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.RepresentationTheory.Homological.ContCohomology.LowDegree | {
"line": 75,
"column": 4
} | {
"line": 75,
"column": 15
} | {
"line": 75,
"column": 16
} | [
{
"pp": "k : Type u_1\nG : Type u_2\ninst✝⁴ : Ring k\ninst✝³ : Group G\ninst✝² : TopologicalSpace k\ninst✝¹ : TopologicalSpace G\ninst✝ : IsTopologicalGroup G\nX : TopRep k G\nx✝ : ↥(↑(TopModuleCat.Hom.hom (X.homogeneousCochains.d 0 1))).ker\nx : C(G, ↑X)\nhx' : x ∈ (X.resolution'.X 0).ρ.invariants\nhx✝ : ⟨x, h... | [
"k : Type u_1\nG : Type u_2\ninst✝⁴ : Ring k\ninst✝³ : Group G\ninst✝² : TopologicalSpace k\ninst✝¹ : TopologicalSpace G\ninst✝ : IsTopologicalGroup G\nX : TopRep k G\nx✝ : ↥(↑(TopModuleCat.Hom.hom (X.homogeneousCochains.d 0 1))).ker\nx : C(G, ↑X)\nhx' : x ∈ (X.resolution'.X 0).ρ.invariants\nhx✝ : ⟨x, hx'⟩ ∈ (↑(Top... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RepresentationTheory.Homological.GroupCohomology.Basic | {
"line": 126,
"column": 4
} | {
"line": 126,
"column": 19
} | {
"line": 127,
"column": 4
} | [
{
"pp": "k G : Type u\ninst✝¹ : CommRing k\nn✝ : ℕ\ninst✝ : Group G\nA : Rep k G\nn : ℕ\n⊢ d A n ≫ d A (n + 1) = 0",
"ppTerm": "?m.28",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Pi.Function.module",
"inhomogeneousCochains.d",
"Rep.V",
"Nat.instOne",
"Categor... | [
"k G : Type u\ninst✝¹ : CommRing k\nn✝ : ℕ\ninst✝ : Group G\nA : Rep k G\nn : ℕ\n⊢ ((freeLiftLEquiv k G (Fin n → G) A).toModuleIso.inv ≫\n ((barComplex k G).linearYonedaObj k A).d n (n + 1) ≫ (freeLiftLEquiv k G (Fin (n + 1) → G) A).toModuleIso.hom) ≫\n (freeLiftLEquiv k G (Fin (n + 1) → G) A).toModuleI... | rw [d_eq, d_eq] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RepresentationTheory.Homological.GroupCohomology.Basic | {
"line": 143,
"column": 2
} | {
"line": 143,
"column": 35
} | {
"line": 143,
"column": 36
} | [
{
"pp": "k G : Type u\ninst✝¹ : CommRing k\nn : ℕ\ninst✝ : Group G\nA : Rep k G\n⊢ d A n ≫ d A (n + 1) = 0",
"ppTerm": "?m.25",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"k G : Type u\ninst✝¹ : CommRing k\nn : ℕ\ninst✝ : Group G\nA : Rep k G\n⊢ d A n ≫ d A (n + 1) = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RepresentationTheory.Induced | {
"line": 151,
"column": 4
} | {
"line": 151,
"column": 15
} | {
"line": 151,
"column": 16
} | [
{
"pp": "k : Type u\nG : Type v\nH : Type v'\ninst✝² : CommRing k\ninst✝¹ : Group G\ninst✝ : Group H\nφ : G →* H\nA✝ : Rep k G\nB✝ : Rep k H\nA : Rep k G\nB : Rep k H\nf : ind φ A ⟶ B\nh : H\na : ↑A\n⊢ ((TensorProduct.AlgebraTensorModule.curry\n ((Hom.hom\n ((fun f ↦\n ... | [
"k : Type u\nG : Type v\nH : Type v'\ninst✝² : CommRing k\ninst✝¹ : Group G\ninst✝ : Group H\nφ : G →* H\nA✝ : Rep k G\nB✝ : Rep k H\nA : Rep k G\nB : Rep k H\nf : ind φ A ⟶ B\nh : H\na : ↑A\n⊢ (B.ρ h⁻¹)\n ((Hom.hom f)\n ((Coinvariants.mk (tprod (MonoidHom.comp (Representation.leftRegular k H) φ) A.ρ))\... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RepresentationTheory.Induced | {
"line": 221,
"column": 4
} | {
"line": 221,
"column": 62
} | {
"line": 222,
"column": 6
} | [
{
"pp": "k : Type u\nG✝ : Type v\nH✝ : Type v'\ninst✝⁴ : CommRing k\ninst✝³ : Group G✝\ninst✝² : Group H✝\nφ✝ : G✝ →* H✝\nA✝ : Rep k G✝\nG H : Type u\ninst✝¹ : Group G\ninst✝ : Group H\nφ : G →* H\nA : Rep k G\nB : Rep k H\ns : G\nx : ↑A\ny : ↑B\n⊢ ((TensorProduct.AlgebraTensorModule.curry\n (TensorPro... | [
"k : Type u\nG✝ : Type v\nH✝ : Type v'\ninst✝⁴ : CommRing k\ninst✝³ : Group G✝\ninst✝² : Group H✝\nφ✝ : G✝ →* H✝\nA✝ : Rep k G✝\nG H : Type u\ninst✝¹ : Group G\ninst✝ : Group H\nφ : G →* H\nA : Rep k G\nB : Rep k H\ns : G\nx : ↑A\ny : ↑B\n⊢ (Coinvariants.mk (tprod (MonoidHom.comp (Representation.leftRegular k H) φ)... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RepresentationTheory.Induced | {
"line": 247,
"column": 4
} | {
"line": 248,
"column": 61
} | {
"line": 249,
"column": 8
} | [
{
"pp": "k : Type u\nG✝ : Type v\nH✝ : Type v'\ninst✝⁴ : CommRing k\ninst✝³ : Group G✝\ninst✝² : Group H✝\nφ✝ : G✝ →* H✝\nA✝ : Rep k G✝\nG H : Type u\ninst✝¹ : Group G\ninst✝ : Group H\nφ : G →* H\nA : Rep k G\nB : Rep k H\nh : H\na : ↑A\nb : ↑B\n⊢ (((TensorProduct.AlgebraTensorModule.curry\n (((in... | [
"k : Type u\nG✝ : Type v\nH✝ : Type v'\ninst✝⁴ : CommRing k\ninst✝³ : Group G✝\ninst✝² : Group H✝\nφ✝ : G✝ →* H✝\nA✝ : Rep k G✝\nG H : Type u\ninst✝¹ : Group G\ninst✝ : Group H\nφ : G →* H\nA : Rep k G\nB : Rep k H\nh : H\na : ↑A\nb : ↑B\n⊢ (Coinvariants.mk (tprod (MonoidHom.comp (Representation.leftRegular k H) φ)... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RepresentationTheory.Homological.GroupCohomology.FiniteCyclic | {
"line": 74,
"column": 62
} | {
"line": 75,
"column": 9
} | {
"line": 75,
"column": 10
} | [
{
"pp": "k G : Type u\ninst✝² : CommRing k\ninst✝¹ : CommGroup G\ninst✝ : Fintype G\nA : Rep k G\ng : G\nhg : ∀ (x : G), x ∈ Subgroup.zpowers g\nx✝ : ↑A\n⊢ x✝ ∈ A.ρ.invariants ↔ x✝ ∈ (Hom.hom (A.applyAsHom g - 𝟙 A)).ker",
"ppTerm": "?m.94",
"assigned": true,
"usedConstants": [
"LinearMap.id",... | [
"k G : Type u\ninst✝² : CommRing k\ninst✝¹ : CommGroup G\ninst✝ : Fintype G\nA : Rep k G\ng : G\nhg : ∀ (x : G), x ∈ Subgroup.zpowers g\nx✝ : ↑A\n⊢ (∀ (g : G), (A.ρ g) x✝ = x✝) ↔ (A.ρ g) x✝ = x✝"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RepresentationTheory.Homological.Resolution | {
"line": 411,
"column": 63
} | {
"line": 411,
"column": 74
} | {
"line": 411,
"column": 75
} | [
{
"pp": "k G : Type u\ninst✝¹ : CommRing k\nn : ℕ\ninst✝ : Group G\nm : ℕ\nkey : (barComplex.d k G (m + 1) ≫ barComplex.d k G m) ≫ (diagonalSuccIsoFree k G m).inv = 0\n⊢ (barComplex.d k G (m + 1) ≫ barComplex.d k G m) ≫ (diagonalSuccIsoFree k G m).inv =\n 0 ≫ (diagonalSuccIsoFree k G m).inv",
"ppTerm": "... | [
"k G : Type u\ninst✝¹ : CommRing k\nn : ℕ\ninst✝ : Group G\nm : ℕ\nkey : (barComplex.d k G (m + 1) ≫ barComplex.d k G m) ≫ (diagonalSuccIsoFree k G m).inv = 0\n⊢ barComplex.d k G (m + 1) ≫\n barComplex.d k G m ≫\n ofHom ↑(Representation.leftRegularTensorTrivialIsoFree (Fin m → G)).symm ≫\n Mono... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RepresentationTheory.Homological.Resolution | {
"line": 423,
"column": 43
} | {
"line": 423,
"column": 76
} | {
"line": 423,
"column": 76
} | [
{
"pp": "k G : Type u\ninst✝¹ : CommRing k\nn : ℕ\ninst✝ : Group G\nj : ℕ\n⊢ (diagonalSuccIsoFree k G (j + 1)).inv ≫ (standardComplex k G).d (j + 1) j = d k G j ≫ (diagonalSuccIsoFree k G j).inv",
"ppTerm": "?m.59",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Nat.instOne",
"Cat... | [
"k G : Type u\ninst✝¹ : CommRing k\nn : ℕ\ninst✝ : Group G\nj : ℕ\n⊢ (diagonalSuccIsoFree k G (j + 1)).inv ≫ (standardComplex k G).d (j + 1) j =\n (diagonalSuccIsoFree k G (j + 1)).inv ≫ (standardComplex k G).d (j + 1) j"
] | d_comp_diagonalSuccIsoFree_inv_eq | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RepresentationTheory.Homological.GroupCohomology.Hilbert90 | {
"line": 176,
"column": 4
} | {
"line": 176,
"column": 72
} | {
"line": 177,
"column": 2
} | [
{
"pp": "K L : Type\ninst✝¹⁶ : Field K\ninst✝¹⁵ : Field L\ninst✝¹⁴ : Algebra K L\ninst✝¹³ : FiniteDimensional K L\ninst✝¹² : IsGalois K L\ninst✝¹¹ : IsCyclic Gal(L/K)\ng : Gal(L/K)\nA : Type u_1\nB : Type u_2\ninst✝¹⁰ : CommRing A\ninst✝⁹ : CommRing B\ninst✝⁸ : Algebra A B\ninst✝⁷ : Algebra A L\ninst✝⁶ : Algebr... | [] | exact (algebraMap K L).injective.comp (IsFractionRing.injective A K) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.RepresentationTheory.Homological.GroupHomology.Basic | {
"line": 182,
"column": 2
} | {
"line": 182,
"column": 33
} | {
"line": 182,
"column": 34
} | [
{
"pp": "k G : Type u\ninst✝¹ : CommRing k\ninst✝ : Group G\nA : Rep k G\nn : ℕ\n⊢ d A (n + 1) ≫ d A n = 0",
"ppTerm": "?m.25",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"k G : Type u\ninst✝¹ : CommRing k\ninst✝ : Group G\nA : Rep k G\nn : ℕ\n⊢ d A (n + 1) ≫ d A n = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RepresentationTheory.Homological.GroupCohomology.LowDegree | {
"line": 309,
"column": 2
} | {
"line": 309,
"column": 79
} | {
"line": 309,
"column": 80
} | [
{
"pp": "k G : Type u\ninst✝¹ : CommRing k\ninst✝ : Group G\nA : Rep k G\nf : ↥(cocycles₁ A)\nthis : (A.ρ 1) (f 1) - f (1 * 1) + f 1 = 0\n⊢ f 1 = 0",
