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.UniqueFactorizationDomain.Kaplansky | {
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Mathlib.RepresentationTheory.Homological.GroupHomology.LowDegree | {
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{
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Mathlib.RingTheory.MvPowerSeries.Rename | {
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{
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Mathlib.RepresentationTheory.Homological.GroupHomology.LowDegree | {
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} | {
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{
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Mathlib.RepresentationTheory.Homological.GroupHomology.LowDegree | {
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} | {
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} | {
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{
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... | [] | by | [anonymous] | by |
Mathlib.RingTheory.MvPowerSeries.Rename | {
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} | {
"line": 271,
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} | {
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} | [
{
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Mathlib.RepresentationTheory.Homological.GroupHomology.LowDegree | {
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} | {
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} | {
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{
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Mathlib.RingTheory.MvPowerSeries.Rename | {
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} | {
"line": 270,
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} | {
"line": 271,
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{
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... | [] | by | [anonymous] | by |
Mathlib.RepresentationTheory.Homological.GroupHomology.LowDegree | {
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} | {
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} | {
"line": 386,
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} | [
{
"pp": "k G : Type u\ninst✝¹ : CommRing k\ninst✝ : Group G\nA : Rep k G\ng : G × G\na : ↑A\n⊢ single g a ∈ cycles₂ A ↔ single (g.1 * g.2) ((A.ρ g.1) a) = single g.2 a + single g.1 ((A.ρ g.1) a)",
"ppTerm": "?m.56",
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Mathlib.RingTheory.MvPowerSeries.Rename | {
"line": 276,
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} | {
"line": 276,
"column": 83
} | {
"line": 277,
"column": 2
} | [
{
"pp": "σ : Type u_1\nτ : Type u_2\nR : Type u_4\ninst✝ : CommSemiring R\ne : σ ↪ τ\n⊢ (killCompl e).comp (rename ⇑e) = AlgHom.id R (MvPowerSeries σ R)",
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Mathlib.RepresentationTheory.Homological.GroupHomology.LowDegree | {
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} | {
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} | {
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{
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Mathlib.RepresentationTheory.Homological.GroupHomology.LowDegree | {
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} | {
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"column": 36
} | {
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} | [
{
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Mathlib.RepresentationTheory.Homological.GroupHomology.LowDegree | {
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} | {
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} | {
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{
"pp": "k G : Type u\ninst✝¹ : CommRing k\ninst✝ : Group G\nA : Rep k G\ng : G\na : ↑A\n⊢ single g ((A.ρ g) a) + single g⁻¹ a ∈ boundaries₁ A",
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Mathlib.RingTheory.MvPowerSeries.Rename | {
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} | {
"line": 284,
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} | {
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{
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Mathlib.RingTheory.MvPowerSeries.Rename | {
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} | {
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} | {
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} | [
{
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Mathlib.RepresentationTheory.Homological.GroupHomology.LowDegree | {
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} | {
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} | {
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{
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Mathlib.RepresentationTheory.Homological.GroupHomology.LowDegree | {
"line": 450,
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} | {
"line": 450,
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} | {
"line": 451,
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} | [
{
"pp": "k G : Type u\ninst✝¹ : CommRing k\ninst✝ : Group G\nA : Rep k G\nx : G × G →₀ ↑A\nh : x ∈ boundaries₂ A\n⊢ x ∈ cycles₂ A",
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Mathlib.RepresentationTheory.Homological.GroupHomology.LowDegree | {
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} | {
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} | {
"line": 471,
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{
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... | [] | by | [anonymous] | by |
Mathlib.RepresentationTheory.Homological.GroupHomology.LowDegree | {
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} | {
"line": 475,
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} | {
"line": 476,
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{
