module string | startPos dict | endPos dict | nextStartPos dict | goals list | goalsAfter list | ppTac string | elaborator string | kind string |
|---|---|---|---|---|---|---|---|---|
Mathlib.CategoryTheory.Monoidal.Category | {
"line": 986,
"column": 2
} | {
"line": 986,
"column": 13
} | {
"line": 986,
"column": 14
} | [
{
"pp": "J : Type u_1\ninst✝² : Category.{v_1, u_1} J\nC : Type u_2\ninst✝¹ : Category.{v_2, u_2} C\ninst✝ : MonoidalCategory C\nF G F' G' : J ⥤ C\nα : F ⟶ F'\nβ : G ⟶ G'\nX Y : J\nf : X ⟶ Y\nX' : J\n⊢ F.map f ▷ G.obj X' ≫ (α.app Y ⊗ₘ β.app X') = (α.app X ⊗ₘ β.app X') ≫ F'.map f ▷ G'.obj X'",
"ppTerm": "?m.... | [
"J : Type u_1\ninst✝² : Category.{v_1, u_1} J\nC : Type u_2\ninst✝¹ : Category.{v_2, u_2} C\ninst✝ : MonoidalCategory C\nF G F' G' : J ⥤ C\nα : F ⟶ F'\nβ : G ⟶ G'\nX Y : J\nf : X ⟶ Y\nX' : J\n⊢ F.map f ▷ G.obj X' ≫ (α.app Y ⊗ₘ β.app X') = (α.app X ⊗ₘ β.app X') ≫ F'.map f ▷ G'.obj X'"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Monoidal.Category | {
"line": 992,
"column": 2
} | {
"line": 992,
"column": 13
} | {
"line": 992,
"column": 14
} | [
{
"pp": "J : Type u_1\ninst✝² : Category.{v_1, u_1} J\nC : Type u_2\ninst✝¹ : Category.{v_2, u_2} C\ninst✝ : MonoidalCategory C\nF G F' G' : J ⥤ C\nα : F ⟶ F'\nβ : G ⟶ G'\nX' Y' : J\nf : X' ⟶ Y'\nX : J\n⊢ F.obj X ◁ G.map f ≫ (α.app X ⊗ₘ β.app Y') = (α.app X ⊗ₘ β.app X') ≫ F'.obj X ◁ G'.map f",
"ppTerm": "?m... | [
"J : Type u_1\ninst✝² : Category.{v_1, u_1} J\nC : Type u_2\ninst✝¹ : Category.{v_2, u_2} C\ninst✝ : MonoidalCategory C\nF G F' G' : J ⥤ C\nα : F ⟶ F'\nβ : G ⟶ G'\nX' Y' : J\nf : X' ⟶ Y'\nX : J\n⊢ F.obj X ◁ G.map f ≫ (α.app X ⊗ₘ β.app Y') = (α.app X ⊗ₘ β.app X') ≫ F'.obj X ◁ G'.map f"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Linear.LinearFunctor | {
"line": 83,
"column": 2
} | {
"line": 84,
"column": 9
} | {
"line": 84,
"column": 10
} | [
{
"pp": "R : Type u_1\ninst✝¹³ : Semiring R\nC : Type u_2\nD : Type u_3\ninst✝¹² : Category.{v_1, u_2} C\ninst✝¹¹ : Category.{v_2, u_3} D\ninst✝¹⁰ : Preadditive C\ninst✝⁹ : Preadditive D\ninst✝⁸ : CategoryTheory.Linear R C\ninst✝⁷ : CategoryTheory.Linear R D\nF : C ⥤ D\ninst✝⁶ : Linear R F\nE : Type u_4\ninst✝⁵... | [
"R : Type u_1\ninst✝¹³ : Semiring R\nC : Type u_2\nD : Type u_3\ninst✝¹² : Category.{v_1, u_2} C\ninst✝¹¹ : Category.{v_2, u_3} D\ninst✝¹⁰ : Preadditive C\ninst✝⁹ : Preadditive D\ninst✝⁸ : CategoryTheory.Linear R C\ninst✝⁷ : CategoryTheory.Linear R D\nF : C ⥤ D\ninst✝⁶ : Linear R F\nE : Type u_4\ninst✝⁵ : Category.... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Monoidal.Preadditive | {
"line": 184,
"column": 92
} | {
"line": 188,
"column": 47
} | {
"line": 190,
"column": 0
} | [
{
"pp": "C : Type u_1\ninst✝⁵ : Category.{v_1, u_1} C\ninst✝⁴ : Preadditive C\ninst✝³ : MonoidalCategory C\ninst✝² : MonoidalPreadditive C\ninst✝¹ : HasFiniteBiproducts C\nJ : Type\ninst✝ : Finite J\nX : C\nf : J → C\nj : J\n⊢ X ◁ biproduct.ι f j ≫ (leftDistributor X f).hom = biproduct.ι (fun j ↦ X ⊗ f j) j",
... | [] | by
classical
cases nonempty_fintype J
simp [leftDistributor_hom, Preadditive.comp_sum, ← whiskerLeft_comp_assoc,
biproduct.ι_π, whiskerLeft_dite, dite_comp] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Monoidal.Functor | {
"line": 225,
"column": 13
} | {
"line": 225,
"column": 22
} | {
"line": 225,
"column": 23
} | [
{
"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 : C ⥤ D\nG : D ⥤ E\ninst✝¹ : F.LaxMon... | [
"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 : C ⥤ D\nG : D ⥤ E\ninst✝¹ : F.LaxMonoidal\ninst✝... | comp_obj, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.CategoryTheory.Monoidal.Functor | {
"line": 227,
"column": 13
} | {
"line": 227,
"column": 22
} | {
"line": 227,
"column": 23
} | [
{
"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 : C ⥤ D\nG : D ⥤ E\ninst✝¹ : F.LaxMon... | [
"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 : C ⥤ D\nG : D ⥤ E\ninst✝¹ : F.LaxMonoidal\ninst✝... | comp_obj, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Algebra.Category.ModuleCat.Monoidal.Basic | {
"line": 470,
"column": 2
} | {
"line": 478,
"column": 37
} | {
"line": 480,
"column": 0
} | [
{
"pp": "R : Type u\ninst✝ : CommRing R\n⊢ MonoidalLinear R (ModuleCat R)",
"ppTerm": "?m.8",
"assigned": true,
"usedConstants": [
"TensorProduct.instDistribMulAction",
"CategoryTheory.MonoidalCategoryStruct.whiskerLeft",
"instHSMul",
"LinearMap.lTensor_smul",
"Category... | [] | refine ⟨?_, ?_⟩
· intros
ext : 1
refine TensorProduct.ext (LinearMap.ext fun x => LinearMap.ext fun y => ?_)
simp [ModuleCat.hom_whiskerLeft]
· intros
ext : 1
refine TensorProduct.ext (LinearMap.ext fun x => LinearMap.ext fun y => ?_)
simp [ModuleCat.hom_whiskerRight] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Category.ModuleCat.Monoidal.Basic | {
"line": 470,
"column": 2
} | {
"line": 478,
"column": 37
} | {
"line": 480,
"column": 0
} | [
{
"pp": "R : Type u\ninst✝ : CommRing R\n⊢ MonoidalLinear R (ModuleCat R)",
"ppTerm": "?m.8",
"assigned": true,
"usedConstants": [
"TensorProduct.instDistribMulAction",
"CategoryTheory.MonoidalCategoryStruct.whiskerLeft",
"instHSMul",
"LinearMap.lTensor_smul",
"Category... | [] | refine ⟨?_, ?_⟩
· intros
ext : 1
refine TensorProduct.ext (LinearMap.ext fun x => LinearMap.ext fun y => ?_)
simp [ModuleCat.hom_whiskerLeft]
· intros
ext : 1
refine TensorProduct.ext (LinearMap.ext fun x => LinearMap.ext fun y => ?_)
simp [ModuleCat.hom_whiskerRight] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Monoidal.Functor | {
"line": 574,
"column": 4
} | {
"line": 574,
"column": 39
} | {
"line": 575,
"column": 4
} | [
{
"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.toOplaxMonoidal = b.toOplaxMonoidal\nx✝¹ x✝ : C\n⊢ μ F x✝¹ x✝ = μ F x✝¹ x✝",
"ppTerm": "?μ",
"assigned": tru... | [
"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.toOplaxMonoidal = b.toOplaxMonoidal\nx✝¹ x✝ : C\n⊢ μ F x✝¹ x✝ ≫ (μIso F x✝¹ x✝).inv = μ F x✝¹ x✝ ≫ (μIso F x✝¹ x✝).inv"
] | rw [← cancel_mono (μIso F _ _).inv] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.CategoryTheory.Limits.Preserves.Shapes.Pullbacks | {
"line": 53,
"column": 8
} | {
"line": 53,
"column": 19
} | {
"line": 53,
"column": 20
} | [
{
"pp": "C : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\nD : Type u₂\ninst✝ : Category.{v₂, u₂} D\nX Y Z : C\nf : X ⟶ Z\ng : Y ⟶ Z\nc : PullbackCone f g\nG : C ⥤ D\n⊢ G.map c.fst ≫ G.map f = G.map c.snd ≫ G.map g",
"ppTerm": "?m.64",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedG... | [
"C : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\nD : Type u₂\ninst✝ : Category.{v₂, u₂} D\nX Y Z : C\nf : X ⟶ Z\ng : Y ⟶ Z\nc : PullbackCone f g\nG : C ⥤ D\n⊢ G.map c.fst ≫ G.map f = G.map c.snd ≫ G.map g"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Monoidal.Functor | {
"line": 915,
"column": 4
} | {
"line": 915,
"column": 15
} | {
"line": 915,
"column": 16
} | [
{
"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... | [
"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✝ : F.OplaxM... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Monoidal.Functor | {
"line": 921,
"column": 4
} | {
"line": 921,
"column": 15
} | {
"line": 921,
"column": 16
} | [
{
"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... | [
"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✝ : F.OplaxM... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Limits.Preserves.Shapes.Pullbacks | {
"line": 181,
"column": 51
} | {
"line": 181,
"column": 62
} | {
"line": 181,
"column": 63
} | [
{
"pp": "C : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\nD : Type u₂\ninst✝ : Category.{v₂, u₂} D\nW X Y : C\nf : W ⟶ X\ng : W ⟶ Y\nc : PushoutCocone f g\nG : C ⥤ D\n⊢ G.map f ≫ G.map c.inl = G.map g ≫ G.map c.inr",
"ppTerm": "?m.64",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"used... | [
"C : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\nD : Type u₂\ninst✝ : Category.{v₂, u₂} D\nW X Y : C\nf : W ⟶ X\ng : W ⟶ Y\nc : PushoutCocone f g\nG : C ⥤ D\n⊢ G.map f ≫ G.map c.inl = G.map g ≫ G.map c.inr"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.MorphismProperty.Composition | {
