module stringlengths 16 90 | startPos dict | endPos dict | goals listlengths 0 96 | ppTac stringlengths 1 14.5k | elaborator stringclasses 365
values | kind stringclasses 368
values |
|---|---|---|---|---|---|---|
Mathlib.AlgebraicGeometry.AffineScheme | {
"line": 593,
"column": 2
} | {
"line": 593,
"column": 39
} | [
{
"pp": "X : Scheme\nU : X.Opens\nhU : IsAffineOpen U\nf : ↑Γ(X, U)\n⊢ hU.fromSpec ''ᵁ PrimeSpectrum.basicOpen f = X.basicOpen f",
"usedConstants": [
"AlgebraicGeometry.Scheme.Hom.opensFunctor",
"Eq.mpr",
"AlgebraicGeometry.Spec",
"AlgebraicGeometry.SheafedSpace.instTopologicalSpaceC... | rw [← hU.fromSpec_preimage_basicOpen] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.AlgebraicGeometry.Gluing | {
"line": 365,
"column": 11
} | {
"line": 365,
"column": 33
} | [
{
"pp": "X : Scheme\n𝒰 : X.OpenCover\ni : (gluedCover 𝒰).J\nx : ↥((gluedCover 𝒰).U i)\nj : (gluedCover 𝒰).J\ny : ↥((gluedCover 𝒰).U j)\nh :\n (ConcreteCategory.hom (((gluedCover 𝒰).ι i).base ≫ (fromGlued 𝒰).base)) x =\n (ConcreteCategory.hom (((gluedCover 𝒰).ι j).base ≫ (fromGlued 𝒰).base)) y\n⊢ ((... | ← Scheme.Hom.comp_base | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.AlgebraicGeometry.Gluing | {
"line": 558,
"column": 6
} | {
"line": 559,
"column": 20
} | [
{
"pp": "case inst\nJ : Type w\ninst✝² : Category.{v, w} J\nF : J ⥤ Scheme\ninst✝¹ : ∀ {i j : J} (f : i ⟶ j), IsOpenImmersion (F.map f)\ninst✝ : (F ⋙ forget).IsLocallyDirected\ni j k : J\nx : ↥(pullback (V F i j).ι (V F i k).ι)\nk₁ : (k : J) × (k ⟶ i) × (k ⟶ j)\nk₂ : (k_1 : J) × (k_1 ⟶ i) × (k_1 ⟶ k)\nl : J\nhl... | simp only [limit.lift_π, PullbackCone.mk_pt, PullbackCone.mk_π_app, α]
infer_instance | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.AlgebraicGeometry.Gluing | {
"line": 558,
"column": 6
} | {
"line": 559,
"column": 20
} | [
{
"pp": "case inst\nJ : Type w\ninst✝² : Category.{v, w} J\nF : J ⥤ Scheme\ninst✝¹ : ∀ {i j : J} (f : i ⟶ j), IsOpenImmersion (F.map f)\ninst✝ : (F ⋙ forget).IsLocallyDirected\ni j k : J\nx : ↥(pullback (V F i j).ι (V F i k).ι)\nk₁ : (k : J) × (k ⟶ i) × (k ⟶ j)\nk₂ : (k_1 : J) × (k_1 ⟶ i) × (k_1 ⟶ k)\nl : J\nhl... | simp only [limit.lift_π, PullbackCone.mk_pt, PullbackCone.mk_π_app, α]
infer_instance | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.AlgebraicGeometry.AffineScheme | {
"line": 1115,
"column": 76
} | {
"line": 1115,
"column": 99
} | [
{
"pp": "X : Scheme\ninst✝ : IsAffine X\ns : Set ↑Γ(X, ⊤)\n⊢ ⇑(hom (X.isoSpec.inv.base ≫ X.toSpecΓ.base)) ⁻¹' PrimeSpectrum.zeroLocus s = PrimeSpectrum.zeroLocus s",
"usedConstants": [
"Eq.mpr",
"AlgebraicGeometry.Spec",
"AlgebraicGeometry.SheafedSpace.instTopologicalSpaceCarrierCarrier",
... | ← Scheme.Hom.comp_base, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.AlgebraicGeometry.Gluing | {
"line": 676,
"column": 8
} | {
"line": 676,
"column": 39
} | [
{
"pp": "J : Type w\ninst✝⁴ : Category.{v, w} J\nF : J ⥤ Scheme\ninst✝³ : ∀ {i j : J} (f : i ⟶ j), IsOpenImmersion (F.map f)\ninst✝² : (F ⋙ forget).IsLocallyDirected\ninst✝¹ : Quiver.IsThin J\ninst✝ : Small.{u, w} J\ni j k : Shrink.{u, w} J\nx : failed to pretty print expression (use 'set_option pp.rawOnError t... | IsOpenImmersion.comp_lift_assoc | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.AlgebraicGeometry.AffineScheme | {
"line": 1303,
"column": 2
} | {
"line": 1303,
"column": 11
} | [
{
"pp": "R : CommRingCat\nx : PrimeSpectrum ↑R\n⊢ CommRingCat.ofHom (algebraMap (↑R) (Localization.AtPrime x.asIdeal)) ≫ (stalkIso R x).inv =\n (Scheme.ΓSpecIso R).inv ≫ (Spec R).presheaf.germ ⊤ x trivial",
"usedConstants": [
"AlgebraicGeometry.Spec",
"AlgebraicGeometry.SheafedSpace.instTopol... | ext s : 2 | _private.Lean.Elab.Tactic.Ext.0.Lean.Elab.Tactic.Ext.evalExt | Lean.Elab.Tactic.Ext.ext |
Mathlib.AlgebraicGeometry.Gluing | {
"line": 679,
"column": 51
} | {
"line": 679,
"column": 82
} | [
{
"pp": "J : Type w\ninst✝⁴ : Category.{v, w} J\nF : J ⥤ Scheme\ninst✝³ : ∀ {i j : J} (f : i ⟶ j), IsOpenImmersion (F.map f)\ninst✝² : (F ⋙ forget).IsLocallyDirected\ninst✝¹ : Quiver.IsThin J\ninst✝ : Small.{u, w} J\ni j k : Shrink.{u, w} J\nx : failed to pretty print expression (use 'set_option pp.rawOnError t... | IsOpenImmersion.comp_lift_assoc | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.AlgebraicGeometry.Limits | {
"line": 402,
"column": 4
} | {
"line": 410,
"column": 67
} | [
{
"pp": "ι : Type u\nf : ι → Scheme\nσ : Type v\ng : σ → Scheme\nX Y : Scheme\n⊢ { I₀ := PUnit.{w + 1} ⊕ PUnit.{w + 1}, X := fun x ↦ Sum.elim (fun x ↦ X) (fun x ↦ Y) x,\n f := fun x ↦ Sum.rec (fun x ↦ coprod.inl) (fun x ↦ coprod.inr) x }.presieve₀ ∈\n (Scheme.precoverage IsOpenImmersion).coverings (X ... | rw [Scheme.presieve₀_mem_precoverage_iff]
refine ⟨fun x ↦ ?_, fun x ↦ x.rec (fun _ ↦ inferInstance) (fun _ ↦ inferInstance)⟩
use ((coprodMk X Y).symm x).elim (fun _ ↦ Sum.inl .unit) (fun _ ↦ Sum.inr .unit)
obtain ⟨x, rfl⟩ := (coprodMk X Y).surjective x
simp only [Sum.elim_inl, Sum.elim_inr, Set.mem_rang... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.AlgebraicGeometry.Limits | {
"line": 402,
"column": 4
} | {
"line": 410,
"column": 67
} | [
{
"pp": "ι : Type u\nf : ι → Scheme\nσ : Type v\ng : σ → Scheme\nX Y : Scheme\n⊢ { I₀ := PUnit.{w + 1} ⊕ PUnit.{w + 1}, X := fun x ↦ Sum.elim (fun x ↦ X) (fun x ↦ Y) x,\n f := fun x ↦ Sum.rec (fun x ↦ coprod.inl) (fun x ↦ coprod.inr) x }.presieve₀ ∈\n (Scheme.precoverage IsOpenImmersion).coverings (X ... | rw [Scheme.presieve₀_mem_precoverage_iff]
refine ⟨fun x ↦ ?_, fun x ↦ x.rec (fun _ ↦ inferInstance) (fun _ ↦ inferInstance)⟩
use ((coprodMk X Y).symm x).elim (fun _ ↦ Sum.inl .unit) (fun _ ↦ Sum.inr .unit)
obtain ⟨x, rfl⟩ := (coprodMk X Y).surjective x
simp only [Sum.elim_inl, Sum.elim_inr, Set.mem_rang... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Geometry.RingedSpace.PresheafedSpace.Gluing | {
"line": 466,
"column": 4
} | {
"line": 471,
"column": 21
} | [
{
"pp": "case w.op.left\nC : Type u\ninst✝¹ : Category.{v, u} C\nD : GlueData C\ninst✝ : HasLimits C\ni : D.J\nU : Opens ↑↑(D.U i)\nj k : D.J\n⊢ (D.diagramOverOpenπ U i ≫ D.ιInvAppπEqMap U ≫ D.ιInvApp U) ≫\n limit.π (D.diagramOverOpen U) (op (WalkingMultispan.left (j, k))) =\n 𝟙 (limit (D.diagramOverOp... | rw [← limit.w (componentwiseDiagram 𝖣.diagram.multispan _)
(Quiver.Hom.op (WalkingMultispan.Hom.fst (j, k))),
← Category.assoc, Category.id_comp]
congr 1
simp_rw [Category.assoc]
apply π_ιInvApp_π | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Geometry.RingedSpace.PresheafedSpace.Gluing | {
"line": 466,
"column": 4
} | {
"line": 471,
"column": 21
} | [
{
"pp": "case w.op.left\nC : Type u\ninst✝¹ : Category.{v, u} C\nD : GlueData C\ninst✝ : HasLimits C\ni : D.J\nU : Opens ↑↑(D.U i)\nj k : D.J\n⊢ (D.diagramOverOpenπ U i ≫ D.ιInvAppπEqMap U ≫ D.ιInvApp U) ≫\n limit.π (D.diagramOverOpen U) (op (WalkingMultispan.left (j, k))) =\n 𝟙 (limit (D.diagramOverOp... | rw [← limit.w (componentwiseDiagram 𝖣.diagram.multispan _)
