module string | startPos dict | endPos dict | nextStartPos dict | goals list | goalsAfter list | ppTac string | elaborator string | kind string |
|---|---|---|---|---|---|---|---|---|
Mathlib.Geometry.RingedSpace.OpenImmersion | {
"line": 903,
"column": 11
} | {
"line": 922,
"column": 61
} | {
"line": 924,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝² : Category.{v, u} C\ninst✝¹ : HasLimits C\nι : Type v\nF : Discrete ι ⥤ SheafedSpace C\ninst✝ : HasColimit F\ni j : Discrete ι\nh : i ≠ j\nU : Opens ↑↑(F.obj i).toPresheafedSpace\n⊢ (Opens.map (colimit.ι (F ⋙ forgetToPresheafedSpace) j).base).obj\n ((Opens.map (preservesColimitI... | [] | by
ext x
apply iff_false_intro
rintro ⟨y, hy, eq⟩
replace eq := ConcreteCategory.congr_arg (preservesColimitIso (SheafedSpace.forget C) F ≪≫
HasColimit.isoOfNatIso Discrete.natIsoFunctor ≪≫ TopCat.sigmaIsoSigma.{v, v} _).hom eq
simp_rw [CategoryTheory.Iso.trans_hom, ← TopCat.comp_app, ← PresheafedSpace.co... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.AlgebraicGeometry.OpenImmersion | {
"line": 746,
"column": 35
} | {
"line": 746,
"column": 54
} | {
"line": 746,
"column": 55
} | [
{
"pp": "X Y U : Scheme\nf : Y ⟶ U\ng : U ⟶ X\nh : IsOpenImmersion g\nV : U.Opens\n⊢ Scheme.Hom.app f V = Scheme.Hom.app f V ≫ Y.presheaf.map (eqToHom ⋯ ≫ (Opens.map f.base).map (eqToHom ⋯).op.unop).op",
"ppTerm": "?m.128",
"assigned": true,
"usedConstants": [
"AlgebraicGeometry.Scheme.Hom.ope... | [
"X Y U : Scheme\nf : Y ⟶ U\ng : U ⟶ X\nh : IsOpenImmersion g\nV : U.Opens\n⊢ Scheme.Hom.app f V = Scheme.Hom.app f V ≫ Y.presheaf.map (eqToHom ⋯ ≫ (Opens.map f.base).map (eqToHom ⋯)).op"
] | Quiver.Hom.unop_op, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.Sheaves.CommRingCat | {
"line": 125,
"column": 4
} | {
"line": 125,
"column": 19
} | {
"line": 126,
"column": 4
} | [
{
"pp": "X : TopCat\nC : Type u\ninst✝ : Category.{v, u} C\nF : Presheaf CommRingCat X\nG : F.SubmonoidPresheaf\nS : (x : ↑X) → Submonoid ↑(F.stalk x)\nU V : (Opens ↑X)ᵒᵖ\ni : U ⟶ V\ns : ↑(F.obj U)\nhs : ∀ (i : ↥(unop U)), (CommRingCat.Hom.hom (F.germ (unop U) ↑i ⋯)) s ∈ S ↑i\nx : ↥(unop V)\n⊢ (ConcreteCategory... | [
"X : TopCat\nC : Type u\ninst✝ : Category.{v, u} C\nF : Presheaf CommRingCat X\nG : F.SubmonoidPresheaf\nS : (x : ↑X) → Submonoid ↑(F.stalk x)\nU V : (Opens ↑X)ᵒᵖ\ni : U ⟶ V\ns : ↑(F.obj U)\nhs : ∀ (i : ↥(unop U)), (CommRingCat.Hom.hom (F.germ (unop U) ↑i ⋯)) s ∈ S ↑i\nx : ↥(unop V)\n⊢ (ConcreteCategory.hom (F.germ... | rw [F.germ_res] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.AlgebraicGeometry.StructureSheaf | {
"line": 583,
"column": 4
} | {
"line": 583,
"column": 58
} | {
"line": 585,
"column": 0
} | [
{
"pp": "case e_a\nR M A : Type u\ninst✝⁴ : CommRing R\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : CommRing A\ninst✝ : Algebra R A\nU : Opens ↑(PrimeSpectrum.Top R)\nx : ↑(PrimeSpectrum.Top R)\nhxU : x ∈ U\nr : R\nm : (structureSheafInType R M).obj.obj (op U)\n⊢ r • m = (algebraMap R ((structureShea... | [] | exact (IsScalarTower.algebraMap_smul Γ(R, U) r m).symm | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.AlgebraicGeometry.Restrict | {
"line": 853,
"column": 2
} | {
"line": 853,
"column": 85
} | {
"line": 854,
"column": 2
} | [
{
"pp": "C : Type u₁\ninst✝ : Category.{v, u₁} C\nX Y : Scheme\nf : X ⟶ Y\nU : Y.Opens\nV : X.Opens\ne : V ≤ f ⁻¹ᵁ U\n⊢ U.topIso.hom ≫ (Arrow.mk (Scheme.Hom.appLE f U V e)).hom =\n (Arrow.mk (Scheme.Hom.appTop (Scheme.Hom.resLE f U V e))).hom ≫ V.topIso.hom",
"ppTerm": "?m.40",
"assigned": true,
... | [
"C : Type u₁\ninst✝ : Category.{v, u₁} C\nX Y : Scheme\nf : X ⟶ Y\nU : Y.Opens\nV : X.Opens\ne : V ≤ f ⁻¹ᵁ U\n⊢ Scheme.Hom.appLE f (U.ι ''ᵁ ⊤) V ⋯ = Scheme.Hom.appTop (Scheme.Hom.resLE f U V e) ≫ X.presheaf.map (eqToHom ⋯)"
] | simp only [Scheme.Opens.topIso_hom, eqToHom_op, Arrow.mk_hom, Scheme.Hom.map_appLE] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.CategoryTheory.LocallyDirected | {
"line": 54,
"column": 2
} | {
"line": 56,
"column": 47
} | {
"line": 58,
"column": 0
} | [
{
"pp": "J : Type u_1\ninst✝ : Category.{v_1, u_1} J\nF : Discrete J ⥤ Type u_2\n⊢ F.IsLocallyDirected",
"ppTerm": "?m.5",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"ULift.casesOn",
"CategoryTheory.CategoryStruct.toQuiver",
"Quiver.Hom",
"congrArg",
"Category... | [] | constructor
rintro ⟨i⟩ ⟨j⟩ ⟨k⟩ ⟨⟨⟨⟩⟩⟩ ⟨⟨⟨⟩⟩⟩
simpa using fun x ↦ ⟨i, 𝟙 _, 𝟙 _, x, by simp⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.LocallyDirected | {
"line": 54,
"column": 2
} | {
"line": 56,
"column": 47
} | {
"line": 58,
"column": 0
} | [
{
"pp": "J : Type u_1\ninst✝ : Category.{v_1, u_1} J\nF : Discrete J ⥤ Type u_2\n⊢ F.IsLocallyDirected",
"ppTerm": "?m.5",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"ULift.casesOn",
"CategoryTheory.CategoryStruct.toQuiver",
"Quiver.Hom",
"congrArg",
"Category... | [] | constructor
rintro ⟨i⟩ ⟨j⟩ ⟨k⟩ ⟨⟨⟨⟩⟩⟩ ⟨⟨⟨⟩⟩⟩
simpa using fun x ↦ ⟨i, 𝟙 _, 𝟙 _, x, by simp⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.GlueData | {
"line": 335,
"column": 2
} | {
"line": 335,
"column": 23
} | {
"line": 336,
"column": 2
} | [
{
"pp": "C : Type u₁\ninst✝³ : Category.{v, u₁} C\nD : GlueData C\ninst✝² : HasMulticoequalizer D.diagram\nF : C ⥤ Type v\ninst✝¹ : PreservesColimit D.diagram.multispan F\ninst✝ : ∀ (i j k : D.J), PreservesLimit (cospan (D.f i j) (D.f i k)) F\nx : F.obj D.glued\n⊢ ∃ i y, (ConcreteCategory.hom (F.map (D.ι i))) y... | [
"C : Type u₁\ninst✝³ : Category.{v, u₁} C\nD : GlueData C\ninst✝² : HasMulticoequalizer D.diagram\nF : C ⥤ Type v\ninst✝¹ : PreservesColimit D.diagram.multispan F\ninst✝ : ∀ (i j k : D.J), PreservesLimit (cospan (D.f i j) (D.f i k)) F\nx : F.obj D.glued\ne : F.obj D.glued ≅ (D.mapGlueData F).glued := D.gluedIso F\n... | let e := D.gluedIso F | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLet___1 | Lean.Parser.Tactic.tacticLet__ |
Mathlib.CategoryTheory.GlueData | {
"line": 410,
"column": 40
} | {
"line": 410,
"column": 45
} | {
"line": 410,
"column": 45
} | [
{
"pp": "C : Type u₁\ninst✝¹ : Category.{v, u₁} C\nC' : Type u₂\ninst✝ : Category.{v, u₂} C'\nD : GlueData' C\ni j k : D.J\nhij : i = j\n⊢ (if h : i = k then D.U i else D.V i k h) = if h : j = k then D.U j else D.V j k h",
"ppTerm": "?m.430",
"assigned": true,
"usedConstants": [
"dite_congr",
... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.CategoryTheory.GlueData | {
"line": 410,
"column": 40
} | {
"line": 410,
"column": 45
} | {
"line": 410,
"column": 45
} | [
{
"pp": "C : Type u₁\ninst✝¹ : Category.{v, u₁} C\nC' : Type u₂\ninst✝ : Category.{v, u₂} C'\nD : GlueData' C\ni j k : D.J\nhij : i = j\n⊢ (if h : i = k then D.U i else D.V i k h) = if h : j = k then D.U j else D.V j k h",
"ppTerm": "?m.430",
"assigned": true,
"usedConstants": [
"dite_congr",
... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.GlueData | {
"line": 410,
"column": 40
} | {
"line": 410,
"column": 45
} | {
"line": 410,
"column": 45
} | [
{
"pp": "C : Type u₁\ninst✝¹ : Category.{v, u₁} C\nC' : Type u₂\ninst✝ : Category.{v, u₂} C'\nD : GlueData' C\ni j k : D.J\nhij : i = j\n⊢ (if h : i = k then D.U i else D.V i k h) = if h : j = k then D.U j else D.V j k h",
"ppTerm": "?m.430",
"assigned": true,
"usedConstants": [
"dite_congr",
... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.GlueData | {
"line": 410,
"column": 61
} | {
"line": 410,
"column": 66
} | {
"line": 410,
"column": 66
} | [
{
"pp": "C : Type u₁\ninst✝¹ : Category.{v, u₁} C\nC' : Type u₂\ninst✝ : Category.{v, u₂} C'\nD : GlueData' C\ni j k : D.J\nhij : i = j\n⊢ (if h : i = j then D.U i else D.V i j h) = if h : j = i then D.U j else D.V j i h",
"ppTerm": "?m.431",
"assigned": true,
"usedConstants": [
