ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- Action.functorCategoryMonoidalAdjunction.counit_app_app
@[simp]
theorem functorCategoryMonoidalAdjunction.counit_app_app (A : SingleObj G ⥤ V) :
    ((functorCategoryMonoidalAdjunction _ _).counit.app A).app PUnit.unit = 𝟙 _ 
Arguments: (V : Type (u + 1)) [CategoryTheory.LargeCategory V] (G : MonCat) [CategoryTheory.MonoidalCategory V] (A : CategoryTheory.SingleObj ↑G ⥤ V)
Goal: ((Action.functorCategoryMonoidalAdjunction V G).counit.app A).app PUnit.unit =
  𝟙
    ((((Action.functorCategoryMonoidalEquivalenceInverse V G).toFunctor ⋙
              (Action.functorCategoryMonoidalEquivalence V G).toFunctor).obj
          A).obj
      PUnit.unit)
-- Action.functorCategoryMonoidalEquivalence.functor_map
@[simp]
theorem functorCategoryMonoidalEquivalence.functor_map {A B : Action V G} (f : A ⟶ B) :
    (functorCategoryMonoidalEquivalence _ _).map f = FunctorCategoryEquivalence.functor.map f 
Arguments: (V : Type (u + 1)) [CategoryTheory.LargeCategory V] (G : MonCat) [CategoryTheory.MonoidalCategory V] {A B : Action V G} (f : A ⟶ B)
Goal: (Action.functorCategoryMonoidalEquivalence V G).map f = Action.FunctorCategoryEquivalence.functor.map f
-- Action.functorCategoryMonoidalEquivalence.inverse_map
@[simp]
theorem functorCategoryMonoidalEquivalence.inverse_map {A B : SingleObj G ⥤ V} (f : A ⟶ B) :
    (functorCategoryMonoidalEquivalenceInverse _ _).map f =
      FunctorCategoryEquivalence.inverse.map f 
Arguments: (V : Type (u + 1)) [CategoryTheory.LargeCategory V] (G : MonCat) [CategoryTheory.MonoidalCategory V] {A B : CategoryTheory.SingleObj ↑G ⥤ V} (f : A ⟶ B)
Goal: (Action.functorCategoryMonoidalEquivalenceInverse V G).map f = Action.FunctorCategoryEquivalence.inverse.map f
-- Action.rightDual_v
@[simp]
theorem rightDual_v [RightRigidCategory V] : Xᘁ.V = X.Vᘁ 
Arguments: {V : Type (u + 1)} [CategoryTheory.LargeCategory V] [CategoryTheory.MonoidalCategory V] {H : Grp} (X : Action V ((CategoryTheory.forget₂ Grp MonCat).obj H)) [CategoryTheory.RightRigidCategory V]
Goal: Xᘁ.V = X.Vᘁ
-- Action.leftDual_v
@[simp]
theorem leftDual_v [LeftRigidCategory V] : (ᘁX).V = ᘁX.V 
Arguments: {V : Type (u + 1)} [CategoryTheory.LargeCategory V] [CategoryTheory.MonoidalCategory V] {H : Grp} (X : Action V ((CategoryTheory.forget₂ Grp MonCat).obj H)) [CategoryTheory.LeftRigidCategory V]
Goal: (ᘁX).V = ᘁX.V
-- Action.rightDual_ρ
@[simp, nolint simpNF]
theorem rightDual_ρ [RightRigidCategory V] (h : H) : Xᘁ.ρ h = (X.ρ (h⁻¹ : H))ᘁ 
Arguments: {V : Type (u + 1)} [CategoryTheory.LargeCategory V] [CategoryTheory.MonoidalCategory V] {H : Grp} (X : Action V ((CategoryTheory.forget₂ Grp MonCat).obj H)) [CategoryTheory.RightRigidCategory V] (h : ↑H)
