file_path
stringlengths 11
79
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stringlengths 2
100
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list | end
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stringclasses 4
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Mathlib/Data/Finset/Fold.lean
|
Finset.fold_ite'
|
[
{
"state_after": "case empty\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.24216\nop : β → β → β\nhc : IsCommutative β op\nha : IsAssociative β op\nf : α → β\nb : β\ns : Finset α\na : α\ng : α → β\nhb : op b b = b\np : α → Prop\ninst✝ : DecidablePred p\n⊢ fold op b (fun i => if p i then f i else g i) ∅ =\n op (fold op b f (filter p ∅)) (fold op b g (filter (fun i => ¬p i) ∅))\n\ncase insert\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.24216\nop : β → β → β\nhc : IsCommutative β op\nha : IsAssociative β op\nf : α → β\nb : β\ns✝ : Finset α\na : α\ng : α → β\nhb : op b b = b\np : α → Prop\ninst✝ : DecidablePred p\nx : α\ns : Finset α\nhx : ¬x ∈ s\nIH :\n fold op b (fun i => if p i then f i else g i) s =\n op (fold op b f (filter p s)) (fold op b g (filter (fun i => ¬p i) s))\n⊢ fold op b (fun i => if p i then f i else g i) (insert x s) =\n op (fold op b f (filter p (insert x s))) (fold op b g (filter (fun i => ¬p i) (insert x s)))",
"state_before": "α : Type u_2\nβ : Type u_1\nγ : Type ?u.24216\nop : β → β → β\nhc : IsCommutative β op\nha : IsAssociative β op\nf : α → β\nb : β\ns : Finset α\na : α\ng : α → β\nhb : op b b = b\np : α → Prop\ninst✝ : DecidablePred p\n⊢ fold op b (fun i => if p i then f i else g i) s =\n op (fold op b f (filter p s)) (fold op b g (filter (fun i => ¬p i) s))",
"tactic": "induction' s using Finset.induction_on with x s hx IH"
},
{
"state_after": "no goals",
"state_before": "case empty\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.24216\nop : β → β → β\nhc : IsCommutative β op\nha : IsAssociative β op\nf : α → β\nb : β\ns : Finset α\na : α\ng : α → β\nhb : op b b = b\np : α → Prop\ninst✝ : DecidablePred p\n⊢ fold op b (fun i => if p i then f i else g i) ∅ =\n op (fold op b f (filter p ∅)) (fold op b g (filter (fun i => ¬p i) ∅))",
"tactic": "simp [hb]"
},
{
"state_after": "case insert\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.24216\nop : β → β → β\nhc : IsCommutative β op\nha : IsAssociative β op\nf : α → β\nb : β\ns✝ : Finset α\na : α\ng : α → β\nhb : op b b = b\np : α → Prop\ninst✝ : DecidablePred p\nx : α\ns : Finset α\nhx : ¬x ∈ s\nIH :\n fold op b (fun i => if p i then f i else g i) s =\n op (fold op b f (filter p s)) (fold op b g (filter (fun i => ¬p i) s))\n⊢ op (if p x then f x else g x) (fold op b (fun i => if p i then f i else g i) s) =\n op (fold op b f (filter p (insert x s))) (fold op b g (filter (fun i => ¬p i) (insert x s)))",
"state_before": "case insert\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.24216\nop : β → β → β\nhc : IsCommutative β op\nha : IsAssociative β op\nf : α → β\nb : β\ns✝ : Finset α\na : α\ng : α → β\nhb : op b b = b\np : α → Prop\ninst✝ : DecidablePred p\nx : α\ns : Finset α\nhx : ¬x ∈ s\nIH :\n fold op b (fun i => if p i then f i else g i) s =\n op (fold op b f (filter p s)) (fold op b g (filter (fun i => ¬p i) s))\n⊢ fold op b (fun i => if p i then f i else g i) (insert x s) =\n op (fold op b f (filter p (insert x s))) (fold op b g (filter (fun i => ¬p i) (insert x s)))",
"tactic": "simp only [Finset.fold_insert hx]"
},
{
"state_after": "case insert.inl\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.24216\nop : β → β → β\nhc : IsCommutative β op\nha : IsAssociative β op\nf : α → β\nb : β\ns✝ : Finset α\na : α\ng : α → β\nhb : op b b = b\np : α → Prop\ninst✝ : DecidablePred p\nx : α\ns : Finset α\nhx : ¬x ∈ s\nIH :\n fold op b (fun i => if p i then f i else g i) s =\n op (fold op b f (filter p s)) (fold op b g (filter (fun i => ¬p i) s))\nh : p x\n⊢ op (f x) (fold op b (fun i => if p i then f i else g i) s) =\n op (fold op b f (filter p (insert x s))) (fold op b g (filter (fun i => ¬p i) (insert x s)))\n\ncase insert.inr\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.24216\nop : β → β → β\nhc : IsCommutative β op\nha : IsAssociative β op\nf : α → β\nb : β\ns✝ : Finset α\na : α\ng : α → β\nhb : op b b = b\np : α → Prop\ninst✝ : DecidablePred p\nx : α\ns : Finset α\nhx : ¬x ∈ s\nIH :\n fold op b (fun i => if p i then f i else g i) s =\n op (fold op b f (filter p s)) (fold op b g (filter (fun i => ¬p i) s))\nh : ¬p x\n⊢ op (g x) (fold op b (fun i => if p i then f i else g i) s) =\n op (fold op b f (filter p (insert x s))) (fold op b g (filter (fun i => ¬p i) (insert x s)))",
"state_before": "case insert\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.24216\nop : β → β → β\nhc : IsCommutative β op\nha : IsAssociative β op\nf : α → β\nb : β\ns✝ : Finset α\na : α\ng : α → β\nhb : op b b = b\np : α → Prop\ninst✝ : DecidablePred p\nx : α\ns : Finset α\nhx : ¬x ∈ s\nIH :\n fold op b (fun i => if p i then f i else g i) s =\n op (fold op b f (filter p s)) (fold op b g (filter (fun i => ¬p i) s))\n⊢ op (if p x then f x else g x) (fold op b (fun i => if p i then f i else g i) s) =\n op (fold op b f (filter p (insert x s))) (fold op b g (filter (fun i => ¬p i) (insert x s)))",
"tactic": "split_ifs with h"
},
{
"state_after": "case insert.inl\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.24216\nop : β → β → β\nhc : IsCommutative β op\nha : IsAssociative β op\nf : α → β\nb : β\ns✝ : Finset α\na : α\ng : α → β\nhb : op b b = b\np : α → Prop\ninst✝ : DecidablePred p\nx : α\ns : Finset α\nhx : ¬x ∈ s\nIH :\n fold op b (fun i => if p i then f i else g i) s =\n op (fold op b f (filter p s)) (fold op b g (filter (fun i => ¬p i) s))\nh : p x\nthis : ¬x ∈ filter p s\n⊢ op (f x) (fold op b (fun i => if p i then f i else g i) s) =\n op (fold op b f (filter p (insert x s))) (fold op b g (filter (fun i => ¬p i) (insert x s)))",
"state_before": "case insert.inl\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.24216\nop : β → β → β\nhc : IsCommutative β op\nha : IsAssociative β op\nf : α → β\nb : β\ns✝ : Finset α\na : α\ng : α → β\nhb : op b b = b\np : α → Prop\ninst✝ : DecidablePred p\nx : α\ns : Finset α\nhx : ¬x ∈ s\nIH :\n fold op b (fun i => if p i then f i else g i) s =\n op (fold op b f (filter p s)) (fold op b g (filter (fun i => ¬p i) s))\nh : p x\n⊢ op (f x) (fold op b (fun i => if p i then f i else g i) s) =\n op (fold op b f (filter p (insert x s))) (fold op b g (filter (fun i => ¬p i) (insert x s)))",
"tactic": "have : x ∉ Finset.filter p s := by simp [hx]"
},
{
"state_after": "no goals",
"state_before": "case insert.inl\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.24216\nop : β → β → β\nhc : IsCommutative β op\nha : IsAssociative β op\nf : α → β\nb : β\ns✝ : Finset α\na : α\ng : α → β\nhb : op b b = b\np : α → Prop\ninst✝ : DecidablePred p\nx : α\ns : Finset α\nhx : ¬x ∈ s\nIH :\n fold op b (fun i => if p i then f i else g i) s =\n op (fold op b f (filter p s)) (fold op b g (filter (fun i => ¬p i) s))\nh : p x\nthis : ¬x ∈ filter p s\n⊢ op (f x) (fold op b (fun i => if p i then f i else g i) s) =\n op (fold op b f (filter p (insert x s))) (fold op b g (filter (fun i => ¬p i) (insert x s)))",
"tactic": "simp [Finset.filter_insert, h, Finset.fold_insert this, ha.assoc, IH]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_2\nβ : Type u_1\nγ : Type ?u.24216\nop : β → β → β\nhc : IsCommutative β op\nha : IsAssociative β op\nf : α → β\nb : β\ns✝ : Finset α\na : α\ng : α → β\nhb : op b b = b\np : α → Prop\ninst✝ : DecidablePred p\nx : α\ns : Finset α\nhx : ¬x ∈ s\nIH :\n fold op b (fun i => if p i then f i else g i) s =\n op (fold op b f (filter p s)) (fold op b g (filter (fun i => ¬p i) s))\nh : p x\n⊢ ¬x ∈ filter p s",
"tactic": "simp [hx]"
},
{
"state_after": "case insert.inr\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.24216\nop : β → β → β\nhc : IsCommutative β op\nha : IsAssociative β op\nf : α → β\nb : β\ns✝ : Finset α\na : α\ng : α → β\nhb : op b b = b\np : α → Prop\ninst✝ : DecidablePred p\nx : α\ns : Finset α\nhx : ¬x ∈ s\nIH :\n fold op b (fun i => if p i then f i else g i) s =\n op (fold op b f (filter p s)) (fold op b g (filter (fun i => ¬p i) s))\nh : ¬p x\nthis : ¬x ∈ filter (fun i => ¬p i) s\n⊢ op (g x) (fold op b (fun i => if p i then f i else g i) s) =\n op (fold op b f (filter p (insert x s))) (fold op b g (filter (fun i => ¬p i) (insert x s)))",
"state_before": "case insert.inr\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.24216\nop : β → β → β\nhc : IsCommutative β op\nha : IsAssociative β op\nf : α → β\nb : β\ns✝ : Finset α\na : α\ng : α → β\nhb : op b b = b\np : α → Prop\ninst✝ : DecidablePred p\nx : α\ns : Finset α\nhx : ¬x ∈ s\nIH :\n fold op b (fun i => if p i then f i else g i) s =\n op (fold op b f (filter p s)) (fold op b g (filter (fun i => ¬p i) s))\nh : ¬p x\n⊢ op (g x) (fold op b (fun i => if p i then f i else g i) s) =\n op (fold op b f (filter p (insert x s))) (fold op b g (filter (fun i => ¬p i) (insert x s)))",
"tactic": "have : x ∉ Finset.filter (fun i => ¬ p i) s := by simp [hx]"
},
{
"state_after": "no goals",
"state_before": "case insert.inr\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.24216\nop : β → β → β\nhc : IsCommutative β op\nha : IsAssociative β op\nf : α → β\nb : β\ns✝ : Finset α\na : α\ng : α → β\nhb : op b b = b\np : α → Prop\ninst✝ : DecidablePred p\nx : α\ns : Finset α\nhx : ¬x ∈ s\nIH :\n fold op b (fun i => if p i then f i else g i) s =\n op (fold op b f (filter p s)) (fold op b g (filter (fun i => ¬p i) s))\nh : ¬p x\nthis : ¬x ∈ filter (fun i => ¬p i) s\n⊢ op (g x) (fold op b (fun i => if p i then f i else g i) s) =\n op (fold op b f (filter p (insert x s))) (fold op b g (filter (fun i => ¬p i) (insert x s)))",
"tactic": "simp [Finset.filter_insert, h, Finset.fold_insert this, IH, ← ha.assoc, hc.comm]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_2\nβ : Type u_1\nγ : Type ?u.24216\nop : β → β → β\nhc : IsCommutative β op\nha : IsAssociative β op\nf : α → β\nb : β\ns✝ : Finset α\na : α\ng : α → β\nhb : op b b = b\np : α → Prop\ninst✝ : DecidablePred p\nx : α\ns : Finset α\nhx : ¬x ∈ s\nIH :\n fold op b (fun i => if p i then f i else g i) s =\n op (fold op b f (filter p s)) (fold op b g (filter (fun i => ¬p i) s))\nh : ¬p x\n⊢ ¬x ∈ filter (fun i => ¬p i) s",
"tactic": "simp [hx]"
}
] |
[
154,
89
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
143,
1
] |
Mathlib/Order/SuccPred/Limit.lean
|
Order.not_isSuccLimit_iff_exists_covby
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝ : LT α\na : α\n⊢ ¬IsSuccLimit a ↔ ∃ b, b ⋖ a",
"tactic": "simp [IsSuccLimit]"
}
] |
[
50,
21
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
49,
1
] |
Mathlib/SetTheory/Ordinal/Arithmetic.lean
|
Ordinal.lsub_not_mem_range
|
[] |
[
1710,
25
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1708,
1
] |
Mathlib/Data/Stream/Init.lean
|
Stream'.corec'_eq
|
[] |
[
395,
17
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
394,
1
] |
Mathlib/RingTheory/Subring/Basic.lean
|
Subring.coe_map
|
[] |
[
599,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
598,
1
] |
Mathlib/Data/Multiset/Basic.lean
|
Multiset.nsmul_replicate
|
[] |
[
961,
49
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
960,
1
] |
Mathlib/GroupTheory/Subsemigroup/Centralizer.lean
|
Set.center_subset_centralizer
|
[] |
[
129,
23
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
128,
1
] |
Mathlib/Combinatorics/SimpleGraph/Connectivity.lean
|
SimpleGraph.Iso.connected_iff
|
[] |
[
1964,
101
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1962,
1
] |
Mathlib/Data/List/BigOperators/Basic.lean
|
List.prod_pos
|
[
{
"state_after": "case nil\nι : Type ?u.157247\nα : Type ?u.157250\nM : Type ?u.157253\nN : Type ?u.157256\nP : Type ?u.157259\nM₀ : Type ?u.157262\nG : Type ?u.157265\nR : Type u_1\ninst✝ : StrictOrderedSemiring R\nl : List R\nh✝ : ∀ (a : R), a ∈ l → 0 < a\nh : ∀ (a : R), a ∈ [] → 0 < a\n⊢ 0 < prod []\n\ncase cons\nι : Type ?u.157247\nα : Type ?u.157250\nM : Type ?u.157253\nN : Type ?u.157256\nP : Type ?u.157259\nM₀ : Type ?u.157262\nG : Type ?u.157265\nR : Type u_1\ninst✝ : StrictOrderedSemiring R\nl✝ : List R\nh✝ : ∀ (a : R), a ∈ l✝ → 0 < a\na : R\nl : List R\nih : (∀ (a : R), a ∈ l → 0 < a) → 0 < prod l\nh : ∀ (a_1 : R), a_1 ∈ a :: l → 0 < a_1\n⊢ 0 < prod (a :: l)",
"state_before": "ι : Type ?u.157247\nα : Type ?u.157250\nM : Type ?u.157253\nN : Type ?u.157256\nP : Type ?u.157259\nM₀ : Type ?u.157262\nG : Type ?u.157265\nR : Type u_1\ninst✝ : StrictOrderedSemiring R\nl : List R\nh : ∀ (a : R), a ∈ l → 0 < a\n⊢ 0 < prod l",
"tactic": "induction' l with a l ih"
},
{
"state_after": "no goals",
"state_before": "case nil\nι : Type ?u.157247\nα : Type ?u.157250\nM : Type ?u.157253\nN : Type ?u.157256\nP : Type ?u.157259\nM₀ : Type ?u.157262\nG : Type ?u.157265\nR : Type u_1\ninst✝ : StrictOrderedSemiring R\nl : List R\nh✝ : ∀ (a : R), a ∈ l → 0 < a\nh : ∀ (a : R), a ∈ [] → 0 < a\n⊢ 0 < prod []",
"tactic": "simp"
},
{
"state_after": "case cons\nι : Type ?u.157247\nα : Type ?u.157250\nM : Type ?u.157253\nN : Type ?u.157256\nP : Type ?u.157259\nM₀ : Type ?u.157262\nG : Type ?u.157265\nR : Type u_1\ninst✝ : StrictOrderedSemiring R\nl✝ : List R\nh✝ : ∀ (a : R), a ∈ l✝ → 0 < a\na : R\nl : List R\nih : (∀ (a : R), a ∈ l → 0 < a) → 0 < prod l\nh : ∀ (a_1 : R), a_1 ∈ a :: l → 0 < a_1\n⊢ 0 < a * prod l",
"state_before": "case cons\nι : Type ?u.157247\nα : Type ?u.157250\nM : Type ?u.157253\nN : Type ?u.157256\nP : Type ?u.157259\nM₀ : Type ?u.157262\nG : Type ?u.157265\nR : Type u_1\ninst✝ : StrictOrderedSemiring R\nl✝ : List R\nh✝ : ∀ (a : R), a ∈ l✝ → 0 < a\na : R\nl : List R\nih : (∀ (a : R), a ∈ l → 0 < a) → 0 < prod l\nh : ∀ (a_1 : R), a_1 ∈ a :: l → 0 < a_1\n⊢ 0 < prod (a :: l)",
"tactic": "rw [prod_cons]"
},
{
"state_after": "no goals",
"state_before": "case cons\nι : Type ?u.157247\nα : Type ?u.157250\nM : Type ?u.157253\nN : Type ?u.157256\nP : Type ?u.157259\nM₀ : Type ?u.157262\nG : Type ?u.157265\nR : Type u_1\ninst✝ : StrictOrderedSemiring R\nl✝ : List R\nh✝ : ∀ (a : R), a ∈ l✝ → 0 < a\na : R\nl : List R\nih : (∀ (a : R), a ∈ l → 0 < a) → 0 < prod l\nh : ∀ (a_1 : R), a_1 ∈ a :: l → 0 < a_1\n⊢ 0 < a * prod l",
"tactic": "exact mul_pos (h _ <| mem_cons_self _ _) (ih fun a ha => h a <| mem_cons_of_mem _ ha)"
}
] |
[
583,
90
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
578,
1
] |
Mathlib/Topology/Constructions.lean
|
isOpenMap_toAdd
|
[] |
[
115,
83
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
115,
1
] |
Mathlib/Topology/Algebra/Group/Basic.lean
|
tendsto_inv_nhdsWithin_Iic
|
[
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nG : Type w\nH : Type x\ninst✝⁶ : TopologicalSpace G\ninst✝⁵ : Group G\ninst✝⁴ : TopologicalGroup G\ninst✝³ : TopologicalSpace α\nf : α → G\ns : Set α\nx : α\ninst✝² : TopologicalSpace H\ninst✝¹ : OrderedCommGroup H\ninst✝ : ContinuousInv H\na : H\n⊢ Tendsto (fun a => a⁻¹) (𝓟 (Iic a)) (𝓟 (Ici a⁻¹))",
"tactic": "simp [tendsto_principal_principal]"
}
] |
[
591,
74
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
590,
1
] |
Mathlib/CategoryTheory/Adjunction/Basic.lean
|
CategoryTheory.Adjunction.homEquiv_naturality_right_symm
|
[
{
"state_after": "C : Type u₁\ninst✝¹ : Category C\nD : Type u₂\ninst✝ : Category D\nF : C ⥤ D\nG : D ⥤ C\nadj : F ⊣ G\nX' X : C\nY Y' : D\nf : X ⟶ G.obj Y\ng : Y ⟶ Y'\n⊢ f ≫ G.map g = ↑(homEquiv adj X Y') (↑(homEquiv adj X Y).symm f ≫ g)",
"state_before": "C : Type u₁\ninst✝¹ : Category C\nD : Type u₂\ninst✝ : Category D\nF : C ⥤ D\nG : D ⥤ C\nadj : F ⊣ G\nX' X : C\nY Y' : D\nf : X ⟶ G.obj Y\ng : Y ⟶ Y'\n⊢ ↑(homEquiv adj X Y').symm (f ≫ G.map g) = ↑(homEquiv adj X Y).symm f ≫ g",
"tactic": "rw [Equiv.symm_apply_eq]"
},
{
"state_after": "no goals",
"state_before": "C : Type u₁\ninst✝¹ : Category C\nD : Type u₂\ninst✝ : Category D\nF : C ⥤ D\nG : D ⥤ C\nadj : F ⊣ G\nX' X : C\nY Y' : D\nf : X ⟶ G.obj Y\ng : Y ⟶ Y'\n⊢ f ≫ G.map g = ↑(homEquiv adj X Y') (↑(homEquiv adj X Y).symm f ≫ g)",
"tactic": "simp only [homEquiv_naturality_right,eq_self_iff_true,Equiv.apply_symm_apply]"
}
] |
[
176,
80
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
173,
1
] |
Std/Data/Nat/Gcd.lean
|
Nat.gcd_mul_dvd_mul_gcd
|
[
{
"state_after": "k m n m' : Nat\nhm' : m' ∣ m\nn' : Nat\nhn' : n' ∣ n\nh : gcd k (m * n) = m' * n'\n⊢ gcd k (m * n) ∣ gcd k m * gcd k n",
"state_before": "k m n : Nat\n⊢ gcd k (m * n) ∣ gcd k m * gcd k n",
"tactic": "let ⟨⟨⟨m', hm'⟩, ⟨n', hn'⟩⟩, (h : gcd k (m * n) = m' * n')⟩ :=\n prod_dvd_and_dvd_of_dvd_prod <| gcd_dvd_right k (m * n)"
},
{
"state_after": "k m n m' : Nat\nhm' : m' ∣ m\nn' : Nat\nhn' : n' ∣ n\nh : gcd k (m * n) = m' * n'\n⊢ m' * n' ∣ gcd k m * gcd k n",
"state_before": "k m n m' : Nat\nhm' : m' ∣ m\nn' : Nat\nhn' : n' ∣ n\nh : gcd k (m * n) = m' * n'\n⊢ gcd k (m * n) ∣ gcd k m * gcd k n",
"tactic": "rw [h]"
},
{
"state_after": "k m n m' : Nat\nhm' : m' ∣ m\nn' : Nat\nhn' : n' ∣ n\nh : gcd k (m * n) = m' * n'\nh' : m' * n' ∣ k\n⊢ m' * n' ∣ gcd k m * gcd k n",
"state_before": "k m n m' : Nat\nhm' : m' ∣ m\nn' : Nat\nhn' : n' ∣ n\nh : gcd k (m * n) = m' * n'\n⊢ m' * n' ∣ gcd k m * gcd k n",
"tactic": "have h' : m' * n' ∣ k := h ▸ gcd_dvd_left .."
},
{
"state_after": "no goals",
"state_before": "k m n m' : Nat\nhm' : m' ∣ m\nn' : Nat\nhn' : n' ∣ n\nh : gcd k (m * n) = m' * n'\nh' : m' * n' ∣ k\n⊢ m' * n' ∣ gcd k m * gcd k n",
"tactic": "exact Nat.mul_dvd_mul\n (dvd_gcd (Nat.dvd_trans (Nat.dvd_mul_right m' n') h') hm')\n (dvd_gcd (Nat.dvd_trans (Nat.dvd_mul_left n' m') h') hn')"
}
] |
[
402,
62
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
395,
1
] |
Mathlib/Algebra/Order/Hom/Basic.lean
|
le_map_mul_map_div
|
[
{
"state_after": "no goals",
"state_before": "ι : Type ?u.2724\nF : Type u_3\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.2736\nδ : Type ?u.2739\ninst✝³ : Group α\ninst✝² : CommSemigroup β\ninst✝¹ : LE β\ninst✝ : SubmultiplicativeHomClass F α β\nf : F\na b : α\n⊢ ↑f a ≤ ↑f b * ↑f (a / b)",
"tactic": "simpa only [mul_comm, div_mul_cancel'] using map_mul_le_mul f (a / b) b"
}
] |
[
133,
74
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
131,
1
] |
Mathlib/Order/Filter/SmallSets.lean
|
Filter.frequently_smallSets_mem
|
[] |
[
84,
57
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
83,
1
] |
Mathlib/Analysis/SpecialFunctions/Trigonometric/Basic.lean
|
Complex.continuousOn_sin
|
[] |
[
60,
30
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
59,
1
] |
Mathlib/GroupTheory/Coset.lean
|
Subgroup.quotientiInfSubgroupOfEmbedding_apply_mk
|
[] |
[
760,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
756,
1
] |
Mathlib/Analysis/Normed/Group/Hom.lean
|
NormedAddGroupHom.coe_comp
|
[] |
[
710,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
708,
1
] |
Mathlib/Data/List/Perm.lean
|
List.mem_permutationsAux_of_perm
|
[
{
"state_after": "α : Type uu\nβ : Type vv\nl₁ l₂ : List α\n⊢ ∀ (ts is l : List α), l ~ is ++ ts → (∃ is' x, l = is' ++ ts) ∨ l ∈ permutationsAux ts is",
"state_before": "α : Type uu\nβ : Type vv\nl₁ l₂ : List α\n⊢ ∀ {ts is l : List α}, l ~ is ++ ts → (∃ is' x, l = is' ++ ts) ∨ l ∈ permutationsAux ts is",
"tactic": "show ∀ (ts is l : List α),\n l ~ is ++ ts → (∃ (is' : _)(_ : is' ~ is), l = is' ++ ts) ∨ l ∈ permutationsAux ts is"
},
{
"state_after": "α : Type uu\nβ : Type vv\nl₁ l₂ : List α\n⊢ ∀ (t : α) (ts is : List α),\n (∀ (l : List α), l ~ t :: is ++ ts → (∃ is' x, l = is' ++ ts) ∨ l ∈ permutationsAux ts (t :: is)) →\n (∀ (l : List α), l ~ [] ++ is → (∃ is' x, l = is' ++ is) ∨ l ∈ permutationsAux is []) →\n ∀ (l : List α), l ~ is ++ t :: ts → (∃ is' x, l = is' ++ t :: ts) ∨ l ∈ permutationsAux (t :: ts) is",
"state_before": "α : Type uu\nβ : Type vv\nl₁ l₂ : List α\n⊢ ∀ (ts is l : List α), l ~ is ++ ts → (∃ is' x, l = is' ++ ts) ∨ l ∈ permutationsAux ts is",
"tactic": "refine' permutationsAux.rec (by simp) _"
},
{
"state_after": "α : Type uu\nβ : Type vv\nl₁ l₂ : List α\nt : α\nts is : List α\nIH1 : ∀ (l : List α), l ~ t :: is ++ ts → (∃ is' x, l = is' ++ ts) ∨ l ∈ permutationsAux ts (t :: is)\nIH2 : ∀ (l : List α), l ~ [] ++ is → (∃ is' x, l = is' ++ is) ∨ l ∈ permutationsAux is []\nl : List α\np : l ~ is ++ t :: ts\n⊢ (∃ is' x, l = is' ++ t :: ts) ∨ l ∈ permutationsAux (t :: ts) is",
"state_before": "α : Type uu\nβ : Type vv\nl₁ l₂ : List α\n⊢ ∀ (t : α) (ts is : List α),\n (∀ (l : List α), l ~ t :: is ++ ts → (∃ is' x, l = is' ++ ts) ∨ l ∈ permutationsAux ts (t :: is)) →\n (∀ (l : List α), l ~ [] ++ is → (∃ is' x, l = is' ++ is) ∨ l ∈ permutationsAux is []) →\n ∀ (l : List α), l ~ is ++ t :: ts → (∃ is' x, l = is' ++ t :: ts) ∨ l ∈ permutationsAux (t :: ts) is",
"tactic": "intro t ts is IH1 IH2 l p"
},
{
"state_after": "α : Type uu\nβ : Type vv\nl₁ l₂ : List α\nt : α\nts is : List α\nIH1 : ∀ (l : List α), l ~ t :: is ++ ts → (∃ is' x, l = is' ++ ts) ∨ l ∈ permutationsAux ts (t :: is)\nIH2 : ∀ (l : List α), l ~ [] ++ is → (∃ is' x, l = is' ++ is) ∨ l ∈ permutationsAux is []\nl : List α\np : l ~ is ++ t :: ts\n⊢ (∃ is' x, l = is' ++ t :: ts) ∨\n l ∈ permutationsAux ts (t :: is) ∨ ∃ l₁ l₂, l₁ ++ l₂ ∈ permutations is ∧ l₂ ≠ [] ∧ l = l₁ ++ t :: l₂ ++ ts",
"state_before": "α : Type uu\nβ : Type vv\nl₁ l₂ : List α\nt : α\nts is : List α\nIH1 : ∀ (l : List α), l ~ t :: is ++ ts → (∃ is' x, l = is' ++ ts) ∨ l ∈ permutationsAux ts (t :: is)\nIH2 : ∀ (l : List α), l ~ [] ++ is → (∃ is' x, l = is' ++ is) ∨ l ∈ permutationsAux is []\nl : List α\np : l ~ is ++ t :: ts\n⊢ (∃ is' x, l = is' ++ t :: ts) ∨ l ∈ permutationsAux (t :: ts) is",
"tactic": "rw [permutationsAux_cons, mem_foldr_permutationsAux2]"
},
{