"ppTerm": "?m.30",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"k G : Type u\ninst✝¹ : CommRing k\ninst✝ : Group G\nA : Rep k G\nf : ↥(cocycles₁ A)\nthis : (A.ρ 1) (f 1) - f (1 * 1) + f 1 = 0\n⊢ f 1 = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RepresentationTheory.Homological.GroupCohomology.LowDegree | {
"line": 504,
"column": 2
} | {
"line": 504,
"column": 51
} | {
"line": 504,
"column": 52
} | [
{
"pp": "G : Type u_1\nA : Type u_2\ninst✝² : Monoid G\ninst✝¹ : AddCommGroup A\ninst✝ : MulAction G A\nf : G → A\nhf : IsCocycle₁ f\n⊢ f 1 = 0",
"ppTerm": "?m.14",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"G : Type u_1\nA : Type u_2\ninst✝² : Monoid G\ninst✝¹ : AddCommGroup A\ninst✝ : MulAction G A\nf : G → A\nhf : IsCocycle₁ f\n⊢ f 1 = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RepresentationTheory.Homological.GroupCohomology.LowDegree | {
"line": 508,
"column": 2
} | {
"line": 508,
"column": 62
} | {
"line": 508,
"column": 63
} | [
{
"pp": "G : Type u_1\nA : Type u_2\ninst✝² : Monoid G\ninst✝¹ : AddCommGroup A\ninst✝ : MulAction G A\nf : G × G → A\nhf : IsCocycle₂ f\ng : G\n⊢ f (1, g) = f (1, 1)",
"ppTerm": "?m.20",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"G : Type u_1\nA : Type u_2\ninst✝² : Monoid G\ninst✝¹ : AddCommGroup A\ninst✝ : MulAction G A\nf : G × G → A\nhf : IsCocycle₂ f\ng : G\n⊢ f (1, g) = f (1, 1)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RepresentationTheory.Homological.GroupCohomology.LowDegree | {
"line": 512,
"column": 2
} | {
"line": 512,
"column": 42
} | {
"line": 512,
"column": 43
} | [
{
"pp": "G : Type u_1\nA : Type u_2\ninst✝² : Monoid G\ninst✝¹ : AddCommGroup A\ninst✝ : MulAction G A\nf : G × G → A\nhf : IsCocycle₂ f\ng : G\n⊢ f (g, 1) = g • f (1, 1)",
"ppTerm": "?m.24",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"G : Type u_1\nA : Type u_2\ninst✝² : Monoid G\ninst✝¹ : AddCommGroup A\ninst✝ : MulAction G A\nf : G × G → A\nhf : IsCocycle₂ f\ng : G\n⊢ f (g, 1) = g • f (1, 1)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RepresentationTheory.Homological.GroupCohomology.LowDegree | {
"line": 636,
"column": 2
} | {
"line": 636,
"column": 51
} | {
"line": 636,
"column": 52
} | [
{
"pp": "G : Type u_1\nM : Type u_2\ninst✝² : Monoid G\ninst✝¹ : CommGroup M\ninst✝ : MulAction G M\nf : G → M\nhf : IsMulCocycle₁ f\n⊢ f 1 = 1",
"ppTerm": "?m.14",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"G : Type u_1\nM : Type u_2\ninst✝² : Monoid G\ninst✝¹ : CommGroup M\ninst✝ : MulAction G M\nf : G → M\nhf : IsMulCocycle₁ f\n⊢ f 1 = 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RepresentationTheory.Homological.GroupCohomology.LowDegree | {
"line": 640,
"column": 2
} | {
"line": 640,
"column": 62
} | {
"line": 640,
"column": 63
} | [
{
"pp": "G : Type u_1\nM : Type u_2\ninst✝² : Monoid G\ninst✝¹ : CommGroup M\ninst✝ : MulAction G M\nf : G × G → M\nhf : IsMulCocycle₂ f\ng : G\n⊢ f (1, g) = f (1, 1)",
"ppTerm": "?m.20",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"G : Type u_1\nM : Type u_2\ninst✝² : Monoid G\ninst✝¹ : CommGroup M\ninst✝ : MulAction G M\nf : G × G → M\nhf : IsMulCocycle₂ f\ng : G\n⊢ f (1, g) = f (1, 1)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RepresentationTheory.Homological.GroupCohomology.LowDegree | {
"line": 644,
"column": 2
} | {
"line": 644,
"column": 42
} | {
"line": 644,
"column": 43
} | [
{
"pp": "G : Type u_1\nM : Type u_2\ninst✝² : Monoid G\ninst✝¹ : CommGroup M\ninst✝ : MulAction G M\nf : G × G → M\nhf : IsMulCocycle₂ f\ng : G\n⊢ f (g, 1) = g • f (1, 1)",
"ppTerm": "?m.24",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"G : Type u_1\nM : Type u_2\ninst✝² : Monoid G\ninst✝¹ : CommGroup M\ninst✝ : MulAction G M\nf : G × G → M\nhf : IsMulCocycle₂ f\ng : G\n⊢ f (g, 1) = g • f (1, 1)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RepresentationTheory.Homological.GroupCohomology.LowDegree | {
"line": 929,
"column": 2
} | {
"line": 929,
"column": 13
} | {
"line": 929,
"column": 14
} | [
{
"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.ρ.invariants), C ((ConcreteCategory.hom (H0Iso A).inv) x)\n⊢ C x",
"ppTerm": "?m.34",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"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.ρ.invariants), C ((ConcreteCategory.hom (H0Iso A).inv) x)\n⊢ C x"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RepresentationTheory.Homological.GroupCohomology.LowDegree | {
"line": 992,
"column": 45
} | {
"line": 992,
"column": 62
} | {
"line": 992,
"column": 63
} | [
{
"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 : ↥(cocycles₁ A)), C ((ConcreteCategory.hom (H1π A)) x)\ny : ↑(cocycles A 1)\n⊢ C ((ConcreteCategory.hom (π A 1)) y)",
"ppTerm": "?m.32",
"assigned": true,
"usedConstants": [
"gr... | [
"k G : Type u\ninst✝¹ : CommRing k\ninst✝ : Group G\nA : Rep k G\nC : ↑(H1 A) → Prop\nx : ↑(H1 A)\nh : ∀ (x : ↥(cocycles₁ A)), C ((ConcreteCategory.hom (H1π A)) x)\ny : ↑(cocycles A 1)\n⊢ C ((ConcreteCategory.hom (π A 1)) y)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RepresentationTheory.Homological.GroupCohomology.LowDegree | {
"line": 1067,
"column": 2
} | {
"line": 1068,
"column": 5
} | {
"line": 1070,
"column": 0
} | [
{
"pp": "k G : Type u\ninst✝¹ : CommRing k\ninst✝ : Group G\nA : Rep k G\nx y : ↥(cocycles₂ A)\n⊢ (ConcreteCategory.hom (H2π A)) x = (ConcreteCategory.hom (H2π A)) y ↔ ⇑x - ⇑y ∈ coboundaries₂ A",
"ppTerm": "?m.37",
"assigned": true,
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"Eq.... | [] | rw [← sub_eq_zero, ← map_sub, H2π_eq_zero_iff]
rfl | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RepresentationTheory.Homological.GroupCohomology.LowDegree | {
"line": 1067,
"column": 2
} | {
"line": 1068,
"column": 5
} | {
"line": 1070,
"column": 0
} | [
{
"pp": "k G : Type u\ninst✝¹ : CommRing k\ninst✝ : Group G\nA : Rep k G\nx y : ↥(cocycles₂ A)\n⊢ (ConcreteCategory.hom (H2π A)) x = (ConcreteCategory.hom (H2π A)) y ↔ ⇑x - ⇑y ∈ coboundaries₂ A",
"ppTerm": "?m.37",
"assigned": true,
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"Eq.... | [] | rw [← sub_eq_zero, ← map_sub, H2π_eq_zero_iff]
rfl | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RepresentationTheory.Homological.GroupCohomology.LowDegree | {
"line": 1073,
"column": 45
} | {
"line": 1073,
"column": 62
} | {
"line": 1073,
"column": 63
} | [
{
"pp": "k G : Type u\ninst✝¹ : CommRing k\ninst✝ : Group G\nA : Rep k G\nC : ↑(H2 A) → Prop\nx : ↑(H2 A)\nh : ∀ (x : ↥(cocycles₂ A)), C ((ConcreteCategory.hom (H2π A)) x)\ny : ↑(cocycles A 2)\n⊢ C ((ConcreteCategory.hom (π A 2)) y)",
"ppTerm": "?m.32",
"assigned": true,
"usedConstants": [
"gr... | [
"k G : Type u\ninst✝¹ : CommRing k\ninst✝ : Group G\nA : Rep k G\nC : ↑(H2 A) → Prop\nx : ↑(H2 A)\nh : ∀ (x : ↥(cocycles₂ A)), C ((ConcreteCategory.hom (H2π A)) x)\ny : ↑(cocycles A 2)\n⊢ C ((ConcreteCategory.hom (π A 2)) y)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RepresentationTheory.Homological.GroupCohomology.LowDegree | {
"line": 1073,
"column": 45
} | {
"line": 1073,
"column": 89
} | {
"line": 1075,
"column": 0
} | [
{
"pp": "k G : Type u\ninst✝¹ : CommRing k\ninst✝ : Group G\nA : Rep k G\nC : ↑(H2 A) → Prop\nx : ↑(H2 A)\nh : ∀ (x : ↥(cocycles₂ A)), C ((ConcreteCategory.hom (H2π A)) x)\ny : ↑(cocycles A 2)\n⊢ C ((ConcreteCategory.hom (π A 2)) y)",
"ppTerm": "?m.32",
"assigned": true,
"usedConstants": [
"Pi... | [] | simpa [H2π] using h ((isoCocycles₂ A).hom y) | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.RepresentationTheory.Homological.GroupCohomology.LowDegree | {
"line": 1073,
"column": 45
} | {
"line": 1073,
"column": 89
} | {
"line": 1075,
"column": 0
} | [
{
"pp": "k G : Type u\ninst✝¹ : CommRing k\ninst✝ : Group G\nA : Rep k G\nC : ↑(H2 A) → Prop\nx : ↑(H2 A)\nh : ∀ (x : ↥(cocycles₂ A)), C ((ConcreteCategory.hom (H2π A)) x)\ny : ↑(cocycles A 2)\n⊢ C ((ConcreteCategory.hom (π A 2)) y)",
"ppTerm": "?m.32",
"assigned": true,
"usedConstants": [
"Pi... | [] | simpa [H2π] using h ((isoCocycles₂ A).hom y) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RepresentationTheory.Homological.GroupCohomology.LowDegree | {
"line": 1073,
"column": 45
} | {
"line": 1073,
"column": 89
} | {
"line": 1075,
"column": 0
} | [
{
"pp": "k G : Type u\ninst✝¹ : CommRing k\ninst✝ : Group G\nA : Rep k G\nC : ↑(H2 A) → Prop\nx : ↑(H2 A)\nh : ∀ (x : ↥(cocycles₂ A)), C ((ConcreteCategory.hom (H2π A)) x)\ny : ↑(cocycles A 2)\n⊢ C ((ConcreteCategory.hom (π A 2)) y)",
