module string | startPos dict | endPos dict | nextStartPos dict | goals list | goalsAfter list | ppTac string | elaborator string | kind string |
|---|---|---|---|---|---|---|---|---|
Mathlib.RepresentationTheory.Coinduced | {
"line": 83,
"column": 4
} | {
"line": 83,
"column": 27
} | {
"line": 83,
"column": 28
} | [
{
"pp": "k : Type u_1\nG : Type u_2\nH : Type u_3\ninst✝⁶ : Semiring k\ninst✝⁵ : Monoid G\ninst✝⁴ : Monoid H\nφ : G →* H\nA : Type u_4\nB : Type u_5\ninst✝³ : AddCommMonoid A\ninst✝² : Module k A\ninst✝¹ : AddCommMonoid B\ninst✝ : Module k B\nσ : Representation k G A\nρ : Representation k G B\nh : H\nx : H → B\... | [
"k : Type u_1\nG : Type u_2\nH : Type u_3\ninst✝⁶ : Semiring k\ninst✝⁵ : Monoid G\ninst✝⁴ : Monoid H\nφ : G →* H\nA : Type u_4\nB : Type u_5\ninst✝³ : AddCommMonoid A\ninst✝² : Module k A\ninst✝¹ : AddCommMonoid B\ninst✝ : Module k B\nσ : Representation k G A\nρ : Representation k G B\nh : H\nx : H → B\nhx : x ∈ co... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RepresentationTheory.Coinduced | {
"line": 95,
"column": 4
} | {
"line": 95,
"column": 19
} | {
"line": 95,
"column": 20
} | [
{
"pp": "k : Type u_1\nG : Type u_2\nH : Type u_3\ninst✝⁶ : Semiring k\ninst✝⁵ : Monoid G\ninst✝⁴ : Monoid H\nφ : G →* H\nA : Type u_4\nB : Type u_5\ninst✝³ : AddCommMonoid A\ninst✝² : Module k A\ninst✝¹ : AddCommMonoid B\ninst✝ : Module k B\nσ : Representation k G A\nρ : Representation k G B\nf : σ.Intertwinin... | [
"k : Type u_1\nG : Type u_2\nH : Type u_3\ninst✝⁶ : Semiring k\ninst✝⁵ : Monoid G\ninst✝⁴ : Monoid H\nφ : G →* H\nA : Type u_4\nB : Type u_5\ninst✝³ : AddCommMonoid A\ninst✝² : Module k A\ninst✝¹ : AddCommMonoid B\ninst✝ : Module k B\nσ : Representation k G A\nρ : Representation k G B\nf : σ.IntertwiningMap ρ\nx : ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RepresentationTheory.Coinduced | {
"line": 188,
"column": 40
} | {
"line": 188,
"column": 51
} | {
"line": 188,
"column": 52
} | [
{
"pp": "k : Type u\nG : Type v\nH : Type w\ninst✝² : CommRing k\ninst✝¹ : Monoid G\ninst✝ : Monoid H\nφ : G →* H\nA : Rep k G\nf g : ↑(coind' φ A)\nhfg : ∀ (h : H), (Hom.hom f).toLinearMap (MonoidAlgebra.single h 1) = (Hom.hom g).toLinearMap (MonoidAlgebra.single h 1)\nh : H\n⊢ ((Hom.hom f).toLinearMap ∘ₗ Mono... | [
"k : Type u\nG : Type v\nH : Type w\ninst✝² : CommRing k\ninst✝¹ : Monoid G\ninst✝ : Monoid H\nφ : G →* H\nA : Rep k G\nf g : ↑(coind' φ A)\nhfg : ∀ (h : H), (Hom.hom f).toLinearMap (MonoidAlgebra.single h 1) = (Hom.hom g).toLinearMap (MonoidAlgebra.single h 1)\nh : H\n⊢ (Hom.hom f) (MonoidAlgebra.single h 1) = (Ho... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RepresentationTheory.Coinduced | {
"line": 220,
"column": 15
} | {
"line": 220,
"column": 26
} | {
"line": 220,
"column": 27
} | [
{
"pp": "k : Type u\nG : Type v\nH : Type w\ninst✝² : CommRing k\ninst✝¹ : Monoid G\ninst✝ : Monoid H\nφ : G →* H\nA : Rep k G\nf : res φ (leftRegular k H) ⟶ A\ng : G\nh : H\n⊢ ?m.240",
"ppTerm": "?m.241",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"k : Type u\nG : Type v\nH : Type w\ninst✝² : CommRing k\ninst✝¹ : Monoid G\ninst✝ : Monoid H\nφ : G →* H\nA : Rep k G\nf : res φ (leftRegular k H) ⟶ A\ng : G\nh : H\n⊢ ?m.240"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RepresentationTheory.Coinduced | {
"line": 248,
