module string | startPos dict | endPos dict | nextStartPos dict | goals list | goalsAfter list | ppTac string | elaborator string | kind string |
|---|---|---|---|---|---|---|---|---|
Mathlib.RingTheory.FormalGroup.Basic | {
"line": 301,
"column": 31
} | {
"line": 301,
"column": 56
} | {
"line": 301,
"column": 57
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\nF : FormalGroup R\nthis : Invertible ((PowerSeries.coeff 1) F.zeroX)\naux₀ : PowerSeries.HasSubst F.zeroX\n⊢ F.zeroX = PowerSeries.subst F.zeroX PowerSeries.X",
"ppTerm": "?m.89",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
... | [
"R : Type u_1\ninst✝ : CommRing R\nF : FormalGroup R\nthis : Invertible ((PowerSeries.coeff 1) F.zeroX)\naux₀ : PowerSeries.HasSubst F.zeroX\n⊢ F.zeroX = F.zeroX"
] | PowerSeries.subst_X aux₀, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.HahnSeries.Summable | {
"line": 741,
"column": 2
} | {
"line": 743,
"column": 46
} | {
"line": 745,
"column": 0
} | [
{
"pp": "Γ : Type u_1\nR : Type u_3\ninst✝³ : AddCommMonoid Γ\ninst✝² : LinearOrder Γ\ninst✝¹ : IsOrderedCancelAddMonoid Γ\ninst✝ : CommRing R\n⊢ powers 0 = single 0 1",
"ppTerm": "?m.26",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Mu... | [] | ext n
rw [powers_of_orderTop_pos (by simp)]
obtain rfl | ha := eq_or_ne n 0 <;> simp [*] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.HahnSeries.Summable | {
"line": 741,
"column": 2
} | {
"line": 743,
"column": 46
} | {
"line": 745,
"column": 0
} | [
{
"pp": "Γ : Type u_1\nR : Type u_3\ninst✝³ : AddCommMonoid Γ\ninst✝² : LinearOrder Γ\ninst✝¹ : IsOrderedCancelAddMonoid Γ\ninst✝ : CommRing R\n⊢ powers 0 = single 0 1",
"ppTerm": "?m.26",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Mu... | [] | ext n
rw [powers_of_orderTop_pos (by simp)]
obtain rfl | ha := eq_or_ne n 0 <;> simp [*] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.HahnSeries.Summable | {
"line": 789,
"column": 44
} | {
"line": 802,
"column": 39
} | {
"line": 804,
"column": 0
} | [
{
"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 : R⟦Γ⟧\nr : R\nhr : r * x.leadingCoeff = 1\noinv : Γ\nhxo : oinv + x.order = 0\n⊢ 0 < (1 - (single oinv) r * x).orderTop",
"ppTerm": "?m.51",
"assigned": true... | [] | by
let y := (x - single x.order x.leadingCoeff)
by_cases hy : y = 0
· have hrx : (single oinv) r * x = 1 := by
rw [eq_of_sub_eq_zero hy, single_mul_single, hxo, hr, single_zero_one]
simp only [hrx, sub_self, orderTop_zero, WithTop.top_pos]
· have hr' : IsRegular r := IsUnit.isRegular <| .of_mul_eq_one... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.HahnSeries.Summable | {
"line": 846,
"column": 12
} | {
"line": 846,
"column": 22
} | {
"line": 846,
"column": 23
} | [
{
"pp": "case mp\nΓ : Type u_1\nR : Type u_3\ninst✝⁴ : AddCommGroup Γ\ninst✝³ : LinearOrder Γ\ninst✝² : IsOrderedAddMonoid Γ\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\nu i : R⟦Γ⟧\nui : u * i = 1\niu : i * u = 1\n⊢ coeff 1 (u.order + i.order) = 1",
"ppTerm": "?mp",
"assigned": true,
"usedConstants": [... | [
"case mp\nΓ : Type u_1\nR : Type u_3\ninst✝⁴ : AddCommGroup Γ\ninst✝³ : LinearOrder Γ\ninst✝² : IsOrderedAddMonoid Γ\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\nu i : R⟦Γ⟧\nui : u * i = 1\niu : i * u = 1\n⊢ (if u.order + i.order = 0 then 1 else 0) = 1"
] | coeff_one, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.HahnSeries.Summable | {
"line": 882,
"column": 4
} | {
"line": 884,
"column": 95
} | {
"line": 885,
"column": 4
} | [
{
"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\nx : R⟦Γ⟧\nx0 : x ≠ 0\n⊢ x * x⁻¹ = 1",
"ppTerm": "?m.37",
"assigned": true,
"usedConstants": [
"Iff.mpr"... | [
"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\nx : R⟦Γ⟧\nx0 : x ≠ 0\nh :\n (1 - (1 - (single (-x.order)) x.leadingCoeff⁻¹ * x)) *\n (SummableFamily.powers (1 - (single (-x.or... | have h :=
SummableFamily.one_sub_self_mul_hsum_powers
(unit_aux x (inv_mul_cancel₀ (leadingCoeff_ne_zero.mpr x0)) _ (neg_add_cancel x.order)) | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.RingTheory.HopfAlgebra.Quotient | {
"line": 54,
"column": 2
} | {
"line": 55,
"column": 82
} | {
"line": 56,
"column": 2
} | [
{
"pp": "R : Type u_1\nA : Type u_2\nB : Type u_3\ninst✝⁴ : CommSemiring R\ninst✝³ : Semiring A\ninst✝² : Semiring B\ninst✝¹ : HopfAlgebra R A\ninst✝ : HopfAlgebraStruct R B\nf : A →ₐc[R] B\nhf : Function.Surjective ⇑f\nhS : antipode R ∘ₗ f.toLinearMap = f.toLinearMap ∘ₗ antipode R\n⊢ HopfAlgebra R B",
"ppT... | [
"case refine_1\nR : Type u_1\nA : Type u_2\nB : Type u_3\ninst✝⁴ : CommSemiring R\ninst✝³ : Semiring A\ninst✝² : Semiring B\ninst✝¹ : HopfAlgebra R A\ninst✝ : HopfAlgebraStruct R B\nf : A →ₐc[R] B\nhf : Function.Surjective ⇑f\nhS : antipode R ∘ₗ f.toLinearMap = f.toLinearMap ∘ₗ antipode R\n⊢ (toConv (antipode R) * ... | refine .ofConvInverse (antipode R) (ofConv_injective ?_) (ofConv_injective ?_) <;>
rw [← LinearMap.cancel_right (show Function.Surjective f.toLinearMap from hf)] | Lean.Parser.Tactic.«_aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tactic_<;>__1» | Lean.Parser.Tactic.«tactic_<;>_» |
Mathlib.Topology.Algebra.ClopenNhdofOne | {
"line": 71,
"column": 4
} | {
"line": 71,
"column": 29
} | {
"line": 72,
"column": 4
} | [
{
"pp": "case a\nG : Type u_1\ninst✝⁴ : Group G\ninst✝³ : TopologicalSpace G\ninst✝² : IsTopologicalGroup G\ninst✝¹ : CompactSpace G\ninst✝ : TotallyDisconnectedSpace G\nH : ClosedSubgroup G\ng : G\nhg : g ∈ sInf {N | IsOpen ↑N ∧ ↑H ≤ N}\nhg_not : g ∉ ↑H\n⊢ False",
"ppTerm": "?a✝",
"assigned": true,
... | [
"case a\nG : Type u_1\ninst✝⁴ : Group G\ninst✝³ : TopologicalSpace G\ninst✝² : IsTopologicalGroup G\ninst✝¹ : CompactSpace G\ninst✝ : TotallyDisconnectedSpace G\nH : ClosedSubgroup G\ng : G\nhg : g ∈ sInf {N | IsOpen ↑N ∧ ↑H ≤ N}\nhg_not : g ∉ ↑H\nU : Set G := (g • ↑H)ᶜ\n⊢ False"
] | let U : Set G := (g • H)ᶜ | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLet___1 | Lean.Parser.Tactic.tacticLet__ |
Mathlib.RingTheory.Ideal.KrullsHeightTheorem | {
"line": 269,
"column": 4
} | {
"line": 269,
"column": 37
} | {
"line": 271,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsNoetherianRing R\nI : Ideal R\nhI : I ≠ ⊤\n⊢ ↑(Cardinal.toENat (Submodule.spanRank I)) ≤ Submodule.spanRank I",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"Semiring.toModule",
"CommSemiring.toSemiring",
"CommRing.to... | [] | exact I.spanRank.ofENat_toENat_le | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.RingTheory.Ideal.KrullsHeightTheorem | {
"line": 269,
"column": 4
} | {
"line": 269,
"column": 37
} | {
"line": 271,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsNoetherianRing R\nI : Ideal R\nhI : I ≠ ⊤\n⊢ ↑(Cardinal.toENat (Submodule.spanRank I)) ≤ Submodule.spanRank I",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"Semiring.toModule",
"CommSemiring.toSemiring",
"CommRing.to... | [] | exact I.spanRank.ofENat_toENat_le | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Ideal.KrullsHeightTheorem | {
"line": 269,
"column": 4
} | {
"line": 269,
"column": 37
} | {
"line": 271,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsNoetherianRing R\nI : Ideal R\nhI : I ≠ ⊤\n⊢ ↑(Cardinal.toENat (Submodule.spanRank I)) ≤ Submodule.spanRank I",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"Semiring.toModule",
"CommSemiring.toSemiring",
"CommRing.to... | [] | exact I.spanRank.ofENat_toENat_le | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Ideal.KrullsHeightTheorem | {
"line": 318,
"column": 23
} | {
"line": 318,
"column": 72
} | {
"line": 318,
"column": 72
} | [
{
"pp": "R : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsNoetherianRing R\np : Ideal R\ninst✝ : p.IsPrime\nI : Ideal R\nhI : p ∈ I.minimalPrimes\nhr : Submodule.spanRank I ≤ ↑p.height\nhs : (Submodule.generators I).Finite\n⊢ p ∈ (span ↑hs.toFinset).minimalPrimes",
"ppTerm": "?m.75",
"assigned": true,
... | [] | by rwa [hs.coe_toFinset, span, I.span_generators] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.Ideal.KrullsHeightTheorem | {
"line": 339,
"column": 4
} | {
"line": 339,
"column": 53
} | {
"line": 340,
"column": 4
} | [
{
"pp": "R : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsNoetherianRing R\nI p : Ideal R\ninst✝ : p.IsPrime\nhrp : I ≤ p\np' : Ideal (R ⧸ I) := map (algebraMap R (R ⧸ I)) p\nthis : p'.IsPrime\ns : Finset (R ⧸ I)\nhps : p' ∈ (span ↑s).minimalPrimes\nhs : ↑s.card = p'.height\nhsp' : ↑s ⊆ ↑p'\nx : R ⧸ I\nhx : x ∈ ↑p... | [
"R : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsNoetherianRing R\nI p : Ideal R\ninst✝ : p.IsPrime\nhrp : I ≤ p\np' : Ideal (R ⧸ I) := map (algebraMap R (R ⧸ I)) p\nthis : p'.IsPrime\ns : Finset (R ⧸ I)\nhps : p' ∈ (span ↑s).minimalPrimes\nhs : ↑s.card = p'.height\nhsp' : ↑s ⊆ ↑p'\nx : R\nhx : (Quotient.mk I) x ∈ ↑p... | obtain ⟨x, rfl⟩ := Ideal.Quotient.mk_surjective x | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.RingTheory.KrullDimension.Regular | {
"line": 68,
"column": 31
} | {
"line": 70,
"column": 94
} | {
"line": 70,
"column": 94
} | [
{
"pp": "R : Type u_1\ninst✝⁴ : CommRing R\ninst✝³ : IsNoetherianRing R\nM : Type u_2\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\ninst✝ : Module.Finite R M\nx : R\nh : x ∈ (annihilator R M).jacobson\na✝ : Nontrivial M\np : LTSeries ↑(support R M)\nhxp : x ∈ (↑(RelSeries.last p)).asIdeal\nq : LTSeries (PrimeS... | [] | simpa using!
