module stringlengths 16 90 | startPos dict | endPos dict | nextStartPos dict | goals listlengths 0 96 | goalsAfter listlengths 0 96 | ppTac stringlengths 1 14.5k | elaborator stringclasses 375
values | kind stringclasses 379
values |
|---|---|---|---|---|---|---|---|---|
Mathlib.RingTheory.Bialgebra.MonoidAlgebra | {
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} | {
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{
"pp": "R : Type u_1\nS : Type u_2\nA : Type u_3\nB : Type u_4\nG : Type u_5\nH : Type u_6\nI : Type u_7\nM : Type u_8\nN : Type u_9\nO : Type u_10\ninst✝⁷ : CommSemiring R\ninst✝⁶ : CommSemiring S\ninst✝⁵ : Semiring A\ninst✝⁴ : Semiring B\ninst✝³ : Bialgebra R A\ninst✝² : Bialgebra R B\ninst✝¹ : AddMonoid M\n... | [] | by | [anonymous] | by |
Mathlib.RepresentationTheory.Homological.GroupHomology.Functoriality | {
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} | {
"line": 811,
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} | {
"line": 812,
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{
"pp": "k G H : Type u\ninst✝² : CommRing k\ninst✝¹ : Group G\ninst✝ : Group H\nA : Rep k G\nB : Rep k H\nf : G →* H\nφ : A ⟶ res f B\n⊢ H2π A ≫ map f φ 2 = mapCycles₂ f φ ≫ H2π B",
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"assigned": true,
"usedConstants": [
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"Submodule",
... | [] | by | [anonymous] | by |
Mathlib.RepresentationTheory.Homological.GroupHomology.Functoriality | {
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"line": 836,
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{
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"usedConstants": [
"ChainComplex",
... | [] | by | [anonymous] | by |
Mathlib.RingTheory.Bialgebra.MonoidAlgebra | {
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} | {
"line": 367,
"column": 70
} | {
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{
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Mathlib.RepresentationTheory.Homological.GroupHomology.Functoriality | {
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{
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"a... | [] | by | [anonymous] | by |
Mathlib.RingTheory.Bialgebra.MonoidAlgebra | {
"line": 374,
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} | {
"line": 374,
"column": 78
} | {
"line": 375,
"column": 2
} | [
{
"pp": "R : Type u_1\nA : Type u_3\nM : Type u_8\ninst✝³ : CommSemiring R\ninst✝² : Semiring A\ninst✝¹ : Bialgebra R A\ninst✝ : AddMonoid M\nm : M\na : A\n⊢ (toMultiplicativeBialgEquiv R A M) (single m a) = MonoidAlgebra.single (Multiplicative.ofAdd m) a",
"ppTerm": "?m.21",
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"used... | [] | by | [anonymous] | by |
Mathlib.RepresentationTheory.Homological.GroupHomology.Functoriality | {
"line": 843,
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} | {
"line": 843,
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} | {
"line": 843,
"column": 21
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{
"pp": "k G H : Type u\ninst✝² : CommRing k\ninst✝¹ : Group G\ninst✝ : Group H\nA : Rep k G\nB : Rep k H\nf : G →* H\nφ : A ⟶ res f B\nn✝ n : ℕ\nx✝¹ x✝ : Rep k G\n⊢ (functor k G n).map 0 = 0",
"ppTerm": "?m.32",
"assigned": true,
"usedConstants": [
"ChainComplex",
"HomologicalComplex.in... | [] | by | [anonymous] | by |
Mathlib.RingTheory.Bialgebra.MonoidAlgebra | {
"line": 386,
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} | {
"line": 386,
"column": 20
} | {
"line": 386,
"column": 21
} | [
{
"pp": "R : Type u_1\nS : Type u_2\nA : Type u_3\nB : Type u_4\nG : Type u_5\nH : Type u_6\nI : Type u_7\nM : Type u_8\nN : Type u_9\nO : Type u_10\ninst✝⁴ : CommSemiring R\ninst✝³ : CommSemiring S\ninst✝² : CommSemiring A\ninst✝¹ : Algebra R A\ninst✝ : AddMonoid M\nf g : Multiplicative M →* A\n⊢ ((lift R A M)... | [] | by | [anonymous] | by |
Mathlib.RingTheory.Bialgebra.MonoidAlgebra | {
"line": 396,
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} | {
"line": 396,
"column": 11
} | {
"line": 396,
"column": 12
} | [
{
"pp": "R : Type u_1\nS : Type u_2\nA : Type u_3\nB : Type u_4\nG : Type u_5\nH : Type u_6\nI : Type u_7\nM : Type u_8\nN : Type u_9\nO : Type u_10\ninst✝³ : CommSemiring R\ninst✝² : CommSemiring S\ninst✝¹ : AddCommMonoid M\ninst✝ : AddCommMonoid N\nf : R →+* S\n⊢ (mapRingHom M f).comp (algebraMap R R[M]) = (a... | [] | by | [anonymous] | by |
