module stringlengths 16 90 | startPos dict | endPos dict | goals listlengths 0 96 | ppTac stringlengths 1 14.5k | elaborator stringclasses 365
values | kind stringclasses 368
values |
|---|---|---|---|---|---|---|
Mathlib.RepresentationTheory.Character | {
"line": 58,
"column": 52
} | {
"line": 58,
"column": 98
} | [
{
"pp": "k : Type u\ninst✝¹ : Field k\nG : Type u\ninst✝ : Monoid G\nV : FDRep k G\ng h : G\n⊢ V.character (h * g) = V.character (g * h)",
"usedConstants": [
"LinearMap.trace",
"MonoidHom.instMonoidHomClass",
"MonoidHom.instFunLike",
"Semiring.toModule",
"HMul.hMul",
"Mon... | simp only [trace_mul_comm, character, map_mul] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RepresentationTheory.Character | {
"line": 157,
"column": 52
} | {
"line": 157,
"column": 98
} | [
{
"pp": "G : Type u_1\nk : Type u_2\nV : Type u_3\ninst✝³ : Monoid G\ninst✝² : Field k\ninst✝¹ : AddCommGroup V\ninst✝ : Module k V\nρ : Representation k G V\ng h : G\n⊢ ρ.character (h * g) = ρ.character (g * h)",
"usedConstants": [
"LinearMap.trace",
"MonoidHom.instMonoidHomClass",
"Repre... | simp only [trace_mul_comm, character, map_mul] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.RepresentationTheory.Character | {
"line": 157,
"column": 52
} | {
"line": 157,
"column": 98
} | [
{
"pp": "G : Type u_1\nk : Type u_2\nV : Type u_3\ninst✝³ : Monoid G\ninst✝² : Field k\ninst✝¹ : AddCommGroup V\ninst✝ : Module k V\nρ : Representation k G V\ng h : G\n⊢ ρ.character (h * g) = ρ.character (g * h)",
"usedConstants": [
"LinearMap.trace",
"MonoidHom.instMonoidHomClass",
"Repre... | simp only [trace_mul_comm, character, map_mul] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RepresentationTheory.Character | {
"line": 157,
"column": 52
} | {
"line": 157,
"column": 98
} | [
{
"pp": "G : Type u_1\nk : Type u_2\nV : Type u_3\ninst✝³ : Monoid G\ninst✝² : Field k\ninst✝¹ : AddCommGroup V\ninst✝ : Module k V\nρ : Representation k G V\ng h : G\n⊢ ρ.character (h * g) = ρ.character (g * h)",
"usedConstants": [
"LinearMap.trace",
"MonoidHom.instMonoidHomClass",
"Repre... | simp only [trace_mul_comm, character, map_mul] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RepresentationTheory.FinGroupCharZero | {
"line": 146,
"column": 36
} | {
"line": 146,
"column": 60
} | [
{
"pp": "k : Type u\ninst✝⁴ : Field k\nG : Type u\ninst✝³ : Group G\ninst✝² : IsAlgClosed k\ninst✝¹ : CharZero k\ninst✝ : Fintype G\nV : FDRep k G\nh : ∑ g, V.character g * V.character g⁻¹ = ↑(Nat.card G)\nthis✝¹ : NeZero ↑(Nat.card G)\nthis✝ : Invertible ↑(Fintype.card G)\nthis : Invertible ↑(Nat.card G)\neq :... | ← Nat.cast_inj (R := k), | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RepresentationTheory.Homological.Resolution | {
"line": 140,
"column": 84
} | {
"line": 146,
"column": 80
} | [
{
"pp": "k G : Type u\ninst✝¹ : CommRing k\nn : ℕ\ninst✝ : Monoid G\n⊢ (compForgetAugmented G).ExtraDegeneracy",
"usedConstants": [
"CategoryTheory.Comma.right",
"Opposite",
"CategoryTheory.typesCartesianMonoidalCategory",
"CategoryTheory.SimplicialObject.const",
"CategoryTheor... | by
refine
ExtraDegeneracy.ofIso (?_ : (Arrow.mk <| terminal.from G).augmentedCechNerve ≅ _)
(extraDegeneracyAugmentedCechNerve G)
exact
Comma.isoMk (CechNerveTerminalFrom.iso G ≪≫ cechNerveTerminalFromIsoCompForget G)
(Iso.refl _) (by ext : 1; exact IsTerminal.hom_ext terminalIsTerminal _ _) | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RepresentationTheory.Homological.GroupCohomology.LowDegree | {
"line": 766,
"column": 2
} | {
"line": 766,
"column": 33
} | [
{
"pp": "k G : Type u\ninst✝¹ : CommRing k\ninst✝ : Group G\nA : Rep k G\n⊢ (KernelFork.ofι (HomologicalComplex.iCycles (inhomogeneousCochains A) 0) ⋯).mapOfIsLimit ⋯.fIsKernel\n (dArrowIso₀₁ A).hom ≫\n (shortComplexH0 A).f =\n iCocycles A 0 ≫ (cochainsIso₀ A).hom",
"usedConstants": [
"... | apply KernelFork.mapOfIsLimit_ι | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.RepresentationTheory.Homological.GroupCohomology.LongExactSequence | {
"line": 48,
"column": 6
} | {
"line": 49,
"column": 79
} | [
{
"pp": "k G : Type u\ninst✝¹ : CommRing k\ninst✝ : Group G\nX : ShortComplex (Rep k G)\nhX : X.ShortExact\ni : ℕ\n⊢ ((X.map (cochainsFunctor k G)).map (HomologicalComplex.eval (ModuleCat k) (ComplexShape.up ℕ) i)).Exact",
"usedConstants": [
"CategoryTheory.Functor.PreservesHomology.preservesRightHomo... | have : LinearMap.range X.f.hom.toLinearMap = LinearMap.ker X.g.hom.toLinearMap :=
(hX.exact.map (forget₂ (Rep k G) (ModuleCat k))).moduleCat_range_eq_ker | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.RepresentationTheory.Homological.GroupHomology.LongExactSequence | {
"line": 48,
"column": 6
} | {
"line": 49,
"column": 79
} | [
{
"pp": "k G : Type u\ninst✝¹ : CommRing k\ninst✝ : Group G\nX : ShortComplex (Rep k G)\nhX : X.ShortExact\nthis : Mono X.f := hX.mono_f\ni : ℕ\n⊢ ((X.map (chainsFunctor k G)).map (HomologicalComplex.eval (ModuleCat k) (ComplexShape.down ℕ) i)).Exact",
"usedConstants": [
"CategoryTheory.Functor.Preser... | have : LinearMap.range X.f.hom.toLinearMap = LinearMap.ker X.g.hom.toLinearMap :=
(hX.exact.map (forget₂ (Rep k G) (ModuleCat k))).moduleCat_range_eq_ker | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.RingTheory.AdicCompletion.LocalRing | {
"line": 57,
"column": 10
} | {
"line": 57,
"column": 19
} | [
{
"pp": "case pos\nR : Type u_1\ninst✝¹ : CommRing R\nm : Ideal R\nhmax : m.IsMaximal\ninst✝ : IsAdicComplete m R\nr : R\nh : r ∉ m\nmapu : ∀ {n : ℕ}, 0 < n → IsUnit ((mk (m ^ n)) r)\ninvSeries' : (n : { n // 0 < n }) → R ⧸ m ^ ↑n\ninvSeries_spec' : ∀ (n : { n // 0 < n }), invSeries' n * (mk (m ^ ↑n)) r = 1\nin... | mul_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.Algebra.Nonarchimedean.AdicTopology | {
"line": 208,
"column": 4
} | {
"line": 208,
"column": 19
} | [
{
"pp": "case mp\nA : Type u_2\ninst✝² : CommRing A\ninst✝¹ : TopologicalSpace A\ninst✝ : IsTopologicalRing A\nh : ∀ (n : ℕ), IsOpen ↑(⊥ ^ n)\n_h' : ∀ s ∈ 𝓝 0, ∃ n, ↑(⊥ ^ n) ⊆ s\n⊢ IsOpen {0}",
"usedConstants": [
"Semiring.toModule",
"IsScalarTower.right",
"congrArg",
"CommSemiring.... | simpa using h 1 | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.RepresentationTheory.Homological.GroupHomology.LowDegree | {
"line": 442,
"column": 2
} | {
"line": 442,
"column": 44
} | [
{
