module stringlengths 16 90 | startPos dict | endPos dict | nextStartPos dict | goals listlengths 0 96 | goalsAfter listlengths 0 96 | ppTac stringlengths 1 14.5k | elaborator stringclasses 371
values | kind stringclasses 375
values |
|---|---|---|---|---|---|---|---|---|
Mathlib.Algebra.SkewMonoidAlgebra.Basic | {
"line": 390,
"column": 2
} | {
"line": 390,
"column": 27
} | {
"line": 390,
"column": 27
} | [
{
"pp": "k : Type u_1\nG : Type u_2\ninst✝ : AddCommMonoid k\nf : SkewMonoidAlgebra k G\n⊢ f.sum single = f",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"AddMonoid.toAddZeroClass",
"AddZeroClass.toAddZero",
"SkewMonoidAlgebra.toFinsupp_injective",
"AddZero.toZero... | [
"k : Type u_1\nG : Type u_2\ninst✝ : AddCommMonoid k\nf : SkewMonoidAlgebra k G\n⊢ (f.sum single).toFinsupp = f.toFinsupp"
] | apply toFinsupp_injective | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.Algebra.Star.UnitaryStarAlgAut | {
"line": 41,
"column": 26
} | {
"line": 41,
"column": 41
} | {
"line": 43,
"column": 0
} | [
{
"pp": "S : Type u_1\nR : Type u_2\ninst✝⁴ : Semiring R\ninst✝³ : StarMul R\ninst✝² : SMul S R\ninst✝¹ : IsScalarTower S R R\ninst✝ : SMulCommClass S R R\ng h : ↥(unitary R)\na✝ : R\n⊢ (let __RingEquiv := MulSemiringAction.toRingEquiv (ConjAct Rˣ) R (ConjAct.toConjAct (toUnits (g * h)));\n { toRingEquiv :... | [] | simp [mul_smul] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Algebra.Symmetrized | {
"line": 255,
"column": 41
} | {
"line": 255,
"column": 50
} | {
"line": 255,
"column": 51
} | [
{
"pp": "α : Type u_1\ninst✝¹ : Semiring α\ninst✝ : Invertible 2\nx✝ : αˢʸᵐ\n⊢ sym (⅟2 * (0 + unsym x✝ * 0)) = 0",
"ppTerm": "?m.307",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Equiv.instEquivLike",
"HMul.hMul",
"MulZeroC... | [
"α : Type u_1\ninst✝¹ : Semiring α\ninst✝ : Invertible 2\nx✝ : αˢʸᵐ\n⊢ sym (⅟2 * (0 + 0)) = 0"
] | mul_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Symmetrized | {
"line": 256,
"column": 8
} | {
"line": 256,
"column": 17
} | {
"line": 256,
"column": 18
} | [
{
"pp": "α : Type u_1\ninst✝¹ : Semiring α\ninst✝ : Invertible 2\nx✝ : αˢʸᵐ\n⊢ sym (⅟2 * 0) = 0",
"ppTerm": "?m.315",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Equiv.instEquivLike",
"HMul.hMul",
"MulZeroClass.toMul",
... | [
"α : Type u_1\ninst✝¹ : Semiring α\ninst✝ : Invertible 2\nx✝ : αˢʸᵐ\n⊢ sym 0 = 0"
] | mul_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Symmetrized | {
"line": 258,
"column": 41
} | {
"line": 258,
"column": 50
} | {
"line": 258,
"column": 51
} | [
{
"pp": "α : Type u_1\ninst✝¹ : Semiring α\ninst✝ : Invertible 2\nx✝ : αˢʸᵐ\n⊢ sym (⅟2 * (unsym x✝ * 0 + 0)) = 0",
"ppTerm": "?m.339",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Equiv.instEquivLike",
"HMul.hMul",
"MulZeroC... | [
"α : Type u_1\ninst✝¹ : Semiring α\ninst✝ : Invertible 2\nx✝ : αˢʸᵐ\n⊢ sym (⅟2 * (0 + 0)) = 0"
] | mul_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Symmetrized | {
"line": 259,
"column": 8
} | {
"line": 259,
"column": 17
} | {
"line": 259,
"column": 18
} | [
{
"pp": "α : Type u_1\ninst✝¹ : Semiring α\ninst✝ : Invertible 2\nx✝ : αˢʸᵐ\n⊢ sym (⅟2 * 0) = 0",
"ppTerm": "?m.347",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Equiv.instEquivLike",
"HMul.hMul",
"MulZeroClass.toMul",
... | [
"α : Type u_1\ninst✝¹ : Semiring α\ninst✝ : Invertible 2\nx✝ : αˢʸᵐ\n⊢ sym 0 = 0"
] | mul_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Tropical.BigOperators | {
"line": 120,
"column": 42
} | {
"line": 120,
"column": 67
} | {
"line": 120,
"column": 68
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝¹ : ConditionallyCompleteLinearOrder R\ninst✝ : Fintype S\nf : S → Tropical (WithTop R)\n⊢ untrop (∑ i, f i) = sInf ((fun i ↦ untrop (f i)) '' ↑univ)",
"ppTerm": "?m.26",
"assigned": true,
"usedConstants": [
"WithTop.instInfSet",
"Eq.mpr",
... | [
"R : Type u_1\nS : Type u_2\ninst✝¹ : ConditionallyCompleteLinearOrder R\ninst✝ : Fintype S\nf : S → Tropical (WithTop R)\n⊢ sInf (untrop ∘ f '' ↑univ) = sInf ((fun i ↦ untrop (f i)) '' ↑univ)"
] | untrop_sum_eq_sInf_image, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Star.LinearMap | {
"line": 301,
"column": 2
} | {
"line": 301,
"column": 18
} | {
"line": 303,
"column": 0
} | [
{
"pp": "R : Type u_4\nV : Type u_5\ninst✝⁵ : CommRing R\ninst✝⁴ : InvolutiveStar R\ninst✝³ : AddCommGroup V\ninst✝² : StarAddMonoid V\ninst✝¹ : Module R V\ninst✝ : StarModule R V\nf : WithConv (End R V)\nx : R\n⊢ IsUnit (star (toConv (x • 1 - (star f).ofConv))).ofConv ↔ IsUnit (star x • 1 - f.ofConv)",
"pp... | [] | simp [one_eq_id] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Topology.Sheaves.SheafOfFunctions | {
"line": 61,
"column": 4
} | {
"line": 61,
"column": 77
} | {
"line": 62,
"column": 4
} | [
{
"pp": "X : TopCat\nT : ↑X → Type u_1\nι : Type u_2\nU : ι → Opens ↑X\nsf : (i : ι) → ToType ((X.presheafToTypes T).obj (Opposite.op (U i)))\nhsf : (X.presheafToTypes T).IsCompatible U sf\nindex : ↥(iSup U) → ι\nindex_spec : ∀ (x : ↥(iSup U)), ↑x ∈ U (index x)\n⊢ ∃! s, (X.presheafToTypes T).IsGluing U sf s",
... | [
"X : TopCat\nT : ↑X → Type u_1\nι : Type u_2\nU : ι → Opens ↑X\nsf : (i : ι) → ToType ((X.presheafToTypes T).obj (Opposite.op (U i)))\nhsf : (X.presheafToTypes T).IsCompatible U sf\nindex : ↥(iSup U) → ι\nindex_spec : ∀ (x : ↥(iSup U)), ↑x ∈ U (index x)\ns : (x : ↥(iSup U)) → T ↑x := fun x ↦ sf (index x) ⟨↑x, ⋯⟩\n⊢... | let s : ∀ x : ↑(iSup U), T x := fun x => sf (index x) ⟨x.1, index_spec x⟩ | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLet___1 | Lean.Parser.Tactic.tacticLet__ |
Mathlib.AlgebraicGeometry.Spec | {
"line": 399,
"column": 4
} | {
"line": 401,
"column": 31
} | {
"line": 402,
"column": 4
} | [
{
"pp": "case h₃\nR S : CommRingCat\nf : R ⟶ S\np : PrimeSpectrum ↑R\ninst✝ : Algebra ↑R ↑S\nx : ↑S\nhx : (toPushforwardStalkAlgHom R S p) x = 0\n⊢ ∃ m, m • x = 0",
"ppTerm": "?h₃",
"assigned": true,
"usedConstants": [
"CategoryTheory.Functor.op",
"CategoryTheory.Functor",
"Lattice... | [
"case h₃\nR S : CommRingCat\nf : R ⟶ S\np : PrimeSpectrum ↑R\ninst✝ : Algebra ↑R ↑S\nx : ↑S\nhx :\n (((TopCat.Presheaf.pushforward CommRingCat (Spec.topMap (CommRingCat.ofHom (algebraMap ↑R ↑S)))).obj\n (structureSheaf ↑S).obj).germ\n ⊤ p trivial).hom'\n ((CommRingCat.ofHom\n ... | rw [toPushforwardStalkAlgHom_apply,
← (toPushforwardStalk (CommRingCat.ofHom (algebraMap ↑R ↑S)) p).hom.map_zero,
toPushforwardStalk] at hx | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.AlgebraicGeometry.Cover.MorphismProperty | {
"line": 103,
"column": 10
} | {
"line": 105,
"column": 9
} | {
"line": 107,
