file_path
stringlengths 11
79
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stringlengths 2
100
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list | end
list | commit
stringclasses 4
values | url
stringclasses 4
values | start
list |
|---|---|---|---|---|---|---|
Mathlib/Data/Set/Sigma.lean
|
Set.mk_sigma_iff
|
[] |
[
67,
10
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
66,
1
] |
Mathlib/GroupTheory/Perm/Basic.lean
|
Equiv.mul_swap_eq_iff
|
[
{
"state_after": "α : Type u\nβ : Type v\ninst✝ : DecidableEq α\ni j : α\nσ : Perm α\nh : σ * swap i j = σ\nswap_id : swap i j = 1\n⊢ i = j",
"state_before": "α : Type u\nβ : Type v\ninst✝ : DecidableEq α\ni j : α\nσ : Perm α\nh : σ * swap i j = σ\n⊢ i = j",
"tactic": "have swap_id : swap i j = 1 := mul_left_cancel (_root_.trans h (one_mul σ).symm)"
},
{
"state_after": "α : Type u\nβ : Type v\ninst✝ : DecidableEq α\ni j : α\nσ : Perm α\nh : σ * swap i j = σ\nswap_id : swap i j = 1\n⊢ ↑1 j = j",
"state_before": "α : Type u\nβ : Type v\ninst✝ : DecidableEq α\ni j : α\nσ : Perm α\nh : σ * swap i j = σ\nswap_id : swap i j = 1\n⊢ i = j",
"tactic": "rw [← swap_apply_right i j, swap_id]"
},
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\ninst✝ : DecidableEq α\ni j : α\nσ : Perm α\nh : σ * swap i j = σ\nswap_id : swap i j = 1\n⊢ ↑1 j = j",
"tactic": "rfl"
},
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\ninst✝ : DecidableEq α\ni j : α\nσ : Perm α\nh : i = j\n⊢ σ * swap i j = σ",
"tactic": "erw [h, swap_self, mul_one]"
}
] |
[
587,
44
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
581,
1
] |
Mathlib/RingTheory/FractionalIdeal.lean
|
FractionalIdeal.isNoetherian
|
[
{
"state_after": "case intro.intro.intro\nR : Type ?u.1863870\ninst✝⁷ : CommRing R\nS : Submonoid R\nP : Type ?u.1864077\ninst✝⁶ : CommRing P\ninst✝⁵ : Algebra R P\nloc : IsLocalization S P\nR₁ : Type u_1\ninst✝⁴ : CommRing R₁\nK : Type u_2\ninst✝³ : Field K\ninst✝² : Algebra R₁ K\nfrac : IsFractionRing R₁ K\ninst✝¹ : IsDomain R₁\ninst✝ : IsNoetherianRing R₁\nd : R₁\nJ : Ideal R₁\nleft✝ : d ≠ 0\n⊢ IsNoetherian R₁ { x // x ∈ ↑(spanSingleton R₁⁰ (↑(algebraMap R₁ K) d)⁻¹ * ↑J) }",
"state_before": "R : Type ?u.1863870\ninst✝⁷ : CommRing R\nS : Submonoid R\nP : Type ?u.1864077\ninst✝⁶ : CommRing P\ninst✝⁵ : Algebra R P\nloc : IsLocalization S P\nR₁ : Type u_1\ninst✝⁴ : CommRing R₁\nK : Type u_2\ninst✝³ : Field K\ninst✝² : Algebra R₁ K\nfrac : IsFractionRing R₁ K\ninst✝¹ : IsDomain R₁\ninst✝ : IsNoetherianRing R₁\nI : FractionalIdeal R₁⁰ K\n⊢ IsNoetherian R₁ { x // x ∈ ↑I }",
"tactic": "obtain ⟨d, J, _, rfl⟩ := exists_eq_spanSingleton_mul I"
},
{
"state_after": "case intro.intro.intro.hI\nR : Type ?u.1863870\ninst✝⁷ : CommRing R\nS : Submonoid R\nP : Type ?u.1864077\ninst✝⁶ : CommRing P\ninst✝⁵ : Algebra R P\nloc : IsLocalization S P\nR₁ : Type u_1\ninst✝⁴ : CommRing R₁\nK : Type u_2\ninst✝³ : Field K\ninst✝² : Algebra R₁ K\nfrac : IsFractionRing R₁ K\ninst✝¹ : IsDomain R₁\ninst✝ : IsNoetherianRing R₁\nd : R₁\nJ : Ideal R₁\nleft✝ : d ≠ 0\n⊢ IsNoetherian R₁ { x // x ∈ ↑↑J }",
"state_before": "case intro.intro.intro\nR : Type ?u.1863870\ninst✝⁷ : CommRing R\nS : Submonoid R\nP : Type ?u.1864077\ninst✝⁶ : CommRing P\ninst✝⁵ : Algebra R P\nloc : IsLocalization S P\nR₁ : Type u_1\ninst✝⁴ : CommRing R₁\nK : Type u_2\ninst✝³ : Field K\ninst✝² : Algebra R₁ K\nfrac : IsFractionRing R₁ K\ninst✝¹ : IsDomain R₁\ninst✝ : IsNoetherianRing R₁\nd : R₁\nJ : Ideal R₁\nleft✝ : d ≠ 0\n⊢ IsNoetherian R₁ { x // x ∈ ↑(spanSingleton R₁⁰ (↑(algebraMap R₁ K) d)⁻¹ * ↑J) }",
"tactic": "apply isNoetherian_spanSingleton_inv_to_map_mul"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.intro.hI\nR : Type ?u.1863870\ninst✝⁷ : CommRing R\nS : Submonoid R\nP : Type ?u.1864077\ninst✝⁶ : CommRing P\ninst✝⁵ : Algebra R P\nloc : IsLocalization S P\nR₁ : Type u_1\ninst✝⁴ : CommRing R₁\nK : Type u_2\ninst✝³ : Field K\ninst✝² : Algebra R₁ K\nfrac : IsFractionRing R₁ K\ninst✝¹ : IsDomain R₁\ninst✝ : IsNoetherianRing R₁\nd : R₁\nJ : Ideal R₁\nleft✝ : d ≠ 0\n⊢ IsNoetherian R₁ { x // x ∈ ↑↑J }",
"tactic": "apply isNoetherian_coeIdeal"
}
] |
[
1599,
30
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1596,
1
] |
Mathlib/Analysis/Seminorm.lean
|
Seminorm.le_def
|
[] |
[
273,
10
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
272,
1
] |
Mathlib/Analysis/SpecialFunctions/Trigonometric/Deriv.lean
|
ContDiff.csin
|
[] |
[
419,
30
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
418,
1
] |
Mathlib/Order/SymmDiff.lean
|
sdiff_symmDiff_right
|
[
{
"state_after": "no goals",
"state_before": "ι : Type ?u.65700\nα : Type u_1\nβ : Type ?u.65706\nπ : ι → Type ?u.65711\ninst✝ : GeneralizedBooleanAlgebra α\na b c d : α\n⊢ b \\ a ∆ b = a ⊓ b",
"tactic": "rw [symmDiff_comm, inf_comm, sdiff_symmDiff_left]"
}
] |
[
438,
52
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
437,
1
] |
Mathlib/Data/Dfinsupp/Basic.lean
|
Dfinsupp.filter_smul
|
[
{
"state_after": "case h\nι : Type u\nγ : Type w\nβ : ι → Type v\nβ₁ : ι → Type v₁\nβ₂ : ι → Type v₂\ninst✝³ : Monoid γ\ninst✝² : (i : ι) → AddMonoid (β i)\ninst✝¹ : (i : ι) → DistribMulAction γ (β i)\np : ι → Prop\ninst✝ : DecidablePred p\nr : γ\nf : Π₀ (i : ι), β i\ni✝ : ι\n⊢ ↑(filter p (r • f)) i✝ = ↑(r • filter p f) i✝",
"state_before": "ι : Type u\nγ : Type w\nβ : ι → Type v\nβ₁ : ι → Type v₁\nβ₂ : ι → Type v₂\ninst✝³ : Monoid γ\ninst✝² : (i : ι) → AddMonoid (β i)\ninst✝¹ : (i : ι) → DistribMulAction γ (β i)\np : ι → Prop\ninst✝ : DecidablePred p\nr : γ\nf : Π₀ (i : ι), β i\n⊢ filter p (r • f) = r • filter p f",
"tactic": "ext"
},
{
"state_after": "no goals",
"state_before": "case h\nι : Type u\nγ : Type w\nβ : ι → Type v\nβ₁ : ι → Type v₁\nβ₂ : ι → Type v₂\ninst✝³ : Monoid γ\ninst✝² : (i : ι) → AddMonoid (β i)\ninst✝¹ : (i : ι) → DistribMulAction γ (β i)\np : ι → Prop\ninst✝ : DecidablePred p\nr : γ\nf : Π₀ (i : ι), β i\ni✝ : ι\n⊢ ↑(filter p (r • f)) i✝ = ↑(r • filter p f) i✝",
"tactic": "simp [smul_apply, smul_ite]"
}
] |
[
443,
30
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
440,
1
] |
Mathlib/LinearAlgebra/Matrix/IsDiag.lean
|
Matrix.IsDiag.add
|
[
{
"state_after": "α : Type u_1\nβ : Type ?u.5295\nR : Type ?u.5298\nn : Type u_2\nm : Type ?u.5304\ninst✝ : AddZeroClass α\nA B : Matrix n n α\nha : IsDiag A\nhb : IsDiag B\ni j : n\nh : i ≠ j\n⊢ (A + B) i j = 0",
"state_before": "α : Type u_1\nβ : Type ?u.5295\nR : Type ?u.5298\nn : Type u_2\nm : Type ?u.5304\ninst✝ : AddZeroClass α\nA B : Matrix n n α\nha : IsDiag A\nhb : IsDiag B\n⊢ IsDiag (A + B)",
"tactic": "intro i j h"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.5295\nR : Type ?u.5298\nn : Type u_2\nm : Type ?u.5304\ninst✝ : AddZeroClass α\nA B : Matrix n n α\nha : IsDiag A\nhb : IsDiag B\ni j : n\nh : i ≠ j\n⊢ (A + B) i j = 0",
"tactic": "simp [ha h, hb h]"
}
] |
[
98,
20
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
95,
1
] |
Mathlib/Data/List/Sigma.lean
|
List.keys_cons
|
[] |
[
56,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
55,
1
] |
Mathlib/RingTheory/OreLocalization/Basic.lean
|
OreLocalization.zero_div_eq_zero
|
[
{
"state_after": "R : Type u_1\ninst✝¹ : Semiring R\nS : Submonoid R\ninst✝ : OreSet S\ns : { x // x ∈ S }\n⊢ ∃ u v, 0 * ↑u = 0 * v ∧ ↑1 * ↑u = ↑s * v",
"state_before": "R : Type u_1\ninst✝¹ : Semiring R\nS : Submonoid R\ninst✝ : OreSet S\ns : { x // x ∈ S }\n⊢ 0 /ₒ s = 0",
"tactic": "rw [OreLocalization.zero_def, oreDiv_eq_iff]"
},
{
"state_after": "no goals",
"state_before": "R : Type u_1\ninst✝¹ : Semiring R\nS : Submonoid R\ninst✝ : OreSet S\ns : { x // x ∈ S }\n⊢ ∃ u v, 0 * ↑u = 0 * v ∧ ↑1 * ↑u = ↑s * v",
"tactic": "exact ⟨s, 1, by simp⟩"
},
{
"state_after": "no goals",
"state_before": "R : Type u_1\ninst✝¹ : Semiring R\nS : Submonoid R\ninst✝ : OreSet S\ns : { x // x ∈ S }\n⊢ 0 * ↑s = 0 * 1 ∧ ↑1 * ↑s = ↑s * 1",
"tactic": "simp"
}
] |
[
671,
24
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
669,
1
] |
Mathlib/Algebra/Module/Submodule/Basic.lean
|
Submodule.toSubMulAction_mono
|
[] |
[
172,
37
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
171,
1
] |
Mathlib/Topology/Instances/ENNReal.lean
|
NNReal.summable_sigma
|
[
{
"state_after": "case mp\nα : Type u_2\nβ✝ : Type ?u.316734\nγ : Type ?u.316737\nβ : α → Type u_1\nf : (x : α) × β x → ℝ≥0\n⊢ Summable f →\n (∀ (x : α), Summable fun y => f { fst := x, snd := y }) ∧ Summable fun x => ∑' (y : β x), f { fst := x, snd := y }\n\ncase mpr\nα : Type u_2\nβ✝ : Type ?u.316734\nγ : Type ?u.316737\nβ : α → Type u_1\nf : (x : α) × β x → ℝ≥0\n⊢ ((∀ (x : α), Summable fun y => f { fst := x, snd := y }) ∧ Summable fun x => ∑' (y : β x), f { fst := x, snd := y }) →\n Summable f",
"state_before": "α : Type u_2\nβ✝ : Type ?u.316734\nγ : Type ?u.316737\nβ : α → Type u_1\nf : (x : α) × β x → ℝ≥0\n⊢ Summable f ↔\n (∀ (x : α), Summable fun y => f { fst := x, snd := y }) ∧ Summable fun x => ∑' (y : β x), f { fst := x, snd := y }",
"tactic": "constructor"
},
{
"state_after": "case mp\nα : Type u_2\nβ✝ : Type ?u.316734\nγ : Type ?u.316737\nβ : α → Type u_1\nf : (x : α) × β x → ℝ≥0\n⊢ (Summable fun a => ↑(f a)) →\n (∀ (x : α), Summable fun a => ↑(f { fst := x, snd := a })) ∧\n Summable fun a => ∑' (a_1 : β a), ↑(f { fst := a, snd := a_1 })",
"state_before": "case mp\nα : Type u_2\nβ✝ : Type ?u.316734\nγ : Type ?u.316737\nβ : α → Type u_1\nf : (x : α) × β x → ℝ≥0\n⊢ Summable f →\n (∀ (x : α), Summable fun y => f { fst := x, snd := y }) ∧ Summable fun x => ∑' (y : β x), f { fst := x, snd := y }",
"tactic": "simp only [← NNReal.summable_coe, NNReal.coe_tsum]"
},
{
"state_after": "no goals",
"state_before": "case mp\nα : Type u_2\nβ✝ : Type ?u.316734\nγ : Type ?u.316737\nβ : α → Type u_1\nf : (x : α) × β x → ℝ≥0\n⊢ (Summable fun a => ↑(f a)) →\n (∀ (x : α), Summable fun a => ↑(f { fst := x, snd := a })) ∧\n Summable fun a => ∑' (a_1 : β a), ↑(f { fst := a, snd := a_1 })",
"tactic": "exact fun h => ⟨h.sigma_factor, h.sigma⟩"
},
{
"state_after": "case mpr.intro\nα : Type u_2\nβ✝ : Type ?u.316734\nγ : Type ?u.316737\nβ : α → Type u_1\nf : (x : α) × β x → ℝ≥0\nh₁ : ∀ (x : α), Summable fun y => f { fst := x, snd := y }\nh₂ : Summable fun x => ∑' (y : β x), f { fst := x, snd := y }\n⊢ Summable f",
"state_before": "case mpr\nα : Type u_2\nβ✝ : Type ?u.316734\nγ : Type ?u.316737\nβ : α → Type u_1\nf : (x : α) × β x → ℝ≥0\n⊢ ((∀ (x : α), Summable fun y => f { fst := x, snd := y }) ∧ Summable fun x => ∑' (y : β x), f { fst := x, snd := y }) →\n Summable f",
"tactic": "rintro ⟨h₁, h₂⟩"
},
{
"state_after": "no goals",
"state_before": "case mpr.intro\nα : Type u_2\nβ✝ : Type ?u.316734\nγ : Type ?u.316737\nβ : α → Type u_1\nf : (x : α) × β x → ℝ≥0\nh₁ : ∀ (x : α), Summable fun y => f { fst := x, snd := y }\nh₂ : Summable fun x => ∑' (y : β x), f { fst := x, snd := y }\n⊢ Summable f",
"tactic": "simpa only [← ENNReal.tsum_coe_ne_top_iff_summable, ENNReal.tsum_sigma',\n ENNReal.coe_tsum (h₁ _)] using h₂"
}
] |
[
1172,
40
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1165,
1
] |
Mathlib/Data/List/Sort.lean
|
List.monotone_iff_ofFn_sorted
|
[] |
[
161,
99
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
161,
1
] |
Mathlib/Topology/Order/Basic.lean
|
tendsto_of_tendsto_of_tendsto_of_le_of_le'
|
[] |
[
964,
42
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
961,
1
] |
Mathlib/Data/List/Indexes.lean
|
List.oldMapIdxCore_append
|
[
{
"state_after": "α : Type u\nβ : Type v\nf : ℕ → α → β\nn : ℕ\nl₁ l₂ : List α\n⊢ List.oldMapIdxCore f n (l₁ ++ l₂) = List.oldMapIdxCore f n l₁ ++ List.oldMapIdxCore f (n + length l₁) l₂",
"state_before": "α : Type u\nβ : Type v\n⊢ ∀ (f : ℕ → α → β) (n : ℕ) (l₁ l₂ : List α),\n List.oldMapIdxCore f n (l₁ ++ l₂) = List.oldMapIdxCore f n l₁ ++ List.oldMapIdxCore f (n + length l₁) l₂",
"tactic": "intros f n l₁ l₂"
},
{
"state_after": "α : Type u\nβ : Type v\nf : ℕ → α → β\nn : ℕ\nl₁ l₂ : List α\nlen : ℕ\ne : length (l₁ ++ l₂) = len\n⊢ List.oldMapIdxCore f n (l₁ ++ l₂) = List.oldMapIdxCore f n l₁ ++ List.oldMapIdxCore f (n + length l₁) l₂",
"state_before": "α : Type u\nβ : Type v\nf : ℕ → α → β\nn : ℕ\nl₁ l₂ : List α\n⊢ List.oldMapIdxCore f n (l₁ ++ l₂) = List.oldMapIdxCore f n l₁ ++ List.oldMapIdxCore f (n + length l₁) l₂",
"tactic": "generalize e : (l₁ ++ l₂).length = len"
},
{
"state_after": "α : Type u\nβ : Type v\nf : ℕ → α → β\nlen : ℕ\n⊢ ∀ (n : ℕ) (l₁ l₂ : List α),\n length (l₁ ++ l₂) = len →\n List.oldMapIdxCore f n (l₁ ++ l₂) = List.oldMapIdxCore f n l₁ ++ List.oldMapIdxCore f (n + length l₁) l₂",
"state_before": "α : Type u\nβ : Type v\nf : ℕ → α → β\nn : ℕ\nl₁ l₂ : List α\nlen : ℕ\ne : length (l₁ ++ l₂) = len\n⊢ List.oldMapIdxCore f n (l₁ ++ l₂) = List.oldMapIdxCore f n l₁ ++ List.oldMapIdxCore f (n + length l₁) l₂",
"tactic": "revert n l₁ l₂"
},
{
"state_after": "case zero\nα : Type u\nβ : Type v\nf : ℕ → α → β\nn : ℕ\nl₁ l₂ : List α\nh : length (l₁ ++ l₂) = Nat.zero\n⊢ List.oldMapIdxCore f n (l₁ ++ l₂) = List.oldMapIdxCore f n l₁ ++ List.oldMapIdxCore f (n + length l₁) l₂\n\ncase succ\nα : Type u\nβ : Type v\nf : ℕ → α → β\nlen : ℕ\nih :\n ∀ (n : ℕ) (l₁ l₂ : List α),\n length (l₁ ++ l₂) = len →\n List.oldMapIdxCore f n (l₁ ++ l₂) = List.oldMapIdxCore f n l₁ ++ List.oldMapIdxCore f (n + length l₁) l₂\nn : ℕ\nl₁ l₂ : List α\nh : length (l₁ ++ l₂) = Nat.succ len\n⊢ List.oldMapIdxCore f n (l₁ ++ l₂) = List.oldMapIdxCore f n l₁ ++ List.oldMapIdxCore f (n + length l₁) l₂",
"state_before": "α : Type u\nβ : Type v\nf : ℕ → α → β\nlen : ℕ\n⊢ ∀ (n : ℕ) (l₁ l₂ : List α),\n length (l₁ ++ l₂) = len →\n List.oldMapIdxCore f n (l₁ ++ l₂) = List.oldMapIdxCore f n l₁ ++ List.oldMapIdxCore f (n + length l₁) l₂",
"tactic": "induction' len with len ih <;> intros n l₁ l₂ h"
},
{
"state_after": "case zero\nα : Type u\nβ : Type v\nf : ℕ → α → β\nn : ℕ\nl₁ l₂ : List α\nh : length (l₁ ++ l₂) = Nat.zero\nl₁_nil : l₁ = []\n⊢ List.oldMapIdxCore f n (l₁ ++ l₂) = List.oldMapIdxCore f n l₁ ++ List.oldMapIdxCore f (n + length l₁) l₂",
"state_before": "case zero\nα : Type u\nβ : Type v\nf : ℕ → α → β\nn : ℕ\nl₁ l₂ : List α\nh : length (l₁ ++ l₂) = Nat.zero\n⊢ List.oldMapIdxCore f n (l₁ ++ l₂) = List.oldMapIdxCore f n l₁ ++ List.oldMapIdxCore f (n + length l₁) l₂",
"tactic": "have l₁_nil : l₁ = [] := by cases l₁; rfl; contradiction"
},
{
"state_after": "case zero\nα : Type u\nβ : Type v\nf : ℕ → α → β\nn : ℕ\nl₁ l₂ : List α\nh : length (l₁ ++ l₂) = Nat.zero\nl₁_nil : l₁ = []\nl₂_nil : l₂ = []\n⊢ List.oldMapIdxCore f n (l₁ ++ l₂) = List.oldMapIdxCore f n l₁ ++ List.oldMapIdxCore f (n + length l₁) l₂",
"state_before": "case zero\nα : Type u\nβ : Type v\nf : ℕ → α → β\nn : ℕ\nl₁ l₂ : List α\nh : length (l₁ ++ l₂) = Nat.zero\nl₁_nil : l₁ = []\n⊢ List.oldMapIdxCore f n (l₁ ++ l₂) = List.oldMapIdxCore f n l₁ ++ List.oldMapIdxCore f (n + length l₁) l₂",
"tactic": "have l₂_nil : l₂ = [] := by cases l₂; rfl; rw [List.length_append] at h; contradiction"
},
{
"state_after": "case zero\nα : Type u\nβ : Type v\nf : ℕ → α → β\nn : ℕ\nl₁ l₂ : List α\nh : length (l₁ ++ l₂) = Nat.zero\nl₁_nil : l₁ = []\nl₂_nil : l₂ = []\n⊢ List.oldMapIdxCore f n ([] ++ []) = List.oldMapIdxCore f n [] ++ List.oldMapIdxCore f (n + length []) []",
"state_before": "case zero\nα : Type u\nβ : Type v\nf : ℕ → α → β\nn : ℕ\nl₁ l₂ : List α\nh : length (l₁ ++ l₂) = Nat.zero\nl₁_nil : l₁ = []\nl₂_nil : l₂ = []\n⊢ List.oldMapIdxCore f n (l₁ ++ l₂) = List.oldMapIdxCore f n l₁ ++ List.oldMapIdxCore f (n + length l₁) l₂",
"tactic": "simp only [l₁_nil, l₂_nil]"
},
{
"state_after": "no goals",
"state_before": "case zero\nα : Type u\nβ : Type v\nf : ℕ → α → β\nn : ℕ\nl₁ l₂ : List α\nh : length (l₁ ++ l₂) = Nat.zero\nl₁_nil : l₁ = []\nl₂_nil : l₂ = []\n⊢ List.oldMapIdxCore f n ([] ++ []) = List.oldMapIdxCore f n [] ++ List.oldMapIdxCore f (n + length []) []",
"tactic": "rfl"
},
{
"state_after": "case nil\nα : Type u\nβ : Type v\nf : ℕ → α → β\nn : ℕ\nl₂ : List α\nh : length ([] ++ l₂) = Nat.zero\n⊢ [] = []\n\ncase cons\nα : Type u\nβ : Type v\nf : ℕ → α → β\nn : ℕ\nl₂ : List α\nhead✝ : α\ntail✝ : List α\nh : length (head✝ :: tail✝ ++ l₂) = Nat.zero\n⊢ head✝ :: tail✝ = []",
"state_before": "α : Type u\nβ : Type v\nf : ℕ → α → β\nn : ℕ\nl₁ l₂ : List α\nh : length (l₁ ++ l₂) = Nat.zero\n⊢ l₁ = []",
"tactic": "cases l₁"
},
{
"state_after": "case cons\nα : Type u\nβ : Type v\nf : ℕ → α → β\nn : ℕ\nl₂ : List α\nhead✝ : α\ntail✝ : List α\nh : length (head✝ :: tail✝ ++ l₂) = Nat.zero\n⊢ head✝ :: tail✝ = []",
"state_before": "case nil\nα : Type u\nβ : Type v\nf : ℕ → α → β\nn : ℕ\nl₂ : List α\nh : length ([] ++ l₂) = Nat.zero\n⊢ [] = []\n\ncase cons\nα : Type u\nβ : Type v\nf : ℕ → α → β\nn : ℕ\nl₂ : List α\nhead✝ : α\ntail✝ : List α\nh : length (head✝ :: tail✝ ++ l₂) = Nat.zero\n⊢ head✝ :: tail✝ = []",
"tactic": "rfl"
},
{
"state_after": "no goals",
"state_before": "case cons\nα : Type u\nβ : Type v\nf : ℕ → α → β\nn : ℕ\nl₂ : List α\nhead✝ : α\ntail✝ : List α\nh : length (head✝ :: tail✝ ++ l₂) = Nat.zero\n⊢ head✝ :: tail✝ = []",
"tactic": "contradiction"
},
{
"state_after": "case nil\nα : Type u\nβ : Type v\nf : ℕ → α → β\nn : ℕ\nl₁ : List α\nl₁_nil : l₁ = []\nh : length (l₁ ++ []) = Nat.zero\n⊢ [] = []\n\ncase cons\nα : Type u\nβ : Type v\nf : ℕ → α → β\nn : ℕ\nl₁ : List α\nl₁_nil : l₁ = []\nhead✝ : α\ntail✝ : List α\nh : length (l₁ ++ head✝ :: tail✝) = Nat.zero\n⊢ head✝ :: tail✝ = []",
"state_before": "α : Type u\nβ : Type v\nf : ℕ → α → β\nn : ℕ\nl₁ l₂ : List α\nh : length (l₁ ++ l₂) = Nat.zero\nl₁_nil : l₁ = []\n⊢ l₂ = []",
"tactic": "cases l₂"
},
{
"state_after": "case cons\nα : Type u\nβ : Type v\nf : ℕ → α → β\nn : ℕ\nl₁ : List α\nl₁_nil : l₁ = []\nhead✝ : α\ntail✝ : List α\nh : length (l₁ ++ head✝ :: tail✝) = Nat.zero\n⊢ head✝ :: tail✝ = []",
"state_before": "case nil\nα : Type u\nβ : Type v\nf : ℕ → α → β\nn : ℕ\nl₁ : List α\nl₁_nil : l₁ = []\nh : length (l₁ ++ []) = Nat.zero\n⊢ [] = []\n\ncase cons\nα : Type u\nβ : Type v\nf : ℕ → α → β\nn : ℕ\nl₁ : List α\nl₁_nil : l₁ = []\nhead✝ : α\ntail✝ : List α\nh : length (l₁ ++ head✝ :: tail✝) = Nat.zero\n⊢ head✝ :: tail✝ = []",
"tactic": "rfl"
},
{
"state_after": "case cons\nα : Type u\nβ : Type v\nf : ℕ → α → β\nn : ℕ\nl₁ : List α\nl₁_nil : l₁ = []\nhead✝ : α\ntail✝ : List α\nh : length l₁ + length (head✝ :: tail✝) = Nat.zero\n⊢ head✝ :: tail✝ = []",
"state_before": "case cons\nα : Type u\nβ : Type v\nf : ℕ → α → β\nn : ℕ\nl₁ : List α\nl₁_nil : l₁ = []\nhead✝ : α\ntail✝ : List α\nh : length (l₁ ++ head✝ :: tail✝) = Nat.zero\n⊢ head✝ :: tail✝ = []",
"tactic": "rw [List.length_append] at h"
},
{
"state_after": "no goals",
"state_before": "case cons\nα : Type u\nβ : Type v\nf : ℕ → α → β\nn : ℕ\nl₁ : List α\nl₁_nil : l₁ = []\nhead✝ : α\ntail✝ : List α\nh : length l₁ + length (head✝ :: tail✝) = Nat.zero\n⊢ head✝ :: tail✝ = []",
"tactic": "contradiction"
},
{
"state_after": "case succ.nil\nα : Type u\nβ : Type v\nf : ℕ → α → β\nlen : ℕ\nih :\n ∀ (n : ℕ) (l₁ l₂ : List α),\n length (l₁ ++ l₂) = len →\n List.oldMapIdxCore f n (l₁ ++ l₂) = List.oldMapIdxCore f n l₁ ++ List.oldMapIdxCore f (n + length l₁) l₂\nn : ℕ\nl₂ : List α\nh : length ([] ++ l₂) = Nat.succ len\n⊢ List.oldMapIdxCore f n ([] ++ l₂) = List.oldMapIdxCore f n [] ++ List.oldMapIdxCore f (n + length []) l₂\n\ncase succ.cons\nα : Type u\nβ : Type v\nf : ℕ → α → β\nlen : ℕ\nih :\n ∀ (n : ℕ) (l₁ l₂ : List α),\n length (l₁ ++ l₂) = len →\n List.oldMapIdxCore f n (l₁ ++ l₂) = List.oldMapIdxCore f n l₁ ++ List.oldMapIdxCore f (n + length l₁) l₂\nn : ℕ\nl₂ : List α\nhead : α\ntail : List α\nh : length (head :: tail ++ l₂) = Nat.succ len\n⊢ List.oldMapIdxCore f n (head :: tail ++ l₂) =\n List.oldMapIdxCore f n (head :: tail) ++ List.oldMapIdxCore f (n + length (head :: tail)) l₂",
"state_before": "case succ\nα : Type u\nβ : Type v\nf : ℕ → α → β\nlen : ℕ\nih :\n ∀ (n : ℕ) (l₁ l₂ : List α),\n length (l₁ ++ l₂) = len →\n List.oldMapIdxCore f n (l₁ ++ l₂) = List.oldMapIdxCore f n l₁ ++ List.oldMapIdxCore f (n + length l₁) l₂\nn : ℕ\nl₁ l₂ : List α\nh : length (l₁ ++ l₂) = Nat.succ len\n⊢ List.oldMapIdxCore f n (l₁ ++ l₂) = List.oldMapIdxCore f n l₁ ++ List.oldMapIdxCore f (n + length l₁) l₂",
"tactic": "cases' l₁ with head tail"
},
{
"state_after": "no goals",
"state_before": "case succ.nil\nα : Type u\nβ : Type v\nf : ℕ → α → β\nlen : ℕ\nih :\n ∀ (n : ℕ) (l₁ l₂ : List α),\n length (l₁ ++ l₂) = len →\n List.oldMapIdxCore f n (l₁ ++ l₂) = List.oldMapIdxCore f n l₁ ++ List.oldMapIdxCore f (n + length l₁) l₂\nn : ℕ\nl₂ : List α\nh : length ([] ++ l₂) = Nat.succ len\n⊢ List.oldMapIdxCore f n ([] ++ l₂) = List.oldMapIdxCore f n [] ++ List.oldMapIdxCore f (n + length []) l₂",
"tactic": "rfl"
},
{
"state_after": "case succ.cons\nα : Type u\nβ : Type v\nf : ℕ → α → β\nlen : ℕ\nih :\n ∀ (n : ℕ) (l₁ l₂ : List α),\n length (l₁ ++ l₂) = len →\n List.oldMapIdxCore f n (l₁ ++ l₂) = List.oldMapIdxCore f n l₁ ++ List.oldMapIdxCore f (n + length l₁) l₂\nn : ℕ\nl₂ : List α\nhead : α\ntail : List α\nh : length (head :: tail ++ l₂) = Nat.succ len\n⊢ List.oldMapIdxCore f (n + 1) (tail ++ l₂) =\n List.oldMapIdxCore f (n + 1) tail ++ List.oldMapIdxCore f (n + Nat.succ (length tail)) l₂",
"state_before": "case succ.cons\nα : Type u\nβ : Type v\nf : ℕ → α → β\nlen : ℕ\nih :\n ∀ (n : ℕ) (l₁ l₂ : List α),\n length (l₁ ++ l₂) = len →\n List.oldMapIdxCore f n (l₁ ++ l₂) = List.oldMapIdxCore f n l₁ ++ List.oldMapIdxCore f (n + length l₁) l₂\nn : ℕ\nl₂ : List α\nhead : α\ntail : List α\nh : length (head :: tail ++ l₂) = Nat.succ len\n⊢ List.oldMapIdxCore f n (head :: tail ++ l₂) =\n List.oldMapIdxCore f n (head :: tail) ++ List.oldMapIdxCore f (n + length (head :: tail)) l₂",
"tactic": "simp only [List.oldMapIdxCore, List.append_eq, length_cons, cons_append,cons.injEq, true_and]"
},
{
"state_after": "case succ.cons\nα : Type u\nβ : Type v\nf : ℕ → α → β\nlen : ℕ\nih :\n ∀ (n : ℕ) (l₁ l₂ : List α),\n length (l₁ ++ l₂) = len →\n List.oldMapIdxCore f n (l₁ ++ l₂) = List.oldMapIdxCore f n l₁ ++ List.oldMapIdxCore f (n + length l₁) l₂\nn : ℕ\nl₂ : List α\nhead : α\ntail : List α\nh : length (head :: tail ++ l₂) = Nat.succ len\nthis : n + Nat.succ (length tail) = n + 1 + length tail\n⊢ List.oldMapIdxCore f (n + 1) (tail ++ l₂) =\n List.oldMapIdxCore f (n + 1) tail ++ List.oldMapIdxCore f (n + Nat.succ (length tail)) l₂\n\ncase this\nα : Type u\nβ : Type v\nf : ℕ → α → β\nlen : ℕ\nih :\n ∀ (n : ℕ) (l₁ l₂ : List α),\n length (l₁ ++ l₂) = len →\n List.oldMapIdxCore f n (l₁ ++ l₂) = List.oldMapIdxCore f n l₁ ++ List.oldMapIdxCore f (n + length l₁) l₂\nn : ℕ\nl₂ : List α\nhead : α\ntail : List α\nh : length (head :: tail ++ l₂) = Nat.succ len\n⊢ n + Nat.succ (length tail) = n + 1 + length tail",
"state_before": "case succ.cons\nα : Type u\nβ : Type v\nf : ℕ → α → β\nlen : ℕ\nih :\n ∀ (n : ℕ) (l₁ l₂ : List α),\n length (l₁ ++ l₂) = len →\n List.oldMapIdxCore f n (l₁ ++ l₂) = List.oldMapIdxCore f n l₁ ++ List.oldMapIdxCore f (n + length l₁) l₂\nn : ℕ\nl₂ : List α\nhead : α\ntail : List α\nh : length (head :: tail ++ l₂) = Nat.succ len\n⊢ List.oldMapIdxCore f (n + 1) (tail ++ l₂) =\n List.oldMapIdxCore f (n + 1) tail ++ List.oldMapIdxCore f (n + Nat.succ (length tail)) l₂",
"tactic": "suffices : n + Nat.succ (length tail) = n + 1 + tail.length"
},
{
"state_after": "case this\nα : Type u\nβ : Type v\nf : ℕ → α → β\nlen : ℕ\nih :\n ∀ (n : ℕ) (l₁ l₂ : List α),\n length (l₁ ++ l₂) = len →\n List.oldMapIdxCore f n (l₁ ++ l₂) = List.oldMapIdxCore f n l₁ ++ List.oldMapIdxCore f (n + length l₁) l₂\nn : ℕ\nl₂ : List α\nhead : α\ntail : List α\nh : length (head :: tail ++ l₂) = Nat.succ len\n⊢ n + Nat.succ (length tail) = n + 1 + length tail",
"state_before": "case succ.cons\nα : Type u\nβ : Type v\nf : ℕ → α → β\nlen : ℕ\nih :\n ∀ (n : ℕ) (l₁ l₂ : List α),\n length (l₁ ++ l₂) = len →\n List.oldMapIdxCore f n (l₁ ++ l₂) = List.oldMapIdxCore f n l₁ ++ List.oldMapIdxCore f (n + length l₁) l₂\nn : ℕ\nl₂ : List α\nhead : α\ntail : List α\nh : length (head :: tail ++ l₂) = Nat.succ len\nthis : n + Nat.succ (length tail) = n + 1 + length tail\n⊢ List.oldMapIdxCore f (n + 1) (tail ++ l₂) =\n List.oldMapIdxCore f (n + 1) tail ++ List.oldMapIdxCore f (n + Nat.succ (length tail)) l₂\n\ncase this\nα : Type u\nβ : Type v\nf : ℕ → α → β\nlen : ℕ\nih :\n ∀ (n : ℕ) (l₁ l₂ : List α),\n length (l₁ ++ l₂) = len →\n List.oldMapIdxCore f n (l₁ ++ l₂) = List.oldMapIdxCore f n l₁ ++ List.oldMapIdxCore f (n + length l₁) l₂\nn : ℕ\nl₂ : List α\nhead : α\ntail : List α\nh : length (head :: tail ++ l₂) = Nat.succ len\n⊢ n + Nat.succ (length tail) = n + 1 + length tail",
"tactic": "{ rw [this]\n apply ih (n + 1) _ _ _\n simp only [cons_append, length_cons, length_append, Nat.succ.injEq] at h\n simp only [length_append, h] }"
},
{
"state_after": "no goals",
"state_before": "case this\nα : Type u\nβ : Type v\nf : ℕ → α → β\nlen : ℕ\nih :\n ∀ (n : ℕ) (l₁ l₂ : List α),\n length (l₁ ++ l₂) = len →\n List.oldMapIdxCore f n (l₁ ++ l₂) = List.oldMapIdxCore f n l₁ ++ List.oldMapIdxCore f (n + length l₁) l₂\nn : ℕ\nl₂ : List α\nhead : α\ntail : List α\nh : length (head :: tail ++ l₂) = Nat.succ len\n⊢ n + Nat.succ (length tail) = n + 1 + length tail",
"tactic": "{ rw [Nat.add_assoc]; simp only [Nat.add_comm] }"
}
] |
[
94,
55
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
76,
11
] |
Mathlib/Analysis/Normed/MulAction.lean
|
nnnorm_smul_le
|
[] |
[
38,
19
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
37,
1
] |
Std/Data/String/Lemmas.lean
|
String.Iterator.forward_eq_nextn
|
[
{
"state_after": "case h.h\nit : Iterator\nn : Nat\n⊢ forward it n = nextn it n",
"state_before": "⊢ forward = nextn",
"tactic": "funext it n"
},
{
"state_after": "no goals",
"state_before": "case h.h\nit : Iterator\nn : Nat\n⊢ forward it n = nextn it n",
"tactic": "induction n generalizing it <;> simp [forward, nextn, *]"
}
] |
[
497,
72
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
496,
9
] |
Mathlib/Data/Num/Lemmas.lean
|
Num.cast_inj
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝ : LinearOrderedSemiring α\nm n : Num\n⊢ ↑m = ↑n ↔ m = n",
"tactic": "rw [← cast_to_nat m, ← cast_to_nat n, Nat.cast_inj, to_nat_inj]"
}
] |
[
877,
66
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
876,
1
] |
Mathlib/Analysis/SpecialFunctions/Log/Base.lean
|
Real.logb_pos
|
[
{
"state_after": "b x y : ℝ\nhb : 1 < b\nhx : 1 < x\n⊢ 1 < x",