⟨mem_support_mono (by simpa [h0] using! q.monotone (Fin.zero_le _)) p.head.2, q.monotone
((Fin.natCast_eq_mk (Nat.lt_of_add_left_lt hi)).trans_le (Nat.le_add_left 1 i)) hxq⟩ | Lean.Elab.Tactic.Simpa.evalSimpaUsingBang | Lean.Parser.Tactic.simpaUsingBang |
Mathlib.RingTheory.KrullDimension.Regular | {
"line": 68,
"column": 31
} | {
"line": 70,
"column": 94
} | {
"line": 70,
"column": 94
} | [
{
"pp": "R : Type u_1\ninst✝⁴ : CommRing R\ninst✝³ : IsNoetherianRing R\nM : Type u_2\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\ninst✝ : Module.Finite R M\nx : R\nh : x ∈ (annihilator R M).jacobson\na✝ : Nontrivial M\np : LTSeries ↑(support R M)\nhxp : x ∈ (↑(RelSeries.last p)).asIdeal\nq : LTSeries (PrimeS... | [] | simpa using!
⟨mem_support_mono (by simpa [h0] using! q.monotone (Fin.zero_le _)) p.head.2, q.monotone
((Fin.natCast_eq_mk (Nat.lt_of_add_left_lt hi)).trans_le (Nat.le_add_left 1 i)) hxq⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.KrullDimension.Regular | {
"line": 68,
"column": 31
} | {
"line": 70,
"column": 94
} | {
"line": 70,
"column": 94
} | [
{
"pp": "R : Type u_1\ninst✝⁴ : CommRing R\ninst✝³ : IsNoetherianRing R\nM : Type u_2\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\ninst✝ : Module.Finite R M\nx : R\nh : x ∈ (annihilator R M).jacobson\na✝ : Nontrivial M\np : LTSeries ↑(support R M)\nhxp : x ∈ (↑(RelSeries.last p)).asIdeal\nq : LTSeries (PrimeS... | [] | simpa using!
⟨mem_support_mono (by simpa [h0] using! q.monotone (Fin.zero_le _)) p.head.2, q.monotone
((Fin.natCast_eq_mk (Nat.lt_of_add_left_lt hi)).trans_le (Nat.le_add_left 1 i)) hxq⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Ideal.KrullsHeightTheorem | {
"line": 476,
"column": 33
} | {
"line": 476,
"column": 61
} | {
"line": 476,
"column": 62
} | [
{
"pp": "case refine_1\nR : 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 (... | [
"case refine_1\nR : 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 (PrimeSpectru... | PrimeSpectrum.comap_asIdeal, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.LocalProperties.Injective | {
"line": 51,
"column": 2
} | {
"line": 51,
"column": 92
} | {
"line": 52,
"column": 2
} | [
{
"pp": "R : Type u\ninst✝¹⁴ : CommRing R\nM : Type v\ninst✝¹³ : AddCommGroup M\ninst✝¹² : Module R M\nS : Submonoid R\ninst✝¹¹ : Small.{v, u} R\ninst✝¹⁰ : IsNoetherianRing R\nRₛ : Type u'\ninst✝⁹ : Small.{v', u'} Rₛ\ninst✝⁸ : CommRing Rₛ\ninst✝⁷ : Algebra R Rₛ\nMₛ : Type v'\ninst✝⁶ : AddCommGroup Mₛ\ninst✝⁵ : ... | [
"R : Type u\ninst✝¹⁴ : CommRing R\nM : Type v\ninst✝¹³ : AddCommGroup M\ninst✝¹² : Module R M\nS : Submonoid R\ninst✝¹¹ : Small.{v, u} R\ninst✝¹⁰ : IsNoetherianRing R\nRₛ : Type u'\ninst✝⁹ : Small.{v', u'} Rₛ\ninst✝⁸ : CommRing Rₛ\ninst✝⁷ : Algebra R Rₛ\nMₛ : Type v'\ninst✝⁶ : AddCommGroup Mₛ\ninst✝⁵ : Module R Mₛ\... | refine ⟨IsLocalization.mk' Rₛ 1 a • mapExtendScalars S (Algebra.linearMap _ _) f _ g', ?_⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.RingTheory.LocalProperties.Injective | {
"line": 87,
"column": 4
} | {
"line": 87,
"column": 52
} | {
"line": 88,
"column": 2
} | [
{
"pp": "R : Type u\ninst✝⁴ : CommRing R\nM : Type v\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : Small.{v, u} R\ninst✝ : IsNoetherianRing R\nH : ∀ (I : Ideal R) (x : I.IsMaximal), Injective (Localization.AtPrime I) (LocalizedModule I.primeCompl M)\nI : Ideal R\nx✝¹ : FinitePresentation R ↥I := finit... | [] | simp [IsLocalizedModule.mapExtendScalars, ← eq'] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.RingTheory.LocalProperties.ProjectiveDimension | {
"line": 66,
"column": 2
} | {
"line": 80,
"column": 32
} | {
"line": 82,
"column": 0
} | [
{
"pp": "R : Type u\ninst✝¹ : CommRing R\ninst✝ : Small.{v, u} R\nS : Submonoid R\nM : ModuleCat R\n⊢ projectiveDimension (M.localizedModule S) ≤ projectiveDimension M",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"WithBot.instPreorder",
"Eq.mpr",
"WithBot.some",
... | [] | have aux (n : ℕ) : projectiveDimension M ≤ n → projectiveDimension (M.localizedModule S) ≤ n := by
simp only [projectiveDimension_le_iff]
intro h
exact ModuleCat.localizedModule_hasProjectiveDimensionLE n S M
refine le_of_forall_ge (fun N ↦ ?_)
induction N with
| bot =>
simp only [le_bot_iff, proj... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.LocalProperties.ProjectiveDimension | {
"line": 66,
"column": 2
} | {
"line": 80,
"column": 32
} | {
"line": 82,
"column": 0
} | [
{
"pp": "R : Type u\ninst✝¹ : CommRing R\ninst✝ : Small.{v, u} R\nS : Submonoid R\nM : ModuleCat R\n⊢ projectiveDimension (M.localizedModule S) ≤ projectiveDimension M",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"WithBot.instPreorder",
"Eq.mpr",
"WithBot.some",
... | [] | have aux (n : ℕ) : projectiveDimension M ≤ n → projectiveDimension (M.localizedModule S) ≤ n := by
simp only [projectiveDimension_le_iff]
intro h
exact ModuleCat.localizedModule_hasProjectiveDimensionLE n S M
refine le_of_forall_ge (fun N ↦ ?_)
induction N with
| bot =>
simp only [le_bot_iff, proj... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.LaurentSeries | {
"line": 296,
"column": 4
} | {
"line": 296,
"column": 18
} | {
"line": 298,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\nx : R⟦X⟧\n⊢ ∃ c, ↑c * x = ↑c * x",
"ppTerm": "?m.212",
"assigned": true,
"usedConstants": [
"HMul.hMul",
"Monoid.toMulOneClass",
"MvPowerSeries.instCommSemiring",
"CommSemiring.toSemiring",
"Membership.mem",
"MulOne.toMul... | [] | exact ⟨1, rfl⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.RingTheory.LocalRing.Etale | {
"line": 115,
"column": 4
} | {
"line": 115,
"column": 53
} | {
"line": 116,
"column": 4
} | [
{
"pp": "case refine_1\nR : Type u_1\nS : Type u_2\ninst✝⁷ : CommRing R\ninst✝⁶ : CommRing S\ninst✝⁵ : Algebra R S\ninst✝⁴ : IsLocalRing S\ninst✝³ : IsLocalRing R\ninst✝² : Module.Finite R S\ninst✝¹ : FaithfulSMul R S\ninst✝ : Etale R S\nthis : Module.Free R S\nx : S ⧸ Ideal.map (algebraMap R S) (maximalIdeal R... | [
"case refine_1\nR : Type u_1\nS : Type u_2\ninst✝⁷ : CommRing R\ninst✝⁶ : CommRing S\ninst✝⁵ : Algebra R S\ninst✝⁴ : IsLocalRing S\ninst✝³ : IsLocalRing R\ninst✝² : Module.Finite R S\ninst✝¹ : FaithfulSMul R S\ninst✝ : Etale R S\nthis : Module.Free R S\nr : R\nx : S\n⊢ (Ideal.quotEquivOfEq ⋯).toAddEquiv\n ((Id... | obtain ⟨x, rfl⟩ := Ideal.Quotient.mk_surjective x | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.RingTheory.Invariant.Profinite | {
"line": 170,
"column": 2
} | {
"line": 170,
"column": 51
} | {
"line": 171,
"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\nP✝ : Ideal A\ninst✝⁹ : TopologicalSpace G\ninst✝⁸ : CompactSpace G\ninst✝⁷ : TotallyDisconnectedSpace G\ninst✝⁶ :... | [
"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\nP✝ : Ideal A\ninst✝⁹ : TopologicalSpace G\ninst✝⁸ : CompactSpace G\ninst✝⁷ : TotallyDisconnectedSpace G\ninst✝⁶ : IsTopologic... | obtain ⟨x, rfl⟩ := Ideal.Quotient.mk_surjective x | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.RingTheory.Invariant.Profinite | {
"line": 202,
"column": 4
} | {
"line": 202,
"column": 53
} | {
"line": 203,
"column": 4
} | [
{
"pp": "case refine_2\nA : 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 ... | [
"case refine_2\nA : 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 : IsTopologi... | obtain ⟨x, rfl⟩ := Ideal.Quotient.mk_surjective x | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.RingTheory.Localization.AtPrime.Extension | {
"line": 85,
"column": 2
} | {
"line": 85,
"column": 51
} | {
"line": 86,
"column": 2
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝⁷ : CommRing R\ninst✝⁶ : CommRing S\ninst✝⁵ : Algebra R S\np : Ideal R\ninst✝⁴ : p.IsPrime\nSₚ : Type u_4\ninst✝³ : CommRing Sₚ\ninst✝² : Algebra S Sₚ\ninst✝¹ : IsLocalization (algebraMapSubmonoid S p.primeCompl) Sₚ\nP : Ideal S\nhPp : P.LiesOver p\ninst✝ : P.IsMaximal\... | [
"R : Type u_1\nS : Type u_2\ninst✝⁷ : CommRing R\ninst✝⁶ : CommRing S\ninst✝⁵ : Algebra R S\np : Ideal R\ninst✝⁴ : p.IsPrime\nSₚ : Type u_4\ninst✝³ : CommRing Sₚ\ninst✝² : Algebra S Sₚ\ninst✝¹ : IsLocalization (algebraMapSubmonoid S p.primeCompl) Sₚ\nP : Ideal S\nhPp : P.LiesOver p\ninst✝ : P.IsMaximal\nx : Sₚ\n⊢ ∃... | obtain ⟨x, rfl⟩ := Ideal.Quotient.mk_surjective x | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.RingTheory.MvPolynomial.EulerIdentity | {
"line": 39,
"column": 4
} | {
"line": 39,
"column": 58
} | {
"line": 40,
"column": 4
} | [
{
