module string | startPos dict | endPos dict | nextStartPos dict | goals list | goalsAfter list | ppTac string | elaborator string | kind string |
|---|---|---|---|---|---|---|---|---|
Mathlib.CategoryTheory.Sites.Hypercover.One | {
"line": 724,
"column": 2
} | {
"line": 724,
"column": 13
} | {
"line": 724,
"column": 14
} | [
{
"pp": "C : Type u\ninst✝ : Category.{v, u} C\nS : C\nE F : PreOneHypercover S\ne : E ≅ F\ni : E.I₀\nhs : (e.hom ≫ e.inv).s₀ = (𝟙 E).s₀\nhh : ∀ (i : E.I₀), (e.hom ≫ e.inv).h₀ i = (𝟙 E).h₀ i ≫ eqToHom ⋯\nh✝ :\n ∃ (hs₁ : ∀ (i j : E.I₀) (k : E.I₁ i j), (e.hom ≫ e.inv).s₁ k = (congrIndexOneOfEq ⋯ ⋯) ((𝟙 E).s₁ ... | [
"C : Type u\ninst✝ : Category.{v, u} C\nS : C\nE F : PreOneHypercover S\ne : E ≅ F\ni : E.I₀\nhs : (e.hom ≫ e.inv).s₀ = (𝟙 E).s₀\nhh : ∀ (i : E.I₀), (e.hom ≫ e.inv).h₀ i = (𝟙 E).h₀ i ≫ eqToHom ⋯\nh✝ :\n ∃ (hs₁ : ∀ (i j : E.I₀) (k : E.I₁ i j), (e.hom ≫ e.inv).s₁ k = (congrIndexOneOfEq ⋯ ⋯) ((𝟙 E).s₁ k)),\n ∀ ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Sites.Hypercover.One | {
"line": 730,
"column": 2
} | {
"line": 730,
"column": 13
} | {
"line": 730,
"column": 14
} | [
{
"pp": "C : Type u\ninst✝ : Category.{v, u} C\nS : C\nE F : PreOneHypercover S\ne : E ≅ F\ni : F.I₀\nhs : (e.inv ≫ e.hom).s₀ = (𝟙 F).s₀\nhh : ∀ (i : F.I₀), (e.inv ≫ e.hom).h₀ i = (𝟙 F).h₀ i ≫ eqToHom ⋯\nh✝ :\n ∃ (hs₁ : ∀ (i j : F.I₀) (k : F.I₁ i j), (e.inv ≫ e.hom).s₁ k = (congrIndexOneOfEq ⋯ ⋯) ((𝟙 F).s₁ ... | [
"C : Type u\ninst✝ : Category.{v, u} C\nS : C\nE F : PreOneHypercover S\ne : E ≅ F\ni : F.I₀\nhs : (e.inv ≫ e.hom).s₀ = (𝟙 F).s₀\nhh : ∀ (i : F.I₀), (e.inv ≫ e.hom).h₀ i = (𝟙 F).h₀ i ≫ eqToHom ⋯\nh✝ :\n ∃ (hs₁ : ∀ (i j : F.I₀) (k : F.I₁ i j), (e.inv ≫ e.hom).s₁ k = (congrIndexOneOfEq ⋯ ⋯) ((𝟙 F).s₁ k)),\n ∀ ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Sites.Hypercover.One | {
"line": 739,
"column": 2
} | {
"line": 739,
"column": 13
} | {
"line": 739,
"column": 14
} | [
{
"pp": "C : Type u\ninst✝ : Category.{v, u} C\nS : C\nE F : PreOneHypercover S\ne : E ≅ F\ni j : E.I₀\nk : E.I₁ i j\nhs : (e.hom ≫ e.inv).s₀ = (𝟙 E).s₀\nw✝¹ : ∀ (i : E.I₀), (e.hom ≫ e.inv).h₀ i = (𝟙 E).h₀ i ≫ eqToHom ⋯\nw✝ : ∀ (i j : E.I₀) (k : E.I₁ i j), (e.hom ≫ e.inv).s₁ k = (congrIndexOneOfEq ⋯ ⋯) ((𝟙 E... | [
"C : Type u\ninst✝ : Category.{v, u} C\nS : C\nE F : PreOneHypercover S\ne : E ≅ F\ni j : E.I₀\nk : E.I₁ i j\nhs : (e.hom ≫ e.inv).s₀ = (𝟙 E).s₀\nw✝¹ : ∀ (i : E.I₀), (e.hom ≫ e.inv).h₀ i = (𝟙 E).h₀ i ≫ eqToHom ⋯\nw✝ : ∀ (i j : E.I₀) (k : E.I₁ i j), (e.hom ≫ e.inv).s₁ k = (congrIndexOneOfEq ⋯ ⋯) ((𝟙 E).s₁ k)\nhh ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Sites.Hypercover.One | {
"line": 748,
"column": 2
} | {
"line": 748,
"column": 13
} | {
"line": 748,
"column": 14
} | [
{
"pp": "C : Type u\ninst✝ : Category.{v, u} C\nS : C\nE F : PreOneHypercover S\ne : E ≅ F\ni j : F.I₀\nk : F.I₁ i j\nhs : (e.inv ≫ e.hom).s₀ = (𝟙 F).s₀\nw✝¹ : ∀ (i : F.I₀), (e.inv ≫ e.hom).h₀ i = (𝟙 F).h₀ i ≫ eqToHom ⋯\nw✝ : ∀ (i j : F.I₀) (k : F.I₁ i j), (e.inv ≫ e.hom).s₁ k = (congrIndexOneOfEq ⋯ ⋯) ((𝟙 F... | [
"C : Type u\ninst✝ : Category.{v, u} C\nS : C\nE F : PreOneHypercover S\ne : E ≅ F\ni j : F.I₀\nk : F.I₁ i j\nhs : (e.inv ≫ e.hom).s₀ = (𝟙 F).s₀\nw✝¹ : ∀ (i : F.I₀), (e.inv ≫ e.hom).h₀ i = (𝟙 F).h₀ i ≫ eqToHom ⋯\nw✝ : ∀ (i j : F.I₀) (k : F.I₁ i j), (e.inv ≫ e.hom).s₁ k = (congrIndexOneOfEq ⋯ ⋯) ((𝟙 F).s₁ k)\nhh ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Sites.Over | {
"line": 62,
"column": 30
} | {
"line": 62,
"column": 41
} | {
"line": 62,
"column": 42
} | [
{
"pp": "C : Type u\ninst✝ : Category.{v, u} C\nX : C\nY : Over X\na✝ b✝ : Presieve Y\nh :\n { toFun := fun S ↦ map (Over.forget X) S, invFun := fun S' ↦ functorPullback (Over.forget X) S', left_inv := ⋯,\n right_inv := ⋯ }\n a✝ ≤\n { toFun := fun S ↦ map (Over.forget X) S, invFun := fun S' ↦ fu... | [
"C : Type u\ninst✝ : Category.{v, u} C\nX : C\nY : Over X\na✝ b✝ : Presieve Y\nh :\n { toFun := fun S ↦ map (Over.forget X) S, invFun := fun S' ↦ functorPullback (Over.forget X) S', left_inv := ⋯,\n right_inv := ⋯ }\n a✝ ≤\n { toFun := fun S ↦ map (Over.forget X) S, invFun := fun S' ↦ functorPullbac... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Sites.Over | {
"line": 95,
"column": 4
} | {
"line": 95,
"column": 24
} | {
"line": 96,
"column": 4
} | [
{
"pp": "C : Type u\ninst✝ : Category.{v, u} C\nX : C\nY : Over X\n⊢ ∀ {a b : Sieve Y},\n { toFun := functorPushforward (Over.forget X), invFun := functorPullback (Over.forget X), left_inv := ⋯,\n right_inv := ⋯ }\n a ≤\n { toFun := functorPushforward (Over.forget X), invFun := fun... | [
"C : Type u\ninst✝ : Category.{v, u} C\nX : C\nY : Over X\n⊢ ∀ {a b : Sieve Y}, functorPushforward (Over.forget X) a ≤ functorPushforward (Over.forget X) b ↔ a ≤ b"
] | rw [Equiv.coe_fn_mk] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.CategoryTheory.Sites.Over | {
"line": 96,
"column": 22
} | {
"line": 96,
"column": 33
} | {
"line": 96,
"column": 34
} | [
{
"pp": "C : Type u\ninst✝ : Category.{v, u} C\nX : C\nY : Over X\na✝ b✝ : Sieve Y\nh : functorPushforward (Over.forget X) a✝ ≤ functorPushforward (Over.forget X) b✝\n⊢ a✝ ≤ b✝",
"ppTerm": "?m.50",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"C : Type u\ninst✝ : Category.{v, u} C\nX : C\nY : Over X\na✝ b✝ : Sieve Y\nh : functorPushforward (Over.forget X) a✝ ≤ functorPushforward (Over.forget X) b✝\n⊢ a✝ ≤ b✝"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Sites.Over | {
"line": 149,
"column": 29
} | {
"line": 149,
"column": 40
} | {
"line": 149,
"column": 41
} | [
{
"pp": "C : Type u\ninst✝ : Category.{v, u} C\nX : C\nY : Over X\nE : PreOneHypercover Y\ni₁ i₂ : E.I₀\nW : Over X\np₁ : W ⟶ E.X i₁\np₂ : W ⟶ E.X i₂\nY✝ : C\nf✝ : Y✝ ⟶ W.left\nk : (E.map (Over.forget X)).I₁ i₁ i₂\nb : Y✝ ⟶ (E.map (Over.forget X)).Y k\nhb₁ : f✝ ≫ Over.Hom.left p₁ = b ≫ (E.map (Over.forget X)).p... | [
"C : Type u\ninst✝ : Category.{v, u} C\nX : C\nY : Over X\nE : PreOneHypercover Y\ni₁ i₂ : E.I₀\nW : Over X\np₁ : W ⟶ E.X i₁\np₂ : W ⟶ E.X i₂\nY✝ : C\nf✝ : Y✝ ⟶ W.left\nk : (E.map (Over.forget X)).I₁ i₁ i₂\nb : Y✝ ⟶ (E.map (Over.forget X)).Y k\nhb₁ : f✝ ≫ Over.Hom.left p₁ = b ≫ (E.map (Over.forget X)).p₁ k\nhb₂ : f... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Sites.Over | {
"line": 185,
"column": 27
} | {
"line": 185,
"column": 38
} | {
"line": 185,
"column": 39
} | [
{
"pp": "C : Type u\ninst✝ : Category.{v, u} C\nX Y : C\nf : X ⟶ Y\nZ : Over X\nS : Sieve Z.left\nW : Over Y\ng : W ⟶ (Over.map f).obj Z\nhg : ((overEquiv ((Over.map f).obj Z)).symm S).arrows g\n⊢ 𝟙 W.left ≫ ((Over.map f).obj (Over.mk (Over.Hom.left g ≫ Z.hom))).hom = W.hom",
"ppTerm": "?m.146",
"assig... | [
"C : Type u\ninst✝ : Category.{v, u} C\nX Y : C\nf : X ⟶ Y\nZ : Over X\nS : Sieve Z.left\nW : Over Y\ng : W ⟶ (Over.map f).obj Z\nhg : ((overEquiv ((Over.map f).obj Z)).symm S).arrows g\n⊢ Over.Hom.left g ≫ Z.hom ≫ f = W.hom"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Sites.Over | {
"line": 252,
"column": 2
} | {
"line": 252,
"column": 60
} | {
"line": 252,
"column": 61
} | [
{
"pp": "C : Type u\ninst✝ : Category.{v, u} C\nJ : GrothendieckTopology C\nX : C\nY : Over X\nS : Sieve Y.left\nhS : S ∈ J Y.left\n⊢ (Sieve.overEquiv Y).symm S ∈ (J.over X) Y",
"ppTerm": "?m.30",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"CategoryTheory.Over",
"OrderIso.apply... | [
"C : Type u\ninst✝ : Category.{v, u} C\nJ : GrothendieckTopology C\nX : C\nY : Over X\nS : Sieve Y.left\nhS : S ∈ J Y.left\n⊢ S ∈ J Y.left"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Category.ModuleCat.Sheaf.Generators | {
"line": 186,
"column": 2
} | {
"line": 186,
"column": 34
} | {
"line": 187,
"column": 2
} | [
{
"pp": "C : Type u'\ninst✝⁷ : Category.{v', u'} C\nJ : GrothendieckTopology C\nR : Sheaf J RingCat\ninst✝⁶ : HasWeakSheafify J AddCommGrpCat\ninst✝⁵ : J.WEqualsLocallyBijective AddCommGrpCat\nM✝ N P : SheafOfModules R\nC' : Type u₁\ninst✝⁴ : Category.{v₁, u₁} C'\nJ' : GrothendieckTopology C'\nS : Sheaf J' Ring... | [
"C : Type u'\ninst✝⁷ : Category.{v', u'} C\nJ : GrothendieckTopology C\nR : Sheaf J RingCat\ninst✝⁶ : HasWeakSheafify J AddCommGrpCat\ninst✝⁵ : J.WEqualsLocallyBijective AddCommGrpCat\nM✝ N P : SheafOfModules R\nC' : Type u₁\ninst✝⁴ : Category.{v₁, u₁} C'\nJ' : GrothendieckTopology C'\nS : Sheaf J' RingCat\ninst✝³ ... | rw [GeneratingSections.map_π_eq] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.CategoryTheory.Sites.Hypercover.One | {
"line": 879,
"column": 6
} | {
"line": 879,
"column": 17
} | {
"line": 879,
"column": 18
} | [
{
"pp": "case h₀\nC : Type u\ninst✝³ : Category.{v, u} C\nA : Type u_1\ninst✝² : Category.{v_1, u_1} A\nS : C\nE : PreOneHypercover S\nF : PreOneHypercover S\nG✝ : PreOneHypercover S\ninst✝¹ : ∀ (i : E.I₀) (j : F.I₀), HasPullback (E.f i) (F.f j)\ninst✝ : ∀ (i j : E.I₀) (k : E.I₁ i j) (a b : F.I₀) (l : F.I₁ a b)... | [
