module string | startPos dict | endPos dict | nextStartPos dict | goals list | goalsAfter list | ppTac string | elaborator string | kind string |
|---|---|---|---|---|---|---|---|---|
Mathlib.Geometry.RingedSpace.SheafedSpace | {
"line": 290,
"column": 27
} | {
"line": 290,
"column": 44
} | {
"line": 290,
"column": 45
} | [
{
"pp": "C : Type u\ninst✝⁶ : Category.{v, u} C\nFC : C → C → Type u_1\nCC : C → Type v\ninst✝⁵ : (X Y : C) → FunLike (FC X Y) (CC X) (CC Y)\ninstCC : ConcreteCategory C FC\ninst✝⁴ : HasColimits C\ninst✝³ : HasLimits C\ninst✝² : PreservesLimits (CategoryTheory.forget C)\ninst✝¹ : PreservesFilteredColimits (Cate... | [
"C : Type u\ninst✝⁶ : Category.{v, u} C\nFC : C → C → Type u_1\nCC : C → Type v\ninst✝⁵ : (X Y : C) → FunLike (FC X Y) (CC X) (CC Y)\ninstCC : ConcreteCategory C FC\ninst✝⁴ : HasColimits C\ninst✝³ : HasLimits C\ninst✝² : PreservesLimits (CategoryTheory.forget C)\ninst✝¹ : PreservesFilteredColimits (CategoryTheory.f... | Category.id_comp, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Geometry.RingedSpace.Basic | {
"line": 188,
"column": 75
} | {
"line": 188,
"column": 93
} | {
"line": 189,
"column": 8
} | [
{
"pp": "case a\nX : RingedSpace\nU V : (Opens ↑↑X.toPresheafedSpace)ᵒᵖ\ni : U ⟶ V\ninst✝ : IsIso i\nf : ↑(X.presheaf.obj U)\nthis :\n X.basicOpen ((ConcreteCategory.hom (X.presheaf.map (𝟙 U))) f) =\n unop U ⊓ X.basicOpen ((ConcreteCategory.hom (X.presheaf.map i)) f)\n⊢ X.basicOpen f ≤ X.basicOpen ((Concre... | [
"case a\nX : RingedSpace\nU V : (Opens ↑↑X.toPresheafedSpace)ᵒᵖ\ni : U ⟶ V\ninst✝ : IsIso i\nf : ↑(X.presheaf.obj U)\nthis :\n X.basicOpen ((ConcreteCategory.hom (𝟙 (X.presheaf.obj U))) f) =\n unop U ⊓ X.basicOpen ((ConcreteCategory.hom (X.presheaf.map i)) f)\n⊢ X.basicOpen f ≤ X.basicOpen ((ConcreteCategory.h... | X.presheaf.map_id, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Geometry.RingedSpace.Basic | {
"line": 195,
"column": 59
} | {
"line": 199,
"column": 37
} | {
"line": 201,
"column": 0
} | [
{
"pp": "X : RingedSpace\nU : Opens ↑↑X.toPresheafedSpace\nf g : ↑(X.presheaf.obj (op U))\n⊢ X.basicOpen (f * g) = X.basicOpen f ⊓ X.basicOpen g",
"ppTerm": "?m.28",
"assigned": true,
"usedConstants": [
"Set.ext",
"RingHom.instRingHomClass",
"SetLike.mem_coe._simp_1",
"False"... | [] | by
ext x
by_cases hx : x ∈ U
· simp [mem_basicOpen (hx := hx)]
· simp [mt (basicOpen_le X _ ·) hx] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Geometry.RingedSpace.PresheafedSpace.HasColimits | {
"line": 319,
"column": 4
} | {
"line": 319,
"column": 57
} | {
"line": 320,
"column": 2
} | [
{
"pp": "case app\nJ : Type u'\ninst✝⁵ : Category.{v', u'} J\nC : Type u\ninst✝⁴ : Category.{v, u} C\ninst✝³ : HasColimitsOfShape J TopCat\ninst✝² : ∀ (X : TopCat), HasLimitsOfShape Jᵒᵖ (Presheaf C X)\ninst✝¹ : HasLimitsOfShape Jᵒᵖ C\nF : J ⥤ PresheafedSpace C\ninst✝ : HasColimit F\nU : Opens ↑↑(Limits.colimit ... | [] | exact ι_preservesColimitIso_inv (forget C) F (unop X) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.AlgebraicGeometry.Spec | {
"line": 367,
"column": 2
} | {
"line": 368,
"column": 31
} | {
"line": 369,
"column": 2
} | [
{
"pp": "R S : CommRingCat\np : PrimeSpectrum ↑R\ninst✝ : Algebra ↑R ↑S\ny :\n ↑(((TopCat.Presheaf.pushforward CommRingCat (Spec.topMap (CommRingCat.ofHom (algebraMap ↑R ↑S)))).obj\n (structureSheaf ↑S).obj).stalk\n p)\nU : TopologicalSpace.Opens ↑(Spec.topObj (CommRingCat.of ↑R))\nhp : p ∈ U\ns ... | [
"R S : CommRingCat\np : PrimeSpectrum ↑R\ninst✝ : Algebra ↑R ↑S\ny :\n ↑(((TopCat.Presheaf.pushforward CommRingCat (Spec.topMap (CommRingCat.ofHom (algebraMap ↑R ↑S)))).obj\n (structureSheaf ↑S).obj).stalk\n p)\nU : TopologicalSpace.Opens ↑(Spec.topObj (CommRingCat.of ↑R))\nhp : p ∈ U\ns :\n ToType\... | set s' := (Spec.topMap (CommRingCat.ofHom (algebraMap R S)) _* (structureSheaf S).1).map
(homOfLE hrU).op s with h | Mathlib.Tactic._aux_Mathlib_Tactic_Set___elabRules_Mathlib_Tactic_setTactic_1 | Mathlib.Tactic.setTactic |
Mathlib.AlgebraicGeometry.Scheme | {
"line": 872,
"column": 8
} | {
"line": 872,
"column": 26
} | {
"line": 872,
"column": 27
} | [
{
"pp": "X : Scheme\nU V : X.Opens\nh : U = V\nW : (Spec Γ(X, V)).Opens\n⊢ Scheme.Spec.map (X.presheaf.map (𝟙 (op U))).op = 𝟙 (Scheme.Spec.obj (op Γ(X, U)))",
"ppTerm": "?m.90",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"AlgebraicGeometry.SheafedSpace.instTopologicalSpaceCarrierCa... | [
"X : Scheme\nU V : X.Opens\nh : U = V\nW : (Spec Γ(X, V)).Opens\n⊢ Scheme.Spec.map (𝟙 Γ(X, U)).op = 𝟙 (Scheme.Spec.obj (op Γ(X, U)))"
] | X.presheaf.map_id, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.AlgebraicGeometry.OpenImmersion | {
"line": 808,
"column": 61
} | {
"line": 812,
"column": 44
} | {
"line": 814,
"column": 0
} | [
{
"pp": "X Y : Scheme\nf : X ⟶ Y\ninst✝ : IsOpenImmersion f\nhf : Scheme.Hom.opensRange f = ⊤\n⊢ IsIso f",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"Set.range_eq_univ",
"Eq.mpr",
"AlgebraicGeometry.SheafedSpace.instTopologicalSpaceCarrierCarrier",
"AlgebraicGeo... | [] | by
rw [isIso_iff_isOpenImmersion_and_epi_base]
refine ⟨inferInstance, ?_⟩
rw [TopCat.epi_iff_surjective, ← Set.range_eq_univ]
exact TopologicalSpace.Opens.ext_iff.mp hf | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.AlgebraicGeometry.StructureSheaf | {
"line": 140,
"column": 4
} | {
"line": 146,
"column": 62
} | {
"line": 147,
"column": 2
} | [
{
"pp": "R 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\nP : ↑(PrimeSpectrum.Top R)\nU : Opens ↑(PrimeSpectrum.Top R)\na b : (x : ↥U) → Localizations A ↑x\nha : a ∈ (sectionsSubmodule A U).carrier\nhb : b ∈ (sectionsSubmodule A U).carr... | [] | obtain ⟨Va, ma, ia, ra, sa, wa⟩ := ha x
obtain ⟨Vb, mb, ib, rb, sb, wb⟩ := hb x
refine ⟨Va ⊓ Vb, ⟨ma, mb⟩, Opens.infLELeft _ _ ≫ ia, ra * rb, sa * sb, fun x ↦ ?_⟩
obtain ⟨hsax, hsa⟩ := wa ⟨x.1, x.2.1⟩
obtain ⟨hsbx, hsb⟩ := wb ⟨x.1, x.2.2⟩
exact ⟨x.1.asIdeal.primeCompl.mul_mem hsax hsbx,
congr(... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.AlgebraicGeometry.StructureSheaf | {
"line": 140,
"column": 4
} | {
"line": 146,
"column": 62
} | {
"line": 147,
"column": 2
} | [
{
"pp": "R 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\nP : ↑(PrimeSpectrum.Top R)\nU : Opens ↑(PrimeSpectrum.Top R)\na b : (x : ↥U) → Localizations A ↑x\nha : a ∈ (sectionsSubmodule A U).carrier\nhb : b ∈ (sectionsSubmodule A U).carr... | [] | obtain ⟨Va, ma, ia, ra, sa, wa⟩ := ha x
obtain ⟨Vb, mb, ib, rb, sb, wb⟩ := hb x
refine ⟨Va ⊓ Vb, ⟨ma, mb⟩, Opens.infLELeft _ _ ≫ ia, ra * rb, sa * sb, fun x ↦ ?_⟩
obtain ⟨hsax, hsa⟩ := wa ⟨x.1, x.2.1⟩
obtain ⟨hsbx, hsb⟩ := wb ⟨x.1, x.2.2⟩
exact ⟨x.1.asIdeal.primeCompl.mul_mem hsax hsbx,
congr(... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.AlgebraicGeometry.StructureSheaf | {
"line": 307,
"column": 32
} | {
"line": 307,
"column": 44
} | {
"line": 309,
"column": 0
} | [
{
"pp": "R M : Type u\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nf : R\nU : Opens ↑(PrimeSpectrum.Top R)\nhu : U ≤ basicOpen f\nx : ↥(unop (op U))\n⊢ ↑(const 0 f U hu) x = ↑0 x",
"ppTerm": "?m.27",
"assigned": true,
"usedConstants": [
"AlgebraicGeometry.StructureSheaf.L... | [] | by simp; rfl | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.AlgebraicGeometry.GammaSpecAdjunction | {
"line": 207,
"column": 60
} | {
"line": 207,
"column": 82
} | {
"line": 207,
"column": 83
} | [
{
"pp": "X : LocallyRingedSpace\nx : ↑X.toTopCat\n⊢ (CommRingCat.ofHom\n (algebraMap (↑(Γ.obj (op X)))\n ((structureSheafInType ↑(Γ.obj (op X)) ↑(Γ.obj (op X))).obj.obj (op (basicOpen 1)))) ≫\n (structurePresheafInCommRingCat ↑(Γ.obj (op X))).germ (basicOpen 1)\n ((Concrete... | [
"X : LocallyRingedSpace\nx : ↑X.toTopCat\n⊢ (CommRingCat.ofHom\n (algebraMap (↑(Γ.obj (op X)))\n ((structureSheafInType ↑(Γ.obj (op X)) ↑(Γ.obj (op X))).obj.obj (op (basicOpen 1)))) ≫\n X.toΓSpecSheafedSpace.hom.c.app (op (basicOpen 1)) ≫\n ((pushforward CommRingCat X.toΓSpecShea... | stalkFunctor_map_germ, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.AlgebraicGeometry.GammaSpecAdjunction | {
"line": 223,
"column": 4
} | {
"line": 223,
"column": 69
} | {
"line": 224,
"column": 4
} | [
{
"pp": "X : LocallyRingedSpace\nr : ↑(Γ.obj (op X))\nx : ↑X.toTopCat\np : PrimeSpectrum ↑(Γ.obj (op X)) := X.toΓSpecFun x\nS : CommRingCat := (structureSheaf ↑(Γ.obj (op X))).presheaf.stalk p\nt : ↑S\nht : IsUnit ((CommRingCat.Hom.hom (PresheafedSpace.Hom.stalkMap X.toΓSpecSheafedSpace.hom x)) t)\n⊢ IsUnit t",... | [
"X : LocallyRingedSpace\nr✝ : ↑(Γ.obj (op X))\nx : ↑X.toTopCat\np : PrimeSpectrum ↑(Γ.obj (op X)) := X.toΓSpecFun x\nS : CommRingCat := (structureSheaf ↑(Γ.obj (op X))).presheaf.stalk p\nt : ↑S\nht : IsUnit ((CommRingCat.Hom.hom (PresheafedSpace.Hom.stalkMap X.toΓSpecSheafedSpace.hom x)) t)\nr : ↑(Γ.obj (op X))\ns ... | obtain ⟨⟨r, s⟩, he⟩ := IsLocalization.surj p.asIdeal.primeCompl t | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.AlgebraicGeometry.AffineScheme | {