"column": 0
} | [
{
"pp": "K : Precoverage Scheme\nX Y Z : Scheme\n𝒰 : Cover K X\nf : X ⟶ Z\ng : Y ⟶ Z\ninst✝ : ∀ (x : 𝒰.I₀), HasPullback (𝒰.f x ≫ f) g\nP Q : MorphismProperty Scheme\nJ : Type u_1\nobj : J → Scheme\nmap : (j : J) → obj j ⟶ X\ncovers : ∀ (x : ↥X), ∃ j y, (map j) y = x\nmap_prop : ∀ (j : J), P (map j)\n⊢ { I₀ :... | [] | by
simp_rw [presieve₀_mem_precoverage_iff, Set.mem_range]
grind | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.AlgebraicGeometry.OpenImmersion | {
"line": 832,
"column": 5
} | {
"line": 835,
"column": 46
} | {
"line": 835,
"column": 46
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\nX✝ Y✝ : Scheme\nf✝ : X✝ ⟶ Y✝\nH : IsOpenImmersion f✝\nP X Y Z : Scheme\nfst : P ⟶ X\nsnd : P ⟶ Y\nf : X ⟶ Z\ng : Y ⟶ Z\nh : IsPullback fst snd f g\ninst✝ : IsOpenImmersion g\np : ↥P\nx : ↥X\nhx : fst p = x\nthis : IsOpenImmersion fst\n⊢ (Z.presheaf.stalkCongr ⋯ ≪... | [] | by
subst hx
simp [← Scheme.Hom.stalkMap_comp, ← Scheme.Hom.stalkMap_comp,
Scheme.Hom.stalkMap_congr_hom _ _ h.w] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.AlgebraicGeometry.StructureSheaf | {
"line": 465,
"column": 4
} | {
"line": 467,
"column": 85
} | {
"line": 468,
"column": 4
} | [
{
"pp": "R M : Type u\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nU : Opens ↑(PrimeSpectrum.Top R)\nhU : IsCompact ↑U\ns : (structureSheafInType R M).obj.obj (op U)\ng : ↥U → R\nhxg : ∀ (x : ↥U), ↑x ∈ basicOpen (g x)\nigU : ∀ (x : ↥U), basicOpen (g x) ≤ U\nf : ↥U → M\nH :\n ∀ (x : ↥U),\n... | [
"R M : Type u\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nU : Opens ↑(PrimeSpectrum.Top R)\nhU : IsCompact ↑U\ns : (structureSheafInType R M).obj.obj (op U)\ng : ↥U → R\nhxg : ∀ (x : ↥U), ↑x ∈ basicOpen (g x)\nigU : ∀ (x : ↥U), basicOpen (g x) ≤ U\nf : ↥U → M\nH :\n ∀ (x : ↥U),\n const (f... | · refine congr((structureSheafInType R M).obj.map (homOfLE ((PrimeSpectrum.basicOpen_mul (g i)
(g j)).trans_le inf_le_left)).op $(H i)).symm.trans (Subtype.ext <| funext fun a ↦ ?_)
exact LocalizedModule.mk_eq.mpr ⟨1, by simp [Submonoid.smul_def, ← smul_assoc]⟩ | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.AlgebraicGeometry.StructureSheaf | {
"line": 484,
"column": 4
} | {
"line": 484,
"column": 45
} | {
"line": 485,
"column": 2
} | [
{
"pp": "case refine_1.e_a\nR M : Type u\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nU : Opens ↑(PrimeSpectrum.Top R)\nhU : IsCompact ↑U\ns : (structureSheafInType R M).obj.obj (op U)\ng : ↥U → R\nhxg : ∀ (x : ↥U), ↑x ∈ basicOpen (g x)\nigU : ∀ (x : ↥U), basicOpen (g x) ≤ U\nf : ↥U → M\nH... | [] | convert! (hn i j).symm using 1 <;> module | Lean.Parser.Tactic.«_aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tactic_<;>__1» | Lean.Parser.Tactic.«tactic_<;>_» |
Mathlib.AlgebraicGeometry.Restrict | {
"line": 561,
"column": 2
} | {
"line": 561,
"column": 34
} | {
"line": 561,
"column": 35
} | [
{
"pp": "X Y : Scheme\nf : X ⟶ Y\nU : Y.Opens\n⊢ IsPullback (f ∣_ U) (f ⁻¹ᵁ U).ι U.ι f",
"ppTerm": "?m.25",
"assigned": true,
"usedConstants": [
"AlgebraicGeometry.PresheafedSpace.carrier",
"TopologicalSpace.Opens.instPartialOrder",
"CommRingCat",
"PartialOrder.toPreorder",
... | [
"case H\nX Y : Scheme\nf : X ⟶ Y\nU : Y.Opens\n⊢ (f ⁻¹ᵁ U).ι ≫ f = f ∣_ U ≫ U.ι",
"case H'\nX Y : Scheme\nf : X ⟶ Y\nU : Y.Opens\n⊢ f ⁻¹ᵁ Scheme.Hom.opensRange U.ι = Scheme.Hom.opensRange (f ⁻¹ᵁ U).ι"
] | apply IsOpenImmersion.isPullback | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.AlgebraicGeometry.Restrict | {
"line": 568,
"column": 47
} | {
"line": 568,
"column": 70
} | {
"line": 568,
"column": 71
} | [
{
"pp": "case refine_1\nX : Scheme\nU V W : X.Opens\nhU : U ≤ W\nhV : V ≤ W\n⊢ V.ι ''ᵁ (Opens.map ((X.homOfLE hV).base ≫ W.ι.base)).obj U = U ⊓ V",
"ppTerm": "?refine_1",
"assigned": true,
"usedConstants": [
"AlgebraicGeometry.Scheme.Hom.opensFunctor",
"Eq.mpr",
"AlgebraicGeometry.... | [
"case refine_1\nX : Scheme\nU V W : X.Opens\nhU : U ≤ W\nhV : V ≤ W\n⊢ V.ι ''ᵁ (Opens.map (X.homOfLE hV ≫ W.ι).base).obj U = U ⊓ V"
] | ← Scheme.Hom.comp_base, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.AlgebraicGeometry.GammaSpecAdjunction | {
"line": 289,
"column": 4
} | {
"line": 289,
"column": 46
} | {
"line": 290,
"column": 4
} | [
{
"pp": "X Y : LocallyRingedSpace\nf : X ⟶ Y\n⊢ X.toΓSpec ≫ (Γ.rightOp ⋙ Spec.toLocallyRingedSpace).map f = (𝟭 LocallyRingedSpace).map f ≫ Y.toΓSpec",
"ppTerm": "?m.29",
"assigned": true,
"usedConstants": [
"Opposite",
"CategoryTheory.CategoryStruct.toQuiver",
"AlgebraicGeometry.L... | [
"case w\nX Y : LocallyRingedSpace\nf : X ⟶ Y\n⊢ ((𝟭 LocallyRingedSpace).obj X).toΓSpec.base ≫ (Spec.locallyRingedSpaceMap (Γ.rightOp.map f).unop).base =\n ((𝟭 LocallyRingedSpace).map f ≫ Y.toΓSpec).base",
"case h\nX Y : LocallyRingedSpace\nf : X ⟶ Y\n⊢ ∀ (r : ↑(unop (Γ.rightOp.obj Y))),\n (Γ.rightOp.map f... | apply LocallyRingedSpace.comp_ring_hom_ext | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.AlgebraicGeometry.GammaSpecAdjunction | {
"line": 317,
"column": 2
} | {
"line": 317,
"column": 44
} | {
"line": 318,
"column": 2
} | [
{
"pp": "R : CommRingCat\n⊢ identityToΓSpec.app (Spec.toLocallyRingedSpace.obj (op R)) ≫\n Spec.toLocallyRingedSpace.map (SpecΓIdentity.inv.app R).op =\n 𝟙 ((𝟭 LocallyRingedSpace).obj (Spec.toLocallyRingedSpace.obj (op R)))",
"ppTerm": "?m.41",
"assigned": true,
"usedConstants": [
"C... | [
"case w\nR : CommRingCat\n⊢ ((𝟭 LocallyRingedSpace).obj (Spec.toLocallyRingedSpace.obj (op R))).toΓSpec.base ≫\n (Spec.locallyRingedSpaceMap (SpecΓIdentity.inv.app R).op.unop).base =\n (𝟙 ((𝟭 LocallyRingedSpace).obj (Spec.toLocallyRingedSpace.obj (op R)))).base",
"case h\nR : CommRingCat\n⊢ ∀ (r : ↑(un... | apply LocallyRingedSpace.comp_ring_hom_ext | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.AlgebraicGeometry.StructureSheaf | {
"line": 1032,
"column": 2
} | {
"line": 1032,
"column": 17
} | {
"line": 1033,
"column": 2
} | [
{
"pp": "R M : Type u\ninst✝⁵ : CommRing R\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\nS : Type u\ninst✝² : CommRing S\nN : Type u\ninst✝¹ : AddCommGroup N\ninst✝ : Module S N\nσ : R →+* S\nf : M →ₛₗ[σ] N\nU : Opens ↑(PrimeSpectrum.Top R)\nV : Opens ↑(PrimeSpectrum.Top S)\nhUV : V.carrier ⊆ PrimeSpectrum.com... | [
"R M : Type u\ninst✝⁵ : CommRing R\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\nS : Type u\ninst✝² : CommRing S\nN : Type u\ninst✝¹ : AddCommGroup N\ninst✝ : Module S N\nσ : R →+* S\nf : M →ₛₗ[σ] N\nU : Opens ↑(PrimeSpectrum.Top R)\nV : Opens ↑(PrimeSpectrum.Top S)\nhUV : V.carrier ⊆ PrimeSpectrum.comap σ ⁻¹' U.c... | refine ⟨hs, ?_⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.AlgebraicGeometry.AffineScheme | {
