module string | startPos dict | endPos dict | nextStartPos dict | goals list | goalsAfter list | ppTac string | elaborator string | kind string |
|---|---|---|---|---|---|---|---|---|
Mathlib.AlgebraicGeometry.IdealSheaf.Basic | {
"line": 810,
"column": 4
} | {
"line": 810,
"column": 21
} | {
"line": 810,
"column": 22
} | [
{
"pp": "X Y : Scheme\nf : X.Hom Y\ninst✝¹ : QuasiCompact f\n𝒰 : X.OpenCover\ninst✝ : Finite 𝒰.I₀\nh✝ : Nonempty 𝒰.I₀\nthis : ∀ (U : ↑Y.affineOpens), (⋃ i, ↑(ker (𝒰.f i ≫ f)).support) ∩ ↑↑U = ↑f.ker.support ∩ ↑↑U\nx : ↥Y\nU : TopologicalSpace.Opens ↥Y\nhU : U ∈ Y.affineOpens\nhxU : x ∈ ↑U\n⊢ x ∈ ⋃ i, ↑(ker ... | [
"X Y : Scheme\nf : X.Hom Y\ninst✝¹ : QuasiCompact f\n𝒰 : X.OpenCover\ninst✝ : Finite 𝒰.I₀\nh✝ : Nonempty 𝒰.I₀\nthis : ∀ (U : ↑Y.affineOpens), (⋃ i, ↑(ker (𝒰.f i ≫ f)).support) ∩ ↑↑U = ↑f.ker.support ∩ ↑↑U\nx : ↥Y\nU : TopologicalSpace.Opens ↥Y\nhU : U ∈ Y.affineOpens\nhxU : x ∈ ↑U\n⊢ (∃ i, x ∈ (ker (𝒰.f i ≫ f)... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.IdealSheaf.Basic | {
"line": 841,
"column": 2
} | {
"line": 841,
"column": 13
} | {
"line": 841,
"column": 14
} | [
{
"pp": "X Y U V : Scheme\nf : X ⟶ Y\nf' : U ⟶ V\niU : U ⟶ X\niV : V ⟶ Y\ninst✝¹ : IsOpenImmersion iV\ninst✝ : QuasiCompact f\nH : IsPullback f' iU iV f\nW : ↑V.affineOpens\nthis✝² : QuasiCompact f'\nthis✝¹ : IsOpenImmersion iU\nx : ↑Γ(V, ↑W)\nthis✝ : iU ''ᵁ f' ⁻¹ᵁ ↑W = f ⁻¹ᵁ iV ''ᵁ ↑W\ne : Γ(X, f ⁻¹ᵁ iV ''ᵁ ↑W... | [
"X Y U V : Scheme\nf : X ⟶ Y\nf' : U ⟶ V\niU : U ⟶ X\niV : V ⟶ Y\ninst✝¹ : IsOpenImmersion iV\ninst✝ : QuasiCompact f\nH : IsPullback f' iU iV f\nW : ↑V.affineOpens\nthis✝² : QuasiCompact f'\nthis✝¹ : IsOpenImmersion iU\nx : ↑Γ(V, ↑W)\nthis✝ : iU ''ᵁ f' ⁻¹ᵁ ↑W = f ⁻¹ᵁ iV ''ᵁ ↑W\ne : Γ(X, f ⁻¹ᵁ iV ''ᵁ ↑W) ≅ Γ(U, f' ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.Morphisms.Affine | {
"line": 81,
"column": 2
} | {
"line": 81,
"column": 51
} | {
"line": 82,
"column": 2
} | [
{
"pp": "X✝ Y Z : Scheme\nf : X✝ ⟶ Y\ng : Y ⟶ Z\nX : Scheme\nr : ↑Γ(X, ⊤)\nU : X.Opens\nhU : IsAffineOpen U\n⊢ ↑((X.basicOpen r).ι ''ᵁ (X.basicOpen r).ι ⁻¹ᵁ U) = ↑(U ⊓ X.basicOpen r)",
"ppTerm": "?m.125",
"assigned": true,
"usedConstants": [
"AlgebraicGeometry.SheafedSpace.instTopologicalSpace... | [
"X✝ Y Z : Scheme\nf : X✝ ⟶ Y\ng : Y ⟶ Z\nX : Scheme\nr : ↑Γ(X, ⊤)\nU : X.Opens\nhU : IsAffineOpen U\n⊢ U.1 ∩ Set.range ⇑(X.basicOpen r).ι = ↑(U ⊓ X.basicOpen r)"
] | refine Set.image_preimage_eq_inter_range.trans ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.AlgebraicGeometry.PullbackCarrier | {
"line": 214,
"column": 8
} | {
"line": 214,
"column": 81
} | {
"line": 214,
"column": 82
} | [
{
"pp": "case h₁\nX Y S : Scheme\nf : X ⟶ S\ng : Y ⟶ S\nt : ↥(pullback f g)\n⊢ Spec.map (ofPointTensor t) ≫ Spec.map (Triplet.ofPoint t).tensorInr ≫ Y.fromSpecResidueField (Triplet.ofPoint t).y =\n Spec.map (Hom.residueFieldMap (pullback.snd f g) t) ≫ Y.fromSpecResidueField ((pullback.snd f g) t)",
"ppTe... | [
"case h₁\nX Y S : Scheme\nf : X ⟶ S\ng : Y ⟶ S\nt : ↥(pullback f g)\n⊢ Spec.map (ofPointTensor t) ≫ Spec.map (Triplet.ofPoint t).tensorInr ≫ Y.fromSpecResidueField (Triplet.ofPoint t).y =\n Spec.map\n (pushout.inr ((residueFieldCongr ⋯).inv ≫ Hom.residueFieldMap f (Triplet.ofPoint t).x)\n ((res... | ← pushout.inr_desc _ _ (residueFieldCongr_inv_residueFieldMap_ofPoint t), | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.AlgebraicGeometry.IdealSheaf.Basic | {
"line": 888,
"column": 33
} | {
"line": 889,
"column": 56
} | {
"line": 889,
"column": 57
} | [
{
"pp": "X Y✝ Z Y : Scheme\nf g : (Over Y)ᵒᵖ\nhfg : f ⟶ g\n⊢ Hom.ker (Opposite.unop f).hom ≤ Hom.ker (Opposite.unop g).hom",
"ppTerm": "?m.36",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"CategoryTheory.Over",
"AlgebraicGeometry.Scheme",
"CategoryTheory.CategoryStruct.toQ... | [
"X Y✝ Z Y : Scheme\nf g : (Over Y)ᵒᵖ\nhfg : f ⟶ g\n⊢ Hom.ker (Opposite.unop f).hom ≤ Hom.ker (Over.Hom.left hfg.unop ≫ (Opposite.unop f).hom)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.Morphisms.Affine | {
"line": 144,
"column": 4
} | {
"line": 144,
"column": 52
} | {
"line": 144,
"column": 53
} | [
{
"pp": "case hs₂\nX : Scheme\nU : X.Opens\ns : Set ↑Γ(X, U)\nhs : Ideal.span s = ⊤\nhs₂ : ∀ i ∈ s, IsAffineOpen (X.basicOpen i)\nj : ↑Γ(X, U)\nhj : j ∈ s\n⊢ IsAffineOpen\n (X.basicOpen ((ConcreteCategory.hom (Scheme.Hom.appIso U.ι ⊤).inv) ((ConcreteCategory.hom U.topIso.inv) j)))",
"ppTerm": "?hs₂",
... | [
"case hs₂\nX : Scheme\nU : X.Opens\ns : Set ↑Γ(X, U)\nhs : Ideal.span s = ⊤\nhs₂ : ∀ i ∈ s, IsAffineOpen (X.basicOpen i)\nj : ↑Γ(X, U)\nhj : j ∈ s\n⊢ IsAffineOpen (X.basicOpen j)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.IdealSheaf.Subscheme | {
"line": 343,
"column": 17
} | {
"line": 343,
"column": 33
} | {
"line": 343,
"column": 33
} | [
{
"pp": "X : Scheme\nI : X.IdealSheafData\nU V : ↑X.affineOpens\nh : U ≤ V\nthis : IsIso (X.homOfLE ⋯)\n⊢ X.homOfLE ⋯ = inv (X.homOfLE ⋯)",
"ppTerm": "?m.84",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"AlgebraicGeometry.SheafedSpace.instTopologicalSpaceCarrierCarrier",
"Algebr... | [
"X : Scheme\nI : X.IdealSheafData\nU V : ↑X.affineOpens\nh : U ≤ V\nthis : IsIso (X.homOfLE ⋯)\n⊢ X.homOfLE ⋯ ≫ X.homOfLE ⋯ = 𝟙 ↑(↑U ⊓ ↑V)"
] | ← hom_comp_eq_id | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.AlgebraicGeometry.Morphisms.Affine | {
"line": 191,
"column": 38
} | {
"line": 191,
"column": 49
} | {
"line": 191,
"column": 50
} | [
{
"pp": "X Y : Scheme\nf : X ⟶ Y\nU : ↥Y → Y.Opens\nhxU : ∀ (x : ↥Y), x ∈ U x\nhU : ∀ (x : ↥Y), IsAffineOpen (U x)\nhfU : ∀ (x : ↥Y), IsAffineOpen (f ⁻¹ᵁ U x)\nx : ↥Y\nx✝ : x ∈ ⊤\n⊢ x ∈ ⨆ i, ↑⟨U i, ⋯⟩",
"ppTerm": "?m.65",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"AlgebraicGeometry.... | [
"X Y : Scheme\nf : X ⟶ Y\nU : ↥Y → Y.Opens\nhxU : ∀ (x : ↥Y), x ∈ U x\nhU : ∀ (x : ↥Y), IsAffineOpen (U x)\nhfU : ∀ (x : ↥Y), IsAffineOpen (f ⁻¹ᵁ U x)\nx : ↥Y\nx✝ : x ∈ ⊤\n⊢ ∃ i, x ∈ U i"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.Morphisms.Affine | {
"line": 221,
"column": 2
} | {
"line": 221,
"column": 53
} | {
"line": 222,
"column": 2
} | [
{
"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\nW : X.Opens\nhW : IsAffineOpen W\n⊢ IsAffineOpen (coprod.desc f g ⁻¹ᵁ W)",
"ppTerm": "?m.27",
"assigned": true,
"usedConstants": [
"AlgebraicGeometry.Presheaf... | [
"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\nW : X.Opens\nhW : IsAffineOpen W\nthis : IsAffine ↑(f ⁻¹ᵁ W)\n⊢ IsAffineOpen (coprod.desc f g ⁻¹ᵁ W)"
] | have : IsAffine (f ⁻¹ᵁ W).toScheme := hW.preimage f | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.AlgebraicGeometry.Morphisms.Affine | {
"line": 228,
"column": 33
} | {
"line": 228,
"column": 70
} | {
"line": 228,
"column": 71
} | [
{
"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\nW : X.Opens\nhW : IsAffineOpen W\nthis✝ : IsAffine ↑(f ⁻¹ᵁ W)\nthis : IsAffine ↑(g ⁻¹ᵁ W)\ni : ↑(f ⁻¹ᵁ W) ⨿ ↑(g ⁻¹ᵁ W) ⟶ U ⨿ V := coprod.map (f ⁻¹ᵁ W).ι (g ⁻¹ᵁ W).ι\nx : ↥U\nhx... | [
"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\nW : X.Opens\nhW : IsAffineOpen W\nthis✝ : IsAffine ↑(f ⁻¹ᵁ W)\nthis : IsAffine ↑(g ⁻¹ᵁ W)\ni : ↑(f ⁻¹ᵁ W) ⨿ ↑(g ⁻¹ᵁ W) ⟶ U ⨿ V := coprod.map (f ⁻¹ᵁ W).ι (g ⁻¹ᵁ W).ι\nx : ↥U\nhx : (coprodMk... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.Morphisms.Affine | {
"line": 230,
"column": 33
} | {
"line": 230,
"column": 70
} | {
"line": 230,
"column": 71
} | [
{
"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\nW : X.Opens\nhW : IsAffineOpen W\nthis✝ : IsAffine ↑(f ⁻¹ᵁ W)\nthis : IsAffine ↑(g ⁻¹ᵁ W)\ni : ↑(f ⁻¹ᵁ W) ⨿ ↑(g ⁻¹ᵁ W) ⟶ U ⨿ V := coprod.map (f ⁻¹ᵁ W).ι (g ⁻¹ᵁ W).ι\nx : ↥V\nhx... | [
"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\nW : X.Opens\nhW : IsAffineOpen W\nthis✝ : IsAffine ↑(f ⁻¹ᵁ W)\nthis : IsAffine ↑(g ⁻¹ᵁ W)\ni : ↑(f ⁻¹ᵁ W) ⨿ ↑(g ⁻¹ᵁ W) ⟶ U ⨿ V := coprod.map (f ⁻¹ᵁ W).ι (g ⁻¹ᵁ W).ι\nx : ↥V\nhx : (coprodMk... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.Morphisms.Affine | {
"line": 234,
"column": 6
} | {
"line": 234,
"column": 46
} | {
"line": 234,
"column": 47
} | [
{
"pp": "case a.inl\nX✝ 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\nW : X.Opens\nhW : IsAffineOpen W\nthis✝ : IsAffine ↑(f ⁻¹ᵁ W)\nthis : IsAffine ↑(g ⁻¹ᵁ W)\ni : ↑(f ⁻¹ᵁ W) ⨿ ↑(g ⁻¹ᵁ W) ⟶ U ⨿ V := coprod.map (f ⁻¹ᵁ W).ι (g ⁻¹ᵁ W).ι... | [
"case a.inl\nX✝ 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\nW : X.Opens\nhW : IsAffineOpen W\nthis✝ : IsAffine ↑(f ⁻¹ᵁ W)\nthis : IsAffine ↑(g ⁻¹ᵁ W)\ni : ↑(f ⁻¹ᵁ W) ⨿ ↑(g ⁻¹ᵁ W) ⟶ U ⨿ V := coprod.map (f ⁻¹ᵁ W).ι (g ⁻¹ᵁ W).ι\nx : ↥U\nhx... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Finiteness.FiniteTypeLocal | {