Goal: Xᘁ.ρ h = X.ρ h⁻¹ᘁ
-- Action.leftDual_ρ
@[simp, nolint simpNF]
theorem leftDual_ρ [LeftRigidCategory V] (h : H) : (ᘁX).ρ h = ᘁX.ρ (h⁻¹ : H) 
Arguments: {V : Type (u + 1)} [CategoryTheory.LargeCategory V] [CategoryTheory.MonoidalCategory V] {H : Grp} (X : Action V ((CategoryTheory.forget₂ Grp MonCat).obj H)) [CategoryTheory.LeftRigidCategory V] (h : ↑H)
Goal: (ᘁX).ρ h = ᘁX.ρ h⁻¹
-- Action.leftRegularTensorIso_hom_hom
@[simps]
noncomputable def leftRegularTensorIso (G : Type u) [Group G] (X : Action (Type u) (MonCat.of G)) :
    leftRegular G ⊗ X ≅ leftRegular G ⊗ Action.mk X.V 1 
Arguments: (G : Type u) [Group G] (X : Action (Type u) (MonCat.of G)) (g : (CategoryTheory.MonoidalCategory.tensorObj (Action.leftRegular G) X).V)
Goal: (Action.leftRegularTensorIso G X).hom.hom g = (g.1, X.ρ g.1⁻¹ g.2)
-- Action.leftRegularTensorIso_inv_hom
@[simps]
noncomputable def leftRegularTensorIso (G : Type u) [Group G] (X : Action (Type u) (MonCat.of G)) :
    leftRegular G ⊗ X ≅ leftRegular G ⊗ Action.mk X.V 1 
Arguments: (G : Type u) [Group G] (X : Action (Type u) (MonCat.of G)) (g : (CategoryTheory.MonoidalCategory.tensorObj (Action.leftRegular G) { V := X.V, ρ := 1 }).V)
Goal: (Action.leftRegularTensorIso G X).inv.hom g = (g.1, X.ρ g.1 g.2)
-- Action.diagonalSucc_inv_hom
@[simps!]
noncomputable def diagonalSucc (G : Type u) [Monoid G] (n : ℕ) :
    diagonal G (n + 1) ≅ leftRegular G ⊗ diagonal G n 
Arguments: (G : Type u) [Monoid G] (n : ℕ)
Goal: ∀ (a : (CategoryTheory.MonoidalCategory.tensorObj (Action.leftRegular G) (Action.diagonal G n)).V),
  (Action.diagonalSucc G n).inv.hom a = (Equiv.piFinSuccAbove (fun a => G) 0).symm a
-- Action.diagonalSucc_hom_hom
@[simps!]
noncomputable def diagonalSucc (G : Type u) [Monoid G] (n : ℕ) :
    diagonal G (n + 1) ≅ leftRegular G ⊗ diagonal G n 
Arguments: (G : Type u) [Monoid G] (n : ℕ)
Goal: ∀ (a : (Action.diagonal G (n + 1)).V), (Action.diagonalSucc G n).hom.hom a = (Equiv.piFinSuccAbove (fun a => G) 0) a
-- CategoryTheory.MonoidalFunctor.mapActionLax_ε_hom
@[simps!]
def mapActionLax (F : LaxMonoidalFunctor V W) (G : MonCat.{u}) :
    LaxMonoidalFunctor (Action V G) (Action W G) 
Arguments: {V : Type (u + 1)} [CategoryTheory.LargeCategory V] {W : Type (u + 1)} [CategoryTheory.LargeCategory W] [CategoryTheory.MonoidalCategory V] [CategoryTheory.MonoidalCategory W] (F : CategoryTheory.LaxMonoidalFunctor V W) (G : MonCat)
Goal: (CategoryTheory.MonoidalFunctor.mapActionLax F G).ε.hom = F.ε
-- CategoryTheory.MonoidalFunctor.mapActionLax_toFunctor_obj_V
@[simps!]
