ 2 ^ g\nhgmod : 2 ^ g % 3 = c % 3\nhp : ¬c + 3 * (k + 1) ≤ 2 ^ g\n⊢ 2 ^ g * 2 ^ 2 % 3 = c * 1 % 3"
},
{
"line": "try (with_reducible rfl)",
"before_state": "case neg.refine_2\nc : ℕ\nh : c = 1 ∨ c = 2\nk g : ℕ\nhkg : c + 3 * k ≤ 2 ^ g\nhgmod : 2 ^ g % 3 = c % 3\nhp : ¬c + 3 * (k + 1) ≤ 2 ^ g\n⊢ 2 ^ g * 2 ^ 2 % 3 = c * 1 % 3",
"after_state": "case neg.refine_2\nc : ℕ\nh : c = 1 ∨ c = 2\nk g : ℕ\nhkg : c + 3 * k ≤ 2 ^ g\nhgmod : 2 ^ g % 3 = c % 3\nhp : ¬c + 3 * (k + 1) ≤ 2 ^ g\n⊢ 2 ^ g * 2 ^ 2 % 3 = c * 1 % 3"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "case neg.refine_2\nc : ℕ\nh : c = 1 ∨ c = 2\nk g : ℕ\nhkg : c + 3 * k ≤ 2 ^ g\nhgmod : 2 ^ g % 3 = c % 3\nhp : ¬c + 3 * (k + 1) ≤ 2 ^ g\n⊢ 2 ^ g * 2 ^ 2 % 3 = c * 1 % 3",
"after_state": "case neg.refine_2\nc : ℕ\nh : c = 1 ∨ c = 2\nk g : ℕ\nhkg : c + 3 * k ≤ 2 ^ g\nhgmod : 2 ^ g % 3 = c % 3\nhp : ¬c + 3 * (k + 1) ≤ 2 ^ g\n⊢ 2 ^ g * 2 ^ 2 % 3 = c * 1 % 3"
},
{
"line": "with_reducible rfl",
"before_state": "case neg.refine_2\nc : ℕ\nh : c = 1 ∨ c = 2\nk g : ℕ\nhkg : c + 3 * k ≤ 2 ^ g\nhgmod : 2 ^ g % 3 = c % 3\nhp : ¬c + 3 * (k + 1) ≤ 2 ^ g\n⊢ 2 ^ g * 2 ^ 2 % 3 = c * 1 % 3",
"after_state": "case neg.refine_2\nc : ℕ\nh : c = 1 ∨ c = 2\nk g : ℕ\nhkg : c + 3 * k ≤ 2 ^ g\nhgmod : 2 ^ g % 3 = c % 3\nhp : ¬c + 3 * (k + 1) ≤ 2 ^ g\n⊢ 2 ^ g * 2 ^ 2 % 3 = c * 1 % 3"
},
{
"line": "rfl",
"before_state": "case neg.refine_2\nc : ℕ\nh : c = 1 ∨ c = 2\nk g : ℕ\nhkg : c + 3 * k ≤ 2 ^ g\nhgmod : 2 ^ g % 3 = c % 3\nhp : ¬c + 3 * (k + 1) ≤ 2 ^ g\n⊢ 2 ^ g * 2 ^ 2 % 3 = c * 1 % 3",
"after_state": "case neg.refine_2\nc : ℕ\nh : c = 1 ∨ c = 2\nk g : ℕ\nhkg : c + 3 * k ≤ 2 ^ g\nhgmod : 2 ^ g % 3 = c % 3\nhp : ¬c + 3 * (k + 1) ≤ 2 ^ g\n⊢ 2 ^ g * 2 ^ 2 % 3 = c * 1 % 3"
},
{
"line": "apply_rfl",
"before_state": "case neg.refine_2\nc : ℕ\nh : c = 1 ∨ c = 2\nk g : ℕ\nhkg : c + 3 * k ≤ 2 ^ g\nhgmod : 2 ^ g % 3 = c % 3\nhp : ¬c + 3 * (k + 1) ≤ 2 ^ g\n⊢ 2 ^ g * 2 ^ 2 % 3 = c * 1 % 3",
"after_state": "case neg.refine_2\nc : ℕ\nh : c = 1 ∨ c = 2\nk g : ℕ\nhkg : c + 3 * k ≤ 2 ^ g\nhgmod : 2 ^ g % 3 = c % 3\nhp : ¬c + 3 * (k + 1) ≤ 2 ^ g\n⊢ 2 ^ g * 2 ^ 2 % 3 = c * 1 % 3"
},
{
"line": "skip",
"before_state": "case neg.refine_2\nc : ℕ\nh : c = 1 ∨ c = 2\nk g : ℕ\nhkg : c + 3 * k ≤ 2 ^ g\nhgmod : 2 ^ g % 3 = c % 3\nhp : ¬c + 3 * (k + 1) ≤ 2 ^ g\n⊢ 2 ^ g * 2 ^ 2 % 3 = c * 1 % 3",
"after_state": "case neg.refine_2\nc : ℕ\nh : c = 1 ∨ c = 2\nk g : ℕ\nhkg : c + 3 * k ≤ 2 ^ g\nhgmod : 2 ^ g % 3 = c % 3\nhp : ¬c + 3 * (k + 1) ≤ 2 ^ g\n⊢ 2 ^ g * 2 ^ 2 % 3 = c * 1 % 3"
},
{
"line": "exact ModEq.mul hgmod rfl",
"before_state": "case neg.refine_2\nc : ℕ\nh : c = 1 ∨ c = 2\nk g : ℕ\nhkg : c + 3 * k ≤ 2 ^ g\nhgmod : 2 ^ g % 3 = c % 3\nhp : ¬c + 3 * (k + 1) ≤ 2 ^ g\n⊢ 2 ^ g * 2 ^ 2 % 3 = c * 1 % 3",
"after_state": "No Goals!"
}
] |
theorem le_pow2_and_pow2_eq_mod3 (a : ℕ) (h : a % 3 = 1 ∨ a % 3 = 2) :
∃ m : ℕ, a ≤ 2 ^ m ∧ 2 ^ m % 3 = a % 3 := by
obtain ⟨m, hm⟩ := le_pow2_and_pow2_eq_mod3' (a % 3) (a / 3) h
use m
constructor
· convert hm.1; exact (mod_add_div a 3).symm
· rw [hm.2, mod_mod _ 3]
|
/root/DuelModelResearch/mathlib4/Archive/MiuLanguage/DecisionSuf.lean
|
{
"open": [
"MiuAtom List Nat"
],
"variables": []
}
|
[
{
"line": "obtain ⟨m, hm⟩ := le_pow2_and_pow2_eq_mod3' (a % 3) (a / 3) h",
"before_state": "a : ℕ\nh : a % 3 = 1 ∨ a % 3 = 2\n⊢ ∃ m, a ≤ 2 ^ m ∧ 2 ^ m % 3 = a % 3",
"after_state": "No Goals!"
}
] |
theorem OxfordInvariants.Week3P1 (n : ℕ) (a : ℕ → ℕ) (a_pos : ∀ i ≤ n, 0 < a i)
(ha : ∀ i, i + 2 ≤ n → a (i + 1) ∣ a i + a (i + 2)) :
∃ b : ℕ, (b : α) = ∑ i ∈ Finset.range n, (a 0 : α) * a n / (a i * a (i + 1)) := by
-- Treat separately `n = 0` and `n ≥ 1`
rcases n with - | n
/- Case `n = 0`
The sum is trivially equal to `0` -/
· exact ⟨0, by rw [Nat.cast_zero, Finset.sum_range_zero]⟩
-- `⟨Claim it, Prove it⟩`
/- Case `n ≥ 1`. We replace `n` by `n + 1` everywhere to make this inequality explicit
Set up the stronger induction hypothesis -/
rsuffices ⟨b, hb, -⟩ :
∃ b : ℕ,
(b : α) = ∑ i ∈ Finset.range (n + 1), (a 0 : α) * a (n + 1) / (a i * a (i + 1)) ∧
a (n + 1) ∣ a n * b - a 0
· exact ⟨b, hb⟩
simp_rw [← @Nat.cast_pos α] at a_pos
/- Declare the induction
`ih` will be the induction hypothesis -/
induction' n with n ih
/- Base case
Claim that the sum equals `1` -/
· refine ⟨1, ?_, ?_⟩
-- Check that this indeed equals the sum
· rw [Nat.cast_one, Finset.sum_range_one]
norm_num
rw [div_self]
exact (mul_pos (a_pos 0 (Nat.zero_le _)) (a_pos 1 (Nat.zero_lt_succ _))).ne'
-- Check the divisibility condition
· rw [mul_one, tsub_self]
exact dvd_zero _
/- Induction step
`b` is the value of the previous sum as a natural, `hb` is the proof that it is indeed the
value, and `han` is the divisibility condition -/
obtain ⟨b, hb, han⟩ :=
ih (fun i hi => ha i <| Nat.le_succ_of_le hi) fun i hi => a_pos i <| Nat.le_succ_of_le hi
specialize ha n le_rfl
have ha₀ : a 0 ≤ a n * b := by
-- Needing this is an artifact of `ℕ`-subtraction.
rw [← @Nat.cast_le α]
rw [Nat.cast_mul]
rw [hb]
rw [← div_le_iff₀' (a_pos _ <| n.le_succ.trans <| Nat.le_succ _)]
rw [← mul_div_mul_right _ _ (a_pos _ <| Nat.le_succ _).ne']
suffices h : ∀ i, i ∈ Finset.range (n + 1) → 0 ≤ (a 0 : α) * a (n + 1) / (a i * a (i + 1)) from
Finset.single_le_sum h (Finset.self_mem_range_succ n)
refine fun i _ ↦ div_nonneg ?_ ?_ <;> refine mul_nonneg ?_ ?_ <;> exact Nat.cast_nonneg _
-- Claim that the sum equals `(aₙ + aₙ₊₂)/aₙ₊₁ * b - (aₙ * b - a₀)/aₙ₊₁`
refine ⟨(a n + a (n + 2)) / a (n + 1) * b - (a n * b - a 0) / a (n + 1), ?_, ?_⟩
-- Check that this indeed equals the sum
· calc
(((a n + a (n + 2)) / a (n + 1) * b - (a n * b - a 0) / a (n + 1) : ℕ) : α) =
((a n + a (n + 2)) / a (n + 1) * b - (a n * b - a 0) / a (n + 1) ) := by
have :((a (n + 1)) : α) ≠ 0 := ne_of_gt <| a_pos (n + 1) <| Nat.le_succ (n + 1)
simp only [← Nat.cast_add]
simp only [← Nat.cast_div ha this]
simp only [← Nat.cast_mul]
simp only [← Nat.cast_sub ha₀]
simp only [← Nat.cast_div han this]
rw [Nat.cast_sub (Nat.div_le_of_le_mul _)]
rw [← mul_assoc]
rw [Nat.mul_div_cancel' ha]
rw [add_mul]
exact tsub_le_self.trans (Nat.le_add_right _ _)
_ = a (n + 2) / a (n + 1) * b + a 0 * a (n + 2) / (a (n + 1) * a (n + 2)) := by
rw [add_div]
rw [add_mul]
rw [sub_div]
rw [mul_div_right_comm]
rw [add_sub_sub_cancel]
rw [mul_div_mul_right _ _ (a_pos _ le_rfl).ne']
_ = ∑ i ∈ Finset.range (n + 2), (a 0 : α) * a (n + 2) / (a i * a (i + 1)) := by
rw [Finset.sum_range_succ]
rw [hb]
rw [Finset.mul_sum]
congr; ext i
rw [← mul_div_assoc]
rw [← mul_div_right_comm]
rw [mul_div_assoc]
rw [mul_div_cancel_right₀ _ (a_pos _ <| Nat.le_succ _).ne']
rw [mul_comm]
-- Check the divisibility condition
· rw [Nat.mul_sub, ← mul_assoc, Nat.mul_div_cancel' ha, add_mul, Nat.mul_div_cancel' han,
add_tsub_tsub_cancel ha₀, add_tsub_cancel_right]
exact dvd_mul_right _ _
|
/root/DuelModelResearch/mathlib4/Archive/OxfordInvariants/Summer2021/Week3P1.lean
|
{
"open": [],
"variables": [
"{α : Type*} [Field α] [LinearOrder α] [IsStrictOrderedRing α]"
]
}
|
[
{
"line": "rcases n with - | n",
"before_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\nn : ℕ\na : ℕ → ℕ\na_pos : ∀ i ≤ n, 0 < a i\nha : ∀ (i : ℕ), i + 2 ≤ n → a (i + 1) ∣ a i + a (i + 2)\n⊢ ∃ b, ↑b = ∑ i ∈ Finset.range n, ↑(a 0) * ↑(a n) / (↑(a i) * ↑(a (i + 1)))",
"after_state": "case zero\nα : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\na_pos : ∀ i ≤ 0, 0 < a i\nha : ∀ (i : ℕ), i + 2 ≤ 0 → a (i + 1) ∣ a i + a (i + 2)\n⊢ ∃ b, ↑b = ∑ i ∈ Finset.range 0, ↑(a 0) * ↑(a 0) / (↑(a i) * ↑(a (i + 1)))\n---\ncase succ\nα : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\na_pos : ∀ i ≤ n + 1, 0 < a i\nha : ∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)\n⊢ ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))"
},
{
"line": "exact\n ⟨0, by rw [Nat.cast_zero, Finset.sum_range_zero]⟩\n -- `⟨Claim it, Prove it⟩`\n /- Case `n ≥ 1`. We replace `n` by `n + 1` everywhere to make this inequality explicit\n Set up the stronger induction hypothesis -/",
"before_state": "case zero\nα : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\na_pos : ∀ i ≤ 0, 0 < a i\nha : ∀ (i : ℕ), i + 2 ≤ 0 → a (i + 1) ∣ a i + a (i + 2)\n⊢ ∃ b, ↑b = ∑ i ∈ Finset.range 0, ↑(a 0) * ↑(a 0) / (↑(a i) * ↑(a (i + 1)))",
"after_state": "No Goals!"
},
{
"line": "rw [Nat.cast_zero, Finset.sum_range_zero]",
"before_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\na_pos : ∀ i ≤ 0, 0 < a i\nha : ∀ (i : ℕ), i + 2 ≤ 0 → a (i + 1) ∣ a i + a (i + 2)\n⊢ ↑0 = ∑ i ∈ Finset.range 0, ↑(a 0) * ↑(a 0) / (↑(a i) * ↑(a (i + 1)))",
"after_state": "No Goals!"
},
{
"line": "rewrite [Nat.cast_zero, Finset.sum_range_zero]",
"before_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\na_pos : ∀ i ≤ 0, 0 < a i\nha : ∀ (i : ℕ), i + 2 ≤ 0 → a (i + 1) ∣ a i + a (i + 2)\n⊢ ↑0 = ∑ i ∈ Finset.range 0, ↑(a 0) * ↑(a 0) / (↑(a i) * ↑(a (i + 1)))",
"after_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\na_pos : ∀ i ≤ 0, 0 < a i\nha : ∀ (i : ℕ), i + 2 ≤ 0 → a (i + 1) ∣ a i + a (i + 2)\n⊢ 0 = 0"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\na_pos : ∀ i ≤ 0, 0 < a i\nha : ∀ (i : ℕ), i + 2 ≤ 0 → a (i + 1) ∣ a i + a (i + 2)\n⊢ 0 = 0",
"after_state": "No Goals!"
},
{
"line": "try (with_reducible rfl)",
"before_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\na_pos : ∀ i ≤ 0, 0 < a i\nha : ∀ (i : ℕ), i + 2 ≤ 0 → a (i + 1) ∣ a i + a (i + 2)\n⊢ 0 = 0",
"after_state": "No Goals!"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\na_pos : ∀ i ≤ 0, 0 < a i\nha : ∀ (i : ℕ), i + 2 ≤ 0 → a (i + 1) ∣ a i + a (i + 2)\n⊢ 0 = 0",
"after_state": "No Goals!"
},
{
"line": "with_reducible rfl",
"before_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\na_pos : ∀ i ≤ 0, 0 < a i\nha : ∀ (i : ℕ), i + 2 ≤ 0 → a (i + 1) ∣ a i + a (i + 2)\n⊢ 0 = 0",
"after_state": "No Goals!"
},
{
"line": "rfl",
"before_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\na_pos : ∀ i ≤ 0, 0 < a i\nha : ∀ (i : ℕ), i + 2 ≤ 0 → a (i + 1) ∣ a i + a (i + 2)\n⊢ 0 = 0",
"after_state": "No Goals!"
},
{
"line": "eq_refl",
"before_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\na_pos : ∀ i ≤ 0, 0 < a i\nha : ∀ (i : ℕ), i + 2 ≤ 0 → a (i + 1) ∣ a i + a (i + 2)\n⊢ 0 = 0",
"after_state": "No Goals!"
},
{
"line": "rsuffices ⟨b, hb, -⟩ :\n ∃ b : ℕ, (b : α) = ∑ i ∈ Finset.range (n + 1), (a 0 : α) * a (n + 1) / (a i * a (i + 1)) ∧ a (n + 1) ∣ a n * b - a 0",
"before_state": "case succ\nα : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\na_pos : ∀ i ≤ n + 1, 0 < a i\nha : ∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)\n⊢ ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))",
"after_state": "case succ.intro.intro\nα : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\na_pos : ∀ i ≤ n + 1, 0 < a i\nha : ∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\n⊢ ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\n---\nα : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\na_pos : ∀ i ≤ n + 1, 0 < a i\nha : ∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)\n⊢ ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0"
},
{
"line": "obtain ⟨b, hb, -⟩ :\n ∃ b : ℕ, (b : α) = ∑ i ∈ Finset.range (n + 1), (a 0 : α) * a (n + 1) / (a i * a (i + 1)) ∧ a (n + 1) ∣ a n * b - a 0",
"before_state": "case succ\nα : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\na_pos : ∀ i ≤ n + 1, 0 < a i\nha : ∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)\n⊢ ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))",
"after_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\na_pos : ∀ i ≤ n + 1, 0 < a i\nha : ∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)\n⊢ ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\n---\ncase succ.intro.intro\nα : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\na_pos : ∀ i ≤ n + 1, 0 < a i\nha : ∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\n⊢ ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))"
},
{
"line": "rotate_left",
"before_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\na_pos : ∀ i ≤ n + 1, 0 < a i\nha : ∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)\n⊢ ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\n---\ncase succ.intro.intro\nα : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\na_pos : ∀ i ≤ n + 1, 0 < a i\nha : ∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\n⊢ ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))",
"after_state": "case succ.intro.intro\nα : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\na_pos : ∀ i ≤ n + 1, 0 < a i\nha : ∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\n⊢ ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\n---\nα : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\na_pos : ∀ i ≤ n + 1, 0 < a i\nha : ∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)\n⊢ ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0"
},
{
"line": "exact ⟨b, hb⟩",
"before_state": "case succ.intro.intro\nα : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\na_pos : ∀ i ≤ n + 1, 0 < a i\nha : ∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\n⊢ ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))",
"after_state": "No Goals!"
},
{
"line": "simp_rw [← @Nat.cast_pos α] at a_pos",
"before_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\na_pos : ∀ i ≤ n + 1, 0 < a i\nha : ∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)\n⊢ ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0",
"after_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nha : ∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)\na_pos : ∀ i ≤ n + 1, 0 < ↑(a i)\n⊢ ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0"
},
{
"line": "simp (failIfUnchanged✝ := false✝) only at a_pos",
"before_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\na_pos : ∀ i ≤ n + 1, 0 < a i\nha : ∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)\n⊢ ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0",
"after_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\na_pos : ∀ i ≤ n + 1, 0 < a i\nha : ∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)\n⊢ ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0"
},
{
"line": "simp only [← @Nat.cast_pos α] at a_pos",
"before_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\na_pos : ∀ i ≤ n + 1, 0 < a i\nha : ∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)\n⊢ ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0",
"after_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nha : ∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)\na_pos : ∀ i ≤ n + 1, 0 < ↑(a i)\n⊢ ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0"
},
{
"line": "induction' n with n ih",
"before_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nha : ∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)\na_pos : ∀ i ≤ n + 1, 0 < ↑(a i)\n⊢ ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0",
"after_state": "case zero\nα : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nha : ∀ (i : ℕ), i + 2 ≤ 0 + 1 → a (i + 1) ∣ a i + a (i + 2)\na_pos : ∀ i ≤ 0 + 1, 0 < ↑(a i)\n⊢ ∃ b, ↑b = ∑ i ∈ Finset.range (0 + 1), ↑(a 0) * ↑(a (0 + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (0 + 1) ∣ a 0 * b - a 0\n---\ncase succ\nα : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\nha : ∀ (i : ℕ), i + 2 ≤ n + 1 + 1 → a (i + 1) ∣ a i + a (i + 2)\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\n⊢ ∃ b,\n ↑b = ∑ i ∈ Finset.range (n + 1 + 1), ↑(a 0) * ↑(a (n + 1 + 1)) / (↑(a i) * ↑(a (i + 1))) ∧\n a (n + 1 + 1) ∣ a (n + 1) * b - a 0"
},
{
"line": "refine\n ⟨1, ?_, ?_⟩\n -- Check that this indeed equals the sum",
"before_state": "case zero\nα : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nha : ∀ (i : ℕ), i + 2 ≤ 0 + 1 → a (i + 1) ∣ a i + a (i + 2)\na_pos : ∀ i ≤ 0 + 1, 0 < ↑(a i)\n⊢ ∃ b, ↑b = ∑ i ∈ Finset.range (0 + 1), ↑(a 0) * ↑(a (0 + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (0 + 1) ∣ a 0 * b - a 0",
"after_state": "case zero.refine_1\nα : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nha : ∀ (i : ℕ), i + 2 ≤ 0 + 1 → a (i + 1) ∣ a i + a (i + 2)\na_pos : ∀ i ≤ 0 + 1, 0 < ↑(a i)\n⊢ ↑1 = ∑ i ∈ Finset.range (0 + 1), ↑(a 0) * ↑(a (0 + 1)) / (↑(a i) * ↑(a (i + 1)))\n---\ncase zero.refine_2\nα : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nha : ∀ (i : ℕ), i + 2 ≤ 0 + 1 → a (i + 1) ∣ a i + a (i + 2)\na_pos : ∀ i ≤ 0 + 1, 0 < ↑(a i)\n⊢ a (0 + 1) ∣ a 0 * 1 - a 0"
},
{
"line": "rw [Nat.cast_one, Finset.sum_range_one]",
"before_state": "case zero.refine_1\nα : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nha : ∀ (i : ℕ), i + 2 ≤ 0 + 1 → a (i + 1) ∣ a i + a (i + 2)\na_pos : ∀ i ≤ 0 + 1, 0 < ↑(a i)\n⊢ ↑1 = ∑ i ∈ Finset.range (0 + 1), ↑(a 0) * ↑(a (0 + 1)) / (↑(a i) * ↑(a (i + 1)))",
"after_state": "case zero.refine_1\nα : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nha : ∀ (i : ℕ), i + 2 ≤ 0 + 1 → a (i + 1) ∣ a i + a (i + 2)\na_pos : ∀ i ≤ 0 + 1, 0 < ↑(a i)\n⊢ 1 = ↑(a 0) * ↑(a (0 + 1)) / (↑(a 0) * ↑(a (0 + 1)))"
},
{
"line": "rewrite [Nat.cast_one, Finset.sum_range_one]",
"before_state": "case zero.refine_1\nα : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nha : ∀ (i : ℕ), i + 2 ≤ 0 + 1 → a (i + 1) ∣ a i + a (i + 2)\na_pos : ∀ i ≤ 0 + 1, 0 < ↑(a i)\n⊢ ↑1 = ∑ i ∈ Finset.range (0 + 1), ↑(a 0) * ↑(a (0 + 1)) / (↑(a i) * ↑(a (i + 1)))",
"after_state": "case zero.refine_1\nα : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nha : ∀ (i : ℕ), i + 2 ≤ 0 + 1 → a (i + 1) ∣ a i + a (i + 2)\na_pos : ∀ i ≤ 0 + 1, 0 < ↑(a i)\n⊢ 1 = ↑(a 0) * ↑(a (0 + 1)) / (↑(a 0) * ↑(a (0 + 1)))"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "case zero.refine_1\nα : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nha : ∀ (i : ℕ), i + 2 ≤ 0 + 1 → a (i + 1) ∣ a i + a (i + 2)\na_pos : ∀ i ≤ 0 + 1, 0 < ↑(a i)\n⊢ 1 = ↑(a 0) * ↑(a (0 + 1)) / (↑(a 0) * ↑(a (0 + 1)))",
"after_state": "case zero.refine_1\nα : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nha : ∀ (i : ℕ), i + 2 ≤ 0 + 1 → a (i + 1) ∣ a i + a (i + 2)\na_pos : ∀ i ≤ 0 + 1, 0 < ↑(a i)\n⊢ 1 = ↑(a 0) * ↑(a (0 + 1)) / (↑(a 0) * ↑(a (0 + 1)))"
},
{
"line": "try (with_reducible rfl)",
"before_state": "case zero.refine_1\nα : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nha : ∀ (i : ℕ), i + 2 ≤ 0 + 1 → a (i + 1) ∣ a i + a (i + 2)\na_pos : ∀ i ≤ 0 + 1, 0 < ↑(a i)\n⊢ 1 = ↑(a 0) * ↑(a (0 + 1)) / (↑(a 0) * ↑(a (0 + 1)))",
"after_state": "case zero.refine_1\nα : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nha : ∀ (i : ℕ), i + 2 ≤ 0 + 1 → a (i + 1) ∣ a i + a (i + 2)\na_pos : ∀ i ≤ 0 + 1, 0 < ↑(a i)\n⊢ 1 = ↑(a 0) * ↑(a (0 + 1)) / (↑(a 0) * ↑(a (0 + 1)))"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "case zero.refine_1\nα : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nha : ∀ (i : ℕ), i + 2 ≤ 0 + 1 → a (i + 1) ∣ a i + a (i + 2)\na_pos : ∀ i ≤ 0 + 1, 0 < ↑(a i)\n⊢ 1 = ↑(a 0) * ↑(a (0 + 1)) / (↑(a 0) * ↑(a (0 + 1)))",
"after_state": "case zero.refine_1\nα : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nha : ∀ (i : ℕ), i + 2 ≤ 0 + 1 → a (i + 1) ∣ a i + a (i + 2)\na_pos : ∀ i ≤ 0 + 1, 0 < ↑(a i)\n⊢ 1 = ↑(a 0) * ↑(a (0 + 1)) / (↑(a 0) * ↑(a (0 + 1)))"
},
{
"line": "with_reducible rfl",
"before_state": "case zero.refine_1\nα : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nha : ∀ (i : ℕ), i + 2 ≤ 0 + 1 → a (i + 1) ∣ a i + a (i + 2)\na_pos : ∀ i ≤ 0 + 1, 0 < ↑(a i)\n⊢ 1 = ↑(a 0) * ↑(a (0 + 1)) / (↑(a 0) * ↑(a (0 + 1)))",
"after_state": "case zero.refine_1\nα : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nha : ∀ (i : ℕ), i + 2 ≤ 0 + 1 → a (i + 1) ∣ a i + a (i + 2)\na_pos : ∀ i ≤ 0 + 1, 0 < ↑(a i)\n⊢ 1 = ↑(a 0) * ↑(a (0 + 1)) / (↑(a 0) * ↑(a (0 + 1)))"
},
{
"line": "rfl",
"before_state": "case zero.refine_1\nα : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nha : ∀ (i : ℕ), i + 2 ≤ 0 + 1 → a (i + 1) ∣ a i + a (i + 2)\na_pos : ∀ i ≤ 0 + 1, 0 < ↑(a i)\n⊢ 1 = ↑(a 0) * ↑(a (0 + 1)) / (↑(a 0) * ↑(a (0 + 1)))",
"after_state": "case zero.refine_1\nα : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nha : ∀ (i : ℕ), i + 2 ≤ 0 + 1 → a (i + 1) ∣ a i + a (i + 2)\na_pos : ∀ i ≤ 0 + 1, 0 < ↑(a i)\n⊢ 1 = ↑(a 0) * ↑(a (0 + 1)) / (↑(a 0) * ↑(a (0 + 1)))"
},
{
"line": "apply_rfl",
"before_state": "case zero.refine_1\nα : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nha : ∀ (i : ℕ), i + 2 ≤ 0 + 1 → a (i + 1) ∣ a i + a (i + 2)\na_pos : ∀ i ≤ 0 + 1, 0 < ↑(a i)\n⊢ 1 = ↑(a 0) * ↑(a (0 + 1)) / (↑(a 0) * ↑(a (0 + 1)))",
"after_state": "case zero.refine_1\nα : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nha : ∀ (i : ℕ), i + 2 ≤ 0 + 1 → a (i + 1) ∣ a i + a (i + 2)\na_pos : ∀ i ≤ 0 + 1, 0 < ↑(a i)\n⊢ 1 = ↑(a 0) * ↑(a (0 + 1)) / (↑(a 0) * ↑(a (0 + 1)))"
},
{
"line": "skip",
"before_state": "case zero.refine_1\nα : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nha : ∀ (i : ℕ), i + 2 ≤ 0 + 1 → a (i + 1) ∣ a i + a (i + 2)\na_pos : ∀ i ≤ 0 + 1, 0 < ↑(a i)\n⊢ 1 = ↑(a 0) * ↑(a (0 + 1)) / (↑(a 0) * ↑(a (0 + 1)))",
"after_state": "case zero.refine_1\nα : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nha : ∀ (i : ℕ), i + 2 ≤ 0 + 1 → a (i + 1) ∣ a i + a (i + 2)\na_pos : ∀ i ≤ 0 + 1, 0 < ↑(a i)\n⊢ 1 = ↑(a 0) * ↑(a (0 + 1)) / (↑(a 0) * ↑(a (0 + 1)))"
},
{
"line": "norm_num",
"before_state": "case zero.refine_1\nα : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nha : ∀ (i : ℕ), i + 2 ≤ 0 + 1 → a (i + 1) ∣ a i + a (i + 2)\na_pos : ∀ i ≤ 0 + 1, 0 < ↑(a i)\n⊢ 1 = ↑(a 0) * ↑(a (0 + 1)) / (↑(a 0) * ↑(a (0 + 1)))",
"after_state": "case zero.refine_1\nα : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nha : ∀ (i : ℕ), i + 2 ≤ 0 + 1 → a (i + 1) ∣ a i + a (i + 2)\na_pos : ∀ i ≤ 0 + 1, 0 < ↑(a i)\n⊢ 1 = ↑(a 0) * ↑(a 1) / (↑(a 0) * ↑(a 1))"
},
{
"line": "rw [div_self]",
"before_state": "case zero.refine_1\nα : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nha : ∀ (i : ℕ), i + 2 ≤ 0 + 1 → a (i + 1) ∣ a i + a (i + 2)\na_pos : ∀ i ≤ 0 + 1, 0 < ↑(a i)\n⊢ 1 = ↑(a 0) * ↑(a 1) / (↑(a 0) * ↑(a 1))",
"after_state": "case zero.refine_1\nα : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nha : ∀ (i : ℕ), i + 2 ≤ 0 + 1 → a (i + 1) ∣ a i + a (i + 2)\na_pos : ∀ i ≤ 0 + 1, 0 < ↑(a i)\n⊢ ↑(a 0) * ↑(a 1) ≠ 0"
},
{
"line": "rewrite [div_self]",
"before_state": "case zero.refine_1\nα : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nha : ∀ (i : ℕ), i + 2 ≤ 0 + 1 → a (i + 1) ∣ a i + a (i + 2)\na_pos : ∀ i ≤ 0 + 1, 0 < ↑(a i)\n⊢ 1 = ↑(a 0) * ↑(a 1) / (↑(a 0) * ↑(a 1))",
"after_state": "case zero.refine_1\nα : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nha : ∀ (i : ℕ), i + 2 ≤ 0 + 1 → a (i + 1) ∣ a i + a (i + 2)\na_pos : ∀ i ≤ 0 + 1, 0 < ↑(a i)\n⊢ 1 = 1\n---\ncase zero.refine_1\nα : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nha : ∀ (i : ℕ), i + 2 ≤ 0 + 1 → a (i + 1) ∣ a i + a (i + 2)\na_pos : ∀ i ≤ 0 + 1, 0 < ↑(a i)\n⊢ ↑(a 0) * ↑(a 1) ≠ 0"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "case zero.refine_1\nα : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nha : ∀ (i : ℕ), i + 2 ≤ 0 + 1 → a (i + 1) ∣ a i + a (i + 2)\na_pos : ∀ i ≤ 0 + 1, 0 < ↑(a i)\n⊢ 1 = 1\n---\ncase zero.refine_1\nα : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nha : ∀ (i : ℕ), i + 2 ≤ 0 + 1 → a (i + 1) ∣ a i + a (i + 2)\na_pos : ∀ i ≤ 0 + 1, 0 < ↑(a i)\n⊢ ↑(a 0) * ↑(a 1) ≠ 0",
"after_state": "case zero.refine_1\nα : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nha : ∀ (i : ℕ), i + 2 ≤ 0 + 1 → a (i + 1) ∣ a i + a (i + 2)\na_pos : ∀ i ≤ 0 + 1, 0 < ↑(a i)\n⊢ ↑(a 0) * ↑(a 1) ≠ 0"
},
{
"line": "try (with_reducible rfl)",
"before_state": "case zero.refine_1\nα : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nha : ∀ (i : ℕ), i + 2 ≤ 0 + 1 → a (i + 1) ∣ a i + a (i + 2)\na_pos : ∀ i ≤ 0 + 1, 0 < ↑(a i)\n⊢ 1 = 1\n---\ncase zero.refine_1\nα : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nha : ∀ (i : ℕ), i + 2 ≤ 0 + 1 → a (i + 1) ∣ a i + a (i + 2)\na_pos : ∀ i ≤ 0 + 1, 0 < ↑(a i)\n⊢ ↑(a 0) * ↑(a 1) ≠ 0",
"after_state": "case zero.refine_1\nα : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nha : ∀ (i : ℕ), i + 2 ≤ 0 + 1 → a (i + 1) ∣ a i + a (i + 2)\na_pos : ∀ i ≤ 0 + 1, 0 < ↑(a i)\n⊢ ↑(a 0) * ↑(a 1) ≠ 0"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "case zero.refine_1\nα : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nha : ∀ (i : ℕ), i + 2 ≤ 0 + 1 → a (i + 1) ∣ a i + a (i + 2)\na_pos : ∀ i ≤ 0 + 1, 0 < ↑(a i)\n⊢ 1 = 1\n---\ncase zero.refine_1\nα : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nha : ∀ (i : ℕ), i + 2 ≤ 0 + 1 → a (i + 1) ∣ a i + a (i + 2)\na_pos : ∀ i ≤ 0 + 1, 0 < ↑(a i)\n⊢ ↑(a 0) * ↑(a 1) ≠ 0",
"after_state": "case zero.refine_1\nα : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nha : ∀ (i : ℕ), i + 2 ≤ 0 + 1 → a (i + 1) ∣ a i + a (i + 2)\na_pos : ∀ i ≤ 0 + 1, 0 < ↑(a i)\n⊢ ↑(a 0) * ↑(a 1) ≠ 0"
},
{
"line": "with_reducible rfl",
"before_state": "case zero.refine_1\nα : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nha : ∀ (i : ℕ), i + 2 ≤ 0 + 1 → a (i + 1) ∣ a i + a (i + 2)\na_pos : ∀ i ≤ 0 + 1, 0 < ↑(a i)\n⊢ 1 = 1\n---\ncase zero.refine_1\nα : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nha : ∀ (i : ℕ), i + 2 ≤ 0 + 1 → a (i + 1) ∣ a i + a (i + 2)\na_pos : ∀ i ≤ 0 + 1, 0 < ↑(a i)\n⊢ ↑(a 0) * ↑(a 1) ≠ 0",
"after_state": "case zero.refine_1\nα : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nha : ∀ (i : ℕ), i + 2 ≤ 0 + 1 → a (i + 1) ∣ a i + a (i + 2)\na_pos : ∀ i ≤ 0 + 1, 0 < ↑(a i)\n⊢ ↑(a 0) * ↑(a 1) ≠ 0"
},
{
"line": "rfl",
"before_state": "case zero.refine_1\nα : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nha : ∀ (i : ℕ), i + 2 ≤ 0 + 1 → a (i + 1) ∣ a i + a (i + 2)\na_pos : ∀ i ≤ 0 + 1, 0 < ↑(a i)\n⊢ 1 = 1\n---\ncase zero.refine_1\nα : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nha : ∀ (i : ℕ), i + 2 ≤ 0 + 1 → a (i + 1) ∣ a i + a (i + 2)\na_pos : ∀ i ≤ 0 + 1, 0 < ↑(a i)\n⊢ ↑(a 0) * ↑(a 1) ≠ 0",
"after_state": "case zero.refine_1\nα : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nha : ∀ (i : ℕ), i + 2 ≤ 0 + 1 → a (i + 1) ∣ a i + a (i + 2)\na_pos : ∀ i ≤ 0 + 1, 0 < ↑(a i)\n⊢ ↑(a 0) * ↑(a 1) ≠ 0"
},
{
"line": "eq_refl",
"before_state": "case zero.refine_1\nα : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nha : ∀ (i : ℕ), i + 2 ≤ 0 + 1 → a (i + 1) ∣ a i + a (i + 2)\na_pos : ∀ i ≤ 0 + 1, 0 < ↑(a i)\n⊢ 1 = 1\n---\ncase zero.refine_1\nα : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nha : ∀ (i : ℕ), i + 2 ≤ 0 + 1 → a (i + 1) ∣ a i + a (i + 2)\na_pos : ∀ i ≤ 0 + 1, 0 < ↑(a i)\n⊢ ↑(a 0) * ↑(a 1) ≠ 0",
"after_state": "case zero.refine_1\nα : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nha : ∀ (i : ℕ), i + 2 ≤ 0 + 1 → a (i + 1) ∣ a i + a (i + 2)\na_pos : ∀ i ≤ 0 + 1, 0 < ↑(a i)\n⊢ ↑(a 0) * ↑(a 1) ≠ 0"
},
{
"line": "exact (mul_pos (a_pos 0 (Nat.zero_le _)) (a_pos 1 (Nat.zero_lt_succ _))).ne'",
"before_state": "case zero.refine_1\nα : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nha : ∀ (i : ℕ), i + 2 ≤ 0 + 1 → a (i + 1) ∣ a i + a (i + 2)\na_pos : ∀ i ≤ 0 + 1, 0 < ↑(a i)\n⊢ ↑(a 0) * ↑(a 1) ≠ 0",
"after_state": "No Goals!"
},
{
"line": "rw [mul_one, tsub_self]",
"before_state": "case zero.refine_2\nα : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nha : ∀ (i : ℕ), i + 2 ≤ 0 + 1 → a (i + 1) ∣ a i + a (i + 2)\na_pos : ∀ i ≤ 0 + 1, 0 < ↑(a i)\n⊢ a (0 + 1) ∣ a 0 * 1 - a 0",
"after_state": "case zero.refine_2\nα : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nha : ∀ (i : ℕ), i + 2 ≤ 0 + 1 → a (i + 1) ∣ a i + a (i + 2)\na_pos : ∀ i ≤ 0 + 1, 0 < ↑(a i)\n⊢ a (0 + 1) ∣ 0"
},
{
"line": "rewrite [mul_one, tsub_self]",
"before_state": "case zero.refine_2\nα : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nha : ∀ (i : ℕ), i + 2 ≤ 0 + 1 → a (i + 1) ∣ a i + a (i + 2)\na_pos : ∀ i ≤ 0 + 1, 0 < ↑(a i)\n⊢ a (0 + 1) ∣ a 0 * 1 - a 0",
"after_state": "case zero.refine_2\nα : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nha : ∀ (i : ℕ), i + 2 ≤ 0 + 1 → a (i + 1) ∣ a i + a (i + 2)\na_pos : ∀ i ≤ 0 + 1, 0 < ↑(a i)\n⊢ a (0 + 1) ∣ 0"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "case zero.refine_2\nα : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nha : ∀ (i : ℕ), i + 2 ≤ 0 + 1 → a (i + 1) ∣ a i + a (i + 2)\na_pos : ∀ i ≤ 0 + 1, 0 < ↑(a i)\n⊢ a (0 + 1) ∣ 0",
"after_state": "case zero.refine_2\nα : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nha : ∀ (i : ℕ), i + 2 ≤ 0 + 1 → a (i + 1) ∣ a i + a (i + 2)\na_pos : ∀ i ≤ 0 + 1, 0 < ↑(a i)\n⊢ a (0 + 1) ∣ 0"
},
{
"line": "try (with_reducible rfl)",
"before_state": "case zero.refine_2\nα : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nha : ∀ (i : ℕ), i + 2 ≤ 0 + 1 → a (i + 1) ∣ a i + a (i + 2)\na_pos : ∀ i ≤ 0 + 1, 0 < ↑(a i)\n⊢ a (0 + 1) ∣ 0",
"after_state": "case zero.refine_2\nα : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nha : ∀ (i : ℕ), i + 2 ≤ 0 + 1 → a (i + 1) ∣ a i + a (i + 2)\na_pos : ∀ i ≤ 0 + 1, 0 < ↑(a i)\n⊢ a (0 + 1) ∣ 0"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "case zero.refine_2\nα : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nha : ∀ (i : ℕ), i + 2 ≤ 0 + 1 → a (i + 1) ∣ a i + a (i + 2)\na_pos : ∀ i ≤ 0 + 1, 0 < ↑(a i)\n⊢ a (0 + 1) ∣ 0",
"after_state": "case zero.refine_2\nα : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nha : ∀ (i : ℕ), i + 2 ≤ 0 + 1 → a (i + 1) ∣ a i + a (i + 2)\na_pos : ∀ i ≤ 0 + 1, 0 < ↑(a i)\n⊢ a (0 + 1) ∣ 0"
},
{
"line": "with_reducible rfl",
"before_state": "case zero.refine_2\nα : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nha : ∀ (i : ℕ), i + 2 ≤ 0 + 1 → a (i + 1) ∣ a i + a (i + 2)\na_pos : ∀ i ≤ 0 + 1, 0 < ↑(a i)\n⊢ a (0 + 1) ∣ 0",
"after_state": "case zero.refine_2\nα : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nha : ∀ (i : ℕ), i + 2 ≤ 0 + 1 → a (i + 1) ∣ a i + a (i + 2)\na_pos : ∀ i ≤ 0 + 1, 0 < ↑(a i)\n⊢ a (0 + 1) ∣ 0"
},
{
"line": "rfl",
"before_state": "case zero.refine_2\nα : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nha : ∀ (i : ℕ), i + 2 ≤ 0 + 1 → a (i + 1) ∣ a i + a (i + 2)\na_pos : ∀ i ≤ 0 + 1, 0 < ↑(a i)\n⊢ a (0 + 1) ∣ 0",
"after_state": "case zero.refine_2\nα : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nha : ∀ (i : ℕ), i + 2 ≤ 0 + 1 → a (i + 1) ∣ a i + a (i + 2)\na_pos : ∀ i ≤ 0 + 1, 0 < ↑(a i)\n⊢ a (0 + 1) ∣ 0"
},
{
"line": "apply_rfl",
"before_state": "case zero.refine_2\nα : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nha : ∀ (i : ℕ), i + 2 ≤ 0 + 1 → a (i + 1) ∣ a i + a (i + 2)\na_pos : ∀ i ≤ 0 + 1, 0 < ↑(a i)\n⊢ a (0 + 1) ∣ 0",
"after_state": "case zero.refine_2\nα : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nha : ∀ (i : ℕ), i + 2 ≤ 0 + 1 → a (i + 1) ∣ a i + a (i + 2)\na_pos : ∀ i ≤ 0 + 1, 0 < ↑(a i)\n⊢ a (0 + 1) ∣ 0"
},
{
"line": "skip",
"before_state": "case zero.refine_2\nα : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nha : ∀ (i : ℕ), i + 2 ≤ 0 + 1 → a (i + 1) ∣ a i + a (i + 2)\na_pos : ∀ i ≤ 0 + 1, 0 < ↑(a i)\n⊢ a (0 + 1) ∣ 0",
"after_state": "case zero.refine_2\nα : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nha : ∀ (i : ℕ), i + 2 ≤ 0 + 1 → a (i + 1) ∣ a i + a (i + 2)\na_pos : ∀ i ≤ 0 + 1, 0 < ↑(a i)\n⊢ a (0 + 1) ∣ 0"
},
{
"line": "exact\n dvd_zero\n _\n /- Induction step\n `b` is the value of the previous sum as a natural, `hb` is the proof that it is indeed the\n value, and `han` is the divisibility condition -/",
"before_state": "case zero.refine_2\nα : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nha : ∀ (i : ℕ), i + 2 ≤ 0 + 1 → a (i + 1) ∣ a i + a (i + 2)\na_pos : ∀ i ≤ 0 + 1, 0 < ↑(a i)\n⊢ a (0 + 1) ∣ 0",
"after_state": "No Goals!"
},
{
"line": "obtain ⟨b, hb, han⟩ := ih (fun i hi => ha i <| Nat.le_succ_of_le hi) fun i hi => a_pos i <| Nat.le_succ_of_le hi",
"before_state": "case succ\nα : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\nha : ∀ (i : ℕ), i + 2 ≤ n + 1 + 1 → a (i + 1) ∣ a i + a (i + 2)\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\n⊢ ∃ b,\n ↑b = ∑ i ∈ Finset.range (n + 1 + 1), ↑(a 0) * ↑(a (n + 1 + 1)) / (↑(a i) * ↑(a (i + 1))) ∧\n a (n + 1 + 1) ∣ a (n + 1) * b - a 0",
"after_state": "case succ.intro.intro\nα : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\nha : ∀ (i : ℕ), i + 2 ≤ n + 1 + 1 → a (i + 1) ∣ a i + a (i + 2)\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\n⊢ ∃ b,\n ↑b = ∑ i ∈ Finset.range (n + 1 + 1), ↑(a 0) * ↑(a (n + 1 + 1)) / (↑(a i) * ↑(a (i + 1))) ∧\n a (n + 1 + 1) ∣ a (n + 1) * b - a 0"
},
{
"line": "specialize ha n le_rfl",
"before_state": "case succ.intro.intro\nα : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\nha : ∀ (i : ℕ), i + 2 ≤ n + 1 + 1 → a (i + 1) ∣ a i + a (i + 2)\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\n⊢ ∃ b,\n ↑b = ∑ i ∈ Finset.range (n + 1 + 1), ↑(a 0) * ↑(a (n + 1 + 1)) / (↑(a i) * ↑(a (i + 1))) ∧\n a (n + 1 + 1) ∣ a (n + 1) * b - a 0",
"after_state": "case succ.intro.intro\nα : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\n⊢ ∃ b,\n ↑b = ∑ i ∈ Finset.range (n + 1 + 1), ↑(a 0) * ↑(a (n + 1 + 1)) / (↑(a i) * ↑(a (i + 1))) ∧\n a (n + 1 + 1) ∣ a (n + 1) * b - a 0"
},
{
"line": "have ha₀ : a 0 ≤ a n * b := by\n -- Needing this is an artifact of `ℕ`-subtraction.\n rw [← @Nat.cast_le α]\n rw [Nat.cast_mul]\n rw [hb]\n rw [← div_le_iff₀' (a_pos _ <| n.le_succ.trans <| Nat.le_succ _)]\n rw [← mul_div_mul_right _ _ (a_pos _ <| Nat.le_succ _).ne']\n suffices h : ∀ i, i ∈ Finset.range (n + 1) → 0 ≤ (a 0 : α) * a (n + 1) / (a i * a (i + 1)) from\n Finset.single_le_sum h (Finset.self_mem_range_succ n)\n refine fun i _ ↦ div_nonneg ?_ ?_ <;> refine mul_nonneg ?_ ?_ <;>\n exact\n Nat.cast_nonneg\n _\n -- Claim that the sum equals `(aₙ + aₙ₊₂)/aₙ₊₁ * b - (aₙ * b - a₀)/aₙ₊₁`",
"before_state": "case succ.intro.intro\nα : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\n⊢ ∃ b,\n ↑b = ∑ i ∈ Finset.range (n + 1 + 1), ↑(a 0) * ↑(a (n + 1 + 1)) / (↑(a i) * ↑(a (i + 1))) ∧\n a (n + 1 + 1) ∣ a (n + 1) * b - a 0",
"after_state": "case succ.intro.intro\nα : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\n⊢ ∃ b,\n ↑b = ∑ i ∈ Finset.range (n + 1 + 1), ↑(a 0) * ↑(a (n + 1 + 1)) / (↑(a i) * ↑(a (i + 1))) ∧\n a (n + 1 + 1) ∣ a (n + 1) * b - a 0"
},
{
"line": "focus\n refine\n no_implicit_lambda%\n (have ha₀ : a 0 ≤ a n * b := ?body✝;\n ?_)\n case body✝ =>\n with_annotate_state\"by\"\n ( rw [← @Nat.cast_le α]\n rw [Nat.cast_mul]\n rw [hb]\n rw [← div_le_iff₀' (a_pos _ <| n.le_succ.trans <| Nat.le_succ _)]\n rw [← mul_div_mul_right _ _ (a_pos _ <| Nat.le_succ _).ne']\n suffices h : ∀ i, i ∈ Finset.range (n + 1) → 0 ≤ (a 0 : α) * a (n + 1) / (a i * a (i + 1)) from\n Finset.single_le_sum h (Finset.self_mem_range_succ n)\n refine fun i _ ↦ div_nonneg ?_ ?_ <;> refine mul_nonneg ?_ ?_ <;>\n exact\n Nat.cast_nonneg\n _\n -- Claim that the sum equals `(aₙ + aₙ₊₂)/aₙ₊₁ * b - (aₙ * b - a₀)/aₙ₊₁`)",
"before_state": "case succ.intro.intro\nα : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\n⊢ ∃ b,\n ↑b = ∑ i ∈ Finset.range (n + 1 + 1), ↑(a 0) * ↑(a (n + 1 + 1)) / (↑(a i) * ↑(a (i + 1))) ∧\n a (n + 1 + 1) ∣ a (n + 1) * b - a 0",
"after_state": "case succ.intro.intro\nα : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\n⊢ ∃ b,\n ↑b = ∑ i ∈ Finset.range (n + 1 + 1), ↑(a 0) * ↑(a (n + 1 + 1)) / (↑(a i) * ↑(a (i + 1))) ∧\n a (n + 1 + 1) ∣ a (n + 1) * b - a 0"
},
{
"line": "refine\n no_implicit_lambda%\n (have ha₀ : a 0 ≤ a n * b := ?body✝;\n ?_)",
"before_state": "case succ.intro.intro\nα : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\n⊢ ∃ b,\n ↑b = ∑ i ∈ Finset.range (n + 1 + 1), ↑(a 0) * ↑(a (n + 1 + 1)) / (↑(a i) * ↑(a (i + 1))) ∧\n a (n + 1 + 1) ∣ a (n + 1) * b - a 0",
"after_state": "case body\nα : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\n⊢ a 0 ≤ a n * b\n---\ncase succ.intro.intro\nα : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\n⊢ ∃ b,\n ↑b = ∑ i ∈ Finset.range (n + 1 + 1), ↑(a 0) * ↑(a (n + 1 + 1)) / (↑(a i) * ↑(a (i + 1))) ∧\n a (n + 1 + 1) ∣ a (n + 1) * b - a 0"
},
{
"line": "case body✝ =>\n with_annotate_state\"by\"\n ( rw [← @Nat.cast_le α]\n rw [Nat.cast_mul]\n rw [hb]\n rw [← div_le_iff₀' (a_pos _ <| n.le_succ.trans <| Nat.le_succ _)]\n rw [← mul_div_mul_right _ _ (a_pos _ <| Nat.le_succ _).ne']\n suffices h : ∀ i, i ∈ Finset.range (n + 1) → 0 ≤ (a 0 : α) * a (n + 1) / (a i * a (i + 1)) from\n Finset.single_le_sum h (Finset.self_mem_range_succ n)\n refine fun i _ ↦ div_nonneg ?_ ?_ <;> refine mul_nonneg ?_ ?_ <;>\n exact\n Nat.cast_nonneg\n _\n -- Claim that the sum equals `(aₙ + aₙ₊₂)/aₙ₊₁ * b - (aₙ * b - a₀)/aₙ₊₁`)",
"before_state": "case body\nα : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\n⊢ a 0 ≤ a n * b\n---\ncase succ.intro.intro\nα : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\n⊢ ∃ b,\n ↑b = ∑ i ∈ Finset.range (n + 1 + 1), ↑(a 0) * ↑(a (n + 1 + 1)) / (↑(a i) * ↑(a (i + 1))) ∧\n a (n + 1 + 1) ∣ a (n + 1) * b - a 0",
"after_state": "case succ.intro.intro\nα : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\n⊢ ∃ b,\n ↑b = ∑ i ∈ Finset.range (n + 1 + 1), ↑(a 0) * ↑(a (n + 1 + 1)) / (↑(a i) * ↑(a (i + 1))) ∧\n a (n + 1 + 1) ∣ a (n + 1) * b - a 0"
},
{
"line": "with_annotate_state\"by\"\n ( rw [← @Nat.cast_le α]\n rw [Nat.cast_mul]\n rw [hb]\n rw [← div_le_iff₀' (a_pos _ <| n.le_succ.trans <| Nat.le_succ _)]\n rw [← mul_div_mul_right _ _ (a_pos _ <| Nat.le_succ _).ne']\n suffices h : ∀ i, i ∈ Finset.range (n + 1) → 0 ≤ (a 0 : α) * a (n + 1) / (a i * a (i + 1)) from\n Finset.single_le_sum h (Finset.self_mem_range_succ n)\n refine fun i _ ↦ div_nonneg ?_ ?_ <;> refine mul_nonneg ?_ ?_ <;>\n exact\n Nat.cast_nonneg\n _\n -- Claim that the sum equals `(aₙ + aₙ₊₂)/aₙ₊₁ * b - (aₙ * b - a₀)/aₙ₊₁`)",
"before_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\n⊢ a 0 ≤ a n * b",
"after_state": "No Goals!"
},
{
"line": "rw [← @Nat.cast_le α]",
"before_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\n⊢ a 0 ≤ a n * b",
"after_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\n⊢ ↑(a 0) ≤ ↑(a n * b)"
},
{
"line": "rewrite [← @Nat.cast_le α]",
"before_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\n⊢ a 0 ≤ a n * b",
"after_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\n⊢ ↑(a 0) ≤ ↑(a n * b)"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\n⊢ ↑(a 0) ≤ ↑(a n * b)",
"after_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\n⊢ ↑(a 0) ≤ ↑(a n * b)"
},
{
"line": "try (with_reducible rfl)",
"before_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\n⊢ ↑(a 0) ≤ ↑(a n * b)",
"after_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\n⊢ ↑(a 0) ≤ ↑(a n * b)"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\n⊢ ↑(a 0) ≤ ↑(a n * b)",
"after_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\n⊢ ↑(a 0) ≤ ↑(a n * b)"
},
{
"line": "with_reducible rfl",
"before_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\n⊢ ↑(a 0) ≤ ↑(a n * b)",
"after_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\n⊢ ↑(a 0) ≤ ↑(a n * b)"
},
{
"line": "rfl",
"before_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\n⊢ ↑(a 0) ≤ ↑(a n * b)",
"after_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\n⊢ ↑(a 0) ≤ ↑(a n * b)"
},
{
"line": "apply_rfl",
"before_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\n⊢ ↑(a 0) ≤ ↑(a n * b)",
"after_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\n⊢ ↑(a 0) ≤ ↑(a n * b)"
},
{
"line": "skip",
"before_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\n⊢ ↑(a 0) ≤ ↑(a n * b)",
"after_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\n⊢ ↑(a 0) ≤ ↑(a n * b)"
},
{
"line": "rw [Nat.cast_mul]",
"before_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\n⊢ ↑(a 0) ≤ ↑(a n * b)",
"after_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\n⊢ ↑(a 0) ≤ ↑(a n) * ↑b"
},
{
"line": "rewrite [Nat.cast_mul]",
"before_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\n⊢ ↑(a 0) ≤ ↑(a n * b)",
"after_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\n⊢ ↑(a 0) ≤ ↑(a n) * ↑b"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\n⊢ ↑(a 0) ≤ ↑(a n) * ↑b",
"after_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\n⊢ ↑(a 0) ≤ ↑(a n) * ↑b"
},
{
"line": "try (with_reducible rfl)",
"before_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\n⊢ ↑(a 0) ≤ ↑(a n) * ↑b",
"after_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\n⊢ ↑(a 0) ≤ ↑(a n) * ↑b"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\n⊢ ↑(a 0) ≤ ↑(a n) * ↑b",
"after_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\n⊢ ↑(a 0) ≤ ↑(a n) * ↑b"
},
{
"line": "with_reducible rfl",
"before_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\n⊢ ↑(a 0) ≤ ↑(a n) * ↑b",
"after_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\n⊢ ↑(a 0) ≤ ↑(a n) * ↑b"
},
{
"line": "rfl",
"before_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\n⊢ ↑(a 0) ≤ ↑(a n) * ↑b",
"after_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\n⊢ ↑(a 0) ≤ ↑(a n) * ↑b"
},
{
"line": "apply_rfl",
"before_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\n⊢ ↑(a 0) ≤ ↑(a n) * ↑b",
"after_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\n⊢ ↑(a 0) ≤ ↑(a n) * ↑b"
},
{
"line": "skip",
"before_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\n⊢ ↑(a 0) ≤ ↑(a n) * ↑b",
"after_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\n⊢ ↑(a 0) ≤ ↑(a n) * ↑b"
},
{
"line": "rw [hb]",
"before_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\n⊢ ↑(a 0) ≤ ↑(a n) * ↑b",
"after_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\n⊢ ↑(a 0) ≤ ↑(a n) * ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))"
},
{
"line": "rewrite [hb]",
"before_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\n⊢ ↑(a 0) ≤ ↑(a n) * ↑b",
"after_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\n⊢ ↑(a 0) ≤ ↑(a n) * ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\n⊢ ↑(a 0) ≤ ↑(a n) * ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))",
"after_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\n⊢ ↑(a 0) ≤ ↑(a n) * ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))"
},
{
"line": "try (with_reducible rfl)",
"before_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\n⊢ ↑(a 0) ≤ ↑(a n) * ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))",
"after_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\n⊢ ↑(a 0) ≤ ↑(a n) * ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\n⊢ ↑(a 0) ≤ ↑(a n) * ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))",
"after_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\n⊢ ↑(a 0) ≤ ↑(a n) * ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))"
},
{
"line": "with_reducible rfl",
"before_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\n⊢ ↑(a 0) ≤ ↑(a n) * ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))",
"after_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\n⊢ ↑(a 0) ≤ ↑(a n) * ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))"
},
{
"line": "rfl",
"before_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\n⊢ ↑(a 0) ≤ ↑(a n) * ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))",
"after_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\n⊢ ↑(a 0) ≤ ↑(a n) * ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))"
},
{
"line": "apply_rfl",
"before_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\n⊢ ↑(a 0) ≤ ↑(a n) * ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))",
"after_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\n⊢ ↑(a 0) ≤ ↑(a n) * ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))"
},
{
"line": "skip",
"before_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\n⊢ ↑(a 0) ≤ ↑(a n) * ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))",
"after_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\n⊢ ↑(a 0) ≤ ↑(a n) * ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))"
},
{
"line": "rw [← div_le_iff₀' (a_pos _ <| n.le_succ.trans <| Nat.le_succ _)]",
"before_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\n⊢ ↑(a 0) ≤ ↑(a n) * ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))",
"after_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\n⊢ ↑(a 0) / ↑(a n) ≤ ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))"
},
{
"line": "rewrite [← div_le_iff₀' (a_pos _ <| n.le_succ.trans <| Nat.le_succ _)]",
"before_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\n⊢ ↑(a 0) ≤ ↑(a n) * ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))",
"after_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\n⊢ ↑(a 0) / ↑(a n) ≤ ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\n⊢ ↑(a 0) / ↑(a n) ≤ ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))",
"after_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\n⊢ ↑(a 0) / ↑(a n) ≤ ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))"
},
{
"line": "try (with_reducible rfl)",
"before_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\n⊢ ↑(a 0) / ↑(a n) ≤ ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))",
"after_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\n⊢ ↑(a 0) / ↑(a n) ≤ ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\n⊢ ↑(a 0) / ↑(a n) ≤ ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))",
"after_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\n⊢ ↑(a 0) / ↑(a n) ≤ ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))"
},
{
"line": "with_reducible rfl",
"before_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\n⊢ ↑(a 0) / ↑(a n) ≤ ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))",
"after_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\n⊢ ↑(a 0) / ↑(a n) ≤ ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))"
},
{
"line": "rfl",
"before_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\n⊢ ↑(a 0) / ↑(a n) ≤ ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))",
"after_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\n⊢ ↑(a 0) / ↑(a n) ≤ ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))"
},
{
"line": "apply_rfl",
"before_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\n⊢ ↑(a 0) / ↑(a n) ≤ ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))",
"after_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\n⊢ ↑(a 0) / ↑(a n) ≤ ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))"
},
{
"line": "skip",
"before_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\n⊢ ↑(a 0) / ↑(a n) ≤ ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))",
"after_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\n⊢ ↑(a 0) / ↑(a n) ≤ ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))"
},
{
"line": "rw [← mul_div_mul_right _ _ (a_pos _ <| Nat.le_succ _).ne']",
"before_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\n⊢ ↑(a 0) / ↑(a n) ≤ ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))",
"after_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\n⊢ ↑(a 0) * ↑(a (n + 1)) / (↑(a n) * ↑(a (n + 1))) ≤\n ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))"
},
{
"line": "rewrite [← mul_div_mul_right _ _ (a_pos _ <| Nat.le_succ _).ne']",
"before_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\n⊢ ↑(a 0) / ↑(a n) ≤ ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))",
"after_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\n⊢ ↑(a 0) * ↑(a (n + 1)) / (↑(a n) * ↑(a (n + 1))) ≤\n ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\n⊢ ↑(a 0) * ↑(a (n + 1)) / (↑(a n) * ↑(a (n + 1))) ≤\n ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))",
"after_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\n⊢ ↑(a 0) * ↑(a (n + 1)) / (↑(a n) * ↑(a (n + 1))) ≤\n ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))"
},
{
"line": "try (with_reducible rfl)",
"before_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\n⊢ ↑(a 0) * ↑(a (n + 1)) / (↑(a n) * ↑(a (n + 1))) ≤\n ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))",
"after_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\n⊢ ↑(a 0) * ↑(a (n + 1)) / (↑(a n) * ↑(a (n + 1))) ≤\n ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\n⊢ ↑(a 0) * ↑(a (n + 1)) / (↑(a n) * ↑(a (n + 1))) ≤\n ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))",
"after_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\n⊢ ↑(a 0) * ↑(a (n + 1)) / (↑(a n) * ↑(a (n + 1))) ≤\n ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))"
},
{
"line": "with_reducible rfl",
"before_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\n⊢ ↑(a 0) * ↑(a (n + 1)) / (↑(a n) * ↑(a (n + 1))) ≤\n ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))",
"after_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\n⊢ ↑(a 0) * ↑(a (n + 1)) / (↑(a n) * ↑(a (n + 1))) ≤\n ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))"
},
{
"line": "rfl",
"before_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\n⊢ ↑(a 0) * ↑(a (n + 1)) / (↑(a n) * ↑(a (n + 1))) ≤\n ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))",
"after_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\n⊢ ↑(a 0) * ↑(a (n + 1)) / (↑(a n) * ↑(a (n + 1))) ≤\n ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))"
},
{
"line": "apply_rfl",
"before_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\n⊢ ↑(a 0) * ↑(a (n + 1)) / (↑(a n) * ↑(a (n + 1))) ≤\n ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))",
"after_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\n⊢ ↑(a 0) * ↑(a (n + 1)) / (↑(a n) * ↑(a (n + 1))) ≤\n ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))"
},
{
"line": "skip",
"before_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\n⊢ ↑(a 0) * ↑(a (n + 1)) / (↑(a n) * ↑(a (n + 1))) ≤\n ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))",
"after_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\n⊢ ↑(a 0) * ↑(a (n + 1)) / (↑(a n) * ↑(a (n + 1))) ≤\n ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))"
},
{
"line": "suffices h : ∀ i, i ∈ Finset.range (n + 1) → 0 ≤ (a 0 : α) * a (n + 1) / (a i * a (i + 1)) from\n Finset.single_le_sum h (Finset.self_mem_range_succ n)",
"before_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\n⊢ ↑(a 0) * ↑(a (n + 1)) / (↑(a n) * ↑(a (n + 1))) ≤\n ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))",
"after_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\n⊢ ∀ i ∈ Finset.range (n + 1), 0 ≤ ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))"
},
{
"line": "refine_lift\n suffices h : ∀ i, i ∈ Finset.range (n + 1) → 0 ≤ (a 0 : α) * a (n + 1) / (a i * a (i + 1)) from\n Finset.single_le_sum h (Finset.self_mem_range_succ n);\n ?_",
"before_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\n⊢ ↑(a 0) * ↑(a (n + 1)) / (↑(a n) * ↑(a (n + 1))) ≤\n ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))",
"after_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\n⊢ ∀ i ∈ Finset.range (n + 1), 0 ≤ ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))"
},
{
"line": "focus\n (refine\n no_implicit_lambda%\n (suffices h : ∀ i, i ∈ Finset.range (n + 1) → 0 ≤ (a 0 : α) * a (n + 1) / (a i * a (i + 1)) from\n Finset.single_le_sum h (Finset.self_mem_range_succ n);\n ?_);\n rotate_right)",
"before_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\n⊢ ↑(a 0) * ↑(a (n + 1)) / (↑(a n) * ↑(a (n + 1))) ≤\n ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))",
"after_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\n⊢ ∀ i ∈ Finset.range (n + 1), 0 ≤ ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))"
},
{
"line": "refine\n no_implicit_lambda%\n (suffices h : ∀ i, i ∈ Finset.range (n + 1) → 0 ≤ (a 0 : α) * a (n + 1) / (a i * a (i + 1)) from\n Finset.single_le_sum h (Finset.self_mem_range_succ n);\n ?_)",
"before_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\n⊢ ↑(a 0) * ↑(a (n + 1)) / (↑(a n) * ↑(a (n + 1))) ≤\n ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))",
"after_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\n⊢ ∀ i ∈ Finset.range (n + 1), 0 ≤ ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))"
},
{
"line": "rotate_right",
"before_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\n⊢ ∀ i ∈ Finset.range (n + 1), 0 ≤ ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))",
"after_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\n⊢ ∀ i ∈ Finset.range (n + 1), 0 ≤ ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))"
},
{
"line": "focus\n refine fun i _ ↦ div_nonneg ?_ ?_ <;> refine mul_nonneg ?_ ?_\n with_annotate_state\"<;>\" skip\n all_goals\n exact\n Nat.cast_nonneg\n _\n -- Claim that the sum equals `(aₙ + aₙ₊₂)/aₙ₊₁ * b - (aₙ * b - a₀)/aₙ₊₁`",
"before_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\n⊢ ∀ i ∈ Finset.range (n + 1), 0 ≤ ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))",
"after_state": "No Goals!"
},
{
"line": "focus\n refine fun i _ ↦ div_nonneg ?_ ?_\n with_annotate_state\"<;>\" skip\n all_goals refine mul_nonneg ?_ ?_",
"before_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\n⊢ ∀ i ∈ Finset.range (n + 1), 0 ≤ ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))",
"after_state": "case refine_1.refine_1\nα : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\ni : ℕ\nx✝ : i ∈ Finset.range (n + 1)\n⊢ 0 ≤ ↑(a 0)\n---\ncase refine_1.refine_2\nα : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\ni : ℕ\nx✝ : i ∈ Finset.range (n + 1)\n⊢ 0 ≤ ↑(a (n + 1))\n---\ncase refine_2.refine_1\nα : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\ni : ℕ\nx✝ : i ∈ Finset.range (n + 1)\n⊢ 0 ≤ ↑(a i)\n---\ncase refine_2.refine_2\nα : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\ni : ℕ\nx✝ : i ∈ Finset.range (n + 1)\n⊢ 0 ≤ ↑(a (i + 1))"
},
{
"line": "refine fun i _ ↦ div_nonneg ?_ ?_",
"before_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\n⊢ ∀ i ∈ Finset.range (n + 1), 0 ≤ ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))",
"after_state": "case refine_1\nα : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\ni : ℕ\nx✝ : i ∈ Finset.range (n + 1)\n⊢ 0 ≤ ↑(a 0) * ↑(a (n + 1))\n---\ncase refine_2\nα : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\ni : ℕ\nx✝ : i ∈ Finset.range (n + 1)\n⊢ 0 ≤ ↑(a i) * ↑(a (i + 1))"
},
{
"line": "with_annotate_state\"<;>\" skip",
"before_state": "case refine_1\nα : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\ni : ℕ\nx✝ : i ∈ Finset.range (n + 1)\n⊢ 0 ≤ ↑(a 0) * ↑(a (n + 1))\n---\ncase refine_2\nα : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\ni : ℕ\nx✝ : i ∈ Finset.range (n + 1)\n⊢ 0 ≤ ↑(a i) * ↑(a (i + 1))",
"after_state": "case refine_1\nα : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\ni : ℕ\nx✝ : i ∈ Finset.range (n + 1)\n⊢ 0 ≤ ↑(a 0) * ↑(a (n + 1))\n---\ncase refine_2\nα : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\ni : ℕ\nx✝ : i ∈ Finset.range (n + 1)\n⊢ 0 ≤ ↑(a i) * ↑(a (i + 1))"
},
{
"line": "skip",
"before_state": "case refine_1\nα : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\ni : ℕ\nx✝ : i ∈ Finset.range (n + 1)\n⊢ 0 ≤ ↑(a 0) * ↑(a (n + 1))\n---\ncase refine_2\nα : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\ni : ℕ\nx✝ : i ∈ Finset.range (n + 1)\n⊢ 0 ≤ ↑(a i) * ↑(a (i + 1))",
"after_state": "case refine_1\nα : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\ni : ℕ\nx✝ : i ∈ Finset.range (n + 1)\n⊢ 0 ≤ ↑(a 0) * ↑(a (n + 1))\n---\ncase refine_2\nα : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\ni : ℕ\nx✝ : i ∈ Finset.range (n + 1)\n⊢ 0 ≤ ↑(a i) * ↑(a (i + 1))"
},
{
"line": "all_goals refine mul_nonneg ?_ ?_",
"before_state": "case refine_1\nα : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\ni : ℕ\nx✝ : i ∈ Finset.range (n + 1)\n⊢ 0 ≤ ↑(a 0) * ↑(a (n + 1))\n---\ncase refine_2\nα : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\ni : ℕ\nx✝ : i ∈ Finset.range (n + 1)\n⊢ 0 ≤ ↑(a i) * ↑(a (i + 1))",
"after_state": "case refine_1.refine_1\nα : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\ni : ℕ\nx✝ : i ∈ Finset.range (n + 1)\n⊢ 0 ≤ ↑(a 0)\n---\ncase refine_1.refine_2\nα : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\ni : ℕ\nx✝ : i ∈ Finset.range (n + 1)\n⊢ 0 ≤ ↑(a (n + 1))\n---\ncase refine_2.refine_1\nα : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\ni : ℕ\nx✝ : i ∈ Finset.range (n + 1)\n⊢ 0 ≤ ↑(a i)\n---\ncase refine_2.refine_2\nα : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\ni : ℕ\nx✝ : i ∈ Finset.range (n + 1)\n⊢ 0 ≤ ↑(a (i + 1))"
},
{
"line": "refine mul_nonneg ?_ ?_",
"before_state": "case refine_1\nα : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\ni : ℕ\nx✝ : i ∈ Finset.range (n + 1)\n⊢ 0 ≤ ↑(a 0) * ↑(a (n + 1))",
"after_state": "case refine_1.refine_1\nα : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\ni : ℕ\nx✝ : i ∈ Finset.range (n + 1)\n⊢ 0 ≤ ↑(a 0)\n---\ncase refine_1.refine_2\nα : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\ni : ℕ\nx✝ : i ∈ Finset.range (n + 1)\n⊢ 0 ≤ ↑(a (n + 1))"
},
{
"line": "refine mul_nonneg ?_ ?_",
"before_state": "case refine_2\nα : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\ni : ℕ\nx✝ : i ∈ Finset.range (n + 1)\n⊢ 0 ≤ ↑(a i) * ↑(a (i + 1))",
"after_state": "case refine_2.refine_1\nα : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\ni : ℕ\nx✝ : i ∈ Finset.range (n + 1)\n⊢ 0 ≤ ↑(a i)\n---\ncase refine_2.refine_2\nα : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\ni : ℕ\nx✝ : i ∈ Finset.range (n + 1)\n⊢ 0 ≤ ↑(a (i + 1))"
},
{
"line": "with_annotate_state\"<;>\" skip",
"before_state": "case refine_1.refine_1\nα : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\ni : ℕ\nx✝ : i ∈ Finset.range (n + 1)\n⊢ 0 ≤ ↑(a 0)\n---\ncase refine_1.refine_2\nα : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\ni : ℕ\nx✝ : i ∈ Finset.range (n + 1)\n⊢ 0 ≤ ↑(a (n + 1))\n---\ncase refine_2.refine_1\nα : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\ni : ℕ\nx✝ : i ∈ Finset.range (n + 1)\n⊢ 0 ≤ ↑(a i)\n---\ncase refine_2.refine_2\nα : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\ni : ℕ\nx✝ : i ∈ Finset.range (n + 1)\n⊢ 0 ≤ ↑(a (i + 1))",
"after_state": "case refine_1.refine_1\nα : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\ni : ℕ\nx✝ : i ∈ Finset.range (n + 1)\n⊢ 0 ≤ ↑(a 0)\n---\ncase refine_1.refine_2\nα : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\ni : ℕ\nx✝ : i ∈ Finset.range (n + 1)\n⊢ 0 ≤ ↑(a (n + 1))\n---\ncase refine_2.refine_1\nα : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\ni : ℕ\nx✝ : i ∈ Finset.range (n + 1)\n⊢ 0 ≤ ↑(a i)\n---\ncase refine_2.refine_2\nα : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\ni : ℕ\nx✝ : i ∈ Finset.range (n + 1)\n⊢ 0 ≤ ↑(a (i + 1))"
},
{
"line": "skip",
"before_state": "case refine_1.refine_1\nα : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\ni : ℕ\nx✝ : i ∈ Finset.range (n + 1)\n⊢ 0 ≤ ↑(a 0)\n---\ncase refine_1.refine_2\nα : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\ni : ℕ\nx✝ : i ∈ Finset.range (n + 1)\n⊢ 0 ≤ ↑(a (n + 1))\n---\ncase refine_2.refine_1\nα : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\ni : ℕ\nx✝ : i ∈ Finset.range (n + 1)\n⊢ 0 ≤ ↑(a i)\n---\ncase refine_2.refine_2\nα : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\ni : ℕ\nx✝ : i ∈ Finset.range (n + 1)\n⊢ 0 ≤ ↑(a (i + 1))",
"after_state": "case refine_1.refine_1\nα : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\ni : ℕ\nx✝ : i ∈ Finset.range (n + 1)\n⊢ 0 ≤ ↑(a 0)\n---\ncase refine_1.refine_2\nα : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\ni : ℕ\nx✝ : i ∈ Finset.range (n + 1)\n⊢ 0 ≤ ↑(a (n + 1))\n---\ncase refine_2.refine_1\nα : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\ni : ℕ\nx✝ : i ∈ Finset.range (n + 1)\n⊢ 0 ≤ ↑(a i)\n---\ncase refine_2.refine_2\nα : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\ni : ℕ\nx✝ : i ∈ Finset.range (n + 1)\n⊢ 0 ≤ ↑(a (i + 1))"
},
{
"line": "all_goals\n exact\n Nat.cast_nonneg\n _\n -- Claim that the sum equals `(aₙ + aₙ₊₂)/aₙ₊₁ * b - (aₙ * b - a₀)/aₙ₊₁`",
"before_state": "case refine_1.refine_1\nα : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\ni : ℕ\nx✝ : i ∈ Finset.range (n + 1)\n⊢ 0 ≤ ↑(a 0)\n---\ncase refine_1.refine_2\nα : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\ni : ℕ\nx✝ : i ∈ Finset.range (n + 1)\n⊢ 0 ≤ ↑(a (n + 1))\n---\ncase refine_2.refine_1\nα : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\ni : ℕ\nx✝ : i ∈ Finset.range (n + 1)\n⊢ 0 ≤ ↑(a i)\n---\ncase refine_2.refine_2\nα : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\ni : ℕ\nx✝ : i ∈ Finset.range (n + 1)\n⊢ 0 ≤ ↑(a (i + 1))",
"after_state": "No Goals!"
},
{
"line": "exact\n Nat.cast_nonneg\n _\n -- Claim that the sum equals `(aₙ + aₙ₊₂)/aₙ₊₁ * b - (aₙ * b - a₀)/aₙ₊₁`",
"before_state": "case refine_1.refine_1\nα : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\ni : ℕ\nx✝ : i ∈ Finset.range (n + 1)\n⊢ 0 ≤ ↑(a 0)",
"after_state": "No Goals!"
},
{
"line": "exact\n Nat.cast_nonneg\n _\n -- Claim that the sum equals `(aₙ + aₙ₊₂)/aₙ₊₁ * b - (aₙ * b - a₀)/aₙ₊₁`",
"before_state": "case refine_1.refine_2\nα : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\ni : ℕ\nx✝ : i ∈ Finset.range (n + 1)\n⊢ 0 ≤ ↑(a (n + 1))",
"after_state": "No Goals!"
},
{
"line": "exact\n Nat.cast_nonneg\n _\n -- Claim that the sum equals `(aₙ + aₙ₊₂)/aₙ₊₁ * b - (aₙ * b - a₀)/aₙ₊₁`",
"before_state": "case refine_2.refine_1\nα : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\ni : ℕ\nx✝ : i ∈ Finset.range (n + 1)\n⊢ 0 ≤ ↑(a i)",
"after_state": "No Goals!"
},
{
"line": "exact\n Nat.cast_nonneg\n _\n -- Claim that the sum equals `(aₙ + aₙ₊₂)/aₙ₊₁ * b - (aₙ * b - a₀)/aₙ₊₁`",
"before_state": "case refine_2.refine_2\nα : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\ni : ℕ\nx✝ : i ∈ Finset.range (n + 1)\n⊢ 0 ≤ ↑(a (i + 1))",
"after_state": "No Goals!"
},
{
"line": "refine\n ⟨(a n + a (n + 2)) / a (n + 1) * b - (a n * b - a 0) / a (n + 1), ?_, ?_⟩\n -- Check that this indeed equals the sum",
"before_state": "case succ.intro.intro\nα : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\n⊢ ∃ b,\n ↑b = ∑ i ∈ Finset.range (n + 1 + 1), ↑(a 0) * ↑(a (n + 1 + 1)) / (↑(a i) * ↑(a (i + 1))) ∧\n a (n + 1 + 1) ∣ a (n + 1) * b - a 0",
"after_state": "case succ.intro.intro.refine_1\nα : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\n⊢ ↑((a n + a (n + 2)) / a (n + 1) * b - (a n * b - a 0) / a (n + 1)) =\n ∑ i ∈ Finset.range (n + 1 + 1), ↑(a 0) * ↑(a (n + 1 + 1)) / (↑(a i) * ↑(a (i + 1)))\n---\ncase succ.intro.intro.refine_2\nα : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\n⊢ a (n + 1 + 1) ∣ a (n + 1) * ((a n + a (n + 2)) / a (n + 1) * b - (a n * b - a 0) / a (n + 1)) - a 0"
},
{
"line": "calc\n (((a n + a (n + 2)) / a (n + 1) * b - (a n * b - a 0) / a (n + 1) : ℕ) : α) =\n ((a n + a (n + 2)) / a (n + 1) * b - (a n * b - a 0) / a (n + 1)) :=\n by\n have : ((a (n + 1)) : α) ≠ 0 := ne_of_gt <| a_pos (n + 1) <| Nat.le_succ (n + 1)\n simp only [← Nat.cast_add]\n simp only [← Nat.cast_div ha this]\n simp only [← Nat.cast_mul]\n simp only [← Nat.cast_sub ha₀]\n simp only [← Nat.cast_div han this]\n rw [Nat.cast_sub (Nat.div_le_of_le_mul _)]\n rw [← mul_assoc]\n rw [Nat.mul_div_cancel' ha]\n rw [add_mul]\n exact tsub_le_self.trans (Nat.le_add_right _ _)\n _ = a (n + 2) / a (n + 1) * b + a 0 * a (n + 2) / (a (n + 1) * a (n + 2)) :=\n by\n rw [add_div]\n rw [add_mul]\n rw [sub_div]\n rw [mul_div_right_comm]\n rw [add_sub_sub_cancel]\n rw [mul_div_mul_right _ _ (a_pos _ le_rfl).ne']\n _ = ∑ i ∈ Finset.range (n + 2), (a 0 : α) * a (n + 2) / (a i * a (i + 1)) :=\n by\n rw [Finset.sum_range_succ]\n rw [hb]\n rw [Finset.mul_sum]\n congr; ext i\n rw [← mul_div_assoc]\n rw [← mul_div_right_comm]\n rw [mul_div_assoc]\n rw [mul_div_cancel_right₀ _ (a_pos _ <| Nat.le_succ _).ne']\n rw [mul_comm]\n -- Check the divisibility condition",
"before_state": "case succ.intro.intro.refine_1\nα : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\n⊢ ↑((a n + a (n + 2)) / a (n + 1) * b - (a n * b - a 0) / a (n + 1)) =\n ∑ i ∈ Finset.range (n + 1 + 1), ↑(a 0) * ↑(a (n + 1 + 1)) / (↑(a i) * ↑(a (i + 1)))",
"after_state": "No Goals!"
},
{
"line": "have : ((a (n + 1)) : α) ≠ 0 := ne_of_gt <| a_pos (n + 1) <| Nat.le_succ (n + 1)",
"before_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\n⊢ ↑((a n + a (n + 2)) / a (n + 1) * b - (a n * b - a 0) / a (n + 1)) =\n (↑(a n) + ↑(a (n + 2))) / ↑(a (n + 1)) * ↑b - (↑(a n) * ↑b - ↑(a 0)) / ↑(a (n + 1))",
"after_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\nthis : ↑(a (n + 1)) ≠ 0\n⊢ ↑((a n + a (n + 2)) / a (n + 1) * b - (a n * b - a 0) / a (n + 1)) =\n (↑(a n) + ↑(a (n + 2))) / ↑(a (n + 1)) * ↑b - (↑(a n) * ↑b - ↑(a 0)) / ↑(a (n + 1))"
},
{
"line": "refine_lift\n have : ((a (n + 1)) : α) ≠ 0 := ne_of_gt <| a_pos (n + 1) <| Nat.le_succ (n + 1);\n ?_",
"before_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\n⊢ ↑((a n + a (n + 2)) / a (n + 1) * b - (a n * b - a 0) / a (n + 1)) =\n (↑(a n) + ↑(a (n + 2))) / ↑(a (n + 1)) * ↑b - (↑(a n) * ↑b - ↑(a 0)) / ↑(a (n + 1))",
"after_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\nthis : ↑(a (n + 1)) ≠ 0\n⊢ ↑((a n + a (n + 2)) / a (n + 1) * b - (a n * b - a 0) / a (n + 1)) =\n (↑(a n) + ↑(a (n + 2))) / ↑(a (n + 1)) * ↑b - (↑(a n) * ↑b - ↑(a 0)) / ↑(a (n + 1))"
},
{
"line": "focus\n (refine\n no_implicit_lambda%\n (have : ((a (n + 1)) : α) ≠ 0 := ne_of_gt <| a_pos (n + 1) <| Nat.le_succ (n + 1);\n ?_);\n rotate_right)",
"before_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\n⊢ ↑((a n + a (n + 2)) / a (n + 1) * b - (a n * b - a 0) / a (n + 1)) =\n (↑(a n) + ↑(a (n + 2))) / ↑(a (n + 1)) * ↑b - (↑(a n) * ↑b - ↑(a 0)) / ↑(a (n + 1))",
"after_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\nthis : ↑(a (n + 1)) ≠ 0\n⊢ ↑((a n + a (n + 2)) / a (n + 1) * b - (a n * b - a 0) / a (n + 1)) =\n (↑(a n) + ↑(a (n + 2))) / ↑(a (n + 1)) * ↑b - (↑(a n) * ↑b - ↑(a 0)) / ↑(a (n + 1))"
},
{
"line": "refine\n no_implicit_lambda%\n (have : ((a (n + 1)) : α) ≠ 0 := ne_of_gt <| a_pos (n + 1) <| Nat.le_succ (n + 1);\n ?_)",
"before_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\n⊢ ↑((a n + a (n + 2)) / a (n + 1) * b - (a n * b - a 0) / a (n + 1)) =\n (↑(a n) + ↑(a (n + 2))) / ↑(a (n + 1)) * ↑b - (↑(a n) * ↑b - ↑(a 0)) / ↑(a (n + 1))",
"after_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\nthis : ↑(a (n + 1)) ≠ 0\n⊢ ↑((a n + a (n + 2)) / a (n + 1) * b - (a n * b - a 0) / a (n + 1)) =\n (↑(a n) + ↑(a (n + 2))) / ↑(a (n + 1)) * ↑b - (↑(a n) * ↑b - ↑(a 0)) / ↑(a (n + 1))"
},
{
"line": "rotate_right",
"before_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\nthis : ↑(a (n + 1)) ≠ 0\n⊢ ↑((a n + a (n + 2)) / a (n + 1) * b - (a n * b - a 0) / a (n + 1)) =\n (↑(a n) + ↑(a (n + 2))) / ↑(a (n + 1)) * ↑b - (↑(a n) * ↑b - ↑(a 0)) / ↑(a (n + 1))",
"after_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\nthis : ↑(a (n + 1)) ≠ 0\n⊢ ↑((a n + a (n + 2)) / a (n + 1) * b - (a n * b - a 0) / a (n + 1)) =\n (↑(a n) + ↑(a (n + 2))) / ↑(a (n + 1)) * ↑b - (↑(a n) * ↑b - ↑(a 0)) / ↑(a (n + 1))"
},
{
"line": "simp only [← Nat.cast_add]",
"before_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\nthis : ↑(a (n + 1)) ≠ 0\n⊢ ↑((a n + a (n + 2)) / a (n + 1) * b - (a n * b - a 0) / a (n + 1)) =\n (↑(a n) + ↑(a (n + 2))) / ↑(a (n + 1)) * ↑b - (↑(a n) * ↑b - ↑(a 0)) / ↑(a (n + 1))",
"after_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\nthis : ↑(a (n + 1)) ≠ 0\n⊢ ↑((a n + a (n + 2)) / a (n + 1) * b - (a n * b - a 0) / a (n + 1)) =\n ↑(a n + a (n + 2)) / ↑(a (n + 1)) * ↑b - (↑(a n) * ↑b - ↑(a 0)) / ↑(a (n + 1))"
},
{
"line": "simp only [← Nat.cast_div ha this]",
"before_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\nthis : ↑(a (n + 1)) ≠ 0\n⊢ ↑((a n + a (n + 2)) / a (n + 1) * b - (a n * b - a 0) / a (n + 1)) =\n ↑(a n + a (n + 2)) / ↑(a (n + 1)) * ↑b - (↑(a n) * ↑b - ↑(a 0)) / ↑(a (n + 1))",
"after_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\nthis : ↑(a (n + 1)) ≠ 0\n⊢ ↑((a n + a (n + 2)) / a (n + 1) * b - (a n * b - a 0) / a (n + 1)) =\n ↑((a n + a (n + 2)) / a (n + 1)) * ↑b - (↑(a n) * ↑b - ↑(a 0)) / ↑(a (n + 1))"
},
{
"line": "simp only [← Nat.cast_mul]",
"before_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\nthis : ↑(a (n + 1)) ≠ 0\n⊢ ↑((a n + a (n + 2)) / a (n + 1) * b - (a n * b - a 0) / a (n + 1)) =\n ↑((a n + a (n + 2)) / a (n + 1)) * ↑b - (↑(a n) * ↑b - ↑(a 0)) / ↑(a (n + 1))",
"after_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\nthis : ↑(a (n + 1)) ≠ 0\n⊢ ↑((a n + a (n + 2)) / a (n + 1) * b - (a n * b - a 0) / a (n + 1)) =\n ↑((a n + a (n + 2)) / a (n + 1) * b) - (↑(a n * b) - ↑(a 0)) / ↑(a (n + 1))"
},
{
"line": "simp only [← Nat.cast_sub ha₀]",
"before_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\nthis : ↑(a (n + 1)) ≠ 0\n⊢ ↑((a n + a (n + 2)) / a (n + 1) * b - (a n * b - a 0) / a (n + 1)) =\n ↑((a n + a (n + 2)) / a (n + 1) * b) - (↑(a n * b) - ↑(a 0)) / ↑(a (n + 1))",
"after_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\nthis : ↑(a (n + 1)) ≠ 0\n⊢ ↑((a n + a (n + 2)) / a (n + 1) * b - (a n * b - a 0) / a (n + 1)) =\n ↑((a n + a (n + 2)) / a (n + 1) * b) - ↑(a n * b - a 0) / ↑(a (n + 1))"
},
{
"line": "simp only [← Nat.cast_div han this]",
"before_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\nthis : ↑(a (n + 1)) ≠ 0\n⊢ ↑((a n + a (n + 2)) / a (n + 1) * b - (a n * b - a 0) / a (n + 1)) =\n ↑((a n + a (n + 2)) / a (n + 1) * b) - ↑(a n * b - a 0) / ↑(a (n + 1))",
"after_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\nthis : ↑(a (n + 1)) ≠ 0\n⊢ ↑((a n + a (n + 2)) / a (n + 1) * b - (a n * b - a 0) / a (n + 1)) =\n ↑((a n + a (n + 2)) / a (n + 1) * b) - ↑((a n * b - a 0) / a (n + 1))"
},
{
"line": "rw [Nat.cast_sub (Nat.div_le_of_le_mul _)]",
"before_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\nthis : ↑(a (n + 1)) ≠ 0\n⊢ ↑((a n + a (n + 2)) / a (n + 1) * b - (a n * b - a 0) / a (n + 1)) =\n ↑((a n + a (n + 2)) / a (n + 1) * b) - ↑((a n * b - a 0) / a (n + 1))",
"after_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\nthis : ↑(a (n + 1)) ≠ 0\n⊢ a n * b - a 0 ≤ a (n + 1) * ((a n + a (n + 2)) / a (n + 1) * b)"
},
{
"line": "rewrite [Nat.cast_sub (Nat.div_le_of_le_mul _)]",
"before_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\nthis : ↑(a (n + 1)) ≠ 0\n⊢ ↑((a n + a (n + 2)) / a (n + 1) * b - (a n * b - a 0) / a (n + 1)) =\n ↑((a n + a (n + 2)) / a (n + 1) * b) - ↑((a n * b - a 0) / a (n + 1))",
"after_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\nthis : ↑(a (n + 1)) ≠ 0\n⊢ ↑((a n + a (n + 2)) / a (n + 1) * b) - ↑((a n * b - a 0) / a (n + 1)) =\n ↑((a n + a (n + 2)) / a (n + 1) * b) - ↑((a n * b - a 0) / a (n + 1))\n---\nα : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\nthis : ↑(a (n + 1)) ≠ 0\n⊢ a n * b - a 0 ≤ a (n + 1) * ((a n + a (n + 2)) / a (n + 1) * b)"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\nthis : ↑(a (n + 1)) ≠ 0\n⊢ ↑((a n + a (n + 2)) / a (n + 1) * b) - ↑((a n * b - a 0) / a (n + 1)) =\n ↑((a n + a (n + 2)) / a (n + 1) * b) - ↑((a n * b - a 0) / a (n + 1))\n---\nα : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\nthis : ↑(a (n + 1)) ≠ 0\n⊢ a n * b - a 0 ≤ a (n + 1) * ((a n + a (n + 2)) / a (n + 1) * b)",
"after_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\nthis : ↑(a (n + 1)) ≠ 0\n⊢ a n * b - a 0 ≤ a (n + 1) * ((a n + a (n + 2)) / a (n + 1) * b)"
},
{
"line": "try (with_reducible rfl)",
"before_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\nthis : ↑(a (n + 1)) ≠ 0\n⊢ ↑((a n + a (n + 2)) / a (n + 1) * b) - ↑((a n * b - a 0) / a (n + 1)) =\n ↑((a n + a (n + 2)) / a (n + 1) * b) - ↑((a n * b - a 0) / a (n + 1))\n---\nα : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\nthis : ↑(a (n + 1)) ≠ 0\n⊢ a n * b - a 0 ≤ a (n + 1) * ((a n + a (n + 2)) / a (n + 1) * b)",
"after_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\nthis : ↑(a (n + 1)) ≠ 0\n⊢ a n * b - a 0 ≤ a (n + 1) * ((a n + a (n + 2)) / a (n + 1) * b)"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\nthis : ↑(a (n + 1)) ≠ 0\n⊢ ↑((a n + a (n + 2)) / a (n + 1) * b) - ↑((a n * b - a 0) / a (n + 1)) =\n ↑((a n + a (n + 2)) / a (n + 1) * b) - ↑((a n * b - a 0) / a (n + 1))\n---\nα : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\nthis : ↑(a (n + 1)) ≠ 0\n⊢ a n * b - a 0 ≤ a (n + 1) * ((a n + a (n + 2)) / a (n + 1) * b)",
"after_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\nthis : ↑(a (n + 1)) ≠ 0\n⊢ a n * b - a 0 ≤ a (n + 1) * ((a n + a (n + 2)) / a (n + 1) * b)"
},
{
"line": "with_reducible rfl",
"before_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\nthis : ↑(a (n + 1)) ≠ 0\n⊢ ↑((a n + a (n + 2)) / a (n + 1) * b) - ↑((a n * b - a 0) / a (n + 1)) =\n ↑((a n + a (n + 2)) / a (n + 1) * b) - ↑((a n * b - a 0) / a (n + 1))\n---\nα : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\nthis : ↑(a (n + 1)) ≠ 0\n⊢ a n * b - a 0 ≤ a (n + 1) * ((a n + a (n + 2)) / a (n + 1) * b)",
"after_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\nthis : ↑(a (n + 1)) ≠ 0\n⊢ a n * b - a 0 ≤ a (n + 1) * ((a n + a (n + 2)) / a (n + 1) * b)"
},
{
"line": "rfl",
"before_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\nthis : ↑(a (n + 1)) ≠ 0\n⊢ ↑((a n + a (n + 2)) / a (n + 1) * b) - ↑((a n * b - a 0) / a (n + 1)) =\n ↑((a n + a (n + 2)) / a (n + 1) * b) - ↑((a n * b - a 0) / a (n + 1))\n---\nα : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\nthis : ↑(a (n + 1)) ≠ 0\n⊢ a n * b - a 0 ≤ a (n + 1) * ((a n + a (n + 2)) / a (n + 1) * b)",
"after_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\nthis : ↑(a (n + 1)) ≠ 0\n⊢ a n * b - a 0 ≤ a (n + 1) * ((a n + a (n + 2)) / a (n + 1) * b)"
},
{
"line": "eq_refl",
"before_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\nthis : ↑(a (n + 1)) ≠ 0\n⊢ ↑((a n + a (n + 2)) / a (n + 1) * b) - ↑((a n * b - a 0) / a (n + 1)) =\n ↑((a n + a (n + 2)) / a (n + 1) * b) - ↑((a n * b - a 0) / a (n + 1))\n---\nα : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\nthis : ↑(a (n + 1)) ≠ 0\n⊢ a n * b - a 0 ≤ a (n + 1) * ((a n + a (n + 2)) / a (n + 1) * b)",
"after_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\nthis : ↑(a (n + 1)) ≠ 0\n⊢ a n * b - a 0 ≤ a (n + 1) * ((a n + a (n + 2)) / a (n + 1) * b)"
},
{
"line": "rw [← mul_assoc]",
"before_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\nthis : ↑(a (n + 1)) ≠ 0\n⊢ a n * b - a 0 ≤ a (n + 1) * ((a n + a (n + 2)) / a (n + 1) * b)",
"after_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\nthis : ↑(a (n + 1)) ≠ 0\n⊢ a n * b - a 0 ≤ a (n + 1) * ((a n + a (n + 2)) / a (n + 1)) * b"
},
{
"line": "rewrite [← mul_assoc]",
"before_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\nthis : ↑(a (n + 1)) ≠ 0\n⊢ a n * b - a 0 ≤ a (n + 1) * ((a n + a (n + 2)) / a (n + 1) * b)",
"after_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\nthis : ↑(a (n + 1)) ≠ 0\n⊢ a n * b - a 0 ≤ a (n + 1) * ((a n + a (n + 2)) / a (n + 1)) * b"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\nthis : ↑(a (n + 1)) ≠ 0\n⊢ a n * b - a 0 ≤ a (n + 1) * ((a n + a (n + 2)) / a (n + 1)) * b",
"after_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\nthis : ↑(a (n + 1)) ≠ 0\n⊢ a n * b - a 0 ≤ a (n + 1) * ((a n + a (n + 2)) / a (n + 1)) * b"
},
{
"line": "try (with_reducible rfl)",
"before_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\nthis : ↑(a (n + 1)) ≠ 0\n⊢ a n * b - a 0 ≤ a (n + 1) * ((a n + a (n + 2)) / a (n + 1)) * b",
"after_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\nthis : ↑(a (n + 1)) ≠ 0\n⊢ a n * b - a 0 ≤ a (n + 1) * ((a n + a (n + 2)) / a (n + 1)) * b"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\nthis : ↑(a (n + 1)) ≠ 0\n⊢ a n * b - a 0 ≤ a (n + 1) * ((a n + a (n + 2)) / a (n + 1)) * b",
"after_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\nthis : ↑(a (n + 1)) ≠ 0\n⊢ a n * b - a 0 ≤ a (n + 1) * ((a n + a (n + 2)) / a (n + 1)) * b"
},
{
"line": "with_reducible rfl",
"before_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\nthis : ↑(a (n + 1)) ≠ 0\n⊢ a n * b - a 0 ≤ a (n + 1) * ((a n + a (n + 2)) / a (n + 1)) * b",
"after_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\nthis : ↑(a (n + 1)) ≠ 0\n⊢ a n * b - a 0 ≤ a (n + 1) * ((a n + a (n + 2)) / a (n + 1)) * b"
},
{
"line": "rfl",
"before_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\nthis : ↑(a (n + 1)) ≠ 0\n⊢ a n * b - a 0 ≤ a (n + 1) * ((a n + a (n + 2)) / a (n + 1)) * b",
"after_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\nthis : ↑(a (n + 1)) ≠ 0\n⊢ a n * b - a 0 ≤ a (n + 1) * ((a n + a (n + 2)) / a (n + 1)) * b"
},
{
"line": "apply_rfl",
"before_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\nthis : ↑(a (n + 1)) ≠ 0\n⊢ a n * b - a 0 ≤ a (n + 1) * ((a n + a (n + 2)) / a (n + 1)) * b",
"after_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\nthis : ↑(a (n + 1)) ≠ 0\n⊢ a n * b - a 0 ≤ a (n + 1) * ((a n + a (n + 2)) / a (n + 1)) * b"
},
{
"line": "skip",
"before_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\nthis : ↑(a (n + 1)) ≠ 0\n⊢ a n * b - a 0 ≤ a (n + 1) * ((a n + a (n + 2)) / a (n + 1)) * b",
"after_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\nthis : ↑(a (n + 1)) ≠ 0\n⊢ a n * b - a 0 ≤ a (n + 1) * ((a n + a (n + 2)) / a (n + 1)) * b"
},
{
"line": "rw [Nat.mul_div_cancel' ha]",
"before_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\nthis : ↑(a (n + 1)) ≠ 0\n⊢ a n * b - a 0 ≤ a (n + 1) * ((a n + a (n + 2)) / a (n + 1)) * b",
"after_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\nthis : ↑(a (n + 1)) ≠ 0\n⊢ a n * b - a 0 ≤ (a n + a (n + 2)) * b"
},
{
"line": "rewrite [Nat.mul_div_cancel' ha]",
"before_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\nthis : ↑(a (n + 1)) ≠ 0\n⊢ a n * b - a 0 ≤ a (n + 1) * ((a n + a (n + 2)) / a (n + 1)) * b",
"after_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\nthis : ↑(a (n + 1)) ≠ 0\n⊢ a n * b - a 0 ≤ (a n + a (n + 2)) * b"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\nthis : ↑(a (n + 1)) ≠ 0\n⊢ a n * b - a 0 ≤ (a n + a (n + 2)) * b",
"after_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\nthis : ↑(a (n + 1)) ≠ 0\n⊢ a n * b - a 0 ≤ (a n + a (n + 2)) * b"
},
{
"line": "try (with_reducible rfl)",
"before_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\nthis : ↑(a (n + 1)) ≠ 0\n⊢ a n * b - a 0 ≤ (a n + a (n + 2)) * b",
"after_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\nthis : ↑(a (n + 1)) ≠ 0\n⊢ a n * b - a 0 ≤ (a n + a (n + 2)) * b"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\nthis : ↑(a (n + 1)) ≠ 0\n⊢ a n * b - a 0 ≤ (a n + a (n + 2)) * b",
"after_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\nthis : ↑(a (n + 1)) ≠ 0\n⊢ a n * b - a 0 ≤ (a n + a (n + 2)) * b"
},
{
"line": "with_reducible rfl",
"before_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\nthis : ↑(a (n + 1)) ≠ 0\n⊢ a n * b - a 0 ≤ (a n + a (n + 2)) * b",
"after_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\nthis : ↑(a (n + 1)) ≠ 0\n⊢ a n * b - a 0 ≤ (a n + a (n + 2)) * b"
},
{
"line": "rfl",
"before_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\nthis : ↑(a (n + 1)) ≠ 0\n⊢ a n * b - a 0 ≤ (a n + a (n + 2)) * b",
"after_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\nthis : ↑(a (n + 1)) ≠ 0\n⊢ a n * b - a 0 ≤ (a n + a (n + 2)) * b"
},
{
"line": "apply_rfl",
"before_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\nthis : ↑(a (n + 1)) ≠ 0\n⊢ a n * b - a 0 ≤ (a n + a (n + 2)) * b",
"after_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\nthis : ↑(a (n + 1)) ≠ 0\n⊢ a n * b - a 0 ≤ (a n + a (n + 2)) * b"
},
{
"line": "skip",
"before_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\nthis : ↑(a (n + 1)) ≠ 0\n⊢ a n * b - a 0 ≤ (a n + a (n + 2)) * b",
"after_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\nthis : ↑(a (n + 1)) ≠ 0\n⊢ a n * b - a 0 ≤ (a n + a (n + 2)) * b"
},
{
"line": "rw [add_mul]",
"before_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\nthis : ↑(a (n + 1)) ≠ 0\n⊢ a n * b - a 0 ≤ (a n + a (n + 2)) * b",
"after_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\nthis : ↑(a (n + 1)) ≠ 0\n⊢ a n * b - a 0 ≤ a n * b + a (n + 2) * b"
},
{
"line": "rewrite [add_mul]",
"before_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\nthis : ↑(a (n + 1)) ≠ 0\n⊢ a n * b - a 0 ≤ (a n + a (n + 2)) * b",
"after_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\nthis : ↑(a (n + 1)) ≠ 0\n⊢ a n * b - a 0 ≤ a n * b + a (n + 2) * b"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\nthis : ↑(a (n + 1)) ≠ 0\n⊢ a n * b - a 0 ≤ a n * b + a (n + 2) * b",
"after_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\nthis : ↑(a (n + 1)) ≠ 0\n⊢ a n * b - a 0 ≤ a n * b + a (n + 2) * b"
},
{
"line": "try (with_reducible rfl)",
"before_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\nthis : ↑(a (n + 1)) ≠ 0\n⊢ a n * b - a 0 ≤ a n * b + a (n + 2) * b",
"after_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\nthis : ↑(a (n + 1)) ≠ 0\n⊢ a n * b - a 0 ≤ a n * b + a (n + 2) * b"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\nthis : ↑(a (n + 1)) ≠ 0\n⊢ a n * b - a 0 ≤ a n * b + a (n + 2) * b",
"after_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\nthis : ↑(a (n + 1)) ≠ 0\n⊢ a n * b - a 0 ≤ a n * b + a (n + 2) * b"
},
{
"line": "with_reducible rfl",
"before_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\nthis : ↑(a (n + 1)) ≠ 0\n⊢ a n * b - a 0 ≤ a n * b + a (n + 2) * b",
"after_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\nthis : ↑(a (n + 1)) ≠ 0\n⊢ a n * b - a 0 ≤ a n * b + a (n + 2) * b"
},
{
"line": "rfl",
"before_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\nthis : ↑(a (n + 1)) ≠ 0\n⊢ a n * b - a 0 ≤ a n * b + a (n + 2) * b",
"after_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\nthis : ↑(a (n + 1)) ≠ 0\n⊢ a n * b - a 0 ≤ a n * b + a (n + 2) * b"
},
{
"line": "apply_rfl",
"before_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\nthis : ↑(a (n + 1)) ≠ 0\n⊢ a n * b - a 0 ≤ a n * b + a (n + 2) * b",
"after_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\nthis : ↑(a (n + 1)) ≠ 0\n⊢ a n * b - a 0 ≤ a n * b + a (n + 2) * b"
},
{
"line": "skip",
"before_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\nthis : ↑(a (n + 1)) ≠ 0\n⊢ a n * b - a 0 ≤ a n * b + a (n + 2) * b",
"after_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\nthis : ↑(a (n + 1)) ≠ 0\n⊢ a n * b - a 0 ≤ a n * b + a (n + 2) * b"
},
{
"line": "exact tsub_le_self.trans (Nat.le_add_right _ _)",
"before_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\nthis : ↑(a (n + 1)) ≠ 0\n⊢ a n * b - a 0 ≤ a n * b + a (n + 2) * b",
"after_state": "No Goals!"
