module string | startPos dict | endPos dict | nextStartPos dict | goals list | goalsAfter list | ppTac string | elaborator string | kind string |
|---|---|---|---|---|---|---|---|---|
Mathlib.AlgebraicGeometry.AffineSpace | {
"line": 433,
"column": 4
} | {
"line": 433,
"column": 87
} | {
"line": 435,
"column": 0
} | [
{
"pp": "case a\nn : Type u\nS : Scheme\ninst✝ : IsEmpty n\n⊢ isomorphisms Scheme (terminal.from (Spec (CommRingCat.of (ULift.{u, 0} ℤ))))",
"ppTerm": "?a✝",
"assigned": true,
"usedConstants": [
"AlgebraicGeometry.Spec",
"AlgebraicGeometry.Scheme",
"AlgebraicGeometry.specULiftZIsTe... | [] | exact isIso_of_isTerminal specULiftZIsTerminal terminalIsTerminal (terminal.from _) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.AlgebraicGeometry.AlgClosed.Basic | {
"line": 82,
"column": 4
} | {
"line": 82,
"column": 92
} | {
"line": 83,
"column": 4
} | [
{
"pp": "X Y : Scheme\nK : Type u\ninst✝² : Field K\ninst✝¹ : IsAlgClosed K\nf : X ⟶ Spec (CommRingCat.of K)\ninst✝ : LocallyOfFiniteType f\nx : ↥X\nhx : IsClosed {x}\np : { p // p ≫ f = 𝟙 (Spec (CommRingCat.of K)) }\npf✝¹ : IsLocalRing ↑(CommRingCat.of K)\nh : IsClosed {↑p (IsLocalRing.closedPoint K)}\npf✝ : ... | [
"X Y : Scheme\nK : Type u\ninst✝² : Field K\ninst✝¹ : IsAlgClosed K\nf : X ⟶ Spec (CommRingCat.of K)\ninst✝ : LocallyOfFiniteType f\nx : ↥X\nhx : IsClosed {x}\np : { p // p ≫ f = 𝟙 (Spec (CommRingCat.of K)) }\npf✝¹ : IsLocalRing ↑(CommRingCat.of K)\nh : IsClosed {↑p (IsLocalRing.closedPoint K)}\npf✝ : IsLocalHom (... | refine (Category.comp_id _).symm.trans (((residueFieldIsoBase f _ h).eq_inv_comp).mp ?_) | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Topology.LocallyFinsupp | {
"line": 165,
"column": 25
} | {
"line": 165,
"column": 92
} | {
"line": 165,
"column": 92
} | [
{
"pp": "X : Type u_1\ninst✝² : TopologicalSpace X\nU : Set X\nY : Type u_2\ninst✝¹ : DecidableEq X\ninst✝ : Zero Y\nx : X\ny : Y\nx✝¹ : X\nx✝ : x✝¹ ∈ univ\n⊢ (univ ∩ Function.support (Pi.single x y)).Finite",
"ppTerm": "?m.39",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
... | [] | by simpa using (finite_singleton x).subset Pi.support_single_subset | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.Spectrum.Prime.ChevalleyComplexity | {
"line": 336,
"column": 2
} | {
"line": 336,
"column": 36
} | {
"line": 338,
"column": 2
} | [
{
"pp": "R₀ : Type u_1\ninst✝² : CommRing R₀\nn : ℕ\nR : Type u_6\ninst✝¹ : CommRing R\ninst✝ : Algebra R₀ R\nc : R\ni : Fin n\ne : InductionObj R n\nhi : c = (e.val i).leadingCoeff\nhc : c ≠ 0\nq₁ : R →ₐ[R₀] Localization.Away c := IsScalarTower.toAlgHom R₀ R (Localization.Away c)\nq₂ : R →ₐ[R₀] R ⧸ Ideal.span ... | [
"R₀ : Type u_1\ninst✝² : CommRing R₀\nn : ℕ\nR : Type u_6\ninst✝¹ : CommRing R\ninst✝ : Algebra R₀ R\nc : R\ni : Fin n\ne : InductionObj R n\nhi : c = (e.val i).leadingCoeff\nhc : c ≠ 0\nq₁ : R →ₐ[R₀] Localization.Away c := IsScalarTower.toAlgHom R₀ R (Localization.Away c)\nq₂ : R →ₐ[R₀] R ⧸ Ideal.span {c} := Ideal... | choose! g₂ hg₂ hq₂g₂ using hT₂span | Mathlib.Tactic.Choose._aux_Mathlib_Tactic_Choose___macroRules_Mathlib_Tactic_Choose_tacticChoose!___Using__1 | Mathlib.Tactic.Choose.tacticChoose!___Using_ |
Mathlib.Topology.LocallyFinsupp | {
"line": 531,
"column": 2
} | {
"line": 533,
"column": 84
} | {
"line": 534,
"column": 2
} | [
{
"pp": "X : Type u_1\ninst✝³ : TopologicalSpace X\nU : Set X\nY : Type u_2\ninst✝² : AddCommGroup Y\ninst✝¹ : LinearOrder Y\ninst✝ : IsOrderedAddMonoid Y\nf₁ f₂ : locallyFinsuppWithin U Y\nx : X\n⊢ (-(f₁ + f₂) ⊔ 0) x ≤ (-f₁ ⊔ 0 + -f₂ ⊔ 0) x",
"ppTerm": "?m.41",
"assigned": true,
"usedConstants": [
... | [
"X : Type u_1\ninst✝³ : TopologicalSpace X\nU : Set X\nY : Type u_2\ninst✝² : AddCommGroup Y\ninst✝¹ : LinearOrder Y\ninst✝ : IsOrderedAddMonoid Y\nf₁ f₂ : locallyFinsuppWithin U Y\nx : X\n⊢ -f₂ x + -f₁ x ≤ max (-f₁ x) 0 + max (-f₂ x) 0 ∧ 0 ≤ max (-f₁ x) 0 + max (-f₂ x) 0"
] | simp only [neg_add_rev, Function.locallyFinsuppWithin.max_apply,
Function.locallyFinsuppWithin.coe_add, Function.locallyFinsuppWithin.coe_neg, Pi.add_apply,
Pi.neg_apply, Function.locallyFinsuppWithin.coe_zero, Pi.zero_apply, sup_le_iff] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.AlgebraicGeometry.SpreadingOut | {
"line": 88,
"column": 2
} | {
"line": 88,
"column": 92
} | {
"line": 90,
"column": 0
} | [
{
"pp": "X : Scheme\nx : ↥X\ninst✝ : X.IsGermInjectiveAt x\nV : X.Opens\nhxV : x ∈ V\nU : X.Opens\nhx : x ∈ U\nhU : IsAffineOpen U\nH : Function.Injective ⇑(ConcreteCategory.hom (X.presheaf.germ U x hx))\nf : ↑Γ(X, U)\nhf : X.basicOpen f ≤ V\nhxf : ↑⟨x, hxV⟩ ∈ X.basicOpen f\n⊢ ∃ U, ∃ (hx : x ∈ U), IsAffineOpen ... | [] | exact ⟨X.basicOpen f, hxf, hU.basicOpen f, hf, injective_germ_basicOpen U hU x hx f hxf H⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.AlgebraicGeometry.Morphisms.Flat | {
"line": 414,
"column": 4
} | {
"line": 414,
"column": 75
} | {
"line": 415,
"column": 4
} | [
{
"pp": "case id_pair\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\nhVU : iSup VX = U... | [
"case left\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\nhVU : iSup VX = UX\nhV : ∀ (i : ... | · simp [show Pairwise.Hom.id_pair i j = 𝟙 (Pairwise.pair i j) from rfl] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.AlgebraicGeometry.SpreadingOut | {
"line": 285,
"column": 2
} | {
"line": 285,
"column": 36
} | {
"line": 286,
"column": 2
} | [
{
"pp": "X : Scheme\nR A : CommRingCat\nx : ↥X\ninst✝ : X.IsGermInjectiveAt x\nU : X.Opens\nhxU : x ∈ U\nφ : A ⟶ X.presheaf.stalk x\nφRA : R ⟶ A\nφRX : R ⟶ Γ(X, U)\nhφRA : (CommRingCat.Hom.hom φRA).FiniteType\ne : φRA ≫ φ = φRX ≫ X.presheaf.germ U x hxU\nV : X.Opens\nhxV : x ∈ V\niVU : V ≤ U\nhV : (CommRingCat.... | [
"X : Scheme\nR A : CommRingCat\nx : ↥X\ninst✝ : X.IsGermInjectiveAt x\nU : X.Opens\nhxU : x ∈ U\nφ : A ⟶ X.presheaf.stalk x\nφRA : R ⟶ A\nφRX : R ⟶ Γ(X, U)\nhφRA : (CommRingCat.Hom.hom φRA).FiniteType\ne : φRA ≫ φ = φRX ≫ X.presheaf.germ U x hxU\nV : X.Opens\nhxV : x ∈ V\niVU : V ≤ U\nhV : (CommRingCat.Hom.hom φ).r... | let f := X.presheaf.germ V' x hxV' | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLet___1 | Lean.Parser.Tactic.tacticLet__ |
Mathlib.AlgebraicGeometry.Birational.RationalMap | {
"line": 627,
"column": 37
} | {
"line": 627,
"column": 86
} | {
"line": 627,
"column": 86
} | [
{
"pp": "X Y : Scheme\ninst✝¹ : IsReduced X\ninst✝ : Y.IsSeparated\nf : X.PartialMap Y\n⊢ (f.toRationalMap.toPartialMap.restrict f.domain ⋯ ⋯).hom = (X.isoOfEq ⋯).hom ≫ f.hom",
"ppTerm": "?m.62",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"AlgebraicGeometry.Scheme.homOfLE",
"Al... | [] | simpa using f.toPartialMap_toRationalMap_restrict | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.AlgebraicGeometry.Birational.RationalMap | {
"line": 627,
"column": 37
} | {
"line": 627,
"column": 86
} | {
"line": 627,
"column": 86
} | [
{
"pp": "X Y : Scheme\ninst✝¹ : IsReduced X\ninst✝ : Y.IsSeparated\nf : X.PartialMap Y\n⊢ (f.toRationalMap.toPartialMap.restrict f.domain ⋯ ⋯).hom = (X.isoOfEq ⋯).hom ≫ f.hom",
"ppTerm": "?m.62",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"AlgebraicGeometry.Scheme.homOfLE",
"Al... | [] | simpa using f.toPartialMap_toRationalMap_restrict | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.AlgebraicGeometry.Birational.RationalMap | {
"line": 627,
"column": 37
} | {
"line": 627,
"column": 86
} | {
"line": 627,
"column": 86
} | [
{
"pp": "X Y : Scheme\ninst✝¹ : IsReduced X\ninst✝ : Y.IsSeparated\nf : X.PartialMap Y\n⊢ (f.toRationalMap.toPartialMap.restrict f.domain ⋯ ⋯).hom = (X.isoOfEq ⋯).hom ≫ f.hom",
"ppTerm": "?m.62",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"AlgebraicGeometry.Scheme.homOfLE",
"Al... | [] | simpa using f.toPartialMap_toRationalMap_restrict | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.AlgebraicGeometry.Birational.Dominant | {
"line": 63,
"column": 2
} | {
"line": 63,
"column": 66
} | {
"line": 64,
"column": 2
} | [
{
"pp": "X Y : Scheme\nf g : X.PartialMap Y\nW : X.Opens\nhW : Dense ↑W\nhWl : W ≤ f.domain\nhWr : W ≤ g.domain\nh : (f.restrict W hW hWl).hom = (g.restrict W hW hWr).hom\n⊢ IsDominant f.hom ↔ IsDominant g.hom",
"ppTerm": "?m.39",
"assigned": true,
"usedConstants": [
"AlgebraicGeometry.Scheme.... | [
