module string | startPos dict | endPos dict | nextStartPos dict | goals list | goalsAfter list | ppTac string | elaborator string | kind string |
|---|---|---|---|---|---|---|---|---|
Mathlib.Probability.Moments.SubGaussian | {
"line": 740,
"column": 8
} | {
"line": 740,
"column": 24
} | {
"line": 741,
"column": 6
} | [
{
"pp": "case c0\nΩ : Type u_1\nmΩ : MeasurableSpace Ω\nμ : Measure Ω\nX Y : Ω → ℝ\ncX cY : ℝ≥0\nhX : HasSubgaussianMGF X cX μ\nhY : HasSubgaussianMGF Y cY μ\nhindep : X ⟂ᵢ[μ] Y\nt : ℝ\n⊢ 0 ≤ mgf Y μ t",
"ppTerm": "?c0",
"assigned": true,
"usedConstants": [
"ProbabilityTheory.mgf_nonneg"
]... | [] | exact mgf_nonneg | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Probability.Moments.SubGaussian | {
"line": 740,
"column": 8
} | {
"line": 740,
"column": 24
} | {
"line": 741,
"column": 6
} | [
{
"pp": "case c0\nΩ : Type u_1\nmΩ : MeasurableSpace Ω\nμ : Measure Ω\nX Y : Ω → ℝ\ncX cY : ℝ≥0\nhX : HasSubgaussianMGF X cX μ\nhY : HasSubgaussianMGF Y cY μ\nhindep : X ⟂ᵢ[μ] Y\nt : ℝ\n⊢ 0 ≤ mgf Y μ t",
"ppTerm": "?c0",
"assigned": true,
"usedConstants": [
"ProbabilityTheory.mgf_nonneg"
]... | [] | exact mgf_nonneg | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Probability.Moments.SubGaussian | {
"line": 740,
"column": 8
} | {
"line": 740,
"column": 24
} | {
"line": 741,
"column": 6
} | [
{
"pp": "case c0\nΩ : Type u_1\nmΩ : MeasurableSpace Ω\nμ : Measure Ω\nX Y : Ω → ℝ\ncX cY : ℝ≥0\nhX : HasSubgaussianMGF X cX μ\nhY : HasSubgaussianMGF Y cY μ\nhindep : X ⟂ᵢ[μ] Y\nt : ℝ\n⊢ 0 ≤ mgf Y μ t",
"ppTerm": "?c0",
"assigned": true,
"usedConstants": [
"ProbabilityTheory.mgf_nonneg"
]... | [] | exact mgf_nonneg | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Probability.Moments.SubGaussian | {
"line": 857,
"column": 53
} | {
"line": 857,
"column": 60
} | {
"line": 859,
"column": 0
} | [
{
"pp": "Ω : Type u_1\nmΩ : MeasurableSpace Ω\nμ : Measure Ω\nX : Ω → ℝ\ninst✝ : IsProbabilityMeasure μ\na b : ℝ\nhm : AEMeasurable X μ\nhb : ∀ᵐ (ω : Ω) ∂μ, X ω ∈ Set.Icc a b\nhc : ∫ (x : Ω), X x ∂μ = 0\nt : ℝ\nht : t < 0\n⊢ rexp ((↑‖-a - -b‖₊ / 2) ^ 2 * (-t) ^ 2 / 2) = rexp ((↑‖b - a‖₊ / 2) ^ 2 * t ^ 2 / 2)",
... | [] | ring_nf | Mathlib.Tactic.RingNF._aux_Mathlib_Tactic_Ring_RingNF___elabRules_Mathlib_Tactic_RingNF_ringNF_1 | Mathlib.Tactic.RingNF.ringNF |
Mathlib.Probability.Moments.SubGaussian | {
"line": 857,
"column": 53
} | {
"line": 857,
"column": 60
} | {
"line": 859,
"column": 0
} | [
{
"pp": "Ω : Type u_1\nmΩ : MeasurableSpace Ω\nμ : Measure Ω\nX : Ω → ℝ\ninst✝ : IsProbabilityMeasure μ\na b : ℝ\nhm : AEMeasurable X μ\nhb : ∀ᵐ (ω : Ω) ∂μ, X ω ∈ Set.Icc a b\nhc : ∫ (x : Ω), X x ∂μ = 0\nt : ℝ\nht : t < 0\n⊢ rexp ((↑‖-a - -b‖₊ / 2) ^ 2 * (-t) ^ 2 / 2) = rexp ((↑‖b - a‖₊ / 2) ^ 2 * t ^ 2 / 2)",
... | [] | ring_nf | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Probability.Moments.SubGaussian | {
"line": 857,
"column": 53
} | {
"line": 857,
"column": 60
} | {
"line": 859,
"column": 0
} | [
{
"pp": "Ω : Type u_1\nmΩ : MeasurableSpace Ω\nμ : Measure Ω\nX : Ω → ℝ\ninst✝ : IsProbabilityMeasure μ\na b : ℝ\nhm : AEMeasurable X μ\nhb : ∀ᵐ (ω : Ω) ∂μ, X ω ∈ Set.Icc a b\nhc : ∫ (x : Ω), X x ∂μ = 0\nt : ℝ\nht : t < 0\n⊢ rexp ((↑‖-a - -b‖₊ / 2) ^ 2 * (-t) ^ 2 / 2) = rexp ((↑‖b - a‖₊ / 2) ^ 2 * t ^ 2 / 2)",
... | [] | ring_nf | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Probability.Moments.SubGaussian | {
"line": 864,
"column": 2
} | {
"line": 864,
"column": 73
} | {
"line": 865,
"column": 2
} | [
{
"pp": "Ω : Type u_1\nmΩ : MeasurableSpace Ω\nμ : Measure Ω\nX : Ω → ℝ\ninst✝ : IsProbabilityMeasure μ\na b : ℝ\nhm : AEMeasurable X μ\nhb : ∀ᵐ (ω : Ω) ∂μ, X ω ∈ Set.Icc a b\n⊢ HasSubgaussianMGF (fun ω ↦ X ω - ∫ (x : Ω), X x ∂μ) ((‖b - ∫ (x : Ω), X x ∂μ - (a - ∫ (x : Ω), X x ∂μ)‖₊ / 2) ^ 2) μ",
"ppTerm": "... | [
"case hb\nΩ : Type u_1\nmΩ : MeasurableSpace Ω\nμ : Measure Ω\nX : Ω → ℝ\ninst✝ : IsProbabilityMeasure μ\na b : ℝ\nhm : AEMeasurable X μ\nhb : ∀ᵐ (ω : Ω) ∂μ, X ω ∈ Set.Icc a b\n⊢ ∀ᵐ (ω : Ω) ∂μ, X ω - ∫ (x : Ω), X x ∂μ ∈ Set.Icc (a - ∫ (x : Ω), X x ∂μ) (b - ∫ (x : Ω), X x ∂μ)",
"case hc\nΩ : Type u_1\nmΩ : Measura... | apply hasSubgaussianMGF_of_mem_Icc_of_integral_eq_zero (hm.sub_const _) | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.RepresentationTheory.Continuous.Basic | {
"line": 60,
"column": 30
} | {
"line": 60,
"column": 38
} | {
"line": 60,
"column": 38
} | [
{
"pp": "R : Type u_1\nG : Type u_2\nV : Type u_3\nW : Type u_4\nU : Type u_5\ninst✝¹³ : Monoid G\ninst✝¹² : Ring R\ninst✝¹¹ : AddCommGroup V\ninst✝¹⁰ : TopologicalSpace V\ninst✝⁹ : IsTopologicalAddGroup V\ninst✝⁸ : Module R V\ninst✝⁷ : AddCommGroup W\ninst✝⁶ : TopologicalSpace W\ninst✝⁵ : IsTopologicalAddGroup... | [
"case ofMonoidHom\nR : Type u_1\nG : Type u_2\nV : Type u_3\nW : Type u_4\nU : Type u_5\ninst✝¹³ : Monoid G\ninst✝¹² : Ring R\ninst✝¹¹ : AddCommGroup V\ninst✝¹⁰ : TopologicalSpace V\ninst✝⁹ : IsTopologicalAddGroup V\ninst✝⁸ : Module R V\ninst✝⁷ : AddCommGroup W\ninst✝⁶ : TopologicalSpace W\ninst✝⁵ : IsTopologicalAd... | cases π₁ | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalCases | Lean.Parser.Tactic.cases |
Mathlib.RepresentationTheory.FiniteIndex | {
"line": 268,
"column": 81
} | {
"line": 270,
"column": 50
} | {
"line": 272,
"column": 0
} | [
{
"pp": "k : Type u\nG : Type v\ninst✝³ : CommRing k\ninst✝² : Group G\nS : Subgroup G\ninst✝¹ : DecidableRel ⇑(QuotientGroup.rightRel S)\ninst✝ : S.FiniteIndex\nA : Rep.{max w u v, u, v} k ↥S\nB : Rep.{max (max u v) w, u, v} k G\nf : A ⟶ res S.subtype B\n⊢ ((coindResAdjunction k S).homEquiv A B).symm f = A.ind... | [] | by
simp [coindResAdjunction, indResHomEquiv, indResAdjunction,
Adjunction.homEquiv_ofNatIsoLeft_symm_apply _] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RepresentationTheory.Homological.GroupCohomology.LowDegree | {
"line": 104,
"column": 2
} | {
"line": 106,
"column": 5
} | {
"line": 108,
"column": 0
} | [
{
"pp": "k G : Type u\ninst✝¹ : CommRing k\ninst✝ : Group G\nA : Rep k G\n⊢ (ModuleCat.Hom.hom (d₀₁ A)).ker = A.ρ.invariants",
"ppTerm": "?m.34",
"assigned": true,
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"Eq.mpr",
"Pi.Function.module",
"Submodule",
"Rep.V",
... | [] | ext x
simp only [LinearMap.mem_ker, mem_invariants, ← @sub_eq_zero _ _ _ x, funext_iff]
rfl | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RepresentationTheory.Homological.GroupCohomology.LowDegree | {
"line": 104,
"column": 2
} | {
"line": 106,
"column": 5
} | {
"line": 108,
"column": 0
} | [
{
"pp": "k G : Type u\ninst✝¹ : CommRing k\ninst✝ : Group G\nA : Rep k G\n⊢ (ModuleCat.Hom.hom (d₀₁ A)).ker = A.ρ.invariants",
"ppTerm": "?m.34",
"assigned": true,
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"Eq.mpr",
"Pi.Function.module",
"Submodule",
"Rep.V",
... | [] | ext x
simp only [LinearMap.mem_ker, mem_invariants, ← @sub_eq_zero _ _ _ x, funext_iff]
rfl | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RepresentationTheory.Homological.GroupCohomology.LowDegree | {
"line": 379,
"column": 2
} | {
"line": 379,
"column": 50
} | {
"line": 380,
"column": 2
} | [
{
"pp": "k G : Type u\ninst✝¹ : CommRing k\ninst✝ : Group G\nA : Rep k G\nf : ↥(cocycles₂ A)\ng : G\n⊢ f (1, g) = f (1, 1)",
"ppTerm": "?m.21",
"assigned": true,
"usedConstants": [
"Pi.Function.module",
"Submodule",
"Rep.V",
"Representation",
"MonoidHom.instFunLike",
... | [
"k G : Type u\ninst✝¹ : CommRing k\ninst✝ : Group G\nA : Rep k G\nf : ↥(cocycles₂ A)\ng : G\nthis : (A.ρ 1) (f (1, g)) + f (1, 1 * g) = f (1 * 1, g) + f (1, 1)\n⊢ f (1, g) = f (1, 1)"
] | have := ((mem_cocycles₂_iff f).1 f.2 1 1 g).symm | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.RingTheory.MvPolynomial.Ideal | {
"line": 91,
"column": 2
} | {
"line": 96,
"column": 79
} | {
"line": 98,
"column": 0
} | [
{
"pp": "σ : Type u_1\nR : Type u_2\ninst✝ : CommSemiring R\nn : ℕ\n⊢ idealOfVars σ R ^ n = restrictSupportIdeal R (⇑degree ⁻¹' Set.Ici n) ⋯",
"ppTerm": "?m.24",
"assigned": true,
"usedConstants": [
"Finsupp.instAddZeroClass",
"Eq.mpr",
"Nat.instCanonicallyOrderedAdd",
"Submo... | [] | rw [idealOfVars_eq_restrictSupportIdeal]
apply Submodule.restrictScalars_injective R
by_cases hn : n = 0
· simp [hn, Set.Ici_zero_eq_univ]
rw [Submodule.restrictScalars_pow hn]