"column": 45
} | {
"line": 248,
"column": 56
} | {
"line": 248,
"column": 57
} | [
{
"pp": "k : Type u\nG : Type v\nH : Type w\ninst✝² : CommRing k\ninst✝¹ : Monoid G\ninst✝ : Monoid H\nφ : G →* H\nA✝ : Rep k G\nB : Rep k H\nA : Rep k G\nf : res φ B ⟶ A\nx✝² : ↑B\nx✝¹ : G\nx✝ : H\n⊢ (LinearMap.pi fun h ↦ (Hom.hom f).toLinearMap ∘ₗ B.ρ h) x✝² (φ x✝¹ * x✝) =\n (A.ρ x✝¹) ((LinearMap.pi fun h ... | [
"k : Type u\nG : Type v\nH : Type w\ninst✝² : CommRing k\ninst✝¹ : Monoid G\ninst✝ : Monoid H\nφ : G →* H\nA✝ : Rep k G\nB : Rep k H\nA : Rep k G\nf : res φ B ⟶ A\nx✝² : ↑B\nx✝¹ : G\nx✝ : H\n⊢ (Hom.hom f) ((B.ρ (φ x✝¹)) ((B.ρ x✝) x✝²)) = (A.ρ x✝¹) ((Hom.hom f) ((B.ρ x✝) x✝²))"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RepresentationTheory.Coinvariants | {
"line": 110,
"column": 4
} | {
"line": 111,
"column": 32
} | {
"line": 111,
"column": 33
} | [
{
"pp": "k : Type u_1\nG : Type u_2\nV : Type u_3\nW : Type u_4\nX : Type u_5\ninst✝⁷ : CommRing k\ninst✝⁶ : Monoid G\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : Module k V\ninst✝³ : AddCommGroup W\ninst✝² : Module k W\ninst✝¹ : AddCommGroup X\ninst✝ : Module k X\nρ : Representation k G V\nτ : Representation k G W\nυ : ... | [
"k : Type u_1\nG : Type u_2\nV : Type u_3\nW : Type u_4\nX : Type u_5\ninst✝⁷ : CommRing k\ninst✝⁶ : Monoid G\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : Module k V\ninst✝³ : AddCommGroup W\ninst✝² : Module k W\ninst✝¹ : AddCommGroup X\ninst✝ : Module k X\nρ : Representation k G V\nτ : Representation k G W\nυ : Representati... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RepresentationTheory.Coinvariants | {
"line": 131,
"column": 34
} | {
"line": 131,
"column": 45
} | {
"line": 131,
"column": 46
} | [
{
"pp": "k : Type u_1\nG : Type u_2\nV : Type u_3\nW : Type u_4\nX : Type u_5\ninst✝⁷ : CommRing k\ninst✝⁶ : Monoid G\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : Module k V\ninst✝³ : AddCommGroup W\ninst✝² : Module k W\ninst✝¹ : AddCommGroup X\ninst✝ : Module k X\nρ : Representation k G V\nτ : Representation k G W\nυ : ... | [
"k : Type u_1\nG : Type u_2\nV : Type u_3\nW : Type u_4\nX : Type u_5\ninst✝⁷ : CommRing k\ninst✝⁶ : Monoid G\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : Module k V\ninst✝³ : AddCommGroup W\ninst✝² : Module k W\ninst✝¹ : AddCommGroup X\ninst✝ : Module k X\nρ : Representation k G V\nτ : Representation k G W\nυ : Representati... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RepresentationTheory.Coinvariants | {
"line": 161,
"column": 4
} | {
"line": 161,
"column": 22
} | {
"line": 161,
"column": 23
} | [
{
"pp": "k : Type u_6\nG : Type u_7\nV : Type u_8\ninst✝⁴ : CommRing k\ninst✝³ : Group G\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\nρ : Representation k G V\nS : Subgroup G\ninst✝ : S.Normal\ng : G\nx✝¹ : V\nx✝ : x✝¹ ∈ Set.range fun gv ↦ ((MonoidHom.comp ρ S.subtype) gv.1) gv.2 - gv.2\ns : ↥S\nx : V\nhs : (... | [
"k : Type u_6\nG : Type u_7\nV : Type u_8\ninst✝⁴ : CommRing k\ninst✝³ : Group G\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\nρ : Representation k G V\nS : Subgroup G\ninst✝ : S.Normal\ng : G\nx✝¹ : V\nx✝ : x✝¹ ∈ Set.range fun gv ↦ ((MonoidHom.comp ρ S.subtype) gv.1) gv.2 - gv.2\ns : ↥S\nx : V\nhs : (fun gv ↦ ((M... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RepresentationTheory.Continuous.TopRep | {
