module stringlengths 16 90 | startPos dict | endPos dict | nextStartPos dict | goals listlengths 0 96 | goalsAfter listlengths 0 96 | ppTac stringlengths 1 14.5k | elaborator stringclasses 375
values | kind stringclasses 379
values |
|---|---|---|---|---|---|---|---|---|
Mathlib.AlgebraicGeometry.AlgClosed.Basic | {
"line": 108,
"column": 4
} | {
"line": 109,
"column": 29
} | {
"line": 110,
"column": 2
} | [
{
"pp": "case refine_1\nX Y : Scheme\nK : Type u\ninst✝⁵ : Field K\ninst✝⁴ : IsAlgClosed K\nf g : X ⟶ Y\ni : Y ⟶ Spec (CommRingCat.of K)\ninst✝³ : IsSeparated i\ninst✝² : LocallyOfFiniteType i\ninst✝¹ : IsReduced X\ninst✝ : LocallyOfFiniteType (f ≫ i)\nS : Set ↥X\nhS : IsLocallyClosed S\nhS' : Dense S\nH : ∀ x ... | [] | rwa [dense_iff_closure_eq, JacobsonSpace.closure_inter_closedPoints_eq_closure hS,
← dense_iff_closure_eq] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.AlgebraicGeometry.AlgClosed.Basic | {
"line": 108,
"column": 4
} | {
"line": 109,
"column": 29
} | {
"line": 110,
"column": 2
} | [
{
"pp": "case refine_1\nX Y : Scheme\nK : Type u\ninst✝⁵ : Field K\ninst✝⁴ : IsAlgClosed K\nf g : X ⟶ Y\ni : Y ⟶ Spec (CommRingCat.of K)\ninst✝³ : IsSeparated i\ninst✝² : LocallyOfFiniteType i\ninst✝¹ : IsReduced X\ninst✝ : LocallyOfFiniteType (f ≫ i)\nS : Set ↥X\nhS : IsLocallyClosed S\nhS' : Dense S\nH : ∀ x ... | [] | rwa [dense_iff_closure_eq, JacobsonSpace.closure_inter_closedPoints_eq_closure hS,
← dense_iff_closure_eq] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Spectrum.Prime.ChevalleyComplexity | {
"line": 201,
"column": 2
} | {
"line": 201,
"column": 17
} | {
"line": 202,
"column": 4
} | [
{
"pp": "case ind\nn : ℕ\nP : (R : Type u) → [inst : CommRing R] → InductionObj R n → Prop\nhP₁ : ∀ (R : Type u) [inst : CommRing R], P R { val := 0 }\nhP₂ :\n ∀ (R : Type u) [inst : CommRing R] (e : InductionObj R n) (i : Fin n),\n (e.val i).Monic → (∀ (j : Fin n), j ≠ i → e.val j = 0) → P R e\nhP₃ :\n ∀ ... | [] | | ind v H_IH => | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | null |
Mathlib.Topology.LocallyFinsupp | {
"line": 288,
"column": 14
} | {
"line": 288,
"column": 16
} | {
"line": 289,
"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... | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.RingTheory.Spectrum.Prime.ChevalleyComplexity | {
"line": 288,
"column": 2
} | {
"line": 288,
"column": 48
} | {
"line": 289,
"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\n⊢ Statement R₀ (Localization.Away c) n\n {\n val :=\n Polynomial.C (IsLocalization.Away.invSelf ... | [
"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)\n⊢ Statement R₀ (Localization.Away c) n\n ... | set q₁ := IsScalarTower.toAlgHom R₀ R (Away c) | Mathlib.Tactic._aux_Mathlib_Tactic_Set___elabRules_Mathlib_Tactic_setTactic_1 | Mathlib.Tactic.setTactic |
Mathlib.Topology.LocallyFinsupp | {
"line": 443,
"column": 14
} | {
"line": 443,
"column": 16
} | {
"line": 444,
"column": 6
} | [
{
"pp": "case right\nX : Type u_1\ninst✝² : TopologicalSpace X\nU : Set X\nY : Type u_2\ninst✝¹ : SemilatticeSup Y\ninst✝ : Zero Y\nD₁ D₂ : locallyFinsuppWithin U Y\nz : X\nhz : z ∈ U\nt₁ : Set X\nht₁ : t₁ ∈ 𝓝 z ∧ (t₁ ∩ D₁.support).Finite\nt₂ : Set X\nht₂ : t₂ ∈ 𝓝 z ∧ (t₂ ∩ D₂.support).Finite\na : X\n⊢ (a ∈ t... | [
"case right\nX : Type u_1\ninst✝² : TopologicalSpace X\nU : Set X\nY : Type u_2\ninst✝¹ : SemilatticeSup Y\ninst✝ : Zero Y\nD₁ D₂ : locallyFinsuppWithin U Y\nz : X\nhz : z ∈ U\nt₁ : Set X\nht₁ : t₁ ∈ 𝓝 z ∧ (t₁ ∩ D₁.support).Finite\nt₂ : Set X\nht₂ : t₂ ∈ 𝓝 z ∧ (t₂ ∩ D₂.support).Finite\na : X\nha : a ∈ t₁ ∩ t₂ ∩ F... | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.Topology.LocallyFinsupp | {
"line": 467,
"column": 14
} | {
"line": 467,
"column": 16
} | {
"line": 468,
"column": 6
} | [
{
"pp": "case right\nX : Type u_1\ninst✝² : TopologicalSpace X\nU : Set X\nY : Type u_2\ninst✝¹ : SemilatticeInf Y\ninst✝ : Zero Y\nD₁ D₂ : locallyFinsuppWithin U Y\nz : X\nhz : z ∈ U\nt₁ : Set X\nht₁ : t₁ ∈ 𝓝 z ∧ (t₁ ∩ D₁.support).Finite\nt₂ : Set X\nht₂ : t₂ ∈ 𝓝 z ∧ (t₂ ∩ D₂.support).Finite\na : X\n⊢ (a ∈ t... | [
"case right\nX : Type u_1\ninst✝² : TopologicalSpace X\nU : Set X\nY : Type u_2\ninst✝¹ : SemilatticeInf Y\ninst✝ : Zero Y\nD₁ D₂ : locallyFinsuppWithin U Y\nz : X\nhz : z ∈ U\nt₁ : Set X\nht₁ : t₁ ∈ 𝓝 z ∧ (t₁ ∩ D₁.support).Finite\nt₂ : Set X\nht₂ : t₂ ∈ 𝓝 z ∧ (t₂ ∩ D₂.support).Finite\na : X\nha : a ∈ t₁ ∩ t₂ ∩ F... | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.AlgebraicGeometry.SpreadingOut | {
"line": 172,
"column": 4
} | {
"line": 172,
"column": 99
} | {
"line": 173,
"column": 4
} | [
{
"pp": "X Y S : Scheme\nf : X ⟶ Y\nsX : X ⟶ S\nsY : Y ⟶ S\nR A : CommRingCat\ninst✝ : IsLocallyNoetherian X\nthis : ∀ (R : CommRingCat), IsNoetherianRing ↑R → (Spec R).IsGermInjective\n⊢ X.IsGermInjective",
"ppTerm": "?m.10",
"assigned": true,
"usedConstants": [
"AlgebraicGeometry.Scheme",
... | [
"X Y S : Scheme\nf : X ⟶ Y\nsX : X ⟶ S\nsY : Y ⟶ S\nR A : CommRingCat\ninst✝ : IsLocallyNoetherian X\nthis : ∀ (R : CommRingCat), IsNoetherianRing ↑R → (Spec R).IsGermInjective\ni : X.affineOpenCover.openCover.I₀\n⊢ IsNoetherianRing ↑(X.affineOpenCover.X i)"
] | refine @Scheme.IsGermInjective.of_openCover _ (X.affineOpenCover.openCover) (fun i ↦ this _ ?_) | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.AlgebraicGeometry.Morphisms.Flat | {
"line": 418,
"column": 4
} | {
"line": 419,
"column": 83
} | {
"line": 420,
"column": 4
} | [
{
"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\ninst✝ : Finite ι\nVX : ι → X.Opens\nhVU : iSup VX = UX\nhV : ∀ (... | [
"case id_single\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 : ∀ ... | obtain ⟨i | ⟨i, j⟩ | ⟨i, j⟩ | ⟨i, j⟩, rfl⟩ :=
(show Function.Surjective Quiver.Hom.op from Quiver.Hom.opEquiv.surjective) f | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.AlgebraicGeometry.Birational.RationalMap | {
"line": 318,
"column": 31
} | {
"line": 318,
"column": 63
} | {
"line": 318,
"column": 63
} | [
{
"pp": "case e'_2\nX 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\nhdense : Dense (↑f.domain ⊓ ↑g.domain)\nthis : (↑(f.domain ⊓ g.domain)).IsGermInjectiveAt ... | [
"case e'_2\nX 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\nhdense : Dense (↑f.domain ⊓ ↑g.domain)\nthis : (↑(f.domain ⊓ g.domain)).IsGermInjectiveAt ⟨x, ⋯⟩\nU : ... | ← cancel_mono (Scheme.Opens.ι _) | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.AlgebraicGeometry.Birational.RationalMap | {
"line": 318,
"column": 31
} | {
"line": 318,
"column": 63
} | {
"line": 318,
"column": 63
} | [
{
"pp": "case e'_3\nX 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\nhdense : Dense (↑f.domain ⊓ ↑g.domain)\nthis : (↑(f.domain ⊓ g.domain)).IsGermInjectiveAt ... | [
"case e'_3\nX 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\nhdense : Dense (↑f.domain ⊓ ↑g.domain)\nthis : (↑(f.domain ⊓ g.domain)).IsGermInjectiveAt ⟨x, ⋯⟩\nU : ... | ← cancel_mono (Scheme.Opens.ι _) | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.AlgebraicGeometry.Birational.RationalMap | {
"line": 463,
"column": 2
} | {
"line": 465,
"column": 57
} | {
"line": 467,
"column": 0
} | [
{
"pp": "X Y S : Scheme\ninst✝² : X.Over S\ninst✝¹ : Y.Over S\nf : X.PartialMap Y\ninst✝ : RationalMap.IsOver S f.toRationalMap\n⊢ ∃ U, ∃ (hU : Dense ↑U) (hU' : U ≤ f.domain), IsOver S (f.restrict U hU hU')",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"AlgebraicGeo... | [] | obtain ⟨f', hf₁, hf₂⟩ := RationalMap.IsOver.exists_partialMap_over (S := S) (f := f.toRationalMap)
obtain ⟨U, hU, hUl, hUr, e⟩ := PartialMap.toRationalMap_eq_iff.mp hf₂
exact ⟨U, hU, hUr, by rw [IsOver, ← e]; infer_instance⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.AlgebraicGeometry.Birational.RationalMap | {
"line": 463,
"column": 2
} | {
"line": 465,
"column": 57
} | {
"line": 467,
"column": 0
} | [
{
"pp": "X Y S : Scheme\ninst✝² : X.Over S\ninst✝¹ : Y.Over S\nf : X.PartialMap Y\ninst✝ : RationalMap.IsOver S f.toRationalMap\n⊢ ∃ U, ∃ (hU : Dense ↑U) (hU' : U ≤ f.domain), IsOver S (f.restrict U hU hU')",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"AlgebraicGeo... | [] | obtain ⟨f', hf₁, hf₂⟩ := RationalMap.IsOver.exists_partialMap_over (S := S) (f := f.toRationalMap)
obtain ⟨U, hU, hUl, hUr, e⟩ := PartialMap.toRationalMap_eq_iff.mp hf₂
