module stringlengths 16 90 | startPos dict | endPos dict | nextStartPos dict | goals listlengths 0 96 | goalsAfter listlengths 0 96 | ppTac stringlengths 1 14.5k | elaborator stringclasses 371
values | kind stringclasses 375
values |
|---|---|---|---|---|---|---|---|---|
Mathlib.CategoryTheory.Limits.ExactFunctor | {
"line": 190,
"column": 9
} | {
"line": 190,
"column": 65
} | {
"line": 190,
"column": 65
} | [
{
"pp": "C : Type u₁\ninst✝³ : Category.{v₁, u₁} C\nD : Type u₂\ninst✝² : Category.{v₂, u₂} D\nF : C ⥤ D\ninst✝¹ : PreservesFiniteLimits F\ninst✝ : PreservesFiniteColimits F\n⊢ exactFunctor C D F",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"CategoryTheory.Limits.P... | [] | simp only [exactFunctor_iff]; constructor <;> assumption | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Limits.Shapes.Kernels | {
"line": 187,
"column": 14
} | {
"line": 187,
"column": 66
} | {
"line": 187,
"column": 66
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : HasZeroMorphisms C\nX Y : C\nf : X ⟶ Y\nW : C\ng : Y ⟶ W\nh : X ⟶ W\nc : KernelFork h\ni : IsLimit c\nhf : Fork.ι c ≫ f = 0\nhfg : f ≫ g = h\ns : Fork f 0\n⊢ i.lift (KernelFork.ofι s.ι ⋯) ≫ Fork.ι (KernelFork.ofι (Fork.ι c) hf) = s.ι",
"ppTerm": "?m.9... | [] | by simp only [KernelFork.ι_ofι, Fork.IsLimit.lift_ι] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Limits.Shapes.Kernels | {
"line": 394,
"column": 2
} | {
"line": 394,
"column": 28
} | {
"line": 396,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : HasZeroMorphisms C\nX Y : C\n⊢ kernelZeroIsoSource.inv ≫ equalizer.ι 0 0 = kernel.lift 0 (𝟙 X) ⋯ ≫ equalizer.ι 0 0",
"ppTerm": "?m.45",
"assigned": true,
"usedConstants": [
"CategoryTheory.Limits.Cone.π",
"CategoryTheory.Functor",... | [] | simp [kernelZeroIsoSource] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.CategoryTheory.Limits.Shapes.Biproducts | {
"line": 585,
"column": 2
} | {
"line": 585,
"column": 28
} | {
"line": 586,
"column": 2
} | [
{
"pp": "J : Type w\nC : Type u\ninst✝³ : Category.{v, u} C\ninst✝² : HasZeroMorphisms C\nf g : J → C\ninst✝¹ : HasBiproduct f\ninst✝ : HasBiproduct g\np : (j : J) → f j ⟶ g j\nj : J\n⊢ ι f j ≫ map p = p j ≫ ι g j",
"ppTerm": "?m.44",
"assigned": true,
"usedConstants": [
"CategoryTheory.Limits... | [
"J : Type w\nC : Type u\ninst✝³ : Category.{v, u} C\ninst✝² : HasZeroMorphisms C\nf g : J → C\ninst✝¹ : HasBiproduct f\ninst✝ : HasBiproduct g\np : (j : J) → f j ⟶ g j\nj : J\n⊢ ι f j ≫ map' p = p j ≫ ι g j"
] | rw [biproduct.map_eq_map'] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.CategoryTheory.Limits.Shapes.Biproducts | {
"line": 682,
"column": 2
} | {
"line": 685,
"column": 36
} | {
"line": 686,
"column": 2
} | [
{
"pp": "case pos\nJ : Type w\nK : Type u_1\nC : Type u\ninst✝³ : Category.{v, u} C\ninst✝² : HasZeroMorphisms C\nf : J → C\ng : K → C\ne : J ≃ K\nw : (j : J) → g (e j) ≅ f j\ninst✝¹ : HasBiproduct f\ninst✝ : HasBiproduct g\nj : J\nk : K\nh : k = e j\n⊢ ι g k ≫ (desc fun k ↦ eqToHom ⋯ ≫ (w (e.symm k)).hom ≫ ι f... | [
"case neg\nJ : Type w\nK : Type u_1\nC : Type u\ninst✝³ : Category.{v, u} C\ninst✝² : HasZeroMorphisms C\nf : J → C\ng : K → C\ne : J ≃ K\nw : (j : J) → g (e j) ≅ f j\ninst✝¹ : HasBiproduct f\ninst✝ : HasBiproduct g\nj : J\nk : K\nh : ¬k = e j\n⊢ ι g k ≫ (desc fun k ↦ eqToHom ⋯ ≫ (w (e.symm k)).hom ≫ ι f (e.symm k)... | · subst h
simp only [ι_desc_assoc, ← eqToHom_iso_hom_naturality_assoc w (e.symm_apply_apply j).symm,
Equiv.symm_apply_apply, eqToHom_comp_ι, Category.assoc, bicone_ι_π_self, Category.comp_id,
lift_π, bicone_ι_π_self_assoc] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.CategoryTheory.Preadditive.Biproducts | {
"line": 582,
"column": 6
} | {
"line": 588,
"column": 18
} | {
"line": 590,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝² : Category.{v, u} C\ninst✝¹ : Preadditive C\nX✝ Y✝ : C\ninst✝ : HasBinaryBiproduct X✝ Y✝\nX Y : C\nb : BinaryBicone X Y\nhb : IsColimit b.inrCokernelCofork\nT : C\nf : X ⟶ T\ng : Y ⟶ T\nm : b.toCocone.pt ⟶ T\nh₁ : BinaryCofan.inl b.toCocone ≫ m = f\nh₂ : BinaryCofan.inr b.toCocone ≫ ... | [] | dsimp at m
have h₁' : b.inl ≫ (m - (b.fst ≫ f + b.snd ≫ g)) = 0 := by simpa using sub_eq_zero.2 h₁
have h₂' : b.inr ≫ (m - (b.fst ≫ f + b.snd ≫ g)) = 0 := by simpa using sub_eq_zero.2 h₂
obtain ⟨q : X ⟶ T, hq : b.fst ≫ q = m - (b.fst ≫ f + b.snd ≫ g)⟩ :=
CokernelCofork.IsColimit.desc' hb _ h₂'... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Preadditive.Biproducts | {
"line": 582,
"column": 6
} | {
"line": 588,
"column": 18
} | {
"line": 590,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝² : Category.{v, u} C\ninst✝¹ : Preadditive C\nX✝ Y✝ : C\ninst✝ : HasBinaryBiproduct X✝ Y✝\nX Y : C\nb : BinaryBicone X Y\nhb : IsColimit b.inrCokernelCofork\nT : C\nf : X ⟶ T\ng : Y ⟶ T\nm : b.toCocone.pt ⟶ T\nh₁ : BinaryCofan.inl b.toCocone ≫ m = f\nh₂ : BinaryCofan.inr b.toCocone ≫ ... | [] | dsimp at m
have h₁' : b.inl ≫ (m - (b.fst ≫ f + b.snd ≫ g)) = 0 := by simpa using sub_eq_zero.2 h₁
have h₂' : b.inr ≫ (m - (b.fst ≫ f + b.snd ≫ g)) = 0 := by simpa using sub_eq_zero.2 h₂
obtain ⟨q : X ⟶ T, hq : b.fst ≫ q = m - (b.fst ≫ f + b.snd ≫ g)⟩ :=
CokernelCofork.IsColimit.desc' hb _ h₂'... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Limits.Shapes.Biproducts | {
"line": 1056,
"column": 8
} | {
"line": 1056,
"column": 52
} | {
"line": 1056,
"column": 53
} | [
{
"pp": "J✝¹ : Type w\nC✝¹ : Type uC\ninst✝¹⁰ : Category.{uC', uC} C✝¹\ninst✝⁹ : HasZeroMorphisms C✝¹\nD✝ : Type uD\ninst✝⁸ : Category.{uD', uD} D✝\ninst✝⁷ : HasZeroMorphisms D✝\nF : J✝¹ → C✝¹\nJ✝ : Type w\nK : Type u_1\nC✝ : Type u\ninst✝⁶ : Category.{v, u} C✝\ninst✝⁵ : HasZeroMorphisms C✝\nJ : Type w\nC : Typ... | [
"J✝¹ : Type w\nC✝¹ : Type uC\ninst✝¹⁰ : Category.{uC', uC} C✝¹\ninst✝⁹ : HasZeroMorphisms C✝¹\nD✝ : Type uD\ninst✝⁸ : Category.{uD', uD} D✝\ninst✝⁷ : HasZeroMorphisms D✝\nF : J✝¹ → C✝¹\nJ✝ : Type w\nK : Type u_1\nC✝ : Type u\ninst✝⁶ : Category.{v, u} C✝\ninst✝⁵ : HasZeroMorphisms C✝\nJ : Type w\nC : Type u\ninst✝⁴ ... | ← biproduct.conePointUniqueUpToIso_hom f hb, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.Limits.Shapes.Biproducts | {
"line": 1059,
"column": 8
} | {
"line": 1059,
"column": 52
} | {
"line": 1059,
"column": 53
} | [
{
"pp": "J✝¹ : Type w\nC✝¹ : Type uC\ninst✝¹⁰ : Category.{uC', uC} C✝¹\ninst✝⁹ : HasZeroMorphisms C✝¹\nD✝ : Type uD\ninst✝⁸ : Category.{uD', uD} D✝\ninst✝⁷ : HasZeroMorphisms D✝\nF : J✝¹ → C✝¹\nJ✝ : Type w\nK : Type u_1\nC✝ : Type u\ninst✝⁶ : Category.{v, u} C✝\ninst✝⁵ : HasZeroMorphisms C✝\nJ : Type w\nC : Typ... | [
"J✝¹ : Type w\nC✝¹ : Type uC\ninst✝¹⁰ : Category.{uC', uC} C✝¹\ninst✝⁹ : HasZeroMorphisms C✝¹\nD✝ : Type uD\ninst✝⁸ : Category.{uD', uD} D✝\ninst✝⁷ : HasZeroMorphisms D✝\nF : J✝¹ → C✝¹\nJ✝ : Type w\nK : Type u_1\nC✝ : Type u\ninst✝⁶ : Category.{v, u} C✝\ninst✝⁵ : HasZeroMorphisms C✝\nJ : Type w\nC : Type u\ninst✝⁴ ... | ← biproduct.conePointUniqueUpToIso_hom f hb, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Order.Antidiag.Pi | {
"line": 115,
"column": 4
} | {
"line": 117,
"column": 40
} | {
"line": 118,
"column": 2
} | [
{
"pp": "case refine_1\nι : Type u_1\nμ : Type u_2\nμ' : Type u_3\ninst✝³ : DecidableEq ι\ninst✝² : AddCommMonoid μ\ninst✝¹ : HasAntidiagonal μ\ninst✝ : DecidableEq μ\nn✝ : μ\ns : Finset ι\nn : μ\ne : ↥s ≃ Fin #s\n⊢ Injective fun f i ↦ if hi : i ∈ s then f (e ⟨i, hi⟩) else 0",
"ppTerm": "?refine_1",
"as... | [] | rintro f g hfg
ext i
simpa using congr_fun hfg (e.symm i) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Order.Antidiag.Pi | {
"line": 115,
"column": 4
} | {
"line": 117,
"column": 40
} | {
"line": 118,
"column": 2
} | [
{
"pp": "case refine_1\nι : Type u_1\nμ : Type u_2\nμ' : Type u_3\ninst✝³ : DecidableEq ι\ninst✝² : AddCommMonoid μ\ninst✝¹ : HasAntidiagonal μ\ninst✝ : DecidableEq μ\nn✝ : μ\ns : Finset ι\nn : μ\ne : ↥s ≃ Fin #s\n⊢ Injective fun f i ↦ if hi : i ∈ s then f (e ⟨i, hi⟩) else 0",
"ppTerm": "?refine_1",
"as... | [] | rintro f g hfg
ext i