"pp": "k G : Type u\ninst✝¹ : CommRing k\ninst✝ : Group G\nA : Rep k G\ng h : G\na : ↑A\n⊢ single (1, h) ((A.ρ g⁻¹) a) - single (g, 1) a ∈ boundaries₂ A",
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Mathlib.RepresentationTheory.Homological.GroupHomology.LowDegree | {
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} | {
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} | {
"line": 481,
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} | [
{
"pp": "k G : Type u\ninst✝¹ : CommRing k\ninst✝ : Group G\nA : Rep k G\ng h : G\na : ↑A\n⊢ single (g, 1) ((A.ρ g) a) - single (1, h) a ∈ boundaries₂ A",
"ppTerm": "?m.37",
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Mathlib.RepresentationTheory.Homological.GroupHomology.LowDegree | {
"line": 485,
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} | {
"line": 485,
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} | {
"line": 486,
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} | [
{
"pp": "k G : Type u\ninst✝¹ : CommRing k\ninst✝ : Group G\nA : Rep k G\ng h : G\na : ↑A\n⊢ single (h, 1) ((A.ρ g⁻¹) a) - single (g * h, 1) a ∈ boundaries₂ A",
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Mathlib.RepresentationTheory.Homological.GroupHomology.LowDegree | {
"line": 514,
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} | {
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} | {
"line": 515,
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{
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"ppTerm": "?m.18",
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Mathlib.RepresentationTheory.Homological.GroupHomology.LowDegree | {
"line": 520,
"column": 29
} | {
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} | {
"line": 521,
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{
"pp": "G : Type u_1\nA : Type u_2\ninst✝² : Group G\ninst✝¹ : AddCommGroup A\ninst✝ : DistribMulAction G A\ng : G\na : A\nha : a ∈ MulAction.fixedPoints G A\n⊢ IsCycle₁ (single g a)",
"ppTerm": "?m.18",
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... | [] | by | [anonymous] | by |
Mathlib.RepresentationTheory.Homological.GroupHomology.LowDegree | {
"line": 525,
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} | {
"line": 525,
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} | {
"line": 526,
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{
"pp": "G : Type u_1\nA : Type u_2\ninst✝² : Group G\ninst✝¹ : AddCommGroup A\ninst✝ : DistribMulAction G A\ng : G × G\na : A\n⊢ IsCycle₂ (single g a) ↔ single g.2 (g.1⁻¹ • a) + single g.1 a = single (g.1 * g.2) a",
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"H... | [] | by | [anonymous] | by |
Mathlib.RingTheory.MvPowerSeries.Rename | {
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} | {
"line": 296,
"column": 75
} | {
"line": 297,
"column": 6
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{
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"ppTerm": "?m.37",
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"usedConstants": [
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"NonAs... | [] | by | [anonymous] | by |
Mathlib.RepresentationTheory.Homological.GroupHomology.LowDegree | {
"line": 531,
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} | {
"line": 531,
"column": 78
} | {
"line": 532,
"column": 2
} | [
{
"pp": "G : Type u_1\nA : Type u_2\ninst✝² : Group G\ninst✝¹ : AddCommGroup A\ninst✝ : DistribMulAction G A\ng : G × G\na : A\n⊢ IsCycle₂ (single g a) ↔ single g.2 a + single g.1 (g.1 • a) = single (g.1 * g.2) (g.1 • a)",
"ppTerm": "?m.52",
"assigned": true,
"usedConstants": [
"Eq.mpr",
... | [] | by | [anonymous] | by |
Mathlib.RepresentationTheory.Homological.GroupHomology.LowDegree | {
"line": 572,
"column": 72
} | {
"line": 572,
"column": 74
} | {
"line": 573,
"column": 2
} | [
{
"pp": "G : Type u_1\nA : Type u_2\ninst✝² : Group G\ninst✝¹ : AddCommGroup A\ninst✝ : DistribMulAction G A\na : A\n⊢ IsBoundary₀ G a ↔ ∃ x, (x.sum fun g z ↦ g • z - z) = a",
"ppTerm": "?m.27",
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"usedConstants": [
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... | [] | by | [anonymous] | by |
Mathlib.RingTheory.UniqueFactorizationDomain.Kaplansky | {
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"column": 74
} | {
"line": 67,
"column": 76
} | {
"line": 68,
"column": 2
} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommSemiring R\ninst✝ : IsDomain R\na : R\nha : a ≠ 0\nH : ∀ (I : Ideal R), I ≠ ⊥ → I.IsPrime → ∃ x ∈ I, Prime x\n⊢ span {a} ∉ kaplanskySet (closure {r | Prime r}).toSubsemigroup",
"ppTerm": "?m.36",
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"CommMonoidWithZero.toCom... | [] | by | [anonymous] | by |
Mathlib.RingTheory.MvPowerSeries.Rename | {
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} | {
"line": 306,
"column": 29
} | {
"line": 307,
"column": 4
} | [
{
"pp": "σ : Type u_1\nτ : Type u_2\nf : σ → τ\ninst✝¹ : TendstoCofinite f\nR : Type u_6\ninst✝ : CommRing R\np : MvPowerSeries σ R\nn : τ →₀ ℕ\nd : σ →₀ ℕ\nhd : (coeff d) p * (coeff n) (d.prod fun s e ↦ X (f s) ^ e) ≠ 0\n⊢ mapDomain f d = n",
"ppTerm": "?m.114",
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"usedConstants": [... | [] | by | Lean.Elab.Tactic.evalWithAnnotateState | by |
Mathlib.RepresentationTheory.Homological.GroupHomology.LowDegree | {
"line": 583,
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} | {
"line": 583,
"column": 95
} | {
"line": 584,
"column": 2
} | [
{
"pp": "G : Type u_1\nA : Type u_2\ninst✝² : Group G\ninst✝¹ : AddCommGroup A\ninst✝ : DistribMulAction G A\nx : G →₀ A\n⊢ IsBoundary₁ x ↔ ∃ y, (y.sum fun g a ↦ single g.2 a - single (g.1 * g.2) (g.1 • a) + single g.1 (g.1 • a)) = x",
"ppTerm": "?m.63",
"assigned": true,
"usedConstants": [
"i... | [] | by | [anonymous] | by |
Mathlib.RepresentationTheory.Homological.GroupHomology.LowDegree | {
"line": 595,
"column": 86