"X Y : Scheme\nf g : X.PartialMap Y\nW : X.Opens\nhW : Dense ↑W\nhWl : W ≤ f.domain\nhWr : W ≤ g.domain\nh : (f.restrict W hW hWl).hom = (g.restrict W hW hWr).hom\ne₁ : IsDominant f.hom ↔ IsDominant (f.restrict W hW hWl).hom\n⊢ IsDominant f.hom ↔ IsDominant g.hom"
] | have e₁ := isDominant_hom_iff_isDominant_restrict_hom f W hW hWl | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.AlgebraicGeometry.Birational.Composition | {
"line": 217,
"column": 40
} | {
"line": 217,
"column": 59
} | {
"line": 217,
"column": 60
} | [
{
"pp": "X Y : Scheme\ninst✝ : IrreducibleSpace ↥X\nf : X ⤏ Y\n⊢ (id X).comp f.representative.toRationalMap = f.representative.toRationalMap",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"AlgebraicGeometry.SheafedSpace.instTopologicalSpaceCarrierCarrier",
"Alg... | [
"X Y : Scheme\ninst✝ : IrreducibleSpace ↥X\nf : X ⤏ Y\n⊢ ((PartialMap.id X).comp f.representative).toRationalMap = f.representative.toRationalMap"
] | toRationalMap_comp, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.AlgebraicGeometry.Morphisms.UniversallyInjective | {
"line": 120,
"column": 4
} | {
"line": 120,
"column": 44
} | {
"line": 121,
"column": 4
} | [
{
"pp": "X Y : Scheme\nf : X ⟶ Y\ntfae_1_iff_4 : UniversallyInjective f ↔ Surjective (pullback.diagonal f)\nh_inj : Injective ⇑f\nhf : ∀ (x : ↥X), (CommRingCat.Hom.hom (Scheme.Hom.residueFieldMap f x)).IsPurelyInseparable\nK : Type u\ninst✝ : Field K\ng₁ g₂ : Spec (CommRingCat.of K) ⟶ X\nhg : (fun g ↦ g ≫ f) g₁... | [
"X Y : Scheme\nf : X ⟶ Y\ntfae_1_iff_4 : UniversallyInjective f ↔ Surjective (pullback.diagonal f)\nh_inj : Injective ⇑f\nhf : ∀ (x : ↥X), (CommRingCat.Hom.hom (Scheme.Hom.residueFieldMap f x)).IsPurelyInseparable\nK : Type u\ninst✝ : Field K\ng₁ g₂ : Spec (CommRingCat.of K) ⟶ X\nhg : (fun g ↦ g ≫ f) g₁ = (fun g ↦ ... | refine ⟨h_inj e, CommRingCat.hom_ext ?_⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.AlgebraicGeometry.Morphisms.UniversallyInjective | {
"line": 165,
"column": 4
} | {
"line": 165,
"column": 69
} | {
"line": 166,
"column": 4
} | [
{
"pp": "case e_a\nX Y : Scheme\nf : X ⟶ Y\ntfae_1_iff_4 : UniversallyInjective f ↔ Surjective (pullback.diagonal f)\ntfae_3_to_2 :\n (Injective ⇑f ∧ ∀ (x : ↥X), (CommRingCat.Hom.hom (Scheme.Hom.residueFieldMap f x)).IsPurelyInseparable) →\n ∀ (K : Type u) [inst : Field K], Injective fun g ↦ g ≫ f\ntfae_2_t... | [
"case e_a\nX Y : Scheme\nf : X ⟶ Y\ntfae_1_iff_4 : UniversallyInjective f ↔ Surjective (pullback.diagonal f)\ntfae_3_to_2 :\n (Injective ⇑f ∧ ∀ (x : ↥X), (CommRingCat.Hom.hom (Scheme.Hom.residueFieldMap f x)).IsPurelyInseparable) →\n ∀ (K : Type u) [inst : Field K], Injective fun g ↦ g ≫ f\ntfae_2_to_4 : (∀ (K ... | rw [← cancel_mono (Scheme.residueFieldCongr (hux ▸ hu).symm).hom] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.AlgebraicGeometry.AffineTransitionLimit | {
"line": 134,
"column": 45
} | {
"line": 134,
"column": 91
} | {
"line": 134,
"column": 91
} | [
{
"pp": "I : Type u\ninst✝² : Category.{u, u} I\nD : I ⥤ Scheme\nc : Cone D\nhc : IsLimit c\ninst✝¹ : IsCofilteredOrEmpty I\ninst✝ : ∀ {i j : I} (f : i ⟶ j), IsAffineHom (D.map f)\nZ : (i : I) → Set ↥(D.obj i)\nhZc : ∀ (i : I), IsClosed (Z i)\nhZne : ∀ (i : I), (Z i).Nonempty\nhZcpt : ∀ (i : I), IsCompact (Z i)... | [] | by simp [le_map_iff_comap_le, le_iSup_of_le i] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.Sets.CompactOpenCovered | {
"line": 82,
"column": 28
} | {
"line": 82,
"column": 35
} | {
"line": 82,
"column": 36
} | [
{
"pp": "case refine_2\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✝ : ∃ V, IsCompact V.carrier ∧ (fun p ↦ f p.fst p.snd) '' V.carrier = U\nV : Opens ((i : ι) × X i)\nhc✝ : IsCompact V.carrier\nheq : (fun p ↦ f p.fst p.snd) '' V.car... | [
"case refine_2\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✝ : ∃ V, IsCompact V.carrier ∧ (fun p ↦ f p.fst p.snd) '' V.carrier = U\nV : Opens ((i : ι) × X i)\nhc✝ : IsCompact V.carrier\nheq : (fun p ↦ f p.fst p.snd) '' V.carrier = U\ns ... | ← heq', | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.AlgebraicGeometry.ColimitsOver | {
"line": 201,
"column": 6
} | {
"line": 202,
"column": 82
} | {
"line": 203,
"column": 4
} | [
{
"pp": "case refine_1.refine_1\nP : MorphismProperty Scheme\ninst✝⁸ : P.IsStableUnderBaseChange\ninst✝⁷ : P.IsMultiplicative\nS : Scheme\nJ : Type u_1\ninst✝⁶ : Category.{v_1, u_1} J\nD : J ⥤ P.Over ⊤ S\n𝒰 : S.OpenCover\ninst✝⁵ : Category.{v_2, ?u.129} 𝒰.I₀\ninst✝⁴ : LocallyDirected 𝒰\nd : ColimitGluingData... | [] | intro i
exact ((d.cocone i).ι.app a).left ≫ colimit.ι d.relativeGluingData.functor i | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.AlgebraicGeometry.ColimitsOver | {
"line": 201,
"column": 6
} | {
"line": 202,
"column": 82
} | {
"line": 203,
"column": 4
} | [
{
"pp": "case refine_1.refine_1\nP : MorphismProperty Scheme\ninst✝⁸ : P.IsStableUnderBaseChange\ninst✝⁷ : P.IsMultiplicative\nS : Scheme\nJ : Type u_1\ninst✝⁶ : Category.{v_1, u_1} J\nD : J ⥤ P.Over ⊤ S\n𝒰 : S.OpenCover\ninst✝⁵ : Category.{v_2, ?u.129} 𝒰.I₀\ninst✝⁴ : LocallyDirected 𝒰\nd : ColimitGluingData... | [] | intro i
exact ((d.cocone i).ι.app a).left ≫ colimit.ι d.relativeGluingData.functor i | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.EffectiveEpi.Preserves | {
"line": 109,
"column": 19
} | {
"line": 111,
"column": 46
} | {
"line": 113,
"column": 0
} | [
{
"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... | [] | by
rw [← isRegularEpi_iff_effectiveEpi]
apply IsRegularEpiCategory.regularEpiOfEpi | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.AlgebraicGeometry.AffineTransitionLimit | {
"line": 831,
"column": 63
} | {
"line": 834,
"column": 70
} | {
"line": 836,
"column": 0
} | [
{
"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 : I), IsAffine (D.obj i)\ns : ↑Γ(c.pt, ⊤)\n⊢ ∃ i t, (ConcreteCategory.hom (Scheme.Hom.appTop (c.π.app i))) t = s",
"ppTerm": "?m.79",
"assigned": true,
"usedConstants": ... | [] | by
have : ∀ i, IsAffine (D.op.obj i).unop := by dsimp; infer_instance
exact ⟨_, (Types.jointly_surjective_of_isColimit
(isColimitOfPreserves (Scheme.Γ ⋙ forget _) hc.op) s).choose_spec⟩ | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.AlgebraicGeometry.EllipticCurve.Affine.Point | {
"line": 174,
"column": 43
} | {
"line": 174,
"column": 64
} | {
"line": 174,
"column": 64
} | [
{
"pp": "R : Type r\nS : Type s\nA F : Type u\nB K : Type v\nL : Type w\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing S\ninst✝⁴ : CommRing A\ninst✝³ : CommRing B\ninst✝² : Field F\ninst✝¹ : Field K\ninst✝ : Field L\nW' : Affine R\nW : Affine F\nf : R →+* S\n⊢ eval₂ (AdjoinRoot.of (W'.map f).polynomial) (AdjoinRoot.ro... | [
"R : Type r\nS : Type s\nA F : Type u\nB K : Type v\nL : Type w\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing S\ninst✝⁴ : CommRing A\ninst✝³ : CommRing B\ninst✝² : Field F\ninst✝¹ : Field K\ninst✝ : Field L\nW' : Affine R\nW : Affine F\nf : R →+* S\n⊢ 0 = 0"
] | AdjoinRoot.eval₂_root | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.AlgebraicGeometry.EllipticCurve.Affine.Point | {
"line": 277,
"column": 2
} | {
"line": 290,
"column": 7
} | {
"line": 292,
"column": 0
} | [
{
"pp": "F : Type u\ninst✝¹ : Field F\nW : Affine F\ninst✝ : DecidableEq F\nx₁ x₂ y₁ y₂ : F\nh₁ : W.Equation x₁ y₁\nh₂ : W.Equation x₂ y₂\nhxy : ¬(x₁ = x₂ ∧ y₁ = W.negY x₂ y₂)\n⊢ XYIdeal W x₂ (C y₂) = XYIdeal W x₂ (linePolynomial x₁ y₁ (W.slope x₁ x₂ y₁ y₂))",
"ppTerm": "?m.22",
"assigned": true,
"u... | [] | have hy₂ : y₂ = (linePolynomial x₁ y₁ <| W.slope x₁ x₂ y₁ y₂).eval x₂ := by
by_cases hx : x₁ = x₂
· have hy : y₁ ≠ W.negY x₂ y₂ := fun h => hxy ⟨hx, h⟩
rcases hx, Y_eq_of_Y_ne h₁ h₂ hx hy with ⟨rfl, rfl⟩
simp [linePolynomial]
· simp [field, linePolynomial, slope_of_X_ne hx]
ring1
nth_rw ... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.AlgebraicGeometry.EllipticCurve.Affine.Point | {
"line": 277,
"column": 2
} | {
"line": 290,
"column": 7
} | {
"line": 292,
"column": 0
} | [
{
"pp": "F : Type u\ninst✝¹ : Field F\nW : Affine F\ninst✝ : DecidableEq F\nx₁ x₂ y₁ y₂ : F\nh₁ : W.Equation x₁ y₁\nh₂ : W.Equation x₂ y₂\nhxy : ¬(x₁ = x₂ ∧ y₁ = W.negY x₂ y₂)\n⊢ XYIdeal W x₂ (C y₂) = XYIdeal W x₂ (linePolynomial x₁ y₁ (W.slope x₁ x₂ y₁ y₂))",
"ppTerm": "?m.22",
"assigned": true,
"u... | [] | have hy₂ : y₂ = (linePolynomial x₁ y₁ <| W.slope x₁ x₂ y₁ y₂).eval x₂ := by
by_cases hx : x₁ = x₂
· have hy : y₁ ≠ W.negY x₂ y₂ := fun h => hxy ⟨hx, h⟩