"pp": "k G : Type u\ninst✝¹ : CommRing k\ninst✝ : Group G\nA : Rep k G\ng : G\na : ↑A\n⊢ single g⁻¹ ((A.ρ g⁻¹) a) + single g a ∈ boundaries₁ A",
"usedConstants": [
"Eq.mpr",
"Submodule",
"Rep.V",
"Representation",
"MonoidHom.instFunLike",
"InvOneClass.toOne",
"Fin... | rw [← d₂₁_single_inv_mul_ρ_add_single g 1] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RepresentationTheory.Homological.GroupHomology.LowDegree | {
"line": 690,
"column": 2
} | {
"line": 690,
"column": 68
} | [
{
"pp": "k G : Type u\ninst✝¹ : CommRing k\ninst✝ : Group G\nA : Rep k G\nx : ↑(shortComplexH0 A).X₂\nhx : (Coinvariants.mk A.ρ) x = 0\n⊢ ∃ x₁, (ConcreteCategory.hom (shortComplexH0 A).f) x₁ = x",
"usedConstants": [
"Submodule",
"Representation.Coinvariants.ker",
"ModuleCat",
"congrA... | rw [Coinvariants.mk_eq_zero, ← range_d₁₀_eq_coinvariantsKer] at hx | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.Coalgebra.GroupLike | {
"line": 113,
"column": 2
} | {
"line": 113,
"column": 23
} | [
{
"pp": "case cons\nR : Type u_2\nA : Type u_3\ninst✝⁵ : CommRing R\ninst✝⁴ : IsDomain R\ninst✝³ : AddCommGroup A\ninst✝² : Module R A\ninst✝¹ : Coalgebra R A\ninst✝ : IsTorsionFree R A\na : A\ns : Finset A\nhas : a ∉ s\nih : ↑s ⊆ {a | IsGroupLikeElem R a} → LinearIndepOn R id ↑s\nhs : IsGroupLikeElem R a ∧ ∀ a... | obtain ⟨ha, hs⟩ := hs | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.RingTheory.DividedPowers.Basic | {
"line": 101,
"column": 18
} | {
"line": 103,
"column": 34
} | [
{
"pp": "A : Type u_1\ninst✝ : CommSemiring A\nI : Ideal A\nx✝ : A\nha : x✝ ∈ ⊥\n⊢ (if x✝ = 0 ∧ 0 = 0 then 1 else 0) = 1",
"usedConstants": [
"Ideal.mem_bot",
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Semiring.toModule",
"instDecidableTrue",
"congrArg",
... | by
rw [mem_bot.mp ha]
simp only [and_self, ite_true] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.DividedPowerAlgebra.Init | {
"line": 143,
"column": 2
} | {
"line": 144,
"column": 40
} | [
{
"pp": "R : Type u_1\nM : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nm : M\n⊢ dp R 0 m = 1",
"usedConstants": [
"Finsupp.instAddZeroClass",
"Eq.mpr",
"MulOne.toOne",
"Nat.instMulZeroClass",
"AddMonoidAlgebra.semiring",
"AlgHom.algHom... | rw [dp_def, ← map_one (mkAlgHom R (Rel R M))]
exact RingQuot.mkAlgHom_rel R Rel.zero | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.DividedPowerAlgebra.Init | {
"line": 143,
"column": 2
} | {
"line": 144,
"column": 40
} | [
{
"pp": "R : Type u_1\nM : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nm : M\n⊢ dp R 0 m = 1",
"usedConstants": [
"Finsupp.instAddZeroClass",
"Eq.mpr",
"MulOne.toOne",
"Nat.instMulZeroClass",
"AddMonoidAlgebra.semiring",
"AlgHom.algHom... | rw [dp_def, ← map_one (mkAlgHom R (Rel R M))]
exact RingQuot.mkAlgHom_rel R Rel.zero | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.DividedPowerAlgebra.Init | {
"line": 266,
"column": 8
} | {
"line": 266,
"column": 83
} | [
{
"pp": "case dp.mem.mem\nR : Type u_3\nM : Type u_4\nι : Type u_5\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nv : ι → M\nhv : Submodule.span R (Set.range v) = ⊤\nm : M\nx : DividedPowerAlgebra R M\nk : ℕ\ni : ι\nn : ι →₀ ℕ\n⊢ (k + n i).choose k • (dp R (k + n i) (v i) * (Finsupp.erase i ... | refine smul_of_tower_mem _ _ (mem_span_of_mem ⟨Finsupp.single i k + n, ?_⟩) | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.RingTheory.DividedPowers.RatAlgebra | {
"line": 106,
"column": 13
} | {
"line": 106,
"column": 22
} | [
{
"pp": "case neg\nA : Type u_1\ninst✝¹ : CommSemiring A\nI : Ideal A\ninst✝ : DecidablePred fun x ↦ x ∈ I\nn : ℕ\nhn_fac : IsUnit ↑(n - 1)!\nhnI : I ^ n = 0\nm : ℕ\nx : A\nhx : x ∈ I\ny : A\nhy : y ∈ I\nhmn : n ≤ m\nh_sub : I ^ m ≤ I ^ n\nhxy : (x + y) ^ m = 0\n⊢ (↑m !)⁻¹ʳ * 0 =\n ∑ x_1 ∈ Finset.antidiagona... | mul_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.DividedPowers.RatAlgebra | {
"line": 144,
"column": 6
} | {
"line": 144,
"column": 15
} | [
{
"pp": "case neg\nA : Type u_1\ninst✝¹ : CommSemiring A\nI : Ideal A\ninst✝ : DecidablePred fun x ↦ x ∈ I\nn : ℕ\nhn_fac : IsUnit ↑(n - 1)!\nhnI : I ^ n = 0\nm k : ℕ\nx : A\nhx : x ∈ I\nhkm : n ≤ m + k\nhxmk : x ^ (m + k) = 0\n⊢ (↑m !)⁻¹ʳ * ((↑k !)⁻¹ʳ * 0) = ↑((m + k).choose m) * ((↑(m + k)!)⁻¹ʳ * 0)",
"us... | mul_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.DividedPowers.RatAlgebra | {
"line": 144,
"column": 16
} | {
"line": 144,
"column": 25
} | [
{
"pp": "case neg\nA : Type u_1\ninst✝¹ : CommSemiring A\nI : Ideal A\ninst✝ : DecidablePred fun x ↦ x ∈ I\nn : ℕ\nhn_fac : IsUnit ↑(n - 1)!\nhnI : I ^ n = 0\nm k : ℕ\nx : A\nhx : x ∈ I\nhkm : n ≤ m + k\nhxmk : x ^ (m + k) = 0\n⊢ (↑m !)⁻¹ʳ * 0 = ↑((m + k).choose m) * ((↑(m + k)!)⁻¹ʳ * 0)",
"usedConstants": ... | mul_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.DividedPowers.RatAlgebra | {
"line": 144,
"column": 26
} | {
"line": 144,
"column": 35
} | [
{
"pp": "case neg\nA : Type u_1\ninst✝¹ : CommSemiring A\nI : Ideal A\ninst✝ : DecidablePred fun x ↦ x ∈ I\nn : ℕ\nhn_fac : IsUnit ↑(n - 1)!\nhnI : I ^ n = 0\nm k : ℕ\nx : A\nhx : x ∈ I\nhkm : n ≤ m + k\nhxmk : x ^ (m + k) = 0\n⊢ 0 = ↑((m + k).choose m) * ((↑(m + k)!)⁻¹ʳ * 0)",
"usedConstants": [
"Eq.... | mul_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.DividedPowers.RatAlgebra | {
"line": 170,
"column": 57
} | {
"line": 170,
"column": 66
} | [
{
"pp": "case neg\nA : Type u_1\ninst✝¹ : CommSemiring A\nI : Ideal A\ninst✝ : DecidablePred fun x ↦ x ∈ I\nn : ℕ\nhn_fac : IsUnit ↑(n - 1)!\nhnI : I ^ n = 0\nm k : ℕ\nhk : k ≠ 0\nx : A\nhx : x ∈ I\nhmk : n ≤ m * k\nhxmk : x ^ (m * k) = 0\n⊢ (↑m !)⁻¹ʳ * (↑k !)⁻¹ʳ ^ m * 0 = ↑(m.uniformBell k) * ((↑(m * k)!)⁻¹ʳ *... | mul_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.DividedPowers.RatAlgebra | {
"line": 170,
"column": 67
} | {
"line": 170,
"column": 76
} | [
{
"pp": "case neg\nA : Type u_1\ninst✝¹ : CommSemiring A\nI : Ideal A\ninst✝ : DecidablePred fun x ↦ x ∈ I\nn : ℕ\nhn_fac : IsUnit ↑(n - 1)!\nhnI : I ^ n = 0\nm k : ℕ\nhk : k ≠ 0\nx : A\nhx : x ∈ I\nhmk : n ≤ m * k\nhxmk : x ^ (m * k) = 0\n⊢ 0 = ↑(m.uniformBell k) * ((↑(m * k)!)⁻¹ʳ * 0)",
"usedConstants": [... | mul_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.DividedPowers.SubDPIdeal | {
"line": 176,
"column": 2
} | {
"line": 181,
"column": 71
} | [
{
"pp": "A : Type u_1\ninst✝ : CommSemiring A\nI : Ideal A\nhI : DividedPowers I\nJ K : Ideal A\nhJ : hI.IsSubDPIdeal J\nhK : hI.IsSubDPIdeal K\n⊢ hI.IsSubDPIdeal (J ⊔ K)",
