file_path
stringlengths 11
79
| full_name
stringlengths 2
100
| traced_tactics
list | end
list | commit
stringclasses 4
values | url
stringclasses 4
values | start
list |
|---|---|---|---|---|---|---|
Mathlib/LinearAlgebra/Prod.lean
|
LinearMap.tailing_le_tunnel
|
[
{
"state_after": "R : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM₂ : Type w\nV₂ : Type w'\nM₃ : Type y\nV₃ : Type y'\nM₄ : Type z\nι : Type x\nM₅ : Type ?u.536871\nM₆ : Type ?u.536874\ninst✝⁴ : Ring R\nN : Type u_1\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : AddCommGroup N\ninst✝ : Module R N\nf : M × N →ₗ[R] M\ni : Injective ↑f\nn : ℕ\n⊢ Submodule.map (comp (comp (Submodule.subtype (tunnel' f i n).fst) ↑(LinearEquiv.symm (tunnel' f i n).snd)) f)\n (Submodule.snd R M N) ≤\n ↑OrderDual.ofDual (↑(tunnel f i) n)",
"state_before": "R : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM₂ : Type w\nV₂ : Type w'\nM₃ : Type y\nV₃ : Type y'\nM₄ : Type z\nι : Type x\nM₅ : Type ?u.536871\nM₆ : Type ?u.536874\ninst✝⁴ : Ring R\nN : Type u_1\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : AddCommGroup N\ninst✝ : Module R N\nf : M × N →ₗ[R] M\ni : Injective ↑f\nn : ℕ\n⊢ tailing f i n ≤ ↑OrderDual.ofDual (↑(tunnel f i) n)",
"tactic": "dsimp [tailing, tunnelAux]"
},
{
"state_after": "R : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM₂ : Type w\nV₂ : Type w'\nM₃ : Type y\nV₃ : Type y'\nM₄ : Type z\nι : Type x\nM₅ : Type ?u.536871\nM₆ : Type ?u.536874\ninst✝⁴ : Ring R\nN : Type u_1\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : AddCommGroup N\ninst✝ : Module R N\nf : M × N →ₗ[R] M\ni : Injective ↑f\nn : ℕ\n⊢ Submodule.map (Submodule.subtype (tunnel' f i n).fst)\n (Submodule.map (↑(LinearEquiv.symm (tunnel' f i n).snd)) (Submodule.map f (Submodule.snd R M N))) ≤\n ↑OrderDual.ofDual (↑(tunnel f i) n)",
"state_before": "R : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM₂ : Type w\nV₂ : Type w'\nM₃ : Type y\nV₃ : Type y'\nM₄ : Type z\nι : Type x\nM₅ : Type ?u.536871\nM₆ : Type ?u.536874\ninst✝⁴ : Ring R\nN : Type u_1\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : AddCommGroup N\ninst✝ : Module R N\nf : M × N →ₗ[R] M\ni : Injective ↑f\nn : ℕ\n⊢ Submodule.map (comp (comp (Submodule.subtype (tunnel' f i n).fst) ↑(LinearEquiv.symm (tunnel' f i n).snd)) f)\n (Submodule.snd R M N) ≤\n ↑OrderDual.ofDual (↑(tunnel f i) n)",
"tactic": "rw [Submodule.map_comp, Submodule.map_comp]"
},
{
"state_after": "no goals",
"state_before": "R : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM₂ : Type w\nV₂ : Type w'\nM₃ : Type y\nV₃ : Type y'\nM₄ : Type z\nι : Type x\nM₅ : Type ?u.536871\nM₆ : Type ?u.536874\ninst✝⁴ : Ring R\nN : Type u_1\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : AddCommGroup N\ninst✝ : Module R N\nf : M × N →ₗ[R] M\ni : Injective ↑f\nn : ℕ\n⊢ Submodule.map (Submodule.subtype (tunnel' f i n).fst)\n (Submodule.map (↑(LinearEquiv.symm (tunnel' f i n).snd)) (Submodule.map f (Submodule.snd R M N))) ≤\n ↑OrderDual.ofDual (↑(tunnel f i) n)",
"tactic": "apply Submodule.map_subtype_le"
}
] |
[
938,
33
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
934,
1
] |
Mathlib/Order/Filter/Basic.lean
|
Filter.eventually_const
|
[
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\nδ : Type ?u.140286\nι : Sort x\nf : Filter α\nt : NeBot f\np : Prop\nh : p\n⊢ (∀ᶠ (x : α) in f, p) ↔ p",
"tactic": "simp [h]"
},
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\nδ : Type ?u.140286\nι : Sort x\nf : Filter α\nt : NeBot f\np : Prop\nh : ¬p\n⊢ (∀ᶠ (x : α) in f, p) ↔ p",
"tactic": "simpa [h] using t.ne"
}
] |
[
1112,
71
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1111,
1
] |
Mathlib/Data/List/Sigma.lean
|
List.kerase_nil
|
[] |
[
400,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
399,
1
] |
Std/Data/List/Lemmas.lean
|
List.isSuffix.isInfix
|
[
{
"state_after": "no goals",
"state_before": "α✝ : Type u_1\nl₁ l₂ : List α✝\nx✝ : l₁ <:+ l₂\nt : List α✝\nh : t ++ l₁ = l₂\n⊢ t ++ l₁ ++ [] = l₂",
"tactic": "rw [h, append_nil]"
}
] |
[
1570,
98
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
1570,
1
] |
Mathlib/CategoryTheory/Monad/Algebra.lean
|
CategoryTheory.Comonad.coalgebra_iso_of_iso
|
[
{
"state_after": "C : Type u₁\ninst✝¹ : Category C\nG : Comonad C\nA B : Coalgebra G\nf : A ⟶ B\ninst✝ : IsIso f.f\n⊢ A.a ≫ G.map f.f ≫ G.map (inv f.f) = A.a",
"state_before": "C : Type u₁\ninst✝¹ : Category C\nG : Comonad C\nA B : Coalgebra G\nf : A ⟶ B\ninst✝ : IsIso f.f\n⊢ B.a ≫ G.map (inv f.f) = inv f.f ≫ A.a",
"tactic": "rw [IsIso.eq_inv_comp f.f, ← f.h_assoc]"
},
{
"state_after": "no goals",
"state_before": "C : Type u₁\ninst✝¹ : Category C\nG : Comonad C\nA B : Coalgebra G\nf : A ⟶ B\ninst✝ : IsIso f.f\n⊢ A.a ≫ G.map f.f ≫ G.map (inv f.f) = A.a",
"tactic": "simp"
},
{
"state_after": "no goals",
"state_before": "C : Type u₁\ninst✝¹ : Category C\nG : Comonad C\nA B : Coalgebra G\nf : A ⟶ B\ninst✝ : IsIso f.f\n⊢ f ≫ Coalgebra.Hom.mk (inv f.f) = 𝟙 A ∧ Coalgebra.Hom.mk (inv f.f) ≫ f = 𝟙 B",
"tactic": "aesop_cat"
}
] |
[
495,
21
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
490,
1
] |
Mathlib/Logic/Relation.lean
|
Relation.TransGen.trans_left
|
[
{
"state_after": "case refl\nα : Type u_1\nβ : Type ?u.18673\nγ : Type ?u.18676\nδ : Type ?u.18679\nr : α → α → Prop\na b c d : α\nhab : TransGen r a b\n⊢ TransGen r a b\n\ncase tail\nα : Type u_1\nβ : Type ?u.18673\nγ : Type ?u.18676\nδ : Type ?u.18679\nr : α → α → Prop\na b c d : α\nhab : TransGen r a b\nb✝ c✝ : α\na✝¹ : ReflTransGen r b b✝\na✝ : r b✝ c✝\na_ih✝ : TransGen r a b✝\n⊢ TransGen r a c✝",
"state_before": "α : Type u_1\nβ : Type ?u.18673\nγ : Type ?u.18676\nδ : Type ?u.18679\nr : α → α → Prop\na b c d : α\nhab : TransGen r a b\nhbc : ReflTransGen r b c\n⊢ TransGen r a c",
"tactic": "induction hbc"
},
{
"state_after": "case tail\nα : Type u_1\nβ : Type ?u.18673\nγ : Type ?u.18676\nδ : Type ?u.18679\nr : α → α → Prop\na b c d : α\nhab : TransGen r a b\nb✝ c✝ : α\na✝¹ : ReflTransGen r b b✝\na✝ : r b✝ c✝\na_ih✝ : TransGen r a b✝\n⊢ TransGen r a c✝",
"state_before": "case refl\nα : Type u_1\nβ : Type ?u.18673\nγ : Type ?u.18676\nδ : Type ?u.18679\nr : α → α → Prop\na b c d : α\nhab : TransGen r a b\n⊢ TransGen r a b\n\ncase tail\nα : Type u_1\nβ : Type ?u.18673\nγ : Type ?u.18676\nδ : Type ?u.18679\nr : α → α → Prop\na b c d : α\nhab : TransGen r a b\nb✝ c✝ : α\na✝¹ : ReflTransGen r b b✝\na✝ : r b✝ c✝\na_ih✝ : TransGen r a b✝\n⊢ TransGen r a c✝",
"tactic": "case refl => assumption"
},
{
"state_after": "no goals",
"state_before": "case tail\nα : Type u_1\nβ : Type ?u.18673\nγ : Type ?u.18676\nδ : Type ?u.18679\nr : α → α → Prop\na b c d : α\nhab : TransGen r a b\nb✝ c✝ : α\na✝¹ : ReflTransGen r b b✝\na✝ : r b✝ c✝\na_ih✝ : TransGen r a b✝\n⊢ TransGen r a c✝",
"tactic": "case tail c d _ hcd hac => exact hac.tail hcd"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.18673\nγ : Type ?u.18676\nδ : Type ?u.18679\nr : α → α → Prop\na b c d : α\nhab : TransGen r a b\n⊢ TransGen r a b",
"tactic": "assumption"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.18673\nγ : Type ?u.18676\nδ : Type ?u.18679\nr : α → α → Prop\na b c✝ d✝ : α\nhab : TransGen r a b\nc d : α\na✝ : ReflTransGen r b c\nhcd : r c d\nhac : TransGen r a c\n⊢ TransGen r a d",
"tactic": "exact hac.tail hcd"
}
] |
[
361,
48
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
358,
1
] |
Mathlib/Data/Polynomial/Laurent.lean
|
Polynomial.toLaurent_X
|
[
{
"state_after": "R : Type u_1\ninst✝ : Semiring R\nthis : X = ↑(monomial 1) 1\n⊢ ↑toLaurent X = T 1",
"state_before": "R : Type u_1\ninst✝ : Semiring R\n⊢ ↑toLaurent X = T 1",
"tactic": "have : (Polynomial.X : R[X]) = monomial 1 1 := by simp [← C_mul_X_pow_eq_monomial]"
},
{
"state_after": "no goals",
"state_before": "R : Type u_1\ninst✝ : Semiring R\nthis : X = ↑(monomial 1) 1\n⊢ ↑toLaurent X = T 1",
"tactic": "simp [this, Polynomial.toLaurent_C_mul_T]"
},
{
"state_after": "no goals",
"state_before": "R : Type u_1\ninst✝ : Semiring R\n⊢ X = ↑(monomial 1) 1",
"tactic": "simp [← C_mul_X_pow_eq_monomial]"
}
] |
[
229,
44
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
227,
1
] |
Std/Data/RBMap/WF.lean
|
Std.RBNode.Balanced.balRight
|
[
{
"state_after": "α✝ : Type u_1\nl : RBNode α✝\nv : α✝\nr : RBNode α✝\ncl : RBColor\nn : Nat\nhl : Balanced l cl (n + 1)\nhr : RedRed True r n\n⊢ RedRed (cl = red)\n (match r with\n | node red b y c => node red l v (node black b y c)\n | r =>\n match l with\n | node black a x b => balance1 (node red a x b) v r\n | node red a x (node black b y c) => node red (balance1 (setRed a) x b) y (node black c v r)\n | l => node red l v r)\n (n + 1)",
"state_before": "α✝ : Type u_1\nl : RBNode α✝\nv : α✝\nr : RBNode α✝\ncl : RBColor\nn : Nat\nhl : Balanced l cl (n + 1)\nhr : RedRed True r n\n⊢ RedRed (cl = red) (balRight l v r) (n + 1)",
"tactic": "unfold balRight"
},
{
"state_after": "case h_1\nα✝ : Type u_1\nl : RBNode α✝\nv : α✝\ncl : RBColor\nn : Nat\nhl : Balanced l cl (n + 1)\nl✝ a✝ : RBNode α✝\nx✝ : α✝\nb✝ : RBNode α✝\nhr : RedRed True (node red a✝ x✝ b✝) n\n⊢ RedRed (cl = red) (node red l v (node black a✝ x✝ b✝)) (n + 1)\n\ncase h_2\nα✝ : Type u_1\nl : RBNode α✝\nv : α✝\nr : RBNode α✝\ncl : RBColor\nn : Nat\nhl : Balanced l cl (n + 1)\nhr : RedRed True r n\nl✝ : RBNode α✝\nx✝ : ∀ (a : RBNode α✝) (x : α✝) (b : RBNode α✝), r = node red a x b → False\n⊢ RedRed (cl = red)\n (match l with\n | node black a x b => balance1 (node red a x b) v r\n | node red a x (node black b y c) => node red (balance1 (setRed a) x b) y (node black c v r)\n | l => node red l v r)\n (n + 1)",
"state_before": "α✝ : Type u_1\nl : RBNode α✝\nv : α✝\nr : RBNode α✝\ncl : RBColor\nn : Nat\nhl : Balanced l cl (n + 1)\nhr : RedRed True r n\n⊢ RedRed (cl = red)\n (match r with\n | node red b y c => node red l v (node black b y c)\n | r =>\n match l with\n | node black a x b => balance1 (node red a x b) v r\n | node red a x (node black b y c) => node red (balance1 (setRed a) x b) y (node black c v r)\n | l => node red l v r)\n (n + 1)",
"tactic": "split"
},
{
"state_after": "no goals",
"state_before": "case h_1\nα✝ : Type u_1\nl : RBNode α✝\nv : α✝\ncl : RBColor\nn : Nat\nhl : Balanced l cl (n + 1)\nl✝ a✝ : RBNode α✝\nx✝ : α✝\nb✝ : RBNode α✝\nhr : RedRed True (node red a✝ x✝ b✝) n\n⊢ RedRed (cl = red) (node red l v (node black a✝ x✝ b✝)) (n + 1)",
"tactic": "next b y c => exact\nlet ⟨cb, cc, hb, hc⟩ := hr.of_red\nmatch cl with\n| red => .redred rfl hl (.black hb hc)\n| black => .balanced (.red hl (.black hb hc))"
},
{
"state_after": "no goals",
"state_before": "α✝ : Type u_1\nl : RBNode α✝\nv : α✝\ncl : RBColor\nn : Nat\nhl : Balanced l cl (n + 1)\nl✝ b : RBNode α✝\ny : α✝\nc : RBNode α✝\nhr : RedRed True (node red b y c) n\n⊢ RedRed (cl = red) (node red l v (node black b y c)) (n + 1)",
"tactic": "exact\nlet ⟨cb, cc, hb, hc⟩ := hr.of_red\nmatch cl with\n| red => .redred rfl hl (.black hb hc)\n| black => .balanced (.red hl (.black hb hc))"
},
{
"state_after": "no goals",
"state_before": "case h_2\nα✝ : Type u_1\nl : RBNode α✝\nv : α✝\nr : RBNode α✝\ncl : RBColor\nn : Nat\nhl : Balanced l cl (n + 1)\nhr : RedRed True r n\nl✝ : RBNode α✝\nx✝ : ∀ (a : RBNode α✝) (x : α✝) (b : RBNode α✝), r = node red a x b → False\n⊢ RedRed (cl = red)\n (match l with\n | node black a x b => balance1 (node red a x b) v r\n | node red a x (node black b y c) => node red (balance1 (setRed a) x b) y (node black c v r)\n | l => node red l v r)\n (n + 1)",
"tactic": "next H => exact match hr with\n| .redred .. => nomatch H _ _ _ rfl\n| .balanced hr => match hl with\n | .black hb hc =>\n let ⟨c, h⟩ := RedRed.balance1 (.redred trivial hb hc) hr; .balanced h\n | .red (.black ha hb) (.black hc hd) =>\n let ⟨c, h⟩ := RedRed.balance1 (.redred trivial ha hb) hc; .redred rfl h (.black hd hr)"
},
{
"state_after": "no goals",
"state_before": "α✝ : Type u_1\nl : RBNode α✝\nv : α✝\nr : RBNode α✝\ncl : RBColor\nn : Nat\nhl : Balanced l cl (n + 1)\nhr : RedRed True r n\nl✝ : RBNode α✝\nH : ∀ (a : RBNode α✝) (x : α✝) (b : RBNode α✝), r = node red a x b → False\n⊢ RedRed (cl = red)\n (match l with\n | node black a x b => balance1 (node red a x b) v r\n | node red a x (node black b y c) => node red (balance1 (setRed a) x b) y (node black c v r)\n | l => node red l v r)\n (n + 1)",
"tactic": "exact match hr with\n| .redred .. => nomatch H _ _ _ rfl\n| .balanced hr => match hl with\n| .black hb hc =>\nlet ⟨c, h⟩ := RedRed.balance1 (.redred trivial hb hc) hr; .balanced h\n| .red (.black ha hb) (.black hc hd) =>\nlet ⟨c, h⟩ := RedRed.balance1 (.redred trivial ha hb) hc; .redred rfl h (.black hd hr)"
}
] |
[
301,
95
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
287,
11
] |
Mathlib/Data/Set/Intervals/Basic.lean
|
Set.Ico_subset_Ico_right
|
[] |
[
456,
26
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
455,
1
] |
Mathlib/RingTheory/Polynomial/Cyclotomic/Basic.lean
|
Polynomial.cyclotomic'_ne_zero
|
[] |
[
107,
34
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
106,
1
] |
Mathlib/GroupTheory/Perm/Basic.lean
|
Equiv.Perm.extendDomain_mul
|
[] |
[
321,
27
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
319,
1
] |
Mathlib/Data/Real/Sqrt.lean
|
Real.nat_sqrt_le_real_sqrt
|
[
{
"state_after": "a : ℕ\n⊢ ↑(Nat.sqrt a) ^ 2 ≤ ↑a",
"state_before": "a : ℕ\n⊢ ↑(Nat.sqrt a) ≤ sqrt ↑a",
"tactic": "rw [Real.le_sqrt (Nat.cast_nonneg _) (Nat.cast_nonneg _)]"
},
{
"state_after": "a : ℕ\n⊢ Nat.sqrt a ^ 2 ≤ a",
"state_before": "a : ℕ\n⊢ ↑(Nat.sqrt a) ^ 2 ≤ ↑a",
"tactic": "norm_cast"
},
{
"state_after": "no goals",
"state_before": "a : ℕ\n⊢ Nat.sqrt a ^ 2 ≤ a",
"tactic": "exact Nat.sqrt_le' a"
}
] |
[
457,
23
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
454,
1
] |
Mathlib/Geometry/Manifold/ChartedSpace.lean
|
groupoid_of_pregroupoid_le
|
[
{
"state_after": "H : Type u\nH' : Type ?u.32686\nM : Type ?u.32689\nM' : Type ?u.32692\nM'' : Type ?u.32695\ninst✝ : TopologicalSpace H\nPG₁ PG₂ : Pregroupoid H\nh : ∀ (f : H → H) (s : Set H), Pregroupoid.property PG₁ f s → Pregroupoid.property PG₂ f s\ne : LocalHomeomorph H H\nhe : e ∈ Pregroupoid.groupoid PG₁\n⊢ e ∈ Pregroupoid.groupoid PG₂",
"state_before": "H : Type u\nH' : Type ?u.32686\nM : Type ?u.32689\nM' : Type ?u.32692\nM'' : Type ?u.32695\ninst✝ : TopologicalSpace H\nPG₁ PG₂ : Pregroupoid H\nh : ∀ (f : H → H) (s : Set H), Pregroupoid.property PG₁ f s → Pregroupoid.property PG₂ f s\n⊢ Pregroupoid.groupoid PG₁ ≤ Pregroupoid.groupoid PG₂",
"tactic": "refine' StructureGroupoid.le_iff.2 fun e he ↦ _"
},
{
"state_after": "H : Type u\nH' : Type ?u.32686\nM : Type ?u.32689\nM' : Type ?u.32692\nM'' : Type ?u.32695\ninst✝ : TopologicalSpace H\nPG₁ PG₂ : Pregroupoid H\nh : ∀ (f : H → H) (s : Set H), Pregroupoid.property PG₁ f s → Pregroupoid.property PG₂ f s\ne : LocalHomeomorph H H\nhe : Pregroupoid.property PG₁ (↑e) e.source ∧ Pregroupoid.property PG₁ (↑(LocalHomeomorph.symm e)) e.target\n⊢ Pregroupoid.property PG₂ (↑e) e.source ∧ Pregroupoid.property PG₂ (↑(LocalHomeomorph.symm e)) e.target",
"state_before": "H : Type u\nH' : Type ?u.32686\nM : Type ?u.32689\nM' : Type ?u.32692\nM'' : Type ?u.32695\ninst✝ : TopologicalSpace H\nPG₁ PG₂ : Pregroupoid H\nh : ∀ (f : H → H) (s : Set H), Pregroupoid.property PG₁ f s → Pregroupoid.property PG₂ f s\ne : LocalHomeomorph H H\nhe : e ∈ Pregroupoid.groupoid PG₁\n⊢ e ∈ Pregroupoid.groupoid PG₂",
"tactic": "rw [mem_groupoid_of_pregroupoid] at he⊢"
},
{
"state_after": "no goals",
"state_before": "H : Type u\nH' : Type ?u.32686\nM : Type ?u.32689\nM' : Type ?u.32692\nM'' : Type ?u.32695\ninst✝ : TopologicalSpace H\nPG₁ PG₂ : Pregroupoid H\nh : ∀ (f : H → H) (s : Set H), Pregroupoid.property PG₁ f s → Pregroupoid.property PG₂ f s\ne : LocalHomeomorph H H\nhe : Pregroupoid.property PG₁ (↑e) e.source ∧ Pregroupoid.property PG₁ (↑(LocalHomeomorph.symm e)) e.target\n⊢ Pregroupoid.property PG₂ (↑e) e.source ∧ Pregroupoid.property PG₂ (↑(LocalHomeomorph.symm e)) e.target",
"tactic": "exact ⟨h _ _ he.1, h _ _ he.2⟩"
}
] |
[
356,
33
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
352,
1
] |
Mathlib/Data/Real/Hyperreal.lean
|
Hyperreal.st_mul
|
[] |
[
657,
27
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
653,
1
] |
Mathlib/RingTheory/WittVector/Defs.lean
|
WittVector.add_coeff_zero
|
[
{
"state_after": "no goals",
"state_before": "p : ℕ\nR : Type u_1\nhp : Fact (Nat.Prime p)\ninst✝ : CommRing R\nx y : 𝕎 R\n⊢ coeff (x + y) 0 = coeff x 0 + coeff y 0",
"tactic": "simp [add_coeff, peval]"
}
] |
[
390,
26
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
389,
1
] |
Mathlib/Algebra/Order/UpperLower.lean
|
IsUpperSet.inv
|
[] |
[
94,
95
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
94,
1
] |
Mathlib/Data/Multiset/Basic.lean
|
Multiset.sizeOf_lt_sizeOf_of_mem
|
[
{
"state_after": "case h\nα : Type u_1\nβ : Type ?u.137282\nγ : Type ?u.137285\ninst✝ : SizeOf α\nx : α\ns : Multiset α\nhx✝ : x ∈ s\nl : List α\nhx : x ∈ Quot.mk Setoid.r l\n⊢ SizeOf.sizeOf x < SizeOf.sizeOf (Quot.mk Setoid.r l)",