rcases hx, Y_eq_of_Y_ne h₁ h₂ hx hy with ⟨rfl, rfl⟩
simp [linePolynomial]
· simp [field, linePolynomial, slope_of_X_ne hx]
ring1
nth_rw ... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.AlgebraicGeometry.EllipticCurve.Affine.Formula | {
"line": 326,
"column": 2
} | {
"line": 326,
"column": 38
} | {
"line": 328,
"column": 0
} | [
{
"pp": "F : Type u\ninst✝¹ : Field F\nW : Affine F\ninst✝ : DecidableEq F\nx₁ x₂ y₁ y₂ : F\nh₁ : W.Equation x₁ y₁\nh₂ : W.Equation x₂ y₂\nhxy : ¬(x₁ = x₂ ∧ y₁ = W.negY x₂ y₂)\n⊢ -((W.addX x₁ x₂ (W.slope x₁ x₂ y₁ y₂) - x₁) * (W.addX x₁ x₂ (W.slope x₁ x₂ y₁ y₂) - x₂) *\n (W.addX x₁ x₂ (W.slope x₁ x₂ y₁ y₂... | [] | rw [neg_eq_zero, sub_self, mul_zero] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.AlgebraicGeometry.EllipticCurve.Affine.Formula | {
"line": 372,
"column": 98
} | {
"line": 375,
"column": 7
} | {
"line": 377,
"column": 0
} | [
{
"pp": "F : Type u\ninst✝¹ : Field F\nW : Affine F\ninst✝ : DecidableEq F\nx₁ x₂ y₁ y₂ : F\nhx : x₁ ≠ x₂\n⊢ let x₃ := W.addX x₁ x₂ (W.slope x₁ x₂ y₁ y₂);\n y₁ * (x₂ - x₃) + y₂ * (x₃ - x₁) + W.negAddY x₁ x₂ y₁ (W.slope x₁ x₂ y₁ y₂) * (x₁ - x₂) = 0",
"ppTerm": "?m.48",
"assigned": true,
"usedConstan... | [] | by
simp_rw [slope_of_X_ne hx, negAddY, addX]
simp [field]
ring1 | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.NumberTheory.EllipticDivisibilitySequence | {
"line": 150,
"column": 89
} | {
"line": 152,
"column": 64
} | {
"line": 154,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\nW : ℤ → R\na b : ℤ\n⊢ atom W (2 * a + 1) (2 * b + 1) = W (a + b + 1) * W (a - b)",
"ppTerm": "?m.49",
"assigned": true,
"usedConstants": [
"Int.instAddCommGroup",
"Distrib.leftDistribClass",
"Eq.mpr",
"IsEllipticNet.atom",
"Non... | [] | by
simp_rw [atom, add_add_add_comm _ (1 : ℤ), ← two_mul, ← mul_add, add_sub_add_comm, sub_self,
add_zero, ← mul_sub, Int.mul_tdiv_cancel_left _ two_ne_zero] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.AlgebraicGeometry.EllipticCurve.Affine.Point | {
"line": 433,
"column": 12
} | {
"line": 437,
"column": 54
} | {
"line": 437,
"column": 55
} | [
{
"pp": "case neg.some.some.inl\nR : Type r\ninst✝¹ : CommRing R\nW' : Affine R\ninst✝ : IsDomain R\np q : R[X]\nhp : ¬p.degree = ⊥\nhq : ¬q.degree = ⊥\ndp : ℕ\nhdp : (p ^ 2).degree = 2 • some dp\nhp' : p.degree = some dp\ndq : ℕ\nhdq : (q ^ 2 * (X ^ 3 + C W'.a₂ * X ^ 2 + C W'.a₄ * X + C W'.a₆)).degree = 2 • so... | [
"case neg.some.some.inl.convert_2\nR : Type r\ninst✝¹ : CommRing R\nW' : Affine R\ninst✝ : IsDomain R\np q : R[X]\nhp : ¬p.degree = ⊥\nhq : ¬q.degree = ⊥\ndp : ℕ\nhdp : (p ^ 2).degree = 2 • some dp\nhp' : p.degree = some dp\ndq : ℕ\nhdq : (q ^ 2 * (X ^ 3 + C W'.a₂ * X ^ 2 + C W'.a₄ * X + C W'.a₆)).degree = 2 • some... | convert!
(degree_sub_eq_right_of_degree_lt <|
(degree_sub_le _ _).trans_lt <|
max_lt_iff.mpr ⟨hdp.trans_lt _, hdpq.trans_lt _⟩).trans
(max_eq_right_of_lt _).symm <;> rw [hdq] | Lean.Parser.Tactic.«_aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tactic_<;>__1» | Lean.Parser.Tactic.«tactic_<;>_» |
Mathlib.NumberTheory.EllipticDivisibilitySequence | {
"line": 268,
"column": 2
} | {
"line": 268,
"column": 9
} | {
"line": 270,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\nW : ℤ → R\nm : ℤ\n⊢ W (m + 1 + (m - 1) + 0) * W (m + 1 - (m - 1)) * W (1 + 0) * W 1 -\n W (m + 1 + 1 + 0) * W (m + 1 - 1) * W (m - 1 + 0) * W (m - 1) +\n W (m - 1 + 1 + 0) * W (m - 1 - 1) * W (m + 1 + 0) * W (m + 1) =\n W (2 * m) * W 2 * W 1 ^ 2 - W (m -... | [] | ring_nf | Mathlib.Tactic.RingNF._aux_Mathlib_Tactic_Ring_RingNF___elabRules_Mathlib_Tactic_RingNF_ringNF_1 | Mathlib.Tactic.RingNF.ringNF |
Mathlib.NumberTheory.EllipticDivisibilitySequence | {
"line": 274,
"column": 2
} | {
"line": 274,
"column": 9
} | {
"line": 276,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\nW : ℤ → R\nm : ℤ\n⊢ W (m + 1 + m + 0) * W (m + 1 - m) * W (1 + 0) * W 1 - W (m + 1 + 1 + 0) * W (m + 1 - 1) * W (m + 0) * W m +\n W (m + 1 + 0) * W (m - 1) * W (m + 1 + 0) * W (m + 1) =\n W (2 * m + 1) * W 1 ^ 3 - W (m + 2) * W m ^ 3 + W (m - 1) * W (m + 1) ^ 3... | [] | ring_nf | Mathlib.Tactic.RingNF._aux_Mathlib_Tactic_Ring_RingNF___elabRules_Mathlib_Tactic_RingNF_ringNF_1 | Mathlib.Tactic.RingNF.ringNF |
Mathlib.AlgebraicGeometry.AffineTransitionLimit | {
"line": 960,
"column": 54
} | {
"line": 966,
"column": 93
} | {
"line": 968,
"column": 0
} | [
{
"pp": "I : Type u\ninst✝⁴ : Category.{u, u} I\nD : I ⥤ Scheme\nc : Cone D\nhc : IsLimit c\ninst✝³ : IsCofiltered I\ninst✝² : ∀ {i j : I} (f : i ⟶ j), IsAffineHom (D.map f)\ninst✝¹ : ∀ (i : I), CompactSpace ↥(D.obj i)\ninst✝ : ∀ (i : I), QuasiSeparatedSpace ↥(D.obj i)\n⊢ Nonempty (IsColimit (Scheme.Γ.mapCocone... | [] | by
have : ReflectsFilteredColimits (forget CommRingCat) :=
⟨fun _ ↦ reflectsColimitsOfShape_of_reflectsIsomorphisms⟩
refine ReflectsColimit.reflects (F := forget _) (Types.FilteredColimit.isColimitOf' _ _ ?_ ?_)
· exact fun s ↦ ⟨.op _, (exists_appTop_π_eq_of_isLimit D c hc s).choose_spec⟩
· exact fun i s t ... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.NumberTheory.EllipticDivisibilitySequence | {
"line": 405,
"column": 77
} | {
"line": 408,
"column": 53
} | {
"line": 410,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\nb c d : R\nn : ℕ\n⊢ preNormEDS b c d ↑n = preNormEDS' b c d n",
"ppTerm": "?m.9",
"assigned": true,
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"Int.cast",
"HMul.hMul",
"AddGroupWithOne.toAddGroup",
"congrArg",
"Co... | [] | by
by_cases hn : n = 0
· simp [hn, preNormEDS]
· simp [preNormEDS, Int.sign_natCast_of_ne_zero hn] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.NumberTheory.EllipticDivisibilitySequence | {
"line": 544,
"column": 46
} | {
"line": 544,
"column": 53
} | {
"line": 544,
"column": 53
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\nb c d : R\nn : ℤ\n⊢ (-preNormEDS (b ^ 4) c d n * if Even n then b else 1) = -(preNormEDS (b ^ 4) c d n * if Even n then b else 1)",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"Int.instAddCommGroup",
"NegZeroClass.toNeg",
"NonU... | [] | neg_mul | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.AlgebraicGeometry.EllipticCurve.NormalForms | {
"line": 517,
"column": 6
} | {
"line": 517,
"column": 31
} | {
"line": 517,
"column": 32
} | [
{
"pp": "R : Type u_1\ninst✝² : CommRing R\nW : WeierstrassCurve R\ninst✝¹ : W.IsCharTwoJNeZeroNF\ninst✝ : CharP R 2\n⊢ W.c₄ = 1",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"CommSemiring.toSemiring",
"AddGroupWithOne.toAddMonoidWithOne",
... | [
"R : Type u_1\ninst✝² : CommRing R\nW : WeierstrassCurve R\ninst✝¹ : W.IsCharTwoJNeZeroNF\ninst✝ : CharP R 2\n⊢ W.b₂ ^ 2 = 1"
] | c₄_of_isCharTwoJNeZeroNF, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.AlgebraicGeometry.EllipticCurve.NormalForms | {
"line": 528,
"column": 2
} | {
"line": 528,
"column": 51
} | {
"line": 530,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝² : CommRing R\nW : WeierstrassCurve R\ninst✝¹ : W.IsCharTwoJNeZeroNF\ninst✝ : CharP R 2\n⊢ -1 ^ 2 * W.a₆ - 8 * 0 ^ 3 - 27 * 0 ^ 2 + 9 * 1 * 0 * 0 = W.a₆",
"ppTerm": "?m.37",
"assigned": true,
"usedConstants": [
"Mathlib.Tactic.Ring.Common.mul_pf_left",
"CharP... | [] | linear_combination -W.a₆ * CharP.cast_eq_zero R 2 | Mathlib.Tactic.LinearCombination._aux_Mathlib_Tactic_LinearCombination___elabRules_Mathlib_Tactic_LinearCombination_linearCombination_1 | Mathlib.Tactic.LinearCombination.linearCombination |
Mathlib.AlgebraicGeometry.EllipticCurve.NormalForms | {
"line": 534,
"column": 2
} | {
"line": 535,
"column": 87
} | {
"line": 537,
"column": 0
} | [
{
"pp": "F : Type u_2\ninst✝³ : Field F\nW : WeierstrassCurve F\ninst✝² : W.IsElliptic\ninst✝¹ : W.IsCharTwoJNeZeroNF\ninst✝ : CharP F 2\n⊢ W.j = 1 / W.a₆",
"ppTerm": "?m.24",
"assigned": true,
"usedConstants": [
"one_pow",
"Units.val",
"Eq.mpr",
"WeierstrassCurve.Δ",
"... | [] | rw [j, Units.val_inv_eq_inv_val, ← div_eq_inv_mul, coe_Δ',
c₄_of_isCharTwoJNeZeroNF_of_char_two, Δ_of_isCharTwoJNeZeroNF_of_char_two, one_pow] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.AlgebraicGeometry.EllipticCurve.NormalForms | {
"line": 534,
"column": 2
} | {
"line": 535,
"column": 87
} | {
"line": 537,
"column": 0
} | [
{
"pp": "F : Type u_2\ninst✝³ : Field F\nW : WeierstrassCurve F\ninst✝² : W.IsElliptic\ninst✝¹ : W.IsCharTwoJNeZeroNF\ninst✝ : CharP F 2\n⊢ W.j = 1 / W.a₆",
