module stringlengths 16 90 | startPos dict | endPos dict | goals listlengths 0 96 | ppTac stringlengths 1 14.5k | elaborator stringclasses 366
values | kind stringclasses 370
values |
|---|---|---|---|---|---|---|
Mathlib.AlgebraicGeometry.IdealSheaf.Basic | {
"line": 718,
"column": 29
} | {
"line": 718,
"column": 40
} | [
{
"pp": "X Y : Scheme\nf : X.Hom Y\ninst✝ : QuasiCompact f\nU✝ U : ↑Y.affineOpens\ns : ↑Γ(Y, ↑U)\nthis : IsLocalization.Away s ↑Γ(Y, Y.basicOpen s)\nx : ↑Γ(Y, ↑U)\nn✝ : ℕ\nhx :\n IsLocalization.mk' (↑Γ(Y, Y.basicOpen s)) x ⟨(fun x ↦ s ^ x) n✝, ⋯⟩ ∈\n (fun U ↦ RingHom.ker (CommRingCat.Hom.hom (f.app ↑U))) (Y... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.IdealSheaf.Basic | {
"line": 734,
"column": 52
} | {
"line": 734,
"column": 63
} | [
{
"pp": "X Y : Scheme\nf : X.Hom Y\ninst✝ : IsEmpty ↥X\nU : ↑Y.affineOpens\nx : ↑Γ(Y, ↑U)\nx✝ : x ∈ ⊤.ideal U\n⊢ x ∈ RingHom.ker (CommRingCat.Hom.hom (f.app ↑U))",
"usedConstants": [
"Eq.mpr",
"Submodule",
"AlgebraicGeometry.SheafedSpace.instTopologicalSpaceCarrierCarrier",
"NonUnita... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.IdealSheaf.Basic | {
"line": 767,
"column": 14
} | {
"line": 767,
"column": 29
} | [
{
"pp": "X Y : Scheme\nf : X.Hom Y\nH : f.ker = ⊤\n⊢ IsEmpty ↥X",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.IdealSheaf.Basic | {
"line": 786,
"column": 2
} | {
"line": 786,
"column": 32
} | [
{
"pp": "case a\nX Y : Scheme\nf : X.Hom Y\ninst✝ : QuasiCompact f\n𝒰 : X.OpenCover\nU : ↑Y.affineOpens\ns : ↑Γ(Y, ↑U)\nhs : s ∈ ⨅ i, (ker (𝒰.f i ≫ f)).ideal U\ni : 𝒰.I₀\nx : ↥(𝒰.X i)\nhxU : (𝒰.f i) x ∈ Opposite.unop (Opposite.op (f ⁻¹ᵁ ↑U))\n⊢ (CommRingCat.Hom.hom (appLE (𝒰.f i ≫ f) (↑U) (𝒰.f i ⁻¹ᵁ f ⁻¹... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.IdealSheaf.Basic | {
"line": 806,
"column": 4
} | {
"line": 806,
"column": 21
} | [
{
"pp": "case h\nX 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... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.IdealSheaf.Basic | {
"line": 815,
"column": 2
} | {
"line": 815,
"column": 32
} | [
{
"pp": "case h\nX Y : Scheme\nf : X.Hom Y\ninst✝ : QuasiCompact f\nU : Y.Opens\nV : ↑(↑U).affineOpens\nx : ↑Γ(↑U, ↑V)\n⊢ x ∈ (Hom.ker (f ∣_ U)).ideal V ↔ x ∈ f.ker.ideal ⟨U.ι ''ᵁ ↑V, ⋯⟩",
"usedConstants": [
"RingHom.ker.congr_simp",
"AlgebraicGeometry.Scheme.Hom.opensFunctor",
"Eq.mpr",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.IdealSheaf.Basic | {
"line": 837,
"column": 2
} | {
"line": 837,
"column": 13
} | [
{
"pp": "case h\nX 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 i... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.Morphisms.Affine | {
"line": 81,
"column": 2
} | {
"line": 81,
"column": 51
} | [
{
"pp": "case h.e'_2.h\nX✝ 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)",
"usedConstants": [
"AlgebraicGeometry.SheafedSpace.instTopologicalSpaceCarrierCarrier",
"Lattice... | refine Set.image_preimage_eq_inter_range.trans ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.AlgebraicGeometry.IdealSheaf.Basic | {
"line": 884,
"column": 33
} | {
"line": 885,
"column": 56
} | [
{
"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",
"usedConstants": [
"Eq.mpr",
"CategoryTheory.Over",
"AlgebraicGeometry.Scheme",
"CategoryTheory.CategoryStruct.toQuiver",
"Quiver.Hom",
"congrArg",... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.IdealSheaf.Basic | {
"line": 893,
"column": 2
} | {
"line": 893,
"column": 13
} | [
{
"pp": "case H\nX : Scheme\ninst✝ : CompactSpace ↥X\n⊢ (Hom.ker X.toSpecΓ).ideal ⟨⊤, ⋯⟩ = ⊥.ideal ⟨⊤, ⋯⟩",
"usedConstants": [
"RingHom.ker.congr_simp",
"AlgebraicGeometry.isAffineOpen_top",
"Eq.mpr",
"RingHom.instRingHomClass",
"AlgebraicGeometry.Spec",
"AlgebraicGeometr... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.PullbackCarrier | {
"line": 213,
"column": 8
} | {
"line": 213,
"column": 81
} | [
{
"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 ((pullback.snd f g) t) =\n Spec.map (Hom.residueFieldMap (pullback.snd f g) t) ≫ Y.fromSpecResidueField ((pullback.snd f g) t)",
