module string | startPos dict | endPos dict | nextStartPos dict | goals list | goalsAfter list | ppTac string | elaborator string | kind string |
|---|---|---|---|---|---|---|---|---|
Mathlib.AlgebraicGeometry.GammaSpecAdjunction | {
"line": 321,
"column": 8
} | {
"line": 322,
"column": 19
} | {
"line": 322,
"column": 19
} | [
{
"pp": "case w\nR : CommRingCat\np : PrimeSpectrum ↑R\nx : ↑R\n⊢ x ∈\n ((TopCat.Hom.hom (Spec.topMap (toSpecΓ R)))\n ((TopCat.Hom.hom (Spec.locallyRingedSpaceObj R).toΓSpecBase) p)).asIdeal ↔\n x ∈ p.asIdeal",
"ppTerm": "?w",
"assigned": true,
"usedConstants": [
"Eq.mpr",
... | [
"case w\nR : CommRingCat\np : PrimeSpectrum ↑R\nx : ↑R\n⊢ x ∈\n ((TopCat.Hom.hom (Spec.topMap (toSpecΓ R)))\n ((TopCat.Hom.hom (Spec.locallyRingedSpaceObj R).toΓSpecBase) p)).asIdeal ↔\n (algebraMap ↑R ↑((structureSheaf ↑R).presheaf.stalk p)) x ∈\n IsLocalRing.maximalIdeal ↑((structureSheaf ↑R... | ← IsLocalization.AtPrime.to_map_mem_maximal_iff ((structureSheaf R).presheaf.stalk p)
p.asIdeal x | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.Limits.Types.Coequalizers | {
"line": 50,
"column": 2
} | {
"line": 53,
"column": 10
} | {
"line": 54,
"column": 2
} | [
{
"pp": "X Y Z : Type u\nf g : X ⟶ Y\nπ : Y ⟶ Z\ne : f ≫ π = g ≫ π\nh : IsColimit (Cofork.ofπ π e)\nU : Set Y\nH : ⇑(hom f) ⁻¹' U = ⇑(hom g) ⁻¹' U\n⊢ ⇑(hom π) ⁻¹' ⇑(hom π) '' U = U",
"ppTerm": "?m.44",
"assigned": true,
"usedConstants": [
"Function.Coequalizer.Rel.casesOn",
"Eq.mpr",
... | [
"X Y Z : Type u\nf g : X ⟶ Y\nπ : Y ⟶ Z\ne : f ≫ π = g ≫ π\nh : IsColimit (Cofork.ofπ π e)\nU : Set Y\nH : ⇑(hom f) ⁻¹' U = ⇑(hom g) ⁻¹' U\nlem : ∀ (x y : (fun X ↦ X) Y), Function.Coequalizer.Rel (⇑(hom f)) (⇑(hom g)) x y → (x ∈ U ↔ y ∈ U)\n⊢ ⇑(hom π) ⁻¹' ⇑(hom π) '' U = U"
] | have lem : ∀ x y, Function.Coequalizer.Rel f g x y → (x ∈ U ↔ y ∈ U) := by
rintro _ _ ⟨x⟩
change x ∈ f ⁻¹' U ↔ x ∈ g ⁻¹' U
rw [H] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.CategoryTheory.Limits.Types.Coequalizers | {
"line": 58,
"column": 8
} | {
"line": 61,
"column": 33
} | {
"line": 61,
"column": 33
} | [
{
"pp": "case mp\nX Y Z : Type u\nf g : X ⟶ Y\nπ : Y ⟶ Z\ne : f ≫ π = g ≫ π\nh : IsColimit (Cofork.ofπ π e)\nU : Set Y\nH : ⇑(hom f) ⁻¹' U = ⇑(hom g) ⁻¹' U\nlem : ∀ (x y : (fun X ↦ X) Y), Function.Coequalizer.Rel (⇑(hom f)) (⇑(hom g)) x y → (x ∈ U ↔ y ∈ U)\neqv : _root_.Equivalence fun x y ↦ x ∈ U ↔ y ∈ U\nx✝ :... | [
"case mp\nX Y Z : Type u\nf g : X ⟶ Y\nπ : Y ⟶ Z\ne : f ≫ π = g ≫ π\nh : IsColimit (Cofork.ofπ π e)\nU : Set Y\nH : ⇑(hom f) ⁻¹' U = ⇑(hom g) ⁻¹' U\nlem : ∀ (x y : (fun X ↦ X) Y), Function.Coequalizer.Rel (⇑(hom f)) (⇑(hom g)) x y → (x ∈ U ↔ y ∈ U)\neqv : _root_.Equivalence fun x y ↦ x ∈ U ↔ y ∈ U\nx✝ : (fun X ↦ X)... | ←
show _ = π from
h.comp_coconePointUniqueUpToIso_inv (coequalizerColimit f g).2
WalkingParallelPair.one | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.AlgebraicGeometry.AffineScheme | {
"line": 794,
"column": 2
} | {
"line": 796,
"column": 53
} | {
"line": 797,
"column": 2
} | [
{
"pp": "X : Scheme\nU : X.Opens\nhU : IsAffineOpen U\nx : ↥U\nhx : IsClosed {↑x}\n⊢ (hU.primeIdealOf x).asIdeal.IsMaximal",
"ppTerm": "?m.25",
"assigned": true,
"usedConstants": [
"AlgebraicGeometry.SheafedSpace.instTopologicalSpaceCarrierCarrier",
"AlgebraicGeometry.PresheafedSpace.car... | [
"X : Scheme\nU : X.Opens\nhU : IsAffineOpen U\nx : ↥U\nhx : IsClosed {↑x}\nhx₀ : IsClosed {x}\n⊢ (hU.primeIdealOf x).asIdeal.IsMaximal"
] | have hx₀ : IsClosed {x} := by
simpa [← Set.image_singleton, Set.preimage_image_eq _ Subtype.val_injective]
using hx.preimage U.isOpenEmbedding'.continuous | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Topology.Gluing | {
"line": 241,
"column": 2
} | {
"line": 241,
"column": 6
} | {
"line": 242,
"column": 2
} | [
{
"pp": "D : GlueData\ni j : D.J\nU : Set ↑(D.U i)\nthis :\n ⇑(ConcreteCategory.hom (D.f j i)) ⁻¹' ⇑(ConcreteCategory.hom (D.ι j)) ⁻¹' ⇑(ConcreteCategory.hom (D.ι i)) '' U =\n ⇑(ConcreteCategory.hom (D.t j i ≫ D.f i j)) ⁻¹' U\n⊢ ⇑(ConcreteCategory.hom (D.ι j)) ⁻¹' ⇑(ConcreteCategory.hom (D.ι i)) '' U =\n ... | [
"D : GlueData\ni j : D.J\nU : Set ↑(D.U i)\nthis :\n ⇑(ConcreteCategory.hom (D.f j i)) ⁻¹' ⇑(ConcreteCategory.hom (D.ι j)) ⁻¹' ⇑(ConcreteCategory.hom (D.ι i)) '' U =\n ⇑(ConcreteCategory.hom (D.t j i ≫ D.f i j)) ⁻¹' U\n⊢ ⇑(ConcreteCategory.hom (D.ι j)) ⁻¹' ⇑(ConcreteCategory.hom (D.ι i)) '' U ∩\n Set.range... | symm | Lean.Elab.Tactic.evalSymm | Lean.Parser.Tactic.symm |
Mathlib.Topology.Gluing | {
"line": 382,
"column": 86
} | {
"line": 389,
"column": 33
} | {
"line": 391,
"column": 0
} | [
{
"pp": "α : Type u\ninst✝ : TopologicalSpace α\nJ : Type u\nU : J → Opens α\n⊢ Function.Injective ⇑(ConcreteCategory.hom (fromOpenSubsetsGlue U))",
"ppTerm": "?m.8",
"assigned": true,
"usedConstants": [
"CategoryTheory.GlueData.t",
"TopCat.GlueData.ofOpenSubsets._proof_3",
"Catego... | [] | by
intro x y e
obtain ⟨i, ⟨x, hx⟩, rfl⟩ := (ofOpenSubsets U).ι_jointly_surjective x
obtain ⟨j, ⟨y, hy⟩, rfl⟩ := (ofOpenSubsets U).ι_jointly_surjective y
rw [ι_fromOpenSubsetsGlue_apply, ι_fromOpenSubsetsGlue_apply] at e
subst e
rw [(ofOpenSubsets U).ι_eq_iff_rel]
exact ⟨⟨⟨x, hx⟩, hy⟩, rfl, rfl⟩ | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.AlgebraicGeometry.AffineScheme | {
"line": 1109,
"column": 4
} | {
"line": 1109,
"column": 8
} | {
"line": 1110,
"column": 4
} | [
{
"pp": "case h\nX : Scheme\ninst✝ : IsAffine X\ns : Set ↥X\nhs : IsClosed s\nZ : Set ↥(Spec Γ(X, ⊤)) := X.toΓSpecFun '' s\nhZ : IsClosed Z\nI : Ideal ↑Γ(X, ⊤)\nhI : Z = PrimeSpectrum.zeroLocus ↑I\n⊢ s = ⇑X.toSpecΓ ⁻¹' X.toΓSpecFun '' s",
"ppTerm": "?h",
"assigned": true,
"usedConstants": [
"A... | [
"case h\nX : Scheme\ninst✝ : IsAffine X\ns : Set ↥X\nhs : IsClosed s\nZ : Set ↥(Spec Γ(X, ⊤)) := ⋯\nhZ : IsClosed Z\nI : Ideal ↑Γ(X, ⊤)\nhI : Z = PrimeSpectrum.zeroLocus ↑I\n⊢ ⇑X.toSpecΓ ⁻¹' X.toΓSpecFun '' s = s"
] | symm | Lean.Elab.Tactic.evalSymm | Lean.Parser.Tactic.symm |
Mathlib.CategoryTheory.Limits.Constructions.ZeroObjects | {
"line": 185,
"column": 2
} | {
"line": 186,
"column": 6
} | {
"line": 188,
"column": 0
} | [
{
"pp": "C : Type u_1\ninst✝³ : Category.{v_1, u_1} C\ninst✝² : HasZeroObject C\ninst✝¹ : HasZeroMorphisms C\nX Y : C\ninst✝ : HasBinaryCoproduct X Y\n⊢ pushout.inl 0 0 ≫ (pushoutZeroZeroIso X Y).hom = coprod.inl",
"ppTerm": "?m.33",
"assigned": true,
"usedConstants": [
"CategoryTheory.Limits.... | [] | dsimp [pushoutZeroZeroIso]
simp | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Limits.Constructions.ZeroObjects | {
"line": 185,
"column": 2
} | {
"line": 186,
"column": 6
} | {
"line": 188,
"column": 0
} | [
{
"pp": "C : Type u_1\ninst✝³ : Category.{v_1, u_1} C\ninst✝² : HasZeroObject C\ninst✝¹ : HasZeroMorphisms C\nX Y : C\ninst✝ : HasBinaryCoproduct X Y\n⊢ pushout.inl 0 0 ≫ (pushoutZeroZeroIso X Y).hom = coprod.inl",
"ppTerm": "?m.33",
"assigned": true,
"usedConstants": [
"CategoryTheory.Limits.... | [] | dsimp [pushoutZeroZeroIso]
simp | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Limits.Constructions.ZeroObjects | {
"line": 192,
"column": 2
} | {
"line": 193,
"column": 6
} | {
"line": 195,
"column": 0
} | [
{
"pp": "C : Type u_1\ninst✝³ : Category.{v_1, u_1} C\ninst✝² : HasZeroObject C\ninst✝¹ : HasZeroMorphisms C\nX Y : C\ninst✝ : HasBinaryCoproduct X Y\n⊢ pushout.inr 0 0 ≫ (pushoutZeroZeroIso X Y).hom = coprod.inr",
"ppTerm": "?m.33",
"assigned": true,
"usedConstants": [
"CategoryTheory.Limits.... | [] | dsimp [pushoutZeroZeroIso]
simp | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Limits.Constructions.ZeroObjects | {
"line": 192,
"column": 2
} | {
"line": 193,
"column": 6
} | {
"line": 195,
"column": 0
} | [
{
"pp": "C : Type u_1\ninst✝³ : Category.{v_1, u_1} C\ninst✝² : HasZeroObject C\ninst✝¹ : HasZeroMorphisms C\nX Y : C\ninst✝ : HasBinaryCoproduct X Y\n⊢ pushout.inr 0 0 ≫ (pushoutZeroZeroIso X Y).hom = coprod.inr",
"ppTerm": "?m.33",
"assigned": true,
"usedConstants": [
"CategoryTheory.Limits.... | [] | dsimp [pushoutZeroZeroIso]
simp | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.AlgebraicGeometry.Gluing | {
"line": 548,
"column": 10
} | {
"line": 548,
"column": 82
} | {
"line": 548,
"column": 82
} | [
{