"line": 79,
"column": 2
} | {
"line": 79,
"column": 56
} | {
"line": 79,
"column": 57
} | [
{
"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 ... | [
"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 S'\nhx : (al... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.Morphisms.Affine | {
"line": 235,
"column": 6
} | {
"line": 235,
"column": 46
} | {
"line": 235,
"column": 47
} | [
{
"pp": "case a.inr\nX✝ 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\nW : X.Opens\nhW : IsAffineOpen W\nthis✝ : IsAffine ↑(f ⁻¹ᵁ W)\nthis : IsAffine ↑(g ⁻¹ᵁ W)\ni : ↑(f ⁻¹ᵁ W) ⨿ ↑(g ⁻¹ᵁ W) ⟶ U ⨿ V := coprod.map (f ⁻¹ᵁ W).ι (g ⁻¹ᵁ W).ι... | [
"case a.inr\nX✝ 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\nW : X.Opens\nhW : IsAffineOpen W\nthis✝ : IsAffine ↑(f ⁻¹ᵁ W)\nthis : IsAffine ↑(g ⁻¹ᵁ W)\ni : ↑(f ⁻¹ᵁ W) ⨿ ↑(g ⁻¹ᵁ W) ⟶ U ⨿ V := coprod.map (f ⁻¹ᵁ W).ι (g ⁻¹ᵁ W).ι\nx : ↥V\nhx... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.Morphisms.Affine | {
"line": 253,
"column": 56
} | {
"line": 253,
"column": 71
} | {
"line": 253,
"column": 72
} | [
{
"pp": "X Y : Scheme\nf : X ⟶ Y\nhf₁ : Topology.IsInducing ⇑f\nhf₂ : IsClosed (Set.range ⇑f)\nx : ↥X\nU : Opens ↥Y\nhU : U ∈ Y.affineOpens\nhxU : f x ∈ ↑U\nV : Opens ↥X\nhV : V ∈ X.affineOpens\nhxV : x ∈ ↑V\nhVU : ↑V ⊆ ↑(f ⁻¹ᵁ U)\nU' : Set ↥Y\nhU' : IsOpen U'\ne : ⇑f ⁻¹' U' = V.carrier\n⊢ ↑(f ⁻¹ᵁ ({ carrier :=... | [
"X Y : Scheme\nf : X ⟶ Y\nhf₁ : Topology.IsInducing ⇑f\nhf₂ : IsClosed (Set.range ⇑f)\nx : ↥X\nU : Opens ↥Y\nhU : U ∈ Y.affineOpens\nhxU : f x ∈ ↑U\nV : Opens ↥X\nhV : V ∈ X.affineOpens\nhxV : x ∈ ↑V\nhVU : ↑V ⊆ ↑(f ⁻¹ᵁ U)\nU' : Set ↥Y\nhU' : IsOpen U'\ne : ⇑f ⁻¹' U' = V.carrier\n⊢ ↑V ⊆ ⇑f ⁻¹' ↑U"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.Morphisms.Affine | {
"line": 258,
"column": 4
} | {
"line": 258,
"column": 15
} | {
"line": 258,
"column": 16
} | [
{
"pp": "X Y : Scheme\nf : X ⟶ Y\nhf₁ : Topology.IsInducing ⇑f\nhf₂ : IsClosed (Set.range ⇑f)\nx : ↥X\nU : Opens ↥Y\nhU : U ∈ Y.affineOpens\nhxU : f x ∈ ↑U\nU' : Y.Opens\nhU'U : U' ≤ U\nhV : f ⁻¹ᵁ U' ∈ X.affineOpens\nhxV : x ∈ ↑(f ⁻¹ᵁ U')\nhVU : ↑(f ⁻¹ᵁ U') ⊆ ↑(f ⁻¹ᵁ U)\nr : ↑Γ(Y, U)\nhrU' : Y.basicOpen r ≤ U'\... | [
"X Y : Scheme\nf : X ⟶ Y\nhf₁ : Topology.IsInducing ⇑f\nhf₂ : IsClosed (Set.range ⇑f)\nx : ↥X\nU : Opens ↥Y\nhU : U ∈ Y.affineOpens\nhxU : f x ∈ ↑U\nU' : Y.Opens\nhU'U : U' ≤ U\nhV : f ⁻¹ᵁ U' ∈ X.affineOpens\nhxV : x ∈ ↑(f ⁻¹ᵁ U')\nhVU : ↑(f ⁻¹ᵁ U') ⊆ ↑(f ⁻¹ᵁ U)\nr : ↑Γ(Y, U)\nhrU' : Y.basicOpen r ≤ U'\nhxr : ↑⟨f x... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.Morphisms.AffineAnd | {
"line": 305,
"column": 4
} | {
"line": 308,
"column": 26
} | {
"line": 309,
"column": 2
} | [
{
"pp": "case refine_1\nQ : {R S : Type u} → [inst : CommRing R] → [inst_1 : CommRing S] → (R →+* S) → Prop\nP : MorphismProperty Scheme\nhP : HasAffineProperty P (affineAnd fun {R S} [CommRing R] [CommRing S] ↦ Q)\nhQi : RingHom.RespectsIso fun {R S} [CommRing R] [CommRing S] ↦ Q\nhQ :\n ∀ {R S T : Type u} [i... | [] | have : (Limits.coprod.desc f g).app W ≫ e.hom ≫ Limits.prod.fst = f.app W := by
simp [e, Scheme.Hom.app_eq_appLE, Scheme.Hom.appLE_comp_appLE]
convert! (hf W hW).2
exact congr(($this).1) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.AlgebraicGeometry.Morphisms.AffineAnd | {
"line": 305,
"column": 4
} | {
"line": 308,
"column": 26
} | {
"line": 309,
"column": 2
} | [
{
"pp": "case refine_1\nQ : {R S : Type u} → [inst : CommRing R] → [inst_1 : CommRing S] → (R →+* S) → Prop\nP : MorphismProperty Scheme\nhP : HasAffineProperty P (affineAnd fun {R S} [CommRing R] [CommRing S] ↦ Q)\nhQi : RingHom.RespectsIso fun {R S} [CommRing R] [CommRing S] ↦ Q\nhQ :\n ∀ {R S T : Type u} [i... | [] | have : (Limits.coprod.desc f g).app W ≫ e.hom ≫ Limits.prod.fst = f.app W := by
simp [e, Scheme.Hom.app_eq_appLE, Scheme.Hom.appLE_comp_appLE]
convert! (hf W hW).2
exact congr(($this).1) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.AlgebraicGeometry.IdealSheaf.Functorial | {
"line": 94,
"column": 4
} | {
"line": 94,
"column": 15
} | {
"line": 94,
"column": 16
} | [
{
"pp": "ZX ZY X Y : Scheme\niX : ZX ⟶ X\niY : ZY ⟶ Y\nZf : ZX ⟶ ZY\nf : X ⟶ Y\ninst✝¹ : IsClosedImmersion iX\ninst✝ : IsClosedImmersion iY\nh : iX ≫ f = Zf ≫ iY\nh' : (Hom.ker iY).comap f = Hom.ker iX\nthis : IsIso (pullback.lift iX Zf h)\n⊢ IsPullback iX Zf f iY",
"ppTerm": "?m.54",
"assigned": false,... | [
"ZX ZY X Y : Scheme\niX : ZX ⟶ X\niY : ZY ⟶ Y\nZf : ZX ⟶ ZY\nf : X ⟶ Y\ninst✝¹ : IsClosedImmersion iX\ninst✝ : IsClosedImmersion iY\nh : iX ≫ f = Zf ≫ iY\nh' : (Hom.ker iY).comap f = Hom.ker iX\nthis : IsIso (pullback.lift iX Zf h)\n⊢ IsPullback iX Zf f iY"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.Morphisms.ClosedImmersion | {
"line": 198,
"column": 30
} | {
"line": 198,
"column": 79
} | {
"line": 198,
"column": 80
} | [
{
"pp": "X Y : Scheme\nf : X ⟶ Y\ninst✝ : IsClosedImmersion f\nH : Scheme.Hom.ker f = ⊥\n⊢ IsIso (Scheme.Hom.imageι f)",
"ppTerm": "?m.30",
"assigned": true,
"usedConstants": [
"AlgebraicGeometry.Scheme",
"AlgebraicGeometry.Scheme.Hom.image",
"CategoryTheory.IsIso",
"Algebrai... | [
"X Y : Scheme\nf : X ⟶ Y\ninst✝ : IsClosedImmersion f\nH : Scheme.Hom.ker f = ⊥\n⊢ IsIso (Scheme.Hom.ker f).subschemeι"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.Morphisms.ClosedImmersion | {
"line": 223,
"column": 4
} | {
"line": 224,
"column": 57
} | {
"line": 225,
"column": 2
} | [
{
"pp": "Z₁ Z₂ X : Scheme\ni₁ : Z₁ ⟶ X\ni₂ : Z₂ ⟶ X\ninst✝¹ : IsClosedImmersion i₁\ninst✝ : IsClosedImmersion i₂\nf : Z₁ ⟶ Z₂\nh✝ : f ≫ i₂ = i₁\nh' : Scheme.Hom.ker i₁ = Scheme.Hom.ker i₂\nf' : MorphismProperty.Over.mk ⊤ i₁ inst✝¹ ⟶ MorphismProperty.Over.mk ⊤ i₂ inst✝ :=\n MorphismProperty.Over.homMk f h✝ True... | [] | rwa [isIso_op_iff, ← isIso_iff_of_reflects_iso _ (MorphismProperty.Over.forget ..),
← isIso_iff_of_reflects_iso _ (Over.forget _)] at h | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticRwa___1 | Lean.Parser.Tactic.tacticRwa__ |
Mathlib.AlgebraicGeometry.Morphisms.ClosedImmersion | {
"line": 223,
"column": 4
} | {
"line": 224,
"column": 57
} | {
"line": 225,
"column": 2
} | [
{
"pp": "Z₁ Z₂ X : Scheme\ni₁ : Z₁ ⟶ X\ni₂ : Z₂ ⟶ X\ninst✝¹ : IsClosedImmersion i₁\ninst✝ : IsClosedImmersion i₂\nf : Z₁ ⟶ Z₂\nh✝ : f ≫ i₂ = i₁\nh' : Scheme.Hom.ker i₁ = Scheme.Hom.ker i₂\nf' : MorphismProperty.Over.mk ⊤ i₁ inst✝¹ ⟶ MorphismProperty.Over.mk ⊤ i₂ inst✝ :=\n MorphismProperty.Over.homMk f h✝ True... | [] | rwa [isIso_op_iff, ← isIso_iff_of_reflects_iso _ (MorphismProperty.Over.forget ..),
← isIso_iff_of_reflects_iso _ (Over.forget _)] at h | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.AlgebraicGeometry.Morphisms.ClosedImmersion | {
"line": 223,
"column": 4
} | {
"line": 224,
"column": 57
} | {
"line": 225,
"column": 2
} | [
{
"pp": "Z₁ Z₂ X : Scheme\ni₁ : Z₁ ⟶ X\ni₂ : Z₂ ⟶ X\ninst✝¹ : IsClosedImmersion i₁\ninst✝ : IsClosedImmersion i₂\nf : Z₁ ⟶ Z₂\nh✝ : f ≫ i₂ = i₁\nh' : Scheme.Hom.ker i₁ = Scheme.Hom.ker i₂\nf' : MorphismProperty.Over.mk ⊤ i₁ inst✝¹ ⟶ MorphismProperty.Over.mk ⊤ i₂ inst✝ :=\n MorphismProperty.Over.homMk f h✝ True... | [] | rwa [isIso_op_iff, ← isIso_iff_of_reflects_iso _ (MorphismProperty.Over.forget ..),
← isIso_iff_of_reflects_iso _ (Over.forget _)] at h | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.AlgebraicGeometry.Morphisms.ClosedImmersion | {
"line": 226,
"column": 2
} | {
"line": 226,
"column": 57
} | {
"line": 226,
"column": 58
} | [
{
"pp": "Z₁ Z₂ X : Scheme\ni₁ : Z₁ ⟶ X\ni₂ : Z₂ ⟶ X\ninst✝¹ : IsClosedImmersion i₁\ninst✝ : IsClosedImmersion i₂\nf : Z₁ ⟶ Z₂\nh : f ≫ i₂ = i₁\nh' : Scheme.Hom.ker i₁ = Scheme.Hom.ker i₂\nf' : MorphismProperty.Over.mk ⊤ i₁ inst✝¹ ⟶ MorphismProperty.Over.mk ⊤ i₂ inst✝ :=\n MorphismProperty.Over.homMk f h True.i... | [
"Z₁ Z₂ X : Scheme\ni₁ : Z₁ ⟶ X\ni₂ : Z₂ ⟶ X\ninst✝¹ : IsClosedImmersion i₁\ninst✝ : IsClosedImmersion i₂\nf : Z₁ ⟶ Z₂\nh : f ≫ i₂ = i₁\nh' : Scheme.Hom.ker i₁ = Scheme.Hom.ker i₂\nf' : MorphismProperty.Over.mk ⊤ i₁ inst✝¹ ⟶ MorphismProperty.Over.mk ⊤ i₂ inst✝ :=\n MorphismProperty.Over.homMk f h True.intro\n⊢ IsIs... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.Morphisms.Separated | {
"line": 118,
"column": 4
} | {
"line": 118,
"column": 20
} | {
"line": 119,
"column": 2
} | [
{
"pp": "case inr\nX Y : Scheme\nf : X ⟶ Y\nh : IsAffineHom f\nthis : ∀ {X Y : Scheme} (f : X ⟶ Y) [h : IsAffineHom f], IsAffine Y → IsSeparated f\nhY : ¬IsAffine Y\nU : ↑Y.affineOpens\nH : IsAffineHom (f ∣_ ↑U)\n⊢ IsSeparated (f ∣_ ↑U)",