Mathlib.RingTheory.Bialgebra.MonoidAlgebra | {
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} | {
"line": 396,
"column": 26
} | {
"line": 396,
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{
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Mathlib.RepresentationTheory.Homological.GroupHomology.Functoriality | {
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} | {
"line": 855,
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} | {
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{
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"ppTerm": "?m.56",
... | [] | by | [anonymous] | by |
Mathlib.RingTheory.Bialgebra.MonoidAlgebra | {
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} | {
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} | {
"line": 397,
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} | [
{
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"ppT... | [] | by | [anonymous] | by |
Mathlib.RepresentationTheory.Homological.GroupHomology.Functoriality | {
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} | {
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{
"pp": "k G H : Type u\ninst✝³ : CommRing k\ninst✝² : Group G\ninst✝¹ : Group H\nA : Rep k G\nB : Rep k H\nf : G →* H\nφ✝ : A ⟶ res f B\nn✝ : ℕ\nS : Subgroup G\ninst✝ : S.Normal\nn : ℕ\nX Y : Rep k G\nφ : X ⟶ Y\n⊢ (functor k G n).map φ ≫ map (QuotientGroup.mk' S) (Y.toCoinvariantsMkQ S) n =\n map (QuotientG... | [] | by | [anonymous] | by |
Mathlib.RingTheory.Bialgebra.MonoidAlgebra | {
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} | {
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} | {
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} | [
{
"pp": "R : Type u_1\nS : Type u_2\nM : Type u_8\ninst✝² : CommSemiring R\ninst✝¹ : CommSemiring S\ninst✝ : AddCommMonoid M\nf : R →+* S\n⊢ (counitAlgHom S S[M]).comp (mapRingHom M f) = f.comp (counitAlgHom R R[M]).toRingHom",
"ppTerm": "?m.45",
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"NonAssocS... | [] | by | [anonymous] | by |
Mathlib.RingTheory.Bialgebra.MonoidAlgebra | {
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} | {
"line": 424,
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} | {
"line": 424,
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} | [
{
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Mathlib.RingTheory.Bialgebra.MonoidAlgebra | {
"line": 440,
"column": 68
} | {
"line": 440,
"column": 70
} | {
"line": 440,
"column": 71
} | [
{
"pp": "R : Type u_11\ninst✝² : CommSemiring R\nA : Type u_12\ninst✝¹ : Semiring A\ninst✝ : Bialgebra R A\nn : ℤ\n⊢ comul (T n) = T n ⊗ₜ[R] T n",
"ppTerm": "?m.25",
"assigned": true,
"usedConstants": [
"LaurentPolynomial.T",
"IsGroupLikeElem.comul_eq_tmul_self",
"NonAssocSemiring.... | [] | by | [anonymous] | by |
Mathlib.RingTheory.Bialgebra.MonoidAlgebra | {
"line": 444,
"column": 54
} | {
"line": 444,
"column": 56
} | {
"line": 445,
"column": 2
} | [
{
"pp": "R : Type u_11\ninst✝² : CommSemiring R\nA : Type u_12\ninst✝¹ : Semiring A\ninst✝ : Bialgebra R A\nn : ℤ\n⊢ counit (T n) = 1",
"ppTerm": "?m.15",
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"usedConstants": [
"LaurentPolynomial.T",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Coalgebra.toCoalgebraSt... | [] | by | [anonymous] | by |
Mathlib.RingTheory.DedekindDomain.SelmerGroup | {
"line": 98,
"column": 60
} | {
"line": 98,
"column": 62
} | {
"line": 99,
"column": 2
} | [
{
"pp": "R : Type u\ninst✝⁴ : CommRing R\ninst✝³ : IsDedekindDomain R\nK : Type v\ninst✝² : Field K\ninst✝¹ : Algebra R K\ninst✝ : IsFractionRing R K\nv : HeightOneSpectrum R\nx : Kˣ\n⊢ ↑(v.valuationOfNeZeroToFun x) = (valuation K v) ↑x",
"ppTerm": "?m.24",
"assigned": true,
"usedConstants": [
... | [] | by | [anonymous] | by |
Mathlib.RingTheory.DedekindDomain.SelmerGroup | {
"line": 111,
"column": 14
} | {
"line": 111,
"column": 16
} | {
"line": 111,
"column": 17
} | [
{
"pp": "R : Type u\ninst✝⁴ : CommRing R\ninst✝³ : IsDedekindDomain R\nK : Type v\ninst✝² : Field K\ninst✝¹ : Algebra R K\ninst✝ : IsFractionRing R K\nv : HeightOneSpectrum R\n⊢ v.valuationOfNeZeroToFun 1 = 1",
"ppTerm": "?m.27",
"assigned": true,
"usedConstants": [
"Units.val",
"Eq.mpr"... | [] | by | [anonymous] | by |
Mathlib.RingTheory.DedekindDomain.SelmerGroup | {
"line": 112,
"column": 18
} | {
"line": 112,
"column": 20
} | {
"line": 113,
"column": 4
} | [
{