"state_after": "case inl.intro.intro\nα : Type uu\nβ : Type vv\nl₁ l₂ : List α\nt : α\nts is : List α\nIH1 : ∀ (l : List α), l ~ t :: is ++ ts → (∃ is' x, l = is' ++ ts) ∨ l ∈ permutationsAux ts (t :: is)\nIH2 : ∀ (l : List α), l ~ [] ++ is → (∃ is' x, l = is' ++ is) ∨ l ∈ permutationsAux is []\nl : List α\np : l ~ is ++ t :: ts\nis' : List α\np' : is' ~ t :: is\ne : l = is' ++ ts\n⊢ (∃ is' x, l = is' ++ t :: ts) ∨\n l ∈ permutationsAux ts (t :: is) ∨ ∃ l₁ l₂, l₁ ++ l₂ ∈ permutations is ∧ l₂ ≠ [] ∧ l = l₁ ++ t :: l₂ ++ ts\n\ncase inr\nα : Type uu\nβ : Type vv\nl₁ l₂ : List α\nt : α\nts is : List α\nIH1 : ∀ (l : List α), l ~ t :: is ++ ts → (∃ is' x, l = is' ++ ts) ∨ l ∈ permutationsAux ts (t :: is)\nIH2 : ∀ (l : List α), l ~ [] ++ is → (∃ is' x, l = is' ++ is) ∨ l ∈ permutationsAux is []\nl : List α\np : l ~ is ++ t :: ts\nm : l ∈ permutationsAux ts (t :: is)\n⊢ (∃ is' x, l = is' ++ t :: ts) ∨\n l ∈ permutationsAux ts (t :: is) ∨ ∃ l₁ l₂, l₁ ++ l₂ ∈ permutations is ∧ l₂ ≠ [] ∧ l = l₁ ++ t :: l₂ ++ ts",
"state_before": "α : Type uu\nβ : Type vv\nl₁ l₂ : List α\nt : α\nts is : List α\nIH1 : ∀ (l : List α), l ~ t :: is ++ ts → (∃ is' x, l = is' ++ ts) ∨ l ∈ permutationsAux ts (t :: is)\nIH2 : ∀ (l : List α), l ~ [] ++ is → (∃ is' x, l = is' ++ is) ∨ l ∈ permutationsAux is []\nl : List α\np : l ~ is ++ t :: ts\n⊢ (∃ is' x, l = is' ++ t :: ts) ∨\n l ∈ permutationsAux ts (t :: is) ∨ ∃ l₁ l₂, l₁ ++ l₂ ∈ permutations is ∧ l₂ ≠ [] ∧ l = l₁ ++ t :: l₂ ++ ts",
"tactic": "rcases IH1 _ (p.trans perm_middle) with (⟨is', p', e⟩ | m)"
},
{
"state_after": "no goals",
"state_before": "α : Type uu\nβ : Type vv\nl₁ l₂ : List α\n⊢ ∀ (is l : List α), l ~ is ++ [] → (∃ is' x, l = is' ++ []) ∨ l ∈ permutationsAux [] is",
"tactic": "simp"
},
{
"state_after": "case inl.intro.intro\nα : Type uu\nβ : Type vv\nl₁ l₂ : List α\nt : α\nts is : List α\nIH1 : ∀ (l : List α), l ~ t :: is ++ ts → (∃ is' x, l = is' ++ ts) ∨ l ∈ permutationsAux ts (t :: is)\nIH2 : ∀ (l : List α), l ~ [] ++ is → (∃ is' x, l = is' ++ is) ∨ l ∈ permutationsAux is []\nl is' : List α\np' : is' ~ t :: is\ne : l = is' ++ ts\n⊢ (∃ is' x, l = is' ++ t :: ts) ∨\n l ∈ permutationsAux ts (t :: is) ∨ ∃ l₁ l₂, l₁ ++ l₂ ∈ permutations is ∧ l₂ ≠ [] ∧ l = l₁ ++ t :: l₂ ++ ts",
"state_before": "case inl.intro.intro\nα : Type uu\nβ : Type vv\nl₁ l₂ : List α\nt : α\nts is : List α\nIH1 : ∀ (l : List α), l ~ t :: is ++ ts → (∃ is' x, l = is' ++ ts) ∨ l ∈ permutationsAux ts (t :: is)\nIH2 : ∀ (l : List α), l ~ [] ++ is → (∃ is' x, l = is' ++ is) ∨ l ∈ permutationsAux is []\nl : List α\np : l ~ is ++ t :: ts\nis' : List α\np' : is' ~ t :: is\ne : l = is' ++ ts\n⊢ (∃ is' x, l = is' ++ t :: ts) ∨\n l ∈ permutationsAux ts (t :: is) ∨ ∃ l₁ l₂, l₁ ++ l₂ ∈ permutations is ∧ l₂ ≠ [] ∧ l = l₁ ++ t :: l₂ ++ ts",
"tactic": "clear p"
},
{
"state_after": "case inl.intro.intro\nα : Type uu\nβ : Type vv\nl₁ l₂ : List α\nt : α\nts is : List α\nIH1 : ∀ (l : List α), l ~ t :: is ++ ts → (∃ is' x, l = is' ++ ts) ∨ l ∈ permutationsAux ts (t :: is)\nIH2 : ∀ (l : List α), l ~ [] ++ is → (∃ is' x, l = is' ++ is) ∨ l ∈ permutationsAux is []\nis' : List α\np' : is' ~ t :: is\n⊢ (∃ is'_1 x, is' ++ ts = is'_1 ++ t :: ts) ∨\n is' ++ ts ∈ permutationsAux ts (t :: is) ∨\n ∃ l₁ l₂, l₁ ++ l₂ ∈ permutations is ∧ l₂ ≠ [] ∧ is' ++ ts = l₁ ++ t :: l₂ ++ ts",
"state_before": "case inl.intro.intro\nα : Type uu\nβ : Type vv\nl₁ l₂ : List α\nt : α\nts is : List α\nIH1 : ∀ (l : List α), l ~ t :: is ++ ts → (∃ is' x, l = is' ++ ts) ∨ l ∈ permutationsAux ts (t :: is)\nIH2 : ∀ (l : List α), l ~ [] ++ is → (∃ is' x, l = is' ++ is) ∨ l ∈ permutationsAux is []\nl is' : List α\np' : is' ~ t :: is\ne : l = is' ++ ts\n⊢ (∃ is' x, l = is' ++ t :: ts) ∨\n l ∈ permutationsAux ts (t :: is) ∨ ∃ l₁ l₂, l₁ ++ l₂ ∈ permutations is ∧ l₂ ≠ [] ∧ l = l₁ ++ t :: l₂ ++ ts",
"tactic": "subst e"
},
{
"state_after": "case inl.intro.intro.intro.intro\nα : Type uu\nβ : Type vv\nl₁✝ l₂✝ : List α\nt : α\nts is : List α\nIH1 : ∀ (l : List α), l ~ t :: is ++ ts → (∃ is' x, l = is' ++ ts) ∨ l ∈ permutationsAux ts (t :: is)\nIH2 : ∀ (l : List α), l ~ [] ++ is → (∃ is' x, l = is' ++ is) ∨ l ∈ permutationsAux is []\nis' : List α\np' : is' ~ t :: is\nl₁ l₂ : List α\ne : is' = l₁ ++ t :: l₂\n⊢ (∃ is'_1 x, is' ++ ts = is'_1 ++ t :: ts) ∨\n is' ++ ts ∈ permutationsAux ts (t :: is) ∨\n ∃ l₁ l₂, l₁ ++ l₂ ∈ permutations is ∧ l₂ ≠ [] ∧ is' ++ ts = l₁ ++ t :: l₂ ++ ts",
"state_before": "case inl.intro.intro\nα : Type uu\nβ : Type vv\nl₁ l₂ : List α\nt : α\nts is : List α\nIH1 : ∀ (l : List α), l ~ t :: is ++ ts → (∃ is' x, l = is' ++ ts) ∨ l ∈ permutationsAux ts (t :: is)\nIH2 : ∀ (l : List α), l ~ [] ++ is → (∃ is' x, l = is' ++ is) ∨ l ∈ permutationsAux is []\nis' : List α\np' : is' ~ t :: is\n⊢ (∃ is'_1 x, is' ++ ts = is'_1 ++ t :: ts) ∨\n is' ++ ts ∈ permutationsAux ts (t :: is) ∨\n ∃ l₁ l₂, l₁ ++ l₂ ∈ permutations is ∧ l₂ ≠ [] ∧ is' ++ ts = l₁ ++ t :: l₂ ++ ts",
"tactic": "rcases mem_split (p'.symm.subset (mem_cons_self _ _)) with ⟨l₁, l₂, e⟩"
},
{
"state_after": "case inl.intro.intro.intro.intro\nα : Type uu\nβ : Type vv\nl₁✝ l₂✝ : List α\nt : α\nts is : List α\nIH1 : ∀ (l : List α), l ~ t :: is ++ ts → (∃ is' x, l = is' ++ ts) ∨ l ∈ permutationsAux ts (t :: is)\nIH2 : ∀ (l : List α), l ~ [] ++ is → (∃ is' x, l = is' ++ is) ∨ l ∈ permutationsAux is []\nl₁ l₂ : List α\np' : l₁ ++ t :: l₂ ~ t :: is\n⊢ (∃ is' x, l₁ ++ t :: l₂ ++ ts = is' ++ t :: ts) ∨\n l₁ ++ t :: l₂ ++ ts ∈ permutationsAux ts (t :: is) ∨\n ∃ l₁_1 l₂_1, l₁_1 ++ l₂_1 ∈ permutations is ∧ l₂_1 ≠ [] ∧ l₁ ++ t :: l₂ ++ ts = l₁_1 ++ t :: l₂_1 ++ ts",
"state_before": "case inl.intro.intro.intro.intro\nα : Type uu\nβ : Type vv\nl₁✝ l₂✝ : List α\nt : α\nts is : List α\nIH1 : ∀ (l : List α), l ~ t :: is ++ ts → (∃ is' x, l = is' ++ ts) ∨ l ∈ permutationsAux ts (t :: is)\nIH2 : ∀ (l : List α), l ~ [] ++ is → (∃ is' x, l = is' ++ is) ∨ l ∈ permutationsAux is []\nis' : List α\np' : is' ~ t :: is\nl₁ l₂ : List α\ne : is' = l₁ ++ t :: l₂\n⊢ (∃ is'_1 x, is' ++ ts = is'_1 ++ t :: ts) ∨\n is' ++ ts ∈ permutationsAux ts (t :: is) ∨\n ∃ l₁ l₂, l₁ ++ l₂ ∈ permutations is ∧ l₂ ≠ [] ∧ is' ++ ts = l₁ ++ t :: l₂ ++ ts",
"tactic": "subst is'"
},
{
"state_after": "case inl.intro.intro.intro.intro\nα : Type uu\nβ : Type vv\nl₁✝ l₂✝ : List α\nt : α\nts is : List α\nIH1 : ∀ (l : List α), l ~ t :: is ++ ts → (∃ is' x, l = is' ++ ts) ∨ l ∈ permutationsAux ts (t :: is)\nIH2 : ∀ (l : List α), l ~ [] ++ is → (∃ is' x, l = is' ++ is) ∨ l ∈ permutationsAux is []\nl₁ l₂ : List α\np' : l₁ ++ t :: l₂ ~ t :: is\np : l₁ ++ l₂ ~ is\n⊢ (∃ is' x, l₁ ++ t :: l₂ ++ ts = is' ++ t :: ts) ∨\n l₁ ++ t :: l₂ ++ ts ∈ permutationsAux ts (t :: is) ∨\n ∃ l₁_1 l₂_1, l₁_1 ++ l₂_1 ∈ permutations is ∧ l₂_1 ≠ [] ∧ l₁ ++ t :: l₂ ++ ts = l₁_1 ++ t :: l₂_1 ++ ts",
"state_before": "case inl.intro.intro.intro.intro\nα : Type uu\nβ : Type vv\nl₁✝ l₂✝ : List α\nt : α\nts is : List α\nIH1 : ∀ (l : List α), l ~ t :: is ++ ts → (∃ is' x, l = is' ++ ts) ∨ l ∈ permutationsAux ts (t :: is)\nIH2 : ∀ (l : List α), l ~ [] ++ is → (∃ is' x, l = is' ++ is) ∨ l ∈ permutationsAux is []\nl₁ l₂ : List α\np' : l₁ ++ t :: l₂ ~ t :: is\n⊢ (∃ is' x, l₁ ++ t :: l₂ ++ ts = is' ++ t :: ts) ∨\n l₁ ++ t :: l₂ ++ ts ∈ permutationsAux ts (t :: is) ∨\n ∃ l₁_1 l₂_1, l₁_1 ++ l₂_1 ∈ permutations is ∧ l₂_1 ≠ [] ∧ l₁ ++ t :: l₂ ++ ts = l₁_1 ++ t :: l₂_1 ++ ts",
"tactic": "have p := (perm_middle.symm.trans p').cons_inv"
},
{
"state_after": "case inl.intro.intro.intro.intro.nil\nα : Type uu\nβ : Type vv\nl₁✝ l₂ : List α\nt : α\nts is : List α\nIH1 : ∀ (l : List α), l ~ t :: is ++ ts → (∃ is' x, l = is' ++ ts) ∨ l ∈ permutationsAux ts (t :: is)\nIH2 : ∀ (l : List α), l ~ [] ++ is → (∃ is' x, l = is' ++ is) ∨ l ∈ permutationsAux is []\nl₁ : List α\np' : l₁ ++ [t] ~ t :: is\np : l₁ ++ [] ~ is\n⊢ (∃ is' x, l₁ ++ [t] ++ ts = is' ++ t :: ts) ∨\n l₁ ++ [t] ++ ts ∈ permutationsAux ts (t :: is) ∨\n ∃ l₁_1 l₂, l₁_1 ++ l₂ ∈ permutations is ∧ l₂ ≠ [] ∧ l₁ ++ [t] ++ ts = l₁_1 ++ t :: l₂ ++ ts\n\ncase inl.intro.intro.intro.intro.cons\nα : Type uu\nβ : Type vv\nl₁✝ l₂ : List α\nt : α\nts is : List α\nIH1 : ∀ (l : List α), l ~ t :: is ++ ts → (∃ is' x, l = is' ++ ts) ∨ l ∈ permutationsAux ts (t :: is)\nIH2 : ∀ (l : List α), l ~ [] ++ is → (∃ is' x, l = is' ++ is) ∨ l ∈ permutationsAux is []\nl₁ : List α\na : α\nl₂' : List α\np' : l₁ ++ t :: a :: l₂' ~ t :: is\np : l₁ ++ a :: l₂' ~ is\n⊢ (∃ is' x, l₁ ++ t :: a :: l₂' ++ ts = is' ++ t :: ts) ∨\n l₁ ++ t :: a :: l₂' ++ ts ∈ permutationsAux ts (t :: is) ∨\n ∃ l₁_1 l₂, l₁_1 ++ l₂ ∈ permutations is ∧ l₂ ≠ [] ∧ l₁ ++ t :: a :: l₂' ++ ts = l₁_1 ++ t :: l₂ ++ ts",
"state_before": "case inl.intro.intro.intro.intro\nα : Type uu\nβ : Type vv\nl₁✝ l₂✝ : List α\nt : α\nts is : List α\nIH1 : ∀ (l : List α), l ~ t :: is ++ ts → (∃ is' x, l = is' ++ ts) ∨ l ∈ permutationsAux ts (t :: is)\nIH2 : ∀ (l : List α), l ~ [] ++ is → (∃ is' x, l = is' ++ is) ∨ l ∈ permutationsAux is []\nl₁ l₂ : List α\np' : l₁ ++ t :: l₂ ~ t :: is\np : l₁ ++ l₂ ~ is\n⊢ (∃ is' x, l₁ ++ t :: l₂ ++ ts = is' ++ t :: ts) ∨\n l₁ ++ t :: l₂ ++ ts ∈ permutationsAux ts (t :: is) ∨\n ∃ l₁_1 l₂_1, l₁_1 ++ l₂_1 ∈ permutations is ∧ l₂_1 ≠ [] ∧ l₁ ++ t :: l₂ ++ ts = l₁_1 ++ t :: l₂_1 ++ ts",
"tactic": "cases' l₂ with a l₂'"
},
{
"state_after": "no goals",
"state_before": "case inl.intro.intro.intro.intro.nil\nα : Type uu\nβ : Type vv\nl₁✝ l₂ : List α\nt : α\nts is : List α\nIH1 : ∀ (l : List α), l ~ t :: is ++ ts → (∃ is' x, l = is' ++ ts) ∨ l ∈ permutationsAux ts (t :: is)\nIH2 : ∀ (l : List α), l ~ [] ++ is → (∃ is' x, l = is' ++ is) ∨ l ∈ permutationsAux is []\nl₁ : List α\np' : l₁ ++ [t] ~ t :: is\np : l₁ ++ [] ~ is\n⊢ (∃ is' x, l₁ ++ [t] ++ ts = is' ++ t :: ts) ∨\n l₁ ++ [t] ++ ts ∈ permutationsAux ts (t :: is) ∨\n ∃ l₁_1 l₂, l₁_1 ++ l₂ ∈ permutations is ∧ l₂ ≠ [] ∧ l₁ ++ [t] ++ ts = l₁_1 ++ t :: l₂ ++ ts",
"tactic": "exact Or.inl ⟨l₁, by simpa using p⟩"
},
{
"state_after": "no goals",
"state_before": "α : Type uu\nβ : Type vv\nl₁✝ l₂ : List α\nt : α\nts is : List α\nIH1 : ∀ (l : List α), l ~ t :: is ++ ts → (∃ is' x, l = is' ++ ts) ∨ l ∈ permutationsAux ts (t :: is)\nIH2 : ∀ (l : List α), l ~ [] ++ is → (∃ is' x, l = is' ++ is) ∨ l ∈ permutationsAux is []\nl₁ : List α\np' : l₁ ++ [t] ~ t :: is\np : l₁ ++ [] ~ is\n⊢ ∃ x, l₁ ++ [t] ++ ts = l₁ ++ t :: ts",
"tactic": "simpa using p"
},
{
"state_after": "no goals",
"state_before": "case inl.intro.intro.intro.intro.cons\nα : Type uu\nβ : Type vv\nl₁✝ l₂ : List α\nt : α\nts is : List α\nIH1 : ∀ (l : List α), l ~ t :: is ++ ts → (∃ is' x, l = is' ++ ts) ∨ l ∈ permutationsAux ts (t :: is)\nIH2 : ∀ (l : List α), l ~ [] ++ is → (∃ is' x, l = is' ++ is) ∨ l ∈ permutationsAux is []\nl₁ : List α\na : α\nl₂' : List α\np' : l₁ ++ t :: a :: l₂' ~ t :: is\np : l₁ ++ a :: l₂' ~ is\n⊢ (∃ is' x, l₁ ++ t :: a :: l₂' ++ ts = is' ++ t :: ts) ∨\n l₁ ++ t :: a :: l₂' ++ ts ∈ permutationsAux ts (t :: is) ∨\n ∃ l₁_1 l₂, l₁_1 ++ l₂ ∈ permutations is ∧ l₂ ≠ [] ∧ l₁ ++ t :: a :: l₂' ++ ts = l₁_1 ++ t :: l₂ ++ ts",
"tactic": "exact Or.inr (Or.inr ⟨l₁, a :: l₂', mem_permutations_of_perm_lemma (IH2 _) p, by simp⟩)"
},
{
"state_after": "no goals",
"state_before": "α : Type uu\nβ : Type vv\nl₁✝ l₂ : List α\nt : α\nts is : List α\nIH1 : ∀ (l : List α), l ~ t :: is ++ ts → (∃ is' x, l = is' ++ ts) ∨ l ∈ permutationsAux ts (t :: is)\nIH2 : ∀ (l : List α), l ~ [] ++ is → (∃ is' x, l = is' ++ is) ∨ l ∈ permutationsAux is []\nl₁ : List α\na : α\nl₂' : List α\np' : l₁ ++ t :: a :: l₂' ~ t :: is\np : l₁ ++ a :: l₂' ~ is\n⊢ a :: l₂' ≠ [] ∧ l₁ ++ t :: a :: l₂' ++ ts = l₁ ++ t :: a :: l₂' ++ ts",
"tactic": "simp"
},
{
"state_after": "no goals",
"state_before": "case inr\nα : Type uu\nβ : Type vv\nl₁ l₂ : List α\nt : α\nts is : List α\nIH1 : ∀ (l : List α), l ~ t :: is ++ ts → (∃ is' x, l = is' ++ ts) ∨ l ∈ permutationsAux ts (t :: is)\nIH2 : ∀ (l : List α), l ~ [] ++ is → (∃ is' x, l = is' ++ is) ∨ l ∈ permutationsAux is []\nl : List α\np : l ~ is ++ t :: ts\nm : l ∈ permutationsAux ts (t :: is)\n⊢ (∃ is' x, l = is' ++ t :: ts) ∨\n l ∈ permutationsAux ts (t :: is) ∨ ∃ l₁ l₂, l₁ ++ l₂ ∈ permutations is ∧ l₂ ≠ [] ∧ l = l₁ ++ t :: l₂ ++ ts",
"tactic": "exact Or.inr (Or.inl m)"
}
] |
[
1260,
28
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1243,
1
] |
Mathlib/Data/Num/Lemmas.lean
|
PosNum.to_nat_inj
|
[
{
"state_after": "no goals",
"state_before": "α : Type ?u.304571\nm n : PosNum\nh : ↑m = ↑n\n⊢ pos m = pos n",
"tactic": "rw [← PosNum.of_to_nat, ← PosNum.of_to_nat, h]"
}
] |
[
532,
91
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
531,
1
] |
Mathlib/Data/List/Basic.lean
|
List.foldl_nil
|
[] |
[
2395,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2394,
1
] |
Mathlib/Data/Polynomial/Eval.lean
|
Polynomial.root_mul_right_of_isRoot
|
[
{
"state_after": "no goals",
"state_before": "R : Type u\nS : Type v\nT : Type w\nι : Type y\na b : R\nm n : ℕ\ninst✝¹ : CommSemiring R\np✝ q✝ : R[X]\nx : R\ninst✝ : CommSemiring S\nf : R →+* S\np q : R[X]\nH : IsRoot p a\n⊢ IsRoot (p * q) a",
"tactic": "rw [IsRoot, eval_mul, IsRoot.def.1 H, zero_mul]"
}
] |
[
1104,
53
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1103,
1
] |
Mathlib/Analysis/Convex/SpecificFunctions/Deriv.lean
|
strictConcaveOn_cos_Icc
|
[
{
"state_after": "x : ℝ\nhx : x ∈ interior (Icc (-(π / 2)) (π / 2))\n⊢ (deriv^[2]) cos x < 0",
"state_before": "⊢ StrictConcaveOn ℝ (Icc (-(π / 2)) (π / 2)) cos",
"tactic": "apply strictConcaveOn_of_deriv2_neg (convex_Icc _ _) continuousOn_cos fun x hx => ?_"
},
{
"state_after": "x : ℝ\nhx : x ∈ Ioo (-(π / 2)) (π / 2)\n⊢ (deriv^[2]) cos x < 0",
"state_before": "x : ℝ\nhx : x ∈ interior (Icc (-(π / 2)) (π / 2))\n⊢ (deriv^[2]) cos x < 0",
"tactic": "rw [interior_Icc] at hx"
},
{
"state_after": "no goals",
"state_before": "x : ℝ\nhx : x ∈ Ioo (-(π / 2)) (π / 2)\n⊢ (deriv^[2]) cos x < 0",
"tactic": "simp [cos_pos_of_mem_Ioo hx]"
}
] |
[
179,
31
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
176,
1
] |
Mathlib/Analysis/Normed/Group/Quotient.lean
|
NormedAddGroupHom.IsQuotient.norm_le
|
[
{
"state_after": "M : Type u_1\nN : Type u_2\ninst✝¹ : SeminormedAddCommGroup M\ninst✝ : SeminormedAddCommGroup N\nf : NormedAddGroupHom M N\nhquot : IsQuotient f\nm : M\n⊢ sInf ((fun m_1 => ‖m + m_1‖) '' ↑(ker f)) ≤ ‖m‖",
"state_before": "M : Type u_1\nN : Type u_2\ninst✝¹ : SeminormedAddCommGroup M\ninst✝ : SeminormedAddCommGroup N\nf : NormedAddGroupHom M N\nhquot : IsQuotient f\nm : M\n⊢ ‖↑f m‖ ≤ ‖m‖",
"tactic": "rw [hquot.norm]"
},
{
"state_after": "case h₁\nM : Type u_1\nN : Type u_2\ninst✝¹ : SeminormedAddCommGroup M\ninst✝ : SeminormedAddCommGroup N\nf : NormedAddGroupHom M N\nhquot : IsQuotient f\nm : M\n⊢ BddBelow ((fun m_1 => ‖m + m_1‖) '' ↑(ker f))\n\ncase h₂\nM : Type u_1\nN : Type u_2\ninst✝¹ : SeminormedAddCommGroup M\ninst✝ : SeminormedAddCommGroup N\nf : NormedAddGroupHom M N\nhquot : IsQuotient f\nm : M\n⊢ ‖m‖ ∈ (fun m_1 => ‖m + m_1‖) '' ↑(ker f)",
"state_before": "M : Type u_1\nN : Type u_2\ninst✝¹ : SeminormedAddCommGroup M\ninst✝ : SeminormedAddCommGroup N\nf : NormedAddGroupHom M N\nhquot : IsQuotient f\nm : M\n⊢ sInf ((fun m_1 => ‖m + m_1‖) '' ↑(ker f)) ≤ ‖m‖",
"tactic": "apply csInf_le"
},
{
"state_after": "case h₁\nM : Type u_1\nN : Type u_2\ninst✝¹ : SeminormedAddCommGroup M\ninst✝ : SeminormedAddCommGroup N\nf : NormedAddGroupHom M N\nhquot : IsQuotient f\nm : M\n⊢ 0 ∈ lowerBounds ((fun m_1 => ‖m + m_1‖) '' ↑(ker f))",
"state_before": "case h₁\nM : Type u_1\nN : Type u_2\ninst✝¹ : SeminormedAddCommGroup M\ninst✝ : SeminormedAddCommGroup N\nf : NormedAddGroupHom M N\nhquot : IsQuotient f\nm : M\n⊢ BddBelow ((fun m_1 => ‖m + m_1‖) '' ↑(ker f))",
"tactic": "use 0"
},
{
"state_after": "case h₁.intro.intro\nM : Type u_1\nN : Type u_2\ninst✝¹ : SeminormedAddCommGroup M\ninst✝ : SeminormedAddCommGroup N\nf : NormedAddGroupHom M N\nhquot : IsQuotient f\nm m' : M\n⊢ 0 ≤ (fun m_1 => ‖m + m_1‖) m'",
"state_before": "case h₁\nM : Type u_1\nN : Type u_2\ninst✝¹ : SeminormedAddCommGroup M\ninst✝ : SeminormedAddCommGroup N\nf : NormedAddGroupHom M N\nhquot : IsQuotient f\nm : M\n⊢ 0 ∈ lowerBounds ((fun m_1 => ‖m + m_1‖) '' ↑(ker f))",
"tactic": "rintro _ ⟨m', -, rfl⟩"
},
{
"state_after": "no goals",
"state_before": "case h₁.intro.intro\nM : Type u_1\nN : Type u_2\ninst✝¹ : SeminormedAddCommGroup M\ninst✝ : SeminormedAddCommGroup N\nf : NormedAddGroupHom M N\nhquot : IsQuotient f\nm m' : M\n⊢ 0 ≤ (fun m_1 => ‖m + m_1‖) m'",
"tactic": "apply norm_nonneg"
},
{
"state_after": "no goals",
"state_before": "case h₂\nM : Type u_1\nN : Type u_2\ninst✝¹ : SeminormedAddCommGroup M\ninst✝ : SeminormedAddCommGroup N\nf : NormedAddGroupHom M N\nhquot : IsQuotient f\nm : M\n⊢ ‖m‖ ∈ (fun m_1 => ‖m + m_1‖) '' ↑(ker f)",
"tactic": "exact ⟨0, f.ker.zero_mem, by simp⟩"
},
{
"state_after": "no goals",
"state_before": "M : Type u_1\nN : Type u_2\ninst✝¹ : SeminormedAddCommGroup M\ninst✝ : SeminormedAddCommGroup N\nf : NormedAddGroupHom M N\nhquot : IsQuotient f\nm : M\n⊢ (fun m_1 => ‖m + m_1‖) 0 = ‖m‖",
"tactic": "simp"
}
] |
[
400,
39
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
393,
1
] |
Mathlib/Topology/Algebra/Order/MonotoneConvergence.lean
|
isGLB_of_tendsto_atBot
|
[] |
[
289,
59
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
286,
1
] |
Mathlib/Analysis/NormedSpace/ENorm.lean
|
ENorm.ext_iff
|
[] |
[
76,
28
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
75,
1
] |
Mathlib/Data/Polynomial/Eval.lean
|
Polynomial.eval₂_list_prod_noncomm
|
[
{
"state_after": "case H0\nR : Type u\nS : Type v\nT : Type w\nι : Type y\na b : R\nm n : ℕ\ninst✝² : Semiring R\np q r : R[X]\ninst✝¹ : Semiring S\nf : R →+* S\nx : S\ninst✝ : Semiring T\nps : List R[X]\nhf✝ : ∀ (p : R[X]), p ∈ ps → ∀ (k : ℕ), Commute (↑f (coeff p k)) x\nhf : ∀ (p : R[X]), p ∈ [] → ∀ (k : ℕ), Commute (↑f (coeff p k)) x\n⊢ eval₂ f x (List.prod []) = List.prod (List.map (eval₂ f x) [])\n\ncase H1\nR : Type u\nS : Type v\nT : Type w\nι : Type y\na b : R\nm n : ℕ\ninst✝² : Semiring R\np✝ q r : R[X]\ninst✝¹ : Semiring S\nf : R →+* S\nx : S\ninst✝ : Semiring T\nps✝ : List R[X]\nhf✝ : ∀ (p : R[X]), p ∈ ps✝ → ∀ (k : ℕ), Commute (↑f (coeff p k)) x\nps : List R[X]\np : R[X]\nihp :\n (∀ (p : R[X]), p ∈ ps → ∀ (k : ℕ), Commute (↑f (coeff p k)) x) →\n eval₂ f x (List.prod ps) = List.prod (List.map (eval₂ f x) ps)\nhf : ∀ (p_1 : R[X]), p_1 ∈ ps ++ [p] → ∀ (k : ℕ), Commute (↑f (coeff p_1 k)) x\n⊢ eval₂ f x (List.prod (ps ++ [p])) = List.prod (List.map (eval₂ f x) (ps ++ [p]))",
"state_before": "R : Type u\nS : Type v\nT : Type w\nι : Type y\na b : R\nm n : ℕ\ninst✝² : Semiring R\np q r : R[X]\ninst✝¹ : Semiring S\nf : R →+* S\nx : S\ninst✝ : Semiring T\nps : List R[X]\nhf : ∀ (p : R[X]), p ∈ ps → ∀ (k : ℕ), Commute (↑f (coeff p k)) x\n⊢ eval₂ f x (List.prod ps) = List.prod (List.map (eval₂ f x) ps)",
"tactic": "induction' ps using List.reverseRecOn with ps p ihp"
},
{
"state_after": "no goals",
"state_before": "case H0\nR : Type u\nS : Type v\nT : Type w\nι : Type y\na b : R\nm n : ℕ\ninst✝² : Semiring R\np q r : R[X]\ninst✝¹ : Semiring S\nf : R →+* S\nx : S\ninst✝ : Semiring T\nps : List R[X]\nhf✝ : ∀ (p : R[X]), p ∈ ps → ∀ (k : ℕ), Commute (↑f (coeff p k)) x\nhf : ∀ (p : R[X]), p ∈ [] → ∀ (k : ℕ), Commute (↑f (coeff p k)) x\n⊢ eval₂ f x (List.prod []) = List.prod (List.map (eval₂ f x) [])",
"tactic": "simp"
},
{
"state_after": "case H1\nR : Type u\nS : Type v\nT : Type w\nι : Type y\na b : R\nm n : ℕ\ninst✝² : Semiring R\np✝ q r : R[X]\ninst✝¹ : Semiring S\nf : R →+* S\nx : S\ninst✝ : Semiring T\nps✝ : List R[X]\nhf✝ : ∀ (p : R[X]), p ∈ ps✝ → ∀ (k : ℕ), Commute (↑f (coeff p k)) x\nps : List R[X]\np : R[X]\nihp :\n (∀ (p : R[X]), p ∈ ps → ∀ (k : ℕ), Commute (↑f (coeff p k)) x) →\n eval₂ f x (List.prod ps) = List.prod (List.map (eval₂ f x) ps)\nhf : (∀ (x_1 : R[X]), x_1 ∈ ps → ∀ (k : ℕ), Commute (↑f (coeff x_1 k)) x) ∧ ∀ (k : ℕ), Commute (↑f (coeff p k)) x\n⊢ eval₂ f x (List.prod (ps ++ [p])) = List.prod (List.map (eval₂ f x) (ps ++ [p]))",
"state_before": "case H1\nR : Type u\nS : Type v\nT : Type w\nι : Type y\na b : R\nm n : ℕ\ninst✝² : Semiring R\np✝ q r : R[X]\ninst✝¹ : Semiring S\nf : R →+* S\nx : S\ninst✝ : Semiring T\nps✝ : List R[X]\nhf✝ : ∀ (p : R[X]), p ∈ ps✝ → ∀ (k : ℕ), Commute (↑f (coeff p k)) x\nps : List R[X]\np : R[X]\nihp :\n (∀ (p : R[X]), p ∈ ps → ∀ (k : ℕ), Commute (↑f (coeff p k)) x) →\n eval₂ f x (List.prod ps) = List.prod (List.map (eval₂ f x) ps)\nhf : ∀ (p_1 : R[X]), p_1 ∈ ps ++ [p] → ∀ (k : ℕ), Commute (↑f (coeff p_1 k)) x\n⊢ eval₂ f x (List.prod (ps ++ [p])) = List.prod (List.map (eval₂ f x) (ps ++ [p]))",
"tactic": "simp only [List.forall_mem_append, List.forall_mem_singleton] at hf"
},
{
"state_after": "no goals",
"state_before": "case H1\nR : Type u\nS : Type v\nT : Type w\nι : Type y\na b : R\nm n : ℕ\ninst✝² : Semiring R\np✝ q r : R[X]\ninst✝¹ : Semiring S\nf : R →+* S\nx : S\ninst✝ : Semiring T\nps✝ : List R[X]\nhf✝ : ∀ (p : R[X]), p ∈ ps✝ → ∀ (k : ℕ), Commute (↑f (coeff p k)) x\nps : List R[X]\np : R[X]\nihp :\n (∀ (p : R[X]), p ∈ ps → ∀ (k : ℕ), Commute (↑f (coeff p k)) x) →\n eval₂ f x (List.prod ps) = List.prod (List.map (eval₂ f x) ps)\nhf : (∀ (x_1 : R[X]), x_1 ∈ ps → ∀ (k : ℕ), Commute (↑f (coeff x_1 k)) x) ∧ ∀ (k : ℕ), Commute (↑f (coeff p k)) x\n⊢ eval₂ f x (List.prod (ps ++ [p])) = List.prod (List.map (eval₂ f x) (ps ++ [p]))",
"tactic": "simp [eval₂_mul_noncomm _ _ hf.2, ihp hf.1]"
}
] |
[
213,
48
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
207,
1
] |
Mathlib/Geometry/Manifold/SmoothManifoldWithCorners.lean
|
extChartAt_source_mem_nhds'
|
[
{
"state_after": "no goals",
"state_before": "𝕜 : Type u_3\nE : Type u_2\nM : Type u_1\nH : Type u_4\nE' : Type ?u.187773\nM' : Type ?u.187776\nH' : Type ?u.187779\ninst✝¹⁰ : NontriviallyNormedField 𝕜\ninst✝⁹ : NormedAddCommGroup E\ninst✝⁸ : NormedSpace 𝕜 E\ninst✝⁷ : TopologicalSpace H\ninst✝⁶ : TopologicalSpace M\nf f' : LocalHomeomorph M H\nI : ModelWithCorners 𝕜 E H\ninst✝⁵ : NormedAddCommGroup E'\ninst✝⁴ : NormedSpace 𝕜 E'\ninst✝³ : TopologicalSpace H'\ninst✝² : TopologicalSpace M'\nI' : ModelWithCorners 𝕜 E' H'\nx : M\ns t : Set M\ninst✝¹ : ChartedSpace H M\ninst✝ : ChartedSpace H' M'\nx' : M\nh : x' ∈ (extChartAt I x).source\n⊢ x' ∈ (chartAt H x).toLocalEquiv.source",
