module stringlengths 16 90 | startPos dict | endPos dict | nextStartPos dict | goals listlengths 0 96 | goalsAfter listlengths 0 96 | ppTac stringlengths 1 14.5k | elaborator stringclasses 375
values | kind stringclasses 379
values |
|---|---|---|---|---|---|---|---|---|
Mathlib.GroupTheory.EckmannHilton | {
"line": 58,
"column": 2
} | {
"line": 58,
"column": 69
} | {
"line": 58,
"column": 70
} | [
{
"pp": "X : Type u\nm₁ m₂ : X → X → X\ne₁ e₂ : X\nh₁ : IsUnital m₁ e₁\nh₂ : IsUnital m₂ e₂\ndistrib : ∀ (a b c d : X), m₁ (m₂ a b) (m₂ c d) = m₂ (m₁ a c) (m₁ b d)\n⊢ e₁ = e₂",
"ppTerm": "?m.9",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"X : Type u\nm₁ m₂ : X → X → X\ne₁ e₂ : X\nh₁ : IsUnital m₁ e₁\nh₂ : IsUnital m₂ e₂\ndistrib : ∀ (a b c d : X), m₁ (m₂ a b) (m₂ c d) = m₂ (m₁ a c) (m₁ b d)\n⊢ e₁ = e₂"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.EckmannHilton | {
"line": 76,
"column": 17
} | {
"line": 76,
"column": 73
} | {
"line": 76,
"column": 74
} | [
{
"pp": "X : Type u\nm₁ m₂ : X → X → X\ne₁ e₂ : X\nh₁ : IsUnital m₁ e₁\nh₂ : IsUnital m₂ e₂\ndistrib : ∀ (a b c d : X), m₁ (m₂ a b) (m₂ c d) = m₂ (m₁ a c) (m₁ b d)\na b : X\n⊢ m₂ a b = m₂ b a",
"ppTerm": "?m.13",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"X : Type u\nm₁ m₂ : X → X → X\ne₁ e₂ : X\nh₁ : IsUnital m₁ e₁\nh₂ : IsUnital m₂ e₂\ndistrib : ∀ (a b c d : X), m₁ (m₂ a b) (m₂ c d) = m₂ (m₁ a c) (m₁ b d)\na b : X\n⊢ m₂ a b = m₂ b a"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.EckmannHilton | {
"line": 83,
"column": 19
} | {
"line": 83,
"column": 75
} | {
"line": 83,
"column": 76
} | [
{
"pp": "X : Type u\nm₁ m₂ : X → X → X\ne₁ e₂ : X\nh₁ : IsUnital m₁ e₁\nh₂ : IsUnital m₂ e₂\ndistrib : ∀ (a b c d : X), m₁ (m₂ a b) (m₂ c d) = m₂ (m₁ a c) (m₁ b d)\na b c : X\n⊢ m₂ (m₂ a b) c = m₂ a (m₂ b c)",
"ppTerm": "?m.14",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedG... | [
"X : Type u\nm₁ m₂ : X → X → X\ne₁ e₂ : X\nh₁ : IsUnital m₁ e₁\nh₂ : IsUnital m₂ e₂\ndistrib : ∀ (a b c d : X), m₁ (m₂ a b) (m₂ c d) = m₂ (m₁ a c) (m₁ b d)\na b c : X\n⊢ m₂ (m₂ a b) c = m₂ a (m₂ b c)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Preadditive.Schur | {
"line": 167,
"column": 17
} | {
"line": 167,
"column": 28
} | {
"line": 167,
"column": 29
} | [
{
"pp": "C : Type u_1\ninst✝⁸ : Category.{v_1, u_1} C\ninst✝⁷ : Preadditive C\n𝕜 : Type u_2\ninst✝⁶ : Field 𝕜\ninst✝⁵ : IsAlgClosed 𝕜\ninst✝⁴ : Linear 𝕜 C\ninst✝³ : HasKernels C\nX Y : C\ninst✝² : FiniteDimensional 𝕜 (X ⟶ X)\ninst✝¹ : Simple X\ninst✝ : Simple Y\nh : Nontrivial (X ⟶ Y)\nf : X ⟶ Y\nnz : f ≠ ... | [
"C : Type u_1\ninst✝⁸ : Category.{v_1, u_1} C\ninst✝⁷ : Preadditive C\n𝕜 : Type u_2\ninst✝⁶ : Field 𝕜\ninst✝⁵ : IsAlgClosed 𝕜\ninst✝⁴ : Linear 𝕜 C\ninst✝³ : HasKernels C\nX Y : C\ninst✝² : FiniteDimensional 𝕜 (X ⟶ X)\ninst✝¹ : Simple X\ninst✝ : Simple Y\nh : Nontrivial (X ⟶ Y)\nf : X ⟶ Y\nnz : f ≠ 0\nfi : IsIs... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Presentable.Adjunction | {
"line": 61,
"column": 4
} | {
"line": 61,
"column": 65
} | {
"line": 62,
"column": 4
} | [
{
"pp": "C : Type u\nD : Type u'\ninst✝⁵ : Category.{v, u} C\ninst✝⁴ : Category.{v', u'} D\nF : C ⥤ D\nG : D ⥤ C\nadj : F ⊣ G\nκ : Cardinal.{w}\ninst✝³ : Fact κ.IsRegular\nP : ObjectProperty C\nhP : P.IsCardinalFilteredGenerator κ\ninst✝² : G.IsCardinalAccessible κ\ninst✝¹ : G.Full\ninst✝ : G.Faithful\nY : D\nt... | [
"C : Type u\nD : Type u'\ninst✝⁵ : Category.{v, u} C\ninst✝⁴ : Category.{v', u'} D\nF : C ⥤ D\nG : D ⥤ C\nadj : F ⊣ G\nκ : Cardinal.{w}\ninst✝³ : Fact κ.IsRegular\nP : ObjectProperty C\nhP : P.IsCardinalFilteredGenerator κ\ninst✝² : G.IsCardinalAccessible κ\ninst✝¹ : G.Full\ninst✝ : G.Faithful\nY : D\nthis : F.IsLe... | obtain ⟨J, _, _, ⟨hY⟩⟩ := hP.exists_colimitsOfShape (G.obj Y) | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.CategoryTheory.Preadditive.Schur | {
"line": 196,
"column": 2
} | {
"line": 196,
"column": 81
} | {
"line": 198,
"column": 0
} | [
{
"pp": "case neg\nC : Type u_1\ninst✝⁸ : Category.{v_1, u_1} C\ninst✝⁷ : Preadditive C\n𝕜 : Type u_2\ninst✝⁶ : Field 𝕜\ninst✝⁵ : IsAlgClosed 𝕜\ninst✝⁴ : Linear 𝕜 C\ninst✝³ : HasKernels C\nX Y : C\ninst✝² : ∀ (X Y : C), FiniteDimensional 𝕜 (X ⟶ Y)\ninst✝¹ : Simple X\ninst✝ : Simple Y\nh : ¬Nonempty (X ≅ Y)... | [] | · exact (finrank_hom_simple_simple_eq_zero_iff 𝕜 X Y).2 (not_nonempty_iff.mp h) | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.CategoryTheory.Monoidal.Bimod | {
"line": 965,
"column": 2
} | {
"line": 965,
"column": 34
} | {
"line": 966,
"column": 2
} | [
{
"pp": "C : Type u₁\ninst✝⁴ : Category.{v₁, u₁} C\ninst✝³ : MonoidalCategory C\ninst✝² : HasCoequalizers C\ninst✝¹ : ∀ (X : C), PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)\ninst✝ : ∀ (X : C), PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)\nV W X Y Z : Mon C\nM : Bimod V W\nN :... | [
"C : Type u₁\ninst✝⁴ : Category.{v₁, u₁} C\ninst✝³ : MonoidalCategory C\ninst✝² : HasCoequalizers C\ninst✝¹ : ∀ (X : C), PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)\ninst✝ : ∀ (X : C), PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)\nV W X Y Z : Mon C\nM : Bimod V W\nN : Bimod W X\n... | dsimp only [AssociatorBimod.hom] | Lean.Elab.Tactic.evalDSimp | Lean.Parser.Tactic.dsimp |
Mathlib.CategoryTheory.Presentable.CardinalDirectedPoset | {
"line": 209,
"column": 6
} | {
"line": 209,
"column": 22
} | {
"line": 209,
"column": 23
} | [
{
"pp": "κ : Cardinal.{u}\ninst✝ : Fact κ.IsRegular\nX : Type u\nhX : Cardinal.mk X = κ\nα : Type u := (S : Set X) × (x : PartialOrder ↑S) × ULift.{u, 0} (PLift (IsCardinalFiltered (↑S) κ))\nthis : (a : α) → PartialOrder ↑a.fst := fun a ↦ a.snd.fst\nι : α → CardinalDirectedPoset κ := fun a ↦ { obj := { carrier ... | [
"κ : Cardinal.{u}\ninst✝ : Fact κ.IsRegular\nX : Type u\nhX : Cardinal.mk X = κ\nα : Type u := (S : Set X) × (x : PartialOrder ↑S) × ULift.{u, 0} (PLift (IsCardinalFiltered (↑S) κ))\nthis : (a : α) → PartialOrder ↑a.fst := fun a ↦ a.snd.fst\nι : α → CardinalDirectedPoset κ := fun a ↦ { obj := { carrier := ↑a.fst, s... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.Category.PartOrdEmb | {
"line": 256,
"column": 12
} | {
"line": 256,
"column": 37
} | {
"line": 256,
"column": 38
} | [
{
"pp": "J : Type u\ninst✝¹ : SmallCategory J\ninst✝ : IsFiltered J\nF : J ⥤ PartOrdEmb\nc : Cocone (F ⋙ forget PartOrdEmb)\nhc : IsColimit c\nx y : CoconePt hc\nj : J\nx₁ y₁ : ↑(F.obj j)\nhx₁ : (ConcreteCategory.hom (c.ι.app j)) x₁ = x\nhy₁ : (ConcreteCategory.hom (c.ι.app j)) y₁ = y\nh₁ : x₁ ≤ y₁\nk : J\ny₂ x... | [
"J : Type u\ninst✝¹ : SmallCategory J\ninst✝ : IsFiltered J\nF : J ⥤ PartOrdEmb\nc : Cocone (F ⋙ forget PartOrdEmb)\nhc : IsColimit c\nx y : CoconePt hc\nj : J\nx₁ y₁ : ↑(F.obj j)\nhx₁ : (ConcreteCategory.hom (c.ι.app j)) x₁ = x\nhy₁ : (ConcreteCategory.hom (c.ι.app j)) y₁ = y\nh₁ : x₁ ≤ y₁\nk : J\ny₂ x₂ : ↑(F.obj ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.Category.PartOrdEmb | {
"line": 257,
"column": 12
} | {
"line": 257,
"column": 37
} | {
"line": 257,
"column": 38
} | [
{
"pp": "J : Type u\ninst✝¹ : SmallCategory J\ninst✝ : IsFiltered J\nF : J ⥤ PartOrdEmb\nc : Cocone (F ⋙ forget PartOrdEmb)\nhc : IsColimit c\nx y : CoconePt hc\nj : J\nx₁ y₁ : ↑(F.obj j)\nhx₁ : (ConcreteCategory.hom (c.ι.app j)) x₁ = x\nhy₁ : (ConcreteCategory.hom (c.ι.app j)) y₁ = y\nh₁ : x₁ ≤ y₁\nk : J\ny₂ x... | [
"J : Type u\ninst✝¹ : SmallCategory J\ninst✝ : IsFiltered J\nF : J ⥤ PartOrdEmb\nc : Cocone (F ⋙ forget PartOrdEmb)\nhc : IsColimit c\nx y : CoconePt hc\nj : J\nx₁ y₁ : ↑(F.obj j)\nhx₁ : (ConcreteCategory.hom (c.ι.app j)) x₁ = x\nhy₁ : (ConcreteCategory.hom (c.ι.app j)) y₁ = y\nh₁ : x₁ ≤ y₁\nk : J\ny₂ x₂ : ↑(F.obj ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.Category.PartOrdEmb | {
"line": 283,
"column": 8
} | {
"line": 283,
"column": 34
} | {
"line": 283,
"column": 35
} | [
{
"pp": "J : Type u\ninst✝¹ : SmallCategory J\ninst✝ : IsFiltered J\nF : J ⥤ PartOrdEmb\nc : Cocone (F ⋙ forget PartOrdEmb)\nhc : IsColimit c\nj : J\nx y : ↑(F.1 j)\nk : J\nx' y' : ↑(F.obj k)\nhx : (ConcreteCategory.hom (c.ι.app k)) x' = { toFun := ⇑(ConcreteCategory.hom (c.ι.app j)), inj' := ⋯ } x\nhy : (Concr... | [
"J : Type u\ninst✝¹ : SmallCategory J\ninst✝ : IsFiltered J\nF : J ⥤ PartOrdEmb\nc : Cocone (F ⋙ forget PartOrdEmb)\nhc : IsColimit c\nj : J\nx y : ↑(F.1 j)\nk : J\nx' y' : ↑(F.obj k)\nhx : (ConcreteCategory.hom (c.ι.app k)) x' = { toFun := ⇑(ConcreteCategory.hom (c.ι.app j)), inj' := ⋯ } x\nhy : (ConcreteCategory.... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Presentable.CardinalDirectedPoset | {
"line": 256,
"column": 21
} | {
"line": 256,
"column": 36
} | {
"line": 256,
"column": 37
} | [
{
"pp": "κ : Cardinal.{u}\ninst✝¹ : Fact κ.IsRegular\nJ : CardinalDirectedPoset κ\nκ' : Cardinal.{u}\ninst✝ : Fact κ'.IsRegular\nhJ : HasCardinalLT (↑J.obj) κ'\nh✝ : κ ≤ κ'\nA : Type u\nx✝¹ : SmallCategory A\nx✝ : IsCardinalFiltered A κ'\nF : A ⥤ CardinalDirectedPoset κ\nc : Cocone F\nthis✝ : IsFiltered A\nthis... | [
"κ : Cardinal.{u}\ninst✝¹ : Fact κ.IsRegular\nJ : CardinalDirectedPoset κ\nκ' : Cardinal.{u}\ninst✝ : Fact κ'.IsRegular\nhJ : HasCardinalLT (↑J.obj) κ'\nh✝ : κ ≤ κ'\nA : Type u\nx✝¹ : SmallCategory A\nx✝ : IsCardinalFiltered A κ'\nF : A ⥤ CardinalDirectedPoset κ\nc : Cocone F\nthis✝ : IsFiltered A\nthis : IsCardina... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Preadditive.Mat | {
"line": 460,
"column": 6
} | {
"line": 460,
"column": 17
} | {
"line": 460,
"column": 18
} | [
{
"pp": "case e_a.e_a\nC : Type u₁\ninst✝⁶ : Category.{v₁, u₁} C\ninst✝⁵ : Preadditive C\nD : Type u₁\ninst✝⁴ : Category.{v₁, u₁} D\ninst✝³ : Preadditive D\ninst✝² : HasFiniteBiproducts D\nF : C ⥤ D\ninst✝¹ : F.Additive\nL : Mat_ C ⥤ D\ninst✝ : L.Additive\nα : embedding C ⋙ L ≅ F\nX✝ Y✝ : Mat_ C\nf : X✝ ⟶ Y✝\nj... | [