"ppTerm": "?m.24",
"assigned": true,
"usedConstants": [
"one_pow",
"Units.val",
"Eq.mpr",
"WeierstrassCurve.Δ",
"... | [] | rw [j, Units.val_inv_eq_inv_val, ← div_eq_inv_mul, coe_Δ',
c₄_of_isCharTwoJNeZeroNF_of_char_two, Δ_of_isCharTwoJNeZeroNF_of_char_two, one_pow] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.AlgebraicGeometry.EllipticCurve.NormalForms | {
"line": 534,
"column": 2
} | {
"line": 535,
"column": 87
} | {
"line": 537,
"column": 0
} | [
{
"pp": "F : Type u_2\ninst✝³ : Field F\nW : WeierstrassCurve F\ninst✝² : W.IsElliptic\ninst✝¹ : W.IsCharTwoJNeZeroNF\ninst✝ : CharP F 2\n⊢ W.j = 1 / W.a₆",
"ppTerm": "?m.24",
"assigned": true,
"usedConstants": [
"one_pow",
"Units.val",
"Eq.mpr",
"WeierstrassCurve.Δ",
"... | [] | rw [j, Units.val_inv_eq_inv_val, ← div_eq_inv_mul, coe_Δ',
c₄_of_isCharTwoJNeZeroNF_of_char_two, Δ_of_isCharTwoJNeZeroNF_of_char_two, one_pow] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.AlgebraicGeometry.EllipticCurve.NormalForms | {
"line": 613,
"column": 2
} | {
"line": 614,
"column": 78
} | {
"line": 616,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝² : CommRing R\nW : WeierstrassCurve R\ninst✝¹ : W.IsCharTwoJEqZeroNF\ninst✝ : CharP R 2\n⊢ W.Δ = W.a₃ ^ 4",
"ppTerm": "?m.21",
"assigned": true,
"usedConstants": [
"Mathlib.Tactic.Ring.Common.mul_pf_left",
"CharP.cast_eq_zero",
"AddGroup.toSubtractionMo... | [] | rw [Δ_of_isCharTwoJEqZeroNF, b₆_of_char_two]
linear_combination (-32 * W.a₄ ^ 3 - 14 * W.a₃ ^ 4) * CharP.cast_eq_zero R 2 | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.AlgebraicGeometry.EllipticCurve.NormalForms | {
"line": 613,
"column": 2
} | {
"line": 614,
"column": 78
} | {
"line": 616,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝² : CommRing R\nW : WeierstrassCurve R\ninst✝¹ : W.IsCharTwoJEqZeroNF\ninst✝ : CharP R 2\n⊢ W.Δ = W.a₃ ^ 4",
"ppTerm": "?m.21",
"assigned": true,
"usedConstants": [
"Mathlib.Tactic.Ring.Common.mul_pf_left",
"CharP.cast_eq_zero",
"AddGroup.toSubtractionMo... | [] | rw [Δ_of_isCharTwoJEqZeroNF, b₆_of_char_two]
linear_combination (-32 * W.a₄ ^ 3 - 14 * W.a₃ ^ 4) * CharP.cast_eq_zero R 2 | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.AlgebraicGeometry.EllipticCurve.IsomOfJ | {
"line": 168,
"column": 4
} | {
"line": 168,
"column": 60
} | {
"line": 169,
"column": 4
} | [
{
"pp": "F : 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 3\ninst✝¹ : E.IsShortNF\ninst✝ : E'.IsShortNF\nha₄ : E.a₄ ≠ 0\nha₄' : E'.a₄ ≠ 0\nthis : NeZero 4\nu : F\nhu : u ^ 4 = E.a₄ / E'.a₄\nr : F\nhr : 1 * r ^ 3 + ... | [
"F : 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 3\ninst✝¹ : E.IsShortNF\ninst✝ : E'.IsShortNF\nha₄ : E.a₄ ≠ 0\nha₄' : E'.a₄ ≠ 0\nthis : NeZero 4\nu : F\nhu : u ^ 4 = E.a₄ / E'.a₄\nr : F\nhr : 1 * r ^ 3 + E.a₄ * r + (... | rw [← pow_ne_zero_iff four_ne_zero, hu, div_ne_zero_iff] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.AlgebraicGeometry.EllipticCurve.Jacobian.Basic | {
"line": 318,
"column": 2
} | {
"line": 320,
"column": 43
} | {
"line": 322,
"column": 0
} | [
{
"pp": "F : Type u\ninst✝ : Field F\nW : Jacobian F\nP : Fin 3 → F\nhPz : P z ≠ 0\n⊢ (eval P) W.polynomialX / P z ^ 4 = Polynomial.evalEval (P x / P z ^ 2) (P y / P z ^ 3) W.toAffine.polynomialX",
"ppTerm": "?m.87",
"assigned": true,
"usedConstants": [
"Mathlib.Tactic.Ring.Common.mul_pf_left"... | [] | linear_combination (norm := (rw [eval_polynomialX, Affine.evalEval_polynomialX]; ring1))
W.a₁ * P y / P z ^ 3 * div_self hPz - 2 * W.a₂ * P x / P z ^ 2 * div_self (pow_ne_zero 2 hPz)
- W.a₄ * div_self (pow_ne_zero 4 hPz) | Mathlib.Tactic.LinearCombination._aux_Mathlib_Tactic_LinearCombination___elabRules_Mathlib_Tactic_LinearCombination_linearCombination_1 | Mathlib.Tactic.LinearCombination.linearCombination |
Mathlib.AlgebraicGeometry.EllipticCurve.Jacobian.Basic | {
"line": 318,
"column": 2
} | {
"line": 320,
"column": 43
} | {
"line": 322,
"column": 0
} | [
{
"pp": "F : Type u\ninst✝ : Field F\nW : Jacobian F\nP : Fin 3 → F\nhPz : P z ≠ 0\n⊢ (eval P) W.polynomialX / P z ^ 4 = Polynomial.evalEval (P x / P z ^ 2) (P y / P z ^ 3) W.toAffine.polynomialX",
"ppTerm": "?m.87",
"assigned": true,
"usedConstants": [
"Mathlib.Tactic.Ring.Common.mul_pf_left"... | [] | linear_combination (norm := (rw [eval_polynomialX, Affine.evalEval_polynomialX]; ring1))
W.a₁ * P y / P z ^ 3 * div_self hPz - 2 * W.a₂ * P x / P z ^ 2 * div_self (pow_ne_zero 2 hPz)
- W.a₄ * div_self (pow_ne_zero 4 hPz) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.AlgebraicGeometry.EllipticCurve.Jacobian.Basic | {
"line": 318,
"column": 2
} | {
"line": 320,
"column": 43
} | {
"line": 322,
"column": 0
} | [
{
"pp": "F : Type u\ninst✝ : Field F\nW : Jacobian F\nP : Fin 3 → F\nhPz : P z ≠ 0\n⊢ (eval P) W.polynomialX / P z ^ 4 = Polynomial.evalEval (P x / P z ^ 2) (P y / P z ^ 3) W.toAffine.polynomialX",
"ppTerm": "?m.87",
"assigned": true,
"usedConstants": [
"Mathlib.Tactic.Ring.Common.mul_pf_left"... | [] | linear_combination (norm := (rw [eval_polynomialX, Affine.evalEval_polynomialX]; ring1))
W.a₁ * P y / P z ^ 3 * div_self hPz - 2 * W.a₂ * P x / P z ^ 2 * div_self (pow_ne_zero 2 hPz)
- W.a₄ * div_self (pow_ne_zero 4 hPz) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.AlgebraicGeometry.EllipticCurve.IsomOfJ | {
"line": 285,
"column": 6
} | {
"line": 285,
"column": 62
} | {
"line": 286,
"column": 6
} | [
{
"pp": "F : Type u_1\ninst✝⁶ : Field F\ninst✝⁵ : IsSepClosed F\nE✝ E'✝ : WeierstrassCurve F\ninst✝⁴ : E✝.IsElliptic\ninst✝³ : E'✝.IsElliptic\np : ℕ\ninst✝² : CharP F p\nhchar2 : 2 ≠ 0\nhchar3 : 3 ≠ 0\nthis✝³ : NeZero 2\nthis✝² : NeZero 4\nthis✝¹ : NeZero 6\nthis✝ : Invertible 2 := invertibleOfNonzero hchar2\nt... | [
"F : Type u_1\ninst✝⁶ : Field F\ninst✝⁵ : IsSepClosed F\nE✝ E'✝ : WeierstrassCurve F\ninst✝⁴ : E✝.IsElliptic\ninst✝³ : E'✝.IsElliptic\np : ℕ\ninst✝² : CharP F p\nhchar2 : 2 ≠ 0\nhchar3 : 3 ≠ 0\nthis✝³ : NeZero 2\nthis✝² : NeZero 4\nthis✝¹ : NeZero 6\nthis✝ : Invertible 2 := invertibleOfNonzero hchar2\nthis : Invert... | rw [← pow_ne_zero_iff four_ne_zero, hu, div_ne_zero_iff] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.AlgebraicGeometry.AffineTransitionLimit | {
"line": 1307,
"column": 52
} | {
"line": 1307,
"column": 74
} | {
"line": 1307,
"column": 74
} | [
{
"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... | [] | simp [Hom.resLE, hak'] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.AlgebraicGeometry.EllipticCurve.Jacobian.Point | {
"line": 147,
"column": 62
} | {
"line": 147,
"column": 67
} | {
"line": 147,
"column": 68
} | [
{
"pp": "case a\nR : Type r\ninst✝ : CommRing R\nW' : Jacobian R\nP : Fin 3 → R\nhP : W'.Equation P\n⊢ P x * P x ^ 2 * P z ^ 2 - 2 * P y * W'.negY P * P z * P z + P x ^ 2 * P x * P z ^ 2 -\n W'.a₁ * P x * W'.negY P * P z ^ 2 * P z -\n W'.a₁ * P y * P x * P z * P z ^ 2... | [
"case a\nR : Type r\ninst✝ : CommRing R\nW' : Jacobian R\nP : Fin 3 → R\nhP : W'.Equation P\n⊢ P x * P x ^ 2 * P z ^ 2 - 2 * P y * W'.negY P * P z * P z + P x ^ 2 * P x * P z ^ 2 -\n W'.a₁ * P x * W'.negY P * P z ^ 2 * P z -\n W'.a₁ * P y * P x * P z * P z ^ 2 +\n ... | dblZ, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.AlgebraicGeometry.EllipticCurve.Jacobian.Formula | {
"line": 207,
"column": 6
} | {
"line": 207,
"column": 11
} | {
"line": 207,
"column": 12
} | [
{
"pp": "R : Type r\ninst✝ : CommRing R\nW' : Jacobian R\nP : Fin 3 → R\nhPz : P z = 0\n⊢ W'.dblZ P = 0",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"HMul.hMul",
"AddGroupWithOne.toAddGroup",
"congrArg",
"CommSemiring.toSemiring",
"Weierstra... | [
"R : Type r\ninst✝ : CommRing R\nW' : Jacobian R\nP : Fin 3 → R\nhPz : P z = 0\n⊢ P z * (P y - W'.negY P) = 0"
] | dblZ, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.AlgebraicGeometry.EllipticCurve.Jacobian.Formula | {
"line": 212,
"column": 6
} | {
"line": 212,
"column": 11
} | {
"line": 212,
"column": 12
} | [
{
"pp": "R : Type r\ninst✝¹ : CommRing R\nW' : Jacobian R\ninst✝ : NoZeroDivisors R\nP Q : Fin 3 → R\nhQz : Q z ≠ 0\nhx : P x * Q z ^ 2 = Q x * P z ^ 2\nhy : P y * Q z ^ 3 = Q y * P z ^ 3\nhy' : P y * Q z ^ 3 = W'.negY Q * P z ^ 3\n⊢ W'.dblZ P = 0",
"ppTerm": "?m.158",
"assigned": true,
"usedConstan... | [
"R : Type r\ninst✝¹ : CommRing R\nW' : Jacobian R\ninst✝ : NoZeroDivisors R\nP Q : Fin 3 → R\nhQz : Q z ≠ 0\nhx : P x * Q z ^ 2 = Q x * P z ^ 2\nhy : P y * Q z ^ 3 = Q y * P z ^ 3\nhy' : P y * Q z ^ 3 = W'.negY Q * P z ^ 3\n⊢ P z * (P y - W'.negY P) = 0"
