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Mathlib.RingTheory.LocalIso | {
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Mathlib.RingTheory.LocalIso | {
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Mathlib.RingTheory.Invariant.Profinite | {
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Mathlib.RingTheory.LocalProperties.Injective | {
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Mathlib.RingTheory.LocalProperties.InjectiveDimension | {
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Mathlib.RingTheory.LocalProperties.InjectiveDimension | {
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Mathlib.RingTheory.LocalProperties.InjectiveDimension | {
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Mathlib.RingTheory.LocalProperties.InjectiveDimension | {
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Mathlib.RingTheory.LocalIso | {
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Mathlib.RingTheory.LocalIso | {
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Mathlib.RingTheory.LocalIso | {
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Mathlib.RingTheory.LocalProperties.InjectiveDimension | {
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Mathlib.RingTheory.LocalProperties.InjectiveDimension | {
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Mathlib.RingTheory.LocalProperties.InjectiveDimension | {
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Mathlib.RingTheory.LocalProperties.InjectiveDimension | {
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Mathlib.RingTheory.LocalProperties.InjectiveDimension | {
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Mathlib.RingTheory.LocalProperties.ProjectiveDimension | {
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Mathlib.RingTheory.LocalProperties.ProjectiveDimension | {
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Mathlib.RingTheory.LocalProperties.ProjectiveDimension | {
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Mathlib.RingTheory.LocalProperties.ProjectiveDimension | {
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Mathlib.RingTheory.LocalProperties.ProjectiveDimension | {
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Mathlib.RingTheory.LocalProperties.ProjectiveDimension | {
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Mathlib.RingTheory.LocalProperties.ProjectiveDimension | {
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Mathlib.RingTheory.LocalProperties.ProjectiveDimension | {
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Mathlib.RingTheory.Invariant.Profinite | {
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Mathlib.RingTheory.Invariant.Profinite | {
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} | [
{
"pp": "A : Type u_1\nB : Type u_2\ninst✝⁶ : CommRing A\ninst✝⁵ : CommRing B\ninst✝⁴ : Algebra A B\nG : Type u\ninst✝³ : Group G\ninst✝² : MulSemiringAction G B\ninst✝¹ : SMulCommClass G A B\ninst✝ : TopologicalSpace G\nQ : Ideal B\nN N' : OpenNormalSubgroup G\ne : N ≤ N'\nx : G ⧸ ↑N.toOpenSubgroup\nhx : x ∈ M... | [] | by | [anonymous] | by |
Mathlib.RingTheory.LocalProperties.Semilocal | {
"line": 54,
"column": 25
} | {
"line": 54,
"column": 27
} | {
"line": 55,
"column": 2
} | [
{
"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] → ... | [] | by | [anonymous] | by |
Mathlib.RingTheory.LocalProperties.ProjectiveDimension | {
"line": 157,
"column": 2
} | {
"line": 169,
"column": 32
} | {
"line": 171,
"column": 0
} | [
{
"pp": "R : Type u\ninst✝³ : CommRing R\ninst✝² : Small.{v, u} R\ninst✝¹ : IsNoetherianRing R\nM : ModuleCat R\ninst✝ : Module.Finite R ↑M\naux : ∀ (n : ℕ), projectiveDimension M ≤ ↑n ↔ ⨆ p, projectiveDimension (M.localizedModule p.asIdeal.primeCompl) ≤ ↑n\nN : WithBot ℕ∞\n⊢ projectiveDimension M ≤ N ↔ ⨆ p, pr... | [] | induction N with
| bot =>
simp only [le_bot_iff, projectiveDimension_eq_bot_iff, ModuleCat.isZero_iff_subsingleton,
iSup_eq_bot, ModuleCat.localizedModule, ← Equiv.subsingleton_congr (equivShrink _)]
refine ⟨fun h p ↦ LocalizedModule.instSubsingleton _, fun h ↦ ?_⟩