"use... | ← pushout.inr_desc _ _ (residueFieldCongr_inv_residueFieldMap_ofPoint t), | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.AlgebraicGeometry.Morphisms.Affine | {
"line": 144,
"column": 4
} | {
"line": 144,
"column": 52
} | [
{
"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)))",
"usedConstants": [
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.Morphisms.Affine | {
"line": 164,
"column": 6
} | {
"line": 164,
"column": 45
} | [
{
"pp": "case of_basicOpenCover\nX✝ Y✝ Z : Scheme\nf✝ : X✝ ⟶ Y✝\ng : Y✝ ⟶ Z\nX Y : Scheme\ninst✝ : IsAffine Y\nf : X ⟶ Y\nS : Finset ↑Γ(Y, ⊤)\nhS' : ∀ (r : ↥S), IsAffine ↑(f ⁻¹ᵁ Y.basicOpen ↑r)\nhS : Ideal.span (⇑(CommRingCat.Hom.hom (Scheme.Hom.appTop f)) '' ↑S) = ⊤\nthis : ∀ (i : ↥S), IsAffineOpen (f ⁻¹ᵁ Y.ba... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.Morphisms.Affine | {
"line": 191,
"column": 38
} | {
"line": 191,
"column": 49
} | [
{
"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, ⋯⟩",
"usedConstants": [
"Eq.mpr",
"AlgebraicGeometry.SheafedSpace.instTopologicalSpaceCarrierCarri... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.Morphisms.Affine | {
"line": 221,
"column": 2
} | {
"line": 221,
"column": 53
} | [
{
"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)",
"usedConstants": [
"AlgebraicGeometry.PresheafedSpace.carrier",
"TopologicalSpace.Ope... | 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
} | [
{
"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... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.Morphisms.Affine | {
"line": 230,
"column": 33
} | {
"line": 230,
"column": 70
} | [
{
"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... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.Morphisms.Affine | {
"line": 234,
"column": 6
} | {
"line": 234,
"column": 46
} | [
{
"pp": "case h.e'_2.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 ⁻... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.Morphisms.Affine | {
"line": 235,
"column": 6
} | {
"line": 235,
"column": 46
} | [
{
"pp": "case h.e'_2.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 ⁻... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.Morphisms.Affine | {
"line": 253,
"column": 56
} | {
"line": 253,
"column": 71
} | [
{
"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 :=... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.Morphisms.Affine | {
"line": 258,
"column": 4
} | {
"line": 258,
"column": 15
} | [
{
"pp": "case h.e'_2\nX 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.basi... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Finiteness.FiniteTypeLocal | {
"line": 79,
"column": 2
} | {
"line": 79,
"column": 56
} | [
{
"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 ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.FiniteStability | {
"line": 72,
"column": 4
} | {
"line": 72,
"column": 15
} | [
{
"pp": "case refine_3\nR : Type w₁\ninst✝⁵ : CommRing R\nA : Type w₂\ninst✝⁴ : CommRing A\ninst✝³ : Algebra R A\nB : Type w₃\ninst✝² : CommRing B\ninst✝¹ : Algebra R B\ninst✝ : FinitePresentation R A\nn : ℕ\nf : MvPolynomial (Fin n) R →ₐ[R] A\nhsurj : Function.Surjective ⇑f\nhfg : (RingHom.ker f.toRingHom).FG\... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.Morphisms.AffineAnd | {
"line": 305,
"column": 4
} | {
"line": 308,
"column": 26
} | [
{
"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(($this).1) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.AlgebraicGeometry.Morphisms.AffineAnd | {
"line": 305,
"column": 4
} | {
"line": 308,
"column": 26
} | [
{
"pp": "case refine_1\nQ : {R S : Type u} → [inst : CommRing R] → [inst_1 : CommRing S] → (R →+* S) → Prop\nP : MorphismProperty Scheme\nhP : HasAffineProperty P (affineAnd fun {R S} [CommRing R] [CommRing S] ↦ Q)\nhQi : RingHom.RespectsIso fun {R S} [CommRing R] [CommRing S] ↦ Q\nhQ :\n ∀ {R S T : Type u} [i... | have : (Limits.coprod.desc f g).app W ≫ e.hom ≫ Limits.prod.fst = f.app W := by