"pp": "case refine_1\nR : Type u_1\nσ : Type u_2\nM : Type u_3\ninst✝¹ : CommSemiring R\nφ : MvPolynomial σ R\ninst✝ : AddCancelCommMonoid M\nw : σ → M\nn n' : M\ni : σ\nh : φ ∈ Submodule.span R ((fun m ↦ AddMonoidAlgebra.single m 1) '' {d | (weight w) d = n})\nh' : n' + w i = n\nm : σ →₀ ℕ\nhm : m ∈ {d | (we... | [
"case refine_1\nR : Type u_1\nσ : Type u_2\nM : Type u_3\ninst✝¹ : CommSemiring R\nφ : MvPolynomial σ R\ninst✝ : AddCancelCommMonoid M\nw : σ → M\nn n' : M\ni : σ\nh : φ ∈ Submodule.span R ((fun m ↦ AddMonoidAlgebra.single m 1) '' {d | (weight w) d = n})\nh' : n' + w i = n\nm : σ →₀ ℕ\nhm : m ∈ {d | (weight w) d = ... | simp_rw [single_eq_monomial, pderiv_monomial, one_mul] | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | Mathlib.Tactic.tacticSimp_rw___ |
Mathlib.RingTheory.Localization.AtPrime.Extension | {
"line": 130,
"column": 9
} | {
"line": 130,
"column": 34
} | {
"line": 130,
"column": 34
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝⁷ : CommRing R\ninst✝⁶ : CommRing S\ninst✝⁵ : Algebra R S\np : Ideal R\ninst✝⁴ : p.IsPrime\nSₚ : Type u_4\ninst✝³ : CommRing Sₚ\ninst✝² : Algebra S Sₚ\ninst✝¹ : IsLocalization (algebraMapSubmonoid S p.primeCompl) Sₚ\nP : Ideal S\nhPp : P.LiesOver p\ninst✝ : P.IsMaximal\... | [
"R : Type u_1\nS : Type u_2\ninst✝⁷ : CommRing R\ninst✝⁶ : CommRing S\ninst✝⁵ : Algebra R S\np : Ideal R\ninst✝⁴ : p.IsPrime\nSₚ : Type u_4\ninst✝³ : CommRing Sₚ\ninst✝² : Algebra S Sₚ\ninst✝¹ : IsLocalization (algebraMapSubmonoid S p.primeCompl) Sₚ\nP : Ideal S\nhPp : P.LiesOver p\ninst✝ : P.IsMaximal\nx : S\ns : ... | RingEquiv.map_ne_zero_iff | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Localization.AtPrime.Extension | {
"line": 133,
"column": 15
} | {
"line": 133,
"column": 24
} | {
"line": 133,
"column": 25
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝⁷ : CommRing R\ninst✝⁶ : CommRing S\ninst✝⁵ : Algebra R S\np : Ideal R\ninst✝⁴ : p.IsPrime\nSₚ : Type u_4\ninst✝³ : CommRing Sₚ\ninst✝² : Algebra S Sₚ\ninst✝¹ : IsLocalization (algebraMapSubmonoid S p.primeCompl) Sₚ\nP : Ideal S\nhPp : P.LiesOver p\ninst✝ : P.IsMaximal\... | [
"R : Type u_1\nS : Type u_2\ninst✝⁷ : CommRing R\ninst✝⁶ : CommRing S\ninst✝⁵ : Algebra R S\np : Ideal R\ninst✝⁴ : p.IsPrime\nSₚ : Type u_4\ninst✝³ : CommRing Sₚ\ninst✝² : Algebra S Sₚ\ninst✝¹ : IsLocalization (algebraMapSubmonoid S p.primeCompl) Sₚ\nP : Ideal S\nhPp : P.LiesOver p\ninst✝ : P.IsMaximal\nx : S\ns : ... | mk'_spec, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Localization.AtPrime.Extension | {
"line": 156,
"column": 2
} | {
"line": 156,
"column": 51
} | {
"line": 157,
"column": 2
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝¹⁷ : CommRing R\ninst✝¹⁶ : CommRing S\ninst✝¹⁵ : Algebra R S\np : Ideal R\ninst✝¹⁴ : p.IsPrime\nRₚ : Type u_3\ninst✝¹³ : CommRing Rₚ\ninst✝¹² : Algebra R Rₚ\ninst✝¹¹ : IsLocalization.AtPrime Rₚ p\ninst✝¹⁰ : IsLocalRing Rₚ\nSₚ : Type u_4\ninst✝⁹ : CommRing Sₚ\ninst✝⁸ : A... | [
"R : Type u_1\nS : Type u_2\ninst✝¹⁷ : CommRing R\ninst✝¹⁶ : CommRing S\ninst✝¹⁵ : Algebra R S\np : Ideal R\ninst✝¹⁴ : p.IsPrime\nRₚ : Type u_3\ninst✝¹³ : CommRing Rₚ\ninst✝¹² : Algebra R Rₚ\ninst✝¹¹ : IsLocalization.AtPrime Rₚ p\ninst✝¹⁰ : IsLocalRing Rₚ\nSₚ : Type u_4\ninst✝⁹ : CommRing Sₚ\ninst✝⁸ : Algebra S Sₚ\... | obtain ⟨x, rfl⟩ := Ideal.Quotient.mk_surjective x | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.RingTheory.MvPolynomial.Symmetric.FundamentalTheorem | {
"line": 299,
"column": 2
} | {
"line": 299,
"column": 93
} | {
"line": 299,
"column": 93
} | [
{
"pp": "case right.inr\nR : Type u_3\ninst✝ : CommRing R\nn : ℕ\np : MvPolynomial (Fin n) R\nhp : p ∈ symmetricSubalgebra (Fin n) R\nh0 : p ≠ 0\n⊢ ⟨p, hp⟩ ∈ (esymmAlgHom (Fin n) R n).range",
"ppTerm": "?right.inr",
"assigned": true,
"usedConstants": [
"Subalgebra.instSetLike",
"Nat.inst... | [
"case right.inr.ind\nR : Type u_3\ninst✝ : CommRing R\nn : ℕ\nt : Lex (Fin n →₀ ℕ)\nih :\n ∀ y < t,\n ∀ (p : MvPolynomial (Fin n) R) (hp : p ∈ symmetricSubalgebra (Fin n) R),\n p ≠ 0 → supDegree (⇑toLex) p = y → ⟨p, hp⟩ ∈ (esymmAlgHom (Fin n) R n).range\np : MvPolynomial (Fin n) R\nhp : p ∈ symmetricSubalg... | induction he : p.supDegree toLex using WellFoundedLT.induction generalizing p with | _ t ih
=> _ | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | Lean.Parser.Tactic.induction |
Mathlib.RingTheory.MvPolynomial.Symmetric.NewtonIdentities | {
"line": 123,
"column": 2
} | {
"line": 123,
"column": 21
} | {
"line": 124,
"column": 2
} | [
{
"pp": "σ : Type u_1\nR : Type u_2\ninst✝² : CommRing R\ninst✝¹ : DecidableEq σ\ninst✝ : Fintype σ\nk : ℕ\nt : Finset σ × σ\nh : t ∈ pairs σ k\n⊢ (-1) ^ #t.1 * ((∏ a ∈ t.1, X a) * X t.2 ^ (k - #t.1)) +\n (-1) ^ #(pairMap σ t).1 * ((∏ a ∈ (pairMap σ t).1, X a) * X (pairMap σ t).2 ^ (k - #(pairMap σ t).1)) ... | [
"σ : Type u_1\nR : Type u_2\ninst✝² : CommRing R\ninst✝¹ : DecidableEq σ\ninst✝ : Fintype σ\nk : ℕ\nt : Finset σ × σ\nh : #t.1 ≤ k ∧ (#t.1 = k → t.2 ∈ t.1)\n⊢ (-1) ^ #t.1 * ((∏ a ∈ t.1, X a) * X t.2 ^ (k - #t.1)) +\n (-1) ^ #(pairMap σ t).1 * ((∏ a ∈ (pairMap σ t).1, X a) * X (pairMap σ t).2 ^ (k - #(pairMap σ... | rw [mem_pairs] at h | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.LaurentSeries | {
"line": 929,
"column": 4
} | {
"line": 929,
"column": 43
} | {
"line": 930,
"column": 4
} | [
{
"pp": "case mpr\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\n⊢ ∃ t,\n (∃ t_1 ∈ nhds 0, (fun x ↦ x.2 - x.1) ⁻¹' t_1 ⊆ t) ∧\n (fun x ↦ ((algebraMap K⟮X⟯ K⸨X⸩) x.1, (algebraMap K⟮X⟯ K⸨X⸩) x.2)) ⁻¹' t ⊆ S",
"ppTerm": "?... | [
"case mpr\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.ofClass Valued.v).ValueGroup₀ˣ\nhd : {y | Valued.v.restrict (y - 0) < ↑d} ⊆ w✝\n⊢ ∃ t,\n (∃ t_1 ∈ nhds 0, (fun x ↦ x.2 - x.1) ⁻¹' t_1 ⊆ t) ∧\n (fun... | obtain ⟨d, hd⟩ := Valued.mem_nhds.mp hT | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.RingTheory.MvPolynomial.Symmetric.NewtonIdentities | {
"line": 262,
"column": 2
} | {
"line": 267,
"column": 81
} | {
"line": 268,
"column": 2
} | [
{
"pp": "σ : Type u_1\ninst✝¹ : Fintype σ\nR : Type u_2\ninst✝ : CommRing R\nk : ℕ\nh : 0 < k\nhesymm :\n ↑k * esymm σ R k =\n (-1) ^ (k + 1) *\n (∑ x ∈ {a ∈ antidiagonal k | a.1 < k} with 0 < x.1, (-1) ^ x.1 * esymm σ R x.1 * psum σ R x.2 +\n ∑ x ∈ {a ∈ antidiagonal k | a.1 < k} with ¬0 < x.1, ... | [
"σ : Type u_1\ninst✝¹ : Fintype σ\nR : Type u_2\ninst✝ : CommRing R\nk : ℕ\nh : 0 < k\nhesymm :\n ↑k * esymm σ R k =\n (-1) ^ (k + 1) *\n (∑ x ∈ {a ∈ antidiagonal k | a.1 < k} with 0 < x.1, (-1) ^ x.1 * esymm σ R x.1 * psum σ R x.2 +\n ∑ x ∈ {a ∈ antidiagonal k | a.1 < k} with ¬0 < x.1, (-1) ^ x.1 *... | have : {a ∈ antidiagonal k | a.fst < k ∧ ¬0 < a.fst} = {(0, k)} := by
ext a
rw [mem_filter, mem_antidiagonal, mem_singleton]
refine ⟨?_, by rintro rfl; lia⟩
rintro ⟨ha, ⟨_, ha0⟩⟩
rw [← ha, Nat.eq_zero_of_not_pos ha0, zero_add, ← Nat.eq_zero_of_not_pos ha0] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.RingTheory.MvPowerSeries.Restricted | {
"line": 93,
"column": 2
} | {
"line": 93,
"column": 79
} | {
"line": 95,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝¹ : NormedRing R\nσ : Type u_2\ninst✝ : IsUltrametricDist R\nc : σ → ℝ\nf g : MvPowerSeries σ R\nhf : Tendsto (fun t ↦ ‖(coeff t) f‖ * t.prod fun x1 x2 ↦ |c| x1 ^ x2) cofinite (𝓝 0)\nhg : Tendsto (fun t ↦ ‖(coeff t) g‖ * t.prod fun x1 x2 ↦ |c| x1 ^ x2) cofinite (𝓝 0)\n⊢ Tendsto (fu... | [] | exact tendsto_antidiagonal (by simp [Finsupp.prod_add_index', pow_add]) hf hg | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.RingTheory.MvPowerSeries.Restricted | {
"line": 90,
"column": 80
} | {
"line": 93,
"column": 79
} | {
"line": 95,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝¹ : NormedRing R\nσ : Type u_2\ninst✝ : IsUltrametricDist R\nc : σ → ℝ\nf g : MvPowerSeries σ R\nhf : IsRestricted c f\nhg : IsRestricted c g\n⊢ IsRestricted c (f * g)",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"Finsupp.instHasAntidiagonal",
"No... | [] | by
classical
rw [← isRestricted_abs_iff, IsRestricted] at *
exact tendsto_antidiagonal (by simp [Finsupp.prod_add_index', pow_add]) hf hg | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.WittVector.StructurePolynomial | {
"line": 254,
"column": 6
} | {
"line": 254,
"column": 19
} | {
"line": 254,
"column": 20
} | [
{
"pp": "case succ\np : ℕ\nidx : Type u_2\nhp : Fact (Nat.Prime p)\nΦ : MvPolynomial idx ℤ\nn k : ℕ\nhk : k ≤ n\nthis : p ^ (n + 1) = p ^ k * p ^ (n - k + 1)\n⊢ ↑(p ^ k * p ^ (n - k + 1)) ∣\n ↑(p ^ k) * ((expand p) (wittStructureInt p Φ k) ^ p ^ (n - k) - wittStructureInt p Φ k ^ p ^ (n + 1 - k))",