} | {
"line": 595,
"column": 88
} | {
"line": 596,
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} | [
{
"pp": "G : Type u_1\nA : Type u_2\ninst✝² : Group G\ninst✝¹ : AddCommGroup A\ninst✝ : DistribMulAction G A\nx : G × G →₀ A\n⊢ IsBoundary₂ x ↔\n ∃ y,\n (y.sum fun g a ↦\n single (g.2.1, g.2.2) a - single (g.1 * g.2.1, g.2.2) (g.1 • a) + single (g.1, g.2.1 * g.2.2) (g.1 • a) -\n sing... | [] | by | [anonymous] | by |
Mathlib.RepresentationTheory.Homological.GroupHomology.LowDegree | {
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} | {
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} | {
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{
"pp": "k✝ G✝ : Type u\ninst✝⁷ : CommRing k✝\ninst✝⁶ : Group G✝\nA✝ : Rep k✝ G✝\nk G A : Type u\ninst✝⁵ : CommRing k\ninst✝⁴ : Group G\ninst✝³ : AddCommGroup A\ninst✝² : Module k A\ninst✝¹ : DistribMulAction G A\ninst✝ : SMulCommClass G k A\nx : A\nhx : IsBoundary₀ G x\n⊢ x ∈ Coinvariants.ker (Representation.o... | [] | by | [anonymous] | by |
Mathlib.RepresentationTheory.Homological.GroupHomology.LowDegree | {
"line": 619,
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} | {
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} | {
"line": 621,
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{
"pp": "k✝ G✝ : Type u\ninst✝⁷ : CommRing k✝\ninst✝⁶ : Group G✝\nA✝ : Rep k✝ G✝\nk G A : Type u\ninst✝⁵ : CommRing k\ninst✝⁴ : Group G\ninst✝³ : AddCommGroup A\ninst✝² : Module k A\ninst✝¹ : DistribMulAction G A\ninst✝ : SMulCommClass G k A\nx : A\nhx : IsBoundary₀ G x\n⊢ x ∈ Coinvariants.ker (Representation.o... | [] | by
rcases (isBoundary₀_iff G x).1 hx with ⟨y, rfl⟩
exact Submodule.finsuppSum_mem _ _ _ _ fun g _ => Coinvariants.mem_ker_of_eq g (y g) _ rfl | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RepresentationTheory.Homological.GroupHomology.LowDegree | {
"line": 626,
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} | {
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} | {
"line": 626,
"column": 67
} | [
{
"pp": "k G A : Type u\ninst✝⁵ : CommRing k\ninst✝⁴ : Group G\ninst✝³ : AddCommGroup A\ninst✝² : Module k A\ninst✝¹ : DistribMulAction G A\ninst✝ : SMulCommClass G k A\nx : A\nhx : x ∈ Coinvariants.ker (Representation.ofDistribMulAction k G A)\nx✝¹ : A\nx✝ : x✝¹ ∈ Set.range fun gv ↦ ((Representation.ofDistribM... | [] | by | [anonymous] | by |
Mathlib.RepresentationTheory.Homological.GroupHomology.LowDegree | {
"line": 626,
"column": 82
} | {
"line": 626,
"column": 84
} | {
"line": 626,
"column": 85
} | [
{
"pp": "k G A : Type u\ninst✝⁵ : CommRing k\ninst✝⁴ : Group G\ninst✝³ : AddCommGroup A\ninst✝² : Module k A\ninst✝¹ : DistribMulAction G A\ninst✝ : SMulCommClass G k A\nx : A\nhx : x ∈ Coinvariants.ker (Representation.ofDistribMulAction k G A)\n⊢ (sum 0 fun g z ↦ g⁻¹ • z - z) = 0",
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"... | [] | by | [anonymous] | by |
Mathlib.RepresentationTheory.Homological.GroupHomology.LowDegree | {
"line": 627,
"column": 44
} | {
"line": 627,
"column": 46
} | {
"line": 627,
"column": 47
} | [
{
"pp": "k G A : Type u\ninst✝⁵ : CommRing k\ninst✝⁴ : Group G\ninst✝³ : AddCommGroup A\ninst✝² : Module k A\ninst✝¹ : DistribMulAction G A\ninst✝ : SMulCommClass G k A\nx : A\nhx : x ∈ Coinvariants.ker (Representation.ofDistribMulAction k G A)\nx✝⁵ x✝⁴ : A\nx✝³ : x✝⁵ ∈ Submodule.span k (Set.range fun gv ↦ ((Re... | [] | by | [anonymous] | by |
Mathlib.RepresentationTheory.Homological.GroupHomology.LowDegree | {
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} | {
"line": 628,
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} | {
"line": 628,
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{
"pp": "k G A : Type u\ninst✝⁵ : CommRing k\ninst✝⁴ : Group G\ninst✝³ : AddCommGroup A\ninst✝² : Module k A\ninst✝¹ : DistribMulAction G A\ninst✝ : SMulCommClass G k A\nx : A\nhx : x ∈ Coinvariants.ker (Representation.ofDistribMulAction k G A)\nr : k\nx✝² : A\nx✝¹ : x✝² ∈ Submodule.span k (Set.range fun gv ↦ (... | [] | by | [anonymous] | by |
Mathlib.RepresentationTheory.Homological.GroupHomology.LowDegree | {
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"column": 18
} | {
"line": 641,
"column": 20
} | {
"line": 642,
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} | [
{
"pp": "k G A : Type u\ninst✝⁵ : CommRing k\ninst✝⁴ : Group G\ninst✝³ : AddCommGroup A\ninst✝² : Module k A\ninst✝¹ : DistribMulAction G A\ninst✝ : SMulCommClass G k A\nx : G →₀ A\nhx : x ∈ cycles₁ (Rep.ofDistribMulAction k G A)\n⊢ IsCycle₁ x",
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"usedConstants": ... | [] | by | [anonymous] | by |
Mathlib.RepresentationTheory.Homological.GroupHomology.LowDegree | {
"line": 689,
"column": 57
} | {
"line": 689,
"column": 59
} | {
"line": 690,
"column": 2
} | [
{
"pp": "k G : Type u\ninst✝¹ : CommRing k\ninst✝ : Group G\nA : Rep k G\n⊢ (shortComplexH0 A).Exact",
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"usedConstants": [
"Eq.mpr",
"Submodule",
"Representation.Coinvariants.ker",
"Rep.V",
"RingHomSurjective.ids",
"Finsupp.modu... | [] | by | [anonymous] | by |
Mathlib.RepresentationTheory.Homological.GroupHomology.LowDegree | {
"line": 699,
"column": 42
} | {
"line": 699,
"column": 44
} | {
"line": 699,
"column": 45
} | [
{
"pp": "k G : Type u\ninst✝¹ : CommRing k\ninst✝ : Group G\nA : Rep k G\n⊢ (ComplexShape.down ℕ).next 0 = 0",
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"assigned": true,
"usedConstants": [
"Nat.instOne",
"congrArg",
"AddRightCancelSemigroup.toAddSemigroup",
"AddCancelMonoid.toAddRightCancelMonoid"... | [] | by | [anonymous] | by |
Mathlib.RepresentationTheory.Homological.GroupHomology.LowDegree | {
"line": 699,
"column": 52
} | {
"line": 699,
"column": 54
} | {
"line": 699,
"column": 55
} | [
{
"pp": "k G : Type u\ninst✝¹ : CommRing k\ninst✝ : Group G\nA : Rep k G\n⊢ (inhomogeneousChains A).d 0 0 = 0",