simpa using congr_fun hfg (e.symm i) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.Fintype.Inv | {
"line": 62,
"column": 95
} | {
"line": 65,
"column": 80
} | {
"line": 67,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝² : Fintype α\ninst✝¹ : DecidableEq β\nf : α → β\nhf : Injective f\ninst✝ : Nonempty α\n⊢ (Set.range f).restrict (invFun f) = hf.invOfMemRange",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"Function.invFun",
"Set.mem_range",
"Fu... | [] | by
ext ⟨b, h⟩
apply hf
simp [hf.left_inv_of_invOfMemRange, @invFun_eq _ _ _ f b (Set.mem_range.mp h)] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.GroupTheory.Perm.Support | {
"line": 297,
"column": 2
} | {
"line": 302,
"column": 9
} | {
"line": 304,
"column": 0
} | [
{
"pp": "α : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nl : List (Perm α)\nx : α\nhx : ∀ f ∈ l, f x = x\n⊢ l.prod x = x",
"ppTerm": "?m.35",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Equiv.instEquivLike",
"HMul.hMul",
"Equiv.Perm.instOne",
"congrArg",
... | [] | induction l with
| nil => rfl
| cons f l ih =>
rw [List.prod_cons, mul_apply, ih, hx]
· simp only [List.mem_cons, true_or]
grind | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | Lean.Parser.Tactic.induction |
Mathlib.GroupTheory.Perm.Support | {
"line": 319,
"column": 4
} | {
"line": 319,
"column": 30
} | {
"line": 321,
"column": 0
} | [
{
"pp": "case neg\nα : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nc : Perm α\ns : Finset α\nhcs : c.support ≤ s\nx : α\nhx' : x ∉ c.support\n⊢ c x ∈ s ↔ x ∈ s",
"ppTerm": "?neg✝",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Equiv.Perm.support",
"Equiv.Perm.notMem_supp... | [] | rw [notMem_support.mp hx'] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.GroupTheory.Perm.Support | {
"line": 319,
"column": 4
} | {
"line": 319,
"column": 30
} | {
"line": 321,
"column": 0
} | [
{
"pp": "case neg\nα : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nc : Perm α\ns : Finset α\nhcs : c.support ≤ s\nx : α\nhx' : x ∉ c.support\n⊢ c x ∈ s ↔ x ∈ s",
"ppTerm": "?neg✝",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Equiv.Perm.support",
"Equiv.Perm.notMem_supp... | [] | rw [notMem_support.mp hx'] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.GroupTheory.Perm.Support | {
"line": 319,
"column": 4
} | {
"line": 319,
"column": 30
} | {
"line": 321,
"column": 0
} | [
{
"pp": "case neg\nα : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nc : Perm α\ns : Finset α\nhcs : c.support ≤ s\nx : α\nhx' : x ∉ c.support\n⊢ c x ∈ s ↔ x ∈ s",
"ppTerm": "?neg✝",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Equiv.Perm.support",
"Equiv.Perm.notMem_supp... | [] | rw [notMem_support.mp hx'] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Combinatorics.Enumerative.Composition | {
"line": 846,
"column": 50
} | {
"line": 846,
"column": 63
} | {
"line": 846,
"column": 64
} | [
{
"pp": "n✝ n : ℕ\ns : Finset (Fin (n - 1))\ni : Fin (n - 1)\nthis : ↑i + 1 ≠ n\n⊢ i ∈ {i | ⟨↑i + 1, ⋯⟩ ∈ {i | ↑i = 0 ∨ ↑i = ↑(Fin.last n) ∨ ∃ j, ∃ (_ : j ∈ s), ↑i = ↑j + 1}.toFinset}.toFinset ↔ i ∈ s",
"ppTerm": "?m.518",
"assigned": true,
"usedConstants": [
"compositionAsSetEquiv._proof_2",
... | [
"n✝ n : ℕ\ns : Finset (Fin (n - 1))\ni : Fin (n - 1)\nthis : ↑i + 1 ≠ n\n⊢ i ∈ {i | ⟨↑i + 1, ⋯⟩ ∈ {i | ↑i = 0 ∨ ↑i = n ∨ ∃ j, ∃ (_ : j ∈ s), ↑i = ↑j + 1}.toFinset}.toFinset ↔ i ∈ s"
] | Fin.val_last, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.GroupTheory.Perm.Support | {
"line": 355,
"column": 69
} | {
"line": 356,
"column": 26
} | {
"line": 358,
"column": 0
} | [
{
"pp": "α : Type u_1\ninst✝² : DecidableEq α\ninst✝¹ : Fintype α\np : α → Prop\ninst✝ : DecidablePred p\nx : α\nu : Perm (Subtype p)\n⊢ x ∈ (ofSubtype u).support ↔ ∃ (hx : p x), ⟨x, hx⟩ ∈ u.support",
"ppTerm": "?m.25",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"Equiv.Perm.support"... | [] | by
simp [support_ofSubtype] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.GroupTheory.Perm.Support | {
"line": 508,
"column": 4
} | {
"line": 508,
"column": 33
} | {
"line": 509,
"column": 4
} | [
{
"pp": "case cons\nα : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nf hd : Perm α\ntl : List (Perm α)\nIH : f ∈ tl → List.Pairwise Disjoint tl → ∀ x ∈ f.support, f x = tl.prod x\nh : f ∈ hd :: tl\nhl : List.Pairwise Disjoint (hd :: tl)\nx : α\nhx : x ∈ f.support\n⊢ f x = (hd :: tl).prod x",
"ppTerm... | [
"case cons\nα : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nf hd : Perm α\ntl : List (Perm α)\nIH : f ∈ tl → List.Pairwise Disjoint tl → ∀ x ∈ f.support, f x = tl.prod x\nh : f ∈ hd :: tl\nhl : (∀ a' ∈ tl, hd.Disjoint a') ∧ List.Pairwise Disjoint tl\nx : α\nhx : x ∈ f.support\n⊢ f x = (hd :: tl).prod x"
] | rw [List.pairwise_cons] at hl | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Data.Fintype.Perm | {
"line": 50,
"column": 2
} | {
"line": 50,
"column": 33
} | {
"line": 51,
"column": 2
} | [
{
"pp": "α : Type u_1\ninst✝ : DecidableEq α\nl : List α\nf : Equiv.Perm α\nh : ∀ (x : α), f x ≠ x → x ∈ l\n⊢ f ∈ permsOfList l",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"Eq.mpr",
"List.mem_of_ne_of_mem",
"MulOne.toOne",
"False",
"Semigr... | [] | induction l generalizing f with | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | null |
Mathlib.GroupTheory.Perm.List | {
"line": 185,
"column": 4
} | {
"line": 192,
"column": 56
} | {
"line": 194,
"column": 0
} | [
{
"pp": "case cons\nα : Type u_1\ninst✝ : DecidableEq α\ni : ℕ\nx : α\nxs : List α\nw : (x :: xs).Nodup\nh : i < (x :: xs).length\n⊢ (x :: xs).formPerm (x :: xs)[i] = (x :: xs)[(i + 1) % (x :: xs).length]",
"ppTerm": "?cons",
"assigned": true,
"usedConstants": [
"IsRightCancelAdd.addRightStric... | [] | have : i ≤ xs.length := by
refine Nat.le_of_lt_succ ?_
simpa using h
rcases this.eq_or_lt with (rfl | hn')
· simp
· rw [formPerm_apply_lt_getElem (x :: xs) w _ (Nat.succ_lt_succ hn')]
congr
rw [Nat.mod_eq_of_lt]; simpa [Nat.succ_eq_add_one] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.GroupTheory.Perm.List | {
"line": 185,
"column": 4
} | {
"line": 192,
"column": 56
} | {
"line": 194,
"column": 0
} | [
{
"pp": "case cons\nα : Type u_1\ninst✝ : DecidableEq α\ni : ℕ\nx : α\nxs : List α\nw : (x :: xs).Nodup\nh : i < (x :: xs).length\n⊢ (x :: xs).formPerm (x :: xs)[i] = (x :: xs)[(i + 1) % (x :: xs).length]",
"ppTerm": "?cons",
"assigned": true,
"usedConstants": [
"IsRightCancelAdd.addRightStric... | [] | have : i ≤ xs.length := by
refine Nat.le_of_lt_succ ?_
simpa using h
rcases this.eq_or_lt with (rfl | hn')
· simp
· rw [formPerm_apply_lt_getElem (x :: xs) w _ (Nat.succ_lt_succ hn')]
congr
rw [Nat.mod_eq_of_lt]; simpa [Nat.succ_eq_add_one] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.GroupTheory.Perm.Sign | {
"line": 106,
"column": 2
} | {
"line": 106,
"column": 33
} | {
"line": 107,
"column": 2
} | [
{
"pp": "case intro\nα : Type u\ninst✝¹ : DecidableEq α\ninst✝ : Finite α\nmotive : Perm α → Prop\nf : Perm α\none : motive 1\nswap_mul : ∀ (f : Perm α) (x y : α), x ≠ y → motive f → motive (swap x y * f)\nval✝ : Fintype α\nl : List (Perm α)\nhl : l.prod = f ∧ ∀ g ∈ l, g.IsSwap\n⊢ motive f",
"ppTerm": "?int... | [] | induction l generalizing f with | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | null |
Mathlib.GroupTheory.Perm.List | {
"line": 326,
"column": 4
} | {
"line": 326,
"column": 63
} | {
"line": 327,
"column": 4
} | [
{
"pp": "case cons\nα : Type u_1\ninst✝ : DecidableEq α\nhd : α\ntl : List α\nhl : (hd :: tl).Nodup\n⊢ (hd :: tl).formPerm = 1 ↔ (hd :: tl).length ≤ 1",
"ppTerm": "?cons",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Equiv.instEquivLike",
"Equiv.Perm.instOne",
"congrArg",
... | [
"case cons\nα : Type u_1\ninst✝ : DecidableEq α\nhd : α\ntl : List α\nhl : (hd :: tl).Nodup\n⊢ (hd :: tl).formPerm = 1 ↔ (hd :: tl).formPerm hd = hd"
] | rw [← formPerm_apply_mem_eq_self_iff _ hl hd mem_cons_self] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.GroupTheory.Perm.Sign | {
"line": 236,
"column": 2