"line": 143,
"column": 24
} | {
"line": 145,
"column": 68
} | {
"line": 147,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝ : Category.{v, u} C\nW : Set (MorphismProperty C)\nh : ∀ W' ∈ W, W'.IsStableUnderComposition\nX✝ Y✝ Z✝ : C\nf : X✝ ⟶ Y✝\ng : Y✝ ⟶ Z✝\nhf : InfSet.sInf W f\nhg : InfSet.sInf W g\n⊢ InfSet.sInf W (f ≫ g)",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"Eq.mpr... | [] | by
rw [sInf_iff] at hf hg ⊢
exact fun W' hW' ↦ (h W' hW').comp_mem _ _ (hf _ hW') (hg _ hW') | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.MorphismProperty.Composition | {
"line": 310,
"column": 6
} | {
"line": 311,
"column": 48
} | {
"line": 313,
"column": 0
} | [
{
"pp": "case of_comp\nC : Type u\ninst✝¹ : Category.{v, u} C\nD : Type u'\ninst✝ : Category.{v', u'} D\nW : MorphismProperty C\nX✝ Y✝ Z✝ : C\nf✝ : X✝ ⟶ Y✝\nx✝ y✝ z✝ : C\ng' : x✝ ⟶ y✝\nf : y✝ ⟶ z✝\nhg' : W g'\nhf : W.multiplicativeClosure' f\nh_rec : ∀ (g : z✝ ⟶ Z✝), W.multiplicativeClosure' g → W.multiplicativ... | [] | rw [Category.assoc]
exact .of_comp g' (f ≫ g) hg' (h_rec g hg) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.MorphismProperty.Composition | {
"line": 310,
"column": 6
} | {
"line": 311,
"column": 48
} | {
"line": 313,
"column": 0
} | [
{
"pp": "case of_comp\nC : Type u\ninst✝¹ : Category.{v, u} C\nD : Type u'\ninst✝ : Category.{v', u'} D\nW : MorphismProperty C\nX✝ Y✝ Z✝ : C\nf✝ : X✝ ⟶ Y✝\nx✝ y✝ z✝ : C\ng' : x✝ ⟶ y✝\nf : y✝ ⟶ z✝\nhg' : W g'\nhf : W.multiplicativeClosure' f\nh_rec : ∀ (g : z✝ ⟶ Z✝), W.multiplicativeClosure' g → W.multiplicativ... | [] | rw [Category.assoc]
exact .of_comp g' (f ≫ g) hg' (h_rec g hg) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.MorphismProperty.Composition | {
"line": 347,
"column": 18
} | {
"line": 347,
"column": 29
} | {
"line": 347,
"column": 30
} | [
{
"pp": "C : Type u\ninst✝ : Category.{v, u} C\nx✝ W' : MorphismProperty C\nh : x✝ ≤ W'\n⊢ x✝.multiplicativeClosure ≤ W'.multiplicativeClosure",
"ppTerm": "?m.10",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"CategoryTheory.MorphismProperty",
"CategoryTheory.MorphismProperty.mul... | [
"C : Type u\ninst✝ : Category.{v, u} C\nx✝ W' : MorphismProperty C\nh : x✝ ≤ W'\n⊢ x✝ ≤ W'.multiplicativeClosure"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.MorphismProperty.Composition | {
"line": 356,
"column": 6
} | {
"line": 356,
"column": 31
} | {
"line": 356,
"column": 32
} | [
{
"pp": "case of_comp\nC : Type u\ninst✝ : Category.{v, u} C\nW : MorphismProperty C\nx y : C\nf✝ : x ⟶ y\nx✝ y✝ z✝ : C\nf : x✝ ⟶ y✝\ng : y✝ ⟶ z✝\nhf : W f\nhg : W.multiplicativeClosure' g\nhr : W.multiplicativeClosure g\n⊢ W.multiplicativeClosure (f ≫ g)",
"ppTerm": "?of_comp",
"assigned": true,
"u... | [] | | of_comp f g hf hg hr => | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | null |
Mathlib.CategoryTheory.MorphismProperty.Composition | {
"line": 364,
"column": 12
} | {
"line": 364,
"column": 23
} | {
"line": 364,
"column": 24
} | [
{
"pp": "case map.id\nC : Type u\ninst✝¹ : Category.{v, u} C\nD : Type u'\ninst✝ : Category.{v', u'} D\nW : MorphismProperty C\nF : C ⥤ D\nX✝¹ Y✝¹ : D\nf : X✝¹ ⟶ Y✝¹\nX✝ Y✝ : C\nf✝ : X✝ ⟶ Y✝\nx : C\n⊢ (W.strictMap F).multiplicativeClosure (F.map (𝟙 x))",
"ppTerm": "?map.id",
"assigned": true,
"used... | [
"case map.id\nC : Type u\ninst✝¹ : Category.{v, u} C\nD : Type u'\ninst✝ : Category.{v', u'} D\nW : MorphismProperty C\nF : C ⥤ D\nX✝¹ Y✝¹ : D\nf : X✝¹ ⟶ Y✝¹\nX✝ Y✝ : C\nf✝ : X✝ ⟶ Y✝\nx : C\n⊢ (W.strictMap F).multiplicativeClosure (𝟙 (F.obj x))"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.MorphismProperty.Composition | {
"line": 366,
"column": 4
} | {
"line": 366,
"column": 15
} | {
"line": 366,
"column": 16
} | [
{
"pp": "case map.comp_of\nC : Type u\ninst✝¹ : Category.{v, u} C\nD : Type u'\ninst✝ : Category.{v', u'} D\nW : MorphismProperty C\nF : C ⥤ D\nX✝¹ Y✝¹ : D\nf : X✝¹ ⟶ Y✝¹\nX✝ Y✝ : C\nf✝¹ : X✝ ⟶ Y✝\nx✝ y✝ z✝ : C\nf✝ : x✝ ⟶ y✝\ng✝ : y✝ ⟶ z✝\nhf : W.multiplicativeClosure f✝\nhg : W g✝\nh : (W.strictMap F).multipli... | [
"case map.comp_of\nC : Type u\ninst✝¹ : Category.{v, u} C\nD : Type u'\ninst✝ : Category.{v', u'} D\nW : MorphismProperty C\nF : C ⥤ D\nX✝¹ Y✝¹ : D\nf : X✝¹ ⟶ Y✝¹\nX✝ Y✝ : C\nf✝¹ : X✝ ⟶ Y✝\nx✝ y✝ z✝ : C\nf✝ : x✝ ⟶ y✝\ng✝ : y✝ ⟶ z✝\nhf : W.multiplicativeClosure f✝\nhg : W g✝\nh : (W.strictMap F).multiplicativeClosur... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.MorphismProperty.Composition | {
"line": 464,
"column": 7
} | {
"line": 464,
"column": 18
} | {
"line": 464,
"column": 19
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\nD : Type u'\ninst✝ : Category.{v', u'} D\nW : MorphismProperty C\nX✝ Y✝ Z✝ : C\nf : X✝ ⟶ Y✝\ng : Y✝ ⟶ Z✝\nhg : IsIso g\nhfg : IsIso (f ≫ g)\n⊢ isomorphisms C f",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"CategoryTh... | [
"C : Type u\ninst✝¹ : Category.{v, u} C\nD : Type u'\ninst✝ : Category.{v', u'} D\nW : MorphismProperty C\nX✝ Y✝ Z✝ : C\nf : X✝ ⟶ Y✝\ng : Y✝ ⟶ Z✝\nhg : IsIso g\nhfg : IsIso (f ≫ g)\n⊢ IsIso f"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.MorphismProperty.Composition | {
"line": 466,
"column": 7
} | {
"line": 466,
"column": 18
} | {
"line": 466,
"column": 19
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\nD : Type u'\ninst✝ : Category.{v', u'} D\nW : MorphismProperty C\nX✝ Y✝ Z✝ : C\nf : X✝ ⟶ Y✝\ng : Y✝ ⟶ Z✝\nhf : IsIso f\nhfg : IsIso (f ≫ g)\n⊢ isomorphisms C g",
"ppTerm": "?m.42",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"CategoryTh... | [
"C : Type u\ninst✝¹ : Category.{v, u} C\nD : Type u'\ninst✝ : Category.{v', u'} D\nW : MorphismProperty C\nX✝ Y✝ Z✝ : C\nf : X✝ ⟶ Y✝\ng : Y✝ ⟶ Z✝\nhf : IsIso f\nhfg : IsIso (f ≫ g)\n⊢ IsIso g"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.MorphismProperty.Composition | {
"line": 470,
"column": 69
} | {
"line": 470,
"column": 80
} | {
"line": 470,
"column": 81
} | [
{
"pp": "C : Type u\ninst✝² : Category.{v, u} C\nD : Type u'\ninst✝¹ : Category.{v', u'} D\nW✝ : MorphismProperty C\nF : C ⥤ D\nW : MorphismProperty D\ninst✝ : W.HasTwoOutOfThreeProperty\nX✝ Y✝ Z✝ : C\nf : X✝ ⟶ Y✝\ng : Y✝ ⟶ Z✝\nhg : W.inverseImage F g\nhfg : W.inverseImage F (f ≫ g)\n⊢ W (F.map f ≫ F.map g)",
... | [
"C : Type u\ninst✝² : Category.{v, u} C\nD : Type u'\ninst✝¹ : Category.{v', u'} D\nW✝ : MorphismProperty C\nF : C ⥤ D\nW : MorphismProperty D\ninst✝ : W.HasTwoOutOfThreeProperty\nX✝ Y✝ Z✝ : C\nf : X✝ ⟶ Y✝\ng : Y✝ ⟶ Z✝\nhg : W.inverseImage F g\nhfg : W.inverseImage F (f ≫ g)\n⊢ W (F.map f ≫ F.map g)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.MorphismProperty.Composition | {
"line": 471,
"column": 67
} | {
"line": 471,
"column": 78
} | {
"line": 471,
"column": 79
} | [
{
"pp": "C : Type u\ninst✝² : Category.{v, u} C\nD : Type u'\ninst✝¹ : Category.{v', u'} D\nW✝ : MorphismProperty C\nF : C ⥤ D\nW : MorphismProperty D\ninst✝ : W.HasTwoOutOfThreeProperty\nX✝ Y✝ Z✝ : C\nf : X✝ ⟶ Y✝\ng : Y✝ ⟶ Z✝\nhf : W.inverseImage F f\nhfg : W.inverseImage F (f ≫ g)\n⊢ W (F.map f ≫ F.map g)",
... | [
"C : Type u\ninst✝² : Category.{v, u} C\nD : Type u'\ninst✝¹ : Category.{v', u'} D\nW✝ : MorphismProperty C\nF : C ⥤ D\nW : MorphismProperty D\ninst✝ : W.HasTwoOutOfThreeProperty\nX✝ Y✝ Z✝ : C\nf : X✝ ⟶ Y✝\ng : Y✝ ⟶ Z✝\nhf : W.inverseImage F f\nhfg : W.inverseImage F (f ≫ g)\n⊢ W (F.map f ≫ F.map g)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.MorphismProperty.Concrete | {
"line": 53,
"column": 14
} | {
"line": 56,
"column": 9
} | {
"line": 57,
"column": 2
} | [
{
"pp": "C : Type u\ninst✝² : Category.{v, u} C\nFC : C → C → Type u_1\nCC : C → Type u_2\ninst✝¹ : (X Y : C) → FunLike (FC X Y) (CC X) (CC Y)\ninst✝ : ConcreteCategory C FC\nX : C\n⊢ MorphismProperty.injective C (𝟙 X)",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"Eq.mpr",
... | [] | by
delta MorphismProperty.injective
convert! injective_id
aesop | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Limits.Shapes.Opposites.Pullbacks | {
"line": 247,
"column": 6
} | {
"line": 247,
"column": 17
} | {
"line": 247,
"column": 18
} | [
{