] | dblZ, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.AlgebraicGeometry.EllipticCurve.Jacobian.Point | {
"line": 152,
"column": 65
} | {
"line": 152,
"column": 70
} | {
"line": 152,
"column": 71
} | [
{
"pp": "case a\nR : Type r\ninst✝ : CommRing R\nW' : Jacobian R\nP : Fin 3 → R\nhP : W'.Equation P\n⊢ -P y * P x ^ 3 * P z ^ 3 + 2 * P y * W'.negY P ^ 2 * P z ^ 3 - 3 * P x ^ 2 * P x * W'.negY P * P z ^ 2 * P z +\n 3 * P x * P y * P x ^ 2 * P z * P z ^ 2 +\n ... | [
"case a\nR : Type r\ninst✝ : CommRing R\nW' : Jacobian R\nP : Fin 3 → R\nhP : W'.Equation P\n⊢ -P y * P x ^ 3 * P z ^ 3 + 2 * P y * W'.negY P ^ 2 * P z ^ 3 - 3 * P x ^ 2 * P x * W'.negY P * P z ^ 2 * P z +\n 3 * P x * P y * P x ^ 2 * P z * P z ^ 2 +\n ... | dblZ, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.AlgebraicGeometry.EllipticCurve.Jacobian.Formula | {
"line": 277,
"column": 48
} | {
"line": 277,
"column": 53
} | {
"line": 277,
"column": 54
} | [
{
"pp": "F : Type u\ninst✝¹ : Field F\nW : Jacobian F\ninst✝ : DecidableEq F\nP Q : Fin 3 → F\nhP : W.Equation P\nhQ : W.Equation Q\nhPz : P z ≠ 0\nhQz : Q z ≠ 0\nhx : P x * Q z ^ 2 = Q x * P z ^ 2\nhy : P y * Q z ^ 3 ≠ W.negY Q * P z ^ 3\n⊢ (W.dblU P ^ 2 - W.a₁ * W.dblU P * P z * (P y - W.negY P) - W.a₂ * P z ... | [
"F : Type u\ninst✝¹ : Field F\nW : Jacobian F\ninst✝ : DecidableEq F\nP Q : Fin 3 → F\nhP : W.Equation P\nhQ : W.Equation Q\nhPz : P z ≠ 0\nhQz : Q z ≠ 0\nhx : P x * Q z ^ 2 = Q x * P z ^ 2\nhy : P y * Q z ^ 3 ≠ W.negY Q * P z ^ 3\n⊢ (W.dblU P ^ 2 - W.a₁ * W.dblU P * P z * (P y - W.negY P) - W.a₂ * P z ^ 2 * (P y -... | dblZ, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.AlgebraicGeometry.EllipticCurve.Jacobian.Formula | {
"line": 314,
"column": 57
} | {
"line": 314,
"column": 62
} | {
"line": 314,
"column": 63
} | [
{
"pp": "F : Type u\ninst✝¹ : Field F\nW : Jacobian F\ninst✝ : DecidableEq F\nP Q : Fin 3 → F\nhP : W.Equation P\nhQ : W.Equation Q\nhPz : P z ≠ 0\nhQz : Q z ≠ 0\nhx : P x * Q z ^ 2 = Q x * P z ^ 2\nhy : P y * Q z ^ 3 ≠ W.negY Q * P z ^ 3\n⊢ (-W.dblU P *\n (W.dblU P ^ 2 - W.a₁ * W.dblU P * P z * (P y -... | [
"F : Type u\ninst✝¹ : Field F\nW : Jacobian F\ninst✝ : DecidableEq F\nP Q : Fin 3 → F\nhP : W.Equation P\nhQ : W.Equation Q\nhPz : P z ≠ 0\nhQz : Q z ≠ 0\nhx : P x * Q z ^ 2 = Q x * P z ^ 2\nhy : P y * Q z ^ 3 ≠ W.negY Q * P z ^ 3\n⊢ (-W.dblU P *\n (W.dblU P ^ 2 - W.a₁ * W.dblU P * P z * (P y - W.negY P) -... | dblZ, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Valuation.ExtendToLocalization | {
"line": 34,
"column": 20
} | {
"line": 34,
"column": 51
} | {
"line": 34,
"column": 52
} | [
{
"pp": "A : Type u_1\ninst✝⁴ : CommRing A\nΓ : Type u_2\ninst✝³ : LinearOrderedCommGroupWithZero Γ\nv : Valuation A Γ\nS : Submonoid A\nhS : S ≤ v.supp.primeCompl\nB : Type u_3\ninst✝² : CommRing B\ninst✝¹ : Algebra A B\ninst✝ : IsLocalization S B\nf : S.LocalizationMap B := IsLocalization.toLocalizationMap S ... | [
"case e'_2.e'_1\nA : Type u_1\ninst✝⁴ : CommRing A\nΓ : Type u_2\ninst✝³ : LinearOrderedCommGroupWithZero Γ\nv : Valuation A Γ\nS : Submonoid A\nhS : S ≤ v.supp.primeCompl\nB : Type u_3\ninst✝² : CommRing B\ninst✝¹ : Algebra A B\ninst✝ : IsLocalization S B\nf : S.LocalizationMap B := IsLocalization.toLocalizationMa... | convert! f.lift_eq (P := Γ) _ 0 | Mathlib.Tactic._aux_Mathlib_Tactic_Convert___macroRules_Mathlib_Tactic_convert!_1 | Mathlib.Tactic.convert! |
Mathlib.RingTheory.Valuation.ExtendToLocalization | {
"line": 42,
"column": 12
} | {
"line": 42,
"column": 19
} | {
"line": 43,
"column": 6
} | [
{
"pp": "case h.left\nA : Type u_1\ninst✝⁴ : CommRing A\nΓ : Type u_2\ninst✝³ : LinearOrderedCommGroupWithZero Γ\nv : Valuation A Γ\nS : Submonoid A\nhS : S ≤ v.supp.primeCompl\nB : Type u_3\ninst✝² : CommRing B\ninst✝¹ : Algebra A B\ninst✝ : IsLocalization S B\nf : S.LocalizationMap B := IsLocalization.toLocal... | [] | ring_nf | Mathlib.Tactic.RingNF._aux_Mathlib_Tactic_Ring_RingNF___elabRules_Mathlib_Tactic_RingNF_ringNF_1 | Mathlib.Tactic.RingNF.ringNF |
Mathlib.RingTheory.Valuation.ExtendToLocalization | {
"line": 42,
"column": 12
} | {
"line": 42,
"column": 19
} | {
"line": 43,
"column": 6
} | [
{
"pp": "case h.right\nA : Type u_1\ninst✝⁴ : CommRing A\nΓ : Type u_2\ninst✝³ : LinearOrderedCommGroupWithZero Γ\nv : Valuation A Γ\nS : Submonoid A\nhS : S ≤ v.supp.primeCompl\nB : Type u_3\ninst✝² : CommRing B\ninst✝¹ : Algebra A B\ninst✝ : IsLocalization S B\nf : S.LocalizationMap B := IsLocalization.toLoca... | [] | ring_nf | Mathlib.Tactic.RingNF._aux_Mathlib_Tactic_Ring_RingNF___elabRules_Mathlib_Tactic_RingNF_ringNF_1 | Mathlib.Tactic.RingNF.ringNF |
Mathlib.RingTheory.Valuation.ExtendToLocalization | {
"line": 46,
"column": 58
} | {
"line": 46,
"column": 69
} | {
"line": 46,
"column": 70
} | [
{
"pp": "case e'_3\nA : Type u_1\ninst✝⁴ : CommRing A\nΓ : Type u_2\ninst✝³ : LinearOrderedCommGroupWithZero Γ\nv : Valuation A Γ\nS : Submonoid A\nhS : S ≤ v.supp.primeCompl\nB : Type u_3\ninst✝² : CommRing B\ninst✝¹ : Algebra A B\ninst✝ : IsLocalization S B\nf : S.LocalizationMap B := IsLocalization.toLocaliz... | [
"case e'_3\nA : Type u_1\ninst✝⁴ : CommRing A\nΓ : Type u_2\ninst✝³ : LinearOrderedCommGroupWithZero Γ\nv : Valuation A Γ\nS : Submonoid A\nhS : S ≤ v.supp.primeCompl\nB : Type u_3\ninst✝² : CommRing B\ninst✝¹ : Algebra A B\ninst✝ : IsLocalization S B\nf : S.LocalizationMap B := IsLocalization.toLocalizationMap S B... | f.mk'_spec, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Valuation.ExtendToLocalization | {
"line": 46,
"column": 70
} | {
"line": 46,
"column": 81
} | {
"line": 47,
"column": 10
} | [
{
"pp": "case e'_3\nA : Type u_1\ninst✝⁴ : CommRing A\nΓ : Type u_2\ninst✝³ : LinearOrderedCommGroupWithZero Γ\nv : Valuation A Γ\nS : Submonoid A\nhS : S ≤ v.supp.primeCompl\nB : Type u_3\ninst✝² : CommRing B\ninst✝¹ : Algebra A B\ninst✝ : IsLocalization S B\nf : S.LocalizationMap B := IsLocalization.toLocaliz... | [
"case e'_3\nA : Type u_1\ninst✝⁴ : CommRing A\nΓ : Type u_2\ninst✝³ : LinearOrderedCommGroupWithZero Γ\nv : Valuation A Γ\nS : Submonoid A\nhS : S ≤ v.supp.primeCompl\nB : Type u_3\ninst✝² : CommRing B\ninst✝¹ : Algebra A B\ninst✝ : IsLocalization S B\nf : S.LocalizationMap B := IsLocalization.toLocalizationMap S B... | f.mk'_spec, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.AlgebraicGeometry.EllipticCurve.Jacobian.Point | {
"line": 215,
"column": 6
} | {
"line": 215,
"column": 64
} | {
"line": 215,
"column": 65
} | [
{
"pp": "R : Type r\ninst✝ : CommRing R\nW' : Jacobian R\nP Q : Fin 3 → R\nh : ¬P ≈ Q\nu v : R\nhu : IsUnit u\nhv : IsUnit v\n⊢ W'.add (u • P) (v • Q) = (u * v) ^ 2 • W'.add P Q",
"ppTerm": "?m.43",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"False",
"WeierstrassCurve.Jacobian.... | [
"R : Type r\ninst✝ : CommRing R\nW' : Jacobian R\nP Q : Fin 3 → R\nh : ¬P ≈ Q\nu v : R\nhu : IsUnit u\nhv : IsUnit v\n⊢ W'.addXYZ (u • P) (v • Q) = (u * v) ^ 2 • W'.add P Q"
] | add_of_not_equiv <| h.comp (smul_equiv_smul P Q hu hv).mp, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.AlgebraicGeometry.EllipticCurve.Jacobian.Point | {
"line": 258,
"column": 14
} | {
"line": 260,
"column": 44
} | {
"line": 262,
"column": 0
} | [
{
"pp": "F : Type u\ninst✝¹ : Field F\nW : Jacobian F\ninst✝ : DecidableEq F\nP Q : Fin 3 → F\nhP : W.Equation P\nhQ : W.Equation Q\nhPz : P z ≠ 0\nhQz : Q z ≠ 0\nhx : P x * Q z ^ 2 = Q x * P z ^ 2\nhy : P y * Q z ^ 3 ≠ W.negY Q * P z ^ 3\n⊢ W.add P Q =\n W.dblZ P •\n ![W.toAffine.addX (P x / P z ^ 2) (... | [] | by
rw [add_of_equiv <| equiv_of_X_eq_of_Y_eq hPz hQz hx <| Y_eq_of_Y_ne' hP hQ hx hy,
dblXYZ_of_Z_ne_zero hP hQ hPz hQz hx hy] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.Algebra.Nonarchimedean.Basic | {
"line": 71,
"column": 8
} | {
"line": 71,
"column": 23
} | {
"line": 72,
"column": 6
} | [
{
"pp": "G : Type u_1\ninst✝⁵ : Group G\ninst✝⁴ : TopologicalSpace G\ninst✝³ : NonarchimedeanGroup G\nH : Type u_2\ninst✝² : Group H\ninst✝¹ : TopologicalSpace H\ninst✝ : IsTopologicalGroup H\nf : G →* H\nemb : IsOpenEmbedding ⇑f\nU : Set H\nhU : U ∈ 𝓝 1\n⊢ U ∈ 𝓝 (f 1)",
"ppTerm": "?m.92",
"assigned":... | [] | rwa [f.map_one] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticRwa___1 | Lean.Parser.Tactic.tacticRwa__ |