apply Module.subsingleton_of_local... | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | Lean.Parser.Tactic.induction |
Mathlib.RingTheory.LocalProperties.ProjectiveDimension | {
"line": 151,
"column": 89
} | {
"line": 151,
"column": 91
} | {
"line": 152,
"column": 2
} | [
{
"pp": "R : Type u\ninst✝³ : CommRing R\ninst✝² : Small.{v, u} R\ninst✝¹ : IsNoetherianRing R\nM : ModuleCat R\ninst✝ : Module.Finite R ↑M\n⊢ projectiveDimension M = ⨆ p, projectiveDimension (M.localizedModule p.asIdeal.primeCompl)",
"ppTerm": "?m.25",
"assigned": true,
"usedConstants": [
"Wi... | [] | by | [anonymous] | by |
Mathlib.RingTheory.LocalRing.Etale | {
"line": 71,
"column": 8
} | {
"line": 71,
"column": 40
} | {
"line": 71,
"column": 41
} | [
{
"pp": "case mp\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✝ : FormallyUnramified R S\nβ : S\nhβ : (ResidueField R)[(residue S) β] = ⊤\ns : S\na✝ : s ∈ ⊤\... | [
"case mp\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✝ : FormallyUnramified R S\nβ : S\nhβ : (aeval ((residue S) β)).range = ⊤\ns : S\na✝ : s ∈ ⊤\n⊢ s ∈ Subalge... | adjoin_singleton_eq_range_aeval, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.LocalRing.Etale | {
"line": 75,
"column": 68
} | {
"line": 75,
"column": 70
} | {
"line": 75,
"column": 71
} | [
{
"pp": "R : 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✝ : FormallyUnramified R S\nβ : S\nhβ : Function.Surjective ⇑(aeval ((residue S) β))\ns : S\na✝ : s ∈ ⊤\... | [] | by | [anonymous] | by |
Mathlib.RingTheory.LocalProperties.Semilocal | {
"line": 72,
"column": 12
} | {
"line": 72,
"column": 14
} | {
"line": 73,
"column": 2
} | [
{
"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] → ... | [] | by | [anonymous] | by |
Mathlib.RingTheory.Invariant.Profinite | {
"line": 140,
"column": 80
} | {
"line": 140,
"column": 82
} | {
"line": 141,
"column": 4
} | [
{
"pp": "A : Type u_1\nB : Type u_2\ninst✝¹³ : CommRing A\ninst✝¹² : CommRing B\ninst✝¹¹ : Algebra A B\nG : Type u\ninst✝¹⁰ : Group G\ninst✝⁹ : MulSemiringAction G B\ninst✝⁸ : SMulCommClass G A B\nP✝ : Ideal A\ninst✝⁷ : TopologicalSpace G\ninst✝⁶ : CompactSpace G\ninst✝⁵ : TotallyDisconnectedSpace G\ninst✝⁴ : I... | [] | by | [anonymous] | by |
Mathlib.RingTheory.LocalProperties.Semilocal | {
"line": 124,
"column": 33
} | {
"line": 124,
"column": 35
} | {
"line": 125,
"column": 4
} | [
{
"pp": "R : Type u_1\ninst✝⁵ : CommRing R\ninst✝⁴ : Finite (MaximalSpectrum R)\nRₚ : (P : Ideal R) → [P.IsMaximal] → Type u_2\ninst✝³ : (P : Ideal R) → [inst : P.IsMaximal] → CommRing (Rₚ P)\ninst✝² : (P : Ideal R) → [inst : P.IsMaximal] → Algebra R (Rₚ P)\ninst✝¹ : ∀ (P : Ideal R) [inst : P.IsMaximal], IsLoca... | [] | by | Lean.Elab.Tactic.evalWithAnnotateState | by |
Mathlib.RingTheory.Invariant.Profinite | {
"line": 148,
"column": 14
} | {
"line": 148,
"column": 16
} | {
"line": 148,
"column": 17
} | [
{
"pp": "A : Type u_1\nB : Type u_2\ninst✝¹³ : CommRing A\ninst✝¹² : CommRing B\ninst✝¹¹ : Algebra A B\nG : Type u\ninst✝¹⁰ : Group G\ninst✝⁹ : MulSemiringAction G B\ninst✝⁸ : SMulCommClass G A B\nP✝ : Ideal A\ninst✝⁷ : TopologicalSpace G\ninst✝⁶ : CompactSpace G\ninst✝⁵ : TotallyDisconnectedSpace G\ninst✝⁴ : I... | [] | by | [anonymous] | by |
Mathlib.RingTheory.LocalRing.Etale | {
"line": 87,
"column": 30
} | {
"line": 87,
"column": 32
} | {
"line": 88,
"column": 6
} | [
{
"pp": "R : 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✝ : FormallyUnramified R S\nβ : S\nhβ_gen : Function.Surjective ⇑(aeval β)\np : R[X]\n⊢ (aeval ((residue... | [] | by | [anonymous] | by |
Mathlib.RingTheory.LocalRing.Etale | {
"line": 64,
"column": 84
} | {
"line": 64,
"column": 86
} | {
"line": 65,
"column": 2
} | [
{
"pp": "R : 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✝ : FormallyUnramified R S\nβ : S\n⊢ (ResidueField R)[(residue S) β] = ⊤ ↔ R[β] = ⊤",
"ppTerm": "?m.... | [] | by | [anonymous] | by |
Mathlib.RingTheory.Invariant.Profinite | {
"line": 149,
"column": 18
} | {
"line": 149,
"column": 20
} | {
"line": 149,
"column": 21
} | [
{