"state_before": "b x y : ℝ\nhb : 1 < b\nhx : 1 < x\n⊢ 0 < logb b x",
"tactic": "rw [logb_pos_iff hb (lt_trans zero_lt_one hx)]"
},
{
"state_after": "no goals",
"state_before": "b x y : ℝ\nhb : 1 < b\nhx : 1 < x\n⊢ 1 < x",
"tactic": "exact hx"
}
] |
[
186,
11
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
184,
1
] |
Mathlib/Data/Set/Basic.lean
|
Set.ite_subset_union
|
[] |
[
2300,
63
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2299,
1
] |
Mathlib/Logic/Equiv/LocalEquiv.lean
|
LocalEquiv.IsImage.iff_preimage_eq
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.27993\nδ : Type ?u.27996\ne : LocalEquiv α β\ne' : LocalEquiv β γ\ns : Set α\nt : Set β\nx : α\ny : β\n⊢ IsImage e s t ↔ e.source ∩ ↑e ⁻¹' t = e.source ∩ s",
"tactic": "simp only [IsImage, ext_iff, mem_inter_iff, mem_preimage, and_congr_right_iff]"
}
] |
[
413,
81
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
412,
1
] |
Mathlib/GroupTheory/Sylow.lean
|
Sylow.exists_subgroup_card_pow_succ
|
[
{
"state_after": "no goals",
"state_before": "G : Type u\nα : Type v\nβ : Type w\ninst✝¹ : Group G\ninst✝ : Fintype G\np n : ℕ\nhp : Fact (Nat.Prime p)\nhdvd : p ^ (n + 1) ∣ Fintype.card G\nH : Subgroup G\nhH : Fintype.card { x // x ∈ H } = p ^ n\ns : ℕ\nhs : Fintype.card G = s * p ^ (n + 1)\n⊢ Fintype.card (G ⧸ H) * Fintype.card { x // x ∈ H } = s * p * Fintype.card { x // x ∈ H }",
"tactic": "rw [← card_eq_card_quotient_mul_card_subgroup H, hH, hs, pow_succ', mul_assoc, mul_comm p]"
},
{
"state_after": "no goals",
"state_before": "G : Type u\nα : Type v\nβ : Type w\ninst✝¹ : Group G\ninst✝ : Fintype G\np n : ℕ\nhp : Fact (Nat.Prime p)\nhdvd : p ^ (n + 1) ∣ Fintype.card G\nH : Subgroup G\nhH : Fintype.card { x // x ∈ H } = p ^ n\ns : ℕ\nhs : Fintype.card G = s * p ^ (n + 1)\nhcard : Fintype.card (G ⧸ H) = s * p\nhm : s * p % p = Fintype.card ({ x // x ∈ normalizer H } ⧸ subgroupOf H (normalizer H)) % p\n⊢ Fintype.card ({ x // x ∈ normalizer H } ⧸ subgroupOf H (normalizer H)) % p = 0",
"tactic": "rwa [Nat.mod_eq_zero_of_dvd (dvd_mul_left _ _), eq_comm] at hm"
},
{
"state_after": "G : Type u\nα : Type v\nβ : Type w\ninst✝¹ : Group G\ninst✝ : Fintype G\np n : ℕ\nhp : Fact (Nat.Prime p)\nhdvd : p ^ (n + 1) ∣ Fintype.card G\nH : Subgroup G\nhH : Fintype.card { x // x ∈ H } = p ^ n\ns : ℕ\nhs : Fintype.card G = s * p ^ (n + 1)\nhcard : Fintype.card (G ⧸ H) = s * p\nhm : s * p % p = Fintype.card ({ x // x ∈ normalizer H } ⧸ subgroupOf H (normalizer H)) % p\nhm' : p ∣ Fintype.card ({ x // x ∈ normalizer H } ⧸ subgroupOf H (normalizer H))\nx : { x // x ∈ normalizer H } ⧸ subgroupOf H (normalizer H)\nhx : orderOf x = p\nhequiv : { x // x ∈ H } ≃ { x // x ∈ subgroupOf H (normalizer H) }\n⊢ Fintype.card\n { x_1 //\n x_1 ∈\n Subgroup.map (Subgroup.subtype (normalizer H))\n (Subgroup.comap (mk' (subgroupOf H (normalizer H))) (zpowers x)) } =\n p ^ (n + 1)",
"state_before": "G : Type u\nα : Type v\nβ : Type w\ninst✝¹ : Group G\ninst✝ : Fintype G\np n : ℕ\nhp : Fact (Nat.Prime p)\nhdvd : p ^ (n + 1) ∣ Fintype.card G\nH : Subgroup G\nhH : Fintype.card { x // x ∈ H } = p ^ n\ns : ℕ\nhs : Fintype.card G = s * p ^ (n + 1)\nhcard : Fintype.card (G ⧸ H) = s * p\nhm : s * p % p = Fintype.card ({ x // x ∈ normalizer H } ⧸ subgroupOf H (normalizer H)) % p\nhm' : p ∣ Fintype.card ({ x // x ∈ normalizer H } ⧸ subgroupOf H (normalizer H))\nx : { x // x ∈ normalizer H } ⧸ subgroupOf H (normalizer H)\nhx : orderOf x = p\nhequiv : { x // x ∈ H } ≃ { x // x ∈ subgroupOf H (normalizer H) }\n⊢ Fintype.card\n { x_1 //\n x_1 ∈\n Subgroup.map (Subgroup.subtype (normalizer H))\n (Subgroup.comap (mk' (subgroupOf H (normalizer H))) (zpowers x)) } =\n p ^ (n + 1)",
"tactic": "show Fintype.card (Subgroup.map H.normalizer.subtype\n (comap (mk' (H.subgroupOf H.normalizer)) (Subgroup.zpowers x))) = p ^ (n + 1)"
},
{
"state_after": "G : Type u\nα : Type v\nβ : Type w\ninst✝¹ : Group G\ninst✝ : Fintype G\np n : ℕ\nhp : Fact (Nat.Prime p)\nhdvd : p ^ (n + 1) ∣ Fintype.card G\nH : Subgroup G\nhH : Fintype.card { x // x ∈ H } = p ^ n\ns : ℕ\nhs : Fintype.card G = s * p ^ (n + 1)\nhcard : Fintype.card (G ⧸ H) = s * p\nhm : s * p % p = Fintype.card ({ x // x ∈ normalizer H } ⧸ subgroupOf H (normalizer H)) % p\nhm' : p ∣ Fintype.card ({ x // x ∈ normalizer H } ⧸ subgroupOf H (normalizer H))\nx : { x // x ∈ normalizer H } ⧸ subgroupOf H (normalizer H)\nhx : orderOf x = p\nhequiv : { x // x ∈ H } ≃ { x // x ∈ subgroupOf H (normalizer H) }\n⊢ Fintype.card ↑(Subtype.val '' ↑(Subgroup.comap (mk' (subgroupOf H (normalizer H))) (zpowers x))) = p ^ (n + 1)",
"state_before": "G : Type u\nα : Type v\nβ : Type w\ninst✝¹ : Group G\ninst✝ : Fintype G\np n : ℕ\nhp : Fact (Nat.Prime p)\nhdvd : p ^ (n + 1) ∣ Fintype.card G\nH : Subgroup G\nhH : Fintype.card { x // x ∈ H } = p ^ n\ns : ℕ\nhs : Fintype.card G = s * p ^ (n + 1)\nhcard : Fintype.card (G ⧸ H) = s * p\nhm : s * p % p = Fintype.card ({ x // x ∈ normalizer H } ⧸ subgroupOf H (normalizer H)) % p\nhm' : p ∣ Fintype.card ({ x // x ∈ normalizer H } ⧸ subgroupOf H (normalizer H))\nx : { x // x ∈ normalizer H } ⧸ subgroupOf H (normalizer H)\nhx : orderOf x = p\nhequiv : { x // x ∈ H } ≃ { x // x ∈ subgroupOf H (normalizer H) }\n⊢ Fintype.card\n { x_1 //\n x_1 ∈\n Subgroup.map (Subgroup.subtype (normalizer H))\n (Subgroup.comap (mk' (subgroupOf H (normalizer H))) (zpowers x)) } =\n p ^ (n + 1)",
"tactic": "suffices Fintype.card (Subtype.val ''\n (Subgroup.comap (mk' (H.subgroupOf H.normalizer)) (zpowers x) : Set H.normalizer)) =\n p ^ (n + 1)\n by convert this using 2"
},
{
"state_after": "G : Type u\nα : Type v\nβ : Type w\ninst✝¹ : Group G\ninst✝ : Fintype G\np n : ℕ\nhp : Fact (Nat.Prime p)\nhdvd : p ^ (n + 1) ∣ Fintype.card G\nH : Subgroup G\nhH : Fintype.card { x // x ∈ H } = p ^ n\ns : ℕ\nhs : Fintype.card G = s * p ^ (n + 1)\nhcard : Fintype.card (G ⧸ H) = s * p\nhm : s * p % p = Fintype.card ({ x // x ∈ normalizer H } ⧸ subgroupOf H (normalizer H)) % p\nhm' : p ∣ Fintype.card ({ x // x ∈ normalizer H } ⧸ subgroupOf H (normalizer H))\nx : { x // x ∈ normalizer H } ⧸ subgroupOf H (normalizer H)\nhx : orderOf x = p\nhequiv : { x // x ∈ H } ≃ { x // x ∈ subgroupOf H (normalizer H) }\n⊢ Fintype.card ↑↑(Subgroup.comap (mk' (subgroupOf H (normalizer H))) (zpowers x)) =\n Fintype.card ({ x // x ∈ subgroupOf H (normalizer H) } × { x_1 // x_1 ∈ zpowers x })",
"state_before": "G : Type u\nα : Type v\nβ : Type w\ninst✝¹ : Group G\ninst✝ : Fintype G\np n : ℕ\nhp : Fact (Nat.Prime p)\nhdvd : p ^ (n + 1) ∣ Fintype.card G\nH : Subgroup G\nhH : Fintype.card { x // x ∈ H } = p ^ n\ns : ℕ\nhs : Fintype.card G = s * p ^ (n + 1)\nhcard : Fintype.card (G ⧸ H) = s * p\nhm : s * p % p = Fintype.card ({ x // x ∈ normalizer H } ⧸ subgroupOf H (normalizer H)) % p\nhm' : p ∣ Fintype.card ({ x // x ∈ normalizer H } ⧸ subgroupOf H (normalizer H))\nx : { x // x ∈ normalizer H } ⧸ subgroupOf H (normalizer H)\nhx : orderOf x = p\nhequiv : { x // x ∈ H } ≃ { x // x ∈ subgroupOf H (normalizer H) }\n⊢ Fintype.card ↑(Subtype.val '' ↑(Subgroup.comap (mk' (subgroupOf H (normalizer H))) (zpowers x))) = p ^ (n + 1)",
"tactic": "rw [Set.card_image_of_injective\n (Subgroup.comap (mk' (H.subgroupOf H.normalizer)) (zpowers x) : Set H.normalizer)\n Subtype.val_injective,\n pow_succ', ← hH, Fintype.card_congr hequiv, ← hx, orderOf_eq_card_zpowers, ←\n Fintype.card_prod]"
},
{
"state_after": "no goals",
"state_before": "G : Type u\nα : Type v\nβ : Type w\ninst✝¹ : Group G\ninst✝ : Fintype G\np n : ℕ\nhp : Fact (Nat.Prime p)\nhdvd : p ^ (n + 1) ∣ Fintype.card G\nH : Subgroup G\nhH : Fintype.card { x // x ∈ H } = p ^ n\ns : ℕ\nhs : Fintype.card G = s * p ^ (n + 1)\nhcard : Fintype.card (G ⧸ H) = s * p\nhm : s * p % p = Fintype.card ({ x // x ∈ normalizer H } ⧸ subgroupOf H (normalizer H)) % p\nhm' : p ∣ Fintype.card ({ x // x ∈ normalizer H } ⧸ subgroupOf H (normalizer H))\nx : { x // x ∈ normalizer H } ⧸ subgroupOf H (normalizer H)\nhx : orderOf x = p\nhequiv : { x // x ∈ H } ≃ { x // x ∈ subgroupOf H (normalizer H) }\n⊢ Fintype.card ↑↑(Subgroup.comap (mk' (subgroupOf H (normalizer H))) (zpowers x)) =\n Fintype.card ({ x // x ∈ subgroupOf H (normalizer H) } × { x_1 // x_1 ∈ zpowers x })",
"tactic": "exact @Fintype.card_congr _ _ (_) (_)\n (preimageMkEquivSubgroupProdSet (H.subgroupOf H.normalizer) (zpowers x))"
},
{
"state_after": "no goals",
"state_before": "G : Type u\nα : Type v\nβ : Type w\ninst✝¹ : Group G\ninst✝ : Fintype G\np n : ℕ\nhp : Fact (Nat.Prime p)\nhdvd : p ^ (n + 1) ∣ Fintype.card G\nH : Subgroup G\nhH : Fintype.card { x // x ∈ H } = p ^ n\ns : ℕ\nhs : Fintype.card G = s * p ^ (n + 1)\nhcard : Fintype.card (G ⧸ H) = s * p\nhm : s * p % p = Fintype.card ({ x // x ∈ normalizer H } ⧸ subgroupOf H (normalizer H)) % p\nhm' : p ∣ Fintype.card ({ x // x ∈ normalizer H } ⧸ subgroupOf H (normalizer H))\nx : { x // x ∈ normalizer H } ⧸ subgroupOf H (normalizer H)\nhx : orderOf x = p\nhequiv : { x // x ∈ H } ≃ { x // x ∈ subgroupOf H (normalizer H) }\nthis : Fintype.card ↑(Subtype.val '' ↑(Subgroup.comap (mk' (subgroupOf H (normalizer H))) (zpowers x))) = p ^ (n + 1)\n⊢ Fintype.card\n { x_1 //\n x_1 ∈\n Subgroup.map (Subgroup.subtype (normalizer H))\n (Subgroup.comap (mk' (subgroupOf H (normalizer H))) (zpowers x)) } =\n p ^ (n + 1)",
"tactic": "convert this using 2"
},
{
"state_after": "G : Type u\nα : Type v\nβ : Type w\ninst✝¹ : Group G\ninst✝ : Fintype G\np n : ℕ\nhp : Fact (Nat.Prime p)\nhdvd : p ^ (n + 1) ∣ Fintype.card G\nH : Subgroup G\nhH : Fintype.card { x // x ∈ H } = p ^ n\ns : ℕ\nhs : Fintype.card G = s * p ^ (n + 1)\nhcard : Fintype.card (G ⧸ H) = s * p\nhm : s * p % p = Fintype.card ({ x // x ∈ normalizer H } ⧸ subgroupOf H (normalizer H)) % p\nhm' : p ∣ Fintype.card ({ x // x ∈ normalizer H } ⧸ subgroupOf H (normalizer H))\nx : { x // x ∈ normalizer H } ⧸ subgroupOf H (normalizer H)\nhx : orderOf x = p\nhequiv : { x // x ∈ H } ≃ { x // x ∈ subgroupOf H (normalizer H) }\ny : G\nhy : y ∈ H\n⊢ y ∈ Subgroup.map (Subgroup.subtype (normalizer H)) (Subgroup.comap (mk' (subgroupOf H (normalizer H))) (zpowers x))",
"state_before": "G : Type u\nα : Type v\nβ : Type w\ninst✝¹ : Group G\ninst✝ : Fintype G\np n : ℕ\nhp : Fact (Nat.Prime p)\nhdvd : p ^ (n + 1) ∣ Fintype.card G\nH : Subgroup G\nhH : Fintype.card { x // x ∈ H } = p ^ n\ns : ℕ\nhs : Fintype.card G = s * p ^ (n + 1)\nhcard : Fintype.card (G ⧸ H) = s * p\nhm : s * p % p = Fintype.card ({ x // x ∈ normalizer H } ⧸ subgroupOf H (normalizer H)) % p\nhm' : p ∣ Fintype.card ({ x // x ∈ normalizer H } ⧸ subgroupOf H (normalizer H))\nx : { x // x ∈ normalizer H } ⧸ subgroupOf H (normalizer H)\nhx : orderOf x = p\nhequiv : { x // x ∈ H } ≃ { x // x ∈ subgroupOf H (normalizer H) }\n⊢ H ≤ Subgroup.map (Subgroup.subtype (normalizer H)) (Subgroup.comap (mk' (subgroupOf H (normalizer H))) (zpowers x))",
"tactic": "intro y hy"
},
{
"state_after": "G : Type u\nα : Type v\nβ : Type w\ninst✝¹ : Group G\ninst✝ : Fintype G\np n : ℕ\nhp : Fact (Nat.Prime p)\nhdvd : p ^ (n + 1) ∣ Fintype.card G\nH : Subgroup G\nhH : Fintype.card { x // x ∈ H } = p ^ n\ns : ℕ\nhs : Fintype.card G = s * p ^ (n + 1)\nhcard : Fintype.card (G ⧸ H) = s * p\nhm : s * p % p = Fintype.card ({ x // x ∈ normalizer H } ⧸ subgroupOf H (normalizer H)) % p\nhm' : p ∣ Fintype.card ({ x // x ∈ normalizer H } ⧸ subgroupOf H (normalizer H))\nx : { x // x ∈ normalizer H } ⧸ subgroupOf H (normalizer H)\nhx : orderOf x = p\nhequiv : { x // x ∈ H } ≃ { x // x ∈ subgroupOf H (normalizer H) }\ny : G\nhy : y ∈ H\n⊢ ∃ x_1, ↑x_1 ∈ zpowers x ∧ ↑x_1 = y",
"state_before": "G : Type u\nα : Type v\nβ : Type w\ninst✝¹ : Group G\ninst✝ : Fintype G\np n : ℕ\nhp : Fact (Nat.Prime p)\nhdvd : p ^ (n + 1) ∣ Fintype.card G\nH : Subgroup G\nhH : Fintype.card { x // x ∈ H } = p ^ n\ns : ℕ\nhs : Fintype.card G = s * p ^ (n + 1)\nhcard : Fintype.card (G ⧸ H) = s * p\nhm : s * p % p = Fintype.card ({ x // x ∈ normalizer H } ⧸ subgroupOf H (normalizer H)) % p\nhm' : p ∣ Fintype.card ({ x // x ∈ normalizer H } ⧸ subgroupOf H (normalizer H))\nx : { x // x ∈ normalizer H } ⧸ subgroupOf H (normalizer H)\nhx : orderOf x = p\nhequiv : { x // x ∈ H } ≃ { x // x ∈ subgroupOf H (normalizer H) }\ny : G\nhy : y ∈ H\n⊢ y ∈ Subgroup.map (Subgroup.subtype (normalizer H)) (Subgroup.comap (mk' (subgroupOf H (normalizer H))) (zpowers x))",
"tactic": "simp only [exists_prop, Subgroup.coeSubtype, mk'_apply, Subgroup.mem_map, Subgroup.mem_comap]"
},
{
"state_after": "G : Type u\nα : Type v\nβ : Type w\ninst✝¹ : Group G\ninst✝ : Fintype G\np n : ℕ\nhp : Fact (Nat.Prime p)\nhdvd : p ^ (n + 1) ∣ Fintype.card G\nH : Subgroup G\nhH : Fintype.card { x // x ∈ H } = p ^ n\ns : ℕ\nhs : Fintype.card G = s * p ^ (n + 1)\nhcard : Fintype.card (G ⧸ H) = s * p\nhm : s * p % p = Fintype.card ({ x // x ∈ normalizer H } ⧸ subgroupOf H (normalizer H)) % p\nhm' : p ∣ Fintype.card ({ x // x ∈ normalizer H } ⧸ subgroupOf H (normalizer H))\nx : { x // x ∈ normalizer H } ⧸ subgroupOf H (normalizer H)\nhx : orderOf x = p\nhequiv : { x // x ∈ H } ≃ { x // x ∈ subgroupOf H (normalizer H) }\ny : G\nhy : y ∈ H\n⊢ (fun x x_1 => x ^ x_1) x 0 = ↑{ val := y, property := (_ : y ∈ normalizer H) }",
"state_before": "G : Type u\nα : Type v\nβ : Type w\ninst✝¹ : Group G\ninst✝ : Fintype G\np n : ℕ\nhp : Fact (Nat.Prime p)\nhdvd : p ^ (n + 1) ∣ Fintype.card G\nH : Subgroup G\nhH : Fintype.card { x // x ∈ H } = p ^ n\ns : ℕ\nhs : Fintype.card G = s * p ^ (n + 1)\nhcard : Fintype.card (G ⧸ H) = s * p\nhm : s * p % p = Fintype.card ({ x // x ∈ normalizer H } ⧸ subgroupOf H (normalizer H)) % p\nhm' : p ∣ Fintype.card ({ x // x ∈ normalizer H } ⧸ subgroupOf H (normalizer H))\nx : { x // x ∈ normalizer H } ⧸ subgroupOf H (normalizer H)\nhx : orderOf x = p\nhequiv : { x // x ∈ H } ≃ { x // x ∈ subgroupOf H (normalizer H) }\ny : G\nhy : y ∈ H\n⊢ ∃ x_1, ↑x_1 ∈ zpowers x ∧ ↑x_1 = y",
"tactic": "refine' ⟨⟨y, le_normalizer hy⟩, ⟨0, _⟩, rfl⟩"
},
{
"state_after": "G : Type u\nα : Type v\nβ : Type w\ninst✝¹ : Group G\ninst✝ : Fintype G\np n : ℕ\nhp : Fact (Nat.Prime p)\nhdvd : p ^ (n + 1) ∣ Fintype.card G\nH : Subgroup G\nhH : Fintype.card { x // x ∈ H } = p ^ n\ns : ℕ\nhs : Fintype.card G = s * p ^ (n + 1)\nhcard : Fintype.card (G ⧸ H) = s * p\nhm : s * p % p = Fintype.card ({ x // x ∈ normalizer H } ⧸ subgroupOf H (normalizer H)) % p\nhm' : p ∣ Fintype.card ({ x // x ∈ normalizer H } ⧸ subgroupOf H (normalizer H))\nx : { x // x ∈ normalizer H } ⧸ subgroupOf H (normalizer H)\nhx : orderOf x = p\nhequiv : { x // x ∈ H } ≃ { x // x ∈ subgroupOf H (normalizer H) }\ny : G\nhy : y ∈ H\n⊢ x ^ 0 = ↑{ val := y, property := (_ : y ∈ normalizer H) }",
"state_before": "G : Type u\nα : Type v\nβ : Type w\ninst✝¹ : Group G\ninst✝ : Fintype G\np n : ℕ\nhp : Fact (Nat.Prime p)\nhdvd : p ^ (n + 1) ∣ Fintype.card G\nH : Subgroup G\nhH : Fintype.card { x // x ∈ H } = p ^ n\ns : ℕ\nhs : Fintype.card G = s * p ^ (n + 1)\nhcard : Fintype.card (G ⧸ H) = s * p\nhm : s * p % p = Fintype.card ({ x // x ∈ normalizer H } ⧸ subgroupOf H (normalizer H)) % p\nhm' : p ∣ Fintype.card ({ x // x ∈ normalizer H } ⧸ subgroupOf H (normalizer H))\nx : { x // x ∈ normalizer H } ⧸ subgroupOf H (normalizer H)\nhx : orderOf x = p\nhequiv : { x // x ∈ H } ≃ { x // x ∈ subgroupOf H (normalizer H) }\ny : G\nhy : y ∈ H\n⊢ (fun x x_1 => x ^ x_1) x 0 = ↑{ val := y, property := (_ : y ∈ normalizer H) }",
"tactic": "dsimp only"
},
{
"state_after": "G : Type u\nα : Type v\nβ : Type w\ninst✝¹ : Group G\ninst✝ : Fintype G\np n : ℕ\nhp : Fact (Nat.Prime p)\nhdvd : p ^ (n + 1) ∣ Fintype.card G\nH : Subgroup G\nhH : Fintype.card { x // x ∈ H } = p ^ n\ns : ℕ\nhs : Fintype.card G = s * p ^ (n + 1)\nhcard : Fintype.card (G ⧸ H) = s * p\nhm : s * p % p = Fintype.card ({ x // x ∈ normalizer H } ⧸ subgroupOf H (normalizer H)) % p\nhm' : p ∣ Fintype.card ({ x // x ∈ normalizer H } ⧸ subgroupOf H (normalizer H))\nx : { x // x ∈ normalizer H } ⧸ subgroupOf H (normalizer H)\nhx : orderOf x = p\nhequiv : { x // x ∈ H } ≃ { x // x ∈ subgroupOf H (normalizer H) }\ny : G\nhy : y ∈ H\n⊢ { val := y, property := (_ : y ∈ normalizer H) } ∈ subgroupOf H (normalizer H)",
"state_before": "G : Type u\nα : Type v\nβ : Type w\ninst✝¹ : Group G\ninst✝ : Fintype G\np n : ℕ\nhp : Fact (Nat.Prime p)\nhdvd : p ^ (n + 1) ∣ Fintype.card G\nH : Subgroup G\nhH : Fintype.card { x // x ∈ H } = p ^ n\ns : ℕ\nhs : Fintype.card G = s * p ^ (n + 1)\nhcard : Fintype.card (G ⧸ H) = s * p\nhm : s * p % p = Fintype.card ({ x // x ∈ normalizer H } ⧸ subgroupOf H (normalizer H)) % p\nhm' : p ∣ Fintype.card ({ x // x ∈ normalizer H } ⧸ subgroupOf H (normalizer H))\nx : { x // x ∈ normalizer H } ⧸ subgroupOf H (normalizer H)\nhx : orderOf x = p\nhequiv : { x // x ∈ H } ≃ { x // x ∈ subgroupOf H (normalizer H) }\ny : G\nhy : y ∈ H\n⊢ x ^ 0 = ↑{ val := y, property := (_ : y ∈ normalizer H) }",
"tactic": "rw [zpow_zero, eq_comm, QuotientGroup.eq_one_iff]"
},
{
"state_after": "no goals",
"state_before": "G : Type u\nα : Type v\nβ : Type w\ninst✝¹ : Group G\ninst✝ : Fintype G\np n : ℕ\nhp : Fact (Nat.Prime p)\nhdvd : p ^ (n + 1) ∣ Fintype.card G\nH : Subgroup G\nhH : Fintype.card { x // x ∈ H } = p ^ n\ns : ℕ\nhs : Fintype.card G = s * p ^ (n + 1)\nhcard : Fintype.card (G ⧸ H) = s * p\nhm : s * p % p = Fintype.card ({ x // x ∈ normalizer H } ⧸ subgroupOf H (normalizer H)) % p\nhm' : p ∣ Fintype.card ({ x // x ∈ normalizer H } ⧸ subgroupOf H (normalizer H))\nx : { x // x ∈ normalizer H } ⧸ subgroupOf H (normalizer H)\nhx : orderOf x = p\nhequiv : { x // x ∈ H } ≃ { x // x ∈ subgroupOf H (normalizer H) }\ny : G\nhy : y ∈ H\n⊢ { val := y, property := (_ : y ∈ normalizer H) } ∈ subgroupOf H (normalizer H)",
"tactic": "simpa using hy"
}
] |
[
629,
20
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
594,
1
] |
Mathlib/GroupTheory/Perm/Support.lean
|
Equiv.Perm.nodup_of_pairwise_disjoint
|
[
{
"state_after": "α : Type u_1\nf g h : Perm α\nl : List (Perm α)\nh1 : ¬1 ∈ l\nh2 : List.Pairwise Disjoint l\n⊢ ∀ {a b : Perm α}, a ∈ l → b ∈ l → Disjoint a b → a ≠ b",
"state_before": "α : Type u_1\nf g h : Perm α\nl : List (Perm α)\nh1 : ¬1 ∈ l\nh2 : List.Pairwise Disjoint l\n⊢ List.Nodup l",
"tactic": "refine' List.Pairwise.imp_of_mem _ h2"
},
{
"state_after": "α : Type u_1\nf g h : Perm α\nl : List (Perm α)\nh1 : ¬1 ∈ l\nh2 : List.Pairwise Disjoint l\nτ σ : Perm α\nh_mem : τ ∈ l\na✝¹ : σ ∈ l\nh_disjoint : Disjoint τ σ\na✝ : τ = σ\n⊢ False",
"state_before": "α : Type u_1\nf g h : Perm α\nl : List (Perm α)\nh1 : ¬1 ∈ l\nh2 : List.Pairwise Disjoint l\n⊢ ∀ {a b : Perm α}, a ∈ l → b ∈ l → Disjoint a b → a ≠ b",
"tactic": "intro τ σ h_mem _ h_disjoint _"
},
{
"state_after": "α : Type u_1\nf g h : Perm α\nl : List (Perm α)\nh1 : ¬1 ∈ l\nh2 : List.Pairwise Disjoint l\nσ : Perm α\na✝ h_mem : σ ∈ l\nh_disjoint : Disjoint σ σ\n⊢ False",
"state_before": "α : Type u_1\nf g h : Perm α\nl : List (Perm α)\nh1 : ¬1 ∈ l\nh2 : List.Pairwise Disjoint l\nτ σ : Perm α\nh_mem : τ ∈ l\na✝¹ : σ ∈ l\nh_disjoint : Disjoint τ σ\na✝ : τ = σ\n⊢ False",
"tactic": "subst τ"
},
{
"state_after": "α : Type u_1\nf g h : Perm α\nl : List (Perm α)\nh1 : ¬1 ∈ l\nh2 : List.Pairwise Disjoint l\nσ : Perm α\na✝ h_mem : σ ∈ l\nh_disjoint : Disjoint σ σ\n⊢ σ = 1",
"state_before": "α : Type u_1\nf g h : Perm α\nl : List (Perm α)\nh1 : ¬1 ∈ l\nh2 : List.Pairwise Disjoint l\nσ : Perm α\na✝ h_mem : σ ∈ l\nh_disjoint : Disjoint σ σ\n⊢ False",
"tactic": "suffices (σ : Perm α) = 1 by\n rw [this] at h_mem\n exact h1 h_mem"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nf g h : Perm α\nl : List (Perm α)\nh1 : ¬1 ∈ l\nh2 : List.Pairwise Disjoint l\nσ : Perm α\na✝ h_mem : σ ∈ l\nh_disjoint : Disjoint σ σ\n⊢ σ = 1",
"tactic": "exact ext fun a => (or_self_iff _).mp (h_disjoint a)"
},
{
"state_after": "α : Type u_1\nf g h : Perm α\nl : List (Perm α)\nh1 : ¬1 ∈ l\nh2 : List.Pairwise Disjoint l\nσ : Perm α\na✝ : σ ∈ l\nh_mem : 1 ∈ l\nh_disjoint : Disjoint σ σ\nthis : σ = 1\n⊢ False",
"state_before": "α : Type u_1\nf g h : Perm α\nl : List (Perm α)\nh1 : ¬1 ∈ l\nh2 : List.Pairwise Disjoint l\nσ : Perm α\na✝ h_mem : σ ∈ l\nh_disjoint : Disjoint σ σ\nthis : σ = 1\n⊢ False",
"tactic": "rw [this] at h_mem"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nf g h : Perm α\nl : List (Perm α)\nh1 : ¬1 ∈ l\nh2 : List.Pairwise Disjoint l\nσ : Perm α\na✝ : σ ∈ l\nh_mem : 1 ∈ l\nh_disjoint : Disjoint σ σ\nthis : σ = 1\n⊢ False",
"tactic": "exact h1 h_mem"
}
] |
[
148,
55
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
140,
1
] |
Mathlib/Topology/LocalExtr.lean
|
IsLocalMax.neg
|
[] |
[
401,
9
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
400,
8
] |
Mathlib/Analysis/Asymptotics/Theta.lean
|
Asymptotics.IsTheta.tendsto_norm_atTop_iff
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.35351\nE : Type ?u.35354\nF : Type ?u.35357\nG : Type ?u.35360\nE' : Type u_2\nF' : Type u_3\nG' : Type ?u.35369\nE'' : Type ?u.35372\nF'' : Type ?u.35375\nG'' : Type ?u.35378\nR : Type ?u.35381\nR' : Type ?u.35384\n𝕜 : Type ?u.35387\n𝕜' : Type ?u.35390\ninst✝¹² : Norm E\ninst✝¹¹ : Norm F\ninst✝¹⁰ : Norm G\ninst✝⁹ : SeminormedAddCommGroup E'\ninst✝⁸ : SeminormedAddCommGroup F'\ninst✝⁷ : SeminormedAddCommGroup G'\ninst✝⁶ : NormedAddCommGroup E''\ninst✝⁵ : NormedAddCommGroup F''\ninst✝⁴ : NormedAddCommGroup G''\ninst✝³ : SeminormedRing R\ninst✝² : SeminormedRing R'\ninst✝¹ : NormedField 𝕜\ninst✝ : NormedField 𝕜'\nc c' c₁ c₂ : ℝ\nf : α → E\ng : α → F\nk : α → G\nf' : α → E'\ng' : α → F'\nk' : α → G'\nf'' : α → E''\ng'' : α → F''\nl l' : Filter α\nh : f' =Θ[l] g'\n⊢ Tendsto (norm ∘ f') l atTop ↔ Tendsto (norm ∘ g') l atTop",
"tactic": "simp only [Function.comp, ← isLittleO_const_left_of_ne (one_ne_zero' ℝ), h.isLittleO_congr_right]"
}
] |
[
213,
100
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
211,
1
] |
Mathlib/Logic/Nonempty.lean
|
nonempty_subtype
|
[] |
[
60,
58
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
59,
1
] |
Mathlib/Data/List/FinRange.lean
|
List.map_coe_finRange
|
[
{
"state_after": "α : Type u\nn : ℕ\n⊢ map (fun a => a) (range n) = range n",
"state_before": "α : Type u\nn : ℕ\n⊢ map Fin.val (finRange n) = range n",
"tactic": "simp_rw [finRange, map_pmap, pmap_eq_map]"
},
{
"state_after": "no goals",
"state_before": "α : Type u\nn : ℕ\n⊢ map (fun a => a) (range n) = range n",
"tactic": "exact List.map_id _"
}
] |
[
30,
22
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
28,
1
] |
Mathlib/Topology/Filter.lean
|
Filter.isOpen_setOf_mem
|
[
{
"state_after": "no goals",
"state_before": "ι : Sort ?u.385\nα : Type u_1\nβ : Type ?u.391\nX : Type ?u.394\nY : Type ?u.397\ns : Set α\n⊢ IsOpen {l | s ∈ l}",
"tactic": "simpa only [Iic_principal] using isOpen_Iic_principal"
}
] |
[
59,
56
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
58,
1
] |
Mathlib/Data/Finset/Lattice.lean
|
Finset.coe_inf_of_nonempty
|
[] |
[
1022,
37
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1020,
1
] |
Mathlib/Data/Int/Bitwise.lean
|
Int.bodd_add_div2
|
[
{
"state_after": "n : ℕ\n⊢ ↑(bif bodd ↑n then 1 else 0) + 2 * div2 ↑n = ↑n",
"state_before": "n : ℕ\n⊢ (bif bodd ↑n then 1 else 0) + 2 * div2 ↑n = ↑n",
"tactic": "rw [show (cond (bodd n) 1 0 : ℤ) = (cond (bodd n) 1 0 : ℕ) by cases bodd n <;> rfl]"
},
{
"state_after": "no goals",
"state_before": "n : ℕ\n⊢ ↑(bif bodd ↑n then 1 else 0) + 2 * div2 ↑n = ↑n",
"tactic": "exact congr_arg ofNat n.bodd_add_div2"
},
{
"state_after": "no goals",
"state_before": "n : ℕ\n⊢ (bif bodd ↑n then 1 else 0) = ↑(bif bodd ↑n then 1 else 0)",
"tactic": "cases bodd n <;> rfl"
},
{
"state_after": "n : ℕ\n⊢ (bif bodd -[n+1] then 1 else 0) + 2 * div2 -[n+1] = -[(bif Nat.bodd n then 1 else 0) + 2 * Nat.div2 n+1]",
"state_before": "n : ℕ\n⊢ (bif bodd -[n+1] then 1 else 0) + 2 * div2 -[n+1] = -[n+1]",
"tactic": "refine' Eq.trans _ (congr_arg negSucc n.bodd_add_div2)"
},
{
"state_after": "n : ℕ\n⊢ (bif !Nat.bodd n then 1 else 0) + 2 * div2 -[n+1] = -[(bif Nat.bodd n then 1 else 0) + 2 * Nat.div2 n+1]",
"state_before": "n : ℕ\n⊢ (bif bodd -[n+1] then 1 else 0) + 2 * div2 -[n+1] = -[(bif Nat.bodd n then 1 else 0) + 2 * Nat.div2 n+1]",
"tactic": "dsimp [bodd]"
},
{
"state_after": "case false\nn : ℕ\n⊢ 1 + 2 * -[Nat.div2 n+1] = -[0 + 2 * Nat.div2 n+1]\n\ncase true\nn : ℕ\n⊢ 0 + 2 * -[Nat.div2 n+1] = -[1 + 2 * Nat.div2 n+1]",
"state_before": "n : ℕ\n⊢ (bif !Nat.bodd n then 1 else 0) + 2 * div2 -[n+1] = -[(bif Nat.bodd n then 1 else 0) + 2 * Nat.div2 n+1]",
"tactic": "cases Nat.bodd n <;> dsimp [cond, not, div2, Int.mul]"
},
{
"state_after": "case false\nn : ℕ\n⊢ -[2 * Nat.div2 n+1] = -[0 + 2 * Nat.div2 n+1]",
"state_before": "case false\nn : ℕ\n⊢ 1 + 2 * -[Nat.div2 n+1] = -[0 + 2 * Nat.div2 n+1]",
"tactic": "change -[2 * Nat.div2 n+1] = _"
},
{
"state_after": "no goals",
"state_before": "case false\nn : ℕ\n⊢ -[2 * Nat.div2 n+1] = -[0 + 2 * Nat.div2 n+1]",
"tactic": "rw [zero_add]"
},
{
"state_after": "case true\nn : ℕ\n⊢ 2 * -[Nat.div2 n+1] = -[2 * Nat.div2 n + 1+1]",
"state_before": "case true\nn : ℕ\n⊢ 0 + 2 * -[Nat.div2 n+1] = -[1 + 2 * Nat.div2 n+1]",
"tactic": "rw [zero_add, add_comm]"
},
{
"state_after": "no goals",
"state_before": "case true\nn : ℕ\n⊢ 2 * -[Nat.div2 n+1] = -[2 * Nat.div2 n + 1+1]",
"tactic": "rfl"
}
] |
[
110,
10
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
100,
1
] |
Mathlib/MeasureTheory/MeasurableSpace.lean
|
MeasurableSet.univ_pi
|
[] |
[
884,
52
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
882,
11
] |
Mathlib/MeasureTheory/Decomposition/SignedHahn.lean
|
MeasureTheory.SignedMeasure.findExistsOneDivLT_spec
|
[
{
"state_after": "α : Type u_1\nβ : Type ?u.2701\ninst✝³ : MeasurableSpace α\nM : Type ?u.2707\ninst✝² : AddCommMonoid M\ninst✝¹ : TopologicalSpace M\ninst✝ : OrderedAddCommMonoid M\ns : SignedMeasure α\ni j : Set α\nhi : ¬restrict s i ≤ restrict 0 i\n⊢ MeasureTheory.SignedMeasure.ExistsOneDivLT s i (Nat.find (_ : ∃ n, MeasureTheory.SignedMeasure.ExistsOneDivLT s i n))",
"state_before": "α : Type u_1\nβ : Type ?u.2701\ninst✝³ : MeasurableSpace α\nM : Type ?u.2707\ninst✝² : AddCommMonoid M\ninst✝¹ : TopologicalSpace M\ninst✝ : OrderedAddCommMonoid M\ns : SignedMeasure α\ni j : Set α\nhi : ¬restrict s i ≤ restrict 0 i\n⊢ MeasureTheory.SignedMeasure.ExistsOneDivLT s i (MeasureTheory.SignedMeasure.findExistsOneDivLT s i)",
