module stringlengths 16 90 | startPos dict | endPos dict | goals listlengths 0 96 | ppTac stringlengths 1 14.5k | elaborator stringclasses 366
values | kind stringclasses 370
values |
|---|---|---|---|---|---|---|
Mathlib.LinearAlgebra.Multilinear.Basic | {
"line": 1106,
"column": 4
} | {
"line": 1114,
"column": 44
} | [
{
"pp": "R : Type uR\nS : Type uS\nι : Type uι\nn : ℕ\nM : Fin n.succ → Type v\nM₁ : ι → Type v₁\nM₁' : ι → Type v₁'\nM₁'' : ι → Type v₁''\nM₂ : Type v₂\nM₃ : Type v₃\nM₄ : Type v₄\nM' : Type v'\ninst✝⁸ : CommSemiring R\ninst✝⁷ : (i : ι) → AddCommMonoid (M₁ i)\ninst✝⁶ : (i : Fin n.succ) → AddCommMonoid (M i)\ni... | intro _ f i f₁ f₂
ext g x
change (g fun j ↦ update f i (f₁ + f₂) j <| x j) =
(g fun j ↦ update f i f₁ j <| x j) + g fun j ↦ update f i f₂ j (x j)
let c : Π (i : ι), (M₁ i →ₗ[R] M₁' i) → M₁' i := fun i f ↦ f (x i)
convert! g.map_update_add (fun j ↦ f j (x j)) i (f₁ (x i)) (f₂ (x i)) with j j j
... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.Multilinear.Basic | {
"line": 1183,
"column": 2
} | {
"line": 1183,
"column": 10
} | [
{
"pp": "case H\nR : Type uR\nι : Type uι\nM₂ : Type v₂\ninst✝³ : CommSemiring R\ninst✝² : AddCommMonoid M₂\ninst✝¹ : Module R M₂\ninst✝ : Finite ι\nf g : MultilinearMap R (fun x ↦ R) M₂\nh : (f fun x ↦ 1) = g fun x ↦ 1\nx : ι → R\nval✝ : Fintype ι\nhf : (f fun i ↦ x i • 1) = (∏ i, x i) • f fun x ↦ 1\nhg : (g f... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.LinearAlgebra.Multilinear.Basic | {
"line": 1366,
"column": 6
} | {
"line": 1366,
"column": 63
} | [
{
"pp": "case neg.inl\nR : Type uR\nι : Type uι\nM₁ : ι → Type v₁\nM₂ : Type v₂\ninst✝⁵ : Semiring R\ninst✝⁴ : (i : ι) → AddCommGroup (M₁ i)\ninst✝³ : AddCommGroup M₂\ninst✝² : (i : ι) → Module R (M₁ i)\ninst✝¹ : Module R M₂\nf : MultilinearMap R M₁ M₂\ninst✝ : LinearOrder ι\na b v : (i : ι) → M₁ i\ns : Finset ... | rw [if_pos rfl, if_pos rfl, s.piecewise_eq_of_mem _ _ hi] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.LinearAlgebra.Multilinear.Basic | {
"line": 1366,
"column": 6
} | {
"line": 1366,
"column": 63
} | [
{
"pp": "case neg.inl\nR : Type uR\nι : Type uι\nM₁ : ι → Type v₁\nM₂ : Type v₂\ninst✝⁵ : Semiring R\ninst✝⁴ : (i : ι) → AddCommGroup (M₁ i)\ninst✝³ : AddCommGroup M₂\ninst✝² : (i : ι) → Module R (M₁ i)\ninst✝¹ : Module R M₂\nf : MultilinearMap R M₁ M₂\ninst✝ : LinearOrder ι\na b v : (i : ι) → M₁ i\ns : Finset ... | rw [if_pos rfl, if_pos rfl, s.piecewise_eq_of_mem _ _ hi] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.Multilinear.Basic | {
"line": 1366,
"column": 6
} | {
"line": 1366,
"column": 63
} | [
{
"pp": "case neg.inl\nR : Type uR\nι : Type uι\nM₁ : ι → Type v₁\nM₂ : Type v₂\ninst✝⁵ : Semiring R\ninst✝⁴ : (i : ι) → AddCommGroup (M₁ i)\ninst✝³ : AddCommGroup M₂\ninst✝² : (i : ι) → Module R (M₁ i)\ninst✝¹ : Module R M₂\nf : MultilinearMap R M₁ M₂\ninst✝ : LinearOrder ι\na b v : (i : ι) → M₁ i\ns : Finset ... | rw [if_pos rfl, if_pos rfl, s.piecewise_eq_of_mem _ _ hi] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Limits.Preserves.Limits | {
"line": 175,
"column": 4
} | {
"line": 175,
"column": 61
} | [
{
"pp": "C : Type u₁\ninst✝⁵ : Category.{v₁, u₁} C\nD : Type u₂\ninst✝⁴ : Category.{v₂, u₂} D\nG : C ⥤ D\nJ : Type w\ninst✝³ : Category.{w', w} J\nF : J ⥤ C\ninst✝² : HasColimit F\ninst✝¹ : HasColimit (F ⋙ G)\ninst✝ : IsIso (colimit.post F G)\n⊢ IsColimit (G.mapCocone (colimit.cocone F))",
"usedConstants": ... | convert! IsColimit.ofPointIso (colimit.isColimit (F ⋙ G)) | Mathlib.Tactic._aux_Mathlib_Tactic_Convert___macroRules_Mathlib_Tactic_convert!_1 | Mathlib.Tactic.convert! |
Mathlib.CategoryTheory.Limits.Types.Filtered | {
"line": 164,
"column": 2
} | {
"line": 165,
"column": 80
} | [
{
"pp": "J : Type v\ninst✝¹ : Category.{w, v} J\nF : J ⥤ Type u\ninst✝ : IsFilteredOrEmpty J\nt : Cocone F\nht : IsColimit t\nj₁ : J\nx₁ : F.obj j₁\nj₂ : J\nx₂ : F.obj j₂\n⊢ ∃ j x₁' x₂', (hom (t.ι.app j)) x₁' = (hom (t.ι.app j₁)) x₁ ∧ (hom (t.ι.app j)) x₂' = (hom (t.ι.app j₂)) x₂",
"usedConstants": [
... | exact ⟨max j₁ j₂, F.map (leftToMax _ _) x₁, F.map (rightToMax _ _) x₂,
congr_hom (t.w (leftToMax j₁ j₂)) x₁, congr_hom (t.w (rightToMax j₁ j₂)) x₂⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.CategoryTheory.Filtered.Basic | {
"line": 349,
"column": 6
} | {
"line": 349,
"column": 68
} | [
{
"pp": "case refine_1\nC : Type u\ninst✝ : Category.{v, u} C\nh : ∀ {J : Type w} [inst : SmallCategory J] [FinCategory J] (F : J ⥤ C), Nonempty (Cocone F)\nthis : Nonempty C\nX Y : C\n⊢ ∃ Z x x, True",
"usedConstants": [
"ULift",
"CategoryTheory.Functor.comp",
"CategoryTheory.ULift.downFu... | obtain ⟨c⟩ := h (ULiftHom.down ⋙ ULift.downFunctor ⋙ pair X Y) | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.CategoryTheory.Filtered.Basic | {
"line": 356,
"column": 6
} | {
"line": 356,
"column": 14
} | [
{
"pp": "case refine_2\nC : Type u\ninst✝ : Category.{v, u} C\nh : ∀ {J : Type w} [inst : SmallCategory J] [FinCategory J] (F : J ⥤ C), Nonempty (Cocone F)\nthis : Nonempty C\nX Y : C\nf g : X ⟶ Y\nc : Cocone (ULiftHom.down ⋙ ULift.downFunctor ⋙ parallelPair f g)\nh₁ :\n (ULiftHom.down ⋙ ULift.downFunctor ⋙ pa... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.CategoryTheory.Functor.Currying | {
"line": 95,
"column": 6
} | {
"line": 97,
"column": 77
} | [
{
"pp": "B : Type u₁\ninst✝⁴ : Category.{v₁, u₁} B\nC : Type u₂\ninst✝³ : Category.{v₂, u₂} C\nD : Type u₃\ninst✝² : Category.{v₃, u₃} D\nE : Type u₄\ninst✝¹ : Category.{v₄, u₄} E\nH : Type u₅\ninst✝ : Category.{v₅, u₅} H\nF : C × D ⥤ E\n⊢ ∀ {X Y : C × D} (f : X ⟶ Y),\n ((curry ⋙ uncurry).obj F).map f ≫ ((fu... | rintro ⟨X₁, X₂⟩ ⟨Y₁, Y₂⟩ ⟨f₁, f₂⟩
dsimp at f₁ f₂ ⊢
simp only [← F.map_comp, prod_comp, Category.comp_id, Category.id_comp] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Functor.Currying | {
"line": 95,
"column": 6
} | {
"line": 97,
"column": 77
} | [
{
"pp": "B : Type u₁\ninst✝⁴ : Category.{v₁, u₁} B\nC : Type u₂\ninst✝³ : Category.{v₂, u₂} C\nD : Type u₃\ninst✝² : Category.{v₃, u₃} D\nE : Type u₄\ninst✝¹ : Category.{v₄, u₄} E\nH : Type u₅\ninst✝ : Category.{v₅, u₅} H\nF : C × D ⥤ E\n⊢ ∀ {X Y : C × D} (f : X ⟶ Y),\n ((curry ⋙ uncurry).obj F).map f ≫ ((fu... | rintro ⟨X₁, X₂⟩ ⟨Y₁, Y₂⟩ ⟨f₁, f₂⟩
dsimp at f₁ f₂ ⊢
simp only [← F.map_comp, prod_comp, Category.comp_id, Category.id_comp] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Limits.Yoneda | {
"line": 109,
"column": 4
} | {
"line": 109,
"column": 99