"pp": "J : 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)\ny₁ : ↥(F.obj k₁.fst)\nhy₁ : (F.map k₁.snd.1) y₁ = ↑((pullbac... | [] | simpa [hy₁, hy₂] using congr($(pullback.condition (f := (V F i j).ι)) x) | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.AlgebraicGeometry.Gluing | {
"line": 548,
"column": 10
} | {
"line": 548,
"column": 82
} | {
"line": 548,
"column": 82
} | [
{
"pp": "J : Type w\ninst✝² : Category.{v, w} J\nF : J ⥤ Scheme\ninst✝¹ : ∀ {i j : J} (f : i ⟶ j), IsOpenImmersion (F.map f)\ninst✝ : (F ⋙ forget).IsLocallyDirected\ni j k : J\nx : ↥(pullback (V F i j).ι (V F i k).ι)\nk₁ : (k : J) × (k ⟶ i) × (k ⟶ j)\ny₁ : ↥(F.obj k₁.fst)\nhy₁ : (F.map k₁.snd.1) y₁ = ↑((pullbac... | [] | simpa [hy₁, hy₂] using congr($(pullback.condition (f := (V F i j).ι)) x) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.AlgebraicGeometry.Gluing | {
"line": 548,
"column": 10
} | {
"line": 548,
"column": 82
} | {
"line": 548,
"column": 82
} | [
{
"pp": "J : 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)\ny₁ : ↥(F.obj k₁.fst)\nhy₁ : (F.map k₁.snd.1) y₁ = ↑((pullbac... | [] | simpa [hy₁, hy₂] using congr($(pullback.condition (f := (V F i j).ι)) x) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Geometry.RingedSpace.PresheafedSpace.Gluing | {
"line": 288,
"column": 10
} | {
"line": 288,
"column": 67
} | {
"line": 289,
"column": 10
} | [
{
"pp": "case h\nC : Type u\ninst✝¹ : Category.{v, u} C\nD : GlueData C\ninst✝ : HasLimits C\ni j k : D.J\nU : Opens ↑↑(D.U i)\n⊢ (pullback.fst (D.f j i) (D.f j k) ≫ D.t j i ≫ D.f i j).c.app (op U) ≫\n invApp (pullback.snd (D.f j i) (D.f j k))\n ((Opens.map (pullback.fst (D.f j i) (D.f j k)).base)... | [
"case h\nC : Type u\ninst✝¹ : Category.{v, u} C\nD : GlueData C\ninst✝ : HasLimits C\ni j k : D.J\nU : Opens ↑↑(D.U i)\n⊢ (pullback.fst (D.f j i) (D.f j k) ≫ D.t j i ≫ D.f i j).c.app (op U) ≫\n invApp (pullback.snd (D.f j i) (D.f j k))\n ((Opens.map (pullback.fst (D.f j i) (D.f j k)).base).1\n ... | rw [eqToHom_map (Opens.map _), eqToHom_op, eqToHom_trans] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.AlgebraicGeometry.Limits | {
"line": 703,
"column": 67
} | {
"line": 703,
"column": 91
} | {
"line": 704,
"column": 2
} | [
{
"pp": "ι : Type u\nf : ι → Scheme\nσ : Type v\ng : σ → Scheme\nX Y : Scheme\nR S : Type u\ninst✝³ : CommRing R\ninst✝² : CommRing S\ninst✝¹ : Finite ↥X\ninst✝ : DiscreteTopology ↥X\nthis : IsAffineOpen (⨆ x, { carrier := {x}, is_open' := ⋯ })\n⊢ IsAffineOpen ⊤",
"ppTerm": "?m.36",
"assigned": true,
... | [] | convert! this; ext; simp | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.AlgebraicGeometry.Limits | {
"line": 703,
"column": 67
} | {
"line": 703,
"column": 91
} | {
"line": 704,
"column": 2
} | [
{
"pp": "ι : Type u\nf : ι → Scheme\nσ : Type v\ng : σ → Scheme\nX Y : Scheme\nR S : Type u\ninst✝³ : CommRing R\ninst✝² : CommRing S\ninst✝¹ : Finite ↥X\ninst✝ : DiscreteTopology ↥X\nthis : IsAffineOpen (⨆ x, { carrier := {x}, is_open' := ⋯ })\n⊢ IsAffineOpen ⊤",
"ppTerm": "?m.36",
"assigned": true,
... | [] | convert! this; ext; simp | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.AlgebraicGeometry.Limits | {
"line": 710,
"column": 2
} | {
"line": 713,
"column": 44
} | {
"line": 715,
"column": 0
} | [
{
"pp": "U X Y : Scheme\nf : U ⟶ X\ng : U ⟶ Y\ninst✝¹ : IsOpenImmersion f\ninst✝ : IsOpenImmersion g\ni : WalkingPair\n⊢ Mono ((span f g ⋙ Scheme.forget).map (WidePushoutShape.Hom.init i))",
"ppTerm": "?m.31",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"AlgebraicGeometry.SheafedSpace... | [] | rw [mono_iff_injective]
cases i
· simpa using! f.isOpenEmbedding.injective
· simpa using! g.isOpenEmbedding.injective | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.AlgebraicGeometry.Limits | {
"line": 710,
"column": 2
} | {
"line": 713,
"column": 44
} | {
"line": 715,
"column": 0
} | [
{
"pp": "U X Y : Scheme\nf : U ⟶ X\ng : U ⟶ Y\ninst✝¹ : IsOpenImmersion f\ninst✝ : IsOpenImmersion g\ni : WalkingPair\n⊢ Mono ((span f g ⋙ Scheme.forget).map (WidePushoutShape.Hom.init i))",
"ppTerm": "?m.31",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"AlgebraicGeometry.SheafedSpace... | [] | rw [mono_iff_injective]
cases i
· simpa using! f.isOpenEmbedding.injective
· simpa using! g.isOpenEmbedding.injective | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.AlgebraicGeometry.Morphisms.Basic | {
"line": 329,
"column": 4
} | {
"line": 332,
"column": 16
} | {
"line": 334,
"column": 0
} | [
{
"pp": "P : MorphismProperty Scheme\ninst✝² : IsZariskiLocalAtSource P\nX Y : Scheme\nf : X ⟶ Y\ninst✝¹ : IsZariskiLocalAtTarget P\ninst✝ : P.RespectsRight IsOpenImmersion\nU : ↥X → Y.Opens\nV : ↥X → X.Opens\nhxU : ∀ (x : ↥X), x ∈ (V x).carrier\ne : ∀ (x : ↥X), V x ≤ f ⁻¹ᵁ U x\nhf : ∀ (x : ↥X), P (Scheme.Hom.r... | [] | rw [eq_top_iff]
rintro x -
simp only [Opens.mem_iSup]
use x, hxU x | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.AlgebraicGeometry.Morphisms.Basic | {
"line": 329,
"column": 4
} | {
"line": 332,
"column": 16
} | {
"line": 334,
"column": 0
} | [
{
"pp": "P : MorphismProperty Scheme\ninst✝² : IsZariskiLocalAtSource P\nX Y : Scheme\nf : X ⟶ Y\ninst✝¹ : IsZariskiLocalAtTarget P\ninst✝ : P.RespectsRight IsOpenImmersion\nU : ↥X → Y.Opens\nV : ↥X → X.Opens\nhxU : ∀ (x : ↥X), x ∈ (V x).carrier\ne : ∀ (x : ↥X), V x ≤ f ⁻¹ᵁ U x\nhf : ∀ (x : ↥X), P (Scheme.Hom.r... | [] | rw [eq_top_iff]
rintro x -
simp only [Opens.mem_iSup]
use x, hxU x | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.AlgebraicGeometry.Morphisms.Basic | {
"line": 389,
"column": 29
} | {
"line": 389,
"column": 48
} | {
"line": 390,
"column": 2
} | [
{
"pp": "case hprecomp\nP : AffineTargetMorphismProperty\nh₁ : ∀ {X Y Z : Scheme} (e : X ≅ Y) (f : Y ⟶ Z) [inst : IsAffine Z], P f → P (e.hom ≫ f)\nh₂ : ∀ {X Y Z : Scheme} (e : Y ≅ Z) (f : X ⟶ Y) [inst : IsAffine Y], P f → P (f ≫ e.hom)\nX Y Z : Scheme\ne : X ≅ Y\nf : Y ⟶ Z\na : IsAffine Z\nh : P f\n⊢ P.toPrope... | [] | exact ⟨a, h₁ e f h⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.AlgebraicGeometry.Pullbacks | {
"line": 672,
"column": 6
} | {
"line": 672,
"column": 10
} | {
"line": 673,
"column": 6
} | [
{
"pp": "X Y Z : Scheme\n𝒰✝ : X.OpenCover\nf✝ : X ⟶ Z\ng : Y ⟶ Z\ninst✝ : ∀ (i : 𝒰✝.I₀), HasPullback (𝒰✝.f i ≫ f✝) g\ns : PullbackCone f✝ g\nf : X ⟶ Y\n𝒰 : Y.OpenCover\n𝒱 : (i : (Precoverage.ZeroHypercover.pullback₁ f 𝒰).I₀) → ((Precoverage.ZeroHypercover.pullback₁ f 𝒰).X i).OpenCover\ni : (openCoverOfBa... | [
"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\nf : X ⟶ Y\n𝒰 : Y.OpenCover\n𝒱 : (i : (Precoverage.ZeroHypercover.pullback₁ f 𝒰).I₀) → ((Precoverage.ZeroHypercover.pullback₁ f 𝒰).X i).OpenCover\ni : (openCoverOfBase 𝒰 f f).I... | symm | Lean.Elab.Tactic.evalSymm | Lean.Parser.Tactic.symm |
Mathlib.AlgebraicGeometry.Morphisms.Basic | {
"line": 622,
"column": 2
} | {
"line": 622,
"column": 33
} | {
"line": 623,
"column": 2
} | [
{
"pp": "P : MorphismProperty Scheme\nQ : AffineTargetMorphismProperty\ninst✝¹ : HasAffineProperty P Q\nhP' : Q.IsStableUnderBaseChange\nX Y S : Scheme\nf : X ⟶ S\ng : Y ⟶ S\ninst✝ : IsAffine S\nH : Q g\nthis✝ : Q.IsLocal := isLocal_affineProperty P\ni : X.affineCover.toPreZeroHypercover.1\ne : pullback (pullba... | [
"P : MorphismProperty Scheme\nQ : AffineTargetMorphismProperty\ninst✝¹ : HasAffineProperty P Q\nhP' : Q.IsStableUnderBaseChange\nX Y S : Scheme\nf : X ⟶ S\ng : Y ⟶ S\ninst✝ : IsAffine S\nH : Q g\nthis✝ : Q.IsLocal := ⋯\ni : X.affineCover.toPreZeroHypercover.1\ne : pullback (pullback.fst f g) (X.affineCover.f i) ≅ p... | apply hP' (.of_hasPullback _ _) | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.AlgebraicGeometry.Morphisms.UnderlyingMap | {
"line": 99,
"column": 2
} | {
"line": 99,
"column": 64
} | {
"line": 100,
"column": 2
} | [
{
"pp": "X Y Z : Scheme\nf✝ : X ⟶ Y\ng : Y ⟶ Z\nthis : (topologically fun {α β} [TopologicalSpace α] [TopologicalSpace β] ↦ Function.Surjective).RespectsIso\nα β : Type u_1\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\nf : α → β\nι : Type u_1\nU : ι → Opens β\nH : IsOpenCover U\nx✝ : Continuous[inst... | [