"ppTerm": "?m.32",
"assigned": true,
"usedConstants": [
"Pi... | [] | simpa [H2π] using h ((isoCocycles₂ A).hom y) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RepresentationTheory.Homological.GroupHomology.FiniteCyclic | {
"line": 65,
"column": 6
} | {
"line": 66,
"column": 83
} | {
"line": 67,
"column": 8
} | [
{
"pp": "case pos\nk G : Type u\ninst✝² : CommRing k\ninst✝¹ : CommGroup G\ninst✝ : Fintype G\nA : Rep k G\ng✝ : G\nhg : ∀ (x : G), x ∈ Subgroup.zpowers g✝\nj : ℕ\na : ↑A\ng : G\nhj : Even (j + 1)\n⊢ (((A.coinvariantsTensorMk ((resolution k g✝⁻¹ ⋯).complex.X (j + 1))).compr₂\n (ModuleCat.Hom.hom\n ... | [
"case pos\nk G : Type u\ninst✝² : CommRing k\ninst✝¹ : CommGroup G\ninst✝ : Fintype G\nA : Rep k G\ng✝ : G\nhg : ∀ (x : G), x ∈ Subgroup.zpowers g✝\nj : ℕ\na : ↑A\ng : G\nhj : Even (j + 1)\n⊢ ∑ x, (A.ρ (x * g⁻¹)) a = ∑ c, (A.ρ (c⁻¹ * g⁻¹)) a"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RepresentationTheory.Homological.GroupCohomology.Functoriality | {
"line": 288,
"column": 4
} | {
"line": 288,
"column": 51
} | {
"line": 288,
"column": 52
} | [
{
"pp": "k G H : Type u\ninst✝² : CommRing k\ninst✝¹ : Group G\ninst✝ : Group H\nA : Rep k H\nB : Rep k G\nf : G →* H\nφ : res f A ⟶ B\nn : ℕ\nx : ↑(shortComplexH1 A).X₁\ng : G\n⊢ (ModuleCat.Hom.hom (Hom.toModuleCatHom φ ≫ (shortComplexH1 B).f)) x g =\n (ModuleCat.Hom.hom ((shortComplexH1 A).f ≫ cochainsMap₁... | [
"k G H : Type u\ninst✝² : CommRing k\ninst✝¹ : Group G\ninst✝ : Group H\nA : Rep k H\nB : Rep k G\nf : G →* H\nφ : res f A ⟶ B\nn : ℕ\nx : ↑(shortComplexH1 A).X₁\ng : G\n⊢ (B.ρ g) ((Hom.hom φ) x) = (Hom.hom φ) ((A.ρ (f g)) x)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RepresentationTheory.Homological.GroupCohomology.Functoriality | {
"line": 292,
"column": 4
} | {
"line": 292,
"column": 65
} | {
"line": 292,
"column": 66
} | [
{
"pp": "k G H : Type u\ninst✝² : CommRing k\ninst✝¹ : Group G\ninst✝ : Group H\nA : Rep k H\nB : Rep k G\nf : G →* H\nφ : res f A ⟶ B\nn : ℕ\nx : ↑(shortComplexH1 A).X₂\ng : G × G\n⊢ (ModuleCat.Hom.hom (cochainsMap₁ f φ ≫ (shortComplexH1 B).g)) x g =\n (ModuleCat.Hom.hom ((shortComplexH1 A).g ≫ cochainsMap₂... | [
"k G H : Type u\ninst✝² : CommRing k\ninst✝¹ : Group G\ninst✝ : Group H\nA : Rep k H\nB : Rep k G\nf : G →* H\nφ : res f A ⟶ B\nn : ℕ\nx : ↑(shortComplexH1 A).X₂\ng : G × G\n⊢ (B.ρ g.1) ((Hom.hom φ) (x (f g.2))) = (Hom.hom φ) ((A.ρ (f g.1)) (x (f g.2)))"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RepresentationTheory.Tannaka | {
"line": 190,
"column": 2
} | {
"line": 190,
"column": 13
} | {
"line": 190,
"column": 14
} | [
{
"pp": "k G : Type u\ninst✝² : CommRing k\ninst✝¹ : Group G\ninst✝ : Finite G\nη₁ η₂ : Aut (forget k G)\nh : η₁.hom.hom.app rightFDRep = η₂.hom.hom.app rightFDRep\nthis : Fintype G\nX : FDRep k G\nv : ↑((forget k G).obj X).obj\nh1 : (forget k G).map (ofRightFDRep X v) ≫ η₁.hom.hom.app X = (forget k G).map (ofR... | [
"k G : Type u\ninst✝² : CommRing k\ninst✝¹ : Group G\ninst✝ : Finite G\nη₁ η₂ : Aut (forget k G)\nh : η₁.hom.hom.app rightFDRep = η₂.hom.hom.app rightFDRep\nthis : Fintype G\nX : FDRep k G\nv : ↑((forget k G).obj X).obj\nh1 : (forget k G).map (ofRightFDRep X v) ≫ η₁.hom.hom.app X = (forget k G).map (ofRightFDRep X ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.MvPolynomial.Ideal | {
"line": 85,
"column": 4
} | {
"line": 85,
"column": 37
} | {
"line": 85,
"column": 38
} | [
{
"pp": "case a\nσ : Type u_1\nR : Type u_2\ninst✝ : CommSemiring R\ni : σ\nc : σ →₀ ℕ\nhx : c + single i 1 ∈ ⇑degree ⁻¹' Set.Ici 1\nhi : i ∈ (c + single i 1).support\n⊢ c + single i 1 ∈ (fun x ↦ (monomial x) 1) ⁻¹' ↑(Submodule.restrictScalars R (idealOfVars σ R))",
"ppTerm": "?a✝",
"assigned": true,
... | [
"case a\nσ : Type u_1\nR : Type u_2\ninst✝ : CommSemiring R\ni : σ\nc : σ →₀ ℕ\nhx : c + single i 1 ∈ ⇑degree ⁻¹' Set.Ici 1\nhi : i ∈ (c + single i 1).support\n⊢ (monomial c) 1 * X i ∈ idealOfVars σ R"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.MvPolynomial.Ideal | {
"line": 123,
"column": 2
} | {
"line": 123,
"column": 17
} | {
"line": 123,
"column": 18
} | [
{
"pp": "case neg\nσ : Type u_1\nR : Type u_2\ninst✝ : CommSemiring R\nn : ℕ\nr : R\nh : ¬r = 0\n⊢ C r ∈ idealOfVars σ R ^ n ↔ r = 0 ∨ n = 0",
"ppTerm": "?neg✝",
"assigned": true,
"usedConstants": [
"Finsupp.instAddZeroClass",
"Eq.mpr",
"False",
"Nat.instMulZeroClass",
... | [
"case neg\nσ : Type u_1\nR : Type u_2\ninst✝ : CommSemiring R\nn : ℕ\nr : R\nh : ¬r = 0\n⊢ C r ∈ idealOfVars σ R ^ n ↔ n = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.MvPowerSeries.Rename | {
"line": 132,
"column": 2
} | {
"line": 132,
"column": 13
} | {
"line": 132,
"column": 14
} | [
{
"pp": "σ : Type u_1\nτ : Type u_2\nR : Type u_4\ninst✝ : CommSemiring R\ne : σ ↪ τ\np : MvPowerSeries σ R\nx : σ →₀ ℕ\n⊢ ∀ b ∈ ⋯.toFinset, b ≠ x → (coeff b) p = 0",
"ppTerm": "?m.39",
"assigned": true,
"usedConstants": [
"Finsupp.mapDomain_tendstoCofinite",
"Eq.mpr",
"Nat.instMul... | [
"σ : Type u_1\nτ : Type u_2\nR : Type u_4\ninst✝ : CommSemiring R\ne : σ ↪ τ\np : MvPowerSeries σ R\nx : σ →₀ ℕ\n⊢ ∀ (b : σ →₀ ℕ), mapDomain (⇑e) b = embDomain e x → ¬b = x → (coeff b) p = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.MvPowerSeries.Rename | {
"line": 161,
"column": 2
} | {
"line": 161,
"column": 28
} | {
"line": 161,
"column": 29
} | [
{
"pp": "σ : Type u_1\nR : Type u_4\ninst✝ : CommSemiring R\nx✝ : MvPowerSeries σ R\ny : σ →₀ ℕ\n⊢ (coeff y) ((rename id) x✝) = (coeff y) ((AlgHom.id R (MvPowerSeries σ R)) x✝)",
"ppTerm": "?m.33",
"assigned": true,
"usedConstants": [
"Finsupp.mapDomain_tendstoCofinite",
"Eq.mpr",
... | [
"σ : Type u_1\nR : Type u_4\ninst✝ : CommSemiring R\nx✝ : MvPowerSeries σ R\ny : σ →₀ ℕ\n⊢ ∑ x ∈ ⋯.toFinset, (coeff x) x✝ = (coeff y) x✝"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.MvPowerSeries.Rename | {
"line": 174,
"column": 2
} | {
"line": 174,
"column": 13
} | {
"line": 174,
"column": 14
} | [
{
"pp": "σ : Type u_1\nτ : Type u_2\nR : Type u_4\ninst✝ : CommSemiring R\ne : σ ↪ τ\na₁✝ a₂✝ : MvPowerSeries σ R\nh : (rename ⇑e) a₁✝ = (rename ⇑e) a₂✝\nx : σ →₀ ℕ\n⊢ (coeff x) a₁✝ = (coeff x) a₂✝",
"ppTerm": "?m.24",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
... | [
"σ : Type u_1\nτ : Type u_2\nR : Type u_4\ninst✝ : CommSemiring R\ne : σ ↪ τ\na₁✝ a₂✝ : MvPowerSeries σ R\nh : (rename ⇑e) a₁✝ = (rename ⇑e) a₂✝\nx : σ →₀ ℕ\n⊢ (coeff x) a₁✝ = (coeff x) a₂✝"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Noetherian.OfPrime | {
"line": 50,
"column": 4
} | {
"line": 59,
"column": 95
} | {
"line": 61,
"column": 0
} | [
{
"pp": "case refine_2\nR : Type u_1\ninst✝ : CommRing R\nI : Ideal R\na : R\nhsup : (I ⊔ span {a}).FG\nhcolon : (Submodule.colon I ↑(span {a})).FG\nw✝¹ : ℕ\nf : Fin w✝¹ → R\nhf : span (Set.range f) = I ⊔ span {a}\nw✝ : ℕ\ni : Fin w✝ → R\nhi : Submodule.span R (Set.range i) = Submodule.colon I ↑(span {a})\nr p ... | [] | · rw [Submodule.mem_sup]
obtain ⟨s, H⟩ := mem_span_range_iff_exists_fun.1 (hf ▸ Ideal.mem_sup_left hy)
simp_rw [← Hf] at H
ring_nf at H
rw [sum_add_distrib, ← sum_mul, add_comm] at H
refine ⟨(∑ k, s k * p k), sum_mem _ (fun _ _ ↦ mul_mem_left _ _ mem_span_range_self),
(∑ k, s k * r... | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.RingTheory.MvPowerSeries.Rename | {
"line": 262,
"column": 2
} | {
"line": 262,
"column": 13
} | {
"line": 262,
"column": 14
} | [
{
"pp": "σ : Type u_1\nτ : Type u_2\nR : Type u_4\ninst✝ : CommSemiring R\ne : σ ↪ τ\nr : R\n⊢ (killCompl e) (C r) = C r",
"ppTerm": "?m.17",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"σ : Type u_1\nτ : Type u_2\nR : Type u_4\ninst✝ : CommSemiring R\ne : σ ↪ τ\nr : R\n⊢ (killCompl e) (C r) = C r"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.UniqueFactorizationDomain.Kaplansky | {
"line": 96,
"column": 21
} | {
"line": 96,
"column": 32
} | {
"line": 96,
"column": 33
} | [
{
"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": ... | [
"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⊢ Associated 1 z"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.PowerSeries.Ideal | {
"line": 110,
"column": 4
} | {
"line": 110,
"column": 24
} | {
"line": 110,
"column": 25
} | [
{
"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... | [
