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.RingTheory.Ideal.KrullsHeightTheorem | {
"line": 244,
"column": 4
} | {
"line": 244,
"column": 15
} | {
"line": 244,
"column": 16
} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsNoetherianRing R\np : Ideal R\ns : Finset R\nhI : p ∈ (span ↑s).minimalPrimes\n⊢ Cardinal.toENat (Submodule.spanRank (span ↑s)) ≤ ↑s.card",
"ppTerm": "?m.26",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Semiring.toModule",
"... | [
"R : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsNoetherianRing R\np : Ideal R\ns : Finset R\nhI : p ∈ (span ↑s).minimalPrimes\n⊢ Submodule.spanRank (span ↑s) ≤ ↑s.card"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Ideal.KrullsHeightTheorem | {
"line": 311,
"column": 10
} | {
"line": 311,
"column": 21
} | {
"line": 311,
"column": 22
} | [
{
"pp": "R : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsNoetherianRing R\np : Ideal R\ninst✝ : p.IsPrime\nn : ℕ∞\nI : Ideal R\nhp : p ∈ I.minimalPrimes\nhI : Submodule.spanRank I ≤ ↑n\n⊢ Cardinal.toENat (Submodule.spanRank I) ≤ n",
"ppTerm": "?m.105",
"assigned": false,
"usedConstants": [],
"used... | [
"R : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsNoetherianRing R\np : Ideal R\ninst✝ : p.IsPrime\nn : ℕ∞\nI : Ideal R\nhp : p ∈ I.minimalPrimes\nhI : Submodule.spanRank I ≤ ↑n\n⊢ Cardinal.toENat (Submodule.spanRank I) ≤ n"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Ideal.KrullsHeightTheorem | {
"line": 326,
"column": 6
} | {
"line": 326,
"column": 60
} | {
"line": 326,
"column": 61
} | [
{
"pp": "case e'_4\nR : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsNoetherianRing R\np : Ideal R\ninst✝ : p.IsPrime\nI : Ideal R\nhI : p ∈ I.minimalPrimes\nhr : Submodule.spanRank I ≤ ↑p.height\nhs : (Submodule.generators I).Finite\n⊢ Cardinal.toENat (Submodule.spanRank I) = ↑(Submodule.spanFinrank I)",
"ppT... | [
"case e'_4\nR : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsNoetherianRing R\np : Ideal R\ninst✝ : p.IsPrime\nI : Ideal R\nhI : p ∈ I.minimalPrimes\nhr : Submodule.spanRank I ≤ ↑p.height\nhs : (Submodule.generators I).Finite\n⊢ Submodule.FG I"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Ideal.KrullsHeightTheorem | {
"line": 341,
"column": 17
} | {
"line": 341,
"column": 54
} | {
"line": 341,
"column": 55
} | [
{
"pp": "R : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsNoetherianRing R\nI p : Ideal R\ninst✝ : p.IsPrime\nhrp : I ≤ p\np' : Ideal (R ⧸ I) := map (algebraMap R (R ⧸ I)) p\nthis : p'.IsPrime\ns : Finset (R ⧸ I)\nhps : p' ∈ (span ↑s).minimalPrimes\nhs : ↑s.card = p'.height\nhsp' : ↑s ⊆ ↑p'\nx : R\nhx : (Quotient.... | [
"R : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsNoetherianRing R\nI p : Ideal R\ninst✝ : p.IsPrime\nhrp : I ≤ p\np' : Ideal (R ⧸ I) := map (algebraMap R (R ⧸ I)) p\nthis : p'.IsPrime\ns : Finset (R ⧸ I)\nhps : p' ∈ (span ↑s).minimalPrimes\nhs : ↑s.card = p'.height\nhsp' : ↑s ⊆ ↑p'\nx : R\nhx : (Quotient.mk I) x ∈ ↑p... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Spectrum.Prime.LTSeries | {
"line": 50,
"column": 71
} | {
"line": 50,
"column": 82
} | {
"line": 50,
"column": 83
} | [
{
"pp": "R : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsNoetherianRing R\ninst✝ : IsLocalRing R\np₁ p₀ : PrimeSpectrum R\nh₀ : p₀ < p₁\nh₁ : p₁ < closedPoint R\nx : R\nhx : x ∈ 𝔪\nhn : x ∉ p₀.asIdeal\ne : ↑(zeroLocus ↑p₀.asIdeal) ≃o PrimeSpectrum (R ⧸ p₀.asIdeal) := p₀.asIdeal.primeSpectrumQuotientOrderIsoZeroL... | [
"R : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsNoetherianRing R\ninst✝ : IsLocalRing R\np₁ p₀ : PrimeSpectrum R\nh₀ : p₀ < p₁\nh₁ : p₁ < closedPoint R\nx : R\nhx : x ∈ 𝔪\nhn : x ∉ p₀.asIdeal\ne : ↑(zeroLocus ↑p₀.asIdeal) ≃o PrimeSpectrum (R ⧸ p₀.asIdeal) := p₀.asIdeal.primeSpectrumQuotientOrderIsoZeroLocus.symm\nq... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Spectrum.Prime.LTSeries | {
"line": 52,
"column": 2
} | {
"line": 52,
"column": 13
} | {
"line": 52,
"column": 14
} | [
{
"pp": "case neg\nR : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsNoetherianRing R\ninst✝ : IsLocalRing R\np₁ p₀ : PrimeSpectrum R\nh₀ : p₀ < p₁\nh₁ : p₁ < closedPoint R\nx : R\nhx : x ∈ 𝔪\nhn : x ∉ p₀.asIdeal\ne : ↑(zeroLocus ↑p₀.asIdeal) ≃o PrimeSpectrum (R ⧸ p₀.asIdeal) := p₀.asIdeal.primeSpectrumQuotientOrd... | [
"case neg\nR : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsNoetherianRing R\ninst✝ : IsLocalRing R\np₁ p₀ : PrimeSpectrum R\nh₀ : p₀ < p₁\nh₁ : p₁ < closedPoint R\nx : R\nhx : x ∈ 𝔪\nhn : x ∉ p₀.asIdeal\ne : ↑(zeroLocus ↑p₀.asIdeal) ≃o PrimeSpectrum (R ⧸ p₀.asIdeal) := p₀.asIdeal.primeSpectrumQuotientOrderIsoZeroLoc... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Spectrum.Prime.LTSeries | {
"line": 104,
"column": 76
} | {
"line": 104,
"column": 90
} | {
"line": 104,
"column": 90
} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsNoetherianRing R\nx : R\nn : ℕ\nhn :\n ∀ (p : LTSeries (PrimeSpectrum R)),\n x ∈ (RelSeries.last p).asIdeal →\n p.length = n →\n ∃ q,\n x ∈ (q.toFun 1).asIdeal ∧\n n = q.length ∧ RelSeries.head p = RelSeries.head q ∧ RelSe... | [] | by simp [← hh] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.KrullDimension.Regular | {
"line": 202,
"column": 4
} | {
"line": 202,
"column": 15
} | {
"line": 202,
"column": 16
} | [
{
"pp": "case nil\nR : Type u_1\ninst✝⁵ : CommRing R\ninst✝⁴ : IsNoetherianRing R\ninst✝³ : IsLocalRing R\nM : Type u_2\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\ninst✝ : Module.Finite R M\nreg : Sequence.IsRegular M []\n⊢ supportDim R (M ⧸ ⊥) + ↑[].length = supportDim R M",
"ppTerm": "?nil",
"assig... | [
"case nil\nR : Type u_1\ninst✝⁵ : CommRing R\ninst✝⁴ : IsNoetherianRing R\ninst✝³ : IsLocalRing R\nM : Type u_2\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\ninst✝ : Module.Finite R M\nreg : Sequence.IsRegular M []\n⊢ supportDim R (M ⧸ ⊥) = supportDim R M"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.KrullDimension.Regular | {
"line": 205,
"column": 6
} | {
"line": 205,
"column": 17
} | {
"line": 205,
"column": 18
} | [
{
"pp": "R : Type u_1\ninst✝⁵ : CommRing R\ninst✝⁴ : IsNoetherianRing R\ninst✝³ : IsLocalRing R\nx : R\nrs' : List R\nih :\n ∀ {M : Type u_2} [inst : AddCommGroup M] [inst_1 : Module R M] [Module.Finite R M],\n Sequence.IsRegular M rs' → supportDim R (M ⧸ ofList rs' • ⊤) + ↑rs'.length = supportDim R M\nM : ... | [
"R : Type u_1\ninst✝⁵ : CommRing R\ninst✝⁴ : IsNoetherianRing R\ninst✝³ : IsLocalRing R\nx : R\nrs' : List R\nih :\n ∀ {M : Type u_2} [inst : AddCommGroup M] [inst_1 : Module R M] [Module.Finite R M],\n Sequence.IsRegular M rs' → supportDim R (M ⧸ ofList rs' • ⊤) + ↑rs'.length = supportDim R M\nM : Type u_2\nin... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.LittleWedderburn | {
"line": 152,
"column": 4
} | {
"line": 152,
"column": 15
} | {
"line": 152,
"column": 16
} | [
{
"pp": "n : ℕ\nIH : ∀ m < n, ∀ (D : Type u_1) [inst : DivisionRing D] [Finite D] (val : Fintype D), card D = m → Subring.center D = ⊤\nD : Type u_1\ninst✝¹ : DivisionRing D\ninst✝ : Finite D\nval✝ : Fintype D\nhn : card D = n\nR : Subring D\nhR : R < ⊤\nx y : D\nhx : x ∈ R\nhy : y ∈ R\nthis : ∀ (g : ↥R), g * ⟨... | [
"n : ℕ\nIH : ∀ m < n, ∀ (D : Type u_1) [inst : DivisionRing D] [Finite D] (val : Fintype D), card D = m → Subring.center D = ⊤\nD : Type u_1\ninst✝¹ : DivisionRing D\ninst✝ : Finite D\nval✝ : Fintype D\nhn : card D = n\nR : Subring D\nhR : R < ⊤\nx y : D\nhx : x ∈ R\nhy : y ∈ R\nthis : ∀ (g : ↥R), g * ⟨y, hy⟩ = ⟨y,... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.LocalIso | {
"line": 78,
"column": 4
} | {
"line": 78,
"column": 15
} | {
"line": 78,
"column": 16
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝⁸ : CommSemiring R\ninst✝⁷ : CommSemiring S\ninst✝⁶ : Algebra R S\nι : Type u_3\nf : ι → S\nh : ⨆ i, PrimeSpectrum.basicOpen (f i) = ⊤\nT : ι → Type u_4\ninst✝⁵ : (i : ι) → CommSemiring (T i)\ninst✝⁴ : (i : ι) → Algebra R (T i)\ninst✝³ : (i : ι) → Algebra S (T i)\ninst✝... | [
"R : Type u_1\nS : Type u_2\ninst✝⁸ : CommSemiring R\ninst✝⁷ : CommSemiring S\ninst✝⁶ : Algebra R S\nι : Type u_3\nf : ι → S\nh : ⨆ i, PrimeSpectrum.basicOpen (f i) = ⊤\nT : ι → Type u_4\ninst✝⁵ : (i : ι) → CommSemiring (T i)\ninst✝⁴ : (i : ι) → Algebra R (T i)\ninst✝³ : (i : ι) → Algebra S (T i)\ninst✝² : ∀ (i : ι... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.LocalProperties.Injective | {
"line": 124,
"column": 2
} | {
"line": 124,
"column": 12
} | {
"line": 125,
"column": 2
} | [
{
"pp": "R : Type u\ninst✝¹² : CommRing R\nM : Type v\ninst✝¹¹ : AddCommGroup M\ninst✝¹⁰ : Module R M\nRₚ : (P : Ideal R) → [P.IsMaximal] → Type u'\ninst✝⁹ : (P : Ideal R) → [inst : P.IsMaximal] → CommRing (Rₚ P)\ninst✝⁸ : ∀ (P : Ideal R) [inst : P.IsMaximal], Small.{v', u'} (Rₚ P)\ninst✝⁷ : (P : Ideal R) → [in... | [