"dite_cond_eq_... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.CategoryTheory.GlueData | {
"line": 410,
"column": 61
} | {
"line": 410,
"column": 66
} | {
"line": 410,
"column": 66
} | [
{
"pp": "C : Type u₁\ninst✝¹ : Category.{v, u₁} C\nC' : Type u₂\ninst✝ : Category.{v, u₂} C'\nD : GlueData' C\ni j k : D.J\nhij : i = j\n⊢ (if h : i = j then D.U i else D.V i j h) = if h : j = i then D.U j else D.V j i h",
"ppTerm": "?m.431",
"assigned": true,
"usedConstants": [
"dite_cond_eq_... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.GlueData | {
"line": 410,
"column": 61
} | {
"line": 410,
"column": 66
} | {
"line": 410,
"column": 66
} | [
{
"pp": "C : Type u₁\ninst✝¹ : Category.{v, u₁} C\nC' : Type u₂\ninst✝ : Category.{v, u₂} C'\nD : GlueData' C\ni j k : D.J\nhij : i = j\n⊢ (if h : i = j then D.U i else D.V i j h) = if h : j = i then D.U j else D.V j i h",
"ppTerm": "?m.431",
"assigned": true,
"usedConstants": [
"dite_cond_eq_... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.GlueData | {
"line": 410,
"column": 82
} | {
"line": 410,
"column": 87
} | {
"line": 410,
"column": 87
} | [
{
"pp": "C : Type u₁\ninst✝¹ : Category.{v, u₁} C\nC' : Type u₂\ninst✝ : Category.{v, u₂} C'\nD : GlueData' C\ni j k : D.J\nhij : i = j\n⊢ D.U i = D.U j",
"ppTerm": "?m.432",
"assigned": true,
"usedConstants": [
"CategoryTheory.GlueData'.U",
"CategoryTheory.GlueData'.J",
"eq_self",... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.CategoryTheory.GlueData | {
"line": 410,
"column": 82
} | {
"line": 410,
"column": 87
} | {
"line": 410,
"column": 87
} | [
{
"pp": "C : Type u₁\ninst✝¹ : Category.{v, u₁} C\nC' : Type u₂\ninst✝ : Category.{v, u₂} C'\nD : GlueData' C\ni j k : D.J\nhij : i = j\n⊢ D.U i = D.U j",
"ppTerm": "?m.432",
"assigned": true,
"usedConstants": [
"CategoryTheory.GlueData'.U",
"CategoryTheory.GlueData'.J",
"eq_self",... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.GlueData | {
"line": 410,
"column": 82
} | {
"line": 410,
"column": 87
} | {
"line": 410,
"column": 87
} | [
{
"pp": "C : Type u₁\ninst✝¹ : Category.{v, u₁} C\nC' : Type u₂\ninst✝ : Category.{v, u₂} C'\nD : GlueData' C\ni j k : D.J\nhij : i = j\n⊢ D.U i = D.U j",
"ppTerm": "?m.432",
"assigned": true,
"usedConstants": [
"CategoryTheory.GlueData'.U",
"CategoryTheory.GlueData'.J",
"eq_self",... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.AlgebraicGeometry.StructureSheaf | {
"line": 1096,
"column": 4
} | {
"line": 1098,
"column": 48
} | {
"line": 1099,
"column": 2
} | [
{
"pp": "R M A : Type u\ninst✝⁹ : CommRing R\ninst✝⁸ : AddCommGroup M\ninst✝⁷ : Module R M\ninst✝⁶ : CommRing A\ninst✝⁵ : Algebra R A\nS✝ : Type u\ninst✝⁴ : CommRing S✝\nN : Type u\ninst✝³ : AddCommGroup N\ninst✝² : Module S✝ N\nσ : R →+* S✝\nf✝ : M →ₛₗ[σ] N\nS : Type u\ninst✝¹ : CommRing S\nP : Type u\ninst✝ :... | [] | dsimp
simp only [comapₗ_eq_localRingHom, PrimeSpectrum.comap_asIdeal]
exact (Localization.localRingHom ..).map_one | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.AlgebraicGeometry.StructureSheaf | {
"line": 1096,
"column": 4
} | {
"line": 1098,
"column": 48
} | {
"line": 1099,
"column": 2
} | [
{
"pp": "R M A : Type u\ninst✝⁹ : CommRing R\ninst✝⁸ : AddCommGroup M\ninst✝⁷ : Module R M\ninst✝⁶ : CommRing A\ninst✝⁵ : Algebra R A\nS✝ : Type u\ninst✝⁴ : CommRing S✝\nN : Type u\ninst✝³ : AddCommGroup N\ninst✝² : Module S✝ N\nσ : R →+* S✝\nf✝ : M →ₛₗ[σ] N\nS : Type u\ninst✝¹ : CommRing S\nP : Type u\ninst✝ :... | [] | dsimp
simp only [comapₗ_eq_localRingHom, PrimeSpectrum.comap_asIdeal]
exact (Localization.localRingHom ..).map_one | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.GlueData | {
"line": 411,
"column": 12
} | {
"line": 411,
"column": 17
} | {
"line": 411,
"column": 17
} | [
{
"pp": "C : Type u₁\ninst✝¹ : Category.{v, u₁} C\nC' : Type u₂\ninst✝ : Category.{v, u₂} C'\nD : GlueData' C\ni j k : D.J\nhij : i = j\n⊢ D.f' i k ≫ eqToHom ⋯ = eqToHom ⋯ ≫ D.f' j k",
"ppTerm": "?m.433",
"assigned": true,
"usedConstants": [
"CategoryTheory.CategoryStruct.toQuiver",
"Cat... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.CategoryTheory.GlueData | {
"line": 411,
"column": 12
} | {
"line": 411,
"column": 17
} | {
"line": 411,
"column": 17
} | [
{
"pp": "C : Type u₁\ninst✝¹ : Category.{v, u₁} C\nC' : Type u₂\ninst✝ : Category.{v, u₂} C'\nD : GlueData' C\ni j k : D.J\nhij : i = j\n⊢ D.f' i k ≫ eqToHom ⋯ = eqToHom ⋯ ≫ D.f' j k",
"ppTerm": "?m.433",
"assigned": true,
"usedConstants": [
"CategoryTheory.CategoryStruct.toQuiver",
"Cat... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.GlueData | {
"line": 411,
"column": 12
} | {
"line": 411,
"column": 17
} | {
"line": 411,
"column": 17
} | [
{
"pp": "C : Type u₁\ninst✝¹ : Category.{v, u₁} C\nC' : Type u₂\ninst✝ : Category.{v, u₂} C'\nD : GlueData' C\ni j k : D.J\nhij : i = j\n⊢ D.f' i k ≫ eqToHom ⋯ = eqToHom ⋯ ≫ D.f' j k",
"ppTerm": "?m.433",
"assigned": true,
"usedConstants": [
"CategoryTheory.CategoryStruct.toQuiver",
"Cat... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.GlueData | {
"line": 411,
"column": 23
} | {
"line": 411,
"column": 28
} | {
"line": 411,
"column": 28
} | [
{
"pp": "C : Type u₁\ninst✝¹ : Category.{v, u₁} C\nC' : Type u₂\ninst✝ : Category.{v, u₂} C'\nD : GlueData' C\ni j k : D.J\nhij : i = j\n⊢ D.f' i j ≫ eqToHom ⋯ = eqToHom ⋯ ≫ D.f' j i",
"ppTerm": "?m.434",
"assigned": true,
"usedConstants": [
"CategoryTheory.CategoryStruct.toQuiver",
"Cat... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.CategoryTheory.GlueData | {
"line": 411,
"column": 23
} | {
"line": 411,
"column": 28
} | {
"line": 411,
"column": 28
} | [
{
"pp": "C : Type u₁\ninst✝¹ : Category.{v, u₁} C\nC' : Type u₂\ninst✝ : Category.{v, u₂} C'\nD : GlueData' C\ni j k : D.J\nhij : i = j\n⊢ D.f' i j ≫ eqToHom ⋯ = eqToHom ⋯ ≫ D.f' j i",
"ppTerm": "?m.434",
"assigned": true,
"usedConstants": [
"CategoryTheory.CategoryStruct.toQuiver",
"Cat... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.GlueData | {
"line": 411,
"column": 23
} | {
"line": 411,
"column": 28
} | {
"line": 411,
"column": 28
} | [
{
"pp": "C : Type u₁\ninst✝¹ : Category.{v, u₁} C\nC' : Type u₂\ninst✝ : Category.{v, u₂} C'\nD : GlueData' C\ni j k : D.J\nhij : i = j\n⊢ D.f' i j ≫ eqToHom ⋯ = eqToHom ⋯ ≫ D.f' j i",
"ppTerm": "?m.434",
"assigned": true,
"usedConstants": [
"CategoryTheory.CategoryStruct.toQuiver",
"Cat... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.GlueData | {
"line": 426,
"column": 38
} | {
"line": 426,
"column": 43
} | {
"line": 426,
"column": 43
} | [
{
"pp": "C : Type u₁\ninst✝¹ : Category.{v, u₁} C\nC' : Type u₂\ninst✝ : Category.{v, u₂} C'\nD : GlueData' C\ni j k : D.J\nhij : ¬i = j\nhik : ¬i = k\nhjk : ¬j = k\nthis : j ≠ i\n⊢ (if h : i = j then D.U i else D.V i j h) = D.V i j hij",
"ppTerm": "?m.456",
"assigned": true,
"usedConstants": [
... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.CategoryTheory.GlueData | {
"line": 426,
"column": 38
} | {
"line": 426,
"column": 43
} | {
"line": 426,
"column": 43
} | [
{
"pp": "C : Type u₁\ninst✝¹ : Category.{v, u₁} C\nC' : Type u₂\ninst✝ : Category.{v, u₂} C'\nD : GlueData' C\ni j k : D.J\nhij : ¬i = j\nhik : ¬i = k\nhjk : ¬j = k\nthis : j ≠ i\n⊢ (if h : i = j then D.U i else D.V i j h) = D.V i j hij",
"ppTerm": "?m.456",
"assigned": true,
"usedConstants": [
... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.GlueData | {
"line": 426,
"column": 38
} | {
"line": 426,
"column": 43
} | {
"line": 426,
"column": 43
} | [
{
"pp": "C : Type u₁\ninst✝¹ : Category.{v, u₁} C\nC' : Type u₂\ninst✝ : Category.{v, u₂} C'\nD : GlueData' C\ni j k : D.J\nhij : ¬i = j\nhik : ¬i = k\nhjk : ¬j = k\nthis : j ≠ i\n⊢ (if h : i = j then D.U i else D.V i j h) = D.V i j hij",
"ppTerm": "?m.456",