"usedConstants": [
"Iff.mpr",
"Eq.mpr",
"Ideal.subset_span",
"Submodule",
"Lattice.toSemilatticeSup",
... | rw [← J.span_eq, ← K.span_eq, ← span_union,
span_isSubDPIdeal_iff (Set.union_subset_iff.mpr ⟨hJ.1, hK.1⟩)]
intro n hn a ha
rcases ha with ha | ha
· exact span_mono Set.subset_union_left (subset_span (hJ.2 n hn ha))
· exact span_mono Set.subset_union_right (subset_span (hK.2 n hn ha)) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.DividedPowers.SubDPIdeal | {
"line": 176,
"column": 2
} | {
"line": 181,
"column": 71
} | [
{
"pp": "A : Type u_1\ninst✝ : CommSemiring A\nI : Ideal A\nhI : DividedPowers I\nJ K : Ideal A\nhJ : hI.IsSubDPIdeal J\nhK : hI.IsSubDPIdeal K\n⊢ hI.IsSubDPIdeal (J ⊔ K)",
"usedConstants": [
"Iff.mpr",
"Eq.mpr",
"Ideal.subset_span",
"Submodule",
"Lattice.toSemilatticeSup",
... | rw [← J.span_eq, ← K.span_eq, ← span_union,
span_isSubDPIdeal_iff (Set.union_subset_iff.mpr ⟨hJ.1, hK.1⟩)]
intro n hn a ha
rcases ha with ha | ha
· exact span_mono Set.subset_union_left (subset_span (hJ.2 n hn ha))
· exact span_mono Set.subset_union_right (subset_span (hK.2 n hn ha)) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.DualNumber | {
"line": 63,
"column": 2
} | {
"line": 63,
"column": 39
} | [
{
"pp": "R : Type u_1\nM : Type u_2\ninst✝⁴ : CommSemiring R\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : Module Rᵐᵒᵖ M\ninst✝ : IsCentralScalar R M\nh : ∀ (I : Ideal R), I.IsMaximal → IsNilpotent I\na : TrivSqZeroExt R M\nha : ¬IsUnit a.fst\nI : Ideal R\nhI : I.IsMaximal\nhaI : a.fst ∈ I\nn : ℕ\nhn ... | exact hn.le (Ideal.pow_mem_pow haI _) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.RingTheory.HahnSeries.HEval | {
"line": 84,
"column": 2
} | {
"line": 84,
"column": 17
} | [
{
"pp": "case h\nΓ : 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⟧\nf : PowerSeries R\nr : R\nn : ℕ\n⊢ (powerSeriesFamily x (r • f)) n = ((HahnSeries.single 0) r... | simp [mul_smul] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.RingTheory.HahnSeries.HEval | {
"line": 98,
"column": 4
} | {
"line": 98,
"column": 29
} | [
{
"pp": "case pos\nΓ : 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\ng : Γ\nh : 0 < x.orderTop\nn : ℕ\nhn : ((powerSeriesFamily x (a * b)) ... | obtain ⟨c, hcn, hc⟩ := he | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.RingTheory.Henselian | {
"line": 157,
"column": 4
} | {
"line": 159,
"column": 33
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\nhR : HenselianLocalRing R\n⊢ maximalIdeal R ≤ ⊥.jacobson",
"usedConstants": [
"Eq.mpr",
"le_sInf_iff",
"Eq.ge",
"Semiring.toModule",
"congrArg",
"CommSemiring.toSemiring",
"Submodule.completeLattice",
"PartialOrder.to... | rw [Ideal.jacobson, le_sInf_iff]
rintro I ⟨-, hI⟩
exact (eq_maximalIdeal hI).ge | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Henselian | {
"line": 157,
"column": 4
} | {
"line": 159,
"column": 33
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\nhR : HenselianLocalRing R\n⊢ maximalIdeal R ≤ ⊥.jacobson",
"usedConstants": [
"Eq.mpr",
"le_sInf_iff",
"Eq.ge",
"Semiring.toModule",
"congrArg",
"CommSemiring.toSemiring",
"Submodule.completeLattice",
"PartialOrder.to... | rw [Ideal.jacobson, le_sInf_iff]
rintro I ⟨-, hI⟩
exact (eq_maximalIdeal hI).ge | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.HahnSeries.Summable | {
"line": 780,
"column": 6
} | {
"line": 780,
"column": 14
} | [
{
"pp": "Γ : Type u_1\nR : Type u_3\ninst✝³ : AddCommMonoid Γ\ninst✝² : LinearOrder Γ\ninst✝¹ : IsOrderedCancelAddMonoid Γ\ninst✝ : CommRing R\nx y : R⟦Γ⟧\nr : R\nhr : r * x.leadingCoeff = 1\nhxy : x = y + (single x.order) x.leadingCoeff\noinv : Γ\nhxo : oinv + x.order = 0\n⊢ 1 - (single oinv) r * (y + (single ... | mul_add, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Henselian | {
"line": 210,
"column": 12
} | {
"line": 210,
"column": 36
} | [
{
"pp": "case succ\nR : Type u_1\ninst✝¹ : CommRing R\nI : Ideal R\ninst✝ : IsAdicComplete I R\nf : R[X]\nx✝ : f.Monic\na₀ : R\nh₁ : Polynomial.eval a₀ f ∈ I\nh₂ : IsUnit ((Ideal.Quotient.mk I) (Polynomial.eval a₀ (derivative f)))\nf' : R[X] := derivative f\nc : ℕ → R := fun n ↦ Nat.recOn n a₀ fun x b ↦ b - Pol... | ← taylor_eval_sub (c n), | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Ideal.MinimalPrime.Colon | {
"line": 52,
"column": 2
} | {
"line": 52,
"column": 47
} | [
{
"pp": "case neg\nR : Type u_1\nM : Type u_2\ninst✝³ : CommSemiring R\ninst✝² : AddCommMonoid M\ninst✝¹ : Module R M\nN : Submodule R M\nI : Ideal R\nx : M\ninst✝ : IsNoetherianRing R\nhx : x ∉ N\nann : Ideal R := N.colon {x}\nhI : I ∈ ann.minimalPrimes\nkey : ∃ n, n ≠ 0 ∧ ∃ J, I ^ n * J ≤ ann ∧ ¬J ≤ I\n⊢ ∃ x'... | obtain ⟨hn0, J, hJ, hJI⟩ := Nat.find_spec key | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.RingTheory.Ideal.AssociatedPrime.Finiteness | {
"line": 141,
"column": 4
} | {
"line": 141,
"column": 40
} | [
{
"pp": "A : Type u\ninst✝³ : CommRing A\ninst✝² : IsNoetherianRing A\nM : Type v\ninst✝¹ : AddCommGroup M\ninst✝ : Module A M\nx✝ : Module.Finite A M\nmotive : (N : Type v) → [inst : AddCommGroup N] → [inst_1 : Module A N] → [Module.Finite A N] → Prop\nsubsingleton :\n ∀ (N : Type v) [inst : AddCommGroup N] [... | exact equiv _ _ Submodule.topEquiv H | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.RingTheory.Ideal.AssociatedPrime.Finiteness | {
"line": 202,
"column": 4
} | {
"line": 202,
"column": 96
} | [
{
"pp": "A : Type u\ninst✝¹ : CommRing A\ninst✝ : IsNoetherianRing A\nI : Ideal A\nh : Disjoint ↑I ↑(nonZeroDivisors A)\nx : A\nhP : I ≤ ⊥.colon {x}\nprime : (⊥.colon {x}).IsPrime\n⊢ ∀ n ∈ I, x • n = 0",
"usedConstants": [
"NonUnitalNonAssocCommRing.toNonUnitalNonAssocCommSemiring",
"Eq.mpr",
... | simpa only [smul_eq_mul, mul_comm x, SetLike.le_def, Submodule.mem_colon_singleton] using hP | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.RingTheory.Ideal.AssociatedPrime.Finiteness | {
"line": 202,
"column": 4
} | {
"line": 202,
"column": 96
} | [
{
"pp": "A : Type u\ninst✝¹ : CommRing A\ninst✝ : IsNoetherianRing A\nI : Ideal A\nh : Disjoint ↑I ↑(nonZeroDivisors A)\nx : A\nhP : I ≤ ⊥.colon {x}\nprime : (⊥.colon {x}).IsPrime\n⊢ ∀ n ∈ I, x • n = 0",
"usedConstants": [
"NonUnitalNonAssocCommRing.toNonUnitalNonAssocCommSemiring",
"Eq.mpr",
... | simpa only [smul_eq_mul, mul_comm x, SetLike.le_def, Submodule.mem_colon_singleton] using hP | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Ideal.AssociatedPrime.Finiteness | {