simp [e, Scheme.Hom.app_eq_appLE, Scheme.Hom.appLE_comp_appLE]
convert! (hf W hW).2
exact congr(($this).1) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Spectrum.Prime.Noetherian | {
"line": 61,
"column": 4
} | {
"line": 62,
"column": 48
} | [
{
"pp": "R : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsArtinianRing R\np : Ideal R\ninst✝ : p.IsPrime\nr : R\nhr : (toPiLocalization R) r = Pi.single { asIdeal := p, isPrime := ⋯ } 1\n⊢ (algebraMap R (Localization p.primeCompl)) r = 1",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Spectrum.Prime.Noetherian | {
"line": 71,
"column": 6
} | {
"line": 72,
"column": 51
} | [
{
"pp": "R : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsArtinianRing R\np : Ideal R\ninst✝ : p.IsPrime\nr : R\nhr : (toPiLocalization R) r = Pi.single { asIdeal := p, isPrime := ⋯ } 1\nthis✝ : (algebraMap R (Localization p.primeCompl)) r = 1\nq : Ideal R\nhq : q.IsPrime\ne : q ≠ p\nthis : { asIdeal := q, isPrime... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.JacobsonSpace | {
"line": 80,
"column": 2
} | {
"line": 80,
"column": 13
} | [
{
"pp": "X : Type u_1\ninst✝¹ : TopologicalSpace X\ninst✝ : JacobsonSpace X\n⊢ closure[inst✝¹] (closedPoints X) = Set.univ",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.JacobsonSpace | {
"line": 207,
"column": 27
} | {
"line": 207,
"column": 38
} | [
{
"pp": "X : Type u_1\nY : Type u_2\ninst✝² : TopologicalSpace X\ninst✝¹ : TopologicalSpace Y\nf : X → Y\ninst✝ : JacobsonSpace X\nS : Set X\nhS : IsLocallyClosed S\nhf₁ : Continuous[inst✝², inst✝¹] f\nhf₂ : IsClosedMap f\nhfS : (f '' S).Finite\nhS'' : S.Nonempty\nhS' : IsIrreducible S\nH₁ : IsIrreducible (S ∩ ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.RingHom.Integral | {
"line": 63,
"column": 74
} | {
"line": 63,
"column": 85
} | [
{
"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 ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.Morphisms.Separated | {
"line": 118,
"column": 4
} | {
"line": 118,
"column": 20
} | [
{
"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)",
"usedConstants": [
"AlgebraicGeometry.PresheafedSpace.carrier",... | exact this _ U.2 | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.AlgebraicGeometry.Morphisms.Integral | {
"line": 108,
"column": 6
} | {
"line": 108,
"column": 27
} | [
{
"pp": "Z S : Scheme\n⊢ ∀ {X Y : Scheme} (f : X ⟶ Y) [IsIntegralHom f], UniversallyClosed f",
"usedConstants": [
"AlgebraicGeometry.IsIntegralHom",
"Eq.mpr",
"AlgebraicGeometry.Scheme",
"CategoryTheory.CategoryStruct.toQuiver",
"Quiver.Hom",
"congrArg",
"id",
... | universallyClosed_eq, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.AlgebraicGeometry.Morphisms.ClosedImmersion | {
"line": 197,
"column": 30
} | {
"line": 197,
"column": 79
} | [
{
"pp": "X Y : Scheme\nf : X ⟶ Y\ninst✝ : IsClosedImmersion f\nH : Scheme.Hom.ker f = ⊥\n⊢ IsIso (Scheme.Hom.imageι f)",
"usedConstants": [
"AlgebraicGeometry.Scheme",
"AlgebraicGeometry.Scheme.Hom.image",
"CategoryTheory.IsIso",
"AlgebraicGeometry.Scheme.Hom.imageι",
"id",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.Morphisms.ClosedImmersion | {
"line": 221,
"column": 4
} | {
"line": 222,
"column": 57
} | [
{
"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": 221,
"column": 4
} | {
"line": 222,
"column": 57
} | [
{
"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": 221,
"column": 4
} | {
"line": 222,
"column": 57
} | [
{
"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": 224,
"column": 2
} | {
"line": 224,
"column": 57
} | [
{
"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... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.Morphisms.UniversallyClosed | {
"line": 103,
"column": 2
} | {
"line": 103,
"column": 42
} | [
{
"pp": "X Y : Scheme\nf : X ⟶ Y\n⊢ IsZariskiLocalAtTarget (topologically @IsClosedMap).universally",
"usedConstants": [
"IsClosedMap",