"assigned": true,
"usedConstants": [
... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.GlueData | {
"line": 427,
"column": 42
} | {
"line": 427,
"column": 47
} | {
"line": 427,
"column": 47
} | [
{
"pp": "C : Type u₁\ninst✝¹ : Category.{v, u₁} C\nC' : Type u₂\ninst✝ : Category.{v, u₂} C'\nD : GlueData' C\ni j k : D.J\nhij : ¬i = j\nhik : ¬i = k\nhjk : ¬j = k\nthis : j ≠ i\n⊢ (if h : i = j then eqToHom ⋯ else eqToHom ⋯ ≫ D.f i j h) ≫ eqToHom ⋯ = eqToHom ⋯ ≫ D.f i j hij",
"ppTerm": "?m.466",
"assi... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.CategoryTheory.GlueData | {
"line": 427,
"column": 63
} | {
"line": 427,
"column": 68
} | {
"line": 427,
"column": 68
} | [
{
"pp": "C : Type u₁\ninst✝¹ : Category.{v, u₁} C\nC' : Type u₂\ninst✝ : Category.{v, u₂} C'\nD : GlueData' C\ni j k : D.J\nhij : ¬i = j\nhik : ¬i = k\nhjk : ¬j = k\nthis : j ≠ i\n⊢ (if h : i = k then eqToHom ⋯ else eqToHom ⋯ ≫ D.f i k h) ≫ eqToHom ⋯ = eqToHom ⋯ ≫ D.f i k hik",
"ppTerm": "?m.468",
"assi... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.CategoryTheory.GlueData | {
"line": 429,
"column": 40
} | {
"line": 429,
"column": 45
} | {
"line": 429,
"column": 45
} | [
{
"pp": "C : Type u₁\ninst✝¹ : Category.{v, u₁} C\nC' : Type u₂\ninst✝ : Category.{v, u₂} C'\nD : GlueData' C\ni j k : D.J\nhij : ¬i = j\nhik : ¬i = k\nhjk : ¬j = k\nthis : j ≠ i\n⊢ D.V j k hjk = if h : j = k then D.U j else D.V j k h",
"ppTerm": "?m.469",
"assigned": true,
"usedConstants": [
... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.CategoryTheory.GlueData | {
"line": 429,
"column": 40
} | {
"line": 429,
"column": 45
} | {
"line": 429,
"column": 45
} | [
{
"pp": "C : Type u₁\ninst✝¹ : Category.{v, u₁} C\nC' : Type u₂\ninst✝ : Category.{v, u₂} C'\nD : GlueData' C\ni j k : D.J\nhij : ¬i = j\nhik : ¬i = k\nhjk : ¬j = k\nthis : j ≠ i\n⊢ D.V j k hjk = if h : j = k then D.U j else D.V j k h",
"ppTerm": "?m.469",
"assigned": true,
"usedConstants": [
... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.GlueData | {
"line": 429,
"column": 40
} | {
"line": 429,
"column": 45
} | {
"line": 429,
"column": 45
} | [
{
"pp": "C : Type u₁\ninst✝¹ : Category.{v, u₁} C\nC' : Type u₂\ninst✝ : Category.{v, u₂} C'\nD : GlueData' C\ni j k : D.J\nhij : ¬i = j\nhik : ¬i = k\nhjk : ¬j = k\nthis : j ≠ i\n⊢ D.V j k hjk = if h : j = k then D.U j else D.V j k h",
"ppTerm": "?m.469",
"assigned": true,
"usedConstants": [
... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.GlueData | {
"line": 429,
"column": 61
} | {
"line": 429,
"column": 66
} | {
"line": 429,
"column": 66
} | [
{
"pp": "C : Type u₁\ninst✝¹ : Category.{v, u₁} C\nC' : Type u₂\ninst✝ : Category.{v, u₂} C'\nD : GlueData' C\ni j k : D.J\nhij : ¬i = j\nhik : ¬i = k\nhjk : ¬j = k\nthis : j ≠ i\n⊢ D.V j i ⋯ = if h : j = i then D.U j else D.V j i h",
"ppTerm": "?m.470",
"assigned": true,
"usedConstants": [
"o... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.CategoryTheory.GlueData | {
"line": 429,
"column": 61
} | {
"line": 429,
"column": 66
} | {
"line": 429,
"column": 66
} | [
{
"pp": "C : Type u₁\ninst✝¹ : Category.{v, u₁} C\nC' : Type u₂\ninst✝ : Category.{v, u₂} C'\nD : GlueData' C\ni j k : D.J\nhij : ¬i = j\nhik : ¬i = k\nhjk : ¬j = k\nthis : j ≠ i\n⊢ D.V j i ⋯ = if h : j = i then D.U j else D.V j i h",
"ppTerm": "?m.470",
"assigned": true,
"usedConstants": [
"o... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.GlueData | {
"line": 429,
"column": 61
} | {
"line": 429,
"column": 66
} | {
"line": 429,
"column": 66
} | [
{
"pp": "C : Type u₁\ninst✝¹ : Category.{v, u₁} C\nC' : Type u₂\ninst✝ : Category.{v, u₂} C'\nD : GlueData' C\ni j k : D.J\nhij : ¬i = j\nhik : ¬i = k\nhjk : ¬j = k\nthis : j ≠ i\n⊢ D.V j i ⋯ = if h : j = i then D.U j else D.V j i h",
"ppTerm": "?m.470",
"assigned": true,
"usedConstants": [
"o... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.GlueData | {
"line": 430,
"column": 22
} | {
"line": 430,
"column": 27
} | {
"line": 430,
"column": 27
} | [
{
"pp": "C : Type u₁\ninst✝¹ : Category.{v, u₁} C\nC' : Type u₂\ninst✝ : Category.{v, u₂} C'\nD : GlueData' C\ni j k : D.J\nhij : ¬i = j\nhik : ¬i = k\nhjk : ¬j = k\nthis : j ≠ i\n⊢ D.f j k hjk ≫ eqToHom ⋯ = eqToHom ⋯ ≫ if h : j = k then eqToHom ⋯ else eqToHom ⋯ ≫ D.f j k h",
"ppTerm": "?m.473",
"assign... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.CategoryTheory.GlueData | {
"line": 430,
"column": 43
} | {
"line": 430,
"column": 48
} | {
"line": 430,
"column": 48
} | [
{
"pp": "C : Type u₁\ninst✝¹ : Category.{v, u₁} C\nC' : Type u₂\ninst✝ : Category.{v, u₂} C'\nD : GlueData' C\ni j k : D.J\nhij : ¬i = j\nhik : ¬i = k\nhjk : ¬j = k\nthis : j ≠ i\n⊢ D.f j i ⋯ ≫ eqToHom ⋯ = eqToHom ⋯ ≫ if h : j = i then eqToHom ⋯ else eqToHom ⋯ ≫ D.f j i h",
"ppTerm": "?m.475",
"assigned... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.AlgebraicGeometry.AffineScheme | {
"line": 1031,
"column": 4
} | {
"line": 1031,
"column": 41
} | {
"line": 1032,
"column": 4
} | [
{
"pp": "X : Scheme\nP : ↑X.affineOpens → Prop\nι : Sort u_2\nU : ι → ↑X.affineOpens\niSup_U : ⨆ i, ↑(U i) = ⊤\nV : ↑X.affineOpens\nbasicOpen : ∀ (U : ↑X.affineOpens) (f : ↑Γ(X, ↑U)), P U → P (X.affineBasicOpen f)\nopenCover :\n ∀ (U : ↑X.affineOpens) (s : Finset ↑Γ(X, ↑U)), Ideal.span ↑s = ⊤ → (∀ (f : ↥s), P ... | [
"X : Scheme\nP : ↑X.affineOpens → Prop\nι : Sort u_2\nU : ι → ↑X.affineOpens\niSup_U : ⨆ i, ↑(U i) = ⊤\nV : ↑X.affineOpens\nbasicOpen : ∀ (U : ↑X.affineOpens) (f : ↑Γ(X, ↑U)), P U → P (X.affineBasicOpen f)\nopenCover :\n ∀ (U : ↑X.affineOpens) (s : Finset ↑Γ(X, ↑U)), Ideal.span ↑s = ⊤ → (∀ (f : ↥s), P (X.affineBas... | convert! basicOpen _ g (hU i) using 1 | Mathlib.Tactic._aux_Mathlib_Tactic_Convert___macroRules_Mathlib_Tactic_convert!_1 | Mathlib.Tactic.convert! |
Mathlib.Topology.Gluing | {
"line": 394,
"column": 6
} | {
"line": 394,
"column": 34
} | {
"line": 394,
"column": 34
} | [
{
"pp": "α : Type u\ninst✝ : TopologicalSpace α\nJ : Type u\nU : J → Opens α\ns : Set ↑(ofOpenSubsets U).glued\nhs : IsOpen s\n⊢ IsOpen[inst✝] (⇑(ConcreteCategory.hom (fromOpenSubsetsGlue U)) '' s)",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"CategoryTheory.GlueData.diagram",
... | [
"α : Type u\ninst✝ : TopologicalSpace α\nJ : Type u\nU : J → Opens α\ns : Set ↑(ofOpenSubsets U).glued\nhs : ∀ (i : (ofOpenSubsets U).J), IsOpen (⇑(ConcreteCategory.hom ((ofOpenSubsets U).ι i)) ⁻¹' s)\n⊢ IsOpen[inst✝] (⇑(ConcreteCategory.hom (fromOpenSubsetsGlue U)) '' s)"
] | (ofOpenSubsets U).isOpen_iff | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.Gluing | {
"line": 401,
"column": 4
} | {
"line": 406,
"column": 67
} | {
"line": 407,
"column": 2
} | [
{
"pp": "case right.left\nα : Type u\ninst✝ : TopologicalSpace α\nJ : Type u\nU : J → Opens α\ns : Set ↑(ofOpenSubsets U).glued\nhs : ∀ (i : (ofOpenSubsets U).J), IsOpen (⇑(ConcreteCategory.hom ((ofOpenSubsets U).ι i)) ⁻¹' s)\ni : (ofOpenSubsets U).J\nx : ↑(of α)\nhx' : x ∈ U i\nhx : (ConcreteCategory.hom ((ofO... | [] | rw [← Set.image_preimage_eq_inter_range]
apply (Opens.isOpenEmbedding (X := TopCat.of α) (U i)).isOpenMap
convert! hs i using 1
rw [← ι_fromOpenSubsetsGlue, coe_comp, Set.preimage_comp]
congr! 1
exact Set.preimage_image_eq _ (fromOpenSubsetsGlue_injective U) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Gluing | {
"line": 401,
"column": 4
} | {
"line": 406,
"column": 67
} | {
"line": 407,
"column": 2
} | [
{
"pp": "case right.left\nα : Type u\ninst✝ : TopologicalSpace α\nJ : Type u\nU : J → Opens α\ns : Set ↑(ofOpenSubsets U).glued\nhs : ∀ (i : (ofOpenSubsets U).J), IsOpen (⇑(ConcreteCategory.hom ((ofOpenSubsets U).ι i)) ⁻¹' s)\ni : (ofOpenSubsets U).J\nx : ↑(of α)\nhx' : x ∈ U i\nhx : (ConcreteCategory.hom ((ofO... | [] | rw [← Set.image_preimage_eq_inter_range]