"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 ∈ I, ∑ i, T... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.PowerSeries.Ideal | {
"line": 120,
"column": 6
} | {
"line": 121,
"column": 74
} | {
"line": 122,
"column": 2
} | [
{
"pp": "case succ\nR : 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✝ → ... | [] | simp [trunc_succ, add_mul, sum_add_distrib, ← sub_sub, IH, pow_succ, mul_assoc, ← hG',
mul_sub, H, mul_sum, monomial_eq_C_mul_X_pow, mul_left_comm (C _)] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.RingTheory.PowerSeries.Ideal | {
"line": 120,
"column": 6
} | {
"line": 121,
"column": 74
} | {
"line": 122,
"column": 2
} | [
{
"pp": "case succ\nR : 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✝ → ... | [] | simp [trunc_succ, add_mul, sum_add_distrib, ← sub_sub, IH, pow_succ, mul_assoc, ← hG',
mul_sub, H, mul_sum, monomial_eq_C_mul_X_pow, mul_left_comm (C _)] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.PowerSeries.Ideal | {
"line": 120,
"column": 6
} | {
"line": 121,
"column": 74
} | {
"line": 122,
"column": 2
} | [
{
"pp": "case succ\nR : 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✝ → ... | [] | simp [trunc_succ, add_mul, sum_add_distrib, ← sub_sub, IH, pow_succ, mul_assoc, ← hG',
mul_sub, H, mul_sum, monomial_eq_C_mul_X_pow, mul_left_comm (C _)] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RepresentationTheory.Homological.GroupHomology.LowDegree | {
"line": 886,
"column": 43
} | {
"line": 886,
"column": 54
} | {
"line": 886,
"column": 55
} | [
{
"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": [
"ModuleCat",
... | [
"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)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.UniqueFactorizationDomain.Kaplansky | {
"line": 93,
"column": 2
} | {
"line": 93,
"column": 45
} | {
"line": 94,
"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\na x b : R\nhsubset : closure {r | Prime r} ≤ closure {r | IsUnit r ∨ Prime r}\nthis : a ∈ closure {r | IsUnit r ∨ Prime r}\n⊢ ∃ f, (∀ b ∈ f, Prime b) ∧ Associated f.prod a",
"ppTerm... | [] | induction this using closure_induction with | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | null |
Mathlib.RingTheory.PowerSeries.Ideal | {
"line": 123,
"column": 11
} | {
"line": 123,
"column": 49
} | {
"line": 123,
"column": 50
} | [
{
"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... | [
"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 ∈ I, ∑ i, T... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.PowerSeries.Ideal | {
"line": 135,
"column": 4
} | {
"line": 135,
"column": 69
} | {
"line": 135,
"column": 70
} | [
{
"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\nr : R\nhr : r ∈ S\n⊢ r ∈ ⇑constantCoeff '' ↑I",
"ppTerm": "?m.48",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"SetLike.mem_co... | [
"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\nr : R\nhr : r ∈ S\n⊢ ∃ x ∈ I, constantCoeff x = r"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RepresentationTheory.Homological.GroupHomology.LowDegree | {
"line": 948,
"column": 43
} | {
"line": 948,
"column": 60
} | {
"line": 948,
"column": 61
} | [
{
"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": [
"Module... | [
"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)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RepresentationTheory.Homological.GroupHomology.LowDegree | {
"line": 986,
"column": 6
} | {
"line": 986,
"column": 29
} | {
"line": 986,
"column": 30
} | [
{
"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)... | [
"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 (H1π A)) ((ModuleCat.Hom.hom (cycles₁IsoOfIsTrivial A).inv) (single (g * h) a)) =\n (ModuleCat.Hom.hom (H1π A)) ((ModuleCat.Hom.hom (cycles₁IsoOfIsTrivial A).inv) (single g a + single h a... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RepresentationTheory.Homological.GroupHomology.LowDegree | {
"line": 1088,
"column": 2
} | {
"line": 1089,
"column": 5
} | {
"line": 1091,
"column": 0
} | [
{
"pp": "k G : Type u\ninst✝¹ : CommRing k\ninst✝ : Group G\nA : Rep k G\nx y : ↥(cycles₂ A)\n⊢ (ConcreteCategory.hom (H2π A)) x = (ConcreteCategory.hom (H2π A)) y ↔ ↑x - ↑y ∈ boundaries₂ A",
"ppTerm": "?m.39",
"assigned": true,
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"Eq.mpr"... | [] | rw [← sub_eq_zero, ← map_sub, H2π_eq_zero_iff]
rfl | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RepresentationTheory.Homological.GroupHomology.LowDegree | {
"line": 1088,
"column": 2
} | {
"line": 1089,
"column": 5
} | {
"line": 1091,
"column": 0
} | [
{
"pp": "k G : Type u\ninst✝¹ : CommRing k\ninst✝ : Group G\nA : Rep k G\nx y : ↥(cycles₂ A)\n⊢ (ConcreteCategory.hom (H2π A)) x = (ConcreteCategory.hom (H2π A)) y ↔ ↑x - ↑y ∈ boundaries₂ A",
"ppTerm": "?m.39",
"assigned": true,
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"Eq.mpr"... | [] | rw [← sub_eq_zero, ← map_sub, H2π_eq_zero_iff]
rfl | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RepresentationTheory.Homological.GroupHomology.LowDegree | {
"line": 1094,
"column": 44
} | {
"line": 1094,
"column": 61
} | {
"line": 1094,
"column": 62
} | [
{
"pp": "k G : Type u\ninst✝¹ : CommRing k\ninst✝ : Group G\nA : Rep k G\nC : ↑(H2 A) → Prop\nx : ↑(H2 A)\nh : ∀ (x : ↥(cycles₂ A)), C ((ConcreteCategory.hom (H2π A)) x)\ny : ↑(cycles A 2)\n⊢ C ((ConcreteCategory.hom (π A 2)) y)",
"ppTerm": "?m.32",
"assigned": true,
"usedConstants": [
"Module... | [
"k G : Type u\ninst✝¹ : CommRing k\ninst✝ : Group G\nA : Rep k G\nC : ↑(H2 A) → Prop\nx : ↑(H2 A)\nh : ∀ (x : ↥(cycles₂ A)), C ((ConcreteCategory.hom (H2π A)) x)\ny : ↑(cycles A 2)\n⊢ C ((ConcreteCategory.hom (π A 2)) y)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.Nonarchimedean.AdicTopology | {
"line": 62,
"column": 61
} | {
"line": 63,
"column": 98
} | {
"line": 64,
"column": 6
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\nI : Ideal R\nthis : ∀ (i j : ℕ), ∃ k, I ^ k ≤ I ^ i ∧ I ^ k ≤ I ^ j\n⊢ ∀ (i j : ℕ), ∃ k, I ^ k • ⊤ ≤ I ^ i • ⊤ ⊓ I ^ j • ⊤",
"ppTerm": "?m.71",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Submodule",
"instHSMul",
"Semiring.toModu... | [] | by
simpa only [smul_eq_mul, mul_top, Algebra.algebraMap_self, map_id, le_inf_iff] using! this | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.Algebra.Nonarchimedean.AdicTopology | {
"line": 104,
"column": 37
} | {
"line": 104,
"column": 48
} | {
"line": 104,
"column": 49
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\nI : Ideal R\nU : Set R\ni : ℕ\nh : id ↑((fun i ↦ toAddSubgroup (I ^ i • ⊤)) i) ⊆ U\n⊢ ↑(I ^ i) ⊆ U",
"ppTerm": "?m.77",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"R : Type u_1\ninst✝ : CommRing R\nI : Ideal R\nU : Set R\ni : ℕ\nh : id ↑((fun i ↦ toAddSubgroup (I ^ i • ⊤)) i) ⊆ U\n⊢ ↑(I ^ i) ⊆ U"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.Nonarchimedean.AdicTopology | {
"line": 131,
"column": 39
} | {
"line": 131,
"column": 68
} | {
"line": 131,
"column": 69
} | [
{
"pp": "R : Type u_1\ninst✝² : CommRing R\nI : Ideal R\nM : Type u_2\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nm : M\ni : ℕ\na : R\na_in : a ∈ ↑(I ^ i • ⊤)\n⊢ a ∈ I ^ i",
"ppTerm": "?m.122",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"R : Type u_1\ninst✝² : CommRing R\nI : Ideal R\nM : Type u_2\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nm : M\ni : ℕ\na : R\na_in : a ∈ ↑(I ^ i • ⊤)\n⊢ a ∈ I ^ i"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.AdicCompletion.Topology | {
"line": 54,
"column": 6
} | {
"line": 54,
"column": 17
} | {
"line": 54,
"column": 18
} | [
{
"pp": "R : Type u_1\ninst✝² : CommRing R\ninst✝¹ : UniformSpace R\ninst✝ : IsUniformAddGroup R\nI : Ideal R\nhI : IsAdic I\nthis✝ : (𝓝 0).IsCountablyGenerated\nthis : (𝓤 R).IsCountablyGenerated\nH : ∀ (f : ℕ → R), (∀ {m n : ℕ}, m ≤ n → f m - f n ∈ I ^ m) → ∃ L, ∀ (n : ℕ), f n - L ∈ I ^ n\nu : ℕ → R\nhu : Ca... | [
"R : Type u_1\ninst✝² : CommRing R\ninst✝¹ : UniformSpace R\ninst✝ : IsUniformAddGroup R\nI : Ideal R\nhI : IsAdic I\nthis✝ : (𝓝 0).IsCountablyGenerated\nthis : (𝓤 R).IsCountablyGenerated\nH : ∀ (f : ℕ → R), (∀ {m n : ℕ}, m ≤ n → f m - f n ∈ I ^ m) → ∃ L, ∀ (n : ℕ), f n - L ∈ I ^ n\nu : ℕ → R\nhu : CauchySeq u\n⊢... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.Nonarchimedean.AdicTopology | {