exact ⟨U, hU, hUr, by rw [IsOver, ← e]; infer_instance⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.AlgebraicGeometry.Morphisms.UniversallyInjective | {
"line": 69,
"column": 19
} | {
"line": 69,
"column": 21
} | {
"line": 69,
"column": 22
} | [
{
"pp": "case a\nX Y : Scheme\nf : X ⟶ Y\nhf : diagonal (@Surjective) f\n⊢ topologically (fun {α β} [TopologicalSpace α] [TopologicalSpace β] x ↦ Injective x) f",
"ppTerm": "?a✝",
"assigned": true,
"usedConstants": [
"AlgebraicGeometry.PresheafedSpace.carrier",
"CommRingCat",
"Comm... | [
"case a\nX Y : Scheme\nf : X ⟶ Y\nhf : diagonal (@Surjective) f\nx₁ : ↥X\n⊢ ∀ ⦃a₂ : ↥X⦄, f x₁ = f a₂ → x₁ = a₂"
] | x₁ | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.AlgebraicGeometry.Cover.QuasiCompact | {
"line": 65,
"column": 2
} | {
"line": 67,
"column": 26
} | {
"line": 69,
"column": 0
} | [
{
"pp": "S : Scheme\n𝒰 : PreZeroHypercover S\ninst✝ : QuasiCompactCover 𝒰\nU : S.Opens\nhU : IsCompact ↑U\n⊢ ∃ n f V, (∀ (i : Fin n), IsAffineOpen (V i)) ∧ ⋃ i, ⇑(𝒰.f (f i)) '' ↑(V i) = ↑U",
"ppTerm": "?m.32",
"assigned": true,
"usedConstants": [
"AlgebraicGeometry.IsAffineOpen.isCompact",
... | [] | obtain ⟨n, a, V, ha, heq⟩ := (isCompactOpenCovered_of_isCompact 𝒰 hU).exists_mem_of_isBasis
(fun i ↦ (𝒰.X i).isBasis_affineOpens) (fun _ _ h ↦ h.isCompact)
exact ⟨n, a, V, ha, heq⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.AlgebraicGeometry.Cover.QuasiCompact | {
"line": 65,
"column": 2
} | {
"line": 67,
"column": 26
} | {
"line": 69,
"column": 0
} | [
{
"pp": "S : Scheme\n𝒰 : PreZeroHypercover S\ninst✝ : QuasiCompactCover 𝒰\nU : S.Opens\nhU : IsCompact ↑U\n⊢ ∃ n f V, (∀ (i : Fin n), IsAffineOpen (V i)) ∧ ⋃ i, ⇑(𝒰.f (f i)) '' ↑(V i) = ↑U",
"ppTerm": "?m.32",
"assigned": true,
"usedConstants": [
"AlgebraicGeometry.IsAffineOpen.isCompact",
... | [] | obtain ⟨n, a, V, ha, heq⟩ := (isCompactOpenCovered_of_isCompact 𝒰 hU).exists_mem_of_isBasis
(fun i ↦ (𝒰.X i).isBasis_affineOpens) (fun _ _ h ↦ h.isCompact)
exact ⟨n, a, V, ha, heq⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.AlgebraicGeometry.Cover.QuasiCompact | {
"line": 109,
"column": 53
} | {
"line": 109,
"column": 70
} | {
"line": 109,
"column": 70
} | [
{
"pp": "S : Scheme\n𝒰✝ 𝒰 : PreZeroHypercover S\nK : Precoverage Scheme\ninst✝ : QuasiCompactCover 𝒰\nT : Scheme\nf : T ⟶ S\nU' : T.Opens\nhU' : IsAffineOpen U'\nU : S.Opens\nhU : IsAffineOpen U\nhsub : ⇑f '' ↑U' ⊆ ↑U\ns : Set 𝒰.I₀\nhf : s.Finite\nV : (i : 𝒰.I₀) → i ∈ s → TopologicalSpace.Opens ((fun x ↦ ↥... | [] | simpa using! hsub | Lean.Elab.Tactic.Simpa.evalSimpaUsingBang | Lean.Parser.Tactic.simpaUsingBang |
Mathlib.AlgebraicGeometry.Cover.QuasiCompact | {
"line": 109,
"column": 53
} | {
"line": 109,
"column": 70
} | {
"line": 109,
"column": 70
} | [
{
"pp": "S : Scheme\n𝒰✝ 𝒰 : PreZeroHypercover S\nK : Precoverage Scheme\ninst✝ : QuasiCompactCover 𝒰\nT : Scheme\nf : T ⟶ S\nU' : T.Opens\nhU' : IsAffineOpen U'\nU : S.Opens\nhU : IsAffineOpen U\nhsub : ⇑f '' ↑U' ⊆ ↑U\ns : Set 𝒰.I₀\nhf : s.Finite\nV : (i : 𝒰.I₀) → i ∈ s → TopologicalSpace.Opens ((fun x ↦ ↥... | [] | simpa using! hsub | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.AlgebraicGeometry.Cover.QuasiCompact | {
"line": 109,
"column": 53
} | {
"line": 109,
"column": 70
} | {
"line": 109,
"column": 70
} | [
{
"pp": "S : Scheme\n𝒰✝ 𝒰 : PreZeroHypercover S\nK : Precoverage Scheme\ninst✝ : QuasiCompactCover 𝒰\nT : Scheme\nf : T ⟶ S\nU' : T.Opens\nhU' : IsAffineOpen U'\nU : S.Opens\nhU : IsAffineOpen U\nhsub : ⇑f '' ↑U' ⊆ ↑U\ns : Set 𝒰.I₀\nhf : s.Finite\nV : (i : 𝒰.I₀) → i ∈ s → TopologicalSpace.Opens ((fun x ↦ ↥... | [] | simpa using! hsub | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Spectrum.Prime.ChevalleyComplexity | {
"line": 492,
"column": 8
} | {
"line": 525,
"column": 58
} | {
"line": 526,
"column": 6
} | [
{
"pp": "R✝ : Type u_2\ninst✝³ : CommRing R✝\nn : ℕ\nR : Type u_2\ninst✝² : CommRing R\nc : InductionObj R n\ni j : Fin n\nhi : (c.val i).Monic\nhle : (c.val i).degree ≤ (c.val j).degree\nhne : i ≠ j\nH :\n ∀ {R₀ : Type u_1} [inst : CommRing R₀] [inst_1 : Algebra R₀ R],\n Statement R₀ R n { val := Function.... | [] | by_cases hi' : c.val i = 1
· gcongr
· exact c.one_le_coeffSubmodule
· refine Submodule.span_le.mpr (Set.union_subset ?_ ?_)
· exact Set.subset_union_left.trans Submodule.subset_span
· refine Set.iUnion_subset fun k ↦ ?_
simp only [update_apply, hi', modB... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.EffectiveEpi.Comp | {
"line": 124,
"column": 19
} | {
"line": 127,
"column": 65
} | {
"line": 129,
"column": 0
} | [
{
"pp": "C : Type u_1\ninst✝² : Category.{v_1, u_1} C\nB B' : C\nα : Type u_2\nX : α → C\nπ : (a : α) → X a ⟶ B\ni : B ⟶ B'\ninst✝¹ : EffectiveEpiFamily X π\ninst✝ : IsIso i\nW✝ : C\ne : (a : α) → X a ⟶ W✝\nh : ∀ {Z : C} (a₁ a₂ : α) (g₁ : Z ⟶ X a₁) (g₂ : Z ⟶ X a₂), g₁ ≫ π a₁ ≫ i = g₂ ≫ π a₂ ≫ i → g₁ ≫ e a₁ = g₂... | [] | by
simp only [Category.assoc] at hm
simp [← EffectiveEpiFamily.uniq X π e
(effectiveEpiFamilyStructCompIso_aux X π i e h) (i ≫ m) hm] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.Spectrum.Prime.ChevalleyComplexity | {
"line": 492,
"column": 8
} | {
"line": 525,
"column": 58
} | {
"line": 526,
"column": 6
} | [
{
"pp": "R✝ : Type u_2\ninst✝³ : CommRing R✝\nn : ℕ\nR : Type u_2\ninst✝² : CommRing R\nc : InductionObj R n\ni j : Fin n\nhi : (c.val i).Monic\nhle : (c.val i).degree ≤ (c.val j).degree\nhne : i ≠ j\nH :\n ∀ {R₀ : Type u_1} [inst : CommRing R₀] [inst_1 : Algebra R₀ R],\n Statement R₀ R n { val := Function.... | [] | by_cases hi' : c.val i = 1
· gcongr
· exact c.one_le_coeffSubmodule
· refine Submodule.span_le.mpr (Set.union_subset ?_ ?_)
· exact Set.subset_union_left.trans Submodule.subset_span
· refine Set.iUnion_subset fun k ↦ ?_
simp only [update_apply, hi', modB... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Spectrum.Prime.ChevalleyComplexity | {
"line": 531,
"column": 8
} | {
"line": 531,
"column": 34
} | {
"line": 532,
"column": 8
} | [
{
"pp": "case hP₃.inr.refine_2.left\nR✝ : Type u_2\ninst✝³ : CommRing R✝\nn : ℕ\nR : Type u_2\ninst✝² : CommRing R\nc : InductionObj R n\ni j : Fin n\nhi : (c.val i).Monic\nhle : (c.val i).degree ≤ (c.val j).degree\nhne : i ≠ j\nH :\n ∀ {R₀ : Type u_1} [inst : CommRing R₀] [inst_1 : Algebra R₀ R],\n Stateme... | [
"case hP₃.inr.refine_2.left\nR✝ : Type u_2\ninst✝³ : CommRing R✝\nn : ℕ\nR : Type u_2\ninst✝² : CommRing R\nc : InductionObj R n\ni j : Fin n\nhi : (c.val i).Monic\nhle : (c.val i).degree ≤ (c.val j).degree\nhne : i ≠ j\nH :\n ∀ {R₀ : Type u_1} [inst : CommRing R₀] [inst_1 : Algebra R₀ R],\n Statement R₀ R n { ... | rw [Submodule.one_eq_span] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.AlgebraicGeometry.EllipticCurve.Weierstrass | {
"line": 170,
"column": 2
} | {
"line": 170,
"column": 80
} | {
"line": 172,
"column": 0
} | [
{
"pp": "R : Type u\ninst✝¹ : CommRing R\nW : WeierstrassCurve R\ninst✝ : CharP R 2\n⊢ -(W.a₁ ^ 2) ^ 2 * W.b₈ - 8 * (W.a₁ * W.a₃) ^ 3 - 27 * (W.a₃ ^ 2) ^ 2 + 9 * W.a₁ ^ 2 * (W.a₁ * W.a₃) * W.a₃ ^ 2 =\n W.a₁ ^ 4 * W.b₈ + W.a₃ ^ 4 + W.a₁ ^ 3 * W.a₃ ^ 3",
"ppTerm": "?m.92",
"assigned": true,
"usedCo... | [] | linear_combination (-W.a₁ ^ 4 * W.b₈ - 14 * W.a₃ ^ 4) * 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.AffineTransitionLimit | {
"line": 898,
"column": 2
} | {
"line": 898,
"column": 78
} | {
"line": 899,
"column": 2
} | [
{
"pp": "I : Type u\ninst✝⁴ : Category.{u, u} I\nD : I ⥤ Scheme\nc : Cone D\nhc : IsLimit c\ninst✝³ : IsCofiltered I\ninst✝² : ∀ {i j : I} (f : i ⟶ j), IsAffineHom (D.map f)\ns : ↑Γ(c.pt, ⊤)\ninst✝¹ : ∀ (i : I), CompactSpace ↥(D.obj i)\ninst✝ : ∀ (i : I), QuasiSeparatedSpace ↥(D.obj i)\nthis : CompactSpace ↥c.p... | [
"I : Type u\ninst✝⁴ : Category.{u, u} I\nD : I ⥤ Scheme\nc : Cone D\nhc : IsLimit c\ninst✝³ : IsCofiltered I\ninst✝² : ∀ {i j : I} (f : i ⟶ j), IsAffineHom (D.map f)\ns : ↑Γ(c.pt, ⊤)\ninst✝¹ : ∀ (i : I), CompactSpace ↥(D.obj i)\ninst✝ : ∀ (i : I), QuasiSeparatedSpace ↥(D.obj i)\nthis : CompactSpace ↥c.pt\ni : ↥c.pt... | choose σi hσiσ hσi using fun x ↦ Set.mem_iUnion₂.mp (hσ.ge (Set.mem_univ x)) | Mathlib.Tactic.Choose._aux_Mathlib_Tactic_Choose___elabRules_Mathlib_Tactic_Choose_choose_1 | Mathlib.Tactic.Choose.choose |