"pp": "C : Type u₁\ninst✝ : Category.{v₁, u₁} C\nX Y Z : C\nf : X ⟶ Z\ng : Y ⟶ Z\n⊢ HasPushout f.op g.op ↔ HasPullback f g",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"CategoryTheory.Limits.WalkingSpan",
"Opposite",
"CategoryTheory.CategoryStruct.toQ... | [
"C : Type u₁\ninst✝ : Category.{v₁, u₁} C\nX Y Z : C\nf : X ⟶ Z\ng : Y ⟶ Z\n⊢ HasColimit (span f.op g.op) ↔ HasPullback f g"
] | HasPushout, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.Limits.Shapes.Opposites.Pullbacks | {
"line": 253,
"column": 6
} | {
"line": 253,
"column": 17
} | {
"line": 253,
"column": 18
} | [
{
"pp": "C : Type u₁\ninst✝ : Category.{v₁, u₁} C\nX Y Z : Cᵒᵖ\nf : X ⟶ Z\ng : Y ⟶ Z\n⊢ HasPushout f.unop g.unop ↔ HasPullback f g",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"CategoryTheory.Limits.WalkingSpan",
"Opposite",
"CategoryTheory.CategoryStru... | [
"C : Type u₁\ninst✝ : Category.{v₁, u₁} C\nX Y Z : Cᵒᵖ\nf : X ⟶ Z\ng : Y ⟶ Z\n⊢ HasColimit (span f.unop g.unop) ↔ HasPullback f g"
] | HasPushout, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Category.ModuleCat.Colimits | {
"line": 54,
"column": 34
} | {
"line": 54,
"column": 49
} | {
"line": 54,
"column": 49
} | [
{
"pp": "R : Type w\ninst✝² : Ring R\nJ : Type u\ninst✝¹ : Category.{v, u} J\nF : J ⥤ ModuleCat R\ninst✝ : HasColimit (F ⋙ forget₂ (ModuleCat R) AddCommGrpCat)\n⊢ ∀ (j : J),\n colimit.ι (F ⋙ forget₂ (ModuleCat R) AddCommGrpCat) j ≫\n colimMap { app := fun j ↦ (F.obj j).smul 1, naturality := ⋯ } =\n ... | [] | simp +instances | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Algebra.Category.ModuleCat.Colimits | {
"line": 54,
"column": 34
} | {
"line": 54,
"column": 49
} | {
"line": 54,
"column": 49
} | [
{
"pp": "R : Type w\ninst✝² : Ring R\nJ : Type u\ninst✝¹ : Category.{v, u} J\nF : J ⥤ ModuleCat R\ninst✝ : HasColimit (F ⋙ forget₂ (ModuleCat R) AddCommGrpCat)\n⊢ ∀ (j : J),\n colimit.ι (F ⋙ forget₂ (ModuleCat R) AddCommGrpCat) j ≫\n colimMap { app := fun j ↦ (F.obj j).smul 1, naturality := ⋯ } =\n ... | [] | simp +instances | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Category.ModuleCat.Colimits | {
"line": 54,
"column": 34
} | {
"line": 54,
"column": 49
} | {
"line": 54,
"column": 49
} | [
{
"pp": "R : Type w\ninst✝² : Ring R\nJ : Type u\ninst✝¹ : Category.{v, u} J\nF : J ⥤ ModuleCat R\ninst✝ : HasColimit (F ⋙ forget₂ (ModuleCat R) AddCommGrpCat)\n⊢ ∀ (j : J),\n colimit.ι (F ⋙ forget₂ (ModuleCat R) AddCommGrpCat) j ≫\n colimMap { app := fun j ↦ (F.obj j).smul 1, naturality := ⋯ } =\n ... | [] | simp +instances | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Category.ModuleCat.Colimits | {
"line": 59,
"column": 47
} | {
"line": 59,
"column": 62
} | {
"line": 59,
"column": 62
} | [
{
"pp": "R : Type w\ninst✝² : Ring R\nJ : Type u\ninst✝¹ : Category.{v, u} J\nF : J ⥤ ModuleCat R\ninst✝ : HasColimit (F ⋙ forget₂ (ModuleCat R) AddCommGrpCat)\nr s : R\nj : J\n⊢ colimit.ι (F ⋙ forget₂ (ModuleCat R) AddCommGrpCat) j ≫\n colimMap { app := fun j ↦ (F.obj j).smul (r * s), naturality := ⋯ } =\... | [] | simp +instances | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Algebra.Category.ModuleCat.Colimits | {
"line": 59,
"column": 47
} | {
"line": 59,
"column": 62
} | {
"line": 59,
"column": 62
} | [
{
"pp": "R : Type w\ninst✝² : Ring R\nJ : Type u\ninst✝¹ : Category.{v, u} J\nF : J ⥤ ModuleCat R\ninst✝ : HasColimit (F ⋙ forget₂ (ModuleCat R) AddCommGrpCat)\nr s : R\nj : J\n⊢ colimit.ι (F ⋙ forget₂ (ModuleCat R) AddCommGrpCat) j ≫\n colimMap { app := fun j ↦ (F.obj j).smul (r * s), naturality := ⋯ } =\... | [] | simp +instances | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Category.ModuleCat.Colimits | {
"line": 59,
"column": 47
} | {
"line": 59,
"column": 62
} | {
"line": 59,
"column": 62
} | [
{
"pp": "R : Type w\ninst✝² : Ring R\nJ : Type u\ninst✝¹ : Category.{v, u} J\nF : J ⥤ ModuleCat R\ninst✝ : HasColimit (F ⋙ forget₂ (ModuleCat R) AddCommGrpCat)\nr s : R\nj : J\n⊢ colimit.ι (F ⋙ forget₂ (ModuleCat R) AddCommGrpCat) j ≫\n colimMap { app := fun j ↦ (F.obj j).smul (r * s), naturality := ⋯ } =\... | [] | simp +instances | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Category.ModuleCat.Colimits | {
"line": 53,
"column": 35
} | {
"line": 53,
"column": 50
} | {
"line": 53,
"column": 50
} | [
{
"pp": "R : Type w\ninst✝² : Ring R\nJ : Type u\ninst✝¹ : Category.{v, u} J\nF : J ⥤ ModuleCat R\ninst✝ : HasColimit (F ⋙ forget₂ (ModuleCat R) AddCommGrpCat)\n⊢ ∀ (j : J),\n colimit.ι (F ⋙ forget₂ (ModuleCat R) AddCommGrpCat) j ≫\n colimMap { app := fun j ↦ (F.obj j).smul 0, naturality := ⋯ } =\n ... | [] | simp +instances | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Algebra.Category.ModuleCat.Colimits | {
"line": 53,
"column": 35
} | {
"line": 53,
"column": 50
} | {
"line": 53,
"column": 50
} | [
{
"pp": "R : Type w\ninst✝² : Ring R\nJ : Type u\ninst✝¹ : Category.{v, u} J\nF : J ⥤ ModuleCat R\ninst✝ : HasColimit (F ⋙ forget₂ (ModuleCat R) AddCommGrpCat)\n⊢ ∀ (j : J),\n colimit.ι (F ⋙ forget₂ (ModuleCat R) AddCommGrpCat) j ≫\n colimMap { app := fun j ↦ (F.obj j).smul 0, naturality := ⋯ } =\n ... | [] | simp +instances | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Category.ModuleCat.Colimits | {
"line": 53,
"column": 35
} | {
"line": 53,
"column": 50
} | {
"line": 53,
"column": 50
} | [
{
"pp": "R : Type w\ninst✝² : Ring R\nJ : Type u\ninst✝¹ : Category.{v, u} J\nF : J ⥤ ModuleCat R\ninst✝ : HasColimit (F ⋙ forget₂ (ModuleCat R) AddCommGrpCat)\n⊢ ∀ (j : J),\n colimit.ι (F ⋙ forget₂ (ModuleCat R) AddCommGrpCat) j ≫\n colimMap { app := fun j ↦ (F.obj j).smul 0, naturality := ⋯ } =\n ... | [] | simp +instances | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Category.Grp.Colimits | {
"line": 91,
"column": 2
} | {
"line": 93,
"column": 6
} | {
"line": 95,
"column": 0
} | [
{
"pp": "J : Type u\ninst✝¹ : Category.{v, u} J\nF : J ⥤ AddCommGrpCat\nc : Cocone F\ninst✝ : DecidableEq J\nj : J\nx : ↑(F.obj j)\n⊢ (desc F c) ((ι F j) x) = (ConcreteCategory.hom (c.ι.app j)) x",
"ppTerm": "?m.33",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"CategoryTheory.Functor"... | [] | dsimp [desc, ι]
erw [QuotientAddGroup.lift_mk']
simp | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Category.Grp.Colimits | {
"line": 91,
"column": 2
} | {
"line": 93,
"column": 6
} | {
"line": 95,
"column": 0
} | [
{
"pp": "J : Type u\ninst✝¹ : Category.{v, u} J\nF : J ⥤ AddCommGrpCat\nc : Cocone F\ninst✝ : DecidableEq J\nj : J\nx : ↑(F.obj j)\n⊢ (desc F c) ((ι F j) x) = (ConcreteCategory.hom (c.ι.app j)) x",
"ppTerm": "?m.33",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"CategoryTheory.Functor"... | [] | dsimp [desc, ι]
erw [QuotientAddGroup.lift_mk']
simp | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Adjunction.Mates | {
"line": 182,
"column": 6
} | {
"line": 182,
"column": 18
} | {
"line": 182,
"column": 19
} | [
{
"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\nG₁ : A ⥤ C\nG₂ : C ⥤ E\nH₁ : B ⥤ D\nH₂ : D ⥤ F... | [
"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\nG₁ : A ⥤ C\nG₂ : C ⥤ E\nH₁ : B ⥤ D\nH₂ : D ⥤ F\nL₁ : A ⥤ B... | L₃.map_comp, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Congruence.Hom | {
"line": 116,
"column": 16
} | {
"line": 116,
"column": 33
} | {
"line": 117,
"column": 2
} | [
{
"pp": "case symm\nM : Type u_1\nN : Type u_2\ninst✝¹ : NonAssocSemiring M\ninst✝ : NonAssocSemiring N\nc : RingCon M\nf : M →+* N\nh✝ : ker f ≤ c\nhf : Surjective ⇑f\nthis : Equivalence (Relation.Map ⇑c.toSetoid ⇑f ⇑f)\ni✝¹ i✝ x✝ y✝ : N\na✝ : RingConGen.Rel (Relation.Map ⇑c ⇑f ⇑f) x✝ y✝\nh : Relation.Map (⇑c)... | [] | exact this.symm h | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.RingTheory.Congruence.Hom | {
"line": 116,
"column": 16
} | {
"line": 116,
"column": 33
} | {
"line": 117,
"column": 2
} | [
{
"pp": "case symm\nM : Type u_1\nN : Type u_2\ninst✝¹ : NonAssocSemiring M\ninst✝ : NonAssocSemiring N\nc : RingCon M\nf : M →+* N\nh✝ : ker f ≤ c\nhf : Surjective ⇑f\nthis : Equivalence (Relation.Map ⇑c.toSetoid ⇑f ⇑f)\ni✝¹ i✝ x✝ y✝ : N\na✝ : RingConGen.Rel (Relation.Map ⇑c ⇑f ⇑f) x✝ y✝\nh : Relation.Map (⇑c)... | [] | exact this.symm h | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Congruence.Hom | {
"line": 116,