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"usedConstants": [
"Rep.V",
"False",
"Nat.instMulZeroClass",
"Finsupp.module",
"Nat.instOne",
"CategoryTheory.Categ... | [] | by | [anonymous] | by |
Mathlib.RepresentationTheory.Homological.GroupHomology.LowDegree | {
"line": 703,
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} | {
"line": 703,
"column": 63
} | {
"line": 704,
"column": 2
} | [
{
"pp": "k G : Type u\ninst✝¹ : CommRing k\ninst✝ : Group G\nA : Rep k G\n⊢ (cyclesIso₀ A).inv ≫ iCycles A 0 = (chainsIso₀ A).inv",
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"usedConstants": [
"CategoryTheory.Category.assoc",
"Rep.V",
"groupHomology.cyclesIso₀._proof_4",
"Finsupp.... | [] | by | [anonymous] | by |
Mathlib.RepresentationTheory.Homological.GroupHomology.LowDegree | {
"line": 717,
"column": 53
} | {
"line": 717,
"column": 55
} | {
"line": 717,
"column": 56
} | [
{
"pp": "k G : Type u\ninst✝¹ : CommRing k\ninst✝ : Group G\nA : Rep k G\n⊢ (ComplexShape.down ℕ).prev 0 = 1",
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"usedConstants": [
"Nat.instOne",
"congrArg",
"AddMonoid.toAddZeroClass",
"AddRightCancelSemigroup.toAddSemigroup",
"AddCa... | [] | by | [anonymous] | by |
Mathlib.RepresentationTheory.Homological.GroupHomology.LowDegree | {
"line": 733,
"column": 18
} | {
"line": 733,
"column": 20
} | {
"line": 733,
"column": 21
} | [
{
"pp": "k G : Type u\ninst✝¹ : CommRing k\ninst✝ : Group G\nA : Rep k G\nx : ↑A\n⊢ (ComplexShape.down ℕ).next 0 = 0",
"ppTerm": "?m.34",
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"usedConstants": [
"Nat.instIsOrderedAddMonoid",
"AddLeftCancelSemigroup.toIsLeftCancelAdd",
"Nat.instOne",
"congrArg",
... | [] | by | [anonymous] | by |
Mathlib.RepresentationTheory.Homological.GroupHomology.LowDegree | {
"line": 733,
"column": 51
} | {
"line": 733,
"column": 53
} | {
"line": 733,
"column": 54
} | [
{
"pp": "k G : Type u\ninst✝¹ : CommRing k\ninst✝ : Group G\nA : Rep k G\nx : ↑A\n⊢ (ConcreteCategory.hom ((inhomogeneousChains A).d 0 0)) ((ConcreteCategory.hom (chainsIso₀ A).inv) x) = 0",
"ppTerm": "?m.35",
"assigned": true,
"usedConstants": [
"Rep.V",
"False",
"Nat.instMulZeroC... | [] | by | [anonymous] | by |
Mathlib.RingTheory.UniqueFactorizationDomain.Kaplansky | {
"line": 90,
"column": 18
} | {
"line": 90,
"column": 20
} | {
"line": 90,
"column": 21
} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommSemiring R\ninst✝ : IsDomain R\nH : ∀ (I : Ideal R), I ≠ ⊥ → I.IsPrime → ∃ x ∈ I, Prime x\na : R\nha : a ≠ 0\nha₂ : ¬↑(span {a}) ∩ ↑(closure {r | Prime r}).toSubsemigroup = ∅\nx : R\nhx : x ∈ ↑(span {a})\nb : R\nhx₂ : a * b ∈ ↑(closure {r | Prime r}).toSubsemigroup\nhb : b * ... | [] | by | [anonymous] | by |
Mathlib.RingTheory.MvPowerSeries.Rename | {
"line": 300,
"column": 58
} | {
"line": 300,
"column": 60
} | {
"line": 301,
"column": 2
} | [
{
"pp": "σ : Type u_1\nτ : Type u_2\nf : σ → τ\ninst✝¹ : TendstoCofinite f\nR : Type u_6\ninst✝ : CommRing R\np : MvPowerSeries σ R\n⊢ (rename f) p = subst (X ∘ f) p",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"Finsupp.mapDomain_tendstoCofinite",
"Eq.mpr",
"NonAssocSe... | [] | by | [anonymous] | by |
Mathlib.RepresentationTheory.Homological.GroupHomology.LowDegree | {
"line": 735,
"column": 67
} | {
"line": 735,
"column": 69
} | {
"line": 735,
"column": 70
} | [
{
"pp": "k G : Type u\ninst✝¹ : CommRing k\ninst✝ : Group G\nA : Rep k G\nx : ↑A\n⊢ (ConcreteCategory.hom (iCycles A 0))\n (cyclesMk 0 0 cyclesMk₀_eq._proof_1 ((ConcreteCategory.hom (chainsIso₀ A).inv) x) ⋯) =\n (ConcreteCategory.hom (iCycles A 0)) ((ConcreteCategory.hom (cyclesIso₀ A).inv) x)",
"pp... | [] | by | [anonymous] | by |
Mathlib.RepresentationTheory.Homological.GroupHomology.LowDegree | {
"line": 748,
"column": 40
} | {
"line": 748,
"column": 42
} | {
"line": 748,
"column": 43
} | [
{
"pp": "k G : Type u\ninst✝¹ : CommRing k\ninst✝ : Group G\nA : Rep k G\n⊢ (ComplexShape.down ℕ).prev 1 = 2",
"ppTerm": "?m.67",
"assigned": true,
"usedConstants": [
"Nat.instOne",
"congrArg",
"AddRightCancelSemigroup.toAddSemigroup",
"AddCancelMonoid.toAddRightCancelMonoid"... | [] | by | [anonymous] | by |
Mathlib.RepresentationTheory.Homological.GroupHomology.LowDegree | {
"line": 748,
"column": 50
} | {
"line": 748,
"column": 52
} | {
"line": 748,
"column": 53
} | [
{
"pp": "k G : Type u\ninst✝¹ : CommRing k\ninst✝ : Group G\nA : Rep k G\n⊢ (ComplexShape.down ℕ).next 1 = 0",
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"assigned": true,
"usedConstants": [
"Nat.instOne",
"congrArg",
"AddRightCancelSemigroup.toAddSemigroup",
"ChainComplex.next_nat_succ",
"Add... | [] | by | [anonymous] | by |
Mathlib.RingTheory.PowerSeries.Ideal | {
"line": 62,
"column": 44
} | {
"line": 62,
"column": 46
} | {
"line": 62,
"column": 47
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\nI : Ideal R⟦X⟧\nhXI : X ∈ I\nthis : ∀ x ∈ I, C (constantCoeff x) ∈ I\n⊢ Ideal.map (C.comp constantCoeff) I ≤ I",
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"assigned": true,
"usedConstants": [
"Eq.mpr",
"RingHom.instRingHomClass",
"Semiring.toModule",
"Comm... | [] | by | [anonymous] | by |
Mathlib.RingTheory.PowerSeries.Ideal | {
"line": 61,
"column": 40
} | {
"line": 61,
"column": 42
} | {
"line": 62,
"column": 2
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\nI : Ideal R⟦X⟧\nhXI : X ∈ I\n⊢ Ideal.map (C.comp constantCoeff) I ≤ I",
"ppTerm": "?m.25",
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"usedConstants": [
"MvPowerSeries.instAddCommGroup",
"Iff.mpr",
"Eq.mpr",
"RingHom.instRingHomClass",
"Submodule.addSu... | [] | by | [anonymous] | by |
Mathlib.RingTheory.PowerSeries.Ideal | {
"line": 72,
"column": 25
} | {
"line": 72,
"column": 27
} | {
"line": 72,
"column": 28
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\nI : Ideal R⟦X⟧\nS : Set R\nhXI : X ∈ I\nhSI : span S = Ideal.map constantCoeff I\n⊢ span {X} ⊔ Ideal.map (C.comp constantCoeff) I ≤ I",