"ppTe... | [
"case succ\np : ℕ\nidx : Type u_2\nhp : Fact (Nat.Prime p)\nΦ : MvPolynomial idx ℤ\nn k : ℕ\nhk : k ≤ n\nthis : p ^ (n + 1) = p ^ k * p ^ (n - k + 1)\n⊢ ↑(p ^ k) * ↑(p ^ (n - k + 1)) ∣\n ↑(p ^ k) * ((expand p) (wittStructureInt p Φ k) ^ p ^ (n - k) - wittStructureInt p Φ k ^ p ^ (n + 1 - k))"
] | Nat.cast_mul, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.WittVector.Verschiebung | {
"line": 177,
"column": 8
} | {
"line": 177,
"column": 20
} | {
"line": 177,
"column": 21
} | [
{
"pp": "case neg\np : ℕ\nhp : Fact (Nat.Prime p)\nx : ℕ → ℤ\nn : ℕ\nhn : ¬n.succ = 0\n⊢ (MvPolynomial.eval x) ((bind₁ verschiebungPoly) (wittPolynomial p ℤ (n + 1))) =\n (MvPolynomial.eval x) ↑p * (MvPolynomial.eval x) (wittPolynomial p ℤ n)",
"ppTerm": "?neg✝",
"assigned": true,
"usedConstants"... | [
"case neg\np : ℕ\nhp : Fact (Nat.Prime p)\nx : ℕ → ℤ\nn : ℕ\nhn : ¬n.succ = 0\n⊢ (MvPolynomial.eval x) ((bind₁ verschiebungPoly) (wittPolynomial p ℤ (n + 1))) =\n ↑p * (MvPolynomial.eval x) (wittPolynomial p ℤ n)"
] | map_natCast, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.WittVector.Teichmuller | {
"line": 115,
"column": 13
} | {
"line": 115,
"column": 35
} | {
"line": 117,
"column": 0
} | [
{
"pp": "case zero\np : ℕ\nR : Type u_1\nhp : Fact (Nat.Prime p)\ninst✝ : CommRing R\n⊢ ((teichmuller p) 0).coeff 0 = coeff 0 0",
"ppTerm": "?zero",
"assigned": true,
"usedConstants": [
"WittVector.instZero",
"Eq.mpr",
"MonoidHom.instFunLike",
"MonoidHom",
"congrArg",
... | [] | · rw [zero_coeff]; rfl | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.RingTheory.WittVector.Teichmuller | {
"line": 115,
"column": 13
} | {
"line": 115,
"column": 35
} | {
"line": 117,
"column": 0
} | [
{
"pp": "case succ\np : ℕ\nR : Type u_1\nhp : Fact (Nat.Prime p)\ninst✝ : CommRing R\nn✝ : ℕ\n⊢ ((teichmuller p) 0).coeff (n✝ + 1) = coeff 0 (n✝ + 1)",
"ppTerm": "?succ",
"assigned": true,
"usedConstants": [
"WittVector.instZero",
"Eq.mpr",
"MonoidHom.instFunLike",
"MonoidHom... | [] | · rw [zero_coeff]; rfl | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.RingTheory.Perfection | {
"line": 816,
"column": 6
} | {
"line": 816,
"column": 35
} | {
"line": 816,
"column": 35
} | [
{
"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 : (val K v O hv p) a✝ ≠ 0 ∧ (val K v O ... | [] | exact mul_ne_zero hfg.1 hfg.2 | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.RingTheory.Polynomial.Hermite.Basic | {
"line": 189,
"column": 4
} | {
"line": 190,
"column": 93
} | {
"line": 191,
"column": 2
} | [
{
"pp": "case inl\nn k : ℕ\nh_le : k ≤ n\nm : ℕ\nhm : n - k = m + m\n⊢ (hermite n).coeff k = (-1) ^ ((n - k) / 2) * ↑(n - k - 1)‼ * ↑(n.choose k)",
"ppTerm": "?inl",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Nat.choose",
"instHDiv",
"HMul.hMul",
"congrArg",
... | [] | rw [(by lia : n = 2 * m + k),
Nat.add_sub_cancel, Nat.mul_div_cancel_left _ (Nat.succ_pos 1), coeff_hermite_explicit] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.Polynomial.HilbertPoly | {
"line": 198,
"column": 2
} | {
"line": 198,
"column": 21
} | {
"line": 198,
"column": 21
} | [
{
"pp": "F : Type u_1\ninst✝¹ : Field F\ninst✝ : CharZero F\np : F[X]\nd : ℕ\n⊢ ∃! h, ∃ N, ∀ n > N, (PowerSeries.coeff n) (↑p * ↑(invOneSubPow F d)) = eval (↑n) h",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"Units.val",
"Polynomial.eval",
"Semiring.toModule",
"H... | [
"case h\nF : Type u_1\ninst✝¹ : Field F\ninst✝ : CharZero F\np : F[X]\nd : ℕ\n⊢ (fun h ↦ ∃ N, ∀ n > N, (PowerSeries.coeff n) (↑p * ↑(invOneSubPow F d)) = eval (↑n) h) (p.hilbertPoly d) ∧\n ∀ (y : F[X]),\n (fun h ↦ ∃ N, ∀ n > N, (PowerSeries.coeff n) (↑p * ↑(invOneSubPow F d)) = eval (↑n) h) y → y = p.hilber... | use hilbertPoly p d | Mathlib.Tactic._aux_Mathlib_Tactic_Use___elabRules_Mathlib_Tactic_useSyntax_1 | Mathlib.Tactic.useSyntax |
Mathlib.RingTheory.Polynomial.ShiftedLegendre | {
"line": 72,
"column": 8
} | {
"line": 72,
"column": 44
} | {
"line": 72,
"column": 45
} | [
{
"pp": "case a\nn x : ℕ\na✝ : x ∈ range (n + 1)\n⊢ ↑((n + x)! / x !) * C (↑(n.choose x) * (-1) ^ x) = ↑n ! * C ((-1) ^ x * ↑(n.choose x) * ↑((n + x).choose n))",
"ppTerm": "?a✝",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Polynomial.C",
"Nat.choose",
"instHDiv",
"... | [
"case a\nn x : ℕ\na✝ : x ∈ range (n + 1)\n⊢ ↑((n + x)! / x !) * C (↑(n.choose x) * (-1) ^ x) = ↑n ! * (C ((-1) ^ x * ↑(n.choose x)) * C ↑((n + x).choose n))"
] | C_mul (b := ((n + x).choose n : ℤ)), | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.PowerSeries.Catalan | {
"line": 52,
"column": 32
} | {
"line": 52,
"column": 42
} | {
"line": 52,
"column": 43
} | [
{
"pp": "case succ\nn : ℕ\n⊢ (coeff (n + 1)) 1 + (coeff (n + 1)) (catalanSeries ^ 2 * X) = (coeff (n + 1)) catalanSeries",
"ppTerm": "?succ",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Semiring.toModule",
"HMul.hMul",
"con... | [
"case succ\nn : ℕ\n⊢ (if n + 1 = 0 then 1 else 0) + (coeff (n + 1)) (catalanSeries ^ 2 * X) = (coeff (n + 1)) catalanSeries"
] | coeff_one, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.RingTheory.PolynomialLaw.Basic | {
"line": 172,
"column": 26
} | {
"line": 172,
"column": 55
} | {
"line": 174,
"column": 0
} | [
{
"pp": "R : Type u\ninst✝⁴ : CommSemiring R\nM : Type u_1\ninst✝³ : AddCommMonoid M\ninst✝² : Module R M\nN : Type u_2\ninst✝¹ : AddCommMonoid N\ninst✝ : Module R N\nr a b : R\nf✝ g✝ f g : M →ₚₗ[R] N\nx✝³ : Type u\nx✝² : CommSemiring x✝³\nx✝¹ : Algebra R x✝³\nx✝ : x✝³ ⊗[R] M\n⊢ (f + g).toFun' x✝³ x✝ = (g + f).... | [] | simp only [add_def, add_comm] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.RingTheory.PolynomialLaw.Basic | {
"line": 219,
"column": 26
} | {
"line": 219,
"column": 55
} | {
"line": 221,
"column": 0
} | [
{
"pp": "R : Type u\ninst✝⁴ : CommRing R\nM : Type u_1\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\nN : Type u_2\ninst✝¹ : AddCommGroup N\ninst✝ : Module R N\nf✝ f g : M →ₚₗ[R] N\nx✝³ : Type u\nx✝² : CommSemiring x✝³\nx✝¹ : Algebra R x✝³\nx✝ : x✝³ ⊗[R] M\n⊢ (f + g).toFun' x✝³ x✝ = (g + f).toFun' x✝³ x✝",
... | [] | simp only [add_def, add_comm] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.RingTheory.PowerSeries.Log | {
"line": 69,
"column": 2
} | {
"line": 69,
"column": 52
} | {
"line": 70,
"column": 2
} | [
{
"pp": "A : Type u_1\ninst✝¹ : CommRing A\ninst✝ : Algebra ℚ A\nn : ℕ\n⊢ (coeff n) ((d⁄dX A) (log A)) = (coeff n) (mk fun n ↦ (algebraMap ℚ A) ((-1) ^ n))",
"ppTerm": "?m.28",
"assigned": true,
"usedConstants": [
"Rat.instOfNat",
"NonAssocSemiring.toAddCommMonoidWithOne",
"RingHom... | [
"A : Type u_1\ninst✝¹ : CommRing A\ninst✝ : Algebra ℚ A\nn : ℕ\nthis : ↑n + 1 = (algebraMap ℚ A) (↑n + 1)\n⊢ (coeff n) ((d⁄dX A) (log A)) = (coeff n) (mk fun n ↦ (algebraMap ℚ A) ((-1) ^ n))"
] | have : (n + 1) = algebraMap ℚ A (n + 1) := by simp | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.RingTheory.PowerSeries.Ideal | {
"line": 172,
"column": 2
} | {
"line": 172,
"column": 21
} | {
"line": 173,
"column": 2
} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommRing R\nP : Ideal R⟦X⟧\ninst✝ : P.IsPrime\n⊢ spanFinrank P ≤ spanFinrank (Ideal.map constantCoeff P) + 1",
"ppTerm": "?m.31",
"assigned": true,
"usedConstants": [
"Semiring.toModule",
"CommSemiring.toSemiring",
"Classical.propDecidable",
"R... | [
"case pos\nR : Type u_1\ninst✝¹ : CommRing R\nP : Ideal R⟦X⟧\ninst✝ : P.IsPrime\nhfg : P.FG\n⊢ spanFinrank P ≤ spanFinrank (Ideal.map constantCoeff P) + 1",
"case neg\nR : Type u_1\ninst✝¹ : CommRing R\nP : Ideal R⟦X⟧\ninst✝ : P.IsPrime\nhfg : ¬P.FG\n⊢ spanFinrank P ≤ spanFinrank (Ideal.map constantCoeff P) + 1"
... | by_cases hfg : P.FG | «_aux_Init_ByCases___macroRules_tacticBy_cases_:__2» | «tacticBy_cases_:_» |
Mathlib.RingTheory.PowerSeries.Schroder | {
"line": 104,
"column": 37
} | {
"line": 104,
"column": 61
} | {
"line": 104,
"column": 61
} | [
{
"pp": "case neg\nn : ℕ\nhn : ¬n = 0\nhn' : 0 < n\n⊢ ∑ i ∈ Icc 0 (n - 1), i.largeSchroder * (n - 1 - i).largeSchroder =\n ∑ i ∈ range (n - 1 + 1), i.largeSchroder * (n - 1 - i).largeSchroder",
"ppTerm": "?neg✝",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Nat.largeSchroder",
... | [
"case neg\nn : ℕ\nhn : ¬n = 0\nhn' : 0 < n\n⊢ ∑ i ∈ range (n - 1 + 1), i.largeSchroder * (n - 1 - i).largeSchroder =\n ∑ i ∈ range (n - 1 + 1), i.largeSchroder * (n - 1 - i).largeSchroder"
] | ← range_succ_eq_Icc_zero | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.PowerSeries.Restricted | {
"line": 129,
"column": 2
} | {
"line": 129,
"column": 62
} | {
"line": 130,
"column": 2
