module string | startPos dict | endPos dict | nextStartPos dict | goals list | goalsAfter list | ppTac string | elaborator string | kind string |
|---|---|---|---|---|---|---|---|---|
Mathlib.RingTheory.Spectrum.Prime.Polynomial | {
"line": 50,
"column": 6
} | {
"line": 50,
"column": 85
} | {
"line": 50,
"column": 85
} | [
{
"pp": "case inr\nR : Type u_1\nA : Type u_2\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : Algebra R A\ninst✝² : Module.Free R A\ninst✝¹ : Module.Finite R A\nf : A\nI : Ideal R\ninst✝ : I.IsPrime\nh✝ : Nontrivial R\nthis : Module.finrank I.ResidueField (I.ResidueField ⊗[R] A) = Module.finrank R A\n⊢ IsNi... | [
"case inr\nR : Type u_1\nA : Type u_2\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : Algebra R A\ninst✝² : Module.Free R A\ninst✝¹ : Module.Finite R A\nf : A\nI : Ideal R\ninst✝ : I.IsPrime\nh✝ : Nontrivial R\nthis : Module.finrank I.ResidueField (I.ResidueField ⊗[R] A) = Module.finrank R A\n⊢ IsNilpotent ((Al... | ← IsNilpotent.map_iff (Algebra.TensorProduct.comm R A I.ResidueField).injective | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Spectrum.Prime.Polynomial | {
"line": 103,
"column": 8
} | {
"line": 103,
"column": 27
} | {
"line": 103,
"column": 28
} | [
{
"pp": "case mpr\nR : Type u_2\nA : Type u_1\ninst✝² : CommRing R\ninst✝¹ : CommRing A\ninst✝ : Algebra R A\nf : A\ns : Set A\nx : PrimeSpectrum R\nq : Ideal ((A ⧸ Ideal.span s) ⊗[R] x.asIdeal.ResidueField)\nhq : q.IsPrime\nhfq : (Ideal.Quotient.mk (Ideal.span s)) f ⊗ₜ[R] 1 ∉ q\nthis : ∀ a ∈ s, (Ideal.Quotient... | [
"case mpr\nR : Type u_2\nA : Type u_1\ninst✝² : CommRing R\ninst✝¹ : CommRing A\ninst✝ : Algebra R A\nf : A\ns : Set A\nx : PrimeSpectrum R\nq : Ideal ((A ⧸ Ideal.span s) ⊗[R] x.asIdeal.ResidueField)\nhq : q.IsPrime\nhfq : (Ideal.Quotient.mk (Ideal.span s)) f ⊗ₜ[R] 1 ∉ q\nthis : ∀ a ∈ s, (Ideal.Quotient.mk (Ideal.s... | ← comap_comp_apply, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Spectrum.Prime.Polynomial | {
"line": 169,
"column": 2
} | {
"line": 169,
"column": 38
} | {
"line": 170,
"column": 2
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\nx : PrimeSpectrum R\n⊢ ∃ a, comap C a = x",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"Polynomial.C",
"CommSemiring.toSemiring",
"Polynomial",
"CommRing.toCommSemiring",
"Exists.intro",
"PrimeSpectrum",
... | [
"R : Type u_1\ninst✝ : CommRing R\nx : PrimeSpectrum R\n⊢ comap C (comap (evalRingHom 0) x) = x"
] | refine ⟨comap (evalRingHom 0) x, ?_⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.RingTheory.Spectrum.Prime.Polynomial | {
"line": 170,
"column": 6
} | {
"line": 170,
"column": 25
} | {
"line": 170,
"column": 26
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\nx : PrimeSpectrum R\n⊢ comap C (comap (evalRingHom 0) x) = x",
"ppTerm": "?m.28",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Polynomial.C",
"PrimeSpectrum.comap_comp_apply",
"congrArg",
"CommSemiring.toSemiring",
"id... | [
"R : Type u_1\ninst✝ : CommRing R\nx : PrimeSpectrum R\n⊢ comap ((evalRingHom 0).comp C) x = x"
] | ← comap_comp_apply, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Spectrum.Prime.Polynomial | {
"line": 234,
"column": 6
} | {
"line": 234,
"column": 25
} | {
"line": 234,
"column": 26
} | [
{
"pp": "R : Type u_2\ninst✝ : CommRing R\nσ : Type u_1\nx : PrimeSpectrum R\n⊢ comap C (comap (eval₂Hom (RingHom.id R) 0) x) = x",
"ppTerm": "?m.33",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Nat.instMulZeroClass",
"PrimeSpectrum.comap_comp_apply",
"congrArg",
"C... | [
"R : Type u_2\ninst✝ : CommRing R\nσ : Type u_1\nx : PrimeSpectrum R\n⊢ comap ((eval₂Hom (RingHom.id R) 0).comp C) x = x"
] | ← comap_comp_apply, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.AlgebraicGeometry.Artinian | {
"line": 69,
"column": 4
} | {
"line": 74,
"column": 55
} | {
"line": 76,
"column": 0
} | [
{
"pp": "X : Scheme\ninst✝ : IsLocallyNoetherian X\nh : topologicalKrullDim ↥X ≤ 0\nU : ↑X.affineOpens\n⊢ IsArtinianRing ↑Γ(X, ↑U)",
"ppTerm": "?m.10",
"assigned": true,
"usedConstants": [
"WithBot.addMonoidWithOne",
"WithBot.instPreorder",
"Eq.mpr",
"AlgebraicGeometry.IsAffi... | [] | have _ : IsNoetherianRing Γ(X, U) := IsLocallyNoetherian.component_noetherian U
rw [isArtinianRing_iff_krullDimLE_zero, Ring.KrullDimLE, Order.krullDimLE_iff, ← ringKrullDim,
Nat.cast_zero, ← PrimeSpectrum.topologicalKrullDim_eq_ringKrullDim Γ(X, U)]
change topologicalKrullDim (Spec Γ(X, U)) ≤ 0
rw [←... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.AlgebraicGeometry.Artinian | {
"line": 69,
"column": 4
} | {
"line": 74,
"column": 55
} | {
"line": 76,
"column": 0
} | [
{
"pp": "X : Scheme\ninst✝ : IsLocallyNoetherian X\nh : topologicalKrullDim ↥X ≤ 0\nU : ↑X.affineOpens\n⊢ IsArtinianRing ↑Γ(X, ↑U)",
"ppTerm": "?m.10",
"assigned": true,
"usedConstants": [
"WithBot.addMonoidWithOne",
"WithBot.instPreorder",
"Eq.mpr",
"AlgebraicGeometry.IsAffi... | [] | have _ : IsNoetherianRing Γ(X, U) := IsLocallyNoetherian.component_noetherian U
rw [isArtinianRing_iff_krullDimLE_zero, Ring.KrullDimLE, Order.krullDimLE_iff, ← ringKrullDim,
Nat.cast_zero, ← PrimeSpectrum.topologicalKrullDim_eq_ringKrullDim Γ(X, U)]
change topologicalKrullDim (Spec Γ(X, U)) ≤ 0
rw [←... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.AlgebraicGeometry.Artinian | {
"line": 102,
"column": 2
} | {
"line": 102,
"column": 79
} | {
"line": 103,
"column": 2
} | [
{
"pp": "X Y : Scheme\nf : X ⟶ Y\ninst✝ : IsLocallyArtinian X\nx : ↥X\nW : X.Opens\nhW1 : IsAffineOpen W\nhW2 : x ∈ W\nright✝ : W.carrier ⊆ ↑⊤\nthis✝ : IsArtinianRing ↑Γ(X, W)\nthis : DiscreteTopology ↥(Spec Γ(X, W))\n⊢ IsOpen {x}",
"ppTerm": "?m.74",
"assigned": true,
"usedConstants": [
"Alge... | [
"X Y : Scheme\nf : X ⟶ Y\ninst✝ : IsLocallyArtinian X\nx : ↥X\nW : X.Opens\nhW1 : IsAffineOpen W\nhW2 : x ∈ W\nright✝ : W.carrier ⊆ ↑⊤\nthis✝¹ : IsArtinianRing ↑Γ(X, W)\nthis✝ : DiscreteTopology ↥(Spec Γ(X, W))\nthis : DiscreteTopology ↥↑W\n⊢ IsOpen {x}"
] | have : DiscreteTopology W := hW1.isoSpec.hom.homeomorph.symm.discreteTopology | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.AlgebraicGeometry.Fiber | {
"line": 165,
"column": 53
} | {
"line": 165,
"column": 58
} | {
"line": 167,
"column": 0
} | [
{
"pp": "X Y : Scheme\nf : X ⟶ Y\nx : ↥X\n⊢ Set.range ⇑(asFiberHom f x) = {asFiber f x}",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"AlgebraicGeometry.Spec",
"CommRingCat.carrier",
"AlgebraicGeometry.PresheafedSpace.carrier",
"congrArg",
"CategoryTheory.Co... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.AlgebraicGeometry.Fiber | {
"line": 165,
"column": 53
} | {
"line": 165,
"column": 58
} | {
"line": 167,
"column": 0
} | [
{
"pp": "X Y : Scheme\nf : X ⟶ Y\nx : ↥X\n⊢ Set.range ⇑(asFiberHom f x) = {asFiber f x}",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"AlgebraicGeometry.Spec",
"CommRingCat.carrier",
"AlgebraicGeometry.PresheafedSpace.carrier",
"congrArg",
"CategoryTheory.Co... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.AlgebraicGeometry.Fiber | {
"line": 165,
"column": 53
} | {
"line": 165,
"column": 58
} | {
"line": 167,