Mathlib.AlgebraicGeometry.EllipticCurve.Jacobian.Point | {
"line": 452,
"column": 16
} | {
"line": 452,
"column": 51
} | {
"line": 452,
"column": 51
} | [
{
"pp": "F : Type u\ninst✝ : Field F\nW : Jacobian F\nP : Fin 3 → F\nhP : ¬W.Nonsingular P\n⊢ (if hP : W.Nonsingular P ∧ P z ≠ 0 then Affine.Point.some (P x / P z ^ 2) (P y / P z ^ 3) ⋯ else 0) = 0",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"instDecidableNot",
... | [
"F : Type u\ninst✝ : Field F\nW : Jacobian F\nP : Fin 3 → F\nhP : ¬W.Nonsingular P\n⊢ 0 = 0"
] | dif_neg <| not_and_of_not_left _ hP | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Valuation.ValuativeRel.Basic | {
"line": 572,
"column": 41
} | {
"line": 572,
"column": 89
} | {
"line": 572,
"column": 89
} | [
{
"pp": "R : Type u_1\ninst✝¹ : Semiring R\ninst✝ : ValuativeRel R\nx x' y y' z : R\nn : ℕ\n⊢ ∀ (x : R) (y : ↥(posSubmonoid R)),\n ValueGroupWithZero.mk x y ^ (n + 1) = ValueGroupWithZero.mk x y ^ n * ValueGroupWithZero.mk x y",
"ppTerm": "?m.164",
"assigned": true,
"usedConstants": [
"Subm... | [] | by simp_rw [HPow.hPow, Pow.pow]; simp [pow_succ] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.Algebra.UniformField | {
"line": 141,
"column": 4
} | {
"line": 141,
"column": 26
} | {
"line": 142,
"column": 4
} | [
{
"pp": "K : Type u_1\ninst✝⁴ : Field K\ninst✝³ : UniformSpace K\ninst✝² : IsTopologicalDivisionRing K\ninst✝¹ : CompletableTopField K\ninst✝ : IsUniformAddGroup K\nx : hat K\nx_ne : x ≠ 0\nthis : T1Space (hat K)\nf : hat K → hat K := fun x ↦ x * x.hatInv\nc : K → hat K := fun x ↦ ↑x\ncont : ContinuousAt f x\nc... | [
"K : Type u_1\ninst✝⁴ : Field K\ninst✝³ : UniformSpace K\ninst✝² : IsTopologicalDivisionRing K\ninst✝¹ : CompletableTopField K\ninst✝ : IsUniformAddGroup K\nx : hat K\nx_ne : x ≠ 0\nthis : T1Space (hat K)\nf : hat K → hat K := fun x ↦ x * x.hatInv\nc : K → hat K := fun x ↦ ↑x\ncont : ContinuousAt f x\nclo : x ∈ clo... | rw [mem_singleton_iff] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.PrincipalIdealDomainOfPrime | {
"line": 44,
"column": 6
} | {
"line": 44,
"column": 88
} | {
"line": 46,
"column": 0
} | [
{
"pp": "case refine_2\nR : Type u_1\ninst✝ : CommSemiring R\nI : Ideal R\na x : R\nhx : I ⊔ span {a} = span {x}\ny : R\nhy : Submodule.colon I ↑(span {a}) = span {y}\n⊢ I * span {y} ≤ I ∧ y * a ∈ I",
"ppTerm": "?refine_2",
"assigned": true,
"usedConstants": [
"Submodule",
"Ideal.mul_le_... | [] | exact ⟨mul_le_right, mem_colon_span_singleton.1 <| hy ▸ mem_span_singleton_self y⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.RingTheory.Valuation.Discrete.Basic | {
"line": 176,
"column": 17
} | {
"line": 176,
"column": 56
} | {
"line": 177,
"column": 4
} | [
{
"pp": "case hH.ofNat\nA : Type u_2\ninst✝ : Ring A\nv : Valuation A (WithZero (Multiplicative ℤ))\nhv : v.IsRankOneDiscrete\nn✝ : (WithZero (Multiplicative ℤ))ˣ\nhπ✝ : (exp 1)⁻¹ ∈ range ⇑v\nx✝ : ∃ k, Units.mk0 (exp 1)⁻¹ ⋯ ^ k = n✝\nπ : A\nhπ : v π = (exp 1)⁻¹\nn : ℕ\nh : Units.mk0 (exp 1)⁻¹ ⋯ ^ Int.ofNat n = ... | [] | refine ⟨1, ?_, π ^ n, ?_⟩ <;> simp [hπ] | Lean.Parser.Tactic.«_aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tactic_<;>__1» | Lean.Parser.Tactic.«tactic_<;>_» |
Mathlib.RingTheory.Valuation.Discrete.Basic | {
"line": 176,
"column": 17
} | {
"line": 176,
"column": 56
} | {
"line": 177,
"column": 4
} | [
{
"pp": "case hH.ofNat\nA : Type u_2\ninst✝ : Ring A\nv : Valuation A (WithZero (Multiplicative ℤ))\nhv : v.IsRankOneDiscrete\nn✝ : (WithZero (Multiplicative ℤ))ˣ\nhπ✝ : (exp 1)⁻¹ ∈ range ⇑v\nx✝ : ∃ k, Units.mk0 (exp 1)⁻¹ ⋯ ^ k = n✝\nπ : A\nhπ : v π = (exp 1)⁻¹\nn : ℕ\nh : Units.mk0 (exp 1)⁻¹ ⋯ ^ Int.ofNat n = ... | [] | refine ⟨1, ?_, π ^ n, ?_⟩ <;> simp [hπ] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Valuation.Discrete.Basic | {
"line": 176,
"column": 17
} | {
"line": 176,
"column": 56
} | {
"line": 177,
"column": 4
} | [
{
"pp": "case hH.ofNat\nA : Type u_2\ninst✝ : Ring A\nv : Valuation A (WithZero (Multiplicative ℤ))\nhv : v.IsRankOneDiscrete\nn✝ : (WithZero (Multiplicative ℤ))ˣ\nhπ✝ : (exp 1)⁻¹ ∈ range ⇑v\nx✝ : ∃ k, Units.mk0 (exp 1)⁻¹ ⋯ ^ k = n✝\nπ : A\nhπ : v π = (exp 1)⁻¹\nn : ℕ\nh : Units.mk0 (exp 1)⁻¹ ⋯ ^ Int.ofNat n = ... | [] | refine ⟨1, ?_, π ^ n, ?_⟩ <;> simp [hπ] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Valuation.Discrete.Basic | {
"line": 330,
"column": 2
} | {
"line": 335,
"column": 52
} | {
"line": 337,
"column": 0
} | [
{
"pp": "Γ : Type u_1\ninst✝¹ : LinearOrderedCommGroupWithZero Γ\nK : Type u_2\ninst✝ : Field K\nv : Valuation K Γ\nhv : v.IsRankOneDiscrete\nπ₁ π₂ : ↥v.valuationSubring\nh1 : v.IsUniformizer ↑π₁\nh2 : v.IsUniformizer ↑π₂\n⊢ Associated π₁ π₂",
"ppTerm": "?m.29",
"assigned": true,
"usedConstants": [
... | [] | have hval : v ((π₁ : K)⁻¹ * π₂) = 1 := by
simp [IsUniformizer.iff.mp h1, IsUniformizer.iff.mp h2]
set p : v.integer := ⟨(π₁.1 : K)⁻¹ * π₂.1, (v.mem_integer_iff _).mpr (le_of_eq hval)⟩ with hp
use ((Integers.isUnit_iff_valuation_eq_one (x := p) <| integer.integers v).mpr hval).unit
apply_fun ((↑) : K₀ → K) usi... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Valuation.Discrete.Basic | {
"line": 330,
"column": 2
} | {
"line": 335,
"column": 52
} | {
"line": 337,
"column": 0
} | [
{
"pp": "Γ : Type u_1\ninst✝¹ : LinearOrderedCommGroupWithZero Γ\nK : Type u_2\ninst✝ : Field K\nv : Valuation K Γ\nhv : v.IsRankOneDiscrete\nπ₁ π₂ : ↥v.valuationSubring\nh1 : v.IsUniformizer ↑π₁\nh2 : v.IsUniformizer ↑π₂\n⊢ Associated π₁ π₂",
"ppTerm": "?m.29",
"assigned": true,
"usedConstants": [
... | [] | have hval : v ((π₁ : K)⁻¹ * π₂) = 1 := by
simp [IsUniformizer.iff.mp h1, IsUniformizer.iff.mp h2]
set p : v.integer := ⟨(π₁.1 : K)⁻¹ * π₂.1, (v.mem_integer_iff _).mpr (le_of_eq hval)⟩ with hp
use ((Integers.isUnit_iff_valuation_eq_one (x := p) <| integer.integers v).mpr hval).unit
apply_fun ((↑) : K₀ → K) usi... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Valuation.Discrete.Basic | {
"line": 381,
"column": 29
} | {
"line": 381,
"column": 44
} | {
"line": 381,
"column": 44
} | [
{
"pp": "case neg\nΓ : Type u_1\ninst✝¹ : LinearOrderedCommGroupWithZero Γ\nK : Type u_2\ninst✝ : Field K\nv : Valuation K Γ\nhv : v.IsRankOneDiscrete\nπ : v.Uniformizer\nx : ↥v.valuationSubring\nhx : x ∈ maximalIdeal ↥v.valuationSubring\nhx₀ : ¬x = 0\nn : ℕ\nu : (↥v.valuationSubring)ˣ\nhu : ↑x = ↑(π.val ^ n * ... | [
"case neg\nΓ : Type u_1\ninst✝¹ : LinearOrderedCommGroupWithZero Γ\nK : Type u_2\ninst✝ : Field K\nv : Valuation K Γ\nhv : v.IsRankOneDiscrete\nπ : v.Uniformizer\nx : ↥v.valuationSubring\nhx : x ∈ maximalIdeal ↥v.valuationSubring\nhx₀ : ¬x = 0\nn : ℕ\nu : (↥v.valuationSubring)ˣ\nhu : x = π.val ^ n * ↑u\n⊢ x ∈ span ... | Subtype.coe_inj | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Valuation.Discrete.Basic | {
"line": 457,
"column": 21
} | {
"line": 457,
"column": 62
} | {
"line": 457,
"column": 62
} | [
{
"pp": "Γ : Type u_1\ninst✝³ : LinearOrderedCommGroupWithZero Γ\nK : Type u_2\ninst✝² : Field K\nv : Valuation K Γ\ninst✝¹ : IsCyclic ↥(MonoidWithZeroHom.ofClass v).valueGroup\ninst✝ : Nontrivial ↥(MonoidWithZeroHom.ofClass v).valueGroup\n⊢ maximalIdeal ↥v.valuationSubring ≠ ⊥",
"ppTerm": "?m.43",
"ass... | [
"Γ : Type u_1\ninst✝³ : LinearOrderedCommGroupWithZero Γ\nK : Type u_2\ninst✝² : Field K\nv : Valuation K Γ\ninst✝¹ : IsCyclic ↥(MonoidWithZeroHom.ofClass v).valueGroup\ninst✝ : Nontrivial ↥(MonoidWithZeroHom.ofClass v).valueGroup\n⊢ ¬IsField ↥v.valuationSubring"
] | rw [ne_eq, ← isField_iff_maximalIdeal_eq] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Topology.Algebra.Valued.ValuedField | {
"line": 79,
"column": 2
} | {
"line": 79,
"column": 49
} | {
"line": 80,
"column": 2
} | [
{
"pp": "K : Type u_1\ninst✝¹ : DivisionRing K\nΓ₀ : Type u_2\ninst✝ : LinearOrderedCommGroupWithZero Γ₀\nv : Valuation K Γ₀\nx y r s : K\ny_ne : y ≠ 0\nhr : r ≠ 0\nhs : s ≠ 0\nh : v (x - y) < min (v s / v r * (v y * v y)) (v y)\n⊢ v (x⁻¹ - y⁻¹) * v r < v s",
"ppTerm": "?m.55",
"assigned": true,
"us... | [
"K : Type u_1\ninst✝¹ : DivisionRing K\nΓ₀ : Type u_2\ninst✝ : LinearOrderedCommGroupWithZero Γ₀\nv : Valuation K Γ₀\nx y r s : K\ny_ne : y ≠ 0\nhr : r ≠ 0\nhs : s ≠ 0\nh : v (x - y) < min (v s / v r * (v y * v y)) (v y)\nhr' : 0 < v r\n⊢ v (x⁻¹ - y⁻¹) * v r < v s"
] | have hr' : 0 < v r := by simp [zero_lt_iff, hr] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.GroupTheory.ArchimedeanDensely | {