"line": 201,
"column": 4
} | {
"line": 201,
"column": 37
} | {
"line": 201,
"column": 38
} | [
{
"pp": "k : Type u\nG : Type v\nX✝ Y✝ : Type w\ninst✝¹² : TopologicalSpace k\ninst✝¹¹ : Ring k\ninst✝¹⁰ : Monoid G\ninst✝⁹ : AddCommGroup X✝\ninst✝⁸ : Module k X✝\ninst✝⁷ : TopologicalSpace X✝\ninst✝⁶ : IsTopologicalAddGroup X✝\ninst✝⁵ : ContinuousSMul k X✝\ninst✝⁴ : AddCommGroup Y✝\ninst✝³ : Module k Y✝\ninst... | [
"k : Type u\nG : Type v\nX✝ Y✝ : Type w\ninst✝¹² : TopologicalSpace k\ninst✝¹¹ : Ring k\ninst✝¹⁰ : Monoid G\ninst✝⁹ : AddCommGroup X✝\ninst✝⁸ : Module k X✝\ninst✝⁷ : TopologicalSpace X✝\ninst✝⁶ : IsTopologicalAddGroup X✝\ninst✝⁵ : ContinuousSMul k X✝\ninst✝⁴ : AddCommGroup Y✝\ninst✝³ : Module k Y✝\ninst✝² : Topolog... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.StrongLaw | {
"line": 799,
"column": 4
} | {
"line": 799,
"column": 20
} | {
"line": 799,
"column": 21
} | [
{
"pp": "Ω : Type u_1\nmΩ : MeasurableSpace Ω\nμ : Measure Ω\nE : Type u_2\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace ℝ E\ninst✝² : CompleteSpace E\ninst✝¹ : MeasurableSpace E\ninst✝ : BorelSpace E\nX : ℕ → Ω → E\nhint : Integrable (X 0) μ\nhindep : Pairwise ((fun x1 x2 ↦ x1 ⟂ᵢ[μ] x2) on X)\nhident : ... | [
"Ω : Type u_1\nmΩ : MeasurableSpace Ω\nμ : Measure Ω\nE : Type u_2\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace ℝ E\ninst✝² : CompleteSpace E\ninst✝¹ : MeasurableSpace E\ninst✝ : BorelSpace E\nX : ℕ → Ω → E\nhint : Integrable (X 0) μ\nhindep : Pairwise ((fun x1 x2 ↦ x1 ⟂ᵢ[μ] x2) on X)\nhident : ∀ (i : ℕ), I... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RepresentationTheory.FinGroupCharZero | {
"line": 114,
"column": 6
} | {
"line": 114,
"column": 22
} | {
"line": 114,
"column": 23
} | [
{
"pp": "case refine_2\nk : Type u\ninst✝⁴ : Field k\nG : Type u\ninst✝³ : Finite G\ninst✝² : Group G\ninst✝¹ : IsAlgClosed k\ninst✝ : NeZero ↑(Nat.card G)\nV : FDRep k G\nh : Module.finrank k (V ⟶ V) = 1\nW : FDRep k G\nf : W ⟶ V\nx✝ : Mono f\nι : Abelian.image f ⟶ V := Abelian.image.ι f\nhf : Abelian.factorTh... | [
"case refine_2\nk : Type u\ninst✝⁴ : Field k\nG : Type u\ninst✝³ : Finite G\ninst✝² : Group G\ninst✝¹ : IsAlgClosed k\ninst✝ : NeZero ↑(Nat.card G)\nV : FDRep k G\nh : Module.finrank k (V ⟶ V) = 1\nW : FDRep k G\nf : W ⟶ V\nx✝ : Mono f\nι : Abelian.image f ⟶ V := Abelian.image.ι f\nhf : Abelian.factorThruImage f ≫ ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RepresentationTheory.Continuous.Basic | {
"line": 317,
"column": 35
} | {
"line": 317,
"column": 76
} | {
"line": 317,
"column": 77
} | [
{
"pp": "case mk''.mk''\nR : Type u_1\nG : Type u_2\nV : Type u_3\nW : Type u_4\ninst✝⁹ : Monoid G\ninst✝⁸ : Ring R\ninst✝⁷ : AddCommGroup V\ninst✝⁶ : TopologicalSpace V\ninst✝⁵ : IsTopologicalAddGroup V\ninst✝⁴ : Module R V\ninst✝³ : AddCommGroup W\ninst✝² : TopologicalSpace W\ninst✝¹ : IsTopologicalAddGroup W... | [
"case mk''.mk''\nR : Type u_1\nG : Type u_2\nV : Type u_3\nW : Type u_4\ninst✝⁹ : Monoid G\ninst✝⁸ : Ring R\ninst✝⁷ : AddCommGroup V\ninst✝⁶ : TopologicalSpace V\ninst✝⁵ : IsTopologicalAddGroup V\ninst✝⁴ : Module R V\ninst✝³ : AddCommGroup W\ninst✝² : TopologicalSpace W\ninst✝¹ : IsTopologicalAddGroup W\ninst✝ : Mo... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RepresentationTheory.FinGroupCharZero | {
"line": 125,
