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
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Mathlib.RingTheory.Ideal.Pure | {
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Mathlib.RingTheory.Ideal.Pure | {
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Mathlib.RingTheory.Ideal.KrullsHeightTheorem | {
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Mathlib.RingTheory.Ideal.KrullsHeightTheorem | {
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Mathlib.RingTheory.Ideal.KrullsHeightTheorem | {
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{
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Mathlib.RingTheory.KrullDimension.LocalRing | {
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{
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Mathlib.RingTheory.Ideal.KrullsHeightTheorem | {
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"column": 10
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Mathlib.RingTheory.KrullDimension.LocalRing | {
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Mathlib.RingTheory.Ideal.KrullsHeightTheorem | {
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} | {
"line": 209,
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{
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Mathlib.RingTheory.KrullDimension.LocalRing | {
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... | [] | by | [anonymous] | by |
Mathlib.RingTheory.KrullDimension.LocalRing | {
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Mathlib.RingTheory.KrullDimension.LocalRing | {
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Mathlib.RingTheory.Ideal.KrullsHeightTheorem | {
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Mathlib.RingTheory.KrullDimension.LocalRing | {
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Mathlib.RingTheory.Spectrum.Prime.Module | {
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Mathlib.RingTheory.Spectrum.Prime.Module | {
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Mathlib.RingTheory.Spectrum.Prime.Module | {
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Mathlib.RingTheory.Spectrum.Prime.Module | {
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{
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Mathlib.RingTheory.Spectrum.Prime.Module | {
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Mathlib.RingTheory.KrullDimension.Module | {
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Mathlib.RingTheory.KrullDimension.Module | {
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Mathlib.RingTheory.KrullDimension.Module | {
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{
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Mathlib.RingTheory.KrullDimension.Module | {
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} | {
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Mathlib.RingTheory.KrullDimension.Module | {
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Mathlib.RingTheory.KrullDimension.Module | {
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} | {
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{
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Mathlib.RingTheory.KrullDimension.Module | {
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} | {
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{
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Mathlib.RingTheory.KrullDimension.Module | {
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"congrArg... | [] | by | [anonymous] | by |
Mathlib.RingTheory.Ideal.KrullsHeightTheorem | {
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} | {
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} | {
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Mathlib.Topology.Algebra.Category.ProfiniteGrp.Limits | {
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{
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Mathlib.RingTheory.KrullDimension.Module | {
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Mathlib.RingTheory.KrullDimension.Module | {
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"Nontrivial... | [] | by | [anonymous] | by |
Mathlib.Topology.Algebra.Category.ProfiniteGrp.Limits | {
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{
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Mathlib.RingTheory.Ideal.KrullsHeightTheorem | {
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{
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Mathlib.RingTheory.Ideal.KrullsHeightTheorem | {