"line": 527,
"column": 4
} | {
"line": 527,
"column": 24
} | {
"line": 527,
"column": 24
} | [
{
"pp": "X Y : Scheme\nU : Y.Opens\nhU : IsAffineOpen U\nf : X ⟶ Y\ninst✝ : IsOpenImmersion f\nhU' : U ≤ Scheme.Hom.opensRange f\n⊢ IsAffineOpen (Scheme.Hom.opensRange f ⊓ U)",
"ppTerm": "?m.45",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"Eq.mpr",
"AlgebraicGeometry.SheafedSp... | [
"X Y : Scheme\nU : Y.Opens\nhU : IsAffineOpen U\nf : X ⟶ Y\ninst✝ : IsOpenImmersion f\nhU' : U ≤ Scheme.Hom.opensRange f\n⊢ IsAffineOpen U"
] | inf_eq_right.mpr hU' | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.AlgebraicGeometry.AffineScheme | {
"line": 585,
"column": 2
} | {
"line": 585,
"column": 39
} | {
"line": 586,
"column": 2
} | [
{
"pp": "X : Scheme\nU : X.Opens\nhU : IsAffineOpen U\nf : ↑Γ(X, U)\n⊢ hU.fromSpec ''ᵁ PrimeSpectrum.basicOpen f = X.basicOpen f",
"ppTerm": "?m.25",
"assigned": true,
"usedConstants": [
"AlgebraicGeometry.Scheme.Hom.opensFunctor",
"Eq.mpr",
"AlgebraicGeometry.Spec",
"Algebra... | [
"X : Scheme\nU : X.Opens\nhU : IsAffineOpen U\nf : ↑Γ(X, U)\n⊢ hU.fromSpec ''ᵁ hU.fromSpec ⁻¹ᵁ X.basicOpen f = X.basicOpen f"
] | rw [← hU.fromSpec_preimage_basicOpen] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.CategoryTheory.GlueData | {
"line": 465,
"column": 8
} | {
"line": 465,
"column": 54
} | {
"line": 466,
"column": 4
} | [
{
"pp": "C : Type u₁\ninst✝¹ : Category.{v, u₁} C\nC' : Type u₂\ninst✝ : Category.{v, u₂} C'\nD : GlueData' C\ni k : D.J\nhik : ¬i = k\n⊢ ((if hij : i = i then\n (pullbackSymmetry (D.f' i i) (D.f' i k)).hom ≫\n pullback.map (D.f' i k) (D.f' i i) (D.f' i k) (D.f' i i) (eqToHom ⋯) (eqToHom ⋯) (eqT... | [] | simp [hik, Ne.symm hik, fst_eq_snd_of_mono_eq] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.CategoryTheory.GlueData | {
"line": 465,
"column": 8
} | {
"line": 465,
"column": 54
} | {
"line": 466,
"column": 4
} | [
{
"pp": "C : Type u₁\ninst✝¹ : Category.{v, u₁} C\nC' : Type u₂\ninst✝ : Category.{v, u₂} C'\nD : GlueData' C\ni k : D.J\nhik : ¬i = k\n⊢ ((if hij : i = i then\n (pullbackSymmetry (D.f' i i) (D.f' i k)).hom ≫\n pullback.map (D.f' i k) (D.f' i i) (D.f' i k) (D.f' i i) (eqToHom ⋯) (eqToHom ⋯) (eqT... | [] | simp [hik, Ne.symm hik, fst_eq_snd_of_mono_eq] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.GlueData | {
"line": 465,
"column": 8
} | {
"line": 465,
"column": 54
} | {
"line": 466,
"column": 4
} | [
{
"pp": "C : Type u₁\ninst✝¹ : Category.{v, u₁} C\nC' : Type u₂\ninst✝ : Category.{v, u₂} C'\nD : GlueData' C\ni k : D.J\nhik : ¬i = k\n⊢ ((if hij : i = i then\n (pullbackSymmetry (D.f' i i) (D.f' i k)).hom ≫\n pullback.map (D.f' i k) (D.f' i i) (D.f' i k) (D.f' i i) (eqToHom ⋯) (eqToHom ⋯) (eqT... | [] | simp [hik, Ne.symm hik, fst_eq_snd_of_mono_eq] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Geometry.RingedSpace.LocallyRingedSpace.HasColimits | {
"line": 91,
"column": 70
} | {
"line": 104,
"column": 22
} | {
"line": 104,
"column": 22
} | [
{
"pp": "ι : Type v\ninst✝ : Small.{u, v} ι\nF : Discrete ι ⥤ LocallyRingedSpace\ns : Cocone F\n⊢ ∀ (x : ↑(coproductCofan F).pt.toTopCat),\n IsLocalHom\n (CommRingCat.Hom.hom\n (PresheafedSpace.Hom.stalkMap (colimit.desc (F ⋙ forgetToSheafedSpace) (forgetToSheafedSpace.mapCocone s)).hom\n ... | [] | by
intro x
obtain ⟨i, y, ⟨⟩⟩ := SheafedSpace.colimit_exists_rep (F ⋙ forgetToSheafedSpace) x
have := PresheafedSpace.stalkMap.comp
(colimit.ι (F ⋙ forgetToSheafedSpace) i).hom
(colimit.desc (F ⋙ forgetToSheafedSpace) (forgetToSheafedSpace.mapCocone s)).hom y
simp only... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.AlgebraicGeometry.AffineScheme | {
"line": 1101,
"column": 76
} | {
"line": 1101,
"column": 99
} | {
"line": 1102,
"column": 4
} | [
{
"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",
"ppTerm": "?m.40",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"AlgebraicGeometry.Spec",
"AlgebraicGeometry.Sheaf... | [
"X : Scheme\ninst✝ : IsAffine X\ns : Set ↑Γ(X, ⊤)\n⊢ ⇑(X.isoSpec.inv ≫ X.toSpecΓ) ⁻¹' PrimeSpectrum.zeroLocus s = PrimeSpectrum.zeroLocus s"
] | ← Scheme.Hom.comp_base, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.AlgebraicGeometry.Gluing | {
"line": 280,
"column": 2
} | {
"line": 285,
"column": 8
} | {
"line": 287,
"column": 0
} | [
{
"pp": "X : Scheme\n𝒰 : X.OpenCover\nx y z : 𝒰.I₀\n⊢ pullback (pullback.fst (𝒰.f x) (𝒰.f y)) (pullback.fst (𝒰.f x) (𝒰.f z)) ⟶\n pullback (pullback.fst (𝒰.f y) (𝒰.f z)) (pullback.fst (𝒰.f y) (𝒰.f x))",
"ppTerm": "?m.72",
"assigned": true,
"usedConstants": [
"CategoryTheory.Limits.... | [] | refine (pullbackRightPullbackFstIso _ _ _).hom ≫ ?_
refine ?_ ≫ (pullbackSymmetry _ _).hom
refine ?_ ≫ (pullbackRightPullbackFstIso _ _ _).inv
refine pullback.map _ _ _ _ (pullbackSymmetry _ _).hom (𝟙 _) (𝟙 _) ?_ ?_
· simp [pullback.condition]
· simp | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.AlgebraicGeometry.Gluing | {
"line": 280,
"column": 2
} | {
"line": 285,
"column": 8
} | {
"line": 287,
"column": 0
} | [
{
"pp": "X : Scheme\n𝒰 : X.OpenCover\nx y z : 𝒰.I₀\n⊢ pullback (pullback.fst (𝒰.f x) (𝒰.f y)) (pullback.fst (𝒰.f x) (𝒰.f z)) ⟶\n pullback (pullback.fst (𝒰.f y) (𝒰.f z)) (pullback.fst (𝒰.f y) (𝒰.f x))",
"ppTerm": "?m.72",
"assigned": true,
"usedConstants": [
"CategoryTheory.Limits.... | [] | refine (pullbackRightPullbackFstIso _ _ _).hom ≫ ?_
refine ?_ ≫ (pullbackSymmetry _ _).hom
refine ?_ ≫ (pullbackRightPullbackFstIso _ _ _).inv
refine pullback.map _ _ _ _ (pullbackSymmetry _ _).hom (𝟙 _) (𝟙 _) ?_ ?_
· simp [pullback.condition]
· simp | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.AlgebraicGeometry.AffineScheme | {
"line": 1289,
"column": 2
} | {
"line": 1289,
"column": 11
} | {
"line": 1290,
"column": 2
} | [
{
"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",
"ppTerm": "?m.40",
"assigned": true,
"usedConstants": [
"AlgebraicGeometry.Spec",
... | [
"R : CommRingCat\nx : PrimeSpectrum ↑R\ns : ↑(CommRingCat.of ↑R)\n⊢ (CommRingCat.Hom.hom (CommRingCat.ofHom (algebraMap (↑R) (Localization.AtPrime x.asIdeal)) ≫ (stalkIso R x).inv)) s =\n (CommRingCat.Hom.hom ((Scheme.ΓSpecIso R).inv ≫ (Spec R).presheaf.germ ⊤ x trivial)) s"
] | ext s : 2 | _private.Lean.Elab.Tactic.Ext.0.Lean.Elab.Tactic.Ext.evalExt | Lean.Elab.Tactic.Ext.ext |
Mathlib.AlgebraicGeometry.Gluing | {
"line": 366,
"column": 11
} | {
"line": 366,
"column": 33
} | {
"line": 366,
"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⊢ ((... | [