} | {
"line": 236,
"column": 28
} | {
"line": 237,
"column": 2
} | [
{
"pp": "n : ℕ\nf g : Perm (Fin n)\na b : Fin n\nhab : ⟨a, b⟩ ∈ finPairsLT n\n⊢ (if f (g a) ≤ f (g b) then -1 else 1) =\n (if\n f (if x : g b < g a then ⟨g a, g b⟩ else ⟨g b, g a⟩).fst ≤\n f (if x : g b < g a then ⟨g a, g b⟩ else ⟨g b, g a⟩).snd then\n -1\n else 1) *\n if... | [
"n : ℕ\nf g : Perm (Fin n)\na b : Fin n\nhab : ⟨a, b⟩.snd < ⟨a, b⟩.fst\n⊢ (if f (g a) ≤ f (g b) then -1 else 1) =\n (if\n f (if x : g b < g a then ⟨g a, g b⟩ else ⟨g b, g a⟩).fst ≤\n f (if x : g b < g a then ⟨g a, g b⟩ else ⟨g b, g a⟩).snd then\n -1\n else 1) *\n if\n ... | rw [mem_finPairsLT] at hab | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.GroupTheory.Perm.Closure | {
"line": 106,
"column": 59
} | {
"line": 106,
"column": 70
} | {
"line": 106,
"column": 71
} | [
{
"pp": "α : Type u_2\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nn : ℕ\nσ : Perm α\nh0 : n.Coprime (orderOf σ)\nh1 : σ.IsCycle\nh2 : σ.support = univ\nx : α\nm : ℕ\nhm : (σ ^ n) ^ m = σ\nh2' : (σ ^ n).support = univ\nh1' : (σ ^ n).IsCycle\n⊢ closure {σ ^ n, swap x ((σ ^ n) x)} ≤ closure {σ, swap x ((σ ^ n) x)}... | [
"α : Type u_2\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nn : ℕ\nσ : Perm α\nh0 : n.Coprime (orderOf σ)\nh1 : σ.IsCycle\nh2 : σ.support = univ\nx : α\nm : ℕ\nhm : (σ ^ n) ^ m = σ\nh2' : (σ ^ n).support = univ\nh1' : (σ ^ n).IsCycle\n⊢ {σ ^ n, swap x ((σ ^ n) x)} ⊆ ↑(closure {σ, swap x ((σ ^ n) x)})"
] | closure_le, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.GroupTheory.Perm.Cycle.Basic | {
"line": 366,
"column": 2
} | {
"line": 366,
"column": 62
} | {
"line": 368,
"column": 0
} | [
{
"pp": "α : Type u_2\nf : Perm α\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nhf : f.IsCycle\n⊢ Fintype.card ↥(Subgroup.zpowers f) = Fintype.card ↥f.support",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"Fintype.card_congr",
"Eq.mpr",
"Equiv.Perm.support",
"Finset... | [] | convert! Fintype.card_congr (IsCycle.zpowersEquivSupport hf) | Mathlib.Tactic._aux_Mathlib_Tactic_Convert___macroRules_Mathlib_Tactic_convert!_1 | Mathlib.Tactic.convert! |
Mathlib.GroupTheory.Perm.Sign | {
"line": 313,
"column": 8
} | {
"line": 313,
"column": 25
} | {
"line": 314,
"column": 6
} | [
{
"pp": "α : Type u\ninst✝ : DecidableEq α\nn : ℕ\nx : α\nl : List α\nf : Perm α\ne : α ≃ Fin n\nh : ∀ (x_1 : α), f x_1 ≠ x_1 → x_1 ∈ x :: l\nhfx : ¬x = f x\nhy : ∀ (y : α), (swap x (f x) * f) y ≠ y → y ∈ l\nx✝ : Fin n\n⊢ ↑(e (if f (e.symm x✝) = x then f x else if e.symm x✝ = x then x else f (e.symm x✝))) =\n ... | [] | split_ifs <;> rfl | Lean.Parser.Tactic.«_aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tactic_<;>__1» | Lean.Parser.Tactic.«tactic_<;>_» |
Mathlib.GroupTheory.Perm.Sign | {
"line": 393,
"column": 20
} | {
"line": 393,
"column": 42
} | {
"line": 393,
"column": 43
} | [
{
"pp": "α : Type u\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nx y : α\nH : x = y\n⊢ sign (swap x y) = if x = y then 1 else -1",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"MonoidHom.instFunLike",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"CommRing.toNonUnitalC... | [] | by simp [H, swap_self] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.GroupTheory.Perm.Finite | {
"line": 243,
"column": 2
} | {
"line": 243,
"column": 51
} | {
"line": 244,
"column": 2
} | [
{
"pp": "α : Type u\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nσ : Perm α\nn : ℕ\nh : n.Coprime (orderOf σ)\n⊢ (σ ^ n).support = σ.support",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"Equiv.Perm.support",
"Finset",
"Equiv.Perm.instPowNat",
"Exists",
"DivI... | [
"α : Type u\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nσ : Perm α\nn : ℕ\nh : n.Coprime (orderOf σ)\nm : ℕ\nhm : (σ ^ n) ^ m = σ\n⊢ (σ ^ n).support = σ.support"
] | obtain ⟨m, hm⟩ := exists_pow_eq_self_of_coprime h | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.GroupTheory.Perm.Cycle.Basic | {
"line": 551,
"column": 6
} | {
"line": 551,
"column": 55
} | {
"line": 552,
"column": 6
} | [
{
"pp": "case intro.mpr\nβ : Type u_3\ninst✝ : Finite β\nf : Perm β\nhf : f.IsCycle\nn : ℕ\nval✝ : Fintype β\nh : n.Coprime (orderOf f)\n⊢ (f ^ n).IsCycle",
"ppTerm": "?intro.mpr",
"assigned": true,
"usedConstants": [
"Equiv.Perm.instPowNat",
"Exists",
"DivInvMonoid.toMonoid",
... | [
"case intro.mpr\nβ : Type u_3\ninst✝ : Finite β\nf : Perm β\nhf : f.IsCycle\nn : ℕ\nval✝ : Fintype β\nh : n.Coprime (orderOf f)\nm : ℕ\nhm : (f ^ n) ^ m = f\n⊢ (f ^ n).IsCycle"
] | obtain ⟨m, hm⟩ := exists_pow_eq_self_of_coprime h | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.GroupTheory.Perm.Sign | {
"line": 567,
"column": 6
} | {
"line": 568,
"column": 66
} | {
"line": 570,
"column": 0
} | [
{
"pp": "case right.swap_mul\nα : Type u\ninst✝³ : DecidableEq α\nβ : Type v\ninst✝² : Fintype α\ninst✝¹ : DecidableEq β\ninst✝ : Fintype β\nσa : Perm α\nσb' : Perm β\nb₁ b₂ : β\nhb : b₁ ≠ b₂\nih : sign (sumCongr 1 σb') = sign σb'\n⊢ sign (sumCongr 1 (swap b₁ b₂ * σb')) = sign (swap b₁ b₂ * σb')",
"ppTerm":... | [] | rw [← one_mul (1 : Perm α), ← sumCongr_mul, sign_mul, sign_mul, ih, sumCongr_one_swap,
sign_swap hb, sign_swap (Sum.inr_injective.ne_iff.mpr hb)] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.GroupTheory.Perm.Sign | {
"line": 567,
"column": 6
} | {
"line": 568,
"column": 66
} | {
"line": 570,
"column": 0
} | [
{
"pp": "case right.swap_mul\nα : Type u\ninst✝³ : DecidableEq α\nβ : Type v\ninst✝² : Fintype α\ninst✝¹ : DecidableEq β\ninst✝ : Fintype β\nσa : Perm α\nσb' : Perm β\nb₁ b₂ : β\nhb : b₁ ≠ b₂\nih : sign (sumCongr 1 σb') = sign σb'\n⊢ sign (sumCongr 1 (swap b₁ b₂ * σb')) = sign (swap b₁ b₂ * σb')",
"ppTerm":... | [] | rw [← one_mul (1 : Perm α), ← sumCongr_mul, sign_mul, sign_mul, ih, sumCongr_one_swap,
sign_swap hb, sign_swap (Sum.inr_injective.ne_iff.mpr hb)] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.GroupTheory.Perm.Sign | {
"line": 567,
"column": 6
} | {
"line": 568,
"column": 66
} | {
"line": 570,
"column": 0
} | [
{
"pp": "case right.swap_mul\nα : Type u\ninst✝³ : DecidableEq α\nβ : Type v\ninst✝² : Fintype α\ninst✝¹ : DecidableEq β\ninst✝ : Fintype β\nσa : Perm α\nσb' : Perm β\nb₁ b₂ : β\nhb : b₁ ≠ b₂\nih : sign (sumCongr 1 σb') = sign σb'\n⊢ sign (sumCongr 1 (swap b₁ b₂ * σb')) = sign (swap b₁ b₂ * σb')",
"ppTerm":... | [] | rw [← one_mul (1 : Perm α), ← sumCongr_mul, sign_mul, sign_mul, ih, sumCongr_one_swap,
sign_swap hb, sign_swap (Sum.inr_injective.ne_iff.mpr hb)] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.GroupTheory.Perm.Cycle.Basic | {
"line": 615,
"column": 6
} | {
"line": 615,
"column": 32
} | {
"line": 616,
"column": 6
} | [
{
"pp": "β : Type u_3\ninst✝ : Finite β\nf : Perm β\nhf : f.IsCycle\nhf' : Nat.Prime (orderOf f)\nn : ℕ\nhn : 0 < n\nhn' : n < orderOf f\nval✝ : Fintype β\n⊢ n.Coprime (orderOf f)",
"ppTerm": "?m.43",
"assigned": true,
"usedConstants": [
"Nat.Coprime.symm",
"DivInvMonoid.toMonoid",
... | [
"β : Type u_3\ninst✝ : Finite β\nf : Perm β\nhf : f.IsCycle\nhf' : Nat.Prime (orderOf f)\nn : ℕ\nhn : 0 < n\nhn' : n < orderOf f\nval✝ : Fintype β\n⊢ (orderOf f).Coprime n"
] | refine Nat.Coprime.symm ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Tactic.NormNum.GCD | {
"line": 213,
"column": 37
} | {
"line": 218,
"column": 84
} | {
"line": 220,
"column": 0
} | [
{
"pp": "n : ℤ\nd : ℕ\nhi : Invertible ↑d\nh : n.natAbs.gcd d = 1\n⊢ IsInt (↑n * ⅟↑d).num n",
"ppTerm": "?m.31",
"assigned": true,
"usedConstants": [
"Nat.gcd",
"Iff.mpr",
"Int.cast",
"Eq.mpr",
"Int.instDiv",
"Rat.num",
"instHDiv",
"Invertible.ne_zero"... | [] | by
constructor
have : 0 < d := Nat.pos_iff_ne_zero.mpr <| by simpa using hi.ne_zero
simp_rw [Rat.mul_num, Rat.den_intCast, invOf_eq_inv,
Rat.inv_natCast_den_of_pos this, Rat.inv_natCast_num_of_pos this,
Rat.num_intCast, one_mul, mul_one, h, Nat.cast_one, Int.ediv_one, Int.cast_id] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.GroupTheory.Perm.Cycle.Basic | {
"line": 745,