Mathlib.RingTheory.Spectrum.Prime.ChevalleyComplexity | {
"line": 759,
"column": 6
} | {
"line": 762,
"column": 14
} | {
"line": 763,
"column": 2
} | [
{
"pp": "case succ\nR : Type u_2\ninst✝ : CommRing R\nn✝ n : ℕ\nIH :\n ∀ {M : Submodule ℤ R},\n 1 ∈ M →\n ∀ (k : ℕ) (d : Multiset (Fin n)) (S : ConstructibleSetData (MvPolynomial (Fin n) R)),\n (∀ C ∈ S, C.n ≤ k) →\n (∀ C ∈ S, ∀ (j : Fin C.n), C.g j ∈ coeffsIn (Fin n) M ⊓ Submodule.rest... | [] | simp only [add_lt_add_iff_right] at ht
change 1 + (B.map Fin.val).count (Fin.mk t ht).val ≤ _
rw [Multiset.count_map_eq_count' _ _ Fin.val_injective]
simp [B] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Spectrum.Prime.ChevalleyComplexity | {
"line": 759,
"column": 6
} | {
"line": 762,
"column": 14
} | {
"line": 763,
"column": 2
} | [
{
"pp": "case succ\nR : Type u_2\ninst✝ : CommRing R\nn✝ n : ℕ\nIH :\n ∀ {M : Submodule ℤ R},\n 1 ∈ M →\n ∀ (k : ℕ) (d : Multiset (Fin n)) (S : ConstructibleSetData (MvPolynomial (Fin n) R)),\n (∀ C ∈ S, C.n ≤ k) →\n (∀ C ∈ S, ∀ (j : Fin C.n), C.g j ∈ coeffsIn (Fin n) M ⊓ Submodule.rest... | [] | simp only [add_lt_add_iff_right] at ht
change 1 + (B.map Fin.val).count (Fin.mk t ht).val ≤ _
rw [Multiset.count_map_eq_count' _ _ Fin.val_injective]
simp [B] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.AlgebraicGeometry.EllipticCurve.Affine.Point | {
"line": 203,
"column": 4
} | {
"line": 203,
"column": 47
} | {
"line": 204,
"column": 4
} | [
{
"pp": "R : Type r\nS : Type s\ninst✝¹ : CommRing R\ninst✝ : CommRing S\nW' : Affine R\nf : R →+* S\nhf : Function.Injective ⇑f\np q : R[X]\nhy : Polynomial.map f p • 1 + Polynomial.map f q • (mk (W'.map f)) Y = 0\nhp : Polynomial.map f p = 0\nhq : Polynomial.map f q = 0\n⊢ p • 1 + q • (mk W') Y = 0",
"ppT... | [
"R : Type r\nS : Type s\ninst✝¹ : CommRing R\ninst✝ : CommRing S\nW' : Affine R\nf : R →+* S\nhf : Function.Injective ⇑f\np q : R[X]\nhy : Polynomial.map f p • 1 + Polynomial.map f q • (mk (W'.map f)) Y = 0\nhp : p = 0\nhq : q = 0\n⊢ p • 1 + q • (mk W') Y = 0"
] | rw [Polynomial.map_eq_zero_iff hf] at hp hq | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.Spectrum.Prime.ChevalleyComplexity | {
"line": 894,
"column": 8
} | {
"line": 899,
"column": 43
} | {
"line": 899,
"column": 43
} | [
{
"pp": "case right\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 (MvPolynomia... | [] | obtain j | j := j
· simp only [MvPolynomial.algebraMap_eq, Sum.elim_inl, comp_apply, σ]
exact degrees_map_le.trans (hS _ hx j)
· refine degrees_sub_le.trans ?_
simp only [degrees_C, Multiset.zero_union]
exact degrees_map_le.trans (hf _) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Spectrum.Prime.ChevalleyComplexity | {
"line": 894,
"column": 8
} | {
"line": 899,
"column": 43
} | {
"line": 899,
"column": 43
} | [
{
"pp": "case right\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 (MvPolynomia... | [] | obtain j | j := j
· simp only [MvPolynomial.algebraMap_eq, Sum.elim_inl, comp_apply, σ]
exact degrees_map_le.trans (hS _ hx j)
· refine degrees_sub_le.trans ?_
simp only [degrees_C, Multiset.zero_union]
exact degrees_map_le.trans (hf _) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.AlgebraicGeometry.AffineTransitionLimit | {
"line": 1019,
"column": 2
} | {
"line": 1019,
"column": 78
} | {
"line": 1020,
"column": 2
} | [
{
"pp": "I : Type u\ninst✝⁵ : Category.{u, u} I\nD : I ⥤ Scheme\nc : Cone D\nhc : IsLimit c\ninst✝⁴ : IsCofiltered I\ninst✝³ : ∀ {i j : I} (f : i ⟶ j), IsAffineHom (D.map f)\ninst✝² : ∀ (i : I), CompactSpace ↥(D.obj i)\ninst✝¹ : ∀ (i : I), QuasiSeparatedSpace ↥(D.obj i)\ninst✝ : c.pt.IsQuasiAffine\ni : ↥c.pt → ... | [
"I : Type u\ninst✝⁵ : Category.{u, u} I\nD : I ⥤ Scheme\nc : Cone D\nhc : IsLimit c\ninst✝⁴ : IsCofiltered I\ninst✝³ : ∀ {i j : I} (f : i ⟶ j), IsAffineHom (D.map f)\ninst✝² : ∀ (i : I), CompactSpace ↥(D.obj i)\ninst✝¹ : ∀ (i : I), QuasiSeparatedSpace ↥(D.obj i)\ninst✝ : c.pt.IsQuasiAffine\ni : ↥c.pt → I\nf : (x : ... | choose σi hσiσ hσi using fun x ↦ Set.mem_iUnion₂.mp (hσ.ge (Set.mem_univ x)) | Mathlib.Tactic.Choose._aux_Mathlib_Tactic_Choose___elabRules_Mathlib_Tactic_Choose_choose_1 | Mathlib.Tactic.Choose.choose |
Mathlib.AlgebraicGeometry.AffineTransitionLimit | {
"line": 1099,
"column": 2
} | {
"line": 1099,
"column": 40
} | {
"line": 1101,
"column": 0
} | [
{
"pp": "case intro.refine_2\nI : Type u\ninst✝⁵ : Category.{u, u} I\nD : I ⥤ Scheme\nc : Cone D\nhc : IsLimit c\ninst✝⁴ : IsCofiltered I\ninst✝³ : ∀ {i j : I} (f : i ⟶ j), IsAffineHom (D.map f)\ninst✝² : ∀ (i : I), CompactSpace ↥(D.obj i)\ninst✝¹ : ∀ (i : I), QuasiSeparatedSpace ↥(D.obj i)\nJ : Type u_1\ninst✝... | [] | · rw [← hVU, ← Hom.comp_preimage, c.w] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.AlgebraicGeometry.EllipticCurve.DivisionPolynomial.Basic | {
"line": 472,
"column": 6
} | {
"line": 472,
"column": 8
} | {
"line": 472,
"column": 9
} | [
{
"pp": "R : Type r\ninst✝ : CommRing R\nW : WeierstrassCurve R\n⊢ W.φ 4 = C X * C W.preΨ₄ ^ 2 * W.ψ₂ ^ 2 - C W.preΨ₄ * W.ψ₂ ^ 4 * C W.Ψ₃ + C W.Ψ₃ ^ 4",
"ppTerm": "?m.105",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Polynomial.C",
"WeierstrassCurve.ψ",
"HMul.hMul",
... | [
"R : Type r\ninst✝ : CommRing R\nW : WeierstrassCurve R\n⊢ C X * W.ψ 4 ^ 2 - W.ψ (4 + 1) * W.ψ (4 - 1) =\n C X * C W.preΨ₄ ^ 2 * W.ψ₂ ^ 2 - C W.preΨ₄ * W.ψ₂ ^ 4 * C W.Ψ₃ + C W.Ψ₃ ^ 4"
] | φ, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.AlgebraicGeometry.EllipticCurve.DivisionPolynomial.Basic | {
"line": 479,
"column": 11
} | {
"line": 479,
"column": 13
} | {
"line": 479,
"column": 14
} | [
{
"pp": "R : Type r\ninst✝ : CommRing R\nW : WeierstrassCurve R\nn : ℤ\n⊢ W.φ (-n) = W.φ n",
"ppTerm": "?m.8",
"assigned": true,
"usedConstants": [
"CommSemiring.toSemiring",
"id",
"Int.instNegInt",
"Int",
"WeierstrassCurve.φ",
"Polynomial",
"CommRing.toComm... | [
"R : Type r\ninst✝ : CommRing R\nW : WeierstrassCurve R\nn : ℤ\n⊢ C X * W.ψ (-n) ^ 2 - W.ψ (-n + 1) * W.ψ (-n - 1) = C X * W.ψ n ^ 2 - W.ψ (n + 1) * W.ψ (n - 1)"
] | φ, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.AlgebraicGeometry.EllipticCurve.DivisionPolynomial.Basic | {
"line": 483,
"column": 11
} | {
"line": 483,
"column": 13
} | {
"line": 483,
"column": 14
} | [
{
"pp": "R : Type r\ninst✝ : CommRing R\nW : WeierstrassCurve R\nn : ℤ\n⊢ (mk W) (W.φ n) = (mk W) (C (W.Φ n))",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"Polynomial.C",
"WeierstrassCurve.Affine.CoordinateRing.mk",
"CommSemiring.toSemiring",
"RingHom",
"id... | [
"R : Type r\ninst✝ : CommRing R\nW : WeierstrassCurve R\nn : ℤ\n⊢ (mk W) (C X * W.ψ n ^ 2 - W.ψ (n + 1) * W.ψ (n - 1)) = (mk W) (C (W.Φ n))"
] | φ, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.AlgebraicGeometry.AffineTransitionLimit | {
"line": 1249,
"column": 52
} | {
"line": 1249,
"column": 54
} | {
"line": 1249,
"column": 54
} | [
{
"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✝ ... | ha | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.AlgebraicGeometry.EllipticCurve.NormalForms | {
"line": 340,
"column": 71
} | {
"line": 342,
"column": 75
} | {
"line": 344,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝² : CommRing R\nW : WeierstrassCurve R\ninst✝¹ : W.IsCharThreeJNeZeroNF\ninst✝ : CharP R 3\n⊢ W.c₆ = -W.a₂ ^ 3",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"Mathlib.Tactic.Ring.Common.mul_pf_left",
"CharP.cast_eq_zero",
"AddGroup.toSubtracti... | [] | by
rw [c₆_of_isCharThreeJNeZeroNF]
linear_combination (-21 * W.a₂ ^ 3 - 288 * W.a₆) * CharP.cast_eq_zero R 3 | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.AlgebraicGeometry.EllipticCurve.DivisionPolynomial.Degree | {
"line": 208,
"column": 11
} | {
"line": 208,
"column": 98
} | {
"line": 209,
"column": 2
} | [
{
"pp": "case one\nR : Type u\ninst✝ : CommRing R\nW : WeierstrassCurve R\ndm : ∀ {m n : ℕ} {p q : R[X]}, p.natDegree ≤ m → q.natDegree ≤ n → (p * q).natDegree ≤ m + n :=\n fun {m n} {p q} ↦ natDegree_mul_le_of_le\ndp : ∀ {m n : ℕ} {p : R[X]}, p.natDegree ≤ m → (p ^ n).natDegree ≤ n * m := fun {m n} {p} ↦ natD... | [] | simpa only [preΨ'_one] using ⟨natDegree_one.le, coeff_one_zero.trans Int.cast_one.symm⟩ | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.AlgebraicGeometry.EllipticCurve.DivisionPolynomial.Degree | {
"line": 208,
"column": 11
} | {
"line": 208,