},
{
"line": "rw [add_div]",
"before_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\n⊢ (↑(a n) + ↑(a (n + 2))) / ↑(a (n + 1)) * ↑b - (↑(a n) * ↑b - ↑(a 0)) / ↑(a (n + 1)) =\n ↑(a (n + 2)) / ↑(a (n + 1)) * ↑b + ↑(a 0) * ↑(a (n + 2)) / (↑(a (n + 1)) * ↑(a (n + 2)))",
"after_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\n⊢ (↑(a n) / ↑(a (n + 1)) + ↑(a (n + 2)) / ↑(a (n + 1))) * ↑b - (↑(a n) * ↑b - ↑(a 0)) / ↑(a (n + 1)) =\n ↑(a (n + 2)) / ↑(a (n + 1)) * ↑b + ↑(a 0) * ↑(a (n + 2)) / (↑(a (n + 1)) * ↑(a (n + 2)))"
},
{
"line": "rewrite [add_div]",
"before_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\n⊢ (↑(a n) + ↑(a (n + 2))) / ↑(a (n + 1)) * ↑b - (↑(a n) * ↑b - ↑(a 0)) / ↑(a (n + 1)) =\n ↑(a (n + 2)) / ↑(a (n + 1)) * ↑b + ↑(a 0) * ↑(a (n + 2)) / (↑(a (n + 1)) * ↑(a (n + 2)))",
"after_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\n⊢ (↑(a n) / ↑(a (n + 1)) + ↑(a (n + 2)) / ↑(a (n + 1))) * ↑b - (↑(a n) * ↑b - ↑(a 0)) / ↑(a (n + 1)) =\n ↑(a (n + 2)) / ↑(a (n + 1)) * ↑b + ↑(a 0) * ↑(a (n + 2)) / (↑(a (n + 1)) * ↑(a (n + 2)))"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\n⊢ (↑(a n) / ↑(a (n + 1)) + ↑(a (n + 2)) / ↑(a (n + 1))) * ↑b - (↑(a n) * ↑b - ↑(a 0)) / ↑(a (n + 1)) =\n ↑(a (n + 2)) / ↑(a (n + 1)) * ↑b + ↑(a 0) * ↑(a (n + 2)) / (↑(a (n + 1)) * ↑(a (n + 2)))",
"after_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\n⊢ (↑(a n) / ↑(a (n + 1)) + ↑(a (n + 2)) / ↑(a (n + 1))) * ↑b - (↑(a n) * ↑b - ↑(a 0)) / ↑(a (n + 1)) =\n ↑(a (n + 2)) / ↑(a (n + 1)) * ↑b + ↑(a 0) * ↑(a (n + 2)) / (↑(a (n + 1)) * ↑(a (n + 2)))"
},
{
"line": "try (with_reducible rfl)",
"before_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\n⊢ (↑(a n) / ↑(a (n + 1)) + ↑(a (n + 2)) / ↑(a (n + 1))) * ↑b - (↑(a n) * ↑b - ↑(a 0)) / ↑(a (n + 1)) =\n ↑(a (n + 2)) / ↑(a (n + 1)) * ↑b + ↑(a 0) * ↑(a (n + 2)) / (↑(a (n + 1)) * ↑(a (n + 2)))",
"after_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\n⊢ (↑(a n) / ↑(a (n + 1)) + ↑(a (n + 2)) / ↑(a (n + 1))) * ↑b - (↑(a n) * ↑b - ↑(a 0)) / ↑(a (n + 1)) =\n ↑(a (n + 2)) / ↑(a (n + 1)) * ↑b + ↑(a 0) * ↑(a (n + 2)) / (↑(a (n + 1)) * ↑(a (n + 2)))"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\n⊢ (↑(a n) / ↑(a (n + 1)) + ↑(a (n + 2)) / ↑(a (n + 1))) * ↑b - (↑(a n) * ↑b - ↑(a 0)) / ↑(a (n + 1)) =\n ↑(a (n + 2)) / ↑(a (n + 1)) * ↑b + ↑(a 0) * ↑(a (n + 2)) / (↑(a (n + 1)) * ↑(a (n + 2)))",
"after_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\n⊢ (↑(a n) / ↑(a (n + 1)) + ↑(a (n + 2)) / ↑(a (n + 1))) * ↑b - (↑(a n) * ↑b - ↑(a 0)) / ↑(a (n + 1)) =\n ↑(a (n + 2)) / ↑(a (n + 1)) * ↑b + ↑(a 0) * ↑(a (n + 2)) / (↑(a (n + 1)) * ↑(a (n + 2)))"
},
{
"line": "with_reducible rfl",
"before_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\n⊢ (↑(a n) / ↑(a (n + 1)) + ↑(a (n + 2)) / ↑(a (n + 1))) * ↑b - (↑(a n) * ↑b - ↑(a 0)) / ↑(a (n + 1)) =\n ↑(a (n + 2)) / ↑(a (n + 1)) * ↑b + ↑(a 0) * ↑(a (n + 2)) / (↑(a (n + 1)) * ↑(a (n + 2)))",
"after_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\n⊢ (↑(a n) / ↑(a (n + 1)) + ↑(a (n + 2)) / ↑(a (n + 1))) * ↑b - (↑(a n) * ↑b - ↑(a 0)) / ↑(a (n + 1)) =\n ↑(a (n + 2)) / ↑(a (n + 1)) * ↑b + ↑(a 0) * ↑(a (n + 2)) / (↑(a (n + 1)) * ↑(a (n + 2)))"
},
{
"line": "rfl",
"before_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\n⊢ (↑(a n) / ↑(a (n + 1)) + ↑(a (n + 2)) / ↑(a (n + 1))) * ↑b - (↑(a n) * ↑b - ↑(a 0)) / ↑(a (n + 1)) =\n ↑(a (n + 2)) / ↑(a (n + 1)) * ↑b + ↑(a 0) * ↑(a (n + 2)) / (↑(a (n + 1)) * ↑(a (n + 2)))",
"after_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\n⊢ (↑(a n) / ↑(a (n + 1)) + ↑(a (n + 2)) / ↑(a (n + 1))) * ↑b - (↑(a n) * ↑b - ↑(a 0)) / ↑(a (n + 1)) =\n ↑(a (n + 2)) / ↑(a (n + 1)) * ↑b + ↑(a 0) * ↑(a (n + 2)) / (↑(a (n + 1)) * ↑(a (n + 2)))"
},
{
"line": "apply_rfl",
"before_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\n⊢ (↑(a n) / ↑(a (n + 1)) + ↑(a (n + 2)) / ↑(a (n + 1))) * ↑b - (↑(a n) * ↑b - ↑(a 0)) / ↑(a (n + 1)) =\n ↑(a (n + 2)) / ↑(a (n + 1)) * ↑b + ↑(a 0) * ↑(a (n + 2)) / (↑(a (n + 1)) * ↑(a (n + 2)))",
"after_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\n⊢ (↑(a n) / ↑(a (n + 1)) + ↑(a (n + 2)) / ↑(a (n + 1))) * ↑b - (↑(a n) * ↑b - ↑(a 0)) / ↑(a (n + 1)) =\n ↑(a (n + 2)) / ↑(a (n + 1)) * ↑b + ↑(a 0) * ↑(a (n + 2)) / (↑(a (n + 1)) * ↑(a (n + 2)))"
},
{
"line": "skip",
"before_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\n⊢ (↑(a n) / ↑(a (n + 1)) + ↑(a (n + 2)) / ↑(a (n + 1))) * ↑b - (↑(a n) * ↑b - ↑(a 0)) / ↑(a (n + 1)) =\n ↑(a (n + 2)) / ↑(a (n + 1)) * ↑b + ↑(a 0) * ↑(a (n + 2)) / (↑(a (n + 1)) * ↑(a (n + 2)))",
"after_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\n⊢ (↑(a n) / ↑(a (n + 1)) + ↑(a (n + 2)) / ↑(a (n + 1))) * ↑b - (↑(a n) * ↑b - ↑(a 0)) / ↑(a (n + 1)) =\n ↑(a (n + 2)) / ↑(a (n + 1)) * ↑b + ↑(a 0) * ↑(a (n + 2)) / (↑(a (n + 1)) * ↑(a (n + 2)))"
},
{
"line": "rw [add_mul]",
"before_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\n⊢ (↑(a n) / ↑(a (n + 1)) + ↑(a (n + 2)) / ↑(a (n + 1))) * ↑b - (↑(a n) * ↑b - ↑(a 0)) / ↑(a (n + 1)) =\n ↑(a (n + 2)) / ↑(a (n + 1)) * ↑b + ↑(a 0) * ↑(a (n + 2)) / (↑(a (n + 1)) * ↑(a (n + 2)))",
"after_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\n⊢ ↑(a n) / ↑(a (n + 1)) * ↑b + ↑(a (n + 2)) / ↑(a (n + 1)) * ↑b - (↑(a n) * ↑b - ↑(a 0)) / ↑(a (n + 1)) =\n ↑(a (n + 2)) / ↑(a (n + 1)) * ↑b + ↑(a 0) * ↑(a (n + 2)) / (↑(a (n + 1)) * ↑(a (n + 2)))"
},
{
"line": "rewrite [add_mul]",
"before_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\n⊢ (↑(a n) / ↑(a (n + 1)) + ↑(a (n + 2)) / ↑(a (n + 1))) * ↑b - (↑(a n) * ↑b - ↑(a 0)) / ↑(a (n + 1)) =\n ↑(a (n + 2)) / ↑(a (n + 1)) * ↑b + ↑(a 0) * ↑(a (n + 2)) / (↑(a (n + 1)) * ↑(a (n + 2)))",
"after_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\n⊢ ↑(a n) / ↑(a (n + 1)) * ↑b + ↑(a (n + 2)) / ↑(a (n + 1)) * ↑b - (↑(a n) * ↑b - ↑(a 0)) / ↑(a (n + 1)) =\n ↑(a (n + 2)) / ↑(a (n + 1)) * ↑b + ↑(a 0) * ↑(a (n + 2)) / (↑(a (n + 1)) * ↑(a (n + 2)))"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\n⊢ ↑(a n) / ↑(a (n + 1)) * ↑b + ↑(a (n + 2)) / ↑(a (n + 1)) * ↑b - (↑(a n) * ↑b - ↑(a 0)) / ↑(a (n + 1)) =\n ↑(a (n + 2)) / ↑(a (n + 1)) * ↑b + ↑(a 0) * ↑(a (n + 2)) / (↑(a (n + 1)) * ↑(a (n + 2)))",
"after_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\n⊢ ↑(a n) / ↑(a (n + 1)) * ↑b + ↑(a (n + 2)) / ↑(a (n + 1)) * ↑b - (↑(a n) * ↑b - ↑(a 0)) / ↑(a (n + 1)) =\n ↑(a (n + 2)) / ↑(a (n + 1)) * ↑b + ↑(a 0) * ↑(a (n + 2)) / (↑(a (n + 1)) * ↑(a (n + 2)))"
},
{
"line": "try (with_reducible rfl)",
"before_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\n⊢ ↑(a n) / ↑(a (n + 1)) * ↑b + ↑(a (n + 2)) / ↑(a (n + 1)) * ↑b - (↑(a n) * ↑b - ↑(a 0)) / ↑(a (n + 1)) =\n ↑(a (n + 2)) / ↑(a (n + 1)) * ↑b + ↑(a 0) * ↑(a (n + 2)) / (↑(a (n + 1)) * ↑(a (n + 2)))",
"after_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\n⊢ ↑(a n) / ↑(a (n + 1)) * ↑b + ↑(a (n + 2)) / ↑(a (n + 1)) * ↑b - (↑(a n) * ↑b - ↑(a 0)) / ↑(a (n + 1)) =\n ↑(a (n + 2)) / ↑(a (n + 1)) * ↑b + ↑(a 0) * ↑(a (n + 2)) / (↑(a (n + 1)) * ↑(a (n + 2)))"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\n⊢ ↑(a n) / ↑(a (n + 1)) * ↑b + ↑(a (n + 2)) / ↑(a (n + 1)) * ↑b - (↑(a n) * ↑b - ↑(a 0)) / ↑(a (n + 1)) =\n ↑(a (n + 2)) / ↑(a (n + 1)) * ↑b + ↑(a 0) * ↑(a (n + 2)) / (↑(a (n + 1)) * ↑(a (n + 2)))",
"after_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\n⊢ ↑(a n) / ↑(a (n + 1)) * ↑b + ↑(a (n + 2)) / ↑(a (n + 1)) * ↑b - (↑(a n) * ↑b - ↑(a 0)) / ↑(a (n + 1)) =\n ↑(a (n + 2)) / ↑(a (n + 1)) * ↑b + ↑(a 0) * ↑(a (n + 2)) / (↑(a (n + 1)) * ↑(a (n + 2)))"
},
{
"line": "with_reducible rfl",
"before_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\n⊢ ↑(a n) / ↑(a (n + 1)) * ↑b + ↑(a (n + 2)) / ↑(a (n + 1)) * ↑b - (↑(a n) * ↑b - ↑(a 0)) / ↑(a (n + 1)) =\n ↑(a (n + 2)) / ↑(a (n + 1)) * ↑b + ↑(a 0) * ↑(a (n + 2)) / (↑(a (n + 1)) * ↑(a (n + 2)))",
"after_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\n⊢ ↑(a n) / ↑(a (n + 1)) * ↑b + ↑(a (n + 2)) / ↑(a (n + 1)) * ↑b - (↑(a n) * ↑b - ↑(a 0)) / ↑(a (n + 1)) =\n ↑(a (n + 2)) / ↑(a (n + 1)) * ↑b + ↑(a 0) * ↑(a (n + 2)) / (↑(a (n + 1)) * ↑(a (n + 2)))"
},
{
"line": "rfl",
"before_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\n⊢ ↑(a n) / ↑(a (n + 1)) * ↑b + ↑(a (n + 2)) / ↑(a (n + 1)) * ↑b - (↑(a n) * ↑b - ↑(a 0)) / ↑(a (n + 1)) =\n ↑(a (n + 2)) / ↑(a (n + 1)) * ↑b + ↑(a 0) * ↑(a (n + 2)) / (↑(a (n + 1)) * ↑(a (n + 2)))",
"after_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\n⊢ ↑(a n) / ↑(a (n + 1)) * ↑b + ↑(a (n + 2)) / ↑(a (n + 1)) * ↑b - (↑(a n) * ↑b - ↑(a 0)) / ↑(a (n + 1)) =\n ↑(a (n + 2)) / ↑(a (n + 1)) * ↑b + ↑(a 0) * ↑(a (n + 2)) / (↑(a (n + 1)) * ↑(a (n + 2)))"
},
{
"line": "apply_rfl",
"before_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\n⊢ ↑(a n) / ↑(a (n + 1)) * ↑b + ↑(a (n + 2)) / ↑(a (n + 1)) * ↑b - (↑(a n) * ↑b - ↑(a 0)) / ↑(a (n + 1)) =\n ↑(a (n + 2)) / ↑(a (n + 1)) * ↑b + ↑(a 0) * ↑(a (n + 2)) / (↑(a (n + 1)) * ↑(a (n + 2)))",
"after_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\n⊢ ↑(a n) / ↑(a (n + 1)) * ↑b + ↑(a (n + 2)) / ↑(a (n + 1)) * ↑b - (↑(a n) * ↑b - ↑(a 0)) / ↑(a (n + 1)) =\n ↑(a (n + 2)) / ↑(a (n + 1)) * ↑b + ↑(a 0) * ↑(a (n + 2)) / (↑(a (n + 1)) * ↑(a (n + 2)))"
},
{
"line": "skip",
"before_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\n⊢ ↑(a n) / ↑(a (n + 1)) * ↑b + ↑(a (n + 2)) / ↑(a (n + 1)) * ↑b - (↑(a n) * ↑b - ↑(a 0)) / ↑(a (n + 1)) =\n ↑(a (n + 2)) / ↑(a (n + 1)) * ↑b + ↑(a 0) * ↑(a (n + 2)) / (↑(a (n + 1)) * ↑(a (n + 2)))",
"after_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\n⊢ ↑(a n) / ↑(a (n + 1)) * ↑b + ↑(a (n + 2)) / ↑(a (n + 1)) * ↑b - (↑(a n) * ↑b - ↑(a 0)) / ↑(a (n + 1)) =\n ↑(a (n + 2)) / ↑(a (n + 1)) * ↑b + ↑(a 0) * ↑(a (n + 2)) / (↑(a (n + 1)) * ↑(a (n + 2)))"
},
{
"line": "rw [sub_div]",
"before_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\n⊢ ↑(a n) / ↑(a (n + 1)) * ↑b + ↑(a (n + 2)) / ↑(a (n + 1)) * ↑b - (↑(a n) * ↑b - ↑(a 0)) / ↑(a (n + 1)) =\n ↑(a (n + 2)) / ↑(a (n + 1)) * ↑b + ↑(a 0) * ↑(a (n + 2)) / (↑(a (n + 1)) * ↑(a (n + 2)))",
"after_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\n⊢ ↑(a n) / ↑(a (n + 1)) * ↑b + ↑(a (n + 2)) / ↑(a (n + 1)) * ↑b - (↑(a n) * ↑b / ↑(a (n + 1)) - ↑(a 0) / ↑(a (n + 1))) =\n ↑(a (n + 2)) / ↑(a (n + 1)) * ↑b + ↑(a 0) * ↑(a (n + 2)) / (↑(a (n + 1)) * ↑(a (n + 2)))"
},
{
"line": "rewrite [sub_div]",
"before_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\n⊢ ↑(a n) / ↑(a (n + 1)) * ↑b + ↑(a (n + 2)) / ↑(a (n + 1)) * ↑b - (↑(a n) * ↑b - ↑(a 0)) / ↑(a (n + 1)) =\n ↑(a (n + 2)) / ↑(a (n + 1)) * ↑b + ↑(a 0) * ↑(a (n + 2)) / (↑(a (n + 1)) * ↑(a (n + 2)))",
"after_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\n⊢ ↑(a n) / ↑(a (n + 1)) * ↑b + ↑(a (n + 2)) / ↑(a (n + 1)) * ↑b - (↑(a n) * ↑b / ↑(a (n + 1)) - ↑(a 0) / ↑(a (n + 1))) =\n ↑(a (n + 2)) / ↑(a (n + 1)) * ↑b + ↑(a 0) * ↑(a (n + 2)) / (↑(a (n + 1)) * ↑(a (n + 2)))"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\n⊢ ↑(a n) / ↑(a (n + 1)) * ↑b + ↑(a (n + 2)) / ↑(a (n + 1)) * ↑b - (↑(a n) * ↑b / ↑(a (n + 1)) - ↑(a 0) / ↑(a (n + 1))) =\n ↑(a (n + 2)) / ↑(a (n + 1)) * ↑b + ↑(a 0) * ↑(a (n + 2)) / (↑(a (n + 1)) * ↑(a (n + 2)))",
"after_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\n⊢ ↑(a n) / ↑(a (n + 1)) * ↑b + ↑(a (n + 2)) / ↑(a (n + 1)) * ↑b - (↑(a n) * ↑b / ↑(a (n + 1)) - ↑(a 0) / ↑(a (n + 1))) =\n ↑(a (n + 2)) / ↑(a (n + 1)) * ↑b + ↑(a 0) * ↑(a (n + 2)) / (↑(a (n + 1)) * ↑(a (n + 2)))"
},
{
"line": "try (with_reducible rfl)",
"before_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\n⊢ ↑(a n) / ↑(a (n + 1)) * ↑b + ↑(a (n + 2)) / ↑(a (n + 1)) * ↑b - (↑(a n) * ↑b / ↑(a (n + 1)) - ↑(a 0) / ↑(a (n + 1))) =\n ↑(a (n + 2)) / ↑(a (n + 1)) * ↑b + ↑(a 0) * ↑(a (n + 2)) / (↑(a (n + 1)) * ↑(a (n + 2)))",
"after_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\n⊢ ↑(a n) / ↑(a (n + 1)) * ↑b + ↑(a (n + 2)) / ↑(a (n + 1)) * ↑b - (↑(a n) * ↑b / ↑(a (n + 1)) - ↑(a 0) / ↑(a (n + 1))) =\n ↑(a (n + 2)) / ↑(a (n + 1)) * ↑b + ↑(a 0) * ↑(a (n + 2)) / (↑(a (n + 1)) * ↑(a (n + 2)))"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\n⊢ ↑(a n) / ↑(a (n + 1)) * ↑b + ↑(a (n + 2)) / ↑(a (n + 1)) * ↑b - (↑(a n) * ↑b / ↑(a (n + 1)) - ↑(a 0) / ↑(a (n + 1))) =\n ↑(a (n + 2)) / ↑(a (n + 1)) * ↑b + ↑(a 0) * ↑(a (n + 2)) / (↑(a (n + 1)) * ↑(a (n + 2)))",
"after_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\n⊢ ↑(a n) / ↑(a (n + 1)) * ↑b + ↑(a (n + 2)) / ↑(a (n + 1)) * ↑b - (↑(a n) * ↑b / ↑(a (n + 1)) - ↑(a 0) / ↑(a (n + 1))) =\n ↑(a (n + 2)) / ↑(a (n + 1)) * ↑b + ↑(a 0) * ↑(a (n + 2)) / (↑(a (n + 1)) * ↑(a (n + 2)))"
},
{
"line": "with_reducible rfl",
"before_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\n⊢ ↑(a n) / ↑(a (n + 1)) * ↑b + ↑(a (n + 2)) / ↑(a (n + 1)) * ↑b - (↑(a n) * ↑b / ↑(a (n + 1)) - ↑(a 0) / ↑(a (n + 1))) =\n ↑(a (n + 2)) / ↑(a (n + 1)) * ↑b + ↑(a 0) * ↑(a (n + 2)) / (↑(a (n + 1)) * ↑(a (n + 2)))",
"after_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\n⊢ ↑(a n) / ↑(a (n + 1)) * ↑b + ↑(a (n + 2)) / ↑(a (n + 1)) * ↑b - (↑(a n) * ↑b / ↑(a (n + 1)) - ↑(a 0) / ↑(a (n + 1))) =\n ↑(a (n + 2)) / ↑(a (n + 1)) * ↑b + ↑(a 0) * ↑(a (n + 2)) / (↑(a (n + 1)) * ↑(a (n + 2)))"
},
{
"line": "rfl",
"before_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\n⊢ ↑(a n) / ↑(a (n + 1)) * ↑b + ↑(a (n + 2)) / ↑(a (n + 1)) * ↑b - (↑(a n) * ↑b / ↑(a (n + 1)) - ↑(a 0) / ↑(a (n + 1))) =\n ↑(a (n + 2)) / ↑(a (n + 1)) * ↑b + ↑(a 0) * ↑(a (n + 2)) / (↑(a (n + 1)) * ↑(a (n + 2)))",
"after_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\n⊢ ↑(a n) / ↑(a (n + 1)) * ↑b + ↑(a (n + 2)) / ↑(a (n + 1)) * ↑b - (↑(a n) * ↑b / ↑(a (n + 1)) - ↑(a 0) / ↑(a (n + 1))) =\n ↑(a (n + 2)) / ↑(a (n + 1)) * ↑b + ↑(a 0) * ↑(a (n + 2)) / (↑(a (n + 1)) * ↑(a (n + 2)))"
},
{
"line": "apply_rfl",
"before_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\n⊢ ↑(a n) / ↑(a (n + 1)) * ↑b + ↑(a (n + 2)) / ↑(a (n + 1)) * ↑b - (↑(a n) * ↑b / ↑(a (n + 1)) - ↑(a 0) / ↑(a (n + 1))) =\n ↑(a (n + 2)) / ↑(a (n + 1)) * ↑b + ↑(a 0) * ↑(a (n + 2)) / (↑(a (n + 1)) * ↑(a (n + 2)))",
"after_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\n⊢ ↑(a n) / ↑(a (n + 1)) * ↑b + ↑(a (n + 2)) / ↑(a (n + 1)) * ↑b - (↑(a n) * ↑b / ↑(a (n + 1)) - ↑(a 0) / ↑(a (n + 1))) =\n ↑(a (n + 2)) / ↑(a (n + 1)) * ↑b + ↑(a 0) * ↑(a (n + 2)) / (↑(a (n + 1)) * ↑(a (n + 2)))"
},
{
"line": "skip",
"before_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\n⊢ ↑(a n) / ↑(a (n + 1)) * ↑b + ↑(a (n + 2)) / ↑(a (n + 1)) * ↑b - (↑(a n) * ↑b / ↑(a (n + 1)) - ↑(a 0) / ↑(a (n + 1))) =\n ↑(a (n + 2)) / ↑(a (n + 1)) * ↑b + ↑(a 0) * ↑(a (n + 2)) / (↑(a (n + 1)) * ↑(a (n + 2)))",
"after_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\n⊢ ↑(a n) / ↑(a (n + 1)) * ↑b + ↑(a (n + 2)) / ↑(a (n + 1)) * ↑b - (↑(a n) * ↑b / ↑(a (n + 1)) - ↑(a 0) / ↑(a (n + 1))) =\n ↑(a (n + 2)) / ↑(a (n + 1)) * ↑b + ↑(a 0) * ↑(a (n + 2)) / (↑(a (n + 1)) * ↑(a (n + 2)))"
},
{
"line": "rw [mul_div_right_comm]",
"before_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\n⊢ ↑(a n) / ↑(a (n + 1)) * ↑b + ↑(a (n + 2)) / ↑(a (n + 1)) * ↑b - (↑(a n) * ↑b / ↑(a (n + 1)) - ↑(a 0) / ↑(a (n + 1))) =\n ↑(a (n + 2)) / ↑(a (n + 1)) * ↑b + ↑(a 0) * ↑(a (n + 2)) / (↑(a (n + 1)) * ↑(a (n + 2)))",
"after_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\n⊢ ↑(a n) / ↑(a (n + 1)) * ↑b + ↑(a (n + 2)) / ↑(a (n + 1)) * ↑b - (↑(a n) / ↑(a (n + 1)) * ↑b - ↑(a 0) / ↑(a (n + 1))) =\n ↑(a (n + 2)) / ↑(a (n + 1)) * ↑b + ↑(a 0) * ↑(a (n + 2)) / (↑(a (n + 1)) * ↑(a (n + 2)))"
},
{
"line": "rewrite [mul_div_right_comm]",
"before_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\n⊢ ↑(a n) / ↑(a (n + 1)) * ↑b + ↑(a (n + 2)) / ↑(a (n + 1)) * ↑b - (↑(a n) * ↑b / ↑(a (n + 1)) - ↑(a 0) / ↑(a (n + 1))) =\n ↑(a (n + 2)) / ↑(a (n + 1)) * ↑b + ↑(a 0) * ↑(a (n + 2)) / (↑(a (n + 1)) * ↑(a (n + 2)))",
"after_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\n⊢ ↑(a n) / ↑(a (n + 1)) * ↑b + ↑(a (n + 2)) / ↑(a (n + 1)) * ↑b - (↑(a n) / ↑(a (n + 1)) * ↑b - ↑(a 0) / ↑(a (n + 1))) =\n ↑(a (n + 2)) / ↑(a (n + 1)) * ↑b + ↑(a 0) * ↑(a (n + 2)) / (↑(a (n + 1)) * ↑(a (n + 2)))"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\n⊢ ↑(a n) / ↑(a (n + 1)) * ↑b + ↑(a (n + 2)) / ↑(a (n + 1)) * ↑b - (↑(a n) / ↑(a (n + 1)) * ↑b - ↑(a 0) / ↑(a (n + 1))) =\n ↑(a (n + 2)) / ↑(a (n + 1)) * ↑b + ↑(a 0) * ↑(a (n + 2)) / (↑(a (n + 1)) * ↑(a (n + 2)))",
"after_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\n⊢ ↑(a n) / ↑(a (n + 1)) * ↑b + ↑(a (n + 2)) / ↑(a (n + 1)) * ↑b - (↑(a n) / ↑(a (n + 1)) * ↑b - ↑(a 0) / ↑(a (n + 1))) =\n ↑(a (n + 2)) / ↑(a (n + 1)) * ↑b + ↑(a 0) * ↑(a (n + 2)) / (↑(a (n + 1)) * ↑(a (n + 2)))"
},
{
"line": "try (with_reducible rfl)",
"before_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\n⊢ ↑(a n) / ↑(a (n + 1)) * ↑b + ↑(a (n + 2)) / ↑(a (n + 1)) * ↑b - (↑(a n) / ↑(a (n + 1)) * ↑b - ↑(a 0) / ↑(a (n + 1))) =\n ↑(a (n + 2)) / ↑(a (n + 1)) * ↑b + ↑(a 0) * ↑(a (n + 2)) / (↑(a (n + 1)) * ↑(a (n + 2)))",
"after_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\n⊢ ↑(a n) / ↑(a (n + 1)) * ↑b + ↑(a (n + 2)) / ↑(a (n + 1)) * ↑b - (↑(a n) / ↑(a (n + 1)) * ↑b - ↑(a 0) / ↑(a (n + 1))) =\n ↑(a (n + 2)) / ↑(a (n + 1)) * ↑b + ↑(a 0) * ↑(a (n + 2)) / (↑(a (n + 1)) * ↑(a (n + 2)))"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\n⊢ ↑(a n) / ↑(a (n + 1)) * ↑b + ↑(a (n + 2)) / ↑(a (n + 1)) * ↑b - (↑(a n) / ↑(a (n + 1)) * ↑b - ↑(a 0) / ↑(a (n + 1))) =\n ↑(a (n + 2)) / ↑(a (n + 1)) * ↑b + ↑(a 0) * ↑(a (n + 2)) / (↑(a (n + 1)) * ↑(a (n + 2)))",
"after_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\n⊢ ↑(a n) / ↑(a (n + 1)) * ↑b + ↑(a (n + 2)) / ↑(a (n + 1)) * ↑b - (↑(a n) / ↑(a (n + 1)) * ↑b - ↑(a 0) / ↑(a (n + 1))) =\n ↑(a (n + 2)) / ↑(a (n + 1)) * ↑b + ↑(a 0) * ↑(a (n + 2)) / (↑(a (n + 1)) * ↑(a (n + 2)))"
},
{
"line": "with_reducible rfl",
"before_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\n⊢ ↑(a n) / ↑(a (n + 1)) * ↑b + ↑(a (n + 2)) / ↑(a (n + 1)) * ↑b - (↑(a n) / ↑(a (n + 1)) * ↑b - ↑(a 0) / ↑(a (n + 1))) =\n ↑(a (n + 2)) / ↑(a (n + 1)) * ↑b + ↑(a 0) * ↑(a (n + 2)) / (↑(a (n + 1)) * ↑(a (n + 2)))",
"after_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\n⊢ ↑(a n) / ↑(a (n + 1)) * ↑b + ↑(a (n + 2)) / ↑(a (n + 1)) * ↑b - (↑(a n) / ↑(a (n + 1)) * ↑b - ↑(a 0) / ↑(a (n + 1))) =\n ↑(a (n + 2)) / ↑(a (n + 1)) * ↑b + ↑(a 0) * ↑(a (n + 2)) / (↑(a (n + 1)) * ↑(a (n + 2)))"
},
{
"line": "rfl",
"before_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\n⊢ ↑(a n) / ↑(a (n + 1)) * ↑b + ↑(a (n + 2)) / ↑(a (n + 1)) * ↑b - (↑(a n) / ↑(a (n + 1)) * ↑b - ↑(a 0) / ↑(a (n + 1))) =\n ↑(a (n + 2)) / ↑(a (n + 1)) * ↑b + ↑(a 0) * ↑(a (n + 2)) / (↑(a (n + 1)) * ↑(a (n + 2)))",
"after_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\n⊢ ↑(a n) / ↑(a (n + 1)) * ↑b + ↑(a (n + 2)) / ↑(a (n + 1)) * ↑b - (↑(a n) / ↑(a (n + 1)) * ↑b - ↑(a 0) / ↑(a (n + 1))) =\n ↑(a (n + 2)) / ↑(a (n + 1)) * ↑b + ↑(a 0) * ↑(a (n + 2)) / (↑(a (n + 1)) * ↑(a (n + 2)))"
},
{
"line": "apply_rfl",
"before_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\n⊢ ↑(a n) / ↑(a (n + 1)) * ↑b + ↑(a (n + 2)) / ↑(a (n + 1)) * ↑b - (↑(a n) / ↑(a (n + 1)) * ↑b - ↑(a 0) / ↑(a (n + 1))) =\n ↑(a (n + 2)) / ↑(a (n + 1)) * ↑b + ↑(a 0) * ↑(a (n + 2)) / (↑(a (n + 1)) * ↑(a (n + 2)))",
"after_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\n⊢ ↑(a n) / ↑(a (n + 1)) * ↑b + ↑(a (n + 2)) / ↑(a (n + 1)) * ↑b - (↑(a n) / ↑(a (n + 1)) * ↑b - ↑(a 0) / ↑(a (n + 1))) =\n ↑(a (n + 2)) / ↑(a (n + 1)) * ↑b + ↑(a 0) * ↑(a (n + 2)) / (↑(a (n + 1)) * ↑(a (n + 2)))"
},
{
"line": "skip",
"before_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\n⊢ ↑(a n) / ↑(a (n + 1)) * ↑b + ↑(a (n + 2)) / ↑(a (n + 1)) * ↑b - (↑(a n) / ↑(a (n + 1)) * ↑b - ↑(a 0) / ↑(a (n + 1))) =\n ↑(a (n + 2)) / ↑(a (n + 1)) * ↑b + ↑(a 0) * ↑(a (n + 2)) / (↑(a (n + 1)) * ↑(a (n + 2)))",
"after_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\n⊢ ↑(a n) / ↑(a (n + 1)) * ↑b + ↑(a (n + 2)) / ↑(a (n + 1)) * ↑b - (↑(a n) / ↑(a (n + 1)) * ↑b - ↑(a 0) / ↑(a (n + 1))) =\n ↑(a (n + 2)) / ↑(a (n + 1)) * ↑b + ↑(a 0) * ↑(a (n + 2)) / (↑(a (n + 1)) * ↑(a (n + 2)))"
},
{
"line": "rw [add_sub_sub_cancel]",
"before_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\n⊢ ↑(a n) / ↑(a (n + 1)) * ↑b + ↑(a (n + 2)) / ↑(a (n + 1)) * ↑b - (↑(a n) / ↑(a (n + 1)) * ↑b - ↑(a 0) / ↑(a (n + 1))) =\n ↑(a (n + 2)) / ↑(a (n + 1)) * ↑b + ↑(a 0) * ↑(a (n + 2)) / (↑(a (n + 1)) * ↑(a (n + 2)))",
"after_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\n⊢ ↑(a (n + 2)) / ↑(a (n + 1)) * ↑b + ↑(a 0) / ↑(a (n + 1)) =\n ↑(a (n + 2)) / ↑(a (n + 1)) * ↑b + ↑(a 0) * ↑(a (n + 2)) / (↑(a (n + 1)) * ↑(a (n + 2)))"
},
{
"line": "rewrite [add_sub_sub_cancel]",
"before_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\n⊢ ↑(a n) / ↑(a (n + 1)) * ↑b + ↑(a (n + 2)) / ↑(a (n + 1)) * ↑b - (↑(a n) / ↑(a (n + 1)) * ↑b - ↑(a 0) / ↑(a (n + 1))) =\n ↑(a (n + 2)) / ↑(a (n + 1)) * ↑b + ↑(a 0) * ↑(a (n + 2)) / (↑(a (n + 1)) * ↑(a (n + 2)))",
"after_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\n⊢ ↑(a (n + 2)) / ↑(a (n + 1)) * ↑b + ↑(a 0) / ↑(a (n + 1)) =\n ↑(a (n + 2)) / ↑(a (n + 1)) * ↑b + ↑(a 0) * ↑(a (n + 2)) / (↑(a (n + 1)) * ↑(a (n + 2)))"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\n⊢ ↑(a (n + 2)) / ↑(a (n + 1)) * ↑b + ↑(a 0) / ↑(a (n + 1)) =\n ↑(a (n + 2)) / ↑(a (n + 1)) * ↑b + ↑(a 0) * ↑(a (n + 2)) / (↑(a (n + 1)) * ↑(a (n + 2)))",
"after_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\n⊢ ↑(a (n + 2)) / ↑(a (n + 1)) * ↑b + ↑(a 0) / ↑(a (n + 1)) =\n ↑(a (n + 2)) / ↑(a (n + 1)) * ↑b + ↑(a 0) * ↑(a (n + 2)) / (↑(a (n + 1)) * ↑(a (n + 2)))"
},
{
"line": "try (with_reducible rfl)",
"before_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\n⊢ ↑(a (n + 2)) / ↑(a (n + 1)) * ↑b + ↑(a 0) / ↑(a (n + 1)) =\n ↑(a (n + 2)) / ↑(a (n + 1)) * ↑b + ↑(a 0) * ↑(a (n + 2)) / (↑(a (n + 1)) * ↑(a (n + 2)))",
"after_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\n⊢ ↑(a (n + 2)) / ↑(a (n + 1)) * ↑b + ↑(a 0) / ↑(a (n + 1)) =\n ↑(a (n + 2)) / ↑(a (n + 1)) * ↑b + ↑(a 0) * ↑(a (n + 2)) / (↑(a (n + 1)) * ↑(a (n + 2)))"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\n⊢ ↑(a (n + 2)) / ↑(a (n + 1)) * ↑b + ↑(a 0) / ↑(a (n + 1)) =\n ↑(a (n + 2)) / ↑(a (n + 1)) * ↑b + ↑(a 0) * ↑(a (n + 2)) / (↑(a (n + 1)) * ↑(a (n + 2)))",
"after_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\n⊢ ↑(a (n + 2)) / ↑(a (n + 1)) * ↑b + ↑(a 0) / ↑(a (n + 1)) =\n ↑(a (n + 2)) / ↑(a (n + 1)) * ↑b + ↑(a 0) * ↑(a (n + 2)) / (↑(a (n + 1)) * ↑(a (n + 2)))"
},
{
"line": "with_reducible rfl",
"before_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\n⊢ ↑(a (n + 2)) / ↑(a (n + 1)) * ↑b + ↑(a 0) / ↑(a (n + 1)) =\n ↑(a (n + 2)) / ↑(a (n + 1)) * ↑b + ↑(a 0) * ↑(a (n + 2)) / (↑(a (n + 1)) * ↑(a (n + 2)))",
"after_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\n⊢ ↑(a (n + 2)) / ↑(a (n + 1)) * ↑b + ↑(a 0) / ↑(a (n + 1)) =\n ↑(a (n + 2)) / ↑(a (n + 1)) * ↑b + ↑(a 0) * ↑(a (n + 2)) / (↑(a (n + 1)) * ↑(a (n + 2)))"
},
{
"line": "rfl",
"before_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\n⊢ ↑(a (n + 2)) / ↑(a (n + 1)) * ↑b + ↑(a 0) / ↑(a (n + 1)) =\n ↑(a (n + 2)) / ↑(a (n + 1)) * ↑b + ↑(a 0) * ↑(a (n + 2)) / (↑(a (n + 1)) * ↑(a (n + 2)))",
"after_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\n⊢ ↑(a (n + 2)) / ↑(a (n + 1)) * ↑b + ↑(a 0) / ↑(a (n + 1)) =\n ↑(a (n + 2)) / ↑(a (n + 1)) * ↑b + ↑(a 0) * ↑(a (n + 2)) / (↑(a (n + 1)) * ↑(a (n + 2)))"
},
{
"line": "apply_rfl",
"before_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\n⊢ ↑(a (n + 2)) / ↑(a (n + 1)) * ↑b + ↑(a 0) / ↑(a (n + 1)) =\n ↑(a (n + 2)) / ↑(a (n + 1)) * ↑b + ↑(a 0) * ↑(a (n + 2)) / (↑(a (n + 1)) * ↑(a (n + 2)))",
"after_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\n⊢ ↑(a (n + 2)) / ↑(a (n + 1)) * ↑b + ↑(a 0) / ↑(a (n + 1)) =\n ↑(a (n + 2)) / ↑(a (n + 1)) * ↑b + ↑(a 0) * ↑(a (n + 2)) / (↑(a (n + 1)) * ↑(a (n + 2)))"
},
{
"line": "skip",
"before_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\n⊢ ↑(a (n + 2)) / ↑(a (n + 1)) * ↑b + ↑(a 0) / ↑(a (n + 1)) =\n ↑(a (n + 2)) / ↑(a (n + 1)) * ↑b + ↑(a 0) * ↑(a (n + 2)) / (↑(a (n + 1)) * ↑(a (n + 2)))",
"after_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\n⊢ ↑(a (n + 2)) / ↑(a (n + 1)) * ↑b + ↑(a 0) / ↑(a (n + 1)) =\n ↑(a (n + 2)) / ↑(a (n + 1)) * ↑b + ↑(a 0) * ↑(a (n + 2)) / (↑(a (n + 1)) * ↑(a (n + 2)))"
},
{
"line": "rw [mul_div_mul_right _ _ (a_pos _ le_rfl).ne']",
"before_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\n⊢ ↑(a (n + 2)) / ↑(a (n + 1)) * ↑b + ↑(a 0) / ↑(a (n + 1)) =\n ↑(a (n + 2)) / ↑(a (n + 1)) * ↑b + ↑(a 0) * ↑(a (n + 2)) / (↑(a (n + 1)) * ↑(a (n + 2)))",
"after_state": "No Goals!"
},
{
"line": "rewrite [mul_div_mul_right _ _ (a_pos _ le_rfl).ne']",
"before_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\n⊢ ↑(a (n + 2)) / ↑(a (n + 1)) * ↑b + ↑(a 0) / ↑(a (n + 1)) =\n ↑(a (n + 2)) / ↑(a (n + 1)) * ↑b + ↑(a 0) * ↑(a (n + 2)) / (↑(a (n + 1)) * ↑(a (n + 2)))",
"after_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\n⊢ ↑(a (n + 2)) / ↑(a (n + 1)) * ↑b + ↑(a 0) / ↑(a (n + 1)) = ↑(a (n + 2)) / ↑(a (n + 1)) * ↑b + ↑(a 0) / ↑(a (n + 1))"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\n⊢ ↑(a (n + 2)) / ↑(a (n + 1)) * ↑b + ↑(a 0) / ↑(a (n + 1)) = ↑(a (n + 2)) / ↑(a (n + 1)) * ↑b + ↑(a 0) / ↑(a (n + 1))",
"after_state": "No Goals!"
},
{
"line": "try (with_reducible rfl)",
"before_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\n⊢ ↑(a (n + 2)) / ↑(a (n + 1)) * ↑b + ↑(a 0) / ↑(a (n + 1)) = ↑(a (n + 2)) / ↑(a (n + 1)) * ↑b + ↑(a 0) / ↑(a (n + 1))",
"after_state": "No Goals!"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\n⊢ ↑(a (n + 2)) / ↑(a (n + 1)) * ↑b + ↑(a 0) / ↑(a (n + 1)) = ↑(a (n + 2)) / ↑(a (n + 1)) * ↑b + ↑(a 0) / ↑(a (n + 1))",
"after_state": "No Goals!"
},
{
"line": "with_reducible rfl",
"before_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\n⊢ ↑(a (n + 2)) / ↑(a (n + 1)) * ↑b + ↑(a 0) / ↑(a (n + 1)) = ↑(a (n + 2)) / ↑(a (n + 1)) * ↑b + ↑(a 0) / ↑(a (n + 1))",
"after_state": "No Goals!"
},
{
"line": "rfl",
"before_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\n⊢ ↑(a (n + 2)) / ↑(a (n + 1)) * ↑b + ↑(a 0) / ↑(a (n + 1)) = ↑(a (n + 2)) / ↑(a (n + 1)) * ↑b + ↑(a 0) / ↑(a (n + 1))",
"after_state": "No Goals!"
},
{
"line": "eq_refl",
"before_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\n⊢ ↑(a (n + 2)) / ↑(a (n + 1)) * ↑b + ↑(a 0) / ↑(a (n + 1)) = ↑(a (n + 2)) / ↑(a (n + 1)) * ↑b + ↑(a 0) / ↑(a (n + 1))",
"after_state": "No Goals!"
},
{
"line": "rw [Finset.sum_range_succ]",
"before_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\n⊢ ↑(a (n + 2)) / ↑(a (n + 1)) * ↑b + ↑(a 0) * ↑(a (n + 2)) / (↑(a (n + 1)) * ↑(a (n + 2))) =\n ∑ i ∈ Finset.range (n + 2), ↑(a 0) * ↑(a (n + 2)) / (↑(a i) * ↑(a (i + 1)))",
"after_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\n⊢ ↑(a (n + 2)) / ↑(a (n + 1)) * ↑b + ↑(a 0) * ↑(a (n + 2)) / (↑(a (n + 1)) * ↑(a (n + 2))) =\n ∑ x ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 2)) / (↑(a x) * ↑(a (x + 1))) +\n ↑(a 0) * ↑(a (n + 2)) / (↑(a (n + 1)) * ↑(a (n + 1 + 1)))"
},
{
"line": "rewrite [Finset.sum_range_succ]",
"before_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\n⊢ ↑(a (n + 2)) / ↑(a (n + 1)) * ↑b + ↑(a 0) * ↑(a (n + 2)) / (↑(a (n + 1)) * ↑(a (n + 2))) =\n ∑ i ∈ Finset.range (n + 2), ↑(a 0) * ↑(a (n + 2)) / (↑(a i) * ↑(a (i + 1)))",
"after_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\n⊢ ↑(a (n + 2)) / ↑(a (n + 1)) * ↑b + ↑(a 0) * ↑(a (n + 2)) / (↑(a (n + 1)) * ↑(a (n + 2))) =\n ∑ x ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 2)) / (↑(a x) * ↑(a (x + 1))) +\n ↑(a 0) * ↑(a (n + 2)) / (↑(a (n + 1)) * ↑(a (n + 1 + 1)))"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\n⊢ ↑(a (n + 2)) / ↑(a (n + 1)) * ↑b + ↑(a 0) * ↑(a (n + 2)) / (↑(a (n + 1)) * ↑(a (n + 2))) =\n ∑ x ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 2)) / (↑(a x) * ↑(a (x + 1))) +\n ↑(a 0) * ↑(a (n + 2)) / (↑(a (n + 1)) * ↑(a (n + 1 + 1)))",
"after_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\n⊢ ↑(a (n + 2)) / ↑(a (n + 1)) * ↑b + ↑(a 0) * ↑(a (n + 2)) / (↑(a (n + 1)) * ↑(a (n + 2))) =\n ∑ x ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 2)) / (↑(a x) * ↑(a (x + 1))) +\n ↑(a 0) * ↑(a (n + 2)) / (↑(a (n + 1)) * ↑(a (n + 1 + 1)))"
},
{
"line": "try (with_reducible rfl)",
"before_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\n⊢ ↑(a (n + 2)) / ↑(a (n + 1)) * ↑b + ↑(a 0) * ↑(a (n + 2)) / (↑(a (n + 1)) * ↑(a (n + 2))) =\n ∑ x ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 2)) / (↑(a x) * ↑(a (x + 1))) +\n ↑(a 0) * ↑(a (n + 2)) / (↑(a (n + 1)) * ↑(a (n + 1 + 1)))",
"after_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\n⊢ ↑(a (n + 2)) / ↑(a (n + 1)) * ↑b + ↑(a 0) * ↑(a (n + 2)) / (↑(a (n + 1)) * ↑(a (n + 2))) =\n ∑ x ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 2)) / (↑(a x) * ↑(a (x + 1))) +\n ↑(a 0) * ↑(a (n + 2)) / (↑(a (n + 1)) * ↑(a (n + 1 + 1)))"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\n⊢ ↑(a (n + 2)) / ↑(a (n + 1)) * ↑b + ↑(a 0) * ↑(a (n + 2)) / (↑(a (n + 1)) * ↑(a (n + 2))) =\n ∑ x ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 2)) / (↑(a x) * ↑(a (x + 1))) +\n ↑(a 0) * ↑(a (n + 2)) / (↑(a (n + 1)) * ↑(a (n + 1 + 1)))",
"after_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\n⊢ ↑(a (n + 2)) / ↑(a (n + 1)) * ↑b + ↑(a 0) * ↑(a (n + 2)) / (↑(a (n + 1)) * ↑(a (n + 2))) =\n ∑ x ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 2)) / (↑(a x) * ↑(a (x + 1))) +\n ↑(a 0) * ↑(a (n + 2)) / (↑(a (n + 1)) * ↑(a (n + 1 + 1)))"
},
{
"line": "with_reducible rfl",
"before_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\n⊢ ↑(a (n + 2)) / ↑(a (n + 1)) * ↑b + ↑(a 0) * ↑(a (n + 2)) / (↑(a (n + 1)) * ↑(a (n + 2))) =\n ∑ x ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 2)) / (↑(a x) * ↑(a (x + 1))) +\n ↑(a 0) * ↑(a (n + 2)) / (↑(a (n + 1)) * ↑(a (n + 1 + 1)))",
"after_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\n⊢ ↑(a (n + 2)) / ↑(a (n + 1)) * ↑b + ↑(a 0) * ↑(a (n + 2)) / (↑(a (n + 1)) * ↑(a (n + 2))) =\n ∑ x ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 2)) / (↑(a x) * ↑(a (x + 1))) +\n ↑(a 0) * ↑(a (n + 2)) / (↑(a (n + 1)) * ↑(a (n + 1 + 1)))"
},
{
"line": "rfl",
"before_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\n⊢ ↑(a (n + 2)) / ↑(a (n + 1)) * ↑b + ↑(a 0) * ↑(a (n + 2)) / (↑(a (n + 1)) * ↑(a (n + 2))) =\n ∑ x ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 2)) / (↑(a x) * ↑(a (x + 1))) +\n ↑(a 0) * ↑(a (n + 2)) / (↑(a (n + 1)) * ↑(a (n + 1 + 1)))",
"after_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\n⊢ ↑(a (n + 2)) / ↑(a (n + 1)) * ↑b + ↑(a 0) * ↑(a (n + 2)) / (↑(a (n + 1)) * ↑(a (n + 2))) =\n ∑ x ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 2)) / (↑(a x) * ↑(a (x + 1))) +\n ↑(a 0) * ↑(a (n + 2)) / (↑(a (n + 1)) * ↑(a (n + 1 + 1)))"
},
{
"line": "apply_rfl",
"before_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\n⊢ ↑(a (n + 2)) / ↑(a (n + 1)) * ↑b + ↑(a 0) * ↑(a (n + 2)) / (↑(a (n + 1)) * ↑(a (n + 2))) =\n ∑ x ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 2)) / (↑(a x) * ↑(a (x + 1))) +\n ↑(a 0) * ↑(a (n + 2)) / (↑(a (n + 1)) * ↑(a (n + 1 + 1)))",
"after_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\n⊢ ↑(a (n + 2)) / ↑(a (n + 1)) * ↑b + ↑(a 0) * ↑(a (n + 2)) / (↑(a (n + 1)) * ↑(a (n + 2))) =\n ∑ x ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 2)) / (↑(a x) * ↑(a (x + 1))) +\n ↑(a 0) * ↑(a (n + 2)) / (↑(a (n + 1)) * ↑(a (n + 1 + 1)))"
},
{
"line": "skip",
"before_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\n⊢ ↑(a (n + 2)) / ↑(a (n + 1)) * ↑b + ↑(a 0) * ↑(a (n + 2)) / (↑(a (n + 1)) * ↑(a (n + 2))) =\n ∑ x ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 2)) / (↑(a x) * ↑(a (x + 1))) +\n ↑(a 0) * ↑(a (n + 2)) / (↑(a (n + 1)) * ↑(a (n + 1 + 1)))",
"after_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\n⊢ ↑(a (n + 2)) / ↑(a (n + 1)) * ↑b + ↑(a 0) * ↑(a (n + 2)) / (↑(a (n + 1)) * ↑(a (n + 2))) =\n ∑ x ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 2)) / (↑(a x) * ↑(a (x + 1))) +\n ↑(a 0) * ↑(a (n + 2)) / (↑(a (n + 1)) * ↑(a (n + 1 + 1)))"
},
{
"line": "rw [hb]",
"before_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\n⊢ ↑(a (n + 2)) / ↑(a (n + 1)) * ↑b + ↑(a 0) * ↑(a (n + 2)) / (↑(a (n + 1)) * ↑(a (n + 2))) =\n ∑ x ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 2)) / (↑(a x) * ↑(a (x + 1))) +\n ↑(a 0) * ↑(a (n + 2)) / (↑(a (n + 1)) * ↑(a (n + 1 + 1)))",
"after_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\n⊢ ↑(a (n + 2)) / ↑(a (n + 1)) * ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) +\n ↑(a 0) * ↑(a (n + 2)) / (↑(a (n + 1)) * ↑(a (n + 2))) =\n ∑ x ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 2)) / (↑(a x) * ↑(a (x + 1))) +\n ↑(a 0) * ↑(a (n + 2)) / (↑(a (n + 1)) * ↑(a (n + 1 + 1)))"
},
{
"line": "rewrite [hb]",
"before_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\n⊢ ↑(a (n + 2)) / ↑(a (n + 1)) * ↑b + ↑(a 0) * ↑(a (n + 2)) / (↑(a (n + 1)) * ↑(a (n + 2))) =\n ∑ x ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 2)) / (↑(a x) * ↑(a (x + 1))) +\n ↑(a 0) * ↑(a (n + 2)) / (↑(a (n + 1)) * ↑(a (n + 1 + 1)))",
"after_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\n⊢ ↑(a (n + 2)) / ↑(a (n + 1)) * ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) +\n ↑(a 0) * ↑(a (n + 2)) / (↑(a (n + 1)) * ↑(a (n + 2))) =\n ∑ x ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 2)) / (↑(a x) * ↑(a (x + 1))) +\n ↑(a 0) * ↑(a (n + 2)) / (↑(a (n + 1)) * ↑(a (n + 1 + 1)))"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\n⊢ ↑(a (n + 2)) / ↑(a (n + 1)) * ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) +\n ↑(a 0) * ↑(a (n + 2)) / (↑(a (n + 1)) * ↑(a (n + 2))) =\n ∑ x ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 2)) / (↑(a x) * ↑(a (x + 1))) +\n ↑(a 0) * ↑(a (n + 2)) / (↑(a (n + 1)) * ↑(a (n + 1 + 1)))",
"after_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\n⊢ ↑(a (n + 2)) / ↑(a (n + 1)) * ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) +\n ↑(a 0) * ↑(a (n + 2)) / (↑(a (n + 1)) * ↑(a (n + 2))) =\n ∑ x ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 2)) / (↑(a x) * ↑(a (x + 1))) +\n ↑(a 0) * ↑(a (n + 2)) / (↑(a (n + 1)) * ↑(a (n + 1 + 1)))"
},
{
"line": "try (with_reducible rfl)",
"before_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\n⊢ ↑(a (n + 2)) / ↑(a (n + 1)) * ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) +\n ↑(a 0) * ↑(a (n + 2)) / (↑(a (n + 1)) * ↑(a (n + 2))) =\n ∑ x ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 2)) / (↑(a x) * ↑(a (x + 1))) +\n ↑(a 0) * ↑(a (n + 2)) / (↑(a (n + 1)) * ↑(a (n + 1 + 1)))",
"after_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\n⊢ ↑(a (n + 2)) / ↑(a (n + 1)) * ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) +\n ↑(a 0) * ↑(a (n + 2)) / (↑(a (n + 1)) * ↑(a (n + 2))) =\n ∑ x ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 2)) / (↑(a x) * ↑(a (x + 1))) +\n ↑(a 0) * ↑(a (n + 2)) / (↑(a (n + 1)) * ↑(a (n + 1 + 1)))"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\n⊢ ↑(a (n + 2)) / ↑(a (n + 1)) * ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) +\n ↑(a 0) * ↑(a (n + 2)) / (↑(a (n + 1)) * ↑(a (n + 2))) =\n ∑ x ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 2)) / (↑(a x) * ↑(a (x + 1))) +\n ↑(a 0) * ↑(a (n + 2)) / (↑(a (n + 1)) * ↑(a (n + 1 + 1)))",
"after_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\n⊢ ↑(a (n + 2)) / ↑(a (n + 1)) * ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) +\n ↑(a 0) * ↑(a (n + 2)) / (↑(a (n + 1)) * ↑(a (n + 2))) =\n ∑ x ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 2)) / (↑(a x) * ↑(a (x + 1))) +\n ↑(a 0) * ↑(a (n + 2)) / (↑(a (n + 1)) * ↑(a (n + 1 + 1)))"
},
{
"line": "with_reducible rfl",
"before_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\n⊢ ↑(a (n + 2)) / ↑(a (n + 1)) * ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) +\n ↑(a 0) * ↑(a (n + 2)) / (↑(a (n + 1)) * ↑(a (n + 2))) =\n ∑ x ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 2)) / (↑(a x) * ↑(a (x + 1))) +\n ↑(a 0) * ↑(a (n + 2)) / (↑(a (n + 1)) * ↑(a (n + 1 + 1)))",
"after_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\n⊢ ↑(a (n + 2)) / ↑(a (n + 1)) * ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) +\n ↑(a 0) * ↑(a (n + 2)) / (↑(a (n + 1)) * ↑(a (n + 2))) =\n ∑ x ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 2)) / (↑(a x) * ↑(a (x + 1))) +\n ↑(a 0) * ↑(a (n + 2)) / (↑(a (n + 1)) * ↑(a (n + 1 + 1)))"
},
{
"line": "rfl",
"before_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\n⊢ ↑(a (n + 2)) / ↑(a (n + 1)) * ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) +\n ↑(a 0) * ↑(a (n + 2)) / (↑(a (n + 1)) * ↑(a (n + 2))) =\n ∑ x ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 2)) / (↑(a x) * ↑(a (x + 1))) +\n ↑(a 0) * ↑(a (n + 2)) / (↑(a (n + 1)) * ↑(a (n + 1 + 1)))",
"after_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\n⊢ ↑(a (n + 2)) / ↑(a (n + 1)) * ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) +\n ↑(a 0) * ↑(a (n + 2)) / (↑(a (n + 1)) * ↑(a (n + 2))) =\n ∑ x ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 2)) / (↑(a x) * ↑(a (x + 1))) +\n ↑(a 0) * ↑(a (n + 2)) / (↑(a (n + 1)) * ↑(a (n + 1 + 1)))"
},
{
"line": "apply_rfl",
"before_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\n⊢ ↑(a (n + 2)) / ↑(a (n + 1)) * ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) +\n ↑(a 0) * ↑(a (n + 2)) / (↑(a (n + 1)) * ↑(a (n + 2))) =\n ∑ x ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 2)) / (↑(a x) * ↑(a (x + 1))) +\n ↑(a 0) * ↑(a (n + 2)) / (↑(a (n + 1)) * ↑(a (n + 1 + 1)))",
"after_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\n⊢ ↑(a (n + 2)) / ↑(a (n + 1)) * ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) +\n ↑(a 0) * ↑(a (n + 2)) / (↑(a (n + 1)) * ↑(a (n + 2))) =\n ∑ x ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 2)) / (↑(a x) * ↑(a (x + 1))) +\n ↑(a 0) * ↑(a (n + 2)) / (↑(a (n + 1)) * ↑(a (n + 1 + 1)))"
},
{
"line": "skip",
"before_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\n⊢ ↑(a (n + 2)) / ↑(a (n + 1)) * ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) +\n ↑(a 0) * ↑(a (n + 2)) / (↑(a (n + 1)) * ↑(a (n + 2))) =\n ∑ x ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 2)) / (↑(a x) * ↑(a (x + 1))) +\n ↑(a 0) * ↑(a (n + 2)) / (↑(a (n + 1)) * ↑(a (n + 1 + 1)))",
"after_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\n⊢ ↑(a (n + 2)) / ↑(a (n + 1)) * ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) +\n ↑(a 0) * ↑(a (n + 2)) / (↑(a (n + 1)) * ↑(a (n + 2))) =\n ∑ x ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 2)) / (↑(a x) * ↑(a (x + 1))) +\n ↑(a 0) * ↑(a (n + 2)) / (↑(a (n + 1)) * ↑(a (n + 1 + 1)))"
},
{
"line": "rw [Finset.mul_sum]",
"before_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\n⊢ ↑(a (n + 2)) / ↑(a (n + 1)) * ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) +\n ↑(a 0) * ↑(a (n + 2)) / (↑(a (n + 1)) * ↑(a (n + 2))) =\n ∑ x ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 2)) / (↑(a x) * ↑(a (x + 1))) +\n ↑(a 0) * ↑(a (n + 2)) / (↑(a (n + 1)) * ↑(a (n + 1 + 1)))",
"after_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\n⊢ ∑ i ∈ Finset.range (n + 1), ↑(a (n + 2)) / ↑(a (n + 1)) * (↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))) +\n ↑(a 0) * ↑(a (n + 2)) / (↑(a (n + 1)) * ↑(a (n + 2))) =\n ∑ x ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 2)) / (↑(a x) * ↑(a (x + 1))) +\n ↑(a 0) * ↑(a (n + 2)) / (↑(a (n + 1)) * ↑(a (n + 1 + 1)))"
},
{
"line": "rewrite [Finset.mul_sum]",
"before_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\n⊢ ↑(a (n + 2)) / ↑(a (n + 1)) * ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) +\n ↑(a 0) * ↑(a (n + 2)) / (↑(a (n + 1)) * ↑(a (n + 2))) =\n ∑ x ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 2)) / (↑(a x) * ↑(a (x + 1))) +\n ↑(a 0) * ↑(a (n + 2)) / (↑(a (n + 1)) * ↑(a (n + 1 + 1)))",
"after_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\n⊢ ∑ i ∈ Finset.range (n + 1), ↑(a (n + 2)) / ↑(a (n + 1)) * (↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))) +\n ↑(a 0) * ↑(a (n + 2)) / (↑(a (n + 1)) * ↑(a (n + 2))) =\n ∑ x ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 2)) / (↑(a x) * ↑(a (x + 1))) +\n ↑(a 0) * ↑(a (n + 2)) / (↑(a (n + 1)) * ↑(a (n + 1 + 1)))"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\n⊢ ∑ i ∈ Finset.range (n + 1), ↑(a (n + 2)) / ↑(a (n + 1)) * (↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))) +\n ↑(a 0) * ↑(a (n + 2)) / (↑(a (n + 1)) * ↑(a (n + 2))) =\n ∑ x ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 2)) / (↑(a x) * ↑(a (x + 1))) +\n ↑(a 0) * ↑(a (n + 2)) / (↑(a (n + 1)) * ↑(a (n + 1 + 1)))",
"after_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\n⊢ ∑ i ∈ Finset.range (n + 1), ↑(a (n + 2)) / ↑(a (n + 1)) * (↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))) +\n ↑(a 0) * ↑(a (n + 2)) / (↑(a (n + 1)) * ↑(a (n + 2))) =\n ∑ x ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 2)) / (↑(a x) * ↑(a (x + 1))) +\n ↑(a 0) * ↑(a (n + 2)) / (↑(a (n + 1)) * ↑(a (n + 1 + 1)))"
},
{
"line": "try (with_reducible rfl)",
"before_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\n⊢ ∑ i ∈ Finset.range (n + 1), ↑(a (n + 2)) / ↑(a (n + 1)) * (↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))) +\n ↑(a 0) * ↑(a (n + 2)) / (↑(a (n + 1)) * ↑(a (n + 2))) =\n ∑ x ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 2)) / (↑(a x) * ↑(a (x + 1))) +\n ↑(a 0) * ↑(a (n + 2)) / (↑(a (n + 1)) * ↑(a (n + 1 + 1)))",
"after_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\n⊢ ∑ i ∈ Finset.range (n + 1), ↑(a (n + 2)) / ↑(a (n + 1)) * (↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))) +\n ↑(a 0) * ↑(a (n + 2)) / (↑(a (n + 1)) * ↑(a (n + 2))) =\n ∑ x ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 2)) / (↑(a x) * ↑(a (x + 1))) +\n ↑(a 0) * ↑(a (n + 2)) / (↑(a (n + 1)) * ↑(a (n + 1 + 1)))"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\n⊢ ∑ i ∈ Finset.range (n + 1), ↑(a (n + 2)) / ↑(a (n + 1)) * (↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))) +\n ↑(a 0) * ↑(a (n + 2)) / (↑(a (n + 1)) * ↑(a (n + 2))) =\n ∑ x ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 2)) / (↑(a x) * ↑(a (x + 1))) +\n ↑(a 0) * ↑(a (n + 2)) / (↑(a (n + 1)) * ↑(a (n + 1 + 1)))",
"after_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\n⊢ ∑ i ∈ Finset.range (n + 1), ↑(a (n + 2)) / ↑(a (n + 1)) * (↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))) +\n ↑(a 0) * ↑(a (n + 2)) / (↑(a (n + 1)) * ↑(a (n + 2))) =\n ∑ x ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 2)) / (↑(a x) * ↑(a (x + 1))) +\n ↑(a 0) * ↑(a (n + 2)) / (↑(a (n + 1)) * ↑(a (n + 1 + 1)))"
},
{
"line": "with_reducible rfl",
"before_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\n⊢ ∑ i ∈ Finset.range (n + 1), ↑(a (n + 2)) / ↑(a (n + 1)) * (↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))) +\n ↑(a 0) * ↑(a (n + 2)) / (↑(a (n + 1)) * ↑(a (n + 2))) =\n ∑ x ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 2)) / (↑(a x) * ↑(a (x + 1))) +\n ↑(a 0) * ↑(a (n + 2)) / (↑(a (n + 1)) * ↑(a (n + 1 + 1)))",
"after_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\n⊢ ∑ i ∈ Finset.range (n + 1), ↑(a (n + 2)) / ↑(a (n + 1)) * (↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))) +\n ↑(a 0) * ↑(a (n + 2)) / (↑(a (n + 1)) * ↑(a (n + 2))) =\n ∑ x ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 2)) / (↑(a x) * ↑(a (x + 1))) +\n ↑(a 0) * ↑(a (n + 2)) / (↑(a (n + 1)) * ↑(a (n + 1 + 1)))"
},
{
"line": "rfl",
"before_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\n⊢ ∑ i ∈ Finset.range (n + 1), ↑(a (n + 2)) / ↑(a (n + 1)) * (↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))) +\n ↑(a 0) * ↑(a (n + 2)) / (↑(a (n + 1)) * ↑(a (n + 2))) =\n ∑ x ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 2)) / (↑(a x) * ↑(a (x + 1))) +\n ↑(a 0) * ↑(a (n + 2)) / (↑(a (n + 1)) * ↑(a (n + 1 + 1)))",
"after_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\n⊢ ∑ i ∈ Finset.range (n + 1), ↑(a (n + 2)) / ↑(a (n + 1)) * (↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))) +\n ↑(a 0) * ↑(a (n + 2)) / (↑(a (n + 1)) * ↑(a (n + 2))) =\n ∑ x ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 2)) / (↑(a x) * ↑(a (x + 1))) +\n ↑(a 0) * ↑(a (n + 2)) / (↑(a (n + 1)) * ↑(a (n + 1 + 1)))"
},
{
"line": "apply_rfl",
"before_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\n⊢ ∑ i ∈ Finset.range (n + 1), ↑(a (n + 2)) / ↑(a (n + 1)) * (↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))) +\n ↑(a 0) * ↑(a (n + 2)) / (↑(a (n + 1)) * ↑(a (n + 2))) =\n ∑ x ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 2)) / (↑(a x) * ↑(a (x + 1))) +\n ↑(a 0) * ↑(a (n + 2)) / (↑(a (n + 1)) * ↑(a (n + 1 + 1)))",
"after_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\n⊢ ∑ i ∈ Finset.range (n + 1), ↑(a (n + 2)) / ↑(a (n + 1)) * (↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))) +\n ↑(a 0) * ↑(a (n + 2)) / (↑(a (n + 1)) * ↑(a (n + 2))) =\n ∑ x ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 2)) / (↑(a x) * ↑(a (x + 1))) +\n ↑(a 0) * ↑(a (n + 2)) / (↑(a (n + 1)) * ↑(a (n + 1 + 1)))"
},
{
"line": "skip",
"before_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\n⊢ ∑ i ∈ Finset.range (n + 1), ↑(a (n + 2)) / ↑(a (n + 1)) * (↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))) +\n ↑(a 0) * ↑(a (n + 2)) / (↑(a (n + 1)) * ↑(a (n + 2))) =\n ∑ x ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 2)) / (↑(a x) * ↑(a (x + 1))) +\n ↑(a 0) * ↑(a (n + 2)) / (↑(a (n + 1)) * ↑(a (n + 1 + 1)))",
"after_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\n⊢ ∑ i ∈ Finset.range (n + 1), ↑(a (n + 2)) / ↑(a (n + 1)) * (↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))) +\n ↑(a 0) * ↑(a (n + 2)) / (↑(a (n + 1)) * ↑(a (n + 2))) =\n ∑ x ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 2)) / (↑(a x) * ↑(a (x + 1))) +\n ↑(a 0) * ↑(a (n + 2)) / (↑(a (n + 1)) * ↑(a (n + 1 + 1)))"
},
{
"line": "congr",
"before_state": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\n⊢ ∑ i ∈ Finset.range (n + 1), ↑(a (n + 2)) / ↑(a (n + 1)) * (↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))) +\n ↑(a 0) * ↑(a (n + 2)) / (↑(a (n + 1)) * ↑(a (n + 2))) =\n ∑ x ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 2)) / (↑(a x) * ↑(a (x + 1))) +\n ↑(a 0) * ↑(a (n + 2)) / (↑(a (n + 1)) * ↑(a (n + 1 + 1)))",
"after_state": "case e_a.e_f\nα : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\n⊢ (fun i => ↑(a (n + 2)) / ↑(a (n + 1)) * (↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))))) = fun x =>\n ↑(a 0) * ↑(a (n + 2)) / (↑(a x) * ↑(a (x + 1)))"
},
{
"line": "ext i",
"before_state": "case e_a.e_f\nα : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\n⊢ (fun i => ↑(a (n + 2)) / ↑(a (n + 1)) * (↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))))) = fun x =>\n ↑(a 0) * ↑(a (n + 2)) / (↑(a x) * ↑(a (x + 1)))",
"after_state": "case e_a.e_f.h\nα : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\ni : ℕ\n⊢ ↑(a (n + 2)) / ↑(a (n + 1)) * (↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))) =\n ↑(a 0) * ↑(a (n + 2)) / (↑(a i) * ↑(a (i + 1)))"
},
{
"line": "rw [← mul_div_assoc]",
"before_state": "case e_a.e_f.h\nα : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\ni : ℕ\n⊢ ↑(a (n + 2)) / ↑(a (n + 1)) * (↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))) =\n ↑(a 0) * ↑(a (n + 2)) / (↑(a i) * ↑(a (i + 1)))",
"after_state": "case e_a.e_f.h\nα : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\ni : ℕ\n⊢ ↑(a (n + 2)) / ↑(a (n + 1)) * (↑(a 0) * ↑(a (n + 1))) / (↑(a i) * ↑(a (i + 1))) =\n ↑(a 0) * ↑(a (n + 2)) / (↑(a i) * ↑(a (i + 1)))"
},
{
"line": "rewrite [← mul_div_assoc]",
"before_state": "case e_a.e_f.h\nα : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\ni : ℕ\n⊢ ↑(a (n + 2)) / ↑(a (n + 1)) * (↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))) =\n ↑(a 0) * ↑(a (n + 2)) / (↑(a i) * ↑(a (i + 1)))",
"after_state": "case e_a.e_f.h\nα : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\ni : ℕ\n⊢ ↑(a (n + 2)) / ↑(a (n + 1)) * (↑(a 0) * ↑(a (n + 1))) / (↑(a i) * ↑(a (i + 1))) =\n ↑(a 0) * ↑(a (n + 2)) / (↑(a i) * ↑(a (i + 1)))"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "case e_a.e_f.h\nα : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\ni : ℕ\n⊢ ↑(a (n + 2)) / ↑(a (n + 1)) * (↑(a 0) * ↑(a (n + 1))) / (↑(a i) * ↑(a (i + 1))) =\n ↑(a 0) * ↑(a (n + 2)) / (↑(a i) * ↑(a (i + 1)))",
"after_state": "case e_a.e_f.h\nα : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\ni : ℕ\n⊢ ↑(a (n + 2)) / ↑(a (n + 1)) * (↑(a 0) * ↑(a (n + 1))) / (↑(a i) * ↑(a (i + 1))) =\n ↑(a 0) * ↑(a (n + 2)) / (↑(a i) * ↑(a (i + 1)))"
},
{
"line": "try (with_reducible rfl)",
"before_state": "case e_a.e_f.h\nα : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\ni : ℕ\n⊢ ↑(a (n + 2)) / ↑(a (n + 1)) * (↑(a 0) * ↑(a (n + 1))) / (↑(a i) * ↑(a (i + 1))) =\n ↑(a 0) * ↑(a (n + 2)) / (↑(a i) * ↑(a (i + 1)))",
"after_state": "case e_a.e_f.h\nα : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\ni : ℕ\n⊢ ↑(a (n + 2)) / ↑(a (n + 1)) * (↑(a 0) * ↑(a (n + 1))) / (↑(a i) * ↑(a (i + 1))) =\n ↑(a 0) * ↑(a (n + 2)) / (↑(a i) * ↑(a (i + 1)))"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "case e_a.e_f.h\nα : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\ni : ℕ\n⊢ ↑(a (n + 2)) / ↑(a (n + 1)) * (↑(a 0) * ↑(a (n + 1))) / (↑(a i) * ↑(a (i + 1))) =\n ↑(a 0) * ↑(a (n + 2)) / (↑(a i) * ↑(a (i + 1)))",
"after_state": "case e_a.e_f.h\nα : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\ni : ℕ\n⊢ ↑(a (n + 2)) / ↑(a (n + 1)) * (↑(a 0) * ↑(a (n + 1))) / (↑(a i) * ↑(a (i + 1))) =\n ↑(a 0) * ↑(a (n + 2)) / (↑(a i) * ↑(a (i + 1)))"
},
{
"line": "with_reducible rfl",
"before_state": "case e_a.e_f.h\nα : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\ni : ℕ\n⊢ ↑(a (n + 2)) / ↑(a (n + 1)) * (↑(a 0) * ↑(a (n + 1))) / (↑(a i) * ↑(a (i + 1))) =\n ↑(a 0) * ↑(a (n + 2)) / (↑(a i) * ↑(a (i + 1)))",
"after_state": "case e_a.e_f.h\nα : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\ni : ℕ\n⊢ ↑(a (n + 2)) / ↑(a (n + 1)) * (↑(a 0) * ↑(a (n + 1))) / (↑(a i) * ↑(a (i + 1))) =\n ↑(a 0) * ↑(a (n + 2)) / (↑(a i) * ↑(a (i + 1)))"
},
{
"line": "rfl",
"before_state": "case e_a.e_f.h\nα : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\ni : ℕ\n⊢ ↑(a (n + 2)) / ↑(a (n + 1)) * (↑(a 0) * ↑(a (n + 1))) / (↑(a i) * ↑(a (i + 1))) =\n ↑(a 0) * ↑(a (n + 2)) / (↑(a i) * ↑(a (i + 1)))",
"after_state": "case e_a.e_f.h\nα : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\ni : ℕ\n⊢ ↑(a (n + 2)) / ↑(a (n + 1)) * (↑(a 0) * ↑(a (n + 1))) / (↑(a i) * ↑(a (i + 1))) =\n ↑(a 0) * ↑(a (n + 2)) / (↑(a i) * ↑(a (i + 1)))"
},
{
"line": "apply_rfl",
"before_state": "case e_a.e_f.h\nα : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\ni : ℕ\n⊢ ↑(a (n + 2)) / ↑(a (n + 1)) * (↑(a 0) * ↑(a (n + 1))) / (↑(a i) * ↑(a (i + 1))) =\n ↑(a 0) * ↑(a (n + 2)) / (↑(a i) * ↑(a (i + 1)))",
"after_state": "case e_a.e_f.h\nα : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\ni : ℕ\n⊢ ↑(a (n + 2)) / ↑(a (n + 1)) * (↑(a 0) * ↑(a (n + 1))) / (↑(a i) * ↑(a (i + 1))) =\n ↑(a 0) * ↑(a (n + 2)) / (↑(a i) * ↑(a (i + 1)))"
},
{
"line": "skip",
"before_state": "case e_a.e_f.h\nα : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\ni : ℕ\n⊢ ↑(a (n + 2)) / ↑(a (n + 1)) * (↑(a 0) * ↑(a (n + 1))) / (↑(a i) * ↑(a (i + 1))) =\n ↑(a 0) * ↑(a (n + 2)) / (↑(a i) * ↑(a (i + 1)))",
"after_state": "case e_a.e_f.h\nα : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\ni : ℕ\n⊢ ↑(a (n + 2)) / ↑(a (n + 1)) * (↑(a 0) * ↑(a (n + 1))) / (↑(a i) * ↑(a (i + 1))) =\n ↑(a 0) * ↑(a (n + 2)) / (↑(a i) * ↑(a (i + 1)))"
},
{
"line": "rw [← mul_div_right_comm]",
"before_state": "case e_a.e_f.h\nα : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\ni : ℕ\n⊢ ↑(a (n + 2)) / ↑(a (n + 1)) * (↑(a 0) * ↑(a (n + 1))) / (↑(a i) * ↑(a (i + 1))) =\n ↑(a 0) * ↑(a (n + 2)) / (↑(a i) * ↑(a (i + 1)))",
"after_state": "case e_a.e_f.h\nα : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\ni : ℕ\n⊢ ↑(a (n + 2)) * (↑(a 0) * ↑(a (n + 1))) / ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) =\n ↑(a 0) * ↑(a (n + 2)) / (↑(a i) * ↑(a (i + 1)))"
},
{
"line": "rewrite [← mul_div_right_comm]",
"before_state": "case e_a.e_f.h\nα : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\ni : ℕ\n⊢ ↑(a (n + 2)) / ↑(a (n + 1)) * (↑(a 0) * ↑(a (n + 1))) / (↑(a i) * ↑(a (i + 1))) =\n ↑(a 0) * ↑(a (n + 2)) / (↑(a i) * ↑(a (i + 1)))",
"after_state": "case e_a.e_f.h\nα : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\ni : ℕ\n⊢ ↑(a (n + 2)) * (↑(a 0) * ↑(a (n + 1))) / ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) =\n ↑(a 0) * ↑(a (n + 2)) / (↑(a i) * ↑(a (i + 1)))"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "case e_a.e_f.h\nα : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\ni : ℕ\n⊢ ↑(a (n + 2)) * (↑(a 0) * ↑(a (n + 1))) / ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) =\n ↑(a 0) * ↑(a (n + 2)) / (↑(a i) * ↑(a (i + 1)))",
"after_state": "case e_a.e_f.h\nα : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\ni : ℕ\n⊢ ↑(a (n + 2)) * (↑(a 0) * ↑(a (n + 1))) / ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) =\n ↑(a 0) * ↑(a (n + 2)) / (↑(a i) * ↑(a (i + 1)))"
},
{
"line": "try (with_reducible rfl)",
"before_state": "case e_a.e_f.h\nα : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\ni : ℕ\n⊢ ↑(a (n + 2)) * (↑(a 0) * ↑(a (n + 1))) / ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) =\n ↑(a 0) * ↑(a (n + 2)) / (↑(a i) * ↑(a (i + 1)))",
"after_state": "case e_a.e_f.h\nα : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\ni : ℕ\n⊢ ↑(a (n + 2)) * (↑(a 0) * ↑(a (n + 1))) / ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) =\n ↑(a 0) * ↑(a (n + 2)) / (↑(a i) * ↑(a (i + 1)))"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "case e_a.e_f.h\nα : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\ni : ℕ\n⊢ ↑(a (n + 2)) * (↑(a 0) * ↑(a (n + 1))) / ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) =\n ↑(a 0) * ↑(a (n + 2)) / (↑(a i) * ↑(a (i + 1)))",
"after_state": "case e_a.e_f.h\nα : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\ni : ℕ\n⊢ ↑(a (n + 2)) * (↑(a 0) * ↑(a (n + 1))) / ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) =\n ↑(a 0) * ↑(a (n + 2)) / (↑(a i) * ↑(a (i + 1)))"
},
{
"line": "with_reducible rfl",
"before_state": "case e_a.e_f.h\nα : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\ni : ℕ\n⊢ ↑(a (n + 2)) * (↑(a 0) * ↑(a (n + 1))) / ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) =\n ↑(a 0) * ↑(a (n + 2)) / (↑(a i) * ↑(a (i + 1)))",
"after_state": "case e_a.e_f.h\nα : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\ni : ℕ\n⊢ ↑(a (n + 2)) * (↑(a 0) * ↑(a (n + 1))) / ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) =\n ↑(a 0) * ↑(a (n + 2)) / (↑(a i) * ↑(a (i + 1)))"
},
{
"line": "rfl",
"before_state": "case e_a.e_f.h\nα : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\ni : ℕ\n⊢ ↑(a (n + 2)) * (↑(a 0) * ↑(a (n + 1))) / ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) =\n ↑(a 0) * ↑(a (n + 2)) / (↑(a i) * ↑(a (i + 1)))",
"after_state": "case e_a.e_f.h\nα : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\ni : ℕ\n⊢ ↑(a (n + 2)) * (↑(a 0) * ↑(a (n + 1))) / ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) =\n ↑(a 0) * ↑(a (n + 2)) / (↑(a i) * ↑(a (i + 1)))"
},
{
"line": "apply_rfl",
"before_state": "case e_a.e_f.h\nα : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\ni : ℕ\n⊢ ↑(a (n + 2)) * (↑(a 0) * ↑(a (n + 1))) / ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) =\n ↑(a 0) * ↑(a (n + 2)) / (↑(a i) * ↑(a (i + 1)))",
"after_state": "case e_a.e_f.h\nα : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\ni : ℕ\n⊢ ↑(a (n + 2)) * (↑(a 0) * ↑(a (n + 1))) / ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) =\n ↑(a 0) * ↑(a (n + 2)) / (↑(a i) * ↑(a (i + 1)))"
},
{
"line": "skip",
"before_state": "case e_a.e_f.h\nα : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\ni : ℕ\n⊢ ↑(a (n + 2)) * (↑(a 0) * ↑(a (n + 1))) / ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) =\n ↑(a 0) * ↑(a (n + 2)) / (↑(a i) * ↑(a (i + 1)))",
"after_state": "case e_a.e_f.h\nα : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\ni : ℕ\n⊢ ↑(a (n + 2)) * (↑(a 0) * ↑(a (n + 1))) / ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) =\n ↑(a 0) * ↑(a (n + 2)) / (↑(a i) * ↑(a (i + 1)))"
},
{
"line": "rw [mul_div_assoc]",
"before_state": "case e_a.e_f.h\nα : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\ni : ℕ\n⊢ ↑(a (n + 2)) * (↑(a 0) * ↑(a (n + 1))) / ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) =\n ↑(a 0) * ↑(a (n + 2)) / (↑(a i) * ↑(a (i + 1)))",
"after_state": "case e_a.e_f.h\nα : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\ni : ℕ\n⊢ ↑(a (n + 2)) * (↑(a 0) * ↑(a (n + 1)) / ↑(a (n + 1))) / (↑(a i) * ↑(a (i + 1))) =\n ↑(a 0) * ↑(a (n + 2)) / (↑(a i) * ↑(a (i + 1)))"
},
{
"line": "rewrite [mul_div_assoc]",
"before_state": "case e_a.e_f.h\nα : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\ni : ℕ\n⊢ ↑(a (n + 2)) * (↑(a 0) * ↑(a (n + 1))) / ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) =\n ↑(a 0) * ↑(a (n + 2)) / (↑(a i) * ↑(a (i + 1)))",
"after_state": "case e_a.e_f.h\nα : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\ni : ℕ\n⊢ ↑(a (n + 2)) * (↑(a 0) * ↑(a (n + 1)) / ↑(a (n + 1))) / (↑(a i) * ↑(a (i + 1))) =\n ↑(a 0) * ↑(a (n + 2)) / (↑(a i) * ↑(a (i + 1)))"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "case e_a.e_f.h\nα : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\ni : ℕ\n⊢ ↑(a (n + 2)) * (↑(a 0) * ↑(a (n + 1)) / ↑(a (n + 1))) / (↑(a i) * ↑(a (i + 1))) =\n ↑(a 0) * ↑(a (n + 2)) / (↑(a i) * ↑(a (i + 1)))",
"after_state": "case e_a.e_f.h\nα : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\ni : ℕ\n⊢ ↑(a (n + 2)) * (↑(a 0) * ↑(a (n + 1)) / ↑(a (n + 1))) / (↑(a i) * ↑(a (i + 1))) =\n ↑(a 0) * ↑(a (n + 2)) / (↑(a i) * ↑(a (i + 1)))"
},
{
"line": "try (with_reducible rfl)",
"before_state": "case e_a.e_f.h\nα : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\ni : ℕ\n⊢ ↑(a (n + 2)) * (↑(a 0) * ↑(a (n + 1)) / ↑(a (n + 1))) / (↑(a i) * ↑(a (i + 1))) =\n ↑(a 0) * ↑(a (n + 2)) / (↑(a i) * ↑(a (i + 1)))",
"after_state": "case e_a.e_f.h\nα : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\ni : ℕ\n⊢ ↑(a (n + 2)) * (↑(a 0) * ↑(a (n + 1)) / ↑(a (n + 1))) / (↑(a i) * ↑(a (i + 1))) =\n ↑(a 0) * ↑(a (n + 2)) / (↑(a i) * ↑(a (i + 1)))"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "case e_a.e_f.h\nα : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\ni : ℕ\n⊢ ↑(a (n + 2)) * (↑(a 0) * ↑(a (n + 1)) / ↑(a (n + 1))) / (↑(a i) * ↑(a (i + 1))) =\n ↑(a 0) * ↑(a (n + 2)) / (↑(a i) * ↑(a (i + 1)))",
"after_state": "case e_a.e_f.h\nα : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\ni : ℕ\n⊢ ↑(a (n + 2)) * (↑(a 0) * ↑(a (n + 1)) / ↑(a (n + 1))) / (↑(a i) * ↑(a (i + 1))) =\n ↑(a 0) * ↑(a (n + 2)) / (↑(a i) * ↑(a (i + 1)))"
},
{
"line": "with_reducible rfl",
"before_state": "case e_a.e_f.h\nα : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\ni : ℕ\n⊢ ↑(a (n + 2)) * (↑(a 0) * ↑(a (n + 1)) / ↑(a (n + 1))) / (↑(a i) * ↑(a (i + 1))) =\n ↑(a 0) * ↑(a (n + 2)) / (↑(a i) * ↑(a (i + 1)))",
"after_state": "case e_a.e_f.h\nα : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\ni : ℕ\n⊢ ↑(a (n + 2)) * (↑(a 0) * ↑(a (n + 1)) / ↑(a (n + 1))) / (↑(a i) * ↑(a (i + 1))) =\n ↑(a 0) * ↑(a (n + 2)) / (↑(a i) * ↑(a (i + 1)))"
},
{
"line": "rfl",
"before_state": "case e_a.e_f.h\nα : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\ni : ℕ\n⊢ ↑(a (n + 2)) * (↑(a 0) * ↑(a (n + 1)) / ↑(a (n + 1))) / (↑(a i) * ↑(a (i + 1))) =\n ↑(a 0) * ↑(a (n + 2)) / (↑(a i) * ↑(a (i + 1)))",
"after_state": "case e_a.e_f.h\nα : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\ni : ℕ\n⊢ ↑(a (n + 2)) * (↑(a 0) * ↑(a (n + 1)) / ↑(a (n + 1))) / (↑(a i) * ↑(a (i + 1))) =\n ↑(a 0) * ↑(a (n + 2)) / (↑(a i) * ↑(a (i + 1)))"
},
{
"line": "apply_rfl",
"before_state": "case e_a.e_f.h\nα : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\ni : ℕ\n⊢ ↑(a (n + 2)) * (↑(a 0) * ↑(a (n + 1)) / ↑(a (n + 1))) / (↑(a i) * ↑(a (i + 1))) =\n ↑(a 0) * ↑(a (n + 2)) / (↑(a i) * ↑(a (i + 1)))",
"after_state": "case e_a.e_f.h\nα : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\ni : ℕ\n⊢ ↑(a (n + 2)) * (↑(a 0) * ↑(a (n + 1)) / ↑(a (n + 1))) / (↑(a i) * ↑(a (i + 1))) =\n ↑(a 0) * ↑(a (n + 2)) / (↑(a i) * ↑(a (i + 1)))"
},
{
"line": "skip",
"before_state": "case e_a.e_f.h\nα : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\ni : ℕ\n⊢ ↑(a (n + 2)) * (↑(a 0) * ↑(a (n + 1)) / ↑(a (n + 1))) / (↑(a i) * ↑(a (i + 1))) =\n ↑(a 0) * ↑(a (n + 2)) / (↑(a i) * ↑(a (i + 1)))",
"after_state": "case e_a.e_f.h\nα : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\ni : ℕ\n⊢ ↑(a (n + 2)) * (↑(a 0) * ↑(a (n + 1)) / ↑(a (n + 1))) / (↑(a i) * ↑(a (i + 1))) =\n ↑(a 0) * ↑(a (n + 2)) / (↑(a i) * ↑(a (i + 1)))"
},
{
"line": "rw [mul_div_cancel_right₀ _ (a_pos _ <| Nat.le_succ _).ne']",
"before_state": "case e_a.e_f.h\nα : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\ni : ℕ\n⊢ ↑(a (n + 2)) * (↑(a 0) * ↑(a (n + 1)) / ↑(a (n + 1))) / (↑(a i) * ↑(a (i + 1))) =\n ↑(a 0) * ↑(a (n + 2)) / (↑(a i) * ↑(a (i + 1)))",
"after_state": "case e_a.e_f.h\nα : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\ni : ℕ\n⊢ ↑(a (n + 2)) * ↑(a 0) / (↑(a i) * ↑(a (i + 1))) = ↑(a 0) * ↑(a (n + 2)) / (↑(a i) * ↑(a (i + 1)))"
},
{
"line": "rewrite [mul_div_cancel_right₀ _ (a_pos _ <| Nat.le_succ _).ne']",
"before_state": "case e_a.e_f.h\nα : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\ni : ℕ\n⊢ ↑(a (n + 2)) * (↑(a 0) * ↑(a (n + 1)) / ↑(a (n + 1))) / (↑(a i) * ↑(a (i + 1))) =\n ↑(a 0) * ↑(a (n + 2)) / (↑(a i) * ↑(a (i + 1)))",
"after_state": "case e_a.e_f.h\nα : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\ni : ℕ\n⊢ ↑(a (n + 2)) * ↑(a 0) / (↑(a i) * ↑(a (i + 1))) = ↑(a 0) * ↑(a (n + 2)) / (↑(a i) * ↑(a (i + 1)))"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "case e_a.e_f.h\nα : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\ni : ℕ\n⊢ ↑(a (n + 2)) * ↑(a 0) / (↑(a i) * ↑(a (i + 1))) = ↑(a 0) * ↑(a (n + 2)) / (↑(a i) * ↑(a (i + 1)))",
"after_state": "case e_a.e_f.h\nα : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\ni : ℕ\n⊢ ↑(a (n + 2)) * ↑(a 0) / (↑(a i) * ↑(a (i + 1))) = ↑(a 0) * ↑(a (n + 2)) / (↑(a i) * ↑(a (i + 1)))"
},
{
"line": "try (with_reducible rfl)",
"before_state": "case e_a.e_f.h\nα : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\ni : ℕ\n⊢ ↑(a (n + 2)) * ↑(a 0) / (↑(a i) * ↑(a (i + 1))) = ↑(a 0) * ↑(a (n + 2)) / (↑(a i) * ↑(a (i + 1)))",
"after_state": "case e_a.e_f.h\nα : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\ni : ℕ\n⊢ ↑(a (n + 2)) * ↑(a 0) / (↑(a i) * ↑(a (i + 1))) = ↑(a 0) * ↑(a (n + 2)) / (↑(a i) * ↑(a (i + 1)))"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "case e_a.e_f.h\nα : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\ni : ℕ\n⊢ ↑(a (n + 2)) * ↑(a 0) / (↑(a i) * ↑(a (i + 1))) = ↑(a 0) * ↑(a (n + 2)) / (↑(a i) * ↑(a (i + 1)))",
"after_state": "case e_a.e_f.h\nα : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\ni : ℕ\n⊢ ↑(a (n + 2)) * ↑(a 0) / (↑(a i) * ↑(a (i + 1))) = ↑(a 0) * ↑(a (n + 2)) / (↑(a i) * ↑(a (i + 1)))"
},
{
"line": "with_reducible rfl",
"before_state": "case e_a.e_f.h\nα : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\ni : ℕ\n⊢ ↑(a (n + 2)) * ↑(a 0) / (↑(a i) * ↑(a (i + 1))) = ↑(a 0) * ↑(a (n + 2)) / (↑(a i) * ↑(a (i + 1)))",
"after_state": "case e_a.e_f.h\nα : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\ni : ℕ\n⊢ ↑(a (n + 2)) * ↑(a 0) / (↑(a i) * ↑(a (i + 1))) = ↑(a 0) * ↑(a (n + 2)) / (↑(a i) * ↑(a (i + 1)))"
},
{
"line": "rfl",
"before_state": "case e_a.e_f.h\nα : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\ni : ℕ\n⊢ ↑(a (n + 2)) * ↑(a 0) / (↑(a i) * ↑(a (i + 1))) = ↑(a 0) * ↑(a (n + 2)) / (↑(a i) * ↑(a (i + 1)))",
"after_state": "case e_a.e_f.h\nα : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\ni : ℕ\n⊢ ↑(a (n + 2)) * ↑(a 0) / (↑(a i) * ↑(a (i + 1))) = ↑(a 0) * ↑(a (n + 2)) / (↑(a i) * ↑(a (i + 1)))"
},
{
"line": "apply_rfl",
"before_state": "case e_a.e_f.h\nα : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\ni : ℕ\n⊢ ↑(a (n + 2)) * ↑(a 0) / (↑(a i) * ↑(a (i + 1))) = ↑(a 0) * ↑(a (n + 2)) / (↑(a i) * ↑(a (i + 1)))",
"after_state": "case e_a.e_f.h\nα : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\ni : ℕ\n⊢ ↑(a (n + 2)) * ↑(a 0) / (↑(a i) * ↑(a (i + 1))) = ↑(a 0) * ↑(a (n + 2)) / (↑(a i) * ↑(a (i + 1)))"
},
{
"line": "skip",
"before_state": "case e_a.e_f.h\nα : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\ni : ℕ\n⊢ ↑(a (n + 2)) * ↑(a 0) / (↑(a i) * ↑(a (i + 1))) = ↑(a 0) * ↑(a (n + 2)) / (↑(a i) * ↑(a (i + 1)))",
"after_state": "case e_a.e_f.h\nα : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\ni : ℕ\n⊢ ↑(a (n + 2)) * ↑(a 0) / (↑(a i) * ↑(a (i + 1))) = ↑(a 0) * ↑(a (n + 2)) / (↑(a i) * ↑(a (i + 1)))"
},
{
"line": "rw [mul_comm]\n -- Check the divisibility condition",
"before_state": "case e_a.e_f.h\nα : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\ni : ℕ\n⊢ ↑(a (n + 2)) * ↑(a 0) / (↑(a i) * ↑(a (i + 1))) = ↑(a 0) * ↑(a (n + 2)) / (↑(a i) * ↑(a (i + 1)))",
"after_state": "No Goals!"