(Quiver.Hom.op (WalkingMultispan.Hom.fst (j, k))),
← Category.assoc, Category.id_comp]
congr 1
simp_rw [Category.assoc]
apply π_ιInvApp_π | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.AlgebraicGeometry.Morphisms.UnderlyingMap | {
"line": 272,
"column": 4
} | {
"line": 273,
"column": 39
} | [
{
"pp": "case hP₁\n⊢ ∀ {α β : Type u_1} [inst : TopologicalSpace α] [inst_1 : TopologicalSpace β] (f : α ≃ₜ β), SpecializingMap ⇑f",
"usedConstants": [
"TopologicalSpace",
"Homeomorph.instEquivLike",
"Homeomorph.isClosedMap",
"IsClosedMap.specializingMap",
"Homeomorph",
"... | introv
exact f.isClosedMap.specializingMap | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.AlgebraicGeometry.Morphisms.UnderlyingMap | {
"line": 272,
"column": 4
} | {
"line": 273,
"column": 39
} | [
{
"pp": "case hP₁\n⊢ ∀ {α β : Type u_1} [inst : TopologicalSpace α] [inst_1 : TopologicalSpace β] (f : α ≃ₜ β), SpecializingMap ⇑f",
"usedConstants": [
"TopologicalSpace",
"Homeomorph.instEquivLike",
"Homeomorph.isClosedMap",
"IsClosedMap.specializingMap",
"Homeomorph",
"... | introv
exact f.isClosedMap.specializingMap | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.AlgebraicGeometry.Morphisms.UnderlyingMap | {
"line": 291,
"column": 4
} | {
"line": 291,
"column": 26
} | [
{
"pp": "case hP₃\nα β : Type u_1\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\nf : α → β\nι : Type u_1\nU : ι → Opens β\nhU : IsOpenCover U\nx✝ : Continuous f\nhsp :\n ∀ (i : ι) (x : ↑(f ⁻¹' (U i).carrier)),\n closure {(U i).carrier.restrictPreimage f x} ⊆ (U i).carrier.restrictPreimage f '' cl... | obtain ⟨i, hi⟩ := this | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.AlgebraicGeometry.Pullbacks | {
"line": 395,
"column": 14
} | {
"line": 395,
"column": 23
} | [
{
"pp": "X Y Z : Scheme\n𝒰 : X.OpenCover\nf : X ⟶ Z\ng : Y ⟶ Z\ninst✝ : ∀ (i : 𝒰.I₀), HasPullback (𝒰.f i ≫ f) g\ns : PullbackCone f g\ni : 𝒰.I₀\n⊢ (gluing 𝒰 f g).ι i ≫ p1 𝒰 f g = pullback.fst (𝒰.f i ≫ f) g ≫ 𝒰.f i",
"usedConstants": [
"AlgebraicGeometry.Scheme.GlueData.ι",
"CategoryTheor... | gluing_ι, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.AlgebraicGeometry.Morphisms.Constructors | {
"line": 215,
"column": 6
} | {
"line": 215,
"column": 37
} | [
{
"pp": "case of_sSup_eq_top.a.refine_1\nP : MorphismProperty Scheme\nhP₂ : ∀ {X Y : Scheme} (f : X ⟶ Y) {ι : Type u} (U : ι → Y.Opens), IsOpenCover U → (∀ (i : ι), P (f ∣_ U i)) → P f\nX Y : Scheme\nf : X ⟶ Y\nι : Type u\nU : ι → Y.Opens\nhU : iSup U = ⊤\nH : ∀ (i : ι), P.universally (f ∣_ U i)\nX' Y' : Scheme... | · exact congr($(h.1.1) ⁻¹ᵁ U i) | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.AlgebraicGeometry.Morphisms.Constructors | {
"line": 394,
"column": 4
} | {
"line": 394,
"column": 49
} | [
{
"pp": "case of_sSup_eq_top\nP : {R S : Type u} → [inst : CommRing R] → [inst_1 : CommRing S] → (R →+* S) → Prop\nhP : RingHom.RespectsIso fun {R S} [CommRing R] [CommRing S] ↦ P\nhP' : (RingHom.toMorphismProperty fun {R S} [CommRing R] [CommRing S] ↦ P).RespectsIso\nthis : (stalkwise fun {R S} [CommRing R] [C... | have hy : f x ∈ iSup U := by rw [hU]; trivial | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.AlgebraicGeometry.Morphisms.RingHomProperties | {
"line": 161,
"column": 4
} | {
"line": 172,
"column": 13
} | [
{
"pp": "case to_basicOpen\nP : {R S : Type u} → [inst : CommRing R] → [inst_1 : CommRing S] → (R →+* S) → Prop\nh₁ : RingHom.RespectsIso fun {R S} [CommRing R] [CommRing S] ↦ P\nh₂ : RingHom.LocalizationAwayPreserves fun {R S} [CommRing R] [CommRing S] ↦ P\nh₃ : RingHom.OfLocalizationSpan fun {R S} [CommRing R... | intro X Y _ f r H
rw [sourceAffineLocally_morphismRestrict]
intro U hU
have : X.basicOpen (f.appLE ⊤ U (by simp) r) = U := by
simp only [Scheme.Hom.appLE, Opens.map_top, CommRingCat.comp_apply]
rw [Scheme.basicOpen_res]
simpa using hU
rw [← f.appLE_congr (by simp [Scheme.Hom.appLE]) rf... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.AlgebraicGeometry.Morphisms.RingHomProperties | {
"line": 161,
"column": 4
} | {
"line": 172,
"column": 13
} | [
{
"pp": "case to_basicOpen\nP : {R S : Type u} → [inst : CommRing R] → [inst_1 : CommRing S] → (R →+* S) → Prop\nh₁ : RingHom.RespectsIso fun {R S} [CommRing R] [CommRing S] ↦ P\nh₂ : RingHom.LocalizationAwayPreserves fun {R S} [CommRing R] [CommRing S] ↦ P\nh₃ : RingHom.OfLocalizationSpan fun {R S} [CommRing R... | intro X Y _ f r H
rw [sourceAffineLocally_morphismRestrict]
intro U hU
have : X.basicOpen (f.appLE ⊤ U (by simp) r) = U := by
simp only [Scheme.Hom.appLE, Opens.map_top, CommRingCat.comp_apply]
rw [Scheme.basicOpen_res]
simpa using hU
rw [← f.appLE_congr (by simp [Scheme.Hom.appLE]) rf... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.AlgebraicGeometry.Morphisms.RingHomProperties | {
"line": 214,
"column": 2
} | {
"line": 217,
"column": 27
} | [
{
"pp": "P : {R S : Type u} → [inst : CommRing R] → [inst_1 : CommRing S] → (R →+* S) → Prop\nX Y : Scheme\nf : X ⟶ Y\nhPa : StableUnderCompositionWithLocalizationAwayTarget fun {R S} [CommRing R] [CommRing S] ↦ P\nhPl : LocalizationAwayPreserves fun {R S} [CommRing R] [CommRing S] ↦ P\nx : ↥X\nU₁ U₂ : ↑Y.affin... | have ers : X.basicOpen s ≤ f ⁻¹ᵁ Y.basicOpen r := by
rw [hBss', hBrr']
apply le_trans (X.basicOpen_le _)
simp [Scheme.Hom.appLE] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.AlgebraicGeometry.Pullbacks | {
"line": 830,
"column": 39
} | {
"line": 830,
"column": 69
} | [
{
"pp": "M S T : Scheme\ninst✝¹ : M.Over S\nf : T ⟶ S\ninst✝ : MonObj (Over.mk (M ↘ S))\n⊢ MonObj (Over.mk (pullback (M ↘ S) f ↘ T))",
"usedConstants": [
"AlgebraicGeometry.Scheme",
"AlgebraicGeometry.Scheme.Pullback.instHasPullbacks",
"inferInstance",
"CategoryTheory.over",
"C... | exact Over.monObjMkPullbackSnd | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.AlgebraicGeometry.Morphisms.RingHomProperties | {
"line": 303,
"column": 2
} | {
"line": 312,
"column": 49
} | [
{
"pp": "P : MorphismProperty Scheme\nQ : {R S : Type u} → [inst : CommRing R] → [inst_1 : CommRing S] → (R →+* S) → Prop\ninst✝¹ : HasRingHomProperty P Q\nX Y Z : Scheme\nf : X ⟶ Y\ng : Y ⟶ Z\ninst✝ : IsOpenImmersion f\nH : P g\n⊢ P (f ≫ g)",
"usedConstants": [
"AlgebraicGeometry.Scheme.Hom.opensFunc... | rw [eq_affineLocally P, affineLocally_iff_affineOpens_le] at H ⊢
intro U V e
have : IsIso (f.appLE (f ''ᵁ V) V.1 (f.preimage_image_eq _).ge) :=
inferInstanceAs (IsIso (f.app _ ≫
X.presheaf.map (eqToHom (f.preimage_image_eq _).symm).op))
rw [← Scheme.Hom.appLE_comp_appLE _ _ _ (f ''ᵁ V) V.1
(Set.imag... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.AlgebraicGeometry.Morphisms.RingHomProperties | {
"line": 303,