"column": 30
} | {
"line": 757,
"column": 32
} | {
"line": 759,
"column": 0
} | [
{
"pp": "α : Type u_2\nf : Perm α\na : α\ns : Finset α\nhf : f.IsCycleOn ↑s\nha : a ∈ s\nn : ℕ\n⊢ (f ^ n) a = a ↔ #s ∣ n",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"Eq.mpr",
"instDecidableNot",
"Equiv.Perm.support",
"MulOne.toOne",
"False... | [] | by
obtain rfl | hs := Finset.eq_singleton_or_nontrivial ha
· rw [coe_singleton, isCycleOn_singleton] at hf
simpa using! IsFixedPt.iterate hf n
classical
have h (x : s) : ¬f x = x := hf.apply_ne hs x.2
have := (hf.isCycle_subtypePerm hs).orderOf
simp only [coe_sort_coe, support_subtypePerm, ne_eq, ... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.GroupTheory.Perm.Cycle.Factors | {
"line": 414,
"column": 67
} | {
"line": 414,
"column": 88
} | {
"line": 414,
"column": 88
} | [
{
"pp": "ι : Type u_1\nα : Type u_2\nβ : Type u_3\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nl✝¹ : List α\nf : Perm α\nh : ∀ {x : α}, f x ≠ x → x ∈ l✝¹\nl✝ : List α\ng : Perm α\nhfg : ∀ {x : α}, g x ≠ x → f.cycleOf x = g.cycleOf x\nx : α\nl : List α\nhg : ∀ {x_1 : α}, g x_1 ≠ x_1 → x_1 ∈ x :: l\nhx : ¬g x = x\... | [
"ι : Type u_1\nα : Type u_2\nβ : Type u_3\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nl✝¹ : List α\nf : Perm α\nh : ∀ {x : α}, f x ≠ x → x ∈ l✝¹\nl✝ : List α\ng : Perm α\nhfg : ∀ {x : α}, g x ≠ x → f.cycleOf x = g.cycleOf x\nx : α\nl : List α\nhg : ∀ {x_1 : α}, g x_1 ≠ x_1 → x_1 ∈ x :: l\nhx : ¬g x = x\nm : List (P... | sameCycle_apply_right | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.GroupTheory.Perm.Fin | {
"line": 164,
"column": 14
} | {
"line": 164,
"column": 28
} | {
"line": 164,
"column": 28
} | [
{
"pp": "case pos\nn : ℕ\ni j : Fin n\ninst✝ : NeZero n\nh : i ≤ j\niin : i ∈ Set.range ⇑(castLEEmb ⋯)\nthis✝ : (castLEEmb ⋯).toEquivRange (i.castLT ⋯) = ⟨i, iin⟩\nch : i = j\nthis : (castLEEmb ⋯).toEquivRange.symm ⟨i, iin⟩ = last ↑j\n⊢ ↑⟨(castLEEmb ⋯) ((finRotate (↑j + 1)) (last ↑j)), ⋯⟩ = 0",
"ppTerm": "?... | [
"case pos\nn : ℕ\ni j : Fin n\ninst✝ : NeZero n\nh : i ≤ j\niin : i ∈ Set.range ⇑(castLEEmb ⋯)\nthis✝ : (castLEEmb ⋯).toEquivRange (i.castLT ⋯) = ⟨i, iin⟩\nch : i = j\nthis : (castLEEmb ⋯).toEquivRange.symm ⟨i, iin⟩ = last ↑j\n⊢ ↑⟨(castLEEmb ⋯) 0, ⋯⟩ = 0"
] | finRotate_last | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.GroupTheory.Perm.Fin | {
"line": 366,
"column": 4
} | {
"line": 367,
"column": 40
} | {
"line": 368,
"column": 4
} | [
{
"pp": "case pos\nn : ℕ\ni j k : Fin n\nh : j < k\nhij : i ≤ j\n⊢ (i.cycleIcc j) k = k",
"ppTerm": "?pos✝",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Nat.instOrderedSub",
"congrArg",
"setOf",
"HSub.hSub",
"Membership.mem",
"Nat.sub_le",
"id",
... | [
"case pos\nn : ℕ\ni j k : Fin n\nh : j < k\nhij : i ≤ j\nkin : k ∈ Set.range ⇑(natAdd_castLEEmb ⋯)\n⊢ (i.cycleIcc j) k = k"
] | have kin : k ∈ Set.range (natAdd_castLEEmb (Nat.sub_le n i)) := by
simp [range_natAdd_castLEEmb]; lia | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.GroupTheory.Perm.Cycle.Factors | {
"line": 814,
"column": 4
} | {
"line": 814,
"column": 19
} | {
"line": 815,
"column": 4
} | [
{
"pp": "case refine_2\nα : Type u_2\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\ng f✝ : Perm α\n⊢ ∀ (σ : Perm α),\n σ.IsCycle → ∀ {f : Perm α}, f ∈ σ.cycleFactorsFinset → (σ * f⁻¹).cycleFactorsFinset = σ.cycleFactorsFinset \\ {f}",
"ppTerm": "?refine_2",
"assigned": true,
"usedConstants": [
... | [
"case refine_2\nα : Type u_2\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\ng f✝ σ : Perm α\nhσ : σ.IsCycle\nf : Perm α\nhf : f ∈ σ.cycleFactorsFinset\n⊢ (σ * f⁻¹).cycleFactorsFinset = σ.cycleFactorsFinset \\ {f}"
] | intro σ hσ f hf | Lean.Elab.Tactic.evalIntro | Lean.Parser.Tactic.intro |
Mathlib.GroupTheory.Perm.Cycle.Type | {
"line": 381,
"column": 4
} | {
"line": 381,
"column": 35
} | {
"line": 382,
"column": 4
} | [
{
"pp": "α : Type u_1\ninst✝¹ : Fintype α\ninst✝ : DecidableEq α\nf : Function.End α\np n : ℕ\nhp : Fact (Nat.Prime p)\nhf : f ^ p ^ n = 1\nσ : α ≃ α := { toFun := f, invFun := f ^ (p ^ n - 1), left_inv := ⋯, right_inv := ⋯ }\n⊢ σ ^ p ^ n = 1",
"ppTerm": "?m.77",
"assigned": true,
"usedConstants": [... | [
"α : Type u_1\ninst✝¹ : Fintype α\ninst✝ : DecidableEq α\nf : Function.End α\np n : ℕ\nhp : Fact (Nat.Prime p)\nhf : f ^ p ^ n = 1\nσ : α ≃ α := { toFun := f, invFun := f ^ (p ^ n - 1), left_inv := ⋯, right_inv := ⋯ }\n⊢ (⇑σ)^[p ^ n] = ⇑1"
] | rw [DFunLike.ext'_iff, coe_pow] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.LinearAlgebra.Multilinear.DFinsupp | {
"line": 282,
"column": 78
} | {
"line": 282,
"column": 88
} | {
"line": 282,
"column": 88
} | [
{
"pp": "ι : Type uι\nκ : ι → Type uκ\nR : Type uR\nι' : Type u_1\ninst✝⁵ : DecidableEq ι\ninst✝⁴ : Fintype ι\ninst✝³ : CommSemiring R\ninst✝² : (i : ι) → Fintype (κ i)\ninst✝¹ : (i : ι) → DecidableEq (κ i)\ninst✝ : DecidableEq ι'\np : ((i : ι) → κ i) × ι'\nr : R\nx : (i : ι) → Π₀ (x : κ i), R\n| (r • freeDFins... | [
"ι : Type uι\nκ : ι → Type uκ\nR : Type uR\nι' : Type u_1\ninst✝⁵ : DecidableEq ι\ninst✝⁴ : Fintype ι\ninst✝³ : CommSemiring R\ninst✝² : (i : ι) → Fintype (κ i)\ninst✝¹ : (i : ι) → DecidableEq (κ i)\ninst✝ : DecidableEq ι'\np : ((i : ι) → κ i) × ι'\nr : R\nx : (i : ι) → Π₀ (x : κ i), R\n| r • (freeDFinsuppEquiv (DF... | smul_apply | Lean.Elab.Tactic.Conv.evalRewrite | null |
Mathlib.LinearAlgebra.Multilinear.DFinsupp | {
"line": 287,
"column": 63
} | {
"line": 289,
"column": 32
} | {
"line": 290,
"column": 4
} | [
{
"pp": "ι : Type uι\nκ : ι → Type uκ\nR : Type uR\nι' : Type u_1\ninst✝⁵ : DecidableEq ι\ninst✝⁴ : Fintype ι\ninst✝³ : CommSemiring R\ninst✝² : (i : ι) → Fintype (κ i)\ninst✝¹ : (i : ι) → DecidableEq (κ i)\ninst✝ : DecidableEq ι'\nr : R\nx : (i : ι) → Π₀ (x : κ i), R\np : (i : ι) → κ i\nj : ι'\nthis : ∀ (l : ι... | [] | by
simpa [freeDFinsuppEquiv_def, MultilinearMap.piRingEquiv, DFinsupp.sigmaCurryEquiv,
fromDFinsuppEquiv_apply] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.LinearAlgebra.Multilinear.Basic | {
"line": 502,
"column": 6
} | {
"line": 502,
"column": 81
} | {
"line": 503,
"column": 4
} | [
{
"pp": "R : Type uR\nι : Type uι\nM₁ : ι → Type v₁\nM₂ : Type v₂\ninst✝⁶ : Semiring R\ninst✝⁵ : (i : ι) → AddCommMonoid (M₁ i)\ninst✝⁴ : AddCommMonoid M₂\ninst✝³ : (i : ι) → Module R (M₁ i)\ninst✝² : Module R M₂\nf : MultilinearMap R M₁ M₂\nα : ι → Type u_1\ng : (i : ι) → α i → M₁ i\ninst✝¹ : DecidableEq ι\nin... | [] | exact le_antisymm this (Nat.succ_le_of_lt (_root_.pos_iff_ne_zero.mpr pos)) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Algebra.Category.Ring.Colimits | {
"line": 192,
"column": 17
} | {
"line": 192,
"column": 57
} | {
"line": 193,
"column": 4
} | [
{
"pp": "J : Type v\ninst✝ : SmallCategory J\nF : J ⥤ RingCat\nj : J\n⊢ coconeFun F j 0 = 0",
"ppTerm": "?m.49",
"assigned": true,
"usedConstants": [
"NonAssocSemiring.toAddCommMonoidWithOne",
"Quot.sound",
"AddMonoid.toAddZeroClass",
"RingCat.ring",
"AddZeroClass.toAdd... | [] | by apply Quot.sound; apply Relation.zero | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Category.Ring.Colimits | {
"line": 492,
"column": 17
} | {
"line": 492,
"column": 57
} | {
"line": 493,
"column": 4
} | [
{
"pp": "J : Type v\ninst✝ : SmallCategory J\nF : J ⥤ CommRingCat\nj : J\n⊢ coconeFun F j 0 = 0",
"ppTerm": "?m.49",
"assigned": true,
"usedConstants": [
"NonAssocSemiring.toAddCommMonoidWithOne",
"CommRingCat.carrier",
"Quot.sound",
"CommSemiring.toSemiring",
"AddMonoi... | [] | by apply Quot.sound; apply Relation.zero | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.LinearAlgebra.Matrix.Determinant.Basic | {
"line": 163,
"column": 28
} | {
"line": 163,
"column": 60
} | {
"line": 164,
"column": 10
} | [
{
"pp": "n : Type u_2\ninst✝² : DecidableEq n\ninst✝¹ : Fintype n\nR : Type v\ninst✝ : CommRing R\nM N : Matrix n n R\nσ : Perm n\nx✝ : σ ∈ univ\nτ : Perm n\nthis : ∏ j, M (τ j) (σ j) = ∏ j, M ((τ * σ⁻¹) j) j\n⊢ ↑↑(sign (τ * σ⁻¹ * σ)) = ↑↑(sign τ)",
"ppTerm": "?m.531",