},
{
"line": "rewrite [mul_comm]",
"before_state": "case e_a.e_f.h\nα : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\ni : ℕ\n⊢ ↑(a (n + 2)) * ↑(a 0) / (↑(a i) * ↑(a (i + 1))) = ↑(a 0) * ↑(a (n + 2)) / (↑(a i) * ↑(a (i + 1)))",
"after_state": "case e_a.e_f.h\nα : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\ni : ℕ\n⊢ ↑(a 0) * ↑(a (n + 2)) / (↑(a i) * ↑(a (i + 1))) = ↑(a 0) * ↑(a (n + 2)) / (↑(a i) * ↑(a (i + 1)))"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "case e_a.e_f.h\nα : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\ni : ℕ\n⊢ ↑(a 0) * ↑(a (n + 2)) / (↑(a i) * ↑(a (i + 1))) = ↑(a 0) * ↑(a (n + 2)) / (↑(a i) * ↑(a (i + 1)))",
"after_state": "No Goals!"
},
{
"line": "try (with_reducible rfl)",
"before_state": "case e_a.e_f.h\nα : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\ni : ℕ\n⊢ ↑(a 0) * ↑(a (n + 2)) / (↑(a i) * ↑(a (i + 1))) = ↑(a 0) * ↑(a (n + 2)) / (↑(a i) * ↑(a (i + 1)))",
"after_state": "No Goals!"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "case e_a.e_f.h\nα : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\ni : ℕ\n⊢ ↑(a 0) * ↑(a (n + 2)) / (↑(a i) * ↑(a (i + 1))) = ↑(a 0) * ↑(a (n + 2)) / (↑(a i) * ↑(a (i + 1)))",
"after_state": "No Goals!"
},
{
"line": "with_reducible rfl",
"before_state": "case e_a.e_f.h\nα : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\ni : ℕ\n⊢ ↑(a 0) * ↑(a (n + 2)) / (↑(a i) * ↑(a (i + 1))) = ↑(a 0) * ↑(a (n + 2)) / (↑(a i) * ↑(a (i + 1)))",
"after_state": "No Goals!"
},
{
"line": "rfl",
"before_state": "case e_a.e_f.h\nα : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\ni : ℕ\n⊢ ↑(a 0) * ↑(a (n + 2)) / (↑(a i) * ↑(a (i + 1))) = ↑(a 0) * ↑(a (n + 2)) / (↑(a i) * ↑(a (i + 1)))",
"after_state": "No Goals!"
},
{
"line": "eq_refl",
"before_state": "case e_a.e_f.h\nα : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\ni : ℕ\n⊢ ↑(a 0) * ↑(a (n + 2)) / (↑(a i) * ↑(a (i + 1))) = ↑(a 0) * ↑(a (n + 2)) / (↑(a i) * ↑(a (i + 1)))",
"after_state": "No Goals!"
},
{
"line": "rw [Nat.mul_sub, ← mul_assoc, Nat.mul_div_cancel' ha, add_mul, Nat.mul_div_cancel' han, add_tsub_tsub_cancel ha₀,\n add_tsub_cancel_right]",
"before_state": "case succ.intro.intro.refine_2\nα : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\n⊢ a (n + 1 + 1) ∣ a (n + 1) * ((a n + a (n + 2)) / a (n + 1) * b - (a n * b - a 0) / a (n + 1)) - a 0",
"after_state": "case succ.intro.intro.refine_2\nα : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\n⊢ a (n + 1 + 1) ∣ a (n + 2) * b"
},
{
"line": "rewrite [Nat.mul_sub, ← mul_assoc, Nat.mul_div_cancel' ha, add_mul, Nat.mul_div_cancel' han, add_tsub_tsub_cancel ha₀,\n add_tsub_cancel_right]",
"before_state": "case succ.intro.intro.refine_2\nα : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\n⊢ a (n + 1 + 1) ∣ a (n + 1) * ((a n + a (n + 2)) / a (n + 1) * b - (a n * b - a 0) / a (n + 1)) - a 0",
"after_state": "case succ.intro.intro.refine_2\nα : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\n⊢ a (n + 1 + 1) ∣ a (n + 2) * b"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "case succ.intro.intro.refine_2\nα : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\n⊢ a (n + 1 + 1) ∣ a (n + 2) * b",
"after_state": "case succ.intro.intro.refine_2\nα : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\n⊢ a (n + 1 + 1) ∣ a (n + 2) * b"
},
{
"line": "try (with_reducible rfl)",
"before_state": "case succ.intro.intro.refine_2\nα : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\n⊢ a (n + 1 + 1) ∣ a (n + 2) * b",
"after_state": "case succ.intro.intro.refine_2\nα : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\n⊢ a (n + 1 + 1) ∣ a (n + 2) * b"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "case succ.intro.intro.refine_2\nα : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\n⊢ a (n + 1 + 1) ∣ a (n + 2) * b",
"after_state": "case succ.intro.intro.refine_2\nα : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\n⊢ a (n + 1 + 1) ∣ a (n + 2) * b"
},
{
"line": "with_reducible rfl",
"before_state": "case succ.intro.intro.refine_2\nα : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\n⊢ a (n + 1 + 1) ∣ a (n + 2) * b",
"after_state": "case succ.intro.intro.refine_2\nα : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\n⊢ a (n + 1 + 1) ∣ a (n + 2) * b"
},
{
"line": "rfl",
"before_state": "case succ.intro.intro.refine_2\nα : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\n⊢ a (n + 1 + 1) ∣ a (n + 2) * b",
"after_state": "case succ.intro.intro.refine_2\nα : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\n⊢ a (n + 1 + 1) ∣ a (n + 2) * b"
},
{
"line": "apply_rfl",
"before_state": "case succ.intro.intro.refine_2\nα : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\n⊢ a (n + 1 + 1) ∣ a (n + 2) * b",
"after_state": "case succ.intro.intro.refine_2\nα : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\n⊢ a (n + 1 + 1) ∣ a (n + 2) * b"
},
{
"line": "skip",
"before_state": "case succ.intro.intro.refine_2\nα : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\n⊢ a (n + 1 + 1) ∣ a (n + 2) * b",
"after_state": "case succ.intro.intro.refine_2\nα : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\n⊢ a (n + 1 + 1) ∣ a (n + 2) * b"
},
{
"line": "exact dvd_mul_right _ _",
"before_state": "case succ.intro.intro.refine_2\nα : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : ℕ → ℕ\nn : ℕ\nih :\n (∀ (i : ℕ), i + 2 ≤ n + 1 → a (i + 1) ∣ a i + a (i + 2)) →\n (∀ i ≤ n + 1, 0 < ↑(a i)) →\n ∃ b, ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1))) ∧ a (n + 1) ∣ a n * b - a 0\na_pos : ∀ i ≤ n + 1 + 1, 0 < ↑(a i)\nb : ℕ\nhb : ↑b = ∑ i ∈ Finset.range (n + 1), ↑(a 0) * ↑(a (n + 1)) / (↑(a i) * ↑(a (i + 1)))\nhan : a (n + 1) ∣ a n * b - a 0\nha : a (n + 1) ∣ a n + a (n + 2)\nha₀ : a 0 ≤ a n * b\n⊢ a (n + 1 + 1) ∣ a (n + 2) * b",
"after_state": "No Goals!"
}
] |
theorem cube_root_of_unity_sum (hω : IsPrimitiveRoot ω 3) : 1 + ω + ω ^ 2 = 0 := by
simpa [cyclotomic_prime, Finset.sum_range_succ] using hω.isRoot_cyclotomic (by decide)
|
/root/DuelModelResearch/mathlib4/Archive/Wiedijk100Theorems/SolutionOfCubicQuartic.lean
|
{
"open": [
"Polynomial"
],
"variables": [
"{K : Type*} [Field K] (a b c d e : K) {ω p q r s t u v w x y : K}"
]
}
|
[
{
"line": "simpa [cyclotomic_prime, Finset.sum_range_succ] using hω.isRoot_cyclotomic (by decide)",
"before_state": "K : Type u_1\ninst✝ : Field K\nω : K\nhω : IsPrimitiveRoot ω 3\n⊢ 1 + ω + ω ^ 2 = 0",
"after_state": "No Goals!"
},
{
"line": "decide",
"before_state": "K : Type u_1\ninst✝ : Field K\nω : K\nhω : IsPrimitiveRoot ω 3\n⊢ 0 < 3",
"after_state": "No Goals!"
}
] |
theorem cubic_eq_zero_iff_of_p_eq_zero (ha : a ≠ 0) (hω : IsPrimitiveRoot ω 3)
(hpz : 3 * a * c - b ^ 2 = 0)
(hq : q = (9 * a * b * c - 2 * b ^ 3 - 27 * a ^ 2 * d) / (54 * a ^ 3)) (hs3 : s ^ 3 = 2 * q)
(x : K) :
a * x ^ 3 + b * x ^ 2 + c * x + d = 0 ↔
x = s - b / (3 * a) ∨ x = s * ω - b / (3 * a) ∨ x = s * ω ^ 2 - b / (3 * a) := by
have h₁ : ∀ x a₁ a₂ a₃ : K, x = a₁ ∨ x = a₂ ∨ x = a₃ ↔ (x - a₁) * (x - a₂) * (x - a₃) = 0 := by
intros; simp only [mul_eq_zero, sub_eq_zero, or_assoc]
have hi2 : (2 : K) ≠ 0 := Invertible.ne_zero _
have hi3 : (3 : K) ≠ 0 := Invertible.ne_zero _
have h54 : (54 : K) = 2 * 3 ^ 3 := by norm_num
have hb2 : b ^ 2 = 3 * a * c := by rw [sub_eq_zero] at hpz; rw [hpz]
have hb3 : b ^ 3 = 3 * a * b * c := by rw [pow_succ, hb2]; ring
have h₂ :=
calc
a * x ^ 3 + b * x ^ 2 + c * x + d =
a * (x + b / (3 * a)) ^ 3 + (c - b ^ 2 / (3 * a)) * x + (d - b ^ 3 * a / (3 * a) ^ 3) := by
field_simp; ring
_ = a * (x + b / (3 * a)) ^ 3 + (d - (9 * a * b * c - 2 * b ^ 3) * a / (3 * a) ^ 3) := by
simp only [hb2]; field_simp [ha]; ring
simp only [hb3]; field_simp [ha]; ring
_ = a * ((x + b / (3 * a)) ^ 3 - s ^ 3) := by rw [hs3, hq]; field_simp [h54]; ring
have h₃ : ∀ x, a * x = 0 ↔ x = 0 := by intro x; simp [ha]
have h₄ : ∀ x : K, x ^ 3 - s ^ 3 = (x - s) * (x - s * ω) * (x - s * ω ^ 2) := by
intro x
calc
x ^ 3 - s ^ 3 = (x - s) * (x ^ 2 + x * s + s ^ 2) := by ring
_ = (x - s) * (x ^ 2 - (ω + ω ^ 2) * x * s + (1 + ω + ω ^ 2) * x * s + s ^ 2) := by ring
_ = (x - s) * (x ^ 2 - (ω + ω ^ 2) * x * s + ω ^ 3 * s ^ 2) := by
rw [hω.pow_eq_one]; simp
rw [cube_root_of_unity_sum hω]; simp
_ = (x - s) * (x - s * ω) * (x - s * ω ^ 2) := by ring
rw [h₁]
rw [h₂]
rw [h₃]
rw [h₄ (x + b / (3 * a))]
ring_nf
|
/root/DuelModelResearch/mathlib4/Archive/Wiedijk100Theorems/SolutionOfCubicQuartic.lean
|
{
"open": [
"Polynomial"
],
"variables": [
"{K : Type*} [Field K] (a b c d e : K) {ω p q r s t u v w x y : K}",
"[Invertible (2 : K)] [Invertible (3 : K)]"
]
}
|
[
{
"line": "have h₁ : ∀ x a₁ a₂ a₃ : K, x = a₁ ∨ x = a₂ ∨ x = a₃ ↔ (x - a₁) * (x - a₂) * (x - a₃) = 0 := by intros;\n simp only [mul_eq_zero, sub_eq_zero, or_assoc]",
"before_state": "K : Type u_1\ninst✝² : Field K\na b c d ω q s : K\ninst✝¹ : Invertible 2\ninst✝ : Invertible 3\nha : a ≠ 0\nhω : IsPrimitiveRoot ω 3\nhpz : 3 * a * c - b ^ 2 = 0\nhq : q = (9 * a * b * c - 2 * b ^ 3 - 27 * a ^ 2 * d) / (54 * a ^ 3)\nhs3 : s ^ 3 = 2 * q\nx : K\n⊢ a * x ^ 3 + b * x ^ 2 + c * x + d = 0 ↔ x = s - b / (3 * a) ∨ x = s * ω - b / (3 * a) ∨ x = s * ω ^ 2 - b / (3 * a)",
"after_state": "K : Type u_1\ninst✝² : Field K\na b c d ω q s : K\ninst✝¹ : Invertible 2\ninst✝ : Invertible 3\nha : a ≠ 0\nhω : IsPrimitiveRoot ω 3\nhpz : 3 * a * c - b ^ 2 = 0\nhq : q = (9 * a * b * c - 2 * b ^ 3 - 27 * a ^ 2 * d) / (54 * a ^ 3)\nhs3 : s ^ 3 = 2 * q\nx : K\nh₁ : ∀ (x a₁ a₂ a₃ : K), x = a₁ ∨ x = a₂ ∨ x = a₃ ↔ (x - a₁) * (x - a₂) * (x - a₃) = 0\n⊢ a * x ^ 3 + b * x ^ 2 + c * x + d = 0 ↔ x = s - b / (3 * a) ∨ x = s * ω - b / (3 * a) ∨ x = s * ω ^ 2 - b / (3 * a)"
},
{
"line": "focus\n refine\n no_implicit_lambda%\n (have h₁ : ∀ x a₁ a₂ a₃ : K, x = a₁ ∨ x = a₂ ∨ x = a₃ ↔ (x - a₁) * (x - a₂) * (x - a₃) = 0 := ?body✝;\n ?_)\n case body✝ => with_annotate_state\"by\" (intros; simp only [mul_eq_zero, sub_eq_zero, or_assoc])",
"before_state": "K : Type u_1\ninst✝² : Field K\na b c d ω q s : K\ninst✝¹ : Invertible 2\ninst✝ : Invertible 3\nha : a ≠ 0\nhω : IsPrimitiveRoot ω 3\nhpz : 3 * a * c - b ^ 2 = 0\nhq : q = (9 * a * b * c - 2 * b ^ 3 - 27 * a ^ 2 * d) / (54 * a ^ 3)\nhs3 : s ^ 3 = 2 * q\nx : K\n⊢ a * x ^ 3 + b * x ^ 2 + c * x + d = 0 ↔ x = s - b / (3 * a) ∨ x = s * ω - b / (3 * a) ∨ x = s * ω ^ 2 - b / (3 * a)",
"after_state": "K : Type u_1\ninst✝² : Field K\na b c d ω q s : K\ninst✝¹ : Invertible 2\ninst✝ : Invertible 3\nha : a ≠ 0\nhω : IsPrimitiveRoot ω 3\nhpz : 3 * a * c - b ^ 2 = 0\nhq : q = (9 * a * b * c - 2 * b ^ 3 - 27 * a ^ 2 * d) / (54 * a ^ 3)\nhs3 : s ^ 3 = 2 * q\nx : K\nh₁ : ∀ (x a₁ a₂ a₃ : K), x = a₁ ∨ x = a₂ ∨ x = a₃ ↔ (x - a₁) * (x - a₂) * (x - a₃) = 0\n⊢ a * x ^ 3 + b * x ^ 2 + c * x + d = 0 ↔ x = s - b / (3 * a) ∨ x = s * ω - b / (3 * a) ∨ x = s * ω ^ 2 - b / (3 * a)"
},
{
"line": "refine\n no_implicit_lambda%\n (have h₁ : ∀ x a₁ a₂ a₃ : K, x = a₁ ∨ x = a₂ ∨ x = a₃ ↔ (x - a₁) * (x - a₂) * (x - a₃) = 0 := ?body✝;\n ?_)",
"before_state": "K : Type u_1\ninst✝² : Field K\na b c d ω q s : K\ninst✝¹ : Invertible 2\ninst✝ : Invertible 3\nha : a ≠ 0\nhω : IsPrimitiveRoot ω 3\nhpz : 3 * a * c - b ^ 2 = 0\nhq : q = (9 * a * b * c - 2 * b ^ 3 - 27 * a ^ 2 * d) / (54 * a ^ 3)\nhs3 : s ^ 3 = 2 * q\nx : K\n⊢ a * x ^ 3 + b * x ^ 2 + c * x + d = 0 ↔ x = s - b / (3 * a) ∨ x = s * ω - b / (3 * a) ∨ x = s * ω ^ 2 - b / (3 * a)",
"after_state": "case body\nK : Type u_1\ninst✝² : Field K\na b c d ω q s : K\ninst✝¹ : Invertible 2\ninst✝ : Invertible 3\nha : a ≠ 0\nhω : IsPrimitiveRoot ω 3\nhpz : 3 * a * c - b ^ 2 = 0\nhq : q = (9 * a * b * c - 2 * b ^ 3 - 27 * a ^ 2 * d) / (54 * a ^ 3)\nhs3 : s ^ 3 = 2 * q\nx : K\n⊢ ∀ (x a₁ a₂ a₃ : K), x = a₁ ∨ x = a₂ ∨ x = a₃ ↔ (x - a₁) * (x - a₂) * (x - a₃) = 0\n---\nK : Type u_1\ninst✝² : Field K\na b c d ω q s : K\ninst✝¹ : Invertible 2\ninst✝ : Invertible 3\nha : a ≠ 0\nhω : IsPrimitiveRoot ω 3\nhpz : 3 * a * c - b ^ 2 = 0\nhq : q = (9 * a * b * c - 2 * b ^ 3 - 27 * a ^ 2 * d) / (54 * a ^ 3)\nhs3 : s ^ 3 = 2 * q\nx : K\nh₁ : ∀ (x a₁ a₂ a₃ : K), x = a₁ ∨ x = a₂ ∨ x = a₃ ↔ (x - a₁) * (x - a₂) * (x - a₃) = 0\n⊢ a * x ^ 3 + b * x ^ 2 + c * x + d = 0 ↔ x = s - b / (3 * a) ∨ x = s * ω - b / (3 * a) ∨ x = s * ω ^ 2 - b / (3 * a)"
},
{
"line": "case body✝ => with_annotate_state\"by\" (intros; simp only [mul_eq_zero, sub_eq_zero, or_assoc])",
"before_state": "case body\nK : Type u_1\ninst✝² : Field K\na b c d ω q s : K\ninst✝¹ : Invertible 2\ninst✝ : Invertible 3\nha : a ≠ 0\nhω : IsPrimitiveRoot ω 3\nhpz : 3 * a * c - b ^ 2 = 0\nhq : q = (9 * a * b * c - 2 * b ^ 3 - 27 * a ^ 2 * d) / (54 * a ^ 3)\nhs3 : s ^ 3 = 2 * q\nx : K\n⊢ ∀ (x a₁ a₂ a₃ : K), x = a₁ ∨ x = a₂ ∨ x = a₃ ↔ (x - a₁) * (x - a₂) * (x - a₃) = 0\n---\nK : Type u_1\ninst✝² : Field K\na b c d ω q s : K\ninst✝¹ : Invertible 2\ninst✝ : Invertible 3\nha : a ≠ 0\nhω : IsPrimitiveRoot ω 3\nhpz : 3 * a * c - b ^ 2 = 0\nhq : q = (9 * a * b * c - 2 * b ^ 3 - 27 * a ^ 2 * d) / (54 * a ^ 3)\nhs3 : s ^ 3 = 2 * q\nx : K\nh₁ : ∀ (x a₁ a₂ a₃ : K), x = a₁ ∨ x = a₂ ∨ x = a₃ ↔ (x - a₁) * (x - a₂) * (x - a₃) = 0\n⊢ a * x ^ 3 + b * x ^ 2 + c * x + d = 0 ↔ x = s - b / (3 * a) ∨ x = s * ω - b / (3 * a) ∨ x = s * ω ^ 2 - b / (3 * a)",
"after_state": "K : Type u_1\ninst✝² : Field K\na b c d ω q s : K\ninst✝¹ : Invertible 2\ninst✝ : Invertible 3\nha : a ≠ 0\nhω : IsPrimitiveRoot ω 3\nhpz : 3 * a * c - b ^ 2 = 0\nhq : q = (9 * a * b * c - 2 * b ^ 3 - 27 * a ^ 2 * d) / (54 * a ^ 3)\nhs3 : s ^ 3 = 2 * q\nx : K\nh₁ : ∀ (x a₁ a₂ a₃ : K), x = a₁ ∨ x = a₂ ∨ x = a₃ ↔ (x - a₁) * (x - a₂) * (x - a₃) = 0\n⊢ a * x ^ 3 + b * x ^ 2 + c * x + d = 0 ↔ x = s - b / (3 * a) ∨ x = s * ω - b / (3 * a) ∨ x = s * ω ^ 2 - b / (3 * a)"
},
{
"line": "with_annotate_state\"by\" (intros; simp only [mul_eq_zero, sub_eq_zero, or_assoc])",
"before_state": "K : Type u_1\ninst✝² : Field K\na b c d ω q s : K\ninst✝¹ : Invertible 2\ninst✝ : Invertible 3\nha : a ≠ 0\nhω : IsPrimitiveRoot ω 3\nhpz : 3 * a * c - b ^ 2 = 0\nhq : q = (9 * a * b * c - 2 * b ^ 3 - 27 * a ^ 2 * d) / (54 * a ^ 3)\nhs3 : s ^ 3 = 2 * q\nx : K\n⊢ ∀ (x a₁ a₂ a₃ : K), x = a₁ ∨ x = a₂ ∨ x = a₃ ↔ (x - a₁) * (x - a₂) * (x - a₃) = 0",
"after_state": "No Goals!"
},
{
"line": "intros",
"before_state": "K : Type u_1\ninst✝² : Field K\na b c d ω q s : K\ninst✝¹ : Invertible 2\ninst✝ : Invertible 3\nha : a ≠ 0\nhω : IsPrimitiveRoot ω 3\nhpz : 3 * a * c - b ^ 2 = 0\nhq : q = (9 * a * b * c - 2 * b ^ 3 - 27 * a ^ 2 * d) / (54 * a ^ 3)\nhs3 : s ^ 3 = 2 * q\nx : K\n⊢ ∀ (x a₁ a₂ a₃ : K), x = a₁ ∨ x = a₂ ∨ x = a₃ ↔ (x - a₁) * (x - a₂) * (x - a₃) = 0",
"after_state": "K : Type u_1\ninst✝² : Field K\na b c d ω q s : K\ninst✝¹ : Invertible 2\ninst✝ : Invertible 3\nha : a ≠ 0\nhω : IsPrimitiveRoot ω 3\nhpz : 3 * a * c - b ^ 2 = 0\nhq : q = (9 * a * b * c - 2 * b ^ 3 - 27 * a ^ 2 * d) / (54 * a ^ 3)\nhs3 : s ^ 3 = 2 * q\nx x✝ a₁✝ a₂✝ a₃✝ : K\n⊢ x✝ = a₁✝ ∨ x✝ = a₂✝ ∨ x✝ = a₃✝ ↔ (x✝ - a₁✝) * (x✝ - a₂✝) * (x✝ - a₃✝) = 0"
},
{
"line": "simp only [mul_eq_zero, sub_eq_zero, or_assoc]",
"before_state": "K : Type u_1\ninst✝² : Field K\na b c d ω q s : K\ninst✝¹ : Invertible 2\ninst✝ : Invertible 3\nha : a ≠ 0\nhω : IsPrimitiveRoot ω 3\nhpz : 3 * a * c - b ^ 2 = 0\nhq : q = (9 * a * b * c - 2 * b ^ 3 - 27 * a ^ 2 * d) / (54 * a ^ 3)\nhs3 : s ^ 3 = 2 * q\nx x✝ a₁✝ a₂✝ a₃✝ : K\n⊢ x✝ = a₁✝ ∨ x✝ = a₂✝ ∨ x✝ = a₃✝ ↔ (x✝ - a₁✝) * (x✝ - a₂✝) * (x✝ - a₃✝) = 0",
"after_state": "No Goals!"
},
{
"line": "have hi2 : (2 : K) ≠ 0 := Invertible.ne_zero _",
"before_state": "K : Type u_1\ninst✝² : Field K\na b c d ω q s : K\ninst✝¹ : Invertible 2\ninst✝ : Invertible 3\nha : a ≠ 0\nhω : IsPrimitiveRoot ω 3\nhpz : 3 * a * c - b ^ 2 = 0\nhq : q = (9 * a * b * c - 2 * b ^ 3 - 27 * a ^ 2 * d) / (54 * a ^ 3)\nhs3 : s ^ 3 = 2 * q\nx : K\nh₁ : ∀ (x a₁ a₂ a₃ : K), x = a₁ ∨ x = a₂ ∨ x = a₃ ↔ (x - a₁) * (x - a₂) * (x - a₃) = 0\n⊢ a * x ^ 3 + b * x ^ 2 + c * x + d = 0 ↔ x = s - b / (3 * a) ∨ x = s * ω - b / (3 * a) ∨ x = s * ω ^ 2 - b / (3 * a)",
"after_state": "K : Type u_1\ninst✝² : Field K\na b c d ω q s : K\ninst✝¹ : Invertible 2\ninst✝ : Invertible 3\nha : a ≠ 0\nhω : IsPrimitiveRoot ω 3\nhpz : 3 * a * c - b ^ 2 = 0\nhq : q = (9 * a * b * c - 2 * b ^ 3 - 27 * a ^ 2 * d) / (54 * a ^ 3)\nhs3 : s ^ 3 = 2 * q\nx : K\nh₁ : ∀ (x a₁ a₂ a₃ : K), x = a₁ ∨ x = a₂ ∨ x = a₃ ↔ (x - a₁) * (x - a₂) * (x - a₃) = 0\nhi2 : 2 ≠ 0\n⊢ a * x ^ 3 + b * x ^ 2 + c * x + d = 0 ↔ x = s - b / (3 * a) ∨ x = s * ω - b / (3 * a) ∨ x = s * ω ^ 2 - b / (3 * a)"
},
{
"line": "refine_lift\n have hi2 : (2 : K) ≠ 0 := Invertible.ne_zero _;\n ?_",
"before_state": "K : Type u_1\ninst✝² : Field K\na b c d ω q s : K\ninst✝¹ : Invertible 2\ninst✝ : Invertible 3\nha : a ≠ 0\nhω : IsPrimitiveRoot ω 3\nhpz : 3 * a * c - b ^ 2 = 0\nhq : q = (9 * a * b * c - 2 * b ^ 3 - 27 * a ^ 2 * d) / (54 * a ^ 3)\nhs3 : s ^ 3 = 2 * q\nx : K\nh₁ : ∀ (x a₁ a₂ a₃ : K), x = a₁ ∨ x = a₂ ∨ x = a₃ ↔ (x - a₁) * (x - a₂) * (x - a₃) = 0\n⊢ a * x ^ 3 + b * x ^ 2 + c * x + d = 0 ↔ x = s - b / (3 * a) ∨ x = s * ω - b / (3 * a) ∨ x = s * ω ^ 2 - b / (3 * a)",
"after_state": "K : Type u_1\ninst✝² : Field K\na b c d ω q s : K\ninst✝¹ : Invertible 2\ninst✝ : Invertible 3\nha : a ≠ 0\nhω : IsPrimitiveRoot ω 3\nhpz : 3 * a * c - b ^ 2 = 0\nhq : q = (9 * a * b * c - 2 * b ^ 3 - 27 * a ^ 2 * d) / (54 * a ^ 3)\nhs3 : s ^ 3 = 2 * q\nx : K\nh₁ : ∀ (x a₁ a₂ a₃ : K), x = a₁ ∨ x = a₂ ∨ x = a₃ ↔ (x - a₁) * (x - a₂) * (x - a₃) = 0\nhi2 : 2 ≠ 0\n⊢ a * x ^ 3 + b * x ^ 2 + c * x + d = 0 ↔ x = s - b / (3 * a) ∨ x = s * ω - b / (3 * a) ∨ x = s * ω ^ 2 - b / (3 * a)"
},
{
"line": "focus\n (refine\n no_implicit_lambda%\n (have hi2 : (2 : K) ≠ 0 := Invertible.ne_zero _;\n ?_);\n rotate_right)",
"before_state": "K : Type u_1\ninst✝² : Field K\na b c d ω q s : K\ninst✝¹ : Invertible 2\ninst✝ : Invertible 3\nha : a ≠ 0\nhω : IsPrimitiveRoot ω 3\nhpz : 3 * a * c - b ^ 2 = 0\nhq : q = (9 * a * b * c - 2 * b ^ 3 - 27 * a ^ 2 * d) / (54 * a ^ 3)\nhs3 : s ^ 3 = 2 * q\nx : K\nh₁ : ∀ (x a₁ a₂ a₃ : K), x = a₁ ∨ x = a₂ ∨ x = a₃ ↔ (x - a₁) * (x - a₂) * (x - a₃) = 0\n⊢ a * x ^ 3 + b * x ^ 2 + c * x + d = 0 ↔ x = s - b / (3 * a) ∨ x = s * ω - b / (3 * a) ∨ x = s * ω ^ 2 - b / (3 * a)",
"after_state": "K : Type u_1\ninst✝² : Field K\na b c d ω q s : K\ninst✝¹ : Invertible 2\ninst✝ : Invertible 3\nha : a ≠ 0\nhω : IsPrimitiveRoot ω 3\nhpz : 3 * a * c - b ^ 2 = 0\nhq : q = (9 * a * b * c - 2 * b ^ 3 - 27 * a ^ 2 * d) / (54 * a ^ 3)\nhs3 : s ^ 3 = 2 * q\nx : K\nh₁ : ∀ (x a₁ a₂ a₃ : K), x = a₁ ∨ x = a₂ ∨ x = a₃ ↔ (x - a₁) * (x - a₂) * (x - a₃) = 0\nhi2 : 2 ≠ 0\n⊢ a * x ^ 3 + b * x ^ 2 + c * x + d = 0 ↔ x = s - b / (3 * a) ∨ x = s * ω - b / (3 * a) ∨ x = s * ω ^ 2 - b / (3 * a)"
},
{
"line": "refine\n no_implicit_lambda%\n (have hi2 : (2 : K) ≠ 0 := Invertible.ne_zero _;\n ?_)",
"before_state": "K : Type u_1\ninst✝² : Field K\na b c d ω q s : K\ninst✝¹ : Invertible 2\ninst✝ : Invertible 3\nha : a ≠ 0\nhω : IsPrimitiveRoot ω 3\nhpz : 3 * a * c - b ^ 2 = 0\nhq : q = (9 * a * b * c - 2 * b ^ 3 - 27 * a ^ 2 * d) / (54 * a ^ 3)\nhs3 : s ^ 3 = 2 * q\nx : K\nh₁ : ∀ (x a₁ a₂ a₃ : K), x = a₁ ∨ x = a₂ ∨ x = a₃ ↔ (x - a₁) * (x - a₂) * (x - a₃) = 0\n⊢ a * x ^ 3 + b * x ^ 2 + c * x + d = 0 ↔ x = s - b / (3 * a) ∨ x = s * ω - b / (3 * a) ∨ x = s * ω ^ 2 - b / (3 * a)",
"after_state": "K : Type u_1\ninst✝² : Field K\na b c d ω q s : K\ninst✝¹ : Invertible 2\ninst✝ : Invertible 3\nha : a ≠ 0\nhω : IsPrimitiveRoot ω 3\nhpz : 3 * a * c - b ^ 2 = 0\nhq : q = (9 * a * b * c - 2 * b ^ 3 - 27 * a ^ 2 * d) / (54 * a ^ 3)\nhs3 : s ^ 3 = 2 * q\nx : K\nh₁ : ∀ (x a₁ a₂ a₃ : K), x = a₁ ∨ x = a₂ ∨ x = a₃ ↔ (x - a₁) * (x - a₂) * (x - a₃) = 0\nhi2 : 2 ≠ 0\n⊢ a * x ^ 3 + b * x ^ 2 + c * x + d = 0 ↔ x = s - b / (3 * a) ∨ x = s * ω - b / (3 * a) ∨ x = s * ω ^ 2 - b / (3 * a)"
},
{
"line": "rotate_right",
"before_state": "K : Type u_1\ninst✝² : Field K\na b c d ω q s : K\ninst✝¹ : Invertible 2\ninst✝ : Invertible 3\nha : a ≠ 0\nhω : IsPrimitiveRoot ω 3\nhpz : 3 * a * c - b ^ 2 = 0\nhq : q = (9 * a * b * c - 2 * b ^ 3 - 27 * a ^ 2 * d) / (54 * a ^ 3)\nhs3 : s ^ 3 = 2 * q\nx : K\nh₁ : ∀ (x a₁ a₂ a₃ : K), x = a₁ ∨ x = a₂ ∨ x = a₃ ↔ (x - a₁) * (x - a₂) * (x - a₃) = 0\nhi2 : 2 ≠ 0\n⊢ a * x ^ 3 + b * x ^ 2 + c * x + d = 0 ↔ x = s - b / (3 * a) ∨ x = s * ω - b / (3 * a) ∨ x = s * ω ^ 2 - b / (3 * a)",
"after_state": "K : Type u_1\ninst✝² : Field K\na b c d ω q s : K\ninst✝¹ : Invertible 2\ninst✝ : Invertible 3\nha : a ≠ 0\nhω : IsPrimitiveRoot ω 3\nhpz : 3 * a * c - b ^ 2 = 0\nhq : q = (9 * a * b * c - 2 * b ^ 3 - 27 * a ^ 2 * d) / (54 * a ^ 3)\nhs3 : s ^ 3 = 2 * q\nx : K\nh₁ : ∀ (x a₁ a₂ a₃ : K), x = a₁ ∨ x = a₂ ∨ x = a₃ ↔ (x - a₁) * (x - a₂) * (x - a₃) = 0\nhi2 : 2 ≠ 0\n⊢ a * x ^ 3 + b * x ^ 2 + c * x + d = 0 ↔ x = s - b / (3 * a) ∨ x = s * ω - b / (3 * a) ∨ x = s * ω ^ 2 - b / (3 * a)"
},
{
"line": "have hi3 : (3 : K) ≠ 0 := Invertible.ne_zero _",
"before_state": "K : Type u_1\ninst✝² : Field K\na b c d ω q s : K\ninst✝¹ : Invertible 2\ninst✝ : Invertible 3\nha : a ≠ 0\nhω : IsPrimitiveRoot ω 3\nhpz : 3 * a * c - b ^ 2 = 0\nhq : q = (9 * a * b * c - 2 * b ^ 3 - 27 * a ^ 2 * d) / (54 * a ^ 3)\nhs3 : s ^ 3 = 2 * q\nx : K\nh₁ : ∀ (x a₁ a₂ a₃ : K), x = a₁ ∨ x = a₂ ∨ x = a₃ ↔ (x - a₁) * (x - a₂) * (x - a₃) = 0\nhi2 : 2 ≠ 0\n⊢ a * x ^ 3 + b * x ^ 2 + c * x + d = 0 ↔ x = s - b / (3 * a) ∨ x = s * ω - b / (3 * a) ∨ x = s * ω ^ 2 - b / (3 * a)",
"after_state": "K : Type u_1\ninst✝² : Field K\na b c d ω q s : K\ninst✝¹ : Invertible 2\ninst✝ : Invertible 3\nha : a ≠ 0\nhω : IsPrimitiveRoot ω 3\nhpz : 3 * a * c - b ^ 2 = 0\nhq : q = (9 * a * b * c - 2 * b ^ 3 - 27 * a ^ 2 * d) / (54 * a ^ 3)\nhs3 : s ^ 3 = 2 * q\nx : K\nh₁ : ∀ (x a₁ a₂ a₃ : K), x = a₁ ∨ x = a₂ ∨ x = a₃ ↔ (x - a₁) * (x - a₂) * (x - a₃) = 0\nhi2 : 2 ≠ 0\nhi3 : 3 ≠ 0\n⊢ a * x ^ 3 + b * x ^ 2 + c * x + d = 0 ↔ x = s - b / (3 * a) ∨ x = s * ω - b / (3 * a) ∨ x = s * ω ^ 2 - b / (3 * a)"
},
{
"line": "refine_lift\n have hi3 : (3 : K) ≠ 0 := Invertible.ne_zero _;\n ?_",
"before_state": "K : Type u_1\ninst✝² : Field K\na b c d ω q s : K\ninst✝¹ : Invertible 2\ninst✝ : Invertible 3\nha : a ≠ 0\nhω : IsPrimitiveRoot ω 3\nhpz : 3 * a * c - b ^ 2 = 0\nhq : q = (9 * a * b * c - 2 * b ^ 3 - 27 * a ^ 2 * d) / (54 * a ^ 3)\nhs3 : s ^ 3 = 2 * q\nx : K\nh₁ : ∀ (x a₁ a₂ a₃ : K), x = a₁ ∨ x = a₂ ∨ x = a₃ ↔ (x - a₁) * (x - a₂) * (x - a₃) = 0\nhi2 : 2 ≠ 0\n⊢ a * x ^ 3 + b * x ^ 2 + c * x + d = 0 ↔ x = s - b / (3 * a) ∨ x = s * ω - b / (3 * a) ∨ x = s * ω ^ 2 - b / (3 * a)",
"after_state": "K : Type u_1\ninst✝² : Field K\na b c d ω q s : K\ninst✝¹ : Invertible 2\ninst✝ : Invertible 3\nha : a ≠ 0\nhω : IsPrimitiveRoot ω 3\nhpz : 3 * a * c - b ^ 2 = 0\nhq : q = (9 * a * b * c - 2 * b ^ 3 - 27 * a ^ 2 * d) / (54 * a ^ 3)\nhs3 : s ^ 3 = 2 * q\nx : K\nh₁ : ∀ (x a₁ a₂ a₃ : K), x = a₁ ∨ x = a₂ ∨ x = a₃ ↔ (x - a₁) * (x - a₂) * (x - a₃) = 0\nhi2 : 2 ≠ 0\nhi3 : 3 ≠ 0\n⊢ a * x ^ 3 + b * x ^ 2 + c * x + d = 0 ↔ x = s - b / (3 * a) ∨ x = s * ω - b / (3 * a) ∨ x = s * ω ^ 2 - b / (3 * a)"
},
{
"line": "focus\n (refine\n no_implicit_lambda%\n (have hi3 : (3 : K) ≠ 0 := Invertible.ne_zero _;\n ?_);\n rotate_right)",
"before_state": "K : Type u_1\ninst✝² : Field K\na b c d ω q s : K\ninst✝¹ : Invertible 2\ninst✝ : Invertible 3\nha : a ≠ 0\nhω : IsPrimitiveRoot ω 3\nhpz : 3 * a * c - b ^ 2 = 0\nhq : q = (9 * a * b * c - 2 * b ^ 3 - 27 * a ^ 2 * d) / (54 * a ^ 3)\nhs3 : s ^ 3 = 2 * q\nx : K\nh₁ : ∀ (x a₁ a₂ a₃ : K), x = a₁ ∨ x = a₂ ∨ x = a₃ ↔ (x - a₁) * (x - a₂) * (x - a₃) = 0\nhi2 : 2 ≠ 0\n⊢ a * x ^ 3 + b * x ^ 2 + c * x + d = 0 ↔ x = s - b / (3 * a) ∨ x = s * ω - b / (3 * a) ∨ x = s * ω ^ 2 - b / (3 * a)",
"after_state": "K : Type u_1\ninst✝² : Field K\na b c d ω q s : K\ninst✝¹ : Invertible 2\ninst✝ : Invertible 3\nha : a ≠ 0\nhω : IsPrimitiveRoot ω 3\nhpz : 3 * a * c - b ^ 2 = 0\nhq : q = (9 * a * b * c - 2 * b ^ 3 - 27 * a ^ 2 * d) / (54 * a ^ 3)\nhs3 : s ^ 3 = 2 * q\nx : K\nh₁ : ∀ (x a₁ a₂ a₃ : K), x = a₁ ∨ x = a₂ ∨ x = a₃ ↔ (x - a₁) * (x - a₂) * (x - a₃) = 0\nhi2 : 2 ≠ 0\nhi3 : 3 ≠ 0\n⊢ a * x ^ 3 + b * x ^ 2 + c * x + d = 0 ↔ x = s - b / (3 * a) ∨ x = s * ω - b / (3 * a) ∨ x = s * ω ^ 2 - b / (3 * a)"
},
{
"line": "refine\n no_implicit_lambda%\n (have hi3 : (3 : K) ≠ 0 := Invertible.ne_zero _;\n ?_)",
"before_state": "K : Type u_1\ninst✝² : Field K\na b c d ω q s : K\ninst✝¹ : Invertible 2\ninst✝ : Invertible 3\nha : a ≠ 0\nhω : IsPrimitiveRoot ω 3\nhpz : 3 * a * c - b ^ 2 = 0\nhq : q = (9 * a * b * c - 2 * b ^ 3 - 27 * a ^ 2 * d) / (54 * a ^ 3)\nhs3 : s ^ 3 = 2 * q\nx : K\nh₁ : ∀ (x a₁ a₂ a₃ : K), x = a₁ ∨ x = a₂ ∨ x = a₃ ↔ (x - a₁) * (x - a₂) * (x - a₃) = 0\nhi2 : 2 ≠ 0\n⊢ a * x ^ 3 + b * x ^ 2 + c * x + d = 0 ↔ x = s - b / (3 * a) ∨ x = s * ω - b / (3 * a) ∨ x = s * ω ^ 2 - b / (3 * a)",
"after_state": "K : Type u_1\ninst✝² : Field K\na b c d ω q s : K\ninst✝¹ : Invertible 2\ninst✝ : Invertible 3\nha : a ≠ 0\nhω : IsPrimitiveRoot ω 3\nhpz : 3 * a * c - b ^ 2 = 0\nhq : q = (9 * a * b * c - 2 * b ^ 3 - 27 * a ^ 2 * d) / (54 * a ^ 3)\nhs3 : s ^ 3 = 2 * q\nx : K\nh₁ : ∀ (x a₁ a₂ a₃ : K), x = a₁ ∨ x = a₂ ∨ x = a₃ ↔ (x - a₁) * (x - a₂) * (x - a₃) = 0\nhi2 : 2 ≠ 0\nhi3 : 3 ≠ 0\n⊢ a * x ^ 3 + b * x ^ 2 + c * x + d = 0 ↔ x = s - b / (3 * a) ∨ x = s * ω - b / (3 * a) ∨ x = s * ω ^ 2 - b / (3 * a)"
},
{
"line": "rotate_right",
"before_state": "K : Type u_1\ninst✝² : Field K\na b c d ω q s : K\ninst✝¹ : Invertible 2\ninst✝ : Invertible 3\nha : a ≠ 0\nhω : IsPrimitiveRoot ω 3\nhpz : 3 * a * c - b ^ 2 = 0\nhq : q = (9 * a * b * c - 2 * b ^ 3 - 27 * a ^ 2 * d) / (54 * a ^ 3)\nhs3 : s ^ 3 = 2 * q\nx : K\nh₁ : ∀ (x a₁ a₂ a₃ : K), x = a₁ ∨ x = a₂ ∨ x = a₃ ↔ (x - a₁) * (x - a₂) * (x - a₃) = 0\nhi2 : 2 ≠ 0\nhi3 : 3 ≠ 0\n⊢ a * x ^ 3 + b * x ^ 2 + c * x + d = 0 ↔ x = s - b / (3 * a) ∨ x = s * ω - b / (3 * a) ∨ x = s * ω ^ 2 - b / (3 * a)",
"after_state": "K : Type u_1\ninst✝² : Field K\na b c d ω q s : K\ninst✝¹ : Invertible 2\ninst✝ : Invertible 3\nha : a ≠ 0\nhω : IsPrimitiveRoot ω 3\nhpz : 3 * a * c - b ^ 2 = 0\nhq : q = (9 * a * b * c - 2 * b ^ 3 - 27 * a ^ 2 * d) / (54 * a ^ 3)\nhs3 : s ^ 3 = 2 * q\nx : K\nh₁ : ∀ (x a₁ a₂ a₃ : K), x = a₁ ∨ x = a₂ ∨ x = a₃ ↔ (x - a₁) * (x - a₂) * (x - a₃) = 0\nhi2 : 2 ≠ 0\nhi3 : 3 ≠ 0\n⊢ a * x ^ 3 + b * x ^ 2 + c * x + d = 0 ↔ x = s - b / (3 * a) ∨ x = s * ω - b / (3 * a) ∨ x = s * ω ^ 2 - b / (3 * a)"
},
{
"line": "have h54 : (54 : K) = 2 * 3 ^ 3 := by norm_num",
"before_state": "K : Type u_1\ninst✝² : Field K\na b c d ω q s : K\ninst✝¹ : Invertible 2\ninst✝ : Invertible 3\nha : a ≠ 0\nhω : IsPrimitiveRoot ω 3\nhpz : 3 * a * c - b ^ 2 = 0\nhq : q = (9 * a * b * c - 2 * b ^ 3 - 27 * a ^ 2 * d) / (54 * a ^ 3)\nhs3 : s ^ 3 = 2 * q\nx : K\nh₁ : ∀ (x a₁ a₂ a₃ : K), x = a₁ ∨ x = a₂ ∨ x = a₃ ↔ (x - a₁) * (x - a₂) * (x - a₃) = 0\nhi2 : 2 ≠ 0\nhi3 : 3 ≠ 0\n⊢ a * x ^ 3 + b * x ^ 2 + c * x + d = 0 ↔ x = s - b / (3 * a) ∨ x = s * ω - b / (3 * a) ∨ x = s * ω ^ 2 - b / (3 * a)",
"after_state": "K : Type u_1\ninst✝² : Field K\na b c d ω q s : K\ninst✝¹ : Invertible 2\ninst✝ : Invertible 3\nha : a ≠ 0\nhω : IsPrimitiveRoot ω 3\nhpz : 3 * a * c - b ^ 2 = 0\nhq : q = (9 * a * b * c - 2 * b ^ 3 - 27 * a ^ 2 * d) / (54 * a ^ 3)\nhs3 : s ^ 3 = 2 * q\nx : K\nh₁ : ∀ (x a₁ a₂ a₃ : K), x = a₁ ∨ x = a₂ ∨ x = a₃ ↔ (x - a₁) * (x - a₂) * (x - a₃) = 0\nhi2 : 2 ≠ 0\nhi3 : 3 ≠ 0\nh54 : 54 = 2 * 3 ^ 3\n⊢ a * x ^ 3 + b * x ^ 2 + c * x + d = 0 ↔ x = s - b / (3 * a) ∨ x = s * ω - b / (3 * a) ∨ x = s * ω ^ 2 - b / (3 * a)"
},
{
"line": "focus\n refine\n no_implicit_lambda%\n (have h54 : (54 : K) = 2 * 3 ^ 3 := ?body✝;\n ?_)\n case body✝ => with_annotate_state\"by\" (norm_num)",
"before_state": "K : Type u_1\ninst✝² : Field K\na b c d ω q s : K\ninst✝¹ : Invertible 2\ninst✝ : Invertible 3\nha : a ≠ 0\nhω : IsPrimitiveRoot ω 3\nhpz : 3 * a * c - b ^ 2 = 0\nhq : q = (9 * a * b * c - 2 * b ^ 3 - 27 * a ^ 2 * d) / (54 * a ^ 3)\nhs3 : s ^ 3 = 2 * q\nx : K\nh₁ : ∀ (x a₁ a₂ a₃ : K), x = a₁ ∨ x = a₂ ∨ x = a₃ ↔ (x - a₁) * (x - a₂) * (x - a₃) = 0\nhi2 : 2 ≠ 0\nhi3 : 3 ≠ 0\n⊢ a * x ^ 3 + b * x ^ 2 + c * x + d = 0 ↔ x = s - b / (3 * a) ∨ x = s * ω - b / (3 * a) ∨ x = s * ω ^ 2 - b / (3 * a)",
"after_state": "K : Type u_1\ninst✝² : Field K\na b c d ω q s : K\ninst✝¹ : Invertible 2\ninst✝ : Invertible 3\nha : a ≠ 0\nhω : IsPrimitiveRoot ω 3\nhpz : 3 * a * c - b ^ 2 = 0\nhq : q = (9 * a * b * c - 2 * b ^ 3 - 27 * a ^ 2 * d) / (54 * a ^ 3)\nhs3 : s ^ 3 = 2 * q\nx : K\nh₁ : ∀ (x a₁ a₂ a₃ : K), x = a₁ ∨ x = a₂ ∨ x = a₃ ↔ (x - a₁) * (x - a₂) * (x - a₃) = 0\nhi2 : 2 ≠ 0\nhi3 : 3 ≠ 0\nh54 : 54 = 2 * 3 ^ 3\n⊢ a * x ^ 3 + b * x ^ 2 + c * x + d = 0 ↔ x = s - b / (3 * a) ∨ x = s * ω - b / (3 * a) ∨ x = s * ω ^ 2 - b / (3 * a)"
},
{
"line": "refine\n no_implicit_lambda%\n (have h54 : (54 : K) = 2 * 3 ^ 3 := ?body✝;\n ?_)",
"before_state": "K : Type u_1\ninst✝² : Field K\na b c d ω q s : K\ninst✝¹ : Invertible 2\ninst✝ : Invertible 3\nha : a ≠ 0\nhω : IsPrimitiveRoot ω 3\nhpz : 3 * a * c - b ^ 2 = 0\nhq : q = (9 * a * b * c - 2 * b ^ 3 - 27 * a ^ 2 * d) / (54 * a ^ 3)\nhs3 : s ^ 3 = 2 * q\nx : K\nh₁ : ∀ (x a₁ a₂ a₃ : K), x = a₁ ∨ x = a₂ ∨ x = a₃ ↔ (x - a₁) * (x - a₂) * (x - a₃) = 0\nhi2 : 2 ≠ 0\nhi3 : 3 ≠ 0\n⊢ a * x ^ 3 + b * x ^ 2 + c * x + d = 0 ↔ x = s - b / (3 * a) ∨ x = s * ω - b / (3 * a) ∨ x = s * ω ^ 2 - b / (3 * a)",
"after_state": "case body\nK : Type u_1\ninst✝² : Field K\na b c d ω q s : K\ninst✝¹ : Invertible 2\ninst✝ : Invertible 3\nha : a ≠ 0\nhω : IsPrimitiveRoot ω 3\nhpz : 3 * a * c - b ^ 2 = 0\nhq : q = (9 * a * b * c - 2 * b ^ 3 - 27 * a ^ 2 * d) / (54 * a ^ 3)\nhs3 : s ^ 3 = 2 * q\nx : K\nh₁ : ∀ (x a₁ a₂ a₃ : K), x = a₁ ∨ x = a₂ ∨ x = a₃ ↔ (x - a₁) * (x - a₂) * (x - a₃) = 0\nhi2 : 2 ≠ 0\nhi3 : 3 ≠ 0\n⊢ 54 = 2 * 3 ^ 3\n---\nK : Type u_1\ninst✝² : Field K\na b c d ω q s : K\ninst✝¹ : Invertible 2\ninst✝ : Invertible 3\nha : a ≠ 0\nhω : IsPrimitiveRoot ω 3\nhpz : 3 * a * c - b ^ 2 = 0\nhq : q = (9 * a * b * c - 2 * b ^ 3 - 27 * a ^ 2 * d) / (54 * a ^ 3)\nhs3 : s ^ 3 = 2 * q\nx : K\nh₁ : ∀ (x a₁ a₂ a₃ : K), x = a₁ ∨ x = a₂ ∨ x = a₃ ↔ (x - a₁) * (x - a₂) * (x - a₃) = 0\nhi2 : 2 ≠ 0\nhi3 : 3 ≠ 0\nh54 : 54 = 2 * 3 ^ 3\n⊢ a * x ^ 3 + b * x ^ 2 + c * x + d = 0 ↔ x = s - b / (3 * a) ∨ x = s * ω - b / (3 * a) ∨ x = s * ω ^ 2 - b / (3 * a)"
},
{
"line": "case body✝ => with_annotate_state\"by\" (norm_num)",
"before_state": "case body\nK : Type u_1\ninst✝² : Field K\na b c d ω q s : K\ninst✝¹ : Invertible 2\ninst✝ : Invertible 3\nha : a ≠ 0\nhω : IsPrimitiveRoot ω 3\nhpz : 3 * a * c - b ^ 2 = 0\nhq : q = (9 * a * b * c - 2 * b ^ 3 - 27 * a ^ 2 * d) / (54 * a ^ 3)\nhs3 : s ^ 3 = 2 * q\nx : K\nh₁ : ∀ (x a₁ a₂ a₃ : K), x = a₁ ∨ x = a₂ ∨ x = a₃ ↔ (x - a₁) * (x - a₂) * (x - a₃) = 0\nhi2 : 2 ≠ 0\nhi3 : 3 ≠ 0\n⊢ 54 = 2 * 3 ^ 3\n---\nK : Type u_1\ninst✝² : Field K\na b c d ω q s : K\ninst✝¹ : Invertible 2\ninst✝ : Invertible 3\nha : a ≠ 0\nhω : IsPrimitiveRoot ω 3\nhpz : 3 * a * c - b ^ 2 = 0\nhq : q = (9 * a * b * c - 2 * b ^ 3 - 27 * a ^ 2 * d) / (54 * a ^ 3)\nhs3 : s ^ 3 = 2 * q\nx : K\nh₁ : ∀ (x a₁ a₂ a₃ : K), x = a₁ ∨ x = a₂ ∨ x = a₃ ↔ (x - a₁) * (x - a₂) * (x - a₃) = 0\nhi2 : 2 ≠ 0\nhi3 : 3 ≠ 0\nh54 : 54 = 2 * 3 ^ 3\n⊢ a * x ^ 3 + b * x ^ 2 + c * x + d = 0 ↔ x = s - b / (3 * a) ∨ x = s * ω - b / (3 * a) ∨ x = s * ω ^ 2 - b / (3 * a)",
"after_state": "K : Type u_1\ninst✝² : Field K\na b c d ω q s : K\ninst✝¹ : Invertible 2\ninst✝ : Invertible 3\nha : a ≠ 0\nhω : IsPrimitiveRoot ω 3\nhpz : 3 * a * c - b ^ 2 = 0\nhq : q = (9 * a * b * c - 2 * b ^ 3 - 27 * a ^ 2 * d) / (54 * a ^ 3)\nhs3 : s ^ 3 = 2 * q\nx : K\nh₁ : ∀ (x a₁ a₂ a₃ : K), x = a₁ ∨ x = a₂ ∨ x = a₃ ↔ (x - a₁) * (x - a₂) * (x - a₃) = 0\nhi2 : 2 ≠ 0\nhi3 : 3 ≠ 0\nh54 : 54 = 2 * 3 ^ 3\n⊢ a * x ^ 3 + b * x ^ 2 + c * x + d = 0 ↔ x = s - b / (3 * a) ∨ x = s * ω - b / (3 * a) ∨ x = s * ω ^ 2 - b / (3 * a)"
},
{
"line": "with_annotate_state\"by\" (norm_num)",
"before_state": "K : Type u_1\ninst✝² : Field K\na b c d ω q s : K\ninst✝¹ : Invertible 2\ninst✝ : Invertible 3\nha : a ≠ 0\nhω : IsPrimitiveRoot ω 3\nhpz : 3 * a * c - b ^ 2 = 0\nhq : q = (9 * a * b * c - 2 * b ^ 3 - 27 * a ^ 2 * d) / (54 * a ^ 3)\nhs3 : s ^ 3 = 2 * q\nx : K\nh₁ : ∀ (x a₁ a₂ a₃ : K), x = a₁ ∨ x = a₂ ∨ x = a₃ ↔ (x - a₁) * (x - a₂) * (x - a₃) = 0\nhi2 : 2 ≠ 0\nhi3 : 3 ≠ 0\n⊢ 54 = 2 * 3 ^ 3",
"after_state": "No Goals!"
},
{
"line": "norm_num",
"before_state": "K : Type u_1\ninst✝² : Field K\na b c d ω q s : K\ninst✝¹ : Invertible 2\ninst✝ : Invertible 3\nha : a ≠ 0\nhω : IsPrimitiveRoot ω 3\nhpz : 3 * a * c - b ^ 2 = 0\nhq : q = (9 * a * b * c - 2 * b ^ 3 - 27 * a ^ 2 * d) / (54 * a ^ 3)\nhs3 : s ^ 3 = 2 * q\nx : K\nh₁ : ∀ (x a₁ a₂ a₃ : K), x = a₁ ∨ x = a₂ ∨ x = a₃ ↔ (x - a₁) * (x - a₂) * (x - a₃) = 0\nhi2 : 2 ≠ 0\nhi3 : 3 ≠ 0\n⊢ 54 = 2 * 3 ^ 3",
"after_state": "No Goals!"
},
{
"line": "have hb2 : b ^ 2 = 3 * a * c := by rw [sub_eq_zero] at hpz; rw [hpz]",
"before_state": "K : Type u_1\ninst✝² : Field K\na b c d ω q s : K\ninst✝¹ : Invertible 2\ninst✝ : Invertible 3\nha : a ≠ 0\nhω : IsPrimitiveRoot ω 3\nhpz : 3 * a * c - b ^ 2 = 0\nhq : q = (9 * a * b * c - 2 * b ^ 3 - 27 * a ^ 2 * d) / (54 * a ^ 3)\nhs3 : s ^ 3 = 2 * q\nx : K\nh₁ : ∀ (x a₁ a₂ a₃ : K), x = a₁ ∨ x = a₂ ∨ x = a₃ ↔ (x - a₁) * (x - a₂) * (x - a₃) = 0\nhi2 : 2 ≠ 0\nhi3 : 3 ≠ 0\nh54 : 54 = 2 * 3 ^ 3\n⊢ a * x ^ 3 + b * x ^ 2 + c * x + d = 0 ↔ x = s - b / (3 * a) ∨ x = s * ω - b / (3 * a) ∨ x = s * ω ^ 2 - b / (3 * a)",
"after_state": "K : Type u_1\ninst✝² : Field K\na b c d ω q s : K\ninst✝¹ : Invertible 2\ninst✝ : Invertible 3\nha : a ≠ 0\nhω : IsPrimitiveRoot ω 3\nhpz : 3 * a * c - b ^ 2 = 0\nhq : q = (9 * a * b * c - 2 * b ^ 3 - 27 * a ^ 2 * d) / (54 * a ^ 3)\nhs3 : s ^ 3 = 2 * q\nx : K\nh₁ : ∀ (x a₁ a₂ a₃ : K), x = a₁ ∨ x = a₂ ∨ x = a₃ ↔ (x - a₁) * (x - a₂) * (x - a₃) = 0\nhi2 : 2 ≠ 0\nhi3 : 3 ≠ 0\nh54 : 54 = 2 * 3 ^ 3\nhb2 : b ^ 2 = 3 * a * c\n⊢ a * x ^ 3 + b * x ^ 2 + c * x + d = 0 ↔ x = s - b / (3 * a) ∨ x = s * ω - b / (3 * a) ∨ x = s * ω ^ 2 - b / (3 * a)"
},
{
"line": "focus\n refine\n no_implicit_lambda%\n (have hb2 : b ^ 2 = 3 * a * c := ?body✝;\n ?_)\n case body✝ => with_annotate_state\"by\" (rw [sub_eq_zero] at hpz; rw [hpz])",
"before_state": "K : Type u_1\ninst✝² : Field K\na b c d ω q s : K\ninst✝¹ : Invertible 2\ninst✝ : Invertible 3\nha : a ≠ 0\nhω : IsPrimitiveRoot ω 3\nhpz : 3 * a * c - b ^ 2 = 0\nhq : q = (9 * a * b * c - 2 * b ^ 3 - 27 * a ^ 2 * d) / (54 * a ^ 3)\nhs3 : s ^ 3 = 2 * q\nx : K\nh₁ : ∀ (x a₁ a₂ a₃ : K), x = a₁ ∨ x = a₂ ∨ x = a₃ ↔ (x - a₁) * (x - a₂) * (x - a₃) = 0\nhi2 : 2 ≠ 0\nhi3 : 3 ≠ 0\nh54 : 54 = 2 * 3 ^ 3\n⊢ a * x ^ 3 + b * x ^ 2 + c * x + d = 0 ↔ x = s - b / (3 * a) ∨ x = s * ω - b / (3 * a) ∨ x = s * ω ^ 2 - b / (3 * a)",
"after_state": "K : Type u_1\ninst✝² : Field K\na b c d ω q s : K\ninst✝¹ : Invertible 2\ninst✝ : Invertible 3\nha : a ≠ 0\nhω : IsPrimitiveRoot ω 3\nhpz : 3 * a * c - b ^ 2 = 0\nhq : q = (9 * a * b * c - 2 * b ^ 3 - 27 * a ^ 2 * d) / (54 * a ^ 3)\nhs3 : s ^ 3 = 2 * q\nx : K\nh₁ : ∀ (x a₁ a₂ a₃ : K), x = a₁ ∨ x = a₂ ∨ x = a₃ ↔ (x - a₁) * (x - a₂) * (x - a₃) = 0\nhi2 : 2 ≠ 0\nhi3 : 3 ≠ 0\nh54 : 54 = 2 * 3 ^ 3\nhb2 : b ^ 2 = 3 * a * c\n⊢ a * x ^ 3 + b * x ^ 2 + c * x + d = 0 ↔ x = s - b / (3 * a) ∨ x = s * ω - b / (3 * a) ∨ x = s * ω ^ 2 - b / (3 * a)"
},
{
"line": "refine\n no_implicit_lambda%\n (have hb2 : b ^ 2 = 3 * a * c := ?body✝;\n ?_)",
"before_state": "K : Type u_1\ninst✝² : Field K\na b c d ω q s : K\ninst✝¹ : Invertible 2\ninst✝ : Invertible 3\nha : a ≠ 0\nhω : IsPrimitiveRoot ω 3\nhpz : 3 * a * c - b ^ 2 = 0\nhq : q = (9 * a * b * c - 2 * b ^ 3 - 27 * a ^ 2 * d) / (54 * a ^ 3)\nhs3 : s ^ 3 = 2 * q\nx : K\nh₁ : ∀ (x a₁ a₂ a₃ : K), x = a₁ ∨ x = a₂ ∨ x = a₃ ↔ (x - a₁) * (x - a₂) * (x - a₃) = 0\nhi2 : 2 ≠ 0\nhi3 : 3 ≠ 0\nh54 : 54 = 2 * 3 ^ 3\n⊢ a * x ^ 3 + b * x ^ 2 + c * x + d = 0 ↔ x = s - b / (3 * a) ∨ x = s * ω - b / (3 * a) ∨ x = s * ω ^ 2 - b / (3 * a)",
"after_state": "case body\nK : Type u_1\ninst✝² : Field K\na b c d ω q s : K\ninst✝¹ : Invertible 2\ninst✝ : Invertible 3\nha : a ≠ 0\nhω : IsPrimitiveRoot ω 3\nhpz : 3 * a * c - b ^ 2 = 0\nhq : q = (9 * a * b * c - 2 * b ^ 3 - 27 * a ^ 2 * d) / (54 * a ^ 3)\nhs3 : s ^ 3 = 2 * q\nx : K\nh₁ : ∀ (x a₁ a₂ a₃ : K), x = a₁ ∨ x = a₂ ∨ x = a₃ ↔ (x - a₁) * (x - a₂) * (x - a₃) = 0\nhi2 : 2 ≠ 0\nhi3 : 3 ≠ 0\nh54 : 54 = 2 * 3 ^ 3\n⊢ b ^ 2 = 3 * a * c\n---\nK : Type u_1\ninst✝² : Field K\na b c d ω q s : K\ninst✝¹ : Invertible 2\ninst✝ : Invertible 3\nha : a ≠ 0\nhω : IsPrimitiveRoot ω 3\nhpz : 3 * a * c - b ^ 2 = 0\nhq : q = (9 * a * b * c - 2 * b ^ 3 - 27 * a ^ 2 * d) / (54 * a ^ 3)\nhs3 : s ^ 3 = 2 * q\nx : K\nh₁ : ∀ (x a₁ a₂ a₃ : K), x = a₁ ∨ x = a₂ ∨ x = a₃ ↔ (x - a₁) * (x - a₂) * (x - a₃) = 0\nhi2 : 2 ≠ 0\nhi3 : 3 ≠ 0\nh54 : 54 = 2 * 3 ^ 3\nhb2 : b ^ 2 = 3 * a * c\n⊢ a * x ^ 3 + b * x ^ 2 + c * x + d = 0 ↔ x = s - b / (3 * a) ∨ x = s * ω - b / (3 * a) ∨ x = s * ω ^ 2 - b / (3 * a)"
},
{
"line": "case body✝ => with_annotate_state\"by\" (rw [sub_eq_zero] at hpz; rw [hpz])",
"before_state": "case body\nK : Type u_1\ninst✝² : Field K\na b c d ω q s : K\ninst✝¹ : Invertible 2\ninst✝ : Invertible 3\nha : a ≠ 0\nhω : IsPrimitiveRoot ω 3\nhpz : 3 * a * c - b ^ 2 = 0\nhq : q = (9 * a * b * c - 2 * b ^ 3 - 27 * a ^ 2 * d) / (54 * a ^ 3)\nhs3 : s ^ 3 = 2 * q\nx : K\nh₁ : ∀ (x a₁ a₂ a₃ : K), x = a₁ ∨ x = a₂ ∨ x = a₃ ↔ (x - a₁) * (x - a₂) * (x - a₃) = 0\nhi2 : 2 ≠ 0\nhi3 : 3 ≠ 0\nh54 : 54 = 2 * 3 ^ 3\n⊢ b ^ 2 = 3 * a * c\n---\nK : Type u_1\ninst✝² : Field K\na b c d ω q s : K\ninst✝¹ : Invertible 2\ninst✝ : Invertible 3\nha : a ≠ 0\nhω : IsPrimitiveRoot ω 3\nhpz : 3 * a * c - b ^ 2 = 0\nhq : q = (9 * a * b * c - 2 * b ^ 3 - 27 * a ^ 2 * d) / (54 * a ^ 3)\nhs3 : s ^ 3 = 2 * q\nx : K\nh₁ : ∀ (x a₁ a₂ a₃ : K), x = a₁ ∨ x = a₂ ∨ x = a₃ ↔ (x - a₁) * (x - a₂) * (x - a₃) = 0\nhi2 : 2 ≠ 0\nhi3 : 3 ≠ 0\nh54 : 54 = 2 * 3 ^ 3\nhb2 : b ^ 2 = 3 * a * c\n⊢ a * x ^ 3 + b * x ^ 2 + c * x + d = 0 ↔ x = s - b / (3 * a) ∨ x = s * ω - b / (3 * a) ∨ x = s * ω ^ 2 - b / (3 * a)",
"after_state": "K : Type u_1\ninst✝² : Field K\na b c d ω q s : K\ninst✝¹ : Invertible 2\ninst✝ : Invertible 3\nha : a ≠ 0\nhω : IsPrimitiveRoot ω 3\nhpz : 3 * a * c - b ^ 2 = 0\nhq : q = (9 * a * b * c - 2 * b ^ 3 - 27 * a ^ 2 * d) / (54 * a ^ 3)\nhs3 : s ^ 3 = 2 * q\nx : K\nh₁ : ∀ (x a₁ a₂ a₃ : K), x = a₁ ∨ x = a₂ ∨ x = a₃ ↔ (x - a₁) * (x - a₂) * (x - a₃) = 0\nhi2 : 2 ≠ 0\nhi3 : 3 ≠ 0\nh54 : 54 = 2 * 3 ^ 3\nhb2 : b ^ 2 = 3 * a * c\n⊢ a * x ^ 3 + b * x ^ 2 + c * x + d = 0 ↔ x = s - b / (3 * a) ∨ x = s * ω - b / (3 * a) ∨ x = s * ω ^ 2 - b / (3 * a)"
},
{
"line": "with_annotate_state\"by\" (rw [sub_eq_zero] at hpz; rw [hpz])",
"before_state": "K : Type u_1\ninst✝² : Field K\na b c d ω q s : K\ninst✝¹ : Invertible 2\ninst✝ : Invertible 3\nha : a ≠ 0\nhω : IsPrimitiveRoot ω 3\nhpz : 3 * a * c - b ^ 2 = 0\nhq : q = (9 * a * b * c - 2 * b ^ 3 - 27 * a ^ 2 * d) / (54 * a ^ 3)\nhs3 : s ^ 3 = 2 * q\nx : K\nh₁ : ∀ (x a₁ a₂ a₃ : K), x = a₁ ∨ x = a₂ ∨ x = a₃ ↔ (x - a₁) * (x - a₂) * (x - a₃) = 0\nhi2 : 2 ≠ 0\nhi3 : 3 ≠ 0\nh54 : 54 = 2 * 3 ^ 3\n⊢ b ^ 2 = 3 * a * c",
"after_state": "No Goals!"
},
{
"line": "rw [sub_eq_zero] at hpz",
"before_state": "K : Type u_1\ninst✝² : Field K\na b c d ω q s : K\ninst✝¹ : Invertible 2\ninst✝ : Invertible 3\nha : a ≠ 0\nhω : IsPrimitiveRoot ω 3\nhpz : 3 * a * c - b ^ 2 = 0\nhq : q = (9 * a * b * c - 2 * b ^ 3 - 27 * a ^ 2 * d) / (54 * a ^ 3)\nhs3 : s ^ 3 = 2 * q\nx : K\nh₁ : ∀ (x a₁ a₂ a₃ : K), x = a₁ ∨ x = a₂ ∨ x = a₃ ↔ (x - a₁) * (x - a₂) * (x - a₃) = 0\nhi2 : 2 ≠ 0\nhi3 : 3 ≠ 0\nh54 : 54 = 2 * 3 ^ 3\n⊢ b ^ 2 = 3 * a * c",
"after_state": "K : Type u_1\ninst✝² : Field K\na b c d ω q s : K\ninst✝¹ : Invertible 2\ninst✝ : Invertible 3\nha : a ≠ 0\nhω : IsPrimitiveRoot ω 3\nhpz : 3 * a * c = b ^ 2\nhq : q = (9 * a * b * c - 2 * b ^ 3 - 27 * a ^ 2 * d) / (54 * a ^ 3)\nhs3 : s ^ 3 = 2 * q\nx : K\nh₁ : ∀ (x a₁ a₂ a₃ : K), x = a₁ ∨ x = a₂ ∨ x = a₃ ↔ (x - a₁) * (x - a₂) * (x - a₃) = 0\nhi2 : 2 ≠ 0\nhi3 : 3 ≠ 0\nh54 : 54 = 2 * 3 ^ 3\n⊢ b ^ 2 = 3 * a * c"
},
{
"line": "rewrite [sub_eq_zero] at hpz",
"before_state": "K : Type u_1\ninst✝² : Field K\na b c d ω q s : K\ninst✝¹ : Invertible 2\ninst✝ : Invertible 3\nha : a ≠ 0\nhω : IsPrimitiveRoot ω 3\nhpz : 3 * a * c - b ^ 2 = 0\nhq : q = (9 * a * b * c - 2 * b ^ 3 - 27 * a ^ 2 * d) / (54 * a ^ 3)\nhs3 : s ^ 3 = 2 * q\nx : K\nh₁ : ∀ (x a₁ a₂ a₃ : K), x = a₁ ∨ x = a₂ ∨ x = a₃ ↔ (x - a₁) * (x - a₂) * (x - a₃) = 0\nhi2 : 2 ≠ 0\nhi3 : 3 ≠ 0\nh54 : 54 = 2 * 3 ^ 3\n⊢ b ^ 2 = 3 * a * c",
"after_state": "K : Type u_1\ninst✝² : Field K\na b c d ω q s : K\ninst✝¹ : Invertible 2\ninst✝ : Invertible 3\nha : a ≠ 0\nhω : IsPrimitiveRoot ω 3\nhpz : 3 * a * c = b ^ 2\nhq : q = (9 * a * b * c - 2 * b ^ 3 - 27 * a ^ 2 * d) / (54 * a ^ 3)\nhs3 : s ^ 3 = 2 * q\nx : K\nh₁ : ∀ (x a₁ a₂ a₃ : K), x = a₁ ∨ x = a₂ ∨ x = a₃ ↔ (x - a₁) * (x - a₂) * (x - a₃) = 0\nhi2 : 2 ≠ 0\nhi3 : 3 ≠ 0\nh54 : 54 = 2 * 3 ^ 3\n⊢ b ^ 2 = 3 * a * c"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "K : Type u_1\ninst✝² : Field K\na b c d ω q s : K\ninst✝¹ : Invertible 2\ninst✝ : Invertible 3\nha : a ≠ 0\nhω : IsPrimitiveRoot ω 3\nhpz : 3 * a * c = b ^ 2\nhq : q = (9 * a * b * c - 2 * b ^ 3 - 27 * a ^ 2 * d) / (54 * a ^ 3)\nhs3 : s ^ 3 = 2 * q\nx : K\nh₁ : ∀ (x a₁ a₂ a₃ : K), x = a₁ ∨ x = a₂ ∨ x = a₃ ↔ (x - a₁) * (x - a₂) * (x - a₃) = 0\nhi2 : 2 ≠ 0\nhi3 : 3 ≠ 0\nh54 : 54 = 2 * 3 ^ 3\n⊢ b ^ 2 = 3 * a * c",
"after_state": "K : Type u_1\ninst✝² : Field K\na b c d ω q s : K\ninst✝¹ : Invertible 2\ninst✝ : Invertible 3\nha : a ≠ 0\nhω : IsPrimitiveRoot ω 3\nhpz : 3 * a * c = b ^ 2\nhq : q = (9 * a * b * c - 2 * b ^ 3 - 27 * a ^ 2 * d) / (54 * a ^ 3)\nhs3 : s ^ 3 = 2 * q\nx : K\nh₁ : ∀ (x a₁ a₂ a₃ : K), x = a₁ ∨ x = a₂ ∨ x = a₃ ↔ (x - a₁) * (x - a₂) * (x - a₃) = 0\nhi2 : 2 ≠ 0\nhi3 : 3 ≠ 0\nh54 : 54 = 2 * 3 ^ 3\n⊢ b ^ 2 = 3 * a * c"
},
{
"line": "try (with_reducible rfl)",
"before_state": "K : Type u_1\ninst✝² : Field K\na b c d ω q s : K\ninst✝¹ : Invertible 2\ninst✝ : Invertible 3\nha : a ≠ 0\nhω : IsPrimitiveRoot ω 3\nhpz : 3 * a * c = b ^ 2\nhq : q = (9 * a * b * c - 2 * b ^ 3 - 27 * a ^ 2 * d) / (54 * a ^ 3)\nhs3 : s ^ 3 = 2 * q\nx : K\nh₁ : ∀ (x a₁ a₂ a₃ : K), x = a₁ ∨ x = a₂ ∨ x = a₃ ↔ (x - a₁) * (x - a₂) * (x - a₃) = 0\nhi2 : 2 ≠ 0\nhi3 : 3 ≠ 0\nh54 : 54 = 2 * 3 ^ 3\n⊢ b ^ 2 = 3 * a * c",
"after_state": "K : Type u_1\ninst✝² : Field K\na b c d ω q s : K\ninst✝¹ : Invertible 2\ninst✝ : Invertible 3\nha : a ≠ 0\nhω : IsPrimitiveRoot ω 3\nhpz : 3 * a * c = b ^ 2\nhq : q = (9 * a * b * c - 2 * b ^ 3 - 27 * a ^ 2 * d) / (54 * a ^ 3)\nhs3 : s ^ 3 = 2 * q\nx : K\nh₁ : ∀ (x a₁ a₂ a₃ : K), x = a₁ ∨ x = a₂ ∨ x = a₃ ↔ (x - a₁) * (x - a₂) * (x - a₃) = 0\nhi2 : 2 ≠ 0\nhi3 : 3 ≠ 0\nh54 : 54 = 2 * 3 ^ 3\n⊢ b ^ 2 = 3 * a * c"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "K : Type u_1\ninst✝² : Field K\na b c d ω q s : K\ninst✝¹ : Invertible 2\ninst✝ : Invertible 3\nha : a ≠ 0\nhω : IsPrimitiveRoot ω 3\nhpz : 3 * a * c = b ^ 2\nhq : q = (9 * a * b * c - 2 * b ^ 3 - 27 * a ^ 2 * d) / (54 * a ^ 3)\nhs3 : s ^ 3 = 2 * q\nx : K\nh₁ : ∀ (x a₁ a₂ a₃ : K), x = a₁ ∨ x = a₂ ∨ x = a₃ ↔ (x - a₁) * (x - a₂) * (x - a₃) = 0\nhi2 : 2 ≠ 0\nhi3 : 3 ≠ 0\nh54 : 54 = 2 * 3 ^ 3\n⊢ b ^ 2 = 3 * a * c",
"after_state": "K : Type u_1\ninst✝² : Field K\na b c d ω q s : K\ninst✝¹ : Invertible 2\ninst✝ : Invertible 3\nha : a ≠ 0\nhω : IsPrimitiveRoot ω 3\nhpz : 3 * a * c = b ^ 2\nhq : q = (9 * a * b * c - 2 * b ^ 3 - 27 * a ^ 2 * d) / (54 * a ^ 3)\nhs3 : s ^ 3 = 2 * q\nx : K\nh₁ : ∀ (x a₁ a₂ a₃ : K), x = a₁ ∨ x = a₂ ∨ x = a₃ ↔ (x - a₁) * (x - a₂) * (x - a₃) = 0\nhi2 : 2 ≠ 0\nhi3 : 3 ≠ 0\nh54 : 54 = 2 * 3 ^ 3\n⊢ b ^ 2 = 3 * a * c"
},
{
"line": "with_reducible rfl",
"before_state": "K : Type u_1\ninst✝² : Field K\na b c d ω q s : K\ninst✝¹ : Invertible 2\ninst✝ : Invertible 3\nha : a ≠ 0\nhω : IsPrimitiveRoot ω 3\nhpz : 3 * a * c = b ^ 2\nhq : q = (9 * a * b * c - 2 * b ^ 3 - 27 * a ^ 2 * d) / (54 * a ^ 3)\nhs3 : s ^ 3 = 2 * q\nx : K\nh₁ : ∀ (x a₁ a₂ a₃ : K), x = a₁ ∨ x = a₂ ∨ x = a₃ ↔ (x - a₁) * (x - a₂) * (x - a₃) = 0\nhi2 : 2 ≠ 0\nhi3 : 3 ≠ 0\nh54 : 54 = 2 * 3 ^ 3\n⊢ b ^ 2 = 3 * a * c",
"after_state": "K : Type u_1\ninst✝² : Field K\na b c d ω q s : K\ninst✝¹ : Invertible 2\ninst✝ : Invertible 3\nha : a ≠ 0\nhω : IsPrimitiveRoot ω 3\nhpz : 3 * a * c = b ^ 2\nhq : q = (9 * a * b * c - 2 * b ^ 3 - 27 * a ^ 2 * d) / (54 * a ^ 3)\nhs3 : s ^ 3 = 2 * q\nx : K\nh₁ : ∀ (x a₁ a₂ a₃ : K), x = a₁ ∨ x = a₂ ∨ x = a₃ ↔ (x - a₁) * (x - a₂) * (x - a₃) = 0\nhi2 : 2 ≠ 0\nhi3 : 3 ≠ 0\nh54 : 54 = 2 * 3 ^ 3\n⊢ b ^ 2 = 3 * a * c"
},
{
"line": "rfl",
"before_state": "K : Type u_1\ninst✝² : Field K\na b c d ω q s : K\ninst✝¹ : Invertible 2\ninst✝ : Invertible 3\nha : a ≠ 0\nhω : IsPrimitiveRoot ω 3\nhpz : 3 * a * c = b ^ 2\nhq : q = (9 * a * b * c - 2 * b ^ 3 - 27 * a ^ 2 * d) / (54 * a ^ 3)\nhs3 : s ^ 3 = 2 * q\nx : K\nh₁ : ∀ (x a₁ a₂ a₃ : K), x = a₁ ∨ x = a₂ ∨ x = a₃ ↔ (x - a₁) * (x - a₂) * (x - a₃) = 0\nhi2 : 2 ≠ 0\nhi3 : 3 ≠ 0\nh54 : 54 = 2 * 3 ^ 3\n⊢ b ^ 2 = 3 * a * c",
"after_state": "K : Type u_1\ninst✝² : Field K\na b c d ω q s : K\ninst✝¹ : Invertible 2\ninst✝ : Invertible 3\nha : a ≠ 0\nhω : IsPrimitiveRoot ω 3\nhpz : 3 * a * c = b ^ 2\nhq : q = (9 * a * b * c - 2 * b ^ 3 - 27 * a ^ 2 * d) / (54 * a ^ 3)\nhs3 : s ^ 3 = 2 * q\nx : K\nh₁ : ∀ (x a₁ a₂ a₃ : K), x = a₁ ∨ x = a₂ ∨ x = a₃ ↔ (x - a₁) * (x - a₂) * (x - a₃) = 0\nhi2 : 2 ≠ 0\nhi3 : 3 ≠ 0\nh54 : 54 = 2 * 3 ^ 3\n⊢ b ^ 2 = 3 * a * c"
},
{
"line": "apply_rfl",
"before_state": "K : Type u_1\ninst✝² : Field K\na b c d ω q s : K\ninst✝¹ : Invertible 2\ninst✝ : Invertible 3\nha : a ≠ 0\nhω : IsPrimitiveRoot ω 3\nhpz : 3 * a * c = b ^ 2\nhq : q = (9 * a * b * c - 2 * b ^ 3 - 27 * a ^ 2 * d) / (54 * a ^ 3)\nhs3 : s ^ 3 = 2 * q\nx : K\nh₁ : ∀ (x a₁ a₂ a₃ : K), x = a₁ ∨ x = a₂ ∨ x = a₃ ↔ (x - a₁) * (x - a₂) * (x - a₃) = 0\nhi2 : 2 ≠ 0\nhi3 : 3 ≠ 0\nh54 : 54 = 2 * 3 ^ 3\n⊢ b ^ 2 = 3 * a * c",
"after_state": "K : Type u_1\ninst✝² : Field K\na b c d ω q s : K\ninst✝¹ : Invertible 2\ninst✝ : Invertible 3\nha : a ≠ 0\nhω : IsPrimitiveRoot ω 3\nhpz : 3 * a * c = b ^ 2\nhq : q = (9 * a * b * c - 2 * b ^ 3 - 27 * a ^ 2 * d) / (54 * a ^ 3)\nhs3 : s ^ 3 = 2 * q\nx : K\nh₁ : ∀ (x a₁ a₂ a₃ : K), x = a₁ ∨ x = a₂ ∨ x = a₃ ↔ (x - a₁) * (x - a₂) * (x - a₃) = 0\nhi2 : 2 ≠ 0\nhi3 : 3 ≠ 0\nh54 : 54 = 2 * 3 ^ 3\n⊢ b ^ 2 = 3 * a * c"
},
{
"line": "skip",
"before_state": "K : Type u_1\ninst✝² : Field K\na b c d ω q s : K\ninst✝¹ : Invertible 2\ninst✝ : Invertible 3\nha : a ≠ 0\nhω : IsPrimitiveRoot ω 3\nhpz : 3 * a * c = b ^ 2\nhq : q = (9 * a * b * c - 2 * b ^ 3 - 27 * a ^ 2 * d) / (54 * a ^ 3)\nhs3 : s ^ 3 = 2 * q\nx : K\nh₁ : ∀ (x a₁ a₂ a₃ : K), x = a₁ ∨ x = a₂ ∨ x = a₃ ↔ (x - a₁) * (x - a₂) * (x - a₃) = 0\nhi2 : 2 ≠ 0\nhi3 : 3 ≠ 0\nh54 : 54 = 2 * 3 ^ 3\n⊢ b ^ 2 = 3 * a * c",
"after_state": "K : Type u_1\ninst✝² : Field K\na b c d ω q s : K\ninst✝¹ : Invertible 2\ninst✝ : Invertible 3\nha : a ≠ 0\nhω : IsPrimitiveRoot ω 3\nhpz : 3 * a * c = b ^ 2\nhq : q = (9 * a * b * c - 2 * b ^ 3 - 27 * a ^ 2 * d) / (54 * a ^ 3)\nhs3 : s ^ 3 = 2 * q\nx : K\nh₁ : ∀ (x a₁ a₂ a₃ : K), x = a₁ ∨ x = a₂ ∨ x = a₃ ↔ (x - a₁) * (x - a₂) * (x - a₃) = 0\nhi2 : 2 ≠ 0\nhi3 : 3 ≠ 0\nh54 : 54 = 2 * 3 ^ 3\n⊢ b ^ 2 = 3 * a * c"
},
{
"line": "rw [hpz]",
"before_state": "K : Type u_1\ninst✝² : Field K\na b c d ω q s : K\ninst✝¹ : Invertible 2\ninst✝ : Invertible 3\nha : a ≠ 0\nhω : IsPrimitiveRoot ω 3\nhpz : 3 * a * c = b ^ 2\nhq : q = (9 * a * b * c - 2 * b ^ 3 - 27 * a ^ 2 * d) / (54 * a ^ 3)\nhs3 : s ^ 3 = 2 * q\nx : K\nh₁ : ∀ (x a₁ a₂ a₃ : K), x = a₁ ∨ x = a₂ ∨ x = a₃ ↔ (x - a₁) * (x - a₂) * (x - a₃) = 0\nhi2 : 2 ≠ 0\nhi3 : 3 ≠ 0\nh54 : 54 = 2 * 3 ^ 3\n⊢ b ^ 2 = 3 * a * c",
"after_state": "No Goals!"
},
{
"line": "rewrite [hpz]",
"before_state": "K : Type u_1\ninst✝² : Field K\na b c d ω q s : K\ninst✝¹ : Invertible 2\ninst✝ : Invertible 3\nha : a ≠ 0\nhω : IsPrimitiveRoot ω 3\nhpz : 3 * a * c = b ^ 2\nhq : q = (9 * a * b * c - 2 * b ^ 3 - 27 * a ^ 2 * d) / (54 * a ^ 3)\nhs3 : s ^ 3 = 2 * q\nx : K\nh₁ : ∀ (x a₁ a₂ a₃ : K), x = a₁ ∨ x = a₂ ∨ x = a₃ ↔ (x - a₁) * (x - a₂) * (x - a₃) = 0\nhi2 : 2 ≠ 0\nhi3 : 3 ≠ 0\nh54 : 54 = 2 * 3 ^ 3\n⊢ b ^ 2 = 3 * a * c",
"after_state": "K : Type u_1\ninst✝² : Field K\na b c d ω q s : K\ninst✝¹ : Invertible 2\ninst✝ : Invertible 3\nha : a ≠ 0\nhω : IsPrimitiveRoot ω 3\nhpz : 3 * a * c = b ^ 2\nhq : q = (9 * a * b * c - 2 * b ^ 3 - 27 * a ^ 2 * d) / (54 * a ^ 3)\nhs3 : s ^ 3 = 2 * q\nx : K\nh₁ : ∀ (x a₁ a₂ a₃ : K), x = a₁ ∨ x = a₂ ∨ x = a₃ ↔ (x - a₁) * (x - a₂) * (x - a₃) = 0\nhi2 : 2 ≠ 0\nhi3 : 3 ≠ 0\nh54 : 54 = 2 * 3 ^ 3\n⊢ b ^ 2 = b ^ 2"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "K : Type u_1\ninst✝² : Field K\na b c d ω q s : K\ninst✝¹ : Invertible 2\ninst✝ : Invertible 3\nha : a ≠ 0\nhω : IsPrimitiveRoot ω 3\nhpz : 3 * a * c = b ^ 2\nhq : q = (9 * a * b * c - 2 * b ^ 3 - 27 * a ^ 2 * d) / (54 * a ^ 3)\nhs3 : s ^ 3 = 2 * q\nx : K\nh₁ : ∀ (x a₁ a₂ a₃ : K), x = a₁ ∨ x = a₂ ∨ x = a₃ ↔ (x - a₁) * (x - a₂) * (x - a₃) = 0\nhi2 : 2 ≠ 0\nhi3 : 3 ≠ 0\nh54 : 54 = 2 * 3 ^ 3\n⊢ b ^ 2 = b ^ 2",
"after_state": "No Goals!"