"case e_a.e_a\nC : Type u₁\ninst✝⁶ : Category.{v₁, u₁} C\ninst✝⁵ : Preadditive C\nD : Type u₁\ninst✝⁴ : Category.{v₁, u₁} D\ninst✝³ : Preadditive D\ninst✝² : HasFiniteBiproducts D\nF : C ⥤ D\ninst✝¹ : F.Additive\nL : Mat_ C ⥤ D\ninst✝ : L.Additive\nα : embedding C ⋙ L ≅ F\nX✝ Y✝ : Mat_ C\nf : X✝ ⟶ Y✝\nj : X✝.ι\nk :... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Presentable.OrthogonalReflection | {
"line": 269,
"column": 2
} | {
"line": 269,
"column": 13
} | {
"line": 269,
"column": 14
} | [
{
"pp": "C : Type u\ninst✝⁴ : Category.{v, u} C\nW : MorphismProperty C\nZ : C\ninst✝³ : HasCoproduct D₁.obj₁\ninst✝² : HasCoproduct D₁.obj₂\ninst✝¹ : HasPushouts C\ninst✝ : HasMulticoequalizer (D₂.multispanIndex W Z)\nX Y : C\nf : X ⟶ Y\nhf : W f\ng₁ g₂ : Y ⟶ Z\nhg : f ≫ g₁ = f ≫ g₂\n⊢ g₁ ≫ toSucc W Z = g₂ ≫ t... | [
"C : Type u\ninst✝⁴ : Category.{v, u} C\nW : MorphismProperty C\nZ : C\ninst✝³ : HasCoproduct D₁.obj₁\ninst✝² : HasCoproduct D₁.obj₂\ninst✝¹ : HasPushouts C\ninst✝ : HasMulticoequalizer (D₂.multispanIndex W Z)\nX Y : C\nf : X ⟶ Y\nhf : W f\ng₁ g₂ : Y ⟶ Z\nhg : f ≫ g₁ = f ≫ g₂\n⊢ g₁ ≫ toSucc W Z = g₂ ≫ toSucc W Z"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Presentable.OrthogonalReflection | {
"line": 288,
"column": 4
} | {
"line": 289,
"column": 65
} | {
"line": 290,
"column": 4
} | [
{
"pp": "case refine_1.h₀\nC : Type u\ninst✝⁴ : Category.{v, u} C\nW : MorphismProperty C\nZ : C\ninst✝³ : HasCoproduct D₁.obj₁\ninst✝² : HasCoproduct D₁.obj₂\ninst✝¹ : HasPushouts C\ninst✝ : HasMulticoequalizer (D₂.multispanIndex W Z)\nT : C\nhT : W.isLocal T\nφ₁ φ₂ : succ W Z ⟶ T\nh : toStep W Z ≫ fromStep W ... | [
"case h₁\nC : Type u\ninst✝⁴ : Category.{v, u} C\nW : MorphismProperty C\nZ : C\ninst✝³ : HasCoproduct D₁.obj₁\ninst✝² : HasCoproduct D₁.obj₂\ninst✝¹ : HasPushouts C\ninst✝ : HasMulticoequalizer (D₂.multispanIndex W Z)\nT : C\nhT : W.isLocal T\nφ₁ φ₂ : succ W Z ⟶ T\nh : toStep W Z ≫ fromStep W Z ≫ φ₁ = toStep W Z ≫... | · apply (hT d.1.1.hom d.1.2).1
simp only [← D₁.ι_comp_t_assoc, pushout.condition_assoc, h] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.CategoryTheory.Presentable.OrthogonalReflection | {
"line": 301,
"column": 6
} | {
"line": 301,
"column": 46
} | {
"line": 302,
"column": 8
} | [
{
"pp": "case refine_1.refine_1\nC : Type u\ninst✝⁴ : Category.{v, u} C\nW : MorphismProperty C\nZ : C\ninst✝³ : HasCoproduct D₁.obj₁\ninst✝² : HasCoproduct D₁.obj₂\ninst✝¹ : HasPushouts C\ninst✝ : HasMulticoequalizer (D₂.multispanIndex W Z)\nx✝ : IsIso (toSucc W Z)\nX Y : C\nf : X ⟶ Y\nhf : W f\ng₁ g₂ : Y ⟶ Z\... | [
"case refine_1.refine_1\nC : Type u\ninst✝⁴ : Category.{v, u} C\nW : MorphismProperty C\nZ : C\ninst✝³ : HasCoproduct D₁.obj₁\ninst✝² : HasCoproduct D₁.obj₂\ninst✝¹ : HasPushouts C\ninst✝ : HasMulticoequalizer (D₂.multispanIndex W Z)\nx✝ : IsIso (toSucc W Z)\nX Y : C\nf : X ⟶ Y\nhf : W f\ng₁ g₂ : Y ⟶ Z\nh : (fun g ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Presentable.OrthogonalReflection | {
"line": 314,
"column": 6
} | {
"line": 319,
"column": 47
} | {
"line": 320,
"column": 4
} | [
{
"pp": "case refine_2.h₀\nC : Type u\ninst✝⁴ : Category.{v, u} C\nW : MorphismProperty C\nZ : C\ninst✝³ : HasCoproduct D₁.obj₁\ninst✝² : HasCoproduct D₁.obj₂\ninst✝¹ : HasPushouts C\ninst✝ : HasMulticoequalizer (D₂.multispanIndex W Z)\nhZ : W.isLocal Z\nf : succ W Z ⟶ Z\nhf : toSucc W Z ≫ f = 𝟙 Z\nd : D₁ W Z\... | [] | simp only [Category.assoc] at hf
simp only [Category.comp_id, ← Category.assoc]
refine D₂.condition _ d.1.2 ?_
rw [Category.assoc, Category.assoc, Category.assoc,
← D₁.ι_comp_t_assoc, pushout.condition_assoc, reassoc_of% hf,
← D₁.ι_comp_t_assoc, pushout.condition] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Presentable.OrthogonalReflection | {
"line": 314,
"column": 6
} | {
"line": 319,
"column": 47
} | {
"line": 320,
"column": 4
} | [
{
"pp": "case refine_2.h₀\nC : Type u\ninst✝⁴ : Category.{v, u} C\nW : MorphismProperty C\nZ : C\ninst✝³ : HasCoproduct D₁.obj₁\ninst✝² : HasCoproduct D₁.obj₂\ninst✝¹ : HasPushouts C\ninst✝ : HasMulticoequalizer (D₂.multispanIndex W Z)\nhZ : W.isLocal Z\nf : succ W Z ⟶ Z\nhf : toSucc W Z ≫ f = 𝟙 Z\nd : D₁ W Z\... | [] | simp only [Category.assoc] at hf
simp only [Category.comp_id, ← Category.assoc]
refine D₂.condition _ d.1.2 ?_
rw [Category.assoc, Category.assoc, Category.assoc,
← D₁.ι_comp_t_assoc, pushout.condition_assoc, reassoc_of% hf,
← D₁.ι_comp_t_assoc, pushout.condition] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Presentable.Directed | {
"line": 303,
"column": 30
} | {
"line": 303,
"column": 41
} | {
"line": 303,
"column": 42
} | [
{
"pp": "J : Type w\ninst✝² : SmallCategory J\nκ : Cardinal.{w}\ninst✝¹ : Fact κ.IsRegular\ninst✝ : IsCardinalFiltered J κ\nhJ : ∀ (e : J), ∃ m x, IsEmpty (m ⟶ e)\nι : Type w\nD : ι → DiagramWithUniqueTerminal J κ\nhι : HasCardinalLT ι κ\nm₀ : ι → J\nt₀ : (i : ι) → (D i).top ⟶ m₀ i\nhm₀ : ∀ (i : ι), IsEmpty (m₀... | [
"J : Type w\ninst✝² : SmallCategory J\nκ : Cardinal.{w}\ninst✝¹ : Fact κ.IsRegular\ninst✝ : IsCardinalFiltered J κ\nhJ : ∀ (e : J), ∃ m x, IsEmpty (m ⟶ e)\nι : Type w\nD : ι → DiagramWithUniqueTerminal J κ\nhι : HasCardinalLT ι κ\nm₀ : ι → J\nt₀ : (i : ι) → (D i).top ⟶ m₀ i\nhm₀ : ∀ (i : ι), IsEmpty (m₀ i ⟶ (D i).t... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Presentable.SharplyLT.Basic | {
"line": 191,
"column": 6
} | {
"line": 191,
"column": 40
} | {
"line": 191,
"column": 41
} | [
{
"pp": "case succ.refine_2\nκ₁ κ₂ : Cardinal.{w}\ninst✝¹ : Fact κ₁.IsRegular\ninst✝ : Fact κ₂.IsRegular\nh₀ : κ₁ < κ₂\nX : Type w\nY : (B : Set X) → HasCardinalLT (↑B) κ₂ → Set (SetCardinalLT κ₁ ↑B)\nhY : ∀ (B : Set X) (hB : HasCardinalLT (↑B) κ₂), HasCardinalLT (↑(Y B hB)) κ₂\nm : (B : Set X) → (hB : HasCardi... | [
"case succ.refine_2\nκ₁ κ₂ : Cardinal.{w}\ninst✝¹ : Fact κ₁.IsRegular\ninst✝ : Fact κ₂.IsRegular\nh₀ : κ₁ < κ₂\nX : Type w\nY : (B : Set X) → HasCardinalLT (↑B) κ₂ → Set (SetCardinalLT κ₁ ↑B)\nhY : ∀ (B : Set X) (hB : HasCardinalLT (↑B) κ₂), HasCardinalLT (↑(Y B hB)) κ₂\nm : (B : Set X) → (hB : HasCardinalLT (↑B) κ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Presentable.SharplyLT.Basic | {
"line": 222,
"column": 6
} | {
"line": 222,
"column": 39
} | {
"line": 222,
"column": 40
} | [
{
"pp": "κ₁ κ₂ : Cardinal.{w}\ninst✝² : Fact κ₁.IsRegular\ninst✝¹ : Fact κ₂.IsRegular\nh₀ : κ₁ < κ₂\nX : Type w\ninst✝ : PartialOrder X\nY : (B : Set X) → HasCardinalLT (↑B) κ₂ → Set (SetCardinalLT κ₁ ↑B)\nhY : ∀ (B : Set X) (hB : HasCardinalLT (↑B) κ₂), HasCardinalLT (↑(Y B hB)) κ₂\nhY' : ∀ (B : Set X) (hB : H... | [
"κ₁ κ₂ : Cardinal.{w}\ninst✝² : Fact κ₁.IsRegular\ninst✝¹ : Fact κ₂.IsRegular\nh₀ : κ₁ < κ₂\nX : Type w\ninst✝ : PartialOrder X\nY : (B : Set X) → HasCardinalLT (↑B) κ₂ → Set (SetCardinalLT κ₁ ↑B)\nhY : ∀ (B : Set X) (hB : HasCardinalLT (↑B) κ₂), HasCardinalLT (↑(Y B hB)) κ₂\nhY' : ∀ (B : Set X) (hB : HasCardinalLT... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Presentable.OrthogonalReflection | {
"line": 427,
"column": 44
} | {
"line": 427,
"column": 55
} | {
"line": 427,
"column": 56
} | [
{
"pp": "C : Type u\ninst✝⁷ : Category.{v, u} C\nW : MorphismProperty C\nZ : C\ninst✝⁶ : HasPushouts C\ninst✝⁵ : ∀ (Z : C), HasCoproduct D₁.obj₁\ninst✝⁴ : ∀ (Z : C), HasCoproduct D₁.obj₂\ninst✝³ : ∀ (Z : C), HasMulticoequalizer (D₂.multispanIndex W Z)\nκ : Cardinal.{w}\ninst✝² : OrderBot κ.ord.ToType\ninst✝¹ : ... | [
"C : Type u\ninst✝⁷ : Category.{v, u} C\nW : MorphismProperty C\nZ : C\ninst✝⁶ : HasPushouts C\ninst✝⁵ : ∀ (Z : C), HasCoproduct D₁.obj₁\ninst✝⁴ : ∀ (Z : C), HasCoproduct D₁.obj₂\ninst✝³ : ∀ (Z : C), HasMulticoequalizer (D₂.multispanIndex W Z)\nκ : Cardinal.{w}\ninst✝² : OrderBot κ.ord.ToType\ninst✝¹ : HasIteration... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Presentable.Directed | {
"line": 315,
"column": 27
} | {
"line": 315,
"column": 53
} | {
"line": 315,
"column": 54
} | [
{
"pp": "J : Type w\ninst✝² : SmallCategory J\nκ : Cardinal.{w}\ninst✝¹ : Fact κ.IsRegular\ninst✝ : IsCardinalFiltered J κ\nhJ : ∀ (e : J), ∃ m x, IsEmpty (m ⟶ e)\nι : Type w\nD : ι → DiagramWithUniqueTerminal J κ\nhι : HasCardinalLT ι κ\nm₀ : ι → J\nt₀ : (i : ι) → (D i).top ⟶ m₀ i\nhm₀ : ∀ (i : ι), IsEmpty (m₀... | [
"J : Type w\ninst✝² : SmallCategory J\nκ : Cardinal.{w}\ninst✝¹ : Fact κ.IsRegular\ninst✝ : IsCardinalFiltered J κ\nhJ : ∀ (e : J), ∃ m x, IsEmpty (m ⟶ e)\nι : Type w\nD : ι → DiagramWithUniqueTerminal J κ\nhι : HasCardinalLT ι κ\nm₀ : ι → J\nt₀ : (i : ι) → (D i).top ⟶ m₀ i\nhm₀ : ∀ (i : ι), IsEmpty (m₀ i ⟶ (D i).t... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Presentable.Directed | {
"line": 313,
"column": 2
} | {
"line": 315,
"column": 88
} | {
"line": 317,
"column": 0
} | [
{
"pp": "J : Type w\ninst✝² : SmallCategory J\nκ : Cardinal.{w}\ninst✝¹ : Fact κ.IsRegular\ninst✝ : IsCardinalFiltered J κ\nhJ : ∀ (e : J), ∃ m x, IsEmpty (m ⟶ e)\nι : Type w\nD : ι → DiagramWithUniqueTerminal J κ\nhι : HasCardinalLT ι κ\nm₀ : ι → J\nt₀ : (i : ι) → (D i).top ⟶ m₀ i\nhm₀ : ∀ (i : ι), IsEmpty (m₀... | [] | exact ⟨c.pt, fun i ↦ u i ≫ c.π ⟨⟩,
fun i ↦ ⟨fun hi ↦ (hm₀ i).false (t₁ i ≫ c.π ⟨⟩ ≫ hi)⟩,
fun i₁ i₂ j h₁ h₂ ↦ by simpa [index, shape] using c.condition ⟨⟨i₁, i₂, j⟩, h₁, h₂⟩⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.CategoryTheory.Presentable.Type | {
"line": 59,
"column": 23
} | {
"line": 59,
"column": 38
} | {
"line": 59,
"column": 39
} | [
{
"pp": "X : Type u\nκ : Cardinal.{u}\nhX : HasCardinalLT X κ\ninst✝ : Fact κ.IsRegular\nJ : Type u\nx✝¹ : SmallCategory J\nx✝ : IsCardinalFiltered J κ\nF : J ⥤ Type u\nc : Cocone F\nhc : IsColimit c\nthis : IsFiltered J\nj : J\nf g : X ⟶ F.obj j\nh : f ≫ c.ι.app j = g ≫ c.ι.app j\nk : ToType X → J\na : (x : To... | [
"X : Type u\nκ : Cardinal.{u}\nhX : HasCardinalLT X κ\ninst✝ : Fact κ.IsRegular\nJ : Type u\nx✝¹ : SmallCategory J\nx✝ : IsCardinalFiltered J κ\nF : J ⥤ Type u\nc : Cocone F\nhc : IsColimit c\nthis : IsFiltered J\nj : J\nf g : X ⟶ F.obj j\nh : f ≫ c.ι.app j = g ≫ c.ι.app j\nk : ToType X → J\na : (x : ToType X) → j ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Presentable.OrthogonalReflection | {