"state_before": "α : Type u_1\nβ : Type ?u.137282\nγ : Type ?u.137285\ninst✝ : SizeOf α\nx : α\ns : Multiset α\nhx : x ∈ s\n⊢ SizeOf.sizeOf x < SizeOf.sizeOf s",
"tactic": "induction' s using Quot.inductionOn with l a b"
},
{
"state_after": "no goals",
"state_before": "case h\nα : Type u_1\nβ : Type ?u.137282\nγ : Type ?u.137285\ninst✝ : SizeOf α\nx : α\ns : Multiset α\nhx✝ : x ∈ s\nl : List α\nhx : x ∈ Quot.mk Setoid.r l\n⊢ SizeOf.sizeOf x < SizeOf.sizeOf (Quot.mk Setoid.r l)",
"tactic": "exact List.sizeOf_lt_sizeOf_of_mem hx"
}
] |
[
1509,
40
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1506,
1
] |
Mathlib/Data/Multiset/Antidiagonal.lean
|
Multiset.antidiagonal_cons
|
[
{
"state_after": "α : Type u_1\nβ : Type ?u.9087\na : α\ns : Multiset α\nl : List α\n⊢ zip (powersetAux' l ++ List.map (cons a) (powersetAux' l))\n (reverse (List.map (cons a) (powersetAux' l)) ++ reverse (powersetAux' l)) ~\n List.map (fun x => Prod.map id (cons a) x) (zip (powersetAux' l) (reverse (powersetAux' l))) ++\n List.map (fun x => Prod.map (cons a) id x) (zip (powersetAux' l) (reverse (powersetAux' l)))",
"state_before": "α : Type u_1\nβ : Type ?u.9087\na : α\ns : Multiset α\nl : List α\n⊢ antidiagonal (a ::ₘ Quotient.mk (isSetoid α) l) =\n map (Prod.map id (cons a)) (antidiagonal (Quotient.mk (isSetoid α) l)) +\n map (Prod.map (cons a) id) (antidiagonal (Quotient.mk (isSetoid α) l))",
"tactic": "simp only [revzip, reverse_append, quot_mk_to_coe, coe_eq_coe, powersetAux'_cons, cons_coe,\n coe_map, antidiagonal_coe', coe_add]"
},
{
"state_after": "α : Type u_1\nβ : Type ?u.9087\na : α\ns : Multiset α\nl : List α\n⊢ zip (powersetAux' l) (reverse (List.map (cons a) (powersetAux' l))) ++\n zip (List.map (cons a) (powersetAux' l)) (reverse (powersetAux' l)) =\n zip (List.map id (powersetAux' l)) (List.map (cons a) (reverse (powersetAux' l))) ++\n zip (List.map (cons a) (powersetAux' l)) (List.map id (reverse (powersetAux' l)))\n\nα : Type u_1\nβ : Type ?u.9087\na : α\ns : Multiset α\nl : List α\n⊢ length (powersetAux' l) = length (reverse (List.map (cons a) (powersetAux' l)))",
"state_before": "α : Type u_1\nβ : Type ?u.9087\na : α\ns : Multiset α\nl : List α\n⊢ zip (powersetAux' l ++ List.map (cons a) (powersetAux' l))\n (reverse (List.map (cons a) (powersetAux' l)) ++ reverse (powersetAux' l)) ~\n List.map (fun x => Prod.map id (cons a) x) (zip (powersetAux' l) (reverse (powersetAux' l))) ++\n List.map (fun x => Prod.map (cons a) id x) (zip (powersetAux' l) (reverse (powersetAux' l)))",
"tactic": "rw [← zip_map, ← zip_map, zip_append, (_ : _ ++ _ = _)]"
},
{
"state_after": "case e_a.e_a\nα : Type u_1\nβ : Type ?u.9087\na : α\ns : Multiset α\nl : List α\n⊢ powersetAux' l = List.map id (powersetAux' l)\n\ncase e_a.e_a\nα : Type u_1\nβ : Type ?u.9087\na : α\ns : Multiset α\nl : List α\n⊢ reverse (List.map (cons a) (powersetAux' l)) = List.map (cons a) (reverse (powersetAux' l))\n\ncase e_a.e_a\nα : Type u_1\nβ : Type ?u.9087\na : α\ns : Multiset α\nl : List α\n⊢ reverse (powersetAux' l) = List.map id (reverse (powersetAux' l))",
"state_before": "α : Type u_1\nβ : Type ?u.9087\na : α\ns : Multiset α\nl : List α\n⊢ zip (powersetAux' l) (reverse (List.map (cons a) (powersetAux' l))) ++\n zip (List.map (cons a) (powersetAux' l)) (reverse (powersetAux' l)) =\n zip (List.map id (powersetAux' l)) (List.map (cons a) (reverse (powersetAux' l))) ++\n zip (List.map (cons a) (powersetAux' l)) (List.map id (reverse (powersetAux' l)))",
"tactic": "congr"
},
{
"state_after": "case e_a.e_a\nα : Type u_1\nβ : Type ?u.9087\na : α\ns : Multiset α\nl : List α\n⊢ reverse (List.map (cons a) (powersetAux' l)) = List.map (cons a) (reverse (powersetAux' l))\n\ncase e_a.e_a\nα : Type u_1\nβ : Type ?u.9087\na : α\ns : Multiset α\nl : List α\n⊢ reverse (powersetAux' l) = List.map id (reverse (powersetAux' l))",
"state_before": "case e_a.e_a\nα : Type u_1\nβ : Type ?u.9087\na : α\ns : Multiset α\nl : List α\n⊢ powersetAux' l = List.map id (powersetAux' l)\n\ncase e_a.e_a\nα : Type u_1\nβ : Type ?u.9087\na : α\ns : Multiset α\nl : List α\n⊢ reverse (List.map (cons a) (powersetAux' l)) = List.map (cons a) (reverse (powersetAux' l))\n\ncase e_a.e_a\nα : Type u_1\nβ : Type ?u.9087\na : α\ns : Multiset α\nl : List α\n⊢ reverse (powersetAux' l) = List.map id (reverse (powersetAux' l))",
"tactic": "simp"
},
{
"state_after": "case e_a.e_a\nα : Type u_1\nβ : Type ?u.9087\na : α\ns : Multiset α\nl : List α\n⊢ reverse (powersetAux' l) = List.map id (reverse (powersetAux' l))",
"state_before": "case e_a.e_a\nα : Type u_1\nβ : Type ?u.9087\na : α\ns : Multiset α\nl : List α\n⊢ reverse (List.map (cons a) (powersetAux' l)) = List.map (cons a) (reverse (powersetAux' l))\n\ncase e_a.e_a\nα : Type u_1\nβ : Type ?u.9087\na : α\ns : Multiset α\nl : List α\n⊢ reverse (powersetAux' l) = List.map id (reverse (powersetAux' l))",
"tactic": "rw [map_reverse]"
},
{
"state_after": "no goals",
"state_before": "case e_a.e_a\nα : Type u_1\nβ : Type ?u.9087\na : α\ns : Multiset α\nl : List α\n⊢ reverse (powersetAux' l) = List.map id (reverse (powersetAux' l))",
"tactic": "simp"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.9087\na : α\ns : Multiset α\nl : List α\n⊢ length (powersetAux' l) = length (reverse (List.map (cons a) (powersetAux' l)))",
"tactic": "simp"
}
] |
[
88,
11
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
80,
1
] |
Mathlib/Algebra/Category/Ring/Instances.lean
|
isLocalRingHom_of_iso
|
[
{
"state_after": "case h.e'_3\nR S : CommRingCat\nf : R ≅ S\na : ↑R\nha : IsUnit (↑f.hom a)\n⊢ a = ↑f.inv (↑f.hom a)",
"state_before": "R S : CommRingCat\nf : R ≅ S\na : ↑R\nha : IsUnit (↑f.hom a)\n⊢ IsUnit a",
"tactic": "convert f.inv.isUnit_map ha"
},
{
"state_after": "no goals",
"state_before": "case h.e'_3\nR S : CommRingCat\nf : R ≅ S\na : ↑R\nha : IsUnit (↑f.hom a)\n⊢ a = ↑f.inv (↑f.hom a)",
"tactic": "exact (RingHom.congr_fun f.hom_inv_id _).symm"
}
] |
[
58,
54
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
55,
1
] |
Mathlib/RingTheory/Congruence.lean
|
RingCon.coe_sub
|
[] |
[
263,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
262,
1
] |
Mathlib/SetTheory/ZFC/Basic.lean
|
Class.ext_iff
|
[] |
[
1470,
14
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1469,
1
] |
Mathlib/CategoryTheory/Generator.lean
|
CategoryTheory.isSeparator_def
|
[
{
"state_after": "C : Type u₁\ninst✝¹ : Category C\nD : Type u₂\ninst✝ : Category D\nX Y : C\nf g : X ⟶ Y\nH : C\nh : H ⟶ X\nhG : IsSeparator H\nhfg : ∀ (h : H ⟶ X), h ≫ f = h ≫ g\nhH : H ∈ {H}\n⊢ h ≫ f = h ≫ g",
"state_before": "C : Type u₁\ninst✝¹ : Category C\nD : Type u₂\ninst✝ : Category D\nG : C\nhG : IsSeparator G\nX Y : C\nf g : X ⟶ Y\nhfg : ∀ (h : G ⟶ X), h ≫ f = h ≫ g\nH : C\nhH : H ∈ {G}\nh : H ⟶ X\n⊢ h ≫ f = h ≫ g",
"tactic": "obtain rfl := Set.mem_singleton_iff.1 hH"
},
{
"state_after": "no goals",
"state_before": "C : Type u₁\ninst✝¹ : Category C\nD : Type u₂\ninst✝ : Category D\nX Y : C\nf g : X ⟶ Y\nH : C\nh : H ⟶ X\nhG : IsSeparator H\nhfg : ∀ (h : H ⟶ X), h ≫ f = h ≫ g\nhH : H ∈ {H}\n⊢ h ≫ f = h ≫ g",
"tactic": "exact hfg h"
}
] |
[
462,
73
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
456,
1
] |
Mathlib/Data/Matrix/Basic.lean
|
Matrix.neg_vecMul
|
[
{
"state_after": "case h\nl : Type ?u.907608\nm : Type u_1\nn : Type u_2\no : Type ?u.907617\nm' : o → Type ?u.907622\nn' : o → Type ?u.907627\nR : Type ?u.907630\nS : Type ?u.907633\nα : Type v\nβ : Type w\nγ : Type ?u.907640\ninst✝¹ : NonUnitalNonAssocRing α\ninst✝ : Fintype m\nv : m → α\nA : Matrix m n α\nx✝ : n\n⊢ vecMul (-v) A x✝ = (-vecMul v A) x✝",
"state_before": "l : Type ?u.907608\nm : Type u_1\nn : Type u_2\no : Type ?u.907617\nm' : o → Type ?u.907622\nn' : o → Type ?u.907627\nR : Type ?u.907630\nS : Type ?u.907633\nα : Type v\nβ : Type w\nγ : Type ?u.907640\ninst✝¹ : NonUnitalNonAssocRing α\ninst✝ : Fintype m\nv : m → α\nA : Matrix m n α\n⊢ vecMul (-v) A = -vecMul v A",
"tactic": "ext"
},
{
"state_after": "no goals",
"state_before": "case h\nl : Type ?u.907608\nm : Type u_1\nn : Type u_2\no : Type ?u.907617\nm' : o → Type ?u.907622\nn' : o → Type ?u.907627\nR : Type ?u.907630\nS : Type ?u.907633\nα : Type v\nβ : Type w\nγ : Type ?u.907640\ninst✝¹ : NonUnitalNonAssocRing α\ninst✝ : Fintype m\nv : m → α\nA : Matrix m n α\nx✝ : n\n⊢ vecMul (-v) A x✝ = (-vecMul v A) x✝",
"tactic": "apply neg_dotProduct"
}
] |
[
1898,
23
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1896,
1
] |
Mathlib/GroupTheory/Subsemigroup/Operations.lean
|
Subsemigroup.mem_map_iff_mem
|
[] |
[
263,
19
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
261,
1
] |
Mathlib/RingTheory/HahnSeries.lean
|
HahnSeries.single_injective
|
[
{
"state_after": "no goals",
"state_before": "Γ : Type u_2\nR : Type u_1\ninst✝¹ : PartialOrder Γ\ninst✝ : Zero R\na✝ b : Γ\nr✝ : R\na : Γ\nr s : R\nrs : ↑(single a) r = ↑(single a) s\n⊢ r = s",
"tactic": "rw [← single_coeff_same a r, ← single_coeff_same a s, rs]"
}
] |
[
192,
77
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
191,
1
] |
Mathlib/Analysis/SpecialFunctions/Arsinh.lean
|
Real.arsinh_surjective
|
[] |
[
137,
28
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
136,
1
] |
Mathlib/SetTheory/Ordinal/FixedPoint.lean
|
Ordinal.lt_nfpFamily
|
[] |
[
75,
16
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
74,
1
] |
Mathlib/Algebra/Periodic.lean
|
Function.Periodic.map_vadd_multiples
|
[
{
"state_after": "case mk.intro\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.136331\nf g : α → β\nc c₁ c₂ x✝ : α\ninst✝ : AddCommMonoid α\nhf : Periodic f c\nx : α\nm : ℕ\n⊢ f ({ val := (fun i => i • c) m, property := (_ : ∃ y, (fun i => i • c) y = (fun i => i • c) m) } +ᵥ x) = f x",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.136331\nf g : α → β\nc c₁ c₂ x✝ : α\ninst✝ : AddCommMonoid α\nhf : Periodic f c\na : { x // x ∈ AddSubmonoid.multiples c }\nx : α\n⊢ f (a +ᵥ x) = f x",
"tactic": "rcases a with ⟨_, m, rfl⟩"
},
{
"state_after": "no goals",
"state_before": "case mk.intro\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.136331\nf g : α → β\nc c₁ c₂ x✝ : α\ninst✝ : AddCommMonoid α\nhf : Periodic f c\nx : α\nm : ℕ\n⊢ f ({ val := (fun i => i • c) m, property := (_ : ∃ y, (fun i => i • c) y = (fun i => i • c) m) } +ᵥ x) = f x",
"tactic": "simp [AddSubmonoid.vadd_def, add_comm _ x, hf.nsmul m x]"
}
] |
[
341,
59
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
338,
1
] |
Mathlib/Analysis/Asymptotics/Asymptotics.lean
|
Asymptotics.isBigO_norm_right
|
[
{
"state_after": "α : Type u_1\nβ : Type ?u.119592\nE : Type u_2\nF : Type ?u.119598\nG : Type ?u.119601\nE' : Type ?u.119604\nF' : Type u_3\nG' : Type ?u.119610\nE'' : Type ?u.119613\nF'' : Type ?u.119616\nG'' : Type ?u.119619\nR : Type ?u.119622\nR' : Type ?u.119625\n𝕜 : Type ?u.119628\n𝕜' : Type ?u.119631\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''\nk'' : α → G''\nl l' : Filter α\nu v : α → ℝ\n⊢ (∃ c, IsBigOWith c l f fun x => ‖g' x‖) ↔ ∃ c, IsBigOWith c l f g'",
"state_before": "α : Type u_1\nβ : Type ?u.119592\nE : Type u_2\nF : Type ?u.119598\nG : Type ?u.119601\nE' : Type ?u.119604\nF' : Type u_3\nG' : Type ?u.119610\nE'' : Type ?u.119613\nF'' : Type ?u.119616\nG'' : Type ?u.119619\nR : Type ?u.119622\nR' : Type ?u.119625\n𝕜 : Type ?u.119628\n𝕜' : Type ?u.119631\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''\nk'' : α → G''\nl l' : Filter α\nu v : α → ℝ\n⊢ (f =O[l] fun x => ‖g' x‖) ↔ f =O[l] g'",
"tactic": "simp only [IsBigO_def]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.119592\nE : Type u_2\nF : Type ?u.119598\nG : Type ?u.119601\nE' : Type ?u.119604\nF' : Type u_3\nG' : Type ?u.119610\nE'' : Type ?u.119613\nF'' : Type ?u.119616\nG'' : Type ?u.119619\nR : Type ?u.119622\nR' : Type ?u.119625\n𝕜 : Type ?u.119628\n𝕜' : Type ?u.119631\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''\nk'' : α → G''\nl l' : Filter α\nu v : α → ℝ\n⊢ (∃ c, IsBigOWith c l f fun x => ‖g' x‖) ↔ ∃ c, IsBigOWith c l f g'",
"tactic": "exact exists_congr fun _ => isBigOWith_norm_right"
}
] |
[
716,
52
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
714,
1
] |
Mathlib/MeasureTheory/Function/ConvergenceInMeasure.lean
|
MeasureTheory.TendstoInMeasure.exists_seq_tendsto_ae
|
[
{
"state_after": "α : Type u_1\nι : Type ?u.20960\nE : Type u_2\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : MetricSpace E\nf : ℕ → α → E\ng : α → E\nhfg : TendstoInMeasure μ f atTop g\nh_lt_ε_real : ∀ (ε : ℝ), 0 < ε → ∃ k, 2 * 2⁻¹ ^ k < ε\n⊢ ∃ ns, StrictMono ns ∧ ∀ᵐ (x : α) ∂μ, Tendsto (fun i => f (ns i) x) atTop (𝓝 (g x))",
"state_before": "α : Type u_1\nι : Type ?u.20960\nE : Type u_2\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : MetricSpace E\nf : ℕ → α → E\ng : α → E\nhfg : TendstoInMeasure μ f atTop g\n⊢ ∃ ns, StrictMono ns ∧ ∀ᵐ (x : α) ∂μ, Tendsto (fun i => f (ns i) x) atTop (𝓝 (g x))",
"tactic": "have h_lt_ε_real : ∀ (ε : ℝ) (_ : 0 < ε), ∃ k : ℕ, 2 * (2 : ℝ)⁻¹ ^ k < ε := by\n intro ε hε\n obtain ⟨k, h_k⟩ : ∃ k : ℕ, (2 : ℝ)⁻¹ ^ k < ε := exists_pow_lt_of_lt_one hε (by norm_num)\n refine' ⟨k + 1, (le_of_eq _).trans_lt h_k⟩\n rw [pow_add]; ring"
},
{
"state_after": "α : Type u_1\nι : Type ?u.20960\nE : Type u_2\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : MetricSpace E\nf : ℕ → α → E\ng : α → E\nhfg : TendstoInMeasure μ f atTop g\nh_lt_ε_real : ∀ (ε : ℝ), 0 < ε → ∃ k, 2 * 2⁻¹ ^ k < ε\nns : ℕ → ℕ := ExistsSeqTendstoAe.seqTendstoAeSeq hfg\n⊢ ∃ ns, StrictMono ns ∧ ∀ᵐ (x : α) ∂μ, Tendsto (fun i => f (ns i) x) atTop (𝓝 (g x))",
"state_before": "α : Type u_1\nι : Type ?u.20960\nE : Type u_2\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : MetricSpace E\nf : ℕ → α → E\ng : α → E\nhfg : TendstoInMeasure μ f atTop g\nh_lt_ε_real : ∀ (ε : ℝ), 0 < ε → ∃ k, 2 * 2⁻¹ ^ k < ε\n⊢ ∃ ns, StrictMono ns ∧ ∀ᵐ (x : α) ∂μ, Tendsto (fun i => f (ns i) x) atTop (𝓝 (g x))",
"tactic": "set ns := ExistsSeqTendstoAe.seqTendstoAeSeq hfg"
},
{
"state_after": "α : Type u_1\nι : Type ?u.20960\nE : Type u_2\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : MetricSpace E\nf : ℕ → α → E\ng : α → E\nhfg : TendstoInMeasure μ f atTop g\nh_lt_ε_real : ∀ (ε : ℝ), 0 < ε → ∃ k, 2 * 2⁻¹ ^ k < ε\nns : ℕ → ℕ := ExistsSeqTendstoAe.seqTendstoAeSeq hfg\n⊢ StrictMono ns ∧ ∀ᵐ (x : α) ∂μ, Tendsto (fun i => f (ns i) x) atTop (𝓝 (g x))",
"state_before": "α : Type u_1\nι : Type ?u.20960\nE : Type u_2\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : MetricSpace E\nf : ℕ → α → E\ng : α → E\nhfg : TendstoInMeasure μ f atTop g\nh_lt_ε_real : ∀ (ε : ℝ), 0 < ε → ∃ k, 2 * 2⁻¹ ^ k < ε\nns : ℕ → ℕ := ExistsSeqTendstoAe.seqTendstoAeSeq hfg\n⊢ ∃ ns, StrictMono ns ∧ ∀ᵐ (x : α) ∂μ, Tendsto (fun i => f (ns i) x) atTop (𝓝 (g x))",
"tactic": "use ns"
},
{
"state_after": "α : Type u_1\nι : Type ?u.20960\nE : Type u_2\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : MetricSpace E\nf : ℕ → α → E\ng : α → E\nhfg : TendstoInMeasure μ f atTop g\nh_lt_ε_real : ∀ (ε : ℝ), 0 < ε → ∃ k, 2 * 2⁻¹ ^ k < ε\nns : ℕ → ℕ := ExistsSeqTendstoAe.seqTendstoAeSeq hfg\nS : ℕ → Set α := fun k => {x | 2⁻¹ ^ k ≤ dist (f (ns k) x) (g x)}\n⊢ StrictMono ns ∧ ∀ᵐ (x : α) ∂μ, Tendsto (fun i => f (ns i) x) atTop (𝓝 (g x))",
"state_before": "α : Type u_1\nι : Type ?u.20960\nE : Type u_2\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : MetricSpace E\nf : ℕ → α → E\ng : α → E\nhfg : TendstoInMeasure μ f atTop g\nh_lt_ε_real : ∀ (ε : ℝ), 0 < ε → ∃ k, 2 * 2⁻¹ ^ k < ε\nns : ℕ → ℕ := ExistsSeqTendstoAe.seqTendstoAeSeq hfg\n⊢ StrictMono ns ∧ ∀ᵐ (x : α) ∂μ, Tendsto (fun i => f (ns i) x) atTop (𝓝 (g x))",
"tactic": "let S := fun k => { x | (2 : ℝ)⁻¹ ^ k ≤ dist (f (ns k) x) (g x) }"
},
{
"state_after": "α : Type u_1\nι : Type ?u.20960\nE : Type u_2\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : MetricSpace E\nf : ℕ → α → E\ng : α → E\nhfg : TendstoInMeasure μ f atTop g\nh_lt_ε_real : ∀ (ε : ℝ), 0 < ε → ∃ k, 2 * 2⁻¹ ^ k < ε\nns : ℕ → ℕ := ExistsSeqTendstoAe.seqTendstoAeSeq hfg\nS : ℕ → Set α := fun k => {x | 2⁻¹ ^ k ≤ dist (f (ns k) x) (g x)}\nhμS_le : ∀ (k : ℕ), ↑↑μ (S k) ≤ 2⁻¹ ^ k\n⊢ StrictMono ns ∧ ∀ᵐ (x : α) ∂μ, Tendsto (fun i => f (ns i) x) atTop (𝓝 (g x))",
"state_before": "α : Type u_1\nι : Type ?u.20960\nE : Type u_2\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : MetricSpace E\nf : ℕ → α → E\ng : α → E\nhfg : TendstoInMeasure μ f atTop g\nh_lt_ε_real : ∀ (ε : ℝ), 0 < ε → ∃ k, 2 * 2⁻¹ ^ k < ε\nns : ℕ → ℕ := ExistsSeqTendstoAe.seqTendstoAeSeq hfg\nS : ℕ → Set α := fun k => {x | 2⁻¹ ^ k ≤ dist (f (ns k) x) (g x)}\n⊢ StrictMono ns ∧ ∀ᵐ (x : α) ∂μ, Tendsto (fun i => f (ns i) x) atTop (𝓝 (g x))",
"tactic": "have hμS_le : ∀ k, μ (S k) ≤ (2 : ℝ≥0∞)⁻¹ ^ k := by\n intro k\n have := ExistsSeqTendstoAe.seqTendstoAeSeq_spec hfg k (ns k) le_rfl\n convert this"
},
{