"column": 98
} | {
"line": 209,
"column": 2
} | [
{
"pp": "case one\nR : Type u\ninst✝ : CommRing R\nW : WeierstrassCurve R\ndm : ∀ {m n : ℕ} {p q : R[X]}, p.natDegree ≤ m → q.natDegree ≤ n → (p * q).natDegree ≤ m + n :=\n fun {m n} {p q} ↦ natDegree_mul_le_of_le\ndp : ∀ {m n : ℕ} {p : R[X]}, p.natDegree ≤ m → (p ^ n).natDegree ≤ n * m := fun {m n} {p} ↦ natD... | [] | simpa only [preΨ'_one] using ⟨natDegree_one.le, coeff_one_zero.trans Int.cast_one.symm⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.AlgebraicGeometry.EllipticCurve.DivisionPolynomial.Degree | {
"line": 208,
"column": 11
} | {
"line": 208,
"column": 98
} | {
"line": 209,
"column": 2
} | [
{
"pp": "case one\nR : Type u\ninst✝ : CommRing R\nW : WeierstrassCurve R\ndm : ∀ {m n : ℕ} {p q : R[X]}, p.natDegree ≤ m → q.natDegree ≤ n → (p * q).natDegree ≤ m + n :=\n fun {m n} {p q} ↦ natDegree_mul_le_of_le\ndp : ∀ {m n : ℕ} {p : R[X]}, p.natDegree ≤ m → (p ^ n).natDegree ≤ n * m := fun {m n} {p} ↦ natD... | [] | simpa only [preΨ'_one] using ⟨natDegree_one.le, coeff_one_zero.trans Int.cast_one.symm⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.AlgebraicGeometry.EllipticCurve.NormalForms | {
"line": 499,
"column": 2
} | {
"line": 500,
"column": 7
} | {
"line": 502,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommRing R\nW : WeierstrassCurve R\ninst✝ : W.IsCharTwoJNeZeroNF\n⊢ W.c₆ = -W.b₂ ^ 3 - 864 * W.a₆",
"ppTerm": "?m.32",
"assigned": true,
"usedConstants": [
"Mathlib.Tactic.Ring.Common.mul_pf_left",
"Mathlib.Tactic.Ring.Common.neg_zero",
"Eq.mpr",
... | [] | rw [c₆, b₄_of_isCharTwoJNeZeroNF, b₆_of_isCharTwoJNeZeroNF]
ring1 | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.AlgebraicGeometry.EllipticCurve.NormalForms | {
"line": 499,
"column": 2
} | {
"line": 500,
"column": 7
} | {
"line": 502,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommRing R\nW : WeierstrassCurve R\ninst✝ : W.IsCharTwoJNeZeroNF\n⊢ W.c₆ = -W.b₂ ^ 3 - 864 * W.a₆",
"ppTerm": "?m.32",
"assigned": true,
"usedConstants": [
"Mathlib.Tactic.Ring.Common.mul_pf_left",
"Mathlib.Tactic.Ring.Common.neg_zero",
"Eq.mpr",
... | [] | rw [c₆, b₄_of_isCharTwoJNeZeroNF, b₆_of_isCharTwoJNeZeroNF]
ring1 | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.AlgebraicGeometry.EllipticCurve.NormalForms | {
"line": 614,
"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⊢ -(64 * W.a₄ ^ 3 + 27 * (W.a₃ ^ 2) ^ 2) = W.a₃ ^ 4",
"ppTerm": "?m.31",
"assigned": true,
"usedConstants": [
"Mathlib.Tactic.Ring.Common.mul_pf_left",
"CharP.cast_eq_zer... | [] | linear_combination (-32 * W.a₄ ^ 3 - 14 * W.a₃ ^ 4) * 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.IsomOfJ | {
"line": 105,
"column": 6
} | {
"line": 106,
"column": 49
} | {
"line": 107,
"column": 6
} | [
{
"pp": "case of_j_eq_zero.of_j_ne_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).IsCharTwoJEqZeroNF\nC' : VariableChange F\ninst✝ : (C' • E').IsC... | [
"case of_j_eq_zero.of_j_ne_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).IsCharTwoJEqZeroNF\nC' : VariableChange F\ninst✝ : (C' • E').IsCharTwoJNeZer... | 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.EllipticCurve.IsomOfJ | {
"line": 120,
"column": 42
} | {
"line": 153,
"column": 26
} | {
"line": 155,
"column": 0
} | [
{
"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.IsCharThreeJNeZeroNF\ninst✝ : E'.IsCharThreeJNeZeroNF\nheq : E.j = E'.j\n⊢ ∃ C, C • E = E'",
"ppTerm": "?m.27",
"assigned": true,
... | [] | by
have h := E.Δ'.ne_zero
rw [E.coe_Δ', Δ_of_isCharThreeJNeZeroNF_of_char_three, mul_ne_zero_iff, neg_ne_zero,
pow_ne_zero_iff three_ne_zero] at h
obtain ⟨ha₂, ha₆⟩ := h
have h := E'.Δ'.ne_zero
rw [E'.coe_Δ', Δ_of_isCharThreeJNeZeroNF_of_char_three, mul_ne_zero_iff, neg_ne_zero,
pow_ne_zero_iff three_... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.AlgebraicGeometry.EllipticCurve.Jacobian.Basic | {
"line": 418,
"column": 10
} | {
"line": 418,
"column": 12
} | {
"line": 418,
"column": 13
} | [
{
"pp": "R : Type r\ninst✝ : CommRing R\nW' : Jacobian R\na b : R\nh :\n b ^ 2 + W'.toAffine.a₁ * a * b + W'.toAffine.a₃ * b -\n (a ^ 3 + W'.toAffine.a₂ * a ^ 2 + W'.toAffine.a₄ * a + W'.toAffine.a₆) =\n 0\n⊢ W'.a₁ * b - (3 * a ^ 2 + 2 * W'.a₂ * a + W'.a₄) = 0 →\n 2 * b + W'.toAffine.a₁ * a + W'.toA... | [
"R : Type r\ninst✝ : CommRing R\nW' : Jacobian R\na b : R\nh :\n b ^ 2 + W'.toAffine.a₁ * a * b + W'.toAffine.a₃ * b -\n (a ^ 3 + W'.toAffine.a₂ * a ^ 2 + W'.toAffine.a₄ * a + W'.toAffine.a₆) =\n 0\nha : W'.a₁ * b - (3 * a ^ 2 + 2 * W'.a₂ * a + W'.a₄) = 0\n⊢ 2 * b + W'.toAffine.a₁ * a + W'.toAffine.a₃ = 0 ... | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.AlgebraicGeometry.EllipticCurve.IsomOfJ | {
"line": 324,
"column": 4
} | {
"line": 325,
"column": 76
} | {
"line": 326,
"column": 4
} | [
{
"pp": "case h.a₆\nF : 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 := ⋯\nthis : Invertib... | [
"case h.a₆\nF : 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\nth... | simp_rw [variableChange_a₆, a₁_of_isShortNF, a₂_of_isShortNF, a₃_of_isShortNF,
Units.val_inv_eq_inv_val, Units.val_mk0, inv_pow, inv_mul_eq_div, hu6] | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | Mathlib.Tactic.tacticSimp_rw___ |
Mathlib.AlgebraicGeometry.EllipticCurve.Jacobian.Basic | {
"line": 428,
"column": 58
} | {
"line": 428,
"column": 92
} | {
"line": 429,
"column": 4
} | [
{
"pp": "F : Type u\ninst✝ : Field F\nW : Jacobian F\nP : Fin 3 → F\nhPz : P z ≠ 0\n⊢ W.Equation P ∧\n ((eval P) W.polynomialX ≠ 0 ∧ P z ^ 4 ≠ 0 ∨\n Polynomial.evalEval (P x / P z ^ 2) (P y / P z ^ 3) W.toAffine.polynomialY ≠ 0) ↔\n W.Equation P ∧ ((eval P) W.polynomialX ≠ 0 ∨ (eval P) W.polynomi... | [
"F : Type u\ninst✝ : Field F\nW : Jacobian F\nP : Fin 3 → F\nhPz : P z ≠ 0\n⊢ W.Equation P ∧\n ((eval P) W.polynomialX ≠ 0 ∨ Polynomial.evalEval (P x / P z ^ 2) (P y / P z ^ 3) W.toAffine.polynomialY ≠ 0) ↔\n W.Equation P ∧ ((eval P) W.polynomialX ≠ 0 ∨ (eval P) W.polynomialY ≠ 0)"
] | and_iff_left <| pow_ne_zero 4 hPz, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.AlgebraicGeometry.AffineTransitionLimit | {
"line": 1328,
"column": 4
} | {
"line": 1336,
"column": 43
} | {
"line": 1337,
"column": 2
} | [
{
"pp": "case refine_1\nI : 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), Comp... | [] | refine Cover.hom_ext (𝒲.pullback₁ (c.π.app l)) _ _ fun j ↦ ?_
rw [← cancel_epi (isPullback_morphismRestrict _ _).flip.isoPullback.hom]
dsimp [𝒲]
simp only [pullback.condition_assoc, IsPullback.isoPullback_hom_snd_assoc,
IsPullback.isoPullback_hom_fst_assoc, hF]
have h : c.π.app l ⁻¹ᵁ D.map flk ⁻... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.AlgebraicGeometry.AffineTransitionLimit | {
"line": 1328,
"column": 4
} | {
"line": 1336,
"column": 43
} | {
"line": 1337,
"column": 2
} | [
{
"pp": "case refine_1\nI : 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), Comp... | [] | refine Cover.hom_ext (𝒲.pullback₁ (c.π.app l)) _ _ fun j ↦ ?_
rw [← cancel_epi (isPullback_morphismRestrict _ _).flip.isoPullback.hom]
dsimp [𝒲]
simp only [pullback.condition_assoc, IsPullback.isoPullback_hom_snd_assoc,
IsPullback.isoPullback_hom_fst_assoc, hF]
have h : c.π.app l ⁻¹ᵁ D.map flk ⁻... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.AlgebraicGeometry.EllipticCurve.DivisionPolynomial.Degree | {
"line": 427,
"column": 2
} | {
"line": 429,
"column": 44
} | {
"line": 431,
"column": 0
} | [
{
"pp": "R : Type u\ninst✝ : CommRing R\nW : WeierstrassCurve R\nn : ℤ\n⊢ (W.Φ n).coeff (n.natAbs ^ 2) = 1",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"_private.Mathlib.AlgebraicGeometry.EllipticCurve.DivisionPolynomial.Degree.0.WeierstrassCurve.natDegree_coeff_Φ_... | [] | induction n using Int.negInduction with
| nat n => exact (W.natDegree_coeff_Φ_ofNat n).right
| neg ih => rw [Φ_neg, Int.natAbs_neg, ih] | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | Lean.Parser.Tactic.induction |
Mathlib.AlgebraicGeometry.EllipticCurve.DivisionPolynomial.Degree | {
"line": 427,
"column": 2
} | {
"line": 429,
"column": 44
} | {
"line": 431,
"column": 0
} | [
{
"pp": "R : Type u\ninst✝ : CommRing R\nW : WeierstrassCurve R\nn : ℤ\n⊢ (W.Φ n).coeff (n.natAbs ^ 2) = 1",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"_private.Mathlib.AlgebraicGeometry.EllipticCurve.DivisionPolynomial.Degree.0.WeierstrassCurve.natDegree_coeff_Φ_... | [] | induction n using Int.negInduction with