"line": 447,
"column": 31
} | {
"line": 447,
"column": 63
} | {
"line": 448,
"column": 2
} | [
{
"pp": "X : Scheme\nU : X.Opens\nhU : IsAffineOpen U\n⊢ Set.range ⇑(hU.isoSpec.inv ≫ U.ι) = ↑U",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"AlgebraicGeometry.IsAffineOpen.isoSpec",
"AlgebraicGeometry.Spec",
"AlgebraicGeometry.SheafedSpace.instTopologicalSpaceCarrierC... | [
"X : Scheme\nU : X.Opens\nhU : IsAffineOpen U\n⊢ Set.range\n (⇑(TopCat.Hom.hom U.ι.base) ∘\n ⇑(TopCat.Hom.hom (↑U).isoSpec.inv.base) ∘ ⇑(TopCat.Hom.hom (Spec.map (X.presheaf.map (eqToHom ⋯).op)).base)) =\n ↑U"
] | dsimp [IsAffineOpen.isoSpec_inv] | Lean.Elab.Tactic.evalDSimp | Lean.Parser.Tactic.dsimp |
Mathlib.AlgebraicGeometry.GammaSpecAdjunction | {
"line": 290,
"column": 4
} | {
"line": 298,
"column": 64
} | {
"line": 299,
"column": 4
} | [
{
"pp": "case w\nX Y : LocallyRingedSpace\nf : X ⟶ Y\n⊢ ((𝟭 LocallyRingedSpace).obj X).toΓSpec.base ≫ (Spec.locallyRingedSpaceMap (Γ.rightOp.map f).unop).base =\n ((𝟭 LocallyRingedSpace).map f ≫ Y.toΓSpec).base",
"ppTerm": "?w",
"assigned": true,
"usedConstants": [
"AlgebraicGeometry.Pres... | [
"case h\nX Y : LocallyRingedSpace\nf : X ⟶ Y\n⊢ ∀ (r : ↑(unop (Γ.rightOp.obj Y))),\n (Γ.rightOp.map f).unop ≫ ((𝟭 LocallyRingedSpace).obj X).presheaf.map (homOfLE ⋯).op =\n CommRingCat.ofHom\n (algebraMap (↑(unop (Γ.rightOp.obj Y)))\n ((structureSheafInType ↑(unop (Γ.rightOp.obj Y)) ↑(u... | · ext1 x
dsimp
change PrimeSpectrum.comap (f.c.app (op ⊤)).hom (X.toΓSpecFun x) = Y.toΓSpecFun (f.base x)
dsimp [toΓSpecFun]
rw [← IsLocalRing.comap_closedPoint (f.stalkMap x).hom, ←
PrimeSpectrum.comap_comp_apply, ← PrimeSpectrum.comap_comp_apply,
← CommRingCat.hom_comp, ← CommR... | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.AlgebraicGeometry.StructureSheaf | {
"line": 1048,
"column": 51
} | {
"line": 1048,
"column": 81
} | {
"line": 1049,
"column": 2
} | [
{
"pp": "R M A : Type u\ninst✝⁷ : CommRing R\ninst✝⁶ : AddCommGroup M\ninst✝⁵ : Module R M\ninst✝⁴ : CommRing A\ninst✝³ : Algebra R A\nS : Type u\ninst✝² : CommRing S\nN : Type u\ninst✝¹ : AddCommGroup N\ninst✝ : Module S N\nσ : R →+* S\nf : M →ₛₗ[σ] N\nU : Opens ↑(PrimeSpectrum.Top R)\nV : Opens ↑(PrimeSpectru... | [] | dsimp [comapFun]; rw [map_add] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.AlgebraicGeometry.StructureSheaf | {
"line": 1048,
"column": 51
} | {
"line": 1048,
"column": 81
} | {
"line": 1049,
"column": 2
} | [
{
"pp": "R M A : Type u\ninst✝⁷ : CommRing R\ninst✝⁶ : AddCommGroup M\ninst✝⁵ : Module R M\ninst✝⁴ : CommRing A\ninst✝³ : Algebra R A\nS : Type u\ninst✝² : CommRing S\nN : Type u\ninst✝¹ : AddCommGroup N\ninst✝ : Module S N\nσ : R →+* S\nf : M →ₛₗ[σ] N\nU : Opens ↑(PrimeSpectrum.Top R)\nV : Opens ↑(PrimeSpectru... | [] | dsimp [comapFun]; rw [map_add] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.AlgebraicGeometry.GammaSpecAdjunction | {
"line": 470,
"column": 28
} | {
"line": 470,
"column": 45
} | {
"line": 470,
"column": 46
} | [
{
"pp": "R : CommRingCat\nthis : (Spec R).toSpecΓ = 𝟙 (Spec R) ≫ inv (Spec.map (Scheme.ΓSpecIso R).inv)\n⊢ Spec.map (Scheme.ΓSpecIso R).hom = (Spec R).toSpecΓ",
"ppTerm": "?m.32",
"assigned": true,
"usedConstants": [
"AlgebraicGeometry.Spec",
"AlgebraicGeometry.SheafedSpace.instTopologi... | [
"R : CommRingCat\nthis : (Spec R).toSpecΓ = inv (Spec.map (Scheme.ΓSpecIso R).inv)\n⊢ Spec.map (Scheme.ΓSpecIso R).hom = (Spec R).toSpecΓ"
] | Category.id_comp, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.Limits.Types.Coequalizers | {
"line": 43,
"column": 20
} | {
"line": 43,
"column": 69
} | {
"line": 43,
"column": 69
} | [
{
"pp": "X Y Z : Type u\nf g : X ⟶ Y\nx✝¹ : Cofork f g\nx✝ : (Cofork.ofπ (↾Function.Coequalizer.mk ⇑(hom f) ⇑(hom g)) ⋯).pt ⟶ x✝¹.pt\nhm : (Cofork.ofπ (↾Function.Coequalizer.mk ⇑(hom f) ⇑(hom g)) ⋯).π ≫ x✝ = x✝¹.π\n⊢ x✝ = ↾Function.Coequalizer.desc ⇑(hom f) ⇑(hom g) ⇑(hom x✝¹.π) ⋯",
"ppTerm": "?m.117",
... | [] | by ext x; exact Quot.inductionOn x (congr_hom hm) | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.AlgebraicGeometry.AffineScheme | {
"line": 732,
"column": 2
} | {
"line": 732,
"column": 46
} | {
"line": 734,
"column": 0
} | [
{
"pp": "X : Scheme\nU : X.Opens\nhU : IsAffineOpen U\nf : ↑Γ(X, U)\nV : X.Opens\ni : V ⟶ U\ne : V = X.basicOpen f\n⊢ IsLocalization.Away f ↑Γ(X, V)",
"ppTerm": "?m.46",
"assigned": true,
"usedConstants": [
"AlgebraicGeometry.SheafedSpace.instTopologicalSpaceCarrierCarrier",
"CommRingCat... | [] | subst e; exact isLocalization_basicOpen hU f | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.AlgebraicGeometry.AffineScheme | {
"line": 732,
"column": 2
} | {
"line": 732,
"column": 46
} | {
"line": 734,
"column": 0
} | [
{
"pp": "X : Scheme\nU : X.Opens\nhU : IsAffineOpen U\nf : ↑Γ(X, U)\nV : X.Opens\ni : V ⟶ U\ne : V = X.basicOpen f\n⊢ IsLocalization.Away f ↑Γ(X, V)",
"ppTerm": "?m.46",
"assigned": true,
"usedConstants": [
"AlgebraicGeometry.SheafedSpace.instTopologicalSpaceCarrierCarrier",
"CommRingCat... | [] | subst e; exact isLocalization_basicOpen hU f | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.AlgebraicGeometry.AffineScheme | {
"line": 971,
"column": 2
} | {
"line": 971,
"column": 82
} | {
"line": 972,
"column": 2
} | [
{
"pp": "case refine_1\nR S : CommRingCat\nf : R ⟶ S\nr : ↑R\n⊢ (Arrow.mk (Spec.map f ∣_ basicOpen r)).left ≅\n (Arrow.mk (Spec.map (CommRingCat.ofHom (Localization.awayMap (CommRingCat.Hom.hom f) r)))).left",
"ppTerm": "?refine_1",
"assigned": true,
"usedConstants": [
"AlgebraicGeometry.ba... | [
"case refine_2\nR S : CommRingCat\nf : R ⟶ S\nr : ↑R\n⊢ (Arrow.mk (Spec.map f ∣_ basicOpen r)).right ≅\n (Arrow.mk (Spec.map (CommRingCat.ofHom (Localization.awayMap (CommRingCat.Hom.hom f) r)))).right",
"case refine_3\nR S : CommRingCat\nf : R ⟶ S\nr : ↑R\n⊢ ((Spec S).isoOfEq ⋯ ≪≫ basicOpenIsoSpecAway ((CommR... | · exact (Spec _).isoOfEq (comap_basicOpen _ _) ≪≫ basicOpenIsoSpecAway (f.hom r) | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.AlgebraicGeometry.Gluing | {
"line": 327,
"column": 38
} | {
"line": 327,
"column": 55
} | {
"line": 327,
"column": 56
} | [
{
"pp": "case h₀\nX : Scheme\n𝒰 : X.OpenCover\nx y z : 𝒰.I₀\n⊢ (gluedCoverT' 𝒰 x y z ≫ gluedCoverT' 𝒰 y z x ≫ gluedCoverT' 𝒰 z x y) ≫\n pullback.fst (pullback.fst (𝒰.f x) (𝒰.f y)) (pullback.fst (𝒰.f x) (𝒰.f z)) =\n 𝟙 (pullback (pullback.fst (𝒰.f x) (𝒰.f y)) (pullback.fst (𝒰.f x) (𝒰.f z))) ... | [
"case h₀\nX : Scheme\n𝒰 : X.OpenCover\nx y z : 𝒰.I₀\n⊢ (gluedCoverT' 𝒰 x y z ≫ gluedCoverT' 𝒰 y z x ≫ gluedCoverT' 𝒰 z x y) ≫\n pullback.fst (pullback.fst (𝒰.f x) (𝒰.f y)) (pullback.fst (𝒰.f x) (𝒰.f z)) =\n pullback.fst (pullback.fst (𝒰.f x) (𝒰.f y)) (pullback.fst (𝒰.f x) (𝒰.f z))"
] | Category.id_comp, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.AlgebraicGeometry.Gluing | {
"line": 327,
"column": 38
} | {
"line": 327,
"column": 55
} | {
"line": 327,
"column": 56
} | [
{
"pp": "case h₁\nX : Scheme\n𝒰 : X.OpenCover\nx y z : 𝒰.I₀\n⊢ (gluedCoverT' 𝒰 x y z ≫ gluedCoverT' 𝒰 y z x ≫ gluedCoverT' 𝒰 z x y) ≫\n pullback.snd (pullback.fst (𝒰.f x) (𝒰.f y)) (pullback.fst (𝒰.f x) (𝒰.f z)) =\n 𝟙 (pullback (pullback.fst (𝒰.f x) (𝒰.f y)) (pullback.fst (𝒰.f x) (𝒰.f z))) ... | [
"case h₁\nX : Scheme\n𝒰 : X.OpenCover\nx y z : 𝒰.I₀\n⊢ (gluedCoverT' 𝒰 x y z ≫ gluedCoverT' 𝒰 y z x ≫ gluedCoverT' 𝒰 z x y) ≫\n pullback.snd (pullback.fst (𝒰.f x) (𝒰.f y)) (pullback.fst (𝒰.f x) (𝒰.f z)) =\n pullback.snd (pullback.fst (𝒰.f x) (𝒰.f y)) (pullback.fst (𝒰.f x) (𝒰.f z))"
] | Category.id_comp, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.AlgebraicGeometry.Limits | {
"line": 192,
"column": 80
} | {
"line": 198,
"column": 37
} | {
"line": 200,
"column": 0
} | [
{
"pp": "σ : Type v\ng : σ → Scheme\ninst✝ : Small.{u, v} σ\ni j : σ\nx : ↥(g i)\ny : ↥(g j)\n⊢ (Sigma.ι g i) x = (Sigma.ι g j) y ↔ ⟨i, x⟩ = ⟨j, y⟩",
"ppTerm": "?m.22",
"assigned": true,
"usedConstants": [
"AlgebraicGeometry.PresheafedSpace.Hom",
"Eq.mpr",
"AlgebraicGeometry.Scheme... | [] | by
refine (Scheme.IsLocallyDirected.ι_eq_ι_iff _).trans ⟨?_, ?_⟩
· rintro ⟨k, ⟨⟨⟨⟩⟩⟩, ⟨⟨⟨⟩⟩⟩, x, rfl, rfl⟩; simp
· simp only [Discrete.functor_obj_eq_as, Sigma.mk.injEq]
rintro ⟨rfl, e⟩
obtain rfl := (heq_eq_eq x y).mp e
exact ⟨⟨i⟩, 𝟙 _, 𝟙 _, x, by simp⟩ | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.AlgebraicGeometry.Limits | {
"line": 277,
"column": 8
} | {
"line": 278,
"column": 78
} | {
"line": 279,
"column": 6
} | [
{
"pp": "ι : Type u\nf : ι → Scheme\nX : Scheme\nα : (i : ι) → f i ⟶ X\ninst✝ : ∀ (i : ι), IsOpenImmersion (α i)\nhα : _root_.Pairwise (Disjoint on fun x ↦ Set.range ⇑(α x))\nix : ι\nx : ↥(f ix)\niy : ι\ny : ↥(f iy)\ne : (⇑(Sigma.desc α) ∘ ⇑(sigmaMk f)) ⟨ix, x⟩ = (⇑(Sigma.desc α) ∘ ⇑(sigmaMk f)) ⟨iy, y⟩\nthis :... | [] | by_contra h