"state_after": "α : Type u_1\nι : Type ?u.20960\nE : Type u_2\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : MetricSpace E\nf : ℕ → α → E\ng : α → E\nhfg : TendstoInMeasure μ f atTop g\nh_lt_ε_real : ∀ (ε : ℝ), 0 < ε → ∃ k, 2 * 2⁻¹ ^ k < ε\nns : ℕ → ℕ := ExistsSeqTendstoAe.seqTendstoAeSeq hfg\nS : ℕ → Set α := fun k => {x | 2⁻¹ ^ k ≤ dist (f (ns k) x) (g x)}\nhμS_le : ∀ (k : ℕ), ↑↑μ (S k) ≤ 2⁻¹ ^ k\ns : Set α := limsup S atTop\nhs : s = limsup S atTop\n⊢ StrictMono ns ∧ ∀ᵐ (x : α) ∂μ, Tendsto (fun i => f (ns i) x) atTop (𝓝 (g x))",
"state_before": "α : Type u_1\nι : Type ?u.20960\nE : Type u_2\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : MetricSpace E\nf : ℕ → α → E\ng : α → E\nhfg : TendstoInMeasure μ f atTop g\nh_lt_ε_real : ∀ (ε : ℝ), 0 < ε → ∃ k, 2 * 2⁻¹ ^ k < ε\nns : ℕ → ℕ := ExistsSeqTendstoAe.seqTendstoAeSeq hfg\nS : ℕ → Set α := fun k => {x | 2⁻¹ ^ k ≤ dist (f (ns k) x) (g x)}\nhμS_le : ∀ (k : ℕ), ↑↑μ (S k) ≤ 2⁻¹ ^ k\n⊢ StrictMono ns ∧ ∀ᵐ (x : α) ∂μ, Tendsto (fun i => f (ns i) x) atTop (𝓝 (g x))",
"tactic": "set s := Filter.atTop.limsup S with hs"
},
{
"state_after": "α : Type u_1\nι : Type ?u.20960\nE : Type u_2\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : MetricSpace E\nf : ℕ → α → E\ng : α → E\nhfg : TendstoInMeasure μ f atTop g\nh_lt_ε_real : ∀ (ε : ℝ), 0 < ε → ∃ k, 2 * 2⁻¹ ^ k < ε\nns : ℕ → ℕ := ExistsSeqTendstoAe.seqTendstoAeSeq hfg\nS : ℕ → Set α := fun k => {x | 2⁻¹ ^ k ≤ dist (f (ns k) x) (g x)}\nhμS_le : ∀ (k : ℕ), ↑↑μ (S k) ≤ 2⁻¹ ^ k\ns : Set α := limsup S atTop\nhs : s = limsup S atTop\nhμs : ↑↑μ s = 0\n⊢ StrictMono ns ∧ ∀ᵐ (x : α) ∂μ, Tendsto (fun i => f (ns i) x) atTop (𝓝 (g x))",
"state_before": "α : Type u_1\nι : Type ?u.20960\nE : Type u_2\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : MetricSpace E\nf : ℕ → α → E\ng : α → E\nhfg : TendstoInMeasure μ f atTop g\nh_lt_ε_real : ∀ (ε : ℝ), 0 < ε → ∃ k, 2 * 2⁻¹ ^ k < ε\nns : ℕ → ℕ := ExistsSeqTendstoAe.seqTendstoAeSeq hfg\nS : ℕ → Set α := fun k => {x | 2⁻¹ ^ k ≤ dist (f (ns k) x) (g x)}\nhμS_le : ∀ (k : ℕ), ↑↑μ (S k) ≤ 2⁻¹ ^ k\ns : Set α := limsup S atTop\nhs : s = limsup S atTop\n⊢ StrictMono ns ∧ ∀ᵐ (x : α) ∂μ, Tendsto (fun i => f (ns i) x) atTop (𝓝 (g x))",
"tactic": "have hμs : μ s = 0 := by\n refine' measure_limsup_eq_zero (ne_of_lt <| lt_of_le_of_lt (ENNReal.tsum_le_tsum hμS_le) _)\n simp only [ENNReal.tsum_geometric, ENNReal.one_sub_inv_two, inv_inv]"
},
{
"state_after": "α : Type u_1\nι : Type ?u.20960\nE : Type u_2\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : MetricSpace E\nf : ℕ → α → E\ng : α → E\nhfg : TendstoInMeasure μ f atTop g\nh_lt_ε_real : ∀ (ε : ℝ), 0 < ε → ∃ k, 2 * 2⁻¹ ^ k < ε\nns : ℕ → ℕ := ExistsSeqTendstoAe.seqTendstoAeSeq hfg\nS : ℕ → Set α := fun k => {x | 2⁻¹ ^ k ≤ dist (f (ns k) x) (g x)}\nhμS_le : ∀ (k : ℕ), ↑↑μ (S k) ≤ 2⁻¹ ^ k\ns : Set α := limsup S atTop\nhs : s = limsup S atTop\nhμs : ↑↑μ s = 0\nh_tendsto : ∀ (x : α), x ∈ sᶜ → Tendsto (fun i => f (ns i) x) atTop (𝓝 (g x))\n⊢ StrictMono ns ∧ ∀ᵐ (x : α) ∂μ, Tendsto (fun i => f (ns i) x) atTop (𝓝 (g x))",
"state_before": "α : Type u_1\nι : Type ?u.20960\nE : Type u_2\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : MetricSpace E\nf : ℕ → α → E\ng : α → E\nhfg : TendstoInMeasure μ f atTop g\nh_lt_ε_real : ∀ (ε : ℝ), 0 < ε → ∃ k, 2 * 2⁻¹ ^ k < ε\nns : ℕ → ℕ := ExistsSeqTendstoAe.seqTendstoAeSeq hfg\nS : ℕ → Set α := fun k => {x | 2⁻¹ ^ k ≤ dist (f (ns k) x) (g x)}\nhμS_le : ∀ (k : ℕ), ↑↑μ (S k) ≤ 2⁻¹ ^ k\ns : Set α := limsup S atTop\nhs : s = limsup S atTop\nhμs : ↑↑μ s = 0\n⊢ StrictMono ns ∧ ∀ᵐ (x : α) ∂μ, Tendsto (fun i => f (ns i) x) atTop (𝓝 (g x))",
"tactic": "have h_tendsto : ∀ x ∈ sᶜ, Tendsto (fun i => f (ns i) x) atTop (𝓝 (g x)) := by\n refine' fun x hx => Metric.tendsto_atTop.mpr fun ε hε => _\n rw [hs, limsup_eq_iInf_iSup_of_nat] at hx\n simp only [Set.iSup_eq_iUnion, Set.iInf_eq_iInter, Set.compl_iInter, Set.compl_iUnion,\n Set.mem_iUnion, Set.mem_iInter, Set.mem_compl_iff, Set.mem_setOf_eq, not_le] at hx\n obtain ⟨N, hNx⟩ := hx\n obtain ⟨k, hk_lt_ε⟩ := h_lt_ε_real ε hε\n refine' ⟨max N (k - 1), fun n hn_ge => lt_of_le_of_lt _ hk_lt_ε⟩\n specialize hNx n ((le_max_left _ _).trans hn_ge)\n have h_inv_n_le_k : (2 : ℝ)⁻¹ ^ n ≤ 2 * (2 : ℝ)⁻¹ ^ k := by\n rw [mul_comm, ← inv_mul_le_iff' (zero_lt_two' ℝ)]\n conv_lhs =>\n congr\n rw [← pow_one (2 : ℝ)⁻¹]\n rw [← pow_add, add_comm]\n exact pow_le_pow_of_le_one (one_div (2 : ℝ) ▸ one_half_pos.le) (inv_le_one one_le_two)\n ((le_tsub_add.trans (add_le_add_right (le_max_right _ _) 1)).trans\n (add_le_add_right hn_ge 1))\n exact le_trans hNx.le h_inv_n_le_k"
},
{
"state_after": "α : Type u_1\nι : Type ?u.20960\nE : Type u_2\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : MetricSpace E\nf : ℕ → α → E\ng : α → E\nhfg : TendstoInMeasure μ f atTop g\nh_lt_ε_real : ∀ (ε : ℝ), 0 < ε → ∃ k, 2 * 2⁻¹ ^ k < ε\nns : ℕ → ℕ := ExistsSeqTendstoAe.seqTendstoAeSeq hfg\nS : ℕ → Set α := fun k => {x | 2⁻¹ ^ k ≤ dist (f (ns k) x) (g x)}\nhμS_le : ∀ (k : ℕ), ↑↑μ (S k) ≤ 2⁻¹ ^ k\ns : Set α := limsup S atTop\nhs : s = limsup S atTop\nhμs : ↑↑μ s = 0\nh_tendsto : ∀ (x : α), x ∈ sᶜ → Tendsto (fun i => f (ns i) x) atTop (𝓝 (g x))\n⊢ StrictMono ns ∧ ↑↑μ {a | ¬Tendsto (fun i => f (ns i) a) atTop (𝓝 (g a))} = 0",
"state_before": "α : Type u_1\nι : Type ?u.20960\nE : Type u_2\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : MetricSpace E\nf : ℕ → α → E\ng : α → E\nhfg : TendstoInMeasure μ f atTop g\nh_lt_ε_real : ∀ (ε : ℝ), 0 < ε → ∃ k, 2 * 2⁻¹ ^ k < ε\nns : ℕ → ℕ := ExistsSeqTendstoAe.seqTendstoAeSeq hfg\nS : ℕ → Set α := fun k => {x | 2⁻¹ ^ k ≤ dist (f (ns k) x) (g x)}\nhμS_le : ∀ (k : ℕ), ↑↑μ (S k) ≤ 2⁻¹ ^ k\ns : Set α := limsup S atTop\nhs : s = limsup S atTop\nhμs : ↑↑μ s = 0\nh_tendsto : ∀ (x : α), x ∈ sᶜ → Tendsto (fun i => f (ns i) x) atTop (𝓝 (g x))\n⊢ StrictMono ns ∧ ∀ᵐ (x : α) ∂μ, Tendsto (fun i => f (ns i) x) atTop (𝓝 (g x))",
"tactic": "rw [ae_iff]"
},
{
"state_after": "α : Type u_1\nι : Type ?u.20960\nE : Type u_2\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : MetricSpace E\nf : ℕ → α → E\ng : α → E\nhfg : TendstoInMeasure μ f atTop g\nh_lt_ε_real : ∀ (ε : ℝ), 0 < ε → ∃ k, 2 * 2⁻¹ ^ k < ε\nns : ℕ → ℕ := ExistsSeqTendstoAe.seqTendstoAeSeq hfg\nS : ℕ → Set α := fun k => {x | 2⁻¹ ^ k ≤ dist (f (ns k) x) (g x)}\nhμS_le : ∀ (k : ℕ), ↑↑μ (S k) ≤ 2⁻¹ ^ k\ns : Set α := limsup S atTop\nhs : s = limsup S atTop\nhμs : ↑↑μ s = 0\nh_tendsto : ∀ (x : α), x ∈ sᶜ → Tendsto (fun i => f (ns i) x) atTop (𝓝 (g x))\nx : α\n⊢ x ∈ {a | ¬Tendsto (fun i => f (ns i) a) atTop (𝓝 (g a))} → x ∈ s",
"state_before": "α : Type u_1\nι : Type ?u.20960\nE : Type u_2\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : MetricSpace E\nf : ℕ → α → E\ng : α → E\nhfg : TendstoInMeasure μ f atTop g\nh_lt_ε_real : ∀ (ε : ℝ), 0 < ε → ∃ k, 2 * 2⁻¹ ^ k < ε\nns : ℕ → ℕ := ExistsSeqTendstoAe.seqTendstoAeSeq hfg\nS : ℕ → Set α := fun k => {x | 2⁻¹ ^ k ≤ dist (f (ns k) x) (g x)}\nhμS_le : ∀ (k : ℕ), ↑↑μ (S k) ≤ 2⁻¹ ^ k\ns : Set α := limsup S atTop\nhs : s = limsup S atTop\nhμs : ↑↑μ s = 0\nh_tendsto : ∀ (x : α), x ∈ sᶜ → Tendsto (fun i => f (ns i) x) atTop (𝓝 (g x))\n⊢ StrictMono ns ∧ ↑↑μ {a | ¬Tendsto (fun i => f (ns i) a) atTop (𝓝 (g a))} = 0",
"tactic": "refine' ⟨ExistsSeqTendstoAe.seqTendstoAeSeq_strictMono hfg, measure_mono_null (fun x => _) hμs⟩"
},
{
"state_after": "α : Type u_1\nι : Type ?u.20960\nE : Type u_2\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : MetricSpace E\nf : ℕ → α → E\ng : α → E\nhfg : TendstoInMeasure μ f atTop g\nh_lt_ε_real : ∀ (ε : ℝ), 0 < ε → ∃ k, 2 * 2⁻¹ ^ k < ε\nns : ℕ → ℕ := ExistsSeqTendstoAe.seqTendstoAeSeq hfg\nS : ℕ → Set α := fun k => {x | 2⁻¹ ^ k ≤ dist (f (ns k) x) (g x)}\nhμS_le : ∀ (k : ℕ), ↑↑μ (S k) ≤ 2⁻¹ ^ k\ns : Set α := limsup S atTop\nhs : s = limsup S atTop\nhμs : ↑↑μ s = 0\nh_tendsto : ∀ (x : α), x ∈ sᶜ → Tendsto (fun i => f (ns i) x) atTop (𝓝 (g x))\nx : α\n⊢ ¬x ∈ s → Tendsto (fun i => f (ns i) x) atTop (𝓝 (g x))",
"state_before": "α : Type u_1\nι : Type ?u.20960\nE : Type u_2\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : MetricSpace E\nf : ℕ → α → E\ng : α → E\nhfg : TendstoInMeasure μ f atTop g\nh_lt_ε_real : ∀ (ε : ℝ), 0 < ε → ∃ k, 2 * 2⁻¹ ^ k < ε\nns : ℕ → ℕ := ExistsSeqTendstoAe.seqTendstoAeSeq hfg\nS : ℕ → Set α := fun k => {x | 2⁻¹ ^ k ≤ dist (f (ns k) x) (g x)}\nhμS_le : ∀ (k : ℕ), ↑↑μ (S k) ≤ 2⁻¹ ^ k\ns : Set α := limsup S atTop\nhs : s = limsup S atTop\nhμs : ↑↑μ s = 0\nh_tendsto : ∀ (x : α), x ∈ sᶜ → Tendsto (fun i => f (ns i) x) atTop (𝓝 (g x))\nx : α\n⊢ x ∈ {a | ¬Tendsto (fun i => f (ns i) a) atTop (𝓝 (g a))} → x ∈ s",
"tactic": "rw [Set.mem_setOf_eq, ← @Classical.not_not (x ∈ s), not_imp_not]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nι : Type ?u.20960\nE : Type u_2\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : MetricSpace E\nf : ℕ → α → E\ng : α → E\nhfg : TendstoInMeasure μ f atTop g\nh_lt_ε_real : ∀ (ε : ℝ), 0 < ε → ∃ k, 2 * 2⁻¹ ^ k < ε\nns : ℕ → ℕ := ExistsSeqTendstoAe.seqTendstoAeSeq hfg\nS : ℕ → Set α := fun k => {x | 2⁻¹ ^ k ≤ dist (f (ns k) x) (g x)}\nhμS_le : ∀ (k : ℕ), ↑↑μ (S k) ≤ 2⁻¹ ^ k\ns : Set α := limsup S atTop\nhs : s = limsup S atTop\nhμs : ↑↑μ s = 0\nh_tendsto : ∀ (x : α), x ∈ sᶜ → Tendsto (fun i => f (ns i) x) atTop (𝓝 (g x))\nx : α\n⊢ ¬x ∈ s → Tendsto (fun i => f (ns i) x) atTop (𝓝 (g x))",
"tactic": "exact h_tendsto x"
},
{
"state_after": "α : Type u_1\nι : Type ?u.20960\nE : Type u_2\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : MetricSpace E\nf : ℕ → α → E\ng : α → E\nhfg : TendstoInMeasure μ f atTop g\nε : ℝ\nhε : 0 < ε\n⊢ ∃ k, 2 * 2⁻¹ ^ k < ε",
"state_before": "α : Type u_1\nι : Type ?u.20960\nE : Type u_2\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : MetricSpace E\nf : ℕ → α → E\ng : α → E\nhfg : TendstoInMeasure μ f atTop g\n⊢ ∀ (ε : ℝ), 0 < ε → ∃ k, 2 * 2⁻¹ ^ k < ε",
"tactic": "intro ε hε"
},
{
"state_after": "case intro\nα : Type u_1\nι : Type ?u.20960\nE : Type u_2\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : MetricSpace E\nf : ℕ → α → E\ng : α → E\nhfg : TendstoInMeasure μ f atTop g\nε : ℝ\nhε : 0 < ε\nk : ℕ\nh_k : 2⁻¹ ^ k < ε\n⊢ ∃ k, 2 * 2⁻¹ ^ k < ε",
"state_before": "α : Type u_1\nι : Type ?u.20960\nE : Type u_2\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : MetricSpace E\nf : ℕ → α → E\ng : α → E\nhfg : TendstoInMeasure μ f atTop g\nε : ℝ\nhε : 0 < ε\n⊢ ∃ k, 2 * 2⁻¹ ^ k < ε",
"tactic": "obtain ⟨k, h_k⟩ : ∃ k : ℕ, (2 : ℝ)⁻¹ ^ k < ε := exists_pow_lt_of_lt_one hε (by norm_num)"
},
{
"state_after": "case intro\nα : Type u_1\nι : Type ?u.20960\nE : Type u_2\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : MetricSpace E\nf : ℕ → α → E\ng : α → E\nhfg : TendstoInMeasure μ f atTop g\nε : ℝ\nhε : 0 < ε\nk : ℕ\nh_k : 2⁻¹ ^ k < ε\n⊢ 2 * 2⁻¹ ^ (k + 1) = 2⁻¹ ^ k",
"state_before": "case intro\nα : Type u_1\nι : Type ?u.20960\nE : Type u_2\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : MetricSpace E\nf : ℕ → α → E\ng : α → E\nhfg : TendstoInMeasure μ f atTop g\nε : ℝ\nhε : 0 < ε\nk : ℕ\nh_k : 2⁻¹ ^ k < ε\n⊢ ∃ k, 2 * 2⁻¹ ^ k < ε",
"tactic": "refine' ⟨k + 1, (le_of_eq _).trans_lt h_k⟩"
},
{
"state_after": "case intro\nα : Type u_1\nι : Type ?u.20960\nE : Type u_2\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : MetricSpace E\nf : ℕ → α → E\ng : α → E\nhfg : TendstoInMeasure μ f atTop g\nε : ℝ\nhε : 0 < ε\nk : ℕ\nh_k : 2⁻¹ ^ k < ε\n⊢ 2 * (2⁻¹ ^ k * 2⁻¹ ^ 1) = 2⁻¹ ^ k",
"state_before": "case intro\nα : Type u_1\nι : Type ?u.20960\nE : Type u_2\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : MetricSpace E\nf : ℕ → α → E\ng : α → E\nhfg : TendstoInMeasure μ f atTop g\nε : ℝ\nhε : 0 < ε\nk : ℕ\nh_k : 2⁻¹ ^ k < ε\n⊢ 2 * 2⁻¹ ^ (k + 1) = 2⁻¹ ^ k",
"tactic": "rw [pow_add]"
},
{
"state_after": "no goals",
"state_before": "case intro\nα : Type u_1\nι : Type ?u.20960\nE : Type u_2\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : MetricSpace E\nf : ℕ → α → E\ng : α → E\nhfg : TendstoInMeasure μ f atTop g\nε : ℝ\nhε : 0 < ε\nk : ℕ\nh_k : 2⁻¹ ^ k < ε\n⊢ 2 * (2⁻¹ ^ k * 2⁻¹ ^ 1) = 2⁻¹ ^ k",
"tactic": "ring"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nι : Type ?u.20960\nE : Type u_2\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : MetricSpace E\nf : ℕ → α → E\ng : α → E\nhfg : TendstoInMeasure μ f atTop g\nε : ℝ\nhε : 0 < ε\n⊢ 2⁻¹ < 1",
"tactic": "norm_num"
},
{
"state_after": "α : Type u_1\nι : Type ?u.20960\nE : Type u_2\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : MetricSpace E\nf : ℕ → α → E\ng : α → E\nhfg : TendstoInMeasure μ f atTop g\nh_lt_ε_real : ∀ (ε : ℝ), 0 < ε → ∃ k, 2 * 2⁻¹ ^ k < ε\nns : ℕ → ℕ := ExistsSeqTendstoAe.seqTendstoAeSeq hfg\nS : ℕ → Set α := fun k => {x | 2⁻¹ ^ k ≤ dist (f (ns k) x) (g x)}\nk : ℕ\n⊢ ↑↑μ (S k) ≤ 2⁻¹ ^ k",
"state_before": "α : Type u_1\nι : Type ?u.20960\nE : Type u_2\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : MetricSpace E\nf : ℕ → α → E\ng : α → E\nhfg : TendstoInMeasure μ f atTop g\nh_lt_ε_real : ∀ (ε : ℝ), 0 < ε → ∃ k, 2 * 2⁻¹ ^ k < ε\nns : ℕ → ℕ := ExistsSeqTendstoAe.seqTendstoAeSeq hfg\nS : ℕ → Set α := fun k => {x | 2⁻¹ ^ k ≤ dist (f (ns k) x) (g x)}\n⊢ ∀ (k : ℕ), ↑↑μ (S k) ≤ 2⁻¹ ^ k",
"tactic": "intro k"
},
{
"state_after": "α : Type u_1\nι : Type ?u.20960\nE : Type u_2\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : MetricSpace E\nf : ℕ → α → E\ng : α → E\nhfg : TendstoInMeasure μ f atTop g\nh_lt_ε_real : ∀ (ε : ℝ), 0 < ε → ∃ k, 2 * 2⁻¹ ^ k < ε\nns : ℕ → ℕ := ExistsSeqTendstoAe.seqTendstoAeSeq hfg\nS : ℕ → Set α := fun k => {x | 2⁻¹ ^ k ≤ dist (f (ns k) x) (g x)}\nk : ℕ\nthis : ↑↑μ {x | 2⁻¹ ^ k ≤ dist (f (ns k) x) (g x)} ≤ 2⁻¹ ^ k\n⊢ ↑↑μ (S k) ≤ 2⁻¹ ^ k",
"state_before": "α : Type u_1\nι : Type ?u.20960\nE : Type u_2\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : MetricSpace E\nf : ℕ → α → E\ng : α → E\nhfg : TendstoInMeasure μ f atTop g\nh_lt_ε_real : ∀ (ε : ℝ), 0 < ε → ∃ k, 2 * 2⁻¹ ^ k < ε\nns : ℕ → ℕ := ExistsSeqTendstoAe.seqTendstoAeSeq hfg\nS : ℕ → Set α := fun k => {x | 2⁻¹ ^ k ≤ dist (f (ns k) x) (g x)}\nk : ℕ\n⊢ ↑↑μ (S k) ≤ 2⁻¹ ^ k",
"tactic": "have := ExistsSeqTendstoAe.seqTendstoAeSeq_spec hfg k (ns k) le_rfl"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nι : Type ?u.20960\nE : Type u_2\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : MetricSpace E\nf : ℕ → α → E\ng : α → E\nhfg : TendstoInMeasure μ f atTop g\nh_lt_ε_real : ∀ (ε : ℝ), 0 < ε → ∃ k, 2 * 2⁻¹ ^ k < ε\nns : ℕ → ℕ := ExistsSeqTendstoAe.seqTendstoAeSeq hfg\nS : ℕ → Set α := fun k => {x | 2⁻¹ ^ k ≤ dist (f (ns k) x) (g x)}\nk : ℕ\nthis : ↑↑μ {x | 2⁻¹ ^ k ≤ dist (f (ns k) x) (g x)} ≤ 2⁻¹ ^ k\n⊢ ↑↑μ (S k) ≤ 2⁻¹ ^ k",
"tactic": "convert this"
},
{
"state_after": "α : Type u_1\nι : Type ?u.20960\nE : Type u_2\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : MetricSpace E\nf : ℕ → α → E\ng : α → E\nhfg : TendstoInMeasure μ f atTop g\nh_lt_ε_real : ∀ (ε : ℝ), 0 < ε → ∃ k, 2 * 2⁻¹ ^ k < ε\nns : ℕ → ℕ := ExistsSeqTendstoAe.seqTendstoAeSeq hfg\nS : ℕ → Set α := fun k => {x | 2⁻¹ ^ k ≤ dist (f (ns k) x) (g x)}\nhμS_le : ∀ (k : ℕ), ↑↑μ (S k) ≤ 2⁻¹ ^ k\ns : Set α := limsup S atTop\nhs : s = limsup S atTop\n⊢ (∑' (a : ℕ), 2⁻¹ ^ a) < ⊤",
"state_before": "α : Type u_1\nι : Type ?u.20960\nE : Type u_2\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : MetricSpace E\nf : ℕ → α → E\ng : α → E\nhfg : TendstoInMeasure μ f atTop g\nh_lt_ε_real : ∀ (ε : ℝ), 0 < ε → ∃ k, 2 * 2⁻¹ ^ k < ε\nns : ℕ → ℕ := ExistsSeqTendstoAe.seqTendstoAeSeq hfg\nS : ℕ → Set α := fun k => {x | 2⁻¹ ^ k ≤ dist (f (ns k) x) (g x)}\nhμS_le : ∀ (k : ℕ), ↑↑μ (S k) ≤ 2⁻¹ ^ k\ns : Set α := limsup S atTop\nhs : s = limsup S atTop\n⊢ ↑↑μ s = 0",
"tactic": "refine' measure_limsup_eq_zero (ne_of_lt <| lt_of_le_of_lt (ENNReal.tsum_le_tsum hμS_le) _)"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nι : Type ?u.20960\nE : Type u_2\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : MetricSpace E\nf : ℕ → α → E\ng : α → E\nhfg : TendstoInMeasure μ f atTop g\nh_lt_ε_real : ∀ (ε : ℝ), 0 < ε → ∃ k, 2 * 2⁻¹ ^ k < ε\nns : ℕ → ℕ := ExistsSeqTendstoAe.seqTendstoAeSeq hfg\nS : ℕ → Set α := fun k => {x | 2⁻¹ ^ k ≤ dist (f (ns k) x) (g x)}\nhμS_le : ∀ (k : ℕ), ↑↑μ (S k) ≤ 2⁻¹ ^ k\ns : Set α := limsup S atTop\nhs : s = limsup S atTop\n⊢ (∑' (a : ℕ), 2⁻¹ ^ a) < ⊤",
"tactic": "simp only [ENNReal.tsum_geometric, ENNReal.one_sub_inv_two, inv_inv]"
},
{
"state_after": "α : Type u_1\nι : Type ?u.20960\nE : Type u_2\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : MetricSpace E\nf : ℕ → α → E\ng : α → E\nhfg : TendstoInMeasure μ f atTop g\nh_lt_ε_real : ∀ (ε : ℝ), 0 < ε → ∃ k, 2 * 2⁻¹ ^ k < ε\nns : ℕ → ℕ := ExistsSeqTendstoAe.seqTendstoAeSeq hfg\nS : ℕ → Set α := fun k => {x | 2⁻¹ ^ k ≤ dist (f (ns k) x) (g x)}\nhμS_le : ∀ (k : ℕ), ↑↑μ (S k) ≤ 2⁻¹ ^ k\ns : Set α := limsup S atTop\nhs : s = limsup S atTop\nhμs : ↑↑μ s = 0\nx : α\nhx : x ∈ sᶜ\nε : ℝ\nhε : ε > 0\n⊢ ∃ N, ∀ (n : ℕ), n ≥ N → dist (f (ns n) x) (g x) < ε",