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\nJ : Type w\ninst✝ : Category.{t, w} J\nF : J ⥤ Cᵒᵖ\nc : Cone F\nhc : (X : C) → IsLimit ((yoneda.obj X).mapCone c)\ns : Cone F\nj : J\n⊢ (Quiver.Hom.op ((hom ((hc (unop s.pt)).lift ((yoneda.obj (unop s.pt)).mapCone s))) (𝟙 (unop s.pt))) ≫\n c.π.app j).unop... | simpa using congr_hom ((hc s.pt.unop).fac ((yoneda.obj s.pt.unop).mapCone s) j) (𝟙 (unop s.pt)) | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.CategoryTheory.Limits.Yoneda | {
"line": 109,
"column": 4
} | {
"line": 109,
"column": 99
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\nJ : Type w\ninst✝ : Category.{t, w} J\nF : J ⥤ Cᵒᵖ\nc : Cone F\nhc : (X : C) → IsLimit ((yoneda.obj X).mapCone c)\ns : Cone F\nj : J\n⊢ (Quiver.Hom.op ((hom ((hc (unop s.pt)).lift ((yoneda.obj (unop s.pt)).mapCone s))) (𝟙 (unop s.pt))) ≫\n c.π.app j).unop... | simpa using congr_hom ((hc s.pt.unop).fac ((yoneda.obj s.pt.unop).mapCone s) j) (𝟙 (unop s.pt)) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Limits.Yoneda | {
"line": 109,
"column": 4
} | {
"line": 109,
"column": 99
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\nJ : Type w\ninst✝ : Category.{t, w} J\nF : J ⥤ Cᵒᵖ\nc : Cone F\nhc : (X : C) → IsLimit ((yoneda.obj X).mapCone c)\ns : Cone F\nj : J\n⊢ (Quiver.Hom.op ((hom ((hc (unop s.pt)).lift ((yoneda.obj (unop s.pt)).mapCone s))) (𝟙 (unop s.pt))) ≫\n c.π.app j).unop... | simpa using congr_hom ((hc s.pt.unop).fac ((yoneda.obj s.pt.unop).mapCone s) j) (𝟙 (unop s.pt)) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.ShrinkYoneda | {
"line": 141,
"column": 83
} | {
"line": 142,
"column": 65
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : LocallySmall.{w, v, u} C\nX Y : C\nf : X ⟶ Y\n⊢ shrinkYonedaEquiv (shrinkYoneda.{w, v, u}.map f) = shrinkYonedaObjObjEquiv.symm f",
"usedConstants": [
"CategoryTheory.instSmallOppositeObjFunctorTypeYoneda",
"CategoryTheory.Functor",
... | by
simp [shrinkYonedaEquiv, shrinkYoneda, shrinkYonedaObjObjEquiv] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Filtered.Basic | {
"line": 895,
"column": 6
} | {
"line": 895,
"column": 68
} | [
{
"pp": "case refine_1\nC : Type u\ninst✝ : Category.{v, u} C\nh : ∀ {J : Type w} [inst : SmallCategory J] [FinCategory J] (F : J ⥤ C), Nonempty (Cone F)\nthis : Nonempty C\nX Y : C\n⊢ ∃ W x x, True",
"usedConstants": [
"ULift",
"CategoryTheory.Functor.comp",
"CategoryTheory.ULift.downFunc... | obtain ⟨c⟩ := h (ULiftHom.down ⋙ ULift.downFunctor ⋙ pair X Y) | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.CategoryTheory.Filtered.Basic | {
"line": 902,
"column": 6
} | {
"line": 902,
"column": 14
} | [
{
"pp": "case refine_2\nC : Type u\ninst✝ : Category.{v, u} C\nh : ∀ {J : Type w} [inst : SmallCategory J] [FinCategory J] (F : J ⥤ C), Nonempty (Cone F)\nthis : Nonempty C\nX Y : C\nf g : X ⟶ Y\nc : Cone (ULiftHom.down ⋙ ULift.downFunctor ⋙ parallelPair f g)\nh₁ :\n ((Functor.const (ULiftHom (ULift.{w, 0} Wal... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.CategoryTheory.Monoidal.NaturalTransformation | {
"line": 105,
"column": 6
} | {
"line": 106,
"column": 50
} | [
{
"pp": "case h₁\nC : Type u₁\ninst✝¹⁶ : Category.{v₁, u₁} C\ninst✝¹⁵ : MonoidalCategory C\nD : Type u₂\ninst✝¹⁴ : Category.{v₂, u₂} D\ninst✝¹³ : MonoidalCategory D\nE : Type u₃\ninst✝¹² : Category.{v₃, u₃} E\ninst✝¹¹ : MonoidalCategory E\nE' : Type u₄\ninst✝¹⁰ : Category.{v₄, u₄} E'\ninst✝⁹ : MonoidalCategory ... | simp only [prod_comp_fst, prod'_μ_fst, prod'_app_fst,
prodMonoidal_tensorHom, IsMonoidal.tensor] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.CategoryTheory.Monoidal.NaturalTransformation | {
"line": 105,
"column": 6
} | {
"line": 106,
"column": 50
} | [
{
"pp": "case h₁\nC : Type u₁\ninst✝¹⁶ : Category.{v₁, u₁} C\ninst✝¹⁵ : MonoidalCategory C\nD : Type u₂\ninst✝¹⁴ : Category.{v₂, u₂} D\ninst✝¹³ : MonoidalCategory D\nE : Type u₃\ninst✝¹² : Category.{v₃, u₃} E\ninst✝¹¹ : MonoidalCategory E\nE' : Type u₄\ninst✝¹⁰ : Category.{v₄, u₄} E'\ninst✝⁹ : MonoidalCategory ... | simp only [prod_comp_fst, prod'_μ_fst, prod'_app_fst,
prodMonoidal_tensorHom, IsMonoidal.tensor] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Monoidal.NaturalTransformation | {
"line": 105,
"column": 6
} | {
"line": 106,
"column": 50
} | [
{
"pp": "case h₁\nC : Type u₁\ninst✝¹⁶ : Category.{v₁, u₁} C\ninst✝¹⁵ : MonoidalCategory C\nD : Type u₂\ninst✝¹⁴ : Category.{v₂, u₂} D\ninst✝¹³ : MonoidalCategory D\nE : Type u₃\ninst✝¹² : Category.{v₃, u₃} E\ninst✝¹¹ : MonoidalCategory E\nE' : Type u₄\ninst✝¹⁰ : Category.{v₄, u₄} E'\ninst✝⁹ : MonoidalCategory ... | simp only [prod_comp_fst, prod'_μ_fst, prod'_app_fst,
prodMonoidal_tensorHom, IsMonoidal.tensor] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Monoidal.NaturalTransformation | {
"line": 141,
"column": 4
} | {
"line": 141,
"column": 16
} | [
{
"pp": "C : Type u₁\ninst✝¹³ : Category.{v₁, u₁} C\ninst✝¹² : MonoidalCategory C\nD : Type u₂\ninst✝¹¹ : Category.{v₂, u₂} D\ninst✝¹⁰ : MonoidalCategory D\nE : Type u₃\ninst✝⁹ : Category.{v₃, u₃} E\ninst✝⁸ : MonoidalCategory E\nE' : Type u₄\ninst✝⁷ : Category.{v₄, u₄} E'\ninst✝⁶ : MonoidalCategory E'\nF₁ F₂ F₃... | rw [comp_id] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.CategoryTheory.Monoidal.Transport | {
"line": 117,
"column": 2
} | {
"line": 124,
"column": 65
} | [
{
"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 : MonoidalCategoryStruct D\nF : D ⥤ C\ninst✝ : F.Faithful\nfData : InducingFunctorData F\nthis : MonoidalCategory D := induced F fData\n⊢ F.CoreMonoidal",
"usedConstants": [
... | exact
{ εIso := fData.εIso
μIso := fData.μIso
μIso_hom_natural_left := fun _ ↦ by simp [fData.whiskerRight_eq]
μIso_hom_natural_right := fun _ ↦ by simp [fData.whiskerLeft_eq]
associativity := fun _ _ _ ↦ by simp [fData.associator_eq]
left_unitality := fun _ ↦ by simp [fData.leftUnitor... | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.CategoryTheory.Monoidal.Functor | {
"line": 204,
"column": 30
} | {
"line": 205,
"column": 45
} | [
{
"pp": "C : Type u₁\ninst✝⁶ : Category.{v₁, u₁} C\ninst✝⁵ : MonoidalCategory C\nD : Type u₂\ninst✝⁴ : Category.{v₂, u₂} D\ninst✝³ : MonoidalCategory D\nE : Type u₃\ninst✝² : Category.{v₃, u₃} E\ninst✝¹ : MonoidalCategory E\nC' : Type u₁'\ninst✝ : Category.{v₁', u₁'} C'\nF : C ⥤ D\nε : 𝟙_ D ⟶ F.obj (𝟙_ C)\nμ ... | by
simp_rw [← id_tensorHom, right_unitality] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Monoidal.Functor | {
"line": 650,
"column": 4
} | {
"line": 651,
"column": 8
} | [
{