"assigned": true,
"usedConstan... | [] | simp only [inv_mul_cancel_right] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.LinearAlgebra.Matrix.SemiringInverse | {
"line": 204,
"column": 11
} | {
"line": 204,
"column": 19
} | {
"line": 204,
"column": 20
} | [
{
"pp": "n : Type u_1\nR : Type u_3\ninst✝² : Fintype n\ninst✝¹ : DecidableEq n\ninst✝ : CommSemiring R\nA B : Matrix n n R\nhAB : A * B = 1\nkey : A * (B * adjp 1 B + detp (-1) B • 1) = A * (B * adjp (-1) B + detp 1 B • 1)\n⊢ detp 1 B • A + adjp (-1) B = detp (-1) B • A + adjp 1 B",
"ppTerm": "?m.80",
... | [
"n : Type u_1\nR : Type u_3\ninst✝² : Fintype n\ninst✝¹ : DecidableEq n\ninst✝ : CommSemiring R\nA B : Matrix n n R\nhAB : A * B = 1\nkey : A * (B * adjp 1 B) + A * detp (-1) B • 1 = A * (B * adjp (-1) B) + A * detp 1 B • 1\n⊢ detp 1 B • A + adjp (-1) B = detp (-1) B • A + adjp 1 B"
] | mul_add, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.LinearAlgebra.Matrix.Determinant.Basic | {
"line": 163,
"column": 28
} | {
"line": 163,
"column": 60
} | {
"line": 164,
"column": 10
} | [
{
"pp": "n : Type u_2\ninst✝² : DecidableEq n\ninst✝¹ : Fintype n\nR : Type v\ninst✝ : CommRing R\nM N : Matrix n n R\nσ : Perm n\nx✝ : σ ∈ univ\nτ : Perm n\nthis : ∏ j, M (τ j) (σ j) = ∏ j, M ((τ * σ⁻¹) j) j\n⊢ ↑↑(sign (τ * σ⁻¹ * σ)) = ↑↑(sign τ)",
"ppTerm": "?m.531",
"assigned": true,
"usedConstan... | [] | simp only [inv_mul_cancel_right] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.Matrix.Determinant.Basic | {
"line": 163,
"column": 28
} | {
"line": 163,
"column": 60
} | {
"line": 164,
"column": 10
} | [
{
"pp": "n : Type u_2\ninst✝² : DecidableEq n\ninst✝¹ : Fintype n\nR : Type v\ninst✝ : CommRing R\nM N : Matrix n n R\nσ : Perm n\nx✝ : σ ∈ univ\nτ : Perm n\nthis : ∏ j, M (τ j) (σ j) = ∏ j, M ((τ * σ⁻¹) j) j\n⊢ ↑↑(sign (τ * σ⁻¹ * σ)) = ↑↑(sign τ)",
"ppTerm": "?m.531",
"assigned": true,
"usedConstan... | [] | simp only [inv_mul_cancel_right] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Filtered.Basic | {
"line": 250,
"column": 4
} | {
"line": 251,
"column": 60
} | {
"line": 252,
"column": 2
} | [
{
"pp": "case empty\nC : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : IsFiltered C\nO : Finset C\nH : Finset ((X : C) ×' (Y : C) ×' (_ : X ∈ O) ×' (_ : Y ∈ O) ×' (X ⟶ Y))\n⊢ ∃ S T, ∀ {X Y : C} (mX : X ∈ O) (mY : Y ∈ O) {f : X ⟶ Y}, ⟨X, ⟨Y, ⟨mX, ⟨mY, f⟩⟩⟩⟩ ∈ ∅ → f ≫ T mY = T mX",
"ppTerm": "?empty",
"assi... | [] | obtain ⟨S, f⟩ := sup_objs_exists O
exact ⟨S, fun mX => (f mX).some, by rintro - - - - - ⟨⟩⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Filtered.Basic | {
"line": 250,
"column": 4
} | {
"line": 251,
"column": 60
} | {
"line": 252,
"column": 2
} | [
{
"pp": "case empty\nC : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : IsFiltered C\nO : Finset C\nH : Finset ((X : C) ×' (Y : C) ×' (_ : X ∈ O) ×' (_ : Y ∈ O) ×' (X ⟶ Y))\n⊢ ∃ S T, ∀ {X Y : C} (mX : X ∈ O) (mY : Y ∈ O) {f : X ⟶ Y}, ⟨X, ⟨Y, ⟨mX, ⟨mY, f⟩⟩⟩⟩ ∈ ∅ → f ≫ T mY = T mX",
"ppTerm": "?empty",
"assi... | [] | obtain ⟨S, f⟩ := sup_objs_exists O
exact ⟨S, fun mX => (f mX).some, by rintro - - - - - ⟨⟩⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.Matrix.Determinant.Basic | {
"line": 423,
"column": 53
} | {
"line": 423,
"column": 62
} | {
"line": 423,
"column": 63
} | [
{
"pp": "case insert\nn : Type u_2\ninst✝² : DecidableEq n\ninst✝¹ : Fintype n\nR : Type v\ninst✝ : CommRing R\nM : Matrix n n R\nj : n\nc : n → R\na : R\nk : n\ns✝ : Finset n\nhk : k ∉ s✝\nh_ind : j ∉ s✝ → (M.updateRow j (a • M j + ∑ k ∈ s✝, c k • M k)).det = a • M.det\nhj : j ∉ insert k s✝\nh : k ≠ j\n⊢ (M.up... | [
"case insert\nn : Type u_2\ninst✝² : DecidableEq n\ninst✝¹ : Fintype n\nR : Type v\ninst✝ : CommRing R\nM : Matrix n n R\nj : n\nc : n → R\na : R\nk : n\ns✝ : Finset n\nhk : k ∉ s✝\nh_ind : j ∉ s✝ → (M.updateRow j (a • M j + ∑ k ∈ s✝, c k • M k)).det = a • M.det\nhj : j ∉ insert k s✝\nh : k ≠ j\n⊢ (M.updateRow j (a... | mul_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.Matrix.Determinant.Basic | {
"line": 499,
"column": 2
} | {
"line": 500,
"column": 48
} | {
"line": 502,
"column": 0
} | [
{
"pp": "m : Type u_1\ninst✝³ : DecidableEq m\ninst✝² : Fintype m\nR : Type v\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\nA : Matrix m m R\nhA : ¬LinearIndependent R fun i ↦ Aᵀ i\n⊢ A.det = 0",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"Pi.Function.module",
"Mathlib.Tactic.Co... | [] | contrapose! hA
exact linearIndependent_cols_of_det_ne_zero hA | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.Matrix.Determinant.Basic | {
"line": 499,
"column": 2
} | {
"line": 500,
"column": 48
} | {
"line": 502,
"column": 0
} | [
{
"pp": "m : Type u_1\ninst✝³ : DecidableEq m\ninst✝² : Fintype m\nR : Type v\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\nA : Matrix m m R\nhA : ¬LinearIndependent R fun i ↦ Aᵀ i\n⊢ A.det = 0",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"Pi.Function.module",
"Mathlib.Tactic.Co... | [] | contrapose! hA
exact linearIndependent_cols_of_det_ne_zero hA | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.Matrix.Determinant.Basic | {
"line": 515,
"column": 4
} | {
"line": 515,
"column": 13
} | {
"line": 515,
"column": 14
} | [
{
"pp": "n : Type u_2\ninst✝³ : DecidableEq n\ninst✝² : Fintype n\nR : Type v\ninst✝¹ : CommRing R\ninst✝ : Nontrivial n\nu v : n → R\ni j : n\nhij : i ≠ j\nuv' : Matrix n n R := ((vecMulVec u v).updateRow i v).updateRow j v\nhuv' : uv'.det = 0\nthis : vecMulVec u v = (uv'.updateRow i (u i • uv' i)).updateRow j... | [
"n : Type u_2\ninst✝³ : DecidableEq n\ninst✝² : Fintype n\nR : Type v\ninst✝¹ : CommRing R\ninst✝ : Nontrivial n\nu v : n → R\ni j : n\nhij : i ≠ j\nuv' : Matrix n n R := ((vecMulVec u v).updateRow i v).updateRow j v\nhuv' : uv'.det = 0\nthis : vecMulVec u v = (uv'.updateRow i (u i • uv' i)).updateRow j (u j • uv'.... | mul_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Category.MonCat.FilteredColimits | {
"line": 253,
"column": 16
} | {
"line": 254,
"column": 54
} | {
"line": 255,
"column": 4
} | [
{
"pp": "J : Type v\ninst✝¹ : SmallCategory J\nF : J ⥤ MonCat\ninst✝ : IsFiltered J\nt : Cocone F\n⊢ (F ⋙ forget MonCat).descColimitType ((F ⋙ forget MonCat).coconeTypesEquiv.symm ((forget MonCat).mapCocone t)) 1 = 1",
"ppTerm": "?m.72",
"assigned": true,
"usedConstants": [
"MonoidHom.instMono... | [] | by
simp [colimit_one_eq F IsFiltered.nonempty.some] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Limits.Types.Limits | {
"line": 48,
"column": 4
} | {
"line": 51,
"column": 81
} | {
"line": 53,
"column": 0
} | [
{
"pp": "case refine_2\nJ : Type v\ninst✝ : Category.{w, v} J\nF : J ⥤ Type u\nc : Cone F\nh : ∀ s ∈ F.sections, ∃! x, ∀ (j : J), (hom (c.π.app j)) x = s j\n⊢ IsLimit c",
"ppTerm": "?refine_2",
"assigned": true,
"usedConstants": [
"CategoryTheory.Limits.Cone.π",
"CategoryTheory.Functor",... | [] | have := fun c y ↦ h _ (sectionOfCone c y).2
choose x hx using fun c y ↦ h _ (sectionOfCone c y).2
exact ⟨fun d ↦ ↾(x d), fun c j ↦ by ext y; exact (hx c y).1 j,
fun c f hf ↦ by ext y; exact (hx c y).2 (f y) (fun j ↦ congr_hom (hf j) y)⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Limits.Types.Limits | {
"line": 48,
"column": 4
} | {
"line": 51,
"column": 81
} | {
"line": 53,
"column": 0
} | [
{
"pp": "case refine_2\nJ : Type v\ninst✝ : Category.{w, v} J\nF : J ⥤ Type u\nc : Cone F\nh : ∀ s ∈ F.sections, ∃! x, ∀ (j : J), (hom (c.π.app j)) x = s j\n⊢ IsLimit c",
"ppTerm": "?refine_2",
"assigned": true,
"usedConstants": [
"CategoryTheory.Limits.Cone.π",
"CategoryTheory.Functor",... | [] | have := fun c y ↦ h _ (sectionOfCone c y).2
choose x hx using fun c y ↦ h _ (sectionOfCone c y).2
exact ⟨fun d ↦ ↾(x d), fun c j ↦ by ext y; exact (hx c y).1 j,
fun c f hf ↦ by ext y; exact (hx c y).2 (f y) (fun j ↦ congr_hom (hf j) y)⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Limits.Types.Images | {
"line": 116,
"column": 61
} | {
"line": 116,
"column": 69
} | {
"line": 116,
"column": 69
} | [
{