"tactic": "rw [findExistsOneDivLT, dif_pos hi]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.2701\ninst✝³ : MeasurableSpace α\nM : Type ?u.2707\ninst✝² : AddCommMonoid M\ninst✝¹ : TopologicalSpace M\ninst✝ : OrderedAddCommMonoid M\ns : SignedMeasure α\ni j : Set α\nhi : ¬restrict s i ≤ restrict 0 i\n⊢ MeasureTheory.SignedMeasure.ExistsOneDivLT s i (Nat.find (_ : ∃ n, MeasureTheory.SignedMeasure.ExistsOneDivLT s i n))",
"tactic": "convert Nat.find_spec (existsNatOneDivLTMeasure_of_not_negative hi)"
}
] |
[
116,
70
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
113,
9
] |
Mathlib/Data/Real/Cardinality.lean
|
Cardinal.increasing_cantorFunction
|
[
{
"state_after": "c : ℝ\nf✝ g✝ : ℕ → Bool\nn✝ : ℕ\nh1 : 0 < c\nh2 : c < 1 / 2\nn : ℕ\nf g : ℕ → Bool\nhn : ∀ (k : ℕ), k < n → f k = g k\nfn : f n = false\ngn : g n = true\nh3 : c < 1\n⊢ cantorFunction c f < cantorFunction c g",
"state_before": "c : ℝ\nf✝ g✝ : ℕ → Bool\nn✝ : ℕ\nh1 : 0 < c\nh2 : c < 1 / 2\nn : ℕ\nf g : ℕ → Bool\nhn : ∀ (k : ℕ), k < n → f k = g k\nfn : f n = false\ngn : g n = true\n⊢ cantorFunction c f < cantorFunction c g",
"tactic": "have h3 : c < 1 := by\n apply h2.trans\n norm_num"
},
{
"state_after": "case zero\nc : ℝ\nf✝¹ g✝¹ : ℕ → Bool\nn✝ : ℕ\nh1 : 0 < c\nh2 : c < 1 / 2\nn : ℕ\nf✝ g✝ : ℕ → Bool\nhn✝ : ∀ (k : ℕ), k < n → f✝ k = g✝ k\nfn✝ : f✝ n = false\ngn✝ : g✝ n = true\nh3 : c < 1\nf g : ℕ → Bool\nhn : ∀ (k : ℕ), k < zero → f k = g k\nfn : f zero = false\ngn : g zero = true\n⊢ cantorFunction c f < cantorFunction c g\n\ncase succ\nc : ℝ\nf✝¹ g✝¹ : ℕ → Bool\nn✝¹ : ℕ\nh1 : 0 < c\nh2 : c < 1 / 2\nn✝ : ℕ\nf✝ g✝ : ℕ → Bool\nhn✝ : ∀ (k : ℕ), k < n✝ → f✝ k = g✝ k\nfn✝ : f✝ n✝ = false\ngn✝ : g✝ n✝ = true\nh3 : c < 1\nn : ℕ\nih :\n ∀ {f g : ℕ → Bool},\n (∀ (k : ℕ), k < n → f k = g k) → f n = false → g n = true → cantorFunction c f < cantorFunction c g\nf g : ℕ → Bool\nhn : ∀ (k : ℕ), k < succ n → f k = g k\nfn : f (succ n) = false\ngn : g (succ n) = true\n⊢ cantorFunction c f < cantorFunction c g",
"state_before": "c : ℝ\nf✝ g✝ : ℕ → Bool\nn✝ : ℕ\nh1 : 0 < c\nh2 : c < 1 / 2\nn : ℕ\nf g : ℕ → Bool\nhn : ∀ (k : ℕ), k < n → f k = g k\nfn : f n = false\ngn : g n = true\nh3 : c < 1\n⊢ cantorFunction c f < cantorFunction c g",
"tactic": "induction' n with n ih generalizing f g"
},
{
"state_after": "case succ\nc : ℝ\nf✝¹ g✝¹ : ℕ → Bool\nn✝¹ : ℕ\nh1 : 0 < c\nh2 : c < 1 / 2\nn✝ : ℕ\nf✝ g✝ : ℕ → Bool\nhn✝ : ∀ (k : ℕ), k < n✝ → f✝ k = g✝ k\nfn✝ : f✝ n✝ = false\ngn✝ : g✝ n✝ = true\nh3 : c < 1\nn : ℕ\nih :\n ∀ {f g : ℕ → Bool},\n (∀ (k : ℕ), k < n → f k = g k) → f n = false → g n = true → cantorFunction c f < cantorFunction c g\nf g : ℕ → Bool\nhn : ∀ (k : ℕ), k < succ n → f k = g k\nfn : f (succ n) = false\ngn : g (succ n) = true\n⊢ ((bif f 0 then 1 else 0) + c * cantorFunction c fun n => f (n + 1)) <\n (bif g 0 then 1 else 0) + c * cantorFunction c fun n => g (n + 1)",
"state_before": "case succ\nc : ℝ\nf✝¹ g✝¹ : ℕ → Bool\nn✝¹ : ℕ\nh1 : 0 < c\nh2 : c < 1 / 2\nn✝ : ℕ\nf✝ g✝ : ℕ → Bool\nhn✝ : ∀ (k : ℕ), k < n✝ → f✝ k = g✝ k\nfn✝ : f✝ n✝ = false\ngn✝ : g✝ n✝ = true\nh3 : c < 1\nn : ℕ\nih :\n ∀ {f g : ℕ → Bool},\n (∀ (k : ℕ), k < n → f k = g k) → f n = false → g n = true → cantorFunction c f < cantorFunction c g\nf g : ℕ → Bool\nhn : ∀ (k : ℕ), k < succ n → f k = g k\nfn : f (succ n) = false\ngn : g (succ n) = true\n⊢ cantorFunction c f < cantorFunction c g",
"tactic": "rw [cantorFunction_succ f (le_of_lt h1) h3, cantorFunction_succ g (le_of_lt h1) h3]"
},
{
"state_after": "case succ\nc : ℝ\nf✝¹ g✝¹ : ℕ → Bool\nn✝¹ : ℕ\nh1 : 0 < c\nh2 : c < 1 / 2\nn✝ : ℕ\nf✝ g✝ : ℕ → Bool\nhn✝ : ∀ (k : ℕ), k < n✝ → f✝ k = g✝ k\nfn✝ : f✝ n✝ = false\ngn✝ : g✝ n✝ = true\nh3 : c < 1\nn : ℕ\nih :\n ∀ {f g : ℕ → Bool},\n (∀ (k : ℕ), k < n → f k = g k) → f n = false → g n = true → cantorFunction c f < cantorFunction c g\nf g : ℕ → Bool\nhn : ∀ (k : ℕ), k < succ n → f k = g k\nfn : f (succ n) = false\ngn : g (succ n) = true\n⊢ ((bif g 0 then 1 else 0) + c * cantorFunction c fun n => f (n + 1)) <\n (bif g 0 then 1 else 0) + c * cantorFunction c fun n => g (n + 1)",
"state_before": "case succ\nc : ℝ\nf✝¹ g✝¹ : ℕ → Bool\nn✝¹ : ℕ\nh1 : 0 < c\nh2 : c < 1 / 2\nn✝ : ℕ\nf✝ g✝ : ℕ → Bool\nhn✝ : ∀ (k : ℕ), k < n✝ → f✝ k = g✝ k\nfn✝ : f✝ n✝ = false\ngn✝ : g✝ n✝ = true\nh3 : c < 1\nn : ℕ\nih :\n ∀ {f g : ℕ → Bool},\n (∀ (k : ℕ), k < n → f k = g k) → f n = false → g n = true → cantorFunction c f < cantorFunction c g\nf g : ℕ → Bool\nhn : ∀ (k : ℕ), k < succ n → f k = g k\nfn : f (succ n) = false\ngn : g (succ n) = true\n⊢ ((bif f 0 then 1 else 0) + c * cantorFunction c fun n => f (n + 1)) <\n (bif g 0 then 1 else 0) + c * cantorFunction c fun n => g (n + 1)",
"tactic": "rw [hn 0 <| zero_lt_succ n]"
},
{
"state_after": "case succ.bc\nc : ℝ\nf✝¹ g✝¹ : ℕ → Bool\nn✝¹ : ℕ\nh1 : 0 < c\nh2 : c < 1 / 2\nn✝ : ℕ\nf✝ g✝ : ℕ → Bool\nhn✝ : ∀ (k : ℕ), k < n✝ → f✝ k = g✝ k\nfn✝ : f✝ n✝ = false\ngn✝ : g✝ n✝ = true\nh3 : c < 1\nn : ℕ\nih :\n ∀ {f g : ℕ → Bool},\n (∀ (k : ℕ), k < n → f k = g k) → f n = false → g n = true → cantorFunction c f < cantorFunction c g\nf g : ℕ → Bool\nhn : ∀ (k : ℕ), k < succ n → f k = g k\nfn : f (succ n) = false\ngn : g (succ n) = true\n⊢ (c * cantorFunction c fun n => f (n + 1)) < c * cantorFunction c fun n => g (n + 1)",
"state_before": "case succ\nc : ℝ\nf✝¹ g✝¹ : ℕ → Bool\nn✝¹ : ℕ\nh1 : 0 < c\nh2 : c < 1 / 2\nn✝ : ℕ\nf✝ g✝ : ℕ → Bool\nhn✝ : ∀ (k : ℕ), k < n✝ → f✝ k = g✝ k\nfn✝ : f✝ n✝ = false\ngn✝ : g✝ n✝ = true\nh3 : c < 1\nn : ℕ\nih :\n ∀ {f g : ℕ → Bool},\n (∀ (k : ℕ), k < n → f k = g k) → f n = false → g n = true → cantorFunction c f < cantorFunction c g\nf g : ℕ → Bool\nhn : ∀ (k : ℕ), k < succ n → f k = g k\nfn : f (succ n) = false\ngn : g (succ n) = true\n⊢ ((bif g 0 then 1 else 0) + c * cantorFunction c fun n => f (n + 1)) <\n (bif g 0 then 1 else 0) + c * cantorFunction c fun n => g (n + 1)",
"tactic": "apply add_lt_add_left"
},
{
"state_after": "case succ.bc\nc : ℝ\nf✝¹ g✝¹ : ℕ → Bool\nn✝¹ : ℕ\nh1 : 0 < c\nh2 : c < 1 / 2\nn✝ : ℕ\nf✝ g✝ : ℕ → Bool\nhn✝ : ∀ (k : ℕ), k < n✝ → f✝ k = g✝ k\nfn✝ : f✝ n✝ = false\ngn✝ : g✝ n✝ = true\nh3 : c < 1\nn : ℕ\nih :\n ∀ {f g : ℕ → Bool},\n (∀ (k : ℕ), k < n → f k = g k) → f n = false → g n = true → cantorFunction c f < cantorFunction c g\nf g : ℕ → Bool\nhn : ∀ (k : ℕ), k < succ n → f k = g k\nfn : f (succ n) = false\ngn : g (succ n) = true\n⊢ (cantorFunction c fun n => f (n + 1)) < cantorFunction c fun n => g (n + 1)",
"state_before": "case succ.bc\nc : ℝ\nf✝¹ g✝¹ : ℕ → Bool\nn✝¹ : ℕ\nh1 : 0 < c\nh2 : c < 1 / 2\nn✝ : ℕ\nf✝ g✝ : ℕ → Bool\nhn✝ : ∀ (k : ℕ), k < n✝ → f✝ k = g✝ k\nfn✝ : f✝ n✝ = false\ngn✝ : g✝ n✝ = true\nh3 : c < 1\nn : ℕ\nih :\n ∀ {f g : ℕ → Bool},\n (∀ (k : ℕ), k < n → f k = g k) → f n = false → g n = true → cantorFunction c f < cantorFunction c g\nf g : ℕ → Bool\nhn : ∀ (k : ℕ), k < succ n → f k = g k\nfn : f (succ n) = false\ngn : g (succ n) = true\n⊢ (c * cantorFunction c fun n => f (n + 1)) < c * cantorFunction c fun n => g (n + 1)",
"tactic": "rw [mul_lt_mul_left h1]"
},
{
"state_after": "no goals",
"state_before": "case succ.bc\nc : ℝ\nf✝¹ g✝¹ : ℕ → Bool\nn✝¹ : ℕ\nh1 : 0 < c\nh2 : c < 1 / 2\nn✝ : ℕ\nf✝ g✝ : ℕ → Bool\nhn✝ : ∀ (k : ℕ), k < n✝ → f✝ k = g✝ k\nfn✝ : f✝ n✝ = false\ngn✝ : g✝ n✝ = true\nh3 : c < 1\nn : ℕ\nih :\n ∀ {f g : ℕ → Bool},\n (∀ (k : ℕ), k < n → f k = g k) → f n = false → g n = true → cantorFunction c f < cantorFunction c g\nf g : ℕ → Bool\nhn : ∀ (k : ℕ), k < succ n → f k = g k\nfn : f (succ n) = false\ngn : g (succ n) = true\n⊢ (cantorFunction c fun n => f (n + 1)) < cantorFunction c fun n => g (n + 1)",
"tactic": "exact ih (fun k hk => hn _ <| Nat.succ_lt_succ hk) fn gn"
},
{
"state_after": "c : ℝ\nf✝ g✝ : ℕ → Bool\nn✝ : ℕ\nh1 : 0 < c\nh2 : c < 1 / 2\nn : ℕ\nf g : ℕ → Bool\nhn : ∀ (k : ℕ), k < n → f k = g k\nfn : f n = false\ngn : g n = true\n⊢ 1 / 2 < 1",
"state_before": "c : ℝ\nf✝ g✝ : ℕ → Bool\nn✝ : ℕ\nh1 : 0 < c\nh2 : c < 1 / 2\nn : ℕ\nf g : ℕ → Bool\nhn : ∀ (k : ℕ), k < n → f k = g k\nfn : f n = false\ngn : g n = true\n⊢ c < 1",
"tactic": "apply h2.trans"
},
{
"state_after": "no goals",
"state_before": "c : ℝ\nf✝ g✝ : ℕ → Bool\nn✝ : ℕ\nh1 : 0 < c\nh2 : c < 1 / 2\nn : ℕ\nf g : ℕ → Bool\nhn : ∀ (k : ℕ), k < n → f k = g k\nfn : f n = false\ngn : g n = true\n⊢ 1 / 2 < 1",
"tactic": "norm_num"
},
{
"state_after": "case zero\nc : ℝ\nf✝¹ g✝¹ : ℕ → Bool\nn✝ : ℕ\nh1 : 0 < c\nh2 : c < 1 / 2\nn : ℕ\nf✝ g✝ : ℕ → Bool\nhn✝ : ∀ (k : ℕ), k < n → f✝ k = g✝ k\nfn✝ : f✝ n = false\ngn✝ : g✝ n = true\nh3 : c < 1\nf g : ℕ → Bool\nhn : ∀ (k : ℕ), k < zero → f k = g k\nfn : f zero = false\ngn : g zero = true\nf_max : ℕ → Bool := fun n => rec false (fun x x => true) n\n⊢ cantorFunction c f < cantorFunction c g",
"state_before": "case zero\nc : ℝ\nf✝¹ g✝¹ : ℕ → Bool\nn✝ : ℕ\nh1 : 0 < c\nh2 : c < 1 / 2\nn : ℕ\nf✝ g✝ : ℕ → Bool\nhn✝ : ∀ (k : ℕ), k < n → f✝ k = g✝ k\nfn✝ : f✝ n = false\ngn✝ : g✝ n = true\nh3 : c < 1\nf g : ℕ → Bool\nhn : ∀ (k : ℕ), k < zero → f k = g k\nfn : f zero = false\ngn : g zero = true\n⊢ cantorFunction c f < cantorFunction c g",
"tactic": "let f_max : ℕ → Bool := fun n => Nat.rec false (fun _ _ => true) n"
},
{
"state_after": "case zero\nc : ℝ\nf✝¹ g✝¹ : ℕ → Bool\nn✝ : ℕ\nh1 : 0 < c\nh2 : c < 1 / 2\nn : ℕ\nf✝ g✝ : ℕ → Bool\nhn✝ : ∀ (k : ℕ), k < n → f✝ k = g✝ k\nfn✝ : f✝ n = false\ngn✝ : g✝ n = true\nh3 : c < 1\nf g : ℕ → Bool\nhn : ∀ (k : ℕ), k < zero → f k = g k\nfn : f zero = false\ngn : g zero = true\nf_max : ℕ → Bool := fun n => rec false (fun x x => true) n\nhf_max : ∀ (n : ℕ), f n = true → f_max n = true\n⊢ cantorFunction c f < cantorFunction c g",
"state_before": "case zero\nc : ℝ\nf✝¹ g✝¹ : ℕ → Bool\nn✝ : ℕ\nh1 : 0 < c\nh2 : c < 1 / 2\nn : ℕ\nf✝ g✝ : ℕ → Bool\nhn✝ : ∀ (k : ℕ), k < n → f✝ k = g✝ k\nfn✝ : f✝ n = false\ngn✝ : g✝ n = true\nh3 : c < 1\nf g : ℕ → Bool\nhn : ∀ (k : ℕ), k < zero → f k = g k\nfn : f zero = false\ngn : g zero = true\nf_max : ℕ → Bool := fun n => rec false (fun x x => true) n\n⊢ cantorFunction c f < cantorFunction c g",
"tactic": "have hf_max : ∀ n, f n → f_max n := by\n intro n hn\n cases n\n rw [fn] at hn\n contradiction\n apply rfl"
},
{
"state_after": "case zero\nc : ℝ\nf✝¹ g✝¹ : ℕ → Bool\nn✝ : ℕ\nh1 : 0 < c\nh2 : c < 1 / 2\nn : ℕ\nf✝ g✝ : ℕ → Bool\nhn✝ : ∀ (k : ℕ), k < n → f✝ k = g✝ k\nfn✝ : f✝ n = false\ngn✝ : g✝ n = true\nh3 : c < 1\nf g : ℕ → Bool\nhn : ∀ (k : ℕ), k < zero → f k = g k\nfn : f zero = false\ngn : g zero = true\nf_max : ℕ → Bool := fun n => rec false (fun x x => true) n\nhf_max : ∀ (n : ℕ), f n = true → f_max n = true\ng_min : ℕ → Bool := fun n => rec true (fun x x => false) n\n⊢ cantorFunction c f < cantorFunction c g",
"state_before": "case zero\nc : ℝ\nf✝¹ g✝¹ : ℕ → Bool\nn✝ : ℕ\nh1 : 0 < c\nh2 : c < 1 / 2\nn : ℕ\nf✝ g✝ : ℕ → Bool\nhn✝ : ∀ (k : ℕ), k < n → f✝ k = g✝ k\nfn✝ : f✝ n = false\ngn✝ : g✝ n = true\nh3 : c < 1\nf g : ℕ → Bool\nhn : ∀ (k : ℕ), k < zero → f k = g k\nfn : f zero = false\ngn : g zero = true\nf_max : ℕ → Bool := fun n => rec false (fun x x => true) n\nhf_max : ∀ (n : ℕ), f n = true → f_max n = true\n⊢ cantorFunction c f < cantorFunction c g",
"tactic": "let g_min : ℕ → Bool := fun n => Nat.rec true (fun _ _ => false) n"
},
{
"state_after": "case zero\nc : ℝ\nf✝¹ g✝¹ : ℕ → Bool\nn✝ : ℕ\nh1 : 0 < c\nh2 : c < 1 / 2\nn : ℕ\nf✝ g✝ : ℕ → Bool\nhn✝ : ∀ (k : ℕ), k < n → f✝ k = g✝ k\nfn✝ : f✝ n = false\ngn✝ : g✝ n = true\nh3 : c < 1\nf g : ℕ → Bool\nhn : ∀ (k : ℕ), k < zero → f k = g k\nfn : f zero = false\ngn : g zero = true\nf_max : ℕ → Bool := fun n => rec false (fun x x => true) n\nhf_max : ∀ (n : ℕ), f n = true → f_max n = true\ng_min : ℕ → Bool := fun n => rec true (fun x x => false) n\nhg_min : ∀ (n : ℕ), g_min n = true → g n = true\n⊢ cantorFunction c f < cantorFunction c g",
"state_before": "case zero\nc : ℝ\nf✝¹ g✝¹ : ℕ → Bool\nn✝ : ℕ\nh1 : 0 < c\nh2 : c < 1 / 2\nn : ℕ\nf✝ g✝ : ℕ → Bool\nhn✝ : ∀ (k : ℕ), k < n → f✝ k = g✝ k\nfn✝ : f✝ n = false\ngn✝ : g✝ n = true\nh3 : c < 1\nf g : ℕ → Bool\nhn : ∀ (k : ℕ), k < zero → f k = g k\nfn : f zero = false\ngn : g zero = true\nf_max : ℕ → Bool := fun n => rec false (fun x x => true) n\nhf_max : ∀ (n : ℕ), f n = true → f_max n = true\ng_min : ℕ → Bool := fun n => rec true (fun x x => false) n\n⊢ cantorFunction c f < cantorFunction c g",
"tactic": "have hg_min : ∀ n, g_min n → g n := by\n intro n hn\n cases n\n rw [gn]\n simp at hn"
},
{
"state_after": "case zero\nc : ℝ\nf✝¹ g✝¹ : ℕ → Bool\nn✝ : ℕ\nh1 : 0 < c\nh2 : c < 1 / 2\nn : ℕ\nf✝ g✝ : ℕ → Bool\nhn✝ : ∀ (k : ℕ), k < n → f✝ k = g✝ k\nfn✝ : f✝ n = false\ngn✝ : g✝ n = true\nh3 : c < 1\nf g : ℕ → Bool\nhn : ∀ (k : ℕ), k < zero → f k = g k\nfn : f zero = false\ngn : g zero = true\nf_max : ℕ → Bool := fun n => rec false (fun x x => true) n\nhf_max : ∀ (n : ℕ), f n = true → f_max n = true\ng_min : ℕ → Bool := fun n => rec true (fun x x => false) n\nhg_min : ∀ (n : ℕ), g_min n = true → g n = true\n⊢ (cantorFunction c fun n => f_max n) < cantorFunction c g",
"state_before": "case zero\nc : ℝ\nf✝¹ g✝¹ : ℕ → Bool\nn✝ : ℕ\nh1 : 0 < c\nh2 : c < 1 / 2\nn : ℕ\nf✝ g✝ : ℕ → Bool\nhn✝ : ∀ (k : ℕ), k < n → f✝ k = g✝ k\nfn✝ : f✝ n = false\ngn✝ : g✝ n = true\nh3 : c < 1\nf g : ℕ → Bool\nhn : ∀ (k : ℕ), k < zero → f k = g k\nfn : f zero = false\ngn : g zero = true\nf_max : ℕ → Bool := fun n => rec false (fun x x => true) n\nhf_max : ∀ (n : ℕ), f n = true → f_max n = true\ng_min : ℕ → Bool := fun n => rec true (fun x x => false) n\nhg_min : ∀ (n : ℕ), g_min n = true → g n = true\n⊢ cantorFunction c f < cantorFunction c g",
"tactic": "apply (cantorFunction_le (le_of_lt h1) h3 hf_max).trans_lt"
},
{
"state_after": "case zero\nc : ℝ\nf✝¹ g✝¹ : ℕ → Bool\nn✝ : ℕ\nh1 : 0 < c\nh2 : c < 1 / 2\nn : ℕ\nf✝ g✝ : ℕ → Bool\nhn✝ : ∀ (k : ℕ), k < n → f✝ k = g✝ k\nfn✝ : f✝ n = false\ngn✝ : g✝ n = true\nh3 : c < 1\nf g : ℕ → Bool\nhn : ∀ (k : ℕ), k < zero → f k = g k\nfn : f zero = false\ngn : g zero = true\nf_max : ℕ → Bool := fun n => rec false (fun x x => true) n\nhf_max : ∀ (n : ℕ), f n = true → f_max n = true\ng_min : ℕ → Bool := fun n => rec true (fun x x => false) n\nhg_min : ∀ (n : ℕ), g_min n = true → g n = true\n⊢ (cantorFunction c fun n => f_max n) < cantorFunction c fun n => g_min n",
"state_before": "case zero\nc : ℝ\nf✝¹ g✝¹ : ℕ → Bool\nn✝ : ℕ\nh1 : 0 < c\nh2 : c < 1 / 2\nn : ℕ\nf✝ g✝ : ℕ → Bool\nhn✝ : ∀ (k : ℕ), k < n → f✝ k = g✝ k\nfn✝ : f✝ n = false\ngn✝ : g✝ n = true\nh3 : c < 1\nf g : ℕ → Bool\nhn : ∀ (k : ℕ), k < zero → f k = g k\nfn : f zero = false\ngn : g zero = true\nf_max : ℕ → Bool := fun n => rec false (fun x x => true) n\nhf_max : ∀ (n : ℕ), f n = true → f_max n = true\ng_min : ℕ → Bool := fun n => rec true (fun x x => false) n\nhg_min : ∀ (n : ℕ), g_min n = true → g n = true\n⊢ (cantorFunction c fun n => f_max n) < cantorFunction c g",
"tactic": "refine' lt_of_lt_of_le _ (cantorFunction_le (le_of_lt h1) h3 hg_min)"
},
{
"state_after": "case h.e'_3\nc : ℝ\nf✝¹ g✝¹ : ℕ → Bool\nn✝ : ℕ\nh1 : 0 < c\nh2 : c < 1 / 2\nn : ℕ\nf✝ g✝ : ℕ → Bool\nhn✝ : ∀ (k : ℕ), k < n → f✝ k = g✝ k\nfn✝ : f✝ n = false\ngn✝ : g✝ n = true\nh3 : c < 1\nf g : ℕ → Bool\nhn : ∀ (k : ℕ), k < zero → f k = g k\nfn : f zero = false\ngn : g zero = true\nf_max : ℕ → Bool := fun n => rec false (fun x x => true) n\nhf_max : ∀ (n : ℕ), f n = true → f_max n = true\ng_min : ℕ → Bool := fun n => rec true (fun x x => false) n\nhg_min : ∀ (n : ℕ), g_min n = true → g n = true\nthis : c / (1 - c) < 1\n⊢ (cantorFunction c fun n => f_max n) = c / (1 - c)\n\ncase h.e'_4\nc : ℝ\nf✝¹ g✝¹ : ℕ → Bool\nn✝ : ℕ\nh1 : 0 < c\nh2 : c < 1 / 2\nn : ℕ\nf✝ g✝ : ℕ → Bool\nhn✝ : ∀ (k : ℕ), k < n → f✝ k = g✝ k\nfn✝ : f✝ n = false\ngn✝ : g✝ n = true\nh3 : c < 1\nf g : ℕ → Bool\nhn : ∀ (k : ℕ), k < zero → f k = g k\nfn : f zero = false\ngn : g zero = true\nf_max : ℕ → Bool := fun n => rec false (fun x x => true) n\nhf_max : ∀ (n : ℕ), f n = true → f_max n = true\ng_min : ℕ → Bool := fun n => rec true (fun x x => false) n\nhg_min : ∀ (n : ℕ), g_min n = true → g n = true\nthis : c / (1 - c) < 1\n⊢ (cantorFunction c fun n => g_min n) = 1",
"state_before": "case zero\nc : ℝ\nf✝¹ g✝¹ : ℕ → Bool\nn✝ : ℕ\nh1 : 0 < c\nh2 : c < 1 / 2\nn : ℕ\nf✝ g✝ : ℕ → Bool\nhn✝ : ∀ (k : ℕ), k < n → f✝ k = g✝ k\nfn✝ : f✝ n = false\ngn✝ : g✝ n = true\nh3 : c < 1\nf g : ℕ → Bool\nhn : ∀ (k : ℕ), k < zero → f k = g k\nfn : f zero = false\ngn : g zero = true\nf_max : ℕ → Bool := fun n => rec false (fun x x => true) n\nhf_max : ∀ (n : ℕ), f n = true → f_max n = true\ng_min : ℕ → Bool := fun n => rec true (fun x x => false) n\nhg_min : ∀ (n : ℕ), g_min n = true → g n = true\nthis : c / (1 - c) < 1\n⊢ (cantorFunction c fun n => f_max n) < cantorFunction c fun n => g_min n",
"tactic": "convert this"
},
{
"state_after": "c : ℝ\nf✝¹ g✝¹ : ℕ → Bool\nn✝¹ : ℕ\nh1 : 0 < c\nh2 : c < 1 / 2\nn✝ : ℕ\nf✝ g✝ : ℕ → Bool\nhn✝¹ : ∀ (k : ℕ), k < n✝ → f✝ k = g✝ k\nfn✝ : f✝ n✝ = false\ngn✝ : g✝ n✝ = true\nh3 : c < 1\nf g : ℕ → Bool\nhn✝ : ∀ (k : ℕ), k < zero → f k = g k\nfn : f zero = false\ngn : g zero = true\nf_max : ℕ → Bool := fun n => rec false (fun x x => true) n\nn : ℕ\nhn : f n = true\n⊢ f_max n = true",
"state_before": "c : ℝ\nf✝¹ g✝¹ : ℕ → Bool\nn✝ : ℕ\nh1 : 0 < c\nh2 : c < 1 / 2\nn : ℕ\nf✝ g✝ : ℕ → Bool\nhn✝ : ∀ (k : ℕ), k < n → f✝ k = g✝ k\nfn✝ : f✝ n = false\ngn✝ : g✝ n = true\nh3 : c < 1\nf g : ℕ → Bool\nhn : ∀ (k : ℕ), k < zero → f k = g k\nfn : f zero = false\ngn : g zero = true\nf_max : ℕ → Bool := fun n => rec false (fun x x => true) n\n⊢ ∀ (n : ℕ), f n = true → f_max n = true",
"tactic": "intro n hn"
},
{
"state_after": "case zero\nc : ℝ\nf✝¹ g✝¹ : ℕ → Bool\nn✝ : ℕ\nh1 : 0 < c\nh2 : c < 1 / 2\nn : ℕ\nf✝ g✝ : ℕ → Bool\nhn✝¹ : ∀ (k : ℕ), k < n → f✝ k = g✝ k\nfn✝ : f✝ n = false\ngn✝ : g✝ n = true\nh3 : c < 1\nf g : ℕ → Bool\nhn✝ : ∀ (k : ℕ), k < zero → f k = g k\nfn : f zero = false\ngn : g zero = true\nf_max : ℕ → Bool := fun n => rec false (fun x x => true) n\nhn : f zero = true\n⊢ f_max zero = true\n\ncase succ\nc : ℝ\nf✝¹ g✝¹ : ℕ → Bool\nn✝¹ : ℕ\nh1 : 0 < c\nh2 : c < 1 / 2\nn : ℕ\nf✝ g✝ : ℕ → Bool\nhn✝¹ : ∀ (k : ℕ), k < n → f✝ k = g✝ k\nfn✝ : f✝ n = false\ngn✝ : g✝ n = true\nh3 : c < 1\nf g : ℕ → Bool\nhn✝ : ∀ (k : ℕ), k < zero → f k = g k\nfn : f zero = false\ngn : g zero = true\nf_max : ℕ → Bool := fun n => rec false (fun x x => true) n\nn✝ : ℕ\nhn : f (succ n✝) = true\n⊢ f_max (succ n✝) = true",
"state_before": "c : ℝ\nf✝¹ g✝¹ : ℕ → Bool\nn✝¹ : ℕ\nh1 : 0 < c\nh2 : c < 1 / 2\nn✝ : ℕ\nf✝ g✝ : ℕ → Bool\nhn✝¹ : ∀ (k : ℕ), k < n✝ → f✝ k = g✝ k\nfn✝ : f✝ n✝ = false\ngn✝ : g✝ n✝ = true\nh3 : c < 1\nf g : ℕ → Bool\nhn✝ : ∀ (k : ℕ), k < zero → f k = g k\nfn : f zero = false\ngn : g zero = true\nf_max : ℕ → Bool := fun n => rec false (fun x x => true) n\nn : ℕ\nhn : f n = true\n⊢ f_max n = true",
"tactic": "cases n"
},
{
"state_after": "case zero\nc : ℝ\nf✝¹ g✝¹ : ℕ → Bool\nn✝ : ℕ\nh1 : 0 < c\nh2 : c < 1 / 2\nn : ℕ\nf✝ g✝ : ℕ → Bool\nhn✝¹ : ∀ (k : ℕ), k < n → f✝ k = g✝ k\nfn✝ : f✝ n = false\ngn✝ : g✝ n = true\nh3 : c < 1\nf g : ℕ → Bool\nhn✝ : ∀ (k : ℕ), k < zero → f k = g k\nfn : f zero = false\ngn : g zero = true\nf_max : ℕ → Bool := fun n => rec false (fun x x => true) n\nhn : false = true\n⊢ f_max zero = true\n\ncase succ\nc : ℝ\nf✝¹ g✝¹ : ℕ → Bool\nn✝¹ : ℕ\nh1 : 0 < c\nh2 : c < 1 / 2\nn : ℕ\nf✝ g✝ : ℕ → Bool\nhn✝¹ : ∀ (k : ℕ), k < n → f✝ k = g✝ k\nfn✝ : f✝ n = false\ngn✝ : g✝ n = true\nh3 : c < 1\nf g : ℕ → Bool\nhn✝ : ∀ (k : ℕ), k < zero → f k = g k\nfn : f zero = false\ngn : g zero = true\nf_max : ℕ → Bool := fun n => rec false (fun x x => true) n\nn✝ : ℕ\nhn : f (succ n✝) = true\n⊢ f_max (succ n✝) = true",
"state_before": "case zero\nc : ℝ\nf✝¹ g✝¹ : ℕ → Bool\nn✝ : ℕ\nh1 : 0 < c\nh2 : c < 1 / 2\nn : ℕ\nf✝ g✝ : ℕ → Bool\nhn✝¹ : ∀ (k : ℕ), k < n → f✝ k = g✝ k\nfn✝ : f✝ n = false\ngn✝ : g✝ n = true\nh3 : c < 1\nf g : ℕ → Bool\nhn✝ : ∀ (k : ℕ), k < zero → f k = g k\nfn : f zero = false\ngn : g zero = true\nf_max : ℕ → Bool := fun n => rec false (fun x x => true) n\nhn : f zero = true\n⊢ f_max zero = true\n\ncase succ\nc : ℝ\nf✝¹ g✝¹ : ℕ → Bool\nn✝¹ : ℕ\nh1 : 0 < c\nh2 : c < 1 / 2\nn : ℕ\nf✝ g✝ : ℕ → Bool\nhn✝¹ : ∀ (k : ℕ), k < n → f✝ k = g✝ k\nfn✝ : f✝ n = false\ngn✝ : g✝ n = true\nh3 : c < 1\nf g : ℕ → Bool\nhn✝ : ∀ (k : ℕ), k < zero → f k = g k\nfn : f zero = false\ngn : g zero = true\nf_max : ℕ → Bool := fun n => rec false (fun x x => true) n\nn✝ : ℕ\nhn : f (succ n✝) = true\n⊢ f_max (succ n✝) = true",
"tactic": "rw [fn] at hn"
},
{
"state_after": "case succ\nc : ℝ\nf✝¹ g✝¹ : ℕ → Bool\nn✝¹ : ℕ\nh1 : 0 < c\nh2 : c < 1 / 2\nn : ℕ\nf✝ g✝ : ℕ → Bool\nhn✝¹ : ∀ (k : ℕ), k < n → f✝ k = g✝ k\nfn✝ : f✝ n = false\ngn✝ : g✝ n = true\nh3 : c < 1\nf g : ℕ → Bool\nhn✝ : ∀ (k : ℕ), k < zero → f k = g k\nfn : f zero = false\ngn : g zero = true\nf_max : ℕ → Bool := fun n => rec false (fun x x => true) n\nn✝ : ℕ\nhn : f (succ n✝) = true\n⊢ f_max (succ n✝) = true",
"state_before": "case zero\nc : ℝ\nf✝¹ g✝¹ : ℕ → Bool\nn✝ : ℕ\nh1 : 0 < c\nh2 : c < 1 / 2\nn : ℕ\nf✝ g✝ : ℕ → Bool\nhn✝¹ : ∀ (k : ℕ), k < n → f✝ k = g✝ k\nfn✝ : f✝ n = false\ngn✝ : g✝ n = true\nh3 : c < 1\nf g : ℕ → Bool\nhn✝ : ∀ (k : ℕ), k < zero → f k = g k\nfn : f zero = false\ngn : g zero = true\nf_max : ℕ → Bool := fun n => rec false (fun x x => true) n\nhn : false = true\n⊢ f_max zero = true\n\ncase succ\nc : ℝ\nf✝¹ g✝¹ : ℕ → Bool\nn✝¹ : ℕ\nh1 : 0 < c\nh2 : c < 1 / 2\nn : ℕ\nf✝ g✝ : ℕ → Bool\nhn✝¹ : ∀ (k : ℕ), k < n → f✝ k = g✝ k\nfn✝ : f✝ n = false\ngn✝ : g✝ n = true\nh3 : c < 1\nf g : ℕ → Bool\nhn✝ : ∀ (k : ℕ), k < zero → f k = g k\nfn : f zero = false\ngn : g zero = true\nf_max : ℕ → Bool := fun n => rec false (fun x x => true) n\nn✝ : ℕ\nhn : f (succ n✝) = true\n⊢ f_max (succ n✝) = true",
"tactic": "contradiction"
},
{
"state_after": "no goals",
"state_before": "case succ\nc : ℝ\nf✝¹ g✝¹ : ℕ → Bool\nn✝¹ : ℕ\nh1 : 0 < c\nh2 : c < 1 / 2\nn : ℕ\nf✝ g✝ : ℕ → Bool\nhn✝¹ : ∀ (k : ℕ), k < n → f✝ k = g✝ k\nfn✝ : f✝ n = false\ngn✝ : g✝ n = true\nh3 : c < 1\nf g : ℕ → Bool\nhn✝ : ∀ (k : ℕ), k < zero → f k = g k\nfn : f zero = false\ngn : g zero = true\nf_max : ℕ → Bool := fun n => rec false (fun x x => true) n\nn✝ : ℕ\nhn : f (succ n✝) = true\n⊢ f_max (succ n✝) = true",
"tactic": "apply rfl"
},
{
"state_after": "c : ℝ\nf✝¹ g✝¹ : ℕ → Bool\nn✝¹ : ℕ\nh1 : 0 < c\nh2 : c < 1 / 2\nn✝ : ℕ\nf✝ g✝ : ℕ → Bool\nhn✝¹ : ∀ (k : ℕ), k < n✝ → f✝ k = g✝ k\nfn✝ : f✝ n✝ = false\ngn✝ : g✝ n✝ = true\nh3 : c < 1\nf g : ℕ → Bool\nhn✝ : ∀ (k : ℕ), k < zero → f k = g k\nfn : f zero = false\ngn : g zero = true\nf_max : ℕ → Bool := fun n => rec false (fun x x => true) n\nhf_max : ∀ (n : ℕ), f n = true → f_max n = true\ng_min : ℕ → Bool := fun n => rec true (fun x x => false) n\nn : ℕ\nhn : g_min n = true\n⊢ g n = true",
"state_before": "c : ℝ\nf✝¹ g✝¹ : ℕ → Bool\nn✝ : ℕ\nh1 : 0 < c\nh2 : c < 1 / 2\nn : ℕ\nf✝ g✝ : ℕ → Bool\nhn✝ : ∀ (k : ℕ), k < n → f✝ k = g✝ k\nfn✝ : f✝ n = false\ngn✝ : g✝ n = true\nh3 : c < 1\nf g : ℕ → Bool\nhn : ∀ (k : ℕ), k < zero → f k = g k\nfn : f zero = false\ngn : g zero = true\nf_max : ℕ → Bool := fun n => rec false (fun x x => true) n\nhf_max : ∀ (n : ℕ), f n = true → f_max n = true\ng_min : ℕ → Bool := fun n => rec true (fun x x => false) n\n⊢ ∀ (n : ℕ), g_min n = true → g n = true",
"tactic": "intro n hn"
},
{
"state_after": "case zero\nc : ℝ\nf✝¹ g✝¹ : ℕ → Bool\nn✝ : ℕ\nh1 : 0 < c\nh2 : c < 1 / 2\nn : ℕ\nf✝ g✝ : ℕ → Bool\nhn✝¹ : ∀ (k : ℕ), k < n → f✝ k = g✝ k\nfn✝ : f✝ n = false\ngn✝ : g✝ n = true\nh3 : c < 1\nf g : ℕ → Bool\nhn✝ : ∀ (k : ℕ), k < zero → f k = g k\nfn : f zero = false\ngn : g zero = true\nf_max : ℕ → Bool := fun n => rec false (fun x x => true) n\nhf_max : ∀ (n : ℕ), f n = true → f_max n = true\ng_min : ℕ → Bool := fun n => rec true (fun x x => false) n\nhn : g_min zero = true\n⊢ g zero = true\n\ncase succ\nc : ℝ\nf✝¹ g✝¹ : ℕ → Bool\nn✝¹ : ℕ\nh1 : 0 < c\nh2 : c < 1 / 2\nn : ℕ\nf✝ g✝ : ℕ → Bool\nhn✝¹ : ∀ (k : ℕ), k < n → f✝ k = g✝ k\nfn✝ : f✝ n = false\ngn✝ : g✝ n = true\nh3 : c < 1\nf g : ℕ → Bool\nhn✝ : ∀ (k : ℕ), k < zero → f k = g k\nfn : f zero = false\ngn : g zero = true\nf_max : ℕ → Bool := fun n => rec false (fun x x => true) n\nhf_max : ∀ (n : ℕ), f n = true → f_max n = true\ng_min : ℕ → Bool := fun n => rec true (fun x x => false) n\nn✝ : ℕ\nhn : g_min (succ n✝) = true\n⊢ g (succ n✝) = true",
"state_before": "c : ℝ\nf✝¹ g✝¹ : ℕ → Bool\nn✝¹ : ℕ\nh1 : 0 < c\nh2 : c < 1 / 2\nn✝ : ℕ\nf✝ g✝ : ℕ → Bool\nhn✝¹ : ∀ (k : ℕ), k < n✝ → f✝ k = g✝ k\nfn✝ : f✝ n✝ = false\ngn✝ : g✝ n✝ = true\nh3 : c < 1\nf g : ℕ → Bool\nhn✝ : ∀ (k : ℕ), k < zero → f k = g k\nfn : f zero = false\ngn : g zero = true\nf_max : ℕ → Bool := fun n => rec false (fun x x => true) n\nhf_max : ∀ (n : ℕ), f n = true → f_max n = true\ng_min : ℕ → Bool := fun n => rec true (fun x x => false) n\nn : ℕ\nhn : g_min n = true\n⊢ g n = true",
"tactic": "cases n"
},
{
"state_after": "case succ\nc : ℝ\nf✝¹ g✝¹ : ℕ → Bool\nn✝¹ : ℕ\nh1 : 0 < c\nh2 : c < 1 / 2\nn : ℕ\nf✝ g✝ : ℕ → Bool\nhn✝¹ : ∀ (k : ℕ), k < n → f✝ k = g✝ k\nfn✝ : f✝ n = false\ngn✝ : g✝ n = true\nh3 : c < 1\nf g : ℕ → Bool\nhn✝ : ∀ (k : ℕ), k < zero → f k = g k\nfn : f zero = false\ngn : g zero = true\nf_max : ℕ → Bool := fun n => rec false (fun x x => true) n\nhf_max : ∀ (n : ℕ), f n = true → f_max n = true\ng_min : ℕ → Bool := fun n => rec true (fun x x => false) n\nn✝ : ℕ\nhn : g_min (succ n✝) = true\n⊢ g (succ n✝) = true",
"state_before": "case zero\nc : ℝ\nf✝¹ g✝¹ : ℕ → Bool\nn✝ : ℕ\nh1 : 0 < c\nh2 : c < 1 / 2\nn : ℕ\nf✝ g✝ : ℕ → Bool\nhn✝¹ : ∀ (k : ℕ), k < n → f✝ k = g✝ k\nfn✝ : f✝ n = false\ngn✝ : g✝ n = true\nh3 : c < 1\nf g : ℕ → Bool\nhn✝ : ∀ (k : ℕ), k < zero → f k = g k\nfn : f zero = false\ngn : g zero = true\nf_max : ℕ → Bool := fun n => rec false (fun x x => true) n\nhf_max : ∀ (n : ℕ), f n = true → f_max n = true\ng_min : ℕ → Bool := fun n => rec true (fun x x => false) n\nhn : g_min zero = true\n⊢ g zero = true\n\ncase succ\nc : ℝ\nf✝¹ g✝¹ : ℕ → Bool\nn✝¹ : ℕ\nh1 : 0 < c\nh2 : c < 1 / 2\nn : ℕ\nf✝ g✝ : ℕ → Bool\nhn✝¹ : ∀ (k : ℕ), k < n → f✝ k = g✝ k\nfn✝ : f✝ n = false\ngn✝ : g✝ n = true\nh3 : c < 1\nf g : ℕ → Bool\nhn✝ : ∀ (k : ℕ), k < zero → f k = g k\nfn : f zero = false\ngn : g zero = true\nf_max : ℕ → Bool := fun n => rec false (fun x x => true) n\nhf_max : ∀ (n : ℕ), f n = true → f_max n = true\ng_min : ℕ → Bool := fun n => rec true (fun x x => false) n\nn✝ : ℕ\nhn : g_min (succ n✝) = true\n⊢ g (succ n✝) = true",
