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.Ideal.HasGoingUp | {
"line": 49,
"column": 45
} | {
"line": 49,
"column": 47
} | {
"line": 50,
"column": 2
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing S\ninst✝⁴ : Algebra R S\ninst✝³ : Algebra.HasGoingUp R S\np q : Ideal R\ninst✝² : q.IsPrime\nP : Ideal S\ninst✝¹ : P.IsPrime\ninst✝ : P.LiesOver p\nhle : p ≤ q\n⊢ ∃ Q, P ≤ Q ∧ Q.IsPrime ∧ Q.LiesOver q",
"ppTerm": "?m.31",
"assig... | [] | by | [anonymous] | by |
Mathlib.RingTheory.Ideal.HasGoingUp | {
"line": 58,
"column": 45
} | {
"line": 58,
"column": 47
} | {
"line": 59,
"column": 2
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing S\ninst✝⁴ : Algebra R S\ninst✝³ : Algebra.HasGoingUp R S\np q : Ideal R\ninst✝² : q.IsPrime\nP : Ideal S\ninst✝¹ : P.IsPrime\ninst✝ : P.LiesOver p\nhpq : p < q\n⊢ ∃ Q, P < Q ∧ Q.IsPrime ∧ Q.LiesOver q",
"ppTerm": "?m.31",
"assig... | [] | by | [anonymous] | by |
Mathlib.RingTheory.HopfAlgebra.MonoidAlgebra | {
"line": 56,
"column": 32
} | {
"line": 56,
"column": 34
} | {
"line": 57,
"column": 4
} | [
{
"pp": "R : Type u_1\nA : Type u_2\ninst✝³ : CommSemiring R\ninst✝² : Semiring A\ninst✝¹ : HopfAlgebra R A\nG : Type u_3\ninst✝ : Group G\n⊢ LinearMap.mul' R A[G] ∘ₗ LinearMap.lTensor A[G] (antipode R) ∘ₗ comul = Algebra.linearMap R A[G] ∘ₗ counit",
"ppTerm": "?m.95",
"assigned": true,
"usedConstan... | [] | by | [anonymous] | by |
Mathlib.RingTheory.HopfAlgebra.MonoidAlgebra | {
"line": 76,
"column": 67
} | {
"line": 76,
"column": 69
} | {
"line": 77,
"column": 2
} | [
{
"pp": "R : Type u_1\nA : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : Semiring A\ninst✝ : HopfAlgebra R A\na : A\n⊢ (antipode R) (C a) = C ((antipode R) a)",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"Int.instAddCommGroup",
"LaurentPolynomial.T",
"AddMonoidAlgebra.in... | [] | by | [anonymous] | by |
Mathlib.RingTheory.HopfAlgebra.MonoidAlgebra | {
"line": 82,
"column": 56
} | {
"line": 82,
"column": 58
} | {
"line": 83,
"column": 2
} | [
{
"pp": "R : Type u_1\nA : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : Semiring A\ninst✝ : HopfAlgebra R A\nn : ℤ\n⊢ (antipode R) (T n) = T (-n)",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"LaurentPolynomial.T",
"AddMonoidAlgebra.instHopfAlgebraStruct",
"AddGroup.toSu... | [] | by | [anonymous] | by |
Mathlib.RingTheory.HopfAlgebra.MonoidAlgebra | {
"line": 89,
"column": 82
} | {
"line": 89,
"column": 84
} | {
"line": 90,
"column": 2
} | [
{
"pp": "R : Type u_1\nA : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : Semiring A\ninst✝ : HopfAlgebra R A\na : A\nn : ℤ\n⊢ (antipode R) (C a * T n) = C ((antipode R) a) * T (-n)",
"ppTerm": "?m.38",
"assigned": true,
"usedConstants": [
"LaurentPolynomial.T",
"AddMonoidAlgebra.instHopfAl... | [] | by | [anonymous] | by |
Mathlib.RingTheory.Ideal.IsAugmentation | {
"line": 42,
"column": 72
} | {
"line": 42,
"column": 74
} | {
"line": 43,
"column": 2
} | [
{
"pp": "R : Type u_1\ninst✝² : CommSemiring R\nA : Type u_2\ninst✝¹ : CommSemiring A\ninst✝ : Algebra R A\nS : Subalgebra R A\nI : Ideal A\n⊢ IsAugmentation (↥S) I ↔ IsCompl (toSubmodule S) (Submodule.restrictScalars R I)",
"ppTerm": "?m.29",
"assigned": true,
"usedConstants": [
"Ideal.IsAugm... | [] | by | [anonymous] | by |
Mathlib.RingTheory.HopfAlgebra.Quotient | {
"line": 58,
"column": 82
} | {
"line": 58,
"column": 84
} | {
"line": 59,
"column": 10
} | [
{
"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⊢ (toConv (antipode R) * toC... | [] | by | [anonymous] | by |
Mathlib.RingTheory.Ideal.HasGoingUp | {
"line": 83,
"column": 39
} | {
"line": 83,
"column": 41
} | {
"line": 84,
"column": 6
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝⁴ : CommRing R\ninst✝³ : CommRing S\ninst✝² : Algebra R S\ninst✝¹ : Algebra.HasGoingUp R S\nl : RelSeries {(a, b) | a < b}\nq : PrimeSpectrum R\nlt : (q, l.head) ∈ {(a, b) | a < b}\nih :\n ∀ (P : Ideal S) [inst : P.IsPrime] [lo : P.LiesOver l.head.asIdeal],\n ∃ L,\n... | [] | by | [anonymous] | by |
Mathlib.RingTheory.HopfAlgebra.Quotient | {
"line": 61,
"column": 80
} | {
"line": 61,
"column": 82
} | {
"line": 62,
"column": 10
} | [
{
"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⊢ (toConv (f.toLinearMap ∘ₗ ... | [] | by | [anonymous] | by |
Mathlib.RingTheory.Ideal.HasGoingUp | {
"line": 85,
"column": 19
} | {
"line": 85,
"column": 21
} | {
"line": 85,
"column": 22
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝⁴ : CommRing R\ninst✝³ : CommRing S\ninst✝² : Algebra R S\ninst✝¹ : Algebra.HasGoingUp R S\nl : RelSeries {(a, b) | a < b}\nq : PrimeSpectrum R\nlt : (q, l.head) ∈ {(a, b) | a < b}\nih :\n ∀ (P : Ideal S) [inst : P.IsPrime] [lo : P.LiesOver l.head.asIdeal],\n ∃ L,\n... | [] | by | [anonymous] | by |
Mathlib.RingTheory.Ideal.HasGoingUp | {
"line": 73,
"column": 79
} | {
"line": 73,
"column": 81
} | {
"line": 74,
"column": 2
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝⁴ : CommRing R\ninst✝³ : CommRing S\ninst✝² : Algebra R S\ninst✝¹ : Algebra.HasGoingUp R S\nl : LTSeries (PrimeSpectrum R)\nP : Ideal S\ninst✝ : P.IsPrime\nlo : P.LiesOver (RelSeries.head l).asIdeal\n⊢ ∃ L,\n L.length = l.length ∧\n RelSeries.head L = { asIdeal ... | [] | by | [anonymous] | by |
Mathlib.RingTheory.HopfAlgebra.Quotient | {
"line": 63,
"column": 64
} | {
"line": 63,
"column": 66
} | {
"line": 64,
"column": 10
} | [
{
"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⊢ (↑f).toLinearMap ∘ₗ (toCon... | [] | by | [anonymous] | by |
Mathlib.RingTheory.HopfAlgebra.Quotient | {
"line": 67,
"column": 82
} | {
"line": 67,