"line": 338,
"column": 74
} | {
"line": 342,
"column": 27
} | {
"line": 343,
"column": 2
} | [
{
"pp": "G : Type u_2\ninst✝³ : AddCommGroup G\ninst✝² : LinearOrder G\ninst✝¹ : IsOrderedAddMonoid G\ninst✝ : Nontrivial G\ng : G\nthis : ({x | 0 ≤ x}.WellFoundedOn fun x1 x2 ↦ x1 < x2) ↔ Nonempty (G ≃+o ℤ)\n⊢ ({x | g ≤ x}.WellFoundedOn fun x1 x2 ↦ x1 < x2) ↔ Nonempty (G ≃+o ℤ)",
"ppTerm": "?m.35",
"as... | [] | by
rw [← this]
refine ⟨fun h ↦ (h.mapsTo (· + g) ?_).mono' ?_, fun h ↦ (h.mapsTo (· - g) ?_).mono' ?_⟩ <;>
· try intro
simp [Function.onFun] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.GroupTheory.ArchimedeanDensely | {
"line": 379,
"column": 85
} | {
"line": 386,
"column": 23
} | {
"line": 388,
"column": 0
} | [
{
"pp": "G : Type u_2\ninst✝³ : CommGroup G\ninst✝² : LinearOrder G\ninst✝¹ : IsOrderedMonoid G\ninst✝ : Nontrivial G\ng : G\n⊢ ({x | g ≤ x}.WellFoundedOn fun x1 x2 ↦ x1 < x2) ↔ Nonempty (G ≃*o Multiplicative ℤ)",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"Additi... | [] | by
let e : G ≃o Additive G := OrderIso.refl G
suffices Set.WellFoundedOn {x : G | g ≤ x} (· < ·) ↔ Set.WellFoundedOn {x | e g ≤ x} (· < ·) by
rw [this, LinearOrderedAddCommGroup.wellFoundedOn_setOf_le_lt_iff_nonempty_discrete,
OrderAddMonoidIso.toMultiplicativeRight.nonempty_congr]
refine ⟨fun h ↦ (h.ma... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.PowerSeries.Basic | {
"line": 199,
"column": 2
} | {
"line": 199,
"column": 24
} | {
"line": 201,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝ : Semiring R\na : R\nn : ℕ\nh : n ≠ 0\n⊢ (coeff n) (C a) = 0",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Semiring.toModule",
"congrArg",
"LinearMap.instFunLike",
"RingHom",
"id",
"PowerSeries.coeff",
... | [] | rw [coeff_C, if_neg h] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.PowerSeries.Basic | {
"line": 199,
"column": 2
} | {
"line": 199,
"column": 24
} | {
"line": 201,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝ : Semiring R\na : R\nn : ℕ\nh : n ≠ 0\n⊢ (coeff n) (C a) = 0",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Semiring.toModule",
"congrArg",
"LinearMap.instFunLike",
"RingHom",
"id",
"PowerSeries.coeff",
... | [] | rw [coeff_C, if_neg h] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.PowerSeries.Basic | {
"line": 199,
"column": 2
} | {
"line": 199,
"column": 24
} | {
"line": 201,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝ : Semiring R\na : R\nn : ℕ\nh : n ≠ 0\n⊢ (coeff n) (C a) = 0",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Semiring.toModule",
"congrArg",
"LinearMap.instFunLike",
"RingHom",
"id",
"PowerSeries.coeff",
... | [] | rw [coeff_C, if_neg h] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.MvPowerSeries.Basic | {
"line": 355,
"column": 12
} | {
"line": 355,
"column": 34
} | {
"line": 358,
"column": 0
} | [
{
"pp": "σ : Type u_1\nR : Type u_2\ninst✝ : Semiring R\nn : σ →₀ ℕ\nh : n ≠ 0\na : R\n⊢ (coeff n) (C a) = 0",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Nat.instMulZeroClass",
"Semiring.toModule",
"congrArg",
"LinearMap.instFunLike",
"Clas... | [] | rw [coeff_C, if_neg h] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.MvPowerSeries.Basic | {
"line": 355,
"column": 12
} | {
"line": 355,
"column": 34
} | {
"line": 358,
"column": 0
} | [
{
"pp": "σ : Type u_1\nR : Type u_2\ninst✝ : Semiring R\nn : σ →₀ ℕ\nh : n ≠ 0\na : R\n⊢ (coeff n) (C a) = 0",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Nat.instMulZeroClass",
"Semiring.toModule",
"congrArg",
"LinearMap.instFunLike",
"Clas... | [] | rw [coeff_C, if_neg h] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.MvPowerSeries.Basic | {
"line": 355,
"column": 12
} | {
"line": 355,
"column": 34
} | {
"line": 358,
"column": 0
} | [
{
"pp": "σ : Type u_1\nR : Type u_2\ninst✝ : Semiring R\nn : σ →₀ ℕ\nh : n ≠ 0\na : R\n⊢ (coeff n) (C a) = 0",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Nat.instMulZeroClass",
"Semiring.toModule",
"congrArg",
"LinearMap.instFunLike",
"Clas... | [] | rw [coeff_C, if_neg h] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.PowerSeries.Basic | {
"line": 696,
"column": 4
} | {
"line": 698,
"column": 23
} | {
"line": 700,
"column": 0
} | [
{
"pp": "case h.right\nA : Type u_2\ninst✝ : CommRing A\nhA : ¬Subsingleton A\na✝ : Nontrivial A\n⊢ Ideal.span {X} < ⊤",
"ppTerm": "?h.right",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Preorder.toLT",
"Dvd.dvd",
"NeZero.o... | [] | · rw [lt_top_iff_ne_top, Ne, Ideal.eq_top_iff_one, Ideal.mem_span_singleton,
X_dvd_iff, constantCoeff_one]
exact one_ne_zero | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Topology.Algebra.Valued.ValuedField | {
"line": 576,
"column": 10
} | {
"line": 576,
"column": 19
} | {
"line": 577,
"column": 8
} | [
{
"pp": "case neg.inl\nK : Type u_1\ninst✝¹ : Field K\nΓ₀ : Type u_2\ninst✝ : LinearOrderedCommGroupWithZero Γ₀\nhv : Valued K Γ₀\nh : Function.Surjective ⇑v\nH : ¬∃ x, v x = 0\n⊢ IsClosed {a | ¬v a = 0}",
"ppTerm": "?neg.inl✝",
"assigned": true,
"usedConstants": [
"GroupWithZero.toMonoidWithZ... | [] | simp at H | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Topology.Algebra.Valued.ValuedField | {
"line": 576,
"column": 10
} | {
"line": 576,
"column": 19
} | {
"line": 577,
"column": 8
} | [
{
"pp": "case neg.inl\nK : Type u_1\ninst✝¹ : Field K\nΓ₀ : Type u_2\ninst✝ : LinearOrderedCommGroupWithZero Γ₀\nhv : Valued K Γ₀\nh : Function.Surjective ⇑v\nH : ¬∃ x, v x = 0\n⊢ IsClosed {a | ¬v a = 0}",
"ppTerm": "?neg.inl✝",
"assigned": true,
"usedConstants": [
"GroupWithZero.toMonoidWithZ... | [] | simp at H | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Algebra.Valued.ValuedField | {
"line": 576,
"column": 10
} | {
"line": 576,
"column": 19
} | {
"line": 577,
"column": 8
} | [
{
"pp": "case neg.inl\nK : Type u_1\ninst✝¹ : Field K\nΓ₀ : Type u_2\ninst✝ : LinearOrderedCommGroupWithZero Γ₀\nhv : Valued K Γ₀\nh : Function.Surjective ⇑v\nH : ¬∃ x, v x = 0\n⊢ IsClosed {a | ¬v a = 0}",
"ppTerm": "?neg.inl✝",
"assigned": true,
"usedConstants": [
"GroupWithZero.toMonoidWithZ... | [] | simp at H | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.MvPowerSeries.PiTopology | {
"line": 238,
"column": 2
} | {
"line": 238,
"column": 16
} | {
"line": 240,
"column": 0
} | [
{
"pp": "σ : Type u_1\nR : Type u_2\ninst✝² : TopologicalSpace R\ninst✝¹ : CommRing R\ninst✝ : DiscreteTopology R\nf : MvPowerSeries σ R\nH : ∀ᶠ (x : ℕ) in atTop, constantCoeff f ^ x = 0\n⊢ IsNilpotent (constantCoeff f)",
"ppTerm": "?m.49",
"assigned": true,
"usedConstants": [
"CommSemiring.to... | [] | exact H.exists | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.RingTheory.PowerSeries.Order | {
"line": 217,
"column": 2
} | {
"line": 225,
"column": 16
} | {
"line": 227,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝ : Semiring R\nφ : R⟦X⟧\n⊢ 1 ≤ φ.order ↔ constantCoeff φ = 0",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"CharP.cast_eq_zero",
"Eq.mpr",
"PowerSeries.coeff_of_lt_order",
"Preorder.toLT",
"Semiring.toModule",
"instCharZeroE... | [] | constructor
· intro h
rw [← coeff_zero_eq_constantCoeff]
apply coeff_of_lt_order
simpa using Order.one_le_iff_pos.mp h
· intro h
refine le_order _ _ fun d hd ↦ ?_
rw [Nat.cast_lt_one] at hd
simp [hd, h] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.PowerSeries.Order | {
"line": 217,
"column": 2
} | {
"line": 225,
"column": 16
} | {
"line": 227,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝ : Semiring R\nφ : R⟦X⟧\n⊢ 1 ≤ φ.order ↔ constantCoeff φ = 0",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"CharP.cast_eq_zero",
"Eq.mpr",
"PowerSeries.coeff_of_lt_order",
"Preorder.toLT",
"Semiring.toModule",
"instCharZeroE... | [] | constructor
· intro h
rw [← coeff_zero_eq_constantCoeff]
apply coeff_of_lt_order
simpa using Order.one_le_iff_pos.mp h
· intro h
refine le_order _ _ fun d hd ↦ ?_
rw [Nat.cast_lt_one] at hd
simp [hd, h] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.PowerSeries.Order | {
"line": 411,
"column": 12
} | {
"line": 411,
"column": 44
} | {
"line": 411,
"column": 45
} | [
{
"pp": "case neg.inr.inl\nR : Type u_1\ninst✝¹ : Semiring R\ninst✝ : NoZeroDivisors R\nφ ψ : R⟦X⟧\nh✝ : φ ≠ 0 ∧ ψ ≠ 0\nij : ℕ × ℕ\nhij : ij ∈ antidiagonal (φ.order.toNat + ψ.order.toNat)\nh : ij ≠ (φ.order.toNat, ψ.order.toNat)\nh' : ij.1 < φ.order.toNat\n⊢ (coeff ij.1) φ * (coeff ij.2) ψ = 0",