"pp": "A : Type u_1\nB : Type u_2\ninst✝¹³ : CommRing A\ninst✝¹² : CommRing B\ninst✝¹¹ : Algebra A B\nG : Type u\ninst✝¹⁰ : Group G\ninst✝⁹ : MulSemiringAction G B\ninst✝⁸ : SMulCommClass G A B\nP✝ : Ideal A\ninst✝⁷ : TopologicalSpace G\ninst✝⁶ : CompactSpace G\ninst✝⁵ : TotallyDisconnectedSpace G\ninst✝⁴ : I... | [] | by | [anonymous] | by |
Mathlib.RingTheory.LocalRing.Etale | {
"line": 94,
"column": 41
} | {
"line": 94,
"column": 43
} | {
"line": 95,
"column": 2
} | [
{
"pp": "R : 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✝ : FormallyUnramified R S\n⊢ ∃ β, R[β] = ⊤",
"ppTerm": "?m.23",
"assigned": true,
"usedCons... | [] | by | [anonymous] | by |
Mathlib.RingTheory.Invariant.Profinite | {
"line": 154,
"column": 64
} | {
"line": 154,
"column": 66
} | {
"line": 155,
"column": 2
} | [
{
"pp": "A : Type u_1\nB : Type u_2\ninst✝¹³ : CommRing A\ninst✝¹² : CommRing B\ninst✝¹¹ : Algebra A B\nG : Type u\ninst✝¹⁰ : Group G\ninst✝⁹ : MulSemiringAction G B\ninst✝⁸ : SMulCommClass G A B\nP✝ : Ideal A\ninst✝⁷ : TopologicalSpace G\ninst✝⁶ : CompactSpace G\ninst✝⁵ : TotallyDisconnectedSpace G\ninst✝⁴ : I... | [] | by | [anonymous] | by |
Mathlib.RingTheory.LocalProperties.Semilocal | {
"line": 132,
"column": 57
} | {
"line": 132,
"column": 59
} | {
"line": 133,
"column": 4
} | [
{
"pp": "R : Type u_1\ninst✝⁵ : CommRing R\ninst✝⁴ : Finite (MaximalSpectrum R)\nRₚ : (P : Ideal R) → [P.IsMaximal] → Type u_2\ninst✝³ : (P : Ideal R) → [inst : P.IsMaximal] → CommRing (Rₚ P)\ninst✝² : (P : Ideal R) → [inst : P.IsMaximal] → Algebra R (Rₚ P)\ninst✝¹ : ∀ (P : Ideal R) [inst : P.IsMaximal], IsLoca... | [] | by | Lean.Elab.Tactic.evalWithAnnotateState | by |
Mathlib.RingTheory.LocalProperties.Semilocal | {
"line": 121,
"column": 30
} | {
"line": 121,
"column": 32
} | {
"line": 122,
"column": 2
} | [
{
"pp": "R : Type u_1\ninst✝⁵ : CommRing R\ninst✝⁴ : Finite (MaximalSpectrum R)\nRₚ : (P : Ideal R) → [P.IsMaximal] → Type u_2\ninst✝³ : (P : Ideal R) → [inst : P.IsMaximal] → CommRing (Rₚ P)\ninst✝² : (P : Ideal R) → [inst : P.IsMaximal] → Algebra R (Rₚ P)\ninst✝¹ : ∀ (P : Ideal R) [inst : P.IsMaximal], IsLoca... | [] | by | [anonymous] | by |
Mathlib.RingTheory.LocalRing.Etale | {
"line": 106,
"column": 58
} | {
"line": 106,
"column": 60
} | {
"line": 107,
"column": 2
} | [
{
"pp": "R : 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\n⊢ Module.finrank R S = Module.finrank (ResidueField R) (ResidueField S)",
"ppTerm": "... | [] | by | [anonymous] | by |
Mathlib.RingTheory.Invariant.Profinite | {
"line": 161,
"column": 66
} | {
"line": 161,
"column": 68
} | {
"line": 162,
"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 :... | [] | by | [anonymous] | by |
Mathlib.RingTheory.LocalRing.Etale | {
"line": 122,
"column": 78
} | {
"line": 122,
"column": 80
} | {
"line": 123,
"column": 2
} | [
{
"pp": "R : 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\nβ : S\nhadj : R[β] = ⊤\n⊢ map (residue R) (minpoly R β) = minpoly (ResidueField R) ((resi... | [] | by | [anonymous] | by |
Mathlib.RingTheory.LocalRing.Etale | {
"line": 143,
"column": 49
} | {
"line": 143,
"column": 51
} | {
"line": 144,
"column": 2
} | [
{
"pp": "R : 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\nβ : S\nhadj : R[β] = ⊤\n⊢ IsUnit ((aeval β) (derivative (minpoly R β)))",
"ppTerm": "... | [] | by | [anonymous] | by |
Mathlib.RingTheory.LaurentSeries | {
"line": 120,
"column": 92
} | {
"line": 120,
"column": 94
} | {
"line": 121,
"column": 4
} | [
{
"pp": "R✝ : Type u_1\nR : Type u_2\nV : Type u_3\ninst✝² : AddCommGroup V\ninst✝¹ : Semiring R\ninst✝ : Module R V\nk : ℕ\nf : V⸨X⸩\n⊢ BddBelow (Function.support fun n ↦ Ring.choose (n + ↑k) k • f.coeff (n + ↑k))",
"ppTerm": "?m.62",
"assigned": true,
"usedConstants": [
"Int.instAddCommGroup... | [] | by | [anonymous] | by |
Mathlib.RingTheory.LaurentSeries | {
"line": 124,
"column": 18
} | {
"line": 124,
"column": 20
} | {
"line": 125,
"column": 4
} | [
{