"tactic": "rwa [← extChartAt_source I]"
}
] |
[
1069,
63
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1067,
1
] |
Mathlib/Combinatorics/SimpleGraph/Connectivity.lean
|
SimpleGraph.reachable_iff_nonempty_univ
|
[] |
[
1848,
33
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1846,
1
] |
Mathlib/MeasureTheory/Integral/Bochner.lean
|
MeasureTheory.integral_trim_ae
|
[
{
"state_after": "α : Type ?u.1721748\nE : Type ?u.1721751\nF : Type u_2\n𝕜 : Type ?u.1721757\ninst✝⁹ : NormedAddCommGroup E\ninst✝⁸ : NormedSpace ℝ E\ninst✝⁷ : CompleteSpace E\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : NormedSpace 𝕜 E\ninst✝⁴ : SMulCommClass ℝ 𝕜 E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace ℝ F\ninst✝¹ : CompleteSpace F\nH : Type ?u.1724428\nβ : Type u_1\nγ : Type ?u.1724434\ninst✝ : NormedAddCommGroup H\nm m0 : MeasurableSpace β\nμ : Measure β\nhm : m ≤ m0\nf : β → F\nhf : AEStronglyMeasurable f (Measure.trim μ hm)\n⊢ (∫ (a : β), AEStronglyMeasurable.mk f hf a ∂μ) = ∫ (a : β), AEStronglyMeasurable.mk f hf a ∂Measure.trim μ hm",
"state_before": "α : Type ?u.1721748\nE : Type ?u.1721751\nF : Type u_2\n𝕜 : Type ?u.1721757\ninst✝⁹ : NormedAddCommGroup E\ninst✝⁸ : NormedSpace ℝ E\ninst✝⁷ : CompleteSpace E\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : NormedSpace 𝕜 E\ninst✝⁴ : SMulCommClass ℝ 𝕜 E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace ℝ F\ninst✝¹ : CompleteSpace F\nH : Type ?u.1724428\nβ : Type u_1\nγ : Type ?u.1724434\ninst✝ : NormedAddCommGroup H\nm m0 : MeasurableSpace β\nμ : Measure β\nhm : m ≤ m0\nf : β → F\nhf : AEStronglyMeasurable f (Measure.trim μ hm)\n⊢ (∫ (x : β), f x ∂μ) = ∫ (x : β), f x ∂Measure.trim μ hm",
"tactic": "rw [integral_congr_ae (ae_eq_of_ae_eq_trim hf.ae_eq_mk), integral_congr_ae hf.ae_eq_mk]"
},
{
"state_after": "no goals",
"state_before": "α : Type ?u.1721748\nE : Type ?u.1721751\nF : Type u_2\n𝕜 : Type ?u.1721757\ninst✝⁹ : NormedAddCommGroup E\ninst✝⁸ : NormedSpace ℝ E\ninst✝⁷ : CompleteSpace E\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : NormedSpace 𝕜 E\ninst✝⁴ : SMulCommClass ℝ 𝕜 E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace ℝ F\ninst✝¹ : CompleteSpace F\nH : Type ?u.1724428\nβ : Type u_1\nγ : Type ?u.1724434\ninst✝ : NormedAddCommGroup H\nm m0 : MeasurableSpace β\nμ : Measure β\nhm : m ≤ m0\nf : β → F\nhf : AEStronglyMeasurable f (Measure.trim μ hm)\n⊢ (∫ (a : β), AEStronglyMeasurable.mk f hf a ∂μ) = ∫ (a : β), AEStronglyMeasurable.mk f hf a ∂Measure.trim μ hm",
"tactic": "exact integral_trim hm hf.stronglyMeasurable_mk"
}
] |
[
1794,
50
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1791,
1
] |
Mathlib/Data/Set/Intervals/Pi.lean
|
Set.image_update_Icc_left
|
[
{
"state_after": "no goals",
"state_before": "ι : Type u_1\nα : ι → Type u_2\ninst✝¹ : DecidableEq ι\ninst✝ : (i : ι) → PartialOrder (α i)\nf : (i : ι) → α i\ni : ι\na : α i\n⊢ update f i '' Icc a (f i) = Icc (update f i a) f",
"tactic": "simpa using image_update_Icc f i a (f i)"
}
] |
[
164,
100
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
163,
1
] |
Mathlib/MeasureTheory/Measure/AEDisjoint.lean
|
MeasureTheory.AEDisjoint.congr
|
[] |
[
92,
68
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
90,
11
] |
Mathlib/Order/Filter/Basic.lean
|
Filter.comap_iSup
|
[
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\nδ : Type ?u.254383\nι✝ : Sort x\nf✝ f₁ f₂ : Filter α\ng g₁ g₂ : Filter β\nm✝ : α → β\nm' : β → γ\ns✝ : Set α\nt : Set β\nι : Sort u_1\nf : ι → Filter β\nm : α → β\ns : Set α\nhs : s ∈ ⨆ (i : ι), comap m (f i)\n⊢ ∀ (i : ι), ∃ t, t ∈ f i ∧ m ⁻¹' t ⊆ s",
"tactic": "simpa only [mem_comap, exists_prop, mem_iSup] using mem_iSup.1 hs"
},
{
"state_after": "α : Type u\nβ : Type v\nγ : Type w\nδ : Type ?u.254383\nι✝ : Sort x\nf✝ f₁ f₂ : Filter α\ng g₁ g₂ : Filter β\nm✝ : α → β\nm' : β → γ\ns✝ : Set α\nt✝ : Set β\nι : Sort u_1\nf : ι → Filter β\nm : α → β\ns : Set α\nhs : s ∈ ⨆ (i : ι), comap m (f i)\nthis : ∀ (i : ι), ∃ t, t ∈ f i ∧ m ⁻¹' t ⊆ s\nt : ι → Set β\nht : ∀ (x : ι), t x ∈ f x ∧ m ⁻¹' t x ⊆ s\n⊢ ∀ (i : ι), m ⁻¹' t i ⊆ s",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\nδ : Type ?u.254383\nι✝ : Sort x\nf✝ f₁ f₂ : Filter α\ng g₁ g₂ : Filter β\nm✝ : α → β\nm' : β → γ\ns✝ : Set α\nt✝ : Set β\nι : Sort u_1\nf : ι → Filter β\nm : α → β\ns : Set α\nhs : s ∈ ⨆ (i : ι), comap m (f i)\nthis : ∀ (i : ι), ∃ t, t ∈ f i ∧ m ⁻¹' t ⊆ s\nt : ι → Set β\nht : ∀ (x : ι), t x ∈ f x ∧ m ⁻¹' t x ⊆ s\n⊢ (m ⁻¹' ⋃ (i : ι), t i) ⊆ s",
"tactic": "rw [preimage_iUnion, iUnion_subset_iff]"
},
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\nδ : Type ?u.254383\nι✝ : Sort x\nf✝ f₁ f₂ : Filter α\ng g₁ g₂ : Filter β\nm✝ : α → β\nm' : β → γ\ns✝ : Set α\nt✝ : Set β\nι : Sort u_1\nf : ι → Filter β\nm : α → β\ns : Set α\nhs : s ∈ ⨆ (i : ι), comap m (f i)\nthis : ∀ (i : ι), ∃ t, t ∈ f i ∧ m ⁻¹' t ⊆ s\nt : ι → Set β\nht : ∀ (x : ι), t x ∈ f x ∧ m ⁻¹' t x ⊆ s\n⊢ ∀ (i : ι), m ⁻¹' t i ⊆ s",
"tactic": "exact fun i => (ht i).2"
}
] |
[
2236,
49
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2227,
1
] |
Mathlib/Data/Multiset/NatAntidiagonal.lean
|
Multiset.Nat.antidiagonal_succ
|
[
{
"state_after": "no goals",
"state_before": "n : ℕ\n⊢ antidiagonal (n + 1) = (0, n + 1) ::ₘ map (Prod.map Nat.succ id) (antidiagonal n)",
"tactic": "simp only [antidiagonal, List.Nat.antidiagonal_succ, coe_map, cons_coe]"
}
] |
[
64,
74
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
62,
1
] |
Mathlib/Data/Complex/Exponential.lean
|
Complex.exp_sub_cosh
|
[] |
[
737,
45
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
736,
1
] |
Mathlib/Algebra/Order/Hom/Monoid.lean
|
OrderMonoidWithZeroHom.coe_monoidWithZeroHom
|
[] |
[
628,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
627,
1
] |
Mathlib/Tactic/NormNum/NatFib.lean
|
Mathlib.Meta.NormNum.isFibAux_two_mul_add_one
|
[
{
"state_after": "no goals",
"state_before": "n a b n' a' b' : ℕ\nH : IsFibAux n a b\nhn : 2 * n + 1 = n'\nh1 : a * a + b * b = a'\nh2 : b * (2 * a + b) = b'\n⊢ fib n' = a'",
"tactic": "rw [← hn, fib_two_mul_add_one, H.1, H.2, pow_two, pow_two, add_comm, h1]"
},
{
"state_after": "no goals",
"state_before": "n a b n' a' b' : ℕ\nH : IsFibAux n a b\nhn : 2 * n + 1 = n'\nh1 : a * a + b * b = a'\nh2 : b * (2 * a + b) = b'\n⊢ fib (n' + 1) = b'",
"tactic": "rw [← hn, fib_two_mul_add_two, H.1, H.2, h2]"
}
] |
[
40,
52
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
36,
1
] |
Mathlib/Algebra/Module/Equiv.lean
|
LinearEquiv.congr_arg
|
[] |
[
246,
22
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
245,
11
] |
Mathlib/GroupTheory/Subgroup/Actions.lean
|
Subgroup.smul_def
|
[] |
[
37,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
36,
1
] |
Mathlib/RingTheory/Localization/Basic.lean
|
Localization.sub_mk
|
[
{
"state_after": "no goals",
"state_before": "R : Type u_1\ninst✝³ : CommRing R\nM : Submonoid R\nS : Type ?u.2970634\ninst✝² : CommRing S\ninst✝¹ : Algebra R S\nP : Type ?u.2970871\ninst✝ : CommRing P\na c : R\nb d : { x // x ∈ M }\n⊢ mk a b + -mk c d = mk a b + mk (-c) d",
"tactic": "rw [neg_mk]"
},
{
"state_after": "case e_x\nR : Type u_1\ninst✝³ : CommRing R\nM : Submonoid R\nS : Type ?u.2970634\ninst✝² : CommRing S\ninst✝¹ : Algebra R S\nP : Type ?u.2970871\ninst✝ : CommRing P\na c : R\nb d : { x // x ∈ M }\n⊢ ↑b * -c + ↑d * a = ↑d * a - ↑b * c",
"state_before": "R : Type u_1\ninst✝³ : CommRing R\nM : Submonoid R\nS : Type ?u.2970634\ninst✝² : CommRing S\ninst✝¹ : Algebra R S\nP : Type ?u.2970871\ninst✝ : CommRing P\na c : R\nb d : { x // x ∈ M }\n⊢ mk (↑b * -c + ↑d * a) (b * d) = mk (↑d * a - ↑b * c) (b * d)",
"tactic": "congr"
},
{
"state_after": "no goals",
"state_before": "case e_x\nR : Type u_1\ninst✝³ : CommRing R\nM : Submonoid R\nS : Type ?u.2970634\ninst✝² : CommRing S\ninst✝¹ : Algebra R S\nP : Type ?u.2970871\ninst✝ : CommRing P\na c : R\nb d : { x // x ∈ M }\n⊢ ↑b * -c + ↑d * a = ↑d * a - ↑b * c",
"tactic": "ring"
}
] |
[
1150,
53
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1144,
1
] |
Mathlib/LinearAlgebra/Prod.lean
|
LinearMap.prodMap_zero
|
[] |
[
362,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
361,
1
] |
Mathlib/Data/Nat/Parity.lean
|
Nat.even_xor_odd
|
[
{
"state_after": "no goals",
"state_before": "m n✝ n : ℕ\n⊢ Xor' (Even n) (Odd n)",
"tactic": "simp [Xor', odd_iff_not_even, Decidable.em (Even n)]"
}
] |
[
75,
55
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
74,
1
] |
Mathlib/Algebra/Order/Monoid/TypeTags.lean
|
Multiplicative.ofAdd_lt
|
[] |
[
155,
10
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
154,
1
] |
Mathlib/Topology/Filter.lean
|
Filter.isTopologicalBasis_Iic_principal
|
[
{
"state_after": "case intro.intro\nι : Sort ?u.640\nα : Type u_1\nβ : Type ?u.646\nX : Type ?u.649\nY : Type ?u.652\ns t : Set α\nl : Filter α\nhl : l ∈ (Iic ∘ 𝓟) s ∩ (Iic ∘ 𝓟) t\n⊢ ∃ t₃, t₃ ∈ range (Iic ∘ 𝓟) ∧ l ∈ t₃ ∧ t₃ ⊆ (Iic ∘ 𝓟) s ∩ (Iic ∘ 𝓟) t",
"state_before": "ι : Sort ?u.640\nα : Type u_1\nβ : Type ?u.646\nX : Type ?u.649\nY : Type ?u.652\n⊢ ∀ (t₁ : Set (Filter α)),\n t₁ ∈ range (Iic ∘ 𝓟) →\n ∀ (t₂ : Set (Filter α)),\n t₂ ∈ range (Iic ∘ 𝓟) → ∀ (x : Filter α), x ∈ t₁ ∩ t₂ → ∃ t₃, t₃ ∈ range (Iic ∘ 𝓟) ∧ x ∈ t₃ ∧ t₃ ⊆ t₁ ∩ t₂",
"tactic": "rintro _ ⟨s, rfl⟩ _ ⟨t, rfl⟩ l hl"
},
{
"state_after": "no goals",
"state_before": "case intro.intro\nι : Sort ?u.640\nα : Type u_1\nβ : Type ?u.646\nX : Type ?u.649\nY : Type ?u.652\ns t : Set α\nl : Filter α\nhl : l ∈ (Iic ∘ 𝓟) s ∩ (Iic ∘ 𝓟) t\n⊢ ∃ t₃, t₃ ∈ range (Iic ∘ 𝓟) ∧ l ∈ t₃ ∧ t₃ ⊆ (Iic ∘ 𝓟) s ∩ (Iic ∘ 𝓟) t",
"tactic": "exact ⟨Iic (𝓟 s) ∩ Iic (𝓟 t), ⟨s ∩ t, by simp⟩, hl, Subset.rfl⟩"
},
{
"state_after": "no goals",
"state_before": "ι : Sort ?u.640\nα : Type u_1\nβ : Type ?u.646\nX : Type ?u.649\nY : Type ?u.652\ns t : Set α\nl : Filter α\nhl : l ∈ (Iic ∘ 𝓟) s ∩ (Iic ∘ 𝓟) t\n⊢ (Iic ∘ 𝓟) (s ∩ t) = Iic (𝓟 s) ∩ Iic (𝓟 t)",
"tactic": "simp"
}
] |
[
69,
29
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
62,
1
] |
Mathlib/GroupTheory/Finiteness.lean
|
AddGroup.fg_iff_mul_fg
|
[] |
[
334,
45
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
332,
1
] |
Mathlib/Analysis/SpecialFunctions/Trigonometric/Basic.lean
|
Real.sin_pos_of_pos_of_lt_pi
|
[
{
"state_after": "no goals",
"state_before": "x : ℝ\nh0x : 0 < x\nhxp : x < π\nhx2 : ¬x ≤ 2\n⊢ 2 + 2 = 4",
"tactic": "norm_num"
}
] |
[
410,
65
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
404,
1
] |
Mathlib/Data/Finsupp/Defs.lean
|
Finsupp.support_add_eq
|
[
{
"state_after": "α : Type u_1\nβ : Type ?u.284137\nγ : Type ?u.284140\nι : Type ?u.284143\nM : Type u_2\nM' : Type ?u.284149\nN : Type ?u.284152\nP : Type ?u.284155\nG : Type ?u.284158\nH : Type ?u.284161\nR : Type ?u.284164\nS : Type ?u.284167\ninst✝¹ : AddZeroClass M\ninst✝ : DecidableEq α\ng₁ g₂ : α →₀ M\nh : Disjoint g₁.support g₂.support\na : α\nha✝ : a ∈ g₁.support ∪ g₂.support\nha : a ∈ g₁.support\nthis : ¬a ∈ g₂.support\n⊢ a ∈ (g₁ + g₂).support",
"state_before": "α : Type u_1\nβ : Type ?u.284137\nγ : Type ?u.284140\nι : Type ?u.284143\nM : Type u_2\nM' : Type ?u.284149\nN : Type ?u.284152\nP : Type ?u.284155\nG : Type ?u.284158\nH : Type ?u.284161\nR : Type ?u.284164\nS : Type ?u.284167\ninst✝¹ : AddZeroClass M\ninst✝ : DecidableEq α\ng₁ g₂ : α →₀ M\nh : Disjoint g₁.support g₂.support\na : α\nha✝ : a ∈ g₁.support ∪ g₂.support\nha : a ∈ g₁.support\n⊢ a ∈ (g₁ + g₂).support",
"tactic": "have : a ∉ g₂.support := disjoint_left.1 h ha"
},
{
"state_after": "α : Type u_1\nβ : Type ?u.284137\nγ : Type ?u.284140\nι : Type ?u.284143\nM : Type u_2\nM' : Type ?u.284149\nN : Type ?u.284152\nP : Type ?u.284155\nG : Type ?u.284158\nH : Type ?u.284161\nR : Type ?u.284164\nS : Type ?u.284167\ninst✝¹ : AddZeroClass M\ninst✝ : DecidableEq α\ng₁ g₂ : α →₀ M\nh : Disjoint g₁.support g₂.support\na : α\nha✝ : a ∈ g₁.support ∪ g₂.support\nha : ↑g₁ a ≠ 0\nthis : ↑g₂ a = 0\n⊢ ↑(g₁ + g₂) a ≠ 0",
"state_before": "α : Type u_1\nβ : Type ?u.284137\nγ : Type ?u.284140\nι : Type ?u.284143\nM : Type u_2\nM' : Type ?u.284149\nN : Type ?u.284152\nP : Type ?u.284155\nG : Type ?u.284158\nH : Type ?u.284161\nR : Type ?u.284164\nS : Type ?u.284167\ninst✝¹ : AddZeroClass M\ninst✝ : DecidableEq α\ng₁ g₂ : α →₀ M\nh : Disjoint g₁.support g₂.support\na : α\nha✝ : a ∈ g₁.support ∪ g₂.support\nha : a ∈ g₁.support\nthis : ¬a ∈ g₂.support\n⊢ a ∈ (g₁ + g₂).support",
"tactic": "simp only [mem_support_iff, not_not] at *"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.284137\nγ : Type ?u.284140\nι : Type ?u.284143\nM : Type u_2\nM' : Type ?u.284149\nN : Type ?u.284152\nP : Type ?u.284155\nG : Type ?u.284158\nH : Type ?u.284161\nR : Type ?u.284164\nS : Type ?u.284167\ninst✝¹ : AddZeroClass M\ninst✝ : DecidableEq α\ng₁ g₂ : α →₀ M\nh : Disjoint g₁.support g₂.support\na : α\nha✝ : a ∈ g₁.support ∪ g₂.support\nha : ↑g₁ a ≠ 0\nthis : ↑g₂ a = 0\n⊢ ↑(g₁ + g₂) a ≠ 0",
"tactic": "simpa only [add_apply, this, add_zero]"
},
{
"state_after": "α : Type u_1\nβ : Type ?u.284137\nγ : Type ?u.284140\nι : Type ?u.284143\nM : Type u_2\nM' : Type ?u.284149\nN : Type ?u.284152\nP : Type ?u.284155\nG : Type ?u.284158\nH : Type ?u.284161\nR : Type ?u.284164\nS : Type ?u.284167\ninst✝¹ : AddZeroClass M\ninst✝ : DecidableEq α\ng₁ g₂ : α →₀ M\nh : Disjoint g₁.support g₂.support\na : α\nha✝ : a ∈ g₁.support ∪ g₂.support\nha : a ∈ g₂.support\nthis : ¬a ∈ g₁.support\n⊢ a ∈ (g₁ + g₂).support",
"state_before": "α : Type u_1\nβ : Type ?u.284137\nγ : Type ?u.284140\nι : Type ?u.284143\nM : Type u_2\nM' : Type ?u.284149\nN : Type ?u.284152\nP : Type ?u.284155\nG : Type ?u.284158\nH : Type ?u.284161\nR : Type ?u.284164\nS : Type ?u.284167\ninst✝¹ : AddZeroClass M\ninst✝ : DecidableEq α\ng₁ g₂ : α →₀ M\nh : Disjoint g₁.support g₂.support\na : α\nha✝ : a ∈ g₁.support ∪ g₂.support\nha : a ∈ g₂.support\n⊢ a ∈ (g₁ + g₂).support",
"tactic": "have : a ∉ g₁.support := disjoint_right.1 h ha"
},
{
"state_after": "α : Type u_1\nβ : Type ?u.284137\nγ : Type ?u.284140\nι : Type ?u.284143\nM : Type u_2\nM' : Type ?u.284149\nN : Type ?u.284152\nP : Type ?u.284155\nG : Type ?u.284158\nH : Type ?u.284161\nR : Type ?u.284164\nS : Type ?u.284167\ninst✝¹ : AddZeroClass M\ninst✝ : DecidableEq α\ng₁ g₂ : α →₀ M\nh : Disjoint g₁.support g₂.support\na : α\nha✝ : a ∈ g₁.support ∪ g₂.support\nha : ↑g₂ a ≠ 0\nthis : ↑g₁ a = 0\n⊢ ↑(g₁ + g₂) a ≠ 0",
"state_before": "α : Type u_1\nβ : Type ?u.284137\nγ : Type ?u.284140\nι : Type ?u.284143\nM : Type u_2\nM' : Type ?u.284149\nN : Type ?u.284152\nP : Type ?u.284155\nG : Type ?u.284158\nH : Type ?u.284161\nR : Type ?u.284164\nS : Type ?u.284167\ninst✝¹ : AddZeroClass M\ninst✝ : DecidableEq α\ng₁ g₂ : α →₀ M\nh : Disjoint g₁.support g₂.support\na : α\nha✝ : a ∈ g₁.support ∪ g₂.support\nha : a ∈ g₂.support\nthis : ¬a ∈ g₁.support\n⊢ a ∈ (g₁ + g₂).support",
"tactic": "simp only [mem_support_iff, not_not] at *"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.284137\nγ : Type ?u.284140\nι : Type ?u.284143\nM : Type u_2\nM' : Type ?u.284149\nN : Type ?u.284152\nP : Type ?u.284155\nG : Type ?u.284158\nH : Type ?u.284161\nR : Type ?u.284164\nS : Type ?u.284167\ninst✝¹ : AddZeroClass M\ninst✝ : DecidableEq α\ng₁ g₂ : α →₀ M\nh : Disjoint g₁.support g₂.support\na : α\nha✝ : a ∈ g₁.support ∪ g₂.support\nha : ↑g₂ a ≠ 0\nthis : ↑g₁ a = 0\n⊢ ↑(g₁ + g₂) a ≠ 0",
"tactic": "simpa only [add_apply, this, zero_add]"
}
] |
[
996,
88
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
987,
1
] |
Mathlib/RingTheory/Polynomial/Basic.lean
|
MvPolynomial.prime_rename_iff
|
[
{
"state_after": "R : Type u\nS : Type ?u.606815\nσ : Type v\nM : Type w\ninst✝³ : CommRing R\ninst✝² : CommRing S\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nr : R\ns : Set σ\np : MvPolynomial (↑s) R\n⊢ Prime p ↔ Prime (↑(rename Subtype.val) p)",
"state_before": "R : Type u\nS : Type ?u.606815\nσ : Type v\nM : Type w\ninst✝³ : CommRing R\ninst✝² : CommRing S\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nr : R\ns : Set σ\np : MvPolynomial (↑s) R\n⊢ Prime (↑(rename Subtype.val) p) ↔ Prime p",
"tactic": "symm"
},
{
"state_after": "R : Type u\nS : Type ?u.606815\nσ : Type v\nM : Type w\ninst✝³ : CommRing R\ninst✝² : CommRing S\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nr : R\ns : Set σ\np : MvPolynomial (↑s) R\neqv : MvPolynomial (↑(sᶜ)) (MvPolynomial (↑s) R) ≃ₐ[R] MvPolynomial σ R :=\n AlgEquiv.trans (AlgEquiv.symm (sumAlgEquiv R ↑(sᶜ) ↑s))\n (renameEquiv R ((Equiv.sumComm ↑(sᶜ) ↑s).trans (Equiv.Set.sumCompl s)))\n⊢ Prime p ↔ Prime (↑(rename Subtype.val) p)",
"state_before": "R : Type u\nS : Type ?u.606815\nσ : Type v\nM : Type w\ninst✝³ : CommRing R\ninst✝² : CommRing S\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nr : R\ns : Set σ\np : MvPolynomial (↑s) R\n⊢ Prime p ↔ Prime (↑(rename Subtype.val) p)",
"tactic": "let eqv :=\n (sumAlgEquiv R (↥(sᶜ)) s).symm.trans\n (renameEquiv R <| (Equiv.sumComm (↥(sᶜ)) s).trans <| Equiv.Set.sumCompl s)"
},
{
"state_after": "R : Type u\nS : Type ?u.606815\nσ : Type v\nM : Type w\ninst✝³ : CommRing R\ninst✝² : CommRing S\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nr : R\ns : Set σ\np : MvPolynomial (↑s) R\neqv : MvPolynomial (↑(sᶜ)) (MvPolynomial (↑s) R) ≃ₐ[R] MvPolynomial σ R :=\n AlgEquiv.trans (AlgEquiv.symm (sumAlgEquiv R ↑(sᶜ) ↑s))\n (renameEquiv R ((Equiv.sumComm ↑(sᶜ) ↑s).trans (Equiv.Set.sumCompl s)))\nthis : ↑(rename Subtype.val) = RingHom.comp (↑↑eqv) C\n⊢ Prime (↑C p) ↔ Prime (↑(rename Subtype.val) p)",
"state_before": "R : Type u\nS : Type ?u.606815\nσ : Type v\nM : Type w\ninst✝³ : CommRing R\ninst✝² : CommRing S\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nr : R\ns : Set σ\np : MvPolynomial (↑s) R\neqv : MvPolynomial (↑(sᶜ)) (MvPolynomial (↑s) R) ≃ₐ[R] MvPolynomial σ R :=\n AlgEquiv.trans (AlgEquiv.symm (sumAlgEquiv R ↑(sᶜ) ↑s))\n (renameEquiv R ((Equiv.sumComm ↑(sᶜ) ↑s).trans (Equiv.Set.sumCompl s)))\nthis : ↑(rename Subtype.val) = RingHom.comp (↑↑eqv) C\n⊢ Prime p ↔ Prime (↑(rename Subtype.val) p)",
"tactic": "rw [← @prime_C_iff (MvPolynomial s R) (↥(sᶜ)) instCommRingMvPolynomialToCommSemiring p]"
},
{
"state_after": "R : Type u\nS : Type ?u.606815\nσ : Type v\nM : Type w\ninst✝³ : CommRing R\ninst✝² : CommRing S\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nr : R\ns : Set σ\np : MvPolynomial (↑s) R\neqv : MvPolynomial (↑(sᶜ)) (MvPolynomial (↑s) R) ≃ₐ[R] MvPolynomial σ R :=\n AlgEquiv.trans (AlgEquiv.symm (sumAlgEquiv R ↑(sᶜ) ↑s))\n (renameEquiv R ((Equiv.sumComm ↑(sᶜ) ↑s).trans (Equiv.Set.sumCompl s)))\nthis : ↑(rename Subtype.val) = RingHom.comp (↑↑eqv) C\n⊢ Prime (↑?e (↑C p)) ↔ Prime (↑(rename Subtype.val) p)\n\ncase e\nR : Type u\nS : Type ?u.606815\nσ : Type v\nM : Type w\ninst✝³ : CommRing R\ninst✝² : CommRing S\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nr : R\ns : Set σ\np : MvPolynomial (↑s) R\neqv : MvPolynomial (↑(sᶜ)) (MvPolynomial (↑s) R) ≃ₐ[R] MvPolynomial σ R :=\n AlgEquiv.trans (AlgEquiv.symm (sumAlgEquiv R ↑(sᶜ) ↑s))\n (renameEquiv R ((Equiv.sumComm ↑(sᶜ) ↑s).trans (Equiv.Set.sumCompl s)))\nthis : ↑(rename Subtype.val) = RingHom.comp (↑↑eqv) C\n⊢ MvPolynomial (↑(sᶜ)) (MvPolynomial (↑s) R) ≃* MvPolynomial σ R\n\nR : Type u\nS : Type ?u.606815\nσ : Type v\nM : Type w\ninst✝³ : CommRing R\ninst✝² : CommRing S\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nr : R\ns : Set σ\np : MvPolynomial (↑s) R\neqv : MvPolynomial (↑(sᶜ)) (MvPolynomial (↑s) R) ≃ₐ[R] MvPolynomial σ R :=\n AlgEquiv.trans (AlgEquiv.symm (sumAlgEquiv R ↑(sᶜ) ↑s))\n (renameEquiv R ((Equiv.sumComm ↑(sᶜ) ↑s).trans (Equiv.Set.sumCompl s)))\nthis : ↑(rename Subtype.val) = RingHom.comp (↑↑eqv) C\n⊢ CommMonoidWithZero (MvPolynomial σ R)",
"state_before": "R : Type u\nS : Type ?u.606815\nσ : Type v\nM : Type w\ninst✝³ : CommRing R\ninst✝² : CommRing S\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nr : R\ns : Set σ\np : MvPolynomial (↑s) R\neqv : MvPolynomial (↑(sᶜ)) (MvPolynomial (↑s) R) ≃ₐ[R] MvPolynomial σ R :=\n AlgEquiv.trans (AlgEquiv.symm (sumAlgEquiv R ↑(sᶜ) ↑s))\n (renameEquiv R ((Equiv.sumComm ↑(sᶜ) ↑s).trans (Equiv.Set.sumCompl s)))\nthis : ↑(rename Subtype.val) = RingHom.comp (↑↑eqv) C\n⊢ Prime (↑C p) ↔ Prime (↑(rename Subtype.val) p)",
"tactic": "rw [@MulEquiv.prime_iff (MvPolynomial (↑(sᶜ)) (MvPolynomial (↑s) R)) (MvPolynomial σ R) (_) (_)]"
},
{
"state_after": "case e\nR : Type u\nS : Type ?u.606815\nσ : Type v\nM : Type w\ninst✝³ : CommRing R\ninst✝² : CommRing S\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nr : R\ns : Set σ\np : MvPolynomial (↑s) R\neqv : MvPolynomial (↑(sᶜ)) (MvPolynomial (↑s) R) ≃ₐ[R] MvPolynomial σ R :=\n AlgEquiv.trans (AlgEquiv.symm (sumAlgEquiv R ↑(sᶜ) ↑s))\n (renameEquiv R ((Equiv.sumComm ↑(sᶜ) ↑s).trans (Equiv.Set.sumCompl s)))\nthis : ↑(rename Subtype.val) = RingHom.comp (↑↑eqv) C\n⊢ MvPolynomial (↑(sᶜ)) (MvPolynomial (↑s) R) ≃* MvPolynomial σ R\n\nR : Type u\nS : Type ?u.606815\nσ : Type v\nM : Type w\ninst✝³ : CommRing R\ninst✝² : CommRing S\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nr : R\ns : Set σ\np : MvPolynomial (↑s) R\neqv : MvPolynomial (↑(sᶜ)) (MvPolynomial (↑s) R) ≃ₐ[R] MvPolynomial σ R :=\n AlgEquiv.trans (AlgEquiv.symm (sumAlgEquiv R ↑(sᶜ) ↑s))\n (renameEquiv R ((Equiv.sumComm ↑(sᶜ) ↑s).trans (Equiv.Set.sumCompl s)))\nthis : ↑(rename Subtype.val) = RingHom.comp (↑↑eqv) C\n⊢ CommMonoidWithZero (MvPolynomial σ R)\n\nR : Type u\nS : Type ?u.606815\nσ : Type v\nM : Type w\ninst✝³ : CommRing R\ninst✝² : CommRing S\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nr : R\ns : Set σ\np : MvPolynomial (↑s) R\neqv : MvPolynomial (↑(sᶜ)) (MvPolynomial (↑s) R) ≃ₐ[R] MvPolynomial σ R :=\n AlgEquiv.trans (AlgEquiv.symm (sumAlgEquiv R ↑(sᶜ) ↑s))\n (renameEquiv R ((Equiv.sumComm ↑(sᶜ) ↑s).trans (Equiv.Set.sumCompl s)))\nthis : ↑(rename Subtype.val) = RingHom.comp (↑↑eqv) C\n⊢ Prime (↑?e (↑C p)) ↔ Prime (↑(rename Subtype.val) p)",