"column": 16
} | {
"line": 116,
"column": 33
} | {
"line": 117,
"column": 2
} | [
{
"pp": "case symm\nM : Type u_1\nN : Type u_2\ninst✝¹ : NonAssocSemiring M\ninst✝ : NonAssocSemiring N\nc : RingCon M\nf : M →+* N\nh✝ : ker f ≤ c\nhf : Surjective ⇑f\nthis : Equivalence (Relation.Map ⇑c.toSetoid ⇑f ⇑f)\ni✝¹ i✝ x✝ y✝ : N\na✝ : RingConGen.Rel (Relation.Map ⇑c ⇑f ⇑f) x✝ y✝\nh : Relation.Map (⇑c)... | [] | exact this.symm h | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Adjunction.Mates | {
"line": 318,
"column": 4
} | {
"line": 318,
"column": 79
} | {
"line": 320,
"column": 0
} | [
{
"pp": "C : Type u₁\nD : Type u₂\ninst✝¹ : Category.{v₁, u₁} C\ninst✝ : Category.{v₂, u₂} D\nL₁ L₂ : C ⥤ D\nR₁ R₂ : D ⥤ C\nadj₁ : L₁ ⊣ R₁\nadj₂ : L₂ ⊣ R₂\nα : R₁ ⟶ R₂\nc : C\n⊢ adj₁.unit.app c ≫ ((conjugateEquiv adj₁ adj₂).toFun ((conjugateEquiv adj₁ adj₂).invFun α)).app (L₁.obj c) =\n adj₂.unit.app c ≫ R₂.... | [] | exact (unit_conjugateEquiv adj₁ adj₂ ((conjugateEquiv adj₁ adj₂).symm α) c) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.CategoryTheory.Adjunction.Mates | {
"line": 411,
"column": 83
} | {
"line": 411,
"column": 94
} | {
"line": 411,
"column": 95
} | [
{
"pp": "C : Type u₁\nD : Type u₂\ninst✝² : Category.{v₁, u₁} C\ninst✝¹ : Category.{v₂, u₂} D\nL₁ L₂ : C ⥤ D\nR₁ R₂ : D ⥤ C\nadj₁ : L₁ ⊣ R₁\nadj₂ : L₂ ⊣ R₂\nα : L₂ ⟶ L₁\ninst✝ : IsIso ((conjugateEquiv adj₁ adj₂) α)\nthis : IsIso ((conjugateEquiv adj₁ adj₂).symm ((conjugateEquiv adj₁ adj₂) α))\n⊢ IsIso α",
"... | [
"C : Type u₁\nD : Type u₂\ninst✝² : Category.{v₁, u₁} C\ninst✝¹ : Category.{v₂, u₂} D\nL₁ L₂ : C ⥤ D\nR₁ R₂ : D ⥤ C\nadj₁ : L₁ ⊣ R₁\nadj₂ : L₂ ⊣ R₂\nα : L₂ ⟶ L₁\ninst✝ : IsIso ((conjugateEquiv adj₁ adj₂) α)\nthis : IsIso ((conjugateEquiv adj₁ adj₂).symm ((conjugateEquiv adj₁ adj₂) α))\n⊢ IsIso α"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Adjunction.Mates | {
"line": 421,
"column": 7
} | {
"line": 421,
"column": 18
} | {
"line": 421,
"column": 19
} | [
{
"pp": "C : Type u₁\nD : Type u₂\ninst✝² : Category.{v₁, u₁} C\ninst✝¹ : Category.{v₂, u₂} D\nL₁ L₂ : C ⥤ D\nR₁ R₂ : D ⥤ C\nadj₁ : L₁ ⊣ R₁\nadj₂ : L₂ ⊣ R₂\nα : R₁ ⟶ R₂\ninst✝ : IsIso ((conjugateEquiv adj₁ adj₂).symm α)\nthis : IsIso ((conjugateEquiv adj₁ adj₂) ((conjugateEquiv adj₁ adj₂).symm α))\n⊢ IsIso α",
... | [
"C : Type u₁\nD : Type u₂\ninst✝² : Category.{v₁, u₁} C\ninst✝¹ : Category.{v₂, u₂} D\nL₁ L₂ : C ⥤ D\nR₁ R₂ : D ⥤ C\nadj₁ : L₁ ⊣ R₁\nadj₂ : L₂ ⊣ R₂\nα : R₁ ⟶ R₂\ninst✝ : IsIso ((conjugateEquiv adj₁ adj₂).symm α)\nthis : IsIso ((conjugateEquiv adj₁ adj₂) ((conjugateEquiv adj₁ adj₂).symm α))\n⊢ IsIso α"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.TensorAlgebra.Basic | {
"line": 139,
"column": 6
} | {
"line": 139,
"column": 75
} | {
"line": 140,
"column": 4
} | [
{
"pp": "R : Type u_1\ninst✝⁴ : CommSemiring R\nM : Type u_2\ninst✝³ : AddCommMonoid M\ninst✝² : Module R M\nA : Type u_3\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nf : M →ₗ[R] A\nx : M\n⊢ ((ringCon R M).liftₐ ((FreeAlgebra.lift R) ⇑f) ⋯) ↑(FreeAlgebra.ι R x) = f x",
"ppTerm": "?m.179",
"assigned": true... | [] | exact (RingCon.liftₐ_mk _ _ _ _).trans (FreeAlgebra.lift_ι_apply f x) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.CategoryTheory.Adjunction.Mates | {
"line": 489,
"column": 2
} | {
"line": 498,
"column": 19
} | {
"line": 500,
"column": 0
} | [
{
"pp": "A : Type u₁\nB : Type u₂\nC : Type u₃\nD : Type u₄\ninst✝³ : Category.{v₁, u₁} A\ninst✝² : Category.{v₂, u₂} B\ninst✝¹ : Category.{v₃, u₃} C\ninst✝ : Category.{v₄, u₄} D\nG : A ⥤ C\nH : B ⥤ D\nL₁ : A ⥤ B\nR₁ : B ⥤ A\nL₂ : C ⥤ D\nR₂ : D ⥤ C\nL₃ : C ⥤ D\nR₃ : D ⥤ C\nadj₁ : L₁ ⊣ R₁\nadj₂ : L₂ ⊣ R₂\nadj₃ :... | [] | ext b
have vcomp := mateEquiv_vcomp adj₁ adj₂ adj₃ α (L₃.leftUnitor.hom ≫ β ≫ L₂.rightUnitor.inv)
unfold vComp hComp at vcomp
have vcompb := congr_app vcomp b
simp only [comp_obj, id_obj, whiskerLeft_comp, assoc, mateEquiv_apply, whiskerLeft_twice,
Iso.hom_inv_id_assoc, whiskerRight_comp, comp_app, Functor.... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Adjunction.Mates | {
"line": 489,
"column": 2
} | {
"line": 498,
"column": 19
} | {
"line": 500,
"column": 0
} | [
{
"pp": "A : Type u₁\nB : Type u₂\nC : Type u₃\nD : Type u₄\ninst✝³ : Category.{v₁, u₁} A\ninst✝² : Category.{v₂, u₂} B\ninst✝¹ : Category.{v₃, u₃} C\ninst✝ : Category.{v₄, u₄} D\nG : A ⥤ C\nH : B ⥤ D\nL₁ : A ⥤ B\nR₁ : B ⥤ A\nL₂ : C ⥤ D\nR₂ : D ⥤ C\nL₃ : C ⥤ D\nR₃ : D ⥤ C\nadj₁ : L₁ ⊣ R₁\nadj₂ : L₂ ⊣ R₂\nadj₃ :... | [] | ext b
have vcomp := mateEquiv_vcomp adj₁ adj₂ adj₃ α (L₃.leftUnitor.hom ≫ β ≫ L₂.rightUnitor.inv)
unfold vComp hComp at vcomp
have vcompb := congr_app vcomp b
simp only [comp_obj, id_obj, whiskerLeft_comp, assoc, mateEquiv_apply, whiskerLeft_twice,
Iso.hom_inv_id_assoc, whiskerRight_comp, comp_app, Functor.... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.TensorAlgebra.Basic | {
"line": 231,
"column": 2
} | {
"line": 231,
"column": 22
} | {
"line": 233,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝² : CommSemiring R\nM : Type u_2\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nx : R\n⊢ algebraMapInv ((algebraMap R (TensorAlgebra R M)) x) = x",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"TensorAlgebra.definition._proof_3._@.Mathlib.LinearAlgebra.Te... | [] | simp [algebraMapInv] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.LinearAlgebra.TensorAlgebra.Basic | {
"line": 231,
"column": 2
} | {
"line": 231,
"column": 22
} | {
"line": 233,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝² : CommSemiring R\nM : Type u_2\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nx : R\n⊢ algebraMapInv ((algebraMap R (TensorAlgebra R M)) x) = x",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"TensorAlgebra.definition._proof_3._@.Mathlib.LinearAlgebra.Te... | [] | simp [algebraMapInv] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.TensorAlgebra.Basic | {
"line": 231,
"column": 2
} | {
"line": 231,
"column": 22
} | {
"line": 233,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝² : CommSemiring R\nM : Type u_2\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nx : R\n⊢ algebraMapInv ((algebraMap R (TensorAlgebra R M)) x) = x",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"TensorAlgebra.definition._proof_3._@.Mathlib.LinearAlgebra.Te... | [] | simp [algebraMapInv] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Coalgebra.Basic | {
"line": 123,
"column": 2
} | {
"line": 123,
"column": 34
} | {
"line": 123,
"column": 35
} | [
{
"pp": "R : Type u\nA : Type v\nι : Type u_1\ninst✝³ : CommSemiring R\ninst✝² : AddCommMonoid A\ninst✝¹ : Module R A\ninst✝ : Coalgebra R A\na : A\nrepr : Repr R a ι\n⊢ ∑ i ∈ repr.index, counit (repr.left i) ⊗ₜ[R] repr.right i = 1 ⊗ₜ[R] a",
"ppTerm": "?m.72",
"assigned": false,
"usedConstants": [],... | [
"R : Type u\nA : Type v\nι : Type u_1\ninst✝³ : CommSemiring R\ninst✝² : AddCommMonoid A\ninst✝¹ : Module R A\ninst✝ : Coalgebra R A\na : A\nrepr : Repr R a ι\n⊢ ∑ i ∈ repr.index, counit (repr.left i) ⊗ₜ[R] repr.right i = 1 ⊗ₜ[R] a"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Coalgebra.Basic | {
"line": 128,
"column": 2
} | {
"line": 128,
"column": 34
} | {
"line": 128,
"column": 35
} | [
{
"pp": "R : Type u\nA : Type v\nι : Type u_1\ninst✝³ : CommSemiring R\ninst✝² : AddCommMonoid A\ninst✝¹ : Module R A\ninst✝ : Coalgebra R A\na : A\nrepr : Repr R a ι\n⊢ ∑ i ∈ repr.index, repr.left i ⊗ₜ[R] counit (repr.right i) = a ⊗ₜ[R] 1",
"ppTerm": "?m.72",
"assigned": false,
"usedConstants": [],... | [
"R : Type u\nA : Type v\nι : Type u_1\ninst✝³ : CommSemiring R\ninst✝² : AddCommMonoid A\ninst✝¹ : Module R A\ninst✝ : Coalgebra R A\na : A\nrepr : Repr R a ι\n⊢ ∑ i ∈ repr.index, repr.left i ⊗ₜ[R] counit (repr.right i) = a ⊗ₜ[R] 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Coalgebra.Basic | {
"line": 137,
"column": 2
} | {
"line": 138,
"column": 44
} | {
"line": 138,
"column": 45
} | [
{
"pp": "R : Type u\nA : Type v\nι : Type u_1\nκ : ι → Type u_2\nΛ : ι → Type u_3\ninst✝³ : CommSemiring R\ninst✝² : AddCommMonoid A\ninst✝¹ : Module R A\ninst✝ : Coalgebra R A\na : A\nrepr : Repr R a ι\na₁ : (i : ι) → Repr R (repr.left i) (κ i)\na₂ : (i : ι) → Repr R (repr.right i) (Λ i)\n⊢ ∑ i ∈ repr.index, ∑... | [