},
{
"line": "try (with_reducible rfl)",
"before_state": "K : Type u_1\ninst✝² : Field K\na b c d ω q s : K\ninst✝¹ : Invertible 2\ninst✝ : Invertible 3\nha : a ≠ 0\nhω : IsPrimitiveRoot ω 3\nhpz : 3 * a * c = b ^ 2\nhq : q = (9 * a * b * c - 2 * b ^ 3 - 27 * a ^ 2 * d) / (54 * a ^ 3)\nhs3 : s ^ 3 = 2 * q\nx : K\nh₁ : ∀ (x a₁ a₂ a₃ : K), x = a₁ ∨ x = a₂ ∨ x = a₃ ↔ (x - a₁) * (x - a₂) * (x - a₃) = 0\nhi2 : 2 ≠ 0\nhi3 : 3 ≠ 0\nh54 : 54 = 2 * 3 ^ 3\n⊢ b ^ 2 = b ^ 2",
"after_state": "No Goals!"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "K : Type u_1\ninst✝² : Field K\na b c d ω q s : K\ninst✝¹ : Invertible 2\ninst✝ : Invertible 3\nha : a ≠ 0\nhω : IsPrimitiveRoot ω 3\nhpz : 3 * a * c = b ^ 2\nhq : q = (9 * a * b * c - 2 * b ^ 3 - 27 * a ^ 2 * d) / (54 * a ^ 3)\nhs3 : s ^ 3 = 2 * q\nx : K\nh₁ : ∀ (x a₁ a₂ a₃ : K), x = a₁ ∨ x = a₂ ∨ x = a₃ ↔ (x - a₁) * (x - a₂) * (x - a₃) = 0\nhi2 : 2 ≠ 0\nhi3 : 3 ≠ 0\nh54 : 54 = 2 * 3 ^ 3\n⊢ b ^ 2 = b ^ 2",
"after_state": "No Goals!"
},
{
"line": "with_reducible rfl",
"before_state": "K : Type u_1\ninst✝² : Field K\na b c d ω q s : K\ninst✝¹ : Invertible 2\ninst✝ : Invertible 3\nha : a ≠ 0\nhω : IsPrimitiveRoot ω 3\nhpz : 3 * a * c = b ^ 2\nhq : q = (9 * a * b * c - 2 * b ^ 3 - 27 * a ^ 2 * d) / (54 * a ^ 3)\nhs3 : s ^ 3 = 2 * q\nx : K\nh₁ : ∀ (x a₁ a₂ a₃ : K), x = a₁ ∨ x = a₂ ∨ x = a₃ ↔ (x - a₁) * (x - a₂) * (x - a₃) = 0\nhi2 : 2 ≠ 0\nhi3 : 3 ≠ 0\nh54 : 54 = 2 * 3 ^ 3\n⊢ b ^ 2 = b ^ 2",
"after_state": "No Goals!"
},
{
"line": "rfl",
"before_state": "K : Type u_1\ninst✝² : Field K\na b c d ω q s : K\ninst✝¹ : Invertible 2\ninst✝ : Invertible 3\nha : a ≠ 0\nhω : IsPrimitiveRoot ω 3\nhpz : 3 * a * c = b ^ 2\nhq : q = (9 * a * b * c - 2 * b ^ 3 - 27 * a ^ 2 * d) / (54 * a ^ 3)\nhs3 : s ^ 3 = 2 * q\nx : K\nh₁ : ∀ (x a₁ a₂ a₃ : K), x = a₁ ∨ x = a₂ ∨ x = a₃ ↔ (x - a₁) * (x - a₂) * (x - a₃) = 0\nhi2 : 2 ≠ 0\nhi3 : 3 ≠ 0\nh54 : 54 = 2 * 3 ^ 3\n⊢ b ^ 2 = b ^ 2",
"after_state": "No Goals!"
},
{
"line": "eq_refl",
"before_state": "K : Type u_1\ninst✝² : Field K\na b c d ω q s : K\ninst✝¹ : Invertible 2\ninst✝ : Invertible 3\nha : a ≠ 0\nhω : IsPrimitiveRoot ω 3\nhpz : 3 * a * c = b ^ 2\nhq : q = (9 * a * b * c - 2 * b ^ 3 - 27 * a ^ 2 * d) / (54 * a ^ 3)\nhs3 : s ^ 3 = 2 * q\nx : K\nh₁ : ∀ (x a₁ a₂ a₃ : K), x = a₁ ∨ x = a₂ ∨ x = a₃ ↔ (x - a₁) * (x - a₂) * (x - a₃) = 0\nhi2 : 2 ≠ 0\nhi3 : 3 ≠ 0\nh54 : 54 = 2 * 3 ^ 3\n⊢ b ^ 2 = b ^ 2",
"after_state": "No Goals!"
},
{
"line": "have hb3 : b ^ 3 = 3 * a * b * c := by rw [pow_succ, hb2]; ring",
"before_state": "K : Type u_1\ninst✝² : Field K\na b c d ω q s : K\ninst✝¹ : Invertible 2\ninst✝ : Invertible 3\nha : a ≠ 0\nhω : IsPrimitiveRoot ω 3\nhpz : 3 * a * c - b ^ 2 = 0\nhq : q = (9 * a * b * c - 2 * b ^ 3 - 27 * a ^ 2 * d) / (54 * a ^ 3)\nhs3 : s ^ 3 = 2 * q\nx : K\nh₁ : ∀ (x a₁ a₂ a₃ : K), x = a₁ ∨ x = a₂ ∨ x = a₃ ↔ (x - a₁) * (x - a₂) * (x - a₃) = 0\nhi2 : 2 ≠ 0\nhi3 : 3 ≠ 0\nh54 : 54 = 2 * 3 ^ 3\nhb2 : b ^ 2 = 3 * a * c\n⊢ a * x ^ 3 + b * x ^ 2 + c * x + d = 0 ↔ x = s - b / (3 * a) ∨ x = s * ω - b / (3 * a) ∨ x = s * ω ^ 2 - b / (3 * a)",
"after_state": "K : Type u_1\ninst✝² : Field K\na b c d ω q s : K\ninst✝¹ : Invertible 2\ninst✝ : Invertible 3\nha : a ≠ 0\nhω : IsPrimitiveRoot ω 3\nhpz : 3 * a * c - b ^ 2 = 0\nhq : q = (9 * a * b * c - 2 * b ^ 3 - 27 * a ^ 2 * d) / (54 * a ^ 3)\nhs3 : s ^ 3 = 2 * q\nx : K\nh₁ : ∀ (x a₁ a₂ a₃ : K), x = a₁ ∨ x = a₂ ∨ x = a₃ ↔ (x - a₁) * (x - a₂) * (x - a₃) = 0\nhi2 : 2 ≠ 0\nhi3 : 3 ≠ 0\nh54 : 54 = 2 * 3 ^ 3\nhb2 : b ^ 2 = 3 * a * c\nhb3 : b ^ 3 = 3 * a * b * c\n⊢ a * x ^ 3 + b * x ^ 2 + c * x + d = 0 ↔ x = s - b / (3 * a) ∨ x = s * ω - b / (3 * a) ∨ x = s * ω ^ 2 - b / (3 * a)"
},
{
"line": "focus\n refine\n no_implicit_lambda%\n (have hb3 : b ^ 3 = 3 * a * b * c := ?body✝;\n ?_)\n case body✝ => with_annotate_state\"by\" (rw [pow_succ, hb2]; ring)",
"before_state": "K : Type u_1\ninst✝² : Field K\na b c d ω q s : K\ninst✝¹ : Invertible 2\ninst✝ : Invertible 3\nha : a ≠ 0\nhω : IsPrimitiveRoot ω 3\nhpz : 3 * a * c - b ^ 2 = 0\nhq : q = (9 * a * b * c - 2 * b ^ 3 - 27 * a ^ 2 * d) / (54 * a ^ 3)\nhs3 : s ^ 3 = 2 * q\nx : K\nh₁ : ∀ (x a₁ a₂ a₃ : K), x = a₁ ∨ x = a₂ ∨ x = a₃ ↔ (x - a₁) * (x - a₂) * (x - a₃) = 0\nhi2 : 2 ≠ 0\nhi3 : 3 ≠ 0\nh54 : 54 = 2 * 3 ^ 3\nhb2 : b ^ 2 = 3 * a * c\n⊢ a * x ^ 3 + b * x ^ 2 + c * x + d = 0 ↔ x = s - b / (3 * a) ∨ x = s * ω - b / (3 * a) ∨ x = s * ω ^ 2 - b / (3 * a)",
"after_state": "K : Type u_1\ninst✝² : Field K\na b c d ω q s : K\ninst✝¹ : Invertible 2\ninst✝ : Invertible 3\nha : a ≠ 0\nhω : IsPrimitiveRoot ω 3\nhpz : 3 * a * c - b ^ 2 = 0\nhq : q = (9 * a * b * c - 2 * b ^ 3 - 27 * a ^ 2 * d) / (54 * a ^ 3)\nhs3 : s ^ 3 = 2 * q\nx : K\nh₁ : ∀ (x a₁ a₂ a₃ : K), x = a₁ ∨ x = a₂ ∨ x = a₃ ↔ (x - a₁) * (x - a₂) * (x - a₃) = 0\nhi2 : 2 ≠ 0\nhi3 : 3 ≠ 0\nh54 : 54 = 2 * 3 ^ 3\nhb2 : b ^ 2 = 3 * a * c\nhb3 : b ^ 3 = 3 * a * b * c\n⊢ a * x ^ 3 + b * x ^ 2 + c * x + d = 0 ↔ x = s - b / (3 * a) ∨ x = s * ω - b / (3 * a) ∨ x = s * ω ^ 2 - b / (3 * a)"
},
{
"line": "refine\n no_implicit_lambda%\n (have hb3 : b ^ 3 = 3 * a * b * c := ?body✝;\n ?_)",
"before_state": "K : Type u_1\ninst✝² : Field K\na b c d ω q s : K\ninst✝¹ : Invertible 2\ninst✝ : Invertible 3\nha : a ≠ 0\nhω : IsPrimitiveRoot ω 3\nhpz : 3 * a * c - b ^ 2 = 0\nhq : q = (9 * a * b * c - 2 * b ^ 3 - 27 * a ^ 2 * d) / (54 * a ^ 3)\nhs3 : s ^ 3 = 2 * q\nx : K\nh₁ : ∀ (x a₁ a₂ a₃ : K), x = a₁ ∨ x = a₂ ∨ x = a₃ ↔ (x - a₁) * (x - a₂) * (x - a₃) = 0\nhi2 : 2 ≠ 0\nhi3 : 3 ≠ 0\nh54 : 54 = 2 * 3 ^ 3\nhb2 : b ^ 2 = 3 * a * c\n⊢ a * x ^ 3 + b * x ^ 2 + c * x + d = 0 ↔ x = s - b / (3 * a) ∨ x = s * ω - b / (3 * a) ∨ x = s * ω ^ 2 - b / (3 * a)",
"after_state": "case body\nK : Type u_1\ninst✝² : Field K\na b c d ω q s : K\ninst✝¹ : Invertible 2\ninst✝ : Invertible 3\nha : a ≠ 0\nhω : IsPrimitiveRoot ω 3\nhpz : 3 * a * c - b ^ 2 = 0\nhq : q = (9 * a * b * c - 2 * b ^ 3 - 27 * a ^ 2 * d) / (54 * a ^ 3)\nhs3 : s ^ 3 = 2 * q\nx : K\nh₁ : ∀ (x a₁ a₂ a₃ : K), x = a₁ ∨ x = a₂ ∨ x = a₃ ↔ (x - a₁) * (x - a₂) * (x - a₃) = 0\nhi2 : 2 ≠ 0\nhi3 : 3 ≠ 0\nh54 : 54 = 2 * 3 ^ 3\nhb2 : b ^ 2 = 3 * a * c\n⊢ b ^ 3 = 3 * a * b * c\n---\nK : Type u_1\ninst✝² : Field K\na b c d ω q s : K\ninst✝¹ : Invertible 2\ninst✝ : Invertible 3\nha : a ≠ 0\nhω : IsPrimitiveRoot ω 3\nhpz : 3 * a * c - b ^ 2 = 0\nhq : q = (9 * a * b * c - 2 * b ^ 3 - 27 * a ^ 2 * d) / (54 * a ^ 3)\nhs3 : s ^ 3 = 2 * q\nx : K\nh₁ : ∀ (x a₁ a₂ a₃ : K), x = a₁ ∨ x = a₂ ∨ x = a₃ ↔ (x - a₁) * (x - a₂) * (x - a₃) = 0\nhi2 : 2 ≠ 0\nhi3 : 3 ≠ 0\nh54 : 54 = 2 * 3 ^ 3\nhb2 : b ^ 2 = 3 * a * c\nhb3 : b ^ 3 = 3 * a * b * c\n⊢ a * x ^ 3 + b * x ^ 2 + c * x + d = 0 ↔ x = s - b / (3 * a) ∨ x = s * ω - b / (3 * a) ∨ x = s * ω ^ 2 - b / (3 * a)"
},
{
"line": "case body✝ => with_annotate_state\"by\" (rw [pow_succ, hb2]; ring)",
"before_state": "case body\nK : Type u_1\ninst✝² : Field K\na b c d ω q s : K\ninst✝¹ : Invertible 2\ninst✝ : Invertible 3\nha : a ≠ 0\nhω : IsPrimitiveRoot ω 3\nhpz : 3 * a * c - b ^ 2 = 0\nhq : q = (9 * a * b * c - 2 * b ^ 3 - 27 * a ^ 2 * d) / (54 * a ^ 3)\nhs3 : s ^ 3 = 2 * q\nx : K\nh₁ : ∀ (x a₁ a₂ a₃ : K), x = a₁ ∨ x = a₂ ∨ x = a₃ ↔ (x - a₁) * (x - a₂) * (x - a₃) = 0\nhi2 : 2 ≠ 0\nhi3 : 3 ≠ 0\nh54 : 54 = 2 * 3 ^ 3\nhb2 : b ^ 2 = 3 * a * c\n⊢ b ^ 3 = 3 * a * b * c\n---\nK : Type u_1\ninst✝² : Field K\na b c d ω q s : K\ninst✝¹ : Invertible 2\ninst✝ : Invertible 3\nha : a ≠ 0\nhω : IsPrimitiveRoot ω 3\nhpz : 3 * a * c - b ^ 2 = 0\nhq : q = (9 * a * b * c - 2 * b ^ 3 - 27 * a ^ 2 * d) / (54 * a ^ 3)\nhs3 : s ^ 3 = 2 * q\nx : K\nh₁ : ∀ (x a₁ a₂ a₃ : K), x = a₁ ∨ x = a₂ ∨ x = a₃ ↔ (x - a₁) * (x - a₂) * (x - a₃) = 0\nhi2 : 2 ≠ 0\nhi3 : 3 ≠ 0\nh54 : 54 = 2 * 3 ^ 3\nhb2 : b ^ 2 = 3 * a * c\nhb3 : b ^ 3 = 3 * a * b * c\n⊢ a * x ^ 3 + b * x ^ 2 + c * x + d = 0 ↔ x = s - b / (3 * a) ∨ x = s * ω - b / (3 * a) ∨ x = s * ω ^ 2 - b / (3 * a)",
"after_state": "K : Type u_1\ninst✝² : Field K\na b c d ω q s : K\ninst✝¹ : Invertible 2\ninst✝ : Invertible 3\nha : a ≠ 0\nhω : IsPrimitiveRoot ω 3\nhpz : 3 * a * c - b ^ 2 = 0\nhq : q = (9 * a * b * c - 2 * b ^ 3 - 27 * a ^ 2 * d) / (54 * a ^ 3)\nhs3 : s ^ 3 = 2 * q\nx : K\nh₁ : ∀ (x a₁ a₂ a₃ : K), x = a₁ ∨ x = a₂ ∨ x = a₃ ↔ (x - a₁) * (x - a₂) * (x - a₃) = 0\nhi2 : 2 ≠ 0\nhi3 : 3 ≠ 0\nh54 : 54 = 2 * 3 ^ 3\nhb2 : b ^ 2 = 3 * a * c\nhb3 : b ^ 3 = 3 * a * b * c\n⊢ a * x ^ 3 + b * x ^ 2 + c * x + d = 0 ↔ x = s - b / (3 * a) ∨ x = s * ω - b / (3 * a) ∨ x = s * ω ^ 2 - b / (3 * a)"
},
{
"line": "with_annotate_state\"by\" (rw [pow_succ, hb2]; ring)",
"before_state": "K : Type u_1\ninst✝² : Field K\na b c d ω q s : K\ninst✝¹ : Invertible 2\ninst✝ : Invertible 3\nha : a ≠ 0\nhω : IsPrimitiveRoot ω 3\nhpz : 3 * a * c - b ^ 2 = 0\nhq : q = (9 * a * b * c - 2 * b ^ 3 - 27 * a ^ 2 * d) / (54 * a ^ 3)\nhs3 : s ^ 3 = 2 * q\nx : K\nh₁ : ∀ (x a₁ a₂ a₃ : K), x = a₁ ∨ x = a₂ ∨ x = a₃ ↔ (x - a₁) * (x - a₂) * (x - a₃) = 0\nhi2 : 2 ≠ 0\nhi3 : 3 ≠ 0\nh54 : 54 = 2 * 3 ^ 3\nhb2 : b ^ 2 = 3 * a * c\n⊢ b ^ 3 = 3 * a * b * c",
"after_state": "No Goals!"
},
{
"line": "rw [pow_succ, hb2]",
"before_state": "K : Type u_1\ninst✝² : Field K\na b c d ω q s : K\ninst✝¹ : Invertible 2\ninst✝ : Invertible 3\nha : a ≠ 0\nhω : IsPrimitiveRoot ω 3\nhpz : 3 * a * c - b ^ 2 = 0\nhq : q = (9 * a * b * c - 2 * b ^ 3 - 27 * a ^ 2 * d) / (54 * a ^ 3)\nhs3 : s ^ 3 = 2 * q\nx : K\nh₁ : ∀ (x a₁ a₂ a₃ : K), x = a₁ ∨ x = a₂ ∨ x = a₃ ↔ (x - a₁) * (x - a₂) * (x - a₃) = 0\nhi2 : 2 ≠ 0\nhi3 : 3 ≠ 0\nh54 : 54 = 2 * 3 ^ 3\nhb2 : b ^ 2 = 3 * a * c\n⊢ b ^ 3 = 3 * a * b * c",
"after_state": "K : Type u_1\ninst✝² : Field K\na b c d ω q s : K\ninst✝¹ : Invertible 2\ninst✝ : Invertible 3\nha : a ≠ 0\nhω : IsPrimitiveRoot ω 3\nhpz : 3 * a * c - b ^ 2 = 0\nhq : q = (9 * a * b * c - 2 * b ^ 3 - 27 * a ^ 2 * d) / (54 * a ^ 3)\nhs3 : s ^ 3 = 2 * q\nx : K\nh₁ : ∀ (x a₁ a₂ a₃ : K), x = a₁ ∨ x = a₂ ∨ x = a₃ ↔ (x - a₁) * (x - a₂) * (x - a₃) = 0\nhi2 : 2 ≠ 0\nhi3 : 3 ≠ 0\nh54 : 54 = 2 * 3 ^ 3\nhb2 : b ^ 2 = 3 * a * c\n⊢ 3 * a * c * b = 3 * a * b * c"
},
{
"line": "rewrite [pow_succ, hb2]",
"before_state": "K : Type u_1\ninst✝² : Field K\na b c d ω q s : K\ninst✝¹ : Invertible 2\ninst✝ : Invertible 3\nha : a ≠ 0\nhω : IsPrimitiveRoot ω 3\nhpz : 3 * a * c - b ^ 2 = 0\nhq : q = (9 * a * b * c - 2 * b ^ 3 - 27 * a ^ 2 * d) / (54 * a ^ 3)\nhs3 : s ^ 3 = 2 * q\nx : K\nh₁ : ∀ (x a₁ a₂ a₃ : K), x = a₁ ∨ x = a₂ ∨ x = a₃ ↔ (x - a₁) * (x - a₂) * (x - a₃) = 0\nhi2 : 2 ≠ 0\nhi3 : 3 ≠ 0\nh54 : 54 = 2 * 3 ^ 3\nhb2 : b ^ 2 = 3 * a * c\n⊢ b ^ 3 = 3 * a * b * c",
"after_state": "K : Type u_1\ninst✝² : Field K\na b c d ω q s : K\ninst✝¹ : Invertible 2\ninst✝ : Invertible 3\nha : a ≠ 0\nhω : IsPrimitiveRoot ω 3\nhpz : 3 * a * c - b ^ 2 = 0\nhq : q = (9 * a * b * c - 2 * b ^ 3 - 27 * a ^ 2 * d) / (54 * a ^ 3)\nhs3 : s ^ 3 = 2 * q\nx : K\nh₁ : ∀ (x a₁ a₂ a₃ : K), x = a₁ ∨ x = a₂ ∨ x = a₃ ↔ (x - a₁) * (x - a₂) * (x - a₃) = 0\nhi2 : 2 ≠ 0\nhi3 : 3 ≠ 0\nh54 : 54 = 2 * 3 ^ 3\nhb2 : b ^ 2 = 3 * a * c\n⊢ 3 * a * c * b = 3 * a * b * c"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "K : Type u_1\ninst✝² : Field K\na b c d ω q s : K\ninst✝¹ : Invertible 2\ninst✝ : Invertible 3\nha : a ≠ 0\nhω : IsPrimitiveRoot ω 3\nhpz : 3 * a * c - b ^ 2 = 0\nhq : q = (9 * a * b * c - 2 * b ^ 3 - 27 * a ^ 2 * d) / (54 * a ^ 3)\nhs3 : s ^ 3 = 2 * q\nx : K\nh₁ : ∀ (x a₁ a₂ a₃ : K), x = a₁ ∨ x = a₂ ∨ x = a₃ ↔ (x - a₁) * (x - a₂) * (x - a₃) = 0\nhi2 : 2 ≠ 0\nhi3 : 3 ≠ 0\nh54 : 54 = 2 * 3 ^ 3\nhb2 : b ^ 2 = 3 * a * c\n⊢ 3 * a * c * b = 3 * a * b * c",
"after_state": "K : Type u_1\ninst✝² : Field K\na b c d ω q s : K\ninst✝¹ : Invertible 2\ninst✝ : Invertible 3\nha : a ≠ 0\nhω : IsPrimitiveRoot ω 3\nhpz : 3 * a * c - b ^ 2 = 0\nhq : q = (9 * a * b * c - 2 * b ^ 3 - 27 * a ^ 2 * d) / (54 * a ^ 3)\nhs3 : s ^ 3 = 2 * q\nx : K\nh₁ : ∀ (x a₁ a₂ a₃ : K), x = a₁ ∨ x = a₂ ∨ x = a₃ ↔ (x - a₁) * (x - a₂) * (x - a₃) = 0\nhi2 : 2 ≠ 0\nhi3 : 3 ≠ 0\nh54 : 54 = 2 * 3 ^ 3\nhb2 : b ^ 2 = 3 * a * c\n⊢ 3 * a * c * b = 3 * a * b * c"
},
{
"line": "try (with_reducible rfl)",
"before_state": "K : Type u_1\ninst✝² : Field K\na b c d ω q s : K\ninst✝¹ : Invertible 2\ninst✝ : Invertible 3\nha : a ≠ 0\nhω : IsPrimitiveRoot ω 3\nhpz : 3 * a * c - b ^ 2 = 0\nhq : q = (9 * a * b * c - 2 * b ^ 3 - 27 * a ^ 2 * d) / (54 * a ^ 3)\nhs3 : s ^ 3 = 2 * q\nx : K\nh₁ : ∀ (x a₁ a₂ a₃ : K), x = a₁ ∨ x = a₂ ∨ x = a₃ ↔ (x - a₁) * (x - a₂) * (x - a₃) = 0\nhi2 : 2 ≠ 0\nhi3 : 3 ≠ 0\nh54 : 54 = 2 * 3 ^ 3\nhb2 : b ^ 2 = 3 * a * c\n⊢ 3 * a * c * b = 3 * a * b * c",
"after_state": "K : Type u_1\ninst✝² : Field K\na b c d ω q s : K\ninst✝¹ : Invertible 2\ninst✝ : Invertible 3\nha : a ≠ 0\nhω : IsPrimitiveRoot ω 3\nhpz : 3 * a * c - b ^ 2 = 0\nhq : q = (9 * a * b * c - 2 * b ^ 3 - 27 * a ^ 2 * d) / (54 * a ^ 3)\nhs3 : s ^ 3 = 2 * q\nx : K\nh₁ : ∀ (x a₁ a₂ a₃ : K), x = a₁ ∨ x = a₂ ∨ x = a₃ ↔ (x - a₁) * (x - a₂) * (x - a₃) = 0\nhi2 : 2 ≠ 0\nhi3 : 3 ≠ 0\nh54 : 54 = 2 * 3 ^ 3\nhb2 : b ^ 2 = 3 * a * c\n⊢ 3 * a * c * b = 3 * a * b * c"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "K : Type u_1\ninst✝² : Field K\na b c d ω q s : K\ninst✝¹ : Invertible 2\ninst✝ : Invertible 3\nha : a ≠ 0\nhω : IsPrimitiveRoot ω 3\nhpz : 3 * a * c - b ^ 2 = 0\nhq : q = (9 * a * b * c - 2 * b ^ 3 - 27 * a ^ 2 * d) / (54 * a ^ 3)\nhs3 : s ^ 3 = 2 * q\nx : K\nh₁ : ∀ (x a₁ a₂ a₃ : K), x = a₁ ∨ x = a₂ ∨ x = a₃ ↔ (x - a₁) * (x - a₂) * (x - a₃) = 0\nhi2 : 2 ≠ 0\nhi3 : 3 ≠ 0\nh54 : 54 = 2 * 3 ^ 3\nhb2 : b ^ 2 = 3 * a * c\n⊢ 3 * a * c * b = 3 * a * b * c",
"after_state": "K : Type u_1\ninst✝² : Field K\na b c d ω q s : K\ninst✝¹ : Invertible 2\ninst✝ : Invertible 3\nha : a ≠ 0\nhω : IsPrimitiveRoot ω 3\nhpz : 3 * a * c - b ^ 2 = 0\nhq : q = (9 * a * b * c - 2 * b ^ 3 - 27 * a ^ 2 * d) / (54 * a ^ 3)\nhs3 : s ^ 3 = 2 * q\nx : K\nh₁ : ∀ (x a₁ a₂ a₃ : K), x = a₁ ∨ x = a₂ ∨ x = a₃ ↔ (x - a₁) * (x - a₂) * (x - a₃) = 0\nhi2 : 2 ≠ 0\nhi3 : 3 ≠ 0\nh54 : 54 = 2 * 3 ^ 3\nhb2 : b ^ 2 = 3 * a * c\n⊢ 3 * a * c * b = 3 * a * b * c"
},
{
"line": "with_reducible rfl",
"before_state": "K : Type u_1\ninst✝² : Field K\na b c d ω q s : K\ninst✝¹ : Invertible 2\ninst✝ : Invertible 3\nha : a ≠ 0\nhω : IsPrimitiveRoot ω 3\nhpz : 3 * a * c - b ^ 2 = 0\nhq : q = (9 * a * b * c - 2 * b ^ 3 - 27 * a ^ 2 * d) / (54 * a ^ 3)\nhs3 : s ^ 3 = 2 * q\nx : K\nh₁ : ∀ (x a₁ a₂ a₃ : K), x = a₁ ∨ x = a₂ ∨ x = a₃ ↔ (x - a₁) * (x - a₂) * (x - a₃) = 0\nhi2 : 2 ≠ 0\nhi3 : 3 ≠ 0\nh54 : 54 = 2 * 3 ^ 3\nhb2 : b ^ 2 = 3 * a * c\n⊢ 3 * a * c * b = 3 * a * b * c",
"after_state": "K : Type u_1\ninst✝² : Field K\na b c d ω q s : K\ninst✝¹ : Invertible 2\ninst✝ : Invertible 3\nha : a ≠ 0\nhω : IsPrimitiveRoot ω 3\nhpz : 3 * a * c - b ^ 2 = 0\nhq : q = (9 * a * b * c - 2 * b ^ 3 - 27 * a ^ 2 * d) / (54 * a ^ 3)\nhs3 : s ^ 3 = 2 * q\nx : K\nh₁ : ∀ (x a₁ a₂ a₃ : K), x = a₁ ∨ x = a₂ ∨ x = a₃ ↔ (x - a₁) * (x - a₂) * (x - a₃) = 0\nhi2 : 2 ≠ 0\nhi3 : 3 ≠ 0\nh54 : 54 = 2 * 3 ^ 3\nhb2 : b ^ 2 = 3 * a * c\n⊢ 3 * a * c * b = 3 * a * b * c"
},
{
"line": "rfl",
"before_state": "K : Type u_1\ninst✝² : Field K\na b c d ω q s : K\ninst✝¹ : Invertible 2\ninst✝ : Invertible 3\nha : a ≠ 0\nhω : IsPrimitiveRoot ω 3\nhpz : 3 * a * c - b ^ 2 = 0\nhq : q = (9 * a * b * c - 2 * b ^ 3 - 27 * a ^ 2 * d) / (54 * a ^ 3)\nhs3 : s ^ 3 = 2 * q\nx : K\nh₁ : ∀ (x a₁ a₂ a₃ : K), x = a₁ ∨ x = a₂ ∨ x = a₃ ↔ (x - a₁) * (x - a₂) * (x - a₃) = 0\nhi2 : 2 ≠ 0\nhi3 : 3 ≠ 0\nh54 : 54 = 2 * 3 ^ 3\nhb2 : b ^ 2 = 3 * a * c\n⊢ 3 * a * c * b = 3 * a * b * c",
"after_state": "K : Type u_1\ninst✝² : Field K\na b c d ω q s : K\ninst✝¹ : Invertible 2\ninst✝ : Invertible 3\nha : a ≠ 0\nhω : IsPrimitiveRoot ω 3\nhpz : 3 * a * c - b ^ 2 = 0\nhq : q = (9 * a * b * c - 2 * b ^ 3 - 27 * a ^ 2 * d) / (54 * a ^ 3)\nhs3 : s ^ 3 = 2 * q\nx : K\nh₁ : ∀ (x a₁ a₂ a₃ : K), x = a₁ ∨ x = a₂ ∨ x = a₃ ↔ (x - a₁) * (x - a₂) * (x - a₃) = 0\nhi2 : 2 ≠ 0\nhi3 : 3 ≠ 0\nh54 : 54 = 2 * 3 ^ 3\nhb2 : b ^ 2 = 3 * a * c\n⊢ 3 * a * c * b = 3 * a * b * c"
},
{
"line": "apply_rfl",
"before_state": "K : Type u_1\ninst✝² : Field K\na b c d ω q s : K\ninst✝¹ : Invertible 2\ninst✝ : Invertible 3\nha : a ≠ 0\nhω : IsPrimitiveRoot ω 3\nhpz : 3 * a * c - b ^ 2 = 0\nhq : q = (9 * a * b * c - 2 * b ^ 3 - 27 * a ^ 2 * d) / (54 * a ^ 3)\nhs3 : s ^ 3 = 2 * q\nx : K\nh₁ : ∀ (x a₁ a₂ a₃ : K), x = a₁ ∨ x = a₂ ∨ x = a₃ ↔ (x - a₁) * (x - a₂) * (x - a₃) = 0\nhi2 : 2 ≠ 0\nhi3 : 3 ≠ 0\nh54 : 54 = 2 * 3 ^ 3\nhb2 : b ^ 2 = 3 * a * c\n⊢ 3 * a * c * b = 3 * a * b * c",
"after_state": "K : Type u_1\ninst✝² : Field K\na b c d ω q s : K\ninst✝¹ : Invertible 2\ninst✝ : Invertible 3\nha : a ≠ 0\nhω : IsPrimitiveRoot ω 3\nhpz : 3 * a * c - b ^ 2 = 0\nhq : q = (9 * a * b * c - 2 * b ^ 3 - 27 * a ^ 2 * d) / (54 * a ^ 3)\nhs3 : s ^ 3 = 2 * q\nx : K\nh₁ : ∀ (x a₁ a₂ a₃ : K), x = a₁ ∨ x = a₂ ∨ x = a₃ ↔ (x - a₁) * (x - a₂) * (x - a₃) = 0\nhi2 : 2 ≠ 0\nhi3 : 3 ≠ 0\nh54 : 54 = 2 * 3 ^ 3\nhb2 : b ^ 2 = 3 * a * c\n⊢ 3 * a * c * b = 3 * a * b * c"
},
{
"line": "skip",
"before_state": "K : Type u_1\ninst✝² : Field K\na b c d ω q s : K\ninst✝¹ : Invertible 2\ninst✝ : Invertible 3\nha : a ≠ 0\nhω : IsPrimitiveRoot ω 3\nhpz : 3 * a * c - b ^ 2 = 0\nhq : q = (9 * a * b * c - 2 * b ^ 3 - 27 * a ^ 2 * d) / (54 * a ^ 3)\nhs3 : s ^ 3 = 2 * q\nx : K\nh₁ : ∀ (x a₁ a₂ a₃ : K), x = a₁ ∨ x = a₂ ∨ x = a₃ ↔ (x - a₁) * (x - a₂) * (x - a₃) = 0\nhi2 : 2 ≠ 0\nhi3 : 3 ≠ 0\nh54 : 54 = 2 * 3 ^ 3\nhb2 : b ^ 2 = 3 * a * c\n⊢ 3 * a * c * b = 3 * a * b * c",
"after_state": "K : Type u_1\ninst✝² : Field K\na b c d ω q s : K\ninst✝¹ : Invertible 2\ninst✝ : Invertible 3\nha : a ≠ 0\nhω : IsPrimitiveRoot ω 3\nhpz : 3 * a * c - b ^ 2 = 0\nhq : q = (9 * a * b * c - 2 * b ^ 3 - 27 * a ^ 2 * d) / (54 * a ^ 3)\nhs3 : s ^ 3 = 2 * q\nx : K\nh₁ : ∀ (x a₁ a₂ a₃ : K), x = a₁ ∨ x = a₂ ∨ x = a₃ ↔ (x - a₁) * (x - a₂) * (x - a₃) = 0\nhi2 : 2 ≠ 0\nhi3 : 3 ≠ 0\nh54 : 54 = 2 * 3 ^ 3\nhb2 : b ^ 2 = 3 * a * c\n⊢ 3 * a * c * b = 3 * a * b * c"
},
{
"line": "ring",
"before_state": "K : Type u_1\ninst✝² : Field K\na b c d ω q s : K\ninst✝¹ : Invertible 2\ninst✝ : Invertible 3\nha : a ≠ 0\nhω : IsPrimitiveRoot ω 3\nhpz : 3 * a * c - b ^ 2 = 0\nhq : q = (9 * a * b * c - 2 * b ^ 3 - 27 * a ^ 2 * d) / (54 * a ^ 3)\nhs3 : s ^ 3 = 2 * q\nx : K\nh₁ : ∀ (x a₁ a₂ a₃ : K), x = a₁ ∨ x = a₂ ∨ x = a₃ ↔ (x - a₁) * (x - a₂) * (x - a₃) = 0\nhi2 : 2 ≠ 0\nhi3 : 3 ≠ 0\nh54 : 54 = 2 * 3 ^ 3\nhb2 : b ^ 2 = 3 * a * c\n⊢ 3 * a * c * b = 3 * a * b * c",
"after_state": "No Goals!"
},
{
"line": "first\n| ring1\n|\n try_this ring_nf\"\\n\\nThe `ring` tactic failed to close the goal. Use `ring_nf` to obtain a normal form.\n \\nNote that `ring` works primarily in *commutative* rings. \\\n If you have a noncommutative ring, abelian group or module, consider using \\\n `noncomm_ring`, `abel` or `module` instead.\"",
"before_state": "K : Type u_1\ninst✝² : Field K\na b c d ω q s : K\ninst✝¹ : Invertible 2\ninst✝ : Invertible 3\nha : a ≠ 0\nhω : IsPrimitiveRoot ω 3\nhpz : 3 * a * c - b ^ 2 = 0\nhq : q = (9 * a * b * c - 2 * b ^ 3 - 27 * a ^ 2 * d) / (54 * a ^ 3)\nhs3 : s ^ 3 = 2 * q\nx : K\nh₁ : ∀ (x a₁ a₂ a₃ : K), x = a₁ ∨ x = a₂ ∨ x = a₃ ↔ (x - a₁) * (x - a₂) * (x - a₃) = 0\nhi2 : 2 ≠ 0\nhi3 : 3 ≠ 0\nh54 : 54 = 2 * 3 ^ 3\nhb2 : b ^ 2 = 3 * a * c\n⊢ 3 * a * c * b = 3 * a * b * c",
"after_state": "No Goals!"
},
{
"line": "ring1",
"before_state": "K : Type u_1\ninst✝² : Field K\na b c d ω q s : K\ninst✝¹ : Invertible 2\ninst✝ : Invertible 3\nha : a ≠ 0\nhω : IsPrimitiveRoot ω 3\nhpz : 3 * a * c - b ^ 2 = 0\nhq : q = (9 * a * b * c - 2 * b ^ 3 - 27 * a ^ 2 * d) / (54 * a ^ 3)\nhs3 : s ^ 3 = 2 * q\nx : K\nh₁ : ∀ (x a₁ a₂ a₃ : K), x = a₁ ∨ x = a₂ ∨ x = a₃ ↔ (x - a₁) * (x - a₂) * (x - a₃) = 0\nhi2 : 2 ≠ 0\nhi3 : 3 ≠ 0\nh54 : 54 = 2 * 3 ^ 3\nhb2 : b ^ 2 = 3 * a * c\n⊢ 3 * a * c * b = 3 * a * b * c",
"after_state": "No Goals!"
},
{
"line": "have h₂ :=\n calc\n a * x ^ 3 + b * x ^ 2 + c * x + d =\n a * (x + b / (3 * a)) ^ 3 + (c - b ^ 2 / (3 * a)) * x + (d - b ^ 3 * a / (3 * a) ^ 3) :=\n by field_simp; ring\n _ = a * (x + b / (3 * a)) ^ 3 + (d - (9 * a * b * c - 2 * b ^ 3) * a / (3 * a) ^ 3) :=\n by\n simp only [hb2]; field_simp [ha]; ring\n simp only [hb3]; field_simp [ha]; ring\n _ = a * ((x + b / (3 * a)) ^ 3 - s ^ 3) := by rw [hs3, hq]; field_simp [h54]; ring",
"before_state": "K : Type u_1\ninst✝² : Field K\na b c d ω q s : K\ninst✝¹ : Invertible 2\ninst✝ : Invertible 3\nha : a ≠ 0\nhω : IsPrimitiveRoot ω 3\nhpz : 3 * a * c - b ^ 2 = 0\nhq : q = (9 * a * b * c - 2 * b ^ 3 - 27 * a ^ 2 * d) / (54 * a ^ 3)\nhs3 : s ^ 3 = 2 * q\nx : K\nh₁ : ∀ (x a₁ a₂ a₃ : K), x = a₁ ∨ x = a₂ ∨ x = a₃ ↔ (x - a₁) * (x - a₂) * (x - a₃) = 0\nhi2 : 2 ≠ 0\nhi3 : 3 ≠ 0\nh54 : 54 = 2 * 3 ^ 3\nhb2 : b ^ 2 = 3 * a * c\nhb3 : b ^ 3 = 3 * a * b * c\n⊢ a * x ^ 3 + b * x ^ 2 + c * x + d = 0 ↔ x = s - b / (3 * a) ∨ x = s * ω - b / (3 * a) ∨ x = s * ω ^ 2 - b / (3 * a)",
"after_state": "K : Type u_1\ninst✝² : Field K\na b c d ω q s : K\ninst✝¹ : Invertible 2\ninst✝ : Invertible 3\nha : a ≠ 0\nhω : IsPrimitiveRoot ω 3\nhpz : 3 * a * c - b ^ 2 = 0\nhq : q = (9 * a * b * c - 2 * b ^ 3 - 27 * a ^ 2 * d) / (54 * a ^ 3)\nhs3 : s ^ 3 = 2 * q\nx : K\nh₁ : ∀ (x a₁ a₂ a₃ : K), x = a₁ ∨ x = a₂ ∨ x = a₃ ↔ (x - a₁) * (x - a₂) * (x - a₃) = 0\nhi2 : 2 ≠ 0\nhi3 : 3 ≠ 0\nh54 : 54 = 2 * 3 ^ 3\nhb2 : b ^ 2 = 3 * a * c\nhb3 : b ^ 3 = 3 * a * b * c\nh₂ : a * x ^ 3 + b * x ^ 2 + c * x + d = a * ((x + b / (3 * a)) ^ 3 - s ^ 3)\n⊢ a * x ^ 3 + b * x ^ 2 + c * x + d = 0 ↔ x = s - b / (3 * a) ∨ x = s * ω - b / (3 * a) ∨ x = s * ω ^ 2 - b / (3 * a)"
},
{
"line": "refine_lift\n have h₂ :=\n calc\n a * x ^ 3 + b * x ^ 2 + c * x + d =\n a * (x + b / (3 * a)) ^ 3 + (c - b ^ 2 / (3 * a)) * x + (d - b ^ 3 * a / (3 * a) ^ 3) :=\n by field_simp; ring\n _ = a * (x + b / (3 * a)) ^ 3 + (d - (9 * a * b * c - 2 * b ^ 3) * a / (3 * a) ^ 3) :=\n by\n simp only [hb2]; field_simp [ha]; ring\n simp only [hb3]; field_simp [ha]; ring\n _ = a * ((x + b / (3 * a)) ^ 3 - s ^ 3) := by rw [hs3, hq]; field_simp [h54]; ring;\n ?_",
"before_state": "K : Type u_1\ninst✝² : Field K\na b c d ω q s : K\ninst✝¹ : Invertible 2\ninst✝ : Invertible 3\nha : a ≠ 0\nhω : IsPrimitiveRoot ω 3\nhpz : 3 * a * c - b ^ 2 = 0\nhq : q = (9 * a * b * c - 2 * b ^ 3 - 27 * a ^ 2 * d) / (54 * a ^ 3)\nhs3 : s ^ 3 = 2 * q\nx : K\nh₁ : ∀ (x a₁ a₂ a₃ : K), x = a₁ ∨ x = a₂ ∨ x = a₃ ↔ (x - a₁) * (x - a₂) * (x - a₃) = 0\nhi2 : 2 ≠ 0\nhi3 : 3 ≠ 0\nh54 : 54 = 2 * 3 ^ 3\nhb2 : b ^ 2 = 3 * a * c\nhb3 : b ^ 3 = 3 * a * b * c\n⊢ a * x ^ 3 + b * x ^ 2 + c * x + d = 0 ↔ x = s - b / (3 * a) ∨ x = s * ω - b / (3 * a) ∨ x = s * ω ^ 2 - b / (3 * a)",
"after_state": "K : Type u_1\ninst✝² : Field K\na b c d ω q s : K\ninst✝¹ : Invertible 2\ninst✝ : Invertible 3\nha : a ≠ 0\nhω : IsPrimitiveRoot ω 3\nhpz : 3 * a * c - b ^ 2 = 0\nhq : q = (9 * a * b * c - 2 * b ^ 3 - 27 * a ^ 2 * d) / (54 * a ^ 3)\nhs3 : s ^ 3 = 2 * q\nx : K\nh₁ : ∀ (x a₁ a₂ a₃ : K), x = a₁ ∨ x = a₂ ∨ x = a₃ ↔ (x - a₁) * (x - a₂) * (x - a₃) = 0\nhi2 : 2 ≠ 0\nhi3 : 3 ≠ 0\nh54 : 54 = 2 * 3 ^ 3\nhb2 : b ^ 2 = 3 * a * c\nhb3 : b ^ 3 = 3 * a * b * c\nh₂ : a * x ^ 3 + b * x ^ 2 + c * x + d = a * ((x + b / (3 * a)) ^ 3 - s ^ 3)\n⊢ a * x ^ 3 + b * x ^ 2 + c * x + d = 0 ↔ x = s - b / (3 * a) ∨ x = s * ω - b / (3 * a) ∨ x = s * ω ^ 2 - b / (3 * a)"
},
{
"line": "focus\n (refine\n no_implicit_lambda%\n (have h₂ :=\n calc\n a * x ^ 3 + b * x ^ 2 + c * x + d =\n a * (x + b / (3 * a)) ^ 3 + (c - b ^ 2 / (3 * a)) * x + (d - b ^ 3 * a / (3 * a) ^ 3) :=\n by field_simp; ring\n _ = a * (x + b / (3 * a)) ^ 3 + (d - (9 * a * b * c - 2 * b ^ 3) * a / (3 * a) ^ 3) :=\n by\n simp only [hb2]; field_simp [ha]; ring\n simp only [hb3]; field_simp [ha]; ring\n _ = a * ((x + b / (3 * a)) ^ 3 - s ^ 3) := by rw [hs3, hq]; field_simp [h54]; ring;\n ?_);\n rotate_right)",
"before_state": "K : Type u_1\ninst✝² : Field K\na b c d ω q s : K\ninst✝¹ : Invertible 2\ninst✝ : Invertible 3\nha : a ≠ 0\nhω : IsPrimitiveRoot ω 3\nhpz : 3 * a * c - b ^ 2 = 0\nhq : q = (9 * a * b * c - 2 * b ^ 3 - 27 * a ^ 2 * d) / (54 * a ^ 3)\nhs3 : s ^ 3 = 2 * q\nx : K\nh₁ : ∀ (x a₁ a₂ a₃ : K), x = a₁ ∨ x = a₂ ∨ x = a₃ ↔ (x - a₁) * (x - a₂) * (x - a₃) = 0\nhi2 : 2 ≠ 0\nhi3 : 3 ≠ 0\nh54 : 54 = 2 * 3 ^ 3\nhb2 : b ^ 2 = 3 * a * c\nhb3 : b ^ 3 = 3 * a * b * c\n⊢ a * x ^ 3 + b * x ^ 2 + c * x + d = 0 ↔ x = s - b / (3 * a) ∨ x = s * ω - b / (3 * a) ∨ x = s * ω ^ 2 - b / (3 * a)",
"after_state": "K : Type u_1\ninst✝² : Field K\na b c d ω q s : K\ninst✝¹ : Invertible 2\ninst✝ : Invertible 3\nha : a ≠ 0\nhω : IsPrimitiveRoot ω 3\nhpz : 3 * a * c - b ^ 2 = 0\nhq : q = (9 * a * b * c - 2 * b ^ 3 - 27 * a ^ 2 * d) / (54 * a ^ 3)\nhs3 : s ^ 3 = 2 * q\nx : K\nh₁ : ∀ (x a₁ a₂ a₃ : K), x = a₁ ∨ x = a₂ ∨ x = a₃ ↔ (x - a₁) * (x - a₂) * (x - a₃) = 0\nhi2 : 2 ≠ 0\nhi3 : 3 ≠ 0\nh54 : 54 = 2 * 3 ^ 3\nhb2 : b ^ 2 = 3 * a * c\nhb3 : b ^ 3 = 3 * a * b * c\nh₂ : a * x ^ 3 + b * x ^ 2 + c * x + d = a * ((x + b / (3 * a)) ^ 3 - s ^ 3)\n⊢ a * x ^ 3 + b * x ^ 2 + c * x + d = 0 ↔ x = s - b / (3 * a) ∨ x = s * ω - b / (3 * a) ∨ x = s * ω ^ 2 - b / (3 * a)"
},
{
"line": "refine\n no_implicit_lambda%\n (have h₂ :=\n calc\n a * x ^ 3 + b * x ^ 2 + c * x + d =\n a * (x + b / (3 * a)) ^ 3 + (c - b ^ 2 / (3 * a)) * x + (d - b ^ 3 * a / (3 * a) ^ 3) :=\n by field_simp; ring\n _ = a * (x + b / (3 * a)) ^ 3 + (d - (9 * a * b * c - 2 * b ^ 3) * a / (3 * a) ^ 3) :=\n by\n simp only [hb2]; field_simp [ha]; ring\n simp only [hb3]; field_simp [ha]; ring\n _ = a * ((x + b / (3 * a)) ^ 3 - s ^ 3) := by rw [hs3, hq]; field_simp [h54]; ring;\n ?_)",
"before_state": "K : Type u_1\ninst✝² : Field K\na b c d ω q s : K\ninst✝¹ : Invertible 2\ninst✝ : Invertible 3\nha : a ≠ 0\nhω : IsPrimitiveRoot ω 3\nhpz : 3 * a * c - b ^ 2 = 0\nhq : q = (9 * a * b * c - 2 * b ^ 3 - 27 * a ^ 2 * d) / (54 * a ^ 3)\nhs3 : s ^ 3 = 2 * q\nx : K\nh₁ : ∀ (x a₁ a₂ a₃ : K), x = a₁ ∨ x = a₂ ∨ x = a₃ ↔ (x - a₁) * (x - a₂) * (x - a₃) = 0\nhi2 : 2 ≠ 0\nhi3 : 3 ≠ 0\nh54 : 54 = 2 * 3 ^ 3\nhb2 : b ^ 2 = 3 * a * c\nhb3 : b ^ 3 = 3 * a * b * c\n⊢ a * x ^ 3 + b * x ^ 2 + c * x + d = 0 ↔ x = s - b / (3 * a) ∨ x = s * ω - b / (3 * a) ∨ x = s * ω ^ 2 - b / (3 * a)",
"after_state": "K : Type u_1\ninst✝² : Field K\na b c d ω q s : K\ninst✝¹ : Invertible 2\ninst✝ : Invertible 3\nha : a ≠ 0\nhω : IsPrimitiveRoot ω 3\nhpz : 3 * a * c - b ^ 2 = 0\nhq : q = (9 * a * b * c - 2 * b ^ 3 - 27 * a ^ 2 * d) / (54 * a ^ 3)\nhs3 : s ^ 3 = 2 * q\nx : K\nh₁ : ∀ (x a₁ a₂ a₃ : K), x = a₁ ∨ x = a₂ ∨ x = a₃ ↔ (x - a₁) * (x - a₂) * (x - a₃) = 0\nhi2 : 2 ≠ 0\nhi3 : 3 ≠ 0\nh54 : 54 = 2 * 3 ^ 3\nhb2 : b ^ 2 = 3 * a * c\nhb3 : b ^ 3 = 3 * a * b * c\nh₂ : a * x ^ 3 + b * x ^ 2 + c * x + d = a * ((x + b / (3 * a)) ^ 3 - s ^ 3)\n⊢ a * x ^ 3 + b * x ^ 2 + c * x + d = 0 ↔ x = s - b / (3 * a) ∨ x = s * ω - b / (3 * a) ∨ x = s * ω ^ 2 - b / (3 * a)"
},
{
"line": "field_simp",
"before_state": "K : Type u_1\ninst✝² : Field K\na b c d ω q s : K\ninst✝¹ : Invertible 2\ninst✝ : Invertible 3\nha : a ≠ 0\nhω : IsPrimitiveRoot ω 3\nhpz : 3 * a * c - b ^ 2 = 0\nhq : q = (9 * a * b * c - 2 * b ^ 3 - 27 * a ^ 2 * d) / (54 * a ^ 3)\nhs3 : s ^ 3 = 2 * q\nx : K\nh₁ : ∀ (x a₁ a₂ a₃ : K), x = a₁ ∨ x = a₂ ∨ x = a₃ ↔ (x - a₁) * (x - a₂) * (x - a₃) = 0\nhi2 : 2 ≠ 0\nhi3 : 3 ≠ 0\nh54 : 54 = 2 * 3 ^ 3\nhb2 : b ^ 2 = 3 * a * c\nhb3 : b ^ 3 = 3 * a * b * c\n⊢ a * x ^ 3 + b * x ^ 2 + c * x + d =\n a * (x + b / (3 * a)) ^ 3 + (c - b ^ 2 / (3 * a)) * x + (d - b ^ 3 * a / (3 * a) ^ 3)",
"after_state": "K : Type u_1\ninst✝² : Field K\na b c d ω q s : K\ninst✝¹ : Invertible 2\ninst✝ : Invertible 3\nha : a ≠ 0\nhω : IsPrimitiveRoot ω 3\nhpz : 3 * a * c - b ^ 2 = 0\nhq : q = (9 * a * b * c - 2 * b ^ 3 - 27 * a ^ 2 * d) / (54 * a ^ 3)\nhs3 : s ^ 3 = 2 * q\nx : K\nh₁ : ∀ (x a₁ a₂ a₃ : K), x = a₁ ∨ x = a₂ ∨ x = a₃ ↔ (x - a₁) * (x - a₂) * (x - a₃) = 0\nhi2 : 2 ≠ 0\nhi3 : 3 ≠ 0\nh54 : 54 = 2 * 3 ^ 3\nhb2 : b ^ 2 = 3 * a * c\nhb3 : b ^ 3 = 3 * a * b * c\n⊢ (a * x ^ 3 + b * x ^ 2 + c * x + d) * ((3 * a) ^ 3 * (3 * a) * (3 * a) ^ 3) =\n (a * (x * (3 * a) + b) ^ 3 * (3 * a) + (c * (3 * a) - b ^ 2) * x * (3 * a) ^ 3) * (3 * a) ^ 3 +\n (d * (3 * a) ^ 3 - b ^ 3 * a) * ((3 * a) ^ 3 * (3 * a))"
},
{
"line": "ring",
"before_state": "K : Type u_1\ninst✝² : Field K\na b c d ω q s : K\ninst✝¹ : Invertible 2\ninst✝ : Invertible 3\nha : a ≠ 0\nhω : IsPrimitiveRoot ω 3\nhpz : 3 * a * c - b ^ 2 = 0\nhq : q = (9 * a * b * c - 2 * b ^ 3 - 27 * a ^ 2 * d) / (54 * a ^ 3)\nhs3 : s ^ 3 = 2 * q\nx : K\nh₁ : ∀ (x a₁ a₂ a₃ : K), x = a₁ ∨ x = a₂ ∨ x = a₃ ↔ (x - a₁) * (x - a₂) * (x - a₃) = 0\nhi2 : 2 ≠ 0\nhi3 : 3 ≠ 0\nh54 : 54 = 2 * 3 ^ 3\nhb2 : b ^ 2 = 3 * a * c\nhb3 : b ^ 3 = 3 * a * b * c\n⊢ (a * x ^ 3 + b * x ^ 2 + c * x + d) * ((3 * a) ^ 3 * (3 * a) * (3 * a) ^ 3) =\n (a * (x * (3 * a) + b) ^ 3 * (3 * a) + (c * (3 * a) - b ^ 2) * x * (3 * a) ^ 3) * (3 * a) ^ 3 +\n (d * (3 * a) ^ 3 - b ^ 3 * a) * ((3 * a) ^ 3 * (3 * a))",
"after_state": "No Goals!"
},
{
"line": "first\n| ring1\n|\n try_this ring_nf\"\\n\\nThe `ring` tactic failed to close the goal. Use `ring_nf` to obtain a normal form.\n \\nNote that `ring` works primarily in *commutative* rings. \\\n If you have a noncommutative ring, abelian group or module, consider using \\\n `noncomm_ring`, `abel` or `module` instead.\"",
"before_state": "K : Type u_1\ninst✝² : Field K\na b c d ω q s : K\ninst✝¹ : Invertible 2\ninst✝ : Invertible 3\nha : a ≠ 0\nhω : IsPrimitiveRoot ω 3\nhpz : 3 * a * c - b ^ 2 = 0\nhq : q = (9 * a * b * c - 2 * b ^ 3 - 27 * a ^ 2 * d) / (54 * a ^ 3)\nhs3 : s ^ 3 = 2 * q\nx : K\nh₁ : ∀ (x a₁ a₂ a₃ : K), x = a₁ ∨ x = a₂ ∨ x = a₃ ↔ (x - a₁) * (x - a₂) * (x - a₃) = 0\nhi2 : 2 ≠ 0\nhi3 : 3 ≠ 0\nh54 : 54 = 2 * 3 ^ 3\nhb2 : b ^ 2 = 3 * a * c\nhb3 : b ^ 3 = 3 * a * b * c\n⊢ (a * x ^ 3 + b * x ^ 2 + c * x + d) * ((3 * a) ^ 3 * (3 * a) * (3 * a) ^ 3) =\n (a * (x * (3 * a) + b) ^ 3 * (3 * a) + (c * (3 * a) - b ^ 2) * x * (3 * a) ^ 3) * (3 * a) ^ 3 +\n (d * (3 * a) ^ 3 - b ^ 3 * a) * ((3 * a) ^ 3 * (3 * a))",
"after_state": "No Goals!"
},
{
"line": "ring1",
"before_state": "K : Type u_1\ninst✝² : Field K\na b c d ω q s : K\ninst✝¹ : Invertible 2\ninst✝ : Invertible 3\nha : a ≠ 0\nhω : IsPrimitiveRoot ω 3\nhpz : 3 * a * c - b ^ 2 = 0\nhq : q = (9 * a * b * c - 2 * b ^ 3 - 27 * a ^ 2 * d) / (54 * a ^ 3)\nhs3 : s ^ 3 = 2 * q\nx : K\nh₁ : ∀ (x a₁ a₂ a₃ : K), x = a₁ ∨ x = a₂ ∨ x = a₃ ↔ (x - a₁) * (x - a₂) * (x - a₃) = 0\nhi2 : 2 ≠ 0\nhi3 : 3 ≠ 0\nh54 : 54 = 2 * 3 ^ 3\nhb2 : b ^ 2 = 3 * a * c\nhb3 : b ^ 3 = 3 * a * b * c\n⊢ (a * x ^ 3 + b * x ^ 2 + c * x + d) * ((3 * a) ^ 3 * (3 * a) * (3 * a) ^ 3) =\n (a * (x * (3 * a) + b) ^ 3 * (3 * a) + (c * (3 * a) - b ^ 2) * x * (3 * a) ^ 3) * (3 * a) ^ 3 +\n (d * (3 * a) ^ 3 - b ^ 3 * a) * ((3 * a) ^ 3 * (3 * a))",
"after_state": "No Goals!"
},
{
"line": "simp only [hb2]",
"before_state": "K : Type u_1\ninst✝² : Field K\na b c d ω q s : K\ninst✝¹ : Invertible 2\ninst✝ : Invertible 3\nha : a ≠ 0\nhω : IsPrimitiveRoot ω 3\nhpz : 3 * a * c - b ^ 2 = 0\nhq : q = (9 * a * b * c - 2 * b ^ 3 - 27 * a ^ 2 * d) / (54 * a ^ 3)\nhs3 : s ^ 3 = 2 * q\nx : K\nh₁ : ∀ (x a₁ a₂ a₃ : K), x = a₁ ∨ x = a₂ ∨ x = a₃ ↔ (x - a₁) * (x - a₂) * (x - a₃) = 0\nhi2 : 2 ≠ 0\nhi3 : 3 ≠ 0\nh54 : 54 = 2 * 3 ^ 3\nhb2 : b ^ 2 = 3 * a * c\nhb3 : b ^ 3 = 3 * a * b * c\n⊢ a * (x + b / (3 * a)) ^ 3 + (c - b ^ 2 / (3 * a)) * x + (d - b ^ 3 * a / (3 * a) ^ 3) =\n a * (x + b / (3 * a)) ^ 3 + (d - (9 * a * b * c - 2 * b ^ 3) * a / (3 * a) ^ 3)",
"after_state": "K : Type u_1\ninst✝² : Field K\na b c d ω q s : K\ninst✝¹ : Invertible 2\ninst✝ : Invertible 3\nha : a ≠ 0\nhω : IsPrimitiveRoot ω 3\nhpz : 3 * a * c - b ^ 2 = 0\nhq : q = (9 * a * b * c - 2 * b ^ 3 - 27 * a ^ 2 * d) / (54 * a ^ 3)\nhs3 : s ^ 3 = 2 * q\nx : K\nh₁ : ∀ (x a₁ a₂ a₃ : K), x = a₁ ∨ x = a₂ ∨ x = a₃ ↔ (x - a₁) * (x - a₂) * (x - a₃) = 0\nhi2 : 2 ≠ 0\nhi3 : 3 ≠ 0\nh54 : 54 = 2 * 3 ^ 3\nhb2 : b ^ 2 = 3 * a * c\nhb3 : b ^ 3 = 3 * a * b * c\n⊢ a * (x + b / (3 * a)) ^ 3 + (c - 3 * a * c / (3 * a)) * x + (d - b ^ 3 * a / (3 * a) ^ 3) =\n a * (x + b / (3 * a)) ^ 3 + (d - (9 * a * b * c - 2 * b ^ 3) * a / (3 * a) ^ 3)"
},
{
"line": "field_simp [ha]",
"before_state": "K : Type u_1\ninst✝² : Field K\na b c d ω q s : K\ninst✝¹ : Invertible 2\ninst✝ : Invertible 3\nha : a ≠ 0\nhω : IsPrimitiveRoot ω 3\nhpz : 3 * a * c - b ^ 2 = 0\nhq : q = (9 * a * b * c - 2 * b ^ 3 - 27 * a ^ 2 * d) / (54 * a ^ 3)\nhs3 : s ^ 3 = 2 * q\nx : K\nh₁ : ∀ (x a₁ a₂ a₃ : K), x = a₁ ∨ x = a₂ ∨ x = a₃ ↔ (x - a₁) * (x - a₂) * (x - a₃) = 0\nhi2 : 2 ≠ 0\nhi3 : 3 ≠ 0\nh54 : 54 = 2 * 3 ^ 3\nhb2 : b ^ 2 = 3 * a * c\nhb3 : b ^ 3 = 3 * a * b * c\n⊢ a * (x + b / (3 * a)) ^ 3 + (c - 3 * a * c / (3 * a)) * x + (d - b ^ 3 * a / (3 * a) ^ 3) =\n a * (x + b / (3 * a)) ^ 3 + (d - (9 * a * b * c - 2 * b ^ 3) * a / (3 * a) ^ 3)",
"after_state": "K : Type u_1\ninst✝² : Field K\na b c d ω q s : K\ninst✝¹ : Invertible 2\ninst✝ : Invertible 3\nha : a ≠ 0\nhω : IsPrimitiveRoot ω 3\nhpz : 3 * a * c - b ^ 2 = 0\nhq : q = (9 * a * b * c - 2 * b ^ 3 - 27 * a ^ 2 * d) / (54 * a ^ 3)\nhs3 : s ^ 3 = 2 * q\nx : K\nh₁ : ∀ (x a₁ a₂ a₃ : K), x = a₁ ∨ x = a₂ ∨ x = a₃ ↔ (x - a₁) * (x - a₂) * (x - a₃) = 0\nhi2 : 2 ≠ 0\nhi3 : 3 ≠ 0\nh54 : 54 = 2 * 3 ^ 3\nhb2 : b ^ 2 = 3 * a * c\nhb3 : b ^ 3 = 3 * a * b * c\n⊢ b ^ 3 = 9 * a * b * c - 2 * b ^ 3"
},
{
"line": "ring",
"before_state": "K : Type u_1\ninst✝² : Field K\na b c d ω q s : K\ninst✝¹ : Invertible 2\ninst✝ : Invertible 3\nha : a ≠ 0\nhω : IsPrimitiveRoot ω 3\nhpz : 3 * a * c - b ^ 2 = 0\nhq : q = (9 * a * b * c - 2 * b ^ 3 - 27 * a ^ 2 * d) / (54 * a ^ 3)\nhs3 : s ^ 3 = 2 * q\nx : K\nh₁ : ∀ (x a₁ a₂ a₃ : K), x = a₁ ∨ x = a₂ ∨ x = a₃ ↔ (x - a₁) * (x - a₂) * (x - a₃) = 0\nhi2 : 2 ≠ 0\nhi3 : 3 ≠ 0\nh54 : 54 = 2 * 3 ^ 3\nhb2 : b ^ 2 = 3 * a * c\nhb3 : b ^ 3 = 3 * a * b * c\n⊢ b ^ 3 = 9 * a * b * c - 2 * b ^ 3",
"after_state": "K : Type u_1\ninst✝² : Field K\na b c d ω q s : K\ninst✝¹ : Invertible 2\ninst✝ : Invertible 3\nha : a ≠ 0\nhω : IsPrimitiveRoot ω 3\nhpz : 3 * a * c - b ^ 2 = 0\nhq : q = (9 * a * b * c - 2 * b ^ 3 - 27 * a ^ 2 * d) / (54 * a ^ 3)\nhs3 : s ^ 3 = 2 * q\nx : K\nh₁ : ∀ (x a₁ a₂ a₃ : K), x = a₁ ∨ x = a₂ ∨ x = a₃ ↔ (x - a₁) * (x - a₂) * (x - a₃) = 0\nhi2 : 2 ≠ 0\nhi3 : 3 ≠ 0\nh54 : 54 = 2 * 3 ^ 3\nhb2 : b ^ 2 = 3 * a * c\nhb3 : b ^ 3 = 3 * a * b * c\n⊢ b ^ 3 = b * a * c * 9 - b ^ 3 * 2"
},
{
"line": "first\n| ring1\n|\n try_this ring_nf\"\\n\\nThe `ring` tactic failed to close the goal. Use `ring_nf` to obtain a normal form.\n \\nNote that `ring` works primarily in *commutative* rings. \\\n If you have a noncommutative ring, abelian group or module, consider using \\\n `noncomm_ring`, `abel` or `module` instead.\"",
"before_state": "K : Type u_1\ninst✝² : Field K\na b c d ω q s : K\ninst✝¹ : Invertible 2\ninst✝ : Invertible 3\nha : a ≠ 0\nhω : IsPrimitiveRoot ω 3\nhpz : 3 * a * c - b ^ 2 = 0\nhq : q = (9 * a * b * c - 2 * b ^ 3 - 27 * a ^ 2 * d) / (54 * a ^ 3)\nhs3 : s ^ 3 = 2 * q\nx : K\nh₁ : ∀ (x a₁ a₂ a₃ : K), x = a₁ ∨ x = a₂ ∨ x = a₃ ↔ (x - a₁) * (x - a₂) * (x - a₃) = 0\nhi2 : 2 ≠ 0\nhi3 : 3 ≠ 0\nh54 : 54 = 2 * 3 ^ 3\nhb2 : b ^ 2 = 3 * a * c\nhb3 : b ^ 3 = 3 * a * b * c\n⊢ b ^ 3 = 9 * a * b * c - 2 * b ^ 3",
"after_state": "K : Type u_1\ninst✝² : Field K\na b c d ω q s : K\ninst✝¹ : Invertible 2\ninst✝ : Invertible 3\nha : a ≠ 0\nhω : IsPrimitiveRoot ω 3\nhpz : 3 * a * c - b ^ 2 = 0\nhq : q = (9 * a * b * c - 2 * b ^ 3 - 27 * a ^ 2 * d) / (54 * a ^ 3)\nhs3 : s ^ 3 = 2 * q\nx : K\nh₁ : ∀ (x a₁ a₂ a₃ : K), x = a₁ ∨ x = a₂ ∨ x = a₃ ↔ (x - a₁) * (x - a₂) * (x - a₃) = 0\nhi2 : 2 ≠ 0\nhi3 : 3 ≠ 0\nh54 : 54 = 2 * 3 ^ 3\nhb2 : b ^ 2 = 3 * a * c\nhb3 : b ^ 3 = 3 * a * b * c\n⊢ b ^ 3 = b * a * c * 9 - b ^ 3 * 2"
},
{
"line": "ring1",
"before_state": "K : Type u_1\ninst✝² : Field K\na b c d ω q s : K\ninst✝¹ : Invertible 2\ninst✝ : Invertible 3\nha : a ≠ 0\nhω : IsPrimitiveRoot ω 3\nhpz : 3 * a * c - b ^ 2 = 0\nhq : q = (9 * a * b * c - 2 * b ^ 3 - 27 * a ^ 2 * d) / (54 * a ^ 3)\nhs3 : s ^ 3 = 2 * q\nx : K\nh₁ : ∀ (x a₁ a₂ a₃ : K), x = a₁ ∨ x = a₂ ∨ x = a₃ ↔ (x - a₁) * (x - a₂) * (x - a₃) = 0\nhi2 : 2 ≠ 0\nhi3 : 3 ≠ 0\nh54 : 54 = 2 * 3 ^ 3\nhb2 : b ^ 2 = 3 * a * c\nhb3 : b ^ 3 = 3 * a * b * c\n⊢ b ^ 3 = 9 * a * b * c - 2 * b ^ 3",
"after_state": "K : Type u_1\ninst✝² : Field K\na b c d ω q s : K\ninst✝¹ : Invertible 2\ninst✝ : Invertible 3\nha : a ≠ 0\nhω : IsPrimitiveRoot ω 3\nhpz : 3 * a * c - b ^ 2 = 0\nhq : q = (9 * a * b * c - 2 * b ^ 3 - 27 * a ^ 2 * d) / (54 * a ^ 3)\nhs3 : s ^ 3 = 2 * q\nx : K\nh₁ : ∀ (x a₁ a₂ a₃ : K), x = a₁ ∨ x = a₂ ∨ x = a₃ ↔ (x - a₁) * (x - a₂) * (x - a₃) = 0\nhi2 : 2 ≠ 0\nhi3 : 3 ≠ 0\nh54 : 54 = 2 * 3 ^ 3\nhb2 : b ^ 2 = 3 * a * c\nhb3 : b ^ 3 = 3 * a * b * c\n⊢ b ^ 3 = 9 * a * b * c - 2 * b ^ 3"
},
{
"line": "try_this ring_nf\"\\n\\nThe `ring` tactic failed to close the goal. Use `ring_nf` to obtain a normal form.\n \\nNote that `ring` works primarily in *commutative* rings. \\\n If you have a noncommutative ring, abelian group or module, consider using \\\n `noncomm_ring`, `abel` or `module` instead.\"",
"before_state": "K : Type u_1\ninst✝² : Field K\na b c d ω q s : K\ninst✝¹ : Invertible 2\ninst✝ : Invertible 3\nha : a ≠ 0\nhω : IsPrimitiveRoot ω 3\nhpz : 3 * a * c - b ^ 2 = 0\nhq : q = (9 * a * b * c - 2 * b ^ 3 - 27 * a ^ 2 * d) / (54 * a ^ 3)\nhs3 : s ^ 3 = 2 * q\nx : K\nh₁ : ∀ (x a₁ a₂ a₃ : K), x = a₁ ∨ x = a₂ ∨ x = a₃ ↔ (x - a₁) * (x - a₂) * (x - a₃) = 0\nhi2 : 2 ≠ 0\nhi3 : 3 ≠ 0\nh54 : 54 = 2 * 3 ^ 3\nhb2 : b ^ 2 = 3 * a * c\nhb3 : b ^ 3 = 3 * a * b * c\n⊢ b ^ 3 = 9 * a * b * c - 2 * b ^ 3",
"after_state": "K : Type u_1\ninst✝² : Field K\na b c d ω q s : K\ninst✝¹ : Invertible 2\ninst✝ : Invertible 3\nha : a ≠ 0\nhω : IsPrimitiveRoot ω 3\nhpz : 3 * a * c - b ^ 2 = 0\nhq : q = (9 * a * b * c - 2 * b ^ 3 - 27 * a ^ 2 * d) / (54 * a ^ 3)\nhs3 : s ^ 3 = 2 * q\nx : K\nh₁ : ∀ (x a₁ a₂ a₃ : K), x = a₁ ∨ x = a₂ ∨ x = a₃ ↔ (x - a₁) * (x - a₂) * (x - a₃) = 0\nhi2 : 2 ≠ 0\nhi3 : 3 ≠ 0\nh54 : 54 = 2 * 3 ^ 3\nhb2 : b ^ 2 = 3 * a * c\nhb3 : b ^ 3 = 3 * a * b * c\n⊢ b ^ 3 = b * a * c * 9 - b ^ 3 * 2"
},
{
"line": "ring_nf",
"before_state": "K : Type u_1\ninst✝² : Field K\na b c d ω q s : K\ninst✝¹ : Invertible 2\ninst✝ : Invertible 3\nha : a ≠ 0\nhω : IsPrimitiveRoot ω 3\nhpz : 3 * a * c - b ^ 2 = 0\nhq : q = (9 * a * b * c - 2 * b ^ 3 - 27 * a ^ 2 * d) / (54 * a ^ 3)\nhs3 : s ^ 3 = 2 * q\nx : K\nh₁ : ∀ (x a₁ a₂ a₃ : K), x = a₁ ∨ x = a₂ ∨ x = a₃ ↔ (x - a₁) * (x - a₂) * (x - a₃) = 0\nhi2 : 2 ≠ 0\nhi3 : 3 ≠ 0\nh54 : 54 = 2 * 3 ^ 3\nhb2 : b ^ 2 = 3 * a * c\nhb3 : b ^ 3 = 3 * a * b * c\n⊢ b ^ 3 = 9 * a * b * c - 2 * b ^ 3",
"after_state": "K : Type u_1\ninst✝² : Field K\na b c d ω q s : K\ninst✝¹ : Invertible 2\ninst✝ : Invertible 3\nha : a ≠ 0\nhω : IsPrimitiveRoot ω 3\nhpz : 3 * a * c - b ^ 2 = 0\nhq : q = (9 * a * b * c - 2 * b ^ 3 - 27 * a ^ 2 * d) / (54 * a ^ 3)\nhs3 : s ^ 3 = 2 * q\nx : K\nh₁ : ∀ (x a₁ a₂ a₃ : K), x = a₁ ∨ x = a₂ ∨ x = a₃ ↔ (x - a₁) * (x - a₂) * (x - a₃) = 0\nhi2 : 2 ≠ 0\nhi3 : 3 ≠ 0\nh54 : 54 = 2 * 3 ^ 3\nhb2 : b ^ 2 = 3 * a * c\nhb3 : b ^ 3 = 3 * a * b * c\n⊢ b ^ 3 = b * a * c * 9 - b ^ 3 * 2"
},
{
"line": "simp only [hb3]",
"before_state": "K : Type u_1\ninst✝² : Field K\na b c d ω q s : K\ninst✝¹ : Invertible 2\ninst✝ : Invertible 3\nha : a ≠ 0\nhω : IsPrimitiveRoot ω 3\nhpz : 3 * a * c - b ^ 2 = 0\nhq : q = (9 * a * b * c - 2 * b ^ 3 - 27 * a ^ 2 * d) / (54 * a ^ 3)\nhs3 : s ^ 3 = 2 * q\nx : K\nh₁ : ∀ (x a₁ a₂ a₃ : K), x = a₁ ∨ x = a₂ ∨ x = a₃ ↔ (x - a₁) * (x - a₂) * (x - a₃) = 0\nhi2 : 2 ≠ 0\nhi3 : 3 ≠ 0\nh54 : 54 = 2 * 3 ^ 3\nhb2 : b ^ 2 = 3 * a * c\nhb3 : b ^ 3 = 3 * a * b * c\n⊢ b ^ 3 = b * a * c * 9 - b ^ 3 * 2",
"after_state": "K : Type u_1\ninst✝² : Field K\na b c d ω q s : K\ninst✝¹ : Invertible 2\ninst✝ : Invertible 3\nha : a ≠ 0\nhω : IsPrimitiveRoot ω 3\nhpz : 3 * a * c - b ^ 2 = 0\nhq : q = (9 * a * b * c - 2 * b ^ 3 - 27 * a ^ 2 * d) / (54 * a ^ 3)\nhs3 : s ^ 3 = 2 * q\nx : K\nh₁ : ∀ (x a₁ a₂ a₃ : K), x = a₁ ∨ x = a₂ ∨ x = a₃ ↔ (x - a₁) * (x - a₂) * (x - a₃) = 0\nhi2 : 2 ≠ 0\nhi3 : 3 ≠ 0\nh54 : 54 = 2 * 3 ^ 3\nhb2 : b ^ 2 = 3 * a * c\nhb3 : b ^ 3 = 3 * a * b * c\n⊢ 3 * a * b * c = b * a * c * 9 - 3 * a * b * c * 2"
},
{
"line": "field_simp [ha]",
"before_state": "K : Type u_1\ninst✝² : Field K\na b c d ω q s : K\ninst✝¹ : Invertible 2\ninst✝ : Invertible 3\nha : a ≠ 0\nhω : IsPrimitiveRoot ω 3\nhpz : 3 * a * c - b ^ 2 = 0\nhq : q = (9 * a * b * c - 2 * b ^ 3 - 27 * a ^ 2 * d) / (54 * a ^ 3)\nhs3 : s ^ 3 = 2 * q\nx : K\nh₁ : ∀ (x a₁ a₂ a₃ : K), x = a₁ ∨ x = a₂ ∨ x = a₃ ↔ (x - a₁) * (x - a₂) * (x - a₃) = 0\nhi2 : 2 ≠ 0\nhi3 : 3 ≠ 0\nh54 : 54 = 2 * 3 ^ 3\nhb2 : b ^ 2 = 3 * a * c\nhb3 : b ^ 3 = 3 * a * b * c\n⊢ 3 * a * b * c = b * a * c * 9 - 3 * a * b * c * 2",
"after_state": "K : Type u_1\ninst✝² : Field K\na b c d ω q s : K\ninst✝¹ : Invertible 2\ninst✝ : Invertible 3\nha : a ≠ 0\nhω : IsPrimitiveRoot ω 3\nhpz : 3 * a * c - b ^ 2 = 0\nhq : q = (9 * a * b * c - 2 * b ^ 3 - 27 * a ^ 2 * d) / (54 * a ^ 3)\nhs3 : s ^ 3 = 2 * q\nx : K\nh₁ : ∀ (x a₁ a₂ a₃ : K), x = a₁ ∨ x = a₂ ∨ x = a₃ ↔ (x - a₁) * (x - a₂) * (x - a₃) = 0\nhi2 : 2 ≠ 0\nhi3 : 3 ≠ 0\nh54 : 54 = 2 * 3 ^ 3\nhb2 : b ^ 2 = 3 * a * c\nhb3 : b ^ 3 = 3 * a * b * c\n⊢ 3 * a * b * c = b * a * c * 9 - 3 * a * b * c * 2"
},
{
"line": "rw [hs3, hq]",
"before_state": "K : Type u_1\ninst✝² : Field K\na b c d ω q s : K\ninst✝¹ : Invertible 2\ninst✝ : Invertible 3\nha : a ≠ 0\nhω : IsPrimitiveRoot ω 3\nhpz : 3 * a * c - b ^ 2 = 0\nhq : q = (9 * a * b * c - 2 * b ^ 3 - 27 * a ^ 2 * d) / (54 * a ^ 3)\nhs3 : s ^ 3 = 2 * q\nx : K\nh₁ : ∀ (x a₁ a₂ a₃ : K), x = a₁ ∨ x = a₂ ∨ x = a₃ ↔ (x - a₁) * (x - a₂) * (x - a₃) = 0\nhi2 : 2 ≠ 0\nhi3 : 3 ≠ 0\nh54 : 54 = 2 * 3 ^ 3\nhb2 : b ^ 2 = 3 * a * c\nhb3 : b ^ 3 = 3 * a * b * c\n⊢ a * (x + b / (3 * a)) ^ 3 + (d - (9 * a * b * c - 2 * b ^ 3) * a / (3 * a) ^ 3) = a * ((x + b / (3 * a)) ^ 3 - s ^ 3)",
"after_state": "K : Type u_1\ninst✝² : Field K\na b c d ω q s : K\ninst✝¹ : Invertible 2\ninst✝ : Invertible 3\nha : a ≠ 0\nhω : IsPrimitiveRoot ω 3\nhpz : 3 * a * c - b ^ 2 = 0\nhq : q = (9 * a * b * c - 2 * b ^ 3 - 27 * a ^ 2 * d) / (54 * a ^ 3)\nhs3 : s ^ 3 = 2 * q\nx : K\nh₁ : ∀ (x a₁ a₂ a₃ : K), x = a₁ ∨ x = a₂ ∨ x = a₃ ↔ (x - a₁) * (x - a₂) * (x - a₃) = 0\nhi2 : 2 ≠ 0\nhi3 : 3 ≠ 0\nh54 : 54 = 2 * 3 ^ 3\nhb2 : b ^ 2 = 3 * a * c\nhb3 : b ^ 3 = 3 * a * b * c\n⊢ a * (x + b / (3 * a)) ^ 3 + (d - (9 * a * b * c - 2 * b ^ 3) * a / (3 * a) ^ 3) =\n a * ((x + b / (3 * a)) ^ 3 - 2 * ((9 * a * b * c - 2 * b ^ 3 - 27 * a ^ 2 * d) / (54 * a ^ 3)))"
},
{
"line": "rewrite [hs3, hq]",
"before_state": "K : Type u_1\ninst✝² : Field K\na b c d ω q s : K\ninst✝¹ : Invertible 2\ninst✝ : Invertible 3\nha : a ≠ 0\nhω : IsPrimitiveRoot ω 3\nhpz : 3 * a * c - b ^ 2 = 0\nhq : q = (9 * a * b * c - 2 * b ^ 3 - 27 * a ^ 2 * d) / (54 * a ^ 3)\nhs3 : s ^ 3 = 2 * q\nx : K\nh₁ : ∀ (x a₁ a₂ a₃ : K), x = a₁ ∨ x = a₂ ∨ x = a₃ ↔ (x - a₁) * (x - a₂) * (x - a₃) = 0\nhi2 : 2 ≠ 0\nhi3 : 3 ≠ 0\nh54 : 54 = 2 * 3 ^ 3\nhb2 : b ^ 2 = 3 * a * c\nhb3 : b ^ 3 = 3 * a * b * c\n⊢ a * (x + b / (3 * a)) ^ 3 + (d - (9 * a * b * c - 2 * b ^ 3) * a / (3 * a) ^ 3) = a * ((x + b / (3 * a)) ^ 3 - s ^ 3)",
"after_state": "K : Type u_1\ninst✝² : Field K\na b c d ω q s : K\ninst✝¹ : Invertible 2\ninst✝ : Invertible 3\nha : a ≠ 0\nhω : IsPrimitiveRoot ω 3\nhpz : 3 * a * c - b ^ 2 = 0\nhq : q = (9 * a * b * c - 2 * b ^ 3 - 27 * a ^ 2 * d) / (54 * a ^ 3)\nhs3 : s ^ 3 = 2 * q\nx : K\nh₁ : ∀ (x a₁ a₂ a₃ : K), x = a₁ ∨ x = a₂ ∨ x = a₃ ↔ (x - a₁) * (x - a₂) * (x - a₃) = 0\nhi2 : 2 ≠ 0\nhi3 : 3 ≠ 0\nh54 : 54 = 2 * 3 ^ 3\nhb2 : b ^ 2 = 3 * a * c\nhb3 : b ^ 3 = 3 * a * b * c\n⊢ a * (x + b / (3 * a)) ^ 3 + (d - (9 * a * b * c - 2 * b ^ 3) * a / (3 * a) ^ 3) =\n a * ((x + b / (3 * a)) ^ 3 - 2 * ((9 * a * b * c - 2 * b ^ 3 - 27 * a ^ 2 * d) / (54 * a ^ 3)))"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "K : Type u_1\ninst✝² : Field K\na b c d ω q s : K\ninst✝¹ : Invertible 2\ninst✝ : Invertible 3\nha : a ≠ 0\nhω : IsPrimitiveRoot ω 3\nhpz : 3 * a * c - b ^ 2 = 0\nhq : q = (9 * a * b * c - 2 * b ^ 3 - 27 * a ^ 2 * d) / (54 * a ^ 3)\nhs3 : s ^ 3 = 2 * q\nx : K\nh₁ : ∀ (x a₁ a₂ a₃ : K), x = a₁ ∨ x = a₂ ∨ x = a₃ ↔ (x - a₁) * (x - a₂) * (x - a₃) = 0\nhi2 : 2 ≠ 0\nhi3 : 3 ≠ 0\nh54 : 54 = 2 * 3 ^ 3\nhb2 : b ^ 2 = 3 * a * c\nhb3 : b ^ 3 = 3 * a * b * c\n⊢ a * (x + b / (3 * a)) ^ 3 + (d - (9 * a * b * c - 2 * b ^ 3) * a / (3 * a) ^ 3) =\n a * ((x + b / (3 * a)) ^ 3 - 2 * ((9 * a * b * c - 2 * b ^ 3 - 27 * a ^ 2 * d) / (54 * a ^ 3)))",
"after_state": "K : Type u_1\ninst✝² : Field K\na b c d ω q s : K\ninst✝¹ : Invertible 2\ninst✝ : Invertible 3\nha : a ≠ 0\nhω : IsPrimitiveRoot ω 3\nhpz : 3 * a * c - b ^ 2 = 0\nhq : q = (9 * a * b * c - 2 * b ^ 3 - 27 * a ^ 2 * d) / (54 * a ^ 3)\nhs3 : s ^ 3 = 2 * q\nx : K\nh₁ : ∀ (x a₁ a₂ a₃ : K), x = a₁ ∨ x = a₂ ∨ x = a₃ ↔ (x - a₁) * (x - a₂) * (x - a₃) = 0\nhi2 : 2 ≠ 0\nhi3 : 3 ≠ 0\nh54 : 54 = 2 * 3 ^ 3\nhb2 : b ^ 2 = 3 * a * c\nhb3 : b ^ 3 = 3 * a * b * c\n⊢ a * (x + b / (3 * a)) ^ 3 + (d - (9 * a * b * c - 2 * b ^ 3) * a / (3 * a) ^ 3) =\n a * ((x + b / (3 * a)) ^ 3 - 2 * ((9 * a * b * c - 2 * b ^ 3 - 27 * a ^ 2 * d) / (54 * a ^ 3)))"
},
{
"line": "try (with_reducible rfl)",
"before_state": "K : Type u_1\ninst✝² : Field K\na b c d ω q s : K\ninst✝¹ : Invertible 2\ninst✝ : Invertible 3\nha : a ≠ 0\nhω : IsPrimitiveRoot ω 3\nhpz : 3 * a * c - b ^ 2 = 0\nhq : q = (9 * a * b * c - 2 * b ^ 3 - 27 * a ^ 2 * d) / (54 * a ^ 3)\nhs3 : s ^ 3 = 2 * q\nx : K\nh₁ : ∀ (x a₁ a₂ a₃ : K), x = a₁ ∨ x = a₂ ∨ x = a₃ ↔ (x - a₁) * (x - a₂) * (x - a₃) = 0\nhi2 : 2 ≠ 0\nhi3 : 3 ≠ 0\nh54 : 54 = 2 * 3 ^ 3\nhb2 : b ^ 2 = 3 * a * c\nhb3 : b ^ 3 = 3 * a * b * c\n⊢ a * (x + b / (3 * a)) ^ 3 + (d - (9 * a * b * c - 2 * b ^ 3) * a / (3 * a) ^ 3) =\n a * ((x + b / (3 * a)) ^ 3 - 2 * ((9 * a * b * c - 2 * b ^ 3 - 27 * a ^ 2 * d) / (54 * a ^ 3)))",
"after_state": "K : Type u_1\ninst✝² : Field K\na b c d ω q s : K\ninst✝¹ : Invertible 2\ninst✝ : Invertible 3\nha : a ≠ 0\nhω : IsPrimitiveRoot ω 3\nhpz : 3 * a * c - b ^ 2 = 0\nhq : q = (9 * a * b * c - 2 * b ^ 3 - 27 * a ^ 2 * d) / (54 * a ^ 3)\nhs3 : s ^ 3 = 2 * q\nx : K\nh₁ : ∀ (x a₁ a₂ a₃ : K), x = a₁ ∨ x = a₂ ∨ x = a₃ ↔ (x - a₁) * (x - a₂) * (x - a₃) = 0\nhi2 : 2 ≠ 0\nhi3 : 3 ≠ 0\nh54 : 54 = 2 * 3 ^ 3\nhb2 : b ^ 2 = 3 * a * c\nhb3 : b ^ 3 = 3 * a * b * c\n⊢ a * (x + b / (3 * a)) ^ 3 + (d - (9 * a * b * c - 2 * b ^ 3) * a / (3 * a) ^ 3) =\n a * ((x + b / (3 * a)) ^ 3 - 2 * ((9 * a * b * c - 2 * b ^ 3 - 27 * a ^ 2 * d) / (54 * a ^ 3)))"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "K : Type u_1\ninst✝² : Field K\na b c d ω q s : K\ninst✝¹ : Invertible 2\ninst✝ : Invertible 3\nha : a ≠ 0\nhω : IsPrimitiveRoot ω 3\nhpz : 3 * a * c - b ^ 2 = 0\nhq : q = (9 * a * b * c - 2 * b ^ 3 - 27 * a ^ 2 * d) / (54 * a ^ 3)\nhs3 : s ^ 3 = 2 * q\nx : K\nh₁ : ∀ (x a₁ a₂ a₃ : K), x = a₁ ∨ x = a₂ ∨ x = a₃ ↔ (x - a₁) * (x - a₂) * (x - a₃) = 0\nhi2 : 2 ≠ 0\nhi3 : 3 ≠ 0\nh54 : 54 = 2 * 3 ^ 3\nhb2 : b ^ 2 = 3 * a * c\nhb3 : b ^ 3 = 3 * a * b * c\n⊢ a * (x + b / (3 * a)) ^ 3 + (d - (9 * a * b * c - 2 * b ^ 3) * a / (3 * a) ^ 3) =\n a * ((x + b / (3 * a)) ^ 3 - 2 * ((9 * a * b * c - 2 * b ^ 3 - 27 * a ^ 2 * d) / (54 * a ^ 3)))",
"after_state": "K : Type u_1\ninst✝² : Field K\na b c d ω q s : K\ninst✝¹ : Invertible 2\ninst✝ : Invertible 3\nha : a ≠ 0\nhω : IsPrimitiveRoot ω 3\nhpz : 3 * a * c - b ^ 2 = 0\nhq : q = (9 * a * b * c - 2 * b ^ 3 - 27 * a ^ 2 * d) / (54 * a ^ 3)\nhs3 : s ^ 3 = 2 * q\nx : K\nh₁ : ∀ (x a₁ a₂ a₃ : K), x = a₁ ∨ x = a₂ ∨ x = a₃ ↔ (x - a₁) * (x - a₂) * (x - a₃) = 0\nhi2 : 2 ≠ 0\nhi3 : 3 ≠ 0\nh54 : 54 = 2 * 3 ^ 3\nhb2 : b ^ 2 = 3 * a * c\nhb3 : b ^ 3 = 3 * a * b * c\n⊢ a * (x + b / (3 * a)) ^ 3 + (d - (9 * a * b * c - 2 * b ^ 3) * a / (3 * a) ^ 3) =\n a * ((x + b / (3 * a)) ^ 3 - 2 * ((9 * a * b * c - 2 * b ^ 3 - 27 * a ^ 2 * d) / (54 * a ^ 3)))"
},
{
"line": "with_reducible rfl",
"before_state": "K : Type u_1\ninst✝² : Field K\na b c d ω q s : K\ninst✝¹ : Invertible 2\ninst✝ : Invertible 3\nha : a ≠ 0\nhω : IsPrimitiveRoot ω 3\nhpz : 3 * a * c - b ^ 2 = 0\nhq : q = (9 * a * b * c - 2 * b ^ 3 - 27 * a ^ 2 * d) / (54 * a ^ 3)\nhs3 : s ^ 3 = 2 * q\nx : K\nh₁ : ∀ (x a₁ a₂ a₃ : K), x = a₁ ∨ x = a₂ ∨ x = a₃ ↔ (x - a₁) * (x - a₂) * (x - a₃) = 0\nhi2 : 2 ≠ 0\nhi3 : 3 ≠ 0\nh54 : 54 = 2 * 3 ^ 3\nhb2 : b ^ 2 = 3 * a * c\nhb3 : b ^ 3 = 3 * a * b * c\n⊢ a * (x + b / (3 * a)) ^ 3 + (d - (9 * a * b * c - 2 * b ^ 3) * a / (3 * a) ^ 3) =\n a * ((x + b / (3 * a)) ^ 3 - 2 * ((9 * a * b * c - 2 * b ^ 3 - 27 * a ^ 2 * d) / (54 * a ^ 3)))",
"after_state": "K : Type u_1\ninst✝² : Field K\na b c d ω q s : K\ninst✝¹ : Invertible 2\ninst✝ : Invertible 3\nha : a ≠ 0\nhω : IsPrimitiveRoot ω 3\nhpz : 3 * a * c - b ^ 2 = 0\nhq : q = (9 * a * b * c - 2 * b ^ 3 - 27 * a ^ 2 * d) / (54 * a ^ 3)\nhs3 : s ^ 3 = 2 * q\nx : K\nh₁ : ∀ (x a₁ a₂ a₃ : K), x = a₁ ∨ x = a₂ ∨ x = a₃ ↔ (x - a₁) * (x - a₂) * (x - a₃) = 0\nhi2 : 2 ≠ 0\nhi3 : 3 ≠ 0\nh54 : 54 = 2 * 3 ^ 3\nhb2 : b ^ 2 = 3 * a * c\nhb3 : b ^ 3 = 3 * a * b * c\n⊢ a * (x + b / (3 * a)) ^ 3 + (d - (9 * a * b * c - 2 * b ^ 3) * a / (3 * a) ^ 3) =\n a * ((x + b / (3 * a)) ^ 3 - 2 * ((9 * a * b * c - 2 * b ^ 3 - 27 * a ^ 2 * d) / (54 * a ^ 3)))"
},
{
"line": "rfl",
"before_state": "K : Type u_1\ninst✝² : Field K\na b c d ω q s : K\ninst✝¹ : Invertible 2\ninst✝ : Invertible 3\nha : a ≠ 0\nhω : IsPrimitiveRoot ω 3\nhpz : 3 * a * c - b ^ 2 = 0\nhq : q = (9 * a * b * c - 2 * b ^ 3 - 27 * a ^ 2 * d) / (54 * a ^ 3)\nhs3 : s ^ 3 = 2 * q\nx : K\nh₁ : ∀ (x a₁ a₂ a₃ : K), x = a₁ ∨ x = a₂ ∨ x = a₃ ↔ (x - a₁) * (x - a₂) * (x - a₃) = 0\nhi2 : 2 ≠ 0\nhi3 : 3 ≠ 0\nh54 : 54 = 2 * 3 ^ 3\nhb2 : b ^ 2 = 3 * a * c\nhb3 : b ^ 3 = 3 * a * b * c\n⊢ a * (x + b / (3 * a)) ^ 3 + (d - (9 * a * b * c - 2 * b ^ 3) * a / (3 * a) ^ 3) =\n a * ((x + b / (3 * a)) ^ 3 - 2 * ((9 * a * b * c - 2 * b ^ 3 - 27 * a ^ 2 * d) / (54 * a ^ 3)))",
"after_state": "K : Type u_1\ninst✝² : Field K\na b c d ω q s : K\ninst✝¹ : Invertible 2\ninst✝ : Invertible 3\nha : a ≠ 0\nhω : IsPrimitiveRoot ω 3\nhpz : 3 * a * c - b ^ 2 = 0\nhq : q = (9 * a * b * c - 2 * b ^ 3 - 27 * a ^ 2 * d) / (54 * a ^ 3)\nhs3 : s ^ 3 = 2 * q\nx : K\nh₁ : ∀ (x a₁ a₂ a₃ : K), x = a₁ ∨ x = a₂ ∨ x = a₃ ↔ (x - a₁) * (x - a₂) * (x - a₃) = 0\nhi2 : 2 ≠ 0\nhi3 : 3 ≠ 0\nh54 : 54 = 2 * 3 ^ 3\nhb2 : b ^ 2 = 3 * a * c\nhb3 : b ^ 3 = 3 * a * b * c\n⊢ a * (x + b / (3 * a)) ^ 3 + (d - (9 * a * b * c - 2 * b ^ 3) * a / (3 * a) ^ 3) =\n a * ((x + b / (3 * a)) ^ 3 - 2 * ((9 * a * b * c - 2 * b ^ 3 - 27 * a ^ 2 * d) / (54 * a ^ 3)))"
},
{
"line": "apply_rfl",
"before_state": "K : Type u_1\ninst✝² : Field K\na b c d ω q s : K\ninst✝¹ : Invertible 2\ninst✝ : Invertible 3\nha : a ≠ 0\nhω : IsPrimitiveRoot ω 3\nhpz : 3 * a * c - b ^ 2 = 0\nhq : q = (9 * a * b * c - 2 * b ^ 3 - 27 * a ^ 2 * d) / (54 * a ^ 3)\nhs3 : s ^ 3 = 2 * q\nx : K\nh₁ : ∀ (x a₁ a₂ a₃ : K), x = a₁ ∨ x = a₂ ∨ x = a₃ ↔ (x - a₁) * (x - a₂) * (x - a₃) = 0\nhi2 : 2 ≠ 0\nhi3 : 3 ≠ 0\nh54 : 54 = 2 * 3 ^ 3\nhb2 : b ^ 2 = 3 * a * c\nhb3 : b ^ 3 = 3 * a * b * c\n⊢ a * (x + b / (3 * a)) ^ 3 + (d - (9 * a * b * c - 2 * b ^ 3) * a / (3 * a) ^ 3) =\n a * ((x + b / (3 * a)) ^ 3 - 2 * ((9 * a * b * c - 2 * b ^ 3 - 27 * a ^ 2 * d) / (54 * a ^ 3)))",
"after_state": "K : Type u_1\ninst✝² : Field K\na b c d ω q s : K\ninst✝¹ : Invertible 2\ninst✝ : Invertible 3\nha : a ≠ 0\nhω : IsPrimitiveRoot ω 3\nhpz : 3 * a * c - b ^ 2 = 0\nhq : q = (9 * a * b * c - 2 * b ^ 3 - 27 * a ^ 2 * d) / (54 * a ^ 3)\nhs3 : s ^ 3 = 2 * q\nx : K\nh₁ : ∀ (x a₁ a₂ a₃ : K), x = a₁ ∨ x = a₂ ∨ x = a₃ ↔ (x - a₁) * (x - a₂) * (x - a₃) = 0\nhi2 : 2 ≠ 0\nhi3 : 3 ≠ 0\nh54 : 54 = 2 * 3 ^ 3\nhb2 : b ^ 2 = 3 * a * c\nhb3 : b ^ 3 = 3 * a * b * c\n⊢ a * (x + b / (3 * a)) ^ 3 + (d - (9 * a * b * c - 2 * b ^ 3) * a / (3 * a) ^ 3) =\n a * ((x + b / (3 * a)) ^ 3 - 2 * ((9 * a * b * c - 2 * b ^ 3 - 27 * a ^ 2 * d) / (54 * a ^ 3)))"
},
{
"line": "skip",
"before_state": "K : Type u_1\ninst✝² : Field K\na b c d ω q s : K\ninst✝¹ : Invertible 2\ninst✝ : Invertible 3\nha : a ≠ 0\nhω : IsPrimitiveRoot ω 3\nhpz : 3 * a * c - b ^ 2 = 0\nhq : q = (9 * a * b * c - 2 * b ^ 3 - 27 * a ^ 2 * d) / (54 * a ^ 3)\nhs3 : s ^ 3 = 2 * q\nx : K\nh₁ : ∀ (x a₁ a₂ a₃ : K), x = a₁ ∨ x = a₂ ∨ x = a₃ ↔ (x - a₁) * (x - a₂) * (x - a₃) = 0\nhi2 : 2 ≠ 0\nhi3 : 3 ≠ 0\nh54 : 54 = 2 * 3 ^ 3\nhb2 : b ^ 2 = 3 * a * c\nhb3 : b ^ 3 = 3 * a * b * c\n⊢ a * (x + b / (3 * a)) ^ 3 + (d - (9 * a * b * c - 2 * b ^ 3) * a / (3 * a) ^ 3) =\n a * ((x + b / (3 * a)) ^ 3 - 2 * ((9 * a * b * c - 2 * b ^ 3 - 27 * a ^ 2 * d) / (54 * a ^ 3)))",
"after_state": "K : Type u_1\ninst✝² : Field K\na b c d ω q s : K\ninst✝¹ : Invertible 2\ninst✝ : Invertible 3\nha : a ≠ 0\nhω : IsPrimitiveRoot ω 3\nhpz : 3 * a * c - b ^ 2 = 0\nhq : q = (9 * a * b * c - 2 * b ^ 3 - 27 * a ^ 2 * d) / (54 * a ^ 3)\nhs3 : s ^ 3 = 2 * q\nx : K\nh₁ : ∀ (x a₁ a₂ a₃ : K), x = a₁ ∨ x = a₂ ∨ x = a₃ ↔ (x - a₁) * (x - a₂) * (x - a₃) = 0\nhi2 : 2 ≠ 0\nhi3 : 3 ≠ 0\nh54 : 54 = 2 * 3 ^ 3\nhb2 : b ^ 2 = 3 * a * c\nhb3 : b ^ 3 = 3 * a * b * c\n⊢ a * (x + b / (3 * a)) ^ 3 + (d - (9 * a * b * c - 2 * b ^ 3) * a / (3 * a) ^ 3) =\n a * ((x + b / (3 * a)) ^ 3 - 2 * ((9 * a * b * c - 2 * b ^ 3 - 27 * a ^ 2 * d) / (54 * a ^ 3)))"
},
{
"line": "field_simp [h54]",
"before_state": "K : Type u_1\ninst✝² : Field K\na b c d ω q s : K\ninst✝¹ : Invertible 2\ninst✝ : Invertible 3\nha : a ≠ 0\nhω : IsPrimitiveRoot ω 3\nhpz : 3 * a * c - b ^ 2 = 0\nhq : q = (9 * a * b * c - 2 * b ^ 3 - 27 * a ^ 2 * d) / (54 * a ^ 3)\nhs3 : s ^ 3 = 2 * q\nx : K\nh₁ : ∀ (x a₁ a₂ a₃ : K), x = a₁ ∨ x = a₂ ∨ x = a₃ ↔ (x - a₁) * (x - a₂) * (x - a₃) = 0\nhi2 : 2 ≠ 0\nhi3 : 3 ≠ 0\nh54 : 54 = 2 * 3 ^ 3\nhb2 : b ^ 2 = 3 * a * c\nhb3 : b ^ 3 = 3 * a * b * c\n⊢ a * (x + b / (3 * a)) ^ 3 + (d - (9 * a * b * c - 2 * b ^ 3) * a / (3 * a) ^ 3) =\n a * ((x + b / (3 * a)) ^ 3 - 2 * ((9 * a * b * c - 2 * b ^ 3 - 27 * a ^ 2 * d) / (54 * a ^ 3)))",
"after_state": "K : Type u_1\ninst✝² : Field K\na b c d ω q s : K\ninst✝¹ : Invertible 2\ninst✝ : Invertible 3\nha : a ≠ 0\nhω : IsPrimitiveRoot ω 3\nhpz : 3 * a * c - b ^ 2 = 0\nhq : q = (9 * a * b * c - 2 * b ^ 3 - 27 * a ^ 2 * d) / (54 * a ^ 3)\nhs3 : s ^ 3 = 2 * q\nx : K\nh₁ : ∀ (x a₁ a₂ a₃ : K), x = a₁ ∨ x = a₂ ∨ x = a₃ ↔ (x - a₁) * (x - a₂) * (x - a₃) = 0\nhi2 : 2 ≠ 0\nhi3 : 3 ≠ 0\nh54 : 54 = 2 * 3 ^ 3\nhb2 : b ^ 2 = 3 * a * c\nhb3 : b ^ 3 = 3 * a * b * c\n⊢ (a * (x * (3 * a) + b) ^ 3 + (d * (3 * a) ^ 3 - (9 * a * b * c - 2 * b ^ 3) * a)) *\n ((3 * a) ^ 3 * (2 * 3 ^ 3 * a ^ 3)) =\n a *\n ((x * (3 * a) + b) ^ 3 * (2 * 3 ^ 3 * a ^ 3) -\n (3 * a) ^ 3 * (2 * (9 * a * b * c - 2 * b ^ 3 - 27 * a ^ 2 * d))) *\n (3 * a) ^ 3"
},
{
"line": "ring",
"before_state": "K : Type u_1\ninst✝² : Field K\na b c d ω q s : K\ninst✝¹ : Invertible 2\ninst✝ : Invertible 3\nha : a ≠ 0\nhω : IsPrimitiveRoot ω 3\nhpz : 3 * a * c - b ^ 2 = 0\nhq : q = (9 * a * b * c - 2 * b ^ 3 - 27 * a ^ 2 * d) / (54 * a ^ 3)\nhs3 : s ^ 3 = 2 * q\nx : K\nh₁ : ∀ (x a₁ a₂ a₃ : K), x = a₁ ∨ x = a₂ ∨ x = a₃ ↔ (x - a₁) * (x - a₂) * (x - a₃) = 0\nhi2 : 2 ≠ 0\nhi3 : 3 ≠ 0\nh54 : 54 = 2 * 3 ^ 3\nhb2 : b ^ 2 = 3 * a * c\nhb3 : b ^ 3 = 3 * a * b * c\n⊢ (a * (x * (3 * a) + b) ^ 3 + (d * (3 * a) ^ 3 - (9 * a * b * c - 2 * b ^ 3) * a)) *\n ((3 * a) ^ 3 * (2 * 3 ^ 3 * a ^ 3)) =\n a *\n ((x * (3 * a) + b) ^ 3 * (2 * 3 ^ 3 * a ^ 3) -\n (3 * a) ^ 3 * (2 * (9 * a * b * c - 2 * b ^ 3 - 27 * a ^ 2 * d))) *\n (3 * a) ^ 3",
"after_state": "No Goals!"