"column": 84
} | {
"line": 68,
"column": 10
} | [
{
"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⊢ (toConv LinearMap.id * toC... | [] | by | [anonymous] | by |
Mathlib.RingTheory.Ideal.HasGoingUp | {
"line": 99,
"column": 64
} | {
"line": 99,
"column": 66
} | {
"line": 100,
"column": 2
} | [
{
"pp": "R : Type u_3\nS : Type u_4\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\n⊢ HasGoingUp R S ↔ SpecializingMap (PrimeSpectrum.comap (algebraMap R S))",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"PrimeSpectrum.mk",
"PrimeSpectrum.ext"... | [] | by | [anonymous] | by |
Mathlib.RingTheory.HopfAlgebra.Quotient | {
"line": 70,
"column": 80
} | {
"line": 70,
"column": 82
} | {
"line": 71,
"column": 10
} | [
{
"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⊢ (toConv f.toLinearMap * to... | [] | by | [anonymous] | by |
Mathlib.RingTheory.Ideal.HasGoingUp | {
"line": 119,
"column": 30
} | {
"line": 119,
"column": 32
} | {
"line": 120,
"column": 2
} | [
{
"pp": "R : Type u_3\nS : Type u_4\ninst✝⁸ : CommRing R\ninst✝⁷ : CommRing S\ninst✝⁶ : Algebra R S\nT : Type u_5\ninst✝⁵ : CommRing T\ninst✝⁴ : Algebra R T\ninst✝³ : Algebra S T\ninst✝² : IsScalarTower R S T\ninst✝¹ : HasGoingUp R S\ninst✝ : HasGoingUp S T\n⊢ HasGoingUp R T",
"ppTerm": "?m.21",
"assign... | [] | by | [anonymous] | by |
Mathlib.RingTheory.Henselian | {
"line": 225,
"column": 44
} | {
"line": 225,
"column": 46
} | {
"line": 225,
"column": 47
} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommRing R\nI : Ideal R\ninst✝ : IsAdicComplete I R\nf : R[X]\nx✝ : f.Monic\na₀ : R\nh₁ : Polynomial.eval a₀ f ∈ I\nh₂ : IsUnit ((Ideal.Quotient.mk I) (Polynomial.eval a₀ (derivative f)))\nf' : R[X] := derivative f\nc : ℕ → R := fun n ↦ Nat.recOn n a₀ fun x b ↦ b - Polynomial.eva... | [] | by | Lean.Elab.Tactic.evalWithAnnotateState | by |
Mathlib.RingTheory.HopfAlgebra.Quotient | {
"line": 72,
"column": 64
} | {
"line": 72,
"column": 66
} | {
"line": 73,
"column": 10
} | [
{
"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⊢ (↑f).toLinearMap ∘ₗ (toCon... | [] | by | [anonymous] | by |
Mathlib.RingTheory.HopfAlgebra.Quotient | {
"line": 53,
"column": 90
} | {
"line": 53,
"column": 92
} | {
"line": 54,
"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... | [] | by | [anonymous] | by |
Mathlib.RingTheory.HopfAlgebra.Quotient | {
"line": 110,
"column": 60
} | {
"line": 110,
"column": 62
} | {
"line": 110,
"column": 63
} | [
{
"pp": "R : Type u_1\nA : Type u_2\ninst✝⁴ : CommRing R\ninst✝³ : Ring A\ninst✝² : HopfAlgebraStruct R A\nI : Ideal A\ninst✝¹ : I.IsTwoSided\ninst✝ : Ideal.IsHopfIdeal R I\n⊢ antipode R ∘ₗ (mkₐ R I).toLinearMap = (mkₐ R I).toLinearMap ∘ₗ antipode R",
"ppTerm": "?m.89",
"assigned": true,
"usedConsta... | [] | by | [anonymous] | by |
Mathlib.RingTheory.Henselian | {
"line": 205,
"column": 53
} | {
"line": 205,
"column": 55
} | {
"line": 206,
"column": 8
} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommRing R\nI : Ideal R\ninst✝ : IsAdicComplete I R\nf : R[X]\nx✝ : f.Monic\na₀ : R\nh₁ : Polynomial.eval a₀ f ∈ I\nh₂ : IsUnit ((Ideal.Quotient.mk I) (Polynomial.eval a₀ (derivative f)))\nf' : R[X] := derivative f\nc : ℕ → R := fun n ↦ Nat.recOn n a₀ fun x b ↦ b - Polynomial.eva... | [] | by | Lean.Elab.Tactic.evalWithAnnotateState | by |
Mathlib.RingTheory.Henselian | {
"line": 230,
"column": 74
} | {
"line": 230,
"column": 76
} | {
"line": 231,
"column": 8
} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommRing R\nI : Ideal R\ninst✝ : IsAdicComplete I R\nf : R[X]\nx✝ : f.Monic\na₀ : R\nh₁ : Polynomial.eval a₀ f ∈ I\nh₂ : IsUnit ((Ideal.Quotient.mk I) (Polynomial.eval a₀ (derivative f)))\nf' : R[X] := derivative f\nc : ℕ → R := fun n ↦ Nat.recOn n a₀ fun x b ↦ b - Polynomial.eva... | [] | by | Lean.Elab.Tactic.evalWithAnnotateState | by |
Mathlib.RingTheory.Henselian | {
"line": 249,
"column": 64
} | {
"line": 249,
"column": 66
} | {
"line": 249,
"column": 67
} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommRing R\nI : Ideal R\ninst✝ : IsAdicComplete I R\nf : R[X]\nx✝ : f.Monic\na₀ : R\nh₁ : Polynomial.eval a₀ f ∈ I\nh₂ : IsUnit ((Ideal.Quotient.mk I) (Polynomial.eval a₀ (derivative f)))\nf' : R[X] := derivative f\nc : ℕ → R := fun n ↦ Nat.recOn n a₀ fun x b ↦ b - Polynomial.eva... | [] | by | [anonymous] | by |
Mathlib.RingTheory.Ideal.KrullsHeightTheorem | {
"line": 56,
"column": 39
} | {
"line": 56,
"column": 41
} | {
"line": 56,
"column": 42
} | [
{
"pp": "R : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsNoetherianRing R\ninst✝ : IsLocalRing R\nI : Ideal R\nhp✝ : maximalIdeal R ∈ I.minimalPrimes\nhp : I.IsMinimalPrime (maximalIdeal R)\nJ : Ideal (R ⧸ I)\nprime : J.IsPrime\nthis : (Ideal.comap (Ideal.Quotient.mk I) J).IsPrime\n⊢ I ≤ RingHom.ker (algebraMap R... | [] | by | [anonymous] | by |
Mathlib.RingTheory.Ideal.KrullsHeightTheorem | {
"line": 52,
"column": 58
} | {
"line": 52,
"column": 60
} | {
"line": 53,
"column": 6
} | [
{
"pp": "R : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsNoetherianRing R\ninst✝ : IsLocalRing R\nI : Ideal R\nhp : maximalIdeal R ∈ I.minimalPrimes\nJ : Ideal (R ⧸ I)\nprime : J.IsPrime\n⊢ (Ideal.comap (algebraMap (?m.34 J prime) (R ⧸ I)) J).IsMaximal",
"ppTerm": "?m.33",
"assigned": true,
"usedConst... | [] | by | [anonymous] | by |
Mathlib.RingTheory.Henselian | {
"line": 173,
"column": 18
} | {
"line": 173,
"column": 20
} | {
"line": 174,
"column": 4
} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommRing R\nI : Ideal R\ninst✝ : IsAdicComplete I R\n⊢ ∀ (f : R[X]),\n f.Monic →\n ∀ (a₀ : R),\n Polynomial.eval a₀ f ∈ I →\n IsUnit ((Ideal.Quotient.mk I) (Polynomial.eval a₀ (derivative f))) → ∃ a, f.IsRoot a ∧ a - a₀ ∈ I",