"X : Scheme\n𝒰 : X.OpenCover\ni : (gluedCover 𝒰).J\nx : ↥((gluedCover 𝒰).U i)\nj : (gluedCover 𝒰).J\ny : ↥((gluedCover 𝒰).U j)\nh : ((gluedCover 𝒰).ι i ≫ fromGlued 𝒰) x = ((gluedCover 𝒰).ι j ≫ fromGlued 𝒰) y\n⊢ ((gluedCover 𝒰).ι i) x = ((gluedCover 𝒰).ι j) y"
] | ← Scheme.Hom.comp_base | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.AlgebraicGeometry.Limits | {
"line": 403,
"column": 4
} | {
"line": 411,
"column": 67
} | {
"line": 414,
"column": 0
} | [
{
"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": 403,
"column": 4
} | {
"line": 411,
"column": 67
} | {
"line": 414,
"column": 0
} | [
{
"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.AlgebraicGeometry.Gluing | {
"line": 680,
"column": 8
} | {
"line": 680,
"column": 39
} | {
"line": 680,
"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... | [
"J : Type w\ninst✝⁴ : Category.{v, w} J\nF : J ⥤ Scheme\ninst✝³ : ∀ {i j : J} (f : i ⟶ j), IsOpenImmersion (F.map f)\ninst✝² : (F ⋙ forget).IsLocallyDirected\ninst✝¹ : Quiver.IsThin J\ninst✝ : Small.{u, w} J\ni j k : Shrink.{u, w} J\nx : failed to pretty print expression (use 'set_option pp.rawOnError true' for raw... | IsOpenImmersion.comp_lift_assoc | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.AlgebraicGeometry.Pullbacks | {
"line": 382,
"column": 14
} | {
"line": 382,
"column": 23
} | {
"line": 382,
"column": 24
} | [
{
"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",
"ppTerm": "?m.152",
"assigned": true,
"usedConstants": [
"AlgebraicGeo... | [
"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⊢ Multicoequalizer.π (gluing 𝒰 f g).diagram i ≫ p1 𝒰 f g = pullback.fst (𝒰.f i ≫ f) g ≫ 𝒰.f i"
] | gluing_ι, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.AlgebraicGeometry.Gluing | {
"line": 683,
"column": 51
} | {
"line": 683,
"column": 82
} | {
"line": 683,
"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... | [
"J : Type w\ninst✝⁴ : Category.{v, w} J\nF : J ⥤ Scheme\ninst✝³ : ∀ {i j : J} (f : i ⟶ j), IsOpenImmersion (F.map f)\ninst✝² : (F ⋙ forget).IsLocallyDirected\ninst✝¹ : Quiver.IsThin J\ninst✝ : Small.{u, w} J\ni j k : Shrink.{u, w} J\nx : failed to pretty print expression (use 'set_option pp.rawOnError true' for raw... | IsOpenImmersion.comp_lift_assoc | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.AlgebraicGeometry.Morphisms.UnderlyingMap | {
"line": 280,
"column": 4
} | {
"line": 281,
"column": 39
} | {
"line": 282,
"column": 2
} | [
{
"pp": "case hP₁\n⊢ ∀ {α β : Type u_1} [inst : TopologicalSpace α] [inst_1 : TopologicalSpace β] (f : α ≃ₜ β), SpecializingMap ⇑f",
"ppTerm": "?hP₁",
"assigned": true,
"usedConstants": [
"TopologicalSpace",
"Homeomorph.instEquivLike",
"Homeomorph.isClosedMap",
"IsClosedMap.s... | [] | introv
exact f.isClosedMap.specializingMap | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.AlgebraicGeometry.Morphisms.UnderlyingMap | {
"line": 280,
"column": 4
} | {
"line": 281,
"column": 39
} | {
"line": 282,
"column": 2
} | [
{
"pp": "case hP₁\n⊢ ∀ {α β : Type u_1} [inst : TopologicalSpace α] [inst_1 : TopologicalSpace β] (f : α ≃ₜ β), SpecializingMap ⇑f",
"ppTerm": "?hP₁",
"assigned": true,
"usedConstants": [
"TopologicalSpace",
"Homeomorph.instEquivLike",
"Homeomorph.isClosedMap",
"IsClosedMap.s... | [] | introv
exact f.isClosedMap.specializingMap | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.AlgebraicGeometry.Morphisms.UnderlyingMap | {
"line": 299,
"column": 4
} | {
"line": 299,
"column": 26
} | {
"line": 300,
"column": 4
} | [
{
"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[inst✝¹, inst✝] f\nhsp :\n ∀ (i : ι) (x : ↑(f ⁻¹' (U i).carrier)),\n closure[instTopologicalSpaceSubtype] {(U i).carrier.restrictPreimage f... | [
"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[inst✝¹, inst✝] f\nhsp :\n ∀ (i : ι) (x : ↑(f ⁻¹' (U i).carrier)),\n closure[instTopologicalSpaceSubtype] {(U i).carrier.restrictPreimage f x} ⊆\n ... | obtain ⟨i, hi⟩ := this | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.AlgebraicGeometry.Pullbacks | {
"line": 818,
"column": 39
} | {
"line": 818,
"column": 69
} | {
"line": 820,
"column": 0
} | [
{
"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))",
"ppTerm": "?m.25",
"assigned": true,
"usedConstants": [
"AlgebraicGeometry.Scheme",
"AlgebraicGeometry.Scheme.Pullback.instHasPullbacks",
"inferIns... | [] | exact Over.monObjMkPullbackSnd | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Geometry.RingedSpace.PresheafedSpace.Gluing | {
"line": 471,
"column": 4
} | {
"line": 476,
"column": 21
} | {
"line": 477,
"column": 2
} | [
{
"pp": "case 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.diagramOverOpen... | [] | 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": 471,
"column": 4
} | {
"line": 476,
"column": 21
} | {
"line": 477,
"column": 2
} | [
{
"pp": "case 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.diagramOverOpen... | [] | 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.Constructors | {
"line": 223,
"column": 6
} | {
"line": 223,
"column": 37
} | {
"line": 224,
"column": 6
} | [
{
"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... | [
"case of_sSup_eq_top.a.refine_2\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\ni₁ : X' ⟶ ... | · exact congr($(h.1.1) ⁻¹ᵁ U i) | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.RingTheory.RingHom.Locally | {
"line": 213,
"column": 42
} | {
"line": 245,
"column": 92
} | {
"line": 247,
"column": 0
} | [
{
"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": 265,
"column": 6
} | {
"line": 267,
"column": 92
} | {
"line": 268,
"column": 4
} | [
{
"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": 265,
"column": 6
} | {
"line": 267,
"column": 92
} | {
"line": 268,
"column": 4
} | [
{
"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.Constructors | {
"line": 383,
"column": 4
} | {
"line": 383,
"column": 49
} | {
"line": 384,
"column": 4
} | [
{
"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... | [
"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] [CommRing S] ↦... | 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": 160,
"column": 4
} | {
"line": 171,
"column": 13
} | {
"line": 172,
"column": 2
} | [
{
"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": 160,
"column": 4
} | {
"line": 171,
"column": 13
} | {
"line": 172,
"column": 2
} | [
{
"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": 212,
"column": 2
} | {
"line": 215,
"column": 27
} | {
"line": 216,
"column": 2
} | [
{
"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... | [
"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.affineOpens\nV₁ V... | 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.Morphisms.QuasiCompact | {