"ppTerm": "?inr",
"assigned": true,
"usedConstants": [
... | [] | exact this _ U.2 | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.AlgebraicGeometry.Morphisms.ClosedImmersion | {
"line": 253,
"column": 4
} | {
"line": 253,
"column": 44
} | {
"line": 254,
"column": 6
} | [
{
"pp": "X Y : Scheme\ninst✝¹ : IsAffine Y\nf : X ⟶ Y\ninst✝ : CompactSpace ↥X\n⊢ Function.Injective ⇑(ConcreteCategory.hom (Scheme.Hom.appTop X.toSpecΓ))",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"AlgebraicGeometry.Spec",
"AlgebraicGeometry.SheafedSpace.i... | [
"X Y : Scheme\ninst✝¹ : IsAffine Y\nf : X ⟶ Y\ninst✝ : CompactSpace ↥X\n⊢ Function.Injective ⇑(ConcreteCategory.hom (Scheme.ΓSpecIso Γ(X, ⊤)).hom)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.Morphisms.Immersion | {
"line": 163,
"column": 72
} | {
"line": 163,
"column": 83
} | {
"line": 163,
"column": 84
} | [
{
"pp": "X✝ Y✝ Z✝ : Scheme\nf✝ : X✝ ⟶ Y✝\nX Y Y' S : Scheme\nf : X ⟶ S\ng : Y ⟶ S\nf' : Y' ⟶ Y\ng' : Y' ⟶ X\nH : IsPullback f' g' g f\nhg : IsImmersion g\nZ : Scheme := pullback f (Scheme.Hom.coborderRange g).ι\n⊢ g' ≫ f = (f' ≫ Scheme.Hom.liftCoborder g) ≫ (Scheme.Hom.coborderRange g).ι",
"ppTerm": "?m.61"... | [
"X✝ Y✝ Z✝ : Scheme\nf✝ : X✝ ⟶ Y✝\nX Y Y' S : Scheme\nf : X ⟶ S\ng : Y ⟶ S\nf' : Y' ⟶ Y\ng' : Y' ⟶ X\nH : IsPullback f' g' g f\nhg : IsImmersion g\nZ : Scheme := pullback f (Scheme.Hom.coborderRange g).ι\n⊢ g' ≫ f = f' ≫ g"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.Morphisms.Immersion | {
"line": 168,
"column": 8
} | {
"line": 168,
"column": 23
} | {
"line": 168,
"column": 24
} | [
{
"pp": "case refine_1\nX✝ Y✝ Z✝ : Scheme\nf✝ : X✝ ⟶ Y✝\nX Y Y' S : Scheme\nf : X ⟶ S\ng : Y ⟶ S\nf' : Y' ⟶ Y\ng' : Y' ⟶ X\nH : IsPullback f' g' g f\nhg : IsImmersion g\nZ : Scheme := pullback f (Scheme.Hom.coborderRange g).ι\ne : Y' ⟶ Z := pullback.lift g' (f' ≫ Scheme.Hom.liftCoborder g) ⋯\n⊢ IsPullback (e ≫ ... | [
"case refine_1\nX✝ Y✝ Z✝ : Scheme\nf✝ : X✝ ⟶ Y✝\nX Y Y' S : Scheme\nf : X ⟶ S\ng : Y ⟶ S\nf' : Y' ⟶ Y\ng' : Y' ⟶ X\nH : IsPullback f' g' g f\nhg : IsImmersion g\nZ : Scheme := pullback f (Scheme.Hom.coborderRange g).ι\ne : Y' ⟶ Z := pullback.lift g' (f' ≫ Scheme.Hom.liftCoborder g) ⋯\n⊢ IsPullback g' f' f g"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.Morphisms.Immersion | {
"line": 170,
"column": 10
} | {
"line": 170,
"column": 21
} | {
"line": 170,
"column": 22
} | [
{
"pp": "X✝ Y✝ Z✝ : Scheme\nf✝ : X✝ ⟶ Y✝\nX Y Y' S : Scheme\nf : X ⟶ S\ng : Y ⟶ S\nf' : Y' ⟶ Y\ng' : Y' ⟶ X\nH : IsPullback f' g' g f\nhg : IsImmersion g\nZ : Scheme := pullback f (Scheme.Hom.coborderRange g).ι\ne : Y' ⟶ Z := pullback.lift g' (f' ≫ Scheme.Hom.liftCoborder g) ⋯\nthis : IsClosedImmersion e\n⊢ g' ... | [
"X✝ Y✝ Z✝ : Scheme\nf✝ : X✝ ⟶ Y✝\nX Y Y' S : Scheme\nf : X ⟶ S\ng : Y ⟶ S\nf' : Y' ⟶ Y\ng' : Y' ⟶ X\nH : IsPullback f' g' g f\nhg : IsImmersion g\nZ : Scheme := pullback f (Scheme.Hom.coborderRange g).ι\ne : Y' ⟶ Z := pullback.lift g' (f' ≫ Scheme.Hom.liftCoborder g) ⋯\nthis : IsClosedImmersion e\n⊢ g' ≫ f = f' ≫ g... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.RingHom.FinitePresentation | {
"line": 35,
"column": 45
} | {
"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",
... | [] | by simpa using! Submodule.fg_bot | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.Spectrum.Prime.Polynomial | {
"line": 60,
"column": 4
} | {
"line": 61,
"column": 65
} | {
"line": 61,
"column": 66
} | [
{
"pp": "case inr.mp\nR : Type u_1\nA : Type u_2\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : Algebra R A\ninst✝² : Module.Free R A\ninst✝¹ : Module.Finite R A\nf : A\nI : Ideal R\ninst✝ : I.IsPrime\nh✝ : Nontrivial R\nthis : Module.finrank I.ResidueField (I.ResidueField ⊗[R] A) = Module.finrank R A\ni :... | [
"case inr.mp\nR : Type u_1\nA : Type u_2\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : Algebra R A\ninst✝² : Module.Free R A\ninst✝¹ : Module.Finite R A\nf : A\nI : Ideal R\ninst✝ : I.IsPrime\nh✝ : Nontrivial R\nthis : Module.finrank I.ResidueField (I.ResidueField ⊗[R] A) = Module.finrank R A\ni : ℕ\nhi : i <... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Spectrum.Prime.Polynomial | {
"line": 68,
"column": 6
} | {
"line": 68,
"column": 79
} | {
"line": 68,
"column": 80
} | [
{
"pp": "case inr.mpr.inr.inl\nR : Type u_1\nA : Type u_2\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : Algebra R A\ninst✝² : Module.Free R A\ninst✝¹ : Module.Finite R A\nf : A\nI : Ideal R\ninst✝ : I.IsPrime\nh✝ : Nontrivial R\nthis : Module.finrank I.ResidueField (I.ResidueField ⊗[R] A) = Module.finrank... | [
"case inr.mpr.inr.inl\nR : Type u_1\nA : Type u_2\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : Algebra R A\ninst✝² : Module.Free R A\ninst✝¹ : Module.Finite R A\nf : A\nI : Ideal R\ninst✝ : I.IsPrime\nh✝ : Nontrivial R\nthis : Module.finrank I.ResidueField (I.ResidueField ⊗[R] A) = Module.finrank R A\nH : ∀ ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Spectrum.Prime.Polynomial | {
"line": 184,
"column": 19
} | {
"line": 184,
"column": 30
} | {
"line": 184,
"column": 31
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\nf g : R[X]\nhg : g.Monic\nt : Finset R\nht : comap C '' (zeroLocus {g} \\ zeroLocus {f}) = (zeroLocus ↑t)ᶜ\nx✝ : ↑↑t\n⊢ IsCompact (zeroLocus {↑x✝})ᶜ",
"ppTerm": "?m.66",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"R : Type u_1\ninst✝ : CommRing R\nf g : R[X]\nhg : g.Monic\nt : Finset R\nht : comap C '' (zeroLocus {g} \\ zeroLocus {f}) = (zeroLocus ↑t)ᶜ\nx✝ : ↑↑t\n⊢ IsCompact (zeroLocus {↑x✝})ᶜ"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.Artinian | {
"line": 103,
"column": 2
} | {
"line": 103,
"column": 13
} | {
"line": 103,
"column": 14
} | [
{
"pp": "X Y : Scheme\nf : X ⟶ Y\ninst✝ : IsLocallyArtinian X\nx : ↥X\nW : X.Opens\nhW1 : IsAffineOpen W\nhW2 : x ∈ W\nright✝ : W.carrier ⊆ ↑⊤\nthis✝¹ : IsArtinianRing ↑Γ(X, W)\nthis✝ : DiscreteTopology ↥(Spec Γ(X, W))\nthis : DiscreteTopology ↥↑W\n⊢ IsOpen {x}",
"ppTerm": "?m.95",
"assigned": false,
... | [
"X Y : Scheme\nf : X ⟶ Y\ninst✝ : IsLocallyArtinian X\nx : ↥X\nW : X.Opens\nhW1 : IsAffineOpen W\nhW2 : x ∈ W\nright✝ : W.carrier ⊆ ↑⊤\nthis✝¹ : IsArtinianRing ↑Γ(X, W)\nthis✝ : DiscreteTopology ↥(Spec Γ(X, W))\nthis : DiscreteTopology ↥↑W\n⊢ IsOpen {x}"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.Artinian | {
"line": 132,
"column": 4
} | {
"line": 132,
"column": 15
} | {
"line": 132,
"column": 16
} | [
{
"pp": "case refine_2\nX : Scheme\n𝒰 : X.OpenCover\nH : ∀ (i : 𝒰.I₀), IsLocallyArtinian (𝒰.X i)\ni : 𝒰.I₀\nx : ↥(𝒰.X i)\n⊢ IsOpen {(𝒰.f i) x}",
"ppTerm": "?refine_2",
"assigned": true,
"usedConstants": [
"AlgebraicGeometry.SheafedSpace.instTopologicalSpaceCarrierCarrier",
"Algebra... | [
"case refine_2\nX : Scheme\n𝒰 : X.OpenCover\nH : ∀ (i : 𝒰.I₀), IsLocallyArtinian (𝒰.X i)\ni : 𝒰.I₀\nx : ↥(𝒰.X i)\n⊢ IsOpen {(𝒰.f i) x}"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.Geometrically.Reduced | {
"line": 84,
"column": 2
} | {
"line": 95,
"column": 66
} | {
"line": 96,
"column": 2
} | [
{
"pp": "X Y : Scheme\nf : X ⟶ Y\ninst✝³ : GeometricallyReduced f\ninst✝² : Flat f\ninst✝¹ : IsReduced Y\ninst✝ : Finite ↑(irreducibleComponents ↥Y)\npt : ↑(irreducibleComponents ↥Y) → CommRingCat := fun Z ↦ Y.presheaf.stalk ⋯.genericPoint\nhpt : ∀ (Z : ↑(irreducibleComponents ↥Y)), IsField ↑(pt Z)\nthis✝¹ : (Z... | [
"X Y : Scheme\nf : X ⟶ Y\ninst✝³ : GeometricallyReduced f\ninst✝² : Flat f\ninst✝¹ : IsReduced Y\ninst✝ : Finite ↑(irreducibleComponents ↥Y)\npt : ↑(irreducibleComponents ↥Y) → CommRingCat := fun Z ↦ Y.presheaf.stalk ⋯.genericPoint\nhpt : ∀ (Z : ↑(irreducibleComponents ↥Y)), IsField ↑(pt Z)\nthis✝¹ : (Z : ↑(irreduc... | have H : IsSchemeTheoreticallyDominant g := by
rw [isSchemeTheoreticallyDominant_iff_isDominant, isDominant_iff, denseRange_iff_closure_range,
Set.eq_univ_iff_forall]
intro y
let z : Z := Sigma.ι (fun Z ↦ Spec (pt Z)) ⟨_, irreducibleComponent_mem_irreducibleComponents y⟩
(IsLocalRing.closedPoint... | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.AlgebraicGeometry.Morphisms.UniversallyOpen | {
"line": 92,
"column": 2
} | {
"line": 92,
"column": 42
} | {
"line": 93,
"column": 2
} | [
{
"pp": "X Y : Scheme\nf : X ⟶ Y\n⊢ IsZariskiLocalAtTarget (topologically @IsOpenMap).universally",
"ppTerm": "?m.4",
"assigned": true,
"usedConstants": [
"IsOpenMap",
"AlgebraicGeometry.topologically",
"AlgebraicGeometry.universally_isZariskiLocalAtTarget"
],
"usedFVars": ... | [