"tactic": "rw [gn]"
},
{
"state_after": "no goals",
"state_before": "case succ\nc : ℝ\nf✝¹ g✝¹ : ℕ → Bool\nn✝¹ : ℕ\nh1 : 0 < c\nh2 : c < 1 / 2\nn : ℕ\nf✝ g✝ : ℕ → Bool\nhn✝¹ : ∀ (k : ℕ), k < n → f✝ k = g✝ k\nfn✝ : f✝ n = false\ngn✝ : g✝ n = true\nh3 : c < 1\nf g : ℕ → Bool\nhn✝ : ∀ (k : ℕ), k < zero → f k = g k\nfn : f zero = false\ngn : g zero = true\nf_max : ℕ → Bool := fun n => rec false (fun x x => true) n\nhf_max : ∀ (n : ℕ), f n = true → f_max n = true\ng_min : ℕ → Bool := fun n => rec true (fun x x => false) n\nn✝ : ℕ\nhn : g_min (succ n✝) = true\n⊢ g (succ n✝) = true",
"tactic": "simp at hn"
},
{
"state_after": "c : ℝ\nf✝¹ g✝¹ : ℕ → Bool\nn✝ : ℕ\nh1 : 0 < c\nh2 : c < 1 / 2\nn : ℕ\nf✝ g✝ : ℕ → Bool\nhn✝ : ∀ (k : ℕ), k < n → f✝ k = g✝ k\nfn✝ : f✝ n = false\ngn✝ : g✝ n = true\nh3 : c < 1\nf g : ℕ → Bool\nhn : ∀ (k : ℕ), k < zero → f k = g k\nfn : f zero = false\ngn : g zero = true\nf_max : ℕ → Bool := fun n => rec false (fun x x => true) n\nhf_max : ∀ (n : ℕ), f n = true → f_max n = true\ng_min : ℕ → Bool := fun n => rec true (fun x x => false) n\nhg_min : ∀ (n : ℕ), g_min n = true → g n = true\n⊢ c + c < 1\n\nc : ℝ\nf✝¹ g✝¹ : ℕ → Bool\nn✝ : ℕ\nh1 : 0 < c\nh2 : c < 1 / 2\nn : ℕ\nf✝ g✝ : ℕ → Bool\nhn✝ : ∀ (k : ℕ), k < n → f✝ k = g✝ k\nfn✝ : f✝ n = false\ngn✝ : g✝ n = true\nh3 : c < 1\nf g : ℕ → Bool\nhn : ∀ (k : ℕ), k < zero → f k = g k\nfn : f zero = false\ngn : g zero = true\nf_max : ℕ → Bool := fun n => rec false (fun x x => true) n\nhf_max : ∀ (n : ℕ), f n = true → f_max n = true\ng_min : ℕ → Bool := fun n => rec true (fun x x => false) n\nhg_min : ∀ (n : ℕ), g_min n = true → g n = true\n⊢ 0 < 1 - c",
"state_before": "c : ℝ\nf✝¹ g✝¹ : ℕ → Bool\nn✝ : ℕ\nh1 : 0 < c\nh2 : c < 1 / 2\nn : ℕ\nf✝ g✝ : ℕ → Bool\nhn✝ : ∀ (k : ℕ), k < n → f✝ k = g✝ k\nfn✝ : f✝ n = false\ngn✝ : g✝ n = true\nh3 : c < 1\nf g : ℕ → Bool\nhn : ∀ (k : ℕ), k < zero → f k = g k\nfn : f zero = false\ngn : g zero = true\nf_max : ℕ → Bool := fun n => rec false (fun x x => true) n\nhf_max : ∀ (n : ℕ), f n = true → f_max n = true\ng_min : ℕ → Bool := fun n => rec true (fun x x => false) n\nhg_min : ∀ (n : ℕ), g_min n = true → g n = true\n⊢ c / (1 - c) < 1",
"tactic": "rw [div_lt_one, lt_sub_iff_add_lt]"
},
{
"state_after": "no goals",
"state_before": "c : ℝ\nf✝¹ g✝¹ : ℕ → Bool\nn✝ : ℕ\nh1 : 0 < c\nh2 : c < 1 / 2\nn : ℕ\nf✝ g✝ : ℕ → Bool\nhn✝ : ∀ (k : ℕ), k < n → f✝ k = g✝ k\nfn✝ : f✝ n = false\ngn✝ : g✝ n = true\nh3 : c < 1\nf g : ℕ → Bool\nhn : ∀ (k : ℕ), k < zero → f k = g k\nfn : f zero = false\ngn : g zero = true\nf_max : ℕ → Bool := fun n => rec false (fun x x => true) n\nhf_max : ∀ (n : ℕ), f n = true → f_max n = true\ng_min : ℕ → Bool := fun n => rec true (fun x x => false) n\nhg_min : ∀ (n : ℕ), g_min n = true → g n = true\n⊢ 0 < 1 - c",
"tactic": "rwa [sub_pos]"
},
{
"state_after": "case h.e'_4\nc : ℝ\nf✝¹ g✝¹ : ℕ → Bool\nn✝ : ℕ\nh1 : 0 < c\nh2 : c < 1 / 2\nn : ℕ\nf✝ g✝ : ℕ → Bool\nhn✝ : ∀ (k : ℕ), k < n → f✝ k = g✝ k\nfn✝ : f✝ n = false\ngn✝ : g✝ n = true\nh3 : c < 1\nf g : ℕ → Bool\nhn : ∀ (k : ℕ), k < zero → f k = g k\nfn : f zero = false\ngn : g zero = true\nf_max : ℕ → Bool := fun n => rec false (fun x x => true) n\nhf_max : ∀ (n : ℕ), f n = true → f_max n = true\ng_min : ℕ → Bool := fun n => rec true (fun x x => false) n\nhg_min : ∀ (n : ℕ), g_min n = true → g n = true\n⊢ 1 = 1 / 2 + 1 / 2",
"state_before": "c : ℝ\nf✝¹ g✝¹ : ℕ → Bool\nn✝ : ℕ\nh1 : 0 < c\nh2 : c < 1 / 2\nn : ℕ\nf✝ g✝ : ℕ → Bool\nhn✝ : ∀ (k : ℕ), k < n → f✝ k = g✝ k\nfn✝ : f✝ n = false\ngn✝ : g✝ n = true\nh3 : c < 1\nf g : ℕ → Bool\nhn : ∀ (k : ℕ), k < zero → f k = g k\nfn : f zero = false\ngn : g zero = true\nf_max : ℕ → Bool := fun n => rec false (fun x x => true) n\nhf_max : ∀ (n : ℕ), f n = true → f_max n = true\ng_min : ℕ → Bool := fun n => rec true (fun x x => false) n\nhg_min : ∀ (n : ℕ), g_min n = true → g n = true\n⊢ c + c < 1",
"tactic": "convert _root_.add_lt_add h2 h2"
},
{
"state_after": "no goals",
"state_before": "case h.e'_4\nc : ℝ\nf✝¹ g✝¹ : ℕ → Bool\nn✝ : ℕ\nh1 : 0 < c\nh2 : c < 1 / 2\nn : ℕ\nf✝ g✝ : ℕ → Bool\nhn✝ : ∀ (k : ℕ), k < n → f✝ k = g✝ k\nfn✝ : f✝ n = false\ngn✝ : g✝ n = true\nh3 : c < 1\nf g : ℕ → Bool\nhn : ∀ (k : ℕ), k < zero → f k = g k\nfn : f zero = false\ngn : g zero = true\nf_max : ℕ → Bool := fun n => rec false (fun x x => true) n\nhf_max : ∀ (n : ℕ), f n = true → f_max n = true\ng_min : ℕ → Bool := fun n => rec true (fun x x => false) n\nhg_min : ∀ (n : ℕ), g_min n = true → g n = true\n⊢ 1 = 1 / 2 + 1 / 2",
"tactic": "norm_num"
},
{
"state_after": "case h.e'_3\nc : ℝ\nf✝¹ g✝¹ : ℕ → Bool\nn✝ : ℕ\nh1 : 0 < c\nh2 : c < 1 / 2\nn : ℕ\nf✝ g✝ : ℕ → Bool\nhn✝ : ∀ (k : ℕ), k < n → f✝ k = g✝ k\nfn✝ : f✝ n = false\ngn✝ : g✝ n = true\nh3 : c < 1\nf g : ℕ → Bool\nhn : ∀ (k : ℕ), k < zero → f k = g k\nfn : f zero = false\ngn : g zero = true\nf_max : ℕ → Bool := fun n => rec false (fun x x => true) n\nhf_max : ∀ (n : ℕ), f n = true → f_max n = true\ng_min : ℕ → Bool := fun n => rec true (fun x x => false) n\nhg_min : ∀ (n : ℕ), g_min n = true → g n = true\nthis : c / (1 - c) < 1\n⊢ ((bif f_max 0 then 1 else 0) + c * cantorFunction c fun n => f_max (n + 1)) = c * ∑' (n : ℕ), c ^ n",
"state_before": "case h.e'_3\nc : ℝ\nf✝¹ g✝¹ : ℕ → Bool\nn✝ : ℕ\nh1 : 0 < c\nh2 : c < 1 / 2\nn : ℕ\nf✝ g✝ : ℕ → Bool\nhn✝ : ∀ (k : ℕ), k < n → f✝ k = g✝ k\nfn✝ : f✝ n = false\ngn✝ : g✝ n = true\nh3 : c < 1\nf g : ℕ → Bool\nhn : ∀ (k : ℕ), k < zero → f k = g k\nfn : f zero = false\ngn : g zero = true\nf_max : ℕ → Bool := fun n => rec false (fun x x => true) n\nhf_max : ∀ (n : ℕ), f n = true → f_max n = true\ng_min : ℕ → Bool := fun n => rec true (fun x x => false) n\nhg_min : ∀ (n : ℕ), g_min n = true → g n = true\nthis : c / (1 - c) < 1\n⊢ (cantorFunction c fun n => f_max n) = c / (1 - c)",
"tactic": "rw [cantorFunction_succ _ (le_of_lt h1) h3, div_eq_mul_inv, ←\n tsum_geometric_of_lt_1 (le_of_lt h1) h3]"
},
{
"state_after": "no goals",
"state_before": "case h.e'_3\nc : ℝ\nf✝¹ g✝¹ : ℕ → Bool\nn✝ : ℕ\nh1 : 0 < c\nh2 : c < 1 / 2\nn : ℕ\nf✝ g✝ : ℕ → Bool\nhn✝ : ∀ (k : ℕ), k < n → f✝ k = g✝ k\nfn✝ : f✝ n = false\ngn✝ : g✝ n = true\nh3 : c < 1\nf g : ℕ → Bool\nhn : ∀ (k : ℕ), k < zero → f k = g k\nfn : f zero = false\ngn : g zero = true\nf_max : ℕ → Bool := fun n => rec false (fun x x => true) n\nhf_max : ∀ (n : ℕ), f n = true → f_max n = true\ng_min : ℕ → Bool := fun n => rec true (fun x x => false) n\nhg_min : ∀ (n : ℕ), g_min n = true → g n = true\nthis : c / (1 - c) < 1\n⊢ ((bif f_max 0 then 1 else 0) + c * cantorFunction c fun n => f_max (n + 1)) = c * ∑' (n : ℕ), c ^ n",
"tactic": "apply zero_add"
},
{
"state_after": "case h.e'_4.refine'_1\nc : ℝ\nf✝¹ g✝¹ : ℕ → Bool\nn✝ : ℕ\nh1 : 0 < c\nh2 : c < 1 / 2\nn : ℕ\nf✝ g✝ : ℕ → Bool\nhn✝ : ∀ (k : ℕ), k < n → f✝ k = g✝ k\nfn✝ : f✝ n = false\ngn✝ : g✝ n = true\nh3 : c < 1\nf g : ℕ → Bool\nhn : ∀ (k : ℕ), k < zero → f k = g k\nfn : f zero = false\ngn : g zero = true\nf_max : ℕ → Bool := fun n => rec false (fun x x => true) n\nhf_max : ∀ (n : ℕ), f n = true → f_max n = true\ng_min : ℕ → Bool := fun n => rec true (fun x x => false) n\nhg_min : ∀ (n : ℕ), g_min n = true → g n = true\nthis : c / (1 - c) < 1\n⊢ ∀ (b' : ℕ), b' ≠ 0 → cantorFunctionAux c (fun n => g_min n) b' = 0\n\ncase h.e'_4.refine'_2\nc : ℝ\nf✝¹ g✝¹ : ℕ → Bool\nn✝ : ℕ\nh1 : 0 < c\nh2 : c < 1 / 2\nn : ℕ\nf✝ g✝ : ℕ → Bool\nhn✝ : ∀ (k : ℕ), k < n → f✝ k = g✝ k\nfn✝ : f✝ n = false\ngn✝ : g✝ n = true\nh3 : c < 1\nf g : ℕ → Bool\nhn : ∀ (k : ℕ), k < zero → f k = g k\nfn : f zero = false\ngn : g zero = true\nf_max : ℕ → Bool := fun n => rec false (fun x x => true) n\nhf_max : ∀ (n : ℕ), f n = true → f_max n = true\ng_min : ℕ → Bool := fun n => rec true (fun x x => false) n\nhg_min : ∀ (n : ℕ), g_min n = true → g n = true\nthis : c / (1 - c) < 1\n⊢ cantorFunctionAux c (fun n => g_min n) 0 = 1",
"state_before": "case h.e'_4\nc : ℝ\nf✝¹ g✝¹ : ℕ → Bool\nn✝ : ℕ\nh1 : 0 < c\nh2 : c < 1 / 2\nn : ℕ\nf✝ g✝ : ℕ → Bool\nhn✝ : ∀ (k : ℕ), k < n → f✝ k = g✝ k\nfn✝ : f✝ n = false\ngn✝ : g✝ n = true\nh3 : c < 1\nf g : ℕ → Bool\nhn : ∀ (k : ℕ), k < zero → f k = g k\nfn : f zero = false\ngn : g zero = true\nf_max : ℕ → Bool := fun n => rec false (fun x x => true) n\nhf_max : ∀ (n : ℕ), f n = true → f_max n = true\ng_min : ℕ → Bool := fun n => rec true (fun x x => false) n\nhg_min : ∀ (n : ℕ), g_min n = true → g n = true\nthis : c / (1 - c) < 1\n⊢ (cantorFunction c fun n => g_min n) = 1",
"tactic": "refine' (tsum_eq_single 0 _).trans _"
},
{
"state_after": "case h.e'_4.refine'_1\nc : ℝ\nf✝¹ g✝¹ : ℕ → Bool\nn✝¹ : ℕ\nh1 : 0 < c\nh2 : c < 1 / 2\nn✝ : ℕ\nf✝ g✝ : ℕ → Bool\nhn✝¹ : ∀ (k : ℕ), k < n✝ → f✝ k = g✝ k\nfn✝ : f✝ n✝ = false\ngn✝ : g✝ n✝ = true\nh3 : c < 1\nf g : ℕ → Bool\nhn✝ : ∀ (k : ℕ), k < zero → f k = g k\nfn : f zero = false\ngn : g zero = true\nf_max : ℕ → Bool := fun n => rec false (fun x x => true) n\nhf_max : ∀ (n : ℕ), f n = true → f_max n = true\ng_min : ℕ → Bool := fun n => rec true (fun x x => false) n\nhg_min : ∀ (n : ℕ), g_min n = true → g n = true\nthis : c / (1 - c) < 1\nn : ℕ\nhn : n ≠ 0\n⊢ cantorFunctionAux c (fun n => g_min n) n = 0",
"state_before": "case h.e'_4.refine'_1\nc : ℝ\nf✝¹ g✝¹ : ℕ → Bool\nn✝ : ℕ\nh1 : 0 < c\nh2 : c < 1 / 2\nn : ℕ\nf✝ g✝ : ℕ → Bool\nhn✝ : ∀ (k : ℕ), k < n → f✝ k = g✝ k\nfn✝ : f✝ n = false\ngn✝ : g✝ n = true\nh3 : c < 1\nf g : ℕ → Bool\nhn : ∀ (k : ℕ), k < zero → f k = g k\nfn : f zero = false\ngn : g zero = true\nf_max : ℕ → Bool := fun n => rec false (fun x x => true) n\nhf_max : ∀ (n : ℕ), f n = true → f_max n = true\ng_min : ℕ → Bool := fun n => rec true (fun x x => false) n\nhg_min : ∀ (n : ℕ), g_min n = true → g n = true\nthis : c / (1 - c) < 1\n⊢ ∀ (b' : ℕ), b' ≠ 0 → cantorFunctionAux c (fun n => g_min n) b' = 0",
"tactic": "intro n hn"
},
{
"state_after": "case h.e'_4.refine'_1.zero\nc : ℝ\nf✝¹ g✝¹ : ℕ → Bool\nn✝ : ℕ\nh1 : 0 < c\nh2 : c < 1 / 2\nn : ℕ\nf✝ g✝ : ℕ → Bool\nhn✝¹ : ∀ (k : ℕ), k < n → f✝ k = g✝ k\nfn✝ : f✝ n = false\ngn✝ : g✝ n = true\nh3 : c < 1\nf g : ℕ → Bool\nhn✝ : ∀ (k : ℕ), k < zero → f k = g k\nfn : f zero = false\ngn : g zero = true\nf_max : ℕ → Bool := fun n => rec false (fun x x => true) n\nhf_max : ∀ (n : ℕ), f n = true → f_max n = true\ng_min : ℕ → Bool := fun n => rec true (fun x x => false) n\nhg_min : ∀ (n : ℕ), g_min n = true → g n = true\nthis : c / (1 - c) < 1\nhn : zero ≠ 0\n⊢ cantorFunctionAux c (fun n => g_min n) zero = 0\n\ncase h.e'_4.refine'_1.succ\nc : ℝ\nf✝¹ g✝¹ : ℕ → Bool\nn✝¹ : ℕ\nh1 : 0 < c\nh2 : c < 1 / 2\nn : ℕ\nf✝ g✝ : ℕ → Bool\nhn✝¹ : ∀ (k : ℕ), k < n → f✝ k = g✝ k\nfn✝ : f✝ n = false\ngn✝ : g✝ n = true\nh3 : c < 1\nf g : ℕ → Bool\nhn✝ : ∀ (k : ℕ), k < zero → f k = g k\nfn : f zero = false\ngn : g zero = true\nf_max : ℕ → Bool := fun n => rec false (fun x x => true) n\nhf_max : ∀ (n : ℕ), f n = true → f_max n = true\ng_min : ℕ → Bool := fun n => rec true (fun x x => false) n\nhg_min : ∀ (n : ℕ), g_min n = true → g n = true\nthis : c / (1 - c) < 1\nn✝ : ℕ\nhn : succ n✝ ≠ 0\n⊢ cantorFunctionAux c (fun n => g_min n) (succ n✝) = 0",
"state_before": "case h.e'_4.refine'_1\nc : ℝ\nf✝¹ g✝¹ : ℕ → Bool\nn✝¹ : ℕ\nh1 : 0 < c\nh2 : c < 1 / 2\nn✝ : ℕ\nf✝ g✝ : ℕ → Bool\nhn✝¹ : ∀ (k : ℕ), k < n✝ → f✝ k = g✝ k\nfn✝ : f✝ n✝ = false\ngn✝ : g✝ n✝ = true\nh3 : c < 1\nf g : ℕ → Bool\nhn✝ : ∀ (k : ℕ), k < zero → f k = g k\nfn : f zero = false\ngn : g zero = true\nf_max : ℕ → Bool := fun n => rec false (fun x x => true) n\nhf_max : ∀ (n : ℕ), f n = true → f_max n = true\ng_min : ℕ → Bool := fun n => rec true (fun x x => false) n\nhg_min : ∀ (n : ℕ), g_min n = true → g n = true\nthis : c / (1 - c) < 1\nn : ℕ\nhn : n ≠ 0\n⊢ cantorFunctionAux c (fun n => g_min n) n = 0",
"tactic": "cases n"
},
{
"state_after": "case h.e'_4.refine'_1.succ\nc : ℝ\nf✝¹ g✝¹ : ℕ → Bool\nn✝¹ : ℕ\nh1 : 0 < c\nh2 : c < 1 / 2\nn : ℕ\nf✝ g✝ : ℕ → Bool\nhn✝¹ : ∀ (k : ℕ), k < n → f✝ k = g✝ k\nfn✝ : f✝ n = false\ngn✝ : g✝ n = true\nh3 : c < 1\nf g : ℕ → Bool\nhn✝ : ∀ (k : ℕ), k < zero → f k = g k\nfn : f zero = false\ngn : g zero = true\nf_max : ℕ → Bool := fun n => rec false (fun x x => true) n\nhf_max : ∀ (n : ℕ), f n = true → f_max n = true\ng_min : ℕ → Bool := fun n => rec true (fun x x => false) n\nhg_min : ∀ (n : ℕ), g_min n = true → g n = true\nthis : c / (1 - c) < 1\nn✝ : ℕ\nhn : succ n✝ ≠ 0\n⊢ cantorFunctionAux c (fun n => g_min n) (succ n✝) = 0",
"state_before": "case h.e'_4.refine'_1.zero\nc : ℝ\nf✝¹ g✝¹ : ℕ → Bool\nn✝ : ℕ\nh1 : 0 < c\nh2 : c < 1 / 2\nn : ℕ\nf✝ g✝ : ℕ → Bool\nhn✝¹ : ∀ (k : ℕ), k < n → f✝ k = g✝ k\nfn✝ : f✝ n = false\ngn✝ : g✝ n = true\nh3 : c < 1\nf g : ℕ → Bool\nhn✝ : ∀ (k : ℕ), k < zero → f k = g k\nfn : f zero = false\ngn : g zero = true\nf_max : ℕ → Bool := fun n => rec false (fun x x => true) n\nhf_max : ∀ (n : ℕ), f n = true → f_max n = true\ng_min : ℕ → Bool := fun n => rec true (fun x x => false) n\nhg_min : ∀ (n : ℕ), g_min n = true → g n = true\nthis : c / (1 - c) < 1\nhn : zero ≠ 0\n⊢ cantorFunctionAux c (fun n => g_min n) zero = 0\n\ncase h.e'_4.refine'_1.succ\nc : ℝ\nf✝¹ g✝¹ : ℕ → Bool\nn✝¹ : ℕ\nh1 : 0 < c\nh2 : c < 1 / 2\nn : ℕ\nf✝ g✝ : ℕ → Bool\nhn✝¹ : ∀ (k : ℕ), k < n → f✝ k = g✝ k\nfn✝ : f✝ n = false\ngn✝ : g✝ n = true\nh3 : c < 1\nf g : ℕ → Bool\nhn✝ : ∀ (k : ℕ), k < zero → f k = g k\nfn : f zero = false\ngn : g zero = true\nf_max : ℕ → Bool := fun n => rec false (fun x x => true) n\nhf_max : ∀ (n : ℕ), f n = true → f_max n = true\ng_min : ℕ → Bool := fun n => rec true (fun x x => false) n\nhg_min : ∀ (n : ℕ), g_min n = true → g n = true\nthis : c / (1 - c) < 1\nn✝ : ℕ\nhn : succ n✝ ≠ 0\n⊢ cantorFunctionAux c (fun n => g_min n) (succ n✝) = 0",
"tactic": "contradiction"
},
{
"state_after": "no goals",
"state_before": "case h.e'_4.refine'_1.succ\nc : ℝ\nf✝¹ g✝¹ : ℕ → Bool\nn✝¹ : ℕ\nh1 : 0 < c\nh2 : c < 1 / 2\nn : ℕ\nf✝ g✝ : ℕ → Bool\nhn✝¹ : ∀ (k : ℕ), k < n → f✝ k = g✝ k\nfn✝ : f✝ n = false\ngn✝ : g✝ n = true\nh3 : c < 1\nf g : ℕ → Bool\nhn✝ : ∀ (k : ℕ), k < zero → f k = g k\nfn : f zero = false\ngn : g zero = true\nf_max : ℕ → Bool := fun n => rec false (fun x x => true) n\nhf_max : ∀ (n : ℕ), f n = true → f_max n = true\ng_min : ℕ → Bool := fun n => rec true (fun x x => false) n\nhg_min : ∀ (n : ℕ), g_min n = true → g n = true\nthis : c / (1 - c) < 1\nn✝ : ℕ\nhn : succ n✝ ≠ 0\n⊢ cantorFunctionAux c (fun n => g_min n) (succ n✝) = 0",
"tactic": "rfl"
},
{
"state_after": "no goals",
"state_before": "case h.e'_4.refine'_2\nc : ℝ\nf✝¹ g✝¹ : ℕ → Bool\nn✝ : ℕ\nh1 : 0 < c\nh2 : c < 1 / 2\nn : ℕ\nf✝ g✝ : ℕ → Bool\nhn✝ : ∀ (k : ℕ), k < n → f✝ k = g✝ k\nfn✝ : f✝ n = false\ngn✝ : g✝ n = true\nh3 : c < 1\nf g : ℕ → Bool\nhn : ∀ (k : ℕ), k < zero → f k = g k\nfn : f zero = false\ngn : g zero = true\nf_max : ℕ → Bool := fun n => rec false (fun x x => true) n\nhf_max : ∀ (n : ℕ), f n = true → f_max n = true\ng_min : ℕ → Bool := fun n => rec true (fun x x => false) n\nhg_min : ∀ (n : ℕ), g_min n = true → g n = true\nthis : c / (1 - c) < 1\n⊢ cantorFunctionAux c (fun n => g_min n) 0 = 1",
"tactic": "exact cantorFunctionAux_zero _"
}
] |
[
166,
59
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
125,
1
] |
Mathlib/Topology/Algebra/WithZeroTopology.lean
|
WithZeroTopology.tendsto_of_ne_zero
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_2\nΓ₀ : Type u_1\ninst✝ : LinearOrderedCommGroupWithZero Γ₀\nγ✝ γ₁ γ₂ : Γ₀\nl : Filter α\nf : α → Γ₀\nγ : Γ₀\nh : γ ≠ 0\n⊢ Tendsto f l (𝓝 γ) ↔ ∀ᶠ (x : α) in l, f x = γ",
"tactic": "rw [nhds_of_ne_zero h, tendsto_pure]"
}
] |
[
124,
39
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
123,
1
] |
Mathlib/Data/Set/Countable.lean
|
Set.exists_seq_iSup_eq_top_iff_countable
|
[
{
"state_after": "case mp\nα : Type u\nβ : Type v\nγ : Type w\nι : Sort x\ninst✝ : CompleteLattice α\np : α → Prop\nh : ∃ x, p x\n⊢ (∃ s, (∀ (n : ℕ), p (s n)) ∧ (⨆ (n : ℕ), s n) = ⊤) → ∃ S, Set.Countable S ∧ (∀ (s : α), s ∈ S → p s) ∧ sSup S = ⊤\n\ncase mpr\nα : Type u\nβ : Type v\nγ : Type w\nι : Sort x\ninst✝ : CompleteLattice α\np : α → Prop\nh : ∃ x, p x\n⊢ (∃ S, Set.Countable S ∧ (∀ (s : α), s ∈ S → p s) ∧ sSup S = ⊤) → ∃ s, (∀ (n : ℕ), p (s n)) ∧ (⨆ (n : ℕ), s n) = ⊤",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\nι : Sort x\ninst✝ : CompleteLattice α\np : α → Prop\nh : ∃ x, p x\n⊢ (∃ s, (∀ (n : ℕ), p (s n)) ∧ (⨆ (n : ℕ), s n) = ⊤) ↔ ∃ S, Set.Countable S ∧ (∀ (s : α), s ∈ S → p s) ∧ sSup S = ⊤",
"tactic": "constructor"
},
{
"state_after": "case mp.intro.intro\nα : Type u\nβ : Type v\nγ : Type w\nι : Sort x\ninst✝ : CompleteLattice α\np : α → Prop\nh : ∃ x, p x\ns : ℕ → α\nhps : ∀ (n : ℕ), p (s n)\nhs : (⨆ (n : ℕ), s n) = ⊤\n⊢ ∃ S, Set.Countable S ∧ (∀ (s : α), s ∈ S → p s) ∧ sSup S = ⊤",
"state_before": "case mp\nα : Type u\nβ : Type v\nγ : Type w\nι : Sort x\ninst✝ : CompleteLattice α\np : α → Prop\nh : ∃ x, p x\n⊢ (∃ s, (∀ (n : ℕ), p (s n)) ∧ (⨆ (n : ℕ), s n) = ⊤) → ∃ S, Set.Countable S ∧ (∀ (s : α), s ∈ S → p s) ∧ sSup S = ⊤",
"tactic": "rintro ⟨s, hps, hs⟩"
},
{
"state_after": "case mp.intro.intro\nα : Type u\nβ : Type v\nγ : Type w\nι : Sort x\ninst✝ : CompleteLattice α\np : α → Prop\nh : ∃ x, p x\ns : ℕ → α\nhps : ∀ (n : ℕ), p (s n)\nhs : (⨆ (n : ℕ), s n) = ⊤\n⊢ sSup (range s) = ⊤",
"state_before": "case mp.intro.intro\nα : Type u\nβ : Type v\nγ : Type w\nι : Sort x\ninst✝ : CompleteLattice α\np : α → Prop\nh : ∃ x, p x\ns : ℕ → α\nhps : ∀ (n : ℕ), p (s n)\nhs : (⨆ (n : ℕ), s n) = ⊤\n⊢ ∃ S, Set.Countable S ∧ (∀ (s : α), s ∈ S → p s) ∧ sSup S = ⊤",
"tactic": "refine' ⟨range s, countable_range s, forall_range_iff.2 hps, _⟩"
},
{
"state_after": "no goals",
"state_before": "case mp.intro.intro\nα : Type u\nβ : Type v\nγ : Type w\nι : Sort x\ninst✝ : CompleteLattice α\np : α → Prop\nh : ∃ x, p x\ns : ℕ → α\nhps : ∀ (n : ℕ), p (s n)\nhs : (⨆ (n : ℕ), s n) = ⊤\n⊢ sSup (range s) = ⊤",
"tactic": "rwa [sSup_range]"
},
{
"state_after": "case mpr.intro.intro.intro\nα : Type u\nβ : Type v\nγ : Type w\nι : Sort x\ninst✝ : CompleteLattice α\np : α → Prop\nh : ∃ x, p x\nS : Set α\nhSc : Set.Countable S\nhps : ∀ (s : α), s ∈ S → p s\nhS : sSup S = ⊤\n⊢ ∃ s, (∀ (n : ℕ), p (s n)) ∧ (⨆ (n : ℕ), s n) = ⊤",
"state_before": "case mpr\nα : Type u\nβ : Type v\nγ : Type w\nι : Sort x\ninst✝ : CompleteLattice α\np : α → Prop\nh : ∃ x, p x\n⊢ (∃ S, Set.Countable S ∧ (∀ (s : α), s ∈ S → p s) ∧ sSup S = ⊤) → ∃ s, (∀ (n : ℕ), p (s n)) ∧ (⨆ (n : ℕ), s n) = ⊤",
"tactic": "rintro ⟨S, hSc, hps, hS⟩"
},
{
"state_after": "case mpr.intro.intro.intro.inl\nα : Type u\nβ : Type v\nγ : Type w\nι : Sort x\ninst✝ : CompleteLattice α\np : α → Prop\nh : ∃ x, p x\nhSc : Set.Countable ∅\nhps : ∀ (s : α), s ∈ ∅ → p s\nhS : sSup ∅ = ⊤\n⊢ ∃ s, (∀ (n : ℕ), p (s n)) ∧ (⨆ (n : ℕ), s n) = ⊤\n\ncase mpr.intro.intro.intro.inr\nα : Type u\nβ : Type v\nγ : Type w\nι : Sort x\ninst✝ : CompleteLattice α\np : α → Prop\nh : ∃ x, p x\nS : Set α\nhSc : Set.Countable S\nhps : ∀ (s : α), s ∈ S → p s\nhS : sSup S = ⊤\nhne : Set.Nonempty S\n⊢ ∃ s, (∀ (n : ℕ), p (s n)) ∧ (⨆ (n : ℕ), s n) = ⊤",
"state_before": "case mpr.intro.intro.intro\nα : Type u\nβ : Type v\nγ : Type w\nι : Sort x\ninst✝ : CompleteLattice α\np : α → Prop\nh : ∃ x, p x\nS : Set α\nhSc : Set.Countable S\nhps : ∀ (s : α), s ∈ S → p s\nhS : sSup S = ⊤\n⊢ ∃ s, (∀ (n : ℕ), p (s n)) ∧ (⨆ (n : ℕ), s n) = ⊤",
"tactic": "rcases eq_empty_or_nonempty S with (rfl | hne)"
},
{
"state_after": "case mpr.intro.intro.intro.inl\nα : Type u\nβ : Type v\nγ : Type w\nι : Sort x\ninst✝ : CompleteLattice α\np : α → Prop\nh : ∃ x, p x\nhSc : Set.Countable ∅\nhps : ∀ (s : α), s ∈ ∅ → p s\nhS : ⊥ = ⊤\n⊢ ∃ s, (∀ (n : ℕ), p (s n)) ∧ (⨆ (n : ℕ), s n) = ⊤",
"state_before": "case mpr.intro.intro.intro.inl\nα : Type u\nβ : Type v\nγ : Type w\nι : Sort x\ninst✝ : CompleteLattice α\np : α → Prop\nh : ∃ x, p x\nhSc : Set.Countable ∅\nhps : ∀ (s : α), s ∈ ∅ → p s\nhS : sSup ∅ = ⊤\n⊢ ∃ s, (∀ (n : ℕ), p (s n)) ∧ (⨆ (n : ℕ), s n) = ⊤",
"tactic": "rw [sSup_empty] at hS"
},
{
"state_after": "case mpr.intro.intro.intro.inl\nα : Type u\nβ : Type v\nγ : Type w\nι : Sort x\ninst✝ : CompleteLattice α\np : α → Prop\nh : ∃ x, p x\nhSc : Set.Countable ∅\nhps : ∀ (s : α), s ∈ ∅ → p s\nhS : ⊥ = ⊤\nthis : Subsingleton α\n⊢ ∃ s, (∀ (n : ℕ), p (s n)) ∧ (⨆ (n : ℕ), s n) = ⊤",
"state_before": "case mpr.intro.intro.intro.inl\nα : Type u\nβ : Type v\nγ : Type w\nι : Sort x\ninst✝ : CompleteLattice α\np : α → Prop\nh : ∃ x, p x\nhSc : Set.Countable ∅\nhps : ∀ (s : α), s ∈ ∅ → p s\nhS : ⊥ = ⊤\n⊢ ∃ s, (∀ (n : ℕ), p (s n)) ∧ (⨆ (n : ℕ), s n) = ⊤",
"tactic": "haveI := subsingleton_of_bot_eq_top hS"
},
{
"state_after": "case mpr.intro.intro.intro.inl.intro\nα : Type u\nβ : Type v\nγ : Type w\nι : Sort x\ninst✝ : CompleteLattice α\np : α → Prop\nhSc : Set.Countable ∅\nhps : ∀ (s : α), s ∈ ∅ → p s\nhS : ⊥ = ⊤\nthis : Subsingleton α\nx : α\nhx : p x\n⊢ ∃ s, (∀ (n : ℕ), p (s n)) ∧ (⨆ (n : ℕ), s n) = ⊤",
"state_before": "case mpr.intro.intro.intro.inl\nα : Type u\nβ : Type v\nγ : Type w\nι : Sort x\ninst✝ : CompleteLattice α\np : α → Prop\nh : ∃ x, p x\nhSc : Set.Countable ∅\nhps : ∀ (s : α), s ∈ ∅ → p s\nhS : ⊥ = ⊤\nthis : Subsingleton α\n⊢ ∃ s, (∀ (n : ℕ), p (s n)) ∧ (⨆ (n : ℕ), s n) = ⊤",
"tactic": "rcases h with ⟨x, hx⟩"
},
{
"state_after": "no goals",
"state_before": "case mpr.intro.intro.intro.inl.intro\nα : Type u\nβ : Type v\nγ : Type w\nι : Sort x\ninst✝ : CompleteLattice α\np : α → Prop\nhSc : Set.Countable ∅\nhps : ∀ (s : α), s ∈ ∅ → p s\nhS : ⊥ = ⊤\nthis : Subsingleton α\nx : α\nhx : p x\n⊢ ∃ s, (∀ (n : ℕ), p (s n)) ∧ (⨆ (n : ℕ), s n) = ⊤",
"tactic": "exact ⟨fun _ => x, fun _ => hx, Subsingleton.elim _ _⟩"
},
{
"state_after": "case mpr.intro.intro.intro.inr.intro\nα : Type u\nβ : Type v\nγ : Type w\nι : Sort x\ninst✝ : CompleteLattice α\np : α → Prop\nh : ∃ x, p x\nS : Set α\nhSc : Set.Countable S\nhps : ∀ (s : α), s ∈ S → p s\nhS : sSup S = ⊤\nhne : Set.Nonempty S\ns : ℕ → ↑S\nhs : Surjective s\n⊢ ∃ s, (∀ (n : ℕ), p (s n)) ∧ (⨆ (n : ℕ), s n) = ⊤",
"state_before": "case mpr.intro.intro.intro.inr\nα : Type u\nβ : Type v\nγ : Type w\nι : Sort x\ninst✝ : CompleteLattice α\np : α → Prop\nh : ∃ x, p x\nS : Set α\nhSc : Set.Countable S\nhps : ∀ (s : α), s ∈ S → p s\nhS : sSup S = ⊤\nhne : Set.Nonempty S\n⊢ ∃ s, (∀ (n : ℕ), p (s n)) ∧ (⨆ (n : ℕ), s n) = ⊤",
"tactic": "rcases(Set.countable_iff_exists_surjective hne).1 hSc with ⟨s, hs⟩"
},
{
"state_after": "case mpr.intro.intro.intro.inr.intro\nα : Type u\nβ : Type v\nγ : Type w\nι : Sort x\ninst✝ : CompleteLattice α\np : α → Prop\nh : ∃ x, p x\nS : Set α\nhSc : Set.Countable S\nhps : ∀ (s : α), s ∈ S → p s\nhS : sSup S = ⊤\nhne : Set.Nonempty S\ns : ℕ → ↑S\nhs : Surjective s\n⊢ (⨆ (n : ℕ), (fun n => ↑(s n)) n) = ⊤",
"state_before": "case mpr.intro.intro.intro.inr.intro\nα : Type u\nβ : Type v\nγ : Type w\nι : Sort x\ninst✝ : CompleteLattice α\np : α → Prop\nh : ∃ x, p x\nS : Set α\nhSc : Set.Countable S\nhps : ∀ (s : α), s ∈ S → p s\nhS : sSup S = ⊤\nhne : Set.Nonempty S\ns : ℕ → ↑S\nhs : Surjective s\n⊢ ∃ s, (∀ (n : ℕ), p (s n)) ∧ (⨆ (n : ℕ), s n) = ⊤",
"tactic": "refine' ⟨fun n => s n, fun n => hps _ (s n).coe_prop, _⟩"
},
{
"state_after": "no goals",
"state_before": "case mpr.intro.intro.intro.inr.intro\nα : Type u\nβ : Type v\nγ : Type w\nι : Sort x\ninst✝ : CompleteLattice α\np : α → Prop\nh : ∃ x, p x\nS : Set α\nhSc : Set.Countable S\nhps : ∀ (s : α), s ∈ S → p s\nhS : sSup S = ⊤\nhne : Set.Nonempty S\ns : ℕ → ↑S\nhs : Surjective s\n⊢ (⨆ (n : ℕ), (fun n => ↑(s n)) n) = ⊤",
"tactic": "rwa [hs.iSup_comp, ← sSup_eq_iSup']"
}
] |
[
171,
42
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
156,
1
] |
Mathlib/Order/Filter/Basic.lean
|
Filter.isCompl_principal
|
[
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\nδ : Type ?u.117337\nι : Sort x\nf g : Filter α\ns✝ t s : Set α\n⊢ 𝓟 s ⊓ 𝓟 (sᶜ) = ⊥",
"tactic": "rw [inf_principal, inter_compl_self, principal_empty]"
},
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\nδ : Type ?u.117337\nι : Sort x\nf g : Filter α\ns✝ t s : Set α\n⊢ 𝓟 s ⊔ 𝓟 (sᶜ) = ⊤",
"tactic": "rw [sup_principal, union_compl_self, principal_univ]"
}
] |
[
995,
57
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
993,
1
] |
Mathlib/CategoryTheory/Monoidal/Opposite.lean
|
CategoryTheory.mop_comp
|
[] |
[
125,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
124,
1
] |
Mathlib/Geometry/Euclidean/Angle/Unoriented/RightAngle.lean
|
InnerProductGeometry.sin_angle_sub_mul_norm_of_inner_eq_zero
|
[
{
"state_after": "V : Type u_1\ninst✝¹ : NormedAddCommGroup V\ninst✝ : InnerProductSpace ℝ V\nx y : V\nh : inner x (-y) = 0\n⊢ Real.sin (angle x (x - y)) * ‖x - y‖ = ‖y‖",
"state_before": "V : Type u_1\ninst✝¹ : NormedAddCommGroup V\ninst✝ : InnerProductSpace ℝ V\nx y : V\nh : inner x y = 0\n⊢ Real.sin (angle x (x - y)) * ‖x - y‖ = ‖y‖",
"tactic": "rw [← neg_eq_zero, ← inner_neg_right] at h"
},
{
"state_after": "no goals",
"state_before": "V : Type u_1\ninst✝¹ : NormedAddCommGroup V\ninst✝ : InnerProductSpace ℝ V\nx y : V\nh : inner x (-y) = 0\n⊢ Real.sin (angle x (x - y)) * ‖x - y‖ = ‖y‖",
"tactic": "rw [sub_eq_add_neg, sin_angle_add_mul_norm_of_inner_eq_zero h, norm_neg]"
}
] |
[
318,
75
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
315,
1
] |
Mathlib/MeasureTheory/Function/L1Space.lean
|
MeasureTheory.Integrable.comp_aemeasurable
|
[] |
[
605,
60
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
603,
1
] |
Mathlib/Data/Int/Cast/Lemmas.lean
|
Int.cast_lt
|
[] |
[
141,
28
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
140,
1
] |
Mathlib/Order/Filter/Ultrafilter.lean
|
Ultrafilter.eq_pure_of_finite_mem
|
[
{
"state_after": "α : Type u\nβ : Type v\nγ : Type ?u.21406\nf g : Ultrafilter α\ns t : Set α\np q : α → Prop\nh : Set.Finite s\nh' : (⋃ (x : α) (_ : x ∈ s), {x}) ∈ f\n⊢ ∃ x, x ∈ s ∧ f = pure x",
"state_before": "α : Type u\nβ : Type v\nγ : Type ?u.21406\nf g : Ultrafilter α\ns t : Set α\np q : α → Prop\nh : Set.Finite s\nh' : s ∈ f\n⊢ ∃ x, x ∈ s ∧ f = pure x",
"tactic": "rw [← biUnion_of_singleton s] at h'"
},
{
"state_after": "case intro.intro\nα : Type u\nβ : Type v\nγ : Type ?u.21406\nf g : Ultrafilter α\ns t : Set α\np q : α → Prop\nh : Set.Finite s\nh' : (⋃ (x : α) (_ : x ∈ s), {x}) ∈ f\na : α\nhas : a ∈ s\nhaf : {a} ∈ f\n⊢ ∃ x, x ∈ s ∧ f = pure x",