simp [← restrictSupport_nsmul, ← degree_preimage_nsmul, hn, Set.Ici_nsmul_eq] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.MvPolynomial.Ideal | {
"line": 91,
"column": 2
} | {
"line": 96,
"column": 79
} | {
"line": 98,
"column": 0
} | [
{
"pp": "σ : Type u_1\nR : Type u_2\ninst✝ : CommSemiring R\nn : ℕ\n⊢ idealOfVars σ R ^ n = restrictSupportIdeal R (⇑degree ⁻¹' Set.Ici n) ⋯",
"ppTerm": "?m.24",
"assigned": true,
"usedConstants": [
"Finsupp.instAddZeroClass",
"Eq.mpr",
"Nat.instCanonicallyOrderedAdd",
"Submo... | [] | rw [idealOfVars_eq_restrictSupportIdeal]
apply Submodule.restrictScalars_injective R
by_cases hn : n = 0
· simp [hn, Set.Ici_zero_eq_univ]
rw [Submodule.restrictScalars_pow hn]
simp [← restrictSupport_nsmul, ← degree_preimage_nsmul, hn, Set.Ici_nsmul_eq] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.MvPowerSeries.Rename | {
"line": 152,
"column": 16
} | {
"line": 152,
"column": 38
} | {
"line": 152,
"column": 38
} | [
{
"pp": "σ : Type u_1\nτ : Type u_2\nγ : Type u_3\nR : Type u_4\nf : σ → τ\ng : τ → γ\ninst✝² : TendstoCofinite f\ninst✝¹ : CommSemiring R\ninst✝ : TendstoCofinite g\np : MvPowerSeries σ R\ny : γ →₀ ℕ\n⊢ ∀ (a : τ →₀ ℕ) (b : σ →₀ ℕ), mapDomain f b = a → (mapDomain (g ∘ f) b = y ↔ mapDomain g a = y)",
"ppTerm... | [] | grind [mapDomain_comp] | Lean.Elab.Tactic.evalGrind | Lean.Parser.Tactic.grind |
Mathlib.RingTheory.AdicCompletion.RingHom | {
"line": 97,
"column": 2
} | {
"line": 97,
"column": 29
} | {
"line": 98,
"column": 2
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝² : NonAssocSemiring R\ninst✝¹ : CommRing S\nI : Ideal S\ninst✝ : IsAdicComplete I S\nf : (n : ℕ) → R →+* S ⧸ I ^ n\nhf : ∀ {m n : ℕ} (hle : m ≤ n), (factorPow I hle).comp (f n) = f m\nF : R →+* S\nhF : ∀ (n : ℕ), (Ideal.Quotient.mk (I ^ n)).comp F = f n\n⊢ ⇑F = ⇑(liftR... | [
"R : Type u_1\nS : Type u_2\ninst✝² : NonAssocSemiring R\ninst✝¹ : CommRing S\nI : Ideal S\ninst✝ : IsAdicComplete I S\nf : (n : ℕ) → R →+* S ⧸ I ^ n\nhf : ∀ {m n : ℕ} (hle : m ≤ n), (factorPow I hle).comp (f n) = f m\nF : R →+* S\nhF : ∀ (n : ℕ), (Ideal.Quotient.mk (I ^ n)).comp F = f n\n⊢ ∀ (n : ℕ) (r : R), (Idea... | apply IsHausdorff.funext' I | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.RingTheory.AdicCompletion.Completeness | {
"line": 112,
"column": 47
} | {
"line": 116,
"column": 78
} | {
"line": 118,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝² : CommRing R\nI : Ideal R\nM : Type u_2\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nn : ℕ\nx : AdicCompletion I M\nhxn : ↑x n = 0\n⊢ (ofPowSMul I M n) (ofValEqZero I hxn) = x",
"ppTerm": "?m.26",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NonAssocSemi... | [] | by
ext i; by_cases! h : n ≤ i
· obtain ⟨k, rfl⟩ := Nat.exists_eq_add_of_le' h
rw [ofPowSMul_val_apply _ rfl, ofValEqZero, ofValEqZeroAux_prop]
rw [ofPowSMul_val_apply_eq_zero _ h.le, ← x.prop h.le, hxn, _root_.map_zero] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.Bialgebra.SymmetricAlgebra | {
"line": 33,
"column": 6
} | {
"line": 35,
"column": 10
} | {
"line": 35,
"column": 10
} | [
{
"pp": "R : Type u_1\ninst✝² : CommSemiring R\nM : Type u_2\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\n⊢ (↑(Algebra.TensorProduct.assoc R R R (SymmetricAlgebra R M) (SymmetricAlgebra R M) (SymmetricAlgebra R M))).comp\n ((Algebra.TensorProduct.map\n (lift\n ((TensorProduct.mk R... | [] | ext x
simp [Algebra.TensorProduct.one_def, TensorProduct.add_tmul, TensorProduct.tmul_add]
abel | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Bialgebra.SymmetricAlgebra | {
"line": 33,
"column": 6
} | {
"line": 35,
"column": 10
} | {
"line": 35,
"column": 10
} | [
{
"pp": "R : Type u_1\ninst✝² : CommSemiring R\nM : Type u_2\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\n⊢ (↑(Algebra.TensorProduct.assoc R R R (SymmetricAlgebra R M) (SymmetricAlgebra R M) (SymmetricAlgebra R M))).comp\n ((Algebra.TensorProduct.map\n (lift\n ((TensorProduct.mk R... | [] | ext x
simp [Algebra.TensorProduct.one_def, TensorProduct.add_tmul, TensorProduct.tmul_add]
abel | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.DedekindDomain.SelmerGroup | {
"line": 112,
"column": 28
} | {
"line": 112,
"column": 44
} | {
"line": 112,
"column": 44
} | [
{
"pp": "R : Type u\ninst✝⁴ : CommRing R\ninst✝³ : IsDedekindDomain R\nK : Type v\ninst✝² : Field K\ninst✝¹ : Algebra R K\ninst✝ : IsFractionRing R K\nv : HeightOneSpectrum R\nx✝¹ x✝ : Kˣ\n⊢ ↑(v.valuationOfNeZeroToFun (x✝¹ * x✝)) = ↑(v.valuationOfNeZeroToFun x✝¹ * v.valuationOfNeZeroToFun x✝)",
"ppTerm": "?... | [
"R : Type u\ninst✝⁴ : CommRing R\ninst✝³ : IsDedekindDomain R\nK : Type v\ninst✝² : Field K\ninst✝¹ : Algebra R K\ninst✝ : IsFractionRing R K\nv : HeightOneSpectrum R\nx✝¹ x✝ : Kˣ\n⊢ ↑(v.valuationOfNeZeroToFun (x✝¹ * x✝)) = ↑(v.valuationOfNeZeroToFun x✝¹) * ↑(v.valuationOfNeZeroToFun x✝)"
] | WithZero.coe_mul | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.DedekindDomain.SelmerGroup | {
"line": 142,
"column": 8
} | {
"line": 144,
"column": 97
} | {
"line": 144,
"column": 97
} | [
{
"pp": "R : Type u\ninst✝⁴ : CommRing R\ninst✝³ : IsDedekindDomain R\nK : Type v\ninst✝² : Field K\ninst✝¹ : Algebra R K\ninst✝ : IsFractionRing R K\nv : HeightOneSpectrum R\nn : ℕ\n⊢ (powMonoidHom n).range ≤ Subgroup.comap v.valuationOfNeZero (AddSubgroup.toSubgroup (AddSubgroup.zmultiples ↑n))",
"ppTerm"... | [] | rintro _ ⟨x, rfl⟩
exact
⟨v.valuationOfNeZero x, by simp only [powMonoidHom_apply, map_pow, Int.toAdd_pow]; rfl⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.DedekindDomain.SelmerGroup | {
"line": 142,
"column": 8
} | {
"line": 144,
"column": 97
} | {
"line": 144,
"column": 97
} | [
{
"pp": "R : Type u\ninst✝⁴ : CommRing R\ninst✝³ : IsDedekindDomain R\nK : Type v\ninst✝² : Field K\ninst✝¹ : Algebra R K\ninst✝ : IsFractionRing R K\nv : HeightOneSpectrum R\nn : ℕ\n⊢ (powMonoidHom n).range ≤ Subgroup.comap v.valuationOfNeZero (AddSubgroup.toSubgroup (AddSubgroup.zmultiples ↑n))",
"ppTerm"... | [] | rintro _ ⟨x, rfl⟩
exact
⟨v.valuationOfNeZero x, by simp only [powMonoidHom_apply, map_pow, Int.toAdd_pow]; rfl⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Regular.IsSMulRegular | {
"line": 86,
"column": 37
} | {
"line": 87,
"column": 70
} | {
"line": 89,
"column": 0
} | [
{
"pp": "R : Type u_1\nM : Type u_3\ninst✝² : Ring R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nN : Submodule R M\nr : R\n⊢ (∀ (x : M), r • N.mkQ x = 0 → N.mkQ x = 0) ↔ ∀ (x : M), r • x ∈ N → x ∈ N",
"ppTerm": "?m.44",
"assigned": true,
"usedConstants": [
"AddGroup.toSubtractionMonoid",
... | [] | by
simp_rw [← map_smul, N.mkQ_apply, Submodule.Quotient.mk_eq_zero] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.Ideal.AssociatedPrime.Localization | {
"line": 66,
"column": 6
} | {
"line": 68,
"column": 62
} | {
"line": 70,
"column": 0
} | [
{
"pp": "case h.refine_2.inr\nR : Type u_1\ninst✝⁹ : CommRing R\nS : Submonoid R\nR' : Type u_2\ninst✝⁸ : CommRing R'\ninst✝⁷ : Algebra R R'\nhSR' : IsLocalization S R'\nM : Type u_3\nM' : Type u_4\ninst✝⁶ : AddCommGroup M\ninst✝⁵ : Module R M\ninst✝⁴ : AddCommGroup M'\ninst✝³ : Module R M'\nf : M →ₗ[R] M'\nins... | [] | · use n
rw [mem_colon_singleton, mul_pow, mul_smul, ← mem_colon_singleton]
exact Ideal.pow_mem_of_mem _ (by simpa using! ht) n hn | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.RingTheory.Ideal.AssociatedPrime.Localization | {
"line": 105,
"column": 4
} | {
"line": 115,
"column": 95
} | {
"line": 117,
"column": 0
} | [
{
"pp": "case refine_2\nR : Type u_1\ninst✝⁹ : CommRing R\nS : Submonoid R\nR' : Type u_2\ninst✝⁸ : CommRing R'\ninst✝⁷ : Algebra R R'\nhSR' : IsLocalization S R'\nM : Type u_3\nM' : Type u_4\ninst✝⁶ : AddCommGroup M\ninst✝⁵ : Module R M\ninst✝⁴ : AddCommGroup M'\ninst✝³ : Module R M'\nf : M →ₗ[R] M'\ninst✝² : ... | [] | simp only [Ideal.mem_radical_iff, mem_bot, mem_colon_singleton, smul_smul] at hr
obtain ⟨k, hk⟩ := hr
have mem : r ^ k * (∏ a, g a).1 ∈ Ideal.comap (algebraMap R R') p := by
rw [hx]
use 1
simp_rw [pow_one, mem_colon_singleton, algebraMap_smul, ← IsLocalizedModule.mk'_smul,
hk, IsLocali... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Ideal.AssociatedPrime.Localization | {
"line": 105,
"column": 4
} | {
"line": 115,
"column": 95
} | {
"line": 117,
"column": 0
} | [
{