| nat n => exact (W.natDegree_coeff_Φ_ofNat n).right
| neg ih => rw [Φ_neg, Int.natAbs_neg, ih] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.AlgebraicGeometry.EllipticCurve.DivisionPolynomial.Degree | {
"line": 427,
"column": 2
} | {
"line": 429,
"column": 44
} | {
"line": 431,
"column": 0
} | [
{
"pp": "R : Type u\ninst✝ : CommRing R\nW : WeierstrassCurve R\nn : ℤ\n⊢ (W.Φ n).coeff (n.natAbs ^ 2) = 1",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"_private.Mathlib.AlgebraicGeometry.EllipticCurve.DivisionPolynomial.Degree.0.WeierstrassCurve.natDegree_coeff_Φ_... | [] | induction n using Int.negInduction with
| nat n => exact (W.natDegree_coeff_Φ_ofNat n).right
| neg ih => rw [Φ_neg, Int.natAbs_neg, ih] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Algebra.Nonarchimedean.Bases | {
"line": 95,
"column": 6
} | {
"line": 96,
"column": 30
} | {
"line": 97,
"column": 2
} | [
{
"pp": "case h.right\nA : Type u_1\nι : Type u_2\ninst✝¹ : Ring A\ninst✝ : Nonempty ι\nB : ι → AddSubgroup A\nhB : RingSubgroupsBasis B\ni : ι\n⊢ ↑(B i) ⊆ (fun x ↦ -x) ⁻¹' ↑(B i)",
"ppTerm": "?h.right",
"assigned": true,
"usedConstants": [
"AddGroupWithOne.toAddGroup",
"Membership.mem",... | [] | intro x x_in
exact (B i).neg_mem x_in | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Algebra.Nonarchimedean.Bases | {
"line": 95,
"column": 6
} | {
"line": 96,
"column": 30
} | {
"line": 97,
"column": 2
} | [
{
"pp": "case h.right\nA : Type u_1\nι : Type u_2\ninst✝¹ : Ring A\ninst✝ : Nonempty ι\nB : ι → AddSubgroup A\nhB : RingSubgroupsBasis B\ni : ι\n⊢ ↑(B i) ⊆ (fun x ↦ -x) ⁻¹' ↑(B i)",
"ppTerm": "?h.right",
"assigned": true,
"usedConstants": [
"AddGroupWithOne.toAddGroup",
"Membership.mem",... | [] | intro x x_in
exact (B i).neg_mem x_in | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Algebra.Nonarchimedean.Bases | {
"line": 172,
"column": 8
} | {
"line": 172,
"column": 24
} | {
"line": 172,
"column": 24
} | [
{
"pp": "case h.right\nA : Type u_1\nι : Type u_2\ninst✝¹ : Ring A\ninst✝ : Nonempty ι\nB : ι → AddSubgroup A\nhB : RingSubgroupsBasis B\na : A\ns : Set A\ni : ι\nhi : {b | b - a ∈ B i} ⊆ s\nb : A\nb_in : b ∈ ↑(B i)\n⊢ (fun y ↦ a + y) b ∈ {b | b - a ∈ B i}",
"ppTerm": "?h.right",
"assigned": true,
"... | [] | simpa using b_in | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.Topology.Algebra.Nonarchimedean.Bases | {
"line": 276,
"column": 6
} | {
"line": 277,
"column": 30
} | {
"line": 278,
"column": 2
} | [
{
"pp": "case h.right\nι : Type u_1\nR : Type u_2\nA : Type u_3\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing A\ninst✝⁴ : Algebra R A\nM : Type u_4\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : TopologicalSpace R\ninst✝ : Nonempty ι\nB : ι → Submodule R M\nhB : SubmodulesBasis B\ni : ι\n⊢ ↑(B i) ⊆ (fun x ↦ ... | [] | intro x x_in
exact (B i).neg_mem x_in | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Algebra.Nonarchimedean.Bases | {
"line": 276,
"column": 6
} | {
"line": 277,
"column": 30
} | {
"line": 278,
"column": 2
} | [
{
"pp": "case h.right\nι : Type u_1\nR : Type u_2\nA : Type u_3\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing A\ninst✝⁴ : Algebra R A\nM : Type u_4\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : TopologicalSpace R\ninst✝ : Nonempty ι\nB : ι → Submodule R M\nhB : SubmodulesBasis B\ni : ι\n⊢ ↑(B i) ⊆ (fun x ↦ ... | [] | intro x x_in
exact (B i).neg_mem x_in | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Valuation.ValuativeRel.Basic | {
"line": 579,
"column": 4
} | {
"line": 582,
"column": 19
} | {
"line": 583,
"column": 2
} | [
{
"pp": "R : Type u_1\ninst✝¹ : Semiring R\ninst✝ : ValuativeRel R\nx x' y y' z : R\na b : ValueGroupWithZero R\n⊢ a ≤ b ∨ b ≤ a",
"ppTerm": "?m.389",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"HMul.hMul",
"congrArg",
"Membership.mem",
"id",
"Subtype",
... | [] | induction a using ValueGroupWithZero.ind
induction b using ValueGroupWithZero.ind
rw [ValueGroupWithZero.mk_le_mk, ValueGroupWithZero.mk_le_mk]
apply vle_total | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Valuation.ValuativeRel.Basic | {
"line": 579,
"column": 4
} | {
"line": 582,
"column": 19
} | {
"line": 583,
"column": 2
} | [
{
"pp": "R : Type u_1\ninst✝¹ : Semiring R\ninst✝ : ValuativeRel R\nx x' y y' z : R\na b : ValueGroupWithZero R\n⊢ a ≤ b ∨ b ≤ a",
"ppTerm": "?m.389",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"HMul.hMul",
"congrArg",
"Membership.mem",
"id",
"Subtype",
... | [] | induction a using ValueGroupWithZero.ind
induction b using ValueGroupWithZero.ind
rw [ValueGroupWithZero.mk_le_mk, ValueGroupWithZero.mk_le_mk]
apply vle_total | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Algebra.Valued.ValuationTopology | {
"line": 234,
"column": 2
} | {
"line": 241,
"column": 92
} | {
"line": 243,
"column": 0
} | [
{
"pp": "R : Type u\ninst✝¹ : Ring R\nΓ₀ : Type v\ninst✝ : LinearOrderedCommGroupWithZero Γ₀\n_i : Valued R Γ₀\nr : (ofClass v).ValueGroup₀\n⊢ IsOpen[_i.toTopologicalSpace] {x | v.restrict x < r}",
"ppTerm": "?m.34",
"assigned": true,
"usedConstants": [
"Filter.instMembership",
"nhds_neB... | [] | rw [isOpen_iff_mem_nhds]
rcases eq_or_ne r 0 with rfl | hr
· simp
intro x hx
rw [mem_nhds]
simp only [ofPred_subset_ofPred]
exact ⟨Units.mk0 _ hr,
fun y hy ↦ (sub_add_cancel y x).symm ▸ (v.restrict.map_add _ x).trans_lt (max_lt hy hx)⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Algebra.Valued.ValuationTopology | {
"line": 234,
"column": 2
} | {
"line": 241,
"column": 92
} | {
"line": 243,
"column": 0
} | [
{
"pp": "R : Type u\ninst✝¹ : Ring R\nΓ₀ : Type v\ninst✝ : LinearOrderedCommGroupWithZero Γ₀\n_i : Valued R Γ₀\nr : (ofClass v).ValueGroup₀\n⊢ IsOpen[_i.toTopologicalSpace] {x | v.restrict x < r}",
"ppTerm": "?m.34",
"assigned": true,
"usedConstants": [
"Filter.instMembership",
"nhds_neB... | [] | rw [isOpen_iff_mem_nhds]
rcases eq_or_ne r 0 with rfl | hr
· simp
intro x hx
rw [mem_nhds]
simp only [ofPred_subset_ofPred]
exact ⟨Units.mk0 _ hr,
fun y hy ↦ (sub_add_cancel y x).symm ▸ (v.restrict.map_add _ x).trans_lt (max_lt hy hx)⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.AlgebraicGeometry.EllipticCurve.Jacobian.Formula | {
"line": 562,
"column": 2
} | {
"line": 563,
"column": 7
} | {
"line": 565,
"column": 0
} | [
{
"pp": "R : Type r\ninst✝ : CommRing R\nW' : Jacobian R\nP Q : Fin 3 → R\nu v : R\n⊢ W'.negAddY (u • P) (v • Q) = ((u * v) ^ 2) ^ 3 * W'.negAddY P Q",
"ppTerm": "?m.48",
"assigned": true,
"usedConstants": [
"Mathlib.Tactic.Ring.Common.mul_pf_left",
"Mathlib.Tactic.Ring.Common.neg_zero",... | [] | simp only [negAddY, smul_fin3_ext]
ring1 | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.AlgebraicGeometry.EllipticCurve.Jacobian.Formula | {
"line": 562,
"column": 2
} | {
"line": 563,
"column": 7
} | {
"line": 565,
"column": 0
} | [
{
"pp": "R : Type r\ninst✝ : CommRing R\nW' : Jacobian R\nP Q : Fin 3 → R\nu v : R\n⊢ W'.negAddY (u • P) (v • Q) = ((u * v) ^ 2) ^ 3 * W'.negAddY P Q",
"ppTerm": "?m.48",
"assigned": true,
"usedConstants": [
"Mathlib.Tactic.Ring.Common.mul_pf_left",
"Mathlib.Tactic.Ring.Common.neg_zero",... | [] | simp only [negAddY, smul_fin3_ext]
ring1 | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Valuation.Discrete.Basic | {
"line": 166,
"column": 2
} | {
"line": 167,
"column": 17
} | {
"line": 168,
"column": 2
} | [
{
"pp": "A : Type u_2\ninst✝ : Ring A\nv : Valuation A (WithZero (Multiplicative ℤ))\nhv : v.IsRankOneDiscrete\nhπ : exp (-1) ∈ range ⇑v\n⊢ Subgroup.genLTOne (MonoidWithZeroHom.ofClass v).valueGroup =\n Units.mk0 (exp (-1)) generator_eq_exp_neg_one_of_mem_range._proof_1",
"ppTerm": "?m.46",
"assigned... | [
"A : Type u_2\ninst✝ : Ring A\nv : Valuation A (WithZero (Multiplicative ℤ))\nhv : v.IsRankOneDiscrete\nhπ : exp (-1) ∈ range ⇑v\n⊢ Units.mk0 (exp (-1)) ⋯ = Subgroup.genLTOne (MonoidWithZeroHom.ofClass v).valueGroup"
] | suffices Units.mk0 (exp (-1)) (by simp) = (Subgroup.genLTOne (valueGroup (.ofClass v))) by
simp [← this] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticSuffices__1 | Lean.Parser.Tactic.tacticSuffices_ |
Mathlib.AlgebraicGeometry.EllipticCurve.Jacobian.Formula | {