"column": 2
} | {
"line": 312,
"column": 49
} | [
{
"pp": "P : MorphismProperty Scheme\nQ : {R S : Type u} → [inst : CommRing R] → [inst_1 : CommRing S] → (R →+* S) → Prop\ninst✝¹ : HasRingHomProperty P Q\nX Y Z : Scheme\nf : X ⟶ Y\ng : Y ⟶ Z\ninst✝ : IsOpenImmersion f\nH : P g\n⊢ P (f ≫ g)",
"usedConstants": [
"AlgebraicGeometry.Scheme.Hom.opensFunc... | rw [eq_affineLocally P, affineLocally_iff_affineOpens_le] at H ⊢
intro U V e
have : IsIso (f.appLE (f ''ᵁ V) V.1 (f.preimage_image_eq _).ge) :=
inferInstanceAs (IsIso (f.app _ ≫
X.presheaf.map (eqToHom (f.preimage_image_eq _).symm).op))
rw [← Scheme.Hom.appLE_comp_appLE _ _ _ (f ''ᵁ V) V.1
(Set.imag... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.AlgebraicGeometry.Morphisms.RingHomProperties | {
"line": 497,
"column": 4
} | {
"line": 497,
"column": 39
} | [
{
"pp": "P : MorphismProperty Scheme\nQ : {R S : Type u} → [inst : CommRing R] → [inst_1 : CommRing S] → (R →+* S) → Prop\ninst✝ : HasRingHomProperty P Q\nhP : RingHom.StableUnderComposition fun {R S} [CommRing R] [CommRing S] ↦ Q\nZ : Scheme\nhZ : IsAffine Z\nX Y : Scheme\nf : X ⟶ Y\ng : Y ⟶ Z\nhf : P f\nhg : ... | wlog hX : IsAffine X generalizing X | Mathlib.Tactic._aux_Mathlib_Tactic_WLOG___elabRules_Mathlib_Tactic_wlog_1 | Mathlib.Tactic.wlog |
Mathlib.AlgebraicGeometry.Morphisms.RingHomProperties | {
"line": 525,
"column": 4
} | {
"line": 525,
"column": 55
} | [
{
"pp": "case inr\nP : MorphismProperty Scheme\nQ : {R S : Type u} → [inst : CommRing R] → [inst_1 : CommRing S] → (R →+* S) → Prop\ninst✝ : HasRingHomProperty P Q\nH :\n ∀ {R S T : Type u} [inst : CommRing R] [inst_1 : CommRing S] [inst_2 : CommRing T] (f : R →+* S) (g : S →+* T),\n Q (g.comp f) → Q g\nZ :... | have H := comp_of_isOpenImmersion P U.1.ι (f ≫ g) h | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.AlgebraicGeometry.Morphisms.QuasiCompact | {
"line": 221,
"column": 44
} | {
"line": 221,
"column": 67
} | [
{
"pp": "case a\nX : Scheme\nU : X.Opens\nx✝ : ∃ R f, Set.range ⇑f = ↑U\nR : CommRingCat\nf : Spec R ⟶ X\nhf : Set.range ⇑f = ↑U\n⊢ Set.range ⇑(ConcreteCategory.hom ((IsOpenImmersion.lift U.ι f ⋯).base ≫ U.ι.base)) = ↑U",
"usedConstants": [
"subset_refl._simp_1",
"Eq.mpr",
"AlgebraicGeomet... | ← Scheme.Hom.comp_base, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.RingHom.Locally | {
"line": 204,
"column": 42
} | {
"line": 236,
"column": 92
} | [
{
"pp": "P : {R S : Type u} → [inst : CommRing R] → [inst_1 : CommRing S] → (R →+* S) → Prop\nhPi : RespectsIso fun {R S} [CommRing R] [CommRing S] ↦ P\nhPl : LocalizationPreserves fun {R S} [CommRing R] [CommRing S] ↦ P\nhPc : StableUnderComposition fun {R S} [CommRing R] [CommRing S] ↦ P\n⊢ StableUnderComposi... | by
classical
intro R S T _ _ _ f g hf hg
rw [locally_iff_finite] at hf hg
obtain ⟨sf, hsfone, hsf⟩ := hf
obtain ⟨sg, hsgone, hsg⟩ := hg
rw [locally_iff_exists hPi]
refine ⟨sf × sg, fun (a, b) ↦ g a * b, ?_,
fun (a, b) ↦ Localization.Away ((algebraMap T (Localization.Away b.val)) (g a.val)),
in... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.RingHom.Locally | {
"line": 256,
"column": 6
} | {
"line": 258,
"column": 92
} | [
{
"pp": "P : {R S : Type u} → [inst : CommRing R] → [inst_1 : CommRing S] → (R →+* S) → Prop\nhPa : StableUnderCompositionWithLocalizationAwayTarget fun {R S} [CommRing R] [CommRing S] ↦ P\nR S T : Type u\ninst✝⁴ : CommRing R\ninst✝³ : CommRing S\ninst✝² : CommRing T\ninst✝¹ : Algebra S T\nt : S\ninst✝ : IsLoca... | apply IsScalarTower.of_algebraMap_eq
intro x
simp [algebraMap_toAlgebra, IsLocalization.Away.map, ← IsScalarTower.algebraMap_apply] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.RingHom.Locally | {
"line": 256,
"column": 6
} | {
"line": 258,
"column": 92
} | [
{
"pp": "P : {R S : Type u} → [inst : CommRing R] → [inst_1 : CommRing S] → (R →+* S) → Prop\nhPa : StableUnderCompositionWithLocalizationAwayTarget fun {R S} [CommRing R] [CommRing S] ↦ P\nR S T : Type u\ninst✝⁴ : CommRing R\ninst✝³ : CommRing S\ninst✝² : CommRing T\ninst✝¹ : Algebra S T\nt : S\ninst✝ : IsLoca... | apply IsScalarTower.of_algebraMap_eq
intro x
simp [algebraMap_toAlgebra, IsLocalization.Away.map, ← IsScalarTower.algebraMap_apply] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.AlgebraicGeometry.Morphisms.QuasiCompact | {
"line": 308,
"column": 7
} | {
"line": 308,
"column": 16
} | [
{
"pp": "case h\nX : Scheme\nU : X.Opens\nhU : IsCompact U.carrier\nx f : ↑Γ(X, U)\nH : (x |_ X.basicOpen f) ⋯ = 0\ns : Set ↑X.affineOpens\nhs : s.Finite\ne : U = ⨆ i, ↑↑i\nh₁ : ∀ (i : ↑s), ↑↑i ≤ U\nn : ↑s → ℕ\nthis : Finite ↑s\nval✝ : Fintype ↑s\ni : ↑s\nhn :\n (ConcreteCategory.hom (X.presheaf.map (homOfLE ⋯... | mul_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Jacobson.Ring | {
"line": 76,
"column": 2
} | {
"line": 76,
"column": 29
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\nh : ∀ (P : Ideal R), P.IsPrime → P.jacobson = P\nI : Ideal R\nhI : I.IsRadical\nx : R\nhx : x ∈ I.jacobson\nP : Ideal R\nhP : P ∈ {J | I ≤ J ∧ J.IsPrime}\n⊢ x ∈ P",
"usedConstants": [
"Semiring.toModule",
"congrArg",
"CommSemiring.toSemiring",
... | rw [Set.mem_setOf_eq] at hP | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.LocalProperties.Reduced | {
"line": 40,
"column": 2
} | {
"line": 40,
"column": 50
} | [
{
"pp": "case eq_zero.succ\nR : Type u_1\nhR : CommRing R\nM : Submonoid R\nS : Type u_1\nhS : CommRing S\ninst✝¹ : Algebra R S\ninst✝ : IsLocalization M S\na✝ : IsReduced R\nx : S\nn : ℕ\ne : x ^ (n + 1) = 0\ny : R\nm : ↥M\nhx : x * (algebraMap R S) ↑m = (algebraMap R S) y\nhx' : (algebraMap R S) 0 = (algebraM... | simp only [mul_assoc, zero_mul, mul_zero] at hm' | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.RingTheory.Jacobson.Ring | {
"line": 291,
"column": 4
} | {
"line": 291,
"column": 85
} | [
{
"pp": "R : Type u_1\ninst✝⁶ : CommRing R\nRₘ : Type u_3\nSₘ : Type u_4\ninst✝⁵ : CommRing Rₘ\ninst✝⁴ : CommRing Sₘ\nP : Ideal R[X]\npX : R[X]\nhpX : pX ∈ P\ninst✝³ : Algebra (R ⧸ comap C P) Rₘ\ninst✝² : IsLocalization.Away (map (Ideal.Quotient.mk (comap C P)) pX).leadingCoeff Rₘ\ninst✝¹ : Algebra (R[X] ⧸ P) S... | refine (hp.symm ▸ this).of_mul_unit φ' p (algebraMap (R[X] ⧸ P) Sₘ (φ q')) q'' ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.RingTheory.Jacobson.Ring | {
"line": 498,
"column": 6
} | {
"line": 499,
"column": 69
} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommRing R\nP : Ideal R[X]\nhP : P.IsMaximal\ninst✝ : Nontrivial R\nhR : IsJacobsonRing R\nhP' : ∀ (x : R), C x ∈ P → x = 0\nP' : Ideal R := comap C P\npX : R[X]\nhpX : pX ∈ P\nhp0 : map (Ideal.Quotient.mk (comap C P)) pX ≠ 0\na : R ⧸ P' := (map (Ideal.Quotient.mk P') pX).leading... | refine RingHom.IsIntegral.trans (algebraMap (R ⧸ P') (Localization M))