"column": 0
} | [
{
"pp": "X Y : Scheme\nf : X ⟶ Y\nx : ↥X\n⊢ Set.range ⇑(asFiberHom f x) = {asFiber f x}",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"AlgebraicGeometry.Spec",
"CommRingCat.carrier",
"AlgebraicGeometry.PresheafedSpace.carrier",
"congrArg",
"CategoryTheory.Co... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.AlgebraicGeometry.Geometrically.Reduced | {
"line": 85,
"column": 70
} | {
"line": 85,
"column": 99
} | {
"line": 86,
"column": 6
} | [
{
"pp": "X Y : Scheme\nf : X ⟶ Y\ninst✝³ : GeometricallyReduced f\ninst✝² : Flat f\ninst✝¹ : IsReduced Y\ninst✝ : Finite ↑(irreducibleComponents ↥Y)\npt : ↑(irreducibleComponents ↥Y) → CommRingCat := fun Z ↦ Y.presheaf.stalk ⋯.genericPoint\nhpt : ∀ (Z : ↑(irreducibleComponents ↥Y)), IsField ↑(pt Z)\nthis✝¹ : (Z... | [
"X Y : Scheme\nf : X ⟶ Y\ninst✝³ : GeometricallyReduced f\ninst✝² : Flat f\ninst✝¹ : IsReduced Y\ninst✝ : Finite ↑(irreducibleComponents ↥Y)\npt : ↑(irreducibleComponents ↥Y) → CommRingCat := fun Z ↦ Y.presheaf.stalk ⋯.genericPoint\nhpt : ∀ (Z : ↑(irreducibleComponents ↥Y)), IsField ↑(pt Z)\nthis✝¹ : (Z : ↑(irreduc... | denseRange_iff_closure_range, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.AlgebraicGeometry.Geometrically.Irreducible | {
"line": 118,
"column": 2
} | {
"line": 119,
"column": 55
} | {
"line": 121,
"column": 0
} | [
{
"pp": "X S : Scheme\nf : X ⟶ S\n⊢ GeometricallyIrreducible f ↔ ∀ (s : ↥S), GeometricallyIrreducible (Scheme.Hom.fiberToSpecResidueField f s)",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"AlgebraicGeometry.Spec",
"AlgebraicGeometry.SheafedSpace.instTopologicalSpaceCarrierCa... | [] | simp only [GeometricallyIrreducible.eq_geometrically,
← geometrically_iff_forall_fiberToSpecResidueField] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.AlgebraicGeometry.Geometrically.Irreducible | {
"line": 118,
"column": 2
} | {
"line": 119,
"column": 55
} | {
"line": 121,
"column": 0
} | [
{
"pp": "X S : Scheme\nf : X ⟶ S\n⊢ GeometricallyIrreducible f ↔ ∀ (s : ↥S), GeometricallyIrreducible (Scheme.Hom.fiberToSpecResidueField f s)",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"AlgebraicGeometry.Spec",
"AlgebraicGeometry.SheafedSpace.instTopologicalSpaceCarrierCa... | [] | simp only [GeometricallyIrreducible.eq_geometrically,
← geometrically_iff_forall_fiberToSpecResidueField] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.AlgebraicGeometry.Geometrically.Irreducible | {
"line": 118,
"column": 2
} | {
"line": 119,
"column": 55
} | {
"line": 121,
"column": 0
} | [
{
"pp": "X S : Scheme\nf : X ⟶ S\n⊢ GeometricallyIrreducible f ↔ ∀ (s : ↥S), GeometricallyIrreducible (Scheme.Hom.fiberToSpecResidueField f s)",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"AlgebraicGeometry.Spec",
"AlgebraicGeometry.SheafedSpace.instTopologicalSpaceCarrierCa... | [] | simp only [GeometricallyIrreducible.eq_geometrically,
← geometrically_iff_forall_fiberToSpecResidueField] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.AlgebraicGeometry.Morphisms.Integral | {
"line": 120,
"column": 8
} | {
"line": 120,
"column": 72
} | {
"line": 120,
"column": 72
} | [
{
"pp": "case inr\nZ S X : Scheme\nR : CommRingCat\nf : X ⟶ Spec R\nthis : ∀ ⦃X : Scheme⦄ (f : X ⟶ Spec R), (∃ S, X = Spec S) → IsIntegralHom f → topologically (@IsClosedMap) f\nhX : ¬∃ S, X = Spec S\nH : IsIntegralHom f\ninst : IsAffine X\n⊢ topologically (@IsClosedMap) (X.isoSpec.inv ≫ f)",
"ppTerm": "?in... | [
"case inr\nZ S X : Scheme\nR : CommRingCat\nf : X ⟶ Spec R\nthis : ∀ ⦃X : Scheme⦄ (f : X ⟶ Spec R), (∃ S, X = Spec S) → IsIntegralHom f → topologically (@IsClosedMap) f\nhX : ¬∃ S, X = Spec S\ninst : IsAffine X\nH : IsIntegralHom (X.isoSpec.inv ≫ f)\n⊢ topologically (@IsClosedMap) (X.isoSpec.inv ≫ f)"
] | ← cancel_left_of_respectsIso (P := @IsIntegralHom) X.isoSpec.inv | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.AlgebraicGeometry.Morphisms.Integral | {
"line": 139,
"column": 8
} | {
"line": 139,
"column": 72
} | {
"line": 139,
"column": 72
} | [
{
"pp": "case inr\nX : Scheme\nR : CommRingCat\nf : X ⟶ Spec R\nH₁ : UniversallyClosed f\nH₂ : IsAffineHom f\nthis :\n ∀ {X : Scheme} (R : CommRingCat) {f : X ⟶ Spec R},\n UniversallyClosed f → IsAffineHom f → (∃ S, X = Spec S) → IsIntegralHom f\nhX : ¬∃ S, X = Spec S\ninst : IsAffine X\n⊢ IsIntegralHom f",... | [
"case inr\nX : Scheme\nR : CommRingCat\nf : X ⟶ Spec R\nH₁ : UniversallyClosed f\nH₂ : IsAffineHom f\nthis :\n ∀ {X : Scheme} (R : CommRingCat) {f : X ⟶ Spec R},\n UniversallyClosed f → IsAffineHom f → (∃ S, X = Spec S) → IsIntegralHom f\nhX : ¬∃ S, X = Spec S\ninst : IsAffine X\n⊢ IsIntegralHom (X.isoSpec.inv ... | ← cancel_left_of_respectsIso (P := @IsIntegralHom) X.isoSpec.inv | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.AlgebraicGeometry.QuasiAffine | {
"line": 64,
"column": 2
} | {
"line": 64,
"column": 40
} | {
"line": 65,
"column": 2
} | [
{
"pp": "X : Scheme\ninst✝ : X.IsQuasiAffine\nU : TopologicalSpace.Opens ↥X\nx : ↥X\nhxU : x ∈ U\nr : ↑Γ(X, ⊤)\nhxr :\n (ConcreteCategory.hom (LocallyRingedSpace.Hom.toShHom (Hom.toLRSHom X.toSpecΓ)).hom.base) x ∈\n ↑(PrimeSpectrum.basicOpen r)\nhrU :\n ↑(PrimeSpectrum.basicOpen r) ⊆\n ⇑(ConcreteCategor... | [
"X : Scheme\ninst✝ : X.IsQuasiAffine\nU : TopologicalSpace.Opens ↥X\nx : ↥X\nhxU : x ∈ U\nr : ↑Γ(X, ⊤)\nhxr :\n (ConcreteCategory.hom (LocallyRingedSpace.Hom.toShHom (Hom.toLRSHom X.toSpecΓ)).hom.base) x ∈\n ↑(PrimeSpectrum.basicOpen r)\nhrU :\n ↑(PrimeSpectrum.basicOpen r) ⊆\n ⇑(ConcreteCategory.hom (Local... | simp_rw [← toSpecΓ_preimage_basicOpen] | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | Mathlib.Tactic.tacticSimp_rw___ |
Mathlib.AlgebraicGeometry.Morphisms.Flat | {
"line": 252,
"column": 61
} | {
"line": 252,
"column": 66
} | {
"line": 252,
"column": 66
} | [
{
"pp": "X✝ Y✝ Z : Scheme\nf✝ : X✝ ⟶ Y✝\nX Y S T : Scheme\nf : T ⟶ S\ng : Y ⟶ X\niX : X ⟶ S\niY : Y ⟶ T\nH : IsPullback g iY iX f\nUS : S.Opens\nUT : T.Opens\nUX : X.Opens\nhUST : UT ≤ f ⁻¹ᵁ US\nhUSX : UX ≤ iX ⁻¹ᵁ US\nUY : Y.Opens\nhUY : UY = g ⁻¹ᵁ UX ⊓ iY ⁻¹ᵁ UT\nι : Type u_1\ninst✝ : Finite ι\nVX : ι → X.Open... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.AlgebraicGeometry.Morphisms.Flat | {
"line": 252,
"column": 61
} | {
"line": 252,
"column": 66
} | {
"line": 252,
"column": 66
} | [
{
"pp": "X✝ Y✝ Z : Scheme\nf✝ : X✝ ⟶ Y✝\nX Y S T : Scheme\nf : T ⟶ S\ng : Y ⟶ X\niX : X ⟶ S\niY : Y ⟶ T\nH : IsPullback g iY iX f\nUS : S.Opens\nUT : T.Opens\nUX : X.Opens\nhUST : UT ≤ f ⁻¹ᵁ US\nhUSX : UX ≤ iX ⁻¹ᵁ US\nUY : Y.Opens\nhUY : UY = g ⁻¹ᵁ UX ⊓ iY ⁻¹ᵁ UT\nι : Type u_1\ninst✝ : Finite ι\nVX : ι → X.Open... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.AlgebraicGeometry.Morphisms.Flat | {
"line": 252,
"column": 61
} | {
"line": 252,
"column": 66
} | {
"line": 252,
"column": 66
} | [
{
"pp": "X✝ Y✝ Z : Scheme\nf✝ : X✝ ⟶ Y✝\nX Y S T : Scheme\nf : T ⟶ S\ng : Y ⟶ X\niX : X ⟶ S\niY : Y ⟶ T\nH : IsPullback g iY iX f\nUS : S.Opens\nUT : T.Opens\nUX : X.Opens\nhUST : UT ≤ f ⁻¹ᵁ US\nhUSX : UX ≤ iX ⁻¹ᵁ US\nUY : Y.Opens\nhUY : UY = g ⁻¹ᵁ UX ⊓ iY ⁻¹ᵁ UT\nι : Type u_1\ninst✝ : Finite ι\nVX : ι → X.Open... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.AlgebraicGeometry.Morphisms.Flat | {