"R : Type u\ninst✝¹² : CommRing R\nM : Type v\ninst✝¹¹ : AddCommGroup M\ninst✝¹⁰ : Module R M\nRₚ : (P : Ideal R) → [P.IsMaximal] → Type u'\ninst✝⁹ : (P : Ideal R) → [inst : P.IsMaximal] → CommRing (Rₚ P)\ninst✝⁸ : ∀ (P : Ideal R) [inst : P.IsMaximal], Small.{v', u'} (Rₚ P)\ninst✝⁷ : (P : Ideal R) → [inst : P.IsMax... | intro P hP | Lean.Elab.Tactic.evalIntro | Lean.Parser.Tactic.intro |
Mathlib.RingTheory.LocalProperties.InjectiveDimension | {
"line": 42,
"column": 4
} | {
"line": 42,
"column": 50
} | {
"line": 43,
"column": 6
} | [
{
"pp": "R : Type u\ninst✝² : CommRing R\ninst✝¹ : Small.{v, u} R\ninst✝ : IsNoetherianRing R\nS : Submonoid R\nX : ModuleCat R\ninj : Module.Injective R ↑X\nx✝ : Small.{v, u} (Localization S) := small_of_surjective Localization.mkHom_surjective\n⊢ Module.Injective (Localization S) ↑((localizedModuleFunctor S).... | [
"R : Type u\ninst✝² : CommRing R\ninst✝¹ : Small.{v, u} R\ninst✝ : IsNoetherianRing R\nS : Submonoid R\nX : ModuleCat R\ninj : Module.Injective R ↑X\nx✝ : Small.{v, u} (Localization S) := small_of_surjective Localization.mkHom_surjective\n⊢ Module.Injective (Localization S) ↑(X.localizedModule S)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.LocalProperties.InjectiveDimension | {
"line": 89,
"column": 15
} | {
"line": 89,
"column": 26
} | {
"line": 89,
"column": 27
} | [
{
"pp": "case coe.coe\nR : Type u\ninst✝² : CommRing R\ninst✝¹ : Small.{v, u} R\ninst✝ : IsNoetherianRing R\nS : Submonoid R\nM : ModuleCat R\naux : ∀ (n : ℕ), injectiveDimension M ≤ ↑n → injectiveDimension (M.localizedModule S) ≤ ↑n\nn : ℕ\n⊢ injectiveDimension M ≤ ↑↑n → injectiveDimension (M.localizedModule S... | [
"case coe.coe\nR : Type u\ninst✝² : CommRing R\ninst✝¹ : Small.{v, u} R\ninst✝ : IsNoetherianRing R\nS : Submonoid R\nM : ModuleCat R\naux : ∀ (n : ℕ), injectiveDimension M ≤ ↑n → injectiveDimension (M.localizedModule S) ≤ ↑n\nn : ℕ\n⊢ injectiveDimension M ≤ ↑n → injectiveDimension (M.localizedModule S) ≤ ↑n"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.LocalProperties.InjectiveDimension | {
"line": 89,
"column": 4
} | {
"line": 89,
"column": 14
} | {
"line": 89,
"column": 15
} | [
{
"pp": "case coe.coe\nR : Type u\ninst✝² : CommRing R\ninst✝¹ : Small.{v, u} R\ninst✝ : IsNoetherianRing R\nS : Submonoid R\nM : ModuleCat R\naux : ∀ (n : ℕ), injectiveDimension M ≤ ↑n → injectiveDimension (M.localizedModule S) ≤ ↑n\nn : ℕ\n⊢ injectiveDimension M ≤ ↑↑n → injectiveDimension (M.localizedModule S... | [] | | coe n => | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | null |
Mathlib.RingTheory.LocalProperties.InjectiveDimension | {
"line": 108,
"column": 8
} | {
"line": 108,
"column": 59
} | {
"line": 108,
"column": 60
} | [
{
"pp": "R : Type u\ninst✝² : CommRing R\ninst✝¹ : Small.{v, u} R\ninst✝ : IsNoetherianRing R\nM : ModuleCat R\nh : ∀ (m : MaximalSpectrum R), Injective (M.localizedModule m.asIdeal.primeCompl)\np : Ideal R\nhp : p.IsMaximal\nthis : Small.{v, u} (Localization.AtPrime p) := small_of_surjective Localization.mkHom... | [
"R : Type u\ninst✝² : CommRing R\ninst✝¹ : Small.{v, u} R\ninst✝ : IsNoetherianRing R\nM : ModuleCat R\nh : ∀ (m : MaximalSpectrum R), Injective (M.localizedModule m.asIdeal.primeCompl)\np : Ideal R\nhp : p.IsMaximal\nthis : Small.{v, u} (Localization.AtPrime p) := small_of_surjective Localization.mkHom_surjective\... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.LocalProperties.InjectiveDimension | {
"line": 156,
"column": 15
} | {
"line": 156,
"column": 26
} | {
"line": 156,
"column": 27
} | [
{
"pp": "case coe.coe\nR : Type u\ninst✝² : CommRing R\ninst✝¹ : Small.{v, u} R\ninst✝ : IsNoetherianRing R\nM : ModuleCat R\naux : ∀ (n : ℕ), injectiveDimension M ≤ ↑n ↔ ⨆ p, injectiveDimension (M.localizedModule p.asIdeal.primeCompl) ≤ ↑n\nn : ℕ\n⊢ injectiveDimension M ≤ ↑↑n ↔ ⨆ p, injectiveDimension (M.local... | [
"case coe.coe\nR : Type u\ninst✝² : CommRing R\ninst✝¹ : Small.{v, u} R\ninst✝ : IsNoetherianRing R\nM : ModuleCat R\naux : ∀ (n : ℕ), injectiveDimension M ≤ ↑n ↔ ⨆ p, injectiveDimension (M.localizedModule p.asIdeal.primeCompl) ≤ ↑n\nn : ℕ\n⊢ injectiveDimension M ≤ ↑n ↔ ∀ (i : PrimeSpectrum R), injectiveDimension (... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.LocalProperties.InjectiveDimension | {
"line": 156,
"column": 4
} | {
"line": 156,
"column": 14
} | {
"line": 156,
"column": 15
} | [
{
"pp": "case coe.coe\nR : Type u\ninst✝² : CommRing R\ninst✝¹ : Small.{v, u} R\ninst✝ : IsNoetherianRing R\nM : ModuleCat R\naux : ∀ (n : ℕ), injectiveDimension M ≤ ↑n ↔ ⨆ p, injectiveDimension (M.localizedModule p.asIdeal.primeCompl) ≤ ↑n\nn : ℕ\n⊢ injectiveDimension M ≤ ↑↑n ↔ ⨆ p, injectiveDimension (M.local... | [] | | coe n => | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | null |
Mathlib.RingTheory.LocalProperties.ProjectiveDimension | {
"line": 39,
"column": 4
} | {
"line": 39,
"column": 50
} | {
"line": 40,
"column": 6
} | [
{
"pp": "R : Type u\ninst✝¹ : CommRing R\ninst✝ : Small.{v, u} R\nS : Submonoid R\nX : ModuleCat R\nproj : Module.Projective R ↑X\nthis : Small.{v, u} (Localization S)\n⊢ Module.Projective (Localization S) ↑((localizedModuleFunctor S).1 X)",
"ppTerm": "?m.48",
"assigned": true,
"usedConstants": [
... | [
"R : Type u\ninst✝¹ : CommRing R\ninst✝ : Small.{v, u} R\nS : Submonoid R\nX : ModuleCat R\nproj : Module.Projective R ↑X\nthis : Small.{v, u} (Localization S)\n⊢ Module.Projective (Localization S) ↑(X.localizedModule S)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.LocalProperties.InjectiveDimension | {
"line": 178,
"column": 15
} | {
"line": 178,
"column": 26
} | {
"line": 178,
"column": 27
} | [
{
"pp": "case coe.coe\nR : Type u\ninst✝² : CommRing R\ninst✝¹ : Small.{v, u} R\ninst✝ : IsNoetherianRing R\nM : ModuleCat R\naux : ∀ (n : ℕ), injectiveDimension M ≤ ↑n ↔ ⨆ m, injectiveDimension (M.localizedModule m.asIdeal.primeCompl) ≤ ↑n\nn : ℕ\n⊢ injectiveDimension M ≤ ↑↑n ↔ ⨆ p, injectiveDimension (M.local... | [
"case coe.coe\nR : Type u\ninst✝² : CommRing R\ninst✝¹ : Small.{v, u} R\ninst✝ : IsNoetherianRing R\nM : ModuleCat R\naux : ∀ (n : ℕ), injectiveDimension M ≤ ↑n ↔ ⨆ m, injectiveDimension (M.localizedModule m.asIdeal.primeCompl) ≤ ↑n\nn : ℕ\n⊢ injectiveDimension M ≤ ↑n ↔\n ∀ (i : MaximalSpectrum R), injectiveDime... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.LocalProperties.InjectiveDimension | {
"line": 178,
"column": 4
} | {
"line": 178,
"column": 14
} | {
"line": 178,
"column": 15
} | [
{
"pp": "case coe.coe\nR : Type u\ninst✝² : CommRing R\ninst✝¹ : Small.{v, u} R\ninst✝ : IsNoetherianRing R\nM : ModuleCat R\naux : ∀ (n : ℕ), injectiveDimension M ≤ ↑n ↔ ⨆ m, injectiveDimension (M.localizedModule m.asIdeal.primeCompl) ≤ ↑n\nn : ℕ\n⊢ injectiveDimension M ≤ ↑↑n ↔ ⨆ p, injectiveDimension (M.local... | [] | | coe n => | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | null |
Mathlib.RingTheory.LocalProperties.ProjectiveDimension | {
"line": 80,
"column": 15
} | {
"line": 80,
"column": 26
} | {
"line": 80,
"column": 27
} | [
{
"pp": "case coe.coe\nR : Type u\ninst✝¹ : CommRing R\ninst✝ : Small.{v, u} R\nS : Submonoid R\nM : ModuleCat R\naux : ∀ (n : ℕ), projectiveDimension M ≤ ↑n → projectiveDimension (M.localizedModule S) ≤ ↑n\nn : ℕ\n⊢ projectiveDimension M ≤ ↑↑n → projectiveDimension (M.localizedModule S) ≤ ↑↑n",
"ppTerm": "... | [
"case coe.coe\nR : Type u\ninst✝¹ : CommRing R\ninst✝ : Small.{v, u} R\nS : Submonoid R\nM : ModuleCat R\naux : ∀ (n : ℕ), projectiveDimension M ≤ ↑n → projectiveDimension (M.localizedModule S) ≤ ↑n\nn : ℕ\n⊢ projectiveDimension M ≤ ↑n → projectiveDimension (M.localizedModule S) ≤ ↑n"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.LocalProperties.ProjectiveDimension | {
"line": 80,
"column": 4
} | {
"line": 80,
"column": 14
} | {
"line": 80,
"column": 15
} | [
{
"pp": "case coe.coe\nR : Type u\ninst✝¹ : CommRing R\ninst✝ : Small.{v, u} R\nS : Submonoid R\nM : ModuleCat R\naux : ∀ (n : ℕ), projectiveDimension M ≤ ↑n → projectiveDimension (M.localizedModule S) ≤ ↑n\nn : ℕ\n⊢ projectiveDimension M ≤ ↑↑n → projectiveDimension (M.localizedModule S) ≤ ↑↑n",
"ppTerm": "... | [] | | coe n => | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | null |
Mathlib.RingTheory.LocalProperties.ProjectiveDimension | {
"line": 100,
"column": 8
} | {
"line": 100,
"column": 49
} | {
"line": 100,
"column": 50
} | [
{
"pp": "R : Type u\ninst✝³ : CommRing R\ninst✝² : Small.{v, u} R\ninst✝¹ : IsNoetherianRing R\nM : ModuleCat R\ninst✝ : Module.Finite R ↑M\nh : ∀ (m : MaximalSpectrum R), Projective (M.localizedModule m.asIdeal.primeCompl)\nthis✝ : Module.FinitePresentation R ↑M\np : Ideal R\nhp : p.IsMaximal\nthis : Small.{v,... | [
"R : Type u\ninst✝³ : CommRing R\ninst✝² : Small.{v, u} R\ninst✝¹ : IsNoetherianRing R\nM : ModuleCat R\ninst✝ : Module.Finite R ↑M\nh : ∀ (m : MaximalSpectrum R), Projective (M.localizedModule m.asIdeal.primeCompl)\nthis✝ : Module.FinitePresentation R ↑M\np : Ideal R\nhp : p.IsMaximal\nthis : Small.{v, u} (Localiz... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.LocalProperties.Semilocal | {