"pp": "R : Type u\ninst✝⁴ : CommRing R\ninst✝³ : IsDedekindDomain R\nK : Type v\ninst✝² : Field K\ninst✝¹ : Algebra R K\ninst✝ : IsFractionRing R K\nv : HeightOneSpectrum R\nx✝¹ x✝ : Kˣ\n⊢ v.valuationOfNeZeroToFun (x✝¹ * x✝) = v.valuationOfNeZeroToFun x✝¹ * v.valuationOfNeZeroToFun x✝",
"ppTerm": "?m.50",... | [] | by | [anonymous] | by |
Mathlib.RingTheory.DedekindDomain.SelmerGroup | {
"line": 122,
"column": 71
} | {
"line": 122,
"column": 73
} | {
"line": 123,
"column": 2
} | [
{
"pp": "R : Type u\ninst✝⁴ : CommRing R\ninst✝³ : IsDedekindDomain R\nK : Type v\ninst✝² : Field K\ninst✝¹ : Algebra R K\ninst✝ : IsFractionRing R K\nv : HeightOneSpectrum R\nx : Rˣ\n⊢ v.valuationOfNeZero ((Units.map ↑(algebraMap R K)) x) = 1",
"ppTerm": "?m.30",
"assigned": true,
"usedConstants": ... | [] | by | [anonymous] | by |
Mathlib.RingTheory.DedekindDomain.SelmerGroup | {
"line": 122,
"column": 71
} | {
"line": 132,
"column": 33
} | {
"line": 134,
"column": 0
} | [
{
"pp": "R : Type u\ninst✝⁴ : CommRing R\ninst✝³ : IsDedekindDomain R\nK : Type v\ninst✝² : Field K\ninst✝¹ : Algebra R K\ninst✝ : IsFractionRing R K\nv : HeightOneSpectrum R\nx : Rˣ\n⊢ v.valuationOfNeZero ((Units.map ↑(algebraMap R K)) x) = 1",
"ppTerm": "?m.30",
"assigned": true,
"usedConstants": ... | [] | by
rw [← WithZero.coe_inj, valuationOfNeZero_eq, Units.coe_map, eq_iff_le_not_lt]
constructor
· exact v.valuation_le_one x
· obtain ⟨x, _, hx, _⟩ := x
change ¬v.valuation K (algebraMap R K x) < 1
apply_fun v.intValuation at hx
rw [map_one, map_mul] at hx
rw [not_lt, ← hx, ← mul_one <| v.valuatio... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.DedekindDomain.SelmerGroup | {
"line": 145,
"column": 34
} | {
"line": 145,
"column": 36
} | {
"line": 145,
"column": 37
} | [
{
"pp": "R : Type u\ninst✝⁴ : CommRing R\ninst✝³ : IsDedekindDomain R\nK : Type v\ninst✝² : Field K\ninst✝¹ : Algebra R K\ninst✝ : IsFractionRing R K\nv : HeightOneSpectrum R\nn : ℕ\nx : Kˣ\n⊢ (fun x ↦ x • ↑n) (v.valuationOfNeZero x) = Multiplicative.toAdd (v.valuationOfNeZero ((powMonoidHom n) x))",
"ppTer... | [] | by | [anonymous] | by |
Mathlib.RingTheory.DedekindDomain.SelmerGroup | {
"line": 142,
"column": 7
} | {
"line": 142,
"column": 9
} | {
"line": 143,
"column": 8
} | [
{
"pp": "R : Type u\ninst✝⁴ : CommRing R\ninst✝³ : IsDedekindDomain R\nK : Type v\ninst✝² : Field K\ninst✝¹ : Algebra R K\ninst✝ : IsFractionRing R K\nv : HeightOneSpectrum R\nn : ℕ\n⊢ (powMonoidHom n).range ≤ Subgroup.comap v.valuationOfNeZero (AddSubgroup.toSubgroup (AddSubgroup.zmultiples ↑n))",
"ppTerm"... | [] | by | [anonymous] | by |
Mathlib.RingTheory.DedekindDomain.SelmerGroup | {
"line": 149,
"column": 84
} | {
"line": 149,
"column": 86
} | {
"line": 151,
"column": 2
} | [
{
"pp": "R : Type u\ninst✝⁴ : CommRing R\ninst✝³ : IsDedekindDomain R\nK : Type v\ninst✝² : Field K\ninst✝¹ : Algebra R K\ninst✝ : IsFractionRing R K\nv : HeightOneSpectrum R\nn : ℕ\nx : Rˣ\n⊢ (v.valuationOfNeZeroMod n) ↑((Units.map ↑(algebraMap R K)) x) = 1",
"ppTerm": "?m.43",
"assigned": true,
"u... | [] | by | [anonymous] | by |
Mathlib.RingTheory.Ideal.AssociatedPrime.Localization | {
"line": 58,
"column": 82
} | {
"line": 58,
"column": 84
} | {
"line": 59,
"column": 8
} | [
{
"pp": "R : Type u_1\ninst✝⁹ : CommRing R\nS : Submonoid R\nR' : Type u_2\ninst✝⁸ : CommRing R'\ninst✝⁷ : Algebra R R'\nhSR' : IsLocalization S R'\nM : Type u_3\nM' : Type u_4\ninst✝⁶ : AddCommGroup M\ninst✝⁵ : Module R M\ninst✝⁴ : AddCommGroup M'\ninst✝³ : Module R M'\nf : M →ₗ[R] M'\ninst✝² : IsLocalizedModu... | [] | by | Lean.Elab.Tactic.evalWithAnnotateState | by |
Mathlib.RingTheory.DedekindDomain.SelmerGroup | {
"line": 166,
"column": 25
} | {
"line": 166,
"column": 27
} | {
"line": 166,
"column": 28
} | [
{
"pp": "R : Type u\ninst✝⁴ : CommRing R\ninst✝³ : IsDedekindDomain R\nK : Type v\ninst✝² : Field K\ninst✝¹ : Algebra R K\ninst✝ : IsFractionRing R K\nv✝ : HeightOneSpectrum R\nS S' : Set (HeightOneSpectrum R)\nn : ℕ\na✝ b✝ : Kˣ ⧸ (powMonoidHom n).range\nhx : a✝ ∈ {x | ∀ v ∉ S, (v.valuationOfNeZeroMod n) x = 1}... | [] | by | [anonymous] | by |
Mathlib.RingTheory.DedekindDomain.SelmerGroup | {
"line": 165,
"column": 18
} | {
"line": 165,
"column": 20
} | {
"line": 165,
"column": 21
} | [
{
"pp": "R : Type u\ninst✝⁴ : CommRing R\ninst✝³ : IsDedekindDomain R\nK : Type v\ninst✝² : Field K\ninst✝¹ : Algebra R K\ninst✝ : IsFractionRing R K\nv : HeightOneSpectrum R\nS S' : Set (HeightOneSpectrum R)\nn : ℕ\nx✝¹ : HeightOneSpectrum R\nx✝ : x✝¹ ∉ S\n⊢ (x✝¹.valuationOfNeZeroMod n) 1 = 1",