"line": 265,
"column": 50
} | {
"line": 265,
"column": 55
} | {
"line": 266,
"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\nhT : (Com... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.AlgebraicGeometry.Morphisms.Flat | {
"line": 268,
"column": 41
} | {
"line": 268,
"column": 46
} | {
"line": 268,
"column": 46
} | [
{
"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... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.AlgebraicGeometry.Morphisms.Flat | {
"line": 268,
"column": 41
} | {
"line": 268,
"column": 46
} | {
"line": 268,
"column": 46
} | [
{
"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... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.AlgebraicGeometry.Morphisms.Flat | {
"line": 268,
"column": 41
} | {
"line": 268,
"column": 46
} | {
"line": 268,
"column": 46
} | [
{
"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... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.AlgebraicGeometry.Morphisms.Flat | {
"line": 291,
"column": 65
} | {
"line": 291,
"column": 70
} | {
"line": 291,
"column": 70
} | [
{
"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✝ : ∀ (... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.AlgebraicGeometry.Morphisms.Flat | {
"line": 291,
"column": 65
} | {
"line": 291,
"column": 70
} | {
"line": 291,
"column": 70
} | [
{
"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✝ : ∀ (... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.AlgebraicGeometry.Morphisms.Flat | {
"line": 291,
"column": 65
} | {
"line": 291,
"column": 70
} | {
"line": 291,
"column": 70
} | [
{
"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✝ : ∀ (... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.AlgebraicGeometry.Morphisms.Flat | {
"line": 336,
"column": 62
} | {
"line": 336,
"column": 67
} | {
"line": 336,
"column": 67
} | [
{
"pp": "X✝ Y✝ Z : Scheme\nf✝ : X✝ ⟶ Y✝\nX Y S T : Scheme\nf : T ⟶ S\ng : Y ⟶ X\niX : X ⟶ S\niY : Y ⟶ T\nH : IsPullback g iY iX f\nUS : S.Opens\nUT : T.Opens\nUX : X.Opens\nhUST : UT ≤ f ⁻¹ᵁ US\nhUSX : UX ≤ iX ⁻¹ᵁ US\nUY : Y.Opens\nhUY : UY = g ⁻¹ᵁ UX ⊓ iY ⁻¹ᵁ UT\nι : Type u\ninst✝ : Finite ι\nVX : ι → X.Opens\... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.AlgebraicGeometry.Morphisms.Flat | {
"line": 336,
"column": 62
} | {
"line": 336,
"column": 67
} | {
"line": 336,
"column": 67
} | [
{
"pp": "X✝ Y✝ Z : Scheme\nf✝ : X✝ ⟶ Y✝\nX Y S T : Scheme\nf : T ⟶ S\ng : Y ⟶ X\niX : X ⟶ S\niY : Y ⟶ T\nH : IsPullback g iY iX f\nUS : S.Opens\nUT : T.Opens\nUX : X.Opens\nhUST : UT ≤ f ⁻¹ᵁ US\nhUSX : UX ≤ iX ⁻¹ᵁ US\nUY : Y.Opens\nhUY : UY = g ⁻¹ᵁ UX ⊓ iY ⁻¹ᵁ UT\nι : Type u\ninst✝ : Finite ι\nVX : ι → X.Opens\... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.AlgebraicGeometry.Morphisms.Flat | {
"line": 336,
"column": 62
} | {
"line": 336,
"column": 67
} | {
"line": 336,
"column": 67
} | [
{
"pp": "X✝ Y✝ Z : Scheme\nf✝ : X✝ ⟶ Y✝\nX Y S T : Scheme\nf : T ⟶ S\ng : Y ⟶ X\niX : X ⟶ S\niY : Y ⟶ T\nH : IsPullback g iY iX f\nUS : S.Opens\nUT : T.Opens\nUX : X.Opens\nhUST : UT ≤ f ⁻¹ᵁ US\nhUSX : UX ≤ iX ⁻¹ᵁ US\nUY : Y.Opens\nhUY : UY = g ⁻¹ᵁ UX ⊓ iY ⁻¹ᵁ UT\nι : Type u\ninst✝ : Finite ι\nVX : ι → X.Opens\... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.AlgebraicGeometry.Morphisms.Flat | {
"line": 338,
"column": 65
} | {
"line": 338,
"column": 70
} | {
"line": 338,
"column": 70
} | [
{
"pp": "X✝ Y✝ Z : Scheme\nf✝ : X✝ ⟶ Y✝\nX Y S T : Scheme\nf : T ⟶ S\ng : Y ⟶ X\niX : X ⟶ S\niY : Y ⟶ T\nH : IsPullback g iY iX f\nUS : S.Opens\nUT : T.Opens\nUX : X.Opens\nhUST : UT ≤ f ⁻¹ᵁ US\nhUSX : UX ≤ iX ⁻¹ᵁ US\nUY : Y.Opens\nhUY : UY = g ⁻¹ᵁ UX ⊓ iY ⁻¹ᵁ UT\nι : Type u\ninst✝ : Finite ι\nVX : ι → X.Opens\... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Topology.LocallyFinsupp | {
"line": 98,
"column": 61
} | {
"line": 98,
"column": 66
} | {
"line": 99,
"column": 2
} | [
{
"pp": "X : Type u_1\ninst✝¹ : TopologicalSpace X\nY : Type u_2\ninst✝ : Zero Y\nf : X → Y\nz : X\nt : Set X\nht : t ∈ 𝓝 z\n⊢ t ∩ support f = Subtype.val '' {i | ↑i ∈ t}",
"ppTerm": "?m.53",
"assigned": true,
"usedConstants": [
"Set.ext",
"Function.mem_support._simp_1",
"Iff.of_e... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Topology.LocallyFinsupp | {
"line": 99,
"column": 63
} | {
"line": 99,
"column": 68
} | {
"line": 100,
"column": 2
} | [
{
"pp": "X : Type u_1\ninst✝¹ : TopologicalSpace X\nY : Type u_2\ninst✝ : Zero Y\nf : X → Y\nz : X\nt : Set X\nht : t ∈ 𝓝 z\naux1 : t ∩ support f = Subtype.val '' {i | ↑i ∈ t}\n⊢ InjOn Subtype.val {i | ↑i ∈ t}",
"ppTerm": "?m.70",
"assigned": true,
"usedConstants": [
"Function.mem_support._si... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Topology.LocallyFinsupp | {
"line": 111,
"column": 88
} | {
"line": 111,
"column": 93
} | {
"line": 112,
"column": 2
} | [
{
"pp": "X : Type u_1\ninst✝¹ : TopologicalSpace X\nY : Type u_2\nW : Set X\ninst✝ : Zero Y\nf : X → Y\nh : LocallyFiniteSupport f\nhW : IsCompact W\nthis : {i | ({↑i} ∩ W).Nonempty}.Finite\nα : Type u_1\ns t : Set α\n⊢ Subtype.val '' {i | ({↑i} ∩ t).Nonempty} = t ∩ s",
"ppTerm": "?m.41",
"assigned": tr... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Topology.LocallyFinsupp | {
"line": 179,
"column": 31
} | {
"line": 179,
"column": 36
} | {
"line": 181,
"column": 0
} | [
{
"pp": "X : Type u_1\ninst✝² : TopologicalSpace X\nY : Type u_2\ninst✝¹ : DecidableEq X\ninst✝ : Zero Y\nx : X\n⊢ single x 0 = 0",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Function.locallyFinsuppWithin.instFunLike",
"congrArg",
"Function.locallyFins... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Topology.LocallyFinsupp | {
"line": 179,
"column": 31
} | {
"line": 179,
"column": 36
} | {
"line": 181,
"column": 0
} | [
{
"pp": "X : Type u_1\ninst✝² : TopologicalSpace X\nY : Type u_2\ninst✝¹ : DecidableEq X\ninst✝ : Zero Y\nx : X\n⊢ single x 0 = 0",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Function.locallyFinsuppWithin.instFunLike",
"congrArg",
"Function.locallyFins... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.LocallyFinsupp | {
"line": 179,
"column": 31
} | {
"line": 179,
"column": 36
} | {
"line": 181,
"column": 0
} | [
{
"pp": "X : Type u_1\ninst✝² : TopologicalSpace X\nY : Type u_2\ninst✝¹ : DecidableEq X\ninst✝ : Zero Y\nx : X\n⊢ single x 0 = 0",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Function.locallyFinsuppWithin.instFunLike",
"congrArg",
"Function.locallyFins... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.LocallyFinsupp | {
"line": 286,
"column": 6
} | {
"line": 286,
"column": 35
} | {
"line": 287,
"column": 6
} | [
{
"pp": "case right\nX : Type u_1\ninst✝¹ : TopologicalSpace X\nU : Set X\nY : Type u_2\ninst✝ : AddMonoid Y\nf g : X → Y\nhf : f ∈ {f | Function.support f ⊆ U ∧ ∀ z ∈ U, ∃ t ∈ 𝓝 z, (t ∩ Function.support f).Finite}\nhg : g ∈ {f | Function.support f ⊆ U ∧ ∀ z ∈ U, ∃ t ∈ 𝓝 z, (t ∩ Function.support f).Finite}\nz... | [