apply (Opens.isOpenEmbedding (X := TopCat.of α) (U i)).isOpenMap
convert! hs i using 1
rw [← ι_fromOpenSubsetsGlue, coe_comp, Set.preimage_comp]
congr! 1
exact Set.preimage_image_eq _ (fromOpenSubsetsGlue_injective U) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.AlgebraicGeometry.Gluing | {
"line": 244,
"column": 2
} | {
"line": 247,
"column": 14
} | {
"line": 248,
"column": 2
} | [
{
"pp": "D : GlueData\ni j : D.J\nx : ↥(D.U i)\ny : ↥(D.U j)\n⊢ (D.ι i) x = (D.ι j) y ↔ D.Rel ⟨i, x⟩ ⟨j, y⟩",
"ppTerm": "?m.28",
"assigned": true,
"usedConstants": [
"CategoryTheory.GlueData.diagram",
"CategoryTheory.Limits.MultispanShape.prod",
"AlgebraicGeometry.Scheme",
"C... | [
"D : GlueData\ni j : D.J\nx : ↥(D.U i)\ny : ↥(D.U j)\n⊢ (D.ι i) x = (D.ι j) y ↔\n (ConcreteCategory.hom\n (D.toLocallyRingedSpaceGlueData.toSheafedSpaceGlueData.toPresheafedSpaceGlueData.toTopGlueData.ι i))\n x =\n (ConcreteCategory.hom\n (D.toLocallyRingedSpaceGlueData.toSheafedSpa... | refine Iff.trans ?_
(TopCat.GlueData.ι_eq_iff_rel
D.toLocallyRingedSpaceGlueData.toSheafedSpaceGlueData.toPresheafedSpaceGlueData.toTopGlueData
i j x y) | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.AlgebraicGeometry.Gluing | {
"line": 352,
"column": 4
} | {
"line": 352,
"column": 24
} | {
"line": 353,
"column": 2
} | [
{
"pp": "case k\nX : Scheme\n𝒰 : X.OpenCover\n⊢ (b : (MultispanShape.prod (gluedCover 𝒰).J).R) → (gluedCover 𝒰).diagram.right b ⟶ X",
"ppTerm": "?k",
"assigned": true,
"usedConstants": [
"CategoryTheory.Limits.MultispanShape.prod",
"AlgebraicGeometry.Scheme",
"CategoryTheory.Pre... | [] | exact fun x => 𝒰.f x | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.AlgebraicGeometry.Gluing | {
"line": 352,
"column": 4
} | {
"line": 352,
"column": 24
} | {
"line": 353,
"column": 2
} | [
{
"pp": "case k\nX : Scheme\n𝒰 : X.OpenCover\n⊢ (b : (MultispanShape.prod (gluedCover 𝒰).J).R) → (gluedCover 𝒰).diagram.right b ⟶ X",
"ppTerm": "?k",
"assigned": true,
"usedConstants": [
"CategoryTheory.Limits.MultispanShape.prod",
"AlgebraicGeometry.Scheme",
"CategoryTheory.Pre... | [] | exact fun x => 𝒰.f x | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.AlgebraicGeometry.Gluing | {
"line": 352,
"column": 4
} | {
"line": 352,
"column": 24
} | {
"line": 353,
"column": 2
} | [
{
"pp": "case k\nX : Scheme\n𝒰 : X.OpenCover\n⊢ (b : (MultispanShape.prod (gluedCover 𝒰).J).R) → (gluedCover 𝒰).diagram.right b ⟶ X",
"ppTerm": "?k",
"assigned": true,
"usedConstants": [
"CategoryTheory.Limits.MultispanShape.prod",
"AlgebraicGeometry.Scheme",
"CategoryTheory.Pre... | [] | exact fun x => 𝒰.f x | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Geometry.RingedSpace.PresheafedSpace.Gluing | {
"line": 333,
"column": 4
} | {
"line": 333,
"column": 43
} | {
"line": 334,
"column": 4
} | [
{
"pp": "case left\nC : Type u\ninst✝¹ : Category.{v, u} C\nD : GlueData C\ninst✝ : HasLimits C\ni : D.J\nU : Opens ↑↑(D.U i)\nj k : D.J\n⊢ (fun x ↦ (ConcreteCategory.hom (D.ι j).base) ((ConcreteCategory.hom (D.f j k).base) x)) ⁻¹' ↑(⋯.functor.obj U) =\n ⇑(ConcreteCategory.hom (colimit.ι D.diagram.multispan ... | [
"case left\nC : Type u\ninst✝¹ : Category.{v, u} C\nD : GlueData C\ninst✝ : HasLimits C\ni : D.J\nU : Opens ↑↑(D.U i)\nj k : D.J\n⊢ ⇑(ConcreteCategory.hom (D.f j k ≫ D.ι j).base) ⁻¹' ↑(⋯.functor.obj U) =\n ⇑(ConcreteCategory.hom (colimit.ι D.diagram.multispan (WalkingMultispan.left (j, k))).base) ⁻¹' ↑(⋯.functor... | change (D.f j k ≫ 𝖣.ι j).base ⁻¹' _ = _ | Lean.Elab.Tactic.evalChange | Lean.Parser.Tactic.change |
Mathlib.AlgebraicGeometry.Limits | {
"line": 620,
"column": 36
} | {
"line": 620,
"column": 41
} | {
"line": 621,
"column": 4
} | [
{
"pp": "case pos\nι : Type u\nf : ι → Scheme\nσ : Type v\ng : σ → Scheme\nX Y : Scheme\nR✝ S : Type u\ninst✝² : CommRing R✝\ninst✝¹ : CommRing S\ni : ι\nR : ι → Type (max u_1 u)\ninst✝ : (i : ι) → CommRing (R i)\nthis : Algebra ((i : ι) → R i) (R i) := (Pi.evalRingHom R i).toAlgebra\nj : ι\nh : j = i\n⊢ (Funct... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.AlgebraicGeometry.Limits | {
"line": 620,
"column": 36
} | {
"line": 620,
"column": 41
} | {
"line": 621,
"column": 4
} | [
{
"pp": "case neg\nι : Type u\nf : ι → Scheme\nσ : Type v\ng : σ → Scheme\nX Y : Scheme\nR✝ S : Type u\ninst✝² : CommRing R✝\ninst✝¹ : CommRing S\ni : ι\nR : ι → Type (max u_1 u)\ninst✝ : (i : ι) → CommRing (R i)\nthis : Algebra ((i : ι) → R i) (R i) := (Pi.evalRingHom R i).toAlgebra\nj : ι\nh : ¬j = i\n⊢ (Func... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.AlgebraicGeometry.Limits | {
"line": 623,
"column": 47
} | {
"line": 623,
"column": 52
} | {
"line": 624,
"column": 6
} | [
{
"pp": "case pos\nι : Type u\nf : ι → Scheme\nσ : Type v\ng : σ → Scheme\nX Y : Scheme\nR✝ S : Type u\ninst✝² : CommRing R✝\ninst✝¹ : CommRing S\ni : ι\nR : ι → Type (max u_1 u)\ninst✝ : (i : ι) → CommRing (R i)\nthis : Algebra ((i : ι) → R i) (R i) := (Pi.evalRingHom R i).toAlgebra\nx y : (a : ι) → R a\ne : (... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.AlgebraicGeometry.Limits | {
"line": 623,
"column": 47
} | {
"line": 623,
"column": 52
} | {
"line": 624,
"column": 6
} | [
{
"pp": "case neg\nι : Type u\nf : ι → Scheme\nσ : Type v\ng : σ → Scheme\nX Y : Scheme\nR✝ S : Type u\ninst✝² : CommRing R✝\ninst✝¹ : CommRing S\ni : ι\nR : ι → Type (max u_1 u)\ninst✝ : (i : ι) → CommRing (R i)\nthis : Algebra ((i : ι) → R i) (R i) := (Pi.evalRingHom R i).toAlgebra\nx y : (a : ι) → R a\ne : (... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.AlgebraicGeometry.Limits | {
"line": 689,
"column": 65
} | {
"line": 689,
"column": 70
} | {
"line": 689,
"column": 70
} | [
{
"pp": "σ : Type v\nX : Scheme\ns : Set σ\nhs : s.Finite\nU : σ → X.Opens\nhU : ∀ i ∈ s, IsAffineOpen (U i)\nhU' : s.Pairwise (Disjoint on U)\nthis : Finite ↑s\ni j : ↑s\ne : i ≠ j\n⊢ ↑i ≠ ↑j",
"ppTerm": "?m.51",
"assigned": true,
"usedConstants": [
"False",
"eq_false",
"congrArg"... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.AlgebraicGeometry.Limits | {
"line": 689,
"column": 65
} | {
"line": 689,
"column": 70
} | {
"line": 689,
"column": 70
} | [
{
"pp": "σ : Type v\nX : Scheme\ns : Set σ\nhs : s.Finite\nU : σ → X.Opens\nhU : ∀ i ∈ s, IsAffineOpen (U i)\nhU' : s.Pairwise (Disjoint on U)\nthis : Finite ↑s\ni j : ↑s\ne : i ≠ j\n⊢ ↑i ≠ ↑j",
"ppTerm": "?m.51",
"assigned": true,
"usedConstants": [
"False",
"eq_false",
"congrArg"... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.AlgebraicGeometry.Limits | {
"line": 689,
"column": 65
} | {
"line": 689,
"column": 70
} | {
"line": 689,
"column": 70
} | [
{
"pp": "σ : Type v\nX : Scheme\ns : Set σ\nhs : s.Finite\nU : σ → X.Opens\nhU : ∀ i ∈ s, IsAffineOpen (U i)\nhU' : s.Pairwise (Disjoint on U)\nthis : Finite ↑s\ni j : ↑s\ne : i ≠ j\n⊢ ↑i ≠ ↑j",
"ppTerm": "?m.51",
"assigned": true,
"usedConstants": [
"False",
"eq_false",
"congrArg"... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.AlgebraicGeometry.Limits | {
"line": 697,
"column": 2
} | {
"line": 697,
"column": 7
} | {
"line": 699,
"column": 0
} | [
{
"pp": "X : Scheme\nU V : X.Opens\nhU : IsAffineOpen U\nhV : IsAffineOpen V\nH : Disjoint U V\n⊢ U ⊔ V = ⨆ i, Sum.elim (fun x ↦ U) (fun x ↦ V) i",
"ppTerm": "?m.73",
"assigned": true,
"usedConstants": [
"Set.ext",
"TopologicalSpace.Opens.coe_iSup",
"SetLike.mem_coe._simp_1",
... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.AlgebraicGeometry.Pullbacks | {
"line": 523,
"column": 62
} | {
"line": 523,
"column": 67
} | {
"line": 524,
"column": 6
} | [
{