"ppTerm": "?m.14",
"as... | [] | by | [anonymous] | by |
Mathlib.RingTheory.Henselian | {
"line": 270,
"column": 67
} | {
"line": 270,
"column": 69
} | {
"line": 270,
"column": 70
} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsLocalRing R\nf : R[X]\na b : R\nha : Polynomial.eval a f = 0\nhb : Polynomial.eval b f = 0\nh : ¬IsUnit (a - b)\nh' : IsUnit (Polynomial.eval a (derivative f))\nc : R\nh✝ :\n c * (b - a) ^ 2 + Polynomial.eval a (derivative f) * (b - a) + Polynomial.eval a f... | [] | by | Lean.Elab.Tactic.evalWithAnnotateState | by |
Mathlib.RingTheory.Ideal.KrullsHeightTheorem | {
"line": 65,
"column": 24
} | {
"line": 65,
"column": 26
} | {
"line": 65,
"column": 27
} | [
{
"pp": "R : Type u_1\ninst✝³ : CommRing R\ninst✝² : IsNoetherianRing R\ninst✝¹ : IsLocalRing R\nI : Ideal R\ninst✝ : Submodule.IsPrincipal I\nhp : IsLocalRing.maximalIdeal R ∈ I.minimalPrimes\nq : Ideal R\nh₁ : q.IsPrime\nh₂ : q < IsLocalRing.maximalIdeal R\nthis : q.height = 0\n⊢ q.height < ↑1",
"ppTerm":... | [] | by | [anonymous] | by |
Mathlib.RingTheory.Ideal.MonicSpan | {
"line": 28,
"column": 41
} | {
"line": 28,
"column": 43
} | {
"line": 29,
"column": 2
} | [
{
"pp": "k : Type u_2\ninst✝ : Field k\nI : Ideal k[X]\nne : I ≠ ⊥\n⊢ ∃ f, f.Monic ∧ I = span {f}",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"Polynomial.instNormalizationMonoid",
"Submodule",
"False",
"IsDomain.to_noZeroDivisors",
"Semiring.toModule",
... | [] | by | [anonymous] | by |
Mathlib.RingTheory.Henselian | {
"line": 272,
"column": 4
} | {
"line": 273,
"column": 9
} | {
"line": 274,
"column": 2
} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsLocalRing R\nf : R[X]\na b : R\nha : Polynomial.eval a f = 0\nhb : Polynomial.eval b f = 0\nh : ¬IsUnit (a - b)\nh' : IsUnit (Polynomial.eval a (derivative f))\nc : R\nh✝ :\n c * (b - a) ^ 2 + Polynomial.eval a (derivative f) * (b - a) + Polynomial.eval a f... | [] | rw [notMem_maximalIdeal, isUnit_iff_exists] at this
grind | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Henselian | {
"line": 272,
"column": 4
} | {
"line": 273,
"column": 9
} | {
"line": 274,
"column": 2
} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsLocalRing R\nf : R[X]\na b : R\nha : Polynomial.eval a f = 0\nhb : Polynomial.eval b f = 0\nh : ¬IsUnit (a - b)\nh' : IsUnit (Polynomial.eval a (derivative f))\nc : R\nh✝ :\n c * (b - a) ^ 2 + Polynomial.eval a (derivative f) * (b - a) + Polynomial.eval a f... | [] | rw [notMem_maximalIdeal, isUnit_iff_exists] at this
grind | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Henselian | {
"line": 271,
"column": 66
} | {
"line": 271,
"column": 68
} | {
"line": 272,
"column": 4
} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsLocalRing R\nf : R[X]\na b : R\nha : Polynomial.eval a f = 0\nhb : Polynomial.eval b f = 0\nh : ¬IsUnit (a - b)\nh' : IsUnit (Polynomial.eval a (derivative f))\nc : R\nh✝ :\n c * (b - a) ^ 2 + Polynomial.eval a (derivative f) * (b - a) + Polynomial.eval a f... | [] | by | [anonymous] | by |
Mathlib.RingTheory.Ideal.MonicSpan | {
"line": 40,
"column": 67
} | {
"line": 40,
"column": 69
} | {
"line": 41,
"column": 4
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\np : Ideal R[X]\nism : (comap C p).IsMaximal\nne : p ≠ Ideal.map C (comap C p)\nq : Ideal R := comap C p\nthis : Field (R ⧸ q) := Quotient.field q\n⊢ Ideal.map (mapRingHom (Ideal.Quotient.mk q)) p ≠ ⊥",
"ppTerm": "?m.97",
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"usedConstants": [... | [] | by | Lean.Elab.Tactic.evalWithAnnotateState | by |
Mathlib.RingTheory.Henselian | {
"line": 268,
"column": 51
} | {
"line": 268,
"column": 53
} | {
"line": 269,
"column": 2
} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsLocalRing R\nf : R[X]\na b : R\nha : Polynomial.eval a f = 0\nhb : Polynomial.eval b f = 0\nh : ¬IsUnit (a - b)\nh' : IsUnit (Polynomial.eval a (derivative f))\n⊢ a = b",
"ppTerm": "?m.27",
"assigned": true,
"usedConstants": [
"Mathlib.Tact... | [] | by | [anonymous] | by |
Mathlib.RingTheory.Ideal.UFD | {
"line": 41,
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} | {
"line": 41,
"column": 47
} | {
"line": 41,
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{
"pp": "R : Type u_1\ninst✝⁶ : CommRing R\ninst✝⁵ : IsDomain R\ninst✝⁴ : WfDvdMonoid R\nx : R\nhx : Prime x\np : Ideal R\ninst✝³ : p.IsPrime\nhxp : x ∉ p\nS : Type u_2\ninst✝² : CommRing S\ninst✝¹ : Algebra R S\ninst✝ : IsLocalization.Away x S\nhp : Submodule.IsPrincipal (map (algebraMap R S) p)\nthis : Disjoi... | [] | by | Lean.Elab.Tactic.evalWithAnnotateState | by |
Mathlib.RingTheory.Ideal.UFD | {
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{
"pp": "R : Type u_1\ninst✝⁶ : CommRing R\ninst✝⁵ : IsDomain R\ninst✝⁴ : WfDvdMonoid R\nx : R\nhx : Prime x\np : Ideal R\ninst✝³ : p.IsPrime\nhxp : x ∉ p\nS : Type u_2\ninst✝² : CommRing S\ninst✝¹ : Algebra R S\ninst✝ : IsLocalization.Away x S\nthis : Disjoint ↑(Submonoid.powers x) ↑p\nhpbot : ¬p = ⊥\nhi : Fun... | [] | by | [anonymous] | by |
Mathlib.RingTheory.Ideal.UFD | {
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} | {
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} | {
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{
"pp": "R : Type u_1\ninst✝⁶ : CommRing R\ninst✝⁵ : IsDomain R\ninst✝⁴ : WfDvdMonoid R\nx : R\nhx : Prime x\np : Ideal R\ninst✝³ : p.IsPrime\nhxp : x ∉ p\nS : Type u_2\ninst✝² : CommRing S\ninst✝¹ : Algebra R S\ninst✝ : IsLocalization.Away x S\nthis : Disjoint ↑(Submonoid.powers x) ↑p\nhpbot : ¬p = ⊥\nhi : Fun... | [] | by | [anonymous] | by |