"R : Type u\nA : Type v\nι : Type u_1\nκ : ι → Type u_2\nΛ : ι → Type u_3\ninst✝³ : CommSemiring R\ninst✝² : AddCommMonoid A\ninst✝¹ : Module R A\ninst✝ : Coalgebra R A\na : A\nrepr : Repr R a ι\na₁ : (i : ι) → Repr R (repr.left i) (κ i)\na₂ : (i : ι) → Repr R (repr.right i) (Λ i)\n⊢ ∑ i ∈ repr.index, ∑ j ∈ (a₁ i).... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Coalgebra.Basic | {
"line": 171,
"column": 2
} | {
"line": 172,
"column": 9
} | {
"line": 172,
"column": 10
} | [
{
"pp": "R : Type u\nA : Type v\nι : Type u_1\ninst✝³ : CommSemiring R\ninst✝² : AddCommMonoid A\ninst✝¹ : Module R A\ninst✝ : Coalgebra R A\na : A\n𝓡 : Repr R a ι\n⊢ ∑ x ∈ 𝓡.index, counit (𝓡.left x) • 𝓡.right x = a",
"ppTerm": "?m.51",
"assigned": false,
"usedConstants": [],
"usedFVars": []... | [
"R : Type u\nA : Type v\nι : Type u_1\ninst✝³ : CommSemiring R\ninst✝² : AddCommMonoid A\ninst✝¹ : Module R A\ninst✝ : Coalgebra R A\na : A\n𝓡 : Repr R a ι\n⊢ ∑ x ∈ 𝓡.index, counit (𝓡.left x) • 𝓡.right x = a"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Coalgebra.Equiv | {
"line": 311,
"column": 4
} | {
"line": 315,
"column": 61
} | {
"line": 315,
"column": 62
} | [
{
"pp": "case e_f\nR : Type u_1\nA : Type u_2\nB : Type u_3\nC : Type u_4\ninst✝⁶ : CommSemiring R\ninst✝⁵ : AddCommMonoid A\ninst✝⁴ : Module R A\ninst✝³ : Coalgebra R A\ninst✝² : AddCommMonoid B\ninst✝¹ : Module R B\ninst✝ : CoalgebraStruct R B\nf : A ≃ₗc[R] B\nx : A\n⊢ (((↑(TensorProduct.assoc R B B B) ∘ₗ\n ... | [
"case e_f\nR : Type u_1\nA : Type u_2\nB : Type u_3\nC : Type u_4\ninst✝⁶ : CommSemiring R\ninst✝⁵ : AddCommMonoid A\ninst✝⁴ : Module R A\ninst✝³ : Coalgebra R A\ninst✝² : AddCommMonoid B\ninst✝¹ : Module R B\ninst✝ : CoalgebraStruct R B\nf : A ≃ₗc[R] B\nx : A\n⊢ ∑ x_1 ∈ (ℛ R x).index,\n ∑ x_2 ∈ (ℛ R ((ℛ R x).... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Category.ModuleCat.ChangeOfRings | {
"line": 774,
"column": 6
} | {
"line": 774,
"column": 35
} | {
"line": 775,
"column": 6
} | [
{
"pp": "case tmul\nR✝ : Type u₁\nS✝ : Type u₂\ninst✝³ : CommRing R✝\ninst✝² : CommRing S✝\nf✝ : R✝ →+* S✝\nR : Type u₁\nS : Type u₂\ninst✝¹ : CommRing R\ninst✝ : CommRing S\nf : R →+* S\nX : ModuleCat R\nY : ModuleCat S\ng : (extendScalars f).obj X ⟶ Y\nm1 : Module R S := Module.compHom S f\nm2 : Module R ↑Y :... | [
"case tmul\nR✝ : Type u₁\nS✝ : Type u₂\ninst✝³ : CommRing R✝\ninst✝² : CommRing S✝\nf✝ : R✝ →+* S✝\nR : Type u₁\nS : Type u₂\ninst✝¹ : CommRing R\ninst✝ : CommRing S\nf : R →+* S\nX : ModuleCat R\nY : ModuleCat S\ng : (extendScalars f).obj X ⟶ Y\nm1 : Module R S := Module.compHom S f\nm2 : Module R ↑Y := Module.com... | erw [TensorProduct.lift.tmul] | Lean.Parser.Tactic._aux_Init_Meta___macroRules_Lean_Parser_Tactic_tacticErw____1 | Lean.Parser.Tactic.tacticErw___ |
Mathlib.Algebra.Category.ModuleCat.ChangeOfRings | {
"line": 921,
"column": 2
} | {
"line": 921,
"column": 58
} | {
"line": 923,
"column": 0
} | [
{
"pp": "R : Type u₁\nS : Type u₂\ninst✝¹ : CommRing R\ninst✝ : CommRing S\nf : R →+* S\nM : ModuleCat S\nm : ↑M\n⊢ (ConcreteCategory.hom ((extendRestrictScalarsAdj f).counit.app M)) (1 ⊗ₜ[R] m) = m",
"ppTerm": "?m.34",
"assigned": true,
"usedConstants": [
"ModuleCat.ExtendRestrictScalarsAdj.C... | [] | apply ExtendRestrictScalarsAdj.Counit.map_apply_one_tmul | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.Algebra.Category.ModuleCat.ChangeOfRings | {
"line": 921,
"column": 2
} | {
"line": 921,
"column": 58
} | {
"line": 923,
"column": 0
} | [
{
"pp": "R : Type u₁\nS : Type u₂\ninst✝¹ : CommRing R\ninst✝ : CommRing S\nf : R →+* S\nM : ModuleCat S\nm : ↑M\n⊢ (ConcreteCategory.hom ((extendRestrictScalarsAdj f).counit.app M)) (1 ⊗ₜ[R] m) = m",
"ppTerm": "?m.34",
"assigned": true,
"usedConstants": [
"ModuleCat.ExtendRestrictScalarsAdj.C... | [] | apply ExtendRestrictScalarsAdj.Counit.map_apply_one_tmul | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Category.ModuleCat.ChangeOfRings | {
"line": 921,
"column": 2
} | {
"line": 921,
"column": 58
} | {
"line": 923,
"column": 0
} | [
{
"pp": "R : Type u₁\nS : Type u₂\ninst✝¹ : CommRing R\ninst✝ : CommRing S\nf : R →+* S\nM : ModuleCat S\nm : ↑M\n⊢ (ConcreteCategory.hom ((extendRestrictScalarsAdj f).counit.app M)) (1 ⊗ₜ[R] m) = m",
"ppTerm": "?m.34",
"assigned": true,
"usedConstants": [
"ModuleCat.ExtendRestrictScalarsAdj.C... | [] | apply ExtendRestrictScalarsAdj.Counit.map_apply_one_tmul | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Adjunction.Unique | {
"line": 67,
"column": 2
} | {
"line": 67,
"column": 63
} | {
"line": 68,
"column": 2
} | [
{
"pp": "C : Type u_1\nD : Type u_2\ninst✝¹ : Category.{v_1, u_1} C\ninst✝ : Category.{v_2, u_2} D\nF F' : C ⥤ D\nG : D ⥤ C\nadj1 : F ⊣ G\nadj2 : F' ⊣ G\nx : D\n⊢ F.map (adj2.unit.app (G.obj x)) ≫ adj1.counit.app (F'.obj (G.obj x)) ≫ adj2.counit.app x = adj1.counit.app x",
"ppTerm": "?m.69",
"assigned":... | [
"C : Type u_1\nD : Type u_2\ninst✝¹ : Category.{v_1, u_1} C\ninst✝ : Category.{v_2, u_2} D\nF F' : C ⥤ D\nG : D ⥤ C\nadj1 : F ⊣ G\nadj2 : F' ⊣ G\nx : D\n⊢ F.map (adj2.unit.app (G.obj x) ≫ G.map (adj2.counit.app x)) ≫ adj1.counit.app x = adj1.counit.app x"
] | rw [← adj1.counit_naturality, ← Category.assoc, ← F.map_comp] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Order.Heyting.Hom | {
"line": 114,
"column": 24
} | {
"line": 114,
"column": 73
} | {
"line": 114,
"column": 74
} | [
{
"pp": "F : Type u_1\nα : Type u_2\nβ : Type u_3\nγ : Type u_4\nδ : Type u_5\ninst✝² : FunLike F α β\ninst✝¹ : HeytingAlgebra α\nx✝ : HeytingAlgebra β\ninst✝ : HeytingHomClass F α β\nf : F\n⊢ f ⊤ = ⊤",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"HeytingHomClass.map_himp",
"... | [] | by rw [← @himp_self α _ ⊥, ← himp_self, map_himp] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Order.Heyting.Hom | {
"line": 132,
"column": 24
} | {
"line": 132,
"column": 73
} | {
"line": 132,
"column": 74
} | [
{
"pp": "F : Type u_1\nα : Type u_2\nβ : Type u_3\nγ : Type u_4\nδ : Type u_5\ninst✝² : FunLike F α β\ninst✝¹ : BiheytingAlgebra α\nx✝ : BiheytingAlgebra β\ninst✝ : BiheytingHomClass F α β\nf : F\n⊢ f ⊤ = ⊤",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Lattice.toSe... | [] | by rw [← @himp_self α _ ⊥, ← himp_self, map_himp] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Order.Heyting.Hom | {
"line": 268,
"column": 17
} | {
"line": 268,
"column": 37
} | {
"line": 268,
"column": 38
} | [
{
"pp": "F : Type u_1\nα : Type u_2\nβ : Type u_3\nγ : Type u_4\nδ : Type u_5\ninst✝⁴ : FunLike F α β\ninst✝³ : HeytingAlgebra α\ninst✝² : HeytingAlgebra β\ninst✝¹ : HeytingAlgebra γ\ninst✝ : HeytingAlgebra δ\nf : HeytingHom α β\nf' : α → β\nh : f' = ⇑f\n⊢ ∀ (a b : α), f' (a ⊔ b) = f' a ⊔ f' b",
"ppTerm": "... | [
"F : Type u_1\nα : Type u_2\nβ : Type u_3\nγ : Type u_4\nδ : Type u_5\ninst✝⁴ : FunLike F α β\ninst✝³ : HeytingAlgebra α\ninst✝² : HeytingAlgebra β\ninst✝¹ : HeytingAlgebra γ\ninst✝ : HeytingAlgebra δ\nf : HeytingHom α β\nf' : α → β\nh : f' = ⇑f\n⊢ ∀ (a b : α), f (a ⊔ b) = f a ⊔ f b"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.Heyting.Hom | {
"line": 269,
"column": 17
} | {
"line": 269,
"column": 37
} | {
"line": 269,
"column": 38
} | [
{
"pp": "F : Type u_1\nα : Type u_2\nβ : Type u_3\nγ : Type u_4\nδ : Type u_5\ninst✝⁴ : FunLike F α β\ninst✝³ : HeytingAlgebra α\ninst✝² : HeytingAlgebra β\ninst✝¹ : HeytingAlgebra γ\ninst✝ : HeytingAlgebra δ\nf : HeytingHom α β\nf' : α → β\nh : f' = ⇑f\n⊢ ∀ (a b : α), f' (a ⊓ b) = f' a ⊓ f' b",
"ppTerm": "... | [
"F : Type u_1\nα : Type u_2\nβ : Type u_3\nγ : Type u_4\nδ : Type u_5\ninst✝⁴ : FunLike F α β\ninst✝³ : HeytingAlgebra α\ninst✝² : HeytingAlgebra β\ninst✝¹ : HeytingAlgebra γ\ninst✝ : HeytingAlgebra δ\nf : HeytingHom α β\nf' : α → β\nh : f' = ⇑f\n⊢ ∀ (a b : α), f (a ⊓ b) = f a ⊓ f b"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.Heyting.Hom | {
"line": 270,
"column": 17
} | {
"line": 270,
"column": 37
} | {
"line": 270,
"column": 38
} | [
{