"line": 65,
"column": 37
} | {
"line": 65,
"column": 48
} | {
"line": 65,
"column": 49
} | [
{
"pp": "R : Type u_1\ninst✝¹¹ : CommSemiring R\ninst✝¹⁰ : Finite (MaximalSpectrum R)\nM : Type u_2\ninst✝⁹ : AddCommMonoid M\ninst✝⁸ : Module R M\nRₚ : (P : Ideal R) → [P.IsMaximal] → Type u_3\ninst✝⁷ : (P : Ideal R) → [inst : P.IsMaximal] → CommSemiring (Rₚ P)\ninst✝⁶ : (P : Ideal R) → [inst : P.IsMaximal] → ... | [
"R : Type u_1\ninst✝¹¹ : CommSemiring R\ninst✝¹⁰ : Finite (MaximalSpectrum R)\nM : Type u_2\ninst✝⁹ : AddCommMonoid M\ninst✝⁸ : Module R M\nRₚ : (P : Ideal R) → [P.IsMaximal] → Type u_3\ninst✝⁷ : (P : Ideal R) → [inst : P.IsMaximal] → CommSemiring (Rₚ P)\ninst✝⁶ : (P : Ideal R) → [inst : P.IsMaximal] → Algebra R (R... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.LocalProperties.ProjectiveDimension | {
"line": 147,
"column": 15
} | {
"line": 147,
"column": 26
} | {
"line": 147,
"column": 27
} | [
{
"pp": "case coe.coe\nR : Type u\ninst✝³ : CommRing R\ninst✝² : Small.{v, u} R\ninst✝¹ : IsNoetherianRing R\nM : ModuleCat R\ninst✝ : Module.Finite R ↑M\naux : ∀ (n : ℕ), projectiveDimension M ≤ ↑n ↔ ⨆ p, projectiveDimension (M.localizedModule p.asIdeal.primeCompl) ≤ ↑n\nn : ℕ\n⊢ projectiveDimension M ≤ ↑↑n ↔ ... | [
"case coe.coe\nR : Type u\ninst✝³ : CommRing R\ninst✝² : Small.{v, u} R\ninst✝¹ : IsNoetherianRing R\nM : ModuleCat R\ninst✝ : Module.Finite R ↑M\naux : ∀ (n : ℕ), projectiveDimension M ≤ ↑n ↔ ⨆ p, projectiveDimension (M.localizedModule p.asIdeal.primeCompl) ≤ ↑n\nn : ℕ\n⊢ projectiveDimension M ≤ ↑n ↔\n ∀ (i : P... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.LocalProperties.ProjectiveDimension | {
"line": 147,
"column": 4
} | {
"line": 147,
"column": 14
} | {
"line": 147,
"column": 15
} | [
{
"pp": "case coe.coe\nR : Type u\ninst✝³ : CommRing R\ninst✝² : Small.{v, u} R\ninst✝¹ : IsNoetherianRing R\nM : ModuleCat R\ninst✝ : Module.Finite R ↑M\naux : ∀ (n : ℕ), projectiveDimension M ≤ ↑n ↔ ⨆ p, projectiveDimension (M.localizedModule p.asIdeal.primeCompl) ≤ ↑n\nn : ℕ\n⊢ projectiveDimension M ≤ ↑↑n ↔ ... | [] | | coe n => | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | null |
Mathlib.RingTheory.LocalProperties.ProjectiveDimension | {
"line": 169,
"column": 15
} | {
"line": 169,
"column": 26
} | {
"line": 169,
"column": 27
} | [
{
"pp": "case coe.coe\nR : Type u\ninst✝³ : CommRing R\ninst✝² : Small.{v, u} R\ninst✝¹ : IsNoetherianRing R\nM : ModuleCat R\ninst✝ : Module.Finite R ↑M\naux : ∀ (n : ℕ), projectiveDimension M ≤ ↑n ↔ ⨆ p, projectiveDimension (M.localizedModule p.asIdeal.primeCompl) ≤ ↑n\nn : ℕ\n⊢ projectiveDimension M ≤ ↑↑n ↔ ... | [
"case coe.coe\nR : Type u\ninst✝³ : CommRing R\ninst✝² : Small.{v, u} R\ninst✝¹ : IsNoetherianRing R\nM : ModuleCat R\ninst✝ : Module.Finite R ↑M\naux : ∀ (n : ℕ), projectiveDimension M ≤ ↑n ↔ ⨆ p, projectiveDimension (M.localizedModule p.asIdeal.primeCompl) ≤ ↑n\nn : ℕ\n⊢ projectiveDimension M ≤ ↑n ↔\n ∀ (i : M... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.LocalProperties.ProjectiveDimension | {
"line": 169,
"column": 4
} | {
"line": 169,
"column": 14
} | {
"line": 169,
"column": 15
} | [
{
"pp": "case coe.coe\nR : Type u\ninst✝³ : CommRing R\ninst✝² : Small.{v, u} R\ninst✝¹ : IsNoetherianRing R\nM : ModuleCat R\ninst✝ : Module.Finite R ↑M\naux : ∀ (n : ℕ), projectiveDimension M ≤ ↑n ↔ ⨆ p, projectiveDimension (M.localizedModule p.asIdeal.primeCompl) ≤ ↑n\nn : ℕ\n⊢ projectiveDimension M ≤ ↑↑n ↔ ... | [] | | coe n => | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | null |
Mathlib.RingTheory.LocalRing.NonLocalRing | {
"line": 51,
"column": 45
} | {
"line": 51,
"column": 56
} | {
"line": 51,
"column": 57
} | [
{
"pp": "ι : Type u_1\ninst✝² : Nontrivial ι\nR : ι → Type u_2\ninst✝¹ : (i : ι) → Semiring (R i)\ninst✝ : ∀ (i : ι), Nontrivial (R i)\ni₁ i₂ : ι\nhi : i₁ ≠ i₂\nh : IsUnit fun i ↦ if i = i₁ then 0 else 1\n⊢ IsUnit 0",
"ppTerm": "?m.49",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Mul... | [
"ι : Type u_1\ninst✝² : Nontrivial ι\nR : ι → Type u_2\ninst✝¹ : (i : ι) → Semiring (R i)\ninst✝ : ∀ (i : ι), Nontrivial (R i)\ni₁ i₂ : ι\nhi : i₁ ≠ i₂\nh : IsUnit fun i ↦ if i = i₁ then 0 else 1\n⊢ False"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.LocalRing.NonLocalRing | {
"line": 53,
"column": 45
} | {
"line": 53,
"column": 66
} | {
"line": 53,
"column": 67
} | [
{
"pp": "ι : Type u_1\ninst✝² : Nontrivial ι\nR : ι → Type u_2\ninst✝¹ : (i : ι) → Semiring (R i)\ninst✝ : ∀ (i : ι), Nontrivial (R i)\ni₁ i₂ : ι\nhi : i₁ ≠ i₂\nha : ¬IsUnit fun i ↦ if i = i₁ then 0 else 1\nh : IsUnit fun i ↦ if i = i₁ then 1 else 0\n⊢ IsUnit 0",
"ppTerm": "?m.83",
"assigned": true,
... | [
"ι : Type u_1\ninst✝² : Nontrivial ι\nR : ι → Type u_2\ninst✝¹ : (i : ι) → Semiring (R i)\ninst✝ : ∀ (i : ι), Nontrivial (R i)\ni₁ i₂ : ι\nhi : i₁ ≠ i₂\nha : ¬IsUnit fun i ↦ if i = i₁ then 0 else 1\nh : IsUnit fun i ↦ if i = i₁ then 1 else 0\n⊢ False"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.LocalRing.NonLocalRing | {
"line": 60,
"column": 43
} | {
"line": 60,
"column": 54
} | {
"line": 60,
"column": 55
} | [
{
"pp": "R₁ : Type u_1\nR₂ : Type u_2\ninst✝³ : Semiring R₁\ninst✝² : Semiring R₂\ninst✝¹ : Nontrivial R₁\ninst✝ : Nontrivial R₂\nh : IsUnit (1, 0)\n⊢ IsUnit 0",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"MulOne.toOne",
"False",
"NeZero.one",
"Is... | [
"R₁ : Type u_1\nR₂ : Type u_2\ninst✝³ : Semiring R₁\ninst✝² : Semiring R₂\ninst✝¹ : Nontrivial R₁\ninst✝ : Nontrivial R₂\nh : IsUnit (1, 0)\n⊢ False"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.LocalRing.NonLocalRing | {
"line": 62,
"column": 43
} | {
"line": 62,
"column": 54
} | {
"line": 62,
"column": 55
} | [
{
"pp": "R₁ : Type u_1\nR₂ : Type u_2\ninst✝³ : Semiring R₁\ninst✝² : Semiring R₂\ninst✝¹ : Nontrivial R₁\ninst✝ : Nontrivial R₂\nha : ¬IsUnit (1, 0)\nh : IsUnit (0, 1)\n⊢ IsUnit 0",
"ppTerm": "?m.26",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"MulOne.toOne",
"False",
"N... | [
"R₁ : Type u_1\nR₂ : Type u_2\ninst✝³ : Semiring R₁\ninst✝² : Semiring R₂\ninst✝¹ : Nontrivial R₁\ninst✝ : Nontrivial R₂\nha : ¬IsUnit (1, 0)\nh : IsUnit (0, 1)\n⊢ False"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.LocalRing.Pullback | {
"line": 83,
"column": 13
} | {
"line": 83,
"column": 34
} | {
"line": 83,
"column": 35
} | [
{
"pp": "R : Type u_1\nS : Type u_2\nT : Type u_3\ninst✝² : Ring R\ninst✝¹ : Ring S\ninst✝ : Semiring T\nf : R →+* T\ng : S →+* T\nh : Function.Surjective ⇑g\nr : R\n⊢ ∃ a, (f.pullbackFst g) a = r",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Subring.instSetLike",
... | [
"R : Type u_1\nS : Type u_2\nT : Type u_3\ninst✝² : Ring R\ninst✝¹ : Ring S\ninst✝ : Semiring T\nf : R →+* T\ng : S →+* T\nh : Function.Surjective ⇑g\nr : R\n⊢ ∃ x, f r = g x"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.LocalRing.Pullback | {
"line": 87,
"column": 13
} | {
"line": 87,
"column": 34
} | {
"line": 87,
"column": 35
} | [
{
"pp": "R : Type u_1\nS : Type u_2\nT : Type u_3\ninst✝² : Ring R\ninst✝¹ : Ring S\ninst✝ : Semiring T\nf : R →+* T\ng : S →+* T\nh : Function.Surjective ⇑f\ns : S\n⊢ ∃ a, (f.pullbackSnd g) a = s",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"RingHom.pullbackSnd",
"Eq.mpr",
... | [
"R : Type u_1\nS : Type u_2\nT : Type u_3\ninst✝² : Ring R\ninst✝¹ : Ring S\ninst✝ : Semiring T\nf : R →+* T\ng : S →+* T\nh : Function.Surjective ⇑f\ns : S\n⊢ ∃ a, f a = g s"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.LocalRing.Pullback | {
"line": 96,
"column": 67
} | {
"line": 96,
"column": 78
} | {
"line": 96,
"column": 79
} | [
{
"pp": "R : Type u_1\nS : Type u_2\nT : Type u_3\ninst✝² : Ring R\ninst✝¹ : Ring S\ninst✝ : Semiring T\nf : R →+* T\ng : S →+* T\ns : S\nhs : s ∈ ker g\n⊢ (0, s) ∈ f.pullback g",
"ppTerm": "?m.115",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"RingHom.instRingHomClass",
"Subrin... | [
"R : Type u_1\nS : Type u_2\nT : Type u_3\ninst✝² : Ring R\ninst✝¹ : Ring S\ninst✝ : Semiring T\nf : R →+* T\ng : S →+* T\ns : S\nhs : s ∈ ker g\n⊢ 0 = g s"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Localization.Rat | {
"line": 28,
"column": 8
} | {
"line": 28,
"column": 77
} | {
"line": 28,
"column": 78
} | [
{
"pp": "q : ℚ\n⊢ IsRelPrime q.num ↑⟨↑q.den, ⋯⟩",
"ppTerm": "?m.29",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Nat.Coprime",
"False",
"Int.instIsStrictOrderedRing",
"Rat.num",
"congrArg",
"CommSemiring.toSemiring",
"Int.instLinearOrder",
"I... | [
"q : ℚ\n⊢ q.num.natAbs.Coprime q.den"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Localization.Rat | {
"line": 32,
"column": 2
} | {
"line": 32,
"column": 41
} | {
"line": 32,
"column": 42
} | [
{
"pp": "q : ℚ\n⊢ (↑(IsFractionRing.den ℤ q)).natAbs = q.den",