"column": 10
} | {
"line": 125,
"column": 68
} | {
"line": 125,
"column": 69
} | [
{
"pp": "case mp\nk : Type u\ninst✝⁴ : Field k\nG : Type u\ninst✝³ : Group G\ninst✝² : IsAlgClosed k\ninst✝¹ : CharZero k\ninst✝ : Fintype G\nV : FDRep k G\nthis : Invertible ↑(Nat.card G)\nh : Simple V\n⊢ ↑(Nat.card G) = ∑ g, V.character g * V.character g⁻¹",
"ppTerm": "?mp",
"assigned": true,
"use... | [
"case mp\nk : Type u\ninst✝⁴ : Field k\nG : Type u\ninst✝³ : Group G\ninst✝² : IsAlgClosed k\ninst✝¹ : CharZero k\ninst✝ : Fintype G\nV : FDRep k G\nthis : Invertible ↑(Nat.card G)\nh : Simple V\n⊢ ↑(Fintype.card G) = ∑ g, V.character g * V.character g⁻¹"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RepresentationTheory.Continuous.Basic | {
"line": 343,
"column": 2
} | {
"line": 343,
"column": 13
} | {
"line": 343,
"column": 14
} | [
{
"pp": "case mk''.mk''\nR : Type u_1\nG : Type u_2\nV : Type u_3\nW : Type u_4\ninst✝⁹ : Monoid G\ninst✝⁸ : Ring R\ninst✝⁷ : AddCommGroup V\ninst✝⁶ : TopologicalSpace V\ninst✝⁵ : IsTopologicalAddGroup V\ninst✝⁴ : Module R V\ninst✝³ : AddCommGroup W\ninst✝² : TopologicalSpace W\ninst✝¹ : IsTopologicalAddGroup W... | [
"case mk''.mk''\nR : Type u_1\nG : Type u_2\nV : Type u_3\nW : Type u_4\ninst✝⁹ : Monoid G\ninst✝⁸ : Ring R\ninst✝⁷ : AddCommGroup V\ninst✝⁶ : TopologicalSpace V\ninst✝⁵ : IsTopologicalAddGroup V\ninst✝⁴ : Module R V\ninst✝³ : AddCommGroup W\ninst✝² : TopologicalSpace W\ninst✝¹ : IsTopologicalAddGroup W\ninst✝ : Mo... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RepresentationTheory.Rep.Basic | {
"line": 126,
"column": 2
} | {
"line": 126,
"column": 13
} | {
"line": 126,
"column": 14
} | [
{
"pp": "k : Type u\nG : Type v\ninst✝¹ : Semiring k\ninst✝ : Monoid G\nA B : Rep k G\nf : A ⟶ B\ng : G\na : ↑A\n⊢ (Hom.hom f) ((A.ρ g) a) = (B.ρ g) ((Hom.hom f) a)",
"ppTerm": "?m.49",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"k : Type u\nG : Type v\ninst✝¹ : Semiring k\ninst✝ : Monoid G\nA B : Rep k G\nf : A ⟶ B\ng : G\na : ↑A\n⊢ (Hom.hom f) ((A.ρ g) a) = (B.ρ g) ((Hom.hom f) a)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RepresentationTheory.Rep.Basic | {
"line": 500,
"column": 40
} | {
"line": 500,
"column": 51
} | {
"line": 500,
"column": 52
} | [
{
"pp": "k : Type u\nG : Type v\ninst✝¹ : Ring k\ninst✝ : Monoid G\nA B C : Rep k G\nX✝ Y✝ : Action (ModuleCat k) G\nf : X✝ ⟶ Y✝\ng : G\n⊢ ModuleCat.Hom.hom f.hom ∘ₗ (X✝.V.endRingEquiv.toMonoidHom.comp X✝.ρ) g =\n (Y✝.V.endRingEquiv.toMonoidHom.comp Y✝.ρ) g ∘ₗ ModuleCat.Hom.hom f.hom",
"ppTerm": "?m.88",... | [
"k : Type u\nG : Type v\ninst✝¹ : Ring k\ninst✝ : Monoid G\nA B C : Rep k G\nX✝ Y✝ : Action (ModuleCat k) G\nf : X✝ ⟶ Y✝\ng : G\n⊢ ModuleCat.Hom.hom f.hom ∘ₗ ModuleCat.Hom.hom (X✝.ρ g) = ModuleCat.Hom.hom (Y✝.ρ g) ∘ₗ ModuleCat.Hom.hom f.hom"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RepresentationTheory.Rep.Basic | {
"line": 787,
"column": 15
} | {
"line": 787,
"column": 26
} | {
"line": 787,
"column": 27
} | [
{
"pp": "k : Type u\nG✝ : Type v\ninst✝² : CommRing k\ninst✝¹ : Monoid G✝\nG : Type v\ninst✝ : Group G\nA✝ B✝ C✝ : Rep k G\nA B C : Rep k G\nf : A ⊗ B ⟶ C\ng : G\nx : ↑B\ny : ↑A\n⊢ ?m.292",