exact Set.disjoint_iff_forall_ne.mp (hα h) ⟨x, rfl⟩ ⟨y, this.symm⟩ rfl | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.AlgebraicGeometry.Limits | {
"line": 277,
"column": 8
} | {
"line": 278,
"column": 78
} | {
"line": 279,
"column": 6
} | [
{
"pp": "ι : Type u\nf : ι → Scheme\nX : Scheme\nα : (i : ι) → f i ⟶ X\ninst✝ : ∀ (i : ι), IsOpenImmersion (α i)\nhα : _root_.Pairwise (Disjoint on fun x ↦ Set.range ⇑(α x))\nix : ι\nx : ↥(f ix)\niy : ι\ny : ↥(f iy)\ne : (⇑(Sigma.desc α) ∘ ⇑(sigmaMk f)) ⟨ix, x⟩ = (⇑(Sigma.desc α) ∘ ⇑(sigmaMk f)) ⟨iy, y⟩\nthis :... | [] | by_contra h
exact Set.disjoint_iff_forall_ne.mp (hα h) ⟨x, rfl⟩ ⟨y, this.symm⟩ rfl | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.AlgebraicGeometry.Gluing | {
"line": 692,
"column": 4
} | {
"line": 692,
"column": 24
} | {
"line": 693,
"column": 4
} | [
{
"pp": "J : Type w\ninst✝⁴ : Category.{v, w} J\nF : J ⥤ Scheme\ninst✝³ : ∀ {i j : J} (f : i ⟶ j), IsOpenImmersion (F.map f)\ninst✝² : (F ⋙ forget).IsLocallyDirected\ninst✝¹ : Quiver.IsThin J\ninst✝ : Small.{u, w} J\ni j k : Shrink.{u, w} J\nx : failed to pretty print expression (use 'set_option pp.rawOnError t... | [
"J : Type w\ninst✝⁴ : Category.{v, w} J\nF : J ⥤ Scheme\ninst✝³ : ∀ {i j : J} (f : i ⟶ j), IsOpenImmersion (F.map f)\ninst✝² : (F ⋙ forget).IsLocallyDirected\ninst✝¹ : Quiver.IsThin J\ninst✝ : Small.{u, w} J\ni j k : Shrink.{u, w} J\nx : failed to pretty print expression (use 'set_option pp.rawOnError true' for raw... | generalize_proofs h₂ | Batteries.Tactic._aux_Batteries_Tactic_GeneralizeProofs___elabRules_Batteries_Tactic_generalizeProofsElab_1 | Batteries.Tactic.generalizeProofsElab |
Mathlib.AlgebraicGeometry.Pullbacks | {
"line": 593,
"column": 20
} | {
"line": 593,
"column": 37
} | {
"line": 593,
"column": 38
} | [
{
"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\ni : 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 (... | [
"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\ni : 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.f i))\n ... | Category.id_comp, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.AlgebraicGeometry.Morphisms.Basic | {
"line": 207,
"column": 26
} | {
"line": 217,
"column": 52
} | {
"line": 219,
"column": 0
} | [
{
"pp": "P : MorphismProperty Scheme\ninst✝ : IsZariskiLocalAtTarget P\nX Y X' Y' : Scheme\nf : X ⟶ X'\ng : Y ⟶ Y'\nhf : P f\nhg : P g\n⊢ P (coprod.map f g)",
"ppTerm": "?m.22",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"CategoryTheory.Limits.pullback",
"CategoryTheory.Morphis... | [] | by
refine IsZariskiLocalAtTarget.of_openCover (coprodOpenCover.{_, 0} _ _) ?_
rintro (⟨⟨⟩⟩ | ⟨⟨⟩⟩)
· rw [← MorphismProperty.cancel_left_of_respectsIso P
(isPullback_inl_inl_coprodMap f g).flip.isoPullback.hom]
convert! hf
simp [Scheme.Cover.pullbackHom, coprodOpenCover]
· rw [← MorphismProperty.ca... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.AlgebraicGeometry.Morphisms.Basic | {
"line": 241,
"column": 4
} | {
"line": 243,
"column": 35
} | {
"line": 244,
"column": 4
} | [
{
"pp": "case refine_1\nP : MorphismProperty Scheme\ninst✝ : P.RespectsIso\nrestrict : ∀ {X Y : Scheme} (f : X ⟶ Y) (U : X.Opens), P f → P (U.ι ≫ f)\nof_sSup_eq_top :\n ∀ {X Y : Scheme} (f : X ⟶ Y) {ι : Type u} (U : ι → X.Opens), iSup U = ⊤ → (∀ (i : ι), P ((U i).ι ≫ f)) → P f\nX Y : Scheme\nf : X ⟶ Y\n𝒰 : Sc... | [
"case refine_1\nP : MorphismProperty Scheme\ninst✝ : P.RespectsIso\nrestrict : ∀ {X Y : Scheme} (f : X ⟶ Y) (U : X.Opens), P f → P (U.ι ≫ f)\nof_sSup_eq_top :\n ∀ {X Y : Scheme} (f : X ⟶ Y) {ι : Type u} (U : ι → X.Opens), iSup U = ⊤ → (∀ (i : ι), P ((U i).ι ≫ f)) → P f\nX Y : Scheme\nf : X ⟶ Y\n𝒰 : Scheme.zariski... | rw [← IsOpenImmersion.isoOfRangeEq_hom_fac (𝒰.f i) (Scheme.Opens.ι _)
(congr_arg Opens.carrier (𝒰.f i).opensRange.opensRange_ι.symm), Category.assoc,
P.cancel_left_of_respectsIso] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.AlgebraicGeometry.Pullbacks | {
"line": 678,
"column": 4
} | {
"line": 678,
"column": 50
} | {
"line": 679,
"column": 2
} | [
{
"pp": "case h₁\nX 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\nf : X ⟶ Y\n𝒰 : Y.OpenCover\n𝒱 : (i : (Precoverage.ZeroHypercover.pullback₁ f 𝒰).I₀) → ((Precoverage.ZeroHypercover.pullback₁ f 𝒰).X i).OpenCover\ni : (open... | [] | · simp [pullback.condition, Cover.pullbackHom] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.AlgebraicGeometry.Gluing | {
"line": 848,
"column": 16
} | {
"line": 848,
"column": 33
} | {
"line": 848,
"column": 34
} | [
{
"pp": "case h\nJ : Type w\ninst✝⁴ : Category.{v, w} J\nF : J ⥤ Scheme\ninst✝³ : ∀ {i j : J} (f : i ⟶ j), IsOpenImmersion (F.map f)\ninst✝² : (F ⋙ forget).IsLocallyDirected\ninst✝¹ : Quiver.IsThin J\ninst✝ : Small.{u, w} J\nx : ↥(Limits.colimit F)\ni : (glueData F).J\nxi : ↥((glueData F).U i)\nh : ((glueData F... | [
"case h\nJ : Type w\ninst✝⁴ : Category.{v, w} J\nF : J ⥤ Scheme\ninst✝³ : ∀ {i j : J} (f : i ⟶ j), IsOpenImmersion (F.map f)\ninst✝² : (F ⋙ forget).IsLocallyDirected\ninst✝¹ : Quiver.IsThin J\ninst✝ : Small.{u, w} J\nx : ↥(Limits.colimit F)\ni : (glueData F).J\nxi : ↥((glueData F).U i)\nh : ((glueData F).ι i) xi = ... | colimit.cocone_x, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Geometry.RingedSpace.PresheafedSpace.Gluing | {
"line": 492,
"column": 16
} | {
"line": 492,
"column": 69
} | {
"line": 492,
"column": 69
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\nD : GlueData C\ninst✝ : HasLimits C\ni : D.J\nU : Opens ↑↑(D.U i)\n⊢ IsIso ((D.ι i).c.app (op (⋯.functor.obj U)))",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"CategoryTheory.Limits.limit.π",
"CategoryTheory.GlueData.diagram",... | [
"C : Type u\ninst✝¹ : Category.{v, u} C\nD : GlueData C\ninst✝ : HasLimits C\ni : D.J\nU : Opens ↑↑(D.U i)\n⊢ IsIso\n ((colimitPresheafObjIsoComponentwiseLimit D.diagram.multispan (⋯.functor.obj U)).hom ≫\n limit.π (componentwiseDiagram D.diagram.multispan (⋯.functor.obj U)) (op (WalkingMultispan.right i)))... | erw [← colimitPresheafObjIsoComponentwiseLimit_hom_π] | Lean.Parser.Tactic._aux_Init_Meta___macroRules_Lean_Parser_Tactic_tacticErw____1 | Lean.Parser.Tactic.tacticErw___ |
Mathlib.RingTheory.RingHom.Locally | {
"line": 191,
"column": 6
} | {
"line": 191,
"column": 57
} | {
"line": 192,
"column": 6
} | [
{
"pp": "case refine_2\nP : {R S : Type u} → [inst : CommRing R] → [inst_1 : CommRing S] → (R →+* S) → Prop\nhPi : RespectsIso fun {R S} [CommRing R] [CommRing S] ↦ P\nR S T : Type u\nx✝³ : CommRing R\nx✝² : CommRing S\nx✝¹ : CommRing T\nf : R →+* S\ne : S ≃+* T\nx✝ : Locally (fun {R S} [CommRing R] [CommRing S... | [
"case refine_2\nP : {R S : Type u} → [inst : CommRing R] → [inst_1 : CommRing S] → (R →+* S) → Prop\nhPi : RespectsIso fun {R S} [CommRing R] [CommRing S] ↦ P\nR S T : Type u\nx✝³ : CommRing R\nx✝² : CommRing S\nx✝¹ : CommRing T\nf : R →+* S\ne : S ≃+* T\nx✝ : Locally (fun {R S} [CommRing R] [CommRing S] ↦ P) f\ns ... | rw [← RingHom.comp_assoc, this, RingHom.comp_assoc] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.RingHom.Locally | {
"line": 337,
"column": 4
} | {
"line": 337,
"column": 86
} | {
"line": 339,
"column": 0
} | [
{
"pp": "case refine_3\nP : {R S : Type u} → [inst : CommRing R] → [inst_1 : CommRing S] → (R →+* S) → Prop\nhPl : LocalizationAwayPreserves fun {R S} [CommRing R] [CommRing S] ↦ P\nR S : Type u\ninst✝⁷ : CommRing R\ninst✝⁶ : CommRing S\nf : R →+* S\nr : R\nR' S' : Type u\ninst✝⁵ : CommRing R'\ninst✝⁴ : CommRin... | [] | exact hPl ((algebraMap _ (Localization.Away a.val)).comp f) r R' (Sₐ a) (hs _ a.2) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.AlgebraicGeometry.Morphisms.RingHomProperties | {
"line": 339,
"column": 4
} | {
"line": 345,
"column": 65
} | {
"line": 347,
"column": 0
} | [
{
"pp": "case hU\nP : MorphismProperty Scheme\nQ : {R S : Type u} → [inst : CommRing R] → [inst_1 : CommRing S] → (R →+* S) → Prop\ninst✝² : HasRingHomProperty P Q\nX Y : Scheme\nf : X ⟶ Y\ninst✝¹ : IsAffine Y\n𝒰 : X.OpenCover\ninst✝ : ∀ (i : 𝒰.I₀), IsAffine (𝒰.X i)\nH : ∀ (i : 𝒰.I₀), Q (CommRingCat.Hom.hom... | [] | specialize H i
rw [← (isLocal_ringHomProperty P).respectsIso.cancel_right_isIso _
((IsOpenImmersion.isoOfRangeEq (𝒰.f i) (S i).1.ι
Subtype.range_coe.symm).inv.app _), ← CommRingCat.hom_comp, ← Scheme.Hom.comp_appTop,
IsOpenImmersion.isoOfRangeEq_inv_fac_assoc, Scheme.Hom.comp_appTop,
Scheme... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.AlgebraicGeometry.Morphisms.RingHomProperties | {
"line": 339,
"column": 4
} | {
"line": 345,
"column": 65
} | {
"line": 347,
"column": 0
} | [
{
"pp": "case hU\nP : MorphismProperty Scheme\nQ : {R S : Type u} → [inst : CommRing R] → [inst_1 : CommRing S] → (R →+* S) → Prop\ninst✝² : HasRingHomProperty P Q\nX Y : Scheme\nf : X ⟶ Y\ninst✝¹ : IsAffine Y\n𝒰 : X.OpenCover\ninst✝ : ∀ (i : 𝒰.I₀), IsAffine (𝒰.X i)\nH : ∀ (i : 𝒰.I₀), Q (CommRingCat.Hom.hom... | [] | specialize H i
rw [← (isLocal_ringHomProperty P).respectsIso.cancel_right_isIso _