"ppTerm": "?m.9",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"q : ℚ\n⊢ (↑(IsFractionRing.den ℤ q)).natAbs = q.den"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.LaurentSeries | {
"line": 661,
"column": 4
} | {
"line": 661,
"column": 69
} | {
"line": 661,
"column": 70
} | [
{
"pp": "case h.refine_2\nK : Type u_2\ninst✝ : Field K\nuK : UniformSpace K\nd : ℤ\nS : Set (K × K)\nhS : S ∈ uniformity K\nγ : (WithZero (Multiplicative ℤ))ˣ := Units.mk0 (exp (-(d + 1))) ⋯\nx✝ : K⸨X⸩ × K⸨X⸩\nhP : x✝ ∈ {P | Valued.v (P.2 - P.1) < ↑γ}\n⊢ (x✝.1.coeff d, x✝.2.coeff d) ∈ S",
"ppTerm": "?h.ref... | [
"case h.refine_2\nK : Type u_2\ninst✝ : Field K\nuK : UniformSpace K\nd : ℤ\nS : Set (K × K)\nhS : S ∈ uniformity K\nγ : (WithZero (Multiplicative ℤ))ˣ := Units.mk0 (exp (-(d + 1))) ⋯\nx✝ : K⸨X⸩ × K⸨X⸩\nhP : x✝ ∈ {P | Valued.v (P.2 - P.1) < ↑γ}\n⊢ (x✝.1.coeff d, x✝.1.coeff d) ∈ S"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Localization.AtPrime.Extension | {
"line": 224,
"column": 4
} | {
"line": 224,
"column": 44
} | {
"line": 224,
"column": 45
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝¹⁷ : CommRing R\ninst✝¹⁶ : CommRing S\ninst✝¹⁵ : Algebra R S\np : Ideal R\ninst✝¹⁴ : p.IsPrime\nRₚ : Type u_3\ninst✝¹³ : CommRing Rₚ\ninst✝¹² : Algebra R Rₚ\ninst✝¹¹ : IsLocalization.AtPrime Rₚ p\ninst✝¹⁰ : IsLocalRing Rₚ\nSₚ : Type u_4\ninst✝⁹ : CommRing Sₚ\ninst✝⁸ : A... | [
"R : Type u_1\nS : Type u_2\ninst✝¹⁷ : CommRing R\ninst✝¹⁶ : CommRing S\ninst✝¹⁵ : Algebra R S\np : Ideal R\ninst✝¹⁴ : p.IsPrime\nRₚ : Type u_3\ninst✝¹³ : CommRing Rₚ\ninst✝¹² : Algebra R Rₚ\ninst✝¹¹ : IsLocalization.AtPrime Rₚ p\ninst✝¹⁰ : IsLocalRing Rₚ\nSₚ : Type u_4\ninst✝⁹ : CommRing Sₚ\ninst✝⁸ : Algebra S Sₚ\... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.MvPolynomial.IrreducibleQuadratic | {
"line": 84,
"column": 2
} | {
"line": 84,
"column": 20
} | {
"line": 84,
"column": 21
} | [
{
"pp": "n : Type u_3\nR : Type u_4\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\ni : n\nS : Type (max u_4 u_3) := MvPolynomial { j // j ≠ i } R\ne : MvPolynomial n R ≃ₐ[R] S[X] :=\n (renameEquiv R (Equiv.optionSubtypeNe i).symm).trans (optionEquivLeft R { b // b ≠ i })\nhe : (↑e.symm).comp Polynomial.CAlgHom = re... | [
"n : Type u_3\nR : Type u_4\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\ni : n\nS : Type (max u_4 u_3) := MvPolynomial { j // j ≠ i } R\ne : MvPolynomial n R ≃ₐ[R] S[X] :=\n (renameEquiv R (Equiv.optionSubtypeNe i).symm).trans (optionEquivLeft R { b // b ≠ i })\nhe : (↑e.symm).comp Polynomial.CAlgHom = rename Subtype... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.MvPolynomial.IrreducibleQuadratic | {
"line": 98,
"column": 33
} | {
"line": 98,
"column": 44
} | {
"line": 98,
"column": 45
} | [
{
"pp": "n : Type u_1\nR : Type u_2\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\nf : MvPolynomial n R\nnontrivial : f.support.Nontrivial\nd : n →₀ ℕ\nhd : d ∈ f.support\ni : n\nhdi : d i = 1\ndisjoint : (↑f.support).PairwiseDisjoint Finsupp.support\nisPrimitive : ∀ (r : R), (∀ (d : n →₀ ℕ), r ∣ coeff d f) → IsUnit... | [
"n : Type u_1\nR : Type u_2\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\nf : MvPolynomial n R\nnontrivial : f.support.Nontrivial\nd : n →₀ ℕ\nhd : d ∈ f.support\ni : n\nhdi : d i = 1\ndisjoint : (↑f.support).PairwiseDisjoint Finsupp.support\nisPrimitive : ∀ (r : R), (∀ (d : n →₀ ℕ), r ∣ coeff d f) → IsUnit r\n⊢ ¬coeff... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.MvPolynomial.IrreducibleQuadratic | {
"line": 112,
"column": 6
} | {
"line": 112,
"column": 49
} | {
"line": 112,
"column": 50
} | [
{
"pp": "n : Type u_1\nR : Type u_2\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\nf : MvPolynomial n R\nnontrivial : f.support.Nontrivial\nd : n →₀ ℕ\nhd : d ∈ f.support\ni : n\nhdi : d i = 1\ndisjoint : (↑f.support).PairwiseDisjoint Finsupp.support\nisPrimitive : ∀ (r : R), (∀ (d : n →₀ ℕ), r ∣ coeff d f) → IsUnit... | [
"n : Type u_1\nR : Type u_2\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\nf : MvPolynomial n R\nnontrivial : f.support.Nontrivial\nd : n →₀ ℕ\nhd : d ∈ f.support\ni : n\nhdi : d i = 1\ndisjoint : (↑f.support).PairwiseDisjoint Finsupp.support\nisPrimitive : ∀ (r : R), (∀ (d : n →₀ ℕ), r ∣ coeff d f) → IsUnit r\nhfd : co... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.MvPolynomial.IrreducibleQuadratic | {
"line": 119,
"column": 38
} | {
"line": 119,
"column": 63
} | {
"line": 119,
"column": 64
} | [
{
"pp": "n : Type u_1\nR : Type u_2\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\nf : MvPolynomial n R\nnontrivial : f.support.Nontrivial\nd : n →₀ ℕ\nhd : d ∈ f.support\ni : n\nhdi : d i = 1\ndisjoint : (↑f.support).PairwiseDisjoint Finsupp.support\nisPrimitive : ∀ (r : R), (∀ (d : n →₀ ℕ), r ∣ coeff d f) → IsUnit... | [
"n : Type u_1\nR : Type u_2\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\nf : MvPolynomial n R\nnontrivial : f.support.Nontrivial\nd : n →₀ ℕ\nhd : d ∈ f.support\ni : n\nhdi : d i = 1\ndisjoint : (↑f.support).PairwiseDisjoint Finsupp.support\nisPrimitive : ∀ (r : R), (∀ (d : n →₀ ℕ), r ∣ coeff d f) → IsUnit r\nhfd : co... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.MvPolynomial.IrreducibleQuadratic | {
"line": 121,
"column": 6
} | {
"line": 121,
"column": 17
} | {
"line": 121,
"column": 18
} | [
{
"pp": "n : Type u_1\nR : Type u_2\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\nf : MvPolynomial n R\nnontrivial : f.support.Nontrivial\nd : n →₀ ℕ\nhd : d ∈ f.support\ni : n\nhdi : d i = 1\ndisjoint : (↑f.support).PairwiseDisjoint Finsupp.support\nisPrimitive : ∀ (r : R), (∀ (d : n →₀ ℕ), r ∣ coeff d f) → IsUnit... | [
"n : Type u_1\nR : Type u_2\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\nf : MvPolynomial n R\nnontrivial : f.support.Nontrivial\nd : n →₀ ℕ\nhd : d ∈ f.support\ni : n\nhdi : d i = 1\ndisjoint : (↑f.support).PairwiseDisjoint Finsupp.support\nisPrimitive : ∀ (r : R), (∀ (d : n →₀ ℕ), r ∣ coeff d f) → IsUnit r\nhfd : co... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.MvPolynomial.IrreducibleQuadratic | {
"line": 148,
"column": 6
} | {
"line": 148,
"column": 61
} | {
"line": 148,
"column": 62
} | [
{
"pp": "n : Type u_1\nR : Type u_2\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\np : MvPolynomial n R\nhp : p.totalDegree = 1\nhp' : ∀ (x : R), (∀ (i : n →₀ ℕ), x ∣ coeff i p) → IsUnit x\na b : MvPolynomial n R\nhab : p = a * b\nhle : a.totalDegree ≤ b.totalDegree\nha₀ : a ≠ 0\nhb₀ : b ≠ 0\n⊢ a.totalDegree + b.tot... | [
"n : Type u_1\nR : Type u_2\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\np : MvPolynomial n R\nhp : p.totalDegree = 1\nhp' : ∀ (x : R), (∀ (i : n →₀ ℕ), x ∣ coeff i p) → IsUnit x\na b : MvPolynomial n R\nhab : p = a * b\nhle : a.totalDegree ≤ b.totalDegree\nha₀ : a ≠ 0\nhb₀ : b ≠ 0\n⊢ a.totalDegree + b.totalDegree = 1... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.LaurentSeries | {
"line": 818,
"column": 4
} | {
"line": 818,
"column": 15
} | {
"line": 818,
"column": 16
} | [
{
"pp": "case h\nK : Type u_2\ninst✝ : Field K\nF : K⟦X⟧\nη : (WithZero (Multiplicative ℤ))ˣ\nh_neg : 1 < η\n⊢ (PowerSeries.idealX K).intValuation (F - ↑0) < ↑η",
"ppTerm": "?h",
"assigned": true,
"usedConstants": [
"MvPowerSeries.instAddCommGroup",
"Units.val",
"Eq.mpr",
"In... | [
"case h\nK : Type u_2\ninst✝ : Field K\nF : K⟦X⟧\nη : (WithZero (Multiplicative ℤ))ˣ\nh_neg : 1 < η\n⊢ (PowerSeries.idealX K).intValuation F < ↑η"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.LaurentSeries | {
"line": 826,
"column": 6
} | {
"line": 826,
"column": 80
} | {
"line": 827,
"column": 8
} | [
{
"pp": "K : Type u_2\ninst✝ : Field K\nF : K⟦X⟧\nη : (WithZero (Multiplicative ℤ))ˣ\nh_neg : Multiplicative.toAdd (unzero ⋯) ≤ 0\nd : ℕ\nhd : Multiplicative.toAdd (unzero ⋯) = -↑d\n⊢ ∀ n < d + 1, (PowerSeries.coeff n) (F - ↑((trunc (d + 1)) F)) = 0",
"ppTerm": "?m.178",
"assigned": true,
"usedConst... | [
"K : Type u_2\ninst✝ : Field K\nF : K⟦X⟧\nη : (WithZero (Multiplicative ℤ))ˣ\nh_neg : Multiplicative.toAdd (unzero ⋯) ≤ 0\nd : ℕ\nhd : Multiplicative.toAdd (unzero ⋯) = -↑d\n⊢ ∀ n < d + 1, (PowerSeries.coeff n) F = if n < d + 1 then (PowerSeries.coeff n) F else 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.MvPolynomial.IrreducibleQuadratic | {
"line": 196,
"column": 4
} | {
"line": 196,
"column": 32
} | {
"line": 196,
"column": 33
} | [
{
"pp": "case hp'\nn : Type u_1\nR : Type u_2\ninst✝¹ : CommRing R\nc : n →₀ R\ninst✝ : IsDomain R\nhc_nonempty : c.support.Nonempty\nhc_gcd : ∀ (r : R), (∀ (i : n), r ∣ c i) → IsUnit r\nr : R\nhr : ∀ (i : n →₀ ℕ), r ∣ coeff i (sumSMulX c)\ni : n\n⊢ r ∣ c i",
"ppTerm": "?hp'",
"assigned": false,
"us... | [
"case hp'\nn : Type u_1\nR : Type u_2\ninst✝¹ : CommRing R\nc : n →₀ R\ninst✝ : IsDomain R\nhc_nonempty : c.support.Nonempty\nhc_gcd : ∀ (r : R), (∀ (i : n), r ∣ c i) → IsUnit r\nr : R\nhr : ∀ (i : n →₀ ℕ), r ∣ coeff i (sumSMulX c)\ni : n\n⊢ r ∣ c i"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.LaurentSeries | {