"state_before": "R : Type u\nS : Type ?u.606815\nσ : Type v\nM : Type w\ninst✝³ : CommRing R\ninst✝² : CommRing S\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nr : R\ns : Set σ\np : MvPolynomial (↑s) R\neqv : MvPolynomial (↑(sᶜ)) (MvPolynomial (↑s) R) ≃ₐ[R] MvPolynomial σ R :=\n AlgEquiv.trans (AlgEquiv.symm (sumAlgEquiv R ↑(sᶜ) ↑s))\n (renameEquiv R ((Equiv.sumComm ↑(sᶜ) ↑s).trans (Equiv.Set.sumCompl s)))\nthis : ↑(rename Subtype.val) = RingHom.comp (↑↑eqv) C\n⊢ Prime (↑?e (↑C p)) ↔ Prime (↑(rename Subtype.val) p)\n\ncase e\nR : Type u\nS : Type ?u.606815\nσ : Type v\nM : Type w\ninst✝³ : CommRing R\ninst✝² : CommRing S\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nr : R\ns : Set σ\np : MvPolynomial (↑s) R\neqv : MvPolynomial (↑(sᶜ)) (MvPolynomial (↑s) R) ≃ₐ[R] MvPolynomial σ R :=\n AlgEquiv.trans (AlgEquiv.symm (sumAlgEquiv R ↑(sᶜ) ↑s))\n (renameEquiv R ((Equiv.sumComm ↑(sᶜ) ↑s).trans (Equiv.Set.sumCompl s)))\nthis : ↑(rename Subtype.val) = RingHom.comp (↑↑eqv) C\n⊢ MvPolynomial (↑(sᶜ)) (MvPolynomial (↑s) R) ≃* MvPolynomial σ R\n\nR : Type u\nS : Type ?u.606815\nσ : Type v\nM : Type w\ninst✝³ : CommRing R\ninst✝² : CommRing S\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nr : R\ns : Set σ\np : MvPolynomial (↑s) R\neqv : MvPolynomial (↑(sᶜ)) (MvPolynomial (↑s) R) ≃ₐ[R] MvPolynomial σ R :=\n AlgEquiv.trans (AlgEquiv.symm (sumAlgEquiv R ↑(sᶜ) ↑s))\n (renameEquiv R ((Equiv.sumComm ↑(sᶜ) ↑s).trans (Equiv.Set.sumCompl s)))\nthis : ↑(rename Subtype.val) = RingHom.comp (↑↑eqv) C\n⊢ CommMonoidWithZero (MvPolynomial σ R)",
"tactic": "rotate_left"
},
{
"state_after": "R : Type u\nS : Type ?u.606815\nσ : Type v\nM : Type w\ninst✝³ : CommRing R\ninst✝² : CommRing S\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nr : R\ns : Set σ\np : MvPolynomial (↑s) R\neqv : MvPolynomial (↑(sᶜ)) (MvPolynomial (↑s) R) ≃ₐ[R] MvPolynomial σ R :=\n AlgEquiv.trans (AlgEquiv.symm (sumAlgEquiv R ↑(sᶜ) ↑s))\n (renameEquiv R ((Equiv.sumComm ↑(sᶜ) ↑s).trans (Equiv.Set.sumCompl s)))\nthis : ↑(rename Subtype.val) = RingHom.comp (↑↑eqv) C\n⊢ Prime (↑(AlgEquiv.toMulEquiv eqv) (↑C p)) ↔ Prime (↑(rename Subtype.val) p)",
"state_before": "case e\nR : Type u\nS : Type ?u.606815\nσ : Type v\nM : Type w\ninst✝³ : CommRing R\ninst✝² : CommRing S\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nr : R\ns : Set σ\np : MvPolynomial (↑s) R\neqv : MvPolynomial (↑(sᶜ)) (MvPolynomial (↑s) R) ≃ₐ[R] MvPolynomial σ R :=\n AlgEquiv.trans (AlgEquiv.symm (sumAlgEquiv R ↑(sᶜ) ↑s))\n (renameEquiv R ((Equiv.sumComm ↑(sᶜ) ↑s).trans (Equiv.Set.sumCompl s)))\nthis : ↑(rename Subtype.val) = RingHom.comp (↑↑eqv) C\n⊢ MvPolynomial (↑(sᶜ)) (MvPolynomial (↑s) R) ≃* MvPolynomial σ R\n\nR : Type u\nS : Type ?u.606815\nσ : Type v\nM : Type w\ninst✝³ : CommRing R\ninst✝² : CommRing S\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nr : R\ns : Set σ\np : MvPolynomial (↑s) R\neqv : MvPolynomial (↑(sᶜ)) (MvPolynomial (↑s) R) ≃ₐ[R] MvPolynomial σ R :=\n AlgEquiv.trans (AlgEquiv.symm (sumAlgEquiv R ↑(sᶜ) ↑s))\n (renameEquiv R ((Equiv.sumComm ↑(sᶜ) ↑s).trans (Equiv.Set.sumCompl s)))\nthis : ↑(rename Subtype.val) = RingHom.comp (↑↑eqv) C\n⊢ CommMonoidWithZero (MvPolynomial σ R)\n\nR : Type u\nS : Type ?u.606815\nσ : Type v\nM : Type w\ninst✝³ : CommRing R\ninst✝² : CommRing S\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nr : R\ns : Set σ\np : MvPolynomial (↑s) R\neqv : MvPolynomial (↑(sᶜ)) (MvPolynomial (↑s) R) ≃ₐ[R] MvPolynomial σ R :=\n AlgEquiv.trans (AlgEquiv.symm (sumAlgEquiv R ↑(sᶜ) ↑s))\n (renameEquiv R ((Equiv.sumComm ↑(sᶜ) ↑s).trans (Equiv.Set.sumCompl s)))\nthis : ↑(rename Subtype.val) = RingHom.comp (↑↑eqv) C\n⊢ Prime (↑?e (↑C p)) ↔ Prime (↑(rename Subtype.val) p)",
"tactic": "exact eqv.toMulEquiv"
},
{
"state_after": "case h.e'_2.h.e'_3\nR : Type u\nS : Type ?u.606815\nσ : Type v\nM : Type w\ninst✝³ : CommRing R\ninst✝² : CommRing S\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nr : R\ns : Set σ\np : MvPolynomial (↑s) R\neqv : MvPolynomial (↑(sᶜ)) (MvPolynomial (↑s) R) ≃ₐ[R] MvPolynomial σ R :=\n AlgEquiv.trans (AlgEquiv.symm (sumAlgEquiv R ↑(sᶜ) ↑s))\n (renameEquiv R ((Equiv.sumComm ↑(sᶜ) ↑s).trans (Equiv.Set.sumCompl s)))\nthis : ↑(rename Subtype.val) = RingHom.comp (↑↑eqv) C\n⊢ ↑(rename Subtype.val) p = ↑(AlgEquiv.toMulEquiv eqv) (↑C p)",
"state_before": "R : Type u\nS : Type ?u.606815\nσ : Type v\nM : Type w\ninst✝³ : CommRing R\ninst✝² : CommRing S\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nr : R\ns : Set σ\np : MvPolynomial (↑s) R\neqv : MvPolynomial (↑(sᶜ)) (MvPolynomial (↑s) R) ≃ₐ[R] MvPolynomial σ R :=\n AlgEquiv.trans (AlgEquiv.symm (sumAlgEquiv R ↑(sᶜ) ↑s))\n (renameEquiv R ((Equiv.sumComm ↑(sᶜ) ↑s).trans (Equiv.Set.sumCompl s)))\nthis : ↑(rename Subtype.val) = RingHom.comp (↑↑eqv) C\n⊢ Prime (↑(AlgEquiv.toMulEquiv eqv) (↑C p)) ↔ Prime (↑(rename Subtype.val) p)",
"tactic": "convert Iff.rfl"
},
{
"state_after": "no goals",
"state_before": "case h.e'_2.h.e'_3\nR : Type u\nS : Type ?u.606815\nσ : Type v\nM : Type w\ninst✝³ : CommRing R\ninst✝² : CommRing S\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nr : R\ns : Set σ\np : MvPolynomial (↑s) R\neqv : MvPolynomial (↑(sᶜ)) (MvPolynomial (↑s) R) ≃ₐ[R] MvPolynomial σ R :=\n AlgEquiv.trans (AlgEquiv.symm (sumAlgEquiv R ↑(sᶜ) ↑s))\n (renameEquiv R ((Equiv.sumComm ↑(sᶜ) ↑s).trans (Equiv.Set.sumCompl s)))\nthis : ↑(rename Subtype.val) = RingHom.comp (↑↑eqv) C\n⊢ ↑(rename Subtype.val) p = ↑(AlgEquiv.toMulEquiv eqv) (↑C p)",
"tactic": "apply RingHom.congr_fun this p"
},
{
"state_after": "case hC\nR : Type u\nS : Type ?u.606815\nσ : Type v\nM : Type w\ninst✝³ : CommRing R\ninst✝² : CommRing S\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nr : R\ns : Set σ\np : MvPolynomial (↑s) R\neqv : MvPolynomial (↑(sᶜ)) (MvPolynomial (↑s) R) ≃ₐ[R] MvPolynomial σ R :=\n AlgEquiv.trans (AlgEquiv.symm (sumAlgEquiv R ↑(sᶜ) ↑s))\n (renameEquiv R ((Equiv.sumComm ↑(sᶜ) ↑s).trans (Equiv.Set.sumCompl s)))\n⊢ ∀ (r : R), ↑↑(rename Subtype.val) (↑C r) = ↑(RingHom.comp (↑↑eqv) C) (↑C r)\n\ncase hX\nR : Type u\nS : Type ?u.606815\nσ : Type v\nM : Type w\ninst✝³ : CommRing R\ninst✝² : CommRing S\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nr : R\ns : Set σ\np : MvPolynomial (↑s) R\neqv : MvPolynomial (↑(sᶜ)) (MvPolynomial (↑s) R) ≃ₐ[R] MvPolynomial σ R :=\n AlgEquiv.trans (AlgEquiv.symm (sumAlgEquiv R ↑(sᶜ) ↑s))\n (renameEquiv R ((Equiv.sumComm ↑(sᶜ) ↑s).trans (Equiv.Set.sumCompl s)))\n⊢ ∀ (i : ↑s), ↑↑(rename Subtype.val) (X i) = ↑(RingHom.comp (↑↑eqv) C) (X i)",
"state_before": "R : Type u\nS : Type ?u.606815\nσ : Type v\nM : Type w\ninst✝³ : CommRing R\ninst✝² : CommRing S\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nr : R\ns : Set σ\np : MvPolynomial (↑s) R\neqv : MvPolynomial (↑(sᶜ)) (MvPolynomial (↑s) R) ≃ₐ[R] MvPolynomial σ R :=\n AlgEquiv.trans (AlgEquiv.symm (sumAlgEquiv R ↑(sᶜ) ↑s))\n (renameEquiv R ((Equiv.sumComm ↑(sᶜ) ↑s).trans (Equiv.Set.sumCompl s)))\n⊢ ↑(rename Subtype.val) = RingHom.comp (↑↑eqv) C",
"tactic": "apply ringHom_ext"
},
{
"state_after": "case hC\nR : Type u\nS : Type ?u.606815\nσ : Type v\nM : Type w\ninst✝³ : CommRing R\ninst✝² : CommRing S\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nr : R\ns : Set σ\np : MvPolynomial (↑s) R\neqv : MvPolynomial (↑(sᶜ)) (MvPolynomial (↑s) R) ≃ₐ[R] MvPolynomial σ R :=\n AlgEquiv.trans (AlgEquiv.symm (sumAlgEquiv R ↑(sᶜ) ↑s))\n (renameEquiv R ((Equiv.sumComm ↑(sᶜ) ↑s).trans (Equiv.Set.sumCompl s)))\nr✝ : R\n⊢ ↑↑(rename Subtype.val) (↑C r✝) = ↑(RingHom.comp (↑↑eqv) C) (↑C r✝)",
"state_before": "case hC\nR : Type u\nS : Type ?u.606815\nσ : Type v\nM : Type w\ninst✝³ : CommRing R\ninst✝² : CommRing S\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nr : R\ns : Set σ\np : MvPolynomial (↑s) R\neqv : MvPolynomial (↑(sᶜ)) (MvPolynomial (↑s) R) ≃ₐ[R] MvPolynomial σ R :=\n AlgEquiv.trans (AlgEquiv.symm (sumAlgEquiv R ↑(sᶜ) ↑s))\n (renameEquiv R ((Equiv.sumComm ↑(sᶜ) ↑s).trans (Equiv.Set.sumCompl s)))\n⊢ ∀ (r : R), ↑↑(rename Subtype.val) (↑C r) = ↑(RingHom.comp (↑↑eqv) C) (↑C r)",
"tactic": "intro"
},
{
"state_after": "case hC\nR : Type u\nS : Type ?u.606815\nσ : Type v\nM : Type w\ninst✝³ : CommRing R\ninst✝² : CommRing S\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nr : R\ns : Set σ\np : MvPolynomial (↑s) R\neqv : MvPolynomial (↑(sᶜ)) (MvPolynomial (↑s) R) ≃ₐ[R] MvPolynomial σ R :=\n AlgEquiv.trans (AlgEquiv.symm (sumAlgEquiv R ↑(sᶜ) ↑s))\n (renameEquiv R ((Equiv.sumComm ↑(sᶜ) ↑s).trans (Equiv.Set.sumCompl s)))\nr✝ : R\n⊢ ↑(rename Subtype.val) (↑C r✝) =\n ↑(rename (↑(Equiv.Set.sumCompl s) ∘ Sum.swap)) (↑(AlgEquiv.symm (sumAlgEquiv R ↑(sᶜ) ↑s)) (↑C (↑C r✝)))",
"state_before": "case hC\nR : Type u\nS : Type ?u.606815\nσ : Type v\nM : Type w\ninst✝³ : CommRing R\ninst✝² : CommRing S\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nr : R\ns : Set σ\np : MvPolynomial (↑s) R\neqv : MvPolynomial (↑(sᶜ)) (MvPolynomial (↑s) R) ≃ₐ[R] MvPolynomial σ R :=\n AlgEquiv.trans (AlgEquiv.symm (sumAlgEquiv R ↑(sᶜ) ↑s))\n (renameEquiv R ((Equiv.sumComm ↑(sᶜ) ↑s).trans (Equiv.Set.sumCompl s)))\nr✝ : R\n⊢ ↑↑(rename Subtype.val) (↑C r✝) = ↑(RingHom.comp (↑↑eqv) C) (↑C r✝)",
"tactic": "dsimp"
},
{
"state_after": "no goals",
"state_before": "case hC\nR : Type u\nS : Type ?u.606815\nσ : Type v\nM : Type w\ninst✝³ : CommRing R\ninst✝² : CommRing S\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nr : R\ns : Set σ\np : MvPolynomial (↑s) R\neqv : MvPolynomial (↑(sᶜ)) (MvPolynomial (↑s) R) ≃ₐ[R] MvPolynomial σ R :=\n AlgEquiv.trans (AlgEquiv.symm (sumAlgEquiv R ↑(sᶜ) ↑s))\n (renameEquiv R ((Equiv.sumComm ↑(sᶜ) ↑s).trans (Equiv.Set.sumCompl s)))\nr✝ : R\n⊢ ↑(rename Subtype.val) (↑C r✝) =\n ↑(rename (↑(Equiv.Set.sumCompl s) ∘ Sum.swap)) (↑(AlgEquiv.symm (sumAlgEquiv R ↑(sᶜ) ↑s)) (↑C (↑C r✝)))",
"tactic": "erw [iterToSum_C_C, rename_C, rename_C]"
},
{
"state_after": "case hX\nR : Type u\nS : Type ?u.606815\nσ : Type v\nM : Type w\ninst✝³ : CommRing R\ninst✝² : CommRing S\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nr : R\ns : Set σ\np : MvPolynomial (↑s) R\neqv : MvPolynomial (↑(sᶜ)) (MvPolynomial (↑s) R) ≃ₐ[R] MvPolynomial σ R :=\n AlgEquiv.trans (AlgEquiv.symm (sumAlgEquiv R ↑(sᶜ) ↑s))\n (renameEquiv R ((Equiv.sumComm ↑(sᶜ) ↑s).trans (Equiv.Set.sumCompl s)))\ni✝ : ↑s\n⊢ ↑↑(rename Subtype.val) (X i✝) = ↑(RingHom.comp (↑↑eqv) C) (X i✝)",
"state_before": "case hX\nR : Type u\nS : Type ?u.606815\nσ : Type v\nM : Type w\ninst✝³ : CommRing R\ninst✝² : CommRing S\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nr : R\ns : Set σ\np : MvPolynomial (↑s) R\neqv : MvPolynomial (↑(sᶜ)) (MvPolynomial (↑s) R) ≃ₐ[R] MvPolynomial σ R :=\n AlgEquiv.trans (AlgEquiv.symm (sumAlgEquiv R ↑(sᶜ) ↑s))\n (renameEquiv R ((Equiv.sumComm ↑(sᶜ) ↑s).trans (Equiv.Set.sumCompl s)))\n⊢ ∀ (i : ↑s), ↑↑(rename Subtype.val) (X i) = ↑(RingHom.comp (↑↑eqv) C) (X i)",
"tactic": "intro"
},
{
"state_after": "case hX\nR : Type u\nS : Type ?u.606815\nσ : Type v\nM : Type w\ninst✝³ : CommRing R\ninst✝² : CommRing S\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nr : R\ns : Set σ\np : MvPolynomial (↑s) R\neqv : MvPolynomial (↑(sᶜ)) (MvPolynomial (↑s) R) ≃ₐ[R] MvPolynomial σ R :=\n AlgEquiv.trans (AlgEquiv.symm (sumAlgEquiv R ↑(sᶜ) ↑s))\n (renameEquiv R ((Equiv.sumComm ↑(sᶜ) ↑s).trans (Equiv.Set.sumCompl s)))\ni✝ : ↑s\n⊢ ↑(rename Subtype.val) (X i✝) =\n ↑(rename (↑(Equiv.Set.sumCompl s) ∘ Sum.swap)) (↑(AlgEquiv.symm (sumAlgEquiv R ↑(sᶜ) ↑s)) (↑C (X i✝)))",
"state_before": "case hX\nR : Type u\nS : Type ?u.606815\nσ : Type v\nM : Type w\ninst✝³ : CommRing R\ninst✝² : CommRing S\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nr : R\ns : Set σ\np : MvPolynomial (↑s) R\neqv : MvPolynomial (↑(sᶜ)) (MvPolynomial (↑s) R) ≃ₐ[R] MvPolynomial σ R :=\n AlgEquiv.trans (AlgEquiv.symm (sumAlgEquiv R ↑(sᶜ) ↑s))\n (renameEquiv R ((Equiv.sumComm ↑(sᶜ) ↑s).trans (Equiv.Set.sumCompl s)))\ni✝ : ↑s\n⊢ ↑↑(rename Subtype.val) (X i✝) = ↑(RingHom.comp (↑↑eqv) C) (X i✝)",
"tactic": "dsimp"
},
{
"state_after": "case hX\nR : Type u\nS : Type ?u.606815\nσ : Type v\nM : Type w\ninst✝³ : CommRing R\ninst✝² : CommRing S\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nr : R\ns : Set σ\np : MvPolynomial (↑s) R\neqv : MvPolynomial (↑(sᶜ)) (MvPolynomial (↑s) R) ≃ₐ[R] MvPolynomial σ R :=\n AlgEquiv.trans (AlgEquiv.symm (sumAlgEquiv R ↑(sᶜ) ↑s))\n (renameEquiv R ((Equiv.sumComm ↑(sᶜ) ↑s).trans (Equiv.Set.sumCompl s)))\ni✝ : ↑s\n⊢ X ↑i✝ = X ((↑(Equiv.Set.sumCompl s) ∘ Sum.swap) (Sum.inr i✝))",
"state_before": "case hX\nR : Type u\nS : Type ?u.606815\nσ : Type v\nM : Type w\ninst✝³ : CommRing R\ninst✝² : CommRing S\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nr : R\ns : Set σ\np : MvPolynomial (↑s) R\neqv : MvPolynomial (↑(sᶜ)) (MvPolynomial (↑s) R) ≃ₐ[R] MvPolynomial σ R :=\n AlgEquiv.trans (AlgEquiv.symm (sumAlgEquiv R ↑(sᶜ) ↑s))\n (renameEquiv R ((Equiv.sumComm ↑(sᶜ) ↑s).trans (Equiv.Set.sumCompl s)))\ni✝ : ↑s\n⊢ ↑(rename Subtype.val) (X i✝) =\n ↑(rename (↑(Equiv.Set.sumCompl s) ∘ Sum.swap)) (↑(AlgEquiv.symm (sumAlgEquiv R ↑(sᶜ) ↑s)) (↑C (X i✝)))",
"tactic": "erw [iterToSum_C_X, rename_X, rename_X]"
},
{
"state_after": "no goals",
"state_before": "case hX\nR : Type u\nS : Type ?u.606815\nσ : Type v\nM : Type w\ninst✝³ : CommRing R\ninst✝² : CommRing S\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nr : R\ns : Set σ\np : MvPolynomial (↑s) R\neqv : MvPolynomial (↑(sᶜ)) (MvPolynomial (↑s) R) ≃ₐ[R] MvPolynomial σ R :=\n AlgEquiv.trans (AlgEquiv.symm (sumAlgEquiv R ↑(sᶜ) ↑s))\n (renameEquiv R ((Equiv.sumComm ↑(sᶜ) ↑s).trans (Equiv.Set.sumCompl s)))\ni✝ : ↑s\n⊢ X ↑i✝ = X ((↑(Equiv.Set.sumCompl s) ∘ Sum.swap) (Sum.inr i✝))",
"tactic": "rfl"
}
] |
[
841,
35
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
820,
1
] |
Mathlib/LinearAlgebra/AnnihilatingPolynomial.lean
|
Polynomial.span_singleton_annIdealGenerator
|
[
{
"state_after": "case pos\n𝕜 : Type u_1\nA : Type u_2\ninst✝² : Field 𝕜\ninst✝¹ : Ring A\ninst✝ : Algebra 𝕜 A\na : A\nh : annIdealGenerator 𝕜 a = 0\n⊢ Ideal.span {annIdealGenerator 𝕜 a} = annIdeal 𝕜 a\n\ncase neg\n𝕜 : Type u_1\nA : Type u_2\ninst✝² : Field 𝕜\ninst✝¹ : Ring A\ninst✝ : Algebra 𝕜 A\na : A\nh : ¬annIdealGenerator 𝕜 a = 0\n⊢ Ideal.span {annIdealGenerator 𝕜 a} = annIdeal 𝕜 a",
"state_before": "𝕜 : Type u_1\nA : Type u_2\ninst✝² : Field 𝕜\ninst✝¹ : Ring A\ninst✝ : Algebra 𝕜 A\na : A\n⊢ Ideal.span {annIdealGenerator 𝕜 a} = annIdeal 𝕜 a",
"tactic": "by_cases h : annIdealGenerator 𝕜 a = 0"
},
{
"state_after": "no goals",
"state_before": "case pos\n𝕜 : Type u_1\nA : Type u_2\ninst✝² : Field 𝕜\ninst✝¹ : Ring A\ninst✝ : Algebra 𝕜 A\na : A\nh : annIdealGenerator 𝕜 a = 0\n⊢ Ideal.span {annIdealGenerator 𝕜 a} = annIdeal 𝕜 a",
"tactic": "rw [h, annIdealGenerator_eq_zero_iff.mp h, Set.singleton_zero, Ideal.span_zero]"
},
{
"state_after": "case neg.h2\n𝕜 : Type u_1\nA : Type u_2\ninst✝² : Field 𝕜\ninst✝¹ : Ring A\ninst✝ : Algebra 𝕜 A\na : A\nh : ¬annIdealGenerator 𝕜 a = 0\n⊢ IsUnit (↑C (leadingCoeff (IsPrincipal.generator (annIdeal 𝕜 a)))⁻¹)",
"state_before": "case neg\n𝕜 : Type u_1\nA : Type u_2\ninst✝² : Field 𝕜\ninst✝¹ : Ring A\ninst✝ : Algebra 𝕜 A\na : A\nh : ¬annIdealGenerator 𝕜 a = 0\n⊢ Ideal.span {annIdealGenerator 𝕜 a} = annIdeal 𝕜 a",
"tactic": "rw [annIdealGenerator, Ideal.span_singleton_mul_right_unit, Ideal.span_singleton_generator]"
},
{
"state_after": "case neg.h2\n𝕜 : Type u_1\nA : Type u_2\ninst✝² : Field 𝕜\ninst✝¹ : Ring A\ninst✝ : Algebra 𝕜 A\na : A\nh : ¬annIdealGenerator 𝕜 a = 0\n⊢ IsUnit (leadingCoeff (IsPrincipal.generator (annIdeal 𝕜 a)))⁻¹",
"state_before": "case neg.h2\n𝕜 : Type u_1\nA : Type u_2\ninst✝² : Field 𝕜\ninst✝¹ : Ring A\ninst✝ : Algebra 𝕜 A\na : A\nh : ¬annIdealGenerator 𝕜 a = 0\n⊢ IsUnit (↑C (leadingCoeff (IsPrincipal.generator (annIdeal 𝕜 a)))⁻¹)",
"tactic": "apply Polynomial.isUnit_C.mpr"
},
{
"state_after": "case neg.h2.hx\n𝕜 : Type u_1\nA : Type u_2\ninst✝² : Field 𝕜\ninst✝¹ : Ring A\ninst✝ : Algebra 𝕜 A\na : A\nh : ¬annIdealGenerator 𝕜 a = 0\n⊢ (leadingCoeff (IsPrincipal.generator (annIdeal 𝕜 a)))⁻¹ ≠ 0",
"state_before": "case neg.h2\n𝕜 : Type u_1\nA : Type u_2\ninst✝² : Field 𝕜\ninst✝¹ : Ring A\ninst✝ : Algebra 𝕜 A\na : A\nh : ¬annIdealGenerator 𝕜 a = 0\n⊢ IsUnit (leadingCoeff (IsPrincipal.generator (annIdeal 𝕜 a)))⁻¹",
"tactic": "apply IsUnit.mk0"
},
{
"state_after": "case neg.h2.hx\n𝕜 : Type u_1\nA : Type u_2\ninst✝² : Field 𝕜\ninst✝¹ : Ring A\ninst✝ : Algebra 𝕜 A\na : A\nh : ¬annIdealGenerator 𝕜 a = 0\n⊢ ¬leadingCoeff (IsPrincipal.generator (annIdeal 𝕜 a)) = 0",
"state_before": "case neg.h2.hx\n𝕜 : Type u_1\nA : Type u_2\ninst✝² : Field 𝕜\ninst✝¹ : Ring A\ninst✝ : Algebra 𝕜 A\na : A\nh : ¬annIdealGenerator 𝕜 a = 0\n⊢ (leadingCoeff (IsPrincipal.generator (annIdeal 𝕜 a)))⁻¹ ≠ 0",
"tactic": "apply inv_eq_zero.not.mpr"
},
{
"state_after": "case neg.h2.hx\n𝕜 : Type u_1\nA : Type u_2\ninst✝² : Field 𝕜\ninst✝¹ : Ring A\ninst✝ : Algebra 𝕜 A\na : A\nh : ¬annIdealGenerator 𝕜 a = 0\n⊢ ¬IsPrincipal.generator (annIdeal 𝕜 a) = 0",
"state_before": "case neg.h2.hx\n𝕜 : Type u_1\nA : Type u_2\ninst✝² : Field 𝕜\ninst✝¹ : Ring A\ninst✝ : Algebra 𝕜 A\na : A\nh : ¬annIdealGenerator 𝕜 a = 0\n⊢ ¬leadingCoeff (IsPrincipal.generator (annIdeal 𝕜 a)) = 0",
"tactic": "apply Polynomial.leadingCoeff_eq_zero.not.mpr"
},
{
"state_after": "no goals",
"state_before": "case neg.h2.hx\n𝕜 : Type u_1\nA : Type u_2\ninst✝² : Field 𝕜\ninst✝¹ : Ring A\ninst✝ : Algebra 𝕜 A\na : A\nh : ¬annIdealGenerator 𝕜 a = 0\n⊢ ¬IsPrincipal.generator (annIdeal 𝕜 a) = 0",
"tactic": "apply (mul_ne_zero_iff.mp h).1"
}
] |
[
113,
35
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
104,
1
] |
Mathlib/Topology/Category/CompHaus/Basic.lean
|
CompHaus.isIso_of_bijective
|
[
{
"state_after": "X✝ : Type ?u.4524\ninst✝² : TopologicalSpace X✝\ninst✝¹ : CompactSpace X✝\ninst✝ : T2Space X✝\nX Y : CompHaus\nf : X ⟶ Y\nbij : Function.Bijective ((forget CompHaus).map f)\nE : (forget CompHaus).obj X ≃ (forget CompHaus).obj Y := Equiv.ofBijective ((forget CompHaus).map f) bij\n⊢ IsIso f",
"state_before": "X✝ : Type ?u.4524\ninst✝² : TopologicalSpace X✝\ninst✝¹ : CompactSpace X✝\ninst✝ : T2Space X✝\nX Y : CompHaus\nf : X ⟶ Y\nbij : Function.Bijective ((forget CompHaus).map f)\n⊢ IsIso f",
"tactic": "let E := Equiv.ofBijective _ bij"
},
{
"state_after": "X✝ : Type ?u.4524\ninst✝² : TopologicalSpace X✝\ninst✝¹ : CompactSpace X✝\ninst✝ : T2Space X✝\nX Y : CompHaus\nf : X ⟶ Y\nbij : Function.Bijective ((forget CompHaus).map f)\nE : (forget CompHaus).obj X ≃ (forget CompHaus).obj Y := Equiv.ofBijective ((forget CompHaus).map f) bij\nhE : Continuous ↑E.symm\n⊢ IsIso f",
"state_before": "X✝ : Type ?u.4524\ninst✝² : TopologicalSpace X✝\ninst✝¹ : CompactSpace X✝\ninst✝ : T2Space X✝\nX Y : CompHaus\nf : X ⟶ Y\nbij : Function.Bijective ((forget CompHaus).map f)\nE : (forget CompHaus).obj X ≃ (forget CompHaus).obj Y := Equiv.ofBijective ((forget CompHaus).map f) bij\n⊢ IsIso f",
"tactic": "have hE : Continuous E.symm := by\n rw [continuous_iff_isClosed]\n intro S hS\n rw [← E.image_eq_preimage]\n exact isClosedMap f S hS"
},
{
"state_after": "case refine'_1\nX✝ : Type ?u.4524\ninst✝² : TopologicalSpace X✝\ninst✝¹ : CompactSpace X✝\ninst✝ : T2Space X✝\nX Y : CompHaus\nf : X ⟶ Y\nbij : Function.Bijective ((forget CompHaus).map f)\nE : (forget CompHaus).obj X ≃ (forget CompHaus).obj Y := Equiv.ofBijective ((forget CompHaus).map f) bij\nhE : Continuous ↑E.symm\n⊢ f ≫ ContinuousMap.mk ↑E.symm = 𝟙 X\n\ncase refine'_2\nX✝ : Type ?u.4524\ninst✝² : TopologicalSpace X✝\ninst✝¹ : CompactSpace X✝\ninst✝ : T2Space X✝\nX Y : CompHaus\nf : X ⟶ Y\nbij : Function.Bijective ((forget CompHaus).map f)\nE : (forget CompHaus).obj X ≃ (forget CompHaus).obj Y := Equiv.ofBijective ((forget CompHaus).map f) bij\nhE : Continuous ↑E.symm\n⊢ ContinuousMap.mk ↑E.symm ≫ f = 𝟙 Y",
"state_before": "X✝ : Type ?u.4524\ninst✝² : TopologicalSpace X✝\ninst✝¹ : CompactSpace X✝\ninst✝ : T2Space X✝\nX Y : CompHaus\nf : X ⟶ Y\nbij : Function.Bijective ((forget CompHaus).map f)\nE : (forget CompHaus).obj X ≃ (forget CompHaus).obj Y := Equiv.ofBijective ((forget CompHaus).map f) bij\nhE : Continuous ↑E.symm\n⊢ IsIso f",
"tactic": "refine' ⟨⟨⟨E.symm, hE⟩, _, _⟩⟩"
},
{
"state_after": "X✝ : Type ?u.4524\ninst✝² : TopologicalSpace X✝\ninst✝¹ : CompactSpace X✝\ninst✝ : T2Space X✝\nX Y : CompHaus\nf : X ⟶ Y\nbij : Function.Bijective ((forget CompHaus).map f)\nE : (forget CompHaus).obj X ≃ (forget CompHaus).obj Y := Equiv.ofBijective ((forget CompHaus).map f) bij\n⊢ ∀ (s : Set ((forget CompHaus).obj X)), IsClosed s → IsClosed (↑E.symm ⁻¹' s)",
"state_before": "X✝ : Type ?u.4524\ninst✝² : TopologicalSpace X✝\ninst✝¹ : CompactSpace X✝\ninst✝ : T2Space X✝\nX Y : CompHaus\nf : X ⟶ Y\nbij : Function.Bijective ((forget CompHaus).map f)\nE : (forget CompHaus).obj X ≃ (forget CompHaus).obj Y := Equiv.ofBijective ((forget CompHaus).map f) bij\n⊢ Continuous ↑E.symm",
"tactic": "rw [continuous_iff_isClosed]"
},
{
"state_after": "X✝ : Type ?u.4524\ninst✝² : TopologicalSpace X✝\ninst✝¹ : CompactSpace X✝\ninst✝ : T2Space X✝\nX Y : CompHaus\nf : X ⟶ Y\nbij : Function.Bijective ((forget CompHaus).map f)\nE : (forget CompHaus).obj X ≃ (forget CompHaus).obj Y := Equiv.ofBijective ((forget CompHaus).map f) bij\nS : Set ((forget CompHaus).obj X)\nhS : IsClosed S\n⊢ IsClosed (↑E.symm ⁻¹' S)",
"state_before": "X✝ : Type ?u.4524\ninst✝² : TopologicalSpace X✝\ninst✝¹ : CompactSpace X✝\ninst✝ : T2Space X✝\nX Y : CompHaus\nf : X ⟶ Y\nbij : Function.Bijective ((forget CompHaus).map f)\nE : (forget CompHaus).obj X ≃ (forget CompHaus).obj Y := Equiv.ofBijective ((forget CompHaus).map f) bij\n⊢ ∀ (s : Set ((forget CompHaus).obj X)), IsClosed s → IsClosed (↑E.symm ⁻¹' s)",
"tactic": "intro S hS"
},
{