"X Y : Scheme\nf : X ⟶ Y\n⊢ ∀ {X Y : Scheme} (f : X ⟶ Y) {ι : Type u_1} (U : ι → Y.Opens),\n IsOpenCover U → (∀ (i : ι), topologically (@IsOpenMap) (f ∣_ U i)) → topologically (@IsOpenMap) f"
] | apply universally_isZariskiLocalAtTarget | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.AlgebraicGeometry.Morphisms.SchemeTheoreticallyDominant | {
"line": 110,
"column": 8
} | {
"line": 110,
"column": 38
} | {
"line": 110,
"column": 39
} | [
{
"pp": "X Y : Scheme\nZ : Scheme\nS : Scheme\nf✝ : X ⟶ S\ng✝ : Y ⟶ S\nf : X ⟶ S\ng : Y ⟶ S\ninst✝² : IsSchemeTheoreticallyDominant f\ninst✝¹ : QuasiCompact f\ninst✝ : Flat g\nh𝒰 : TopologicalSpace.IsOpenCover fun V ↦ (↑V).1 :=\n TopologicalSpace.Opens.IsBasis.isOpenCover_mem_and_le (Scheme.isBasis_affineOpen... | [
"X Y : Scheme\nZ : Scheme\nS : Scheme\nf✝ : X ⟶ S\ng✝ : Y ⟶ S\nf : X ⟶ S\ng : Y ⟶ S\ninst✝² : IsSchemeTheoreticallyDominant f\ninst✝¹ : QuasiCompact f\ninst✝ : Flat g\nh𝒰 : TopologicalSpace.IsOpenCover fun V ↦ (↑V).1 :=\n TopologicalSpace.Opens.IsBasis.isOpenCover_mem_and_le (Scheme.isBasis_affineOpens Y)\n (T... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.Geometrically.Irreducible | {
"line": 94,
"column": 2
} | {
"line": 94,
"column": 36
} | {
"line": 95,
"column": 4
} | [
{
"pp": "X S : Scheme\nf : X ⟶ S\ninst✝¹ : GeometricallyIrreducible f\ninst✝ : IrreducibleSpace ↥S\nhf : IsOpenMap ⇑f\n⊢ IrreducibleSpace ↥X",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"AlgebraicGeometry.SheafedSpace.instTopologicalSpaceCarrierCarrier",
"Alg... | [
"X S : Scheme\nf : X ⟶ S\ninst✝¹ : GeometricallyIrreducible f\ninst✝ : IrreducibleSpace ↥S\nhf : IsOpenMap ⇑f\n⊢ IsIrreducible Set.univ"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.Morphisms.SchemeTheoreticallyDominant | {
"line": 112,
"column": 2
} | {
"line": 112,
"column": 67
} | {
"line": 112,
"column": 68
} | [
{
"pp": "X Y : Scheme\nZ : Scheme\nS : Scheme\nf✝ : X ⟶ S\ng✝ : Y ⟶ S\nf : X ⟶ S\ng : Y ⟶ S\ninst✝² : IsSchemeTheoreticallyDominant f\ninst✝¹ : QuasiCompact f\ninst✝ : Flat g\nh𝒰 : TopologicalSpace.IsOpenCover fun V ↦ (↑V).1 := ⋯\nV : TopologicalSpace.Opens ↥Y\nU : TopologicalSpace.Opens ↥S\nhU : U ∈ S.affineO... | [
"X Y : Scheme\nZ : Scheme\nS : Scheme\nf✝ : X ⟶ S\ng✝ : Y ⟶ S\nf : X ⟶ S\ng : Y ⟶ S\ninst✝² : IsSchemeTheoreticallyDominant f\ninst✝¹ : QuasiCompact f\ninst✝ : Flat g\nh𝒰 : TopologicalSpace.IsOpenCover fun V ↦ (↑V).1 :=\n TopologicalSpace.Opens.IsBasis.isOpenCover_mem_and_le (Scheme.isBasis_affineOpens Y)\n (T... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.RingHom.Integral | {
"line": 63,
"column": 74
} | {
"line": 63,
"column": 85
} | {
"line": 63,
"column": 86
} | [
{
"pp": "R S : Type u_1\ninst✝¹ : CommRing R\ninst✝ : CommRing S\nf : R →+* S\ns : Set R\nhs : Ideal.span s = ⊤\nH : ∀ (r : ↑s), (fun {R S} [CommRing R] [CommRing S] x ↦ x.IsIntegral) (Localization.awayMap f ↑r)\nr : S\nthis✝² : Algebra R S := f.toAlgebra\nt : R\nht : t ∈ s\nthis✝¹ : Algebra (Localization.Away ... | [
"R S : Type u_1\ninst✝¹ : CommRing R\ninst✝ : CommRing S\nf : R →+* S\ns : Set R\nhs : Ideal.span s = ⊤\nH : ∀ (r : ↑s), (fun {R S} [CommRing R] [CommRing S] x ↦ x.IsIntegral) (Localization.awayMap f ↑r)\nr : S\nthis✝² : Algebra R S := f.toAlgebra\nt : R\nht : t ∈ s\nthis✝¹ : Algebra (Localization.Away t) (Localiza... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.Morphisms.Integral | {
"line": 108,
"column": 6
} | {
"line": 108,
"column": 27
} | {
"line": 108,
"column": 28
} | [
{
"pp": "Z S : Scheme\n⊢ ∀ {X Y : Scheme} (f : X ⟶ Y) [IsIntegralHom f], UniversallyClosed f",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"AlgebraicGeometry.IsIntegralHom",
"Eq.mpr",
"AlgebraicGeometry.Scheme",
"CategoryTheory.CategoryStruct.toQuiver",
"Qui... | [
"Z S : Scheme\n⊢ ∀ {X Y : Scheme} (f : X ⟶ Y) [IsIntegralHom f], (topologically @IsClosedMap).universally f"
] | universallyClosed_eq, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.AlgebraicGeometry.Morphisms.UniversallyClosed | {
"line": 103,
"column": 2
} | {
"line": 103,
"column": 42
} | {
"line": 104,
"column": 2
} | [
{
"pp": "X Y : Scheme\nf : X ⟶ Y\n⊢ IsZariskiLocalAtTarget (topologically @IsClosedMap).universally",
"ppTerm": "?m.4",
"assigned": true,
"usedConstants": [
"IsClosedMap",
"AlgebraicGeometry.topologically",
"AlgebraicGeometry.universally_isZariskiLocalAtTarget"
],
"usedFVar... | [
"X Y : Scheme\nf : X ⟶ Y\n⊢ ∀ {X Y : Scheme} (f : X ⟶ Y) {ι : Type u_1} (U : ι → Y.Opens),\n IsOpenCover U → (∀ (i : ι), topologically (@IsClosedMap) (f ∣_ U i)) → topologically (@IsClosedMap) f"
] | apply universally_isZariskiLocalAtTarget | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.AlgebraicGeometry.Morphisms.UniversallyClosed | {
"line": 134,
"column": 39
} | {
"line": 134,
"column": 54
} | {
"line": 134,
"column": 55
} | [
{
"pp": "X : Scheme\nK : Type u\ninst✝¹ : Field K\nf : X ⟶ Spec (CommRingCat.of K)\ninst✝ : UniversallyClosed f\n𝒰 : X.OpenCover := X.affineCover\nU : 𝒰.I₀ → X.Opens := fun i ↦ Scheme.Hom.opensRange (𝒰.f i)\nT : Scheme := Spec (CommRingCat.of (MvPolynomial 𝒰.I₀ K))\nq : T ⟶ Spec (CommRingCat.of K) := Spec.m... | [
"X : Scheme\nK : Type u\ninst✝¹ : Field K\nf : X ⟶ Spec (CommRingCat.of K)\ninst✝ : UniversallyClosed f\n𝒰 : X.OpenCover := X.affineCover\nU : 𝒰.I₀ → X.Opens := fun i ↦ Scheme.Hom.opensRange (𝒰.f i)\nT : Scheme := Spec (CommRingCat.of (MvPolynomial 𝒰.I₀ K))\nq : T ⟶ Spec (CommRingCat.of K) := Spec.map (CommRing... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.Morphisms.UniversallyClosed | {
"line": 142,
"column": 2
} | {
"line": 142,
"column": 37
} | {
"line": 143,
"column": 2
} | [
{
"pp": "X : Scheme\nK : Type u\ninst✝¹ : Field K\nf : X ⟶ Spec (CommRingCat.of K)\ninst✝ : UniversallyClosed f\n𝒰 : X.OpenCover := X.affineCover\nU : 𝒰.I₀ → X.Opens := fun i ↦ Scheme.Hom.opensRange (𝒰.f i)\nT : Scheme := Spec (CommRingCat.of (MvPolynomial 𝒰.I₀ K))\nq : T ⟶ Spec (CommRingCat.of K) := Spec.m... | [
"X : Scheme\nK : Type u\ninst✝¹ : Field K\nf : X ⟶ Spec (CommRingCat.of K)\ninst✝ : UniversallyClosed f\n𝒰 : X.OpenCover := X.affineCover\nU : 𝒰.I₀ → X.Opens := fun i ↦ Scheme.Hom.opensRange (𝒰.f i)\nT : Scheme := Spec (CommRingCat.of (MvPolynomial 𝒰.I₀ K))\nq : T ⟶ Spec (CommRingCat.of K) := Spec.map (CommRing... | have h : t' ∉ fT '' Z := hU'le ht'g | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.AlgebraicGeometry.Morphisms.Finite | {
"line": 202,
"column": 2
} | {
"line": 202,
"column": 65
} | {
"line": 203,
"column": 4
} | [
{
"pp": "X Y : Scheme\nf : X ⟶ Y\ninst✝¹ : JacobsonSpace ↥Y\ninst✝ : LocallyOfFiniteType f\nx : ↥X\nhx : x ∈ closedPoints ↥X\nthis✝ : IsClosedImmersion (X.fromSpecResidueField x)\nthis : IsFinite (X.fromSpecResidueField x ≫ f)\n⊢ x ∈ ⇑f ⁻¹' closedPoints ↥Y",
"ppTerm": "?m.40",
"assigned": true,
"use... | [
"X Y : Scheme\nf : X ⟶ Y\ninst✝¹ : JacobsonSpace ↥Y\ninst✝ : LocallyOfFiniteType f\nx : ↥X\nhx : x ∈ closedPoints ↥X\nthis✝ : IsClosedImmersion (X.fromSpecResidueField x)\nthis : IsFinite (X.fromSpecResidueField x ≫ f)\n⊢ IsClosed {f x}"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.Morphisms.Finite | {
"line": 211,
"column": 4
} | {
"line": 211,
"column": 15
} | {
"line": 211,
"column": 16
} | [
{
"pp": "case mpr\nX : Scheme\ninst✝ : JacobsonSpace ↥X\nx : ↥X\nH : LocallyOfFiniteType (X.fromSpecResidueField x)\n⊢ IsClosed {x}",
"ppTerm": "?mpr",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"case mpr\nX : Scheme\ninst✝ : JacobsonSpace ↥X\nx : ↥X\nH : LocallyOfFiniteType (X.fromSpecResidueField x)\n⊢ IsClosed {x}"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.Morphisms.UniversallyClosed | {
"line": 152,
"column": 44
} | {
"line": 152,
"column": 59
} | {
"line": 152,
"column": 60
} | [
{
"pp": "X : Scheme\nK : Type u\ninst✝¹ : Field K\nf : X ⟶ Spec (CommRingCat.of K)\ninst✝ : UniversallyClosed f\n𝒰 : X.OpenCover := X.affineCover\nU : 𝒰.I₀ → X.Opens := fun i ↦ Scheme.Hom.opensRange (𝒰.f i)\nT : Scheme := Spec (CommRingCat.of (MvPolynomial 𝒰.I₀ K))\nq : T ⟶ Spec (CommRingCat.of K) := Spec.m... | [
"X : Scheme\nK : Type u\ninst✝¹ : Field K\nf : X ⟶ Spec (CommRingCat.of K)\ninst✝ : UniversallyClosed f\n𝒰 : X.OpenCover := X.affineCover\nU : 𝒰.I₀ → X.Opens := fun i ↦ Scheme.Hom.opensRange (𝒰.f i)\nT : Scheme := Spec (CommRingCat.of (MvPolynomial 𝒰.I₀ K))\nq : T ⟶ Spec (CommRingCat.of K) := Spec.map (CommRing... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Spectrum.Prime.ChevalleyComplexity | {
"line": 244,
"column": 30
} | {
"line": 244,
"column": 41
} | {
"line": 244,
"column": 42
} | [
{