"line": 207,
"column": 44
} | {
"line": 207,
"column": 67
} | {
"line": 207,
"column": 68
} | [
{
"pp": "X : 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",
"ppTerm": "?m.132",
"assigned": true,
"usedConstants": [
"subset_refl._simp_1",
... | [
"X : 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 ⇑(IsOpenImmersion.lift U.ι f ⋯ ≫ U.ι) = ↑U"
] | ← Scheme.Hom.comp_base, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.AlgebraicGeometry.Morphisms.RingHomProperties | {
"line": 301,
"column": 2
} | {
"line": 310,
"column": 49
} | {
"line": 312,
"column": 0
} | [
{
"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)",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
... | [] | 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": 301,
"column": 2
} | {
"line": 310,
"column": 49
} | {
"line": 312,
"column": 0
} | [
{
"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)",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
... | [] | 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.QuasiCompact | {
"line": 293,
"column": 7
} | {
"line": 293,
"column": 16
} | {
"line": 293,
"column": 17
} | [
{
"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 ⋯... | [
"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 : (ConcreteCategory.hom (X.presheaf.map (homOfLE ⋯).op)) (f ^ (Fi... | mul_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.LocalProperties.Reduced | {
"line": 40,
"column": 2
} | {
"line": 40,
"column": 50
} | {
"line": 41,
"column": 2
} | [
{
"pp": "case 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 = (algebraMap R S) ... | [
"case 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 = (algebraMap R S) (y ^ n.succ)... | simp only [mul_assoc, zero_mul, mul_zero] at hm' | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.RingTheory.Ideal.Height | {
"line": 221,
"column": 35
} | {
"line": 221,
"column": 53
} | {
"line": 221,
"column": 54
} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommRing R\nI : Ideal R\ninst✝ : I.IsPrime\n⊢ I.primeHeight = 0 ↔ I ∈ minimalPrimes R",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"PrimeSpectrum.mk",
"congrArg",
"CommSemiring.toSemiring",
"PartialOrder.toPreorder"... | [
"R : Type u_1\ninst✝¹ : CommRing R\nI : Ideal R\ninst✝ : I.IsPrime\n⊢ Order.height { asIdeal := I, isPrime := inst✝ } = 0 ↔ I ∈ minimalPrimes R"
] | Ideal.primeHeight, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.AlgebraicGeometry.Morphisms.RingHomProperties | {
"line": 495,
"column": 4
} | {
"line": 495,
"column": 39
} | {
"line": 496,
"column": 4
} | [
{
"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 : ... | [
"case inr\nP : 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 : P ... | wlog hX : IsAffine X generalizing X | Mathlib.Tactic._aux_Mathlib_Tactic_WLOG___elabRules_Mathlib_Tactic_wlog_1 | Mathlib.Tactic.wlog |
Mathlib.RingTheory.Ideal.Height | {
"line": 368,
"column": 73
} | {
"line": 368,
"column": 91
} | {
"line": 369,
"column": 4
} | [
{
"pp": "R : Type u_1\ninst✝⁴ : CommRing R\nS : Submonoid R\nA : Type u_2\ninst✝³ : CommRing A\ninst✝² : Algebra R A\ninst✝¹ : IsLocalization S A\nJ : Ideal A\ninst✝ : J.IsPrime\n⊢ J.primeHeight = (comap (algebraMap R A) J).primeHeight",
"ppTerm": "?m.41",
"assigned": true,
"usedConstants": [
... | [
"R : Type u_1\ninst✝⁴ : CommRing R\nS : Submonoid R\nA : Type u_2\ninst✝³ : CommRing A\ninst✝² : Algebra R A\ninst✝¹ : IsLocalization S A\nJ : Ideal A\ninst✝ : J.IsPrime\n⊢ Order.height { asIdeal := J, isPrime := inst✝ } = (comap (algebraMap R A) J).primeHeight"
] | Ideal.primeHeight, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Ideal.Height | {
"line": 369,
"column": 4
} | {
"line": 369,
"column": 22
} | {
"line": 369,
"column": 23
} | [
{
"pp": "R : Type u_1\ninst✝⁴ : CommRing R\nS : Submonoid R\nA : Type u_2\ninst✝³ : CommRing A\ninst✝² : Algebra R A\ninst✝¹ : IsLocalization S A\nJ : Ideal A\ninst✝ : J.IsPrime\n⊢ Order.height { asIdeal := J, isPrime := inst✝ } = (comap (algebraMap R A) J).primeHeight",
"ppTerm": "?m.46",
"assigned": t... | [
"R : Type u_1\ninst✝⁴ : CommRing R\nS : Submonoid R\nA : Type u_2\ninst✝³ : CommRing A\ninst✝² : Algebra R A\ninst✝¹ : IsLocalization S A\nJ : Ideal A\ninst✝ : J.IsPrime\n⊢ Order.height { asIdeal := J, isPrime := inst✝ } = Order.height { asIdeal := comap (algebraMap R A) J, isPrime := ⋯ }"
] | Ideal.primeHeight, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.AlgebraicGeometry.Morphisms.RingHomProperties | {
"line": 523,
"column": 4
} | {
"line": 523,
"column": 55
} | {
"line": 524,
"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\nZ :... | [
"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 : Scheme\nhZ... | 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.Properties | {
"line": 198,
"column": 4
} | {
"line": 202,
"column": 13
} | {
"line": 203,
"column": 2
} | [
{
"pp": "case 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\nH :\n (ConcreteCategory.hom (X.sheaf.presheaf.germ (f ⁻¹ᵁ Scheme.Hom.opensRange f) x ⋯))\n ((ConcreteCategory.hom (Scheme.Hom.app f (Scheme.Hom.ope... | [] | · 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": 204,
"column": 8
} | {
"line": 204,
"column": 32
} | {
"line": 204,
"column": 33
} | [
{
"pp": "case 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",
"ppTerm": "?h₃",
"assigned": true,
"usedConstants": [
"AlgebraicGeometry.S... | [
"case h₃\nX : Scheme\nR : CommRingCat\nhX : IsReduced (Spec R)\ns : ↑Γ(Spec R, ⊤)\nhs : PrimeSpectrum.basicOpen ((ConcreteCategory.hom (Scheme.ΓSpecIso R).hom) s) = ⊥\nx : ↥(Spec R)\nhx : x ∈ ⊤\n⊢ (ConcreteCategory.hom ((Spec R).sheaf.presheaf.germ ⊤ x hx)) s = 0"
] | basicOpen_eq_of_affine', | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.AlgebraicGeometry.IdealSheaf.Subscheme | {
"line": 68,
"column": 2
} | {
"line": 69,
"column": 53
} | {
"line": 70,
"column": 2
} | [
{
"pp": "X : Scheme\nI : X.IdealSheafData\nU : ↑X.affineOpens\n⊢ RingHom.ker (CommRingCat.Hom.hom (Hom.appTop (I.glueDataObjι U))) =\n Ideal.comap (CommRingCat.Hom.hom (↑U).topIso.hom) (I.ideal U)",
"ppTerm": "?m.32",
"assigned": true,
"usedConstants": [
"Opposite",
"CommRingCat.carri... | [
"X : Scheme\nI : X.IdealSheafData\nU : ↑X.affineOpens\nφ : Γ(X, ↑U) ⟶ CommRingCat.of (↑Γ(X, ↑U) ⧸ I.ideal U) := CommRingCat.ofHom (Ideal.Quotient.mk (I.ideal U))\n⊢ RingHom.ker (CommRingCat.Hom.hom (Hom.appTop (I.glueDataObjι U))) =\n Ideal.comap (CommRingCat.Hom.hom (↑U).topIso.hom) (I.ideal U)"
] | let φ : Γ(X, U) ⟶ CommRingCat.of (Γ(X, U) ⧸ I.ideal U) :=