"ppTerm": "?m.293",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"k : Type u\nG✝ : Type v\ninst✝² : CommRing k\ninst✝¹ : Monoid G✝\nG : Type v\ninst✝ : Group G\nA✝ B✝ C✝ : Rep k G\nA B C : Rep k G\nf : A ⊗ B ⟶ C\ng : G\nx : ↑B\ny : ↑A\n⊢ ?m.292"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RepresentationTheory.Rep.Basic | {
"line": 791,
"column": 4
} | {
"line": 791,
"column": 15
} | {
"line": 791,
"column": 16
} | [
{
"pp": "k : Type u\nG✝ : Type v\ninst✝² : CommRing k\ninst✝¹ : Monoid G✝\nG : Type v\ninst✝ : Group G\nA✝ B✝ C✝ : Rep k G\nA B C : Rep k G\nf : B ⟶ A.ihom.obj C\ng : G\nx : ↑A\ny : ↑B\n⊢ ((TensorProduct.uncurry (RingHom.id k) ↑A ↑B ↑C) (Hom.hom f).flip ∘ₗ (A.ρ.tprod B.ρ) g) (x ⊗ₜ[k] y) =\n (C.ρ g ∘ₗ (Tensor... | [
"k : Type u\nG✝ : Type v\ninst✝² : CommRing k\ninst✝¹ : Monoid G✝\nG : Type v\ninst✝ : Group G\nA✝ B✝ C✝ : Rep k G\nA B C : Rep k G\nf : B ⟶ A.ihom.obj C\ng : G\nx : ↑A\ny : ↑B\n⊢ ((Hom.hom f) ((B.ρ g) y)) ((A.ρ g) x) = (C.ρ g) (((Hom.hom f) y) x)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RepresentationTheory.Rep.Basic | {
"line": 983,
"column": 45
} | {
"line": 983,
"column": 56
} | {
"line": 983,
"column": 57
} | [
{
"pp": "k : Type u\nG : Type v\ninst✝¹ : CommRing k\ninst✝ : Monoid G\nX Y Z : Action (Type u) G\n⊢ Hom.hom\n (ofHom (δ (X ⊗ Y) Z) ≫\n ofHom (δ X Y) ▷ (linearization k G).obj Z ≫\n (α_ ((linearization k G).obj X) ((linearization k G).obj Y) ((linearization k G).obj Z)).hom) =\n Hom.hom ... | [
"k : Type u\nG : Type v\ninst✝¹ : CommRing k\ninst✝ : Monoid G\nX Y Z : Action (Type u) G\n⊢ ((↑(Representation.TensorProduct.assoc (Representation.linearize k G X) (Representation.linearize k G Y)\n (Representation.linearize k G Z))).comp\n (Representation.IntertwiningMap.rTensor (Represent... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RepresentationTheory.FiniteIndex | {
"line": 94,
"column": 2
} | {
"line": 94,
"column": 13
} | {
"line": 94,
"column": 14
} | [
{
"pp": "k : Type u\nG : Type v\ninst✝² : CommRing k\ninst✝¹ : Group G\nS : Subgroup G\ninst✝ : DecidableRel ⇑(QuotientGroup.rightRel S)\nA : Rep.{w, u, v} k ↥S\ng₁ g₂ g₃ : G\na : ↑A\n⊢ (A.indToCoindAux (g₁ * g₂⁻¹)) a g₃ = (A.indToCoindAux g₁) a (g₃ * g₂)",
"ppTerm": "?m.35",
"assigned": false,
"use... | [
"k : Type u\nG : Type v\ninst✝² : CommRing k\ninst✝¹ : Group G\nS : Subgroup G\ninst✝ : DecidableRel ⇑(QuotientGroup.rightRel S)\nA : Rep.{w, u, v} k ↥S\ng₁ g₂ g₃ : G\na : ↑A\n⊢ (A.indToCoindAux (g₁ * g₂⁻¹)) a g₃ = (A.indToCoindAux g₁) a (g₃ * g₂)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RepresentationTheory.FiniteIndex | {
"line": 125,
"column": 21
} | {
"line": 125,
"column": 83
} | {
"line": 126,
"column": 6
} | [
{
"pp": "k : Type u\nG : Type v\ninst✝³ : CommRing k\ninst✝² : Group G\nS : Subgroup G\ninst✝¹ : DecidableRel ⇑(QuotientGroup.rightRel S)\nA : Rep.{w, u, v} k ↥S\ninst✝ : S.FiniteIndex\nx✝¹ x✝ : ↑(coind.{u, v, v, w} S.subtype A)\n⊢ ∑ g, g.liftOn (fun g ↦ (IndV.mk S.subtype A.ρ g) (↑(x✝¹ + x✝) g)) ⋯ =\n ∑ g, ... | [
"k : Type u\nG : Type v\ninst✝³ : CommRing k\ninst✝² : Group G\nS : Subgroup G\ninst✝¹ : DecidableRel ⇑(QuotientGroup.rightRel S)\nA : Rep.{w, u, v} k ↥S\ninst✝ : S.FiniteIndex\nx✝¹ x✝ : ↑(coind.{u, v, v, w} S.subtype A)\n⊢ ∑ x,\n x.liftOn\n (fun g ↦\n (Coinvariants.mk (tprod (MonoidHom.comp (R... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RepresentationTheory.FiniteIndex | {
"line": 127,
"column": 22
} | {
"line": 127,
"column": 51
} | {
"line": 127,