} | [
{
"pp": "R : Type u_1\ninst✝¹ : NormedRing R\nc : ℝ\ninst✝ : IsUltrametricDist R\nf g : R⟦X⟧\na : ℝ\nha : 1 ≤ a\nb : ℝ\nhb : 1 ≤ b\nfBound1 : ∀ (a_1 : ℕ), ‖(coeff a_1) f‖ * |c| ^ a_1 ≤ a\ngBound1 : ∀ (a : ℕ), ‖(coeff a) g‖ * |c| ^ a ≤ b\nhf : ∀ (ε : ℝ), 0 < ε → ∃ N, ∀ (n : ℕ), N ≤ n → ‖(coeff n) f‖ * |c| ^ n < ... | [
"R : Type u_1\ninst✝¹ : NormedRing R\nc : ℝ\ninst✝ : IsUltrametricDist R\nf g : R⟦X⟧\na : ℝ\nha : 1 ≤ a\nb : ℝ\nhb : 1 ≤ b\nfBound1 : ∀ (a_1 : ℕ), ‖(coeff a_1) f‖ * |c| ^ a_1 ≤ a\ngBound1 : ∀ (a : ℕ), ‖(coeff a) g‖ * |c| ^ a ≤ b\nhf : ∀ (ε : ℝ), 0 < ε → ∃ N, ∀ (n : ℕ), N ≤ n → ‖(coeff n) f‖ * |c| ^ n < ε\nhg : ∀ (ε... | obtain ⟨Nf, fBound2⟩ := (hf (ε / (max a b))) (by positivity) | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.RingTheory.PolynomialLaw.Basic | {
"line": 495,
"column": 4
} | {
"line": 495,
"column": 31
} | {
"line": 496,
"column": 4
} | [
{
"pp": "R : Type u\ninst✝⁸ : CommSemiring R\nM : Type u_1\ninst✝⁷ : AddCommMonoid M\ninst✝⁶ : Module R M\nN : Type u_2\ninst✝⁵ : AddCommMonoid N\ninst✝⁴ : Module R N\nS : Type v\ninst✝³ : CommSemiring S\ninst✝² : Algebra R S\nf : M →ₚₗ[R] N\nT : Type w\ninst✝¹ : CommSemiring T\ninst✝ : Algebra R T\nh : S →ₐ[R]... | [
"R : Type u\ninst✝⁸ : CommSemiring R\nM : Type u_1\ninst✝⁷ : AddCommMonoid M\ninst✝⁶ : Module R M\nN : Type u_2\ninst✝⁵ : AddCommMonoid N\ninst✝⁴ : Module R N\nS : Type v\ninst✝³ : CommSemiring S\ninst✝² : Algebra R S\nf : M →ₚₗ[R] N\nT : Type w\ninst✝¹ : CommSemiring T\ninst✝ : Algebra R T\nh : S →ₐ[R] T\nt : S ⊗[... | simp only [range_φ] at hx ⊢ | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.RingTheory.PowerSeries.WeierstrassPreparation | {
"line": 204,
"column": 2
} | {
"line": 207,
"column": 20
} | {
"line": 209,
"column": 0
} | [
{
"pp": "A : Type u_1\ninst✝ : CommRing A\ng : A⟦X⟧\nI : Ideal A\ni : ℕ\n⊢ ((trunc ((map (Ideal.Quotient.mk I)) g).order.toNat) g).coeff i ∈ I",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"PowerSeries.coeff_of_lt_order_toNat",
"Eq.mpr",
"RingHom.instRingHomClass",
... | [] | rw [coeff_trunc]
split_ifs with h
· simpa [← RingHom.mem_ker] using coeff_of_lt_order_toNat _ h
· exact zero_mem _ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.PowerSeries.WeierstrassPreparation | {
"line": 204,
"column": 2
} | {
"line": 207,
"column": 20
} | {
"line": 209,
"column": 0
} | [
{
"pp": "A : Type u_1\ninst✝ : CommRing A\ng : A⟦X⟧\nI : Ideal A\ni : ℕ\n⊢ ((trunc ((map (Ideal.Quotient.mk I)) g).order.toNat) g).coeff i ∈ I",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"PowerSeries.coeff_of_lt_order_toNat",
"Eq.mpr",
"RingHom.instRingHomClass",
... | [] | rw [coeff_trunc]
split_ifs with h
· simpa [← RingHom.mem_ker] using coeff_of_lt_order_toNat _ h
· exact zero_mem _ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.PowerSeries.WeierstrassPreparation | {
"line": 329,
"column": 4
} | {
"line": 329,
"column": 19
} | {
"line": 330,
"column": 2
} | [
{
"pp": "A : Type u_1\ninst✝¹ : CommRing A\ng : A⟦X⟧\nI : Ideal A\nH : g.IsWeierstrassDivisorAt I\ninst✝ : IsHausdorff I A\nq : A⟦X⟧\nr : A[X]\nhdeg : r.degree < ↑((map (Ideal.Quotient.mk I)) g).order.toNat\nheq : r = 0\nthis : ∀ (k i : ℕ), (coeff i) q ∈ I ^ k\nhq : q = 0\n⊢ q = 0 ∧ r = 0",
"ppTerm": "?m.14... | [] | exact ⟨hq, heq⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.RingTheory.RegularLocalRing.Defs | {
"line": 124,
"column": 4
} | {
"line": 125,
"column": 91
} | {
"line": 126,
"column": 4
} | [
{
"pp": "case pos\nR : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsDedekindDomain R\np : Ideal R\nhp : p.IsPrime\neqbot : p = ⊥\n⊢ IsRegularLocalRing (Localization.AtPrime p)",
"ppTerm": "?pos✝",
"assigned": true,
"usedConstants": [
"RingHom.instRingHomClass",
"Semiring.toModule",
"Al... | [
"case pos\nR : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsDedekindDomain R\np : Ideal R\nhp : p.IsPrime\neqbot : p = ⊥\nthis : Field (Localization.AtPrime p) := ⋯.toField\n⊢ IsRegularLocalRing (Localization.AtPrime p)"
] | let : Field (Localization.AtPrime p) := IsField.toField <| by
simp [isField_iff_maximalIdeal_eq, ← Localization.AtPrime.map_eq_maximalIdeal, eqbot] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLet___1 | Lean.Parser.Tactic.tacticLet__ |
Mathlib.RingTheory.PowerSeries.WeierstrassPreparation | {
"line": 479,
"column": 24
} | {
"line": 479,
"column": 71
} | {
"line": 479,
"column": 71
} | [
{
"pp": "A : Type u_1\ninst✝¹ : CommRing A\ng✝ : A⟦X⟧\nI✝ : Ideal A\ng : A[X]\nI : Ideal A\nH : g.IsDistinguishedAt I\ninst✝ : IsAdicComplete I A\nh✝ : Nontrivial A\nhI : I ≠ ⊤\nthis : Nontrivial (A ⧸ I)\nf : A[X]\n⊢ f /ₘ g * g = f - f %ₘ g",
"ppTerm": "?m.262",
"assigned": true,
"usedConstants": [
... | [] | rw [Polynomial.modByMonic_eq_sub_mul_div]; ring | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.PowerSeries.WeierstrassPreparation | {
"line": 479,
"column": 24
} | {
"line": 479,
"column": 71
} | {
"line": 479,
"column": 71
} | [
{
"pp": "A : Type u_1\ninst✝¹ : CommRing A\ng✝ : A⟦X⟧\nI✝ : Ideal A\ng : A[X]\nI : Ideal A\nH : g.IsDistinguishedAt I\ninst✝ : IsAdicComplete I A\nh✝ : Nontrivial A\nhI : I ≠ ⊤\nthis : Nontrivial (A ⧸ I)\nf : A[X]\n⊢ f /ₘ g * g = f - f %ₘ g",
"ppTerm": "?m.262",
"assigned": true,
"usedConstants": [
... | [] | rw [Polynomial.modByMonic_eq_sub_mul_div]; ring | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.PowerSeries.WeierstrassPreparation | {
"line": 741,
"column": 26
} | {
"line": 741,
"column": 67
} | {
"line": 741,
"column": 68
} | [
{
"pp": "A : Type u_1\ninst✝¹ : CommRing A\ninst✝ : IsLocalRing A\ng q : A⟦X⟧\nr : A[X]\nhg : (map (IsLocalRing.residue A)) g ≠ 0\nn : ℕ := ((map (IsLocalRing.residue A)) g).order.toNat\nH1 : r.degree < ↑n\nH2 :\n 1 =\n (coeff n) ((map (IsLocalRing.residue A)) g * (map (IsLocalRing.residue A)) q) + (IsLocal... | [
"A : Type u_1\ninst✝¹ : CommRing A\ninst✝ : IsLocalRing A\ng q : A⟦X⟧\nr : A[X]\nhg : (map (IsLocalRing.residue A)) g ≠ 0\nn : ℕ := ((map (IsLocalRing.residue A)) g).order.toNat\nH1 : r.degree < ↑n\nH2 : 1 = (coeff n) ((map (IsLocalRing.residue A)) g * (map (IsLocalRing.residue A)) q) + (IsLocalRing.residue A) 0\n⊢... | Polynomial.coeff_eq_zero_of_degree_lt H1, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.RingTheory.SimpleModule.Isotypic | {
"line": 225,
"column": 2
} | {
"line": 225,
"column": 44
} | {
"line": 226,
"column": 2
} | [
{
"pp": "R : Type u_2\nM : Type u\ninst✝⁴ : Ring R\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\nN : Submodule R M\ninst✝¹ : IsSimpleModule R ↥N\ns : Set (Submodule R M)\ninst✝ : IsSemisimpleModule R M\nhs : sSup s = ⊤\nthis : Nontrivial ↥N\nw✝ : Submodule R M\ncompl : IsCompl N w✝\nm : Submodule R M\nhm : m ∈... | [
"R : Type u_2\nM : Type u\ninst✝⁴ : Ring R\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\nN : Submodule R M\ninst✝¹ : IsSimpleModule R ↥N\ns : Set (Submodule R M)\ninst✝ : IsSemisimpleModule R M\nhs : sSup s = ⊤\nthis : Nontrivial ↥N\nw✝ : Submodule R M\ncompl : IsCompl N w✝\nm : Submodule R M\nhm : m ∈ s\nne : N.p... | have ⟨S, ⟨e⟩⟩ := linearEquiv_of_ne_zero ne | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.RingTheory.SimpleModule.Isotypic | {
"line": 313,
"column": 37
} | {
"line": 313,
"column": 59
} | {
"line": 313,
"column": 59
} | [
{
"pp": "R : Type u_2\nM : Type u\nN : Type u_3\ninst✝⁵ : Ring R\ninst✝⁴ : AddCommGroup M\ninst✝³ : AddCommGroup N\ninst✝² : Module R M\ninst✝¹ : Module R N\nS : Submodule R M\ninst✝ : IsSimpleModule R ↥S\nf : M →ₗ[R] N\n| map f S.subtype.range",
"ppTerm": "?m.56",
"assigned": true,
"usedConstants":... | [
"R : Type u_2\nM : Type u\nN : Type u_3\ninst✝⁵ : Ring R\ninst✝⁴ : AddCommGroup M\ninst✝³ : AddCommGroup N\ninst✝² : Module R M\ninst✝¹ : Module R N\nS : Submodule R M\ninst✝ : IsSimpleModule R ↥S\nf : M →ₗ[R] N\n| (f ∘ₗ S.subtype).range"
] | ← LinearMap.range_comp | Lean.Elab.Tactic.Conv.evalRewrite | null |
Mathlib.RingTheory.SimpleModule.Isotypic | {
"line": 399,
"column": 6
} | {
"line": 399,
"column": 57
} | {
"line": 400,
"column": 6
} | [
{
"pp": "R₀ : Type u_1\nR : Type u_2\nM : Type u\nN : Type u_3\nS : Type u_4\ninst✝⁹ : CommSemiring R₀\ninst✝⁸ : Ring R\ninst✝⁷ : Algebra R₀ R\ninst✝⁶ : AddCommGroup M\ninst✝⁵ : AddCommGroup N\ninst✝⁴ : AddCommGroup S\ninst✝³ : Module R M\ninst✝² : Module R N\ninst✝¹ : Module R S\ninst✝ : IsSimpleModule R S\ns ... | [