"state_before": "α : Type u_1\nι : Type ?u.20960\nE : Type u_2\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : MetricSpace E\nf : ℕ → α → E\ng : α → E\nhfg : TendstoInMeasure μ f atTop g\nh_lt_ε_real : ∀ (ε : ℝ), 0 < ε → ∃ k, 2 * 2⁻¹ ^ k < ε\nns : ℕ → ℕ := ExistsSeqTendstoAe.seqTendstoAeSeq hfg\nS : ℕ → Set α := fun k => {x | 2⁻¹ ^ k ≤ dist (f (ns k) x) (g x)}\nhμS_le : ∀ (k : ℕ), ↑↑μ (S k) ≤ 2⁻¹ ^ k\ns : Set α := limsup S atTop\nhs : s = limsup S atTop\nhμs : ↑↑μ s = 0\n⊢ ∀ (x : α), x ∈ sᶜ → Tendsto (fun i => f (ns i) x) atTop (𝓝 (g x))",
"tactic": "refine' fun x hx => Metric.tendsto_atTop.mpr fun ε hε => _"
},
{
"state_after": "α : Type u_1\nι : Type ?u.20960\nE : Type u_2\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : MetricSpace E\nf : ℕ → α → E\ng : α → E\nhfg : TendstoInMeasure μ f atTop g\nh_lt_ε_real : ∀ (ε : ℝ), 0 < ε → ∃ k, 2 * 2⁻¹ ^ k < ε\nns : ℕ → ℕ := ExistsSeqTendstoAe.seqTendstoAeSeq hfg\nS : ℕ → Set α := fun k => {x | 2⁻¹ ^ k ≤ dist (f (ns k) x) (g x)}\nhμS_le : ∀ (k : ℕ), ↑↑μ (S k) ≤ 2⁻¹ ^ k\ns : Set α := limsup S atTop\nhs : s = limsup S atTop\nhμs : ↑↑μ s = 0\nx : α\nhx : x ∈ (⨅ (n : ℕ), ⨆ (i : ℕ) (_ : i ≥ n), S i)ᶜ\nε : ℝ\nhε : ε > 0\n⊢ ∃ N, ∀ (n : ℕ), n ≥ N → dist (f (ns n) x) (g x) < ε",
"state_before": "α : Type u_1\nι : Type ?u.20960\nE : Type u_2\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : MetricSpace E\nf : ℕ → α → E\ng : α → E\nhfg : TendstoInMeasure μ f atTop g\nh_lt_ε_real : ∀ (ε : ℝ), 0 < ε → ∃ k, 2 * 2⁻¹ ^ k < ε\nns : ℕ → ℕ := ExistsSeqTendstoAe.seqTendstoAeSeq hfg\nS : ℕ → Set α := fun k => {x | 2⁻¹ ^ k ≤ dist (f (ns k) x) (g x)}\nhμS_le : ∀ (k : ℕ), ↑↑μ (S k) ≤ 2⁻¹ ^ k\ns : Set α := limsup S atTop\nhs : s = limsup S atTop\nhμs : ↑↑μ s = 0\nx : α\nhx : x ∈ sᶜ\nε : ℝ\nhε : ε > 0\n⊢ ∃ N, ∀ (n : ℕ), n ≥ N → dist (f (ns n) x) (g x) < ε",
"tactic": "rw [hs, limsup_eq_iInf_iSup_of_nat] at hx"
},
{
"state_after": "α : Type u_1\nι : Type ?u.20960\nE : Type u_2\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : MetricSpace E\nf : ℕ → α → E\ng : α → E\nhfg : TendstoInMeasure μ f atTop g\nh_lt_ε_real : ∀ (ε : ℝ), 0 < ε → ∃ k, 2 * 2⁻¹ ^ k < ε\nns : ℕ → ℕ := ExistsSeqTendstoAe.seqTendstoAeSeq hfg\nS : ℕ → Set α := fun k => {x | 2⁻¹ ^ k ≤ dist (f (ns k) x) (g x)}\nhμS_le : ∀ (k : ℕ), ↑↑μ (S k) ≤ 2⁻¹ ^ k\ns : Set α := limsup S atTop\nhs : s = limsup S atTop\nhμs : ↑↑μ s = 0\nx : α\nε : ℝ\nhε : ε > 0\nhx : ∃ i, ∀ (i_1 : ℕ), i_1 ≥ i → dist (f (ExistsSeqTendstoAe.seqTendstoAeSeq hfg i_1) x) (g x) < 2⁻¹ ^ i_1\n⊢ ∃ N, ∀ (n : ℕ), n ≥ N → dist (f (ns n) x) (g x) < ε",
"state_before": "α : Type u_1\nι : Type ?u.20960\nE : Type u_2\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : MetricSpace E\nf : ℕ → α → E\ng : α → E\nhfg : TendstoInMeasure μ f atTop g\nh_lt_ε_real : ∀ (ε : ℝ), 0 < ε → ∃ k, 2 * 2⁻¹ ^ k < ε\nns : ℕ → ℕ := ExistsSeqTendstoAe.seqTendstoAeSeq hfg\nS : ℕ → Set α := fun k => {x | 2⁻¹ ^ k ≤ dist (f (ns k) x) (g x)}\nhμS_le : ∀ (k : ℕ), ↑↑μ (S k) ≤ 2⁻¹ ^ k\ns : Set α := limsup S atTop\nhs : s = limsup S atTop\nhμs : ↑↑μ s = 0\nx : α\nhx : x ∈ (⨅ (n : ℕ), ⨆ (i : ℕ) (_ : i ≥ n), S i)ᶜ\nε : ℝ\nhε : ε > 0\n⊢ ∃ N, ∀ (n : ℕ), n ≥ N → dist (f (ns n) x) (g x) < ε",
"tactic": "simp only [Set.iSup_eq_iUnion, Set.iInf_eq_iInter, Set.compl_iInter, Set.compl_iUnion,\n Set.mem_iUnion, Set.mem_iInter, Set.mem_compl_iff, Set.mem_setOf_eq, not_le] at hx"
},
{
"state_after": "case intro\nα : Type u_1\nι : Type ?u.20960\nE : Type u_2\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : MetricSpace E\nf : ℕ → α → E\ng : α → E\nhfg : TendstoInMeasure μ f atTop g\nh_lt_ε_real : ∀ (ε : ℝ), 0 < ε → ∃ k, 2 * 2⁻¹ ^ k < ε\nns : ℕ → ℕ := ExistsSeqTendstoAe.seqTendstoAeSeq hfg\nS : ℕ → Set α := fun k => {x | 2⁻¹ ^ k ≤ dist (f (ns k) x) (g x)}\nhμS_le : ∀ (k : ℕ), ↑↑μ (S k) ≤ 2⁻¹ ^ k\ns : Set α := limsup S atTop\nhs : s = limsup S atTop\nhμs : ↑↑μ s = 0\nx : α\nε : ℝ\nhε : ε > 0\nN : ℕ\nhNx : ∀ (i : ℕ), i ≥ N → dist (f (ExistsSeqTendstoAe.seqTendstoAeSeq hfg i) x) (g x) < 2⁻¹ ^ i\n⊢ ∃ N, ∀ (n : ℕ), n ≥ N → dist (f (ns n) x) (g x) < ε",
"state_before": "α : Type u_1\nι : Type ?u.20960\nE : Type u_2\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : MetricSpace E\nf : ℕ → α → E\ng : α → E\nhfg : TendstoInMeasure μ f atTop g\nh_lt_ε_real : ∀ (ε : ℝ), 0 < ε → ∃ k, 2 * 2⁻¹ ^ k < ε\nns : ℕ → ℕ := ExistsSeqTendstoAe.seqTendstoAeSeq hfg\nS : ℕ → Set α := fun k => {x | 2⁻¹ ^ k ≤ dist (f (ns k) x) (g x)}\nhμS_le : ∀ (k : ℕ), ↑↑μ (S k) ≤ 2⁻¹ ^ k\ns : Set α := limsup S atTop\nhs : s = limsup S atTop\nhμs : ↑↑μ s = 0\nx : α\nε : ℝ\nhε : ε > 0\nhx : ∃ i, ∀ (i_1 : ℕ), i_1 ≥ i → dist (f (ExistsSeqTendstoAe.seqTendstoAeSeq hfg i_1) x) (g x) < 2⁻¹ ^ i_1\n⊢ ∃ N, ∀ (n : ℕ), n ≥ N → dist (f (ns n) x) (g x) < ε",
"tactic": "obtain ⟨N, hNx⟩ := hx"
},
{
"state_after": "case intro.intro\nα : Type u_1\nι : Type ?u.20960\nE : Type u_2\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : MetricSpace E\nf : ℕ → α → E\ng : α → E\nhfg : TendstoInMeasure μ f atTop g\nh_lt_ε_real : ∀ (ε : ℝ), 0 < ε → ∃ k, 2 * 2⁻¹ ^ k < ε\nns : ℕ → ℕ := ExistsSeqTendstoAe.seqTendstoAeSeq hfg\nS : ℕ → Set α := fun k => {x | 2⁻¹ ^ k ≤ dist (f (ns k) x) (g x)}\nhμS_le : ∀ (k : ℕ), ↑↑μ (S k) ≤ 2⁻¹ ^ k\ns : Set α := limsup S atTop\nhs : s = limsup S atTop\nhμs : ↑↑μ s = 0\nx : α\nε : ℝ\nhε : ε > 0\nN : ℕ\nhNx : ∀ (i : ℕ), i ≥ N → dist (f (ExistsSeqTendstoAe.seqTendstoAeSeq hfg i) x) (g x) < 2⁻¹ ^ i\nk : ℕ\nhk_lt_ε : 2 * 2⁻¹ ^ k < ε\n⊢ ∃ N, ∀ (n : ℕ), n ≥ N → dist (f (ns n) x) (g x) < ε",
"state_before": "case intro\nα : Type u_1\nι : Type ?u.20960\nE : Type u_2\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : MetricSpace E\nf : ℕ → α → E\ng : α → E\nhfg : TendstoInMeasure μ f atTop g\nh_lt_ε_real : ∀ (ε : ℝ), 0 < ε → ∃ k, 2 * 2⁻¹ ^ k < ε\nns : ℕ → ℕ := ExistsSeqTendstoAe.seqTendstoAeSeq hfg\nS : ℕ → Set α := fun k => {x | 2⁻¹ ^ k ≤ dist (f (ns k) x) (g x)}\nhμS_le : ∀ (k : ℕ), ↑↑μ (S k) ≤ 2⁻¹ ^ k\ns : Set α := limsup S atTop\nhs : s = limsup S atTop\nhμs : ↑↑μ s = 0\nx : α\nε : ℝ\nhε : ε > 0\nN : ℕ\nhNx : ∀ (i : ℕ), i ≥ N → dist (f (ExistsSeqTendstoAe.seqTendstoAeSeq hfg i) x) (g x) < 2⁻¹ ^ i\n⊢ ∃ N, ∀ (n : ℕ), n ≥ N → dist (f (ns n) x) (g x) < ε",
"tactic": "obtain ⟨k, hk_lt_ε⟩ := h_lt_ε_real ε hε"
},
{
"state_after": "case intro.intro\nα : Type u_1\nι : Type ?u.20960\nE : Type u_2\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : MetricSpace E\nf : ℕ → α → E\ng : α → E\nhfg : TendstoInMeasure μ f atTop g\nh_lt_ε_real : ∀ (ε : ℝ), 0 < ε → ∃ k, 2 * 2⁻¹ ^ k < ε\nns : ℕ → ℕ := ExistsSeqTendstoAe.seqTendstoAeSeq hfg\nS : ℕ → Set α := fun k => {x | 2⁻¹ ^ k ≤ dist (f (ns k) x) (g x)}\nhμS_le : ∀ (k : ℕ), ↑↑μ (S k) ≤ 2⁻¹ ^ k\ns : Set α := limsup S atTop\nhs : s = limsup S atTop\nhμs : ↑↑μ s = 0\nx : α\nε : ℝ\nhε : ε > 0\nN : ℕ\nhNx : ∀ (i : ℕ), i ≥ N → dist (f (ExistsSeqTendstoAe.seqTendstoAeSeq hfg i) x) (g x) < 2⁻¹ ^ i\nk : ℕ\nhk_lt_ε : 2 * 2⁻¹ ^ k < ε\nn : ℕ\nhn_ge : n ≥ max N (k - 1)\n⊢ dist (f (ns n) x) (g x) ≤ 2 * 2⁻¹ ^ k",
"state_before": "case intro.intro\nα : Type u_1\nι : Type ?u.20960\nE : Type u_2\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : MetricSpace E\nf : ℕ → α → E\ng : α → E\nhfg : TendstoInMeasure μ f atTop g\nh_lt_ε_real : ∀ (ε : ℝ), 0 < ε → ∃ k, 2 * 2⁻¹ ^ k < ε\nns : ℕ → ℕ := ExistsSeqTendstoAe.seqTendstoAeSeq hfg\nS : ℕ → Set α := fun k => {x | 2⁻¹ ^ k ≤ dist (f (ns k) x) (g x)}\nhμS_le : ∀ (k : ℕ), ↑↑μ (S k) ≤ 2⁻¹ ^ k\ns : Set α := limsup S atTop\nhs : s = limsup S atTop\nhμs : ↑↑μ s = 0\nx : α\nε : ℝ\nhε : ε > 0\nN : ℕ\nhNx : ∀ (i : ℕ), i ≥ N → dist (f (ExistsSeqTendstoAe.seqTendstoAeSeq hfg i) x) (g x) < 2⁻¹ ^ i\nk : ℕ\nhk_lt_ε : 2 * 2⁻¹ ^ k < ε\n⊢ ∃ N, ∀ (n : ℕ), n ≥ N → dist (f (ns n) x) (g x) < ε",
"tactic": "refine' ⟨max N (k - 1), fun n hn_ge => lt_of_le_of_lt _ hk_lt_ε⟩"
},
{
"state_after": "case intro.intro\nα : Type u_1\nι : Type ?u.20960\nE : Type u_2\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : MetricSpace E\nf : ℕ → α → E\ng : α → E\nhfg : TendstoInMeasure μ f atTop g\nh_lt_ε_real : ∀ (ε : ℝ), 0 < ε → ∃ k, 2 * 2⁻¹ ^ k < ε\nns : ℕ → ℕ := ExistsSeqTendstoAe.seqTendstoAeSeq hfg\nS : ℕ → Set α := fun k => {x | 2⁻¹ ^ k ≤ dist (f (ns k) x) (g x)}\nhμS_le : ∀ (k : ℕ), ↑↑μ (S k) ≤ 2⁻¹ ^ k\ns : Set α := limsup S atTop\nhs : s = limsup S atTop\nhμs : ↑↑μ s = 0\nx : α\nε : ℝ\nhε : ε > 0\nN k : ℕ\nhk_lt_ε : 2 * 2⁻¹ ^ k < ε\nn : ℕ\nhn_ge : n ≥ max N (k - 1)\nhNx : dist (f (ExistsSeqTendstoAe.seqTendstoAeSeq hfg n) x) (g x) < 2⁻¹ ^ n\n⊢ dist (f (ns n) x) (g x) ≤ 2 * 2⁻¹ ^ k",
"state_before": "case intro.intro\nα : Type u_1\nι : Type ?u.20960\nE : Type u_2\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : MetricSpace E\nf : ℕ → α → E\ng : α → E\nhfg : TendstoInMeasure μ f atTop g\nh_lt_ε_real : ∀ (ε : ℝ), 0 < ε → ∃ k, 2 * 2⁻¹ ^ k < ε\nns : ℕ → ℕ := ExistsSeqTendstoAe.seqTendstoAeSeq hfg\nS : ℕ → Set α := fun k => {x | 2⁻¹ ^ k ≤ dist (f (ns k) x) (g x)}\nhμS_le : ∀ (k : ℕ), ↑↑μ (S k) ≤ 2⁻¹ ^ k\ns : Set α := limsup S atTop\nhs : s = limsup S atTop\nhμs : ↑↑μ s = 0\nx : α\nε : ℝ\nhε : ε > 0\nN : ℕ\nhNx : ∀ (i : ℕ), i ≥ N → dist (f (ExistsSeqTendstoAe.seqTendstoAeSeq hfg i) x) (g x) < 2⁻¹ ^ i\nk : ℕ\nhk_lt_ε : 2 * 2⁻¹ ^ k < ε\nn : ℕ\nhn_ge : n ≥ max N (k - 1)\n⊢ dist (f (ns n) x) (g x) ≤ 2 * 2⁻¹ ^ k",
"tactic": "specialize hNx n ((le_max_left _ _).trans hn_ge)"
},
{
"state_after": "case intro.intro\nα : Type u_1\nι : Type ?u.20960\nE : Type u_2\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : MetricSpace E\nf : ℕ → α → E\ng : α → E\nhfg : TendstoInMeasure μ f atTop g\nh_lt_ε_real : ∀ (ε : ℝ), 0 < ε → ∃ k, 2 * 2⁻¹ ^ k < ε\nns : ℕ → ℕ := ExistsSeqTendstoAe.seqTendstoAeSeq hfg\nS : ℕ → Set α := fun k => {x | 2⁻¹ ^ k ≤ dist (f (ns k) x) (g x)}\nhμS_le : ∀ (k : ℕ), ↑↑μ (S k) ≤ 2⁻¹ ^ k\ns : Set α := limsup S atTop\nhs : s = limsup S atTop\nhμs : ↑↑μ s = 0\nx : α\nε : ℝ\nhε : ε > 0\nN k : ℕ\nhk_lt_ε : 2 * 2⁻¹ ^ k < ε\nn : ℕ\nhn_ge : n ≥ max N (k - 1)\nhNx : dist (f (ExistsSeqTendstoAe.seqTendstoAeSeq hfg n) x) (g x) < 2⁻¹ ^ n\nh_inv_n_le_k : 2⁻¹ ^ n ≤ 2 * 2⁻¹ ^ k\n⊢ dist (f (ns n) x) (g x) ≤ 2 * 2⁻¹ ^ k",
"state_before": "case intro.intro\nα : Type u_1\nι : Type ?u.20960\nE : Type u_2\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : MetricSpace E\nf : ℕ → α → E\ng : α → E\nhfg : TendstoInMeasure μ f atTop g\nh_lt_ε_real : ∀ (ε : ℝ), 0 < ε → ∃ k, 2 * 2⁻¹ ^ k < ε\nns : ℕ → ℕ := ExistsSeqTendstoAe.seqTendstoAeSeq hfg\nS : ℕ → Set α := fun k => {x | 2⁻¹ ^ k ≤ dist (f (ns k) x) (g x)}\nhμS_le : ∀ (k : ℕ), ↑↑μ (S k) ≤ 2⁻¹ ^ k\ns : Set α := limsup S atTop\nhs : s = limsup S atTop\nhμs : ↑↑μ s = 0\nx : α\nε : ℝ\nhε : ε > 0\nN k : ℕ\nhk_lt_ε : 2 * 2⁻¹ ^ k < ε\nn : ℕ\nhn_ge : n ≥ max N (k - 1)\nhNx : dist (f (ExistsSeqTendstoAe.seqTendstoAeSeq hfg n) x) (g x) < 2⁻¹ ^ n\n⊢ dist (f (ns n) x) (g x) ≤ 2 * 2⁻¹ ^ k",
"tactic": "have h_inv_n_le_k : (2 : ℝ)⁻¹ ^ n ≤ 2 * (2 : ℝ)⁻¹ ^ k := by\n rw [mul_comm, ← inv_mul_le_iff' (zero_lt_two' ℝ)]\n conv_lhs =>\n congr\n rw [← pow_one (2 : ℝ)⁻¹]\n rw [← pow_add, add_comm]\n exact pow_le_pow_of_le_one (one_div (2 : ℝ) ▸ one_half_pos.le) (inv_le_one one_le_two)\n ((le_tsub_add.trans (add_le_add_right (le_max_right _ _) 1)).trans\n (add_le_add_right hn_ge 1))"
},
{
"state_after": "no goals",
"state_before": "case intro.intro\nα : Type u_1\nι : Type ?u.20960\nE : Type u_2\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : MetricSpace E\nf : ℕ → α → E\ng : α → E\nhfg : TendstoInMeasure μ f atTop g\nh_lt_ε_real : ∀ (ε : ℝ), 0 < ε → ∃ k, 2 * 2⁻¹ ^ k < ε\nns : ℕ → ℕ := ExistsSeqTendstoAe.seqTendstoAeSeq hfg\nS : ℕ → Set α := fun k => {x | 2⁻¹ ^ k ≤ dist (f (ns k) x) (g x)}\nhμS_le : ∀ (k : ℕ), ↑↑μ (S k) ≤ 2⁻¹ ^ k\ns : Set α := limsup S atTop\nhs : s = limsup S atTop\nhμs : ↑↑μ s = 0\nx : α\nε : ℝ\nhε : ε > 0\nN k : ℕ\nhk_lt_ε : 2 * 2⁻¹ ^ k < ε\nn : ℕ\nhn_ge : n ≥ max N (k - 1)\nhNx : dist (f (ExistsSeqTendstoAe.seqTendstoAeSeq hfg n) x) (g x) < 2⁻¹ ^ n\nh_inv_n_le_k : 2⁻¹ ^ n ≤ 2 * 2⁻¹ ^ k\n⊢ dist (f (ns n) x) (g x) ≤ 2 * 2⁻¹ ^ k",
"tactic": "exact le_trans hNx.le h_inv_n_le_k"
},
{
"state_after": "α : Type u_1\nι : Type ?u.20960\nE : Type u_2\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : MetricSpace E\nf : ℕ → α → E\ng : α → E\nhfg : TendstoInMeasure μ f atTop g\nh_lt_ε_real : ∀ (ε : ℝ), 0 < ε → ∃ k, 2 * 2⁻¹ ^ k < ε\nns : ℕ → ℕ := ExistsSeqTendstoAe.seqTendstoAeSeq hfg\nS : ℕ → Set α := fun k => {x | 2⁻¹ ^ k ≤ dist (f (ns k) x) (g x)}\nhμS_le : ∀ (k : ℕ), ↑↑μ (S k) ≤ 2⁻¹ ^ k\ns : Set α := limsup S atTop\nhs : s = limsup S atTop\nhμs : ↑↑μ s = 0\nx : α\nε : ℝ\nhε : ε > 0\nN k : ℕ\nhk_lt_ε : 2 * 2⁻¹ ^ k < ε\nn : ℕ\nhn_ge : n ≥ max N (k - 1)\nhNx : dist (f (ExistsSeqTendstoAe.seqTendstoAeSeq hfg n) x) (g x) < 2⁻¹ ^ n\n⊢ 2⁻¹ * 2⁻¹ ^ n ≤ 2⁻¹ ^ k",
"state_before": "α : Type u_1\nι : Type ?u.20960\nE : Type u_2\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : MetricSpace E\nf : ℕ → α → E\ng : α → E\nhfg : TendstoInMeasure μ f atTop g\nh_lt_ε_real : ∀ (ε : ℝ), 0 < ε → ∃ k, 2 * 2⁻¹ ^ k < ε\nns : ℕ → ℕ := ExistsSeqTendstoAe.seqTendstoAeSeq hfg\nS : ℕ → Set α := fun k => {x | 2⁻¹ ^ k ≤ dist (f (ns k) x) (g x)}\nhμS_le : ∀ (k : ℕ), ↑↑μ (S k) ≤ 2⁻¹ ^ k\ns : Set α := limsup S atTop\nhs : s = limsup S atTop\nhμs : ↑↑μ s = 0\nx : α\nε : ℝ\nhε : ε > 0\nN k : ℕ\nhk_lt_ε : 2 * 2⁻¹ ^ k < ε\nn : ℕ\nhn_ge : n ≥ max N (k - 1)\nhNx : dist (f (ExistsSeqTendstoAe.seqTendstoAeSeq hfg n) x) (g x) < 2⁻¹ ^ n\n⊢ 2⁻¹ ^ n ≤ 2 * 2⁻¹ ^ k",
"tactic": "rw [mul_comm, ← inv_mul_le_iff' (zero_lt_two' ℝ)]"
},
{
"state_after": "α : Type u_1\nι : Type ?u.20960\nE : Type u_2\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : MetricSpace E\nf : ℕ → α → E\ng : α → E\nhfg : TendstoInMeasure μ f atTop g\nh_lt_ε_real : ∀ (ε : ℝ), 0 < ε → ∃ k, 2 * 2⁻¹ ^ k < ε\nns : ℕ → ℕ := ExistsSeqTendstoAe.seqTendstoAeSeq hfg\nS : ℕ → Set α := fun k => {x | 2⁻¹ ^ k ≤ dist (f (ns k) x) (g x)}\nhμS_le : ∀ (k : ℕ), ↑↑μ (S k) ≤ 2⁻¹ ^ k\ns : Set α := limsup S atTop\nhs : s = limsup S atTop\nhμs : ↑↑μ s = 0\nx : α\nε : ℝ\nhε : ε > 0\nN k : ℕ\nhk_lt_ε : 2 * 2⁻¹ ^ k < ε\nn : ℕ\nhn_ge : n ≥ max N (k - 1)\nhNx : dist (f (ExistsSeqTendstoAe.seqTendstoAeSeq hfg n) x) (g x) < 2⁻¹ ^ n\n⊢ 2⁻¹ ^ 1 * 2⁻¹ ^ n ≤ 2⁻¹ ^ k",
"state_before": "α : Type u_1\nι : Type ?u.20960\nE : Type u_2\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : MetricSpace E\nf : ℕ → α → E\ng : α → E\nhfg : TendstoInMeasure μ f atTop g\nh_lt_ε_real : ∀ (ε : ℝ), 0 < ε → ∃ k, 2 * 2⁻¹ ^ k < ε\nns : ℕ → ℕ := ExistsSeqTendstoAe.seqTendstoAeSeq hfg\nS : ℕ → Set α := fun k => {x | 2⁻¹ ^ k ≤ dist (f (ns k) x) (g x)}\nhμS_le : ∀ (k : ℕ), ↑↑μ (S k) ≤ 2⁻¹ ^ k\ns : Set α := limsup S atTop\nhs : s = limsup S atTop\nhμs : ↑↑μ s = 0\nx : α\nε : ℝ\nhε : ε > 0\nN k : ℕ\nhk_lt_ε : 2 * 2⁻¹ ^ k < ε\nn : ℕ\nhn_ge : n ≥ max N (k - 1)\nhNx : dist (f (ExistsSeqTendstoAe.seqTendstoAeSeq hfg n) x) (g x) < 2⁻¹ ^ n\n⊢ 2⁻¹ * 2⁻¹ ^ n ≤ 2⁻¹ ^ k",
"tactic": "conv_lhs =>\n congr\n rw [← pow_one (2 : ℝ)⁻¹]"
},
{
"state_after": "α : Type u_1\nι : Type ?u.20960\nE : Type u_2\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : MetricSpace E\nf : ℕ → α → E\ng : α → E\nhfg : TendstoInMeasure μ f atTop g\nh_lt_ε_real : ∀ (ε : ℝ), 0 < ε → ∃ k, 2 * 2⁻¹ ^ k < ε\nns : ℕ → ℕ := ExistsSeqTendstoAe.seqTendstoAeSeq hfg\nS : ℕ → Set α := fun k => {x | 2⁻¹ ^ k ≤ dist (f (ns k) x) (g x)}\nhμS_le : ∀ (k : ℕ), ↑↑μ (S k) ≤ 2⁻¹ ^ k\ns : Set α := limsup S atTop\nhs : s = limsup S atTop\nhμs : ↑↑μ s = 0\nx : α\nε : ℝ\nhε : ε > 0\nN k : ℕ\nhk_lt_ε : 2 * 2⁻¹ ^ k < ε\nn : ℕ\nhn_ge : n ≥ max N (k - 1)\nhNx : dist (f (ExistsSeqTendstoAe.seqTendstoAeSeq hfg n) x) (g x) < 2⁻¹ ^ n\n⊢ 2⁻¹ ^ (n + 1) ≤ 2⁻¹ ^ k",
"state_before": "α : Type u_1\nι : Type ?u.20960\nE : Type u_2\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : MetricSpace E\nf : ℕ → α → E\ng : α → E\nhfg : TendstoInMeasure μ f atTop g\nh_lt_ε_real : ∀ (ε : ℝ), 0 < ε → ∃ k, 2 * 2⁻¹ ^ k < ε\nns : ℕ → ℕ := ExistsSeqTendstoAe.seqTendstoAeSeq hfg\nS : ℕ → Set α := fun k => {x | 2⁻¹ ^ k ≤ dist (f (ns k) x) (g x)}\nhμS_le : ∀ (k : ℕ), ↑↑μ (S k) ≤ 2⁻¹ ^ k\ns : Set α := limsup S atTop\nhs : s = limsup S atTop\nhμs : ↑↑μ s = 0\nx : α\nε : ℝ\nhε : ε > 0\nN k : ℕ\nhk_lt_ε : 2 * 2⁻¹ ^ k < ε\nn : ℕ\nhn_ge : n ≥ max N (k - 1)\nhNx : dist (f (ExistsSeqTendstoAe.seqTendstoAeSeq hfg n) x) (g x) < 2⁻¹ ^ n\n⊢ 2⁻¹ ^ 1 * 2⁻¹ ^ n ≤ 2⁻¹ ^ k",
"tactic": "rw [← pow_add, add_comm]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nι : Type ?u.20960\nE : Type u_2\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : MetricSpace E\nf : ℕ → α → E\ng : α → E\nhfg : TendstoInMeasure μ f atTop g\nh_lt_ε_real : ∀ (ε : ℝ), 0 < ε → ∃ k, 2 * 2⁻¹ ^ k < ε\nns : ℕ → ℕ := ExistsSeqTendstoAe.seqTendstoAeSeq hfg\nS : ℕ → Set α := fun k => {x | 2⁻¹ ^ k ≤ dist (f (ns k) x) (g x)}\nhμS_le : ∀ (k : ℕ), ↑↑μ (S k) ≤ 2⁻¹ ^ k\ns : Set α := limsup S atTop\nhs : s = limsup S atTop\nhμs : ↑↑μ s = 0\nx : α\nε : ℝ\nhε : ε > 0\nN k : ℕ\nhk_lt_ε : 2 * 2⁻¹ ^ k < ε\nn : ℕ\nhn_ge : n ≥ max N (k - 1)\nhNx : dist (f (ExistsSeqTendstoAe.seqTendstoAeSeq hfg n) x) (g x) < 2⁻¹ ^ n\n⊢ 2⁻¹ ^ (n + 1) ≤ 2⁻¹ ^ k",