"X Y Z : Scheme\nf✝ : X ⟶ Y\ng : Y ⟶ Z\nthis : (topologically fun {α β} [TopologicalSpace α] [TopologicalSpace β] ↦ Function.Surjective).RespectsIso\nα β : Type u_1\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\nf : α → β\nι : Type u_1\nU : ι → Opens β\nH : IsOpenCover U\nx✝ : Continuous[inst✝¹, inst✝] f... | obtain ⟨i, hxi⟩ : ∃ i, x ∈ U i := by simpa using congr(x ∈ $H) | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.AlgebraicGeometry.Morphisms.UnderlyingMap | {
"line": 100,
"column": 2
} | {
"line": 100,
"column": 38
} | {
"line": 101,
"column": 2
} | [
{
"pp": "X Y Z : Scheme\nf✝ : X ⟶ Y\ng : Y ⟶ Z\nthis : (topologically fun {α β} [TopologicalSpace α] [TopologicalSpace β] ↦ Function.Surjective).RespectsIso\nα β : Type u_1\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\nf : α → β\nι : Type u_1\nU : ι → Opens β\nH : IsOpenCover U\nx✝ : Continuous[inst... | [
"X Y Z : Scheme\nf✝ : X ⟶ Y\ng : Y ⟶ Z\nthis : (topologically fun {α β} [TopologicalSpace α] [TopologicalSpace β] ↦ Function.Surjective).RespectsIso\nα β : Type u_1\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\nf : α → β\nι : Type u_1\nU : ι → Opens β\nH : IsOpenCover U\nx✝ : Continuous[inst✝¹, inst✝] f... | obtain ⟨⟨y, _⟩, hy⟩ := hf i ⟨x, hxi⟩ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.RingTheory.Localization.Away.Lemmas | {
"line": 39,
"column": 30
} | {
"line": 39,
"column": 41
} | {
"line": 39,
"column": 42
} | [
{
"pp": "R : Type u_1\ninst✝³ : CommRing R\ns : Set R\nhsone : Ideal.span s = ⊤\nRₜ : ↑s → Type u_2\ninst✝² : (t : ↑s) → CommRing (Rₜ t)\ninst✝¹ : (t : ↑s) → Algebra R (Rₜ t)\ninst✝ : ∀ (t : ↑s), Away (↑t) (Rₜ t)\np : (t : ↑s) → Set (Rₜ t)\nhtone : ∀ (r : ↑s), Ideal.span (p r) = ⊤\n⊢ (Ideal.span (Set.range (mul... | [
"R : Type u_1\ninst✝³ : CommRing R\ns : Set R\nhsone : Ideal.span s = ⊤\nRₜ : ↑s → Type u_2\ninst✝² : (t : ↑s) → CommRing (Rₜ t)\ninst✝¹ : (t : ↑s) → Algebra R (Rₜ t)\ninst✝ : ∀ (t : ↑s), Away (↑t) (Rₜ t)\np : (t : ↑s) → Set (Rₜ t)\nhtone : ∀ (r : ↑s), Ideal.span (p r) = ⊤\n⊢ ⊤ ≤ (Ideal.span (Set.range (mulNumerato... | eq_top_iff, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Localization.Away.Lemmas | {
"line": 55,
"column": 8
} | {
"line": 55,
"column": 30
} | {
"line": 55,
"column": 30
} | [
{
"pp": "R : Type u_1\ninst✝³ : CommRing R\ns : Set R\nhsone : Ideal.span s = ⊤\nRₜ : ↑s → Type u_2\ninst✝² : (t : ↑s) → CommRing (Rₜ t)\ninst✝¹ : (t : ↑s) → Algebra R (Rₜ t)\ninst✝ : ∀ (t : ↑s), Away (↑t) (Rₜ t)\np : (t : ↑s) → Set (Rₜ t)\nhtone : ∀ (r : ↑s), Ideal.span (p r) = ⊤\na : R\nha : a ∈ s\nthis : IsL... | [
"R : Type u_1\ninst✝³ : CommRing R\ns : Set R\nhsone : Ideal.span s = ⊤\nRₜ : ↑s → Type u_2\ninst✝² : (t : ↑s) → CommRing (Rₜ t)\ninst✝¹ : (t : ↑s) → Algebra R (Rₜ t)\ninst✝ : ∀ (t : ↑s), Away (↑t) (Rₜ t)\np : (t : ↑s) → Set (Rₜ t)\nhtone : ∀ (r : ↑s), Ideal.span (p r) = ⊤\na : R\nha : a ∈ s\nthis : IsLocalization ... | IsLocalization.mk'_one | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.AlgebraicGeometry.Morphisms.UnderlyingMap | {
"line": 265,
"column": 6
} | {
"line": 265,
"column": 33
} | {
"line": 265,
"column": 33
} | [
{
"pp": "X : Scheme\nU : X.Opens\nhU : Dense ↑U\n⊢ DenseRange ⇑U.ι",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"AlgebraicGeometry.SheafedSpace.instTopologicalSpaceCarrierCarrier",
"AlgebraicGeometry.PresheafedSpace.carrier",
"congrArg",
"Category... | [] | simpa [DenseRange] using hU | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.AlgebraicGeometry.Morphisms.UnderlyingMap | {
"line": 265,
"column": 6
} | {
"line": 265,
"column": 33
} | {
"line": 265,
"column": 33
} | [
{
"pp": "X : Scheme\nU : X.Opens\nhU : Dense ↑U\n⊢ DenseRange ⇑U.ι",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"AlgebraicGeometry.SheafedSpace.instTopologicalSpaceCarrierCarrier",
"AlgebraicGeometry.PresheafedSpace.carrier",
"congrArg",
"Category... | [] | simpa [DenseRange] using hU | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.AlgebraicGeometry.Morphisms.UnderlyingMap | {
"line": 265,
"column": 6
} | {
"line": 265,
"column": 33
} | {
"line": 265,
"column": 33
} | [
{
"pp": "X : Scheme\nU : X.Opens\nhU : Dense ↑U\n⊢ DenseRange ⇑U.ι",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"AlgebraicGeometry.SheafedSpace.instTopologicalSpaceCarrierCarrier",
"AlgebraicGeometry.PresheafedSpace.carrier",
"congrArg",
"Category... | [] | simpa [DenseRange] using hU | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Localization.Away.Lemmas | {
"line": 78,
"column": 50
} | {
"line": 88,
"column": 23
} | {
"line": 90,
"column": 0
} | [
{
"pp": "R : Type u_2\nS : Type u_3\ninst✝³ : CommSemiring R\ninst✝² : CommSemiring S\ninst✝¹ : Algebra R S\nx : R\ninst✝ : IsLocalization.Away x S\nI J : Ideal R\nhle : map (algebraMap R S) I ≤ map (algebraMap R S) J\nhxJ : ∀ (y : R), x * y ∈ J → y ∈ J\n⊢ I ≤ J",
"ppTerm": "?m.41",
"assigned": true,
... | [] | by
intro y hy
have hin : algebraMap R S y ∈ I.map (algebraMap R S) := Ideal.mem_map_of_mem (algebraMap R S) hy
grw [hle, IsLocalization.algebraMap_mem_map_algebraMap_iff (Submonoid.powers x)] at hin
obtain ⟨m, ⟨n, hn, rfl⟩, h⟩ := hin
dsimp at h
induction n with
| zero => simpa using h
| succ n ih =>
... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.RingHom.Locally | {
"line": 71,
"column": 6
} | {
"line": 71,
"column": 17
} | {
"line": 71,
"column": 18
} | [
{
"pp": "P : {R S : Type u} → [inst : CommRing R] → [inst_1 : CommRing S] → (R →+* S) → Prop\nR S : Type u\ninst✝¹ : CommRing R\ninst✝ : CommRing S\nf : R →+* S\nx✝ : Locally (fun {R S} [CommRing R] [CommRing S] ↦ P) f\ns : Set S\nhs : Ideal.span s = ⊤\nh : ∀ t ∈ s, (fun {R S} [CommRing R] [CommRing S] ↦ P) ((a... | [
"P : {R S : Type u} → [inst : CommRing R] → [inst_1 : CommRing S] → (R →+* S) → Prop\nR S : Type u\ninst✝¹ : CommRing R\ninst✝ : CommRing S\nf : R →+* S\nx✝ : Locally (fun {R S} [CommRing R] [CommRing S] ↦ P) f\ns : Set S\nhs : Ideal.span s = ⊤\nh : ∀ t ∈ s, (fun {R S} [CommRing R] [CommRing S] ↦ P) ((algebraMap S ... | eq_top_iff, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.AlgebraicGeometry.Morphisms.Constructors | {
"line": 115,
"column": 2
} | {
"line": 115,
"column": 76
} | {
"line": 116,
"column": 2
} | [
{
"pp": "P : MorphismProperty Scheme\nQ : AffineTargetMorphismProperty\ninst✝² : HasAffineProperty P Q\nX Y U✝ V✝ : Scheme\nf : X ⟶ Y\ng : U✝ ⟶ Y\ninst✝¹ : IsAffine U✝\ninst✝ : IsOpenImmersion g\niV : V✝ ⟶ X\nf' : V✝ ⟶ U✝\nh : IsPullback iV f' f g\nH : P.diagonal f\nthis : Q.IsLocal := isLocal_affineProperty P\... | [
"P : MorphismProperty Scheme\nQ : AffineTargetMorphismProperty\ninst✝² : HasAffineProperty P Q\nX Y U✝ V✝ : Scheme\nf : X ⟶ Y\ng : U✝ ⟶ Y\ninst✝¹ : IsAffine U✝\ninst✝ : IsOpenImmersion g\niV : V✝ ⟶ X\nf' : V✝ ⟶ U✝\nh : IsPullback iV f' f g\nH : P.diagonal f\nthis : Q.IsLocal := isLocal_affineProperty P\nU V : Schem... | rw [← Q.cancel_left_of_respectsIso (pullbackDiagonalMapIso f _ f₁ f₂).hom] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.RingHom.Locally | {
"line": 223,
"column": 8
} | {
"line": 223,
"column": 19
} | {
"line": 223,
"column": 20
} | [
{
"pp": "case refine_1\nP : {R S : Type u} → [inst : CommRing R] → [inst_1 : CommRing S] → (R →+* S) → Prop\nhPi : RespectsIso fun {R S} [CommRing R] [CommRing S] ↦ P\nhPl : LocalizationPreserves fun {R S} [CommRing R] [CommRing S] ↦ P\nhPc : StableUnderComposition fun {R S} [CommRing R] [CommRing S] ↦ P\nR S T... | [
"case refine_1\nP : {R S : Type u} → [inst : CommRing R] → [inst_1 : CommRing S] → (R →+* S) → Prop\nhPi : RespectsIso fun {R S} [CommRing R] [CommRing S] ↦ P\nhPl : LocalizationPreserves fun {R S} [CommRing R] [CommRing S] ↦ P\nhPc : StableUnderComposition fun {R S} [CommRing R] [CommRing S] ↦ P\nR S T : Type u\ni... | eq_top_iff, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.RingHom.Locally | {
"line": 297,
"column": 31
} | {
"line": 297,
"column": 42
} | {
"line": 297,
"column": 43
} | [
{
"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\nhPb : IsStableUnderBaseChange 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 : ... | [
"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\nhPb : IsStableUnderBaseChange 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 R S\... | eq_top_iff, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.AlgebraicGeometry.Morphisms.QuasiCompact | {
"line": 112,
"column": 4
} | {
"line": 113,
"column": 27
} | {
"line": 113,
"column": 27
} | [
{