"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 ... | [
"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 u) [inst : C... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.QuasiAffine | {
"line": 148,
"column": 2
} | {
"line": 148,
"column": 13
} | {
"line": 148,
"column": 14
} | [
{
"pp": "X Y : Scheme\nf : X ⟶ Y\ninst✝² : IsAffineHom f\ninst✝¹ : X.IsQuasiAffine\ninst✝ : Y.IsQuasiAffine\n⊢ Spec.map (Hom.appTop f) ⁻¹ᵁ Hom.opensRange Y.toSpecΓ = Hom.opensRange X.toSpecΓ",
"ppTerm": "?m.30",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"X Y : Scheme\nf : X ⟶ Y\ninst✝² : IsAffineHom f\ninst✝¹ : X.IsQuasiAffine\ninst✝ : Y.IsQuasiAffine\n⊢ Spec.map (Hom.appTop f) ⁻¹ᵁ Hom.opensRange Y.toSpecΓ = Hom.opensRange X.toSpecΓ"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Limits.Types.ColimitTypeFiltered | {
"line": 48,
"column": 29
} | {
"line": 48,
"column": 40
} | {
"line": 48,
"column": 41
} | [
{
"pp": "J : Type u\ninst✝¹ : Category.{v, u} J\ninst✝ : IsFiltered J\nF : J ⥤ Type w₀\nx✝ y✝ x y : (j : J) × F.obj j\nf : x.fst ⟶ y.fst\nh : y.snd = (ConcreteCategory.hom (F.map f)) x.snd\n⊢ (ConcreteCategory.hom (F.map f)) x.snd = (ConcreteCategory.hom (F.map (𝟙 y.fst))) y.snd",
"ppTerm": "?m.92",
"a... | [
"J : Type u\ninst✝¹ : Category.{v, u} J\ninst✝ : IsFiltered J\nF : J ⥤ Type w₀\nx✝ y✝ x y : (j : J) × F.obj j\nf : x.fst ⟶ y.fst\nh : y.snd = (ConcreteCategory.hom (F.map f)) x.snd\n⊢ (ConcreteCategory.hom (F.map f)) x.snd = y.snd"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Limits.Types.ColimitTypeFiltered | {
"line": 44,
"column": 4
} | {
"line": 59,
"column": 42
} | {
"line": 60,
"column": 2
} | [
{
"pp": "case mp\nJ : Type u\ninst✝¹ : Category.{v, u} J\ninst✝ : IsFiltered J\nF : J ⥤ Type w₀\nx y : (j : J) × F.obj j\n⊢ Relation.EqvGen F.ColimitTypeRel x y →\n ∃ k f g, (ConcreteCategory.hom (F.map f)) x.snd = (ConcreteCategory.hom (F.map g)) y.snd",
"ppTerm": "?mp",
"assigned": true,
"usedC... | [] | intro h
induction h with
| rel x y h =>
obtain ⟨f, h⟩ := h
exact ⟨y.1, f, 𝟙 _, by simpa using h.symm⟩
| refl x => exact ⟨x.1, 𝟙 _, 𝟙 _, rfl⟩
| symm _ _ _ h =>
obtain ⟨k, f, g, h⟩ := h
exact ⟨k, g, f, h.symm⟩
| trans x y z _ _ h h' =>
obtain ⟨k, f, g, h⟩ := h
ob... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Limits.Types.ColimitTypeFiltered | {
"line": 44,
"column": 4
} | {
"line": 59,
"column": 42
} | {
"line": 60,
"column": 2
} | [
{
"pp": "case mp\nJ : Type u\ninst✝¹ : Category.{v, u} J\ninst✝ : IsFiltered J\nF : J ⥤ Type w₀\nx y : (j : J) × F.obj j\n⊢ Relation.EqvGen F.ColimitTypeRel x y →\n ∃ k f g, (ConcreteCategory.hom (F.map f)) x.snd = (ConcreteCategory.hom (F.map g)) y.snd",
"ppTerm": "?mp",
"assigned": true,
"usedC... | [] | intro h
induction h with
| rel x y h =>
obtain ⟨f, h⟩ := h
exact ⟨y.1, f, 𝟙 _, by simpa using h.symm⟩
| refl x => exact ⟨x.1, 𝟙 _, 𝟙 _, rfl⟩
| symm _ _ _ h =>
obtain ⟨k, f, g, h⟩ := h
exact ⟨k, g, f, h.symm⟩
| trans x y z _ _ h h' =>
obtain ⟨k, f, g, h⟩ := h
ob... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Spectrum.Prime.ChevalleyComplexity | {
"line": 266,
"column": 17
} | {
"line": 266,
"column": 28
} | {
"line": 266,
"column": 29
} | [
{
"pp": "case refine_1\nn : 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 :\... | [
"case refine_1\nn : 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 : Ty... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Spectrum.Prime.ChevalleyComplexity | {
"line": 299,
"column": 4
} | {
"line": 299,
"column": 68
} | {
"line": 300,
"column": 4
} | [
{
"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 ... | [
"case refine_1\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.sp... | refine Fintype.sum_strictMono <| Pi.lt_def.2 ⟨fun j ↦ ?_, i, ?_⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.RingTheory.Spectrum.Prime.ChevalleyComplexity | {
"line": 304,
"column": 51
} | {
"line": 304,
"column": 67
} | {
"line": 304,
"column": 68
} | [
{
"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... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.AlgClosed.Basic | {
"line": 109,
"column": 51
} | {
"line": 109,
"column": 62
} | {
"line": 109,
"column": 63
} | [
{
"pp": "X Y : Scheme\nK : Type u\ninst✝⁵ : Field K\ninst✝⁴ : IsAlgClosed K\nf g : X ⟶ Y\ni : Y ⟶ Spec (CommRingCat.of K)\ninst✝³ : IsSeparated i\ninst✝² : LocallyOfFiniteType i\ninst✝¹ : IsReduced X\ninst✝ : LocallyOfFiniteType (f ≫ i)\nS : Set ↥X\nhS : IsLocallyClosed S\nhS' : Dense S\nH : ∀ x ∈ S, IsClosed {... | [
"X Y : Scheme\nK : Type u\ninst✝⁵ : Field K\ninst✝⁴ : IsAlgClosed K\nf g : X ⟶ Y\ni : Y ⟶ Spec (CommRingCat.of K)\ninst✝³ : IsSeparated i\ninst✝² : LocallyOfFiniteType i\ninst✝¹ : IsReduced X\ninst✝ : LocallyOfFiniteType (f ≫ i)\nS : Set ↥X\nhS : IsLocallyClosed S\nhS' : Dense S\nH : ∀ x ∈ S, IsClosed {x} → f x = g... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.LocallyFinsupp | {
"line": 113,
"column": 2
} | {
"line": 113,
"column": 37
} | {
"line": 115,
"column": 0
} | [
{
"pp": "X : Type u_1\ninst✝¹ : TopologicalSpace X\nY : Type u_2\nW : Set X\ninst✝ : Zero Y\nf : X → Y\nh : LocallyFiniteSupport f\nhW : IsCompact W\nthis : {i | ({↑i} ∩ W).Nonempty}.Finite\nlem : ∀ {α : Type u_1} (s t : Set α), Subtype.val '' {i | ({↑i} ∩ t).Nonempty} = t ∩ s\n⊢ (Subtype.val '' {i | ({↑i} ∩ W)... | [] | exact Finite.image Subtype.val this | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Topology.LocallyFinsupp | {
"line": 165,
"column": 28
} | {
"line": 165,
"column": 39
} | {
"line": 165,
"column": 40
} | [
{
"pp": "X : Type u_1\ninst✝² : TopologicalSpace X\nU : Set X\nY : Type u_2\ninst✝¹ : DecidableEq X\ninst✝ : Zero Y\nx : X\ny : Y\nx✝¹ : X\nx✝ : x✝¹ ∈ univ\n⊢ (univ ∩ Function.support (Pi.single x y)).Finite",
"ppTerm": "?m.39",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
... | [
"X : Type u_1\ninst✝² : TopologicalSpace X\nU : Set X\nY : Type u_2\ninst✝¹ : DecidableEq X\ninst✝ : Zero Y\nx : X\ny : Y\nx✝¹ : X\nx✝ : x✝¹ ∈ univ\n⊢ (Function.support (Pi.single x y)).Finite"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.LocallyFinsupp | {
"line": 382,
"column": 4
} | {
"line": 382,
"column": 67
} | {
"line": 383,
"column": 4
} | [
{
"pp": "case pos\nX : Type u_1\ninst✝ : TopologicalSpace X\nU : Set X\nι : Type u_3\nF : ι → locallyFinsuppWithin U ℤ\nthis : Function.support F = Function.support fun i ↦ ⇑(F i)\nh : (Function.support F).Finite\n⊢ ∑ n ∈ h.toFinset, ⇑(F n) = ∑ᶠ (i : ι), ⇑(F i)",
"ppTerm": "?pos✝",
"assigned": true,
... | [
"case pos\nX : Type u_1\ninst✝ : TopologicalSpace X\nU : Set X\nι : Type u_3\nF : ι → locallyFinsuppWithin U ℤ\nthis : Function.support F = Function.support fun i ↦ ⇑(F i)\nh : (Function.support F).Finite\nh₂ : (Function.support fun i ↦ ⇑(F i)).Finite\n⊢ ∑ n ∈ h.toFinset, ⇑(F n) = ∑ᶠ (i : ι), ⇑(F i)"
] | have h₂ : (fun i ↦ (F i : X → ℤ)).support.Finite := by simp_all | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Topology.LocallyFinsupp | {
"line": 378,
"column": 2
} | {
"line": 384,
"column": 41
} | {
"line": 386,
"column": 0
} | [
{
"pp": "X : Type u_1\ninst✝ : TopologicalSpace X\nU : Set X\nι : Type u_3\nF : ι → locallyFinsuppWithin U ℤ\n⊢ ⇑(∑ᶠ (i : ι), F i) = ∑ᶠ (i : ι), ⇑(F i)",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Int.instAddCommMonoid",
"Pi.addCommMonoid",
"Function.l... | [] | have : F.support = (fun i ↦ (F i : X → ℤ)).support := by
simp [Set.ext_iff, DFunLike.ext_iff, funext_iff]
by_cases h : F.support.Finite
· rw [finsum_eq_sum F h, Function.locallyFinsuppWithin.coe_sum]
have h₂ : (fun i ↦ (F i : X → ℤ)).support.Finite := by simp_all
simp_all [finsum_eq_sum _ h₂]
· simp_a... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.LocallyFinsupp | {
"line": 378,
"column": 2
} | {
"line": 384,
"column": 41
} | {
"line": 386,
"column": 0
} | [
{
"pp": "X : Type u_1\ninst✝ : TopologicalSpace X\nU : Set X\nι : Type u_3\nF : ι → locallyFinsuppWithin U ℤ\n⊢ ⇑(∑ᶠ (i : ι), F i) = ∑ᶠ (i : ι), ⇑(F i)",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Int.instAddCommMonoid",
"Pi.addCommMonoid",
"Function.l... | [] | have : F.support = (fun i ↦ (F i : X → ℤ)).support := by
simp [Set.ext_iff, DFunLike.ext_iff, funext_iff]
by_cases h : F.support.Finite
· rw [finsum_eq_sum F h, Function.locallyFinsuppWithin.coe_sum]
have h₂ : (fun i ↦ (F i : X → ℤ)).support.Finite := by simp_all
simp_all [finsum_eq_sum _ h₂]
· simp_a... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.LocallyFinsupp | {
"line": 547,
"column": 4
} | {
"line": 547,
"column": 38
} | {
"line": 547,
"column": 39
} | [
{