"state_before": "α : Type u\nβ : Type v\nγ : Type ?u.21406\nf g : Ultrafilter α\ns t : Set α\np q : α → Prop\nh : Set.Finite s\nh' : (⋃ (x : α) (_ : x ∈ s), {x}) ∈ f\n⊢ ∃ x, x ∈ s ∧ f = pure x",
"tactic": "rcases(Ultrafilter.finite_biUnion_mem_iff h).mp h' with ⟨a, has, haf⟩"
},
{
"state_after": "no goals",
"state_before": "case intro.intro\nα : Type u\nβ : Type v\nγ : Type ?u.21406\nf g : Ultrafilter α\ns t : Set α\np q : α → Prop\nh : Set.Finite s\nh' : (⋃ (x : α) (_ : x ∈ s), {x}) ∈ f\na : α\nhas : a ∈ s\nhaf : {a} ∈ f\n⊢ ∃ x, x ∈ s ∧ f = pure x",
"tactic": "exact ⟨a, has, eq_of_le (Filter.le_pure_iff.2 haf)⟩"
}
] |
[
322,
54
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
319,
1
] |
Mathlib/Topology/ContinuousFunction/Bounded.lean
|
BoundedContinuousFunction.nnnorm_coeFn_eq
|
[] |
[
1607,
77
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1607,
1
] |
Mathlib/Algebra/RingQuot.lean
|
RingQuot.mkRingHom_surjective
|
[
{
"state_after": "R : Type u₁\ninst✝³ : Semiring R\nS : Type u₂\ninst✝² : CommSemiring S\nA : Type u₃\ninst✝¹ : Semiring A\ninst✝ : Algebra S A\nr✝ r : R → R → Prop\n⊢ Function.Surjective fun x => { toQuot := Quot.mk (Rel r) x }",
"state_before": "R : Type u₁\ninst✝³ : Semiring R\nS : Type u₂\ninst✝² : CommSemiring S\nA : Type u₃\ninst✝¹ : Semiring A\ninst✝ : Algebra S A\nr✝ r : R → R → Prop\n⊢ Function.Surjective ↑(mkRingHom r)",
"tactic": "simp [mkRingHom_def]"
},
{
"state_after": "case mk.mk\nR : Type u₁\ninst✝³ : Semiring R\nS : Type u₂\ninst✝² : CommSemiring S\nA : Type u₃\ninst✝¹ : Semiring A\ninst✝ : Algebra S A\nr✝ r : R → R → Prop\ntoQuot✝ : Quot (Rel r)\na✝ : R\n⊢ ∃ a, (fun x => { toQuot := Quot.mk (Rel r) x }) a = { toQuot := Quot.mk (Rel r) a✝ }",
"state_before": "R : Type u₁\ninst✝³ : Semiring R\nS : Type u₂\ninst✝² : CommSemiring S\nA : Type u₃\ninst✝¹ : Semiring A\ninst✝ : Algebra S A\nr✝ r : R → R → Prop\n⊢ Function.Surjective fun x => { toQuot := Quot.mk (Rel r) x }",
"tactic": "rintro ⟨⟨⟩⟩"
},
{
"state_after": "no goals",
"state_before": "case mk.mk\nR : Type u₁\ninst✝³ : Semiring R\nS : Type u₂\ninst✝² : CommSemiring S\nA : Type u₃\ninst✝¹ : Semiring A\ninst✝ : Algebra S A\nr✝ r : R → R → Prop\ntoQuot✝ : Quot (Rel r)\na✝ : R\n⊢ ∃ a, (fun x => { toQuot := Quot.mk (Rel r) x }) a = { toQuot := Quot.mk (Rel r) a✝ }",
"tactic": "simp"
}
] |
[
408,
7
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
405,
1
] |
Mathlib/Analysis/NormedSpace/FiniteDimension.lean
|
isOpen_setOf_nat_le_rank
|
[
{
"state_after": "𝕜 : Type u\ninst✝¹⁰ : NontriviallyNormedField 𝕜\nE : Type v\ninst✝⁹ : NormedAddCommGroup E\ninst✝⁸ : NormedSpace 𝕜 E\nF : Type w\ninst✝⁷ : NormedAddCommGroup F\ninst✝⁶ : NormedSpace 𝕜 F\nF' : Type x\ninst✝⁵ : AddCommGroup F'\ninst✝⁴ : Module 𝕜 F'\ninst✝³ : TopologicalSpace F'\ninst✝² : TopologicalAddGroup F'\ninst✝¹ : ContinuousSMul 𝕜 F'\ninst✝ : CompleteSpace 𝕜\nn : ℕ\n⊢ IsOpen (⋃ (i : Finset E) (_ : Finset.card i = n), {x | LinearIndependent 𝕜 fun x_1 => ↑↑x ↑x_1})",
"state_before": "𝕜 : Type u\ninst✝¹⁰ : NontriviallyNormedField 𝕜\nE : Type v\ninst✝⁹ : NormedAddCommGroup E\ninst✝⁸ : NormedSpace 𝕜 E\nF : Type w\ninst✝⁷ : NormedAddCommGroup F\ninst✝⁶ : NormedSpace 𝕜 F\nF' : Type x\ninst✝⁵ : AddCommGroup F'\ninst✝⁴ : Module 𝕜 F'\ninst✝³ : TopologicalSpace F'\ninst✝² : TopologicalAddGroup F'\ninst✝¹ : ContinuousSMul 𝕜 F'\ninst✝ : CompleteSpace 𝕜\nn : ℕ\n⊢ IsOpen {f | ↑n ≤ LinearMap.rank ↑f}",
"tactic": "simp only [LinearMap.le_rank_iff_exists_linearIndependent_finset, setOf_exists, ← exists_prop]"
},
{
"state_after": "𝕜 : Type u\ninst✝¹⁰ : NontriviallyNormedField 𝕜\nE : Type v\ninst✝⁹ : NormedAddCommGroup E\ninst✝⁸ : NormedSpace 𝕜 E\nF : Type w\ninst✝⁷ : NormedAddCommGroup F\ninst✝⁶ : NormedSpace 𝕜 F\nF' : Type x\ninst✝⁵ : AddCommGroup F'\ninst✝⁴ : Module 𝕜 F'\ninst✝³ : TopologicalSpace F'\ninst✝² : TopologicalAddGroup F'\ninst✝¹ : ContinuousSMul 𝕜 F'\ninst✝ : CompleteSpace 𝕜\nn : ℕ\nt : Finset E\nx✝ : t ∈ fun i => Finset.card i = n\n⊢ IsOpen {x | LinearIndependent 𝕜 fun x_1 => ↑↑x ↑x_1}",
"state_before": "𝕜 : Type u\ninst✝¹⁰ : NontriviallyNormedField 𝕜\nE : Type v\ninst✝⁹ : NormedAddCommGroup E\ninst✝⁸ : NormedSpace 𝕜 E\nF : Type w\ninst✝⁷ : NormedAddCommGroup F\ninst✝⁶ : NormedSpace 𝕜 F\nF' : Type x\ninst✝⁵ : AddCommGroup F'\ninst✝⁴ : Module 𝕜 F'\ninst✝³ : TopologicalSpace F'\ninst✝² : TopologicalAddGroup F'\ninst✝¹ : ContinuousSMul 𝕜 F'\ninst✝ : CompleteSpace 𝕜\nn : ℕ\n⊢ IsOpen (⋃ (i : Finset E) (_ : Finset.card i = n), {x | LinearIndependent 𝕜 fun x_1 => ↑↑x ↑x_1})",
"tactic": "refine' isOpen_biUnion fun t _ => _"
},
{
"state_after": "𝕜 : Type u\ninst✝¹⁰ : NontriviallyNormedField 𝕜\nE : Type v\ninst✝⁹ : NormedAddCommGroup E\ninst✝⁸ : NormedSpace 𝕜 E\nF : Type w\ninst✝⁷ : NormedAddCommGroup F\ninst✝⁶ : NormedSpace 𝕜 F\nF' : Type x\ninst✝⁵ : AddCommGroup F'\ninst✝⁴ : Module 𝕜 F'\ninst✝³ : TopologicalSpace F'\ninst✝² : TopologicalAddGroup F'\ninst✝¹ : ContinuousSMul 𝕜 F'\ninst✝ : CompleteSpace 𝕜\nn : ℕ\nt : Finset E\nx✝ : t ∈ fun i => Finset.card i = n\nthis : Continuous fun f x => ↑f ↑x\n⊢ IsOpen {x | LinearIndependent 𝕜 fun x_1 => ↑↑x ↑x_1}",
"state_before": "𝕜 : Type u\ninst✝¹⁰ : NontriviallyNormedField 𝕜\nE : Type v\ninst✝⁹ : NormedAddCommGroup E\ninst✝⁸ : NormedSpace 𝕜 E\nF : Type w\ninst✝⁷ : NormedAddCommGroup F\ninst✝⁶ : NormedSpace 𝕜 F\nF' : Type x\ninst✝⁵ : AddCommGroup F'\ninst✝⁴ : Module 𝕜 F'\ninst✝³ : TopologicalSpace F'\ninst✝² : TopologicalAddGroup F'\ninst✝¹ : ContinuousSMul 𝕜 F'\ninst✝ : CompleteSpace 𝕜\nn : ℕ\nt : Finset E\nx✝ : t ∈ fun i => Finset.card i = n\n⊢ IsOpen {x | LinearIndependent 𝕜 fun x_1 => ↑↑x ↑x_1}",
"tactic": "have : Continuous fun f : E →L[𝕜] F => fun x : (t : Set E) => f x :=\n continuous_pi fun x => (ContinuousLinearMap.apply 𝕜 F (x : E)).continuous"
},
{
"state_after": "no goals",
"state_before": "𝕜 : Type u\ninst✝¹⁰ : NontriviallyNormedField 𝕜\nE : Type v\ninst✝⁹ : NormedAddCommGroup E\ninst✝⁸ : NormedSpace 𝕜 E\nF : Type w\ninst✝⁷ : NormedAddCommGroup F\ninst✝⁶ : NormedSpace 𝕜 F\nF' : Type x\ninst✝⁵ : AddCommGroup F'\ninst✝⁴ : Module 𝕜 F'\ninst✝³ : TopologicalSpace F'\ninst✝² : TopologicalAddGroup F'\ninst✝¹ : ContinuousSMul 𝕜 F'\ninst✝ : CompleteSpace 𝕜\nn : ℕ\nt : Finset E\nx✝ : t ∈ fun i => Finset.card i = n\nthis : Continuous fun f x => ↑f ↑x\n⊢ IsOpen {x | LinearIndependent 𝕜 fun x_1 => ↑↑x ↑x_1}",
"tactic": "exact isOpen_setOf_linearIndependent.preimage this"
}
] |
[
271,
53
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
265,
1
] |
Mathlib/Data/IsROrC/Basic.lean
|
IsROrC.conj_I
|
[] |
[
350,
19
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
349,
1
] |
Mathlib/FieldTheory/Tower.lean
|
FiniteDimensional.Subalgebra.isSimpleOrder_of_finrank_prime
|
[
{
"state_after": "F : Type u\nK✝ : Type v\nA✝ : Type w\ninst✝⁹ : Field F\ninst✝⁸ : DivisionRing K✝\ninst✝⁷ : AddCommGroup A✝\ninst✝⁶ : Algebra F K✝\ninst✝⁵ : Module K✝ A✝\ninst✝⁴ : Module F A✝\ninst✝³ : IsScalarTower F K✝ A✝\nA : Type u_1\ninst✝² : Ring A\ninst✝¹ : IsDomain A\ninst✝ : Algebra F A\nhp : Nat.Prime (finrank F A)\nK : Subalgebra F A\nthis : FiniteDimensional F A\n⊢ K = ⊥ ∨ K = ⊤",
"state_before": "F : Type u\nK✝ : Type v\nA✝ : Type w\ninst✝⁹ : Field F\ninst✝⁸ : DivisionRing K✝\ninst✝⁷ : AddCommGroup A✝\ninst✝⁶ : Algebra F K✝\ninst✝⁵ : Module K✝ A✝\ninst✝⁴ : Module F A✝\ninst✝³ : IsScalarTower F K✝ A✝\nA : Type u_1\ninst✝² : Ring A\ninst✝¹ : IsDomain A\ninst✝ : Algebra F A\nhp : Nat.Prime (finrank F A)\nK : Subalgebra F A\n⊢ K = ⊥ ∨ K = ⊤",
"tactic": "haveI : FiniteDimensional _ _ := finiteDimensional_of_finrank hp.pos"
},
{
"state_after": "F : Type u\nK✝ : Type v\nA✝ : Type w\ninst✝⁹ : Field F\ninst✝⁸ : DivisionRing K✝\ninst✝⁷ : AddCommGroup A✝\ninst✝⁶ : Algebra F K✝\ninst✝⁵ : Module K✝ A✝\ninst✝⁴ : Module F A✝\ninst✝³ : IsScalarTower F K✝ A✝\nA : Type u_1\ninst✝² : Ring A\ninst✝¹ : IsDomain A\ninst✝ : Algebra F A\nhp : Nat.Prime (finrank F A)\nK : Subalgebra F A\nthis✝ : FiniteDimensional F A\nthis : DivisionRing { x // x ∈ K } := divisionRingOfFiniteDimensional F { x // x ∈ K }\n⊢ K = ⊥ ∨ K = ⊤",
"state_before": "F : Type u\nK✝ : Type v\nA✝ : Type w\ninst✝⁹ : Field F\ninst✝⁸ : DivisionRing K✝\ninst✝⁷ : AddCommGroup A✝\ninst✝⁶ : Algebra F K✝\ninst✝⁵ : Module K✝ A✝\ninst✝⁴ : Module F A✝\ninst✝³ : IsScalarTower F K✝ A✝\nA : Type u_1\ninst✝² : Ring A\ninst✝¹ : IsDomain A\ninst✝ : Algebra F A\nhp : Nat.Prime (finrank F A)\nK : Subalgebra F A\nthis : FiniteDimensional F A\n⊢ K = ⊥ ∨ K = ⊤",
"tactic": "letI := divisionRingOfFiniteDimensional F K"
},
{
"state_after": "case refine'_1\nF : Type u\nK✝ : Type v\nA✝ : Type w\ninst✝⁹ : Field F\ninst✝⁸ : DivisionRing K✝\ninst✝⁷ : AddCommGroup A✝\ninst✝⁶ : Algebra F K✝\ninst✝⁵ : Module K✝ A✝\ninst✝⁴ : Module F A✝\ninst✝³ : IsScalarTower F K✝ A✝\nA : Type u_1\ninst✝² : Ring A\ninst✝¹ : IsDomain A\ninst✝ : Algebra F A\nhp : Nat.Prime (finrank F A)\nK : Subalgebra F A\nthis✝ : FiniteDimensional F A\nthis : DivisionRing { x // x ∈ K } := divisionRingOfFiniteDimensional F { x // x ∈ K }\n⊢ finrank F { x // x ∈ K } = 1 → K = ⊥\n\ncase refine'_2\nF : Type u\nK✝ : Type v\nA✝ : Type w\ninst✝⁹ : Field F\ninst✝⁸ : DivisionRing K✝\ninst✝⁷ : AddCommGroup A✝\ninst✝⁶ : Algebra F K✝\ninst✝⁵ : Module K✝ A✝\ninst✝⁴ : Module F A✝\ninst✝³ : IsScalarTower F K✝ A✝\nA : Type u_1\ninst✝² : Ring A\ninst✝¹ : IsDomain A\ninst✝ : Algebra F A\nhp : Nat.Prime (finrank F A)\nK : Subalgebra F A\nthis✝ : FiniteDimensional F A\nthis : DivisionRing { x // x ∈ K } := divisionRingOfFiniteDimensional F { x // x ∈ K }\nh : finrank F { x // x ∈ K } = finrank F A\n⊢ K = ⊤",
"state_before": "F : Type u\nK✝ : Type v\nA✝ : Type w\ninst✝⁹ : Field F\ninst✝⁸ : DivisionRing K✝\ninst✝⁷ : AddCommGroup A✝\ninst✝⁶ : Algebra F K✝\ninst✝⁵ : Module K✝ A✝\ninst✝⁴ : Module F A✝\ninst✝³ : IsScalarTower F K✝ A✝\nA : Type u_1\ninst✝² : Ring A\ninst✝¹ : IsDomain A\ninst✝ : Algebra F A\nhp : Nat.Prime (finrank F A)\nK : Subalgebra F A\nthis✝ : FiniteDimensional F A\nthis : DivisionRing { x // x ∈ K } := divisionRingOfFiniteDimensional F { x // x ∈ K }\n⊢ K = ⊥ ∨ K = ⊤",
"tactic": "refine' (hp.eq_one_or_self_of_dvd _ ⟨_, (finrank_mul_finrank F K A).symm⟩).imp _ fun h => _"
},
{
"state_after": "no goals",
"state_before": "case refine'_1\nF : Type u\nK✝ : Type v\nA✝ : Type w\ninst✝⁹ : Field F\ninst✝⁸ : DivisionRing K✝\ninst✝⁷ : AddCommGroup A✝\ninst✝⁶ : Algebra F K✝\ninst✝⁵ : Module K✝ A✝\ninst✝⁴ : Module F A✝\ninst✝³ : IsScalarTower F K✝ A✝\nA : Type u_1\ninst✝² : Ring A\ninst✝¹ : IsDomain A\ninst✝ : Algebra F A\nhp : Nat.Prime (finrank F A)\nK : Subalgebra F A\nthis✝ : FiniteDimensional F A\nthis : DivisionRing { x // x ∈ K } := divisionRingOfFiniteDimensional F { x // x ∈ K }\n⊢ finrank F { x // x ∈ K } = 1 → K = ⊥",
"tactic": "exact Subalgebra.eq_bot_of_finrank_one"
},
{
"state_after": "no goals",
"state_before": "case refine'_2\nF : Type u\nK✝ : Type v\nA✝ : Type w\ninst✝⁹ : Field F\ninst✝⁸ : DivisionRing K✝\ninst✝⁷ : AddCommGroup A✝\ninst✝⁶ : Algebra F K✝\ninst✝⁵ : Module K✝ A✝\ninst✝⁴ : Module F A✝\ninst✝³ : IsScalarTower F K✝ A✝\nA : Type u_1\ninst✝² : Ring A\ninst✝¹ : IsDomain A\ninst✝ : Algebra F A\nhp : Nat.Prime (finrank F A)\nK : Subalgebra F A\nthis✝ : FiniteDimensional F A\nthis : DivisionRing { x // x ∈ K } := divisionRingOfFiniteDimensional F { x // x ∈ K }\nh : finrank F { x // x ∈ K } = finrank F A\n⊢ K = ⊤",
"tactic": "exact\n Algebra.toSubmodule_eq_top.1 (eq_top_of_finrank_eq <| K.finrank_toSubmodule.trans h)"
}
] |
[
147,
97
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
136,
1
] |
Mathlib/Analysis/InnerProductSpace/Projection.lean
|
Submodule.sup_orthogonal_of_completeSpace
|
[
{
"state_after": "case h.e'_2.h.e'_4\n𝕜 : Type u_2\nE : Type u_1\nF : Type ?u.764822\ninst✝⁵ : IsROrC 𝕜\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedAddCommGroup F\ninst✝² : InnerProductSpace 𝕜 E\ninst✝¹ : InnerProductSpace ℝ F\nK : Submodule 𝕜 E\ninst✝ : CompleteSpace { x // x ∈ K }\n⊢ Kᗮ = Kᗮ ⊓ ⊤",
"state_before": "𝕜 : Type u_2\nE : Type u_1\nF : Type ?u.764822\ninst✝⁵ : IsROrC 𝕜\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedAddCommGroup F\ninst✝² : InnerProductSpace 𝕜 E\ninst✝¹ : InnerProductSpace ℝ F\nK : Submodule 𝕜 E\ninst✝ : CompleteSpace { x // x ∈ K }\n⊢ K ⊔ Kᗮ = ⊤",
"tactic": "convert Submodule.sup_orthogonal_inf_of_completeSpace (le_top : K ≤ ⊤) using 2"
},
{
"state_after": "no goals",
"state_before": "case h.e'_2.h.e'_4\n𝕜 : Type u_2\nE : Type u_1\nF : Type ?u.764822\ninst✝⁵ : IsROrC 𝕜\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedAddCommGroup F\ninst✝² : InnerProductSpace 𝕜 E\ninst✝¹ : InnerProductSpace ℝ F\nK : Submodule 𝕜 E\ninst✝ : CompleteSpace { x // x ∈ K }\n⊢ Kᗮ = Kᗮ ⊓ ⊤",
"tactic": "simp"
}
] |
[
751,
7
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
749,
1
] |
Mathlib/CategoryTheory/Limits/Shapes/Pullbacks.lean
|
CategoryTheory.Limits.cospanCompIso_inv_app_one
|
[] |
[
327,
97
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
327,
1
] |
Mathlib/Geometry/Euclidean/Sphere/Basic.lean
|
EuclideanGeometry.cospherical_iff_exists_sphere
|
[
{
"state_after": "case refine'_1\nV : Type ?u.6464\nP : Type u_1\ninst✝ : MetricSpace P\nps : Set P\nh : Cospherical ps\n⊢ ∃ s, ps ⊆ Metric.sphere s.center s.radius\n\ncase refine'_2\nV : Type ?u.6464\nP : Type u_1\ninst✝ : MetricSpace P\nps : Set P\nh : ∃ s, ps ⊆ Metric.sphere s.center s.radius\n⊢ Cospherical ps",
"state_before": "V : Type ?u.6464\nP : Type u_1\ninst✝ : MetricSpace P\nps : Set P\n⊢ Cospherical ps ↔ ∃ s, ps ⊆ Metric.sphere s.center s.radius",
"tactic": "refine' ⟨fun h => _, fun h => _⟩"
},
{
"state_after": "case refine'_1.intro.intro\nV : Type ?u.6464\nP : Type u_1\ninst✝ : MetricSpace P\nps : Set P\nc : P\nr : ℝ\nh : ∀ (p : P), p ∈ ps → dist p c = r\n⊢ ∃ s, ps ⊆ Metric.sphere s.center s.radius",
"state_before": "case refine'_1\nV : Type ?u.6464\nP : Type u_1\ninst✝ : MetricSpace P\nps : Set P\nh : Cospherical ps\n⊢ ∃ s, ps ⊆ Metric.sphere s.center s.radius",
"tactic": "rcases h with ⟨c, r, h⟩"
},
{
"state_after": "no goals",
"state_before": "case refine'_1.intro.intro\nV : Type ?u.6464\nP : Type u_1\ninst✝ : MetricSpace P\nps : Set P\nc : P\nr : ℝ\nh : ∀ (p : P), p ∈ ps → dist p c = r\n⊢ ∃ s, ps ⊆ Metric.sphere s.center s.radius",
"tactic": "exact ⟨⟨c, r⟩, h⟩"
},
{
"state_after": "case refine'_2.intro\nV : Type ?u.6464\nP : Type u_1\ninst✝ : MetricSpace P\nps : Set P\ns : Sphere P\nh : ps ⊆ Metric.sphere s.center s.radius\n⊢ Cospherical ps",
"state_before": "case refine'_2\nV : Type ?u.6464\nP : Type u_1\ninst✝ : MetricSpace P\nps : Set P\nh : ∃ s, ps ⊆ Metric.sphere s.center s.radius\n⊢ Cospherical ps",
"tactic": "rcases h with ⟨s, h⟩"
},
{
"state_after": "no goals",
"state_before": "case refine'_2.intro\nV : Type ?u.6464\nP : Type u_1\ninst✝ : MetricSpace P\nps : Set P\ns : Sphere P\nh : ps ⊆ Metric.sphere s.center s.radius\n⊢ Cospherical ps",
"tactic": "exact ⟨s.center, s.radius, h⟩"
}
] |
[
167,
34
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
161,
1
] |
Mathlib/LinearAlgebra/Finsupp.lean
|
Finsupp.supported_iInter
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_2\nM : Type u_3\nN : Type ?u.111089\nP : Type ?u.111092\nR : Type u_4\nS : Type ?u.111098\ninst✝⁷ : Semiring R\ninst✝⁶ : Semiring S\ninst✝⁵ : AddCommMonoid M\ninst✝⁴ : Module R M\ninst✝³ : AddCommMonoid N\ninst✝² : Module R N\ninst✝¹ : AddCommMonoid P\ninst✝ : Module R P\nι : Type u_1\ns : ι → Set α\nx : α →₀ M\n⊢ x ∈ supported M R (⋂ (i : ι), s i) ↔ x ∈ ⨅ (i : ι), supported M R (s i)",
"tactic": "simp [mem_supported, subset_iInter_iff]"
}
] |
[
306,
68
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
304,
1
] |
Mathlib/GroupTheory/Perm/Cycle/Basic.lean
|
Equiv.Perm.isCycleOn_of_subsingleton
|
[] |
[
776,
82
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
775,
1
] |
Mathlib/Topology/Constructions.lean
|
isOpenMap_snd
|
[] |
[
730,
55
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
729,
1
] |
Mathlib/Order/InitialSeg.lean
|
wellFounded_iff_principalSeg
|
[] |
[
455,
88
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
452,
1
] |
Mathlib/Data/Set/Image.lean
|
Set.range_ite_subset'
|
[
{
"state_after": "case pos\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.93171\nι : Sort ?u.93174\nι' : Sort ?u.93177\nf✝ : ι → α\ns t : Set α\np : Prop\ninst✝ : Decidable p\nf g : α → β\nh : p\n⊢ range (if p then f else g) ⊆ range f ∪ range g\n\ncase neg\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.93171\nι : Sort ?u.93174\nι' : Sort ?u.93177\nf✝ : ι → α\ns t : Set α\np : Prop\ninst✝ : Decidable p\nf g : α → β\nh : ¬p\n⊢ range (if p then f else g) ⊆ range f ∪ range g",
"state_before": "α : Type u_2\nβ : Type u_1\nγ : Type ?u.93171\nι : Sort ?u.93174\nι' : Sort ?u.93177\nf✝ : ι → α\ns t : Set α\np : Prop\ninst✝ : Decidable p\nf g : α → β\n⊢ range (if p then f else g) ⊆ range f ∪ range g",
"tactic": "by_cases h : p"
},
{
"state_after": "case pos\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.93171\nι : Sort ?u.93174\nι' : Sort ?u.93177\nf✝ : ι → α\ns t : Set α\np : Prop\ninst✝ : Decidable p\nf g : α → β\nh : p\n⊢ range f ⊆ range f ∪ range g",
"state_before": "case pos\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.93171\nι : Sort ?u.93174\nι' : Sort ?u.93177\nf✝ : ι → α\ns t : Set α\np : Prop\ninst✝ : Decidable p\nf g : α → β\nh : p\n⊢ range (if p then f else g) ⊆ range f ∪ range g",
"tactic": "rw [if_pos h]"
},
{
"state_after": "no goals",
"state_before": "case pos\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.93171\nι : Sort ?u.93174\nι' : Sort ?u.93177\nf✝ : ι → α\ns t : Set α\np : Prop\ninst✝ : Decidable p\nf g : α → β\nh : p\n⊢ range f ⊆ range f ∪ range g",
"tactic": "exact subset_union_left _ _"
},
{
"state_after": "case neg\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.93171\nι : Sort ?u.93174\nι' : Sort ?u.93177\nf✝ : ι → α\ns t : Set α\np : Prop\ninst✝ : Decidable p\nf g : α → β\nh : ¬p\n⊢ range g ⊆ range f ∪ range g",
"state_before": "case neg\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.93171\nι : Sort ?u.93174\nι' : Sort ?u.93177\nf✝ : ι → α\ns t : Set α\np : Prop\ninst✝ : Decidable p\nf g : α → β\nh : ¬p\n⊢ range (if p then f else g) ⊆ range f ∪ range g",
"tactic": "rw [if_neg h]"
},
{
"state_after": "no goals",
"state_before": "case neg\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.93171\nι : Sort ?u.93174\nι' : Sort ?u.93177\nf✝ : ι → α\ns t : Set α\np : Prop\ninst✝ : Decidable p\nf g : α → β\nh : ¬p\n⊢ range g ⊆ range f ∪ range g",
"tactic": "exact subset_union_right _ _"
}
] |
[
1090,
33
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1084,
1
] |
Mathlib/Data/Real/Sqrt.lean
|
Real.sqrt_mul_self_eq_abs
|
[
{
"state_after": "no goals",
"state_before": "x✝ y x : ℝ\n⊢ sqrt (x * x) = abs x",
"tactic": "rw [← abs_mul_abs_self x, sqrt_mul_self (abs_nonneg _)]"
}
] |
[
247,
58
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
246,
1
] |
Mathlib/Algebra/Algebra/Hom.lean
|
AlgHom.coe_fn_inj
|
[] |
[
199,
20
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
198,
1
] |
Mathlib/Analysis/SpecialFunctions/Trigonometric/Basic.lean
|
Real.cos_mem_Icc
|
[] |
[
632,
35
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
631,
1
] |
Mathlib/CategoryTheory/Idempotents/HomologicalComplex.lean
|
CategoryTheory.Idempotents.Karoubi.HomologicalComplex.p_comp_d
|
[] |
[
43,
44
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
42,
1
] |
Mathlib/Data/Polynomial/FieldDivision.lean
|
Polynomial.rootSet_monomial
|
[
{
"state_after": "no goals",
"state_before": "R : Type u\nS : Type v\nk : Type y\nA : Type z\na✝ b : R\nn✝ : ℕ\ninst✝³ : Field R\np q : R[X]\ninst✝² : CommRing S\ninst✝¹ : IsDomain S\ninst✝ : Algebra R S\nn : ℕ\nhn : n ≠ 0\na : R\nha : a ≠ 0\n⊢ rootSet (↑(monomial n) a) S = {0}",
"tactic": "rw [rootSet, map_monomial, roots_monomial ((_root_.map_ne_zero (algebraMap R S)).2 ha),\n Multiset.toFinset_nsmul _ _ hn, Multiset.toFinset_singleton, Finset.coe_singleton]"
}
] |
[
371,
87
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
368,
1
] |
Mathlib/Topology/ContinuousOn.lean
|
ContinuousWithinAt.tendsto
|
[] |
[
528,
4
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
526,
1
] |
Mathlib/GroupTheory/Subsemigroup/Center.lean
|
Set.div_mem_center₀
|
[
{
"state_after": "M : Type u_1\ninst✝ : GroupWithZero M\na b : M\nha : a ∈ center M\nhb : b ∈ center M\n⊢ a * b⁻¹ ∈ center M",
"state_before": "M : Type u_1\ninst✝ : GroupWithZero M\na b : M\nha : a ∈ center M\nhb : b ∈ center M\n⊢ a / b ∈ center M",
"tactic": "rw [div_eq_mul_inv]"
},
{
"state_after": "no goals",
"state_before": "M : Type u_1\ninst✝ : GroupWithZero M\na b : M\nha : a ∈ center M\nhb : b ∈ center M\n⊢ a * b⁻¹ ∈ center M",
"tactic": "exact mul_mem_center ha (inv_mem_center₀ hb)"
}
] |
[
127,
47
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
124,
1
] |
Mathlib/Data/Finset/Pointwise.lean
|
Finset.image_div
|
[] |
[
1180,
36
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1179,
1
] |
Mathlib/Data/Nat/Parity.lean
|
Nat.even_xor_odd'
|
[
{
"state_after": "case inl.intro\nm n k : ℕ\n⊢ Xor' (k + k = 2 * k) (k + k = 2 * k + 1)\n\ncase inr.intro\nm n k : ℕ\n⊢ Xor' (2 * k + 1 = 2 * k) (2 * k + 1 = 2 * k + 1)",
"state_before": "m n✝ n : ℕ\n⊢ ∃ k, Xor' (n = 2 * k) (n = 2 * k + 1)",
"tactic": "rcases even_or_odd n with (⟨k, rfl⟩ | ⟨k, rfl⟩) <;> use k"
},
{
"state_after": "no goals",
"state_before": "case inl.intro\nm n k : ℕ\n⊢ Xor' (k + k = 2 * k) (k + k = 2 * k + 1)",
"tactic": "simpa only [← two_mul, eq_self_iff_true, xor_true] using (succ_ne_self (2 * k)).symm"
},
{
"state_after": "no goals",
"state_before": "case inr.intro\nm n k : ℕ\n⊢ Xor' (2 * k + 1 = 2 * k) (2 * k + 1 = 2 * k + 1)",
"tactic": "simpa only [xor_true, xor_comm] using (succ_ne_self _)"
}
] |
[
89,
59
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
86,
1
] |
Mathlib/Data/Set/Function.lean
|
Set.MapsTo.restrict_inj
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.34549\nι : Sort ?u.34552\nπ : α → Type ?u.34557\ns s₁ s₂ : Set α\nt t₁ t₂ : Set β\np : Set γ\nf f₁ f₂ f₃ : α → β\ng g₁ g₂ : β → γ\nf' f₁' f₂' : β → α\ng' : γ → β\na : α\nb : β\nh : MapsTo f s t\n⊢ Injective (restrict f s t h) ↔ InjOn f s",
"tactic": "rw [h.restrict_eq_codRestrict, injective_codRestrict, injOn_iff_injective]"
}
] |
[
691,
77
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
690,
1
] |
Mathlib/Data/Polynomial/Degree/Definitions.lean
|
Polynomial.natDegree_X_pow_add_C
|
[
{
"state_after": "case pos\nR : Type u\nS : Type v\na b c d : R\nn✝ m : ℕ\ninst✝¹ : Nontrivial R\ninst✝ : Semiring R\nn : ℕ\nr : R\nhn : n = 0\n⊢ natDegree (X ^ n + ↑C r) = n\n\ncase neg\nR : Type u\nS : Type v\na b c d : R\nn✝ m : ℕ\ninst✝¹ : Nontrivial R\ninst✝ : Semiring R\nn : ℕ\nr : R\nhn : ¬n = 0\n⊢ natDegree (X ^ n + ↑C r) = n",
"state_before": "R : Type u\nS : Type v\na b c d : R\nn✝ m : ℕ\ninst✝¹ : Nontrivial R\ninst✝ : Semiring R\nn : ℕ\nr : R\n⊢ natDegree (X ^ n + ↑C r) = n",
"tactic": "by_cases hn : n = 0"
},
{
"state_after": "no goals",
"state_before": "case pos\nR : Type u\nS : Type v\na b c d : R\nn✝ m : ℕ\ninst✝¹ : Nontrivial R\ninst✝ : Semiring R\nn : ℕ\nr : R\nhn : n = 0\n⊢ natDegree (X ^ n + ↑C r) = n",
"tactic": "rw [hn, pow_zero, ← C_1, ← RingHom.map_add, natDegree_C]"
},
{
"state_after": "no goals",
"state_before": "case neg\nR : Type u\nS : Type v\na b c d : R\nn✝ m : ℕ\ninst✝¹ : Nontrivial R\ninst✝ : Semiring R\nn : ℕ\nr : R\nhn : ¬n = 0\n⊢ natDegree (X ^ n + ↑C r) = n",
"tactic": "exact natDegree_eq_of_degree_eq_some (degree_X_pow_add_C (pos_iff_ne_zero.mpr hn) r)"
}
] |
[
1404,
89
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1401,
1
] |
Mathlib/LinearAlgebra/Orientation.lean
|
Orientation.map_apply
|
[] |
[
78,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
75,
1
] |
Mathlib/Algebra/Star/Subalgebra.lean
|
StarSubalgebra.adjoin_le
|
[] |
[
478,
28
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
477,
1
] |
src/lean/Init/Data/Nat/Basic.lean
|
Nat.mul_pred_left
|
[
{
"state_after": "no goals",
"state_before": "n m : Nat\n⊢ pred n * m = n * m - m",
"tactic": "cases n with\n| zero => simp\n| succ n => rw [Nat.pred_succ, succ_mul, Nat.add_sub_cancel]"
},
{
"state_after": "no goals",
"state_before": "case zero\nm : Nat\n⊢ pred zero * m = zero * m - m",
"tactic": "simp"
},
{
"state_after": "no goals",
"state_before": "case succ\nm n : Nat\n⊢ pred (succ n) * m = succ n * m - m",
"tactic": "rw [Nat.pred_succ, succ_mul, Nat.add_sub_cancel]"
}
] |
[
701,
63
] |
d5348dfac847a56a4595fb6230fd0708dcb4e7e9
|
https://github.com/leanprover/lean4
|
[
698,
1
] |
Mathlib/Algebra/Algebra/Hom.lean
|
AlgHom.map_list_prod
|
[] |
[
427,
30
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
426,
1
] |
Mathlib/NumberTheory/ArithmeticFunction.lean
|
Nat.ArithmeticFunction.pmul_comm
|
[
{
"state_after": "case h\nR : Type u_1\ninst✝ : CommMonoidWithZero R\nf g : ArithmeticFunction R\nx✝ : ℕ\n⊢ ↑(pmul f g) x✝ = ↑(pmul g f) x✝",
"state_before": "R : Type u_1\ninst✝ : CommMonoidWithZero R\nf g : ArithmeticFunction R\n⊢ pmul f g = pmul g f",
"tactic": "ext"
},
{
"state_after": "no goals",
"state_before": "case h\nR : Type u_1\ninst✝ : CommMonoidWithZero R\nf g : ArithmeticFunction R\nx✝ : ℕ\n⊢ ↑(pmul f g) x✝ = ↑(pmul g f) x✝",
"tactic": "simp [mul_comm]"
}
] |
[
518,
18
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
516,
1
] |
Mathlib/RingTheory/Subring/Basic.lean
|
Subring.mem_iInf
|
[
{
"state_after": "no goals",
"state_before": "R : Type u\nS✝ : Type v\nT : Type w\ninst✝² : Ring R\ninst✝¹ : Ring S✝\ninst✝ : Ring T\nι : Sort u_1\nS : ι → Subring R\nx : R\n⊢ (x ∈ ⨅ (i : ι), S i) ↔ ∀ (i : ι), x ∈ S i",
"tactic": "simp only [iInf, mem_sInf, Set.forall_range_iff]"
}
] |
[
741,
51
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
740,
1
] |
Mathlib/Data/List/Basic.lean
|
List.zipRight'_nil_right
|
[] |
[
4078,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
4077,
1
] |
Mathlib/Data/Polynomial/Eval.lean
|
Polynomial.add_comp
|
[] |
[
581,
16
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
580,
1
] |
Mathlib/RingTheory/PowerSeries/Basic.lean
|
PowerSeries.coeff_C
|
[
{
"state_after": "no goals",
"state_before": "R : Type u_1\ninst✝ : Semiring R\nn : ℕ\na : R\n⊢ ↑(coeff R n) (↑(C R) a) = if n = 0 then a else 0",
"tactic": "rw [← monomial_zero_eq_C_apply, coeff_monomial]"
}
] |
[
1414,
50