"state_after": "X✝ : Type ?u.4524\ninst✝² : TopologicalSpace X✝\ninst✝¹ : CompactSpace X✝\ninst✝ : T2Space X✝\nX Y : CompHaus\nf : X ⟶ Y\nbij : Function.Bijective ((forget CompHaus).map f)\nE : (forget CompHaus).obj X ≃ (forget CompHaus).obj Y := Equiv.ofBijective ((forget CompHaus).map f) bij\nS : Set ((forget CompHaus).obj X)\nhS : IsClosed S\n⊢ IsClosed (↑E '' S)",
"state_before": "X✝ : Type ?u.4524\ninst✝² : TopologicalSpace X✝\ninst✝¹ : CompactSpace X✝\ninst✝ : T2Space X✝\nX Y : CompHaus\nf : X ⟶ Y\nbij : Function.Bijective ((forget CompHaus).map f)\nE : (forget CompHaus).obj X ≃ (forget CompHaus).obj Y := Equiv.ofBijective ((forget CompHaus).map f) bij\nS : Set ((forget CompHaus).obj X)\nhS : IsClosed S\n⊢ IsClosed (↑E.symm ⁻¹' S)",
"tactic": "rw [← E.image_eq_preimage]"
},
{
"state_after": "no goals",
"state_before": "X✝ : Type ?u.4524\ninst✝² : TopologicalSpace X✝\ninst✝¹ : CompactSpace X✝\ninst✝ : T2Space X✝\nX Y : CompHaus\nf : X ⟶ Y\nbij : Function.Bijective ((forget CompHaus).map f)\nE : (forget CompHaus).obj X ≃ (forget CompHaus).obj Y := Equiv.ofBijective ((forget CompHaus).map f) bij\nS : Set ((forget CompHaus).obj X)\nhS : IsClosed S\n⊢ IsClosed (↑E '' S)",
"tactic": "exact isClosedMap f S hS"
},
{
"state_after": "case refine'_1.w\nX✝ : Type ?u.4524\ninst✝² : TopologicalSpace X✝\ninst✝¹ : CompactSpace X✝\ninst✝ : T2Space X✝\nX Y : CompHaus\nf : X ⟶ Y\nbij : Function.Bijective ((forget CompHaus).map f)\nE : (forget CompHaus).obj X ≃ (forget CompHaus).obj Y := Equiv.ofBijective ((forget CompHaus).map f) bij\nhE : Continuous ↑E.symm\nx : (forget CompHaus).obj X\n⊢ (forget CompHaus).map (f ≫ ContinuousMap.mk ↑E.symm) x = (forget CompHaus).map (𝟙 X) x",
"state_before": "case refine'_1\nX✝ : Type ?u.4524\ninst✝² : TopologicalSpace X✝\ninst✝¹ : CompactSpace X✝\ninst✝ : T2Space X✝\nX Y : CompHaus\nf : X ⟶ Y\nbij : Function.Bijective ((forget CompHaus).map f)\nE : (forget CompHaus).obj X ≃ (forget CompHaus).obj Y := Equiv.ofBijective ((forget CompHaus).map f) bij\nhE : Continuous ↑E.symm\n⊢ f ≫ ContinuousMap.mk ↑E.symm = 𝟙 X",
"tactic": "ext x"
},
{
"state_after": "no goals",
"state_before": "case refine'_1.w\nX✝ : Type ?u.4524\ninst✝² : TopologicalSpace X✝\ninst✝¹ : CompactSpace X✝\ninst✝ : T2Space X✝\nX Y : CompHaus\nf : X ⟶ Y\nbij : Function.Bijective ((forget CompHaus).map f)\nE : (forget CompHaus).obj X ≃ (forget CompHaus).obj Y := Equiv.ofBijective ((forget CompHaus).map f) bij\nhE : Continuous ↑E.symm\nx : (forget CompHaus).obj X\n⊢ (forget CompHaus).map (f ≫ ContinuousMap.mk ↑E.symm) x = (forget CompHaus).map (𝟙 X) x",
"tactic": "apply E.symm_apply_apply"
},
{
"state_after": "case refine'_2.w\nX✝ : Type ?u.4524\ninst✝² : TopologicalSpace X✝\ninst✝¹ : CompactSpace X✝\ninst✝ : T2Space X✝\nX Y : CompHaus\nf : X ⟶ Y\nbij : Function.Bijective ((forget CompHaus).map f)\nE : (forget CompHaus).obj X ≃ (forget CompHaus).obj Y := Equiv.ofBijective ((forget CompHaus).map f) bij\nhE : Continuous ↑E.symm\nx : (forget CompHaus).obj Y\n⊢ (forget CompHaus).map (ContinuousMap.mk ↑E.symm ≫ f) x = (forget CompHaus).map (𝟙 Y) x",
"state_before": "case refine'_2\nX✝ : Type ?u.4524\ninst✝² : TopologicalSpace X✝\ninst✝¹ : CompactSpace X✝\ninst✝ : T2Space X✝\nX Y : CompHaus\nf : X ⟶ Y\nbij : Function.Bijective ((forget CompHaus).map f)\nE : (forget CompHaus).obj X ≃ (forget CompHaus).obj Y := Equiv.ofBijective ((forget CompHaus).map f) bij\nhE : Continuous ↑E.symm\n⊢ ContinuousMap.mk ↑E.symm ≫ f = 𝟙 Y",
"tactic": "ext x"
},
{
"state_after": "no goals",
"state_before": "case refine'_2.w\nX✝ : Type ?u.4524\ninst✝² : TopologicalSpace X✝\ninst✝¹ : CompactSpace X✝\ninst✝ : T2Space X✝\nX Y : CompHaus\nf : X ⟶ Y\nbij : Function.Bijective ((forget CompHaus).map f)\nE : (forget CompHaus).obj X ≃ (forget CompHaus).obj Y := Equiv.ofBijective ((forget CompHaus).map f) bij\nhE : Continuous ↑E.symm\nx : (forget CompHaus).obj Y\n⊢ (forget CompHaus).map (ContinuousMap.mk ↑E.symm ≫ f) x = (forget CompHaus).map (𝟙 Y) x",
"tactic": "apply E.apply_symm_apply"
}
] |
[
133,
29
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
121,
1
] |
Mathlib/RingTheory/PowerSeries/Basic.lean
|
PowerSeries.order_le
|
[
{
"state_after": "R : Type u_1\ninst✝ : Semiring R\nφ : PowerSeries R\nn : ℕ\nh : ↑(coeff R n) φ ≠ 0\nx✝ : ∃ n, ↑(coeff R n) φ ≠ 0\n⊢ order φ ≤ ↑n",
"state_before": "R : Type u_1\ninst✝ : Semiring R\nφ : PowerSeries R\nn : ℕ\nh : ↑(coeff R n) φ ≠ 0\n⊢ order φ ≤ ↑n",
"tactic": "have _ : ∃ n, coeff R n φ ≠ 0 := Exists.intro n h"
},
{
"state_after": "R : Type u_1\ninst✝ : Semiring R\nφ : PowerSeries R\nn : ℕ\nh : ↑(coeff R n) φ ≠ 0\nx✝ : ∃ n, ↑(coeff R n) φ ≠ 0\n⊢ ↑(Nat.find (_ : ∃ n, ↑(coeff R n) φ ≠ 0)) ≤ ↑n\n\ncase hnc\nR : Type u_1\ninst✝ : Semiring R\nφ : PowerSeries R\nn : ℕ\nh : ↑(coeff R n) φ ≠ 0\nx✝ : ∃ n, ↑(coeff R n) φ ≠ 0\n⊢ ¬φ = 0",
"state_before": "R : Type u_1\ninst✝ : Semiring R\nφ : PowerSeries R\nn : ℕ\nh : ↑(coeff R n) φ ≠ 0\nx✝ : ∃ n, ↑(coeff R n) φ ≠ 0\n⊢ order φ ≤ ↑n",
"tactic": "rw [order, dif_neg]"
},
{
"state_after": "R : Type u_1\ninst✝ : Semiring R\nφ : PowerSeries R\nn : ℕ\nh : ↑(coeff R n) φ ≠ 0\nx✝ : ∃ n, ↑(coeff R n) φ ≠ 0\n⊢ ∃ m, m ≤ n ∧ ↑(coeff R m) φ ≠ 0",
"state_before": "R : Type u_1\ninst✝ : Semiring R\nφ : PowerSeries R\nn : ℕ\nh : ↑(coeff R n) φ ≠ 0\nx✝ : ∃ n, ↑(coeff R n) φ ≠ 0\n⊢ ↑(Nat.find (_ : ∃ n, ↑(coeff R n) φ ≠ 0)) ≤ ↑n",
"tactic": "simp only [PartENat.coe_le_coe, Nat.find_le_iff]"
},
{
"state_after": "no goals",
"state_before": "R : Type u_1\ninst✝ : Semiring R\nφ : PowerSeries R\nn : ℕ\nh : ↑(coeff R n) φ ≠ 0\nx✝ : ∃ n, ↑(coeff R n) φ ≠ 0\n⊢ ∃ m, m ≤ n ∧ ↑(coeff R m) φ ≠ 0",
"tactic": "exact ⟨n, le_rfl, h⟩"
},
{
"state_after": "no goals",
"state_before": "case hnc\nR : Type u_1\ninst✝ : Semiring R\nφ : PowerSeries R\nn : ℕ\nh : ↑(coeff R n) φ ≠ 0\nx✝ : ∃ n, ↑(coeff R n) φ ≠ 0\n⊢ ¬φ = 0",
"tactic": "exact exists_coeff_ne_zero_iff_ne_zero.mp ⟨n, h⟩"
}
] |
[
2268,
53
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2263,
1
] |
Mathlib/Algebra/Algebra/Subalgebra/Basic.lean
|
Subalgebra.iSupLift_of_mem
|
[
{
"state_after": "R : Type u\nA : Type v\nB : Type w\ninst✝⁵ : CommSemiring R\ninst✝⁴ : Semiring A\ninst✝³ : Algebra R A\ninst✝² : Semiring B\ninst✝¹ : Algebra R B\nS : Subalgebra R A\nι : Type u_1\ninst✝ : Nonempty ι\nK : ι → Subalgebra R A\ndir : Directed (fun x x_1 => x ≤ x_1) K\nf : (i : ι) → { x // x ∈ K i } →ₐ[R] B\nhf : ∀ (i j : ι) (h : K i ≤ K j), f i = AlgHom.comp (f j) (inclusion h)\nT : Subalgebra R A\nhT : T = iSup K\ni : ι\nx : { x // x ∈ T }\nhx : ↑x ∈ K i\n⊢ Set.iUnionLift (fun i => ↑(K i)) (fun i x => ↑(f i) x)\n (_ :\n ∀ (i j : ι) (x : A) (hxi : x ∈ (fun i => ↑(K i)) i) (hxj : x ∈ (fun i => ↑(K i)) j),\n (fun i x => ↑(f i) x) i { val := x, property := hxi } = (fun i x => ↑(f i) x) j { val := x, property := hxj })\n ↑T (_ : ↑T ⊆ ⋃ (i : ι), ↑(K i)) x =\n ↑(f i) { val := ↑x, property := hx }",
"state_before": "R : Type u\nA : Type v\nB : Type w\ninst✝⁵ : CommSemiring R\ninst✝⁴ : Semiring A\ninst✝³ : Algebra R A\ninst✝² : Semiring B\ninst✝¹ : Algebra R B\nS : Subalgebra R A\nι : Type u_1\ninst✝ : Nonempty ι\nK : ι → Subalgebra R A\ndir : Directed (fun x x_1 => x ≤ x_1) K\nf : (i : ι) → { x // x ∈ K i } →ₐ[R] B\nhf : ∀ (i j : ι) (h : K i ≤ K j), f i = AlgHom.comp (f j) (inclusion h)\nT : Subalgebra R A\nhT : T = iSup K\ni : ι\nx : { x // x ∈ T }\nhx : ↑x ∈ K i\n⊢ ↑(iSupLift K dir f hf T hT) x = ↑(f i) { val := ↑x, property := hx }",
"tactic": "dsimp [iSupLift, inclusion]"
},
{
"state_after": "no goals",
"state_before": "R : Type u\nA : Type v\nB : Type w\ninst✝⁵ : CommSemiring R\ninst✝⁴ : Semiring A\ninst✝³ : Algebra R A\ninst✝² : Semiring B\ninst✝¹ : Algebra R B\nS : Subalgebra R A\nι : Type u_1\ninst✝ : Nonempty ι\nK : ι → Subalgebra R A\ndir : Directed (fun x x_1 => x ≤ x_1) K\nf : (i : ι) → { x // x ∈ K i } →ₐ[R] B\nhf : ∀ (i j : ι) (h : K i ≤ K j), f i = AlgHom.comp (f j) (inclusion h)\nT : Subalgebra R A\nhT : T = iSup K\ni : ι\nx : { x // x ∈ T }\nhx : ↑x ∈ K i\n⊢ Set.iUnionLift (fun i => ↑(K i)) (fun i x => ↑(f i) x)\n (_ :\n ∀ (i j : ι) (x : A) (hxi : x ∈ (fun i => ↑(K i)) i) (hxj : x ∈ (fun i => ↑(K i)) j),\n (fun i x => ↑(f i) x) i { val := x, property := hxi } = (fun i x => ↑(f i) x) j { val := x, property := hxj })\n ↑T (_ : ↑T ⊆ ⋃ (i : ι), ↑(K i)) x =\n ↑(f i) { val := ↑x, property := hx }",
"tactic": "rw [Set.iUnionLift_of_mem]"
}
] |
[
1232,
29
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1229,
1
] |
Mathlib/Geometry/Manifold/LocalInvariantProperties.lean
|
StructureGroupoid.LocalInvariantProp.right_invariance
|
[
{
"state_after": "H : Type u_1\nM : Type ?u.6200\nH' : Type u_2\nM' : Type ?u.6206\nX : Type ?u.6209\ninst✝⁶ : TopologicalSpace H\ninst✝⁵ : TopologicalSpace M\ninst✝⁴ : ChartedSpace H M\ninst✝³ : TopologicalSpace H'\ninst✝² : TopologicalSpace M'\ninst✝¹ : ChartedSpace H' M'\ninst✝ : TopologicalSpace X\nG : StructureGroupoid H\nG' : StructureGroupoid H'\nP : (H → H') → Set H → H → Prop\ns✝ t u : Set H\nx✝ : H\nhG : LocalInvariantProp G G' P\ns : Set H\nx : H\nf : H → H'\ne : LocalHomeomorph H H\nhe : e ∈ G\nhxe : x ∈ e.source\nh : P (f ∘ ↑(LocalHomeomorph.symm e)) (↑(LocalHomeomorph.symm e) ⁻¹' s) (↑e x)\n⊢ P f s x",
"state_before": "H : Type u_1\nM : Type ?u.6200\nH' : Type u_2\nM' : Type ?u.6206\nX : Type ?u.6209\ninst✝⁶ : TopologicalSpace H\ninst✝⁵ : TopologicalSpace M\ninst✝⁴ : ChartedSpace H M\ninst✝³ : TopologicalSpace H'\ninst✝² : TopologicalSpace M'\ninst✝¹ : ChartedSpace H' M'\ninst✝ : TopologicalSpace X\nG : StructureGroupoid H\nG' : StructureGroupoid H'\nP : (H → H') → Set H → H → Prop\ns✝ t u : Set H\nx✝ : H\nhG : LocalInvariantProp G G' P\ns : Set H\nx : H\nf : H → H'\ne : LocalHomeomorph H H\nhe : e ∈ G\nhxe : x ∈ e.source\n⊢ P (f ∘ ↑(LocalHomeomorph.symm e)) (↑(LocalHomeomorph.symm e) ⁻¹' s) (↑e x) ↔ P f s x",
"tactic": "refine' ⟨fun h ↦ _, hG.right_invariance' he hxe⟩"
},
{
"state_after": "H : Type u_1\nM : Type ?u.6200\nH' : Type u_2\nM' : Type ?u.6206\nX : Type ?u.6209\ninst✝⁶ : TopologicalSpace H\ninst✝⁵ : TopologicalSpace M\ninst✝⁴ : ChartedSpace H M\ninst✝³ : TopologicalSpace H'\ninst✝² : TopologicalSpace M'\ninst✝¹ : ChartedSpace H' M'\ninst✝ : TopologicalSpace X\nG : StructureGroupoid H\nG' : StructureGroupoid H'\nP : (H → H') → Set H → H → Prop\ns✝ t u : Set H\nx✝ : H\nhG : LocalInvariantProp G G' P\ns : Set H\nx : H\nf : H → H'\ne : LocalHomeomorph H H\nhe : e ∈ G\nhxe : x ∈ e.source\nh : P (f ∘ ↑(LocalHomeomorph.symm e)) (↑(LocalHomeomorph.symm e) ⁻¹' s) (↑e x)\nthis :\n P ((f ∘ ↑(LocalHomeomorph.symm e)) ∘ ↑(LocalHomeomorph.symm (LocalHomeomorph.symm e)))\n (↑(LocalHomeomorph.symm (LocalHomeomorph.symm e)) ⁻¹' (↑(LocalHomeomorph.symm e) ⁻¹' s))\n (↑(LocalHomeomorph.symm e) (↑e x))\n⊢ P f s x",
"state_before": "H : Type u_1\nM : Type ?u.6200\nH' : Type u_2\nM' : Type ?u.6206\nX : Type ?u.6209\ninst✝⁶ : TopologicalSpace H\ninst✝⁵ : TopologicalSpace M\ninst✝⁴ : ChartedSpace H M\ninst✝³ : TopologicalSpace H'\ninst✝² : TopologicalSpace M'\ninst✝¹ : ChartedSpace H' M'\ninst✝ : TopologicalSpace X\nG : StructureGroupoid H\nG' : StructureGroupoid H'\nP : (H → H') → Set H → H → Prop\ns✝ t u : Set H\nx✝ : H\nhG : LocalInvariantProp G G' P\ns : Set H\nx : H\nf : H → H'\ne : LocalHomeomorph H H\nhe : e ∈ G\nhxe : x ∈ e.source\nh : P (f ∘ ↑(LocalHomeomorph.symm e)) (↑(LocalHomeomorph.symm e) ⁻¹' s) (↑e x)\n⊢ P f s x",
"tactic": "have := hG.right_invariance' (G.symm he) (e.mapsTo hxe) h"
},
{
"state_after": "H : Type u_1\nM : Type ?u.6200\nH' : Type u_2\nM' : Type ?u.6206\nX : Type ?u.6209\ninst✝⁶ : TopologicalSpace H\ninst✝⁵ : TopologicalSpace M\ninst✝⁴ : ChartedSpace H M\ninst✝³ : TopologicalSpace H'\ninst✝² : TopologicalSpace M'\ninst✝¹ : ChartedSpace H' M'\ninst✝ : TopologicalSpace X\nG : StructureGroupoid H\nG' : StructureGroupoid H'\nP : (H → H') → Set H → H → Prop\ns✝ t u : Set H\nx✝ : H\nhG : LocalInvariantProp G G' P\ns : Set H\nx : H\nf : H → H'\ne : LocalHomeomorph H H\nhe : e ∈ G\nhxe : x ∈ e.source\nh : P (f ∘ ↑(LocalHomeomorph.symm e)) (↑(LocalHomeomorph.symm e) ⁻¹' s) (↑e x)\nthis : P ((f ∘ ↑(LocalHomeomorph.symm e)) ∘ ↑e) (↑e ⁻¹' (↑(LocalHomeomorph.symm e) ⁻¹' s)) x\n⊢ P f s x",
"state_before": "H : Type u_1\nM : Type ?u.6200\nH' : Type u_2\nM' : Type ?u.6206\nX : Type ?u.6209\ninst✝⁶ : TopologicalSpace H\ninst✝⁵ : TopologicalSpace M\ninst✝⁴ : ChartedSpace H M\ninst✝³ : TopologicalSpace H'\ninst✝² : TopologicalSpace M'\ninst✝¹ : ChartedSpace H' M'\ninst✝ : TopologicalSpace X\nG : StructureGroupoid H\nG' : StructureGroupoid H'\nP : (H → H') → Set H → H → Prop\ns✝ t u : Set H\nx✝ : H\nhG : LocalInvariantProp G G' P\ns : Set H\nx : H\nf : H → H'\ne : LocalHomeomorph H H\nhe : e ∈ G\nhxe : x ∈ e.source\nh : P (f ∘ ↑(LocalHomeomorph.symm e)) (↑(LocalHomeomorph.symm e) ⁻¹' s) (↑e x)\nthis :\n P ((f ∘ ↑(LocalHomeomorph.symm e)) ∘ ↑(LocalHomeomorph.symm (LocalHomeomorph.symm e)))\n (↑(LocalHomeomorph.symm (LocalHomeomorph.symm e)) ⁻¹' (↑(LocalHomeomorph.symm e) ⁻¹' s))\n (↑(LocalHomeomorph.symm e) (↑e x))\n⊢ P f s x",
"tactic": "rw [e.symm_symm, e.left_inv hxe] at this"
},
{
"state_after": "case refine'_1\nH : Type u_1\nM : Type ?u.6200\nH' : Type u_2\nM' : Type ?u.6206\nX : Type ?u.6209\ninst✝⁶ : TopologicalSpace H\ninst✝⁵ : TopologicalSpace M\ninst✝⁴ : ChartedSpace H M\ninst✝³ : TopologicalSpace H'\ninst✝² : TopologicalSpace M'\ninst✝¹ : ChartedSpace H' M'\ninst✝ : TopologicalSpace X\nG : StructureGroupoid H\nG' : StructureGroupoid H'\nP : (H → H') → Set H → H → Prop\ns✝ t u : Set H\nx✝ : H\nhG : LocalInvariantProp G G' P\ns : Set H\nx : H\nf : H → H'\ne : LocalHomeomorph H H\nhe : e ∈ G\nhxe : x ∈ e.source\nh : P (f ∘ ↑(LocalHomeomorph.symm e)) (↑(LocalHomeomorph.symm e) ⁻¹' s) (↑e x)\nthis : P ((f ∘ ↑(LocalHomeomorph.symm e)) ∘ ↑e) (↑e ⁻¹' (↑(LocalHomeomorph.symm e) ⁻¹' s)) x\n⊢ (f ∘ ↑(LocalHomeomorph.symm e)) ∘ ↑e =ᶠ[𝓝 x] f\n\ncase refine'_2\nH : Type u_1\nM : Type ?u.6200\nH' : Type u_2\nM' : Type ?u.6206\nX : Type ?u.6209\ninst✝⁶ : TopologicalSpace H\ninst✝⁵ : TopologicalSpace M\ninst✝⁴ : ChartedSpace H M\ninst✝³ : TopologicalSpace H'\ninst✝² : TopologicalSpace M'\ninst✝¹ : ChartedSpace H' M'\ninst✝ : TopologicalSpace X\nG : StructureGroupoid H\nG' : StructureGroupoid H'\nP : (H → H') → Set H → H → Prop\ns✝ t u : Set H\nx✝ : H\nhG : LocalInvariantProp G G' P\ns : Set H\nx : H\nf : H → H'\ne : LocalHomeomorph H H\nhe : e ∈ G\nhxe : x ∈ e.source\nh : P (f ∘ ↑(LocalHomeomorph.symm e)) (↑(LocalHomeomorph.symm e) ⁻¹' s) (↑e x)\nthis : P ((f ∘ ↑(LocalHomeomorph.symm e)) ∘ ↑e) (↑e ⁻¹' (↑(LocalHomeomorph.symm e) ⁻¹' s)) x\n⊢ ↑e ⁻¹' (↑(LocalHomeomorph.symm e) ⁻¹' s) =ᶠ[𝓝 x] s",
"state_before": "H : Type u_1\nM : Type ?u.6200\nH' : Type u_2\nM' : Type ?u.6206\nX : Type ?u.6209\ninst✝⁶ : TopologicalSpace H\ninst✝⁵ : TopologicalSpace M\ninst✝⁴ : ChartedSpace H M\ninst✝³ : TopologicalSpace H'\ninst✝² : TopologicalSpace M'\ninst✝¹ : ChartedSpace H' M'\ninst✝ : TopologicalSpace X\nG : StructureGroupoid H\nG' : StructureGroupoid H'\nP : (H → H') → Set H → H → Prop\ns✝ t u : Set H\nx✝ : H\nhG : LocalInvariantProp G G' P\ns : Set H\nx : H\nf : H → H'\ne : LocalHomeomorph H H\nhe : e ∈ G\nhxe : x ∈ e.source\nh : P (f ∘ ↑(LocalHomeomorph.symm e)) (↑(LocalHomeomorph.symm e) ⁻¹' s) (↑e x)\nthis : P ((f ∘ ↑(LocalHomeomorph.symm e)) ∘ ↑e) (↑e ⁻¹' (↑(LocalHomeomorph.symm e) ⁻¹' s)) x\n⊢ P f s x",
"tactic": "refine' hG.congr _ ((hG.congr_set _).mp this)"
},
{
"state_after": "case refine'_1\nH : Type u_1\nM : Type ?u.6200\nH' : Type u_2\nM' : Type ?u.6206\nX : Type ?u.6209\ninst✝⁶ : TopologicalSpace H\ninst✝⁵ : TopologicalSpace M\ninst✝⁴ : ChartedSpace H M\ninst✝³ : TopologicalSpace H'\ninst✝² : TopologicalSpace M'\ninst✝¹ : ChartedSpace H' M'\ninst✝ : TopologicalSpace X\nG : StructureGroupoid H\nG' : StructureGroupoid H'\nP : (H → H') → Set H → H → Prop\ns✝ t u : Set H\nx✝ : H\nhG : LocalInvariantProp G G' P\ns : Set H\nx : H\nf : H → H'\ne : LocalHomeomorph H H\nhe : e ∈ G\nhxe : x ∈ e.source\nh : P (f ∘ ↑(LocalHomeomorph.symm e)) (↑(LocalHomeomorph.symm e) ⁻¹' s) (↑e x)\nthis : P ((f ∘ ↑(LocalHomeomorph.symm e)) ∘ ↑e) (↑e ⁻¹' (↑(LocalHomeomorph.symm e) ⁻¹' s)) x\nx' : H\nhx' : x' ∈ e.source\n⊢ ((f ∘ ↑(LocalHomeomorph.symm e)) ∘ ↑e) x' = f x'",
"state_before": "case refine'_1\nH : Type u_1\nM : Type ?u.6200\nH' : Type u_2\nM' : Type ?u.6206\nX : Type ?u.6209\ninst✝⁶ : TopologicalSpace H\ninst✝⁵ : TopologicalSpace M\ninst✝⁴ : ChartedSpace H M\ninst✝³ : TopologicalSpace H'\ninst✝² : TopologicalSpace M'\ninst✝¹ : ChartedSpace H' M'\ninst✝ : TopologicalSpace X\nG : StructureGroupoid H\nG' : StructureGroupoid H'\nP : (H → H') → Set H → H → Prop\ns✝ t u : Set H\nx✝ : H\nhG : LocalInvariantProp G G' P\ns : Set H\nx : H\nf : H → H'\ne : LocalHomeomorph H H\nhe : e ∈ G\nhxe : x ∈ e.source\nh : P (f ∘ ↑(LocalHomeomorph.symm e)) (↑(LocalHomeomorph.symm e) ⁻¹' s) (↑e x)\nthis : P ((f ∘ ↑(LocalHomeomorph.symm e)) ∘ ↑e) (↑e ⁻¹' (↑(LocalHomeomorph.symm e) ⁻¹' s)) x\n⊢ (f ∘ ↑(LocalHomeomorph.symm e)) ∘ ↑e =ᶠ[𝓝 x] f",
"tactic": "refine' eventually_of_mem (e.open_source.mem_nhds hxe) fun x' hx' ↦ _"
},
{
"state_after": "no goals",
"state_before": "case refine'_1\nH : Type u_1\nM : Type ?u.6200\nH' : Type u_2\nM' : Type ?u.6206\nX : Type ?u.6209\ninst✝⁶ : TopologicalSpace H\ninst✝⁵ : TopologicalSpace M\ninst✝⁴ : ChartedSpace H M\ninst✝³ : TopologicalSpace H'\ninst✝² : TopologicalSpace M'\ninst✝¹ : ChartedSpace H' M'\ninst✝ : TopologicalSpace X\nG : StructureGroupoid H\nG' : StructureGroupoid H'\nP : (H → H') → Set H → H → Prop\ns✝ t u : Set H\nx✝ : H\nhG : LocalInvariantProp G G' P\ns : Set H\nx : H\nf : H → H'\ne : LocalHomeomorph H H\nhe : e ∈ G\nhxe : x ∈ e.source\nh : P (f ∘ ↑(LocalHomeomorph.symm e)) (↑(LocalHomeomorph.symm e) ⁻¹' s) (↑e x)\nthis : P ((f ∘ ↑(LocalHomeomorph.symm e)) ∘ ↑e) (↑e ⁻¹' (↑(LocalHomeomorph.symm e) ⁻¹' s)) x\nx' : H\nhx' : x' ∈ e.source\n⊢ ((f ∘ ↑(LocalHomeomorph.symm e)) ∘ ↑e) x' = f x'",
"tactic": "simp_rw [Function.comp_apply, e.left_inv hx']"
},
{
"state_after": "case refine'_2\nH : Type u_1\nM : Type ?u.6200\nH' : Type u_2\nM' : Type ?u.6206\nX : Type ?u.6209\ninst✝⁶ : TopologicalSpace H\ninst✝⁵ : TopologicalSpace M\ninst✝⁴ : ChartedSpace H M\ninst✝³ : TopologicalSpace H'\ninst✝² : TopologicalSpace M'\ninst✝¹ : ChartedSpace H' M'\ninst✝ : TopologicalSpace X\nG : StructureGroupoid H\nG' : StructureGroupoid H'\nP : (H → H') → Set H → H → Prop\ns✝ t u : Set H\nx✝ : H\nhG : LocalInvariantProp G G' P\ns : Set H\nx : H\nf : H → H'\ne : LocalHomeomorph H H\nhe : e ∈ G\nhxe : x ∈ e.source\nh : P (f ∘ ↑(LocalHomeomorph.symm e)) (↑(LocalHomeomorph.symm e) ⁻¹' s) (↑e x)\nthis : P ((f ∘ ↑(LocalHomeomorph.symm e)) ∘ ↑e) (↑e ⁻¹' (↑(LocalHomeomorph.symm e) ⁻¹' s)) x\n⊢ ∀ᶠ (x : H) in 𝓝 x, x ∈ ↑e ⁻¹' (↑(LocalHomeomorph.symm e) ⁻¹' s) ↔ x ∈ s",
"state_before": "case refine'_2\nH : Type u_1\nM : Type ?u.6200\nH' : Type u_2\nM' : Type ?u.6206\nX : Type ?u.6209\ninst✝⁶ : TopologicalSpace H\ninst✝⁵ : TopologicalSpace M\ninst✝⁴ : ChartedSpace H M\ninst✝³ : TopologicalSpace H'\ninst✝² : TopologicalSpace M'\ninst✝¹ : ChartedSpace H' M'\ninst✝ : TopologicalSpace X\nG : StructureGroupoid H\nG' : StructureGroupoid H'\nP : (H → H') → Set H → H → Prop\ns✝ t u : Set H\nx✝ : H\nhG : LocalInvariantProp G G' P\ns : Set H\nx : H\nf : H → H'\ne : LocalHomeomorph H H\nhe : e ∈ G\nhxe : x ∈ e.source\nh : P (f ∘ ↑(LocalHomeomorph.symm e)) (↑(LocalHomeomorph.symm e) ⁻¹' s) (↑e x)\nthis : P ((f ∘ ↑(LocalHomeomorph.symm e)) ∘ ↑e) (↑e ⁻¹' (↑(LocalHomeomorph.symm e) ⁻¹' s)) x\n⊢ ↑e ⁻¹' (↑(LocalHomeomorph.symm e) ⁻¹' s) =ᶠ[𝓝 x] s",
"tactic": "rw [eventuallyEq_set]"
},
{
"state_after": "case refine'_2\nH : Type u_1\nM : Type ?u.6200\nH' : Type u_2\nM' : Type ?u.6206\nX : Type ?u.6209\ninst✝⁶ : TopologicalSpace H\ninst✝⁵ : TopologicalSpace M\ninst✝⁴ : ChartedSpace H M\ninst✝³ : TopologicalSpace H'\ninst✝² : TopologicalSpace M'\ninst✝¹ : ChartedSpace H' M'\ninst✝ : TopologicalSpace X\nG : StructureGroupoid H\nG' : StructureGroupoid H'\nP : (H → H') → Set H → H → Prop\ns✝ t u : Set H\nx✝ : H\nhG : LocalInvariantProp G G' P\ns : Set H\nx : H\nf : H → H'\ne : LocalHomeomorph H H\nhe : e ∈ G\nhxe : x ∈ e.source\nh : P (f ∘ ↑(LocalHomeomorph.symm e)) (↑(LocalHomeomorph.symm e) ⁻¹' s) (↑e x)\nthis : P ((f ∘ ↑(LocalHomeomorph.symm e)) ∘ ↑e) (↑e ⁻¹' (↑(LocalHomeomorph.symm e) ⁻¹' s)) x\nx' : H\nhx' : x' ∈ e.source\n⊢ x' ∈ ↑e ⁻¹' (↑(LocalHomeomorph.symm e) ⁻¹' s) ↔ x' ∈ s",
"state_before": "case refine'_2\nH : Type u_1\nM : Type ?u.6200\nH' : Type u_2\nM' : Type ?u.6206\nX : Type ?u.6209\ninst✝⁶ : TopologicalSpace H\ninst✝⁵ : TopologicalSpace M\ninst✝⁴ : ChartedSpace H M\ninst✝³ : TopologicalSpace H'\ninst✝² : TopologicalSpace M'\ninst✝¹ : ChartedSpace H' M'\ninst✝ : TopologicalSpace X\nG : StructureGroupoid H\nG' : StructureGroupoid H'\nP : (H → H') → Set H → H → Prop\ns✝ t u : Set H\nx✝ : H\nhG : LocalInvariantProp G G' P\ns : Set H\nx : H\nf : H → H'\ne : LocalHomeomorph H H\nhe : e ∈ G\nhxe : x ∈ e.source\nh : P (f ∘ ↑(LocalHomeomorph.symm e)) (↑(LocalHomeomorph.symm e) ⁻¹' s) (↑e x)\nthis : P ((f ∘ ↑(LocalHomeomorph.symm e)) ∘ ↑e) (↑e ⁻¹' (↑(LocalHomeomorph.symm e) ⁻¹' s)) x\n⊢ ∀ᶠ (x : H) in 𝓝 x, x ∈ ↑e ⁻¹' (↑(LocalHomeomorph.symm e) ⁻¹' s) ↔ x ∈ s",
"tactic": "refine' eventually_of_mem (e.open_source.mem_nhds hxe) fun x' hx' ↦ _"
},
{
"state_after": "no goals",
"state_before": "case refine'_2\nH : Type u_1\nM : Type ?u.6200\nH' : Type u_2\nM' : Type ?u.6206\nX : Type ?u.6209\ninst✝⁶ : TopologicalSpace H\ninst✝⁵ : TopologicalSpace M\ninst✝⁴ : ChartedSpace H M\ninst✝³ : TopologicalSpace H'\ninst✝² : TopologicalSpace M'\ninst✝¹ : ChartedSpace H' M'\ninst✝ : TopologicalSpace X\nG : StructureGroupoid H\nG' : StructureGroupoid H'\nP : (H → H') → Set H → H → Prop\ns✝ t u : Set H\nx✝ : H\nhG : LocalInvariantProp G G' P\ns : Set H\nx : H\nf : H → H'\ne : LocalHomeomorph H H\nhe : e ∈ G\nhxe : x ∈ e.source\nh : P (f ∘ ↑(LocalHomeomorph.symm e)) (↑(LocalHomeomorph.symm e) ⁻¹' s) (↑e x)\nthis : P ((f ∘ ↑(LocalHomeomorph.symm e)) ∘ ↑e) (↑e ⁻¹' (↑(LocalHomeomorph.symm e) ⁻¹' s)) x\nx' : H\nhx' : x' ∈ e.source\n⊢ x' ∈ ↑e ⁻¹' (↑(LocalHomeomorph.symm e) ⁻¹' s) ↔ x' ∈ s",