CommRingCat.ofHom (Ideal.Quotient.mk (I.ideal U)) | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLet___1 | Lean.Parser.Tactic.tacticLet__ |
Mathlib.AlgebraicGeometry.IdealSheaf.Subscheme | {
"line": 94,
"column": 43
} | {
"line": 94,
"column": 66
} | {
"line": 94,
"column": 67
} | [
{
"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",
"ppTerm": "?m.32",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Al... | [
"X : Scheme\nI : X.IdealSheafData\nU : ↑X.affineOpens\n⊢ ⇑((IsAffineOpen.isoSpec ⋯).inv ≫ (↑U).ι) '' PrimeSpectrum.zeroLocus ↑(I.ideal U) = X.zeroLocus ↑(I.ideal U) ∩ ↑↑U"
] | ← Scheme.Hom.comp_base, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.AlgebraicGeometry.Morphisms.QuasiSeparated | {
"line": 429,
"column": 2
} | {
"line": 429,
"column": 39
} | {
"line": 430,
"column": 2
} | [
{
"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",
"ppTerm": "?m.117",
"assigned": true,
"usedConstants": [
"Opposite",
"CommRingCat.carrier",
"AlgebraicGeome... | [
"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⊢ (f |_ X.basicOpen s) ⋯ = (0 |_ X.basicOpen s) ⋯"
] | apply exists_of_res_eq_of_qcqs hU hU' | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.AlgebraicGeometry.IdealSheaf.Basic | {
"line": 584,
"column": 8
} | {
"line": 584,
"column": 26
} | {
"line": 585,
"column": 8
} | [
{
"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 ... | [
"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 f)\nx : ↑Γ(X... | dsimp only at hx ⊢ | Lean.Elab.Tactic.evalDSimp | Lean.Parser.Tactic.dsimp |
Mathlib.AlgebraicGeometry.Stalk | {
"line": 57,
"column": 2
} | {
"line": 59,
"column": 31
} | {
"line": 61,
"column": 0
} | [
{
"pp": "X : Scheme\nU V : X.Opens\nhU : IsAffineOpen U\nhV : IsAffineOpen V\nx : ↥X\nhxU : x ∈ U\nhxV : x ∈ V\nU' : Opens ↥X\nh₁ : U' ∈ X.affineOpens\nh₂ : x ∈ U'\nh₃ : U' ≤ U ⊓ V\n⊢ fromSpecStalk h₁ h₂ = hV.fromSpecStalk hxV",
"ppTerm": "?m.67",
"assigned": true,
"usedConstants": [
"Eq.mpr",... | [] | · delta fromSpecStalk
rw [← hV.map_fromSpec h₁ (homOfLE <| h₃.trans inf_le_right).op, ← Spec.map_comp_assoc,
TopCat.Presheaf.germ_res] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.AlgebraicGeometry.IdealSheaf.Basic | {
"line": 710,
"column": 28
} | {
"line": 710,
"column": 47
} | {
"line": 710,
"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 ∈ RingHom.ker (CommRingCat.Hom.hom (f.app ↑U))\n⊢ x ∈\n RingHom.ker\n ((CommRingCat.Hom.hom (X.presheaf.map ((Opens.map f.base).map (homOfLE ⋯).op.unop).op)).comp\n (CommRin... | [
"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 ∈ RingHom.ker (CommRingCat.Hom.hom (f.app ↑U))\n⊢ x ∈\n Ideal.comap (CommRingCat.Hom.hom (f.app ↑U))\n (RingHom.ker (CommRingCat.Hom.hom (X.presheaf.map ((Opens.map f.base).map (homOfLE ⋯).op.... | ← RingHom.comap_ker | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.AlgebraicGeometry.IdealSheaf.Basic | {
"line": 734,
"column": 55
} | {
"line": 734,
"column": 74
} | {
"line": 734,
"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)))",
"ppTerm": "?m.30",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Rin... | [
"X Y Z : Scheme\nf : X ⟶ Y\ng : Y.Hom Z\nU : ↑Z.affineOpens\n⊢ RingHom.ker (CommRingCat.Hom.hom (g.app ↑U)) ≤\n Ideal.comap (CommRingCat.Hom.hom (g.app ↑U)) (RingHom.ker (CommRingCat.Hom.hom (app f (g ⁻¹ᵁ ↑U))))"
] | ← RingHom.comap_ker | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.AlgebraicGeometry.Morphisms.Affine | {
"line": 64,
"column": 28
} | {
"line": 64,
"column": 46
} | {
"line": 64,
"column": 46
} | [
{
"pp": "X 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)",
"ppTerm": "?m.30",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"AlgebraicGeometry.SheafedSpace.instTop... | [
"X 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).obj ((Opens.map g.base).obj U))"
] | Opens.map_comp_obj | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.AlgebraicGeometry.IdealSheaf.Basic | {
"line": 801,
"column": 62
} | {
"line": 813,
"column": 38
} | {
"line": 815,
"column": 0
} | [
{
"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",
"ppTerm": "?m.24",
"assigned": true,
"usedConstants": [
"Set.ext",
"Eq.mpr",
"SetLike.mem_coe._simp_1",
"AlgebraicGeometr... | [] | 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": 856,
"column": 49
} | {
"line": 856,
"column": 72
} | {
"line": 856,
"column": 73
} | [
{
"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... | [
"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).support\ninst :... | ← Scheme.Hom.comp_base, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.AlgebraicGeometry.IdealSheaf.Subscheme | {
"line": 388,
"column": 2
} | {
"line": 399,
"column": 18
} | {
"line": 401,
"column": 0
} | [
{
"pp": "X : Scheme\nI : X.IdealSheafData\n⊢ Set.range ⇑I.gluedTo = ↑I.support",
"ppTerm": "?m.7",
"assigned": true,
"usedConstants": [
"AlgebraicGeometry.Scheme.GlueData.ι",
"Iff.mpr",
"Eq.mpr",
"AlgebraicGeometry.Scheme.IdealSheafData.support",
"AlgebraicGeometry.Sche... | [] | 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
} | {
"line": 401,
"column": 0
} | [
{
"pp": "X : Scheme\nI : X.IdealSheafData\n⊢ Set.range ⇑I.gluedTo = ↑I.support",
"ppTerm": "?m.7",
"assigned": true,
"usedConstants": [
"AlgebraicGeometry.Scheme.GlueData.ι",
"Iff.mpr",
"Eq.mpr",
"AlgebraicGeometry.Scheme.IdealSheafData.support",
"AlgebraicGeometry.Sche... | [] | 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
} | {
"line": 407,
"column": 67
} | [
{
"pp": "X : 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",
"ppTerm": "?m.47",
"assigned": true,
"usedConstants": [
"AlgebraicGeometry.Scheme.GlueData.ι",
"Eq.mpr",
... | [
"X : Scheme\nI : X.IdealSheafData\nU : ↑X.affineOpens\n⊢ Set.range ⇑(I.glueData.ι U ≫ I.gluedTo) = ⇑I.gluedTo '' ⇑I.gluedTo ⁻¹' ↑↑U"
] | ← Scheme.Hom.comp_base, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Finiteness.FiniteTypeLocal | {
"line": 48,
"column": 2
} | {
"line": 48,
"column": 49
} | {
"line": 49,
"column": 2
} | [
{
"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... | [
"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... | let y := IsLocalization.commonDenomOfFinset M s | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLet___1 | Lean.Parser.Tactic.tacticLet__ |
Mathlib.AlgebraicGeometry.Morphisms.AffineAnd | {
"line": 85,
"column": 4
} | {
"line": 91,
"column": 31
} | {
"line": 92,
"column": 4
} | [
{
"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] ↦... | [
"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] ↦ Q\nX Y : Sc... | 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.RingTheory.Finiteness.FiniteTypeLocal | {