"pp": "F : ℕᵒᵖ ⥤ Type u\nc : Cone F\nhc : IsLimit c\nhF : ∀ (n : ℕ), Function.Surjective ⇑(ConcreteCategory.hom (F.map (homOfLE ⋯).op))\ni : c.pt ≅ (limitCone F).pt := hc.conePointUniqueUpToIso (limitConeIsLimit F)\n⊢ c.π.app (Opposite.op 0) = i.hom ≫ (limitCone F).π.app (Opposite.op 0)",
"ppTerm": "?m.12... | [
"F : ℕᵒᵖ ⥤ Type u\nc : Cone F\nhc : IsLimit c\nhF : ∀ (n : ℕ), Function.Surjective ⇑(ConcreteCategory.hom (F.map (homOfLE ⋯).op))\ni : c.pt ≅ (limitCone F).pt := hc.conePointUniqueUpToIso (limitConeIsLimit F)\n⊢ c.π.app (Opposite.op 0) = (hc.conePointUniqueUpToIso (limitConeIsLimit F)).hom ≫ ↾fun u ↦ ↑u (Opposite.o... | simp [i] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.LinearAlgebra.Matrix.Determinant.Basic | {
"line": 744,
"column": 59
} | {
"line": 744,
"column": 79
} | {
"line": 745,
"column": 4
} | [
{
"pp": "case refine_2\nR : Type v\ninst✝ : CommRing R\nn : ℕ\nA : Matrix (Fin n.succ) (Fin n.succ) R\ni✝ : Fin n.succ\nx✝ : i✝ ∈ univ\ni : Fin n\nthis : (-1) ^ ↑i = ↑↑(sign i.cycleRange)\n⊢ ∑ y, sign (decomposeFin.symm (i.succ, y)) • ∏ x, A ((decomposeFin.symm (i.succ, y)) x) x =\n -1 * (↑↑(sign i.cycleRang... | [
"case refine_2\nR : Type v\ninst✝ : CommRing R\nn : ℕ\nA : Matrix (Fin n.succ) (Fin n.succ) R\ni✝ : Fin n.succ\nx✝ : i✝ ∈ univ\ni : Fin n\nthis : (-1) ^ ↑i = ↑↑(sign i.cycleRange)\n⊢ ∑ y, sign (decomposeFin.symm (i.succ, y)) • ∏ x, A ((decomposeFin.symm (i.succ, y)) x) x =\n -1 * (A i.succ 0 * (↑↑(sign i.cycleRa... | mul_left_comm (ε _), | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.Limits.Creates | {
"line": 674,
"column": 4
} | {
"line": 674,
"column": 36
} | {
"line": 675,
"column": 4
} | [
{
"pp": "C : Type u₁\ninst✝⁴ : Category.{v₁, u₁} C\nD : Type u₂\ninst✝³ : Category.{v₂, u₂} D\nJ : Type w\ninst✝² : Category.{w', w} J\nK : J ⥤ C\nE : Type u₃\nℰ : Category.{v₃, u₃} E\nF : C ⥤ D\nG : D ⥤ E\ninst✝¹ : CreatesLimit K F\ninst✝ : CreatesLimit (K ⋙ F) G\nc : Cone (K ⋙ F ⋙ G)\nt : IsLimit c\n⊢ Liftabl... | [
"C : Type u₁\ninst✝⁴ : Category.{v₁, u₁} C\nD : Type u₂\ninst✝³ : Category.{v₂, u₂} D\nJ : Type w\ninst✝² : Category.{w', w} J\nK : J ⥤ C\nE : Type u₃\nℰ : Category.{v₃, u₃} E\nF : C ⥤ D\nG : D ⥤ E\ninst✝¹ : CreatesLimit K F\ninst✝ : CreatesLimit (K ⋙ F) G\nc : Cone (K ⋙ F ⋙ G)\nt : IsLimit c\nc' : Cone ((K ⋙ F) ⋙ ... | let c' : Cone ((K ⋙ F) ⋙ G) := c | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLet___1 | Lean.Parser.Tactic.tacticLet__ |
Mathlib.CategoryTheory.Monoidal.Category | {
"line": 496,
"column": 2
} | {
"line": 496,
"column": 19
} | {
"line": 498,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝² : Category.{v, u} C\ninst✝¹ : MonoidalCategory C\nP : Prop\ninst✝ : Decidable P\nX Y Z : C\nf : P → (Y ⟶ Z)\nf' : ¬P → (Y ⟶ Z)\n⊢ (X ◁ if h : P then f h else f' h) = if h : P then X ◁ f h else X ◁ f' h",
"ppTerm": "?m.34",
"assigned": true,
"usedConstants": [
"Eq.mp... | [] | split_ifs <;> rfl | Lean.Parser.Tactic.«_aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tactic_<;>__1» | Lean.Parser.Tactic.«tactic_<;>_» |
Mathlib.CategoryTheory.Monoidal.Category | {
"line": 496,
"column": 2
} | {
"line": 496,
"column": 19
} | {
"line": 498,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝² : Category.{v, u} C\ninst✝¹ : MonoidalCategory C\nP : Prop\ninst✝ : Decidable P\nX Y Z : C\nf : P → (Y ⟶ Z)\nf' : ¬P → (Y ⟶ Z)\n⊢ (X ◁ if h : P then f h else f' h) = if h : P then X ◁ f h else X ◁ f' h",
"ppTerm": "?m.34",
"assigned": true,
"usedConstants": [
"Eq.mp... | [] | split_ifs <;> rfl | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Monoidal.Category | {
"line": 496,
"column": 2
} | {
"line": 496,
"column": 19
} | {
"line": 498,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝² : Category.{v, u} C\ninst✝¹ : MonoidalCategory C\nP : Prop\ninst✝ : Decidable P\nX Y Z : C\nf : P → (Y ⟶ Z)\nf' : ¬P → (Y ⟶ Z)\n⊢ (X ◁ if h : P then f h else f' h) = if h : P then X ◁ f h else X ◁ f' h",
"ppTerm": "?m.34",
"assigned": true,
"usedConstants": [
"Eq.mp... | [] | split_ifs <;> rfl | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Monoidal.Category | {
"line": 501,
"column": 2
} | {
"line": 501,
"column": 19
} | {
"line": 503,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝² : Category.{v, u} C\ninst✝¹ : MonoidalCategory C\nP : Prop\ninst✝ : Decidable P\nX Y : C\nf : P → (X ⟶ Y)\nf' : ¬P → (X ⟶ Y)\nZ : C\n⊢ (if h : P then f h else f' h) ▷ Z = if h : P then f h ▷ Z else f' h ▷ Z",
"ppTerm": "?m.32",
"assigned": true,
"usedConstants": [
"... | [] | split_ifs <;> rfl | Lean.Parser.Tactic.«_aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tactic_<;>__1» | Lean.Parser.Tactic.«tactic_<;>_» |
Mathlib.CategoryTheory.Monoidal.Category | {
"line": 501,
"column": 2
} | {
"line": 501,
"column": 19
} | {
"line": 503,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝² : Category.{v, u} C\ninst✝¹ : MonoidalCategory C\nP : Prop\ninst✝ : Decidable P\nX Y : C\nf : P → (X ⟶ Y)\nf' : ¬P → (X ⟶ Y)\nZ : C\n⊢ (if h : P then f h else f' h) ▷ Z = if h : P then f h ▷ Z else f' h ▷ Z",
"ppTerm": "?m.32",
"assigned": true,
"usedConstants": [
"... | [] | split_ifs <;> rfl | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Monoidal.Category | {
"line": 501,
"column": 2
} | {
"line": 501,
"column": 19
} | {
"line": 503,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝² : Category.{v, u} C\ninst✝¹ : MonoidalCategory C\nP : Prop\ninst✝ : Decidable P\nX Y : C\nf : P → (X ⟶ Y)\nf' : ¬P → (X ⟶ Y)\nZ : C\n⊢ (if h : P then f h else f' h) ▷ Z = if h : P then f h ▷ Z else f' h ▷ Z",
"ppTerm": "?m.32",
"assigned": true,
"usedConstants": [
"... | [] | split_ifs <;> rfl | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Monoidal.Category | {
"line": 505,
"column": 48
} | {
"line": 505,
"column": 65
} | {
"line": 507,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝² : Category.{v, u} C\ninst✝¹ : MonoidalCategory C\nP : Prop\ninst✝ : Decidable P\nW X Y Z : C\nf : W ⟶ X\ng : P → (Y ⟶ Z)\ng' : ¬P → (Y ⟶ Z)\n⊢ (f ⊗ₘ if h : P then g h else g' h) = if h : P then f ⊗ₘ g h else f ⊗ₘ g' h",
"ppTerm": "?m.47",
"assigned": true,
"usedConstants"... | [] | split_ifs <;> rfl | Lean.Parser.Tactic.«_aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tactic_<;>__1» | Lean.Parser.Tactic.«tactic_<;>_» |
Mathlib.CategoryTheory.Monoidal.Category | {
"line": 505,
"column": 48
} | {
"line": 505,
"column": 65
} | {
"line": 507,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝² : Category.{v, u} C\ninst✝¹ : MonoidalCategory C\nP : Prop\ninst✝ : Decidable P\nW X Y Z : C\nf : W ⟶ X\ng : P → (Y ⟶ Z)\ng' : ¬P → (Y ⟶ Z)\n⊢ (f ⊗ₘ if h : P then g h else g' h) = if h : P then f ⊗ₘ g h else f ⊗ₘ g' h",
"ppTerm": "?m.47",
"assigned": true,
"usedConstants"... | [] | split_ifs <;> rfl | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Monoidal.Category | {
"line": 505,
"column": 48
} | {
"line": 505,
"column": 65
} | {
"line": 507,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝² : Category.{v, u} C\ninst✝¹ : MonoidalCategory C\nP : Prop\ninst✝ : Decidable P\nW X Y Z : C\nf : W ⟶ X\ng : P → (Y ⟶ Z)\ng' : ¬P → (Y ⟶ Z)\n⊢ (f ⊗ₘ if h : P then g h else g' h) = if h : P then f ⊗ₘ g h else f ⊗ₘ g' h",
"ppTerm": "?m.47",
"assigned": true,
"usedConstants"... | [] | split_ifs <;> rfl | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Monoidal.Category | {
"line": 509,
"column": 48
} | {
"line": 509,
"column": 65
} | {
"line": 511,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝² : Category.{v, u} C\ninst✝¹ : MonoidalCategory C\nP : Prop\ninst✝ : Decidable P\nW X Y Z : C\nf : W ⟶ X\ng : P → (Y ⟶ Z)\ng' : ¬P → (Y ⟶ Z)\n⊢ (if h : P then g h else g' h) ⊗ₘ f = if h : P then g h ⊗ₘ f else g' h ⊗ₘ f",
"ppTerm": "?m.47",
"assigned": true,
"usedConstants"... | [] | split_ifs <;> rfl | Lean.Parser.Tactic.«_aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tactic_<;>__1» | Lean.Parser.Tactic.«tactic_<;>_» |
Mathlib.CategoryTheory.Monoidal.Category | {
"line": 509,
"column": 48
} | {
"line": 509,
"column": 65
} | {