"ppTerm": "... | [] | by | [anonymous] | by |
Mathlib.RingTheory.DedekindDomain.SelmerGroup | {
"line": 167,
"column": 22
} | {
"line": 167,
"column": 24
} | {
"line": 167,
"column": 25
} | [
{
"pp": "R : Type u\ninst✝⁴ : CommRing R\ninst✝³ : IsDedekindDomain R\nK : Type v\ninst✝² : Field K\ninst✝¹ : Algebra R K\ninst✝ : IsFractionRing R K\nv✝ : HeightOneSpectrum R\nS S' : Set (HeightOneSpectrum R)\nn : ℕ\nx✝ : Kˣ ⧸ (powMonoidHom n).range\nhx : x✝ ∈ {x | ∀ v ∉ S, (v.valuationOfNeZeroMod n) x = 1}\nv... | [] | by | [anonymous] | by |
Mathlib.RingTheory.Regular.IsSMulRegular | {
"line": 80,
"column": 28
} | {
"line": 80,
"column": 30
} | {
"line": 81,
"column": 6
} | [
{
"pp": "R : Type u_1\nM : Type u_3\ninst✝² : Ring R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nN : Submodule R M\nr : R\n⊢ (∀ (a : M) (b : a ∈ N), r • ⟨a, b⟩ = 0 → ⟨a, b⟩ = 0) ↔ ∀ x ∈ N, r • x = 0 → x = 0",
"ppTerm": "?m.40",
"assigned": true,
"usedConstants": [
"SMulMemClass.smul_mem",
... | [] | by | [anonymous] | by |
Mathlib.RingTheory.DedekindDomain.SelmerGroup | {
"line": 180,
"column": 18
} | {
"line": 180,
"column": 20
} | {
"line": 180,
"column": 21
} | [
{
"pp": "R : Type u\ninst✝⁴ : CommRing R\ninst✝³ : IsDedekindDomain R\nK : Type v\ninst✝² : Field K\ninst✝¹ : Algebra R K\ninst✝ : IsFractionRing R K\nv : HeightOneSpectrum R\nS S' : Set (HeightOneSpectrum R)\nn : ℕ\nx y : ↥selmerGroup\n⊢ (fun v ↦ ((↑v).valuationOfNeZeroMod n) ↑(x * y)) =\n (fun v ↦ ((↑v).va... | [] | by | [anonymous] | by |
Mathlib.RingTheory.Regular.IsSMulRegular | {
"line": 86,
"column": 37
} | {
"line": 86,
"column": 39
} | {
"line": 87,
"column": 6
} | [
{
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Mathlib.RingTheory.Regular.IsSMulRegular | {
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Mathlib.RingTheory.Regular.IsSMulRegular | {
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Mathlib.RingTheory.Ideal.AssociatedPrime.Localization | {
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Mathlib.RingTheory.Regular.IsSMulRegular | {
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Mathlib.RingTheory.DedekindDomain.SelmerGroup | {
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Mathlib.RingTheory.Regular.IsSMulRegular | {
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Mathlib.RingTheory.Ideal.AssociatedPrime.Localization | {
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Mathlib.RingTheory.DedekindDomain.SelmerGroup | {
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Mathlib.RingTheory.Ideal.AssociatedPrime.Localization | {
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Mathlib.RingTheory.Ideal.AssociatedPrime.Localization | {
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Mathlib.RingTheory.Regular.IsSMulRegular | {
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Mathlib.RingTheory.Ideal.AssociatedPrime.Localization | {
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Mathlib.RingTheory.Ideal.AssociatedPrime.Localization | {
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Mathlib.RingTheory.DedekindDomain.SelmerGroup | {
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Mathlib.RingTheory.DedekindDomain.SelmerGroup | {
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Mathlib.RingTheory.DedekindDomain.SelmerGroup | {
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Mathlib.RingTheory.DedekindDomain.SelmerGroup | {
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Mathlib.RingTheory.Regular.LinearMap | {
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Mathlib.RingTheory.Regular.LinearMap | {
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Mathlib.RingTheory.Ideal.AssociatedPrime.Localization | {
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Mathlib.RingTheory.Regular.LinearMap | {
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Mathlib.RingTheory.Ideal.AssociatedPrime.Localization | {
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Mathlib.RingTheory.Ideal.AssociatedPrime.Localization | {
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Mathlib.RingTheory.Ideal.AssociatedPrime.Localization | {
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Mathlib.RingTheory.Regular.LinearMap | {
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Mathlib.RingTheory.Ideal.AssociatedPrime.Localization | {