def mapActionLax (F : LaxMonoidalFunctor V W) (G : MonCat.{u}) :
    LaxMonoidalFunctor (Action V G) (Action W G) 
Arguments: {V : Type (u + 1)} [CategoryTheory.LargeCategory V] {W : Type (u + 1)} [CategoryTheory.LargeCategory W] [CategoryTheory.MonoidalCategory V] [CategoryTheory.MonoidalCategory W] (F : CategoryTheory.LaxMonoidalFunctor V W) (G : MonCat)
Goal: ∀ (a : Action V G), ((CategoryTheory.MonoidalFunctor.mapActionLax F G).obj a).V = F.obj a.V
-- CategoryTheory.MonoidalFunctor.mapActionLax_toFunctor_obj_ρ_apply
@[simps!]
def mapActionLax (F : LaxMonoidalFunctor V W) (G : MonCat.{u}) :
    LaxMonoidalFunctor (Action V G) (Action W G) 
Arguments: {V : Type (u + 1)} [CategoryTheory.LargeCategory V] {W : Type (u + 1)} [CategoryTheory.LargeCategory W] [CategoryTheory.MonoidalCategory V] [CategoryTheory.MonoidalCategory W] (F : CategoryTheory.LaxMonoidalFunctor V W) (G : MonCat)
Goal: ∀ (a : Action V G) (g : ↑G), ((CategoryTheory.MonoidalFunctor.mapActionLax F G).obj a).ρ g = F.map (a.ρ g)
-- CategoryTheory.MonoidalFunctor.mapActionLax_μ_hom
@[simps!]
def mapActionLax (F : LaxMonoidalFunctor V W) (G : MonCat.{u}) :
    LaxMonoidalFunctor (Action V G) (Action W G) 
Arguments: {V : Type (u + 1)} [CategoryTheory.LargeCategory V] {W : Type (u + 1)} [CategoryTheory.LargeCategory W] [CategoryTheory.MonoidalCategory V] [CategoryTheory.MonoidalCategory W] (F : CategoryTheory.LaxMonoidalFunctor V W) (G : MonCat) (X Y : Action V G)
Goal: ((CategoryTheory.MonoidalFunctor.mapActionLax F G).μ X Y).hom = F.μ X.V Y.V
-- CategoryTheory.MonoidalFunctor.mapActionLax_toFunctor_map_hom
@[simps!]
def mapActionLax (F : LaxMonoidalFunctor V W) (G : MonCat.{u}) :
    LaxMonoidalFunctor (Action V G) (Action W G) 
Arguments: {V : Type (u + 1)} [CategoryTheory.LargeCategory V] {W : Type (u + 1)} [CategoryTheory.LargeCategory W] [CategoryTheory.MonoidalCategory V] [CategoryTheory.MonoidalCategory W] (F : CategoryTheory.LaxMonoidalFunctor V W) (G : MonCat)
Goal: ∀ {X Y : Action V G} (a : X ⟶ Y), ((CategoryTheory.MonoidalFunctor.mapActionLax F G).map a).hom = F.map a.hom
-- CategoryTheory.MonoidalFunctor.mapAction_toLaxMonoidalFunctor_toFunctor_obj_ρ_apply
@[simps!]
def mapAction : MonoidalFunctor (Action V G) (Action W G) 
Arguments: {V : Type (u + 1)} [CategoryTheory.LargeCategory V] {W : Type (u + 1)} [CategoryTheory.LargeCategory W] [CategoryTheory.MonoidalCategory V] [CategoryTheory.MonoidalCategory W] (F : CategoryTheory.MonoidalFunctor V W) (G : MonCat)
Goal: ∀ (a : Action V G) (g : ↑G), ((F.mapAction G).obj a).ρ g = F.map (a.ρ g)
-- CategoryTheory.MonoidalFunctor.mapAction_toLaxMonoidalFunctor_μ_hom
@[simps!]
def mapAction : MonoidalFunctor (Action V G) (Action W G) 
Arguments: {V : Type (u + 1)} [CategoryTheory.LargeCategory V] {W : Type (u + 1)} [CategoryTheory.LargeCategory W] [CategoryTheory.MonoidalCategory V] [CategoryTheory.MonoidalCategory W] (F : CategoryTheory.MonoidalFunctor V W) (G : MonCat) (X Y : Action V G)
Goal: ((F.mapAction G).μ X Y).hom = F.μ X.V Y.V
-- CategoryTheory.MonoidalFunctor.mapAction_toLaxMonoidalFunctor_ε_hom
@[simps!]