"line": 202,
"column": 4
} | {
"line": 202,
"column": 96
} | [
{
"pp": "A : Type u\ninst✝¹ : CommRing A\ninst✝ : IsNoetherianRing A\nI : Ideal A\nh : Disjoint ↑I ↑(nonZeroDivisors A)\nx : A\nhP : I ≤ ⊥.colon {x}\nprime : (⊥.colon {x}).IsPrime\n⊢ ∀ n ∈ I, x • n = 0",
"usedConstants": [
"NonUnitalNonAssocCommRing.toNonUnitalNonAssocCommSemiring",
"Eq.mpr",
... | simpa only [smul_eq_mul, mul_comm x, SetLike.le_def, Submodule.mem_colon_singleton] using hP | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Ideal.Height | {
"line": 190,
"column": 6
} | {
"line": 190,
"column": 24
} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommRing R\nI : Ideal R\ninst✝ : I.IsPrime\n⊢ I.primeHeight = 0 ↔ I ∈ minimalPrimes R",
"usedConstants": [
"Eq.mpr",
"PrimeSpectrum.mk",
"congrArg",
"CommSemiring.toSemiring",
"PartialOrder.toPreorder",
"NonUnitalNonAssocSemiring.toMulZeroC... | Ideal.primeHeight, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Ideal.Height | {
"line": 228,
"column": 20
} | {
"line": 228,
"column": 38
} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsLocalRing R\n⊢ ↑(maximalIdeal R).primeHeight = Order.krullDim (PrimeSpectrum R)",
"usedConstants": [
"Eq.mpr",
"PrimeSpectrum.mk",
"WithBot.some",
"WithBot",
"congrArg",
"PartialOrder.toPreorder",
"IsLocalRing.ma... | Ideal.primeHeight, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Ideal.Height | {
"line": 316,
"column": 15
} | {
"line": 316,
"column": 33
} | [
{
"pp": "R : Type u_1\ninst✝⁴ : CommRing R\nS : Submonoid R\nA : Type u_2\ninst✝³ : CommRing A\ninst✝² : Algebra R A\ninst✝¹ : IsLocalization S A\nJ : Ideal A\ninst✝ : J.IsPrime\n⊢ J.primeHeight = (comap (algebraMap R A) J).primeHeight",
"usedConstants": [
"Eq.mpr",
"PrimeSpectrum.mk",
"Ri... | Ideal.primeHeight, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Ideal.Height | {
"line": 316,
"column": 34
} | {
"line": 316,
"column": 52
} | [
{
"pp": "R : Type u_1\ninst✝⁴ : CommRing R\nS : Submonoid R\nA : Type u_2\ninst✝³ : CommRing A\ninst✝² : Algebra R A\ninst✝¹ : IsLocalization S A\nJ : Ideal A\ninst✝ : J.IsPrime\n⊢ Order.height { asIdeal := J, isPrime := inst✝ } = (comap (algebraMap R A) J).primeHeight",
"usedConstants": [
"Eq.mpr",
... | Ideal.primeHeight, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.IdealFilter.Basic | {
"line": 144,
"column": 2
} | {
"line": 144,
"column": 36
} | [
{
"pp": "A : Type u_1\ninst✝ : Ring A\nF : IdealFilter A\nI J K : Ideal A\nhI : F.IsTorsionQuot I K\nhJ : F.IsTorsionQuot J K\nx : A\nhx : x ∈ K\nI' : Ideal A\nhI'F : I' ∈ F\nhI'x : I' ≤ Submodule.colon I {x}\n⊢ ∃ I_1 ∈ F, I_1 ≤ Submodule.colon (I ⊓ J) {x}",
"usedConstants": []
}
] | obtain ⟨J', hJ'F, hJ'x⟩ := hJ x hx | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.Topology.Algebra.Category.ProfiniteGrp.Limits | {
"line": 123,
"column": 2
} | {
"line": 124,
"column": 71
} | [
{
"pp": "P : ProfiniteGrp.{u}\nx : ↑P.toProfinite.toTop\nh : x ∈ (Hom.hom P.toLimit).ker\nxne1 : ¬x = 1\n⊢ False",
"usedConstants": [
"Iff.mpr",
"InvOneClass.toOne",
"DivInvOneMonoid.toInvOneClass",
"Compl.compl",
"Profinite.instTotallyDisconnectedSpaceCarrierToTop",
"Pro... | rcases exist_openNormalSubgroup_sub_open_nhds_of_one (isOpen_compl_singleton)
(Set.mem_compl_singleton_iff.mpr fun a => xne1 a.symm) with ⟨H, hH⟩ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRCases | Lean.Parser.Tactic.rcases |
Mathlib.RingTheory.KrullDimension.NonZeroDivisors | {
"line": 55,
"column": 19
} | {
"line": 55,
"column": 45
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\nr : R\nhr : r ∈ R⁰\nhr' : ¬Ideal.span {r} = ⊤\nthis✝¹ : Nonempty ↑(PrimeSpectrum.zeroLocus ↑(Ideal.span {r}))\nthis✝ : Nontrivial (R ⧸ Ideal.span {r})\nthis : Nontrivial R\nl : LTSeries ↑(PrimeSpectrum.zeroLocus ↑(Ideal.span {r}))\np : Ideal R\nhp : p ∈ ⊥.minimalPrimes... | simpa [← h] using l.head.2 | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.RingTheory.KrullDimension.NonZeroDivisors | {
"line": 55,
"column": 19
} | {
"line": 55,
"column": 45
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\nr : R\nhr : r ∈ R⁰\nhr' : ¬Ideal.span {r} = ⊤\nthis✝¹ : Nonempty ↑(PrimeSpectrum.zeroLocus ↑(Ideal.span {r}))\nthis✝ : Nontrivial (R ⧸ Ideal.span {r})\nthis : Nontrivial R\nl : LTSeries ↑(PrimeSpectrum.zeroLocus ↑(Ideal.span {r}))\np : Ideal R\nhp : p ∈ ⊥.minimalPrimes... | simpa [← h] using l.head.2 | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.KrullDimension.NonZeroDivisors | {
"line": 55,
"column": 19
} | {
"line": 55,
"column": 45
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\nr : R\nhr : r ∈ R⁰\nhr' : ¬Ideal.span {r} = ⊤\nthis✝¹ : Nonempty ↑(PrimeSpectrum.zeroLocus ↑(Ideal.span {r}))\nthis✝ : Nontrivial (R ⧸ Ideal.span {r})\nthis : Nontrivial R\nl : LTSeries ↑(PrimeSpectrum.zeroLocus ↑(Ideal.span {r}))\np : Ideal R\nhp : p ∈ ⊥.minimalPrimes... | simpa [← h] using l.head.2 | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.KrullDimension.NonZeroDivisors | {
"line": 111,
"column": 4
} | {
"line": 111,
"column": 80
} | [
{
"pp": "case inr\nR : Type u_1\ninst✝ : CommRing R\nσ : Type u_3\na✝ : Nontrivial R\nh✝ : Infinite σ\nn : ℕ\nι : Fin (n + 1) → σ := ⇑(Infinite.natEmbedding σ) ∘ Fin.val\nthis : Function.Surjective (Function.invFun ι)\n⊢ ↑↑n ≤ ringKrullDim (MvPolynomial (Fin (n + 1)) R)",
"usedConstants": [
"WithBot.i... | refine le_trans ?_ (ringKrullDim_add_natCard_le_ringKrullDim_mvPolynomial _) | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.RingTheory.KrullDimension.PID | {
"line": 54,
"column": 59
} | {
"line": 54,
"column": 77
} | [
{
"pp": "case refine_2\nR : Type u_1\ninst✝³ : CommRing R\ninst✝² : IsDomain R\ninst✝¹ : IsPrincipalIdealRing R\nm : Ideal R\ninst✝ : m.IsMaximal\nh : ¬IsField R\n⊢ 0 < m.primeHeight",
"usedConstants": [
"Eq.mpr",
"PrimeSpectrum.mk",
"Preorder.toLT",
"instCompleteLinearOrderENat",
... | Ideal.primeHeight, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.KrullDimension.Regular | {
"line": 40,
"column": 2
} | {
"line": 40,
"column": 9
} | [
{
"pp": "case inr\nR : Type u_1\ninst✝² : CommRing R\nM : Type u_2\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nq : LTSeries ↑(support R M)\nhmm : (↑(RelSeries.last q)).asIdeal.IsMaximal\nhm : (↑(RelSeries.last q)).asIdeal ≤ (↑(RelSeries.last q)).asIdeal\n⊢ ∃ p, q.length ≤ p.length ∧ (↑(RelSeries.last p)).asId... | · use q | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.RingTheory.KrullDimension.Regular | {