"pp": "X Y Z : Scheme\n𝒰✝ : X.OpenCover\nf✝ : X ⟶ Z\ng✝ : Y ⟶ Z\ninst✝ : ∀ (i : 𝒰✝.I₀), HasPullback (𝒰✝.f i ≫ f✝) g✝\ns : PullbackCone f✝ g✝\n𝒰 : X.OpenCover\nf : X ⟶ Z\ng : Y ⟶ Z\ni :\n { I₀ := 𝒰.I₀, X := fun i ↦ pullback (𝒰.f i ≫ f) g,\n f := fun i ↦ pullback.map (𝒰.f i ≫ f) g f g (𝒰.f i) (𝟙 ... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.AlgebraicGeometry.Pullbacks | {
"line": 583,
"column": 2
} | {
"line": 591,
"column": 100
} | {
"line": 592,
"column": 2
} | [
{
"pp": "X Y Z : Scheme\n𝒰✝ : X.OpenCover\nf✝ : X ⟶ Z\ng✝ : Y ⟶ Z\ninst✝ : ∀ (i : 𝒰✝.I₀), HasPullback (𝒰✝.f i ≫ f✝) g✝\ns : PullbackCone f✝ g✝\n𝒰 : Z.OpenCover\nf : X ⟶ Z\ng : Y ⟶ Z\n⊢ (pullback f g).OpenCover",
"ppTerm": "?m.34",
"assigned": true,
"usedConstants": [
"CategoryTheory.Limits... | [
"X Y Z : Scheme\n𝒰✝ : X.OpenCover\nf✝ : X ⟶ Z\ng✝ : Y ⟶ Z\ninst✝ : ∀ (i : 𝒰✝.I₀), HasPullback (𝒰✝.f i ≫ f✝) g✝\ns : PullbackCone f✝ g✝\n𝒰 : Z.OpenCover\nf : X ⟶ Z\ng : Y ⟶ Z\n⊢ ∀ (i : 𝒰.I₀),\n pullback.map (pullback.snd f (𝒰.f i)) (pullback.snd g (𝒰.f i)) f g (pullback.fst f (𝒰.f i)) (pullback.fst g (𝒰.... | apply
(openCoverOfBase' 𝒰 f g).copy 𝒰.I₀
(fun i =>
pullback (pullback.snd _ _ : pullback f (𝒰.f i) ⟶ _)
(pullback.snd _ _ : pullback g (𝒰.f i) ⟶ _))
(fun i =>
pullback.map _ _ _ _ (pullback.fst _ _) (pullback.fst _ _) (𝒰.f i)
pullback.condition.symm pullback.cond... | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.AlgebraicGeometry.Morphisms.Basic | {
"line": 438,
"column": 56
} | {
"line": 438,
"column": 77
} | {
"line": 438,
"column": 77
} | [
{
"pp": "P : AffineTargetMorphismProperty\ninst✝² : P.toProperty.RespectsIso\nH : ∀ ⦃X Y S : Scheme⦄ [inst : IsAffine S] [inst_1 : IsAffine X] (f : X ⟶ S) (g : Y ⟶ S), P g → P (pullback.fst f g)\nZ X Y S : Scheme\ninst✝¹ : IsAffine S\ninst✝ : IsAffine X\nf : X ⟶ S\ng : Y ⟶ S\nf' : Z ⟶ Y\ng' : Z ⟶ X\nh : IsPullb... | [
"P : AffineTargetMorphismProperty\ninst✝² : P.toProperty.RespectsIso\nH : ∀ ⦃X Y S : Scheme⦄ [inst : IsAffine S] [inst_1 : IsAffine X] (f : X ⟶ S) (g : Y ⟶ S), P g → P (pullback.fst f g)\nZ X Y S : Scheme\ninst✝¹ : IsAffine S\ninst✝ : IsAffine X\nf : X ⟶ S\ng : Y ⟶ S\nf' : Z ⟶ Y\ng' : Z ⟶ X\nh : IsPullback g' f' f ... | h.isoPullback_inv_fst | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.AlgebraicGeometry.Morphisms.UnderlyingMap | {
"line": 236,
"column": 22
} | {
"line": 236,
"column": 51
} | {
"line": 236,
"column": 52
} | [
{
"pp": "X Y Z : Scheme\nf : X ⟶ Y\ng : Y ⟶ Z\nH : DenseRange ⇑(f ≫ g)\n⊢ DenseRange ⇑g",
"ppTerm": "?m.25",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"AlgebraicGeometry.SheafedSpace.instTopologicalSpaceCarrierCarrier",
"AlgebraicGeometry.Scheme",
"AlgebraicGeometry.Pres... | [
"X Y Z : Scheme\nf : X ⟶ Y\ng : Y ⟶ Z\nH : closure (Set.range ⇑(f ≫ g)) = Set.univ\n⊢ closure (Set.range ⇑g) = Set.univ"
] | denseRange_iff_closure_range, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.RingHom.Locally | {
"line": 69,
"column": 94
} | {
"line": 73,
"column": 34
} | {
"line": 75,
"column": 0
} | [
{
"pp": "P : {R S : Type u} → [inst : CommRing R] → [inst_1 : CommRing S] → (R →+* S) → Prop\nR S : Type u\ninst✝¹ : CommRing R\ninst✝ : CommRing S\nf : R →+* S\n⊢ Locally (fun {R S} [CommRing R] [CommRing S] ↦ P) f ↔\n Ideal.span {g | P ((algebraMap S (Localization.Away g)).comp f)} = ⊤",
"ppTerm": "?m.... | [] | by
refine ⟨fun ⟨s, hs, h⟩ ↦ ?_, fun h ↦ ⟨_, h, fun g hg ↦ hg⟩⟩
rw [eq_top_iff, ← hs, Ideal.span_le]
intro g hg
exact Ideal.subset_span (h _ hg) | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.AlgebraicGeometry.Morphisms.Constructors | {
"line": 152,
"column": 2
} | {
"line": 152,
"column": 90
} | {
"line": 153,
"column": 2
} | [
{
"pp": "Q : AffineTargetMorphismProperty\ninst✝² : Q.IsLocal\nX Y : Scheme\nf : X ⟶ Y\n𝒰 : X.OpenCover\ninst✝¹ : ∀ (i : 𝒰.I₀), IsAffine (𝒰.X i)\ninst✝ : IsAffine Y\nh𝒰 : ∀ (i j : 𝒰.I₀), Q (pullback.mapDesc (𝒰.f i) (𝒰.f j) f)\n𝒱 : (pullback f f).OpenCover := Scheme.Pullback.openCoverOfLeftRight 𝒰 𝒰 f ... | [
"Q : AffineTargetMorphismProperty\ninst✝² : Q.IsLocal\nX Y : Scheme\nf : X ⟶ Y\n𝒰 : X.OpenCover\ninst✝¹ : ∀ (i : 𝒰.I₀), IsAffine (𝒰.X i)\ninst✝ : IsAffine Y\nh𝒰 : ∀ (i j : 𝒰.I₀), Q (pullback.mapDesc (𝒰.f i) (𝒰.f j) f)\n𝒱 : (pullback f f).OpenCover := Scheme.Pullback.openCoverOfLeftRight 𝒰 𝒰 f f\ni1 : ∀ (i... | rw [← Q.cancel_left_of_respectsIso this.isoPullback.hom, IsPullback.isoPullback_hom_snd] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.AlgebraicGeometry.Morphisms.Constructors | {
"line": 214,
"column": 2
} | {
"line": 215,
"column": 44
} | {
"line": 216,
"column": 2
} | [
{
"pp": "case restrict\nP : MorphismProperty Scheme\nhP₂ : ∀ {X Y : Scheme} (f : X ⟶ Y) {ι : Type u} (U : ι → Y.Opens), IsOpenCover U → (∀ (i : ι), P (f ∣_ U i)) → P f\n⊢ ∀ {X Y : Scheme} (f : X ⟶ Y) (U : Y.Opens), P.universally f → P.universally (f ∣_ U)",
"ppTerm": "?restrict",
"assigned": true,
"... | [
"case of_sSup_eq_top\nP : MorphismProperty Scheme\nhP₂ : ∀ {X Y : Scheme} (f : X ⟶ Y) {ι : Type u} (U : ι → Y.Opens), IsOpenCover U → (∀ (i : ι), P (f ∣_ U i)) → P f\n⊢ ∀ {X Y : Scheme} (f : X ⟶ Y) {ι : Type u} (U : ι → Y.Opens),\n iSup U = ⊤ → (∀ (i : ι), P.universally (f ∣_ U i)) → P.universally f"
] | · exact fun {X Y} f U => P.universally.of_isPullback
(isPullback_morphismRestrict f U).flip | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.AlgebraicGeometry.Morphisms.QuasiCompact | {
"line": 88,
"column": 2
} | {
"line": 88,
"column": 19
} | {
"line": 89,
"column": 2
} | [
{
"pp": "case e'_1\nX : Scheme\nU : X.Opens\n⊢ IsCompact ↑U ↔ IsCompact U.carrier ∧ IsOpen U.carrier",
"ppTerm": "?e'_1",
"assigned": true,
"usedConstants": [
"AlgebraicGeometry.SheafedSpace.instTopologicalSpaceCarrierCarrier",
"AlgebraicGeometry.PresheafedSpace.carrier",
"and_true... | [
"case e'_2\nX : Scheme\nU : X.Opens\ns : Set ↑X.affineOpens\n⊢ U = ⨆ i ∈ s, ↑i ↔ U.carrier = ⋃ i ∈ s, ↑↑i"
] | · simp [U.isOpen] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.AlgebraicGeometry.Morphisms.QuasiCompact | {
"line": 249,
"column": 2
} | {
"line": 249,
"column": 96
} | {
"line": 250,
"column": 2
} | [
{
"pp": "X : Scheme\nU : X.Opens\nhU : IsAffineOpen U\nx f : ↑Γ(X, U)\nH : (x |_ X.basicOpen f) ⋯ = 0\n⊢ ∃ n, f ^ n * x = 0",
"ppTerm": "?m.51",
"assigned": true,
"usedConstants": [
"RingHom.instRingHomClass",
"AlgebraicGeometry.SheafedSpace.instTopologicalSpaceCarrierCarrier",
"Co... | [
"X : Scheme\nU : X.Opens\nhU : IsAffineOpen U\nx f : ↑Γ(X, U)\nH : (x |_ X.basicOpen f) ⋯ = (CommRingCat.Hom.hom (X.presheaf.map (homOfLE ⋯).op)) 0\n⊢ ∃ n, f ^ n * x = 0"
] | rw [← map_zero (X.presheaf.map (homOfLE <| X.basicOpen_le f : X.basicOpen f ⟶ U).op).hom] at H | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.AlgebraicGeometry.Morphisms.QuasiCompact | {
"line": 305,
"column": 4
} | {
"line": 305,
"column": 38
} | {
"line": 306,
"column": 4
} | [
{
"pp": "X : Scheme\nU : X.Opens\nhU : IsCompact ↑U\nf : ↑Γ(X, U)\nhf : X.basicOpen f = ⊥\ne : X.basicOpen f ≤ ⊥\n⊢ ((1 |_ ⊥) ⋯ |_ X.basicOpen f) e = 0",
"ppTerm": "?m.85",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"AlgebraicGeometry.SheafedSpace.instTopologicalSpaceCarrierCarrier",... | [
"X : Scheme\nU : X.Opens\nhU : IsCompact ↑U\nf : ↑Γ(X, U)\nhf : X.basicOpen f = ⊥\ne : X.basicOpen f ≤ ⊥\n⊢ (0 |_ X.basicOpen f) e = 0"
] | rw [Subsingleton.eq_zero (1 |_ ⊥)] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.Ideal.Height | {
"line": 431,
"column": 40
} | {
"line": 431,
"column": 72
} | {
"line": 431,
"column": 72
} | [
{
"pp": "R : Type u_1\ninst✝⁴ : CommRing R\nS : Type u_2\ninst✝³ : CommRing S\ninst✝² : Algebra R S\nM : Submonoid R\ninst✝¹ : IsLocalization M S\np : Ideal R\ninst✝ : p.IsPrime\nh : Disjoint ↑M ↑p\nP : Ideal S := Ideal.map (algebraMap R S) p\nthis✝¹ : P.IsPrime\nthis✝ : IsLocalization.AtPrime (Localization.AtP... | [