"case h₀\nC : Type u\ninst✝³ : Category.{v, u} C\nA : Type u_1\ninst✝² : Category.{v_1, u_1} A\nS : C\nE : PreOneHypercover S\nF : PreOneHypercover S\nG✝ : PreOneHypercover S\ninst✝¹ : ∀ (i : E.I₀) (j : F.I₀), HasPullback (E.f i) (F.f j)\ninst✝ : ∀ (i j : E.I₀) (k : E.I₁ i j) (a b : F.I₀) (l : F.I₁ a b), HasPullbac... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Sites.Hypercover.One | {
"line": 880,
"column": 6
} | {
"line": 880,
"column": 17
} | {
"line": 880,
"column": 18
} | [
{
"pp": "case h₁\nC : Type u\ninst✝³ : Category.{v, u} C\nA : Type u_1\ninst✝² : Category.{v_1, u_1} A\nS : C\nE : PreOneHypercover S\nF : PreOneHypercover S\nG✝ : PreOneHypercover S\ninst✝¹ : ∀ (i : E.I₀) (j : F.I₀), HasPullback (E.f i) (F.f j)\ninst✝ : ∀ (i j : E.I₀) (k : E.I₁ i j) (a b : F.I₀) (l : F.I₁ a b)... | [
"case h₁\nC : Type u\ninst✝³ : Category.{v, u} C\nA : Type u_1\ninst✝² : Category.{v_1, u_1} A\nS : C\nE : PreOneHypercover S\nF : PreOneHypercover S\nG✝ : PreOneHypercover S\ninst✝¹ : ∀ (i : E.I₀) (j : F.I₀), HasPullback (E.f i) (F.f j)\ninst✝ : ∀ (i j : E.I₀) (k : E.I₁ i j) (a b : F.I₀) (l : F.I₁ a b), HasPullbac... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Sites.Hypercover.One | {
"line": 883,
"column": 6
} | {
"line": 883,
"column": 17
} | {
"line": 883,
"column": 18
} | [
{
"pp": "case h₀\nC : Type u\ninst✝³ : Category.{v, u} C\nA : Type u_1\ninst✝² : Category.{v_1, u_1} A\nS : C\nE : PreOneHypercover S\nF : PreOneHypercover S\nG✝ : PreOneHypercover S\ninst✝¹ : ∀ (i : E.I₀) (j : F.I₀), HasPullback (E.f i) (F.f j)\ninst✝ : ∀ (i j : E.I₀) (k : E.I₁ i j) (a b : F.I₀) (l : F.I₁ a b)... | [
"case h₀\nC : Type u\ninst✝³ : Category.{v, u} C\nA : Type u_1\ninst✝² : Category.{v_1, u_1} A\nS : C\nE : PreOneHypercover S\nF : PreOneHypercover S\nG✝ : PreOneHypercover S\ninst✝¹ : ∀ (i : E.I₀) (j : F.I₀), HasPullback (E.f i) (F.f j)\ninst✝ : ∀ (i j : E.I₀) (k : E.I₁ i j) (a b : F.I₀) (l : F.I₁ a b), HasPullbac... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Sites.Hypercover.One | {
"line": 884,
"column": 6
} | {
"line": 884,
"column": 17
} | {
"line": 884,
"column": 18
} | [
{
"pp": "case h₁\nC : Type u\ninst✝³ : Category.{v, u} C\nA : Type u_1\ninst✝² : Category.{v_1, u_1} A\nS : C\nE : PreOneHypercover S\nF : PreOneHypercover S\nG✝ : PreOneHypercover S\ninst✝¹ : ∀ (i : E.I₀) (j : F.I₀), HasPullback (E.f i) (F.f j)\ninst✝ : ∀ (i j : E.I₀) (k : E.I₁ i j) (a b : F.I₀) (l : F.I₁ a b)... | [
"case h₁\nC : Type u\ninst✝³ : Category.{v, u} C\nA : Type u_1\ninst✝² : Category.{v_1, u_1} A\nS : C\nE : PreOneHypercover S\nF : PreOneHypercover S\nG✝ : PreOneHypercover S\ninst✝¹ : ∀ (i : E.I₀) (j : F.I₀), HasPullback (E.f i) (F.f j)\ninst✝ : ∀ (i j : E.I₀) (k : E.I₁ i j) (a b : F.I₀) (l : F.I₁ a b), HasPullbac... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Category.ModuleCat.Sheaf.LocallyFree | {
"line": 89,
"column": 2
} | {
"line": 90,
"column": 16
} | {
"line": 92,
"column": 0
} | [
{
"pp": "C : Type u₁\ninst✝² : Category.{v₁, u₁} C\nJ : GrothendieckTopology C\nR : Sheaf J RingCat\ninst✝¹ : HasWeakSheafify J AddCommGrpCat\ninst✝ : J.WEqualsLocallyBijective AddCommGrpCat\nI : Type u\n⊢ IsIso (free.generatingSections I).π",
"ppTerm": "?m.31",
"assigned": true,
"usedConstants": [
... | [] | rw [free.generatingSections_π]
infer_instance | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Category.ModuleCat.Sheaf.LocallyFree | {
"line": 89,
"column": 2
} | {
"line": 90,
"column": 16
} | {
"line": 92,
"column": 0
} | [
{
"pp": "C : Type u₁\ninst✝² : Category.{v₁, u₁} C\nJ : GrothendieckTopology C\nR : Sheaf J RingCat\ninst✝¹ : HasWeakSheafify J AddCommGrpCat\ninst✝ : J.WEqualsLocallyBijective AddCommGrpCat\nI : Type u\n⊢ IsIso (free.generatingSections I).π",
"ppTerm": "?m.31",
"assigned": true,
"usedConstants": [
... | [] | rw [free.generatingSections_π]
infer_instance | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Sites.Over | {
"line": 564,
"column": 4
} | {
"line": 565,
"column": 82
} | {
"line": 567,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\nJ : GrothendieckTopology C\nA : Type u'\ninst✝ : Category.{v', u'} A\nX : C\nf : Over X\n⊢ ∀ {X_1 : Over f.left} {S : Sieve X_1},\n Sieve.functorPushforward f.iteratedSliceEquiv.inverse S ∈\n ((J.over X).over f) (f.iteratedSliceEquiv.inverse.obj X_1) ↔\... | [] | simp [GrothendieckTopology.mem_over_iff, Sieve.overEquiv,
← Over.iteratedSliceBackward_forget_forget f, Sieve.functorPushforward_comp] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.CategoryTheory.Sites.Over | {
"line": 564,
"column": 4
} | {
"line": 565,
"column": 82
} | {
"line": 567,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\nJ : GrothendieckTopology C\nA : Type u'\ninst✝ : Category.{v', u'} A\nX : C\nf : Over X\n⊢ ∀ {X_1 : Over f.left} {S : Sieve X_1},\n Sieve.functorPushforward f.iteratedSliceEquiv.inverse S ∈\n ((J.over X).over f) (f.iteratedSliceEquiv.inverse.obj X_1) ↔\... | [] | simp [GrothendieckTopology.mem_over_iff, Sieve.overEquiv,
← Over.iteratedSliceBackward_forget_forget f, Sieve.functorPushforward_comp] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Sites.Over | {
"line": 564,
"column": 4
} | {
"line": 565,
"column": 82
} | {
"line": 567,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\nJ : GrothendieckTopology C\nA : Type u'\ninst✝ : Category.{v', u'} A\nX : C\nf : Over X\n⊢ ∀ {X_1 : Over f.left} {S : Sieve X_1},\n Sieve.functorPushforward f.iteratedSliceEquiv.inverse S ∈\n ((J.over X).over f) (f.iteratedSliceEquiv.inverse.obj X_1) ↔\... | [] | simp [GrothendieckTopology.mem_over_iff, Sieve.overEquiv,
← Over.iteratedSliceBackward_forget_forget f, Sieve.functorPushforward_comp] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Limits.Final.Type | {
"line": 49,
"column": 2
} | {
"line": 49,
"column": 39
} | {
"line": 50,
"column": 2
} | [
{
"pp": "C : Type u₁\nD : Type u₂\ninst✝² : Category.{v₁, u₁} C\ninst✝¹ : Category.{v₂, u₂} D\nF : C ⥤ D\nP : D ⥤ Type w\ninst✝ : F.Initial\n⊢ Function.Bijective F.sectionsPrecomp",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"CategoryTheory.Functor.comp",
"Set.Elem",
"... | [
"case refine_1\nC : Type u₁\nD : Type u₂\ninst✝² : Category.{v₁, u₁} C\ninst✝¹ : Category.{v₂, u₂} D\nF : C ⥤ D\nP : D ⥤ Type w\ninst✝ : F.Initial\ns₁ s₂ : ↑P.sections\nh : F.sectionsPrecomp s₁ = F.sectionsPrecomp s₂\n⊢ s₁ = s₂",
"case refine_2\nC : Type u₁\nD : Type u₂\ninst✝² : Category.{v₁, u₁} C\ninst✝¹ : Cat... | refine ⟨fun s₁ s₂ h ↦ ?_, fun t ↦ ?_⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.CategoryTheory.Limits.Final.Type | {
"line": 70,
"column": 6
} | {
"line": 70,
"column": 17
} | {
"line": 70,
"column": 18
} | [
{
"pp": "case refine_2.refine_2\nC : Type u₁\nD : Type u₂\ninst✝² : Category.{v₁, u₁} C\ninst✝¹ : Category.{v₂, u₂} D\nF : C ⥤ D\nP : D ⥤ Type w\ninst✝ : F.Initial\nt : ↑(F ⋙ P).sections\nval : (Y : D) → P.obj Y\nhval : ∀ (Y : D) (j : CostructuredArrow F Y), (ConcreteCategory.hom (P.map j.hom)) (↑t j.left) = va... | [
"case refine_2.refine_2\nC : Type u₁\nD : Type u₂\ninst✝² : Category.{v₁, u₁} C\ninst✝¹ : Category.{v₂, u₂} D\nF : C ⥤ D\nP : D ⥤ Type w\ninst✝ : F.Initial\nt : ↑(F ⋙ P).sections\nval : (Y : D) → P.obj Y\nhval : ∀ (Y : D) (j : CostructuredArrow F Y), (ConcreteCategory.hom (P.map j.hom)) (↑t j.left) = val Y\nX : C\n... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Category.ModuleCat.Sheaf.PullbackFree | {
"line": 66,
"column": 4
} | {
"line": 66,
"column": 59
} | {
"line": 68,
"column": 0
} | [
{
"pp": "C : Type u₁\ninst✝² : Category.{v₁, u₁} C\nD : Type u₂\ninst✝¹ : Category.{v₂, u₂} D\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nF : C ⥤ D\nS : Sheaf J RingCat\nR : Sheaf K RingCat\ninst✝ : F.IsContinuous J K\nφ : S ⟶ (F.sheafPushforwardContinuous RingCat J K).obj R\nX✝ Y✝ : Cᵒᵖ\nf : X✝ ⟶ ... | [] | exact ConcreteCategory.congr_hom (φ.hom.naturality f) _ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.CategoryTheory.Simple | {
"line": 81,
"column": 10
} | {
"line": 81,
"column": 21
} | {
"line": 81,
"column": 22
} | [
{
"pp": "C : Type u\ninst✝² : Category.{v, u} C\ninst✝¹ : HasZeroMorphisms C\nX Y : C\ninst✝ : Simple Y\ni : X ≅ Y\nY✝ : C\nf : Y✝ ⟶ X\nm : Mono f\nh : f ≠ 0\nw : f ≫ i.hom = 0\n⊢ f = 0",
"ppTerm": "?m.94",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"C : Type u\ninst✝² : Category.{v, u} C\ninst✝¹ : HasZeroMorphisms C\nX Y : C\ninst✝ : Simple Y\ni : X ≅ Y\nY✝ : C\nf : Y✝ ⟶ X\nm : Mono f\nh : f ≠ 0\nw : f ≫ i.hom = 0\n⊢ f = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Simple | {
"line": 133,
"column": 2
} | {
"line": 133,
"column": 44
} | {
"line": 133,
"column": 45
} | [
{