Mathlib.RingTheory.Ideal.UFD | {
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} | {
"line": 36,
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} | {
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{
"pp": "R : Type u_1\ninst✝⁶ : CommRing R\ninst✝⁵ : IsDomain R\ninst✝⁴ : WfDvdMonoid R\nx : R\nhx : Prime x\np : Ideal R\ninst✝³ : p.IsPrime\nhxp : x ∉ p\nS : Type u_2\ninst✝² : CommRing S\ninst✝¹ : Algebra R S\ninst✝ : IsLocalization.Away x S\nhp : Submodule.IsPrincipal (map (algebraMap R S) p)\n⊢ Submodule.I... | [] | by | [anonymous] | by |
Mathlib.RingTheory.Ideal.UFD | {
"line": 62,
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} | {
"line": 62,
"column": 72
} | {
"line": 63,
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{
"pp": "R : Type u_1\ninst✝³ : CommRing R\ninst✝² : IsDomain R\ninst✝¹ : UniqueFactorizationMonoid R\np : Ideal R\ninst✝ : p.IsPrime\nhph : p.height = 1\n⊢ Submodule.IsPrincipal p",
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"assigned": true,
"usedConstants": [
"Submodule",
"Semiring.toModule",
"CommSemi... | [] | by | [anonymous] | by |
Mathlib.RingTheory.Ideal.UFD | {
"line": 81,
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} | {
"line": 81,
"column": 9
} | {
"line": 81,
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{
"pp": "R : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : IsNoetherianRing R\nh : ∀ (p : Ideal R) [p.IsPrime], p.height = 1 → Submodule.IsPrincipal p\nI : Ideal R\nhIn : I ≠ ⊥\na✝ : I.IsPrime\nx : R\nhxI : x ∈ I\nhx0 : x ≠ 0\np : Ideal R\nhpmin : p ∈ (Ideal.span {x}).minimalPrimes\nhpl : p ≤ I\nt... | [] | by | [anonymous] | by |
Mathlib.RingTheory.Ideal.UFD | {
"line": 71,
"column": 35
} | {
"line": 71,
"column": 37
} | {
"line": 72,
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{
"pp": "R : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : IsNoetherianRing R\nh : ∀ (p : Ideal R) [p.IsPrime], p.height = 1 → Submodule.IsPrincipal p\n⊢ UniqueFactorizationMonoid R",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"Eq.mpr",
"Submod... | [] | by | [anonymous] | by |
Mathlib.RingTheory.Ideal.UFD | {
"line": 100,
"column": 18
} | {
"line": 100,
"column": 78
} | {
"line": 101,
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{
"pp": "R : Type u_1\ninst✝⁵ : CommRing R\ninst✝⁴ : IsDomain R\ninst✝³ : IsNoetherianRing R\nx : R\nhx : Prime x\nS : Type u_2\ninst✝² : CommRing S\ninst✝¹ : Algebra R S\ninst✝ : IsLocalization.Away x S\nthis : IsDomain S\nx✝ : UniqueFactorizationMonoid S\np : Ideal R\nhp : p.IsPrime\nh1 : p.height = 1\nhxp : ... | [] | rwa [← Ideal.disjoint_powers_iff_notMem_of_isPrime x] at hxp | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticRwa___1 | Lean.Parser.Tactic.tacticRwa__ |
Mathlib.RingTheory.Ideal.UFD | {
"line": 100,
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} | {
"line": 100,
"column": 78
} | {
"line": 101,
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{
"pp": "R : Type u_1\ninst✝⁵ : CommRing R\ninst✝⁴ : IsDomain R\ninst✝³ : IsNoetherianRing R\nx : R\nhx : Prime x\nS : Type u_2\ninst✝² : CommRing S\ninst✝¹ : Algebra R S\ninst✝ : IsLocalization.Away x S\nthis : IsDomain S\nx✝ : UniqueFactorizationMonoid S\np : Ideal R\nhp : p.IsPrime\nh1 : p.height = 1\nhxp : ... | [] | rwa [← Ideal.disjoint_powers_iff_notMem_of_isPrime x] at hxp | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Ideal.UFD | {
"line": 100,
"column": 18
} | {
"line": 100,
"column": 78
} | {
"line": 101,
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{
"pp": "R : Type u_1\ninst✝⁵ : CommRing R\ninst✝⁴ : IsDomain R\ninst✝³ : IsNoetherianRing R\nx : R\nhx : Prime x\nS : Type u_2\ninst✝² : CommRing S\ninst✝¹ : Algebra R S\ninst✝ : IsLocalization.Away x S\nthis : IsDomain S\nx✝ : UniqueFactorizationMonoid S\np : Ideal R\nhp : p.IsPrime\nh1 : p.height = 1\nhxp : ... | [] | rwa [← Ideal.disjoint_powers_iff_notMem_of_isPrime x] at hxp | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Ideal.UFD | {
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} | {
"line": 100,
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} | {
"line": 100,
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{
"pp": "R : Type u_1\ninst✝⁵ : CommRing R\ninst✝⁴ : IsDomain R\ninst✝³ : IsNoetherianRing R\nx : R\nhx : Prime x\nS : Type u_2\ninst✝² : CommRing S\ninst✝¹ : Algebra R S\ninst✝ : IsLocalization.Away x S\nthis : IsDomain S\nx✝ : UniqueFactorizationMonoid S\np : Ideal R\nhp : p.IsPrime\nh1 : p.height = 1\nhxp : ... | [] | by | [anonymous] | by |
Mathlib.RingTheory.Ideal.MonicSpan | {
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} | {
"line": 37,
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} | {
"line": 38,
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{
"pp": "R : Type u_1\ninst✝ : CommRing R\np : Ideal R[X]\nism : (comap C p).IsMaximal\nne : p ≠ Ideal.map C (comap C p)\n⊢ ∃ f, f.Monic ∧ p = Ideal.map C (comap C p) ⊔ span {f}",
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"usedConstants": [
"Polynomial.ker_mapRingHom",
"Eq.mpr",
"Polyno... | [] | by | [anonymous] | by |
Mathlib.RingTheory.Ideal.UFD | {
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} | {
"line": 93,
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} | {
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{
"pp": "R : Type u_1\ninst✝⁵ : CommRing R\ninst✝⁴ : IsDomain R\ninst✝³ : IsNoetherianRing R\nx : R\nhx : Prime x\nS : Type u_2\ninst✝² : CommRing S\ninst✝¹ : Algebra R S\ninst✝ : IsLocalization.Away x S\n⊢ UniqueFactorizationMonoid R ↔ UniqueFactorizationMonoid S",