"pp": "R✝ : Type u_1\nR : Type u_2\nV : Type u_3\ninst✝² : AddCommGroup V\ninst✝¹ : Semiring R\ninst✝ : Module R V\nk : ℕ\nf g : V⸨X⸩\n⊢ ofSuppBddBelow (fun n ↦ Ring.choose (n + ↑k) k • (f + g).coeff (n + ↑k)) ⋯ =\n ofSuppBddBelow (fun n ↦ Ring.choose (n + ↑k) k • f.coeff (n + ↑k)) ⋯ +\n ofSuppBddBelo... | [] | by | [anonymous] | by |
Mathlib.RingTheory.LaurentSeries | {
"line": 127,
"column": 19
} | {
"line": 127,
"column": 21
} | {
"line": 128,
"column": 4
} | [
{
"pp": "R✝ : Type u_1\nR : Type u_2\nV : Type u_3\ninst✝² : AddCommGroup V\ninst✝¹ : Semiring R\ninst✝ : Module R V\nk : ℕ\nr : R\nf : V⸨X⸩\n⊢ ofSuppBddBelow (fun n ↦ Ring.choose (n + ↑k) k • (r • f).coeff (n + ↑k)) ⋯ =\n (RingHom.id R) r • ofSuppBddBelow (fun n ↦ Ring.choose (n + ↑k) k • f.coeff (n + ↑k)) ... | [] | by | [anonymous] | by |
Mathlib.RingTheory.LaurentSeries | {
"line": 139,
"column": 82
} | {
"line": 139,
"column": 84
} | {
"line": 140,
"column": 2
} | [
{
"pp": "R : Type u_1\ninst✝² : Semiring R\nV : Type u_2\ninst✝¹ : AddCommGroup V\ninst✝ : Module R V\n⊢ hasseDeriv R 0 = LinearMap.id",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"LaurentSeries.hasseDeriv",
"CharP.cast_eq_zero",
"LinearMap.id",
"MulOne.toOne",
... | [] | by | [anonymous] | by |
Mathlib.RingTheory.LocalRing.NonLocalRing | {
"line": 51,
"column": 42
} | {
"line": 51,
"column": 44
} | {
"line": 51,
"column": 45
} | [
{
"pp": "ι : Type u_1\ninst✝² : Nontrivial ι\nR : ι → Type u_2\ninst✝¹ : (i : ι) → Semiring (R i)\ninst✝ : ∀ (i : ι), Nontrivial (R i)\ni₁ i₂ : ι\nhi : i₁ ≠ i₂\nh : IsUnit fun i ↦ if i = i₁ then 0 else 1\n⊢ IsUnit 0",
"ppTerm": "?m.49",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Non... | [] | by | [anonymous] | by |
Mathlib.RingTheory.LocalRing.NonLocalRing | {
"line": 53,
"column": 42
} | {
"line": 53,
"column": 44
} | {
"line": 53,
"column": 45
} | [
{
"pp": "ι : Type u_1\ninst✝² : Nontrivial ι\nR : ι → Type u_2\ninst✝¹ : (i : ι) → Semiring (R i)\ninst✝ : ∀ (i : ι), Nontrivial (R i)\ni₁ i₂ : ι\nhi : i₁ ≠ i₂\nha : ¬IsUnit fun i ↦ if i = i₁ then 0 else 1\nh : IsUnit fun i ↦ if i = i₁ then 1 else 0\n⊢ IsUnit 0",
"ppTerm": "?m.83",
"assigned": true,
... | [] | by | [anonymous] | by |
Mathlib.RingTheory.LaurentSeries | {
"line": 144,
"column": 82
} | {
"line": 144,
"column": 84
} | {
"line": 145,
"column": 2
} | [
{
"pp": "R : Type u_1\ninst✝² : Semiring R\nV : Type u_2\ninst✝¹ : AddCommGroup V\ninst✝ : Module R V\nk : ℕ\nn : ℤ\nx : V\n⊢ (hasseDeriv R k) ((single (n + ↑k)) x) = (single n) (Ring.choose (n + ↑k) k • x)",
"ppTerm": "?m.35",
"assigned": true,
"usedConstants": [
"LaurentSeries.hasseDeriv",
... | [] | by | [anonymous] | by |
Mathlib.RingTheory.LaurentSeries | {
"line": 153,
"column": 76
} | {
"line": 153,
"column": 78
} | {
"line": 154,
"column": 2
} | [
{
"pp": "R : Type u_1\ninst✝² : Semiring R\nV : Type u_2\ninst✝¹ : AddCommGroup V\ninst✝ : Module R V\nk : ℕ\nn : ℤ\nx : V\n⊢ (hasseDeriv R k) ((single n) x) = (single (n - ↑k)) (Ring.choose n k • x)",
"ppTerm": "?m.31",
"assigned": true,
"usedConstants": [
"LaurentSeries.hasseDeriv",
"E... | [] | by | [anonymous] | by |
Mathlib.RingTheory.LocalRing.NonLocalRing | {
"line": 54,
"column": 35
} | {
"line": 54,
"column": 37
} | {
"line": 54,
"column": 38
} | [
{
"pp": "ι : Type u_1\ninst✝² : Nontrivial ι\nR : ι → Type u_2\ninst✝¹ : (i : ι) → Semiring (R i)\ninst✝ : ∀ (i : ι), Nontrivial (R i)\ni₁ i₂ : ι\nhi : i₁ ≠ i₂\nha : ¬IsUnit fun i ↦ if i = i₁ then 0 else 1\nhb : ¬IsUnit fun i ↦ if i = i₁ then 1 else 0\n⊢ ((fun i ↦ if i = i₁ then 0 else 1) + fun i ↦ if i = i₁ th... | [] | by | [anonymous] | by |
Mathlib.RingTheory.LocalRing.NonLocalRing | {
"line": 47,
"column": 79
} | {
"line": 47,
"column": 81
} | {
"line": 48,
"column": 2
} | [
{
"pp": "ι : Type u_1\ninst✝² : Nontrivial ι\nR : ι → Type u_2\ninst✝¹ : (i : ι) → Semiring (R i)\ninst✝ : ∀ (i : ι), Nontrivial (R i)\n⊢ ¬IsLocalRing ((i : ι) → R i)",
"ppTerm": "?m.5",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Ring... | [] | by | [anonymous] | by |
Mathlib.RingTheory.LocalRing.NonLocalRing | {
"line": 60,
"column": 40
} | {
"line": 60,
"column": 42
} | {
"line": 60,
"column": 43
} | [
{