"ppTerm": "... | [
"case neg.inr.inl\nR : Type u_1\ninst✝¹ : Semiring R\ninst✝ : NoZeroDivisors R\nφ ψ : R⟦X⟧\nh✝ : φ ≠ 0 ∧ ψ ≠ 0\nij : ℕ × ℕ\nhij : ij ∈ antidiagonal (φ.order.toNat + ψ.order.toNat)\nh : ij ≠ (φ.order.toNat, ψ.order.toNat)\nh' : ij.1 < φ.order.toNat\n⊢ 0 * (coeff ij.2) ψ = 0"
] | coeff_of_lt_order_toNat ij.1 h', | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.MvPowerSeries.Substitution | {
"line": 137,
"column": 2
} | {
"line": 137,
"column": 79
} | {
"line": 138,
"column": 2
} | [
{
"pp": "σ : Type u_1\nR : Type u_3\ninst✝ : CommRing R\na : σ → R\n⊢ HasSubst (a • X)",
"ppTerm": "?m.24",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"CommRing",
"instHSMul",
"HMul.hMul",
"Algebra.algebraMap",
"CommSemiring.toSemiring",
"HEq.refl",
... | [
"σ : Type u_1\nR : Type u_3\ninst✝ : CommRing R\na : σ → R\n⊢ HSMul.hSMul a = HMul.hMul fun s ↦ (algebraMap R (MvPowerSeries σ R)) (a s)"
] | convert! HasSubst.X.mul_left (fun s ↦ algebraMap R (MvPowerSeries σ R) (a s)) | Mathlib.Tactic._aux_Mathlib_Tactic_Convert___macroRules_Mathlib_Tactic_convert!_1 | Mathlib.Tactic.convert! |
Mathlib.RingTheory.MvPowerSeries.Substitution | {
"line": 364,
"column": 22
} | {
"line": 364,
"column": 41
} | {
"line": 364,
"column": 41
} | [
{
"pp": "σ : Type u_1\nR : Type u_3\ninst✝² : CommRing R\nτ : Type u_4\nS : Type u_5\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nf : MvPowerSeries σ R\nn : τ →₀ ℕ\n⊢ HasSubst 0",
"ppTerm": "?m.55",
"assigned": true,
"usedConstants": [
"False",
"Nat.instMulZeroClass",
"MvPowerSeries.... | [] | simp [hasSubst_def] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.RingTheory.MvPowerSeries.Substitution | {
"line": 364,
"column": 22
} | {
"line": 364,
"column": 41
} | {
"line": 364,
"column": 41
} | [
{
"pp": "σ : Type u_1\nR : Type u_3\ninst✝² : CommRing R\nτ : Type u_4\nS : Type u_5\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nf : MvPowerSeries σ R\nn : τ →₀ ℕ\n⊢ HasSubst 0",
"ppTerm": "?m.55",
"assigned": true,
"usedConstants": [
"False",
"Nat.instMulZeroClass",
"MvPowerSeries.... | [] | simp [hasSubst_def] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.MvPowerSeries.Substitution | {
"line": 364,
"column": 22
} | {
"line": 364,
"column": 41
} | {
"line": 364,
"column": 41
} | [
{
"pp": "σ : Type u_1\nR : Type u_3\ninst✝² : CommRing R\nτ : Type u_4\nS : Type u_5\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nf : MvPowerSeries σ R\nn : τ →₀ ℕ\n⊢ HasSubst 0",
"ppTerm": "?m.55",
"assigned": true,
"usedConstants": [
"False",
"Nat.instMulZeroClass",
"MvPowerSeries.... | [] | simp [hasSubst_def] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.MvPowerSeries.Substitution | {
"line": 440,
"column": 2
} | {
"line": 441,
"column": 38
} | {
"line": 442,
"column": 2
} | [
{
"pp": "case neg\nσ : Type u_1\nR : Type u_3\ninst✝² : CommRing R\nτ : Type u_4\nS : Type u_5\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\na : σ → MvPowerSeries τ S\nha : HasSubst a\nf : MvPowerSeries σ R\nhf : IsNilpotent (constantCoeff f)\nd : σ →₀ ℕ\nhd : ¬d = 0\ni : σ\nhi : i ∈ d.support\n⊢ IsNilpotent ((coe... | [
"case neg\nσ : Type u_1\nR : Type u_3\ninst✝² : CommRing R\nτ : Type u_4\nS : Type u_5\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\na : σ → MvPowerSeries τ S\nha : HasSubst a\nf : MvPowerSeries σ R\nhf : IsNilpotent (constantCoeff f)\nd : σ →₀ ℕ\nhd : ¬d = 0\ni : σ\nhi : i ∈ d.support\n⊢ IsNilpotent\n (((algebraMa... | rw [Finsupp.prod, map_prod, ← Finset.prod_erase_mul _ _ hi, ← algebraMap_smul S,
smul_eq_mul, ← mul_assoc, map_pow] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.Real.Cardinality | {
"line": 177,
"column": 6
} | {
"line": 182,
"column": 30
} | {
"line": 184,
"column": 0
} | [
{
"pp": "case true\nc : ℝ\nh1 : 0 < c\nh2 : c < 1 / 2\nf g : ℕ → Bool\nh : ¬f = g\nthis : ∃ n, f n ≠ g n\nn : ℕ := Nat.find this\nhn : ∀ k < n, f k = g k\nfn : f n = true\n⊢ ¬cantorFunction c f = cantorFunction c g",
"ppTerm": "?true",
"assigned": true,
"usedConstants": [
"Cardinal.cantorFunct... | [] | apply _root_.ne_of_gt
refine increasing_cantorFunction h1 h2 (fun k hk => (hn k hk).symm) ?_ fn
apply Bool.eq_false_of_not_eq_true
rw [← fn]
apply Ne.symm
exact Nat.find_spec this | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Real.Cardinality | {
"line": 177,
"column": 6
} | {
"line": 182,
"column": 30
} | {
"line": 184,
"column": 0
} | [
{
"pp": "case true\nc : ℝ\nh1 : 0 < c\nh2 : c < 1 / 2\nf g : ℕ → Bool\nh : ¬f = g\nthis : ∃ n, f n ≠ g n\nn : ℕ := Nat.find this\nhn : ∀ k < n, f k = g k\nfn : f n = true\n⊢ ¬cantorFunction c f = cantorFunction c g",
"ppTerm": "?true",
"assigned": true,
"usedConstants": [
"Cardinal.cantorFunct... | [] | apply _root_.ne_of_gt
refine increasing_cantorFunction h1 h2 (fun k hk => (hn k hk).symm) ?_ fn
apply Bool.eq_false_of_not_eq_true
rw [← fn]
apply Ne.symm
exact Nat.find_spec this | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.MvPowerSeries.Substitution | {
"line": 549,
"column": 76
} | {
"line": 576,
"column": 53
} | {
"line": 578,
"column": 0
} | [
{
"pp": "σ : Type u_1\nR : Type u_3\ninst✝³ : CommRing R\nτ : Type u_4\nS : Type u_5\ninst✝² : CommRing S\ninst✝¹ : Algebra R S\na : σ → MvPowerSeries τ S\nf : MvPowerSeries σ R\ninst✝ : Finite τ\nk : ℕ\nha : HasSubst a\nha₁ : ∀ (i : σ), constantCoeff (a i) = 0\n⊢ (truncTotal k) (subst a f) = (truncTotal k) (su... | [] | by
ext d
by_cases hd : d.degree < k
· simp_rw [coeff_truncTotal _ hd, coeff_subst ha]
have h1 := coeff_subst_finite ha f d
have h2 := coeff_subst_finite ha (∑ i ∈ range k, (homogeneousComponent i) f) d
rw [finsum_eq_sum _ h1, finsum_eq_sum _ h2]
have : h2.toFinset ⊆ h1.toFinset := by simp +context... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.NumberTheory.LSeries.Basic | {
"line": 355,
"column": 6
} | {
"line": 355,
"column": 13
} | {
"line": 356,
"column": 2
} | [
{
"pp": "case refine_2\nf : ℕ → ℂ\nx : ℝ\ns : ℂ\nhs : x < s.re\nC : ℝ\nhC : ∀ (n : ℕ), n ≠ 0 → ‖f n‖ ≤ C * ↑n ^ (x - 1)\nhC₀ : 0 ≤ C\nhsx : -s.re + x - 1 < -1\nn : ℕ\nhn : 1 ≤ n\n⊢ ↑n ^ (-(s.re + (1 - x))) = ↑n ^ (-s.re + x - 1)",
"ppTerm": "?refine_2",
"assigned": true,
"usedConstants": [
"Ma... | [] | ring_nf | Mathlib.Tactic.RingNF._aux_Mathlib_Tactic_Ring_RingNF___elabRules_Mathlib_Tactic_RingNF_ringNF_1 | Mathlib.Tactic.RingNF.ringNF |
Mathlib.NumberTheory.ArithmeticFunction.LFunction | {
"line": 336,
"column": 2
} | {
"line": 336,
"column": 96
} | {
"line": 337,
"column": 2
} | [
{
"pp": "ι : Type u_1\nR : Type u_2\ninst✝ : CommSemiring R\nq : ι → ℕ\nhq : Northcott q\nf : ι → PowerSeries R\nhf : ∀ (i : ι), PowerSeries.constantCoeff (f i) = 1\nn : ℕ\n⊢ ∀ (n : ℕ), ∀ᶠ (i : ι) in cofinite, ((ofPowerSeries (q i)) (f i)) n = 1 n",
"ppTerm": "?m.44",
"assigned": true,
"usedConstant... | [
"ι : Type u_1\nR : Type u_2\ninst✝ : CommSemiring R\nq : ι → ℕ\nhq : Northcott q\nf : ι → PowerSeries R\nhf : ∀ (i : ι), PowerSeries.constantCoeff (f i) = 1\nn✝ n : ℕ\ni : ι\nhi : n + 1 ≤ q i\n⊢ ((ofPowerSeries (q i)) (f i)) n = 1 n"
] | refine fun n ↦ (tendsto_atTop.mp ((northcott_iff_tendsto q).mp hq) (n + 1)).mono fun i hi ↦ ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Analysis.Convex.EGauge | {
"line": 225,
"column": 15
} | {
"line": 225,
"column": 56
} | {
"line": 226,
"column": 2
} | [
{
"pp": "case inl\n𝕜 : Type u_1\ninst✝⁴ : NormedDivisionRing 𝕜\nE : Type u_2\ninst✝³ : AddCommGroup E\ninst✝² : Module 𝕜 E\nF : Type u_3\ninst✝¹ : AddCommGroup F\ninst✝ : Module 𝕜 F\nU : Set E\nV : Set F\nhU : Balanced 𝕜 U\nhV : Balanced 𝕜 V\na : E\nb : F\nr : ℝ≥0∞\nx : 𝕜\nhx : a ∈ x • U\nhxr : ‖x‖ₑ < r\... | [] | exact ⟨y, ⟨hU.smul_mono hle hx, hy⟩, hyr⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Analysis.Convex.EGauge | {
"line": 225,
"column": 15
} | {
"line": 225,
"column": 56
} | {
"line": 226,
"column": 2
} | [
{
"pp": "case inl\n𝕜 : Type u_1\ninst✝⁴ : NormedDivisionRing 𝕜\nE : Type u_2\ninst✝³ : AddCommGroup E\ninst✝² : Module 𝕜 E\nF : Type u_3\ninst✝¹ : AddCommGroup F\ninst✝ : Module 𝕜 F\nU : Set E\nV : Set F\nhU : Balanced 𝕜 U\nhV : Balanced 𝕜 V\na : E\nb : F\nr : ℝ≥0∞\nx : 𝕜\nhx : a ∈ x • U\nhxr : ‖x‖ₑ < r\... | [] | exact ⟨y, ⟨hU.smul_mono hle hx, hy⟩, hyr⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
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