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Mathlib.RingTheory.Ideal.AssociatedPrime.Localization | {
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Mathlib.RingTheory.DividedPowers.Basic | {
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Mathlib.RingTheory.DividedPowers.Basic | {
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Mathlib.RingTheory.DividedPowers.Basic | {
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Mathlib.RingTheory.DividedPowers.Basic | {
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Mathlib.RingTheory.Regular.LinearMap | {
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"R : Type u_1\nM : Type u_2\nN : Type u_3\ninst✝⁷ : CommRing R\ninst✝⁶ : AddCommGroup M\ninst✝⁵ : AddCommGroup N\ninst✝⁴ : Module R M\ninst✝³ : Module R N\ninst✝² : IsNoetherianRing R\ninst✝¹ : Module.Finite R M\ninst✝ : Module.Finite R N\nhom0 : Subsingleton (N →ₗ[R] M)\nh✝ : Nontrivial M\nh : ∀ r ∈ annihilator R ... | simp only [AddHom.toFun_eq_coe, coe_toAddHom, RingHom.id_apply] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.RingTheory.DividedPowers.Basic | {
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} | {
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} | {
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{
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"usedConstan... | [] | by | [anonymous] | by |
Mathlib.RingTheory.Regular.LinearMap | {
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} | {
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} | {
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{
"pp": "R : Type u_1\nM : Type u_2\nN : Type u_3\ninst✝⁷ : CommRing R\ninst✝⁶ : AddCommGroup M\ninst✝⁵ : AddCommGroup N\ninst✝⁴ : Module R M\ninst✝³ : Module R N\ninst✝² : IsNoetherianRing R\ninst✝¹ : Module.Finite R M\ninst✝ : Module.Finite R N\nhom0 : Subsingleton (N →ₗ[R] M)\nh✝ : Nontrivial M\nh : ∀ r ∈ an... | [] | by | [anonymous] | by |
Mathlib.RingTheory.DividedPowers.Basic | {
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} | {
"line": 118,
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} | {
"line": 119,
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} | [
{
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Mathlib.RingTheory.Depth.Rees | {
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} | {
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} | {
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{
"pp": "R : Type u\ninst✝² : CommRing R\nM : Type u_1\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nI : Ideal R\nr : R\nhr : r ∈ I\nhI : I • ⊤ ≠ ⊤\n⊢ I • ⊤ ≠ ⊤",
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"Iff.mpr",
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... | [] | by | [anonymous] | by |
Mathlib.RingTheory.DividedPowers.Basic | {
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} | {
"line": 124,
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} | {
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{
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"ppTerm": "?m.176",
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"usedConstants": [
"Ideal.mem_bo... | [] | by | [anonymous] | by |
Mathlib.RingTheory.Regular.LinearMap | {
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} | {
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} | {
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{
"pp": "R : Type u_1\nM : Type u_2\nN : Type u_3\ninst✝⁷ : CommRing R\ninst✝⁶ : AddCommGroup M\ninst✝⁵ : AddCommGroup N\ninst✝⁴ : Module R M\ninst✝³ : Module R N\ninst✝² : IsNoetherianRing R\ninst✝¹ : Module.Finite R M\ninst✝ : Module.Finite R N\nhom0 : Subsingleton (N →ₗ[R] M)\nh✝ : Nontrivial M\nh : ∀ r ∈ an... | [] | by | [anonymous] | by |
Mathlib.RingTheory.DividedPowers.Basic | {
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} | {
"line": 131,
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} | {
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{
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"ppTerm": "?m.213",
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"Ideal... | [] | by | [anonymous] | by |
Mathlib.RingTheory.DividedPowers.Basic | {
"line": 140,
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} | {
"line": 140,
"column": 42
} | {
"line": 141,
"column": 2
} | [
{
"pp": "A : Type u_1\ninst✝¹ : CommSemiring A\ninst✝ : DecidableEq A\nn : ℕ\na : A\n⊢ (dividedPowersBot A).dpow n a = if a = 0 ∧ n = 0 then 1 else 0",