"line": 466,
"column": 37
} | {
"line": 466,
"column": 48
} | {
"line": 466,
"column": 49
} | [
{
"pp": "C : Type u\ninst✝⁴ : Category.{v, u} C\nW : MorphismProperty C\nκ : Cardinal.{w}\ninst✝³ : Fact κ.IsRegular\ninst✝² : IsSmall.{w, v, u} W\ninst✝¹ : LocallySmall.{w, v, u} C\nhW : ∀ ⦃X Y : C⦄ (f : X ⟶ Y), W f → IsCardinalPresentable X κ ∧ IsCardinalPresentable Y κ\ninst✝ : HasColimitsOfSize.{w, w, v, u}... | [
"C : Type u\ninst✝⁴ : Category.{v, u} C\nW : MorphismProperty C\nκ : Cardinal.{w}\ninst✝³ : Fact κ.IsRegular\ninst✝² : IsSmall.{w, v, u} W\ninst✝¹ : LocallySmall.{w, v, u} C\nhW : ∀ ⦃X Y : C⦄ (f : X ⟶ Y), W f → IsCardinalPresentable X κ ∧ IsCardinalPresentable Y κ\ninst✝ : HasColimitsOfSize.{w, w, v, u} C\n⊢ ¬κ = 0... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Presentable.Directed | {
"line": 364,
"column": 4
} | {
"line": 364,
"column": 33
} | {
"line": 365,
"column": 4
} | [
{
"pp": "case inr\nJ : Type w\ninst✝¹ : SmallCategory J\nκ : Cardinal.{w}\ninst✝ : Fact κ.IsRegular\nι : Type w\nD : ι → DiagramWithUniqueTerminal J κ\nhι : HasCardinalLT ι κ\nm : J\nu : (i : ι) → (D i).top ⟶ m\nhD : ∀ {i : ι}, ¬(D i).P m\nf : m ⟶ m\nhf : ∃ i, Arrow.mk f = Arrow.mk ((D i.fst).isTerminal.lift ⋯ ... | [
"case inr\nJ : Type w\ninst✝¹ : SmallCategory J\nκ : Cardinal.{w}\ninst✝ : Fact κ.IsRegular\nι : Type w\nD : ι → DiagramWithUniqueTerminal J κ\nhι : HasCardinalLT ι κ\nm : J\nu : (i : ι) → (D i).top ⟶ m\nhD : ∀ {i : ι}, ¬(D i).P m\nf : m ⟶ m\ni : ι\nj : J\nhj : (D i).P j\nhi : Arrow.mk f = Arrow.mk ((D ⟨i, ⟨j, hj⟩⟩... | obtain ⟨⟨i, j, hj⟩, hi⟩ := hf | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.CategoryTheory.RegularCategory.Basic | {
"line": 119,
"column": 6
} | {
"line": 119,
"column": 70
} | {
"line": 119,
"column": 71
} | [
{
"pp": "case refine_1\nC : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : Regular C\nX Y : C\nf : X ⟶ Y\nm : coequalizer (pullback.fst f f) (pullback.snd f f) ⟶ Y := coequalizer.desc f ⋯\ne : X ⟶ coequalizer (pullback.fst f f) (pullback.snd f f) := coequalizer.π (pullback.fst f f) (pullback.snd f f)\nk₁ : pullbac... | [
"case refine_1\nC : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : Regular C\nX Y : C\nf : X ⟶ Y\nm : coequalizer (pullback.fst f f) (pullback.snd f f) ⟶ Y := coequalizer.desc f ⋯\ne : X ⟶ coequalizer (pullback.fst f f) (pullback.snd f f) := coequalizer.π (pullback.fst f f) (pullback.snd f f)\nk₁ : pullback m m ⟶ coeq... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.RepresentedBy | {
"line": 95,
"column": 2
} | {
"line": 95,
"column": 32
} | {
"line": 95,
"column": 33
} | [
{
"pp": "C : Type u\ninst✝ : Category.{v, u} C\nF : Cᵒᵖ ⥤ Type w\nX : C\nR : F.RepresentableBy X\nx✝¹ : Cᵒᵖ\nx✝ : (uliftYoneda.{w, v, u}.obj X).obj x✝¹\n⊢ (ConcreteCategory.hom ((uliftYonedaEquiv.symm { down := R.homEquiv (𝟙 X) }).app x✝¹)).toFun x✝ =\n (ConcreteCategory.hom\n (((equivUliftYonedaIs... | [
"C : Type u\ninst✝ : Category.{v, u} C\nF : Cᵒᵖ ⥤ Type w\nX : C\nR : F.RepresentableBy X\nx✝¹ : Cᵒᵖ\nx✝ : (uliftYoneda.{w, v, u}.obj X).obj x✝¹\n⊢ (ConcreteCategory.hom (F.map x✝.down.op)) (R.homEquiv (𝟙 X)) = R.homEquiv x✝.down"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.RepresentedBy | {
"line": 110,
"column": 14
} | {
"line": 110,
"column": 25
} | {
"line": 110,
"column": 26
} | [
{
"pp": "C : Type u\ninst✝ : Category.{v, u} C\nF : Cᵒᵖ ⥤ Type w\nX : C\nx : F.obj (op X)\nF' : Cᵒᵖ ⥤ Type w\ne : F ≅ F'\nh : F'.IsRepresentedBy ((ConcreteCategory.hom (e.hom.app (op X))) x)\n⊢ F.IsRepresentedBy x",
"ppTerm": "?m.33",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
... | [
"C : Type u\ninst✝ : Category.{v, u} C\nF : Cᵒᵖ ⥤ Type w\nX : C\nx : F.obj (op X)\nF' : Cᵒᵖ ⥤ Type w\ne : F ≅ F'\nh : F'.IsRepresentedBy ((ConcreteCategory.hom (e.hom.app (op X))) x)\n⊢ F.IsRepresentedBy x"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.RegularCategory.Basic | {
"line": 198,
"column": 2
} | {
"line": 199,
"column": 44
} | {
"line": 200,
"column": 4
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : Regular C\nA B : C\nf : A ⟶ B\nA' : Subobject A\nB' : Subobject B\n⊢ IsPullback (frobeniusMorphism f A' B' ≫ ((«exists» f).obj A' ⊓ B').ofLE B' ⋯)\n ((A' ⊓ (Subobject.pullback f).obj B').ofLE A' ⋯) B'.arrow\n ((imageFactorisation f A').F.e ≫ ((«exis... | [
"C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : Regular C\nA B : C\nf : A ⟶ B\nA' : Subobject A\nB' : Subobject B\n⊢ IsPullback ((A' ⊓ (Subobject.pullback f).obj B').ofLE ((Subobject.pullback f).obj B') ⋯)\n ((A' ⊓ (Subobject.pullback f).obj B').ofLE A' ⋯) ((Subobject.pullback f).obj B').arrow A'.arrow"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Presentable.Directed | {
"line": 463,
"column": 9
} | {
"line": 463,
"column": 20
} | {
"line": 463,
"column": 21
} | [
{
"pp": "J : Type w\ninst✝² : SmallCategory J\nκ : Cardinal.{w}\ninst✝¹ : Fact κ.IsRegular\ninst✝ : IsCardinalFiltered J κ\nhJ : ∀ (e : J), ∃ m x, IsEmpty (m ⟶ e)\nthis✝¹ : IsCardinalFiltered (DiagramWithUniqueTerminal J κ) κ\nthis✝ : IsFiltered J\nthis : IsFiltered (DiagramWithUniqueTerminal J κ)\nj : J\nD : D... | [
"J : Type w\ninst✝² : SmallCategory J\nκ : Cardinal.{w}\ninst✝¹ : Fact κ.IsRegular\ninst✝ : IsCardinalFiltered J κ\nhJ : ∀ (e : J), ∃ m x, IsEmpty (m ⟶ e)\nthis✝¹ : IsCardinalFiltered (DiagramWithUniqueTerminal J κ) κ\nthis✝ : IsFiltered J\nthis : IsFiltered (DiagramWithUniqueTerminal J κ)\nj : J\nD : DiagramWithUn... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Sites.Coherent.CoherentTopology | {
"line": 95,
"column": 6
} | {
"line": 95,
"column": 39
} | {
"line": 96,
"column": 6
} | [
{
"pp": "case mp.of\nC : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\ninst✝ : Precoherent C\nX : C\nS : Sieve X\nY : C\nT : Presieve Y\nhS : T ∈ (coherentCoverage C).coverings Y\n⊢ ∃ α, ∃ (_ : Finite α), ∃ Y_1 π, EffectiveEpiFamily Y_1 π ∧ ∀ (a : α), (Sieve.generate T).arrows (π a)",
"ppTerm": "?mp.of",
"a... | [
"case mp.of\nC : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\ninst✝ : Precoherent C\nX : C\nS : Sieve X\nY : C\nT : Presieve Y\na : Type\nh : Finite a\nY' : a → C\nπ : (a : a) → Y' a ⟶ Y\nh' : T = Presieve.ofArrows Y' π\nright✝ : EffectiveEpiFamily Y' π\n⊢ ∃ α, ∃ (_ : Finite α), ∃ Y_1 π, EffectiveEpiFamily Y_1 π ∧ ∀ (... | obtain ⟨a, h, Y', π, h', _⟩ := hS | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.CategoryTheory.Sites.Coherent.Comparison | {
"line": 42,
"column": 4
} | {
"line": 42,
"column": 15
} | {
"line": 42,
"column": 16
} | [
{
"pp": "C : Type u_1\ninst✝² : Category.{v_1, u_1} C\ninst✝¹ : Precoherent C\ninst✝ : HasFiniteCoproducts C\nX Y Z : C\nf : X ⟶ Y\ng : Z ⟶ Y\nx✝ : EffectiveEpi g\nhp : EffectiveEpi g → ∃ β, ∃ (_ : Finite β), ∃ X₂ π₂, EffectiveEpiFamily X₂ π₂ ∧ ∃ ι, ∀ (b : β), ι b ≫ g = π₂ b ≫ f\nβ : Type\nw✝ : Finite β\nX₂ : β... | [
"C : Type u_1\ninst✝² : Category.{v_1, u_1} C\ninst✝¹ : Precoherent C\ninst✝ : HasFiniteCoproducts C\nX Y Z : C\nf : X ⟶ Y\ng : Z ⟶ Y\nx✝ : EffectiveEpi g\nhp : EffectiveEpi g → ∃ β, ∃ (_ : Finite β), ∃ X₂ π₂, EffectiveEpiFamily X₂ π₂ ∧ ∃ ι, ∀ (b : β), ι b ≫ g = π₂ b ≫ f\nβ : Type\nw✝ : Finite β\nX₂ : β → C\nπ₂ : (... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Sites.Coherent.RegularTopology | {
"line": 82,
"column": 44
} | {
"line": 82,
"column": 55
} | {
"line": 82,
"column": 56
} | [
{
"pp": "C : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\ninst✝ : Preregular C\nX : C\nS✝ : Sieve X\nY : C\nR S : Sieve Y\na✝¹ : (regularCoverage C).Saturate Y R\na✝ : ∀ ⦃Y_1 : C⦄ ⦃f : Y_1 ⟶ Y⦄, R.arrows f → (regularCoverage C).Saturate Y_1 (Sieve.pullback f S)\nb : ∀ ⦃Y_1 : C⦄ ⦃f : Y_1 ⟶ Y⦄, R.arrows f → ∃ Y_2 π,... | [
"C : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\ninst✝ : Preregular C\nX : C\nS✝ : Sieve X\nY : C\nR S : Sieve Y\na✝¹ : (regularCoverage C).Saturate Y R\na✝ : ∀ ⦃Y_1 : C⦄ ⦃f : Y_1 ⟶ Y⦄, R.arrows f → (regularCoverage C).Saturate Y_1 (Sieve.pullback f S)\nb : ∀ ⦃Y_1 : C⦄ ⦃f : Y_1 ⟶ Y⦄, R.arrows f → ∃ Y_2 π, EffectiveEp... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Sites.Coherent.Comparison | {
"line": 94,
"column": 56
} | {
"line": 94,
"column": 67
} | {
"line": 94,
"column": 68
} | [
{
"pp": "case mk\nC : Type u_1\ninst✝² : Category.{v_1, u_1} C\ninst✝¹ : Preregular C\ninst✝ : FinitaryPreExtensive C\nB : C\nS : Sieve B\nY✝ : C\nI : Type\nw✝ : Finite I\nX : I → C\nf : (a : I) → X a ⟶ Y✝\nhT : EffectiveEpiFamily X f\nR Y : C\ni✝¹ : Unit\nψ : R ⟶ ∐ fun i ↦ X i\nQ : C\ni✝ : I\ne : Q ⟶ X i✝\n⊢ P... | [
"case mk\nC : Type u_1\ninst✝² : Category.{v_1, u_1} C\ninst✝¹ : Preregular C\ninst✝ : FinitaryPreExtensive C\nB : C\nS : Sieve B\nY✝ : C\nI : Type\nw✝ : Finite I\nX : I → C\nf : (a : I) → X a ⟶ Y✝\nhT : EffectiveEpiFamily X f\nR Y : C\ni✝¹ : Unit\nψ : R ⟶ ∐ fun i ↦ X i\nQ : C\ni✝ : I\ne : Q ⟶ X i✝\n⊢ Presieve.ofAr... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Sites.Coherent.RegularSheaves | {
"line": 108,
"column": 4
} | {
"line": 108,
"column": 32
} | {
"line": 108,
"column": 33
} | [
{
"pp": "case refine_2\nC : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\nP : Cᵒᵖ ⥤ Type u_4\nX B : C\nπ : X ⟶ B\ninst✝ : EffectiveEpi π\nc : PullbackCone π π\nhP :\n ∀ (y : P.obj (op X)),\n (ConcreteCategory.hom (P.map c.fst.op)) y = (ConcreteCategory.hom (P.map c.snd.op)) y →\n ∃! x, (ConcreteCategory.ho... | [
"case refine_2\nC : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\nP : Cᵒᵖ ⥤ Type u_4\nX B : C\nπ : X ⟶ B\ninst✝ : EffectiveEpi π\nc : PullbackCone π π\nhP :\n ∀ (y : P.obj (op X)),\n (ConcreteCategory.hom (P.map c.fst.op)) y = (ConcreteCategory.hom (P.map c.snd.op)) y →\n ∃! x, (ConcreteCategory.hom (P.map π.o... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Sites.Coherent.RegularSheaves | {