"pp": "R₁ : Type u_1\nR₂ : Type u_2\ninst✝³ : Semiring R₁\ninst✝² : Semiring R₂\ninst✝¹ : Nontrivial R₁\ninst✝ : Nontrivial R₂\nh : IsUnit (1, 0)\n⊢ IsUnit 0",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"Eq.mpr",
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Mathlib.RingTheory.LocalRing.NonLocalRing | {
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} | {
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} | {
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Mathlib.RingTheory.LocalRing.MaximalIdeal.Square | {
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Mathlib.RingTheory.LocalRing.NonLocalRing | {
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Mathlib.RingTheory.LocalRing.NonLocalRing | {
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} | {
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} | {
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{
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Mathlib.RingTheory.LocalRing.MaximalIdeal.Square | {
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} | {
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} | {
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} | [
{
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Mathlib.RingTheory.LocalRing.NonLocalRing | {
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Mathlib.RingTheory.LaurentSeries | {
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{
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"... | [] | by | [anonymous] | by |
Mathlib.RingTheory.LaurentSeries | {
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} | {
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... | [] | by | [anonymous] | by |
Mathlib.RingTheory.LaurentSeries | {
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} | {
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} | {
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{
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Mathlib.RingTheory.LocalRing.NonLocalRing | {
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} | {
"line": 88,
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} | {
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{
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Mathlib.RingTheory.LaurentSeries | {
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} | {
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{
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... | [] | by | [anonymous] | by |
Mathlib.RingTheory.LaurentSeries | {
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} | {
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} | {
"line": 191,
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{
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Mathlib.RingTheory.LaurentSeries | {
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{
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Mathlib.RingTheory.LaurentSeries | {
"line": 205,
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} | {
"line": 205,
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} | {
"line": 206,
"column": 2
} | [
{
"pp": "R : Type u_1\ninst✝ : Semiring R\nx : R⟦X⟧\nn : ℕ\n⊢ ((ofPowerSeries ℤ R) x).coeff ↑n = (PowerSeries.coeff n) x",
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Mathlib.RingTheory.LaurentSeries | {
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} | {
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} | {
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{
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Mathlib.RingTheory.LaurentSeries | {
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} | {
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} | {
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{
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Mathlib.RingTheory.LaurentSeries | {
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} | {
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} | {
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{
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Mathlib.RingTheory.Invariant.Profinite | {