"tactic": "exact pow_le_pow_of_le_one (one_div (2 : ℝ) ▸ one_half_pos.le) (inv_le_one one_le_two)\n ((le_tsub_add.trans (add_le_add_right (le_max_right _ _) 1)).trans\n (add_le_add_right hn_ge 1))"
}
] |
[
245,
20
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
196,
1
] |
Mathlib/Data/Nat/Dist.lean
|
Nat.dist_eq_max_sub_min
|
[
{
"state_after": "i j : ℕ\nh : i < j\n⊢ dist i j = max i j - min i j",
"state_before": "i j : ℕ\n⊢ i < j → dist i j = max i j - min i j",
"tactic": "intro h"
},
{
"state_after": "no goals",
"state_before": "i j : ℕ\nh : i < j\n⊢ dist i j = max i j - min i j",
"tactic": "rw [max_eq_right_of_lt h, min_eq_left_of_lt h, dist_eq_sub_of_le (Nat.le_of_lt h)]"
},
{
"state_after": "i j : ℕ\nh : i ≥ j\n⊢ dist i j = max i j - min i j",
"state_before": "i j : ℕ\n⊢ i ≥ j → dist i j = max i j - min i j",
"tactic": "intro h"
},
{
"state_after": "no goals",
"state_before": "i j : ℕ\nh : i ≥ j\n⊢ dist i j = max i j - min i j",
"tactic": "rw [max_eq_left h, min_eq_right h, dist_eq_sub_of_le_right h]"
}
] |
[
114,
78
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
111,
1
] |
Mathlib/Topology/MetricSpace/Basic.lean
|
Metric.uniformEmbedding_iff
|
[
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nX : Type ?u.62248\nι : Type ?u.62251\ninst✝¹ : PseudoMetricSpace α\nx y z : α\nδ ε ε₁ ε₂ : ℝ\ns : Set α\ninst✝ : PseudoMetricSpace β\nf : α → β\n⊢ UniformEmbedding f ↔\n Function.Injective f ∧\n UniformContinuous f ∧ ∀ (δ : ℝ), δ > 0 → ∃ ε, ε > 0 ∧ ∀ {a b : α}, dist (f a) (f b) < ε → dist a b < δ",
"tactic": "rw [uniformEmbedding_iff, and_comm, uniformInducing_iff]"
}
] |
[
846,
59
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
843,
8
] |
Mathlib/Analysis/InnerProductSpace/Projection.lean
|
orthogonalProjectionFn_norm_sq
|
[
{
"state_after": "𝕜 : Type u_2\nE : Type u_1\nF : Type ?u.463344\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 }\nv : E\np : E := orthogonalProjectionFn K v\n⊢ ‖v‖ * ‖v‖ = ‖v - p‖ * ‖v - p‖ + ‖p‖ * ‖p‖",
"state_before": "𝕜 : Type u_2\nE : Type u_1\nF : Type ?u.463344\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 }\nv : E\n⊢ ‖v‖ * ‖v‖ =\n ‖v - orthogonalProjectionFn K v‖ * ‖v - orthogonalProjectionFn K v‖ +\n ‖orthogonalProjectionFn K v‖ * ‖orthogonalProjectionFn K v‖",
"tactic": "set p := orthogonalProjectionFn K v"
},
{
"state_after": "𝕜 : Type u_2\nE : Type u_1\nF : Type ?u.463344\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 }\nv : E\np : E := orthogonalProjectionFn K v\nh' : inner (v - p) p = 0\n⊢ ‖v‖ * ‖v‖ = ‖v - p‖ * ‖v - p‖ + ‖p‖ * ‖p‖",
"state_before": "𝕜 : Type u_2\nE : Type u_1\nF : Type ?u.463344\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 }\nv : E\np : E := orthogonalProjectionFn K v\n⊢ ‖v‖ * ‖v‖ = ‖v - p‖ * ‖v - p‖ + ‖p‖ * ‖p‖",
"tactic": "have h' : ⟪v - p, p⟫ = 0 :=\n orthogonalProjectionFn_inner_eq_zero _ _ (orthogonalProjectionFn_mem v)"
},
{
"state_after": "no goals",
"state_before": "𝕜 : Type u_2\nE : Type u_1\nF : Type ?u.463344\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 }\nv : E\np : E := orthogonalProjectionFn K v\nh' : inner (v - p) p = 0\n⊢ ‖v‖ * ‖v‖ = ‖v - p‖ * ‖v - p‖ + ‖p‖ * ‖p‖",
"tactic": "convert norm_add_sq_eq_norm_sq_add_norm_sq_of_inner_eq_zero (v - p) p h' using 2 <;> simp"
}
] |
[
438,
92
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
431,
1
] |
Mathlib/Analysis/Convex/Between.lean
|
sbtw_const_vadd_iff
|
[
{
"state_after": "no goals",
"state_before": "R : Type u_1\nV : Type u_2\nV' : Type ?u.117591\nP : Type u_3\nP' : Type ?u.117597\ninst✝⁶ : OrderedRing R\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : Module R V\ninst✝³ : AddTorsor V P\ninst✝² : AddCommGroup V'\ninst✝¹ : Module R V'\ninst✝ : AddTorsor V' P'\nx y z : P\nv : V\n⊢ Sbtw R (v +ᵥ x) (v +ᵥ y) (v +ᵥ z) ↔ Sbtw R x y z",
"tactic": "rw [Sbtw, Sbtw, wbtw_const_vadd_iff, (AddAction.injective v).ne_iff,\n (AddAction.injective v).ne_iff]"
}
] |
[
214,
36
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
211,
1
] |
Mathlib/Analysis/Calculus/FDerivSymmetric.lean
|
second_derivative_symmetric_of_eventually
|
[
{
"state_after": "case intro.intro\nE : Type u_1\nF : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\ns : Set E\ns_conv : Convex ℝ s\nf✝ : E → F\nf'✝ : E → E →L[ℝ] F\nf''✝ : E →L[ℝ] E →L[ℝ] F\nhf✝ : ∀ (x : E), x ∈ interior s → HasFDerivAt f✝ (f'✝ x) x\nx : E\nxs : x ∈ s\nhx✝ : HasFDerivWithinAt f'✝ f''✝ (interior s) x\nf : E → F\nf' : E → E →L[ℝ] F\nf'' : E →L[ℝ] E →L[ℝ] F\nhf : ∀ᶠ (y : E) in 𝓝 x, HasFDerivAt f (f' y) y\nhx : HasFDerivAt f' f'' x\nv w : E\nε : ℝ\nεpos : ε > 0\nhε : Metric.ball x ε ⊆ {x | (fun y => HasFDerivAt f (f' y) y) x}\n⊢ ↑(↑f'' v) w = ↑(↑f'' w) v",
"state_before": "E : Type u_1\nF : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\ns : Set E\ns_conv : Convex ℝ s\nf✝ : E → F\nf'✝ : E → E →L[ℝ] F\nf''✝ : E →L[ℝ] E →L[ℝ] F\nhf✝ : ∀ (x : E), x ∈ interior s → HasFDerivAt f✝ (f'✝ x) x\nx : E\nxs : x ∈ s\nhx✝ : HasFDerivWithinAt f'✝ f''✝ (interior s) x\nf : E → F\nf' : E → E →L[ℝ] F\nf'' : E →L[ℝ] E →L[ℝ] F\nhf : ∀ᶠ (y : E) in 𝓝 x, HasFDerivAt f (f' y) y\nhx : HasFDerivAt f' f'' x\nv w : E\n⊢ ↑(↑f'' v) w = ↑(↑f'' w) v",
"tactic": "rcases Metric.mem_nhds_iff.1 hf with ⟨ε, εpos, hε⟩"
},
{
"state_after": "case intro.intro\nE : Type u_1\nF : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\ns : Set E\ns_conv : Convex ℝ s\nf✝ : E → F\nf'✝ : E → E →L[ℝ] F\nf''✝ : E →L[ℝ] E →L[ℝ] F\nhf✝ : ∀ (x : E), x ∈ interior s → HasFDerivAt f✝ (f'✝ x) x\nx : E\nxs : x ∈ s\nhx✝ : HasFDerivWithinAt f'✝ f''✝ (interior s) x\nf : E → F\nf' : E → E →L[ℝ] F\nf'' : E →L[ℝ] E →L[ℝ] F\nhf : ∀ᶠ (y : E) in 𝓝 x, HasFDerivAt f (f' y) y\nhx : HasFDerivAt f' f'' x\nv w : E\nε : ℝ\nεpos : ε > 0\nhε : Metric.ball x ε ⊆ {x | (fun y => HasFDerivAt f (f' y) y) x}\nA : Set.Nonempty (interior (Metric.ball x ε))\n⊢ ↑(↑f'' v) w = ↑(↑f'' w) v",
"state_before": "case intro.intro\nE : Type u_1\nF : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\ns : Set E\ns_conv : Convex ℝ s\nf✝ : E → F\nf'✝ : E → E →L[ℝ] F\nf''✝ : E →L[ℝ] E →L[ℝ] F\nhf✝ : ∀ (x : E), x ∈ interior s → HasFDerivAt f✝ (f'✝ x) x\nx : E\nxs : x ∈ s\nhx✝ : HasFDerivWithinAt f'✝ f''✝ (interior s) x\nf : E → F\nf' : E → E →L[ℝ] F\nf'' : E →L[ℝ] E →L[ℝ] F\nhf : ∀ᶠ (y : E) in 𝓝 x, HasFDerivAt f (f' y) y\nhx : HasFDerivAt f' f'' x\nv w : E\nε : ℝ\nεpos : ε > 0\nhε : Metric.ball x ε ⊆ {x | (fun y => HasFDerivAt f (f' y) y) x}\n⊢ ↑(↑f'' v) w = ↑(↑f'' w) v",
"tactic": "have A : (interior (Metric.ball x ε)).Nonempty := by\n rwa [Metric.isOpen_ball.interior_eq, Metric.nonempty_ball]"
},
{
"state_after": "no goals",
"state_before": "case intro.intro\nE : Type u_1\nF : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\ns : Set E\ns_conv : Convex ℝ s\nf✝ : E → F\nf'✝ : E → E →L[ℝ] F\nf''✝ : E →L[ℝ] E →L[ℝ] F\nhf✝ : ∀ (x : E), x ∈ interior s → HasFDerivAt f✝ (f'✝ x) x\nx : E\nxs : x ∈ s\nhx✝ : HasFDerivWithinAt f'✝ f''✝ (interior s) x\nf : E → F\nf' : E → E →L[ℝ] F\nf'' : E →L[ℝ] E →L[ℝ] F\nhf : ∀ᶠ (y : E) in 𝓝 x, HasFDerivAt f (f' y) y\nhx : HasFDerivAt f' f'' x\nv w : E\nε : ℝ\nεpos : ε > 0\nhε : Metric.ball x ε ⊆ {x | (fun y => HasFDerivAt f (f' y) y) x}\nA : Set.Nonempty (interior (Metric.ball x ε))\n⊢ ↑(↑f'' v) w = ↑(↑f'' w) v",
"tactic": "exact\n Convex.second_derivative_within_at_symmetric (convex_ball x ε) A\n (fun y hy => hε (interior_subset hy)) (Metric.mem_ball_self εpos) hx.hasFDerivWithinAt v w"
},
{
"state_after": "no goals",
"state_before": "E : Type u_1\nF : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\ns : Set E\ns_conv : Convex ℝ s\nf✝ : E → F\nf'✝ : E → E →L[ℝ] F\nf''✝ : E →L[ℝ] E →L[ℝ] F\nhf✝ : ∀ (x : E), x ∈ interior s → HasFDerivAt f✝ (f'✝ x) x\nx : E\nxs : x ∈ s\nhx✝ : HasFDerivWithinAt f'✝ f''✝ (interior s) x\nf : E → F\nf' : E → E →L[ℝ] F\nf'' : E →L[ℝ] E →L[ℝ] F\nhf : ∀ᶠ (y : E) in 𝓝 x, HasFDerivAt f (f' y) y\nhx : HasFDerivAt f' f'' x\nv w : E\nε : ℝ\nεpos : ε > 0\nhε : Metric.ball x ε ⊆ {x | (fun y => HasFDerivAt f (f' y) y) x}\n⊢ Set.Nonempty (interior (Metric.ball x ε))",
"tactic": "rwa [Metric.isOpen_ball.interior_eq, Metric.nonempty_ball]"
}
] |
[
319,
97
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
311,
1
] |
Mathlib/MeasureTheory/Measure/AEDisjoint.lean
|
MeasureTheory.AEDisjoint.measure_diff_left
|
[] |
[
134,
48
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
133,
1
] |
Mathlib/Topology/MetricSpace/Lipschitz.lean
|
LipschitzWith.comap_cobounded_le
|
[] |
[
389,
31
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
387,
1
] |
Mathlib/Topology/SubsetProperties.lean
|
Filter.comap_cocompact_le
|
[
{
"state_after": "α : Type u\nβ : Type v\nι : Type ?u.89720\nπ : ι → Type ?u.89725\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\ns t : Set α\nf : α → β\nhf : Continuous f\n⊢ ∀ (i' : Set α), IsCompact i' → ∃ i, IsCompact i ∧ f ⁻¹' iᶜ ⊆ i'ᶜ",
"state_before": "α : Type u\nβ : Type v\nι : Type ?u.89720\nπ : ι → Type ?u.89725\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\ns t : Set α\nf : α → β\nhf : Continuous f\n⊢ comap f (cocompact β) ≤ cocompact α",
"tactic": "rw [(Filter.hasBasis_cocompact.comap f).le_basis_iff Filter.hasBasis_cocompact]"
},
{
"state_after": "α : Type u\nβ : Type v\nι : Type ?u.89720\nπ : ι → Type ?u.89725\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\ns t✝ : Set α\nf : α → β\nhf : Continuous f\nt : Set α\nht : IsCompact t\n⊢ ∃ i, IsCompact i ∧ f ⁻¹' iᶜ ⊆ tᶜ",
"state_before": "α : Type u\nβ : Type v\nι : Type ?u.89720\nπ : ι → Type ?u.89725\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\ns t : Set α\nf : α → β\nhf : Continuous f\n⊢ ∀ (i' : Set α), IsCompact i' → ∃ i, IsCompact i ∧ f ⁻¹' iᶜ ⊆ i'ᶜ",
"tactic": "intro t ht"
},
{
"state_after": "α : Type u\nβ : Type v\nι : Type ?u.89720\nπ : ι → Type ?u.89725\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\ns t✝ : Set α\nf : α → β\nhf : Continuous f\nt : Set α\nht : IsCompact t\n⊢ f ⁻¹' (f '' t)ᶜ ⊆ tᶜ",
"state_before": "α : Type u\nβ : Type v\nι : Type ?u.89720\nπ : ι → Type ?u.89725\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\ns t✝ : Set α\nf : α → β\nhf : Continuous f\nt : Set α\nht : IsCompact t\n⊢ ∃ i, IsCompact i ∧ f ⁻¹' iᶜ ⊆ tᶜ",
"tactic": "refine' ⟨f '' t, ht.image hf, _⟩"
},
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nι : Type ?u.89720\nπ : ι → Type ?u.89725\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\ns t✝ : Set α\nf : α → β\nhf : Continuous f\nt : Set α\nht : IsCompact t\n⊢ f ⁻¹' (f '' t)ᶜ ⊆ tᶜ",
"tactic": "simpa using t.subset_preimage_image f"
}
] |
[
844,
40
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
839,
1
] |
Mathlib/LinearAlgebra/Basic.lean
|
Submodule.comap_mono
|
[] |
[
812,
16
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
811,
1
] |
Mathlib/Analysis/SpecialFunctions/Polynomials.lean
|
Polynomial.tendsto_atTop_iff_leadingCoeff_nonneg
|
[
{
"state_after": "𝕜 : Type u_1\ninst✝¹ : NormedLinearOrderedField 𝕜\nP Q : 𝕜[X]\ninst✝ : OrderTopology 𝕜\nh : Tendsto (fun x => eval x P) atTop atTop\n⊢ 0 < degree P ∧ 0 ≤ leadingCoeff P",
"state_before": "𝕜 : Type u_1\ninst✝¹ : NormedLinearOrderedField 𝕜\nP Q : 𝕜[X]\ninst✝ : OrderTopology 𝕜\n⊢ Tendsto (fun x => eval x P) atTop atTop ↔ 0 < degree P ∧ 0 ≤ leadingCoeff P",
"tactic": "refine' ⟨fun h => _, fun h => tendsto_atTop_of_leadingCoeff_nonneg P h.1 h.2⟩"
},
{
"state_after": "𝕜 : Type u_1\ninst✝¹ : NormedLinearOrderedField 𝕜\nP Q : 𝕜[X]\ninst✝ : OrderTopology 𝕜\nh : Tendsto (fun x => eval x P) atTop atTop\nthis : Tendsto (fun x => leadingCoeff P * x ^ natDegree P) atTop atTop\n⊢ 0 < degree P ∧ 0 ≤ leadingCoeff P",
"state_before": "𝕜 : Type u_1\ninst✝¹ : NormedLinearOrderedField 𝕜\nP Q : 𝕜[X]\ninst✝ : OrderTopology 𝕜\nh : Tendsto (fun x => eval x P) atTop atTop\n⊢ 0 < degree P ∧ 0 ≤ leadingCoeff P",
"tactic": "have : Tendsto (fun x => P.leadingCoeff * x ^ P.natDegree) atTop atTop :=\n (isEquivalent_atTop_lead P).tendsto_atTop h"
},
{
"state_after": "𝕜 : Type u_1\ninst✝¹ : NormedLinearOrderedField 𝕜\nP Q : 𝕜[X]\ninst✝ : OrderTopology 𝕜\nh : Tendsto (fun x => eval x P) atTop atTop\nthis : 0 < degree P ∧ 0 < leadingCoeff P\n⊢ 0 < degree P ∧ 0 ≤ leadingCoeff P",
"state_before": "𝕜 : Type u_1\ninst✝¹ : NormedLinearOrderedField 𝕜\nP Q : 𝕜[X]\ninst✝ : OrderTopology 𝕜\nh : Tendsto (fun x => eval x P) atTop atTop\nthis : Tendsto (fun x => leadingCoeff P * x ^ natDegree P) atTop atTop\n⊢ 0 < degree P ∧ 0 ≤ leadingCoeff P",
"tactic": "rw [tendsto_const_mul_pow_atTop_iff, ← pos_iff_ne_zero, natDegree_pos_iff_degree_pos] at this"
},
{
"state_after": "no goals",
"state_before": "𝕜 : Type u_1\ninst✝¹ : NormedLinearOrderedField 𝕜\nP Q : 𝕜[X]\ninst✝ : OrderTopology 𝕜\nh : Tendsto (fun x => eval x P) atTop atTop\nthis : 0 < degree P ∧ 0 < leadingCoeff P\n⊢ 0 < degree P ∧ 0 ≤ leadingCoeff P",
"tactic": "exact ⟨this.1, this.2.le⟩"
}
] |
[
73,
28
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
67,
1
] |
Mathlib/RingTheory/FractionalIdeal.lean
|
FractionalIdeal.mapEquiv_refl
|
[
{
"state_after": "no goals",
"state_before": "R : Type u_2\ninst✝⁶ : CommRing R\nS : Submonoid R\nP : Type u_1\ninst✝⁵ : CommRing P\ninst✝⁴ : Algebra R P\nloc : IsLocalization S P\nP' : Type ?u.654570\ninst✝³ : CommRing P'\ninst✝² : Algebra R P'\nloc' : IsLocalization S P'\nP'' : Type ?u.655023\ninst✝¹ : CommRing P''\ninst✝ : Algebra R P''\nloc'' : IsLocalization S P''\nI J : FractionalIdeal S P\ng : P →ₐ[R] P'\nx : FractionalIdeal S P\n⊢ ↑(mapEquiv AlgEquiv.refl) x = ↑(RingEquiv.refl (FractionalIdeal S P)) x",
"tactic": "simp"
}
] |
[
839,
33
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
838,
1
] |
Mathlib/Analysis/SpecialFunctions/Log/Base.lean
|
Real.logb_rpow
|
[
{
"state_after": "b x y : ℝ\nb_pos : 0 < b\nb_ne_one : b ≠ 1\n⊢ log b ≠ 0",
"state_before": "b x y : ℝ\nb_pos : 0 < b\nb_ne_one : b ≠ 1\n⊢ logb b (b ^ x) = x",
"tactic": "rw [logb, div_eq_iff, log_rpow b_pos]"
},
{
"state_after": "no goals",
"state_before": "b x y : ℝ\nb_pos : 0 < b\nb_ne_one : b ≠ 1\n⊢ log b ≠ 0",
"tactic": "exact log_b_ne_zero b_pos b_ne_one"
}
] |
[
92,
37
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
90,
1
] |
Mathlib/Algebra/Ring/Divisibility.lean
|
dvd_add_self_left
|
[] |
[
140,
29
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
139,
1
] |
Std/Data/String/Lemmas.lean
|
String.Iterator.ValidFor.next
|
[
{
"state_after": "case refl\nl : List Char\nc : Char\nr : List Char\nh : ValidFor l (c :: r) { s := { data := List.reverse l ++ c :: r }, i := { byteIdx := utf8Len (List.reverse l) } }\n⊢ ValidFor (c :: l) r\n (Iterator.next { s := { data := List.reverse l ++ c :: r }, i := { byteIdx := utf8Len (List.reverse l) } })",
"state_before": "l : List Char\nc : Char\nr : List Char\nit : Iterator\nh : ValidFor l (c :: r) it\n⊢ ValidFor (c :: l) r (Iterator.next it)",
"tactic": "cases h.out'"
},
{
"state_after": "case refl\nl : List Char\nc : Char\nr : List Char\nh : ValidFor l (c :: r) { s := { data := List.reverse l ++ c :: r }, i := { byteIdx := utf8Len (List.reverse l) } }\n⊢ ValidFor (c :: l) r\n { s := { data := List.reverse l ++ c :: r }, i := { byteIdx := utf8Len (List.reverse l) + csize c } }",
"state_before": "case refl\nl : List Char\nc : Char\nr : List Char\nh : ValidFor l (c :: r) { s := { data := List.reverse l ++ c :: r }, i := { byteIdx := utf8Len (List.reverse l) } }\n⊢ ValidFor (c :: l) r\n (Iterator.next { s := { data := List.reverse l ++ c :: r }, i := { byteIdx := utf8Len (List.reverse l) } })",
"tactic": "simp only [Iterator.next, next_of_valid l.reverse c r]"
},
{
"state_after": "case refl\nl : List Char\nc : Char\nr : List Char\nh : ValidFor l (c :: r) { s := { data := List.reverse l ++ c :: r }, i := { byteIdx := utf8Len (List.reverse l) } }\n⊢ ValidFor (c :: l) r { s := { data := List.reverseAux l (c :: r) }, i := { byteIdx := utf8Len l + csize c } }",
"state_before": "case refl\nl : List Char\nc : Char\nr : List Char\nh : ValidFor l (c :: r) { s := { data := List.reverse l ++ c :: r }, i := { byteIdx := utf8Len (List.reverse l) } }\n⊢ ValidFor (c :: l) r\n { s := { data := List.reverse l ++ c :: r }, i := { byteIdx := utf8Len (List.reverse l) + csize c } }",
"tactic": "rw [← List.reverseAux_eq, utf8Len_reverse]"
},
{
"state_after": "no goals",
"state_before": "case refl\nl : List Char\nc : Char\nr : List Char\nh : ValidFor l (c :: r) { s := { data := List.reverse l ++ c :: r }, i := { byteIdx := utf8Len (List.reverse l) } }\n⊢ ValidFor (c :: l) r { s := { data := List.reverseAux l (c :: r) }, i := { byteIdx := utf8Len l + csize c } }",
"tactic": "constructor"
}
] |
[
552,
60
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
548,
1
] |
Mathlib/Data/String/Basic.lean
|
String.Iterator.hasNext_cons_addChar
|
[
{
"state_after": "no goals",
"state_before": "c : Char\ncs : List Char\ni : Pos\n⊢ hasNext { s := { data := c :: cs }, i := i + c } = hasNext { s := { data := cs }, i := i }",
"tactic": "simp [hasNext, Nat.add_lt_add_iff_lt_right]"
}
] |
[
25,
46
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
23,
1
] |
Mathlib/GroupTheory/GroupAction/SubMulAction.lean