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{
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Mathlib.RingTheory.KrullDimension.PID | {
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} | {
"line": 28,
"column": 19
} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsPrincipalIdealRing R\nI : Ideal R\nhI : I.IsPrime\nP : Ideal R\nhlt : P < I\nhP : P.IsPrime\nthis : IsPrincipalIdealRing (R ⧸ P)\n⊢ RingHom.ker (Ideal.Quotient.mk P) ≤ I",
"ppTerm": "?m.104",
"assigned": true,
"usedConstants": [
"Eq.mpr",
... | [] | by | [anonymous] | by |
Mathlib.RingTheory.KrullDimension.PID | {
"line": 26,
"column": 52
} | {
"line": 26,
"column": 54
} | {
"line": 27,
"column": 4
} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsPrincipalIdealRing R\nI : Ideal R\nhI : I.IsPrime\nP : Ideal R\nhlt : P < I\nhP : P.IsPrime\nthis : IsPrincipalIdealRing (R ⧸ P)\n⊢ (Ideal.map (Ideal.Quotient.mk P) I).IsMaximal",
"ppTerm": "?m.85",
"assigned": true,
"usedConstants": [
"Eq.... | [] | by | Lean.Elab.Tactic.evalWithAnnotateState | by |
Mathlib.RingTheory.KrullDimension.PID | {
"line": 21,
"column": 54
} | {
"line": 21,
"column": 56
} | {
"line": 22,
"column": 2
} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsPrincipalIdealRing R\n⊢ Ring.KrullDimLE 1 R",
"ppTerm": "?m.5",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"Eq.mpr",
"RingHom.instRingHomClass",
"minimalPrimes_eq_minimals",
"Preorder.toLT",
"Lattice.toSem... | [] | by | [anonymous] | by |
Mathlib.RingTheory.KrullDimension.PID | {
"line": 44,
"column": 4
} | {
"line": 44,
"column": 45
} | {
"line": 46,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : IsPrincipalIdealRing R\nh : ¬IsField R\nh' : ringKrullDim R < 1\nh'' : Ring.KrullDimLE 0 R\n⊢ IsField R",
"ppTerm": "?m.50",
"assigned": true,
"usedConstants": [
"CommRing.toCommSemiring",
"Ring.KrullDimLE.isField_o... | [] | exact Ring.KrullDimLE.isField_of_isDomain | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.RingTheory.KrullDimension.PID | {
"line": 38,
"column": 71
} | {
"line": 38,
"column": 73
} | {
"line": 39,
"column": 2
} | [
{
"pp": "R : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : IsPrincipalIdealRing R\nh : ¬IsField R\n⊢ ringKrullDim R = 1",
"ppTerm": "?m.8",
"assigned": true,
"usedConstants": [
"WithBot.addMonoidWithOne",
"WithBot.instPreorder",
"Eq.mpr",
"WithBot",
"Ring.... | [] | by | [anonymous] | by |
Mathlib.RingTheory.KrullDimension.PID | {
"line": 51,
"column": 45
} | {
"line": 51,
"column": 47
} | {
"line": 51,
"column": 48
} | [
{
"pp": "R : Type u_1\ninst✝³ : CommRing R\ninst✝² : IsDomain R\ninst✝¹ : IsPrincipalIdealRing R\nm : Ideal R\ninst✝ : m.IsMaximal\nh✝ : ¬IsField R\nh : ↑m.height ≤ 1\n⊢ m.height ≤ 1",
"ppTerm": "?m.26",
"assigned": true,
"usedConstants": [
"WithBot.addMonoidWithOne",
"WithBot.instPreord... | [] | by | [anonymous] | by |
Mathlib.RingTheory.KrullDimension.PID | {
"line": 49,
"column": 20
} | {
"line": 49,
"column": 22
} | {
"line": 50,
"column": 2
} | [
{
"pp": "R : Type u_1\ninst✝³ : CommRing R\ninst✝² : IsDomain R\ninst✝¹ : IsPrincipalIdealRing R\nm : Ideal R\ninst✝ : m.IsMaximal\nh : ¬IsField R\n⊢ m.height = 1",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"WithBot.addMonoidWithOne",
"WithBot.instPreorder",
"Eq.mpr",... | [] | by | [anonymous] | by |
Mathlib.Topology.Algebra.Category.ProfiniteGrp.Limits | {
"line": 97,
"column": 78
} | {
"line": 97,
"column": 80
} | {
"line": 98,
"column": 2
} | [
{
"pp": "P : ProfiniteGrp.{u}\n⊢ DenseRange ⇑(Hom.hom P.toLimit)",
"ppTerm": "?m.5",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"isOpen_pi_iff",
"Eq.mpr",
"iInf",
"MonoidHom.instFunLike",
"FiniteGrp",
"ProfiniteGrp.limitConePtAux",
"Pi.topological... | [] | by | [anonymous] | by |
Mathlib.Topology.Algebra.Category.ProfiniteGrp.Limits | {
"line": 123,
"column": 6
} | {
"line": 123,
"column": 26
} | {
"line": 123,
"column": 27
} | [
{
"pp": "P : ProfiniteGrp.{u}\nthis : IsClosed (Set.range ⇑(Hom.hom P.toLimit))\n⊢ Function.Surjective ⇑(Hom.hom P.toLimit)",
"ppTerm": "?m.30",
"assigned": true,
"usedConstants": [
"Set.range_eq_univ",
"Eq.mpr",
"congrArg",
"ProfiniteGrp.toLimit",
"Set.univ",
"Pa... | [
"P : ProfiniteGrp.{u}\nthis : IsClosed (Set.range ⇑(Hom.hom P.toLimit))\n⊢ Set.range ⇑(Hom.hom P.toLimit) = Set.univ"
] | ← Set.range_eq_univ, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.Algebra.Category.ProfiniteGrp.Limits | {
"line": 120,
"column": 87
} | {
"line": 120,
"column": 89
} | {
"line": 121,