"pp": "case refine_2\nR : Type u_1\ninst✝⁹ : CommRing R\nS : Submonoid R\nR' : Type u_2\ninst✝⁸ : CommRing R'\ninst✝⁷ : Algebra R R'\nhSR' : IsLocalization S R'\nM : Type u_3\nM' : Type u_4\ninst✝⁶ : AddCommGroup M\ninst✝⁵ : Module R M\ninst✝⁴ : AddCommGroup M'\ninst✝³ : Module R M'\nf : M →ₗ[R] M'\ninst✝² : ... | [] | simp only [Ideal.mem_radical_iff, mem_bot, mem_colon_singleton, smul_smul] at hr
obtain ⟨k, hk⟩ := hr
have mem : r ^ k * (∏ a, g a).1 ∈ Ideal.comap (algebraMap R R') p := by
rw [hx]
use 1
simp_rw [pow_one, mem_colon_singleton, algebraMap_smul, ← IsLocalizedModule.mk'_smul,
hk, IsLocali... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Regular.RegularSequence | {
"line": 82,
"column": 53
} | {
"line": 82,
"column": 62
} | {
"line": 82,
"column": 62
} | [
{
"pp": "R : Type u_1\nS : Type u_2\nM : Type u_3\nM₂ : Type u_4\nM₃ : Type u_5\nM₄ : Type u_6\ninst✝⁴ : CommRing R\ninst✝³ : AddCommGroup M\ninst✝² : AddCommGroup M₂\ninst✝¹ : Module R M\ninst✝ : Module R M₂\nr : R\nrs : List R\n⊢ Ideal.ofList rs • (r • ⊤).mkQ.range = Ideal.ofList rs • ⊤",
"ppTerm": "?m.23... | [
"R : Type u_1\nS : Type u_2\nM : Type u_3\nM₂ : Type u_4\nM₃ : Type u_5\nM₄ : Type u_6\ninst✝⁴ : CommRing R\ninst✝³ : AddCommGroup M\ninst✝² : AddCommGroup M₂\ninst✝¹ : Module R M\ninst✝ : Module R M₂\nr : R\nrs : List R\n⊢ Ideal.ofList rs • ⊤ = Ideal.ofList rs • ⊤"
] | range_mkQ | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Regular.RegularSequence | {
"line": 90,
"column": 61
} | {
"line": 90,
"column": 70
} | {
"line": 90,
"column": 70
} | [
{
"pp": "R : Type u_1\nS : Type u_2\nM : Type u_3\nM₂ : Type u_4\nM₃ : Type u_5\nM₄ : Type u_6\ninst✝⁴ : CommRing R\ninst✝³ : AddCommGroup M\ninst✝² : AddCommGroup M₂\ninst✝¹ : Module R M\ninst✝ : Module R M₂\nr : R\nrs : List R\n⊢ r • (Ideal.ofList rs • ⊤).mkQ.range = r • ⊤",
"ppTerm": "?m.235",
"assig... | [
"R : Type u_1\nS : Type u_2\nM : Type u_3\nM₂ : Type u_4\nM₃ : Type u_5\nM₄ : Type u_6\ninst✝⁴ : CommRing R\ninst✝³ : AddCommGroup M\ninst✝² : AddCommGroup M₂\ninst✝¹ : Module R M\ninst✝ : Module R M₂\nr : R\nrs : List R\n⊢ r • ⊤ = r • ⊤"
] | range_mkQ | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Depth.Rees | {
"line": 136,
"column": 10
} | {
"line": 136,
"column": 88
} | {
"line": 137,
"column": 8
} | [
{
"pp": "R : Type u\ninst✝³ : CommRing R\ninst✝² : Small.{v, u} R\ninst✝¹ : IsNoetherianRing R\nI : Ideal R\nN : ModuleCat R\nNfin : Module.Finite R ↑N\nNsupp : Module.support R ↑N ⊆ PrimeSpectrum.zeroLocus ↑I\nn : ℕ\nih :\n ∀ (M : ModuleCat R) [Module.Finite R ↑M],\n I • ⊤ < ⊤ →\n ∀ (rs : List R),\n ... | [] | exact (smul_top_quotSMulTop_ne_top_of_smul_top_ne_top mem.1 smul_lt.ne).lt_top | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.RingTheory.DividedPowers.RatAlgebra | {
"line": 91,
"column": 2
} | {
"line": 91,
"column": 9
} | {
"line": 92,
"column": 2
} | [
{
"pp": "A : Type u_1\ninst✝¹ : CommSemiring A\nI : Ideal A\ninst✝ : DecidablePred fun x ↦ x ∈ I\nn : ℕ\nhn_fac : IsUnit ↑(n - 1)!\nm : ℕ\nhmn : m < n\nx y : A\nhx : x ∈ I\nhy : y ∈ I\nk : ℕ × ℕ\nhk : k ∈ Finset.antidiagonal m\n⊢ m.choose k.1 • (x ^ k.1 * y ^ k.2) = ↑m ! * ((↑k.1!)⁻¹ʳ * x ^ k.1 * ((↑k.2!)⁻¹ʳ * ... | [
"A : Type u_1\ninst✝¹ : CommSemiring A\nI : Ideal A\ninst✝ : DecidablePred fun x ↦ x ∈ I\nn : ℕ\nhn_fac : IsUnit ↑(n - 1)!\nm : ℕ\nhmn : m < n\nx y : A\nhx : x ∈ I\nhy : y ∈ I\nk : ℕ × ℕ\nhk : k ∈ Finset.antidiagonal m\n⊢ x ^ k.1 * y ^ k.2 * ↑(m.choose k.1) = x ^ k.1 * y ^ k.2 * ↑m ! * (↑k.1!)⁻¹ʳ * (↑k.2!)⁻¹ʳ"
] | ring_nf | Mathlib.Tactic.RingNF._aux_Mathlib_Tactic_Ring_RingNF___elabRules_Mathlib_Tactic_RingNF_ringNF_1 | Mathlib.Tactic.RingNF.ringNF |
Mathlib.RingTheory.DividedPowerAlgebra.Init | {
"line": 287,
"column": 6
} | {
"line": 290,
"column": 49
} | {
"line": 292,
"column": 0
} | [
{
"pp": "case dp.add\nR : Type u_4\nM : Type u_5\nι : Type u_6\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nv : ι → M\nhv : Submodule.span R (Set.range v) = ⊤\nm✝ m n : M\nhm_mem : m ∈ Submodule.span R (Set.range v)\nhn_mem : n ∈ Submodule.span R (Set.range v)\nhm :\n ∀ (x : DividedPowerA... | [] | rw [dp_add, mul_sum]
apply sum_mem (fun c hc ↦ ?_)
rw [← mul_assoc]
exact hn (x * dp R c.1 m) c.2 (hm x c.1 hx) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.DividedPowerAlgebra.Init | {
"line": 287,
"column": 6
} | {
"line": 290,
"column": 49
} | {
"line": 292,
"column": 0
} | [
{
"pp": "case dp.add\nR : Type u_4\nM : Type u_5\nι : Type u_6\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nv : ι → M\nhv : Submodule.span R (Set.range v) = ⊤\nm✝ m n : M\nhm_mem : m ∈ Submodule.span R (Set.range v)\nhn_mem : n ∈ Submodule.span R (Set.range v)\nhm :\n ∀ (x : DividedPowerA... | [] | rw [dp_add, mul_sum]
apply sum_mem (fun c hc ↦ ?_)
rw [← mul_assoc]
exact hn (x * dp R c.1 m) c.2 (hm x c.1 hx) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.DividedPowers.Padic | {
"line": 117,
"column": 25
} | {
"line": 125,
"column": 14
} | {
"line": 127,
"column": 0
} | [
{
"pp": "p : ℕ\nhp : Fact (Nat.Prime p)\nn : ℕ\nx : ℤ_[p]\nhx : x ∈ Ideal.span {↑p}\n⊢ ‖dpow' p n ↑x‖ ≤ 1",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"Norm.norm",
"Eq.mpr",
"NegZeroClass.toNeg",
"MulOne.toOne",
"Int.instIsStrictOrderedRing",
"Real.pa... | [] | by
unfold dpow'
by_cases hn : n = 0
· simp [hn]
· apply le_trans (dpow'_norm_le_of_ne_zero p hn hx)
rw [← zpow_neg_one, ← zpow_zero ↑p]
gcongr
· exact_mod_cast Nat.Prime.one_le hp.elim
· norm_num | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.DividedPowerAlgebra.Init | {
"line": 362,
"column": 6
} | {
"line": 362,
"column": 11
} | {
"line": 362,
"column": 12
} | [
{
"pp": "R : Type u_2\nM : Type u_3\ninst✝⁴ : CommSemiring R\ninst✝³ : AddCommMonoid M\ninst✝² : Module R M\nA : Type u_4\ninst✝¹ : CommSemiring A\ninst✝ : Algebra R A\nI : Ideal A\nhI : DividedPowers I\ng : M →ₗ[R] A\nhg : ∀ (m : M), g m ∈ I\np : MvPolynomial (ℕ × M) R\n⊢ (lift hI g hg) ↑p = (aeval fun nm ↦ hI... | [
"R : Type u_2\nM : Type u_3\ninst✝⁴ : CommSemiring R\ninst✝³ : AddCommMonoid M\ninst✝² : Module R M\nA : Type u_4\ninst✝¹ : CommSemiring A\ninst✝ : Algebra R A\nI : Ideal A\nhI : DividedPowers I\ng : M →ₗ[R] A\nhg : ∀ (m : M), g m ∈ I\np : MvPolynomial (ℕ × M) R\n⊢ (lift' ⋯ ⋯ ⋯ ⋯) ↑p = (aeval fun nm ↦ hI.dpow nm.1 ... | lift, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.DividedPowers.RatAlgebra | {
"line": 161,
"column": 4
} | {
"line": 161,
"column": 11
} | {
"line": 163,
"column": 0
} | [
{
"pp": "A : Type u_1\ninst✝¹ : CommSemiring A\nI : Ideal A\ninst✝ : DecidablePred fun x ↦ x ∈ I\nn : ℕ\nhn_fac : IsUnit ↑(n - 1)!\nm k : ℕ\nhk : k ≠ 0\nhkm : m * k < n\nx : A\nhx : x ∈ I\nhmn : m < n\nhm0 : ¬m = 0\nhkn : k < n\n⊢ ↑(m.uniformBell k) * ↑k ! ^ m * ↑m ! = ↑k ! ^ m * (↑m ! * ↑(m.uniformBell k))",
... | [] | ring_nf | Mathlib.Tactic.RingNF._aux_Mathlib_Tactic_Ring_RingNF___elabRules_Mathlib_Tactic_RingNF_ringNF_1 | Mathlib.Tactic.RingNF.ringNF |
Mathlib.RingTheory.DividedPowerAlgebra.Init | {
"line": 366,
"column": 56
} | {
"line": 366,
"column": 61
} | {
"line": 366,
"column": 62
} | [
{
"pp": "R : Type u_2\nM : Type u_3\ninst✝⁴ : CommSemiring R\ninst✝³ : AddCommMonoid M\ninst✝² : Module R M\nA : Type u_4\ninst✝¹ : CommSemiring A\ninst✝ : Algebra R A\nI : Ideal A\nhI : DividedPowers I\ng : M →ₗ[R] A\nhg : ∀ (m : M), g m ∈ I\nn : ℕ\nm : M\n⊢ (lift hI g hg) (dp R n m) = hI.dpow n (g m)",
"p... | [
"R : Type u_2\nM : Type u_3\ninst✝⁴ : CommSemiring R\ninst✝³ : AddCommMonoid M\ninst✝² : Module R M\nA : Type u_4\ninst✝¹ : CommSemiring A\ninst✝ : Algebra R A\nI : Ideal A\nhI : DividedPowers I\ng : M →ₗ[R] A\nhg : ∀ (m : M), g m ∈ I\nn : ℕ\nm : M\n⊢ (lift' ⋯ ⋯ ⋯ ⋯) (dp R n m) = hI.dpow n (g m)"
] | lift, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Extension.ExtendScalars | {
"line": 151,
"column": 13
} | {
"line": 151,
"column": 52
} | {
"line": 151,
"column": 53
} | [
{
"pp": "R : Type u\nS : Type v\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nP : Extension R S\n⊢ h1Cotangentι ∘ₗ H1Cotangent.map (defaultHom R S P) =\n ↑P.cotangentExtendScalarsEquiv ∘ₗ\n h1Cotangentι ∘ₗ\n H1Cotangent.map\n ((defaultHom P.Ring S P.extendScalars).comp\... | [