|
SubMulAction.stabilizer_of_subMul
|
[
{
"state_after": "S : Type u'\nT : Type u''\nR : Type u\nM : Type v\ninst✝¹ : Group R\ninst✝ : MulAction R M\np : SubMulAction R M\nm : { x // x ∈ p }\n⊢ (MulAction.stabilizer R m).toSubmonoid = (MulAction.stabilizer R ↑m).toSubmonoid",
"state_before": "S : Type u'\nT : Type u''\nR : Type u\nM : Type v\ninst✝¹ : Group R\ninst✝ : MulAction R M\np : SubMulAction R M\nm : { x // x ∈ p }\n⊢ MulAction.stabilizer R m = MulAction.stabilizer R ↑m",
"tactic": "rw [← Subgroup.toSubmonoid_eq]"
},
{
"state_after": "no goals",
"state_before": "S : Type u'\nT : Type u''\nR : Type u\nM : Type v\ninst✝¹ : Group R\ninst✝ : MulAction R M\np : SubMulAction R M\nm : { x // x ∈ p }\n⊢ (MulAction.stabilizer R m).toSubmonoid = (MulAction.stabilizer R ↑m).toSubmonoid",
"tactic": "exact stabilizer_of_subMul.submonoid m"
}
] |
[
323,
41
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
320,
1
] |
Mathlib/Analysis/SpecialFunctions/Trigonometric/ArctanDeriv.lean
|
DifferentiableWithinAt.arctan
|
[] |
[
195,
53
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
193,
1
] |
Mathlib/MeasureTheory/Integral/IntegrableOn.lean
|
MeasureTheory.IntegrableOn.integrable_indicator
|
[] |
[
268,
36
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
266,
1
] |
Mathlib/Algebra/Free.lean
|
FreeSemigroup.map_of
|
[] |
[
570,
52
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
570,
1
] |
Mathlib/Data/Finset/Basic.lean
|
Finset.coe_inter
|
[] |
[
1598,
29
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1597,
1
] |
Mathlib/MeasureTheory/Integral/IntegralEqImproper.lean
|
MeasureTheory.integrableOn_Ioc_of_interval_integral_norm_bounded_right
|
[] |
[
614,
81
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
611,
1
] |
Mathlib/Order/LiminfLimsup.lean
|
Filter.SupHom.apply_blimsup_le
|
[
{
"state_after": "α : Type u_2\nβ : Type u_3\nγ : Type u_1\nι : Type ?u.152425\ninst✝¹ : CompleteLattice α\nf g✝ : Filter β\np q : β → Prop\nu v : β → α\ninst✝ : CompleteLattice γ\ng : sSupHom α γ\n⊢ ↑g (⨅ (s : Set β) (_ : s ∈ f), ⨆ (b : β) (_ : p b ∧ b ∈ s), u b) ≤\n ⨅ (s : Set β) (_ : s ∈ f), ⨆ (b : β) (_ : p b ∧ b ∈ s), ↑g (u b)",
"state_before": "α : Type u_2\nβ : Type u_3\nγ : Type u_1\nι : Type ?u.152425\ninst✝¹ : CompleteLattice α\nf g✝ : Filter β\np q : β → Prop\nu v : β → α\ninst✝ : CompleteLattice γ\ng : sSupHom α γ\n⊢ ↑g (blimsup u f p) ≤ blimsup (↑g ∘ u) f p",
"tactic": "simp only [blimsup_eq_iInf_biSup, Function.comp]"
},
{
"state_after": "α : Type u_2\nβ : Type u_3\nγ : Type u_1\nι : Type ?u.152425\ninst✝¹ : CompleteLattice α\nf g✝ : Filter β\np q : β → Prop\nu v : β → α\ninst✝ : CompleteLattice γ\ng : sSupHom α γ\n⊢ (⨅ (i : Set β) (_ : i ∈ f), ↑g (⨆ (b : β) (_ : p b ∧ b ∈ i), u b)) ≤\n ⨅ (s : Set β) (_ : s ∈ f), ⨆ (b : β) (_ : p b ∧ b ∈ s), ↑g (u b)",
"state_before": "α : Type u_2\nβ : Type u_3\nγ : Type u_1\nι : Type ?u.152425\ninst✝¹ : CompleteLattice α\nf g✝ : Filter β\np q : β → Prop\nu v : β → α\ninst✝ : CompleteLattice γ\ng : sSupHom α γ\n⊢ ↑g (⨅ (s : Set β) (_ : s ∈ f), ⨆ (b : β) (_ : p b ∧ b ∈ s), u b) ≤\n ⨅ (s : Set β) (_ : s ∈ f), ⨆ (b : β) (_ : p b ∧ b ∈ s), ↑g (u b)",
"tactic": "refine' ((OrderHomClass.mono g).map_iInf₂_le _).trans _"
},
{
"state_after": "no goals",
"state_before": "α : Type u_2\nβ : Type u_3\nγ : Type u_1\nι : Type ?u.152425\ninst✝¹ : CompleteLattice α\nf g✝ : Filter β\np q : β → Prop\nu v : β → α\ninst✝ : CompleteLattice γ\ng : sSupHom α γ\n⊢ (⨅ (i : Set β) (_ : i ∈ f), ↑g (⨆ (b : β) (_ : p b ∧ b ∈ i), u b)) ≤\n ⨅ (s : Set β) (_ : s ∈ f), ⨆ (b : β) (_ : p b ∧ b ∈ s), ↑g (u b)",
"tactic": "simp only [_root_.map_iSup, le_refl]"
}
] |
[
966,
39
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
962,
1
] |
Mathlib/Topology/LocallyFinite.lean
|
LocallyFinite.eventually_smallSets
|
[] |
[
71,
38
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
69,
11
] |
Mathlib/Algebra/BigOperators/Basic.lean
|
Finset.prod_range_add
|
[
{
"state_after": "case zero\nι : Type ?u.441582\nβ : Type u\nα : Type v\nγ : Type w\ns s₁ s₂ : Finset α\na : α\nf✝ g : α → β\ninst✝ : CommMonoid β\nf : ℕ → β\nn : ℕ\n⊢ ∏ x in range (n + Nat.zero), f x = (∏ x in range n, f x) * ∏ x in range Nat.zero, f (n + x)\n\ncase succ\nι : Type ?u.441582\nβ : Type u\nα : Type v\nγ : Type w\ns s₁ s₂ : Finset α\na : α\nf✝ g : α → β\ninst✝ : CommMonoid β\nf : ℕ → β\nn m : ℕ\nhm : ∏ x in range (n + m), f x = (∏ x in range n, f x) * ∏ x in range m, f (n + x)\n⊢ ∏ x in range (n + Nat.succ m), f x = (∏ x in range n, f x) * ∏ x in range (Nat.succ m), f (n + x)",
"state_before": "ι : Type ?u.441582\nβ : Type u\nα : Type v\nγ : Type w\ns s₁ s₂ : Finset α\na : α\nf✝ g : α → β\ninst✝ : CommMonoid β\nf : ℕ → β\nn m : ℕ\n⊢ ∏ x in range (n + m), f x = (∏ x in range n, f x) * ∏ x in range m, f (n + x)",
"tactic": "induction' m with m hm"
},
{
"state_after": "no goals",
"state_before": "case zero\nι : Type ?u.441582\nβ : Type u\nα : Type v\nγ : Type w\ns s₁ s₂ : Finset α\na : α\nf✝ g : α → β\ninst✝ : CommMonoid β\nf : ℕ → β\nn : ℕ\n⊢ ∏ x in range (n + Nat.zero), f x = (∏ x in range n, f x) * ∏ x in range Nat.zero, f (n + x)",
"tactic": "simp"
},
{
"state_after": "no goals",
"state_before": "case succ\nι : Type ?u.441582\nβ : Type u\nα : Type v\nγ : Type w\ns s₁ s₂ : Finset α\na : α\nf✝ g : α → β\ninst✝ : CommMonoid β\nf : ℕ → β\nn m : ℕ\nhm : ∏ x in range (n + m), f x = (∏ x in range n, f x) * ∏ x in range m, f (n + x)\n⊢ ∏ x in range (n + Nat.succ m), f x = (∏ x in range n, f x) * ∏ x in range (Nat.succ m), f (n + x)",
"tactic": "erw [Nat.add_succ, prod_range_succ, prod_range_succ, hm, mul_assoc]"
}
] |
[
1249,
72
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1245,
1
] |
Mathlib/RingTheory/SimpleModule.lean
|
LinearMap.surjective_of_ne_zero
|
[] |
[
151,
42
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
149,
1
] |
Mathlib/Topology/Covering.lean
|
isCoveringMap_iff_isCoveringMapOn_univ
|
[
{
"state_after": "no goals",
"state_before": "E : Type u_1\nX : Type u_2\ninst✝¹ : TopologicalSpace E\ninst✝ : TopologicalSpace X\nf : E → X\ns : Set X\n⊢ IsCoveringMap f ↔ IsCoveringMapOn f Set.univ",
"tactic": "simp only [IsCoveringMap, IsCoveringMapOn, Set.mem_univ, forall_true_left]"
}
] |
[
144,
77
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
143,
1
] |
Mathlib/Probability/Kernel/Basic.lean
|
ProbabilityTheory.kernel.measurable_coe
|
[] |
[
214,
57
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
212,
11
] |
Mathlib/LinearAlgebra/AffineSpace/FiniteDimensional.lean
|
Affine.Simplex.span_eq_top
|
[
{
"state_after": "no goals",
"state_before": "k : Type u_1\nV : Type u_2\nP : Type ?u.206684\nι : Type ?u.206687\ninst✝⁴ : DivisionRing k\ninst✝³ : AddCommGroup V\ninst✝² : Module k V\ninst✝¹ : AffineSpace V P\ninst✝ : FiniteDimensional k V\nn : ℕ\nT : Simplex k V n\nhrank : finrank k V = n\n⊢ affineSpan k (Set.range T.points) = ⊤",
"tactic": "rw [AffineIndependent.affineSpan_eq_top_iff_card_eq_finrank_add_one T.Independent,\n Fintype.card_fin, hrank]"
}
] |
[
280,
29
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
277,
1
] |
Mathlib/Data/Seq/Computation.lean
|
Computation.liftRel_think_right
|
[
{
"state_after": "α : Type u\nβ : Type v\nγ : Type w\nR : α → β → Prop\nca : Computation α\ncb : Computation β\n⊢ LiftRel (swap R) (think cb) ca ↔ LiftRel (swap R) cb ca",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\nR : α → β → Prop\nca : Computation α\ncb : Computation β\n⊢ LiftRel R ca (think cb) ↔ LiftRel R ca cb",
"tactic": "rw [← LiftRel.swap R, ← LiftRel.swap R]"
},
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\nR : α → β → Prop\nca : Computation α\ncb : Computation β\n⊢ LiftRel (swap R) (think cb) ca ↔ LiftRel (swap R) cb ca",
"tactic": "apply liftRel_think_left"
}
] |
[
1211,
69
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1209,
1
] |
Mathlib/RingTheory/MvPolynomial/Symmetric.lean
|
MvPolynomial.IsSymmetric.zero
|
[] |
[
115,
37
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
114,
1
] |
Mathlib/Algebra/Algebra/Equiv.lean
|
AlgEquiv.symm_toEquiv_eq_symm
|
[] |
[
341,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
340,
1
] |
Mathlib/Data/Fintype/Card.lean
|
Finite.injective_iff_bijective
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.27086\nγ : Type ?u.27089\ninst✝ : Finite α\nf : α → α\n⊢ Injective f ↔ Bijective f",
"tactic": "simp [Bijective, injective_iff_surjective]"
}
] |
[
634,
45
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
633,
1
] |
Mathlib/Analysis/Convex/Basic.lean
|
Convex.openSegment_subset
|
[] |
[
75,
68
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
73,
1
] |
Mathlib/Combinatorics/SimpleGraph/Basic.lean
|
SimpleGraph.Embedding.apply_mem_neighborSet_iff
|
[] |
[
1786,
16
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1785,
1
] |
Mathlib/Analysis/SpecialFunctions/Integrals.lean
|
integral_log_of_pos
|
[] |
[
491,
43
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
489,
1
] |
Mathlib/FieldTheory/Separable.lean
|
Polynomial.separable_prod_X_sub_C_iff
|
[
{
"state_after": "no goals",
"state_before": "F : Type u\ninst✝² : Field F\nK : Type v\ninst✝¹ : Field K\nι : Type u_1\ninst✝ : Fintype ι\nf : ι → F\n⊢ (∀ (x : ι), x ∈ univ → ∀ (y : ι), y ∈ univ → f x = f y → x = y) ↔ Function.Injective f",
"tactic": "simp_rw [mem_univ, true_imp_iff, Function.Injective]"
}
] |
[
309,
95
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
307,
1
] |
Mathlib/Algebra/Order/CompleteField.lean
|
LinearOrderedField.cutMap_nonempty
|
[] |
[
144,
38
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
143,
1
] |
Mathlib/Data/Finset/LocallyFinite.lean
|
Finset.Ico_filter_lt_of_right_le
|
[] |
[
322,
63
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
320,
1
] |
Mathlib/MeasureTheory/Measure/FiniteMeasure.lean
|
MeasureTheory.FiniteMeasure.mass_zero_iff
|
[
{
"state_after": "Ω : Type u_1\ninst✝ : MeasurableSpace Ω\nμ : FiniteMeasure Ω\nμ_mass : mass μ = 0\n⊢ μ = 0",
"state_before": "Ω : Type u_1\ninst✝ : MeasurableSpace Ω\nμ : FiniteMeasure Ω\n⊢ mass μ = 0 ↔ μ = 0",
"tactic": "refine' ⟨fun μ_mass => _, fun hμ => by simp only [hμ, zero_mass]⟩"
},
{
"state_after": "case a\nΩ : Type u_1\ninst✝ : MeasurableSpace Ω\nμ : FiniteMeasure Ω\nμ_mass : mass μ = 0\n⊢ ↑μ = ↑0",
"state_before": "Ω : Type u_1\ninst✝ : MeasurableSpace Ω\nμ : FiniteMeasure Ω\nμ_mass : mass μ = 0\n⊢ μ = 0",
"tactic": "apply toMeasure_injective"
},
{
"state_after": "case a\nΩ : Type u_1\ninst✝ : MeasurableSpace Ω\nμ : FiniteMeasure Ω\nμ_mass : mass μ = 0\n⊢ ↑↑↑μ univ = 0",
"state_before": "case a\nΩ : Type u_1\ninst✝ : MeasurableSpace Ω\nμ : FiniteMeasure Ω\nμ_mass : mass μ = 0\n⊢ ↑μ = ↑0",
"tactic": "apply Measure.measure_univ_eq_zero.mp"
},
{
"state_after": "no goals",
"state_before": "case a\nΩ : Type u_1\ninst✝ : MeasurableSpace Ω\nμ : FiniteMeasure Ω\nμ_mass : mass μ = 0\n⊢ ↑↑↑μ univ = 0",
"tactic": "rwa [← ennreal_mass, ENNReal.coe_eq_zero]"
},
{
"state_after": "no goals",
"state_before": "Ω : Type u_1\ninst✝ : MeasurableSpace Ω\nμ : FiniteMeasure Ω\nhμ : μ = 0\n⊢ mass μ = 0",
"tactic": "simp only [hμ, zero_mass]"
}
] |
[
182,
44
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
178,
1
] |
Std/Data/List/Lemmas.lean
|
List.diff_cons
|
[
{
"state_after": "α : Type u_1\ninst✝ : DecidableEq α\nl₁ l₂ : List α\na : α\n⊢ (if a ∈ l₁ then List.diff (List.erase l₁ a) l₂ else List.diff l₁ l₂) = List.diff (List.erase l₁ a) l₂",
"state_before": "α : Type u_1\ninst✝ : DecidableEq α\nl₁ l₂ : List α\na : α\n⊢ List.diff l₁ (a :: l₂) = List.diff (List.erase l₁ a) l₂",
"tactic": "simp [List.diff]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝ : DecidableEq α\nl₁ l₂ : List α\na : α\n⊢ (if a ∈ l₁ then List.diff (List.erase l₁ a) l₂ else List.diff l₁ l₂) = List.diff (List.erase l₁ a) l₂",
"tactic": "split <;> simp [*, erase_of_not_mem]"
}
] |
[
1495,
57
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
1494,
9
] |
Mathlib/GroupTheory/Subgroup/Basic.lean
|
Subgroup.toSubmonoid_mono
|
[] |
[
473,
34
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
472,
1
] |
src/lean/Init/Control/Lawful.lean
|
pure_id_seq
|
[
{
"state_after": "no goals",
"state_before": "f : Type u_1 → Type u_2\nα : Type u_1\ninst✝¹ : Applicative f\ninst✝ : LawfulApplicative f\nx : f α\n⊢ (Seq.seq (pure id) fun x_1 => x) = x",
"tactic": "simp [pure_seq]"
}
] |
[
44,
18
] |
d5348dfac847a56a4595fb6230fd0708dcb4e7e9
|
https://github.com/leanprover/lean4
|
[
43,
9
] |
Mathlib/SetTheory/Cardinal/Basic.lean
|
Cardinal.inductionOn
|
[] |
[
129,
27
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
128,
1
] |
Mathlib/Order/Directed.lean
|
directedOn_of_inf_mem
|
[] |
[
166,
46
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
164,
1
] |
Mathlib/Algebra/Algebra/Subalgebra/Basic.lean
|
Subalgebra.centralizer_le
|
[] |
[
1411,
27
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1410,
1
] |
Mathlib/Data/Real/CauSeq.lean
|
CauSeq.of_near
|
[
{
"state_after": "case intro\nα : Type u_1\nβ : Type u_2\ninst✝² : LinearOrderedField α\ninst✝¹ : Ring β\nabv : β → α\ninst✝ : IsAbsoluteValue abv\nf : ℕ → β\ng : CauSeq β abv\nh : ∀ (ε : α), ε > 0 → ∃ i, ∀ (j : ℕ), j ≥ i → abv (f j - ↑g j) < ε\nε : α\nε0 : ε > 0\ni : ℕ\nhi : ∀ (j : ℕ), j ≥ i → abv (f j - ↑g j) < ε / 2 / 2 ∧ ∀ (k : ℕ), k ≥ j → abv (↑g k - ↑g j) < ε / 2\nj : ℕ\nij : j ≥ i\nh₁ : abv (f i - ↑g i) < ε / 2 / 2\nh₂ : ∀ (k : ℕ), k ≥ i → abv (↑g k - ↑g i) < ε / 2\n⊢ abv (f j - f i) < ε",
"state_before": "α : Type u_1\nβ : Type u_2\ninst✝² : LinearOrderedField α\ninst✝¹ : Ring β\nabv : β → α\ninst✝ : IsAbsoluteValue abv\nf : ℕ → β\ng : CauSeq β abv\nh : ∀ (ε : α), ε > 0 → ∃ i, ∀ (j : ℕ), j ≥ i → abv (f j - ↑g j) < ε\nε : α\nε0 : ε > 0\ni : ℕ\nhi : ∀ (j : ℕ), j ≥ i → abv (f j - ↑g j) < ε / 2 / 2 ∧ ∀ (k : ℕ), k ≥ j → abv (↑g k - ↑g j) < ε / 2\nj : ℕ\nij : j ≥ i\n⊢ abv (f j - f i) < ε",
"tactic": "cases' hi _ le_rfl with h₁ h₂"
},
{
"state_after": "case intro\nα : Type u_1\nβ : Type u_2\ninst✝² : LinearOrderedField α\ninst✝¹ : Ring β\nabv : β → α\ninst✝ : IsAbsoluteValue abv\nf : ℕ → β\ng : CauSeq β abv\nh : ∀ (ε : α), ε > 0 → ∃ i, ∀ (j : ℕ), j ≥ i → abv (f j - ↑g j) < ε\nε : α\nε0 : ε > 0\ni : ℕ\nhi : ∀ (j : ℕ), j ≥ i → abv (f j - ↑g j) < ε / 2 / 2 ∧ ∀ (k : ℕ), k ≥ j → abv (↑g k - ↑g j) < ε / 2\nj : ℕ\nij : j ≥ i\nh₁ : abv (↑g i - f i) < ε / 2 / 2\nh₂ : ∀ (k : ℕ), k ≥ i → abv (↑g k - ↑g i) < ε / 2\n⊢ abv (f j - f i) < ε",
"state_before": "case intro\nα : Type u_1\nβ : Type u_2\ninst✝² : LinearOrderedField α\ninst✝¹ : Ring β\nabv : β → α\ninst✝ : IsAbsoluteValue abv\nf : ℕ → β\ng : CauSeq β abv\nh : ∀ (ε : α), ε > 0 → ∃ i, ∀ (j : ℕ), j ≥ i → abv (f j - ↑g j) < ε\nε : α\nε0 : ε > 0\ni : ℕ\nhi : ∀ (j : ℕ), j ≥ i → abv (f j - ↑g j) < ε / 2 / 2 ∧ ∀ (k : ℕ), k ≥ j → abv (↑g k - ↑g j) < ε / 2\nj : ℕ\nij : j ≥ i\nh₁ : abv (f i - ↑g i) < ε / 2 / 2\nh₂ : ∀ (k : ℕ), k ≥ i → abv (↑g k - ↑g i) < ε / 2\n⊢ abv (f j - f i) < ε",
"tactic": "rw [abv_sub abv] at h₁"
},
{
"state_after": "case intro\nα : Type u_1\nβ : Type u_2\ninst✝² : LinearOrderedField α\ninst✝¹ : Ring β\nabv : β → α\ninst✝ : IsAbsoluteValue abv\nf : ℕ → β\ng : CauSeq β abv\nh : ∀ (ε : α), ε > 0 → ∃ i, ∀ (j : ℕ), j ≥ i → abv (f j - ↑g j) < ε\nε : α\nε0 : ε > 0\ni : ℕ\nhi : ∀ (j : ℕ), j ≥ i → abv (f j - ↑g j) < ε / 2 / 2 ∧ ∀ (k : ℕ), k ≥ j → abv (↑g k - ↑g j) < ε / 2\nj : ℕ\nij : j ≥ i\nh₁ : abv (↑g i - f i) < ε / 2 / 2\nh₂ : ∀ (k : ℕ), k ≥ i → abv (↑g k - ↑g i) < ε / 2\nthis : abv (f j - ↑g j + (↑g i - f i)) < ε / 2 / 2 + ε / 2 / 2\n⊢ abv (f j - f i) < ε",
"state_before": "case intro\nα : Type u_1\nβ : Type u_2\ninst✝² : LinearOrderedField α\ninst✝¹ : Ring β\nabv : β → α\ninst✝ : IsAbsoluteValue abv\nf : ℕ → β\ng : CauSeq β abv\nh : ∀ (ε : α), ε > 0 → ∃ i, ∀ (j : ℕ), j ≥ i → abv (f j - ↑g j) < ε\nε : α\nε0 : ε > 0\ni : ℕ\nhi : ∀ (j : ℕ), j ≥ i → abv (f j - ↑g j) < ε / 2 / 2 ∧ ∀ (k : ℕ), k ≥ j → abv (↑g k - ↑g j) < ε / 2\nj : ℕ\nij : j ≥ i\nh₁ : abv (↑g i - f i) < ε / 2 / 2\nh₂ : ∀ (k : ℕ), k ≥ i → abv (↑g k - ↑g i) < ε / 2\n⊢ abv (f j - f i) < ε",
"tactic": "have := lt_of_le_of_lt (abv_add abv _ _) (add_lt_add (hi _ ij).1 h₁)"
},
{
"state_after": "case intro\nα : Type u_1\nβ : Type u_2\ninst✝² : LinearOrderedField α\ninst✝¹ : Ring β\nabv : β → α\ninst✝ : IsAbsoluteValue abv\nf : ℕ → β\ng : CauSeq β abv\nh : ∀ (ε : α), ε > 0 → ∃ i, ∀ (j : ℕ), j ≥ i → abv (f j - ↑g j) < ε\nε : α\nε0 : ε > 0\ni : ℕ\nhi : ∀ (j : ℕ), j ≥ i → abv (f j - ↑g j) < ε / 2 / 2 ∧ ∀ (k : ℕ), k ≥ j → abv (↑g k - ↑g j) < ε / 2\nj : ℕ\nij : j ≥ i\nh₁ : abv (↑g i - f i) < ε / 2 / 2\nh₂ : ∀ (k : ℕ), k ≥ i → abv (↑g k - ↑g i) < ε / 2\nthis✝ : abv (f j - ↑g j + (↑g i - f i)) < ε / 2 / 2 + ε / 2 / 2\nthis : abv (f j - ↑g j + (↑g i - f i) + (↑g j - ↑g i)) < ε / 2 / 2 + ε / 2 / 2 + ε / 2\n⊢ abv (f j - f i) < ε",