"R : Type u_1\ninst✝⁴ : CommRing R\nS : Type u_2\ninst✝³ : CommRing S\ninst✝² : Algebra R S\nM : Submonoid R\ninst✝¹ : IsLocalization M S\np : Ideal R\ninst✝ : p.IsPrime\nh : Disjoint ↑M ↑p\nP : Ideal S := Ideal.map (algebraMap R S) p\nthis✝¹ : P.IsPrime\nthis✝ : IsLocalization.AtPrime (Localization.AtPrime (Ideal.... | AtPrime.ringKrullDim_eq_height p | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Ideal.Height | {
"line": 529,
"column": 2
} | {
"line": 533,
"column": 60
} | {
"line": 535,
"column": 0
} | [
{
"pp": "case a\nR : Type u_1\ninst✝¹ : CommRing R\ninst✝ : Nontrivial R\n⊢ ⨆ I, ⨆ (_ : I ≠ ⊤), I.height ≤ ⨆ I, ⨆ (_ : I.IsMaximal), I.height",
"ppTerm": "?a✝",
"assigned": true,
"usedConstants": [
"instCompleteLinearOrderENat",
"Semiring.toModule",
"CommSemiring.toSemiring",
... | [] | · refine iSup_mono' fun I => ?_
obtain rfl | I_top := eq_or_ne I ⊤
· exact ⟨⊥, by grind [iSup_le_iff, Ideal.IsPrime.ne_top]⟩
· obtain ⟨M, hM, hIM⟩ := exists_le_maximal I I_top
exact ⟨M, iSup_mono' (fun hI ↦ ⟨hM, height_mono hIM⟩)⟩ | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.AlgebraicGeometry.Morphisms.QuasiSeparated | {
"line": 319,
"column": 4
} | {
"line": 319,
"column": 26
} | {
"line": 320,
"column": 2
} | [
{
"pp": "case h\nX : Scheme\nU : X.Opens\nhU : IsCompact U.carrier\nhU'✝ : IsQuasiSeparated ⊥.carrier\nf : ↑Γ(X, ⊥)\nx : ↑Γ(X, X.basicOpen f)\n⊢ X.basicOpen f ≤ ⊥",
"ppTerm": "?h",
"assigned": true,
"usedConstants": [
"AlgebraicGeometry.SheafedSpace.instTopologicalSpaceCarrierCarrier",
"... | [] | exact X.basicOpen_le f | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.AlgebraicGeometry.Morphisms.RingHomProperties | {
"line": 564,
"column": 2
} | {
"line": 586,
"column": 17
} | {
"line": 587,
"column": 2
} | [
{
"pp": "case inr\nP : MorphismProperty Scheme\nQ : {R S : Type u} → [inst : CommRing R] → [inst_1 : CommRing S] → (R →+* S) → Prop\ninst✝¹ : HasRingHomProperty P Q\nhQ : RingHom.StableUnderCompositionWithLocalizationAwaySource fun {R S} [CommRing R] [CommRing S] ↦ Q\nX Y : Scheme\ninst✝ : IsAffine Y\nU : Y.Ope... | [
"P : MorphismProperty Scheme\nQ : {R S : Type u} → [inst : CommRing R] → [inst_1 : CommRing S] → (R →+* S) → Prop\ninst✝¹ : HasRingHomProperty P Q\nhQ : RingHom.StableUnderCompositionWithLocalizationAwaySource fun {R S} [CommRing R] [CommRing S] ↦ Q\nX Y : Scheme\ninst✝ : IsAffine Y\nU : Y.Opens\nf : X ⟶ ↑U\nhf : P... | · obtain ⟨(Us : Set Y.Opens), hUs, heq⟩ := Opens.isBasis_iff_cover.mp (isBasis_basicOpen Y) U
let V (s : Us) : X.Opens := f ⁻¹ᵁ U.ι ⁻¹ᵁ s
rw [IsZariskiLocalAtSource.iff_of_iSup_eq_top (P := P) V]
· intro s
let f' : (V s).toScheme ⟶ U.ι ⁻¹ᵁ s := f ∣_ U.ι ⁻¹ᵁ s
have hf' : P f' := IsZariskiLocalAtT... | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.AlgebraicGeometry.Properties | {
"line": 343,
"column": 2
} | {
"line": 351,
"column": 15
} | {
"line": 353,
"column": 0
} | [
{
"pp": "X : Scheme\ninst✝ : IsIntegral X\nU V : X.Opens\ni : U ⟶ V\nH : Nonempty ↥↑U\n⊢ Function.Injective ⇑(ConcreteCategory.hom (X.presheaf.map i.op))",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"AlgebraicGeometry.irreducibleSpace_of_isIntegral",
"AddGroup.toSubtractionM... | [] | rw [injective_iff_map_eq_zero]
intro x hx
rw [← basicOpen_eq_bot_iff] at hx ⊢
rw [Scheme.basicOpen_res] at hx
revert hx
contrapose!
simp_rw [Ne, ← Opens.not_nonempty_iff_eq_bot, Classical.not_not]
apply nonempty_preirreducible_inter U.isOpen (RingedSpace.basicOpen _ _).isOpen
simpa using H | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.AlgebraicGeometry.Properties | {
"line": 343,
"column": 2
} | {
"line": 351,
"column": 15
} | {
"line": 353,
"column": 0
} | [
{
"pp": "X : Scheme\ninst✝ : IsIntegral X\nU V : X.Opens\ni : U ⟶ V\nH : Nonempty ↥↑U\n⊢ Function.Injective ⇑(ConcreteCategory.hom (X.presheaf.map i.op))",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"AlgebraicGeometry.irreducibleSpace_of_isIntegral",
"AddGroup.toSubtractionM... | [] | rw [injective_iff_map_eq_zero]
intro x hx
rw [← basicOpen_eq_bot_iff] at hx ⊢
rw [Scheme.basicOpen_res] at hx
revert hx
contrapose!
simp_rw [Ne, ← Opens.not_nonempty_iff_eq_bot, Classical.not_not]
apply nonempty_preirreducible_inter U.isOpen (RingedSpace.basicOpen _ _).isOpen
simpa using H | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.AlgebraicGeometry.IdealSheaf.Basic | {
"line": 272,
"column": 30
} | {
"line": 272,
"column": 35
} | {
"line": 272,
"column": 35
} | [
{
"pp": "X : Scheme\nI J : X.IdealSheafData\nι : Type u_1\nU : ι → ↑X.affineOpens\nhU : ⨆ i, ↑(U i) = ⊤\nH : ∀ (i : ι), I.ideal (U i) = J.ideal (U i)\n⊢ ∀ (i : ι), I.ideal (U i) ≤ J.ideal (U i)",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"Semiring.toModule",
"Opposite",
... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.AlgebraicGeometry.IdealSheaf.Basic | {
"line": 272,
"column": 30
} | {
"line": 272,
"column": 35
} | {
"line": 272,
"column": 35
} | [
{
"pp": "X : Scheme\nI J : X.IdealSheafData\nι : Type u_1\nU : ι → ↑X.affineOpens\nhU : ⨆ i, ↑(U i) = ⊤\nH : ∀ (i : ι), I.ideal (U i) = J.ideal (U i)\n⊢ ∀ (i : ι), I.ideal (U i) ≤ J.ideal (U i)",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"Semiring.toModule",
"Opposite",
... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.AlgebraicGeometry.IdealSheaf.Basic | {
"line": 272,
"column": 30
} | {
"line": 272,
"column": 35
} | {
"line": 272,
"column": 35
} | [
{
"pp": "X : Scheme\nI J : X.IdealSheafData\nι : Type u_1\nU : ι → ↑X.affineOpens\nhU : ⨆ i, ↑(U i) = ⊤\nH : ∀ (i : ι), I.ideal (U i) = J.ideal (U i)\n⊢ ∀ (i : ι), I.ideal (U i) ≤ J.ideal (U i)",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"Semiring.toModule",
"Opposite",
... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.AlgebraicGeometry.IdealSheaf.Basic | {
"line": 272,
"column": 75
} | {
"line": 272,
"column": 80
} | {
"line": 272,
"column": 80
} | [
{
"pp": "X : Scheme\nI J : X.IdealSheafData\nι : Type u_1\nU : ι → ↑X.affineOpens\nhU : ⨆ i, ↑(U i) = ⊤\nH : ∀ (i : ι), I.ideal (U i) = J.ideal (U i)\n⊢ ∀ (i : ι), J.ideal (U i) ≤ I.ideal (U i)",
"ppTerm": "?m.29",
"assigned": true,
"usedConstants": [
"Semiring.toModule",
"Opposite",
... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.AlgebraicGeometry.IdealSheaf.Basic | {
"line": 272,
"column": 75
} | {
"line": 272,
"column": 80
} | {
"line": 272,
"column": 80
} | [
{
"pp": "X : Scheme\nI J : X.IdealSheafData\nι : Type u_1\nU : ι → ↑X.affineOpens\nhU : ⨆ i, ↑(U i) = ⊤\nH : ∀ (i : ι), I.ideal (U i) = J.ideal (U i)\n⊢ ∀ (i : ι), J.ideal (U i) ≤ I.ideal (U i)",
"ppTerm": "?m.29",
"assigned": true,
"usedConstants": [
"Semiring.toModule",
"Opposite",
... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.AlgebraicGeometry.IdealSheaf.Basic | {
"line": 272,
"column": 75
} | {
"line": 272,
"column": 80
} | {
"line": 272,
"column": 80
} | [
{
"pp": "X : Scheme\nI J : X.IdealSheafData\nι : Type u_1\nU : ι → ↑X.affineOpens\nhU : ⨆ i, ↑(U i) = ⊤\nH : ∀ (i : ι), I.ideal (U i) = J.ideal (U i)\n⊢ ∀ (i : ι), J.ideal (U i) ≤ I.ideal (U i)",
"ppTerm": "?m.29",
"assigned": true,
"usedConstants": [
"Semiring.toModule",
"Opposite",
... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.AlgebraicGeometry.Morphisms.RingHomProperties | {
"line": 680,
"column": 2
} | {
"line": 707,
"column": 67
} | {
"line": 709,
"column": 0
} | [
{
"pp": "P : MorphismProperty Scheme\nQ : {R S : Type u} → [inst : CommRing R] → [inst_1 : CommRing S] → (R →+* S) → Prop\ninst✝ : HasRingHomProperty P Q\nX Y : Scheme\nf : X ⟶ Y\nhQ : OfLocalizationPrime fun {R S} [CommRing R] [CommRing S] ↦ Q\nH : ∀ (x : ↥X), Q (CommRingCat.Hom.hom (Scheme.Hom.stalkMap f x))\... | [] | have hQi := (HasRingHomProperty.isLocal_ringHomProperty P).respectsIso
wlog hY : IsAffine Y generalizing X Y f