(IsLocalization.map (Localization M') φ M.le_comap_map) ?_ ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.AlgebraicGeometry.Properties | {
"line": 185,
"column": 4
} | {
"line": 189,
"column": 13
} | [
{
"pp": "case h.h₂\nX✝ X Y : Scheme\nf : X ⟶ Y\ninst✝ : IsOpenImmersion f\nhX : IsReduced Y\ns : ↑Γ(Y, Scheme.Hom.opensRange f)\nhs : Y.basicOpen s = ⊥\nx : ↥X\nthis : IsReduced X\nH :\n (ConcreteCategory.hom (X.sheaf.presheaf.germ (f ⁻¹ᵁ Scheme.Hom.opensRange f) x ⋯))\n ((ConcreteCategory.hom (Scheme.Hom... | · have H : (X.presheaf.germ _ x _).hom _ = 0 := H
rw [← Scheme.Hom.germ_stalkMap_apply f ⟨_, _⟩ x] at H
apply_fun inv <| f.stalkMap x at H
rw [← CommRingCat.comp_apply, CategoryTheory.IsIso.hom_inv_id, map_zero] at H
exact H | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.AlgebraicGeometry.Properties | {
"line": 191,
"column": 8
} | {
"line": 191,
"column": 32
} | [
{
"pp": "case h.h₃\nX : Scheme\nR : CommRingCat\nhX : IsReduced (Spec R)\ns : ↑Γ(Spec R, ⊤)\nhs : (Spec R).basicOpen s = ⊥\nx : ↥(Spec R)\nhx : x ∈ ⊤\n⊢ (ConcreteCategory.hom ((Spec R).sheaf.presheaf.germ ⊤ x hx)) s = 0",
"usedConstants": [
"AlgebraicGeometry.Spec",
"AlgebraicGeometry.SheafedSpa... | basicOpen_eq_of_affine', | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.AlgebraicGeometry.Morphisms.QuasiSeparated | {
"line": 440,
"column": 2
} | {
"line": 440,
"column": 39
} | [
{
"pp": "X : Scheme\nU : Opens ↥X\nhU : IsCompact U.carrier\nhU' : IsQuasiSeparated U.carrier\nf s : ↑Γ(X, U)\nhf : (f |_ X.basicOpen s) ⋯ = 0\n⊢ ∃ n, s ^ n * f = s ^ n * 0",
"usedConstants": [
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"Opposite",
"CommRingCat.carrier",
"CommR... | apply exists_of_res_eq_of_qcqs hU hU' | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.AlgebraicGeometry.IdealSheaf.Subscheme | {
"line": 92,
"column": 43
} | {
"line": 92,
"column": 66
} | [
{
"pp": "X : Scheme\nI : X.IdealSheafData\nU : ↑X.affineOpens\n⊢ ⇑(ConcreteCategory.hom ((IsAffineOpen.isoSpec ⋯).inv.base ≫ (↑U).ι.base)) '' PrimeSpectrum.zeroLocus ↑(I.ideal U) =\n X.zeroLocus ↑(I.ideal U) ∩ ↑↑U",
"usedConstants": [
"Eq.mpr",
"AlgebraicGeometry.IsAffineOpen.isoSpec",
... | ← Scheme.Hom.comp_base, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.AlgebraicGeometry.Morphisms.Affine | {
"line": 66,
"column": 28
} | {
"line": 66,
"column": 46
} | [
{
"pp": "case isAffine_preimage\nX Y Z : Scheme\nf : X ⟶ Y\ng : Y ⟶ Z\ninst✝¹ : IsAffineHom f\ninst✝ : IsAffineHom g\nU : Z.Opens\nhU : IsAffineOpen U\n⊢ IsAffineOpen ((Opens.map (f.base ≫ g.base)).obj U)",
"usedConstants": [
"Eq.mpr",
"AlgebraicGeometry.SheafedSpace.instTopologicalSpaceCarrierC... | Opens.map_comp_obj | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.AlgebraicGeometry.IdealSheaf.Subscheme | {
"line": 388,
"column": 2
} | {
"line": 399,
"column": 18
} | [
{
"pp": "X : Scheme\nI : X.IdealSheafData\n⊢ Set.range ⇑I.gluedTo = ↑I.support",
"usedConstants": [
"AlgebraicGeometry.Scheme.GlueData.ι",
"Iff.mpr",
"Eq.mpr",
"AlgebraicGeometry.Scheme.IdealSheafData.support",
"AlgebraicGeometry.Scheme.Hom.toLRSHom",
"AlgebraicGeometry.S... | refine subset_antisymm (Set.range_subset_iff.mpr fun x ↦ ?_) ?_
· obtain ⟨ix, x : I.glueDataObj ix, rfl⟩ :=
I.glueData.toGlueData.ι_jointly_surjective forget x
change (I.glueData.ι _ ≫ I.gluedTo) x ∈ I.support
rw [ι_gluedTo]
exact ((I.range_glueDataObjι_ι_eq_support_inter ix).le ⟨_, rfl⟩).1
· intr... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.AlgebraicGeometry.IdealSheaf.Subscheme | {
"line": 388,
"column": 2
} | {
"line": 399,
"column": 18
} | [
{
"pp": "X : Scheme\nI : X.IdealSheafData\n⊢ Set.range ⇑I.gluedTo = ↑I.support",
"usedConstants": [
"AlgebraicGeometry.Scheme.GlueData.ι",
"Iff.mpr",
"Eq.mpr",
"AlgebraicGeometry.Scheme.IdealSheafData.support",
"AlgebraicGeometry.Scheme.Hom.toLRSHom",
"AlgebraicGeometry.S... | refine subset_antisymm (Set.range_subset_iff.mpr fun x ↦ ?_) ?_
· obtain ⟨ix, x : I.glueDataObj ix, rfl⟩ :=
I.glueData.toGlueData.ι_jointly_surjective forget x
change (I.glueData.ι _ ≫ I.gluedTo) x ∈ I.support
rw [ι_gluedTo]
exact ((I.range_glueDataObjι_ι_eq_support_inter ix).le ⟨_, rfl⟩).1
· intr... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.AlgebraicGeometry.IdealSheaf.Subscheme | {
"line": 407,
"column": 43
} | {
"line": 407,
"column": 66
} | [
{
"pp": "case a\nX : Scheme\nI : X.IdealSheafData\nU : ↑X.affineOpens\n⊢ Set.range ⇑(ConcreteCategory.hom ((I.glueData.ι U).base ≫ I.gluedTo.base)) = ⇑I.gluedTo '' (⇑I.gluedTo ⁻¹' ↑↑U)",
"usedConstants": [
"AlgebraicGeometry.Scheme.GlueData.ι",
"Eq.mpr",
"AlgebraicGeometry.Scheme",
"... | ← Scheme.Hom.comp_base, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.AlgebraicGeometry.IdealSheaf.Basic | {
"line": 585,
"column": 8
} | {
"line": 585,
"column": 26
} | [
{
"pp": "case a\nX : Scheme\nI : X.IdealSheafData\nZ : Closeds ↥X\nU : ↑X.affineOpens\nf : ↑Γ(X, ↑U)\nF : Γ(X, ↑U) ⟶ Γ(X, X.basicOpen f) := X.presheaf.map (homOfLE ⋯).op\nthis✝ : Algebra ↑Γ(X, ↑U) ↑Γ(X, ↑(X.affineBasicOpen f)) := (CommRingCat.Hom.hom F).toAlgebra\nthis : IsLocalization.Away f ↑Γ(X, X.basicOpen ... | dsimp only at hx ⊢ | Lean.Elab.Tactic.evalDSimp | Lean.Parser.Tactic.dsimp |
Mathlib.AlgebraicGeometry.IdealSheaf.Subscheme | {
"line": 720,
"column": 25
} | {
"line": 720,
"column": 48
} | [
{
"pp": "X Y : Scheme\nf : X ⟶ Y\nU : ↑Y.affineOpens\ninst✝ : QuasiCompact f\n⊢ closure (Set.range ⇑f) ⊆ closure (Set.range ⇑(ConcreteCategory.hom ((Hom.toImage f).base ≫ (Hom.imageι f).base)))",
"usedConstants": [
"Eq.mpr",
"AlgebraicGeometry.SheafedSpace.instTopologicalSpaceCarrierCarrier",
... | ← Scheme.Hom.comp_base, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.AlgebraicGeometry.Morphisms.AffineAnd | {
"line": 85,
"column": 4
} | {
"line": 91,
"column": 31
} | [
{
"pp": "Q : {R S : Type u} → [inst : CommRing R] → [inst_1 : CommRing S] → (R →+* S) → Prop\nhPi : RingHom.RespectsIso fun {R S} [CommRing R] [CommRing S] ↦ Q\nhQl : RingHom.LocalizationAwayPreserves fun {R S} [CommRing R] [CommRing S] ↦ Q\nhQs : RingHom.OfLocalizationSpan fun {R S} [CommRing R] [CommRing S] ↦... | haveI : IsAffine X := by
apply isAffine_of_isAffineOpen_basicOpen (f.appTop '' s)
· apply_fun Ideal.map (f.appTop).hom at hs
rwa [Ideal.map_span, Ideal.map_top] at hs
· rintro - ⟨r, hr, rfl⟩
simp_rw [Scheme.preimage_basicOpen] at hf
exact (hf ⟨r, hr⟩).left | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHaveI___1 | Lean.Parser.Tactic.tacticHaveI__ |
Mathlib.AlgebraicGeometry.IdealSheaf.Basic | {
"line": 711,
"column": 28
} | {
"line": 711,
"column": 47
} | [
{
"pp": "case refine_1.a\nX Y : Scheme\nf : X.Hom Y\ninst✝ : QuasiCompact f\nU✝ U : ↑Y.affineOpens\ns x : ↑Γ(Y, ↑U)\nhx : x ∈ (fun U ↦ RingHom.ker (CommRingCat.Hom.hom (f.app ↑U))) U\n⊢ x ∈\n RingHom.ker\n ((CommRingCat.Hom.hom (X.presheaf.map ((Opens.map f.base).map (homOfLE ⋯).op.unop).op)).comp\n ... | ← RingHom.comap_ker | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.AlgebraicGeometry.IdealSheaf.Basic | {