"line": 511,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝² : Category.{v, u} C\ninst✝¹ : MonoidalCategory C\nP : Prop\ninst✝ : Decidable P\nW X Y Z : C\nf : W ⟶ X\ng : P → (Y ⟶ Z)\ng' : ¬P → (Y ⟶ Z)\n⊢ (if h : P then g h else g' h) ⊗ₘ f = if h : P then g h ⊗ₘ f else g' h ⊗ₘ f",
"ppTerm": "?m.47",
"assigned": true,
"usedConstants"... | [] | split_ifs <;> rfl | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Monoidal.Category | {
"line": 509,
"column": 48
} | {
"line": 509,
"column": 65
} | {
"line": 511,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝² : Category.{v, u} C\ninst✝¹ : MonoidalCategory C\nP : Prop\ninst✝ : Decidable P\nW X Y Z : C\nf : W ⟶ X\ng : P → (Y ⟶ Z)\ng' : ¬P → (Y ⟶ Z)\n⊢ (if h : P then g h else g' h) ⊗ₘ f = if h : P then g h ⊗ₘ f else g' h ⊗ₘ f",
"ppTerm": "?m.47",
"assigned": true,
"usedConstants"... | [] | split_ifs <;> rfl | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Monoidal.Functor | {
"line": 124,
"column": 2
} | {
"line": 125,
"column": 22
} | {
"line": 127,
"column": 0
} | [
{
"pp": "C : Type u₁\ninst✝⁴ : Category.{v₁, u₁} C\ninst✝³ : MonoidalCategory C\nD : Type u₂\ninst✝² : Category.{v₂, u₂} D\ninst✝¹ : MonoidalCategory D\nF : C ⥤ D\ninst✝ : F.LaxMonoidal\nX Y Z : C\n⊢ F.obj X ◁ μ F Y Z ≫ μ F X (Y ⊗ Z) ≫ F.map (α_ X Y Z).inv =\n (α_ (F.obj X) (F.obj Y) (F.obj Z)).inv ≫ μ F X Y... | [] | rw [Iso.eq_inv_comp, ← associativity_assoc, ← F.map_comp, Iso.hom_inv_id,
F.map_id, comp_id] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.CategoryTheory.Monoidal.Functor | {
"line": 124,
"column": 2
} | {
"line": 125,
"column": 22
} | {
"line": 127,
"column": 0
} | [
{
"pp": "C : Type u₁\ninst✝⁴ : Category.{v₁, u₁} C\ninst✝³ : MonoidalCategory C\nD : Type u₂\ninst✝² : Category.{v₂, u₂} D\ninst✝¹ : MonoidalCategory D\nF : C ⥤ D\ninst✝ : F.LaxMonoidal\nX Y Z : C\n⊢ F.obj X ◁ μ F Y Z ≫ μ F X (Y ⊗ Z) ≫ F.map (α_ X Y Z).inv =\n (α_ (F.obj X) (F.obj Y) (F.obj Z)).inv ≫ μ F X Y... | [] | rw [Iso.eq_inv_comp, ← associativity_assoc, ← F.map_comp, Iso.hom_inv_id,
F.map_id, comp_id] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Monoidal.Functor | {
"line": 124,
"column": 2
} | {
"line": 125,
"column": 22
} | {
"line": 127,
"column": 0
} | [
{
"pp": "C : Type u₁\ninst✝⁴ : Category.{v₁, u₁} C\ninst✝³ : MonoidalCategory C\nD : Type u₂\ninst✝² : Category.{v₂, u₂} D\ninst✝¹ : MonoidalCategory D\nF : C ⥤ D\ninst✝ : F.LaxMonoidal\nX Y Z : C\n⊢ F.obj X ◁ μ F Y Z ≫ μ F X (Y ⊗ Z) ≫ F.map (α_ X Y Z).inv =\n (α_ (F.obj X) (F.obj Y) (F.obj Z)).inv ≫ μ F X Y... | [] | rw [Iso.eq_inv_comp, ← associativity_assoc, ← F.map_comp, Iso.hom_inv_id,
F.map_id, comp_id] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Monoidal.Functor | {
"line": 489,
"column": 59
} | {
"line": 489,
"column": 74
} | {
"line": 489,
"column": 74
} | [
{
"pp": "C : Type u₁\ninst✝⁴ : Category.{v₁, u₁} C\ninst✝³ : MonoidalCategory C\nD : Type u₂\ninst✝² : Category.{v₂, u₂} D\ninst✝¹ : MonoidalCategory D\nF : C ⥤ D\ninst✝ : F.Monoidal\nX Y Z : C\n⊢ (α_ (F.obj X) (F.obj Y) (F.obj Z)).hom ≫ (α_ (F.obj X) (F.obj Y) (F.obj Z)).inv =\n (α_ (F.obj X) (F.obj Y) (F.o... | [
"C : Type u₁\ninst✝⁴ : Category.{v₁, u₁} C\ninst✝³ : MonoidalCategory C\nD : Type u₂\ninst✝² : Category.{v₂, u₂} D\ninst✝¹ : MonoidalCategory D\nF : C ⥤ D\ninst✝ : F.Monoidal\nX Y Z : C\n⊢ (μ F X Y ▷ F.obj Z ≫ μ F (X ⊗ Y) Z ≫ F.map (α_ X Y Z).hom ≫ δ F X (Y ⊗ Z) ≫ F.obj X ◁ δ F Y Z) ≫\n (α_ (F.obj X) (F.obj Y)... | map_associator' | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.Monoidal.Functor | {
"line": 558,
"column": 4
} | {
"line": 558,
"column": 24
} | {
"line": 559,
"column": 2
} | [
{
"pp": "case μ\nC : Type u₁\ninst✝³ : Category.{v₁, u₁} C\ninst✝² : MonoidalCategory C\nD : Type u₂\ninst✝¹ : Category.{v₂, u₂} D\ninst✝ : MonoidalCategory D\nF : C ⥤ D\na b : F.Monoidal\neq : a.toLaxMonoidal = b.toLaxMonoidal\n⊢ μ F = μ F",
"ppTerm": "?μ",
"assigned": true,
"usedConstants": [
... | [] | exact congr(($eq).μ) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.CategoryTheory.Monoidal.Functor | {
"line": 558,
"column": 4
} | {
"line": 558,
"column": 24
} | {
"line": 559,
"column": 2
} | [
{
"pp": "case μ\nC : Type u₁\ninst✝³ : Category.{v₁, u₁} C\ninst✝² : MonoidalCategory C\nD : Type u₂\ninst✝¹ : Category.{v₂, u₂} D\ninst✝ : MonoidalCategory D\nF : C ⥤ D\na b : F.Monoidal\neq : a.toLaxMonoidal = b.toLaxMonoidal\n⊢ μ F = μ F",
"ppTerm": "?μ",
"assigned": true,
"usedConstants": [
... | [] | exact congr(($eq).μ) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Monoidal.Functor | {
"line": 558,
"column": 4
} | {
"line": 558,
"column": 24
} | {
"line": 559,
"column": 2
} | [
{
"pp": "case μ\nC : Type u₁\ninst✝³ : Category.{v₁, u₁} C\ninst✝² : MonoidalCategory C\nD : Type u₂\ninst✝¹ : Category.{v₂, u₂} D\ninst✝ : MonoidalCategory D\nF : C ⥤ D\na b : F.Monoidal\neq : a.toLaxMonoidal = b.toLaxMonoidal\n⊢ μ F = μ F",
"ppTerm": "?μ",
"assigned": true,
"usedConstants": [
... | [] | exact congr(($eq).μ) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Category.ModuleCat.Monoidal.Basic | {
"line": 124,
"column": 2
} | {
"line": 124,
"column": 34
} | {
"line": 125,
"column": 2
} | [
{
"pp": "R : Type u\ninst✝ : CommSemiring R\nW : SemimoduleCat R\nX : SemimoduleCat R\nY : SemimoduleCat R\nZ : SemimoduleCat R\n⊢ Hom.hom\n (whiskerRight (associator W X Y).hom Z ≫\n (associator W (tensorObj X Y) Z).hom ≫ whiskerLeft W (associator X Y Z).hom) =\n Hom.hom ((associator (tensorObj ... | [
"R : Type u\ninst✝ : CommSemiring R\nW : SemimoduleCat R\nX : SemimoduleCat R\nY : SemimoduleCat R\nZ : SemimoduleCat R\n⊢ ∀ (w : ↑W) (x : ↑X) (y : ↑Y) (z : ↑Z),\n (Hom.hom\n (whiskerRight (associator W X Y).hom Z ≫\n (associator W (tensorObj X Y) Z).hom ≫ whiskerLeft W (associator X Y Z).hom... | apply TensorProduct.ext_fourfold | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.Tactic.CategoryTheory.Monoidal.PureCoherence | {
"line": 224,
"column": 6
} | {
"line": 224,
"column": 49
} | {
"line": 226,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : MonoidalCategory C\nf g f' : C\nη θ : f ≅ g\nη_f : 𝟙_ C ⊗ f ≅ f'\nη_g : 𝟙_ C ⊗ g ≅ f'\nh_η : 𝟙_ C ◁ η ≪≫ η_g = η_f\nh_θ : 𝟙_ C ◁ θ ≪≫ η_g = η_f\n⊢ (λ_ f).inv ≫ η_f.hom ≫ η_g.inv ≫ (λ_ g).hom = θ.hom",
"ppTerm": "?m.105",
"assigned": true,
... | [] | simp [← reassoc_of% (congrArg Iso.hom h_θ)] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Tactic.CategoryTheory.Monoidal.PureCoherence | {
"line": 224,
"column": 6
} | {
"line": 224,
"column": 49
} | {
"line": 226,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : MonoidalCategory C\nf g f' : C\nη θ : f ≅ g\nη_f : 𝟙_ C ⊗ f ≅ f'\nη_g : 𝟙_ C ⊗ g ≅ f'\nh_η : 𝟙_ C ◁ η ≪≫ η_g = η_f\nh_θ : 𝟙_ C ◁ θ ≪≫ η_g = η_f\n⊢ (λ_ f).inv ≫ η_f.hom ≫ η_g.inv ≫ (λ_ g).hom = θ.hom",
"ppTerm": "?m.105",
"assigned": true,
... | [] | simp [← reassoc_of% (congrArg Iso.hom h_θ)] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Tactic.CategoryTheory.Monoidal.PureCoherence | {
"line": 224,
"column": 6
} | {
"line": 224,
"column": 49
} | {
"line": 226,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : MonoidalCategory C\nf g f' : C\nη θ : f ≅ g\nη_f : 𝟙_ C ⊗ f ≅ f'\nη_g : 𝟙_ C ⊗ g ≅ f'\nh_η : 𝟙_ C ◁ η ≪≫ η_g = η_f\nh_θ : 𝟙_ C ◁ θ ≪≫ η_g = η_f\n⊢ (λ_ f).inv ≫ η_f.hom ≫ η_g.inv ≫ (λ_ g).hom = θ.hom",
"ppTerm": "?m.105",
"assigned": true,
... | [] | simp [← reassoc_of% (congrArg Iso.hom h_θ)] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Monoidal.Functor | {
"line": 1322,
"column": 4
} | {
"line": 1322,
"column": 40
} | {
"line": 1323,
"column": 2
} | [
{
"pp": "case e_a\nC : Type u₁\ninst✝⁷ : Category.{v₁, u₁} C\ninst✝⁶ : MonoidalCategory C\nD : Type u₂\ninst✝⁵ : Category.{v₂, u₂} D\ninst✝⁴ : MonoidalCategory D\nE : Type u₃\ninst✝³ : Category.{v₃, u₃} E\ninst✝² : MonoidalCategory E\nC' : Type u₁'\ninst✝¹ : Category.{v₁', u₁'} C'\nF G : C ⥤ D\ninst✝ : F.Monoid... | [] | simp [← id_tensorHom, -tensorHom_id] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Algebra.Ring.BooleanRing | {
"line": 92,