"line": 857,
"column": 10
} | {
"line": 857,
"column": 27
} | {
"line": 857,
"column": 28
} | [
{
"pp": "case h\nK : Type u_2\ninst✝ : Field K\nf : K⸨X⸩\nγ : (WithZero (Multiplicative ℤ))ˣ\nF : K⟦X⟧ := f.powerSeriesPart\nhF : F = f.powerSeriesPart\nord_nonpos : HahnSeries.order f < 0\nη : (WithZero (Multiplicative ℤ))ˣ := Units.mk0 (exp (HahnSeries.order f)) ⋯\nhη : η = Units.mk0 (exp (HahnSeries.order f)... | [
"case h\nK : Type u_2\ninst✝ : Field K\nf : K⸨X⸩\nγ : (WithZero (Multiplicative ℤ))ˣ\nF : K⟦X⟧ := f.powerSeriesPart\nhF : F = f.powerSeriesPart\nord_nonpos : HahnSeries.order f < 0\nη : (WithZero (Multiplicative ℤ))ˣ := Units.mk0 (exp (HahnSeries.order f)) ⋯\nhη : η = Units.mk0 (exp (HahnSeries.order f)) ⋯\nP : K[X... | ← inv_eq_one_div, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.MvPolynomial.IrreducibleQuadratic | {
"line": 235,
"column": 4
} | {
"line": 235,
"column": 24
} | {
"line": 235,
"column": 25
} | [
{
"pp": "case isPrimitive\nn : Type u_1\nR : Type u_2\ninst✝¹ : CommRing R\nc : n →₀ R\ninst✝ : IsDomain R\nhc : c.support.Nontrivial\nh_dvd : ∀ (r : R), (∀ (i : n), r ∣ c i) → IsUnit r\nι : n ↪ n ⊕ n →₀ ℕ := { toFun := fun i ↦ Finsupp.single (Sum.inl i) 1 + Finsupp.single (Sum.inr i) 1, inj' := ⋯ }\naux : sumS... | [
"case isPrimitive\nn : Type u_1\nR : Type u_2\ninst✝¹ : CommRing R\nc : n →₀ R\ninst✝ : IsDomain R\nhc : c.support.Nontrivial\nh_dvd : ∀ (r : R), (∀ (i : n), r ∣ c i) → IsUnit r\nι : n ↪ n ⊕ n →₀ ℕ := { toFun := fun i ↦ Finsupp.single (Sum.inl i) 1 + Finsupp.single (Sum.inr i) 1, inj' := ⋯ }\naux : sumSMulXSMulY c ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.MvPolynomial.Symmetric.NewtonIdentities | {
"line": 243,
"column": 4
} | {
"line": 243,
"column": 15
} | {
"line": 243,
"column": 16
} | [
{
"pp": "σ : Type u_1\ninst✝¹ : Fintype σ\nR : Type u_2\ninst✝ : CommRing R\nk : ℕ := Fintype.card σ\nthis : (-1) ^ (k + 1) * ∑ a ∈ antidiagonal k, (-1) ^ a.1 * esymm σ R a.1 * psum σ R a.2 = 0\n⊢ ∑ a ∈ antidiagonal (Fintype.card σ), (-1) ^ a.1 * esymm σ R a.1 * psum σ R a.2 = 0",
"ppTerm": "?m.103",
"a... | [
"σ : Type u_1\ninst✝¹ : Fintype σ\nR : Type u_2\ninst✝ : CommRing R\nk : ℕ := Fintype.card σ\nthis : (-1) ^ (k + 1) * ∑ a ∈ antidiagonal k, (-1) ^ a.1 * esymm σ R a.1 * psum σ R a.2 = 0\n⊢ ∑ a ∈ antidiagonal (Fintype.card σ), (-1) ^ a.1 * esymm σ R a.1 * psum σ R a.2 = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.MvPowerSeries.Expand | {
"line": 141,
"column": 57
} | {
"line": 148,
"column": 23
} | {
"line": 150,
"column": 0
} | [
{
"pp": "σ : Type u_1\nR : Type u_3\ninst✝ : CommRing R\np : ℕ\nhp : p ≠ 0\nφ : MvPowerSeries σ R\n⊢ Function.support ((expand p hp) φ) ⊆ (fun x ↦ p • x) '' Function.support φ",
"ppTerm": "?m.26",
"assigned": true,
"usedConstants": [
"MvPowerSeries.expand",
"Finsupp.instFunLike",
"... | [] | by
intro d hd
have : ∀ i, p ∣ d i := fun _ => by_contra fun hc => hd (coeff_expand_of_not_dvd p hp φ hc)
let m := d.mapRange (fun n => n / p) (Nat.zero_div p)
have eq_aux : p • m = d := (Finsupp.ext fun a => Nat.eq_mul_of_div_eq_right (this a) rfl).symm
rw [Function.mem_support, ← eq_aux, ← coeff_apply (expan... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.MvPowerSeries.Expand | {
"line": 155,
"column": 2
} | {
"line": 155,
"column": 28
} | {
"line": 156,
"column": 2
} | [
{
"pp": "σ : Type u_1\nR : Type u_3\ninst✝ : CommRing R\np : ℕ\nhp : p ≠ 0\nφ : MvPowerSeries σ R\nd : σ →₀ ℕ\nhd : d ∈ (fun x ↦ p • x) '' Function.support φ\n⊢ d ∈ Function.support ((expand p hp) φ)",
"ppTerm": "?m.38",
"assigned": true,
"usedConstants": [
"MvPowerSeries.expand",
"Nat.i... | [
"σ : Type u_1\nR : Type u_3\ninst✝ : CommRing R\np : ℕ\nhp : p ≠ 0\nφ : MvPowerSeries σ R\nd n : σ →₀ ℕ\nhn₁ : n ∈ Function.support φ\nhn₂ : (fun x ↦ p • x) n = d\n⊢ d ∈ Function.support ((expand p hp) φ)"
] | obtain ⟨n, hn₁, hn₂⟩ := hd | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.RingTheory.MvPowerSeries.Expand | {
"line": 165,
"column": 4
} | {
"line": 165,
"column": 20
} | {
"line": 165,
"column": 21
} | [
{
"pp": "case pos\nσ : Type u_1\nR : Type u_3\ninst✝ : CommRing R\np : ℕ\nhp : p ≠ 0\nφ : MvPowerSeries σ R\nhφ : φ = 0\n⊢ ((expand p hp) φ).order = p • φ.order",
"ppTerm": "?pos✝",
"assigned": true,
"usedConstants": [
"MvPowerSeries.expand",
"Eq.mpr",
"NonAssocSemiring.toAddCommMo... | [
"case pos\nσ : Type u_1\nR : Type u_3\ninst✝ : CommRing R\np : ℕ\nhp : p ≠ 0\nφ : MvPowerSeries σ R\nhφ : φ = 0\n⊢ ⊤ = ↑p * ⊤"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Morita.Matrix | {
"line": 170,
"column": 25
} | {
"line": 170,
"column": 55
} | {
"line": 170,
"column": 56
} | [
{
"pp": "R : Type u\nι : Type v\ninst✝² : Ring R\ninst✝¹ : Fintype ι\ninst✝ : DecidableEq ι\nM : ModuleCat (Matrix ι ι R)\nj : ι\nv : ι → ↥(toModuleCatObj R (↑M) j)\ni : ι\n⊢ ?m.237",
"ppTerm": "?m.242",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"R : Type u\nι : Type v\ninst✝² : Ring R\ninst✝¹ : Fintype ι\ninst✝ : DecidableEq ι\nM : ModuleCat (Matrix ι ι R)\nj : ι\nv : ι → ↥(toModuleCatObj R (↑M) j)\ni : ι\n⊢ ?m.237"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.MvPowerSeries.Restricted | {
"line": 42,
"column": 2
} | {
"line": 42,
"column": 28
} | {
"line": 42,
"column": 29
} | [
{
"pp": "R : Type u_1\ninst✝ : NormedRing R\nσ : Type u_2\nc : σ → ℝ\n⊢ IsRestricted c 0",
"ppTerm": "?m.6",
"assigned": true,
"usedConstants": [
"Norm.norm",
"Eq.mpr",
"Nat.instMulZeroClass",
"Real",
"MvPowerSeries.instZero",
"Semiring.toModule",
"NormedRin... | [
"R : Type u_1\ninst✝ : NormedRing R\nσ : Type u_2\nc : σ → ℝ\n⊢ Tendsto (fun t ↦ 0) cofinite (𝓝 0)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.MvPowerSeries.Restricted | {
"line": 54,
"column": 2
} | {
"line": 54,
"column": 40
} | {
"line": 54,
"column": 41
} | [
{
"pp": "R : Type u_1\ninst✝ : NormedRing R\nσ : Type u_2\nc : σ → ℝ\na : R\n⊢ IsRestricted c (C a)",
"ppTerm": "?m.8",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"R : Type u_1\ninst✝ : NormedRing R\nσ : Type u_2\nc : σ → ℝ\na : R\n⊢ IsRestricted c (C a)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.MvPowerSeries.Expand | {
"line": 201,
"column": 4
} | {
"line": 214,
"column": 20
} | {
"line": 215,
"column": 2
} | [
{
"pp": "case pos\nσ : Type u_1\nR : Type u_3\ninst✝¹ : CommRing R\np : ℕ\nhp : p ≠ 0\ninst✝ : DecidableEq σ\nn : σ →₀ ℕ\nφ : MvPowerSeries σ R\nd : σ →₀ ℕ\nh : ∀ (i : σ), p ∣ d i\n⊢ MvPolynomial.coeff d ((trunc' R (p • n)) ((expand p hp) φ)) =\n MvPolynomial.coeff d ((MvPolynomial.expand p) ((trunc' R n) φ)... | [] | obtain ⟨m, hm⟩ : ∃ m, p • m = d := ⟨d.mapRange (fun a ↦ a / p) (by simp),
by ext i; simp [(Nat.mul_div_cancel' (h i))]⟩
by_cases h_le : m ≤ n
· rw [← hm, coeff_trunc', if_pos (nsmul_le_nsmul_right h_le p), coeff_expand_smul,
MvPolynomial.coeff_expand_smul _ hp, coeff_trunc', if_pos h_le]
· hav... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.MvPowerSeries.Expand | {
"line": 201,
"column": 4
} | {
"line": 214,
"column": 20
} | {
"line": 215,
"column": 2
} | [
{
"pp": "case pos\nσ : Type u_1\nR : Type u_3\ninst✝¹ : CommRing R\np : ℕ\nhp : p ≠ 0\ninst✝ : DecidableEq σ\nn : σ →₀ ℕ\nφ : MvPowerSeries σ R\nd : σ →₀ ℕ\nh : ∀ (i : σ), p ∣ d i\n⊢ MvPolynomial.coeff d ((trunc' R (p • n)) ((expand p hp) φ)) =\n MvPolynomial.coeff d ((MvPolynomial.expand p) ((trunc' R n) φ)... | [] | obtain ⟨m, hm⟩ : ∃ m, p • m = d := ⟨d.mapRange (fun a ↦ a / p) (by simp),
by ext i; simp [(Nat.mul_div_cancel' (h i))]⟩
by_cases h_le : m ≤ n
· rw [← hm, coeff_trunc', if_pos (nsmul_le_nsmul_right h_le p), coeff_expand_smul,
MvPolynomial.coeff_expand_smul _ hp, coeff_trunc', if_pos h_le]
· hav... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.MvPowerSeries.Restricted | {
"line": 67,
"column": 2
} | {
"line": 67,
"column": 28
} | {
"line": 67,
"column": 29
} | [
{
"pp": "R : Type u_1\ninst✝ : NormedRing R\nσ : Type u_2\nc : σ → ℝ\nf : MvPowerSeries σ R\nhf : Tendsto (fun t ↦ ‖(coeff t) f‖ * t.prod fun x1 x2 ↦ |c| x1 ^ x2) cofinite (𝓝 0)\n⊢ Tendsto (fun t ↦ ‖(coeff t) (-f)‖ * t.prod fun x1 x2 ↦ |c| x1 ^ x2) cofinite (𝓝 0)",
"ppTerm": "?m.39",
"assigned": true,... | [
"R : Type u_1\ninst✝ : NormedRing R\nσ : Type u_2\nc : σ → ℝ\nf : MvPowerSeries σ R\nhf : Tendsto (fun t ↦ ‖(coeff t) f‖ * t.prod fun x1 x2 ↦ |c| x1 ^ x2) cofinite (𝓝 0)\n⊢ Tendsto (fun t ↦ ‖(coeff t) f‖ * t.prod fun x1 x2 ↦ |c x1| ^ x2) cofinite (𝓝 0)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Nilpotent.GeometricallyReduced | {
"line": 69,
"column": 4
} | {
"line": 69,
"column": 99
} | {
"line": 71,
"column": 0
} | [
{
"pp": "case refine_2\nk : Type u_3\nA : Type u_4\ninst✝² : Field k\ninst✝¹ : Ring A\ninst✝ : Algebra k A\ne : (p : Ideal k) → [inst : p.IsPrime] → AlgebraicClosure k ≃ₐ[k] AlgebraicClosure p.ResidueField :=\n fun p [p.IsPrime] ↦\n have this := ⋯;\n IsAlgClosure.equiv k (AlgebraicClosure k) (AlgebraicCl... | [] | exact isReduced_of_injective _ (Algebra.TensorProduct.congr (e p).symm AlgEquiv.refl).injective | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.RingTheory.Nilpotent.GeometricallyReduced | {