"column": 2
} | [
{
"pp": "P : ProfiniteGrp.{u}\n⊢ Function.Surjective ⇑(Hom.hom P.toLimit)",
"ppTerm": "?m.4",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"Set.range_eq_univ",
"Eq.mpr",
"MulOne.toOne",
"Pi.t2Space",
"ProfiniteGrp.limitConePtAux",
"Pi.topologicalSpace",
... | [] | by | [anonymous] | by |
Mathlib.Topology.Algebra.Category.ProfiniteGrp.Limits | {
"line": 127,
"column": 85
} | {
"line": 127,
"column": 87
} | {
"line": 128,
"column": 2
} | [
{
"pp": "P : ProfiniteGrp.{u}\n⊢ Function.Injective ⇑(Hom.hom P.toLimit)",
"ppTerm": "?m.4",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"Eq.mpr",
"MulOne.toOne",
"False",
"MonoidHom.instFunLike",
"InvOneClass.toOne",
"ProfiniteGrp.limitConePtAux",
... | [] | by | [anonymous] | by |
Mathlib.Topology.Algebra.Category.ProfiniteGrp.Limits | {
"line": 145,
"column": 69
} | {
"line": 145,
"column": 71
} | {
"line": 146,
"column": 2
} | [
{
"pp": "P : ProfiniteGrp.{u}\n⊢ IsIso P.toLimit",
"ppTerm": "?m.5",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"CategoryTheory.IsIso",
"congrArg",
"CategoryTheory.ConcreteCategory.hom",
"ProfiniteGrp.toLimit",
"ProfiniteGrp.toLimit_surjective",
"Partial... | [] | by | [anonymous] | by |
Mathlib.RingTheory.Ideal.KrullsHeightTheorem | {
"line": 187,
"column": 36
} | {
"line": 187,
"column": 38
} | {
"line": 188,
"column": 2
} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsNoetherianRing R\nI p : Ideal R\nhp : p ∈ I.minimalPrimes\n⊢ p.height ≤ Cardinal.toENat (Submodule.spanRank I)",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"Ideal.radical_mono",
"Ideal.span_le",
"Iff.mpr",
"LE... | [] | by | [anonymous] | by |
Mathlib.RingTheory.Ideal.KrullsHeightTheorem | {
"line": 241,
"column": 25
} | {
"line": 241,
"column": 27
} | {
"line": 242,
"column": 2
} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsNoetherianRing R\np : Ideal R\ns : Finset R\nhI : p ∈ (span ↑s).minimalPrimes\n⊢ p.height ≤ ↑s.card",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Semiring.toModule",
"ENat.instNatCast",
"Cardinal",
... | [] | by | [anonymous] | by |
Mathlib.RingTheory.Ideal.KrullsHeightTheorem | {
"line": 250,
"column": 63
} | {
"line": 250,
"column": 65
} | {
"line": 250,
"column": 66
} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsNoetherianRing R\np : Ideal R\ns : Set R\nhs : s.Finite\nhI : p ∈ (span s).minimalPrimes\n⊢ p ∈ (span ↑hs.toFinset).minimalPrimes",
"ppTerm": "?m.24",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Ideal.minimalPrimes",
"congrA... | [] | by | [anonymous] | by |
Mathlib.RingTheory.Ideal.KrullsHeightTheorem | {
"line": 248,
"column": 26
} | {
"line": 248,
"column": 28
} | {
"line": 249,
"column": 2
} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsNoetherianRing R\np : Ideal R\ns : Set R\nhs : s.Finite\nhI : p ∈ (span s).minimalPrimes\n⊢ p.height ≤ ↑s.ncard",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Ideal.minimalPrimes",
"ENat.instNatCast",
... | [] | by | [anonymous] | by |
Mathlib.RingTheory.Ideal.KrullsHeightTheorem | {
"line": 255,
"column": 36
} | {
"line": 255,
"column": 38
} | {
"line": 256,
"column": 2
} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsNoetherianRing R\nI : Ideal R\nhI : I ≠ ⊤\n⊢ I.height ≤ Cardinal.toENat (Submodule.spanRank I)",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"iInf",
"instCompleteLinearOrderENat",
"Semiring.toModule",... | [] | by | [anonymous] | by |
Mathlib.RingTheory.Ideal.KrullsHeightTheorem | {
"line": 262,
"column": 46
} | {
"line": 262,
"column": 48
} | {
"line": 263,
"column": 4
} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsNoetherianRing R\nI : Ideal R\nhI : I ≠ ⊤\n⊢ ↑(Submodule.spanFinrank I) = Cardinal.toENat (Submodule.spanRank I)",
"ppTerm": "?m.28",
"assigned": true,
"usedConstants": [
"Submodule.fg_iff_spanRank_eq_spanFinrank",
"Iff.mpr",
"E... | [] | by | Lean.Elab.Tactic.evalWithAnnotateState | by |
Mathlib.RingTheory.Ideal.KrullsHeightTheorem | {
"line": 261,
"column": 32
} | {
"line": 261,
"column": 34
} | {
"line": 262,
"column": 2
} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsNoetherianRing R\nI : Ideal R\nhI : I ≠ ⊤\n⊢ I.height ≤ ↑(Submodule.spanFinrank I)",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"Submodule.fg_iff_spanRank_eq_spanFinrank",
"Iff.mpr",
"Eq.mpr",
"NonAssocSemirin... | [] | by | [anonymous] | by |
Mathlib.RingTheory.Ideal.KrullsHeightTheorem | {
"line": 267,
"column": 29
} | {
"line": 267,
"column": 31
} | {
"line": 268,
"column": 2
} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsNoetherianRing R\nI : Ideal R\nhI : I ≠ ⊤\n⊢ ↑I.height ≤ Submodule.spanRank I",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"Semiring.toModule",
"Cardinal",
"CommSemiring.toSemiring",
"Cardinal.commSemiring",
... | [] | by | [anonymous] | by |
Mathlib.RingTheory.Ideal.KrullsHeightTheorem | {
"line": 276,
"column": 51
} | {
"line": 276,
"column": 53
} | {
"line": 277,
"column": 2
} | [
{
"pp": "R : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsNoetherianRing R\ninst✝ : IsLocalRing R\n⊢ FiniteRingKrullDim R",
"ppTerm": "?m.4",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"Eq.mpr",
"WithBot.some",
"WithBot",
"Semiring.toModule",
"instTopENat",
... | [] | by | [anonymous] | by |
Mathlib.RingTheory.Ideal.KrullsHeightTheorem | {
"line": 287,
"column": 60
} | {
"line": 287,
"column": 62
} | {
"line": 288,
"column": 2
} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsNoetherianRing R\nI : Ideal R\nhI : I ≠ ⊤\n⊢ ∃ J ≤ I, Submodule.spanRank J = ↑I.height ∧ J.height = I.height",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"Ideal.height_le_spanRank",
"Eq.mpr",
"False... | [] | by | [anonymous] | by |
Mathlib.RingTheory.KrullDimension.Polynomial | {
"line": 33,
"column": 62
} | {
"line": 33,
"column": 64
} | {
"line": 34,
"column": 2
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\n⊢ ringKrullDim R[X] ≤ 2 * ringKrullDim R + 1",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"WithBot.instPreorder",
"Eq.mpr",
"Polynomial.C",
"RingHom.instRingHomClass",
"WithBot",
"Ring.krullDimLE_iff",
... | [] | by | [anonymous] | by |
Mathlib.RingTheory.Ideal.KrullsHeightTheorem | {
"line": 311,
"column": 7
} | {
"line": 311,
"column": 9
} | {
"line": 311,
"column": 10
} | [
{
"pp": "R : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsNoetherianRing R\np : Ideal R\ninst✝ : p.IsPrime\nn : ℕ∞\nI : Ideal R\nhp : p ∈ I.minimalPrimes\nhI : Submodule.spanRank I ≤ ↑n\n⊢ Cardinal.toENat (Submodule.spanRank I) ≤ n",
"ppTerm": "?m.105",
"assigned": true,
"usedConstants": [
"RingH... | [] | by | [anonymous] | by |
Mathlib.RingTheory.Ideal.KrullsHeightTheorem | {
"line": 302,
"column": 85
} | {
"line": 302,
"column": 87
} | {
"line": 303,
"column": 2
} | [
{
"pp": "R : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsNoetherianRing R\np : Ideal R\ninst✝ : p.IsPrime\nn : ℕ∞\n⊢ p.height ≤ n ↔ ∃ I, p ∈ I.minimalPrimes ∧ Submodule.spanRank I ≤ ↑n",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"RingHom.instRingHomClass",
"Mu... | [] | by | [anonymous] | by |
Mathlib.RingTheory.KrullDimension.Polynomial | {
"line": 58,
"column": 88
} | {
"line": 58,
"column": 90
} | {
"line": 59,
"column": 4
} | [
{
"pp": "R : Type u_1\ninst✝⁴ : CommRing R\ninst✝³ : IsNoetherianRing R\np : Ideal R\ninst✝² : p.IsMaximal\nP : Ideal R[X]\ninst✝¹ : P.IsMaximal\ninst✝ : P.LiesOver p\nx✝ : Field (R ⧸ p) := Quotient.field p\nh : (Ideal.map (Ideal.Quotient.mk (Ideal.map (algebraMap R R[X]) p)) P).height = 1\n⊢ P.height = p.heigh... | [] | by | [anonymous] | by |
Mathlib.RingTheory.Ideal.KrullsHeightTheorem | {
"line": 319,
"column": 23
} | {
"line": 319,
"column": 25
} | {
"line": 319,
"column": 26
} | [
{
"pp": "R : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsNoetherianRing R\np : Ideal R\ninst✝ : p.IsPrime\nI : Ideal R\nhI : p ∈ I.minimalPrimes\nhr : Submodule.spanRank I ≤ ↑p.height\nhs : (Submodule.generators I).Finite\n⊢ p ∈ (span ↑hs.toFinset).minimalPrimes",
"ppTerm": "?m.75",
"assigned": true,
... | [] | by | [anonymous] | by |
Mathlib.RingTheory.Ideal.KrullsHeightTheorem | {
"line": 323,
"column": 4
} | {
"line": 323,
"column": 38
} | {
"line": 324,
"column": 2
} | [
{
"pp": "case refine_1\nR : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsNoetherianRing R\np : Ideal R\ninst✝ : p.IsPrime\nI : Ideal R\nhI : p ∈ I.minimalPrimes\nhr : ↑(Submodule.spanFinrank I) ≤ ↑p.height\nhs : (Submodule.generators I).Finite\n⊢ ↑(Submodule.spanFinrank I) ≤ p.height",
"ppTerm": "?refine_1",
... | [] | exact Cardinal.nat_le_ofENat.mp hr | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.RingTheory.Ideal.KrullsHeightTheorem | {
"line": 316,
"column": 70
} | {
"line": 316,
"column": 72
} | {
"line": 317,
"column": 2
} | [
{
"pp": "R : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsNoetherianRing R\np : Ideal R\ninst✝ : p.IsPrime\n⊢ ∃ s, p ∈ (span ↑s).minimalPrimes ∧ ↑s.card = p.height",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"Submodule.fg_iff_spanRank_eq_spanFinrank",
"Iff.mpr",