"pp": "C : Type u\ninst✝² : Category.{v, u} C\ninst✝¹ : HasZeroMorphisms C\nX : C\ninst✝ : Simple X\n⊢ ¬IsZero X",
"ppTerm": "?m.7",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"_private.Mathlib.CategoryTheory.Simple.0.CategoryTheory.Simple.not_isZero._simp_1_1",
"CategoryTheo... | [
"C : Type u\ninst✝² : Category.{v, u} C\ninst✝¹ : HasZeroMorphisms C\nX : C\ninst✝ : Simple X\n⊢ ¬𝟙 X = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Simple | {
"line": 205,
"column": 4
} | {
"line": 205,
"column": 74
} | {
"line": 205,
"column": 75
} | [
{
"pp": "case mp\nC : Type u\ninst✝² : Category.{v, u} C\ninst✝¹ : Preadditive C\ninst✝ : HasBinaryBiproducts C\nX Y : C\nh : (biprod.snd ≫ biprod.inr) ≫ biprod.snd = 0 ≫ biprod.snd\n⊢ IsZero Y",
"ppTerm": "?mp",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"case mp\nC : Type u\ninst✝² : Category.{v, u} C\ninst✝¹ : Preadditive C\ninst✝ : HasBinaryBiproducts C\nX Y : C\nh : (biprod.snd ≫ biprod.inr) ≫ biprod.snd = 0 ≫ biprod.snd\n⊢ IsZero Y"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Sheaves.Presheaf | {
"line": 338,
"column": 42
} | {
"line": 338,
"column": 78
} | {
"line": 338,
"column": 78
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : HasColimits C\nX Y : TopCat\nf : X ⟶ Y\nℱ : Presheaf C Y\nU V : Opens ↑X\nHU : IsOpen (⇑(ConcreteCategory.hom f) '' ↑U)\nHV : IsOpen (⇑(ConcreteCategory.hom f) '' ↑V)\nle : U ≤ V\nj : CostructuredArrow (Opens.map f).op (op V)\neq :\n ((LeftExtension.mk (... | [
"C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : HasColimits C\nX Y : TopCat\nf : X ⟶ Y\nℱ : Presheaf C Y\nU V : Opens ↑X\nHU : IsOpen (⇑(ConcreteCategory.hom f) '' ↑U)\nHV : IsOpen (⇑(ConcreteCategory.hom f) '' ↑V)\nle : U ≤ V\nj : CostructuredArrow (Opens.map f).op (op V)\neq :\n ((LeftExtension.mk ((Opens.map f... | Limits.coconeOfDiagramTerminal_ι_app | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.Sheaves.SheafCondition.Sites | {
"line": 198,
"column": 2
} | {
"line": 198,
"column": 82
} | {
"line": 200,
"column": 0
} | [
{
"pp": "case refine_2\nC : Type u\ninst✝ : Category.{v, u} C\nX Y : TopCat\nf : X ⟶ Y\nF : TopCat.Presheaf C Y\nU : Opens ↑X\n⊢ Nonempty (StructuredArrow U (Opens.map f))",
"ppTerm": "?refine_2",
"assigned": true,
"usedConstants": [
"Lattice.toSemilatticeSup",
"CategoryTheory.CategorySt... | [] | · exact ⟨StructuredArrow.mk <| show U ⟶ (Opens.map f).obj ⊤ from homOfLE le_top⟩ | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Topology.Sheaves.SheafCondition.PairwiseIntersections | {
"line": 128,
"column": 20
} | {
"line": 128,
"column": 31
} | {
"line": 128,
"column": 32
} | [
{
"pp": "C : Type u_1\ninst✝ : Category.{v_1, u_1} C\nX : TopCat\nι : Type u_2\nU : ι → Opens ↑X\nV : OpensLeCover U\ni : ι\na : (Functor.fromPUnit V).obj { as := PUnit.unit } ⟶ (pairwiseToOpensLeCover U).obj (single i)\ni' : ι\nb : (Functor.fromPUnit V).obj { as := PUnit.unit } ⟶ (pairwiseToOpensLeCover U).obj... | [
"C : Type u_1\ninst✝ : Category.{v_1, u_1} C\nX : TopCat\nι : Type u_2\nU : ι → Opens ↑X\nV : OpensLeCover U\ni : ι\na : (Functor.fromPUnit V).obj { as := PUnit.unit } ⟶ (pairwiseToOpensLeCover U).obj (single i)\ni' : ι\nb : (Functor.fromPUnit V).obj { as := PUnit.unit } ⟶ (pairwiseToOpensLeCover U).obj (single i')... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Sheaves.SheafCondition.PairwiseIntersections | {
"line": 123,
"column": 6
} | {
"line": 212,
"column": 47
} | {
"line": 212,
"column": 47
} | [
{
"pp": "C : Type u_1\ninst✝ : Category.{v_1, u_1} C\nX : TopCat\nι : Type u_2\nU : ι → Opens ↑X\nV : OpensLeCover U\nA B : StructuredArrow V (pairwiseToOpensLeCover U)\n⊢ ∃ l, List.IsChain Zag (A :: l) ∧ (A :: l).getLast ⋯ = B",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"List.ge... | [] | rcases A with ⟨⟨⟨⟩⟩, ⟨i⟩ | ⟨i, j⟩, a⟩ <;> rcases B with ⟨⟨⟨⟩⟩, ⟨i'⟩ | ⟨i', j'⟩, b⟩
· refine
⟨[{ left := ⟨⟨⟩⟩
right := pair i i'
hom := ObjectProperty.homMk (homOfLE
(by simpa using le_inf a.hom.le b.hom.le)) }, _], ?_, rfl⟩
exact
List.IsChain... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Sheaves.SheafCondition.PairwiseIntersections | {
"line": 123,
"column": 6
} | {
"line": 212,
"column": 47
} | {
"line": 212,
"column": 47
} | [
{
"pp": "C : Type u_1\ninst✝ : Category.{v_1, u_1} C\nX : TopCat\nι : Type u_2\nU : ι → Opens ↑X\nV : OpensLeCover U\nA B : StructuredArrow V (pairwiseToOpensLeCover U)\n⊢ ∃ l, List.IsChain Zag (A :: l) ∧ (A :: l).getLast ⋯ = B",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"List.ge... | [] | rcases A with ⟨⟨⟨⟩⟩, ⟨i⟩ | ⟨i, j⟩, a⟩ <;> rcases B with ⟨⟨⟨⟩⟩, ⟨i'⟩ | ⟨i', j'⟩, b⟩
· refine
⟨[{ left := ⟨⟨⟩⟩
right := pair i i'
hom := ObjectProperty.homMk (homOfLE
(by simpa using le_inf a.hom.le b.hom.le)) }, _], ?_, rfl⟩
exact
List.IsChain... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Algebra.Module.Basic | {
"line": 52,
"column": 23
} | {
"line": 52,
"column": 34
} | {
"line": 52,
"column": 35
} | [
{
"pp": "R : Type u_1\nM : Type u_2\ninst✝⁴ : Ring R\ninst✝³ : TopologicalSpace M\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\ninst✝ : ContinuousConstSMul R M\n⊢ Continuous[inst✝³, inst✝³] fun a ↦ -a",
"ppTerm": "?m.10",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals":... | [
"R : Type u_1\nM : Type u_2\ninst✝⁴ : Ring R\ninst✝³ : TopologicalSpace M\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\ninst✝ : ContinuousConstSMul R M\n⊢ Continuous[inst✝³, inst✝³] fun a ↦ -a"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.Module.Basic | {
"line": 96,
"column": 4
} | {
"line": 96,
"column": 20
} | {
"line": 96,
"column": 21
} | [
{
"pp": "case refine_2\nR : Type u_1\nM : Type u_2\ninst✝¹⁰ : Ring R\ninst✝⁹ : TopologicalSpace R\ninst✝⁸ : TopologicalSpace M\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : ContinuousAdd M\ninst✝⁵ : Module R M\ninst✝⁴ : ContinuousSMul R M\ninst✝³ : IsDomain R\ninst✝² : Nontrivial M\ninst✝¹ : (𝓝[≠] 0).NeBot\ninst✝ : IsTor... | [
"case refine_2\nR : Type u_1\nM : Type u_2\ninst✝¹⁰ : Ring R\ninst✝⁹ : TopologicalSpace R\ninst✝⁸ : TopologicalSpace M\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : ContinuousAdd M\ninst✝⁵ : Module R M\ninst✝⁴ : ContinuousSMul R M\ninst✝³ : IsDomain R\ninst✝² : Nontrivial M\ninst✝¹ : (𝓝[≠] 0).NeBot\ninst✝ : IsTorsionFree R M... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.Module.Basic | {
"line": 233,
"column": 6
} | {
"line": 233,
"column": 22
} | {
"line": 233,
"column": 22
} | [
{
"pp": "ι : Type u_1\nR : Type u_2\nM : ι → Type u_3\ninst✝⁴ : Semiring R\ninst✝³ : (i : ι) → AddCommMonoid (M i)\ninst✝² : (i : ι) → Module R (M i)\ninst✝¹ : (i : ι) → TopologicalSpace (M i)\ninst✝ : DecidableEq ι\ns : (i : ι) → Submodule R (M i)\n⊢ closure[Pi.topologicalSpace] ↑(⨆ i, map (LinearMap.single R ... | [
"ι : Type u_1\nR : Type u_2\nM : ι → Type u_3\ninst✝⁴ : Semiring R\ninst✝³ : (i : ι) → AddCommMonoid (M i)\ninst✝² : (i : ι) → Module R (M i)\ninst✝¹ : (i : ι) → TopologicalSpace (M i)\ninst✝ : DecidableEq ι\ns : (i : ι) → Submodule R (M i)\n⊢ closure[Pi.topologicalSpace] ↑(⨆ i, map (LinearMap.single R M i) (s i)) ... | ← closure_pi_set | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.Sheaves.Stalks | {
"line": 303,
"column": 2
} | {
"line": 303,
"column": 13
} | {
"line": 303,
"column": 14
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : HasColimits C\nX Y : TopCat\nf : X ⟶ Y\nF : Presheaf C Y\nV : (Opens ↑Y)ᵒᵖ\nx : ↑X\nhx : (ConcreteCategory.hom f) x ∈ unop V\n⊢ ((pullbackPushforwardAdjunction C f).unit.app F).app V ≫ germToPullbackStalk C f F ((Opens.map f).obj (unop V)) x hx =\n F.g... | [
"C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : HasColimits C\nX Y : TopCat\nf : X ⟶ Y\nF : Presheaf C Y\nV : (Opens ↑Y)ᵒᵖ\nx : ↑X\nhx : (ConcreteCategory.hom f) x ∈ unop V\n⊢ ((pullbackPushforwardAdjunction C f).unit.app F).app V ≫ germToPullbackStalk C f F ((Opens.map f).obj (unop V)) x hx =\n F.germ (unop V)... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Sheaves.SheafCondition.PairwiseIntersections | {
"line": 392,
"column": 2
} | {
"line": 425,
"column": 44
} | {
"line": 427,
"column": 0
} | [
{
"pp": "C : Type u_1\ninst✝ : Category.{v_1, u_1} C\nX : TopCat\nF : Sheaf C X\nU V : Opens ↑X\ns : PullbackCone (F.obj.map (homOfLE ⋯).op) (F.obj.map (homOfLE ⋯).op)\n⊢ IsLimit (F.interUnionPullbackCone U V)",
"ppTerm": "?m.64",
"assigned": true,
"usedConstants": [
"Set.ext",
"Category... | [] | let ι : ULift.{w} WalkingPair → Opens X := fun ⟨j⟩ => WalkingPair.casesOn j U V
have hι : U ⊔ V = iSup ι := by
ext
rw [Opens.coe_iSup, Set.mem_iUnion]
constructor
· rintro (h | h)
exacts [⟨⟨WalkingPair.left⟩, h⟩, ⟨⟨WalkingPair.right⟩, h⟩]
· rintro ⟨⟨_ | _⟩, h⟩
exacts [Or.inl h, Or.inr ... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Sheaves.SheafCondition.PairwiseIntersections | {