"ppTerm": "?m.63",
"assigned": true,
"usedConstants": [
"SetLike.mem_coe._simp_1",
"Lattice.toSemilatticeSup"... | [] | by | [anonymous] | by |
Mathlib.RepresentationTheory.Homological.GroupHomology.LowDegree | {
"line": 762,
"column": 42
} | {
"line": 762,
"column": 44
} | {
"line": 763,
"column": 2
} | [
{
"pp": "k G : Type u\ninst✝¹ : CommRing k\ninst✝ : Group G\nA : Rep k G\n⊢ (isoCycles₁ A).hom ≫ (shortComplexH1 A).moduleCatLeftHomologyData.i = iCycles A 1 ≫ (chainsIso₁ A).hom",
"ppTerm": "?m.46",
"assigned": true,
"usedConstants": [
"groupHomology.isoCycles₁",
"Submodule",
"Rep... | [] | by | [anonymous] | by |
Mathlib.RepresentationTheory.Homological.GroupHomology.LowDegree | {
"line": 775,
"column": 78
} | {
"line": 775,
"column": 80
} | {
"line": 776,
"column": 2
} | [
{
"pp": "k G : Type u\ninst✝¹ : CommRing k\ninst✝ : Group G\nA : Rep k G\n⊢ toCycles A 2 1 ≫ (isoCycles₁ A).hom = (chainsIso₂ A).hom ≫ (shortComplexH1 A).moduleCatLeftHomologyData.f'",
"ppTerm": "?m.48",
"assigned": true,
"usedConstants": [
"groupHomology.isoCycles₁",
"CategoryTheory.Cat... | [] | by | [anonymous] | by |
Mathlib.RepresentationTheory.Homological.GroupHomology.LowDegree | {
"line": 781,
"column": 18
} | {
"line": 781,
"column": 20
} | {
"line": 781,
"column": 21
} | [
{
"pp": "k G : Type u\ninst✝¹ : CommRing k\ninst✝ : Group G\nA : Rep k G\nx : ↥(cycles₁ A)\n⊢ (ComplexShape.down ℕ).next 1 = 0",
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"assigned": true,
"usedConstants": [
"Nat.instIsOrderedAddMonoid",
"AddLeftCancelSemigroup.toIsLeftCancelAdd",
"Nat.instOne",
"c... | [] | by | [anonymous] | by |
Mathlib.RingTheory.PowerSeries.Ideal | {
"line": 70,
"column": 36
} | {
"line": 70,
"column": 38
} | {
"line": 71,
"column": 2
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\nI : Ideal R⟦X⟧\nS : Set R\nhXI : X ∈ I\nhSI : span S = Ideal.map constantCoeff I\n⊢ I = span (insert X (⇑C '' S))",
"ppTerm": "?m.31",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"Eq.mpr",
"Submodule",
"RingHom.instRingHomClass",... | [] | by | [anonymous] | by |
Mathlib.RingTheory.UniqueFactorizationDomain.Kaplansky | {
"line": 96,
"column": 18
} | {
"line": 96,
"column": 20
} | {
"line": 96,
"column": 21
} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommSemiring R\ninst✝ : IsDomain R\nH : ∀ (I : Ideal R), I ≠ ⊥ → I.IsPrime → ∃ x ∈ I, Prime x\na x b : R\nhsubset : closure {r | Prime r} ≤ closure {r | IsUnit r ∨ Prime r}\nz : R\nh : IsUnit z\n⊢ (∀ b ∈ ∅, Prime b) ∧ Associated ∅.prod z",
"ppTerm": "?m.165",
"assigned": ... | [] | by | [anonymous] | by |
Mathlib.RepresentationTheory.Homological.GroupHomology.LowDegree | {
"line": 781,
"column": 51
} | {
"line": 781,
"column": 53
} | {
"line": 782,
"column": 6
} | [
{
"pp": "k G : Type u\ninst✝¹ : CommRing k\ninst✝ : Group G\nA : Rep k G\nx : ↥(cycles₁ A)\n⊢ (ConcreteCategory.hom ((inhomogeneousChains A).d 1 0)) ((ConcreteCategory.hom (chainsIso₁ A).inv) ↑x) = 0",
"ppTerm": "?m.39",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Submodule",
"... | [] | by | [anonymous] | by |
Mathlib.RingTheory.PowerSeries.Ideal | {
"line": 81,
"column": 50
} | {
"line": 81,
"column": 52
} | {
"line": 82,
"column": 4
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\nI : Ideal R⟦X⟧\nhI : X ∈ I\nhfg : I.FG\n⊢ (Ideal.map constantCoeff I).FG",
"ppTerm": "?m.62",
"assigned": true,
"usedConstants": [
"Submodule",
"Semiring.toModule",
"CommSemiring.toSemiring",
"Submodule.FG.map",
"RingHom",
... | [] | by | Lean.Elab.Tactic.evalWithAnnotateState | by |
Mathlib.RingTheory.UniqueFactorizationDomain.Kaplansky | {
"line": 97,
"column": 20
} | {
"line": 97,
"column": 22
} | {
"line": 97,
"column": 23
} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommSemiring R\ninst✝ : IsDomain R\nH : ∀ (I : Ideal R), I ≠ ⊥ → I.IsPrime → ∃ x ∈ I, Prime x\na x b : R\nhsubset : closure {r | Prime r} ≤ closure {r | IsUnit r ∨ Prime r}\nz : R\nh : Prime z\n⊢ (∀ b ∈ {z}, Prime b) ∧ Associated {z}.prod z",
"ppTerm": "?m.186",
"assigned... | [] | by | [anonymous] | by |
Mathlib.RepresentationTheory.Homological.GroupHomology.LowDegree | {
"line": 784,
"column": 67
} | {
"line": 784,
"column": 69
} | {
"line": 785,
"column": 4
} | [
{
"pp": "k G : Type u\ninst✝¹ : CommRing k\ninst✝ : Group G\nA : Rep k G\nx : ↥(cycles₁ A)\n⊢ (ConcreteCategory.hom (iCycles A 1))\n (cyclesMk 1 0 cyclesMk₁_eq._proof_1 ((ConcreteCategory.hom (chainsIso₁ A).inv) ↑x) ⋯) =\n (ConcreteCategory.hom (iCycles A 1)) ((ConcreteCategory.hom (isoCycles₁ A).inv) x... | [] | by | [anonymous] | by |
Mathlib.RepresentationTheory.Homological.GroupHomology.LowDegree | {
"line": 800,
"column": 40
} | {
"line": 800,
"column": 42
} | {
"line": 800,
"column": 43
} | [
{
"pp": "k G : Type u\ninst✝¹ : CommRing k\ninst✝ : Group G\nA : Rep k G\n⊢ (ComplexShape.down ℕ).prev 2 = 3",
"ppTerm": "?m.67",
"assigned": true,
"usedConstants": [
"Nat.instOne",
"congrArg",
"AddRightCancelSemigroup.toAddSemigroup",
"AddCancelMonoid.toAddRightCancelMonoid"... | [] | by | [anonymous] | by |
Mathlib.RepresentationTheory.Homological.GroupHomology.LowDegree | {
"line": 800,
"column": 50
} | {
"line": 800,
"column": 52
} | {
"line": 800,
"column": 53
} | [
{
"pp": "k G : Type u\ninst✝¹ : CommRing k\ninst✝ : Group G\nA : Rep k G\n⊢ (ComplexShape.down ℕ).next 2 = 1",
"ppTerm": "?m.68",
"assigned": true,
"usedConstants": [
"Nat.instOne",
"congrArg",
"AddRightCancelSemigroup.toAddSemigroup",
"ChainComplex.next_nat_succ",
"Add... | [] | by | [anonymous] | by |
Mathlib.RingTheory.UniqueFactorizationDomain.Kaplansky | {
"line": 98,
"column": 21
} | {
"line": 98,
"column": 23
} | {
"line": 98,
"column": 24
} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommSemiring R\ninst✝ : IsDomain R\nH : ∀ (I : Ideal R), I ≠ ⊥ → I.IsPrime → ∃ x ∈ I, Prime x\na x b : R\nhsubset : closure {r | Prime r} ≤ closure {r | IsUnit r ∨ Prime r}\n⊢ (∀ b ∈ ∅, Prime b) ∧ Associated ∅.prod 1",