"AlgebraicGeometry.topologically",
"AlgebraicGeometry.universally_isZariskiLocalAtTarget"
]
}
] | apply universally_isZariskiLocalAtTarget | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.AlgebraicGeometry.Morphisms.ClosedImmersion | {
"line": 246,
"column": 12
} | {
"line": 247,
"column": 50
} | [
{
"pp": "X Y : Scheme\ninst✝¹ : IsAffine Y\nf : X ⟶ Y\ninst✝ : CompactSpace ↥X\nhfinj : Function.Injective ⇑(ConcreteCategory.hom (Scheme.Hom.appTop f))\nthis : Scheme.Hom.ker f = ⊥\n⊢ DenseRange ⇑f",
"usedConstants": [
"AlgebraicGeometry.SheafedSpace.instTopologicalSpaceCarrierCarrier",
"Algebr... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.Morphisms.ClosedImmersion | {
"line": 251,
"column": 4
} | {
"line": 251,
"column": 44
} | [
{
"pp": "X Y : Scheme\ninst✝¹ : IsAffine Y\nf : X ⟶ Y\ninst✝ : CompactSpace ↥X\n⊢ Function.Injective ⇑(ConcreteCategory.hom (Scheme.Hom.appTop X.toSpecΓ))",
"usedConstants": [
"Eq.mpr",
"AlgebraicGeometry.Spec",
"AlgebraicGeometry.SheafedSpace.instTopologicalSpaceCarrierCarrier",
"La... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.Morphisms.UniversallyClosed | {
"line": 134,
"column": 39
} | {
"line": 134,
"column": 54
} | [
{
"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... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.Morphisms.UniversallyClosed | {
"line": 142,
"column": 2
} | {
"line": 142,
"column": 37
} | [
{
"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... | 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": 203,
"column": 2
} | {
"line": 203,
"column": 65
} | [
{
"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",
"usedConstants": [
"Eq.mpr",
"Algebra... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.Morphisms.Finite | {
"line": 212,
"column": 4
} | {
"line": 212,
"column": 15
} | [
{
"pp": "case mpr\nX : Scheme\ninst✝ : JacobsonSpace ↥X\nx : ↥X\nH : LocallyOfFiniteType (X.fromSpecResidueField x)\n⊢ IsClosed {x}",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.Morphisms.UniversallyClosed | {
"line": 152,
"column": 44
} | {
"line": 152,
"column": 59
} | [
{
"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... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.RingHom.FinitePresentation | {
"line": 35,
"column": 48
} | {
"line": 35,
"column": 59
} | [
{
"pp": "R✝ S✝ : Type u_1\ninst✝¹ : CommRing R✝\ninst✝ : CommRing S✝\ne : R✝ ≃+* S✝\n⊢ (ker e.toRingHom).FG",
"usedConstants": [
"Eq.mpr",
"RingHom.instRingHomClass",
"Semiring.toModule",
"congrArg",
"CommSemiring.toSemiring",
"RingHom",
"id",
"Bot.bot",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Spectrum.Prime.Polynomial | {
"line": 59,
"column": 4
} | {
"line": 60,
"column": 65
} | [
{
"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 :... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Spectrum.Prime.Polynomial | {
"line": 67,
"column": 6
} | {
"line": 67,
"column": 79
} | [
{
"pp": "case inr.mpr.a.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.finra... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.IdealSheaf.Functorial | {
"line": 94,
"column": 4
} | {
"line": 94,
"column": 15
} | [
{
"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",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.IdealSheaf.Functorial | {
"line": 209,
"column": 61
} | {
"line": 209,
"column": 72
} | [
{
"pp": "X Y Z : Scheme\nI : X.IdealSheafData\nJ : Y.IdealSheafData\nf : X ⟶ Y\nH : J ≤ I.map f\n⊢ Hom.ker J.subschemeι ≤ Hom.ker (I.subschemeι ≫ f)",
"usedConstants": [
"Eq.mpr",
"AlgebraicGeometry.Scheme",
"congrArg",
"PartialOrder.toPreorder",
"AlgebraicGeometry.Scheme.Ideal... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.Morphisms.Immersion | {
"line": 65,
"column": 47
} | {
"line": 65,
"column": 58
} | [
{
"pp": "X Y Z : Scheme\nf✝ f : X ⟶ Y\ninst✝ : IsImmersion f\n⊢ Set.range ⇑f ⊆ Set.range ⇑(coborderRange f).ι",
"usedConstants": [
"Eq.mpr",
"AlgebraicGeometry.SheafedSpace.instTopologicalSpaceCarrierCarrier",