def mapAction : MonoidalFunctor (Action V G) (Action W G) 
Arguments: {V : Type (u + 1)} [CategoryTheory.LargeCategory V] {W : Type (u + 1)} [CategoryTheory.LargeCategory W] [CategoryTheory.MonoidalCategory V] [CategoryTheory.MonoidalCategory W] (F : CategoryTheory.MonoidalFunctor V W) (G : MonCat)
Goal: (F.mapAction G).ε.hom = F.ε
-- CategoryTheory.MonoidalFunctor.mapAction_toLaxMonoidalFunctor_toFunctor_map_hom
@[simps!]
def mapAction : MonoidalFunctor (Action V G) (Action W G) 
Arguments: {V : Type (u + 1)} [CategoryTheory.LargeCategory V] {W : Type (u + 1)} [CategoryTheory.LargeCategory W] [CategoryTheory.MonoidalCategory V] [CategoryTheory.MonoidalCategory W] (F : CategoryTheory.MonoidalFunctor V W) (G : MonCat)
Goal: ∀ {X Y : Action V G} (a : X ⟶ Y), ((F.mapAction G).map a).hom = F.map a.hom
-- CategoryTheory.MonoidalFunctor.mapAction_toLaxMonoidalFunctor_toFunctor_obj_V
@[simps!]
def mapAction : MonoidalFunctor (Action V G) (Action W G) 
Arguments: {V : Type (u + 1)} [CategoryTheory.LargeCategory V] {W : Type (u + 1)} [CategoryTheory.LargeCategory W] [CategoryTheory.MonoidalCategory V] [CategoryTheory.MonoidalCategory W] (F : CategoryTheory.MonoidalFunctor V W) (G : MonCat)
Goal: ∀ (a : Action V G), ((F.mapAction G).obj a).V = F.obj a.V
-- CategoryTheory.MonoidalFunctor.mapAction_ε_inv_hom
@[simp]
theorem mapAction_ε_inv_hom : (inv (F.mapAction G).ε).hom = inv F.ε 
Arguments: {V : Type (u + 1)} [CategoryTheory.LargeCategory V] {W : Type (u + 1)} [CategoryTheory.LargeCategory W] [CategoryTheory.MonoidalCategory V] [CategoryTheory.MonoidalCategory W] (F : CategoryTheory.MonoidalFunctor V W) (G : MonCat)
Goal: (CategoryTheory.inv (F.mapAction G).ε).hom = CategoryTheory.inv F.ε
-- CategoryTheory.MonoidalFunctor.mapAction_μ_inv_hom
@[simp]
theorem mapAction_μ_inv_hom (X Y : Action V G) :
    (inv ((F.mapAction G).μ X Y)).hom = inv (F.μ X.V Y.V) 
Arguments: {V : Type (u + 1)} [CategoryTheory.LargeCategory V] {W : Type (u + 1)} [CategoryTheory.LargeCategory W] [CategoryTheory.MonoidalCategory V] [CategoryTheory.MonoidalCategory W] (F : CategoryTheory.MonoidalFunctor V W) (G : MonCat) (X Y : Action V G)
Goal: (CategoryTheory.inv ((F.mapAction G).μ X Y)).hom = CategoryTheory.inv (F.μ X.V Y.V)
-- MeasureTheory.Integrable.fin_nat_prod
theorem Integrable.fin_nat_prod {n : ℕ} {E : Fin n → Type*}
    [∀ i, MeasureSpace (E i)] [∀ i, SigmaFinite (volume : Measure (E i))]
    {f : (i : Fin n) → E i → 𝕜} (hf : ∀ i, Integrable (f i)) :
    Integrable (fun (x : (i : Fin n) → E i) ↦ ∏ i, f i (x i)) 