"tactic": "simp_rw [mem_preimage, e.left_inv hx']"
}
] |
[
155,
43
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
145,
1
] |
Mathlib/RingTheory/DedekindDomain/Factorization.lean
|
Associates.finite_factors
|
[
{
"state_after": "R : Type u_1\ninst✝⁵ : CommRing R\ninst✝⁴ : IsDomain R\ninst✝³ : IsDedekindDomain R\nK : Type ?u.7591\ninst✝² : Field K\ninst✝¹ : Algebra R K\ninst✝ : IsFractionRing R K\nv : HeightOneSpectrum R\nI : Ideal R\nhI : I ≠ 0\nh_supp : {v | ¬↑(count (Associates.mk v.asIdeal) (factors (Associates.mk I))) = 0} = {v | v.asIdeal ∣ I}\n⊢ ∀ᶠ (v : HeightOneSpectrum R) in Filter.cofinite, ↑(count (Associates.mk v.asIdeal) (factors (Associates.mk I))) = 0",
"state_before": "R : Type u_1\ninst✝⁵ : CommRing R\ninst✝⁴ : IsDomain R\ninst✝³ : IsDedekindDomain R\nK : Type ?u.7591\ninst✝² : Field K\ninst✝¹ : Algebra R K\ninst✝ : IsFractionRing R K\nv : HeightOneSpectrum R\nI : Ideal R\nhI : I ≠ 0\n⊢ ∀ᶠ (v : HeightOneSpectrum R) in Filter.cofinite, ↑(count (Associates.mk v.asIdeal) (factors (Associates.mk I))) = 0",
"tactic": "have h_supp : {v : HeightOneSpectrum R | ¬((Associates.mk v.asIdeal).count\n (Associates.mk I).factors : ℤ) = 0} = {v : HeightOneSpectrum R | v.asIdeal ∣ I} := by\n ext v\n simp_rw [Int.coe_nat_eq_zero]\n exact Associates.count_ne_zero_iff_dvd hI v.irreducible"
},
{
"state_after": "R : Type u_1\ninst✝⁵ : CommRing R\ninst✝⁴ : IsDomain R\ninst✝³ : IsDedekindDomain R\nK : Type ?u.7591\ninst✝² : Field K\ninst✝¹ : Algebra R K\ninst✝ : IsFractionRing R K\nv : HeightOneSpectrum R\nI : Ideal R\nhI : I ≠ 0\nh_supp : {v | ¬↑(count (Associates.mk v.asIdeal) (factors (Associates.mk I))) = 0} = {v | v.asIdeal ∣ I}\n⊢ Set.Finite {v | v.asIdeal ∣ I}",
"state_before": "R : Type u_1\ninst✝⁵ : CommRing R\ninst✝⁴ : IsDomain R\ninst✝³ : IsDedekindDomain R\nK : Type ?u.7591\ninst✝² : Field K\ninst✝¹ : Algebra R K\ninst✝ : IsFractionRing R K\nv : HeightOneSpectrum R\nI : Ideal R\nhI : I ≠ 0\nh_supp : {v | ¬↑(count (Associates.mk v.asIdeal) (factors (Associates.mk I))) = 0} = {v | v.asIdeal ∣ I}\n⊢ ∀ᶠ (v : HeightOneSpectrum R) in Filter.cofinite, ↑(count (Associates.mk v.asIdeal) (factors (Associates.mk I))) = 0",
"tactic": "rw [Filter.eventually_cofinite, h_supp]"
},
{
"state_after": "no goals",
"state_before": "R : Type u_1\ninst✝⁵ : CommRing R\ninst✝⁴ : IsDomain R\ninst✝³ : IsDedekindDomain R\nK : Type ?u.7591\ninst✝² : Field K\ninst✝¹ : Algebra R K\ninst✝ : IsFractionRing R K\nv : HeightOneSpectrum R\nI : Ideal R\nhI : I ≠ 0\nh_supp : {v | ¬↑(count (Associates.mk v.asIdeal) (factors (Associates.mk I))) = 0} = {v | v.asIdeal ∣ I}\n⊢ Set.Finite {v | v.asIdeal ∣ I}",
"tactic": "exact Ideal.finite_factors hI"
},
{
"state_after": "case h\nR : Type u_1\ninst✝⁵ : CommRing R\ninst✝⁴ : IsDomain R\ninst✝³ : IsDedekindDomain R\nK : Type ?u.7591\ninst✝² : Field K\ninst✝¹ : Algebra R K\ninst✝ : IsFractionRing R K\nv✝ : HeightOneSpectrum R\nI : Ideal R\nhI : I ≠ 0\nv : HeightOneSpectrum R\n⊢ v ∈ {v | ¬↑(count (Associates.mk v.asIdeal) (factors (Associates.mk I))) = 0} ↔ v ∈ {v | v.asIdeal ∣ I}",
"state_before": "R : Type u_1\ninst✝⁵ : CommRing R\ninst✝⁴ : IsDomain R\ninst✝³ : IsDedekindDomain R\nK : Type ?u.7591\ninst✝² : Field K\ninst✝¹ : Algebra R K\ninst✝ : IsFractionRing R K\nv : HeightOneSpectrum R\nI : Ideal R\nhI : I ≠ 0\n⊢ {v | ¬↑(count (Associates.mk v.asIdeal) (factors (Associates.mk I))) = 0} = {v | v.asIdeal ∣ I}",
"tactic": "ext v"
},
{
"state_after": "case h\nR : Type u_1\ninst✝⁵ : CommRing R\ninst✝⁴ : IsDomain R\ninst✝³ : IsDedekindDomain R\nK : Type ?u.7591\ninst✝² : Field K\ninst✝¹ : Algebra R K\ninst✝ : IsFractionRing R K\nv✝ : HeightOneSpectrum R\nI : Ideal R\nhI : I ≠ 0\nv : HeightOneSpectrum R\n⊢ v ∈ {v | ¬count (Associates.mk v.asIdeal) (factors (Associates.mk I)) = 0} ↔ v ∈ {v | v.asIdeal ∣ I}",
"state_before": "case h\nR : Type u_1\ninst✝⁵ : CommRing R\ninst✝⁴ : IsDomain R\ninst✝³ : IsDedekindDomain R\nK : Type ?u.7591\ninst✝² : Field K\ninst✝¹ : Algebra R K\ninst✝ : IsFractionRing R K\nv✝ : HeightOneSpectrum R\nI : Ideal R\nhI : I ≠ 0\nv : HeightOneSpectrum R\n⊢ v ∈ {v | ¬↑(count (Associates.mk v.asIdeal) (factors (Associates.mk I))) = 0} ↔ v ∈ {v | v.asIdeal ∣ I}",
"tactic": "simp_rw [Int.coe_nat_eq_zero]"
},
{
"state_after": "no goals",
"state_before": "case h\nR : Type u_1\ninst✝⁵ : CommRing R\ninst✝⁴ : IsDomain R\ninst✝³ : IsDedekindDomain R\nK : Type ?u.7591\ninst✝² : Field K\ninst✝¹ : Algebra R K\ninst✝ : IsFractionRing R K\nv✝ : HeightOneSpectrum R\nI : Ideal R\nhI : I ≠ 0\nv : HeightOneSpectrum R\n⊢ v ∈ {v | ¬count (Associates.mk v.asIdeal) (factors (Associates.mk I)) = 0} ↔ v ∈ {v | v.asIdeal ∣ I}",
"tactic": "exact Associates.count_ne_zero_iff_dvd hI v.irreducible"
}
] |
[
69,
32
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
60,
1
] |
Mathlib/Topology/Algebra/Group/Basic.lean
|
isClosedMap_div_left
|
[] |
[
1169,
37
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1168,
1
] |
Mathlib/RingTheory/NonUnitalSubsemiring/Basic.lean
|
NonUnitalSubsemiring.coe_comap
|
[] |
[
278,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
277,
1
] |
Mathlib/Algebra/Order/Ring/Defs.lean
|
one_lt_mul_of_lt_of_le
|
[] |
[
322,
69
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
321,
1
] |
Mathlib/Data/Set/Image.lean
|
Set.disjoint_preimage_iff
|
[] |
[
1613,
49
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1611,
1
] |
Mathlib/Order/WithBot.lean
|
WithTop.coe_max
|
[] |
[
1246,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1245,
1
] |
Mathlib/Analysis/SpecialFunctions/Integrals.lean
|
IntervalIntegrable.log
|
[] |
[
215,
45
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
212,
1
] |
Mathlib/Logic/Unique.lean
|
Unique.bijective
|
[
{
"state_after": "A : Sort u_1\nB : Sort u_2\ninst✝¹ : Unique A\ninst✝ : Unique B\nf : A → B\n⊢ ∃ g, Function.LeftInverse g f ∧ Function.RightInverse g f",
"state_before": "A : Sort u_1\nB : Sort u_2\ninst✝¹ : Unique A\ninst✝ : Unique B\nf : A → B\n⊢ Function.Bijective f",
"tactic": "rw [Function.bijective_iff_has_inverse]"
},
{
"state_after": "no goals",
"state_before": "A : Sort u_1\nB : Sort u_2\ninst✝¹ : Unique A\ninst✝ : Unique B\nf : A → B\n⊢ ∃ g, Function.LeftInverse g f ∧ Function.RightInverse g f",
"tactic": "refine' ⟨default, _, _⟩ <;> intro x <;> simp"
}
] |
[
259,
47
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
257,
1
] |
Mathlib/Algebra/CharZero/Lemmas.lean
|
add_self_eq_zero
|
[
{
"state_after": "no goals",
"state_before": "R : Type u_1\ninst✝² : NonAssocSemiring R\ninst✝¹ : NoZeroDivisors R\ninst✝ : CharZero R\na✝ a : R\n⊢ a + a = 0 ↔ a = 0",
"tactic": "simp only [(two_mul a).symm, mul_eq_zero, two_ne_zero, false_or_iff]"
}
] |
[
74,
71
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
73,
1
] |
Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean
|
MeasureTheory.Measure.ext_of_Ico_finite
|
[
{
"state_after": "α✝ : Type ?u.676748\nβ : Type ?u.676751\nγ : Type ?u.676754\nγ₂ : Type ?u.676757\nδ : Type ?u.676760\nι : Sort y\ns t u : Set α✝\ninst✝²² : TopologicalSpace α✝\ninst✝²¹ : MeasurableSpace α✝\ninst✝²⁰ : OpensMeasurableSpace α✝\ninst✝¹⁹ : TopologicalSpace β\ninst✝¹⁸ : MeasurableSpace β\ninst✝¹⁷ : OpensMeasurableSpace β\ninst✝¹⁶ : TopologicalSpace γ\ninst✝¹⁵ : MeasurableSpace γ\ninst✝¹⁴ : BorelSpace γ\ninst✝¹³ : TopologicalSpace γ₂\ninst✝¹² : MeasurableSpace γ₂\ninst✝¹¹ : BorelSpace γ₂\ninst✝¹⁰ : MeasurableSpace δ\nα' : Type ?u.676853\ninst✝⁹ : TopologicalSpace α'\ninst✝⁸ : MeasurableSpace α'\ninst✝⁷ : LinearOrder α✝\ninst✝⁶ : OrderClosedTopology α✝\na b x : α✝\nα : Type u_1\ninst✝⁵ : TopologicalSpace α\nm : MeasurableSpace α\ninst✝⁴ : SecondCountableTopology α\ninst✝³ : LinearOrder α\ninst✝² : OrderTopology α\ninst✝¹ : BorelSpace α\nμ ν : Measure α\ninst✝ : IsFiniteMeasure μ\nhμν : ↑↑μ univ = ↑↑ν univ\nh : ∀ ⦃a b : α⦄, a < b → ↑↑μ (Ico a b) = ↑↑ν (Ico a b)\n⊢ ∀ (s : Set α), s ∈ {S | ∃ l u, l < u ∧ Ico l u = S} → ↑↑μ s = ↑↑ν s",
"state_before": "α✝ : Type ?u.676748\nβ : Type ?u.676751\nγ : Type ?u.676754\nγ₂ : Type ?u.676757\nδ : Type ?u.676760\nι : Sort y\ns t u : Set α✝\ninst✝²² : TopologicalSpace α✝\ninst✝²¹ : MeasurableSpace α✝\ninst✝²⁰ : OpensMeasurableSpace α✝\ninst✝¹⁹ : TopologicalSpace β\ninst✝¹⁸ : MeasurableSpace β\ninst✝¹⁷ : OpensMeasurableSpace β\ninst✝¹⁶ : TopologicalSpace γ\ninst✝¹⁵ : MeasurableSpace γ\ninst✝¹⁴ : BorelSpace γ\ninst✝¹³ : TopologicalSpace γ₂\ninst✝¹² : MeasurableSpace γ₂\ninst✝¹¹ : BorelSpace γ₂\ninst✝¹⁰ : MeasurableSpace δ\nα' : Type ?u.676853\ninst✝⁹ : TopologicalSpace α'\ninst✝⁸ : MeasurableSpace α'\ninst✝⁷ : LinearOrder α✝\ninst✝⁶ : OrderClosedTopology α✝\na b x : α✝\nα : Type u_1\ninst✝⁵ : TopologicalSpace α\nm : MeasurableSpace α\ninst✝⁴ : SecondCountableTopology α\ninst✝³ : LinearOrder α\ninst✝² : OrderTopology α\ninst✝¹ : BorelSpace α\nμ ν : Measure α\ninst✝ : IsFiniteMeasure μ\nhμν : ↑↑μ univ = ↑↑ν univ\nh : ∀ ⦃a b : α⦄, a < b → ↑↑μ (Ico a b) = ↑↑ν (Ico a b)\n⊢ μ = ν",
"tactic": "refine'\n ext_of_generate_finite _ (BorelSpace.measurable_eq.trans (borel_eq_generateFrom_Ico α))\n (isPiSystem_Ico (id : α → α) id) _ hμν"
},
{
"state_after": "case intro.intro.intro\nα✝ : Type ?u.676748\nβ : Type ?u.676751\nγ : Type ?u.676754\nγ₂ : Type ?u.676757\nδ : Type ?u.676760\nι : Sort y\ns t u : Set α✝\ninst✝²² : TopologicalSpace α✝\ninst✝²¹ : MeasurableSpace α✝\ninst✝²⁰ : OpensMeasurableSpace α✝\ninst✝¹⁹ : TopologicalSpace β\ninst✝¹⁸ : MeasurableSpace β\ninst✝¹⁷ : OpensMeasurableSpace β\ninst✝¹⁶ : TopologicalSpace γ\ninst✝¹⁵ : MeasurableSpace γ\ninst✝¹⁴ : BorelSpace γ\ninst✝¹³ : TopologicalSpace γ₂\ninst✝¹² : MeasurableSpace γ₂\ninst✝¹¹ : BorelSpace γ₂\ninst✝¹⁰ : MeasurableSpace δ\nα' : Type ?u.676853\ninst✝⁹ : TopologicalSpace α'\ninst✝⁸ : MeasurableSpace α'\ninst✝⁷ : LinearOrder α✝\ninst✝⁶ : OrderClosedTopology α✝\na✝ b✝ x : α✝\nα : Type u_1\ninst✝⁵ : TopologicalSpace α\nm : MeasurableSpace α\ninst✝⁴ : SecondCountableTopology α\ninst✝³ : LinearOrder α\ninst✝² : OrderTopology α\ninst✝¹ : BorelSpace α\nμ ν : Measure α\ninst✝ : IsFiniteMeasure μ\nhμν : ↑↑μ univ = ↑↑ν univ\nh : ∀ ⦃a b : α⦄, a < b → ↑↑μ (Ico a b) = ↑↑ν (Ico a b)\na b : α\nhlt : a < b\n⊢ ↑↑μ (Ico a b) = ↑↑ν (Ico a b)",
"state_before": "α✝ : Type ?u.676748\nβ : Type ?u.676751\nγ : Type ?u.676754\nγ₂ : Type ?u.676757\nδ : Type ?u.676760\nι : Sort y\ns t u : Set α✝\ninst✝²² : TopologicalSpace α✝\ninst✝²¹ : MeasurableSpace α✝\ninst✝²⁰ : OpensMeasurableSpace α✝\ninst✝¹⁹ : TopologicalSpace β\ninst✝¹⁸ : MeasurableSpace β\ninst✝¹⁷ : OpensMeasurableSpace β\ninst✝¹⁶ : TopologicalSpace γ\ninst✝¹⁵ : MeasurableSpace γ\ninst✝¹⁴ : BorelSpace γ\ninst✝¹³ : TopologicalSpace γ₂\ninst✝¹² : MeasurableSpace γ₂\ninst✝¹¹ : BorelSpace γ₂\ninst✝¹⁰ : MeasurableSpace δ\nα' : Type ?u.676853\ninst✝⁹ : TopologicalSpace α'\ninst✝⁸ : MeasurableSpace α'\ninst✝⁷ : LinearOrder α✝\ninst✝⁶ : OrderClosedTopology α✝\na b x : α✝\nα : Type u_1\ninst✝⁵ : TopologicalSpace α\nm : MeasurableSpace α\ninst✝⁴ : SecondCountableTopology α\ninst✝³ : LinearOrder α\ninst✝² : OrderTopology α\ninst✝¹ : BorelSpace α\nμ ν : Measure α\ninst✝ : IsFiniteMeasure μ\nhμν : ↑↑μ univ = ↑↑ν univ\nh : ∀ ⦃a b : α⦄, a < b → ↑↑μ (Ico a b) = ↑↑ν (Ico a b)\n⊢ ∀ (s : Set α), s ∈ {S | ∃ l u, l < u ∧ Ico l u = S} → ↑↑μ s = ↑↑ν s",
"tactic": "rintro - ⟨a, b, hlt, rfl⟩"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.intro\nα✝ : Type ?u.676748\nβ : Type ?u.676751\nγ : Type ?u.676754\nγ₂ : Type ?u.676757\nδ : Type ?u.676760\nι : Sort y\ns t u : Set α✝\ninst✝²² : TopologicalSpace α✝\ninst✝²¹ : MeasurableSpace α✝\ninst✝²⁰ : OpensMeasurableSpace α✝\ninst✝¹⁹ : TopologicalSpace β\ninst✝¹⁸ : MeasurableSpace β\ninst✝¹⁷ : OpensMeasurableSpace β\ninst✝¹⁶ : TopologicalSpace γ\ninst✝¹⁵ : MeasurableSpace γ\ninst✝¹⁴ : BorelSpace γ\ninst✝¹³ : TopologicalSpace γ₂\ninst✝¹² : MeasurableSpace γ₂\ninst✝¹¹ : BorelSpace γ₂\ninst✝¹⁰ : MeasurableSpace δ\nα' : Type ?u.676853\ninst✝⁹ : TopologicalSpace α'\ninst✝⁸ : MeasurableSpace α'\ninst✝⁷ : LinearOrder α✝\ninst✝⁶ : OrderClosedTopology α✝\na✝ b✝ x : α✝\nα : Type u_1\ninst✝⁵ : TopologicalSpace α\nm : MeasurableSpace α\ninst✝⁴ : SecondCountableTopology α\ninst✝³ : LinearOrder α\ninst✝² : OrderTopology α\ninst✝¹ : BorelSpace α\nμ ν : Measure α\ninst✝ : IsFiniteMeasure μ\nhμν : ↑↑μ univ = ↑↑ν univ\nh : ∀ ⦃a b : α⦄, a < b → ↑↑μ (Ico a b) = ↑↑ν (Ico a b)\na b : α\nhlt : a < b\n⊢ ↑↑μ (Ico a b) = ↑↑ν (Ico a b)",
"tactic": "exact h hlt"
}
] |
[
674,
16
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
666,
1
] |
Mathlib/CategoryTheory/Monoidal/Braided.lean
|
CategoryTheory.associator_monoidal_aux
|
[
{
"state_after": "C : Type u₁\ninst✝⁸ : Category C\ninst✝⁷ : MonoidalCategory C\ninst✝⁶ : BraidedCategory C\nD : Type u₂\ninst✝⁵ : Category D\ninst✝⁴ : MonoidalCategory D\ninst✝³ : BraidedCategory D\nE : Type u₃\ninst✝² : Category E\ninst✝¹ : MonoidalCategory E\ninst✝ : BraidedCategory E\nW X Y Z : C\n⊢ (𝟙 W ⊗ (β_ X (Y ⊗ Z)).hom) ≫ (𝟙 W ⊗ (α_ Y Z X).hom) ≫ (α_ W Y (Z ⊗ X)).inv ≫ ((β_ W Y).hom ⊗ 𝟙 (Z ⊗ X)) =\n (((((𝟙 W ⊗ (α_ X Y Z).inv) ≫ (α_ W (X ⊗ Y) Z).inv ≫ ((α_ W X Y).inv ⊗ 𝟙 Z)) ≫ ((β_ (W ⊗ X) Y).hom ⊗ 𝟙 Z)) ≫\n ((α_ Y W X).inv ⊗ 𝟙 Z)) ≫\n (α_ (Y ⊗ W) X Z).hom) ≫\n (𝟙 (Y ⊗ W) ⊗ (β_ X Z).hom)",
"state_before": "C : Type u₁\ninst✝⁸ : Category C\ninst✝⁷ : MonoidalCategory C\ninst✝⁶ : BraidedCategory C\nD : Type u₂\ninst✝⁵ : Category D\ninst✝⁴ : MonoidalCategory D\ninst✝³ : BraidedCategory D\nE : Type u₃\ninst✝² : Category E\ninst✝¹ : MonoidalCategory E\ninst✝ : BraidedCategory E\nW X Y Z : C\n⊢ (𝟙 W ⊗ (β_ X (Y ⊗ Z)).hom) ≫ (𝟙 W ⊗ (α_ Y Z X).hom) ≫ (α_ W Y (Z ⊗ X)).inv ≫ ((β_ W Y).hom ⊗ 𝟙 (Z ⊗ X)) =\n (α_ W X (Y ⊗ Z)).inv ≫\n (α_ (W ⊗ X) Y Z).inv ≫\n ((β_ (W ⊗ X) Y).hom ⊗ 𝟙 Z) ≫ ((α_ Y W X).inv ⊗ 𝟙 Z) ≫ (α_ (Y ⊗ W) X Z).hom ≫ (𝟙 (Y ⊗ W) ⊗ (β_ X Z).hom)",
"tactic": "slice_rhs 1 2 => rw [← pentagon_inv]"
},
{
"state_after": "C : Type u₁\ninst✝⁸ : Category C\ninst✝⁷ : MonoidalCategory C\ninst✝⁶ : BraidedCategory C\nD : Type u₂\ninst✝⁵ : Category D\ninst✝⁴ : MonoidalCategory D\ninst✝³ : BraidedCategory D\nE : Type u₃\ninst✝² : Category E\ninst✝¹ : MonoidalCategory E\ninst✝ : BraidedCategory E\nW X Y Z : C\n⊢ (𝟙 W ⊗ (β_ X (Y ⊗ Z)).hom) ≫ (𝟙 W ⊗ (α_ Y Z X).hom) ≫ (α_ W Y (Z ⊗ X)).inv ≫ ((β_ W Y).hom ⊗ 𝟙 (Z ⊗ X)) =\n (𝟙 W ⊗ (α_ X Y Z).inv) ≫\n (α_ W (X ⊗ Y) Z).inv ≫\n ((((𝟙 W ⊗ (β_ X Y).hom) ⊗ 𝟙 Z) ≫ ((α_ W Y X).inv ⊗ 𝟙 Z) ≫ (((β_ W Y).hom ⊗ 𝟙 X) ⊗ 𝟙 Z)) ≫\n (α_ (Y ⊗ W) X Z).hom) ≫\n (𝟙 (Y ⊗ W) ⊗ (β_ X Z).hom)",
"state_before": "C : Type u₁\ninst✝⁸ : Category C\ninst✝⁷ : MonoidalCategory C\ninst✝⁶ : BraidedCategory C\nD : Type u₂\ninst✝⁵ : Category D\ninst✝⁴ : MonoidalCategory D\ninst✝³ : BraidedCategory D\nE : Type u₃\ninst✝² : Category E\ninst✝¹ : MonoidalCategory E\ninst✝ : BraidedCategory E\nW X Y Z : C\n⊢ (𝟙 W ⊗ (β_ X (Y ⊗ Z)).hom) ≫ (𝟙 W ⊗ (α_ Y Z X).hom) ≫ (α_ W Y (Z ⊗ X)).inv ≫ ((β_ W Y).hom ⊗ 𝟙 (Z ⊗ X)) =\n (((((𝟙 W ⊗ (α_ X Y Z).inv) ≫ (α_ W (X ⊗ Y) Z).inv ≫ ((α_ W X Y).inv ⊗ 𝟙 Z)) ≫ ((β_ (W ⊗ X) Y).hom ⊗ 𝟙 Z)) ≫\n ((α_ Y W X).inv ⊗ 𝟙 Z)) ≫\n (α_ (Y ⊗ W) X Z).hom) ≫\n (𝟙 (Y ⊗ W) ⊗ (β_ X Z).hom)",
"tactic": "slice_rhs 3 5 => rw [← tensor_comp, ← tensor_comp, hexagon_reverse, tensor_comp, tensor_comp]"
},
{
"state_after": "C : Type u₁\ninst✝⁸ : Category C\ninst✝⁷ : MonoidalCategory C\ninst✝⁶ : BraidedCategory C\nD : Type u₂\ninst✝⁵ : Category D\ninst✝⁴ : MonoidalCategory D\ninst✝³ : BraidedCategory D\nE : Type u₃\ninst✝² : Category E\ninst✝¹ : MonoidalCategory E\ninst✝ : BraidedCategory E\nW X Y Z : C\n⊢ (𝟙 W ⊗ (β_ X (Y ⊗ Z)).hom) ≫ (𝟙 W ⊗ (α_ Y Z X).hom) ≫ (α_ W Y (Z ⊗ X)).inv ≫ ((β_ W Y).hom ⊗ 𝟙 (Z ⊗ X)) =\n (𝟙 W ⊗ (α_ X Y Z).inv) ≫\n (α_ W (X ⊗ Y) Z).inv ≫\n ((𝟙 W ⊗ (β_ X Y).hom) ⊗ 𝟙 Z) ≫\n ((α_ W Y X).inv ⊗ 𝟙 Z) ≫ ((α_ (W ⊗ Y) X Z).hom ≫ ((β_ W Y).hom ⊗ 𝟙 X ⊗ 𝟙 Z)) ≫ (𝟙 (Y ⊗ W) ⊗ (β_ X Z).hom)",
"state_before": "C : Type u₁\ninst✝⁸ : Category C\ninst✝⁷ : MonoidalCategory C\ninst✝⁶ : BraidedCategory C\nD : Type u₂\ninst✝⁵ : Category D\ninst✝⁴ : MonoidalCategory D\ninst✝³ : BraidedCategory D\nE : Type u₃\ninst✝² : Category E\ninst✝¹ : MonoidalCategory E\ninst✝ : BraidedCategory E\nW X Y Z : C\n⊢ (𝟙 W ⊗ (β_ X (Y ⊗ Z)).hom) ≫ (𝟙 W ⊗ (α_ Y Z X).hom) ≫ (α_ W Y (Z ⊗ X)).inv ≫ ((β_ W Y).hom ⊗ 𝟙 (Z ⊗ X)) =\n (𝟙 W ⊗ (α_ X Y Z).inv) ≫\n (α_ W (X ⊗ Y) Z).inv ≫\n ((((𝟙 W ⊗ (β_ X Y).hom) ⊗ 𝟙 Z) ≫ ((α_ W Y X).inv ⊗ 𝟙 Z) ≫ (((β_ W Y).hom ⊗ 𝟙 X) ⊗ 𝟙 Z)) ≫\n (α_ (Y ⊗ W) X Z).hom) ≫\n (𝟙 (Y ⊗ W) ⊗ (β_ X Z).hom)",
"tactic": "slice_rhs 5 6 => rw [associator_naturality]"
},
{
"state_after": "C : Type u₁\ninst✝⁸ : Category C\ninst✝⁷ : MonoidalCategory C\ninst✝⁶ : BraidedCategory C\nD : Type u₂\ninst✝⁵ : Category D\ninst✝⁴ : MonoidalCategory D\ninst✝³ : BraidedCategory D\nE : Type u₃\ninst✝² : Category E\ninst✝¹ : MonoidalCategory E\ninst✝ : BraidedCategory E\nW X Y Z : C\n⊢ (𝟙 W ⊗ (β_ X (Y ⊗ Z)).hom) ≫ (𝟙 W ⊗ (α_ Y Z X).hom) ≫ (α_ W Y (Z ⊗ X)).inv ≫ ((β_ W Y).hom ⊗ 𝟙 (Z ⊗ X)) =\n (𝟙 W ⊗ (α_ X Y Z).inv) ≫\n (α_ W (X ⊗ Y) Z).inv ≫\n ((𝟙 W ⊗ (β_ X Y).hom) ⊗ 𝟙 Z) ≫\n ((α_ W Y X).inv ⊗ 𝟙 Z) ≫ (α_ (W ⊗ Y) X Z).hom ≫ (𝟙 (W ⊗ Y) ⊗ (β_ X Z).hom) ≫ ((β_ W Y).hom ⊗ 𝟙 (Z ⊗ X))",
"state_before": "C : Type u₁\ninst✝⁸ : Category C\ninst✝⁷ : MonoidalCategory C\ninst✝⁶ : BraidedCategory C\nD : Type u₂\ninst✝⁵ : Category D\ninst✝⁴ : MonoidalCategory D\ninst✝³ : BraidedCategory D\nE : Type u₃\ninst✝² : Category E\ninst✝¹ : MonoidalCategory E\ninst✝ : BraidedCategory E\nW X Y Z : C\n⊢ (𝟙 W ⊗ (β_ X (Y ⊗ Z)).hom) ≫ (𝟙 W ⊗ (α_ Y Z X).hom) ≫ (α_ W Y (Z ⊗ X)).inv ≫ ((β_ W Y).hom ⊗ 𝟙 (Z ⊗ X)) =\n (𝟙 W ⊗ (α_ X Y Z).inv) ≫\n (α_ W (X ⊗ Y) Z).inv ≫\n ((𝟙 W ⊗ (β_ X Y).hom) ⊗ 𝟙 Z) ≫\n ((α_ W Y X).inv ⊗ 𝟙 Z) ≫ ((α_ (W ⊗ Y) X Z).hom ≫ ((β_ W Y).hom ⊗ 𝟙 X ⊗ 𝟙 Z)) ≫ (𝟙 (Y ⊗ W) ⊗ (β_ X Z).hom)",
"tactic": "slice_rhs 6 7 => rw [tensor_id, tensor_id_comp_id_tensor, ← id_tensor_comp_tensor_id]"
},
{
"state_after": "C : Type u₁\ninst✝⁸ : Category C\ninst✝⁷ : MonoidalCategory C\ninst✝⁶ : BraidedCategory C\nD : Type u₂\ninst✝⁵ : Category D\ninst✝⁴ : MonoidalCategory D\ninst✝³ : BraidedCategory D\nE : Type u₃\ninst✝² : Category E\ninst✝¹ : MonoidalCategory E\ninst✝ : BraidedCategory E\nW X Y Z : C\n⊢ (𝟙 W ⊗ (β_ X (Y ⊗ Z)).hom) ≫ (𝟙 W ⊗ (α_ Y Z X).hom) ≫ (α_ W Y (Z ⊗ X)).inv ≫ ((β_ W Y).hom ⊗ 𝟙 (Z ⊗ X)) =\n (𝟙 W ⊗ (α_ X Y Z).inv) ≫\n (((((𝟙 W ⊗ (β_ X Y).hom ⊗ 𝟙 Z) ≫ (α_ W (Y ⊗ X) Z).inv) ≫ ((α_ W Y X).inv ⊗ 𝟙 Z)) ≫ (α_ (W ⊗ Y) X Z).hom) ≫\n (𝟙 (W ⊗ Y) ⊗ (β_ X Z).hom)) ≫\n ((β_ W Y).hom ⊗ 𝟙 (Z ⊗ X))",
"state_before": "C : Type u₁\ninst✝⁸ : Category C\ninst✝⁷ : MonoidalCategory C\ninst✝⁶ : BraidedCategory C\nD : Type u₂\ninst✝⁵ : Category D\ninst✝⁴ : MonoidalCategory D\ninst✝³ : BraidedCategory D\nE : Type u₃\ninst✝² : Category E\ninst✝¹ : MonoidalCategory E\ninst✝ : BraidedCategory E\nW X Y Z : C\n⊢ (𝟙 W ⊗ (β_ X (Y ⊗ Z)).hom) ≫ (𝟙 W ⊗ (α_ Y Z X).hom) ≫ (α_ W Y (Z ⊗ X)).inv ≫ ((β_ W Y).hom ⊗ 𝟙 (Z ⊗ X)) =\n (𝟙 W ⊗ (α_ X Y Z).inv) ≫\n (α_ W (X ⊗ Y) Z).inv ≫\n ((𝟙 W ⊗ (β_ X Y).hom) ⊗ 𝟙 Z) ≫\n ((α_ W Y X).inv ⊗ 𝟙 Z) ≫ (α_ (W ⊗ Y) X Z).hom ≫ (𝟙 (W ⊗ Y) ⊗ (β_ X Z).hom) ≫ ((β_ W Y).hom ⊗ 𝟙 (Z ⊗ X))",
"tactic": "slice_rhs 2 3 => rw [← associator_inv_naturality]"
},
{
"state_after": "C : Type u₁\ninst✝⁸ : Category C\ninst✝⁷ : MonoidalCategory C\ninst✝⁶ : BraidedCategory C\nD : Type u₂\ninst✝⁵ : Category D\ninst✝⁴ : MonoidalCategory D\ninst✝³ : BraidedCategory D\nE : Type u₃\ninst✝² : Category E\ninst✝¹ : MonoidalCategory E\ninst✝ : BraidedCategory E\nW X Y Z : C\n⊢ (𝟙 W ⊗ (β_ X (Y ⊗ Z)).hom) ≫ (𝟙 W ⊗ (α_ Y Z X).hom) ≫ (α_ W Y (Z ⊗ X)).inv ≫ ((β_ W Y).hom ⊗ 𝟙 (Z ⊗ X)) =\n (𝟙 W ⊗ (α_ X Y Z).inv) ≫\n (𝟙 W ⊗ (β_ X Y).hom ⊗ 𝟙 Z) ≫\n (((𝟙 W ⊗ (α_ Y X Z).hom) ≫ (α_ W Y (X ⊗ Z)).inv) ≫ (𝟙 (W ⊗ Y) ⊗ (β_ X Z).hom)) ≫ ((β_ W Y).hom ⊗ 𝟙 (Z ⊗ X))",
"state_before": "C : Type u₁\ninst✝⁸ : Category C\ninst✝⁷ : MonoidalCategory C\ninst✝⁶ : BraidedCategory C\nD : Type u₂\ninst✝⁵ : Category D\ninst✝⁴ : MonoidalCategory D\ninst✝³ : BraidedCategory D\nE : Type u₃\ninst✝² : Category E\ninst✝¹ : MonoidalCategory E\ninst✝ : BraidedCategory E\nW X Y Z : C\n⊢ (𝟙 W ⊗ (β_ X (Y ⊗ Z)).hom) ≫ (𝟙 W ⊗ (α_ Y Z X).hom) ≫ (α_ W Y (Z ⊗ X)).inv ≫ ((β_ W Y).hom ⊗ 𝟙 (Z ⊗ X)) =\n (𝟙 W ⊗ (α_ X Y Z).inv) ≫\n (((((𝟙 W ⊗ (β_ X Y).hom ⊗ 𝟙 Z) ≫ (α_ W (Y ⊗ X) Z).inv) ≫ ((α_ W Y X).inv ⊗ 𝟙 Z)) ≫ (α_ (W ⊗ Y) X Z).hom) ≫\n (𝟙 (W ⊗ Y) ⊗ (β_ X Z).hom)) ≫\n ((β_ W Y).hom ⊗ 𝟙 (Z ⊗ X))",
"tactic": "slice_rhs 3 5 => rw [pentagon_inv_inv_hom]"
},
{
"state_after": "C : Type u₁\ninst✝⁸ : Category C\ninst✝⁷ : MonoidalCategory C\ninst✝⁶ : BraidedCategory C\nD : Type u₂\ninst✝⁵ : Category D\ninst✝⁴ : MonoidalCategory D\ninst✝³ : BraidedCategory D\nE : Type u₃\ninst✝² : Category E\ninst✝¹ : MonoidalCategory E\ninst✝ : BraidedCategory E\nW X Y Z : C\n⊢ (𝟙 W ⊗ (β_ X (Y ⊗ Z)).hom) ≫ (𝟙 W ⊗ (α_ Y Z X).hom) ≫ (α_ W Y (Z ⊗ X)).inv ≫ ((β_ W Y).hom ⊗ 𝟙 (Z ⊗ X)) =\n (𝟙 W ⊗ (α_ X Y Z).inv) ≫\n (𝟙 W ⊗ (β_ X Y).hom ⊗ 𝟙 Z) ≫\n (𝟙 W ⊗ (α_ Y X Z).hom) ≫ ((𝟙 W ⊗ 𝟙 Y ⊗ (β_ X Z).hom) ≫ (α_ W Y (Z ⊗ X)).inv) ≫ ((β_ W Y).hom ⊗ 𝟙 (Z ⊗ X))",
"state_before": "C : Type u₁\ninst✝⁸ : Category C\ninst✝⁷ : MonoidalCategory C\ninst✝⁶ : BraidedCategory C\nD : Type u₂\ninst✝⁵ : Category D\ninst✝⁴ : MonoidalCategory D\ninst✝³ : BraidedCategory D\nE : Type u₃\ninst✝² : Category E\ninst✝¹ : MonoidalCategory E\ninst✝ : BraidedCategory E\nW X Y Z : C\n⊢ (𝟙 W ⊗ (β_ X (Y ⊗ Z)).hom) ≫ (𝟙 W ⊗ (α_ Y Z X).hom) ≫ (α_ W Y (Z ⊗ X)).inv ≫ ((β_ W Y).hom ⊗ 𝟙 (Z ⊗ X)) =\n (𝟙 W ⊗ (α_ X Y Z).inv) ≫\n (𝟙 W ⊗ (β_ X Y).hom ⊗ 𝟙 Z) ≫\n (((𝟙 W ⊗ (α_ Y X Z).hom) ≫ (α_ W Y (X ⊗ Z)).inv) ≫ (𝟙 (W ⊗ Y) ⊗ (β_ X Z).hom)) ≫ ((β_ W Y).hom ⊗ 𝟙 (Z ⊗ X))",
"tactic": "slice_rhs 4 5 => rw [← tensor_id, ← associator_inv_naturality]"
},
{
"state_after": "C : Type u₁\ninst✝⁸ : Category C\ninst✝⁷ : MonoidalCategory C\ninst✝⁶ : BraidedCategory C\nD : Type u₂\ninst✝⁵ : Category D\ninst✝⁴ : MonoidalCategory D\ninst✝³ : BraidedCategory D\nE : Type u₃\ninst✝² : Category E\ninst✝¹ : MonoidalCategory E\ninst✝ : BraidedCategory E\nW X Y Z : C\n⊢ (𝟙 W ⊗ (β_ X (Y ⊗ Z)).hom) ≫ (𝟙 W ⊗ (α_ Y Z X).hom) ≫ (α_ W Y (Z ⊗ X)).inv ≫ ((β_ W Y).hom ⊗ 𝟙 (Z ⊗ X)) =\n (𝟙 W ⊗ (α_ X Y Z).inv) ≫\n (((𝟙 W ⊗ (α_ X Y Z).hom) ≫ (𝟙 W ⊗ (β_ X (Y ⊗ Z)).hom) ≫ (𝟙 W ⊗ (α_ Y Z X).hom)) ≫ (α_ W Y (Z ⊗ X)).inv) ≫\n ((β_ W Y).hom ⊗ 𝟙 (Z ⊗ X))",