"line": 731,
"column": 2
} | {
"line": 731,
"column": 51
} | {
"line": 732,
"column": 2
} | [
{
"pp": "R : Type r\nS : Type s\ninst✝¹ : CommRing R\ninst✝ : CommRing S\nW' : Jacobian R\nf : R →+* S\nP : Fin 3 → R\n⊢ (W'.map f).negDblY (⇑f ∘ P) = f (W'.negDblY P)",
"ppTerm": "?m.21",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NegZeroClass.toNeg",
"WeierstrassCurve.Jacobi... | [
"R : Type r\nS : Type s\ninst✝¹ : CommRing R\ninst✝ : CommRing S\nW' : Jacobian R\nf : R →+* S\nP : Fin 3 → R\n⊢ -f (W'.dblU P) * (f (W'.dblX P) - (⇑f ∘ P) x * ((⇑f ∘ P) y - f (W'.negY P)) ^ 2) +\n (⇑f ∘ P) y * ((⇑f ∘ P) y - f (W'.negY P)) ^ 3 =\n f (-W'.dblU P * (W'.dblX P - P x * (P y - W'.negY P) ^ 2) + ... | simp only [negDblY, map_dblU, map_dblX, map_negY] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Topology.Algebra.Valued.WithVal | {
"line": 550,
"column": 97
} | {
"line": 550,
"column": 100
} | {
"line": 551,
"column": 4
} | [
{
"pp": "case h\nR : Type u_4\nΓ₀ : Type u_5\nΓ₀' : Type u_6\ninst✝² : Ring R\ninst✝¹ : LinearOrderedCommGroupWithZero Γ₀\ninst✝ : LinearOrderedCommGroupWithZero Γ₀'\nv : Valuation R Γ₀\nw : Valuation R Γ₀'\nhval : Valued R Γ₀'\nhv : Valued.v = w\nh : v.IsEquiv w\nγ : (ofClass Valued.v).ValueGroup₀ˣ\nr s : R\nh... | [
"case h\nR : Type u_4\nΓ₀ : Type u_5\nΓ₀' : Type u_6\ninst✝² : Ring R\ninst✝¹ : LinearOrderedCommGroupWithZero Γ₀\ninst✝ : LinearOrderedCommGroupWithZero Γ₀'\nv : Valuation R Γ₀\nw : Valuation R Γ₀'\nhval : Valued R Γ₀'\nhv : Valued.v = w\nh : v.IsEquiv w\nγ : (ofClass Valued.v).ValueGroup₀ˣ\nr s : R\nhr₀ : 0 < Val... | hv, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.Algebra.Valued.WithVal | {
"line": 574,
"column": 47
} | {
"line": 574,
"column": 50
} | {
"line": 574,
"column": 51
} | [
{
"pp": "case h\nR : Type u_4\nΓ₀ : Type u_5\nΓ₀' : Type u_6\ninst✝² : Ring R\ninst✝¹ : LinearOrderedCommGroupWithZero Γ₀\ninst✝ : LinearOrderedCommGroupWithZero Γ₀'\nv : Valuation R Γ₀\nw : Valuation R Γ₀'\nhval : Valued R Γ₀'\nhv : Valued.v = w\nh : w.IsEquiv v\nγ : (ofClass Valued.v).ValueGroup₀ˣ\nr s : With... | [
"case h\nR : Type u_4\nΓ₀ : Type u_5\nΓ₀' : Type u_6\ninst✝² : Ring R\ninst✝¹ : LinearOrderedCommGroupWithZero Γ₀\ninst✝ : LinearOrderedCommGroupWithZero Γ₀'\nv : Valuation R Γ₀\nw : Valuation R Γ₀'\nhval : Valued R Γ₀'\nhv : Valued.v = w\nh : w.IsEquiv v\nγ : (ofClass Valued.v).ValueGroup₀ˣ\nr s : WithVal v\nhr₀ :... | hv, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.Algebra.Valued.WithVal | {
"line": 580,
"column": 26
} | {
"line": 580,
"column": 29
} | {
"line": 580,
"column": 30
} | [
{
"pp": "case h\nR : Type u_4\nΓ₀ : Type u_5\nΓ₀' : Type u_6\ninst✝² : Ring R\ninst✝¹ : LinearOrderedCommGroupWithZero Γ₀\ninst✝ : LinearOrderedCommGroupWithZero Γ₀'\nv : Valuation R Γ₀\nw : Valuation R Γ₀'\nhval : Valued R Γ₀'\nhv : Valued.v = w\nh : w.IsEquiv v\nγ : (ofClass Valued.v).ValueGroup₀ˣ\nr s : With... | [
"case h\nR : Type u_4\nΓ₀ : Type u_5\nΓ₀' : Type u_6\ninst✝² : Ring R\ninst✝¹ : LinearOrderedCommGroupWithZero Γ₀\ninst✝ : LinearOrderedCommGroupWithZero Γ₀'\nv : Valuation R Γ₀\nw : Valuation R Γ₀'\nhval : Valued R Γ₀'\nhv : Valued.v = w\nh : w.IsEquiv v\nγ : (ofClass Valued.v).ValueGroup₀ˣ\nr s : WithVal v\nhr₀ :... | hv, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Valuation.Discrete.IsDiscreteValuationRing | {
"line": 90,
"column": 46
} | {
"line": 90,
"column": 61
} | {
"line": 90,
"column": 62
} | [
{
"pp": "A : Type u_1\nK : Type u_2\ninst✝⁵ : CommRing A\ninst✝⁴ : IsDomain A\ninst✝³ : IsDiscreteValuationRing A\ninst✝² : Field K\ninst✝¹ : Algebra A K\ninst✝ : IsFractionRing A K\nx : K\nπ : A\nhπ : Irreducible π\nn : ℕ\nu : Aˣ\nha : ¬↑u * π ^ n = 0\nm : ℕ\nw : Aˣ\nh_frac : (algebraMap A K) (↑u * π ^ n) / (a... | [
"A : Type u_1\nK : Type u_2\ninst✝⁵ : CommRing A\ninst✝⁴ : IsDomain A\ninst✝³ : IsDiscreteValuationRing A\ninst✝² : Field K\ninst✝¹ : Algebra A K\ninst✝ : IsFractionRing A K\nx : K\nπ : A\nhπ : Irreducible π\nn : ℕ\nu : Aˣ\nha : ¬↑u * π ^ n = 0\nm : ℕ\nw : Aˣ\nh_frac : (algebraMap A K) (↑u * π ^ n) / (algebraMap A ... | ← zpow_natCast, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Valuation.Discrete.IsDiscreteValuationRing | {
"line": 90,
"column": 62
} | {
"line": 90,
"column": 77
} | {
"line": 90,
"column": 78
} | [
{
"pp": "A : Type u_1\nK : Type u_2\ninst✝⁵ : CommRing A\ninst✝⁴ : IsDomain A\ninst✝³ : IsDiscreteValuationRing A\ninst✝² : Field K\ninst✝¹ : Algebra A K\ninst✝ : IsFractionRing A K\nx : K\nπ : A\nhπ : Irreducible π\nn : ℕ\nu : Aˣ\nha : ¬↑u * π ^ n = 0\nm : ℕ\nw : Aˣ\nh_frac : (algebraMap A K) (↑u * π ^ n) / (a... | [
"A : Type u_1\nK : Type u_2\ninst✝⁵ : CommRing A\ninst✝⁴ : IsDomain A\ninst✝³ : IsDiscreteValuationRing A\ninst✝² : Field K\ninst✝¹ : Algebra A K\ninst✝ : IsFractionRing A K\nx : K\nπ : A\nhπ : Irreducible π\nn : ℕ\nu : Aˣ\nha : ¬↑u * π ^ n = 0\nm : ℕ\nw : Aˣ\nh_frac : (algebraMap A K) (↑u * π ^ n) / (algebraMap A ... | ← zpow_natCast, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Valuation.Discrete.IsDiscreteValuationRing | {
"line": 69,
"column": 2
} | {
"line": 104,
"column": 20
} | {
"line": 106,
"column": 0
} | [
{
"pp": "A : Type u_1\nK : Type u_2\ninst✝⁵ : CommRing A\ninst✝⁴ : IsDomain A\ninst✝³ : IsDiscreteValuationRing A\ninst✝² : Field K\ninst✝¹ : Algebra A K\ninst✝ : IsFractionRing A K\nx : K\nH : (valuation K (maximalIdeal A)) x ≤ 1\n⊢ ∃ a, (algebraMap A K) a = x",
"ppTerm": "?m.29",
"assigned": true,
... | [] | obtain ⟨π, hπ⟩ := exists_irreducible A
obtain ⟨a, b, hb, h_frac⟩ := IsFractionRing.div_surjective A x
by_cases ha : a = 0
· rw [← h_frac]
use 0
rw [ha, map_zero, zero_div]
· rw [← h_frac] at H
obtain ⟨n, u, rfl⟩ := eq_unit_mul_pow_irreducible ha hπ
obtain ⟨m, w, rfl⟩ := eq_unit_mul_pow_irreducib... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Valuation.Discrete.IsDiscreteValuationRing | {
"line": 69,
"column": 2
} | {
"line": 104,
"column": 20
} | {
"line": 106,
"column": 0
} | [
{
"pp": "A : Type u_1\nK : Type u_2\ninst✝⁵ : CommRing A\ninst✝⁴ : IsDomain A\ninst✝³ : IsDiscreteValuationRing A\ninst✝² : Field K\ninst✝¹ : Algebra A K\ninst✝ : IsFractionRing A K\nx : K\nH : (valuation K (maximalIdeal A)) x ≤ 1\n⊢ ∃ a, (algebraMap A K) a = x",
"ppTerm": "?m.29",
"assigned": true,
... | [] | obtain ⟨π, hπ⟩ := exists_irreducible A
obtain ⟨a, b, hb, h_frac⟩ := IsFractionRing.div_surjective A x
by_cases ha : a = 0
· rw [← h_frac]
use 0
rw [ha, map_zero, zero_div]
· rw [← h_frac] at H
obtain ⟨n, u, rfl⟩ := eq_unit_mul_pow_irreducible ha hπ
obtain ⟨m, w, rfl⟩ := eq_unit_mul_pow_irreducib... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Algebra.Valued.ValuedField | {
"line": 149,
"column": 52
} | {
"line": 149,
"column": 82
} | {
"line": 149,
"column": 82
} | [
{
"pp": "K : Type u_1\ninst✝¹ : DivisionRing K\nΓ₀ : Type u_2\ninst✝ : LinearOrderedCommGroupWithZero Γ₀\nhv : Valued K Γ₀\nhsurj : Function.Surjective ⇑v\nγ : Γ₀\nhγ : γ ≠ 0\nx : K\nhx : v x = γ\n⊢ (restrict₀ (ofClass v)) x ≠ 0",
"ppTerm": "?m.122",
"assigned": true,
"usedConstants": [
"Group... | [] | simp [restrict₀_apply, hx, hγ] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Topology.Algebra.Valued.ValuedField | {
"line": 149,
"column": 52
} | {
"line": 149,
"column": 82
} | {
"line": 149,
"column": 82
} | [
{
"pp": "K : Type u_1\ninst✝¹ : DivisionRing K\nΓ₀ : Type u_2\ninst✝ : LinearOrderedCommGroupWithZero Γ₀\nhv : Valued K Γ₀\nhsurj : Function.Surjective ⇑v\nγ : Γ₀\nhγ : γ ≠ 0\nx : K\nhx : v x = γ\n⊢ (restrict₀ (ofClass v)) x ≠ 0",
"ppTerm": "?m.122",
"assigned": true,
"usedConstants": [
"Group... | [] | simp [restrict₀_apply, hx, hγ] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Algebra.Valued.ValuedField | {
"line": 149,
"column": 52
} | {
"line": 149,
"column": 82
} | {
"line": 149,
"column": 82
} | [
{
"pp": "K : Type u_1\ninst✝¹ : DivisionRing K\nΓ₀ : Type u_2\ninst✝ : LinearOrderedCommGroupWithZero Γ₀\nhv : Valued K Γ₀\nhsurj : Function.Surjective ⇑v\nγ : Γ₀\nhγ : γ ≠ 0\nx : K\nhx : v x = γ\n⊢ (restrict₀ (ofClass v)) x ≠ 0",
"ppTerm": "?m.122",
"assigned": true,
"usedConstants": [
"Group... | [] | simp [restrict₀_apply, hx, hγ] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Algebra.Valued.ValuedField | {
"line": 286,
"column": 46
} | {
"line": 286,
"column": 48
} | {
"line": 287,