"line": 174,
"column": 6
} | {
"line": 174,
"column": 17
} | {
"line": 174,
"column": 18
} | [
{
"pp": "case mp.right\nR : Type u_1\ninst✝¹ : CommRing R\ntop : TopologicalSpace R\ninst✝ : IsTopologicalRing R\nJ : Ideal R\nH : top = J.adicTopology\nthis : TopologicalSpace R := J.adicTopology\ns : Set R\nhs : s ∈ 𝓝 0\n⊢ ∃ n, ↑(J ^ n) ⊆ s",
"ppTerm": "?mp.right",
"assigned": false,
"usedConstan... | [
"case mp.right\nR : Type u_1\ninst✝¹ : CommRing R\ntop : TopologicalSpace R\ninst✝ : IsTopologicalRing R\nJ : Ideal R\nH : top = J.adicTopology\nthis : TopologicalSpace R := J.adicTopology\ns : Set R\nhs : s ∈ 𝓝 0\n⊢ ∃ n, ↑(J ^ n) ⊆ s"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.AdicCompletion.Topology | {
"line": 68,
"column": 31
} | {
"line": 68,
"column": 42
} | {
"line": 68,
"column": 43
} | [
{
"pp": "R : Type u_1\ninst✝² : CommRing R\ninst✝¹ : UniformSpace R\ninst✝ : IsUniformAddGroup R\nI : Ideal R\nhI : IsAdic I\nthis✝ : (𝓝 0).IsCountablyGenerated\nthis : (𝓤 R).IsCountablyGenerated\nH : CompleteSpace R\nf : ℕ → R\nhf : ∀ {m n : ℕ}, m ≤ n → f m - f n ∈ I ^ m\ni : ℕ\nx✝ : True\nm : ℕ\nhm : i ≤ m\... | [
"R : Type u_1\ninst✝² : CommRing R\ninst✝¹ : UniformSpace R\ninst✝ : IsUniformAddGroup R\nI : Ideal R\nhI : IsAdic I\nthis✝ : (𝓝 0).IsCountablyGenerated\nthis : (𝓤 R).IsCountablyGenerated\nH : CompleteSpace R\nf : ℕ → R\nhf : ∀ {m n : ℕ}, m ≤ n → f m - f n ∈ I ^ m\ni : ℕ\nx✝ : True\nm : ℕ\nhm : i ≤ m\nn : ℕ\nhn :... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.AdicCompletion.Topology | {
"line": 71,
"column": 6
} | {
"line": 71,
"column": 34
} | {
"line": 71,
"column": 35
} | [
{
"pp": "R : Type u_1\ninst✝² : CommRing R\ninst✝¹ : UniformSpace R\ninst✝ : IsUniformAddGroup R\nI : Ideal R\nhI : IsAdic I\nthis✝ : (𝓝 0).IsCountablyGenerated\nthis : (𝓤 R).IsCountablyGenerated\nH : CompleteSpace R\nf : ℕ → R\nhf : ∀ {m n : ℕ}, m ≤ n → f m - f n ∈ I ^ m\nL : R\nhL : Filter.map f Filter.atTo... | [
"R : Type u_1\ninst✝² : CommRing R\ninst✝¹ : UniformSpace R\ninst✝ : IsUniformAddGroup R\nI : Ideal R\nhI : IsAdic I\nthis✝ : (𝓝 0).IsCountablyGenerated\nthis : (𝓤 R).IsCountablyGenerated\nH : CompleteSpace R\nf : ℕ → R\nhf : ∀ {m n : ℕ}, m ≤ n → f m - f n ∈ I ^ m\nL : R\nhL : Filter.map f Filter.atTop ≤ 𝓝 L\ni ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.AdicCompletion.Topology | {
"line": 72,
"column": 4
} | {
"line": 72,
"column": 15
} | {
"line": 72,
"column": 16
} | [
{
"pp": "case mpr\nR : Type u_1\ninst✝² : CommRing R\ninst✝¹ : UniformSpace R\ninst✝ : IsUniformAddGroup R\nI : Ideal R\nhI : IsAdic I\nthis✝ : (𝓝 0).IsCountablyGenerated\nthis : (𝓤 R).IsCountablyGenerated\nH : CompleteSpace R\nf : ℕ → R\nhf : ∀ {m n : ℕ}, m ≤ n → f m - f n ∈ I ^ m\nL : R\nhL : Filter.map f F... | [
"case mpr\nR : Type u_1\ninst✝² : CommRing R\ninst✝¹ : UniformSpace R\ninst✝ : IsUniformAddGroup R\nI : Ideal R\nhI : IsAdic I\nthis✝ : (𝓝 0).IsCountablyGenerated\nthis : (𝓤 R).IsCountablyGenerated\nH : CompleteSpace R\nf : ℕ → R\nhf : ∀ {m n : ℕ}, m ≤ n → f m - f n ∈ I ^ m\nL : R\nhL : Filter.map f Filter.atTop ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.AdicCompletion.Topology | {
"line": 90,
"column": 4
} | {
"line": 90,
"column": 15
} | {
"line": 90,
"column": 16
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝¹ : CommRing R\ninst✝ : CommRing S\nI : Ideal R\ne : R ≃+* S\nthis✝ : WithIdeal R := { i := I }\nthis : WithIdeal S := { i := Ideal.map e I }\n⊢ IsUniformEmbedding ⇑↑(WithIdeal.uniformEquiv e ⋯)",
"ppTerm": "?m.66",
"assigned": true,
"usedConstants": [
... | [
"R : Type u_1\nS : Type u_2\ninst✝¹ : CommRing R\ninst✝ : CommRing S\nI : Ideal R\ne : R ≃+* S\nthis✝ : WithIdeal R := { i := I }\nthis : WithIdeal S := { i := Ideal.map e I }\n⊢ IsUniformEmbedding ⇑(WithIdeal.uniformEquiv e ⋯)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.MvPowerSeries.Equiv | {
"line": 82,
"column": 2
} | {
"line": 82,
"column": 15
} | {
"line": 83,
"column": 2
} | [
{
"pp": "case pos\nσ : Type u_1\nR : Type u_2\ninst✝ : CommSemiring R\nx : Option σ →₀ ℕ\nr : R\nn : ℕ\ny : σ →₀ ℕ\nh1 : optionElim n y = x\nh2 : n = x none\n⊢ r = (coeff y) ((monomial x.some) r)",
"ppTerm": "?pos✝",
"assigned": true,
"usedConstants": [
"Nat.instMulZeroClass",
"Semiring.... | [
"case neg\nσ : Type u_1\nR : Type u_2\ninst✝ : CommSemiring R\nx : Option σ →₀ ℕ\nr : R\nn : ℕ\ny : σ →₀ ℕ\nh1 : optionElim n y = x\nh2 : ¬n = x none\n⊢ r = (coeff y) 0",
"case pos\nσ : Type u_1\nR : Type u_2\ninst✝ : CommSemiring R\nx : Option σ →₀ ℕ\nr : R\nn : ℕ\ny : σ →₀ ℕ\nh1 : ¬optionElim n y = x\nh3 : n = ... | · simp [← h1] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Topology.Algebra.Nonarchimedean.AdicTopology | {
"line": 214,
"column": 4
} | {
"line": 214,
"column": 15
} | {
"line": 214,
"column": 16
} | [
{
"pp": "case mp\nA : Type u_2\ninst✝² : CommRing A\ninst✝¹ : TopologicalSpace A\ninst✝ : IsTopologicalRing A\nh : ∀ (n : ℕ), IsOpen[inst✝¹] ↑(⊥ ^ n)\n_h' : ∀ s ∈ 𝓝 0, ∃ n, ↑(⊥ ^ n) ⊆ s\n⊢ IsOpen[inst✝¹] {0}",
"ppTerm": "?mp",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGo... | [
"case mp\nA : Type u_2\ninst✝² : CommRing A\ninst✝¹ : TopologicalSpace A\ninst✝ : IsTopologicalRing A\nh : ∀ (n : ℕ), IsOpen[inst✝¹] ↑(⊥ ^ n)\n_h' : ∀ s ∈ 𝓝 0, ∃ n, ↑(⊥ ^ n) ⊆ s\n⊢ IsOpen[inst✝¹] {0}"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.Nonarchimedean.AdicTopology | {
"line": 262,
"column": 2
} | {
"line": 262,
"column": 29
} | {
"line": 262,
"column": 30
} | [
{
"pp": "R : Type u_1\ninst✝³ : CommRing R\ninst✝² : WithIdeal R\nS : Type u_2\ninst✝¹ : CommRing S\ninst✝ : WithIdeal S\nf : R →+* S\nhf : Ideal.map f i ≤ i\nn : ℕ\nx✝ : True\n⊢ Ideal.map f (i ^ n) ≤ i ^ n",
"ppTerm": "?m.92",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"RingHom.inst... | [
"R : Type u_1\ninst✝³ : CommRing R\ninst✝² : WithIdeal R\nS : Type u_2\ninst✝¹ : CommRing S\ninst✝ : WithIdeal S\nf : R →+* S\nhf : Ideal.map f i ≤ i\nn : ℕ\nx✝ : True\n⊢ Ideal.map f i ^ n ≤ i ^ n"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.MvPowerSeries.Equiv | {
"line": 128,
"column": 4
} | {
"line": 128,
"column": 63
} | {
"line": 129,
"column": 6
} | [
{
"pp": "σ : Type u_1\nR : Type u_2\ninst✝ : CommSemiring R\n⊢ ∀ (r : R),\n optionFunLeft σ R ((algebraMap R (MvPowerSeries (Option σ) R)) r) =\n (algebraMap R (PowerSeries (MvPowerSeries σ R))) r",
"ppTerm": "?m.29",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Algebra.algebr... | [
"σ : Type u_1\nR : Type u_2\ninst✝ : CommSemiring R\n⊢ ∀ (r : R), optionFunLeft σ R (C r) = C (C r)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.MvPowerSeries.Equiv | {
"line": 145,
"column": 2
} | {
"line": 145,
"column": 66
} | {
"line": 146,
"column": 4
} | [
{
"pp": "σ : Type u_1\nR : Type u_2\ninst✝ : CommSemiring R\ni : σ\nthis : optionElim 0 (single i 1) = single (Option.some i) 1\n⊢ (optionEquivLeft σ R) (X (Option.some i)) = PowerSeries.C (X i)",
"ppTerm": "?m.34",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}... | [
"σ : Type u_1\nR : Type u_2\ninst✝ : CommSemiring R\ni : σ\nthis : optionElim 0 (single i 1) = single (Option.some i) 1\n⊢ (optionEquivLeft σ R) (X (Option.some i)) = PowerSeries.C (X i)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.MvPowerSeries.Equiv | {
"line": 150,
"column": 2
} | {
"line": 150,
"column": 60
} | {
"line": 151,
"column": 4
} | [
{
"pp": "σ : Type u_1\nR : Type u_2\ninst✝ : CommSemiring R\n⊢ (optionEquivLeft σ R) (X none) = PowerSeries.X",
"ppTerm": "?m.10",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"σ : Type u_1\nR : Type u_2\ninst✝ : CommSemiring R\n⊢ (optionEquivLeft σ R) (X none) = PowerSeries.X"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.MvPowerSeries.Equiv | {
"line": 155,
"column": 2
} | {
"line": 155,
"column": 13
} | {
"line": 155,
"column": 14
} | [
{
"pp": "σ : Type u_1\nR : Type u_2\ninst✝ : CommSemiring R\nr : R\n⊢ (optionEquivLeft σ R) (C r) = PowerSeries.C (C r)",
"ppTerm": "?m.15",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"σ : Type u_1\nR : Type u_2\ninst✝ : CommSemiring R\nr : R\n⊢ (optionEquivLeft σ R) (C r) = PowerSeries.C (C r)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.AdicCompletion.LocalRing | {
"line": 74,
"column": 81
} | {
"line": 74,
"column": 92
} | {
"line": 74,
"column": 93
} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommRing R\nI m : Ideal R\ninst✝ : m.IsMaximal\nle : I ≤ m\nfg : I.FG\nmapeq :\n Ideal.map (algebraMap R (AdicCompletion I R)) m = comap (evalOneₐ I).toRingHom (Ideal.map (Ideal.Quotient.mk I) m)\n⊢ RingHom.ker (Ideal.Quotient.mk I) ≤ m",
"ppTerm": "?m.151",
"assigned": ... | [
"R : Type u_1\ninst✝¹ : CommRing R\nI m : Ideal R\ninst✝ : m.IsMaximal\nle : I ≤ m\nfg : I.FG\nmapeq :\n Ideal.map (algebraMap R (AdicCompletion I R)) m = comap (evalOneₐ I).toRingHom (Ideal.map (Ideal.Quotient.mk I) m)\n⊢ I ≤ m"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.AdicCompletion.LocalRing | {
"line": 107,
"column": 6
} | {
"line": 107,
"column": 26
} | {
"line": 107,
"column": 27
} | [
{
"pp": "R : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsNoetherianRing R\ninst✝ : IsLocalRing R\nx : AdicCompletion (maximalIdeal R) R\nthis : (eval (maximalIdeal R) R 1).ker = maximalIdeal R • ⊤\n⊢ x ∈ maximalIdeal (AdicCompletion (maximalIdeal R) R) ↔ ↑x 1 = 0",
"ppTerm": "?m.69",
"assigned": true,
... | [
"R : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsNoetherianRing R\ninst✝ : IsLocalRing R\nx : AdicCompletion (maximalIdeal R) R\nthis : (eval (maximalIdeal R) R 1).ker = maximalIdeal R • ⊤\n⊢ x ∈ Ideal.map (algebraMap R (AdicCompletion (maximalIdeal R) R)) (maximalIdeal R) ↔ ↑x 1 = 0"
] | maximalIdeal_eq_map, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.AdicCompletion.LocalRing | {
"line": 147,
"column": 4
} | {
"line": 147,
"column": 35
} | {
"line": 147,
"column": 36
} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsLocalRing R\nfg : (maximalIdeal R).FG\nthis : IsLocalRing (AdicCompletion (maximalIdeal R) R)\nx : IsLocalRing.ResidueField (AdicCompletion (maximalIdeal R) R)\ny : AdicCompletion (maximalIdeal R) R\nhy : (residue (AdicCompletion (maximalIdeal R) R)) y = x\n... | [
"R : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsLocalRing R\nfg : (maximalIdeal R).FG\nthis : IsLocalRing (AdicCompletion (maximalIdeal R) R)\nx : IsLocalRing.ResidueField (AdicCompletion (maximalIdeal R) R)\ny : AdicCompletion (maximalIdeal R) R\nhy : (residue (AdicCompletion (maximalIdeal R) R)) y = x\nz : R\nhz : ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.MvPowerSeries.Equiv | {
"line": 289,
"column": 6
} | {
"line": 289,
"column": 55
} | {
"line": 290,
"column": 8
} | [
{
"pp": "σ✝ : Type u_1\nR✝ : Type u_2\nn : ℕ\ninst✝³ : CommRing R✝\ninst✝² : Finite σ✝\nσ : Type u_3\nR : Type u_4\ninst✝¹ : Finite σ\ninst✝ : CommRing R\nm✝ n✝ : ℕ\nh : m✝ ≤ n✝\nx✝ : MvPowerSeries σ R\n⊢ ((Ideal.Quotient.factorₐ (MvPolynomial σ R) ⋯).comp (truncTotalAlgHom σ R n✝)) x✝ = (truncTotalAlgHom σ R m... | [
"σ✝ : Type u_1\nR✝ : Type u_2\nn : ℕ\ninst✝³ : CommRing R✝\ninst✝² : Finite σ✝\nσ : Type u_3\nR : Type u_4\ninst✝¹ : Finite σ\ninst✝ : CommRing R\nm✝ n✝ : ℕ\nh : m✝ ≤ n✝\nx✝ : MvPowerSeries σ R\n⊢ (truncTotal n✝) x✝ - (truncTotal m✝) x✝ ∈ MvPolynomial.idealOfVars σ R ^ m✝"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.AdicCompletion.LocalRing | {
"line": 174,
"column": 4
} | {
"line": 174,
"column": 44
} | {
"line": 174,
"column": 45
} | [
{
"pp": "R : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsNoetherianRing R\ninst✝ : IsLocalRing R\nfg : (maximalIdeal R).FG\ncomapeq :\n comap (algebraMap R (AdicCompletion (maximalIdeal R) R)) (maximalIdeal (AdicCompletion (maximalIdeal R) R)) =\n maximalIdeal R\nf : (maximalIdeal R).Cotangent →ₗ[R] (maximalI... | [
"R : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsNoetherianRing R\ninst✝ : IsLocalRing R\nfg : (maximalIdeal R).FG\ncomapeq :\n comap (algebraMap R (AdicCompletion (maximalIdeal R) R)) (maximalIdeal (AdicCompletion (maximalIdeal R) R)) =\n maximalIdeal R\nf : (maximalIdeal R).Cotangent →ₗ[R] (maximalIdeal (AdicCo... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RepresentationTheory.Homological.GroupHomology.Functoriality | {
"line": 169,
"column": 51
} | {
"line": 170,
"column": 23
} | {
"line": 172,
"column": 0
} | [
{
"pp": "k G H : Type u\ninst✝² : CommRing k\ninst✝¹ : Group G\ninst✝ : Group H\nA : Rep k G\nB : Rep k H\nf : G →* H\nφ : A ⟶ res f B\nn : ℕ\n⊢ π A n ≫ map f φ n = cyclesMap f φ n ≫ π B n",
"ppTerm": "?m.50",
"assigned": true,
"usedConstants": [
"HomologicalComplex.homologyπ",
"Nat.inst... | [] | by
simp [map, cyclesMap] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.MvPowerSeries.Equiv | {
"line": 310,
"column": 4
} | {
"line": 311,
"column": 42
} | {
"line": 312,
"column": 2
} | [
{
"pp": "σ : Type u_1\nR : Type u_2\ninst✝¹ : CommRing R\ninst✝ : Finite σ\np : MvPolynomial σ R\nn : ℕ\nthis : p - (truncTotal n) ↑p ∈ MvPolynomial.idealOfVars σ R ^ n\n⊢ ↑((AdicCompletion.of (MvPolynomial.idealOfVars σ R) (MvPolynomial σ R)) p) n = ↑((toAdicCompletion σ R) ↑p) n",
"ppTerm": "?m.52",
"... | [] | simpa [toAdicCompletion, AdicCompletion.liftAlgHom, AdicCompletion.liftRingHom,
Ideal.Quotient.mk_eq_mk_iff_sub_mem] | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.RingTheory.MvPowerSeries.Equiv | {
"line": 310,
"column": 4
} | {
"line": 311,
"column": 42
} | {
"line": 312,
"column": 2
} | [
{
"pp": "σ : Type u_1\nR : Type u_2\ninst✝¹ : CommRing R\ninst✝ : Finite σ\np : MvPolynomial σ R\nn : ℕ\nthis : p - (truncTotal n) ↑p ∈ MvPolynomial.idealOfVars σ R ^ n\n⊢ ↑((AdicCompletion.of (MvPolynomial.idealOfVars σ R) (MvPolynomial σ R)) p) n = ↑((toAdicCompletion σ R) ↑p) n",
"ppTerm": "?m.52",
"... | [] | simpa [toAdicCompletion, AdicCompletion.liftAlgHom, AdicCompletion.liftRingHom,
Ideal.Quotient.mk_eq_mk_iff_sub_mem] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.MvPowerSeries.Equiv | {
"line": 310,
"column": 4
} | {
"line": 311,
"column": 42
} | {
"line": 312,
"column": 2
} | [
{
"pp": "σ : Type u_1\nR : Type u_2\ninst✝¹ : CommRing R\ninst✝ : Finite σ\np : MvPolynomial σ R\nn : ℕ\nthis : p - (truncTotal n) ↑p ∈ MvPolynomial.idealOfVars σ R ^ n\n⊢ ↑((AdicCompletion.of (MvPolynomial.idealOfVars σ R) (MvPolynomial σ R)) p) n = ↑((toAdicCompletion σ R) ↑p) n",
"ppTerm": "?m.52",
"... | [] | simpa [toAdicCompletion, AdicCompletion.liftAlgHom, AdicCompletion.liftRingHom,
Ideal.Quotient.mk_eq_mk_iff_sub_mem] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.MvPowerSeries.Equiv | {
"line": 308,
"column": 2
} | {
"line": 312,
"column": 97
} | {
"line": 314,
"column": 0
} | [
{
"pp": "σ : Type u_1\nR : Type u_2\ninst✝¹ : CommRing R\ninst✝ : Finite σ\np : MvPolynomial σ R\n⊢ (toAdicCompletion σ R) ↑p = (AdicCompletion.of (MvPolynomial.idealOfVars σ R) (MvPolynomial σ R)) p",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"MvPowerSeries.truncTotal",
"I... | [] | symm; ext n
suffices p - (truncTotal n) p ∈ MvPolynomial.idealOfVars σ R ^ n by
simpa [toAdicCompletion, AdicCompletion.liftAlgHom, AdicCompletion.liftRingHom,
Ideal.Quotient.mk_eq_mk_iff_sub_mem]
exact (MvPolynomial.mem_pow_idealOfVars_iff' ..).mpr fun x hx ↦ by simp [coeff_truncTotal _ hx] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.MvPowerSeries.Equiv | {
"line": 308,
"column": 2
} | {
"line": 312,
"column": 97
} | {
"line": 314,
"column": 0
} | [
{
"pp": "σ : Type u_1\nR : Type u_2\ninst✝¹ : CommRing R\ninst✝ : Finite σ\np : MvPolynomial σ R\n⊢ (toAdicCompletion σ R) ↑p = (AdicCompletion.of (MvPolynomial.idealOfVars σ R) (MvPolynomial σ R)) p",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"MvPowerSeries.truncTotal",
"I... | [] | symm; ext n
suffices p - (truncTotal n) p ∈ MvPolynomial.idealOfVars σ R ^ n by
simpa [toAdicCompletion, AdicCompletion.liftAlgHom, AdicCompletion.liftRingHom,
Ideal.Quotient.mk_eq_mk_iff_sub_mem]
exact (MvPolynomial.mem_pow_idealOfVars_iff' ..).mpr fun x hx ↦ by simp [coeff_truncTotal _ hx] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.AdicCompletion.LocalRing | {
"line": 188,
"column": 4
} | {
"line": 189,
"column": 11
} | {
"line": 189,
"column": 12
} | [
{
"pp": "case h\nR : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsNoetherianRing R\ninst✝ : IsLocalRing R\nfg : (maximalIdeal R).FG\ncomapeq :\n comap (algebraMap R (AdicCompletion (maximalIdeal R) R)) (maximalIdeal (AdicCompletion (maximalIdeal R) R)) =\n maximalIdeal R\nf : (maximalIdeal R).Cotangent →ₗ[R] (... | [