((IsOpenImmersion.isoOfRangeEq (𝒰.f i) (S i).1.ι
Subtype.range_coe.symm).inv.app _), ← CommRingCat.hom_comp, ← Scheme.Hom.comp_appTop,
IsOpenImmersion.isoOfRangeEq_inv_fac_assoc, Scheme.Hom.comp_appTop,
Scheme... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.AlgebraicGeometry.Morphisms.QuasiSeparated | {
"line": 100,
"column": 4
} | {
"line": 101,
"column": 82
} | {
"line": 103,
"column": 0
} | [
{
"pp": "case mpr\nX Y : Scheme\ninst✝² : IsAffine Y\nf : X ⟶ Y\nU₁ U₂ : Scheme\nf₁ : U₁ ⟶ X\nf₂ : U₂ ⟶ X\ninst✝¹ : IsAffine U₁\ninst✝ : IsAffine U₂\nh₁ : IsOpenImmersion f₁\nh₂ : IsOpenImmersion f₂\ng : pullback f₁ f₂ ⟶ X := pullback.fst f₁ f₂ ≫ f₁\ne : ↥(pullback f₁ f₂) ≃ₜ ↑(Set.range ⇑f₁ ∩ Set.range ⇑f₂)\nH ... | [] | exact @Homeomorph.compactSpace _ _ _ _
(H ⟨_, isAffineOpen_opensRange f₁⟩ ⟨_, isAffineOpen_opensRange f₂⟩) e.symm | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.RingTheory.Ideal.Height | {
"line": 100,
"column": 35
} | {
"line": 100,
"column": 52
} | {
"line": 100,
"column": 52
} | [
{
"pp": "R : Type u_1\ninst✝² : CommRing R\np : Ideal R\ninst✝¹ : p.IsPrime\ninst✝ : p.FiniteHeight\nn : ℕ\nhn : ↑n = p.primeHeight\n⊢ ∃ l, RelSeries.last l = { asIdeal := p, isPrime := ⋯ } ∧ ↑l.length = p.primeHeight",
"ppTerm": "?m.58",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"P... | [
"R : Type u_1\ninst✝² : CommRing R\np : Ideal R\ninst✝¹ : p.IsPrime\ninst✝ : p.FiniteHeight\nn : ℕ\nhn : ↑n = Order.height { asIdeal := p, isPrime := inst✝¹ }\n⊢ ∃ l, RelSeries.last l = { asIdeal := p, isPrime := ⋯ } ∧ ↑l.length = Order.height { asIdeal := p, isPrime := inst✝¹ }"
] | Ideal.primeHeight | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Ideal.Height | {
"line": 168,
"column": 35
} | {
"line": 168,
"column": 52
} | {
"line": 168,
"column": 52
} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommRing R\nI : Ideal R\ninst✝ : I.IsPrime\n⊢ ↑I.primeHeight ≤ ringKrullDim R",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"WithBot.instPreorder",
"Eq.mpr",
"PrimeSpectrum.mk",
"WithBot.some",
"WithBot",
"congrArg",
... | [
"R : Type u_1\ninst✝¹ : CommRing R\nI : Ideal R\ninst✝ : I.IsPrime\n⊢ ↑(Order.height { asIdeal := I, isPrime := inst✝ }) ≤ ringKrullDim R"
] | Ideal.primeHeight | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Ideal.Height | {
"line": 288,
"column": 35
} | {
"line": 288,
"column": 52
} | {
"line": 288,
"column": 52
} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsLocalRing R\n⊢ ↑(maximalIdeal R).primeHeight = ringKrullDim R",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"PrimeSpectrum.mk",
"WithBot.some",
"WithBot",
"congrArg",
"PartialOrder.toPreor... | [
"R : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsLocalRing R\n⊢ ↑(Order.height { asIdeal := maximalIdeal R, isPrime := ⋯ }) = ringKrullDim R"
] | Ideal.primeHeight | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Ideal.Height | {
"line": 492,
"column": 6
} | {
"line": 492,
"column": 48
} | {
"line": 493,
"column": 2
} | [
{
"pp": "case neg\nR : Type u_1\ninst✝¹ : CommRing R\ninst✝ : Nontrivial R\nI : Ideal R\nh : ¬I = ⊤\n⊢ ↑(⨆ (_ : I ≠ ⊤), I.height) ≤ ringKrullDim R",
"ppTerm": "?neg✝",
"assigned": true,
"usedConstants": [
"instCompleteLatticeWithBot",
"False",
"WithBot.some",
"WithBot",
... | [] | simp [h, height_le_ringKrullDim_of_ne_top] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.RingTheory.Ideal.Height | {
"line": 492,
"column": 6
} | {
"line": 492,
"column": 48
} | {
"line": 493,
"column": 2
} | [
{
"pp": "case neg\nR : Type u_1\ninst✝¹ : CommRing R\ninst✝ : Nontrivial R\nI : Ideal R\nh : ¬I = ⊤\n⊢ ↑(⨆ (_ : I ≠ ⊤), I.height) ≤ ringKrullDim R",
"ppTerm": "?neg✝",
"assigned": true,
"usedConstants": [
"instCompleteLatticeWithBot",
"False",
"WithBot.some",
"WithBot",
... | [] | simp [h, height_le_ringKrullDim_of_ne_top] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Ideal.Height | {
"line": 492,
"column": 6
} | {
"line": 492,
"column": 48
} | {
"line": 493,
"column": 2
} | [
{
"pp": "case neg\nR : Type u_1\ninst✝¹ : CommRing R\ninst✝ : Nontrivial R\nI : Ideal R\nh : ¬I = ⊤\n⊢ ↑(⨆ (_ : I ≠ ⊤), I.height) ≤ ringKrullDim R",
"ppTerm": "?neg✝",
"assigned": true,
"usedConstants": [
"instCompleteLatticeWithBot",
"False",
"WithBot.some",
"WithBot",
... | [] | simp [h, height_le_ringKrullDim_of_ne_top] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.AlgebraicGeometry.IdealSheaf.Basic | {
"line": 131,
"column": 29
} | {
"line": 131,
"column": 52
} | {
"line": 132,
"column": 2
} | [
{
"pp": "X : Scheme\n⊢ ∀ (U : ↑X.affineOpens) (f : ↑Γ(X, ↑U)),\n Ideal.map (CommRingCat.Hom.hom (X.presheaf.map (homOfLE ⋯).op)) (⊤ U) = ⊤ (X.affineBasicOpen f)",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"RingHom.instRingHomClass",
"AlgebraicGeometry.SheafedSpace.instTo... | [] | by simp [Ideal.map_top] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.AlgebraicGeometry.IdealSheaf.Subscheme | {
"line": 132,
"column": 2
} | {
"line": 139,
"column": 5
} | {
"line": 141,
"column": 0
} | [
{
"pp": "X : Scheme\nI : X.IdealSheafData\nV : ↑X.affineOpens\nf : ↑Γ(X, ↑V)\n⊢ Hom.opensRange (I.glueDataObjMap ⋯) = I.glueDataObjι V ⁻¹ᵁ (↑V).ι ⁻¹ᵁ X.basicOpen f",
"ppTerm": "?m.51",
"assigned": true,
"usedConstants": [
"Ideal.Quotient.commSemiring",
"Eq.mpr",
"AlgebraicGeometry.... | [] | letI := (Ideal.quotientMap _ _ (I.ideal_le_comap_ideal (X.affineBasicOpen_le f))).toAlgebra
let f' : Γ(X, V) ⧸ I.ideal V := Ideal.Quotient.mk _ f
have := I.isLocalization_away (X.affineBasicOpen_le f) f rfl
ext1
refine (localization_away_comap_range _ f').trans ?_
rw [← comap_basicOpen, ← V.2.fromSpec_preimag... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.AlgebraicGeometry.IdealSheaf.Subscheme | {
"line": 132,
"column": 2
} | {
"line": 139,
"column": 5
} | {
"line": 141,
"column": 0
} | [
{
"pp": "X : Scheme\nI : X.IdealSheafData\nV : ↑X.affineOpens\nf : ↑Γ(X, ↑V)\n⊢ Hom.opensRange (I.glueDataObjMap ⋯) = I.glueDataObjι V ⁻¹ᵁ (↑V).ι ⁻¹ᵁ X.basicOpen f",
"ppTerm": "?m.51",
"assigned": true,
"usedConstants": [
"Ideal.Quotient.commSemiring",
"Eq.mpr",
"AlgebraicGeometry.... | [] | letI := (Ideal.quotientMap _ _ (I.ideal_le_comap_ideal (X.affineBasicOpen_le f))).toAlgebra
let f' : Γ(X, V) ⧸ I.ideal V := Ideal.Quotient.mk _ f
have := I.isLocalization_away (X.affineBasicOpen_le f) f rfl
ext1
refine (localization_away_comap_range _ f').trans ?_
rw [← comap_basicOpen, ← V.2.fromSpec_preimag... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Geometry.RingedSpace.LocallyRingedSpace.ResidueField | {
"line": 78,
"column": 66
} | {
"line": 80,
"column": 50
} | {
"line": 82,
"column": 0
} | [
{
"pp": "X : LocallyRingedSpace\nU : Opens ↑X.toTopCat\nx : ↥U\nf : ↑(X.presheaf.obj (op U))\n⊢ (ConcreteCategory.hom (X.evaluation x)) f = 0 ↔ ↑x ∉ X.toRingedSpace.basicOpen f",
"ppTerm": "?m.22",
"assigned": true,
"usedConstants": [
"not_iff_not",
"Eq.mpr",
"AlgebraicGeometry.She... | [] | by
rw [X.toRingedSpace.mem_basicOpen f x.1 x.2, ← not_iff_not, not_not]
exact (IsLocalRing.residue_ne_zero_iff_isUnit _) | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.AlgebraicGeometry.IdealSheaf.Basic | {
"line": 767,
"column": 34
} | {
"line": 767,
"column": 75
} | {
"line": 768,
"column": 4
} | [
{
"pp": "X Y : Scheme\nf : X ⟶ Y\nx : ↥X\nU : TopologicalSpace.Opens ↥Y\nhU : U ∈ Y.affineOpens\nhxU : f x ∈ ↑U\ns : ↑Γ(Y, U)\nhs : s ∈ (ker f).ideal ⟨U, hU⟩\nhxs : f x ∈ Y.basicOpen s\nthis : x ∈ X.basicOpen ((ConcreteCategory.hom (app f U)) s)\n⊢ False",
"ppTerm": "?m.133",
"assigned": true,
"used... | [
"X Y : Scheme\nf : X ⟶ Y\nx : ↥X\nU : TopologicalSpace.Opens ↥Y\nhU : U ∈ Y.affineOpens\nhxU : f x ∈ ↑U\ns : ↑Γ(Y, U)\nhs : s ∈ (ker f).ideal ⟨U, hU⟩\nhxs : f x ∈ Y.basicOpen s\nthis : x ∈ X.basicOpen 0\n⊢ False"
] | RingHom.mem_ker.mp (f.ideal_ker_le _ hs), | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.AlgebraicGeometry.IdealSheaf.Basic | {
"line": 812,
"column": 2
} | {
"line": 813,
"column": 38
} | {
"line": 815,
"column": 0
} | [
{
"pp": "case inr\nX Y : Scheme\nf : X.Hom Y\ninst✝¹ : QuasiCompact f\n𝒰 : X.OpenCover\ninst✝ : Finite 𝒰.I₀\nh✝ : Nonempty 𝒰.I₀\nU : ↑Y.affineOpens\n⊢ (⋃ i, ↑(ker (𝒰.f i ≫ f)).support) ∩ ↑↑U = ↑f.ker.support ∩ ↑↑U",
"ppTerm": "?inr",
"assigned": true,
"usedConstants": [
"AlgebraicGeometry.... | [] | simp only [Set.iUnion_inter, coe_support_inter, ← f.iInf_ker_openCover_map_comp_apply 𝒰,
Scheme.zeroLocus_iInf_of_nonempty] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.AlgebraicGeometry.IdealSheaf.Subscheme | {
"line": 293,
"column": 4
} | {
"line": 294,
"column": 10
} | {
"line": 295,
"column": 4
} | [
{
"pp": "case h₀\nX : Scheme\nI : X.IdealSheafData\ni j k : ↑X.affineOpens\n⊢ (pullback.lift (I.glueDataT'Aux (i, j).1 (i, j).2 (i, k).2 (j, k).2 ⋯)\n (I.glueDataT'Aux (i, j).1 (i, j).2 (i, k).2 (j, i).2 ⋯) ⋯ ≫\n pullback.snd (pullback.fst (I.glueDataObjι (j, k).1) (X.homOfLE ⋯))\n (pul... | [