"R₀ : Type u_1\nR : Type u_2\nM : Type u\nN : Type u_3\nS : Type u_4\ninst✝⁹ : CommSemiring R₀\ninst✝⁸ : Ring R\ninst✝⁷ : Algebra R₀ R\ninst✝⁶ : AddCommGroup M\ninst✝⁵ : AddCommGroup N\ninst✝⁴ : AddCommGroup S\ninst✝³ : Module R M\ninst✝² : Module R N\ninst✝¹ : Module R S\ninst✝ : IsSimpleModule R S\ns : Set ↑(isot... | simp_rw [CompleteSublattice.coe_iSup, iSup₂_le_iff] | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | Mathlib.Tactic.tacticSimp_rw___ |
Mathlib.RingTheory.PowerSeries.WeierstrassPreparation | {
"line": 773,
"column": 4
} | {
"line": 773,
"column": 90
} | {
"line": 775,
"column": 0
} | [
{
"pp": "case refine_2\nA : Type u_1\ninst✝¹ : CommRing A\ninst✝ : IsLocalRing A\ng q : A⟦X⟧\nr : A[X]\nhg : (map (IsLocalRing.residue A)) g ≠ 0\nH✝ : (X ^ ((map (IsLocalRing.residue A)) g).order.toNat).IsWeierstrassDivision g q r\nn : ℕ := ((map (IsLocalRing.residue A)) g).order.toNat\nH : (X ^ n).IsWeierstras... | [] | simp_rw [← this, f, Polynomial.coe_sub, Polynomial.coe_pow, Polynomial.coe_X, sub_mul] | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | Mathlib.Tactic.tacticSimp_rw___ |
Mathlib.RingTheory.TwoSidedIdeal.BigOperators | {
"line": 65,
"column": 6
} | {
"line": 65,
"column": 76
} | {
"line": 66,
"column": 4
} | [
{
"pp": "case cons.inl\nR : Type u_1\ninst✝ : Ring R\nI : TwoSidedIdeal R\nι : Type u_2\nf : ι → R\nx : ι\nl : List ι\nih : (∃ x ∈ l, f x ∈ I) → (List.map f l).prod ∈ I\nh : f x ∈ I\n⊢ (List.map f (x :: l)).prod ∈ I",
"ppTerm": "?cons.inl",
"assigned": true,
"usedConstants": [
"TwoSidedIdeal.m... | [] | simpa only [List.map_cons, List.prod_cons] using I.mul_mem_right _ _ h | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.RingTheory.TwoSidedIdeal.BigOperators | {
"line": 65,
"column": 6
} | {
"line": 65,
"column": 76
} | {
"line": 66,
"column": 4
} | [
{
"pp": "case cons.inl\nR : Type u_1\ninst✝ : Ring R\nI : TwoSidedIdeal R\nι : Type u_2\nf : ι → R\nx : ι\nl : List ι\nih : (∃ x ∈ l, f x ∈ I) → (List.map f l).prod ∈ I\nh : f x ∈ I\n⊢ (List.map f (x :: l)).prod ∈ I",
"ppTerm": "?cons.inl",
"assigned": true,
"usedConstants": [
"TwoSidedIdeal.m... | [] | simpa only [List.map_cons, List.prod_cons] using I.mul_mem_right _ _ h | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.TwoSidedIdeal.BigOperators | {
"line": 65,
"column": 6
} | {
"line": 65,
"column": 76
} | {
"line": 66,
"column": 4
} | [
{
"pp": "case cons.inl\nR : Type u_1\ninst✝ : Ring R\nI : TwoSidedIdeal R\nι : Type u_2\nf : ι → R\nx : ι\nl : List ι\nih : (∃ x ∈ l, f x ∈ I) → (List.map f l).prod ∈ I\nh : f x ∈ I\n⊢ (List.map f (x :: l)).prod ∈ I",
"ppTerm": "?cons.inl",
"assigned": true,
"usedConstants": [
"TwoSidedIdeal.m... | [] | simpa only [List.map_cons, List.prod_cons] using I.mul_mem_right _ _ h | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.WittVector.MulCoeff | {
"line": 142,
"column": 2
} | {
"line": 142,
"column": 84
} | {
"line": 143,
"column": 2
} | [
{
"pp": "p n : ℕ\n⊢ wittPolyProd p (n + 1) =\n -(↑p ^ (n + 1) * X (0, n + 1)) * (↑p ^ (n + 1) * X (1, n + 1)) +\n ↑p ^ (n + 1) * X (0, n + 1) * (rename (Prod.mk 1)) (wittPolynomial p ℤ (n + 1)) +\n ↑p ^ (n + 1) * X (1, n + 1) * (rename (Prod.mk 0)) (wittPolynomial p ℤ (n + 1)) +\n remain... | [
"p n : ℕ\nmvpz : ↑p ^ (n + 1) = C (↑p ^ (n + 1))\n⊢ wittPolyProd p (n + 1) =\n -(↑p ^ (n + 1) * X (0, n + 1)) * (↑p ^ (n + 1) * X (1, n + 1)) +\n ↑p ^ (n + 1) * X (0, n + 1) * (rename (Prod.mk 1)) (wittPolynomial p ℤ (n + 1)) +\n ↑p ^ (n + 1) * X (1, n + 1) * (rename (Prod.mk 0)) (wittPolynomial ... | have mvpz : (p : 𝕄) ^ (n + 1) = MvPolynomial.C ((p : ℤ) ^ (n + 1)) := by norm_cast | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.RingTheory.WittVector.FrobeniusFractionField | {
"line": 201,
"column": 4
} | {
"line": 202,
"column": 39
} | {
"line": 203,
"column": 4
} | [
{
"pp": "case zero\np : ℕ\nhp : Fact (Nat.Prime p)\nk : Type u_1\ninst✝² : Field k\ninst✝¹ : CharP k p\ninst✝ : IsAlgClosed k\na₁ a₂ : 𝕎 k\nha₁ : a₁.coeff 0 ≠ 0\nha₂ : a₂.coeff 0 ≠ 0\n⊢ (frobenius (frobeniusRotation p ha₁ ha₂) * a₁).coeff 0 = (frobeniusRotation p ha₁ ha₂ * a₂).coeff 0",
"ppTerm": "?zero",
... | [
"case zero\np : ℕ\nhp : Fact (Nat.Prime p)\nk : Type u_1\ninst✝² : Field k\ninst✝¹ : CharP k p\ninst✝ : IsAlgClosed k\na₁ a₂ : 𝕎 k\nha₁ : a₁.coeff 0 ≠ 0\nha₂ : a₂.coeff 0 ≠ 0\n⊢ solution p a₁ a₂ ^ p * a₁.coeff 0 = solution p a₁ a₂ * a₂.coeff 0"
] | simp only [WittVector.mul_coeff_zero, WittVector.coeff_frobenius_charP, frobeniusRotation,
coeff_mk, frobeniusRotationCoeff] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.SetTheory.Descriptive.Tree | {
"line": 42,
"column": 2
} | {
"line": 42,
"column": 33
} | {
"line": 43,
"column": 2
} | [
{
"pp": "A : Type u_1\nT : ↥(tree A)\nx y : List A\nh : x ++ y ∈ T\n⊢ x ∈ T",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"List.append_assoc",
"congrArg",
"Membership.mem",
"Eq.mp",
"id",
"Subtype",
"List.rec",
"List.append_... | [] | induction y generalizing x with | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | null |
Mathlib.SetTheory.Ordinal.CantorNormalForm | {
"line": 110,
"column": 12
} | {
"line": 110,
"column": 14
} | {
"line": 110,
"column": 15
} | [
{
"pp": "case refine_2\nb o✝ o : Ordinal.{u_1}\n⊢ o ≠ 0 →\n foldr (fun p r ↦ b ^ p.1 * p.2 + r) 0 (CNF b (o % b ^ log b o)) = o % b ^ log b o →\n foldr (fun p r ↦ b ^ p.1 * p.2 + r) 0 (CNF b o) = o",
"ppTerm": "?refine_2",
"assigned": true,
"usedConstants": [
"Ne",
"Ordinal.zero"... | [
"case refine_2\nb o✝ o : Ordinal.{u_1}\nho : o ≠ 0\n⊢ foldr (fun p r ↦ b ^ p.1 * p.2 + r) 0 (CNF b (o % b ^ log b o)) = o % b ^ log b o →\n foldr (fun p r ↦ b ^ p.1 * p.2 + r) 0 (CNF b o) = o"
] | ho | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.SetTheory.Ordinal.CantorNormalForm | {
"line": 111,
"column": 24
} | {
"line": 111,
"column": 35
} | {
"line": 111,
"column": 36
} | [
{
"pp": "case refine_2\nb o✝ o : Ordinal.{u_1}\nho : o ≠ 0\nIH : foldr (fun p r ↦ b ^ p.1 * p.2 + r) 0 (CNF b (o % b ^ log b o)) = o % b ^ log b o\n⊢ foldr (fun p r ↦ b ^ p.1 * p.2 + r) 0 ((log b o, o / b ^ log b o) :: CNF b (o % b ^ log b o)) = o",
"ppTerm": "?refine_2",
"assigned": true,
"usedCons... | [
"case refine_2\nb o✝ o : Ordinal.{u_1}\nho : o ≠ 0\nIH : foldr (fun p r ↦ b ^ p.1 * p.2 + r) 0 (CNF b (o % b ^ log b o)) = o % b ^ log b o\n⊢ b ^ (log b o, o / b ^ log b o).1 * (log b o, o / b ^ log b o).2 +\n foldr (fun p r ↦ b ^ p.1 * p.2 + r) 0 (CNF b (o % b ^ log b o)) =\n o"
] | foldr_cons, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.SetTheory.Ordinal.CantorNormalForm | {
"line": 194,
"column": 2
} | {
"line": 194,
"column": 57
} | {
"line": 196,
"column": 0
} | [
{
"pp": "b o e : Ordinal.{u_1}\nh : e ∉ map Prod.fst (CNF b o)\n⊢ (coeff b o) e = 0",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"List.mem_toFinset",
"Finsupp.instFunLike",
"Eq.mpr",
"Ordinal.instLinearOrder",
"LinearOrder.toDecidableEq",
"congrArg",
... | [] | rwa [← notMem_support_iff, support_coeff, mem_toFinset] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticRwa___1 | Lean.Parser.Tactic.tacticRwa__ |
Mathlib.SetTheory.Ordinal.CantorNormalForm | {
"line": 194,
"column": 2
} | {
"line": 194,
"column": 57
} | {
"line": 196,
"column": 0
} | [
{
"pp": "b o e : Ordinal.{u_1}\nh : e ∉ map Prod.fst (CNF b o)\n⊢ (coeff b o) e = 0",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"List.mem_toFinset",
"Finsupp.instFunLike",
"Eq.mpr",
"Ordinal.instLinearOrder",
"LinearOrder.toDecidableEq",
"congrArg",
... | [] | rwa [← notMem_support_iff, support_coeff, mem_toFinset] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.SetTheory.Ordinal.CantorNormalForm | {
"line": 194,
"column": 2
} | {
"line": 194,
"column": 57
} | {
"line": 196,
"column": 0
} | [
{
"pp": "b o e : Ordinal.{u_1}\nh : e ∉ map Prod.fst (CNF b o)\n⊢ (coeff b o) e = 0",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"List.mem_toFinset",
"Finsupp.instFunLike",
"Eq.mpr",
"Ordinal.instLinearOrder",
"LinearOrder.toDecidableEq",
"congrArg",
... | [] | rwa [← notMem_support_iff, support_coeff, mem_toFinset] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Order.SuccPred | {
"line": 32,
"column": 4
} | {
"line": 32,
"column": 40
} | {
"line": 33,
"column": 4
} | [
{
"pp": "case pos\nα : Type u_1\ninst✝³ : LinearOrder α\ninst✝² : TopologicalSpace α\ninst✝¹ : OrderTopology α\na : α\ninst✝ : SuccOrder α\nha : ¬IsSuccPrelimit a\nb : α\nhb : b ⋖ a\nha' : IsMax a\n⊢ IsOpen[inst✝²] {a}",
"ppTerm": "?pos✝",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"... | [
"α : Type u_1\ninst✝³ : LinearOrder α\ninst✝² : TopologicalSpace α\ninst✝¹ : OrderTopology α\na : α\ninst✝ : SuccOrder α\nha : ¬IsSuccPrelimit a\nb : α\nhb : b ⋖ a\nha' : IsMax a\n⊢ {a} = Ioi b"