"line": 392,
"column": 2
} | {
"line": 425,
"column": 44
} | {
"line": 427,
"column": 0
} | [
{
"pp": "C : Type u_1\ninst✝ : Category.{v_1, u_1} C\nX : TopCat\nF : Sheaf C X\nU V : Opens ↑X\ns : PullbackCone (F.obj.map (homOfLE ⋯).op) (F.obj.map (homOfLE ⋯).op)\n⊢ IsLimit (F.interUnionPullbackCone U V)",
"ppTerm": "?m.64",
"assigned": true,
"usedConstants": [
"Set.ext",
"Category... | [] | let ι : ULift.{w} WalkingPair → Opens X := fun ⟨j⟩ => WalkingPair.casesOn j U V
have hι : U ⊔ V = iSup ι := by
ext
rw [Opens.coe_iSup, Set.mem_iUnion]
constructor
· rintro (h | h)
exacts [⟨⟨WalkingPair.left⟩, h⟩, ⟨⟨WalkingPair.right⟩, h⟩]
· rintro ⟨⟨_ | _⟩, h⟩
exacts [Or.inl h, Or.inr ... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Sheaves.Stalks | {
"line": 502,
"column": 2
} | {
"line": 502,
"column": 31
} | {
"line": 502,
"column": 32
} | [
{
"pp": "C : Type u\ninst✝³ : Category.{v, u} C\ninst✝² : HasColimits C\nX : TopCat\nFC : C → C → Type u_1\nCC : C → Type v\ninst✝¹ : (X Y : C) → FunLike (FC X Y) (CC X) (CC Y)\ninstCC : ConcreteCategory C FC\ninst✝ : PreservesFilteredColimits (forget C)\nB : Set (Opens ↑X)\nhB : Opens.IsBasis B\nF G : Presheaf... | [
"C : Type u\ninst✝³ : Category.{v, u} C\ninst✝² : HasColimits C\nX : TopCat\nFC : C → C → Type u_1\nCC : C → Type v\ninst✝¹ : (X Y : C) → FunLike (FC X Y) (CC X) (CC Y)\ninstCC : ConcreteCategory C FC\ninst✝ : PreservesFilteredColimits (forget C)\nB : Set (Opens ↑X)\nhB : Opens.IsBasis B\nF G : Presheaf C X\nα : F ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.Module.ContinuousLinearMap.Basic | {
"line": 413,
"column": 47
} | {
"line": 413,
"column": 58
} | {
"line": 413,
"column": 59
} | [
{
"pp": "R₁ : Type u_1\nR₂ : Type u_2\nR₃ : Type u_3\ninst✝¹⁷ : Semiring R₁\ninst✝¹⁶ : Semiring R₂\ninst✝¹⁵ : Semiring R₃\nσ₁₂ : R₁ →+* R₂\nσ₂₃ : R₂ →+* R₃\nσ₁₃ : R₁ →+* R₃\nM₁ : Type u_4\ninst✝¹⁴ : TopologicalSpace M₁\ninst✝¹³ : AddCommMonoid M₁\nM'₁ : Type u_5\ninst✝¹² : TopologicalSpace M'₁\ninst✝¹¹ : AddCom... | [
"R₁ : Type u_1\nR₂ : Type u_2\nR₃ : Type u_3\ninst✝¹⁷ : Semiring R₁\ninst✝¹⁶ : Semiring R₂\ninst✝¹⁵ : Semiring R₃\nσ₁₂ : R₁ →+* R₂\nσ₂₃ : R₂ →+* R₃\nσ₁₃ : R₁ →+* R₃\nM₁ : Type u_4\ninst✝¹⁴ : TopologicalSpace M₁\ninst✝¹³ : AddCommMonoid M₁\nM'₁ : Type u_5\ninst✝¹² : TopologicalSpace M'₁\ninst✝¹¹ : AddCommMonoid M'₁\... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.Module.ContinuousLinearMap.Basic | {
"line": 508,
"column": 2
} | {
"line": 508,
"column": 57
} | {
"line": 508,
"column": 58
} | [
{
"pp": "R : Type u_9\nE : Type u_10\nF : Type u_11\ninst✝⁶ : Semiring R\ninst✝⁵ : TopologicalSpace E\ninst✝⁴ : AddCommMonoid E\ninst✝³ : Module R E\ninst✝² : TopologicalSpace F\ninst✝¹ : AddCommMonoid F\ninst✝ : Module R F\nf : E →L[R] F\ng : F →L[R] E\nhinv : g ∘SL f = ContinuousLinearMap.id R E\n⊢ Function.L... | [
"R : Type u_9\nE : Type u_10\nF : Type u_11\ninst✝⁶ : Semiring R\ninst✝⁵ : TopologicalSpace E\ninst✝⁴ : AddCommMonoid E\ninst✝³ : Module R E\ninst✝² : TopologicalSpace F\ninst✝¹ : AddCommMonoid F\ninst✝ : Module R F\nf : E →L[R] F\ng : F →L[R] E\nhinv : g ∘SL f = ContinuousLinearMap.id R E\n⊢ Function.LeftInverse ⇑... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.Module.ContinuousLinearMap.Basic | {
"line": 734,
"column": 38
} | {
"line": 734,
"column": 79
} | {
"line": 734,
"column": 80
} | [
{
"pp": "M₁ : Type u_4\ninst✝⁸ : TopologicalSpace M₁\ninst✝⁷ : AddCommMonoid M₁\nM₂ : Type u_6\ninst✝⁶ : TopologicalSpace M₂\ninst✝⁵ : AddCommMonoid M₂\nR : Type u_9\ninst✝⁴ : DivisionSemiring R\ninst✝³ : Module R M₁\ninst✝² : Module R M₂\ninst✝¹ : TopologicalSpace R\ninst✝ : ContinuousSMul R M₂\nf : M₁ →L[R] R... | [
"M₁ : Type u_4\ninst✝⁸ : TopologicalSpace M₁\ninst✝⁷ : AddCommMonoid M₁\nM₂ : Type u_6\ninst✝⁶ : TopologicalSpace M₂\ninst✝⁵ : AddCommMonoid M₂\nR : Type u_9\ninst✝⁴ : DivisionSemiring R\ninst✝³ : Module R M₁\ninst✝² : Module R M₂\ninst✝¹ : TopologicalSpace R\ninst✝ : ContinuousSMul R M₂\nf : M₁ →L[R] R\nhf : f ≠ 0... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.Module.ContinuousLinearMap.Basic | {
"line": 923,
"column": 18
} | {
"line": 923,
"column": 22
} | {
"line": 923,
"column": 23
} | [
{
"pp": "case succ\nR : Type u_1\ninst✝² : Ring R\ninst✝¹ : TopologicalSpace R\ninst✝ : IsTopologicalRing R\nc : R\nn : ℕ\nihn : toSpanSingleton R c ^ n = toSpanSingleton R (c ^ n)\n⊢ toSpanSingleton R c ^ n * toSpanSingleton R c = toSpanSingleton R (c ^ (n + 1))",
"ppTerm": "?succ",
"assigned": true,
... | [
"case succ\nR : Type u_1\ninst✝² : Ring R\ninst✝¹ : TopologicalSpace R\ninst✝ : IsTopologicalRing R\nc : R\nn : ℕ\nihn : toSpanSingleton R c ^ n = toSpanSingleton R (c ^ n)\n⊢ toSpanSingleton R (c ^ n) * toSpanSingleton R c = toSpanSingleton R (c ^ (n + 1))"
] | ihn, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Category.ModuleCat.Topology.Homology | {
"line": 119,
"column": 8
} | {
"line": 119,
"column": 56
} | {
"line": 119,
"column": 57
} | [
{
"pp": "R : Type u\ninst✝¹ : Ring R\ninst✝ : TopologicalSpace R\nM N : TopModuleCat R\nφ : M ⟶ N\nS : ShortComplex (TopModuleCat R)\nD₁ : S.LeftHomologyData :=\n { K := ker S.g, H := coker ((isLimitKer S.g).lift (KernelFork.ofι S.f ⋯)), i := kerι S.g,\n π := cokerπ ((isLimitKer S.g).lift (KernelFork.ofι S.... | [
"R : Type u\ninst✝¹ : Ring R\ninst✝ : TopologicalSpace R\nM N : TopModuleCat R\nφ : M ⟶ N\nS : ShortComplex (TopModuleCat R)\nD₁ : S.LeftHomologyData :=\n { K := ker S.g, H := coker ((isLimitKer S.g).lift (KernelFork.ofι S.f ⋯)), i := kerι S.g,\n π := cokerπ ((isLimitKer S.g).lift (KernelFork.ofι S.f ⋯)), wi :=... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Category.ModuleCat.Topology.Homology | {
"line": 123,
"column": 6
} | {
"line": 123,
"column": 81
} | {
"line": 123,
"column": 82
} | [
{
"pp": "case refine_2\nR : Type u\ninst✝¹ : Ring R\ninst✝ : TopologicalSpace R\nM N : TopModuleCat R\nφ : M ⟶ N\nS : ShortComplex (TopModuleCat R)\nD₁ : S.LeftHomologyData :=\n { K := ker S.g, H := coker ((isLimitKer S.g).lift (KernelFork.ofι S.f ⋯)), i := kerι S.g,\n π := cokerπ ((isLimitKer S.g).lift (Ke... | [
"case refine_2\nR : Type u\ninst✝¹ : Ring R\ninst✝ : TopologicalSpace R\nM N : TopModuleCat R\nφ : M ⟶ N\nS : ShortComplex (TopModuleCat R)\nD₁ : S.LeftHomologyData :=\n { K := ker S.g, H := coker ((isLimitKer S.g).lift (KernelFork.ofι S.f ⋯)), i := kerι S.g,\n π := cokerπ ((isLimitKer S.g).lift (KernelFork.ofι... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Group.WithOne.Basic | {
"line": 110,
"column": 32
} | {
"line": 110,
"column": 55
} | {
"line": 110,
"column": 56
} | [
{
"pp": "α : Type u\nβ : Type v\ninst✝¹ : Mul α\ninst✝ : Mul β\nf : α →ₙ* β\nhf : Function.Injective ⇑f\na₁ a₂ : α\nH : (mapMulHom f) ↑a₁ = (mapMulHom f) ↑a₂\n⊢ ↑a₁ = ↑a₂",
"ppTerm": "?m.24",
"assigned": true,
"usedConstants": [
"WithOne",
"Eq.mpr",
"WithOne.coe_inj._simp_2",
... | [
"α : Type u\nβ : Type v\ninst✝¹ : Mul α\ninst✝ : Mul β\nf : α →ₙ* β\nhf : Function.Injective ⇑f\na₁ a₂ : α\nH : (mapMulHom f) ↑a₁ = (mapMulHom f) ↑a₂\n⊢ a₁ = a₂"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Group.WithOne.Basic | {
"line": 110,
"column": 29
} | {
"line": 110,
"column": 57
} | {
"line": 112,
"column": 0
} | [
{
"pp": "α : Type u\nβ : Type v\ninst✝¹ : Mul α\ninst✝ : Mul β\nf : α →ₙ* β\nhf : Function.Injective ⇑f\na₁ a₂ : α\nH : (mapMulHom f) ↑a₁ = (mapMulHom f) ↑a₂\n⊢ ↑a₁ = ↑a₂",
"ppTerm": "?m.24",
"assigned": true,
"usedConstants": [
"MulHom",
"WithOne",
"Eq.mpr",
"WithOne.coe_inj... | [] | by simpa [hf.eq_iff] using H | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.Algebra.Module.Equiv | {
"line": 649,
"column": 16
} | {
"line": 649,
"column": 27
} | {
"line": 649,
"column": 28
} | [
{
"pp": "R₁ : Type u_1\nR₂ : Type u_2\nR₃ : Type u_3\ninst✝²¹ : Semiring R₁\ninst✝²⁰ : Semiring R₂\ninst✝¹⁹ : Semiring R₃\nσ₁₂ : R₁ →+* R₂\nσ₂₁ : R₂ →+* R₁\ninst✝¹⁸ : RingHomInvPair σ₁₂ σ₂₁\ninst✝¹⁷ : RingHomInvPair σ₂₁ σ₁₂\nσ₂₃ : R₂ →+* R₃\nσ₃₂ : R₃ →+* R₂\ninst✝¹⁶ : RingHomInvPair σ₂₃ σ₃₂\ninst✝¹⁵ : RingHomIn... | [
"R₁ : Type u_1\nR₂ : Type u_2\nR₃ : Type u_3\ninst✝²¹ : Semiring R₁\ninst✝²⁰ : Semiring R₂\ninst✝¹⁹ : Semiring R₃\nσ₁₂ : R₁ →+* R₂\nσ₂₁ : R₂ →+* R₁\ninst✝¹⁸ : RingHomInvPair σ₁₂ σ₂₁\ninst✝¹⁷ : RingHomInvPair σ₂₁ σ₁₂\nσ₂₃ : R₂ →+* R₃\nσ₃₂ : R₃ →+* R₂\ninst✝¹⁶ : RingHomInvPair σ₂₃ σ₃₂\ninst✝¹⁵ : RingHomInvPair σ₃₂ σ₂... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.Module.Equiv | {
"line": 649,
"column": 56
} | {
"line": 649,
"column": 67
} | {
"line": 649,
"column": 68
} | [
{
"pp": "R₁ : Type u_1\nR₂ : Type u_2\nR₃ : Type u_3\ninst✝²¹ : Semiring R₁\ninst✝²⁰ : Semiring R₂\ninst✝¹⁹ : Semiring R₃\nσ₁₂ : R₁ →+* R₂\nσ₂₁ : R₂ →+* R₁\ninst✝¹⁸ : RingHomInvPair σ₁₂ σ₂₁\ninst✝¹⁷ : RingHomInvPair σ₂₁ σ₁₂\nσ₂₃ : R₂ →+* R₃\nσ₃₂ : R₃ →+* R₂\ninst✝¹⁶ : RingHomInvPair σ₂₃ σ₃₂\ninst✝¹⁵ : RingHomIn... | [