· rw [IsZariskiLocalAtTarget.iff_of_iSup_eq_top (P := P) _ (iSup_affineOpens_eq_top _)]
intro U
refine this (fun x ↦ ?_) U.2
exact (hQi.arrow_mk_iso_iff (AlgebraicGeometry.morphismRestrictStalk... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.AlgebraicGeometry.Morphisms.RingHomProperties | {
"line": 680,
"column": 2
} | {
"line": 707,
"column": 67
} | {
"line": 709,
"column": 0
} | [
{
"pp": "P : MorphismProperty Scheme\nQ : {R S : Type u} → [inst : CommRing R] → [inst_1 : CommRing S] → (R →+* S) → Prop\ninst✝ : HasRingHomProperty P Q\nX Y : Scheme\nf : X ⟶ Y\nhQ : OfLocalizationPrime fun {R S} [CommRing R] [CommRing S] ↦ Q\nH : ∀ (x : ↥X), Q (CommRingCat.Hom.hom (Scheme.Hom.stalkMap f x))\... | [] | have hQi := (HasRingHomProperty.isLocal_ringHomProperty P).respectsIso
wlog hY : IsAffine Y generalizing X Y f
· rw [IsZariskiLocalAtTarget.iff_of_iSup_eq_top (P := P) _ (iSup_affineOpens_eq_top _)]
intro U
refine this (fun x ↦ ?_) U.2
exact (hQi.arrow_mk_iso_iff (AlgebraicGeometry.morphismRestrictStalk... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.AlgebraicGeometry.ResidueField | {
"line": 217,
"column": 62
} | {
"line": 219,
"column": 6
} | {
"line": 221,
"column": 0
} | [
{
"pp": "X : Scheme\nx y : ↥X\nh : x = y\n⊢ X.residue x ≫ (residueFieldCongr h).hom = (X.presheaf.stalkCongr ⋯).hom ≫ X.residue y",
"ppTerm": "?m.39",
"assigned": true,
"usedConstants": [
"AlgebraicGeometry.Scheme.residueFieldCongr",
"AlgebraicGeometry.PresheafedSpace.carrier",
"Ca... | [] | by
subst h
simp | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.AlgebraicGeometry.PullbackCarrier | {
"line": 57,
"column": 28
} | {
"line": 57,
"column": 33
} | {
"line": 59,
"column": 0
} | [
{
"pp": "case mk.mk\nX Y S : Scheme\nf : X ⟶ S\ng : Y ⟶ S\nx✝¹ : ↥X\ny✝¹ : ↥Y\ns✝¹ : ↥S\nhx✝¹ : f x✝¹ = s✝¹\nhy✝¹ : g y✝¹ = s✝¹\nx✝ : ↥X\ny✝ : ↥Y\ns✝ : ↥S\nhx✝ : f x✝ = s✝\nhy✝ : g y✝ = s✝\nex : { x := x✝¹, y := y✝¹, s := s✝¹, hx := hx✝¹, hy := hy✝¹ }.x = { x := x✝, y := y✝, s := s✝, hx := hx✝, hy := hy✝ }.x\ne... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.AlgebraicGeometry.PullbackCarrier | {
"line": 317,
"column": 2
} | {
"line": 317,
"column": 93
} | {
"line": 318,
"column": 2
} | [
{
"pp": "X Y S : Scheme\nf : X ⟶ S\ng : Y ⟶ S\ninst✝² : Nonempty ↥X\ninst✝¹ : Nonempty ↥Y\ninst✝ : Subsingleton ↥S\nthis : Nonempty ↥S\n⊢ Nonempty ↥(pullback f g)",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"AlgebraicGeometry.Scheme.isEmpty_pullback_iff",
"Eq.mpr",
"C... | [
"X Y S : Scheme\nf : X ⟶ S\ng : Y ⟶ S\ninst✝² : Nonempty ↥X\ninst✝¹ : Nonempty ↥Y\ninst✝ : Subsingleton ↥S\nthis : Nonempty ↥S\n⊢ ∃ x ∈ Set.range ⇑f, x ∈ Set.range ⇑g"
] | rw [← not_isEmpty_iff, AlgebraicGeometry.Scheme.isEmpty_pullback_iff, Set.not_disjoint_iff] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.AlgebraicGeometry.IdealSheaf.Basic | {
"line": 895,
"column": 58
} | {
"line": 897,
"column": 81
} | {
"line": 899,
"column": 0
} | [
{
"pp": "X : Scheme\ninst✝ : CompactSpace ↥X\n⊢ Hom.ker X.toSpecΓ = ⊥",
"ppTerm": "?m.7",
"assigned": true,
"usedConstants": [
"RingHom.ker.congr_simp",
"CategoryTheory.Iso.commRingCatIsoToRingEquiv",
"AlgebraicGeometry.isAffineOpen_top",
"Eq.mpr",
"RingHom.instRingHomC... | [] | by
apply IdealSheafData.ext_of_isAffine
simpa using! RingHom.ker_coe_equiv (ΓSpecIso Γ(X, ⊤)).commRingCatIsoToRingEquiv | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.RingHom.EssFiniteType | {
"line": 25,
"column": 2
} | {
"line": 26,
"column": 40
} | {
"line": 28,
"column": 0
} | [
{
"pp": "R : Type u_1\nS : Type u_2\nT : Type u_3\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : CommRing T\nf : R →+* S\ng : S →+* T\nhf : f.EssFiniteType\nhg : g.EssFiniteType\n⊢ (g.comp f).EssFiniteType",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"Algebra.algebraMap",
... | [] | algebraize [f, g, g.comp f]
exact Algebra.EssFiniteType.comp R S T | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.RingHom.EssFiniteType | {
"line": 25,
"column": 2
} | {
"line": 26,
"column": 40
} | {
"line": 28,
"column": 0
} | [
{
"pp": "R : Type u_1\nS : Type u_2\nT : Type u_3\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : CommRing T\nf : R →+* S\ng : S →+* T\nhf : f.EssFiniteType\nhg : g.EssFiniteType\n⊢ (g.comp f).EssFiniteType",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"Algebra.algebraMap",
... | [] | algebraize [f, g, g.comp f]
exact Algebra.EssFiniteType.comp R S T | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Finiteness.FiniteTypeLocal | {
"line": 53,
"column": 10
} | {
"line": 53,
"column": 18
} | {
"line": 53,
"column": 18
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝⁷ : CommRing R\ninst✝⁶ : CommRing S\ninst✝⁵ : Algebra R S\nS' : Type u_4\ninst✝⁴ : CommRing S'\ninst✝³ : Algebra S S'\ninst✝² : Algebra R S'\ninst✝¹ : IsScalarTower R S S'\nM : Submonoid S\ninst✝ : IsLocalization M S'\nx : S\ns : Finset S'\nA : Subalgebra R S\nhA₁ : ↑(f... | [
"R : Type u_1\nS : Type u_2\ninst✝⁷ : CommRing R\ninst✝⁶ : CommRing S\ninst✝⁵ : Algebra R S\nS' : Type u_4\ninst✝⁴ : CommRing S'\ninst✝³ : Algebra S S'\ninst✝² : Algebra R S'\ninst✝¹ : IsScalarTower R S S'\nM : Submonoid S\ninst✝ : IsLocalization M S'\nx : S\ns : Finset S'\nA : Subalgebra R S\nhA₁ : ↑(finsetInteger... | rw [hx₁] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.AlgebraicGeometry.IdealSheaf.Subscheme | {
"line": 504,
"column": 14
} | {
"line": 508,
"column": 60
} | {
"line": 510,
"column": 0
} | [
{
"pp": "X : Scheme\nI : X.IdealSheafData\nx : ↥I.subscheme\n⊢ x ∈ Set.range ⇑(I.glueData.ι (Cover.idx (X.openCoverOfIsOpenCover (fun i ↦ ↑i) ⋯) ↑x) ≫ I.subschemeIso.inv)",
"ppTerm": "?m.42",
"assigned": true,
"usedConstants": [
"AlgebraicGeometry.Scheme.GlueData.ι",
"AlgebraicGeometry.i... | [] | by
let U := (X.openCoverOfIsOpenCover _ (iSup_affineOpens_eq_top X)).idx x.1
obtain ⟨⟨y, hy : y ∈ U.1⟩, rfl : y = x.1⟩ :=
(X.openCoverOfIsOpenCover _ (iSup_affineOpens_eq_top X)).covers x.1
exact (I.opensRange_glueData_ι_subschemeIso_inv U).ge hy | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.AlgebraicGeometry.Morphisms.Affine | {
"line": 271,
"column": 2
} | {
"line": 271,
"column": 34
} | {
"line": 272,
"column": 2
} | [
{
"pp": "X Y Z : Scheme\nf : X ⟶ Z\ng : Y ⟶ Z\nU : X.Opens\nhU : IsAffineOpen U\nV : Y.Opens\nhV : IsCompact ↑V\nW : Z.Opens\nhW : IsAffineOpen W\nhUW : U ≤ f ⁻¹ᵁ W\nhVW : V ≤ g ⁻¹ᵁ W\nthis : IsAffine ↑U\n⊢ IsCompact (↑(pullback.fst f g ⁻¹ᵁ U) ⊓ ↑(pullback.snd f g ⁻¹ᵁ V))",
"ppTerm": "?m.89",
"assigned"... | [
"X Y Z : Scheme\nf : X ⟶ Z\ng : Y ⟶ Z\nU : X.Opens\nhU : IsAffineOpen U\nV : Y.Opens\nhV : IsCompact ↑V\nW : Z.Opens\nhW : IsAffineOpen W\nhUW : U ≤ f ⁻¹ᵁ W\nhVW : V ≤ g ⁻¹ᵁ W\nthis✝ : IsAffine ↑U\nthis : IsAffine ↑W\n⊢ IsCompact (↑(pullback.fst f g ⁻¹ᵁ U) ⊓ ↑(pullback.snd f g ⁻¹ᵁ V))"
] | have : IsAffine W.toScheme := hW | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.AlgebraicGeometry.IdealSheaf.Subscheme | {
"line": 666,
"column": 4
} | {
"line": 668,
"column": 93
} | {
"line": 669,
"column": 4
} | [
{
"pp": "X Y : Scheme\nf : X ⟶ Y\nU : ↑Y.affineOpens\ne : Γ(X, f ⁻¹ᵁ ↑U) ≅ Γ(pullback f (↑U).ι, ⊤) :=\n Functor.mapIso X.presheaf (eqToIso ⋯).op ≪≫ Hom.appIso (pullback.fst f (↑U).ι) ⊤\n⊢ (↑U).topIso.hom ≫ Hom.app f ↑U ≫ e.hom =\n Hom.appTop (IsAffineOpen.isoSpec ⋯).inv ≫ Hom.appTop (pullback.snd f (↑U).ι ≫... | [
"X Y : Scheme\nf : X ⟶ Y\nU : ↑Y.affineOpens\ne : Γ(X, f ⁻¹ᵁ ↑U) ≅ Γ(pullback f (↑U).ι, ⊤) :=\n Functor.mapIso X.presheaf (eqToIso ⋯).op ≪≫ Hom.appIso (pullback.fst f (↑U).ι) ⊤\n⊢ Hom.appLE (pullback.snd f (↑U).ι ≫ (↑U).ι) ((↑U).ι ''ᵁ ⊤) ⊤ ⋯ = Hom.appLE (pullback.snd f (↑U).ι ≫ 𝟙 ↑↑U) ⊤ ⊤ ⋯"