"ppTerm": "?m.193",
"assigned": true,
"usedConsta... | [] | by | [anonymous] | by |
Mathlib.RepresentationTheory.Homological.GroupHomology.LowDegree | {
"line": 814,
"column": 42
} | {
"line": 814,
"column": 44
} | {
"line": 815,
"column": 2
} | [
{
"pp": "k G : Type u\ninst✝¹ : CommRing k\ninst✝ : Group G\nA : Rep k G\n⊢ (isoCycles₂ A).hom ≫ (shortComplexH2 A).moduleCatLeftHomologyData.i = iCycles A 2 ≫ (chainsIso₂ A).hom",
"ppTerm": "?m.46",
"assigned": true,
"usedConstants": [
"Submodule",
"Rep.V",
"CategoryTheory.Functor... | [] | by | [anonymous] | by |
Mathlib.RingTheory.PowerSeries.Ideal | {
"line": 78,
"column": 61
} | {
"line": 78,
"column": 63
} | {
"line": 79,
"column": 2
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\nI : Ideal R⟦X⟧\nhI : X ∈ I\n⊢ spanFinrank I ≤ spanFinrank (Ideal.map constantCoeff I) + 1",
"ppTerm": "?m.33",
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"usedConstants": [
"Set.ncard_image_le",
"Eq.mpr",
"Submodule",
"Nat.zero_le",
"le_refl",
"S... | [] | by | [anonymous] | by |
Mathlib.RingTheory.PowerSeries.Ideal | {
"line": 78,
"column": 61
} | {
"line": 89,
"column": 32
} | {
"line": 91,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\nI : Ideal R⟦X⟧\nhI : X ∈ I\n⊢ spanFinrank I ≤ spanFinrank (Ideal.map constantCoeff I) + 1",
"ppTerm": "?m.33",
"assigned": true,
"usedConstants": [
"Set.ncard_image_le",
"Eq.mpr",
"Submodule",
"Nat.zero_le",
"le_refl",
"S... | [] | by
by_cases hfg : I.FG
swap; · exact spanFinrank_of_not_fg hfg ▸ Nat.zero_le _
replace hfg : (Ideal.map constantCoeff I).FG := by
have : RingHomSurjective (constantCoeff (R := R)) := ⟨constantCoeff_surj⟩
exact map_eq_submodule_map constantCoeff I ▸ Submodule.FG.map _ hfg
nth_rw 1 [eq_span_insert_X_of_X_... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RepresentationTheory.Homological.GroupHomology.LowDegree | {
"line": 827,
"column": 78
} | {
"line": 827,
"column": 80
} | {
"line": 828,
"column": 2
} | [
{
"pp": "k G : Type u\ninst✝¹ : CommRing k\ninst✝ : Group G\nA : Rep k G\n⊢ toCycles A 3 2 ≫ (isoCycles₂ A).hom = (chainsIso₃ A).hom ≫ (shortComplexH2 A).moduleCatLeftHomologyData.f'",
"ppTerm": "?m.48",
"assigned": true,
"usedConstants": [
"groupHomology.d₃₂",
"CategoryTheory.Category.a... | [] | by | [anonymous] | by |
Mathlib.RepresentationTheory.Homological.GroupHomology.LowDegree | {
"line": 833,
"column": 18
} | {
"line": 833,
"column": 20
} | {
"line": 833,
"column": 21
} | [
{
"pp": "k G : Type u\ninst✝¹ : CommRing k\ninst✝ : Group G\nA : Rep k G\nx : ↥(cycles₂ A)\n⊢ (ComplexShape.down ℕ).next 2 = 1",
"ppTerm": "?m.38",
"assigned": true,
"usedConstants": [
"Nat.instIsOrderedAddMonoid",
"AddLeftCancelSemigroup.toIsLeftCancelAdd",
"Nat.instOne",
"c... | [] | by | [anonymous] | by |
Mathlib.RingTheory.UniqueFactorizationDomain.Kaplansky | {
"line": 102,
"column": 23
} | {
"line": 102,
"column": 25
} | {
"line": 102,
"column": 26
} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommSemiring R\ninst✝ : IsDomain R\nH : ∀ (I : Ideal R), I ≠ ⊥ → I.IsPrime → ∃ x ∈ I, Prime x\na x b : R\nhsubset : closure {r | Prime r} ≤ closure {r | IsUnit r ∨ Prime r}\nz₁ z₂ : R\nhz₁ : z₁ ∈ closure {r | IsUnit r ∨ Prime r}\nhz₂ : z₂ ∈ closure {r | IsUnit r ∨ Prime r}\nS₁ : ... | [] | by | [anonymous] | by |
Mathlib.RepresentationTheory.Homological.GroupHomology.LowDegree | {
"line": 833,
"column": 51
} | {
"line": 833,
"column": 53
} | {
"line": 834,
"column": 6
} | [
{
"pp": "k G : Type u\ninst✝¹ : CommRing k\ninst✝ : Group G\nA : Rep k G\nx : ↥(cycles₂ A)\n⊢ (ConcreteCategory.hom ((inhomogeneousChains A).d 2 1)) ((ConcreteCategory.hom (chainsIso₂ A).inv) ↑x) = 0",
"ppTerm": "?m.39",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Submodule",
"... | [] | by | [anonymous] | by |
Mathlib.RingTheory.PowerSeries.Ideal | {
"line": 109,
"column": 85
} | {
"line": 109,
"column": 87
} | {
"line": 110,
"column": 4
} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommRing R\nI : Ideal R⟦X⟧\ninst✝ : I.IsPrime\nhXI : X ∉ I\nn✝ : ℕ\nF : Fin n✝ → R⟦X⟧\nhJI : Submodule.span R⟦X⟧ (Set.range F) ≤ I\nhJ : FG (Submodule.span R⟦X⟧ (Set.range F))\nh' : Ideal.map constantCoeff I = span (Set.range (⇑constantCoeff ∘ F))\nT : R⟦X⟧ → Fin n✝ → R\nhT : ∀ g... | [] | by | Lean.Elab.Tactic.evalWithAnnotateState | by |
Mathlib.RepresentationTheory.Homological.GroupHomology.LowDegree | {
"line": 836,
"column": 67
} | {
"line": 836,
"column": 69
} | {
"line": 837,
"column": 4
} | [
{
"pp": "k G : Type u\ninst✝¹ : CommRing k\ninst✝ : Group G\nA : Rep k G\nx : ↥(cycles₂ A)\n⊢ (ConcreteCategory.hom (iCycles A 2))\n (cyclesMk 2 1 cyclesMk₂_eq._proof_1 ((ConcreteCategory.hom (chainsIso₂ A).inv) ↑x) ⋯) =\n (ConcreteCategory.hom (iCycles A 2)) ((ConcreteCategory.hom (isoCycles₂ A).inv) x... | [] | by | [anonymous] | by |
Mathlib.RepresentationTheory.Homological.GroupHomology.LowDegree | {
"line": 865,
"column": 79
} | {
"line": 865,
"column": 81
} | {
"line": 866,
"column": 2
} | [
{
"pp": "k G : Type u\ninst✝¹ : CommRing k\ninst✝ : Group G\nA : Rep k G\n⊢ π A 0 ≫ (H0Iso A).hom = (cyclesIso₀ A).hom ≫ (coinvariantsMk k G).app A",
"ppTerm": "?m.44",
"assigned": true,
"usedConstants": [
"CategoryTheory.Category.assoc",
"Rep.coinvariantsFunctor",
"Rep.V",
"... | [] | by | [anonymous] | by |
Mathlib.RepresentationTheory.Homological.GroupHomology.LowDegree | {
"line": 875,
"column": 58
} | {
"line": 875,
"column": 60
} | {
"line": 876,
"column": 2
} | [
{
"pp": "k G : Type u\ninst✝¹ : CommRing k\ninst✝ : Group G\nA : Rep k G\n⊢ H0π A ≫ (H0Iso A).hom = (coinvariantsMk k G).app A",
"ppTerm": "?m.29",
"assigned": true,
"usedConstants": [
"CategoryTheory.Category.assoc",
"Rep.coinvariantsFunctor",
"Rep.V",
"CategoryTheory.Catego... | [] | by | [anonymous] | by |
Mathlib.RepresentationTheory.Homological.GroupHomology.LowDegree | {
"line": 880,
"column": 42