"pp": "C : Type u₁\ninst✝⁶ : Category.{v₁, u₁} C\ninst✝⁵ : MonoidalCategory C\nD : Type u₂\ninst✝⁴ : Category.{v₂, u₂} D\ninst✝³ : MonoidalCategory D\nE : Type u₃\ninst✝² : Category.{v₃, u₃} E\ninst✝¹ : MonoidalCategory E\nC' : Type u₁'\ninst✝ : Category.{v₁', u₁'} C'\nF : C ⥤ D\nεIso : 𝟙_ D ≅ F.obj (𝟙_ C)\... | rw [← cancel_mono (ρ_ (F.obj X)).inv, Iso.hom_inv_id, oplax_right_unitality]
simp | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Monoidal.Functor | {
"line": 650,
"column": 4
} | {
"line": 651,
"column": 8
} | [
{
"pp": "C : Type u₁\ninst✝⁶ : Category.{v₁, u₁} C\ninst✝⁵ : MonoidalCategory C\nD : Type u₂\ninst✝⁴ : Category.{v₂, u₂} D\ninst✝³ : MonoidalCategory D\nE : Type u₃\ninst✝² : Category.{v₃, u₃} E\ninst✝¹ : MonoidalCategory E\nC' : Type u₁'\ninst✝ : Category.{v₁', u₁'} C'\nF : C ⥤ D\nεIso : 𝟙_ D ≅ F.obj (𝟙_ C)\... | rw [← cancel_mono (ρ_ (F.obj X)).inv, Iso.hom_inv_id, oplax_right_unitality]
simp | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Monoidal.Functor | {
"line": 669,
"column": 8
} | {
"line": 669,
"column": 38
} | [
{
"pp": "C : Type u₁\ninst✝⁶ : Category.{v₁, u₁} C\ninst✝⁵ : MonoidalCategory C\nD : Type u₂\ninst✝⁴ : Category.{v₂, u₂} D\ninst✝³ : MonoidalCategory D\nE : Type u₃\ninst✝² : Category.{v₃, u₃} E\ninst✝¹ : MonoidalCategory E\nC' : Type u₁'\ninst✝ : Category.{v₁', u₁'} C'\nF : C ⥤ D\nh : F.CoreMonoidal\nX✝ Y✝ : C... | ← cancel_epi (h.μIso _ _).hom, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.Monoidal.Functor | {
"line": 672,
"column": 8
} | {
"line": 672,
"column": 38
} | [
{
"pp": "C : Type u₁\ninst✝⁶ : Category.{v₁, u₁} C\ninst✝⁵ : MonoidalCategory C\nD : Type u₂\ninst✝⁴ : Category.{v₂, u₂} D\ninst✝³ : MonoidalCategory D\nE : Type u₃\ninst✝² : Category.{v₃, u₃} E\ninst✝¹ : MonoidalCategory E\nC' : Type u₁'\ninst✝ : Category.{v₁', u₁'} C'\nF : C ⥤ D\nh : F.CoreMonoidal\nX✝ Y✝ x✝¹... | ← cancel_epi (h.μIso _ _).hom, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.Monoidal.Functor | {
"line": 868,
"column": 13
} | {
"line": 873,
"column": 51
} | [
{
"pp": "C : Type u₁\ninst✝⁸ : Category.{v₁, u₁} C\ninst✝⁷ : MonoidalCategory C\nD : Type u₂\ninst✝⁶ : Category.{v₂, u₂} D\ninst✝⁵ : MonoidalCategory D\nE : Type u₃\ninst✝⁴ : Category.{v₃, u₃} E\ninst✝³ : MonoidalCategory E\nC' : Type u₁'\ninst✝² : Category.{v₁', u₁'} C'\nF✝ F : C ⥤ D\nG : C ⥤ E\ninst✝¹ : F.Mon... | by
ext
· simp only [CategoryTheory.prod_comp_fst, prod'_μ_fst, prod'_δ_fst, μ_δ,
prod'_obj, prodMonoidal_tensorObj, prod_id]
· simp only [CategoryTheory.prod_comp_snd, prod'_μ_snd, prod'_δ_snd, μ_δ,
prod'_obj, prodMonoidal_tensorObj, prod_id] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Tactic.CategoryTheory.Monoidal.PureCoherence | {
"line": 64,
"column": 2
} | {
"line": 64,
"column": 10
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : MonoidalCategory C\np f g h pf : C\nη : f ≅ g\nθ : g ≅ h\nη_f : p ⊗ f ≅ pf\nη_g : p ⊗ g ≅ pf\nη_h : p ⊗ h ≅ pf\nih_η : p ◁ η ≪≫ η_g = η_f\nih_θ : p ◁ θ ≪≫ η_h = η_g\n⊢ p ◁ (η ≪≫ θ) ≪≫ η_h = η_f",
"usedConstants": [
"CategoryTheory.MonoidalCatego... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Tactic.CategoryTheory.Monoidal.PureCoherence | {
"line": 64,
"column": 2
} | {
"line": 64,
"column": 10
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : MonoidalCategory C\np f g h pf : C\nη : f ≅ g\nθ : g ≅ h\nη_f : p ⊗ f ≅ pf\nη_g : p ⊗ g ≅ pf\nη_h : p ⊗ h ≅ pf\nih_η : p ◁ η ≪≫ η_g = η_f\nih_θ : p ◁ θ ≪≫ η_h = η_g\n⊢ p ◁ (η ≪≫ θ) ≪≫ η_h = η_f",
"usedConstants": [
"CategoryTheory.MonoidalCatego... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Tactic.CategoryTheory.Monoidal.PureCoherence | {
"line": 64,
"column": 2
} | {
"line": 64,
"column": 10
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : MonoidalCategory C\np f g h pf : C\nη : f ≅ g\nθ : g ≅ h\nη_f : p ⊗ f ≅ pf\nη_g : p ⊗ g ≅ pf\nη_h : p ⊗ h ≅ pf\nih_η : p ◁ η ≪≫ η_g = η_f\nih_θ : p ◁ θ ≪≫ η_h = η_g\n⊢ p ◁ (η ≪≫ θ) ≪≫ η_h = η_f",
"usedConstants": [
"CategoryTheory.MonoidalCatego... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Tactic.CategoryTheory.Monoidal.PureCoherence | {
"line": 63,
"column": 33
} | {
"line": 64,
"column": 10
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : MonoidalCategory C\np f g h pf : C\nη : f ≅ g\nθ : g ≅ h\nη_f : p ⊗ f ≅ pf\nη_g : p ⊗ g ≅ pf\nη_h : p ⊗ h ≅ pf\nih_η : p ◁ η ≪≫ η_g = η_f\nih_θ : p ◁ θ ≪≫ η_h = η_g\n⊢ p ◁ (η ≪≫ θ) ≪≫ η_h = η_f",
"usedConstants": [
"CategoryTheory.MonoidalCatego... | by
simp_all | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Monoidal.Functor | {
"line": 939,
"column": 58
} | {
"line": 947,
"column": 71
} | [
{
"pp": "C : 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 : C ⥤ D\nG : D ⥤ C\nadj : F ⊣ G\ninst... | by
rw [homEquiv_counit, homEquiv_unit, MonoidalCategory.whiskerLeft_comp, homEquiv_unit,
homEquiv_counit, map_comp, map_comp, map_comp, map_comp, map_comp, map_comp,
assoc, assoc, assoc, assoc, assoc, counit_naturality, counit_naturality_assoc,
counit_naturality_assoc, left_triangle_components_ass... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Tactic.CategoryTheory.Monoidal.Normalize | {
"line": 120,
"column": 2
} | {
"line": 120,
"column": 10
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : MonoidalCategory C\nf g h i j : C\nα : f ≅ g\nη : g ⟶ h\nηs : h ⟶ i\nηs₁ : h ⊗ j ⟶ i ⊗ j\nη₁ : g ⊗ j ⟶ h ⊗ j\nη₂ : g ⊗ j ⟶ i ⊗ j\nη₃ : f ⊗ j ⟶ i ⊗ j\ne_ηs₁ : ηs ▷ j = ηs₁\ne_η₁ : η ▷ j = η₁\ne_η₂ : η₁ ≫ ηs₁ = η₂\ne_η₃ : (α ▷ᵢ j).hom ≫ η₂ = η₃\n⊢ (α.hom ≫ ... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Tactic.CategoryTheory.Monoidal.Normalize | {
"line": 120,
"column": 2
} | {
"line": 120,
"column": 10
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : MonoidalCategory C\nf g h i j : C\nα : f ≅ g\nη : g ⟶ h\nηs : h ⟶ i\nηs₁ : h ⊗ j ⟶ i ⊗ j\nη₁ : g ⊗ j ⟶ h ⊗ j\nη₂ : g ⊗ j ⟶ i ⊗ j\nη₃ : f ⊗ j ⟶ i ⊗ j\ne_ηs₁ : ηs ▷ j = ηs₁\ne_η₁ : η ▷ j = η₁\ne_η₂ : η₁ ≫ ηs₁ = η₂\ne_η₃ : (α ▷ᵢ j).hom ≫ η₂ = η₃\n⊢ (α.hom ≫ ... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Tactic.CategoryTheory.Monoidal.Normalize | {
"line": 120,
"column": 2
} | {
"line": 120,
"column": 10
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : MonoidalCategory C\nf g h i j : C\nα : f ≅ g\nη : g ⟶ h\nηs : h ⟶ i\nηs₁ : h ⊗ j ⟶ i ⊗ j\nη₁ : g ⊗ j ⟶ h ⊗ j\nη₂ : g ⊗ j ⟶ i ⊗ j\nη₃ : f ⊗ j ⟶ i ⊗ j\ne_ηs₁ : ηs ▷ j = ηs₁\ne_η₁ : η ▷ j = η₁\ne_η₂ : η₁ ≫ ηs₁ = η₂\ne_η₃ : (α ▷ᵢ j).hom ≫ η₂ = η₃\n⊢ (α.hom ≫ ... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Tactic.CategoryTheory.Monoidal.Normalize | {
"line": 119,
"column": 33
} | {
"line": 120,
"column": 10