"case h\nR : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsNoetherianRing R\ninst✝ : IsLocalRing R\nfg : (maximalIdeal R).FG\ncomapeq :\n comap (algebraMap R (AdicCompletion (maximalIdeal R) R)) (maximalIdeal (AdicCompletion (maximalIdeal R) R)) =\n maximalIdeal R\nf : (maximalIdeal R).Cotangent →ₗ[R] (maximalIdeal... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.AdicCompletion.LocalRing | {
"line": 196,
"column": 4
} | {
"line": 196,
"column": 52
} | {
"line": 196,
"column": 53
} | [
{
"pp": "R : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsNoetherianRing R\ninst✝ : IsLocalRing R\nfg : (maximalIdeal R).FG\ncomapeq :\n comap (algebraMap R (AdicCompletion (maximalIdeal R) R)) (maximalIdeal (AdicCompletion (maximalIdeal R) R)) =\n maximalIdeal R\nf : (maximalIdeal R).Cotangent →ₗ[R] (maximalI... | [
"R : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsNoetherianRing R\ninst✝ : IsLocalRing R\nfg : (maximalIdeal R).FG\ncomapeq :\n comap (algebraMap R (AdicCompletion (maximalIdeal R) R)) (maximalIdeal (AdicCompletion (maximalIdeal R) R)) =\n maximalIdeal R\nf : (maximalIdeal R).Cotangent →ₗ[R] (maximalIdeal (AdicCo... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RepresentationTheory.Homological.GroupHomology.Functoriality | {
"line": 230,
"column": 2
} | {
"line": 231,
"column": 15
} | {
"line": 233,
"column": 0
} | [
{
"pp": "k G H : Type u\ninst✝² : CommRing k\ninst✝¹ : Group G\ninst✝ : Group H\nA : Rep k G\nB : Rep k H\nf : G →* H\nφ : A ⟶ res f B\ng✝ : Fin 0 → G\nx✝ : ↑(ModuleCat.of k ↑A)\n⊢ (ModuleCat.Hom.hom (ModuleCat.ofHom (lsingle g✝) ≫ (chainsMap f φ).f 0 ≫ (chainsIso₀ B).hom)) x✝ =\n (ModuleCat.Hom.hom (ModuleC... | [] | simp [chainsMap_f, Unique.eq_default (α := Fin 0 → G), Unique.eq_default (α := Fin 0 → H),
chainsIso₀] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.RingTheory.AdicCompletion.Noetherian | {
"line": 24,
"column": 69
} | {
"line": 24,
"column": 94
} | {
"line": 24,
"column": 95
} | [
{
"pp": "R : Type u_1\ninst✝⁴ : CommRing R\nI : Ideal R\nM : Type u_2\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : IsNoetherianRing R\ninst✝ : Module.Finite R M\nh : I ≤ ⊥.jacobson\nx : M\nhx : ∀ (n : ℕ), x ≡ 0 [SMOD I ^ n • ⊤]\n⊢ x ∈ ⨅ i, I ^ i • ⊤",
"ppTerm": "?m.38",
"assigned": true,
... | [
"R : Type u_1\ninst✝⁴ : CommRing R\nI : Ideal R\nM : Type u_2\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : IsNoetherianRing R\ninst✝ : Module.Finite R M\nh : I ≤ ⊥.jacobson\nx : M\nhx : ∀ (n : ℕ), x ≡ 0 [SMOD I ^ n • ⊤]\n⊢ ∀ (i : ℕ), x ∈ I ^ i • ⊤"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.AdicCompletion.Noetherian | {
"line": 33,
"column": 65
} | {
"line": 33,
"column": 90
} | {
"line": 33,
"column": 91
} | [
{
"pp": "R : Type u_1\ninst✝⁶ : CommRing R\nI : Ideal R\nM : Type u_2\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : Module R M\ninst✝³ : IsNoetherianRing R\ninst✝² : Module.Finite R M\ninst✝¹ : IsDomain R\ninst✝ : IsTorsionFree R M\nh : I ≠ ⊤\nx : M\nhx : ∀ (n : ℕ), x ≡ 0 [SMOD I ^ n • ⊤]\n⊢ x ∈ ⨅ i, I ^ i • ⊤",
"ppTe... | [
"R : Type u_1\ninst✝⁶ : CommRing R\nI : Ideal R\nM : Type u_2\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : Module R M\ninst✝³ : IsNoetherianRing R\ninst✝² : Module.Finite R M\ninst✝¹ : IsDomain R\ninst✝ : IsTorsionFree R M\nh : I ≠ ⊤\nx : M\nhx : ∀ (n : ℕ), x ≡ 0 [SMOD I ^ n • ⊤]\n⊢ ∀ (i : ℕ), x ∈ I ^ i • ⊤"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.AdicCompletion.Noetherian | {
"line": 43,
"column": 4
} | {
"line": 43,
"column": 22
} | {
"line": 44,
"column": 4
} | [
{
"pp": "case h\nR : Type u_1\ninst✝⁷ : CommRing R\nI : Ideal R\nM : Type u_2\ninst✝⁶ : AddCommGroup M\ninst✝⁵ : Module R M\ninst✝⁴ : IsNoetherianRing R\ninst✝³ : Module.Finite R M\nA : Type u_3\ninst✝² : CommRing A\ninst✝¹ : IsArtinianRing A\ninst✝ : IsLocalRing A\nf : ℕ → A\nhf : ∀ {m n : ℕ}, m ≤ n → f m ≡ f ... | [
"case pos\nR : Type u_1\ninst✝⁷ : CommRing R\nI : Ideal R\nM : Type u_2\ninst✝⁶ : AddCommGroup M\ninst✝⁵ : Module R M\ninst✝⁴ : IsNoetherianRing R\ninst✝³ : Module.Finite R M\nA : Type u_3\ninst✝² : CommRing A\ninst✝¹ : IsArtinianRing A\ninst✝ : IsLocalRing A\nf : ℕ → A\nhf : ∀ {m n : ℕ}, m ≤ n → f m ≡ f n [SMOD ma... | by_cases h : m ≤ n | «_aux_Init_ByCases___macroRules_tacticBy_cases_:__2» | «tacticBy_cases_:_» |
Mathlib.RingTheory.AdicCompletion.Completeness | {
"line": 73,
"column": 2
} | {
"line": 73,
"column": 13
} | {
"line": 73,
"column": 14
} | [
{
"pp": "R : Type u_1\ninst✝² : CommRing R\nI : Ideal R\nM : Type u_2\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\na b : ℕ\nh : a ≤ b\nx : AdicCompletion I ↥(I ^ b • ⊤)\nz : ↥(I ^ b • ⊤)\n⊢ (LinearMap.reduceModIdeal (I ^ a) (I ^ b • ⊤).subtype) (Submodule.Quotient.mk z) = 0",
"ppTerm": "?m.59",
"assign... | [
"R : Type u_1\ninst✝² : CommRing R\nI : Ideal R\nM : Type u_2\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\na b : ℕ\nh : a ≤ b\nx : AdicCompletion I ↥(I ^ b • ⊤)\nz : ↥(I ^ b • ⊤)\n⊢ ↑z ∈ I ^ a • ⊤"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RepresentationTheory.Homological.GroupHomology.Functoriality | {
"line": 303,
"column": 4
} | {
"line": 303,
"column": 50
} | {
"line": 303,
"column": 51
} | [
{
"pp": "k G H : Type u\ninst✝² : CommRing k\ninst✝¹ : Group G\ninst✝ : Group H\nA : Rep k G\nB : Rep k H\nf : G →* H\nφ : A ⟶ res f B\nn : ℕ\na✝ : G × G\nx✝ : ↑A\n⊢ (ModuleCat.Hom.hom (chainsMap₂ f φ ≫ d₂₁ B) ∘ₗ lsingle a✝) x✝ =\n (ModuleCat.Hom.hom (d₂₁ A ≫ chainsMap₁ f φ) ∘ₗ lsingle a✝) x✝",
"ppTerm":... | [
"k G H : Type u\ninst✝² : CommRing k\ninst✝¹ : Group G\ninst✝ : Group H\nA : Rep k G\nB : Rep k H\nf : G →* H\nφ : A ⟶ res f B\nn : ℕ\na✝ : G × G\nx✝ : ↑A\n⊢ single (f a✝.2) ((B.ρ (f a✝.1⁻¹)) ((Hom.hom φ) x✝)) = single (f a✝.2) ((Hom.hom φ) ((A.ρ a✝.1⁻¹) x✝))"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RepresentationTheory.Homological.GroupHomology.Functoriality | {
"line": 307,
"column": 4
} | {
"line": 307,
"column": 32
} | {
"line": 307,
"column": 33
} | [
{
"pp": "k G H : Type u\ninst✝² : CommRing k\ninst✝¹ : Group G\ninst✝ : Group H\nA : Rep k G\nB : Rep k H\nf : G →* H\nφ : A ⟶ res f B\nn : ℕ\na✝ : G\nx✝ : ↑A\n⊢ (ModuleCat.Hom.hom (chainsMap₁ f φ ≫ d₁₀ B) ∘ₗ lsingle a✝) x✝ =\n (ModuleCat.Hom.hom (d₁₀ A ≫ Hom.toModuleCatHom φ) ∘ₗ lsingle a✝) x✝",
"ppTerm... | [
"k G H : Type u\ninst✝² : CommRing k\ninst✝¹ : Group G\ninst✝ : Group H\nA : Rep k G\nB : Rep k H\nf : G →* H\nφ : A ⟶ res f B\nn : ℕ\na✝ : G\nx✝ : ↑A\n⊢ (B.ρ (f a✝⁻¹)) ((Hom.hom φ) x✝) = (Hom.hom φ) ((A.ρ a✝⁻¹) x✝)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.AdicCompletion.Completeness | {
"line": 86,
"column": 2
} | {
"line": 86,
"column": 65
} | {
"line": 87,
"column": 4
} | [
{
"pp": "R : Type u_1\ninst✝² : CommRing R\nI : Ideal R\nM : Type u_2\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\na b c : ℕ\nx : AdicCompletion I M\nh : c = b + a\nha : ↑x a = 0\n⊢ ∃ t, (powSMulQuotInclusion I M h ⊤) t = ↑x c",
"ppTerm": "?m.33",
"assigned": true,
"usedConstants": [
"Eq.mpr"... | [
"R : Type u_1\ninst✝² : CommRing R\nI : Ideal R\nM : Type u_2\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\na b c : ℕ\nx : AdicCompletion I M\nh : c = b + a\nha : ↑x a = 0\n⊢ ↑x c ∈ I ^ a • ⊤"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.MvPowerSeries.Equiv | {
"line": 424,
"column": 4
} | {
"line": 424,
"column": 15
} | {
"line": 424,
"column": 16
} | [
{
"pp": "σ : Type u_2\nR : Type u_4\ninst✝ : CommRing R\nf : R⟦X⟧\nhf : constantCoeff f = 0\nd : σ →₀ ℕ\ns : σ\n⊢ s ∉ ↑d.support → s ∉ {s | (MvPowerSeries.coeff d) ((PowerSeries.toMvPowerSeries s) f) ≠ 0}",
"ppTerm": "?m.32",
"assigned": true,
"usedConstants": [
"Finsupp.instFunLike",
"E... | [
"σ : Type u_2\nR : Type u_4\ninst✝ : CommRing R\nf : R⟦X⟧\nhf : constantCoeff f = 0\nd : σ →₀ ℕ\ns : σ\n⊢ d s = 0 → (MvPowerSeries.coeff d) ((PowerSeries.toMvPowerSeries s) f) = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.AdicCompletion.RingHom | {
"line": 134,
"column": 2
} | {
"line": 134,
"column": 13
} | {
"line": 134,
"column": 14
} | [
{
"pp": "R : Type u_3\nS : Type u_4\nA : Type u_5\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing S\ninst✝³ : Algebra R S\nI : Ideal S\ninst✝² : IsAdicComplete I S\ninst✝¹ : CommRing A\ninst✝ : Algebra R A\nf g : A →ₐ[R] S\nH : ∀ (n : ℕ), (Quotient.mkₐ R (I ^ n)).comp f = (Quotient.mkₐ R (I ^ n)).comp g\nx : A\nn : ℕ\n... | [