"line": 92,
"column": 2
} | {
"line": 92,
"column": 21
} | {
"line": 93,
"column": 2
} | [
{
"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",
"ppTerm": "?m.71",
"assigned": true,
"usedConstants": [
"Lattice... | [
"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) = ⊤\n⊢ FiniteType R S"
] | choose t ht using h | Mathlib.Tactic.Choose._aux_Mathlib_Tactic_Choose___elabRules_Mathlib_Tactic_Choose_choose_1 | Mathlib.Tactic.Choose.choose |
Mathlib.AlgebraicGeometry.IdealSheaf.Subscheme | {
"line": 714,
"column": 25
} | {
"line": 714,
"column": 48
} | {
"line": 714,
"column": 49
} | [
{
"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)))",
"ppTerm": "?m.63",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"AlgebraicGeometry.Sheafed... | [
"X Y : Scheme\nf : X ⟶ Y\nU : ↑Y.affineOpens\ninst✝ : QuasiCompact f\n⊢ closure (Set.range ⇑f) ⊆ closure (Set.range ⇑(Hom.toImage f ≫ Hom.imageι f))"
] | ← Scheme.Hom.comp_base, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Finiteness.FiniteTypeLocal | {
"line": 162,
"column": 2
} | {
"line": 162,
"column": 39
} | {
"line": 163,
"column": 2
} | [
{
"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 :... | [
"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 : ↥s) → Algeb... | 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": 73,
"column": 2
} | {
"line": 74,
"column": 67
} | {
"line": 75,
"column": 2
} | [
{
"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... | [
"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✝⁴ : Algebra R S := f.t... | 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.AlgebraicGeometry.Morphisms.AffineAnd | {
"line": 240,
"column": 2
} | {
"line": 241,
"column": 7
} | {
"line": 242,
"column": 2
} | [
{
"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... | [
"case refine_2\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.OfLocalizationSpan fun {... | · rw [eq_targetAffineLocally P, targetAffineLocally_affineAnd_iff hQi]
lia | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.RingTheory.RingHom.Finite | {
"line": 125,
"column": 2
} | {
"line": 125,
"column": 39
} | {
"line": 126,
"column": 2
} | [
{
"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... | [
"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), IsLocalization (Sub... | 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
} | {
"line": 128,
"column": 0
} | [
{
"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.AlgebraicGeometry.Morphisms.FiniteType | {
"line": 165,
"column": 6
} | {
"line": 166,
"column": 92
} | {
"line": 167,
"column": 6
} | [
{
"pp": "X : Scheme\nP : MorphismProperty Scheme\nhP : P ≤ @LocallyOfFiniteType\ninst✝ : P.RespectsIso\nF : P.CostructuredArrow ⊤ Scheme.Spec X ⥤ CommRingCatᵒᵖ :=\n MorphismProperty.CostructuredArrow.forget P ⊤ Scheme.Spec X ⋙ CostructuredArrow.proj Scheme.Spec X\nQ' : ObjectProperty CommRingCat := fun S ↦ ∃ R... | [
"X : Scheme\nP : MorphismProperty Scheme\nhP : P ≤ @LocallyOfFiniteType\ninst✝ : P.RespectsIso\nF : P.CostructuredArrow ⊤ Scheme.Spec X ⥤ CommRingCatᵒᵖ :=\n MorphismProperty.CostructuredArrow.forget P ⊤ Scheme.Spec X ⋙ CostructuredArrow.proj Scheme.Spec X\nQ' : ObjectProperty CommRingCat := fun S ↦ ∃ R ∈ Set.range... | obtain ⟨_, ⟨_, ⟨f, rfl⟩, rfl⟩, hqf, hfU⟩ :=
PrimeSpectrum.isBasis_basic_opens.exists_subset_of_mem_open hqU (S.hom ⁻¹ᵁ U).isOpen | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.AlgebraicGeometry.Morphisms.FiniteType | {
"line": 161,
"column": 4
} | {
"line": 169,
"column": 58
} | {
"line": 170,
"column": 4
} | [
{
"pp": "case refine_1\nX : Scheme\nP : MorphismProperty Scheme\nhP : P ≤ @LocallyOfFiniteType\ninst✝ : P.RespectsIso\nF : P.CostructuredArrow ⊤ Scheme.Spec X ⥤ CommRingCatᵒᵖ :=\n MorphismProperty.CostructuredArrow.forget P ⊤ Scheme.Spec X ⋙ CostructuredArrow.proj Scheme.Spec X\nQ' : ObjectProperty CommRingCat... | [
"case refine_1\nX : Scheme\nP : MorphismProperty Scheme\nhP : P ≤ @LocallyOfFiniteType\ninst✝ : P.RespectsIso\nF : P.CostructuredArrow ⊤ Scheme.Spec X ⥤ CommRingCatᵒᵖ :=\n MorphismProperty.CostructuredArrow.forget P ⊤ Scheme.Spec X ⋙ CostructuredArrow.proj Scheme.Spec X\nQ' : ObjectProperty CommRingCat := fun S ↦ ... | have (q : Spec (F.obj S).unop) : ∃ f, q ∈ PrimeSpectrum.basicOpen f ∧
Q' Γ(Spec (F.obj S).unop, PrimeSpectrum.basicOpen f) := by
obtain ⟨_, ⟨U, hU, rfl⟩, hqU, -⟩ :=
X.isBasis_affineOpens.exists_subset_of_mem_open (Set.mem_univ <| S.hom q) isOpen_univ
obtain ⟨_, ⟨_, ⟨f, rfl⟩, rfl⟩, hqf, hfU⟩ ... | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.AlgebraicGeometry.Morphisms.Immersion | {
"line": 97,
"column": 43
} | {
"line": 97,
"column": 66
} | {
"line": 97,
"column": 67
} | [
{
"pp": "case e'_3\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.base... | [
"case e'_3\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 ⇑(Scheme.Hom.liftCoborder f ≫ (Scheme.Hom.coborderRange f).ι) =\n ⇑(Scheme.Hom.coborderRange f).ι '' Su... | ← Scheme.Hom.comp_base, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.AlgebraicGeometry.Morphisms.Immersion | {
"line": 251,
"column": 6
} | {
"line": 251,
"column": 64
} | {
"line": 251,
"column": 64
} | [
{
"pp": "case 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))",
"ppTerm": "?hf",
"assigned": true,
"usedConstants": [
"AlgebraicGeometry.Scheme.Ho... | [
"case hf\nX Y : Scheme\nf : X ⟶ Y\ninst✝¹ : IsImmersion f\ninst✝ : QuasiCompact f\nU : ↑(↑(Hom.coborderRange f)).affineOpens\n⊢ Mono (X.presheaf.map (eqToHom ⋯).op)"
] | ← ConcreteCategory.mono_iff_injective_of_preservesPullback | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.AlgebraicGeometry.Morphisms.ClosedImmersion | {
"line": 156,
"column": 5
} | {
"line": 156,
"column": 54
} | {
"line": 156,
"column": 54
} | [
{
"pp": "X Y : Scheme\nf : X ⟶ Y\ninst✝ : IsClosedImmersion f\n⊢ IsClosedImmersion (Scheme.Hom.toImage f ≫ Scheme.Hom.imageι f)",
"ppTerm": "?m.22",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"AlgebraicGeometry.Scheme",
"AlgebraicGeometry.Scheme.Hom.image",
"AlgebraicGeom... | [] | by rw [Scheme.Hom.toImage_imageι]; infer_instance | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.AlgebraicGeometry.Morphisms.ClosedImmersion | {
"line": 274,
"column": 63
} | {
"line": 274,
"column": 81
} | {
"line": 274,
"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... | [
"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\nres : (i :... | Opens.map_comp_obj | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.AlgebraicGeometry.Morphisms.ClosedImmersion | {
"line": 375,