"case h₁\nX : Scheme\nI : X.IdealSheafData\ni j k : ↑X.affineOpens\n⊢ (pullback.lift (I.glueDataT'Aux (i, j).1 (i, j).2 (i, k).2 (j, k).2 ⋯)\n (I.glueDataT'Aux (i, j).1 (i, j).2 (i, k).2 (j, i).2 ⋯) ⋯ ≫\n pullback.snd (pullback.fst (I.glueDataObjι (j, k).1) (X.homOfLE ⋯))\n (pullback.fst (I... | · rw [← cancel_mono (glueDataObjι _ _)]
simp | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.AlgebraicGeometry.PullbackCarrier | {
"line": 241,
"column": 4
} | {
"line": 241,
"column": 99
} | {
"line": 242,
"column": 2
} | [
{
"pp": "case h₀\nX Y S : Scheme\nf : X ⟶ S\ng : Y ⟶ S\nT : Triplet f g\np : ↥(Spec T.tensor)\n⊢ Spec.map (Hom.residueFieldMap T.SpecTensorTo p) ≫\n Spec.map (ofPointTensor (T.SpecTensorTo p)) ≫ (ofPoint (T.SpecTensorTo p)).SpecTensorTo ≫ pullback.fst f g =\n (Spec T.tensor).fromSpecResidueField p ≫ T.S... | [] | rw [← Hom.SpecMap_residueFieldMap_fromSpecResidueField_assoc, ofPointTensor_SpecTensorTo_assoc] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.AlgebraicGeometry.PullbackCarrier | {
"line": 244,
"column": 4
} | {
"line": 244,
"column": 99
} | {
"line": 246,
"column": 0
} | [
{
"pp": "case h₁\nX Y S : Scheme\nf : X ⟶ S\ng : Y ⟶ S\nT : Triplet f g\np : ↥(Spec T.tensor)\n⊢ Spec.map (Hom.residueFieldMap T.SpecTensorTo p) ≫\n Spec.map (ofPointTensor (T.SpecTensorTo p)) ≫ (ofPoint (T.SpecTensorTo p)).SpecTensorTo ≫ pullback.snd f g =\n (Spec T.tensor).fromSpecResidueField p ≫ T.S... | [] | rw [← Hom.SpecMap_residueFieldMap_fromSpecResidueField_assoc, ofPointTensor_SpecTensorTo_assoc] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.AlgebraicGeometry.PullbackCarrier | {
"line": 255,
"column": 4
} | {
"line": 255,
"column": 13
} | {
"line": 257,
"column": 0
} | [
{
"pp": "case mpr\nX Y S : Scheme\nf : X ⟶ S\ng : Y ⟶ S\nT : Triplet f g\nsnd✝¹ snd✝ : ↥(Spec T.tensor)\ne : (Spec.map (Triplet.tensorCongr ⋯).inv) ⟨T, snd✝¹⟩.snd = ⟨T, snd✝⟩.snd\n⊢ ⟨T, snd✝¹⟩ = ⟨T, snd✝⟩",
"ppTerm": "?mpr",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"AlgebraicGeomet... | [] | simpa [e] | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.AlgebraicGeometry.IdealSheaf.Basic | {
"line": 846,
"column": 4
} | {
"line": 880,
"column": 67
} | {
"line": 881,
"column": 2
} | [
{
"pp": "case a\nX Y : Scheme\nf : X ⟶ Y\ninst✝ : QuasiCompact f\n⊢ ↑(ker f).support ⊆ closure (Set.range ⇑f)",
"ppTerm": "?a✝",
"assigned": true,
"usedConstants": [
"PrimeSpectrum.closure_range_comap",
"AlgebraicGeometry.Scheme.Hom.opensFunctor",
"Iff.mpr",
"AlgebraicGeometr... | [] | wlog hY : ∃ S, Y = Spec S
· intro x hx
let 𝒰 := Y.affineCover
obtain ⟨i, x, rfl⟩ := 𝒰.exists_eq x
have inst : QuasiCompact (𝒰.pullbackHom f i) :=
MorphismProperty.pullback_snd _ _ inferInstance
have := this (𝒰.pullbackHom f i) ⟨_, rfl⟩
((coe_support_inter _ ⟨⊤, isAffineOp... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.AlgebraicGeometry.IdealSheaf.Basic | {
"line": 846,
"column": 4
} | {
"line": 880,
"column": 67
} | {
"line": 881,
"column": 2
} | [
{
"pp": "case a\nX Y : Scheme\nf : X ⟶ Y\ninst✝ : QuasiCompact f\n⊢ ↑(ker f).support ⊆ closure (Set.range ⇑f)",
"ppTerm": "?a✝",
"assigned": true,
"usedConstants": [
"PrimeSpectrum.closure_range_comap",
"AlgebraicGeometry.Scheme.Hom.opensFunctor",
"Iff.mpr",
"AlgebraicGeometr... | [] | wlog hY : ∃ S, Y = Spec S
· intro x hx
let 𝒰 := Y.affineCover
obtain ⟨i, x, rfl⟩ := 𝒰.exists_eq x
have inst : QuasiCompact (𝒰.pullbackHom f i) :=
MorphismProperty.pullback_snd _ _ inferInstance
have := this (𝒰.pullbackHom f i) ⟨_, rfl⟩
((coe_support_inter _ ⟨⊤, isAffineOp... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.AlgebraicGeometry.PullbackCarrier | {
"line": 357,
"column": 2
} | {
"line": 364,
"column": 10
} | {
"line": 365,
"column": 2
} | [
{
"pp": "case mp\nX Y S : Scheme\nf : X ⟶ S\ng : Y ⟶ S\nX' Y' S' : Scheme\nf' : X' ⟶ S'\ng' : Y' ⟶ S'\ni₁ : X ⟶ X'\ni₂ : Y ⟶ Y'\ni₃ : S ⟶ S'\ne₁ : f ≫ i₃ = i₁ ≫ f'\ne₂ : g ≫ i₃ = i₂ ≫ g'\ninst✝ : Mono i₃\nz : ↥(pullback f' g')\n⊢ z ∈ Set.range ⇑(pullback.map f g f' g' i₁ i₂ i₃ e₁ e₂) →\n z ∈ ⇑(pullback.fst f... | [
"case mpr\nX Y S : Scheme\nf : X ⟶ S\ng : Y ⟶ S\nX' Y' S' : Scheme\nf' : X' ⟶ S'\ng' : Y' ⟶ S'\ni₁ : X ⟶ X'\ni₂ : Y ⟶ Y'\ni₃ : S ⟶ S'\ne₁ : f ≫ i₃ = i₁ ≫ f'\ne₂ : g ≫ i₃ = i₂ ≫ g'\ninst✝ : Mono i₃\nz : ↥(pullback f' g')\n⊢ z ∈ ⇑(pullback.fst f' g') ⁻¹' Set.range ⇑i₁ ∩ ⇑(pullback.snd f' g') ⁻¹' Set.range ⇑i₂ →\n ... | · rintro ⟨t, rfl⟩
constructor
· use pullback.fst f g t
rw [← Scheme.Hom.comp_apply, ← Scheme.Hom.comp_apply]
simp
· use pullback.snd f g t
rw [← Scheme.Hom.comp_apply, ← Scheme.Hom.comp_apply]
simp | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.RingTheory.Finiteness.FiniteTypeLocal | {
"line": 74,
"column": 85
} | {
"line": 79,
"column": 58
} | {
"line": 81,
"column": 0
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝⁷ : CommRing R\ninst✝⁶ : CommRing S\ninst✝⁵ : Algebra R S\nM : Submonoid R\nS' : Type u_4\ninst✝⁴ : CommRing S'\ninst✝³ : Algebra S S'\ninst✝² : Algebra R S'\ninst✝¹ : IsScalarTower R S S'\ninst✝ : IsLocalization (Submonoid.map (algebraMap R S) M) S'\nx : S\ns : Finset ... | [] | by
obtain ⟨⟨_, a, ha, rfl⟩, e⟩ :=
IsLocalization.exists_smul_mem_of_mem_adjoin (M.map (algebraMap R S)) x s (Algebra.adjoin R _)
Algebra.subset_adjoin (by rintro _ ⟨a, _, rfl⟩; exact Subalgebra.algebraMap_mem _ a) hx
refine ⟨⟨a, ha⟩, ?_⟩
simpa only [Submonoid.smul_def, algebraMap_smul] using e | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.AlgebraicGeometry.Morphisms.Affine | {
"line": 220,
"column": 2
} | {
"line": 235,
"column": 49
} | {
"line": 237,
"column": 0
} | [
{
"pp": "X✝ Y Z : Scheme\nf✝ : X✝ ⟶ Y\ng✝ : Y ⟶ Z\nU V X : Scheme\nf : U ⟶ X\ng : V ⟶ X\ninst✝¹ : IsAffineHom f\ninst✝ : IsAffineHom g\n⊢ IsAffineHom (coprod.desc f g)",
"ppTerm": "?m.21",
"assigned": true,
"usedConstants": [
"AlgebraicGeometry.PresheafedSpace.Hom",
"Eq.mpr",
"Alge... | [] | refine ⟨fun W hW ↦ ?_⟩
have : IsAffine (f ⁻¹ᵁ W).toScheme := hW.preimage f
have : IsAffine (g ⁻¹ᵁ W).toScheme := hW.preimage g
let i : (f ⁻¹ᵁ W).toScheme ⨿ (g ⁻¹ᵁ W).toScheme ⟶ U ⨿ V := coprod.map (f ⁻¹ᵁ W).ι (g ⁻¹ᵁ W).ι
convert! isAffineOpen_opensRange i
apply le_antisymm
· intro x hx
obtain ⟨(x | x), ... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.AlgebraicGeometry.Morphisms.Affine | {
"line": 220,
"column": 2
} | {
"line": 235,
"column": 49
} | {
"line": 237,
"column": 0
} | [
{
"pp": "X✝ Y Z : Scheme\nf✝ : X✝ ⟶ Y\ng✝ : Y ⟶ Z\nU V X : Scheme\nf : U ⟶ X\ng : V ⟶ X\ninst✝¹ : IsAffineHom f\ninst✝ : IsAffineHom g\n⊢ IsAffineHom (coprod.desc f g)",
"ppTerm": "?m.21",
"assigned": true,
"usedConstants": [
"AlgebraicGeometry.PresheafedSpace.Hom",
"Eq.mpr",
"Alge... | [] | refine ⟨fun W hW ↦ ?_⟩
have : IsAffine (f ⁻¹ᵁ W).toScheme := hW.preimage f
have : IsAffine (g ⁻¹ᵁ W).toScheme := hW.preimage g
let i : (f ⁻¹ᵁ W).toScheme ⨿ (g ⁻¹ᵁ W).toScheme ⟶ U ⨿ V := coprod.map (f ⁻¹ᵁ W).ι (g ⁻¹ᵁ W).ι
convert! isAffineOpen_opensRange i
apply le_antisymm
· intro x hx
obtain ⟨(x | x), ... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.AlgebraicGeometry.IdealSheaf.Subscheme | {
"line": 682,
"column": 85
} | {
"line": 686,
"column": 55
} | {
"line": 686,
"column": 55
} | [
{
"pp": "X Y : Scheme\nf : X ⟶ Y\nU : ↑Y.affineOpens\n⊢ ∀ (x y : (Precoverage.ZeroHypercover.pullback₁ f (Y.openCoverOfIsOpenCover (fun i ↦ ↑i) ⋯)).I₀),\n pullback.fst ((Precoverage.ZeroHypercover.pullback₁ f (Y.openCoverOfIsOpenCover (fun i ↦ ↑i) ⋯)).f x)\n ((Precoverage.ZeroHypercover.pullback₁ f ... | [] | by
intro U V
rw [← cancel_mono f.imageι]
simp [IdealSheafData.glueDataObjι, Scheme.Hom.liftQuotient_comp_assoc,
← pullback.condition, ← pullback.condition_assoc] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.AlgebraicGeometry.IdealSheaf.Functorial | {
"line": 75,
"column": 61
} | {
"line": 78,
"column": 97
} | {
"line": 80,
"column": 0
} | [
{
"pp": "X Y : Scheme\nI : Y.IdealSheafData\nf : X ⟶ Y\n⊢ (I.comap f).support = I.support.preimage ⋯",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"CategoryTheory.Limits.pullback",
"AlgebraicGeometry.Scheme.IdealSheafData.support",
"AlgebraicGeometry.She... | [] | by
ext1
rw [comap, Scheme.Hom.support_ker, Pullback.range_fst, range_subschemeι,
TopologicalSpace.Closeds.coe_preimage, (I.support.isClosed.preimage f.continuous).closure_eq] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.AlgebraicGeometry.IdealSheaf.Functorial | {
"line": 104,
"column": 35
} | {
"line": 114,
"column": 29
} | {
"line": 116,
"column": 0
} | [
{
"pp": "X Y : Scheme\nI : X.IdealSheafData\nf : X ⟶ Y\nJ : Y.IdealSheafData\n⊢ J ≤ I.map f ↔ J.comap f ≤ I",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"AlgebraicGeometry.Scheme.IdealSheafData.inclusion",
"Eq.mpr",
"CategoryTheory.Category.assoc",
"CategoryTheor... | [] | by
constructor
· intro H
rw [← I.ker_subschemeι, ← pullback.lift_fst (f := f) (g := J.subschemeι) I.subschemeι