"Eq.mpr"... | [] | by | [anonymous] | by |
Mathlib.RingTheory.KrullDimension.Polynomial | {
"line": 63,
"column": 79
} | {
"line": 63,
"column": 81
} | {
"line": 64,
"column": 4
} | [
{
"pp": "R : Type u_1\ninst✝⁴ : CommRing R\ninst✝³ : IsNoetherianRing R\np : Ideal R\ninst✝² : p.IsMaximal\nP : Ideal R[X]\ninst✝¹ : P.IsMaximal\ninst✝ : P.LiesOver p\nx✝ : Field (R ⧸ p) := Quotient.field p\ne : R[X] ⧸ Ideal.map (algebraMap R R[X]) p ≃+* (R ⧸ p)[X] := p.polynomialQuotientEquivQuotientPolynomial... | [] | by | Lean.Elab.Tactic.evalWithAnnotateState | by |
Mathlib.RingTheory.Ideal.KrullsHeightTheorem | {
"line": 341,
"column": 14
} | {
"line": 341,
"column": 16
} | {
"line": 341,
"column": 17
} | [
{
"pp": "R : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsNoetherianRing R\nI p : Ideal R\ninst✝ : p.IsPrime\nhrp : I ≤ p\np' : Ideal (R ⧸ I) := map (algebraMap R (R ⧸ I)) p\nthis : p'.IsPrime\ns : Finset (R ⧸ I)\nhps : p' ∈ (span ↑s).minimalPrimes\nhs : ↑s.card = p'.height\nhsp' : ↑s ⊆ ↑p'\nx : R\nhx : (Quotient.... | [] | by | [anonymous] | by |
Mathlib.RingTheory.Ideal.KrullsHeightTheorem | {
"line": 338,
"column": 49
} | {
"line": 338,
"column": 51
} | {
"line": 339,
"column": 4
} | [
{
"pp": "R : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsNoetherianRing R\nI p : Ideal R\ninst✝ : p.IsPrime\nhrp : I ≤ p\np' : Ideal (R ⧸ I) := map (algebraMap R (R ⧸ I)) p\nthis : p'.IsPrime\ns : Finset (R ⧸ I)\nhps : p' ∈ (span ↑s).minimalPrimes\nhs : ↑s.card = p'.height\nhsp' : ↑s ⊆ ↑p'\n⊢ Set.SurjOn ⇑(Quotien... | [] | by | Lean.Elab.Tactic.evalWithAnnotateState | by |
Mathlib.RingTheory.KrullDimension.Polynomial | {
"line": 56,
"column": 62
} | {
"line": 56,
"column": 64
} | {
"line": 57,
"column": 2
} | [
{
"pp": "R : Type u_1\ninst✝⁴ : CommRing R\ninst✝³ : IsNoetherianRing R\np : Ideal R\ninst✝² : p.IsMaximal\nP : Ideal R[X]\ninst✝¹ : P.IsMaximal\ninst✝ : P.LiesOver p\n⊢ P.height = p.height + 1",
"ppTerm": "?m.25",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Polynomial.C",
"Rin... | [] | by | [anonymous] | by |
Mathlib.RingTheory.KrullDimension.Polynomial | {
"line": 73,
"column": 80
} | {
"line": 73,
"column": 82
} | {
"line": 74,
"column": 2
} | [
{
"pp": "R : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsNoetherianRing R\np : Ideal R\ninst✝ : p.IsMaximal\n⊢ (Ideal.map C p).height = p.height",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"Polynomial.C",
"Module.Flat.of_free",
"CommRing",
"Semiring.toModule",
... | [] | by | [anonymous] | by |
Mathlib.RingTheory.KrullDimension.Polynomial | {
"line": 83,
"column": 22
} | {
"line": 83,
"column": 24
} | {
"line": 83,
"column": 25
} | [
{
"pp": "R : Type u_1\ninst✝³ : CommRing R\ninst✝² : IsNoetherianRing R\np : Ideal R\nP : Ideal R[X]\ninst✝¹ : P.IsMaximal\ninst✝ : P.LiesOver p\n⊢ p.IsPrime",
"ppTerm": "?m.26",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"CommSemiring.toSemiring",
"Polynomial... | [] | by | Lean.Elab.Tactic.evalWithAnnotateState | by |
Mathlib.RingTheory.Ideal.KrullsHeightTheorem | {
"line": 348,
"column": 80
} | {
"line": 348,
"column": 82
} | {
"line": 349,
"column": 6
} | [
{
"pp": "R : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsNoetherianRing R\nI p : Ideal R\ninst✝ : p.IsPrime\nhrp : I ≤ p\np' : Ideal (R ⧸ I) := map (algebraMap R (R ⧸ I)) p\nthis✝ : p'.IsPrime\ns : Finset (R ⧸ I)\nhps : p' ∈ (span ↑s).minimalPrimes\nhs : ↑s.card = p'.height\nhsp' : ↑s ⊆ ↑p'\nthis : Set.SurjOn ⇑(Q... | [] | by | Lean.Elab.Tactic.evalWithAnnotateState | by |
Mathlib.RingTheory.KrullDimension.Polynomial | {
"line": 86,
"column": 25
} | {
"line": 86,
"column": 27
} | {
"line": 87,
"column": 4
} | [
{
"pp": "R : Type u_1\ninst✝³ : CommRing R\ninst✝² : IsNoetherianRing R\np : Ideal R\nP : Ideal R[X]\ninst✝¹ : P.IsMaximal\ninst✝ : P.LiesOver p\nthis : p.IsPrime\nRₚ : Type u_1 := Localization.AtPrime p\np' : Ideal Rₚ := Ideal.map (algebraMap R Rₚ) p\np'_def : p' = Ideal.map (algebraMap R Rₚ) p\n⊢ p'.IsMaximal... | [] | by | Lean.Elab.Tactic.evalWithAnnotateState | by |
Mathlib.RingTheory.Ideal.KrullsHeightTheorem | {
"line": 345,
"column": 33
} | {
"line": 345,
"column": 35
} | {
"line": 346,
"column": 4
} | [
{
"pp": "R : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsNoetherianRing R\nI p : Ideal R\ninst✝ : p.IsPrime\nhrp : I ≤ p\np' : Ideal (R ⧸ I) := map (algebraMap R (R ⧸ I)) p\nthis✝ : p'.IsPrime\ns : Finset (R ⧸ I)\nhps : p' ∈ (span ↑s).minimalPrimes\nhs : ↑s.card = p'.height\nhsp' : ↑s ⊆ ↑p'\nthis : Set.SurjOn ⇑(Q... | [] | by | [anonymous] | by |