"column": 22
} | {
"line": 375,
"column": 33
} | {
"line": 376,
"column": 2
} | [
{
"pp": "X✝ 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 ⇑(ConcreteCate... | [
"X✝ 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\nha : IsAffine Y\nhsurj : Function.Surjective ⇑(ConcreteCatego... | ⟨ha, hsurj⟩ | Lean.Elab.Tactic.evalIntro | Lean.Parser.Term.anonymousCtor |
Mathlib.AlgebraicGeometry.Morphisms.FinitePresentation | {
"line": 60,
"column": 8
} | {
"line": 60,
"column": 40
} | {
"line": 60,
"column": 41
} | [
{
"pp": "X✝ 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",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"RingHom.FinitePresentation",
"Al... | [
"X✝ Y✝ : Scheme\nf✝ : X✝ ⟶ Y✝\nX Y : Scheme\nf : X ⟶ Y\n⊢ (∀ {U : Y.Opens},\n IsAffineOpen U →\n ∀ {V : X.Opens},\n IsAffineOpen V → ∀ (e : V ≤ f ⁻¹ᵁ U), (CommRingCat.Hom.hom (Scheme.Hom.appLE f U V e)).FinitePresentation) ↔\n affineLocally (fun {R S} [CommRing R] [CommRing S] ↦ RingHom.Fini... | locallyOfFinitePresentation_iff, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.AlgebraicGeometry.Noetherian | {
"line": 148,
"column": 4
} | {
"line": 148,
"column": 41
} | {
"line": 151,
"column": 0
} | [
{
"pp": "case mpr.hS'\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))",
"ppTerm": "?mpr.hS'",
"assigned": true,
"usedConstants": [
"Alge... | [] | exact IsOpenImmersion.ΓIsoTop (𝒰.f i) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.AlgebraicGeometry.Morphisms.UniversallyOpen | {
"line": 114,
"column": 4
} | {
"line": 114,
"column": 41
} | {
"line": 115,
"column": 4
} | [
{
"pp": "case inr\nX Y : Scheme\nf : X ⟶ Y\ninst✝ : LocallyOfFinitePresentation f\nhf : GeneralizingMap ⇑f\nthis :\n ∀ {X Y : Scheme} (f : X ⟶ Y) [LocallyOfFinitePresentation f],\n GeneralizingMap ⇑f →\n (∃ R, Y = Spec R) → topologically (fun {α β} [TopologicalSpace α] [TopologicalSpace β] ↦ IsOpenMap)... | [
"case inr\nX Y : Scheme\nf : X ⟶ Y\ninst✝ : LocallyOfFinitePresentation f\nhf : GeneralizingMap ⇑f\nthis :\n ∀ {X Y : Scheme} (f : X ⟶ Y) [LocallyOfFinitePresentation f],\n GeneralizingMap ⇑f →\n (∃ R, Y = Spec R) → topologically (fun {α β} [TopologicalSpace α] [TopologicalSpace β] ↦ IsOpenMap) f\nhY : ¬∃ ... | dsimp only [Scheme.Cover.pullbackHom] | Lean.Elab.Tactic.evalDSimp | Lean.Parser.Tactic.dsimp |
Mathlib.AlgebraicGeometry.Morphisms.Integral | {
"line": 146,
"column": 2
} | {
"line": 149,
"column": 80
} | {
"line": 151,
"column": 0
} | [
{
"pp": "R 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.mapRingHom (Com... | [] | 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.RingTheory.Spectrum.Prime.ChevalleyComplexity | {
"line": 239,
"column": 6
} | {
"line": 239,
"column": 36
} | {
"line": 240,
"column": 6
} | [
{
"pp": "n : ℕ\nP : (R : Type u) → [inst : CommRing R] → InductionObj R n → Prop\nhP₁ : ∀ (R : Type u) [inst : CommRing R], P R { val := 0 }\nhP₂ :\n ∀ (R : Type u) [inst : CommRing R] (e : InductionObj R n) (i : Fin n),\n (e.val i).Monic → (∀ (j : Fin n), j ≠ i → e.val j = 0) → P R e\nhP₃ :\n ∀ (R : Type ... | [
"n : ℕ\nP : (R : Type u) → [inst : CommRing R] → InductionObj R n → Prop\nhP₁ : ∀ (R : Type u) [inst : CommRing R], P R { val := 0 }\nhP₂ :\n ∀ (R : Type u) [inst : CommRing R] (e : InductionObj R n) (i : Fin n),\n (e.val i).Monic → (∀ (j : Fin n), j ≠ i → e.val j = 0) → P R e\nhP₃ :\n ∀ (R : Type u) [inst : C... | refine ⟨i, hi', fun j hj ↦ ?_⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.AlgebraicGeometry.Morphisms.Flat | {
"line": 131,
"column": 48
} | {
"line": 131,
"column": 71
} | {
"line": 131,
"column": 72
} | [
{
"pp": "case inr\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 = Spec... | [
"case inr\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 = Spec R\n𝒰 : Y.O... | ← Scheme.Hom.comp_base, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.AlgebraicGeometry.QuasiAffine | {
"line": 118,
"column": 2
} | {
"line": 128,
"column": 38
} | {
"line": 129,
"column": 2
} | [
{
"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))",
"ppTerm": "?m.30",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"AlgebraicGeometry.SheafedS... | [
"X Y : Scheme\nf : X ⟶ Y\ninst✝¹ : IsAffineHom f\ninst✝ : Y.IsQuasiAffine\nthis✝¹ : CompactSpace ↥X\nthis✝ : X.IsQuasiAffine\nthis :\n ∀ (r : ↑Γ(Y, ⊤)),\n IsPushout (Hom.appTop f) (Y.presheaf.map (homOfLE ⋯).op) (X.presheaf.map (homOfLE ⋯).op)\n (Hom.appLE f (Y.basicOpen r) (X.basicOpen ((ConcreteCategory.... | 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.AlgebraicGeometry.Morphisms.Flat | {
"line": 270,
"column": 2
} | {
"line": 271,
"column": 85
} | {
"line": 273,
"column": 2
} | [
{
"pp": "X Y S T : Scheme\nf : T ⟶ S\ng : Y ⟶ X\niX : X ⟶ S\niY : Y ⟶ T\nH : IsPullback g iY iX f\nUS : S.Opens\nUT : T.Opens\nUX : X.Opens\nhUST : UT ≤ f ⁻¹ᵁ US\nhUSX : UX ≤ iX ⁻¹ᵁ US\nUY : Y.Opens\nhUY : UY = g ⁻¹ᵁ UX ⊓ iY ⁻¹ᵁ UT\nι : Type u_1\ninst✝ : Finite ι\nVX : ι → X.Opens\nhVU : iSup VX = UX\nhV : ∀ (i... | [
"X Y S T : Scheme\nf : T ⟶ S\ng : Y ⟶ X\niX : X ⟶ S\niY : Y ⟶ T\nH : IsPullback g iY iX f\nUS : S.Opens\nUT : T.Opens\nUX : X.Opens\nhUST : UT ≤ f ⁻¹ᵁ US\nhUSX : UX ≤ iX ⁻¹ᵁ US\nUY : Y.Opens\nhUY : UY = g ⁻¹ᵁ UX ⊓ iY ⁻¹ᵁ UT\nι : Type u_1\ninst✝ : Finite ι\nVX : ι → X.Opens\nhVU : iSup VX = UX\nhV : ∀ (i : ι), Mono ... | let ψY : Γ(Y, UY) →+* Π i, Γ(Y, g ⁻¹ᵁ VX i ⊓ iY ⁻¹ᵁ UT) := RingHom.pi fun i ↦
(Y.presheaf.map (homOfLE (by subst hUY hVU; gcongr; exact le_iSup _ _)).op).hom | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLet___1 | Lean.Parser.Tactic.tacticLet__ |
Mathlib.AlgebraicGeometry.AffineSpace | {
"line": 181,
"column": 8
} | {
"line": 181,
"column": 29
} | {
"line": 182,
"column": 8
} | [
{
"pp": "n : Type u\nS : Scheme\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) ∘ X) ≫\n 𝔸(n; S).toSpecΓ ≫\n Spec.map (CommRingCat.ofHom (eval₂Hom (CommRingCat.Hom.... | [
"n : Type u\nS : Scheme\ninst✝ : IsAffine S\n⊢ Scheme.Hom.appTop\n (homOfVector (Spec.map (CommRingCat.ofHom C) ≫ S.isoSpec.inv)\n (⇑(ConcreteCategory.hom (Scheme.ΓSpecIso (CommRingCat.of (MvPolynomial n ↑Γ(S, ⊤)))).inv) ∘ X) ≫\n 𝔸(n; S).toSpecΓ ≫\n Spec.map (CommRingCat.ofHom (eval₂H... | apply ext_of_isAffine | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.CategoryTheory.Limits.Shapes.Pullback.Connected | {
"line": 46,
"column": 8
} | {
"line": 46,
"column": 16
} | {
"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... | [
"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\nj : I\nf :... | this _ j | Lean.Elab.Tactic.evalRewriteSeq | null |
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