"line": 735,
"column": 55
} | {
"line": 735,
"column": 74
} | [
{
"pp": "X Y Z : Scheme\nf : X ⟶ Y\ng : Y.Hom Z\nU : ↑Z.affineOpens\n⊢ RingHom.ker (CommRingCat.Hom.hom (g.app ↑U)) ≤\n RingHom.ker ((CommRingCat.Hom.hom (app f (g ⁻¹ᵁ ↑U))).comp (CommRingCat.Hom.hom (g.app ↑U)))",
"usedConstants": [
"Eq.mpr",
"RingHom.instRingHomClass",
"AlgebraicGeome... | ← RingHom.comap_ker | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.AlgebraicGeometry.Morphisms.AffineAnd | {
"line": 240,
"column": 2
} | {
"line": 241,
"column": 7
} | [
{
"pp": "case refine_1\nQ : {R S : Type u} → [inst : CommRing R] → [inst_1 : CommRing S] → (R →+* S) → Prop\nP : MorphismProperty Scheme\nhQi : RingHom.RespectsIso fun {R S} [CommRing R] [CommRing S] ↦ Q\nhQl : RingHom.LocalizationAwayPreserves fun {R S} [CommRing R] [CommRing S] ↦ Q\nhQs : RingHom.OfLocalizati... | · rw [eq_targetAffineLocally P, targetAffineLocally_affineAnd_iff hQi]
lia | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.AlgebraicGeometry.IdealSheaf.Basic | {
"line": 802,
"column": 62
} | {
"line": 814,
"column": 38
} | [
{
"pp": "X Y : Scheme\nf : X.Hom Y\ninst✝¹ : QuasiCompact f\n𝒰 : X.OpenCover\ninst✝ : Finite 𝒰.I₀\n⊢ ⋃ i, ↑(ker (𝒰.f i ≫ f)).support = ↑f.ker.support",
"usedConstants": [
"Set.ext",
"Eq.mpr",
"SetLike.mem_coe._simp_1",
"AlgebraicGeometry.Scheme.IdealSheafData.support",
"Alge... | by
cases isEmpty_or_nonempty 𝒰.I₀
· have : IsEmpty X := Function.isEmpty 𝒰.idx
simp [ker_eq_top_of_isEmpty]
suffices ∀ U : Y.affineOpens,
(⋃ i, (𝒰.f i ≫ f).ker.support) ∩ U = (f.ker.support ∩ U : Set Y) by
ext x
obtain ⟨_, ⟨U, hU, rfl⟩, hxU, -⟩ :=
Y.isBasis_affineOpens.exists_subset_of_... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.AlgebraicGeometry.IdealSheaf.Basic | {
"line": 857,
"column": 49
} | {
"line": 857,
"column": 72
} | [
{
"pp": "case a.inr.refine_2\nX Y : Scheme\nf : X ⟶ Y\ninst✝ : QuasiCompact f\nthis✝¹ : ∀ {X Y : Scheme} (f : X ⟶ Y) [QuasiCompact f], (∃ S, Y = Spec S) → ↑(ker f).support ⊆ closure (Set.range ⇑f)\nhY : ¬∃ S, Y = Spec S\n𝒰 : Y.OpenCover := Y.affineCover\ni : 𝒰.I₀\nx : ↥(𝒰.X i)\nhx : (𝒰.f i) x ∈ ↑(ker f).sup... | ← Scheme.Hom.comp_base, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.RingHom.Finite | {
"line": 73,
"column": 2
} | {
"line": 74,
"column": 67
} | [
{
"pp": "R S : Type u_5\ninst✝⁷ : CommRing R\ninst✝⁶ : CommRing S\nf : R →+* S\nM : Submonoid R\nR' S' : Type u_5\ninst✝⁵ : CommRing R'\ninst✝⁴ : CommRing S'\ninst✝³ : Algebra R R'\ninst✝² : Algebra S S'\ninst✝¹ : IsLocalization M R'\ninst✝ : IsLocalization (Submonoid.map f M) S'\nhf : f.Finite\nthis✝³ : Algebr... | have : IsLocalization (Algebra.algebraMapSubmonoid S M) S' := by
rwa [Algebra.algebraMapSubmonoid, RingHom.algebraMap_toAlgebra] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.RingTheory.Finiteness.FiniteTypeLocal | {
"line": 48,
"column": 2
} | {
"line": 48,
"column": 49
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝⁷ : CommRing R\ninst✝⁶ : CommRing S\ninst✝⁵ : Algebra R S\nS' : Type u_4\ninst✝⁴ : CommRing S'\ninst✝³ : Algebra S S'\ninst✝² : Algebra R S'\ninst✝¹ : IsScalarTower R S S'\nM : Submonoid S\ninst✝ : IsLocalization M S'\nx : S\ns : Finset S'\nA : Subalgebra R S\nhA₁ : ↑(f... | let y := IsLocalization.commonDenomOfFinset M s | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLet___1 | Lean.Parser.Tactic.tacticLet__ |
Mathlib.RingTheory.RingHom.Finite | {
"line": 125,
"column": 2
} | {
"line": 125,
"column": 39
} | [
{
"pp": "R S : Type u_5\ninst✝¹ : CommRing R\ninst✝ : CommRing S\nf : R →+* S\ns : Finset R\nhs : Ideal.span ↑s = ⊤\nthis✝² : Algebra R S := f.toAlgebra\nthis✝¹ : (r : ↥s) → Algebra (Localization.Away ↑r) (Localization.Away (f ↑r)) :=\n fun r ↦ (Localization.awayMap f ↑r).toAlgebra\nthis✝ : ∀ (r : ↥s), IsLocal... | simp_rw [Submonoid.map_powers] at hn₂ | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | Mathlib.Tactic.tacticSimp_rw___ |
Mathlib.RingTheory.RingHom.Finite | {
"line": 127,
"column": 2
} | {
"line": 127,
"column": 68
} | [
{
"pp": "case h\nR S : Type u_5\ninst✝¹ : CommRing R\ninst✝ : CommRing S\nf : R →+* S\ns : Finset R\nhs : Ideal.span ↑s = ⊤\nthis✝² : Algebra R S := ⋯\nthis✝¹ : (r : ↥s) → Algebra (Localization.Away ↑r) (Localization.Away (f ↑r)) := ⋯\nthis✝ : ∀ (r : ↥s), IsLocalization (Submonoid.map (algebraMap R S) (Submonoi... | exact le_iSup (fun x : s => Submodule.span R (sf x : Set S)) r hn₂ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.RingTheory.Finiteness.FiniteTypeLocal | {
"line": 92,
"column": 2
} | {
"line": 92,
"column": 21
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\ns✝ : Set S\nhs✝ : Ideal.span s✝ = ⊤\ns : Finset S\nh₁ : ↑s ⊆ s✝\nhs : Ideal.span ↑s = ⊤\nh : ∀ (r : ↥s), ⊤.FG\n⊢ FiniteType R S",
"usedConstants": [
"Lattice.toSemilatticeSup",
"CompleteLattice.to... | choose t ht using h | Mathlib.Tactic.Choose._aux_Mathlib_Tactic_Choose___elabRules_Mathlib_Tactic_Choose_choose_1 | Mathlib.Tactic.Choose.choose |
Mathlib.RingTheory.Finiteness.FiniteTypeLocal | {
"line": 162,
"column": 2
} | {
"line": 162,
"column": 39
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\ns✝ : Set R\nhs✝ : Ideal.span s✝ = ⊤\ns : Finset R\nh₁ : ↑s ⊆ s✝\nhs : Ideal.span ↑s = ⊤\nh : ∀ (i : ↥s), FiniteType (Localization.Away ↑i) (Localization.Away ↑i ⊗[R] S)\nf : R →+* S := algebraMap R S\nthis : (r :... | simp_rw [Submonoid.map_powers] at hn₂ | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | Mathlib.Tactic.tacticSimp_rw___ |
Mathlib.AlgebraicGeometry.Morphisms.Integral | {
"line": 150,
"column": 2
} | {
"line": 153,
"column": 80
} | [
{
"pp": "case h\nR S : CommRingCat\nφ : R ⟶ S\nH₁ : UniversallyClosed (Spec.map φ)\nH₂ : IsAffineHom (Spec.map φ)\nalgInst✝¹ : Algebra ↑R ↑S := φ.hom'.toAlgebra\nalgInst✝ : Algebra (Polynomial ↑R) (Polynomial ↑S) := (Polynomial.mapRingHom φ.hom').toAlgebra\n⊢ IsClosedMap (PrimeSpectrum.comap (Polynomial.mapRing... | exact H₁.universally_isClosedMap (Spec.map (CommRingCat.ofHom Polynomial.C))
(Spec.map (CommRingCat.ofHom Polynomial.C)) (Spec.map _)
(isPullback_SpecMap_of_isPushout _ _ _ _
(CommRingCat.isPushout_of_isPushout R S (Polynomial R) (Polynomial S))).flip | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.AlgebraicGeometry.Morphisms.ClosedImmersion | {
"line": 158,
"column": 5
} | {
"line": 158,
"column": 54
} | [
{
"pp": "X Y : Scheme\nf : X ⟶ Y\ninst✝ : IsClosedImmersion f\n⊢ IsClosedImmersion (Scheme.Hom.toImage f ≫ Scheme.Hom.imageι f)",
"usedConstants": [
"Eq.mpr",
"AlgebraicGeometry.Scheme",
"AlgebraicGeometry.Scheme.Hom.image",
"AlgebraicGeometry.Scheme.Hom.imageι",