"column": 4
} | [
{
"pp": "K : Type u_1\ninst✝¹ : Field K\nΓ₀ : Type u_2\ninst✝ : LinearOrderedCommGroupWithZero Γ₀\nhv : Valued K Γ₀\nx₀ : hat K\nh : x₀ ≠ 0\npreimage_one : ⇑v ⁻¹' {1} ∈ 𝓝 1\nV : Set (hat K)\nV_in : V ∈ 𝓝 1\nhV : ∀ (x : K), ↑x ∈ V → v x = 1\nV' : Set (hat K)\nV'_in : V' ∈ 𝓝 1\nzeroV' : 0 ∉ V'\nhV' : ∀ x ∈ V',... | [
"K : Type u_1\ninst✝¹ : Field K\nΓ₀ : Type u_2\ninst✝ : LinearOrderedCommGroupWithZero Γ₀\nhv : Valued K Γ₀\nx₀ : hat K\nh : x₀ ≠ 0\npreimage_one : ⇑v ⁻¹' {1} ∈ 𝓝 1\nV : Set (hat K)\nV_in : V ∈ 𝓝 1\nhV : ∀ (x : K), ↑x ∈ V → v x = 1\nV' : Set (hat K)\nV'_in : V' ∈ 𝓝 1\nzeroV' : 0 ∉ V'\nhV' : ∀ x ∈ V', ∀ y ∈ V', x... | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.GroupTheory.ArchimedeanDensely | {
"line": 467,
"column": 12
} | {
"line": 467,
"column": 14
} | {
"line": 467,
"column": 15
} | [
{
"pp": "case refine_2\nG₀ : Type u_2\ninst✝¹ : LinearOrderedCommGroupWithZero G₀\ninst✝ : Nontrivial G₀ˣ\ng : G₀\nhg : g ≠ 0\nhg' : g⁻¹ ≠ 0\nh : ({x | x ≤ g} \\ {0}).WellFoundedOn fun x1 x2 ↦ x1 > x2\na : G₀\n⊢ g⁻¹ ≤ a → ∀ (b : G₀), g⁻¹ ≤ b → a < b → b⁻¹ < a⁻¹",
"ppTerm": "?refine_2",
"assigned": true,... | [
"case refine_2\nG₀ : Type u_2\ninst✝¹ : LinearOrderedCommGroupWithZero G₀\ninst✝ : Nontrivial G₀ˣ\ng : G₀\nhg : g ≠ 0\nhg' : g⁻¹ ≠ 0\nh : ({x | x ≤ g} \\ {0}).WellFoundedOn fun x1 x2 ↦ x1 > x2\na : G₀\nha : g⁻¹ ≤ a\n⊢ ∀ (b : G₀), g⁻¹ ≤ b → a < b → b⁻¹ < a⁻¹"
] | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.RingTheory.PowerSeries.Basic | {
"line": 211,
"column": 2
} | {
"line": 213,
"column": 45
} | {
"line": 215,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝ : Semiring R\n⊢ Subsingleton R⟦X⟧ ↔ Subsingleton R",
"ppTerm": "?m.1",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"subsingleton_iff",
"congrArg",
"RingHom",
"inferInstance",
"PowerSeries.C_injective",
"Eq.mp",
"id",
... | [] | refine ⟨fun h ↦ ?_, fun _ ↦ inferInstance⟩
rw [subsingleton_iff] at h ⊢
exact fun a b ↦ C_injective (h (C a) (C b)) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.PowerSeries.Basic | {
"line": 211,
"column": 2
} | {
"line": 213,
"column": 45
} | {
"line": 215,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝ : Semiring R\n⊢ Subsingleton R⟦X⟧ ↔ Subsingleton R",
"ppTerm": "?m.1",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"subsingleton_iff",
"congrArg",
"RingHom",
"inferInstance",
"PowerSeries.C_injective",
"Eq.mp",
"id",
... | [] | refine ⟨fun h ↦ ?_, fun _ ↦ inferInstance⟩
rw [subsingleton_iff] at h ⊢
exact fun a b ↦ C_injective (h (C a) (C b)) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.MvPowerSeries.Basic | {
"line": 289,
"column": 2
} | {
"line": 291,
"column": 94
} | {
"line": 293,
"column": 0
} | [
{
"pp": "σ : Type u_1\nR : Type u_2\ninst✝ : Semiring R\nφ₁ φ₂ φ₃ : MvPowerSeries σ R\nn : σ →₀ ℕ\n⊢ (coeff n) (φ₁ * φ₂ * φ₃) = (coeff n) (φ₁ * (φ₂ * φ₃))",
"ppTerm": "?m.27",
"assigned": true,
"usedConstants": [
"Finsupp.instHasAntidiagonal",
"Eq.mpr",
"Finset.mul_sum",
"Nat... | [] | simp only [coeff_mul, Finset.sum_mul, Finset.mul_sum, Finset.sum_sigma']
apply Finset.sum_nbij' (fun ⟨⟨_i, j⟩, ⟨k, l⟩⟩ ↦ ⟨(k, l + j), (l, j)⟩)
(fun ⟨⟨i, _j⟩, ⟨k, l⟩⟩ ↦ ⟨(i + k, l), (i, k)⟩) <;> aesop (add simp [add_assoc, mul_assoc]) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.MvPowerSeries.Basic | {
"line": 289,
"column": 2
} | {
"line": 291,
"column": 94
} | {
"line": 293,
"column": 0
} | [
{
"pp": "σ : Type u_1\nR : Type u_2\ninst✝ : Semiring R\nφ₁ φ₂ φ₃ : MvPowerSeries σ R\nn : σ →₀ ℕ\n⊢ (coeff n) (φ₁ * φ₂ * φ₃) = (coeff n) (φ₁ * (φ₂ * φ₃))",
"ppTerm": "?m.27",
"assigned": true,
"usedConstants": [
"Finsupp.instHasAntidiagonal",
"Eq.mpr",
"Finset.mul_sum",
"Nat... | [] | simp only [coeff_mul, Finset.sum_mul, Finset.mul_sum, Finset.sum_sigma']
apply Finset.sum_nbij' (fun ⟨⟨_i, j⟩, ⟨k, l⟩⟩ ↦ ⟨(k, l + j), (l, j)⟩)
(fun ⟨⟨i, _j⟩, ⟨k, l⟩⟩ ↦ ⟨(i + k, l), (i, k)⟩) <;> aesop (add simp [add_assoc, mul_assoc]) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.PowerSeries.Basic | {
"line": 593,
"column": 15
} | {
"line": 593,
"column": 34
} | {
"line": 593,
"column": 35
} | [
{
"pp": "R : Type u_1\ninst✝ : CommSemiring R\nf : R⟦X⟧\na b : R\nn : ℕ\n⊢ b ^ n * (a ^ n * (coeff n) f) = a ^ n * b ^ n * (coeff n) f",
"ppTerm": "?m.29",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Semiring.toModule",
"HMul.hMul",
"Monoid.toMulOneClass",
"CommSemi... | [
"R : Type u_1\ninst✝ : CommSemiring R\nf : R⟦X⟧\na b : R\nn : ℕ\n⊢ b ^ n * (a ^ n * (coeff n) f) = b ^ n * a ^ n * (coeff n) f"
] | mul_comm _ (b ^ n), | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.DedekindDomain.AdicValuation | {
"line": 515,
"column": 2
} | {
"line": 517,
"column": 72
} | {
"line": 518,
"column": 2
} | [
{
"pp": "R : Type u_1\ninst✝⁴ : CommRing R\ninst✝³ : IsDedekindDomain R\nK : Type u_2\ninst✝² : Field K\ninst✝¹ : Algebra R K\ninst✝ : IsFractionRing R K\nv : HeightOneSpectrum R\nx : K\n⊢ ∃ n d, x * (algebraMap R K) ↑d = (algebraMap R K) n ∨ x * (algebraMap R K) n = (algebraMap R K) ↑d",
"ppTerm": "?m.42",... | [
"case inl\nR : Type u_1\ninst✝⁴ : CommRing R\ninst✝³ : IsDedekindDomain R\nK : Type u_2\ninst✝² : Field K\ninst✝¹ : Algebra R K\ninst✝ : IsFractionRing R K\nv : HeightOneSpectrum R\nx : K\nr : ↥(valuationSubringAtPrime K v)\nhr : (algebraMap (↥(valuationSubringAtPrime K v)) K) r = x\nn : R\nd : ↥v.asIdeal.primeComp... | obtain (⟨r, hr⟩ | ⟨r, hr⟩) :=
ValuationRing.isInteger_or_isInteger (valuationSubringAtPrime K v) x
<;> obtain ⟨⟨n, d⟩, hnd⟩ := IsLocalization.surj v.asIdeal.primeCompl r | Lean.Parser.Tactic.«_aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tactic_<;>__1» | Lean.Parser.Tactic.«tactic_<;>_» |
Mathlib.Topology.Algebra.Valued.ValuedField | {
"line": 581,
"column": 10
} | {
"line": 587,
"column": 55
} | {
"line": 588,
"column": 4
} | [
{
"pp": "case neg.inr\nK : Type u_1\ninst✝¹ : Field K\nΓ₀ : Type u_2\ninst✝ : LinearOrderedCommGroupWithZero Γ₀\nhv : Valued K Γ₀\nh : Function.Surjective ⇑v\nγ : Γ₀\nH : ¬∃ x, v x = γ\nhγ : γ ≠ 0\n⊢ IsClosed {a | ¬v a = γ}",
"ppTerm": "?neg.inr✝",
"assigned": true,
"usedConstants": [
"isClose... | [] | obtain ⟨r, hr⟩ := h γ
have hr' : restrict₀ (.ofClass (valuedCompletion (K := K)).v) r ≠ 0 := by
rw [ne_eq, ← embedding_inj, embedding_restrict₀ r]
simpa [hr]
convert! isClosed_univ.sdiff (isOpen_sphere (hat K) hr') using 1
ext x
simp [← hr, ← v.restrict_de... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Algebra.Valued.ValuedField | {
"line": 581,
"column": 10
} | {
"line": 587,
"column": 55
} | {
"line": 588,
"column": 4
} | [
{
"pp": "case neg.inr\nK : Type u_1\ninst✝¹ : Field K\nΓ₀ : Type u_2\ninst✝ : LinearOrderedCommGroupWithZero Γ₀\nhv : Valued K Γ₀\nh : Function.Surjective ⇑v\nγ : Γ₀\nH : ¬∃ x, v x = γ\nhγ : γ ≠ 0\n⊢ IsClosed {a | ¬v a = γ}",
"ppTerm": "?neg.inr✝",
"assigned": true,
"usedConstants": [
"isClose... | [] | obtain ⟨r, hr⟩ := h γ
have hr' : restrict₀ (.ofClass (valuedCompletion (K := K)).v) r ≠ 0 := by
rw [ne_eq, ← embedding_inj, embedding_restrict₀ r]
simpa [hr]
convert! isClosed_univ.sdiff (isOpen_sphere (hat K) hr') using 1
ext x
simp [← hr, ← v.restrict_de... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.Finset.Finsupp | {
"line": 64,
"column": 40
} | {
"line": 75,
"column": 26
} | {
"line": 77,
"column": 0
} | [
{
"pp": "ι : Type u_1\nα : Type u_2\ninst✝ : Zero α\ns : Finset ι\nf : ι →₀ α\nt : ι →₀ Finset α\nht : t.support ⊆ s\n⊢ f ∈ s.finsupp ⇑t ↔ ∀ (i : ι), f i ∈ t i",
"ppTerm": "?m.21",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"Finsupp.mem_support_iff",
"Finsupp.instFunLike",
... | [] | by
refine
mem_finsupp_iff.trans
(forall_and.symm.trans <|
forall_congr' fun i =>
⟨fun h => ?_, fun h =>
⟨fun hi => ht <| mem_support_iff.2 fun H => mem_support_iff.1 hi ?_, fun _ => h⟩⟩)
· by_cases hi : i ∈ s
· exact h.2 hi
· rw [notMem_support_iff.1 (mt h.1 hi), notM... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.Instances.ENat | {
"line": 69,
"column": 54
} | {
"line": 69,
"column": 56
} | {
"line": 69,
"column": 57
} | [
{
"pp": "n a : ℕ\n⊢ n + 1 ≤ a → ↑n < ↑a",
"ppTerm": "?m.38",
"assigned": true,
"usedConstants": [
"Preorder.toLE",
"instOfNatNat",
"LE.le",
"instHAdd",
"HAdd.hAdd",
"Nat.instPreorder",
"Nat",
"instAddNat",
"OfNat.ofNat"
],
"usedFVars": [
... | [