"state_before": "case intro\nα : Type u_1\nβ : Type u_2\ninst✝² : LinearOrderedField α\ninst✝¹ : Ring β\nabv : β → α\ninst✝ : IsAbsoluteValue abv\nf : ℕ → β\ng : CauSeq β abv\nh : ∀ (ε : α), ε > 0 → ∃ i, ∀ (j : ℕ), j ≥ i → abv (f j - ↑g j) < ε\nε : α\nε0 : ε > 0\ni : ℕ\nhi : ∀ (j : ℕ), j ≥ i → abv (f j - ↑g j) < ε / 2 / 2 ∧ ∀ (k : ℕ), k ≥ j → abv (↑g k - ↑g j) < ε / 2\nj : ℕ\nij : j ≥ i\nh₁ : abv (↑g i - f i) < ε / 2 / 2\nh₂ : ∀ (k : ℕ), k ≥ i → abv (↑g k - ↑g i) < ε / 2\nthis : abv (f j - ↑g j + (↑g i - f i)) < ε / 2 / 2 + ε / 2 / 2\n⊢ abv (f j - f i) < ε",
"tactic": "have := lt_of_le_of_lt (abv_add abv _ _) (add_lt_add this (h₂ _ ij))"
},
{
"state_after": "no goals",
"state_before": "case intro\nα : Type u_1\nβ : Type u_2\ninst✝² : LinearOrderedField α\ninst✝¹ : Ring β\nabv : β → α\ninst✝ : IsAbsoluteValue abv\nf : ℕ → β\ng : CauSeq β abv\nh : ∀ (ε : α), ε > 0 → ∃ i, ∀ (j : ℕ), j ≥ i → abv (f j - ↑g j) < ε\nε : α\nε0 : ε > 0\ni : ℕ\nhi : ∀ (j : ℕ), j ≥ i → abv (f j - ↑g j) < ε / 2 / 2 ∧ ∀ (k : ℕ), k ≥ j → abv (↑g k - ↑g j) < ε / 2\nj : ℕ\nij : j ≥ i\nh₁ : abv (↑g i - f i) < ε / 2 / 2\nh₂ : ∀ (k : ℕ), k ≥ i → abv (↑g k - ↑g i) < ε / 2\nthis✝ : abv (f j - ↑g j + (↑g i - f i)) < ε / 2 / 2 + ε / 2 / 2\nthis : abv (f j - ↑g j + (↑g i - f i) + (↑g j - ↑g i)) < ε / 2 / 2 + ε / 2 / 2 + ε / 2\n⊢ abv (f j - f i) < ε",
"tactic": "rwa [add_halves, add_halves, add_right_comm, sub_add_sub_cancel, sub_add_sub_cancel] at this"
}
] |
[
519,
100
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
511,
1
] |
Mathlib/NumberTheory/PellMatiyasevic.lean
|
Pell.strictMono_x
|
[
{
"state_after": "a : ℕ\na1 : 1 < a\nm n : ℕ\nh : m < n + 1\nthis : xn a1 m ≤ xn a1 n\n⊢ xn a1 m < xn a1 (n + 1)",
"state_before": "a : ℕ\na1 : 1 < a\nm n : ℕ\nh : m < n + 1\n⊢ xn a1 m < xn a1 (n + 1)",
"tactic": "have : xn a1 m ≤ xn a1 n :=\n Or.elim (lt_or_eq_of_le <| Nat.le_of_succ_le_succ h) (fun hl => le_of_lt <| strictMono_x hl)\n fun e => by rw [e]"
},
{
"state_after": "a : ℕ\na1 : 1 < a\nm n : ℕ\nh : m < n + 1\nthis : xn a1 m ≤ xn a1 n\n⊢ xn a1 m < xn a1 n * a + Pell.d a1 * yn a1 n",
"state_before": "a : ℕ\na1 : 1 < a\nm n : ℕ\nh : m < n + 1\nthis : xn a1 m ≤ xn a1 n\n⊢ xn a1 m < xn a1 (n + 1)",
"tactic": "simp"
},
{
"state_after": "a : ℕ\na1 : 1 < a\nm n : ℕ\nh : m < n + 1\nthis : xn a1 m ≤ xn a1 n\n⊢ xn a1 n < xn a1 n * a",
"state_before": "a : ℕ\na1 : 1 < a\nm n : ℕ\nh : m < n + 1\nthis : xn a1 m ≤ xn a1 n\n⊢ xn a1 m < xn a1 n * a + Pell.d a1 * yn a1 n",
"tactic": "refine' lt_of_lt_of_le (lt_of_le_of_lt this _) (Nat.le_add_right _ _)"
},
{
"state_after": "a : ℕ\na1 : 1 < a\nm n : ℕ\nh : m < n + 1\nthis : xn a1 m ≤ xn a1 n\nt : xn a1 n * 1 < xn a1 n * a\n⊢ xn a1 n < xn a1 n * a",
"state_before": "a : ℕ\na1 : 1 < a\nm n : ℕ\nh : m < n + 1\nthis : xn a1 m ≤ xn a1 n\n⊢ xn a1 n < xn a1 n * a",
"tactic": "have t := Nat.mul_lt_mul_of_pos_left a1 (x_pos a1 n)"
},
{
"state_after": "no goals",
"state_before": "a : ℕ\na1 : 1 < a\nm n : ℕ\nh : m < n + 1\nthis : xn a1 m ≤ xn a1 n\nt : xn a1 n * 1 < xn a1 n * a\n⊢ xn a1 n < xn a1 n * a",
"tactic": "rwa [mul_one] at t"
},
{
"state_after": "no goals",
"state_before": "a : ℕ\na1 : 1 < a\nm n : ℕ\nh : m < n + 1\ne : m = n\n⊢ xn a1 m ≤ xn a1 n",
"tactic": "rw [e]"
}
] |
[
423,
23
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
415,
1
] |
Mathlib/Data/List/Nodup.lean
|
List.Nodup.erase
|
[] |
[
317,
37
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
316,
1
] |
Mathlib/Data/Matrix/Hadamard.lean
|
Matrix.dotProduct_vecMul_hadamard
|
[
{
"state_after": "α : Type u_3\nβ : Type ?u.28934\nγ : Type ?u.28937\nm : Type u_1\nn : Type u_2\nR : Type ?u.28946\nA B C : Matrix m n α\ninst✝⁶ : Fintype m\ninst✝⁵ : Fintype n\ninst✝⁴ : Semiring α\ninst✝³ : Semiring R\ninst✝² : Module R α\ninst✝¹ : DecidableEq m\ninst✝ : DecidableEq n\nv : m → α\nw : n → α\n⊢ vecMul v (A ⊙ B) ⬝ᵥ w = ∑ y : n, ∑ x : m, ((diagonal v ⬝ A) ⊙ (B ⬝ diagonal w)) x y",
"state_before": "α : Type u_3\nβ : Type ?u.28934\nγ : Type ?u.28937\nm : Type u_1\nn : Type u_2\nR : Type ?u.28946\nA B C : Matrix m n α\ninst✝⁶ : Fintype m\ninst✝⁵ : Fintype n\ninst✝⁴ : Semiring α\ninst✝³ : Semiring R\ninst✝² : Module R α\ninst✝¹ : DecidableEq m\ninst✝ : DecidableEq n\nv : m → α\nw : n → α\n⊢ vecMul v (A ⊙ B) ⬝ᵥ w = trace (diagonal v ⬝ A ⬝ (B ⬝ diagonal w)ᵀ)",
"tactic": "rw [← sum_hadamard_eq, Finset.sum_comm]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_3\nβ : Type ?u.28934\nγ : Type ?u.28937\nm : Type u_1\nn : Type u_2\nR : Type ?u.28946\nA B C : Matrix m n α\ninst✝⁶ : Fintype m\ninst✝⁵ : Fintype n\ninst✝⁴ : Semiring α\ninst✝³ : Semiring R\ninst✝² : Module R α\ninst✝¹ : DecidableEq m\ninst✝ : DecidableEq n\nv : m → α\nw : n → α\n⊢ vecMul v (A ⊙ B) ⬝ᵥ w = ∑ y : n, ∑ x : m, ((diagonal v ⬝ A) ⊙ (B ⬝ diagonal w)) x y",
"tactic": "simp [dotProduct, vecMul, Finset.sum_mul, mul_assoc]"
}
] |
[
158,
55
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
155,
1
] |
Mathlib/Analysis/Convex/Segment.lean
|
Prod.segment_subset
|
[
{
"state_after": "case intro.intro.intro.intro.intro\n𝕜 : Type u_3\nE : Type u_1\nF : Type u_2\nG : Type ?u.301724\nι : Type ?u.301727\nπ : ι → Type ?u.301732\ninst✝⁴ : OrderedSemiring 𝕜\ninst✝³ : AddCommMonoid E\ninst✝² : AddCommMonoid F\ninst✝¹ : Module 𝕜 E\ninst✝ : Module 𝕜 F\nx y z : E × F\na b : 𝕜\nha : 0 ≤ a\nhb : 0 ≤ b\nhab : a + b = 1\nhz : a • x + b • y = z\n⊢ z ∈ [x.fst-[𝕜]y.fst] ×ˢ [x.snd-[𝕜]y.snd]",
"state_before": "𝕜 : Type u_3\nE : Type u_1\nF : Type u_2\nG : Type ?u.301724\nι : Type ?u.301727\nπ : ι → Type ?u.301732\ninst✝⁴ : OrderedSemiring 𝕜\ninst✝³ : AddCommMonoid E\ninst✝² : AddCommMonoid F\ninst✝¹ : Module 𝕜 E\ninst✝ : Module 𝕜 F\nx y : E × F\n⊢ [x-[𝕜]y] ⊆ [x.fst-[𝕜]y.fst] ×ˢ [x.snd-[𝕜]y.snd]",
"tactic": "rintro z ⟨a, b, ha, hb, hab, hz⟩"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.intro.intro.intro\n𝕜 : Type u_3\nE : Type u_1\nF : Type u_2\nG : Type ?u.301724\nι : Type ?u.301727\nπ : ι → Type ?u.301732\ninst✝⁴ : OrderedSemiring 𝕜\ninst✝³ : AddCommMonoid E\ninst✝² : AddCommMonoid F\ninst✝¹ : Module 𝕜 E\ninst✝ : Module 𝕜 F\nx y z : E × F\na b : 𝕜\nha : 0 ≤ a\nhb : 0 ≤ b\nhab : a + b = 1\nhz : a • x + b • y = z\n⊢ z ∈ [x.fst-[𝕜]y.fst] ×ˢ [x.snd-[𝕜]y.snd]",
"tactic": "exact ⟨⟨a, b, ha, hb, hab, congr_arg Prod.fst hz⟩, a, b, ha, hb, hab, congr_arg Prod.snd hz⟩"
}
] |
[
594,
95
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
592,
1
] |
Mathlib/Topology/Bases.lean
|
TopologicalSpace.isTopologicalBasis_opens
|
[
{
"state_after": "no goals",
"state_before": "α : Type u\nt : TopologicalSpace α\n⊢ ∀ (u : Set α), u ∈ {U | IsOpen U} → IsOpen u",
"tactic": "tauto"
},
{
"state_after": "no goals",
"state_before": "α : Type u\nt : TopologicalSpace α\n⊢ ∀ (a : α) (u : Set α), a ∈ u → IsOpen u → ∃ v, v ∈ {U | IsOpen U} ∧ a ∈ v ∧ v ⊆ u",
"tactic": "tauto"
}
] |
[
242,
59
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
241,
1
] |
Mathlib/Geometry/Euclidean/Angle/Unoriented/RightAngle.lean
|
EuclideanGeometry.tan_angle_mul_dist_of_angle_eq_pi_div_two
|
[
{
"state_after": "V : Type u_1\nP : Type u_2\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\np₁ p₂ p₃ : P\nh : inner (p₂ -ᵥ p₃) (p₁ -ᵥ p₂) = 0\nh0 : p₁ = p₂ ∨ p₃ ≠ p₂\n⊢ Real.tan (∠ p₂ p₃ p₁) * dist p₃ p₂ = dist p₁ p₂",
"state_before": "V : Type u_1\nP : Type u_2\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\np₁ p₂ p₃ : P\nh : ∠ p₁ p₂ p₃ = π / 2\nh0 : p₁ = p₂ ∨ p₃ ≠ p₂\n⊢ Real.tan (∠ p₂ p₃ p₁) * dist p₃ p₂ = dist p₁ p₂",
"tactic": "rw [angle, ← inner_eq_zero_iff_angle_eq_pi_div_two, real_inner_comm, ← neg_eq_zero, ←\n inner_neg_left, neg_vsub_eq_vsub_rev] at h"
},
{
"state_after": "V : Type u_1\nP : Type u_2\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\np₁ p₂ p₃ : P\nh : inner (p₂ -ᵥ p₃) (p₁ -ᵥ p₂) = 0\nh0 : p₂ -ᵥ p₃ ≠ 0 ∨ p₁ -ᵥ p₂ = 0\n⊢ Real.tan (∠ p₂ p₃ p₁) * dist p₃ p₂ = dist p₁ p₂",
"state_before": "V : Type u_1\nP : Type u_2\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\np₁ p₂ p₃ : P\nh : inner (p₂ -ᵥ p₃) (p₁ -ᵥ p₂) = 0\nh0 : p₁ = p₂ ∨ p₃ ≠ p₂\n⊢ Real.tan (∠ p₂ p₃ p₁) * dist p₃ p₂ = dist p₁ p₂",
"tactic": "rw [ne_comm, ← @vsub_ne_zero V, ← @vsub_eq_zero_iff_eq V, or_comm] at h0"
},
{
"state_after": "no goals",
"state_before": "V : Type u_1\nP : Type u_2\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\np₁ p₂ p₃ : P\nh : inner (p₂ -ᵥ p₃) (p₁ -ᵥ p₂) = 0\nh0 : p₂ -ᵥ p₃ ≠ 0 ∨ p₁ -ᵥ p₂ = 0\n⊢ Real.tan (∠ p₂ p₃ p₁) * dist p₃ p₂ = dist p₁ p₂",
"tactic": "rw [angle, dist_eq_norm_vsub V p₁ p₂, dist_eq_norm_vsub' V p₃ p₂, ← vsub_add_vsub_cancel p₁ p₂ p₃,\n add_comm, tan_angle_add_mul_norm_of_inner_eq_zero h h0]"
}
] |
[
489,
60
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
483,
1
] |
Mathlib/RingTheory/PowerSeries/Basic.lean
|
PowerSeries.coeff_one_X
|
[
{
"state_after": "no goals",
"state_before": "R : Type u_1\ninst✝ : Semiring R\n⊢ ↑(coeff R 1) X = 1",
"tactic": "rw [coeff_X, if_pos rfl]"
}
] |
[
1441,
87
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1441,
1
] |
Mathlib/Order/CompleteBooleanAlgebra.lean
|
iSup_inf_of_monotone
|
[
{
"state_after": "α : Type u\nβ : Type v\nι✝ : Sort w\nκ : ι✝ → Sort ?u.12298\ninst✝² : Frame α\ns t : Set α\na b : α\nι : Type u_1\ninst✝¹ : Preorder ι\ninst✝ : IsDirected ι fun x x_1 => x ≤ x_1\nf g : ι → α\nhf : Monotone f\nhg : Monotone g\n⊢ ((⨆ (i : ι), f i) ⊓ ⨆ (i : ι), g i) ≤ ⨆ (i : ι), f i ⊓ g i",
"state_before": "α : Type u\nβ : Type v\nι✝ : Sort w\nκ : ι✝ → Sort ?u.12298\ninst✝² : Frame α\ns t : Set α\na b : α\nι : Type u_1\ninst✝¹ : Preorder ι\ninst✝ : IsDirected ι fun x x_1 => x ≤ x_1\nf g : ι → α\nhf : Monotone f\nhg : Monotone g\n⊢ (⨆ (i : ι), f i ⊓ g i) = (⨆ (i : ι), f i) ⊓ ⨆ (i : ι), g i",
"tactic": "refine' (le_iSup_inf_iSup f g).antisymm _"
},
{
"state_after": "α : Type u\nβ : Type v\nι✝ : Sort w\nκ : ι✝ → Sort ?u.12298\ninst✝² : Frame α\ns t : Set α\na b : α\nι : Type u_1\ninst✝¹ : Preorder ι\ninst✝ : IsDirected ι fun x x_1 => x ≤ x_1\nf g : ι → α\nhf : Monotone f\nhg : Monotone g\n⊢ (⨆ (i : ι × ι), f i.fst ⊓ g i.snd) ≤ ⨆ (i : ι), f i ⊓ g i",
"state_before": "α : Type u\nβ : Type v\nι✝ : Sort w\nκ : ι✝ → Sort ?u.12298\ninst✝² : Frame α\ns t : Set α\na b : α\nι : Type u_1\ninst✝¹ : Preorder ι\ninst✝ : IsDirected ι fun x x_1 => x ≤ x_1\nf g : ι → α\nhf : Monotone f\nhg : Monotone g\n⊢ ((⨆ (i : ι), f i) ⊓ ⨆ (i : ι), g i) ≤ ⨆ (i : ι), f i ⊓ g i",
"tactic": "rw [iSup_inf_iSup]"
},
{
"state_after": "α : Type u\nβ : Type v\nι✝ : Sort w\nκ : ι✝ → Sort ?u.12298\ninst✝² : Frame α\ns t : Set α\na b : α\nι : Type u_1\ninst✝¹ : Preorder ι\ninst✝ : IsDirected ι fun x x_1 => x ≤ x_1\nf g : ι → α\nhf : Monotone f\nhg : Monotone g\ni : ι × ι\n⊢ ∃ i', f i.fst ⊓ g i.snd ≤ f i' ⊓ g i'",
"state_before": "α : Type u\nβ : Type v\nι✝ : Sort w\nκ : ι✝ → Sort ?u.12298\ninst✝² : Frame α\ns t : Set α\na b : α\nι : Type u_1\ninst✝¹ : Preorder ι\ninst✝ : IsDirected ι fun x x_1 => x ≤ x_1\nf g : ι → α\nhf : Monotone f\nhg : Monotone g\n⊢ (⨆ (i : ι × ι), f i.fst ⊓ g i.snd) ≤ ⨆ (i : ι), f i ⊓ g i",
"tactic": "refine' iSup_mono' fun i => _"
},
{
"state_after": "no goals",
"state_before": "case intro.intro\nα : Type u\nβ : Type v\nι✝ : Sort w\nκ : ι✝ → Sort ?u.12298\ninst✝² : Frame α\ns t : Set α\na b : α\nι : Type u_1\ninst✝¹ : Preorder ι\ninst✝ : IsDirected ι fun x x_1 => x ≤ x_1\nf g : ι → α\nhf : Monotone f\nhg : Monotone g\ni : ι × ι\nj : ι\nh₁ : i.fst ≤ j\nh₂ : i.snd ≤ j\n⊢ ∃ i', f i.fst ⊓ g i.snd ≤ f i' ⊓ g i'",
"tactic": "exact ⟨j, inf_le_inf (hf h₁) (hg h₂)⟩"
}
] |
[
164,
40
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
158,
1
] |
Mathlib/Combinatorics/SimpleGraph/Regularity/Equitabilise.lean
|
Finpartition.card_filter_equitabilise_small
|
[
{
"state_after": "α : Type u_1\ninst✝ : DecidableEq α\ns t : Finset α\nm n a b : ℕ\nP : Finpartition s\nh : a * m + b * (m + 1) = card s\nhm : m ≠ 0\n⊢ card (filter (fun u => card u = m) (equitabilise h).parts) * m + b * (m + 1) = a * m + b * (m + 1)",
"state_before": "α : Type u_1\ninst✝ : DecidableEq α\ns t : Finset α\nm n a b : ℕ\nP : Finpartition s\nh : a * m + b * (m + 1) = card s\nhm : m ≠ 0\n⊢ card (filter (fun u => card u = m) (equitabilise h).parts) = a",
"tactic": "refine' (mul_eq_mul_right_iff.1 <| (add_left_inj (b * (m + 1))).1 _).resolve_right hm"
},
{
"state_after": "α : Type u_1\ninst✝ : DecidableEq α\ns t : Finset α\nm n a b : ℕ\nP : Finpartition s\nh : a * m + b * (m + 1) = card s\nhm : m ≠ 0\n⊢ card (filter (fun u => card u = m) (equitabilise h).parts) * m + b * (m + 1) =\n Finset.sum (equitabilise h).parts fun i => card i",
"state_before": "α : Type u_1\ninst✝ : DecidableEq α\ns t : Finset α\nm n a b : ℕ\nP : Finpartition s\nh : a * m + b * (m + 1) = card s\nhm : m ≠ 0\n⊢ card (filter (fun u => card u = m) (equitabilise h).parts) * m + b * (m + 1) = a * m + b * (m + 1)",
"tactic": "rw [h, ← (P.equitabilise h).sum_card_parts]"
},
{
"state_after": "α : Type u_1\ninst✝ : DecidableEq α\ns t : Finset α\nm n a b : ℕ\nP : Finpartition s\nh : a * m + b * (m + 1) = card s\nhm : m ≠ 0\nhunion :\n (equitabilise h).parts =\n filter (fun u => card u = m) (equitabilise h).parts ∪ filter (fun u => card u = m + 1) (equitabilise h).parts\n⊢ card (filter (fun u => card u = m) (equitabilise h).parts) * m + b * (m + 1) =\n Finset.sum (equitabilise h).parts fun i => card i",
"state_before": "α : Type u_1\ninst✝ : DecidableEq α\ns t : Finset α\nm n a b : ℕ\nP : Finpartition s\nh : a * m + b * (m + 1) = card s\nhm : m ≠ 0\n⊢ card (filter (fun u => card u = m) (equitabilise h).parts) * m + b * (m + 1) =\n Finset.sum (equitabilise h).parts fun i => card i",
"tactic": "have hunion :\n (P.equitabilise h).parts =\n ((P.equitabilise h).parts.filter fun u => u.card = m) ∪\n (P.equitabilise h).parts.filter fun u => u.card = m + 1 := by\n rw [← filter_or, filter_true_of_mem]\n exact fun x => card_eq_of_mem_parts_equitabilise"
},
{
"state_after": "α : Type u_1\ninst✝ : DecidableEq α\ns t : Finset α\nm n a b : ℕ\nP : Finpartition s\nh : a * m + b * (m + 1) = card s\nhm : m ≠ 0\nhunion :\n (equitabilise h).parts =\n filter (fun u => card u = m) (equitabilise h).parts ∪ filter (fun u => card u = m + 1) (equitabilise h).parts\n⊢ card (filter (fun u => card u = m) (equitabilise h).parts) * m + b * (m + 1) =\n Finset.sum\n (filter (fun u => card u = m) (equitabilise h).parts ∪ filter (fun u => card u = m + 1) (equitabilise h).parts)\n fun i => card i",
"state_before": "α : Type u_1\ninst✝ : DecidableEq α\ns t : Finset α\nm n a b : ℕ\nP : Finpartition s\nh : a * m + b * (m + 1) = card s\nhm : m ≠ 0\nhunion :\n (equitabilise h).parts =\n filter (fun u => card u = m) (equitabilise h).parts ∪ filter (fun u => card u = m + 1) (equitabilise h).parts\n⊢ card (filter (fun u => card u = m) (equitabilise h).parts) * m + b * (m + 1) =\n Finset.sum (equitabilise h).parts fun i => card i",
"tactic": "nth_rw 2 [hunion]"
},
{
"state_after": "α : Type u_1\ninst✝ : DecidableEq α\ns t : Finset α\nm n a b : ℕ\nP : Finpartition s\nh : a * m + b * (m + 1) = card s\nhm : m ≠ 0\nhunion :\n (equitabilise h).parts =\n filter (fun u => card u = m) (equitabilise h).parts ∪ filter (fun u => card u = m + 1) (equitabilise h).parts\n⊢ Disjoint (filter (fun u => card u = m) (equitabilise h).parts)\n (filter (fun u => card u = m + 1) (equitabilise h).parts)",
"state_before": "α : Type u_1\ninst✝ : DecidableEq α\ns t : Finset α\nm n a b : ℕ\nP : Finpartition s\nh : a * m + b * (m + 1) = card s\nhm : m ≠ 0\nhunion :\n (equitabilise h).parts =\n filter (fun u => card u = m) (equitabilise h).parts ∪ filter (fun u => card u = m + 1) (equitabilise h).parts\n⊢ card (filter (fun u => card u = m) (equitabilise h).parts) * m + b * (m + 1) =\n Finset.sum\n (filter (fun u => card u = m) (equitabilise h).parts ∪ filter (fun u => card u = m + 1) (equitabilise h).parts)\n fun i => card i",
"tactic": "rw [sum_union, sum_const_nat fun x hx => (mem_filter.1 hx).2,\n sum_const_nat fun x hx => (mem_filter.1 hx).2, P.card_filter_equitabilise_big]"
},
{
"state_after": "α : Type u_1\ninst✝ : DecidableEq α\ns t : Finset α\nm n a b : ℕ\nP : Finpartition s\nh : a * m + b * (m + 1) = card s\nhm : m ≠ 0\nhunion :\n (equitabilise h).parts =\n filter (fun u => card u = m) (equitabilise h).parts ∪ filter (fun u => card u = m + 1) (equitabilise h).parts\n⊢ Disjoint (fun u => card u = m) fun u => card u = m + 1",
"state_before": "α : Type u_1\ninst✝ : DecidableEq α\ns t : Finset α\nm n a b : ℕ\nP : Finpartition s\nh : a * m + b * (m + 1) = card s\nhm : m ≠ 0\nhunion :\n (equitabilise h).parts =\n filter (fun u => card u = m) (equitabilise h).parts ∪ filter (fun u => card u = m + 1) (equitabilise h).parts\n⊢ Disjoint (filter (fun u => card u = m) (equitabilise h).parts)\n (filter (fun u => card u = m + 1) (equitabilise h).parts)",
"tactic": "refine' disjoint_filter_filter' _ _ _"
},
{
"state_after": "α : Type u_1\ninst✝ : DecidableEq α\ns t : Finset α\nm n a b : ℕ\nP : Finpartition s\nh✝ : a * m + b * (m + 1) = card s\nhm : m ≠ 0\nhunion :\n (equitabilise h✝).parts =\n filter (fun u => card u = m) (equitabilise h✝).parts ∪ filter (fun u => card u = m + 1) (equitabilise h✝).parts\nx : Finset α → Prop\nha : x ≤ fun u => card u = m\nhb : x ≤ fun u => card u = m + 1\ni : Finset α\nh : x i\n⊢ ⊥ i",