"CategoryTheor... | by rw [Scheme.Hom.toImage_imageι]; infer_instance | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.AlgebraicGeometry.Morphisms.ClosedImmersion | {
"line": 275,
"column": 63
} | {
"line": 275,
"column": 81
} | [
{
"pp": "X Y : Scheme\ninst✝¹ : IsAffine Y\nf : X ⟶ Y\ninst✝ : CompactSpace ↥X\nhfopen : IsOpenMap ⇑f\nhfinj₁ : Function.Injective ⇑f\nhfinj₂ : Function.Injective ⇑(ConcreteCategory.hom (Scheme.Hom.appTop f))\nx : ↥X\nφ : Γ(Y, ⊤) ⟶ Γ(X, ⊤) := Scheme.Hom.appTop f\n𝒰 : X.OpenCover := X.affineCover.finiteSubcover... | Opens.map_comp_obj | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.AlgebraicGeometry.Morphisms.ClosedImmersion | {
"line": 376,
"column": 22
} | {
"line": 376,
"column": 33
} | [
{
"pp": "case hP'.H\nX✝ Y✝ Z : Scheme\nthis :\n AffineTargetMorphismProperty.IsLocal fun X x f [IsAffine x] ↦\n IsAffine X ∧ Function.Surjective ⇑(ConcreteCategory.hom (Scheme.Hom.appTop f))\nX Y S : Scheme\ninst✝¹ : IsAffine S\ninst✝ : IsAffine X\nf : X ⟶ S\ng : Y ⟶ S\n⊢ IsAffine Y ∧ Function.Surjective ⇑(... | ⟨ha, hsurj⟩ | Lean.Elab.Tactic.evalIntro | Lean.Parser.Term.anonymousCtor |
Mathlib.AlgebraicGeometry.Morphisms.FinitePresentation | {
"line": 60,
"column": 8
} | {
"line": 60,
"column": 40
} | [
{
"pp": "case h\nX✝ Y✝ : Scheme\nf✝ : X✝ ⟶ Y✝\nX Y : Scheme\nf : X ⟶ Y\n⊢ LocallyOfFinitePresentation f ↔ affineLocally (fun {R S} [CommRing R] [CommRing S] ↦ RingHom.FinitePresentation) f",
"usedConstants": [
"Eq.mpr",
"RingHom.FinitePresentation",
"AlgebraicGeometry.SheafedSpace.instTopo... | locallyOfFinitePresentation_iff, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.AlgebraicGeometry.Morphisms.Immersion | {
"line": 97,
"column": 43
} | {
"line": 97,
"column": 66
} | [
{
"pp": "case h.e'_3.a\nX 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 ⇑(ConcreteCategory.hom ((Scheme.Hom.liftCoborder f).base ≫ (Scheme.Hom.coborderRange f).ι.... | ← Scheme.Hom.comp_base, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.AlgebraicGeometry.Morphisms.Immersion | {
"line": 252,
"column": 6
} | {
"line": 252,
"column": 64
} | [
{
"pp": "case h.h.hf\nX Y : Scheme\nf : X ⟶ Y\ninst✝¹ : IsImmersion f\ninst✝ : QuasiCompact f\nU : ↑(↑(Hom.coborderRange f)).affineOpens\n⊢ Function.Injective ⇑(CommRingCat.Hom.hom (X.presheaf.map (eqToHom ⋯).op))",
"usedConstants": [
"AlgebraicGeometry.Scheme.Hom.opensFunctor",
"Eq.mpr",
... | ← ConcreteCategory.mono_iff_injective_of_preservesPullback | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.AlgebraicGeometry.QuasiAffine | {
"line": 119,
"column": 2
} | {
"line": 129,
"column": 38
} | [
{
"pp": "X Y : Scheme\nf : X ⟶ Y\ninst✝¹ : IsAffineHom f\ninst✝ : Y.IsQuasiAffine\nthis✝ : CompactSpace ↥X\nthis : X.IsQuasiAffine\n⊢ IsPullback f X.toSpecΓ Y.toSpecΓ (Spec.map (Hom.appTop f))",
"usedConstants": [
"Eq.mpr",
"AlgebraicGeometry.SheafedSpace.instTopologicalSpaceCarrierCarrier",
... | have (r : Γ(Y, ⊤)) :
IsPushout f.appTop (Y.presheaf.map (homOfLE le_top).op)
(X.presheaf.map (homOfLE le_top).op) (f.appLE (Y.basicOpen r)
(X.basicOpen (f.appTop r)) (Scheme.preimage_basicOpen_top ..).ge) := by
have := isLocalization_basicOpen_of_qcqs isCompact_univ isQuasiSeparated_univ r
... | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.CategoryTheory.Limits.Shapes.Pullback.Connected | {
"line": 46,
"column": 8
} | {
"line": 46,
"column": 16
} | [
{
"pp": "I : Type u_1\nC : Type u_2\ninst✝² : Category.{v_1, u_1} I\ninst✝¹ : IsConnected I\ninst✝ : Category.{v_2, u_2} C\nF G : I ⥤ C\nα : F ⟶ G\ncF : Cone F\ncG : Cone G\nf✝ : (Cone.postcompose α).obj cF ⟶ cG\nhf : ∀ (i : I), IsPullback (cF.π.app i) f✝.hom (α.app i) (cG.π.app i)\nhcG : IsLimit cG\ns : Cone F... | this _ j | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.Limits.Shapes.Pullback.Connected | {
"line": 74,
"column": 8
} | {
"line": 74,
"column": 16
} | [
{
"pp": "I : Type u_1\nC : Type u_2\ninst✝² : Category.{v_1, u_1} I\ninst✝¹ : IsConnected I\ninst✝ : Category.{v_2, u_2} C\nF G : I ⥤ C\nα : F ⟶ G\ncF : Cocone F\ncG : Cocone G\nf✝ : cF ⟶ (Cocone.precompose α).obj cG\nhf : ∀ (i : I), IsPushout (cF.ι.app i) (α.app i) f✝.hom (cG.ι.app i)\nhcF : IsColimit cF\ns : ... | this _ j | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.AlgebraicGeometry.Noetherian | {
"line": 148,
"column": 4
} | {
"line": 148,
"column": 41
} | [
{
"pp": "case mpr.hS'.f.e\nX : Scheme\n𝒰 : X.OpenCover\ninst✝ : ∀ (i : 𝒰.I₀), IsAffine (𝒰.X i)\nhCNoeth : ∀ (i : 𝒰.I₀), IsNoetherianRing ↑Γ(𝒰.X i, ⊤)\nfS : 𝒰.I₀ → ↑X.affineOpens := ⋯\ni : 𝒰.I₀\n⊢ Γ(𝒰.X i, ⊤) ≅ Γ(X, ↑(fS i))",
"usedConstants": [
"AlgebraicGeometry.Scheme",
"CategoryTheory... | exact IsOpenImmersion.ΓIsoTop (𝒰.f i) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.AlgebraicGeometry.Noetherian | {
"line": 305,
"column": 67
} | {
"line": 313,
"column": 43
} | [
{
"pp": "X : Scheme\n𝒰 : X.OpenCover\ninst✝¹ : Finite 𝒰.I₀\ninst✝ : ∀ (i : 𝒰.I₀), IsAffine (𝒰.X i)\n⊢ IsNoetherian X ↔ ∀ (i : 𝒰.I₀), IsNoetherianRing ↑Γ(𝒰.X i, ⊤)",
"usedConstants": [
"Iff.mpr",
"Eq.mpr",
"AlgebraicGeometry.IsNoetherian.toIsLocallyNoetherian",
"AlgebraicGeometr... | by
constructor
· intro h i
apply (isLocallyNoetherian_iff_of_affine_openCover _).mp
exact h.toIsLocallyNoetherian
· intro hNoeth
convert IsNoetherian.mk
· exact (isLocallyNoetherian_iff_of_affine_openCover _).mpr hNoeth
· exact Scheme.OpenCover.compactSpace 𝒰 | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.AlgebraicGeometry.AffineSpace | {
"line": 184,
"column": 8
} | {
"line": 184,
"column": 29
} | [
{
"pp": "n : Type v\nS X : Scheme\ninst✝¹ : X.Over S\ninst✝ : IsAffine S\n⊢ homOfVector (Spec.map (CommRingCat.ofHom C) ≫ S.isoSpec.inv)\n (⇑(ConcreteCategory.hom (Scheme.ΓSpecIso (CommRingCat.of (MvPolynomial n ↑Γ(S, ⊤)))).inv) ∘ MvPolynomial.X) ≫\n 𝔸(n; S).toSpecΓ ≫\n Spec.map (CommRingCat... | apply ext_of_isAffine | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.AlgebraicGeometry.AffineSpace | {
"line": 282,
"column": 42
} | {
"line": 282,
"column": 59
} | [
{
"pp": "case a.h\nn : Type v\nS T : Scheme\nf : S ⟶ T\ni : n\n⊢ (ConcreteCategory.hom (Scheme.Hom.appTop (map n f))) (coord T i) = (toSpecMvPolyIntEquiv n) (toSpecMvPoly n S) i",
"usedConstants": [
"Eq.mpr",
"AlgebraicGeometry.Spec",
"Nat.instMulZeroClass",
"AlgebraicGeometry.Sheafe... | map_appTop_coord, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.AlgebraicGeometry.RelativeGluing | {
"line": 46,
"column": 6
} | {
"line": 47,
"column": 85
} | [
{
"pp": "case h.refine_2.a\nJ : Type u_1\ninst✝² : Category.{u_2, u_1} J\nF G : J ⥤ Scheme\ns : F ⟶ G\ninst✝¹ : Quiver.IsThin J\nhs : NatTrans.Equifibered s\nH : ∀ {i j : J} (hij : i ⟶ j), Function.Injective ⇑(F.map hij)\ninst✝ : (G ⋙ forget).IsLocallyDirected\ni j k : J\nfi : i ⟶ k\nfj : j ⟶ k\nxi : (F ⋙ forge... | simp only [Functor.comp_map, Scheme.forget_map, ← Scheme.Hom.comp_apply,