"AlgebraicGeometry.PresheafedSpace.carrier",
"congrArg",
"CategoryTh... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Spectrum.Prime.Polynomial | {
"line": 183,
"column": 19
} | {
"line": 183,
"column": 30
} | [
{
"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",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.Morphisms.Immersion | {
"line": 98,
"column": 45
} | {
"line": 98,
"column": 56
} | [
{
"pp": "X Y Z : Scheme\nf : X ⟶ Y\ninst✝ : IsImmersion f\nthis✝ : IsPreimmersion (Scheme.Hom.liftCoborder f ≫ (Scheme.Hom.coborderRange f).ι)\nthis : IsPreimmersion (Scheme.Hom.liftCoborder f)\n⊢ Set.range ⇑f ⊆ Set.range ⇑(Scheme.Hom.coborderRange f).ι",
"usedConstants": [
"Eq.mpr",
"AlgebraicG... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.Morphisms.Immersion | {
"line": 163,
"column": 72
} | {
"line": 163,
"column": 83
} | [
{
"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",
"usedConstants": ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.Morphisms.Immersion | {
"line": 168,
"column": 8
} | {
"line": 168,
"column": 23
} | [
{
"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 ≫ ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.Morphisms.Immersion | {
"line": 170,
"column": 10
} | {
"line": 170,
"column": 21
} | [
{
"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' ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.QuasiAffine | {
"line": 147,
"column": 2
} | {
"line": 147,
"column": 13
} | [
{
"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",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.AlgClosed.Basic | {
"line": 108,
"column": 51
} | {
"line": 108,
"column": 62
} | [
{
"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 {... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.HopkinsLevitzki | {
"line": 182,
"column": 7
} | {
"line": 182,
"column": 59
} | [
{
"pp": "R : Type u_3\ninst✝¹ : CommRing R\ninst✝ : IsNoetherianRing R\n⊢ IsArtinianRing R ↔ Ring.KrullDimLE 0 R",
"usedConstants": [
"Eq.mpr",
"IsArtinianRing",
"congrArg",
"CommSemiring.toSemiring",
"isArtinianRing_iff_isNoetherianRing_krullDimLE_zero",
"id",
"IsN... | isArtinianRing_iff_isNoetherianRing_krullDimLE_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.AlgebraicGeometry.Artinian | {
"line": 103,
"column": 2
} | {
"line": 103,
"column": 13
} | [
{
"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}",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.Artinian | {
"line": 132,
"column": 4
} | {
"line": 132,
"column": 15
} | [
{
"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}",
"usedConstants": [
"AlgebraicGeometry.SheafedSpace.instTopologicalSpaceCarrierCarrier",
"AlgebraicGeometry.Scheme",
"CategoryTheory.PreZero... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.SpreadingOut | {
"line": 115,
"column": 2
} | {
"line": 115,
"column": 13
} | [
{
"pp": "X Y : Scheme\nf : X ⟶ Y\ninst✝ : IsOpenImmersion f\nV : ↑X.affineOpens\nhU : IsAffineOpen (f ''ᵁ ↑V)\nhU' : f ''ᵁ ↑V ≤ Scheme.Hom.opensRange f\nhV : (IsOpenImmersion.affineOpensEquiv f) V = ⟨⟨f ''ᵁ ↑V, hU⟩, hU'⟩\ny : ↥X\nhy : y ∈ ↑↑V\nH✝ : Y.IsGermInjectiveAt (f y)\ne : f y = f y\nH :\n MorphismProper... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.SpreadingOut | {
"line": 180,
"column": 2
} | {
"line": 180,
"column": 83
} | [
{
"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) ... | 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
} | [
{
"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",
"usedConstants": [
"AlgebraicGeometry.PresheafedSpace.carrier",
"CategoryTheory.ConcreteCategory.hom",
"CommRingCat",
"TopCat.instCatego... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.SpreadingOut | {
"line": 251,
"column": 14
} | {
"line": 251,
"column": 25
} | [
{
"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.... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.SpreadingOut | {
"line": 373,
"column": 4
} | {
"line": 373,
"column": 67