"pp": "F : Type u_1\nα : Type u_2\nβ : Type u_3\nγ : Type u_4\nδ : Type u_5\ninst✝⁴ : FunLike F α β\ninst✝³ : HeytingAlgebra α\ninst✝² : HeytingAlgebra β\ninst✝¹ : HeytingAlgebra γ\ninst✝ : HeytingAlgebra δ\nf : HeytingHom α β\nf' : α → β\nh : f' = ⇑f\n⊢ f' ⊥ = ⊥",
"ppTerm": "?m.43",
"assigned": true,... | [
"F : Type u_1\nα : Type u_2\nβ : Type u_3\nγ : Type u_4\nδ : Type u_5\ninst✝⁴ : FunLike F α β\ninst✝³ : HeytingAlgebra α\ninst✝² : HeytingAlgebra β\ninst✝¹ : HeytingAlgebra γ\ninst✝ : HeytingAlgebra δ\nf : HeytingHom α β\nf' : α → β\nh : f' = ⇑f\n⊢ f ⊥ = ⊥"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.Heyting.Hom | {
"line": 271,
"column": 18
} | {
"line": 271,
"column": 38
} | {
"line": 271,
"column": 39
} | [
{
"pp": "F : Type u_1\nα : Type u_2\nβ : Type u_3\nγ : Type u_4\nδ : Type u_5\ninst✝⁴ : FunLike F α β\ninst✝³ : HeytingAlgebra α\ninst✝² : HeytingAlgebra β\ninst✝¹ : HeytingAlgebra γ\ninst✝ : HeytingAlgebra δ\nf : HeytingHom α β\nf' : α → β\nh : f' = ⇑f\n⊢ ∀ (a b : α), f' (a ⇨ b) = f' a ⇨ f' b",
"ppTerm": "... | [
"F : Type u_1\nα : Type u_2\nβ : Type u_3\nγ : Type u_4\nδ : Type u_5\ninst✝⁴ : FunLike F α β\ninst✝³ : HeytingAlgebra α\ninst✝² : HeytingAlgebra β\ninst✝¹ : HeytingAlgebra γ\ninst✝ : HeytingAlgebra δ\nf : HeytingHom α β\nf' : α → β\nh : f' = ⇑f\n⊢ ∀ (a b : α), f (a ⇨ b) = f a ⇨ f b"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.Heyting.Hom | {
"line": 373,
"column": 17
} | {
"line": 373,
"column": 37
} | {
"line": 373,
"column": 38
} | [
{
"pp": "F : Type u_1\nα : Type u_2\nβ : Type u_3\nγ : Type u_4\nδ : Type u_5\ninst✝⁴ : FunLike F α β\ninst✝³ : CoheytingAlgebra α\ninst✝² : CoheytingAlgebra β\ninst✝¹ : CoheytingAlgebra γ\ninst✝ : CoheytingAlgebra δ\nf : CoheytingHom α β\nf' : α → β\nh : f' = ⇑f\n⊢ ∀ (a b : α), f' (a ⊔ b) = f' a ⊔ f' b",
"... | [
"F : Type u_1\nα : Type u_2\nβ : Type u_3\nγ : Type u_4\nδ : Type u_5\ninst✝⁴ : FunLike F α β\ninst✝³ : CoheytingAlgebra α\ninst✝² : CoheytingAlgebra β\ninst✝¹ : CoheytingAlgebra γ\ninst✝ : CoheytingAlgebra δ\nf : CoheytingHom α β\nf' : α → β\nh : f' = ⇑f\n⊢ ∀ (a b : α), f (a ⊔ b) = f a ⊔ f b"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.Heyting.Hom | {
"line": 374,
"column": 17
} | {
"line": 374,
"column": 37
} | {
"line": 374,
"column": 38
} | [
{
"pp": "F : Type u_1\nα : Type u_2\nβ : Type u_3\nγ : Type u_4\nδ : Type u_5\ninst✝⁴ : FunLike F α β\ninst✝³ : CoheytingAlgebra α\ninst✝² : CoheytingAlgebra β\ninst✝¹ : CoheytingAlgebra γ\ninst✝ : CoheytingAlgebra δ\nf : CoheytingHom α β\nf' : α → β\nh : f' = ⇑f\n⊢ ∀ (a b : α), f' (a ⊓ b) = f' a ⊓ f' b",
"... | [
"F : Type u_1\nα : Type u_2\nβ : Type u_3\nγ : Type u_4\nδ : Type u_5\ninst✝⁴ : FunLike F α β\ninst✝³ : CoheytingAlgebra α\ninst✝² : CoheytingAlgebra β\ninst✝¹ : CoheytingAlgebra γ\ninst✝ : CoheytingAlgebra δ\nf : CoheytingHom α β\nf' : α → β\nh : f' = ⇑f\n⊢ ∀ (a b : α), f (a ⊓ b) = f a ⊓ f b"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.Heyting.Hom | {
"line": 375,
"column": 17
} | {
"line": 375,
"column": 37
} | {
"line": 375,
"column": 38
} | [
{
"pp": "F : Type u_1\nα : Type u_2\nβ : Type u_3\nγ : Type u_4\nδ : Type u_5\ninst✝⁴ : FunLike F α β\ninst✝³ : CoheytingAlgebra α\ninst✝² : CoheytingAlgebra β\ninst✝¹ : CoheytingAlgebra γ\ninst✝ : CoheytingAlgebra δ\nf : CoheytingHom α β\nf' : α → β\nh : f' = ⇑f\n⊢ f' ⊤ = ⊤",
"ppTerm": "?m.43",
"assign... | [
"F : Type u_1\nα : Type u_2\nβ : Type u_3\nγ : Type u_4\nδ : Type u_5\ninst✝⁴ : FunLike F α β\ninst✝³ : CoheytingAlgebra α\ninst✝² : CoheytingAlgebra β\ninst✝¹ : CoheytingAlgebra γ\ninst✝ : CoheytingAlgebra δ\nf : CoheytingHom α β\nf' : α → β\nh : f' = ⇑f\n⊢ f ⊤ = ⊤"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.Heyting.Hom | {
"line": 376,
"column": 19
} | {
"line": 376,
"column": 39
} | {
"line": 376,
"column": 40
} | [
{
"pp": "F : Type u_1\nα : Type u_2\nβ : Type u_3\nγ : Type u_4\nδ : Type u_5\ninst✝⁴ : FunLike F α β\ninst✝³ : CoheytingAlgebra α\ninst✝² : CoheytingAlgebra β\ninst✝¹ : CoheytingAlgebra γ\ninst✝ : CoheytingAlgebra δ\nf : CoheytingHom α β\nf' : α → β\nh : f' = ⇑f\n⊢ ∀ (a b : α), f' (a \\ b) = f' a \\ f' b",
... | [
"F : Type u_1\nα : Type u_2\nβ : Type u_3\nγ : Type u_4\nδ : Type u_5\ninst✝⁴ : FunLike F α β\ninst✝³ : CoheytingAlgebra α\ninst✝² : CoheytingAlgebra β\ninst✝¹ : CoheytingAlgebra γ\ninst✝ : CoheytingAlgebra δ\nf : CoheytingHom α β\nf' : α → β\nh : f' = ⇑f\n⊢ ∀ (a b : α), f (a \\ b) = f a \\ f b"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.Heyting.Hom | {
"line": 478,
"column": 17
} | {
"line": 478,
"column": 37
} | {
"line": 478,
"column": 38
} | [
{
"pp": "F : Type u_1\nα : Type u_2\nβ : Type u_3\nγ : Type u_4\nδ : Type u_5\ninst✝⁴ : FunLike F α β\ninst✝³ : BiheytingAlgebra α\ninst✝² : BiheytingAlgebra β\ninst✝¹ : BiheytingAlgebra γ\ninst✝ : BiheytingAlgebra δ\nf : BiheytingHom α β\nf' : α → β\nh : f' = ⇑f\n⊢ ∀ (a b : α), f' (a ⊔ b) = f' a ⊔ f' b",
"... | [
"F : Type u_1\nα : Type u_2\nβ : Type u_3\nγ : Type u_4\nδ : Type u_5\ninst✝⁴ : FunLike F α β\ninst✝³ : BiheytingAlgebra α\ninst✝² : BiheytingAlgebra β\ninst✝¹ : BiheytingAlgebra γ\ninst✝ : BiheytingAlgebra δ\nf : BiheytingHom α β\nf' : α → β\nh : f' = ⇑f\n⊢ ∀ (a b : α), f (a ⊔ b) = f a ⊔ f b"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.Heyting.Hom | {
"line": 479,
"column": 17
} | {
"line": 479,
"column": 37
} | {
"line": 479,
"column": 38
} | [
{
"pp": "F : Type u_1\nα : Type u_2\nβ : Type u_3\nγ : Type u_4\nδ : Type u_5\ninst✝⁴ : FunLike F α β\ninst✝³ : BiheytingAlgebra α\ninst✝² : BiheytingAlgebra β\ninst✝¹ : BiheytingAlgebra γ\ninst✝ : BiheytingAlgebra δ\nf : BiheytingHom α β\nf' : α → β\nh : f' = ⇑f\n⊢ ∀ (a b : α), f' (a ⊓ b) = f' a ⊓ f' b",
"... | [
"F : Type u_1\nα : Type u_2\nβ : Type u_3\nγ : Type u_4\nδ : Type u_5\ninst✝⁴ : FunLike F α β\ninst✝³ : BiheytingAlgebra α\ninst✝² : BiheytingAlgebra β\ninst✝¹ : BiheytingAlgebra γ\ninst✝ : BiheytingAlgebra δ\nf : BiheytingHom α β\nf' : α → β\nh : f' = ⇑f\n⊢ ∀ (a b : α), f (a ⊓ b) = f a ⊓ f b"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.Heyting.Hom | {
"line": 480,
"column": 18
} | {
"line": 480,
"column": 38
} | {
"line": 480,
"column": 39
} | [
{
"pp": "F : Type u_1\nα : Type u_2\nβ : Type u_3\nγ : Type u_4\nδ : Type u_5\ninst✝⁴ : FunLike F α β\ninst✝³ : BiheytingAlgebra α\ninst✝² : BiheytingAlgebra β\ninst✝¹ : BiheytingAlgebra γ\ninst✝ : BiheytingAlgebra δ\nf : BiheytingHom α β\nf' : α → β\nh : f' = ⇑f\n⊢ ∀ (a b : α), f' (a ⇨ b) = f' a ⇨ f' b",
"... | [
"F : Type u_1\nα : Type u_2\nβ : Type u_3\nγ : Type u_4\nδ : Type u_5\ninst✝⁴ : FunLike F α β\ninst✝³ : BiheytingAlgebra α\ninst✝² : BiheytingAlgebra β\ninst✝¹ : BiheytingAlgebra γ\ninst✝ : BiheytingAlgebra δ\nf : BiheytingHom α β\nf' : α → β\nh : f' = ⇑f\n⊢ ∀ (a b : α), f (a ⇨ b) = f a ⇨ f b"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.Heyting.Hom | {
"line": 481,
"column": 19
} | {
"line": 481,
"column": 39
} | {
"line": 481,
"column": 40
} | [
{
"pp": "F : Type u_1\nα : Type u_2\nβ : Type u_3\nγ : Type u_4\nδ : Type u_5\ninst✝⁴ : FunLike F α β\ninst✝³ : BiheytingAlgebra α\ninst✝² : BiheytingAlgebra β\ninst✝¹ : BiheytingAlgebra γ\ninst✝ : BiheytingAlgebra δ\nf : BiheytingHom α β\nf' : α → β\nh : f' = ⇑f\n⊢ ∀ (a b : α), f' (a \\ b) = f' a \\ f' b",
... | [
"F : Type u_1\nα : Type u_2\nβ : Type u_3\nγ : Type u_4\nδ : Type u_5\ninst✝⁴ : FunLike F α β\ninst✝³ : BiheytingAlgebra α\ninst✝² : BiheytingAlgebra β\ninst✝¹ : BiheytingAlgebra γ\ninst✝ : BiheytingAlgebra δ\nf : BiheytingHom α β\nf' : α → β\nh : f' = ⇑f\n⊢ ∀ (a b : α), f (a \\ b) = f a \\ f b"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Monoidal.Comon_ | {
"line": 232,
"column": 41
} | {
"line": 232,
"column": 65
} | {
"line": 233,
"column": 2
} | [
{
"pp": "C : Type u₁\ninst✝⁴ : Category.{v₁, u₁} C\ninst✝³ : MonoidalCategory C\nM N O : C\ninst✝² : ComonObj M\ninst✝¹ : ComonObj N\ninst✝ : ComonObj O\nA : Comon C\n⊢ ε ≫ Δ = Δ ≫ (ε ⊗ₘ ε)",
"ppTerm": "?m.44",
"assigned": true,
"usedConstants": [
"CategoryTheory.ComonObj.comul",
"Catego... | [] | simp [unitors_inv_equal] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.CategoryTheory.Monoidal.Comon_ | {
"line": 232,
"column": 41
} | {
"line": 232,
"column": 65
} | {
"line": 233,
"column": 2