} | {
"line": 880,
"column": 44
} | {
"line": 881,
"column": 2
} | [
{
"pp": "k G : Type u\ninst✝¹ : CommRing k\ninst✝ : Group G\nA : Rep k G\n⊢ (cyclesIso₀ A).hom ≫ H0π A = π A 0",
"ppTerm": "?m.27",
"assigned": true,
"usedConstants": [
"Rep.V",
"CategoryTheory.CategoryStruct.toQuiver",
"Quiver.Hom",
"ModuleCat",
"congrArg",
"Comm... | [] | by | [anonymous] | by |
Mathlib.RingTheory.PowerSeries.Ideal | {
"line": 116,
"column": 66
} | {
"line": 116,
"column": 68
} | {
"line": 117,
"column": 4
} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommRing R\nI : Ideal R⟦X⟧\ninst✝ : I.IsPrime\nhXI : X ∉ I\nn✝ : ℕ\nF : Fin n✝ → R⟦X⟧\nhJI : Submodule.span R⟦X⟧ (Set.range F) ≤ I\nhJ : FG (Submodule.span R⟦X⟧ (Set.range F))\nh' : Ideal.map constantCoeff I = span (Set.range (⇑constantCoeff ∘ F))\nT : R⟦X⟧ → Fin n✝ → R\nhT : ∀ g... | [] | by | Lean.Elab.Tactic.evalWithAnnotateState | by |
Mathlib.RepresentationTheory.Homological.GroupHomology.LowDegree | {
"line": 886,
"column": 40
} | {
"line": 886,
"column": 42
} | {
"line": 886,
"column": 43
} | [
{
"pp": "k G : Type u\ninst✝¹ : CommRing k\ninst✝ : Group G\nA : Rep k G\nC : ↑(H0 A) → Prop\nx : ↑(H0 A)\nh : ∀ (x : ↑A), C ((ConcreteCategory.hom (H0π A)) x)\ny : ↑(cycles A 0)\n⊢ C ((ConcreteCategory.hom (π A 0)) y)",
"ppTerm": "?m.28",
"assigned": true,
"usedConstants": [
"Rep.V",
"M... | [] | by | [anonymous] | by |
Mathlib.RingTheory.UniqueFactorizationDomain.Kaplansky | {
"line": 82,
"column": 93
} | {
"line": 82,
"column": 95
} | {
"line": 83,
"column": 2
} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommSemiring R\ninst✝ : IsDomain R\nH : ∀ (I : Ideal R), I ≠ ⊥ → I.IsPrime → ∃ x ∈ I, Prime x\n⊢ UniqueFactorizationMonoid R",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"CommMonoidWithZero.toCommMonoid",
"Iff.mpr",
"Eq.mpr",
"Idea... | [] | by | [anonymous] | by |
Mathlib.RepresentationTheory.Homological.GroupHomology.LowDegree | {
"line": 895,
"column": 45
} | {
"line": 895,
"column": 47
} | {
"line": 895,
"column": 48
} | [
{
"pp": "k G : Type u\ninst✝² : CommRing k\ninst✝¹ : Group G\nA : Rep k G\ninst✝ : A.IsTrivial\n⊢ (ComplexShape.down ℕ).prev 0 = 1",
"ppTerm": "?m.49",
"assigned": true,
"usedConstants": [
"Nat.instOne",
"congrArg",
"AddMonoid.toAddZeroClass",
"AddRightCancelSemigroup.toAddSe... | [] | by | [anonymous] | by |
Mathlib.RingTheory.PowerSeries.Ideal | {
"line": 122,
"column": 34
} | {
"line": 122,
"column": 36
} | {
"line": 123,
"column": 4
} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommRing R\nI : Ideal R⟦X⟧\ninst✝ : I.IsPrime\nhXI : X ∉ I\nn : ℕ\nF : Fin n → R⟦X⟧\nhJI : Submodule.span R⟦X⟧ (Set.range F) ≤ I\nhJ : FG (Submodule.span R⟦X⟧ (Set.range F))\nh' : Ideal.map constantCoeff I = span (Set.range (⇑constantCoeff ∘ F))\nT : R⟦X⟧ → Fin n → R\nhT : ∀ g ∈ ... | [] | by | Lean.Elab.Tactic.evalWithAnnotateState | by |
Mathlib.RepresentationTheory.Homological.GroupHomology.LowDegree | {
"line": 895,
"column": 57
} | {
"line": 895,
"column": 59
} | {
"line": 896,
"column": 4
} | [
{
"pp": "k G : Type u\ninst✝² : CommRing k\ninst✝¹ : Group G\nA : Rep k G\ninst✝ : A.IsTrivial\n⊢ (inhomogeneousChains A).d 1 0 = 0",
"ppTerm": "?m.50",
"assigned": true,
"usedConstants": [
"dite_cond_eq_true",
"Finsupp.instFunLike",
"LinearMap.id",
"Finsupp.smulZeroClass",
... | [] | by | [anonymous] | by |
Mathlib.RepresentationTheory.Homological.GroupHomology.LowDegree | {
"line": 906,
"column": 61
} | {
"line": 906,
"column": 63
} | {
"line": 907,
"column": 2
} | [
{
"pp": "k G : Type u\ninst✝² : CommRing k\ninst✝¹ : Group G\nA : Rep k G\ninst✝ : A.IsTrivial\n⊢ π A 0 ≫ (H0IsoOfIsTrivial A).hom = (cyclesIso₀ A).hom",
"ppTerm": "?m.36",
"assigned": true,
"usedConstants": [
"Rep.V",
"groupHomology.cyclesIso₀._proof_4",
"groupHomology.cyclesIso₀.... | [] | by | [anonymous] | by |
Mathlib.RingTheory.PowerSeries.Ideal | {
"line": 98,
"column": 76
} | {
"line": 98,
"column": 78
} | {
"line": 99,
"column": 2
} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommRing R\nI : Ideal R⟦X⟧\ninst✝ : I.IsPrime\nJ : Ideal R⟦X⟧\nhJI : J ≤ I\nhXI : X ∉ I\nhJ : J.FG\nh' : Ideal.map constantCoeff I ≤ Ideal.map constantCoeff J\n⊢ I = J",
"ppTerm": "?m.38",
"assigned": true,
"usedConstants": [
"MvPowerSeries.instAddCommGroup",
... | [] | by | [anonymous] | by |
Mathlib.RingTheory.PowerSeries.Ideal | {
"line": 133,
"column": 37
} | {
"line": 133,
"column": 39
} | {
"line": 134,
"column": 4
} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommRing R\nI : Ideal R⟦X⟧\ninst✝ : I.IsPrime\nhI : X ∉ I\nS : Set R\nhSI : span S = Ideal.map constantCoeff I\nhS : S.Finite\n⊢ SurjOn (⇑constantCoeff) (↑I) S",
"ppTerm": "?m.43",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Ideal.subset_span",
"... | [] | by | Lean.Elab.Tactic.evalWithAnnotateState | by |
Mathlib.RepresentationTheory.Homological.GroupHomology.LowDegree | {
"line": 930,
"column": 77
} | {
"line": 930,
"column": 79
} | {
"line": 931,
"column": 2
} | [
{
"pp": "k G : Type u\ninst✝¹ : CommRing k\ninst✝ : Group G\nA : Rep k G\nx : ↥(cycles₁ A)\n⊢ (ConcreteCategory.hom (H1π A)) x = 0 ↔ ↑x ∈ boundaries₁ A",
"ppTerm": "?m.25",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Submodule",
"Rep.V",
"Submodule.Quotient.instZeroQuotie... | [] | by | [anonymous] | by |
Mathlib.RepresentationTheory.Homological.GroupHomology.LowDegree | {
"line": 941,
"column": 53
} | {
"line": 941,
"column": 55
} | {
"line": 942,
"column": 2
} | [
{
"pp": "k G : Type u\ninst✝¹ : CommRing k\ninst✝ : Group G\nA : Rep k G\nx y : ↥(cycles₁ A)\n⊢ (ConcreteCategory.hom (H1π A)) x = (ConcreteCategory.hom (H1π A)) y ↔ ↑x - ↑y ∈ boundaries₁ A",
"ppTerm": "?m.39",
"assigned": true,
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"Eq.mpr"... | [] | by | [anonymous] | by |
Mathlib.RingTheory.PowerSeries.Ideal | {
"line": 144,
"column": 66
} | {
"line": 144,
"column": 68
} | {
"line": 144,
"column": 69
} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommRing R\nI : Ideal R⟦X⟧\ninst✝ : I.IsPrime\nhI : X ∉ I\nS : Set R\nhSI : span S = Ideal.map constantCoeff I\nhS : S.Finite\nthis : SurjOn (⇑constantCoeff) (↑I) S\nT : Set R⟦X⟧\nhTI : T ⊆ ↑I\nhinj : InjOn (⇑constantCoeff) T\nhT : ⇑constantCoeff '' T = S\nf : R⟦X⟧\nhf : constant... | [] | by | [anonymous] | by |
Mathlib.RepresentationTheory.Homological.GroupHomology.LowDegree | {
"line": 948,
"column": 40
} | {
"line": 948,
"column": 42
} | {
"line": 948,
"column": 43
} | [
{
"pp": "k G : Type u\ninst✝¹ : CommRing k\ninst✝ : Group G\nA : Rep k G\nC : ↑(H1 A) → Prop\nx : ↑(H1 A)\nh : ∀ (x : ↥(cycles₁ A)), C ((ConcreteCategory.hom (H1π A)) x)\ny : ↑(cycles A 1)\n⊢ C ((ConcreteCategory.hom (π A 1)) y)",
"ppTerm": "?m.32",
"assigned": true,
"usedConstants": [
"groupH... | [] | by | [anonymous] | by |
Mathlib.RepresentationTheory.Homological.GroupHomology.LowDegree | {
"line": 962,
"column": 56
} | {
"line": 962,
"column": 58
} | {
"line": 963,
"column": 2
} | [
{
"pp": "k G : Type u\ninst✝¹ : CommRing k\ninst✝ : Group G\nA : Rep k G\n⊢ π A 1 ≫ (H1Iso A).hom = (isoCycles₁ A).hom ≫ (shortComplexH1 A).moduleCatLeftHomologyData.π",
"ppTerm": "?m.46",
"assigned": true,
"usedConstants": [
"Submodule",
"Rep.V",
"CategoryTheory.Functor",
"C... | [] | by | [anonymous] | by |
Mathlib.RepresentationTheory.Homological.GroupHomology.LowDegree | {
"line": 986,
"column": 68
} | {
"line": 986,
"column": 70
} | {
"line": 987,
"column": 8
} | [
{
"pp": "k G : Type u\ninst✝² : CommRing k\ninst✝¹ : Group G\nA : Rep k G\ninst✝ : A.IsTrivial\ng h : G\na : ↑A\n⊢ (ModuleCat.Hom.hom (d₂₁ A)) (single (g, h) a) =\n ↑((ModuleCat.Hom.hom (cycles₁IsoOfIsTrivial A).inv) (single g a + single h a)) -\n ↑((ModuleCat.Hom.hom (cycles₁IsoOfIsTrivial A).inv) (sin... | [] | by | [anonymous] | by |
Mathlib.RepresentationTheory.Homological.GroupHomology.LowDegree | {
"line": 985,
"column": 77
} | {
"line": 985,
"column": 79
} | {
"line": 986,
"column": 6
} | [
{
"pp": "k G : Type u\ninst✝² : CommRing k\ninst✝¹ : Group G\nA : Rep k G\ninst✝ : A.IsTrivial\ng h : G\na : ↑A\n⊢ (Multiplicative.toAdd\n (Multiplicative.ofAdd\n (↑(ModuleCat.Hom.hom (H1π A) ∘ₗ\n ModuleCat.Hom.hom (cycles₁IsoOfIsTrivial A).inv ∘ₗ lsingle (g * h))).toIntLinearMap)... | [] | by | [anonymous] | by |
Mathlib.RepresentationTheory.Homological.GroupHomology.LowDegree | {
"line": 1002,
"column": 90
} | {
"line": 1002,
"column": 92
} | {
"line": 1003,
"column": 6
} | [
{
"pp": "k G : Type u\ninst✝² : CommRing k\ninst✝¹ : Group G\nA : Rep k G\ninst✝ : A.IsTrivial\nx✝¹ : ↥(cycles₁ A).toAddSubgroup\ny : G →₀ ↑A\nhy : y ∈ (cycles₁ A).toAddSubgroup\nx✝ : ⟨y, hy⟩ ∈ (shortComplexH1 A).moduleCatToCycles.range.toAddSubgroup\nz : ↑(shortComplexH1 A).X₁\nhz : (shortComplexH1 A).moduleCa... | [] | by | [anonymous] | by |
Mathlib.RepresentationTheory.Homological.GroupHomology.LowDegree | {
"line": 1012,
"column": 54
} | {
"line": 1012,
"column": 56
} | {
"line": 1013,
"column": 2
} | [
{
"pp": "k G : Type u\ninst✝² : CommRing k\ninst✝¹ : Group G\nA : Rep k G\ninst✝ : A.IsTrivial\ng : G\na : ↑A\n⊢ (H1ToTensorOfIsTrivial A)\n ((ConcreteCategory.hom (H1π A)) ((ConcreteCategory.hom (cycles₁IsoOfIsTrivial A).inv) (single g a))) =\n Additive.ofMul (Abelianization.of g) ⊗ₜ[ℤ] a",
"ppTerm... | [] | by | [anonymous] | by |
Mathlib.RepresentationTheory.Homological.GroupHomology.LowDegree | {
"line": 1027,
"column": 50
} | {
"line": 1027,
"column": 52
} | {
"line": 1028,
"column": 6
} | [
{
"pp": "k G : Type u\ninst✝² : CommRing k\ninst✝¹ : Group G\nA : Rep k G\ninst✝ : A.IsTrivial\n⊢ ((TensorProduct.mk ℤ (Additive (Abelianization G)) ↑A).compr₂ₛₗ\n (H1ToTensorOfIsTrivial A ∘ₗ TensorProduct.lift (mkH1OfIsTrivial A))).toAddMonoidHom =\n ((TensorProduct.mk ℤ (Additive (Abelianization G))... | [] | by | [anonymous] | by |
Mathlib.RingTheory.PowerSeries.Ideal | {
"line": 148,
"column": 51
} | {
"line": 148,
"column": 53
} | {
"line": 148,
"column": 54
} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommRing R\nI : Ideal R⟦X⟧\ninst✝ : I.IsPrime\nhI : X ∉ I\nS : Set R\nhSI : span S = Ideal.map constantCoeff I\nhS : S.Finite\nthis : SurjOn (⇑constantCoeff) (↑I) S\nT : Set R⟦X⟧\nhTI : T ⊆ ↑I\nhinj : InjOn (⇑constantCoeff) T\nhT : ⇑constantCoeff '' T = S\nf : R⟦X⟧\nhf : constant... | [] | by | [anonymous] | by |
Mathlib.RepresentationTheory.Homological.GroupHomology.LowDegree | {
"line": 1034,
"column": 85
} | {
"line": 1034,
"column": 87
} | {
"line": 1035,
"column": 8
} | [
{
"pp": "k G : Type u\ninst✝² : CommRing k\ninst✝¹ : Group G\nA : Rep k G\ninst✝ : A.IsTrivial\n⊢ (fun f ↦ f.comp ↑(cycles₁IsoOfIsTrivial A).symm.toLinearEquiv.toAddEquiv)\n (((fun f ↦ f.comp ↑(H1Iso A).symm.toLinearEquiv.toAddEquiv)\n (TensorProduct.lift (mkH1OfIsTrivial A) ∘ₗ H1ToTensorOfIsTri... | [] | by | [anonymous] | by |
Mathlib.RepresentationTheory.Homological.GroupHomology.LowDegree | {
"line": 1046,
"column": 54
} | {
"line": 1046,
"column": 56
} | {
"line": 1047,
"column": 2
} | [
{
"pp": "k G : Type u\ninst✝² : CommRing k\ninst✝¹ : Group G\nA : Rep k G\ninst✝ : A.IsTrivial\ng : G\na : ↑A\n⊢ (H1AddEquivOfIsTrivial A)\n ((ConcreteCategory.hom (H1π A)) ((ConcreteCategory.hom (cycles₁IsoOfIsTrivial A).inv) (single g a))) =\n Additive.ofMul (Abelianization.of g) ⊗ₜ[ℤ] a",
"ppTerm... | [] | by | [anonymous] | by |
Mathlib.RepresentationTheory.Homological.GroupHomology.LowDegree | {
"line": 1052,
"column": 61
} | {
"line": 1052,
"column": 63
} | {
"line": 1053,
"column": 2
} | [
{
"pp": "k G : Type u\ninst✝² : CommRing k\ninst✝¹ : Group G\nA : Rep k G\ninst✝ : A.IsTrivial\ng : G\na : ↑A\n⊢ (H1AddEquivOfIsTrivial A).symm (Additive.ofMul (Abelianization.of g) ⊗ₜ[ℤ] a) =\n (ConcreteCategory.hom (H1π A)) ((ConcreteCategory.hom (cycles₁IsoOfIsTrivial A).inv) (single g a))",
"ppTerm":... | [] | by | [anonymous] | by |
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