Arguments: {𝕜 : Type u_1} [RCLike 𝕜] {n : ℕ} {E : Fin n → Type u_2} [(i : Fin n) → MeasureTheory.MeasureSpace (E i)] [∀ (i : Fin n), MeasureTheory.SigmaFinite MeasureTheory.volume] {f : (i : Fin n) → E i → 𝕜} (hf : ∀ (i : Fin n), MeasureTheory.Integrable (f i) MeasureTheory.volume)
Goal: MeasureTheory.Integrable (fun x => ∏ i : Fin n, f i (x i)) MeasureTheory.volume
-- MeasureTheory.Integrable.fintype_prod_dep
theorem Integrable.fintype_prod_dep {ι : Type*} [Fintype ι] {E : ι → Type*}
    {f : (i : ι) → E i → 𝕜} [∀ i, MeasureSpace (E i)] [∀ i, SigmaFinite (volume : Measure (E i))]
    (hf : ∀ i, Integrable (f i)) :
    Integrable (fun (x : (i : ι) → E i) ↦ ∏ i, f i (x i)) 
Arguments: {𝕜 : Type u_1} [RCLike 𝕜] {ι : Type u_2} [Fintype ι] {E : ι → Type u_3} {f : (i : ι) → E i → 𝕜} [(i : ι) → MeasureTheory.MeasureSpace (E i)] [∀ (i : ι), MeasureTheory.SigmaFinite MeasureTheory.volume] (hf : ∀ (i : ι), MeasureTheory.Integrable (f i) MeasureTheory.volume)
Goal: MeasureTheory.Integrable (fun x => ∏ i : ι, f i (x i)) MeasureTheory.volume
-- MeasureTheory.Integrable.fintype_prod
theorem Integrable.fintype_prod {ι : Type*} [Fintype ι] {E : Type*}
    {f : ι → E → 𝕜} [MeasureSpace E] [SigmaFinite (volume : Measure E)]
    (hf : ∀ i, Integrable (f i)) :
    Integrable (fun (x : ι → E) ↦ ∏ i, f i (x i)) 
Arguments: {𝕜 : Type u_1} [RCLike 𝕜] {ι : Type u_2} [Fintype ι] {E : Type u_3} {f : ι → E → 𝕜} [MeasureTheory.MeasureSpace E] [MeasureTheory.SigmaFinite MeasureTheory.volume] (hf : ∀ (i : ι), MeasureTheory.Integrable (f i) MeasureTheory.volume)
Goal: MeasureTheory.Integrable (fun x => ∏ i : ι, f i (x i)) MeasureTheory.volume
-- MeasureTheory.integral_fin_nat_prod_eq_prod
theorem integral_fin_nat_prod_eq_prod {n : ℕ} {E : Fin n → Type*}
    [∀ i, MeasureSpace (E i)] [∀ i, SigmaFinite (volume : Measure (E i))]
    (f : (i : Fin n) → E i → 𝕜) :
    ∫ x : (i : Fin n) → E i, ∏ i, f i (x i) = ∏ i, ∫ x, f i x 
Arguments: {𝕜 : Type u_1} [RCLike 𝕜] {n : ℕ} {E : Fin n → Type u_2} [(i : Fin n) → MeasureTheory.MeasureSpace (E i)] [∀ (i : Fin n), MeasureTheory.SigmaFinite MeasureTheory.volume] (f : (i : Fin n) → E i → 𝕜)
Goal: ∫ (x : (i : Fin n) → E i), ∏ i : Fin n, f i (x i) = ∏ i : Fin n, ∫ (x : E i), f i x
-- MeasureTheory.integral_fintype_prod_eq_prod
theorem integral_fintype_prod_eq_prod (ι : Type*) [Fintype ι] {E : ι → Type*}
    (f : (i : ι) → E i → 𝕜) [∀ i, MeasureSpace (E i)] [∀ i, SigmaFinite (volume : Measure (E i))] :
    ∫ x : (i : ι) → E i, ∏ i, f i (x i) = ∏ i, ∫ x, f i x 