"pp": "X : Scheme\nU : X.Opens\nhU : IsCompact ↑U\nf : ↑Γ(X, U)\ns : Set ↑X.affineOpens\nhs : s.Finite\ne : U = ⨆ i ∈ s, ↑i\nV : ↑s\n⊢ ↑↑V ⊓ X.basicOpen f ∈ X.affineOpens",
"ppTerm": "?m.56",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"AlgebraicGeometry.SheafedSpace.instTopological... | [] | rw [← X.basicOpen_res _ (homOfLE ((le_iSup₂ V.1 V.2).trans_eq e.symm)).op]
exact V.1.2.basicOpen _ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.AlgebraicGeometry.Morphisms.QuasiCompact | {
"line": 112,
"column": 4
} | {
"line": 113,
"column": 27
} | {
"line": 113,
"column": 27
} | [
{
"pp": "X : Scheme\nU : X.Opens\nhU : IsCompact ↑U\nf : ↑Γ(X, U)\ns : Set ↑X.affineOpens\nhs : s.Finite\ne : U = ⨆ i ∈ s, ↑i\nV : ↑s\n⊢ ↑↑V ⊓ X.basicOpen f ∈ X.affineOpens",
"ppTerm": "?m.56",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"AlgebraicGeometry.SheafedSpace.instTopological... | [] | rw [← X.basicOpen_res _ (homOfLE ((le_iSup₂ V.1 V.2).trans_eq e.symm)).op]
exact V.1.2.basicOpen _ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.AlgebraicGeometry.Morphisms.QuasiCompact | {
"line": 225,
"column": 2
} | {
"line": 230,
"column": 25
} | {
"line": 231,
"column": 2
} | [
{
"pp": "case inr\nX : Scheme\nS : CommRingCat\nf : X ⟶ Spec S\ninst✝ : QuasiCompact f\nZ : Set ↥X\nhZ : IsClosed Z\nH : StableUnderSpecialization (⇑f '' Z)\nthis :\n ∀ {X : Scheme} (S : CommRingCat) (f : X ⟶ Spec S) [QuasiCompact f] (Z : Set ↥X),\n IsClosed Z → StableUnderSpecialization (⇑f '' Z) → (∃ R, X... | [
"X : Scheme\nS : CommRingCat\nf : X ⟶ Spec S\ninst✝ : QuasiCompact f\nZ : Set ↥X\nhZ : IsClosed Z\nH : StableUnderSpecialization (⇑f '' Z)\nhX : ∃ R, X = Spec R\n⊢ IsClosed (⇑f '' Z)"
] | · obtain ⟨R, g, hg⟩ := compactSpace_iff_exists.mp (QuasiCompact.compactSpace_of_compactSpace f)
have inst : QuasiCompact (g ≫ f) := HasAffineProperty.iff_of_isAffine.mpr (by infer_instance)
have := this _ (g ≫ f) (g ⁻¹' Z) (hZ.preimage g.continuous)
simp_rw [Scheme.Hom.comp_base, TopCat.comp_app, ← Set.imag... | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.RingTheory.Ideal.Height | {
"line": 169,
"column": 2
} | {
"line": 169,
"column": 34
} | {
"line": 171,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommRing R\nI : Ideal R\ninst✝ : I.IsPrime\n⊢ ↑(Order.height { asIdeal := I, isPrime := inst✝ }) ≤ ringKrullDim R",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"PrimeSpectrum.mk",
"PartialOrder.toPreorder",
"Order.height_le_krullDim",
... | [] | exact Order.height_le_krullDim _ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.RingTheory.Ideal.Height | {
"line": 362,
"column": 2
} | {
"line": 362,
"column": 63
} | {
"line": 363,
"column": 2
} | [
{
"pp": "R : Type u_2\ninst✝ : CommRing R\nn : WithBot ℕ∞\n⊢ ringKrullDim R ≤ n ↔ ∀ ⦃p : Ideal R⦄, p.IsPrime → ↑p.height ≤ n",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"WithBot.instSupSet",
"WithBot.instPreorder",
"Eq.mpr",
"instCompleteLatticeWithBot",
"... | [
"R : Type u_2\ninst✝ : CommRing R\nn : WithBot ℕ∞\n⊢ (∀ (i : PrimeSpectrum R), ↑(Order.height i) ≤ n) ↔ ∀ ⦃p : Ideal R⦄, p.IsPrime → ↑p.height ≤ n"
] | rw [ringKrullDim, Order.krullDim_eq_iSup_height, iSup_le_iff] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.AlgebraicGeometry.Morphisms.RingHomProperties | {
"line": 512,
"column": 4
} | {
"line": 512,
"column": 35
} | {
"line": 513,
"column": 4
} | [
{
"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\nX ... | [
"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\nX Y Z : Scheme... | rw [morphismRestrict_comp] at H | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.AlgebraicGeometry.Morphisms.QuasiSeparated | {
"line": 300,
"column": 2
} | {
"line": 300,
"column": 12
} | {
"line": 301,
"column": 2
} | [
{
"pp": "case h\nX : Scheme\nS : ↑X.affineOpens\nU₁ U₂ : X.Opens\nn₁ n₂ : ℕ\ny₁ : ↑Γ(X, U₁)\ny₂ : ↑Γ(X, U₂)\nf : ↑Γ(X, U₁ ⊔ U₂)\nx : ↑Γ(X, X.basicOpen f)\nh₁ : ↑S ≤ U₁\nh₂ : ↑S ≤ U₂\ne₁ :\n (y₁ |_ X.basicOpen ((f |_ U₁) ⋯)) ⋯ =\n ((f |_ U₁) ⋯ |_ X.basicOpen ((f |_ U₁) ⋯)) ⋯ ^ n₁ * (x |_ X.basicOpen ((f |_ U... | [
"case h\nX : Scheme\nS : ↑X.affineOpens\nU₁ U₂ : X.Opens\nn₁ n₂ : ℕ\ny₁ : ↑Γ(X, U₁)\ny₂ : ↑Γ(X, U₂)\nf : ↑Γ(X, U₁ ⊔ U₂)\nx : ↑Γ(X, X.basicOpen f)\nh₁ : ↑S ≤ U₁\nh₂ : ↑S ≤ U₂\ne₁ :\n (y₁ |_ X.basicOpen ((f |_ U₁) ⋯)) ⋯ =\n ((f |_ U₁) ⋯ |_ X.basicOpen ((f |_ U₁) ⋯)) ⋯ ^ n₁ * (x |_ X.basicOpen ((f |_ U₁) ⋯)) ⋯\ne₂... | intro m hm | Lean.Elab.Tactic.evalIntro | Lean.Parser.Tactic.intro |
Mathlib.RingTheory.Ideal.Height | {
"line": 563,
"column": 2
} | {
"line": 563,
"column": 59
} | {
"line": 564,
"column": 2
} | [
{
"pp": "R : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\np : Ideal R\ninst✝ : p.IsPrime\nh1 : p.height = 1\nx : R\nhx : x ∈ p\nhxp : Prime x\nthis : (span {x}).IsPrime\n⊢ p = span {x}",
"ppTerm": "?m.31",
"assigned": true,
"usedConstants": [
"False",
"Semiring.toModule",
"i... | [
"R : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\np : Ideal R\ninst✝ : p.IsPrime\nh1 : p.height = 1\nx : R\nhx : x ∈ p\nhxp : Prime x\nthis✝ : (span {x}).IsPrime\nthis : p.FiniteHeight\n⊢ p = span {x}"
] | have : p.FiniteHeight := by simp [p.finiteHeight_iff, h1] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.AlgebraicGeometry.Morphisms.QuasiSeparated | {
"line": 329,
"column": 4
} | {
"line": 331,
"column": 41
} | {
"line": 332,
"column": 4
} | [
{
"pp": "case refine_2\nX : Scheme\nU✝ : X.Opens\nhU✝ : IsCompact U✝.carrier\nS : X.Opens\nhS : IsCompact S.carrier\nU : ↑X.affineOpens\nhU :\n IsQuasiSeparated S.carrier →\n ∀ (f : ↑Γ(X, S)) (x : ↑Γ(X, X.basicOpen f)),\n ∃ n y,\n (ConcreteCategory.hom (X.presheaf.map (homOfLE ⋯).op)) y =\n ... | [
"case refine_2\nX : Scheme\nU✝ : X.Opens\nhU✝ : IsCompact U✝.carrier\nS : X.Opens\nhS : IsCompact S.carrier\nU : ↑X.affineOpens\nhU :\n IsQuasiSeparated S.carrier →\n ∀ (f : ↑Γ(X, S)) (x : ↑Γ(X, X.basicOpen f)),\n ∃ n y,\n (ConcreteCategory.hom (X.presheaf.map (homOfLE ⋯).op)) y =\n (Concre... | obtain ⟨n₂, y₂, hy₂⟩ :=
exists_eq_pow_mul_of_isAffineOpen X _ U.2 (X.presheaf.map (homOfLE le_sup_right).op f)
(X.presheaf.map (homOfLE _).op x) | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.AlgebraicGeometry.Properties | {
"line": 368,
"column": 4
} | {
"line": 368,
"column": 36
} | {
"line": 368,
"column": 37
} | [
{
"pp": "R : CommRingCat\nx : ↥(Spec R)\n⊢ height x = height (OrderDual.ofDual ((specOrderIsoPrimeSpectrum R) x))",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"AlgebraicGeometry.specOrderIsoPrimeSpectrum_apply",
"OrderDual.instLE",
"OrderDual.toDual",
"Eq.mpr",
... | [
"R : CommRingCat\nx : ↥(Spec R)\n⊢ height x = height (OrderDual.ofDual (OrderDual.toDual x))"
] | specOrderIsoPrimeSpectrum_apply, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.AlgebraicGeometry.Morphisms.OpenImmersion | {
"line": 44,
"column": 4
} | {
"line": 44,
"column": 22
} | {
"line": 45,
"column": 4
} | [
{
"pp": "case mp\nR S : CommRingCat\nf : R ⟶ S\nhf : Function.Surjective ⇑(CommRingCat.Hom.hom f)\nH : IsOpenImmersion (Spec.map f)\ne : ↑R\nhe : IsIdempotentElem e\nhe' : Set.range (PrimeSpectrum.comap (CommRingCat.Hom.hom f)) = PrimeSpectrum.zeroLocus {e}\n⊢ ∃ e, IsIdempotentElem e ∧ RingHom.ker (CommRingCat.... | [
"case mp\nR S : CommRingCat\nf : R ⟶ S\nhf : Function.Surjective ⇑(CommRingCat.Hom.hom f)\nH : IsOpenImmersion (Spec.map f)\ne : ↑R\nhe : IsIdempotentElem e\nhe' : Set.range (PrimeSpectrum.comap (CommRingCat.Hom.hom f)) = PrimeSpectrum.zeroLocus {e}\n⊢ RingHom.ker (CommRingCat.Hom.hom f) = Ideal.span {e}"
] | refine ⟨e, he, ?_⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.AlgebraicGeometry.Morphisms.OpenImmersion | {
"line": 39,
"column": 2
} | {
"line": 65,
"column": 51
} | {
"line": 67,
"column": 0
} | [
{
"pp": "R S : CommRingCat\nf : R ⟶ S\nhf : Function.Surjective ⇑(CommRingCat.Hom.hom f)\n⊢ IsOpenImmersion (Spec.map f) ↔ ∃ e, IsIdempotentElem e ∧ RingHom.ker (CommRingCat.Hom.hom f) = Ideal.span {e}",
"ppTerm": "?m.33",
"assigned": true,
"usedConstants": [
"RingHom.ker.congr_simp",
"I... | [] | constructor
· intro H
obtain ⟨e, he, he'⟩ := PrimeSpectrum.isClopen_iff_zeroLocus.mp
⟨PrimeSpectrum.isClosed_range_comap_of_surjective _ _ hf,
(Spec.map f).isOpenEmbedding.isOpen_range⟩
refine ⟨e, he, ?_⟩
let φ : R ⟶ _ := (CommRingCat.ofHom (Ideal.Quotient.mk (.span {e})))