"line": 127,
"column": 4
} | {
"line": 127,
"column": 32
} | {
"line": 127,
"column": 33
} | [
{
"pp": "case refine_1\nC : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\nP : Cᵒᵖ ⥤ Type u_4\nX B : C\nπ : X ⟶ B\ninst✝ : EffectiveEpi π\nc : PullbackCone π π\nhP :\n ∀ (b : (fun X ↦ X) ↑{x | (ConcreteCategory.hom (P.map c.fst.op)) x = (ConcreteCategory.hom (P.map c.snd.op)) x}),\n ∃! a, (ConcreteCategory.hom (... | [
"case refine_1\nC : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\nP : Cᵒᵖ ⥤ Type u_4\nX B : C\nπ : X ⟶ B\ninst✝ : EffectiveEpi π\nc : PullbackCone π π\nhP :\n ∀ (b : (fun X ↦ X) ↑{x | (ConcreteCategory.hom (P.map c.fst.op)) x = (ConcreteCategory.hom (P.map c.snd.op)) x}),\n ∃! a, (ConcreteCategory.hom (mapToEqualiz... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Sites.Coherent.RegularSheaves | {
"line": 131,
"column": 4
} | {
"line": 131,
"column": 32
} | {
"line": 131,
"column": 33
} | [
{
"pp": "case refine_2\nC : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\nP : Cᵒᵖ ⥤ Type u_4\nX B : C\nπ : X ⟶ B\ninst✝ : EffectiveEpi π\nc : PullbackCone π π\nhP :\n ∀ (b : (fun X ↦ X) ↑{x | (ConcreteCategory.hom (P.map c.fst.op)) x = (ConcreteCategory.hom (P.map c.snd.op)) x}),\n ∃! a, (ConcreteCategory.hom (... | [
"case refine_2\nC : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\nP : Cᵒᵖ ⥤ Type u_4\nX B : C\nπ : X ⟶ B\ninst✝ : EffectiveEpi π\nc : PullbackCone π π\nhP :\n ∀ (b : (fun X ↦ X) ↑{x | (ConcreteCategory.hom (P.map c.fst.op)) x = (ConcreteCategory.hom (P.map c.snd.op)) x}),\n ∃! a, (ConcreteCategory.hom (mapToEqualiz... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Sites.Coherent.SheafComparison | {
"line": 81,
"column": 6
} | {
"line": 81,
"column": 17
} | {
"line": 81,
"column": 18
} | [
{
"pp": "case refine_2.refine_1\nC : Type u_1\nD : Type u_2\ninst✝⁷ : Category.{v_1, u_1} C\ninst✝⁶ : Category.{v_2, u_2} D\nF : C ⥤ D\ninst✝⁵ : F.PreservesFiniteEffectiveEpiFamilies\ninst✝⁴ : F.ReflectsFiniteEffectiveEpiFamilies\ninst✝³ : F.Full\ninst✝² : F.Faithful\ninst✝¹ : F.EffectivelyEnough\ninst✝ : Preco... | [
"case refine_2.refine_1\nC : Type u_1\nD : Type u_2\ninst✝⁷ : Category.{v_1, u_1} C\ninst✝⁶ : Category.{v_2, u_2} D\nF : C ⥤ D\ninst✝⁵ : F.PreservesFiniteEffectiveEpiFamilies\ninst✝⁴ : F.ReflectsFiniteEffectiveEpiFamilies\ninst✝³ : F.Full\ninst✝² : F.Faithful\ninst✝¹ : F.EffectivelyEnough\ninst✝ : Precoherent D\nX ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Sites.EpiMono | {
"line": 127,
"column": 2
} | {
"line": 128,
"column": 18
} | {
"line": 129,
"column": 2
} | [
{
"pp": "case mp\nC : Type u\ninst✝⁸ : Category.{v, u} C\nJ : GrothendieckTopology C\nA : Type u'\ninst✝⁷ : Category.{v', u'} A\nFA : A → A → Type u_1\nCA : A → Type w\ninst✝⁶ : (X Y : A) → FunLike (FA X Y) (CA X) (CA Y)\ninst✝⁵ : ConcreteCategory A FA\ninst✝⁴ : HasFunctorialSurjectiveInjectiveFactorization A\n... | [
"case mpr\nC : Type u\ninst✝⁸ : Category.{v, u} C\nJ : GrothendieckTopology C\nA : Type u'\ninst✝⁷ : Category.{v', u'} A\nFA : A → A → Type u_1\nCA : A → Type w\ninst✝⁶ : (X Y : A) → FunLike (FA X Y) (CA X) (CA Y)\ninst✝⁵ : ConcreteCategory A FA\ninst✝⁴ : HasFunctorialSurjectiveInjectiveFactorization A\ninst✝³ : J.... | · intro
infer_instance | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.CategoryTheory.Sites.Coherent.RegularSheaves | {
"line": 151,
"column": 4
} | {
"line": 151,
"column": 32
} | {
"line": 151,
"column": 33
} | [
{
"pp": "case refine_1\nC : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\nP : Cᵒᵖ ⥤ Type u_4\nX B : C\nπ : X ⟶ B\ninst✝ : EffectiveEpi π\nc : PullbackCone π π\nhc : IsLimit c\nthis : HasPullback π π\nhP :\n ∀\n (b :\n (fun X ↦ X)\n ↑{x |\n (ConcreteCategory.hom (P.map (pullback.fst π π).o... | [
"case refine_1\nC : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\nP : Cᵒᵖ ⥤ Type u_4\nX B : C\nπ : X ⟶ B\ninst✝ : EffectiveEpi π\nc : PullbackCone π π\nhc : IsLimit c\nthis : HasPullback π π\nhP :\n ∀\n (b :\n (fun X ↦ X)\n ↑{x |\n (ConcreteCategory.hom (P.map (pullback.fst π π).op)) x =\n ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Sites.Coherent.SheafComparison | {
"line": 296,
"column": 2
} | {
"line": 296,
"column": 75
} | {
"line": 297,
"column": 2
} | [
{
"pp": "C : Type u_1\ninst✝⁵ : Category.{v_1, u_1} C\nA : Type u₃\ninst✝⁴ : Category.{v₃, u₃} A\nF : Cᵒᵖ ⥤ A\nB : Type u₄\ninst✝³ : Category.{v₄, u₄} B\ns : A ⥤ B\ninst✝² : Preregular C\ninst✝¹ : FinitaryExtensive C\nh : ∀ {Y X : C} (f : Y ⟶ X) [EffectiveEpi f], HasPullback f f\ninst✝ : ReflectsFiniteLimits s\... | [
"case refine_1\nC : Type u_1\ninst✝⁵ : Category.{v_1, u_1} C\nA : Type u₃\ninst✝⁴ : Category.{v₃, u₃} A\nF : Cᵒᵖ ⥤ A\nB : Type u₄\ninst✝³ : Category.{v₄, u₄} B\ns : A ⥤ B\ninst✝² : Preregular C\ninst✝¹ : FinitaryExtensive C\nh : ∀ {Y X : C} (f : Y ⟶ X) [EffectiveEpi f], HasPullback f f\ninst✝ : ReflectsFiniteLimits... | refine ⟨⟨fun n ↦ ⟨fun {K} ↦ ⟨fun {c} hc ↦ ?_⟩⟩⟩, fun _ _ π _ c hc ↦ ⟨?_⟩⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.CategoryTheory.Sites.Coherent.RegularSheaves | {
"line": 152,
"column": 4
} | {
"line": 152,
"column": 32
} | {
"line": 152,
"column": 33
} | [
{
"pp": "case refine_2\nC : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\nP : Cᵒᵖ ⥤ Type u_4\nX B : C\nπ : X ⟶ B\ninst✝ : EffectiveEpi π\nc : PullbackCone π π\nhc : IsLimit c\nthis : HasPullback π π\nhP :\n ∀\n (b :\n (fun X ↦ X)\n ↑{x |\n (ConcreteCategory.hom (P.map (pullback.fst π π).o... | [
"case refine_2\nC : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\nP : Cᵒᵖ ⥤ Type u_4\nX B : C\nπ : X ⟶ B\ninst✝ : EffectiveEpi π\nc : PullbackCone π π\nhc : IsLimit c\nthis : HasPullback π π\nhP :\n ∀\n (b :\n (fun X ↦ X)\n ↑{x |\n (ConcreteCategory.hom (P.map (pullback.fst π π).op)) x =\n ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Sites.Coherent.RegularSheaves | {
"line": 202,
"column": 2
} | {
"line": 202,
"column": 38
} | {
"line": 203,
"column": 2
} | [
{
"pp": "C : Type u_1\ninst✝ : Category.{v_1, u_1} C\nX B : C\nπ : X ⟶ B\nc : PullbackCone π π\nhc : IsLimit c\n⊢ (parallelPair (ObjectProperty.homMk (Over.homMk c.fst ⋯)).op (ObjectProperty.homMk (Over.homMk c.snd ⋯)).op).Initial",
"ppTerm": "?m.142",
"assigned": true,
"usedConstants": [
"Uni... | [
"case h₁\nC : Type u_1\ninst✝ : Category.{v_1, u_1} C\nX B : C\nπ : X ⟶ B\nc : PullbackCone π π\nhc : IsLimit c\n⊢ ∀ (Z : (Sieve.ofArrows (fun x ↦ X) fun x ↦ π).arrows.categoryᵒᵖ),\n Nonempty (op ((Sieve.ofArrows (fun x ↦ X) fun x ↦ π).arrows.categoryMk π ⋯) ⟶ Z)",
"case h₂\nC : Type u_1\ninst✝ : Category.{v_1... | apply Limits.parallelPair_initial_mk | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.CategoryTheory.Sites.Coherent.SequentialLimit | {
"line": 103,
"column": 19
} | {
"line": 103,
"column": 48
} | {
"line": 103,
"column": 49
} | [
{
"pp": "C : Type u\ninst✝³ : Category.{v, u} C\ninst✝² : Preregular C\ninst✝¹ : FinitaryExtensive C\nF : ℕᵒᵖ ⥤ Sheaf (coherentTopology C) (Type v)\nc : Cone F\nhc : IsLimit c\nhF : ∀ (n : ℕ), Sheaf.IsLocallySurjective (F.map (homOfLE ⋯).op)\ninst✝ : HasLimitsOfShape ℕᵒᵖ C\nh : ∀ (G : ℕᵒᵖ ⥤ C), (∀ (n : ℕ), Effe... | [
"C : Type u\ninst✝³ : Category.{v, u} C\ninst✝² : Preregular C\ninst✝¹ : FinitaryExtensive C\nF : ℕᵒᵖ ⥤ Sheaf (coherentTopology C) (Type v)\nc : Cone F\nhc : IsLimit c\nhF : ∀ (n : ℕ), Sheaf.IsLocallySurjective (F.map (homOfLE ⋯).op)\ninst✝ : HasLimitsOfShape ℕᵒᵖ C\nh : ∀ (G : ℕᵒᵖ ⥤ C), (∀ (n : ℕ), EffectiveEpi (G.... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Sites.Coherent.RegularSheaves | {
"line": 240,
"column": 2
} | {
"line": 240,
"column": 17
} | {
"line": 242,
"column": 0
} | [
{
"pp": "case zero\nC : Type u_1\nD : Type u_2\nE : Type u_3\ninst✝² : Category.{v_1, u_1} C\ninst✝¹ : Category.{v_2, u_2} D\ninst✝ : Category.{v_3, u_3} E\nP : Cᵒᵖ ⥤ D\nX B : C\nπ : X ⟶ B\nc : PullbackCone π π\nhc : IsLimit c\nS : Presieve B := (Sieve.ofArrows (fun x ↦ X) fun x ↦ π).arrows\nX' : S.category := ... | [] | all_goals aesop | Lean.Elab.Tactic.evalAllGoals | Lean.Parser.Tactic.allGoals |
Mathlib.CategoryTheory.Sites.Coherent.RegularSheaves | {
"line": 268,
"column": 4
} | {
"line": 268,
"column": 15
} | {
"line": 268,
"column": 16
} | [
{
"pp": "case refine_1\nC : Type u_1\ninst✝² : Category.{v_1, u_1} C\nX : C\ninst✝¹ : Projective X\nF : Cᵒᵖ ⥤ Type u_4\nY : C\nf : Y ⟶ X\nhf : EffectiveEpi f\ninst✝ : (ofArrows (fun x ↦ Y) fun x ↦ f).regular\nx : Unit → F.obj (op Y)\nhx : Arrows.Compatible F (fun x ↦ f) x\nx✝ : Unit\n⊢ (ConcreteCategory.hom (F.... | [
"case refine_1\nC : Type u_1\ninst✝² : Category.{v_1, u_1} C\nX : C\ninst✝¹ : Projective X\nF : Cᵒᵖ ⥤ Type u_4\nY : C\nf : Y ⟶ X\nhf : EffectiveEpi f\ninst✝ : (ofArrows (fun x ↦ Y) fun x ↦ f).regular\nx : Unit → F.obj (op Y)\nhx : Arrows.Compatible F (fun x ↦ f) x\nx✝ : Unit\n⊢ (ConcreteCategory.hom (F.map f.op)) (... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Sites.Descent.IsStack | {
"line": 68,
"column": 2
} | {
"line": 68,
"column": 13
} | {
"line": 68,
"column": 14
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\nF : LocallyDiscrete Cᵒᵖ ⥤ᵖ Cat\nJ : GrothendieckTopology C\ninst✝ : F.IsStack J\nS : C\nR : Presieve S\nhR : Sieve.generate R ∈ J S\n⊢ F.IsStackFor R",
"ppTerm": "?m.18",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []... | [
"C : Type u\ninst✝¹ : Category.{v, u} C\nF : LocallyDiscrete Cᵒᵖ ⥤ᵖ Cat\nJ : GrothendieckTopology C\ninst✝ : F.IsStack J\nS : C\nR : Presieve S\nhR : Sieve.generate R ∈ J S\n⊢ F.IsStackFor R"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Sites.Descent.DescentDataAsCoalgebra | {
"line": 164,
"column": 10
} | {
"line": 164,
"column": 52
} | {
"line": 165,
"column": 10
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\nF : LocallyDiscrete Cᵒᵖ ⥤ᵖ Adj Cat\nι : Type u_1\ninst✝ : Unique ι\nX S : C\nf : X ⟶ S\nD : F.DescentDataAsCoalgebra fun x ↦ f\ni₁ i₂ : ι\n⊢ ((𝟭 (F.DescentDataAsCoalgebra fun x ↦ f)).obj D).hom i₁ i₂ ≫\n (F.map f.op.toLoc).l.toFunctor.map ((F.map f.op.toLoc... | [