"case right\nX : Type u_1\ninst✝¹ : TopologicalSpace X\nU : Set X\nY : Type u_2\ninst✝ : AddMonoid Y\nf g : X → Y\nhf : f ∈ {f | Function.support f ⊆ U ∧ ∀ z ∈ U, ∃ t ∈ 𝓝 z, (t ∩ Function.support f).Finite}\nhg : g ∈ {f | Function.support f ⊆ U ∧ ∀ z ∈ U, ∃ t ∈ 𝓝 z, (t ∩ Function.support f).Finite}\nz : X\nhz : z... | obtain ⟨t₂, ht₂⟩ := hg.2 z hz | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.Topology.LocallyFinsupp | {
"line": 622,
"column": 2
} | {
"line": 622,
"column": 7
} | {
"line": 624,
"column": 0
} | [
{
"pp": "X : Type u_1\ninst✝¹ : TopologicalSpace X\nY : Type u_2\ninst✝ : Zero Y\nU V : Set X\nhV : V ⊆ U\na✝ : X\n⊢ (if a✝ ∈ V then 0 a✝ else 0) = 0 a✝",
"ppTerm": "?m.37",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"False",
"Decidable.casesOn",
"Function.locallyFinsuppW... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Topology.LocallyFinsupp | {
"line": 650,
"column": 4
} | {
"line": 650,
"column": 9
} | {
"line": 651,
"column": 2
} | [
{
"pp": "X : Type u_1\ninst✝² : TopologicalSpace X\nY : Type u_2\ninst✝¹ : DecidableEq X\ninst✝ : AddCommMonoid Y\nU : Set X\nF : locallyFinsuppWithin U Y\nh : F.support.Finite\na✝ : X\n⊢ a✝ ∉ ↑h.toFinset → a✝ ∉ Function.support fun x ↦ restrict (single x (F x)) ⋯",
"ppTerm": "?m.56",
"assigned": true,
... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Topology.LocallyFinsupp | {
"line": 654,
"column": 4
} | {
"line": 654,
"column": 9
} | {
"line": 655,
"column": 2
} | [
{
"pp": "case pos\nX : Type u_1\ninst✝² : TopologicalSpace X\nY : Type u_2\ninst✝¹ : DecidableEq X\ninst✝ : AddCommMonoid Y\nU : Set X\nF : locallyFinsuppWithin U Y\nh : F.support.Finite\nthis : (Function.support fun x ↦ restrict (single x (F x)) ⋯) ⊆ ↑h.toFinset\nz : X\nhz : z ∉ U\n⊢ (∑ i ∈ h.toFinset, restric... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Topology.LocallyFinsupp | {
"line": 654,
"column": 4
} | {
"line": 654,
"column": 9
} | {
"line": 655,
"column": 2
} | [
{
"pp": "case pos\nX : Type u_1\ninst✝² : TopologicalSpace X\nY : Type u_2\ninst✝¹ : DecidableEq X\ninst✝ : AddCommMonoid Y\nU : Set X\nF : locallyFinsuppWithin U Y\nh : F.support.Finite\nthis : (Function.support fun x ↦ restrict (single x (F x)) ⋯) ⊆ ↑h.toFinset\nz : X\nhz : z ∉ U\n⊢ (∑ i ∈ h.toFinset, restric... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.LocallyFinsupp | {
"line": 654,
"column": 4
} | {
"line": 654,
"column": 9
} | {
"line": 655,
"column": 2
} | [
{
"pp": "case pos\nX : Type u_1\ninst✝² : TopologicalSpace X\nY : Type u_2\ninst✝¹ : DecidableEq X\ninst✝ : AddCommMonoid Y\nU : Set X\nF : locallyFinsuppWithin U Y\nh : F.support.Finite\nthis : (Function.support fun x ↦ restrict (single x (F x)) ⋯) ⊆ ↑h.toFinset\nz : X\nhz : z ∉ U\n⊢ (∑ i ∈ h.toFinset, restric... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.LocallyFinsupp | {
"line": 657,
"column": 4
} | {
"line": 657,
"column": 9
} | {
"line": 658,
"column": 2
} | [
{
"pp": "case pos\nX : Type u_1\ninst✝² : TopologicalSpace X\nY : Type u_2\ninst✝¹ : DecidableEq X\ninst✝ : AddCommMonoid Y\nU : Set X\nF : locallyFinsuppWithin U Y\nh : F.support.Finite\nthis : (Function.support fun x ↦ restrict (single x (F x)) ⋯) ⊆ ↑h.toFinset\nz : X\nhz✝ : ¬z ∉ U\nhz : z ∈ F.support\n⊢ (if ... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Topology.LocallyFinsupp | {
"line": 657,
"column": 4
} | {
"line": 657,
"column": 9
} | {
"line": 658,
"column": 2
} | [
{
"pp": "case pos\nX : Type u_1\ninst✝² : TopologicalSpace X\nY : Type u_2\ninst✝¹ : DecidableEq X\ninst✝ : AddCommMonoid Y\nU : Set X\nF : locallyFinsuppWithin U Y\nh : F.support.Finite\nthis : (Function.support fun x ↦ restrict (single x (F x)) ⋯) ⊆ ↑h.toFinset\nz : X\nhz✝ : ¬z ∉ U\nhz : z ∈ F.support\n⊢ (if ... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.LocallyFinsupp | {
"line": 657,
"column": 4
} | {
"line": 657,
"column": 9
} | {
"line": 658,
"column": 2
} | [
{
"pp": "case pos\nX : Type u_1\ninst✝² : TopologicalSpace X\nY : Type u_2\ninst✝¹ : DecidableEq X\ninst✝ : AddCommMonoid Y\nU : Set X\nF : locallyFinsuppWithin U Y\nh : F.support.Finite\nthis : (Function.support fun x ↦ restrict (single x (F x)) ⋯) ⊆ ↑h.toFinset\nz : X\nhz✝ : ¬z ∉ U\nhz : z ∈ F.support\n⊢ (if ... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.LocallyFinsupp | {
"line": 658,
"column": 4
} | {
"line": 658,
"column": 9
} | {
"line": 660,
"column": 0
} | [
{
"pp": "case neg\nX : Type u_1\ninst✝² : TopologicalSpace X\nY : Type u_2\ninst✝¹ : DecidableEq X\ninst✝ : AddCommMonoid Y\nU : Set X\nF : locallyFinsuppWithin U Y\nh : F.support.Finite\nthis : (Function.support fun x ↦ restrict (single x (F x)) ⋯) ⊆ ↑h.toFinset\nz : X\nhz✝ : ¬z ∉ U\nhz : z ∉ F.support\n⊢ (if ... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Topology.LocallyFinsupp | {
"line": 658,
"column": 4
} | {
"line": 658,
"column": 9
} | {
"line": 660,
"column": 0
} | [
{
"pp": "case neg\nX : Type u_1\ninst✝² : TopologicalSpace X\nY : Type u_2\ninst✝¹ : DecidableEq X\ninst✝ : AddCommMonoid Y\nU : Set X\nF : locallyFinsuppWithin U Y\nh : F.support.Finite\nthis : (Function.support fun x ↦ restrict (single x (F x)) ⋯) ⊆ ↑h.toFinset\nz : X\nhz✝ : ¬z ∉ U\nhz : z ∉ F.support\n⊢ (if ... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.LocallyFinsupp | {
"line": 658,
"column": 4
} | {
"line": 658,
"column": 9
} | {
"line": 660,
"column": 0
} | [
{
"pp": "case neg\nX : Type u_1\ninst✝² : TopologicalSpace X\nY : Type u_2\ninst✝¹ : DecidableEq X\ninst✝ : AddCommMonoid Y\nU : Set X\nF : locallyFinsuppWithin U Y\nh : F.support.Finite\nthis : (Function.support fun x ↦ restrict (single x (F x)) ⋯) ⊆ ↑h.toFinset\nz : X\nhz✝ : ¬z ∉ U\nhz : z ∉ F.support\n⊢ (if ... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.LocallyFinsupp | {
"line": 684,
"column": 2
} | {
"line": 684,
"column": 7
} | {
"line": 686,
"column": 0
} | [
{
"pp": "X : Type u_1\ninst✝ : TopologicalSpace X\nU V : Set X\nD : locallyFinsuppWithin U ℤ\nh : V ⊆ U\nx : X\n⊢ (if x ∈ V then (D x)⁺ else 0) = (if x ∈ V then D x else 0)⁺",
"ppTerm": "?m.34",
"assigned": true,
"usedConstants": [
"Int.instAddCommGroup",
"Eq.mpr",
"Decidable.cases... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Topology.LocallyFinsupp | {
"line": 693,
"column": 2
} | {
"line": 693,
"column": 7
} | {
"line": 695,
"column": 0
} | [
{
"pp": "X : Type u_1\ninst✝ : TopologicalSpace X\nU V : Set X\nD : locallyFinsuppWithin U ℤ\nh : V ⊆ U\nx : X\n⊢ (if x ∈ V then (D x)⁻ else 0) = (if x ∈ V then D x else 0)⁻",
"ppTerm": "?m.34",
"assigned": true,
"usedConstants": [
"Int.instAddCommGroup",
"AddGroup.toSubtractionMonoid",
... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.RingTheory.Spectrum.Prime.ChevalleyComplexity | {
"line": 367,
"column": 17
} | {
"line": 367,
"column": 34
} | {
"line": 367,
"column": 35
} | [
{
"pp": "case e'_3\nR₀ : Type u_1\ninst✝² : CommRing R₀\nn : ℕ\nR : Type u_6\ninst✝¹ : CommRing R\ninst✝ : Algebra R₀ R\nc : R\ni : Fin n\ne : InductionObj R n\nhi : c = (e.val i).leadingCoeff\nhc : c ≠ 0\nq₁ : R →ₐ[R₀] Localization.Away c := IsScalarTower.toAlgHom R₀ R (Localization.Away c)\nq₂ : R →ₐ[R₀] R ⧸ ... | [