"R : Type u\nS : Type v\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nP : Extension R S\n⊢ ↑P.cotangentExtendScalarsEquiv.symm ∘ₗ h1Cotangentι ∘ₗ H1Cotangent.map (defaultHom R S P) =\n h1Cotangentι ∘ₗ\n H1Cotangent.map\n ((defaultHom P.Ring S P.extendScalars).comp\n ((Gener... | ← LinearEquiv.toLinearMap_symm_comp_eq, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.DividedPowers.SubDPIdeal | {
"line": 568,
"column": 2
} | {
"line": 568,
"column": 65
} | {
"line": 569,
"column": 2
} | [
{
"pp": "A : Type u_1\ninst✝¹ : CommRing A\nI : Ideal A\nhI : DividedPowers I\nB : Type u_2\ninst✝ : CommRing B\nf : A →+* B\nhIf : hI.IsSubDPIdeal (RingHom.ker f ⊓ I)\nn : ℕ\na : A\nha : a ∈ I\nh : ∃ a_1, f ↑a_1 = f a\n⊢ hI.dpow n ↑(Classical.choose h) - hI.dpow n a ∈ RingHom.ker f",
"ppTerm": "?m.105",
... | [
"A : Type u_1\ninst✝¹ : CommRing A\nI : Ideal A\nhI : DividedPowers I\nB : Type u_2\ninst✝ : CommRing B\nf : A →+* B\nhIf : hI.IsSubDPIdeal (RingHom.ker f ⊓ I)\nn : ℕ\na : A\nha : a ∈ I\nh : ∃ a_1, f ↑a_1 = f a\n⊢ ↑(Classical.choose h) - a ∈ RingHom.ker f"
] | apply (hI.isSubDPIdeal_inf_iff.mp hIf) (Submodule.coe_mem _) ha | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.RingTheory.Grassmannian | {
"line": 136,
"column": 10
} | {
"line": 136,
"column": 54
} | {
"line": 137,
"column": 8
} | [
{
"pp": "R : Type u\ninst✝⁶ : CommRing R\nM : Type v\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : Module R M\nk : ℕ\nA : Type w\ninst✝³ : CommRing A\ninst✝² : Algebra R A\nB : Type w\ninst✝¹ : CommRing B\ninst✝ : Algebra R B\nf : A →ₐ[R] B\nN : G(k, A ⊗[R] M; A)\nthis✝ : Algebra A B := f.toAlgebra\nthis : IsScalarTower R... | [] | simpa using Module.rankAtStalk_baseChange .. | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.RingTheory.Grassmannian | {
"line": 136,
"column": 10
} | {
"line": 136,
"column": 54
} | {
"line": 137,
"column": 8
} | [
{
"pp": "R : Type u\ninst✝⁶ : CommRing R\nM : Type v\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : Module R M\nk : ℕ\nA : Type w\ninst✝³ : CommRing A\ninst✝² : Algebra R A\nB : Type w\ninst✝¹ : CommRing B\ninst✝ : Algebra R B\nf : A →ₐ[R] B\nN : G(k, A ⊗[R] M; A)\nthis✝ : Algebra A B := f.toAlgebra\nthis : IsScalarTower R... | [] | simpa using Module.rankAtStalk_baseChange .. | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Grassmannian | {
"line": 136,
"column": 10
} | {
"line": 136,
"column": 54
} | {
"line": 137,
"column": 8
} | [
{
"pp": "R : Type u\ninst✝⁶ : CommRing R\nM : Type v\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : Module R M\nk : ℕ\nA : Type w\ninst✝³ : CommRing A\ninst✝² : Algebra R A\nB : Type w\ninst✝¹ : CommRing B\ninst✝ : Algebra R B\nf : A →ₐ[R] B\nN : G(k, A ⊗[R] M; A)\nthis✝ : Algebra A B := f.toAlgebra\nthis : IsScalarTower R... | [] | simpa using Module.rankAtStalk_baseChange .. | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.HahnSeries.HEval | {
"line": 118,
"column": 66
} | {
"line": 148,
"column": 36
} | {
"line": 150,
"column": 0
} | [
{
"pp": "Γ : Type u_1\nR : Type u_3\nV : Type u_4\ninst✝⁵ : AddCommMonoid Γ\ninst✝⁴ : LinearOrder Γ\ninst✝³ : IsOrderedCancelAddMonoid Γ\ninst✝² : CommRing R\ninst✝¹ : CommRing V\ninst✝ : Algebra R V\nx : V⟦Γ⟧\na b : PowerSeries R\n⊢ (powerSeriesFamily x (a * b)).hsum = ((powerSeriesFamily x a).mul (powerSeries... | [] | by
by_cases h : 0 < x.orderTop;
· ext g
simp only [coeff_hsum_eq_sum, smulFamily_toFun, h, powers_of_orderTop_pos,
HahnSeries.coeff_smul, mul_toFun, Algebra.mul_smul_comm, Algebra.smul_mul_assoc]
rw [sum_subset (support_powerSeriesFamily_subset a b g)
(fun i hi his ↦ by simpa [h, PowerSeries.coe... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.HahnSeries.HEval | {
"line": 177,
"column": 4
} | {
"line": 178,
"column": 55
} | {
"line": 179,
"column": 2
} | [
{
"pp": "Γ : Type u_1\nΓ' : Type u_2\nR : Type u_3\nV : Type u_4\nα : Type u_5\nβ : Type u_6\nσ : Type u_7\ninst✝³ : AddCommMonoid Γ\ninst✝² : LinearOrder Γ\ninst✝¹ : IsOrderedCancelAddMonoid Γ\ninst✝ : CommRing R\nx : R⟦Γ⟧\n⊢ (powerSeriesFamily x 0).hsum = 0",
"ppTerm": "?m.77",
"assigned": true,
"... | [] | simp only [hsum, smulFamily_toFun, map_zero, zero_smul,
coeff_zero, finsum_zero, mk_eq_zero, Pi.zero_def] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.RingTheory.HahnSeries.HEval | {
"line": 177,
"column": 4
} | {
"line": 178,
"column": 55
} | {
"line": 179,
"column": 2
} | [
{
"pp": "Γ : Type u_1\nΓ' : Type u_2\nR : Type u_3\nV : Type u_4\nα : Type u_5\nβ : Type u_6\nσ : Type u_7\ninst✝³ : AddCommMonoid Γ\ninst✝² : LinearOrder Γ\ninst✝¹ : IsOrderedCancelAddMonoid Γ\ninst✝ : CommRing R\nx : R⟦Γ⟧\n⊢ (powerSeriesFamily x 0).hsum = 0",
"ppTerm": "?m.77",
"assigned": true,
"... | [] | simp only [hsum, smulFamily_toFun, map_zero, zero_smul,
coeff_zero, finsum_zero, mk_eq_zero, Pi.zero_def] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.HahnSeries.HEval | {
"line": 177,
"column": 4
} | {
"line": 178,
"column": 55
} | {
"line": 179,
"column": 2
} | [
{
"pp": "Γ : Type u_1\nΓ' : Type u_2\nR : Type u_3\nV : Type u_4\nα : Type u_5\nβ : Type u_6\nσ : Type u_7\ninst✝³ : AddCommMonoid Γ\ninst✝² : LinearOrder Γ\ninst✝¹ : IsOrderedCancelAddMonoid Γ\ninst✝ : CommRing R\nx : R⟦Γ⟧\n⊢ (powerSeriesFamily x 0).hsum = 0",
"ppTerm": "?m.77",
"assigned": true,
"... | [] | simp only [hsum, smulFamily_toFun, map_zero, zero_smul,
coeff_zero, finsum_zero, mk_eq_zero, Pi.zero_def] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.HahnSeries.Summable | {
"line": 188,
"column": 6
} | {
"line": 188,
"column": 78
} | {
"line": 189,
"column": 6
} | [
{
"pp": "Γ : Type u_1\nΓ' : Type u_2\nR : Type u_3\nV : Type u_4\nα : Type u_5\nβ : Type u_6\ninst✝¹ : PartialOrder Γ\ninst✝ : AddCommMonoid R\ns : SummableFamily Γ R α\ng : Γ\n⊢ g ∉ ⋃ a, (s a).support → g ∉ Function.support fun g ↦ ∑ᶠ (i : α), (s i).coeff g",
"ppTerm": "?m.32",
"assigned": true,
"u... | [
"Γ : Type u_1\nΓ' : Type u_2\nR : Type u_3\nV : Type u_4\nα : Type u_5\nβ : Type u_6\ninst✝¹ : PartialOrder Γ\ninst✝ : AddCommMonoid R\ns : SummableFamily Γ R α\ng : Γ\n⊢ (∀ (x : α), g ∉ (s x).support) → ∑ᶠ (i : α), (s i).coeff g = 0"
] | rw [Set.mem_iUnion, not_exists, Function.mem_support, Classical.not_not] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.HahnSeries.HahnEmbedding | {
"line": 76,
"column": 2
} | {
"line": 101,
"column": 65
} | {
"line": 102,
"column": 0
} | [
{
"pp": "M : Type u_1\ninst✝² : AddCommGroup M\ninst✝¹ : LinearOrder M\ninst✝ : IsOrderedAddMonoid M\n⊢ ∃ f,\n Function.Injective ⇑f ∧\n ∀ (a : M), ArchimedeanClass.mk a = (FiniteArchimedeanClass.withTopOrderIso M) (ofLex (f a)).orderTop",
"ppTerm": "?m.35",
"assigned": true,
"usedConstants"... | [] | let f₁ := DivisibleHull.coeOrderAddMonoidHom M
have hf₁ : Function.Injective f₁ := DivisibleHull.coe_injective
have hf₁class (a : M) : mk a = (DivisibleHull.archimedeanClassOrderIso M).symm (mk (f₁ a)) := by
simp [f₁]
obtain ⟨f₂', hf₂', hf₂class'⟩ := hahnEmbedding_isOrderedModule_rat (DivisibleHull M)
let f... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.HahnSeries.HahnEmbedding | {
"line": 76,
"column": 2
} | {
"line": 101,
"column": 65
} | {
"line": 102,
"column": 0
} | [
{
"pp": "M : Type u_1\ninst✝² : AddCommGroup M\ninst✝¹ : LinearOrder M\ninst✝ : IsOrderedAddMonoid M\n⊢ ∃ f,\n Function.Injective ⇑f ∧\n ∀ (a : M), ArchimedeanClass.mk a = (FiniteArchimedeanClass.withTopOrderIso M) (ofLex (f a)).orderTop",
"ppTerm": "?m.35",
"assigned": true,
"usedConstants"... | [] | let f₁ := DivisibleHull.coeOrderAddMonoidHom M
have hf₁ : Function.Injective f₁ := DivisibleHull.coe_injective
have hf₁class (a : M) : mk a = (DivisibleHull.archimedeanClassOrderIso M).symm (mk (f₁ a)) := by
simp [f₁]
obtain ⟨f₂', hf₂', hf₂class'⟩ := hahnEmbedding_isOrderedModule_rat (DivisibleHull M)
let f... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.HopfAlgebra.GroupLike | {
"line": 28,
"column": 2
} | {
"line": 28,
"column": 97
} | {
"line": 30,
"column": 0
} | [
{
"pp": "R : Type u_1\nA : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : Semiring A\ninst✝ : HopfAlgebra R A\na : A\nha : IsGroupLikeElem R a\n⊢ (antipode R) a * a = 1",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"IsGroupLikeElem.comul_eq_tmul_self",
"NonAssocSemiring.toAddCom... | [] | simpa [ha, -mul_antipode_lTensor_comul_apply] using mul_antipode_rTensor_comul_apply (R := R) a | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.RingTheory.HopfAlgebra.GroupLike | {
"line": 28,
"column": 2
} | {
"line": 28,