"column": 52
} | [
{
"pp": "k : Type u\nG : Type v\ninst✝³ : CommRing k\ninst✝² : Group G\nS : Subgroup G\ninst✝¹ : DecidableRel ⇑(QuotientGroup.rightRel S)\nA : Rep.{w, u, v} k ↥S\ninst✝ : S.FiniteIndex\nx✝¹ : k\nx✝ : ↑(coind.{u, v, v, w} S.subtype A)\n⊢ ∑ g, g.liftOn (fun g ↦ (IndV.mk S.subtype A.ρ g) (↑(x✝¹ • x✝) g)) ⋯ =\n ... | [
"k : Type u\nG : Type v\ninst✝³ : CommRing k\ninst✝² : Group G\nS : Subgroup G\ninst✝¹ : DecidableRel ⇑(QuotientGroup.rightRel S)\nA : Rep.{w, u, v} k ↥S\ninst✝ : S.FiniteIndex\nx✝¹ : k\nx✝ : ↑(coind.{u, v, v, w} S.subtype A)\n⊢ ∑ x,\n x.liftOn\n (fun g ↦\n x✝¹ •\n (Coinvariants.mk (... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RepresentationTheory.FiniteIndex | {
"line": 156,
"column": 4
} | {
"line": 156,
"column": 15
} | {
"line": 156,
"column": 16
} | [
{
"pp": "case h₀.h\nk : Type u\nG : Type v\ninst✝³ : CommRing k\ninst✝² : Group G\nS : Subgroup G\ninst✝¹ : DecidableRel ⇑(QuotientGroup.rightRel S)\nA : Rep.{w, u, v} k ↥S\ninst✝ : S.FiniteIndex\ng : ↑(coind.{u, v, v, w} S.subtype A)\na b : G\na✝ : ⟦b⟧ ∈ Finset.univ\nhb : ⟦b⟧ ≠ ⟦a⟧\n⊢ ↑(A.indToCoind (⟦b⟧.liftO... | [
"case h₀.h\nk : Type u\nG : Type v\ninst✝³ : CommRing k\ninst✝² : Group G\nS : Subgroup G\ninst✝¹ : DecidableRel ⇑(QuotientGroup.rightRel S)\nA : Rep.{w, u, v} k ↥S\ninst✝ : S.FiniteIndex\ng : ↑(coind.{u, v, v, w} S.subtype A)\na b : G\na✝ : ⟦b⟧ ∈ Finset.univ\nhb : ⟦b⟧ ≠ ⟦a⟧\n⊢ (A.indToCoindAux b) (↑g b) a = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RepresentationTheory.Homological.FiniteCyclic | {
"line": 61,
"column": 4
} | {
"line": 61,
"column": 15
} | {
"line": 61,
"column": 16
} | [
{
"pp": "case h\nk : Type u_1\nG : Type u_2\ninst✝⁴ : CommRing k\ninst✝³ : Group G\nV : Type u_4\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\nρ : Representation k G V\ng : G\ninst✝ : Finite G\nhg : ∀ (x : G), x ∈ Subgroup.zpowers g\nα : V\nn : ℕ\na✝ : (fun gv ↦ (ρ gv.1) gv.2 - gv.2) ((fun x ↦ g ^ x) n, α) ∈ ↑... | [
"case h\nk : Type u_1\nG : Type u_2\ninst✝⁴ : CommRing k\ninst✝³ : Group G\nV : Type u_4\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\nρ : Representation k G V\ng : G\ninst✝ : Finite G\nhg : ∀ (x : G), x ∈ Subgroup.zpowers g\nα : V\nn : ℕ\na✝ : (fun gv ↦ (ρ gv.1) gv.2 - gv.2) ((fun x ↦ g ^ x) n, α) ∈ ↑(ρ g - Linea... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RepresentationTheory.Homological.FiniteCyclic | {
"line": 63,
"column": 4
} | {
"line": 63,
"column": 15
} | {
"line": 63,
"column": 16
} | [
{
"pp": "case refine_2\nk : Type u_1\nG : Type u_2\ninst✝⁴ : CommRing k\ninst✝³ : Group G\nV : Type u_4\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\nρ : Representation k G V\ng : G\ninst✝ : Finite G\nhg : ∀ (x : G), x ∈ Subgroup.zpowers g\ny : V\n⊢ (ρ g - LinearMap.id) y ∈ Coinvariants.ker ρ",
"ppTerm": "... | [
"case refine_2\nk : Type u_1\nG : Type u_2\ninst✝⁴ : CommRing k\ninst✝³ : Group G\nV : Type u_4\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\nρ : Representation k G V\ng : G\ninst✝ : Finite G\nhg : ∀ (x : G), x ∈ Subgroup.zpowers g\ny : V\n⊢ (ρ g) y - y ∈ Coinvariants.ker ρ"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RepresentationTheory.FiniteIndex | {
"line": 167,
"column": 4
} | {
"line": 167,
"column": 15
} | {
"line": 167,
"column": 16
} | [
{