} | [
{
"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... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.FunctionField | {
"line": 45,
"column": 65
} | {
"line": 45,
"column": 76
} | [
{
"pp": "X : Scheme\ninst✝ : IrreducibleSpace ↥X\nU : X.Opens\nh : Nonempty ↥↑U\n⊢ (Set.univ ∩ ↑U).Nonempty",
"usedConstants": [
"Eq.mpr",
"AlgebraicGeometry.SheafedSpace.instTopologicalSpaceCarrierCarrier",
"AlgebraicGeometry.PresheafedSpace.carrier",
"congrArg",
"CommRingCat"... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.AffineSpace | {
"line": 60,
"column": 6
} | {
"line": 60,
"column": 45
} | [
{
"pp": "case hf.hC.a\nn : Type v\nR : CommRingCat\nf g : ℤ[n] ⟶ R\nh : ∀ (i : n), (ConcreteCategory.hom f) (X i) = (ConcreteCategory.hom g) (X i)\nthis :\n (CommRingCat.Hom.hom f).comp (mapEquiv n ULift.ringEquiv.symm).toRingHom =\n (CommRingCat.Hom.hom g).comp (mapEquiv n ULift.ringEquiv.symm).toRingHom\n... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.AffineSpace | {
"line": 61,
"column": 6
} | {
"line": 61,
"column": 45
} | [
{
"pp": "case hf.hX\nn : Type v\nR : CommRingCat\nf g : ℤ[n] ⟶ R\nh : ∀ (i : n), (ConcreteCategory.hom f) (X i) = (ConcreteCategory.hom g) (X i)\nthis :\n (CommRingCat.Hom.hom f).comp (mapEquiv n ULift.ringEquiv.symm).toRingHom =\n (CommRingCat.Hom.hom g).comp (mapEquiv n ULift.ringEquiv.symm).toRingHom\nx ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.FunctionField | {
"line": 129,
"column": 68
} | {
"line": 129,
"column": 79
} | [
{
"pp": "X✝ : Scheme\nX : Scheme\ninst✝ : IsIntegral X\nU : X.Opens\nhU : IsAffineOpen U\nh : Nonempty ↥↑U\n⊢ (Set.univ ∩ ↑U).Nonempty",
"usedConstants": [
"Eq.mpr",
"AlgebraicGeometry.SheafedSpace.instTopologicalSpaceCarrierCarrier",
"AlgebraicGeometry.PresheafedSpace.carrier",
"con... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.AffineSpace | {
"line": 64,
"column": 4
} | {
"line": 64,
"column": 15
} | [
{
"pp": "case hX\nn : Type v\nR : CommRingCat\nf g : ℤ[n] ⟶ R\nh : ∀ (i : n), (ConcreteCategory.hom f) (X i) = (ConcreteCategory.hom g) (X i)\ni✝ : n\n⊢ ((CommRingCat.Hom.hom f).comp (mapEquiv n ULift.ringEquiv.symm).toRingHom) (X i✝) =\n ((CommRingCat.Hom.hom g).comp (mapEquiv n ULift.ringEquiv.symm).toRing... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.FunctionField | {
"line": 148,
"column": 67
} | {
"line": 148,
"column": 78
} | [
{
"pp": "X : Scheme\ninst✝¹ : IsIntegral X\nU : X.Opens\nhU : IsAffineOpen U\ninst✝ : Nonempty ↥↑U\n⊢ (Set.univ ∩ ↑U).Nonempty",
"usedConstants": [
"Eq.mpr",
"AlgebraicGeometry.SheafedSpace.instTopologicalSpaceCarrierCarrier",
"AlgebraicGeometry.PresheafedSpace.carrier",
"congrArg",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.Birational.RationalMap | {
"line": 70,
"column": 4
} | {
"line": 70,
"column": 15
} | [
{
"pp": "case mpr.e_hom\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",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.Birational.RationalMap | {
"line": 260,
"column": 4
} | {
"line": 261,
"column": 84
} | [
{
"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 ⊓ ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.Birational.RationalMap | {
"line": 274,
"column": 2
} | {
"line": 274,
"column": 13
} | [
{
"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... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.Birational.RationalMap | {
"line": 402,
"column": 36
} | {
"line": 402,
"column": 47
} | [
{
"pp": "X Y S : Scheme\ninst✝¹ : X.Over S\ninst✝ : Y.Over S\nf : X.PartialMap Y\ne✝ : f.toRationalMap.compHom (Y ↘ S) = Hom.toRationalMap (X ↘ S)\nU : X.Opens\nhU : Dense ↑U\nhUl : U ≤ ((fun x ↦ x.compHom (Y ↘ S)) f).domain\nhUr : U ≤ (Hom.toPartialMap (X ↘ S)).domain\ne : (((fun x ↦ x.compHom (Y ↘ S)) f).rest... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.Birational.RationalMap | {
"line": 412,
"column": 51
} | {
"line": 412,
"column": 62
} | [
{
"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... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.Birational.RationalMap | {