Mathlib.RingTheory.KrullDimension.Polynomial | {
"line": 90,
"column": 60
} | {
"line": 90,
"column": 62
} | {
"line": 91,
"column": 4
} | [
{
"pp": "R : Type u_1\ninst✝³ : CommRing R\ninst✝² : IsNoetherianRing R\np : Ideal R\nP : Ideal R[X]\ninst✝¹ : P.IsMaximal\ninst✝ : P.LiesOver p\nthis✝ : p.IsPrime\nRₚ : Type u_1 := Localization.AtPrime p\np' : Ideal Rₚ := Ideal.map (algebraMap R Rₚ) p\np'_def : p' = Ideal.map (algebraMap R Rₚ) p\nthis : p'.IsM... | [] | by | Lean.Elab.Tactic.evalWithAnnotateState | by |
Mathlib.RingTheory.Ideal.KrullsHeightTheorem | {
"line": 332,
"column": 65
} | {
"line": 332,
"column": 67
} | {
"line": 333,
"column": 2
} | [
{
"pp": "R : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsNoetherianRing R\nI p : Ideal R\ninst✝ : p.IsPrime\nhrp : I ≤ p\n⊢ p.height ≤ (map (Quotient.mk I) p).height + ↑(Submodule.spanFinrank I)",
"ppTerm": "?m.32",
"assigned": true,
"usedConstants": [
"_private.Mathlib.RingTheory.Ideal.KrullsHe... | [] | by | [anonymous] | by |
Mathlib.RingTheory.Ideal.KrullsHeightTheorem | {
"line": 360,
"column": 55
} | {
"line": 360,
"column": 57
} | {
"line": 361,
"column": 2
} | [
{
"pp": "R : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsNoetherianRing R\np I : Ideal R\ninst✝ : p.IsPrime\nh : I ≤ p\n⊢ ↑p.height ≤ ringKrullDim (R ⧸ I) + ↑(Submodule.spanFinrank I)",
"ppTerm": "?m.24",
"assigned": true,
"usedConstants": [
"WithBot.instPreorder",
"Ideal.Quotient.commSemi... | [] | by | [anonymous] | by |
Mathlib.RingTheory.KrullDimension.Polynomial | {
"line": 97,
"column": 37
} | {
"line": 97,
"column": 39
} | {
"line": 98,
"column": 4
} | [
{
"pp": "R : Type u_1\ninst✝³ : CommRing R\ninst✝² : IsNoetherianRing R\np : Ideal R\nP : Ideal R[X]\ninst✝¹ : P.IsMaximal\ninst✝ : P.LiesOver p\nthis✝³ : p.IsPrime\nRₚ : Type u_1 := Localization.AtPrime p\np' : Ideal Rₚ := Ideal.map (algebraMap R Rₚ) p\np'_def : p' = Ideal.map (algebraMap R Rₚ) p\nthis✝² : p'.... | [] | by | Lean.Elab.Tactic.evalWithAnnotateState | by |
Mathlib.RingTheory.Ideal.KrullsHeightTheorem | {
"line": 370,
"column": 61
} | {
"line": 370,
"column": 63
} | {
"line": 371,
"column": 2
} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsNoetherianRing R\nI : Ideal R\nh : I ≤ Ring.jacobson R\n⊢ ringKrullDim R ≤ ringKrullDim (R ⧸ I) + ↑(Submodule.spanFinrank I)",
"ppTerm": "?m.21",
"assigned": true,
"usedConstants": [
"CharP.cast_eq_zero",
"Nontrivial",
"WithBot.... | [] | by | [anonymous] | by |
Mathlib.RingTheory.KrullDimension.Polynomial | {
"line": 100,
"column": 37
} | {
"line": 100,
"column": 39
} | {
"line": 101,
"column": 4
} | [
{
"pp": "R : Type u_1\ninst✝³ : CommRing R\ninst✝² : IsNoetherianRing R\np : Ideal R\nP : Ideal R[X]\ninst✝¹ : P.IsMaximal\ninst✝ : P.LiesOver p\nthis✝³ : p.IsPrime\nRₚ : Type u_1 := Localization.AtPrime p\np' : Ideal Rₚ := Ideal.map (algebraMap R Rₚ) p\np'_def : p' = Ideal.map (algebraMap R Rₚ) p\nthis✝² : p'.... | [] | by | Lean.Elab.Tactic.evalWithAnnotateState | by |
Mathlib.RingTheory.Ideal.KrullsHeightTheorem | {
"line": 380,
"column": 83
} | {
"line": 380,
"column": 85
} | {
"line": 381,
"column": 2
} | [
{
"pp": "R : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsNoetherianRing R\ns : Set R\np : Ideal R\ninst✝ : p.IsPrime\nhrm : s ⊆ ↑p\n⊢ p.height ≤ (map (Quotient.mk (span s)) p).height + s.encard",
"ppTerm": "?m.29",
"assigned": true,
"usedConstants": [
"Ideal.span_le",
"Eq.mpr",
"le_r... | [] | by | [anonymous] | by |
Mathlib.RingTheory.Ideal.KrullsHeightTheorem | {
"line": 387,
"column": 82
} | {
"line": 387,
"column": 84
} | {
"line": 388,
"column": 2
} | [
{
"pp": "R : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsNoetherianRing R\np : Ideal R\ninst✝ : p.IsPrime\ns : Set R\nhs : s ⊆ ↑p\n⊢ ↑p.height ≤ ringKrullDim (R ⧸ span s) + ↑s.encard",
"ppTerm": "?m.21",
"assigned": true,
"usedConstants": [
"WithBot.instPreorder",
"Ideal.Quotient.commSemir... | [] | by | [anonymous] | by |
Mathlib.RingTheory.Ideal.KrullsHeightTheorem | {
"line": 394,
"column": 65
} | {
"line": 394,
"column": 67
} | {
"line": 394,
"column": 68
} | [
{
"pp": "R : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsNoetherianRing R\nr : R\np : Ideal R\ninst✝ : p.IsPrime\nhrm : r ∈ p\n⊢ {r} ⊆ ↑p",
"ppTerm": "?m.43",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"SetLike.mem_coe._simp_1",
"Semiring.toModule",
"CommSemiring.toSemiring... | [] | by | [anonymous] | by |
Mathlib.RingTheory.Ideal.KrullsHeightTheorem | {
"line": 393,
"column": 62
} | {
"line": 393,
"column": 64
} | {
"line": 394,