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : MonoidalCategory C\nf g h i j : C\nα : f ≅ g\nη : g ⟶ h\nηs : h ⟶ i\nηs₁ : h ⊗ j ⟶ i ⊗ j\nη₁ : g ⊗ j ⟶ h ⊗ j\nη₂ : g ⊗ j ⟶ i ⊗ j\nη₃ : f ⊗ j ⟶ i ⊗ j\ne_ηs₁ : ηs ▷ j = ηs₁\ne_η₁ : η ▷ j = η₁\ne_η₂ : η₁ ≫ ηs₁ = η₂\ne_η₃ : (α ▷ᵢ j).hom ≫ η₂ = η₃\n⊢ (α.hom ≫ ... | by
simp_all | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Tactic.CategoryTheory.Monoidal.Normalize | {
"line": 167,
"column": 2
} | {
"line": 167,
"column": 10
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : MonoidalCategory C\nf f' g h i : C\nα : f' ≅ g\nη : g ⟶ h\nηs : h ⟶ i\nηs₁ : h ⊗ f ⟶ i ⊗ f\nη₁ : g ⊗ f ⟶ h ⊗ f\nη₂ : g ⊗ f ⟶ i ⊗ f\nη₃ : f' ⊗ f ⟶ i ⊗ f\ne_ηs₁ : ηs ▷ f = ηs₁\ne_η₁ : η ▷ f = η₁\ne_η₂ : η₁ ≫ ηs₁ = η₂\ne_η₃ : (α ▷ᵢ f).hom ≫ η₂ = η₃\n⊢ (α.hom... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Tactic.CategoryTheory.Monoidal.Normalize | {
"line": 167,
"column": 2
} | {
"line": 167,
"column": 10
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : MonoidalCategory C\nf f' g h i : C\nα : f' ≅ g\nη : g ⟶ h\nηs : h ⟶ i\nηs₁ : h ⊗ f ⟶ i ⊗ f\nη₁ : g ⊗ f ⟶ h ⊗ f\nη₂ : g ⊗ f ⟶ i ⊗ f\nη₃ : f' ⊗ f ⟶ i ⊗ f\ne_ηs₁ : ηs ▷ f = ηs₁\ne_η₁ : η ▷ f = η₁\ne_η₂ : η₁ ≫ ηs₁ = η₂\ne_η₃ : (α ▷ᵢ f).hom ≫ η₂ = η₃\n⊢ (α.hom... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Tactic.CategoryTheory.Monoidal.Normalize | {
"line": 167,
"column": 2
} | {
"line": 167,
"column": 10
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : MonoidalCategory C\nf f' g h i : C\nα : f' ≅ g\nη : g ⟶ h\nηs : h ⟶ i\nηs₁ : h ⊗ f ⟶ i ⊗ f\nη₁ : g ⊗ f ⟶ h ⊗ f\nη₂ : g ⊗ f ⟶ i ⊗ f\nη₃ : f' ⊗ f ⟶ i ⊗ f\ne_ηs₁ : ηs ▷ f = ηs₁\ne_η₁ : η ▷ f = η₁\ne_η₂ : η₁ ≫ ηs₁ = η₂\ne_η₃ : (α ▷ᵢ f).hom ≫ η₂ = η₃\n⊢ (α.hom... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Tactic.CategoryTheory.Monoidal.Normalize | {
"line": 166,
"column": 33
} | {
"line": 167,
"column": 10
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : MonoidalCategory C\nf f' g h i : C\nα : f' ≅ g\nη : g ⟶ h\nηs : h ⟶ i\nηs₁ : h ⊗ f ⟶ i ⊗ f\nη₁ : g ⊗ f ⟶ h ⊗ f\nη₂ : g ⊗ f ⟶ i ⊗ f\nη₃ : f' ⊗ f ⟶ i ⊗ f\ne_ηs₁ : ηs ▷ f = ηs₁\ne_η₁ : η ▷ f = η₁\ne_η₂ : η₁ ≫ ηs₁ = η₂\ne_η₃ : (α ▷ᵢ f).hom ≫ η₂ = η₃\n⊢ (α.hom... | by
simp_all | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Tactic.CategoryTheory.Monoidal.Normalize | {
"line": 179,
"column": 2
} | {
"line": 179,
"column": 10
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : MonoidalCategory C\nf f' g g' h i : C\nη : f ⟶ g\nηs : f' ⟶ g'\nθ : h ⟶ i\nηθ : f' ⊗ h ⟶ g' ⊗ i\nη₁ : f ⊗ f' ⊗ h ⟶ g ⊗ g' ⊗ i\nηθ₁ : f ⊗ f' ⊗ h ⟶ (g ⊗ g') ⊗ i\nηθ₂ : (f ⊗ f') ⊗ h ⟶ (g ⊗ g') ⊗ i\ne_ηθ : ηs ⊗ₘ θ = ηθ\ne_η₁ : (Iso.refl f).hom ≫ η ≫ (Iso.refl... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Tactic.CategoryTheory.Monoidal.Normalize | {
"line": 179,
"column": 2
} | {
"line": 179,
"column": 10
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : MonoidalCategory C\nf f' g g' h i : C\nη : f ⟶ g\nηs : f' ⟶ g'\nθ : h ⟶ i\nηθ : f' ⊗ h ⟶ g' ⊗ i\nη₁ : f ⊗ f' ⊗ h ⟶ g ⊗ g' ⊗ i\nηθ₁ : f ⊗ f' ⊗ h ⟶ (g ⊗ g') ⊗ i\nηθ₂ : (f ⊗ f') ⊗ h ⟶ (g ⊗ g') ⊗ i\ne_ηθ : ηs ⊗ₘ θ = ηθ\ne_η₁ : (Iso.refl f).hom ≫ η ≫ (Iso.refl... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Tactic.CategoryTheory.Monoidal.Normalize | {
"line": 179,
"column": 2
} | {
"line": 179,
"column": 10
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : MonoidalCategory C\nf f' g g' h i : C\nη : f ⟶ g\nηs : f' ⟶ g'\nθ : h ⟶ i\nηθ : f' ⊗ h ⟶ g' ⊗ i\nη₁ : f ⊗ f' ⊗ h ⟶ g ⊗ g' ⊗ i\nηθ₁ : f ⊗ f' ⊗ h ⟶ (g ⊗ g') ⊗ i\nηθ₂ : (f ⊗ f') ⊗ h ⟶ (g ⊗ g') ⊗ i\ne_ηθ : ηs ⊗ₘ θ = ηθ\ne_η₁ : (Iso.refl f).hom ≫ η ≫ (Iso.refl... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Tactic.CategoryTheory.Monoidal.Normalize | {
"line": 178,
"column": 28
} | {
"line": 179,
"column": 10
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : MonoidalCategory C\nf f' g g' h i : C\nη : f ⟶ g\nηs : f' ⟶ g'\nθ : h ⟶ i\nηθ : f' ⊗ h ⟶ g' ⊗ i\nη₁ : f ⊗ f' ⊗ h ⟶ g ⊗ g' ⊗ i\nηθ₁ : f ⊗ f' ⊗ h ⟶ (g ⊗ g') ⊗ i\nηθ₂ : (f ⊗ f') ⊗ h ⟶ (g ⊗ g') ⊗ i\ne_ηθ : ηs ⊗ₘ θ = ηθ\ne_η₁ : (Iso.refl f).hom ≫ η ≫ (Iso.refl... | by
simp_all | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.Coalgebra.Hom | {
"line": 186,
"column": 27
} | {
"line": 186,
"column": 35
} | [
{
"pp": "case h\nR : Type u_1\nA : Type u_2\nB : Type u_3\nC : Type u_4\nD : Type u_5\ninst✝¹² : CommSemiring R\ninst✝¹¹ : AddCommMonoid A\ninst✝¹⁰ : Module R A\ninst✝⁹ : AddCommMonoid B\ninst✝⁸ : Module R B\ninst✝⁷ : AddCommMonoid C\ninst✝⁶ : Module R C\ninst✝⁵ : AddCommMonoid D\ninst✝⁴ : Module R D\ninst✝³ : ... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.RingTheory.Coalgebra.Equiv | {
"line": 176,
"column": 6
} | {
"line": 177,
"column": 95
} | [
{
"pp": "R : Type u_1\nA : Type u_2\nB : Type u_3\nC : Type u_4\ninst✝⁹ : CommSemiring R\ninst✝⁸ : AddCommMonoid A\ninst✝⁷ : AddCommMonoid B\ninst✝⁶ : AddCommMonoid C\ninst✝⁵ : Module R A\ninst✝⁴ : Module R B\ninst✝³ : Module R C\ninst✝² : CoalgebraStruct R A\ninst✝¹ : CoalgebraStruct R B\ninst✝ : CoalgebraStru... | simp only [TensorProduct.congr, toCoalgHom_eq_coe, CoalgHom.toLinearMap_eq_coe,
LinearEquiv.ofLinear_toLinearMap, ← LinearMap.comp_assoc, CoalgHomClass.map_comp_comul] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.RingTheory.Bialgebra.Hom | {
"line": 120,
"column": 4
} | {
"line": 120,
"column": 12
} | [
{
"pp": "R : Type u_1\nA : Type u_2\nB : Type u_3\nC : Type u_4\nD : Type u_5\ninst✝¹² : CommSemiring R\ninst✝¹¹ : Semiring A\ninst✝¹⁰ : Semiring B\ninst✝⁹ : Semiring C\ninst✝⁸ : Semiring D\ninst✝⁷ : Algebra R A\ninst✝⁶ : Algebra R B\ninst✝⁵ : Algebra R C\ninst✝⁴ : Algebra R D\ninst✝³ : CoalgebraStruct R A\nins... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.RingTheory.Bialgebra.Hom | {
"line": 227,
"column": 19
} | {
"line": 227,
"column": 27
} | [
{
"pp": "R : Type u_1\nA : Type u_2\nB : Type u_3\nC : Type u_4\nD : Type u_5\ninst✝¹² : CommSemiring R\ninst✝¹¹ : Semiring A\ninst✝¹⁰ : Semiring B\ninst✝⁹ : Semiring C\ninst✝⁸ : Semiring D\ninst✝⁷ : Algebra R A\ninst✝⁶ : Algebra R B\ninst✝⁵ : Algebra R C\ninst✝⁴ : Algebra R D\ninst✝³ : CoalgebraStruct R A\nins... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.RingTheory.Bialgebra.Hom | {
"line": 227,
"column": 19
} | {
"line": 227,
"column": 27
} | [
{
"pp": "R : Type u_1\nA : Type u_2\nB : Type u_3\nC : Type u_4\nD : Type u_5\ninst✝¹² : CommSemiring R\ninst✝¹¹ : Semiring A\ninst✝¹⁰ : Semiring B\ninst✝⁹ : Semiring C\ninst✝⁸ : Semiring D\ninst✝⁷ : Algebra R A\ninst✝⁶ : Algebra R B\ninst✝⁵ : Algebra R C\ninst✝⁴ : Algebra R D\ninst✝³ : CoalgebraStruct R A\nins... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Bialgebra.Hom | {