"n a : ℕ\nha : n + 1 ≤ a\n⊢ ↑n < ↑a"
] | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.RingTheory.MvPowerSeries.Order | {
"line": 524,
"column": 41
} | {
"line": 526,
"column": 55
} | {
"line": 528,
"column": 0
} | [
{
"pp": "σ : Type u_1\nR : Type u_3\ninst✝ : Ring R\nf g : MvPowerSeries σ R\nd : σ →₀ ℕ\nh : ↑(degree d) < g.order\n⊢ (coeff d) ((1 - g) * f) = (coeff d) f",
"ppTerm": "?m.32",
"assigned": true,
"usedConstants": [
"Finsupp.instAddZeroClass",
"NonAssocSemiring.toAddCommMonoidWithOne",
... | [] | by
rw [degree_eq_weight_one] at h
exact coeff_mul_right_one_sub_of_lt_weightedOrder _ h | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.DedekindDomain.AdicValuation | {
"line": 954,
"column": 61
} | {
"line": 956,
"column": 28
} | {
"line": 958,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝⁴ : CommRing R\ninst✝³ : IsDedekindDomain R\nK : Type u_2\ninst✝² : Field K\ninst✝¹ : Algebra R K\ninst✝ : IsFractionRing R K\nv : HeightOneSpectrum R\nr : R\n⊢ ↑r ∈ adicCompletionIntegers K v",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"Algebra.cast",... | [] | by
rw [mem_adicCompletionIntegers, valuedAdicCompletion_eq_valuation]
exact valuation_le_one v r | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.PowerSeries.Trunc | {
"line": 228,
"column": 4
} | {
"line": 228,
"column": 27
} | {
"line": 228,
"column": 28
} | [
{
"pp": "R : Type u_2\ninst✝ : CommSemiring R\nn a b : ℕ\nf g : R⟦X⟧\nha : n < a\nhb : n < b\n⊢ (coeff n) ↑((trunc n.succ) (↑((trunc n.succ) f) * ↑((trunc n.succ) ↑((trunc b) g)))) = (coeff n) (f * g)",
"ppTerm": "?m.52",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Semiring.toModule"... | [
"R : Type u_2\ninst✝ : CommSemiring R\nn a b : ℕ\nf g : R⟦X⟧\nha : n < a\nhb : n < b\n⊢ (coeff n) ↑((trunc n.succ) (↑((trunc n.succ) f) * ↑((trunc n.succ) g))) = (coeff n) (f * g)"
] | trunc_trunc_of_le g hb, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.SetTheory.Cardinal.Continuum | {
"line": 182,
"column": 2
} | {
"line": 183,
"column": 33
} | {
"line": 184,
"column": 2
} | [
{
"pp": "case a\nx : Cardinal.{u_1}\nh₁ : 2 ≤ x\nh₂ : x ≤ 𝔠\n⊢ x ^ ℵ₀ ≤ 𝔠",
"ppTerm": "?a✝",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Cardinal.power_le_power_right",
"Cardinal.instPowCardinal",
"Cardinal",
"congrArg",
"PartialOrder.toPreorder",
"Pre... | [
"case a\nx : Cardinal.{u_1}\nh₁ : 2 ≤ x\nh₂ : x ≤ 𝔠\n⊢ 𝔠 ≤ x ^ ℵ₀"
] | · rw [← continuum_power_aleph0]
exact power_le_power_right h₂ | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.RingTheory.MvPowerSeries.Substitution | {
"line": 225,
"column": 2
} | {
"line": 225,
"column": 78
} | {
"line": 225,
"column": 79
} | [
{
"pp": "σ : Type u_1\nR : Type u_3\ninst✝⁶ : CommRing R\nτ : Type u_4\nS : Type u_5\ninst✝⁵ : CommRing S\ninst✝⁴ : Algebra R S\na : σ → MvPowerSeries τ S\ninst✝³ : UniformSpace R\ninst✝² : DiscreteUniformity R\ninst✝¹ : UniformSpace S\ninst✝ : DiscreteUniformity S\nha : HasSubst a\n⊢ ⇑(aeval ⋯) = eval₂ (algebr... | [
"case e'_4\nσ : Type u_1\nR : Type u_3\ninst✝⁶ : CommRing R\nτ : Type u_4\nS : Type u_5\ninst✝⁵ : CommRing S\ninst✝⁴ : Algebra R S\na : σ → MvPowerSeries τ S\ninst✝³ : UniformSpace R\ninst✝² : DiscreteUniformity R\ninst✝¹ : UniformSpace S\ninst✝ : DiscreteUniformity S\nha : HasSubst a\n⊢ ⊥ = inst✝³",
"case e'_7\n... | convert! coe_aeval (R := R) (hasSubst_iff_hasEval_of_discreteTopology.mp ha) | Mathlib.Tactic._aux_Mathlib_Tactic_Convert___macroRules_Mathlib_Tactic_convert!_1 | Mathlib.Tactic.convert! |
Mathlib.NumberTheory.ArithmeticFunction.LFunction | {
"line": 105,
"column": 59
} | {
"line": 105,
"column": 79
} | {
"line": 105,
"column": 79
} | [
{
"pp": "case pos\nR : Type u_1\ninst✝ : CommSemiring R\nq : ℕ\nf g : PowerSeries R\nhq : 1 < q\nk : ℕ\nhs :\n Finset.map ({ toFun := fun k ↦ q ^ k, inj' := ⋯ }.prodMap { toFun := fun k ↦ q ^ k, inj' := ⋯ })\n (Finset.antidiagonal k) ⊆\n (q ^ k).divisorsAntidiagonal\ni j : ℕ\nhab : q ^ (i + j) = q ^ k ... | [
"case pos\nR : Type u_1\ninst✝ : CommSemiring R\nq : ℕ\nf g : PowerSeries R\nhq : 1 < q\nk : ℕ\nhs :\n Finset.map ({ toFun := fun k ↦ q ^ k, inj' := ⋯ }.prodMap { toFun := fun k ↦ q ^ k, inj' := ⋯ })\n (Finset.antidiagonal k) ⊆\n (q ^ k).divisorsAntidiagonal\ni j : ℕ\nhab : i + j = k ∧ q ^ k ≠ 0\nh :\n (q... | Nat.pow_right_inj hq | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.ArithmeticFunction.LFunction | {
"line": 301,
"column": 2
} | {
"line": 301,
"column": 12
} | {
"line": 302,
"column": 2
} | [
{
"pp": "ι : Type u_1\nR : Type u_2\ninst✝ : CommSemiring R\nf : ι → ArithmeticFunction R\nthis✝ : UniformSpace R := ⊥\nthis : IsUniformInducing DFunLike.coe\nn : ℕ\ns : Set ι\nhs : s ∈ cofinite\nhs' : ∀ y ∈ s, ∀ i ∈ Set.Iic n, (f y) i = 1 i\nt : Finset ι := ⋯.toFinset\nu v : Finset ι\nhu : ∀ i ∈ u \\ t, i ∈ s\... | [
"ι : Type u_1\nR : Type u_2\ninst✝ : CommSemiring R\nf : ι → ArithmeticFunction R\nthis✝ : UniformSpace R := ⊥\nthis : IsUniformInducing DFunLike.coe\nn : ℕ\ns : Set ι\nhs : s ∈ cofinite\nhs' : ∀ y ∈ s, ∀ i ∈ Set.Iic n, (f y) i = 1 i\nt : Finset ι := ⋯.toFinset\nu v : Finset ι\nhu : ∀ i ∈ u \\ t, i ∈ s\nhv : ∀ i ∈ ... | intro w hw | Lean.Elab.Tactic.evalIntro | Lean.Parser.Tactic.intro |
Mathlib.Analysis.Calculus.TangentCone.Basic | {
"line": 50,
"column": 95
} | {
"line": 54,
"column": 16
} | {
"line": 56,
"column": 0
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝² : AddCommGroup E\ninst✝¹ : SMul 𝕜 E\ninst✝ : TopologicalSpace E\ns : Set E\nx : E\nι : Sort u_3\np : ι → Prop\nU : ι → Set E\nh : (𝓝 0).HasBasis p U\n⊢ tangentConeAt 𝕜 s x = ⋂ i, ⋂ (_ : p i), closure[inst✝] (univ • (U i ∩ (fun x_2 ↦ x + x_2) ⁻¹' s))",
"ppTerm"... | [] | by
ext y
simp only [tangentConeAt_def, mem_ofPred_eq, mem_iInter₂, ← map₂_smul, ← map_prod_eq_map₂,
((nhdsWithin_hasBasis h _).top_prod.map _).clusterPt_iff_forall_mem_closure, image_prod,
image2_smul] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Calculus.TangentCone.Real | {
"line": 37,
"column": 49
} | {
"line": 37,
"column": 51
} | {
"line": 38,
"column": 2
} | [
{
"pp": "E : Type u_1\ninst✝³ : AddCommGroup E\ninst✝² : Module ℝ E\ninst✝¹ : TopologicalSpace E\ninst✝ : ContinuousSMul ℝ E\ns : Set E\nx y : E\nh : openSegment ℝ x y ⊆ s\na : ℝ≥0\n⊢ a ∈ Ioo 0 1 → x + id a • (y - x) ∈ s",
"ppTerm": "?m.75",
"assigned": true,
"usedConstants": [
"Lattice.toSemi... | [
"E : Type u_1\ninst✝³ : AddCommGroup E\ninst✝² : Module ℝ E\ninst✝¹ : TopologicalSpace E\ninst✝ : ContinuousSMul ℝ E\ns : Set E\nx y : E\nh : openSegment ℝ x y ⊆ s\na : ℝ≥0\nha : a ∈ Ioo 0 1\n⊢ x + id a • (y - x) ∈ s"
] | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.Analysis.Calculus.Deriv.Basic | {
"line": 355,
"column": 2
} | {
"line": 355,
"column": 58
} | {
"line": 357,
"column": 0
} | [
{
"pp": "𝕜 : Type u\ninst✝³ : NontriviallyNormedField 𝕜\nF : Type v\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace 𝕜 F\nf : 𝕜 → F\nf' : F\nx : 𝕜\ninst✝ : PartialOrder 𝕜\n⊢ HasDerivWithinAt f f' (Iio x) x ↔ HasDerivWithinAt f f' (Iic x) x",
"ppTerm": "?m.21",
"assigned": true,
"usedConsta... | [] | rw [← Iic_sdiff_right, hasDerivWithinAt_sdiff_singleton] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.Calculus.Deriv.Basic | {
"line": 355,
"column": 2
} | {
"line": 355,
"column": 58
} | {
"line": 357,
"column": 0
} | [
{
"pp": "𝕜 : Type u\ninst✝³ : NontriviallyNormedField 𝕜\nF : Type v\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace 𝕜 F\nf : 𝕜 → F\nf' : F\nx : 𝕜\ninst✝ : PartialOrder 𝕜\n⊢ HasDerivWithinAt f f' (Iio x) x ↔ HasDerivWithinAt f f' (Iic x) x",
"ppTerm": "?m.21",
"assigned": true,
"usedConsta... | [] | rw [← Iic_sdiff_right, hasDerivWithinAt_sdiff_singleton] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Calculus.Deriv.Basic | {
"line": 355,
"column": 2
} | {
"line": 355,
"column": 58
} | {
"line": 357,
"column": 0
} | [
{
"pp": "𝕜 : Type u\ninst✝³ : NontriviallyNormedField 𝕜\nF : Type v\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace 𝕜 F\nf : 𝕜 → F\nf' : F\nx : 𝕜\ninst✝ : PartialOrder 𝕜\n⊢ HasDerivWithinAt f f' (Iio x) x ↔ HasDerivWithinAt f f' (Iic x) x",
"ppTerm": "?m.21",
"assigned": true,
"usedConsta... | [] | rw [← Iic_sdiff_right, hasDerivWithinAt_sdiff_singleton] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
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