have : IsOpenI... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.AlgebraicGeometry.Morphisms.OpenImmersion | {
"line": 39,
"column": 2
} | {
"line": 65,
"column": 51
} | {
"line": 67,
"column": 0
} | [
{
"pp": "R S : CommRingCat\nf : R ⟶ S\nhf : Function.Surjective ⇑(CommRingCat.Hom.hom f)\n⊢ IsOpenImmersion (Spec.map f) ↔ ∃ e, IsIdempotentElem e ∧ RingHom.ker (CommRingCat.Hom.hom f) = Ideal.span {e}",
"ppTerm": "?m.33",
"assigned": true,
"usedConstants": [
"RingHom.ker.congr_simp",
"I... | [] | constructor
· intro H
obtain ⟨e, he, he'⟩ := PrimeSpectrum.isClopen_iff_zeroLocus.mp
⟨PrimeSpectrum.isClosed_range_comap_of_surjective _ _ hf,
(Spec.map f).isOpenEmbedding.isOpen_range⟩
refine ⟨e, he, ?_⟩
let φ : R ⟶ _ := (CommRingCat.ofHom (Ideal.Quotient.mk (.span {e})))
have : IsOpenI... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.AlgebraicGeometry.Stalk | {
"line": 312,
"column": 27
} | {
"line": 317,
"column": 29
} | {
"line": 319,
"column": 0
} | [
{
"pp": "X : Scheme\nR : CommRingCat\ninst✝ : IsLocalRing ↑R\nf : Spec R ⟶ X\nU : X.Opens\nhU : f (closedPoint ↑R) ∈ U\n⊢ X.presheaf.germ U (f (closedPoint ↑R)) hU ≫ stalkClosedPointTo f =\n Hom.app f U ≫ (Functor.mapIso (Spec R).presheaf (eqToIso ⋯).op ≪≫ ΓSpecIso R).hom",
"ppTerm": "?m.69",
"assign... | [] | by
rw [stalkClosedPointTo, Scheme.Hom.germ_stalkMap_assoc, Iso.trans_hom]
congr 1
rw [← Iso.eq_comp_inv, Category.assoc, ΓSpecIso_hom_stalkClosedPointIso_inv]
simp only [Functor.mapIso_hom, Iso.op_hom, eqToIso.hom,
TopCat.Presheaf.germ_res] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.AlgebraicGeometry.IdealSheaf.Basic | {
"line": 708,
"column": 4
} | {
"line": 711,
"column": 33
} | {
"line": 712,
"column": 2
} | [
{
"pp": "case refine_1.a\nX Y : Scheme\nf : X.Hom Y\ninst✝ : QuasiCompact f\nU✝ U : ↑Y.affineOpens\ns : ↑Γ(Y, ↑U)\n⊢ Ideal.map (CommRingCat.Hom.hom (Y.presheaf.map (homOfLE ⋯).op)) (RingHom.ker (CommRingCat.Hom.hom (f.app ↑U))) ≤\n RingHom.ker (CommRingCat.Hom.hom (f.app ↑(Y.affineBasicOpen s)))",
"ppTer... | [] | refine Ideal.map_le_iff_le_comap.mpr fun x hx ↦ ?_
simp_rw [RingHom.comap_ker, ← CommRingCat.hom_comp, Scheme.affineBasicOpen_coe, f.naturality,
CommRingCat.hom_comp, ← RingHom.comap_ker]
exact Ideal.ker_le_comap _ hx | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.AlgebraicGeometry.IdealSheaf.Basic | {
"line": 708,
"column": 4
} | {
"line": 711,
"column": 33
} | {
"line": 712,
"column": 2
} | [
{
"pp": "case refine_1.a\nX Y : Scheme\nf : X.Hom Y\ninst✝ : QuasiCompact f\nU✝ U : ↑Y.affineOpens\ns : ↑Γ(Y, ↑U)\n⊢ Ideal.map (CommRingCat.Hom.hom (Y.presheaf.map (homOfLE ⋯).op)) (RingHom.ker (CommRingCat.Hom.hom (f.app ↑U))) ≤\n RingHom.ker (CommRingCat.Hom.hom (f.app ↑(Y.affineBasicOpen s)))",
"ppTer... | [] | refine Ideal.map_le_iff_le_comap.mpr fun x hx ↦ ?_
simp_rw [RingHom.comap_ker, ← CommRingCat.hom_comp, Scheme.affineBasicOpen_coe, f.naturality,
CommRingCat.hom_comp, ← RingHom.comap_ker]
exact Ideal.ker_le_comap _ hx | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.AlgebraicGeometry.Stalk | {
"line": 337,
"column": 2
} | {
"line": 340,
"column": 84
} | {
"line": 342,
"column": 0
} | [
{
"pp": "X : Scheme\nx : ↥X\n⊢ stalkClosedPointTo (X.fromSpecStalk x) = (X.presheaf.stalkCongr ⋯).hom",
"ppTerm": "?m.30",
"assigned": true,
"usedConstants": [
"AlgebraicGeometry.PresheafedSpace.Hom",
"Eq.mpr",
"AlgebraicGeometry.Spec",
"AlgebraicGeometry.Scheme",
"Oppo... | [] | refine TopCat.Presheaf.stalk_hom_ext _ fun U hxU ↦ ?_
simp only [TopCat.Presheaf.stalkCongr_hom, TopCat.Presheaf.germ_stalkSpecializes]
have : X.fromSpecStalk x = Spec.map (𝟙 (X.presheaf.stalk x)) ≫ X.fromSpecStalk x := by simp
convert! germ_stalkClosedPointTo_Spec_fromSpecStalk (𝟙 (X.presheaf.stalk x)) U hxU | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.AlgebraicGeometry.Stalk | {
"line": 337,
"column": 2
} | {
"line": 340,
"column": 84
} | {
"line": 342,
"column": 0
} | [
{
"pp": "X : Scheme\nx : ↥X\n⊢ stalkClosedPointTo (X.fromSpecStalk x) = (X.presheaf.stalkCongr ⋯).hom",
"ppTerm": "?m.30",
"assigned": true,
"usedConstants": [
"AlgebraicGeometry.PresheafedSpace.Hom",
"Eq.mpr",
"AlgebraicGeometry.Spec",
"AlgebraicGeometry.Scheme",
"Oppo... | [] | refine TopCat.Presheaf.stalk_hom_ext _ fun U hxU ↦ ?_
simp only [TopCat.Presheaf.stalkCongr_hom, TopCat.Presheaf.germ_stalkSpecializes]
have : X.fromSpecStalk x = Spec.map (𝟙 (X.presheaf.stalk x)) ≫ X.fromSpecStalk x := by simp
convert! germ_stalkClosedPointTo_Spec_fromSpecStalk (𝟙 (X.presheaf.stalk x)) U hxU | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.AlgebraicGeometry.IdealSheaf.Basic | {
"line": 727,
"column": 6
} | {
"line": 728,
"column": 79
} | {
"line": 729,
"column": 4
} | [
{
"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 RingHom.ker (CommRingCat.Hom.hom (f.app ↑(Y.affineBasicOpen... | [] | · simp only [Scheme.affineBasicOpen_coe, RingHom.mem_ker] at hx
rw [← IsLocalization.mk'_spec' (M := .powers s), map_mul, hx, mul_zero] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.AlgebraicGeometry.Stalk | {
"line": 389,
"column": 4
} | {
"line": 389,
"column": 8
} | {
"line": 390,
"column": 4
} | [
{
"pp": "X Y : Scheme\nf✝ : X ⟶ Y\nU V : X.Opens\nhU : IsAffineOpen U\nhV : IsAffineOpen V\nR : CommRingCat\ninst✝ : IsLocalRing ↑R\nx : ↥X\nf : X.presheaf.stalk x ⟶ R\nhf : IsLocalHom (CommRingCat.Hom.hom f)\n⊢ (fun f ↦ ⟨f (closedPoint ↑R), ⟨Scheme.stalkClosedPointTo f, ⋯⟩⟩)\n ((fun xf ↦ Spec.map ↑xf.snd ... | [
"X Y : Scheme\nf✝ : X ⟶ Y\nU V : X.Opens\nhU : IsAffineOpen U\nhV : IsAffineOpen V\nR : CommRingCat\ninst✝ : IsLocalRing ↑R\nx : ↥X\nf : X.presheaf.stalk x ⟶ R\nhf : IsLocalHom (CommRingCat.Hom.hom f)\n⊢ ⟨x, ⟨f, hf⟩⟩ =\n (fun f ↦ ⟨f (closedPoint ↑R), ⟨Scheme.stalkClosedPointTo f, ⋯⟩⟩)\n ((fun xf ↦ Spec.map ... | symm | Lean.Elab.Tactic.evalSymm | Lean.Parser.Tactic.symm |
Mathlib.AlgebraicGeometry.IdealSheaf.Basic | {
"line": 863,
"column": 10
} | {
"line": 863,
"column": 78
} | {
"line": 864,
"column": 8
} | [
{
"pp": "case a.inr.refine_1.h\nX Y : Scheme\nf : X ⟶ Y\ninst✝ : QuasiCompact f\nthis : ∀ {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... | [] | exact (ConcreteCategory.bijective_of_isIso ((𝒰.f i).appIso ⊤).inv).2 | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.AlgebraicGeometry.IdealSheaf.Basic | {
"line": 863,
"column": 10
} | {
"line": 863,
"column": 78
} | {
"line": 864,
"column": 8
} | [
{
"pp": "case a.inr.refine_1.h\nX Y : Scheme\nf : X ⟶ Y\ninst✝ : QuasiCompact f\nthis : ∀ {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... | [] | exact (ConcreteCategory.bijective_of_isIso ((𝒰.f i).appIso ⊤).inv).2 | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.AlgebraicGeometry.IdealSheaf.Basic | {
"line": 863,
"column": 10
} | {
"line": 863,
"column": 78
} | {
"line": 864,
"column": 8
} | [
{
"pp": "case a.inr.refine_1.h\nX Y : Scheme\nf : X ⟶ Y\ninst✝ : QuasiCompact f\nthis : ∀ {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... | [] | exact (ConcreteCategory.bijective_of_isIso ((𝒰.f i).appIso ⊤).inv).2 | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.RingHom.EssFiniteType | {
"line": 45,
"column": 42
} | {
"line": 47,
"column": 18
} | {
"line": 49,
"column": 0
} | [
{
"pp": "R S T : Type u_4\nx✝⁴ : CommRing R\nx✝³ : CommRing S\nx✝² : CommRing T\nx✝¹ : Algebra R S\nx✝ : Algebra R T\nh : (algebraMap R T).EssFiniteType\n⊢ (algebraMap S (TensorProduct R S T)).EssFiniteType",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"RingHom.essFiniteType_algebr... | [] | by
rw [essFiniteType_algebraMap] at h ⊢
infer_instance | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.Finiteness.FiniteTypeLocal | {
"line": 58,
"column": 2
} | {
"line": 58,
"column": 45
} | {
"line": 59,
"column": 2
} | [
{
"pp": "case h\nR : 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\nh... | [
"R : Type u_1\nS : Type u_2\ninst✝⁷ : CommRing R\ninst✝⁶ : CommRing S\ninst✝⁵ : Algebra R S\nS' : Type u_4\ninst✝⁴ : CommRing S'\ninst✝³ : Algebra S S'\ninst✝² : Algebra R S'\ninst✝¹ : IsScalarTower R S S'\nM : Submonoid S\ninst✝ : IsLocalization M S'\nx : S\ns : Finset S'\nA : Subalgebra R S\nhA₁ : ↑(finsetInteger... | convert! A.mul_mem hx' (hA₂ a.prop) using 1 | Mathlib.Tactic._aux_Mathlib_Tactic_Convert___macroRules_Mathlib_Tactic_convert!_1 | Mathlib.Tactic.convert! |