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... | [] | by | [anonymous] | by |
Mathlib.RingTheory.IdealFilter.Basic | {
"line": 110,
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} | {
"line": 110,
"column": 21
} | {
"line": 110,
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{
"pp": "A : Type u_1\ninst✝ : Ring A\nF : IdealFilter A\nL K : Ideal A\nh : F.IsTorsionQuot (L ⊓ K) K\nk : A\nhk : k ∈ K\nI : Ideal A\nhI : I ∈ F\nhI_le : I ≤ Submodule.colon (L ⊓ K) {k}\nhcol : Submodule.colon (L ⊓ K) {k} = Submodule.colon L {k}\n⊢ I ≤ Submodule.colon L {k}",
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"assig... | [] | by | [anonymous] | by |
Mathlib.RingTheory.IdealFilter.Basic | {
"line": 110,
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} | {
"line": 110,
"column": 21
} | {
"line": 110,
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{
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"assigned":... | [] | by | [anonymous] | by |
Mathlib.RingTheory.IdealFilter.Basic | {
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} | {
"line": 104,
"column": 57
} | {
"line": 105,
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{
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"Submodule",
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Mathlib.RingTheory.IdealFilter.Basic | {
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} | {
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} | {
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{
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"usedConstants": [
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... | [] | by | [anonymous] | by |
Mathlib.RingTheory.IdealFilter.Basic | {
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} | {
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} | {
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{
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... | [] | by | [anonymous] | by |
Mathlib.RingTheory.IdealFilter.Topology | {
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} | {
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} | {
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{
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Mathlib.RingTheory.IdealFilter.Topology | {
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{
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Mathlib.RingTheory.IdealFilter.Topology | {
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} | {
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{
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"le_refl",
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... | [] | by | [anonymous] | by |
Mathlib.RingTheory.IdealFilter.Topology | {
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} | {
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} | {
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{
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"assigned": true,
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... | [] | by | [anonymous] | by |
Mathlib.RingTheory.IdealFilter.Topology | {
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} | {
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} | {
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{
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"usedConstants": [
"AddGroup.toSubtractionMonoid",
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Mathlib.RingTheory.IdealFilter.Topology | {
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} | {
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} | {
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{
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... | [] | by | [anonymous] | by |
Mathlib.RingTheory.IdealFilter.Topology | {
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} | {
"line": 66,
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} | {
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{
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"usedConstants": [
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"HMul.hMu... | [] | by | [anonymous] | by |
Mathlib.RingTheory.IdealFilter.Topology | {
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} | {
"line": 69,
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} | {
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{
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Mathlib.RingTheory.IdealFilter.Basic | {
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} | {
"line": 141,
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} | {
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{
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... | [] | by | [anonymous] | by |
Mathlib.RingTheory.IdealFilter.Topology | {
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} | {
"line": 79,
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} | {
"line": 79,
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{
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"ppTerm": "?m.58",
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"usedConstants": [
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... | [] | by | [anonymous] | by |
Mathlib.RingTheory.IdealFilter.Basic | {
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} | {
"line": 148,
"column": 70
} | {
"line": 149,
"column": 2
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{