"pp": "case hx\nk : Type u\nG : Type v\ninst✝³ : CommRing k\ninst✝² : Group G\nS : Subgroup G\ninst✝¹ : DecidableRel ⇑(QuotientGroup.rightRel S)\nA : Rep.{w, u, v} k ↥S\ninst✝ : S.FiniteIndex\ng : G\na : ↑A\nx : G\nhx : x ∉ MulAction.orbit (↥S) g\n⊢ x ∉\n Function.support\n ↑(A.indToCoind\n (... | [
"case hx\nk : Type u\nG : Type v\ninst✝³ : CommRing k\ninst✝² : Group G\nS : Subgroup G\ninst✝¹ : DecidableRel ⇑(QuotientGroup.rightRel S)\nA : Rep.{w, u, v} k ↥S\ninst✝ : S.FiniteIndex\ng : G\na : ↑A\nx : G\nhx : x ∉ MulAction.orbit (↥S) g\n⊢ (A.indToCoindAux g) a x = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RepresentationTheory.Homological.FiniteCyclic | {
"line": 80,
"column": 4
} | {
"line": 81,
"column": 11
} | {
"line": 81,
"column": 12
} | [
{
"pp": "case refine_1\nk : Type u_1\nG : Type u_2\ninst✝² : CommRing k\ninst✝¹ : Group G\ninst✝ : Finite G\nthis : Fintype G\ng : G\ny : MonoidAlgebra k G\n⊢ (fun gv ↦ ((leftRegular k G) gv.1) gv.2 - gv.2) (g, y) ∈\n ↑((linearCombination k fun x ↦ 1) ∘ₗ ↑(MonoidAlgebra.coeffLinearEquiv k)).ker",
"ppTerm... | [
"case refine_1\nk : Type u_1\nG : Type u_2\ninst✝² : CommRing k\ninst✝¹ : Group G\ninst✝ : Finite G\nthis : Fintype G\ng : G\ny : MonoidAlgebra k G\n⊢ ∑ x, y.coeff (g⁻¹ * x) = ∑ x, y.coeff x"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RepresentationTheory.Homological.ContCohomology.LowDegree | {
"line": 42,
"column": 2
} | {
"line": 42,
"column": 18
} | {
"line": 42,
"column": 19
} | [
{
"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\nσ : ↑(X.homogeneousCochains.X 0).toModuleCat\ng : G\nhσ : ↑σ g⁻¹ = ↑σ 1\n⊢ (X.ρ g) (↑σ 1) = ↑σ 1",
"ppTerm": "?m.237",
"assigned... | [
"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\nσ : ↑(X.homogeneousCochains.X 0).toModuleCat\ng : G\nhσ : ↑σ g⁻¹ = ↑σ 1\n⊢ (X.ρ g) (↑σ 1) = ↑σ 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RepresentationTheory.Homological.GroupCohomology.Basic | {
"line": 127,
"column": 4
} | {
"line": 127,
"column": 19
} | {
"line": 128,
"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": 144,
"column": 2
} | {
"line": 144,
"column": 35
} | {
"line": 144,
"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.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.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.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.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.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": 177,
"column": 4
} | {
"line": 177,
"column": 72
} | {
"line": 178,
"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.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": 922,
"column": 2
} | {
"line": 922,
"column": 13
} | {
"line": 922,
"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": 985,
"column": 45
} | {
"line": 985,
"column": 62
} | {
"line": 985,
"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": 1060,
"column": 2
} | {
"line": 1061,
"column": 5
} | {
"line": 1063,
"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": 1060,
"column": 2
} | {
"line": 1061,
"column": 5
} | {
"line": 1063,
"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": 1066,
"column": 45
} | {
"line": 1066,
"column": 62
} | {
"line": 1066,
"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": 1066,
"column": 45
} | {
"line": 1066,
"column": 89
} | {
"line": 1068,