"R₁ : Type u_1\nR₂ : Type u_2\nR₃ : Type u_3\ninst✝²¹ : Semiring R₁\ninst✝²⁰ : Semiring R₂\ninst✝¹⁹ : Semiring R₃\nσ₁₂ : R₁ →+* R₂\nσ₂₁ : R₂ →+* R₁\ninst✝¹⁸ : RingHomInvPair σ₁₂ σ₂₁\ninst✝¹⁷ : RingHomInvPair σ₂₁ σ₁₂\nσ₂₃ : R₂ →+* R₃\nσ₃₂ : R₃ →+* R₂\ninst✝¹⁶ : RingHomInvPair σ₂₃ σ₃₂\ninst✝¹⁵ : RingHomInvPair σ₃₂ σ₂... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.Module.Equiv | {
"line": 1268,
"column": 57
} | {
"line": 1268,
"column": 81
} | {
"line": 1268,
"column": 82
} | [
{
"pp": "R : Type u_1\nM : Type u_2\nM₂ : Type u_3\ninst✝⁶ : TopologicalSpace M\ninst✝⁵ : TopologicalSpace M₂\ninst✝⁴ : Semiring R\ninst✝³ : AddCommMonoid M\ninst✝² : Module R M\ninst✝¹ : AddCommMonoid M₂\ninst✝ : Module R M₂\nf : M →L[R] M₂\nhf : f.inverse.IsInvertible\nH : ¬f.IsInvertible\n⊢ Subsingleton M₂ ∧... | [
"R : Type u_1\nM : Type u_2\nM₂ : Type u_3\ninst✝⁶ : TopologicalSpace M\ninst✝⁵ : TopologicalSpace M₂\ninst✝⁴ : Semiring R\ninst✝³ : AddCommMonoid M\ninst✝² : Module R M\ninst✝¹ : AddCommMonoid M₂\ninst✝ : Module R M₂\nf : M →L[R] M₂\nhf : f.inverse.IsInvertible\nH : ¬f.IsInvertible\n⊢ Subsingleton M₂ ∧ Subsingleto... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Limits.Preserves.Over | {
"line": 35,
"column": 31
} | {
"line": 35,
"column": 42
} | {
"line": 35,
"column": 43
} | [
{
"pp": "C : Type u_1\ninst✝ : Category.{v_1, u_1} C\nX : C\nJ : Type u_3\nhJ : Category.{u_2, u_3} J\nhJ' : IsCofiltered J\nF : J ⥤ Over X\nc : Cone F\nhc : IsLimit c\ns : Cone (F ⋙ Over.forget X)\ni j k : J\ne : j ⟶ k\n⊢ Over.Hom.left (((Functor.const J).obj (Over.mk (s.π.app i ≫ (F.obj i).hom))).map e ≫ Over... | [
"C : Type u_1\ninst✝ : Category.{v_1, u_1} C\nX : C\nJ : Type u_3\nhJ : Category.{u_2, u_3} J\nhJ' : IsCofiltered J\nF : J ⥤ Over X\nc : Cone F\nhc : IsLimit c\ns : Cone (F ⋙ Over.forget X)\ni j k : J\ne : j ⟶ k\n⊢ s.π.app k = s.π.app j ≫ Over.Hom.left (F.map e)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Limits.Preserves.Over | {
"line": 49,
"column": 31
} | {
"line": 49,
"column": 42
} | {
"line": 49,
"column": 43
} | [
{
"pp": "C : Type u_1\ninst✝ : Category.{v_1, u_1} C\nX : C\nJ : Type u_3\nhJ : Category.{u_2, u_3} J\nhJ' : IsFiltered J\nF : J ⥤ Under X\nc : Cocone F\nhc : IsColimit c\ns : Cocone (F ⋙ Under.forget X)\ni j k : J\ne : j ⟶ k\n⊢ Under.Hom.right (F.map e ≫ Under.homMk (s.ι.app k) ⋯) =\n Under.Hom.right (Under... | [
"C : Type u_1\ninst✝ : Category.{v_1, u_1} C\nX : C\nJ : Type u_3\nhJ : Category.{u_2, u_3} J\nhJ' : IsFiltered J\nF : J ⥤ Under X\nc : Cocone F\nhc : IsColimit c\ns : Cocone (F ⋙ Under.forget X)\ni j k : J\ne : j ⟶ k\n⊢ Under.Hom.right (F.map e) ≫ s.ι.app k = s.ι.app j"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Presentable.IsCardinalFiltered | {
"line": 52,
"column": 2
} | {
"line": 53,
"column": 42
} | {
"line": 53,
"column": 43
} | [
{
"pp": "T : Type u\nκ : Cardinal.{w}\nhT : HasCardinalLT T κ\nhκ : Cardinal.aleph0 ≤ κ\n⊢ HasCardinalLT (Arrow (WalkingParallelFamily T)) κ",
"ppTerm": "?m.4",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"HasCardinalLT",
"Cardinal",
"CategoryTheory.Limits.WalkingParallelF... | [
"T : Type u\nκ : Cardinal.{w}\nhT : HasCardinalLT T κ\nhκ : Cardinal.aleph0 ≤ κ\n⊢ HasCardinalLT T κ"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Presentable.IsCardinalFiltered | {
"line": 90,
"column": 44
} | {
"line": 90,
"column": 55
} | {
"line": 90,
"column": 56
} | [
{
"pp": "J : Type u\ninst✝¹ : Category.{v, u} J\nκ : Cardinal.{w}\nhκ : Fact κ.IsRegular\ninst✝ : IsCardinalFiltered J κ\nK : Type u'\nS : K → J\nhS : HasCardinalLT K κ\n⊢ HasCardinalLT (Arrow (Discrete K)) κ",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"HasCardina... | [
"J : Type u\ninst✝¹ : Category.{v, u} J\nκ : Cardinal.{w}\nhκ : Fact κ.IsRegular\ninst✝ : IsCardinalFiltered J κ\nK : Type u'\nS : K → J\nhS : HasCardinalLT K κ\n⊢ HasCardinalLT K κ"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Presentable.IsCardinalFiltered | {
"line": 96,
"column": 44
} | {
"line": 96,
"column": 55
} | {
"line": 96,
"column": 56
} | [
{
"pp": "J : Type u\ninst✝¹ : Category.{v, u} J\nκ : Cardinal.{w}\nhκ : Fact κ.IsRegular\ninst✝ : IsCardinalFiltered J κ\nK : Type u'\nS : K → J\nhS : HasCardinalLT K κ\nk : K\n⊢ HasCardinalLT (Arrow (Discrete K)) κ",
"ppTerm": "?m.33",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Has... | [
"J : Type u\ninst✝¹ : Category.{v, u} J\nκ : Cardinal.{w}\nhκ : Fact κ.IsRegular\ninst✝ : IsCardinalFiltered J κ\nK : Type u'\nS : K → J\nhS : HasCardinalLT K κ\nk : K\n⊢ HasCardinalLT K κ"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Presentable.IsCardinalFiltered | {
"line": 138,
"column": 20
} | {
"line": 138,
"column": 35
} | {
"line": 138,
"column": 36
} | [
{
"pp": "J : Type u\ninst✝¹ : Category.{v, u} J\nκ : Cardinal.{w}\nhκ : Fact κ.IsRegular\ninst✝ : IsCardinalFiltered J κ\nι : Type v'\nj : J\nk : ι → J\nf : (i : ι) → j ⟶ k i\nhι : HasCardinalLT ι κ\nφ : ι → (j ⟶ max k hι) := fun i ↦ f i ≫ toMax k hι i\n⊢ ∀ (i : ι), f i ≫ (fun i ↦ toMax k hι i ≫ coeqHom φ hι) i... | [
"J : Type u\ninst✝¹ : Category.{v, u} J\nκ : Cardinal.{w}\nhκ : Fact κ.IsRegular\ninst✝ : IsCardinalFiltered J κ\nι : Type v'\nj : J\nk : ι → J\nf : (i : ι) → j ⟶ k i\nhι : HasCardinalLT ι κ\nφ : ι → (j ⟶ max k hι) := fun i ↦ f i ≫ toMax k hι i\n⊢ ∀ (i : ι), f i ≫ toMax k hι i ≫ coeqHom (fun i ↦ f i ≫ toMax k hι i)... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Presentable.IsCardinalFiltered | {
"line": 181,
"column": 34
} | {
"line": 181,
"column": 85
} | {
"line": 182,
"column": 8
} | [
{
"pp": "J : Type w\ninst✝¹ : Preorder J\nκ : Cardinal.{w}\ninst✝ : Fact κ.IsRegular\nh : ∀ ⦃K : Type w⦄ (s : K → J), Cardinal.mk K < κ → ∃ j, ∀ (k : K), s k ≤ j\nA : Type w\nx✝ : SmallCategory A\nF : A ⥤ J\nhA : HasCardinalLT (Arrow A) κ\n⊢ Cardinal.mk A < κ",
"ppTerm": "?m.23",
"assigned": false,
... | [
"J : Type w\ninst✝¹ : Preorder J\nκ : Cardinal.{w}\ninst✝ : Fact κ.IsRegular\nh : ∀ ⦃K : Type w⦄ (s : K → J), Cardinal.mk K < κ → ∃ j, ∀ (k : K), s k ≤ j\nA : Type w\nx✝ : SmallCategory A\nF : A ⥤ J\nhA : HasCardinalLT (Arrow A) κ\n⊢ Cardinal.mk A < κ"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Presentable.IsCardinalFiltered | {
"line": 213,
"column": 18
} | {
"line": 213,
"column": 33
} | {
"line": 213,
"column": 34
} | [
{
"pp": "J : Type u\ninst✝² : Category.{v, u} J\nκ : Cardinal.{w}\ninst✝¹ : Fact κ.IsRegular\ninst✝ : IsCardinalFiltered J κ\nj₀ : J\nA : Type w\nx✝ : SmallCategory A\nF : A ⥤ Under j₀\nhA : HasCardinalLT (Arrow A) κ\nthis : IsFiltered J\nc : Cocone (F ⋙ Under.forget j₀) := cocone (F ⋙ Under.forget j₀) hA\nx : ... | [
"J : Type u\ninst✝² : Category.{v, u} J\nκ : Cardinal.{w}\ninst✝¹ : Fact κ.IsRegular\ninst✝ : IsCardinalFiltered J κ\nj₀ : J\nA : Type w\nx✝ : SmallCategory A\nF : A ⥤ Under j₀\nhA : HasCardinalLT (Arrow A) κ\nthis : IsFiltered J\nc : Cocone (F ⋙ Under.forget j₀) := cocone (F ⋙ Under.forget j₀) hA\nx : A → (j₀ ⟶ Is... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Presentable.IsCardinalFiltered | {
"line": 233,
"column": 12
} | {
"line": 233,
"column": 23
} | {
"line": 233,
"column": 24
} | [
{
"pp": "case h₁\nJ₁ : Type u\nJ₂ : Type u'\ninst✝⁵ : Category.{v, u} J₁\ninst✝⁴ : Category.{v', u'} J₂\nκ : Cardinal.{w}\ninst✝³ : Fact κ.IsRegular\ninst✝² : IsCardinalFiltered J₁ κ\ninst✝¹ : IsCardinalFiltered J₂ κ\nA✝ : Type w\ninst✝ : SmallCategory A✝\nF : A✝ ⥤ J₁ × J₂\nhC : HasCardinalLT (Arrow A✝) κ\nc₁ :... | [
"case h₁\nJ₁ : Type u\nJ₂ : Type u'\ninst✝⁵ : Category.{v, u} J₁\ninst✝⁴ : Category.{v', u'} J₂\nκ : Cardinal.{w}\ninst✝³ : Fact κ.IsRegular\ninst✝² : IsCardinalFiltered J₁ κ\ninst✝¹ : IsCardinalFiltered J₂ κ\nA✝ : Type w\ninst✝ : SmallCategory A✝\nF : A✝ ⥤ J₁ × J₂\nhC : HasCardinalLT (Arrow A✝) κ\nc₁ : Cocone (F ⋙... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Presentable.IsCardinalFiltered | {
"line": 234,
"column": 12
} | {
"line": 234,
"column": 23
} | {
"line": 234,
"column": 24
} | [
{
"pp": "case h₂\nJ₁ : Type u\nJ₂ : Type u'\ninst✝⁵ : Category.{v, u} J₁\ninst✝⁴ : Category.{v', u'} J₂\nκ : Cardinal.{w}\ninst✝³ : Fact κ.IsRegular\ninst✝² : IsCardinalFiltered J₁ κ\ninst✝¹ : IsCardinalFiltered J₂ κ\nA✝ : Type w\ninst✝ : SmallCategory A✝\nF : A✝ ⥤ J₁ × J₂\nhC : HasCardinalLT (Arrow A✝) κ\nc₁ :... | [
"case h₂\nJ₁ : Type u\nJ₂ : Type u'\ninst✝⁵ : Category.{v, u} J₁\ninst✝⁴ : Category.{v', u'} J₂\nκ : Cardinal.{w}\ninst✝³ : Fact κ.IsRegular\ninst✝² : IsCardinalFiltered J₁ κ\ninst✝¹ : IsCardinalFiltered J₂ κ\nA✝ : Type w\ninst✝ : SmallCategory A✝\nF : A✝ ⥤ J₁ × J₂\nhC : HasCardinalLT (Arrow A✝) κ\nc₁ : Cocone (F ⋙... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Presentable.Basic | {
"line": 77,
"column": 4
} | {
"line": 77,
"column": 64