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"Order.IsPFilter.of_def",
"Semiring.toModule",
"IdealFilter.IsTorsionQuot",
"Set.ofPred",
"Order.PF... | [] | by | [anonymous] | by |
Mathlib.RingTheory.IdealFilter.Topology | {
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} | {
"line": 81,
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} | {
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{
"pp": "A : Type u_1\ninst✝ : Ring A\nF : IdealFilter A\nx✝ : ∃ B, B.sets = {x | ∃ I ∈ F, ↑I = x}\nB : RingFilterBasis A\nhB : B.sets = {x | ∃ I ∈ F, ↑I = x}\nI : Ideal A\nhI : I ∈ F\na : A\nJ : Ideal A\nhJ : J ∈ F\nhbasis : ↑J ∈ B\nhsub : ↑J ⊆ (fun x ↦ x * a) ⁻¹' ↑I\nx : A\nhx : x ∈ J\n⊢ x ∈ Submodule.colon I... | [] | by | [anonymous] | by |
Mathlib.RingTheory.IdealFilter.Topology | {
"line": 77,
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} | {
"line": 77,
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} | {
"line": 78,
"column": 2
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{
"pp": "A : Type u_1\ninst✝ : Ring A\nF : IdealFilter A\n⊢ F.IsUniform ↔ ∃ B, B.sets = {x | ∃ I ∈ F, ↑I = x}",
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"Submodule",
"IdealFilter.IsUniform",
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"Submodule.colon",
"E... | [] | by | [anonymous] | by |
Mathlib.RingTheory.IdealFilter.Basic | {
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} | {
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} | {
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{
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Mathlib.RingTheory.IdealFilter.Basic | {
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} | {
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{
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Mathlib.RingTheory.IdealFilter.Topology | {
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} | {
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} | {
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{
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} | {
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} | {
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} | [
{
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"usedConstants": [
"Filter.instMembership",
"Semiring.toModule",
"instVAddOfAdd",
"congrArg",
"PartialOrder... | [] | by | [anonymous] | by |
Mathlib.Topology.Algebra.ClopenNhdofOne | {
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} | {
"line": 33,
"column": 53
} | {
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{
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"usedConstants": [
"Subgroup.isOpen_of_isClosed_of_finiteIndex",
... | [] | by | [anonymous] | by |
Mathlib.Topology.Algebra.ClopenNhdofOne | {
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} | {
"line": 52,
"column": 24
} | {
"line": 52,
"column": 25
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{
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"ppTerm": "?m.38",
"assigned": true,
"usedConstants": [
"le... | [] | by | [anonymous] | by |
Mathlib.Topology.Algebra.ClopenNhdofOne | {
"line": 50,
"column": 51
} | {
"line": 50,
"column": 53
} | {
"line": 51,
"column": 2
} | [
{
"pp": "G : Type u_1\ninst✝⁴ : Group G\ninst✝³ : TopologicalSpace G\ninst✝² : IsTopologicalGroup G\ninst✝¹ : CompactSpace G\ninst✝ : TotallyDisconnectedSpace G\nU : Set G\nUOpen : IsOpen U\neinU : 1 ∈ U\n⊢ ∃ H, ↑H ⊆ U",
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"assigned": true,
"usedConstants": [
"Filter.instMembe... | [] | by | [anonymous] | by |
Mathlib.Topology.Algebra.ClopenNhdofOne | {
"line": 74,
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} | {
"line": 74,
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} | {
"line": 75,
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{
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"ppTerm"... | [] | by | Lean.Elab.Tactic.evalWithAnnotateState | by |
Mathlib.RingTheory.Ideal.Pure | {
"line": 46,
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} | {
"line": 46,
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} | {
"line": 47,
"column": 6
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{
"pp": "R : Type u_1\ninst✝ : CommRing R\nI J : Ideal R\n⊢ I • ⊤ ≤ Submodule.comap (Submodule.subtype J) I",
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"assigned": true,
"usedConstants": [
"NonAssocSemiring.toAddCommMonoidWithOne",
"Submodule",
"RingHomSurjective.ids",
"Submodule.module._proof_1",
... | [] | by | [anonymous] | by |
Mathlib.Topology.Algebra.ClopenNhdofOne | {
"line": 80,
"column": 29
} | {
"line": 80,
"column": 31
} | {
"line": 81,
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} | [
{
"pp": "G : 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\nUOpen : IsOpen U\neinU : 1 ∈ U\nN : Open... | [] | by | Lean.Elab.Tactic.evalWithAnnotateState | by |
Mathlib.RingTheory.Ideal.Pure | {
"line": 49,
"column": 94
} | {
"line": 49,
"column": 96
} | {
"line": 50,
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} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\nI J : Ideal R\nf : ↥J ⧸ I • ⊤ →ₗ[R] R ⧸ I := (I • ⊤).mapQ I (Submodule.subtype J) ⋯\n⊢ LinearMap.lTensor (R ⧸ I) (Submodule.subtype J) =\n ↑(TensorProduct.rid R (R ⧸ I)).symm ∘ₗ f ∘ₗ ↑(quotTensorEquivQuotSMul (↥J) I)",
"ppTerm": "?m.161",
"assigned": true,
... | [] | by | Lean.Elab.Tactic.evalWithAnnotateState | by |
Mathlib.Topology.Algebra.ClopenNhdofOne | {
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"line": 66,