"column": 97
} | {
"line": 30,
"column": 0
} | [
{
"pp": "R : Type u_1\nA : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : Semiring A\ninst✝ : HopfAlgebra R A\na : A\nha : IsGroupLikeElem R a\n⊢ (antipode R) a * a = 1",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"IsGroupLikeElem.comul_eq_tmul_self",
"NonAssocSemiring.toAddCom... | [] | simpa [ha, -mul_antipode_lTensor_comul_apply] using mul_antipode_rTensor_comul_apply (R := R) a | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.HopfAlgebra.GroupLike | {
"line": 28,
"column": 2
} | {
"line": 28,
"column": 97
} | {
"line": 30,
"column": 0
} | [
{
"pp": "R : Type u_1\nA : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : Semiring A\ninst✝ : HopfAlgebra R A\na : A\nha : IsGroupLikeElem R a\n⊢ (antipode R) a * a = 1",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"IsGroupLikeElem.comul_eq_tmul_self",
"NonAssocSemiring.toAddCom... | [] | simpa [ha, -mul_antipode_lTensor_comul_apply] using mul_antipode_rTensor_comul_apply (R := R) a | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.HahnSeries.Summable | {
"line": 817,
"column": 4
} | {
"line": 824,
"column": 26
} | {
"line": 826,
"column": 0
} | [
{
"pp": "case inr\nΓ : Type u_1\nR : Type u_3\ninst✝³ : AddCommMonoid Γ\ninst✝² : LinearOrder Γ\ninst✝¹ : IsOrderedCancelAddMonoid Γ\ninst✝ : CommRing R\nx : R⟦Γ⟧\nh : 0 < (x - 1).orderTop\nh✝ : Nontrivial R\n⊢ IsUnit x",
"ppTerm": "?inr",
"assigned": true,
"usedConstants": [
"Iff.mpr",
... | [] | refine isUnit_of_isUnit_leadingCoeff_AddUnitOrder ?_ ?_
· rw [(x.orderTop_self_sub_one_pos_iff.mp h).2]
exact isUnit_one
· have := (x.orderTop_self_sub_one_pos_iff.mp h).1
rw [← order_eq_orderTop_of_ne_zero
(fun h ↦ WithTop.top_ne_zero (orderTop_eq_top.mpr h ▸ this)), WithTop.coe_eq_zero] at... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.HahnSeries.Summable | {
"line": 817,
"column": 4
} | {
"line": 824,
"column": 26
} | {
"line": 826,
"column": 0
} | [
{
"pp": "case inr\nΓ : Type u_1\nR : Type u_3\ninst✝³ : AddCommMonoid Γ\ninst✝² : LinearOrder Γ\ninst✝¹ : IsOrderedCancelAddMonoid Γ\ninst✝ : CommRing R\nx : R⟦Γ⟧\nh : 0 < (x - 1).orderTop\nh✝ : Nontrivial R\n⊢ IsUnit x",
"ppTerm": "?inr",
"assigned": true,
"usedConstants": [
"Iff.mpr",
... | [] | refine isUnit_of_isUnit_leadingCoeff_AddUnitOrder ?_ ?_
· rw [(x.orderTop_self_sub_one_pos_iff.mp h).2]
exact isUnit_one
· have := (x.orderTop_self_sub_one_pos_iff.mp h).1
rw [← order_eq_orderTop_of_ne_zero
(fun h ↦ WithTop.top_ne_zero (orderTop_eq_top.mpr h ▸ this)), WithTop.coe_eq_zero] at... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.HahnSeries.Summable | {
"line": 870,
"column": 2
} | {
"line": 870,
"column": 33
} | {
"line": 871,
"column": 2
} | [
{
"pp": "Γ : Type u_1\nR : Type u_3\ninst✝³ : AddCommGroup Γ\ninst✝² : LinearOrder Γ\ninst✝¹ : IsOrderedAddMonoid Γ\ninst✝ : Field R\na : Γ\nr : R\n⊢ ((single a) r)⁻¹ = (single (-a)) r⁻¹",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"NegZeroClass.toNeg",
"ZeroHom.funLike",
... | [
"case inl\nΓ : Type u_1\nR : Type u_3\ninst✝³ : AddCommGroup Γ\ninst✝² : LinearOrder Γ\ninst✝¹ : IsOrderedAddMonoid Γ\ninst✝ : Field R\na : Γ\n⊢ ((single a) 0)⁻¹ = (single (-a)) 0⁻¹",
"case inr\nΓ : Type u_1\nR : Type u_3\ninst✝³ : AddCommGroup Γ\ninst✝² : LinearOrder Γ\ninst✝¹ : IsOrderedAddMonoid Γ\ninst✝ : Fie... | obtain rfl | hr := eq_or_ne r 0 | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.RingTheory.HahnSeries.Summable | {
"line": 890,
"column": 22
} | {
"line": 890,
"column": 69
} | {
"line": 891,
"column": 2
} | [
{
"pp": "Γ : Type u_1\nΓ' : Type u_2\nR : Type u_3\nV : Type u_4\nα : Type u_5\nβ : Type u_6\ninst✝³ : AddCommGroup Γ\ninst✝² : LinearOrder Γ\ninst✝¹ : IsOrderedAddMonoid Γ\ninst✝ : Field R\nq : ℚ\nx : R⟦Γ⟧\n⊢ q • x = ↑q * x",
"ppTerm": "?m.118",
"assigned": true,
"usedConstants": [
"NonAssocS... | [] | ext; simp [← single_zero_ratCast, Rat.smul_def] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.HahnSeries.Summable | {
"line": 890,
"column": 22
} | {
"line": 890,
"column": 69
} | {
"line": 891,
"column": 2
} | [
{
"pp": "Γ : Type u_1\nΓ' : Type u_2\nR : Type u_3\nV : Type u_4\nα : Type u_5\nβ : Type u_6\ninst✝³ : AddCommGroup Γ\ninst✝² : LinearOrder Γ\ninst✝¹ : IsOrderedAddMonoid Γ\ninst✝ : Field R\nq : ℚ\nx : R⟦Γ⟧\n⊢ q • x = ↑q * x",
"ppTerm": "?m.118",
"assigned": true,
"usedConstants": [
"NonAssocS... | [] | ext; simp [← single_zero_ratCast, Rat.smul_def] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.IdealFilter.Basic | {
"line": 125,
"column": 42
} | {
"line": 125,
"column": 57
} | {
"line": 125,
"column": 57
} | [
{
"pp": "A : Type u_1\ninst✝ : Ring A\nF : IdealFilter A\nI : Ideal A\nx : A\nhx : x ∈ I\nJ : Ideal A\nhJ : J ∈ ↑F\n⊢ ⊤ = Submodule.colon I {x}",
"ppTerm": "?m.35",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Submodule",
"SetLike.mem_coe._simp_1",
"Submodule.colon",
... | [] | simpa [eq_comm] | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.RingTheory.IdealFilter.Basic | {
"line": 125,
"column": 42
} | {
"line": 125,
"column": 57
} | {
"line": 125,
"column": 57
} | [
{
"pp": "A : Type u_1\ninst✝ : Ring A\nF : IdealFilter A\nI : Ideal A\nx : A\nhx : x ∈ I\nJ : Ideal A\nhJ : J ∈ ↑F\n⊢ ⊤ = Submodule.colon I {x}",
"ppTerm": "?m.35",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Submodule",
"SetLike.mem_coe._simp_1",
"Submodule.colon",
... | [] | simpa [eq_comm] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.IdealFilter.Basic | {
"line": 125,
"column": 42
} | {
"line": 125,
"column": 57
} | {
"line": 125,
"column": 57
} | [
{
"pp": "A : Type u_1\ninst✝ : Ring A\nF : IdealFilter A\nI : Ideal A\nx : A\nhx : x ∈ I\nJ : Ideal A\nhJ : J ∈ ↑F\n⊢ ⊤ = Submodule.colon I {x}",
"ppTerm": "?m.35",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Submodule",
"SetLike.mem_coe._simp_1",
"Submodule.colon",
... | [] | simpa [eq_comm] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.IdealFilter.Basic | {
"line": 177,
"column": 4
} | {
"line": 185,
"column": 43
} | {
"line": 186,
"column": 2
} | [
{
"pp": "case mp\nA : Type u_1\ninst✝ : Ring A\nF : IdealFilter A\n⊢ F.IsGabriel → F.IsUniform ∧ F • F = F",
"ppTerm": "?mp",
"assigned": true,
"usedConstants": [
"Set.ext",
"OrderDual.toDual",
"IdealFilter.IsUniform",
"Submodule.colon",
"Semiring.toModule",
"Equi... | [] | intro hF
refine ⟨hF.toIsUniform, ?_⟩
ext I
constructor <;> intro hI
· rcases hI with ⟨J, hJ, htors⟩
refine hF.gabriel_closed I ⟨J, hJ, fun x hx ↦ ?_⟩
rcases htors x hx with ⟨K, hK, hincl⟩
exact Order.PFilter.mem_of_le hincl hK
· exact ⟨I, hI, isTorsionQuot_self F I⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.IdealFilter.Basic | {
"line": 177,
"column": 4
} | {
"line": 185,
"column": 43
} | {
"line": 186,
"column": 2
} | [
{
"pp": "case mp\nA : Type u_1\ninst✝ : Ring A\nF : IdealFilter A\n⊢ F.IsGabriel → F.IsUniform ∧ F • F = F",
"ppTerm": "?mp",
"assigned": true,
"usedConstants": [
"Set.ext",
"OrderDual.toDual",
"IdealFilter.IsUniform",
"Submodule.colon",
"Semiring.toModule",
"Equi... | [] | intro hF
refine ⟨hF.toIsUniform, ?_⟩
ext I
constructor <;> intro hI
· rcases hI with ⟨J, hJ, htors⟩
refine hF.gabriel_closed I ⟨J, hJ, fun x hx ↦ ?_⟩
rcases htors x hx with ⟨K, hK, hincl⟩
exact Order.PFilter.mem_of_le hincl hK
· exact ⟨I, hI, isTorsionQuot_self F I⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Ideal.KrullsHeightTheorem | {
"line": 287,
"column": 2
} | {
"line": 296,
"column": 32
} | {
"line": 298,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsNoetherianRing R\nI : Ideal R\nhI : I ≠ ⊤\n⊢ ∃ J ≤ I, Submodule.spanRank J = ↑I.height ∧ J.height = I.height",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"Ideal.height_le_spanRank",
"Eq.mpr",
"False... | [] | obtain ⟨J, hJ₁, hJ₂, hJ₃⟩ := exists_spanRank_le_and_le_height_of_le_height I _
(ENat.coe_toNat_le_self I.height)
rw [ENat.coe_toNat_eq_self.mpr (Ideal.height_ne_top hI)] at hJ₃
refine ⟨J, hJ₁, le_antisymm ?_ (le_trans ?_ (J.height_le_spanRank ?_)),
le_antisymm (Ideal.height_mono hJ₁) hJ₃⟩
· convert! hJ₂
... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Ideal.KrullsHeightTheorem | {
"line": 287,
"column": 2
} | {
"line": 296,
"column": 32
} | {