((I.subschemeι ≫ f).toImage ≫ inclusion H) (by simp)]
exact Hom.le_ker_comp _ _
· intro H
have : (inclusion H ≫ (J.comapIso f).hom ≫ pullback.snd _ _) ≫ J.subschemeι =
I.subsche... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.AlgebraicGeometry.Morphisms.Immersion | {
"line": 111,
"column": 2
} | {
"line": 129,
"column": 62
} | {
"line": 131,
"column": 0
} | [
{
"pp": "X Y Z : Scheme\nf : X ⟶ Y\n⊢ IsZariskiLocalAtTarget @IsImmersion",
"ppTerm": "?m.3",
"assigned": true,
"usedConstants": [
"IsLocallyClosed.image",
"Set.restrictPreimage",
"Homeomorph.isInducing",
"Iff.mpr",
"Set.range_eq_univ",
"Set.range_comp",
"Eq... | [] | suffices IsZariskiLocalAtTarget
(topologically fun {X Y} _ _ f ↦ IsLocallyClosed (Set.range f)) from
isImmersion_eq_inf ▸ inferInstance
apply +allowSynthFailures topologically_isZariskiLocalAtTarget'
· refine { precomp := ?_, postcomp := ?_ }
· intro X Y Z i hi f hf
change IsIso i at hi
ch... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.AlgebraicGeometry.Morphisms.Immersion | {
"line": 111,
"column": 2
} | {
"line": 129,
"column": 62
} | {
"line": 131,
"column": 0
} | [
{
"pp": "X Y Z : Scheme\nf : X ⟶ Y\n⊢ IsZariskiLocalAtTarget @IsImmersion",
"ppTerm": "?m.3",
"assigned": true,
"usedConstants": [
"IsLocallyClosed.image",
"Set.restrictPreimage",
"Homeomorph.isInducing",
"Iff.mpr",
"Set.range_eq_univ",
"Set.range_comp",
"Eq... | [] | suffices IsZariskiLocalAtTarget
(topologically fun {X Y} _ _ f ↦ IsLocallyClosed (Set.range f)) from
isImmersion_eq_inf ▸ inferInstance
apply +allowSynthFailures topologically_isZariskiLocalAtTarget'
· refine { precomp := ?_, postcomp := ?_ }
· intro X Y Z i hi f hf
change IsIso i at hi
ch... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.RingHom.FinitePresentation | {
"line": 35,
"column": 48
} | {
"line": 35,
"column": 77
} | {
"line": 37,
"column": 0
} | [
{
"pp": "R✝ S✝ : Type u_1\ninst✝¹ : CommRing R✝\ninst✝ : CommRing S✝\ne : R✝ ≃+* S✝\n⊢ (ker e.toRingHom).FG",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"RingHom.instRingHomClass",
"Semiring.toModule",
"congrArg",
"CommSemiring.toSemiring",
... | [] | simpa using! Submodule.fg_bot | Lean.Elab.Tactic.Simpa.evalSimpaUsingBang | Lean.Parser.Tactic.simpaUsingBang |
Mathlib.RingTheory.RingHom.FinitePresentation | {
"line": 35,
"column": 48
} | {
"line": 35,
"column": 77
} | {
"line": 37,
"column": 0
} | [
{
"pp": "R✝ S✝ : Type u_1\ninst✝¹ : CommRing R✝\ninst✝ : CommRing S✝\ne : R✝ ≃+* S✝\n⊢ (ker e.toRingHom).FG",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"RingHom.instRingHomClass",
"Semiring.toModule",
"congrArg",
"CommSemiring.toSemiring",
... | [] | simpa using! Submodule.fg_bot | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.RingHom.FinitePresentation | {
"line": 35,
"column": 48
} | {
"line": 35,
"column": 77
} | {
"line": 37,
"column": 0
} | [
{
"pp": "R✝ S✝ : Type u_1\ninst✝¹ : CommRing R✝\ninst✝ : CommRing S✝\ne : R✝ ≃+* S✝\n⊢ (ker e.toRingHom).FG",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"RingHom.instRingHomClass",
"Semiring.toModule",
"congrArg",
"CommSemiring.toSemiring",
... | [] | simpa using! Submodule.fg_bot | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Spectrum.Prime.ConstructibleSet | {
"line": 125,
"column": 4
} | {
"line": 132,
"column": 73
} | {
"line": 133,
"column": 2
} | [
{
"pp": "case sdiff\nR : Type u_1\ninst✝ : CommSemiring R\ns✝ : Set (PrimeSpectrum R)\ni : R\ns : Set R\nhs : s.Finite\n⊢ ∃ S, S.toSet = ↑(basicOpen i) \\ ⋃ j ∈ s, ↑(basicOpen j)",
"ppTerm": "?sdiff",
"assigned": true,
"usedConstants": [
"Set.ext",
"Eq.mpr",
"PrimeSpectrum.BasicCon... | [] | have : Finite s := hs
refine ⟨{⟨i, Nat.card s, fun i ↦ ((Finite.equivFin s).symm i).1⟩}, ?_⟩
simp only [ConstructibleSetData.toSet, Finset.mem_singleton, BasicConstructibleSetData.toSet,
Set.iUnion_iUnion_eq_left, basicOpen_eq_zeroLocus_compl, ← Set.compl_iInter₂,
compl_sdiff_compl, ← zeroLocus_iU... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Spectrum.Prime.ConstructibleSet | {
"line": 125,
"column": 4
} | {
"line": 132,
"column": 73
} | {
"line": 133,
"column": 2
} | [
{
"pp": "case sdiff\nR : Type u_1\ninst✝ : CommSemiring R\ns✝ : Set (PrimeSpectrum R)\ni : R\ns : Set R\nhs : s.Finite\n⊢ ∃ S, S.toSet = ↑(basicOpen i) \\ ⋃ j ∈ s, ↑(basicOpen j)",
"ppTerm": "?sdiff",
"assigned": true,
"usedConstants": [
"Set.ext",
"Eq.mpr",
"PrimeSpectrum.BasicCon... | [] | have : Finite s := hs
refine ⟨{⟨i, Nat.card s, fun i ↦ ((Finite.equivFin s).symm i).1⟩}, ?_⟩
simp only [ConstructibleSetData.toSet, Finset.mem_singleton, BasicConstructibleSetData.toSet,
Set.iUnion_iUnion_eq_left, basicOpen_eq_zeroLocus_compl, ← Set.compl_iInter₂,
compl_sdiff_compl, ← zeroLocus_iU... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.AlgebraicGeometry.Geometrically.Basic | {
"line": 58,
"column": 4
} | {
"line": 58,
"column": 53
} | {
"line": 60,
"column": 0
} | [
{
"pp": "case refine_2\nP : ObjectProperty Scheme\nX Y : Scheme\nf : X ⟶ Y\nhf : MorphismProperty.universally (fun X Y x ↦ IsIntegral Y → Subsingleton ↥Y → P X) f\naK : Type u\ny : Field aK\nZ : Spec (of aK) ⟶ Y\nW : Scheme\nfst : W ⟶ X\nsnd : W ⟶ Spec (of aK)\nh : IsPullback fst snd f Z\n⊢ P W",
"ppTerm": ... | [] | exact hf _ _ _ h.flip inferInstance inferInstance | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.AlgebraicGeometry.Geometrically.Basic | {
"line": 58,
"column": 4
} | {
"line": 58,
"column": 53
} | {
"line": 60,
"column": 0
} | [
{
"pp": "case refine_2\nP : ObjectProperty Scheme\nX Y : Scheme\nf : X ⟶ Y\nhf : MorphismProperty.universally (fun X Y x ↦ IsIntegral Y → Subsingleton ↥Y → P X) f\naK : Type u\ny : Field aK\nZ : Spec (of aK) ⟶ Y\nW : Scheme\nfst : W ⟶ X\nsnd : W ⟶ Spec (of aK)\nh : IsPullback fst snd f Z\n⊢ P W",
"ppTerm": ... | [] | exact hf _ _ _ h.flip inferInstance inferInstance | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.AlgebraicGeometry.Geometrically.Basic | {
"line": 58,
"column": 4
} | {
"line": 58,
"column": 53
} | {
"line": 60,
"column": 0
} | [
{
"pp": "case refine_2\nP : ObjectProperty Scheme\nX Y : Scheme\nf : X ⟶ Y\nhf : MorphismProperty.universally (fun X Y x ↦ IsIntegral Y → Subsingleton ↥Y → P X) f\naK : Type u\ny : Field aK\nZ : Spec (of aK) ⟶ Y\nW : Scheme\nfst : W ⟶ X\nsnd : W ⟶ Spec (of aK)\nh : IsPullback fst snd f Z\n⊢ P W",
"ppTerm": ... | [] | exact hf _ _ _ h.flip inferInstance inferInstance | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.AlgebraicGeometry.Noetherian | {
"line": 215,
"column": 2
} | {
"line": 215,
"column": 32
} | {
"line": 217,
"column": 0
} | [
{
"pp": "case Kf\nX Z : Scheme\ninst✝¹ : IsLocallyNoetherian X\nf : Z ⟶ X\ninst✝ : IsOpenImmersion f\nU : X.Opens\nhU : IsAffineOpen U\n⊢ ↑U ∩ Set.range ⇑f ⊆ Set.range ⇑f",
"ppTerm": "?Kf",
"assigned": true,
"usedConstants": [
"AlgebraicGeometry.PresheafedSpace.carrier",
"CategoryTheory.... | [] | · exact Set.inter_subset_right | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.AlgebraicGeometry.Morphisms.SchemeTheoreticallyDominant | {
"line": 78,
"column": 42
} | {
"line": 78,
"column": 64
} | {
"line": 78,
"column": 65
} | [
{
"pp": "X Y : Scheme\nf : X ⟶ Y\ninst✝¹ : IsSchemeTheoreticallyDominant f\ninst✝ : QuasiCompact f\nU : Y.Opens\nhU : IsAffineOpen U\n⊢ RingHom.ker (ConcreteCategory.hom (app f U)) = ⊥",
"ppTerm": "?m.49",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"RingHom.instRingHomClass",
"... | [
"X Y : Scheme\nf : X ⟶ Y\ninst✝¹ : IsSchemeTheoreticallyDominant f\ninst✝ : QuasiCompact f\nU : Y.Opens\nhU : IsAffineOpen U\n⊢ (ker f).ideal ⟨U, hU⟩ = ⊥"
] | ← f.ker_apply ⟨U, hU⟩, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.AlgebraicGeometry.Morphisms.Flat | {
"line": 267,
"column": 2
} | {
"line": 267,
"column": 66
} | {
"line": 268,
"column": 2
} | [
{
"pp": "X Y S T : Scheme\nf : T ⟶ S\ng : Y ⟶ X\niX : X ⟶ S\niY : Y ⟶ T\nH : IsPullback g iY iX f\nUS : S.Opens\nUT : T.Opens\nUX : X.Opens\nhUST : UT ≤ f ⁻¹ᵁ US\nhUSX : UX ≤ iX ⁻¹ᵁ US\nUY : Y.Opens\nhUY : UY = g ⁻¹ᵁ UX ⊓ iY ⁻¹ᵁ UT\nι : Type u_1\ninst✝ : Finite ι\nVX : ι → X.Opens\nhVU : iSup VX = UX\nhV : ∀ (i... | [
"X Y S T : Scheme\nf : T ⟶ S\ng : Y ⟶ X\niX : X ⟶ S\niY : Y ⟶ T\nH : IsPullback g iY iX f\nUS : S.Opens\nUT : T.Opens\nUX : X.Opens\nhUST : UT ≤ f ⁻¹ᵁ US\nhUSX : UX ≤ iX ⁻¹ᵁ US\nUY : Y.Opens\nhUY : UY = g ⁻¹ᵁ UX ⊓ iY ⁻¹ᵁ UT\nι : Type u_1\ninst✝ : Finite ι\nVX : ι → X.Opens\nhVU : iSup VX = UX\nhV : ∀ (i : ι), Mono ... | algebraize [(iX.appLE US UX hUSX).hom, (f.appLE US UT hUST).hom] | Mathlib.Tactic._aux_Mathlib_Tactic_Algebraize___elabRules_Mathlib_Tactic_tacticAlgebraize___1 | Mathlib.Tactic.tacticAlgebraize__ |
Mathlib.RingTheory.Spectrum.Prime.ChevalleyComplexity | {
"line": 245,
"column": 6
} | {
"line": 263,
"column": 78
} | {
"line": 264,
"column": 4
} | [
{