} | {
"line": 79,
"column": 0
} | [
{
"pp": "C : Type u₁\ninst✝⁴ : Category.{v₁, u₁} C\nD : Type u₂\ninst✝³ : Category.{v₂, u₂} D\nF : C ⥤ D\nκ : Cardinal.{w}\ninst✝² : Fact κ.IsRegular\ninst✝¹ : F.IsCardinalAccessible κ\nκ' : Cardinal.{w}\ninst✝ : Fact κ'.IsRegular\nh : κ ≤ κ'\nJ : Type w\nx✝¹ : SmallCategory J\nx✝ : IsCardinalFiltered J κ'\nthi... | [] | exact F.preservesColimitsOfShape_of_isCardinalAccessible κ J | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.CategoryTheory.Presentable.IsCardinalFiltered | {
"line": 271,
"column": 4
} | {
"line": 273,
"column": 77
} | {
"line": 274,
"column": 4
} | [
{
"pp": "J : Type u\ninst✝¹ : Category.{v, u} J\nκ : Cardinal.{w}\ninst✝ : Fact κ.IsRegular\nh₁ : ∀ ⦃ι : Type w⦄ (j : ι → J), HasCardinalLT ι κ → ∃ k, ∀ (i : ι), Nonempty (j i ⟶ k)\nh₂ : ∀ ⦃ι : Type w⦄ ⦃j k : J⦄ (f : ι → (j ⟶ k)), HasCardinalLT ι κ → ∃ l a b, ∀ (i : ι), f i ≫ a = b\nι : Type w\nj : ι → J\nk : J... | [
"J : Type u\ninst✝¹ : Category.{v, u} J\nκ : Cardinal.{w}\ninst✝ : Fact κ.IsRegular\nh₁ : ∀ ⦃ι : Type w⦄ (j : ι → J), HasCardinalLT ι κ → ∃ k, ∀ (i : ι), Nonempty (j i ⟶ k)\nh₂ : ∀ ⦃ι : Type w⦄ ⦃j k : J⦄ (f : ι → (j ⟶ k)), HasCardinalLT ι κ → ∃ l a b, ∀ (i : ι), f i ≫ a = b\nι : Type w\nj : ι → J\nk : J\nf₁ f₂ : (i... | obtain ⟨l, a, b, hl⟩ := h₂ (Sum.elim (fun (_ : PUnit.{w + 1}) ↦ f₁ i)
(fun (_ : PUnit.{w + 1}) ↦ f₂ i))
(hasCardinalLT_of_finite _ _ (Cardinal.IsRegular.aleph0_le Fact.out)) | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.CategoryTheory.Presentable.IsCardinalFiltered | {
"line": 303,
"column": 29
} | {
"line": 303,
"column": 40
} | {
"line": 303,
"column": 41
} | [
{
"pp": "J : Type u\ninst✝¹ : Category.{v, u} J\nκ : Cardinal.{w}\ninst✝ : Fact κ.IsRegular\nx✝³ :\n (∀ ⦃ι : Type w⦄ (j : ι → J), HasCardinalLT ι κ → ∃ k, ∀ (i : ι), Nonempty (j i ⟶ k)) ∧\n ∀ ⦃ι : Type w⦄ ⦃j k : J⦄ (f : ι → (j ⟶ k)), HasCardinalLT ι κ → ∃ l a b, ∀ (i : ι), f i ≫ a = b\nh₁ : ∀ ⦃ι : Type w⦄ (... | [
"J : Type u\ninst✝¹ : Category.{v, u} J\nκ : Cardinal.{w}\ninst✝ : Fact κ.IsRegular\nx✝³ :\n (∀ ⦃ι : Type w⦄ (j : ι → J), HasCardinalLT ι κ → ∃ k, ∀ (i : ι), Nonempty (j i ⟶ k)) ∧\n ∀ ⦃ι : Type w⦄ ⦃j k : J⦄ (f : ι → (j ⟶ k)), HasCardinalLT ι κ → ∃ l a b, ∀ (i : ι), f i ≫ a = b\nh₁ : ∀ ⦃ι : Type w⦄ (j : ι → J), ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Presentable.Basic | {
"line": 237,
"column": 2
} | {
"line": 238,
"column": 18
} | {
"line": 240,
"column": 0
} | [
{
"pp": "case mpr\nC : Type u₁\ninst✝³ : Category.{v₁, u₁} C\nX : C\nκ : Cardinal.{w}\ninst✝² : Fact κ.IsRegular\nC' : Type u₃\ninst✝¹ : Category.{v₃, u₃} C'\nF : C ⥤ C'\ninst✝ : F.IsEquivalence\n⊢ IsCardinalPresentable X κ → IsCardinalPresentable (F.obj X) κ",
"ppTerm": "?mpr",
"assigned": true,
"u... | [] | · intro
infer_instance | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Algebra.Module.SnakeLemma | {
"line": 197,
"column": 42
} | {
"line": 197,
"column": 87
} | {
"line": 197,
"column": 88
} | [
{
"pp": "R : Type u_1\ninst✝¹⁸ : CommRing R\nM₁ : Type u_2\nM₂ : Type u_3\nM₃ : Type u_4\nN₁ : Type u_5\nN₂ : Type u_6\nN₃ : Type u_7\ninst✝¹⁷ : AddCommGroup M₁\ninst✝¹⁶ : Module R M₁\ninst✝¹⁵ : AddCommGroup M₂\ninst✝¹⁴ : Module R M₂\ninst✝¹³ : AddCommGroup M₃\ninst✝¹² : Module R M₃\ninst✝¹¹ : AddCommGroup N₁\n... | [
"R : Type u_1\ninst✝¹⁸ : CommRing R\nM₁ : Type u_2\nM₂ : Type u_3\nM₃ : Type u_4\nN₁ : Type u_5\nN₂ : Type u_6\nN₃ : Type u_7\ninst✝¹⁷ : AddCommGroup M₁\ninst✝¹⁶ : Module R M₁\ninst✝¹⁵ : AddCommGroup M₂\ninst✝¹⁴ : Module R M₂\ninst✝¹³ : AddCommGroup M₃\ninst✝¹² : Module R M₃\ninst✝¹¹ : AddCommGroup N₁\ninst✝¹⁰ : Mo... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Limits.MorphismProperty | {
"line": 106,
"column": 4
} | {
"line": 107,
"column": 42
} | {
"line": 107,
"column": 43
} | [
{
"pp": "T : Type u_1\ninst✝⁵ : Category.{v_1, u_1} T\nP : MorphismProperty T\nA : Type u_2\ninst✝⁴ : Category.{v_2, u_2} A\nL : A ⥤ T\ninst✝³ : L.Faithful\ninst✝² : L.Full\nY : A\ninst✝¹ : P.ContainsIdentities\ninst✝ : P.RespectsIso\nX : CostructuredArrow L (L.obj Y)\np : (MorphismProperty.costructuredArrowObj... | [
"T : Type u_1\ninst✝⁵ : Category.{v_1, u_1} T\nP : MorphismProperty T\nA : Type u_2\ninst✝⁴ : Category.{v_2, u_2} A\nL : A ⥤ T\ninst✝³ : L.Faithful\ninst✝² : L.Full\nY : A\ninst✝¹ : P.ContainsIdentities\ninst✝ : P.RespectsIso\nX : CostructuredArrow L (L.obj Y)\np : (MorphismProperty.costructuredArrowObj L P).limits... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Limits.Constructions.Over.Products | {
"line": 74,
"column": 8
} | {
"line": 74,
"column": 19
} | {
"line": 74,
"column": 20
} | [
{
"pp": "J : Type w\nC : Type u\ninst✝ : Category.{v, u} C\nX✝ Y Z : C\nf : Y ⟶ X✝\ng : Z ⟶ X✝\nX : BinaryFan (Over.mk f) (Over.mk g)\n⊢ (Iso.refl\n (({ obj := fun c ↦ PullbackCone.mk (Over.Hom.left c.fst) (Over.Hom.left c.snd) ⋯,\n map := fun {c₁ c₂} a ↦ { hom := Over.Hom.left a.h... | [
"J : Type w\nC : Type u\ninst✝ : Category.{v, u} C\nX✝ Y Z : C\nf : Y ⟶ X✝\ng : Z ⟶ X✝\nX : BinaryFan (Over.mk f) (Over.mk g)\n⊢ X.pt.hom = Over.Hom.left X.fst ≫ f"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Limits.MorphismProperty | {
"line": 147,
"column": 10
} | {
"line": 147,
"column": 21
} | {
"line": 147,
"column": 22
} | [
{
"pp": "T : Type u_1\ninst✝⁷ : Category.{v_1, u_1} T\nP : MorphismProperty T\nA : Type u_2\ninst✝⁶ : Category.{v_2, u_2} A\nL : A ⥤ T\ninst✝⁵ : HasPullbacks A\ninst✝⁴ : HasPullbacks T\ninst✝³ : PreservesLimitsOfShape WalkingCospan L\nX : T\ninst✝² : P.IsStableUnderComposition\ninst✝¹ : P.IsStableUnderBaseChang... | [
"T : Type u_1\ninst✝⁷ : Category.{v_1, u_1} T\nP : MorphismProperty T\nA : Type u_2\ninst✝⁶ : Category.{v_2, u_2} A\nL : A ⥤ T\ninst✝⁵ : HasPullbacks A\ninst✝⁴ : HasPullbacks T\ninst✝³ : PreservesLimitsOfShape WalkingCospan L\nX : T\ninst✝² : P.IsStableUnderComposition\ninst✝¹ : P.IsStableUnderBaseChange\ninst✝ : P... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Limits.MorphismProperty | {
"line": 188,
"column": 4
} | {
"line": 189,
"column": 40
} | {
"line": 189,
"column": 41
} | [
{
"pp": "T : Type u_1\ninst✝⁵ : Category.{v_1, u_1} T\nP : MorphismProperty T\nA : Type u_2\ninst✝⁴ : Category.{v_2, u_2} A\nL : A ⥤ T\ninst✝³ : L.Faithful\ninst✝² : L.Full\nY : A\ninst✝¹ : P.ContainsIdentities\ninst✝ : P.RespectsIso\nX : StructuredArrow (L.obj Y) L\np : (MorphismProperty.structuredArrowObj L P... | [
"T : Type u_1\ninst✝⁵ : Category.{v_1, u_1} T\nP : MorphismProperty T\nA : Type u_2\ninst✝⁴ : Category.{v_2, u_2} A\nL : A ⥤ T\ninst✝³ : L.Faithful\ninst✝² : L.Full\nY : A\ninst✝¹ : P.ContainsIdentities\ninst✝ : P.RespectsIso\nX : StructuredArrow (L.obj Y) L\np : (MorphismProperty.structuredArrowObj L P).colimitsOf... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Limits.MorphismProperty | {
"line": 364,
"column": 2
} | {
"line": 366,
"column": 50
} | {
"line": 368,
"column": 0
} | [
{
"pp": "T : Type u_1\ninst✝⁴ : Category.{v_1, u_1} T\nP : MorphismProperty T\nX : T\ninst✝³ : HasPushouts T\ninst✝² : P.IsStableUnderComposition\ninst✝¹ : P.IsStableUnderCobaseChange\ninst✝ : P.HasOfPrecompProperty P\n⊢ HasPushouts (P.Under ⊤ X)",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants... | [] | apply +allowSynthFailures hasColimitsOfShape_of_closedUnderColimitsOfShape
· exact inferInstanceAs (HasColimitsOfShape WalkingSpan (Under X))
· apply Under.closedUnderColimitsOfShape_pushout | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Limits.MorphismProperty | {
"line": 364,
"column": 2
} | {
"line": 366,
"column": 50
} | {
"line": 368,
"column": 0
} | [
{
"pp": "T : Type u_1\ninst✝⁴ : Category.{v_1, u_1} T\nP : MorphismProperty T\nX : T\ninst✝³ : HasPushouts T\ninst✝² : P.IsStableUnderComposition\ninst✝¹ : P.IsStableUnderCobaseChange\ninst✝ : P.HasOfPrecompProperty P\n⊢ HasPushouts (P.Under ⊤ X)",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants... | [] | apply +allowSynthFailures hasColimitsOfShape_of_closedUnderColimitsOfShape
· exact inferInstanceAs (HasColimitsOfShape WalkingSpan (Under X))
· apply Under.closedUnderColimitsOfShape_pushout | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Limits.Constructions.Over.Products | {
"line": 135,
"column": 15
} | {
"line": 135,
"column": 26
} | {
"line": 135,
"column": 27
} | [
{
"pp": "J : Type w\nC : Type u\ninst✝ : Category.{v, u} C\nX✝ Y Z : C\nf : X✝ ⟶ Y\ng : X✝ ⟶ Z\nX : BinaryCofan (Under.mk f) (Under.mk g)\n⊢ (f ≫ Under.Hom.right X.inl) ≫ 𝟙 X.pt.right = X.pt.hom",
"ppTerm": "?m.341",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"CategoryTheory.instCat... | [
"J : Type w\nC : Type u\ninst✝ : Category.{v, u} C\nX✝ Y Z : C\nf : X✝ ⟶ Y\ng : X✝ ⟶ Z\nX : BinaryCofan (Under.mk f) (Under.mk g)\n⊢ f ≫ Under.Hom.right X.inl = X.pt.hom"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Limits.Constructions.Over.Products | {