} | [
{
"pp": "C : Type u₁\ninst✝⁴ : Category.{v₁, u₁} C\ninst✝³ : MonoidalCategory C\nM N O : C\ninst✝² : ComonObj M\ninst✝¹ : ComonObj N\ninst✝ : ComonObj O\nA : Comon C\n⊢ ε ≫ Δ = Δ ≫ (ε ⊗ₘ ε)",
"ppTerm": "?m.44",
"assigned": true,
"usedConstants": [
"CategoryTheory.ComonObj.comul",
"Catego... | [] | simp [unitors_inv_equal] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Monoidal.Comon_ | {
"line": 232,
"column": 41
} | {
"line": 232,
"column": 65
} | {
"line": 233,
"column": 2
} | [
{
"pp": "C : Type u₁\ninst✝⁴ : Category.{v₁, u₁} C\ninst✝³ : MonoidalCategory C\nM N O : C\ninst✝² : ComonObj M\ninst✝¹ : ComonObj N\ninst✝ : ComonObj O\nA : Comon C\n⊢ ε ≫ Δ = Δ ≫ (ε ⊗ₘ ε)",
"ppTerm": "?m.44",
"assigned": true,
"usedConstants": [
"CategoryTheory.ComonObj.comul",
"Catego... | [] | simp [unitors_inv_equal] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Monoidal.Comon_ | {
"line": 254,
"column": 8
} | {
"line": 254,
"column": 25
} | {
"line": 254,
"column": 26
} | [
{
"pp": "C : Type u₁\ninst✝⁴ : Category.{v₁, u₁} C\ninst✝³ : MonoidalCategory C\nM N O : C\ninst✝² : ComonObj M\ninst✝¹ : ComonObj N\ninst✝ : ComonObj O\nA : Comon C\n⊢ op A.X ◁ ε.op ≫ Δ.op = (ρ_ (op A.X)).hom",
"ppTerm": "?m.72",
"assigned": true,
"usedConstants": [
"CategoryTheory.ComonObj.c... | [
"C : Type u₁\ninst✝⁴ : Category.{v₁, u₁} C\ninst✝³ : MonoidalCategory C\nM N O : C\ninst✝² : ComonObj M\ninst✝¹ : ComonObj N\ninst✝ : ComonObj O\nA : Comon C\n⊢ (A.X ◁ ε).op ≫ Δ.op = (ρ_ (op A.X)).hom"
] | ← op_whiskerLeft, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.Monoidal.Comon_ | {
"line": 257,
"column": 59
} | {
"line": 257,
"column": 76
} | {
"line": 257,
"column": 77
} | [
{
"pp": "C : Type u₁\ninst✝⁴ : Category.{v₁, u₁} C\ninst✝³ : MonoidalCategory C\nM N O : C\ninst✝² : ComonObj M\ninst✝¹ : ComonObj N\ninst✝ : ComonObj O\nA : Comon C\n⊢ (Δ ≫ Δ ▷ A.X).op = (α_ A.X A.X A.X).inv.op ≫ op A.X ◁ Δ.op ≫ Δ.op",
"ppTerm": "?m.118",
"assigned": true,
"usedConstants": [
... | [
"C : Type u₁\ninst✝⁴ : Category.{v₁, u₁} C\ninst✝³ : MonoidalCategory C\nM N O : C\ninst✝² : ComonObj M\ninst✝¹ : ComonObj N\ninst✝ : ComonObj O\nA : Comon C\n⊢ (Δ ≫ Δ ▷ A.X).op = (α_ A.X A.X A.X).inv.op ≫ (A.X ◁ Δ).op ≫ Δ.op"
] | ← op_whiskerLeft, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.UniqueFactorizationDomain.Multiplicity | {
"line": 168,
"column": 4
} | {
"line": 168,
"column": 24
} | {
"line": 168,
"column": 25
} | [
{
"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\nthis : StrongNormalizationMonoid R := UniqueFactorizationMonoid.strongNormalizationMonoid\nq : R\nhq : q ∈ normal... | [
"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\nthis : StrongNormalizationMonoid R := UniqueFactorizationMonoid.strongNormalizationMonoid\nq : R\nhq : q ∈ normalizedFactors ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Localization.Away.Basic | {
"line": 96,
"column": 2
} | {
"line": 96,
"column": 13
} | {
"line": 96,
"column": 14
} | [
{
"pp": "case h\nR : Type u_1\ninst✝³ : CommSemiring R\nS : Type u_2\ninst✝² : CommSemiring S\ninst✝¹ : Algebra R S\nx : R\ninst✝ : Away x S\nz : S\na : R\nn : ℕ\nh : z * (algebraMap R S) ↑(a, ⟨(fun x_1 ↦ x ^ x_1) n, ⋯⟩).2 = (algebraMap R S) (a, ⟨(fun x_1 ↦ x ^ x_1) n, ⋯⟩).1\n⊢ z * (algebraMap R S) x ^ n = (alg... | [
"case h\nR : Type u_1\ninst✝³ : CommSemiring R\nS : Type u_2\ninst✝² : CommSemiring S\ninst✝¹ : Algebra R S\nx : R\ninst✝ : Away x S\nz : S\na : R\nn : ℕ\nh : z * (algebraMap R S) ↑(a, ⟨(fun x_1 ↦ x ^ x_1) n, ⋯⟩).2 = (algebraMap R S) (a, ⟨(fun x_1 ↦ x ^ x_1) n, ⋯⟩).1\n⊢ z * (algebraMap R S) x ^ n = (algebraMap R S)... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Localization.Away.Basic | {
"line": 115,
"column": 4
} | {
"line": 115,
"column": 15
} | {
"line": 115,
"column": 16
} | [
{
"pp": "case h\nR : Type u_1\ninst✝² : CommSemiring R\nS : Type u_2\ninst✝¹ : CommSemiring S\ninst✝ : Algebra R S\nr : R\nmap_unit : IsUnit ((algebraMap R S) r)\nsurj : ∀ (s : S), ∃ n a, s * (algebraMap R S) r ^ n = (algebraMap R S) a\nexists_of_eq : ∀ (a b : R), (algebraMap R S) a = (algebraMap R S) b → ∃ n, ... | [
"case h\nR : Type u_1\ninst✝² : CommSemiring R\nS : Type u_2\ninst✝¹ : CommSemiring S\ninst✝ : Algebra R S\nr : R\nmap_unit : IsUnit ((algebraMap R S) r)\nsurj : ∀ (s : S), ∃ n a, s * (algebraMap R S) r ^ n = (algebraMap R S) a\nexists_of_eq : ∀ (a b : R), (algebraMap R S) a = (algebraMap R S) b → ∃ n, r ^ n * a = ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Localization.Away.Basic | {
"line": 294,
"column": 4
} | {
"line": 294,
"column": 53
} | {
"line": 295,
"column": 4
} | [
{
"pp": "case refine_3\nR : Type u_1\ninst✝⁸ : CommSemiring R\nS : Type u_2\ninst✝⁷ : CommSemiring S\ninst✝⁶ : Algebra R S\nT : Type u_5\ninst✝⁵ : CommSemiring T\ninst✝⁴ : Algebra S T\ninst✝³ : Algebra R T\ninst✝² : IsScalarTower R S T\nx y : R\ninst✝¹ : Away x S\ninst✝ : Away ((algebraMap R S) y) T\na b : R\nh... | [
"case refine_3\nR : Type u_1\ninst✝⁸ : CommSemiring R\nS : Type u_2\ninst✝⁷ : CommSemiring S\ninst✝⁶ : Algebra R S\nT : Type u_5\ninst✝⁵ : CommSemiring T\ninst✝⁴ : Algebra S T\ninst✝³ : Algebra R T\ninst✝² : IsScalarTower R S T\nx y : R\ninst✝¹ : Away x S\ninst✝ : Away ((algebraMap R S) y) T\na b : R\nh : (algebraM... | simp only [← map_pow, ← map_mul, ← map_mul] at hn | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.RingTheory.TwoSidedIdeal.Operations | {
"line": 63,
"column": 34
} | {
"line": 63,
"column": 45
} | {
"line": 63,
"column": 46
} | [
{
"pp": "R : Type u_1\ninst✝ : NonUnitalNonAssocRing R\ns : Set R\nx : R\nhx : x ∈ s\n⊢ x - 0 ∈ s",
"ppTerm": "?m.27",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"sub_zero",
"NonUnitalNonAssocRing.toAddCommGroup",
"HSub.hSub",
"NonUnitalNonAssocSem... | [
"R : Type u_1\ninst✝ : NonUnitalNonAssocRing R\ns : Set R\nx : R\nhx : x ∈ s\n⊢ x ∈ s"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.TwoSidedIdeal.Operations | {
"line": 108,
"column": 20
} | {
"line": 108,
"column": 31
} | {
"line": 108,
"column": 32
} | [
{
"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 :... | [
"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 : R) (hx : x ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.TwoSidedIdeal.Operations | {
"line": 185,
"column": 2
} | {
"line": 189,
"column": 24
} | {
"line": 190,
"column": 2
} | [
{
"pp": "R : Type u_1\ninst✝ : NonUnitalRing R\ns : Set R\nh_left : ∀ (x y : R), y ∈ s → x * y ∈ s\nh_right : ∀ (y x : R), y ∈ s → y * x ∈ s\nz : R\nh_left' : ∀ {x y : R}, y ∈ closure s → x * y ∈ closure s\n⊢ z ∈ span s ↔ z ∈ closure s",
"ppTerm": "?m.54",
"assigned": true,
"usedConstants": [
... | [
"R : Type u_1\ninst✝ : NonUnitalRing R\ns : Set R\nh_left : ∀ (x y : R), y ∈ s → x * y ∈ s\nh_right : ∀ (y x : R), y ∈ s → y * x ∈ s\nz : R\nh_left' : ∀ {x y : R}, y ∈ closure s → x * y ∈ closure s\nh_right' : ∀ {y x : R}, y ∈ closure s → y * x ∈ closure s\n⊢ z ∈ span s ↔ z ∈ closure s"
] | have h_right' {y x} (hy : y ∈ closure s) : y * x ∈ closure s := by
have := (AddMonoidHom.mulRight x).map_closure s ▸ mem_map_of_mem _ hy
refine closure_mono ?_ this
rintro - ⟨y, hy, rfl⟩
exact h_right y x hy | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.RingTheory.TwoSidedIdeal.Operations | {
"line": 196,
"column": 32
} | {
"line": 196,
"column": 78
} | {
"line": 196,
"column": 79
} | [
{
"pp": "R : Type u_1\ninst✝ : NonUnitalRing R\ns : Set R\nh_left : ∀ (x y : R), y ∈ s → x * y ∈ s\nh_right : ∀ (y x : R), y ∈ s → y * x ∈ s\nz : R\nh_left' : ∀ {x y : R}, y ∈ closure s → x * y ∈ closure s\nh_right' : ∀ {y x : R}, y ∈ closure s → y * x ∈ closure s\nI : TwoSidedIdeal R := mk' ↑(closure s) ⋯ ⋯ ⋯ ... | [
"R : Type u_1\ninst✝ : NonUnitalRing R\ns : Set R\nh_left : ∀ (x y : R), y ∈ s → x * y ∈ s\nh_right : ∀ (y x : R), y ∈ s → y * x ∈ s\nz : R\nh_left' : ∀ {x y : R}, y ∈ closure s → x * y ∈ closure s\nh_right' : ∀ {y x : R}, y ∈ closure s → y * x ∈ closure s\nI : TwoSidedIdeal R := mk' ↑(closure s) ⋯ ⋯ ⋯ ⋯ ⋯\nh : ∀ (... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Localization.Away.Basic | {
"line": 512,
"column": 2
} | {
"line": 512,
"column": 42
} | {
"line": 513,