Mathlib.RingTheory.Finiteness.FiniteTypeLocal | {
"line": 121,
"column": 4
} | {
"line": 123,
"column": 60
} | {
"line": 124,
"column": 4
} | [
{
"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 = ⊤\nt : (r : ↥s) → Finset (Localization.Away ↑r)\nht : ∀ (r : ↥s), adjoin R ↑(t r) = ⊤\nl : ↑↑s →₀ S\nhl : (Finsupp.linearCombi... | [
"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 = ⊤\nt : (r : ↥s) → Finset (Localization.Away ↑r)\nht : ∀ (r : ↥s), adjoin R ↑(t r) = ⊤\nl : ↑↑s →₀ S\nhl : (Finsupp.linearCombination S Sub... | rw [show ∀ A : Set S, (∃ n, (r : S) ^ n • x ∈ Algebra.adjoin R A) ↔
(∃ m : (Submonoid.powers (r : S)), (m : S) • x ∈ Algebra.adjoin R A) by
{ exact fun _ => by simp [Submonoid.mem_powers_iff] }] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.AlgebraicGeometry.Morphisms.Affine | {
"line": 309,
"column": 4
} | {
"line": 309,
"column": 63
} | {
"line": 311,
"column": 0
} | [
{
"pp": "case mpr\nX Y : Scheme\ninst✝² : IsAffine Y\nf : X ⟶ Y\nH : ∀ (U V : X.Opens), IsAffineOpen U → IsAffineOpen V → IsAffineOpen (U ⊓ V)\nU₁ U₂ : Scheme\nf₁ : U₁ ⟶ X\nf₂ : U₂ ⟶ X\ninst✝¹ : IsAffine U₁\ninst✝ : IsAffine U₂\nh₁ : IsOpenImmersion f₁\nh₂ : IsOpenImmersion f₂\nthis : IsAffine ↑(Scheme.Hom.open... | [] | exact .of_isIso (pullback.fst f₁ f₂ ≫ f₁).isoOpensRange.hom | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.RingTheory.RingHom.Finite | {
"line": 65,
"column": 2
} | {
"line": 76,
"column": 32
} | {
"line": 78,
"column": 0
} | [
{
"pp": "⊢ LocalizationPreserves @Finite",
"ppTerm": "?m.8",
"assigned": true,
"usedConstants": [
"Algebra.algebraMapSubmonoid._proof_1",
"Eq.mpr",
"RingHom.instRingHomClass",
"CommRing",
"RingHom.LocalizationPreserves._proof_2",
"IsLocalization.map",
"RingH... | [] | introv R hf
letI := f.toAlgebra
letI := ((algebraMap S S').comp f).toAlgebra
let f' : R' →+* S' := IsLocalization.map S' f (Submonoid.le_comap_map M)
letI := f'.toAlgebra
have : IsScalarTower R R' S' := IsScalarTower.of_algebraMap_eq'
(IsLocalization.map_comp M.le_comap_map).symm
have : IsScalarTower R ... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.RingHom.Finite | {
"line": 65,
"column": 2
} | {
"line": 76,
"column": 32
} | {
"line": 78,
"column": 0
} | [
{
"pp": "⊢ LocalizationPreserves @Finite",
"ppTerm": "?m.8",
"assigned": true,
"usedConstants": [
"Algebra.algebraMapSubmonoid._proof_1",
"Eq.mpr",
"RingHom.instRingHomClass",
"CommRing",
"RingHom.LocalizationPreserves._proof_2",
"IsLocalization.map",
"RingH... | [] | introv R hf
letI := f.toAlgebra
letI := ((algebraMap S S').comp f).toAlgebra
let f' : R' →+* S' := IsLocalization.map S' f (Submonoid.le_comap_map M)
letI := f'.toAlgebra
have : IsScalarTower R R' S' := IsScalarTower.of_algebraMap_eq'
(IsLocalization.map_comp M.le_comap_map).symm
have : IsScalarTower R ... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.AlgebraicGeometry.Morphisms.ClosedImmersion | {
"line": 63,
"column": 2
} | {
"line": 63,
"column": 82
} | {
"line": 65,
"column": 0
} | [
{
"pp": "X Y : Scheme\nf : X ⟶ Y\n⊢ IsClosedImmersion f ↔ IsPreimmersion f ∧ IsClosed (Set.range ⇑f)",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"AlgebraicGeometry.SheafedSpace.instTopologicalSpaceCarrierCarrier",
"AlgebraicGeometry.SurjectiveOnStalks",
... | [] | rw [isClosedImmersion_iff, isPreimmersion_iff, and_assoc, isClosedEmbedding_iff] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.AlgebraicGeometry.Morphisms.ClosedImmersion | {
"line": 63,
"column": 2
} | {
"line": 63,
"column": 82
} | {
"line": 65,
"column": 0
} | [
{
"pp": "X Y : Scheme\nf : X ⟶ Y\n⊢ IsClosedImmersion f ↔ IsPreimmersion f ∧ IsClosed (Set.range ⇑f)",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"AlgebraicGeometry.SheafedSpace.instTopologicalSpaceCarrierCarrier",
"AlgebraicGeometry.SurjectiveOnStalks",
... | [] | rw [isClosedImmersion_iff, isPreimmersion_iff, and_assoc, isClosedEmbedding_iff] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.AlgebraicGeometry.Morphisms.ClosedImmersion | {
"line": 63,
"column": 2
} | {
"line": 63,
"column": 82
} | {
"line": 65,
"column": 0
} | [
{
"pp": "X Y : Scheme\nf : X ⟶ Y\n⊢ IsClosedImmersion f ↔ IsPreimmersion f ∧ IsClosed (Set.range ⇑f)",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"AlgebraicGeometry.SheafedSpace.instTopologicalSpaceCarrierCarrier",
"AlgebraicGeometry.SurjectiveOnStalks",
... | [] | rw [isClosedImmersion_iff, isPreimmersion_iff, and_assoc, isClosedEmbedding_iff] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Spectrum.Prime.ConstructibleSet | {
"line": 52,
"column": 47
} | {
"line": 52,
"column": 60
} | {
"line": 54,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝ : CommSemiring R\n⊢ map (RingHom.id R) = id",
"ppTerm": "?m.8",
"assigned": true,
"usedConstants": [
"congrArg",
"CommSemiring.toSemiring",
"PrimeSpectrum.BasicConstructibleSetData",
"id",
"PrimeSpectrum.BasicConstructibleSetData.map_id",
... | [] | ext : 1; simp | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Spectrum.Prime.ConstructibleSet | {
"line": 52,
"column": 47
} | {
"line": 52,
"column": 60
} | {
"line": 54,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝ : CommSemiring R\n⊢ map (RingHom.id R) = id",
"ppTerm": "?m.8",
"assigned": true,
"usedConstants": [
"congrArg",
"CommSemiring.toSemiring",
"PrimeSpectrum.BasicConstructibleSetData",
"id",
"PrimeSpectrum.BasicConstructibleSetData.map_id",
... | [] | ext : 1; simp | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.AlgebraicGeometry.Noetherian | {
"line": 136,
"column": 4
} | {
"line": 136,
"column": 42
} | {
"line": 137,
"column": 4
} | [
{
"pp": "case mp\nX : Scheme\n𝒰 : X.OpenCover\ninst✝ : ∀ (i : 𝒰.I₀), IsAffine (𝒰.X i)\nh : IsLocallyNoetherian X\ni : 𝒰.I₀\n⊢ IsNoetherianRing ↑Γ(𝒰.X i, ⊤)",
"ppTerm": "?mp",
"assigned": true,
"usedConstants": [
"AlgebraicGeometry.Scheme",
"CategoryTheory.PreZeroHypercover.f",
... | [
"case mp\nX : Scheme\n𝒰 : X.OpenCover\ninst✝ : ∀ (i : 𝒰.I₀), IsAffine (𝒰.X i)\nh : IsLocallyNoetherian X\ni : 𝒰.I₀\nU : X.Opens := Scheme.Hom.opensRange (𝒰.f i)\n⊢ IsNoetherianRing ↑Γ(𝒰.X i, ⊤)"
] | let U := Scheme.Hom.opensRange (𝒰.f i) | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLet___1 | Lean.Parser.Tactic.tacticLet__ |
Mathlib.AlgebraicGeometry.Geometrically.Basic | {
"line": 73,
"column": 2
} | {
"line": 80,
"column": 95
} | {
"line": 82,
"column": 0
} | [
{
"pp": "P : ObjectProperty Scheme\ninst✝ : P.IsClosedUnderIsomorphisms\n⊢ IsZariskiLocalAtTarget (geometrically P)",
"ppTerm": "?m.4",
"assigned": true,
"usedConstants": [
"AlgebraicGeometry.IsIntegral",
"Iff.mpr",
"Eq.mpr",
"CategoryTheory.MorphismProperty",
"Algebrai... | [] | rw [geometrically_eq_universally]
refine universally_isZariskiLocalAtTarget _ fun {X} Y f ι U hU H _ _ ↦ ?_
obtain ⟨y⟩ := (inferInstance : Nonempty Y)
obtain ⟨i, hy⟩ := hU.exists_mem y
have heq : U i = ⊤ := eq_top_iff.mpr fun z _ ↦ by rwa [Subsingleton.elim z y]
let e : ↑(U i) ≅ Y := Y.isoOfEq heq ≪≫ Y.topIso... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.AlgebraicGeometry.Geometrically.Basic | {
"line": 73,
"column": 2
} | {
"line": 80,
"column": 95
} | {
"line": 82,
"column": 0
} | [
{
"pp": "P : ObjectProperty Scheme\ninst✝ : P.IsClosedUnderIsomorphisms\n⊢ IsZariskiLocalAtTarget (geometrically P)",
"ppTerm": "?m.4",
"assigned": true,
"usedConstants": [
"AlgebraicGeometry.IsIntegral",
"Iff.mpr",
"Eq.mpr",
"CategoryTheory.MorphismProperty",
"Algebrai... | [] | rw [geometrically_eq_universally]
refine universally_isZariskiLocalAtTarget _ fun {X} Y f ι U hU H _ _ ↦ ?_
obtain ⟨y⟩ := (inferInstance : Nonempty Y)
obtain ⟨i, hy⟩ := hU.exists_mem y
have heq : U i = ⊤ := eq_top_iff.mpr fun z _ ↦ by rwa [Subsingleton.elim z y]
let e : ↑(U i) ≅ Y := Y.isoOfEq heq ≪≫ Y.topIso... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.AlgebraicGeometry.Noetherian | {
"line": 225,
"column": 2
} | {
"line": 232,
"column": 33
} | {
"line": 233,
"column": 2
} | [
{
"pp": "X : Scheme\ninst✝ : IsLocallyNoetherian X\nU V : ↑X.affineOpens\nhInd : Topology.IsInducing ⇑(IsAffineOpen.fromSpec ⋯)\n⊢ IsCompact (⇑(IsAffineOpen.fromSpec ⋯) ⁻¹' (↑↑U ∩ ↑↑V))",