"line": 372,
"column": 32
} | {
"line": 372,
"column": 43
} | {
"line": 372,
"column": 44
} | [
{
"pp": "C : Type u\ninst✝ : Category.{v, u} C\nB : C\nF : Discrete PEmpty.{1} ⥤ Over B\ns : Cone F\nm : s.pt ⟶ { pt := mk (𝟙 B), π := { app := fun p ↦ p.as.elim, naturality := ⋯ } }.pt\nx✝ :\n ∀ (j : Discrete PEmpty.{1}),\n m ≫ { pt := mk (𝟙 B), π := { app := fun p ↦ p.as.elim, naturality := ⋯ } }.π.app ... | [
"C : Type u\ninst✝ : Category.{v, u} C\nB : C\nF : Discrete PEmpty.{1} ⥤ Over B\ns : Cone F\nm : s.pt ⟶ { pt := mk (𝟙 B), π := { app := fun p ↦ p.as.elim, naturality := ⋯ } }.pt\nx✝ :\n ∀ (j : Discrete PEmpty.{1}),\n m ≫ { pt := mk (𝟙 B), π := { app := fun p ↦ p.as.elim, naturality := ⋯ } }.π.app j = s.π.app ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.CharP.CharAndCard | {
"line": 89,
"column": 50
} | {
"line": 89,
"column": 66
} | {
"line": 89,
"column": 67
} | [
{
"pp": "R : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : Fintype R\np f : ℕ\nhp : Fact (Nat.Prime p)\nhR : Fintype.card R = p ^ f\nh0 : f = 0\n⊢ Fintype.card R ≤ 1",
"ppTerm": "?m.18",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"R : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : Fintype R\np f : ℕ\nhp : Fact (Nat.Prime p)\nhR : Fintype.card R = p ^ f\nh0 : f = 0\n⊢ Fintype.card R ≤ 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.CharP.CharAndCard | {
"line": 91,
"column": 8
} | {
"line": 91,
"column": 28
} | {
"line": 91,
"column": 29
} | [
{
"pp": "R : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : Fintype R\np f : ℕ\nhp : Fact (Nat.Prime p)\nhR : Fintype.card R = p ^ f\nhf : f ≠ 0\n⊢ ↑p = 0",
"ppTerm": "?m.28",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"R : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : Fintype R\np f : ℕ\nhp : Fact (Nat.Prime p)\nhR : Fintype.card R = p ^ f\nhf : f ≠ 0\n⊢ ↑p = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.MorphismProperty.OverAdjunction | {
"line": 196,
"column": 6
} | {
"line": 196,
"column": 17
} | {
"line": 196,
"column": 18
} | [
{
"pp": "case hfg\nT : Type u_1\ninst✝⁶ : Category.{v_1, u_1} T\nP Q : MorphismProperty T\ninst✝⁵ : Q.IsMultiplicative\nX Y Z : T\ninst✝⁴ : P.IsStableUnderComposition\ninst✝³ : Q.IsStableUnderBaseChange\nf : X ⟶ Y\ninst✝² : P.HasPullbacksAlong f\ninst✝¹ : P.IsStableUnderBaseChangeAlong f\ninst✝ : Q.HasOfPostcom... | [
"case hfg\nT : Type u_1\ninst✝⁶ : Category.{v_1, u_1} T\nP Q : MorphismProperty T\ninst✝⁵ : Q.IsMultiplicative\nX Y Z : T\ninst✝⁴ : P.IsStableUnderComposition\ninst✝³ : Q.IsStableUnderBaseChange\nf : X ⟶ Y\ninst✝² : P.HasPullbacksAlong f\ninst✝¹ : P.IsStableUnderBaseChangeAlong f\ninst✝ : Q.HasOfPostcompProperty Q\... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.CharP.Invertible | {
"line": 50,
"column": 71
} | {
"line": 50,
"column": 82
} | {
"line": 50,
"column": 83
} | [
{
"pp": "R : Type u_1\ninst✝¹ : Ring R\np : ℕ\ninst✝ : CharP R p\nn : ℕ\nthis : ↑(↑n * n.gcdA p + ↑p * n.gcdB p) = ↑↑(n.gcd p)\n⊢ ↑n * ↑(n.gcdA p) = ↑(n.gcd p)",
"ppTerm": "?m.24",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"R : Type u_1\ninst✝¹ : Ring R\np : ℕ\ninst✝ : CharP R p\nn : ℕ\nthis : ↑(↑n * n.gcdA p + ↑p * n.gcdB p) = ↑↑(n.gcd p)\n⊢ ↑n * ↑(n.gcdA p) = ↑(n.gcd p)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.CharP.LinearMaps | {
"line": 51,
"column": 50
} | {
"line": 51,
"column": 93
} | {
"line": 51,
"column": 94
} | [
{
"pp": "R : Type u_1\nM : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\np : ℕ\nhchar : CharP R p\nhtorsion : ∃ x, Ideal.torsionOf R M x = ⊥\nn : ℕ\nexact : ↑n = ↑n • 1\nh : ∀ (x : M), (↑n • 1) x = 0 x\nx : M\nhx : Ideal.torsionOf R M x = ⊥\n⊢ ↑n = 0",
"ppTerm": "?m.106",
... | [
"R : Type u_1\nM : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\np : ℕ\nhchar : CharP R p\nhtorsion : ∃ x, Ideal.torsionOf R M x = ⊥\nn : ℕ\nexact : ↑n = ↑n • 1\nh : ∀ (x : M), (↑n • 1) x = 0 x\nx : M\nhx : Ideal.torsionOf R M x = ⊥\n⊢ ↑n = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.MorphismProperty.OverAdjunction | {
"line": 355,
"column": 6
} | {
"line": 355,
"column": 17
} | {
"line": 355,
"column": 18
} | [
{
"pp": "case hfg\nT : Type u_1\ninst✝⁶ : Category.{v_1, u_1} T\nP Q : MorphismProperty T\ninst✝⁵ : Q.IsMultiplicative\nX Y Z : T\ninst✝⁴ : P.IsStableUnderComposition\ninst✝³ : Q.IsStableUnderCobaseChange\nf : X ⟶ Y\ninst✝² : P.HasPushoutsAlong f\ninst✝¹ : P.IsStableUnderCobaseChangeAlong f\ninst✝ : Q.HasOfPrec... | [
"case hfg\nT : Type u_1\ninst✝⁶ : Category.{v_1, u_1} T\nP Q : MorphismProperty T\ninst✝⁵ : Q.IsMultiplicative\nX Y Z : T\ninst✝⁴ : P.IsStableUnderComposition\ninst✝³ : Q.IsStableUnderCobaseChange\nf : X ⟶ Y\ninst✝² : P.HasPushoutsAlong f\ninst✝¹ : P.IsStableUnderCobaseChangeAlong f\ninst✝ : Q.HasOfPrecompProperty ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.CharP.LocalRing | {
"line": 58,
"column": 17
} | {
"line": 58,
"column": 37
} | {
"line": 58,
"column": 38
} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsLocalRing R\nq : ℕ\nchar_R_q : CharP R q\nq_pos : ¬q = 0\nK : Type u_1 := IsLocalRing.ResidueField R\nRM_char : CharP K (ringChar K)\nr : ℕ := ringChar K\nn : ℕ := q.factorization r\nr_prime : Nat.Prime r\na : ℕ := q / r ^ n\nq_eq_a_mul_rn : q = a * r ^ n\nr... | [
"R : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsLocalRing R\nq : ℕ\nchar_R_q : CharP R q\nq_pos : ¬q = 0\nK : Type u_1 := IsLocalRing.ResidueField R\nRM_char : CharP K (ringChar K)\nr : ℕ := ringChar K\nn : ℕ := q.factorization r\nr_prime : Nat.Prime r\na : ℕ := q / r ^ n\nq_eq_a_mul_rn : q = a * r ^ n\nr_ne_dvd_a : ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Flat.Equalizer | {
"line": 217,
"column": 8
} | {
"line": 217,
"column": 19
} | {
"line": 217,
"column": 20
} | [
{
"pp": "R : Type u_1\ninst✝⁹ : CommRing R\nM : Type u_3\ninst✝⁸ : AddCommGroup M\ninst✝⁷ : Module R M\nN : Type u_4\nP : Type u_5\ninst✝⁶ : AddCommGroup N\ninst✝⁵ : AddCommGroup P\ninst✝⁴ : Module R N\ninst✝³ : Module R P\ninst✝² : Module.Flat R P\nf : N →ₗ[R] P\nhf : Function.Surjective ⇑f\ng : M →ₗ[R] N\nhg ... | [
"R : Type u_1\ninst✝⁹ : CommRing R\nM : Type u_3\ninst✝⁸ : AddCommGroup M\ninst✝⁷ : Module R M\nN : Type u_4\nP : Type u_5\ninst✝⁶ : AddCommGroup N\ninst✝⁵ : AddCommGroup P\ninst✝⁴ : Module R N\ninst✝³ : Module R P\ninst✝² : Module.Flat R P\nf : N →ₗ[R] P\nhf : Function.Surjective ⇑f\ng : M →ₗ[R] N\nhg : Function.I... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Flat.Equalizer | {
"line": 223,
"column": 2
} | {
"line": 223,
"column": 13
} | {
"line": 223,
"column": 14
} | [
{
"pp": "R : Type u_1\ninst✝⁹ : CommRing R\nM : Type u_3\ninst✝⁸ : AddCommGroup M\ninst✝⁷ : Module R M\nN : Type u_4\nP : Type u_5\ninst✝⁶ : AddCommGroup N\ninst✝⁵ : AddCommGroup P\ninst✝⁴ : Module R N\ninst✝³ : Module R P\ninst✝² : Module.Flat R P\nf : N →ₗ[R] P\nhf : Function.Surjective ⇑f\ng : M →ₗ[R] N\nhg ... | [
"R : Type u_1\ninst✝⁹ : CommRing R\nM : Type u_3\ninst✝⁸ : AddCommGroup M\ninst✝⁷ : Module R M\nN : Type u_4\nP : Type u_5\ninst✝⁶ : AddCommGroup N\ninst✝⁵ : AddCommGroup P\ninst✝⁴ : Module R N\ninst✝³ : Module R P\ninst✝² : Module.Flat R P\nf : N →ₗ[R] P\nhf : Function.Surjective ⇑f\ng : M →ₗ[R] N\nhg : Function.I... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.CharP.MixedCharZero | {
"line": 170,
"column": 2
} | {
"line": 170,
"column": 36
} | {
"line": 170,
"column": 37
} | [
{
"pp": "case refine_2\nR : Type u_1\ninst✝¹ : CommRing R\ninst✝ : Algebra ℚ R\nI : Ideal R\na b : ℕ\nh_ab : ↑a = ↑b\nhI : a ≠ b\n⊢ ↑a - ↑b ≠ 0",
"ppTerm": "?refine_2",
"assigned": true,
"usedConstants": [
"Rat.instSub",
"Eq.mpr",
"GroupWithZero.toMonoidWithZero",
"_private.M... | [
"case refine_2\nR : Type u_1\ninst✝¹ : CommRing R\ninst✝ : Algebra ℚ R\nI : Ideal R\na b : ℕ\nh_ab : ↑a = ↑b\nhI : a ≠ b\n⊢ ¬↑a = ↑b"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Flat.Equalizer | {
"line": 334,
"column": 8
} | {
"line": 334,
"column": 21
} | {
"line": 335,
"column": 8
} | [
{
"pp": "case tmul.tmul\nR : Type u_1\nS : Type u_2\ninst✝¹¹ : CommRing R\ninst✝¹⁰ : CommRing S\ninst✝⁹ : Algebra R S\nT : Type u_3\ninst✝⁸ : CommRing T\ninst✝⁷ : Algebra R T\ninst✝⁶ : Algebra S T\ninst✝⁵ : IsScalarTower R S T\nA : Type u_4\nB : Type u_5\ninst✝⁴ : CommRing A\ninst✝³ : CommRing B\ninst✝² : Algeb... | [] | | tmul x z => | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | null |
Mathlib.Algebra.CharZero.Quotient | {
"line": 50,
"column": 75
} | {
"line": 53,
"column": 5
} | {
"line": 55,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝¹ : DivisionRing R\ninst✝ : CharZero R\np r : R\nn : ℕ\nhn : n ≠ 0\n⊢ n • r ∈ zmultiples p ↔ ∃ k, r - ↑k • (p / ↑n) ∈ zmultiples p",
"ppTerm": "?m.32",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"Int.cast",