] | convert! isOpen_Ioi (a := b) using 1 | Mathlib.Tactic._aux_Mathlib_Tactic_Convert___macroRules_Mathlib_Tactic_convert!_1 | Mathlib.Tactic.convert! |
Mathlib.Topology.Order.SuccPred | {
"line": 35,
"column": 4
} | {
"line": 35,
"column": 60
} | {
"line": 36,
"column": 4
} | [
{
"pp": "case neg\nα : Type u_1\ninst✝³ : LinearOrder α\ninst✝² : TopologicalSpace α\ninst✝¹ : OrderTopology α\na : α\ninst✝ : SuccOrder α\nha : ¬IsSuccPrelimit a\nb : α\nhb : b ⋖ a\nha' : ¬IsMax a\n⊢ IsOpen[inst✝²] {a}",
"ppTerm": "?neg✝",
"assigned": true,
"usedConstants": [
"Eq.mpr",
... | [
"α : Type u_1\ninst✝³ : LinearOrder α\ninst✝² : TopologicalSpace α\ninst✝¹ : OrderTopology α\na : α\ninst✝ : SuccOrder α\nha : ¬IsSuccPrelimit a\nb : α\nhb : b ⋖ a\nha' : ¬IsMax a\n⊢ {a} = Ioo b (Order.succ a)"
] | convert! isOpen_Ioo (a := b) (b := Order.succ a) using 1 | Mathlib.Tactic._aux_Mathlib_Tactic_Convert___macroRules_Mathlib_Tactic_convert!_1 | Mathlib.Tactic.convert! |
Mathlib.Topology.Order.SuccPred | {
"line": 75,
"column": 4
} | {
"line": 75,
"column": 38
} | {
"line": 77,
"column": 0
} | [
{
"pp": "case neg\nα : Type u_1\ninst✝⁴ : LinearOrder α\ninst✝³ : TopologicalSpace α\ninst✝² : OrderTopology α\ninst✝¹ : SuccOrder α\ninst✝ : NoMaxOrder α\ns : Set α\no : α\nho : o ∈ s\nho' : ¬IsSuccLimit o\n⊢ s ∈ 𝓝 o ↔ IsSuccLimit o → ∃ a < o, Ioo a o ⊆ s",
"ppTerm": "?neg✝",
"assigned": true,
"us... | [] | simp [nhds_eq_pure.2 ho', ho, ho'] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Topology.Order.SuccPred | {
"line": 75,
"column": 4
} | {
"line": 75,
"column": 38
} | {
"line": 77,
"column": 0
} | [
{
"pp": "case neg\nα : Type u_1\ninst✝⁴ : LinearOrder α\ninst✝³ : TopologicalSpace α\ninst✝² : OrderTopology α\ninst✝¹ : SuccOrder α\ninst✝ : NoMaxOrder α\ns : Set α\no : α\nho : o ∈ s\nho' : ¬IsSuccLimit o\n⊢ s ∈ 𝓝 o ↔ IsSuccLimit o → ∃ a < o, Ioo a o ⊆ s",
"ppTerm": "?neg✝",
"assigned": true,
"us... | [] | simp [nhds_eq_pure.2 ho', ho, ho'] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Order.SuccPred | {
"line": 75,
"column": 4
} | {
"line": 75,
"column": 38
} | {
"line": 77,
"column": 0
} | [
{
"pp": "case neg\nα : Type u_1\ninst✝⁴ : LinearOrder α\ninst✝³ : TopologicalSpace α\ninst✝² : OrderTopology α\ninst✝¹ : SuccOrder α\ninst✝ : NoMaxOrder α\ns : Set α\no : α\nho : o ∈ s\nho' : ¬IsSuccLimit o\n⊢ s ∈ 𝓝 o ↔ IsSuccLimit o → ∃ a < o, Ioo a o ⊆ s",
"ppTerm": "?neg✝",
"assigned": true,
"us... | [] | simp [nhds_eq_pure.2 ho', ho, ho'] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.SetTheory.Ordinal.Notation | {
"line": 278,
"column": 6
} | {
"line": 278,
"column": 12
} | {
"line": 278,
"column": 12
} | [
{
"pp": "e : ONote\nn : ℕ+\na₁ a₂ : ONote\nh : a₁ < a₂\n⊢ e.oadd n a₁ < e.oadd n a₂",
"ppTerm": "?m.10",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Preorder.toLT",
"Ordinal.partialOrder",
"congrArg",
"ONote.oadd",
"PartialOrder.toPreorder",
"ONote.lt_de... | [
"e : ONote\nn : ℕ+\na₁ a₂ : ONote\nh : a₁ < a₂\n⊢ (e.oadd n a₁).repr < (e.oadd n a₂).repr"
] | lt_def | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.SetTheory.Ordinal.Veblen | {
"line": 134,
"column": 2
} | {
"line": 134,
"column": 37
} | {
"line": 136,
"column": 0
} | [
{
"pp": "f : Ordinal.{u} → Ordinal.{u}\nhf : IsNormal f\n⊢ veblenWith f 1 = deriv f",
"ppTerm": "?m.8",
"assigned": true,
"usedConstants": [
"congrArg",
"AddMonoid.toAddZeroClass",
"Eq.mp",
"Ordinal.veblenWith_add_one",
"zero_add",
"Ordinal.addMonoidWithOne",
... | [] | simpa using veblenWith_add_one hf 0 | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.SetTheory.Ordinal.Veblen | {
"line": 134,
"column": 2
} | {
"line": 134,
"column": 37
} | {
"line": 136,
"column": 0
} | [
{
"pp": "f : Ordinal.{u} → Ordinal.{u}\nhf : IsNormal f\n⊢ veblenWith f 1 = deriv f",
"ppTerm": "?m.8",
"assigned": true,
"usedConstants": [
"congrArg",
"AddMonoid.toAddZeroClass",
"Eq.mp",
"Ordinal.veblenWith_add_one",
"zero_add",
"Ordinal.addMonoidWithOne",
... | [] | simpa using veblenWith_add_one hf 0 | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.SetTheory.Ordinal.Veblen | {
"line": 134,
"column": 2
} | {
"line": 134,
"column": 37
} | {
"line": 136,
"column": 0
} | [
{
"pp": "f : Ordinal.{u} → Ordinal.{u}\nhf : IsNormal f\n⊢ veblenWith f 1 = deriv f",
"ppTerm": "?m.8",
"assigned": true,
"usedConstants": [
"congrArg",
"AddMonoid.toAddZeroClass",
"Eq.mp",
"Ordinal.veblenWith_add_one",
"zero_add",
"Ordinal.addMonoidWithOne",
... | [] | simpa using veblenWith_add_one hf 0 | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.SetTheory.ZFC.PSet | {
"line": 406,
"column": 64
} | {
"line": 411,
"column": 29
} | {
"line": 411,
"column": 29
} | [
{
"pp": "α : Type u\nA : α → PSet.{u}\ny : PSet.{u}\nx✝ : ∃ z ∈ mk α A, y ∈ z\nβ : Type u\nB : β → PSet.{u}\na : (mk α A).Type\ne : (mk β B).Equiv (A a)\nb : (mk β B).Type\nyb : y.Equiv ((mk β B).Func b)\n⊢ y ∈ ⋃₀ mk α A",
"ppTerm": "?m.84",
"assigned": true,
"usedConstants": [
"_private.Mathl... | [] | by
rw [← eta (A a)] at e
exact
let ⟨βt, _⟩ := e
let ⟨c, bc⟩ := βt b
⟨⟨a, c⟩, yb.trans bc⟩ | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.SetTheory.Ordinal.CantorNormalForm | {
"line": 355,
"column": 2
} | {
"line": 369,
"column": 27
} | {
"line": 371,
"column": 0
} | [
{
"pp": "b : Ordinal.{u_1}\nhb : 1 < b\nf : Ordinal.{u_1} →₀ Ordinal.{u_1}\nhf : ∀ (e : Ordinal.{u_1}), f e < b\n⊢ coeff b (eval b f) = f",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"Ordinal.CNF.eval",
"Finsupp.instFunLike",
"Eq.mpr",
"LT.lt.pos",
"Ordinal... | [] | induction f using Finsupp.induction_on_max with
| zero => simp
| single_add e x f hf' hx IH =>
have IH' (e') : f e' < b := by
by_cases he' : e' ∈ f.support
· apply (hf e').trans_eq'
rw [add_apply, single_eq_of_ne, zero_add]
exact (hf' _ he').ne
· rw [notMem_support_iff.1 he']
... | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | Lean.Parser.Tactic.induction |
Mathlib.SetTheory.Ordinal.CantorNormalForm | {
"line": 355,
"column": 2
} | {
"line": 369,
"column": 27
} | {
"line": 371,
"column": 0
} | [
{
"pp": "b : Ordinal.{u_1}\nhb : 1 < b\nf : Ordinal.{u_1} →₀ Ordinal.{u_1}\nhf : ∀ (e : Ordinal.{u_1}), f e < b\n⊢ coeff b (eval b f) = f",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"Ordinal.CNF.eval",
"Finsupp.instFunLike",
"Eq.mpr",
"LT.lt.pos",
"Ordinal... | [] | induction f using Finsupp.induction_on_max with
| zero => simp
| single_add e x f hf' hx IH =>
have IH' (e') : f e' < b := by
by_cases he' : e' ∈ f.support
· apply (hf e').trans_eq'
rw [add_apply, single_eq_of_ne, zero_add]
exact (hf' _ he').ne
· rw [notMem_support_iff.1 he']
... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.SetTheory.Ordinal.CantorNormalForm | {
"line": 355,
"column": 2
} | {
"line": 369,
"column": 27
} | {
"line": 371,
"column": 0
} | [
{
"pp": "b : Ordinal.{u_1}\nhb : 1 < b\nf : Ordinal.{u_1} →₀ Ordinal.{u_1}\nhf : ∀ (e : Ordinal.{u_1}), f e < b\n⊢ coeff b (eval b f) = f",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"Ordinal.CNF.eval",
"Finsupp.instFunLike",
"Eq.mpr",
"LT.lt.pos",
"Ordinal... | [] | induction f using Finsupp.induction_on_max with
| zero => simp
| single_add e x f hf' hx IH =>
have IH' (e') : f e' < b := by
by_cases he' : e' ∈ f.support
· apply (hf e').trans_eq'
rw [add_apply, single_eq_of_ne, zero_add]
exact (hf' _ he').ne
· rw [notMem_support_iff.1 he']
... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.SetTheory.ZFC.Cardinal | {
"line": 78,
"column": 33
} | {
"line": 78,
"column": 47
} | {
"line": 78,
"column": 48
} | [
{
"pp": "α : Type u\ninst✝ : Small.{v, u} α\nf : α → ZFSet.{v}\n⊢ lift.{max u (v + 1), max u v} (lift.{u, v} (range f).card) ≤ lift.{max u (v + 1), max u v} (lift.{v, u} #α)",
"ppTerm": "?m.9",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Cardinal",
"Cardinal.lift_lift",
"... | [
"α : Type u\ninst✝ : Small.{v, u} α\nf : α → ZFSet.{v}\n⊢ lift.{max u (v + 1), max u v} (lift.{u, v} (range f).card) ≤ lift.{max v u (v + 1), u} #α"
] | lift_lift.{v}, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.SetTheory.ZFC.Basic | {
"line": 429,
"column": 4
} | {
"line": 429,
"column": 27
} | {
"line": 429,
"column": 28
} | [
{
"pp": "α β : Type u\nA : α → PSet.{u}\nB : β → PSet.{u}\nαβ : ∀ (a : α), ∃ b, (A a).Equiv (B b)\na : (PSet.mk α A).Type\nc : ((PSet.mk α A).Func a).Type\nb : β\nhb : (A a).Equiv (B b)\nx✝ : PSet.{u}\nea : A a = x✝\n⊢ ∃ b, ((⋃₀ PSet.mk α A).Func ⟨a, c⟩).Equiv ((⋃₀ PSet.mk β B).Func b)",
"ppTerm": "?m.40",
... | [