"line": 227,
"column": 19
} | {
"line": 227,
"column": 27
} | [
{
"pp": "R : Type u_1\nA : Type u_2\nB : Type u_3\nC : Type u_4\nD : Type u_5\ninst✝¹² : CommSemiring R\ninst✝¹¹ : Semiring A\ninst✝¹⁰ : Semiring B\ninst✝⁹ : Semiring C\ninst✝⁸ : Semiring D\ninst✝⁷ : Algebra R A\ninst✝⁶ : Algebra R B\ninst✝⁵ : Algebra R C\ninst✝⁴ : Algebra R D\ninst✝³ : CoalgebraStruct R A\nins... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Bialgebra.Hom | {
"line": 228,
"column": 27
} | {
"line": 228,
"column": 35
} | [
{
"pp": "R : Type u_1\nA : Type u_2\nB : Type u_3\nC : Type u_4\nD : Type u_5\ninst✝¹² : CommSemiring R\ninst✝¹¹ : Semiring A\ninst✝¹⁰ : Semiring B\ninst✝⁹ : Semiring C\ninst✝⁸ : Semiring D\ninst✝⁷ : Algebra R A\ninst✝⁶ : Algebra R B\ninst✝⁵ : Algebra R C\ninst✝⁴ : Algebra R D\ninst✝³ : CoalgebraStruct R A\nins... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.RingTheory.Coalgebra.Basic | {
"line": 401,
"column": 2
} | {
"line": 402,
"column": 66
} | [
{
"pp": "R : Type u\nι : Type v\nA : Type w\ninst✝³ : CommSemiring R\ninst✝² : AddCommMonoid A\ninst✝¹ : Module R A\ninst✝ : CoalgebraStruct R A\ni : ι\n⊢ comul ∘ₗ lapply i = TensorProduct.map (lapply i) (lapply i) ∘ₗ comul",
"usedConstants": [
"Finsupp.instFunLike",
"LinearMap.id",
"Eq.mp... | ext j; have := eq_or_ne i j
aesop (add simp [TensorProduct.map_map, proj_comp_single, diag]) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Coalgebra.Basic | {
"line": 401,
"column": 2
} | {
"line": 402,
"column": 66
} | [
{
"pp": "R : Type u\nι : Type v\nA : Type w\ninst✝³ : CommSemiring R\ninst✝² : AddCommMonoid A\ninst✝¹ : Module R A\ninst✝ : CoalgebraStruct R A\ni : ι\n⊢ comul ∘ₗ lapply i = TensorProduct.map (lapply i) (lapply i) ∘ₗ comul",
"usedConstants": [
"Finsupp.instFunLike",
"LinearMap.id",
"Eq.mp... | ext j; have := eq_or_ne i j
aesop (add simp [TensorProduct.map_map, proj_comp_single, diag]) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Coalgebra.Basic | {
"line": 574,
"column": 33
} | {
"line": 579,
"column": 65
} | [
{
"pp": "R : Type u_1\nA : Type u_2\nB : Type u_3\ninst✝³ : CommSemiring R\ninst✝² : AddCommMonoid B\ninst✝¹ : Module R B\ninst✝ : Coalgebra R B\ne : A ≃ B\nthis✝ : AddCommMonoid A := e.addCommMonoid\nthis : Module R A := Equiv.module R e\n⊢ LinearMap.lTensor A counit ∘ₗ comul = (TensorProduct.mk R A R).flip 1"... | by
ext
apply (TensorProduct.map_bijective (g := .id) (e.linearEquiv R).bijective
Function.bijective_id).injective
simpa +instances [coalgebraStruct, LinearMap.comp_assoc, TensorProduct.map_map,
LinearMap.lTensor] using Coalgebra.lTensor_counit_comul _ | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Adjunction.Mates | {
"line": 175,
"column": 19
} | {
"line": 175,
"column": 66
} | [
{
"pp": "case a.a\nA : 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\nG₁ : A ⥤ C\nG₂ : C ⥤ E\nH₁ : B ⥤ D\n... | rw [← assoc, ← assoc, ← unit_naturality (adj₃)] | Lean.Parser.Tactic.Conv._aux_Init_Conv___macroRules_Lean_Parser_Tactic_Conv_convRw___1 | Lean.Parser.Tactic.Conv.convRw__ |
Mathlib.CategoryTheory.Adjunction.Mates | {
"line": 175,
"column": 19
} | {
"line": 175,
"column": 66
} | [
{
"pp": "case a.a\nA : 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\nG₁ : A ⥤ C\nG₂ : C ⥤ E\nH₁ : B ⥤ D\n... | rw [← assoc, ← assoc, ← unit_naturality (adj₃)] | Lean.Elab.Tactic.Conv.evalConvSeq1Indented | Lean.Parser.Tactic.Conv.convSeq1Indented |
Mathlib.CategoryTheory.Adjunction.Mates | {
"line": 175,
"column": 19
} | {
"line": 175,
"column": 66
} | [
{
"pp": "case a.a\nA : 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\nG₁ : A ⥤ C\nG₂ : C ⥤ E\nH₁ : B ⥤ D\n... | rw [← assoc, ← assoc, ← unit_naturality (adj₃)] | Lean.Elab.Tactic.Conv.evalConvSeq | Lean.Parser.Tactic.Conv.convSeq |
Mathlib.CategoryTheory.Category.Bipointed | {
"line": 73,
"column": 4
} | {
"line": 73,
"column": 50
} | [
{
"pp": "X Y Z : Bipointed\nf : X.Hom Y\ng : Y.Hom Z\n⊢ (g.toFun ∘ f.toFun) X.toProd.2 = Z.toProd.2",
"usedConstants": [
"Eq.mpr",
"congrArg",
"Bipointed.toProd",
"Function.comp",
"Bipointed.Hom.map_snd",
"id",
"Bipointed.X",
"Eq.refl",
"Function.comp_ap... | rw [Function.comp_apply, f.map_snd, g.map_snd] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.CategoryTheory.Category.Bipointed | {
"line": 73,
"column": 4
} | {
"line": 73,
"column": 50
} | [
{
"pp": "X Y Z : Bipointed\nf : X.Hom Y\ng : Y.Hom Z\n⊢ (g.toFun ∘ f.toFun) X.toProd.2 = Z.toProd.2",
"usedConstants": [
"Eq.mpr",
"congrArg",
"Bipointed.toProd",
"Function.comp",
"Bipointed.Hom.map_snd",
"id",
"Bipointed.X",
"Eq.refl",
"Function.comp_ap... | rw [Function.comp_apply, f.map_snd, g.map_snd] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Category.Bipointed | {
"line": 73,
"column": 4
} | {
"line": 73,
"column": 50
} | [
{
"pp": "X Y Z : Bipointed\nf : X.Hom Y\ng : Y.Hom Z\n⊢ (g.toFun ∘ f.toFun) X.toProd.2 = Z.toProd.2",
"usedConstants": [
"Eq.mpr",
"congrArg",
"Bipointed.toProd",
"Function.comp",
"Bipointed.Hom.map_snd",
"id",
"Bipointed.X",
"Eq.refl",
"Function.comp_ap... | rw [Function.comp_apply, f.map_snd, g.map_snd] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.GroupTheory.MonoidLocalization.Away | {
"line": 62,
"column": 6
} | {
"line": 62,
"column": 27
} | [
{
"pp": "M : Type u_1\ninst✝² : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst✝¹ : CommMonoid N\nP : Type u_3\ninst✝ : CommMonoid P\ng : M →* P\nhg✝ : ∀ (y : ↥S), IsUnit (g ↑y)\nx : M\nF : AwayMap x N\nhg : IsUnit (g x)\ny : ↥(powers x)\nn : ℕ\nhn : (fun x_1 ↦ x ^ x_1) n = ↑y\n⊢ IsUnit (g x ^ n)",
"usedC... | exact IsUnit.pow n hg | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.RingTheory.UniqueFactorizationDomain.Multiplicity | {
"line": 159,
"column": 4
} | {
"line": 159,
"column": 12
} | [
{
"pp": "case pos\nR : Type u_2\ninst✝¹ : CommMonoidWithZero R\ninst✝ : UniqueFactorizationMonoid R\na b : R\nha : a ≠ 0\nh : ∀ (p : R), Prime p → emultiplicity p a ≤ emultiplicity p b\nhb : b = 0\n⊢ a ∣ b",
"usedConstants": [
"Dvd.dvd",
"congrArg",
"semigroupDvd",
"SemigroupWithZero... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.RingTheory.UniqueFactorizationDomain.Multiplicity | {
"line": 159,
"column": 4
} | {
"line": 159,
"column": 12
} | [
{
"pp": "case pos\nR : Type u_2\ninst✝¹ : CommMonoidWithZero R\ninst✝ : UniqueFactorizationMonoid R\na b : R\nha : a ≠ 0\nh : ∀ (p : R), Prime p → emultiplicity p a ≤ emultiplicity p b\nhb : b = 0\n⊢ a ∣ b",
"usedConstants": [
"Dvd.dvd",
"congrArg",