"line": 69,
"column": 4
} | {
"line": 69,
"column": 99
} | {
"line": 71,
"column": 0
} | [
{
"pp": "case refine_2\nk : Type u_3\nA : Type u_4\ninst✝² : Field k\ninst✝¹ : Ring A\ninst✝ : Algebra k A\ne : (p : Ideal k) → [inst : p.IsPrime] → AlgebraicClosure k ≃ₐ[k] AlgebraicClosure p.ResidueField :=\n fun p [p.IsPrime] ↦\n have this := ⋯;\n IsAlgClosure.equiv k (AlgebraicClosure k) (AlgebraicCl... | [] | exact isReduced_of_injective _ (Algebra.TensorProduct.congr (e p).symm AlgEquiv.refl).injective | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Nilpotent.GeometricallyReduced | {
"line": 69,
"column": 4
} | {
"line": 69,
"column": 99
} | {
"line": 71,
"column": 0
} | [
{
"pp": "case refine_2\nk : Type u_3\nA : Type u_4\ninst✝² : Field k\ninst✝¹ : Ring A\ninst✝ : Algebra k A\ne : (p : Ideal k) → [inst : p.IsPrime] → AlgebraicClosure k ≃ₐ[k] AlgebraicClosure p.ResidueField :=\n fun p [p.IsPrime] ↦\n have this := ⋯;\n IsAlgClosure.equiv k (AlgebraicClosure k) (AlgebraicCl... | [] | exact isReduced_of_injective _ (Algebra.TensorProduct.congr (e p).symm AlgEquiv.refl).injective | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Nilpotent.GeometricallyReduced | {
"line": 64,
"column": 2
} | {
"line": 69,
"column": 99
} | {
"line": 71,
"column": 0
} | [
{
"pp": "k : Type u_3\nA : Type u_4\ninst✝² : Field k\ninst✝¹ : Ring A\ninst✝ : Algebra k A\n⊢ IsGeometricallyReduced k A ↔ IsReduced (AlgebraicClosure k ⊗[k] A)",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"AlgEquiv.instEquivLike",
"isReduced_of_injective",
"Algebra.t... | [] | let e (p : Ideal k) [p.IsPrime] : AlgebraicClosure k ≃ₐ[k] AlgebraicClosure p.ResidueField :=
have := p.algEquivResidueFieldOfField.isAlgebraic
IsAlgClosure.equiv k _ _
refine ⟨fun ⟨h⟩ ↦ ?_, fun h ↦ ⟨fun p hp ↦ ?_⟩⟩
· exact isReduced_of_injective _ (Algebra.TensorProduct.congr (e ⊥) AlgEquiv.refl).injective... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Nilpotent.GeometricallyReduced | {
"line": 64,
"column": 2
} | {
"line": 69,
"column": 99
} | {
"line": 71,
"column": 0
} | [
{
"pp": "k : Type u_3\nA : Type u_4\ninst✝² : Field k\ninst✝¹ : Ring A\ninst✝ : Algebra k A\n⊢ IsGeometricallyReduced k A ↔ IsReduced (AlgebraicClosure k ⊗[k] A)",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"AlgEquiv.instEquivLike",
"isReduced_of_injective",
"Algebra.t... | [] | let e (p : Ideal k) [p.IsPrime] : AlgebraicClosure k ≃ₐ[k] AlgebraicClosure p.ResidueField :=
have := p.algEquivResidueFieldOfField.isAlgebraic
IsAlgClosure.equiv k _ _
refine ⟨fun ⟨h⟩ ↦ ?_, fun h ↦ ⟨fun p hp ↦ ?_⟩⟩
· exact isReduced_of_injective _ (Algebra.TensorProduct.congr (e ⊥) AlgEquiv.refl).injective... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.MvPowerSeries.Restricted | {
"line": 79,
"column": 4
} | {
"line": 79,
"column": 15
} | {
"line": 79,
"column": 16
} | [
{
"pp": "case inr\nM : Type u_3\nS : Type u_4\ninst✝³ : AddMonoid M\ninst✝² : Finset.HasAntidiagonal M\ninst✝¹ : NormedRing S\ninst✝ : IsUltrametricDist S\nC : M → ℝ\nhC : ∀ (a b : M), C (a + b) = C a * C b\nf g : M → S\nhf : Tendsto (fun i ↦ ‖f i‖ * C i) cofinite (𝓝 0)\nhg : Tendsto (fun i ↦ ‖g i‖ * C i) cofi... | [
"case inr\nM : Type u_3\nS : Type u_4\ninst✝³ : AddMonoid M\ninst✝² : Finset.HasAntidiagonal M\ninst✝¹ : NormedRing S\ninst✝ : IsUltrametricDist S\nC : M → ℝ\nhC : ∀ (a b : M), C (a + b) = C a * C b\nf g : M → S\nhf : Tendsto (fun i ↦ ‖f i‖ * C i) cofinite (𝓝 0)\nhg : Tendsto (fun i ↦ ‖g i‖ * C i) cofinite (𝓝 0)\... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.MvPowerSeries.Restricted | {
"line": 79,
"column": 51
} | {
"line": 79,
"column": 62
} | {
"line": 79,
"column": 63
} | [
{
"pp": "M : Type u_3\nS : Type u_4\ninst✝³ : AddMonoid M\ninst✝² : Finset.HasAntidiagonal M\ninst✝¹ : NormedRing S\ninst✝ : IsUltrametricDist S\nC : M → ℝ\nhC : ∀ (a b : M), C (a + b) = C a * C b\nf g : M → S\nhf : Tendsto (fun i ↦ ‖f i‖ * C i) cofinite (𝓝 0)\nhg : Tendsto (fun i ↦ ‖g i‖ * C i) cofinite (𝓝 0... | [
"M : Type u_3\nS : Type u_4\ninst✝³ : AddMonoid M\ninst✝² : Finset.HasAntidiagonal M\ninst✝¹ : NormedRing S\ninst✝ : IsUltrametricDist S\nC : M → ℝ\nhC : ∀ (a b : M), C (a + b) = C a * C b\nf g : M → S\nhf : Tendsto (fun i ↦ ‖f i‖ * C i) cofinite (𝓝 0)\nhg : Tendsto (fun i ↦ ‖g i‖ * C i) cofinite (𝓝 0)\nthis :\n ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.MvPowerSeries.Restricted | {
"line": 80,
"column": 10
} | {
"line": 80,
"column": 21
} | {
"line": 80,
"column": 22
} | [
{
"pp": "M : Type u_3\nS : Type u_4\ninst✝³ : AddMonoid M\ninst✝² : Finset.HasAntidiagonal M\ninst✝¹ : NormedRing S\ninst✝ : IsUltrametricDist S\nC : M → ℝ\nhC : ∀ (a b : M), C (a + b) = C a * C b\nf g : M → S\nhf : Tendsto (fun i ↦ ‖f i‖ * C i) cofinite (𝓝 0)\nhg : Tendsto (fun i ↦ ‖g i‖ * C i) cofinite (𝓝 0... | [
"M : Type u_3\nS : Type u_4\ninst✝³ : AddMonoid M\ninst✝² : Finset.HasAntidiagonal M\ninst✝¹ : NormedRing S\ninst✝ : IsUltrametricDist S\nC : M → ℝ\nhC : ∀ (a b : M), C (a + b) = C a * C b\nf g : M → S\nhf : Tendsto (fun i ↦ ‖f i‖ * C i) cofinite (𝓝 0)\nhg : Tendsto (fun i ↦ ‖g i‖ * C i) cofinite (𝓝 0)\nthis :\n ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.NoetherNormalization | {
"line": 88,
"column": 27
} | {
"line": 88,
"column": 38
} | {
"line": 88,
"column": 39
} | [
{
"pp": "k : Type u_1\ninst✝ : Field k\nn : ℕ\nf : MvPolynomial (Fin (n + 1)) k\nv w : Fin (n + 1) →₀ ℕ\n⊢ (T1 f (-1)).comp (T1 f 1) = AlgHom.id k (MvPolynomial (Fin (n + 1)) k)",
"ppTerm": "?m.59",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NegZeroClass.toNeg",
"AddMonoidAlge... | [
"k : Type u_1\ninst✝ : Field k\nn : ℕ\nf : MvPolynomial (Fin (n + 1)) k\nv w : Fin (n + 1) →₀ ℕ\n⊢ (bind₁ fun i ↦ if i = 0 then MvPolynomial.X 0 else MvPolynomial.X i + -MvPolynomial.X 0 ^ up ^ ↑i).comp\n (bind₁ fun i ↦ if i = 0 then MvPolynomial.X 0 else MvPolynomial.X i + MvPolynomial.X 0 ^ up ^ ↑i) =\n A... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.NoetherNormalization | {
"line": 111,
"column": 2
} | {
"line": 112,
"column": 24
} | {
"line": 112,
"column": 25
} | [
{
"pp": "k : Type u_1\ninst✝ : Field k\nn : ℕ\nf : MvPolynomial (Fin (n + 1)) k\nv : Fin (n + 1) →₀ ℕ\na : k\nha : a ≠ 0\nh : ∀ (i : Fin n), (Polynomial.C (MvPolynomial.X i) + Polynomial.X ^ r i.succ) ^ v i.succ ≠ 0\n⊢ 0 +\n ((Polynomial.X ^ v 0).natDegree +\n ∑ i, ((Polynomial.C (MvPolynomial.X i) ... | [
"k : Type u_1\ninst✝ : Field k\nn : ℕ\nf : MvPolynomial (Fin (n + 1)) k\nv : Fin (n + 1) →₀ ℕ\na : k\nha : a ≠ 0\nh : ∀ (i : Fin n), (Polynomial.C (MvPolynomial.X i) + Polynomial.X ^ r i.succ) ^ v i.succ ≠ 0\n⊢ ∑ x, v x.succ * (Polynomial.C (MvPolynomial.X x) + Polynomial.X ^ up ^ ↑x.succ).natDegree =\n ∑ i, up ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.WittVector.WittPolynomial | {
"line": 123,
"column": 48
} | {
"line": 129,
"column": 37
} | {
"line": 131,
"column": 0
} | [
{
"pp": "p : ℕ\nR : Type u_1\ninst✝ : CommRing R\nhp : Fact (Nat.Prime p)\nn : ℕ\n⊢ constantCoeff (W_ R n) = 0",
"ppTerm": "?m.9",
"assigned": true,
"usedConstants": [
"Finsupp.instAddZeroClass",
"Eq.mpr",
"RingHom.instRingHomClass",
"wittPolynomial",
"pow_pos",
"... | [] | by
simp only [wittPolynomial, map_sum, constantCoeff_monomial]
rw [sum_eq_zero]
rintro i _
rw [if_neg]
rw [Finsupp.single_eq_zero]
exact ne_of_gt (pow_pos hp.1.pos _) | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.NoetherNormalization | {
"line": 154,
"column": 4
} | {
"line": 155,
"column": 11
} | {
"line": 155,
"column": 12
} | [
{
"pp": "k : Type u_1\ninst✝ : Field k\nn : ℕ\nf : MvPolynomial (Fin (n + 1)) k\nfne : f ≠ 0\nv : Fin (n + 1) →₀ ℕ\nvin : v ∈ f.support\nh : (Fin (n + 1) →₀ ℕ) → MvPolynomial (Fin (n + 1)) k := fun w ↦ (MvPolynomial.monomial w) (MvPolynomial.coeff w f)\nvs :\n ∀ x' ∈ f.support,\n ((MvPolynomial.finSuccEquiv... | [
"k : Type u_1\ninst✝ : Field k\nn : ℕ\nf : MvPolynomial (Fin (n + 1)) k\nfne : f ≠ 0\nv : Fin (n + 1) →₀ ℕ\nvin : v ∈ f.support\nh : (Fin (n + 1) →₀ ℕ) → MvPolynomial (Fin (n + 1)) k := fun w ↦ (MvPolynomial.monomial w) (MvPolynomial.coeff w f)\nvs :\n ∀ x' ∈ f.support,\n ((MvPolynomial.finSuccEquiv k n) ((T f)... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.NoetherNormalization | {
"line": 161,
"column": 28
} | {
"line": 161,
"column": 43
} | {
"line": 161,
"column": 44
} | [
{
"pp": "k : Type u_1\ninst✝ : Field k\nn : ℕ\nf : MvPolynomial (Fin (n + 1)) k\nfne : f ≠ 0\nv : Fin (n + 1) →₀ ℕ\nvin : v ∈ f.support\nh : (Fin (n + 1) →₀ ℕ) → MvPolynomial (Fin (n + 1)) k := fun w ↦ (MvPolynomial.monomial w) (MvPolynomial.coeff w f)\nvs :\n ∀ x ∈ f.support \\ {v},\n ((MvPolynomial.finSuc... | [