"case e'_3\nR₀ : Type u_1\ninst✝² : CommRing R₀\nn : ℕ\nR : Type u_6\ninst✝¹ : CommRing R\ninst✝ : Algebra R₀ R\nc : R\ni : Fin n\ne : InductionObj R n\nhi : c = (e.val i).leadingCoeff\nhc : c ≠ 0\nq₁ : R →ₐ[R₀] Localization.Away c := IsScalarTower.toAlgHom R₀ R (Localization.Away c)\nq₂ : R →ₐ[R₀] R ⧸ Ideal.span {... | ← Finset.mem_coe, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.AlgebraicGeometry.FunctionField | {
"line": 98,
"column": 35
} | {
"line": 99,
"column": 68
} | {
"line": 101,
"column": 0
} | [
{
"pp": "X : Scheme\ninst✝¹ : IrreducibleSpace ↥X\nU : X.Opens\ninst✝ : Nonempty ↥↑U\nx : ↥X\nhx : x ∈ U\nf : ↑Γ(X, U)\n⊢ (algebraMap ↑(X.presheaf.stalk x) ↑X.functionField) ((ConcreteCategory.hom (X.presheaf.germ U x hx)) f) =\n (ConcreteCategory.hom (X.germToFunctionField U)) f",
"ppTerm": "?m.38",
... | [] | by
simp [RingHom.algebraMap_toAlgebra, ← ConcreteCategory.comp_apply] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.AlgebraicGeometry.Birational.RationalMap | {
"line": 297,
"column": 2
} | {
"line": 314,
"column": 86
} | {
"line": 316,
"column": 0
} | [
{
"pp": "X Y : Scheme\ninst✝¹ : IrreducibleSpace ↥X\nx : ↥X\ninst✝ : X.IsGermInjectiveAt x\nf g : X.PartialMap Y\nhxf : x ∈ f.domain\nhxg : x ∈ g.domain\nH : f.fromSpecStalkOfMem hxf = g.fromSpecStalkOfMem hxg\n⊢ f.equiv g",
"ppTerm": "?m.26",
"assigned": true,
"usedConstants": [
"AlgebraicGeo... | [] | have hdense : Dense ((f.domain ⊓ g.domain) : Set X) :=
f.dense_domain.inter_of_isOpen_left g.dense_domain f.domain.2
have := (isGermInjectiveAt_iff_of_isOpenImmersion (f := (f.domain ⊓ g.domain).ι)
(x := ⟨x, hxf, hxg⟩)).mp ‹_›
have := spread_out_unique_of_isGermInjective' (X := (f.domain ⊓ g.domain).toSchem... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.AlgebraicGeometry.Birational.RationalMap | {
"line": 297,
"column": 2
} | {
"line": 314,
"column": 86
} | {
"line": 316,
"column": 0
} | [
{
"pp": "X Y : Scheme\ninst✝¹ : IrreducibleSpace ↥X\nx : ↥X\ninst✝ : X.IsGermInjectiveAt x\nf g : X.PartialMap Y\nhxf : x ∈ f.domain\nhxg : x ∈ g.domain\nH : f.fromSpecStalkOfMem hxf = g.fromSpecStalkOfMem hxg\n⊢ f.equiv g",
"ppTerm": "?m.26",
"assigned": true,
"usedConstants": [
"AlgebraicGeo... | [] | have hdense : Dense ((f.domain ⊓ g.domain) : Set X) :=
f.dense_domain.inter_of_isOpen_left g.dense_domain f.domain.2
have := (isGermInjectiveAt_iff_of_isOpenImmersion (f := (f.domain ⊓ g.domain).ι)
(x := ⟨x, hxf, hxg⟩)).mp ‹_›
have := spread_out_unique_of_isGermInjective' (X := (f.domain ⊓ g.domain).toSchem... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Spectrum.Prime.ChevalleyComplexity | {
"line": 390,
"column": 17
} | {
"line": 390,
"column": 34
} | {
"line": 390,
"column": 35
} | [
{
"pp": "case e'_3\nR₀ : Type u_1\ninst✝² : CommRing R₀\nn : ℕ\nR : Type u_6\ninst✝¹ : CommRing R\ninst✝ : Algebra R₀ R\nc : R\ni : Fin n\ne : InductionObj R n\nhi : c = (e.val i).leadingCoeff\nhc : c ≠ 0\nq₁ : R →ₐ[R₀] Localization.Away c := IsScalarTower.toAlgHom R₀ R (Localization.Away c)\nq₂ : R →ₐ[R₀] R ⧸ ... | [
"case e'_3\nR₀ : Type u_1\ninst✝² : CommRing R₀\nn : ℕ\nR : Type u_6\ninst✝¹ : CommRing R\ninst✝ : Algebra R₀ R\nc : R\ni : Fin n\ne : InductionObj R n\nhi : c = (e.val i).leadingCoeff\nhc : c ≠ 0\nq₁ : R →ₐ[R₀] Localization.Away c := IsScalarTower.toAlgHom R₀ R (Localization.Away c)\nq₂ : R →ₐ[R₀] R ⧸ Ideal.span {... | ← Finset.mem_coe, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.AlgebraicGeometry.Morphisms.Flat | {
"line": 457,
"column": 54
} | {
"line": 457,
"column": 59
} | {
"line": 458,
"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\ninst✝ : Flat f\nhUS : IsAffineOpen US\nhUT : IsAffineOpen UT\nhUX : IsCompact ↑... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.AlgebraicGeometry.Morphisms.Flat | {
"line": 502,
"column": 54
} | {
"line": 502,
"column": 59
} | {
"line": 503,
"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\ninst✝ : Flat f\nhUS : IsAffineOpen US\nhUT : IsCompact ↑UT\nhUT' : IsQuasiSepar... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.RingTheory.Spectrum.Prime.ChevalleyComplexity | {
"line": 420,
"column": 10
} | {
"line": 420,
"column": 39
} | {
"line": 421,
"column": 10
} | [
{
"pp": "case ht\nR₀ : Type u_1\ninst✝² : CommRing R₀\nn : ℕ\nR : Type u_6\ninst✝¹ : CommRing R\ninst✝ : Algebra R₀ R\nc : R\ni : Fin n\ne : InductionObj R n\nhi : c = (e.val i).leadingCoeff\nhc : c ≠ 0\nq₁ : R →ₐ[R₀] Localization.Away c := ⋯\nq₂ : R →ₐ[R₀] R ⧸ Ideal.span {c} := ⋯\nq₂_surjective : Surjective ⇑q... | [
"case h\nR₀ : Type u_1\ninst✝² : CommRing R₀\nn : ℕ\nR : Type u_6\ninst✝¹ : CommRing R\ninst✝ : Algebra R₀ R\nc : R\ni : Fin n\ne : InductionObj R n\nhi : c = (e.val i).leadingCoeff\nhc : c ≠ 0\nq₁ : R →ₐ[R₀] Localization.Away c := ⋯\nq₂ : R →ₐ[R₀] R ⧸ Ideal.span {c} := ⋯\nq₂_surjective : Surjective ⇑q₂\ne₁ : Induc... | · exact one_le_coeffSubmodule | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.RingTheory.Spectrum.Prime.ChevalleyComplexity | {
"line": 427,
"column": 6
} | {
"line": 427,
"column": 35
} | {
"line": 428,
"column": 6
} | [
{
"pp": "case ha\nR₀ : Type u_1\ninst✝² : CommRing R₀\nn : ℕ\nR : Type u_6\ninst✝¹ : CommRing R\ninst✝ : Algebra R₀ R\nc : R\ni : Fin n\ne : InductionObj R n\nhi : c = (e.val i).leadingCoeff\nhc : c ≠ 0\nq₁ : R →ₐ[R₀] Localization.Away c := ⋯\nq₂ : R →ₐ[R₀] R ⧸ Ideal.span {c} := ⋯\nq₂_surjective : Surjective ⇑q... | [
"case ht\nR₀ : Type u_1\ninst✝² : CommRing R₀\nn : ℕ\nR : Type u_6\ninst✝¹ : CommRing R\ninst✝ : Algebra R₀ R\nc : R\ni : Fin n\ne : InductionObj R n\nhi : c = (e.val i).leadingCoeff\nhc : c ≠ 0\nq₁ : R →ₐ[R₀] Localization.Away c := ⋯\nq₂ : R →ₐ[R₀] R ⧸ Ideal.span {c} := ⋯\nq₂_surjective : Surjective ⇑q₂\ne₁ : Indu... | · exact one_le_coeffSubmodule | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Topology.Sets.CompactOpenCovered | {
"line": 51,
"column": 48
} | {
"line": 51,
"column": 53
} | {
"line": 52,
"column": 2
} | [
{
"pp": "case refine_1.inl\nS : Type u_1\nι : Type u_2\nX : ι → Type u_3\nf : (i : ι) → X i → S\ninst✝¹ : (i : ι) → TopologicalSpace (X i)\nU : Set S\ninst✝ : Unique ι\nx✝ : IsCompactOpenCovered f U\ns : Set ι\nhs : s.Finite\nV : (i : ι) → i ∈ s → Opens (X i)\nhc : ∀ (i : ι) (h : i ∈ s), IsCompact (V i h).carri... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Topology.Sets.CompactOpenCovered | {
"line": 51,
"column": 48
} | {
"line": 51,
"column": 53
} | {
"line": 52,
"column": 2
} | [
{
"pp": "case refine_1.inr\nS : Type u_1\nι : Type u_2\nX : ι → Type u_3\nf : (i : ι) → X i → S\ninst✝¹ : (i : ι) → TopologicalSpace (X i)\nU : Set S\ninst✝ : Unique ι\nx✝ : IsCompactOpenCovered f U\ns : Set ι\nhs : s.Finite\nV : (i : ι) → i ∈ s → Opens (X i)\nhc : ∀ (i : ι) (h : i ∈ s), IsCompact (V i h).carri... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Topology.Sets.CompactOpenCovered | {
"line": 76,
"column": 6
} | {
"line": 76,
"column": 11
} | {
"line": 77,
"column": 2
} | [
{
"pp": "case refine_1.refine_3\nS : Type u_1\nι : Type u_2\nX : ι → Type u_3\nf : (i : ι) → X i → S\ninst✝ : (i : ι) → TopologicalSpace (X i)\nU : Set S\nx✝ : IsCompactOpenCovered f U\ns : Set ι\nhs : s.Finite\nV : (i : ι) → i ∈ s → Opens (X i)\nhc : ∀ (i : ι) (h : i ∈ s), IsCompact (V i h).carrier\nhU : ⋃ i, ... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Topology.Sets.CompactOpenCovered | {