"column": 61
} | {
"line": 92,
"column": 69
} | {
"line": 92,
"column": 70
} | [
{
"pp": "α : Type u_1\ninst✝ : BooleanRing α\na b : α\n⊢ a * (a + b) + b * (a + b) = a * a + a * b + (b * a + b * b)",
"ppTerm": "?m.156",
"assigned": true,
"usedConstants": [
"Distrib.leftDistribClass",
"Eq.mpr",
"HMul.hMul",
"congrArg",
"id",
"Distrib.toAdd",
... | [
"α : Type u_1\ninst✝ : BooleanRing α\na b : α\n⊢ a * a + a * b + b * (a + b) = a * a + a * b + (b * a + b * b)"
] | mul_add, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Ring.BooleanRing | {
"line": 101,
"column": 53
} | {
"line": 101,
"column": 61
} | {
"line": 101,
"column": 62
} | [
{
"pp": "α : Type u_1\ninst✝ : BooleanRing α\na : α\n⊢ a * (1 + a) = 0",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"Distrib.leftDistribClass",
"Eq.mpr",
"HMul.hMul",
"congrArg",
"AddGroupWithOne.toAddMonoidWithOne",
"id",
"Distrib.toAdd",
... | [
"α : Type u_1\ninst✝ : BooleanRing α\na : α\n⊢ a * 1 + a * a = 0"
] | mul_add, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Ring.BooleanRing | {
"line": 195,
"column": 6
} | {
"line": 195,
"column": 14
} | {
"line": 195,
"column": 15
} | [
{
"pp": "α : Type u_1\ninst✝ : BooleanRing α\na b : α\n⊢ a * (a + b + a * b) = a",
"ppTerm": "?m.7",
"assigned": true,
"usedConstants": [
"Distrib.leftDistribClass",
"Eq.mpr",
"HMul.hMul",
"congrArg",
"CommSemiring.toSemiring",
"BooleanRing.toCommRing",
"id"... | [
"α : Type u_1\ninst✝ : BooleanRing α\na b : α\n⊢ a * (a + b) + a * (a * b) = a"
] | mul_add, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Ring.BooleanRing | {
"line": 195,
"column": 15
} | {
"line": 195,
"column": 23
} | {
"line": 195,
"column": 24
} | [
{
"pp": "α : Type u_1\ninst✝ : BooleanRing α\na b : α\n⊢ a * (a + b) + a * (a * b) = a",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"Distrib.leftDistribClass",
"Eq.mpr",
"HMul.hMul",
"congrArg",
"CommSemiring.toSemiring",
"BooleanRing.toCommRing",
... | [
"α : Type u_1\ninst✝ : BooleanRing α\na b : α\n⊢ a * a + a * b + a * (a * b) = a"
] | mul_add, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.Category.Pointed | {
"line": 74,
"column": 22
} | {
"line": 74,
"column": 75
} | {
"line": 74,
"column": 75
} | [
{
"pp": "X Y Z : Pointed\nf : X.Hom Y\ng : Y.Hom Z\n⊢ (g.toFun ∘ f.toFun) X.point = Z.point",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"Function.comp",
"id",
"Pointed.Hom.map_point",
"Pointed.point",
"Eq.refl",
"Fun... | [] | by rw [Function.comp_apply, f.map_point, g.map_point] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.Coalgebra.TensorProduct | {
"line": 144,
"column": 21
} | {
"line": 144,
"column": 32
} | {
"line": 144,
"column": 33
} | [
{
"pp": "case e'_2.tmul.tmul\nR : Type u_1\nS : Type u_2\nA : Type u_3\nB : Type u_4\ninst✝¹⁰ : CommSemiring R\ninst✝⁹ : CommSemiring S\ninst✝⁸ : AddCommMonoid A\ninst✝⁷ : AddCommMonoid B\ninst✝⁶ : Algebra R S\ninst✝⁵ : Module R A\ninst✝⁴ : Module S A\ninst✝³ : Module R B\ninst✝² : IsScalarTower R S A\ninst✝¹ :... | [
"case e'_2.tmul.tmul\nR : Type u_1\nS : Type u_2\nA : Type u_3\nB : Type u_4\ninst✝¹⁰ : CommSemiring R\ninst✝⁹ : CommSemiring S\ninst✝⁸ : AddCommMonoid A\ninst✝⁷ : AddCommMonoid B\ninst✝⁶ : Algebra R S\ninst✝⁵ : Module R A\ninst✝⁴ : Module S A\ninst✝³ : Module R B\ninst✝² : IsScalarTower R S A\ninst✝¹ : Coalgebra R... | smul_assoc, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Coalgebra.TensorProduct | {
"line": 158,
"column": 21
} | {
"line": 158,
"column": 32
} | {
"line": 158,
"column": 33
} | [
{
"pp": "case e'_2.tmul.tmul\nR : Type u_1\nS : Type u_2\nA : Type u_3\nB : Type u_4\ninst✝¹⁰ : CommSemiring R\ninst✝⁹ : CommSemiring S\ninst✝⁸ : AddCommMonoid A\ninst✝⁷ : AddCommMonoid B\ninst✝⁶ : Algebra R S\ninst✝⁵ : Module R A\ninst✝⁴ : Module S A\ninst✝³ : Module R B\ninst✝² : IsScalarTower R S A\ninst✝¹ :... | [
"case e'_2.tmul.tmul\nR : Type u_1\nS : Type u_2\nA : Type u_3\nB : Type u_4\ninst✝¹⁰ : CommSemiring R\ninst✝⁹ : CommSemiring S\ninst✝⁸ : AddCommMonoid A\ninst✝⁷ : AddCommMonoid B\ninst✝⁶ : Algebra R S\ninst✝⁵ : Module R A\ninst✝⁴ : Module S A\ninst✝³ : Module R B\ninst✝² : IsScalarTower R S A\ninst✝¹ : Coalgebra R... | smul_assoc, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.Adjunction.Mates | {
"line": 216,
"column": 2
} | {
"line": 228,
"column": 57
} | {
"line": 230,
"column": 0
} | [
{
"pp": "A : Type u₁\nB : Type u₂\nC : Type u₃\nD : Type u₄\nE : Type u₅\nF : Type u₆\ninst✝⁵ : Category.{v₁, u₁} A\ninst✝⁴ : Category.{v₂, u₂} B\ninst✝³ : Category.{v₃, u₃} C\ninst✝² : Category.{v₄, u₄} D\ninst✝¹ : Category.{v₅, u₅} E\ninst✝ : Category.{v₆, u₆} F\nG : A ⥤ D\nH : B ⥤ E\nK : C ⥤ F\nL₁ : A ⥤ B\nR... | [] | unfold vComp hComp mateEquiv Adjunction.comp
ext c
simp only [comp_obj, whiskerRight_comp, assoc, mk'_unit, whiskerLeft_comp, mk'_counit,
whiskerRight_twice, Iso.inv_hom_id_assoc, Equiv.coe_fn_mk, comp_app, id_obj,
rightUnitor_inv_app, Functor.whiskerLeft_app, Functor.whiskerRight_app, map_id,
associato... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Adjunction.Mates | {
"line": 216,
"column": 2
} | {
"line": 228,
"column": 57
} | {
"line": 230,
"column": 0
} | [
{
"pp": "A : Type u₁\nB : Type u₂\nC : Type u₃\nD : Type u₄\nE : Type u₅\nF : Type u₆\ninst✝⁵ : Category.{v₁, u₁} A\ninst✝⁴ : Category.{v₂, u₂} B\ninst✝³ : Category.{v₃, u₃} C\ninst✝² : Category.{v₄, u₄} D\ninst✝¹ : Category.{v₅, u₅} E\ninst✝ : Category.{v₆, u₆} F\nG : A ⥤ D\nH : B ⥤ E\nK : C ⥤ F\nL₁ : A ⥤ B\nR... | [] | unfold vComp hComp mateEquiv Adjunction.comp
ext c
simp only [comp_obj, whiskerRight_comp, assoc, mk'_unit, whiskerLeft_comp, mk'_counit,
whiskerRight_twice, Iso.inv_hom_id_assoc, Equiv.coe_fn_mk, comp_app, id_obj,
rightUnitor_inv_app, Functor.whiskerLeft_app, Functor.whiskerRight_app, map_id,
associato... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Adjunction.Mates | {
"line": 329,
"column": 2
} | {
"line": 329,
"column": 31
} | {
"line": 331,
"column": 0
} | [
{
"pp": "C : Type u₁\nD : Type u₂\ninst✝¹ : Category.{v₁, u₁} C\ninst✝ : Category.{v₂, u₂} D\nL₁ : C ⥤ D\nR₁ : D ⥤ C\nadj₁ : L₁ ⊣ R₁\n⊢ 𝟙 R₁ = (conjugateEquiv adj₁ adj₁) (𝟙 L₁)",
"ppTerm": "?m.41",
"assigned": true,
"usedConstants": [
"CategoryTheory.Functor",
"Equiv.instEquivLike",
... | [] | simp only [conjugateEquiv_id] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.RingTheory.Localization.Away.Basic | {
"line": 125,
"column": 4
} | {
"line": 127,
"column": 35
} | {
"line": 128,
"column": 2
} | [
{
"pp": "case refine_2\nR : Type u_1\ninst✝³ : CommSemiring R\nS : Type u_2\ninst✝² : CommSemiring S\ninst✝¹ : Algebra R S\nr : R\ninst✝ : Away r S\nu : Rˣ\ns : S\n⊢ ∃ n a, s * (algebraMap R S) (r * ↑u) ^ n = (algebraMap R S) a",
"ppTerm": "?refine_2",
"assigned": true,
"usedConstants": [
"Uni... | [] | obtain ⟨n, a, hn⟩ := surj r s
use n, a * u ^ n
simp [mul_pow, ← mul_assoc, hn] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Localization.Away.Basic | {
"line": 125,
"column": 4
} | {
"line": 127,
"column": 35
} | {
"line": 128,
"column": 2
} | [
{
"pp": "case refine_2\nR : Type u_1\ninst✝³ : CommSemiring R\nS : Type u_2\ninst✝² : CommSemiring S\ninst✝¹ : Algebra R S\nr : R\ninst✝ : Away r S\nu : Rˣ\ns : S\n⊢ ∃ n a, s * (algebraMap R S) (r * ↑u) ^ n = (algebraMap R S) a",
"ppTerm": "?refine_2",
"assigned": true,
"usedConstants": [
"Uni... | [] | obtain ⟨n, a, hn⟩ := surj r s
use n, a * u ^ n
simp [mul_pow, ← mul_assoc, hn] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Jacobson.Radical | {
"line": 158,
"column": 4
} | {
"line": 158,
"column": 22
} | {
"line": 159,
"column": 2
} | [
{
"pp": "R : Type u_1\ninst✝¹ : Ring R\nI : Ideal R\ninst✝ : I.IsTwoSided\nf : R ⧸ I →ₛₗ[Ideal.Quotient.mk I] R ⧸ I := { toAddHom := AddHom.id (R ⧸ I), map_smul' := ⋯ }\n⊢ ↑(Submodule.map f (Module.jacobson R (R ⧸ I))) = ↑(Module.jacobson R (R ⧸ I))",
"ppTerm": "?m.75",
"assigned": true,
"usedConsta... | [] | apply Set.image_id | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
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