"line": 204,
"column": 4
} | {
"line": 206,
"column": 62
} | [
{
"pp": "case cons\nR : 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 = supportDi... | simp [supportDim_eq_of_equiv (Submodule.quotOfListConsSMulTopEquivQuotSMulTopInner M x _),
← supportDim_quotSMulTop_succ_eq_supportDim ((isRegular_cons_iff M _ _).mp reg).1 mem,
← ih ((isRegular_cons_iff M _ _).mp reg).2, ← add_assoc] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.RingTheory.Length | {
"line": 180,
"column": 6
} | {
"line": 180,
"column": 85
} | [
{
"pp": "case pos\nR : Type u_1\nM : Type u_2\ninst✝⁶ : Ring R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : Module R M\nN : Type u_3\nP : Type u_4\ninst✝³ : AddCommGroup N\ninst✝² : AddCommGroup P\ninst✝¹ : Module R N\ninst✝ : Module R P\nf : N →ₗ[R] M\ng : M →ₗ[R] P\nhf : Function.Injective ⇑f\nhg : Function.Surjective ... | simp_rw [r, RelSeries.smash_length, Nat.cast_add, s', t', RelSeries.map_length] | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | Mathlib.Tactic.tacticSimp_rw___ |
Mathlib.RingTheory.Length | {
"line": 217,
"column": 4
} | {
"line": 217,
"column": 11
} | [
{
"pp": "case h_option\nR : Type u_1\ninst✝ : Ring R\n⊢ ∀ (α : Type u_5) [inst : Fintype α],\n (∀ (M : α → Type u_6) [inst_1 : (i : α) → AddCommGroup (M i)] [inst_2 : (i : α) → Module R (M i)],\n length R ((i : α) → M i) = ∑ i, length R (M i)) →\n ∀ (M : Option α → Type u_6) [inst_1 : (i : Option... | intro ι | Lean.Elab.Tactic.evalIntro | null |
Mathlib.RingTheory.LocalProperties.Semilocal | {
"line": 73,
"column": 2
} | {
"line": 74,
"column": 90
} | [
{
"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] → ... | simp_rw [← Module.Finite.iff_fg] at ⊢ H
exact .of_isLocalized_maximal _ _ _ (fun P ↦ N.toLocalized' (Rₚ P) P.primeCompl (f P)) H | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.LocalProperties.Semilocal | {
"line": 73,
"column": 2
} | {
"line": 74,
"column": 90
} | [
{
"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] → ... | simp_rw [← Module.Finite.iff_fg] at ⊢ H
exact .of_isLocalized_maximal _ _ _ (fun P ↦ N.toLocalized' (Rₚ P) P.primeCompl (f P)) H | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Localization.AtPrime.Extension | {
"line": 74,
"column": 2
} | {
"line": 77,
"column": 45
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝¹⁵ : CommRing R\ninst✝¹⁴ : CommRing S\ninst✝¹³ : Algebra R S\np : Ideal R\ninst✝¹² : p.IsPrime\nRₚ : Type u_3\ninst✝¹¹ : CommRing Rₚ\ninst✝¹⁰ : Algebra R Rₚ\ninst✝⁹ : IsLocalization.AtPrime Rₚ p\ninst✝⁸ : IsLocalRing Rₚ\nSₚ : Type u_4\ninst✝⁷ : CommRing Sₚ\ninst✝⁶ : Alg... | rw [liesOver_iff, eq_comm, ← map_eq_maximalIdeal p, over_def P p]
exact under_map_eq_map_under _
(over_def P p ▸ map_eq_maximalIdeal p Rₚ ▸ maximalIdeal.isMaximal Rₚ)
(isPrime_map_of_liesOver S p Sₚ P).ne_top | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Localization.AtPrime.Extension | {
"line": 74,
"column": 2
} | {
"line": 77,
"column": 45
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝¹⁵ : CommRing R\ninst✝¹⁴ : CommRing S\ninst✝¹³ : Algebra R S\np : Ideal R\ninst✝¹² : p.IsPrime\nRₚ : Type u_3\ninst✝¹¹ : CommRing Rₚ\ninst✝¹⁰ : Algebra R Rₚ\ninst✝⁹ : IsLocalization.AtPrime Rₚ p\ninst✝⁸ : IsLocalRing Rₚ\nSₚ : Type u_4\ninst✝⁷ : CommRing Sₚ\ninst✝⁶ : Alg... | rw [liesOver_iff, eq_comm, ← map_eq_maximalIdeal p, over_def P p]
exact under_map_eq_map_under _
(over_def P p ▸ map_eq_maximalIdeal p Rₚ ▸ maximalIdeal.isMaximal Rₚ)
(isPrime_map_of_liesOver S p Sₚ P).ne_top | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.MvPolynomial.EulerIdentity | {
"line": 68,
"column": 4
} | {
"line": 68,
"column": 56
} | [
{
"pp": "case refine_1.e_a\nR : Type u_1\nσ : Type u_2\ninst✝¹ : CommSemiring R\nφ : MvPolynomial σ R\ninst✝ : Fintype σ\nn : ℕ\nw : σ → ℕ\nh : φ ∈ Submodule.span R ((fun i ↦ single i 1) '' {d | (weight w) d = n})\nm : σ →₀ ℕ\nhm : m ∈ {d | (weight w) d = n}\n⊢ ∑ x, m x * w x = n",
"usedConstants": [
... | rwa [Set.mem_setOf, weight_apply, sum_fintype] at hm | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticRwa___1 | Lean.Parser.Tactic.tacticRwa__ |
Mathlib.RingTheory.MvPolynomial.Symmetric.FundamentalTheorem | {
"line": 181,
"column": 34
} | {
"line": 181,
"column": 68
} | [
{
"pp": "case refine_3\nR : Type u_3\nm : ℕ\ninst✝¹ : CommSemiring R\ninst✝ : Nontrivial R\ni : ℕ\nhim : i < m\nthis :\n supDegree (⇑toLex) (esymm (Fin m) R (i + 1)) = supDegree (⇑toLex) ((monomial (∑ j ∈ Iic ⟨i, him⟩, fun₀ | j => 1)) 1) ∧\n leadingCoeff (⇑toLex) (esymm (Fin m) R (i + 1)) =\n leadingCo... | ← Finsupp.indicator_eq_sum_single, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.MvPolynomial.Symmetric.FundamentalTheorem | {
"line": 312,
"column": 6
} | {
"line": 312,
"column": 23
} | [
{
"pp": "case right.inr.ind.inr.refine_2\nR : Type u_3\ninst✝ : CommRing R\nn : ℕ\np : MvPolynomial (Fin n) R\nhp : p ∈ symmetricSubalgebra (Fin n) R\nh0 : p ≠ 0\nih :\n ∀ y < supDegree (⇑toLex) p,\n ∀ (p : MvPolynomial (Fin n) R) (hp : p ∈ symmetricSubalgebra (Fin n) R),\n p ≠ 0 → supDegree (⇑toLex) p... | rwa [sub_ne_zero] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticRwa___1 | Lean.Parser.Tactic.tacticRwa__ |
Mathlib.RingTheory.MvPolynomial.Symmetric.FundamentalTheorem | {
"line": 312,
"column": 6
} | {
"line": 312,
"column": 23
} | [
{
"pp": "case right.inr.ind.inr.refine_2\nR : Type u_3\ninst✝ : CommRing R\nn : ℕ\np : MvPolynomial (Fin n) R\nhp : p ∈ symmetricSubalgebra (Fin n) R\nh0 : p ≠ 0\nih :\n ∀ y < supDegree (⇑toLex) p,\n ∀ (p : MvPolynomial (Fin n) R) (hp : p ∈ symmetricSubalgebra (Fin n) R),\n p ≠ 0 → supDegree (⇑toLex) p... | rwa [sub_ne_zero] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.MvPolynomial.Symmetric.FundamentalTheorem | {
"line": 312,
"column": 6
} | {
"line": 312,
"column": 23
} | [
{
"pp": "case right.inr.ind.inr.refine_2\nR : Type u_3\ninst✝ : CommRing R\nn : ℕ\np : MvPolynomial (Fin n) R\nhp : p ∈ symmetricSubalgebra (Fin n) R\nh0 : p ≠ 0\nih :\n ∀ y < supDegree (⇑toLex) p,\n ∀ (p : MvPolynomial (Fin n) R) (hp : p ∈ symmetricSubalgebra (Fin n) R),\n p ≠ 0 → supDegree (⇑toLex) p... | rwa [sub_ne_zero] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.MvPowerSeries.Expand | {
"line": 49,
"column": 2
} | {
"line": 52,
"column": 8