"ppTerm": "?m.35",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"AlgebraicGeometry.Spec",
"A... | [
"X : Scheme\ninst✝ : IsLocallyNoetherian X\nU V : ↑X.affineOpens\nhInd : Topology.IsInducing ⇑(IsAffineOpen.fromSpec ⋯)\n⊢ ↑↑U ∩ ↑↑V ⊆ Set.range ⇑(IsAffineOpen.fromSpec ⋯)"
] | · rw [← Set.preimage_inter_range, IsAffineOpen.range_fromSpec, Set.inter_comm]
apply hInd.isCompact_preimage'
· apply (noetherianSpace_set_iff _).mp
· convert! noetherianSpace_of_isAffineOpen U.1 U.2
apply IsLocallyNoetherian.component_noetherian
· exact Set.inter_subset_left
· rw [IsAff... | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.AlgebraicGeometry.QuasiAffine | {
"line": 105,
"column": 48
} | {
"line": 108,
"column": 41
} | {
"line": 110,
"column": 0
} | [
{
"pp": "X✝ Y : Scheme\nf : X✝ ⟶ Y\nX : Scheme\ninst✝ : X.IsQuasiAffine\nx : ↥X\nx✝ : x ∈ ⊤\n⊢ x ∈ ⨆ i, X.basicOpen ↑i",
"ppTerm": "?m.22",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"AlgebraicGeometry.SheafedSpace.instTopologicalSpaceCarrierCarrier",
"Lattice.toSemilatticeSup... | [] | by
obtain ⟨_, ⟨_, ⟨r, hr, rfl⟩, rfl⟩, hxr, -⟩ :=
(IsQuasiAffine.isBasis_basicOpen X).exists_subset_of_mem_open (Set.mem_univ x) isOpen_univ
exact Opens.mem_iSup.mpr ⟨⟨r, hr⟩, hxr⟩ | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Limits.Shapes.Pullback.Connected | {
"line": 72,
"column": 76
} | {
"line": 72,
"column": 84
} | {
"line": 72,
"column": 84
} | [
{
"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 : ... | [
"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 : Cocone G\nj✝... | Cocone.w | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.Limits.Types.ColimitTypeFiltered | {
"line": 117,
"column": 2
} | {
"line": 120,
"column": 66
} | {
"line": 122,
"column": 0
} | [
{
"pp": "case mpr\nJ : Type u\ninst✝¹ : Category.{v, u} J\ninst✝ : IsFiltered J\nF : J ⥤ Type w₀\nc : F.CoconeTypes\n⊢ (∀ (j : J) (x x' : F.obj j),\n c.ι j x = c.ι j x' → ∃ k f, (ConcreteCategory.hom (F.map f)) x = (ConcreteCategory.hom (F.map f)) x') →\n ∀ (j j' : J) (x : F.obj j) (x' : F.obj j'),\n ... | [] | · intro h j j' x x' eq
obtain ⟨k, g, eq⟩ := h (max j j') (F.map (leftToMax _ _) x)
(F.map (rightToMax _ _) x') (by simpa only [c.ι_naturality_apply])
exact ⟨k, leftToMax _ _ ≫ g, rightToMax _ _ ≫ g, by simp [eq]⟩ | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.RingTheory.Spectrum.Prime.ChevalleyComplexity | {
"line": 356,
"column": 36
} | {
"line": 357,
"column": 66
} | {
"line": 358,
"column": 4
} | [
{
"pp": "R₀ : Type u_1\ninst✝² : CommRing R₀\nn : ℕ\nR : Type u_6\ninst✝¹ : CommRing R\ninst✝ : Algebra R₀ R\nc : R\ni : Fin n\ne : InductionObj R n\nhi : c = (e.val i).leadingCoeff\nhc : c ≠ 0\nq₁ : R →ₐ[R₀] Localization.Away c := IsScalarTower.toAlgHom R₀ R (Localization.Away c)\nq₂ : R →ₐ[R₀] R ⧸ Ideal.span ... | [] | by
simpa using (Set.biUnion_union (SetLike.coe S₁) S₂ _).symm | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.AlgebraicGeometry.FunctionField | {
"line": 74,
"column": 2
} | {
"line": 74,
"column": 6
} | {
"line": 75,
"column": 2
} | [
{
"pp": "X Y : Scheme\nf : X ⟶ Y\ninst✝¹ : IsOpenImmersion f\nhX : IrreducibleSpace ↥X\ninst✝ : IrreducibleSpace ↥Y\n⊢ Set.univ = closure (⇑f '' Set.univ)",
"ppTerm": "?m.96",
"assigned": true,
"usedConstants": [
"AlgebraicGeometry.SheafedSpace.instTopologicalSpaceCarrierCarrier",
"Algeb... | [
"X Y : Scheme\nf : X ⟶ Y\ninst✝¹ : IsOpenImmersion f\nhX : IrreducibleSpace ↥X\ninst✝ : IrreducibleSpace ↥Y\n⊢ closure (⇑f '' Set.univ) = Set.univ"
] | symm | Lean.Elab.Tactic.evalSymm | Lean.Parser.Tactic.symm |
Mathlib.AlgebraicGeometry.Cover.Directed | {
"line": 80,
"column": 49
} | {
"line": 84,
"column": 32
} | {
"line": 86,
"column": 0
} | [
{
"pp": "P : MorphismProperty Scheme\nX : Scheme\n𝒰 : Cover (precoverage P) X\ninst✝¹ : Category.{v_1, u_1} 𝒰.I₀\ninst✝ : 𝒰.LocallyDirected\ni j : 𝒰.I₀\nxi : ↥(𝒰.X i)\nxj : ↥(𝒰.X j)\nh : (𝒰.f i) xi = (𝒰.f j) xj\n⊢ ∃ k fi fj xk, (𝒰.trans fi) xk = xi ∧ (𝒰.trans fj) xk = xj",
"ppTerm": "?m.62",
"... | [] | by
obtain ⟨z, rfl, rfl⟩ := Scheme.Pullback.exists_preimage_pullback xi xj h
obtain ⟨k, fi, fj, xk, rfl⟩ := 𝒰.exists_lift_trans_eq z
use k, fi, fj, xk
simp [← Scheme.Hom.comp_apply] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.AlgebraicGeometry.Morphisms.UniversallyInjective | {
"line": 131,
"column": 4
} | {
"line": 131,
"column": 74
} | {
"line": 132,
"column": 2
} | [
{
"pp": "X Y : Scheme\nf : X ⟶ Y\ntfae_1_iff_4 : UniversallyInjective f ↔ Surjective (pullback.diagonal f)\ntfae_3_to_2 :\n (Injective ⇑f ∧ ∀ (x : ↥X), (CommRingCat.Hom.hom (Scheme.Hom.residueFieldMap f x)).IsPurelyInseparable) →\n ∀ (K : Type u) [inst : Field K], Injective fun g ↦ g ≫ f\nh : ∀ (K : Type u)... | [] | rw [← Scheme.Hom.comp_apply, ← hφ₂, Scheme.fromSpecResidueField_apply] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.AlgebraicGeometry.Cover.Over | {
"line": 138,
"column": 4
} | {
"line": 150,
"column": 45
} | {
"line": 152,
"column": 0
} | [
{
"pp": "P : MorphismProperty Scheme\nS : Scheme\ninst✝⁸ : P.IsStableUnderBaseChange\ninst✝⁷ : IsJointlySurjectivePreserving P\nX W : Scheme\n𝒰 : Cover (precoverage P) X\nf : W ⟶ X\ninst✝⁶ : W.Over S\ninst✝⁵ : X.Over S\ninst✝⁴ : Cover.Over S 𝒰\ninst✝³ : Hom.IsOver f S\nQ : MorphismProperty Scheme\ninst✝² : Q.... | [] | rw [presieve₀_mem_precoverage_iff]
refine ⟨fun x ↦ ?_, fun j ↦ ?_⟩
· obtain ⟨i, hy⟩ := Cover.exists_eq (𝒰.pullback₁ f) x
use i
exact (mem_range_iff_of_surjective ((𝒰.pullback₁ f).f i) _
((PreservesPullback.iso (MorphismProperty.Over.forget Q _ _ ⋙ Over.forget S)
(f.asOverProp S) ... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.AlgebraicGeometry.Cover.Over | {
"line": 138,
"column": 4
} | {
"line": 150,
"column": 45
} | {
"line": 152,
"column": 0
} | [
{
"pp": "P : MorphismProperty Scheme\nS : Scheme\ninst✝⁸ : P.IsStableUnderBaseChange\ninst✝⁷ : IsJointlySurjectivePreserving P\nX W : Scheme\n𝒰 : Cover (precoverage P) X\nf : W ⟶ X\ninst✝⁶ : W.Over S\ninst✝⁵ : X.Over S\ninst✝⁴ : Cover.Over S 𝒰\ninst✝³ : Hom.IsOver f S\nQ : MorphismProperty Scheme\ninst✝² : Q.... | [] | rw [presieve₀_mem_precoverage_iff]
refine ⟨fun x ↦ ?_, fun j ↦ ?_⟩
· obtain ⟨i, hy⟩ := Cover.exists_eq (𝒰.pullback₁ f) x
use i
exact (mem_range_iff_of_surjective ((𝒰.pullback₁ f).f i) _
((PreservesPullback.iso (MorphismProperty.Over.forget Q _ _ ⋙ Over.forget S)
(f.asOverProp S) ... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Spectrum.Prime.ChevalleyComplexity | {
"line": 458,
"column": 4
} | {
"line": 458,
"column": 75
} | {
"line": 459,
"column": 4
} | [
{
"pp": "case hP₂\nR✝ : Type u_2\ninst✝² : CommRing R✝\nn : ℕ\nR : Type u_2\ninst✝¹ : CommRing R\ng : InductionObj R n\ni : Fin n\nhi : (g.val i).Monic\nhi_min : ∀ (j : Fin n), j ≠ i → g.val j = 0\nR₀✝ : Type u_1\nR₀ : CommRing R₀✝\ninst✝ : Algebra R₀✝ R\nf : R[X]\nM : Type u_2 := R[X] ⧸ Ideal.span {g.val i}\nt... | [
"case hP₂\nR✝ : Type u_2\ninst✝² : CommRing R✝\nn : ℕ\nR : Type u_2\ninst✝¹ : CommRing R\ng : InductionObj R n\ni : Fin n\nhi : (g.val i).Monic\nhi_min : ∀ (j : Fin n), j ≠ i → g.val j = 0\nR₀✝ : Type u_1\nR₀ : CommRing R₀✝\ninst✝ : Algebra R₀✝ R\nf : R[X]\nM : Type u_2 := R[X] ⧸ Ideal.span {g.val i}\nthis✝ : Modul... | have : Module.Finite R M := .of_basis (AdjoinRoot.powerBasis' hi).basis | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.AlgebraicGeometry.Cover.Over | {
"line": 190,
"column": 19
} | {
"line": 190,
"column": 84
} | {
"line": 190,
"column": 85
} | [
{
"pp": "P : MorphismProperty Scheme\nS : Scheme\ninst✝⁸ : P.IsStableUnderBaseChange\ninst✝⁷ : IsJointlySurjectivePreserving P\nX W : Scheme\n𝒰 : Cover (precoverage P) X\nf : W ⟶ X\ninst✝⁶ : W.Over S\ninst✝⁵ : X.Over S\ninst✝⁴ : Cover.Over S 𝒰\ninst✝³ : Hom.IsOver f S\nQ : MorphismProperty Scheme\ninst✝² : Q.... | [] | by exact (pullback.snd ((𝒰.f j).asOverProp S) (f.asOverProp S)).w | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.Sets.CompactOpenCovered | {
"line": 152,
"column": 19
} | {
"line": 152,
"column": 28
} | {
"line": 152,
"column": 29
} | [
{
"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)\nB : (i : ι) → Set (Opens (X i))\nhB : ∀ (i : ι), IsBasis (B i)\nhBc : ∀ (i : ι), ∀ U ∈ B i, IsCompact U.carrier\nU : Set S\ns : Set ι\nhs : s.Finite\nV : (i : ι) → i ∈ s → Opens (X i)\nhc : ∀ ... | [