"C : Type u\ninst✝¹ : Category.{v, u} C\nF : LocallyDiscrete Cᵒᵖ ⥤ᵖ Adj Cat\nι : Type u_1\ninst✝ : Unique ι\nX S : C\nf : X ⟶ S\nD : F.DescentDataAsCoalgebra fun x ↦ f\ni₂ : ι\n⊢ ((𝟭 (F.DescentDataAsCoalgebra fun x ↦ f)).obj D).hom default i₂ ≫\n (F.map f.op.toLoc).l.toFunctor.map ((F.map f.op.toLoc).r.toFunc... | obtain rfl := Subsingleton.elim i₁ default | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.CategoryTheory.Sites.Descent.DescentDataAsCoalgebra | {
"line": 167,
"column": 6
} | {
"line": 169,
"column": 10
} | {
"line": 169,
"column": 10
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\nF : LocallyDiscrete Cᵒᵖ ⥤ᵖ Adj Cat\nι : Type u_1\ninst✝ : Unique ι\nX S : C\nf : X ⟶ S\nD₁ D₂ : F.DescentDataAsCoalgebra fun x ↦ f\nα : D₁ ⟶ D₂\n⊢ (𝟭 (F.DescentDataAsCoalgebra fun x ↦ f)).map α ≫ (isoMk (fun i ↦ eqToIso ⋯) ⋯).hom =\n (isoMk (fun i ↦ eqToIso ⋯... | [] | ext i
obtain rfl := Subsingleton.elim i default
simp | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Sites.Descent.DescentDataAsCoalgebra | {
"line": 167,
"column": 6
} | {
"line": 169,
"column": 10
} | {
"line": 169,
"column": 10
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\nF : LocallyDiscrete Cᵒᵖ ⥤ᵖ Adj Cat\nι : Type u_1\ninst✝ : Unique ι\nX S : C\nf : X ⟶ S\nD₁ D₂ : F.DescentDataAsCoalgebra fun x ↦ f\nα : D₁ ⟶ D₂\n⊢ (𝟭 (F.DescentDataAsCoalgebra fun x ↦ f)).map α ≫ (isoMk (fun i ↦ eqToIso ⋯) ⋯).hom =\n (isoMk (fun i ↦ eqToIso ⋯... | [] | ext i
obtain rfl := Subsingleton.elim i default
simp | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Sites.Finite | {
"line": 44,
"column": 2
} | {
"line": 44,
"column": 13
} | {
"line": 44,
"column": 14
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\nX : C\nι : Type u_1\ninst✝ : Finite ι\nY : ι → C\nf : (i : ι) → Y i ⟶ X\n⊢ ofArrows Y f ∈ (finite C).coverings X",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"CategoryTheory.CategoryStruct.toQuiver",
"Quiver.Ho... | [
"C : Type u\ninst✝¹ : Category.{v, u} C\nX : C\nι : Type u_1\ninst✝ : Finite ι\nY : ι → C\nf : (i : ι) → Y i ⟶ X\n⊢ (Set.range fun i ↦ ⟨Y i, f i⟩).Finite"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Sites.Finite | {
"line": 61,
"column": 29
} | {
"line": 61,
"column": 40
} | {
"line": 61,
"column": 41
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : HasPullbacks C\nX Y : C\nu : Y ⟶ X\ns : Presieve X\nhs : s ∈ (Precoverage.finite C).coverings X\n⊢ pullbackArrows u s ∈ (Precoverage.finite C).coverings Y",
"ppTerm": "?m.35",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Categor... | [
"C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : HasPullbacks C\nX Y : C\nu : Y ⟶ X\ns : Presieve X\nhs : s ∈ (Precoverage.finite C).coverings X\n⊢ ((fun f ↦ ⟨Limits.pullback f.snd u, pullback.snd f.snd u⟩) '' s.uncurry).Finite"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Sites.Finite | {
"line": 62,
"column": 31
} | {
"line": 62,
"column": 42
} | {
"line": 62,
"column": 43
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : HasPullbacks C\nX : C\ns : Presieve X\nt : ⦃Y : C⦄ → (f : Y ⟶ X) → s f → Presieve Y\nhs : s ∈ (Precoverage.finite C).coverings X\nht : ∀ ⦃Y : C⦄ (f : Y ⟶ X) (H : s f), t f H ∈ (Precoverage.finite C).coverings Y\n⊢ s.bind t ∈ (Precoverage.finite C).coverin... | [
"C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : HasPullbacks C\nX : C\ns : Presieve X\nt : ⦃Y : C⦄ → (f : Y ⟶ X) → s f → Presieve Y\nhs : s ∈ (Precoverage.finite C).coverings X\nht : ∀ ⦃Y : C⦄ (f : Y ⟶ X) (H : s f), t f H ∈ (Precoverage.finite C).coverings Y\n⊢ (⋃ i, ⋃ (h : i ∈ s.uncurry), (Sigma.map id fun Z g ↦ ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Sites.GlobalSections | {
"line": 152,
"column": 4
} | {
"line": 152,
"column": 78
} | {
"line": 152,
"column": 79
} | [
{
"pp": "C : Type u\ninst✝³ : Category.{v, u} C\nJ : GrothendieckTopology C\nA : Type u₂\ninst✝² : Category.{v₂, u₂} A\ninst✝¹ : HasWeakSheafify J A\ninst✝ : HasGlobalSectionsFunctor J A\nF : Sheaf J A\nc : Cone F.obj\nf : c.pt ⟶ F.coneΓ.pt\nhf : (Functor.const Cᵒᵖ).map f ≫ F.coneΓ.π = c.π\n⊢ f = ΓHomEquiv c.π"... | [
"C : Type u\ninst✝³ : Category.{v, u} C\nJ : GrothendieckTopology C\nA : Type u₂\ninst✝² : Category.{v₂, u₂} A\ninst✝¹ : HasWeakSheafify J A\ninst✝ : HasGlobalSectionsFunctor J A\nF : Sheaf J A\nc : Cone F.obj\nf : c.pt ⟶ F.coneΓ.pt\nhf : (Functor.const Cᵒᵖ).map f ≫ F.coneΓ.π = c.π\n⊢ f = ΓHomEquiv c.π"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Sites.Descent.DescentDataPrime | {
"line": 203,
"column": 2
} | {
"line": 203,
"column": 13
} | {
"line": 203,
"column": 14
} | [
{
"pp": "C : Type u\ninst✝ : Category.{v, u} C\nF : Pseudofunctor (LocallyDiscrete Cᵒᵖ) Cat\nι : Type t\nS : C\nX : ι → C\nf : (i : ι) → X i ⟶ S\nsq : (i j : ι) → ChosenPullback (f i) (f j)\nsq₃ : (i₁ i₂ i₃ : ι) → ChosenPullback₃ (sq i₁ i₂) (sq i₂ i₃) (sq i₁ i₃)\nD : F.DescentData' sq sq₃\ni₁ i₂ : ι\n⊢ IsIso (D... | [
"C : Type u\ninst✝ : Category.{v, u} C\nF : Pseudofunctor (LocallyDiscrete Cᵒᵖ) Cat\nι : Type t\nS : C\nX : ι → C\nf : (i : ι) → X i ⟶ S\nsq : (i j : ι) → ChosenPullback (f i) (f j)\nsq₃ : (i₁ i₂ i₃ : ι) → ChosenPullback₃ (sq i₁ i₂) (sq i₂ i₃) (sq i₁ i₃)\nD : F.DescentData' sq sq₃\ni₁ i₂ : ι\n⊢ IsIso (D.hom i₁ i₂)"... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Sites.Hypercover.Homotopy | {
"line": 203,
"column": 4
} | {
"line": 203,
"column": 85
} | {
"line": 204,
"column": 4
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\nS : C\nE : PreOneHypercover S\nF : PreOneHypercover S\ninst✝ : HasPullbacks C\nf g : E.Hom F\ni✝ j✝ : (cylinder f g).I₀\nk : (cylinder f g).I₁ i✝ j✝\n⊢ pullback.snd\n (pullback.map (cylinderf f g i✝.snd) (cylinderf f g j✝.snd) (E.f i✝.fst) (E.f j✝.fst)\n ... | [
"C : Type u\ninst✝¹ : Category.{v, u} C\nS : C\nE : PreOneHypercover S\nF : PreOneHypercover S\ninst✝ : HasPullbacks C\nf g : E.Hom F\ni✝ j✝ : (cylinder f g).I₀\nk : (cylinder f g).I₁ i✝ j✝\nthis : E.p₁ k.down = pullback.lift (E.p₁ k.down) (E.p₂ k.down) ⋯ ≫ pullback.fst (E.f i✝.fst) (E.f j✝.fst)\n⊢ pullback.snd\n ... | have : E.p₁ k.down = pullback.lift _ _ (E.w k.down) ≫ pullback.fst _ _ := by simp | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.CategoryTheory.Sites.Descent.Precoverage | {
"line": 214,
"column": 6
} | {
"line": 214,
"column": 17
} | {
"line": 214,
"column": 18
} | [
{
"pp": "case e'_6\nC : Type u\ninst✝¹ : Category.{v, u} C\nF : LocallyDiscrete Cᵒᵖ ⥤ᵖ Cat\nJ : GrothendieckTopology C\ninst✝ : F.IsPrestack J\nι : Type t\nS : C\nX : ι → C\nf : (i : ι) → X i ⟶ S\nι' : Type t'\nX' : ι' → C\nf' : (j : ι') → X' j ⟶ S\nα : ι' → ι\np' : (j : ι') → X' j ⟶ X (α j)\nw : ∀ (j : ι'), p'... | [
"case e'_6\nC : Type u\ninst✝¹ : Category.{v, u} C\nF : LocallyDiscrete Cᵒᵖ ⥤ᵖ Cat\nJ : GrothendieckTopology C\ninst✝ : F.IsPrestack J\nι : Type t\nS : C\nX : ι → C\nf : (i : ι) → X i ⟶ S\nι' : Type t'\nX' : ι' → C\nf' : (j : ι') → X' j ⟶ S\nα : ι' → ι\np' : (j : ι') → X' j ⟶ X (α j)\nw : ∀ (j : ι'), p' j ≫ f (α j)... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Sites.Descent.Precoverage | {
"line": 220,
"column": 44
} | {
"line": 220,
"column": 55
} | {
"line": 220,
"column": 56
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\nF : LocallyDiscrete Cᵒᵖ ⥤ᵖ Cat\nJ : GrothendieckTopology C\ninst✝ : F.IsPrestack J\nι : Type t\nS : C\nX : ι → C\nf : (i : ι) → X i ⟶ S\nι' : Type t'\nX' : ι' → C\nf' : (j : ι') → X' j ⟶ S\nα : ι' → ι\np' : (j : ι') → X' j ⟶ X (α j)\nw : ∀ (j : ι'), p' j ≫ f (α j... | [
"C : Type u\ninst✝¹ : Category.{v, u} C\nF : LocallyDiscrete Cᵒᵖ ⥤ᵖ Cat\nJ : GrothendieckTopology C\ninst✝ : F.IsPrestack J\nι : Type t\nS : C\nX : ι → C\nf : (i : ι) → X i ⟶ S\nι' : Type t'\nX' : ι' → C\nf' : (j : ι') → X' j ⟶ S\nα : ι' → ι\np' : (j : ι') → X' j ⟶ X (α j)\nw : ∀ (j : ι'), p' j ≫ f (α j) = f' j\nhf... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Sites.Descent.Precoverage | {
"line": 228,
"column": 49
} | {
"line": 228,
"column": 60
} | {
"line": 228,
"column": 61
} | [
{
"pp": "C : Type u\ninst✝ : Category.{v, u} C\nF : LocallyDiscrete Cᵒᵖ ⥤ᵖ Cat\nι : Type t\nS : C\nX : ι → C\nf : (i : ι) → X i ⟶ S\nι' : Type t'\nX' : ι' → C\nf' : (j : ι') → X' j ⟶ S\nα : ι' → ι\np' : (j : ι') → X' j ⟶ X (α j)\nw : ∀ (j : ι'), p' j ≫ f (α j) = f' j\nD₁ D₂ : F.DescentData f\nφ : (pullFunctor F... | [
"C : Type u\ninst✝ : Category.{v, u} C\nF : LocallyDiscrete Cᵒᵖ ⥤ᵖ Cat\nι : Type t\nS : C\nX : ι → C\nf : (i : ι) → X i ⟶ S\nι' : Type t'\nX' : ι' → C\nf' : (j : ι') → X' j ⟶ S\nα : ι' → ι\np' : (j : ι') → X' j ⟶ X (α j)\nw : ∀ (j : ι'), p' j ≫ f (α j) = f' j\nD₁ D₂ : F.DescentData f\nφ : (pullFunctor F ⋯).obj D₁ ⟶... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Sites.Descent.Precoverage | {
"line": 229,
"column": 49
} | {
"line": 229,
"column": 60
} | {
"line": 229,
"column": 61
} | [
{
"pp": "C : Type u\ninst✝ : Category.{v, u} C\nF : LocallyDiscrete Cᵒᵖ ⥤ᵖ Cat\nι : Type t\nS : C\nX : ι → C\nf : (i : ι) → X i ⟶ S\nι' : Type t'\nX' : ι' → C\nf' : (j : ι') → X' j ⟶ S\nα : ι' → ι\np' : (j : ι') → X' j ⟶ X (α j)\nw : ∀ (j : ι'), p' j ≫ f (α j) = f' j\nD₁ D₂ : F.DescentData f\nφ : (pullFunctor F... | [
"C : Type u\ninst✝ : Category.{v, u} C\nF : LocallyDiscrete Cᵒᵖ ⥤ᵖ Cat\nι : Type t\nS : C\nX : ι → C\nf : (i : ι) → X i ⟶ S\nι' : Type t'\nX' : ι' → C\nf' : (j : ι') → X' j ⟶ S\nα : ι' → ι\np' : (j : ι') → X' j ⟶ X (α j)\nw : ∀ (j : ι'), p' j ≫ f (α j) = f' j\nD₁ D₂ : F.DescentData f\nφ : (pullFunctor F ⋯).obj D₁ ⟶... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Sites.Hypercover.Subcanonical | {
"line": 119,
"column": 4
} | {
"line": 119,
"column": 42
} | {
"line": 119,
"column": 43
} | [
{
"pp": "C : Type u\ninst✝³ : Category.{v, u} C\nJ : Precoverage C\ninst✝² : J.toGrothendieck.Subcanonical\ninst✝¹ : Limits.HasPullbacks C\ninst✝ : J.IsStableUnderBaseChange\nP X Y Z : C\nfst : P ⟶ X\nsnd : P ⟶ Y\nf : X ⟶ Z\ng : Y ⟶ Z\n𝒰 : J.ZeroHypercover X\nH : ∀ (i : 𝒰.I₀), IsPullback (pullback.snd fst (𝒰... | [