},
{
"line": "first\n| ring1\n|\n try_this ring_nf\"\\n\\nThe `ring` tactic failed to close the goal. Use `ring_nf` to obtain a normal form.\n \\nNote that `ring` works primarily in *commutative* rings. \\\n If you have a noncommutative ring, abelian group or module, consider using \\\n `noncomm_ring`, `abel` or `module` instead.\"",
"before_state": "K : Type u_1\ninst✝² : Field K\na b c d ω q s : K\ninst✝¹ : Invertible 2\ninst✝ : Invertible 3\nha : a ≠ 0\nhω : IsPrimitiveRoot ω 3\nhpz : 3 * a * c - b ^ 2 = 0\nhq : q = (9 * a * b * c - 2 * b ^ 3 - 27 * a ^ 2 * d) / (54 * a ^ 3)\nhs3 : s ^ 3 = 2 * q\nx : K\nh₁ : ∀ (x a₁ a₂ a₃ : K), x = a₁ ∨ x = a₂ ∨ x = a₃ ↔ (x - a₁) * (x - a₂) * (x - a₃) = 0\nhi2 : 2 ≠ 0\nhi3 : 3 ≠ 0\nh54 : 54 = 2 * 3 ^ 3\nhb2 : b ^ 2 = 3 * a * c\nhb3 : b ^ 3 = 3 * a * b * c\n⊢ (a * (x * (3 * a) + b) ^ 3 + (d * (3 * a) ^ 3 - (9 * a * b * c - 2 * b ^ 3) * a)) *\n ((3 * a) ^ 3 * (2 * 3 ^ 3 * a ^ 3)) =\n a *\n ((x * (3 * a) + b) ^ 3 * (2 * 3 ^ 3 * a ^ 3) -\n (3 * a) ^ 3 * (2 * (9 * a * b * c - 2 * b ^ 3 - 27 * a ^ 2 * d))) *\n (3 * a) ^ 3",
"after_state": "No Goals!"
},
{
"line": "ring1",
"before_state": "K : Type u_1\ninst✝² : Field K\na b c d ω q s : K\ninst✝¹ : Invertible 2\ninst✝ : Invertible 3\nha : a ≠ 0\nhω : IsPrimitiveRoot ω 3\nhpz : 3 * a * c - b ^ 2 = 0\nhq : q = (9 * a * b * c - 2 * b ^ 3 - 27 * a ^ 2 * d) / (54 * a ^ 3)\nhs3 : s ^ 3 = 2 * q\nx : K\nh₁ : ∀ (x a₁ a₂ a₃ : K), x = a₁ ∨ x = a₂ ∨ x = a₃ ↔ (x - a₁) * (x - a₂) * (x - a₃) = 0\nhi2 : 2 ≠ 0\nhi3 : 3 ≠ 0\nh54 : 54 = 2 * 3 ^ 3\nhb2 : b ^ 2 = 3 * a * c\nhb3 : b ^ 3 = 3 * a * b * c\n⊢ (a * (x * (3 * a) + b) ^ 3 + (d * (3 * a) ^ 3 - (9 * a * b * c - 2 * b ^ 3) * a)) *\n ((3 * a) ^ 3 * (2 * 3 ^ 3 * a ^ 3)) =\n a *\n ((x * (3 * a) + b) ^ 3 * (2 * 3 ^ 3 * a ^ 3) -\n (3 * a) ^ 3 * (2 * (9 * a * b * c - 2 * b ^ 3 - 27 * a ^ 2 * d))) *\n (3 * a) ^ 3",
"after_state": "No Goals!"
},
{
"line": "rotate_right",
"before_state": "K : Type u_1\ninst✝² : Field K\na b c d ω q s : K\ninst✝¹ : Invertible 2\ninst✝ : Invertible 3\nha : a ≠ 0\nhω : IsPrimitiveRoot ω 3\nhpz : 3 * a * c - b ^ 2 = 0\nhq : q = (9 * a * b * c - 2 * b ^ 3 - 27 * a ^ 2 * d) / (54 * a ^ 3)\nhs3 : s ^ 3 = 2 * q\nx : K\nh₁ : ∀ (x a₁ a₂ a₃ : K), x = a₁ ∨ x = a₂ ∨ x = a₃ ↔ (x - a₁) * (x - a₂) * (x - a₃) = 0\nhi2 : 2 ≠ 0\nhi3 : 3 ≠ 0\nh54 : 54 = 2 * 3 ^ 3\nhb2 : b ^ 2 = 3 * a * c\nhb3 : b ^ 3 = 3 * a * b * c\nh₂ : a * x ^ 3 + b * x ^ 2 + c * x + d = a * ((x + b / (3 * a)) ^ 3 - s ^ 3)\n⊢ a * x ^ 3 + b * x ^ 2 + c * x + d = 0 ↔ x = s - b / (3 * a) ∨ x = s * ω - b / (3 * a) ∨ x = s * ω ^ 2 - b / (3 * a)",
"after_state": "K : Type u_1\ninst✝² : Field K\na b c d ω q s : K\ninst✝¹ : Invertible 2\ninst✝ : Invertible 3\nha : a ≠ 0\nhω : IsPrimitiveRoot ω 3\nhpz : 3 * a * c - b ^ 2 = 0\nhq : q = (9 * a * b * c - 2 * b ^ 3 - 27 * a ^ 2 * d) / (54 * a ^ 3)\nhs3 : s ^ 3 = 2 * q\nx : K\nh₁ : ∀ (x a₁ a₂ a₃ : K), x = a₁ ∨ x = a₂ ∨ x = a₃ ↔ (x - a₁) * (x - a₂) * (x - a₃) = 0\nhi2 : 2 ≠ 0\nhi3 : 3 ≠ 0\nh54 : 54 = 2 * 3 ^ 3\nhb2 : b ^ 2 = 3 * a * c\nhb3 : b ^ 3 = 3 * a * b * c\nh₂ : a * x ^ 3 + b * x ^ 2 + c * x + d = a * ((x + b / (3 * a)) ^ 3 - s ^ 3)\n⊢ a * x ^ 3 + b * x ^ 2 + c * x + d = 0 ↔ x = s - b / (3 * a) ∨ x = s * ω - b / (3 * a) ∨ x = s * ω ^ 2 - b / (3 * a)"
},
{
"line": "have h₃ : ∀ x, a * x = 0 ↔ x = 0 := by intro x; simp [ha]",
"before_state": "K : Type u_1\ninst✝² : Field K\na b c d ω q s : K\ninst✝¹ : Invertible 2\ninst✝ : Invertible 3\nha : a ≠ 0\nhω : IsPrimitiveRoot ω 3\nhpz : 3 * a * c - b ^ 2 = 0\nhq : q = (9 * a * b * c - 2 * b ^ 3 - 27 * a ^ 2 * d) / (54 * a ^ 3)\nhs3 : s ^ 3 = 2 * q\nx : K\nh₁ : ∀ (x a₁ a₂ a₃ : K), x = a₁ ∨ x = a₂ ∨ x = a₃ ↔ (x - a₁) * (x - a₂) * (x - a₃) = 0\nhi2 : 2 ≠ 0\nhi3 : 3 ≠ 0\nh54 : 54 = 2 * 3 ^ 3\nhb2 : b ^ 2 = 3 * a * c\nhb3 : b ^ 3 = 3 * a * b * c\nh₂ : a * x ^ 3 + b * x ^ 2 + c * x + d = a * ((x + b / (3 * a)) ^ 3 - s ^ 3)\n⊢ a * x ^ 3 + b * x ^ 2 + c * x + d = 0 ↔ x = s - b / (3 * a) ∨ x = s * ω - b / (3 * a) ∨ x = s * ω ^ 2 - b / (3 * a)",
"after_state": "K : Type u_1\ninst✝² : Field K\na b c d ω q s : K\ninst✝¹ : Invertible 2\ninst✝ : Invertible 3\nha : a ≠ 0\nhω : IsPrimitiveRoot ω 3\nhpz : 3 * a * c - b ^ 2 = 0\nhq : q = (9 * a * b * c - 2 * b ^ 3 - 27 * a ^ 2 * d) / (54 * a ^ 3)\nhs3 : s ^ 3 = 2 * q\nx : K\nh₁ : ∀ (x a₁ a₂ a₃ : K), x = a₁ ∨ x = a₂ ∨ x = a₃ ↔ (x - a₁) * (x - a₂) * (x - a₃) = 0\nhi2 : 2 ≠ 0\nhi3 : 3 ≠ 0\nh54 : 54 = 2 * 3 ^ 3\nhb2 : b ^ 2 = 3 * a * c\nhb3 : b ^ 3 = 3 * a * b * c\nh₂ : a * x ^ 3 + b * x ^ 2 + c * x + d = a * ((x + b / (3 * a)) ^ 3 - s ^ 3)\nh₃ : ∀ (x : K), a * x = 0 ↔ x = 0\n⊢ a * x ^ 3 + b * x ^ 2 + c * x + d = 0 ↔ x = s - b / (3 * a) ∨ x = s * ω - b / (3 * a) ∨ x = s * ω ^ 2 - b / (3 * a)"
},
{
"line": "focus\n refine\n no_implicit_lambda%\n (have h₃ : ∀ x, a * x = 0 ↔ x = 0 := ?body✝;\n ?_)\n case body✝ => with_annotate_state\"by\" (intro x; simp [ha])",
"before_state": "K : Type u_1\ninst✝² : Field K\na b c d ω q s : K\ninst✝¹ : Invertible 2\ninst✝ : Invertible 3\nha : a ≠ 0\nhω : IsPrimitiveRoot ω 3\nhpz : 3 * a * c - b ^ 2 = 0\nhq : q = (9 * a * b * c - 2 * b ^ 3 - 27 * a ^ 2 * d) / (54 * a ^ 3)\nhs3 : s ^ 3 = 2 * q\nx : K\nh₁ : ∀ (x a₁ a₂ a₃ : K), x = a₁ ∨ x = a₂ ∨ x = a₃ ↔ (x - a₁) * (x - a₂) * (x - a₃) = 0\nhi2 : 2 ≠ 0\nhi3 : 3 ≠ 0\nh54 : 54 = 2 * 3 ^ 3\nhb2 : b ^ 2 = 3 * a * c\nhb3 : b ^ 3 = 3 * a * b * c\nh₂ : a * x ^ 3 + b * x ^ 2 + c * x + d = a * ((x + b / (3 * a)) ^ 3 - s ^ 3)\n⊢ a * x ^ 3 + b * x ^ 2 + c * x + d = 0 ↔ x = s - b / (3 * a) ∨ x = s * ω - b / (3 * a) ∨ x = s * ω ^ 2 - b / (3 * a)",
"after_state": "K : Type u_1\ninst✝² : Field K\na b c d ω q s : K\ninst✝¹ : Invertible 2\ninst✝ : Invertible 3\nha : a ≠ 0\nhω : IsPrimitiveRoot ω 3\nhpz : 3 * a * c - b ^ 2 = 0\nhq : q = (9 * a * b * c - 2 * b ^ 3 - 27 * a ^ 2 * d) / (54 * a ^ 3)\nhs3 : s ^ 3 = 2 * q\nx : K\nh₁ : ∀ (x a₁ a₂ a₃ : K), x = a₁ ∨ x = a₂ ∨ x = a₃ ↔ (x - a₁) * (x - a₂) * (x - a₃) = 0\nhi2 : 2 ≠ 0\nhi3 : 3 ≠ 0\nh54 : 54 = 2 * 3 ^ 3\nhb2 : b ^ 2 = 3 * a * c\nhb3 : b ^ 3 = 3 * a * b * c\nh₂ : a * x ^ 3 + b * x ^ 2 + c * x + d = a * ((x + b / (3 * a)) ^ 3 - s ^ 3)\nh₃ : ∀ (x : K), a * x = 0 ↔ x = 0\n⊢ a * x ^ 3 + b * x ^ 2 + c * x + d = 0 ↔ x = s - b / (3 * a) ∨ x = s * ω - b / (3 * a) ∨ x = s * ω ^ 2 - b / (3 * a)"
},
{
"line": "refine\n no_implicit_lambda%\n (have h₃ : ∀ x, a * x = 0 ↔ x = 0 := ?body✝;\n ?_)",
"before_state": "K : Type u_1\ninst✝² : Field K\na b c d ω q s : K\ninst✝¹ : Invertible 2\ninst✝ : Invertible 3\nha : a ≠ 0\nhω : IsPrimitiveRoot ω 3\nhpz : 3 * a * c - b ^ 2 = 0\nhq : q = (9 * a * b * c - 2 * b ^ 3 - 27 * a ^ 2 * d) / (54 * a ^ 3)\nhs3 : s ^ 3 = 2 * q\nx : K\nh₁ : ∀ (x a₁ a₂ a₃ : K), x = a₁ ∨ x = a₂ ∨ x = a₃ ↔ (x - a₁) * (x - a₂) * (x - a₃) = 0\nhi2 : 2 ≠ 0\nhi3 : 3 ≠ 0\nh54 : 54 = 2 * 3 ^ 3\nhb2 : b ^ 2 = 3 * a * c\nhb3 : b ^ 3 = 3 * a * b * c\nh₂ : a * x ^ 3 + b * x ^ 2 + c * x + d = a * ((x + b / (3 * a)) ^ 3 - s ^ 3)\n⊢ a * x ^ 3 + b * x ^ 2 + c * x + d = 0 ↔ x = s - b / (3 * a) ∨ x = s * ω - b / (3 * a) ∨ x = s * ω ^ 2 - b / (3 * a)",
"after_state": "case body\nK : Type u_1\ninst✝² : Field K\na b c d ω q s : K\ninst✝¹ : Invertible 2\ninst✝ : Invertible 3\nha : a ≠ 0\nhω : IsPrimitiveRoot ω 3\nhpz : 3 * a * c - b ^ 2 = 0\nhq : q = (9 * a * b * c - 2 * b ^ 3 - 27 * a ^ 2 * d) / (54 * a ^ 3)\nhs3 : s ^ 3 = 2 * q\nx : K\nh₁ : ∀ (x a₁ a₂ a₃ : K), x = a₁ ∨ x = a₂ ∨ x = a₃ ↔ (x - a₁) * (x - a₂) * (x - a₃) = 0\nhi2 : 2 ≠ 0\nhi3 : 3 ≠ 0\nh54 : 54 = 2 * 3 ^ 3\nhb2 : b ^ 2 = 3 * a * c\nhb3 : b ^ 3 = 3 * a * b * c\nh₂ : a * x ^ 3 + b * x ^ 2 + c * x + d = a * ((x + b / (3 * a)) ^ 3 - s ^ 3)\n⊢ ∀ (x : K), a * x = 0 ↔ x = 0\n---\nK : Type u_1\ninst✝² : Field K\na b c d ω q s : K\ninst✝¹ : Invertible 2\ninst✝ : Invertible 3\nha : a ≠ 0\nhω : IsPrimitiveRoot ω 3\nhpz : 3 * a * c - b ^ 2 = 0\nhq : q = (9 * a * b * c - 2 * b ^ 3 - 27 * a ^ 2 * d) / (54 * a ^ 3)\nhs3 : s ^ 3 = 2 * q\nx : K\nh₁ : ∀ (x a₁ a₂ a₃ : K), x = a₁ ∨ x = a₂ ∨ x = a₃ ↔ (x - a₁) * (x - a₂) * (x - a₃) = 0\nhi2 : 2 ≠ 0\nhi3 : 3 ≠ 0\nh54 : 54 = 2 * 3 ^ 3\nhb2 : b ^ 2 = 3 * a * c\nhb3 : b ^ 3 = 3 * a * b * c\nh₂ : a * x ^ 3 + b * x ^ 2 + c * x + d = a * ((x + b / (3 * a)) ^ 3 - s ^ 3)\nh₃ : ∀ (x : K), a * x = 0 ↔ x = 0\n⊢ a * x ^ 3 + b * x ^ 2 + c * x + d = 0 ↔ x = s - b / (3 * a) ∨ x = s * ω - b / (3 * a) ∨ x = s * ω ^ 2 - b / (3 * a)"
},
{
"line": "case body✝ => with_annotate_state\"by\" (intro x; simp [ha])",
"before_state": "case body\nK : Type u_1\ninst✝² : Field K\na b c d ω q s : K\ninst✝¹ : Invertible 2\ninst✝ : Invertible 3\nha : a ≠ 0\nhω : IsPrimitiveRoot ω 3\nhpz : 3 * a * c - b ^ 2 = 0\nhq : q = (9 * a * b * c - 2 * b ^ 3 - 27 * a ^ 2 * d) / (54 * a ^ 3)\nhs3 : s ^ 3 = 2 * q\nx : K\nh₁ : ∀ (x a₁ a₂ a₃ : K), x = a₁ ∨ x = a₂ ∨ x = a₃ ↔ (x - a₁) * (x - a₂) * (x - a₃) = 0\nhi2 : 2 ≠ 0\nhi3 : 3 ≠ 0\nh54 : 54 = 2 * 3 ^ 3\nhb2 : b ^ 2 = 3 * a * c\nhb3 : b ^ 3 = 3 * a * b * c\nh₂ : a * x ^ 3 + b * x ^ 2 + c * x + d = a * ((x + b / (3 * a)) ^ 3 - s ^ 3)\n⊢ ∀ (x : K), a * x = 0 ↔ x = 0\n---\nK : Type u_1\ninst✝² : Field K\na b c d ω q s : K\ninst✝¹ : Invertible 2\ninst✝ : Invertible 3\nha : a ≠ 0\nhω : IsPrimitiveRoot ω 3\nhpz : 3 * a * c - b ^ 2 = 0\nhq : q = (9 * a * b * c - 2 * b ^ 3 - 27 * a ^ 2 * d) / (54 * a ^ 3)\nhs3 : s ^ 3 = 2 * q\nx : K\nh₁ : ∀ (x a₁ a₂ a₃ : K), x = a₁ ∨ x = a₂ ∨ x = a₃ ↔ (x - a₁) * (x - a₂) * (x - a₃) = 0\nhi2 : 2 ≠ 0\nhi3 : 3 ≠ 0\nh54 : 54 = 2 * 3 ^ 3\nhb2 : b ^ 2 = 3 * a * c\nhb3 : b ^ 3 = 3 * a * b * c\nh₂ : a * x ^ 3 + b * x ^ 2 + c * x + d = a * ((x + b / (3 * a)) ^ 3 - s ^ 3)\nh₃ : ∀ (x : K), a * x = 0 ↔ x = 0\n⊢ a * x ^ 3 + b * x ^ 2 + c * x + d = 0 ↔ x = s - b / (3 * a) ∨ x = s * ω - b / (3 * a) ∨ x = s * ω ^ 2 - b / (3 * a)",
"after_state": "K : Type u_1\ninst✝² : Field K\na b c d ω q s : K\ninst✝¹ : Invertible 2\ninst✝ : Invertible 3\nha : a ≠ 0\nhω : IsPrimitiveRoot ω 3\nhpz : 3 * a * c - b ^ 2 = 0\nhq : q = (9 * a * b * c - 2 * b ^ 3 - 27 * a ^ 2 * d) / (54 * a ^ 3)\nhs3 : s ^ 3 = 2 * q\nx : K\nh₁ : ∀ (x a₁ a₂ a₃ : K), x = a₁ ∨ x = a₂ ∨ x = a₃ ↔ (x - a₁) * (x - a₂) * (x - a₃) = 0\nhi2 : 2 ≠ 0\nhi3 : 3 ≠ 0\nh54 : 54 = 2 * 3 ^ 3\nhb2 : b ^ 2 = 3 * a * c\nhb3 : b ^ 3 = 3 * a * b * c\nh₂ : a * x ^ 3 + b * x ^ 2 + c * x + d = a * ((x + b / (3 * a)) ^ 3 - s ^ 3)\nh₃ : ∀ (x : K), a * x = 0 ↔ x = 0\n⊢ a * x ^ 3 + b * x ^ 2 + c * x + d = 0 ↔ x = s - b / (3 * a) ∨ x = s * ω - b / (3 * a) ∨ x = s * ω ^ 2 - b / (3 * a)"
},
{
"line": "with_annotate_state\"by\" (intro x; simp [ha])",
"before_state": "K : Type u_1\ninst✝² : Field K\na b c d ω q s : K\ninst✝¹ : Invertible 2\ninst✝ : Invertible 3\nha : a ≠ 0\nhω : IsPrimitiveRoot ω 3\nhpz : 3 * a * c - b ^ 2 = 0\nhq : q = (9 * a * b * c - 2 * b ^ 3 - 27 * a ^ 2 * d) / (54 * a ^ 3)\nhs3 : s ^ 3 = 2 * q\nx : K\nh₁ : ∀ (x a₁ a₂ a₃ : K), x = a₁ ∨ x = a₂ ∨ x = a₃ ↔ (x - a₁) * (x - a₂) * (x - a₃) = 0\nhi2 : 2 ≠ 0\nhi3 : 3 ≠ 0\nh54 : 54 = 2 * 3 ^ 3\nhb2 : b ^ 2 = 3 * a * c\nhb3 : b ^ 3 = 3 * a * b * c\nh₂ : a * x ^ 3 + b * x ^ 2 + c * x + d = a * ((x + b / (3 * a)) ^ 3 - s ^ 3)\n⊢ ∀ (x : K), a * x = 0 ↔ x = 0",
"after_state": "No Goals!"
},
{
"line": "intro x",
"before_state": "K : Type u_1\ninst✝² : Field K\na b c d ω q s : K\ninst✝¹ : Invertible 2\ninst✝ : Invertible 3\nha : a ≠ 0\nhω : IsPrimitiveRoot ω 3\nhpz : 3 * a * c - b ^ 2 = 0\nhq : q = (9 * a * b * c - 2 * b ^ 3 - 27 * a ^ 2 * d) / (54 * a ^ 3)\nhs3 : s ^ 3 = 2 * q\nx : K\nh₁ : ∀ (x a₁ a₂ a₃ : K), x = a₁ ∨ x = a₂ ∨ x = a₃ ↔ (x - a₁) * (x - a₂) * (x - a₃) = 0\nhi2 : 2 ≠ 0\nhi3 : 3 ≠ 0\nh54 : 54 = 2 * 3 ^ 3\nhb2 : b ^ 2 = 3 * a * c\nhb3 : b ^ 3 = 3 * a * b * c\nh₂ : a * x ^ 3 + b * x ^ 2 + c * x + d = a * ((x + b / (3 * a)) ^ 3 - s ^ 3)\n⊢ ∀ (x : K), a * x = 0 ↔ x = 0",
"after_state": "K : Type u_1\ninst✝² : Field K\na b c d ω q s : K\ninst✝¹ : Invertible 2\ninst✝ : Invertible 3\nha : a ≠ 0\nhω : IsPrimitiveRoot ω 3\nhpz : 3 * a * c - b ^ 2 = 0\nhq : q = (9 * a * b * c - 2 * b ^ 3 - 27 * a ^ 2 * d) / (54 * a ^ 3)\nhs3 : s ^ 3 = 2 * q\nx✝ : K\nh₁ : ∀ (x a₁ a₂ a₃ : K), x = a₁ ∨ x = a₂ ∨ x = a₃ ↔ (x - a₁) * (x - a₂) * (x - a₃) = 0\nhi2 : 2 ≠ 0\nhi3 : 3 ≠ 0\nh54 : 54 = 2 * 3 ^ 3\nhb2 : b ^ 2 = 3 * a * c\nhb3 : b ^ 3 = 3 * a * b * c\nh₂ : a * x✝ ^ 3 + b * x✝ ^ 2 + c * x✝ + d = a * ((x✝ + b / (3 * a)) ^ 3 - s ^ 3)\nx : K\n⊢ a * x = 0 ↔ x = 0"
},
{
"line": "simp [ha]",
"before_state": "K : Type u_1\ninst✝² : Field K\na b c d ω q s : K\ninst✝¹ : Invertible 2\ninst✝ : Invertible 3\nha : a ≠ 0\nhω : IsPrimitiveRoot ω 3\nhpz : 3 * a * c - b ^ 2 = 0\nhq : q = (9 * a * b * c - 2 * b ^ 3 - 27 * a ^ 2 * d) / (54 * a ^ 3)\nhs3 : s ^ 3 = 2 * q\nx✝ : K\nh₁ : ∀ (x a₁ a₂ a₃ : K), x = a₁ ∨ x = a₂ ∨ x = a₃ ↔ (x - a₁) * (x - a₂) * (x - a₃) = 0\nhi2 : 2 ≠ 0\nhi3 : 3 ≠ 0\nh54 : 54 = 2 * 3 ^ 3\nhb2 : b ^ 2 = 3 * a * c\nhb3 : b ^ 3 = 3 * a * b * c\nh₂ : a * x✝ ^ 3 + b * x✝ ^ 2 + c * x✝ + d = a * ((x✝ + b / (3 * a)) ^ 3 - s ^ 3)\nx : K\n⊢ a * x = 0 ↔ x = 0",
"after_state": "No Goals!"
},
{
"line": "have h₄ : ∀ x : K, x ^ 3 - s ^ 3 = (x - s) * (x - s * ω) * (x - s * ω ^ 2) :=\n by\n intro x\n calc\n x ^ 3 - s ^ 3 = (x - s) * (x ^ 2 + x * s + s ^ 2) := by ring\n _ = (x - s) * (x ^ 2 - (ω + ω ^ 2) * x * s + (1 + ω + ω ^ 2) * x * s + s ^ 2) := by ring\n _ = (x - s) * (x ^ 2 - (ω + ω ^ 2) * x * s + ω ^ 3 * s ^ 2) :=\n by\n rw [hω.pow_eq_one]; simp\n rw [cube_root_of_unity_sum hω]; simp\n _ = (x - s) * (x - s * ω) * (x - s * ω ^ 2) := by ring",
"before_state": "K : Type u_1\ninst✝² : Field K\na b c d ω q s : K\ninst✝¹ : Invertible 2\ninst✝ : Invertible 3\nha : a ≠ 0\nhω : IsPrimitiveRoot ω 3\nhpz : 3 * a * c - b ^ 2 = 0\nhq : q = (9 * a * b * c - 2 * b ^ 3 - 27 * a ^ 2 * d) / (54 * a ^ 3)\nhs3 : s ^ 3 = 2 * q\nx : K\nh₁ : ∀ (x a₁ a₂ a₃ : K), x = a₁ ∨ x = a₂ ∨ x = a₃ ↔ (x - a₁) * (x - a₂) * (x - a₃) = 0\nhi2 : 2 ≠ 0\nhi3 : 3 ≠ 0\nh54 : 54 = 2 * 3 ^ 3\nhb2 : b ^ 2 = 3 * a * c\nhb3 : b ^ 3 = 3 * a * b * c\nh₂ : a * x ^ 3 + b * x ^ 2 + c * x + d = a * ((x + b / (3 * a)) ^ 3 - s ^ 3)\nh₃ : ∀ (x : K), a * x = 0 ↔ x = 0\n⊢ a * x ^ 3 + b * x ^ 2 + c * x + d = 0 ↔ x = s - b / (3 * a) ∨ x = s * ω - b / (3 * a) ∨ x = s * ω ^ 2 - b / (3 * a)",
"after_state": "K : Type u_1\ninst✝² : Field K\na b c d ω q s : K\ninst✝¹ : Invertible 2\ninst✝ : Invertible 3\nha : a ≠ 0\nhω : IsPrimitiveRoot ω 3\nhpz : 3 * a * c - b ^ 2 = 0\nhq : q = (9 * a * b * c - 2 * b ^ 3 - 27 * a ^ 2 * d) / (54 * a ^ 3)\nhs3 : s ^ 3 = 2 * q\nx : K\nh₁ : ∀ (x a₁ a₂ a₃ : K), x = a₁ ∨ x = a₂ ∨ x = a₃ ↔ (x - a₁) * (x - a₂) * (x - a₃) = 0\nhi2 : 2 ≠ 0\nhi3 : 3 ≠ 0\nh54 : 54 = 2 * 3 ^ 3\nhb2 : b ^ 2 = 3 * a * c\nhb3 : b ^ 3 = 3 * a * b * c\nh₂ : a * x ^ 3 + b * x ^ 2 + c * x + d = a * ((x + b / (3 * a)) ^ 3 - s ^ 3)\nh₃ : ∀ (x : K), a * x = 0 ↔ x = 0\nh₄ : ∀ (x : K), x ^ 3 - s ^ 3 = (x - s) * (x - s * ω) * (x - s * ω ^ 2)\n⊢ a * x ^ 3 + b * x ^ 2 + c * x + d = 0 ↔ x = s - b / (3 * a) ∨ x = s * ω - b / (3 * a) ∨ x = s * ω ^ 2 - b / (3 * a)"
},
{
"line": "focus\n refine\n no_implicit_lambda%\n (have h₄ : ∀ x : K, x ^ 3 - s ^ 3 = (x - s) * (x - s * ω) * (x - s * ω ^ 2) := ?body✝;\n ?_)\n case body✝ =>\n with_annotate_state\"by\"\n ( intro x\n calc\n x ^ 3 - s ^ 3 = (x - s) * (x ^ 2 + x * s + s ^ 2) := by ring\n _ = (x - s) * (x ^ 2 - (ω + ω ^ 2) * x * s + (1 + ω + ω ^ 2) * x * s + s ^ 2) := by ring\n _ = (x - s) * (x ^ 2 - (ω + ω ^ 2) * x * s + ω ^ 3 * s ^ 2) :=\n by\n rw [hω.pow_eq_one]; simp\n rw [cube_root_of_unity_sum hω]; simp\n _ = (x - s) * (x - s * ω) * (x - s * ω ^ 2) := by ring)",
"before_state": "K : Type u_1\ninst✝² : Field K\na b c d ω q s : K\ninst✝¹ : Invertible 2\ninst✝ : Invertible 3\nha : a ≠ 0\nhω : IsPrimitiveRoot ω 3\nhpz : 3 * a * c - b ^ 2 = 0\nhq : q = (9 * a * b * c - 2 * b ^ 3 - 27 * a ^ 2 * d) / (54 * a ^ 3)\nhs3 : s ^ 3 = 2 * q\nx : K\nh₁ : ∀ (x a₁ a₂ a₃ : K), x = a₁ ∨ x = a₂ ∨ x = a₃ ↔ (x - a₁) * (x - a₂) * (x - a₃) = 0\nhi2 : 2 ≠ 0\nhi3 : 3 ≠ 0\nh54 : 54 = 2 * 3 ^ 3\nhb2 : b ^ 2 = 3 * a * c\nhb3 : b ^ 3 = 3 * a * b * c\nh₂ : a * x ^ 3 + b * x ^ 2 + c * x + d = a * ((x + b / (3 * a)) ^ 3 - s ^ 3)\nh₃ : ∀ (x : K), a * x = 0 ↔ x = 0\n⊢ a * x ^ 3 + b * x ^ 2 + c * x + d = 0 ↔ x = s - b / (3 * a) ∨ x = s * ω - b / (3 * a) ∨ x = s * ω ^ 2 - b / (3 * a)",
"after_state": "K : Type u_1\ninst✝² : Field K\na b c d ω q s : K\ninst✝¹ : Invertible 2\ninst✝ : Invertible 3\nha : a ≠ 0\nhω : IsPrimitiveRoot ω 3\nhpz : 3 * a * c - b ^ 2 = 0\nhq : q = (9 * a * b * c - 2 * b ^ 3 - 27 * a ^ 2 * d) / (54 * a ^ 3)\nhs3 : s ^ 3 = 2 * q\nx : K\nh₁ : ∀ (x a₁ a₂ a₃ : K), x = a₁ ∨ x = a₂ ∨ x = a₃ ↔ (x - a₁) * (x - a₂) * (x - a₃) = 0\nhi2 : 2 ≠ 0\nhi3 : 3 ≠ 0\nh54 : 54 = 2 * 3 ^ 3\nhb2 : b ^ 2 = 3 * a * c\nhb3 : b ^ 3 = 3 * a * b * c\nh₂ : a * x ^ 3 + b * x ^ 2 + c * x + d = a * ((x + b / (3 * a)) ^ 3 - s ^ 3)\nh₃ : ∀ (x : K), a * x = 0 ↔ x = 0\nh₄ : ∀ (x : K), x ^ 3 - s ^ 3 = (x - s) * (x - s * ω) * (x - s * ω ^ 2)\n⊢ a * x ^ 3 + b * x ^ 2 + c * x + d = 0 ↔ x = s - b / (3 * a) ∨ x = s * ω - b / (3 * a) ∨ x = s * ω ^ 2 - b / (3 * a)"
},
{
"line": "refine\n no_implicit_lambda%\n (have h₄ : ∀ x : K, x ^ 3 - s ^ 3 = (x - s) * (x - s * ω) * (x - s * ω ^ 2) := ?body✝;\n ?_)",
"before_state": "K : Type u_1\ninst✝² : Field K\na b c d ω q s : K\ninst✝¹ : Invertible 2\ninst✝ : Invertible 3\nha : a ≠ 0\nhω : IsPrimitiveRoot ω 3\nhpz : 3 * a * c - b ^ 2 = 0\nhq : q = (9 * a * b * c - 2 * b ^ 3 - 27 * a ^ 2 * d) / (54 * a ^ 3)\nhs3 : s ^ 3 = 2 * q\nx : K\nh₁ : ∀ (x a₁ a₂ a₃ : K), x = a₁ ∨ x = a₂ ∨ x = a₃ ↔ (x - a₁) * (x - a₂) * (x - a₃) = 0\nhi2 : 2 ≠ 0\nhi3 : 3 ≠ 0\nh54 : 54 = 2 * 3 ^ 3\nhb2 : b ^ 2 = 3 * a * c\nhb3 : b ^ 3 = 3 * a * b * c\nh₂ : a * x ^ 3 + b * x ^ 2 + c * x + d = a * ((x + b / (3 * a)) ^ 3 - s ^ 3)\nh₃ : ∀ (x : K), a * x = 0 ↔ x = 0\n⊢ a * x ^ 3 + b * x ^ 2 + c * x + d = 0 ↔ x = s - b / (3 * a) ∨ x = s * ω - b / (3 * a) ∨ x = s * ω ^ 2 - b / (3 * a)",
"after_state": "case body\nK : Type u_1\ninst✝² : Field K\na b c d ω q s : K\ninst✝¹ : Invertible 2\ninst✝ : Invertible 3\nha : a ≠ 0\nhω : IsPrimitiveRoot ω 3\nhpz : 3 * a * c - b ^ 2 = 0\nhq : q = (9 * a * b * c - 2 * b ^ 3 - 27 * a ^ 2 * d) / (54 * a ^ 3)\nhs3 : s ^ 3 = 2 * q\nx : K\nh₁ : ∀ (x a₁ a₂ a₃ : K), x = a₁ ∨ x = a₂ ∨ x = a₃ ↔ (x - a₁) * (x - a₂) * (x - a₃) = 0\nhi2 : 2 ≠ 0\nhi3 : 3 ≠ 0\nh54 : 54 = 2 * 3 ^ 3\nhb2 : b ^ 2 = 3 * a * c\nhb3 : b ^ 3 = 3 * a * b * c\nh₂ : a * x ^ 3 + b * x ^ 2 + c * x + d = a * ((x + b / (3 * a)) ^ 3 - s ^ 3)\nh₃ : ∀ (x : K), a * x = 0 ↔ x = 0\n⊢ ∀ (x : K), x ^ 3 - s ^ 3 = (x - s) * (x - s * ω) * (x - s * ω ^ 2)\n---\nK : Type u_1\ninst✝² : Field K\na b c d ω q s : K\ninst✝¹ : Invertible 2\ninst✝ : Invertible 3\nha : a ≠ 0\nhω : IsPrimitiveRoot ω 3\nhpz : 3 * a * c - b ^ 2 = 0\nhq : q = (9 * a * b * c - 2 * b ^ 3 - 27 * a ^ 2 * d) / (54 * a ^ 3)\nhs3 : s ^ 3 = 2 * q\nx : K\nh₁ : ∀ (x a₁ a₂ a₃ : K), x = a₁ ∨ x = a₂ ∨ x = a₃ ↔ (x - a₁) * (x - a₂) * (x - a₃) = 0\nhi2 : 2 ≠ 0\nhi3 : 3 ≠ 0\nh54 : 54 = 2 * 3 ^ 3\nhb2 : b ^ 2 = 3 * a * c\nhb3 : b ^ 3 = 3 * a * b * c\nh₂ : a * x ^ 3 + b * x ^ 2 + c * x + d = a * ((x + b / (3 * a)) ^ 3 - s ^ 3)\nh₃ : ∀ (x : K), a * x = 0 ↔ x = 0\nh₄ : ∀ (x : K), x ^ 3 - s ^ 3 = (x - s) * (x - s * ω) * (x - s * ω ^ 2)\n⊢ a * x ^ 3 + b * x ^ 2 + c * x + d = 0 ↔ x = s - b / (3 * a) ∨ x = s * ω - b / (3 * a) ∨ x = s * ω ^ 2 - b / (3 * a)"
},
{
"line": "case body✝ =>\n with_annotate_state\"by\"\n ( intro x\n calc\n x ^ 3 - s ^ 3 = (x - s) * (x ^ 2 + x * s + s ^ 2) := by ring\n _ = (x - s) * (x ^ 2 - (ω + ω ^ 2) * x * s + (1 + ω + ω ^ 2) * x * s + s ^ 2) := by ring\n _ = (x - s) * (x ^ 2 - (ω + ω ^ 2) * x * s + ω ^ 3 * s ^ 2) :=\n by\n rw [hω.pow_eq_one]; simp\n rw [cube_root_of_unity_sum hω]; simp\n _ = (x - s) * (x - s * ω) * (x - s * ω ^ 2) := by ring)",
"before_state": "case body\nK : Type u_1\ninst✝² : Field K\na b c d ω q s : K\ninst✝¹ : Invertible 2\ninst✝ : Invertible 3\nha : a ≠ 0\nhω : IsPrimitiveRoot ω 3\nhpz : 3 * a * c - b ^ 2 = 0\nhq : q = (9 * a * b * c - 2 * b ^ 3 - 27 * a ^ 2 * d) / (54 * a ^ 3)\nhs3 : s ^ 3 = 2 * q\nx : K\nh₁ : ∀ (x a₁ a₂ a₃ : K), x = a₁ ∨ x = a₂ ∨ x = a₃ ↔ (x - a₁) * (x - a₂) * (x - a₃) = 0\nhi2 : 2 ≠ 0\nhi3 : 3 ≠ 0\nh54 : 54 = 2 * 3 ^ 3\nhb2 : b ^ 2 = 3 * a * c\nhb3 : b ^ 3 = 3 * a * b * c\nh₂ : a * x ^ 3 + b * x ^ 2 + c * x + d = a * ((x + b / (3 * a)) ^ 3 - s ^ 3)\nh₃ : ∀ (x : K), a * x = 0 ↔ x = 0\n⊢ ∀ (x : K), x ^ 3 - s ^ 3 = (x - s) * (x - s * ω) * (x - s * ω ^ 2)\n---\nK : Type u_1\ninst✝² : Field K\na b c d ω q s : K\ninst✝¹ : Invertible 2\ninst✝ : Invertible 3\nha : a ≠ 0\nhω : IsPrimitiveRoot ω 3\nhpz : 3 * a * c - b ^ 2 = 0\nhq : q = (9 * a * b * c - 2 * b ^ 3 - 27 * a ^ 2 * d) / (54 * a ^ 3)\nhs3 : s ^ 3 = 2 * q\nx : K\nh₁ : ∀ (x a₁ a₂ a₃ : K), x = a₁ ∨ x = a₂ ∨ x = a₃ ↔ (x - a₁) * (x - a₂) * (x - a₃) = 0\nhi2 : 2 ≠ 0\nhi3 : 3 ≠ 0\nh54 : 54 = 2 * 3 ^ 3\nhb2 : b ^ 2 = 3 * a * c\nhb3 : b ^ 3 = 3 * a * b * c\nh₂ : a * x ^ 3 + b * x ^ 2 + c * x + d = a * ((x + b / (3 * a)) ^ 3 - s ^ 3)\nh₃ : ∀ (x : K), a * x = 0 ↔ x = 0\nh₄ : ∀ (x : K), x ^ 3 - s ^ 3 = (x - s) * (x - s * ω) * (x - s * ω ^ 2)\n⊢ a * x ^ 3 + b * x ^ 2 + c * x + d = 0 ↔ x = s - b / (3 * a) ∨ x = s * ω - b / (3 * a) ∨ x = s * ω ^ 2 - b / (3 * a)",
"after_state": "K : Type u_1\ninst✝² : Field K\na b c d ω q s : K\ninst✝¹ : Invertible 2\ninst✝ : Invertible 3\nha : a ≠ 0\nhω : IsPrimitiveRoot ω 3\nhpz : 3 * a * c - b ^ 2 = 0\nhq : q = (9 * a * b * c - 2 * b ^ 3 - 27 * a ^ 2 * d) / (54 * a ^ 3)\nhs3 : s ^ 3 = 2 * q\nx : K\nh₁ : ∀ (x a₁ a₂ a₃ : K), x = a₁ ∨ x = a₂ ∨ x = a₃ ↔ (x - a₁) * (x - a₂) * (x - a₃) = 0\nhi2 : 2 ≠ 0\nhi3 : 3 ≠ 0\nh54 : 54 = 2 * 3 ^ 3\nhb2 : b ^ 2 = 3 * a * c\nhb3 : b ^ 3 = 3 * a * b * c\nh₂ : a * x ^ 3 + b * x ^ 2 + c * x + d = a * ((x + b / (3 * a)) ^ 3 - s ^ 3)\nh₃ : ∀ (x : K), a * x = 0 ↔ x = 0\nh₄ : ∀ (x : K), x ^ 3 - s ^ 3 = (x - s) * (x - s * ω) * (x - s * ω ^ 2)\n⊢ a * x ^ 3 + b * x ^ 2 + c * x + d = 0 ↔ x = s - b / (3 * a) ∨ x = s * ω - b / (3 * a) ∨ x = s * ω ^ 2 - b / (3 * a)"
},
{
"line": "with_annotate_state\"by\"\n ( intro x\n calc\n x ^ 3 - s ^ 3 = (x - s) * (x ^ 2 + x * s + s ^ 2) := by ring\n _ = (x - s) * (x ^ 2 - (ω + ω ^ 2) * x * s + (1 + ω + ω ^ 2) * x * s + s ^ 2) := by ring\n _ = (x - s) * (x ^ 2 - (ω + ω ^ 2) * x * s + ω ^ 3 * s ^ 2) :=\n by\n rw [hω.pow_eq_one]; simp\n rw [cube_root_of_unity_sum hω]; simp\n _ = (x - s) * (x - s * ω) * (x - s * ω ^ 2) := by ring)",
"before_state": "K : Type u_1\ninst✝² : Field K\na b c d ω q s : K\ninst✝¹ : Invertible 2\ninst✝ : Invertible 3\nha : a ≠ 0\nhω : IsPrimitiveRoot ω 3\nhpz : 3 * a * c - b ^ 2 = 0\nhq : q = (9 * a * b * c - 2 * b ^ 3 - 27 * a ^ 2 * d) / (54 * a ^ 3)\nhs3 : s ^ 3 = 2 * q\nx : K\nh₁ : ∀ (x a₁ a₂ a₃ : K), x = a₁ ∨ x = a₂ ∨ x = a₃ ↔ (x - a₁) * (x - a₂) * (x - a₃) = 0\nhi2 : 2 ≠ 0\nhi3 : 3 ≠ 0\nh54 : 54 = 2 * 3 ^ 3\nhb2 : b ^ 2 = 3 * a * c\nhb3 : b ^ 3 = 3 * a * b * c\nh₂ : a * x ^ 3 + b * x ^ 2 + c * x + d = a * ((x + b / (3 * a)) ^ 3 - s ^ 3)\nh₃ : ∀ (x : K), a * x = 0 ↔ x = 0\n⊢ ∀ (x : K), x ^ 3 - s ^ 3 = (x - s) * (x - s * ω) * (x - s * ω ^ 2)",
"after_state": "No Goals!"
},
{
"line": "intro x",
"before_state": "K : Type u_1\ninst✝² : Field K\na b c d ω q s : K\ninst✝¹ : Invertible 2\ninst✝ : Invertible 3\nha : a ≠ 0\nhω : IsPrimitiveRoot ω 3\nhpz : 3 * a * c - b ^ 2 = 0\nhq : q = (9 * a * b * c - 2 * b ^ 3 - 27 * a ^ 2 * d) / (54 * a ^ 3)\nhs3 : s ^ 3 = 2 * q\nx : K\nh₁ : ∀ (x a₁ a₂ a₃ : K), x = a₁ ∨ x = a₂ ∨ x = a₃ ↔ (x - a₁) * (x - a₂) * (x - a₃) = 0\nhi2 : 2 ≠ 0\nhi3 : 3 ≠ 0\nh54 : 54 = 2 * 3 ^ 3\nhb2 : b ^ 2 = 3 * a * c\nhb3 : b ^ 3 = 3 * a * b * c\nh₂ : a * x ^ 3 + b * x ^ 2 + c * x + d = a * ((x + b / (3 * a)) ^ 3 - s ^ 3)\nh₃ : ∀ (x : K), a * x = 0 ↔ x = 0\n⊢ ∀ (x : K), x ^ 3 - s ^ 3 = (x - s) * (x - s * ω) * (x - s * ω ^ 2)",
"after_state": "K : Type u_1\ninst✝² : Field K\na b c d ω q s : K\ninst✝¹ : Invertible 2\ninst✝ : Invertible 3\nha : a ≠ 0\nhω : IsPrimitiveRoot ω 3\nhpz : 3 * a * c - b ^ 2 = 0\nhq : q = (9 * a * b * c - 2 * b ^ 3 - 27 * a ^ 2 * d) / (54 * a ^ 3)\nhs3 : s ^ 3 = 2 * q\nx✝ : K\nh₁ : ∀ (x a₁ a₂ a₃ : K), x = a₁ ∨ x = a₂ ∨ x = a₃ ↔ (x - a₁) * (x - a₂) * (x - a₃) = 0\nhi2 : 2 ≠ 0\nhi3 : 3 ≠ 0\nh54 : 54 = 2 * 3 ^ 3\nhb2 : b ^ 2 = 3 * a * c\nhb3 : b ^ 3 = 3 * a * b * c\nh₂ : a * x✝ ^ 3 + b * x✝ ^ 2 + c * x✝ + d = a * ((x✝ + b / (3 * a)) ^ 3 - s ^ 3)\nh₃ : ∀ (x : K), a * x = 0 ↔ x = 0\nx : K\n⊢ x ^ 3 - s ^ 3 = (x - s) * (x - s * ω) * (x - s * ω ^ 2)"
},
{
"line": "calc\n x ^ 3 - s ^ 3 = (x - s) * (x ^ 2 + x * s + s ^ 2) := by ring\n _ = (x - s) * (x ^ 2 - (ω + ω ^ 2) * x * s + (1 + ω + ω ^ 2) * x * s + s ^ 2) := by ring\n _ = (x - s) * (x ^ 2 - (ω + ω ^ 2) * x * s + ω ^ 3 * s ^ 2) :=\n by\n rw [hω.pow_eq_one]; simp\n rw [cube_root_of_unity_sum hω]; simp\n _ = (x - s) * (x - s * ω) * (x - s * ω ^ 2) := by ring",
"before_state": "K : Type u_1\ninst✝² : Field K\na b c d ω q s : K\ninst✝¹ : Invertible 2\ninst✝ : Invertible 3\nha : a ≠ 0\nhω : IsPrimitiveRoot ω 3\nhpz : 3 * a * c - b ^ 2 = 0\nhq : q = (9 * a * b * c - 2 * b ^ 3 - 27 * a ^ 2 * d) / (54 * a ^ 3)\nhs3 : s ^ 3 = 2 * q\nx✝ : K\nh₁ : ∀ (x a₁ a₂ a₃ : K), x = a₁ ∨ x = a₂ ∨ x = a₃ ↔ (x - a₁) * (x - a₂) * (x - a₃) = 0\nhi2 : 2 ≠ 0\nhi3 : 3 ≠ 0\nh54 : 54 = 2 * 3 ^ 3\nhb2 : b ^ 2 = 3 * a * c\nhb3 : b ^ 3 = 3 * a * b * c\nh₂ : a * x✝ ^ 3 + b * x✝ ^ 2 + c * x✝ + d = a * ((x✝ + b / (3 * a)) ^ 3 - s ^ 3)\nh₃ : ∀ (x : K), a * x = 0 ↔ x = 0\nx : K\n⊢ x ^ 3 - s ^ 3 = (x - s) * (x - s * ω) * (x - s * ω ^ 2)",
"after_state": "No Goals!"
},
{
"line": "ring",
"before_state": "K : Type u_1\ninst✝² : Field K\na b c d ω q s : K\ninst✝¹ : Invertible 2\ninst✝ : Invertible 3\nha : a ≠ 0\nhω : IsPrimitiveRoot ω 3\nhpz : 3 * a * c - b ^ 2 = 0\nhq : q = (9 * a * b * c - 2 * b ^ 3 - 27 * a ^ 2 * d) / (54 * a ^ 3)\nhs3 : s ^ 3 = 2 * q\nx✝ : K\nh₁ : ∀ (x a₁ a₂ a₃ : K), x = a₁ ∨ x = a₂ ∨ x = a₃ ↔ (x - a₁) * (x - a₂) * (x - a₃) = 0\nhi2 : 2 ≠ 0\nhi3 : 3 ≠ 0\nh54 : 54 = 2 * 3 ^ 3\nhb2 : b ^ 2 = 3 * a * c\nhb3 : b ^ 3 = 3 * a * b * c\nh₂ : a * x✝ ^ 3 + b * x✝ ^ 2 + c * x✝ + d = a * ((x✝ + b / (3 * a)) ^ 3 - s ^ 3)\nh₃ : ∀ (x : K), a * x = 0 ↔ x = 0\nx : K\n⊢ x ^ 3 - s ^ 3 = (x - s) * (x ^ 2 + x * s + s ^ 2)",
"after_state": "No Goals!"
},
{
"line": "first\n| ring1\n|\n try_this ring_nf\"\\n\\nThe `ring` tactic failed to close the goal. Use `ring_nf` to obtain a normal form.\n \\nNote that `ring` works primarily in *commutative* rings. \\\n If you have a noncommutative ring, abelian group or module, consider using \\\n `noncomm_ring`, `abel` or `module` instead.\"",
"before_state": "K : Type u_1\ninst✝² : Field K\na b c d ω q s : K\ninst✝¹ : Invertible 2\ninst✝ : Invertible 3\nha : a ≠ 0\nhω : IsPrimitiveRoot ω 3\nhpz : 3 * a * c - b ^ 2 = 0\nhq : q = (9 * a * b * c - 2 * b ^ 3 - 27 * a ^ 2 * d) / (54 * a ^ 3)\nhs3 : s ^ 3 = 2 * q\nx✝ : K\nh₁ : ∀ (x a₁ a₂ a₃ : K), x = a₁ ∨ x = a₂ ∨ x = a₃ ↔ (x - a₁) * (x - a₂) * (x - a₃) = 0\nhi2 : 2 ≠ 0\nhi3 : 3 ≠ 0\nh54 : 54 = 2 * 3 ^ 3\nhb2 : b ^ 2 = 3 * a * c\nhb3 : b ^ 3 = 3 * a * b * c\nh₂ : a * x✝ ^ 3 + b * x✝ ^ 2 + c * x✝ + d = a * ((x✝ + b / (3 * a)) ^ 3 - s ^ 3)\nh₃ : ∀ (x : K), a * x = 0 ↔ x = 0\nx : K\n⊢ x ^ 3 - s ^ 3 = (x - s) * (x ^ 2 + x * s + s ^ 2)",
"after_state": "No Goals!"
},
{
"line": "ring1",
"before_state": "K : Type u_1\ninst✝² : Field K\na b c d ω q s : K\ninst✝¹ : Invertible 2\ninst✝ : Invertible 3\nha : a ≠ 0\nhω : IsPrimitiveRoot ω 3\nhpz : 3 * a * c - b ^ 2 = 0\nhq : q = (9 * a * b * c - 2 * b ^ 3 - 27 * a ^ 2 * d) / (54 * a ^ 3)\nhs3 : s ^ 3 = 2 * q\nx✝ : K\nh₁ : ∀ (x a₁ a₂ a₃ : K), x = a₁ ∨ x = a₂ ∨ x = a₃ ↔ (x - a₁) * (x - a₂) * (x - a₃) = 0\nhi2 : 2 ≠ 0\nhi3 : 3 ≠ 0\nh54 : 54 = 2 * 3 ^ 3\nhb2 : b ^ 2 = 3 * a * c\nhb3 : b ^ 3 = 3 * a * b * c\nh₂ : a * x✝ ^ 3 + b * x✝ ^ 2 + c * x✝ + d = a * ((x✝ + b / (3 * a)) ^ 3 - s ^ 3)\nh₃ : ∀ (x : K), a * x = 0 ↔ x = 0\nx : K\n⊢ x ^ 3 - s ^ 3 = (x - s) * (x ^ 2 + x * s + s ^ 2)",
"after_state": "No Goals!"
},
{
"line": "ring",
"before_state": "K : Type u_1\ninst✝² : Field K\na b c d ω q s : K\ninst✝¹ : Invertible 2\ninst✝ : Invertible 3\nha : a ≠ 0\nhω : IsPrimitiveRoot ω 3\nhpz : 3 * a * c - b ^ 2 = 0\nhq : q = (9 * a * b * c - 2 * b ^ 3 - 27 * a ^ 2 * d) / (54 * a ^ 3)\nhs3 : s ^ 3 = 2 * q\nx✝ : K\nh₁ : ∀ (x a₁ a₂ a₃ : K), x = a₁ ∨ x = a₂ ∨ x = a₃ ↔ (x - a₁) * (x - a₂) * (x - a₃) = 0\nhi2 : 2 ≠ 0\nhi3 : 3 ≠ 0\nh54 : 54 = 2 * 3 ^ 3\nhb2 : b ^ 2 = 3 * a * c\nhb3 : b ^ 3 = 3 * a * b * c\nh₂ : a * x✝ ^ 3 + b * x✝ ^ 2 + c * x✝ + d = a * ((x✝ + b / (3 * a)) ^ 3 - s ^ 3)\nh₃ : ∀ (x : K), a * x = 0 ↔ x = 0\nx : K\n⊢ (x - s) * (x ^ 2 + x * s + s ^ 2) = (x - s) * (x ^ 2 - (ω + ω ^ 2) * x * s + (1 + ω + ω ^ 2) * x * s + s ^ 2)",
"after_state": "No Goals!"
},
{
"line": "first\n| ring1\n|\n try_this ring_nf\"\\n\\nThe `ring` tactic failed to close the goal. Use `ring_nf` to obtain a normal form.\n \\nNote that `ring` works primarily in *commutative* rings. \\\n If you have a noncommutative ring, abelian group or module, consider using \\\n `noncomm_ring`, `abel` or `module` instead.\"",
"before_state": "K : Type u_1\ninst✝² : Field K\na b c d ω q s : K\ninst✝¹ : Invertible 2\ninst✝ : Invertible 3\nha : a ≠ 0\nhω : IsPrimitiveRoot ω 3\nhpz : 3 * a * c - b ^ 2 = 0\nhq : q = (9 * a * b * c - 2 * b ^ 3 - 27 * a ^ 2 * d) / (54 * a ^ 3)\nhs3 : s ^ 3 = 2 * q\nx✝ : K\nh₁ : ∀ (x a₁ a₂ a₃ : K), x = a₁ ∨ x = a₂ ∨ x = a₃ ↔ (x - a₁) * (x - a₂) * (x - a₃) = 0\nhi2 : 2 ≠ 0\nhi3 : 3 ≠ 0\nh54 : 54 = 2 * 3 ^ 3\nhb2 : b ^ 2 = 3 * a * c\nhb3 : b ^ 3 = 3 * a * b * c\nh₂ : a * x✝ ^ 3 + b * x✝ ^ 2 + c * x✝ + d = a * ((x✝ + b / (3 * a)) ^ 3 - s ^ 3)\nh₃ : ∀ (x : K), a * x = 0 ↔ x = 0\nx : K\n⊢ (x - s) * (x ^ 2 + x * s + s ^ 2) = (x - s) * (x ^ 2 - (ω + ω ^ 2) * x * s + (1 + ω + ω ^ 2) * x * s + s ^ 2)",
"after_state": "No Goals!"
},
{
"line": "ring1",
"before_state": "K : Type u_1\ninst✝² : Field K\na b c d ω q s : K\ninst✝¹ : Invertible 2\ninst✝ : Invertible 3\nha : a ≠ 0\nhω : IsPrimitiveRoot ω 3\nhpz : 3 * a * c - b ^ 2 = 0\nhq : q = (9 * a * b * c - 2 * b ^ 3 - 27 * a ^ 2 * d) / (54 * a ^ 3)\nhs3 : s ^ 3 = 2 * q\nx✝ : K\nh₁ : ∀ (x a₁ a₂ a₃ : K), x = a₁ ∨ x = a₂ ∨ x = a₃ ↔ (x - a₁) * (x - a₂) * (x - a₃) = 0\nhi2 : 2 ≠ 0\nhi3 : 3 ≠ 0\nh54 : 54 = 2 * 3 ^ 3\nhb2 : b ^ 2 = 3 * a * c\nhb3 : b ^ 3 = 3 * a * b * c\nh₂ : a * x✝ ^ 3 + b * x✝ ^ 2 + c * x✝ + d = a * ((x✝ + b / (3 * a)) ^ 3 - s ^ 3)\nh₃ : ∀ (x : K), a * x = 0 ↔ x = 0\nx : K\n⊢ (x - s) * (x ^ 2 + x * s + s ^ 2) = (x - s) * (x ^ 2 - (ω + ω ^ 2) * x * s + (1 + ω + ω ^ 2) * x * s + s ^ 2)",
"after_state": "No Goals!"
},
{
"line": "rw [hω.pow_eq_one]",
"before_state": "K : Type u_1\ninst✝² : Field K\na b c d ω q s : K\ninst✝¹ : Invertible 2\ninst✝ : Invertible 3\nha : a ≠ 0\nhω : IsPrimitiveRoot ω 3\nhpz : 3 * a * c - b ^ 2 = 0\nhq : q = (9 * a * b * c - 2 * b ^ 3 - 27 * a ^ 2 * d) / (54 * a ^ 3)\nhs3 : s ^ 3 = 2 * q\nx✝ : K\nh₁ : ∀ (x a₁ a₂ a₃ : K), x = a₁ ∨ x = a₂ ∨ x = a₃ ↔ (x - a₁) * (x - a₂) * (x - a₃) = 0\nhi2 : 2 ≠ 0\nhi3 : 3 ≠ 0\nh54 : 54 = 2 * 3 ^ 3\nhb2 : b ^ 2 = 3 * a * c\nhb3 : b ^ 3 = 3 * a * b * c\nh₂ : a * x✝ ^ 3 + b * x✝ ^ 2 + c * x✝ + d = a * ((x✝ + b / (3 * a)) ^ 3 - s ^ 3)\nh₃ : ∀ (x : K), a * x = 0 ↔ x = 0\nx : K\n⊢ (x - s) * (x ^ 2 - (ω + ω ^ 2) * x * s + (1 + ω + ω ^ 2) * x * s + s ^ 2) =\n (x - s) * (x ^ 2 - (ω + ω ^ 2) * x * s + ω ^ 3 * s ^ 2)",
"after_state": "K : Type u_1\ninst✝² : Field K\na b c d ω q s : K\ninst✝¹ : Invertible 2\ninst✝ : Invertible 3\nha : a ≠ 0\nhω : IsPrimitiveRoot ω 3\nhpz : 3 * a * c - b ^ 2 = 0\nhq : q = (9 * a * b * c - 2 * b ^ 3 - 27 * a ^ 2 * d) / (54 * a ^ 3)\nhs3 : s ^ 3 = 2 * q\nx✝ : K\nh₁ : ∀ (x a₁ a₂ a₃ : K), x = a₁ ∨ x = a₂ ∨ x = a₃ ↔ (x - a₁) * (x - a₂) * (x - a₃) = 0\nhi2 : 2 ≠ 0\nhi3 : 3 ≠ 0\nh54 : 54 = 2 * 3 ^ 3\nhb2 : b ^ 2 = 3 * a * c\nhb3 : b ^ 3 = 3 * a * b * c\nh₂ : a * x✝ ^ 3 + b * x✝ ^ 2 + c * x✝ + d = a * ((x✝ + b / (3 * a)) ^ 3 - s ^ 3)\nh₃ : ∀ (x : K), a * x = 0 ↔ x = 0\nx : K\n⊢ (x - s) * (x ^ 2 - (ω + ω ^ 2) * x * s + (1 + ω + ω ^ 2) * x * s + s ^ 2) =\n (x - s) * (x ^ 2 - (ω + ω ^ 2) * x * s + 1 * s ^ 2)"
},
{
"line": "rewrite [hω.pow_eq_one]",
"before_state": "K : Type u_1\ninst✝² : Field K\na b c d ω q s : K\ninst✝¹ : Invertible 2\ninst✝ : Invertible 3\nha : a ≠ 0\nhω : IsPrimitiveRoot ω 3\nhpz : 3 * a * c - b ^ 2 = 0\nhq : q = (9 * a * b * c - 2 * b ^ 3 - 27 * a ^ 2 * d) / (54 * a ^ 3)\nhs3 : s ^ 3 = 2 * q\nx✝ : K\nh₁ : ∀ (x a₁ a₂ a₃ : K), x = a₁ ∨ x = a₂ ∨ x = a₃ ↔ (x - a₁) * (x - a₂) * (x - a₃) = 0\nhi2 : 2 ≠ 0\nhi3 : 3 ≠ 0\nh54 : 54 = 2 * 3 ^ 3\nhb2 : b ^ 2 = 3 * a * c\nhb3 : b ^ 3 = 3 * a * b * c\nh₂ : a * x✝ ^ 3 + b * x✝ ^ 2 + c * x✝ + d = a * ((x✝ + b / (3 * a)) ^ 3 - s ^ 3)\nh₃ : ∀ (x : K), a * x = 0 ↔ x = 0\nx : K\n⊢ (x - s) * (x ^ 2 - (ω + ω ^ 2) * x * s + (1 + ω + ω ^ 2) * x * s + s ^ 2) =\n (x - s) * (x ^ 2 - (ω + ω ^ 2) * x * s + ω ^ 3 * s ^ 2)",
"after_state": "K : Type u_1\ninst✝² : Field K\na b c d ω q s : K\ninst✝¹ : Invertible 2\ninst✝ : Invertible 3\nha : a ≠ 0\nhω : IsPrimitiveRoot ω 3\nhpz : 3 * a * c - b ^ 2 = 0\nhq : q = (9 * a * b * c - 2 * b ^ 3 - 27 * a ^ 2 * d) / (54 * a ^ 3)\nhs3 : s ^ 3 = 2 * q\nx✝ : K\nh₁ : ∀ (x a₁ a₂ a₃ : K), x = a₁ ∨ x = a₂ ∨ x = a₃ ↔ (x - a₁) * (x - a₂) * (x - a₃) = 0\nhi2 : 2 ≠ 0\nhi3 : 3 ≠ 0\nh54 : 54 = 2 * 3 ^ 3\nhb2 : b ^ 2 = 3 * a * c\nhb3 : b ^ 3 = 3 * a * b * c\nh₂ : a * x✝ ^ 3 + b * x✝ ^ 2 + c * x✝ + d = a * ((x✝ + b / (3 * a)) ^ 3 - s ^ 3)\nh₃ : ∀ (x : K), a * x = 0 ↔ x = 0\nx : K\n⊢ (x - s) * (x ^ 2 - (ω + ω ^ 2) * x * s + (1 + ω + ω ^ 2) * x * s + s ^ 2) =\n (x - s) * (x ^ 2 - (ω + ω ^ 2) * x * s + 1 * s ^ 2)"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "K : Type u_1\ninst✝² : Field K\na b c d ω q s : K\ninst✝¹ : Invertible 2\ninst✝ : Invertible 3\nha : a ≠ 0\nhω : IsPrimitiveRoot ω 3\nhpz : 3 * a * c - b ^ 2 = 0\nhq : q = (9 * a * b * c - 2 * b ^ 3 - 27 * a ^ 2 * d) / (54 * a ^ 3)\nhs3 : s ^ 3 = 2 * q\nx✝ : K\nh₁ : ∀ (x a₁ a₂ a₃ : K), x = a₁ ∨ x = a₂ ∨ x = a₃ ↔ (x - a₁) * (x - a₂) * (x - a₃) = 0\nhi2 : 2 ≠ 0\nhi3 : 3 ≠ 0\nh54 : 54 = 2 * 3 ^ 3\nhb2 : b ^ 2 = 3 * a * c\nhb3 : b ^ 3 = 3 * a * b * c\nh₂ : a * x✝ ^ 3 + b * x✝ ^ 2 + c * x✝ + d = a * ((x✝ + b / (3 * a)) ^ 3 - s ^ 3)\nh₃ : ∀ (x : K), a * x = 0 ↔ x = 0\nx : K\n⊢ (x - s) * (x ^ 2 - (ω + ω ^ 2) * x * s + (1 + ω + ω ^ 2) * x * s + s ^ 2) =\n (x - s) * (x ^ 2 - (ω + ω ^ 2) * x * s + 1 * s ^ 2)",
"after_state": "K : Type u_1\ninst✝² : Field K\na b c d ω q s : K\ninst✝¹ : Invertible 2\ninst✝ : Invertible 3\nha : a ≠ 0\nhω : IsPrimitiveRoot ω 3\nhpz : 3 * a * c - b ^ 2 = 0\nhq : q = (9 * a * b * c - 2 * b ^ 3 - 27 * a ^ 2 * d) / (54 * a ^ 3)\nhs3 : s ^ 3 = 2 * q\nx✝ : K\nh₁ : ∀ (x a₁ a₂ a₃ : K), x = a₁ ∨ x = a₂ ∨ x = a₃ ↔ (x - a₁) * (x - a₂) * (x - a₃) = 0\nhi2 : 2 ≠ 0\nhi3 : 3 ≠ 0\nh54 : 54 = 2 * 3 ^ 3\nhb2 : b ^ 2 = 3 * a * c\nhb3 : b ^ 3 = 3 * a * b * c\nh₂ : a * x✝ ^ 3 + b * x✝ ^ 2 + c * x✝ + d = a * ((x✝ + b / (3 * a)) ^ 3 - s ^ 3)\nh₃ : ∀ (x : K), a * x = 0 ↔ x = 0\nx : K\n⊢ (x - s) * (x ^ 2 - (ω + ω ^ 2) * x * s + (1 + ω + ω ^ 2) * x * s + s ^ 2) =\n (x - s) * (x ^ 2 - (ω + ω ^ 2) * x * s + 1 * s ^ 2)"
},
{
"line": "try (with_reducible rfl)",
"before_state": "K : Type u_1\ninst✝² : Field K\na b c d ω q s : K\ninst✝¹ : Invertible 2\ninst✝ : Invertible 3\nha : a ≠ 0\nhω : IsPrimitiveRoot ω 3\nhpz : 3 * a * c - b ^ 2 = 0\nhq : q = (9 * a * b * c - 2 * b ^ 3 - 27 * a ^ 2 * d) / (54 * a ^ 3)\nhs3 : s ^ 3 = 2 * q\nx✝ : K\nh₁ : ∀ (x a₁ a₂ a₃ : K), x = a₁ ∨ x = a₂ ∨ x = a₃ ↔ (x - a₁) * (x - a₂) * (x - a₃) = 0\nhi2 : 2 ≠ 0\nhi3 : 3 ≠ 0\nh54 : 54 = 2 * 3 ^ 3\nhb2 : b ^ 2 = 3 * a * c\nhb3 : b ^ 3 = 3 * a * b * c\nh₂ : a * x✝ ^ 3 + b * x✝ ^ 2 + c * x✝ + d = a * ((x✝ + b / (3 * a)) ^ 3 - s ^ 3)\nh₃ : ∀ (x : K), a * x = 0 ↔ x = 0\nx : K\n⊢ (x - s) * (x ^ 2 - (ω + ω ^ 2) * x * s + (1 + ω + ω ^ 2) * x * s + s ^ 2) =\n (x - s) * (x ^ 2 - (ω + ω ^ 2) * x * s + 1 * s ^ 2)",
"after_state": "K : Type u_1\ninst✝² : Field K\na b c d ω q s : K\ninst✝¹ : Invertible 2\ninst✝ : Invertible 3\nha : a ≠ 0\nhω : IsPrimitiveRoot ω 3\nhpz : 3 * a * c - b ^ 2 = 0\nhq : q = (9 * a * b * c - 2 * b ^ 3 - 27 * a ^ 2 * d) / (54 * a ^ 3)\nhs3 : s ^ 3 = 2 * q\nx✝ : K\nh₁ : ∀ (x a₁ a₂ a₃ : K), x = a₁ ∨ x = a₂ ∨ x = a₃ ↔ (x - a₁) * (x - a₂) * (x - a₃) = 0\nhi2 : 2 ≠ 0\nhi3 : 3 ≠ 0\nh54 : 54 = 2 * 3 ^ 3\nhb2 : b ^ 2 = 3 * a * c\nhb3 : b ^ 3 = 3 * a * b * c\nh₂ : a * x✝ ^ 3 + b * x✝ ^ 2 + c * x✝ + d = a * ((x✝ + b / (3 * a)) ^ 3 - s ^ 3)\nh₃ : ∀ (x : K), a * x = 0 ↔ x = 0\nx : K\n⊢ (x - s) * (x ^ 2 - (ω + ω ^ 2) * x * s + (1 + ω + ω ^ 2) * x * s + s ^ 2) =\n (x - s) * (x ^ 2 - (ω + ω ^ 2) * x * s + 1 * s ^ 2)"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "K : Type u_1\ninst✝² : Field K\na b c d ω q s : K\ninst✝¹ : Invertible 2\ninst✝ : Invertible 3\nha : a ≠ 0\nhω : IsPrimitiveRoot ω 3\nhpz : 3 * a * c - b ^ 2 = 0\nhq : q = (9 * a * b * c - 2 * b ^ 3 - 27 * a ^ 2 * d) / (54 * a ^ 3)\nhs3 : s ^ 3 = 2 * q\nx✝ : K\nh₁ : ∀ (x a₁ a₂ a₃ : K), x = a₁ ∨ x = a₂ ∨ x = a₃ ↔ (x - a₁) * (x - a₂) * (x - a₃) = 0\nhi2 : 2 ≠ 0\nhi3 : 3 ≠ 0\nh54 : 54 = 2 * 3 ^ 3\nhb2 : b ^ 2 = 3 * a * c\nhb3 : b ^ 3 = 3 * a * b * c\nh₂ : a * x✝ ^ 3 + b * x✝ ^ 2 + c * x✝ + d = a * ((x✝ + b / (3 * a)) ^ 3 - s ^ 3)\nh₃ : ∀ (x : K), a * x = 0 ↔ x = 0\nx : K\n⊢ (x - s) * (x ^ 2 - (ω + ω ^ 2) * x * s + (1 + ω + ω ^ 2) * x * s + s ^ 2) =\n (x - s) * (x ^ 2 - (ω + ω ^ 2) * x * s + 1 * s ^ 2)",
"after_state": "K : Type u_1\ninst✝² : Field K\na b c d ω q s : K\ninst✝¹ : Invertible 2\ninst✝ : Invertible 3\nha : a ≠ 0\nhω : IsPrimitiveRoot ω 3\nhpz : 3 * a * c - b ^ 2 = 0\nhq : q = (9 * a * b * c - 2 * b ^ 3 - 27 * a ^ 2 * d) / (54 * a ^ 3)\nhs3 : s ^ 3 = 2 * q\nx✝ : K\nh₁ : ∀ (x a₁ a₂ a₃ : K), x = a₁ ∨ x = a₂ ∨ x = a₃ ↔ (x - a₁) * (x - a₂) * (x - a₃) = 0\nhi2 : 2 ≠ 0\nhi3 : 3 ≠ 0\nh54 : 54 = 2 * 3 ^ 3\nhb2 : b ^ 2 = 3 * a * c\nhb3 : b ^ 3 = 3 * a * b * c\nh₂ : a * x✝ ^ 3 + b * x✝ ^ 2 + c * x✝ + d = a * ((x✝ + b / (3 * a)) ^ 3 - s ^ 3)\nh₃ : ∀ (x : K), a * x = 0 ↔ x = 0\nx : K\n⊢ (x - s) * (x ^ 2 - (ω + ω ^ 2) * x * s + (1 + ω + ω ^ 2) * x * s + s ^ 2) =\n (x - s) * (x ^ 2 - (ω + ω ^ 2) * x * s + 1 * s ^ 2)"
},
{
"line": "with_reducible rfl",
"before_state": "K : Type u_1\ninst✝² : Field K\na b c d ω q s : K\ninst✝¹ : Invertible 2\ninst✝ : Invertible 3\nha : a ≠ 0\nhω : IsPrimitiveRoot ω 3\nhpz : 3 * a * c - b ^ 2 = 0\nhq : q = (9 * a * b * c - 2 * b ^ 3 - 27 * a ^ 2 * d) / (54 * a ^ 3)\nhs3 : s ^ 3 = 2 * q\nx✝ : K\nh₁ : ∀ (x a₁ a₂ a₃ : K), x = a₁ ∨ x = a₂ ∨ x = a₃ ↔ (x - a₁) * (x - a₂) * (x - a₃) = 0\nhi2 : 2 ≠ 0\nhi3 : 3 ≠ 0\nh54 : 54 = 2 * 3 ^ 3\nhb2 : b ^ 2 = 3 * a * c\nhb3 : b ^ 3 = 3 * a * b * c\nh₂ : a * x✝ ^ 3 + b * x✝ ^ 2 + c * x✝ + d = a * ((x✝ + b / (3 * a)) ^ 3 - s ^ 3)\nh₃ : ∀ (x : K), a * x = 0 ↔ x = 0\nx : K\n⊢ (x - s) * (x ^ 2 - (ω + ω ^ 2) * x * s + (1 + ω + ω ^ 2) * x * s + s ^ 2) =\n (x - s) * (x ^ 2 - (ω + ω ^ 2) * x * s + 1 * s ^ 2)",
"after_state": "K : Type u_1\ninst✝² : Field K\na b c d ω q s : K\ninst✝¹ : Invertible 2\ninst✝ : Invertible 3\nha : a ≠ 0\nhω : IsPrimitiveRoot ω 3\nhpz : 3 * a * c - b ^ 2 = 0\nhq : q = (9 * a * b * c - 2 * b ^ 3 - 27 * a ^ 2 * d) / (54 * a ^ 3)\nhs3 : s ^ 3 = 2 * q\nx✝ : K\nh₁ : ∀ (x a₁ a₂ a₃ : K), x = a₁ ∨ x = a₂ ∨ x = a₃ ↔ (x - a₁) * (x - a₂) * (x - a₃) = 0\nhi2 : 2 ≠ 0\nhi3 : 3 ≠ 0\nh54 : 54 = 2 * 3 ^ 3\nhb2 : b ^ 2 = 3 * a * c\nhb3 : b ^ 3 = 3 * a * b * c\nh₂ : a * x✝ ^ 3 + b * x✝ ^ 2 + c * x✝ + d = a * ((x✝ + b / (3 * a)) ^ 3 - s ^ 3)\nh₃ : ∀ (x : K), a * x = 0 ↔ x = 0\nx : K\n⊢ (x - s) * (x ^ 2 - (ω + ω ^ 2) * x * s + (1 + ω + ω ^ 2) * x * s + s ^ 2) =\n (x - s) * (x ^ 2 - (ω + ω ^ 2) * x * s + 1 * s ^ 2)"
},
{
"line": "rfl",
"before_state": "K : Type u_1\ninst✝² : Field K\na b c d ω q s : K\ninst✝¹ : Invertible 2\ninst✝ : Invertible 3\nha : a ≠ 0\nhω : IsPrimitiveRoot ω 3\nhpz : 3 * a * c - b ^ 2 = 0\nhq : q = (9 * a * b * c - 2 * b ^ 3 - 27 * a ^ 2 * d) / (54 * a ^ 3)\nhs3 : s ^ 3 = 2 * q\nx✝ : K\nh₁ : ∀ (x a₁ a₂ a₃ : K), x = a₁ ∨ x = a₂ ∨ x = a₃ ↔ (x - a₁) * (x - a₂) * (x - a₃) = 0\nhi2 : 2 ≠ 0\nhi3 : 3 ≠ 0\nh54 : 54 = 2 * 3 ^ 3\nhb2 : b ^ 2 = 3 * a * c\nhb3 : b ^ 3 = 3 * a * b * c\nh₂ : a * x✝ ^ 3 + b * x✝ ^ 2 + c * x✝ + d = a * ((x✝ + b / (3 * a)) ^ 3 - s ^ 3)\nh₃ : ∀ (x : K), a * x = 0 ↔ x = 0\nx : K\n⊢ (x - s) * (x ^ 2 - (ω + ω ^ 2) * x * s + (1 + ω + ω ^ 2) * x * s + s ^ 2) =\n (x - s) * (x ^ 2 - (ω + ω ^ 2) * x * s + 1 * s ^ 2)",
"after_state": "K : Type u_1\ninst✝² : Field K\na b c d ω q s : K\ninst✝¹ : Invertible 2\ninst✝ : Invertible 3\nha : a ≠ 0\nhω : IsPrimitiveRoot ω 3\nhpz : 3 * a * c - b ^ 2 = 0\nhq : q = (9 * a * b * c - 2 * b ^ 3 - 27 * a ^ 2 * d) / (54 * a ^ 3)\nhs3 : s ^ 3 = 2 * q\nx✝ : K\nh₁ : ∀ (x a₁ a₂ a₃ : K), x = a₁ ∨ x = a₂ ∨ x = a₃ ↔ (x - a₁) * (x - a₂) * (x - a₃) = 0\nhi2 : 2 ≠ 0\nhi3 : 3 ≠ 0\nh54 : 54 = 2 * 3 ^ 3\nhb2 : b ^ 2 = 3 * a * c\nhb3 : b ^ 3 = 3 * a * b * c\nh₂ : a * x✝ ^ 3 + b * x✝ ^ 2 + c * x✝ + d = a * ((x✝ + b / (3 * a)) ^ 3 - s ^ 3)\nh₃ : ∀ (x : K), a * x = 0 ↔ x = 0\nx : K\n⊢ (x - s) * (x ^ 2 - (ω + ω ^ 2) * x * s + (1 + ω + ω ^ 2) * x * s + s ^ 2) =\n (x - s) * (x ^ 2 - (ω + ω ^ 2) * x * s + 1 * s ^ 2)"
},
{
"line": "apply_rfl",
"before_state": "K : Type u_1\ninst✝² : Field K\na b c d ω q s : K\ninst✝¹ : Invertible 2\ninst✝ : Invertible 3\nha : a ≠ 0\nhω : IsPrimitiveRoot ω 3\nhpz : 3 * a * c - b ^ 2 = 0\nhq : q = (9 * a * b * c - 2 * b ^ 3 - 27 * a ^ 2 * d) / (54 * a ^ 3)\nhs3 : s ^ 3 = 2 * q\nx✝ : K\nh₁ : ∀ (x a₁ a₂ a₃ : K), x = a₁ ∨ x = a₂ ∨ x = a₃ ↔ (x - a₁) * (x - a₂) * (x - a₃) = 0\nhi2 : 2 ≠ 0\nhi3 : 3 ≠ 0\nh54 : 54 = 2 * 3 ^ 3\nhb2 : b ^ 2 = 3 * a * c\nhb3 : b ^ 3 = 3 * a * b * c\nh₂ : a * x✝ ^ 3 + b * x✝ ^ 2 + c * x✝ + d = a * ((x✝ + b / (3 * a)) ^ 3 - s ^ 3)\nh₃ : ∀ (x : K), a * x = 0 ↔ x = 0\nx : K\n⊢ (x - s) * (x ^ 2 - (ω + ω ^ 2) * x * s + (1 + ω + ω ^ 2) * x * s + s ^ 2) =\n (x - s) * (x ^ 2 - (ω + ω ^ 2) * x * s + 1 * s ^ 2)",
"after_state": "K : Type u_1\ninst✝² : Field K\na b c d ω q s : K\ninst✝¹ : Invertible 2\ninst✝ : Invertible 3\nha : a ≠ 0\nhω : IsPrimitiveRoot ω 3\nhpz : 3 * a * c - b ^ 2 = 0\nhq : q = (9 * a * b * c - 2 * b ^ 3 - 27 * a ^ 2 * d) / (54 * a ^ 3)\nhs3 : s ^ 3 = 2 * q\nx✝ : K\nh₁ : ∀ (x a₁ a₂ a₃ : K), x = a₁ ∨ x = a₂ ∨ x = a₃ ↔ (x - a₁) * (x - a₂) * (x - a₃) = 0\nhi2 : 2 ≠ 0\nhi3 : 3 ≠ 0\nh54 : 54 = 2 * 3 ^ 3\nhb2 : b ^ 2 = 3 * a * c\nhb3 : b ^ 3 = 3 * a * b * c\nh₂ : a * x✝ ^ 3 + b * x✝ ^ 2 + c * x✝ + d = a * ((x✝ + b / (3 * a)) ^ 3 - s ^ 3)\nh₃ : ∀ (x : K), a * x = 0 ↔ x = 0\nx : K\n⊢ (x - s) * (x ^ 2 - (ω + ω ^ 2) * x * s + (1 + ω + ω ^ 2) * x * s + s ^ 2) =\n (x - s) * (x ^ 2 - (ω + ω ^ 2) * x * s + 1 * s ^ 2)"
},
{
"line": "skip",
"before_state": "K : Type u_1\ninst✝² : Field K\na b c d ω q s : K\ninst✝¹ : Invertible 2\ninst✝ : Invertible 3\nha : a ≠ 0\nhω : IsPrimitiveRoot ω 3\nhpz : 3 * a * c - b ^ 2 = 0\nhq : q = (9 * a * b * c - 2 * b ^ 3 - 27 * a ^ 2 * d) / (54 * a ^ 3)\nhs3 : s ^ 3 = 2 * q\nx✝ : K\nh₁ : ∀ (x a₁ a₂ a₃ : K), x = a₁ ∨ x = a₂ ∨ x = a₃ ↔ (x - a₁) * (x - a₂) * (x - a₃) = 0\nhi2 : 2 ≠ 0\nhi3 : 3 ≠ 0\nh54 : 54 = 2 * 3 ^ 3\nhb2 : b ^ 2 = 3 * a * c\nhb3 : b ^ 3 = 3 * a * b * c\nh₂ : a * x✝ ^ 3 + b * x✝ ^ 2 + c * x✝ + d = a * ((x✝ + b / (3 * a)) ^ 3 - s ^ 3)\nh₃ : ∀ (x : K), a * x = 0 ↔ x = 0\nx : K\n⊢ (x - s) * (x ^ 2 - (ω + ω ^ 2) * x * s + (1 + ω + ω ^ 2) * x * s + s ^ 2) =\n (x - s) * (x ^ 2 - (ω + ω ^ 2) * x * s + 1 * s ^ 2)",
"after_state": "K : Type u_1\ninst✝² : Field K\na b c d ω q s : K\ninst✝¹ : Invertible 2\ninst✝ : Invertible 3\nha : a ≠ 0\nhω : IsPrimitiveRoot ω 3\nhpz : 3 * a * c - b ^ 2 = 0\nhq : q = (9 * a * b * c - 2 * b ^ 3 - 27 * a ^ 2 * d) / (54 * a ^ 3)\nhs3 : s ^ 3 = 2 * q\nx✝ : K\nh₁ : ∀ (x a₁ a₂ a₃ : K), x = a₁ ∨ x = a₂ ∨ x = a₃ ↔ (x - a₁) * (x - a₂) * (x - a₃) = 0\nhi2 : 2 ≠ 0\nhi3 : 3 ≠ 0\nh54 : 54 = 2 * 3 ^ 3\nhb2 : b ^ 2 = 3 * a * c\nhb3 : b ^ 3 = 3 * a * b * c\nh₂ : a * x✝ ^ 3 + b * x✝ ^ 2 + c * x✝ + d = a * ((x✝ + b / (3 * a)) ^ 3 - s ^ 3)\nh₃ : ∀ (x : K), a * x = 0 ↔ x = 0\nx : K\n⊢ (x - s) * (x ^ 2 - (ω + ω ^ 2) * x * s + (1 + ω + ω ^ 2) * x * s + s ^ 2) =\n (x - s) * (x ^ 2 - (ω + ω ^ 2) * x * s + 1 * s ^ 2)"
},
{
"line": "simp",
"before_state": "K : Type u_1\ninst✝² : Field K\na b c d ω q s : K\ninst✝¹ : Invertible 2\ninst✝ : Invertible 3\nha : a ≠ 0\nhω : IsPrimitiveRoot ω 3\nhpz : 3 * a * c - b ^ 2 = 0\nhq : q = (9 * a * b * c - 2 * b ^ 3 - 27 * a ^ 2 * d) / (54 * a ^ 3)\nhs3 : s ^ 3 = 2 * q\nx✝ : K\nh₁ : ∀ (x a₁ a₂ a₃ : K), x = a₁ ∨ x = a₂ ∨ x = a₃ ↔ (x - a₁) * (x - a₂) * (x - a₃) = 0\nhi2 : 2 ≠ 0\nhi3 : 3 ≠ 0\nh54 : 54 = 2 * 3 ^ 3\nhb2 : b ^ 2 = 3 * a * c\nhb3 : b ^ 3 = 3 * a * b * c\nh₂ : a * x✝ ^ 3 + b * x✝ ^ 2 + c * x✝ + d = a * ((x✝ + b / (3 * a)) ^ 3 - s ^ 3)\nh₃ : ∀ (x : K), a * x = 0 ↔ x = 0\nx : K\n⊢ (x - s) * (x ^ 2 - (ω + ω ^ 2) * x * s + (1 + ω + ω ^ 2) * x * s + s ^ 2) =\n (x - s) * (x ^ 2 - (ω + ω ^ 2) * x * s + 1 * s ^ 2)",
"after_state": "K : Type u_1\ninst✝² : Field K\na b c d ω q s : K\ninst✝¹ : Invertible 2\ninst✝ : Invertible 3\nha : a ≠ 0\nhω : IsPrimitiveRoot ω 3\nhpz : 3 * a * c - b ^ 2 = 0\nhq : q = (9 * a * b * c - 2 * b ^ 3 - 27 * a ^ 2 * d) / (54 * a ^ 3)\nhs3 : s ^ 3 = 2 * q\nx✝ : K\nh₁ : ∀ (x a₁ a₂ a₃ : K), x = a₁ ∨ x = a₂ ∨ x = a₃ ↔ (x - a₁) * (x - a₂) * (x - a₃) = 0\nhi2 : 2 ≠ 0\nhi3 : 3 ≠ 0\nh54 : 54 = 2 * 3 ^ 3\nhb2 : b ^ 2 = 3 * a * c\nhb3 : b ^ 3 = 3 * a * b * c\nh₂ : a * x✝ ^ 3 + b * x✝ ^ 2 + c * x✝ + d = a * ((x✝ + b / (3 * a)) ^ 3 - s ^ 3)\nh₃ : ∀ (x : K), a * x = 0 ↔ x = 0\nx : K\n⊢ ((1 + ω + ω ^ 2 = 0 ∨ x = 0) ∨ s = 0) ∨ x - s = 0"
},
{
"line": "rw [cube_root_of_unity_sum hω]",
"before_state": "K : Type u_1\ninst✝² : Field K\na b c d ω q s : K\ninst✝¹ : Invertible 2\ninst✝ : Invertible 3\nha : a ≠ 0\nhω : IsPrimitiveRoot ω 3\nhpz : 3 * a * c - b ^ 2 = 0\nhq : q = (9 * a * b * c - 2 * b ^ 3 - 27 * a ^ 2 * d) / (54 * a ^ 3)\nhs3 : s ^ 3 = 2 * q\nx✝ : K\nh₁ : ∀ (x a₁ a₂ a₃ : K), x = a₁ ∨ x = a₂ ∨ x = a₃ ↔ (x - a₁) * (x - a₂) * (x - a₃) = 0\nhi2 : 2 ≠ 0\nhi3 : 3 ≠ 0\nh54 : 54 = 2 * 3 ^ 3\nhb2 : b ^ 2 = 3 * a * c\nhb3 : b ^ 3 = 3 * a * b * c\nh₂ : a * x✝ ^ 3 + b * x✝ ^ 2 + c * x✝ + d = a * ((x✝ + b / (3 * a)) ^ 3 - s ^ 3)\nh₃ : ∀ (x : K), a * x = 0 ↔ x = 0\nx : K\n⊢ ((1 + ω + ω ^ 2 = 0 ∨ x = 0) ∨ s = 0) ∨ x - s = 0",
"after_state": "K : Type u_1\ninst✝² : Field K\na b c d ω q s : K\ninst✝¹ : Invertible 2\ninst✝ : Invertible 3\nha : a ≠ 0\nhω : IsPrimitiveRoot ω 3\nhpz : 3 * a * c - b ^ 2 = 0\nhq : q = (9 * a * b * c - 2 * b ^ 3 - 27 * a ^ 2 * d) / (54 * a ^ 3)\nhs3 : s ^ 3 = 2 * q\nx✝ : K\nh₁ : ∀ (x a₁ a₂ a₃ : K), x = a₁ ∨ x = a₂ ∨ x = a₃ ↔ (x - a₁) * (x - a₂) * (x - a₃) = 0\nhi2 : 2 ≠ 0\nhi3 : 3 ≠ 0\nh54 : 54 = 2 * 3 ^ 3\nhb2 : b ^ 2 = 3 * a * c\nhb3 : b ^ 3 = 3 * a * b * c\nh₂ : a * x✝ ^ 3 + b * x✝ ^ 2 + c * x✝ + d = a * ((x✝ + b / (3 * a)) ^ 3 - s ^ 3)\nh₃ : ∀ (x : K), a * x = 0 ↔ x = 0\nx : K\n⊢ ((1 + ω + ω ^ 2 = 0 ∨ x = 0) ∨ s = 0) ∨ x - s = 0"
},
{
"line": "rewrite [cube_root_of_unity_sum hω]",
"before_state": "K : Type u_1\ninst✝² : Field K\na b c d ω q s : K\ninst✝¹ : Invertible 2\ninst✝ : Invertible 3\nha : a ≠ 0\nhω : IsPrimitiveRoot ω 3\nhpz : 3 * a * c - b ^ 2 = 0\nhq : q = (9 * a * b * c - 2 * b ^ 3 - 27 * a ^ 2 * d) / (54 * a ^ 3)\nhs3 : s ^ 3 = 2 * q\nx✝ : K\nh₁ : ∀ (x a₁ a₂ a₃ : K), x = a₁ ∨ x = a₂ ∨ x = a₃ ↔ (x - a₁) * (x - a₂) * (x - a₃) = 0\nhi2 : 2 ≠ 0\nhi3 : 3 ≠ 0\nh54 : 54 = 2 * 3 ^ 3\nhb2 : b ^ 2 = 3 * a * c\nhb3 : b ^ 3 = 3 * a * b * c\nh₂ : a * x✝ ^ 3 + b * x✝ ^ 2 + c * x✝ + d = a * ((x✝ + b / (3 * a)) ^ 3 - s ^ 3)\nh₃ : ∀ (x : K), a * x = 0 ↔ x = 0\nx : K\n⊢ ((1 + ω + ω ^ 2 = 0 ∨ x = 0) ∨ s = 0) ∨ x - s = 0",
"after_state": "K : Type u_1\ninst✝² : Field K\na b c d ω q s : K\ninst✝¹ : Invertible 2\ninst✝ : Invertible 3\nha : a ≠ 0\nhω : IsPrimitiveRoot ω 3\nhpz : 3 * a * c - b ^ 2 = 0\nhq : q = (9 * a * b * c - 2 * b ^ 3 - 27 * a ^ 2 * d) / (54 * a ^ 3)\nhs3 : s ^ 3 = 2 * q\nx✝ : K\nh₁ : ∀ (x a₁ a₂ a₃ : K), x = a₁ ∨ x = a₂ ∨ x = a₃ ↔ (x - a₁) * (x - a₂) * (x - a₃) = 0\nhi2 : 2 ≠ 0\nhi3 : 3 ≠ 0\nh54 : 54 = 2 * 3 ^ 3\nhb2 : b ^ 2 = 3 * a * c\nhb3 : b ^ 3 = 3 * a * b * c\nh₂ : a * x✝ ^ 3 + b * x✝ ^ 2 + c * x✝ + d = a * ((x✝ + b / (3 * a)) ^ 3 - s ^ 3)\nh₃ : ∀ (x : K), a * x = 0 ↔ x = 0\nx : K\n⊢ ((1 + ω + ω ^ 2 = 0 ∨ x = 0) ∨ s = 0) ∨ x - s = 0"
},
{
"line": "ring",
"before_state": "K : Type u_1\ninst✝² : Field K\na b c d ω q s : K\ninst✝¹ : Invertible 2\ninst✝ : Invertible 3\nha : a ≠ 0\nhω : IsPrimitiveRoot ω 3\nhpz : 3 * a * c - b ^ 2 = 0\nhq : q = (9 * a * b * c - 2 * b ^ 3 - 27 * a ^ 2 * d) / (54 * a ^ 3)\nhs3 : s ^ 3 = 2 * q\nx✝ : K\nh₁ : ∀ (x a₁ a₂ a₃ : K), x = a₁ ∨ x = a₂ ∨ x = a₃ ↔ (x - a₁) * (x - a₂) * (x - a₃) = 0\nhi2 : 2 ≠ 0\nhi3 : 3 ≠ 0\nh54 : 54 = 2 * 3 ^ 3\nhb2 : b ^ 2 = 3 * a * c\nhb3 : b ^ 3 = 3 * a * b * c\nh₂ : a * x✝ ^ 3 + b * x✝ ^ 2 + c * x✝ + d = a * ((x✝ + b / (3 * a)) ^ 3 - s ^ 3)\nh₃ : ∀ (x : K), a * x = 0 ↔ x = 0\nx : K\n⊢ (x - s) * (x ^ 2 - (ω + ω ^ 2) * x * s + ω ^ 3 * s ^ 2) = (x - s) * (x - s * ω) * (x - s * ω ^ 2)",
"after_state": "No Goals!"