"line": 466,
"column": 4
} | {
"line": 466,
"column": 49
} | [
{
"pp": "X Y Z S : Scheme\nsX : X ⟶ S\nsY : Y ⟶ S\ninst✝¹ : IsIntegral X\ninst✝ : LocallyOfFiniteType sY\nf : X.PartialMap Y\nhf : f.toRationalMap.compHom sY = Hom.toRationalMap sX\n⊢ (↑⟨f.toRationalMap, hf⟩).fromFunctionField ≫ sY = X.fromSpecStalk (genericPoint ↥X) ≫ sX",
"usedConstants": [
"Algebra... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Spectrum.Prime.ChevalleyComplexity | {
"line": 245,
"column": 30
} | {
"line": 245,
"column": 41
} | [
{
"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 ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.Birational.RationalMap | {
"line": 556,
"column": 37
} | {
"line": 556,
"column": 48
} | [
{
"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",
"usedConstants": [
"Eq.mpr",
"AlgebraicGeometry.Scheme.homOfLE",
"AlgebraicGeometry.Scheme",
"AlgebraicGeom... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.Cover.Directed | {
"line": 232,
"column": 42
} | {
"line": 232,
"column": 53
} | [
{
"pp": "P : MorphismProperty Scheme\nX : Scheme\n𝒰 : X.OpenCover\ninst✝¹ : Category.{v_1, ?u.58344} 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.... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Spectrum.Prime.ChevalleyComplexity | {
"line": 267,
"column": 17
} | {
"line": 267,
"column": 28
} | [
{
"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 :\... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.Cover.Directed | {
"line": 312,
"column": 4
} | {
"line": 312,
"column": 15
} | [
{
"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, ⋯⟩)",
"usedConstants": [
"CategoryTheory.PreZeroHypercover.mk",
"TopologicalSpace.Opens.mem_top",
"Eq.mpr",
"SetLike.mem_co... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.RelativeGluing | {
"line": 174,
"column": 2
} | {
"line": 180,
"column": 8
} | [
{
"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",
"usedConstants": [
... | · 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.RingTheory.Spectrum.Prime.ChevalleyComplexity | {
"line": 300,
"column": 4
} | {
"line": 300,
"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 ... | 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": 305,
"column": 51
} | {
"line": 305,
"column": 67
} | [
{
"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 ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Spectrum.Prime.ChevalleyComplexity | {
"line": 358,
"column": 8
} | {
"line": 358,
"column": 19
} | [
{
"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 ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Sets.CompactOpenCovered | {
"line": 72,
"column": 8
} | {
"line": 72,
"column": 23
} | [
{
"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),... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Sets.CompactOpenCovered | {
"line": 90,
"column": 22
} | {
"line": 90,
"column": 33
} | [
{
"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... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Sets.CompactOpenCovered | {
"line": 115,
"column": 6
} | {
"line": 115,
"column": 17
} | [
{
"pp": "case refine_1.refine_2\nS : 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)\ninst✝ : TopologicalSpace S\nU : Set S\nhU : IsCompact U\nUs : (x : S) → x ∈ U → Set S\nhU' : ∀ (x : S) (a : x ∈ U), Us x a ⊆ U\nhUx : ∀ (x : S) (a : x ∈ U), x ∈ Us x a... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.ColimitsOver | {
"line": 144,
"column": 4
} | {
"line": 144,
"column": 39
} | [
{
"pp": "case refine_1\nP : MorphismProperty Scheme\ninst✝⁵ : P.IsStableUnderBaseChange\ninst✝⁴ : P.IsMultiplicative\nS : Scheme\nJ : Type u_1\ninst✝³ : Category.{v_1, u_1} J\nD : J ⥤ P.Over ⊤ S\n𝒰 : S.OpenCover\ninst✝² : Category.{v_2, u_2} 𝒰.I₀\ninst✝¹ : LocallyDirected 𝒰\nd : ColimitGluingData D 𝒰\ninst✝... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Sets.CompactOpenCovered | {
"line": 164,