"pp": "case pos\nX : 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\nn : ℕ\nf : locallyFinsuppWithin U Y\nx : X\nh : f x < 0\n⊢ max (n • f x) 0 = n • max (f x) 0",
"ppTerm": "?pos✝",
"assigned": true,
"u... | [
"case pos\nX : 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\nn : ℕ\nf : locallyFinsuppWithin U Y\nx : X\nh : f x < 0\n⊢ n • f x ≤ 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.LocallyFinsupp | {
"line": 548,
"column": 4
} | {
"line": 548,
"column": 28
} | {
"line": 548,
"column": 29
} | [
{
"pp": "case neg\nX : 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\nn : ℕ\nf : locallyFinsuppWithin U Y\nx : X\nh : ¬f x < 0\n⊢ max (n • f x) 0 = n • max (f x) 0",
"ppTerm": "?neg✝",
"assigned": true,
"... | [
"case neg\nX : 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\nn : ℕ\nf : locallyFinsuppWithin U Y\nx : X\nh : ¬f x < 0\n⊢ 0 ≤ n • f x"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.LocallyFinsupp | {
"line": 560,
"column": 4
} | {
"line": 560,
"column": 38
} | {
"line": 560,
"column": 39
} | [
{
"pp": "case pos\nX : 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\nn : ℕ\nf : locallyFinsuppWithin U Y\nx : X\nh : -f x < 0\n⊢ max (-(n • f x)) 0 = n • max (-f x) 0",
"ppTerm": "?pos✝",
"assigned": true,
... | [
"case pos\nX : 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\nn : ℕ\nf : locallyFinsuppWithin U Y\nx : X\nh : -f x < 0\n⊢ 0 ≤ n • f x"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.LocallyFinsupp | {
"line": 561,
"column": 4
} | {
"line": 561,
"column": 28
} | {
"line": 561,
"column": 29
} | [
{
"pp": "case neg\nX : 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\nn : ℕ\nf : locallyFinsuppWithin U Y\nx : X\nh : ¬-f x < 0\n⊢ max (-(n • f x)) 0 = n • max (-f x) 0",
"ppTerm": "?neg✝",
"assigned": true,
... | [
"case neg\nX : 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\nn : ℕ\nf : locallyFinsuppWithin U Y\nx : X\nh : ¬-f x < 0\n⊢ n • f x ≤ 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Spectrum.Prime.ChevalleyComplexity | {
"line": 357,
"column": 8
} | {
"line": 357,
"column": 19
} | {
"line": 357,
"column": 20
} | [
{
"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... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.SpreadingOut | {
"line": 180,
"column": 2
} | {
"line": 180,
"column": 83
} | {
"line": 181,
"column": 2
} | [
{
"pp": "X Y S : Scheme\nf✝ : X ⟶ Y\nsX : X ⟶ S\nsY : Y ⟶ S\nR✝ A : CommRingCat\ninst✝ : IsLocallyNoetherian X\nR : CommRingCat\nhR : IsNoetherianRing ↑R\nI : Ideal ↑R\nhI : I.IsPrime\nJ : Ideal ↑R := RingHom.ker (algebraMap (↑R) (Localization.AtPrime I))\nhJ : ∀ (x : ↑R), x ∈ J ↔ ∃ y, ↑y * x = 0\nf : (x : ↑R) ... | [
"X Y S : Scheme\nf✝ : X ⟶ Y\nsX : X ⟶ S\nsY : Y ⟶ S\nR✝ A : CommRingCat\ninst✝ : IsLocallyNoetherian X\nR : CommRingCat\nhR : IsNoetherianRing ↑R\nI : Ideal ↑R\nhI : I.IsPrime\nJ : Ideal ↑R := RingHom.ker (algebraMap (↑R) (Localization.AtPrime I))\nhJ : ∀ (x : ↑R), x ∈ J ↔ ∃ y, ↑y * x = 0\nf : (x : ↑R) → x ∈ J → ↥I... | rw [pow_one, mul_comm, ← smul_eq_mul, ← Submodule.mem_annihilator_span_singleton] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.AlgebraicGeometry.SpreadingOut | {
"line": 235,
"column": 4
} | {
"line": 235,
"column": 15
} | {
"line": 235,
"column": 16
} | [
{
"pp": "case e\nX Y : Scheme\nx : ↥X\ninst✝ : X.IsGermInjectiveAt x\nf g : X ⟶ Y\ne : X.fromSpecStalk x ≫ f = X.fromSpecStalk x ≫ g\n⊢ f x = g x",
"ppTerm": "?e",
"assigned": true,
"usedConstants": [
"AlgebraicGeometry.PresheafedSpace.carrier",
"CategoryTheory.ConcreteCategory.hom",
... | [
"case e\nX Y : Scheme\nx : ↥X\ninst✝ : X.IsGermInjectiveAt x\nf g : X ⟶ Y\ne : X.fromSpecStalk x ≫ f = X.fromSpecStalk x ≫ g\n⊢ f x = g x"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.SpreadingOut | {
"line": 251,
"column": 14
} | {
"line": 251,
"column": 25
} | {
"line": 251,
"column": 26
} | [
{
"pp": "X : Scheme\nR A : CommRingCat\nU : X.Opens\nx : ↥X\nhxU : x ∈ U\nφ : A ⟶ X.presheaf.stalk x\nφRA : R ⟶ A\nφRX : R ⟶ Γ(X, U)\ne : φRA ≫ φ = φRX ≫ X.presheaf.germ U x hxU\nthis : Algebra ↑R ↑A := (CommRingCat.Hom.hom φRA).toAlgebra\ns : Finset ↑A\nhs : Algebra.adjoin ↑R ↑s = ⊤\nW : ↑A → TopologicalSpace.... | [
"X : Scheme\nR A : CommRingCat\nU : X.Opens\nx : ↥X\nhxU : x ∈ U\nφ : A ⟶ X.presheaf.stalk x\nφRA : R ⟶ A\nφRX : R ⟶ Γ(X, U)\ne : φRA ≫ φ = φRX ≫ X.presheaf.germ U x hxU\nthis : Algebra ↑R ↑A := (CommRingCat.Hom.hom φRA).toAlgebra\ns : Finset ↑A\nhs : Algebra.adjoin ↑R ↑s = ⊤\nW : ↑A → TopologicalSpace.Opens ↥X\nhx... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.Morphisms.Flat | {
"line": 417,
"column": 6
} | {
"line": 417,
"column": 17
} | {
"line": 417,
"column": 18
} | [
{
"pp": "case right\nX 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\ninst✝ : Finite ι\nVX : ι → X.Opens\nhVU : iSup VX = UX\... | [
"case right\nX 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\ninst✝ : Finite ι\nVX : ι → X.Opens\nhVU : iSup VX = UX\nhV : ∀ (i :... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.SpreadingOut | {
"line": 373,
"column": 4
} | {
"line": 373,
"column": 67
} | {
"line": 373,
"column": 68
} | [
{
"pp": "case refine_3\nX Y S : Scheme\nsX : X ⟶ S\nsY : Y ⟶ S\ninst✝¹ : LocallyOfFiniteType sY\nx : ↥X\ninst✝ : X.IsGermInjectiveAt x\nφ : Spec (X.presheaf.stalk x) ⟶ Y\nh : φ ≫ sY = X.fromSpecStalk x ≫ sX\nthis :\n ∃ U,\n ∃ (hxU : x ∈ U),\n ∃ f,\n Spec.map (Scheme.stalkClosedPointTo φ) ≫ Y.fro... | [
"case refine_3\nX Y S : Scheme\nsX : X ⟶ S\nsY : Y ⟶ S\ninst✝¹ : LocallyOfFiniteType sY\nx : ↥X\ninst✝ : X.IsGermInjectiveAt x\nφ : Spec (X.presheaf.stalk x) ⟶ Y\nh : φ ≫ sY = X.fromSpecStalk x ≫ sX\nthis :\n ∃ U,\n ∃ (hxU : x ∈ U),\n ∃ f,\n Spec.map (Scheme.stalkClosedPointTo φ) ≫ Y.fromSpecStalk (... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.FunctionField | {
"line": 45,
"column": 65
} | {
"line": 45,
"column": 76
} | {
"line": 45,
"column": 77
} | [
{
"pp": "X : Scheme\ninst✝ : IrreducibleSpace ↥X\nU : X.Opens\nh : Nonempty ↥↑U\n⊢ (Set.univ ∩ ↑U).Nonempty",
"ppTerm": "?m.36",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"AlgebraicGeometry.SheafedSpace.instTopologicalSpaceCarrierCarrier",
"AlgebraicGeometry.PresheafedSpace.ca... | [
"X : Scheme\ninst✝ : IrreducibleSpace ↥X\nU : X.Opens\nh : Nonempty ↥↑U\n⊢ (↑U).Nonempty"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.FunctionField | {
"line": 136,
"column": 68
} | {
"line": 136,
"column": 79
} | {
"line": 136,
"column": 80
} | [
{
"pp": "X✝ : Scheme\nX : Scheme\ninst✝ : IsIntegral X\nU : X.Opens\nhU : IsAffineOpen U\nh : Nonempty ↥↑U\n⊢ (Set.univ ∩ ↑U).Nonempty",
"ppTerm": "?m.35",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"AlgebraicGeometry.SheafedSpace.instTopologicalSpaceCarrierCarrier",
"Algebraic... | [
"X✝ : Scheme\nX : Scheme\ninst✝ : IsIntegral X\nU : X.Opens\nhU : IsAffineOpen U\nh : Nonempty ↥↑U\n⊢ (↑U).Nonempty"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.FunctionField | {
"line": 155,
"column": 67
} | {
"line": 155,
"column": 78
} | {
"line": 155,
"column": 79
} | [
{
"pp": "X : Scheme\ninst✝¹ : IsIntegral X\nU : X.Opens\nhU : IsAffineOpen U\ninst✝ : Nonempty ↥↑U\n⊢ (Set.univ ∩ ↑U).Nonempty",
"ppTerm": "?m.37",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"AlgebraicGeometry.SheafedSpace.instTopologicalSpaceCarrierCarrier",
"AlgebraicGeometry... | [
"X : Scheme\ninst✝¹ : IsIntegral X\nU : X.Opens\nhU : IsAffineOpen U\ninst✝ : Nonempty ↥↑U\n⊢ (↑U).Nonempty"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.FunctionField | {
"line": 202,
"column": 2
} | {
"line": 202,
"column": 13
} | {
"line": 202,
"column": 14
} | [
{
"pp": "X : Scheme\ninst✝ : IsIntegral X\nf : ↑X.functionField\nhf : f ≠ 0\nU : Opens ↥X\nhU : genericPoint ↥X ∈ U\ng : ToType (X.presheaf.obj (op U))\nhg : (ConcreteCategory.hom (X.presheaf.germ U (genericPoint ↥X) hU)) g = f\nA : Opens ↥X\nhA : A ∈ X.affineOpens\nhxA : genericPoint ↥X ∈ ↑A\nhAU : ↑A ⊆ ↑U\nth... | [
"X : Scheme\ninst✝ : IsIntegral X\nf : ↑X.functionField\nhf : f ≠ 0\nU : Opens ↥X\nhU : genericPoint ↥X ∈ U\ng : ToType (X.presheaf.obj (op U))\nhg : (ConcreteCategory.hom (X.presheaf.germ U (genericPoint ↥X) hU)) g = f\nA : Opens ↥X\nhA : A ∈ X.affineOpens\nhxA : genericPoint ↥X ∈ ↑A\nhAU : ↑A ⊆ ↑U\nthis✝ : Nonemp... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.Birational.RationalMap | {
"line": 70,
"column": 4
} | {
"line": 70,
"column": 15
} | {
"line": 70,
"column": 16
} | [
{