},
{
"line": "first\n| ring1\n|\n try_this ring_nf\"\\n\\nThe `ring` tactic failed to close the goal. Use `ring_nf` to obtain a normal form.\n \\nNote that `ring` works primarily in *commutative* rings. \\\n If you have a noncommutative ring, abelian group or module, consider using \\\n `noncomm_ring`, `abel` or `module` instead.\"",
"before_state": "K : Type u_1\ninst✝² : Field K\na b c d ω q s : K\ninst✝¹ : Invertible 2\ninst✝ : Invertible 3\nha : a ≠ 0\nhω : IsPrimitiveRoot ω 3\nhpz : 3 * a * c - b ^ 2 = 0\nhq : q = (9 * a * b * c - 2 * b ^ 3 - 27 * a ^ 2 * d) / (54 * a ^ 3)\nhs3 : s ^ 3 = 2 * q\nx✝ : K\nh₁ : ∀ (x a₁ a₂ a₃ : K), x = a₁ ∨ x = a₂ ∨ x = a₃ ↔ (x - a₁) * (x - a₂) * (x - a₃) = 0\nhi2 : 2 ≠ 0\nhi3 : 3 ≠ 0\nh54 : 54 = 2 * 3 ^ 3\nhb2 : b ^ 2 = 3 * a * c\nhb3 : b ^ 3 = 3 * a * b * c\nh₂ : a * x✝ ^ 3 + b * x✝ ^ 2 + c * x✝ + d = a * ((x✝ + b / (3 * a)) ^ 3 - s ^ 3)\nh₃ : ∀ (x : K), a * x = 0 ↔ x = 0\nx : K\n⊢ (x - s) * (x ^ 2 - (ω + ω ^ 2) * x * s + ω ^ 3 * s ^ 2) = (x - s) * (x - s * ω) * (x - s * ω ^ 2)",
"after_state": "No Goals!"
},
{
"line": "ring1",
"before_state": "K : Type u_1\ninst✝² : Field K\na b c d ω q s : K\ninst✝¹ : Invertible 2\ninst✝ : Invertible 3\nha : a ≠ 0\nhω : IsPrimitiveRoot ω 3\nhpz : 3 * a * c - b ^ 2 = 0\nhq : q = (9 * a * b * c - 2 * b ^ 3 - 27 * a ^ 2 * d) / (54 * a ^ 3)\nhs3 : s ^ 3 = 2 * q\nx✝ : K\nh₁ : ∀ (x a₁ a₂ a₃ : K), x = a₁ ∨ x = a₂ ∨ x = a₃ ↔ (x - a₁) * (x - a₂) * (x - a₃) = 0\nhi2 : 2 ≠ 0\nhi3 : 3 ≠ 0\nh54 : 54 = 2 * 3 ^ 3\nhb2 : b ^ 2 = 3 * a * c\nhb3 : b ^ 3 = 3 * a * b * c\nh₂ : a * x✝ ^ 3 + b * x✝ ^ 2 + c * x✝ + d = a * ((x✝ + b / (3 * a)) ^ 3 - s ^ 3)\nh₃ : ∀ (x : K), a * x = 0 ↔ x = 0\nx : K\n⊢ (x - s) * (x ^ 2 - (ω + ω ^ 2) * x * s + ω ^ 3 * s ^ 2) = (x - s) * (x - s * ω) * (x - s * ω ^ 2)",
"after_state": "No Goals!"
},
{
"line": "rw [h₁]",
"before_state": "K : Type u_1\ninst✝² : Field K\na b c d ω q s : K\ninst✝¹ : Invertible 2\ninst✝ : Invertible 3\nha : a ≠ 0\nhω : IsPrimitiveRoot ω 3\nhpz : 3 * a * c - b ^ 2 = 0\nhq : q = (9 * a * b * c - 2 * b ^ 3 - 27 * a ^ 2 * d) / (54 * a ^ 3)\nhs3 : s ^ 3 = 2 * q\nx : K\nh₁ : ∀ (x a₁ a₂ a₃ : K), x = a₁ ∨ x = a₂ ∨ x = a₃ ↔ (x - a₁) * (x - a₂) * (x - a₃) = 0\nhi2 : 2 ≠ 0\nhi3 : 3 ≠ 0\nh54 : 54 = 2 * 3 ^ 3\nhb2 : b ^ 2 = 3 * a * c\nhb3 : b ^ 3 = 3 * a * b * c\nh₂ : a * x ^ 3 + b * x ^ 2 + c * x + d = a * ((x + b / (3 * a)) ^ 3 - s ^ 3)\nh₃ : ∀ (x : K), a * x = 0 ↔ x = 0\nh₄ : ∀ (x : K), x ^ 3 - s ^ 3 = (x - s) * (x - s * ω) * (x - s * ω ^ 2)\n⊢ a * x ^ 3 + b * x ^ 2 + c * x + d = 0 ↔ x = s - b / (3 * a) ∨ x = s * ω - b / (3 * a) ∨ x = s * ω ^ 2 - b / (3 * a)",
"after_state": "K : Type u_1\ninst✝² : Field K\na b c d ω q s : K\ninst✝¹ : Invertible 2\ninst✝ : Invertible 3\nha : a ≠ 0\nhω : IsPrimitiveRoot ω 3\nhpz : 3 * a * c - b ^ 2 = 0\nhq : q = (9 * a * b * c - 2 * b ^ 3 - 27 * a ^ 2 * d) / (54 * a ^ 3)\nhs3 : s ^ 3 = 2 * q\nx : K\nh₁ : ∀ (x a₁ a₂ a₃ : K), x = a₁ ∨ x = a₂ ∨ x = a₃ ↔ (x - a₁) * (x - a₂) * (x - a₃) = 0\nhi2 : 2 ≠ 0\nhi3 : 3 ≠ 0\nh54 : 54 = 2 * 3 ^ 3\nhb2 : b ^ 2 = 3 * a * c\nhb3 : b ^ 3 = 3 * a * b * c\nh₂ : a * x ^ 3 + b * x ^ 2 + c * x + d = a * ((x + b / (3 * a)) ^ 3 - s ^ 3)\nh₃ : ∀ (x : K), a * x = 0 ↔ x = 0\nh₄ : ∀ (x : K), x ^ 3 - s ^ 3 = (x - s) * (x - s * ω) * (x - s * ω ^ 2)\n⊢ a * x ^ 3 + b * x ^ 2 + c * x + d = 0 ↔\n (x - (s - b / (3 * a))) * (x - (s * ω - b / (3 * a))) * (x - (s * ω ^ 2 - b / (3 * a))) = 0"
},
{
"line": "rewrite [h₁]",
"before_state": "K : Type u_1\ninst✝² : Field K\na b c d ω q s : K\ninst✝¹ : Invertible 2\ninst✝ : Invertible 3\nha : a ≠ 0\nhω : IsPrimitiveRoot ω 3\nhpz : 3 * a * c - b ^ 2 = 0\nhq : q = (9 * a * b * c - 2 * b ^ 3 - 27 * a ^ 2 * d) / (54 * a ^ 3)\nhs3 : s ^ 3 = 2 * q\nx : K\nh₁ : ∀ (x a₁ a₂ a₃ : K), x = a₁ ∨ x = a₂ ∨ x = a₃ ↔ (x - a₁) * (x - a₂) * (x - a₃) = 0\nhi2 : 2 ≠ 0\nhi3 : 3 ≠ 0\nh54 : 54 = 2 * 3 ^ 3\nhb2 : b ^ 2 = 3 * a * c\nhb3 : b ^ 3 = 3 * a * b * c\nh₂ : a * x ^ 3 + b * x ^ 2 + c * x + d = a * ((x + b / (3 * a)) ^ 3 - s ^ 3)\nh₃ : ∀ (x : K), a * x = 0 ↔ x = 0\nh₄ : ∀ (x : K), x ^ 3 - s ^ 3 = (x - s) * (x - s * ω) * (x - s * ω ^ 2)\n⊢ a * x ^ 3 + b * x ^ 2 + c * x + d = 0 ↔ x = s - b / (3 * a) ∨ x = s * ω - b / (3 * a) ∨ x = s * ω ^ 2 - b / (3 * a)",
"after_state": "K : Type u_1\ninst✝² : Field K\na b c d ω q s : K\ninst✝¹ : Invertible 2\ninst✝ : Invertible 3\nha : a ≠ 0\nhω : IsPrimitiveRoot ω 3\nhpz : 3 * a * c - b ^ 2 = 0\nhq : q = (9 * a * b * c - 2 * b ^ 3 - 27 * a ^ 2 * d) / (54 * a ^ 3)\nhs3 : s ^ 3 = 2 * q\nx : K\nh₁ : ∀ (x a₁ a₂ a₃ : K), x = a₁ ∨ x = a₂ ∨ x = a₃ ↔ (x - a₁) * (x - a₂) * (x - a₃) = 0\nhi2 : 2 ≠ 0\nhi3 : 3 ≠ 0\nh54 : 54 = 2 * 3 ^ 3\nhb2 : b ^ 2 = 3 * a * c\nhb3 : b ^ 3 = 3 * a * b * c\nh₂ : a * x ^ 3 + b * x ^ 2 + c * x + d = a * ((x + b / (3 * a)) ^ 3 - s ^ 3)\nh₃ : ∀ (x : K), a * x = 0 ↔ x = 0\nh₄ : ∀ (x : K), x ^ 3 - s ^ 3 = (x - s) * (x - s * ω) * (x - s * ω ^ 2)\n⊢ a * x ^ 3 + b * x ^ 2 + c * x + d = 0 ↔\n (x - (s - b / (3 * a))) * (x - (s * ω - b / (3 * a))) * (x - (s * ω ^ 2 - b / (3 * a))) = 0"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "K : Type u_1\ninst✝² : Field K\na b c d ω q s : K\ninst✝¹ : Invertible 2\ninst✝ : Invertible 3\nha : a ≠ 0\nhω : IsPrimitiveRoot ω 3\nhpz : 3 * a * c - b ^ 2 = 0\nhq : q = (9 * a * b * c - 2 * b ^ 3 - 27 * a ^ 2 * d) / (54 * a ^ 3)\nhs3 : s ^ 3 = 2 * q\nx : K\nh₁ : ∀ (x a₁ a₂ a₃ : K), x = a₁ ∨ x = a₂ ∨ x = a₃ ↔ (x - a₁) * (x - a₂) * (x - a₃) = 0\nhi2 : 2 ≠ 0\nhi3 : 3 ≠ 0\nh54 : 54 = 2 * 3 ^ 3\nhb2 : b ^ 2 = 3 * a * c\nhb3 : b ^ 3 = 3 * a * b * c\nh₂ : a * x ^ 3 + b * x ^ 2 + c * x + d = a * ((x + b / (3 * a)) ^ 3 - s ^ 3)\nh₃ : ∀ (x : K), a * x = 0 ↔ x = 0\nh₄ : ∀ (x : K), x ^ 3 - s ^ 3 = (x - s) * (x - s * ω) * (x - s * ω ^ 2)\n⊢ a * x ^ 3 + b * x ^ 2 + c * x + d = 0 ↔\n (x - (s - b / (3 * a))) * (x - (s * ω - b / (3 * a))) * (x - (s * ω ^ 2 - b / (3 * a))) = 0",
"after_state": "K : Type u_1\ninst✝² : Field K\na b c d ω q s : K\ninst✝¹ : Invertible 2\ninst✝ : Invertible 3\nha : a ≠ 0\nhω : IsPrimitiveRoot ω 3\nhpz : 3 * a * c - b ^ 2 = 0\nhq : q = (9 * a * b * c - 2 * b ^ 3 - 27 * a ^ 2 * d) / (54 * a ^ 3)\nhs3 : s ^ 3 = 2 * q\nx : K\nh₁ : ∀ (x a₁ a₂ a₃ : K), x = a₁ ∨ x = a₂ ∨ x = a₃ ↔ (x - a₁) * (x - a₂) * (x - a₃) = 0\nhi2 : 2 ≠ 0\nhi3 : 3 ≠ 0\nh54 : 54 = 2 * 3 ^ 3\nhb2 : b ^ 2 = 3 * a * c\nhb3 : b ^ 3 = 3 * a * b * c\nh₂ : a * x ^ 3 + b * x ^ 2 + c * x + d = a * ((x + b / (3 * a)) ^ 3 - s ^ 3)\nh₃ : ∀ (x : K), a * x = 0 ↔ x = 0\nh₄ : ∀ (x : K), x ^ 3 - s ^ 3 = (x - s) * (x - s * ω) * (x - s * ω ^ 2)\n⊢ a * x ^ 3 + b * x ^ 2 + c * x + d = 0 ↔\n (x - (s - b / (3 * a))) * (x - (s * ω - b / (3 * a))) * (x - (s * ω ^ 2 - b / (3 * a))) = 0"
},
{
"line": "try (with_reducible rfl)",
"before_state": "K : Type u_1\ninst✝² : Field K\na b c d ω q s : K\ninst✝¹ : Invertible 2\ninst✝ : Invertible 3\nha : a ≠ 0\nhω : IsPrimitiveRoot ω 3\nhpz : 3 * a * c - b ^ 2 = 0\nhq : q = (9 * a * b * c - 2 * b ^ 3 - 27 * a ^ 2 * d) / (54 * a ^ 3)\nhs3 : s ^ 3 = 2 * q\nx : K\nh₁ : ∀ (x a₁ a₂ a₃ : K), x = a₁ ∨ x = a₂ ∨ x = a₃ ↔ (x - a₁) * (x - a₂) * (x - a₃) = 0\nhi2 : 2 ≠ 0\nhi3 : 3 ≠ 0\nh54 : 54 = 2 * 3 ^ 3\nhb2 : b ^ 2 = 3 * a * c\nhb3 : b ^ 3 = 3 * a * b * c\nh₂ : a * x ^ 3 + b * x ^ 2 + c * x + d = a * ((x + b / (3 * a)) ^ 3 - s ^ 3)\nh₃ : ∀ (x : K), a * x = 0 ↔ x = 0\nh₄ : ∀ (x : K), x ^ 3 - s ^ 3 = (x - s) * (x - s * ω) * (x - s * ω ^ 2)\n⊢ a * x ^ 3 + b * x ^ 2 + c * x + d = 0 ↔\n (x - (s - b / (3 * a))) * (x - (s * ω - b / (3 * a))) * (x - (s * ω ^ 2 - b / (3 * a))) = 0",
"after_state": "K : Type u_1\ninst✝² : Field K\na b c d ω q s : K\ninst✝¹ : Invertible 2\ninst✝ : Invertible 3\nha : a ≠ 0\nhω : IsPrimitiveRoot ω 3\nhpz : 3 * a * c - b ^ 2 = 0\nhq : q = (9 * a * b * c - 2 * b ^ 3 - 27 * a ^ 2 * d) / (54 * a ^ 3)\nhs3 : s ^ 3 = 2 * q\nx : K\nh₁ : ∀ (x a₁ a₂ a₃ : K), x = a₁ ∨ x = a₂ ∨ x = a₃ ↔ (x - a₁) * (x - a₂) * (x - a₃) = 0\nhi2 : 2 ≠ 0\nhi3 : 3 ≠ 0\nh54 : 54 = 2 * 3 ^ 3\nhb2 : b ^ 2 = 3 * a * c\nhb3 : b ^ 3 = 3 * a * b * c\nh₂ : a * x ^ 3 + b * x ^ 2 + c * x + d = a * ((x + b / (3 * a)) ^ 3 - s ^ 3)\nh₃ : ∀ (x : K), a * x = 0 ↔ x = 0\nh₄ : ∀ (x : K), x ^ 3 - s ^ 3 = (x - s) * (x - s * ω) * (x - s * ω ^ 2)\n⊢ a * x ^ 3 + b * x ^ 2 + c * x + d = 0 ↔\n (x - (s - b / (3 * a))) * (x - (s * ω - b / (3 * a))) * (x - (s * ω ^ 2 - b / (3 * a))) = 0"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "K : Type u_1\ninst✝² : Field K\na b c d ω q s : K\ninst✝¹ : Invertible 2\ninst✝ : Invertible 3\nha : a ≠ 0\nhω : IsPrimitiveRoot ω 3\nhpz : 3 * a * c - b ^ 2 = 0\nhq : q = (9 * a * b * c - 2 * b ^ 3 - 27 * a ^ 2 * d) / (54 * a ^ 3)\nhs3 : s ^ 3 = 2 * q\nx : K\nh₁ : ∀ (x a₁ a₂ a₃ : K), x = a₁ ∨ x = a₂ ∨ x = a₃ ↔ (x - a₁) * (x - a₂) * (x - a₃) = 0\nhi2 : 2 ≠ 0\nhi3 : 3 ≠ 0\nh54 : 54 = 2 * 3 ^ 3\nhb2 : b ^ 2 = 3 * a * c\nhb3 : b ^ 3 = 3 * a * b * c\nh₂ : a * x ^ 3 + b * x ^ 2 + c * x + d = a * ((x + b / (3 * a)) ^ 3 - s ^ 3)\nh₃ : ∀ (x : K), a * x = 0 ↔ x = 0\nh₄ : ∀ (x : K), x ^ 3 - s ^ 3 = (x - s) * (x - s * ω) * (x - s * ω ^ 2)\n⊢ a * x ^ 3 + b * x ^ 2 + c * x + d = 0 ↔\n (x - (s - b / (3 * a))) * (x - (s * ω - b / (3 * a))) * (x - (s * ω ^ 2 - b / (3 * a))) = 0",
"after_state": "K : Type u_1\ninst✝² : Field K\na b c d ω q s : K\ninst✝¹ : Invertible 2\ninst✝ : Invertible 3\nha : a ≠ 0\nhω : IsPrimitiveRoot ω 3\nhpz : 3 * a * c - b ^ 2 = 0\nhq : q = (9 * a * b * c - 2 * b ^ 3 - 27 * a ^ 2 * d) / (54 * a ^ 3)\nhs3 : s ^ 3 = 2 * q\nx : K\nh₁ : ∀ (x a₁ a₂ a₃ : K), x = a₁ ∨ x = a₂ ∨ x = a₃ ↔ (x - a₁) * (x - a₂) * (x - a₃) = 0\nhi2 : 2 ≠ 0\nhi3 : 3 ≠ 0\nh54 : 54 = 2 * 3 ^ 3\nhb2 : b ^ 2 = 3 * a * c\nhb3 : b ^ 3 = 3 * a * b * c\nh₂ : a * x ^ 3 + b * x ^ 2 + c * x + d = a * ((x + b / (3 * a)) ^ 3 - s ^ 3)\nh₃ : ∀ (x : K), a * x = 0 ↔ x = 0\nh₄ : ∀ (x : K), x ^ 3 - s ^ 3 = (x - s) * (x - s * ω) * (x - s * ω ^ 2)\n⊢ a * x ^ 3 + b * x ^ 2 + c * x + d = 0 ↔\n (x - (s - b / (3 * a))) * (x - (s * ω - b / (3 * a))) * (x - (s * ω ^ 2 - b / (3 * a))) = 0"
},
{
"line": "with_reducible rfl",
"before_state": "K : Type u_1\ninst✝² : Field K\na b c d ω q s : K\ninst✝¹ : Invertible 2\ninst✝ : Invertible 3\nha : a ≠ 0\nhω : IsPrimitiveRoot ω 3\nhpz : 3 * a * c - b ^ 2 = 0\nhq : q = (9 * a * b * c - 2 * b ^ 3 - 27 * a ^ 2 * d) / (54 * a ^ 3)\nhs3 : s ^ 3 = 2 * q\nx : K\nh₁ : ∀ (x a₁ a₂ a₃ : K), x = a₁ ∨ x = a₂ ∨ x = a₃ ↔ (x - a₁) * (x - a₂) * (x - a₃) = 0\nhi2 : 2 ≠ 0\nhi3 : 3 ≠ 0\nh54 : 54 = 2 * 3 ^ 3\nhb2 : b ^ 2 = 3 * a * c\nhb3 : b ^ 3 = 3 * a * b * c\nh₂ : a * x ^ 3 + b * x ^ 2 + c * x + d = a * ((x + b / (3 * a)) ^ 3 - s ^ 3)\nh₃ : ∀ (x : K), a * x = 0 ↔ x = 0\nh₄ : ∀ (x : K), x ^ 3 - s ^ 3 = (x - s) * (x - s * ω) * (x - s * ω ^ 2)\n⊢ a * x ^ 3 + b * x ^ 2 + c * x + d = 0 ↔\n (x - (s - b / (3 * a))) * (x - (s * ω - b / (3 * a))) * (x - (s * ω ^ 2 - b / (3 * a))) = 0",
"after_state": "K : Type u_1\ninst✝² : Field K\na b c d ω q s : K\ninst✝¹ : Invertible 2\ninst✝ : Invertible 3\nha : a ≠ 0\nhω : IsPrimitiveRoot ω 3\nhpz : 3 * a * c - b ^ 2 = 0\nhq : q = (9 * a * b * c - 2 * b ^ 3 - 27 * a ^ 2 * d) / (54 * a ^ 3)\nhs3 : s ^ 3 = 2 * q\nx : K\nh₁ : ∀ (x a₁ a₂ a₃ : K), x = a₁ ∨ x = a₂ ∨ x = a₃ ↔ (x - a₁) * (x - a₂) * (x - a₃) = 0\nhi2 : 2 ≠ 0\nhi3 : 3 ≠ 0\nh54 : 54 = 2 * 3 ^ 3\nhb2 : b ^ 2 = 3 * a * c\nhb3 : b ^ 3 = 3 * a * b * c\nh₂ : a * x ^ 3 + b * x ^ 2 + c * x + d = a * ((x + b / (3 * a)) ^ 3 - s ^ 3)\nh₃ : ∀ (x : K), a * x = 0 ↔ x = 0\nh₄ : ∀ (x : K), x ^ 3 - s ^ 3 = (x - s) * (x - s * ω) * (x - s * ω ^ 2)\n⊢ a * x ^ 3 + b * x ^ 2 + c * x + d = 0 ↔\n (x - (s - b / (3 * a))) * (x - (s * ω - b / (3 * a))) * (x - (s * ω ^ 2 - b / (3 * a))) = 0"
},
{
"line": "rfl",
"before_state": "K : Type u_1\ninst✝² : Field K\na b c d ω q s : K\ninst✝¹ : Invertible 2\ninst✝ : Invertible 3\nha : a ≠ 0\nhω : IsPrimitiveRoot ω 3\nhpz : 3 * a * c - b ^ 2 = 0\nhq : q = (9 * a * b * c - 2 * b ^ 3 - 27 * a ^ 2 * d) / (54 * a ^ 3)\nhs3 : s ^ 3 = 2 * q\nx : K\nh₁ : ∀ (x a₁ a₂ a₃ : K), x = a₁ ∨ x = a₂ ∨ x = a₃ ↔ (x - a₁) * (x - a₂) * (x - a₃) = 0\nhi2 : 2 ≠ 0\nhi3 : 3 ≠ 0\nh54 : 54 = 2 * 3 ^ 3\nhb2 : b ^ 2 = 3 * a * c\nhb3 : b ^ 3 = 3 * a * b * c\nh₂ : a * x ^ 3 + b * x ^ 2 + c * x + d = a * ((x + b / (3 * a)) ^ 3 - s ^ 3)\nh₃ : ∀ (x : K), a * x = 0 ↔ x = 0\nh₄ : ∀ (x : K), x ^ 3 - s ^ 3 = (x - s) * (x - s * ω) * (x - s * ω ^ 2)\n⊢ a * x ^ 3 + b * x ^ 2 + c * x + d = 0 ↔\n (x - (s - b / (3 * a))) * (x - (s * ω - b / (3 * a))) * (x - (s * ω ^ 2 - b / (3 * a))) = 0",
"after_state": "K : Type u_1\ninst✝² : Field K\na b c d ω q s : K\ninst✝¹ : Invertible 2\ninst✝ : Invertible 3\nha : a ≠ 0\nhω : IsPrimitiveRoot ω 3\nhpz : 3 * a * c - b ^ 2 = 0\nhq : q = (9 * a * b * c - 2 * b ^ 3 - 27 * a ^ 2 * d) / (54 * a ^ 3)\nhs3 : s ^ 3 = 2 * q\nx : K\nh₁ : ∀ (x a₁ a₂ a₃ : K), x = a₁ ∨ x = a₂ ∨ x = a₃ ↔ (x - a₁) * (x - a₂) * (x - a₃) = 0\nhi2 : 2 ≠ 0\nhi3 : 3 ≠ 0\nh54 : 54 = 2 * 3 ^ 3\nhb2 : b ^ 2 = 3 * a * c\nhb3 : b ^ 3 = 3 * a * b * c\nh₂ : a * x ^ 3 + b * x ^ 2 + c * x + d = a * ((x + b / (3 * a)) ^ 3 - s ^ 3)\nh₃ : ∀ (x : K), a * x = 0 ↔ x = 0\nh₄ : ∀ (x : K), x ^ 3 - s ^ 3 = (x - s) * (x - s * ω) * (x - s * ω ^ 2)\n⊢ a * x ^ 3 + b * x ^ 2 + c * x + d = 0 ↔\n (x - (s - b / (3 * a))) * (x - (s * ω - b / (3 * a))) * (x - (s * ω ^ 2 - b / (3 * a))) = 0"
},
{
"line": "apply_rfl",
"before_state": "K : Type u_1\ninst✝² : Field K\na b c d ω q s : K\ninst✝¹ : Invertible 2\ninst✝ : Invertible 3\nha : a ≠ 0\nhω : IsPrimitiveRoot ω 3\nhpz : 3 * a * c - b ^ 2 = 0\nhq : q = (9 * a * b * c - 2 * b ^ 3 - 27 * a ^ 2 * d) / (54 * a ^ 3)\nhs3 : s ^ 3 = 2 * q\nx : K\nh₁ : ∀ (x a₁ a₂ a₃ : K), x = a₁ ∨ x = a₂ ∨ x = a₃ ↔ (x - a₁) * (x - a₂) * (x - a₃) = 0\nhi2 : 2 ≠ 0\nhi3 : 3 ≠ 0\nh54 : 54 = 2 * 3 ^ 3\nhb2 : b ^ 2 = 3 * a * c\nhb3 : b ^ 3 = 3 * a * b * c\nh₂ : a * x ^ 3 + b * x ^ 2 + c * x + d = a * ((x + b / (3 * a)) ^ 3 - s ^ 3)\nh₃ : ∀ (x : K), a * x = 0 ↔ x = 0\nh₄ : ∀ (x : K), x ^ 3 - s ^ 3 = (x - s) * (x - s * ω) * (x - s * ω ^ 2)\n⊢ a * x ^ 3 + b * x ^ 2 + c * x + d = 0 ↔\n (x - (s - b / (3 * a))) * (x - (s * ω - b / (3 * a))) * (x - (s * ω ^ 2 - b / (3 * a))) = 0",
"after_state": "K : Type u_1\ninst✝² : Field K\na b c d ω q s : K\ninst✝¹ : Invertible 2\ninst✝ : Invertible 3\nha : a ≠ 0\nhω : IsPrimitiveRoot ω 3\nhpz : 3 * a * c - b ^ 2 = 0\nhq : q = (9 * a * b * c - 2 * b ^ 3 - 27 * a ^ 2 * d) / (54 * a ^ 3)\nhs3 : s ^ 3 = 2 * q\nx : K\nh₁ : ∀ (x a₁ a₂ a₃ : K), x = a₁ ∨ x = a₂ ∨ x = a₃ ↔ (x - a₁) * (x - a₂) * (x - a₃) = 0\nhi2 : 2 ≠ 0\nhi3 : 3 ≠ 0\nh54 : 54 = 2 * 3 ^ 3\nhb2 : b ^ 2 = 3 * a * c\nhb3 : b ^ 3 = 3 * a * b * c\nh₂ : a * x ^ 3 + b * x ^ 2 + c * x + d = a * ((x + b / (3 * a)) ^ 3 - s ^ 3)\nh₃ : ∀ (x : K), a * x = 0 ↔ x = 0\nh₄ : ∀ (x : K), x ^ 3 - s ^ 3 = (x - s) * (x - s * ω) * (x - s * ω ^ 2)\n⊢ a * x ^ 3 + b * x ^ 2 + c * x + d = 0 ↔\n (x - (s - b / (3 * a))) * (x - (s * ω - b / (3 * a))) * (x - (s * ω ^ 2 - b / (3 * a))) = 0"
},
{
"line": "skip",
"before_state": "K : Type u_1\ninst✝² : Field K\na b c d ω q s : K\ninst✝¹ : Invertible 2\ninst✝ : Invertible 3\nha : a ≠ 0\nhω : IsPrimitiveRoot ω 3\nhpz : 3 * a * c - b ^ 2 = 0\nhq : q = (9 * a * b * c - 2 * b ^ 3 - 27 * a ^ 2 * d) / (54 * a ^ 3)\nhs3 : s ^ 3 = 2 * q\nx : K\nh₁ : ∀ (x a₁ a₂ a₃ : K), x = a₁ ∨ x = a₂ ∨ x = a₃ ↔ (x - a₁) * (x - a₂) * (x - a₃) = 0\nhi2 : 2 ≠ 0\nhi3 : 3 ≠ 0\nh54 : 54 = 2 * 3 ^ 3\nhb2 : b ^ 2 = 3 * a * c\nhb3 : b ^ 3 = 3 * a * b * c\nh₂ : a * x ^ 3 + b * x ^ 2 + c * x + d = a * ((x + b / (3 * a)) ^ 3 - s ^ 3)\nh₃ : ∀ (x : K), a * x = 0 ↔ x = 0\nh₄ : ∀ (x : K), x ^ 3 - s ^ 3 = (x - s) * (x - s * ω) * (x - s * ω ^ 2)\n⊢ a * x ^ 3 + b * x ^ 2 + c * x + d = 0 ↔\n (x - (s - b / (3 * a))) * (x - (s * ω - b / (3 * a))) * (x - (s * ω ^ 2 - b / (3 * a))) = 0",
"after_state": "K : Type u_1\ninst✝² : Field K\na b c d ω q s : K\ninst✝¹ : Invertible 2\ninst✝ : Invertible 3\nha : a ≠ 0\nhω : IsPrimitiveRoot ω 3\nhpz : 3 * a * c - b ^ 2 = 0\nhq : q = (9 * a * b * c - 2 * b ^ 3 - 27 * a ^ 2 * d) / (54 * a ^ 3)\nhs3 : s ^ 3 = 2 * q\nx : K\nh₁ : ∀ (x a₁ a₂ a₃ : K), x = a₁ ∨ x = a₂ ∨ x = a₃ ↔ (x - a₁) * (x - a₂) * (x - a₃) = 0\nhi2 : 2 ≠ 0\nhi3 : 3 ≠ 0\nh54 : 54 = 2 * 3 ^ 3\nhb2 : b ^ 2 = 3 * a * c\nhb3 : b ^ 3 = 3 * a * b * c\nh₂ : a * x ^ 3 + b * x ^ 2 + c * x + d = a * ((x + b / (3 * a)) ^ 3 - s ^ 3)\nh₃ : ∀ (x : K), a * x = 0 ↔ x = 0\nh₄ : ∀ (x : K), x ^ 3 - s ^ 3 = (x - s) * (x - s * ω) * (x - s * ω ^ 2)\n⊢ a * x ^ 3 + b * x ^ 2 + c * x + d = 0 ↔\n (x - (s - b / (3 * a))) * (x - (s * ω - b / (3 * a))) * (x - (s * ω ^ 2 - b / (3 * a))) = 0"
},
{
"line": "rw [h₂]",
"before_state": "K : Type u_1\ninst✝² : Field K\na b c d ω q s : K\ninst✝¹ : Invertible 2\ninst✝ : Invertible 3\nha : a ≠ 0\nhω : IsPrimitiveRoot ω 3\nhpz : 3 * a * c - b ^ 2 = 0\nhq : q = (9 * a * b * c - 2 * b ^ 3 - 27 * a ^ 2 * d) / (54 * a ^ 3)\nhs3 : s ^ 3 = 2 * q\nx : K\nh₁ : ∀ (x a₁ a₂ a₃ : K), x = a₁ ∨ x = a₂ ∨ x = a₃ ↔ (x - a₁) * (x - a₂) * (x - a₃) = 0\nhi2 : 2 ≠ 0\nhi3 : 3 ≠ 0\nh54 : 54 = 2 * 3 ^ 3\nhb2 : b ^ 2 = 3 * a * c\nhb3 : b ^ 3 = 3 * a * b * c\nh₂ : a * x ^ 3 + b * x ^ 2 + c * x + d = a * ((x + b / (3 * a)) ^ 3 - s ^ 3)\nh₃ : ∀ (x : K), a * x = 0 ↔ x = 0\nh₄ : ∀ (x : K), x ^ 3 - s ^ 3 = (x - s) * (x - s * ω) * (x - s * ω ^ 2)\n⊢ a * x ^ 3 + b * x ^ 2 + c * x + d = 0 ↔\n (x - (s - b / (3 * a))) * (x - (s * ω - b / (3 * a))) * (x - (s * ω ^ 2 - b / (3 * a))) = 0",
"after_state": "K : Type u_1\ninst✝² : Field K\na b c d ω q s : K\ninst✝¹ : Invertible 2\ninst✝ : Invertible 3\nha : a ≠ 0\nhω : IsPrimitiveRoot ω 3\nhpz : 3 * a * c - b ^ 2 = 0\nhq : q = (9 * a * b * c - 2 * b ^ 3 - 27 * a ^ 2 * d) / (54 * a ^ 3)\nhs3 : s ^ 3 = 2 * q\nx : K\nh₁ : ∀ (x a₁ a₂ a₃ : K), x = a₁ ∨ x = a₂ ∨ x = a₃ ↔ (x - a₁) * (x - a₂) * (x - a₃) = 0\nhi2 : 2 ≠ 0\nhi3 : 3 ≠ 0\nh54 : 54 = 2 * 3 ^ 3\nhb2 : b ^ 2 = 3 * a * c\nhb3 : b ^ 3 = 3 * a * b * c\nh₂ : a * x ^ 3 + b * x ^ 2 + c * x + d = a * ((x + b / (3 * a)) ^ 3 - s ^ 3)\nh₃ : ∀ (x : K), a * x = 0 ↔ x = 0\nh₄ : ∀ (x : K), x ^ 3 - s ^ 3 = (x - s) * (x - s * ω) * (x - s * ω ^ 2)\n⊢ a * ((x + b / (3 * a)) ^ 3 - s ^ 3) = 0 ↔\n (x - (s - b / (3 * a))) * (x - (s * ω - b / (3 * a))) * (x - (s * ω ^ 2 - b / (3 * a))) = 0"
},
{
"line": "rewrite [h₂]",
"before_state": "K : Type u_1\ninst✝² : Field K\na b c d ω q s : K\ninst✝¹ : Invertible 2\ninst✝ : Invertible 3\nha : a ≠ 0\nhω : IsPrimitiveRoot ω 3\nhpz : 3 * a * c - b ^ 2 = 0\nhq : q = (9 * a * b * c - 2 * b ^ 3 - 27 * a ^ 2 * d) / (54 * a ^ 3)\nhs3 : s ^ 3 = 2 * q\nx : K\nh₁ : ∀ (x a₁ a₂ a₃ : K), x = a₁ ∨ x = a₂ ∨ x = a₃ ↔ (x - a₁) * (x - a₂) * (x - a₃) = 0\nhi2 : 2 ≠ 0\nhi3 : 3 ≠ 0\nh54 : 54 = 2 * 3 ^ 3\nhb2 : b ^ 2 = 3 * a * c\nhb3 : b ^ 3 = 3 * a * b * c\nh₂ : a * x ^ 3 + b * x ^ 2 + c * x + d = a * ((x + b / (3 * a)) ^ 3 - s ^ 3)\nh₃ : ∀ (x : K), a * x = 0 ↔ x = 0\nh₄ : ∀ (x : K), x ^ 3 - s ^ 3 = (x - s) * (x - s * ω) * (x - s * ω ^ 2)\n⊢ a * x ^ 3 + b * x ^ 2 + c * x + d = 0 ↔\n (x - (s - b / (3 * a))) * (x - (s * ω - b / (3 * a))) * (x - (s * ω ^ 2 - b / (3 * a))) = 0",
"after_state": "K : Type u_1\ninst✝² : Field K\na b c d ω q s : K\ninst✝¹ : Invertible 2\ninst✝ : Invertible 3\nha : a ≠ 0\nhω : IsPrimitiveRoot ω 3\nhpz : 3 * a * c - b ^ 2 = 0\nhq : q = (9 * a * b * c - 2 * b ^ 3 - 27 * a ^ 2 * d) / (54 * a ^ 3)\nhs3 : s ^ 3 = 2 * q\nx : K\nh₁ : ∀ (x a₁ a₂ a₃ : K), x = a₁ ∨ x = a₂ ∨ x = a₃ ↔ (x - a₁) * (x - a₂) * (x - a₃) = 0\nhi2 : 2 ≠ 0\nhi3 : 3 ≠ 0\nh54 : 54 = 2 * 3 ^ 3\nhb2 : b ^ 2 = 3 * a * c\nhb3 : b ^ 3 = 3 * a * b * c\nh₂ : a * x ^ 3 + b * x ^ 2 + c * x + d = a * ((x + b / (3 * a)) ^ 3 - s ^ 3)\nh₃ : ∀ (x : K), a * x = 0 ↔ x = 0\nh₄ : ∀ (x : K), x ^ 3 - s ^ 3 = (x - s) * (x - s * ω) * (x - s * ω ^ 2)\n⊢ a * ((x + b / (3 * a)) ^ 3 - s ^ 3) = 0 ↔\n (x - (s - b / (3 * a))) * (x - (s * ω - b / (3 * a))) * (x - (s * ω ^ 2 - b / (3 * a))) = 0"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "K : Type u_1\ninst✝² : Field K\na b c d ω q s : K\ninst✝¹ : Invertible 2\ninst✝ : Invertible 3\nha : a ≠ 0\nhω : IsPrimitiveRoot ω 3\nhpz : 3 * a * c - b ^ 2 = 0\nhq : q = (9 * a * b * c - 2 * b ^ 3 - 27 * a ^ 2 * d) / (54 * a ^ 3)\nhs3 : s ^ 3 = 2 * q\nx : K\nh₁ : ∀ (x a₁ a₂ a₃ : K), x = a₁ ∨ x = a₂ ∨ x = a₃ ↔ (x - a₁) * (x - a₂) * (x - a₃) = 0\nhi2 : 2 ≠ 0\nhi3 : 3 ≠ 0\nh54 : 54 = 2 * 3 ^ 3\nhb2 : b ^ 2 = 3 * a * c\nhb3 : b ^ 3 = 3 * a * b * c\nh₂ : a * x ^ 3 + b * x ^ 2 + c * x + d = a * ((x + b / (3 * a)) ^ 3 - s ^ 3)\nh₃ : ∀ (x : K), a * x = 0 ↔ x = 0\nh₄ : ∀ (x : K), x ^ 3 - s ^ 3 = (x - s) * (x - s * ω) * (x - s * ω ^ 2)\n⊢ a * ((x + b / (3 * a)) ^ 3 - s ^ 3) = 0 ↔\n (x - (s - b / (3 * a))) * (x - (s * ω - b / (3 * a))) * (x - (s * ω ^ 2 - b / (3 * a))) = 0",
"after_state": "K : Type u_1\ninst✝² : Field K\na b c d ω q s : K\ninst✝¹ : Invertible 2\ninst✝ : Invertible 3\nha : a ≠ 0\nhω : IsPrimitiveRoot ω 3\nhpz : 3 * a * c - b ^ 2 = 0\nhq : q = (9 * a * b * c - 2 * b ^ 3 - 27 * a ^ 2 * d) / (54 * a ^ 3)\nhs3 : s ^ 3 = 2 * q\nx : K\nh₁ : ∀ (x a₁ a₂ a₃ : K), x = a₁ ∨ x = a₂ ∨ x = a₃ ↔ (x - a₁) * (x - a₂) * (x - a₃) = 0\nhi2 : 2 ≠ 0\nhi3 : 3 ≠ 0\nh54 : 54 = 2 * 3 ^ 3\nhb2 : b ^ 2 = 3 * a * c\nhb3 : b ^ 3 = 3 * a * b * c\nh₂ : a * x ^ 3 + b * x ^ 2 + c * x + d = a * ((x + b / (3 * a)) ^ 3 - s ^ 3)\nh₃ : ∀ (x : K), a * x = 0 ↔ x = 0\nh₄ : ∀ (x : K), x ^ 3 - s ^ 3 = (x - s) * (x - s * ω) * (x - s * ω ^ 2)\n⊢ a * ((x + b / (3 * a)) ^ 3 - s ^ 3) = 0 ↔\n (x - (s - b / (3 * a))) * (x - (s * ω - b / (3 * a))) * (x - (s * ω ^ 2 - b / (3 * a))) = 0"
},
{
"line": "try (with_reducible rfl)",
"before_state": "K : Type u_1\ninst✝² : Field K\na b c d ω q s : K\ninst✝¹ : Invertible 2\ninst✝ : Invertible 3\nha : a ≠ 0\nhω : IsPrimitiveRoot ω 3\nhpz : 3 * a * c - b ^ 2 = 0\nhq : q = (9 * a * b * c - 2 * b ^ 3 - 27 * a ^ 2 * d) / (54 * a ^ 3)\nhs3 : s ^ 3 = 2 * q\nx : K\nh₁ : ∀ (x a₁ a₂ a₃ : K), x = a₁ ∨ x = a₂ ∨ x = a₃ ↔ (x - a₁) * (x - a₂) * (x - a₃) = 0\nhi2 : 2 ≠ 0\nhi3 : 3 ≠ 0\nh54 : 54 = 2 * 3 ^ 3\nhb2 : b ^ 2 = 3 * a * c\nhb3 : b ^ 3 = 3 * a * b * c\nh₂ : a * x ^ 3 + b * x ^ 2 + c * x + d = a * ((x + b / (3 * a)) ^ 3 - s ^ 3)\nh₃ : ∀ (x : K), a * x = 0 ↔ x = 0\nh₄ : ∀ (x : K), x ^ 3 - s ^ 3 = (x - s) * (x - s * ω) * (x - s * ω ^ 2)\n⊢ a * ((x + b / (3 * a)) ^ 3 - s ^ 3) = 0 ↔\n (x - (s - b / (3 * a))) * (x - (s * ω - b / (3 * a))) * (x - (s * ω ^ 2 - b / (3 * a))) = 0",
"after_state": "K : Type u_1\ninst✝² : Field K\na b c d ω q s : K\ninst✝¹ : Invertible 2\ninst✝ : Invertible 3\nha : a ≠ 0\nhω : IsPrimitiveRoot ω 3\nhpz : 3 * a * c - b ^ 2 = 0\nhq : q = (9 * a * b * c - 2 * b ^ 3 - 27 * a ^ 2 * d) / (54 * a ^ 3)\nhs3 : s ^ 3 = 2 * q\nx : K\nh₁ : ∀ (x a₁ a₂ a₃ : K), x = a₁ ∨ x = a₂ ∨ x = a₃ ↔ (x - a₁) * (x - a₂) * (x - a₃) = 0\nhi2 : 2 ≠ 0\nhi3 : 3 ≠ 0\nh54 : 54 = 2 * 3 ^ 3\nhb2 : b ^ 2 = 3 * a * c\nhb3 : b ^ 3 = 3 * a * b * c\nh₂ : a * x ^ 3 + b * x ^ 2 + c * x + d = a * ((x + b / (3 * a)) ^ 3 - s ^ 3)\nh₃ : ∀ (x : K), a * x = 0 ↔ x = 0\nh₄ : ∀ (x : K), x ^ 3 - s ^ 3 = (x - s) * (x - s * ω) * (x - s * ω ^ 2)\n⊢ a * ((x + b / (3 * a)) ^ 3 - s ^ 3) = 0 ↔\n (x - (s - b / (3 * a))) * (x - (s * ω - b / (3 * a))) * (x - (s * ω ^ 2 - b / (3 * a))) = 0"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "K : Type u_1\ninst✝² : Field K\na b c d ω q s : K\ninst✝¹ : Invertible 2\ninst✝ : Invertible 3\nha : a ≠ 0\nhω : IsPrimitiveRoot ω 3\nhpz : 3 * a * c - b ^ 2 = 0\nhq : q = (9 * a * b * c - 2 * b ^ 3 - 27 * a ^ 2 * d) / (54 * a ^ 3)\nhs3 : s ^ 3 = 2 * q\nx : K\nh₁ : ∀ (x a₁ a₂ a₃ : K), x = a₁ ∨ x = a₂ ∨ x = a₃ ↔ (x - a₁) * (x - a₂) * (x - a₃) = 0\nhi2 : 2 ≠ 0\nhi3 : 3 ≠ 0\nh54 : 54 = 2 * 3 ^ 3\nhb2 : b ^ 2 = 3 * a * c\nhb3 : b ^ 3 = 3 * a * b * c\nh₂ : a * x ^ 3 + b * x ^ 2 + c * x + d = a * ((x + b / (3 * a)) ^ 3 - s ^ 3)\nh₃ : ∀ (x : K), a * x = 0 ↔ x = 0\nh₄ : ∀ (x : K), x ^ 3 - s ^ 3 = (x - s) * (x - s * ω) * (x - s * ω ^ 2)\n⊢ a * ((x + b / (3 * a)) ^ 3 - s ^ 3) = 0 ↔\n (x - (s - b / (3 * a))) * (x - (s * ω - b / (3 * a))) * (x - (s * ω ^ 2 - b / (3 * a))) = 0",
"after_state": "K : Type u_1\ninst✝² : Field K\na b c d ω q s : K\ninst✝¹ : Invertible 2\ninst✝ : Invertible 3\nha : a ≠ 0\nhω : IsPrimitiveRoot ω 3\nhpz : 3 * a * c - b ^ 2 = 0\nhq : q = (9 * a * b * c - 2 * b ^ 3 - 27 * a ^ 2 * d) / (54 * a ^ 3)\nhs3 : s ^ 3 = 2 * q\nx : K\nh₁ : ∀ (x a₁ a₂ a₃ : K), x = a₁ ∨ x = a₂ ∨ x = a₃ ↔ (x - a₁) * (x - a₂) * (x - a₃) = 0\nhi2 : 2 ≠ 0\nhi3 : 3 ≠ 0\nh54 : 54 = 2 * 3 ^ 3\nhb2 : b ^ 2 = 3 * a * c\nhb3 : b ^ 3 = 3 * a * b * c\nh₂ : a * x ^ 3 + b * x ^ 2 + c * x + d = a * ((x + b / (3 * a)) ^ 3 - s ^ 3)\nh₃ : ∀ (x : K), a * x = 0 ↔ x = 0\nh₄ : ∀ (x : K), x ^ 3 - s ^ 3 = (x - s) * (x - s * ω) * (x - s * ω ^ 2)\n⊢ a * ((x + b / (3 * a)) ^ 3 - s ^ 3) = 0 ↔\n (x - (s - b / (3 * a))) * (x - (s * ω - b / (3 * a))) * (x - (s * ω ^ 2 - b / (3 * a))) = 0"
},
{
"line": "with_reducible rfl",
"before_state": "K : Type u_1\ninst✝² : Field K\na b c d ω q s : K\ninst✝¹ : Invertible 2\ninst✝ : Invertible 3\nha : a ≠ 0\nhω : IsPrimitiveRoot ω 3\nhpz : 3 * a * c - b ^ 2 = 0\nhq : q = (9 * a * b * c - 2 * b ^ 3 - 27 * a ^ 2 * d) / (54 * a ^ 3)\nhs3 : s ^ 3 = 2 * q\nx : K\nh₁ : ∀ (x a₁ a₂ a₃ : K), x = a₁ ∨ x = a₂ ∨ x = a₃ ↔ (x - a₁) * (x - a₂) * (x - a₃) = 0\nhi2 : 2 ≠ 0\nhi3 : 3 ≠ 0\nh54 : 54 = 2 * 3 ^ 3\nhb2 : b ^ 2 = 3 * a * c\nhb3 : b ^ 3 = 3 * a * b * c\nh₂ : a * x ^ 3 + b * x ^ 2 + c * x + d = a * ((x + b / (3 * a)) ^ 3 - s ^ 3)\nh₃ : ∀ (x : K), a * x = 0 ↔ x = 0\nh₄ : ∀ (x : K), x ^ 3 - s ^ 3 = (x - s) * (x - s * ω) * (x - s * ω ^ 2)\n⊢ a * ((x + b / (3 * a)) ^ 3 - s ^ 3) = 0 ↔\n (x - (s - b / (3 * a))) * (x - (s * ω - b / (3 * a))) * (x - (s * ω ^ 2 - b / (3 * a))) = 0",
"after_state": "K : Type u_1\ninst✝² : Field K\na b c d ω q s : K\ninst✝¹ : Invertible 2\ninst✝ : Invertible 3\nha : a ≠ 0\nhω : IsPrimitiveRoot ω 3\nhpz : 3 * a * c - b ^ 2 = 0\nhq : q = (9 * a * b * c - 2 * b ^ 3 - 27 * a ^ 2 * d) / (54 * a ^ 3)\nhs3 : s ^ 3 = 2 * q\nx : K\nh₁ : ∀ (x a₁ a₂ a₃ : K), x = a₁ ∨ x = a₂ ∨ x = a₃ ↔ (x - a₁) * (x - a₂) * (x - a₃) = 0\nhi2 : 2 ≠ 0\nhi3 : 3 ≠ 0\nh54 : 54 = 2 * 3 ^ 3\nhb2 : b ^ 2 = 3 * a * c\nhb3 : b ^ 3 = 3 * a * b * c\nh₂ : a * x ^ 3 + b * x ^ 2 + c * x + d = a * ((x + b / (3 * a)) ^ 3 - s ^ 3)\nh₃ : ∀ (x : K), a * x = 0 ↔ x = 0\nh₄ : ∀ (x : K), x ^ 3 - s ^ 3 = (x - s) * (x - s * ω) * (x - s * ω ^ 2)\n⊢ a * ((x + b / (3 * a)) ^ 3 - s ^ 3) = 0 ↔\n (x - (s - b / (3 * a))) * (x - (s * ω - b / (3 * a))) * (x - (s * ω ^ 2 - b / (3 * a))) = 0"
},
{
"line": "rfl",
"before_state": "K : Type u_1\ninst✝² : Field K\na b c d ω q s : K\ninst✝¹ : Invertible 2\ninst✝ : Invertible 3\nha : a ≠ 0\nhω : IsPrimitiveRoot ω 3\nhpz : 3 * a * c - b ^ 2 = 0\nhq : q = (9 * a * b * c - 2 * b ^ 3 - 27 * a ^ 2 * d) / (54 * a ^ 3)\nhs3 : s ^ 3 = 2 * q\nx : K\nh₁ : ∀ (x a₁ a₂ a₃ : K), x = a₁ ∨ x = a₂ ∨ x = a₃ ↔ (x - a₁) * (x - a₂) * (x - a₃) = 0\nhi2 : 2 ≠ 0\nhi3 : 3 ≠ 0\nh54 : 54 = 2 * 3 ^ 3\nhb2 : b ^ 2 = 3 * a * c\nhb3 : b ^ 3 = 3 * a * b * c\nh₂ : a * x ^ 3 + b * x ^ 2 + c * x + d = a * ((x + b / (3 * a)) ^ 3 - s ^ 3)\nh₃ : ∀ (x : K), a * x = 0 ↔ x = 0\nh₄ : ∀ (x : K), x ^ 3 - s ^ 3 = (x - s) * (x - s * ω) * (x - s * ω ^ 2)\n⊢ a * ((x + b / (3 * a)) ^ 3 - s ^ 3) = 0 ↔\n (x - (s - b / (3 * a))) * (x - (s * ω - b / (3 * a))) * (x - (s * ω ^ 2 - b / (3 * a))) = 0",
"after_state": "K : Type u_1\ninst✝² : Field K\na b c d ω q s : K\ninst✝¹ : Invertible 2\ninst✝ : Invertible 3\nha : a ≠ 0\nhω : IsPrimitiveRoot ω 3\nhpz : 3 * a * c - b ^ 2 = 0\nhq : q = (9 * a * b * c - 2 * b ^ 3 - 27 * a ^ 2 * d) / (54 * a ^ 3)\nhs3 : s ^ 3 = 2 * q\nx : K\nh₁ : ∀ (x a₁ a₂ a₃ : K), x = a₁ ∨ x = a₂ ∨ x = a₃ ↔ (x - a₁) * (x - a₂) * (x - a₃) = 0\nhi2 : 2 ≠ 0\nhi3 : 3 ≠ 0\nh54 : 54 = 2 * 3 ^ 3\nhb2 : b ^ 2 = 3 * a * c\nhb3 : b ^ 3 = 3 * a * b * c\nh₂ : a * x ^ 3 + b * x ^ 2 + c * x + d = a * ((x + b / (3 * a)) ^ 3 - s ^ 3)\nh₃ : ∀ (x : K), a * x = 0 ↔ x = 0\nh₄ : ∀ (x : K), x ^ 3 - s ^ 3 = (x - s) * (x - s * ω) * (x - s * ω ^ 2)\n⊢ a * ((x + b / (3 * a)) ^ 3 - s ^ 3) = 0 ↔\n (x - (s - b / (3 * a))) * (x - (s * ω - b / (3 * a))) * (x - (s * ω ^ 2 - b / (3 * a))) = 0"
},
{
"line": "apply_rfl",
"before_state": "K : Type u_1\ninst✝² : Field K\na b c d ω q s : K\ninst✝¹ : Invertible 2\ninst✝ : Invertible 3\nha : a ≠ 0\nhω : IsPrimitiveRoot ω 3\nhpz : 3 * a * c - b ^ 2 = 0\nhq : q = (9 * a * b * c - 2 * b ^ 3 - 27 * a ^ 2 * d) / (54 * a ^ 3)\nhs3 : s ^ 3 = 2 * q\nx : K\nh₁ : ∀ (x a₁ a₂ a₃ : K), x = a₁ ∨ x = a₂ ∨ x = a₃ ↔ (x - a₁) * (x - a₂) * (x - a₃) = 0\nhi2 : 2 ≠ 0\nhi3 : 3 ≠ 0\nh54 : 54 = 2 * 3 ^ 3\nhb2 : b ^ 2 = 3 * a * c\nhb3 : b ^ 3 = 3 * a * b * c\nh₂ : a * x ^ 3 + b * x ^ 2 + c * x + d = a * ((x + b / (3 * a)) ^ 3 - s ^ 3)\nh₃ : ∀ (x : K), a * x = 0 ↔ x = 0\nh₄ : ∀ (x : K), x ^ 3 - s ^ 3 = (x - s) * (x - s * ω) * (x - s * ω ^ 2)\n⊢ a * ((x + b / (3 * a)) ^ 3 - s ^ 3) = 0 ↔\n (x - (s - b / (3 * a))) * (x - (s * ω - b / (3 * a))) * (x - (s * ω ^ 2 - b / (3 * a))) = 0",
"after_state": "K : Type u_1\ninst✝² : Field K\na b c d ω q s : K\ninst✝¹ : Invertible 2\ninst✝ : Invertible 3\nha : a ≠ 0\nhω : IsPrimitiveRoot ω 3\nhpz : 3 * a * c - b ^ 2 = 0\nhq : q = (9 * a * b * c - 2 * b ^ 3 - 27 * a ^ 2 * d) / (54 * a ^ 3)\nhs3 : s ^ 3 = 2 * q\nx : K\nh₁ : ∀ (x a₁ a₂ a₃ : K), x = a₁ ∨ x = a₂ ∨ x = a₃ ↔ (x - a₁) * (x - a₂) * (x - a₃) = 0\nhi2 : 2 ≠ 0\nhi3 : 3 ≠ 0\nh54 : 54 = 2 * 3 ^ 3\nhb2 : b ^ 2 = 3 * a * c\nhb3 : b ^ 3 = 3 * a * b * c\nh₂ : a * x ^ 3 + b * x ^ 2 + c * x + d = a * ((x + b / (3 * a)) ^ 3 - s ^ 3)\nh₃ : ∀ (x : K), a * x = 0 ↔ x = 0\nh₄ : ∀ (x : K), x ^ 3 - s ^ 3 = (x - s) * (x - s * ω) * (x - s * ω ^ 2)\n⊢ a * ((x + b / (3 * a)) ^ 3 - s ^ 3) = 0 ↔\n (x - (s - b / (3 * a))) * (x - (s * ω - b / (3 * a))) * (x - (s * ω ^ 2 - b / (3 * a))) = 0"
},
{
"line": "skip",
"before_state": "K : Type u_1\ninst✝² : Field K\na b c d ω q s : K\ninst✝¹ : Invertible 2\ninst✝ : Invertible 3\nha : a ≠ 0\nhω : IsPrimitiveRoot ω 3\nhpz : 3 * a * c - b ^ 2 = 0\nhq : q = (9 * a * b * c - 2 * b ^ 3 - 27 * a ^ 2 * d) / (54 * a ^ 3)\nhs3 : s ^ 3 = 2 * q\nx : K\nh₁ : ∀ (x a₁ a₂ a₃ : K), x = a₁ ∨ x = a₂ ∨ x = a₃ ↔ (x - a₁) * (x - a₂) * (x - a₃) = 0\nhi2 : 2 ≠ 0\nhi3 : 3 ≠ 0\nh54 : 54 = 2 * 3 ^ 3\nhb2 : b ^ 2 = 3 * a * c\nhb3 : b ^ 3 = 3 * a * b * c\nh₂ : a * x ^ 3 + b * x ^ 2 + c * x + d = a * ((x + b / (3 * a)) ^ 3 - s ^ 3)\nh₃ : ∀ (x : K), a * x = 0 ↔ x = 0\nh₄ : ∀ (x : K), x ^ 3 - s ^ 3 = (x - s) * (x - s * ω) * (x - s * ω ^ 2)\n⊢ a * ((x + b / (3 * a)) ^ 3 - s ^ 3) = 0 ↔\n (x - (s - b / (3 * a))) * (x - (s * ω - b / (3 * a))) * (x - (s * ω ^ 2 - b / (3 * a))) = 0",
"after_state": "K : Type u_1\ninst✝² : Field K\na b c d ω q s : K\ninst✝¹ : Invertible 2\ninst✝ : Invertible 3\nha : a ≠ 0\nhω : IsPrimitiveRoot ω 3\nhpz : 3 * a * c - b ^ 2 = 0\nhq : q = (9 * a * b * c - 2 * b ^ 3 - 27 * a ^ 2 * d) / (54 * a ^ 3)\nhs3 : s ^ 3 = 2 * q\nx : K\nh₁ : ∀ (x a₁ a₂ a₃ : K), x = a₁ ∨ x = a₂ ∨ x = a₃ ↔ (x - a₁) * (x - a₂) * (x - a₃) = 0\nhi2 : 2 ≠ 0\nhi3 : 3 ≠ 0\nh54 : 54 = 2 * 3 ^ 3\nhb2 : b ^ 2 = 3 * a * c\nhb3 : b ^ 3 = 3 * a * b * c\nh₂ : a * x ^ 3 + b * x ^ 2 + c * x + d = a * ((x + b / (3 * a)) ^ 3 - s ^ 3)\nh₃ : ∀ (x : K), a * x = 0 ↔ x = 0\nh₄ : ∀ (x : K), x ^ 3 - s ^ 3 = (x - s) * (x - s * ω) * (x - s * ω ^ 2)\n⊢ a * ((x + b / (3 * a)) ^ 3 - s ^ 3) = 0 ↔\n (x - (s - b / (3 * a))) * (x - (s * ω - b / (3 * a))) * (x - (s * ω ^ 2 - b / (3 * a))) = 0"
},
{
"line": "rw [h₃]",
"before_state": "K : Type u_1\ninst✝² : Field K\na b c d ω q s : K\ninst✝¹ : Invertible 2\ninst✝ : Invertible 3\nha : a ≠ 0\nhω : IsPrimitiveRoot ω 3\nhpz : 3 * a * c - b ^ 2 = 0\nhq : q = (9 * a * b * c - 2 * b ^ 3 - 27 * a ^ 2 * d) / (54 * a ^ 3)\nhs3 : s ^ 3 = 2 * q\nx : K\nh₁ : ∀ (x a₁ a₂ a₃ : K), x = a₁ ∨ x = a₂ ∨ x = a₃ ↔ (x - a₁) * (x - a₂) * (x - a₃) = 0\nhi2 : 2 ≠ 0\nhi3 : 3 ≠ 0\nh54 : 54 = 2 * 3 ^ 3\nhb2 : b ^ 2 = 3 * a * c\nhb3 : b ^ 3 = 3 * a * b * c\nh₂ : a * x ^ 3 + b * x ^ 2 + c * x + d = a * ((x + b / (3 * a)) ^ 3 - s ^ 3)\nh₃ : ∀ (x : K), a * x = 0 ↔ x = 0\nh₄ : ∀ (x : K), x ^ 3 - s ^ 3 = (x - s) * (x - s * ω) * (x - s * ω ^ 2)\n⊢ a * ((x + b / (3 * a)) ^ 3 - s ^ 3) = 0 ↔\n (x - (s - b / (3 * a))) * (x - (s * ω - b / (3 * a))) * (x - (s * ω ^ 2 - b / (3 * a))) = 0",
"after_state": "K : Type u_1\ninst✝² : Field K\na b c d ω q s : K\ninst✝¹ : Invertible 2\ninst✝ : Invertible 3\nha : a ≠ 0\nhω : IsPrimitiveRoot ω 3\nhpz : 3 * a * c - b ^ 2 = 0\nhq : q = (9 * a * b * c - 2 * b ^ 3 - 27 * a ^ 2 * d) / (54 * a ^ 3)\nhs3 : s ^ 3 = 2 * q\nx : K\nh₁ : ∀ (x a₁ a₂ a₃ : K), x = a₁ ∨ x = a₂ ∨ x = a₃ ↔ (x - a₁) * (x - a₂) * (x - a₃) = 0\nhi2 : 2 ≠ 0\nhi3 : 3 ≠ 0\nh54 : 54 = 2 * 3 ^ 3\nhb2 : b ^ 2 = 3 * a * c\nhb3 : b ^ 3 = 3 * a * b * c\nh₂ : a * x ^ 3 + b * x ^ 2 + c * x + d = a * ((x + b / (3 * a)) ^ 3 - s ^ 3)\nh₃ : ∀ (x : K), a * x = 0 ↔ x = 0\nh₄ : ∀ (x : K), x ^ 3 - s ^ 3 = (x - s) * (x - s * ω) * (x - s * ω ^ 2)\n⊢ (x + b / (3 * a)) ^ 3 - s ^ 3 = 0 ↔\n (x - (s - b / (3 * a))) * (x - (s * ω - b / (3 * a))) * (x - (s * ω ^ 2 - b / (3 * a))) = 0"
},
{
"line": "rewrite [h₃]",
"before_state": "K : Type u_1\ninst✝² : Field K\na b c d ω q s : K\ninst✝¹ : Invertible 2\ninst✝ : Invertible 3\nha : a ≠ 0\nhω : IsPrimitiveRoot ω 3\nhpz : 3 * a * c - b ^ 2 = 0\nhq : q = (9 * a * b * c - 2 * b ^ 3 - 27 * a ^ 2 * d) / (54 * a ^ 3)\nhs3 : s ^ 3 = 2 * q\nx : K\nh₁ : ∀ (x a₁ a₂ a₃ : K), x = a₁ ∨ x = a₂ ∨ x = a₃ ↔ (x - a₁) * (x - a₂) * (x - a₃) = 0\nhi2 : 2 ≠ 0\nhi3 : 3 ≠ 0\nh54 : 54 = 2 * 3 ^ 3\nhb2 : b ^ 2 = 3 * a * c\nhb3 : b ^ 3 = 3 * a * b * c\nh₂ : a * x ^ 3 + b * x ^ 2 + c * x + d = a * ((x + b / (3 * a)) ^ 3 - s ^ 3)\nh₃ : ∀ (x : K), a * x = 0 ↔ x = 0\nh₄ : ∀ (x : K), x ^ 3 - s ^ 3 = (x - s) * (x - s * ω) * (x - s * ω ^ 2)\n⊢ a * ((x + b / (3 * a)) ^ 3 - s ^ 3) = 0 ↔\n (x - (s - b / (3 * a))) * (x - (s * ω - b / (3 * a))) * (x - (s * ω ^ 2 - b / (3 * a))) = 0",
"after_state": "K : Type u_1\ninst✝² : Field K\na b c d ω q s : K\ninst✝¹ : Invertible 2\ninst✝ : Invertible 3\nha : a ≠ 0\nhω : IsPrimitiveRoot ω 3\nhpz : 3 * a * c - b ^ 2 = 0\nhq : q = (9 * a * b * c - 2 * b ^ 3 - 27 * a ^ 2 * d) / (54 * a ^ 3)\nhs3 : s ^ 3 = 2 * q\nx : K\nh₁ : ∀ (x a₁ a₂ a₃ : K), x = a₁ ∨ x = a₂ ∨ x = a₃ ↔ (x - a₁) * (x - a₂) * (x - a₃) = 0\nhi2 : 2 ≠ 0\nhi3 : 3 ≠ 0\nh54 : 54 = 2 * 3 ^ 3\nhb2 : b ^ 2 = 3 * a * c\nhb3 : b ^ 3 = 3 * a * b * c\nh₂ : a * x ^ 3 + b * x ^ 2 + c * x + d = a * ((x + b / (3 * a)) ^ 3 - s ^ 3)\nh₃ : ∀ (x : K), a * x = 0 ↔ x = 0\nh₄ : ∀ (x : K), x ^ 3 - s ^ 3 = (x - s) * (x - s * ω) * (x - s * ω ^ 2)\n⊢ (x + b / (3 * a)) ^ 3 - s ^ 3 = 0 ↔\n (x - (s - b / (3 * a))) * (x - (s * ω - b / (3 * a))) * (x - (s * ω ^ 2 - b / (3 * a))) = 0"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "K : Type u_1\ninst✝² : Field K\na b c d ω q s : K\ninst✝¹ : Invertible 2\ninst✝ : Invertible 3\nha : a ≠ 0\nhω : IsPrimitiveRoot ω 3\nhpz : 3 * a * c - b ^ 2 = 0\nhq : q = (9 * a * b * c - 2 * b ^ 3 - 27 * a ^ 2 * d) / (54 * a ^ 3)\nhs3 : s ^ 3 = 2 * q\nx : K\nh₁ : ∀ (x a₁ a₂ a₃ : K), x = a₁ ∨ x = a₂ ∨ x = a₃ ↔ (x - a₁) * (x - a₂) * (x - a₃) = 0\nhi2 : 2 ≠ 0\nhi3 : 3 ≠ 0\nh54 : 54 = 2 * 3 ^ 3\nhb2 : b ^ 2 = 3 * a * c\nhb3 : b ^ 3 = 3 * a * b * c\nh₂ : a * x ^ 3 + b * x ^ 2 + c * x + d = a * ((x + b / (3 * a)) ^ 3 - s ^ 3)\nh₃ : ∀ (x : K), a * x = 0 ↔ x = 0\nh₄ : ∀ (x : K), x ^ 3 - s ^ 3 = (x - s) * (x - s * ω) * (x - s * ω ^ 2)\n⊢ (x + b / (3 * a)) ^ 3 - s ^ 3 = 0 ↔\n (x - (s - b / (3 * a))) * (x - (s * ω - b / (3 * a))) * (x - (s * ω ^ 2 - b / (3 * a))) = 0",
"after_state": "K : Type u_1\ninst✝² : Field K\na b c d ω q s : K\ninst✝¹ : Invertible 2\ninst✝ : Invertible 3\nha : a ≠ 0\nhω : IsPrimitiveRoot ω 3\nhpz : 3 * a * c - b ^ 2 = 0\nhq : q = (9 * a * b * c - 2 * b ^ 3 - 27 * a ^ 2 * d) / (54 * a ^ 3)\nhs3 : s ^ 3 = 2 * q\nx : K\nh₁ : ∀ (x a₁ a₂ a₃ : K), x = a₁ ∨ x = a₂ ∨ x = a₃ ↔ (x - a₁) * (x - a₂) * (x - a₃) = 0\nhi2 : 2 ≠ 0\nhi3 : 3 ≠ 0\nh54 : 54 = 2 * 3 ^ 3\nhb2 : b ^ 2 = 3 * a * c\nhb3 : b ^ 3 = 3 * a * b * c\nh₂ : a * x ^ 3 + b * x ^ 2 + c * x + d = a * ((x + b / (3 * a)) ^ 3 - s ^ 3)\nh₃ : ∀ (x : K), a * x = 0 ↔ x = 0\nh₄ : ∀ (x : K), x ^ 3 - s ^ 3 = (x - s) * (x - s * ω) * (x - s * ω ^ 2)\n⊢ (x + b / (3 * a)) ^ 3 - s ^ 3 = 0 ↔\n (x - (s - b / (3 * a))) * (x - (s * ω - b / (3 * a))) * (x - (s * ω ^ 2 - b / (3 * a))) = 0"
},
{
"line": "try (with_reducible rfl)",
"before_state": "K : Type u_1\ninst✝² : Field K\na b c d ω q s : K\ninst✝¹ : Invertible 2\ninst✝ : Invertible 3\nha : a ≠ 0\nhω : IsPrimitiveRoot ω 3\nhpz : 3 * a * c - b ^ 2 = 0\nhq : q = (9 * a * b * c - 2 * b ^ 3 - 27 * a ^ 2 * d) / (54 * a ^ 3)\nhs3 : s ^ 3 = 2 * q\nx : K\nh₁ : ∀ (x a₁ a₂ a₃ : K), x = a₁ ∨ x = a₂ ∨ x = a₃ ↔ (x - a₁) * (x - a₂) * (x - a₃) = 0\nhi2 : 2 ≠ 0\nhi3 : 3 ≠ 0\nh54 : 54 = 2 * 3 ^ 3\nhb2 : b ^ 2 = 3 * a * c\nhb3 : b ^ 3 = 3 * a * b * c\nh₂ : a * x ^ 3 + b * x ^ 2 + c * x + d = a * ((x + b / (3 * a)) ^ 3 - s ^ 3)\nh₃ : ∀ (x : K), a * x = 0 ↔ x = 0\nh₄ : ∀ (x : K), x ^ 3 - s ^ 3 = (x - s) * (x - s * ω) * (x - s * ω ^ 2)\n⊢ (x + b / (3 * a)) ^ 3 - s ^ 3 = 0 ↔\n (x - (s - b / (3 * a))) * (x - (s * ω - b / (3 * a))) * (x - (s * ω ^ 2 - b / (3 * a))) = 0",
"after_state": "K : Type u_1\ninst✝² : Field K\na b c d ω q s : K\ninst✝¹ : Invertible 2\ninst✝ : Invertible 3\nha : a ≠ 0\nhω : IsPrimitiveRoot ω 3\nhpz : 3 * a * c - b ^ 2 = 0\nhq : q = (9 * a * b * c - 2 * b ^ 3 - 27 * a ^ 2 * d) / (54 * a ^ 3)\nhs3 : s ^ 3 = 2 * q\nx : K\nh₁ : ∀ (x a₁ a₂ a₃ : K), x = a₁ ∨ x = a₂ ∨ x = a₃ ↔ (x - a₁) * (x - a₂) * (x - a₃) = 0\nhi2 : 2 ≠ 0\nhi3 : 3 ≠ 0\nh54 : 54 = 2 * 3 ^ 3\nhb2 : b ^ 2 = 3 * a * c\nhb3 : b ^ 3 = 3 * a * b * c\nh₂ : a * x ^ 3 + b * x ^ 2 + c * x + d = a * ((x + b / (3 * a)) ^ 3 - s ^ 3)\nh₃ : ∀ (x : K), a * x = 0 ↔ x = 0\nh₄ : ∀ (x : K), x ^ 3 - s ^ 3 = (x - s) * (x - s * ω) * (x - s * ω ^ 2)\n⊢ (x + b / (3 * a)) ^ 3 - s ^ 3 = 0 ↔\n (x - (s - b / (3 * a))) * (x - (s * ω - b / (3 * a))) * (x - (s * ω ^ 2 - b / (3 * a))) = 0"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "K : Type u_1\ninst✝² : Field K\na b c d ω q s : K\ninst✝¹ : Invertible 2\ninst✝ : Invertible 3\nha : a ≠ 0\nhω : IsPrimitiveRoot ω 3\nhpz : 3 * a * c - b ^ 2 = 0\nhq : q = (9 * a * b * c - 2 * b ^ 3 - 27 * a ^ 2 * d) / (54 * a ^ 3)\nhs3 : s ^ 3 = 2 * q\nx : K\nh₁ : ∀ (x a₁ a₂ a₃ : K), x = a₁ ∨ x = a₂ ∨ x = a₃ ↔ (x - a₁) * (x - a₂) * (x - a₃) = 0\nhi2 : 2 ≠ 0\nhi3 : 3 ≠ 0\nh54 : 54 = 2 * 3 ^ 3\nhb2 : b ^ 2 = 3 * a * c\nhb3 : b ^ 3 = 3 * a * b * c\nh₂ : a * x ^ 3 + b * x ^ 2 + c * x + d = a * ((x + b / (3 * a)) ^ 3 - s ^ 3)\nh₃ : ∀ (x : K), a * x = 0 ↔ x = 0\nh₄ : ∀ (x : K), x ^ 3 - s ^ 3 = (x - s) * (x - s * ω) * (x - s * ω ^ 2)\n⊢ (x + b / (3 * a)) ^ 3 - s ^ 3 = 0 ↔\n (x - (s - b / (3 * a))) * (x - (s * ω - b / (3 * a))) * (x - (s * ω ^ 2 - b / (3 * a))) = 0",
"after_state": "K : Type u_1\ninst✝² : Field K\na b c d ω q s : K\ninst✝¹ : Invertible 2\ninst✝ : Invertible 3\nha : a ≠ 0\nhω : IsPrimitiveRoot ω 3\nhpz : 3 * a * c - b ^ 2 = 0\nhq : q = (9 * a * b * c - 2 * b ^ 3 - 27 * a ^ 2 * d) / (54 * a ^ 3)\nhs3 : s ^ 3 = 2 * q\nx : K\nh₁ : ∀ (x a₁ a₂ a₃ : K), x = a₁ ∨ x = a₂ ∨ x = a₃ ↔ (x - a₁) * (x - a₂) * (x - a₃) = 0\nhi2 : 2 ≠ 0\nhi3 : 3 ≠ 0\nh54 : 54 = 2 * 3 ^ 3\nhb2 : b ^ 2 = 3 * a * c\nhb3 : b ^ 3 = 3 * a * b * c\nh₂ : a * x ^ 3 + b * x ^ 2 + c * x + d = a * ((x + b / (3 * a)) ^ 3 - s ^ 3)\nh₃ : ∀ (x : K), a * x = 0 ↔ x = 0\nh₄ : ∀ (x : K), x ^ 3 - s ^ 3 = (x - s) * (x - s * ω) * (x - s * ω ^ 2)\n⊢ (x + b / (3 * a)) ^ 3 - s ^ 3 = 0 ↔\n (x - (s - b / (3 * a))) * (x - (s * ω - b / (3 * a))) * (x - (s * ω ^ 2 - b / (3 * a))) = 0"
},
{
"line": "with_reducible rfl",
"before_state": "K : Type u_1\ninst✝² : Field K\na b c d ω q s : K\ninst✝¹ : Invertible 2\ninst✝ : Invertible 3\nha : a ≠ 0\nhω : IsPrimitiveRoot ω 3\nhpz : 3 * a * c - b ^ 2 = 0\nhq : q = (9 * a * b * c - 2 * b ^ 3 - 27 * a ^ 2 * d) / (54 * a ^ 3)\nhs3 : s ^ 3 = 2 * q\nx : K\nh₁ : ∀ (x a₁ a₂ a₃ : K), x = a₁ ∨ x = a₂ ∨ x = a₃ ↔ (x - a₁) * (x - a₂) * (x - a₃) = 0\nhi2 : 2 ≠ 0\nhi3 : 3 ≠ 0\nh54 : 54 = 2 * 3 ^ 3\nhb2 : b ^ 2 = 3 * a * c\nhb3 : b ^ 3 = 3 * a * b * c\nh₂ : a * x ^ 3 + b * x ^ 2 + c * x + d = a * ((x + b / (3 * a)) ^ 3 - s ^ 3)\nh₃ : ∀ (x : K), a * x = 0 ↔ x = 0\nh₄ : ∀ (x : K), x ^ 3 - s ^ 3 = (x - s) * (x - s * ω) * (x - s * ω ^ 2)\n⊢ (x + b / (3 * a)) ^ 3 - s ^ 3 = 0 ↔\n (x - (s - b / (3 * a))) * (x - (s * ω - b / (3 * a))) * (x - (s * ω ^ 2 - b / (3 * a))) = 0",
"after_state": "K : Type u_1\ninst✝² : Field K\na b c d ω q s : K\ninst✝¹ : Invertible 2\ninst✝ : Invertible 3\nha : a ≠ 0\nhω : IsPrimitiveRoot ω 3\nhpz : 3 * a * c - b ^ 2 = 0\nhq : q = (9 * a * b * c - 2 * b ^ 3 - 27 * a ^ 2 * d) / (54 * a ^ 3)\nhs3 : s ^ 3 = 2 * q\nx : K\nh₁ : ∀ (x a₁ a₂ a₃ : K), x = a₁ ∨ x = a₂ ∨ x = a₃ ↔ (x - a₁) * (x - a₂) * (x - a₃) = 0\nhi2 : 2 ≠ 0\nhi3 : 3 ≠ 0\nh54 : 54 = 2 * 3 ^ 3\nhb2 : b ^ 2 = 3 * a * c\nhb3 : b ^ 3 = 3 * a * b * c\nh₂ : a * x ^ 3 + b * x ^ 2 + c * x + d = a * ((x + b / (3 * a)) ^ 3 - s ^ 3)\nh₃ : ∀ (x : K), a * x = 0 ↔ x = 0\nh₄ : ∀ (x : K), x ^ 3 - s ^ 3 = (x - s) * (x - s * ω) * (x - s * ω ^ 2)\n⊢ (x + b / (3 * a)) ^ 3 - s ^ 3 = 0 ↔\n (x - (s - b / (3 * a))) * (x - (s * ω - b / (3 * a))) * (x - (s * ω ^ 2 - b / (3 * a))) = 0"
},
{
"line": "rfl",
"before_state": "K : Type u_1\ninst✝² : Field K\na b c d ω q s : K\ninst✝¹ : Invertible 2\ninst✝ : Invertible 3\nha : a ≠ 0\nhω : IsPrimitiveRoot ω 3\nhpz : 3 * a * c - b ^ 2 = 0\nhq : q = (9 * a * b * c - 2 * b ^ 3 - 27 * a ^ 2 * d) / (54 * a ^ 3)\nhs3 : s ^ 3 = 2 * q\nx : K\nh₁ : ∀ (x a₁ a₂ a₃ : K), x = a₁ ∨ x = a₂ ∨ x = a₃ ↔ (x - a₁) * (x - a₂) * (x - a₃) = 0\nhi2 : 2 ≠ 0\nhi3 : 3 ≠ 0\nh54 : 54 = 2 * 3 ^ 3\nhb2 : b ^ 2 = 3 * a * c\nhb3 : b ^ 3 = 3 * a * b * c\nh₂ : a * x ^ 3 + b * x ^ 2 + c * x + d = a * ((x + b / (3 * a)) ^ 3 - s ^ 3)\nh₃ : ∀ (x : K), a * x = 0 ↔ x = 0\nh₄ : ∀ (x : K), x ^ 3 - s ^ 3 = (x - s) * (x - s * ω) * (x - s * ω ^ 2)\n⊢ (x + b / (3 * a)) ^ 3 - s ^ 3 = 0 ↔\n (x - (s - b / (3 * a))) * (x - (s * ω - b / (3 * a))) * (x - (s * ω ^ 2 - b / (3 * a))) = 0",
"after_state": "K : Type u_1\ninst✝² : Field K\na b c d ω q s : K\ninst✝¹ : Invertible 2\ninst✝ : Invertible 3\nha : a ≠ 0\nhω : IsPrimitiveRoot ω 3\nhpz : 3 * a * c - b ^ 2 = 0\nhq : q = (9 * a * b * c - 2 * b ^ 3 - 27 * a ^ 2 * d) / (54 * a ^ 3)\nhs3 : s ^ 3 = 2 * q\nx : K\nh₁ : ∀ (x a₁ a₂ a₃ : K), x = a₁ ∨ x = a₂ ∨ x = a₃ ↔ (x - a₁) * (x - a₂) * (x - a₃) = 0\nhi2 : 2 ≠ 0\nhi3 : 3 ≠ 0\nh54 : 54 = 2 * 3 ^ 3\nhb2 : b ^ 2 = 3 * a * c\nhb3 : b ^ 3 = 3 * a * b * c\nh₂ : a * x ^ 3 + b * x ^ 2 + c * x + d = a * ((x + b / (3 * a)) ^ 3 - s ^ 3)\nh₃ : ∀ (x : K), a * x = 0 ↔ x = 0\nh₄ : ∀ (x : K), x ^ 3 - s ^ 3 = (x - s) * (x - s * ω) * (x - s * ω ^ 2)\n⊢ (x + b / (3 * a)) ^ 3 - s ^ 3 = 0 ↔\n (x - (s - b / (3 * a))) * (x - (s * ω - b / (3 * a))) * (x - (s * ω ^ 2 - b / (3 * a))) = 0"
},
{
"line": "apply_rfl",
"before_state": "K : Type u_1\ninst✝² : Field K\na b c d ω q s : K\ninst✝¹ : Invertible 2\ninst✝ : Invertible 3\nha : a ≠ 0\nhω : IsPrimitiveRoot ω 3\nhpz : 3 * a * c - b ^ 2 = 0\nhq : q = (9 * a * b * c - 2 * b ^ 3 - 27 * a ^ 2 * d) / (54 * a ^ 3)\nhs3 : s ^ 3 = 2 * q\nx : K\nh₁ : ∀ (x a₁ a₂ a₃ : K), x = a₁ ∨ x = a₂ ∨ x = a₃ ↔ (x - a₁) * (x - a₂) * (x - a₃) = 0\nhi2 : 2 ≠ 0\nhi3 : 3 ≠ 0\nh54 : 54 = 2 * 3 ^ 3\nhb2 : b ^ 2 = 3 * a * c\nhb3 : b ^ 3 = 3 * a * b * c\nh₂ : a * x ^ 3 + b * x ^ 2 + c * x + d = a * ((x + b / (3 * a)) ^ 3 - s ^ 3)\nh₃ : ∀ (x : K), a * x = 0 ↔ x = 0\nh₄ : ∀ (x : K), x ^ 3 - s ^ 3 = (x - s) * (x - s * ω) * (x - s * ω ^ 2)\n⊢ (x + b / (3 * a)) ^ 3 - s ^ 3 = 0 ↔\n (x - (s - b / (3 * a))) * (x - (s * ω - b / (3 * a))) * (x - (s * ω ^ 2 - b / (3 * a))) = 0",
"after_state": "K : Type u_1\ninst✝² : Field K\na b c d ω q s : K\ninst✝¹ : Invertible 2\ninst✝ : Invertible 3\nha : a ≠ 0\nhω : IsPrimitiveRoot ω 3\nhpz : 3 * a * c - b ^ 2 = 0\nhq : q = (9 * a * b * c - 2 * b ^ 3 - 27 * a ^ 2 * d) / (54 * a ^ 3)\nhs3 : s ^ 3 = 2 * q\nx : K\nh₁ : ∀ (x a₁ a₂ a₃ : K), x = a₁ ∨ x = a₂ ∨ x = a₃ ↔ (x - a₁) * (x - a₂) * (x - a₃) = 0\nhi2 : 2 ≠ 0\nhi3 : 3 ≠ 0\nh54 : 54 = 2 * 3 ^ 3\nhb2 : b ^ 2 = 3 * a * c\nhb3 : b ^ 3 = 3 * a * b * c\nh₂ : a * x ^ 3 + b * x ^ 2 + c * x + d = a * ((x + b / (3 * a)) ^ 3 - s ^ 3)\nh₃ : ∀ (x : K), a * x = 0 ↔ x = 0\nh₄ : ∀ (x : K), x ^ 3 - s ^ 3 = (x - s) * (x - s * ω) * (x - s * ω ^ 2)\n⊢ (x + b / (3 * a)) ^ 3 - s ^ 3 = 0 ↔\n (x - (s - b / (3 * a))) * (x - (s * ω - b / (3 * a))) * (x - (s * ω ^ 2 - b / (3 * a))) = 0"
},
{
"line": "skip",
"before_state": "K : Type u_1\ninst✝² : Field K\na b c d ω q s : K\ninst✝¹ : Invertible 2\ninst✝ : Invertible 3\nha : a ≠ 0\nhω : IsPrimitiveRoot ω 3\nhpz : 3 * a * c - b ^ 2 = 0\nhq : q = (9 * a * b * c - 2 * b ^ 3 - 27 * a ^ 2 * d) / (54 * a ^ 3)\nhs3 : s ^ 3 = 2 * q\nx : K\nh₁ : ∀ (x a₁ a₂ a₃ : K), x = a₁ ∨ x = a₂ ∨ x = a₃ ↔ (x - a₁) * (x - a₂) * (x - a₃) = 0\nhi2 : 2 ≠ 0\nhi3 : 3 ≠ 0\nh54 : 54 = 2 * 3 ^ 3\nhb2 : b ^ 2 = 3 * a * c\nhb3 : b ^ 3 = 3 * a * b * c\nh₂ : a * x ^ 3 + b * x ^ 2 + c * x + d = a * ((x + b / (3 * a)) ^ 3 - s ^ 3)\nh₃ : ∀ (x : K), a * x = 0 ↔ x = 0\nh₄ : ∀ (x : K), x ^ 3 - s ^ 3 = (x - s) * (x - s * ω) * (x - s * ω ^ 2)\n⊢ (x + b / (3 * a)) ^ 3 - s ^ 3 = 0 ↔\n (x - (s - b / (3 * a))) * (x - (s * ω - b / (3 * a))) * (x - (s * ω ^ 2 - b / (3 * a))) = 0",
"after_state": "K : Type u_1\ninst✝² : Field K\na b c d ω q s : K\ninst✝¹ : Invertible 2\ninst✝ : Invertible 3\nha : a ≠ 0\nhω : IsPrimitiveRoot ω 3\nhpz : 3 * a * c - b ^ 2 = 0\nhq : q = (9 * a * b * c - 2 * b ^ 3 - 27 * a ^ 2 * d) / (54 * a ^ 3)\nhs3 : s ^ 3 = 2 * q\nx : K\nh₁ : ∀ (x a₁ a₂ a₃ : K), x = a₁ ∨ x = a₂ ∨ x = a₃ ↔ (x - a₁) * (x - a₂) * (x - a₃) = 0\nhi2 : 2 ≠ 0\nhi3 : 3 ≠ 0\nh54 : 54 = 2 * 3 ^ 3\nhb2 : b ^ 2 = 3 * a * c\nhb3 : b ^ 3 = 3 * a * b * c\nh₂ : a * x ^ 3 + b * x ^ 2 + c * x + d = a * ((x + b / (3 * a)) ^ 3 - s ^ 3)\nh₃ : ∀ (x : K), a * x = 0 ↔ x = 0\nh₄ : ∀ (x : K), x ^ 3 - s ^ 3 = (x - s) * (x - s * ω) * (x - s * ω ^ 2)\n⊢ (x + b / (3 * a)) ^ 3 - s ^ 3 = 0 ↔\n (x - (s - b / (3 * a))) * (x - (s * ω - b / (3 * a))) * (x - (s * ω ^ 2 - b / (3 * a))) = 0"
},
{
"line": "rw [h₄ (x + b / (3 * a))]",
"before_state": "K : Type u_1\ninst✝² : Field K\na b c d ω q s : K\ninst✝¹ : Invertible 2\ninst✝ : Invertible 3\nha : a ≠ 0\nhω : IsPrimitiveRoot ω 3\nhpz : 3 * a * c - b ^ 2 = 0\nhq : q = (9 * a * b * c - 2 * b ^ 3 - 27 * a ^ 2 * d) / (54 * a ^ 3)\nhs3 : s ^ 3 = 2 * q\nx : K\nh₁ : ∀ (x a₁ a₂ a₃ : K), x = a₁ ∨ x = a₂ ∨ x = a₃ ↔ (x - a₁) * (x - a₂) * (x - a₃) = 0\nhi2 : 2 ≠ 0\nhi3 : 3 ≠ 0\nh54 : 54 = 2 * 3 ^ 3\nhb2 : b ^ 2 = 3 * a * c\nhb3 : b ^ 3 = 3 * a * b * c\nh₂ : a * x ^ 3 + b * x ^ 2 + c * x + d = a * ((x + b / (3 * a)) ^ 3 - s ^ 3)\nh₃ : ∀ (x : K), a * x = 0 ↔ x = 0\nh₄ : ∀ (x : K), x ^ 3 - s ^ 3 = (x - s) * (x - s * ω) * (x - s * ω ^ 2)\n⊢ (x + b / (3 * a)) ^ 3 - s ^ 3 = 0 ↔\n (x - (s - b / (3 * a))) * (x - (s * ω - b / (3 * a))) * (x - (s * ω ^ 2 - b / (3 * a))) = 0",
"after_state": "K : Type u_1\ninst✝² : Field K\na b c d ω q s : K\ninst✝¹ : Invertible 2\ninst✝ : Invertible 3\nha : a ≠ 0\nhω : IsPrimitiveRoot ω 3\nhpz : 3 * a * c - b ^ 2 = 0\nhq : q = (9 * a * b * c - 2 * b ^ 3 - 27 * a ^ 2 * d) / (54 * a ^ 3)\nhs3 : s ^ 3 = 2 * q\nx : K\nh₁ : ∀ (x a₁ a₂ a₃ : K), x = a₁ ∨ x = a₂ ∨ x = a₃ ↔ (x - a₁) * (x - a₂) * (x - a₃) = 0\nhi2 : 2 ≠ 0\nhi3 : 3 ≠ 0\nh54 : 54 = 2 * 3 ^ 3\nhb2 : b ^ 2 = 3 * a * c\nhb3 : b ^ 3 = 3 * a * b * c\nh₂ : a * x ^ 3 + b * x ^ 2 + c * x + d = a * ((x + b / (3 * a)) ^ 3 - s ^ 3)\nh₃ : ∀ (x : K), a * x = 0 ↔ x = 0\nh₄ : ∀ (x : K), x ^ 3 - s ^ 3 = (x - s) * (x - s * ω) * (x - s * ω ^ 2)\n⊢ (x + b / (3 * a) - s) * (x + b / (3 * a) - s * ω) * (x + b / (3 * a) - s * ω ^ 2) = 0 ↔\n (x - (s - b / (3 * a))) * (x - (s * ω - b / (3 * a))) * (x - (s * ω ^ 2 - b / (3 * a))) = 0"
},
{
"line": "rewrite [h₄ (x + b / (3 * a))]",
"before_state": "K : Type u_1\ninst✝² : Field K\na b c d ω q s : K\ninst✝¹ : Invertible 2\ninst✝ : Invertible 3\nha : a ≠ 0\nhω : IsPrimitiveRoot ω 3\nhpz : 3 * a * c - b ^ 2 = 0\nhq : q = (9 * a * b * c - 2 * b ^ 3 - 27 * a ^ 2 * d) / (54 * a ^ 3)\nhs3 : s ^ 3 = 2 * q\nx : K\nh₁ : ∀ (x a₁ a₂ a₃ : K), x = a₁ ∨ x = a₂ ∨ x = a₃ ↔ (x - a₁) * (x - a₂) * (x - a₃) = 0\nhi2 : 2 ≠ 0\nhi3 : 3 ≠ 0\nh54 : 54 = 2 * 3 ^ 3\nhb2 : b ^ 2 = 3 * a * c\nhb3 : b ^ 3 = 3 * a * b * c\nh₂ : a * x ^ 3 + b * x ^ 2 + c * x + d = a * ((x + b / (3 * a)) ^ 3 - s ^ 3)\nh₃ : ∀ (x : K), a * x = 0 ↔ x = 0\nh₄ : ∀ (x : K), x ^ 3 - s ^ 3 = (x - s) * (x - s * ω) * (x - s * ω ^ 2)\n⊢ (x + b / (3 * a)) ^ 3 - s ^ 3 = 0 ↔\n (x - (s - b / (3 * a))) * (x - (s * ω - b / (3 * a))) * (x - (s * ω ^ 2 - b / (3 * a))) = 0",
"after_state": "K : Type u_1\ninst✝² : Field K\na b c d ω q s : K\ninst✝¹ : Invertible 2\ninst✝ : Invertible 3\nha : a ≠ 0\nhω : IsPrimitiveRoot ω 3\nhpz : 3 * a * c - b ^ 2 = 0\nhq : q = (9 * a * b * c - 2 * b ^ 3 - 27 * a ^ 2 * d) / (54 * a ^ 3)\nhs3 : s ^ 3 = 2 * q\nx : K\nh₁ : ∀ (x a₁ a₂ a₃ : K), x = a₁ ∨ x = a₂ ∨ x = a₃ ↔ (x - a₁) * (x - a₂) * (x - a₃) = 0\nhi2 : 2 ≠ 0\nhi3 : 3 ≠ 0\nh54 : 54 = 2 * 3 ^ 3\nhb2 : b ^ 2 = 3 * a * c\nhb3 : b ^ 3 = 3 * a * b * c\nh₂ : a * x ^ 3 + b * x ^ 2 + c * x + d = a * ((x + b / (3 * a)) ^ 3 - s ^ 3)\nh₃ : ∀ (x : K), a * x = 0 ↔ x = 0\nh₄ : ∀ (x : K), x ^ 3 - s ^ 3 = (x - s) * (x - s * ω) * (x - s * ω ^ 2)\n⊢ (x + b / (3 * a) - s) * (x + b / (3 * a) - s * ω) * (x + b / (3 * a) - s * ω ^ 2) = 0 ↔\n (x - (s - b / (3 * a))) * (x - (s * ω - b / (3 * a))) * (x - (s * ω ^ 2 - b / (3 * a))) = 0"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "K : Type u_1\ninst✝² : Field K\na b c d ω q s : K\ninst✝¹ : Invertible 2\ninst✝ : Invertible 3\nha : a ≠ 0\nhω : IsPrimitiveRoot ω 3\nhpz : 3 * a * c - b ^ 2 = 0\nhq : q = (9 * a * b * c - 2 * b ^ 3 - 27 * a ^ 2 * d) / (54 * a ^ 3)\nhs3 : s ^ 3 = 2 * q\nx : K\nh₁ : ∀ (x a₁ a₂ a₃ : K), x = a₁ ∨ x = a₂ ∨ x = a₃ ↔ (x - a₁) * (x - a₂) * (x - a₃) = 0\nhi2 : 2 ≠ 0\nhi3 : 3 ≠ 0\nh54 : 54 = 2 * 3 ^ 3\nhb2 : b ^ 2 = 3 * a * c\nhb3 : b ^ 3 = 3 * a * b * c\nh₂ : a * x ^ 3 + b * x ^ 2 + c * x + d = a * ((x + b / (3 * a)) ^ 3 - s ^ 3)\nh₃ : ∀ (x : K), a * x = 0 ↔ x = 0\nh₄ : ∀ (x : K), x ^ 3 - s ^ 3 = (x - s) * (x - s * ω) * (x - s * ω ^ 2)\n⊢ (x + b / (3 * a) - s) * (x + b / (3 * a) - s * ω) * (x + b / (3 * a) - s * ω ^ 2) = 0 ↔\n (x - (s - b / (3 * a))) * (x - (s * ω - b / (3 * a))) * (x - (s * ω ^ 2 - b / (3 * a))) = 0",
"after_state": "K : Type u_1\ninst✝² : Field K\na b c d ω q s : K\ninst✝¹ : Invertible 2\ninst✝ : Invertible 3\nha : a ≠ 0\nhω : IsPrimitiveRoot ω 3\nhpz : 3 * a * c - b ^ 2 = 0\nhq : q = (9 * a * b * c - 2 * b ^ 3 - 27 * a ^ 2 * d) / (54 * a ^ 3)\nhs3 : s ^ 3 = 2 * q\nx : K\nh₁ : ∀ (x a₁ a₂ a₃ : K), x = a₁ ∨ x = a₂ ∨ x = a₃ ↔ (x - a₁) * (x - a₂) * (x - a₃) = 0\nhi2 : 2 ≠ 0\nhi3 : 3 ≠ 0\nh54 : 54 = 2 * 3 ^ 3\nhb2 : b ^ 2 = 3 * a * c\nhb3 : b ^ 3 = 3 * a * b * c\nh₂ : a * x ^ 3 + b * x ^ 2 + c * x + d = a * ((x + b / (3 * a)) ^ 3 - s ^ 3)\nh₃ : ∀ (x : K), a * x = 0 ↔ x = 0\nh₄ : ∀ (x : K), x ^ 3 - s ^ 3 = (x - s) * (x - s * ω) * (x - s * ω ^ 2)\n⊢ (x + b / (3 * a) - s) * (x + b / (3 * a) - s * ω) * (x + b / (3 * a) - s * ω ^ 2) = 0 ↔\n (x - (s - b / (3 * a))) * (x - (s * ω - b / (3 * a))) * (x - (s * ω ^ 2 - b / (3 * a))) = 0"
},
{
"line": "try (with_reducible rfl)",
"before_state": "K : Type u_1\ninst✝² : Field K\na b c d ω q s : K\ninst✝¹ : Invertible 2\ninst✝ : Invertible 3\nha : a ≠ 0\nhω : IsPrimitiveRoot ω 3\nhpz : 3 * a * c - b ^ 2 = 0\nhq : q = (9 * a * b * c - 2 * b ^ 3 - 27 * a ^ 2 * d) / (54 * a ^ 3)\nhs3 : s ^ 3 = 2 * q\nx : K\nh₁ : ∀ (x a₁ a₂ a₃ : K), x = a₁ ∨ x = a₂ ∨ x = a₃ ↔ (x - a₁) * (x - a₂) * (x - a₃) = 0\nhi2 : 2 ≠ 0\nhi3 : 3 ≠ 0\nh54 : 54 = 2 * 3 ^ 3\nhb2 : b ^ 2 = 3 * a * c\nhb3 : b ^ 3 = 3 * a * b * c\nh₂ : a * x ^ 3 + b * x ^ 2 + c * x + d = a * ((x + b / (3 * a)) ^ 3 - s ^ 3)\nh₃ : ∀ (x : K), a * x = 0 ↔ x = 0\nh₄ : ∀ (x : K), x ^ 3 - s ^ 3 = (x - s) * (x - s * ω) * (x - s * ω ^ 2)\n⊢ (x + b / (3 * a) - s) * (x + b / (3 * a) - s * ω) * (x + b / (3 * a) - s * ω ^ 2) = 0 ↔\n (x - (s - b / (3 * a))) * (x - (s * ω - b / (3 * a))) * (x - (s * ω ^ 2 - b / (3 * a))) = 0",
"after_state": "K : Type u_1\ninst✝² : Field K\na b c d ω q s : K\ninst✝¹ : Invertible 2\ninst✝ : Invertible 3\nha : a ≠ 0\nhω : IsPrimitiveRoot ω 3\nhpz : 3 * a * c - b ^ 2 = 0\nhq : q = (9 * a * b * c - 2 * b ^ 3 - 27 * a ^ 2 * d) / (54 * a ^ 3)\nhs3 : s ^ 3 = 2 * q\nx : K\nh₁ : ∀ (x a₁ a₂ a₃ : K), x = a₁ ∨ x = a₂ ∨ x = a₃ ↔ (x - a₁) * (x - a₂) * (x - a₃) = 0\nhi2 : 2 ≠ 0\nhi3 : 3 ≠ 0\nh54 : 54 = 2 * 3 ^ 3\nhb2 : b ^ 2 = 3 * a * c\nhb3 : b ^ 3 = 3 * a * b * c\nh₂ : a * x ^ 3 + b * x ^ 2 + c * x + d = a * ((x + b / (3 * a)) ^ 3 - s ^ 3)\nh₃ : ∀ (x : K), a * x = 0 ↔ x = 0\nh₄ : ∀ (x : K), x ^ 3 - s ^ 3 = (x - s) * (x - s * ω) * (x - s * ω ^ 2)\n⊢ (x + b / (3 * a) - s) * (x + b / (3 * a) - s * ω) * (x + b / (3 * a) - s * ω ^ 2) = 0 ↔\n (x - (s - b / (3 * a))) * (x - (s * ω - b / (3 * a))) * (x - (s * ω ^ 2 - b / (3 * a))) = 0"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "K : Type u_1\ninst✝² : Field K\na b c d ω q s : K\ninst✝¹ : Invertible 2\ninst✝ : Invertible 3\nha : a ≠ 0\nhω : IsPrimitiveRoot ω 3\nhpz : 3 * a * c - b ^ 2 = 0\nhq : q = (9 * a * b * c - 2 * b ^ 3 - 27 * a ^ 2 * d) / (54 * a ^ 3)\nhs3 : s ^ 3 = 2 * q\nx : K\nh₁ : ∀ (x a₁ a₂ a₃ : K), x = a₁ ∨ x = a₂ ∨ x = a₃ ↔ (x - a₁) * (x - a₂) * (x - a₃) = 0\nhi2 : 2 ≠ 0\nhi3 : 3 ≠ 0\nh54 : 54 = 2 * 3 ^ 3\nhb2 : b ^ 2 = 3 * a * c\nhb3 : b ^ 3 = 3 * a * b * c\nh₂ : a * x ^ 3 + b * x ^ 2 + c * x + d = a * ((x + b / (3 * a)) ^ 3 - s ^ 3)\nh₃ : ∀ (x : K), a * x = 0 ↔ x = 0\nh₄ : ∀ (x : K), x ^ 3 - s ^ 3 = (x - s) * (x - s * ω) * (x - s * ω ^ 2)\n⊢ (x + b / (3 * a) - s) * (x + b / (3 * a) - s * ω) * (x + b / (3 * a) - s * ω ^ 2) = 0 ↔\n (x - (s - b / (3 * a))) * (x - (s * ω - b / (3 * a))) * (x - (s * ω ^ 2 - b / (3 * a))) = 0",