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{
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Mathlib.RingTheory.LaurentSeries | {
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} | {
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} | {
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} | [
{
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Mathlib.RingTheory.LaurentSeries | {
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} | {
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} | {
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{
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Mathlib.RingTheory.LaurentSeries | {
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} | {
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} | {
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{
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Mathlib.RingTheory.LaurentSeries | {
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} | {
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} | {
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{
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Mathlib.RingTheory.LaurentSeries | {
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} | {
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} | {
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Mathlib.RingTheory.LaurentSeries | {
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} | {
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{
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Mathlib.RingTheory.LaurentSeries | {
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} | {
"line": 352,
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} | {
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Mathlib.RingTheory.LaurentSeries | {
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} | {
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} | {
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} | [
{
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Mathlib.RingTheory.LaurentSeries | {
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} | {
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Mathlib.RingTheory.LaurentSeries | {
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} | {
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} | {
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{
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Mathlib.RingTheory.LaurentSeries | {
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} | {
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} | {
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} | [
{
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Mathlib.RingTheory.LaurentSeries | {
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} | {
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} | {
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Mathlib.RingTheory.LaurentSeries | {
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{
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Mathlib.RingTheory.LaurentSeries | {
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} | {
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} | {
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{
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Mathlib.RingTheory.LaurentSeries | {
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} | {
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Mathlib.RingTheory.LocalRing.Subring | {
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Mathlib.RingTheory.LaurentSeries | {
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} | [
{
"pp": "K : Type u_2\ninst✝ : Field K\nP : K[X]\nhP : ¬P = 0\n⊢ Ideal.span {P} ≠ 0 ∧ Ideal.span {Polynomial.X} ≠ 0",
"ppTerm": "?m.78",
"assigned": true,
"usedConstants": [
"False",
"Semiring.toModule",
"eq_false",
"congrArg",
"Field.instIsLocalRing",
"_private.M... | [] | by | Lean.Elab.Tactic.evalWithAnnotateState | by |
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