"pp": "case mpr\nX Y : Scheme\nU : X.Opens\nhU : Dense ↑U\nf : ↑U ⟶ Y\nhV : Dense ↑U\ng : ↑U ⟶ Y\ne :\n { domain := U, dense_domain := hU, hom := f }.hom =\n (X.isoOfEq ⋯).hom ≫ { domain := U, dense_domain := hV, hom := g }.hom\n⊢ f = g",
"ppTerm": "?mpr",
"assigned": false,
"usedConstants": [... | [
"case mpr\nX Y : Scheme\nU : X.Opens\nhU : Dense ↑U\nf : ↑U ⟶ Y\nhV : Dense ↑U\ng : ↑U ⟶ Y\ne :\n { domain := U, dense_domain := hU, hom := f }.hom =\n (X.isoOfEq ⋯).hom ≫ { domain := U, dense_domain := hV, hom := g }.hom\n⊢ f = g"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.Morphisms.Flat | {
"line": 425,
"column": 8
} | {
"line": 425,
"column": 28
} | {
"line": 425,
"column": 29
} | [
{
"pp": "case refine_2.pair\nX 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\ninst✝ : Finite ι\nVX : ι → X.Opens\nhVU : iSup ... | [
"case refine_2.pair\nX 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\ninst✝ : Finite ι\nVX : ι → X.Opens\nhVU : iSup VX = UX\nhV ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.Birational.RationalMap | {
"line": 313,
"column": 4
} | {
"line": 314,
"column": 84
} | {
"line": 314,
"column": 85
} | [
{
"pp": "case refine_1\nX Y : Scheme\ninst✝¹ : IrreducibleSpace ↥X\nx : ↥X\ninst✝ : X.IsGermInjectiveAt x\nf g : X.PartialMap Y\nhxf : x ∈ f.domain\nhxg : x ∈ g.domain\nhdense : Dense (↑f.domain ⊓ ↑g.domain)\nH :\n (f.restrict (f.domain ⊓ g.domain) hdense ⋯).fromSpecStalkOfMem ⋯ =\n (g.restrict (f.domain ⊓ ... | [
"case refine_1\nX Y : Scheme\ninst✝¹ : IrreducibleSpace ↥X\nx : ↥X\ninst✝ : X.IsGermInjectiveAt x\nf g : X.PartialMap Y\nhxf : x ∈ f.domain\nhxg : x ∈ g.domain\nhdense : Dense (↑f.domain ⊓ ↑g.domain)\nH :\n (f.restrict (f.domain ⊓ g.domain) hdense ⋯).fromSpecStalkOfMem ⋯ =\n (g.restrict (f.domain ⊓ g.domain) hd... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.Birational.RationalMap | {
"line": 327,
"column": 2
} | {
"line": 327,
"column": 13
} | {
"line": 327,
"column": 14
} | [
{
"pp": "X Y S : Scheme\ninst✝⁵ : X.Over S\ninst✝⁴ : Y.Over S\ninst✝³ : IsReduced X\ninst✝² : IsSeparated (Y ↘ S)\nf g : X.PartialMap Y\ninst✝¹ : IsOver S f\ninst✝ : IsOver S g\nW : X.Opens\nhW : Dense ↑W\nhWl : W ≤ f.domain\nhWr : W ≤ g.domain\nx✝ : f.equiv g\nV : X.Opens\nhV : Dense ↑V\nhVl : V ≤ f.domain\nhV... | [
"X Y S : Scheme\ninst✝⁵ : X.Over S\ninst✝⁴ : Y.Over S\ninst✝³ : IsReduced X\ninst✝² : IsSeparated (Y ↘ S)\nf g : X.PartialMap Y\ninst✝¹ : IsOver S f\ninst✝ : IsOver S g\nW : X.Opens\nhW : Dense ↑W\nhWl : W ≤ f.domain\nhWr : W ≤ g.domain\nx✝ : f.equiv g\nV : X.Opens\nhV : Dense ↑V\nhVl : V ≤ f.domain\nhVr : V ≤ g.do... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.Birational.RationalMap | {
"line": 479,
"column": 51
} | {
"line": 479,
"column": 62
} | {
"line": 479,
"column": 63
} | [
{
"pp": "X Y S : Scheme\ninst✝³ : X.Over S\ninst✝² : Y.Over S\ninst✝¹ : IsReduced X\ninst✝ : S.IsSeparated\nf : X.PartialMap Y\nx✝ : RationalMap.IsOver S f.toRationalMap\nU : X.Opens\nhU : Dense ↑U\nhU' : U ≤ f.domain\nH : IsOver S (f.restrict U hU hU')\nthis : IsDominant (X.homOfLE hU')\n⊢ X.homOfLE hU' ≫ (f.c... | [
"X Y S : Scheme\ninst✝³ : X.Over S\ninst✝² : Y.Over S\ninst✝¹ : IsReduced X\ninst✝ : S.IsSeparated\nf : X.PartialMap Y\nx✝ : RationalMap.IsOver S f.toRationalMap\nU : X.Opens\nhU : Dense ↑U\nhU' : U ≤ f.domain\nH : IsOver S (f.restrict U hU hU')\nthis : IsDominant (X.homOfLE hU')\n⊢ X.homOfLE hU' ≫ f.hom ≫ Y ↘ S = ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.Birational.RationalMap | {
"line": 627,
"column": 37
} | {
"line": 627,
"column": 48
} | {
"line": 627,
"column": 49
} | [
{
"pp": "X Y : Scheme\ninst✝¹ : IsReduced X\ninst✝ : Y.IsSeparated\nf : X.PartialMap Y\n⊢ (f.toRationalMap.toPartialMap.restrict f.domain ⋯ ⋯).hom = (X.isoOfEq ⋯).hom ≫ f.hom",
"ppTerm": "?m.62",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"AlgebraicGeometry.Scheme.homOfLE",
"Al... | [
"X Y : Scheme\ninst✝¹ : IsReduced X\ninst✝ : Y.IsSeparated\nf : X.PartialMap Y\n⊢ X.homOfLE ⋯ ≫ f.toRationalMap.toPartialMap.hom = f.hom"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.Birational.Birational | {
"line": 68,
"column": 4
} | {
"line": 68,
"column": 15
} | {
"line": 68,
"column": 16
} | [
{
"pp": "case mpr\nX Y : Scheme\nU₁ : X.Opens\nhU₁ : Dense ↑U₁\nV₂ : Y.Opens\nhU₂✝ : Dense ↑V₂\nhV₁ : Dense ↑U₁\ng : ↑U₁ ≅ ↑V₂\nhU₂ : Dense ↑V₂\nf : ↑U₁ ≅ ↑V₂\ne : f = X.isoOfEq ⋯ ≪≫ g ≪≫ Y.isoOfEq ⋯\n⊢ { source := U₁, dense_source := hU₁, target := V₂, dense_target := hU₂, iso := f } =\n { source := U₁, den... | [
"case mpr\nX Y : Scheme\nU₁ : X.Opens\nhU₁ : Dense ↑U₁\nV₂ : Y.Opens\nhU₂✝ : Dense ↑V₂\nhV₁ : Dense ↑U₁\ng : ↑U₁ ≅ ↑V₂\nhU₂ : Dense ↑V₂\nf : ↑U₁ ≅ ↑V₂\ne : f = X.isoOfEq ⋯ ≪≫ g ≪≫ Y.isoOfEq ⋯\n⊢ f = g"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.Birational.Birational | {
"line": 97,
"column": 2
} | {
"line": 97,
"column": 46
} | {
"line": 97,
"column": 47
} | [
{
"pp": "X Y S : Scheme\nsX : X ⟶ S\nsY : Y ⟶ S\nf : X.PartialIso Y\nhf : IsOver sX sY f\n⊢ IsOver sY sX f.symm",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"AlgebraicGeometry.Scheme",
"CategoryTheory.CategoryStruct.toQuiver",
"Quiver.Hom",
"Algeb... | [
"X Y S : Scheme\nsX : X ⟶ S\nsY : Y ⟶ S\nf : X.PartialIso Y\nhf : IsOver sX sY f\n⊢ f.source.ι ≫ sX = f.iso.hom ≫ f.target.ι ≫ sY"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.Birational.Composition | {
"line": 53,
"column": 4
} | {
"line": 53,
"column": 45
} | {
"line": 54,
"column": 6
} | [
{
"pp": "X Y Z : Scheme\ninst✝² : PreirreducibleSpace ↥X\ninst✝¹ : Nonempty ↥Y\nf : X.PartialMap Y\ninst✝ : IsDominant f.hom\ng : Y.PartialMap Z\n⊢ (f.domain.ι ''ᵁ f.hom ⁻¹ᵁ g.domain).carrier.Nonempty",
"ppTerm": "?m.96",
"assigned": true,
"usedConstants": [
"AlgebraicGeometry.Scheme.Hom.opens... | [
"X Y Z : Scheme\ninst✝² : PreirreducibleSpace ↥X\ninst✝¹ : Nonempty ↥Y\nf : X.PartialMap Y\ninst✝ : IsDominant f.hom\ng : Y.PartialMap Z\n⊢ (⇑f.hom ⁻¹' ↑g.domain).Nonempty"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.Birational.Composition | {
"line": 71,
"column": 8
} | {
"line": 71,
"column": 49
} | {
"line": 72,
"column": 10
} | [
{
"pp": "X Y Z : Scheme\ninst✝² : PreirreducibleSpace ↥X\ninst✝¹ : Nonempty ↥Y\nf : X.PartialMap Y\ninst✝ : IsDominant f.hom\ng : Y.PartialMap Z\nV : Y.Opens\nhV : Dense ↑V\nhV' : V ≤ g.domain\n⊢ (f.domain.ι ''ᵁ f.hom ⁻¹ᵁ V).carrier.Nonempty",
"ppTerm": "?m.98",
"assigned": true,
"usedConstants": [
... | [
"X Y Z : Scheme\ninst✝² : PreirreducibleSpace ↥X\ninst✝¹ : Nonempty ↥Y\nf : X.PartialMap Y\ninst✝ : IsDominant f.hom\ng : Y.PartialMap Z\nV : Y.Opens\nhV : Dense ↑V\nhV' : V ≤ g.domain\n⊢ (⇑f.hom ⁻¹' ↑V).Nonempty"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.Birational.Composition | {
"line": 86,
"column": 31
} | {
"line": 86,
"column": 42
} | {
"line": 86,
"column": 43
} | [
{
"pp": "X Y Z : Scheme\ninst✝³ : PreirreducibleSpace ↥X\ninst✝² : Nonempty ↥Y\nf₁ f₂ : X.PartialMap Y\ninst✝¹ : IsDominant f₁.hom\ninst✝ : IsDominant f₂.hom\ng : Y.PartialMap Z\nW : X.Opens\nhW : Dense ↑W\nhW₁ : W ≤ f₁.domain\nhW₂ : W ≤ f₂.domain\ne : (f₁.restrict W hW hW₁).hom = (f₂.restrict W hW hW₂).hom\n⊢ ... | [
"X Y Z : Scheme\ninst✝³ : PreirreducibleSpace ↥X\ninst✝² : Nonempty ↥Y\nf₁ f₂ : X.PartialMap Y\ninst✝¹ : IsDominant f₁.hom\ninst✝ : IsDominant f₂.hom\ng : Y.PartialMap Z\nW : X.Opens\nhW : Dense ↑W\nhW₁ : W ≤ f₁.domain\nhW₂ : W ≤ f₂.domain\ne : (f₁.restrict W hW hW₁).hom = (f₂.restrict W hW hW₂).hom\n⊢ X.homOfLE hW... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.Birational.Composition | {
"line": 97,
"column": 31
} | {
"line": 97,
"column": 42
} | {
"line": 97,
"column": 43
} | [
{
"pp": "X Y Z : Scheme\ninst✝² : PreirreducibleSpace ↥X\ninst✝¹ : Nonempty ↥Y\nf : X.PartialMap Y\ninst✝ : IsDominant f.hom\ng₁ g₂ : Y.PartialMap Z\nW : Y.Opens\nhW : Dense ↑W\nhW₁ : W ≤ g₁.domain\nhW₂ : W ≤ g₂.domain\ne : (g₁.restrict W hW hW₁).hom = (g₂.restrict W hW hW₂).hom\n⊢ (g₁.restrict W hW hW₁).hom = ... | [
"X Y Z : Scheme\ninst✝² : PreirreducibleSpace ↥X\ninst✝¹ : Nonempty ↥Y\nf : X.PartialMap Y\ninst✝ : IsDominant f.hom\ng₁ g₂ : Y.PartialMap Z\nW : Y.Opens\nhW : Dense ↑W\nhW₁ : W ≤ g₁.domain\nhW₂ : W ≤ g₂.domain\ne : (g₁.restrict W hW hW₁).hom = (g₂.restrict W hW hW₂).hom\n⊢ Y.homOfLE hW₁ ≫ g₁.hom = Y.homOfLE hW₂ ≫ ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.Morphisms.UniversallyInjective | {
"line": 115,
"column": 4
} | {
"line": 115,
"column": 81
} | {
"line": 116,
"column": 4
} | [
{
"pp": "X Y : Scheme\nf : X ⟶ Y\ntfae_1_iff_4 : UniversallyInjective f ↔ Surjective (pullback.diagonal f)\nh_inj : Injective ⇑f\nhf : ∀ (x : ↥X), (CommRingCat.Hom.hom (Scheme.Hom.residueFieldMap f x)).IsPurelyInseparable\nK : Type u\ninst✝ : Field K\ng₁ g₂ : Spec (CommRingCat.of K) ⟶ X\nhg : (fun g ↦ g ≫ f) g₁... | [