"line": 298,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsNoetherianRing R\nI : Ideal R\nhI : I ≠ ⊤\n⊢ ∃ J ≤ I, Submodule.spanRank J = ↑I.height ∧ J.height = I.height",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"Ideal.height_le_spanRank",
"Eq.mpr",
"False... | [] | obtain ⟨J, hJ₁, hJ₂, hJ₃⟩ := exists_spanRank_le_and_le_height_of_le_height I _
(ENat.coe_toNat_le_self I.height)
rw [ENat.coe_toNat_eq_self.mpr (Ideal.height_ne_top hI)] at hJ₃
refine ⟨J, hJ₁, le_antisymm ?_ (le_trans ?_ (J.height_le_spanRank ?_)),
le_antisymm (Ideal.height_mono hJ₁) hJ₃⟩
· convert! hJ₂
... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.KrullDimension.Regular | {
"line": 83,
"column": 2
} | {
"line": 83,
"column": 33
} | {
"line": 84,
"column": 2
} | [
{
"pp": "R : Type u_1\ninst✝³ : CommRing R\nM : Type u_2\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\ninst✝ : Module.Finite R M\nx : R\nhn : ∀ p ∈ (annihilator R M).minimalPrimes, x ∉ p\na✝ : Nontrivial M\n⊢ supportDim R (QuotSMulTop x M) + 1 ≤ supportDim R M",
"ppTerm": "?m.42",
"assigned": true,
... | [
"R : Type u_1\ninst✝³ : CommRing R\nM : Type u_2\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\ninst✝ : Module.Finite R M\nx : R\nhn : ∀ p ∈ (annihilator R M).minimalPrimes, x ∉ p\na✝¹ : Nontrivial M\na✝ : Nontrivial (QuotSMulTop x M)\n⊢ supportDim R (QuotSMulTop x M) + 1 ≤ supportDim R M"
] | nontriviality (QuotSMulTop x M) | Mathlib.Tactic.Nontriviality.elabNontriviality | Mathlib.Tactic.Nontriviality.nontriviality |
Mathlib.RingTheory.KrullDimension.Regular | {
"line": 203,
"column": 38
} | {
"line": 203,
"column": 74
} | {
"line": 203,
"column": 74
} | [
{
"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 : ... | [] | simp [span_singleton_eq_top.mpr isu] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.RingTheory.KrullDimension.Regular | {
"line": 203,
"column": 38
} | {
"line": 203,
"column": 74
} | {
"line": 203,
"column": 74
} | [
{
"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 : ... | [] | simp [span_singleton_eq_top.mpr isu] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.KrullDimension.Regular | {
"line": 203,
"column": 38
} | {
"line": 203,
"column": 74
} | {
"line": 203,
"column": 74
} | [
{
"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 : ... | [] | simp [span_singleton_eq_top.mpr isu] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Ideal.KrullsHeightTheorem | {
"line": 483,
"column": 2
} | {
"line": 483,
"column": 99
} | {
"line": 484,
"column": 2
} | [
{
"pp": "R : Type u_1\ninst✝⁸ : CommRing R\ninst✝⁷ : IsNoetherianRing R\nS : Type u_2\ninst✝⁶ : CommRing S\ninst✝⁵ : Algebra R S\ninst✝⁴ : IsNoetherianRing S\ninst✝³ : Algebra.HasGoingDown R S\np : Ideal R\ninst✝² : p.IsPrime\nP : Ideal S\ninst✝¹ : P.IsPrime\ninst✝ : P.LiesOver p\nlp : LTSeries (PrimeSpectrum R... | [
"R : Type u_1\ninst✝⁸ : CommRing R\ninst✝⁷ : IsNoetherianRing R\nS : Type u_2\ninst✝⁶ : CommRing S\ninst✝⁵ : Algebra R S\ninst✝⁴ : IsNoetherianRing S\ninst✝³ : Algebra.HasGoingDown R S\np : Ideal R\ninst✝² : p.IsPrime\nP : Ideal S\ninst✝¹ : P.IsPrime\ninst✝ : P.LiesOver p\nlp : LTSeries (PrimeSpectrum R)\nhlp : Rel... | rw [← hlenp, ← hlenq, ← Nat.cast_add, ← this, (⟨P, ‹_›⟩ : PrimeSpectrum S).height_eq_orderHeight] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.LocalIso | {
"line": 142,
"column": 6
} | {
"line": 142,
"column": 20
} | {
"line": 142,
"column": 21
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝⁸ : CommSemiring R\ninst✝⁷ : CommSemiring S\ninst✝⁶ : Algebra R S\nT : Type u_3\ninst✝⁵ : CommSemiring T\ninst✝⁴ : Algebra S T\ninst✝³ : Algebra R T\ninst✝² : IsScalarTower R S T\ninst✝¹ : IsLocalIso R S\ninst✝ : IsLocalIso S T\ns : Set S := {g | IsStandardOpenImmersion... | [
"R : Type u_1\nS : Type u_2\ninst✝⁸ : CommSemiring R\ninst✝⁷ : CommSemiring S\ninst✝⁶ : Algebra R S\nT : Type u_3\ninst✝⁵ : CommSemiring T\ninst✝⁴ : Algebra S T\ninst✝³ : Algebra R T\ninst✝² : IsScalarTower R S T\ninst✝¹ : IsLocalIso R S\ninst✝ : IsLocalIso S T\ns : Set S := {g | IsStandardOpenImmersion R (Localiza... | Ideal.map_top, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.LocalIso | {
"line": 145,
"column": 2
} | {
"line": 146,
"column": 29
} | {
"line": 147,
"column": 2
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝⁸ : CommSemiring R\ninst✝⁷ : CommSemiring S\ninst✝⁶ : Algebra R S\nT : Type u_3\ninst✝⁵ : CommSemiring T\ninst✝⁴ : Algebra S T\ninst✝³ : Algebra R T\ninst✝² : IsScalarTower R S T\ninst✝¹ : IsLocalIso R S\ninst✝ : IsLocalIso S T\ns : Set S := {g | IsStandardOpenImmersion... | [
"R : Type u_1\nS : Type u_2\ninst✝⁸ : CommSemiring R\ninst✝⁷ : CommSemiring S\ninst✝⁶ : Algebra R S\nT : Type u_3\ninst✝⁵ : CommSemiring T\ninst✝⁴ : Algebra S T\ninst✝³ : Algebra R T\ninst✝² : IsScalarTower R S T\ninst✝¹ : IsLocalIso R S\ninst✝ : IsLocalIso S T\ns : Set S := {g | IsStandardOpenImmersion R (Localiza... | simp only [Set.mem_image, Set.mem_setOf_eq, SetLike.mem_coe, forall_exists_index, and_imp,
forall_apply_eq_imp_iff₂] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.RingTheory.Invariant.Profinite | {
"line": 93,
"column": 2
} | {
"line": 93,
"column": 14
} | {
"line": 94,
"column": 2
} | [
{
"pp": "A : Type u_1\nB : Type u_2\ninst✝¹⁵ : CommRing A\ninst✝¹⁴ : CommRing B\ninst✝¹³ : Algebra A B\nG : Type u\ninst✝¹² : Group G\ninst✝¹¹ : MulSemiringAction G B\ninst✝¹⁰ : SMulCommClass G A B\ninst✝⁹ : TopologicalSpace G\ninst✝⁸ : CompactSpace G\ninst✝⁷ : TotallyDisconnectedSpace G\ninst✝⁶ : IsTopological... | [
"A : Type u_1\nB : Type u_2\ninst✝¹⁵ : CommRing A\ninst✝¹⁴ : CommRing B\ninst✝¹³ : Algebra A B\nG : Type u\ninst✝¹² : Group G\ninst✝¹¹ : MulSemiringAction G B\ninst✝¹⁰ : SMulCommClass G A B\ninst✝⁹ : TopologicalSpace G\ninst✝⁸ : CompactSpace G\ninst✝⁷ : TotallyDisconnectedSpace G\ninst✝⁶ : IsTopologicalGroup G\nins... | rw [(s N).2] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.LaurentSeries | {
"line": 243,
"column": 6
} | {
"line": 243,
"column": 28
} | {
"line": 243,
"column": 29
} | [
{
"pp": "case pos\nR : Type u_1\ninst✝ : Semiring R\nx : R⸨X⸩\nn : ℤ\nh : order x ≤ n\n⊢ (PowerSeries.coeff (n - order x).natAbs) x.powerSeriesPart = x.coeff n",
"ppTerm": "?pos✝",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"HahnSeries.order",
"Semiring.toModule",
"congrA... | [
"case pos\nR : Type u_1\ninst✝ : Semiring R\nx : R⸨X⸩\nn : ℤ\nh : order x ≤ n\n⊢ x.coeff (order x + ↑(n - order x).natAbs) = x.coeff n"
] | powerSeriesPart_coeff, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.LaurentSeries | {
"line": 566,
"column": 12
} | {
"line": 566,
"column": 38
} | {
"line": 566,
"column": 39
} | [
{
"pp": "case neg\nK : Type u_2\ninst✝ : Field K\nD : ℤ\nf : K⸨X⸩\nh_val_f : ∀ n < D, f.coeff n = 0\nF : K⟦X⟧ := f.powerSeriesPart\nord_nonpos : HahnSeries.order f ≤ 0\ns : ℕ\nhs : HahnSeries.order f = -↑s\nhDs : 0 < D + ↑s\nd : ℕ\nhd : D + ↑s = ↑d\nn : ℕ\nhn : n < d\n⊢ (PowerSeries.coeff n) F = 0",
"ppTerm... | [
"case neg\nK : Type u_2\ninst✝ : Field K\nD : ℤ\nf : K⸨X⸩\nh_val_f : ∀ n < D, f.coeff n = 0\nF : K⟦X⟧ := f.powerSeriesPart\nord_nonpos : HahnSeries.order f ≤ 0\ns : ℕ\nhs : HahnSeries.order f = -↑s\nhDs : 0 < D + ↑s\nd : ℕ\nhd : D + ↑s = ↑d\nn : ℕ\nhn : n < d\n⊢ f.coeff (HahnSeries.order f + ↑n) = 0"
] | powerSeriesPart_coeff f n, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Localization.Pi | {
"line": 93,
"column": 4
} | {
"line": 93,
"column": 43
} | {
"line": 94,
"column": 4
} | [
{
"pp": "case pos\nι : Type u_1\nR : ι → Type u_2\nS : ι → Type u_3\ninst✝⁵ : (i : ι) → CommSemiring (R i)\ninst✝⁴ : (i : ι) → CommSemiring (S i)\ninst✝³ : (i : ι) → Algebra (R i) (S i)\nM : Submonoid ((i : ι) → R i)\ninst✝² : ∀ (i : ι), IsLocalization (Submonoid.map (Pi.evalRingHom R i) M) (S i)\ninst✝¹ : ∀ (i... | [
"case pos\nι : Type u_1\nR : ι → Type u_2\nS : ι → Type u_3\ninst✝⁵ : (i : ι) → CommSemiring (R i)\ninst✝⁴ : (i : ι) → CommSemiring (S i)\ninst✝³ : (i : ι) → Algebra (R i) (S i)\nM : Submonoid ((i : ι) → R i)\ninst✝² : ∀ (i : ι), IsLocalization (Submonoid.map (Pi.evalRingHom R i) M) (S i)\ninst✝¹ : ∀ (i : ι), Ring.... | have := uniqueOfZeroMem h₀ (S := (S i)) | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.RingTheory.LaurentSeries | {
"line": 582,
"column": 12
} | {
"line": 582,
"column": 38
} | {
"line": 582,
"column": 39
} | [
{