"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)\nB : (i : ι) → Set (Opens (X i))\nhB : ∀ (i : ι), IsBasis (B i)\nhBc : ∀ (i : ι), ∀ U ∈ B i, IsCompact U.carrier\nU : Set S\ns : Set ι\nhs : s.Finite\nV : (i : ι) → i ∈ s → Opens (X i)\nhc : ∀ (i : ι) (h :... | coe_sSup, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.AlgebraicGeometry.ColimitsOver | {
"line": 219,
"column": 68
} | {
"line": 219,
"column": 76
} | {
"line": 219,
"column": 76
} | [
{
"pp": "case refine_2\nP : MorphismProperty Scheme\ninst✝⁸ : P.IsStableUnderBaseChange\ninst✝⁷ : P.IsMultiplicative\nS : Scheme\nJ : Type u_1\ninst✝⁶ : Category.{v_1, u_1} J\nD : J ⥤ P.Over ⊤ S\n𝒰 : S.OpenCover\ninst✝⁵ : Category.{v_2, ?u.266} 𝒰.I₀\ninst✝⁴ : LocallyDirected 𝒰\nd : ColimitGluingData D 𝒰\nin... | [
"case refine_2\nP : MorphismProperty Scheme\ninst✝⁸ : P.IsStableUnderBaseChange\ninst✝⁷ : P.IsMultiplicative\nS : Scheme\nJ : Type u_1\ninst✝⁶ : Category.{v_1, u_1} J\nD : J ⥤ P.Over ⊤ S\n𝒰 : S.OpenCover\ninst✝⁵ : Category.{v_2, ?u.266} 𝒰.I₀\ninst✝⁴ : LocallyDirected 𝒰\nd : ColimitGluingData D 𝒰\ninst✝³ :\n ∀ ... | Cocone.w | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.AlgebraicGeometry.AffineTransitionLimit | {
"line": 512,
"column": 73
} | {
"line": 512,
"column": 91
} | {
"line": 513,
"column": 2
} | [
{
"pp": "I : Type u\ninst✝³ : Category.{u, u} I\nS X : Scheme\nD : I ⥤ Scheme\nt : D ⟶ (Functor.const I).obj S\nf : X ⟶ S\ninst✝² : ∀ (i : I), CompactSpace ↥(D.obj i)\ninst✝¹ : IsCofiltered I\ninst✝ : ∀ {i j : I} (f : i ⟶ j), IsAffineHom (D.map f)\nA : ExistsHomHomCompEqCompAux D t f\nW : (pullback.diagonalObj ... | [] | ext <;> simp [hab] | Lean.Parser.Tactic.«_aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tactic_<;>__1» | Lean.Parser.Tactic.«tactic_<;>_» |
Mathlib.AlgebraicGeometry.EllipticCurve.Weierstrass | {
"line": 452,
"column": 44
} | {
"line": 453,
"column": 28
} | {
"line": 455,
"column": 0
} | [
{
"pp": "R : Type u\ninst✝² : CommRing R\nW : WeierstrassCurve R\ninst✝¹ : W.IsElliptic\nA : Type v\ninst✝ : CommRing A\nf : R →+* A\n⊢ ↑(W.map f).Δ' = f ↑W.Δ'",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"Units.val",
"Eq.mpr",
"WeierstrassCurve.Δ",
"congrArg",
... | [] | by
rw [coe_Δ', map_Δ, coe_Δ'] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.AlgebraicGeometry.EllipticCurve.Affine.Basic | {
"line": 132,
"column": 72
} | {
"line": 132,
"column": 78
} | {
"line": 133,
"column": 2
} | [
{
"pp": "case inl\nR : Type r\ninst✝¹ : CommRing R\nW : Affine R\ninst✝ : IsDomain R\nh✝ : ¬Irreducible W.polynomial\nf g : R[X]\nh0 : 3 = f.degree + g.degree\nh1 : { a := 0, b := 0, c := W.a₁, d := W.a₃ }.toPoly.degree = (f + g).degree\nh : f.degree = 0 ∧ g.degree = 3\n⊢ 1 < 3",
"ppTerm": "?inl✝",
"ass... | [] | decide | Lean.Elab.Tactic.evalDecide | Lean.Parser.Tactic.decide |
Mathlib.AlgebraicGeometry.EllipticCurve.Affine.Basic | {
"line": 132,
"column": 72
} | {
"line": 132,
"column": 78
} | {
"line": 133,
"column": 2
} | [
{
"pp": "case inl\nR : Type r\ninst✝¹ : CommRing R\nW : Affine R\ninst✝ : IsDomain R\nh✝ : ¬Irreducible W.polynomial\nf g : R[X]\nh0 : 3 = f.degree + g.degree\nh1 : { a := 0, b := 0, c := W.a₁, d := W.a₃ }.toPoly.degree = (f + g).degree\nh : f.degree = 0 ∧ g.degree = 3\n⊢ 0 < 3",
"ppTerm": "?inl",
"assi... | [] | decide | Lean.Elab.Tactic.evalDecide | Lean.Parser.Tactic.decide |
Mathlib.AlgebraicGeometry.EllipticCurve.Affine.Basic | {
"line": 132,
"column": 72
} | {
"line": 132,
"column": 78
} | {
"line": 133,
"column": 2
} | [
{
"pp": "case inr.inl\nR : Type r\ninst✝¹ : CommRing R\nW : Affine R\ninst✝ : IsDomain R\nh✝ : ¬Irreducible W.polynomial\nf g : R[X]\nh0 : 3 = f.degree + g.degree\nh1 : { a := 0, b := 0, c := W.a₁, d := W.a₃ }.toPoly.degree = (f + g).degree\nh : f.degree = 1 ∧ g.degree = 2\n⊢ 1 < 2",
"ppTerm": "?inr.inl✝",
... | [] | decide | Lean.Elab.Tactic.evalDecide | Lean.Parser.Tactic.decide |
Mathlib.AlgebraicGeometry.EllipticCurve.Affine.Basic | {
"line": 132,
"column": 72
} | {
"line": 132,
"column": 78
} | {
"line": 133,
"column": 2
} | [
{
"pp": "case inr.inl\nR : Type r\ninst✝¹ : CommRing R\nW : Affine R\ninst✝ : IsDomain R\nh✝ : ¬Irreducible W.polynomial\nf g : R[X]\nh0 : 3 = f.degree + g.degree\nh1 : { a := 0, b := 0, c := W.a₁, d := W.a₃ }.toPoly.degree = (f + g).degree\nh : f.degree = 1 ∧ g.degree = 2\n⊢ 1 < 2",
"ppTerm": "?inr.inl",
... | [] | decide | Lean.Elab.Tactic.evalDecide | Lean.Parser.Tactic.decide |
Mathlib.AlgebraicGeometry.EllipticCurve.Affine.Basic | {
"line": 133,
"column": 71
} | {
"line": 133,
"column": 77
} | {
"line": 135,
"column": 0
} | [
{
"pp": "case inr.inr.inl\nR : Type r\ninst✝¹ : CommRing R\nW : Affine R\ninst✝ : IsDomain R\nh✝ : ¬Irreducible W.polynomial\nf g : R[X]\nh0 : 3 = f.degree + g.degree\nh1 : { a := 0, b := 0, c := W.a₁, d := W.a₃ }.toPoly.degree = (f + g).degree\nh : f.degree = 2 ∧ g.degree = 1\n⊢ 1 < 2",
"ppTerm": "?inr.inr... | [] | decide | Lean.Elab.Tactic.evalDecide | Lean.Parser.Tactic.decide |
Mathlib.AlgebraicGeometry.EllipticCurve.Affine.Basic | {
"line": 133,
"column": 71
} | {
"line": 133,
"column": 77
} | {
"line": 135,
"column": 0
} | [
{
"pp": "case inr.inr.inl\nR : Type r\ninst✝¹ : CommRing R\nW : Affine R\ninst✝ : IsDomain R\nh✝ : ¬Irreducible W.polynomial\nf g : R[X]\nh0 : 3 = f.degree + g.degree\nh1 : { a := 0, b := 0, c := W.a₁, d := W.a₃ }.toPoly.degree = (f + g).degree\nh : f.degree = 2 ∧ g.degree = 1\n⊢ 1 < 2",
"ppTerm": "?inr.inr... | [] | decide | Lean.Elab.Tactic.evalDecide | Lean.Parser.Tactic.decide |
Mathlib.AlgebraicGeometry.EllipticCurve.Affine.Basic | {
"line": 133,
"column": 71
} | {
"line": 133,
"column": 77
} | {
"line": 135,
"column": 0
} | [
{
"pp": "case inr.inr.inr\nR : Type r\ninst✝¹ : CommRing R\nW : Affine R\ninst✝ : IsDomain R\nh✝ : ¬Irreducible W.polynomial\nf g : R[X]\nh0 : 3 = f.degree + g.degree\nh1 : { a := 0, b := 0, c := W.a₁, d := W.a₃ }.toPoly.degree = (f + g).degree\nh : f.degree = 3 ∧ g.degree = 0\n⊢ 1 < 3",
"ppTerm": "?inr.inr... | [] | decide | Lean.Elab.Tactic.evalDecide | Lean.Parser.Tactic.decide |
Mathlib.AlgebraicGeometry.EllipticCurve.Affine.Basic | {
"line": 133,
"column": 71
} | {
"line": 133,
"column": 77
} | {
"line": 135,
"column": 0
} | [
{
"pp": "case inr.inr.inr\nR : Type r\ninst✝¹ : CommRing R\nW : Affine R\ninst✝ : IsDomain R\nh✝ : ¬Irreducible W.polynomial\nf g : R[X]\nh0 : 3 = f.degree + g.degree\nh1 : { a := 0, b := 0, c := W.a₁, d := W.a₃ }.toPoly.degree = (f + g).degree\nh : f.degree = 3 ∧ g.degree = 0\n⊢ 0 < 3",
"ppTerm": "?inr.inr... | [] | decide | Lean.Elab.Tactic.evalDecide | Lean.Parser.Tactic.decide |
Mathlib.AlgebraicGeometry.AffineTransitionLimit | {
"line": 882,
"column": 6
} | {
"line": 883,
"column": 84
} | {
"line": 884,
"column": 6
} | [
{
"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... | [
"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.pt\ni : ↥c.pt... | simp_rw [Scheme.Hom.appLE, ConcreteCategory.comp_apply, ht, TopCat.Presheaf.restrictOpen,
TopCat.Presheaf.restrict, ← ConcreteCategory.comp_apply, ← Functor.map_comp] | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | Mathlib.Tactic.tacticSimp_rw___ |
Mathlib.RingTheory.Spectrum.Prime.ChevalleyComplexity | {
"line": 847,
"column": 10
} | {
"line": 847,
"column": 37
} | {
"line": 847,
"column": 37
} | [
{
"pp": "R : Type u_2\ninst✝ : CommRing R\nm n : ℕ\nf : MvPolynomial (Fin n) R →ₐ[R] MvPolynomial (Fin m) R\nk : ℕ\nd : Multiset (Fin m)\nS : ConstructibleSetData (MvPolynomial (Fin m) R)\nhSn : ∀ C ∈ S, C.n ≤ k\nhS : ∀ C ∈ S, ∀ (j : Fin C.n), (C.g j).degrees ≤ d\nhf : ∀ (i : Fin n), (f (MvPolynomial.X i)).degr... | [] | by simp [commAlgEquiv_C, σ] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.AlgebraicGeometry.EllipticCurve.Affine.AddSubMap | {
"line": 92,
"column": 4
} | {
"line": 92,
"column": 43
} | {
"line": 94,
"column": 0
} | [
{
"pp": "case «2»\nR : Type u_1\ninst✝ : CommRing R\nW : WeierstrassCurve R\n⊢ (t ^ 2 - C 4 * s * u).IsHomogeneous 2",
"ppTerm": "?«2»",
"assigned": true,
"usedConstants": [
"Finsupp.instAddZeroClass",
"Nat.instMulZeroClass",
"AddMonoidAlgebra.semiring",
"HMul.hMul",
"C... | [] | exact isHomogeneous_X_pow .. |>.sub CXY | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
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