"line": 76,
"column": 6
} | {
"line": 76,
"column": 11
} | {
"line": 77,
"column": 2
} | [
{
"pp": "case refine_1.refine_3\nS : Type u_1\nι : Type u_2\nX : ι → Type u_3\nf : (i : ι) → X i → S\ninst✝ : (i : ι) → TopologicalSpace (X i)\nU : Set S\nx✝ : IsCompactOpenCovered f U\ns : Set ι\nhs : s.Finite\nV : (i : ι) → i ∈ s → Opens (X i)\nhc : ∀ (i : ι) (h : i ∈ s), IsCompact (V i h).carrier\nhU : ⋃ i, ... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Sets.CompactOpenCovered | {
"line": 76,
"column": 6
} | {
"line": 76,
"column": 11
} | {
"line": 77,
"column": 2
} | [
{
"pp": "case refine_1.refine_3\nS : Type u_1\nι : Type u_2\nX : ι → Type u_3\nf : (i : ι) → X i → S\ninst✝ : (i : ι) → TopologicalSpace (X i)\nU : Set S\nx✝ : IsCompactOpenCovered f U\ns : Set ι\nhs : s.Finite\nV : (i : ι) → i ∈ s → Opens (X i)\nhc : ∀ (i : ι) (h : i ∈ s), IsCompact (V i h).carrier\nhU : ⋃ i, ... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Sets.CompactOpenCovered | {
"line": 103,
"column": 40
} | {
"line": 103,
"column": 45
} | {
"line": 103,
"column": 45
} | [
{
"pp": "S : Type u_1\nι : Type u_2\nX : ι → Type u_3\nf : (i : ι) → X i → S\ninst✝¹ : (i : ι) → TopologicalSpace (X i)\ninst✝ : TopologicalSpace S\nU : Set S\nhU : IsCompact U\ns : Set (Opens S)\nhs : ⋃ t ∈ s, ↑t = U\nH : ∀ t ∈ s, IsCompactOpenCovered f ↑t\nt : Finset ↑s\nht : U ⊆ ⋃ i ∈ t, ↑↑i\nx : S\nh : x ∈ ... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Topology.Sets.CompactOpenCovered | {
"line": 103,
"column": 40
} | {
"line": 103,
"column": 45
} | {
"line": 103,
"column": 45
} | [
{
"pp": "S : Type u_1\nι : Type u_2\nX : ι → Type u_3\nf : (i : ι) → X i → S\ninst✝¹ : (i : ι) → TopologicalSpace (X i)\ninst✝ : TopologicalSpace S\nU : Set S\nhU : IsCompact U\ns : Set (Opens S)\nhs : ⋃ t ∈ s, ↑t = U\nH : ∀ t ∈ s, IsCompactOpenCovered f ↑t\nt : Finset ↑s\nht : U ⊆ ⋃ i ∈ t, ↑↑i\nx : S\nh : x ∈ ... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Sets.CompactOpenCovered | {
"line": 103,
"column": 40
} | {
"line": 103,
"column": 45
} | {
"line": 103,
"column": 45
} | [
{
"pp": "S : Type u_1\nι : Type u_2\nX : ι → Type u_3\nf : (i : ι) → X i → S\ninst✝¹ : (i : ι) → TopologicalSpace (X i)\ninst✝ : TopologicalSpace S\nU : Set S\nhU : IsCompact U\ns : Set (Opens S)\nhs : ⋃ t ∈ s, ↑t = U\nH : ∀ t ∈ s, IsCompactOpenCovered f ↑t\nt : Finset ↑s\nht : U ⊆ ⋃ i ∈ t, ↑↑i\nx : S\nh : x ∈ ... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Sets.CompactOpenCovered | {
"line": 151,
"column": 34
} | {
"line": 151,
"column": 39
} | {
"line": 151,
"column": 39
} | [
{
"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 : ∀ ... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Topology.Sets.CompactOpenCovered | {
"line": 151,
"column": 34
} | {
"line": 151,
"column": 39
} | {
"line": 151,
"column": 39
} | [
{
"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 : ∀ ... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Sets.CompactOpenCovered | {
"line": 151,
"column": 34
} | {
"line": 151,
"column": 39
} | {
"line": 151,
"column": 39
} | [
{
"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 : ∀ ... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Sets.CompactOpenCovered | {
"line": 160,
"column": 32
} | {
"line": 164,
"column": 29
} | {
"line": 166,
"column": 0
} | [
{
"pp": "S : Type u_1\nι : Type u_2\nX : ι → Type u_3\nf : (i : ι) → X i → S\ninst✝² : (i : ι) → TopologicalSpace (X i)\ninst✝¹ : Finite ι\ninst✝ : TopologicalSpace S\nhf : ∀ (i : ι), IsSpectralMap (f i)\nU : Set S\nhs : ∀ x ∈ U, ∃ i, x ∈ Set.range (f i)\nhU : IsOpen U\nhc : IsCompact U\n⊢ IsCompactOpenCovered ... | [] | by
refine ⟨.univ, Set.finite_univ, fun i _ ↦ ⟨f i ⁻¹' U, hU.preimage (hf i).1⟩,
fun i _ ↦ hc.preimage_of_isOpen (hf i) hU, subset_antisymm (by simp) fun x hx ↦ ?_⟩
obtain ⟨i, y, rfl⟩ := hs x hx
simpa using ⟨i, y, hx, rfl⟩ | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.AlgebraicGeometry.AffineTransitionLimit | {
"line": 313,
"column": 2
} | {
"line": 313,
"column": 71
} | {
"line": 314,
"column": 2
} | [
{
"pp": "I : Type u\ninst✝² : Category.{u, u} I\nD : I ⥤ Scheme\nc : Cone D\nhc : IsLimit c\ninst✝¹ : IsCofiltered I\ninst✝ : ∀ {i j : I} (f : i ⟶ j), IsAffineHom (D.map f)\n⊢ TopologicalSpace.Opens.IsBasis {x | ∃ i V, ∃ (_ : IsAffineOpen V), c.π.app i ⁻¹ᵁ V = x}",
"ppTerm": "?m.78",
"assigned": true,
... | [
"I : Type u\ninst✝² : Category.{u, u} I\nD : I ⥤ Scheme\nc : Cone D\nhc : IsLimit c\ninst✝¹ : IsCofiltered I\ninst✝ : ∀ {i j : I} (f : i ⟶ j), IsAffineHom (D.map f)\nU : TopologicalSpace.Opens ↥c.pt\nx : ↥c.pt\nhxU : x ∈ U\n⊢ ∃ U' ∈ {x | ∃ i V, ∃ (_ : IsAffineOpen V), c.π.app i ⁻¹ᵁ V = x}, x ∈ U' ∧ U' ≤ U"
] | refine TopologicalSpace.Opens.isBasis_iff_nbhd.mpr fun {U x} hxU ↦ ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.CategoryTheory.EffectiveEpi.Comp | {
"line": 47,
"column": 4
} | {
"line": 47,
"column": 9
} | {
"line": 48,
"column": 2
} | [
{
"pp": "C : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\nα : Type u_2\nB : C\nX Y : α → C\nf : (a : α) → X a ⟶ B\ng : (a : α) → Y a ⟶ X a\ni : (a : α) → X a ⟶ Y a\nhi : ∀ (a : α), i a ≫ g a = 𝟙 (X a)\ninst✝ : EffectiveEpiFamily X f\nW✝ : C\ne : (a : α) → Y a ⟶ W✝\nw : ∀ {Z : C} (a₁ a₂ : α) (g₁ : Z ⟶ Y a₁) (g₂ : ... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.CategoryTheory.EffectiveEpi.Preserves | {
"line": 110,
"column": 8
} | {
"line": 110,
"column": 39
} | {
"line": 110,
"column": 39
} | [
{
"pp": "C : Type u_1\ninst✝⁴ : Category.{v_1, u_1} C\nD : Type u_2\ninst✝³ : Category.{v_2, u_2} D\ninst✝² : IsRegularEpiCategory D\nF : C ⥤ D\ninst✝¹ : F.PreservesEpimorphisms\ninst✝ : HasPullbacks D\nX✝ Y✝ : C\nx✝¹ : X✝ ⟶ Y✝\nx✝ : EffectiveEpi x✝¹\n⊢ EffectiveEpi (F.map x✝¹)",
"ppTerm": "?m.20",
"ass... | [
"C : Type u_1\ninst✝⁴ : Category.{v_1, u_1} C\nD : Type u_2\ninst✝³ : Category.{v_2, u_2} D\ninst✝² : IsRegularEpiCategory D\nF : C ⥤ D\ninst✝¹ : F.PreservesEpimorphisms\ninst✝ : HasPullbacks D\nX✝ Y✝ : C\nx✝¹ : X✝ ⟶ Y✝\nx✝ : EffectiveEpi x✝¹\n⊢ IsRegularEpi (F.map x✝¹)"
] | ← isRegularEpi_iff_effectiveEpi | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.AlgebraicGeometry.EllipticCurve.Weierstrass | {
"line": 318,
"column": 2
} | {
"line": 318,
"column": 15
} | {
"line": 319,
"column": 2
} | [
{
"pp": "R : Type u\ninst✝¹ : CommRing R\nW : WeierstrassCurve R\ninst✝ : CharP R 2\n⊢ { a := 4, b := W.b₂, c := 2 * W.b₄, d := W.b₆ } = { a := 0, b := W.b₂, c := 0, d := W.b₆ }",
"ppTerm": "?m.22",
"assigned": true,
"usedConstants": [
"WeierstrassCurve.b₄._proof_1",
"HMul.hMul",
"... | [
"case a\nR : Type u\ninst✝¹ : CommRing R\nW : WeierstrassCurve R\ninst✝ : CharP R 2\n⊢ 4 = 0",
"case c\nR : Type u\ninst✝¹ : CommRing R\nW : WeierstrassCurve R\ninst✝ : CharP R 2\n⊢ 2 * W.b₄ = 0"
] | ext <;> dsimp | Lean.Parser.Tactic.«_aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tactic_<;>__1» | Lean.Parser.Tactic.«tactic_<;>_» |
Mathlib.AlgebraicGeometry.EllipticCurve.Weierstrass | {
"line": 333,
"column": 2
} | {
"line": 333,