] | simp only [Opens.topIso_hom, eqToHom_op, Hom.app_eq_appLE, Iso.trans_hom, Functor.mapIso_hom,
Iso.op_hom, eqToIso.hom, Hom.appIso_hom, Hom.appLE_map, Hom.map_appLE, Hom.appLE_comp_appLE,
Opens.map_top, e, pullback.condition, IsAffineOpen.toSpecΓ_isoSpec_inv, Category.assoc] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.AlgebraicGeometry.Morphisms.AffineAnd | {
"line": 140,
"column": 2
} | {
"line": 140,
"column": 7
} | {
"line": 142,
"column": 0
} | [
{
"pp": "Q : {R S : Type u} → [inst : CommRing R] → [inst_1 : CommRing S] → (R →+* S) → Prop\nhQi : RingHom.RespectsIso fun {R S} [CommRing R] [CommRing S] ↦ Q\nX Y : Scheme\nf : X ⟶ Y\n⊢ (∀ (U : Y.Opens), IsAffineOpen U → IsAffineOpen (f ⁻¹ᵁ U) ∧ Q (CommRingCat.Hom.hom (Scheme.Hom.app f U))) ↔\n (∀ (U : Y.O... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.AlgebraicGeometry.IdealSheaf.Subscheme | {
"line": 711,
"column": 8
} | {
"line": 711,
"column": 37
} | {
"line": 711,
"column": 38
} | [
{
"pp": "X Y : Scheme\nf : X ⟶ Y\nU : ↑Y.affineOpens\ninst✝ : QuasiCompact f\n⊢ DenseRange ⇑(Hom.toImage f)",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"AlgebraicGeometry.SheafedSpace.instTopologicalSpaceCarrierCarrier",
"AlgebraicGeometry.Scheme.Hom.image",... | [
"X Y : Scheme\nf : X ⟶ Y\nU : ↑Y.affineOpens\ninst✝ : QuasiCompact f\n⊢ closure (Set.range ⇑(Hom.toImage f)) = Set.univ"
] | denseRange_iff_closure_range, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.AlgebraicGeometry.Morphisms.AffineAnd | {
"line": 244,
"column": 4
} | {
"line": 244,
"column": 9
} | {
"line": 246,
"column": 0
} | [
{
"pp": "case refine_2\nQ : {R S : Type u} → [inst : CommRing R] → [inst_1 : CommRing S] → (R →+* S) → Prop\nP : MorphismProperty Scheme\nhQi : RingHom.RespectsIso fun {R S} [CommRing R] [CommRing S] ↦ Q\nhQl : RingHom.LocalizationAwayPreserves fun {R S} [CommRing R] [CommRing S] ↦ Q\nhQs : RingHom.OfLocalizati... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.AlgebraicGeometry.Morphisms.FiniteType | {
"line": 136,
"column": 2
} | {
"line": 136,
"column": 71
} | {
"line": 137,
"column": 2
} | [
{
"pp": "R : CommRingCat\ninst✝¹ : JacobsonSpace ↥(Spec R)\nS : CommRingCat\nφ : R ⟶ S\ninst✝ : LocallyOfFiniteType (Spec.map φ)\n⊢ JacobsonSpace ↥(Spec S)",
"ppTerm": "?m.312",
"assigned": true,
"usedConstants": [
"AlgebraicGeometry.Spec",
"CommRingCat.Hom.hom",
"CommRingCat.carri... | [
"R : CommRingCat\ninst✝¹ : JacobsonSpace ↥(Spec R)\nS : CommRingCat\nφ : R ⟶ S\ninst✝ : LocallyOfFiniteType (Spec.map φ)\nthis : (CommRingCat.Hom.hom φ).FiniteType\n⊢ JacobsonSpace ↥(Spec S)"
] | have : RingHom.FiniteType φ.hom := HasRingHomProperty.Spec_iff.mp ‹_› | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.AlgebraicGeometry.Morphisms.AffineAnd | {
"line": 280,
"column": 2
} | {
"line": 280,
"column": 45
} | {
"line": 282,
"column": 0
} | [
{
"pp": "Q Q' : {R S : Type u} → [inst : CommRing R] → [inst_1 : CommRing S] → (R →+* S) → Prop\nP P' : MorphismProperty Scheme\nhP : HasAffineProperty P (affineAnd fun {R S} [CommRing R] [CommRing S] ↦ Q)\nhP' : HasAffineProperty P' (affineAnd fun {R S} [CommRing R] [CommRing S] ↦ Q')\nhQQ' : ∀ {R S : Type u} ... | [] | exact targetAffineLocally_affineAnd_le hQQ' | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.AlgebraicGeometry.Morphisms.Immersion | {
"line": 65,
"column": 44
} | {
"line": 65,
"column": 75
} | {
"line": 65,
"column": 75
} | [
{
"pp": "X Y Z : Scheme\nf✝ f : X ⟶ Y\ninst✝ : IsImmersion f\n⊢ Set.range ⇑f ⊆ Set.range ⇑(coborderRange f).ι",
"ppTerm": "?m.21",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"AlgebraicGeometry.SheafedSpace.instTopologicalSpaceCarrierCarrier",
"AlgebraicGeometry.PresheafedSpace.... | [] | by simpa using! subset_coborder | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.AlgebraicGeometry.Morphisms.Immersion | {
"line": 98,
"column": 42
} | {
"line": 98,
"column": 73
} | {
"line": 98,
"column": 73
} | [
{
"pp": "X Y Z : Scheme\nf : X ⟶ Y\ninst✝ : IsImmersion f\nthis✝ : IsPreimmersion (Scheme.Hom.liftCoborder f ≫ (Scheme.Hom.coborderRange f).ι)\nthis : IsPreimmersion (Scheme.Hom.liftCoborder f)\n⊢ Set.range ⇑f ⊆ Set.range ⇑(Scheme.Hom.coborderRange f).ι",
"ppTerm": "?m.196",
"assigned": true,
"usedC... | [] | by simpa using! subset_coborder | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.AlgebraicGeometry.Morphisms.Separated | {
"line": 180,
"column": 2
} | {
"line": 182,
"column": 48
} | {
"line": 183,
"column": 2
} | [
{
"pp": "X Y : Scheme\nf : X ⟶ Y\n𝒰 : Y.OpenCover\n𝒱 : (i : 𝒰.I₀) → (pullback f (𝒰.f i)).OpenCover\nx : ↥X\n⊢ (pullback.diagonal f) x ∈ ↑(diagonalCoverDiagonalRange f 𝒰 𝒱)",
"ppTerm": "?m.44",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"AlgebraicGeometry.SheafedSpace.instTopolo... | [
"X Y : Scheme\nf : X ⟶ Y\n𝒰 : Y.OpenCover\n𝒱 : (i : 𝒰.I₀) → (pullback f (𝒰.f i)).OpenCover\nx : ↥X\n⊢ ∃ a b y, ((diagonalCover f 𝒰 𝒱).f ⟨a, (b, b)⟩) y = (pullback.diagonal f) x"
] | simp only [diagonalCoverDiagonalRange, openCoverOfBase_I₀, openCoverOfBase_X,
openCoverOfLeftRight_I₀, Opens.iSup_mk, Opens.carrier_eq_coe, Hom.coe_opensRange, Opens.coe_mk,
Set.mem_iUnion, Set.mem_range, Sigma.exists] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.RingTheory.Spectrum.Prime.ConstructibleSet | {
"line": 149,
"column": 11
} | {
"line": 149,
"column": 28
} | {
"line": 149,
"column": 29
} | [
{
"pp": "R : Type u\ninst✝ : CommRing R\ns : ConstructibleSetData R\nhs : IsConstructible s.toSet\n⊢ ⋃ i,\n Set.range\n (comap\n ((Pi.evalRingHom (fun i ↦ Localization.Away ((Ideal.Quotient.mk (Ideal.span (Set.range (↑i).g))) (↑i).f))\n i).comp\n (algebraMap R\n ... | [
"R : Type u\ninst✝ : CommRing R\ns : ConstructibleSetData R\nhs : IsConstructible s.toSet\n⊢ ⋃ i,\n Set.range\n (comap\n ((Pi.evalRingHom (fun i ↦ Localization.Away ((Ideal.Quotient.mk (Ideal.span (Set.range (↑i).g))) (↑i).f))\n i).comp\n (algebraMap R\n (... | ← Finset.mem_coe, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.RingTheory.Spectrum.Prime.ConstructibleSet | {
"line": 149,
"column": 2
} | {
"line": 149,
"column": 51
} | {
"line": 150,
"column": 2
} | [
{
"pp": "R : Type u\ninst✝ : CommRing R\ns : ConstructibleSetData R\nhs : IsConstructible s.toSet\n⊢ ⋃ i,\n Set.range\n (comap\n ((Pi.evalRingHom (fun i ↦ Localization.Away ((Ideal.Quotient.mk (Ideal.span (Set.range (↑i).g))) (↑i).f))\n i).comp\n (algebraMap R\n ... | [
"R : Type u\ninst✝ : CommRing R\ns : ConstructibleSetData R\nhs : IsConstructible s.toSet\n⊢ ⋃ i,\n Set.range\n (comap\n ((Pi.evalRingHom (fun i ↦ Localization.Away ((Ideal.Quotient.mk (Ideal.span (Set.range (↑i).g))) (↑i).f))\n i).comp\n (algebraMap R\n (... | simp_rw [← Finset.mem_coe, Set.biUnion_eq_iUnion] | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | Mathlib.Tactic.tacticSimp_rw___ |
Mathlib.AlgebraicGeometry.Morphisms.Separated | {
"line": 207,
"column": 4
} | {
"line": 207,
"column": 86
} | {
"line": 208,
"column": 4
} | [
{
"pp": "X Y : Scheme\nf : X ⟶ Y\n𝒰 : Y.OpenCover\n𝒱 : (i : 𝒰.I₀) → (pullback f (𝒰.f i)).OpenCover\ninst✝¹ : ∀ (i : 𝒰.I₀), IsAffine (𝒰.X i)\ninst✝ : ∀ (i : 𝒰.I₀) (j : (𝒱 i).I₀), IsAffine ((𝒱 i).X j)\nU : (i : 𝒰.I₀) × (𝒱 i).I₀ → (↑(diagonalCoverDiagonalRange f 𝒰 𝒱)).Opens :=\n fun i ↦ (diagonalCove... | [
"X Y : Scheme\nf : X ⟶ Y\n𝒰 : Y.OpenCover\n𝒱 : (i : 𝒰.I₀) → (pullback f (𝒰.f i)).OpenCover\ninst✝¹ : ∀ (i : 𝒰.I₀), IsAffine (𝒰.X i)\ninst✝ : ∀ (i : 𝒰.I₀) (j : (𝒱 i).I₀), IsAffine ((𝒱 i).X j)\nU : (i : 𝒰.I₀) × (𝒱 i).I₀ → (↑(diagonalCoverDiagonalRange f 𝒰 𝒱)).Opens :=\n fun i ↦ (diagonalCoverDiagonalRan... | rw [Scheme.Hom.image_preimage_eq_opensRange_inf, inf_eq_right, Opens.opensRange_ι] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
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