"Eq.mpr",
"Int.cast_natCast",
... | [] | by
rw [← natCast_zsmul r, zsmul_mem_zmultiples_iff_exists_sub_div (Int.natCast_ne_zero.mpr hn),
Int.cast_natCast]
rfl | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.FieldTheory.IntermediateField.Basic | {
"line": 906,
"column": 4
} | {
"line": 907,
"column": 50
} | {
"line": 908,
"column": 2
} | [
{
"pp": "case refine_1\nK : Type u_1\nL : Type u_2\ninst✝² : Field K\ninst✝¹ : Field L\ninst✝ : Algebra K L\nF E : IntermediateField K L\nh : F ≤ E\nx : L\nhx : x ∈ lift (restrict h)\n⊢ x ∈ F",
"ppTerm": "?refine_1",
"assigned": true,
"usedConstants": [
"IntermediateField.lift",
"Interme... | [] | let y : E := ⟨x, lift_le (restrict h) hx⟩
exact (mem_restrict h y).1 ((mem_lift y).1 hx) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.FieldTheory.IntermediateField.Basic | {
"line": 906,
"column": 4
} | {
"line": 907,
"column": 50
} | {
"line": 908,
"column": 2
} | [
{
"pp": "case refine_1\nK : Type u_1\nL : Type u_2\ninst✝² : Field K\ninst✝¹ : Field L\ninst✝ : Algebra K L\nF E : IntermediateField K L\nh : F ≤ E\nx : L\nhx : x ∈ lift (restrict h)\n⊢ x ∈ F",
"ppTerm": "?refine_1",
"assigned": true,
"usedConstants": [
"IntermediateField.lift",
"Interme... | [] | let y : E := ⟨x, lift_le (restrict h) hx⟩
exact (mem_restrict h y).1 ((mem_lift y).1 hx) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.Seq.Defs | {
"line": 147,
"column": 2
} | {
"line": 147,
"column": 13
} | {
"line": 147,
"column": 14
} | [
{
"pp": "α : Type u\nx : α\ns : Seq α\nh : cons x s = nil\n⊢ False",
"ppTerm": "?m.7",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"α : Type u\nx : α\ns : Seq α\nh : cons x s = nil\n⊢ False"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.Seq.Defs | {
"line": 167,
"column": 4
} | {
"line": 167,
"column": 16
} | {
"line": 168,
"column": 4
} | [
{
"pp": "case mpr\nα : Type u\nx x' : α\ns s' : Seq α\n⊢ x = x' ∧ s = s' → cons x s = cons x' s'",
"ppTerm": "?mpr",
"assigned": true,
"usedConstants": [
"Stream'.Seq",
"_private.Mathlib.Data.Seq.Defs.0.Stream'.Seq.cons_eq_cons.match_1_1",
"And",
"Stream'.Seq.cons",
"Eq... | [
"case mpr\nα : Type u\nx x' : α\ns s' : Seq α\nleft✝ : x = x'\nright✝ : s = s'\n⊢ cons x s = cons x' s'"
] | intro ⟨_, _⟩ | Lean.Elab.Tactic.evalIntro | null |
Mathlib.Data.Seq.Defs | {
"line": 167,
"column": 4
} | {
"line": 167,
"column": 16
} | {
"line": 168,
"column": 4
} | [
{
"pp": "case mpr\nα : Type u\nx x' : α\ns s' : Seq α\n⊢ x = x' ∧ s = s' → cons x s = cons x' s'",
"ppTerm": "?mpr",
"assigned": true,
"usedConstants": [
"Stream'.Seq",
"_private.Mathlib.Data.Seq.Defs.0.Stream'.Seq.cons_eq_cons.match_1_1",
"And",
"Stream'.Seq.cons",
"Eq... | [
"case mpr\nα : Type u\nx x' : α\ns s' : Seq α\nleft✝ : x = x'\nright✝ : s = s'\n⊢ cons x s = cons x' s'"
] | intro ⟨_, _⟩ | Lean.Elab.Tactic.evalIntro | Lean.Parser.Tactic.intro |
Mathlib.Data.Stream.Init | {
"line": 244,
"column": 21
} | {
"line": 244,
"column": 39
} | {
"line": 244,
"column": 40
} | [
{
"pp": "case succ\nα : Type u\nf : α → α\na : α\nn : ℕ\nih : (iterate f a).get (n + 1) = (iterate f (f a)).get n\n⊢ (iterate f a).get (n + 1 + 1) = (iterate f (f a)).get (n + 1)",
"ppTerm": "?succ",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"id",
"instOfNatN... | [
"case succ\nα : Type u\nf : α → α\na : α\nn : ℕ\nih : (iterate f a).get (n + 1) = (iterate f (f a)).get n\n⊢ f ((iterate f a).get (n + 1)) = (iterate f (f a)).get (n + 1)"
] | get_succ_iterate', | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Data.Stream.Init | {
"line": 461,
"column": 21
} | {
"line": 461,
"column": 32
} | {
"line": 461,
"column": 33
} | [
{
"pp": "α : Type u\na : Stream' α\nb : α\nx : List α\nih : ∀ (n : ℕ) (h : n < x.length), (x ++ₛ a).get n = x[n]\nn : ℕ\nh : n + 1 < (b :: x).length\n⊢ n < x.length",
"ppTerm": "?m.47",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"α : Type u\na : Stream' α\nb : α\nx : List α\nih : ∀ (n : ℕ) (h : n < x.length), (x ++ₛ a).get n = x[n]\nn : ℕ\nh : n + 1 < (b :: x).length\n⊢ n < x.length"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.Stream.Init | {
"line": 469,
"column": 66
} | {
"line": 469,
"column": 77
} | {
"line": 469,
"column": 78
} | [
{
"pp": "α : Type u\nx : List α\na b : Stream' α\nn : ℕ\nh : (x ++ₛ a).get (x.length + n) = (x ++ₛ b).get (x.length + n)\n⊢ a.get n = b.get n",
"ppTerm": "?m.28",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"α : Type u\nx : List α\na b : Stream' α\nn : ℕ\nh : (x ++ₛ a).get (x.length + n) = (x ++ₛ b).get (x.length + n)\n⊢ a.get n = b.get n"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.Seq.Defs | {
"line": 544,
"column": 4
} | {
"line": 550,
"column": 29
} | {
"line": 551,
"column": 2
} | [
{
"pp": "case zero\nα : Type u\nC : Seq α → Prop\na : α\nh1 : ∀ (b : α) (s' : Seq α), a = b ∨ C s' → C (cons b s')\ns : Seq α\ne : some a = ↑s 0\n⊢ C s",
"ppTerm": "?zero",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Stream'.Seq",
"congrArg",
"Stream'.Seq.destruct_eq_cons... | [] | have TH : s = cons a (tail s) := by
apply destruct_eq_cons
unfold destruct get? Functor.map
rw [← e]
rfl
rw [TH]
apply h1 _ _ (Or.inl rfl) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.Seq.Defs | {
"line": 544,
"column": 4
} | {
"line": 550,
"column": 29
} | {
"line": 551,
"column": 2
} | [
{
"pp": "case zero\nα : Type u\nC : Seq α → Prop\na : α\nh1 : ∀ (b : α) (s' : Seq α), a = b ∨ C s' → C (cons b s')\ns : Seq α\ne : some a = ↑s 0\n⊢ C s",
"ppTerm": "?zero",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Stream'.Seq",
"congrArg",
"Stream'.Seq.destruct_eq_cons... | [] | have TH : s = cons a (tail s) := by
apply destruct_eq_cons
unfold destruct get? Functor.map
rw [← e]
rfl
rw [TH]
apply h1 _ _ (Or.inl rfl) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.Seq.Defs | {
"line": 568,
"column": 4
} | {
"line": 568,
"column": 29
} | {
"line": 568,
"column": 29
} | [
{
"pp": "α : Type u\nβ : Type v\nγ : Type w\nl : List α\nn : ℕ\nh : l.length ≤ n\n⊢ l.length ≤ n + 1",
"ppTerm": "?m.24",
"assigned": true,
"usedConstants": [
"Nat.le_succ_of_le",
"List.length"
],
"usedFVars": [
"α",
"l",
"n",
"h"
],
"usedGoals": [... | [] | exact Nat.le_succ_of_le h | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Data.Stream.Init | {
"line": 576,
"column": 14
} | {
"line": 576,
"column": 25
} | {
"line": 576,
"column": 26
} | [
{
"pp": "α : Type u\nm n : ℕ\na : Stream' α\nh : take m a <+: take n a\n⊢ m ≤ n",
"ppTerm": "?m.9",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"α : Type u\nm n : ℕ\na : Stream' α\nh : take m a <+: take n a\n⊢ m ≤ n"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.Seq.Basic | {
"line": 93,
"column": 4
} | {
"line": 93,
"column": 41
} | {
"line": 93,
"column": 42
} | [
{
"pp": "case pos\nα : Type u\ns : Seq α\nn : ℕ\nh : s.Terminates\n⊢ s.length' ≤ ↑n ↔ s.TerminatedAt n",
"ppTerm": "?pos✝",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"instCharZeroENat",
"instAddMonoidWithOneENat",
"Stream'.Seq.length'",
"Stream'.Seq.TerminatedAt",
... | [
"case pos\nα : Type u\ns : Seq α\nn : ℕ\nh : s.Terminates\n⊢ s.length h ≤ n ↔ s.TerminatedAt n"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.Seq.Basic | {
"line": 94,
"column": 4
} | {
"line": 94,
"column": 45
} | {
"line": 94,
"column": 46
} | [
{
"pp": "case neg\nα : Type u\ns : Seq α\nn : ℕ\nh : ¬s.Terminates\n⊢ s.length' ≤ ↑n ↔ s.TerminatedAt n",
"ppTerm": "?neg✝",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"ENat.coe_ne_top._simp_1",
"False",
"Stream'.Seq.length'",
"Stream'.Seq.TerminatedAt",
"ENat... | [
"case neg\nα : Type u\ns : Seq α\nn : ℕ\nh : ¬s.Terminates\n⊢ ¬s.TerminatedAt n"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.Seq.Basic | {
"line": 117,
"column": 4
} | {
"line": 117,
"column": 41
} | {
"line": 117,
"column": 42
} | [
{
"pp": "case pos\nα : Type u\ns : Seq α\nn : ℕ\nh : s.Terminates\n⊢ ↑n < s.length' ↔ ∃ a, a ∈ s.get? n",
"ppTerm": "?pos✝",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"instCharZeroENat",
"instAddMonoidWithOneENat",
"Stream'.Seq.length'",
"ENat.instNatCast",
"... | [
"case pos\nα : Type u\ns : Seq α\nn : ℕ\nh : s.Terminates\n⊢ n < s.length h ↔ ∃ a, s.get? n = some a"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.Stream.Init | {
"line": 598,
"column": 12
} | {
"line": 598,
"column": 30
} | {
"line": 598,
"column": 31
} | [
{
"pp": "case zero\nα : Type u\ns₁ s₂ : Stream' α\nh : ∀ (n : ℕ), take n s₁ = take n s₂\n⊢ s₁.get 0 = s₂.get 0",
"ppTerm": "?zero",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"case zero\nα : Type u\ns₁ s₂ : Stream' α\nh : ∀ (n : ℕ), take n s₁ = take n s₂\n⊢ s₁.get 0 = s₂.get 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
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