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1413,
1
] |
Mathlib/CategoryTheory/Limits/HasLimits.lean
|
CategoryTheory.Limits.hasLimitOfEquivalenceComp
|
[
{
"state_after": "J : Type u₁\ninst✝³ : Category J\nK : Type u₂\ninst✝² : Category K\nC : Type u\ninst✝¹ : Category C\nF : J ⥤ C\ne : K ≌ J\ninst✝ : HasLimit (e.functor ⋙ F)\nthis : HasLimit (e.inverse ⋙ e.functor ⋙ F)\n⊢ HasLimit F",
"state_before": "J : Type u₁\ninst✝³ : Category J\nK : Type u₂\ninst✝² : Category K\nC : Type u\ninst✝¹ : Category C\nF : J ⥤ C\ne : K ≌ J\ninst✝ : HasLimit (e.functor ⋙ F)\n⊢ HasLimit F",
"tactic": "haveI : HasLimit (e.inverse ⋙ e.functor ⋙ F) := Limits.hasLimitEquivalenceComp e.symm"
},
{
"state_after": "no goals",
"state_before": "J : Type u₁\ninst✝³ : Category J\nK : Type u₂\ninst✝² : Category K\nC : Type u\ninst✝¹ : Category C\nF : J ⥤ C\ne : K ≌ J\ninst✝ : HasLimit (e.functor ⋙ F)\nthis : HasLimit (e.inverse ⋙ e.functor ⋙ F)\n⊢ HasLimit F",
"tactic": "apply hasLimitOfIso (e.invFunIdAssoc F)"
}
] |
[
515,
42
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
513,
1
] |
Mathlib/Algebra/Algebra/Unitization.lean
|
Unitization.inr_neg
|
[] |
[
296,
24
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
295,
1
] |
Mathlib/Data/Set/UnionLift.lean
|
Set.liftCover_of_mem
|
[] |
[
176,
53
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
174,
1
] |
Mathlib/Data/Set/Lattice.lean
|
Set.iUnion_eq_dif
|
[] |
[
251,
16
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
249,
1
] |
Mathlib/Data/List/Cycle.lean
|
List.nextOr_singleton
|
[] |
[
50,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
49,
1
] |
Mathlib/Data/Set/Intervals/Pi.lean
|
Set.image_mulSingle_Ico
|
[] |
[
207,
27
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
205,
1
] |
Std/Data/Int/Lemmas.lean
|
Int.sub_lt_sub_of_le_of_lt
|
[] |
[
1130,
54
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
1128,
11
] |
Mathlib/Algebra/Invertible.lean
|
mul_invOf_self
|
[] |
[
115,
28
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
114,
1
] |
Mathlib/GroupTheory/Complement.lean
|
Subgroup.isComplement'_of_coprime
|
[] |
[
550,
90
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
547,
1
] |
Mathlib/Analysis/Convex/Function.lean
|
ConvexOn.comp_linearMap
|
[
{
"state_after": "no goals",
"state_before": "𝕜 : Type u_2\nE : Type u_4\nF : Type u_1\nα : Type ?u.262486\nβ : Type u_3\nι : Type ?u.262492\ninst✝⁷ : OrderedSemiring 𝕜\ninst✝⁶ : AddCommMonoid E\ninst✝⁵ : AddCommMonoid F\ninst✝⁴ : OrderedAddCommMonoid α\ninst✝³ : OrderedAddCommMonoid β\ninst✝² : Module 𝕜 E\ninst✝¹ : Module 𝕜 F\ninst✝ : SMul 𝕜 β\nf : F → β\ns : Set F\nhf : ConvexOn 𝕜 s f\ng : E →ₗ[𝕜] F\nx : E\nhx : x ∈ ↑g ⁻¹' s\ny : E\nhy : y ∈ ↑g ⁻¹' s\na b : 𝕜\nha : 0 ≤ a\nhb : 0 ≤ b\nhab : a + b = 1\n⊢ f (↑g (a • x + b • y)) = f (a • ↑g x + b • ↑g y)",
"tactic": "rw [g.map_add, g.map_smul, g.map_smul]"
}
] |
[
481,
61
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
476,
1
] |
Mathlib/Topology/Sets/Closeds.lean
|
TopologicalSpace.Clopens.clopen
|
[] |
[
291,
12
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
290,
1
] |
Mathlib/Analysis/BoxIntegral/Box/Basic.lean
|
BoxIntegral.Box.Ioo_subset_Icc
|
[] |
[
455,
40
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
454,
11
] |
Mathlib/RingTheory/Subring/Basic.lean
|
Subring.comap_iInf
|
[] |
[
1077,
26
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1075,
1
] |
Mathlib/Topology/Instances/ENNReal.lean
|
ENNReal.continuousAt_coe_iff
|
[] |
[
99,
23
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
97,
1
] |
Mathlib/Order/Bounds/Basic.lean
|
bddAbove_empty
|
[
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\nι : Sort x\ninst✝² : Preorder α\ninst✝¹ : Preorder β\ns t : Set α\na b : α\ninst✝ : Nonempty α\n⊢ BddAbove ∅",
"tactic": "simp only [BddAbove, upperBounds_empty, univ_nonempty]"
}
] |
[
875,
57
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
874,
1
] |
Mathlib/Combinatorics/Young/YoungDiagram.lean
|
YoungDiagram.ofRowLens_to_rowLens_eq_self
|
[
{
"state_after": "case cells.a.mk\nμ : YoungDiagram\ni j : ℕ\n⊢ (i, j) ∈ (ofRowLens (rowLens μ) (_ : List.Sorted (fun x x_1 => x ≥ x_1) (rowLens μ))).cells ↔ (i, j) ∈ μ.cells",
"state_before": "μ : YoungDiagram\n⊢ ofRowLens (rowLens μ) (_ : List.Sorted (fun x x_1 => x ≥ x_1) (rowLens μ)) = μ",
"tactic": "ext ⟨i, j⟩"
},
{
"state_after": "case cells.a.mk\nμ : YoungDiagram\ni j : ℕ\n⊢ (∃ h, j < rowLen μ i) ↔ (i, j) ∈ μ",
"state_before": "case cells.a.mk\nμ : YoungDiagram\ni j : ℕ\n⊢ (i, j) ∈ (ofRowLens (rowLens μ) (_ : List.Sorted (fun x x_1 => x ≥ x_1) (rowLens μ))).cells ↔ (i, j) ∈ μ.cells",
"tactic": "simp only [mem_cells, mem_ofRowLens, length_rowLens, get_rowLens]"
},
{
"state_after": "no goals",
"state_before": "case cells.a.mk\nμ : YoungDiagram\ni j : ℕ\n⊢ (∃ h, j < rowLen μ i) ↔ (i, j) ∈ μ",
"tactic": "simpa [← mem_iff_lt_colLen, mem_iff_lt_rowLen] using j.zero_le.trans_lt"
}
] |
[
519,
74
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
516,
1
] |
Mathlib/MeasureTheory/Measure/MeasureSpace.lean
|
MeasureTheory.ae_restrict_iff'
|
[
{
"state_after": "α : Type u_1\nβ : Type ?u.572648\nγ : Type ?u.572651\nδ : Type ?u.572654\nι : Type ?u.572657\nR : Type ?u.572660\nR' : Type ?u.572663\nm0 : MeasurableSpace α\ninst✝¹ : MeasurableSpace β\ninst✝ : MeasurableSpace γ\nμ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α\ns s' t : Set α\np : α → Prop\nhs : MeasurableSet s\n⊢ ↑↑μ ({x | p x}ᶜ ∩ s) = 0 ↔ ↑↑μ ({x | x ∈ s → p x}ᶜ) = 0",
"state_before": "α : Type u_1\nβ : Type ?u.572648\nγ : Type ?u.572651\nδ : Type ?u.572654\nι : Type ?u.572657\nR : Type ?u.572660\nR' : Type ?u.572663\nm0 : MeasurableSpace α\ninst✝¹ : MeasurableSpace β\ninst✝ : MeasurableSpace γ\nμ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α\ns s' t : Set α\np : α → Prop\nhs : MeasurableSet s\n⊢ (∀ᵐ (x : α) ∂Measure.restrict μ s, p x) ↔ ∀ᵐ (x : α) ∂μ, x ∈ s → p x",
"tactic": "simp only [ae_iff, ← compl_setOf, restrict_apply_eq_zero' hs]"
},
{
"state_after": "α : Type u_1\nβ : Type ?u.572648\nγ : Type ?u.572651\nδ : Type ?u.572654\nι : Type ?u.572657\nR : Type ?u.572660\nR' : Type ?u.572663\nm0 : MeasurableSpace α\ninst✝¹ : MeasurableSpace β\ninst✝ : MeasurableSpace γ\nμ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α\ns s' t : Set α\np : α → Prop\nhs : MeasurableSet s\n⊢ (↑↑μ ({x | p x}ᶜ ∩ s) = 0) = (↑↑μ ({x | x ∈ s → p x}ᶜ) = 0)",
"state_before": "α : Type u_1\nβ : Type ?u.572648\nγ : Type ?u.572651\nδ : Type ?u.572654\nι : Type ?u.572657\nR : Type ?u.572660\nR' : Type ?u.572663\nm0 : MeasurableSpace α\ninst✝¹ : MeasurableSpace β\ninst✝ : MeasurableSpace γ\nμ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α\ns s' t : Set α\np : α → Prop\nhs : MeasurableSet s\n⊢ ↑↑μ ({x | p x}ᶜ ∩ s) = 0 ↔ ↑↑μ ({x | x ∈ s → p x}ᶜ) = 0",
"tactic": "rw [iff_iff_eq]"
},
{
"state_after": "case e_a.e_a.h\nα : Type u_1\nβ : Type ?u.572648\nγ : Type ?u.572651\nδ : Type ?u.572654\nι : Type ?u.572657\nR : Type ?u.572660\nR' : Type ?u.572663\nm0 : MeasurableSpace α\ninst✝¹ : MeasurableSpace β\ninst✝ : MeasurableSpace γ\nμ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α\ns s' t : Set α\np : α → Prop\nhs : MeasurableSet s\nx : α\n⊢ x ∈ {x | p x}ᶜ ∩ s ↔ x ∈ {x | x ∈ s → p x}ᶜ",
"state_before": "α : Type u_1\nβ : Type ?u.572648\nγ : Type ?u.572651\nδ : Type ?u.572654\nι : Type ?u.572657\nR : Type ?u.572660\nR' : Type ?u.572663\nm0 : MeasurableSpace α\ninst✝¹ : MeasurableSpace β\ninst✝ : MeasurableSpace γ\nμ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α\ns s' t : Set α\np : α → Prop\nhs : MeasurableSet s\n⊢ (↑↑μ ({x | p x}ᶜ ∩ s) = 0) = (↑↑μ ({x | x ∈ s → p x}ᶜ) = 0)",
"tactic": "congr with x"
},
{
"state_after": "no goals",
"state_before": "case e_a.e_a.h\nα : Type u_1\nβ : Type ?u.572648\nγ : Type ?u.572651\nδ : Type ?u.572654\nι : Type ?u.572657\nR : Type ?u.572660\nR' : Type ?u.572663\nm0 : MeasurableSpace α\ninst✝¹ : MeasurableSpace β\ninst✝ : MeasurableSpace γ\nμ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α\ns s' t : Set α\np : α → Prop\nhs : MeasurableSet s\nx : α\n⊢ x ∈ {x | p x}ᶜ ∩ s ↔ x ∈ {x | x ∈ s → p x}ᶜ",
"tactic": "simp [and_comm]"
}
] |
[
2812,
49
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2809,
1
] |
Mathlib/MeasureTheory/Decomposition/Lebesgue.lean
|
MeasureTheory.Measure.singularPart_withDensity
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.55804\nm : MeasurableSpace α\nμ ν✝ ν : Measure α\nf : α → ℝ≥0∞\nhf : Measurable f\n⊢ withDensity ν f = 0 + withDensity ν f",
"tactic": "rw [zero_add]"
}
] |
[
305,
60
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
302,
1
] |
Mathlib/Data/Real/ENNReal.lean
|
ENNReal.toReal_iSup
|
[
{
"state_after": "no goals",
"state_before": "α : Type ?u.843793\nβ : Type ?u.843796\na b c d : ℝ≥0∞\nr p q : ℝ≥0\nι : Sort u_1\nf g : ι → ℝ≥0∞\nhf : ∀ (i : ι), f i ≠ ⊤\n⊢ ENNReal.toReal (iSup f) = ⨆ (i : ι), ENNReal.toReal (f i)",
"tactic": "simp only [ENNReal.toReal, toNNReal_iSup hf, NNReal.coe_iSup]"
}
] |
[
2390,
64
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2389,
1
] |
Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean
|
MeasureTheory.StronglyMeasurable.finStronglyMeasurable_of_set_sigmaFinite
|
[
{
"state_after": "α : Type u_2\nβ : Type u_1\nγ : Type ?u.51589\nι : Type ?u.51592\ninst✝² : Countable ι\nf g : α → β\ninst✝¹ : TopologicalSpace β\ninst✝ : Zero β\nm : MeasurableSpace α\nμ : Measure α\nhf_meas : StronglyMeasurable f\nt : Set α\nht : MeasurableSet t\nhft_zero : ∀ (x : α), x ∈ tᶜ → f x = 0\nhtμ this : SigmaFinite (Measure.restrict μ t)\n⊢ FinStronglyMeasurable f μ",
"state_before": "α : Type u_2\nβ : Type u_1\nγ : Type ?u.51589\nι : Type ?u.51592\ninst✝² : Countable ι\nf g : α → β\ninst✝¹ : TopologicalSpace β\ninst✝ : Zero β\nm : MeasurableSpace α\nμ : Measure α\nhf_meas : StronglyMeasurable f\nt : Set α\nht : MeasurableSet t\nhft_zero : ∀ (x : α), x ∈ tᶜ → f x = 0\nhtμ : SigmaFinite (Measure.restrict μ t)\n⊢ FinStronglyMeasurable f μ",
"tactic": "haveI : SigmaFinite (μ.restrict t) := htμ"
},
{
"state_after": "α : Type u_2\nβ : Type u_1\nγ : Type ?u.51589\nι : Type ?u.51592\ninst✝² : Countable ι\nf g : α → β\ninst✝¹ : TopologicalSpace β\ninst✝ : Zero β\nm : MeasurableSpace α\nμ : Measure α\nhf_meas : StronglyMeasurable f\nt : Set α\nht : MeasurableSet t\nhft_zero : ∀ (x : α), x ∈ tᶜ → f x = 0\nhtμ this : SigmaFinite (Measure.restrict μ t)\nS : ℕ → Set α := spanningSets (Measure.restrict μ t)\n⊢ FinStronglyMeasurable f μ",
"state_before": "α : Type u_2\nβ : Type u_1\nγ : Type ?u.51589\nι : Type ?u.51592\ninst✝² : Countable ι\nf g : α → β\ninst✝¹ : TopologicalSpace β\ninst✝ : Zero β\nm : MeasurableSpace α\nμ : Measure α\nhf_meas : StronglyMeasurable f\nt : Set α\nht : MeasurableSet t\nhft_zero : ∀ (x : α), x ∈ tᶜ → f x = 0\nhtμ this : SigmaFinite (Measure.restrict μ t)\n⊢ FinStronglyMeasurable f μ",
"tactic": "let S := spanningSets (μ.restrict t)"
},
{
"state_after": "α : Type u_2\nβ : Type u_1\nγ : Type ?u.51589\nι : Type ?u.51592\ninst✝² : Countable ι\nf g : α → β\ninst✝¹ : TopologicalSpace β\ninst✝ : Zero β\nm : MeasurableSpace α\nμ : Measure α\nhf_meas : StronglyMeasurable f\nt : Set α\nht : MeasurableSet t\nhft_zero : ∀ (x : α), x ∈ tᶜ → f x = 0\nhtμ this : SigmaFinite (Measure.restrict μ t)\nS : ℕ → Set α := spanningSets (Measure.restrict μ t)\nhS_meas : ∀ (n : ℕ), MeasurableSet (S n)\n⊢ FinStronglyMeasurable f μ",
"state_before": "α : Type u_2\nβ : Type u_1\nγ : Type ?u.51589\nι : Type ?u.51592\ninst✝² : Countable ι\nf g : α → β\ninst✝¹ : TopologicalSpace β\ninst✝ : Zero β\nm : MeasurableSpace α\nμ : Measure α\nhf_meas : StronglyMeasurable f\nt : Set α\nht : MeasurableSet t\nhft_zero : ∀ (x : α), x ∈ tᶜ → f x = 0\nhtμ this : SigmaFinite (Measure.restrict μ t)\nS : ℕ → Set α := spanningSets (Measure.restrict μ t)\n⊢ FinStronglyMeasurable f μ",
"tactic": "have hS_meas : ∀ n, MeasurableSet (S n) := measurable_spanningSets (μ.restrict t)"
},
{
"state_after": "α : Type u_2\nβ : Type u_1\nγ : Type ?u.51589\nι : Type ?u.51592\ninst✝² : Countable ι\nf g : α → β\ninst✝¹ : TopologicalSpace β\ninst✝ : Zero β\nm : MeasurableSpace α\nμ : Measure α\nhf_meas : StronglyMeasurable f\nt : Set α\nht : MeasurableSet t\nhft_zero : ∀ (x : α), x ∈ tᶜ → f x = 0\nhtμ this : SigmaFinite (Measure.restrict μ t)\nS : ℕ → Set α := spanningSets (Measure.restrict μ t)\nhS_meas : ∀ (n : ℕ), MeasurableSet (S n)\nf_approx : ℕ → α →ₛ β := StronglyMeasurable.approx hf_meas\n⊢ FinStronglyMeasurable f μ",
"state_before": "α : Type u_2\nβ : Type u_1\nγ : Type ?u.51589\nι : Type ?u.51592\ninst✝² : Countable ι\nf g : α → β\ninst✝¹ : TopologicalSpace β\ninst✝ : Zero β\nm : MeasurableSpace α\nμ : Measure α\nhf_meas : StronglyMeasurable f\nt : Set α\nht : MeasurableSet t\nhft_zero : ∀ (x : α), x ∈ tᶜ → f x = 0\nhtμ this : SigmaFinite (Measure.restrict μ t)\nS : ℕ → Set α := spanningSets (Measure.restrict μ t)\nhS_meas : ∀ (n : ℕ), MeasurableSet (S n)\n⊢ FinStronglyMeasurable f μ",
"tactic": "let f_approx := hf_meas.approx"
},
{
"state_after": "α : Type u_2\nβ : Type u_1\nγ : Type ?u.51589\nι : Type ?u.51592\ninst✝² : Countable ι\nf g : α → β\ninst✝¹ : TopologicalSpace β\ninst✝ : Zero β\nm : MeasurableSpace α\nμ : Measure α\nhf_meas : StronglyMeasurable f\nt : Set α\nht : MeasurableSet t\nhft_zero : ∀ (x : α), x ∈ tᶜ → f x = 0\nhtμ this : SigmaFinite (Measure.restrict μ t)\nS : ℕ → Set α := spanningSets (Measure.restrict μ t)\nhS_meas : ∀ (n : ℕ), MeasurableSet (S n)\nf_approx : ℕ → α →ₛ β := StronglyMeasurable.approx hf_meas\nfs : ℕ → α →ₛ β := fun n => SimpleFunc.restrict (f_approx n) (S n ∩ t)\n⊢ FinStronglyMeasurable f μ",
"state_before": "α : Type u_2\nβ : Type u_1\nγ : Type ?u.51589\nι : Type ?u.51592\ninst✝² : Countable ι\nf g : α → β\ninst✝¹ : TopologicalSpace β\ninst✝ : Zero β\nm : MeasurableSpace α\nμ : Measure α\nhf_meas : StronglyMeasurable f\nt : Set α\nht : MeasurableSet t\nhft_zero : ∀ (x : α), x ∈ tᶜ → f x = 0\nhtμ this : SigmaFinite (Measure.restrict μ t)\nS : ℕ → Set α := spanningSets (Measure.restrict μ t)\nhS_meas : ∀ (n : ℕ), MeasurableSet (S n)\nf_approx : ℕ → α →ₛ β := StronglyMeasurable.approx hf_meas\n⊢ FinStronglyMeasurable f μ",
"tactic": "let fs n := SimpleFunc.restrict (f_approx n) (S n ∩ t)"
},
{
"state_after": "α : Type u_2\nβ : Type u_1\nγ : Type ?u.51589\nι : Type ?u.51592\ninst✝² : Countable ι\nf g : α → β\ninst✝¹ : TopologicalSpace β\ninst✝ : Zero β\nm : MeasurableSpace α\nμ : Measure α\nhf_meas : StronglyMeasurable f\nt : Set α\nht : MeasurableSet t\nhft_zero : ∀ (x : α), x ∈ tᶜ → f x = 0\nhtμ this : SigmaFinite (Measure.restrict μ t)\nS : ℕ → Set α := spanningSets (Measure.restrict μ t)\nhS_meas : ∀ (n : ℕ), MeasurableSet (S n)\nf_approx : ℕ → α →ₛ β := StronglyMeasurable.approx hf_meas\nfs : ℕ → α →ₛ β := fun n => SimpleFunc.restrict (f_approx n) (S n ∩ t)\nh_fs_t_compl : ∀ (n : ℕ) (x : α), ¬x ∈ t → ↑(fs n) x = 0\n⊢ FinStronglyMeasurable f μ",
"state_before": "α : Type u_2\nβ : Type u_1\nγ : Type ?u.51589\nι : Type ?u.51592\ninst✝² : Countable ι\nf g : α → β\ninst✝¹ : TopologicalSpace β\ninst✝ : Zero β\nm : MeasurableSpace α\nμ : Measure α\nhf_meas : StronglyMeasurable f\nt : Set α\nht : MeasurableSet t\nhft_zero : ∀ (x : α), x ∈ tᶜ → f x = 0\nhtμ this : SigmaFinite (Measure.restrict μ t)\nS : ℕ → Set α := spanningSets (Measure.restrict μ t)\nhS_meas : ∀ (n : ℕ), MeasurableSet (S n)\nf_approx : ℕ → α →ₛ β := StronglyMeasurable.approx hf_meas\nfs : ℕ → α →ₛ β := fun n => SimpleFunc.restrict (f_approx n) (S n ∩ t)\n⊢ FinStronglyMeasurable f μ",
"tactic": "have h_fs_t_compl : ∀ n, ∀ x, x ∉ t → fs n x = 0 := by\n intro n x hxt\n rw [SimpleFunc.restrict_apply _ ((hS_meas n).inter ht)]\n refine' Set.indicator_of_not_mem _ _\n simp [hxt]"
},
{
"state_after": "case refine'_1\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.51589\nι : Type ?u.51592\ninst✝² : Countable ι\nf g : α → β\ninst✝¹ : TopologicalSpace β\ninst✝ : Zero β\nm : MeasurableSpace α\nμ : Measure α\nhf_meas : StronglyMeasurable f\nt : Set α\nht : MeasurableSet t\nhft_zero : ∀ (x : α), x ∈ tᶜ → f x = 0\nhtμ this : SigmaFinite (Measure.restrict μ t)\nS : ℕ → Set α := spanningSets (Measure.restrict μ t)\nhS_meas : ∀ (n : ℕ), MeasurableSet (S n)\nf_approx : ℕ → α →ₛ β := StronglyMeasurable.approx hf_meas\nfs : ℕ → α →ₛ β := fun n => SimpleFunc.restrict (f_approx n) (S n ∩ t)\nh_fs_t_compl : ∀ (n : ℕ) (x : α), ¬x ∈ t → ↑(fs n) x = 0\n⊢ ∀ (n : ℕ), ↑↑μ (support ↑(fs n)) < ⊤\n\ncase refine'_2\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.51589\nι : Type ?u.51592\ninst✝² : Countable ι\nf g : α → β\ninst✝¹ : TopologicalSpace β\ninst✝ : Zero β\nm : MeasurableSpace α\nμ : Measure α\nhf_meas : StronglyMeasurable f\nt : Set α\nht : MeasurableSet t\nhft_zero : ∀ (x : α), x ∈ tᶜ → f x = 0\nhtμ this : SigmaFinite (Measure.restrict μ t)\nS : ℕ → Set α := spanningSets (Measure.restrict μ t)\nhS_meas : ∀ (n : ℕ), MeasurableSet (S n)\nf_approx : ℕ → α →ₛ β := StronglyMeasurable.approx hf_meas\nfs : ℕ → α →ₛ β := fun n => SimpleFunc.restrict (f_approx n) (S n ∩ t)\nh_fs_t_compl : ∀ (n : ℕ) (x : α), ¬x ∈ t → ↑(fs n) x = 0\nx : α\n⊢ Tendsto (fun n => ↑(fs n) x) atTop (𝓝 (f x))",
"state_before": "α : Type u_2\nβ : Type u_1\nγ : Type ?u.51589\nι : Type ?u.51592\ninst✝² : Countable ι\nf g : α → β\ninst✝¹ : TopologicalSpace β\ninst✝ : Zero β\nm : MeasurableSpace α\nμ : Measure α\nhf_meas : StronglyMeasurable f\nt : Set α\nht : MeasurableSet t\nhft_zero : ∀ (x : α), x ∈ tᶜ → f x = 0\nhtμ this : SigmaFinite (Measure.restrict μ t)\nS : ℕ → Set α := spanningSets (Measure.restrict μ t)\nhS_meas : ∀ (n : ℕ), MeasurableSet (S n)\nf_approx : ℕ → α →ₛ β := StronglyMeasurable.approx hf_meas\nfs : ℕ → α →ₛ β := fun n => SimpleFunc.restrict (f_approx n) (S n ∩ t)\nh_fs_t_compl : ∀ (n : ℕ) (x : α), ¬x ∈ t → ↑(fs n) x = 0\n⊢ FinStronglyMeasurable f μ",
"tactic": "refine' ⟨fs, _, fun x => _⟩"
},
{
"state_after": "α : Type u_2\nβ : Type u_1\nγ : Type ?u.51589\nι : Type ?u.51592\ninst✝² : Countable ι\nf g : α → β\ninst✝¹ : TopologicalSpace β\ninst✝ : Zero β\nm : MeasurableSpace α\nμ : Measure α\nhf_meas : StronglyMeasurable f\nt : Set α\nht : MeasurableSet t\nhft_zero : ∀ (x : α), x ∈ tᶜ → f x = 0\nhtμ this : SigmaFinite (Measure.restrict μ t)\nS : ℕ → Set α := spanningSets (Measure.restrict μ t)\nhS_meas : ∀ (n : ℕ), MeasurableSet (S n)\nf_approx : ℕ → α →ₛ β := StronglyMeasurable.approx hf_meas\nfs : ℕ → α →ₛ β := fun n => SimpleFunc.restrict (f_approx n) (S n ∩ t)\nn : ℕ\nx : α\nhxt : ¬x ∈ t\n⊢ ↑(fs n) x = 0",
"state_before": "α : Type u_2\nβ : Type u_1\nγ : Type ?u.51589\nι : Type ?u.51592\ninst✝² : Countable ι\nf g : α → β\ninst✝¹ : TopologicalSpace β\ninst✝ : Zero β\nm : MeasurableSpace α\nμ : Measure α\nhf_meas : StronglyMeasurable f\nt : Set α\nht : MeasurableSet t\nhft_zero : ∀ (x : α), x ∈ tᶜ → f x = 0\nhtμ this : SigmaFinite (Measure.restrict μ t)\nS : ℕ → Set α := spanningSets (Measure.restrict μ t)\nhS_meas : ∀ (n : ℕ), MeasurableSet (S n)\nf_approx : ℕ → α →ₛ β := StronglyMeasurable.approx hf_meas\nfs : ℕ → α →ₛ β := fun n => SimpleFunc.restrict (f_approx n) (S n ∩ t)\n⊢ ∀ (n : ℕ) (x : α), ¬x ∈ t → ↑(fs n) x = 0",
"tactic": "intro n x hxt"
},
{
"state_after": "α : Type u_2\nβ : Type u_1\nγ : Type ?u.51589\nι : Type ?u.51592\ninst✝² : Countable ι\nf g : α → β\ninst✝¹ : TopologicalSpace β\ninst✝ : Zero β\nm : MeasurableSpace α\nμ : Measure α\nhf_meas : StronglyMeasurable f\nt : Set α\nht : MeasurableSet t\nhft_zero : ∀ (x : α), x ∈ tᶜ → f x = 0\nhtμ this : SigmaFinite (Measure.restrict μ t)\nS : ℕ → Set α := spanningSets (Measure.restrict μ t)\nhS_meas : ∀ (n : ℕ), MeasurableSet (S n)\nf_approx : ℕ → α →ₛ β := StronglyMeasurable.approx hf_meas\nfs : ℕ → α →ₛ β := fun n => SimpleFunc.restrict (f_approx n) (S n ∩ t)\nn : ℕ\nx : α\nhxt : ¬x ∈ t\n⊢ indicator (S n ∩ t) (↑(f_approx n)) x = 0",
"state_before": "α : Type u_2\nβ : Type u_1\nγ : Type ?u.51589\nι : Type ?u.51592\ninst✝² : Countable ι\nf g : α → β\ninst✝¹ : TopologicalSpace β\ninst✝ : Zero β\nm : MeasurableSpace α\nμ : Measure α\nhf_meas : StronglyMeasurable f\nt : Set α\nht : MeasurableSet t\nhft_zero : ∀ (x : α), x ∈ tᶜ → f x = 0\nhtμ this : SigmaFinite (Measure.restrict μ t)\nS : ℕ → Set α := spanningSets (Measure.restrict μ t)\nhS_meas : ∀ (n : ℕ), MeasurableSet (S n)\nf_approx : ℕ → α →ₛ β := StronglyMeasurable.approx hf_meas\nfs : ℕ → α →ₛ β := fun n => SimpleFunc.restrict (f_approx n) (S n ∩ t)\nn : ℕ\nx : α\nhxt : ¬x ∈ t\n⊢ ↑(fs n) x = 0",
"tactic": "rw [SimpleFunc.restrict_apply _ ((hS_meas n).inter ht)]"
},
{
"state_after": "α : Type u_2\nβ : Type u_1\nγ : Type ?u.51589\nι : Type ?u.51592\ninst✝² : Countable ι\nf g : α → β\ninst✝¹ : TopologicalSpace β\ninst✝ : Zero β\nm : MeasurableSpace α\nμ : Measure α\nhf_meas : StronglyMeasurable f\nt : Set α\nht : MeasurableSet t\nhft_zero : ∀ (x : α), x ∈ tᶜ → f x = 0\nhtμ this : SigmaFinite (Measure.restrict μ t)\nS : ℕ → Set α := spanningSets (Measure.restrict μ t)\nhS_meas : ∀ (n : ℕ), MeasurableSet (S n)\nf_approx : ℕ → α →ₛ β := StronglyMeasurable.approx hf_meas\nfs : ℕ → α →ₛ β := fun n => SimpleFunc.restrict (f_approx n) (S n ∩ t)\nn : ℕ\nx : α\nhxt : ¬x ∈ t\n⊢ ¬x ∈ S n ∩ t",
"state_before": "α : Type u_2\nβ : Type u_1\nγ : Type ?u.51589\nι : Type ?u.51592\ninst✝² : Countable ι\nf g : α → β\ninst✝¹ : TopologicalSpace β\ninst✝ : Zero β\nm : MeasurableSpace α\nμ : Measure α\nhf_meas : StronglyMeasurable f\nt : Set α\nht : MeasurableSet t\nhft_zero : ∀ (x : α), x ∈ tᶜ → f x = 0\nhtμ this : SigmaFinite (Measure.restrict μ t)\nS : ℕ → Set α := spanningSets (Measure.restrict μ t)\nhS_meas : ∀ (n : ℕ), MeasurableSet (S n)\nf_approx : ℕ → α →ₛ β := StronglyMeasurable.approx hf_meas\nfs : ℕ → α →ₛ β := fun n => SimpleFunc.restrict (f_approx n) (S n ∩ t)\nn : ℕ\nx : α\nhxt : ¬x ∈ t\n⊢ indicator (S n ∩ t) (↑(f_approx n)) x = 0",
"tactic": "refine' Set.indicator_of_not_mem _ _"
},
{
"state_after": "no goals",
"state_before": "α : Type u_2\nβ : Type u_1\nγ : Type ?u.51589\nι : Type ?u.51592\ninst✝² : Countable ι\nf g : α → β\ninst✝¹ : TopologicalSpace β\ninst✝ : Zero β\nm : MeasurableSpace α\nμ : Measure α\nhf_meas : StronglyMeasurable f\nt : Set α\nht : MeasurableSet t\nhft_zero : ∀ (x : α), x ∈ tᶜ → f x = 0\nhtμ this : SigmaFinite (Measure.restrict μ t)\nS : ℕ → Set α := spanningSets (Measure.restrict μ t)\nhS_meas : ∀ (n : ℕ), MeasurableSet (S n)\nf_approx : ℕ → α →ₛ β := StronglyMeasurable.approx hf_meas\nfs : ℕ → α →ₛ β := fun n => SimpleFunc.restrict (f_approx n) (S n ∩ t)\nn : ℕ\nx : α\nhxt : ¬x ∈ t\n⊢ ¬x ∈ S n ∩ t",
"tactic": "simp [hxt]"
},
{
"state_after": "case refine'_1\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.51589\nι : Type ?u.51592\ninst✝² : Countable ι\nf g : α → β\ninst✝¹ : TopologicalSpace β\ninst✝ : Zero β\nm : MeasurableSpace α\nμ : Measure α\nhf_meas : StronglyMeasurable f\nt : Set α\nht : MeasurableSet t\nhft_zero : ∀ (x : α), x ∈ tᶜ → f x = 0\nhtμ this : SigmaFinite (Measure.restrict μ t)\nS : ℕ → Set α := spanningSets (Measure.restrict μ t)\nhS_meas : ∀ (n : ℕ), MeasurableSet (S n)\nf_approx : ℕ → α →ₛ β := StronglyMeasurable.approx hf_meas\nfs : ℕ → α →ₛ β := fun n => SimpleFunc.restrict (f_approx n) (S n ∩ t)\nh_fs_t_compl : ∀ (n : ℕ) (x : α), ¬x ∈ t → ↑(fs n) x = 0\n⊢ ∀ (n : ℕ),\n ↑↑μ\n (⋃ (y : β) (_ :\n y ∈\n Finset.filter (fun y => y ≠ 0)\n (SimpleFunc.range\n (SimpleFunc.restrict (StronglyMeasurable.approx hf_meas n)\n (spanningSets (Measure.restrict μ t) n ∩ t)))),\n ↑(SimpleFunc.restrict (StronglyMeasurable.approx hf_meas n) (spanningSets (Measure.restrict μ t) n ∩ t)) ⁻¹'\n {y}) <\n ⊤",
"state_before": "case refine'_1\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.51589\nι : Type ?u.51592\ninst✝² : Countable ι\nf g : α → β\ninst✝¹ : TopologicalSpace β\ninst✝ : Zero β\nm : MeasurableSpace α\nμ : Measure α\nhf_meas : StronglyMeasurable f\nt : Set α\nht : MeasurableSet t\nhft_zero : ∀ (x : α), x ∈ tᶜ → f x = 0\nhtμ this : SigmaFinite (Measure.restrict μ t)\nS : ℕ → Set α := spanningSets (Measure.restrict μ t)\nhS_meas : ∀ (n : ℕ), MeasurableSet (S n)\nf_approx : ℕ → α →ₛ β := StronglyMeasurable.approx hf_meas\nfs : ℕ → α →ₛ β := fun n => SimpleFunc.restrict (f_approx n) (S n ∩ t)\nh_fs_t_compl : ∀ (n : ℕ) (x : α), ¬x ∈ t → ↑(fs n) x = 0\n⊢ ∀ (n : ℕ), ↑↑μ (support ↑(fs n)) < ⊤",
"tactic": "simp_rw [SimpleFunc.support_eq]"
},
{
"state_after": "case refine'_1\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.51589\nι : Type ?u.51592\ninst✝² : Countable ι\nf g : α → β\ninst✝¹ : TopologicalSpace β\ninst✝ : Zero β\nm : MeasurableSpace α\nμ : Measure α\nhf_meas : StronglyMeasurable f\nt : Set α\nht : MeasurableSet t\nhft_zero : ∀ (x : α), x ∈ tᶜ → f x = 0\nhtμ this : SigmaFinite (Measure.restrict μ t)\nS : ℕ → Set α := spanningSets (Measure.restrict μ t)\nhS_meas : ∀ (n : ℕ), MeasurableSet (S n)\nf_approx : ℕ → α →ₛ β := StronglyMeasurable.approx hf_meas\nfs : ℕ → α →ₛ β := fun n => SimpleFunc.restrict (f_approx n) (S n ∩ t)\nh_fs_t_compl : ∀ (n : ℕ) (x : α), ¬x ∈ t → ↑(fs n) x = 0\nn : ℕ\n⊢ ∑ p in\n Finset.filter (fun y => y ≠ 0)\n (SimpleFunc.range\n (SimpleFunc.restrict (StronglyMeasurable.approx hf_meas n) (spanningSets (Measure.restrict μ t) n ∩ t))),\n ↑↑μ\n (↑(SimpleFunc.restrict (StronglyMeasurable.approx hf_meas n) (spanningSets (Measure.restrict μ t) n ∩ t)) ⁻¹'\n {p}) <\n ⊤",
"state_before": "case refine'_1\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.51589\nι : Type ?u.51592\ninst✝² : Countable ι\nf g : α → β\ninst✝¹ : TopologicalSpace β\ninst✝ : Zero β\nm : MeasurableSpace α\nμ : Measure α\nhf_meas : StronglyMeasurable f\nt : Set α\nht : MeasurableSet t\nhft_zero : ∀ (x : α), x ∈ tᶜ → f x = 0\nhtμ this : SigmaFinite (Measure.restrict μ t)\nS : ℕ → Set α := spanningSets (Measure.restrict μ t)\nhS_meas : ∀ (n : ℕ), MeasurableSet (S n)\nf_approx : ℕ → α →ₛ β := StronglyMeasurable.approx hf_meas\nfs : ℕ → α →ₛ β := fun n => SimpleFunc.restrict (f_approx n) (S n ∩ t)\nh_fs_t_compl : ∀ (n : ℕ) (x : α), ¬x ∈ t → ↑(fs n) x = 0\n⊢ ∀ (n : ℕ),\n ↑↑μ\n (⋃ (y : β) (_ :\n y ∈\n Finset.filter (fun y => y ≠ 0)\n (SimpleFunc.range\n (SimpleFunc.restrict (StronglyMeasurable.approx hf_meas n)\n (spanningSets (Measure.restrict μ t) n ∩ t)))),\n ↑(SimpleFunc.restrict (StronglyMeasurable.approx hf_meas n) (spanningSets (Measure.restrict μ t) n ∩ t)) ⁻¹'\n {y}) <\n ⊤",