"column": 2
} | [
{
"pp": "R : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsNoetherianRing R\nr : R\np : Ideal R\ninst✝ : p.IsPrime\nhrm : r ∈ p\n⊢ p.height ≤ (map (Quotient.mk (span {r})) p).height + 1",
"ppTerm": "?m.34",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"SetLike.mem_coe._simp_1",
"Set.... | [] | by | [anonymous] | by |
Mathlib.RingTheory.KrullDimension.Polynomial | {
"line": 82,
"column": 31
} | {
"line": 82,
"column": 33
} | {
"line": 83,
"column": 2
} | [
{
"pp": "R : Type u_1\ninst✝³ : CommRing R\ninst✝² : IsNoetherianRing R\np : Ideal R\nP : Ideal R[X]\ninst✝¹ : P.IsMaximal\ninst✝ : P.LiesOver p\n⊢ P.height = p.height + 1",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"Eq.mpr",
"Polynomial.C",
"NonAssoc... | [] | by | [anonymous] | by |
Mathlib.RingTheory.Ideal.KrullsHeightTheorem | {
"line": 399,
"column": 67
} | {
"line": 399,
"column": 69
} | {
"line": 399,
"column": 70
} | [
{
"pp": "R : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsNoetherianRing R\nr : R\np : Ideal R\ninst✝ : p.IsPrime\nhrp : r ∈ p\n⊢ {r} ⊆ ↑?m.31",
"ppTerm": "?m.36",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"SetLike.mem_coe._simp_1",
"Semiring.toModule",
"CommSemiring.toSemi... | [] | by | [anonymous] | by |
Mathlib.RingTheory.Ideal.KrullsHeightTheorem | {
"line": 398,
"column": 66
} | {
"line": 398,
"column": 68
} | {
"line": 399,
"column": 2
} | [
{
"pp": "R : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsNoetherianRing R\nr : R\np : Ideal R\ninst✝ : p.IsPrime\nhrp : r ∈ p\n⊢ ↑p.height ≤ ringKrullDim (R ⧸ span {r}) + 1",
"ppTerm": "?m.26",
"assigned": true,
"usedConstants": [
"WithBot.instPreorder",
"Ideal.Quotient.commSemiring",
... | [] | by | [anonymous] | by |
Mathlib.RingTheory.KrullDimension.Polynomial | {
"line": 107,
"column": 83
} | {
"line": 107,
"column": 85
} | {
"line": 108,
"column": 2
} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsNoetherianRing R\n⊢ ringKrullDim R[X] = ringKrullDim R + 1",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"Nontrivial",
"Iff.mpr",
"WithBot.instPreorder",
"ringKrullDim_succ_le_ringKrullDim_polynomial",
"E... | [] | by | [anonymous] | by |
Mathlib.RingTheory.Ideal.KrullsHeightTheorem | {
"line": 403,
"column": 67
} | {
"line": 403,
"column": 69
} | {
"line": 404,
"column": 2
} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsNoetherianRing R\ns : Set R\nhs : s ⊆ ↑(Ring.jacobson R)\n⊢ ringKrullDim R ≤ ringKrullDim (R ⧸ Ideal.span s) + ↑s.encard",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"WithBot.instPreorder",
"Ideal.Quotient.commSemiring",
... | [] | by | [anonymous] | by |
Mathlib.RingTheory.Ideal.KrullsHeightTheorem | {
"line": 410,
"column": 75
} | {
"line": 410,
"column": 77
} | {
"line": 411,
"column": 2
} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsNoetherianRing R\ns : Finset R\nhs : ↑s ⊆ ↑(Ring.jacobson R)\n⊢ ringKrullDim R ≤ ringKrullDim (R ⧸ Ideal.span ↑s) + ↑s.card",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"WithBot.instPreorder",
"Ideal.Quotient.commSemiring... | [] | by | [anonymous] | by |
Mathlib.RingTheory.KrullDimension.Polynomial | {
"line": 121,
"column": 69
} | {
"line": 121,
"column": 71
} | {
"line": 122,
"column": 2
} | [
{
"pp": "R : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsNoetherianRing R\nι : Type u_2\ninst✝ : Finite ι\n⊢ ringKrullDim (MvPolynomial ι R) = ringKrullDim R + ↑(Nat.card ι)",
"ppTerm": "?m.10",
"assigned": true,
"usedConstants": [
"CharP.cast_eq_zero",
"WithBot.addMonoidWithOne",
"E... | [] | by | [anonymous] | by |
Mathlib.RingTheory.KrullDimension.Regular | {
"line": 36,
"column": 80
} | {
"line": 36,
"column": 82
} | {
"line": 37,
"column": 2
} | [
{
"pp": "R : Type u_1\ninst✝² : CommRing R\nM : Type u_2\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nq : LTSeries ↑(support R M)\n⊢ ∃ p, q.length ≤ p.length ∧ (↑(RelSeries.last p)).asIdeal.IsMaximal",
"ppTerm": "?m.30",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"PrimeSpectrum.mk",... | [] | by | [anonymous] | by |
Mathlib.RingTheory.Spectrum.Prime.LTSeries | {
"line": 50,
"column": 68
} | {
"line": 50,
"column": 70
} | {
"line": 50,
"column": 71
} | [
{
"pp": "R : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsNoetherianRing R\ninst✝ : IsLocalRing R\np₁ p₀ : PrimeSpectrum R\nh₀ : p₀ < p₁\nh₁ : p₁ < closedPoint R\nx : R\nhx : x ∈ 𝔪\nhn : x ∉ p₀.asIdeal\ne : ↑(zeroLocus ↑p₀.asIdeal) ≃o PrimeSpectrum (R ⧸ p₀.asIdeal) := p₀.asIdeal.primeSpectrumQuotientOrderIsoZeroL... | [] | by | [anonymous] | by |