Category.assoc, ← Functor.map_comp, show flj ≫ fj = fli ≫ fi by subsingleton] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.RingTheory.Spectrum.Prime.ChevalleyComplexity | {
"line": 239,
"column": 6
} | {
"line": 239,
"column": 36
} | [
{
"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 ... | refine ⟨i, hi', fun j hj ↦ ?_⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.RingTheory.Spectrum.Prime.ChevalleyComplexity | {
"line": 336,
"column": 2
} | {
"line": 336,
"column": 69
} | [
{
"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 ... | rw [coeffSubmodule_mapRingHom_comp, ← Submodule.map_pow] at hT₂span | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Topology.Sets.CompactOpenCovered | {
"line": 119,
"column": 41
} | {
"line": 120,
"column": 90
} | [
{
"pp": "S : Type u_1\nι : Type u_2\nX : ι → Type u_3\nf : (i : ι) → X i → S\ninst✝ : (i : ι) → TopologicalSpace (X i)\ni : ι\nV : Opens (X i)\nhV : IsCompact ↑V\n⊢ IsCompactOpenCovered f (f i '' ↑V)",
"usedConstants": [
"Iff.mpr",
"Iff.of_eq",
"congrArg",
"TopologicalSpace.Opens",
... | by
refine ⟨{i}, Set.finite_singleton i, fun j hj ↦ hj ▸ V, by rintro i rfl; simpa, by simp⟩ | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.AlgebraicGeometry.Cover.QuasiCompact | {
"line": 103,
"column": 26
} | {
"line": 103,
"column": 62
} | [
{
"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 le_trans hle inf_le_left | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.AlgebraicGeometry.Cover.QuasiCompact | {
"line": 103,
"column": 26
} | {
"line": 103,
"column": 62
} | [
{
"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 le_trans hle inf_le_left | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.AlgebraicGeometry.Cover.QuasiCompact | {
"line": 103,
"column": 26
} | {
"line": 103,
"column": 62
} | [
{
"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 le_trans hle inf_le_left | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.AlgebraicGeometry.Morphisms.Flat | {
"line": 126,
"column": 48
} | {
"line": 126,
"column": 71
} | [
{
"pp": "case inr.a\nX Y : Scheme\nf : X ⟶ Y\ninst✝² : Flat f\ninst✝¹ : QuasiCompact f\ninst✝ : Surjective f\ns : Set ↥Y\nhs : IsOpen (⇑f ⁻¹' s)\nthis :\n ∀ {X Y : Scheme} (f : X ⟶ Y) [Flat f] [QuasiCompact f] [Surjective f] (s : Set ↥Y),\n IsOpen (⇑f ⁻¹' s) → (∃ R, Y = Spec R) → IsOpen s\nhY : ¬∃ R, Y = Sp... | ← Scheme.Hom.comp_base, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.AlgebraicGeometry.Morphisms.Flat | {
"line": 286,
"column": 2
} | {
"line": 290,
"column": 60
} | [
{
"pp": "case intro\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_1\ninst✝ : Finite ι\nVX : ι → X.Opens\nhVU : iSup VX = U... | have hφ : Function.Injective φ := by
dsimp [φ]
refine .comp ?_ (Algebra.TensorProduct.piRight _ Γ(S, US) _ _).injective
exact .piMap fun i ↦ (hV _).comp <| CommRingCat.isPushout_tensorProduct _ _ _
|>.flip.isoPushout.commRingCatIsoToRingEquiv.injective | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.AlgebraicGeometry.EllipticCurve.Weierstrass | {
"line": 304,
"column": 78
} | {
"line": 306,
"column": 7
} | [
{
"pp": "R : Type u\ninst✝ : CommRing R\nW : WeierstrassCurve R\n⊢ W.twoTorsionPolynomial.discr = 16 * W.Δ",
"usedConstants": [
"Mathlib.Tactic.Ring.mul_pp_pf_overlap",
"WeierstrassCurve.b₄._proof_1",
"WeierstrassCurve.Δ",
"Mathlib.Tactic.Ring.pow_one",
"NegZeroClass.toNeg",
... | by
simp only [b₂, b₄, b₆, b₈, Δ, twoTorsionPolynomial, Cubic.discr]
ring1 | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.AlgebraicGeometry.EllipticCurve.Weierstrass | {
"line": 400,
"column": 44
} | {
"line": 401,
"column": 48
} | [
{
"pp": "R : Type u\ninst✝¹ : CommRing R\nW : WeierstrassCurve R\ninst✝ : W.IsElliptic\nh : W.c₄ = 0\n⊢ W.j = 0",
"usedConstants": [
"Eq.mpr",
"Nat.instMulZeroClass",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"CommRing.toNonUnitalCommRing",
"congrArg",
"CommSemirin... | by
rw [j_eq_zero_iff', h, zero_pow three_ne_zero] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.AlgebraicGeometry.AffineTransitionLimit | {
"line": 778,
"column": 2
} | {
"line": 778,
"column": 53
} | [
{
"pp": "I : Type u\ninst✝² : Category.{u, u} I\nD : I ⥤ Scheme\nc : Cone D\nhc : IsLimit c\ninst✝¹ : IsCofiltered I\ninst✝ : ∀ {i j : I} (f : i ⟶ j), IsAffineHom (D.map f)\ni : I\nU : (D.obj i).Opens\nhU : IsCompact ↑U\ns : ↑Γ(D.obj i, U)\nhs : (ConcreteCategory.hom (Scheme.Hom.app (c.π.app i) U)) s = 0\nthis ... | have H : (D.map (𝟙 _) ⁻¹ᵁ U).ι ''ᵁ ⊤ ≤ U := by simp | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.AlgebraicGeometry.EllipticCurve.Affine.Basic | {
"line": 157,
"column": 6
} | {
"line": 157,
"column": 20
} | [
{
"pp": "R : Type r\ninst✝ : CommRing R\nW : Affine R\nx y : R\n⊢ W.Equation x y ↔ y ^ 2 + W.a₁ * x * y + W.a₃ * y = x ^ 3 + W.a₂ * x ^ 2 + W.a₄ * x + W.a₆",
"usedConstants": [
"Eq.mpr",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"HMul.hMul",
"CommRing.toNonUnitalCommRing",
... | equation_iff', | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.AlgebraicGeometry.EllipticCurve.Affine.Basic | {
"line": 165,
"column": 6
} | {
"line": 165,
"column": 20
} | [
{
"pp": "R : Type r\ninst✝ : CommRing R\nW : Affine R\nx y : R\n⊢ W.Equation x y ↔ (toAffine ({ u := 1, r := x, s := 0, t := y } • W)).Equation 0 0",
"usedConstants": [
"Eq.mpr",
"instHSMul",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"HMul.hMul",
"CommRing.toNonUnitalCom... | equation_iff', | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.AlgebraicGeometry.EllipticCurve.Affine.Basic | {
"line": 215,
"column": 19
} | {
"line": 215,
"column": 33
} | [
{
"pp": "R : Type r\ninst✝ : CommRing R\nW : Affine R\nx y : R\n⊢ W.Equation x y ∧ (evalEval x y W.polynomialX ≠ 0 ∨ evalEval x y W.polynomialY ≠ 0) ↔\n W.Equation x y ∧ (W.a₁ * y - (3 * x ^ 2 + 2 * W.a₂ * x + W.a₄) ≠ 0 ∨ 2 * y + W.a₁ * x + W.a₃ ≠ 0)",
"usedConstants": [
"Eq.mpr",
"NonUnitalC... | equation_iff', | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.AlgebraicGeometry.AffineTransitionLimit | {
"line": 879,
"column": 2
} | {
"line": 881,
"column": 97
} | [
{
"pp": "I : Type u\ninst✝⁴ : Category.{u, u} I\nD : I ⥤ Scheme\nc : Cone D\nhc : IsLimit c\ninst✝³ : IsCofiltered I\ninst✝² : ∀ {i j : I} (f : i ⟶ j), IsAffineHom (D.map f)\ns : ↑Γ(c.pt, ⊤)\ninst✝¹ : ∀ (i : I), CompactSpace ↥(D.obj i)\ninst✝ : ∀ (i : I), QuasiSeparatedSpace ↥(D.obj i)\nthis : CompactSpace ↥c.p... | obtain ⟨k, fk, hk⟩ := IsCofiltered.inf_exists S
(σ.attach.image₂ (fun (x y : σ) ↦ ⟨j x.1 y.1, i x.1, hjS x.2 y.2, hiS x.2, fjx x y⟩) σ.attach ∪
σ.attach.image₂ (fun (x y : σ) ↦ ⟨j x.1 y.1, i y.1, hjS x.2 y.2, hiS y.2, fjy x y⟩) σ.attach) | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.LinearAlgebra.FreeModule.Norm | {
"line": 65,
"column": 83
} | {
"line": 72,
"column": 70
} | [