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Mathlib.RingTheory.DividedPowers.Basic | {
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} | {
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} | {
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{
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Mathlib.RingTheory.DividedPowers.Basic | {
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} | {
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} | {
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{
"pp": "A : Type u_1\ninst✝ : CommSemiring A\nI : Ideal A\nhI hI' : DividedPowers I\nh : (fun h ↦ h.dpow) hI = (fun h ↦ h.dpow) hI'\n⊢ hI = hI'",
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Mathlib.RingTheory.DividedPowers.Basic | {
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} | {
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} | {
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} | [
{
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"con... | [] | by | [anonymous] | by |
Mathlib.RingTheory.DividedPowers.Basic | {
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} | {
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"column": 77
} | {
"line": 189,
"column": 2
} | [
{
"pp": "A : Type u_1\ninst✝ : CommSemiring A\na b : A\ndp : ℕ → A → A\ndp_add : ∀ (n : ℕ), dp n (a + b) = ∑ k ∈ antidiagonal n, dp k.1 a * dp k.2 b\n⊢ (PowerSeries.mk fun n ↦ dp n (a + b)) = (PowerSeries.mk fun n ↦ dp n a) * PowerSeries.mk fun n ↦ dp n b",
"ppTerm": "?m.38",
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"used... | [] | by | [anonymous] | by |
Mathlib.RingTheory.DividedPowers.Basic | {
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} | {
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"column": 49
} | {
"line": 203,
"column": 2
} | [
{
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... | [] | by | [anonymous] | by |
Mathlib.RingTheory.DividedPowers.Basic | {
"line": 206,
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} | {
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} | {
"line": 207,
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} | [
{
"pp": "A : Type u_1\ninst✝ : CommSemiring A\nI : Ideal A\na b : A\nhI : DividedPowers I\nn : ℕ\nha : a ∈ I\n⊢ hI.dpow n (a * b) = hI.dpow n a * b ^ n",
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"Eq.mpr",
"HMul.hMul",
"CommSemiring.toNonUnitalCommSemiring",
"co... | [] | by | [anonymous] | by |
Mathlib.RingTheory.DividedPowers.Basic | {
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} | {
"line": 210,
"column": 49
} | {
"line": 211,
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} | [
{
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"usedConstants": [
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"instHSMul",
"instSMulOfMul",
"HMul.hMul",
"congrA... | [] | by | [anonymous] | by |
Mathlib.RingTheory.Depth.Rees | {
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} | {
"line": 87,
"column": 77
} | {
"line": 87,
"column": 78
} | [
{
"pp": "R : Type u\ninst✝⁴ : CommRing R\ninst✝³ : Small.{v, u} R\ninst✝² : IsNoetherianRing R\nI : Ideal R\nN : ModuleCat R\ninst✝¹ : Module.Finite R ↑N\nh_supp : (Module.annihilator R ↑N).radical = I.radical\nn : ℕ\nih :\n ∀ (M : ModuleCat R) [Module.Finite R ↑M],\n I • ⊤ < ⊤ →\n (∀ i < n, Subsinglet... | [] | by | [anonymous] | by |
Mathlib.RingTheory.DividedPowers.Basic | {
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} | {
"line": 214,
"column": 41
} | {
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} | [
{
"pp": "A : Type u_1\ninst✝ : CommSemiring A\nI : Ideal A\na : A\nhI : DividedPowers I\nn : ℕ\nha : a ∈ I\n⊢ ↑n ! * hI.dpow n a = a ^ n",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"MulOne.toOne",
"Semigroup.t... | [] | by | [anonymous] | by |
Mathlib.RingTheory.DividedPowers.Basic | {
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"column": 65
} | {
"line": 223,
"column": 67
} | {
"line": 224,
"column": 2
} | [
{
"pp": "A : Type u_1\ninst✝ : CommSemiring A\nI : Ideal A\nhI : DividedPowers I\nn : ℕ\nhn : n ≠ 0\n⊢ hI.dpow n 0 = 0",
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"usedConstants": [
"Eq.mpr",
"HMul.hMul",
"MulZeroClass.toMul",
"congrArg",
"CommSemiring.toSemiring",
"Mu... | [] | by | [anonymous] | by |
Mathlib.RingTheory.DividedPowers.Basic | {