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} | {
"line": 67,
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{
"pp": "G : 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\n⊢ ↑H = sInf {N | IsOpen ↑N ∧ ↑H ≤ N}",
"ppTerm": "?m.16",
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"usedConstants": [
"Iff.mpr",
... | [] | by | [anonymous] | by |
Mathlib.RingTheory.Ideal.KrullsHeightTheorem | {
"line": 63,
"column": 47
} | {
"line": 63,
"column": 49
} | {
"line": 64,
"column": 2
} | [
{
"pp": "R : Type u_1\ninst✝³ : CommRing R\ninst✝² : IsNoetherianRing R\ninst✝¹ : IsLocalRing R\nI : Ideal R\ninst✝ : Submodule.IsPrincipal I\nhp : IsLocalRing.maximalIdeal R ∈ I.minimalPrimes\n⊢ (IsLocalRing.maximalIdeal R).height ≤ 1",
"ppTerm": "?m.24",
"assigned": true,
"usedConstants": [
... | [] | by | [anonymous] | by |
Mathlib.RingTheory.Ideal.Pure | {
"line": 44,
"column": 70
} | {
"line": 44,
"column": 72
} | {
"line": 45,
"column": 2
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\nI J : Ideal R\n⊢ Function.Injective ⇑(LinearMap.lTensor (R ⧸ I) (Submodule.subtype J)) ↔ I ⊓ J = I * J",
"ppTerm": "?m.30",
"assigned": true,
"usedConstants": [
"Ideal.Quotient.isScalarTower",
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWit... | [] | by | [anonymous] | by |
Mathlib.RingTheory.Ideal.Pure | {
"line": 61,
"column": 21
} | {
"line": 61,
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} | {
"line": 62,
"column": 2
} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommRing R\nI J : Ideal R\ninst✝ : I.Pure\n⊢ I ⊓ J = I * J",
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"assigned": true,
"usedConstants": [
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"Semiring... | [] | by | [anonymous] | by |
Mathlib.RingTheory.Ideal.Pure | {
"line": 68,
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} | {
"line": 68,
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} | {
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{
"pp": "R : Type u_1\ninst✝¹ : CommRing R\nI : Ideal R\ninst✝ : I.Pure\n⊢ IsIdempotentElem I",
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"Semiring.toModule",
"HMul.hMul",
"IsScalarTower.right",
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"CommSemiring.toSemiring",
... | [] | by | [anonymous] | by |
Mathlib.RingTheory.Ideal.Pure | {
"line": 76,
"column": 80
} | {
"line": 76,
"column": 82
} | {
"line": 76,
"column": 83
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\ne : R\nhe : IsIdempotentElem e\nh : FG (R ∙ e)\n⊢ e + (1 - e) = 1",
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"assigned": true,
"usedConstants": [
"AddGroupWithOne.toAddGroup",
"congrArg",
"CommSemiring.toSemiring",
"AddGroupWithOne.toAddMonoidWithOne",
... | [] | by | [anonymous] | by |
Mathlib.RingTheory.Ideal.Pure | {
"line": 77,
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} | {
"line": 77,
"column": 9
} | {
"line": 77,
"column": 10
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\ne : R\nhe : IsIdempotentElem e\nh : FG (R ∙ e)\n⊢ e * (1 - e) = 0",
"ppTerm": "?m.83",
"assigned": true,
"usedConstants": [
"_private.Mathlib.RingTheory.Ideal.Pure.0.Ideal.Pure.of_isIdempotentElem._proof_1_1"
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"usedFVars": [
"R",
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Mathlib.RingTheory.Ideal.KrullsHeightTheorem | {
"line": 109,
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} | {
"line": 109,
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} | {
"line": 110,
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} | [
{
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"assigned": true,
"usedConstants": [
"Eq.mpr",
"IsLocalization.minimalPrimes_map",
"inst... | [] | by | [anonymous] | by |
Mathlib.RingTheory.Ideal.KrullsHeightTheorem | {
"line": 134,
"column": 55
} | {
"line": 134,
"column": 57
} | {
"line": 134,
"column": 58
} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsNoetherianRing R\nx : R\nhx' : ¬IsUnit x\n⊢ span {x} ≠ ⊤",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Semiring.toModule",
"congrArg",
"CommSemiring.toSemiring",
"Ideal.span_singleton_eq_top._s... | [] | by | [anonymous] | by |
Mathlib.RingTheory.Ideal.KrullsHeightTheorem | {
"line": 133,
"column": 29
} | {
"line": 133,
"column": 31
} | {
"line": 134,
"column": 2
} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsNoetherianRing R\nx : R\nhx' : ¬IsUnit x\n⊢ (span {x}).height ≤ 1",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Semiring.toModule",
"instAddMonoidWithOneENat",
"instIsPrincipalSpanSingletonSet",
... | [] | by | [anonymous] | by |
Mathlib.RingTheory.Ideal.Pure | {
"line": 72,
"column": 14
} | {
"line": 72,
"column": 16
} | {
"line": 73,
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{
"pp": "R : Type u_1\ninst✝ : CommRing R\nI : Ideal R\nh : I.FG\nh' : IsIdempotentElem I\n⊢ I.Pure",
"ppTerm": "?m.8",
"assigned": true,
"usedConstants": [
"LinearMap.id",
"LinearMap.fst",
"AlgEquiv.prodQuotientOfIsIdempotentElem",
"Submodule",
"Submodule.Quotient.addC... | [] | by | [anonymous] | by |
Mathlib.RingTheory.Ideal.KrullsHeightTheorem | {
"line": 157,
"column": 60
} | {
"line": 157,
"column": 62
} | {
"line": 157,
"column": 63
} | [
{
"pp": "R : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsNoetherianRing R\nq p : Ideal R\ninst✝ : q.IsPrime\nhqp : q < p\nx : R\ns : Set R\nhp : p ∈ (span (insert x s)).minimalPrimes\nt : Set R\nhtq : t ⊆ ↑q\nhsp : s ⊆ ↑(span (insert x t)).radical\nf : R →+* R ⧸ span t := Quotient.mk (span t)\nhf : Function.Surje... | [] | by | [anonymous] | by |