"after_state": "K : Type u_1\ninst✝² : Field K\na b c d ω q s : K\ninst✝¹ : Invertible 2\ninst✝ : Invertible 3\nha : a ≠ 0\nhω : IsPrimitiveRoot ω 3\nhpz : 3 * a * c - b ^ 2 = 0\nhq : q = (9 * a * b * c - 2 * b ^ 3 - 27 * a ^ 2 * d) / (54 * a ^ 3)\nhs3 : s ^ 3 = 2 * q\nx : K\nh₁ : ∀ (x a₁ a₂ a₃ : K), x = a₁ ∨ x = a₂ ∨ x = a₃ ↔ (x - a₁) * (x - a₂) * (x - a₃) = 0\nhi2 : 2 ≠ 0\nhi3 : 3 ≠ 0\nh54 : 54 = 2 * 3 ^ 3\nhb2 : b ^ 2 = 3 * a * c\nhb3 : b ^ 3 = 3 * a * b * c\nh₂ : a * x ^ 3 + b * x ^ 2 + c * x + d = a * ((x + b / (3 * a)) ^ 3 - s ^ 3)\nh₃ : ∀ (x : K), a * x = 0 ↔ x = 0\nh₄ : ∀ (x : K), x ^ 3 - s ^ 3 = (x - s) * (x - s * ω) * (x - s * ω ^ 2)\n⊢ (x + b / (3 * a) - s) * (x + b / (3 * a) - s * ω) * (x + b / (3 * a) - s * ω ^ 2) = 0 ↔\n (x - (s - b / (3 * a))) * (x - (s * ω - b / (3 * a))) * (x - (s * ω ^ 2 - b / (3 * a))) = 0"
},
{
"line": "with_reducible rfl",
"before_state": "K : Type u_1\ninst✝² : Field K\na b c d ω q s : K\ninst✝¹ : Invertible 2\ninst✝ : Invertible 3\nha : a ≠ 0\nhω : IsPrimitiveRoot ω 3\nhpz : 3 * a * c - b ^ 2 = 0\nhq : q = (9 * a * b * c - 2 * b ^ 3 - 27 * a ^ 2 * d) / (54 * a ^ 3)\nhs3 : s ^ 3 = 2 * q\nx : K\nh₁ : ∀ (x a₁ a₂ a₃ : K), x = a₁ ∨ x = a₂ ∨ x = a₃ ↔ (x - a₁) * (x - a₂) * (x - a₃) = 0\nhi2 : 2 ≠ 0\nhi3 : 3 ≠ 0\nh54 : 54 = 2 * 3 ^ 3\nhb2 : b ^ 2 = 3 * a * c\nhb3 : b ^ 3 = 3 * a * b * c\nh₂ : a * x ^ 3 + b * x ^ 2 + c * x + d = a * ((x + b / (3 * a)) ^ 3 - s ^ 3)\nh₃ : ∀ (x : K), a * x = 0 ↔ x = 0\nh₄ : ∀ (x : K), x ^ 3 - s ^ 3 = (x - s) * (x - s * ω) * (x - s * ω ^ 2)\n⊢ (x + b / (3 * a) - s) * (x + b / (3 * a) - s * ω) * (x + b / (3 * a) - s * ω ^ 2) = 0 ↔\n (x - (s - b / (3 * a))) * (x - (s * ω - b / (3 * a))) * (x - (s * ω ^ 2 - b / (3 * a))) = 0",
"after_state": "K : Type u_1\ninst✝² : Field K\na b c d ω q s : K\ninst✝¹ : Invertible 2\ninst✝ : Invertible 3\nha : a ≠ 0\nhω : IsPrimitiveRoot ω 3\nhpz : 3 * a * c - b ^ 2 = 0\nhq : q = (9 * a * b * c - 2 * b ^ 3 - 27 * a ^ 2 * d) / (54 * a ^ 3)\nhs3 : s ^ 3 = 2 * q\nx : K\nh₁ : ∀ (x a₁ a₂ a₃ : K), x = a₁ ∨ x = a₂ ∨ x = a₃ ↔ (x - a₁) * (x - a₂) * (x - a₃) = 0\nhi2 : 2 ≠ 0\nhi3 : 3 ≠ 0\nh54 : 54 = 2 * 3 ^ 3\nhb2 : b ^ 2 = 3 * a * c\nhb3 : b ^ 3 = 3 * a * b * c\nh₂ : a * x ^ 3 + b * x ^ 2 + c * x + d = a * ((x + b / (3 * a)) ^ 3 - s ^ 3)\nh₃ : ∀ (x : K), a * x = 0 ↔ x = 0\nh₄ : ∀ (x : K), x ^ 3 - s ^ 3 = (x - s) * (x - s * ω) * (x - s * ω ^ 2)\n⊢ (x + b / (3 * a) - s) * (x + b / (3 * a) - s * ω) * (x + b / (3 * a) - s * ω ^ 2) = 0 ↔\n (x - (s - b / (3 * a))) * (x - (s * ω - b / (3 * a))) * (x - (s * ω ^ 2 - b / (3 * a))) = 0"
},
{
"line": "rfl",
"before_state": "K : Type u_1\ninst✝² : Field K\na b c d ω q s : K\ninst✝¹ : Invertible 2\ninst✝ : Invertible 3\nha : a ≠ 0\nhω : IsPrimitiveRoot ω 3\nhpz : 3 * a * c - b ^ 2 = 0\nhq : q = (9 * a * b * c - 2 * b ^ 3 - 27 * a ^ 2 * d) / (54 * a ^ 3)\nhs3 : s ^ 3 = 2 * q\nx : K\nh₁ : ∀ (x a₁ a₂ a₃ : K), x = a₁ ∨ x = a₂ ∨ x = a₃ ↔ (x - a₁) * (x - a₂) * (x - a₃) = 0\nhi2 : 2 ≠ 0\nhi3 : 3 ≠ 0\nh54 : 54 = 2 * 3 ^ 3\nhb2 : b ^ 2 = 3 * a * c\nhb3 : b ^ 3 = 3 * a * b * c\nh₂ : a * x ^ 3 + b * x ^ 2 + c * x + d = a * ((x + b / (3 * a)) ^ 3 - s ^ 3)\nh₃ : ∀ (x : K), a * x = 0 ↔ x = 0\nh₄ : ∀ (x : K), x ^ 3 - s ^ 3 = (x - s) * (x - s * ω) * (x - s * ω ^ 2)\n⊢ (x + b / (3 * a) - s) * (x + b / (3 * a) - s * ω) * (x + b / (3 * a) - s * ω ^ 2) = 0 ↔\n (x - (s - b / (3 * a))) * (x - (s * ω - b / (3 * a))) * (x - (s * ω ^ 2 - b / (3 * a))) = 0",
"after_state": "K : Type u_1\ninst✝² : Field K\na b c d ω q s : K\ninst✝¹ : Invertible 2\ninst✝ : Invertible 3\nha : a ≠ 0\nhω : IsPrimitiveRoot ω 3\nhpz : 3 * a * c - b ^ 2 = 0\nhq : q = (9 * a * b * c - 2 * b ^ 3 - 27 * a ^ 2 * d) / (54 * a ^ 3)\nhs3 : s ^ 3 = 2 * q\nx : K\nh₁ : ∀ (x a₁ a₂ a₃ : K), x = a₁ ∨ x = a₂ ∨ x = a₃ ↔ (x - a₁) * (x - a₂) * (x - a₃) = 0\nhi2 : 2 ≠ 0\nhi3 : 3 ≠ 0\nh54 : 54 = 2 * 3 ^ 3\nhb2 : b ^ 2 = 3 * a * c\nhb3 : b ^ 3 = 3 * a * b * c\nh₂ : a * x ^ 3 + b * x ^ 2 + c * x + d = a * ((x + b / (3 * a)) ^ 3 - s ^ 3)\nh₃ : ∀ (x : K), a * x = 0 ↔ x = 0\nh₄ : ∀ (x : K), x ^ 3 - s ^ 3 = (x - s) * (x - s * ω) * (x - s * ω ^ 2)\n⊢ (x + b / (3 * a) - s) * (x + b / (3 * a) - s * ω) * (x + b / (3 * a) - s * ω ^ 2) = 0 ↔\n (x - (s - b / (3 * a))) * (x - (s * ω - b / (3 * a))) * (x - (s * ω ^ 2 - b / (3 * a))) = 0"
},
{
"line": "apply_rfl",
"before_state": "K : Type u_1\ninst✝² : Field K\na b c d ω q s : K\ninst✝¹ : Invertible 2\ninst✝ : Invertible 3\nha : a ≠ 0\nhω : IsPrimitiveRoot ω 3\nhpz : 3 * a * c - b ^ 2 = 0\nhq : q = (9 * a * b * c - 2 * b ^ 3 - 27 * a ^ 2 * d) / (54 * a ^ 3)\nhs3 : s ^ 3 = 2 * q\nx : K\nh₁ : ∀ (x a₁ a₂ a₃ : K), x = a₁ ∨ x = a₂ ∨ x = a₃ ↔ (x - a₁) * (x - a₂) * (x - a₃) = 0\nhi2 : 2 ≠ 0\nhi3 : 3 ≠ 0\nh54 : 54 = 2 * 3 ^ 3\nhb2 : b ^ 2 = 3 * a * c\nhb3 : b ^ 3 = 3 * a * b * c\nh₂ : a * x ^ 3 + b * x ^ 2 + c * x + d = a * ((x + b / (3 * a)) ^ 3 - s ^ 3)\nh₃ : ∀ (x : K), a * x = 0 ↔ x = 0\nh₄ : ∀ (x : K), x ^ 3 - s ^ 3 = (x - s) * (x - s * ω) * (x - s * ω ^ 2)\n⊢ (x + b / (3 * a) - s) * (x + b / (3 * a) - s * ω) * (x + b / (3 * a) - s * ω ^ 2) = 0 ↔\n (x - (s - b / (3 * a))) * (x - (s * ω - b / (3 * a))) * (x - (s * ω ^ 2 - b / (3 * a))) = 0",
"after_state": "K : Type u_1\ninst✝² : Field K\na b c d ω q s : K\ninst✝¹ : Invertible 2\ninst✝ : Invertible 3\nha : a ≠ 0\nhω : IsPrimitiveRoot ω 3\nhpz : 3 * a * c - b ^ 2 = 0\nhq : q = (9 * a * b * c - 2 * b ^ 3 - 27 * a ^ 2 * d) / (54 * a ^ 3)\nhs3 : s ^ 3 = 2 * q\nx : K\nh₁ : ∀ (x a₁ a₂ a₃ : K), x = a₁ ∨ x = a₂ ∨ x = a₃ ↔ (x - a₁) * (x - a₂) * (x - a₃) = 0\nhi2 : 2 ≠ 0\nhi3 : 3 ≠ 0\nh54 : 54 = 2 * 3 ^ 3\nhb2 : b ^ 2 = 3 * a * c\nhb3 : b ^ 3 = 3 * a * b * c\nh₂ : a * x ^ 3 + b * x ^ 2 + c * x + d = a * ((x + b / (3 * a)) ^ 3 - s ^ 3)\nh₃ : ∀ (x : K), a * x = 0 ↔ x = 0\nh₄ : ∀ (x : K), x ^ 3 - s ^ 3 = (x - s) * (x - s * ω) * (x - s * ω ^ 2)\n⊢ (x + b / (3 * a) - s) * (x + b / (3 * a) - s * ω) * (x + b / (3 * a) - s * ω ^ 2) = 0 ↔\n (x - (s - b / (3 * a))) * (x - (s * ω - b / (3 * a))) * (x - (s * ω ^ 2 - b / (3 * a))) = 0"
},
{
"line": "skip",
"before_state": "K : Type u_1\ninst✝² : Field K\na b c d ω q s : K\ninst✝¹ : Invertible 2\ninst✝ : Invertible 3\nha : a ≠ 0\nhω : IsPrimitiveRoot ω 3\nhpz : 3 * a * c - b ^ 2 = 0\nhq : q = (9 * a * b * c - 2 * b ^ 3 - 27 * a ^ 2 * d) / (54 * a ^ 3)\nhs3 : s ^ 3 = 2 * q\nx : K\nh₁ : ∀ (x a₁ a₂ a₃ : K), x = a₁ ∨ x = a₂ ∨ x = a₃ ↔ (x - a₁) * (x - a₂) * (x - a₃) = 0\nhi2 : 2 ≠ 0\nhi3 : 3 ≠ 0\nh54 : 54 = 2 * 3 ^ 3\nhb2 : b ^ 2 = 3 * a * c\nhb3 : b ^ 3 = 3 * a * b * c\nh₂ : a * x ^ 3 + b * x ^ 2 + c * x + d = a * ((x + b / (3 * a)) ^ 3 - s ^ 3)\nh₃ : ∀ (x : K), a * x = 0 ↔ x = 0\nh₄ : ∀ (x : K), x ^ 3 - s ^ 3 = (x - s) * (x - s * ω) * (x - s * ω ^ 2)\n⊢ (x + b / (3 * a) - s) * (x + b / (3 * a) - s * ω) * (x + b / (3 * a) - s * ω ^ 2) = 0 ↔\n (x - (s - b / (3 * a))) * (x - (s * ω - b / (3 * a))) * (x - (s * ω ^ 2 - b / (3 * a))) = 0",
"after_state": "K : Type u_1\ninst✝² : Field K\na b c d ω q s : K\ninst✝¹ : Invertible 2\ninst✝ : Invertible 3\nha : a ≠ 0\nhω : IsPrimitiveRoot ω 3\nhpz : 3 * a * c - b ^ 2 = 0\nhq : q = (9 * a * b * c - 2 * b ^ 3 - 27 * a ^ 2 * d) / (54 * a ^ 3)\nhs3 : s ^ 3 = 2 * q\nx : K\nh₁ : ∀ (x a₁ a₂ a₃ : K), x = a₁ ∨ x = a₂ ∨ x = a₃ ↔ (x - a₁) * (x - a₂) * (x - a₃) = 0\nhi2 : 2 ≠ 0\nhi3 : 3 ≠ 0\nh54 : 54 = 2 * 3 ^ 3\nhb2 : b ^ 2 = 3 * a * c\nhb3 : b ^ 3 = 3 * a * b * c\nh₂ : a * x ^ 3 + b * x ^ 2 + c * x + d = a * ((x + b / (3 * a)) ^ 3 - s ^ 3)\nh₃ : ∀ (x : K), a * x = 0 ↔ x = 0\nh₄ : ∀ (x : K), x ^ 3 - s ^ 3 = (x - s) * (x - s * ω) * (x - s * ω ^ 2)\n⊢ (x + b / (3 * a) - s) * (x + b / (3 * a) - s * ω) * (x + b / (3 * a) - s * ω ^ 2) = 0 ↔\n (x - (s - b / (3 * a))) * (x - (s * ω - b / (3 * a))) * (x - (s * ω ^ 2 - b / (3 * a))) = 0"
},
{
"line": "ring_nf",
"before_state": "K : Type u_1\ninst✝² : Field K\na b c d ω q s : K\ninst✝¹ : Invertible 2\ninst✝ : Invertible 3\nha : a ≠ 0\nhω : IsPrimitiveRoot ω 3\nhpz : 3 * a * c - b ^ 2 = 0\nhq : q = (9 * a * b * c - 2 * b ^ 3 - 27 * a ^ 2 * d) / (54 * a ^ 3)\nhs3 : s ^ 3 = 2 * q\nx : K\nh₁ : ∀ (x a₁ a₂ a₃ : K), x = a₁ ∨ x = a₂ ∨ x = a₃ ↔ (x - a₁) * (x - a₂) * (x - a₃) = 0\nhi2 : 2 ≠ 0\nhi3 : 3 ≠ 0\nh54 : 54 = 2 * 3 ^ 3\nhb2 : b ^ 2 = 3 * a * c\nhb3 : b ^ 3 = 3 * a * b * c\nh₂ : a * x ^ 3 + b * x ^ 2 + c * x + d = a * ((x + b / (3 * a)) ^ 3 - s ^ 3)\nh₃ : ∀ (x : K), a * x = 0 ↔ x = 0\nh₄ : ∀ (x : K), x ^ 3 - s ^ 3 = (x - s) * (x - s * ω) * (x - s * ω ^ 2)\n⊢ (x + b / (3 * a) - s) * (x + b / (3 * a) - s * ω) * (x + b / (3 * a) - s * ω ^ 2) = 0 ↔\n (x - (s - b / (3 * a))) * (x - (s * ω - b / (3 * a))) * (x - (s * ω ^ 2 - b / (3 * a))) = 0",
"after_state": "No Goals!"
}
] |
theorem quartic_depressed_eq_zero_iff
(hq_nonzero : q ≠ 0)
(hu : u ^ 3 - p * u ^ 2 - 4 * r * u + 4 * p * r - q ^ 2 = 0)
(hs : s ^ 2 = u - p)
(hv : v ^ 2 = 4 * s ^ 2 - 8 * (u - q / s))
(hw : w ^ 2 = 4 * s ^ 2 - 8 * (u + q / s))
(x : K) :
x ^ 4 + p * x ^ 2 + q * x + r = 0 ↔
x = (-2 * s - v) / 4 ∨ x = (-2 * s + v) / 4 ∨ x = (2 * s - w) / 4 ∨ x = (2 * s + w) / 4 := by
have hi2 : (2 : K) ≠ 0 := Invertible.ne_zero _
have h4 : (4 : K) = 2 ^ 2 := by norm_num
have hs_nonzero : s ≠ 0 := by
contrapose! hq_nonzero with hs0
linear_combination (exp := 2) -hu + (4 * r - u ^ 2) * hs + (u ^ 2 * s - 4 * r * s) * hs0
calc
_ ↔ 4 * (x ^ 4 + p * x ^ 2 + q * x + r) = 0 := by simp [h4, hi2]
_ ↔ (2 * (x * x) + 2 * s * x + (u - q / s)) * (2 * (x * x) + -(2 * s) * x + (u + q / s)) =
0 := by
apply Eq.congr_left
field_simp
linear_combination -hu + (-x ^ 2 * s ^ 2 - x ^ 2 * p + x ^ 2 * u) * hw +
(x ^ 2 * w ^ 2 + 8 * x ^ 2 * u + 8 * x ^ 2 * q / s - u ^ 2 + 4 * r) * hs
_ ↔ _ := by
have hv' : discrim 2 (2 * s) (u - q / s) = v * v := by rw [discrim]; linear_combination -hv
have hw' : discrim 2 (-(2 * s)) (u + q / s) = w * w := by rw [discrim]; linear_combination -hw
rw [mul_eq_zero]
rw [quadratic_eq_zero_iff hi2 hv']
rw [quadratic_eq_zero_iff hi2 hw']
simp [(by norm_num : (2 : K) * 2 = 4), or_assoc, or_comm]
|
/root/DuelModelResearch/mathlib4/Archive/Wiedijk100Theorems/SolutionOfCubicQuartic.lean
|
{
"open": [
"Polynomial"
],
"variables": [
"{K : Type*} [Field K] (a b c d e : K) {ω p q r s t u v w x y : K}",
"[Invertible (2 : K)] [Invertible (3 : K)]",
"[Invertible (2 : K)]"
]
}
|
[
{
"line": "have hi2 : (2 : K) ≠ 0 := Invertible.ne_zero _",
"before_state": "K : Type u_1\ninst✝³ : Field K\np q r s u v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nhq_nonzero : q ≠ 0\nhu : u ^ 3 - p * u ^ 2 - 4 * r * u + 4 * p * r - q ^ 2 = 0\nhs : s ^ 2 = u - p\nhv : v ^ 2 = 4 * s ^ 2 - 8 * (u - q / s)\nhw : w ^ 2 = 4 * s ^ 2 - 8 * (u + q / s)\nx : K\n⊢ x ^ 4 + p * x ^ 2 + q * x + r = 0 ↔\n x = (-2 * s - v) / 4 ∨ x = (-2 * s + v) / 4 ∨ x = (2 * s - w) / 4 ∨ x = (2 * s + w) / 4",
"after_state": "K : Type u_1\ninst✝³ : Field K\np q r s u v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nhq_nonzero : q ≠ 0\nhu : u ^ 3 - p * u ^ 2 - 4 * r * u + 4 * p * r - q ^ 2 = 0\nhs : s ^ 2 = u - p\nhv : v ^ 2 = 4 * s ^ 2 - 8 * (u - q / s)\nhw : w ^ 2 = 4 * s ^ 2 - 8 * (u + q / s)\nx : K\nhi2 : 2 ≠ 0\n⊢ x ^ 4 + p * x ^ 2 + q * x + r = 0 ↔\n x = (-2 * s - v) / 4 ∨ x = (-2 * s + v) / 4 ∨ x = (2 * s - w) / 4 ∨ x = (2 * s + w) / 4"
},
{
"line": "refine_lift\n have hi2 : (2 : K) ≠ 0 := Invertible.ne_zero _;\n ?_",
"before_state": "K : Type u_1\ninst✝³ : Field K\np q r s u v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nhq_nonzero : q ≠ 0\nhu : u ^ 3 - p * u ^ 2 - 4 * r * u + 4 * p * r - q ^ 2 = 0\nhs : s ^ 2 = u - p\nhv : v ^ 2 = 4 * s ^ 2 - 8 * (u - q / s)\nhw : w ^ 2 = 4 * s ^ 2 - 8 * (u + q / s)\nx : K\n⊢ x ^ 4 + p * x ^ 2 + q * x + r = 0 ↔\n x = (-2 * s - v) / 4 ∨ x = (-2 * s + v) / 4 ∨ x = (2 * s - w) / 4 ∨ x = (2 * s + w) / 4",
"after_state": "K : Type u_1\ninst✝³ : Field K\np q r s u v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nhq_nonzero : q ≠ 0\nhu : u ^ 3 - p * u ^ 2 - 4 * r * u + 4 * p * r - q ^ 2 = 0\nhs : s ^ 2 = u - p\nhv : v ^ 2 = 4 * s ^ 2 - 8 * (u - q / s)\nhw : w ^ 2 = 4 * s ^ 2 - 8 * (u + q / s)\nx : K\nhi2 : 2 ≠ 0\n⊢ x ^ 4 + p * x ^ 2 + q * x + r = 0 ↔\n x = (-2 * s - v) / 4 ∨ x = (-2 * s + v) / 4 ∨ x = (2 * s - w) / 4 ∨ x = (2 * s + w) / 4"
},
{
"line": "focus\n (refine\n no_implicit_lambda%\n (have hi2 : (2 : K) ≠ 0 := Invertible.ne_zero _;\n ?_);\n rotate_right)",
"before_state": "K : Type u_1\ninst✝³ : Field K\np q r s u v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nhq_nonzero : q ≠ 0\nhu : u ^ 3 - p * u ^ 2 - 4 * r * u + 4 * p * r - q ^ 2 = 0\nhs : s ^ 2 = u - p\nhv : v ^ 2 = 4 * s ^ 2 - 8 * (u - q / s)\nhw : w ^ 2 = 4 * s ^ 2 - 8 * (u + q / s)\nx : K\n⊢ x ^ 4 + p * x ^ 2 + q * x + r = 0 ↔\n x = (-2 * s - v) / 4 ∨ x = (-2 * s + v) / 4 ∨ x = (2 * s - w) / 4 ∨ x = (2 * s + w) / 4",
"after_state": "K : Type u_1\ninst✝³ : Field K\np q r s u v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nhq_nonzero : q ≠ 0\nhu : u ^ 3 - p * u ^ 2 - 4 * r * u + 4 * p * r - q ^ 2 = 0\nhs : s ^ 2 = u - p\nhv : v ^ 2 = 4 * s ^ 2 - 8 * (u - q / s)\nhw : w ^ 2 = 4 * s ^ 2 - 8 * (u + q / s)\nx : K\nhi2 : 2 ≠ 0\n⊢ x ^ 4 + p * x ^ 2 + q * x + r = 0 ↔\n x = (-2 * s - v) / 4 ∨ x = (-2 * s + v) / 4 ∨ x = (2 * s - w) / 4 ∨ x = (2 * s + w) / 4"
},
{
"line": "refine\n no_implicit_lambda%\n (have hi2 : (2 : K) ≠ 0 := Invertible.ne_zero _;\n ?_)",
"before_state": "K : Type u_1\ninst✝³ : Field K\np q r s u v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nhq_nonzero : q ≠ 0\nhu : u ^ 3 - p * u ^ 2 - 4 * r * u + 4 * p * r - q ^ 2 = 0\nhs : s ^ 2 = u - p\nhv : v ^ 2 = 4 * s ^ 2 - 8 * (u - q / s)\nhw : w ^ 2 = 4 * s ^ 2 - 8 * (u + q / s)\nx : K\n⊢ x ^ 4 + p * x ^ 2 + q * x + r = 0 ↔\n x = (-2 * s - v) / 4 ∨ x = (-2 * s + v) / 4 ∨ x = (2 * s - w) / 4 ∨ x = (2 * s + w) / 4",
"after_state": "K : Type u_1\ninst✝³ : Field K\np q r s u v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nhq_nonzero : q ≠ 0\nhu : u ^ 3 - p * u ^ 2 - 4 * r * u + 4 * p * r - q ^ 2 = 0\nhs : s ^ 2 = u - p\nhv : v ^ 2 = 4 * s ^ 2 - 8 * (u - q / s)\nhw : w ^ 2 = 4 * s ^ 2 - 8 * (u + q / s)\nx : K\nhi2 : 2 ≠ 0\n⊢ x ^ 4 + p * x ^ 2 + q * x + r = 0 ↔\n x = (-2 * s - v) / 4 ∨ x = (-2 * s + v) / 4 ∨ x = (2 * s - w) / 4 ∨ x = (2 * s + w) / 4"
},
{
"line": "rotate_right",
"before_state": "K : Type u_1\ninst✝³ : Field K\np q r s u v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nhq_nonzero : q ≠ 0\nhu : u ^ 3 - p * u ^ 2 - 4 * r * u + 4 * p * r - q ^ 2 = 0\nhs : s ^ 2 = u - p\nhv : v ^ 2 = 4 * s ^ 2 - 8 * (u - q / s)\nhw : w ^ 2 = 4 * s ^ 2 - 8 * (u + q / s)\nx : K\nhi2 : 2 ≠ 0\n⊢ x ^ 4 + p * x ^ 2 + q * x + r = 0 ↔\n x = (-2 * s - v) / 4 ∨ x = (-2 * s + v) / 4 ∨ x = (2 * s - w) / 4 ∨ x = (2 * s + w) / 4",
"after_state": "K : Type u_1\ninst✝³ : Field K\np q r s u v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nhq_nonzero : q ≠ 0\nhu : u ^ 3 - p * u ^ 2 - 4 * r * u + 4 * p * r - q ^ 2 = 0\nhs : s ^ 2 = u - p\nhv : v ^ 2 = 4 * s ^ 2 - 8 * (u - q / s)\nhw : w ^ 2 = 4 * s ^ 2 - 8 * (u + q / s)\nx : K\nhi2 : 2 ≠ 0\n⊢ x ^ 4 + p * x ^ 2 + q * x + r = 0 ↔\n x = (-2 * s - v) / 4 ∨ x = (-2 * s + v) / 4 ∨ x = (2 * s - w) / 4 ∨ x = (2 * s + w) / 4"
},
{
"line": "have h4 : (4 : K) = 2 ^ 2 := by norm_num",
"before_state": "K : Type u_1\ninst✝³ : Field K\np q r s u v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nhq_nonzero : q ≠ 0\nhu : u ^ 3 - p * u ^ 2 - 4 * r * u + 4 * p * r - q ^ 2 = 0\nhs : s ^ 2 = u - p\nhv : v ^ 2 = 4 * s ^ 2 - 8 * (u - q / s)\nhw : w ^ 2 = 4 * s ^ 2 - 8 * (u + q / s)\nx : K\nhi2 : 2 ≠ 0\n⊢ x ^ 4 + p * x ^ 2 + q * x + r = 0 ↔\n x = (-2 * s - v) / 4 ∨ x = (-2 * s + v) / 4 ∨ x = (2 * s - w) / 4 ∨ x = (2 * s + w) / 4",
"after_state": "K : Type u_1\ninst✝³ : Field K\np q r s u v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nhq_nonzero : q ≠ 0\nhu : u ^ 3 - p * u ^ 2 - 4 * r * u + 4 * p * r - q ^ 2 = 0\nhs : s ^ 2 = u - p\nhv : v ^ 2 = 4 * s ^ 2 - 8 * (u - q / s)\nhw : w ^ 2 = 4 * s ^ 2 - 8 * (u + q / s)\nx : K\nhi2 : 2 ≠ 0\nh4 : 4 = 2 ^ 2\n⊢ x ^ 4 + p * x ^ 2 + q * x + r = 0 ↔\n x = (-2 * s - v) / 4 ∨ x = (-2 * s + v) / 4 ∨ x = (2 * s - w) / 4 ∨ x = (2 * s + w) / 4"
},
{
"line": "focus\n refine\n no_implicit_lambda%\n (have h4 : (4 : K) = 2 ^ 2 := ?body✝;\n ?_)\n case body✝ => with_annotate_state\"by\" (norm_num)",
"before_state": "K : Type u_1\ninst✝³ : Field K\np q r s u v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nhq_nonzero : q ≠ 0\nhu : u ^ 3 - p * u ^ 2 - 4 * r * u + 4 * p * r - q ^ 2 = 0\nhs : s ^ 2 = u - p\nhv : v ^ 2 = 4 * s ^ 2 - 8 * (u - q / s)\nhw : w ^ 2 = 4 * s ^ 2 - 8 * (u + q / s)\nx : K\nhi2 : 2 ≠ 0\n⊢ x ^ 4 + p * x ^ 2 + q * x + r = 0 ↔\n x = (-2 * s - v) / 4 ∨ x = (-2 * s + v) / 4 ∨ x = (2 * s - w) / 4 ∨ x = (2 * s + w) / 4",
"after_state": "K : Type u_1\ninst✝³ : Field K\np q r s u v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nhq_nonzero : q ≠ 0\nhu : u ^ 3 - p * u ^ 2 - 4 * r * u + 4 * p * r - q ^ 2 = 0\nhs : s ^ 2 = u - p\nhv : v ^ 2 = 4 * s ^ 2 - 8 * (u - q / s)\nhw : w ^ 2 = 4 * s ^ 2 - 8 * (u + q / s)\nx : K\nhi2 : 2 ≠ 0\nh4 : 4 = 2 ^ 2\n⊢ x ^ 4 + p * x ^ 2 + q * x + r = 0 ↔\n x = (-2 * s - v) / 4 ∨ x = (-2 * s + v) / 4 ∨ x = (2 * s - w) / 4 ∨ x = (2 * s + w) / 4"
},
{
"line": "refine\n no_implicit_lambda%\n (have h4 : (4 : K) = 2 ^ 2 := ?body✝;\n ?_)",
"before_state": "K : Type u_1\ninst✝³ : Field K\np q r s u v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nhq_nonzero : q ≠ 0\nhu : u ^ 3 - p * u ^ 2 - 4 * r * u + 4 * p * r - q ^ 2 = 0\nhs : s ^ 2 = u - p\nhv : v ^ 2 = 4 * s ^ 2 - 8 * (u - q / s)\nhw : w ^ 2 = 4 * s ^ 2 - 8 * (u + q / s)\nx : K\nhi2 : 2 ≠ 0\n⊢ x ^ 4 + p * x ^ 2 + q * x + r = 0 ↔\n x = (-2 * s - v) / 4 ∨ x = (-2 * s + v) / 4 ∨ x = (2 * s - w) / 4 ∨ x = (2 * s + w) / 4",
"after_state": "case body\nK : Type u_1\ninst✝³ : Field K\np q r s u v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nhq_nonzero : q ≠ 0\nhu : u ^ 3 - p * u ^ 2 - 4 * r * u + 4 * p * r - q ^ 2 = 0\nhs : s ^ 2 = u - p\nhv : v ^ 2 = 4 * s ^ 2 - 8 * (u - q / s)\nhw : w ^ 2 = 4 * s ^ 2 - 8 * (u + q / s)\nx : K\nhi2 : 2 ≠ 0\n⊢ 4 = 2 ^ 2\n---\nK : Type u_1\ninst✝³ : Field K\np q r s u v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nhq_nonzero : q ≠ 0\nhu : u ^ 3 - p * u ^ 2 - 4 * r * u + 4 * p * r - q ^ 2 = 0\nhs : s ^ 2 = u - p\nhv : v ^ 2 = 4 * s ^ 2 - 8 * (u - q / s)\nhw : w ^ 2 = 4 * s ^ 2 - 8 * (u + q / s)\nx : K\nhi2 : 2 ≠ 0\nh4 : 4 = 2 ^ 2\n⊢ x ^ 4 + p * x ^ 2 + q * x + r = 0 ↔\n x = (-2 * s - v) / 4 ∨ x = (-2 * s + v) / 4 ∨ x = (2 * s - w) / 4 ∨ x = (2 * s + w) / 4"
},
{
"line": "case body✝ => with_annotate_state\"by\" (norm_num)",
"before_state": "case body\nK : Type u_1\ninst✝³ : Field K\np q r s u v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nhq_nonzero : q ≠ 0\nhu : u ^ 3 - p * u ^ 2 - 4 * r * u + 4 * p * r - q ^ 2 = 0\nhs : s ^ 2 = u - p\nhv : v ^ 2 = 4 * s ^ 2 - 8 * (u - q / s)\nhw : w ^ 2 = 4 * s ^ 2 - 8 * (u + q / s)\nx : K\nhi2 : 2 ≠ 0\n⊢ 4 = 2 ^ 2\n---\nK : Type u_1\ninst✝³ : Field K\np q r s u v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nhq_nonzero : q ≠ 0\nhu : u ^ 3 - p * u ^ 2 - 4 * r * u + 4 * p * r - q ^ 2 = 0\nhs : s ^ 2 = u - p\nhv : v ^ 2 = 4 * s ^ 2 - 8 * (u - q / s)\nhw : w ^ 2 = 4 * s ^ 2 - 8 * (u + q / s)\nx : K\nhi2 : 2 ≠ 0\nh4 : 4 = 2 ^ 2\n⊢ x ^ 4 + p * x ^ 2 + q * x + r = 0 ↔\n x = (-2 * s - v) / 4 ∨ x = (-2 * s + v) / 4 ∨ x = (2 * s - w) / 4 ∨ x = (2 * s + w) / 4",
"after_state": "K : Type u_1\ninst✝³ : Field K\np q r s u v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nhq_nonzero : q ≠ 0\nhu : u ^ 3 - p * u ^ 2 - 4 * r * u + 4 * p * r - q ^ 2 = 0\nhs : s ^ 2 = u - p\nhv : v ^ 2 = 4 * s ^ 2 - 8 * (u - q / s)\nhw : w ^ 2 = 4 * s ^ 2 - 8 * (u + q / s)\nx : K\nhi2 : 2 ≠ 0\nh4 : 4 = 2 ^ 2\n⊢ x ^ 4 + p * x ^ 2 + q * x + r = 0 ↔\n x = (-2 * s - v) / 4 ∨ x = (-2 * s + v) / 4 ∨ x = (2 * s - w) / 4 ∨ x = (2 * s + w) / 4"
},
{
"line": "with_annotate_state\"by\" (norm_num)",
"before_state": "K : Type u_1\ninst✝³ : Field K\np q r s u v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nhq_nonzero : q ≠ 0\nhu : u ^ 3 - p * u ^ 2 - 4 * r * u + 4 * p * r - q ^ 2 = 0\nhs : s ^ 2 = u - p\nhv : v ^ 2 = 4 * s ^ 2 - 8 * (u - q / s)\nhw : w ^ 2 = 4 * s ^ 2 - 8 * (u + q / s)\nx : K\nhi2 : 2 ≠ 0\n⊢ 4 = 2 ^ 2",
"after_state": "No Goals!"
},
{
"line": "norm_num",
"before_state": "K : Type u_1\ninst✝³ : Field K\np q r s u v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nhq_nonzero : q ≠ 0\nhu : u ^ 3 - p * u ^ 2 - 4 * r * u + 4 * p * r - q ^ 2 = 0\nhs : s ^ 2 = u - p\nhv : v ^ 2 = 4 * s ^ 2 - 8 * (u - q / s)\nhw : w ^ 2 = 4 * s ^ 2 - 8 * (u + q / s)\nx : K\nhi2 : 2 ≠ 0\n⊢ 4 = 2 ^ 2",
"after_state": "No Goals!"
},
{
"line": "have hs_nonzero : s ≠ 0 := by\n contrapose! hq_nonzero with hs0\n linear_combination (exp := 2) -hu + (4 * r - u ^ 2) * hs + (u ^ 2 * s - 4 * r * s) * hs0",
"before_state": "K : Type u_1\ninst✝³ : Field K\np q r s u v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nhq_nonzero : q ≠ 0\nhu : u ^ 3 - p * u ^ 2 - 4 * r * u + 4 * p * r - q ^ 2 = 0\nhs : s ^ 2 = u - p\nhv : v ^ 2 = 4 * s ^ 2 - 8 * (u - q / s)\nhw : w ^ 2 = 4 * s ^ 2 - 8 * (u + q / s)\nx : K\nhi2 : 2 ≠ 0\nh4 : 4 = 2 ^ 2\n⊢ x ^ 4 + p * x ^ 2 + q * x + r = 0 ↔\n x = (-2 * s - v) / 4 ∨ x = (-2 * s + v) / 4 ∨ x = (2 * s - w) / 4 ∨ x = (2 * s + w) / 4",
"after_state": "K : Type u_1\ninst✝³ : Field K\np q r s u v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nhq_nonzero : q ≠ 0\nhu : u ^ 3 - p * u ^ 2 - 4 * r * u + 4 * p * r - q ^ 2 = 0\nhs : s ^ 2 = u - p\nhv : v ^ 2 = 4 * s ^ 2 - 8 * (u - q / s)\nhw : w ^ 2 = 4 * s ^ 2 - 8 * (u + q / s)\nx : K\nhi2 : 2 ≠ 0\nh4 : 4 = 2 ^ 2\nhs_nonzero : s ≠ 0\n⊢ x ^ 4 + p * x ^ 2 + q * x + r = 0 ↔\n x = (-2 * s - v) / 4 ∨ x = (-2 * s + v) / 4 ∨ x = (2 * s - w) / 4 ∨ x = (2 * s + w) / 4"
},
{
"line": "focus\n refine\n no_implicit_lambda%\n (have hs_nonzero : s ≠ 0 := ?body✝;\n ?_)\n case body✝ =>\n with_annotate_state\"by\"\n ( contrapose! hq_nonzero with hs0\n linear_combination (exp := 2) -hu + (4 * r - u ^ 2) * hs + (u ^ 2 * s - 4 * r * s) * hs0)",
"before_state": "K : Type u_1\ninst✝³ : Field K\np q r s u v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nhq_nonzero : q ≠ 0\nhu : u ^ 3 - p * u ^ 2 - 4 * r * u + 4 * p * r - q ^ 2 = 0\nhs : s ^ 2 = u - p\nhv : v ^ 2 = 4 * s ^ 2 - 8 * (u - q / s)\nhw : w ^ 2 = 4 * s ^ 2 - 8 * (u + q / s)\nx : K\nhi2 : 2 ≠ 0\nh4 : 4 = 2 ^ 2\n⊢ x ^ 4 + p * x ^ 2 + q * x + r = 0 ↔\n x = (-2 * s - v) / 4 ∨ x = (-2 * s + v) / 4 ∨ x = (2 * s - w) / 4 ∨ x = (2 * s + w) / 4",
"after_state": "K : Type u_1\ninst✝³ : Field K\np q r s u v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nhq_nonzero : q ≠ 0\nhu : u ^ 3 - p * u ^ 2 - 4 * r * u + 4 * p * r - q ^ 2 = 0\nhs : s ^ 2 = u - p\nhv : v ^ 2 = 4 * s ^ 2 - 8 * (u - q / s)\nhw : w ^ 2 = 4 * s ^ 2 - 8 * (u + q / s)\nx : K\nhi2 : 2 ≠ 0\nh4 : 4 = 2 ^ 2\nhs_nonzero : s ≠ 0\n⊢ x ^ 4 + p * x ^ 2 + q * x + r = 0 ↔\n x = (-2 * s - v) / 4 ∨ x = (-2 * s + v) / 4 ∨ x = (2 * s - w) / 4 ∨ x = (2 * s + w) / 4"
},
{
"line": "refine\n no_implicit_lambda%\n (have hs_nonzero : s ≠ 0 := ?body✝;\n ?_)",
"before_state": "K : Type u_1\ninst✝³ : Field K\np q r s u v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nhq_nonzero : q ≠ 0\nhu : u ^ 3 - p * u ^ 2 - 4 * r * u + 4 * p * r - q ^ 2 = 0\nhs : s ^ 2 = u - p\nhv : v ^ 2 = 4 * s ^ 2 - 8 * (u - q / s)\nhw : w ^ 2 = 4 * s ^ 2 - 8 * (u + q / s)\nx : K\nhi2 : 2 ≠ 0\nh4 : 4 = 2 ^ 2\n⊢ x ^ 4 + p * x ^ 2 + q * x + r = 0 ↔\n x = (-2 * s - v) / 4 ∨ x = (-2 * s + v) / 4 ∨ x = (2 * s - w) / 4 ∨ x = (2 * s + w) / 4",
"after_state": "case body\nK : Type u_1\ninst✝³ : Field K\np q r s u v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nhq_nonzero : q ≠ 0\nhu : u ^ 3 - p * u ^ 2 - 4 * r * u + 4 * p * r - q ^ 2 = 0\nhs : s ^ 2 = u - p\nhv : v ^ 2 = 4 * s ^ 2 - 8 * (u - q / s)\nhw : w ^ 2 = 4 * s ^ 2 - 8 * (u + q / s)\nx : K\nhi2 : 2 ≠ 0\nh4 : 4 = 2 ^ 2\n⊢ s ≠ 0\n---\nK : Type u_1\ninst✝³ : Field K\np q r s u v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nhq_nonzero : q ≠ 0\nhu : u ^ 3 - p * u ^ 2 - 4 * r * u + 4 * p * r - q ^ 2 = 0\nhs : s ^ 2 = u - p\nhv : v ^ 2 = 4 * s ^ 2 - 8 * (u - q / s)\nhw : w ^ 2 = 4 * s ^ 2 - 8 * (u + q / s)\nx : K\nhi2 : 2 ≠ 0\nh4 : 4 = 2 ^ 2\nhs_nonzero : s ≠ 0\n⊢ x ^ 4 + p * x ^ 2 + q * x + r = 0 ↔\n x = (-2 * s - v) / 4 ∨ x = (-2 * s + v) / 4 ∨ x = (2 * s - w) / 4 ∨ x = (2 * s + w) / 4"
},
{
"line": "case body✝ =>\n with_annotate_state\"by\"\n ( contrapose! hq_nonzero with hs0\n linear_combination (exp := 2) -hu + (4 * r - u ^ 2) * hs + (u ^ 2 * s - 4 * r * s) * hs0)",
"before_state": "case body\nK : Type u_1\ninst✝³ : Field K\np q r s u v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nhq_nonzero : q ≠ 0\nhu : u ^ 3 - p * u ^ 2 - 4 * r * u + 4 * p * r - q ^ 2 = 0\nhs : s ^ 2 = u - p\nhv : v ^ 2 = 4 * s ^ 2 - 8 * (u - q / s)\nhw : w ^ 2 = 4 * s ^ 2 - 8 * (u + q / s)\nx : K\nhi2 : 2 ≠ 0\nh4 : 4 = 2 ^ 2\n⊢ s ≠ 0\n---\nK : Type u_1\ninst✝³ : Field K\np q r s u v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nhq_nonzero : q ≠ 0\nhu : u ^ 3 - p * u ^ 2 - 4 * r * u + 4 * p * r - q ^ 2 = 0\nhs : s ^ 2 = u - p\nhv : v ^ 2 = 4 * s ^ 2 - 8 * (u - q / s)\nhw : w ^ 2 = 4 * s ^ 2 - 8 * (u + q / s)\nx : K\nhi2 : 2 ≠ 0\nh4 : 4 = 2 ^ 2\nhs_nonzero : s ≠ 0\n⊢ x ^ 4 + p * x ^ 2 + q * x + r = 0 ↔\n x = (-2 * s - v) / 4 ∨ x = (-2 * s + v) / 4 ∨ x = (2 * s - w) / 4 ∨ x = (2 * s + w) / 4",
"after_state": "K : Type u_1\ninst✝³ : Field K\np q r s u v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nhq_nonzero : q ≠ 0\nhu : u ^ 3 - p * u ^ 2 - 4 * r * u + 4 * p * r - q ^ 2 = 0\nhs : s ^ 2 = u - p\nhv : v ^ 2 = 4 * s ^ 2 - 8 * (u - q / s)\nhw : w ^ 2 = 4 * s ^ 2 - 8 * (u + q / s)\nx : K\nhi2 : 2 ≠ 0\nh4 : 4 = 2 ^ 2\nhs_nonzero : s ≠ 0\n⊢ x ^ 4 + p * x ^ 2 + q * x + r = 0 ↔\n x = (-2 * s - v) / 4 ∨ x = (-2 * s + v) / 4 ∨ x = (2 * s - w) / 4 ∨ x = (2 * s + w) / 4"
},
{
"line": "with_annotate_state\"by\"\n ( contrapose! hq_nonzero with hs0\n linear_combination (exp := 2) -hu + (4 * r - u ^ 2) * hs + (u ^ 2 * s - 4 * r * s) * hs0)",
"before_state": "K : Type u_1\ninst✝³ : Field K\np q r s u v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nhq_nonzero : q ≠ 0\nhu : u ^ 3 - p * u ^ 2 - 4 * r * u + 4 * p * r - q ^ 2 = 0\nhs : s ^ 2 = u - p\nhv : v ^ 2 = 4 * s ^ 2 - 8 * (u - q / s)\nhw : w ^ 2 = 4 * s ^ 2 - 8 * (u + q / s)\nx : K\nhi2 : 2 ≠ 0\nh4 : 4 = 2 ^ 2\n⊢ s ≠ 0",
"after_state": "No Goals!"
},
{
"line": "contrapose! hq_nonzero with hs0",
"before_state": "K : Type u_1\ninst✝³ : Field K\np q r s u v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nhq_nonzero : q ≠ 0\nhu : u ^ 3 - p * u ^ 2 - 4 * r * u + 4 * p * r - q ^ 2 = 0\nhs : s ^ 2 = u - p\nhv : v ^ 2 = 4 * s ^ 2 - 8 * (u - q / s)\nhw : w ^ 2 = 4 * s ^ 2 - 8 * (u + q / s)\nx : K\nhi2 : 2 ≠ 0\nh4 : 4 = 2 ^ 2\n⊢ s ≠ 0",
"after_state": "K : Type u_1\ninst✝³ : Field K\np q r s u v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nhu : u ^ 3 - p * u ^ 2 - 4 * r * u + 4 * p * r - q ^ 2 = 0\nhs : s ^ 2 = u - p\nhv : v ^ 2 = 4 * s ^ 2 - 8 * (u - q / s)\nhw : w ^ 2 = 4 * s ^ 2 - 8 * (u + q / s)\nx : K\nhi2 : 2 ≠ 0\nh4 : 4 = 2 ^ 2\nhs0 : s = 0\n⊢ q = 0"
},
{
"line": "revert hq_nonzero",
"before_state": "K : Type u_1\ninst✝³ : Field K\np q r s u v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nhq_nonzero : q ≠ 0\nhu : u ^ 3 - p * u ^ 2 - 4 * r * u + 4 * p * r - q ^ 2 = 0\nhs : s ^ 2 = u - p\nhv : v ^ 2 = 4 * s ^ 2 - 8 * (u - q / s)\nhw : w ^ 2 = 4 * s ^ 2 - 8 * (u + q / s)\nx : K\nhi2 : 2 ≠ 0\nh4 : 4 = 2 ^ 2\n⊢ s ≠ 0",
"after_state": "K : Type u_1\ninst✝³ : Field K\np q r s u v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nhu : u ^ 3 - p * u ^ 2 - 4 * r * u + 4 * p * r - q ^ 2 = 0\nhs : s ^ 2 = u - p\nhv : v ^ 2 = 4 * s ^ 2 - 8 * (u - q / s)\nhw : w ^ 2 = 4 * s ^ 2 - 8 * (u + q / s)\nx : K\nhi2 : 2 ≠ 0\nh4 : 4 = 2 ^ 2\n⊢ q ≠ 0 → s ≠ 0"
},
{
"line": "contrapose!",
"before_state": "K : Type u_1\ninst✝³ : Field K\np q r s u v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nhu : u ^ 3 - p * u ^ 2 - 4 * r * u + 4 * p * r - q ^ 2 = 0\nhs : s ^ 2 = u - p\nhv : v ^ 2 = 4 * s ^ 2 - 8 * (u - q / s)\nhw : w ^ 2 = 4 * s ^ 2 - 8 * (u + q / s)\nx : K\nhi2 : 2 ≠ 0\nh4 : 4 = 2 ^ 2\n⊢ q ≠ 0 → s ≠ 0",
"after_state": "K : Type u_1\ninst✝³ : Field K\np q r s u v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nhu : u ^ 3 - p * u ^ 2 - 4 * r * u + 4 * p * r - q ^ 2 = 0\nhs : s ^ 2 = u - p\nhv : v ^ 2 = 4 * s ^ 2 - 8 * (u - q / s)\nhw : w ^ 2 = 4 * s ^ 2 - 8 * (u + q / s)\nx : K\nhi2 : 2 ≠ 0\nh4 : 4 = 2 ^ 2\n⊢ s = 0 → q = 0"
},
{
"line": "contrapose",
"before_state": "K : Type u_1\ninst✝³ : Field K\np q r s u v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nhu : u ^ 3 - p * u ^ 2 - 4 * r * u + 4 * p * r - q ^ 2 = 0\nhs : s ^ 2 = u - p\nhv : v ^ 2 = 4 * s ^ 2 - 8 * (u - q / s)\nhw : w ^ 2 = 4 * s ^ 2 - 8 * (u + q / s)\nx : K\nhi2 : 2 ≠ 0\nh4 : 4 = 2 ^ 2\n⊢ q ≠ 0 → s ≠ 0",
"after_state": "K : Type u_1\ninst✝³ : Field K\np q r s u v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nhu : u ^ 3 - p * u ^ 2 - 4 * r * u + 4 * p * r - q ^ 2 = 0\nhs : s ^ 2 = u - p\nhv : v ^ 2 = 4 * s ^ 2 - 8 * (u - q / s)\nhw : w ^ 2 = 4 * s ^ 2 - 8 * (u + q / s)\nx : K\nhi2 : 2 ≠ 0\nh4 : 4 = 2 ^ 2\n⊢ ¬s ≠ 0 → ¬q ≠ 0"
},
{
"line": "refine mtr✝ ?_",
"before_state": "K : Type u_1\ninst✝³ : Field K\np q r s u v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nhu : u ^ 3 - p * u ^ 2 - 4 * r * u + 4 * p * r - q ^ 2 = 0\nhs : s ^ 2 = u - p\nhv : v ^ 2 = 4 * s ^ 2 - 8 * (u - q / s)\nhw : w ^ 2 = 4 * s ^ 2 - 8 * (u + q / s)\nx : K\nhi2 : 2 ≠ 0\nh4 : 4 = 2 ^ 2\n⊢ q ≠ 0 → s ≠ 0",
"after_state": "K : Type u_1\ninst✝³ : Field K\np q r s u v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nhu : u ^ 3 - p * u ^ 2 - 4 * r * u + 4 * p * r - q ^ 2 = 0\nhs : s ^ 2 = u - p\nhv : v ^ 2 = 4 * s ^ 2 - 8 * (u - q / s)\nhw : w ^ 2 = 4 * s ^ 2 - 8 * (u + q / s)\nx : K\nhi2 : 2 ≠ 0\nh4 : 4 = 2 ^ 2\n⊢ ¬s ≠ 0 → ¬q ≠ 0"
},
{
"line": "try push_neg",
"before_state": "K : Type u_1\ninst✝³ : Field K\np q r s u v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nhu : u ^ 3 - p * u ^ 2 - 4 * r * u + 4 * p * r - q ^ 2 = 0\nhs : s ^ 2 = u - p\nhv : v ^ 2 = 4 * s ^ 2 - 8 * (u - q / s)\nhw : w ^ 2 = 4 * s ^ 2 - 8 * (u + q / s)\nx : K\nhi2 : 2 ≠ 0\nh4 : 4 = 2 ^ 2\n⊢ ¬s ≠ 0 → ¬q ≠ 0",
"after_state": "K : Type u_1\ninst✝³ : Field K\np q r s u v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nhu : u ^ 3 - p * u ^ 2 - 4 * r * u + 4 * p * r - q ^ 2 = 0\nhs : s ^ 2 = u - p\nhv : v ^ 2 = 4 * s ^ 2 - 8 * (u - q / s)\nhw : w ^ 2 = 4 * s ^ 2 - 8 * (u + q / s)\nx : K\nhi2 : 2 ≠ 0\nh4 : 4 = 2 ^ 2\n⊢ s = 0 → q = 0"
},
{
"line": "first\n| push_neg\n| skip",
"before_state": "K : Type u_1\ninst✝³ : Field K\np q r s u v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nhu : u ^ 3 - p * u ^ 2 - 4 * r * u + 4 * p * r - q ^ 2 = 0\nhs : s ^ 2 = u - p\nhv : v ^ 2 = 4 * s ^ 2 - 8 * (u - q / s)\nhw : w ^ 2 = 4 * s ^ 2 - 8 * (u + q / s)\nx : K\nhi2 : 2 ≠ 0\nh4 : 4 = 2 ^ 2\n⊢ ¬s ≠ 0 → ¬q ≠ 0",
"after_state": "K : Type u_1\ninst✝³ : Field K\np q r s u v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nhu : u ^ 3 - p * u ^ 2 - 4 * r * u + 4 * p * r - q ^ 2 = 0\nhs : s ^ 2 = u - p\nhv : v ^ 2 = 4 * s ^ 2 - 8 * (u - q / s)\nhw : w ^ 2 = 4 * s ^ 2 - 8 * (u + q / s)\nx : K\nhi2 : 2 ≠ 0\nh4 : 4 = 2 ^ 2\n⊢ s = 0 → q = 0"
},
{
"line": "push_neg",
"before_state": "K : Type u_1\ninst✝³ : Field K\np q r s u v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nhu : u ^ 3 - p * u ^ 2 - 4 * r * u + 4 * p * r - q ^ 2 = 0\nhs : s ^ 2 = u - p\nhv : v ^ 2 = 4 * s ^ 2 - 8 * (u - q / s)\nhw : w ^ 2 = 4 * s ^ 2 - 8 * (u + q / s)\nx : K\nhi2 : 2 ≠ 0\nh4 : 4 = 2 ^ 2\n⊢ ¬s ≠ 0 → ¬q ≠ 0",
"after_state": "K : Type u_1\ninst✝³ : Field K\np q r s u v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nhu : u ^ 3 - p * u ^ 2 - 4 * r * u + 4 * p * r - q ^ 2 = 0\nhs : s ^ 2 = u - p\nhv : v ^ 2 = 4 * s ^ 2 - 8 * (u - q / s)\nhw : w ^ 2 = 4 * s ^ 2 - 8 * (u + q / s)\nx : K\nhi2 : 2 ≠ 0\nh4 : 4 = 2 ^ 2\n⊢ s = 0 → q = 0"
},
{
"line": "intro hs0",
"before_state": "K : Type u_1\ninst✝³ : Field K\np q r s u v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nhu : u ^ 3 - p * u ^ 2 - 4 * r * u + 4 * p * r - q ^ 2 = 0\nhs : s ^ 2 = u - p\nhv : v ^ 2 = 4 * s ^ 2 - 8 * (u - q / s)\nhw : w ^ 2 = 4 * s ^ 2 - 8 * (u + q / s)\nx : K\nhi2 : 2 ≠ 0\nh4 : 4 = 2 ^ 2\n⊢ s = 0 → q = 0",
"after_state": "K : Type u_1\ninst✝³ : Field K\np q r s u v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nhu : u ^ 3 - p * u ^ 2 - 4 * r * u + 4 * p * r - q ^ 2 = 0\nhs : s ^ 2 = u - p\nhv : v ^ 2 = 4 * s ^ 2 - 8 * (u - q / s)\nhw : w ^ 2 = 4 * s ^ 2 - 8 * (u + q / s)\nx : K\nhi2 : 2 ≠ 0\nh4 : 4 = 2 ^ 2\nhs0 : s = 0\n⊢ q = 0"
},
{
"line": "linear_combination (exp := 2) -hu + (4 * r - u ^ 2) * hs + (u ^ 2 * s - 4 * r * s) * hs0",
"before_state": "K : Type u_1\ninst✝³ : Field K\np q r s u v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nhu : u ^ 3 - p * u ^ 2 - 4 * r * u + 4 * p * r - q ^ 2 = 0\nhs : s ^ 2 = u - p\nhv : v ^ 2 = 4 * s ^ 2 - 8 * (u - q / s)\nhw : w ^ 2 = 4 * s ^ 2 - 8 * (u + q / s)\nx : K\nhi2 : 2 ≠ 0\nh4 : 4 = 2 ^ 2\nhs0 : s = 0\n⊢ q = 0",
"after_state": "No Goals!"
},
{
"line": "calc\n _ ↔ 4 * (x ^ 4 + p * x ^ 2 + q * x + r) = 0 := by simp [h4, hi2]\n _ ↔ (2 * (x * x) + 2 * s * x + (u - q / s)) * (2 * (x * x) + -(2 * s) * x + (u + q / s)) = 0 :=\n by\n apply Eq.congr_left\n field_simp\n linear_combination\n -hu + (-x ^ 2 * s ^ 2 - x ^ 2 * p + x ^ 2 * u) * hw +\n (x ^ 2 * w ^ 2 + 8 * x ^ 2 * u + 8 * x ^ 2 * q / s - u ^ 2 + 4 * r) * hs\n _ ↔ _ := by\n have hv' : discrim 2 (2 * s) (u - q / s) = v * v := by rw [discrim]; linear_combination -hv\n have hw' : discrim 2 (-(2 * s)) (u + q / s) = w * w := by rw [discrim]; linear_combination -hw\n rw [mul_eq_zero]\n rw [quadratic_eq_zero_iff hi2 hv']\n rw [quadratic_eq_zero_iff hi2 hw']\n simp [(by norm_num : (2 : K) * 2 = 4), or_assoc, or_comm]",
"before_state": "K : Type u_1\ninst✝³ : Field K\np q r s u v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nhq_nonzero : q ≠ 0\nhu : u ^ 3 - p * u ^ 2 - 4 * r * u + 4 * p * r - q ^ 2 = 0\nhs : s ^ 2 = u - p\nhv : v ^ 2 = 4 * s ^ 2 - 8 * (u - q / s)\nhw : w ^ 2 = 4 * s ^ 2 - 8 * (u + q / s)\nx : K\nhi2 : 2 ≠ 0\nh4 : 4 = 2 ^ 2\nhs_nonzero : s ≠ 0\n⊢ x ^ 4 + p * x ^ 2 + q * x + r = 0 ↔\n x = (-2 * s - v) / 4 ∨ x = (-2 * s + v) / 4 ∨ x = (2 * s - w) / 4 ∨ x = (2 * s + w) / 4",
"after_state": "No Goals!"
},
{
"line": "simp [h4, hi2]",
"before_state": "K : Type u_1\ninst✝³ : Field K\np q r s u v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nhq_nonzero : q ≠ 0\nhu : u ^ 3 - p * u ^ 2 - 4 * r * u + 4 * p * r - q ^ 2 = 0\nhs : s ^ 2 = u - p\nhv : v ^ 2 = 4 * s ^ 2 - 8 * (u - q / s)\nhw : w ^ 2 = 4 * s ^ 2 - 8 * (u + q / s)\nx : K\nhi2 : 2 ≠ 0\nh4 : 4 = 2 ^ 2\nhs_nonzero : s ≠ 0\n⊢ x ^ 4 + p * x ^ 2 + q * x + r = 0 ↔ 4 * (x ^ 4 + p * x ^ 2 + q * x + r) = 0",
"after_state": "No Goals!"
},
{
"line": "apply Eq.congr_left",
"before_state": "K : Type u_1\ninst✝³ : Field K\np q r s u v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nhq_nonzero : q ≠ 0\nhu : u ^ 3 - p * u ^ 2 - 4 * r * u + 4 * p * r - q ^ 2 = 0\nhs : s ^ 2 = u - p\nhv : v ^ 2 = 4 * s ^ 2 - 8 * (u - q / s)\nhw : w ^ 2 = 4 * s ^ 2 - 8 * (u + q / s)\nx : K\nhi2 : 2 ≠ 0\nh4 : 4 = 2 ^ 2\nhs_nonzero : s ≠ 0\n⊢ 4 * (x ^ 4 + p * x ^ 2 + q * x + r) = 0 ↔\n (2 * (x * x) + 2 * s * x + (u - q / s)) * (2 * (x * x) + -(2 * s) * x + (u + q / s)) = 0",
"after_state": "case h\nK : Type u_1\ninst✝³ : Field K\np q r s u v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nhq_nonzero : q ≠ 0\nhu : u ^ 3 - p * u ^ 2 - 4 * r * u + 4 * p * r - q ^ 2 = 0\nhs : s ^ 2 = u - p\nhv : v ^ 2 = 4 * s ^ 2 - 8 * (u - q / s)\nhw : w ^ 2 = 4 * s ^ 2 - 8 * (u + q / s)\nx : K\nhi2 : 2 ≠ 0\nh4 : 4 = 2 ^ 2\nhs_nonzero : s ≠ 0\n⊢ 4 * (x ^ 4 + p * x ^ 2 + q * x + r) =\n (2 * (x * x) + 2 * s * x + (u - q / s)) * (2 * (x * x) + -(2 * s) * x + (u + q / s))"
},
{
"line": "field_simp",
"before_state": "case h\nK : Type u_1\ninst✝³ : Field K\np q r s u v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nhq_nonzero : q ≠ 0\nhu : u ^ 3 - p * u ^ 2 - 4 * r * u + 4 * p * r - q ^ 2 = 0\nhs : s ^ 2 = u - p\nhv : v ^ 2 = 4 * s ^ 2 - 8 * (u - q / s)\nhw : w ^ 2 = 4 * s ^ 2 - 8 * (u + q / s)\nx : K\nhi2 : 2 ≠ 0\nh4 : 4 = 2 ^ 2\nhs_nonzero : s ≠ 0\n⊢ 4 * (x ^ 4 + p * x ^ 2 + q * x + r) =\n (2 * (x * x) + 2 * s * x + (u - q / s)) * (2 * (x * x) + -(2 * s) * x + (u + q / s))",
"after_state": "case h\nK : Type u_1\ninst✝³ : Field K\np q r s u v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nhq_nonzero : q ≠ 0\nhu : u ^ 3 - p * u ^ 2 - 4 * r * u + 4 * p * r - q ^ 2 = 0\nhs : s ^ 2 = u - p\nhv : v ^ 2 = 4 * s ^ 2 - 8 * (u - q / s)\nhw : w ^ 2 = 4 * s ^ 2 - 8 * (u + q / s)\nx : K\nhi2 : 2 ≠ 0\nh4 : 4 = 2 ^ 2\nhs_nonzero : s ≠ 0\n⊢ 4 * (x ^ 4 + p * x ^ 2 + q * x + r) * (s * s) =\n ((2 * (x * x) + 2 * s * x) * s + (u * s - q)) * ((2 * (x * x) + -(2 * s * x)) * s + (u * s + q))"
},
{
"line": "linear_combination\n -hu + (-x ^ 2 * s ^ 2 - x ^ 2 * p + x ^ 2 * u) * hw +\n (x ^ 2 * w ^ 2 + 8 * x ^ 2 * u + 8 * x ^ 2 * q / s - u ^ 2 + 4 * r) * hs",
"before_state": "case h\nK : Type u_1\ninst✝³ : Field K\np q r s u v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nhq_nonzero : q ≠ 0\nhu : u ^ 3 - p * u ^ 2 - 4 * r * u + 4 * p * r - q ^ 2 = 0\nhs : s ^ 2 = u - p\nhv : v ^ 2 = 4 * s ^ 2 - 8 * (u - q / s)\nhw : w ^ 2 = 4 * s ^ 2 - 8 * (u + q / s)\nx : K\nhi2 : 2 ≠ 0\nh4 : 4 = 2 ^ 2\nhs_nonzero : s ≠ 0\n⊢ 4 * (x ^ 4 + p * x ^ 2 + q * x + r) * (s * s) =\n ((2 * (x * x) + 2 * s * x) * s + (u * s - q)) * ((2 * (x * x) + -(2 * s * x)) * s + (u * s + q))",
"after_state": "No Goals!"
},
{
"line": "have hv' : discrim 2 (2 * s) (u - q / s) = v * v := by rw [discrim]; linear_combination -hv",
"before_state": "K : Type u_1\ninst✝³ : Field K\np q r s u v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nhq_nonzero : q ≠ 0\nhu : u ^ 3 - p * u ^ 2 - 4 * r * u + 4 * p * r - q ^ 2 = 0\nhs : s ^ 2 = u - p\nhv : v ^ 2 = 4 * s ^ 2 - 8 * (u - q / s)\nhw : w ^ 2 = 4 * s ^ 2 - 8 * (u + q / s)\nx : K\nhi2 : 2 ≠ 0\nh4 : 4 = 2 ^ 2\nhs_nonzero : s ≠ 0\n⊢ (2 * (x * x) + 2 * s * x + (u - q / s)) * (2 * (x * x) + -(2 * s) * x + (u + q / s)) = 0 ↔\n x = (-2 * s - v) / 4 ∨ x = (-2 * s + v) / 4 ∨ x = (2 * s - w) / 4 ∨ x = (2 * s + w) / 4",
"after_state": "K : Type u_1\ninst✝³ : Field K\np q r s u v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nhq_nonzero : q ≠ 0\nhu : u ^ 3 - p * u ^ 2 - 4 * r * u + 4 * p * r - q ^ 2 = 0\nhs : s ^ 2 = u - p\nhv : v ^ 2 = 4 * s ^ 2 - 8 * (u - q / s)\nhw : w ^ 2 = 4 * s ^ 2 - 8 * (u + q / s)\nx : K\nhi2 : 2 ≠ 0\nh4 : 4 = 2 ^ 2\nhs_nonzero : s ≠ 0\nhv' : discrim 2 (2 * s) (u - q / s) = v * v\n⊢ (2 * (x * x) + 2 * s * x + (u - q / s)) * (2 * (x * x) + -(2 * s) * x + (u + q / s)) = 0 ↔\n x = (-2 * s - v) / 4 ∨ x = (-2 * s + v) / 4 ∨ x = (2 * s - w) / 4 ∨ x = (2 * s + w) / 4"
},
{
"line": "focus\n refine\n no_implicit_lambda%\n (have hv' : discrim 2 (2 * s) (u - q / s) = v * v := ?body✝;\n ?_)\n case body✝ => with_annotate_state\"by\" (rw [discrim]; linear_combination -hv)",
"before_state": "K : Type u_1\ninst✝³ : Field K\np q r s u v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nhq_nonzero : q ≠ 0\nhu : u ^ 3 - p * u ^ 2 - 4 * r * u + 4 * p * r - q ^ 2 = 0\nhs : s ^ 2 = u - p\nhv : v ^ 2 = 4 * s ^ 2 - 8 * (u - q / s)\nhw : w ^ 2 = 4 * s ^ 2 - 8 * (u + q / s)\nx : K\nhi2 : 2 ≠ 0\nh4 : 4 = 2 ^ 2\nhs_nonzero : s ≠ 0\n⊢ (2 * (x * x) + 2 * s * x + (u - q / s)) * (2 * (x * x) + -(2 * s) * x + (u + q / s)) = 0 ↔\n x = (-2 * s - v) / 4 ∨ x = (-2 * s + v) / 4 ∨ x = (2 * s - w) / 4 ∨ x = (2 * s + w) / 4",
"after_state": "K : Type u_1\ninst✝³ : Field K\np q r s u v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nhq_nonzero : q ≠ 0\nhu : u ^ 3 - p * u ^ 2 - 4 * r * u + 4 * p * r - q ^ 2 = 0\nhs : s ^ 2 = u - p\nhv : v ^ 2 = 4 * s ^ 2 - 8 * (u - q / s)\nhw : w ^ 2 = 4 * s ^ 2 - 8 * (u + q / s)\nx : K\nhi2 : 2 ≠ 0\nh4 : 4 = 2 ^ 2\nhs_nonzero : s ≠ 0\nhv' : discrim 2 (2 * s) (u - q / s) = v * v\n⊢ (2 * (x * x) + 2 * s * x + (u - q / s)) * (2 * (x * x) + -(2 * s) * x + (u + q / s)) = 0 ↔\n x = (-2 * s - v) / 4 ∨ x = (-2 * s + v) / 4 ∨ x = (2 * s - w) / 4 ∨ x = (2 * s + w) / 4"
},
{
"line": "refine\n no_implicit_lambda%\n (have hv' : discrim 2 (2 * s) (u - q / s) = v * v := ?body✝;\n ?_)",
"before_state": "K : Type u_1\ninst✝³ : Field K\np q r s u v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nhq_nonzero : q ≠ 0\nhu : u ^ 3 - p * u ^ 2 - 4 * r * u + 4 * p * r - q ^ 2 = 0\nhs : s ^ 2 = u - p\nhv : v ^ 2 = 4 * s ^ 2 - 8 * (u - q / s)\nhw : w ^ 2 = 4 * s ^ 2 - 8 * (u + q / s)\nx : K\nhi2 : 2 ≠ 0\nh4 : 4 = 2 ^ 2\nhs_nonzero : s ≠ 0\n⊢ (2 * (x * x) + 2 * s * x + (u - q / s)) * (2 * (x * x) + -(2 * s) * x + (u + q / s)) = 0 ↔\n x = (-2 * s - v) / 4 ∨ x = (-2 * s + v) / 4 ∨ x = (2 * s - w) / 4 ∨ x = (2 * s + w) / 4",
"after_state": "case body\nK : Type u_1\ninst✝³ : Field K\np q r s u v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nhq_nonzero : q ≠ 0\nhu : u ^ 3 - p * u ^ 2 - 4 * r * u + 4 * p * r - q ^ 2 = 0\nhs : s ^ 2 = u - p\nhv : v ^ 2 = 4 * s ^ 2 - 8 * (u - q / s)\nhw : w ^ 2 = 4 * s ^ 2 - 8 * (u + q / s)\nx : K\nhi2 : 2 ≠ 0\nh4 : 4 = 2 ^ 2\nhs_nonzero : s ≠ 0\n⊢ discrim 2 (2 * s) (u - q / s) = v * v\n---\nK : Type u_1\ninst✝³ : Field K\np q r s u v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nhq_nonzero : q ≠ 0\nhu : u ^ 3 - p * u ^ 2 - 4 * r * u + 4 * p * r - q ^ 2 = 0\nhs : s ^ 2 = u - p\nhv : v ^ 2 = 4 * s ^ 2 - 8 * (u - q / s)\nhw : w ^ 2 = 4 * s ^ 2 - 8 * (u + q / s)\nx : K\nhi2 : 2 ≠ 0\nh4 : 4 = 2 ^ 2\nhs_nonzero : s ≠ 0\nhv' : discrim 2 (2 * s) (u - q / s) = v * v\n⊢ (2 * (x * x) + 2 * s * x + (u - q / s)) * (2 * (x * x) + -(2 * s) * x + (u + q / s)) = 0 ↔\n x = (-2 * s - v) / 4 ∨ x = (-2 * s + v) / 4 ∨ x = (2 * s - w) / 4 ∨ x = (2 * s + w) / 4"
},
{
"line": "case body✝ => with_annotate_state\"by\" (rw [discrim]; linear_combination -hv)",
"before_state": "case body\nK : Type u_1\ninst✝³ : Field K\np q r s u v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nhq_nonzero : q ≠ 0\nhu : u ^ 3 - p * u ^ 2 - 4 * r * u + 4 * p * r - q ^ 2 = 0\nhs : s ^ 2 = u - p\nhv : v ^ 2 = 4 * s ^ 2 - 8 * (u - q / s)\nhw : w ^ 2 = 4 * s ^ 2 - 8 * (u + q / s)\nx : K\nhi2 : 2 ≠ 0\nh4 : 4 = 2 ^ 2\nhs_nonzero : s ≠ 0\n⊢ discrim 2 (2 * s) (u - q / s) = v * v\n---\nK : Type u_1\ninst✝³ : Field K\np q r s u v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nhq_nonzero : q ≠ 0\nhu : u ^ 3 - p * u ^ 2 - 4 * r * u + 4 * p * r - q ^ 2 = 0\nhs : s ^ 2 = u - p\nhv : v ^ 2 = 4 * s ^ 2 - 8 * (u - q / s)\nhw : w ^ 2 = 4 * s ^ 2 - 8 * (u + q / s)\nx : K\nhi2 : 2 ≠ 0\nh4 : 4 = 2 ^ 2\nhs_nonzero : s ≠ 0\nhv' : discrim 2 (2 * s) (u - q / s) = v * v\n⊢ (2 * (x * x) + 2 * s * x + (u - q / s)) * (2 * (x * x) + -(2 * s) * x + (u + q / s)) = 0 ↔\n x = (-2 * s - v) / 4 ∨ x = (-2 * s + v) / 4 ∨ x = (2 * s - w) / 4 ∨ x = (2 * s + w) / 4",
"after_state": "K : Type u_1\ninst✝³ : Field K\np q r s u v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nhq_nonzero : q ≠ 0\nhu : u ^ 3 - p * u ^ 2 - 4 * r * u + 4 * p * r - q ^ 2 = 0\nhs : s ^ 2 = u - p\nhv : v ^ 2 = 4 * s ^ 2 - 8 * (u - q / s)\nhw : w ^ 2 = 4 * s ^ 2 - 8 * (u + q / s)\nx : K\nhi2 : 2 ≠ 0\nh4 : 4 = 2 ^ 2\nhs_nonzero : s ≠ 0\nhv' : discrim 2 (2 * s) (u - q / s) = v * v\n⊢ (2 * (x * x) + 2 * s * x + (u - q / s)) * (2 * (x * x) + -(2 * s) * x + (u + q / s)) = 0 ↔\n x = (-2 * s - v) / 4 ∨ x = (-2 * s + v) / 4 ∨ x = (2 * s - w) / 4 ∨ x = (2 * s + w) / 4"
},
{
"line": "with_annotate_state\"by\" (rw [discrim]; linear_combination -hv)",
"before_state": "K : Type u_1\ninst✝³ : Field K\np q r s u v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nhq_nonzero : q ≠ 0\nhu : u ^ 3 - p * u ^ 2 - 4 * r * u + 4 * p * r - q ^ 2 = 0\nhs : s ^ 2 = u - p\nhv : v ^ 2 = 4 * s ^ 2 - 8 * (u - q / s)\nhw : w ^ 2 = 4 * s ^ 2 - 8 * (u + q / s)\nx : K\nhi2 : 2 ≠ 0\nh4 : 4 = 2 ^ 2\nhs_nonzero : s ≠ 0\n⊢ discrim 2 (2 * s) (u - q / s) = v * v",
"after_state": "No Goals!"
},
{
"line": "rw [discrim]",
"before_state": "K : Type u_1\ninst✝³ : Field K\np q r s u v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nhq_nonzero : q ≠ 0\nhu : u ^ 3 - p * u ^ 2 - 4 * r * u + 4 * p * r - q ^ 2 = 0\nhs : s ^ 2 = u - p\nhv : v ^ 2 = 4 * s ^ 2 - 8 * (u - q / s)\nhw : w ^ 2 = 4 * s ^ 2 - 8 * (u + q / s)\nx : K\nhi2 : 2 ≠ 0\nh4 : 4 = 2 ^ 2\nhs_nonzero : s ≠ 0\n⊢ discrim 2 (2 * s) (u - q / s) = v * v",
"after_state": "K : Type u_1\ninst✝³ : Field K\np q r s u v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nhq_nonzero : q ≠ 0\nhu : u ^ 3 - p * u ^ 2 - 4 * r * u + 4 * p * r - q ^ 2 = 0\nhs : s ^ 2 = u - p\nhv : v ^ 2 = 4 * s ^ 2 - 8 * (u - q / s)\nhw : w ^ 2 = 4 * s ^ 2 - 8 * (u + q / s)\nx : K\nhi2 : 2 ≠ 0\nh4 : 4 = 2 ^ 2\nhs_nonzero : s ≠ 0\n⊢ (2 * s) ^ 2 - 4 * 2 * (u - q / s) = v * v"
},
{
"line": "rewrite [discrim]",
"before_state": "K : Type u_1\ninst✝³ : Field K\np q r s u v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nhq_nonzero : q ≠ 0\nhu : u ^ 3 - p * u ^ 2 - 4 * r * u + 4 * p * r - q ^ 2 = 0\nhs : s ^ 2 = u - p\nhv : v ^ 2 = 4 * s ^ 2 - 8 * (u - q / s)\nhw : w ^ 2 = 4 * s ^ 2 - 8 * (u + q / s)\nx : K\nhi2 : 2 ≠ 0\nh4 : 4 = 2 ^ 2\nhs_nonzero : s ≠ 0\n⊢ discrim 2 (2 * s) (u - q / s) = v * v",
"after_state": "K : Type u_1\ninst✝³ : Field K\np q r s u v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nhq_nonzero : q ≠ 0\nhu : u ^ 3 - p * u ^ 2 - 4 * r * u + 4 * p * r - q ^ 2 = 0\nhs : s ^ 2 = u - p\nhv : v ^ 2 = 4 * s ^ 2 - 8 * (u - q / s)\nhw : w ^ 2 = 4 * s ^ 2 - 8 * (u + q / s)\nx : K\nhi2 : 2 ≠ 0\nh4 : 4 = 2 ^ 2\nhs_nonzero : s ≠ 0\n⊢ (2 * s) ^ 2 - 4 * 2 * (u - q / s) = v * v"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "K : Type u_1\ninst✝³ : Field K\np q r s u v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nhq_nonzero : q ≠ 0\nhu : u ^ 3 - p * u ^ 2 - 4 * r * u + 4 * p * r - q ^ 2 = 0\nhs : s ^ 2 = u - p\nhv : v ^ 2 = 4 * s ^ 2 - 8 * (u - q / s)\nhw : w ^ 2 = 4 * s ^ 2 - 8 * (u + q / s)\nx : K\nhi2 : 2 ≠ 0\nh4 : 4 = 2 ^ 2\nhs_nonzero : s ≠ 0\n⊢ (2 * s) ^ 2 - 4 * 2 * (u - q / s) = v * v",
"after_state": "K : Type u_1\ninst✝³ : Field K\np q r s u v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nhq_nonzero : q ≠ 0\nhu : u ^ 3 - p * u ^ 2 - 4 * r * u + 4 * p * r - q ^ 2 = 0\nhs : s ^ 2 = u - p\nhv : v ^ 2 = 4 * s ^ 2 - 8 * (u - q / s)\nhw : w ^ 2 = 4 * s ^ 2 - 8 * (u + q / s)\nx : K\nhi2 : 2 ≠ 0\nh4 : 4 = 2 ^ 2\nhs_nonzero : s ≠ 0\n⊢ (2 * s) ^ 2 - 4 * 2 * (u - q / s) = v * v"
},
{
"line": "try (with_reducible rfl)",
"before_state": "K : Type u_1\ninst✝³ : Field K\np q r s u v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nhq_nonzero : q ≠ 0\nhu : u ^ 3 - p * u ^ 2 - 4 * r * u + 4 * p * r - q ^ 2 = 0\nhs : s ^ 2 = u - p\nhv : v ^ 2 = 4 * s ^ 2 - 8 * (u - q / s)\nhw : w ^ 2 = 4 * s ^ 2 - 8 * (u + q / s)\nx : K\nhi2 : 2 ≠ 0\nh4 : 4 = 2 ^ 2\nhs_nonzero : s ≠ 0\n⊢ (2 * s) ^ 2 - 4 * 2 * (u - q / s) = v * v",
"after_state": "K : Type u_1\ninst✝³ : Field K\np q r s u v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nhq_nonzero : q ≠ 0\nhu : u ^ 3 - p * u ^ 2 - 4 * r * u + 4 * p * r - q ^ 2 = 0\nhs : s ^ 2 = u - p\nhv : v ^ 2 = 4 * s ^ 2 - 8 * (u - q / s)\nhw : w ^ 2 = 4 * s ^ 2 - 8 * (u + q / s)\nx : K\nhi2 : 2 ≠ 0\nh4 : 4 = 2 ^ 2\nhs_nonzero : s ≠ 0\n⊢ (2 * s) ^ 2 - 4 * 2 * (u - q / s) = v * v"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "K : Type u_1\ninst✝³ : Field K\np q r s u v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nhq_nonzero : q ≠ 0\nhu : u ^ 3 - p * u ^ 2 - 4 * r * u + 4 * p * r - q ^ 2 = 0\nhs : s ^ 2 = u - p\nhv : v ^ 2 = 4 * s ^ 2 - 8 * (u - q / s)\nhw : w ^ 2 = 4 * s ^ 2 - 8 * (u + q / s)\nx : K\nhi2 : 2 ≠ 0\nh4 : 4 = 2 ^ 2\nhs_nonzero : s ≠ 0\n⊢ (2 * s) ^ 2 - 4 * 2 * (u - q / s) = v * v",
"after_state": "K : Type u_1\ninst✝³ : Field K\np q r s u v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nhq_nonzero : q ≠ 0\nhu : u ^ 3 - p * u ^ 2 - 4 * r * u + 4 * p * r - q ^ 2 = 0\nhs : s ^ 2 = u - p\nhv : v ^ 2 = 4 * s ^ 2 - 8 * (u - q / s)\nhw : w ^ 2 = 4 * s ^ 2 - 8 * (u + q / s)\nx : K\nhi2 : 2 ≠ 0\nh4 : 4 = 2 ^ 2\nhs_nonzero : s ≠ 0\n⊢ (2 * s) ^ 2 - 4 * 2 * (u - q / s) = v * v"
},
{
"line": "with_reducible rfl",
"before_state": "K : Type u_1\ninst✝³ : Field K\np q r s u v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nhq_nonzero : q ≠ 0\nhu : u ^ 3 - p * u ^ 2 - 4 * r * u + 4 * p * r - q ^ 2 = 0\nhs : s ^ 2 = u - p\nhv : v ^ 2 = 4 * s ^ 2 - 8 * (u - q / s)\nhw : w ^ 2 = 4 * s ^ 2 - 8 * (u + q / s)\nx : K\nhi2 : 2 ≠ 0\nh4 : 4 = 2 ^ 2\nhs_nonzero : s ≠ 0\n⊢ (2 * s) ^ 2 - 4 * 2 * (u - q / s) = v * v",
"after_state": "K : Type u_1\ninst✝³ : Field K\np q r s u v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nhq_nonzero : q ≠ 0\nhu : u ^ 3 - p * u ^ 2 - 4 * r * u + 4 * p * r - q ^ 2 = 0\nhs : s ^ 2 = u - p\nhv : v ^ 2 = 4 * s ^ 2 - 8 * (u - q / s)\nhw : w ^ 2 = 4 * s ^ 2 - 8 * (u + q / s)\nx : K\nhi2 : 2 ≠ 0\nh4 : 4 = 2 ^ 2\nhs_nonzero : s ≠ 0\n⊢ (2 * s) ^ 2 - 4 * 2 * (u - q / s) = v * v"
},
{
"line": "rfl",
"before_state": "K : Type u_1\ninst✝³ : Field K\np q r s u v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nhq_nonzero : q ≠ 0\nhu : u ^ 3 - p * u ^ 2 - 4 * r * u + 4 * p * r - q ^ 2 = 0\nhs : s ^ 2 = u - p\nhv : v ^ 2 = 4 * s ^ 2 - 8 * (u - q / s)\nhw : w ^ 2 = 4 * s ^ 2 - 8 * (u + q / s)\nx : K\nhi2 : 2 ≠ 0\nh4 : 4 = 2 ^ 2\nhs_nonzero : s ≠ 0\n⊢ (2 * s) ^ 2 - 4 * 2 * (u - q / s) = v * v",
"after_state": "K : Type u_1\ninst✝³ : Field K\np q r s u v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nhq_nonzero : q ≠ 0\nhu : u ^ 3 - p * u ^ 2 - 4 * r * u + 4 * p * r - q ^ 2 = 0\nhs : s ^ 2 = u - p\nhv : v ^ 2 = 4 * s ^ 2 - 8 * (u - q / s)\nhw : w ^ 2 = 4 * s ^ 2 - 8 * (u + q / s)\nx : K\nhi2 : 2 ≠ 0\nh4 : 4 = 2 ^ 2\nhs_nonzero : s ≠ 0\n⊢ (2 * s) ^ 2 - 4 * 2 * (u - q / s) = v * v"
},
{
"line": "apply_rfl",
"before_state": "K : Type u_1\ninst✝³ : Field K\np q r s u v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nhq_nonzero : q ≠ 0\nhu : u ^ 3 - p * u ^ 2 - 4 * r * u + 4 * p * r - q ^ 2 = 0\nhs : s ^ 2 = u - p\nhv : v ^ 2 = 4 * s ^ 2 - 8 * (u - q / s)\nhw : w ^ 2 = 4 * s ^ 2 - 8 * (u + q / s)\nx : K\nhi2 : 2 ≠ 0\nh4 : 4 = 2 ^ 2\nhs_nonzero : s ≠ 0\n⊢ (2 * s) ^ 2 - 4 * 2 * (u - q / s) = v * v",
"after_state": "K : Type u_1\ninst✝³ : Field K\np q r s u v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nhq_nonzero : q ≠ 0\nhu : u ^ 3 - p * u ^ 2 - 4 * r * u + 4 * p * r - q ^ 2 = 0\nhs : s ^ 2 = u - p\nhv : v ^ 2 = 4 * s ^ 2 - 8 * (u - q / s)\nhw : w ^ 2 = 4 * s ^ 2 - 8 * (u + q / s)\nx : K\nhi2 : 2 ≠ 0\nh4 : 4 = 2 ^ 2\nhs_nonzero : s ≠ 0\n⊢ (2 * s) ^ 2 - 4 * 2 * (u - q / s) = v * v"
},
{
"line": "skip",
"before_state": "K : Type u_1\ninst✝³ : Field K\np q r s u v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nhq_nonzero : q ≠ 0\nhu : u ^ 3 - p * u ^ 2 - 4 * r * u + 4 * p * r - q ^ 2 = 0\nhs : s ^ 2 = u - p\nhv : v ^ 2 = 4 * s ^ 2 - 8 * (u - q / s)\nhw : w ^ 2 = 4 * s ^ 2 - 8 * (u + q / s)\nx : K\nhi2 : 2 ≠ 0\nh4 : 4 = 2 ^ 2\nhs_nonzero : s ≠ 0\n⊢ (2 * s) ^ 2 - 4 * 2 * (u - q / s) = v * v",
"after_state": "K : Type u_1\ninst✝³ : Field K\np q r s u v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nhq_nonzero : q ≠ 0\nhu : u ^ 3 - p * u ^ 2 - 4 * r * u + 4 * p * r - q ^ 2 = 0\nhs : s ^ 2 = u - p\nhv : v ^ 2 = 4 * s ^ 2 - 8 * (u - q / s)\nhw : w ^ 2 = 4 * s ^ 2 - 8 * (u + q / s)\nx : K\nhi2 : 2 ≠ 0\nh4 : 4 = 2 ^ 2\nhs_nonzero : s ≠ 0\n⊢ (2 * s) ^ 2 - 4 * 2 * (u - q / s) = v * v"
},
{
"line": "linear_combination -hv",
"before_state": "K : Type u_1\ninst✝³ : Field K\np q r s u v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nhq_nonzero : q ≠ 0\nhu : u ^ 3 - p * u ^ 2 - 4 * r * u + 4 * p * r - q ^ 2 = 0\nhs : s ^ 2 = u - p\nhv : v ^ 2 = 4 * s ^ 2 - 8 * (u - q / s)\nhw : w ^ 2 = 4 * s ^ 2 - 8 * (u + q / s)\nx : K\nhi2 : 2 ≠ 0\nh4 : 4 = 2 ^ 2\nhs_nonzero : s ≠ 0\n⊢ (2 * s) ^ 2 - 4 * 2 * (u - q / s) = v * v",
"after_state": "No Goals!"