"column": 15
} | {
"line": 334,
"column": 2
} | [
{
"pp": "R : Type u\ninst✝¹ : CommRing R\nW : WeierstrassCurve R\ninst✝ : CharP R 3\n⊢ { a := 4, b := W.b₂, c := 2 * W.b₄, d := W.b₆ } = { a := 1, b := W.b₂, c := -W.b₄, d := W.b₆ }",
"ppTerm": "?m.24",
"assigned": true,
"usedConstants": [
"WeierstrassCurve.b₄._proof_1",
"NegZeroClass.to... | [
"case a\nR : Type u\ninst✝¹ : CommRing R\nW : WeierstrassCurve R\ninst✝ : CharP R 3\n⊢ 4 = 1",
"case c\nR : Type u\ninst✝¹ : CommRing R\nW : WeierstrassCurve R\ninst✝ : CharP R 3\n⊢ 2 * W.b₄ = -W.b₄"
] | ext <;> dsimp | Lean.Parser.Tactic.«_aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tactic_<;>__1» | Lean.Parser.Tactic.«tactic_<;>_» |
Mathlib.AlgebraicGeometry.EllipticCurve.Affine.Basic | {
"line": 162,
"column": 16
} | {
"line": 162,
"column": 41
} | {
"line": 162,
"column": 42
} | [
{
"pp": "R : Type r\ninst✝ : CommRing R\nW : Affine R\n⊢ evalEval 0 0 W.polynomial = 0 ↔ W.a₆ = 0",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NegZeroClass.toNeg",
"congrArg",
"CommSemiring.toSemiring",
"id",
"SubtractionMonoid.toSubNegZero... | [
"R : Type r\ninst✝ : CommRing R\nW : Affine R\n⊢ -W.a₆ = 0 ↔ W.a₆ = 0"
] | evalEval_polynomial_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Spectrum.Prime.ChevalleyComplexity | {
"line": 832,
"column": 8
} | {
"line": 832,
"column": 42
} | {
"line": 832,
"column": 43
} | [
{
"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... | [
"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)).degrees ≤ d\ng :... | range_comap_of_surjective _ _ hg', | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Spectrum.Prime.ChevalleyComplexity | {
"line": 901,
"column": 8
} | {
"line": 903,
"column": 43
} | {
"line": 903,
"column": 43
} | [
{
"pp": "case right.inr\nR : 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 (MvPolyn... | [] | · refine degrees_sub_le.trans ?_
simp only [degrees_C, Multiset.zero_union]
exact degrees_map_le.trans (hf _) | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.NumberTheory.EllipticDivisibilitySequence | {
"line": 165,
"column": 31
} | {
"line": 165,
"column": 54
} | {
"line": 165,
"column": 55
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\nW : ℤ → R\na b c : ℤ\n⊢ W a * W 0 * atom W b c - atom W a b * atom W a c + atom W a c * atom W a b = W a * W 0 * atom W b c",
"ppTerm": "?m.21",
"assigned": true,
"usedConstants": [
"NonUnitalNonAssocCommRing.toNonUnitalNonAssocCommSemiring",
"E... | [
"R : Type u_1\ninst✝ : CommRing R\nW : ℤ → R\na b c : ℤ\n⊢ W a * W 0 * atom W b c - atom W a c * atom W a b + atom W a c * atom W a b = W a * W 0 * atom W b c"
] | mul_comm <| atom W a b, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.NumberTheory.EllipticDivisibilitySequence | {
"line": 503,
"column": 2
} | {
"line": 504,
"column": 7
} | {
"line": 506,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\nb c d : R\nk : ℤ\n⊢ preNormEDS (b ^ 4) c d k * complEDS₂ b c d k = preNormEDS (b ^ 4) c d (2 * k) * if Even k then 1 else b",
"ppTerm": "?m.54",
"assigned": true,
"usedConstants": [
"Mathlib.Tactic.Ring.Common.mul_pf_left",
"Mathlib.Tactic.Ring.... | [] | rw [complEDS₂, preNormEDS_even]
ring1 | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.EllipticDivisibilitySequence | {
"line": 503,
"column": 2
} | {
"line": 504,
"column": 7
} | {
"line": 506,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\nb c d : R\nk : ℤ\n⊢ preNormEDS (b ^ 4) c d k * complEDS₂ b c d k = preNormEDS (b ^ 4) c d (2 * k) * if Even k then 1 else b",
"ppTerm": "?m.54",
"assigned": true,
"usedConstants": [
"Mathlib.Tactic.Ring.Common.mul_pf_left",
"Mathlib.Tactic.Ring.... | [] | rw [complEDS₂, preNormEDS_even]
ring1 | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.AlgebraicGeometry.EllipticCurve.DivisionPolynomial.Degree | {
"line": 70,
"column": 41
} | {
"line": 72,
"column": 17
} | {
"line": 74,
"column": 0
} | [
{
"pp": "R : Type u\ninst✝ : CommRing R\nW : WeierstrassCurve R\n⊢ W.Ψ₂Sq.coeff 3 = 4",
"ppTerm": "?m.10",
"assigned": true,
"usedConstants": [
"one_pow",
"Eq.mpr",
"Polynomial.C",
"NonAssocSemiring.toAddCommMonoidWithOne",
"MulOne.toOne",
"le_refl",
"False"... | [] | by
rw [Ψ₂Sq]
compute_degree! | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.AlgebraicGeometry.EllipticCurve.NormalForms | {
"line": 149,
"column": 31
} | {
"line": 149,
"column": 51
} | {
"line": 149,
"column": 52
} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommRing R\nW : WeierstrassCurve R\ninst✝ : W.IsCharNeTwoNF\n⊢ -(4 * W.a₂) ^ 3 + 36 * (4 * W.a₂) * W.b₄ - 216 * W.b₆ = -64 * W.a₂ ^ 3 + 288 * W.a₂ * W.a₄ - 864 * W.a₆",
"ppTerm": "?m.62",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NegZeroClass.toNeg",... | [
"R : Type u_1\ninst✝¹ : CommRing R\nW : WeierstrassCurve R\ninst✝ : W.IsCharNeTwoNF\n⊢ -(4 * W.a₂) ^ 3 + 36 * (4 * W.a₂) * (2 * W.a₄) - 216 * W.b₆ = -64 * W.a₂ ^ 3 + 288 * W.a₂ * W.a₄ - 864 * W.a₆"
] | b₄_of_isCharNeTwoNF, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.AlgebraicGeometry.EllipticCurve.NormalForms | {
"line": 155,
"column": 30
} | {
"line": 155,
"column": 50
} | {
"line": 155,
"column": 51
} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommRing R\nW : WeierstrassCurve R\ninst✝ : W.IsCharNeTwoNF\n⊢ -(4 * W.a₂) ^ 2 * W.b₈ - 8 * W.b₄ ^ 3 - 27 * W.b₆ ^ 2 + 9 * (4 * W.a₂) * W.b₄ * W.b₆ =\n -64 * W.a₂ ^ 3 * W.a₆ + 16 * W.a₂ ^ 2 * W.a₄ ^ 2 - 64 * W.a₄ ^ 3 - 432 * W.a₆ ^ 2 + 288 * W.a₂ * W.a₄ * W.a₆",
"ppTerm": ... | [
"R : Type u_1\ninst✝¹ : CommRing R\nW : WeierstrassCurve R\ninst✝ : W.IsCharNeTwoNF\n⊢ -(4 * W.a₂) ^ 2 * W.b₈ - 8 * (2 * W.a₄) ^ 3 - 27 * W.b₆ ^ 2 + 9 * (4 * W.a₂) * (2 * W.a₄) * W.b₆ =\n -64 * W.a₂ ^ 3 * W.a₆ + 16 * W.a₂ ^ 2 * W.a₄ ^ 2 - 64 * W.a₄ ^ 3 - 432 * W.a₆ ^ 2 + 288 * W.a₂ * W.a₄ * W.a₆"
] | b₄_of_isCharNeTwoNF, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.AlgebraicGeometry.EllipticCurve.IsomOfJ | {
"line": 100,
"column": 6
} | {
"line": 101,
"column": 49
} | {
"line": 102,
"column": 6
} | [
{
"pp": "case of_j_ne_zero.of_j_eq_zero\nF : Type u_1\ninst✝⁶ : Field F\ninst✝⁵ : IsSepClosed F\nE E' : WeierstrassCurve F\ninst✝⁴ : E.IsElliptic\ninst✝³ : E'.IsElliptic\ninst✝² : CharP F 2\nheq : E.j = E'.j\nC : VariableChange F\ninst✝¹ : (C • E).IsCharTwoJNeZeroNF\nC' : VariableChange F\ninst✝ : (C' • E').IsC... | [
"case of_j_ne_zero.of_j_eq_zero\nF : Type u_1\ninst✝⁶ : Field F\ninst✝⁵ : IsSepClosed F\nE E' : WeierstrassCurve F\ninst✝⁴ : E.IsElliptic\ninst✝³ : E'.IsElliptic\ninst✝² : CharP F 2\nheq : E.j = E'.j\nC : VariableChange F\ninst✝¹ : (C • E).IsCharTwoJNeZeroNF\nC' : VariableChange F\ninst✝ : (C' • E').IsCharTwoJEqZer... | rw [variableChange_j, heq, ← variableChange_j E' C',
j_of_isCharTwoJEqZeroNF_of_char_two] at h | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.AlgebraicGeometry.AffineTransitionLimit | {
"line": 1244,
"column": 2
} | {
"line": 1255,
"column": 55
} | {
"line": 1259,
"column": 2
} | [
{
"pp": "I : Type u\ninst✝⁵ : Category.{u, u} I\nS X : Scheme\nD : I ⥤ Scheme\nt : D ⟶ (Functor.const I).obj S\nf : X ⟶ S\nc : Cone D\nhc : IsLimit c\ninst✝⁴ : IsCofiltered I\ninst✝³ : LocallyOfFinitePresentation f\ninst✝² : ∀ {i j : I} (f : i ⟶ j), IsAffineHom (D.map f)\ninst✝¹ : ∀ (i : I), CompactSpace ↥(D.ob... | [