"state_before": "C : Type u₁\ninst✝⁸ : Category C\ninst✝⁷ : MonoidalCategory C\ninst✝⁶ : BraidedCategory C\nD : Type u₂\ninst✝⁵ : Category D\ninst✝⁴ : MonoidalCategory D\ninst✝³ : BraidedCategory D\nE : Type u₃\ninst✝² : Category E\ninst✝¹ : MonoidalCategory E\ninst✝ : BraidedCategory E\nW X Y Z : C\n⊢ (𝟙 W ⊗ (β_ X (Y ⊗ Z)).hom) ≫ (𝟙 W ⊗ (α_ Y Z X).hom) ≫ (α_ W Y (Z ⊗ X)).inv ≫ ((β_ W Y).hom ⊗ 𝟙 (Z ⊗ X)) =\n (𝟙 W ⊗ (α_ X Y Z).inv) ≫\n (𝟙 W ⊗ (β_ X Y).hom ⊗ 𝟙 Z) ≫\n (𝟙 W ⊗ (α_ Y X Z).hom) ≫ ((𝟙 W ⊗ 𝟙 Y ⊗ (β_ X Z).hom) ≫ (α_ W Y (Z ⊗ X)).inv) ≫ ((β_ W Y).hom ⊗ 𝟙 (Z ⊗ X))",
"tactic": "slice_rhs 2 4 => rw [← tensor_comp, ← tensor_comp, ← hexagon_forward, tensor_comp, tensor_comp]"
},
{
"state_after": "no goals",
"state_before": "C : Type u₁\ninst✝⁸ : Category C\ninst✝⁷ : MonoidalCategory C\ninst✝⁶ : BraidedCategory C\nD : Type u₂\ninst✝⁵ : Category D\ninst✝⁴ : MonoidalCategory D\ninst✝³ : BraidedCategory D\nE : Type u₃\ninst✝² : Category E\ninst✝¹ : MonoidalCategory E\ninst✝ : BraidedCategory E\nW X Y Z : C\n⊢ (𝟙 W ⊗ (β_ X (Y ⊗ Z)).hom) ≫ (𝟙 W ⊗ (α_ Y Z X).hom) ≫ (α_ W Y (Z ⊗ X)).inv ≫ ((β_ W Y).hom ⊗ 𝟙 (Z ⊗ X)) =\n (𝟙 W ⊗ (α_ X Y Z).inv) ≫\n (((𝟙 W ⊗ (α_ X Y Z).hom) ≫ (𝟙 W ⊗ (β_ X (Y ⊗ Z)).hom) ≫ (𝟙 W ⊗ (α_ Y Z X).hom)) ≫ (α_ W Y (Z ⊗ X)).inv) ≫\n ((β_ W Y).hom ⊗ 𝟙 (Z ⊗ X))",
"tactic": "simp"
}
] |
[
618,
7
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
603,
1
] |
Mathlib/Analysis/InnerProductSpace/Basic.lean
|
inner_eq_sum_norm_sq_div_four
|
[
{
"state_after": "𝕜 : Type u_1\nE : Type u_2\nF : Type ?u.2659185\ninst✝⁴ : IsROrC 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace 𝕜 E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : InnerProductSpace ℝ F\ndec_E : DecidableEq E\nx y : E\n⊢ ↑((‖x + y‖ * ‖x + y‖ - ‖x - y‖ * ‖x - y‖) / 4) + ↑((‖x - I • y‖ * ‖x - I • y‖ - ‖x + I • y‖ * ‖x + I • y‖) / 4) * I =\n (↑‖x + y‖ ^ 2 - ↑‖x - y‖ ^ 2 + (↑‖x - I • y‖ ^ 2 - ↑‖x + I • y‖ ^ 2) * I) / 4",
"state_before": "𝕜 : Type u_1\nE : Type u_2\nF : Type ?u.2659185\ninst✝⁴ : IsROrC 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace 𝕜 E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : InnerProductSpace ℝ F\ndec_E : DecidableEq E\nx y : E\n⊢ inner x y = (↑‖x + y‖ ^ 2 - ↑‖x - y‖ ^ 2 + (↑‖x - I • y‖ ^ 2 - ↑‖x + I • y‖ ^ 2) * I) / 4",
"tactic": "rw [← re_add_im ⟪x, y⟫, re_inner_eq_norm_add_mul_self_sub_norm_sub_mul_self_div_four,\n im_inner_eq_norm_sub_i_smul_mul_self_sub_norm_add_i_smul_mul_self_div_four]"
},
{
"state_after": "𝕜 : Type u_1\nE : Type u_2\nF : Type ?u.2659185\ninst✝⁴ : IsROrC 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace 𝕜 E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : InnerProductSpace ℝ F\ndec_E : DecidableEq E\nx y : E\n⊢ (↑‖x + y‖ * ↑‖x + y‖ - ↑‖x - y‖ * ↑‖x - y‖) / 4 +\n (↑‖x - I • y‖ * ↑‖x - I • y‖ - ↑‖x + I • y‖ * ↑‖x + I • y‖) / 4 * I =\n (↑‖x + y‖ ^ 2 - ↑‖x - y‖ ^ 2 + (↑‖x - I • y‖ ^ 2 - ↑‖x + I • y‖ ^ 2) * I) / 4",
"state_before": "𝕜 : Type u_1\nE : Type u_2\nF : Type ?u.2659185\ninst✝⁴ : IsROrC 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace 𝕜 E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : InnerProductSpace ℝ F\ndec_E : DecidableEq E\nx y : E\n⊢ ↑((‖x + y‖ * ‖x + y‖ - ‖x - y‖ * ‖x - y‖) / 4) + ↑((‖x - I • y‖ * ‖x - I • y‖ - ‖x + I • y‖ * ‖x + I • y‖) / 4) * I =\n (↑‖x + y‖ ^ 2 - ↑‖x - y‖ ^ 2 + (↑‖x - I • y‖ ^ 2 - ↑‖x + I • y‖ ^ 2) * I) / 4",
"tactic": "push_cast"
},
{
"state_after": "no goals",
"state_before": "𝕜 : Type u_1\nE : Type u_2\nF : Type ?u.2659185\ninst✝⁴ : IsROrC 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace 𝕜 E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : InnerProductSpace ℝ F\ndec_E : DecidableEq E\nx y : E\n⊢ (↑‖x + y‖ * ↑‖x + y‖ - ↑‖x - y‖ * ↑‖x - y‖) / 4 +\n (↑‖x - I • y‖ * ↑‖x - I • y‖ - ↑‖x + I • y‖ * ↑‖x + I • y‖) / 4 * I =\n (↑‖x + y‖ ^ 2 - ↑‖x - y‖ ^ 2 + (↑‖x - I • y‖ ^ 2 - ↑‖x + I • y‖ ^ 2) * I) / 4",
"tactic": "simp only [sq, ← mul_div_right_comm, ← add_div]"
}
] |
[
1171,
50
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1165,
1
] |
Mathlib/Order/ModularLattice.lean
|
Disjoint.disjoint_sup_left_of_disjoint_sup_right
|
[
{
"state_after": "α : Type u_1\ninst✝² : Lattice α\ninst✝¹ : OrderBot α\ninst✝ : IsModularLattice α\na b c : α\nh : Disjoint b c\nhsup : Disjoint a (b ⊔ c)\n⊢ Disjoint c (b ⊔ a)",
"state_before": "α : Type u_1\ninst✝² : Lattice α\ninst✝¹ : OrderBot α\ninst✝ : IsModularLattice α\na b c : α\nh : Disjoint b c\nhsup : Disjoint a (b ⊔ c)\n⊢ Disjoint (a ⊔ b) c",
"tactic": "rw [disjoint_comm, sup_comm]"
},
{
"state_after": "α : Type u_1\ninst✝² : Lattice α\ninst✝¹ : OrderBot α\ninst✝ : IsModularLattice α\na b c : α\nh : Disjoint b c\nhsup : Disjoint a (b ⊔ c)\n⊢ Disjoint (c ⊔ b) a",
"state_before": "α : Type u_1\ninst✝² : Lattice α\ninst✝¹ : OrderBot α\ninst✝ : IsModularLattice α\na b c : α\nh : Disjoint b c\nhsup : Disjoint a (b ⊔ c)\n⊢ Disjoint c (b ⊔ a)",
"tactic": "apply Disjoint.disjoint_sup_right_of_disjoint_sup_left h.symm"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝² : Lattice α\ninst✝¹ : OrderBot α\ninst✝ : IsModularLattice α\na b c : α\nh : Disjoint b c\nhsup : Disjoint a (b ⊔ c)\n⊢ Disjoint (c ⊔ b) a",
"tactic": "rwa [sup_comm, disjoint_comm] at hsup"
}
] |
[
408,
40
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
403,
1
] |
Mathlib/Logic/Equiv/Defs.lean
|
Equiv.eq_symm_apply
|
[] |
[
335,
48
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
334,
1
] |
Mathlib/Data/List/Basic.lean
|
List.intersperse_cons_cons
|
[] |
[
2848,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2846,
1
] |
Mathlib/Algebra/CharP/Two.lean
|
CharTwo.list_sum_mul_self
|
[
{
"state_after": "no goals",
"state_before": "R : Type u_1\nι : Type ?u.14336\ninst✝¹ : CommSemiring R\ninst✝ : CharP R 2\nl : List R\n⊢ List.sum l * List.sum l = List.sum (List.map (fun x => x * x) l)",
"tactic": "simp_rw [← pow_two, list_sum_sq]"
}
] |
[
104,
35
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
103,
1
] |
Mathlib/Data/Set/Intervals/Monoid.lean
|
Set.image_add_const_Ioo
|
[] |
[
107,
31
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
106,
1
] |
Mathlib/Data/Set/Image.lean
|
Set.preimage_setOf_eq
|
[] |
[
116,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
115,
1
] |
Mathlib/Analysis/BoxIntegral/Partition/Measure.lean
|
BoxIntegral.Box.measurableSet_coe
|
[
{
"state_after": "ι : Type u_1\nI : Box ι\ninst✝ : Countable ι\n⊢ MeasurableSet (pi univ fun i => Ioc (lower I i) (upper I i))",
"state_before": "ι : Type u_1\nI : Box ι\ninst✝ : Countable ι\n⊢ MeasurableSet ↑I",
"tactic": "rw [coe_eq_pi]"
},
{
"state_after": "no goals",
"state_before": "ι : Type u_1\nI : Box ι\ninst✝ : Countable ι\n⊢ MeasurableSet (pi univ fun i => Ioc (lower I i) (upper I i))",
"tactic": "exact MeasurableSet.univ_pi fun i => measurableSet_Ioc"
}
] |
[
62,
57
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
60,
1
] |
Mathlib/Data/Polynomial/RingDivision.lean
|
Polynomial.Monic.irreducible_of_degree_eq_one
|
[] |
[
433,
46
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
432,
1
] |
Mathlib/LinearAlgebra/Matrix/SesquilinearForm.lean
|
Matrix.toMatrix₂Aux_toLinearMap₂'Aux
|
[
{
"state_after": "case a.h\nR : Type u_3\nR₁ : Type u_4\nR₂ : Type u_5\nM : Type ?u.137024\nM₁ : Type ?u.137027\nM₂ : Type ?u.137030\nM₁' : Type ?u.137033\nM₂' : Type ?u.137036\nn : Type u_1\nm : Type u_2\nn' : Type ?u.137045\nm' : Type ?u.137048\nι : Type ?u.137051\ninst✝¹⁰ : CommRing R\ninst✝⁹ : CommRing R₁\ninst✝⁸ : CommRing R₂\ninst✝⁷ : AddCommMonoid M₁\ninst✝⁶ : Module R₁ M₁\ninst✝⁵ : AddCommMonoid M₂\ninst✝⁴ : Module R₂ M₂\ninst✝³ : Fintype n\ninst✝² : Fintype m\ninst✝¹ : DecidableEq n\ninst✝ : DecidableEq m\nσ₁ : R₁ →+* R\nσ₂ : R₂ →+* R\nf : Matrix n m R\ni : n\nj : m\n⊢ ↑(toMatrix₂Aux (fun i => ↑(LinearMap.stdBasis R₁ (fun x => R₁) i) 1) fun j =>\n ↑(LinearMap.stdBasis R₂ (fun x => R₂) j) 1)\n (toLinearMap₂'Aux σ₁ σ₂ f) i j =\n f i j",
"state_before": "R : Type u_3\nR₁ : Type u_4\nR₂ : Type u_5\nM : Type ?u.137024\nM₁ : Type ?u.137027\nM₂ : Type ?u.137030\nM₁' : Type ?u.137033\nM₂' : Type ?u.137036\nn : Type u_1\nm : Type u_2\nn' : Type ?u.137045\nm' : Type ?u.137048\nι : Type ?u.137051\ninst✝¹⁰ : CommRing R\ninst✝⁹ : CommRing R₁\ninst✝⁸ : CommRing R₂\ninst✝⁷ : AddCommMonoid M₁\ninst✝⁶ : Module R₁ M₁\ninst✝⁵ : AddCommMonoid M₂\ninst✝⁴ : Module R₂ M₂\ninst✝³ : Fintype n\ninst✝² : Fintype m\ninst✝¹ : DecidableEq n\ninst✝ : DecidableEq m\nσ₁ : R₁ →+* R\nσ₂ : R₂ →+* R\nf : Matrix n m R\n⊢ ↑(toMatrix₂Aux (fun i => ↑(LinearMap.stdBasis R₁ (fun x => R₁) i) 1) fun j =>\n ↑(LinearMap.stdBasis R₂ (fun x => R₂) j) 1)\n (toLinearMap₂'Aux σ₁ σ₂ f) =\n f",
"tactic": "ext (i j)"
},
{
"state_after": "no goals",
"state_before": "case a.h\nR : Type u_3\nR₁ : Type u_4\nR₂ : Type u_5\nM : Type ?u.137024\nM₁ : Type ?u.137027\nM₂ : Type ?u.137030\nM₁' : Type ?u.137033\nM₂' : Type ?u.137036\nn : Type u_1\nm : Type u_2\nn' : Type ?u.137045\nm' : Type ?u.137048\nι : Type ?u.137051\ninst✝¹⁰ : CommRing R\ninst✝⁹ : CommRing R₁\ninst✝⁸ : CommRing R₂\ninst✝⁷ : AddCommMonoid M₁\ninst✝⁶ : Module R₁ M₁\ninst✝⁵ : AddCommMonoid M₂\ninst✝⁴ : Module R₂ M₂\ninst✝³ : Fintype n\ninst✝² : Fintype m\ninst✝¹ : DecidableEq n\ninst✝ : DecidableEq m\nσ₁ : R₁ →+* R\nσ₂ : R₂ →+* R\nf : Matrix n m R\ni : n\nj : m\n⊢ ↑(toMatrix₂Aux (fun i => ↑(LinearMap.stdBasis R₁ (fun x => R₁) i) 1) fun j =>\n ↑(LinearMap.stdBasis R₂ (fun x => R₂) j) 1)\n (toLinearMap₂'Aux σ₁ σ₂ f) i j =\n f i j",
"tactic": "simp_rw [LinearMap.toMatrix₂Aux_apply, Matrix.toLinearMap₂'Aux_stdBasis]"
}
] |
[
140,
75
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
135,
1
] |
Mathlib/GroupTheory/OrderOfElement.lean
|
orderOf_eq_one_iff
|
[
{
"state_after": "no goals",
"state_before": "G : Type u\nA : Type v\nx y : G\na b : A\nn m : ℕ\ninst✝¹ : Monoid G\ninst✝ : AddMonoid A\n⊢ orderOf x = 1 ↔ x = 1",
"tactic": "rw [orderOf, minimalPeriod_eq_one_iff_isFixedPt, IsFixedPt, mul_one]"
}
] |
[
227,
71
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
226,
1
] |
Mathlib/MeasureTheory/Measure/MeasureSpace.lean
|
MeasureTheory.union_ae_eq_left_iff_ae_subset
|
[
{
"state_after": "α : Type u_1\nβ : Type ?u.30507\nγ : Type ?u.30510\nδ : Type ?u.30513\nι : Type ?u.30516\nR : Type ?u.30519\nR' : Type ?u.30522\nm : MeasurableSpace α\nμ μ₁ μ₂ : Measure α\ns s₁ s₂ t : Set α\n⊢ s ∪ t =ᵐ[μ] s ↔ ↑↑μ (t \\ s) = 0",
"state_before": "α : Type u_1\nβ : Type ?u.30507\nγ : Type ?u.30510\nδ : Type ?u.30513\nι : Type ?u.30516\nR : Type ?u.30519\nR' : Type ?u.30522\nm : MeasurableSpace α\nμ μ₁ μ₂ : Measure α\ns s₁ s₂ t : Set α\n⊢ s ∪ t =ᵐ[μ] s ↔ t ≤ᵐ[μ] s",
"tactic": "rw [ae_le_set]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.30507\nγ : Type ?u.30510\nδ : Type ?u.30513\nι : Type ?u.30516\nR : Type ?u.30519\nR' : Type ?u.30522\nm : MeasurableSpace α\nμ μ₁ μ₂ : Measure α\ns s₁ s₂ t : Set α\n⊢ s ∪ t =ᵐ[μ] s ↔ ↑↑μ (t \\ s) = 0",
"tactic": "refine'\n ⟨fun h => by simpa only [union_diff_left] using (ae_eq_set.mp h).1, fun h =>\n eventuallyLE_antisymm_iff.mpr\n ⟨by rwa [ae_le_set, union_diff_left],\n HasSubset.Subset.eventuallyLE <| subset_union_left s t⟩⟩"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.30507\nγ : Type ?u.30510\nδ : Type ?u.30513\nι : Type ?u.30516\nR : Type ?u.30519\nR' : Type ?u.30522\nm : MeasurableSpace α\nμ μ₁ μ₂ : Measure α\ns s₁ s₂ t : Set α\nh : s ∪ t =ᵐ[μ] s\n⊢ ↑↑μ (t \\ s) = 0",
"tactic": "simpa only [union_diff_left] using (ae_eq_set.mp h).1"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.30507\nγ : Type ?u.30510\nδ : Type ?u.30513\nι : Type ?u.30516\nR : Type ?u.30519\nR' : Type ?u.30522\nm : MeasurableSpace α\nμ μ₁ μ₂ : Measure α\ns s₁ s₂ t : Set α\nh : ↑↑μ (t \\ s) = 0\n⊢ s ∪ t ≤ᵐ[μ] s",
"tactic": "rwa [ae_le_set, union_diff_left]"
}
] |
[
303,
67
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
297,
1
] |
Mathlib/Computability/TMToPartrec.lean
|
Turing.PartrecToTM2.trNat_zero
|
[
{
"state_after": "⊢ trNum 0 = []",
"state_before": "⊢ trNat 0 = []",
"tactic": "rw [trNat, Nat.cast_zero]"
},
{
"state_after": "no goals",
"state_before": "⊢ trNum 0 = []",
"tactic": "rfl"
}
] |
[
1202,
71
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1202,
1
] |
Mathlib/NumberTheory/Padics/PadicVal.lean
|
padicValInt_dvd_iff
|
[
{
"state_after": "no goals",
"state_before": "p : ℕ\nhp : Fact (Nat.Prime p)\nn : ℕ\na : ℤ\n⊢ ↑p ^ n ∣ a ↔ a = 0 ∨ n ≤ padicValInt p a",
"tactic": "rw [padicValInt, ← Int.natAbs_eq_zero, ← padicValNat_dvd_iff, ← Int.coe_nat_dvd_left,\n Int.coe_nat_pow]"
}
] |
[
518,
21
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
516,
1
] |
Mathlib/Data/Finset/Basic.lean
|
Finset.toList_toFinset
|
[
{
"state_after": "case a\nα : Type u_1\nβ : Type ?u.480823\nγ : Type ?u.480826\ninst✝ : DecidableEq α\ns : Finset α\na✝ : α\n⊢ a✝ ∈ List.toFinset (toList s) ↔ a✝ ∈ s",
"state_before": "α : Type u_1\nβ : Type ?u.480823\nγ : Type ?u.480826\ninst✝ : DecidableEq α\ns : Finset α\n⊢ List.toFinset (toList s) = s",
"tactic": "ext"
},
{
"state_after": "no goals",
"state_before": "case a\nα : Type u_1\nβ : Type ?u.480823\nγ : Type ?u.480826\ninst✝ : DecidableEq α\ns : Finset α\na✝ : α\n⊢ a✝ ∈ List.toFinset (toList s) ↔ a✝ ∈ s",
"tactic": "simp"
}
] |
[
3389,
7
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
3387,
1
] |
Mathlib/SetTheory/Ordinal/FixedPoint.lean
|
Ordinal.nfp_id
|
[
{
"state_after": "f : Ordinal → Ordinal\na : Ordinal\n⊢ (sup fun n => id a) = id a",
"state_before": "f : Ordinal → Ordinal\na : Ordinal\n⊢ nfp id a = id a",
"tactic": "simp_rw [← sup_iterate_eq_nfp, iterate_id]"
},
{
"state_after": "no goals",
"state_before": "f : Ordinal → Ordinal\na : Ordinal\n⊢ (sup fun n => id a) = id a",
"tactic": "exact sup_const a"
}
] |
[
465,
22
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
462,
1
] |
Mathlib/Order/SymmDiff.lean
|
symmDiff_eq_Xor'
|
[] |
[
91,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
90,
1
] |
Mathlib/RingTheory/QuotientNilpotent.lean
|
Ideal.IsNilpotent.induction_on
|
[
{
"state_after": "case intro\nR : Type ?u.1313\nS : Type u_1\ninst✝² : CommSemiring R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nI : Ideal S\nP : ⦃S : Type u_1⦄ → [inst : CommRing S] → Ideal S → Prop\nh₁ : ∀ ⦃S : Type u_1⦄ [inst : CommRing S] (I : Ideal S), I ^ 2 = ⊥ → P I\nh₂ : ∀ ⦃S : Type u_1⦄ [inst : CommRing S] (I J : Ideal S), I ≤ J → P I → P (map (Quotient.mk I) J) → P J\nn : ℕ\nhI : I ^ n = ⊥\n⊢ P I",
"state_before": "R : Type ?u.1313\nS : Type u_1\ninst✝² : CommSemiring R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nI : Ideal S\nhI : IsNilpotent I\nP : ⦃S : Type u_1⦄ → [inst : CommRing S] → Ideal S → Prop\nh₁ : ∀ ⦃S : Type u_1⦄ [inst : CommRing S] (I : Ideal S), I ^ 2 = ⊥ → P I\nh₂ : ∀ ⦃S : Type u_1⦄ [inst : CommRing S] (I J : Ideal S), I ≤ J → P I → P (map (Quotient.mk I) J) → P J\n⊢ P I",
"tactic": "obtain ⟨n, hI : I ^ n = ⊥⟩ := hI"
},
{
"state_after": "case intro.h\nR : Type ?u.1313\nS✝ : Type u_1\ninst✝⁴ : CommSemiring R\ninst✝³ : CommRing S✝\ninst✝² : Algebra R S✝\nI✝ : Ideal S✝\nP : ⦃S : Type u_1⦄ → [inst : CommRing S] → Ideal S → Prop\nh₁ : ∀ ⦃S : Type u_1⦄ [inst : CommRing S] (I : Ideal S), I ^ 2 = ⊥ → P I\nh₂ : ∀ ⦃S : Type u_1⦄ [inst : CommRing S] (I J : Ideal S), I ≤ J → P I → P (map (Quotient.mk I) J) → P J\nn✝ : ℕ\nhI✝ : I✝ ^ n✝ = ⊥\nn : ℕ\nH : ∀ (m : ℕ), m < n → ∀ {S : Type u_1} [inst : CommRing S] [inst_1 : Algebra R S] (I : Ideal S), I ^ m = ⊥ → P I\nS : Type u_1\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nI : Ideal S\nhI : I ^ n = ⊥\n⊢ P I",
"state_before": "case intro\nR : Type ?u.1313\nS : Type u_1\ninst✝² : CommSemiring R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nI : Ideal S\nP : ⦃S : Type u_1⦄ → [inst : CommRing S] → Ideal S → Prop\nh₁ : ∀ ⦃S : Type u_1⦄ [inst : CommRing S] (I : Ideal S), I ^ 2 = ⊥ → P I\nh₂ : ∀ ⦃S : Type u_1⦄ [inst : CommRing S] (I J : Ideal S), I ≤ J → P I → P (map (Quotient.mk I) J) → P J\nn : ℕ\nhI : I ^ n = ⊥\n⊢ P I",
"tactic": "induction' n using Nat.strong_induction_on with n H generalizing S"
},
{
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{
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{
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{
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{
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"tactic": "rw [← Ideal.zero_eq_bot, zero_pow]"
},
{
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"tactic": "exact zero_lt_two"
},
{
"state_after": "case neg.zero\nR : Type ?u.1313\nS✝ : Type u_1\ninst✝⁴ : CommSemiring R\ninst✝³ : CommRing S✝\ninst✝² : Algebra R S✝\nI✝ : Ideal S✝\nP : ⦃S : Type u_1⦄ → [inst : CommRing S] → Ideal S → Prop\nh₁ : ∀ ⦃S : Type u_1⦄ [inst : CommRing S] (I : Ideal S), I ^ 2 = ⊥ → P I\nh₂ : ∀ ⦃S : Type u_1⦄ [inst : CommRing S] (I J : Ideal S), I ≤ J → P I → P (map (Quotient.mk I) J) → P J\nn : ℕ\nhI✝ : I✝ ^ n = ⊥\nS : Type u_1\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nI : Ideal S\nhI' : ¬I = ⊥\nH : ∀ (m : ℕ), m < Nat.zero → ∀ {S : Type u_1} [inst : CommRing S] [inst_1 : Algebra R S] (I : Ideal S), I ^ m = ⊥ → P I\nhI : ⊤ = ⊥\n⊢ P I",
"state_before": "case neg.zero\nR : Type ?u.1313\nS✝ : Type u_1\ninst✝⁴ : CommSemiring R\ninst✝³ : CommRing S✝\ninst✝² : Algebra R S✝\nI✝ : Ideal S✝\nP : ⦃S : Type u_1⦄ → [inst : CommRing S] → Ideal S → Prop\nh₁ : ∀ ⦃S : Type u_1⦄ [inst : CommRing S] (I : Ideal S), I ^ 2 = ⊥ → P I\nh₂ : ∀ ⦃S : Type u_1⦄ [inst : CommRing S] (I J : Ideal S), I ≤ J → P I → P (map (Quotient.mk I) J) → P J\nn : ℕ\nhI✝ : I✝ ^ n = ⊥\nS : Type u_1\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nI : Ideal S\nhI' : ¬I = ⊥\nH : ∀ (m : ℕ), m < Nat.zero → ∀ {S : Type u_1} [inst : CommRing S] [inst_1 : Algebra R S] (I : Ideal S), I ^ m = ⊥ → P I\nhI : I ^ Nat.zero = ⊥\n⊢ P I",
"tactic": "rw [pow_zero, Ideal.one_eq_top] at hI"
},
{
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"tactic": "haveI := subsingleton_of_bot_eq_top hI.symm"
},
{
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"tactic": "exact (hI' (Subsingleton.elim _ _)).elim"
},
{
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"tactic": "rw [pow_one] at hI"
},
{
"state_after": "no goals",
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"tactic": "exact (hI' hI).elim"
},
{
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"tactic": "apply H n.succ _ (I ^ 2)"
},
{
"state_after": "case neg.succ.succ.a\nR : Type ?u.1313\nS✝ : Type u_1\ninst✝⁴ : CommSemiring R\ninst✝³ : CommRing S✝\ninst✝² : Algebra R S✝\nI✝ : Ideal S✝\nP : ⦃S : Type u_1⦄ → [inst : CommRing S] → Ideal S → Prop\nh₁ : ∀ ⦃S : Type u_1⦄ [inst : CommRing S] (I : Ideal S), I ^ 2 = ⊥ → P I\nh₂ : ∀ ⦃S : Type u_1⦄ [inst : CommRing S] (I J : Ideal S), I ≤ J → P I → P (map (Quotient.mk I) J) → P J\nn✝ : ℕ\nhI✝ : I✝ ^ n✝ = ⊥\nS : Type u_1\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nI : Ideal S\nhI' : ¬I = ⊥\nn : ℕ\nH :\n ∀ (m : ℕ),\n m < Nat.succ (Nat.succ n) →\n ∀ {S : Type u_1} [inst : CommRing S] [inst_1 : Algebra R S] (I : Ideal S), I ^ m = ⊥ → P I\nhI : I ^ Nat.succ (Nat.succ n) = ⊥\n⊢ I ^ (2 * (n + 1)) ≤ I ^ (n + 1 + 1)",
"state_before": "case neg.succ.succ.a\nR : Type ?u.1313\nS✝ : Type u_1\ninst✝⁴ : CommSemiring R\ninst✝³ : CommRing S✝\ninst✝² : Algebra R S✝\nI✝ : Ideal S✝\nP : ⦃S : Type u_1⦄ → [inst : CommRing S] → Ideal S → Prop\nh₁ : ∀ ⦃S : Type u_1⦄ [inst : CommRing S] (I : Ideal S), I ^ 2 = ⊥ → P I\nh₂ : ∀ ⦃S : Type u_1⦄ [inst : CommRing S] (I J : Ideal S), I ≤ J → P I → P (map (Quotient.mk I) J) → P J\nn✝ : ℕ\nhI✝ : I✝ ^ n✝ = ⊥\nS : Type u_1\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nI : Ideal S\nhI' : ¬I = ⊥\nn : ℕ\nH :\n ∀ (m : ℕ),\n m < Nat.succ (Nat.succ n) →\n ∀ {S : Type u_1} [inst : CommRing S] [inst_1 : Algebra R S] (I : Ideal S), I ^ m = ⊥ → P I\nhI : I ^ Nat.succ (Nat.succ n) = ⊥\n⊢ (I ^ 2) ^ Nat.succ n = ⊥",
"tactic": "rw [← pow_mul, eq_bot_iff, ← hI, Nat.succ_eq_add_one, Nat.succ_eq_add_one]"
},
{
"state_after": "no goals",
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"tactic": "apply Ideal.pow_le_pow (by linarith)"
},
{
"state_after": "no goals",
"state_before": "R : Type ?u.1313\nS✝ : Type u_1\ninst✝⁴ : CommSemiring R\ninst✝³ : CommRing S✝\ninst✝² : Algebra R S✝\nI✝ : Ideal S✝\nP : ⦃S : Type u_1⦄ → [inst : CommRing S] → Ideal S → Prop\nh₁ : ∀ ⦃S : Type u_1⦄ [inst : CommRing S] (I : Ideal S), I ^ 2 = ⊥ → P I\nh₂ : ∀ ⦃S : Type u_1⦄ [inst : CommRing S] (I J : Ideal S), I ≤ J → P I → P (map (Quotient.mk I) J) → P J\nn✝ : ℕ\nhI✝ : I✝ ^ n✝ = ⊥\nS : Type u_1\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nI : Ideal S\nhI' : ¬I = ⊥\nn : ℕ\nH :\n ∀ (m : ℕ),\n m < Nat.succ (Nat.succ n) →\n ∀ {S : Type u_1} [inst : CommRing S] [inst_1 : Algebra R S] (I : Ideal S), I ^ m = ⊥ → P I\nhI : I ^ Nat.succ (Nat.succ n) = ⊥\n⊢ n + 1 + 1 ≤ 2 * (n + 1)",
"tactic": "linarith"
},
{
"state_after": "no goals",
"state_before": "R : Type ?u.1313\nS✝ : Type u_1\ninst✝⁴ : CommSemiring R\ninst✝³ : CommRing S✝\ninst✝² : Algebra R S✝\nI✝ : Ideal S✝\nP : ⦃S : Type u_1⦄ → [inst : CommRing S] → Ideal S → Prop\nh₁ : ∀ ⦃S : Type u_1⦄ [inst : CommRing S] (I : Ideal S), I ^ 2 = ⊥ → P I\nh₂ : ∀ ⦃S : Type u_1⦄ [inst : CommRing S] (I J : Ideal S), I ≤ J → P I → P (map (Quotient.mk I) J) → P J\nn✝ : ℕ\nhI✝ : I✝ ^ n✝ = ⊥\nS : Type u_1\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nI : Ideal S\nhI' : ¬I = ⊥\nn : ℕ\nH :\n ∀ (m : ℕ),\n m < Nat.succ (Nat.succ n) →\n ∀ {S : Type u_1} [inst : CommRing S] [inst_1 : Algebra R S] (I : Ideal S), I ^ m = ⊥ → P I\nhI : I ^ Nat.succ (Nat.succ n) = ⊥\n⊢ Nat.succ n < Nat.succ (Nat.succ n)",
"tactic": "exact n.succ.lt_succ_self"
},
{