"C : Type u\ninst✝³ : Category.{v, u} C\nJ : Precoverage C\ninst✝² : J.toGrothendieck.Subcanonical\ninst✝¹ : Limits.HasPullbacks C\ninst✝ : J.IsStableUnderBaseChange\nP X Y Z : C\nfst : P ⟶ X\nsnd : P ⟶ Y\nf : X ⟶ Z\ng : Y ⟶ Z\n𝒰 : J.ZeroHypercover X\nH : ∀ (i : 𝒰.I₀), IsPullback (pullback.snd fst (𝒰.f i)) (pull... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Sites.Hypercover.Subcanonical | {
"line": 127,
"column": 4
} | {
"line": 127,
"column": 70
} | {
"line": 127,
"column": 71
} | [
{
"pp": "C : Type u\ninst✝³ : Category.{v, u} C\nJ : Precoverage C\ninst✝² : J.toGrothendieck.Subcanonical\ninst✝¹ : Limits.HasPullbacks C\ninst✝ : J.IsStableUnderBaseChange\nP X Y Z : C\nfst : P ⟶ X\nsnd : P ⟶ Y\nf : X ⟶ Z\ng : Y ⟶ Z\n𝒰 : J.ZeroHypercover X\nH : ∀ (i : 𝒰.I₀), IsPullback (pullback.snd fst (𝒰... | [
"C : Type u\ninst✝³ : Category.{v, u} C\nJ : Precoverage C\ninst✝² : J.toGrothendieck.Subcanonical\ninst✝¹ : Limits.HasPullbacks C\ninst✝ : J.IsStableUnderBaseChange\nP X Y Z : C\nfst : P ⟶ X\nsnd : P ⟶ Y\nf : X ⟶ Z\ng : Y ⟶ Z\n𝒰 : J.ZeroHypercover X\nH : ∀ (i : 𝒰.I₀), IsPullback (pullback.snd fst (𝒰.f i)) (pull... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Sites.Descent.Precoverage | {
"line": 250,
"column": 12
} | {
"line": 250,
"column": 23
} | {
"line": 250,
"column": 24
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\nF : LocallyDiscrete Cᵒᵖ ⥤ᵖ Cat\nJ : GrothendieckTopology C\ninst✝ : F.IsPrestack J\nι : Type t\nS : C\nX : ι → C\nf : (i : ι) → X i ⟶ S\nι' : Type t'\nX' : ι' → C\nf' : (j : ι') → X' j ⟶ S\nα : ι' → ι\np' : (j : ι') → X' j ⟶ X (α j)\nw : ∀ (j : ι'), p' j ≫ f (α j... | [
"C : Type u\ninst✝¹ : Category.{v, u} C\nF : LocallyDiscrete Cᵒᵖ ⥤ᵖ Cat\nJ : GrothendieckTopology C\ninst✝ : F.IsPrestack J\nι : Type t\nS : C\nX : ι → C\nf : (i : ι) → X i ⟶ S\nι' : Type t'\nX' : ι' → C\nf' : (j : ι') → X' j ⟶ S\nα : ι' → ι\np' : (j : ι') → X' j ⟶ X (α j)\nw : ∀ (j : ι'), p' j ≫ f (α j) = f' j\nhf... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Sites.Descent.Precoverage | {
"line": 259,
"column": 10
} | {
"line": 259,
"column": 21
} | {
"line": 259,
"column": 22
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\nF : LocallyDiscrete Cᵒᵖ ⥤ᵖ Cat\nJ : GrothendieckTopology C\ninst✝ : F.IsPrestack J\nι : Type t\nS : C\nX : ι → C\nf : (i : ι) → X i ⟶ S\nι' : Type t'\nX' : ι' → C\nf' : (j : ι') → X' j ⟶ S\nα : ι' → ι\np' : (j : ι') → X' j ⟶ X (α j)\nw : ∀ (j : ι'), p' j ≫ f (α j... | [
"C : Type u\ninst✝¹ : Category.{v, u} C\nF : LocallyDiscrete Cᵒᵖ ⥤ᵖ Cat\nJ : GrothendieckTopology C\ninst✝ : F.IsPrestack J\nι : Type t\nS : C\nX : ι → C\nf : (i : ι) → X i ⟶ S\nι' : Type t'\nX' : ι' → C\nf' : (j : ι') → X' j ⟶ S\nα : ι' → ι\np' : (j : ι') → X' j ⟶ X (α j)\nw : ∀ (j : ι'), p' j ≫ f (α j) = f' j\nhf... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Sites.Hypercover.Subcanonical | {
"line": 133,
"column": 4
} | {
"line": 133,
"column": 15
} | {
"line": 133,
"column": 16
} | [
{
"pp": "C : Type u\ninst✝³ : Category.{v, u} C\nJ : Precoverage C\ninst✝² : J.toGrothendieck.Subcanonical\ninst✝¹ : Limits.HasPullbacks C\ninst✝ : J.IsStableUnderBaseChange\nP X Y Z : C\nfst : P ⟶ X\nsnd : P ⟶ Y\nf : X ⟶ Z\ng : Y ⟶ Z\n𝒰 : J.ZeroHypercover X\nH : ∀ (i : 𝒰.I₀), IsPullback (pullback.snd fst (𝒰... | [
"C : Type u\ninst✝³ : Category.{v, u} C\nJ : Precoverage C\ninst✝² : J.toGrothendieck.Subcanonical\ninst✝¹ : Limits.HasPullbacks C\ninst✝ : J.IsStableUnderBaseChange\nP X Y Z : C\nfst : P ⟶ X\nsnd : P ⟶ Y\nf : X ⟶ Z\ng : Y ⟶ Z\n𝒰 : J.ZeroHypercover X\nH : ∀ (i : 𝒰.I₀), IsPullback (pullback.snd fst (𝒰.f i)) (pull... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Sites.Descent.Precoverage | {
"line": 264,
"column": 12
} | {
"line": 264,
"column": 23
} | {
"line": 264,
"column": 24
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\nF : LocallyDiscrete Cᵒᵖ ⥤ᵖ Cat\nJ : GrothendieckTopology C\ninst✝ : F.IsPrestack J\nι : Type t\nS : C\nX : ι → C\nf : (i : ι) → X i ⟶ S\nι' : Type t'\nX' : ι' → C\nf' : (j : ι') → X' j ⟶ S\nα : ι' → ι\np' : (j : ι') → X' j ⟶ X (α j)\nw : ∀ (j : ι'), p' j ≫ f (α j... | [
"C : Type u\ninst✝¹ : Category.{v, u} C\nF : LocallyDiscrete Cᵒᵖ ⥤ᵖ Cat\nJ : GrothendieckTopology C\ninst✝ : F.IsPrestack J\nι : Type t\nS : C\nX : ι → C\nf : (i : ι) → X i ⟶ S\nι' : Type t'\nX' : ι' → C\nf' : (j : ι') → X' j ⟶ S\nα : ι' → ι\np' : (j : ι') → X' j ⟶ X (α j)\nw : ∀ (j : ι'), p' j ≫ f (α j) = f' j\nhf... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Sites.Descent.Precoverage | {
"line": 265,
"column": 2
} | {
"line": 267,
"column": 35
} | {
"line": 267,
"column": 36
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\nF : LocallyDiscrete Cᵒᵖ ⥤ᵖ Cat\nJ : GrothendieckTopology C\ninst✝ : F.IsPrestack J\nι : Type t\nS : C\nX : ι → C\nf : (i : ι) → X i ⟶ S\nι' : Type t'\nX' : ι' → C\nf' : (j : ι') → X' j ⟶ S\nα : ι' → ι\np' : (j : ι') → X' j ⟶ X (α j)\nw : ∀ (j : ι'), p' j ≫ f (α j... | [
"C : Type u\ninst✝¹ : Category.{v, u} C\nF : LocallyDiscrete Cᵒᵖ ⥤ᵖ Cat\nJ : GrothendieckTopology C\ninst✝ : F.IsPrestack J\nι : Type t\nS : C\nX : ι → C\nf : (i : ι) → X i ⟶ S\nι' : Type t'\nX' : ι' → C\nf' : (j : ι') → X' j ⟶ S\nα : ι' → ι\np' : (j : ι') → X' j ⟶ X (α j)\nw : ∀ (j : ι'), p' j ≫ f (α j) = f' j\nhf... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Sites.SheafHom | {
"line": 54,
"column": 4
} | {
"line": 54,
"column": 28
} | {
"line": 54,
"column": 29
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\nJ : GrothendieckTopology C\nA : Type u'\ninst✝ : Category.{v', u'} A\nF G : Cᵒᵖ ⥤ A\nX : C\nφ : (Over.forget (unop (op X))).op ⋙ F ⟶ (Over.forget (unop (op X))).op ⋙ G\nY : Over (unop (op X))\n⊢ ((ConcreteCategory.hom (↾(Over.map (𝟙 (op X)).unop).op.whiskerLeft)... | [
"C : Type u\ninst✝¹ : Category.{v, u} C\nJ : GrothendieckTopology C\nA : Type u'\ninst✝ : Category.{v', u'} A\nF G : Cᵒᵖ ⥤ A\nX : C\nφ : (Over.forget (unop (op X))).op ⋙ F ⟶ (Over.forget (unop (op X))).op ⋙ G\nY : Over (unop (op X))\n⊢ φ.app (op ((Over.map (𝟙 X)).obj Y)) = φ.app (op Y)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Sites.SheafHom | {
"line": 58,
"column": 4
} | {
"line": 58,
"column": 30
} | {
"line": 58,
"column": 31
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\nJ : GrothendieckTopology C\nA : Type u'\ninst✝ : Category.{v', u'} A\nF G : Cᵒᵖ ⥤ A\nX Y Z : C\nf : Y ⟶ X\ng : Z ⟶ Y\nφ : (Over.forget (unop (op X))).op ⋙ F ⟶ (Over.forget (unop (op X))).op ⋙ G\nW : Over (unop (op Z))\n⊢ ((ConcreteCategory.hom (↾(Over.map (op f ≫... | [
"C : Type u\ninst✝¹ : Category.{v, u} C\nJ : GrothendieckTopology C\nA : Type u'\ninst✝ : Category.{v', u'} A\nF G : Cᵒᵖ ⥤ A\nX Y Z : C\nf : Y ⟶ X\ng : Z ⟶ Y\nφ : (Over.forget (unop (op X))).op ⋙ F ⟶ (Over.forget (unop (op X))).op ⋙ G\nW : Over (unop (op Z))\n⊢ φ.app (op ((Over.map (g ≫ f)).obj W)) = φ.app (op ((Ov... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Sites.MayerVietorisSquare | {
"line": 132,
"column": 37
} | {
"line": 132,
"column": 76
} | {
"line": 132,
"column": 77
} | [
{
"pp": "C : Type u\ninst✝³ : Category.{v, u} C\nJ : GrothendieckTopology C\ninst✝² : HasWeakSheafify J (Type v)\nsq : Square C\ninst✝¹ : Mono sq.f₂₄\ninst✝ : Mono sq.f₃₄\nh₁ : sq.IsPullback\nh₂ : Sieve.ofTwoArrows sq.f₂₄ sq.f₃₄ ∈ J sq.X₄\nthis : Mono sq.f₁₃\nF : Sheaf J (Type v)\ns : PullbackCone (sq.op.map F.... | [
"C : Type u\ninst✝³ : Category.{v, u} C\nJ : GrothendieckTopology C\ninst✝² : HasWeakSheafify J (Type v)\nsq : Square C\ninst✝¹ : Mono sq.f₂₄\ninst✝ : Mono sq.f₃₄\nh₁ : sq.IsPullback\nh₂ : Sieve.ofTwoArrows sq.f₂₄ sq.f₃₄ ∈ J sq.X₄\nthis : Mono sq.f₁₃\nF : Sheaf J (Type v)\ns : PullbackCone (sq.op.map F.obj).f₂₄ (sq... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Sites.MayerVietorisSquare | {
"line": 135,
"column": 12
} | {
"line": 135,
"column": 23
} | {
"line": 135,
"column": 24
} | [
{
"pp": "case left.right\nC : Type u\ninst✝³ : Category.{v, u} C\nJ : GrothendieckTopology C\ninst✝² : HasWeakSheafify J (Type v)\nsq : Square C\ninst✝¹ : Mono sq.f₂₄\ninst✝ : Mono sq.f₃₄\nh₁ : sq.IsPullback\nh₂ : Sieve.ofTwoArrows sq.f₂₄ sq.f₃₄ ∈ J sq.X₄\nthis : Mono sq.f₁₃\nF : Sheaf J (Type v)\ns : PullbackC... | [
"case left.right\nC : Type u\ninst✝³ : Category.{v, u} C\nJ : GrothendieckTopology C\ninst✝² : HasWeakSheafify J (Type v)\nsq : Square C\ninst✝¹ : Mono sq.f₂₄\ninst✝ : Mono sq.f₃₄\nh₁ : sq.IsPullback\nh₂ : Sieve.ofTwoArrows sq.f₂₄ sq.f₃₄ ∈ J sq.X₄\nthis : Mono sq.f₁₃\nF : Sheaf J (Type v)\ns : PullbackCone (sq.op.m... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Sites.MayerVietorisSquare | {
"line": 137,
"column": 12
} | {
"line": 137,
"column": 23
} | {
"line": 137,
"column": 24
} | [
{
"pp": "case right.left\nC : Type u\ninst✝³ : Category.{v, u} C\nJ : GrothendieckTopology C\ninst✝² : HasWeakSheafify J (Type v)\nsq : Square C\ninst✝¹ : Mono sq.f₂₄\ninst✝ : Mono sq.f₃₄\nh₁ : sq.IsPullback\nh₂ : Sieve.ofTwoArrows sq.f₂₄ sq.f₃₄ ∈ J sq.X₄\nthis : Mono sq.f₁₃\nF : Sheaf J (Type v)\ns : PullbackC... | [
"case right.left\nC : Type u\ninst✝³ : Category.{v, u} C\nJ : GrothendieckTopology C\ninst✝² : HasWeakSheafify J (Type v)\nsq : Square C\ninst✝¹ : Mono sq.f₂₄\ninst✝ : Mono sq.f₃₄\nh₁ : sq.IsPullback\nh₂ : Sieve.ofTwoArrows sq.f₂₄ sq.f₃₄ ∈ J sq.X₄\nthis : Mono sq.f₁₃\nF : Sheaf J (Type v)\ns : PullbackCone (sq.op.m... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Sites.MayerVietorisSquare | {
"line": 138,
"column": 37
} | {
"line": 138,
"column": 76
} | {
"line": 138,
"column": 77
} | [
{
"pp": "C : Type u\ninst✝³ : Category.{v, u} C\nJ : GrothendieckTopology C\ninst✝² : HasWeakSheafify J (Type v)\nsq : Square C\ninst✝¹ : Mono sq.f₂₄\ninst✝ : Mono sq.f₃₄\nh₁ : sq.IsPullback\nh₂ : Sieve.ofTwoArrows sq.f₂₄ sq.f₃₄ ∈ J sq.X₄\nthis : Mono sq.f₁₃\nF : Sheaf J (Type v)\ns : PullbackCone (sq.op.map F.... | [