"I : Type u\ninst✝⁵ : Category.{u, u} I\nS X : Scheme\nD : I ⥤ Scheme\nt : D ⟶ (Functor.const I).obj S\nf : X ⟶ S\nc : Cone D\nhc : IsLimit c\ninst✝⁴ : IsCofiltered I\ninst✝³ : LocallyOfFinitePresentation f\ninst✝² : ∀ {i j : I} (f : i ⟶ j), IsAffineHom (D.map f)\ninst✝¹ : ∀ (i : I), CompactSpace ↥(D.obj i)\ninst✝ ... | obtain ⟨i', fi'i, hi'⟩ : ∃ (i' : I) (fi'i : i' ⟶ i),
∀ j, D.map fi'i ⁻¹ᵁ 𝒱 j ≤ t.app i' ⁻¹ᵁ j.1.1.2.1.2.1 := by
choose k fk hk using fun j ↦ exists_map_preimage_le_map_preimage D c hc (h𝒱𝒰 j).1.isCompact
(V := t.app i ⁻¹ᵁ j.1.1.2.1.2.1) (by
rw [← Hom.comp_preimage, ← NatTrans.comp_app, ha]
... | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.AlgebraicGeometry.EllipticCurve.DivisionPolynomial.Degree | {
"line": 266,
"column": 32
} | {
"line": 266,
"column": 37
} | {
"line": 268,
"column": 0
} | [
{
"pp": "case neg.zero\nR : Type u\ninst✝¹ : CommRing R\nW : WeierstrassCurve R\ninst✝ : Nontrivial R\nh : ↑0 ≠ 0\nhn : ¬2 < 0\n⊢ W.preΨ' 0 ≠ 0",
"ppTerm": "?neg.zero✝",
"assigned": true,
"usedConstants": [
"False",
"congrArg",
"CommSemiring.toSemiring",
"False.elim",
"... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.AlgebraicGeometry.EllipticCurve.DivisionPolynomial.Degree | {
"line": 266,
"column": 32
} | {
"line": 266,
"column": 37
} | {
"line": 268,
"column": 0
} | [
{
"pp": "case neg.succ.zero\nR : Type u\ninst✝¹ : CommRing R\nW : WeierstrassCurve R\ninst✝ : Nontrivial R\nh : ↑(0 + 1) ≠ 0\nhn : ¬2 < 0 + 1\n⊢ W.preΨ' (0 + 1) ≠ 0",
"ppTerm": "?neg.succ.zero✝",
"assigned": true,
"usedConstants": [
"False",
"Polynomial.instOne",
"NeZero.one",
... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.AlgebraicGeometry.EllipticCurve.DivisionPolynomial.Degree | {
"line": 266,
"column": 32
} | {
"line": 266,
"column": 37
} | {
"line": 268,
"column": 0
} | [
{
"pp": "case neg.succ.succ\nR : Type u\ninst✝¹ : CommRing R\nW : WeierstrassCurve R\ninst✝ : Nontrivial R\nn✝ : ℕ\nh : ↑(n✝ + 1 + 1) ≠ 0\nhn : ¬2 < n✝ + 1 + 1\n⊢ W.preΨ' (n✝ + 1 + 1) ≠ 0",
"ppTerm": "?neg.succ.succ✝",
"assigned": true,
"usedConstants": [
"IsRightCancelAdd.addRightStrictMono_o... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.AlgebraicGeometry.EllipticCurve.Jacobian.Basic | {
"line": 537,
"column": 74
} | {
"line": 538,
"column": 53
} | {
"line": 540,
"column": 0
} | [
{
"pp": "R : Type r\ninst✝¹ : CommRing R\nW' : Jacobian R\nS : Type s\ninst✝ : CommRing S\nf : R →+* S\n⊢ (W'.map f).polynomialX = (MvPolynomial.map f) W'.polynomialX",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"Derivation",
"Finsupp.instAddZeroClass",
"WeierstrassCur... | [] | by
simp only [polynomialX, map_polynomial, pderiv_map] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.AlgebraicGeometry.EllipticCurve.Jacobian.Basic | {
"line": 585,
"column": 57
} | {
"line": 586,
"column": 89
} | {
"line": 588,
"column": 0
} | [
{
"pp": "R : Type r\ninst✝¹⁰ : CommRing R\nW' : Jacobian R\nS : Type s\ninst✝⁹ : CommRing S\nA : Type u\ninst✝⁸ : CommRing A\nB : Type v\ninst✝⁷ : CommRing B\ninst✝⁶ : Algebra R S\ninst✝⁵ : Algebra R A\ninst✝⁴ : Algebra S A\ninst✝³ : IsScalarTower R S A\ninst✝² : Algebra R B\ninst✝¹ : Algebra S B\ninst✝ : IsSca... | [] | by
rw [← RingHom.coe_coe, ← map_nonsingular _ hf, AlgHom.toRingHom_eq_coe, map_baseChange] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Data.Int.WithZero | {
"line": 48,
"column": 21
} | {
"line": 48,
"column": 41
} | {
"line": 48,
"column": 42
} | [
{
"pp": "e : ℝ≥0\nhe : e ≠ 0\n⊢ (if hx : 1 = 0 then 0 else e ^ toAdd (unzero hx)) = 1",
"ppTerm": "?m.36",
"assigned": true,
"usedConstants": [
"WithZero.instNontrivial",
"Eq.mpr",
"MulOne.toOne",
"Int.instInhabited",
"NeZero.one",
"Equiv.instEquivLike",
"NN... | [
"e : ℝ≥0\nhe : e ≠ 0\n⊢ e ^ toAdd (unzero ⋯) = 1"
] | dif_neg one_ne_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.AlgebraicGeometry.EllipticCurve.Jacobian.Point | {
"line": 156,
"column": 2
} | {
"line": 156,
"column": 73
} | {
"line": 157,
"column": 2
} | [
{
"pp": "R : Type r\ninst✝ : CommRing R\nW' : Jacobian R\nP : Fin 3 → R\nhP : W'.Equation P\n⊢ W'.addY P (W'.neg P) = -W'.dblZ P ^ 3",
"ppTerm": "?m.25",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NegZeroClass.toNeg",
"WeierstrassCurve.Jacobian.addY",
"congrArg",
"... | [
"R : Type r\ninst✝ : CommRing R\nW' : Jacobian R\nP : Fin 3 → R\nhP : W'.Equation P\n⊢ -![W'.dblZ P ^ 2, W'.dblZ P ^ 3, 0] y = -W'.dblZ P ^ 3"
] | rw [addY, addX_neg hP, negAddY_neg hP, addZ_neg, negY_of_Z_eq_zero rfl] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.AlgebraicGeometry.EllipticCurve.Jacobian.Formula | {
"line": 560,
"column": 21
} | {
"line": 560,
"column": 84
} | {
"line": 560,
"column": 84
} | [
{
"pp": "F : Type u\ninst✝ : Field F\nW : Jacobian F\nP Q : Fin 3 → F\nhPz : P z ≠ 0\nhQz : Q z ≠ 0\n⊢ W.negAddY P Q = W.negAddY P Q * (P z * Q z) ^ 3 / (P z * Q z) ^ 3",
"ppTerm": "?m.226",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"IsDomain.to_noZeroDivisors",
"instHDiv",
... | [
"F : Type u\ninst✝ : Field F\nW : Jacobian F\nP Q : Fin 3 → F\nhPz : P z ≠ 0\nhQz : Q z ≠ 0\n⊢ W.negAddY P Q = W.negAddY P Q"
] | mul_div_cancel_right₀ _ <| pow_ne_zero 3 <| mul_ne_zero hPz hQz | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.Algebra.Valued.ValuationTopology | {
"line": 92,
"column": 38
} | {
"line": 92,
"column": 43
} | {
"line": 92,
"column": 43
} | [
{
"pp": "R : Type u\ninst✝¹ : Ring R\nΓ₀ : Type v\ninst✝ : LinearOrderedCommGroupWithZero Γ₀\nv : Valuation R Γ₀\nx : R\nγ : (ofClass v).ValueGroup₀ˣ\nγx : Γ₀ˣ\nHx : v x = ↑γx\n⊢ ¬v x = 0",
"ppTerm": "?m.387",
"assigned": true,
"usedConstants": [
"Units.val",
"GroupWithZero.toMonoidWithZ... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Topology.Algebra.Valued.ValuationTopology | {
"line": 109,
"column": 38
} | {
"line": 109,
"column": 43
} | {
"line": 109,
"column": 43
} | [
{
"pp": "R : Type u\ninst✝¹ : Ring R\nΓ₀ : Type v\ninst✝ : LinearOrderedCommGroupWithZero Γ₀\nv : Valuation R Γ₀\nx : R\nγ : (ofClass v).ValueGroup₀ˣ\nγx : Γ₀ˣ\nHx : v x = ↑γx\n⊢ ¬v x = 0",
"ppTerm": "?m.580",
"assigned": true,
"usedConstants": [
"Units.val",
"GroupWithZero.toMonoidWithZ... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.RingTheory.Valuation.ValuativeRel.Basic | {
"line": 551,
"column": 4
} | {
"line": 551,
"column": 69
} | {
"line": 552,
"column": 4
} | [
{
"pp": "R : Type u_1\ninst✝¹ : Semiring R\ninst✝ : ValuativeRel R\nx x' y y' z x✝¹ : R\nx✝ : ↥(posSubmonoid R)\n⊢ 0 * ValueGroupWithZero.mk x✝¹ x✝ = 0",
"ppTerm": "?m.357",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Submonoid.mul",
"HMul.hMul",
"congrArg",
"Member... | [
"R : Type u_1\ninst✝¹ : Semiring R\ninst✝ : ValuativeRel R\nx x' y y' z x✝¹ : R\nx✝ : ↥(posSubmonoid R)\n⊢ ValueGroupWithZero.mk (0 * x✝¹) (1 * x✝) = ValueGroupWithZero.mk 0 1"
] | rw [← ValueGroupWithZero.mk_zero 1, ValueGroupWithZero.mk_mul_mk] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
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