"state_after": "case neg.succ.succ.a.a\nR : Type ?u.1313\nS✝ : Type u_1\ninst✝⁴ : CommSemiring R\ninst✝³ : CommRing S✝\ninst✝² : Algebra R S✝\nI✝ : Ideal S✝\nP : ⦃S : Type u_1⦄ → [inst : CommRing S] → Ideal S → Prop\nh₁ : ∀ ⦃S : Type u_1⦄ [inst : CommRing S] (I : Ideal S), I ^ 2 = ⊥ → P I\nh₂ : ∀ ⦃S : Type u_1⦄ [inst : CommRing S] (I J : Ideal S), I ≤ J → P I → P (map (Quotient.mk I) J) → P J\nn✝ : ℕ\nhI✝ : I✝ ^ n✝ = ⊥\nS : Type u_1\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nI : Ideal S\nhI' : ¬I = ⊥\nn : ℕ\nH :\n ∀ (m : ℕ),\n m < Nat.succ (Nat.succ n) →\n ∀ {S : Type u_1} [inst : CommRing S] [inst_1 : Algebra R S] (I : Ideal S), I ^ m = ⊥ → P I\nhI : I ^ Nat.succ (Nat.succ n) = ⊥\n⊢ map (Quotient.mk (I ^ 2)) I ^ 2 = ⊥",
"state_before": "case neg.succ.succ.a\nR : Type ?u.1313\nS✝ : Type u_1\ninst✝⁴ : CommSemiring R\ninst✝³ : CommRing S✝\ninst✝² : Algebra R S✝\nI✝ : Ideal S✝\nP : ⦃S : Type u_1⦄ → [inst : CommRing S] → Ideal S → Prop\nh₁ : ∀ ⦃S : Type u_1⦄ [inst : CommRing S] (I : Ideal S), I ^ 2 = ⊥ → P I\nh₂ : ∀ ⦃S : Type u_1⦄ [inst : CommRing S] (I J : Ideal S), I ≤ J → P I → P (map (Quotient.mk I) J) → P J\nn✝ : ℕ\nhI✝ : I✝ ^ n✝ = ⊥\nS : Type u_1\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nI : Ideal S\nhI' : ¬I = ⊥\nn : ℕ\nH :\n ∀ (m : ℕ),\n m < Nat.succ (Nat.succ n) →\n ∀ {S : Type u_1} [inst : CommRing S] [inst_1 : Algebra R S] (I : Ideal S), I ^ m = ⊥ → P I\nhI : I ^ Nat.succ (Nat.succ n) = ⊥\n⊢ P (map (Quotient.mk (I ^ 2)) I)",
"tactic": "apply h₁"
},
{
"state_after": "no goals",
"state_before": "case neg.succ.succ.a.a\nR : Type ?u.1313\nS✝ : Type u_1\ninst✝⁴ : CommSemiring R\ninst✝³ : CommRing S✝\ninst✝² : Algebra R S✝\nI✝ : Ideal S✝\nP : ⦃S : Type u_1⦄ → [inst : CommRing S] → Ideal S → Prop\nh₁ : ∀ ⦃S : Type u_1⦄ [inst : CommRing S] (I : Ideal S), I ^ 2 = ⊥ → P I\nh₂ : ∀ ⦃S : Type u_1⦄ [inst : CommRing S] (I J : Ideal S), I ≤ J → P I → P (map (Quotient.mk I) J) → P J\nn✝ : ℕ\nhI✝ : I✝ ^ n✝ = ⊥\nS : Type u_1\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nI : Ideal S\nhI' : ¬I = ⊥\nn : ℕ\nH :\n ∀ (m : ℕ),\n m < Nat.succ (Nat.succ n) →\n ∀ {S : Type u_1} [inst : CommRing S] [inst_1 : Algebra R S] (I : Ideal S), I ^ m = ⊥ → P I\nhI : I ^ Nat.succ (Nat.succ n) = ⊥\n⊢ map (Quotient.mk (I ^ 2)) I ^ 2 = ⊥",
"tactic": "rw [← Ideal.map_pow, Ideal.map_quotient_self]"
}
] |
[
55,
50
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
29,
1
] |
Mathlib/MeasureTheory/Group/FundamentalDomain.lean
|
MeasureTheory.IsFundamentalDomain.set_lintegral_eq
|
[
{
"state_after": "no goals",
"state_before": "G : Type u_1\nH : Type ?u.260361\nα : Type u_2\nβ : Type ?u.260367\nE : Type ?u.260370\ninst✝¹⁰ : Group G\ninst✝⁹ : Group H\ninst✝⁸ : MulAction G α\ninst✝⁷ : MeasurableSpace α\ninst✝⁶ : MulAction H β\ninst✝⁵ : MeasurableSpace β\ninst✝⁴ : NormedAddCommGroup E\ns t : Set α\nμ : Measure α\ninst✝³ : MeasurableSpace G\ninst✝² : MeasurableSMul G α\ninst✝¹ : SMulInvariantMeasure G α μ\ninst✝ : Countable G\nν : Measure α\nhs : IsFundamentalDomain G s\nht : IsFundamentalDomain G t\nf : α → ℝ≥0∞\nhf : ∀ (g : G) (x : α), f (g • x) = f x\n⊢ (∑' (g : G), ∫⁻ (x : α) in s ∩ g • t, f x ∂μ) = ∑' (g : G), ∫⁻ (x : α) in g • t ∩ s, f (g⁻¹ • x) ∂μ",
"tactic": "simp only [hf, inter_comm]"
}
] |
[
335,
66
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
329,
11
] |
Mathlib/Analysis/SpecialFunctions/Trigonometric/Basic.lean
|
Complex.sin_int_mul_two_pi_sub
|
[] |
[
1197,
44
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1196,
1
] |
Mathlib/GroupTheory/GroupAction/SubMulAction.lean
|
SubMulAction.coe_copy
|
[] |
[
148,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
147,
1
] |
Mathlib/CategoryTheory/Preadditive/Generator.lean
|
CategoryTheory.isSeparator_iff_faithful_preadditiveCoyonedaObj
|
[
{
"state_after": "C : Type u\ninst✝¹ : Category C\ninst✝ : Preadditive C\nG : C\n⊢ Faithful (preadditiveCoyonedaObj G.op ⋙ forget₂ (ModuleCat (End G.op)) AddCommGroupCat) ↔\n Faithful (preadditiveCoyonedaObj G.op)",
"state_before": "C : Type u\ninst✝¹ : Category C\ninst✝ : Preadditive C\nG : C\n⊢ IsSeparator G ↔ Faithful (preadditiveCoyonedaObj G.op)",
"tactic": "rw [isSeparator_iff_faithful_preadditiveCoyoneda, preadditiveCoyoneda_obj]"
},
{
"state_after": "no goals",
"state_before": "C : Type u\ninst✝¹ : Category C\ninst✝ : Preadditive C\nG : C\n⊢ Faithful (preadditiveCoyonedaObj G.op ⋙ forget₂ (ModuleCat (End G.op)) AddCommGroupCat) ↔\n Faithful (preadditiveCoyonedaObj G.op)",
"tactic": "exact ⟨fun h => Faithful.of_comp _ (forget₂ _ AddCommGroupCat.{v}), fun h => Faithful.comp _ _⟩"
}
] |
[
67,
98
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
64,
1
] |
Mathlib/Data/Real/Sqrt.lean
|
Real.sqrt_eq_cases
|
[
{
"state_after": "case mp\nx y : ℝ\n⊢ sqrt x = y → y * y = x ∧ 0 ≤ y ∨ x < 0 ∧ y = 0\n\ncase mpr\nx y : ℝ\n⊢ y * y = x ∧ 0 ≤ y ∨ x < 0 ∧ y = 0 → sqrt x = y",
"state_before": "x y : ℝ\n⊢ sqrt x = y ↔ y * y = x ∧ 0 ≤ y ∨ x < 0 ∧ y = 0",
"tactic": "constructor"
},
{
"state_after": "case mp\nx : ℝ\n⊢ sqrt x * sqrt x = x ∧ 0 ≤ sqrt x ∨ x < 0 ∧ sqrt x = 0",
"state_before": "case mp\nx y : ℝ\n⊢ sqrt x = y → y * y = x ∧ 0 ≤ y ∨ x < 0 ∧ y = 0",
"tactic": "rintro rfl"
},
{
"state_after": "case mp.inl\nx : ℝ\nhle : 0 ≤ x\n⊢ sqrt x * sqrt x = x ∧ 0 ≤ sqrt x ∨ x < 0 ∧ sqrt x = 0\n\ncase mp.inr\nx : ℝ\nhlt : x < 0\n⊢ sqrt x * sqrt x = x ∧ 0 ≤ sqrt x ∨ x < 0 ∧ sqrt x = 0",
"state_before": "case mp\nx : ℝ\n⊢ sqrt x * sqrt x = x ∧ 0 ≤ sqrt x ∨ x < 0 ∧ sqrt x = 0",
"tactic": "cases' le_or_lt 0 x with hle hlt"
},
{
"state_after": "no goals",
"state_before": "case mp.inl\nx : ℝ\nhle : 0 ≤ x\n⊢ sqrt x * sqrt x = x ∧ 0 ≤ sqrt x ∨ x < 0 ∧ sqrt x = 0",
"tactic": "exact Or.inl ⟨mul_self_sqrt hle, sqrt_nonneg x⟩"
},
{
"state_after": "no goals",
"state_before": "case mp.inr\nx : ℝ\nhlt : x < 0\n⊢ sqrt x * sqrt x = x ∧ 0 ≤ sqrt x ∨ x < 0 ∧ sqrt x = 0",
"tactic": "exact Or.inr ⟨hlt, sqrt_eq_zero_of_nonpos hlt.le⟩"
},
{
"state_after": "case mpr.inl.intro\ny : ℝ\nhy : 0 ≤ y\n⊢ sqrt (y * y) = y\n\ncase mpr.inr.intro\nx : ℝ\nhx : x < 0\n⊢ sqrt x = 0",
"state_before": "case mpr\nx y : ℝ\n⊢ y * y = x ∧ 0 ≤ y ∨ x < 0 ∧ y = 0 → sqrt x = y",
"tactic": "rintro (⟨rfl, hy⟩ | ⟨hx, rfl⟩)"
},
{
"state_after": "no goals",
"state_before": "case mpr.inl.intro\ny : ℝ\nhy : 0 ≤ y\n⊢ sqrt (y * y) = y\n\ncase mpr.inr.intro\nx : ℝ\nhx : x < 0\n⊢ sqrt x = 0",
"tactic": "exacts [sqrt_mul_self hy, sqrt_eq_zero_of_nonpos hx.le]"
}
] |
[
216,
60
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
209,
1
] |
Mathlib/Order/Disjoint.lean
|
Disjoint.le_bot
|
[] |
[
133,
25
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
132,
1
] |
Mathlib/MeasureTheory/Measure/MeasureSpace.lean
|
MeasureTheory.Measure.exists_eq_disjoint_finiteSpanningSetsIn
|
[] |
[
4092,
27
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
4086,
1
] |
Mathlib/Dynamics/Circle/RotationNumber/TranslationNumber.lean
|
CircleDeg1Lift.translationNumber_pow
|
[
{
"state_after": "no goals",
"state_before": "f g : CircleDeg1Lift\n⊢ τ (f ^ 0) = ↑0 * τ f",
"tactic": "simp"
},
{
"state_after": "no goals",
"state_before": "f g : CircleDeg1Lift\nn : ℕ\n⊢ τ (f ^ (n + 1)) = ↑(n + 1) * τ f",
"tactic": "rw [pow_succ', translationNumber_mul_of_commute (Commute.pow_self f n),\n translationNumber_pow n, Nat.cast_add_one, add_mul, one_mul]"
}
] |
[
749,
67
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
745,
1
] |
Mathlib/Topology/Inseparable.lean
|
specializes_of_nhdsWithin
|
[] |
[
184,
29
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
179,
1
] |
Mathlib/Algebra/BigOperators/Basic.lean
|
Finset.prod_congr
|
[
{
"state_after": "ι : Type ?u.280993\nβ : Type u\nα : Type v\nγ : Type w\ns s₁ s₂ : Finset α\na : α\nf g : α → β\ninst✝ : CommMonoid β\nh : s₁ = s₂\n⊢ (∀ (x : α), x ∈ s₂ → f x = g x) → Finset.prod s₂ f = Finset.prod s₂ g",
"state_before": "ι : Type ?u.280993\nβ : Type u\nα : Type v\nγ : Type w\ns s₁ s₂ : Finset α\na : α\nf g : α → β\ninst✝ : CommMonoid β\nh : s₁ = s₂\n⊢ (∀ (x : α), x ∈ s₂ → f x = g x) → Finset.prod s₁ f = Finset.prod s₂ g",
"tactic": "rw [h]"
},
{
"state_after": "no goals",
"state_before": "ι : Type ?u.280993\nβ : Type u\nα : Type v\nγ : Type w\ns s₁ s₂ : Finset α\na : α\nf g : α → β\ninst✝ : CommMonoid β\nh : s₁ = s₂\n⊢ (∀ (x : α), x ∈ s₂ → f x = g x) → Finset.prod s₂ f = Finset.prod s₂ g",
"tactic": "exact fold_congr"
}
] |
[
380,
27
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
379,
1
] |
Mathlib/Topology/ContinuousOn.lean
|
nhdsWithin_inter
|
[
{
"state_after": "α : Type u_1\nβ : Type ?u.17060\nγ : Type ?u.17063\nδ : Type ?u.17066\ninst✝ : TopologicalSpace α\na : α\ns t : Set α\n⊢ 𝓝 a ⊓ 𝓟 (s ∩ t) = 𝓝 a ⊓ 𝓟 s ⊓ (𝓝 a ⊓ 𝓟 t)",
"state_before": "α : Type u_1\nβ : Type ?u.17060\nγ : Type ?u.17063\nδ : Type ?u.17066\ninst✝ : TopologicalSpace α\na : α\ns t : Set α\n⊢ 𝓝[s ∩ t] a = 𝓝[s] a ⊓ 𝓝[t] a",
"tactic": "delta nhdsWithin"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.17060\nγ : Type ?u.17063\nδ : Type ?u.17066\ninst✝ : TopologicalSpace α\na : α\ns t : Set α\n⊢ 𝓝 a ⊓ 𝓟 (s ∩ t) = 𝓝 a ⊓ 𝓟 s ⊓ (𝓝 a ⊓ 𝓟 t)",
"tactic": "rw [inf_left_comm, inf_assoc, inf_principal, ← inf_assoc, inf_idem]"
}
] |
[
263,
70
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
261,
1
] |
Mathlib/RingTheory/SimpleModule.lean
|
LinearMap.surjective_or_eq_zero
|
[
{
"state_after": "R : Type u_1\ninst✝⁵ : Ring R\nM : Type u_3\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\nm : Submodule R M\nN : Type u_2\ninst✝² : AddCommGroup N\ninst✝¹ : Module R N\ninst✝ : IsSimpleModule R N\nf : M →ₗ[R] N\n⊢ range f = ⊥ ∨ range f = ⊤",
"state_before": "R : Type u_1\ninst✝⁵ : Ring R\nM : Type u_3\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\nm : Submodule R M\nN : Type u_2\ninst✝² : AddCommGroup N\ninst✝¹ : Module R N\ninst✝ : IsSimpleModule R N\nf : M →ₗ[R] N\n⊢ Function.Surjective ↑f ∨ f = 0",
"tactic": "rw [← range_eq_top, ← range_eq_bot, or_comm]"
},
{
"state_after": "no goals",
"state_before": "R : Type u_1\ninst✝⁵ : Ring R\nM : Type u_3\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\nm : Submodule R M\nN : Type u_2\ninst✝² : AddCommGroup N\ninst✝¹ : Module R N\ninst✝ : IsSimpleModule R N\nf : M →ₗ[R] N\n⊢ range f = ⊥ ∨ range f = ⊤",
"tactic": "apply eq_bot_or_eq_top"
}
] |
[
146,
25
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
143,
1
] |
Mathlib/Analysis/Calculus/Series.lean
|
tendstoUniformlyOn_tsum
|
[
{
"state_after": "α : Type u_1\nβ : Type u_2\n𝕜 : Type ?u.260\nE : Type ?u.263\nF : Type u_3\ninst✝⁴ : IsROrC 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : CompleteSpace F\nu : α → ℝ\nf : α → β → F\nhu : Summable u\ns : Set β\nhfu : ∀ (n : α) (x : β), x ∈ s → ‖f n x‖ ≤ u n\nε : ℝ\nεpos : ε > 0\n⊢ ∀ᶠ (n : Finset α) in atTop, ∀ (x : β), x ∈ s → dist (∑' (n : α), f n x) (∑ n in n, f n x) < ε",
"state_before": "α : Type u_1\nβ : Type u_2\n𝕜 : Type ?u.260\nE : Type ?u.263\nF : Type u_3\ninst✝⁴ : IsROrC 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : CompleteSpace F\nu : α → ℝ\nf : α → β → F\nhu : Summable u\ns : Set β\nhfu : ∀ (n : α) (x : β), x ∈ s → ‖f n x‖ ≤ u n\n⊢ TendstoUniformlyOn (fun t x => ∑ n in t, f n x) (fun x => ∑' (n : α), f n x) atTop s",
"tactic": "refine' tendstoUniformlyOn_iff.2 fun ε εpos => _"
},
{
"state_after": "case h\nα : Type u_1\nβ : Type u_2\n𝕜 : Type ?u.260\nE : Type ?u.263\nF : Type u_3\ninst✝⁴ : IsROrC 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : CompleteSpace F\nu : α → ℝ\nf : α → β → F\nhu : Summable u\ns : Set β\nhfu : ∀ (n : α) (x : β), x ∈ s → ‖f n x‖ ≤ u n\nε : ℝ\nεpos : ε > 0\nt : Finset α\nht : (∑' (b : { x // ¬x ∈ t }), u ↑b) < ε\nx : β\nhx : x ∈ s\n⊢ dist (∑' (n : α), f n x) (∑ n in t, f n x) < ε",
"state_before": "α : Type u_1\nβ : Type u_2\n𝕜 : Type ?u.260\nE : Type ?u.263\nF : Type u_3\ninst✝⁴ : IsROrC 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : CompleteSpace F\nu : α → ℝ\nf : α → β → F\nhu : Summable u\ns : Set β\nhfu : ∀ (n : α) (x : β), x ∈ s → ‖f n x‖ ≤ u n\nε : ℝ\nεpos : ε > 0\n⊢ ∀ᶠ (n : Finset α) in atTop, ∀ (x : β), x ∈ s → dist (∑' (n : α), f n x) (∑ n in n, f n x) < ε",
"tactic": "filter_upwards [(tendsto_order.1 (tendsto_tsum_compl_atTop_zero u)).2 _ εpos]with t ht x hx"
},
{
"state_after": "case h\nα : Type u_1\nβ : Type u_2\n𝕜 : Type ?u.260\nE : Type ?u.263\nF : Type u_3\ninst✝⁴ : IsROrC 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : CompleteSpace F\nu : α → ℝ\nf : α → β → F\nhu : Summable u\ns : Set β\nhfu : ∀ (n : α) (x : β), x ∈ s → ‖f n x‖ ≤ u n\nε : ℝ\nεpos : ε > 0\nt : Finset α\nht : (∑' (b : { x // ¬x ∈ t }), u ↑b) < ε\nx : β\nhx : x ∈ s\nA : Summable fun n => ‖f n x‖\n⊢ dist (∑' (n : α), f n x) (∑ n in t, f n x) < ε",
"state_before": "case h\nα : Type u_1\nβ : Type u_2\n𝕜 : Type ?u.260\nE : Type ?u.263\nF : Type u_3\ninst✝⁴ : IsROrC 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : CompleteSpace F\nu : α → ℝ\nf : α → β → F\nhu : Summable u\ns : Set β\nhfu : ∀ (n : α) (x : β), x ∈ s → ‖f n x‖ ≤ u n\nε : ℝ\nεpos : ε > 0\nt : Finset α\nht : (∑' (b : { x // ¬x ∈ t }), u ↑b) < ε\nx : β\nhx : x ∈ s\n⊢ dist (∑' (n : α), f n x) (∑ n in t, f n x) < ε",
"tactic": "have A : Summable fun n => ‖f n x‖ :=\n summable_of_nonneg_of_le (fun n => norm_nonneg _) (fun n => hfu n x hx) hu"
},
{
"state_after": "case h\nα : Type u_1\nβ : Type u_2\n𝕜 : Type ?u.260\nE : Type ?u.263\nF : Type u_3\ninst✝⁴ : IsROrC 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : CompleteSpace F\nu : α → ℝ\nf : α → β → F\nhu : Summable u\ns : Set β\nhfu : ∀ (n : α) (x : β), x ∈ s → ‖f n x‖ ≤ u n\nε : ℝ\nεpos : ε > 0\nt : Finset α\nht : (∑' (b : { x // ¬x ∈ t }), u ↑b) < ε\nx : β\nhx : x ∈ s\nA : Summable fun n => ‖f n x‖\n⊢ ‖∑' (x_1 : { x // ¬x ∈ t }), f (↑x_1) x‖ < ε",
"state_before": "case h\nα : Type u_1\nβ : Type u_2\n𝕜 : Type ?u.260\nE : Type ?u.263\nF : Type u_3\ninst✝⁴ : IsROrC 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : CompleteSpace F\nu : α → ℝ\nf : α → β → F\nhu : Summable u\ns : Set β\nhfu : ∀ (n : α) (x : β), x ∈ s → ‖f n x‖ ≤ u n\nε : ℝ\nεpos : ε > 0\nt : Finset α\nht : (∑' (b : { x // ¬x ∈ t }), u ↑b) < ε\nx : β\nhx : x ∈ s\nA : Summable fun n => ‖f n x‖\n⊢ dist (∑' (n : α), f n x) (∑ n in t, f n x) < ε",
"tactic": "rw [dist_eq_norm, ← sum_add_tsum_subtype_compl (summable_of_summable_norm A) t, add_sub_cancel']"
},
{
"state_after": "α : Type u_1\nβ : Type u_2\n𝕜 : Type ?u.260\nE : Type ?u.263\nF : Type u_3\ninst✝⁴ : IsROrC 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : CompleteSpace F\nu : α → ℝ\nf : α → β → F\nhu : Summable u\ns : Set β\nhfu : ∀ (n : α) (x : β), x ∈ s → ‖f n x‖ ≤ u n\nε : ℝ\nεpos : ε > 0\nt : Finset α\nht : (∑' (b : { x // ¬x ∈ t }), u ↑b) < ε\nx : β\nhx : x ∈ s\nA : Summable fun n => ‖f n x‖\n⊢ ‖∑' (x_1 : { x // ¬x ∈ t }), f (↑x_1) x‖ ≤ ∑' (b : { x // ¬x ∈ t }), u ↑b",
"state_before": "case h\nα : Type u_1\nβ : Type u_2\n𝕜 : Type ?u.260\nE : Type ?u.263\nF : Type u_3\ninst✝⁴ : IsROrC 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : CompleteSpace F\nu : α → ℝ\nf : α → β → F\nhu : Summable u\ns : Set β\nhfu : ∀ (n : α) (x : β), x ∈ s → ‖f n x‖ ≤ u n\nε : ℝ\nεpos : ε > 0\nt : Finset α\nht : (∑' (b : { x // ¬x ∈ t }), u ↑b) < ε\nx : β\nhx : x ∈ s\nA : Summable fun n => ‖f n x‖\n⊢ ‖∑' (x_1 : { x // ¬x ∈ t }), f (↑x_1) x‖ < ε",
"tactic": "apply lt_of_le_of_lt _ ht"
},
{
"state_after": "α : Type u_1\nβ : Type u_2\n𝕜 : Type ?u.260\nE : Type ?u.263\nF : Type u_3\ninst✝⁴ : IsROrC 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : CompleteSpace F\nu : α → ℝ\nf : α → β → F\nhu : Summable u\ns : Set β\nhfu : ∀ (n : α) (x : β), x ∈ s → ‖f n x‖ ≤ u n\nε : ℝ\nεpos : ε > 0\nt : Finset α\nht : (∑' (b : { x // ¬x ∈ t }), u ↑b) < ε\nx : β\nhx : x ∈ s\nA : Summable fun n => ‖f n x‖\n⊢ (∑' (i : ↑fun x => x ∈ t → False), ‖f (↑i) x‖) ≤ ∑' (b : { x // ¬x ∈ t }), u ↑b",
"state_before": "α : Type u_1\nβ : Type u_2\n𝕜 : Type ?u.260\nE : Type ?u.263\nF : Type u_3\ninst✝⁴ : IsROrC 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : CompleteSpace F\nu : α → ℝ\nf : α → β → F\nhu : Summable u\ns : Set β\nhfu : ∀ (n : α) (x : β), x ∈ s → ‖f n x‖ ≤ u n\nε : ℝ\nεpos : ε > 0\nt : Finset α\nht : (∑' (b : { x // ¬x ∈ t }), u ↑b) < ε\nx : β\nhx : x ∈ s\nA : Summable fun n => ‖f n x‖\n⊢ ‖∑' (x_1 : { x // ¬x ∈ t }), f (↑x_1) x‖ ≤ ∑' (b : { x // ¬x ∈ t }), u ↑b",
"tactic": "apply (norm_tsum_le_tsum_norm (A.subtype _)).trans"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type u_2\n𝕜 : Type ?u.260\nE : Type ?u.263\nF : Type u_3\ninst✝⁴ : IsROrC 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : CompleteSpace F\nu : α → ℝ\nf : α → β → F\nhu : Summable u\ns : Set β\nhfu : ∀ (n : α) (x : β), x ∈ s → ‖f n x‖ ≤ u n\nε : ℝ\nεpos : ε > 0\nt : Finset α\nht : (∑' (b : { x // ¬x ∈ t }), u ↑b) < ε\nx : β\nhx : x ∈ s\nA : Summable fun n => ‖f n x‖\n⊢ (∑' (i : ↑fun x => x ∈ t → False), ‖f (↑i) x‖) ≤ ∑' (b : { x // ¬x ∈ t }), u ↑b",
"tactic": "exact tsum_le_tsum (fun n => hfu _ _ hx) (A.subtype _) (hu.subtype _)"
}
] |
[
53,
72
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
42,
1
] |
Mathlib/Data/Matrix/Basic.lean
|
Matrix.transposeAddEquiv_symm
|
[] |
[
2037,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2036,
1
] |
Mathlib/CategoryTheory/Abelian/Exact.lean
|
CategoryTheory.Abelian.exact_epi_comp_iff
|
[
{
"state_after": "C : Type u₁\ninst✝² : Category C\ninst✝¹ : Abelian C\nX Y Z : C\nf : X ⟶ Y\ng : Y ⟶ Z\nW : C\nh : W ⟶ X\ninst✝ : Epi h\nhfg : Exact (h ≫ f) g\n⊢ Exact f g",
"state_before": "C : Type u₁\ninst✝² : Category C\ninst✝¹ : Abelian C\nX Y Z : C\nf : X ⟶ Y\ng : Y ⟶ Z\nW : C\nh : W ⟶ X\ninst✝ : Epi h\n⊢ Exact (h ≫ f) g ↔ Exact f g",
"tactic": "refine' ⟨fun hfg => _, fun h => exact_epi_comp h⟩"
},
{
"state_after": "C : Type u₁\ninst✝² : Category C\ninst✝¹ : Abelian C\nX Y Z : C\nf : X ⟶ Y\ng : Y ⟶ Z\nW : C\nh : W ⟶ X\ninst✝ : Epi h\nhfg : Exact (h ≫ f) g\nhc : IsColimit\n (CokernelCofork.ofπ (Cofork.π (colimit.cocone (parallelPair (h ≫ f) 0)))\n (_ : f ≫ Cofork.π (colimit.cocone (parallelPair (h ≫ f) 0)) = 0)) :=\n isCokernelOfComp h (h ≫ f) (colimit.isColimit (parallelPair (h ≫ f) 0))\n (_ : f ≫ Cofork.π (colimit.cocone (parallelPair (h ≫ f) 0)) = 0) (_ : h ≫ f = h ≫ f)\n⊢ Exact f g",
"state_before": "C : Type u₁\ninst✝² : Category C\ninst✝¹ : Abelian C\nX Y Z : C\nf : X ⟶ Y\ng : Y ⟶ Z\nW : C\nh : W ⟶ X\ninst✝ : Epi h\nhfg : Exact (h ≫ f) g\n⊢ Exact f g",
"tactic": "let hc := isCokernelOfComp _ _ (colimit.isColimit (parallelPair (h ≫ f) 0))\n (by rw [← cancel_epi h, ← Category.assoc, CokernelCofork.condition, comp_zero]) rfl"
},
{
"state_after": "C : Type u₁\ninst✝² : Category C\ninst✝¹ : Abelian C\nX Y Z : C\nf : X ⟶ Y\ng : Y ⟶ Z\nW : C\nh : W ⟶ X\ninst✝ : Epi h\nhfg : Exact (h ≫ f) g\nhc : IsColimit\n (CokernelCofork.ofπ (Cofork.π (colimit.cocone (parallelPair (h ≫ f) 0)))\n (_ : f ≫ Cofork.π (colimit.cocone (parallelPair (h ≫ f) 0)) = 0)) :=\n isCokernelOfComp h (h ≫ f) (colimit.isColimit (parallelPair (h ≫ f) 0))\n (_ : f ≫ Cofork.π (colimit.cocone (parallelPair (h ≫ f) 0)) = 0) (_ : h ≫ f = h ≫ f)\n⊢ f ≫ g = 0",
"state_before": "C : Type u₁\ninst✝² : Category C\ninst✝¹ : Abelian C\nX Y Z : C\nf : X ⟶ Y\ng : Y ⟶ Z\nW : C\nh : W ⟶ X\ninst✝ : Epi h\nhfg : Exact (h ≫ f) g\nhc : IsColimit\n (CokernelCofork.ofπ (Cofork.π (colimit.cocone (parallelPair (h ≫ f) 0)))\n (_ : f ≫ Cofork.π (colimit.cocone (parallelPair (h ≫ f) 0)) = 0)) :=\n isCokernelOfComp h (h ≫ f) (colimit.isColimit (parallelPair (h ≫ f) 0))\n (_ : f ≫ Cofork.π (colimit.cocone (parallelPair (h ≫ f) 0)) = 0) (_ : h ≫ f = h ≫ f)\n⊢ Exact f g",
"tactic": "refine' (exact_iff' _ _ (limit.isLimit _) hc).2 ⟨_, ((exact_iff _ _).1 hfg).2⟩"
},
{
"state_after": "no goals",
"state_before": "C : Type u₁\ninst✝² : Category C\ninst✝¹ : Abelian C\nX Y Z : C\nf : X ⟶ Y\ng : Y ⟶ Z\nW : C\nh : W ⟶ X\ninst✝ : Epi h\nhfg : Exact (h ≫ f) g\nhc : IsColimit\n (CokernelCofork.ofπ (Cofork.π (colimit.cocone (parallelPair (h ≫ f) 0)))\n (_ : f ≫ Cofork.π (colimit.cocone (parallelPair (h ≫ f) 0)) = 0)) :=\n isCokernelOfComp h (h ≫ f) (colimit.isColimit (parallelPair (h ≫ f) 0))\n (_ : f ≫ Cofork.π (colimit.cocone (parallelPair (h ≫ f) 0)) = 0) (_ : h ≫ f = h ≫ f)\n⊢ f ≫ g = 0",
"tactic": "exact zero_of_epi_comp h (by rw [← hfg.1, Category.assoc])"
},
{
"state_after": "no goals",
"state_before": "C : Type u₁\ninst✝² : Category C\ninst✝¹ : Abelian C\nX Y Z : C\nf : X ⟶ Y\ng : Y ⟶ Z\nW : C\nh : W ⟶ X\ninst✝ : Epi h\nhfg : Exact (h ≫ f) g\n⊢ f ≫ Cofork.π (colimit.cocone (parallelPair (h ≫ f) 0)) = 0",
"tactic": "rw [← cancel_epi h, ← Category.assoc, CokernelCofork.condition, comp_zero]"
},
{
"state_after": "no goals",
"state_before": "C : Type u₁\ninst✝² : Category C\ninst✝¹ : Abelian C\nX Y Z : C\nf : X ⟶ Y\ng : Y ⟶ Z\nW : C\nh : W ⟶ X\ninst✝ : Epi h\nhfg : Exact (h ≫ f) g\nhc : IsColimit\n (CokernelCofork.ofπ (Cofork.π (colimit.cocone (parallelPair (h ≫ f) 0)))\n (_ : f ≫ Cofork.π (colimit.cocone (parallelPair (h ≫ f) 0)) = 0)) :=\n isCokernelOfComp h (h ≫ f) (colimit.isColimit (parallelPair (h ≫ f) 0))\n (_ : f ≫ Cofork.π (colimit.cocone (parallelPair (h ≫ f) 0)) = 0) (_ : h ≫ f = h ≫ f)\n⊢ h ≫ f ≫ g = 0",
"tactic": "rw [← hfg.1, Category.assoc]"
}
] |
[
124,
61
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
119,
1
] |
Mathlib/Data/Finset/Sups.lean
|
Finset.mem_disjSups
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝³ : DecidableEq α\ninst✝² : SemilatticeSup α\ninst✝¹ : OrderBot α\ninst✝ : DecidableRel Disjoint\ns s₁ s₂ t t₁ t₂ u : Finset α\na b c : α\n⊢ c ∈ s ○ t ↔ ∃ a, a ∈ s ∧ ∃ b, b ∈ t ∧ Disjoint a b ∧ a ⊔ b = c",
"tactic": "simp [disjSups, and_assoc]"
}
] |
[
438,
29
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
437,
1
] |
Mathlib/Topology/Constructions.lean
|
continuous_curry
|
[] |
[
503,
30
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
502,
1
] |
Mathlib/Analysis/SpecialFunctions/Trigonometric/Deriv.lean
|
ContDiffAt.sinh
|
[] |
[
1175,
42
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1173,
1
] |
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