"pp": "case neg\nK : Type u_2\ninst✝ : Field K\nD : ℤ\nf : K⸨X⸩\nh_val_f : ∀ n < D, f.coeff n = 0\nF : K⟦X⟧ := f.powerSeriesPart\nord_nonpos : 0 < HahnSeries.order f\ns : ℕ\nhs : HahnSeries.order f = ↑s\nhDs : 0 < D - ↑s\nd : ℕ\nhd : D - ↑s = ↑d\nn : ℕ\nhn : n < d\n⊢ (PowerSeries.coeff n) F = 0",
"ppTerm"... | [
"case neg\nK : Type u_2\ninst✝ : Field K\nD : ℤ\nf : K⸨X⸩\nh_val_f : ∀ n < D, f.coeff n = 0\nF : K⟦X⟧ := f.powerSeriesPart\nord_nonpos : 0 < HahnSeries.order f\ns : ℕ\nhs : HahnSeries.order f = ↑s\nhDs : 0 < D - ↑s\nd : ℕ\nhd : D - ↑s = ↑d\nn : ℕ\nhn : n < d\n⊢ f.coeff (HahnSeries.order f + ↑n) = 0"
] | powerSeriesPart_coeff f n, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.LaurentSeries | {
"line": 642,
"column": 4
} | {
"line": 642,
"column": 59
} | {
"line": 644,
"column": 0
} | [
{
"pp": "case h\nK : Type u_2\ninst✝ : Field K\nn : WithZero (Multiplicative ℤ)\nhn0 : ¬n = 0\n⊢ Valued.v ((single (-n.log)) 1) = n",
"ppTerm": "?h",
"assigned": true,
"usedConstants": [
"Int.instAddCommMonoid",
"LinearOrderedCommGroupWithZero.toLinearOrderedCommMonoidWithZero",
"Z... | [] | simp [LaurentSeries.valuation_single_zpow, exp_log hn0] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.RingTheory.MvPolynomial.Symmetric.FundamentalTheorem | {
"line": 81,
"column": 30
} | {
"line": 81,
"column": 44
} | {
"line": 81,
"column": 45
} | [
{
"pp": "case e'_3.e'_6.h\ni n m : ℕ\nhin : i < n\nhim : i + 1 < m\nt : Fin n → ℕ\na✝ : Fin n\n⊢ i + 1 ≤ ↑a✝ ↔ a✝ ≠ ⟨i, hin⟩ ∧ i ≤ ↑a✝",
"ppTerm": "?e'_3.e'_6.h",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"Fin.mk",
"id",
"Ne",
"instOfNatNat",
... | [
"case e'_3.e'_6.h\ni n m : ℕ\nhin : i < n\nhim : i + 1 < m\nt : Fin n → ℕ\na✝ : Fin n\n⊢ i < ↑a✝ ↔ a✝ ≠ ⟨i, hin⟩ ∧ i ≤ ↑a✝"
] | i.succ_le_iff, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.RingTheory.MvPolynomial.Symmetric.FundamentalTheorem | {
"line": 110,
"column": 46
} | {
"line": 110,
"column": 52
} | {
"line": 110,
"column": 53
} | [
{
"pp": "case refine_1\nn m : ℕ\nhmn : m ≤ n\ns : Fin m → ℕ\nhs : Antitone s\nx✝¹ : Fin m\ni✝ : ℕ\nthis : i✝ ≤ m - 1\ni : ℕ\nhi : i + 1 < m\nx✝ : i✝ ≤ i\nih : ∀ (hi : i + 1 < m), (accumulate n m) (invAccumulate n m s) ⟨i + 1, hi⟩ = s ⟨i + 1, hi⟩\nhim : i < m\n⊢ invAccumulate n m s ⟨i, ⋯⟩ + (accumulate n m) (inv... | [
"case refine_1\nn m : ℕ\nhmn : m ≤ n\ns : Fin m → ℕ\nhs : Antitone s\nx✝¹ : Fin m\ni✝ : ℕ\nthis : i✝ ≤ m - 1\ni : ℕ\nhi : i + 1 < m\nx✝ : i✝ ≤ i\nih : ∀ (hi : i + 1 < m), (accumulate n m) (invAccumulate n m s) ⟨i + 1, hi⟩ = s ⟨i + 1, hi⟩\nhim : i < m\n⊢ invAccumulate n m s ⟨i, ⋯⟩ + s ⟨i + 1, hi⟩ = s ⟨i, him⟩"
] | ih hi, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Nilpotent.GeometricallyReduced | {
"line": 73,
"column": 2
} | {
"line": 76,
"column": 79
} | {
"line": 78,
"column": 0
} | [
{
"pp": "k✝ : Type u_1\nA✝ : Type u_2\ninst✝⁹ : Field k✝\ninst✝⁸ : Ring A✝\ninst✝⁷ : Algebra k✝ A✝\nk : Type u_3\nA : Type u_4\nK : Type u_5\ninst✝⁶ : Field k\ninst✝⁵ : Ring A\ninst✝⁴ : Algebra k A\ninst✝³ : Field K\ninst✝² : Algebra k K\ninst✝¹ : Algebra.IsAlgebraic k K\ninst✝ : IsGeometricallyReduced k A\n⊢ I... | [] | have := (isGeometricallyReduced_field_iff k A).mp ‹_›
exact isReduced_of_injective
(Algebra.TensorProduct.map ((IsAlgClosed.lift : K →ₐ[k] AlgebraicClosure k)) 1)
(Module.Flat.rTensor_preserves_injective_linearMap _ (RingHom.injective _)) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Nilpotent.GeometricallyReduced | {
"line": 73,
"column": 2
} | {
"line": 76,
"column": 79
} | {
"line": 78,
"column": 0
} | [
{
"pp": "k✝ : Type u_1\nA✝ : Type u_2\ninst✝⁹ : Field k✝\ninst✝⁸ : Ring A✝\ninst✝⁷ : Algebra k✝ A✝\nk : Type u_3\nA : Type u_4\nK : Type u_5\ninst✝⁶ : Field k\ninst✝⁵ : Ring A\ninst✝⁴ : Algebra k A\ninst✝³ : Field K\ninst✝² : Algebra k K\ninst✝¹ : Algebra.IsAlgebraic k K\ninst✝ : IsGeometricallyReduced k A\n⊢ I... | [] | have := (isGeometricallyReduced_field_iff k A).mp ‹_›
exact isReduced_of_injective
(Algebra.TensorProduct.map ((IsAlgClosed.lift : K →ₐ[k] AlgebraicClosure k)) 1)
(Module.Flat.rTensor_preserves_injective_linearMap _ (RingHom.injective _)) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Morita.Matrix | {
"line": 210,
"column": 4
} | {
"line": 211,
"column": 98
} | {
"line": 212,
"column": 4
} | [
{
"pp": "R : Type u\nι : Type v\ninst✝⁴ : Ring R\ninst✝³ : Fintype ι\ninst✝² : DecidableEq ι\nR₀ : Type u_1\ninst✝¹ : CommRing R₀\ninst✝ : Algebra R₀ R\ni : ι\nX Y : ModuleCat R\nf : X ⟶ Y\nr : R₀\nv : ι → ↑X\n⊢ (ModuleCat.Hom.hom ((ModuleCat.matrixEquivalence R i).functor.map (r • f))) v =\n (ModuleCat.Hom.... | [
"R : Type u\nι : Type v\ninst✝⁴ : Ring R\ninst✝³ : Fintype ι\ninst✝² : DecidableEq ι\nR₀ : Type u_1\ninst✝¹ : CommRing R₀\ninst✝ : Algebra R₀ R\ni : ι\nX Y : ModuleCat R\nf : X ⟶ Y\nr : R₀\nv : ι → ↑X\n⊢ (LinearMap.mapMatrixModule ι (r • ModuleCat.Hom.hom f)) v = r • (LinearMap.mapMatrixModule ι (ModuleCat.Hom.hom ... | simp only [ModuleCat.matrixEquivalence_functor, ModuleCat.toMatrixModCat_obj_carrier,
ModuleCat.toMatrixModCat_map, ModuleCat.hom_smul, ModuleCat.hom_ofHom, LinearMap.smul_apply] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.RingTheory.LaurentSeries | {
"line": 1090,
"column": 4
} | {
"line": 1092,
"column": 14
} | {
"line": 1093,
"column": 2
} | [
{
"pp": "case h.right\nK : Type u_2\ninst✝ : Field K\na : adicCompletion K⟮X⟯ (idealX K)\nthis : ∀ (s : Set (adicCompletion K⟮X⟯ (idealX K))), s ∈ 𝓝 0 ↔ ∃ γ, {x | Valued.v.restrict x < ↑γ} ⊆ s\nha : a = 0\nS : Set (WithZero (Multiplicative ℤ))\nγ : WithZero (Multiplicative ℤ)\nγ_ne_zero : γ ≠ 0\nγ_le : Set.Iio... | [] | · refine Set.Subset.trans (fun a _ ↦ ?_) (Set.preimage_mono γ_le)
rw [Set.mem_preimage, Set.mem_Iio, ← Valued.valuedCompletion_apply a]
simp_all | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.RingTheory.Teichmuller | {
"line": 69,
"column": 92
} | {
"line": 70,
"column": 36
} | {
"line": 72,
"column": 0
} | [
{
"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✝ : IsPrecomplete I R\nx : Perfection (R ⧸ I) p\ny : R\nn : ℕ\nh : (Ideal.Quotient.mk I) y = (coeff (R ⧸ I) p n) x\nthis : x.teichmullerAux (n + 1) ≡ ⋯.choose [SMOD I ^ (n + 1)]\n⊢ Quotient... | [] | by
simp [SModEq.idealQuotientMk, h] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.WittVector.Frobenius | {
"line": 118,
"column": 2
} | {
"line": 119,
"column": 62
} | {
"line": 121,
"column": 0
} | [
{
"pp": "p : ℕ\nhp : Fact (Nat.Prime p)\nn j : ℕ\nhj : j < p ^ n\n⊢ p ^ (n - v p (j + 1)) ∣ (p ^ n).choose (j + 1)",
"ppTerm": "?m.35",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"le_refl",
"Nat.Prime",
"Nat.choose",
"ENat.instNatCast",
"congrArg",
"Nat.... | [] | apply pow_dvd_of_le_emultiplicity
rw [hp.out.emultiplicity_choose_prime_pow hj j.succ_ne_zero] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.WittVector.Frobenius | {
"line": 118,
"column": 2
} | {
"line": 119,
"column": 62
} | {
"line": 121,
"column": 0
} | [
{
"pp": "p : ℕ\nhp : Fact (Nat.Prime p)\nn j : ℕ\nhj : j < p ^ n\n⊢ p ^ (n - v p (j + 1)) ∣ (p ^ n).choose (j + 1)",
"ppTerm": "?m.35",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"le_refl",
"Nat.Prime",
"Nat.choose",
"ENat.instNatCast",
"congrArg",
"Nat.... | [] | apply pow_dvd_of_le_emultiplicity
rw [hp.out.emultiplicity_choose_prime_pow hj j.succ_ne_zero] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.WittVector.IsPoly | {
"line": 180,
"column": 2
} | {
"line": 180,
"column": 27
} | {
"line": 181,
"column": 2
} | [
{
"pp": "p : ℕ\ninst✝ : Fact (Nat.Prime p)\nf g : ⦃R : Type u⦄ → [CommRing R] → 𝕎 R → 𝕎 R\nh : ∀ (R : Type u) [_Rcr : CommRing R] (x : 𝕎 R) (n : ℕ), (ghostComponent n) (f x) = (ghostComponent n) (g x)\nφ : ℕ → MvPolynomial ℕ ℤ\nhf : ∀ ⦃R : Type u⦄ [inst : CommRing R] (x : 𝕎 R), (f x).coeff = fun n ↦ (aeval ... | [
"p : ℕ\ninst✝ : Fact (Nat.Prime p)\nf g : ⦃R : Type u⦄ → [CommRing R] → 𝕎 R → 𝕎 R\nh : ∀ (R : Type u) [_Rcr : CommRing R] (x : 𝕎 R) (n : ℕ), (ghostComponent n) (f x) = (ghostComponent n) (g x)\nφ : ℕ → MvPolynomial ℕ ℤ\nhf : ∀ ⦃R : Type u⦄ [inst : CommRing R] (x : 𝕎 R), (f x).coeff = fun n ↦ (aeval x.coeff) (φ ... | apply MvPolynomial.funext | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.RingTheory.WittVector.Verschiebung | {
"line": 44,
"column": 74
} | {
"line": 45,
"column": 29
} | {
"line": 47,
"column": 0
} | [
{