Arguments: {𝕜 : Type u_1} [RCLike 𝕜] (ι : Type u_2) [Fintype ι] {E : ι → Type u_3} (f : (i : ι) → E i → 𝕜) [(i : ι) → MeasureTheory.MeasureSpace (E i)] [∀ (i : ι), MeasureTheory.SigmaFinite MeasureTheory.volume]
Goal: ∫ (x : (i : ι) → E i), ∏ i : ι, f i (x i) = ∏ i : ι, ∫ (x : E i), f i x
-- MeasureTheory.integral_fintype_prod_eq_pow
theorem integral_fintype_prod_eq_pow {E : Type*} (ι : Type*) [Fintype ι] (f : E → 𝕜)
    [MeasureSpace E] [SigmaFinite (volume : Measure E)] :
    ∫ x : ι → E, ∏ i, f (x i) = (∫ x, f x) ^ (card ι) 
Arguments: {𝕜 : Type u_1} [RCLike 𝕜] {E : Type u_2} (ι : Type u_3) [Fintype ι] (f : E → 𝕜) [MeasureTheory.MeasureSpace E] [MeasureTheory.SigmaFinite MeasureTheory.volume]
Goal: ∫ (x : ι → E), ∏ i : ι, f (x i) = (∫ (x : E), f x) ^ Fintype.card ι
-- groupCohomology.dZero_apply
@[simps]
def dZero : A →ₗ[k] G → A 
Arguments: {k G : Type u} [CommRing k] [Group G] (A : Rep k G) (m : CoeSort.coe A) (g : G)
Goal: (groupCohomology.dZero A) m g = (A.ρ g) m - m
-- groupCohomology.dZero_ker_eq_invariants
theorem dZero_ker_eq_invariants : LinearMap.ker (dZero A) = invariants A.ρ 
Arguments: {k G : Type u} [CommRing k] [Group G] (A : Rep k G)
Goal: LinearMap.ker (groupCohomology.dZero A) = A.ρ.invariants
-- groupCohomology.dZero_eq_zero
@[simp] theorem dZero_eq_zero [A.IsTrivial] : dZero A = 0 
Arguments: {k G : Type u} [CommRing k] [Group G] (A : Rep k G) [A.IsTrivial]
Goal: groupCohomology.dZero A = 0
-- groupCohomology.dOne_apply
@[simps]
def dOne : (G → A) →ₗ[k] G × G → A 
Arguments: {k G : Type u} [CommRing k] [Group G] (A : Rep k G) (f : G → CoeSort.coe A) (g : G × G)
Goal: (groupCohomology.dOne A) f g = (A.ρ g.1) (f g.2) - f (g.1 * g.2) + f g.1
-- groupCohomology.dTwo_apply
@[simps]
def dTwo : (G × G → A) →ₗ[k] G × G × G → A 
Arguments: {k G : Type u} [CommRing k] [Group G] (A : Rep k G) (f : G × G → CoeSort.coe A) (g : G × G × G)
Goal: (groupCohomology.dTwo A) f g =
  (A.ρ g.1) (f (g.2.1, g.2.2)) - f (g.1 * g.2.1, g.2.2) + f (g.1, g.2.1 * g.2.2) - f (g.1, g.2.1)
-- groupCohomology.dZero_comp_eq
theorem dZero_comp_eq : dZero A ∘ₗ (zeroCochainsLequiv A) =
    oneCochainsLequiv A ∘ₗ (inhomogeneousCochains A).d 0 1 
Arguments: {k G : Type u} [CommRing k] [Group G] (A : Rep k G)
Goal: groupCohomology.dZero A ∘ₗ ↑(groupCohomology.zeroCochainsLequiv A) =
  ↑(groupCohomology.oneCochainsLequiv A) ∘ₗ (groupCohomology.inhomogeneousCochains A).d 0 1
-- groupCohomology.dOne_comp_eq
theorem dOne_comp_eq : dOne A ∘ₗ oneCochainsLequiv A =
    twoCochainsLequiv A ∘ₗ (inhomogeneousCochains A).d 1 2 
Arguments: {k G : Type u} [CommRing k] [Group G] (A : Rep k G)