"case mk\nα β : Type u\nA : α → PSet.{u}\nB : β → PSet.{u}\nαβ : ∀ (a : α), ∃ b, (A a).Equiv (B b)\na : (PSet.mk α A).Type\nc : ((PSet.mk α A).Func a).Type\nb : β\nhb : (A a).Equiv (B b)\nγ : Type u\nΓ : γ → PSet.{u}\nA_ih✝ : ∀ (a_1 : γ), A a = Γ a_1 → ∃ b, ((⋃₀ PSet.mk α A).Func ⟨a, c⟩).Equiv ((⋃₀ PSet.mk β B).Fun... | induction ea : A a with | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | null |
Mathlib.SetTheory.Ordinal.Notation | {
"line": 474,
"column": 44
} | {
"line": 504,
"column": 91
} | {
"line": 506,
"column": 0
} | [
{
"pp": "e₁ : ONote\nn₁ : ℕ+\na₁ e₂ : ONote\nn₂ : ℕ+\na₂ : ONote\nh₁ : (e₁.oadd n₁ a₁).NF\nh₂ : (e₂.oadd n₂ a₂).NF\n⊢ (e₁.oadd n₁ a₁ - e₂.oadd n₂ a₂).repr = (e₁.oadd n₁ a₁).repr - (e₂.oadd n₂ a₂).repr",
"ppTerm": "?m.42",
"assigned": true,
"usedConstants": [
"PNat.val",
"Iff.mpr",
... | [] | by
haveI := h₁.snd; haveI := h₂.snd; have h' := repr_sub a₁ a₂
conv_lhs at h' => dsimp [HSub.hSub, Sub.sub, sub]
conv_lhs => dsimp only [HSub.hSub, Sub.sub]; dsimp only [sub]
have ee := @cmp_compares _ _ h₁.fst h₂.fst
cases h : cmp e₁ e₂ <;> simp only [h] at ee
· rw [Ordinal.sub_eq_zero_iff_le.2... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Tactic.Algebra.Lemmas | {
"line": 184,
"column": 62
} | {
"line": 185,
"column": 19
} | {
"line": 188,
"column": 0
} | [
{
"pp": "R : Type u_1\nA : Type u_2\nsR : CommSemiring R\nsA : CommSemiring A\nsAlg : Algebra R A\nr s t : R\nh : r + s = t\n⊢ (algebraMap R A) r + (algebraMap R A) s = (algebraMap R A) t",
"ppTerm": "?m.25",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoi... | [] | by
rw [← map_add, h] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.SetTheory.Ordinal.Notation | {
"line": 529,
"column": 4
} | {
"line": 538,
"column": 29
} | {
"line": 540,
"column": 0
} | [
{
"pp": "e₁ : ONote\nn₁ : ℕ+\na₁ : ONote\nb₁ : Ordinal.{0}\nh₁ : (e₁.oadd n₁ a₁).NFBelow b₁\ne₂ : ONote\nn₂ : ℕ+\na₂ : ONote\nb₂ : Ordinal.{0}\nh₂ : (e₂.oadd n₂ a₂).NFBelow b₂\n⊢ (e₁.oadd n₁ a₁ * e₂.oadd n₂ a₂).NFBelow (e₁.repr + b₂)",
"ppTerm": "?m.28",
"assigned": true,
"usedConstants": [
"O... | [] | have IH := oadd_mul_nfBelow h₁ h₂.snd
by_cases e0 : e₂ = 0 <;> simp only [e0, oadd_mul, ↓reduceIte]
· apply NFBelow.oadd h₁.fst h₁.snd
grw [← h₂.lt.pos, add_zero]
· haveI := h₁.fst
haveI := h₂.fst
apply NFBelow.oadd
· infer_instance
· rwa [repr_add]
· grw [repr_add, h₂.lt... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.SetTheory.Ordinal.Notation | {
"line": 529,
"column": 4
} | {
"line": 538,
"column": 29
} | {
"line": 540,
"column": 0
} | [
{
"pp": "e₁ : ONote\nn₁ : ℕ+\na₁ : ONote\nb₁ : Ordinal.{0}\nh₁ : (e₁.oadd n₁ a₁).NFBelow b₁\ne₂ : ONote\nn₂ : ℕ+\na₂ : ONote\nb₂ : Ordinal.{0}\nh₂ : (e₂.oadd n₂ a₂).NFBelow b₂\n⊢ (e₁.oadd n₁ a₁ * e₂.oadd n₂ a₂).NFBelow (e₁.repr + b₂)",
"ppTerm": "?m.28",
"assigned": true,
"usedConstants": [
"O... | [] | have IH := oadd_mul_nfBelow h₁ h₂.snd
by_cases e0 : e₂ = 0 <;> simp only [e0, oadd_mul, ↓reduceIte]
· apply NFBelow.oadd h₁.fst h₁.snd
grw [← h₂.lt.pos, add_zero]
· haveI := h₁.fst
haveI := h₂.fst
apply NFBelow.oadd
· infer_instance
· rwa [repr_add]
· grw [repr_add, h₂.lt... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.SetTheory.ZFC.Ordinal | {
"line": 209,
"column": 47
} | {
"line": 211,
"column": 54
} | {
"line": 213,
"column": 0
} | [
{
"pp": "x : ZFSet.{u}\nh : x.IsOrdinal\n⊢ ∀ {a b : ↥x},\n Subrel (fun x1 x2 ↦ x1 ∈ x2) (fun x_1 ↦ x_1 ∈ x) a b ∨ a = b ∨ Subrel (fun x1 x2 ↦ x1 ∈ x2) (fun x_1 ↦ x_1 ∈ x) b a",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"ZFSet",
"Membersh... | [] | by
intro ⟨a, ha⟩ ⟨b, hb⟩
simpa using mem_trichotomous (h.mem ha) (h.mem hb) | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.SetTheory.Ordinal.Notation | {
"line": 571,
"column": 6
} | {
"line": 571,
"column": 62
} | {
"line": 573,
"column": 0
} | [
{
"pp": "case neg.e_a.e_a\ne₁ : ONote\nn₁ : ℕ+\na₁ e₂ : ONote\nn₂ : ℕ+\na₂ : ONote\nh₁ : (e₁.oadd n₁ a₁).NF\nh₂ : (e₂.oadd n₂ a₂).NF\nIH : ((e₁.oadd n₁ a₁).mul a₂).repr = (e₁.oadd n₁ a₁).repr * a₂.repr\nao : a₁.repr + ω ^ e₁.repr * ↑↑n₁ = ω ^ e₁.repr * ↑↑n₁\ne0 : ¬e₂ = 0\nthis✝¹ : e₁.NF\nthis✝ : e₂.NF\nthis : ¬... | [] | simpa using! opow_dvd_opow ω (one_le_iff_ne_zero.2 this) | Lean.Elab.Tactic.Simpa.evalSimpaUsingBang | Lean.Parser.Tactic.simpaUsingBang |
Mathlib.SetTheory.Ordinal.Notation | {
"line": 823,
"column": 2
} | {
"line": 824,
"column": 64
} | {
"line": 825,
"column": 2
} | [
{
"pp": "a0 a' : ONote\nN0 : a0.NF\nNa' : a'.NF\nm : ℕ\nd : ω ∣ a'.repr\ne0 : a0.repr ≠ 0\nh : a'.repr + ↑m < ω ^ a0.repr\nn : ℕ+\nk : ℕ\nR' : Ordinal.{0} := (opowAux 0 a0 (a0.oadd n a' * ↑m) k m).repr\n⊢ (k ≠ 0 → R' < (ω ^ a0.repr) ^ succ ↑k) ∧\n (ω ^ a0.repr) ^ ↑k * (ω ^ a0.repr * ↑↑n + a'.repr) + R' = (ω ... | [
"a0 a' : ONote\nN0 : a0.NF\nNa' : a'.NF\nm : ℕ\nd : ω ∣ a'.repr\ne0 : a0.repr ≠ 0\nh : a'.repr + ↑m < ω ^ a0.repr\nn : ℕ+\nk : ℕ\nR' : Ordinal.{0} := (opowAux 0 a0 (a0.oadd n a' * ↑m) k m).repr\nNo : (a0.oadd n a').NF\n⊢ (k ≠ 0 → R' < (ω ^ a0.repr) ^ succ ↑k) ∧\n (ω ^ a0.repr) ^ ↑k * (ω ^ a0.repr * ↑↑n + a'.repr... | haveI No : NF (oadd a0 n a') :=
N0.oadd n (Na'.below_of_lt' <| lt_of_le_of_lt le_self_add h) | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHaveI___1 | Lean.Parser.Tactic.tacticHaveI__ |
Mathlib.Tactic.ComputeAsymptotics.Multiseries.Majorized | {
"line": 107,
"column": 2
} | {
"line": 116,
"column": 95
} | {
"line": 118,
"column": 0
} | [
{
"pp": "f g b : ℝ → ℝ\nf_exp g_exp : ℝ\nhf : Majorized f b f_exp\nhg : Majorized g b g_exp\nh_pos : ∀ᶠ (t : ℝ) in atTop, 0 < b t\n⊢ Majorized (f * g) b (f_exp + g_exp)",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"Mathlib.Tactic.Ring.Common.mul_pf_left",
"NormedCommRing.toN... | [] | simp only [Majorized] at *
intro exp h_exp
let ε := (exp - f_exp - g_exp) / 2
specialize hf (f_exp + ε) (by dsimp [ε]; linarith)
specialize hg (g_exp + ε) (by dsimp [ε]; linarith)
apply (hf.mul hg).trans_eventuallyEq (g₁ := fun t ↦ b t ^ (f_exp + ε) * b t ^ (g_exp + ε))
apply h_pos.mono
intro t hx
simp ... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Tactic.ComputeAsymptotics.Multiseries.Majorized | {
"line": 107,
"column": 2
} | {
"line": 116,
"column": 95
} | {
"line": 118,
"column": 0
} | [
{
"pp": "f g b : ℝ → ℝ\nf_exp g_exp : ℝ\nhf : Majorized f b f_exp\nhg : Majorized g b g_exp\nh_pos : ∀ᶠ (t : ℝ) in atTop, 0 < b t\n⊢ Majorized (f * g) b (f_exp + g_exp)",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"Mathlib.Tactic.Ring.Common.mul_pf_left",
"NormedCommRing.toN... | [] | simp only [Majorized] at *
intro exp h_exp
let ε := (exp - f_exp - g_exp) / 2
specialize hf (f_exp + ε) (by dsimp [ε]; linarith)
specialize hg (g_exp + ε) (by dsimp [ε]; linarith)
apply (hf.mul hg).trans_eventuallyEq (g₁ := fun t ↦ b t ^ (f_exp + ε) * b t ^ (g_exp + ε))
apply h_pos.mono
intro t hx
simp ... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.SetTheory.Ordinal.Notation | {
"line": 860,
"column": 6
} | {
"line": 860,
"column": 71
} | {
"line": 861,
"column": 6
} | [
{
"pp": "case hbc\na0 a' : ONote\nN0 : a0.NF\nNa' : a'.NF\nm : ℕ\nd : ω ∣ a'.repr\ne0✝ : a0.repr ≠ 0\nh : a'.repr + ↑m < ω ^ a0.repr\nn : ℕ+\nNo : (a0.oadd n a').NF\nk : ℕ\nR' : Ordinal.{0} := (opowAux 0 a0 (a0.oadd n a' * ↑m) (k + 1) m).repr\nR : Ordinal.{0} := (opowAux 0 a0 (a0.oadd n a' * ↑m) k m).repr\nω0 :... | [
"case hbc.refine_2\na0 a' : ONote\nN0 : a0.NF\nNa' : a'.NF\nm : ℕ\nd : ω ∣ a'.repr\ne0✝ : a0.repr ≠ 0\nh : a'.repr + ↑m < ω ^ a0.repr\nn : ℕ+\nNo : (a0.oadd n a').NF\nk : ℕ\nR' : Ordinal.{0} := (opowAux 0 a0 (a0.oadd n a' * ↑m) (k + 1) m).repr\nR : Ordinal.{0} := (opowAux 0 a0 (a0.oadd n a' * ↑m) k m).repr\nω0 : Or... | have : _ < ω ^ (repr a0 + repr a0) := (No.below_of_lt ?_).repr_lt | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Tactic.ComputeAsymptotics.Multiseries.Basis | {
"line": 152,
"column": 2
} | {
"line": 156,
"column": 85
} | {
"line": 158,
"column": 0
} | [
{
"pp": "basis_hd : ℝ → ℝ\nbasis_tl : Basis\nh_basis : WellFormedBasis (basis_hd :: basis_tl)\n⊢ WellFormedBasis (basis_hd :: basis_tl ++ [Real.log ∘ (basis_hd :: basis_tl).getLast ⋯])",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"List.getLast",
"Eq.mpr",
"Real",
... | [] | apply h_basis.push
· simp [Real.tendsto_log_atTop.comp, h_basis.right]
· intro g hg
simpa [List.getLast_of_getLast?_eq_some hg] using Real.isLittleO_log_id_atTop.comp_tendsto <|
Real.tendsto_log_atTop.comp <| h_basis.tendsto_atTop <| List.mem_of_getLast? hg | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
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