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"column": 77
} | {
"line": 233,
"column": 79
} | {
"line": 234,
"column": 4
} | [
{
"pp": "A : Type u_1\ninst✝ : CommSemiring A\nI : Ideal A\na : A\nn : ℕ\nhn : n ≠ 0\nhnI : ∀ {y : A}, y ∈ I → n • y = 0\nhI : DividedPowers I\nha : a ∈ I\n⊢ ↑n ! * hI.dpow n a = n • ↑(n - 1)! * hI.dpow n a",
"ppTerm": "?m.56",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NonAssocSemi... | [] | by | Lean.Elab.Tactic.evalWithAnnotateState | by |
Mathlib.RingTheory.DividedPowers.Basic | {
"line": 232,
"column": 55
} | {
"line": 232,
"column": 57
} | {
"line": 233,
"column": 2
} | [
{
"pp": "A : Type u_1\ninst✝ : CommSemiring A\nI : Ideal A\na : A\nn : ℕ\nhn : n ≠ 0\nhnI : ∀ {y : A}, y ∈ I → n • y = 0\nhI : DividedPowers I\nha : a ∈ I\n⊢ a ^ n = 0",
"ppTerm": "?m.30",
"assigned": true,
"usedConstants": [
"DividedPowers.dpow_mem",
"Eq.mpr",
"NonAssocSemiring.to... | [] | by | [anonymous] | by |
Mathlib.RingTheory.Depth.Rees | {
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"column": 81
} | {
"line": 88,
"column": 83
} | {
"line": 88,
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} | [
{
"pp": "R : Type u\ninst✝⁴ : CommRing R\ninst✝³ : Small.{v, u} R\ninst✝² : IsNoetherianRing R\nI : Ideal R\nN : ModuleCat R\ninst✝¹ : Module.Finite R ↑N\nh_supp : (Module.annihilator R ↑N).radical = I.radical\nn : ℕ\nih :\n ∀ (M : ModuleCat R) [Module.Finite R ↑M],\n I • ⊤ < ⊤ →\n (∀ i < n, Subsinglet... | [] | by | [anonymous] | by |
Mathlib.RingTheory.Regular.LinearMap | {
"line": 94,
"column": 26
} | {
"line": 94,
"column": 28
} | {
"line": 95,
"column": 6
} | [
{
"pp": "R : Type u_1\nM : Type u_2\nN : Type u_3\ninst✝⁷ : CommRing R\ninst✝⁶ : AddCommGroup M\ninst✝⁵ : AddCommGroup N\ninst✝⁴ : Module R M\ninst✝³ : Module R N\ninst✝² : IsNoetherianRing R\ninst✝¹ : Module.Finite R M\ninst✝ : Module.Finite R N\nhom0 : Subsingleton (N →ₗ[R] M)\nh✝ : Nontrivial M\nh : ∀ r ∈ an... | [] | by | Lean.Elab.Tactic.evalWithAnnotateState | by |
Mathlib.RingTheory.Depth.Rees | {
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} | {
"line": 83,
"column": 96
} | {
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{
"pp": "R : Type u\ninst✝⁴ : CommRing R\ninst✝³ : Small.{v, u} R\ninst✝² : IsNoetherianRing R\nI : Ideal R\nN : ModuleCat R\ninst✝¹ : Module.Finite R ↑N\nh_supp : (Module.annihilator R ↑N).radical = I.radical\nn : ℕ\nih :\n ∀ (M : ModuleCat R) [Module.Finite R ↑M],\n I • ⊤ < ⊤ →\n (∀ i < n, Subsinglet... | [] | by | Lean.Elab.Tactic.evalWithAnnotateState | by |
Mathlib.RingTheory.DividedPowers.Basic | {
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"column": 33
} | {
"line": 243,
"column": 35
} | {
"line": 244,
"column": 2
} | [
{
"pp": "A : Type u_1\ninst✝ : CommSemiring A\nI : Ideal A\na : A\nhI : DividedPowers I\nJ : Ideal A\nhJ : DividedPowers J\nn : ℕ\nha : a ∈ I • J\n⊢ hI.dpow n a = hJ.dpow n a",
"ppTerm": "?m.21",
"assigned": true,
"usedConstants": [
"Mathlib.Tactic.Ring.Common.mul_pf_left",
"Eq.mpr",
... | [] | by | [anonymous] | by |
Mathlib.RingTheory.Regular.LinearMap | {
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"column": 82
} | {
"line": 47,
"column": 84
} | {
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} | [
{
"pp": "R : Type u_1\nM : Type u_2\nN : Type u_3\ninst✝⁷ : CommRing R\ninst✝⁶ : AddCommGroup M\ninst✝⁵ : AddCommGroup N\ninst✝⁴ : Module R M\ninst✝³ : Module R N\ninst✝² : IsNoetherianRing R\ninst✝¹ : Module.Finite R M\ninst✝ : Module.Finite R N\n⊢ Subsingleton (N →ₗ[R] M) ↔ ∃ r ∈ annihilator R N, IsSMulRegula... | [] | by | [anonymous] | by |
Mathlib.RingTheory.DividedPowerAlgebra.Init | {
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"column": 70
} | {
"line": 113,
"column": 72
} | {
"line": 114,
"column": 2
} | [
{
"pp": "R : Type u_2\nM : Type u_3\ninst✝² : CommSemiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\na : R\n⊢ ↑(C a) = (algebraMap R (DividedPowerAlgebra R M)) a",
"ppTerm": "?m.16",
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"usedConstants": [
"Finsupp.instAddZeroClass",
"Eq.mpr",
"Nat.instMulZeroC... | [] | by | [anonymous] | by |
Mathlib.RingTheory.DividedPowerAlgebra.Init | {
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"column": 84
} | {
"line": 118,
"column": 86
} | {
"line": 119,
"column": 2
} | [
{
"pp": "R : Type u_2\nM : Type u_3\ninst✝² : CommSemiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\na : R\n⊢ (RingCon.mkₐ R (ringCon R M)) (C a) = (algebraMap R (DividedPowerAlgebra R M)) a",