"column": 2
} | {
"line": 164,
"column": 13
} | [
{
"pp": "S : Type u_1\nι : Type u_2\nX : ι → Type u_3\nf : (i : ι) → X i → S\ninst✝² : (i : ι) → TopologicalSpace (X i)\ninst✝¹ : Finite ι\ninst✝ : TopologicalSpace S\nhf : ∀ (i : ι), IsSpectralMap (f i)\nU : Set S\nhs : ∀ x ∈ U, ∃ i, x ∈ Set.range (f i)\nhU : IsOpen U\nhc : IsCompact U\ni : ι\ny : X i\nhx : f ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Sets.CompactOpenCovered | {
"line": 173,
"column": 2
} | {
"line": 173,
"column": 13
} | [
{
"pp": "S : Type u_1\nι : Type u_2\nX : ι → Type u_3\nf : (i : ι) → X i → S\ninst✝² : (i : ι) → TopologicalSpace (X i)\ninst✝¹ : TopologicalSpace S\ninst✝ : ∀ (i : ι), PrespectralSpace (X i)\nhfc : ∀ (i : ι), Continuous (f i)\nh : ∀ (i : ι), IsOpenMap (f i)\nU : Set S\nhs : ∀ x ∈ U, ∃ i, x ∈ Set.range (f i)\nh... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.Cover.QuasiCompact | {
"line": 55,
"column": 2
} | {
"line": 55,
"column": 13
} | [
{
"pp": "S : Scheme\n𝒰 : PreZeroHypercover S\ninst✝ : QuasiCompactCover 𝒰\nU : S.Opens\nhU : IsCompact ↑U\nUs : Set (TopologicalSpace.Opens ↥S)\nhUs : Us ⊆ S.affineOpens\nhUf : Us.Finite\nhUc : U = sSup Us\n⊢ ∀ t ∈ SetLike.coe '' Us, IsCompactOpenCovered (fun x ↦ ⇑(𝒰.f x)) t",
"usedConstants": [
"E... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.Cover.QuasiCompact | {
"line": 103,
"column": 26
} | {
"line": 103,
"column": 37
} | [
{
"pp": "S : Scheme\n𝒰✝ 𝒰 : PreZeroHypercover S\nK : Precoverage Scheme\ninst✝ : QuasiCompactCover 𝒰\nT : Scheme\nf : T ⟶ S\nU' : T.Opens\nhU' : IsAffineOpen U'\nthis :\n ∀ {U' : T.Opens},\n IsAffineOpen U' →\n (∃ U, IsAffineOpen U ∧ ⇑f '' ↑U' ⊆ ↑U) →\n IsCompactOpenCovered (fun x ↦ ⇑((PreZer... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.Cover.QuasiCompact | {
"line": 108,
"column": 53
} | {
"line": 108,
"column": 64
} | [
{
"pp": "S : Scheme\n𝒰✝ 𝒰 : PreZeroHypercover S\nK : Precoverage Scheme\ninst✝ : QuasiCompactCover 𝒰\nT : Scheme\nf : T ⟶ S\nU' : T.Opens\nhU' : IsAffineOpen U'\nU : S.Opens\nhU : IsAffineOpen U\nhsub : ⇑f '' ↑U' ⊆ ↑U\ns : Set 𝒰.I₀\nhf : s.Finite\nV : (i : 𝒰.I₀) → i ∈ s → TopologicalSpace.Opens ((fun x ↦ ↥... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.Cover.QuasiCompact | {
"line": 109,
"column": 6
} | {
"line": 109,
"column": 67
} | [
{
"pp": "S : Scheme\n𝒰✝ 𝒰 : PreZeroHypercover S\nK : Precoverage Scheme\ninst✝ : QuasiCompactCover 𝒰\nT : Scheme\nf : T ⟶ S\nU' : T.Opens\nhU' : IsAffineOpen U'\nU : S.Opens\nhU : IsAffineOpen U\nhsub : ⇑f '' ↑U' ⊆ ↑U\ns : Set 𝒰.I₀\nhf : s.Finite\nV : (i : 𝒰.I₀) → i ∈ s → TopologicalSpace.Opens ((fun x ↦ ↥... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.Cover.QuasiCompact | {
"line": 137,
"column": 4
} | {
"line": 137,
"column": 95
} | [
{
"pp": "S : Scheme\n𝒰✝¹ 𝒰✝ : PreZeroHypercover S\nK : Precoverage Scheme\nX : Scheme\n𝒰 : PreZeroHypercover X\ninst✝¹ : QuasiCompactCover 𝒰\nf : (x : 𝒰.I₀) → PreZeroHypercover (𝒰.X x)\ninst✝ : ∀ (x : 𝒰.I₀), QuasiCompactCover (f x)\nU : X.Opens\nhU✝ : IsAffineOpen U\ns : Set 𝒰.I₀\nhs : s.Finite\nV : (i ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Spectrum.Prime.ChevalleyComplexity | {
"line": 388,
"column": 6
} | {
"line": 389,
"column": 71
} | [
{
"pp": "case h.e'_2.h.e'_4.h.e'_4\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... | · rw [Set.preimage_diff, preimage_comap_zeroLocus, preimage_comap_zeroLocus,
Set.image_singleton, Set.range_comp, AlgHom.toRingHom_eq_coe] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.CategoryTheory.EffectiveEpi.Comp | {
"line": 86,
"column": 9
} | {
"line": 86,
"column": 28
} | [
{
"pp": "C✝ : Type u_1\ninst✝³ : Category.{v_1, u_1} C✝\nC : Type u_2\ninst✝² : Category.{v_2, u_2} C\nI : Type u_3\nZ Y : I → C\nX : C\ng : (i : I) → Z i ⟶ Y i\nf : (i : I) → Y i ⟶ X\ninst✝¹ : EffectiveEpiFamily Z fun i ↦ g i ≫ f i\ninst✝ : ∀ (i : I), Epi (g i)\nW : C\nφ : (a : I) → Y a ⟶ W\nh : ∀ {Z : C} (a₁ ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.