"semigroupDvd",
"SemigroupWithZero... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.UniqueFactorizationDomain.Multiplicity | {
"line": 159,
"column": 4
} | {
"line": 159,
"column": 12
} | [
{
"pp": "case pos\nR : Type u_2\ninst✝¹ : CommMonoidWithZero R\ninst✝ : UniqueFactorizationMonoid R\na b : R\nha : a ≠ 0\nh : ∀ (p : R), Prime p → emultiplicity p a ≤ emultiplicity p b\nhb : b = 0\n⊢ a ∣ b",
"usedConstants": [
"Dvd.dvd",
"congrArg",
"semigroupDvd",
"SemigroupWithZero... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.TwoSidedIdeal.Operations | {
"line": 108,
"column": 68
} | {
"line": 108,
"column": 76
} | [
{
"pp": "R : Type u_1\ninst✝ : NonUnitalNonAssocRing R\ns : Set R\np : (x : R) → x ∈ span s → Prop\nmem : ∀ (x : R) (h : x ∈ s), p x ⋯\nzero : p 0 ⋯\nadd : ∀ (x y : R) (hx : x ∈ span s) (hy : y ∈ span s), p x hx → p y hy → p (x + y) ⋯\nneg : ∀ (x : R) (hx : x ∈ span s), p x hx → p (-x) ⋯\nleft_absorb : ∀ (a x :... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.RingTheory.TwoSidedIdeal.Operations | {
"line": 108,
"column": 68
} | {
"line": 108,
"column": 76
} | [
{
"pp": "R : Type u_1\ninst✝ : NonUnitalNonAssocRing R\ns : Set R\np : (x : R) → x ∈ span s → Prop\nmem : ∀ (x : R) (h : x ∈ s), p x ⋯\nzero : p 0 ⋯\nadd : ∀ (x y : R) (hx : x ∈ span s) (hy : y ∈ span s), p x hx → p y hy → p (x + y) ⋯\nneg : ∀ (x : R) (hx : x ∈ span s), p x hx → p (-x) ⋯\nleft_absorb : ∀ (a x :... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.TwoSidedIdeal.Operations | {
"line": 108,
"column": 68
} | {
"line": 108,
"column": 76
} | [
{
"pp": "R : Type u_1\ninst✝ : NonUnitalNonAssocRing R\ns : Set R\np : (x : R) → x ∈ span s → Prop\nmem : ∀ (x : R) (h : x ∈ s), p x ⋯\nzero : p 0 ⋯\nadd : ∀ (x y : R) (hx : x ∈ span s) (hy : y ∈ span s), p x hx → p y hy → p (x + y) ⋯\nneg : ∀ (x : R) (hx : x ∈ span s), p x hx → p (-x) ⋯\nleft_absorb : ∀ (a x :... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Jacobson.Ideal | {
"line": 168,
"column": 13
} | {
"line": 168,
"column": 28
} | [
{
"pp": "case mp\nR : Type u\ninst✝ : Ring R\nI : Ideal R\nh : I.jacobson = I\nx : R\nhx : x ∉ I.jacobson\n⊢ ∃ M, (I ≤ M ∧ M.IsMaximal) ∧ x ∉ M",
"usedConstants": [
"Semiring.toModule",
"congrArg",
"PartialOrder.toPreorder",
"setOf",
"Preorder.toLE",
"Membership.mem",
... | Ideal.jacobson, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Jacobson.Ideal | {
"line": 173,
"column": 8
} | {
"line": 173,
"column": 23
} | [
{
"pp": "case mpr\nR : Type u\ninst✝ : Ring R\nI : Ideal R\nh : ∀ x ∉ I, ∃ M, (I ≤ M ∧ M.IsMaximal) ∧ x ∉ M\nx : R\nhx : x ∉ I\n⊢ x ∉ I.jacobson",
"usedConstants": [
"Eq.mpr",
"Submodule",
"Semiring.toModule",
"congrArg",
"PartialOrder.toPreorder",
"setOf",
"Preorde... | Ideal.jacobson, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Localization.Away.Basic | {
"line": 575,
"column": 2
} | {
"line": 576,
"column": 73
} | [
{
"pp": "R : Type u_1\ninst✝ : CommSemiring R\ns : Finset R\nf : (a : ↑↑s) → Away ↑a\nh : ∀ (a b : ↑↑s), (Away.awayToAwayRight ↑a ↑b) (f a) = (Away.awayToAwayLeft ↑b ↑a) (f b)\nmem : 1 ∈ Ideal.span ↑s\nspan_eq : Ideal.span ↑s = ⊤\nn : ↑↑s → ℕ\nr✝ : ↑↑s → R\neq✝ : ∀ (a : ↑↑s), f a * (algebraMap R (Away ↑a)) ↑a ^... | let N' := (s ×ˢ s).attach.sup fun a ↦ N'
⟨_, (Finset.mem_product.mp a.2).1⟩ ⟨_, (Finset.mem_product.mp a.2).2⟩ | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLet___1 | Lean.Parser.Tactic.tacticLet__ |
Mathlib.RingTheory.Localization.Away.Basic | {
"line": 698,
"column": 2
} | {
"line": 717,
"column": 56
} | [
{
"pp": "R : Type u_3\ninst✝⁵ : CommRing R\nx : R\nB : Type u_4\ninst✝⁴ : CommRing B\ninst✝³ : Algebra R B\ninst✝² : IsLocalization.Away x B\ninst✝¹ : IsDomain R\ninst✝ : WfDvdMonoid R\nb : B\nhb : b ≠ 0\nhx : Irreducible x\n⊢ ∃ a n, ¬x ∣ a ∧ selfZPow x B n * (algebraMap R B) a = b",
"usedConstants": [
... | obtain ⟨⟨a₀, y⟩, H⟩ := surj (Submonoid.powers x) b
obtain ⟨d, hy⟩ := (Submonoid.mem_powers_iff y.1 x).mp y.2
have ha₀ : a₀ ≠ 0 := by
haveI := isDomain_of_le_nonZeroDivisors B
(powers_le_nonZeroDivisors_of_noZeroDivisors hx.ne_zero)
simp only [← hy, map_pow] at H
apply ((injective_iff_map_eq_zero' ... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Localization.Away.Basic | {
"line": 698,
"column": 2
} | {
"line": 717,
"column": 56
} | [
{
"pp": "R : Type u_3\ninst✝⁵ : CommRing R\nx : R\nB : Type u_4\ninst✝⁴ : CommRing B\ninst✝³ : Algebra R B\ninst✝² : IsLocalization.Away x B\ninst✝¹ : IsDomain R\ninst✝ : WfDvdMonoid R\nb : B\nhb : b ≠ 0\nhx : Irreducible x\n⊢ ∃ a n, ¬x ∣ a ∧ selfZPow x B n * (algebraMap R B) a = b",
"usedConstants": [
... | obtain ⟨⟨a₀, y⟩, H⟩ := surj (Submonoid.powers x) b
obtain ⟨d, hy⟩ := (Submonoid.mem_powers_iff y.1 x).mp y.2
have ha₀ : a₀ ≠ 0 := by
haveI := isDomain_of_le_nonZeroDivisors B
(powers_le_nonZeroDivisors_of_noZeroDivisors hx.ne_zero)
simp only [← hy, map_pow] at H
apply ((injective_iff_map_eq_zero' ... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Localization.Integer | {
"line": 65,
"column": 12
} | {
"line": 65,
"column": 37
} | [
{
"pp": "case h\nR : Type u_1\ninst✝² : CommSemiring R\nS : Type u_2\ninst✝¹ : CommSemiring S\ninst✝ : Algebra R S\na : R\nb : S\nb' : R\nhb : (algebraMap R S) b' = b\n⊢ (algebraMap R S) (a * b') = a • (algebraMap R S) b'",
"usedConstants": [
"Eq.mpr",
"instHSMul",
"HMul.hMul",
"Alge... | (algebraMap R S).map_mul, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.Monoidal.Mon | {
"line": 396,
"column": 19
} | {
"line": 397,
"column": 31
} | [
{
"pp": "case a.a\nC : Type u₁\ninst✝⁴ : Category.{v₁, u₁} C\ninst✝³ : MonoidalCategory C\ninst✝² : BraidedCategory C\nM N : C\ninst✝¹ : MonObj M\ninst✝ : MonObj N\n| (μ ▷ M ⊗ₘ μ ▷ N) ≫ (μ ⊗ₘ μ)",
"usedConstants": [
"CategoryTheory.MonoidalCategoryStruct.whiskerLeft",
"CategoryTheory.MonObj.mul_... | rw [tensorHom_comp_tensorHom, mul_assoc, mul_assoc, ← tensorHom_comp_tensorHom,
← tensorHom_comp_tensorHom] | Lean.Parser.Tactic.Conv._aux_Init_Conv___macroRules_Lean_Parser_Tactic_Conv_convRw___1 | Lean.Parser.Tactic.Conv.convRw__ |
Mathlib.CategoryTheory.Monoidal.Mon | {
"line": 396,
"column": 19
} | {
"line": 397,
"column": 31
} | [
{
"pp": "case a.a\nC : Type u₁\ninst✝⁴ : Category.{v₁, u₁} C\ninst✝³ : MonoidalCategory C\ninst✝² : BraidedCategory C\nM N : C\ninst✝¹ : MonObj M\ninst✝ : MonObj N\n| (μ ▷ M ⊗ₘ μ ▷ N) ≫ (μ ⊗ₘ μ)",
"usedConstants": [
"CategoryTheory.MonoidalCategoryStruct.whiskerLeft",
"CategoryTheory.MonObj.mul_... | rw [tensorHom_comp_tensorHom, mul_assoc, mul_assoc, ← tensorHom_comp_tensorHom,