},
{
"line": "have hw' : discrim 2 (-(2 * s)) (u + q / s) = w * w := by rw [discrim]; linear_combination -hw",
"before_state": "K : Type u_1\ninst✝³ : Field K\np q r s u v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nhq_nonzero : q ≠ 0\nhu : u ^ 3 - p * u ^ 2 - 4 * r * u + 4 * p * r - q ^ 2 = 0\nhs : s ^ 2 = u - p\nhv : v ^ 2 = 4 * s ^ 2 - 8 * (u - q / s)\nhw : w ^ 2 = 4 * s ^ 2 - 8 * (u + q / s)\nx : K\nhi2 : 2 ≠ 0\nh4 : 4 = 2 ^ 2\nhs_nonzero : s ≠ 0\nhv' : discrim 2 (2 * s) (u - q / s) = v * v\n⊢ (2 * (x * x) + 2 * s * x + (u - q / s)) * (2 * (x * x) + -(2 * s) * x + (u + q / s)) = 0 ↔\n x = (-2 * s - v) / 4 ∨ x = (-2 * s + v) / 4 ∨ x = (2 * s - w) / 4 ∨ x = (2 * s + w) / 4",
"after_state": "K : Type u_1\ninst✝³ : Field K\np q r s u v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nhq_nonzero : q ≠ 0\nhu : u ^ 3 - p * u ^ 2 - 4 * r * u + 4 * p * r - q ^ 2 = 0\nhs : s ^ 2 = u - p\nhv : v ^ 2 = 4 * s ^ 2 - 8 * (u - q / s)\nhw : w ^ 2 = 4 * s ^ 2 - 8 * (u + q / s)\nx : K\nhi2 : 2 ≠ 0\nh4 : 4 = 2 ^ 2\nhs_nonzero : s ≠ 0\nhv' : discrim 2 (2 * s) (u - q / s) = v * v\nhw' : discrim 2 (-(2 * s)) (u + q / s) = w * w\n⊢ (2 * (x * x) + 2 * s * x + (u - q / s)) * (2 * (x * x) + -(2 * s) * x + (u + q / s)) = 0 ↔\n x = (-2 * s - v) / 4 ∨ x = (-2 * s + v) / 4 ∨ x = (2 * s - w) / 4 ∨ x = (2 * s + w) / 4"
},
{
"line": "focus\n refine\n no_implicit_lambda%\n (have hw' : discrim 2 (-(2 * s)) (u + q / s) = w * w := ?body✝;\n ?_)\n case body✝ => with_annotate_state\"by\" (rw [discrim]; linear_combination -hw)",
"before_state": "K : Type u_1\ninst✝³ : Field K\np q r s u v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nhq_nonzero : q ≠ 0\nhu : u ^ 3 - p * u ^ 2 - 4 * r * u + 4 * p * r - q ^ 2 = 0\nhs : s ^ 2 = u - p\nhv : v ^ 2 = 4 * s ^ 2 - 8 * (u - q / s)\nhw : w ^ 2 = 4 * s ^ 2 - 8 * (u + q / s)\nx : K\nhi2 : 2 ≠ 0\nh4 : 4 = 2 ^ 2\nhs_nonzero : s ≠ 0\nhv' : discrim 2 (2 * s) (u - q / s) = v * v\n⊢ (2 * (x * x) + 2 * s * x + (u - q / s)) * (2 * (x * x) + -(2 * s) * x + (u + q / s)) = 0 ↔\n x = (-2 * s - v) / 4 ∨ x = (-2 * s + v) / 4 ∨ x = (2 * s - w) / 4 ∨ x = (2 * s + w) / 4",
"after_state": "K : Type u_1\ninst✝³ : Field K\np q r s u v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nhq_nonzero : q ≠ 0\nhu : u ^ 3 - p * u ^ 2 - 4 * r * u + 4 * p * r - q ^ 2 = 0\nhs : s ^ 2 = u - p\nhv : v ^ 2 = 4 * s ^ 2 - 8 * (u - q / s)\nhw : w ^ 2 = 4 * s ^ 2 - 8 * (u + q / s)\nx : K\nhi2 : 2 ≠ 0\nh4 : 4 = 2 ^ 2\nhs_nonzero : s ≠ 0\nhv' : discrim 2 (2 * s) (u - q / s) = v * v\nhw' : discrim 2 (-(2 * s)) (u + q / s) = w * w\n⊢ (2 * (x * x) + 2 * s * x + (u - q / s)) * (2 * (x * x) + -(2 * s) * x + (u + q / s)) = 0 ↔\n x = (-2 * s - v) / 4 ∨ x = (-2 * s + v) / 4 ∨ x = (2 * s - w) / 4 ∨ x = (2 * s + w) / 4"
},
{
"line": "refine\n no_implicit_lambda%\n (have hw' : discrim 2 (-(2 * s)) (u + q / s) = w * w := ?body✝;\n ?_)",
"before_state": "K : Type u_1\ninst✝³ : Field K\np q r s u v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nhq_nonzero : q ≠ 0\nhu : u ^ 3 - p * u ^ 2 - 4 * r * u + 4 * p * r - q ^ 2 = 0\nhs : s ^ 2 = u - p\nhv : v ^ 2 = 4 * s ^ 2 - 8 * (u - q / s)\nhw : w ^ 2 = 4 * s ^ 2 - 8 * (u + q / s)\nx : K\nhi2 : 2 ≠ 0\nh4 : 4 = 2 ^ 2\nhs_nonzero : s ≠ 0\nhv' : discrim 2 (2 * s) (u - q / s) = v * v\n⊢ (2 * (x * x) + 2 * s * x + (u - q / s)) * (2 * (x * x) + -(2 * s) * x + (u + q / s)) = 0 ↔\n x = (-2 * s - v) / 4 ∨ x = (-2 * s + v) / 4 ∨ x = (2 * s - w) / 4 ∨ x = (2 * s + w) / 4",
"after_state": "case body\nK : Type u_1\ninst✝³ : Field K\np q r s u v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nhq_nonzero : q ≠ 0\nhu : u ^ 3 - p * u ^ 2 - 4 * r * u + 4 * p * r - q ^ 2 = 0\nhs : s ^ 2 = u - p\nhv : v ^ 2 = 4 * s ^ 2 - 8 * (u - q / s)\nhw : w ^ 2 = 4 * s ^ 2 - 8 * (u + q / s)\nx : K\nhi2 : 2 ≠ 0\nh4 : 4 = 2 ^ 2\nhs_nonzero : s ≠ 0\nhv' : discrim 2 (2 * s) (u - q / s) = v * v\n⊢ discrim 2 (-(2 * s)) (u + q / s) = w * w\n---\nK : Type u_1\ninst✝³ : Field K\np q r s u v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nhq_nonzero : q ≠ 0\nhu : u ^ 3 - p * u ^ 2 - 4 * r * u + 4 * p * r - q ^ 2 = 0\nhs : s ^ 2 = u - p\nhv : v ^ 2 = 4 * s ^ 2 - 8 * (u - q / s)\nhw : w ^ 2 = 4 * s ^ 2 - 8 * (u + q / s)\nx : K\nhi2 : 2 ≠ 0\nh4 : 4 = 2 ^ 2\nhs_nonzero : s ≠ 0\nhv' : discrim 2 (2 * s) (u - q / s) = v * v\nhw' : discrim 2 (-(2 * s)) (u + q / s) = w * w\n⊢ (2 * (x * x) + 2 * s * x + (u - q / s)) * (2 * (x * x) + -(2 * s) * x + (u + q / s)) = 0 ↔\n x = (-2 * s - v) / 4 ∨ x = (-2 * s + v) / 4 ∨ x = (2 * s - w) / 4 ∨ x = (2 * s + w) / 4"
},
{
"line": "case body✝ => with_annotate_state\"by\" (rw [discrim]; linear_combination -hw)",
"before_state": "case body\nK : Type u_1\ninst✝³ : Field K\np q r s u v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nhq_nonzero : q ≠ 0\nhu : u ^ 3 - p * u ^ 2 - 4 * r * u + 4 * p * r - q ^ 2 = 0\nhs : s ^ 2 = u - p\nhv : v ^ 2 = 4 * s ^ 2 - 8 * (u - q / s)\nhw : w ^ 2 = 4 * s ^ 2 - 8 * (u + q / s)\nx : K\nhi2 : 2 ≠ 0\nh4 : 4 = 2 ^ 2\nhs_nonzero : s ≠ 0\nhv' : discrim 2 (2 * s) (u - q / s) = v * v\n⊢ discrim 2 (-(2 * s)) (u + q / s) = w * w\n---\nK : Type u_1\ninst✝³ : Field K\np q r s u v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nhq_nonzero : q ≠ 0\nhu : u ^ 3 - p * u ^ 2 - 4 * r * u + 4 * p * r - q ^ 2 = 0\nhs : s ^ 2 = u - p\nhv : v ^ 2 = 4 * s ^ 2 - 8 * (u - q / s)\nhw : w ^ 2 = 4 * s ^ 2 - 8 * (u + q / s)\nx : K\nhi2 : 2 ≠ 0\nh4 : 4 = 2 ^ 2\nhs_nonzero : s ≠ 0\nhv' : discrim 2 (2 * s) (u - q / s) = v * v\nhw' : discrim 2 (-(2 * s)) (u + q / s) = w * w\n⊢ (2 * (x * x) + 2 * s * x + (u - q / s)) * (2 * (x * x) + -(2 * s) * x + (u + q / s)) = 0 ↔\n x = (-2 * s - v) / 4 ∨ x = (-2 * s + v) / 4 ∨ x = (2 * s - w) / 4 ∨ x = (2 * s + w) / 4",
"after_state": "K : Type u_1\ninst✝³ : Field K\np q r s u v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nhq_nonzero : q ≠ 0\nhu : u ^ 3 - p * u ^ 2 - 4 * r * u + 4 * p * r - q ^ 2 = 0\nhs : s ^ 2 = u - p\nhv : v ^ 2 = 4 * s ^ 2 - 8 * (u - q / s)\nhw : w ^ 2 = 4 * s ^ 2 - 8 * (u + q / s)\nx : K\nhi2 : 2 ≠ 0\nh4 : 4 = 2 ^ 2\nhs_nonzero : s ≠ 0\nhv' : discrim 2 (2 * s) (u - q / s) = v * v\nhw' : discrim 2 (-(2 * s)) (u + q / s) = w * w\n⊢ (2 * (x * x) + 2 * s * x + (u - q / s)) * (2 * (x * x) + -(2 * s) * x + (u + q / s)) = 0 ↔\n x = (-2 * s - v) / 4 ∨ x = (-2 * s + v) / 4 ∨ x = (2 * s - w) / 4 ∨ x = (2 * s + w) / 4"
},
{
"line": "with_annotate_state\"by\" (rw [discrim]; linear_combination -hw)",
"before_state": "K : Type u_1\ninst✝³ : Field K\np q r s u v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nhq_nonzero : q ≠ 0\nhu : u ^ 3 - p * u ^ 2 - 4 * r * u + 4 * p * r - q ^ 2 = 0\nhs : s ^ 2 = u - p\nhv : v ^ 2 = 4 * s ^ 2 - 8 * (u - q / s)\nhw : w ^ 2 = 4 * s ^ 2 - 8 * (u + q / s)\nx : K\nhi2 : 2 ≠ 0\nh4 : 4 = 2 ^ 2\nhs_nonzero : s ≠ 0\nhv' : discrim 2 (2 * s) (u - q / s) = v * v\n⊢ discrim 2 (-(2 * s)) (u + q / s) = w * w",
"after_state": "No Goals!"
},
{
"line": "rw [discrim]",
"before_state": "K : Type u_1\ninst✝³ : Field K\np q r s u v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nhq_nonzero : q ≠ 0\nhu : u ^ 3 - p * u ^ 2 - 4 * r * u + 4 * p * r - q ^ 2 = 0\nhs : s ^ 2 = u - p\nhv : v ^ 2 = 4 * s ^ 2 - 8 * (u - q / s)\nhw : w ^ 2 = 4 * s ^ 2 - 8 * (u + q / s)\nx : K\nhi2 : 2 ≠ 0\nh4 : 4 = 2 ^ 2\nhs_nonzero : s ≠ 0\nhv' : discrim 2 (2 * s) (u - q / s) = v * v\n⊢ discrim 2 (-(2 * s)) (u + q / s) = w * w",
"after_state": "K : Type u_1\ninst✝³ : Field K\np q r s u v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nhq_nonzero : q ≠ 0\nhu : u ^ 3 - p * u ^ 2 - 4 * r * u + 4 * p * r - q ^ 2 = 0\nhs : s ^ 2 = u - p\nhv : v ^ 2 = 4 * s ^ 2 - 8 * (u - q / s)\nhw : w ^ 2 = 4 * s ^ 2 - 8 * (u + q / s)\nx : K\nhi2 : 2 ≠ 0\nh4 : 4 = 2 ^ 2\nhs_nonzero : s ≠ 0\nhv' : discrim 2 (2 * s) (u - q / s) = v * v\n⊢ (-(2 * s)) ^ 2 - 4 * 2 * (u + q / s) = w * w"
},
{
"line": "rewrite [discrim]",
"before_state": "K : Type u_1\ninst✝³ : Field K\np q r s u v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nhq_nonzero : q ≠ 0\nhu : u ^ 3 - p * u ^ 2 - 4 * r * u + 4 * p * r - q ^ 2 = 0\nhs : s ^ 2 = u - p\nhv : v ^ 2 = 4 * s ^ 2 - 8 * (u - q / s)\nhw : w ^ 2 = 4 * s ^ 2 - 8 * (u + q / s)\nx : K\nhi2 : 2 ≠ 0\nh4 : 4 = 2 ^ 2\nhs_nonzero : s ≠ 0\nhv' : discrim 2 (2 * s) (u - q / s) = v * v\n⊢ discrim 2 (-(2 * s)) (u + q / s) = w * w",
"after_state": "K : Type u_1\ninst✝³ : Field K\np q r s u v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nhq_nonzero : q ≠ 0\nhu : u ^ 3 - p * u ^ 2 - 4 * r * u + 4 * p * r - q ^ 2 = 0\nhs : s ^ 2 = u - p\nhv : v ^ 2 = 4 * s ^ 2 - 8 * (u - q / s)\nhw : w ^ 2 = 4 * s ^ 2 - 8 * (u + q / s)\nx : K\nhi2 : 2 ≠ 0\nh4 : 4 = 2 ^ 2\nhs_nonzero : s ≠ 0\nhv' : discrim 2 (2 * s) (u - q / s) = v * v\n⊢ (-(2 * s)) ^ 2 - 4 * 2 * (u + q / s) = w * w"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "K : Type u_1\ninst✝³ : Field K\np q r s u v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nhq_nonzero : q ≠ 0\nhu : u ^ 3 - p * u ^ 2 - 4 * r * u + 4 * p * r - q ^ 2 = 0\nhs : s ^ 2 = u - p\nhv : v ^ 2 = 4 * s ^ 2 - 8 * (u - q / s)\nhw : w ^ 2 = 4 * s ^ 2 - 8 * (u + q / s)\nx : K\nhi2 : 2 ≠ 0\nh4 : 4 = 2 ^ 2\nhs_nonzero : s ≠ 0\nhv' : discrim 2 (2 * s) (u - q / s) = v * v\n⊢ (-(2 * s)) ^ 2 - 4 * 2 * (u + q / s) = w * w",
"after_state": "K : Type u_1\ninst✝³ : Field K\np q r s u v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nhq_nonzero : q ≠ 0\nhu : u ^ 3 - p * u ^ 2 - 4 * r * u + 4 * p * r - q ^ 2 = 0\nhs : s ^ 2 = u - p\nhv : v ^ 2 = 4 * s ^ 2 - 8 * (u - q / s)\nhw : w ^ 2 = 4 * s ^ 2 - 8 * (u + q / s)\nx : K\nhi2 : 2 ≠ 0\nh4 : 4 = 2 ^ 2\nhs_nonzero : s ≠ 0\nhv' : discrim 2 (2 * s) (u - q / s) = v * v\n⊢ (-(2 * s)) ^ 2 - 4 * 2 * (u + q / s) = w * w"
},
{
"line": "try (with_reducible rfl)",
"before_state": "K : Type u_1\ninst✝³ : Field K\np q r s u v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nhq_nonzero : q ≠ 0\nhu : u ^ 3 - p * u ^ 2 - 4 * r * u + 4 * p * r - q ^ 2 = 0\nhs : s ^ 2 = u - p\nhv : v ^ 2 = 4 * s ^ 2 - 8 * (u - q / s)\nhw : w ^ 2 = 4 * s ^ 2 - 8 * (u + q / s)\nx : K\nhi2 : 2 ≠ 0\nh4 : 4 = 2 ^ 2\nhs_nonzero : s ≠ 0\nhv' : discrim 2 (2 * s) (u - q / s) = v * v\n⊢ (-(2 * s)) ^ 2 - 4 * 2 * (u + q / s) = w * w",
"after_state": "K : Type u_1\ninst✝³ : Field K\np q r s u v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nhq_nonzero : q ≠ 0\nhu : u ^ 3 - p * u ^ 2 - 4 * r * u + 4 * p * r - q ^ 2 = 0\nhs : s ^ 2 = u - p\nhv : v ^ 2 = 4 * s ^ 2 - 8 * (u - q / s)\nhw : w ^ 2 = 4 * s ^ 2 - 8 * (u + q / s)\nx : K\nhi2 : 2 ≠ 0\nh4 : 4 = 2 ^ 2\nhs_nonzero : s ≠ 0\nhv' : discrim 2 (2 * s) (u - q / s) = v * v\n⊢ (-(2 * s)) ^ 2 - 4 * 2 * (u + q / s) = w * w"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "K : Type u_1\ninst✝³ : Field K\np q r s u v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nhq_nonzero : q ≠ 0\nhu : u ^ 3 - p * u ^ 2 - 4 * r * u + 4 * p * r - q ^ 2 = 0\nhs : s ^ 2 = u - p\nhv : v ^ 2 = 4 * s ^ 2 - 8 * (u - q / s)\nhw : w ^ 2 = 4 * s ^ 2 - 8 * (u + q / s)\nx : K\nhi2 : 2 ≠ 0\nh4 : 4 = 2 ^ 2\nhs_nonzero : s ≠ 0\nhv' : discrim 2 (2 * s) (u - q / s) = v * v\n⊢ (-(2 * s)) ^ 2 - 4 * 2 * (u + q / s) = w * w",
"after_state": "K : Type u_1\ninst✝³ : Field K\np q r s u v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nhq_nonzero : q ≠ 0\nhu : u ^ 3 - p * u ^ 2 - 4 * r * u + 4 * p * r - q ^ 2 = 0\nhs : s ^ 2 = u - p\nhv : v ^ 2 = 4 * s ^ 2 - 8 * (u - q / s)\nhw : w ^ 2 = 4 * s ^ 2 - 8 * (u + q / s)\nx : K\nhi2 : 2 ≠ 0\nh4 : 4 = 2 ^ 2\nhs_nonzero : s ≠ 0\nhv' : discrim 2 (2 * s) (u - q / s) = v * v\n⊢ (-(2 * s)) ^ 2 - 4 * 2 * (u + q / s) = w * w"
},
{
"line": "with_reducible rfl",
"before_state": "K : Type u_1\ninst✝³ : Field K\np q r s u v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nhq_nonzero : q ≠ 0\nhu : u ^ 3 - p * u ^ 2 - 4 * r * u + 4 * p * r - q ^ 2 = 0\nhs : s ^ 2 = u - p\nhv : v ^ 2 = 4 * s ^ 2 - 8 * (u - q / s)\nhw : w ^ 2 = 4 * s ^ 2 - 8 * (u + q / s)\nx : K\nhi2 : 2 ≠ 0\nh4 : 4 = 2 ^ 2\nhs_nonzero : s ≠ 0\nhv' : discrim 2 (2 * s) (u - q / s) = v * v\n⊢ (-(2 * s)) ^ 2 - 4 * 2 * (u + q / s) = w * w",
"after_state": "K : Type u_1\ninst✝³ : Field K\np q r s u v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nhq_nonzero : q ≠ 0\nhu : u ^ 3 - p * u ^ 2 - 4 * r * u + 4 * p * r - q ^ 2 = 0\nhs : s ^ 2 = u - p\nhv : v ^ 2 = 4 * s ^ 2 - 8 * (u - q / s)\nhw : w ^ 2 = 4 * s ^ 2 - 8 * (u + q / s)\nx : K\nhi2 : 2 ≠ 0\nh4 : 4 = 2 ^ 2\nhs_nonzero : s ≠ 0\nhv' : discrim 2 (2 * s) (u - q / s) = v * v\n⊢ (-(2 * s)) ^ 2 - 4 * 2 * (u + q / s) = w * w"
},
{
"line": "rfl",
"before_state": "K : Type u_1\ninst✝³ : Field K\np q r s u v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nhq_nonzero : q ≠ 0\nhu : u ^ 3 - p * u ^ 2 - 4 * r * u + 4 * p * r - q ^ 2 = 0\nhs : s ^ 2 = u - p\nhv : v ^ 2 = 4 * s ^ 2 - 8 * (u - q / s)\nhw : w ^ 2 = 4 * s ^ 2 - 8 * (u + q / s)\nx : K\nhi2 : 2 ≠ 0\nh4 : 4 = 2 ^ 2\nhs_nonzero : s ≠ 0\nhv' : discrim 2 (2 * s) (u - q / s) = v * v\n⊢ (-(2 * s)) ^ 2 - 4 * 2 * (u + q / s) = w * w",
"after_state": "K : Type u_1\ninst✝³ : Field K\np q r s u v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nhq_nonzero : q ≠ 0\nhu : u ^ 3 - p * u ^ 2 - 4 * r * u + 4 * p * r - q ^ 2 = 0\nhs : s ^ 2 = u - p\nhv : v ^ 2 = 4 * s ^ 2 - 8 * (u - q / s)\nhw : w ^ 2 = 4 * s ^ 2 - 8 * (u + q / s)\nx : K\nhi2 : 2 ≠ 0\nh4 : 4 = 2 ^ 2\nhs_nonzero : s ≠ 0\nhv' : discrim 2 (2 * s) (u - q / s) = v * v\n⊢ (-(2 * s)) ^ 2 - 4 * 2 * (u + q / s) = w * w"
},
{
"line": "apply_rfl",
"before_state": "K : Type u_1\ninst✝³ : Field K\np q r s u v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nhq_nonzero : q ≠ 0\nhu : u ^ 3 - p * u ^ 2 - 4 * r * u + 4 * p * r - q ^ 2 = 0\nhs : s ^ 2 = u - p\nhv : v ^ 2 = 4 * s ^ 2 - 8 * (u - q / s)\nhw : w ^ 2 = 4 * s ^ 2 - 8 * (u + q / s)\nx : K\nhi2 : 2 ≠ 0\nh4 : 4 = 2 ^ 2\nhs_nonzero : s ≠ 0\nhv' : discrim 2 (2 * s) (u - q / s) = v * v\n⊢ (-(2 * s)) ^ 2 - 4 * 2 * (u + q / s) = w * w",
"after_state": "K : Type u_1\ninst✝³ : Field K\np q r s u v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nhq_nonzero : q ≠ 0\nhu : u ^ 3 - p * u ^ 2 - 4 * r * u + 4 * p * r - q ^ 2 = 0\nhs : s ^ 2 = u - p\nhv : v ^ 2 = 4 * s ^ 2 - 8 * (u - q / s)\nhw : w ^ 2 = 4 * s ^ 2 - 8 * (u + q / s)\nx : K\nhi2 : 2 ≠ 0\nh4 : 4 = 2 ^ 2\nhs_nonzero : s ≠ 0\nhv' : discrim 2 (2 * s) (u - q / s) = v * v\n⊢ (-(2 * s)) ^ 2 - 4 * 2 * (u + q / s) = w * w"
},
{
"line": "skip",
"before_state": "K : Type u_1\ninst✝³ : Field K\np q r s u v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nhq_nonzero : q ≠ 0\nhu : u ^ 3 - p * u ^ 2 - 4 * r * u + 4 * p * r - q ^ 2 = 0\nhs : s ^ 2 = u - p\nhv : v ^ 2 = 4 * s ^ 2 - 8 * (u - q / s)\nhw : w ^ 2 = 4 * s ^ 2 - 8 * (u + q / s)\nx : K\nhi2 : 2 ≠ 0\nh4 : 4 = 2 ^ 2\nhs_nonzero : s ≠ 0\nhv' : discrim 2 (2 * s) (u - q / s) = v * v\n⊢ (-(2 * s)) ^ 2 - 4 * 2 * (u + q / s) = w * w",
"after_state": "K : Type u_1\ninst✝³ : Field K\np q r s u v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nhq_nonzero : q ≠ 0\nhu : u ^ 3 - p * u ^ 2 - 4 * r * u + 4 * p * r - q ^ 2 = 0\nhs : s ^ 2 = u - p\nhv : v ^ 2 = 4 * s ^ 2 - 8 * (u - q / s)\nhw : w ^ 2 = 4 * s ^ 2 - 8 * (u + q / s)\nx : K\nhi2 : 2 ≠ 0\nh4 : 4 = 2 ^ 2\nhs_nonzero : s ≠ 0\nhv' : discrim 2 (2 * s) (u - q / s) = v * v\n⊢ (-(2 * s)) ^ 2 - 4 * 2 * (u + q / s) = w * w"
},
{
"line": "linear_combination -hw",
"before_state": "K : Type u_1\ninst✝³ : Field K\np q r s u v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nhq_nonzero : q ≠ 0\nhu : u ^ 3 - p * u ^ 2 - 4 * r * u + 4 * p * r - q ^ 2 = 0\nhs : s ^ 2 = u - p\nhv : v ^ 2 = 4 * s ^ 2 - 8 * (u - q / s)\nhw : w ^ 2 = 4 * s ^ 2 - 8 * (u + q / s)\nx : K\nhi2 : 2 ≠ 0\nh4 : 4 = 2 ^ 2\nhs_nonzero : s ≠ 0\nhv' : discrim 2 (2 * s) (u - q / s) = v * v\n⊢ (-(2 * s)) ^ 2 - 4 * 2 * (u + q / s) = w * w",
"after_state": "No Goals!"
},
{
"line": "rw [mul_eq_zero]",
"before_state": "K : Type u_1\ninst✝³ : Field K\np q r s u v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nhq_nonzero : q ≠ 0\nhu : u ^ 3 - p * u ^ 2 - 4 * r * u + 4 * p * r - q ^ 2 = 0\nhs : s ^ 2 = u - p\nhv : v ^ 2 = 4 * s ^ 2 - 8 * (u - q / s)\nhw : w ^ 2 = 4 * s ^ 2 - 8 * (u + q / s)\nx : K\nhi2 : 2 ≠ 0\nh4 : 4 = 2 ^ 2\nhs_nonzero : s ≠ 0\nhv' : discrim 2 (2 * s) (u - q / s) = v * v\nhw' : discrim 2 (-(2 * s)) (u + q / s) = w * w\n⊢ (2 * (x * x) + 2 * s * x + (u - q / s)) * (2 * (x * x) + -(2 * s) * x + (u + q / s)) = 0 ↔\n x = (-2 * s - v) / 4 ∨ x = (-2 * s + v) / 4 ∨ x = (2 * s - w) / 4 ∨ x = (2 * s + w) / 4",
"after_state": "K : Type u_1\ninst✝³ : Field K\np q r s u v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nhq_nonzero : q ≠ 0\nhu : u ^ 3 - p * u ^ 2 - 4 * r * u + 4 * p * r - q ^ 2 = 0\nhs : s ^ 2 = u - p\nhv : v ^ 2 = 4 * s ^ 2 - 8 * (u - q / s)\nhw : w ^ 2 = 4 * s ^ 2 - 8 * (u + q / s)\nx : K\nhi2 : 2 ≠ 0\nh4 : 4 = 2 ^ 2\nhs_nonzero : s ≠ 0\nhv' : discrim 2 (2 * s) (u - q / s) = v * v\nhw' : discrim 2 (-(2 * s)) (u + q / s) = w * w\n⊢ 2 * (x * x) + 2 * s * x + (u - q / s) = 0 ∨ 2 * (x * x) + -(2 * s) * x + (u + q / s) = 0 ↔\n x = (-2 * s - v) / 4 ∨ x = (-2 * s + v) / 4 ∨ x = (2 * s - w) / 4 ∨ x = (2 * s + w) / 4"
},
{
"line": "rewrite [mul_eq_zero]",
"before_state": "K : Type u_1\ninst✝³ : Field K\np q r s u v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nhq_nonzero : q ≠ 0\nhu : u ^ 3 - p * u ^ 2 - 4 * r * u + 4 * p * r - q ^ 2 = 0\nhs : s ^ 2 = u - p\nhv : v ^ 2 = 4 * s ^ 2 - 8 * (u - q / s)\nhw : w ^ 2 = 4 * s ^ 2 - 8 * (u + q / s)\nx : K\nhi2 : 2 ≠ 0\nh4 : 4 = 2 ^ 2\nhs_nonzero : s ≠ 0\nhv' : discrim 2 (2 * s) (u - q / s) = v * v\nhw' : discrim 2 (-(2 * s)) (u + q / s) = w * w\n⊢ (2 * (x * x) + 2 * s * x + (u - q / s)) * (2 * (x * x) + -(2 * s) * x + (u + q / s)) = 0 ↔\n x = (-2 * s - v) / 4 ∨ x = (-2 * s + v) / 4 ∨ x = (2 * s - w) / 4 ∨ x = (2 * s + w) / 4",
"after_state": "K : Type u_1\ninst✝³ : Field K\np q r s u v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nhq_nonzero : q ≠ 0\nhu : u ^ 3 - p * u ^ 2 - 4 * r * u + 4 * p * r - q ^ 2 = 0\nhs : s ^ 2 = u - p\nhv : v ^ 2 = 4 * s ^ 2 - 8 * (u - q / s)\nhw : w ^ 2 = 4 * s ^ 2 - 8 * (u + q / s)\nx : K\nhi2 : 2 ≠ 0\nh4 : 4 = 2 ^ 2\nhs_nonzero : s ≠ 0\nhv' : discrim 2 (2 * s) (u - q / s) = v * v\nhw' : discrim 2 (-(2 * s)) (u + q / s) = w * w\n⊢ 2 * (x * x) + 2 * s * x + (u - q / s) = 0 ∨ 2 * (x * x) + -(2 * s) * x + (u + q / s) = 0 ↔\n x = (-2 * s - v) / 4 ∨ x = (-2 * s + v) / 4 ∨ x = (2 * s - w) / 4 ∨ x = (2 * s + w) / 4"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "K : Type u_1\ninst✝³ : Field K\np q r s u v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nhq_nonzero : q ≠ 0\nhu : u ^ 3 - p * u ^ 2 - 4 * r * u + 4 * p * r - q ^ 2 = 0\nhs : s ^ 2 = u - p\nhv : v ^ 2 = 4 * s ^ 2 - 8 * (u - q / s)\nhw : w ^ 2 = 4 * s ^ 2 - 8 * (u + q / s)\nx : K\nhi2 : 2 ≠ 0\nh4 : 4 = 2 ^ 2\nhs_nonzero : s ≠ 0\nhv' : discrim 2 (2 * s) (u - q / s) = v * v\nhw' : discrim 2 (-(2 * s)) (u + q / s) = w * w\n⊢ 2 * (x * x) + 2 * s * x + (u - q / s) = 0 ∨ 2 * (x * x) + -(2 * s) * x + (u + q / s) = 0 ↔\n x = (-2 * s - v) / 4 ∨ x = (-2 * s + v) / 4 ∨ x = (2 * s - w) / 4 ∨ x = (2 * s + w) / 4",
"after_state": "K : Type u_1\ninst✝³ : Field K\np q r s u v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nhq_nonzero : q ≠ 0\nhu : u ^ 3 - p * u ^ 2 - 4 * r * u + 4 * p * r - q ^ 2 = 0\nhs : s ^ 2 = u - p\nhv : v ^ 2 = 4 * s ^ 2 - 8 * (u - q / s)\nhw : w ^ 2 = 4 * s ^ 2 - 8 * (u + q / s)\nx : K\nhi2 : 2 ≠ 0\nh4 : 4 = 2 ^ 2\nhs_nonzero : s ≠ 0\nhv' : discrim 2 (2 * s) (u - q / s) = v * v\nhw' : discrim 2 (-(2 * s)) (u + q / s) = w * w\n⊢ 2 * (x * x) + 2 * s * x + (u - q / s) = 0 ∨ 2 * (x * x) + -(2 * s) * x + (u + q / s) = 0 ↔\n x = (-2 * s - v) / 4 ∨ x = (-2 * s + v) / 4 ∨ x = (2 * s - w) / 4 ∨ x = (2 * s + w) / 4"
},
{
"line": "try (with_reducible rfl)",
"before_state": "K : Type u_1\ninst✝³ : Field K\np q r s u v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nhq_nonzero : q ≠ 0\nhu : u ^ 3 - p * u ^ 2 - 4 * r * u + 4 * p * r - q ^ 2 = 0\nhs : s ^ 2 = u - p\nhv : v ^ 2 = 4 * s ^ 2 - 8 * (u - q / s)\nhw : w ^ 2 = 4 * s ^ 2 - 8 * (u + q / s)\nx : K\nhi2 : 2 ≠ 0\nh4 : 4 = 2 ^ 2\nhs_nonzero : s ≠ 0\nhv' : discrim 2 (2 * s) (u - q / s) = v * v\nhw' : discrim 2 (-(2 * s)) (u + q / s) = w * w\n⊢ 2 * (x * x) + 2 * s * x + (u - q / s) = 0 ∨ 2 * (x * x) + -(2 * s) * x + (u + q / s) = 0 ↔\n x = (-2 * s - v) / 4 ∨ x = (-2 * s + v) / 4 ∨ x = (2 * s - w) / 4 ∨ x = (2 * s + w) / 4",
"after_state": "K : Type u_1\ninst✝³ : Field K\np q r s u v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nhq_nonzero : q ≠ 0\nhu : u ^ 3 - p * u ^ 2 - 4 * r * u + 4 * p * r - q ^ 2 = 0\nhs : s ^ 2 = u - p\nhv : v ^ 2 = 4 * s ^ 2 - 8 * (u - q / s)\nhw : w ^ 2 = 4 * s ^ 2 - 8 * (u + q / s)\nx : K\nhi2 : 2 ≠ 0\nh4 : 4 = 2 ^ 2\nhs_nonzero : s ≠ 0\nhv' : discrim 2 (2 * s) (u - q / s) = v * v\nhw' : discrim 2 (-(2 * s)) (u + q / s) = w * w\n⊢ 2 * (x * x) + 2 * s * x + (u - q / s) = 0 ∨ 2 * (x * x) + -(2 * s) * x + (u + q / s) = 0 ↔\n x = (-2 * s - v) / 4 ∨ x = (-2 * s + v) / 4 ∨ x = (2 * s - w) / 4 ∨ x = (2 * s + w) / 4"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "K : Type u_1\ninst✝³ : Field K\np q r s u v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nhq_nonzero : q ≠ 0\nhu : u ^ 3 - p * u ^ 2 - 4 * r * u + 4 * p * r - q ^ 2 = 0\nhs : s ^ 2 = u - p\nhv : v ^ 2 = 4 * s ^ 2 - 8 * (u - q / s)\nhw : w ^ 2 = 4 * s ^ 2 - 8 * (u + q / s)\nx : K\nhi2 : 2 ≠ 0\nh4 : 4 = 2 ^ 2\nhs_nonzero : s ≠ 0\nhv' : discrim 2 (2 * s) (u - q / s) = v * v\nhw' : discrim 2 (-(2 * s)) (u + q / s) = w * w\n⊢ 2 * (x * x) + 2 * s * x + (u - q / s) = 0 ∨ 2 * (x * x) + -(2 * s) * x + (u + q / s) = 0 ↔\n x = (-2 * s - v) / 4 ∨ x = (-2 * s + v) / 4 ∨ x = (2 * s - w) / 4 ∨ x = (2 * s + w) / 4",
"after_state": "K : Type u_1\ninst✝³ : Field K\np q r s u v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nhq_nonzero : q ≠ 0\nhu : u ^ 3 - p * u ^ 2 - 4 * r * u + 4 * p * r - q ^ 2 = 0\nhs : s ^ 2 = u - p\nhv : v ^ 2 = 4 * s ^ 2 - 8 * (u - q / s)\nhw : w ^ 2 = 4 * s ^ 2 - 8 * (u + q / s)\nx : K\nhi2 : 2 ≠ 0\nh4 : 4 = 2 ^ 2\nhs_nonzero : s ≠ 0\nhv' : discrim 2 (2 * s) (u - q / s) = v * v\nhw' : discrim 2 (-(2 * s)) (u + q / s) = w * w\n⊢ 2 * (x * x) + 2 * s * x + (u - q / s) = 0 ∨ 2 * (x * x) + -(2 * s) * x + (u + q / s) = 0 ↔\n x = (-2 * s - v) / 4 ∨ x = (-2 * s + v) / 4 ∨ x = (2 * s - w) / 4 ∨ x = (2 * s + w) / 4"
},
{
"line": "with_reducible rfl",
"before_state": "K : Type u_1\ninst✝³ : Field K\np q r s u v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nhq_nonzero : q ≠ 0\nhu : u ^ 3 - p * u ^ 2 - 4 * r * u + 4 * p * r - q ^ 2 = 0\nhs : s ^ 2 = u - p\nhv : v ^ 2 = 4 * s ^ 2 - 8 * (u - q / s)\nhw : w ^ 2 = 4 * s ^ 2 - 8 * (u + q / s)\nx : K\nhi2 : 2 ≠ 0\nh4 : 4 = 2 ^ 2\nhs_nonzero : s ≠ 0\nhv' : discrim 2 (2 * s) (u - q / s) = v * v\nhw' : discrim 2 (-(2 * s)) (u + q / s) = w * w\n⊢ 2 * (x * x) + 2 * s * x + (u - q / s) = 0 ∨ 2 * (x * x) + -(2 * s) * x + (u + q / s) = 0 ↔\n x = (-2 * s - v) / 4 ∨ x = (-2 * s + v) / 4 ∨ x = (2 * s - w) / 4 ∨ x = (2 * s + w) / 4",
"after_state": "K : Type u_1\ninst✝³ : Field K\np q r s u v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nhq_nonzero : q ≠ 0\nhu : u ^ 3 - p * u ^ 2 - 4 * r * u + 4 * p * r - q ^ 2 = 0\nhs : s ^ 2 = u - p\nhv : v ^ 2 = 4 * s ^ 2 - 8 * (u - q / s)\nhw : w ^ 2 = 4 * s ^ 2 - 8 * (u + q / s)\nx : K\nhi2 : 2 ≠ 0\nh4 : 4 = 2 ^ 2\nhs_nonzero : s ≠ 0\nhv' : discrim 2 (2 * s) (u - q / s) = v * v\nhw' : discrim 2 (-(2 * s)) (u + q / s) = w * w\n⊢ 2 * (x * x) + 2 * s * x + (u - q / s) = 0 ∨ 2 * (x * x) + -(2 * s) * x + (u + q / s) = 0 ↔\n x = (-2 * s - v) / 4 ∨ x = (-2 * s + v) / 4 ∨ x = (2 * s - w) / 4 ∨ x = (2 * s + w) / 4"
},
{
"line": "rfl",
"before_state": "K : Type u_1\ninst✝³ : Field K\np q r s u v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nhq_nonzero : q ≠ 0\nhu : u ^ 3 - p * u ^ 2 - 4 * r * u + 4 * p * r - q ^ 2 = 0\nhs : s ^ 2 = u - p\nhv : v ^ 2 = 4 * s ^ 2 - 8 * (u - q / s)\nhw : w ^ 2 = 4 * s ^ 2 - 8 * (u + q / s)\nx : K\nhi2 : 2 ≠ 0\nh4 : 4 = 2 ^ 2\nhs_nonzero : s ≠ 0\nhv' : discrim 2 (2 * s) (u - q / s) = v * v\nhw' : discrim 2 (-(2 * s)) (u + q / s) = w * w\n⊢ 2 * (x * x) + 2 * s * x + (u - q / s) = 0 ∨ 2 * (x * x) + -(2 * s) * x + (u + q / s) = 0 ↔\n x = (-2 * s - v) / 4 ∨ x = (-2 * s + v) / 4 ∨ x = (2 * s - w) / 4 ∨ x = (2 * s + w) / 4",
"after_state": "K : Type u_1\ninst✝³ : Field K\np q r s u v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nhq_nonzero : q ≠ 0\nhu : u ^ 3 - p * u ^ 2 - 4 * r * u + 4 * p * r - q ^ 2 = 0\nhs : s ^ 2 = u - p\nhv : v ^ 2 = 4 * s ^ 2 - 8 * (u - q / s)\nhw : w ^ 2 = 4 * s ^ 2 - 8 * (u + q / s)\nx : K\nhi2 : 2 ≠ 0\nh4 : 4 = 2 ^ 2\nhs_nonzero : s ≠ 0\nhv' : discrim 2 (2 * s) (u - q / s) = v * v\nhw' : discrim 2 (-(2 * s)) (u + q / s) = w * w\n⊢ 2 * (x * x) + 2 * s * x + (u - q / s) = 0 ∨ 2 * (x * x) + -(2 * s) * x + (u + q / s) = 0 ↔\n x = (-2 * s - v) / 4 ∨ x = (-2 * s + v) / 4 ∨ x = (2 * s - w) / 4 ∨ x = (2 * s + w) / 4"
},
{
"line": "apply_rfl",
"before_state": "K : Type u_1\ninst✝³ : Field K\np q r s u v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nhq_nonzero : q ≠ 0\nhu : u ^ 3 - p * u ^ 2 - 4 * r * u + 4 * p * r - q ^ 2 = 0\nhs : s ^ 2 = u - p\nhv : v ^ 2 = 4 * s ^ 2 - 8 * (u - q / s)\nhw : w ^ 2 = 4 * s ^ 2 - 8 * (u + q / s)\nx : K\nhi2 : 2 ≠ 0\nh4 : 4 = 2 ^ 2\nhs_nonzero : s ≠ 0\nhv' : discrim 2 (2 * s) (u - q / s) = v * v\nhw' : discrim 2 (-(2 * s)) (u + q / s) = w * w\n⊢ 2 * (x * x) + 2 * s * x + (u - q / s) = 0 ∨ 2 * (x * x) + -(2 * s) * x + (u + q / s) = 0 ↔\n x = (-2 * s - v) / 4 ∨ x = (-2 * s + v) / 4 ∨ x = (2 * s - w) / 4 ∨ x = (2 * s + w) / 4",
"after_state": "K : Type u_1\ninst✝³ : Field K\np q r s u v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nhq_nonzero : q ≠ 0\nhu : u ^ 3 - p * u ^ 2 - 4 * r * u + 4 * p * r - q ^ 2 = 0\nhs : s ^ 2 = u - p\nhv : v ^ 2 = 4 * s ^ 2 - 8 * (u - q / s)\nhw : w ^ 2 = 4 * s ^ 2 - 8 * (u + q / s)\nx : K\nhi2 : 2 ≠ 0\nh4 : 4 = 2 ^ 2\nhs_nonzero : s ≠ 0\nhv' : discrim 2 (2 * s) (u - q / s) = v * v\nhw' : discrim 2 (-(2 * s)) (u + q / s) = w * w\n⊢ 2 * (x * x) + 2 * s * x + (u - q / s) = 0 ∨ 2 * (x * x) + -(2 * s) * x + (u + q / s) = 0 ↔\n x = (-2 * s - v) / 4 ∨ x = (-2 * s + v) / 4 ∨ x = (2 * s - w) / 4 ∨ x = (2 * s + w) / 4"
},
{
"line": "skip",
"before_state": "K : Type u_1\ninst✝³ : Field K\np q r s u v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nhq_nonzero : q ≠ 0\nhu : u ^ 3 - p * u ^ 2 - 4 * r * u + 4 * p * r - q ^ 2 = 0\nhs : s ^ 2 = u - p\nhv : v ^ 2 = 4 * s ^ 2 - 8 * (u - q / s)\nhw : w ^ 2 = 4 * s ^ 2 - 8 * (u + q / s)\nx : K\nhi2 : 2 ≠ 0\nh4 : 4 = 2 ^ 2\nhs_nonzero : s ≠ 0\nhv' : discrim 2 (2 * s) (u - q / s) = v * v\nhw' : discrim 2 (-(2 * s)) (u + q / s) = w * w\n⊢ 2 * (x * x) + 2 * s * x + (u - q / s) = 0 ∨ 2 * (x * x) + -(2 * s) * x + (u + q / s) = 0 ↔\n x = (-2 * s - v) / 4 ∨ x = (-2 * s + v) / 4 ∨ x = (2 * s - w) / 4 ∨ x = (2 * s + w) / 4",
"after_state": "K : Type u_1\ninst✝³ : Field K\np q r s u v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nhq_nonzero : q ≠ 0\nhu : u ^ 3 - p * u ^ 2 - 4 * r * u + 4 * p * r - q ^ 2 = 0\nhs : s ^ 2 = u - p\nhv : v ^ 2 = 4 * s ^ 2 - 8 * (u - q / s)\nhw : w ^ 2 = 4 * s ^ 2 - 8 * (u + q / s)\nx : K\nhi2 : 2 ≠ 0\nh4 : 4 = 2 ^ 2\nhs_nonzero : s ≠ 0\nhv' : discrim 2 (2 * s) (u - q / s) = v * v\nhw' : discrim 2 (-(2 * s)) (u + q / s) = w * w\n⊢ 2 * (x * x) + 2 * s * x + (u - q / s) = 0 ∨ 2 * (x * x) + -(2 * s) * x + (u + q / s) = 0 ↔\n x = (-2 * s - v) / 4 ∨ x = (-2 * s + v) / 4 ∨ x = (2 * s - w) / 4 ∨ x = (2 * s + w) / 4"
},
{
"line": "rw [quadratic_eq_zero_iff hi2 hv']",
"before_state": "K : Type u_1\ninst✝³ : Field K\np q r s u v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nhq_nonzero : q ≠ 0\nhu : u ^ 3 - p * u ^ 2 - 4 * r * u + 4 * p * r - q ^ 2 = 0\nhs : s ^ 2 = u - p\nhv : v ^ 2 = 4 * s ^ 2 - 8 * (u - q / s)\nhw : w ^ 2 = 4 * s ^ 2 - 8 * (u + q / s)\nx : K\nhi2 : 2 ≠ 0\nh4 : 4 = 2 ^ 2\nhs_nonzero : s ≠ 0\nhv' : discrim 2 (2 * s) (u - q / s) = v * v\nhw' : discrim 2 (-(2 * s)) (u + q / s) = w * w\n⊢ 2 * (x * x) + 2 * s * x + (u - q / s) = 0 ∨ 2 * (x * x) + -(2 * s) * x + (u + q / s) = 0 ↔\n x = (-2 * s - v) / 4 ∨ x = (-2 * s + v) / 4 ∨ x = (2 * s - w) / 4 ∨ x = (2 * s + w) / 4",
"after_state": "K : Type u_1\ninst✝³ : Field K\np q r s u v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nhq_nonzero : q ≠ 0\nhu : u ^ 3 - p * u ^ 2 - 4 * r * u + 4 * p * r - q ^ 2 = 0\nhs : s ^ 2 = u - p\nhv : v ^ 2 = 4 * s ^ 2 - 8 * (u - q / s)\nhw : w ^ 2 = 4 * s ^ 2 - 8 * (u + q / s)\nx : K\nhi2 : 2 ≠ 0\nh4 : 4 = 2 ^ 2\nhs_nonzero : s ≠ 0\nhv' : discrim 2 (2 * s) (u - q / s) = v * v\nhw' : discrim 2 (-(2 * s)) (u + q / s) = w * w\n⊢ (x = (-(2 * s) + v) / (2 * 2) ∨ x = (-(2 * s) - v) / (2 * 2)) ∨ 2 * (x * x) + -(2 * s) * x + (u + q / s) = 0 ↔\n x = (-2 * s - v) / 4 ∨ x = (-2 * s + v) / 4 ∨ x = (2 * s - w) / 4 ∨ x = (2 * s + w) / 4"
},
{
"line": "rewrite [quadratic_eq_zero_iff hi2 hv']",
"before_state": "K : Type u_1\ninst✝³ : Field K\np q r s u v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nhq_nonzero : q ≠ 0\nhu : u ^ 3 - p * u ^ 2 - 4 * r * u + 4 * p * r - q ^ 2 = 0\nhs : s ^ 2 = u - p\nhv : v ^ 2 = 4 * s ^ 2 - 8 * (u - q / s)\nhw : w ^ 2 = 4 * s ^ 2 - 8 * (u + q / s)\nx : K\nhi2 : 2 ≠ 0\nh4 : 4 = 2 ^ 2\nhs_nonzero : s ≠ 0\nhv' : discrim 2 (2 * s) (u - q / s) = v * v\nhw' : discrim 2 (-(2 * s)) (u + q / s) = w * w\n⊢ 2 * (x * x) + 2 * s * x + (u - q / s) = 0 ∨ 2 * (x * x) + -(2 * s) * x + (u + q / s) = 0 ↔\n x = (-2 * s - v) / 4 ∨ x = (-2 * s + v) / 4 ∨ x = (2 * s - w) / 4 ∨ x = (2 * s + w) / 4",
"after_state": "K : Type u_1\ninst✝³ : Field K\np q r s u v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nhq_nonzero : q ≠ 0\nhu : u ^ 3 - p * u ^ 2 - 4 * r * u + 4 * p * r - q ^ 2 = 0\nhs : s ^ 2 = u - p\nhv : v ^ 2 = 4 * s ^ 2 - 8 * (u - q / s)\nhw : w ^ 2 = 4 * s ^ 2 - 8 * (u + q / s)\nx : K\nhi2 : 2 ≠ 0\nh4 : 4 = 2 ^ 2\nhs_nonzero : s ≠ 0\nhv' : discrim 2 (2 * s) (u - q / s) = v * v\nhw' : discrim 2 (-(2 * s)) (u + q / s) = w * w\n⊢ (x = (-(2 * s) + v) / (2 * 2) ∨ x = (-(2 * s) - v) / (2 * 2)) ∨ 2 * (x * x) + -(2 * s) * x + (u + q / s) = 0 ↔\n x = (-2 * s - v) / 4 ∨ x = (-2 * s + v) / 4 ∨ x = (2 * s - w) / 4 ∨ x = (2 * s + w) / 4"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "K : Type u_1\ninst✝³ : Field K\np q r s u v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nhq_nonzero : q ≠ 0\nhu : u ^ 3 - p * u ^ 2 - 4 * r * u + 4 * p * r - q ^ 2 = 0\nhs : s ^ 2 = u - p\nhv : v ^ 2 = 4 * s ^ 2 - 8 * (u - q / s)\nhw : w ^ 2 = 4 * s ^ 2 - 8 * (u + q / s)\nx : K\nhi2 : 2 ≠ 0\nh4 : 4 = 2 ^ 2\nhs_nonzero : s ≠ 0\nhv' : discrim 2 (2 * s) (u - q / s) = v * v\nhw' : discrim 2 (-(2 * s)) (u + q / s) = w * w\n⊢ (x = (-(2 * s) + v) / (2 * 2) ∨ x = (-(2 * s) - v) / (2 * 2)) ∨ 2 * (x * x) + -(2 * s) * x + (u + q / s) = 0 ↔\n x = (-2 * s - v) / 4 ∨ x = (-2 * s + v) / 4 ∨ x = (2 * s - w) / 4 ∨ x = (2 * s + w) / 4",
"after_state": "K : Type u_1\ninst✝³ : Field K\np q r s u v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nhq_nonzero : q ≠ 0\nhu : u ^ 3 - p * u ^ 2 - 4 * r * u + 4 * p * r - q ^ 2 = 0\nhs : s ^ 2 = u - p\nhv : v ^ 2 = 4 * s ^ 2 - 8 * (u - q / s)\nhw : w ^ 2 = 4 * s ^ 2 - 8 * (u + q / s)\nx : K\nhi2 : 2 ≠ 0\nh4 : 4 = 2 ^ 2\nhs_nonzero : s ≠ 0\nhv' : discrim 2 (2 * s) (u - q / s) = v * v\nhw' : discrim 2 (-(2 * s)) (u + q / s) = w * w\n⊢ (x = (-(2 * s) + v) / (2 * 2) ∨ x = (-(2 * s) - v) / (2 * 2)) ∨ 2 * (x * x) + -(2 * s) * x + (u + q / s) = 0 ↔\n x = (-2 * s - v) / 4 ∨ x = (-2 * s + v) / 4 ∨ x = (2 * s - w) / 4 ∨ x = (2 * s + w) / 4"
},
{
"line": "try (with_reducible rfl)",
"before_state": "K : Type u_1\ninst✝³ : Field K\np q r s u v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nhq_nonzero : q ≠ 0\nhu : u ^ 3 - p * u ^ 2 - 4 * r * u + 4 * p * r - q ^ 2 = 0\nhs : s ^ 2 = u - p\nhv : v ^ 2 = 4 * s ^ 2 - 8 * (u - q / s)\nhw : w ^ 2 = 4 * s ^ 2 - 8 * (u + q / s)\nx : K\nhi2 : 2 ≠ 0\nh4 : 4 = 2 ^ 2\nhs_nonzero : s ≠ 0\nhv' : discrim 2 (2 * s) (u - q / s) = v * v\nhw' : discrim 2 (-(2 * s)) (u + q / s) = w * w\n⊢ (x = (-(2 * s) + v) / (2 * 2) ∨ x = (-(2 * s) - v) / (2 * 2)) ∨ 2 * (x * x) + -(2 * s) * x + (u + q / s) = 0 ↔\n x = (-2 * s - v) / 4 ∨ x = (-2 * s + v) / 4 ∨ x = (2 * s - w) / 4 ∨ x = (2 * s + w) / 4",
"after_state": "K : Type u_1\ninst✝³ : Field K\np q r s u v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nhq_nonzero : q ≠ 0\nhu : u ^ 3 - p * u ^ 2 - 4 * r * u + 4 * p * r - q ^ 2 = 0\nhs : s ^ 2 = u - p\nhv : v ^ 2 = 4 * s ^ 2 - 8 * (u - q / s)\nhw : w ^ 2 = 4 * s ^ 2 - 8 * (u + q / s)\nx : K\nhi2 : 2 ≠ 0\nh4 : 4 = 2 ^ 2\nhs_nonzero : s ≠ 0\nhv' : discrim 2 (2 * s) (u - q / s) = v * v\nhw' : discrim 2 (-(2 * s)) (u + q / s) = w * w\n⊢ (x = (-(2 * s) + v) / (2 * 2) ∨ x = (-(2 * s) - v) / (2 * 2)) ∨ 2 * (x * x) + -(2 * s) * x + (u + q / s) = 0 ↔\n x = (-2 * s - v) / 4 ∨ x = (-2 * s + v) / 4 ∨ x = (2 * s - w) / 4 ∨ x = (2 * s + w) / 4"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "K : Type u_1\ninst✝³ : Field K\np q r s u v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nhq_nonzero : q ≠ 0\nhu : u ^ 3 - p * u ^ 2 - 4 * r * u + 4 * p * r - q ^ 2 = 0\nhs : s ^ 2 = u - p\nhv : v ^ 2 = 4 * s ^ 2 - 8 * (u - q / s)\nhw : w ^ 2 = 4 * s ^ 2 - 8 * (u + q / s)\nx : K\nhi2 : 2 ≠ 0\nh4 : 4 = 2 ^ 2\nhs_nonzero : s ≠ 0\nhv' : discrim 2 (2 * s) (u - q / s) = v * v\nhw' : discrim 2 (-(2 * s)) (u + q / s) = w * w\n⊢ (x = (-(2 * s) + v) / (2 * 2) ∨ x = (-(2 * s) - v) / (2 * 2)) ∨ 2 * (x * x) + -(2 * s) * x + (u + q / s) = 0 ↔\n x = (-2 * s - v) / 4 ∨ x = (-2 * s + v) / 4 ∨ x = (2 * s - w) / 4 ∨ x = (2 * s + w) / 4",
"after_state": "K : Type u_1\ninst✝³ : Field K\np q r s u v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nhq_nonzero : q ≠ 0\nhu : u ^ 3 - p * u ^ 2 - 4 * r * u + 4 * p * r - q ^ 2 = 0\nhs : s ^ 2 = u - p\nhv : v ^ 2 = 4 * s ^ 2 - 8 * (u - q / s)\nhw : w ^ 2 = 4 * s ^ 2 - 8 * (u + q / s)\nx : K\nhi2 : 2 ≠ 0\nh4 : 4 = 2 ^ 2\nhs_nonzero : s ≠ 0\nhv' : discrim 2 (2 * s) (u - q / s) = v * v\nhw' : discrim 2 (-(2 * s)) (u + q / s) = w * w\n⊢ (x = (-(2 * s) + v) / (2 * 2) ∨ x = (-(2 * s) - v) / (2 * 2)) ∨ 2 * (x * x) + -(2 * s) * x + (u + q / s) = 0 ↔\n x = (-2 * s - v) / 4 ∨ x = (-2 * s + v) / 4 ∨ x = (2 * s - w) / 4 ∨ x = (2 * s + w) / 4"
},
{
"line": "with_reducible rfl",
"before_state": "K : Type u_1\ninst✝³ : Field K\np q r s u v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nhq_nonzero : q ≠ 0\nhu : u ^ 3 - p * u ^ 2 - 4 * r * u + 4 * p * r - q ^ 2 = 0\nhs : s ^ 2 = u - p\nhv : v ^ 2 = 4 * s ^ 2 - 8 * (u - q / s)\nhw : w ^ 2 = 4 * s ^ 2 - 8 * (u + q / s)\nx : K\nhi2 : 2 ≠ 0\nh4 : 4 = 2 ^ 2\nhs_nonzero : s ≠ 0\nhv' : discrim 2 (2 * s) (u - q / s) = v * v\nhw' : discrim 2 (-(2 * s)) (u + q / s) = w * w\n⊢ (x = (-(2 * s) + v) / (2 * 2) ∨ x = (-(2 * s) - v) / (2 * 2)) ∨ 2 * (x * x) + -(2 * s) * x + (u + q / s) = 0 ↔\n x = (-2 * s - v) / 4 ∨ x = (-2 * s + v) / 4 ∨ x = (2 * s - w) / 4 ∨ x = (2 * s + w) / 4",
"after_state": "K : Type u_1\ninst✝³ : Field K\np q r s u v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nhq_nonzero : q ≠ 0\nhu : u ^ 3 - p * u ^ 2 - 4 * r * u + 4 * p * r - q ^ 2 = 0\nhs : s ^ 2 = u - p\nhv : v ^ 2 = 4 * s ^ 2 - 8 * (u - q / s)\nhw : w ^ 2 = 4 * s ^ 2 - 8 * (u + q / s)\nx : K\nhi2 : 2 ≠ 0\nh4 : 4 = 2 ^ 2\nhs_nonzero : s ≠ 0\nhv' : discrim 2 (2 * s) (u - q / s) = v * v\nhw' : discrim 2 (-(2 * s)) (u + q / s) = w * w\n⊢ (x = (-(2 * s) + v) / (2 * 2) ∨ x = (-(2 * s) - v) / (2 * 2)) ∨ 2 * (x * x) + -(2 * s) * x + (u + q / s) = 0 ↔\n x = (-2 * s - v) / 4 ∨ x = (-2 * s + v) / 4 ∨ x = (2 * s - w) / 4 ∨ x = (2 * s + w) / 4"
},
{
"line": "rfl",
"before_state": "K : Type u_1\ninst✝³ : Field K\np q r s u v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nhq_nonzero : q ≠ 0\nhu : u ^ 3 - p * u ^ 2 - 4 * r * u + 4 * p * r - q ^ 2 = 0\nhs : s ^ 2 = u - p\nhv : v ^ 2 = 4 * s ^ 2 - 8 * (u - q / s)\nhw : w ^ 2 = 4 * s ^ 2 - 8 * (u + q / s)\nx : K\nhi2 : 2 ≠ 0\nh4 : 4 = 2 ^ 2\nhs_nonzero : s ≠ 0\nhv' : discrim 2 (2 * s) (u - q / s) = v * v\nhw' : discrim 2 (-(2 * s)) (u + q / s) = w * w\n⊢ (x = (-(2 * s) + v) / (2 * 2) ∨ x = (-(2 * s) - v) / (2 * 2)) ∨ 2 * (x * x) + -(2 * s) * x + (u + q / s) = 0 ↔\n x = (-2 * s - v) / 4 ∨ x = (-2 * s + v) / 4 ∨ x = (2 * s - w) / 4 ∨ x = (2 * s + w) / 4",
"after_state": "K : Type u_1\ninst✝³ : Field K\np q r s u v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nhq_nonzero : q ≠ 0\nhu : u ^ 3 - p * u ^ 2 - 4 * r * u + 4 * p * r - q ^ 2 = 0\nhs : s ^ 2 = u - p\nhv : v ^ 2 = 4 * s ^ 2 - 8 * (u - q / s)\nhw : w ^ 2 = 4 * s ^ 2 - 8 * (u + q / s)\nx : K\nhi2 : 2 ≠ 0\nh4 : 4 = 2 ^ 2\nhs_nonzero : s ≠ 0\nhv' : discrim 2 (2 * s) (u - q / s) = v * v\nhw' : discrim 2 (-(2 * s)) (u + q / s) = w * w\n⊢ (x = (-(2 * s) + v) / (2 * 2) ∨ x = (-(2 * s) - v) / (2 * 2)) ∨ 2 * (x * x) + -(2 * s) * x + (u + q / s) = 0 ↔\n x = (-2 * s - v) / 4 ∨ x = (-2 * s + v) / 4 ∨ x = (2 * s - w) / 4 ∨ x = (2 * s + w) / 4"
},
{
"line": "apply_rfl",
"before_state": "K : Type u_1\ninst✝³ : Field K\np q r s u v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nhq_nonzero : q ≠ 0\nhu : u ^ 3 - p * u ^ 2 - 4 * r * u + 4 * p * r - q ^ 2 = 0\nhs : s ^ 2 = u - p\nhv : v ^ 2 = 4 * s ^ 2 - 8 * (u - q / s)\nhw : w ^ 2 = 4 * s ^ 2 - 8 * (u + q / s)\nx : K\nhi2 : 2 ≠ 0\nh4 : 4 = 2 ^ 2\nhs_nonzero : s ≠ 0\nhv' : discrim 2 (2 * s) (u - q / s) = v * v\nhw' : discrim 2 (-(2 * s)) (u + q / s) = w * w\n⊢ (x = (-(2 * s) + v) / (2 * 2) ∨ x = (-(2 * s) - v) / (2 * 2)) ∨ 2 * (x * x) + -(2 * s) * x + (u + q / s) = 0 ↔\n x = (-2 * s - v) / 4 ∨ x = (-2 * s + v) / 4 ∨ x = (2 * s - w) / 4 ∨ x = (2 * s + w) / 4",
"after_state": "K : Type u_1\ninst✝³ : Field K\np q r s u v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nhq_nonzero : q ≠ 0\nhu : u ^ 3 - p * u ^ 2 - 4 * r * u + 4 * p * r - q ^ 2 = 0\nhs : s ^ 2 = u - p\nhv : v ^ 2 = 4 * s ^ 2 - 8 * (u - q / s)\nhw : w ^ 2 = 4 * s ^ 2 - 8 * (u + q / s)\nx : K\nhi2 : 2 ≠ 0\nh4 : 4 = 2 ^ 2\nhs_nonzero : s ≠ 0\nhv' : discrim 2 (2 * s) (u - q / s) = v * v\nhw' : discrim 2 (-(2 * s)) (u + q / s) = w * w\n⊢ (x = (-(2 * s) + v) / (2 * 2) ∨ x = (-(2 * s) - v) / (2 * 2)) ∨ 2 * (x * x) + -(2 * s) * x + (u + q / s) = 0 ↔\n x = (-2 * s - v) / 4 ∨ x = (-2 * s + v) / 4 ∨ x = (2 * s - w) / 4 ∨ x = (2 * s + w) / 4"
},
{
"line": "skip",
"before_state": "K : Type u_1\ninst✝³ : Field K\np q r s u v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nhq_nonzero : q ≠ 0\nhu : u ^ 3 - p * u ^ 2 - 4 * r * u + 4 * p * r - q ^ 2 = 0\nhs : s ^ 2 = u - p\nhv : v ^ 2 = 4 * s ^ 2 - 8 * (u - q / s)\nhw : w ^ 2 = 4 * s ^ 2 - 8 * (u + q / s)\nx : K\nhi2 : 2 ≠ 0\nh4 : 4 = 2 ^ 2\nhs_nonzero : s ≠ 0\nhv' : discrim 2 (2 * s) (u - q / s) = v * v\nhw' : discrim 2 (-(2 * s)) (u + q / s) = w * w\n⊢ (x = (-(2 * s) + v) / (2 * 2) ∨ x = (-(2 * s) - v) / (2 * 2)) ∨ 2 * (x * x) + -(2 * s) * x + (u + q / s) = 0 ↔\n x = (-2 * s - v) / 4 ∨ x = (-2 * s + v) / 4 ∨ x = (2 * s - w) / 4 ∨ x = (2 * s + w) / 4",
"after_state": "K : Type u_1\ninst✝³ : Field K\np q r s u v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nhq_nonzero : q ≠ 0\nhu : u ^ 3 - p * u ^ 2 - 4 * r * u + 4 * p * r - q ^ 2 = 0\nhs : s ^ 2 = u - p\nhv : v ^ 2 = 4 * s ^ 2 - 8 * (u - q / s)\nhw : w ^ 2 = 4 * s ^ 2 - 8 * (u + q / s)\nx : K\nhi2 : 2 ≠ 0\nh4 : 4 = 2 ^ 2\nhs_nonzero : s ≠ 0\nhv' : discrim 2 (2 * s) (u - q / s) = v * v\nhw' : discrim 2 (-(2 * s)) (u + q / s) = w * w\n⊢ (x = (-(2 * s) + v) / (2 * 2) ∨ x = (-(2 * s) - v) / (2 * 2)) ∨ 2 * (x * x) + -(2 * s) * x + (u + q / s) = 0 ↔\n x = (-2 * s - v) / 4 ∨ x = (-2 * s + v) / 4 ∨ x = (2 * s - w) / 4 ∨ x = (2 * s + w) / 4"
},
{
"line": "rw [quadratic_eq_zero_iff hi2 hw']",
"before_state": "K : Type u_1\ninst✝³ : Field K\np q r s u v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nhq_nonzero : q ≠ 0\nhu : u ^ 3 - p * u ^ 2 - 4 * r * u + 4 * p * r - q ^ 2 = 0\nhs : s ^ 2 = u - p\nhv : v ^ 2 = 4 * s ^ 2 - 8 * (u - q / s)\nhw : w ^ 2 = 4 * s ^ 2 - 8 * (u + q / s)\nx : K\nhi2 : 2 ≠ 0\nh4 : 4 = 2 ^ 2\nhs_nonzero : s ≠ 0\nhv' : discrim 2 (2 * s) (u - q / s) = v * v\nhw' : discrim 2 (-(2 * s)) (u + q / s) = w * w\n⊢ (x = (-(2 * s) + v) / (2 * 2) ∨ x = (-(2 * s) - v) / (2 * 2)) ∨ 2 * (x * x) + -(2 * s) * x + (u + q / s) = 0 ↔\n x = (-2 * s - v) / 4 ∨ x = (-2 * s + v) / 4 ∨ x = (2 * s - w) / 4 ∨ x = (2 * s + w) / 4",
"after_state": "K : Type u_1\ninst✝³ : Field K\np q r s u v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nhq_nonzero : q ≠ 0\nhu : u ^ 3 - p * u ^ 2 - 4 * r * u + 4 * p * r - q ^ 2 = 0\nhs : s ^ 2 = u - p\nhv : v ^ 2 = 4 * s ^ 2 - 8 * (u - q / s)\nhw : w ^ 2 = 4 * s ^ 2 - 8 * (u + q / s)\nx : K\nhi2 : 2 ≠ 0\nh4 : 4 = 2 ^ 2\nhs_nonzero : s ≠ 0\nhv' : discrim 2 (2 * s) (u - q / s) = v * v\nhw' : discrim 2 (-(2 * s)) (u + q / s) = w * w\n⊢ (x = (-(2 * s) + v) / (2 * 2) ∨ x = (-(2 * s) - v) / (2 * 2)) ∨\n x = (- -(2 * s) + w) / (2 * 2) ∨ x = (- -(2 * s) - w) / (2 * 2) ↔\n x = (-2 * s - v) / 4 ∨ x = (-2 * s + v) / 4 ∨ x = (2 * s - w) / 4 ∨ x = (2 * s + w) / 4"
},
{
"line": "rewrite [quadratic_eq_zero_iff hi2 hw']",
"before_state": "K : Type u_1\ninst✝³ : Field K\np q r s u v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nhq_nonzero : q ≠ 0\nhu : u ^ 3 - p * u ^ 2 - 4 * r * u + 4 * p * r - q ^ 2 = 0\nhs : s ^ 2 = u - p\nhv : v ^ 2 = 4 * s ^ 2 - 8 * (u - q / s)\nhw : w ^ 2 = 4 * s ^ 2 - 8 * (u + q / s)\nx : K\nhi2 : 2 ≠ 0\nh4 : 4 = 2 ^ 2\nhs_nonzero : s ≠ 0\nhv' : discrim 2 (2 * s) (u - q / s) = v * v\nhw' : discrim 2 (-(2 * s)) (u + q / s) = w * w\n⊢ (x = (-(2 * s) + v) / (2 * 2) ∨ x = (-(2 * s) - v) / (2 * 2)) ∨ 2 * (x * x) + -(2 * s) * x + (u + q / s) = 0 ↔\n x = (-2 * s - v) / 4 ∨ x = (-2 * s + v) / 4 ∨ x = (2 * s - w) / 4 ∨ x = (2 * s + w) / 4",
"after_state": "K : Type u_1\ninst✝³ : Field K\np q r s u v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nhq_nonzero : q ≠ 0\nhu : u ^ 3 - p * u ^ 2 - 4 * r * u + 4 * p * r - q ^ 2 = 0\nhs : s ^ 2 = u - p\nhv : v ^ 2 = 4 * s ^ 2 - 8 * (u - q / s)\nhw : w ^ 2 = 4 * s ^ 2 - 8 * (u + q / s)\nx : K\nhi2 : 2 ≠ 0\nh4 : 4 = 2 ^ 2\nhs_nonzero : s ≠ 0\nhv' : discrim 2 (2 * s) (u - q / s) = v * v\nhw' : discrim 2 (-(2 * s)) (u + q / s) = w * w\n⊢ (x = (-(2 * s) + v) / (2 * 2) ∨ x = (-(2 * s) - v) / (2 * 2)) ∨\n x = (- -(2 * s) + w) / (2 * 2) ∨ x = (- -(2 * s) - w) / (2 * 2) ↔\n x = (-2 * s - v) / 4 ∨ x = (-2 * s + v) / 4 ∨ x = (2 * s - w) / 4 ∨ x = (2 * s + w) / 4"
},
{
"line": "with_annotate_state\"]\" (try (with_reducible rfl))",
"before_state": "K : Type u_1\ninst✝³ : Field K\np q r s u v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nhq_nonzero : q ≠ 0\nhu : u ^ 3 - p * u ^ 2 - 4 * r * u + 4 * p * r - q ^ 2 = 0\nhs : s ^ 2 = u - p\nhv : v ^ 2 = 4 * s ^ 2 - 8 * (u - q / s)\nhw : w ^ 2 = 4 * s ^ 2 - 8 * (u + q / s)\nx : K\nhi2 : 2 ≠ 0\nh4 : 4 = 2 ^ 2\nhs_nonzero : s ≠ 0\nhv' : discrim 2 (2 * s) (u - q / s) = v * v\nhw' : discrim 2 (-(2 * s)) (u + q / s) = w * w\n⊢ (x = (-(2 * s) + v) / (2 * 2) ∨ x = (-(2 * s) - v) / (2 * 2)) ∨\n x = (- -(2 * s) + w) / (2 * 2) ∨ x = (- -(2 * s) - w) / (2 * 2) ↔\n x = (-2 * s - v) / 4 ∨ x = (-2 * s + v) / 4 ∨ x = (2 * s - w) / 4 ∨ x = (2 * s + w) / 4",
"after_state": "K : Type u_1\ninst✝³ : Field K\np q r s u v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nhq_nonzero : q ≠ 0\nhu : u ^ 3 - p * u ^ 2 - 4 * r * u + 4 * p * r - q ^ 2 = 0\nhs : s ^ 2 = u - p\nhv : v ^ 2 = 4 * s ^ 2 - 8 * (u - q / s)\nhw : w ^ 2 = 4 * s ^ 2 - 8 * (u + q / s)\nx : K\nhi2 : 2 ≠ 0\nh4 : 4 = 2 ^ 2\nhs_nonzero : s ≠ 0\nhv' : discrim 2 (2 * s) (u - q / s) = v * v\nhw' : discrim 2 (-(2 * s)) (u + q / s) = w * w\n⊢ (x = (-(2 * s) + v) / (2 * 2) ∨ x = (-(2 * s) - v) / (2 * 2)) ∨\n x = (- -(2 * s) + w) / (2 * 2) ∨ x = (- -(2 * s) - w) / (2 * 2) ↔\n x = (-2 * s - v) / 4 ∨ x = (-2 * s + v) / 4 ∨ x = (2 * s - w) / 4 ∨ x = (2 * s + w) / 4"
},
{
"line": "try (with_reducible rfl)",
"before_state": "K : Type u_1\ninst✝³ : Field K\np q r s u v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nhq_nonzero : q ≠ 0\nhu : u ^ 3 - p * u ^ 2 - 4 * r * u + 4 * p * r - q ^ 2 = 0\nhs : s ^ 2 = u - p\nhv : v ^ 2 = 4 * s ^ 2 - 8 * (u - q / s)\nhw : w ^ 2 = 4 * s ^ 2 - 8 * (u + q / s)\nx : K\nhi2 : 2 ≠ 0\nh4 : 4 = 2 ^ 2\nhs_nonzero : s ≠ 0\nhv' : discrim 2 (2 * s) (u - q / s) = v * v\nhw' : discrim 2 (-(2 * s)) (u + q / s) = w * w\n⊢ (x = (-(2 * s) + v) / (2 * 2) ∨ x = (-(2 * s) - v) / (2 * 2)) ∨\n x = (- -(2 * s) + w) / (2 * 2) ∨ x = (- -(2 * s) - w) / (2 * 2) ↔\n x = (-2 * s - v) / 4 ∨ x = (-2 * s + v) / 4 ∨ x = (2 * s - w) / 4 ∨ x = (2 * s + w) / 4",
"after_state": "K : Type u_1\ninst✝³ : Field K\np q r s u v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nhq_nonzero : q ≠ 0\nhu : u ^ 3 - p * u ^ 2 - 4 * r * u + 4 * p * r - q ^ 2 = 0\nhs : s ^ 2 = u - p\nhv : v ^ 2 = 4 * s ^ 2 - 8 * (u - q / s)\nhw : w ^ 2 = 4 * s ^ 2 - 8 * (u + q / s)\nx : K\nhi2 : 2 ≠ 0\nh4 : 4 = 2 ^ 2\nhs_nonzero : s ≠ 0\nhv' : discrim 2 (2 * s) (u - q / s) = v * v\nhw' : discrim 2 (-(2 * s)) (u + q / s) = w * w\n⊢ (x = (-(2 * s) + v) / (2 * 2) ∨ x = (-(2 * s) - v) / (2 * 2)) ∨\n x = (- -(2 * s) + w) / (2 * 2) ∨ x = (- -(2 * s) - w) / (2 * 2) ↔\n x = (-2 * s - v) / 4 ∨ x = (-2 * s + v) / 4 ∨ x = (2 * s - w) / 4 ∨ x = (2 * s + w) / 4"
},
{
"line": "first\n| (with_reducible rfl)\n| skip",
"before_state": "K : Type u_1\ninst✝³ : Field K\np q r s u v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nhq_nonzero : q ≠ 0\nhu : u ^ 3 - p * u ^ 2 - 4 * r * u + 4 * p * r - q ^ 2 = 0\nhs : s ^ 2 = u - p\nhv : v ^ 2 = 4 * s ^ 2 - 8 * (u - q / s)\nhw : w ^ 2 = 4 * s ^ 2 - 8 * (u + q / s)\nx : K\nhi2 : 2 ≠ 0\nh4 : 4 = 2 ^ 2\nhs_nonzero : s ≠ 0\nhv' : discrim 2 (2 * s) (u - q / s) = v * v\nhw' : discrim 2 (-(2 * s)) (u + q / s) = w * w\n⊢ (x = (-(2 * s) + v) / (2 * 2) ∨ x = (-(2 * s) - v) / (2 * 2)) ∨\n x = (- -(2 * s) + w) / (2 * 2) ∨ x = (- -(2 * s) - w) / (2 * 2) ↔\n x = (-2 * s - v) / 4 ∨ x = (-2 * s + v) / 4 ∨ x = (2 * s - w) / 4 ∨ x = (2 * s + w) / 4",
"after_state": "K : Type u_1\ninst✝³ : Field K\np q r s u v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nhq_nonzero : q ≠ 0\nhu : u ^ 3 - p * u ^ 2 - 4 * r * u + 4 * p * r - q ^ 2 = 0\nhs : s ^ 2 = u - p\nhv : v ^ 2 = 4 * s ^ 2 - 8 * (u - q / s)\nhw : w ^ 2 = 4 * s ^ 2 - 8 * (u + q / s)\nx : K\nhi2 : 2 ≠ 0\nh4 : 4 = 2 ^ 2\nhs_nonzero : s ≠ 0\nhv' : discrim 2 (2 * s) (u - q / s) = v * v\nhw' : discrim 2 (-(2 * s)) (u + q / s) = w * w\n⊢ (x = (-(2 * s) + v) / (2 * 2) ∨ x = (-(2 * s) - v) / (2 * 2)) ∨\n x = (- -(2 * s) + w) / (2 * 2) ∨ x = (- -(2 * s) - w) / (2 * 2) ↔\n x = (-2 * s - v) / 4 ∨ x = (-2 * s + v) / 4 ∨ x = (2 * s - w) / 4 ∨ x = (2 * s + w) / 4"
},
{
"line": "with_reducible rfl",
"before_state": "K : Type u_1\ninst✝³ : Field K\np q r s u v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nhq_nonzero : q ≠ 0\nhu : u ^ 3 - p * u ^ 2 - 4 * r * u + 4 * p * r - q ^ 2 = 0\nhs : s ^ 2 = u - p\nhv : v ^ 2 = 4 * s ^ 2 - 8 * (u - q / s)\nhw : w ^ 2 = 4 * s ^ 2 - 8 * (u + q / s)\nx : K\nhi2 : 2 ≠ 0\nh4 : 4 = 2 ^ 2\nhs_nonzero : s ≠ 0\nhv' : discrim 2 (2 * s) (u - q / s) = v * v\nhw' : discrim 2 (-(2 * s)) (u + q / s) = w * w\n⊢ (x = (-(2 * s) + v) / (2 * 2) ∨ x = (-(2 * s) - v) / (2 * 2)) ∨\n x = (- -(2 * s) + w) / (2 * 2) ∨ x = (- -(2 * s) - w) / (2 * 2) ↔\n x = (-2 * s - v) / 4 ∨ x = (-2 * s + v) / 4 ∨ x = (2 * s - w) / 4 ∨ x = (2 * s + w) / 4",
"after_state": "K : Type u_1\ninst✝³ : Field K\np q r s u v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nhq_nonzero : q ≠ 0\nhu : u ^ 3 - p * u ^ 2 - 4 * r * u + 4 * p * r - q ^ 2 = 0\nhs : s ^ 2 = u - p\nhv : v ^ 2 = 4 * s ^ 2 - 8 * (u - q / s)\nhw : w ^ 2 = 4 * s ^ 2 - 8 * (u + q / s)\nx : K\nhi2 : 2 ≠ 0\nh4 : 4 = 2 ^ 2\nhs_nonzero : s ≠ 0\nhv' : discrim 2 (2 * s) (u - q / s) = v * v\nhw' : discrim 2 (-(2 * s)) (u + q / s) = w * w\n⊢ (x = (-(2 * s) + v) / (2 * 2) ∨ x = (-(2 * s) - v) / (2 * 2)) ∨\n x = (- -(2 * s) + w) / (2 * 2) ∨ x = (- -(2 * s) - w) / (2 * 2) ↔\n x = (-2 * s - v) / 4 ∨ x = (-2 * s + v) / 4 ∨ x = (2 * s - w) / 4 ∨ x = (2 * s + w) / 4"
},
{
"line": "rfl",
"before_state": "K : Type u_1\ninst✝³ : Field K\np q r s u v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nhq_nonzero : q ≠ 0\nhu : u ^ 3 - p * u ^ 2 - 4 * r * u + 4 * p * r - q ^ 2 = 0\nhs : s ^ 2 = u - p\nhv : v ^ 2 = 4 * s ^ 2 - 8 * (u - q / s)\nhw : w ^ 2 = 4 * s ^ 2 - 8 * (u + q / s)\nx : K\nhi2 : 2 ≠ 0\nh4 : 4 = 2 ^ 2\nhs_nonzero : s ≠ 0\nhv' : discrim 2 (2 * s) (u - q / s) = v * v\nhw' : discrim 2 (-(2 * s)) (u + q / s) = w * w\n⊢ (x = (-(2 * s) + v) / (2 * 2) ∨ x = (-(2 * s) - v) / (2 * 2)) ∨\n x = (- -(2 * s) + w) / (2 * 2) ∨ x = (- -(2 * s) - w) / (2 * 2) ↔\n x = (-2 * s - v) / 4 ∨ x = (-2 * s + v) / 4 ∨ x = (2 * s - w) / 4 ∨ x = (2 * s + w) / 4",
"after_state": "K : Type u_1\ninst✝³ : Field K\np q r s u v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nhq_nonzero : q ≠ 0\nhu : u ^ 3 - p * u ^ 2 - 4 * r * u + 4 * p * r - q ^ 2 = 0\nhs : s ^ 2 = u - p\nhv : v ^ 2 = 4 * s ^ 2 - 8 * (u - q / s)\nhw : w ^ 2 = 4 * s ^ 2 - 8 * (u + q / s)\nx : K\nhi2 : 2 ≠ 0\nh4 : 4 = 2 ^ 2\nhs_nonzero : s ≠ 0\nhv' : discrim 2 (2 * s) (u - q / s) = v * v\nhw' : discrim 2 (-(2 * s)) (u + q / s) = w * w\n⊢ (x = (-(2 * s) + v) / (2 * 2) ∨ x = (-(2 * s) - v) / (2 * 2)) ∨\n x = (- -(2 * s) + w) / (2 * 2) ∨ x = (- -(2 * s) - w) / (2 * 2) ↔\n x = (-2 * s - v) / 4 ∨ x = (-2 * s + v) / 4 ∨ x = (2 * s - w) / 4 ∨ x = (2 * s + w) / 4"
},
{
"line": "apply_rfl",
"before_state": "K : Type u_1\ninst✝³ : Field K\np q r s u v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nhq_nonzero : q ≠ 0\nhu : u ^ 3 - p * u ^ 2 - 4 * r * u + 4 * p * r - q ^ 2 = 0\nhs : s ^ 2 = u - p\nhv : v ^ 2 = 4 * s ^ 2 - 8 * (u - q / s)\nhw : w ^ 2 = 4 * s ^ 2 - 8 * (u + q / s)\nx : K\nhi2 : 2 ≠ 0\nh4 : 4 = 2 ^ 2\nhs_nonzero : s ≠ 0\nhv' : discrim 2 (2 * s) (u - q / s) = v * v\nhw' : discrim 2 (-(2 * s)) (u + q / s) = w * w\n⊢ (x = (-(2 * s) + v) / (2 * 2) ∨ x = (-(2 * s) - v) / (2 * 2)) ∨\n x = (- -(2 * s) + w) / (2 * 2) ∨ x = (- -(2 * s) - w) / (2 * 2) ↔\n x = (-2 * s - v) / 4 ∨ x = (-2 * s + v) / 4 ∨ x = (2 * s - w) / 4 ∨ x = (2 * s + w) / 4",
"after_state": "K : Type u_1\ninst✝³ : Field K\np q r s u v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nhq_nonzero : q ≠ 0\nhu : u ^ 3 - p * u ^ 2 - 4 * r * u + 4 * p * r - q ^ 2 = 0\nhs : s ^ 2 = u - p\nhv : v ^ 2 = 4 * s ^ 2 - 8 * (u - q / s)\nhw : w ^ 2 = 4 * s ^ 2 - 8 * (u + q / s)\nx : K\nhi2 : 2 ≠ 0\nh4 : 4 = 2 ^ 2\nhs_nonzero : s ≠ 0\nhv' : discrim 2 (2 * s) (u - q / s) = v * v\nhw' : discrim 2 (-(2 * s)) (u + q / s) = w * w\n⊢ (x = (-(2 * s) + v) / (2 * 2) ∨ x = (-(2 * s) - v) / (2 * 2)) ∨\n x = (- -(2 * s) + w) / (2 * 2) ∨ x = (- -(2 * s) - w) / (2 * 2) ↔\n x = (-2 * s - v) / 4 ∨ x = (-2 * s + v) / 4 ∨ x = (2 * s - w) / 4 ∨ x = (2 * s + w) / 4"
},
{
"line": "skip",
"before_state": "K : Type u_1\ninst✝³ : Field K\np q r s u v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nhq_nonzero : q ≠ 0\nhu : u ^ 3 - p * u ^ 2 - 4 * r * u + 4 * p * r - q ^ 2 = 0\nhs : s ^ 2 = u - p\nhv : v ^ 2 = 4 * s ^ 2 - 8 * (u - q / s)\nhw : w ^ 2 = 4 * s ^ 2 - 8 * (u + q / s)\nx : K\nhi2 : 2 ≠ 0\nh4 : 4 = 2 ^ 2\nhs_nonzero : s ≠ 0\nhv' : discrim 2 (2 * s) (u - q / s) = v * v\nhw' : discrim 2 (-(2 * s)) (u + q / s) = w * w\n⊢ (x = (-(2 * s) + v) / (2 * 2) ∨ x = (-(2 * s) - v) / (2 * 2)) ∨\n x = (- -(2 * s) + w) / (2 * 2) ∨ x = (- -(2 * s) - w) / (2 * 2) ↔\n x = (-2 * s - v) / 4 ∨ x = (-2 * s + v) / 4 ∨ x = (2 * s - w) / 4 ∨ x = (2 * s + w) / 4",
"after_state": "K : Type u_1\ninst✝³ : Field K\np q r s u v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nhq_nonzero : q ≠ 0\nhu : u ^ 3 - p * u ^ 2 - 4 * r * u + 4 * p * r - q ^ 2 = 0\nhs : s ^ 2 = u - p\nhv : v ^ 2 = 4 * s ^ 2 - 8 * (u - q / s)\nhw : w ^ 2 = 4 * s ^ 2 - 8 * (u + q / s)\nx : K\nhi2 : 2 ≠ 0\nh4 : 4 = 2 ^ 2\nhs_nonzero : s ≠ 0\nhv' : discrim 2 (2 * s) (u - q / s) = v * v\nhw' : discrim 2 (-(2 * s)) (u + q / s) = w * w\n⊢ (x = (-(2 * s) + v) / (2 * 2) ∨ x = (-(2 * s) - v) / (2 * 2)) ∨\n x = (- -(2 * s) + w) / (2 * 2) ∨ x = (- -(2 * s) - w) / (2 * 2) ↔\n x = (-2 * s - v) / 4 ∨ x = (-2 * s + v) / 4 ∨ x = (2 * s - w) / 4 ∨ x = (2 * s + w) / 4"
},
{
"line": "simp [(by norm_num : (2 : K) * 2 = 4), or_assoc, or_comm]",
"before_state": "K : Type u_1\ninst✝³ : Field K\np q r s u v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nhq_nonzero : q ≠ 0\nhu : u ^ 3 - p * u ^ 2 - 4 * r * u + 4 * p * r - q ^ 2 = 0\nhs : s ^ 2 = u - p\nhv : v ^ 2 = 4 * s ^ 2 - 8 * (u - q / s)\nhw : w ^ 2 = 4 * s ^ 2 - 8 * (u + q / s)\nx : K\nhi2 : 2 ≠ 0\nh4 : 4 = 2 ^ 2\nhs_nonzero : s ≠ 0\nhv' : discrim 2 (2 * s) (u - q / s) = v * v\nhw' : discrim 2 (-(2 * s)) (u + q / s) = w * w\n⊢ (x = (-(2 * s) + v) / (2 * 2) ∨ x = (-(2 * s) - v) / (2 * 2)) ∨\n x = (- -(2 * s) + w) / (2 * 2) ∨ x = (- -(2 * s) - w) / (2 * 2) ↔\n x = (-2 * s - v) / 4 ∨ x = (-2 * s + v) / 4 ∨ x = (2 * s - w) / 4 ∨ x = (2 * s + w) / 4",
"after_state": "No Goals!"
},
{
"line": "norm_num",
"before_state": "K : Type u_1\ninst✝³ : Field K\np q r s u v w : K\ninst✝² : Invertible 2\ninst✝¹ : Invertible 3\ninst✝ : Invertible 2\nhq_nonzero : q ≠ 0\nhu : u ^ 3 - p * u ^ 2 - 4 * r * u + 4 * p * r - q ^ 2 = 0\nhs : s ^ 2 = u - p\nhv : v ^ 2 = 4 * s ^ 2 - 8 * (u - q / s)\nhw : w ^ 2 = 4 * s ^ 2 - 8 * (u + q / s)\nx : K\nhi2 : 2 ≠ 0\nh4 : 4 = 2 ^ 2\nhs_nonzero : s ≠ 0\nhv' : discrim 2 (2 * s) (u - q / s) = v * v\nhw' : discrim 2 (-(2 * s)) (u + q / s) = w * w\n⊢ 2 * 2 = 4",
"after_state": "No Goals!"
}
] |
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