"pp": "n : ℕ\nP : (R : Type u) → [inst : CommRing R] → InductionObj R n → Prop\nhP₁ : ∀ (R : Type u) [inst : CommRing R], P R { val := 0 }\nhP₂ :\n ∀ (R : Type u) [inst : CommRing R] (e : InductionObj R n) (i : Fin n),\n (e.val i).Monic → (∀ (j : Fin n), j ≠ i → e.val j = 0) → P R e\nhP₃ :\n ∀ (R : Type ... | [] | rw [hv, Prod.Lex.lt_iff']
constructor
· intro j
simp only [coe_mapRingHom, InductionObj.ofLex_degree_fst, Pi.smul_apply,
comp_apply, smul_eq_mul]
refine ((degree_mul_le _ _).trans (add_le_add degree_C_le degree_map_le)).trans ?_
simp
rw [lt_iff_le_not_ge]
simp o... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Spectrum.Prime.ChevalleyComplexity | {
"line": 245,
"column": 6
} | {
"line": 263,
"column": 78
} | {
"line": 264,
"column": 4
} | [
{
"pp": "n : ℕ\nP : (R : Type u) → [inst : CommRing R] → InductionObj R n → Prop\nhP₁ : ∀ (R : Type u) [inst : CommRing R], P R { val := 0 }\nhP₂ :\n ∀ (R : Type u) [inst : CommRing R] (e : InductionObj R n) (i : Fin n),\n (e.val i).Monic → (∀ (j : Fin n), j ≠ i → e.val j = 0) → P R e\nhP₃ :\n ∀ (R : Type ... | [] | rw [hv, Prod.Lex.lt_iff']
constructor
· intro j
simp only [coe_mapRingHom, InductionObj.ofLex_degree_fst, Pi.smul_apply,
comp_apply, smul_eq_mul]
refine ((degree_mul_le _ _).trans (add_le_add degree_C_le degree_map_le)).trans ?_
simp
rw [lt_iff_le_not_ge]
simp o... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Spectrum.Prime.ChevalleyComplexity | {
"line": 308,
"column": 2
} | {
"line": 308,
"column": 35
} | {
"line": 309,
"column": 2
} | [
{
"pp": "R₀ : Type u_1\ninst✝² : CommRing R₀\nn : ℕ\nR : Type u_6\ninst✝¹ : CommRing R\ninst✝ : Algebra R₀ R\nc : R\ni : Fin n\ne : InductionObj R n\nhi : c = (e.val i).leadingCoeff\nhc : c ≠ 0\nq₁ : R →ₐ[R₀] Localization.Away c := IsScalarTower.toAlgHom R₀ R (Localization.Away c)\nq₂ : R →ₐ[R₀] R ⧸ Ideal.span ... | [
"R₀ : Type u_1\ninst✝² : CommRing R₀\nn : ℕ\nR : Type u_6\ninst✝¹ : CommRing R\ninst✝ : Algebra R₀ R\nc : R\ni : Fin n\ne : InductionObj R n\nhi : c = (e.val i).leadingCoeff\nhc : c ≠ 0\nq₁ : R →ₐ[R₀] Localization.Away c := IsScalarTower.toAlgHom R₀ R (Localization.Away c)\nq₂ : R →ₐ[R₀] R ⧸ Ideal.span {c} := Ideal... | simp only [forall_and] at hT₁ hT₂ | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Topology.LocallyFinsupp | {
"line": 223,
"column": 4
} | {
"line": 223,
"column": 57
} | {
"line": 224,
"column": 4
} | [
{
"pp": "case mp\nX : Type u_1\ninst✝² : TopologicalSpace X\nU : Set X\nY : Type u_2\ninst✝¹ : Zero Y\ninst✝ : T1Space X\nD : locallyFinsuppWithin U Y\nx : X\n⊢ x ∈ D.support → x ∈ {x | D x = 0}ᶜ ∩ U",
"ppTerm": "?mp",
"assigned": true,
"usedConstants": [
"Function.locallyFinsuppWithin.instFun... | [
"case mpr\nX : Type u_1\ninst✝² : TopologicalSpace X\nU : Set X\nY : Type u_2\ninst✝¹ : Zero Y\ninst✝ : T1Space X\nD : locallyFinsuppWithin U Y\nx : X\n⊢ x ∈ {x | D x = 0}ᶜ ∩ U → x ∈ D.support"
] | · exact fun hx ↦ ⟨by tauto, D.supportWithinDomain hx⟩ | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Topology.LocallyFinsupp | {
"line": 529,
"column": 9
} | {
"line": 529,
"column": 25
} | {
"line": 530,
"column": 2
} | [
{
"pp": "X : Type u_1\ninst✝³ : TopologicalSpace X\nU : Set X\nY : Type u_2\ninst✝² : AddCommGroup Y\ninst✝¹ : LinearOrder Y\ninst✝ : IsOrderedAddMonoid Y\nf₁ f₂ : locallyFinsuppWithin U Y\n⊢ (f₁ + f₂)⁻ ≤ f₁⁻ + f₂⁻",
"ppTerm": "?m.25",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Latt... | [
"X : Type u_1\ninst✝³ : TopologicalSpace X\nU : Set X\nY : Type u_2\ninst✝² : AddCommGroup Y\ninst✝¹ : LinearOrder Y\ninst✝ : IsOrderedAddMonoid Y\nf₁ f₂ : locallyFinsuppWithin U Y\n⊢ -(f₁ + f₂) ⊔ 0 ≤ f₁⁻ + f₂⁻"
] | rw [negPart_def] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Topology.LocallyFinsupp | {
"line": 529,
"column": 9
} | {
"line": 529,
"column": 25
} | {
"line": 530,
"column": 2
} | [
{
"pp": "X : Type u_1\ninst✝³ : TopologicalSpace X\nU : Set X\nY : Type u_2\ninst✝² : AddCommGroup Y\ninst✝¹ : LinearOrder Y\ninst✝ : IsOrderedAddMonoid Y\nf₁ f₂ : locallyFinsuppWithin U Y\n⊢ (f₁ + f₂)⁻ ≤ f₁⁻ + f₂⁻",
"ppTerm": "?m.25",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Latt... | [
"X : Type u_1\ninst✝³ : TopologicalSpace X\nU : Set X\nY : Type u_2\ninst✝² : AddCommGroup Y\ninst✝¹ : LinearOrder Y\ninst✝ : IsOrderedAddMonoid Y\nf₁ f₂ : locallyFinsuppWithin U Y\n⊢ -(f₁ + f₂) ⊔ 0 ≤ f₁⁻ + f₂⁻"
] | rw [negPart_def] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.LocallyFinsupp | {
"line": 529,
"column": 9
} | {
"line": 529,
"column": 25
} | {
"line": 530,
"column": 2
} | [
{
"pp": "X : Type u_1\ninst✝³ : TopologicalSpace X\nU : Set X\nY : Type u_2\ninst✝² : AddCommGroup Y\ninst✝¹ : LinearOrder Y\ninst✝ : IsOrderedAddMonoid Y\nf₁ f₂ : locallyFinsuppWithin U Y\n⊢ (f₁ + f₂)⁻ ≤ f₁⁻ + f₂⁻",
"ppTerm": "?m.25",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Latt... | [
"X : Type u_1\ninst✝³ : TopologicalSpace X\nU : Set X\nY : Type u_2\ninst✝² : AddCommGroup Y\ninst✝¹ : LinearOrder Y\ninst✝ : IsOrderedAddMonoid Y\nf₁ f₂ : locallyFinsuppWithin U Y\n⊢ -(f₁ + f₂) ⊔ 0 ≤ f₁⁻ + f₂⁻"
] | rw [negPart_def] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.LocallyFinsupp | {
"line": 529,
"column": 9
} | {
"line": 529,
"column": 25
} | {
"line": 530,
"column": 2
} | [
{
"pp": "X : Type u_1\ninst✝³ : TopologicalSpace X\nU : Set X\nY : Type u_2\ninst✝² : AddCommGroup Y\ninst✝¹ : LinearOrder Y\ninst✝ : IsOrderedAddMonoid Y\nf₁ f₂ : locallyFinsuppWithin U Y\n⊢ -(f₁ + f₂) ⊔ 0 ≤ f₁⁻ + f₂⁻",
"ppTerm": "?m.30",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"... | [
"X : Type u_1\ninst✝³ : TopologicalSpace X\nU : Set X\nY : Type u_2\ninst✝² : AddCommGroup Y\ninst✝¹ : LinearOrder Y\ninst✝ : IsOrderedAddMonoid Y\nf₁ f₂ : locallyFinsuppWithin U Y\n⊢ -(f₁ + f₂) ⊔ 0 ≤ -f₁ ⊔ 0 + f₂⁻"
] | rw [negPart_def] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Topology.LocallyFinsupp | {
"line": 529,
"column": 9
} | {
"line": 529,
"column": 25
} | {
"line": 530,
"column": 2
} | [
{
"pp": "X : Type u_1\ninst✝³ : TopologicalSpace X\nU : Set X\nY : Type u_2\ninst✝² : AddCommGroup Y\ninst✝¹ : LinearOrder Y\ninst✝ : IsOrderedAddMonoid Y\nf₁ f₂ : locallyFinsuppWithin U Y\n⊢ -(f₁ + f₂) ⊔ 0 ≤ f₁⁻ + f₂⁻",
"ppTerm": "?m.30",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"... | [
"X : Type u_1\ninst✝³ : TopologicalSpace X\nU : Set X\nY : Type u_2\ninst✝² : AddCommGroup Y\ninst✝¹ : LinearOrder Y\ninst✝ : IsOrderedAddMonoid Y\nf₁ f₂ : locallyFinsuppWithin U Y\n⊢ -(f₁ + f₂) ⊔ 0 ≤ -f₁ ⊔ 0 + f₂⁻"
] | rw [negPart_def] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.LocallyFinsupp | {
"line": 529,
"column": 9
} | {
"line": 529,
"column": 25
} | {
"line": 530,
"column": 2
} | [
{
"pp": "X : Type u_1\ninst✝³ : TopologicalSpace X\nU : Set X\nY : Type u_2\ninst✝² : AddCommGroup Y\ninst✝¹ : LinearOrder Y\ninst✝ : IsOrderedAddMonoid Y\nf₁ f₂ : locallyFinsuppWithin U Y\n⊢ -(f₁ + f₂) ⊔ 0 ≤ f₁⁻ + f₂⁻",
"ppTerm": "?m.30",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"... | [
"X : Type u_1\ninst✝³ : TopologicalSpace X\nU : Set X\nY : Type u_2\ninst✝² : AddCommGroup Y\ninst✝¹ : LinearOrder Y\ninst✝ : IsOrderedAddMonoid Y\nf₁ f₂ : locallyFinsuppWithin U Y\n⊢ -(f₁ + f₂) ⊔ 0 ≤ -f₁ ⊔ 0 + f₂⁻"
] | rw [negPart_def] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.LocallyFinsupp | {
"line": 529,
"column": 9
} | {
"line": 529,
"column": 25
} | {
"line": 530,
"column": 2
} | [
{
"pp": "X : Type u_1\ninst✝³ : TopologicalSpace X\nU : Set X\nY : Type u_2\ninst✝² : AddCommGroup Y\ninst✝¹ : LinearOrder Y\ninst✝ : IsOrderedAddMonoid Y\nf₁ f₂ : locallyFinsuppWithin U Y\n⊢ -(f₁ + f₂) ⊔ 0 ≤ -f₁ ⊔ 0 + f₂⁻",
"ppTerm": "?m.35",
"assigned": true,
"usedConstants": [
"Eq.mpr",
... | [
"X : Type u_1\ninst✝³ : TopologicalSpace X\nU : Set X\nY : Type u_2\ninst✝² : AddCommGroup Y\ninst✝¹ : LinearOrder Y\ninst✝ : IsOrderedAddMonoid Y\nf₁ f₂ : locallyFinsuppWithin U Y\n⊢ -(f₁ + f₂) ⊔ 0 ≤ -f₁ ⊔ 0 + -f₂ ⊔ 0"
] | rw [negPart_def] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Topology.LocallyFinsupp | {
"line": 529,
"column": 9
} | {
"line": 529,
"column": 25
} | {
"line": 530,
"column": 2
} | [
{
"pp": "X : Type u_1\ninst✝³ : TopologicalSpace X\nU : Set X\nY : Type u_2\ninst✝² : AddCommGroup Y\ninst✝¹ : LinearOrder Y\ninst✝ : IsOrderedAddMonoid Y\nf₁ f₂ : locallyFinsuppWithin U Y\n⊢ -(f₁ + f₂) ⊔ 0 ≤ -f₁ ⊔ 0 + f₂⁻",
"ppTerm": "?m.35",
"assigned": true,
"usedConstants": [
"Eq.mpr",
... | [
"X : Type u_1\ninst✝³ : TopologicalSpace X\nU : Set X\nY : Type u_2\ninst✝² : AddCommGroup Y\ninst✝¹ : LinearOrder Y\ninst✝ : IsOrderedAddMonoid Y\nf₁ f₂ : locallyFinsuppWithin U Y\n⊢ -(f₁ + f₂) ⊔ 0 ≤ -f₁ ⊔ 0 + -f₂ ⊔ 0"
] | rw [negPart_def] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.LocallyFinsupp | {
"line": 529,
"column": 9
} | {
"line": 529,
"column": 25
} | {
"line": 530,
"column": 2
} | [
{
"pp": "X : Type u_1\ninst✝³ : TopologicalSpace X\nU : Set X\nY : Type u_2\ninst✝² : AddCommGroup Y\ninst✝¹ : LinearOrder Y\ninst✝ : IsOrderedAddMonoid Y\nf₁ f₂ : locallyFinsuppWithin U Y\n⊢ -(f₁ + f₂) ⊔ 0 ≤ -f₁ ⊔ 0 + f₂⁻",