"state_before": "α : Type u_1\ninst✝ : DecidableEq α\ns t : Finset α\nm n a b : ℕ\nP : Finpartition s\nh : a * m + b * (m + 1) = card s\nhm : m ≠ 0\nhunion :\n (equitabilise h).parts =\n filter (fun u => card u = m) (equitabilise h).parts ∪ filter (fun u => card u = m + 1) (equitabilise h).parts\n⊢ Disjoint (fun u => card u = m) fun u => card u = m + 1",
"tactic": "intro x ha hb i h"
},
{
"state_after": "α : Type u_1\ninst✝ : DecidableEq α\ns t : Finset α\nm n a b : ℕ\nP : Finpartition s\nh✝ : a * m + b * (m + 1) = card s\nhm : m ≠ 0\nhunion :\n (equitabilise h✝).parts =\n filter (fun u => card u = m) (equitabilise h✝).parts ∪ filter (fun u => card u = m + 1) (equitabilise h✝).parts\nx : Finset α → Prop\nha : x ≤ fun u => card u = m\nhb : x ≤ fun u => card u = m + 1\ni : Finset α\nh : x i\n⊢ succ m = m",
"state_before": "α : Type u_1\ninst✝ : DecidableEq α\ns t : Finset α\nm n a b : ℕ\nP : Finpartition s\nh✝ : a * m + b * (m + 1) = card s\nhm : m ≠ 0\nhunion :\n (equitabilise h✝).parts =\n filter (fun u => card u = m) (equitabilise h✝).parts ∪ filter (fun u => card u = m + 1) (equitabilise h✝).parts\nx : Finset α → Prop\nha : x ≤ fun u => card u = m\nhb : x ≤ fun u => card u = m + 1\ni : Finset α\nh : x i\n⊢ ⊥ i",
"tactic": "apply succ_ne_self m _"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝ : DecidableEq α\ns t : Finset α\nm n a b : ℕ\nP : Finpartition s\nh✝ : a * m + b * (m + 1) = card s\nhm : m ≠ 0\nhunion :\n (equitabilise h✝).parts =\n filter (fun u => card u = m) (equitabilise h✝).parts ∪ filter (fun u => card u = m + 1) (equitabilise h✝).parts\nx : Finset α → Prop\nha : x ≤ fun u => card u = m\nhb : x ≤ fun u => card u = m + 1\ni : Finset α\nh : x i\n⊢ succ m = m",
"tactic": "exact (hb i h).symm.trans (ha i h)"
},
{
"state_after": "α : Type u_1\ninst✝ : DecidableEq α\ns t : Finset α\nm n a b : ℕ\nP : Finpartition s\nh : a * m + b * (m + 1) = card s\nhm : m ≠ 0\n⊢ ∀ (x : Finset α), x ∈ (equitabilise h).parts → card x = m ∨ card x = m + 1",
"state_before": "α : Type u_1\ninst✝ : DecidableEq α\ns t : Finset α\nm n a b : ℕ\nP : Finpartition s\nh : a * m + b * (m + 1) = card s\nhm : m ≠ 0\n⊢ (equitabilise h).parts =\n filter (fun u => card u = m) (equitabilise h).parts ∪ filter (fun u => card u = m + 1) (equitabilise h).parts",
"tactic": "rw [← filter_or, filter_true_of_mem]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝ : DecidableEq α\ns t : Finset α\nm n a b : ℕ\nP : Finpartition s\nh : a * m + b * (m + 1) = card s\nhm : m ≠ 0\n⊢ ∀ (x : Finset α), x ∈ (equitabilise h).parts → card x = m ∨ card x = m + 1",
"tactic": "exact fun x => card_eq_of_mem_parts_equitabilise"
}
] |
[
188,
37
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
172,
1
] |
Mathlib/Topology/MetricSpace/Basic.lean
|
Metric.sphere_disjoint_ball
|
[] |
[
550,
69
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
549,
1
] |
Mathlib/Data/Polynomial/AlgebraMap.lean
|
Polynomial.aeval_eq_sum_range'
|
[
{
"state_after": "R : Type u\nS : Type v\nT : Type w\nA : Type z\nA' : Type ?u.1288464\nB' : Type ?u.1288467\na b : R\nn✝ : ℕ\ninst✝⁸ : CommSemiring A'\ninst✝⁷ : Semiring B'\ninst✝⁶ : CommSemiring R\np✝ q : R[X]\ninst✝⁵ : Semiring A\ninst✝⁴ : Algebra R A\nB : Type ?u.1288680\ninst✝³ : Semiring B\ninst✝² : Algebra R B\nx✝ : A\ninst✝¹ : Semiring S\nf : R →+* S\ninst✝ : Algebra R S\np : R[X]\nn : ℕ\nhn : natDegree p < n\nx : S\n⊢ ↑(aeval x) p = ∑ x_1 in range n, ↑(algebraMap R S) (coeff p x_1) * x ^ x_1",
"state_before": "R : Type u\nS : Type v\nT : Type w\nA : Type z\nA' : Type ?u.1288464\nB' : Type ?u.1288467\na b : R\nn✝ : ℕ\ninst✝⁸ : CommSemiring A'\ninst✝⁷ : Semiring B'\ninst✝⁶ : CommSemiring R\np✝ q : R[X]\ninst✝⁵ : Semiring A\ninst✝⁴ : Algebra R A\nB : Type ?u.1288680\ninst✝³ : Semiring B\ninst✝² : Algebra R B\nx✝ : A\ninst✝¹ : Semiring S\nf : R →+* S\ninst✝ : Algebra R S\np : R[X]\nn : ℕ\nhn : natDegree p < n\nx : S\n⊢ ↑(aeval x) p = ∑ i in range n, coeff p i • x ^ i",
"tactic": "simp_rw [Algebra.smul_def]"
},
{
"state_after": "no goals",
"state_before": "R : Type u\nS : Type v\nT : Type w\nA : Type z\nA' : Type ?u.1288464\nB' : Type ?u.1288467\na b : R\nn✝ : ℕ\ninst✝⁸ : CommSemiring A'\ninst✝⁷ : Semiring B'\ninst✝⁶ : CommSemiring R\np✝ q : R[X]\ninst✝⁵ : Semiring A\ninst✝⁴ : Algebra R A\nB : Type ?u.1288680\ninst✝³ : Semiring B\ninst✝² : Algebra R B\nx✝ : A\ninst✝¹ : Semiring S\nf : R →+* S\ninst✝ : Algebra R S\np : R[X]\nn : ℕ\nhn : natDegree p < n\nx : S\n⊢ ↑(aeval x) p = ∑ x_1 in range n, ↑(algebraMap R S) (coeff p x_1) * x ^ x_1",
"tactic": "exact eval₂_eq_sum_range' (algebraMap R S) hn x"
}
] |
[
364,
50
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
361,
1
] |
Mathlib/LinearAlgebra/AffineSpace/AffineEquiv.lean
|
AffineEquiv.pointReflection_fixed_iff_of_module
|
[] |
[
608,
100
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
606,
1
] |
Mathlib/Order/Filter/Pi.lean
|
Filter.pi_inj
|
[
{
"state_after": "ι : Type u_2\nα : ι → Type u_1\nf f₁ f₂ : (i : ι) → Filter (α i)\ns : (i : ι) → Set (α i)\ninst✝ : ∀ (i : ι), NeBot (f₁ i)\nh : pi f₁ = pi f₂\n⊢ f₁ = f₂",
"state_before": "ι : Type u_2\nα : ι → Type u_1\nf f₁ f₂ : (i : ι) → Filter (α i)\ns : (i : ι) → Set (α i)\ninst✝ : ∀ (i : ι), NeBot (f₁ i)\n⊢ pi f₁ = pi f₂ ↔ f₁ = f₂",
"tactic": "refine' ⟨fun h => _, congr_arg pi⟩"
},
{
"state_after": "ι : Type u_2\nα : ι → Type u_1\nf f₁ f₂ : (i : ι) → Filter (α i)\ns : (i : ι) → Set (α i)\ninst✝ : ∀ (i : ι), NeBot (f₁ i)\nh : pi f₁ = pi f₂\nhle : f₁ ≤ f₂\n⊢ f₁ = f₂",
"state_before": "ι : Type u_2\nα : ι → Type u_1\nf f₁ f₂ : (i : ι) → Filter (α i)\ns : (i : ι) → Set (α i)\ninst✝ : ∀ (i : ι), NeBot (f₁ i)\nh : pi f₁ = pi f₂\n⊢ f₁ = f₂",
"tactic": "have hle : f₁ ≤ f₂ := pi_le_pi.1 h.le"
},
{
"state_after": "ι : Type u_2\nα : ι → Type u_1\nf f₁ f₂ : (i : ι) → Filter (α i)\ns : (i : ι) → Set (α i)\ninst✝ : ∀ (i : ι), NeBot (f₁ i)\nh : pi f₁ = pi f₂\nhle : f₁ ≤ f₂\nthis : ∀ (i : ι), NeBot (f₂ i)\n⊢ f₁ = f₂",
"state_before": "ι : Type u_2\nα : ι → Type u_1\nf f₁ f₂ : (i : ι) → Filter (α i)\ns : (i : ι) → Set (α i)\ninst✝ : ∀ (i : ι), NeBot (f₁ i)\nh : pi f₁ = pi f₂\nhle : f₁ ≤ f₂\n⊢ f₁ = f₂",
"tactic": "haveI : ∀ i, NeBot (f₂ i) := fun i => neBot_of_le (hle i)"
},
{
"state_after": "no goals",
"state_before": "ι : Type u_2\nα : ι → Type u_1\nf f₁ f₂ : (i : ι) → Filter (α i)\ns : (i : ι) → Set (α i)\ninst✝ : ∀ (i : ι), NeBot (f₁ i)\nh : pi f₁ = pi f₂\nhle : f₁ ≤ f₂\nthis : ∀ (i : ι), NeBot (f₂ i)\n⊢ f₁ = f₂",
"tactic": "exact hle.antisymm (pi_le_pi.1 h.ge)"
}
] |
[
187,
39
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
183,
1
] |
Mathlib/Data/Set/Basic.lean
|
Set.mem_of_mem_of_subset
|
[] |
[
248,
7
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
247,
1
] |
Mathlib/Data/Nat/PartENat.lean
|
PartENat.casesOn'
|
[] |
[
147,
20
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
145,
11
] |
Mathlib/Data/Nat/Digits.lean
|
Nat.ofDigits_digits
|
[
{
"state_after": "case zero\nn✝ n : ℕ\n⊢ ofDigits zero (digits zero n) = n\n\ncase succ\nn✝ n b : ℕ\n⊢ ofDigits (succ b) (digits (succ b) n) = n",
"state_before": "n✝ b n : ℕ\n⊢ ofDigits b (digits b n) = n",
"tactic": "cases' b with b"
},
{
"state_after": "case zero.zero\nn : ℕ\n⊢ ofDigits zero (digits zero zero) = zero\n\ncase zero.succ\nn✝ n : ℕ\n⊢ ofDigits zero (digits zero (succ n)) = succ n",
"state_before": "case zero\nn✝ n : ℕ\n⊢ ofDigits zero (digits zero n) = n",
"tactic": "cases' n with n"
},
{
"state_after": "no goals",
"state_before": "case zero.zero\nn : ℕ\n⊢ ofDigits zero (digits zero zero) = zero",
"tactic": "rfl"
},
{
"state_after": "case zero.succ\nn✝ n : ℕ\n⊢ ofDigits 0 [n + 1] = n + 1",
"state_before": "case zero.succ\nn✝ n : ℕ\n⊢ ofDigits zero (digits zero (succ n)) = succ n",
"tactic": "change ofDigits 0 [n + 1] = n + 1"
},
{
"state_after": "no goals",
"state_before": "case zero.succ\nn✝ n : ℕ\n⊢ ofDigits 0 [n + 1] = n + 1",
"tactic": "dsimp [ofDigits]"
},
{
"state_after": "case succ.zero\nn✝ n : ℕ\n⊢ ofDigits (succ zero) (digits (succ zero) n) = n\n\ncase succ.succ\nn✝ n b : ℕ\n⊢ ofDigits (succ (succ b)) (digits (succ (succ b)) n) = n",
"state_before": "case succ\nn✝ n b : ℕ\n⊢ ofDigits (succ b) (digits (succ b) n) = n",
"tactic": "cases' b with b"
},
{
"state_after": "case succ.zero.zero\nn : ℕ\n⊢ ofDigits (succ zero) (digits (succ zero) zero) = zero\n\ncase succ.zero.succ\nn✝ n : ℕ\nih : ofDigits (succ zero) (digits (succ zero) n) = n\n⊢ ofDigits (succ zero) (digits (succ zero) (succ n)) = succ n",
"state_before": "case succ.zero\nn✝ n : ℕ\n⊢ ofDigits (succ zero) (digits (succ zero) n) = n",
"tactic": "induction' n with n ih"
},
{
"state_after": "no goals",
"state_before": "case succ.zero.zero\nn : ℕ\n⊢ ofDigits (succ zero) (digits (succ zero) zero) = zero",
"tactic": "rfl"
},
{
"state_after": "case succ.zero.succ\nn✝ n : ℕ\nih : ofDigits 1 (digits 1 n) = n\n⊢ ofDigits 1 (digits 1 (succ n)) = succ n",
"state_before": "case succ.zero.succ\nn✝ n : ℕ\nih : ofDigits (succ zero) (digits (succ zero) n) = n\n⊢ ofDigits (succ zero) (digits (succ zero) (succ n)) = succ n",
"tactic": "rw[show succ zero = 1 by rfl] at ih ⊢"
},
{
"state_after": "no goals",
"state_before": "case succ.zero.succ\nn✝ n : ℕ\nih : ofDigits 1 (digits 1 n) = n\n⊢ ofDigits 1 (digits 1 (succ n)) = succ n",
"tactic": "simp only [ih, add_comm 1, ofDigits_one_cons, Nat.cast_id, digits_one_succ]"
},
{
"state_after": "no goals",
"state_before": "n✝ n : ℕ\nih : ofDigits 1 (digits 1 n) = n\n⊢ succ zero = 1",
"tactic": "rfl"
},
{
"state_after": "n✝ n b : ℕ\n⊢ ∀ (n : ℕ),\n (∀ (m : ℕ), m < n → ofDigits (succ (succ b)) (digits (succ (succ b)) m) = m) →\n ofDigits (succ (succ b)) (digits (succ (succ b)) n) = n",
"state_before": "case succ.succ\nn✝ n b : ℕ\n⊢ ofDigits (succ (succ b)) (digits (succ (succ b)) n) = n",
"tactic": "apply Nat.strongInductionOn n _"
},
{
"state_after": "n b : ℕ\n⊢ ∀ (n : ℕ),\n (∀ (m : ℕ), m < n → ofDigits (succ (succ b)) (digits (succ (succ b)) m) = m) →\n ofDigits (succ (succ b)) (digits (succ (succ b)) n) = n",
"state_before": "n✝ n b : ℕ\n⊢ ∀ (n : ℕ),\n (∀ (m : ℕ), m < n → ofDigits (succ (succ b)) (digits (succ (succ b)) m) = m) →\n ofDigits (succ (succ b)) (digits (succ (succ b)) n) = n",
"tactic": "clear n"
},
{
"state_after": "n✝ b n : ℕ\nh : ∀ (m : ℕ), m < n → ofDigits (succ (succ b)) (digits (succ (succ b)) m) = m\n⊢ ofDigits (succ (succ b)) (digits (succ (succ b)) n) = n",
"state_before": "n b : ℕ\n⊢ ∀ (n : ℕ),\n (∀ (m : ℕ), m < n → ofDigits (succ (succ b)) (digits (succ (succ b)) m) = m) →\n ofDigits (succ (succ b)) (digits (succ (succ b)) n) = n",
"tactic": "intro n h"
},
{
"state_after": "case zero\nn b : ℕ\nh : ∀ (m : ℕ), m < zero → ofDigits (succ (succ b)) (digits (succ (succ b)) m) = m\n⊢ ofDigits (succ (succ b)) (digits (succ (succ b)) zero) = zero\n\ncase succ\nn b n✝ : ℕ\nh : ∀ (m : ℕ), m < succ n✝ → ofDigits (succ (succ b)) (digits (succ (succ b)) m) = m\n⊢ ofDigits (succ (succ b)) (digits (succ (succ b)) (succ n✝)) = succ n✝",
"state_before": "n✝ b n : ℕ\nh : ∀ (m : ℕ), m < n → ofDigits (succ (succ b)) (digits (succ (succ b)) m) = m\n⊢ ofDigits (succ (succ b)) (digits (succ (succ b)) n) = n",
"tactic": "cases n"
},
{
"state_after": "case zero\nn b : ℕ\nh : ∀ (m : ℕ), m < zero → ofDigits (succ (succ b)) (digits (succ (succ b)) m) = m\n⊢ ofDigits (succ (succ b)) [] = zero",
"state_before": "case zero\nn b : ℕ\nh : ∀ (m : ℕ), m < zero → ofDigits (succ (succ b)) (digits (succ (succ b)) m) = m\n⊢ ofDigits (succ (succ b)) (digits (succ (succ b)) zero) = zero",
"tactic": "rw [digits_zero]"
},
{
"state_after": "no goals",
"state_before": "case zero\nn b : ℕ\nh : ∀ (m : ℕ), m < zero → ofDigits (succ (succ b)) (digits (succ (succ b)) m) = m\n⊢ ofDigits (succ (succ b)) [] = zero",
"tactic": "rfl"
},
{
"state_after": "case succ\nn b n✝ : ℕ\nh : ∀ (m : ℕ), m < succ n✝ → ofDigits (succ (succ b)) (digits (succ (succ b)) m) = m\n⊢ ofDigits (b + 1 + 1) ((n✝ + 1) % (b + 2) :: digits (b + 2) ((n✝ + 1) / (b + 2))) = n✝ + 1",
"state_before": "case succ\nn b n✝ : ℕ\nh : ∀ (m : ℕ), m < succ n✝ → ofDigits (succ (succ b)) (digits (succ (succ b)) m) = m\n⊢ ofDigits (succ (succ b)) (digits (succ (succ b)) (succ n✝)) = succ n✝",
"tactic": "simp only [Nat.succ_eq_add_one, digits_add_two_add_one]"
},
{
"state_after": "case succ\nn b n✝ : ℕ\nh : ∀ (m : ℕ), m < succ n✝ → ofDigits (succ (succ b)) (digits (succ (succ b)) m) = m\n⊢ (n✝ + 1) % (b + 2) + (b + 1 + 1) * ofDigits (b + 1 + 1) (digits (b + 2) ((n✝ + 1) / (b + 2))) = n✝ + 1",
"state_before": "case succ\nn b n✝ : ℕ\nh : ∀ (m : ℕ), m < succ n✝ → ofDigits (succ (succ b)) (digits (succ (succ b)) m) = m\n⊢ ofDigits (b + 1 + 1) ((n✝ + 1) % (b + 2) :: digits (b + 2) ((n✝ + 1) / (b + 2))) = n✝ + 1",
"tactic": "dsimp [ofDigits]"
},
{
"state_after": "case succ\nn b n✝ : ℕ\nh : ∀ (m : ℕ), m < succ n✝ → ofDigits (succ (succ b)) (digits (succ (succ b)) m) = m\n⊢ (n✝ + 1) % (b + 2) + (b + 1 + 1) * ((n✝ + 1) / (b + 2)) = n✝ + 1",
"state_before": "case succ\nn b n✝ : ℕ\nh : ∀ (m : ℕ), m < succ n✝ → ofDigits (succ (succ b)) (digits (succ (succ b)) m) = m\n⊢ (n✝ + 1) % (b + 2) + (b + 1 + 1) * ofDigits (b + 1 + 1) (digits (b + 2) ((n✝ + 1) / (b + 2))) = n✝ + 1",
"tactic": "rw [h _ (Nat.div_lt_self' _ b)]"
},
{
"state_after": "no goals",
"state_before": "case succ\nn b n✝ : ℕ\nh : ∀ (m : ℕ), m < succ n✝ → ofDigits (succ (succ b)) (digits (succ (succ b)) m) = m\n⊢ (n✝ + 1) % (b + 2) + (b + 1 + 1) * ((n✝ + 1) / (b + 2)) = n✝ + 1",
"tactic": "rw [Nat.mod_add_div]"
}
] |
[
276,
29
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
256,
1
] |
Mathlib/MeasureTheory/Function/SimpleFunc.lean
|
MeasureTheory.SimpleFunc.inv_apply
|
[] |
[
530,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
529,
1
] |
Mathlib/Order/Monotone/Monovary.lean
|
AntivaryOn.dual_right
|
[] |
[
244,
8
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
243,
1
] |
Mathlib/Data/List/Permutation.lean
|
List.map_permutationsAux2
|
[
{
"state_after": "α : Type u_1\nβ : Type u_2\nt : α\nts ys : List α\nf : List α → β\n⊢ (permutationsAux2 (id t) ts (map f []) ys ?f').snd = (permutationsAux2 t ts [] ys f).snd\n\ncase f'\nα : Type u_1\nβ : Type u_2\nt : α\nts ys : List α\nf : List α → β\n⊢ List α → β\n\ncase f'\nα : Type u_1\nβ : Type u_2\nt : α\nts ys : List α\nf : List α → β\n⊢ List α → β\n\ncase f'\nα : Type u_1\nβ : Type u_2\nt : α\nts ys : List α\nf : List α → β\n⊢ List α → β\n\ncase H\nα : Type u_1\nβ : Type u_2\nt : α\nts ys : List α\nf : List α → β\n⊢ ∀ (a : List α), f (id a) = ?f' (map id a)",
"state_before": "α : Type u_1\nβ : Type u_2\nt : α\nts ys : List α\nf : List α → β\n⊢ map f (permutationsAux2 t ts [] ys id).snd = (permutationsAux2 t ts [] ys f).snd",
"tactic": "rw [map_permutationsAux2' id, map_id, map_id]"
},
{
"state_after": "case H\nα : Type u_1\nβ : Type u_2\nt : α\nts ys : List α\nf : List α → β\n⊢ ∀ (a : List α), f (id a) = f (map id a)",
"state_before": "α : Type u_1\nβ : Type u_2\nt : α\nts ys : List α\nf : List α → β\n⊢ (permutationsAux2 (id t) ts (map f []) ys ?f').snd = (permutationsAux2 t ts [] ys f).snd\n\ncase f'\nα : Type u_1\nβ : Type u_2\nt : α\nts ys : List α\nf : List α → β\n⊢ List α → β\n\ncase f'\nα : Type u_1\nβ : Type u_2\nt : α\nts ys : List α\nf : List α → β\n⊢ List α → β\n\ncase f'\nα : Type u_1\nβ : Type u_2\nt : α\nts ys : List α\nf : List α → β\n⊢ List α → β\n\ncase H\nα : Type u_1\nβ : Type u_2\nt : α\nts ys : List α\nf : List α → β\n⊢ ∀ (a : List α), f (id a) = ?f' (map id a)",
"tactic": "rfl"
},
{
"state_after": "no goals",
"state_before": "case H\nα : Type u_1\nβ : Type u_2\nt : α\nts ys : List α\nf : List α → β\n⊢ ∀ (a : List α), f (id a) = f (map id a)",
"tactic": "simp"
}
] |
[
108,
7
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
105,
1
] |
Mathlib/Algebra/Group/Defs.lean
|
mul_inv_cancel_right
|
[
{
"state_after": "no goals",
"state_before": "G : Type u_1\ninst✝ : Group G\na✝ b✝ c a b : G\n⊢ a * b * b⁻¹ = a",
"tactic": "rw [mul_assoc, mul_right_inv, mul_one]"
}
] |
[
1119,
44
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1118,
1
] |
Mathlib/Logic/Basic.lean
|
ball_or_left
|
[] |
[
1109,
54
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1108,
1
] |
Mathlib/Data/Real/ENNReal.lean
|
ENNReal.pow_ne_zero
|
[
{
"state_after": "no goals",
"state_before": "α : Type ?u.108457\nβ : Type ?u.108460\na b c d : ℝ≥0∞\nr p q : ℝ≥0\n⊢ a ≠ 0 → ∀ (n : ℕ), a ^ n ≠ 0",
"tactic": "simpa only [pos_iff_ne_zero] using ENNReal.pow_pos"
}
] |
[
751,
53
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
750,
11
] |
Mathlib/RingTheory/DedekindDomain/Ideal.lean
|
IsDedekindDomain.quotientEquivPiFactors_mk
|
[] |
[
1350,
35
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1348,
1
] |
Mathlib/MeasureTheory/Measure/MeasureSpace.lean
|
MeasureTheory.Measure.volume_subtype_coe_le_volume
|
[] |
[
1462,
36
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1460,
1
] |
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