"k : Type u_1\ninst✝ : Field k\nn : ℕ\nf : MvPolynomial (Fin (n + 1)) k\nfne : f ≠ 0\nv : Fin (n + 1) →₀ ℕ\nvin : v ∈ f.support\nh : (Fin (n + 1) →₀ ℕ) → MvPolynomial (Fin (n + 1)) k := fun w ↦ (MvPolynomial.monomial w) (MvPolynomial.coeff w f)\nvs :\n ∀ x ∈ f.support \\ {v},\n ((MvPolynomial.finSuccEquiv k n) ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Teichmuller | {
"line": 47,
"column": 35
} | {
"line": 47,
"column": 46
} | {
"line": 47,
"column": 47
} | [
{
"pp": "p : ℕ\ninst✝² : Fact (Nat.Prime p)\nR : Type u_1\ninst✝¹ : CommRing R\nI : Ideal R\ninst✝ : CharP (R ⧸ I) p\nx : Perfection (R ⧸ I) p\n⊢ ∀ (n : ℕ), x.teichmullerAux n ≡ x.teichmullerAux (n + 1) [SMOD I ^ n • ⊤]",
"ppTerm": "?m.25",
"assigned": true,
"usedConstants": [
"Eq.mpr",
... | [
"p : ℕ\ninst✝² : Fact (Nat.Prime p)\nR : Type u_1\ninst✝¹ : CommRing R\nI : Ideal R\ninst✝ : CharP (R ⧸ I) p\nx : Perfection (R ⧸ I) p\n⊢ ∀ (n : ℕ), x.teichmullerAux n ≡ x.teichmullerAux (n + 1) [SMOD I ^ n]"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.NoetherNormalization | {
"line": 163,
"column": 9
} | {
"line": 163,
"column": 68
} | {
"line": 163,
"column": 69
} | [
{
"pp": "k : Type u_1\ninst✝ : Field k\nn : ℕ\nf : MvPolynomial (Fin (n + 1)) k\nfne : f ≠ 0\nv : Fin (n + 1) →₀ ℕ\nvin : v ∈ f.support\nh : (Fin (n + 1) →₀ ℕ) → MvPolynomial (Fin (n + 1)) k := fun w ↦ (MvPolynomial.monomial w) (MvPolynomial.coeff w f)\nvs :\n ∀ x ∈ f.support \\ {v},\n ((MvPolynomial.finSuc... | [
"k : Type u_1\ninst✝ : Field k\nn : ℕ\nf : MvPolynomial (Fin (n + 1)) k\nfne : f ≠ 0\nv : Fin (n + 1) →₀ ℕ\nvin : v ∈ f.support\nh : (Fin (n + 1) →₀ ℕ) → MvPolynomial (Fin (n + 1)) k := fun w ↦ (MvPolynomial.monomial w) (MvPolynomial.coeff w f)\nvs :\n ∀ x ∈ f.support \\ {v},\n ((MvPolynomial.finSuccEquiv k n) ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.NoetherNormalization | {
"line": 168,
"column": 2
} | {
"line": 168,
"column": 41
} | {
"line": 168,
"column": 42
} | [
{
"pp": "k : Type u_1\ninst✝ : Field k\nn : ℕ\nf : MvPolynomial (Fin (n + 1)) k\nfne : f ≠ 0\nv : Fin (n + 1) →₀ ℕ\nvin : v ∈ f.support\nh : (Fin (n + 1) →₀ ℕ) → MvPolynomial (Fin (n + 1)) k := fun w ↦ (MvPolynomial.monomial w) (MvPolynomial.coeff w f)\nvs :\n ∀ x ∈ f.support \\ {v},\n ((MvPolynomial.finSuc... | [
"k : Type u_1\ninst✝ : Field k\nn : ℕ\nf : MvPolynomial (Fin (n + 1)) k\nfne : f ≠ 0\nv : Fin (n + 1) →₀ ℕ\nvin : v ∈ f.support\nh : (Fin (n + 1) →₀ ℕ) → MvPolynomial (Fin (n + 1)) k := fun w ↦ (MvPolynomial.monomial w) (MvPolynomial.coeff w f)\nvs :\n ∀ x ∈ f.support \\ {v},\n ((MvPolynomial.finSuccEquiv k n) ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Teichmuller | {
"line": 167,
"column": 2
} | {
"line": 167,
"column": 70
} | {
"line": 168,
"column": 2
} | [
{
"pp": "p : ℕ\ninst✝³ : Fact (Nat.Prime p)\nR : Type u_1\ninst✝² : CommRing R\nI : Ideal R\ninst✝¹ : CharP (R ⧸ I) p\ninst✝ : IsAdicComplete I R\nx : Perfection (R ⧸ I) p\n⊢ (Ideal.Quotient.mk I) ((teichmuller p I) x) = (coeff (R ⧸ I) p 0) x",
"ppTerm": "?m.30",
"assigned": true,
"usedConstants": [... | [
"p : ℕ\ninst✝³ : Fact (Nat.Prime p)\nR : Type u_1\ninst✝² : CommRing R\nI : Ideal R\ninst✝¹ : CharP (R ⧸ I) p\ninst✝ : IsAdicComplete I R\nx : Perfection (R ⧸ I) p\nthis : (teichmuller p I) x ≡ Quotient.out ((coeff (R ⧸ I) p 0) x) ^ p ^ 0 [SMOD I ^ (0 + 1)]\n⊢ (Ideal.Quotient.mk I) ((teichmuller p I) x) = (coeff (R... | have := teichmuller_sModEq <| Ideal.Quotient.mk_out <| coeff _ p 0 x | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.RingTheory.Teichmuller | {
"line": 169,
"column": 2
} | {
"line": 169,
"column": 38
} | {
"line": 169,
"column": 39
} | [
{
"pp": "p : ℕ\ninst✝³ : Fact (Nat.Prime p)\nR : Type u_1\ninst✝² : CommRing R\nI : Ideal R\ninst✝¹ : CharP (R ⧸ I) p\ninst✝ : IsAdicComplete I R\nx : Perfection (R ⧸ I) p\nthis : (teichmuller p I) x ≡ Quotient.out ((coeff (R ⧸ I) p 0) x) ^ p ^ 0 [SMOD I]\n⊢ (Ideal.Quotient.mk I) ((teichmuller p I) x) = (coeff ... | [
"p : ℕ\ninst✝³ : Fact (Nat.Prime p)\nR : Type u_1\ninst✝² : CommRing R\nI : Ideal R\ninst✝¹ : CharP (R ⧸ I) p\ninst✝ : IsAdicComplete I R\nx : Perfection (R ⧸ I) p\nthis : (teichmuller p I) x ≡ Quotient.out ((coeff (R ⧸ I) p 0) x) ^ p ^ 0 [SMOD I]\n⊢ (Ideal.Quotient.mk I) ((teichmuller₀ p I) x) = (coeff (R ⧸ I) p 0... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.NoetherNormalization | {
"line": 215,
"column": 13
} | {
"line": 217,
"column": 26
} | {
"line": 219,
"column": 0
} | [
{
"pp": "k : Type u_1\ninst✝ : Field k\nn : ℕ\nf : MvPolynomial (Fin (n + 1)) k\nv w : Fin (n + 1) →₀ ℕ\nI : Ideal (MvPolynomial (Fin (n + 1)) k)\n⊢ Ideal.map ((T f).symm.toRingEquiv.toRingHom.comp ↑(T f)) I = Ideal.map (↑(T f).symm) (Ideal.map (T f) I)",
"ppTerm": "?m.140",
"assigned": true,
"usedC... | [] | by
rw [← Ideal.map_map, Ideal.map_coe, Ideal.map_coe]
exact congrArg _ rfl | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.WittVector.WittPolynomial | {
"line": 268,
"column": 2
} | {
"line": 268,
"column": 50
} | {
"line": 270,
"column": 0
} | [
{
"pp": "p : ℕ\nR : Type u_1\ninst✝¹ : CommRing R\ninst✝ : Invertible ↑p\nk : ℕ\n⊢ X k - ∑ i ∈ range k, C (↑p ^ i) * xInTermsOfW p R i ^ p ^ (k - i) +\n ∑ x ∈ range k, C ↑p ^ x * (bind₁ (xInTermsOfW p R)) (X x) ^ p ^ (k - x) =\n X k",
"ppTerm": "?m.79",
"assigned": true,
"usedConstants": [
... | [] | simp only [C_pow, bind₁_X_right, sub_add_cancel] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.RingTheory.WittVector.Basic | {
"line": 279,
"column": 4
} | {
"line": 279,
"column": 15
} | {
"line": 279,
"column": 16
} | [
{
"pp": "case refine_1\np : ℕ\nR : Type u_1\nS : Type u_2\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Fact (Nat.Prime p)\nf : R →+* S\nx : 𝕎 R\nn : ℕ\nh : ((map f) x).coeff n = coeff 0 n\n⊢ f (x.coeff n) = 0",
"ppTerm": "?refine_1",
"assigned": false,
"usedConstants": [],
"usedFVars": []... | [
"case refine_1\np : ℕ\nR : Type u_1\nS : Type u_2\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Fact (Nat.Prime p)\nf : R →+* S\nx : 𝕎 R\nn : ℕ\nh : ((map f) x).coeff n = coeff 0 n\n⊢ f (x.coeff n) = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.WittVector.Basic | {
"line": 281,
"column": 4
} | {
"line": 281,
"column": 15
} | {
"line": 281,
"column": 16
} | [
{
"pp": "case refine_2\np : ℕ\nR : Type u_1\nS : Type u_2\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Fact (Nat.Prime p)\nf : R →+* S\nx : 𝕎 R\nh : ∀ (n : ℕ), f (x.coeff n) = 0\nn : ℕ\n⊢ ((map f) x).coeff n = coeff 0 n",
"ppTerm": "?refine_2",
"assigned": true,
"usedConstants": [
"Witt... | [
"case refine_2\np : ℕ\nR : Type u_1\nS : Type u_2\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Fact (Nat.Prime p)\nf : R →+* S\nx : 𝕎 R\nh : ∀ (n : ℕ), f (x.coeff n) = 0\nn : ℕ\n⊢ f (x.coeff n) = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.WittVector.StructurePolynomial | {
"line": 235,
"column": 81
} | {
"line": 261,
"column": 29
} | {
"line": 263,
"column": 0
} | [
{
"pp": "p : ℕ\nidx : Type u_2\nhp : Fact (Nat.Prime p)\nΦ : MvPolynomial idx ℤ\nn : ℕ\nIH : ∀ m < n, (map (Int.castRingHom ℚ)) (wittStructureInt p Φ m) = wittStructureRat p ((map (Int.castRingHom ℚ)) Φ) m\n⊢ C ↑(p ^ n) ∣\n (bind₁ fun b ↦ (rename fun i ↦ (b, i)) (W_ ℤ n)) Φ -\n ∑ i ∈ Finset.range n, C (... | [] | by
rcases n with - | n
· simp
-- prepare a useful equation for rewriting
have key := bind₁_rename_expand_wittPolynomial Φ n IH
apply_fun map (Int.castRingHom (ZMod (p ^ (n + 1)))) at key
conv_lhs at key => simp only [map_bind₁, map_rename, map_expand, map_wittPolynomial]
-- clean up and massage
rw [C_dv... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.WittVector.IsPoly | {
"line": 124,
"column": 2
} | {
"line": 125,
"column": 58
} | {
"line": 125,
"column": 59
} | [
{
"pp": "p : ℕ\nidx : Type u_1\ninst✝ : Fact (Nat.Prime p)\nf g : ℕ → MvPolynomial (idx × ℕ) ℤ\nn : ℕ\nh :\n (bind₁ (⇑(MvPolynomial.map (Int.castRingHom ℚ)) ∘ fun n ↦ (bind₁ f) (wittPolynomial p ℤ n))) (xInTermsOfW p ℚ n) =\n (bind₁ (⇑(MvPolynomial.map (Int.castRingHom ℚ)) ∘ fun n ↦ (bind₁ g) (wittPolynomia... | [
"p : ℕ\nidx : Type u_1\ninst✝ : Fact (Nat.Prime p)\nf g : ℕ → MvPolynomial (idx × ℕ) ℤ\nn : ℕ\nh :\n (bind₁ (⇑(MvPolynomial.map (Int.castRingHom ℚ)) ∘ fun n ↦ (bind₁ f) (wittPolynomial p ℤ n))) (xInTermsOfW p ℚ n) =\n (bind₁ (⇑(MvPolynomial.map (Int.castRingHom ℚ)) ∘ fun n ↦ (bind₁ g) (wittPolynomial p ℤ n))) (... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.WittVector.IsPoly | {
"line": 134,
"column": 2
} | {
"line": 135,
"column": 58
} | {
"line": 135,
"column": 59
} | [
{
"pp": "p : ℕ\ninst✝ : Fact (Nat.Prime p)\nf g : ℕ → MvPolynomial ℕ ℤ\nn : ℕ\nh :\n (bind₁ (⇑(MvPolynomial.map (Int.castRingHom ℚ)) ∘ fun n ↦ (bind₁ f) (wittPolynomial p ℤ n))) (xInTermsOfW p ℚ n) =\n (bind₁ (⇑(MvPolynomial.map (Int.castRingHom ℚ)) ∘ fun n ↦ (bind₁ g) (wittPolynomial p ℤ n))) (xInTermsOfW ... | [
"p : ℕ\ninst✝ : Fact (Nat.Prime p)\nf g : ℕ → MvPolynomial ℕ ℤ\nn : ℕ\nh :\n (bind₁ (⇑(MvPolynomial.map (Int.castRingHom ℚ)) ∘ fun n ↦ (bind₁ f) (wittPolynomial p ℤ n))) (xInTermsOfW p ℚ n) =\n (bind₁ (⇑(MvPolynomial.map (Int.castRingHom ℚ)) ∘ fun n ↦ (bind₁ g) (wittPolynomial p ℤ n))) (xInTermsOfW p ℚ n)\n⊢ (M... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Perfection | {