"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": 1066,
"column": 45
} | {
"line": 1066,
"column": 89
} | {
"line": 1068,
"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": 1066,
"column": 45
} | {
"line": 1066,
"column": 89
} | {
"line": 1068,
"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.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.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.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": 122,
"column": 2
} | {
"line": 122,
"column": 17
} | {
"line": 122,
"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.RepresentationTheory.Tannaka | {
"line": 188,
"column": 2
} | {
"line": 188,
"column": 13
} | {
"line": 188,
"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.MvPowerSeries.Equiv | {
"line": 86,
"column": 6
} | {
"line": 86,
"column": 55
} | {
"line": 87,
"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.MvPowerSeries.Equiv | {
"line": 106,
"column": 4
} | {
"line": 107,
"column": 42
} | {
"line": 108,
"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": 106,
"column": 4
} | {
"line": 107,
"column": 42
} | {
"line": 108,
"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": 106,
"column": 4
} | {
"line": 107,
"column": 42
} | {
"line": 108,
"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.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.Equiv | {
"line": 104,
"column": 2
} | {
"line": 108,
"column": 97
} | {
"line": 110,
"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": 104,
"column": 2
} | {
"line": 108,
"column": 97
} | {
"line": 110,
"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.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.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.MvPowerSeries.Equiv | {
"line": 220,
"column": 4
} | {
"line": 220,
"column": 15
} | {
"line": 220,
"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.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.RepresentationTheory.Homological.GroupHomology.LowDegree | {
"line": 878,
"column": 43
} | {
"line": 878,
"column": 54
} | {
"line": 878,
"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.RepresentationTheory.Homological.GroupHomology.LowDegree | {
"line": 940,
"column": 43
} | {
"line": 940,
"column": 60
} | {
"line": 940,
"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": 978,
"column": 6
} | {
"line": 978,
"column": 29
} | {
"line": 978,
"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": 1080,
"column": 2
} | {
"line": 1081,
"column": 5
} | {
"line": 1083,
"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": 1080,
"column": 2
} | {
"line": 1081,
"column": 5
} | {
"line": 1083,
"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": 1086,
"column": 44
} | {
"line": 1086,
"column": 61
} | {
"line": 1086,
"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": 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.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.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.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.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.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.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.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 |
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