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!"
}
] |
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