} | [
{
"pp": "σ : Type u_1\nR : Type u_3\ninst✝ : CommRing R\np : ℕ\nhp : p ≠ 0\nd : σ →₀ ℕ\nr : R\n⊢ (expand p hp) ((monomial d) r) = (monomial (p • d)) r",
"usedConstants": [
"MvPowerSeries.expand",
"Finsupp.instFunLike",
"Eq.mpr",
"Finsupp.smulZeroClass",
"MulOne.toOne",
"N... | rw [expand, substAlgHom_monomial (HasSubst.X_pow hp), monomial_eq', Finsupp.prod,
Finsupp.prod_of_support_subset _ Finsupp.support_smul]
· simp [pow_mul, algebraMap_apply, Algebra.algebraMap_self]
· simp | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.MvPowerSeries.Expand | {
"line": 49,
"column": 2
} | {
"line": 52,
"column": 8
} | [
{
"pp": "σ : Type u_1\nR : Type u_3\ninst✝ : CommRing R\np : ℕ\nhp : p ≠ 0\nd : σ →₀ ℕ\nr : R\n⊢ (expand p hp) ((monomial d) r) = (monomial (p • d)) r",
"usedConstants": [
"MvPowerSeries.expand",
"Finsupp.instFunLike",
"Eq.mpr",
"Finsupp.smulZeroClass",
"MulOne.toOne",
"N... | rw [expand, substAlgHom_monomial (HasSubst.X_pow hp), monomial_eq', Finsupp.prod,
Finsupp.prod_of_support_subset _ Finsupp.support_smul]
· simp [pow_mul, algebraMap_apply, Algebra.algebraMap_self]
· simp | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.MvPowerSeries.Expand | {
"line": 245,
"column": 8
} | {
"line": 245,
"column": 19
} | [
{
"pp": "case h\nσ : Type u_1\nR : Type u_3\ninst✝¹ : CommRing R\np : ℕ\nhp : p ≠ 0\ninst✝ : ExpChar R p\nf : MvPowerSeries σ R\nn : σ →₀ ℕ\nthis :\n ↑((MvPolynomial.map (frobenius R p)) ((MvPolynomial.expand p) ((trunc' R (p • n)) f))) =\n (map (frobenius R p)) ↑((MvPolynomial.expand p) ((trunc' R (p • n))... | trunc'_map, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.MvPowerSeries.Expand | {
"line": 247,
"column": 8
} | {
"line": 247,
"column": 19
} | [
{
"pp": "case h\nσ : Type u_1\nR : Type u_3\ninst✝¹ : CommRing R\np : ℕ\nhp : p ≠ 0\ninst✝ : ExpChar R p\nf : MvPowerSeries σ R\nn : σ →₀ ℕ\nthis :\n ↑((MvPolynomial.map (frobenius R p)) ((MvPolynomial.expand p) ((trunc' R (p • n)) f))) =\n (map (frobenius R p)) ↑((MvPolynomial.expand p) ((trunc' R (p • n))... | trunc'_map, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.MvPowerSeries.Rename | {
"line": 275,
"column": 38
} | {
"line": 279,
"column": 50
} | [
{
"pp": "σ : Type u_1\nτ : Type u_2\nR : Type u_4\ninst✝ : CommSemiring R\ne : σ ↪ τ\nt : τ\nh : t ∉ Set.range ⇑e\n⊢ (killCompl e) (X t) = 0",
"usedConstants": [
"Set.singleton_subset_iff",
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"False",
"Nat.instMulZeroClass",
... | by
replace h : single t 1 ∉ Set.range (embDomain e) := by
rwa [mem_range_embDomain_iff, support_single_ne_zero _ (by simp), Finset.coe_singleton,
Set.singleton_subset_iff]
simpa using killCompl_monomial_eq_zero (1 : R) h | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.LaurentSeries | {
"line": 600,
"column": 4
} | {
"line": 600,
"column": 80
} | [
{
"pp": "case coe.ofAdd\nK : Type u_2\ninst✝ : Field K\nf : K⸨X⸩\nD : ℤ\nhD : ↑(Multiplicative.ofAdd D) ≠ 0\n⊢ Valued.v f ≤ ↑(Multiplicative.ofAdd D) ↔ ∀ n < -(↑(Multiplicative.ofAdd D)).log, f.coeff n = 0",
"usedConstants": [
"Int.instAddCommGroup",
"Eq.mpr",
"Int.instAddCommMonoid",
... | rw [← exp, ← neg_neg D, valuation_le_iff_coeff_lt_eq_zero, log_exp, neg_neg] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.Morita.Matrix | {
"line": 197,
"column": 4
} | {
"line": 198,
"column": 80
} | [
{
"pp": "case hf.h\nR : Type u\nι : Type v\ninst✝² : Ring R\ninst✝¹ : Fintype ι\ninst✝ : DecidableEq ι\ni : ι\nX : ModuleCat R\nx : ι → ↑((𝟭 (ModuleCat R)).obj X)\n⊢ (↑(toModuleCatFromModuleCatLinearEquiv R ((toMatrixModCat R ι).obj X) i).symm ∘ₗ\n LinearMap.mapMatrixModule ι (Hom.hom ((unitIso R i).inv... | simp [unitIso, toModuleCatFromModuleCatLinearEquiv, fromModuleCatToModuleCatLinearEquiv,
fromModuleCatToModuleCatLinearEquivtoModuleCatObj, Finset.univ_sum_single] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.RingTheory.NoetherNormalization | {
"line": 291,
"column": 2
} | {
"line": 293,
"column": 86
} | [
{
"pp": "k : Type u_2\nR : Type u_3\ninst✝² : Field k\ninst✝¹ : CommRing R\ninst✝ : Nontrivial R\na : Algebra k R\nfin : Algebra.FiniteType k R\ns : ℕ\ng : MvPolynomial (Fin s) k →ₐ[k] R\ninj : Function.Injective ⇑g\nint : g.IsIntegral\n⊢ ∃ s g, Function.Injective ⇑g ∧ g.Finite",
"usedConstants": [
"E... | have h : algebraMap k R = g.toRingHom.comp (algebraMap k (MvPolynomial (Fin s) k)) := by
algebraize [g.toRingHom]
rw [IsScalarTower.algebraMap_eq k (MvPolynomial (Fin s) k), algebraMap_toAlgebra'] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.RingTheory.LaurentSeries | {
"line": 747,
"column": 50
} | {
"line": 747,
"column": 94
} | [
{
"pp": "K : Type u_2\ninst✝ : Field K\nℱ : Filter K⸨X⸩\nhℱ : Cauchy ℱ\nN₁ : ℤ\nhN₁ : ∀ᶠ (f : K⸨X⸩) in ℱ, ∀ n < N₁, f.coeff n = 0\nN₂ : ℤ\nhN₂ : ∀ n < N₂, coeff hℱ n = 0\nx✝ : K⸨X⸩\nhf : x✝ ∈ {x | (fun f ↦ ∀ n < N₁, f.coeff n = 0) x}\nd : ℤ\nhd : d < min N₁ N₂\n⊢ coeff hℱ d = 0",
"usedConstants": [
"E... | hN₂ d (lt_of_lt_of_le hd (min_le_right _ _)) | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Teichmuller | {
"line": 167,
"column": 2
} | {
"line": 169,
"column": 43
} | [
{
"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",
"usedConstants": [
"Ideal.Quotient.commSemiring",
"... | have := teichmuller_sModEq <| Ideal.Quotient.mk_out <| coeff _ p 0 x
simp_rw [zero_add, pow_one] at this
simpa [SModEq.idealQuotientMk] using this | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Teichmuller | {
"line": 167,
"column": 2
} | {
"line": 169,
"column": 43
} | [
{
"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",
"usedConstants": [
"Ideal.Quotient.commSemiring",
"... | have := teichmuller_sModEq <| Ideal.Quotient.mk_out <| coeff _ p 0 x
simp_rw [zero_add, pow_one] at this
simpa [SModEq.idealQuotientMk] using this | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.LaurentSeries | {
"line": 878,
"column": 2
} | {
"line": 894,
"column": 43
} | [
{
"pp": "K : Type u_2\ninst✝ : Field K\n⊢ DenseRange ⇑(algebraMap K⟮X⟯ K⸨X⸩)",
"usedConstants": [
"Int.instAddCommGroup",
"RatFunc.instFaithfulSMulPolynomialLaurentSeries",
"Filter.instMembership",
"Multiplicative.group",
"AddGroup.toSubtractionMonoid",
"Set.ext",
"... | rw [denseRange_iff_closure_range]
ext f
simp only [UniformSpace.mem_closure_iff_symm_ball, Set.mem_univ, iff_true, Set.Nonempty,
Set.mem_inter_iff, Set.mem_range, exists_exists_eq_and]
intro V hV h_symm