"R : Type u_3\nS : Type u_4\nA : Type u_5\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing S\ninst✝³ : Algebra R S\nI : Ideal S\ninst✝² : IsAdicComplete I S\ninst✝¹ : CommRing A\ninst✝ : Algebra R A\nf g : A →ₐ[R] S\nH : ∀ (n : ℕ), (Quotient.mkₐ R (I ^ n)).comp f = (Quotient.mkₐ R (I ^ n)).comp g\nx : A\nn : ℕ\n⊢ (Ideal.Quo... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.AdicCompletion.RingHom | {
"line": 159,
"column": 4
} | {
"line": 159,
"column": 15
} | {
"line": 159,
"column": 16
} | [
{
"pp": "case refine_2\nR : Type u_1\nS : Type u_2\ninst✝¹ : NonAssocSemiring R\ninst✝ : CommRing S\nI : Ideal S\na : ℕ → ℕ\nha : StrictMono a\nf : (n : ℕ) → R →+* S ⧸ I ^ a n\nm n : ℕ\nhle : m ≤ n\nx : R\ns : ℕ\nhf : ∀ {m : ℕ} (x : R), ((factorPow I ⋯).comp (f (m + 1))) x = (f m) x\n⊢ (f s) x = (Submodule.fact... | [
"case refine_2\nR : Type u_1\nS : Type u_2\ninst✝¹ : NonAssocSemiring R\ninst✝ : CommRing S\nI : Ideal S\na : ℕ → ℕ\nha : StrictMono a\nf : (n : ℕ) → R →+* S ⧸ I ^ a n\nm n : ℕ\nhle : m ≤ n\nx : R\ns : ℕ\nhf : ∀ {m : ℕ} (x : R), ((factorPow I ⋯).comp (f (m + 1))) x = (f m) x\n⊢ (f s) x = (factor ⋯) ((f (s + 1)) x)"... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RepresentationTheory.Homological.GroupHomology.Functoriality | {
"line": 399,
"column": 2
} | {
"line": 399,
"column": 63
} | {
"line": 400,
"column": 4
} | [
{
"pp": "k G H : Type u\ninst✝² : CommRing k\ninst✝¹ : Group G\ninst✝ : Group H\nA : Rep k G\nB : Rep k H\nφ : A ⟶ res 1 B\nx : ↑(ModuleCat.of k ↥(cycles₁ A))\n⊢ ↑((ModuleCat.Hom.hom (mapCycles₁ 1 φ)) x) ∈ boundaries₁ B",
"ppTerm": "?m.85",
"assigned": true,
"usedConstants": [
"Eq.mpr",
... | [
"k G H : Type u\ninst✝² : CommRing k\ninst✝¹ : Group G\ninst✝ : Group H\nA : Rep k G\nB : Rep k H\nφ : A ⟶ res 1 B\nx : ↑(ModuleCat.of k ↥(cycles₁ A))\n⊢ ((↑x).sum fun a b ↦ single 1 ((Hom.hom φ) b)) ∈ boundaries₁ B"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RepresentationTheory.Homological.GroupHomology.Functoriality | {
"line": 436,
"column": 2
} | {
"line": 438,
"column": 9
} | {
"line": 438,
"column": 10
} | [
{
"pp": "k G H : Type u\ninst✝⁴ : CommRing k\ninst✝³ : Group G\ninst✝² : Group H\nA : Rep k G\nB : Rep k H\nf : G →* H\nφ : A ⟶ res f B\nn : ℕ\nS : Subgroup G\ninst✝¹ : S.Normal\ninst✝ : Representation.IsTrivial (MonoidHom.comp A.ρ S.subtype)\nx : G ⧸ S →₀ ↑(A.ofQuotient S)\nhx : x ∈ cycles₁ (A.ofQuotient S)\ns... | [
"k G H : Type u\ninst✝⁴ : CommRing k\ninst✝³ : Group G\ninst✝² : Group H\nA : Rep k G\nB : Rep k H\nf : G →* H\nφ : A ⟶ res f B\nn : ℕ\nS : Subgroup G\ninst✝¹ : S.Normal\ninst✝ : Representation.IsTrivial (MonoidHom.comp A.ρ S.subtype)\nx : G ⧸ S →₀ ↑(A.ofQuotient S)\nhx : x ∈ cycles₁ (A.ofQuotient S)\ns : G ⧸ S → G... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.AdicCompletion.Completeness | {
"line": 145,
"column": 2
} | {
"line": 149,
"column": 89
} | {
"line": 150,
"column": 2
} | [
{
"pp": "R : Type u_1\ninst✝⁴ : CommRing R\nI : Ideal R\nM : Type u_2\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\nι : Type u_3\ninst✝¹ : DecidableEq ι\ninst✝ : Fintype ι\nf : ι → R\ni✝ : ι\na✝ : AdicCauchySequence I M\nn✝ : ℕ\n⊢ ↑(((((lsum (AdicCompletion I R)) fun i ↦ (algebraMap R (AdicCompletion I R)) (f ... | [
"R : Type u_1\ninst✝⁴ : CommRing R\nI : Ideal R\nM : Type u_2\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\nι : Type u_3\ninst✝¹ : DecidableEq ι\ninst✝ : Fintype ι\nf : ι → R\ni✝ : ι\na✝ : AdicCauchySequence I M\nn✝ : ℕ\n⊢ (Ideal.Quotient.mk (I ^ n✝ • ⊤)) (f i✝) • Submodule.Quotient.mk (↑a✝ n✝) = f i✝ • Submodule.... | simp only [algebraMap_apply, Algebra.algebraMap_self, RingHom.id_apply, LinearMap.coe_comp,
coe_lsum, LinearMap.coe_smul, LinearMap.id_coe, LinearEquiv.coe_coe, Function.comp_apply,
finsuppLEquivDirectSum_symm_lof, Pi.smul_apply, id_eq, smul_zero, sum_single_index, smul_eval,
mapQ_eq_factor, factor_eq_facto... | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.RingTheory.AdicCompletion.Completeness | {
"line": 205,
"column": 6
} | {
"line": 206,
"column": 13
} | {
"line": 206,
"column": 14
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\nI : Ideal R\nfg : I.FG\nx : AdicCompletion I R\neq : I ^ 1 * ⊤ = I\n⊢ Function.Injective ⇑(Ideal.Quotient.factor ⋯)",
"ppTerm": "?m.101",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"RingHom.instRingHomClass",
"Semiring.toModule",
... | [
"R : Type u_1\ninst✝ : CommRing R\nI : Ideal R\nfg : I.FG\nx : AdicCompletion I R\neq : I ^ 1 * ⊤ = I\n⊢ Ideal.map (Ideal.Quotient.mk (I ^ 1 * ⊤)) I = ⊥"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.AdicCompletion.Completeness | {
"line": 207,
"column": 4
} | {
"line": 207,
"column": 61
} | {
"line": 207,
"column": 62
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\nI : Ideal R\nfg : I.FG\nx : AdicCompletion I R\neq : I ^ 1 * ⊤ = I\nthis : Function.Injective ⇑(Ideal.Quotient.factor ⋯)\n⊢ x ∈ RingHom.ker (evalOneₐ I).toRingHom ↔ x ∈ (eval I R 1).ker",
"ppTerm": "?m.102",
"assigned": true,
"usedConstants": [
"Eq.mp... | [
"R : Type u_1\ninst✝ : CommRing R\nI : Ideal R\nfg : I.FG\nx : AdicCompletion I R\neq : I ^ 1 * ⊤ = I\nthis : Function.Injective ⇑(Ideal.Quotient.factor ⋯)\n⊢ (Ideal.Quotient.factor ⋯) ((eval I R 1) x) = 0 ↔ (eval I R 1) x = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Artinian.Algebra | {
"line": 56,
"column": 7
} | {
"line": 56,
"column": 45
} | {
"line": 56,
"column": 46
} | [
{
"pp": "R : Type u_1\nA : Type u_2\ninst✝⁴ : CommRing R\ninst✝³ : IsArtinianRing R\ninst✝² : Ring A\ninst✝¹ : Algebra R A\ninst✝ : Algebra.IsIntegral R A\nx✝ : A\n⊢ x✝ ∈ IsUnit.submonoid A ↔ x✝ ∈ A⁰",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Monoid.toMulOneClas... | [
"R : Type u_1\nA : Type u_2\ninst✝⁴ : CommRing R\ninst✝³ : IsArtinianRing R\ninst✝² : Ring A\ninst✝¹ : Algebra R A\ninst✝ : Algebra.IsIntegral R A\nx✝ : A\n⊢ IsUnit x✝ ↔ x✝ ∈ A⁰"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Coalgebra.GroupLike | {
"line": 49,
"column": 14
} | {
"line": 49,
"column": 25
} | {
"line": 49,
"column": 26
} | [
{
"pp": "R : Type u_2\nA : Type u_3\ninst✝⁴ : CommSemiring R\ninst✝³ : AddCommMonoid A\ninst✝² : Module R A\ninst✝¹ : Coalgebra R A\ninst✝ : Nontrivial R\nha : IsGroupLikeElem R 0\n⊢ False",
"ppTerm": "?m.29",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"R : Type u_2\nA : Type u_3\ninst✝⁴ : CommSemiring R\ninst✝³ : AddCommMonoid A\ninst✝² : Module R A\ninst✝¹ : Coalgebra R A\ninst✝ : Nontrivial R\nha : IsGroupLikeElem R 0\n⊢ False"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Coalgebra.GroupLike | {
"line": 114,
"column": 2
} | {
"line": 115,
"column": 62
} | {
"line": 116,
"column": 2
} | [
{
"pp": "case cons\nR : Type u_2\nA : Type u_3\ninst✝⁵ : CommRing R\ninst✝⁴ : IsDomain R\ninst✝³ : AddCommGroup A\ninst✝² : Module R A\ninst✝¹ : Coalgebra R A\ninst✝ : IsTorsionFree R A\na : A\ns : Finset A\nhas : a ∉ s\nih : ↑s ⊆ {a | IsGroupLikeElem R a} → LinearIndepOn R id ↑s\nhs : ↑(Finset.cons a s has) ⊆ ... | [
"case cons\nR : Type u_2\nA : Type u_3\ninst✝⁵ : CommRing R\ninst✝⁴ : IsDomain R\ninst✝³ : AddCommGroup A\ninst✝² : Module R A\ninst✝¹ : Coalgebra R A\ninst✝ : IsTorsionFree R A\na : A\ns : Finset A\nhas : a ∉ s\nih : ↑s ⊆ {a | IsGroupLikeElem R a} → LinearIndepOn R id ↑s\nhs : IsGroupLikeElem R a ∧ ∀ a ∈ s, IsGrou... | simp only [Finset.cons_eq_insert, Finset.coe_insert, Set.subset_def, Set.mem_insert_iff,
Finset.mem_coe, Set.mem_ofPred_eq, forall_eq_or_imp] at hs | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.RingTheory.Coalgebra.GroupLike | {
"line": 144,
"column": 6
} | {
"line": 144,
"column": 21
} | {
"line": 144,
"column": 22
} | [
{
"pp": "R : Type u_2\nA : Type u_3\ninst✝⁵ : CommRing R\ninst✝⁴ : IsDomain R\ninst✝³ : AddCommGroup A\ninst✝² : Module R A\ninst✝¹ : Coalgebra R A\ninst✝ : IsTorsionFree R A\na : A\ns : Finset A\nhas : a ∉ s\nha : IsGroupLikeElem R a\nhs : ∀ a ∈ s, IsGroupLikeElem R a\nd : R\nc : A → R\nhc : ∑ a ∈ s, c a • a =... | [
"R : Type u_2\nA : Type u_3\ninst✝⁵ : CommRing R\ninst✝⁴ : IsDomain R\ninst✝³ : AddCommGroup A\ninst✝² : Module R A\ninst✝¹ : Coalgebra R A\ninst✝ : IsTorsionFree R A\na : A\ns : Finset A\nhas : a ∉ s\nha : IsGroupLikeElem R a\nhs : ∀ a ∈ s, IsGroupLikeElem R a\nd : R\nc : A → R\nhc : ∑ a ∈ s, c a • a = d • a\nih :... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
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