"line": 119,
"column": 2
} | {
"line": 119,
"column": 33
} | {
"line": 120,
"column": 2
} | [
{
"pp": "M : Type u_1\ninst✝ : CommMonoid M\np : ℕ\nf : Perfection M p\nn m : ℕ\n⊢ (coeffMonoidHom M p n) ((⇑(powMulEquiv (Perfection M p) p).symm)^[m] f) = (coeffMonoidHom M p (n + m)) f",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"Function.iterate_succ_apply'",
"Eq.mpr",
... | [] | induction m generalizing n with | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | null |
Mathlib.RingTheory.WittVector.StructurePolynomial | {
"line": 313,
"column": 4
} | {
"line": 314,
"column": 33
} | {
"line": 314,
"column": 34
} | [
{
"pp": "case py₁\np : ℕ\nidx : Type u_2\nhp : Fact (Nat.Prime p)\nΦ : MvPolynomial idx ℤ\nφ : ℕ → MvPolynomial (idx × ℕ) ℤ\nn : ℕ\nh :\n (map (Int.castRingHom ℚ)) ((bind₁ φ) (W_ ℤ n)) =\n (map (Int.castRingHom ℚ)) ((bind₁ fun i ↦ (rename (Prod.mk i)) (W_ ℤ n)) Φ)\n⊢ (bind₁ fun k ↦ (map (Int.castRingHom ℚ))... | [
"case py₁\np : ℕ\nidx : Type u_2\nhp : Fact (Nat.Prime p)\nΦ : MvPolynomial idx ℤ\nφ : ℕ → MvPolynomial (idx × ℕ) ℤ\nn : ℕ\nh :\n (map (Int.castRingHom ℚ)) ((bind₁ φ) (W_ ℤ n)) =\n (map (Int.castRingHom ℚ)) ((bind₁ fun i ↦ (rename (Prod.mk i)) (W_ ℤ n)) Φ)\n⊢ (bind₁ fun k ↦ (map (Int.castRingHom ℚ)) (φ k)) (W_ ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Perfection | {
"line": 236,
"column": 2
} | {
"line": 236,
"column": 39
} | {
"line": 236,
"column": 40
} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommSemiring R\np : ℕ\nhp : Fact (Nat.Prime p)\ninst✝ : CharP R p\nx✝ : Perfection R p\n⊢ (pthRoot R p) x✝ = ↑(frobeniusEquiv (Perfection R p) p).symm x✝",
"ppTerm": "?m.31",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"frobeniusEquiv_apply",
"Non... | [
"R : Type u_1\ninst✝¹ : CommSemiring R\np : ℕ\nhp : Fact (Nat.Prime p)\ninst✝ : CharP R p\nx✝ : Perfection R p\n⊢ (frobenius (Perfection R p) p) ((pthRoot R p) x✝) = x✝"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.WittVector.Identities | {
"line": 77,
"column": 4
} | {
"line": 77,
"column": 34
} | {
"line": 77,
"column": 35
} | [
{
"pp": "case pos\np : ℕ\nR : Type u_1\nhp : Fact (Nat.Prime p)\ninst✝¹ : CommRing R\ninst✝ : CharP R p\ni : ℕ\nhi : i = 1\n⊢ (↑p).coeff i = 1",
"ppTerm": "?pos✝",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"WittVector.instNatCast",
"congrArg",
"AddGroupWithOne.toAddMonoi... | [
"case pos\np : ℕ\nR : Type u_1\nhp : Fact (Nat.Prime p)\ninst✝¹ : CommRing R\ninst✝ : CharP R p\ni : ℕ\nhi : i = 1\n⊢ (↑p).coeff 1 = 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.WittVector.Identities | {
"line": 78,
"column": 4
} | {
"line": 78,
"column": 30
} | {
"line": 78,
"column": 31
} | [
{
"pp": "case neg\np : ℕ\nR : Type u_1\nhp : Fact (Nat.Prime p)\ninst✝¹ : CommRing R\ninst✝ : CharP R p\ni : ℕ\nhi : ¬i = 1\n⊢ (↑p).coeff i = 0",
"ppTerm": "?neg✝",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"case neg\np : ℕ\nR : Type u_1\nhp : Fact (Nat.Prime p)\ninst✝¹ : CommRing R\ninst✝ : CharP R p\ni : ℕ\nhi : ¬i = 1\n⊢ (↑p).coeff i = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.WittVector.Identities | {
"line": 90,
"column": 2
} | {
"line": 90,
"column": 47
} | {
"line": 90,
"column": 48
} | [
{
"pp": "p : ℕ\nR : Type u_1\nhp : Fact (Nat.Prime p)\ninst✝² : CommRing R\ninst✝¹ : Nontrivial R\ninst✝ : CharP R p\nh : ↑p = 0\n⊢ False",
"ppTerm": "?m.6",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"p : ℕ\nR : Type u_1\nhp : Fact (Nat.Prime p)\ninst✝² : CommRing R\ninst✝¹ : Nontrivial R\ninst✝ : CharP R p\nh : ↑p = 0\n⊢ False"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.WittVector.Identities | {
"line": 93,
"column": 2
} | {
"line": 93,
"column": 13
} | {
"line": 93,
"column": 14
} | [
{
"pp": "p : ℕ\nR : Type u_1\nhp : Fact (Nat.Prime p)\ninst✝² : CommRing R\ninst✝¹ : Nontrivial R\ninst✝ : CharP R p\n⊢ ↑p ≠ 0",
"ppTerm": "?m.6",
"assigned": true,
"usedConstants": [
"WittVector.instZero",
"WittVector.instCommRing",
"CommSemiring.toSemiring",
"AddGroupWithOn... | [
"p : ℕ\nR : Type u_1\nhp : Fact (Nat.Prime p)\ninst✝² : CommRing R\ninst✝¹ : Nontrivial R\ninst✝ : CharP R p\n⊢ ¬↑p = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.WittVector.Identities | {
"line": 127,
"column": 6
} | {
"line": 127,
"column": 17
} | {
"line": 127,
"column": 18
} | [
{
"pp": "case succ.succ\np : ℕ\nR : Type u_1\nhp : Fact (Nat.Prime p)\ninst✝¹ : CommRing R\ninst✝ : CharP R p\nx : 𝕎 R\nn : ℕ\nih : ∀ {m : ℕ}, m < n → (x * ↑p ^ n).coeff m = 0\nm' : ℕ\nh : m' + 1 < n + 1\n⊢ m' < n",
"ppTerm": "?succ.succ✝",
"assigned": false,
"usedConstants": [],
"usedFVars": [... | [
"case succ.succ\np : ℕ\nR : Type u_1\nhp : Fact (Nat.Prime p)\ninst✝¹ : CommRing R\ninst✝ : CharP R p\nx : 𝕎 R\nn : ℕ\nih : ∀ {m : ℕ}, m < n → (x * ↑p ^ n).coeff m = 0\nm' : ℕ\nh : m' + 1 < n + 1\n⊢ m' < n"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.WittVector.Identities | {
"line": 164,
"column": 6
} | {
"line": 164,
"column": 17
} | {
"line": 164,
"column": 18
} | [
{
"pp": "case succ.succ\np : ℕ\nR : Type u_1\nhp : Fact (Nat.Prime p)\ninst✝ : CommRing R\nx : 𝕎 R\nn : ℕ\nih : ∀ {m : ℕ}, m < n → ((⇑verschiebung)^[n] x).coeff m = 0\nm' : ℕ\nh : m' + 1 < n + 1\n⊢ m' < n",
"ppTerm": "?succ.succ✝",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"u... | [
"case succ.succ\np : ℕ\nR : Type u_1\nhp : Fact (Nat.Prime p)\ninst✝ : CommRing R\nx : 𝕎 R\nn : ℕ\nih : ∀ {m : ℕ}, m < n → ((⇑verschiebung)^[n] x).coeff m = 0\nm' : ℕ\nh : m' + 1 < n + 1\n⊢ m' < n"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.WittVector.IsPoly | {
"line": 361,
"column": 2
} | {
"line": 366,
"column": 22
} | {
"line": 368,
"column": 0
} | [
{
"pp": "p : ℕ\nR S : Type u\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Fact (Nat.Prime p)\nf : ⦃R : Type u⦄ → [CommRing R] → 𝕎 R → 𝕎 R → 𝕎 R\nhf : IsPoly₂ p f\ng : R →+* S\nx y : 𝕎 R\n⊢ (WittVector.map g) (f x y) = f ((WittVector.map g) x) ((WittVector.map g) y)",
"ppTerm": "?m.30",
"assign... | [] | obtain ⟨φ, hf⟩ := hf
ext n
simp +unfoldPartialApp only [map_coeff, hf, map_aeval, peval, uncurry]
apply eval₂Hom_congr (RingHom.ext_int _ _) _ rfl
ext ⟨i, k⟩
fin_cases i <;> simp | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.WittVector.IsPoly | {
"line": 361,
"column": 2
} | {
"line": 366,
"column": 22
} | {
"line": 368,
"column": 0
} | [
{
"pp": "p : ℕ\nR S : Type u\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Fact (Nat.Prime p)\nf : ⦃R : Type u⦄ → [CommRing R] → 𝕎 R → 𝕎 R → 𝕎 R\nhf : IsPoly₂ p f\ng : R →+* S\nx y : 𝕎 R\n⊢ (WittVector.map g) (f x y) = f ((WittVector.map g) x) ((WittVector.map g) y)",
"ppTerm": "?m.30",
"assign... | [] | obtain ⟨φ, hf⟩ := hf
ext n
simp +unfoldPartialApp only [map_coeff, hf, map_aeval, peval, uncurry]
apply eval₂Hom_congr (RingHom.ext_int _ _) _ rfl
ext ⟨i, k⟩
fin_cases i <;> simp | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.WittVector.Truncated | {
"line": 109,
"column": 2
} | {
"line": 109,
"column": 30
} | {
"line": 109,
"column": 31
} | [
{
"pp": "p n : ℕ\nR : Type u_1\ninst✝ : CommRing R\nx y : TruncatedWittVector p n R\nh : ∀ (n_1 : ℕ), x.out.coeff n_1 = y.out.coeff n_1\ni : Fin n\n⊢ coeff i x = coeff i y",
"ppTerm": "?m.29",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"p n : ℕ\nR : Type u_1\ninst✝ : CommRing R\nx y : TruncatedWittVector p n R\nh : ∀ (n_1 : ℕ), x.out.coeff n_1 = y.out.coeff n_1\ni : Fin n\n⊢ coeff i x = coeff i y"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.WittVector.Complete | {
"line": 71,
"column": 6
} | {
"line": 71,
"column": 17
} | {
"line": 71,
"column": 18
} | [
{
"pp": "p : ℕ\nhp : Fact (Nat.Prime p)\nk : Type u_1\ninst✝² : CommRing k\ninst✝¹ : CharP k p\ninst✝ : PerfectRing k p\nx : 𝕎 k\nh : x.coeff 0 = 0\n⊢ x = verschiebung (x.shift 1)",
"ppTerm": "?m.85",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"p : ℕ\nhp : Fact (Nat.Prime p)\nk : Type u_1\ninst✝² : CommRing k\ninst✝¹ : CharP k p\ninst✝ : PerfectRing k p\nx : 𝕎 k\nh : x.coeff 0 = 0\n⊢ x = verschiebung (x.shift 1)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.WittVector.Complete | {
"line": 89,
"column": 6
} | {
"line": 89,
"column": 17
} | {
"line": 89,
"column": 18
} | [
{
"pp": "p : ℕ\nhp : Fact (Nat.Prime p)\nk : Type u_1\ninst✝² : CommRing k\ninst✝¹ : CharP k p\ninst✝ : PerfectRing k p\nx : 𝕎 k\nn : ℕ\nh : ∀ m < n, x.coeff m = 0\n⊢ x = (⇑verschiebung)^[n] (x.shift n)",
"ppTerm": "?m.100",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoal... | [
"p : ℕ\nhp : Fact (Nat.Prime p)\nk : Type u_1\ninst✝² : CommRing k\ninst✝¹ : CharP k p\ninst✝ : PerfectRing k p\nx : 𝕎 k\nn : ℕ\nh : ∀ m < n, x.coeff m = 0\n⊢ x = (⇑verschiebung)^[n] (x.shift n)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
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