"tactic": "refine' fun n => (measure_biUnion_finset_le _ _).trans_lt _"
},
{
"state_after": "case refine'_1\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.51589\nι : Type ?u.51592\ninst✝² : Countable ι\nf g : α → β\ninst✝¹ : TopologicalSpace β\ninst✝ : Zero β\nm : MeasurableSpace α\nμ : Measure α\nhf_meas : StronglyMeasurable f\nt : Set α\nht : MeasurableSet t\nhft_zero : ∀ (x : α), x ∈ tᶜ → f x = 0\nhtμ this : SigmaFinite (Measure.restrict μ t)\nS : ℕ → Set α := spanningSets (Measure.restrict μ t)\nhS_meas : ∀ (n : ℕ), MeasurableSet (S n)\nf_approx : ℕ → α →ₛ β := StronglyMeasurable.approx hf_meas\nfs : ℕ → α →ₛ β := fun n => SimpleFunc.restrict (f_approx n) (S n ∩ t)\nh_fs_t_compl : ∀ (n : ℕ) (x : α), ¬x ∈ t → ↑(fs n) x = 0\nn : ℕ\ny : β\nhy :\n y ∈\n Finset.filter (fun y => y ≠ 0)\n (SimpleFunc.range\n (SimpleFunc.restrict (StronglyMeasurable.approx hf_meas n) (spanningSets (Measure.restrict μ t) n ∩ t)))\n⊢ ↑↑μ\n (↑(SimpleFunc.restrict (StronglyMeasurable.approx hf_meas n) (spanningSets (Measure.restrict μ t) n ∩ t)) ⁻¹'\n {y}) <\n ⊤",
"state_before": "case refine'_1\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.51589\nι : Type ?u.51592\ninst✝² : Countable ι\nf g : α → β\ninst✝¹ : TopologicalSpace β\ninst✝ : Zero β\nm : MeasurableSpace α\nμ : Measure α\nhf_meas : StronglyMeasurable f\nt : Set α\nht : MeasurableSet t\nhft_zero : ∀ (x : α), x ∈ tᶜ → f x = 0\nhtμ this : SigmaFinite (Measure.restrict μ t)\nS : ℕ → Set α := spanningSets (Measure.restrict μ t)\nhS_meas : ∀ (n : ℕ), MeasurableSet (S n)\nf_approx : ℕ → α →ₛ β := StronglyMeasurable.approx hf_meas\nfs : ℕ → α →ₛ β := fun n => SimpleFunc.restrict (f_approx n) (S n ∩ t)\nh_fs_t_compl : ∀ (n : ℕ) (x : α), ¬x ∈ t → ↑(fs n) x = 0\nn : ℕ\n⊢ ∑ p in\n Finset.filter (fun y => y ≠ 0)\n (SimpleFunc.range\n (SimpleFunc.restrict (StronglyMeasurable.approx hf_meas n) (spanningSets (Measure.restrict μ t) n ∩ t))),\n ↑↑μ\n (↑(SimpleFunc.restrict (StronglyMeasurable.approx hf_meas n) (spanningSets (Measure.restrict μ t) n ∩ t)) ⁻¹'\n {p}) <\n ⊤",
"tactic": "refine' ENNReal.sum_lt_top_iff.mpr fun y hy => _"
},
{
"state_after": "case refine'_1\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.51589\nι : Type ?u.51592\ninst✝² : Countable ι\nf g : α → β\ninst✝¹ : TopologicalSpace β\ninst✝ : Zero β\nm : MeasurableSpace α\nμ : Measure α\nhf_meas : StronglyMeasurable f\nt : Set α\nht : MeasurableSet t\nhft_zero : ∀ (x : α), x ∈ tᶜ → f x = 0\nhtμ this : SigmaFinite (Measure.restrict μ t)\nS : ℕ → Set α := spanningSets (Measure.restrict μ t)\nhS_meas : ∀ (n : ℕ), MeasurableSet (S n)\nf_approx : ℕ → α →ₛ β := StronglyMeasurable.approx hf_meas\nfs : ℕ → α →ₛ β := fun n => SimpleFunc.restrict (f_approx n) (S n ∩ t)\nh_fs_t_compl : ∀ (n : ℕ) (x : α), ¬x ∈ t → ↑(fs n) x = 0\nn : ℕ\ny : β\nhy :\n y ∈\n Finset.filter (fun y => y ≠ 0)\n (SimpleFunc.range\n (SimpleFunc.restrict (StronglyMeasurable.approx hf_meas n) (spanningSets (Measure.restrict μ t) n ∩ t)))\n⊢ ↑↑μ (S n ∩ t ∩ ↑(StronglyMeasurable.approx hf_meas n) ⁻¹' {y}) < ⊤\n\ncase refine'_1\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.51589\nι : Type ?u.51592\ninst✝² : Countable ι\nf g : α → β\ninst✝¹ : TopologicalSpace β\ninst✝ : Zero β\nm : MeasurableSpace α\nμ : Measure α\nhf_meas : StronglyMeasurable f\nt : Set α\nht : MeasurableSet t\nhft_zero : ∀ (x : α), x ∈ tᶜ → f x = 0\nhtμ this : SigmaFinite (Measure.restrict μ t)\nS : ℕ → Set α := spanningSets (Measure.restrict μ t)\nhS_meas : ∀ (n : ℕ), MeasurableSet (S n)\nf_approx : ℕ → α →ₛ β := StronglyMeasurable.approx hf_meas\nfs : ℕ → α →ₛ β := fun n => SimpleFunc.restrict (f_approx n) (S n ∩ t)\nh_fs_t_compl : ∀ (n : ℕ) (x : α), ¬x ∈ t → ↑(fs n) x = 0\nn : ℕ\ny : β\nhy :\n y ∈\n Finset.filter (fun y => y ≠ 0)\n (SimpleFunc.range\n (SimpleFunc.restrict (StronglyMeasurable.approx hf_meas n) (spanningSets (Measure.restrict μ t) n ∩ t)))\n⊢ y ≠ 0",
"state_before": "case refine'_1\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.51589\nι : Type ?u.51592\ninst✝² : Countable ι\nf g : α → β\ninst✝¹ : TopologicalSpace β\ninst✝ : Zero β\nm : MeasurableSpace α\nμ : Measure α\nhf_meas : StronglyMeasurable f\nt : Set α\nht : MeasurableSet t\nhft_zero : ∀ (x : α), x ∈ tᶜ → f x = 0\nhtμ this : SigmaFinite (Measure.restrict μ t)\nS : ℕ → Set α := spanningSets (Measure.restrict μ t)\nhS_meas : ∀ (n : ℕ), MeasurableSet (S n)\nf_approx : ℕ → α →ₛ β := StronglyMeasurable.approx hf_meas\nfs : ℕ → α →ₛ β := fun n => SimpleFunc.restrict (f_approx n) (S n ∩ t)\nh_fs_t_compl : ∀ (n : ℕ) (x : α), ¬x ∈ t → ↑(fs n) x = 0\nn : ℕ\ny : β\nhy :\n y ∈\n Finset.filter (fun y => y ≠ 0)\n (SimpleFunc.range\n (SimpleFunc.restrict (StronglyMeasurable.approx hf_meas n) (spanningSets (Measure.restrict μ t) n ∩ t)))\n⊢ ↑↑μ\n (↑(SimpleFunc.restrict (StronglyMeasurable.approx hf_meas n) (spanningSets (Measure.restrict μ t) n ∩ t)) ⁻¹'\n {y}) <\n ⊤",
"tactic": "rw [SimpleFunc.restrict_preimage_singleton _ ((hS_meas n).inter ht)]"
},
{
"state_after": "case refine'_1\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.51589\nι : Type ?u.51592\ninst✝² : Countable ι\nf g : α → β\ninst✝¹ : TopologicalSpace β\ninst✝ : Zero β\nm : MeasurableSpace α\nμ : Measure α\nhf_meas : StronglyMeasurable f\nt : Set α\nht : MeasurableSet t\nhft_zero : ∀ (x : α), x ∈ tᶜ → f x = 0\nhtμ this : SigmaFinite (Measure.restrict μ t)\nS : ℕ → Set α := spanningSets (Measure.restrict μ t)\nhS_meas : ∀ (n : ℕ), MeasurableSet (S n)\nf_approx : ℕ → α →ₛ β := StronglyMeasurable.approx hf_meas\nfs : ℕ → α →ₛ β := fun n => SimpleFunc.restrict (f_approx n) (S n ∩ t)\nh_fs_t_compl : ∀ (n : ℕ) (x : α), ¬x ∈ t → ↑(fs n) x = 0\nn : ℕ\ny : β\nhy :\n y ∈\n Finset.filter (fun y => y ≠ 0)\n (SimpleFunc.range\n (SimpleFunc.restrict (StronglyMeasurable.approx hf_meas n) (spanningSets (Measure.restrict μ t) n ∩ t)))\n⊢ y ≠ 0\n\ncase refine'_1\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.51589\nι : Type ?u.51592\ninst✝² : Countable ι\nf g : α → β\ninst✝¹ : TopologicalSpace β\ninst✝ : Zero β\nm : MeasurableSpace α\nμ : Measure α\nhf_meas : StronglyMeasurable f\nt : Set α\nht : MeasurableSet t\nhft_zero : ∀ (x : α), x ∈ tᶜ → f x = 0\nhtμ this : SigmaFinite (Measure.restrict μ t)\nS : ℕ → Set α := spanningSets (Measure.restrict μ t)\nhS_meas : ∀ (n : ℕ), MeasurableSet (S n)\nf_approx : ℕ → α →ₛ β := StronglyMeasurable.approx hf_meas\nfs : ℕ → α →ₛ β := fun n => SimpleFunc.restrict (f_approx n) (S n ∩ t)\nh_fs_t_compl : ∀ (n : ℕ) (x : α), ¬x ∈ t → ↑(fs n) x = 0\nn : ℕ\ny : β\nhy :\n y ∈\n Finset.filter (fun y => y ≠ 0)\n (SimpleFunc.range\n (SimpleFunc.restrict (StronglyMeasurable.approx hf_meas n) (spanningSets (Measure.restrict μ t) n ∩ t)))\n⊢ ↑↑μ (S n ∩ t ∩ ↑(StronglyMeasurable.approx hf_meas n) ⁻¹' {y}) < ⊤",
"state_before": "case refine'_1\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.51589\nι : Type ?u.51592\ninst✝² : Countable ι\nf g : α → β\ninst✝¹ : TopologicalSpace β\ninst✝ : Zero β\nm : MeasurableSpace α\nμ : Measure α\nhf_meas : StronglyMeasurable f\nt : Set α\nht : MeasurableSet t\nhft_zero : ∀ (x : α), x ∈ tᶜ → f x = 0\nhtμ this : SigmaFinite (Measure.restrict μ t)\nS : ℕ → Set α := spanningSets (Measure.restrict μ t)\nhS_meas : ∀ (n : ℕ), MeasurableSet (S n)\nf_approx : ℕ → α →ₛ β := StronglyMeasurable.approx hf_meas\nfs : ℕ → α →ₛ β := fun n => SimpleFunc.restrict (f_approx n) (S n ∩ t)\nh_fs_t_compl : ∀ (n : ℕ) (x : α), ¬x ∈ t → ↑(fs n) x = 0\nn : ℕ\ny : β\nhy :\n y ∈\n Finset.filter (fun y => y ≠ 0)\n (SimpleFunc.range\n (SimpleFunc.restrict (StronglyMeasurable.approx hf_meas n) (spanningSets (Measure.restrict μ t) n ∩ t)))\n⊢ ↑↑μ (S n ∩ t ∩ ↑(StronglyMeasurable.approx hf_meas n) ⁻¹' {y}) < ⊤\n\ncase refine'_1\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.51589\nι : Type ?u.51592\ninst✝² : Countable ι\nf g : α → β\ninst✝¹ : TopologicalSpace β\ninst✝ : Zero β\nm : MeasurableSpace α\nμ : Measure α\nhf_meas : StronglyMeasurable f\nt : Set α\nht : MeasurableSet t\nhft_zero : ∀ (x : α), x ∈ tᶜ → f x = 0\nhtμ this : SigmaFinite (Measure.restrict μ t)\nS : ℕ → Set α := spanningSets (Measure.restrict μ t)\nhS_meas : ∀ (n : ℕ), MeasurableSet (S n)\nf_approx : ℕ → α →ₛ β := StronglyMeasurable.approx hf_meas\nfs : ℕ → α →ₛ β := fun n => SimpleFunc.restrict (f_approx n) (S n ∩ t)\nh_fs_t_compl : ∀ (n : ℕ) (x : α), ¬x ∈ t → ↑(fs n) x = 0\nn : ℕ\ny : β\nhy :\n y ∈\n Finset.filter (fun y => y ≠ 0)\n (SimpleFunc.range\n (SimpleFunc.restrict (StronglyMeasurable.approx hf_meas n) (spanningSets (Measure.restrict μ t) n ∩ t)))\n⊢ y ≠ 0",
"tactic": "swap"
},
{
"state_after": "case refine'_1\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.51589\nι : Type ?u.51592\ninst✝² : Countable ι\nf g : α → β\ninst✝¹ : TopologicalSpace β\ninst✝ : Zero β\nm : MeasurableSpace α\nμ : Measure α\nhf_meas : StronglyMeasurable f\nt : Set α\nht : MeasurableSet t\nhft_zero : ∀ (x : α), x ∈ tᶜ → f x = 0\nhtμ this : SigmaFinite (Measure.restrict μ t)\nS : ℕ → Set α := spanningSets (Measure.restrict μ t)\nhS_meas : ∀ (n : ℕ), MeasurableSet (S n)\nf_approx : ℕ → α →ₛ β := StronglyMeasurable.approx hf_meas\nfs : ℕ → α →ₛ β := fun n => SimpleFunc.restrict (f_approx n) (S n ∩ t)\nh_fs_t_compl : ∀ (n : ℕ) (x : α), ¬x ∈ t → ↑(fs n) x = 0\nn : ℕ\ny : β\nhy :\n y ∈\n Finset.filter (fun y => y ≠ 0)\n (SimpleFunc.range\n (SimpleFunc.restrict (StronglyMeasurable.approx hf_meas n) (spanningSets (Measure.restrict μ t) n ∩ t)))\n⊢ ↑↑μ (S n ∩ t) < ⊤",
"state_before": "case refine'_1\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.51589\nι : Type ?u.51592\ninst✝² : Countable ι\nf g : α → β\ninst✝¹ : TopologicalSpace β\ninst✝ : Zero β\nm : MeasurableSpace α\nμ : Measure α\nhf_meas : StronglyMeasurable f\nt : Set α\nht : MeasurableSet t\nhft_zero : ∀ (x : α), x ∈ tᶜ → f x = 0\nhtμ this : SigmaFinite (Measure.restrict μ t)\nS : ℕ → Set α := spanningSets (Measure.restrict μ t)\nhS_meas : ∀ (n : ℕ), MeasurableSet (S n)\nf_approx : ℕ → α →ₛ β := StronglyMeasurable.approx hf_meas\nfs : ℕ → α →ₛ β := fun n => SimpleFunc.restrict (f_approx n) (S n ∩ t)\nh_fs_t_compl : ∀ (n : ℕ) (x : α), ¬x ∈ t → ↑(fs n) x = 0\nn : ℕ\ny : β\nhy :\n y ∈\n Finset.filter (fun y => y ≠ 0)\n (SimpleFunc.range\n (SimpleFunc.restrict (StronglyMeasurable.approx hf_meas n) (spanningSets (Measure.restrict μ t) n ∩ t)))\n⊢ ↑↑μ (S n ∩ t ∩ ↑(StronglyMeasurable.approx hf_meas n) ⁻¹' {y}) < ⊤",
"tactic": "refine' (measure_mono (Set.inter_subset_left _ _)).trans_lt _"
},
{
"state_after": "case refine'_1\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.51589\nι : Type ?u.51592\ninst✝² : Countable ι\nf g : α → β\ninst✝¹ : TopologicalSpace β\ninst✝ : Zero β\nm : MeasurableSpace α\nμ : Measure α\nhf_meas : StronglyMeasurable f\nt : Set α\nht : MeasurableSet t\nhft_zero : ∀ (x : α), x ∈ tᶜ → f x = 0\nhtμ this : SigmaFinite (Measure.restrict μ t)\nS : ℕ → Set α := spanningSets (Measure.restrict μ t)\nhS_meas : ∀ (n : ℕ), MeasurableSet (S n)\nf_approx : ℕ → α →ₛ β := StronglyMeasurable.approx hf_meas\nfs : ℕ → α →ₛ β := fun n => SimpleFunc.restrict (f_approx n) (S n ∩ t)\nh_fs_t_compl : ∀ (n : ℕ) (x : α), ¬x ∈ t → ↑(fs n) x = 0\nn : ℕ\ny : β\nhy :\n y ∈\n Finset.filter (fun y => y ≠ 0)\n (SimpleFunc.range\n (SimpleFunc.restrict (StronglyMeasurable.approx hf_meas n) (spanningSets (Measure.restrict μ t) n ∩ t)))\nh_lt_top : ↑↑(Measure.restrict μ t) (spanningSets (Measure.restrict μ t) n) < ⊤\n⊢ ↑↑μ (S n ∩ t) < ⊤",
"state_before": "case refine'_1\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.51589\nι : Type ?u.51592\ninst✝² : Countable ι\nf g : α → β\ninst✝¹ : TopologicalSpace β\ninst✝ : Zero β\nm : MeasurableSpace α\nμ : Measure α\nhf_meas : StronglyMeasurable f\nt : Set α\nht : MeasurableSet t\nhft_zero : ∀ (x : α), x ∈ tᶜ → f x = 0\nhtμ this : SigmaFinite (Measure.restrict μ t)\nS : ℕ → Set α := spanningSets (Measure.restrict μ t)\nhS_meas : ∀ (n : ℕ), MeasurableSet (S n)\nf_approx : ℕ → α →ₛ β := StronglyMeasurable.approx hf_meas\nfs : ℕ → α →ₛ β := fun n => SimpleFunc.restrict (f_approx n) (S n ∩ t)\nh_fs_t_compl : ∀ (n : ℕ) (x : α), ¬x ∈ t → ↑(fs n) x = 0\nn : ℕ\ny : β\nhy :\n y ∈\n Finset.filter (fun y => y ≠ 0)\n (SimpleFunc.range\n (SimpleFunc.restrict (StronglyMeasurable.approx hf_meas n) (spanningSets (Measure.restrict μ t) n ∩ t)))\n⊢ ↑↑μ (S n ∩ t) < ⊤",
"tactic": "have h_lt_top := measure_spanningSets_lt_top (μ.restrict t) n"
},
{
"state_after": "no goals",
"state_before": "case refine'_1\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.51589\nι : Type ?u.51592\ninst✝² : Countable ι\nf g : α → β\ninst✝¹ : TopologicalSpace β\ninst✝ : Zero β\nm : MeasurableSpace α\nμ : Measure α\nhf_meas : StronglyMeasurable f\nt : Set α\nht : MeasurableSet t\nhft_zero : ∀ (x : α), x ∈ tᶜ → f x = 0\nhtμ this : SigmaFinite (Measure.restrict μ t)\nS : ℕ → Set α := spanningSets (Measure.restrict μ t)\nhS_meas : ∀ (n : ℕ), MeasurableSet (S n)\nf_approx : ℕ → α →ₛ β := StronglyMeasurable.approx hf_meas\nfs : ℕ → α →ₛ β := fun n => SimpleFunc.restrict (f_approx n) (S n ∩ t)\nh_fs_t_compl : ∀ (n : ℕ) (x : α), ¬x ∈ t → ↑(fs n) x = 0\nn : ℕ\ny : β\nhy :\n y ∈\n Finset.filter (fun y => y ≠ 0)\n (SimpleFunc.range\n (SimpleFunc.restrict (StronglyMeasurable.approx hf_meas n) (spanningSets (Measure.restrict μ t) n ∩ t)))\nh_lt_top : ↑↑(Measure.restrict μ t) (spanningSets (Measure.restrict μ t) n) < ⊤\n⊢ ↑↑μ (S n ∩ t) < ⊤",
"tactic": "rwa [Measure.restrict_apply' ht] at h_lt_top"
},
{
"state_after": "case refine'_1\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.51589\nι : Type ?u.51592\ninst✝² : Countable ι\nf g : α → β\ninst✝¹ : TopologicalSpace β\ninst✝ : Zero β\nm : MeasurableSpace α\nμ : Measure α\nhf_meas : StronglyMeasurable f\nt : Set α\nht : MeasurableSet t\nhft_zero : ∀ (x : α), x ∈ tᶜ → f x = 0\nhtμ this✝ : SigmaFinite (Measure.restrict μ t)\nS : ℕ → Set α := spanningSets (Measure.restrict μ t)\nhS_meas : ∀ (n : ℕ), MeasurableSet (S n)\nf_approx : ℕ → α →ₛ β := StronglyMeasurable.approx hf_meas\nfs : ℕ → α →ₛ β := fun n => SimpleFunc.restrict (f_approx n) (S n ∩ t)\nh_fs_t_compl : ∀ (n : ℕ) (x : α), ¬x ∈ t → ↑(fs n) x = 0\nn : ℕ\ny : β\nhy :\n y ∈\n Finset.filter (fun y => y ≠ 0)\n (SimpleFunc.range\n (SimpleFunc.restrict (StronglyMeasurable.approx hf_meas n) (spanningSets (Measure.restrict μ t) n ∩ t)))\nthis : (y : β) → Decidable (y = 0) := fun y => Classical.propDecidable (y = 0)\n⊢ y ≠ 0",
"state_before": "case refine'_1\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.51589\nι : Type ?u.51592\ninst✝² : Countable ι\nf g : α → β\ninst✝¹ : TopologicalSpace β\ninst✝ : Zero β\nm : MeasurableSpace α\nμ : Measure α\nhf_meas : StronglyMeasurable f\nt : Set α\nht : MeasurableSet t\nhft_zero : ∀ (x : α), x ∈ tᶜ → f x = 0\nhtμ this : SigmaFinite (Measure.restrict μ t)\nS : ℕ → Set α := spanningSets (Measure.restrict μ t)\nhS_meas : ∀ (n : ℕ), MeasurableSet (S n)\nf_approx : ℕ → α →ₛ β := StronglyMeasurable.approx hf_meas\nfs : ℕ → α →ₛ β := fun n => SimpleFunc.restrict (f_approx n) (S n ∩ t)\nh_fs_t_compl : ∀ (n : ℕ) (x : α), ¬x ∈ t → ↑(fs n) x = 0\nn : ℕ\ny : β\nhy :\n y ∈\n Finset.filter (fun y => y ≠ 0)\n (SimpleFunc.range\n (SimpleFunc.restrict (StronglyMeasurable.approx hf_meas n) (spanningSets (Measure.restrict μ t) n ∩ t)))\n⊢ y ≠ 0",
"tactic": "letI : (y : β) → Decidable (y = 0) := fun y => Classical.propDecidable _"
},
{
"state_after": "case refine'_1\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.51589\nι : Type ?u.51592\ninst✝² : Countable ι\nf g : α → β\ninst✝¹ : TopologicalSpace β\ninst✝ : Zero β\nm : MeasurableSpace α\nμ : Measure α\nhf_meas : StronglyMeasurable f\nt : Set α\nht : MeasurableSet t\nhft_zero : ∀ (x : α), x ∈ tᶜ → f x = 0\nhtμ this✝ : SigmaFinite (Measure.restrict μ t)\nS : ℕ → Set α := spanningSets (Measure.restrict μ t)\nhS_meas : ∀ (n : ℕ), MeasurableSet (S n)\nf_approx : ℕ → α →ₛ β := StronglyMeasurable.approx hf_meas\nfs : ℕ → α →ₛ β := fun n => SimpleFunc.restrict (f_approx n) (S n ∩ t)\nh_fs_t_compl : ∀ (n : ℕ) (x : α), ¬x ∈ t → ↑(fs n) x = 0\nn : ℕ\ny : β\nhy :\n y ∈\n SimpleFunc.range\n (SimpleFunc.restrict (StronglyMeasurable.approx hf_meas n) (spanningSets (Measure.restrict μ t) n ∩ t)) ∧\n y ≠ 0\nthis : (y : β) → Decidable (y = 0) := fun y => Classical.propDecidable (y = 0)\n⊢ y ≠ 0",
"state_before": "case refine'_1\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.51589\nι : Type ?u.51592\ninst✝² : Countable ι\nf g : α → β\ninst✝¹ : TopologicalSpace β\ninst✝ : Zero β\nm : MeasurableSpace α\nμ : Measure α\nhf_meas : StronglyMeasurable f\nt : Set α\nht : MeasurableSet t\nhft_zero : ∀ (x : α), x ∈ tᶜ → f x = 0\nhtμ this✝ : SigmaFinite (Measure.restrict μ t)\nS : ℕ → Set α := spanningSets (Measure.restrict μ t)\nhS_meas : ∀ (n : ℕ), MeasurableSet (S n)\nf_approx : ℕ → α →ₛ β := StronglyMeasurable.approx hf_meas\nfs : ℕ → α →ₛ β := fun n => SimpleFunc.restrict (f_approx n) (S n ∩ t)\nh_fs_t_compl : ∀ (n : ℕ) (x : α), ¬x ∈ t → ↑(fs n) x = 0\nn : ℕ\ny : β\nhy :\n y ∈\n Finset.filter (fun y => y ≠ 0)\n (SimpleFunc.range\n (SimpleFunc.restrict (StronglyMeasurable.approx hf_meas n) (spanningSets (Measure.restrict μ t) n ∩ t)))\nthis : (y : β) → Decidable (y = 0) := fun y => Classical.propDecidable (y = 0)\n⊢ y ≠ 0",
"tactic": "rw [Finset.mem_filter] at hy"
},
{
"state_after": "no goals",
"state_before": "case refine'_1\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.51589\nι : Type ?u.51592\ninst✝² : Countable ι\nf g : α → β\ninst✝¹ : TopologicalSpace β\ninst✝ : Zero β\nm : MeasurableSpace α\nμ : Measure α\nhf_meas : StronglyMeasurable f\nt : Set α\nht : MeasurableSet t\nhft_zero : ∀ (x : α), x ∈ tᶜ → f x = 0\nhtμ this✝ : SigmaFinite (Measure.restrict μ t)\nS : ℕ → Set α := spanningSets (Measure.restrict μ t)\nhS_meas : ∀ (n : ℕ), MeasurableSet (S n)\nf_approx : ℕ → α →ₛ β := StronglyMeasurable.approx hf_meas\nfs : ℕ → α →ₛ β := fun n => SimpleFunc.restrict (f_approx n) (S n ∩ t)\nh_fs_t_compl : ∀ (n : ℕ) (x : α), ¬x ∈ t → ↑(fs n) x = 0\nn : ℕ\ny : β\nhy :\n y ∈\n SimpleFunc.range\n (SimpleFunc.restrict (StronglyMeasurable.approx hf_meas n) (spanningSets (Measure.restrict μ t) n ∩ t)) ∧\n y ≠ 0\nthis : (y : β) → Decidable (y = 0) := fun y => Classical.propDecidable (y = 0)\n⊢ y ≠ 0",
"tactic": "exact hy.2"
},
{
"state_after": "case pos\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.51589\nι : Type ?u.51592\ninst✝² : Countable ι\nf g : α → β\ninst✝¹ : TopologicalSpace β\ninst✝ : Zero β\nm : MeasurableSpace α\nμ : Measure α\nhf_meas : StronglyMeasurable f\nt : Set α\nht : MeasurableSet t\nhft_zero : ∀ (x : α), x ∈ tᶜ → f x = 0\nhtμ this : SigmaFinite (Measure.restrict μ t)\nS : ℕ → Set α := spanningSets (Measure.restrict μ t)\nhS_meas : ∀ (n : ℕ), MeasurableSet (S n)\nf_approx : ℕ → α →ₛ β := StronglyMeasurable.approx hf_meas\nfs : ℕ → α →ₛ β := fun n => SimpleFunc.restrict (f_approx n) (S n ∩ t)\nh_fs_t_compl : ∀ (n : ℕ) (x : α), ¬x ∈ t → ↑(fs n) x = 0\nx : α\nhxt : x ∈ t\n⊢ Tendsto (fun n => ↑(fs n) x) atTop (𝓝 (f x))\n\ncase neg\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.51589\nι : Type ?u.51592\ninst✝² : Countable ι\nf g : α → β\ninst✝¹ : TopologicalSpace β\ninst✝ : Zero β\nm : MeasurableSpace α\nμ : Measure α\nhf_meas : StronglyMeasurable f\nt : Set α\nht : MeasurableSet t\nhft_zero : ∀ (x : α), x ∈ tᶜ → f x = 0\nhtμ this : SigmaFinite (Measure.restrict μ t)\nS : ℕ → Set α := spanningSets (Measure.restrict μ t)\nhS_meas : ∀ (n : ℕ), MeasurableSet (S n)\nf_approx : ℕ → α →ₛ β := StronglyMeasurable.approx hf_meas\nfs : ℕ → α →ₛ β := fun n => SimpleFunc.restrict (f_approx n) (S n ∩ t)\nh_fs_t_compl : ∀ (n : ℕ) (x : α), ¬x ∈ t → ↑(fs n) x = 0\nx : α\nhxt : ¬x ∈ t\n⊢ Tendsto (fun n => ↑(fs n) x) atTop (𝓝 (f x))",
"state_before": "case refine'_2\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.51589\nι : Type ?u.51592\ninst✝² : Countable ι\nf g : α → β\ninst✝¹ : TopologicalSpace β\ninst✝ : Zero β\nm : MeasurableSpace α\nμ : Measure α\nhf_meas : StronglyMeasurable f\nt : Set α\nht : MeasurableSet t\nhft_zero : ∀ (x : α), x ∈ tᶜ → f x = 0\nhtμ this : SigmaFinite (Measure.restrict μ t)\nS : ℕ → Set α := spanningSets (Measure.restrict μ t)\nhS_meas : ∀ (n : ℕ), MeasurableSet (S n)\nf_approx : ℕ → α →ₛ β := StronglyMeasurable.approx hf_meas\nfs : ℕ → α →ₛ β := fun n => SimpleFunc.restrict (f_approx n) (S n ∩ t)\nh_fs_t_compl : ∀ (n : ℕ) (x : α), ¬x ∈ t → ↑(fs n) x = 0\nx : α\n⊢ Tendsto (fun n => ↑(fs n) x) atTop (𝓝 (f x))",
"tactic": "by_cases hxt : x ∈ t"
},
{
"state_after": "case neg\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.51589\nι : Type ?u.51592\ninst✝² : Countable ι\nf g : α → β\ninst✝¹ : TopologicalSpace β\ninst✝ : Zero β\nm : MeasurableSpace α\nμ : Measure α\nhf_meas : StronglyMeasurable f\nt : Set α\nht : MeasurableSet t\nhft_zero : ∀ (x : α), x ∈ tᶜ → f x = 0\nhtμ this : SigmaFinite (Measure.restrict μ t)\nS : ℕ → Set α := spanningSets (Measure.restrict μ t)\nhS_meas : ∀ (n : ℕ), MeasurableSet (S n)\nf_approx : ℕ → α →ₛ β := StronglyMeasurable.approx hf_meas\nfs : ℕ → α →ₛ β := fun n => SimpleFunc.restrict (f_approx n) (S n ∩ t)\nh_fs_t_compl : ∀ (n : ℕ) (x : α), ¬x ∈ t → ↑(fs n) x = 0\nx : α\nhxt : ¬x ∈ t\n⊢ Tendsto (fun n => ↑(fs n) x) atTop (𝓝 (f x))\n\ncase pos\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.51589\nι : Type ?u.51592\ninst✝² : Countable ι\nf g : α → β\ninst✝¹ : TopologicalSpace β\ninst✝ : Zero β\nm : MeasurableSpace α\nμ : Measure α\nhf_meas : StronglyMeasurable f\nt : Set α\nht : MeasurableSet t\nhft_zero : ∀ (x : α), x ∈ tᶜ → f x = 0\nhtμ this : SigmaFinite (Measure.restrict μ t)\nS : ℕ → Set α := spanningSets (Measure.restrict μ t)\nhS_meas : ∀ (n : ℕ), MeasurableSet (S n)\nf_approx : ℕ → α →ₛ β := StronglyMeasurable.approx hf_meas\nfs : ℕ → α →ₛ β := fun n => SimpleFunc.restrict (f_approx n) (S n ∩ t)\nh_fs_t_compl : ∀ (n : ℕ) (x : α), ¬x ∈ t → ↑(fs n) x = 0\nx : α\nhxt : x ∈ t\n⊢ Tendsto (fun n => ↑(fs n) x) atTop (𝓝 (f x))",
"state_before": "case pos\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.51589\nι : Type ?u.51592\ninst✝² : Countable ι\nf g : α → β\ninst✝¹ : TopologicalSpace β\ninst✝ : Zero β\nm : MeasurableSpace α\nμ : Measure α\nhf_meas : StronglyMeasurable f\nt : Set α\nht : MeasurableSet t\nhft_zero : ∀ (x : α), x ∈ tᶜ → f x = 0\nhtμ this : SigmaFinite (Measure.restrict μ t)\nS : ℕ → Set α := spanningSets (Measure.restrict μ t)\nhS_meas : ∀ (n : ℕ), MeasurableSet (S n)\nf_approx : ℕ → α →ₛ β := StronglyMeasurable.approx hf_meas\nfs : ℕ → α →ₛ β := fun n => SimpleFunc.restrict (f_approx n) (S n ∩ t)\nh_fs_t_compl : ∀ (n : ℕ) (x : α), ¬x ∈ t → ↑(fs n) x = 0\nx : α\nhxt : x ∈ t\n⊢ Tendsto (fun n => ↑(fs n) x) atTop (𝓝 (f x))\n\ncase neg\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.51589\nι : Type ?u.51592\ninst✝² : Countable ι\nf g : α → β\ninst✝¹ : TopologicalSpace β\ninst✝ : Zero β\nm : MeasurableSpace α\nμ : Measure α\nhf_meas : StronglyMeasurable f\nt : Set α\nht : MeasurableSet t\nhft_zero : ∀ (x : α), x ∈ tᶜ → f x = 0\nhtμ this : SigmaFinite (Measure.restrict μ t)\nS : ℕ → Set α := spanningSets (Measure.restrict μ t)\nhS_meas : ∀ (n : ℕ), MeasurableSet (S n)\nf_approx : ℕ → α →ₛ β := StronglyMeasurable.approx hf_meas\nfs : ℕ → α →ₛ β := fun n => SimpleFunc.restrict (f_approx n) (S n ∩ t)\nh_fs_t_compl : ∀ (n : ℕ) (x : α), ¬x ∈ t → ↑(fs n) x = 0\nx : α\nhxt : ¬x ∈ t\n⊢ Tendsto (fun n => ↑(fs n) x) atTop (𝓝 (f x))",
"tactic": "swap"
},
{
"state_after": "case pos\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.51589\nι : Type ?u.51592\ninst✝² : Countable ι\nf g : α → β\ninst✝¹ : TopologicalSpace β\ninst✝ : Zero β\nm : MeasurableSpace α\nμ : Measure α\nhf_meas : StronglyMeasurable f\nt : Set α\nht : MeasurableSet t\nhft_zero : ∀ (x : α), x ∈ tᶜ → f x = 0\nhtμ this : SigmaFinite (Measure.restrict μ t)\nS : ℕ → Set α := spanningSets (Measure.restrict μ t)\nhS_meas : ∀ (n : ℕ), MeasurableSet (S n)\nf_approx : ℕ → α →ₛ β := StronglyMeasurable.approx hf_meas\nfs : ℕ → α →ₛ β := fun n => SimpleFunc.restrict (f_approx n) (S n ∩ t)\nh_fs_t_compl : ∀ (n : ℕ) (x : α), ¬x ∈ t → ↑(fs n) x = 0\nx : α\nhxt : x ∈ t\nh : Tendsto (fun n => ↑(f_approx n) x) atTop (𝓝 (f x))\n⊢ Tendsto (fun n => ↑(fs n) x) atTop (𝓝 (f x))",
"state_before": "case pos\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.51589\nι : Type ?u.51592\ninst✝² : Countable ι\nf g : α → β\ninst✝¹ : TopologicalSpace β\ninst✝ : Zero β\nm : MeasurableSpace α\nμ : Measure α\nhf_meas : StronglyMeasurable f\nt : Set α\nht : MeasurableSet t\nhft_zero : ∀ (x : α), x ∈ tᶜ → f x = 0\nhtμ this : SigmaFinite (Measure.restrict μ t)\nS : ℕ → Set α := spanningSets (Measure.restrict μ t)\nhS_meas : ∀ (n : ℕ), MeasurableSet (S n)\nf_approx : ℕ → α →ₛ β := StronglyMeasurable.approx hf_meas\nfs : ℕ → α →ₛ β := fun n => SimpleFunc.restrict (f_approx n) (S n ∩ t)\nh_fs_t_compl : ∀ (n : ℕ) (x : α), ¬x ∈ t → ↑(fs n) x = 0\nx : α\nhxt : x ∈ t\n⊢ Tendsto (fun n => ↑(fs n) x) atTop (𝓝 (f x))",
"tactic": "have h : Tendsto (fun n => (f_approx n) x) atTop (𝓝 (f x)) := hf_meas.tendsto_approx x"
},
{
"state_after": "case pos.intro\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.51589\nι : Type ?u.51592\ninst✝² : Countable ι\nf g : α → β\ninst✝¹ : TopologicalSpace β\ninst✝ : Zero β\nm : MeasurableSpace α\nμ : Measure α\nhf_meas : StronglyMeasurable f\nt : Set α\nht : MeasurableSet t\nhft_zero : ∀ (x : α), x ∈ tᶜ → f x = 0\nhtμ this : SigmaFinite (Measure.restrict μ t)\nS : ℕ → Set α := spanningSets (Measure.restrict μ t)\nhS_meas : ∀ (n : ℕ), MeasurableSet (S n)\nf_approx : ℕ → α →ₛ β := StronglyMeasurable.approx hf_meas\nfs : ℕ → α →ₛ β := fun n => SimpleFunc.restrict (f_approx n) (S n ∩ t)\nh_fs_t_compl : ∀ (n : ℕ) (x : α), ¬x ∈ t → ↑(fs n) x = 0\nx : α\nhxt : x ∈ t\nh✝ : Tendsto (fun n => ↑(f_approx n) x) atTop (𝓝 (f x))\nh : ∀ (s : Set β), s ∈ 𝓝 (f x) → ∃ a, ∀ (b : ℕ), b ≥ a → ↑(f_approx b) x ∈ s\nn₁ : ℕ\nhn₁ : ∀ (m_1 : ℕ), n₁ ≤ m_1 → ↑(fs m_1) x = ↑(f_approx m_1) x\n⊢ ∀ (s : Set β), s ∈ 𝓝 (f x) → ∃ a, ∀ (b : ℕ), b ≥ a → ↑(fs b) x ∈ s",