Goal: groupCohomology.dOne A ∘ₗ ↑(groupCohomology.oneCochainsLequiv A) =
  ↑(groupCohomology.twoCochainsLequiv A) ∘ₗ (groupCohomology.inhomogeneousCochains A).d 1 2
-- groupCohomology.dTwo_comp_eq
theorem dTwo_comp_eq :
    dTwo A ∘ₗ twoCochainsLequiv A = threeCochainsLequiv A ∘ₗ (inhomogeneousCochains A).d 2 3 
Arguments: {k G : Type u} [CommRing k] [Group G] (A : Rep k G)
Goal: groupCohomology.dTwo A ∘ₗ ↑(groupCohomology.twoCochainsLequiv A) =
  ↑(groupCohomology.threeCochainsLequiv A) ∘ₗ (groupCohomology.inhomogeneousCochains A).d 2 3
-- groupCohomology.dOne_comp_dZero
theorem dOne_comp_dZero : dOne A ∘ₗ dZero A = 0 
Arguments: {k G : Type u} [CommRing k] [Group G] (A : Rep k G)
Goal: groupCohomology.dOne A ∘ₗ groupCohomology.dZero A = 0
-- groupCohomology.dTwo_comp_dOne
theorem dTwo_comp_dOne : dTwo A ∘ₗ dOne A = 0 
Arguments: {k G : Type u} [CommRing k] [Group G] (A : Rep k G)
Goal: groupCohomology.dTwo A ∘ₗ groupCohomology.dOne A = 0
-- groupCohomology.mem_oneCocycles_def
theorem mem_oneCocycles_def (f : G → A) :
    f ∈ oneCocycles A ↔ ∀ g h : G, A.ρ g (f h) - f (g * h) + f g = 0 
Arguments: {k G : Type u} [CommRing k] [Group G] {A : Rep k G} (f : G → CoeSort.coe A)
Goal: f ∈ groupCohomology.oneCocycles A ↔ ∀ (g h : G), (A.ρ g) (f h) - f (g * h) + f g = 0
-- groupCohomology.mem_oneCocycles_iff
theorem mem_oneCocycles_iff (f : G → A) :
    f ∈ oneCocycles A ↔ ∀ g h : G, f (g * h) = A.ρ g (f h) + f g 
Arguments: {k G : Type u} [CommRing k] [Group G] {A : Rep k G} (f : G → CoeSort.coe A)
Goal: f ∈ groupCohomology.oneCocycles A ↔ ∀ (g h : G), f (g * h) = (A.ρ g) (f h) + f g
-- groupCohomology.oneCocycles_map_one
@[simp] theorem oneCocycles_map_one (f : oneCocycles A) : f.1 1 = 0 
Arguments: {k G : Type u} [CommRing k] [Group G] {A : Rep k G} (f : ↥(groupCohomology.oneCocycles A))
Goal: ↑f 1 = 0
-- groupCohomology.oneCocycles_map_inv
@[simp] theorem oneCocycles_map_inv (f : oneCocycles A) (g : G) :
    A.ρ g (f.1 g⁻¹) = - f.1 g 
Arguments: {k G : Type u} [CommRing k] [Group G] {A : Rep k G} (f : ↥(groupCohomology.oneCocycles A)) (g : G)
Goal: (A.ρ g) (↑f g⁻¹) = -↑f g
-- groupCohomology.oneCocycles_map_mul_of_isTrivial
theorem oneCocycles_map_mul_of_isTrivial [A.IsTrivial] (f : oneCocycles A) (g h : G) :
    f.1 (g * h) = f.1 g + f.1 h 
Arguments: {k G : Type u} [CommRing k] [Group G] {A : Rep k G} [A.IsTrivial] (f : ↥(groupCohomology.oneCocycles A)) (g h : G)
Goal: ↑f (g * h) = ↑f g + ↑f h
-- groupCohomology.mem_oneCocycles_of_addMonoidHom