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"assigned": true,
"usedConstants": [
"Finsupp.instAddZeroClass",
"Eq.... | [] | by | [anonymous] | by |
Mathlib.RingTheory.Depth.Rees | {
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} | {
"line": 68,
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} | {
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{
"pp": "R : Type u\ninst✝⁴ : CommRing R\ninst✝³ : Small.{v, u} R\ninst✝² : IsNoetherianRing R\nI : Ideal R\nn : ℕ\nM : ModuleCat R\ninst✝¹ : Module.Finite R ↑M\nsmul_lt : I • ⊤ < ⊤\nN : ModuleCat R\ninst✝ : Module.Finite R ↑N\nh_supp : Module.support R ↑N = PrimeSpectrum.zeroLocus ↑I\nh_ext : ∀ i < n, Subsingl... | [] | by | [anonymous] | by |
Mathlib.RingTheory.DividedPowerAlgebra.Init | {
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} | {
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} | {
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{
"pp": "R : Type u_2\nM : Type u_3\ninst✝² : CommSemiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nP : DividedPowerAlgebra R M → Prop\nf : DividedPowerAlgebra R M\nh_C : ∀ (a : R), P ↑(C a)\nh_add : ∀ (f g : DividedPowerAlgebra R M), P f → P g → P (f + g)\nh_dp : ∀ (f : DividedPowerAlgebra R M) (n : ℕ)... | [] | by | [anonymous] | by |
Mathlib.RingTheory.DividedPowerAlgebra.Init | {
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} | {
"line": 153,
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} | {
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{
"pp": "R : Type u_2\nM : Type u_3\ninst✝² : CommSemiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nm : M\n⊢ dp R 0 m = 1",
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"Eq.mpr",
"RingCon.toCon",
"MulOne.toOne",
"Nat.instMulZeroClass",
"AddMonoidAlgeb... | [] | by | [anonymous] | by |
Mathlib.RingTheory.DividedPowerAlgebra.Init | {
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} | {
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} | {
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{
"pp": "R : Type u_2\nM : Type u_3\ninst✝² : CommSemiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nr : R\nn : ℕ\nm : M\n⊢ dp R n (r • m) = r ^ n • dp R n m",
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"usedConstants": [
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"RingCon.toCon",
"Nat.instMulZeroClass",
"A... | [] | by | [anonymous] | by |
Mathlib.RingTheory.DividedPowers.Basic | {
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"column": 80
} | {
"line": 260,
"column": 82
} | {
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{
"pp": "A : Type u_1\ninst✝ : CommSemiring A\nI : Ideal A\na : A\nhI : DividedPowers I\nι : Type u_2\ns : Finset ι\nn : ι → ℕ\nha : a ∈ I\n⊢ ∏ i ∈ s, hI.dpow (n i) a = ↑(multinomial s n) * hI.dpow (s.sum n) a",
"ppTerm": "?m.26",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NonAssocS... | [] | by | [anonymous] | by |
Mathlib.RingTheory.DividedPowerAlgebra.Init | {
"line": 161,
"column": 69
} | {
"line": 161,
"column": 71
} | {
"line": 162,
"column": 2
} | [
{
"pp": "R : Type u_2\nM : Type u_3\ninst✝² : CommSemiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nn : ℕ\n⊢ dp R n 0 = if n = 0 then 1 else 0",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"AddMonoidAlgebra.instAddMonoid",
"Nat.instMulZeroClass",
... | [] | by | [anonymous] | by |
Mathlib.RingTheory.DividedPowerAlgebra.Init | {
"line": 169,
"column": 72
} | {
"line": 169,
"column": 74
} | {
"line": 170,
"column": 2
} | [
{
"pp": "R : Type u_2\nM : Type u_3\ninst✝² : CommSemiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nn : ℕ\nhn : n ≠ 0\n⊢ dp R n 0 = 0",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"AddMonoidAlgebra.instAddMonoid",
"Nat.instMulZeroClass",
"AddMono... | [] | by | [anonymous] | by |
Mathlib.RingTheory.DividedPowerAlgebra.Init | {
"line": 173,
"column": 63
} | {
"line": 173,
"column": 65
} | {
"line": 174,
"column": 2
} | [
{
"pp": "R : Type u_2\nM : Type u_3\ninst✝² : CommSemiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nn p : ℕ\nm : M\n⊢ dp R n m * dp R p m = (n + p).choose n • dp R (n + p) m",
"ppTerm": "?m.32",
"assigned": true,
"usedConstants": [
"RingCon.toCon",
"AddMonoidAlgebra.instAddMonoid... | [] | by | [anonymous] | by |
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