"column": 2
} | [
{
"pp": "S : Type u_2\ninst✝¹ : CommSemiring S\nR : Type u_4\ninst✝ : CommRing R\nf : R →+* S\na b : R\nh : a ∣ b\nH : Function.Injective ⇑(awayMap f a)\n⊢ Function.Injective ⇑(awayMap f b)",
"ppTerm": "?m.22",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"_private.Mathlib.RingTheory.L... | [
"S : Type u_2\ninst✝¹ : CommSemiring S\nR : Type u_4\ninst✝ : CommRing R\nf : R →+* S\na b : R\nh : a ∣ b\nH : ∀ (a_1 : R), f a_1 = 0 → ∃ n, a ^ n * a_1 = 0\n⊢ ∀ (a : R), f a = 0 → ∃ n, b ^ n * a = 0"
] | simp only [awayMap_injective_iff] at H ⊢ | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.RingTheory.Localization.Away.Basic | {
"line": 559,
"column": 39
} | {
"line": 559,
"column": 67
} | {
"line": 559,
"column": 68
} | [
{
"pp": "R : Type u_1\ninst✝ : CommSemiring R\ns : Set R\nspan_eq : Ideal.span s = ⊤\nx y : R\neq : (algebraMap R ((a : ↑s) → Away ↑a)) x = (algebraMap R ((a : ↑s) → Away ↑a)) y\nthis : Module.eqIdeal R x y = ⊤\n⊢ x = y",
"ppTerm": "?m.37",
"assigned": false,
"usedConstants": [],
"usedFVars": []... | [
"R : Type u_1\ninst✝ : CommSemiring R\ns : Set R\nspan_eq : Ideal.span s = ⊤\nx y : R\neq : (algebraMap R ((a : ↑s) → Away ↑a)) x = (algebraMap R ((a : ↑s) → Away ↑a)) y\nthis : Module.eqIdeal R x y = ⊤\n⊢ x = y"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.TwoSidedIdeal.Operations | {
"line": 351,
"column": 14
} | {
"line": 351,
"column": 25
} | {
"line": 351,
"column": 26
} | [
{
"pp": "R : Type u_1\ninst✝ : Ring R\nI✝ I : TwoSidedIdeal R\na✝ x✝ : R\n⊢ a✝ ∈ asIdeal I → a✝ * x✝ ∈ asIdeal I",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Semiring.toModule",
"HMul.hMul",
"Ring.toNonAssocRing",
"TwoSidedIdeal.asIdeal",
"... | [
"R : Type u_1\ninst✝ : Ring R\nI✝ I : TwoSidedIdeal R\na✝ x✝ : R\n⊢ a✝ ∈ I → a✝ * x✝ ∈ I"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.TwoSidedIdeal.Operations | {
"line": 362,
"column": 2
} | {
"line": 362,
"column": 79
} | {
"line": 363,
"column": 4
} | [
{
"pp": "R : Type u_1\ninst✝ : Ring R\nI : TwoSidedIdeal R\nx : Rᵐᵒᵖ\n⊢ x ∈ asIdealOpposite I ↔ MulOpposite.unop x ∈ I",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Submodule",
"TwoSidedIdeal.asIdealOpposite",
"instHSMul",
"Semiring.toModule",
... | [
"R : Type u_1\ninst✝ : Ring R\nI : TwoSidedIdeal R\nx : Rᵐᵒᵖ\n⊢ I.ringCon 0 (MulOpposite.unop x) ↔ I.ringCon (MulOpposite.unop x) 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Jacobson.Radical | {
"line": 53,
"column": 4
} | {
"line": 53,
"column": 24
} | {
"line": 53,
"column": 25
} | [
{
"pp": "case inr\nR : Type u_1\nR₂ : Type u_2\nM : Type u_3\nM₂ : Type u_4\ninst✝⁶ : Ring R\ninst✝⁵ : Ring R₂\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\ninst✝² : AddCommGroup M₂\ninst✝¹ : Module R₂ M₂\nτ₁₂ : R →+* R₂\ninst✝ : RingHomSurjective τ₁₂\nf : M →ₛₗ[τ₁₂] M₂\nS : ↑{m | IsCoatom m}\nm : M\nhm : m ∈ ... | [
"case inr\nR : Type u_1\nR₂ : Type u_2\nM : Type u_3\nM₂ : Type u_4\ninst✝⁶ : Ring R\ninst✝⁵ : Ring R₂\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\ninst✝² : AddCommGroup M₂\ninst✝¹ : Module R₂ M₂\nτ₁₂ : R →+* R₂\ninst✝ : RingHomSurjective τ₁₂\nf : M →ₛₗ[τ₁₂] M₂\nS : ↑{m | IsCoatom m}\nm : M\nhm : m ∈ jacobson R M... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Jacobson.Ideal | {
"line": 126,
"column": 2
} | {
"line": 126,
"column": 13
} | {
"line": 126,
"column": 14
} | [
{
"pp": "case h\nR : Type u\ninst✝ : Ring R\nI : Ideal R\nr : R\nh : r ∈ I.jacobson\ns : R\nhs : s * 1 * r + s - 1 ∈ I\n⊢ s * r + s - 1 ∈ I",
"ppTerm": "?h",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"case h\nR : Type u\ninst✝ : Ring R\nI : Ideal R\nr : R\nh : r ∈ I.jacobson\ns : R\nhs : s * 1 * r + s - 1 ∈ I\n⊢ s * r + s - 1 ∈ I"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Jacobson.Ideal | {
"line": 244,
"column": 23
} | {
"line": 244,
"column": 39
} | {
"line": 244,
"column": 40
} | [
{
"pp": "R : Type u\nS : Type v\ninst✝² : Ring R\ninst✝¹ : Ring S\nI✝ I : Ideal R\ninst✝ : I.IsTwoSided\nx r : R\nxJ : x ∈ I.jacobson\n𝔪 : Ideal R\n𝔪_mem : 𝔪 ∈ {J | I ≤ J ∧ J.IsMaximal}\nr𝔪 : r ∉ 𝔪\n𝔪₀ : Ideal R := Submodule.comap (DistribSMul.toLinearMap R R (MulOpposite.op r)) 𝔪\nthis : x ∈ 𝔪₀\n⊢ x * ... | [
"R : Type u\nS : Type v\ninst✝² : Ring R\ninst✝¹ : Ring S\nI✝ I : Ideal R\ninst✝ : I.IsTwoSided\nx r : R\nxJ : x ∈ I.jacobson\n𝔪 : Ideal R\n𝔪_mem : 𝔪 ∈ {J | I ≤ J ∧ J.IsMaximal}\nr𝔪 : r ∉ 𝔪\n𝔪₀ : Ideal R := Submodule.comap (DistribSMul.toLinearMap R R (MulOpposite.op r)) 𝔪\nthis : x ∈ 𝔪₀\n⊢ x * r ∈ 𝔪"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Jacobson.Ideal | {
"line": 249,
"column": 24
} | {
"line": 249,
"column": 40
} | {
"line": 249,
"column": 41
} | [
{
"pp": "R : Type u\nS : Type v\ninst✝² : Ring R\ninst✝¹ : Ring S\nI✝ I : Ideal R\ninst✝ : I.IsTwoSided\nx r : R\nxJ : x ∈ I.jacobson\n𝔪 : Ideal R\n𝔪_mem : 𝔪 ∈ {J | I ≤ J ∧ J.IsMaximal}\nr𝔪 : r ∉ 𝔪\n𝔪₀ : Ideal R := Submodule.comap (DistribSMul.toLinearMap R R (MulOpposite.op r)) 𝔪\nI𝔪₀ : I ≤ 𝔪₀\nh : 1 ... | [
"R : Type u\nS : Type v\ninst✝² : Ring R\ninst✝¹ : Ring S\nI✝ I : Ideal R\ninst✝ : I.IsTwoSided\nx r : R\nxJ : x ∈ I.jacobson\n𝔪 : Ideal R\n𝔪_mem : 𝔪 ∈ {J | I ≤ J ∧ J.IsMaximal}\nr𝔪 : r ∉ 𝔪\n𝔪₀ : Ideal R := Submodule.comap (DistribSMul.toLinearMap R R (MulOpposite.op r)) 𝔪\nI𝔪₀ : I ≤ 𝔪₀\nh : 1 ∈ 𝔪₀\n⊢ r ∈... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Jacobson.Ideal | {
"line": 264,
"column": 6
} | {
"line": 264,
"column": 17
} | {
"line": 264,
"column": 18
} | [
{
"pp": "R : Type u\nS : Type v\ninst✝² : Ring R\ninst✝¹ : Ring S\nI✝ I : Ideal R\ninst✝ : I.IsTwoSided\nx r : R\nxJ : x ∈ I.jacobson\n𝔪 : Ideal R\n𝔪_mem : 𝔪 ∈ {J | I ≤ J ∧ J.IsMaximal}\nr𝔪 : r ∉ 𝔪\n𝔪₀ : Ideal R := Submodule.comap (DistribSMul.toLinearMap R R (MulOpposite.op r)) 𝔪\nI𝔪₀ : I ≤ 𝔪₀\nJ : Id... | [
"R : Type u\nS : Type v\ninst✝² : Ring R\ninst✝¹ : Ring S\nI✝ I : Ideal R\ninst✝ : I.IsTwoSided\nx r : R\nxJ : x ∈ I.jacobson\n𝔪 : Ideal R\n𝔪_mem : 𝔪 ∈ {J | I ≤ J ∧ J.IsMaximal}\nr𝔪 : r ∉ 𝔪\n𝔪₀ : Ideal R := Submodule.comap (DistribSMul.toLinearMap R R (MulOpposite.op r)) 𝔪\nI𝔪₀ : I ≤ 𝔪₀\nJ : Ideal R\nb : R... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Jacobson.Ideal | {
"line": 307,
"column": 4
} | {
"line": 307,
"column": 15
} | {
"line": 307,
"column": 16
} | [
{
"pp": "case mp\nR : Type u\ninst✝ : CommRing R\nI : Ideal R\nhf : Function.Surjective ⇑(Quotient.mk I)\nh : (map (Quotient.mk I) I).jacobson = map (Quotient.mk I) I\n⊢ ⊥.jacobson = ⊥",
"ppTerm": "?mp",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"case mp\nR : Type u\ninst✝ : CommRing R\nI : Ideal R\nhf : Function.Surjective ⇑(Quotient.mk I)\nh : (map (Quotient.mk I) I).jacobson = map (Quotient.mk I) I\n⊢ ⊥.jacobson = ⊥"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Jacobson.Ideal | {
"line": 311,
"column": 4
} | {
"line": 311,
"column": 15
} | {
"line": 311,
"column": 16
} | [
{
"pp": "case mpr\nR : Type u\ninst✝ : CommRing R\nI : Ideal R\nhf : Function.Surjective ⇑(Quotient.mk I)\nh : (RingHom.ker (Quotient.mk I)).jacobson = RingHom.ker (Quotient.mk I)\n⊢ I.jacobson = I",
"ppTerm": "?mpr",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
... | [
"case mpr\nR : Type u\ninst✝ : CommRing R\nI : Ideal R\nhf : Function.Surjective ⇑(Quotient.mk I)\nh : (RingHom.ker (Quotient.mk I)).jacobson = RingHom.ker (Quotient.mk I)\n⊢ I.jacobson = I"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Jacobson.Ideal | {
"line": 323,
"column": 4
} | {
"line": 323,
"column": 15
} | {
"line": 323,
"column": 16
} | [
{
"pp": "case mp\nR : Type u\ninst✝ : CommRing R\nI : Ideal R\nhf : Function.Surjective ⇑(Quotient.mk I)\nh : I.radical = I.jacobson\nthis : (map (Quotient.mk I) I).radical = (map (Quotient.mk I) I).jacobson\n⊢ ⊥.radical = ⊥.jacobson",
"ppTerm": "?mp",
"assigned": false,
"usedConstants": [],
"us... | [
"case mp\nR : Type u\ninst✝ : CommRing R\nI : Ideal R\nhf : Function.Surjective ⇑(Quotient.mk I)\nh : I.radical = I.jacobson\nthis : (map (Quotient.mk I) I).radical = (map (Quotient.mk I) I).jacobson\n⊢ ⊥.radical = ⊥.jacobson"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
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