rw [uniformity_eq_comap_neg_add_nhds_zero_swapped] at hV
obtain ⟨T, hT₀, hT₁⟩ := hV
obtain ⟨γ, hγ⟩ :=... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.LaurentSeries | {
"line": 878,
"column": 2
} | {
"line": 894,
"column": 43
} | [
{
"pp": "K : Type u_2\ninst✝ : Field K\n⊢ DenseRange ⇑(algebraMap K⟮X⟯ K⸨X⸩)",
"usedConstants": [
"Int.instAddCommGroup",
"RatFunc.instFaithfulSMulPolynomialLaurentSeries",
"Filter.instMembership",
"Multiplicative.group",
"AddGroup.toSubtractionMonoid",
"Set.ext",
"... | rw [denseRange_iff_closure_range]
ext f
simp only [UniformSpace.mem_closure_iff_symm_ball, Set.mem_univ, iff_true, Set.Nonempty,
Set.mem_inter_iff, Set.mem_range, exists_exists_eq_and]
intro V hV h_symm
rw [uniformity_eq_comap_neg_add_nhds_zero_swapped] at hV
obtain ⟨T, hT₀, hT₁⟩ := hV
obtain ⟨γ, hγ⟩ :=... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.WittVector.Defs | {
"line": 231,
"column": 4
} | {
"line": 232,
"column": 75
} | [
{
"pp": "case a.h\np : ℕ\nhp : Fact (Nat.Prime p)\nn : ℕ\nIH : ∀ m < n, 0 < m → (bind₁ fun k ↦ 1) (xInTermsOfW p ℚ m) = 0\nhn : 0 < n\n⊢ C ⅟↑p ^ n - C ↑p ^ 0 * (bind₁ fun k ↦ 1) (xInTermsOfW p ℚ 0) ^ p ^ (n - 0) * C ⅟↑p ^ n = 0",
"usedConstants": [
"one_pow",
"Finsupp.instAddZeroClass",
"E... | simp only [one_mul, pow_zero]
simp only [one_pow, one_mul, xInTermsOfW_zero, sub_self, bind₁_X_right] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.WittVector.Defs | {
"line": 231,
"column": 4
} | {
"line": 232,
"column": 75
} | [
{
"pp": "case a.h\np : ℕ\nhp : Fact (Nat.Prime p)\nn : ℕ\nIH : ∀ m < n, 0 < m → (bind₁ fun k ↦ 1) (xInTermsOfW p ℚ m) = 0\nhn : 0 < n\n⊢ C ⅟↑p ^ n - C ↑p ^ 0 * (bind₁ fun k ↦ 1) (xInTermsOfW p ℚ 0) ^ p ^ (n - 0) * C ⅟↑p ^ n = 0",
"usedConstants": [
"one_pow",
"Finsupp.instAddZeroClass",
"E... | simp only [one_mul, pow_zero]
simp only [one_pow, one_mul, xInTermsOfW_zero, sub_self, bind₁_X_right] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.WittVector.Defs | {
"line": 233,
"column": 2
} | {
"line": 235,
"column": 91
} | [
{
"pp": "case a.h.h₀\np : ℕ\nhp : Fact (Nat.Prime p)\nn : ℕ\nIH : ∀ m < n, 0 < m → (bind₁ fun k ↦ 1) (xInTermsOfW p ℚ m) = 0\nhn : 0 < n\n⊢ ∀ b ∈ Finset.range n, b ≠ 0 → C ↑p ^ b * (bind₁ fun k ↦ 1) (xInTermsOfW p ℚ b) ^ p ^ (n - b) = 0",
"usedConstants": [
"Finsupp.instAddZeroClass",
"Eq.mpr",
... | · intro i hin hi0
rw [Finset.mem_range] at hin
rw [IH _ hin (Nat.pos_of_ne_zero hi0), zero_pow (pow_ne_zero _ hp.1.ne_zero), mul_zero] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.RingTheory.LaurentSeries | {
"line": 925,
"column": 12
} | {
"line": 925,
"column": 35
} | [
{
"pp": "K : Type u_2\ninst✝ : Field K\nS : Set (K⟮X⟯ × K⟮X⟯)\nT : Set (K⸨X⸩ × K⸨X⸩)\npre_T : (fun x ↦ ((algebraMap K⟮X⟯ K⸨X⸩) x.1, (algebraMap K⟮X⟯ K⸨X⸩) x.2)) ⁻¹' T ⊆ S\nR : Set K⸨X⸩\nhR : R ∈ nhds 0\npre_R : (fun x ↦ x.2 - x.1) ⁻¹' R ⊆ T\nd : (MonoidWithZeroHom.ValueGroup₀ Valued.v)ˣ\nhd : {y | Valued.v.rest... | Valuation.restrict_def, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.LaurentSeries | {
"line": 936,
"column": 6
} | {
"line": 936,
"column": 36
} | [
{
"pp": "case h.mpr.refine_1\nK : Type u_2\ninst✝ : Field K\nS : Set (K⟮X⟯ × K⟮X⟯)\nw✝ : Set K⟮X⟯\nhT : w✝ ∈ nhds 0\npre_T : (fun x ↦ x.2 - x.1) ⁻¹' w✝ ⊆ S\nd : (MonoidWithZeroHom.ValueGroup₀ Valued.v)ˣ\nhd : {y | Valued.v.restrict (y - 0) < ↑d} ⊆ w✝\nX : Set K⸨X⸩ := {f | Valued.v f < embedding ↑d}\nX_def : X =... | refine ⟨?_, Set.Subset.refl _⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.RingTheory.LaurentSeries | {
"line": 940,
"column": 14
} | {
"line": 940,
"column": 37
} | [
{
"pp": "K : Type u_2\ninst✝ : Field K\nS : Set (K⟮X⟯ × K⟮X⟯)\nw✝ : Set K⟮X⟯\nhT : w✝ ∈ nhds 0\npre_T : (fun x ↦ x.2 - x.1) ⁻¹' w✝ ⊆ S\nd : (MonoidWithZeroHom.ValueGroup₀ Valued.v)ˣ\nhd : {y | Valued.v.restrict (y - 0) < ↑d} ⊆ w✝\nX : Set K⸨X⸩ := {f | Valued.v f < embedding ↑d}\nX_def : X = {f | Valued.v f < em... | Valuation.restrict_def, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.LaurentSeries | {
"line": 1014,
"column": 16
} | {
"line": 1018,
"column": 38
} | [
{
"pp": "R : Type u_1\nK : Type u_2\ninst✝ : Field K\n⊢ ∀ (x y : RatFuncAdicCompl K), (comparePkg K).toFun (x * y) = (comparePkg K).toFun x * (comparePkg K).toFun y",
"usedConstants": [
"Int.instAddCommGroup",
"Eq.mpr",
"LaurentSeries.extensionAsRingHom",
"Int.instAddCommMonoid",
... | by
intro x y
rw [Equiv.toFun_as_coe, UniformEquiv.coe_toEquiv, comparePkg_eq_extension,
(extensionAsRingHom K (continuous_coe' _)).map_mul]
simp [← comparePkg_eq_extension] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.LaurentSeries | {
"line": 1081,
"column": 13
} | {
"line": 1081,
"column": 36
} | [
{
"pp": "K : 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 γ ⊆ S\nx : ad... | Valuation.restrict_def, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.LaurentSeries | {
"line": 1142,
"column": 2
} | {
"line": 1143,
"column": 45
} | [
{
"pp": "K : Type u_2\ninst✝ : Field K\nF : K⟦X⟧\n⊢ (LaurentSeriesRingEquiv K) ((ofPowerSeries ℤ K) F) ∈ adicCompletionIntegers K⟮X⟯ (idealX K)",
"usedConstants": [
"Int.instAddCommGroup",
"Eq.mpr",
"Int.instAddCommMonoid",
"LinearOrderedCommGroupWithZero.toLinearOrderedCommMonoidWit... | simp only [mem_adicCompletionIntegers, LaurentSeriesRingEquiv_def,
valuation_compare, val_le_one_iff_eq_coe] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.RingTheory.WittVector.StructurePolynomial | {
"line": 328,
"column": 2
} | {
"line": 328,
"column": 46
} | [
{
"pp": "p : ℕ\nR : Type u_1\nidx : Type u_2\ninst✝ : CommRing R\nhp : Fact (Nat.Prime p)\nΦ : MvPolynomial idx ℤ\nn : ℕ\n⊢ (fun i ↦ (rename (Prod.mk i)) (W_ R n)) = fun i ↦ (map (Int.castRingHom R)) ((rename (Prod.mk i)) (W_ ℤ n))",
"usedConstants": [
"Finsupp.instAddZeroClass",
"wittPolynomial... | · simp only [map_rename, map_wittPolynomial] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.RingTheory.WittVector.Verschiebung | {
"line": 66,
"column": 43
} | {
"line": 66,
"column": 52
} | [
{
"pp": "p : ℕ\nR : Type u_1\ninst✝ : CommRing R\nhp : Fact (Nat.Prime p)\nx : 𝕎 R\nn : ℕ\n⊢ ∑ k ∈ Finset.range (n + 1), ↑p ^ (k + 1) * x.verschiebungFun.coeff (k + 1) ^ p ^ (n + 1 - (k + 1)) + ↑p ^ 0 * 0 =\n ↑p * ∑ i ∈ Finset.range (n + 1), ↑p ^ i * x.coeff i ^ p ^ (n - i)",
"usedConstants": [
"E... | mul_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.WittVector.Verschiebung | {