← tensorHom_comp_tensorHom] | Lean.Elab.Tactic.Conv.evalConvSeq1Indented | Lean.Parser.Tactic.Conv.convSeq1Indented |
Mathlib.CategoryTheory.Monoidal.Mon | {
"line": 396,
"column": 19
} | {
"line": 397,
"column": 31
} | [
{
"pp": "case a.a\nC : Type u₁\ninst✝⁴ : Category.{v₁, u₁} C\ninst✝³ : MonoidalCategory C\ninst✝² : BraidedCategory C\nM N : C\ninst✝¹ : MonObj M\ninst✝ : MonObj N\n| (μ ▷ M ⊗ₘ μ ▷ N) ≫ (μ ⊗ₘ μ)",
"usedConstants": [
"CategoryTheory.MonoidalCategoryStruct.whiskerLeft",
"CategoryTheory.MonObj.mul_... | rw [tensorHom_comp_tensorHom, mul_assoc, mul_assoc, ← tensorHom_comp_tensorHom,
← tensorHom_comp_tensorHom] | Lean.Elab.Tactic.Conv.evalConvSeq | Lean.Parser.Tactic.Conv.convSeq |
Mathlib.CategoryTheory.Limits.Opposites | {
"line": 166,
"column": 4
} | {
"line": 167,
"column": 65
} | [
{
"pp": "C : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\nJ : Type u₂\ninst✝ : Category.{v₂, u₂} J\nF : Jᵒᵖ ⥤ Cᵒᵖ\nc : Cocone F.unop\nhc : IsColimit c\ns : Cone F\nm : s.pt ⟶ (coneOfCoconeUnop c).pt\nw : ∀ (j : Jᵒᵖ), m ≫ (coneOfCoconeUnop c).π.app j = s.π.app j\n⊢ m = (hc.desc (coconeUnopOfCone s)).op",
"usedCons... | refine Quiver.Hom.unop_inj (hc.hom_ext fun j => Quiver.Hom.op_inj ?_)
simpa only [Quiver.Hom.unop_op, IsColimit.fac] using w (op j) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Limits.Opposites | {
"line": 166,
"column": 4
} | {
"line": 167,
"column": 65
} | [
{
"pp": "C : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\nJ : Type u₂\ninst✝ : Category.{v₂, u₂} J\nF : Jᵒᵖ ⥤ Cᵒᵖ\nc : Cocone F.unop\nhc : IsColimit c\ns : Cone F\nm : s.pt ⟶ (coneOfCoconeUnop c).pt\nw : ∀ (j : Jᵒᵖ), m ≫ (coneOfCoconeUnop c).π.app j = s.π.app j\n⊢ m = (hc.desc (coconeUnopOfCone s)).op",
"usedCons... | refine Quiver.Hom.unop_inj (hc.hom_ext fun j => Quiver.Hom.op_inj ?_)
simpa only [Quiver.Hom.unop_op, IsColimit.fac] using w (op j) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Limits.Constructions.BinaryProducts | {
"line": 103,
"column": 8
} | {
"line": 105,
"column": 50
} | [
{
"pp": "C : Type u\ninst✝⁵ : Category.{v, u} C\nD : Type u'\ninst✝⁴ : Category.{v', u'} D\nF : C ⥤ D\ninst✝³ : HasTerminal C\ninst✝² : HasPullbacks C\ninst✝¹ : PreservesLimitsOfShape (Discrete PEmpty.{1}) F\ninst✝ : PreservesLimitsOfShape WalkingCospan F\nK : Discrete WalkingPair ⥤ C\n⊢ IsLimit (F.mapCone (lim... | apply
isBinaryProductOfIsTerminalIsPullback _ _ (isLimitOfHasTerminalOfPreservesLimit F)
apply isLimitOfHasPullbackOfPreservesLimit | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Limits.Constructions.BinaryProducts | {
"line": 103,
"column": 8
} | {
"line": 105,
"column": 50
} | [
{
"pp": "C : Type u\ninst✝⁵ : Category.{v, u} C\nD : Type u'\ninst✝⁴ : Category.{v', u'} D\nF : C ⥤ D\ninst✝³ : HasTerminal C\ninst✝² : HasPullbacks C\ninst✝¹ : PreservesLimitsOfShape (Discrete PEmpty.{1}) F\ninst✝ : PreservesLimitsOfShape WalkingCospan F\nK : Discrete WalkingPair ⥤ C\n⊢ IsLimit (F.mapCone (lim... | apply
isBinaryProductOfIsTerminalIsPullback _ _ (isLimitOfHasTerminalOfPreservesLimit F)
apply isLimitOfHasPullbackOfPreservesLimit | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Limits.Shapes.Opposites.Pullbacks | {
"line": 315,
"column": 6
} | {
"line": 315,
"column": 18
} | [
{
"pp": "C : Type u₁\ninst✝ : Category.{v₁, u₁} C\nX Y Z : C\nf : X ⟶ Y\ng : X ⟶ Z\n⊢ HasPullback f.op g.op ↔ HasPushout f g",
"usedConstants": [
"Eq.mpr",
"Opposite",
"CategoryTheory.CategoryStruct.toQuiver",
"congrArg",
"Quiver.Hom.op",
"CategoryTheory.Limits.HasLimit",... | HasPullback, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.Limits.Shapes.Opposites.Pullbacks | {
"line": 321,
"column": 6
} | {
"line": 321,
"column": 18
} | [
{
"pp": "C : Type u₁\ninst✝ : Category.{v₁, u₁} C\nX Y Z : Cᵒᵖ\nf : X ⟶ Y\ng : X ⟶ Z\n⊢ HasPullback f.unop g.unop ↔ HasPushout f g",
"usedConstants": [
"Eq.mpr",
"Opposite",
"CategoryTheory.CategoryStruct.toQuiver",
"congrArg",
"Quiver.Hom.unop",
"CategoryTheory.Limits.Ha... | HasPullback, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Module.LocalizedModule.Basic | {
"line": 569,
"column": 4
} | {
"line": 570,
"column": 30
} | [
{
"pp": "R : Type u_1\ninst✝¹¹ : CommSemiring R\nS : Submonoid R\nM : Type u_2\nM' : Type u_3\nM'' : Type u_4\ninst✝¹⁰ : AddCommMonoid M\ninst✝⁹ : AddCommMonoid M'\ninst✝⁸ : AddCommMonoid M''\nA : Type u_5\ninst✝⁷ : CommSemiring A\ninst✝⁶ : Algebra R A\ninst✝⁵ : Module A M'\ninst✝⁴ : IsLocalization S A\ninst✝³ ... | exact ⟨c, by simpa only [Submonoid.smul_def, map_smul, e.symm_apply_apply]
using congr(e.symm $hc)⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Algebra.Module.LocalizedModule.Basic | {
"line": 681,
"column": 18
} | {
"line": 681,
"column": 58
} | [
{
"pp": "case h.h.h.e_6.h\nR : Type u_1\ninst✝⁴ : CommSemiring R\nS : Submonoid R\nM : Type u_2\nM'' : Type u_4\ninst✝³ : AddCommMonoid M\ninst✝² : AddCommMonoid M''\ninst✝¹ : Module R M\ninst✝ : Module R M''\ng : M →ₗ[R] M''\nh : ∀ (x : ↥S), IsUnit ((algebraMap R (End R M'')) ↑x)\nl : LocalizedModule S M →ₗ[R]... | simp only [one_smul, Submonoid.smul_def] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Algebra.Category.Ring.Constructions | {
"line": 424,
"column": 4
} | {
"line": 424,
"column": 78
} | [
{
"pp": "case lift\nA B C : CommRingCat\nf : A ⟶ C\ng : B ⟶ C\ns : PullbackCone f g\nx : ↑s.pt\n⊢ ((Hom.hom s.fst).prod (Hom.hom s.snd)) x ∈\n ((Hom.hom f).comp (RingHom.fst ↑A ↑B)).eqLocus ((Hom.hom g).comp (RingHom.snd ↑A ↑B))",
"usedConstants": [
"CommRingCat.Hom.hom",
"CommRingCat.carrier... | exact congr_arg (fun f : s.pt →+* C => f x) (congrArg Hom.hom s.condition) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.CategoryTheory.MorphismProperty.Composition | {
"line": 174,
"column": 2
} | {
"line": 174,
"column": 34
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\nD : Type u'\ninst✝ : Category.{v', u'} D\nF₁ F₂ : C ⥤ D\napp : (X : C) → F₁.obj X ⟶ F₂.obj X\nX Y : C\ne : X ≅ Y\nhe : F₁.map e.hom ≫ app Y = app X ≫ F₂.map e.hom\n⊢ F₁.map e.inv ≫ app X = app Y ≫ F₂.map e.inv",
"usedConstants": [
"Eq.mpr",
"Categ... | rw [← cancel_epi (F₁.map e.hom)] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.CategoryTheory.MorphismProperty.Composition | {
"line": 275,
"column": 6
} | {
"line": 275,
"column": 47
} | [
{
"pp": "case comp_of\nC : Type u\ninst✝¹ : Category.{v, u} C\nD : Type u'\ninst✝ : Category.{v', u'} D\nW : MorphismProperty C\nX✝ Y✝ Z✝ : C\ng✝ : Y✝ ⟶ Z✝\nx✝ y✝ z✝ : C\nf' : x✝ ⟶ y✝\ng : y✝ ⟶ z✝\nhf' : W.multiplicativeClosure f'\nhg : W g\nh_rec : ∀ (f : X✝ ⟶ x✝), W.multiplicativeClosure f → W.multiplicativeC... | exact .comp_of (f ≫ f') g (h_rec f hf) hg | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.CategoryTheory.Bicategory.Functor.Pseudofunctor | {