{
"pp": "S : Type u_2\nι : Type u_3\ninst✝⁶ : CommRing S\ninst✝⁵ : IsDomain S\nF : Type u_4\ninst✝⁴ : Field F\ninst✝³ : Algebra F[X] S\ninst✝² : Finite ι\ninst✝¹ : Algebra F S\ninst✝ : IsScalarTower F F[X] S\nb : Basis ι F[X] S\nf : S\nhf : f ≠ 0\n⊢ finrank F (S ⧸ span {f}) = ((Algebra.norm F[X]) f).natDegree",... | by
haveI := Fintype.ofFinite ι
have h := span_singleton_eq_bot.not.2 hf
rw [natDegree_eq_of_degree_eq
(degree_eq_degree_of_associated <| associated_norm_prod_smith b hf)]
rw [natDegree_prod _ _ fun i _ => smithCoeffs_ne_zero b _ h i, finrank_quotient_eq_sum F h b]
congr with i
exact (AdjoinRoot.powerB... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.ClassGroup | {
"line": 338,
"column": 72
} | {
"line": 341,
"column": 59
} | [
{
"pp": "R : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : IsDedekindDomain R\n⊢ Function.Surjective ⇑mk0",
"usedConstants": [
"ClassGroup.mk0_integralRep",
"Units.val",
"Eq.mpr",
"MonoidHom.range",
"FractionRing.field",
"ClassGroup.Quot_mk_eq_mk",
"Fr... | by
rintro ⟨I⟩
refine ⟨⟨ClassGroup.integralRep I.1, ClassGroup.integralRep_mem_nonZeroDivisors I.ne_zero⟩, ?_⟩
rw [ClassGroup.mk0_integralRep, ClassGroup.Quot_mk_eq_mk] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.NumberTheory.EllipticDivisibilitySequence | {
"line": 216,
"column": 13
} | {
"line": 216,
"column": 27
} | [
{
"pp": "case nat.succ.succ.succ\nR : Type u\ninst✝ : CommRing R\nb c d : R\nm : ℕ\n⊢ preNormEDS b c d (2 * ↑(m + 1 + 1 + 1)) =\n preNormEDS b c d (↑(m + 1 + 1 + 1) - 1) ^ 2 * preNormEDS b c d ↑(m + 1 + 1 + 1) *\n preNormEDS b c d (↑(m + 1 + 1 + 1) + 2) -\n preNormEDS b c d (↑(m + 1 + 1 + 1) - 2)... | Nat.cast_succ, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.NumberTheory.EllipticDivisibilitySequence | {
"line": 221,
"column": 4
} | {
"line": 222,
"column": 9
} | [
{
"pp": "case neg\nR : Type u\ninst✝ : CommRing R\nb c d : R\nih :\n ∀ (n : ℕ),\n preNormEDS b c d (2 * ↑n) =\n preNormEDS b c d (↑n - 1) ^ 2 * preNormEDS b c d ↑n * preNormEDS b c d (↑n + 2) -\n preNormEDS b c d (↑n - 2) * preNormEDS b c d ↑n * preNormEDS b c d (↑n + 1) ^ 2\nm : ℕ\n⊢ preNormEDS... | simp_rw [mul_neg, ← sub_neg_eq_add, ← neg_sub', ← neg_add', preNormEDS_neg, ih]
ring1 | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.EllipticDivisibilitySequence | {
"line": 221,
"column": 4
} | {
"line": 222,
"column": 9
} | [
{
"pp": "case neg\nR : Type u\ninst✝ : CommRing R\nb c d : R\nih :\n ∀ (n : ℕ),\n preNormEDS b c d (2 * ↑n) =\n preNormEDS b c d (↑n - 1) ^ 2 * preNormEDS b c d ↑n * preNormEDS b c d (↑n + 2) -\n preNormEDS b c d (↑n - 2) * preNormEDS b c d ↑n * preNormEDS b c d (↑n + 1) ^ 2\nm : ℕ\n⊢ preNormEDS... | simp_rw [mul_neg, ← sub_neg_eq_add, ← neg_sub', ← neg_add', preNormEDS_neg, ih]
ring1 | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.AlgebraicGeometry.AffineTransitionLimit | {
"line": 976,
"column": 4
} | {
"line": 976,
"column": 64
} | [
{
"pp": "I : Type u\ninst✝⁵ : Category.{u, u} I\nD : I ⥤ Scheme\nc : Cone D\nhc : IsLimit c\ninst✝⁴ : IsCofiltered I\ninst✝³ : ∀ {i j : I} (f : i ⟶ j), IsAffineHom (D.map f)\ninst✝² : ∀ (i : I), CompactSpace ↥(D.obj i)\ninst✝¹ : ∀ (i : I), QuasiSeparatedSpace ↥(D.obj i)\ninst✝ : c.pt.IsQuasiAffine\nx : ↥c.pt\ni... | obtain ⟨k, fki, fkj, -⟩ := IsCofilteredOrEmpty.cone_objs i j | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.NumberTheory.EllipticDivisibilitySequence | {
"line": 231,
"column": 13
} | {
"line": 231,
"column": 27
} | [
{
"pp": "case nat.succ.succ\nR : Type u\ninst✝ : CommRing R\nb c d : R\nn✝ : ℕ\n⊢ preNormEDS b c d (2 * ↑(n✝ + 1 + 1) + 1) =\n (preNormEDS b c d (↑(n✝ + 1 + 1) + 2) * preNormEDS b c d ↑(n✝ + 1 + 1) ^ 3 * if Even ↑(n✝ + 1 + 1) then b else 1) -\n preNormEDS b c d (↑(n✝ + 1 + 1) - 1) * preNormEDS b c d (↑(... | Nat.cast_succ, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.NumberTheory.EllipticDivisibilitySequence | {
"line": 237,
"column": 13
} | {
"line": 237,
"column": 27
} | [
{
"pp": "case neg.succ\nR : Type u\ninst✝ : CommRing R\nb c d : R\nih :\n ∀ (n : ℕ),\n preNormEDS b c d (2 * ↑n + 1) =\n (preNormEDS b c d (↑n + 2) * preNormEDS b c d ↑n ^ 3 * if Even ↑n then b else 1) -\n preNormEDS b c d (↑n - 1) * preNormEDS b c d (↑n + 1) ^ 3 * if Even ↑n then 1 else b\nm : ... | Nat.cast_succ, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.AlgebraicGeometry.EllipticCurve.Affine.Formula | {
"line": 338,
"column": 33
} | {
"line": 338,
"column": 42
} | [
{
"pp": "case pos\nF : Type u\ninst✝¹ : Field F\nW : Affine F\ninst✝ : DecidableEq F\nx₁ x₂ y₁ y₂ : F\nh₁ : W.Nonsingular x₁ y₁\nh₂ : W.Nonsingular x₂ y₂\nhxy : ¬(x₁ = x₂ ∧ y₁ = W.negY x₂ y₂)\nhx₁ : W.addX x₁ x₂ (W.slope x₁ x₂ y₁ y₂) = x₁\n⊢ W.Nonsingular x₁ (W.slope x₁ x₂ y₁ y₂ * 0 + y₁)",
"usedConstants":... | mul_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.EllipticDivisibilitySequence | {
"line": 473,
"column": 13
} | {
"line": 473,
"column": 27
} | [
{
"pp": "case neg.succ\nR : Type u\ninst✝ : CommRing R\nb c d : R\nk : ℤ\nih :\n ∀ (n : ℕ),\n complEDS b c d k (2 * ↑n + 1) =\n complEDS b c d k ↑n ^ 2 * normEDS b c d ((↑n + 1) * k + 1) * normEDS b c d ((↑n + 1) * k - 1) -\n complEDS b c d k (↑n + 1) ^ 2 * normEDS b c d (↑n * k + 1) * normEDS b... | Nat.cast_succ, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.AlgebraicGeometry.EllipticCurve.Affine.Formula | {
"line": 471,
"column": 25
} | {
"line": 471,
"column": 37
} | [
{
"pp": "R : Type r\nS : Type s\nF : Type u\nK : Type v\ninst✝¹² : CommRing R\ninst✝¹¹ : CommRing S\ninst✝¹⁰ : Field F\ninst✝⁹ : Field K\nW' : Affine R\ninst✝⁸ : Algebra R S\ninst✝⁷ : DecidableEq F\ninst✝⁶ : DecidableEq K\ninst✝⁵ : Algebra R F\ninst✝⁴ : Algebra S F\ninst✝³ : IsScalarTower R S F\ninst✝² : Algebr... | ← map_slope, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.AlgebraicGeometry.EllipticCurve.DivisionPolynomial.Basic | {
"line": 467,
"column": 2
} | {
"line": 467,
"column": 25
} | [
{
"pp": "R : Type r\ninst✝ : CommRing R\nW : WeierstrassCurve R\n⊢ W.φ 3 = C X * C W.Ψ₃ ^ 2 - C W.preΨ₄ * W.ψ₂ ^ 2",
"usedConstants": [
"Polynomial.C",
"Semigroup.toMul",
"WeierstrassCurve.ψ",
"HMul.hMul",
"CommRing.toNonUnitalCommRing",
"Monoid.toMulOneClass",
"con... | simp [φ, mul_assoc, sq] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.AlgebraicGeometry.EllipticCurve.DivisionPolynomial.Basic | {
"line": 467,
"column": 2
} | {
"line": 467,
"column": 25
} | [
{
"pp": "R : Type r\ninst✝ : CommRing R\nW : WeierstrassCurve R\n⊢ W.φ 3 = C X * C W.Ψ₃ ^ 2 - C W.preΨ₄ * W.ψ₂ ^ 2",
"usedConstants": [
"Polynomial.C",
"Semigroup.toMul",
"WeierstrassCurve.ψ",
"HMul.hMul",
"CommRing.toNonUnitalCommRing",
"Monoid.toMulOneClass",
"con... | simp [φ, mul_assoc, sq] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
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