"state_before": "case pos.intro\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.51589\nι : Type ?u.51592\ninst✝² : Countable ι\nf g : α → β\ninst✝¹ : TopologicalSpace β\ninst✝ : Zero β\nm : MeasurableSpace α\nμ : Measure α\nhf_meas : StronglyMeasurable f\nt : Set α\nht : MeasurableSet t\nhft_zero : ∀ (x : α), x ∈ tᶜ → f x = 0\nhtμ this : SigmaFinite (Measure.restrict μ t)\nS : ℕ → Set α := spanningSets (Measure.restrict μ t)\nhS_meas : ∀ (n : ℕ), MeasurableSet (S n)\nf_approx : ℕ → α →ₛ β := StronglyMeasurable.approx hf_meas\nfs : ℕ → α →ₛ β := fun n => SimpleFunc.restrict (f_approx n) (S n ∩ t)\nh_fs_t_compl : ∀ (n : ℕ) (x : α), ¬x ∈ t → ↑(fs n) x = 0\nx : α\nhxt : x ∈ t\nh : Tendsto (fun n => ↑(f_approx n) x) atTop (𝓝 (f x))\nn₁ : ℕ\nhn₁ : ∀ (m_1 : ℕ), n₁ ≤ m_1 → ↑(fs m_1) x = ↑(f_approx m_1) x\n⊢ Tendsto (fun n => ↑(fs n) x) atTop (𝓝 (f x))",
"tactic": "rw [tendsto_atTop'] at h ⊢"
},
{
"state_after": "case pos.intro\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.51589\nι : Type ?u.51592\ninst✝² : Countable ι\nf g : α → β\ninst✝¹ : TopologicalSpace β\ninst✝ : Zero β\nm : MeasurableSpace α\nμ : Measure α\nhf_meas : StronglyMeasurable f\nt : Set α\nht : MeasurableSet t\nhft_zero : ∀ (x : α), x ∈ tᶜ → f x = 0\nhtμ this : SigmaFinite (Measure.restrict μ t)\nS : ℕ → Set α := spanningSets (Measure.restrict μ t)\nhS_meas : ∀ (n : ℕ), MeasurableSet (S n)\nf_approx : ℕ → α →ₛ β := StronglyMeasurable.approx hf_meas\nfs : ℕ → α →ₛ β := fun n => SimpleFunc.restrict (f_approx n) (S n ∩ t)\nh_fs_t_compl : ∀ (n : ℕ) (x : α), ¬x ∈ t → ↑(fs n) x = 0\nx : α\nhxt : x ∈ t\nh✝ : Tendsto (fun n => ↑(f_approx n) x) atTop (𝓝 (f x))\nh : ∀ (s : Set β), s ∈ 𝓝 (f x) → ∃ a, ∀ (b : ℕ), b ≥ a → ↑(f_approx b) x ∈ s\nn₁ : ℕ\nhn₁ : ∀ (m_1 : ℕ), n₁ ≤ m_1 → ↑(fs m_1) x = ↑(f_approx m_1) x\ns : Set β\nhs : s ∈ 𝓝 (f x)\n⊢ ∃ a, ∀ (b : ℕ), b ≥ a → ↑(fs b) x ∈ s",
"state_before": "case pos.intro\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.51589\nι : Type ?u.51592\ninst✝² : Countable ι\nf g : α → β\ninst✝¹ : TopologicalSpace β\ninst✝ : Zero β\nm : MeasurableSpace α\nμ : Measure α\nhf_meas : StronglyMeasurable f\nt : Set α\nht : MeasurableSet t\nhft_zero : ∀ (x : α), x ∈ tᶜ → f x = 0\nhtμ this : SigmaFinite (Measure.restrict μ t)\nS : ℕ → Set α := spanningSets (Measure.restrict μ t)\nhS_meas : ∀ (n : ℕ), MeasurableSet (S n)\nf_approx : ℕ → α →ₛ β := StronglyMeasurable.approx hf_meas\nfs : ℕ → α →ₛ β := fun n => SimpleFunc.restrict (f_approx n) (S n ∩ t)\nh_fs_t_compl : ∀ (n : ℕ) (x : α), ¬x ∈ t → ↑(fs n) x = 0\nx : α\nhxt : x ∈ t\nh✝ : Tendsto (fun n => ↑(f_approx n) x) atTop (𝓝 (f x))\nh : ∀ (s : Set β), s ∈ 𝓝 (f x) → ∃ a, ∀ (b : ℕ), b ≥ a → ↑(f_approx b) x ∈ s\nn₁ : ℕ\nhn₁ : ∀ (m_1 : ℕ), n₁ ≤ m_1 → ↑(fs m_1) x = ↑(f_approx m_1) x\n⊢ ∀ (s : Set β), s ∈ 𝓝 (f x) → ∃ a, ∀ (b : ℕ), b ≥ a → ↑(fs b) x ∈ s",
"tactic": "intro s hs"
},
{
"state_after": "case pos.intro.intro\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.51589\nι : Type ?u.51592\ninst✝² : Countable ι\nf g : α → β\ninst✝¹ : TopologicalSpace β\ninst✝ : Zero β\nm : MeasurableSpace α\nμ : Measure α\nhf_meas : StronglyMeasurable f\nt : Set α\nht : MeasurableSet t\nhft_zero : ∀ (x : α), x ∈ tᶜ → f x = 0\nhtμ this : SigmaFinite (Measure.restrict μ t)\nS : ℕ → Set α := spanningSets (Measure.restrict μ t)\nhS_meas : ∀ (n : ℕ), MeasurableSet (S n)\nf_approx : ℕ → α →ₛ β := StronglyMeasurable.approx hf_meas\nfs : ℕ → α →ₛ β := fun n => SimpleFunc.restrict (f_approx n) (S n ∩ t)\nh_fs_t_compl : ∀ (n : ℕ) (x : α), ¬x ∈ t → ↑(fs n) x = 0\nx : α\nhxt : x ∈ t\nh✝ : Tendsto (fun n => ↑(f_approx n) x) atTop (𝓝 (f x))\nh : ∀ (s : Set β), s ∈ 𝓝 (f x) → ∃ a, ∀ (b : ℕ), b ≥ a → ↑(f_approx b) x ∈ s\nn₁ : ℕ\nhn₁ : ∀ (m_1 : ℕ), n₁ ≤ m_1 → ↑(fs m_1) x = ↑(f_approx m_1) x\ns : Set β\nhs : s ∈ 𝓝 (f x)\nn₂ : ℕ\nhn₂ : ∀ (b : ℕ), b ≥ n₂ → ↑(f_approx b) x ∈ s\n⊢ ∃ a, ∀ (b : ℕ), b ≥ a → ↑(fs b) x ∈ s",
"state_before": "case pos.intro\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.51589\nι : Type ?u.51592\ninst✝² : Countable ι\nf g : α → β\ninst✝¹ : TopologicalSpace β\ninst✝ : Zero β\nm : MeasurableSpace α\nμ : Measure α\nhf_meas : StronglyMeasurable f\nt : Set α\nht : MeasurableSet t\nhft_zero : ∀ (x : α), x ∈ tᶜ → f x = 0\nhtμ this : SigmaFinite (Measure.restrict μ t)\nS : ℕ → Set α := spanningSets (Measure.restrict μ t)\nhS_meas : ∀ (n : ℕ), MeasurableSet (S n)\nf_approx : ℕ → α →ₛ β := StronglyMeasurable.approx hf_meas\nfs : ℕ → α →ₛ β := fun n => SimpleFunc.restrict (f_approx n) (S n ∩ t)\nh_fs_t_compl : ∀ (n : ℕ) (x : α), ¬x ∈ t → ↑(fs n) x = 0\nx : α\nhxt : x ∈ t\nh✝ : Tendsto (fun n => ↑(f_approx n) x) atTop (𝓝 (f x))\nh : ∀ (s : Set β), s ∈ 𝓝 (f x) → ∃ a, ∀ (b : ℕ), b ≥ a → ↑(f_approx b) x ∈ s\nn₁ : ℕ\nhn₁ : ∀ (m_1 : ℕ), n₁ ≤ m_1 → ↑(fs m_1) x = ↑(f_approx m_1) x\ns : Set β\nhs : s ∈ 𝓝 (f x)\n⊢ ∃ a, ∀ (b : ℕ), b ≥ a → ↑(fs b) x ∈ s",
"tactic": "obtain ⟨n₂, hn₂⟩ := h s hs"
},
{
"state_after": "case pos.intro.intro\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.51589\nι : Type ?u.51592\ninst✝² : Countable ι\nf g : α → β\ninst✝¹ : TopologicalSpace β\ninst✝ : Zero β\nm✝ : MeasurableSpace α\nμ : Measure α\nhf_meas : StronglyMeasurable f\nt : Set α\nht : MeasurableSet t\nhft_zero : ∀ (x : α), x ∈ tᶜ → f x = 0\nhtμ this : SigmaFinite (Measure.restrict μ t)\nS : ℕ → Set α := spanningSets (Measure.restrict μ t)\nhS_meas : ∀ (n : ℕ), MeasurableSet (S n)\nf_approx : ℕ → α →ₛ β := StronglyMeasurable.approx hf_meas\nfs : ℕ → α →ₛ β := fun n => SimpleFunc.restrict (f_approx n) (S n ∩ t)\nh_fs_t_compl : ∀ (n : ℕ) (x : α), ¬x ∈ t → ↑(fs n) x = 0\nx : α\nhxt : x ∈ t\nh✝ : Tendsto (fun n => ↑(f_approx n) x) atTop (𝓝 (f x))\nh : ∀ (s : Set β), s ∈ 𝓝 (f x) → ∃ a, ∀ (b : ℕ), b ≥ a → ↑(f_approx b) x ∈ s\nn₁ : ℕ\nhn₁ : ∀ (m : ℕ), n₁ ≤ m → ↑(fs m) x = ↑(f_approx m) x\ns : Set β\nhs : s ∈ 𝓝 (f x)\nn₂ : ℕ\nhn₂ : ∀ (b : ℕ), b ≥ n₂ → ↑(f_approx b) x ∈ s\nm : ℕ\nhm : m ≥ max n₁ n₂\n⊢ ↑(fs m) x ∈ s",
"state_before": "case pos.intro.intro\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.51589\nι : Type ?u.51592\ninst✝² : Countable ι\nf g : α → β\ninst✝¹ : TopologicalSpace β\ninst✝ : Zero β\nm : MeasurableSpace α\nμ : Measure α\nhf_meas : StronglyMeasurable f\nt : Set α\nht : MeasurableSet t\nhft_zero : ∀ (x : α), x ∈ tᶜ → f x = 0\nhtμ this : SigmaFinite (Measure.restrict μ t)\nS : ℕ → Set α := spanningSets (Measure.restrict μ t)\nhS_meas : ∀ (n : ℕ), MeasurableSet (S n)\nf_approx : ℕ → α →ₛ β := StronglyMeasurable.approx hf_meas\nfs : ℕ → α →ₛ β := fun n => SimpleFunc.restrict (f_approx n) (S n ∩ t)\nh_fs_t_compl : ∀ (n : ℕ) (x : α), ¬x ∈ t → ↑(fs n) x = 0\nx : α\nhxt : x ∈ t\nh✝ : Tendsto (fun n => ↑(f_approx n) x) atTop (𝓝 (f x))\nh : ∀ (s : Set β), s ∈ 𝓝 (f x) → ∃ a, ∀ (b : ℕ), b ≥ a → ↑(f_approx b) x ∈ s\nn₁ : ℕ\nhn₁ : ∀ (m_1 : ℕ), n₁ ≤ m_1 → ↑(fs m_1) x = ↑(f_approx m_1) x\ns : Set β\nhs : s ∈ 𝓝 (f x)\nn₂ : ℕ\nhn₂ : ∀ (b : ℕ), b ≥ n₂ → ↑(f_approx b) x ∈ s\n⊢ ∃ a, ∀ (b : ℕ), b ≥ a → ↑(fs b) x ∈ s",
"tactic": "refine' ⟨max n₁ n₂, fun m hm => _⟩"
},
{
"state_after": "case pos.intro.intro\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.51589\nι : Type ?u.51592\ninst✝² : Countable ι\nf g : α → β\ninst✝¹ : TopologicalSpace β\ninst✝ : Zero β\nm✝ : MeasurableSpace α\nμ : Measure α\nhf_meas : StronglyMeasurable f\nt : Set α\nht : MeasurableSet t\nhft_zero : ∀ (x : α), x ∈ tᶜ → f x = 0\nhtμ this : SigmaFinite (Measure.restrict μ t)\nS : ℕ → Set α := spanningSets (Measure.restrict μ t)\nhS_meas : ∀ (n : ℕ), MeasurableSet (S n)\nf_approx : ℕ → α →ₛ β := StronglyMeasurable.approx hf_meas\nfs : ℕ → α →ₛ β := fun n => SimpleFunc.restrict (f_approx n) (S n ∩ t)\nh_fs_t_compl : ∀ (n : ℕ) (x : α), ¬x ∈ t → ↑(fs n) x = 0\nx : α\nhxt : x ∈ t\nh✝ : Tendsto (fun n => ↑(f_approx n) x) atTop (𝓝 (f x))\nh : ∀ (s : Set β), s ∈ 𝓝 (f x) → ∃ a, ∀ (b : ℕ), b ≥ a → ↑(f_approx b) x ∈ s\nn₁ : ℕ\nhn₁ : ∀ (m : ℕ), n₁ ≤ m → ↑(fs m) x = ↑(f_approx m) x\ns : Set β\nhs : s ∈ 𝓝 (f x)\nn₂ : ℕ\nhn₂ : ∀ (b : ℕ), b ≥ n₂ → ↑(f_approx b) x ∈ s\nm : ℕ\nhm : m ≥ max n₁ n₂\n⊢ ↑(f_approx m) x ∈ s",
"state_before": "case pos.intro.intro\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.51589\nι : Type ?u.51592\ninst✝² : Countable ι\nf g : α → β\ninst✝¹ : TopologicalSpace β\ninst✝ : Zero β\nm✝ : MeasurableSpace α\nμ : Measure α\nhf_meas : StronglyMeasurable f\nt : Set α\nht : MeasurableSet t\nhft_zero : ∀ (x : α), x ∈ tᶜ → f x = 0\nhtμ this : SigmaFinite (Measure.restrict μ t)\nS : ℕ → Set α := spanningSets (Measure.restrict μ t)\nhS_meas : ∀ (n : ℕ), MeasurableSet (S n)\nf_approx : ℕ → α →ₛ β := StronglyMeasurable.approx hf_meas\nfs : ℕ → α →ₛ β := fun n => SimpleFunc.restrict (f_approx n) (S n ∩ t)\nh_fs_t_compl : ∀ (n : ℕ) (x : α), ¬x ∈ t → ↑(fs n) x = 0\nx : α\nhxt : x ∈ t\nh✝ : Tendsto (fun n => ↑(f_approx n) x) atTop (𝓝 (f x))\nh : ∀ (s : Set β), s ∈ 𝓝 (f x) → ∃ a, ∀ (b : ℕ), b ≥ a → ↑(f_approx b) x ∈ s\nn₁ : ℕ\nhn₁ : ∀ (m : ℕ), n₁ ≤ m → ↑(fs m) x = ↑(f_approx m) x\ns : Set β\nhs : s ∈ 𝓝 (f x)\nn₂ : ℕ\nhn₂ : ∀ (b : ℕ), b ≥ n₂ → ↑(f_approx b) x ∈ s\nm : ℕ\nhm : m ≥ max n₁ n₂\n⊢ ↑(fs m) x ∈ s",
"tactic": "rw [hn₁ m ((le_max_left _ _).trans hm.le)]"
},
{
"state_after": "no goals",
"state_before": "case pos.intro.intro\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.51589\nι : Type ?u.51592\ninst✝² : Countable ι\nf g : α → β\ninst✝¹ : TopologicalSpace β\ninst✝ : Zero β\nm✝ : MeasurableSpace α\nμ : Measure α\nhf_meas : StronglyMeasurable f\nt : Set α\nht : MeasurableSet t\nhft_zero : ∀ (x : α), x ∈ tᶜ → f x = 0\nhtμ this : SigmaFinite (Measure.restrict μ t)\nS : ℕ → Set α := spanningSets (Measure.restrict μ t)\nhS_meas : ∀ (n : ℕ), MeasurableSet (S n)\nf_approx : ℕ → α →ₛ β := StronglyMeasurable.approx hf_meas\nfs : ℕ → α →ₛ β := fun n => SimpleFunc.restrict (f_approx n) (S n ∩ t)\nh_fs_t_compl : ∀ (n : ℕ) (x : α), ¬x ∈ t → ↑(fs n) x = 0\nx : α\nhxt : x ∈ t\nh✝ : Tendsto (fun n => ↑(f_approx n) x) atTop (𝓝 (f x))\nh : ∀ (s : Set β), s ∈ 𝓝 (f x) → ∃ a, ∀ (b : ℕ), b ≥ a → ↑(f_approx b) x ∈ s\nn₁ : ℕ\nhn₁ : ∀ (m : ℕ), n₁ ≤ m → ↑(fs m) x = ↑(f_approx m) x\ns : Set β\nhs : s ∈ 𝓝 (f x)\nn₂ : ℕ\nhn₂ : ∀ (b : ℕ), b ≥ n₂ → ↑(f_approx b) x ∈ s\nm : ℕ\nhm : m ≥ max n₁ n₂\n⊢ ↑(f_approx m) x ∈ s",
"tactic": "exact hn₂ m ((le_max_right _ _).trans hm.le)"
},
{
"state_after": "case neg\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.51589\nι : Type ?u.51592\ninst✝² : Countable ι\nf g : α → β\ninst✝¹ : TopologicalSpace β\ninst✝ : Zero β\nm : MeasurableSpace α\nμ : Measure α\nhf_meas : StronglyMeasurable f\nt : Set α\nht : MeasurableSet t\nhft_zero : ∀ (x : α), x ∈ tᶜ → f x = 0\nhtμ this : SigmaFinite (Measure.restrict μ t)\nS : ℕ → Set α := spanningSets (Measure.restrict μ t)\nhS_meas : ∀ (n : ℕ), MeasurableSet (S n)\nf_approx : ℕ → α →ₛ β := StronglyMeasurable.approx hf_meas\nfs : ℕ → α →ₛ β := fun n => SimpleFunc.restrict (f_approx n) (S n ∩ t)\nh_fs_t_compl : ∀ (n : ℕ) (x : α), ¬x ∈ t → ↑(fs n) x = 0\nx : α\nhxt : ¬x ∈ t\n⊢ Tendsto (fun n => 0) atTop (𝓝 0)",
"state_before": "case neg\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.51589\nι : Type ?u.51592\ninst✝² : Countable ι\nf g : α → β\ninst✝¹ : TopologicalSpace β\ninst✝ : Zero β\nm : MeasurableSpace α\nμ : Measure α\nhf_meas : StronglyMeasurable f\nt : Set α\nht : MeasurableSet t\nhft_zero : ∀ (x : α), x ∈ tᶜ → f x = 0\nhtμ this : SigmaFinite (Measure.restrict μ t)\nS : ℕ → Set α := spanningSets (Measure.restrict μ t)\nhS_meas : ∀ (n : ℕ), MeasurableSet (S n)\nf_approx : ℕ → α →ₛ β := StronglyMeasurable.approx hf_meas\nfs : ℕ → α →ₛ β := fun n => SimpleFunc.restrict (f_approx n) (S n ∩ t)\nh_fs_t_compl : ∀ (n : ℕ) (x : α), ¬x ∈ t → ↑(fs n) x = 0\nx : α\nhxt : ¬x ∈ t\n⊢ Tendsto (fun n => ↑(fs n) x) atTop (𝓝 (f x))",
"tactic": "rw [funext fun n => h_fs_t_compl n x hxt, hft_zero x hxt]"
},
{
"state_after": "no goals",
"state_before": "case neg\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.51589\nι : Type ?u.51592\ninst✝² : Countable ι\nf g : α → β\ninst✝¹ : TopologicalSpace β\ninst✝ : Zero β\nm : MeasurableSpace α\nμ : Measure α\nhf_meas : StronglyMeasurable f\nt : Set α\nht : MeasurableSet t\nhft_zero : ∀ (x : α), x ∈ tᶜ → f x = 0\nhtμ this : SigmaFinite (Measure.restrict μ t)\nS : ℕ → Set α := spanningSets (Measure.restrict μ t)\nhS_meas : ∀ (n : ℕ), MeasurableSet (S n)\nf_approx : ℕ → α →ₛ β := StronglyMeasurable.approx hf_meas\nfs : ℕ → α →ₛ β := fun n => SimpleFunc.restrict (f_approx n) (S n ∩ t)\nh_fs_t_compl : ∀ (n : ℕ) (x : α), ¬x ∈ t → ↑(fs n) x = 0\nx : α\nhxt : ¬x ∈ t\n⊢ Tendsto (fun n => 0) atTop (𝓝 0)",
"tactic": "exact tendsto_const_nhds"
},
{
"state_after": "case intro\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.51589\nι : Type ?u.51592\ninst✝² : Countable ι\nf g : α → β\ninst✝¹ : TopologicalSpace β\ninst✝ : Zero β\nm✝ : MeasurableSpace α\nμ : Measure α\nhf_meas : StronglyMeasurable f\nt : Set α\nht : MeasurableSet t\nhft_zero : ∀ (x : α), x ∈ tᶜ → f x = 0\nhtμ this : SigmaFinite (Measure.restrict μ t)\nS : ℕ → Set α := spanningSets (Measure.restrict μ t)\nhS_meas : ∀ (n : ℕ), MeasurableSet (S n)\nf_approx : ℕ → α →ₛ β := StronglyMeasurable.approx hf_meas\nfs : ℕ → α →ₛ β := fun n => SimpleFunc.restrict (f_approx n) (S n ∩ t)\nh_fs_t_compl : ∀ (n : ℕ) (x : α), ¬x ∈ t → ↑(fs n) x = 0\nx : α\nhxt : x ∈ t\nh : Tendsto (fun n => ↑(f_approx n) x) atTop (𝓝 (f x))\nn : ℕ\nhn : ∀ (m : ℕ), n ≤ m → x ∈ S m ∩ t\nm : ℕ\nhnm : n ≤ m\n⊢ ↑(fs m) x = ↑(f_approx m) x",
"state_before": "case intro\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.51589\nι : Type ?u.51592\ninst✝² : Countable ι\nf g : α → β\ninst✝¹ : TopologicalSpace β\ninst✝ : Zero β\nm : MeasurableSpace α\nμ : Measure α\nhf_meas : StronglyMeasurable f\nt : Set α\nht : MeasurableSet t\nhft_zero : ∀ (x : α), x ∈ tᶜ → f x = 0\nhtμ this : SigmaFinite (Measure.restrict μ t)\nS : ℕ → Set α := spanningSets (Measure.restrict μ t)\nhS_meas : ∀ (n : ℕ), MeasurableSet (S n)\nf_approx : ℕ → α →ₛ β := StronglyMeasurable.approx hf_meas\nfs : ℕ → α →ₛ β := fun n => SimpleFunc.restrict (f_approx n) (S n ∩ t)\nh_fs_t_compl : ∀ (n : ℕ) (x : α), ¬x ∈ t → ↑(fs n) x = 0\nx : α\nhxt : x ∈ t\nh : Tendsto (fun n => ↑(f_approx n) x) atTop (𝓝 (f x))\nn : ℕ\nhn : ∀ (m : ℕ), n ≤ m → x ∈ S m ∩ t\n⊢ ∃ n, ∀ (m_1 : ℕ), n ≤ m_1 → ↑(fs m_1) x = ↑(f_approx m_1) x",
"tactic": "refine' ⟨n, fun m hnm => _⟩"
},
{
"state_after": "no goals",
"state_before": "case intro\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.51589\nι : Type ?u.51592\ninst✝² : Countable ι\nf g : α → β\ninst✝¹ : TopologicalSpace β\ninst✝ : Zero β\nm✝ : MeasurableSpace α\nμ : Measure α\nhf_meas : StronglyMeasurable f\nt : Set α\nht : MeasurableSet t\nhft_zero : ∀ (x : α), x ∈ tᶜ → f x = 0\nhtμ this : SigmaFinite (Measure.restrict μ t)\nS : ℕ → Set α := spanningSets (Measure.restrict μ t)\nhS_meas : ∀ (n : ℕ), MeasurableSet (S n)\nf_approx : ℕ → α →ₛ β := StronglyMeasurable.approx hf_meas\nfs : ℕ → α →ₛ β := fun n => SimpleFunc.restrict (f_approx n) (S n ∩ t)\nh_fs_t_compl : ∀ (n : ℕ) (x : α), ¬x ∈ t → ↑(fs n) x = 0\nx : α\nhxt : x ∈ t\nh : Tendsto (fun n => ↑(f_approx n) x) atTop (𝓝 (f x))\nn : ℕ\nhn : ∀ (m : ℕ), n ≤ m → x ∈ S m ∩ t\nm : ℕ\nhnm : n ≤ m\n⊢ ↑(fs m) x = ↑(f_approx m) x",
"tactic": "simp_rw [SimpleFunc.restrict_apply _ ((hS_meas m).inter ht),\n Set.indicator_of_mem (hn m hnm)]"
},
{
"state_after": "case intro\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.51589\nι : Type ?u.51592\ninst✝² : Countable ι\nf g : α → β\ninst✝¹ : TopologicalSpace β\ninst✝ : Zero β\nm : MeasurableSpace α\nμ : Measure α\nhf_meas : StronglyMeasurable f\nt : Set α\nht : MeasurableSet t\nhft_zero : ∀ (x : α), x ∈ tᶜ → f x = 0\nhtμ this : SigmaFinite (Measure.restrict μ t)\nS : ℕ → Set α := spanningSets (Measure.restrict μ t)\nhS_meas : ∀ (n : ℕ), MeasurableSet (S n)\nf_approx : ℕ → α →ₛ β := StronglyMeasurable.approx hf_meas\nfs : ℕ → α →ₛ β := fun n => SimpleFunc.restrict (f_approx n) (S n ∩ t)\nh_fs_t_compl : ∀ (n : ℕ) (x : α), ¬x ∈ t → ↑(fs n) x = 0\nx : α\nhxt : x ∈ t\nh : Tendsto (fun n => ↑(f_approx n) x) atTop (𝓝 (f x))\nn : ℕ\nhn : ∀ (m : ℕ), n ≤ m → x ∈ S m\n⊢ ∃ n, ∀ (m : ℕ), n ≤ m → x ∈ S m ∩ t\n\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.51589\nι : Type ?u.51592\ninst✝² : Countable ι\nf g : α → β\ninst✝¹ : TopologicalSpace β\ninst✝ : Zero β\nm : MeasurableSpace α\nμ : Measure α\nhf_meas : StronglyMeasurable f\nt : Set α\nht : MeasurableSet t\nhft_zero : ∀ (x : α), x ∈ tᶜ → f x = 0\nhtμ this : SigmaFinite (Measure.restrict μ t)\nS : ℕ → Set α := spanningSets (Measure.restrict μ t)\nhS_meas : ∀ (n : ℕ), MeasurableSet (S n)\nf_approx : ℕ → α →ₛ β := StronglyMeasurable.approx hf_meas\nfs : ℕ → α →ₛ β := fun n => SimpleFunc.restrict (f_approx n) (S n ∩ t)\nh_fs_t_compl : ∀ (n : ℕ) (x : α), ¬x ∈ t → ↑(fs n) x = 0\nx : α\nhxt : x ∈ t\nh : Tendsto (fun n => ↑(f_approx n) x) atTop (𝓝 (f x))\n⊢ ∃ n, ∀ (m : ℕ), n ≤ m → x ∈ S m",
"state_before": "α : Type u_2\nβ : Type u_1\nγ : Type ?u.51589\nι : Type ?u.51592\ninst✝² : Countable ι\nf g : α → β\ninst✝¹ : TopologicalSpace β\ninst✝ : Zero β\nm : MeasurableSpace α\nμ : Measure α\nhf_meas : StronglyMeasurable f\nt : Set α\nht : MeasurableSet t\nhft_zero : ∀ (x : α), x ∈ tᶜ → f x = 0\nhtμ this : SigmaFinite (Measure.restrict μ t)\nS : ℕ → Set α := spanningSets (Measure.restrict μ t)\nhS_meas : ∀ (n : ℕ), MeasurableSet (S n)\nf_approx : ℕ → α →ₛ β := StronglyMeasurable.approx hf_meas\nfs : ℕ → α →ₛ β := fun n => SimpleFunc.restrict (f_approx n) (S n ∩ t)\nh_fs_t_compl : ∀ (n : ℕ) (x : α), ¬x ∈ t → ↑(fs n) x = 0\nx : α\nhxt : x ∈ t\nh : Tendsto (fun n => ↑(f_approx n) x) atTop (𝓝 (f x))\n⊢ ∃ n, ∀ (m : ℕ), n ≤ m → x ∈ S m ∩ t",
"tactic": "rsuffices ⟨n, hn⟩ : ∃ n, ∀ m, n ≤ m → x ∈ S m"
},
{
"state_after": "case intro\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.51589\nι : Type ?u.51592\ninst✝² : Countable ι\nf g : α → β\ninst✝¹ : TopologicalSpace β\ninst✝ : Zero β\nm : MeasurableSpace α\nμ : Measure α\nhf_meas : StronglyMeasurable f\nt : Set α\nht : MeasurableSet t\nhft_zero : ∀ (x : α), x ∈ tᶜ → f x = 0\nhtμ this : SigmaFinite (Measure.restrict μ t)\nS : ℕ → Set α := spanningSets (Measure.restrict μ t)\nhS_meas : ∀ (n : ℕ), MeasurableSet (S n)\nf_approx : ℕ → α →ₛ β := StronglyMeasurable.approx hf_meas\nfs : ℕ → α →ₛ β := fun n => SimpleFunc.restrict (f_approx n) (S n ∩ t)\nh_fs_t_compl : ∀ (n : ℕ) (x : α), ¬x ∈ t → ↑(fs n) x = 0\nx : α\nhxt : x ∈ t\nh : Tendsto (fun n => ↑(f_approx n) x) atTop (𝓝 (f x))\nn : ℕ\nhn : x ∈ S n\n⊢ ∃ n, ∀ (m : ℕ), n ≤ m → x ∈ S m\n\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.51589\nι : Type ?u.51592\ninst✝² : Countable ι\nf g : α → β\ninst✝¹ : TopologicalSpace β\ninst✝ : Zero β\nm : MeasurableSpace α\nμ : Measure α\nhf_meas : StronglyMeasurable f\nt : Set α\nht : MeasurableSet t\nhft_zero : ∀ (x : α), x ∈ tᶜ → f x = 0\nhtμ this : SigmaFinite (Measure.restrict μ t)\nS : ℕ → Set α := spanningSets (Measure.restrict μ t)\nhS_meas : ∀ (n : ℕ), MeasurableSet (S n)\nf_approx : ℕ → α →ₛ β := StronglyMeasurable.approx hf_meas\nfs : ℕ → α →ₛ β := fun n => SimpleFunc.restrict (f_approx n) (S n ∩ t)\nh_fs_t_compl : ∀ (n : ℕ) (x : α), ¬x ∈ t → ↑(fs n) x = 0\nx : α\nhxt : x ∈ t\nh : Tendsto (fun n => ↑(f_approx n) x) atTop (𝓝 (f x))\n⊢ ∃ n, x ∈ S n",
"state_before": "α : Type u_2\nβ : Type u_1\nγ : Type ?u.51589\nι : Type ?u.51592\ninst✝² : Countable ι\nf g : α → β\ninst✝¹ : TopologicalSpace β\ninst✝ : Zero β\nm : MeasurableSpace α\nμ : Measure α\nhf_meas : StronglyMeasurable f\nt : Set α\nht : MeasurableSet t\nhft_zero : ∀ (x : α), x ∈ tᶜ → f x = 0\nhtμ this : SigmaFinite (Measure.restrict μ t)\nS : ℕ → Set α := spanningSets (Measure.restrict μ t)\nhS_meas : ∀ (n : ℕ), MeasurableSet (S n)\nf_approx : ℕ → α →ₛ β := StronglyMeasurable.approx hf_meas\nfs : ℕ → α →ₛ β := fun n => SimpleFunc.restrict (f_approx n) (S n ∩ t)\nh_fs_t_compl : ∀ (n : ℕ) (x : α), ¬x ∈ t → ↑(fs n) x = 0\nx : α\nhxt : x ∈ t\nh : Tendsto (fun n => ↑(f_approx n) x) atTop (𝓝 (f x))\n⊢ ∃ n, ∀ (m : ℕ), n ≤ m → x ∈ S m",
"tactic": "rsuffices ⟨n, hn⟩ : ∃ n, x ∈ S n"
},
{
"state_after": "α : Type u_2\nβ : Type u_1\nγ : Type ?u.51589\nι : Type ?u.51592\ninst✝² : Countable ι\nf g : α → β\ninst✝¹ : TopologicalSpace β\ninst✝ : Zero β\nm : MeasurableSpace α\nμ : Measure α\nhf_meas : StronglyMeasurable f\nt : Set α\nht : MeasurableSet t\nhft_zero : ∀ (x : α), x ∈ tᶜ → f x = 0\nhtμ this : SigmaFinite (Measure.restrict μ t)\nS : ℕ → Set α := spanningSets (Measure.restrict μ t)\nhS_meas : ∀ (n : ℕ), MeasurableSet (S n)\nf_approx : ℕ → α →ₛ β := StronglyMeasurable.approx hf_meas\nfs : ℕ → α →ₛ β := fun n => SimpleFunc.restrict (f_approx n) (S n ∩ t)\nh_fs_t_compl : ∀ (n : ℕ) (x : α), ¬x ∈ t → ↑(fs n) x = 0\nx : α\nhxt : x ∈ t\nh : Tendsto (fun n => ↑(f_approx n) x) atTop (𝓝 (f x))\n⊢ x ∈ univ",
"state_before": "α : Type u_2\nβ : Type u_1\nγ : Type ?u.51589\nι : Type ?u.51592\ninst✝² : Countable ι\nf g : α → β\ninst✝¹ : TopologicalSpace β\ninst✝ : Zero β\nm : MeasurableSpace α\nμ : Measure α\nhf_meas : StronglyMeasurable f\nt : Set α\nht : MeasurableSet t\nhft_zero : ∀ (x : α), x ∈ tᶜ → f x = 0\nhtμ this : SigmaFinite (Measure.restrict μ t)\nS : ℕ → Set α := spanningSets (Measure.restrict μ t)\nhS_meas : ∀ (n : ℕ), MeasurableSet (S n)\nf_approx : ℕ → α →ₛ β := StronglyMeasurable.approx hf_meas\nfs : ℕ → α →ₛ β := fun n => SimpleFunc.restrict (f_approx n) (S n ∩ t)\nh_fs_t_compl : ∀ (n : ℕ) (x : α), ¬x ∈ t → ↑(fs n) x = 0\nx : α\nhxt : x ∈ t\nh : Tendsto (fun n => ↑(f_approx n) x) atTop (𝓝 (f x))\n⊢ ∃ n, x ∈ S n",
"tactic": "rw [← Set.mem_iUnion, iUnion_spanningSets (μ.restrict t)]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_2\nβ : Type u_1\nγ : Type ?u.51589\nι : Type ?u.51592\ninst✝² : Countable ι\nf g : α → β\ninst✝¹ : TopologicalSpace β\ninst✝ : Zero β\nm : MeasurableSpace α\nμ : Measure α\nhf_meas : StronglyMeasurable f\nt : Set α\nht : MeasurableSet t\nhft_zero : ∀ (x : α), x ∈ tᶜ → f x = 0\nhtμ this : SigmaFinite (Measure.restrict μ t)\nS : ℕ → Set α := spanningSets (Measure.restrict μ t)\nhS_meas : ∀ (n : ℕ), MeasurableSet (S n)\nf_approx : ℕ → α →ₛ β := StronglyMeasurable.approx hf_meas\nfs : ℕ → α →ₛ β := fun n => SimpleFunc.restrict (f_approx n) (S n ∩ t)\nh_fs_t_compl : ∀ (n : ℕ) (x : α), ¬x ∈ t → ↑(fs n) x = 0\nx : α\nhxt : x ∈ t\nh : Tendsto (fun n => ↑(f_approx n) x) atTop (𝓝 (f x))\n⊢ x ∈ univ",
"tactic": "trivial"
},
{
"state_after": "no goals",
"state_before": "case intro\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.51589\nι : Type ?u.51592\ninst✝² : Countable ι\nf g : α → β\ninst✝¹ : TopologicalSpace β\ninst✝ : Zero β\nm : MeasurableSpace α\nμ : Measure α\nhf_meas : StronglyMeasurable f\nt : Set α\nht : MeasurableSet t\nhft_zero : ∀ (x : α), x ∈ tᶜ → f x = 0\nhtμ this : SigmaFinite (Measure.restrict μ t)\nS : ℕ → Set α := spanningSets (Measure.restrict μ t)\nhS_meas : ∀ (n : ℕ), MeasurableSet (S n)\nf_approx : ℕ → α →ₛ β := StronglyMeasurable.approx hf_meas\nfs : ℕ → α →ₛ β := fun n => SimpleFunc.restrict (f_approx n) (S n ∩ t)\nh_fs_t_compl : ∀ (n : ℕ) (x : α), ¬x ∈ t → ↑(fs n) x = 0\nx : α\nhxt : x ∈ t\nh : Tendsto (fun n => ↑(f_approx n) x) atTop (𝓝 (f x))\nn : ℕ\nhn : ∀ (m : ℕ), n ≤ m → x ∈ S m\n⊢ ∃ n, ∀ (m : ℕ), n ≤ m → x ∈ S m ∩ t",
"tactic": "exact ⟨n, fun m hnm => Set.mem_inter (hn m hnm) hxt⟩"
},
{
"state_after": "no goals",
"state_before": "case intro\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.51589\nι : Type ?u.51592\ninst✝² : Countable ι\nf g : α → β\ninst✝¹ : TopologicalSpace β\ninst✝ : Zero β\nm : MeasurableSpace α\nμ : Measure α\nhf_meas : StronglyMeasurable f\nt : Set α\nht : MeasurableSet t\nhft_zero : ∀ (x : α), x ∈ tᶜ → f x = 0\nhtμ this : SigmaFinite (Measure.restrict μ t)\nS : ℕ → Set α := spanningSets (Measure.restrict μ t)\nhS_meas : ∀ (n : ℕ), MeasurableSet (S n)\nf_approx : ℕ → α →ₛ β := StronglyMeasurable.approx hf_meas\nfs : ℕ → α →ₛ β := fun n => SimpleFunc.restrict (f_approx n) (S n ∩ t)\nh_fs_t_compl : ∀ (n : ℕ) (x : α), ¬x ∈ t → ↑(fs n) x = 0\nx : α\nhxt : x ∈ t\nh : Tendsto (fun n => ↑(f_approx n) x) atTop (𝓝 (f x))\nn : ℕ\nhn : x ∈ S n\n⊢ ∃ n, ∀ (m : ℕ), n ≤ m → x ∈ S m",
"tactic": "exact ⟨n, fun m hnm => monotone_spanningSets (μ.restrict t) hnm hn⟩"
}
] |
[
344,
49
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
297,
1
] |
Mathlib/Algebra/Order/Pointwise.lean
|
csInf_inv
|
[
{
"state_after": "α : Type u_1\ninst✝³ : ConditionallyCompleteLattice α\ninst✝² : Group α\ninst✝¹ : CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1\ninst✝ : CovariantClass α α (swap fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1\ns t : Set α\nhs₀ : Set.Nonempty s\nhs₁ : BddAbove s\n⊢ sInf (Inv.inv '' s) = (sSup s)⁻¹",
"state_before": "α : Type u_1\ninst✝³ : ConditionallyCompleteLattice α\ninst✝² : Group α\ninst✝¹ : CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1\ninst✝ : CovariantClass α α (swap fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1\ns t : Set α\nhs₀ : Set.Nonempty s\nhs₁ : BddAbove s\n⊢ sInf s⁻¹ = (sSup s)⁻¹",
"tactic": "rw [← image_inv]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝³ : ConditionallyCompleteLattice α\ninst✝² : Group α\ninst✝¹ : CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1\ninst✝ : CovariantClass α α (swap fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1\ns t : Set α\nhs₀ : Set.Nonempty s\nhs₁ : BddAbove s\n⊢ sInf (Inv.inv '' s) = (sSup s)⁻¹",
"tactic": "exact ((OrderIso.inv α).map_csSup' hs₀ hs₁).symm"
}
] |
[
142,
51
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
140,
1
] |
Mathlib/GroupTheory/Subgroup/Basic.lean
|
MonoidHom.coe_range
|
[] |
[
2592,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2591,
1
] |
Mathlib/Geometry/Manifold/SmoothManifoldWithCorners.lean
|
LocalHomeomorph.extend_coe_symm
|
[] |
[
791,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
790,
1
] |
Mathlib/Init/Function.lean
|
Function.leftInverse_of_surjective_of_rightInverse
|
[
{
"state_after": "no goals",
"state_before": "α : Sort u₁\nβ : Sort u₂\nφ : Sort u₃\nδ : Sort u₄\nζ : Sort u₅\nf : α → β\ng : β → α\nsurjf : Surjective f\nrfg : RightInverse f g\ny : β\nx : α\nhx : f x = y\n⊢ f (g y) = y",
"tactic": "rw [← hx, rfg]"
}
] |
[
120,
20
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
116,
1
] |
Std/Data/Int/Lemmas.lean
|
Int.lt_succ
|
[] |
[
642,
55
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
642,
1
] |
Mathlib/Order/Filter/AtTopBot.lean
|
Filter.atTop_basis_Ioi
|
[] |
[
259,
61
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
256,
1
] |
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