"C : Type u\ninst✝³ : Category.{v, u} C\nJ : GrothendieckTopology C\ninst✝² : HasWeakSheafify J (Type v)\nsq : Square C\ninst✝¹ : Mono sq.f₂₄\ninst✝ : Mono sq.f₃₄\nh₁ : sq.IsPullback\nh₂ : Sieve.ofTwoArrows sq.f₂₄ sq.f₃₄ ∈ J sq.X₄\nthis : Mono sq.f₁₃\nF : Sheaf J (Type v)\ns : PullbackCone (sq.op.map F.obj).f₂₄ (sq... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Sites.Descent.Precoverage | {
"line": 404,
"column": 4
} | {
"line": 405,
"column": 11
} | {
"line": 405,
"column": 12
} | [
{
"pp": "C : Type u\ninst✝⁴ : Category.{v, u} C\nF : LocallyDiscrete Cᵒᵖ ⥤ᵖ Cat\ninst✝³ : HasPullbacks C\nJ : Precoverage C\ninst✝² : J.HasIsos\ninst✝¹ : J.IsStableUnderBaseChange\ninst✝ : J.IsStableUnderComposition\nhF : ∀ (S : C), ∀ R ∈ J.coverings S, F.IsPrestackFor R\nS : C\nM N : ↑(F.obj { as := op S })\nX... | [
"C : Type u\ninst✝⁴ : Category.{v, u} C\nF : LocallyDiscrete Cᵒᵖ ⥤ᵖ Cat\ninst✝³ : HasPullbacks C\nJ : Precoverage C\ninst✝² : J.HasIsos\ninst✝¹ : J.IsStableUnderBaseChange\ninst✝ : J.IsStableUnderComposition\nhF : ∀ (S : C), ∀ R ∈ J.coverings S, F.IsPrestackFor R\nS : C\nM N : ↑(F.obj { as := op S })\nX : Over S\nR... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Sites.NonabelianCohomology.H1 | {
"line": 138,
"column": 2
} | {
"line": 138,
"column": 13
} | {
"line": 138,
"column": 14
} | [
{
"pp": "C : Type u\ninst✝ : Category.{v, u} C\nG : Cᵒᵖ ⥤ GrpCat\nI : Type w'\nU : I → C\nγ : OneCocycle G U\ni : I\nT : C\na : T ⟶ U i\n⊢ γ.ev i i a a = 1",
"ppTerm": "?m.17",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"C : Type u\ninst✝ : Category.{v, u} C\nG : Cᵒᵖ ⥤ GrpCat\nI : Type w'\nU : I → C\nγ : OneCocycle G U\ni : I\nT : C\na : T ⟶ U i\n⊢ γ.ev i i a a = 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Sites.SheafHom | {
"line": 191,
"column": 59
} | {
"line": 191,
"column": 70
} | {
"line": 191,
"column": 71
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\nA : Type u'\ninst✝ : Category.{v', u'} A\nF G : Cᵒᵖ ⥤ A\nX : C\nS : Sieve X\nhG : ⦃Y : C⦄ → (f : Y ⟶ X) → IsLimit (G.mapCone (Sieve.pullback f S).arrows.cocone.op)\nx : Presieve.FamilyOfElements (presheafHom F G) S.arrows\nhx : x.Compatible\nY₁ Y₂ : Over X\nφ : Y... | [
"C : Type u\ninst✝¹ : Category.{v, u} C\nA : Type u'\ninst✝ : Category.{v', u'} A\nF G : Cᵒᵖ ⥤ A\nX : C\nS : Sieve X\nhG : ⦃Y : C⦄ → (f : Y ⟶ X) → IsLimit (G.mapCone (Sieve.pullback f S).arrows.cocone.op)\nx : Presieve.FamilyOfElements (presheafHom F G) S.arrows\nhx : x.Compatible\nY₁ Y₂ : Over X\nφ : Y₂ ⟶ Y₁\nZ : ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Sites.SheafHom | {
"line": 198,
"column": 43
} | {
"line": 198,
"column": 54
} | {
"line": 198,
"column": 55
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\nA : Type u'\ninst✝ : Category.{v', u'} A\nF G : Cᵒᵖ ⥤ A\nX : C\nS : Sieve X\nhG : ⦃Y : C⦄ → (f : Y ⟶ X) → IsLimit (G.mapCone (Sieve.pullback f S).arrows.cocone.op)\nx : Presieve.FamilyOfElements (presheafHom F G) S.arrows\nhx : x.Compatible\nY : C\ng : Y ⟶ X\nhg ... | [
"C : Type u\ninst✝¹ : Category.{v, u} C\nA : Type u'\ninst✝ : Category.{v', u'} A\nF G : Cᵒᵖ ⥤ A\nX : C\nS : Sieve X\nhG : ⦃Y : C⦄ → (f : Y ⟶ X) → IsLimit (G.mapCone (Sieve.pullback f S).arrows.cocone.op)\nx : Presieve.FamilyOfElements (presheafHom F G) S.arrows\nhx : x.Compatible\nY : C\ng : Y ⟶ X\nhg : S.arrows g... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Sites.Point.OfIsCofiltered | {
"line": 77,
"column": 2
} | {
"line": 77,
"column": 69
} | {
"line": 77,
"column": 70
} | [
{
"pp": "C : Type u\ninst✝⁴ : Category.{v, u} C\ninst✝³ : LocallySmall.{w, v, u} C\nN : Type u'\ninst✝² : Category.{v', u'} N\np : N ⥤ C\ninst✝¹ : InitiallySmall N\ninst✝ : IsCofiltered N\nU : N\nX : C\nf₁ f₂ : p.obj U ⟶ X\nhf : fiberMk f₁ = fiberMk f₂\nV : Nᵒᵖ\ng : op U ⟶ V\nhg :\n (hom ((p.op ⋙ shrinkYoneda.... | [
"C : Type u\ninst✝⁴ : Category.{v, u} C\ninst✝³ : LocallySmall.{w, v, u} C\nN : Type u'\ninst✝² : Category.{v', u'} N\np : N ⥤ C\ninst✝¹ : InitiallySmall N\ninst✝ : IsCofiltered N\nU : N\nX : C\nf₁ f₂ : p.obj U ⟶ X\nhf : fiberMk f₁ = fiberMk f₂\nV : Nᵒᵖ\ng : op U ⟶ V\nhg :\n (hom ((p.op ⋙ shrinkYoneda.{w, v, u}.ob... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Sites.Point.Map | {
"line": 125,
"column": 2
} | {
"line": 125,
"column": 62
} | {
"line": 126,
"column": 4
} | [
{
"pp": "C : Type u\ninst✝⁵ : Category.{v, u} C\nD : Type u'\ninst✝⁴ : Category.{v', u'} D\nJ : GrothendieckTopology C\nΦ : J.Point\nF : C ⥤ D\nK : GrothendieckTopology D\ninst✝³ : F.IsCocontinuous J K\ninst✝² : LocallySmall.{w, v', u'} D\nA : Type u''\ninst✝¹ : Category.{v'', u''} A\ninst✝ : HasColimitsOfSize.... | [
"C : Type u\ninst✝⁵ : Category.{v, u} C\nD : Type u'\ninst✝⁴ : Category.{v', u'} D\nJ : GrothendieckTopology C\nΦ : J.Point\nF : C ⥤ D\nK : GrothendieckTopology D\ninst✝³ : F.IsCocontinuous J K\ninst✝² : LocallySmall.{w, v', u'} D\nA : Type u''\ninst✝¹ : Category.{v'', u''} A\ninst✝ : HasColimitsOfSize.{w, w, v'', ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Sites.Point.OfIsCofiltered | {
"line": 91,
"column": 2
} | {
"line": 91,
"column": 13
} | {
"line": 91,
"column": 14
} | [
{
"pp": "C : Type u\ninst✝³ : Category.{v, u} C\ninst✝² : LocallySmall.{w, v, u} C\nN : Type u'\ninst✝¹ : Category.{v', u'} N\np : N ⥤ C\ninst✝ : InitiallySmall N\nU V : N\ng : V ⟶ U\n⊢ fiberMk (p.map g) = fiberMk (𝟙 (p.obj U))",
"ppTerm": "?m.39",
"assigned": false,
"usedConstants": [],
"usedF... | [
"C : Type u\ninst✝³ : Category.{v, u} C\ninst✝² : LocallySmall.{w, v, u} C\nN : Type u'\ninst✝¹ : Category.{v', u'} N\np : N ⥤ C\ninst✝ : InitiallySmall N\nU V : N\ng : V ⟶ U\n⊢ fiberMk (p.map g) = fiberMk (𝟙 (p.obj U))"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Sites.Point.Presheaf | {
"line": 42,
"column": 29
} | {
"line": 42,
"column": 40
} | {
"line": 42,
"column": 41
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : LocallySmall.{w, v, u} C\nX U : C\nR : Sieve U\nhR : R ∈ ⊥ U\nx : (shrinkYoneda.{w, v, u}.flip.obj (op X)).obj U\n⊢ R = ⊤",
"ppTerm": "?m.43",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : LocallySmall.{w, v, u} C\nX U : C\nR : Sieve U\nhR : R ∈ ⊥ U\nx : (shrinkYoneda.{w, v, u}.flip.obj (op X)).obj U\n⊢ R = ⊤"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Sites.Point.Presheaf | {
"line": 99,
"column": 4
} | {
"line": 100,
"column": 11
} | {
"line": 100,
"column": 12
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : LocallySmall.{w, v, u} C\nX : C\nS : Sieve X\nhS :\n ∀ (Φ : (pointsBot C).FullSubcategory) (x : Φ.obj.fiber.obj X),\n ∃ Y g, ∃ (_ : S.arrows g), ∃ y, (ConcreteCategory.hom (Φ.obj.fiber.map g)) y = x\nY : C\na : Y ⟶ X\nha : S.arrows a\nb : X ⟶ Y\nhb :\... | [
"C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : LocallySmall.{w, v, u} C\nX : C\nS : Sieve X\nhS :\n ∀ (Φ : (pointsBot C).FullSubcategory) (x : Φ.obj.fiber.obj X),\n ∃ Y g, ∃ (_ : S.arrows g), ∃ y, (ConcreteCategory.hom (Φ.obj.fiber.map g)) y = x\nY : C\na : Y ⟶ X\nha : S.arrows a\nb : X ⟶ Y\nhb :\n (Concrete... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Sites.Point.OfIsCofiltered | {
"line": 118,
"column": 47
} | {
"line": 118,
"column": 58
} | {
"line": 118,
"column": 59
} | [
{
"pp": "C : Type u\ninst✝⁴ : Category.{v, u} C\ninst✝³ : LocallySmall.{w, v, u} C\nN : Type u'\ninst✝² : Category.{v', u'} N\np : N ⥤ C\ninst✝¹ : InitiallySmall N\nJ : GrothendieckTopology C\ninst✝ : IsCofiltered N\nX : C\nV U : N\nf : p.obj U ⟶ X\nφ₁ : ((functor p).obj V).fst ⟶ ⟨X, fiberMk f⟩.fst\nhφ₁ : (hom ... | [
"C : Type u\ninst✝⁴ : Category.{v, u} C\ninst✝³ : LocallySmall.{w, v, u} C\nN : Type u'\ninst✝² : Category.{v', u'} N\np : N ⥤ C\ninst✝¹ : InitiallySmall N\nJ : GrothendieckTopology C\ninst✝ : IsCofiltered N\nX : C\nV U : N\nf : p.obj U ⟶ X\nφ₁ : ((functor p).obj V).fst ⟶ ⟨X, fiberMk f⟩.fst\nhφ₁ : (hom ((fiber p).m... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Sites.RegularEpi | {
"line": 62,
"column": 8
} | {
"line": 62,
"column": 38
} | {
"line": 62,
"column": 39
} | [
{
"pp": "C : Type u_1\nD : Type u_2\ninst✝⁶ : Category.{u_3, u_1} C\ninst✝⁵ : Category.{u_4, u_2} D\nJ : GrothendieckTopology C\ninst✝⁴ : HasPullbacks D\ninst✝³ : HasPushouts D\ninst✝² : IsRegularEpiCategory D\nh : ∀ {F G : Sheaf J D} (f : F ⟶ G) [Epi f], ∃ I p i, Epi p ∧ Mono i ∧ p ≫ i = f.hom\ninst✝¹ : HasShe... | [
"C : Type u_1\nD : Type u_2\ninst✝⁶ : Category.{u_3, u_1} C\ninst✝⁵ : Category.{u_4, u_2} D\nJ : GrothendieckTopology C\ninst✝⁴ : HasPullbacks D\ninst✝³ : HasPushouts D\ninst✝² : IsRegularEpiCategory D\nh : ∀ {F G : Sheaf J D} (f : F ⟶ G) [Epi f], ∃ I p i, Epi p ∧ Mono i ∧ p ≫ i = f.hom\ninst✝¹ : HasSheafify J D\ni... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Subfunctor.Finite | {
"line": 134,
"column": 2
} | {
"line": 134,
"column": 43
} | {
"line": 134,
"column": 44
} | [
{
"pp": "C : Type u\ninst✝ : Category.{v, u} C\nF : Cᵒᵖ ⥤ Type w\nι : Type w'\nX : ι → Cᵒᵖ\nx : (i : ι) → F.obj (X i)\nh : PresheafIsGeneratedBy F x\nF' : Cᵒᵖ ⥤ Type w\nf : F ⟶ F'\n⊢ (Subfunctor.range f).IsGeneratedBy fun i ↦ (ConcreteCategory.hom (f.app (X i))) (x i)",
"ppTerm": "?m.33",
"assigned": tr... | [
"C : Type u\ninst✝ : Category.{v, u} C\nF : Cᵒᵖ ⥤ Type w\nι : Type w'\nX : ι → Cᵒᵖ\nx : (i : ι) → F.obj (X i)\nh : PresheafIsGeneratedBy F x\nF' : Cᵒᵖ ⥤ Type w\nf : F ⟶ F'\n⊢ (⊤.image f).IsGeneratedBy fun i ↦ (ConcreteCategory.hom (f.app (X i))) (x i)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Subfunctor.Finite | {
"line": 137,
"column": 2
} | {
"line": 137,
"column": 46
} | {
"line": 137,
"column": 47
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\nF : Cᵒᵖ ⥤ Type w\nι : Type w'\nX : ι → Cᵒᵖ\nx : (i : ι) → F.obj (X i)\nh : PresheafIsGeneratedBy F x\nF' : Cᵒᵖ ⥤ Type w\nf : F ⟶ F'\ninst✝ : Epi f\n⊢ PresheafIsGeneratedBy F' fun i ↦ (ConcreteCategory.hom (f.app (X i))) (x i)",
"ppTerm": "?m.32",
"assigne... | [