"X Y : Scheme\nf : X ⟶ Y\ntfae_1_iff_4 : UniversallyInjective f ↔ Surjective (pullback.diagonal f)\nh_inj : Injective ⇑f\nhf : ∀ (x : ↥X), (CommRingCat.Hom.hom (Scheme.Hom.residueFieldMap f x)).IsPurelyInseparable\nK : Type u\ninst✝ : Field K\ng₁ g₂ : Spec (CommRingCat.of K) ⟶ X\nhg : (fun g ↦ g ≫ f) g₁ = (fun g ↦ ... | rw [← f.residueFieldMap_congr'_assoc (h_inj e), CommRingCat.hom_ext_iff] at h | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.AlgebraicGeometry.Morphisms.UniversallyInjective | {
"line": 160,
"column": 6
} | {
"line": 160,
"column": 67
} | {
"line": 160,
"column": 68
} | [
{
"pp": "X Y : Scheme\nf : X ⟶ Y\ntfae_1_iff_4 : UniversallyInjective f ↔ Surjective (pullback.diagonal f)\ntfae_3_to_2 :\n (Injective ⇑f ∧ ∀ (x : ↥X), (CommRingCat.Hom.hom (Scheme.Hom.residueFieldMap f x)).IsPurelyInseparable) →\n ∀ (K : Type u) [inst : Field K], Injective fun g ↦ g ≫ f\ntfae_2_to_4 : (∀ (... | [
"X Y : Scheme\nf : X ⟶ Y\ntfae_1_iff_4 : UniversallyInjective f ↔ Surjective (pullback.diagonal f)\ntfae_3_to_2 :\n (Injective ⇑f ∧ ∀ (x : ↥X), (CommRingCat.Hom.hom (Scheme.Hom.residueFieldMap f x)).IsPurelyInseparable) →\n ∀ (K : Type u) [inst : Field K], Injective fun g ↦ g ≫ f\ntfae_2_to_4 : (∀ (K : Type u) ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.Cover.Directed | {
"line": 236,
"column": 42
} | {
"line": 236,
"column": 53
} | {
"line": 236,
"column": 54
} | [
{
"pp": "P : MorphismProperty Scheme\nX : Scheme\n𝒰 : X.OpenCover\ninst✝¹ : Category.{v_1, ?u.54} 𝒰.I₀\ninst✝ : Cover.LocallyDirected 𝒰\ns : Cocone (Cover.functorOfLocallyDirected 𝒰)\nm : (Cover.coconeOfLocallyDirected 𝒰).pt ⟶ s.pt\nhm : ∀ (j : 𝒰.I₀), (Cover.coconeOfLocallyDirected 𝒰).ι.app j ≫ m = s.ι.a... | [
"P : MorphismProperty Scheme\nX : Scheme\n𝒰 : X.OpenCover\ninst✝¹ : Category.{v_1, ?u.54} 𝒰.I₀\ninst✝ : Cover.LocallyDirected 𝒰\ns : Cocone (Cover.functorOfLocallyDirected 𝒰)\nm : (Cover.coconeOfLocallyDirected 𝒰).pt ⟶ s.pt\nhm : ∀ (j : 𝒰.I₀), (Cover.coconeOfLocallyDirected 𝒰).ι.app j ≫ m = s.ι.app j\nj : 𝒰... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Spectrum.Prime.ChevalleyComplexity | {
"line": 416,
"column": 18
} | {
"line": 416,
"column": 53
} | {
"line": 416,
"column": 54
} | [
{
"pp": "case h\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 := ⋯\nq₂ : R →ₐ[R₀] R ⧸ Ideal.span {c} := ⋯\nq₂_surjective : Surjective ⇑q₂... | [
"case h\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 {c} ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.Cover.Directed | {
"line": 316,
"column": 4
} | {
"line": 316,
"column": 15
} | {
"line": 316,
"column": 16
} | [
{
"pp": "case h\nP : MorphismProperty Scheme\nX : Scheme\nx : ↥X\n⊢ x ∈ Set.range ⇑({ I₀ := ↑X.affineOpens, X := fun U ↦ ↑↑U, f := fun U ↦ (↑U).ι }.f ⟨⋯.choose, ⋯⟩)",
"ppTerm": "?h",
"assigned": true,
"usedConstants": [
"CategoryTheory.PreZeroHypercover.mk",
"TopologicalSpace.Opens.mem_t... | [
"case h\nP : MorphismProperty Scheme\nX : Scheme\nx : ↥X\n⊢ x ∈ ⋯.choose"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Spectrum.Prime.ChevalleyComplexity | {
"line": 424,
"column": 8
} | {
"line": 424,
"column": 19
} | {
"line": 424,
"column": 20
} | [
{
"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... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.AffineTransitionLimit | {
"line": 121,
"column": 6
} | {
"line": 121,
"column": 46
} | {
"line": 121,
"column": 47
} | [
{
"pp": "I : Type u\ninst✝² : Category.{u, u} I\nD : I ⥤ Scheme\nc : Cone D\nhc : IsLimit c\ninst✝¹ : IsCofilteredOrEmpty I\ninst✝ : ∀ {i j : I} (f : i ⟶ j), IsAffineHom (D.map f)\nZ : (i : I) → Set ↥(D.obj i)\nhZc : ∀ (i : I), IsClosed (Z i)\nhZne : ∀ (i : I), (Z i).Nonempty\nhZcpt : ∀ (i : I), IsCompact (Z i)... | [
"I : Type u\ninst✝² : Category.{u, u} I\nD : I ⥤ Scheme\nc : Cone D\nhc : IsLimit c\ninst✝¹ : IsCofilteredOrEmpty I\ninst✝ : ∀ {i j : I} (f : i ⟶ j), IsAffineHom (D.map f)\nZ : (i : I) → Set ↥(D.obj i)\nhZc : ∀ (i : I), IsClosed (Z i)\nhZne : ∀ (i : I), (Z i).Nonempty\nhZcpt : ∀ (i : I), IsCompact (Z i)\nhmapsTo : ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.RelativeGluing | {
"line": 179,
"column": 2
} | {
"line": 185,
"column": 8
} | {
"line": 186,
"column": 2
} | [
{
"pp": "case refine_1\nS : Scheme\n𝒰 : S.OpenCover\ninst✝³ : Category.{u_2, u_1} 𝒰.I₀\ninst✝² : LocallyDirected 𝒰\nd : RelativeGluingData 𝒰\ninst✝¹ : Small.{u, u_1} 𝒰.I₀\ninst✝ : Quiver.IsThin 𝒰.I₀\ni : 𝒰.I₀\n⊢ (s : PullbackCone (𝒰.f i) d.toBase) → s.pt ⟶ d.functor.obj i",
"ppTerm": "?refine_1",
... | [
"case refine_2\nS : Scheme\n𝒰 : S.OpenCover\ninst✝³ : Category.{u_2, u_1} 𝒰.I₀\ninst✝² : LocallyDirected 𝒰\nd : RelativeGluingData 𝒰\ninst✝¹ : Small.{u, u_1} 𝒰.I₀\ninst✝ : Quiver.IsThin 𝒰.I₀\ni : 𝒰.I₀\n⊢ ∀ (s : PullbackCone (𝒰.f i) d.toBase), IsOpenImmersion.lift (colimit.ι d.functor i) s.snd ⋯ ≫ d.natTrans... | · intro s
apply IsOpenImmersion.lift (colimit.ι d.functor i) s.snd
rw [← preimage_toBase_eq_range_ι]
rintro x ⟨x, rfl⟩
use s.fst x
rw [← Scheme.Hom.comp_apply, ← s.condition]
simp | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.AlgebraicGeometry.AffineTransitionLimit | {
"line": 149,
"column": 2
} | {
"line": 149,
"column": 13
} | {
"line": 149,
"column": 14
} | [
{
"pp": "I : Type u\ninst✝² : Category.{u, u} I\nD : I ⥤ Scheme\nc : Cone D\nhc : IsLimit c\ninst✝¹ : IsCofilteredOrEmpty I\ninst✝ : ∀ {i j : I} (f : i ⟶ j), IsAffineHom (D.map f)\nZ : (i : I) → Set ↥(D.obj i)\nhZc : ∀ (i : I), IsClosed (Z i)\nhZne : ∀ (i : I), (Z i).Nonempty\nhZcpt : ∀ (i : I), IsCompact (Z i)... | [
"I : Type u\ninst✝² : Category.{u, u} I\nD : I ⥤ Scheme\nc : Cone D\nhc : IsLimit c\ninst✝¹ : IsCofilteredOrEmpty I\ninst✝ : ∀ {i j : I} (f : i ⟶ j), IsAffineHom (D.map f)\nZ : (i : I) → Set ↥(D.obj i)\nhZc : ∀ (i : I), IsClosed (Z i)\nhZne : ∀ (i : I), (Z i).Nonempty\nhZcpt : ∀ (i : I), IsCompact (Z i)\nhmapsTo : ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.AffineTransitionLimit | {
"line": 194,
"column": 44
} | {
"line": 194,
"column": 55
} | {
"line": 194,
"column": 56
} | [
{
"pp": "I : Type u\ninst✝³ : Category.{u, u} I\nD : I ⥤ Scheme\nc : Cone D\nhc : IsLimit c\ninst✝² : IsCofiltered I\ninst✝¹ : ∀ {i j : I} (f : i ⟶ j), IsAffineHom (D.map f)\ninst✝ : ∀ (i : I), CompactSpace ↥(D.obj i)\ni : I\nU : (D.obj i).Opens\nhU : c.π.app i ⁻¹ᵁ U = ⊤\nH : ∀ (j : I) (fji : j ⟶ i), D.map fji ... | [
"I : Type u\ninst✝³ : Category.{u, u} I\nD : I ⥤ Scheme\nc : Cone D\nhc : IsLimit c\ninst✝² : IsCofiltered I\ninst✝¹ : ∀ {i j : I} (f : i ⟶ j), IsAffineHom (D.map f)\ninst✝ : ∀ (i : I), CompactSpace ↥(D.obj i)\ni : I\nU : (D.obj i).Opens\nhU : c.π.app i ⁻¹ᵁ U = ⊤\nH : ∀ (j : I) (fji : j ⟶ i), D.map fji ⁻¹ᵁ U ≠ ⊤\ns... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Sets.CompactOpenCovered | {
"line": 72,
"column": 8
} | {
"line": 72,
"column": 23
} | {
"line": 72,
"column": 24
} | [
{
"pp": "case pos\nS : Type u_1\nι : Type u_2\nX : ι → Type u_3\nf : (i : ι) → X i → S\ninst✝ : (i : ι) → TopologicalSpace (X i)\nU : Set S\nx✝ : IsCompactOpenCovered f U\ns : Set ι\nhs : s.Finite\nV : (i : ι) → i ∈ s → Opens (X i)\nhc : ∀ (i : ι) (h : i ∈ s), IsCompact (V i h).carrier\nhU : ⋃ i, ⋃ (h : i ∈ s),... | [
"case pos\nS : Type u_1\nι : Type u_2\nX : ι → Type u_3\nf : (i : ι) → X i → S\ninst✝ : (i : ι) → TopologicalSpace (X i)\nU : Set S\nx✝ : IsCompactOpenCovered f U\ns : Set ι\nhs : s.Finite\nV : (i : ι) → i ∈ s → Opens (X i)\nhc : ∀ (i : ι) (h : i ∈ s), IsCompact (V i h).carrier\nhU : ⋃ i, ⋃ (h : i ∈ s), f i '' ↑(V ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Sets.CompactOpenCovered | {
"line": 90,
"column": 22
} | {
"line": 90,
"column": 33
} | {
"line": 90,
"column": 34
} | [
{
"pp": "S : Type u_1\nι : Type u_2\nX : ι → Type u_3\nf : (i : ι) → X i → S\ninst✝¹ : (i : ι) → TopologicalSpace (X i)\nU : Set S\nκ : Type u_4\ninst✝ : Finite κ\ns : κ → Set S\nhs : ⋃ i, s i = U\nH : ∀ (i : κ), IsCompactOpenCovered f (s i)\nV : κ → Opens ((i : ι) × X i)\nhVeq : ∀ (i : κ), IsCompact (V i).carr... | [
"S : Type u_1\nι : Type u_2\nX : ι → Type u_3\nf : (i : ι) → X i → S\ninst✝¹ : (i : ι) → TopologicalSpace (X i)\nU : Set S\nκ : Type u_4\ninst✝ : Finite κ\ns : κ → Set S\nhs : ⋃ i, s i = U\nH : ∀ (i : κ), IsCompactOpenCovered f (s i)\nV : κ → Opens ((i : ι) × X i)\nhVeq : ∀ (i : κ), IsCompact (V i).carrier\nhVc : ∀... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
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