"line": 305,
"column": 87
} | {
"line": 306,
"column": 34
} | [
{
"pp": "B : Type u₁\ninst✝² : Bicategory B\nC : Type u₂\ninst✝¹ : Bicategory C\nD : Type u₃\ninst✝ : Bicategory D\nF✝ : B ⥤ᵖ C\nF : B ⥤ᵒᵖᴸ C\nF' : F.PseudoCore\na✝ b✝ c✝ d✝ : B\nf : a✝ ⟶ b✝\ng : b✝ ⟶ c✝\nh : c✝ ⟶ d✝\n⊢ F.map₂ (α_ f g h).hom =\n F.map₂ (α_ f g h).hom ≫\n F.mapComp f (g ≫ h) ≫ F.map f ◁ ... | ←
F'.mapCompIso_hom f (g ≫ h), | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.Bicategory.Functor.Pseudofunctor | {
"line": 341,
"column": 4
} | {
"line": 341,
"column": 81
} | [
{
"pp": "B : Type u₁\ninst✝² : Bicategory B\nC : Type u₂\ninst✝¹ : Bicategory C\nD : Type u₃\ninst✝ : Bicategory D\nF✝ : B ⥤ᵖ C\nF : B ⥤ᴸ C\nF' : F.PseudoCore\na b c d : B\nf : a ⟶ b\ng : b ⟶ c\nh : c ⟶ d\n⊢ F.map₂ (α_ f g h).hom =\n (F'.mapCompIso (f ≫ g) h).hom ≫\n (F'.mapCompIso f g).hom ▷ F.map h ≫\... | rw [F'.mapCompIso_inv, F'.mapCompIso_inv, ← inv_comp_eq, ← IsIso.inv_comp_eq] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.MvPolynomial.Tower | {
"line": 62,
"column": 4
} | {
"line": 62,
"column": 37
} | [
{
"pp": "R : Type u_1\nA : Type u_2\nB : Type u_3\nσ : Type u_4\ninst✝⁹ : CommSemiring R\ninst✝⁸ : CommSemiring A\ninst✝⁷ : CommSemiring B\ninst✝⁶ : Algebra R A\ninst✝⁵ : Algebra A B\ninst✝⁴ : Algebra R B\ninst✝³ : IsScalarTower R A B\ninst✝² : IsDomain A\ninst✝¹ : Module.IsTorsionFree A B\ninst✝ : Nontrivial B... | iff_false_intro (one_ne_zero' B), | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.WithTerminal.Cone | {
"line": 96,
"column": 34
} | {
"line": 96,
"column": 69
} | [
{
"pp": "case h\nC : Type u₁\ninst✝² : Category.{v₁, u₁} C\nD : Type u₂\ninst✝¹ : Category.{v₂, u₂} D\nJ : Type w\ninst✝ : Category.{w', w} J\nX : C\nK : J ⥤ Over X\nF : C ⥤ D\nt✝ : Cone K\nt : Cone (liftFromOver.obj K)\nx✝¹ x✝ : J\nf : x✝¹ ⟶ x✝\n⊢ Over.Hom.left (((Functor.const J).obj (Over.mk (t.π.app star)))... | simpa using (t.w (incl.map f)).symm | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.CategoryTheory.IsConnected | {
"line": 418,
"column": 2
} | {
"line": 418,
"column": 19
} | [
{
"pp": "case h\nJ : Type u₁\ninst✝ : Category.{v₁, u₁} J\nh : ∀ (j₁ j₂ : J), Zigzag j₁ j₂\n⊢ ∀ {α : Type u₁} (F : J → α), (∀ {j₁ j₂ : J} (x : j₁ ⟶ j₂), F j₁ = F j₂) → ∀ (j j' : J), F j = F j'",
"usedConstants": []
}
] | intro α F hF j j' | Lean.Elab.Tactic.evalIntro | Lean.Parser.Tactic.intro |
Mathlib.CategoryTheory.Elements | {
"line": 58,
"column": 2
} | {
"line": 58,
"column": 10
} | [
{
"pp": "case mk.mk.refl\nC : Type u\ninst✝ : Category.{v, u} C\nF : C ⥤ Type w\nfst✝ : C\nsnd✝¹ snd✝ : F.obj fst✝\nh₂ : (ConcreteCategory.hom (F.map (eqToHom ⋯))) ⟨fst✝, snd✝¹⟩.snd = ⟨fst✝, snd✝⟩.snd\n⊢ ⟨fst✝, snd✝¹⟩ = ⟨fst✝, snd✝⟩",
"usedConstants": [
"CategoryTheory.Functor.Elements",
"congrA... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.CategoryTheory.Limits.Shapes.Grothendieck | {
"line": 97,
"column": 28
} | {
"line": 97,
"column": 54
} | [
{
"pp": "case w.h\nC : Type u₁\ninst✝³ : Category.{v₁, u₁} C\nF : C ⥤ Cat\nH : Type u₂\ninst✝² : Category.{v₂, u₂} H\nG✝ : Grothendieck F ⥤ H\ninst✝¹ : ∀ {X Y : C} (f : X ⟶ Y), HasColimit ((F.map f).toFunctor ⋙ Grothendieck.ι F Y ⋙ G✝)\ninst✝ : ∀ (c : C), HasColimitsOfShape (↑(F.obj c)) H\nG : Grothendieck F ⥤ ... | apply Functor.map_id colim | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.RingTheory.FinitePresentation | {
"line": 351,
"column": 8
} | {
"line": 357,
"column": 82
} | [
{
"pp": "case refine_4\nR : Type w₁\nA : Type w₂\ninst✝³ : CommRing R\ninst✝² : CommRing A\ninst✝¹ : Algebra R A\nn : ℕ\nf : MvPolynomial (Fin n) R →ₐ[R] A\nhf : Surjective ⇑f\ninst✝ : FinitePresentation R A\nm : ℕ\nf' : MvPolynomial (Fin m) R →ₐ[R] A\nhf' : Surjective ⇑f'\ns : Finset (MvPolynomial (Fin m) R)\n... | intro p₁ p₂ _ _ h₁ h₂
obtain ⟨_, ⟨x₁, rfl⟩, y₁, hy₁, rfl⟩ := AddSubmonoid.mem_sup.mp h₁
obtain ⟨_, ⟨x₂, rfl⟩, y₂, hy₂, rfl⟩ := AddSubmonoid.mem_sup.mp h₂
rw [mul_add, add_mul, add_assoc, ← map_mul]
apply AddSubmonoid.add_mem_sup
· exact Set.mem_range_self _
· exact add_me... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.FinitePresentation | {
"line": 351,
"column": 8
} | {
"line": 357,
"column": 82
} | [
{
"pp": "case refine_4\nR : Type w₁\nA : Type w₂\ninst✝³ : CommRing R\ninst✝² : CommRing A\ninst✝¹ : Algebra R A\nn : ℕ\nf : MvPolynomial (Fin n) R →ₐ[R] A\nhf : Surjective ⇑f\ninst✝ : FinitePresentation R A\nm : ℕ\nf' : MvPolynomial (Fin m) R →ₐ[R] A\nhf' : Surjective ⇑f'\ns : Finset (MvPolynomial (Fin m) R)\n... | intro p₁ p₂ _ _ h₁ h₂
obtain ⟨_, ⟨x₁, rfl⟩, y₁, hy₁, rfl⟩ := AddSubmonoid.mem_sup.mp h₁
obtain ⟨_, ⟨x₂, rfl⟩, y₂, hy₂, rfl⟩ := AddSubmonoid.mem_sup.mp h₂
rw [mul_add, add_mul, add_assoc, ← map_mul]
apply AddSubmonoid.add_mem_sup
· exact Set.mem_range_self _
· exact add_me... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Grothendieck | {
"line": 208,
"column": 4
} | {
"line": 208,
"column": 12
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\nD : Type u₁\ninst✝ : Category.{v₁, u₁} D\nF : C ⥤ Cat\nX Y : Grothendieck F\ne₁ : X.base ≅ Y.base\ne₂ : (F.map e₁.hom).toFunctor.obj X.fiber ≅ Y.fiber\nthis :\n ((F.mapIso e₁).inv ≫ (F.mapIso e₁).hom).toFunctor.map e₂.inv =\n eqToHom ⋯ ≫ (𝟙 (F.obj Y.base)).t... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.RingTheory.FinitePresentation | {
"line": 440,
"column": 2
} | {
"line": 441,
"column": 46
} | [
{
"pp": "A : Type u_1\nB : Type u_2\nC : Type u_3\ninst✝² : CommRing A\ninst✝¹ : CommRing B\ninst✝ : CommRing C\ng : B →+* C\nf : A →+* B\nhg : g.FinitePresentation\nhf : f.FinitePresentation\n⊢ (g.comp f).FinitePresentation",
"usedConstants": [
"Algebra.algebraMap",
"CommSemiring.toSemiring",
... | algebraize [f, g, g.comp f]
exact Algebra.FinitePresentation.trans A B C | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.FinitePresentation | {
"line": 440,
"column": 2
} | {
"line": 441,
"column": 46
} | [
{
"pp": "A : Type u_1\nB : Type u_2\nC : Type u_3\ninst✝² : CommRing A\ninst✝¹ : CommRing B\ninst✝ : CommRing C\ng : B →+* C\nf : A →+* B\nhg : g.FinitePresentation\nhf : f.FinitePresentation\n⊢ (g.comp f).FinitePresentation",
"usedConstants": [
"Algebra.algebraMap",
"CommSemiring.toSemiring",
... | algebraize [f, g, g.comp f]
exact Algebra.FinitePresentation.trans A B C | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
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