"line": 116,
"column": 4
} | {
"line": 116,
"column": 18
} | [
{
"pp": "p : ℕ\nR : Type u_1\nS : Type u_2\ninst✝¹ : CommRing R\ninst✝ : CommRing S\nhp : Fact (Nat.Prime p)\n⊢ failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)",
"usedConstants": [
"WittVector"
]
}
] | ghost_calc _ _ | WittVector.Tactic._aux_Mathlib_RingTheory_WittVector_IsPoly___elabRules_WittVector_Tactic_ghostCalc_1 | WittVector.Tactic.ghostCalc |
Mathlib.RingTheory.WittVector.MulP | {
"line": 53,
"column": 46
} | {
"line": 53,
"column": 54
} | [
{
"pp": "case succ\np : ℕ\nR : Type u_1\nhp : Fact (Nat.Prime p)\ninst✝ : CommRing R\nx : 𝕎 R\nn : ℕ\nih : ∀ (k : ℕ), (x * ↑n).coeff k = (aeval x.coeff) (wittMulN p n k)\nk : ℕ\n⊢ (x * (↑n + 1)).coeff k = (aeval x.coeff) ((fun k ↦ (bind₁ (Function.uncurry ![wittMulN p n, X])) (wittAdd p k)) k)",
"usedConst... | mul_add, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.WittVector.Frobenius | {
"line": 156,
"column": 36
} | {
"line": 156,
"column": 44
} | [
{
"pp": "p : ℕ\nhp : Fact (Nat.Prime p)\nn : ℕ\nh1 : ↑p ^ n * ⅟↑p ^ n = 1\ni : ℕ\nhi : i < n\n⊢ C (↑p ^ i) *\n (∑ k ∈ range (p ^ (n - i)),\n (C ↑p * (MvPolynomial.map (Int.castRingHom ℚ)) (frobeniusPolyAux p i)) ^ (k + 1) *\n (X i ^ p) ^ (p ^ (n - i) - (k + 1)) *\n ... | mul_add, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.WittVector.Frobenius | {
"line": 236,
"column": 17
} | {
"line": 236,
"column": 31
} | [
{
"pp": "p : ℕ\nR : Type u_1\nhp : Fact (Nat.Prime p)\ninst✝ : CommRing R\n⊢ failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)",
"usedConstants": [
"WittVector"
]
}
] | ghost_calc _ _ | WittVector.Tactic._aux_Mathlib_RingTheory_WittVector_IsPoly___elabRules_WittVector_Tactic_ghostCalc_1 | WittVector.Tactic.ghostCalc |
Mathlib.RingTheory.WittVector.Frobenius | {
"line": 235,
"column": 17
} | {
"line": 235,
"column": 31
} | [
{
"pp": "p : ℕ\nR : Type u_1\nhp : Fact (Nat.Prime p)\ninst✝ : CommRing R\n⊢ failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)",
"usedConstants": [
"WittVector"
]
}
] | ghost_calc _ _ | WittVector.Tactic._aux_Mathlib_RingTheory_WittVector_IsPoly___elabRules_WittVector_Tactic_ghostCalc_1 | WittVector.Tactic.ghostCalc |
Mathlib.RingTheory.Perfection | {
"line": 755,
"column": 12
} | {
"line": 755,
"column": 21
} | [
{
"pp": "case pos\nK : Type u₁\ninst✝⁴ : Field K\nv : Valuation K ℝ≥0\nO : Type u₂\ninst✝³ : CommRing O\ninst✝² : Algebra O K\nhv : v.Integers O\np : ℕ\ninst✝¹ : Fact (Nat.Prime p)\ninst✝ : Fact ¬IsUnit ↑p\nf g : PreTilt O p\nhf : ¬f = 0\nhg : g = 0\n⊢ valAux K v O p (f * 0) = valAux K v O p f * valAux K v O p ... | mul_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Perfection | {
"line": 823,
"column": 15
} | {
"line": 823,
"column": 31
} | [
{
"pp": "K : Type u₁\ninst✝⁴ : Field K\nv : Valuation K ℝ≥0\nO : Type u₂\ninst✝³ : CommRing O\ninst✝² : Algebra O K\nhv : v.Integers O\np : ℕ\ninst✝¹ : Fact (Nat.Prime p)\ninst✝ : Fact ¬IsUnit ↑p\nhp : Nat.Prime p\nthis : Nontrivial (PreTilt O p)\na✝ b✝ : PreTilt O p\nhfg : a✝ * b✝ = 0\n⊢ a✝ = 0 ∨ b✝ = 0",
... | ← map_eq_zero hv | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.RingTheory.WittVector.Truncated | {
"line": 103,
"column": 6
} | {
"line": 103,
"column": 9
} | [
{
"pp": "p n : ℕ\nR : Type u_1\ninst✝ : CommRing R\nx : TruncatedWittVector p n R\ni : Fin n\n⊢ x.out.coeff ↑i = coeff i x",
"usedConstants": [
"Eq.mpr",
"TruncatedWittVector.coeff",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"CommRing.toNonUnitalCommRing",
"congrArg",
... | out | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.WittVector.Complete | {
"line": 85,
"column": 4
} | {
"line": 85,
"column": 49
} | [
{
"pp": "case refine_2\np : 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⊢ ∃ c, x = c * ↑p ^ n",
"usedConstants": [
"NonUnitalNonAssocCommRing.toNonUnitalNonAssocCommSemiring",
"NonUnitalC... | use (frobeniusEquiv p k).symm^[n] (x.shift n) | Mathlib.Tactic._aux_Mathlib_Tactic_Use___elabRules_Mathlib_Tactic_useSyntax_1 | Mathlib.Tactic.useSyntax |
Mathlib.RingTheory.Polynomial.Hermite.Basic | {
"line": 90,
"column": 73
} | {
"line": 90,
"column": 82
} | [
{
"pp": "case succ\nn : ℕ\nih : ∀ (k : ℕ), (hermite n).coeff (n + k + 1) = 0\nk : ℕ\nthis : n + k + 1 + 2 = n + (k + 2) + 1\n⊢ 0 - (↑(n + k + 1) + 2) * 0 = 0",
"usedConstants": [
"Eq.mpr",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"HMul.hMul",
"CommRing.toNonUnitalCommRing",
... | mul_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Polynomial.Hermite.Basic | {
"line": 97,
"column": 58
} | {
"line": 97,
"column": 67
} | [
{
"pp": "case succ\nn : ℕ\nih : (hermite n).coeff n = 1\n⊢ 1 - (↑n + 2) * 0 = 1",
"usedConstants": [
"Eq.mpr",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"HMul.hMul",
"CommRing.toNonUnitalCommRing",
"MulZeroClass.toMul",
"congrArg",
"HSub.hSub",
"NonUnit... | mul_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Polynomial.Hermite.Basic | {
"line": 130,
"column": 43
} | {
"line": 130,
"column": 52
} | [
{
"pp": "case succ.succ\nn : ℕ\nih : ∀ {k : ℕ}, Odd (n + k) → (hermite n).coeff k = 0\nk : ℕ\nhnk : Odd (n + 1 + (k + 1))\n⊢ 0 - (↑k + 2) * 0 = 0",
"usedConstants": [
"Eq.mpr",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"HMul.hMul",
"CommRing.toNonUnitalCommRing",
"MulZer... | mul_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Polynomial.Hermite.Basic | {
"line": 173,
"column": 6
} | {
"line": 175,
"column": 58
} | [
{
"pp": "case e_a.e_a.e_a\nn✝ k✝ : ℕ\nhermite_explicit : ℕ → ℕ → ℤ := fun n k ↦ (-1) ^ n * ↑(2 * n - 1)‼ * ↑((2 * n + k).choose k)\nn k : ℕ\n⊢ (2 * n + 1) * (2 * (n + 1) + (k + 1)).choose (k + 1) =\n (2 * n + 1) * (2 * (n + 1) + k).choose k + (2 * n + (k + 2)).choose (k + 2) * (k + 2)",
"usedConstants": ... | rw [(by ring : 2 * (n + 1) + (k + 1) = 2 * n + 1 + (k + 1) + 1),
(by ring : 2 * (n + 1) + k = 2 * n + 1 + (k + 1)),
(by ring : 2 * n + (k + 2) = 2 * n + 1 + (k + 1))] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.Polynomial.HilbertPoly | {
"line": 238,
"column": 12
} | {
"line": 238,
"column": 33
} | [
{
"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⊢ hilbertPoly 0 d = 0",
"usedConstants": [
"Eq.mpr",
"congrArg",
"id",
"Polynomial.hilbertPoly_zero_left",
"Field.toSemifield",
"Polynomial"... | hilbertPoly_zero_left | Lean.Elab.Tactic.evalRewriteSeq | null |
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