Mathlib.RingTheory.Ideal.Pure | {
"line": 83,
"column": 14
} | {
"line": 83,
"column": 16
} | {
"line": 84,
"column": 2
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\nI : Ideal R\nH : ∀ ⦃J : Ideal R⦄, J.FG → I ⊓ J = I * J\n⊢ I.Pure",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Submodule",
"Submodule.Quotient.addCommMonoid",
"Semiring.toModule",
"HMul.hMul",
"IsSc... | [] | by | [anonymous] | by |
Mathlib.RingTheory.Ideal.Pure | {
"line": 92,
"column": 38
} | {
"line": 92,
"column": 40
} | {
"line": 93,
"column": 4
} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommRing R\nI : Ideal R\ninst✝ : I.Pure\nx : R\nhx : x ∈ I\nh : x ∈ I * span {x}\n⊢ ∃ y ∈ I, x = x * y",
"ppTerm": "?m.31",
"assigned": true,
"usedConstants": [
"Semiring.toModule",
"HMul.hMul",
"congrArg",
"CommSemiring.toSemiring",
"Mem... | [] | by | [anonymous] | by |
Mathlib.RingTheory.Ideal.KrullsHeightTheorem | {
"line": 159,
"column": 90
} | {
"line": 159,
"column": 92
} | {
"line": 160,
"column": 6
} | [
{
"pp": "R : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsNoetherianRing R\nq p : Ideal R\ninst✝ : q.IsPrime\nhqp : q < p\nx : R\ns : Set R\nhp : p ∈ (span (insert x s)).minimalPrimes\nt : Set R\nhtq : t ⊆ ↑q\nhsp : s ⊆ ↑(span (insert x t)).radical\nf : R →+* R ⧸ span t := Quotient.mk (span t)\nhf : Function.Surje... | [] | by | [anonymous] | by |
Mathlib.RingTheory.Ideal.Pure | {
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"column": 26
} | {
"line": 91,
"column": 28
} | {
"line": 92,
"column": 2
} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommRing R\nI : Ideal R\ninst✝ : I.Pure\nx : R\nhx : x ∈ I\n⊢ ∃ y ∈ I, x = x * y",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Ideal.subset_span",
"Submodule",
"Semiring.toModule",
"HMul.hMul",
"IsScalarTower.... | [] | by | [anonymous] | by |
Mathlib.RingTheory.Ideal.Pure | {
"line": 105,
"column": 8
} | {
"line": 105,
"column": 57
} | {
"line": 105,
"column": 57
} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommRing R\nI : Ideal R\nh : ∀ x ∈ I, ∃ y ∈ I, x = x * y\np : Ideal R\ninst✝ : p.IsPrime\nhle : I ≤ p\nx : R\nhx : x ∈ I\ny : R\nhy : y ∈ I\nheq : x = x * y\n⊢ IsUnit ((algebraMap R (Localization.AtPrime p)) (1 - y))",
"ppTerm": "?m.74",
"assigned": true,
"usedConstan... | [
"R : Type u_1\ninst✝¹ : CommRing R\nI : Ideal R\nh : ∀ x ∈ I, ∃ y ∈ I, x = x * y\np : Ideal R\ninst✝ : p.IsPrime\nhle : I ≤ p\nx : R\nhx : x ∈ I\ny : R\nhy : y ∈ I\nheq : x = x * y\n⊢ ∃ m ∈ p.primeCompl, 1 - y ∣ m"
] | IsLocalization.algebraMap_isUnit_iff p.primeCompl | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Ideal.KrullsHeightTheorem | {
"line": 162,
"column": 62
} | {
"line": 162,
"column": 64
} | {
"line": 162,
"column": 65
} | [
{
"pp": "R : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsNoetherianRing R\nq p : Ideal R\ninst✝ : q.IsPrime\nhqp : q < p\nx : R\ns : Set R\nhp : p ∈ (span (insert x s)).minimalPrimes\nt : Set R\nhtq : t ⊆ ↑q\nhsp : s ⊆ ↑(span (insert x t)).radical\nf : R →+* R ⧸ span t := Quotient.mk (span t)\nhf : Function.Surje... | [] | by | [anonymous] | by |
Mathlib.RingTheory.Ideal.Pure | {
"line": 106,
"column": 45
} | {
"line": 106,
"column": 47
} | {
"line": 106,
"column": 48
} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommRing R\nI : Ideal R\nh : ∀ x ∈ I, ∃ y ∈ I, x = x * y\np : Ideal R\ninst✝ : p.IsPrime\nhle : I ≤ p\nx : R\nhx : x ∈ I\ny : R\nhy : y ∈ I\nheq : x = x * y\n⊢ 1 - y ∣ 1 - y",
"ppTerm": "?m.104",
"assigned": true,
"usedConstants": [
"Dvd.dvd",
"AddGroupWit... | [] | by | [anonymous] | by |
Mathlib.RingTheory.Ideal.Pure | {
"line": 104,
"column": 67
} | {
"line": 104,
"column": 69
} | {
"line": 105,
"column": 4
} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommRing R\nI : Ideal R\nh : ∀ x ∈ I, ∃ y ∈ I, x = x * y\np : Ideal R\ninst✝ : p.IsPrime\nhle : I ≤ p\nx : R\nhx : x ∈ I\ny : R\nhy : y ∈ I\nheq : x = x * y\n⊢ IsUnit ((algebraMap R (Localization.AtPrime p)) (1 - y))",
"ppTerm": "?m.74",
"assigned": true,
"usedConstan... | [] | by | Lean.Elab.Tactic.evalWithAnnotateState | by |
Mathlib.RingTheory.Ideal.Pure | {
"line": 109,
"column": 34
} | {
"line": 109,
"column": 36
} | {
"line": 109,
"column": 37
} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommRing R\nI : Ideal R\nh : ∀ x ∈ I, ∃ y ∈ I, x = x * y\np : Ideal R\ninst✝ : p.IsPrime\nhle : I ≤ p\nx : R\nhx : x ∈ I\ny : R\nhy : y ∈ I\nheq : x = x * y\nthis : IsUnit ((algebraMap R (Localization.AtPrime p)) (1 - y))\n⊢ x * (1 - y) = 0",
"ppTerm": "?m.129",
"assigned... | [] | by | Lean.Elab.Tactic.evalWithAnnotateState | by |
Mathlib.RingTheory.Ideal.Pure | {
"line": 101,
"column": 64
} | {
"line": 101,
"column": 66
} | {
"line": 102,
"column": 2
} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommRing R\nI : Ideal R\nh : ∀ x ∈ I, ∃ y ∈ I, x = x * y\np : Ideal R\ninst✝ : p.IsPrime\nhle : I ≤ p\n⊢ I ≤ RingHom.ker (algebraMap R (Localization.AtPrime p))",
"ppTerm": "?m.36",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAddComm... | [] | by | [anonymous] | by |
Mathlib.RingTheory.Ideal.Pure | {
"line": 115,
"column": 66
} | {
"line": 115,
"column": 68
} | {
"line": 116,
"column": 2
} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommRing R\nI : Ideal R\ninst✝ : I.Pure\n⊢ RingHom.ker (RingHom.pi fun p ↦ algebraMap R (Localization.AtPrime (↑p).asIdeal)) = I",
"ppTerm": "?m.30",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"PrimeSpectrum.mk",
"Submodule",
"RingHom.instR... | [] | by | [anonymous] | by |
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