"pp": "p : ℕ\nR : Type u_1\ninst✝ : CommRing R\nx : 𝕎 R\nn : ℕ\n⊢ x.verschiebungFun.coeff n = if n = 0 then 0 else x.coeff (n - 1)",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"CommSemiring.toSemiring",
"HSub.hSub",
"instSubNat",
"instOfNatNat",
"CommRin... | [] | by
simp only [verschiebungFun] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.WittVector.Verschiebung | {
"line": 58,
"column": 2
} | {
"line": 59,
"column": 69
} | {
"line": 61,
"column": 0
} | [
{
"pp": "p : ℕ\nR : Type u_1\ninst✝ : CommRing R\nhp : Fact (Nat.Prime p)\nx : 𝕎 R\n⊢ (ghostComponent 0) x.verschiebungFun = 0",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"MulOne.toOne",
"wittPolynomial",
"WittVector.ghostComponent_apply",
"aeva... | [] | rw [ghostComponent_apply, aeval_wittPolynomial, Finset.range_one, Finset.sum_singleton,
verschiebungFun_coeff_zero, pow_zero, pow_zero, pow_one, one_mul] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.WittVector.Verschiebung | {
"line": 58,
"column": 2
} | {
"line": 59,
"column": 69
} | {
"line": 61,
"column": 0
} | [
{
"pp": "p : ℕ\nR : Type u_1\ninst✝ : CommRing R\nhp : Fact (Nat.Prime p)\nx : 𝕎 R\n⊢ (ghostComponent 0) x.verschiebungFun = 0",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"MulOne.toOne",
"wittPolynomial",
"WittVector.ghostComponent_apply",
"aeva... | [] | rw [ghostComponent_apply, aeval_wittPolynomial, Finset.range_one, Finset.sum_singleton,
verschiebungFun_coeff_zero, pow_zero, pow_zero, pow_one, one_mul] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.WittVector.Verschiebung | {
"line": 58,
"column": 2
} | {
"line": 59,
"column": 69
} | {
"line": 61,
"column": 0
} | [
{
"pp": "p : ℕ\nR : Type u_1\ninst✝ : CommRing R\nhp : Fact (Nat.Prime p)\nx : 𝕎 R\n⊢ (ghostComponent 0) x.verschiebungFun = 0",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"MulOne.toOne",
"wittPolynomial",
"WittVector.ghostComponent_apply",
"aeva... | [] | rw [ghostComponent_apply, aeval_wittPolynomial, Finset.range_one, Finset.sum_singleton,
verschiebungFun_coeff_zero, pow_zero, pow_zero, pow_one, one_mul] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.WittVector.Verschiebung | {
"line": 170,
"column": 2
} | {
"line": 170,
"column": 27
} | {
"line": 171,
"column": 2
} | [
{
"pp": "p : ℕ\nhp : Fact (Nat.Prime p)\nn : ℕ\n⊢ (bind₁ verschiebungPoly) (wittPolynomial p ℤ n) = if n = 0 then 0 else ↑p * wittPolynomial p ℤ (n - 1)",
"ppTerm": "?m.27",
"assigned": true,
"usedConstants": [
"wittPolynomial",
"Nat.instMulZeroClass",
"AddMonoidAlgebra.semiring",
... | [
"p : ℕ\nhp : Fact (Nat.Prime p)\nn : ℕ\n⊢ ∀ (x : ℕ → ℤ),\n (MvPolynomial.eval x) ((bind₁ verschiebungPoly) (wittPolynomial p ℤ n)) =\n (MvPolynomial.eval x) (if n = 0 then 0 else ↑p * wittPolynomial p ℤ (n - 1))"
] | apply MvPolynomial.funext | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.RingTheory.Perfection | {
"line": 580,
"column": 4
} | {
"line": 580,
"column": 15
} | {
"line": 581,
"column": 2
} | [
{
"pp": "case pos\nK : Type u₁\ninst✝² : Field K\nv : Valuation K ℝ≥0\nO : Type u₂\ninst✝¹ : CommRing O\ninst✝ : Algebra O K\nhv : v.Integers O\np : ℕ\nx y : ModP O p\nhx0 : x ≠ 0\nhy0 : y ≠ 0\nhxy0 : x + y = 0\n⊢ preVal K v O p (x + y) ≤ max (preVal K v O p x) (preVal K v O p y)",
"ppTerm": "?pos✝",
"a... | [] | simp [hxy0] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.RingTheory.Perfection | {
"line": 580,
"column": 4
} | {
"line": 580,
"column": 15
} | {
"line": 581,
"column": 2
} | [
{
"pp": "case pos\nK : Type u₁\ninst✝² : Field K\nv : Valuation K ℝ≥0\nO : Type u₂\ninst✝¹ : CommRing O\ninst✝ : Algebra O K\nhv : v.Integers O\np : ℕ\nx y : ModP O p\nhx0 : x ≠ 0\nhy0 : y ≠ 0\nhxy0 : x + y = 0\n⊢ preVal K v O p (x + y) ≤ max (preVal K v O p x) (preVal K v O p y)",
"ppTerm": "?pos✝",
"a... | [] | simp [hxy0] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Perfection | {
"line": 580,
"column": 4
} | {
"line": 580,
"column": 15
} | {
"line": 581,
"column": 2
} | [
{
"pp": "case pos\nK : Type u₁\ninst✝² : Field K\nv : Valuation K ℝ≥0\nO : Type u₂\ninst✝¹ : CommRing O\ninst✝ : Algebra O K\nhv : v.Integers O\np : ℕ\nx y : ModP O p\nhx0 : x ≠ 0\nhy0 : y ≠ 0\nhxy0 : x + y = 0\n⊢ preVal K v O p (x + y) ≤ max (preVal K v O p x) (preVal K v O p y)",
"ppTerm": "?pos✝",
"a... | [] | simp [hxy0] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.WittVector.IsPoly | {
"line": 340,
"column": 2
} | {
"line": 340,
"column": 27
} | {
"line": 341,
"column": 2
} | [
{
"pp": "p : ℕ\ninst✝ : Fact (Nat.Prime p)\nf g : ⦃R : Type u⦄ → [CommRing R] → 𝕎 R → 𝕎 R → 𝕎 R\nh : ∀ (R : Type u) [_Rcr : CommRing R] (x y : 𝕎 R) (n : ℕ), (ghostComponent n) (f x y) = (ghostComponent n) (g x y)\nφ : ℕ → MvPolynomial (Fin 2 × ℕ) ℤ\nhf : ∀ ⦃R : Type u⦄ [inst : CommRing R] (x y : 𝕎 R), (f x... | [
"p : ℕ\ninst✝ : Fact (Nat.Prime p)\nf g : ⦃R : Type u⦄ → [CommRing R] → 𝕎 R → 𝕎 R → 𝕎 R\nh : ∀ (R : Type u) [_Rcr : CommRing R] (x y : 𝕎 R) (n : ℕ), (ghostComponent n) (f x y) = (ghostComponent n) (g x y)\nφ : ℕ → MvPolynomial (Fin 2 × ℕ) ℤ\nhf : ∀ ⦃R : Type u⦄ [inst : CommRing R] (x y : 𝕎 R), (f x y).coeff = ... | apply MvPolynomial.funext | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.RingTheory.WittVector.Identities | {
"line": 205,
"column": 70
} | {
"line": 205,
"column": 77
} | {
"line": 207,
"column": 0
} | [
{
"pp": "case succ\np : ℕ\nR : Type u_1\nhp : Fact (Nat.Prime p)\ninst✝¹ : CommRing R\ninst✝ : CharP R p\nx : 𝕎 R\nk i : ℕ\nih : ((⇑frobenius)^[i] x).coeff k = x.coeff k ^ p ^ i\n⊢ (x.coeff k ^ p ^ i) ^ p = x.coeff k ^ p ^ (i + 1)",
"ppTerm": "?succ",
"assigned": true,
"usedConstants": [
"Mat... | [] | ring_nf | Mathlib.Tactic.RingNF._aux_Mathlib_Tactic_Ring_RingNF___elabRules_Mathlib_Tactic_RingNF_ringNF_1 | Mathlib.Tactic.RingNF.ringNF |
Mathlib.RingTheory.WittVector.InitTail | {
"line": 106,
"column": 9
} | {
"line": 106,
"column": 19
} | {
"line": 106,
"column": 20
} | [
{
"pp": "case neg\np : ℕ\nP : ℕ → Prop\nhp : Fact (Nat.Prime p)\nthis : IsPoly p fun {R} [CommRing R] x ↦ select P x + select (fun i ↦ ¬P i) x\nR : Type u_1\nR._inst : CommRing R\nx : 𝕎 R\nn m : ℕ\nx✝ : m ∈ Finset.range (n + 1)\nPm : ¬P m\n⊢ (if P m then X m ^ p ^ (n - m) else 0) + (if ¬P m then X m else 0) ^ ... | [
"case neg\np : ℕ\nP : ℕ → Prop\nhp : Fact (Nat.Prime p)\nthis : IsPoly p fun {R} [CommRing R] x ↦ select P x + select (fun i ↦ ¬P i) x\nR : Type u_1\nR._inst : CommRing R\nx : 𝕎 R\nn m : ℕ\nx✝ : m ∈ Finset.range (n + 1)\nPm : ¬P m\n⊢ 0 + (if ¬P m then X m else 0) ^ p ^ (n - m) = X m ^ p ^ (n - m)"
] | if_neg Pm, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Polynomial.Hermite.Basic | {
"line": 101,
"column": 59
} | {
"line": 106,
"column": 31
} | {
"line": 108,
"column": 0
} | [
{
"pp": "n : ℕ\n⊢ (hermite n).degree = ↑n",
"ppTerm": "?m.4",
"assigned": true,
"usedConstants": [
"WithBot.addMonoidWithOne",
"WithBot.instPreorder",
"Eq.mpr",
"WithBot.zeroLEOneClass",
"_private.Mathlib.RingTheory.Polynomial.Hermite.Basic.0.Polynomial.degree_hermite._... | [] | by
rw [degree_eq_of_le_of_coeff_ne_zero]
· simp_rw [degree_le_iff_coeff_zero, Nat.cast_lt]
rintro m hnm
exact coeff_hermite_of_lt hnm
· simp [coeff_hermite_self n] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.Polynomial.HilbertPoly | {
"line": 69,
"column": 74
} | {
"line": 69,
"column": 98
} | {
"line": 70,
"column": 4
} | [
{
"pp": "F : Type u_1\ninst✝¹ : Field F\ninst✝ : CharZero F\nd k : ℕ\nhne : ↑d ! ≠ 0\n⊢ (ascPochhammer F d).natDegree * (X - C ↑k + 1).natDegree = d",
"ppTerm": "?m.37",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Polynomial.C",
"Polynomial.instOne",
"IsDomain.to_noZeroDi... | [
"F : Type u_1\ninst✝¹ : Field F\ninst✝ : CharZero F\nd k : ℕ\nhne : ↑d ! ≠ 0\n⊢ d * (X - C ↑k + 1).natDegree = d"
] | ascPochhammer_natDegree, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Polynomial.HilbertPoly | {
"line": 238,
"column": 4
} | {
"line": 238,
"column": 34
} | {
"line": 239,
"column": 2
} | [
{
"pp": "case pos\nF : Type u_1\ninst✝¹ : Field F\ninst✝ : CharZero F\np : F[X]\nd : ℕ\nhdp : d ≤ rootMultiplicity 1 p\nhp : p = 0\n⊢ p.hilbertPoly d = 0",
"ppTerm": "?pos✝",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"id",
"Polynomial.hilbertPoly_zero_left",
... | [] | rw [hp, hilbertPoly_zero_left] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
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