"C : Type u\ninst✝¹ : Category.{v, u} C\nF : Cᵒᵖ ⥤ Type w\nι : Type w'\nX : ι → Cᵒᵖ\nx : (i : ι) → F.obj (X i)\nh : PresheafIsGeneratedBy F x\nF' : Cᵒᵖ ⥤ Type w\nf : F ⟶ F'\ninst✝ : Epi f\n⊢ PresheafIsGeneratedBy F' fun i ↦ (ConcreteCategory.hom (f.app (X i))) (x i)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Subfunctor.Subobject | {
"line": 73,
"column": 6
} | {
"line": 73,
"column": 17
} | {
"line": 73,
"column": 18
} | [
{
"pp": "case mp\nC : Type u\ninst✝ : Category.{v, u} C\nF : C ⥤ Type w\nA B : Subfunctor F\nh :\n { toFun := fun A ↦ Subobject.mk A.ι, invFun := fun X ↦ range X.arrow, left_inv := ⋯, right_inv := ⋯ } A ≤\n { toFun := fun A ↦ Subobject.mk A.ι, invFun := fun X ↦ range X.arrow, left_inv := ⋯, right_inv := ⋯ }... | [
"case mp\nC : Type u\ninst✝ : Category.{v, u} C\nF : C ⥤ Type w\nA B : Subfunctor F\nh :\n { toFun := fun A ↦ Subobject.mk A.ι, invFun := fun X ↦ range X.arrow, left_inv := ⋯, right_inv := ⋯ } A ≤\n { toFun := fun A ↦ Subobject.mk A.ι, invFun := fun X ↦ range X.arrow, left_inv := ⋯, right_inv := ⋯ } B\nthis : r... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Sites.Descent.DescentData | {
"line": 457,
"column": 8
} | {
"line": 459,
"column": 40
} | {
"line": 459,
"column": 41
} | [
{
"pp": "C : Type u\ninst✝ : Category.{v, u} C\nF : Pseudofunctor (LocallyDiscrete Cᵒᵖ) Cat\nι : Type t\nS : C\nX : ι → C\nf : (i : ι) → X i ⟶ S\nM N : ↑(F.obj { as := op S })\ng : (F.toDescentData f).obj M ⟶ (F.toDescentData f).obj N\ni₁ i₂ : ι\nZ : Over S\nf₁ : Z ⟶ (fun i ↦ Over.mk (f i)) i₁\nf₂ : Z ⟶ (fun i ... | [
"C : Type u\ninst✝ : Category.{v, u} C\nF : Pseudofunctor (LocallyDiscrete Cᵒᵖ) Cat\nι : Type t\nS : C\nX : ι → C\nf : (i : ι) → X i ⟶ S\nM N : ↑(F.obj { as := op S })\ng : (F.toDescentData f).obj M ⟶ (F.toDescentData f).obj N\ni₁ i₂ : ι\nZ : Over S\nf₁ : Z ⟶ (fun i ↦ Over.mk (f i)) i₁\nf₂ : Z ⟶ (fun i ↦ Over.mk (f... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Sites.Descent.DescentData | {
"line": 573,
"column": 45
} | {
"line": 573,
"column": 56
} | {
"line": 573,
"column": 57
} | [
{
"pp": "C : Type u\ninst✝ : Category.{v, u} C\nF : Pseudofunctor (LocallyDiscrete Cᵒᵖ) Cat\nS : C\nR : Sieve S\nM N : ↑(F.obj { as := op S })\nX : C\ng : X ⟶ S\nf : Over.mk g ⟶ Over.mk (𝟙 S)\nhf : R.arrows (Over.Hom.left f)\n⊢ Over.Hom.left f = Over.Hom.left (Over.homMk g ⋯)",
"ppTerm": "?m.302",
"ass... | [
"C : Type u\ninst✝ : Category.{v, u} C\nF : Pseudofunctor (LocallyDiscrete Cᵒᵖ) Cat\nS : C\nR : Sieve S\nM N : ↑(F.obj { as := op S })\nX : C\ng : X ⟶ S\nf : Over.mk g ⟶ Over.mk (𝟙 S)\nhf : R.arrows (Over.Hom.left f)\n⊢ Over.Hom.left f = g"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Sites.Descent.DescentData | {
"line": 591,
"column": 67
} | {
"line": 591,
"column": 78
} | {
"line": 591,
"column": 79
} | [
{
"pp": "C : Type u\ninst✝ : Category.{v, u} C\nF : Pseudofunctor (LocallyDiscrete Cᵒᵖ) Cat\nS : C\nR : Sieve S\nS₀ : C\nM N : ↑(F.obj { as := op S₀ })\na : S ⟶ S₀\nh :\n Presieve.IsSheafFor (F.presheafHom M N)\n (Sieve.pullback (Over.isoMk (Iso.refl ((Over.map a).obj (Over.mk (𝟙 S))).left) ⋯).inv\n ... | [
"C : Type u\ninst✝ : Category.{v, u} C\nF : Pseudofunctor (LocallyDiscrete Cᵒᵖ) Cat\nS : C\nR : Sieve S\nS₀ : C\nM N : ↑(F.obj { as := op S₀ })\na : S ⟶ S₀\nh :\n Presieve.IsSheafFor (F.presheafHom M N)\n (Sieve.pullback (Over.isoMk (Iso.refl ((Over.map a).obj (Over.mk (𝟙 S))).left) ⋯).inv\n (Sieve.func... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Sites.Descent.DescentData | {
"line": 598,
"column": 4
} | {
"line": 598,
"column": 21
} | {
"line": 598,
"column": 22
} | [
{
"pp": "case refine_2\nC : Type u\ninst✝ : Category.{v, u} C\nF : Pseudofunctor (LocallyDiscrete Cᵒᵖ) Cat\nS : C\nR : Sieve S\nS₀ : C\nM N : ↑(F.obj { as := op S₀ })\na : S ⟶ S₀\nh✝ :\n Presieve.IsSheafFor (F.presheafHom M N)\n (Sieve.pullback (Over.isoMk (Iso.refl ((Over.map a).obj (Over.mk (𝟙 S))).left)... | [
"case refine_2\nC : Type u\ninst✝ : Category.{v, u} C\nF : Pseudofunctor (LocallyDiscrete Cᵒᵖ) Cat\nS : C\nR : Sieve S\nS₀ : C\nM N : ↑(F.obj { as := op S₀ })\na : S ⟶ S₀\nh✝ :\n Presieve.IsSheafFor (F.presheafHom M N)\n (Sieve.pullback (Over.isoMk (Iso.refl ((Over.map a).obj (Over.mk (𝟙 S))).left) ⋯).inv\n ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Sites.Descent.DescentData | {
"line": 608,
"column": 2
} | {
"line": 608,
"column": 13
} | {
"line": 608,
"column": 14
} | [
{
"pp": "C : Type u\ninst✝ : Category.{v, u} C\nF : Pseudofunctor (LocallyDiscrete Cᵒᵖ) Cat\nS₀ : C\nM N : ↑(F.obj { as := op S₀ })\nS : C\na : S ⟶ S₀\nR : Sieve (Over.mk a)\nhF :\n ∀ ⦃S₀_1 : C⦄ (M N : ↑(F.obj { as := op S₀_1 })) (a_1 : (Over.mk a).left ⟶ S₀_1),\n Presieve.IsSheafFor (F.presheafHom M N)\n ... | [
"C : Type u\ninst✝ : Category.{v, u} C\nF : Pseudofunctor (LocallyDiscrete Cᵒᵖ) Cat\nS₀ : C\nM N : ↑(F.obj { as := op S₀ })\nS : C\na : S ⟶ S₀\nR : Sieve (Over.mk a)\nhF :\n ∀ ⦃S₀_1 : C⦄ (M N : ↑(F.obj { as := op S₀_1 })) (a_1 : (Over.mk a).left ⟶ S₀_1),\n Presieve.IsSheafFor (F.presheafHom M N)\n ((Sieve.... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Subobject.Classifier.Defs | {
"line": 542,
"column": 29
} | {
"line": 542,
"column": 40
} | {
"line": 542,
"column": 41
} | [
{
"pp": "C✝ : Type u\ninst✝² : Category.{v, u} C✝\nC : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : HasPullbacks C\nΩ : C\nh : SubobjectRepresentableBy Ω\nX : C\nπ' : X ⟶ underlying.obj h.Ω₀\ns : PullbackCone (π' ≫ h.Ω₀.arrow) h.Ω₀.arrow\nm : s.pt ⟶ X\nhm : m ≫ 𝟙 X = s.fst\nx✝ : m ≫ π' = s.snd\n⊢ m = s.fst",
... | [
"C✝ : Type u\ninst✝² : Category.{v, u} C✝\nC : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : HasPullbacks C\nΩ : C\nh : SubobjectRepresentableBy Ω\nX : C\nπ' : X ⟶ underlying.obj h.Ω₀\ns : PullbackCone (π' ≫ h.Ω₀.arrow) h.Ω₀.arrow\nm : s.pt ⟶ X\nhm : m ≫ 𝟙 X = s.fst\nx✝ : m ≫ π' = s.snd\n⊢ m = s.fst"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Subobject.Classifier.Defs | {
"line": 695,
"column": 4
} | {
"line": 695,
"column": 15
} | {
"line": 695,
"column": 16
} | [
{
"pp": "C : Type u\ninst✝ : Category.{v, u} C\n𝒞 : Classifier C\nΩ₀ Ω : C\neΩ : 𝒞.Ω ≅ Ω\neΩ₀ : 𝒞.Ω₀ ≅ Ω₀\nfrom' : (C_1 : C) → C_1 ⟶ Ω₀\nt : Ω₀ ⟶ Ω\nht : t = eΩ₀.inv ≫ 𝒞.truth ≫ eΩ.hom\nF G : C\nm : F ⟶ G\nx✝ : Mono m\nχ₀' : F ⟶ Ω₀\nχ' : G ⟶ Ω\nhχ' : IsPullback m χ₀' χ' t\nthis : χ' ≫ eΩ.inv = 𝒞.χ m\n⊢ χ' ... | [
"C : Type u\ninst✝ : Category.{v, u} C\n𝒞 : Classifier C\nΩ₀ Ω : C\neΩ : 𝒞.Ω ≅ Ω\neΩ₀ : 𝒞.Ω₀ ≅ Ω₀\nfrom' : (C_1 : C) → C_1 ⟶ Ω₀\nt : Ω₀ ⟶ Ω\nht : t = eΩ₀.inv ≫ 𝒞.truth ≫ eΩ.hom\nF G : C\nm : F ⟶ G\nx✝ : Mono m\nχ₀' : F ⟶ Ω₀\nχ' : G ⟶ Ω\nhχ' : IsPullback m χ₀' χ' t\nthis : χ' ≫ eΩ.inv = 𝒞.χ m\n⊢ χ' = 𝒞.χ m ≫ e... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Subobject.Classifier.Defs | {
"line": 724,
"column": 4
} | {
"line": 724,
"column": 15
} | {
"line": 724,
"column": 16
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\nD : Type u_1\ninst✝ : Category.{v_1, u_1} D\n𝒞₁ : Classifier C\ne : C ≌ D\nF G : D\nm : F ⟶ G\nx✝ : Mono m\nχ₀' : F ⟶ e.functor.obj 𝒞₁.Ω₀\nχ' : G ⟶ e.functor.obj 𝒞₁.Ω\nhχ' : IsPullback m χ₀' χ' (e.functor.map 𝒞₁.truth)\nthis : e.inverse.map χ' ≫ e.unitInv.app... | [
"C : Type u\ninst✝¹ : Category.{v, u} C\nD : Type u_1\ninst✝ : Category.{v_1, u_1} D\n𝒞₁ : Classifier C\ne : C ≌ D\nF G : D\nm : F ⟶ G\nx✝ : Mono m\nχ₀' : F ⟶ e.functor.obj 𝒞₁.Ω₀\nχ' : G ⟶ e.functor.obj 𝒞₁.Ω\nhχ' : IsPullback m χ₀' χ' (e.functor.map 𝒞₁.truth)\nthis : e.inverse.map χ' ≫ e.unitInv.app 𝒞₁.Ω = 𝒞₁... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Topos.Sheaf | {
"line": 120,
"column": 4
} | {
"line": 120,
"column": 15
} | {
"line": 120,
"column": 16
} | [
{
"pp": "case mpr\nC : Type u\ninst✝ : Category.{v, u} C\nF G : Cᵒᵖ ⥤ Type (max u v)\nm : F ⟶ G\nχ' : G ⟶ Functor.sieves C\nX : Cᵒᵖ\nx : G.obj X\nh₁ : ∀ (x : Cᵒᵖ), m.app x ≫ χ'.app x = Types.isTerminalPUnit.from (F.obj x) ≫ ↾fun x_1 ↦ ⊤\nh₂ : ∀ (x : Cᵒᵖ) (x₁ y₁ : F.obj x), (ConcreteCategory.hom (m.app x)) x₁ = ... | [
"case mpr\nC : Type u\ninst✝ : Category.{v, u} C\nF G : Cᵒᵖ ⥤ Type (max u v)\nm : F ⟶ G\nχ' : G ⟶ Functor.sieves C\nX : Cᵒᵖ\nx : G.obj X\nh₁ : ∀ (x : Cᵒᵖ), m.app x ≫ χ'.app x = Types.isTerminalPUnit.from (F.obj x) ≫ ↾fun x_1 ↦ ⊤\nh₂ : ∀ (x : Cᵒᵖ) (x₁ y₁ : F.obj x), (ConcreteCategory.hom (m.app x)) x₁ = (ConcreteCat... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Topos.Sheaf | {
"line": 145,
"column": 2
} | {
"line": 155,
"column": 38
} | {
"line": 157,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\nJ : GrothendieckTopology C\nF G : Cᵒᵖ ⥤ Type (max u v)\nm : F ⟶ G\ninst✝ : Mono m\nhF : Presieve.IsSheaf J F\nhG : Presieve.IsSeparated J G\nX : Cᵒᵖ\nx : G.obj X\n⊢ J.IsClosed ((ConcreteCategory.hom ((χ m).app X)) x)",
"ppTerm": "?m.34",
"assigned": true,... | [] | intro Y f hf
simp only [Presheaf.χ_app, Opposite.op_unop] at hf ⊢
choose a ha using fun Z (g : Z ⟶ Y) (hg : (Sieve.pullback f ((χ m).app X x)).arrows g) => hg
refine ⟨(hF _ hf).amalgamate a ?_, ?_⟩
· introv Y₁ h
apply (mono_iff_injective (m.app (.op Z))).mp inferInstance
simp_rw [NatTrans.naturality_app... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Topos.Sheaf | {
"line": 145,
"column": 2
} | {
"line": 155,
"column": 38
} | {
"line": 157,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\nJ : GrothendieckTopology C\nF G : Cᵒᵖ ⥤ Type (max u v)\nm : F ⟶ G\ninst✝ : Mono m\nhF : Presieve.IsSheaf J F\nhG : Presieve.IsSeparated J G\nX : Cᵒᵖ\nx : G.obj X\n⊢ J.IsClosed ((ConcreteCategory.hom ((χ m).app X)) x)",
"ppTerm": "?m.34",
"assigned": true,... | [] | intro Y f hf
simp only [Presheaf.χ_app, Opposite.op_unop] at hf ⊢
choose a ha using fun Z (g : Z ⟶ Y) (hg : (Sieve.pullback f ((χ m).app X x)).arrows g) => hg
refine ⟨(hF _ hf).amalgamate a ?_, ?_⟩
· introv Y₁ h
apply (mono_iff_injective (m.app (.op Z))).mp inferInstance
simp_rw [NatTrans.naturality_app... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.