Mathlib.RingTheory.Spectrum.Prime.LTSeries | {
"line": 34,
"column": 69
} | {
"line": 34,
"column": 71
} | {
"line": 35,
"column": 2
} | [
{
"pp": "R : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsNoetherianRing R\ninst✝ : IsLocalRing R\np₁ p₀ : PrimeSpectrum R\nh₀ : p₀ < p₁\nh₁ : p₁ < closedPoint R\nx : R\nhx : x ∈ 𝔪\n⊢ ∃ q, x ∈ q.asIdeal ∧ p₀ < q ∧ q.asIdeal < 𝔪",
"ppTerm": "?m.31",
"assigned": true,
"usedConstants": [
"Iff.mpr"... | [] | by | [anonymous] | by |
Mathlib.RingTheory.Spectrum.Prime.LTSeries | {
"line": 64,
"column": 9
} | {
"line": 64,
"column": 11
} | {
"line": 64,
"column": 12
} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsNoetherianRing R\np₁ p₀ p₂ : PrimeSpectrum R\nh₀ : p₀ < p₁\nh₁ : p₁ < p₂\nx : R\nhx : x ∈ p₂.asIdeal\ne : PrimeSpectrum (Localization.AtPrime p₂.asIdeal) ≃o ↑(Set.Iic { asIdeal := p₂.asIdeal, isPrime := ⋯ }) :=\n IsLocalization.AtPrime.primeSpectrumOrderIso... | [] | by | [anonymous] | by |
Mathlib.RingTheory.Spectrum.Prime.LTSeries | {
"line": 64,
"column": 75
} | {
"line": 64,
"column": 77
} | {
"line": 65,
"column": 10
} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsNoetherianRing R\np₁ p₀ p₂ : PrimeSpectrum R\nh₀ : p₀ < p₁\nh₁ : p₁ < p₂\nx : R\nhx : x ∈ p₂.asIdeal\ne : PrimeSpectrum (Localization.AtPrime p₂.asIdeal) ≃o ↑(Set.Iic { asIdeal := p₂.asIdeal, isPrime := ⋯ }) :=\n IsLocalization.AtPrime.primeSpectrumOrderIso... | [] | by | [anonymous] | by |
Mathlib.RingTheory.KrullDimension.Regular | {
"line": 70,
"column": 27
} | {
"line": 70,
"column": 29
} | {
"line": 70,
"column": 30
} | [
{
"pp": "R : Type u_1\ninst✝⁴ : CommRing R\ninst✝³ : IsNoetherianRing R\nM : Type u_2\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\ninst✝ : Module.Finite R M\nx : R\nh : x ∈ (annihilator R M).jacobson\na✝ : Nontrivial M\np : LTSeries ↑(support R M)\nhxp : x ∈ (↑(RelSeries.last p)).asIdeal\nq : LTSeries (PrimeS... | [] | by | [anonymous] | by |
Mathlib.RingTheory.KrullDimension.Regular | {
"line": 69,
"column": 28
} | {
"line": 69,
"column": 30
} | {
"line": 69,
"column": 31
} | [
{
"pp": "R : Type u_1\ninst✝⁴ : CommRing R\ninst✝³ : IsNoetherianRing R\nM : Type u_2\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\ninst✝ : Module.Finite R M\nx : R\nh : x ∈ (annihilator R M).jacobson\na✝ : Nontrivial M\np : LTSeries ↑(support R M)\nhxp : x ∈ (↑(RelSeries.last p)).asIdeal\nq : LTSeries (PrimeS... | [] | by | [anonymous] | by |
Mathlib.RingTheory.KrullDimension.Regular | {
"line": 65,
"column": 13
} | {
"line": 65,
"column": 15
} | {
"line": 66,
"column": 6
} | [
{
"pp": "R : Type u_1\ninst✝⁴ : CommRing R\ninst✝³ : IsNoetherianRing R\nM : Type u_2\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\ninst✝ : Module.Finite R M\nx : R\nh : x ∈ (annihilator R M).jacobson\na✝ : Nontrivial M\np : LTSeries ↑(support R M)\nhxp : x ∈ (↑(RelSeries.last p)).asIdeal\nq : LTSeries (PrimeS... | [] | by | [anonymous] | by |
Mathlib.RingTheory.Ideal.KrullsHeightTheorem | {
"line": 435,
"column": 73
} | {
"line": 435,
"column": 75
} | {
"line": 436,
"column": 4
} | [
{
"pp": "R : Type u_1\ninst✝⁷ : CommRing R\ninst✝⁶ : IsNoetherianRing R\nS : Type u_2\ninst✝⁵ : CommRing S\ninst✝⁴ : Algebra R S\ninst✝³ : IsNoetherianRing S\np : Ideal R\ninst✝² : p.IsPrime\nP : Ideal S\ninst✝¹ : P.IsPrime\ninst✝ : P.LiesOver p\ns : Finset R\nhp : p ∈ (span ↑s).minimalPrimes\nheq : ↑s.card = p... | [] | by | Lean.Elab.Tactic.evalWithAnnotateState | by |
Mathlib.RingTheory.KrullDimension.Regular | {
"line": 72,
"column": 43
} | {
"line": 72,
"column": 45
} | {
"line": 72,
"column": 46
} | [
{
"pp": "R : Type u_1\ninst✝⁴ : CommRing R\ninst✝³ : IsNoetherianRing R\nM : Type u_2\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\ninst✝ : Module.Finite R M\nx : R\nh : x ∈ (annihilator R M).jacobson\na✝ : Nontrivial M\np : LTSeries ↑(support R M)\nhxp : x ∈ (↑(RelSeries.last p)).asIdeal\nq : LTSeries (PrimeS... | [] | by | [anonymous] | by |
Mathlib.RingTheory.Spectrum.Prime.LTSeries | {
"line": 68,
"column": 28
} | {
"line": 68,
"column": 30
} | {
"line": 69,
"column": 4
} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsNoetherianRing R\np₁ p₀ p₂ : PrimeSpectrum R\nh₀✝ : p₀ < p₁\nh₁✝ : p₁ < p₂\nx : R\nhx : x ∈ p₂.asIdeal\ne : PrimeSpectrum (Localization.AtPrime p₂.asIdeal) ≃o ↑(Set.Iic { asIdeal := p₂.asIdeal, isPrime := ⋯ }) :=\n IsLocalization.AtPrime.primeSpectrumOrderI... | [] | by | Lean.Elab.Tactic.evalWithAnnotateState | by |
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