"ppTerm": "?m.35",
"assigned": true,
"usedConstants": [
"Eq.mpr",
... | [
"X : Type u_1\ninst✝³ : TopologicalSpace X\nU : Set X\nY : Type u_2\ninst✝² : AddCommGroup Y\ninst✝¹ : LinearOrder Y\ninst✝ : IsOrderedAddMonoid Y\nf₁ f₂ : locallyFinsuppWithin U Y\n⊢ -(f₁ + f₂) ⊔ 0 ≤ -f₁ ⊔ 0 + -f₂ ⊔ 0"
] | rw [negPart_def] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.LocallyFinsupp | {
"line": 529,
"column": 9
} | {
"line": 529,
"column": 25
} | {
"line": 530,
"column": 2
} | [
{
"pp": "X : Type u_1\ninst✝³ : TopologicalSpace X\nU : Set X\nY : Type u_2\ninst✝² : AddCommGroup Y\ninst✝¹ : LinearOrder Y\ninst✝ : IsOrderedAddMonoid Y\nf₁ f₂ : locallyFinsuppWithin U Y\n⊢ -(f₁ + f₂) ⊔ 0 ≤ -f₁ ⊔ 0 + -f₂ ⊔ 0",
"ppTerm": "?m.40",
"assigned": false,
"usedConstants": [],
"usedFVa... | [
"X : Type u_1\ninst✝³ : TopologicalSpace X\nU : Set X\nY : Type u_2\ninst✝² : AddCommGroup Y\ninst✝¹ : LinearOrder Y\ninst✝ : IsOrderedAddMonoid Y\nf₁ f₂ : locallyFinsuppWithin U Y\n⊢ -(f₁ + f₂) ⊔ 0 ≤ -f₁ ⊔ 0 + -f₂ ⊔ 0"
] | rw [negPart_def] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Topology.LocallyFinsupp | {
"line": 529,
"column": 9
} | {
"line": 529,
"column": 25
} | {
"line": 530,
"column": 2
} | [
{
"pp": "X : Type u_1\ninst✝³ : TopologicalSpace X\nU : Set X\nY : Type u_2\ninst✝² : AddCommGroup Y\ninst✝¹ : LinearOrder Y\ninst✝ : IsOrderedAddMonoid Y\nf₁ f₂ : locallyFinsuppWithin U Y\n⊢ -(f₁ + f₂) ⊔ 0 ≤ -f₁ ⊔ 0 + -f₂ ⊔ 0",
"ppTerm": "?m.40",
"assigned": false,
"usedConstants": [],
"usedFVa... | [
"X : Type u_1\ninst✝³ : TopologicalSpace X\nU : Set X\nY : Type u_2\ninst✝² : AddCommGroup Y\ninst✝¹ : LinearOrder Y\ninst✝ : IsOrderedAddMonoid Y\nf₁ f₂ : locallyFinsuppWithin U Y\n⊢ -(f₁ + f₂) ⊔ 0 ≤ -f₁ ⊔ 0 + -f₂ ⊔ 0"
] | rw [negPart_def] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.LocallyFinsupp | {
"line": 529,
"column": 9
} | {
"line": 529,
"column": 25
} | {
"line": 530,
"column": 2
} | [
{
"pp": "X : Type u_1\ninst✝³ : TopologicalSpace X\nU : Set X\nY : Type u_2\ninst✝² : AddCommGroup Y\ninst✝¹ : LinearOrder Y\ninst✝ : IsOrderedAddMonoid Y\nf₁ f₂ : locallyFinsuppWithin U Y\n⊢ -(f₁ + f₂) ⊔ 0 ≤ -f₁ ⊔ 0 + -f₂ ⊔ 0",
"ppTerm": "?m.40",
"assigned": false,
"usedConstants": [],
"usedFVa... | [
"X : Type u_1\ninst✝³ : TopologicalSpace X\nU : Set X\nY : Type u_2\ninst✝² : AddCommGroup Y\ninst✝¹ : LinearOrder Y\ninst✝ : IsOrderedAddMonoid Y\nf₁ f₂ : locallyFinsuppWithin U Y\n⊢ -(f₁ + f₂) ⊔ 0 ≤ -f₁ ⊔ 0 + -f₂ ⊔ 0"
] | rw [negPart_def] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.AlgebraicGeometry.SpreadingOut | {
"line": 91,
"column": 33
} | {
"line": 98,
"column": 7
} | {
"line": 100,
"column": 0
} | [
{
"pp": "X Y S : Scheme\nf : X ⟶ Y\nsX : X ⟶ S\nsY : Y ⟶ S\nR A : CommRingCat\nx : ↥X\ninst✝¹ : X.IsGermInjectiveAt x\ninst✝ : IsOpenImmersion f\n⊢ Y.IsGermInjectiveAt (f x)",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"AlgebraicGeometry.Scheme.Hom.opensFunctor",
"Eq.mpr",
... | [] | by
obtain ⟨U, hxU, hU, H⟩ := X.exists_germ_injective x
refine ⟨⟨f ''ᵁ U, ⟨x, hxU, rfl⟩, hU.image_of_isOpenImmersion f, ?_⟩⟩
refine ((MorphismProperty.injective CommRingCat).cancel_right_of_respectsIso _
(f.stalkMap x)).mp ?_
refine ((MorphismProperty.injective CommRingCat).cancel_left_of_respectsIso
(f.... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.Spectrum.Prime.ChevalleyComplexity | {
"line": 368,
"column": 8
} | {
"line": 368,
"column": 26
} | {
"line": 369,
"column": 8
} | [
{
"pp": "case e'_3\nR₀ : Type u_1\ninst✝² : CommRing R₀\nn : ℕ\nR : Type u_6\ninst✝¹ : CommRing R\ninst✝ : Algebra R₀ R\nc : R\ni : Fin n\ne : InductionObj R n\nhi : c = (e.val i).leadingCoeff\nhc : c ≠ 0\nq₁ : R →ₐ[R₀] Localization.Away c := IsScalarTower.toAlgHom R₀ R (Localization.Away c)\nq₂ : R →ₐ[R₀] R ⧸ ... | [
"case e'_3.f\nR₀ : Type u_1\ninst✝² : CommRing R₀\nn : ℕ\nR : Type u_6\ninst✝¹ : CommRing R\ninst✝ : Algebra R₀ R\nc : R\ni : Fin n\ne : InductionObj R n\nhi : c = (e.val i).leadingCoeff\nhc : c ≠ 0\nq₁ : R →ₐ[R₀] Localization.Away c := IsScalarTower.toAlgHom R₀ R (Localization.Away c)\nq₂ : R →ₐ[R₀] R ⧸ Ideal.span... | congr! with x hxT₁ | Congr!._aux_Mathlib_Tactic_CongrExclamation___elabRules_Congr!_congr!_1 | Congr!.congr! |
Mathlib.AlgebraicGeometry.Birational.RationalMap | {
"line": 428,
"column": 11
} | {
"line": 428,
"column": 13
} | {
"line": 428,
"column": 14
} | [
{
"pp": "X Y Z S : Scheme\nsX : X ⟶ S\nsY : Y ⟶ S\nf : X ⤏ Y\ng : Y ⟶ Z\nf₁ : X.PartialMap Y\n⊢ ∀ ⦃b : X.PartialMap Y⦄, f₁ ≈ b → f₁.compHom g ≈ b.compHom g",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"AlgebraicGeometry.Scheme.PartialMap"
],
"usedFVars": [
"X",
... | [
"X Y Z S : Scheme\nsX : X ⟶ S\nsY : Y ⟶ S\nf : X ⤏ Y\ng : Y ⟶ Z\nf₁ f₂ : X.PartialMap Y\n⊢ f₁ ≈ f₂ → f₁.compHom g ≈ f₂.compHom g"
] | f₂ | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.AlgebraicGeometry.Morphisms.Flat | {
"line": 437,
"column": 2
} | {
"line": 437,
"column": 77
} | {
"line": 438,
"column": 2
} | [
{
"pp": "X Y S T : Scheme\nf : T ⟶ S\ng : Y ⟶ X\niX : X ⟶ S\niY : Y ⟶ T\nH : IsPullback g iY iX f\nUS : S.Opens\nUT : T.Opens\nUX : X.Opens\nhUST : UT ≤ f ⁻¹ᵁ US\nhUSX : UX ≤ iX ⁻¹ᵁ US\nUY : Y.Opens\nhUY : UY = g ⁻¹ᵁ UX ⊓ iY ⁻¹ᵁ UT\ninst✝ : Flat f\nhUS : IsAffineOpen US\nhUT : IsAffineOpen UT\nhUX : IsCompact ↑... | [
"X Y S T : Scheme\nf : T ⟶ S\ng : Y ⟶ X\niX : X ⟶ S\niY : Y ⟶ T\nH : IsPullback g iY iX f\nUS : S.Opens\nUT : T.Opens\nUX : X.Opens\nhUST : UT ≤ f ⁻¹ᵁ US\nhUSX : UX ≤ iX ⁻¹ᵁ US\nUY : Y.Opens\nhUY : UY = g ⁻¹ᵁ UX ⊓ iY ⁻¹ᵁ UT\ninst✝ : Flat f\nhUS : IsAffineOpen US\nhUT : IsAffineOpen UT\nhUX : IsCompact ↑UX\nI : Set ... | obtain ⟨I, hI, e⟩ := isCompact_iff_finite_and_eq_biUnion_affineOpens.mp hUX | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.AlgebraicGeometry.Morphisms.Flat | {
"line": 456,
"column": 2
} | {
"line": 456,
"column": 77
} | {
"line": 457,
"column": 2
} | [
{
"pp": "X Y S T : Scheme\nf : T ⟶ S\ng : Y ⟶ X\niX : X ⟶ S\niY : Y ⟶ T\nH : IsPullback g iY iX f\nUS : S.Opens\nUT : T.Opens\nUX : X.Opens\nhUST : UT ≤ f ⁻¹ᵁ US\nhUSX : UX ≤ iX ⁻¹ᵁ US\nUY : Y.Opens\nhUY : UY = g ⁻¹ᵁ UX ⊓ iY ⁻¹ᵁ UT\ninst✝ : Flat f\nhUS : IsAffineOpen US\nhUT : IsAffineOpen UT\nhUX : IsCompact ↑... | [
"X Y S T : Scheme\nf : T ⟶ S\ng : Y ⟶ X\niX : X ⟶ S\niY : Y ⟶ T\nH : IsPullback g iY iX f\nUS : S.Opens\nUT : T.Opens\nUX : X.Opens\nhUST : UT ≤ f ⁻¹ᵁ US\nhUSX : UX ≤ iX ⁻¹ᵁ US\nUY : Y.Opens\nhUY : UY = g ⁻¹ᵁ UX ⊓ iY ⁻¹ᵁ UT\ninst✝ : Flat f\nhUS : IsAffineOpen US\nhUT : IsAffineOpen UT\nhUX : IsCompact ↑UX\nhUX' : I... | obtain ⟨I, hI, e⟩ := isCompact_iff_finite_and_eq_biUnion_affineOpens.mp hUX | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.AlgebraicGeometry.Morphisms.Flat | {
"line": 478,
"column": 2
} | {
"line": 478,
"column": 77
} | {
"line": 479,
"column": 2
} | [
{
"pp": "X Y S T : Scheme\nf : T ⟶ S\ng : Y ⟶ X\niX : X ⟶ S\niY : Y ⟶ T\nH : IsPullback g iY iX f\nUS : S.Opens\nUT : T.Opens\nUX : X.Opens\nhUST : UT ≤ f ⁻¹ᵁ US\nhUSX : UX ≤ iX ⁻¹ᵁ US\nUY : Y.Opens\nhUY : UY = g ⁻¹ᵁ UX ⊓ iY ⁻¹ᵁ UT\ninst✝ : Flat iX\nhUS : IsAffineOpen US\nhUT : IsCompact ↑UT\nhUX : IsCompact ↑U... | [
"X Y S T : Scheme\nf : T ⟶ S\ng : Y ⟶ X\niX : X ⟶ S\niY : Y ⟶ T\nH : IsPullback g iY iX f\nUS : S.Opens\nUT : T.Opens\nUX : X.Opens\nhUST : UT ≤ f ⁻¹ᵁ US\nhUSX : UX ≤ iX ⁻¹ᵁ US\nUY : Y.Opens\nhUY : UY = g ⁻¹ᵁ UX ⊓ iY ⁻¹ᵁ UT\ninst✝ : Flat iX\nhUS : IsAffineOpen US\nhUT : IsCompact ↑UT\nhUX : IsCompact ↑UX\nhf : (Com... | obtain ⟨I, hI, e⟩ := isCompact_iff_finite_and_eq_biUnion_affineOpens.mp hUX | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.AlgebraicGeometry.Morphisms.UniversallyInjective | {
"line": 71,
"column": 4
} | {
"line": 72,
"column": 28
} | {
"line": 74,
"column": 0
} | [
{
"pp": "case a\nX Y : Scheme\nf : X ⟶ Y\nhf : diagonal (@Surjective) f\nx₁ x₂ : ↥X\ne : f x₁ = f x₂\nt : ↥X\nht₁ : (pullback.fst f f) ((pullback.diagonal f) t) = x₁\nht₂ : (pullback.snd f f) ((pullback.diagonal f) t) = x₂\n⊢ x₁ = x₂",
"ppTerm": "?a✝",
"assigned": true,
"usedConstants": [
"Eq.... | [] | rw [← ht₁, ← ht₂, ← Scheme.Hom.comp_apply, ← Scheme.Hom.comp_apply, pullback.diagonal_fst,
pullback.diagonal_snd] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.AlgebraicGeometry.AffineTransitionLimit | {
"line": 68,
"column": 68
} | {
"line": 72,
"column": 65
} | {
"line": 72,
"column": 65
} | [
{
"pp": "I : Type u\ninst✝⁴ : Category.{u, u} I\nD : I ⥤ Scheme\nc : Cone D\nhc : IsLimit c\ninst✝³ : IsCofilteredOrEmpty I\ninst✝² : ∀ {i j : I} (f : i ⟶ j), IsAffineHom (D.map f)\ninst✝¹ : ∀ (i : I), Nonempty ↥(D.obj i)\ninst✝ : ∀ (i : I), CompactSpace ↥(D.obj i)\nh✝ : Nonempty I\ni : I\nthis✝ : IsCofiltered ... | [] | by
rw [← D.map_comp, IsCofiltered.infTo_commutes]
· simp [g]
· simp
· exact Finset.mem_image_of_mem _ (Finset.mem_univ _) | [anonymous] | Lean.Parser.Term.byTactic |
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