theorem mem_oneCocycles_of_addMonoidHom [A.IsTrivial] (f : Additive G →+ A) :
    f ∘ Additive.ofMul ∈ oneCocycles A 
Arguments: {k G : Type u} [CommRing k] [Group G] {A : Rep k G} [A.IsTrivial] (f : Additive G →+ CoeSort.coe A)
Goal: ⇑f ∘ ⇑Additive.ofMul ∈ groupCohomology.oneCocycles A
-- groupCohomology.oneCocyclesLequivOfIsTrivial_apply_apply
@[simps] def oneCocyclesLequivOfIsTrivial [hA : A.IsTrivial] :
    oneCocycles A ≃ₗ[k] Additive G →+ A 
Arguments: {k G : Type u} [CommRing k] [Group G] (A : Rep k G) [hA : A.IsTrivial] (f : ↥(groupCohomology.oneCocycles A))
Goal: ∀ (a : Additive G), ((groupCohomology.oneCocyclesLequivOfIsTrivial A) f) a = (↑f ∘ ⇑Additive.toMul) a
-- groupCohomology.oneCocyclesLequivOfIsTrivial_symm_apply_coe
@[simps] def oneCocyclesLequivOfIsTrivial [hA : A.IsTrivial] :
    oneCocycles A ≃ₗ[k] Additive G →+ A 
Arguments: {k G : Type u} [CommRing k] [Group G] (A : Rep k G) [hA : A.IsTrivial] (f : Additive G →+ CoeSort.coe A) (a : Additive G)
Goal: ↑((groupCohomology.oneCocyclesLequivOfIsTrivial A).symm f) a = f a
-- groupCohomology.mem_twoCocycles_def
theorem mem_twoCocycles_def (f : G × G → A) :
    f ∈ twoCocycles A ↔ ∀ g h j : G,
      A.ρ g (f (h, j)) - f (g * h, j) + f (g, h * j) - f (g, h) = 0 
Arguments: {k G : Type u} [CommRing k] [Group G] {A : Rep k G} (f : G × G → CoeSort.coe A)
Goal: f ∈ groupCohomology.twoCocycles A ↔ ∀ (g h j : G), (A.ρ g) (f (h, j)) - f (g * h, j) + f (g, h * j) - f (g, h) = 0
-- groupCohomology.mem_twoCocycles_iff
theorem mem_twoCocycles_iff (f : G × G → A) :
    f ∈ twoCocycles A ↔ ∀ g h j : G,
      f (g * h, j) + f (g, h) =
        A.ρ g (f (h, j)) + f (g, h * j) 
Arguments: {k G : Type u} [CommRing k] [Group G] {A : Rep k G} (f : G × G → CoeSort.coe A)
Goal: f ∈ groupCohomology.twoCocycles A ↔ ∀ (g h j : G), f (g * h, j) + f (g, h) = (A.ρ g) (f (h, j)) + f (g, h * j)
-- groupCohomology.twoCocycles_map_one_fst
theorem twoCocycles_map_one_fst (f : twoCocycles A) (g : G) :
    f.1 (1, g) = f.1 (1, 1) 
Arguments: {k G : Type u} [CommRing k] [Group G] {A : Rep k G} (f : ↥(groupCohomology.twoCocycles A)) (g : G)
Goal: ↑f (1, g) = ↑f (1, 1)
-- groupCohomology.twoCocycles_map_one_snd
theorem twoCocycles_map_one_snd (f : twoCocycles A) (g : G) :
    f.1 (g, 1) = A.ρ g (f.1 (1, 1)) 
Arguments: {k G : Type u} [CommRing k] [Group G] {A : Rep k G} (f : ↥(groupCohomology.twoCocycles A)) (g : G)
Goal: ↑f (g, 1) = (A.ρ g) (↑f (1, 1))
-- groupCohomology.twoCocycles_ρ_map_inv_sub_map_inv
lemma twoCocycles_ρ_map_inv_sub_map_inv (f : twoCocycles A) (g : G) :
    A.ρ g (f.1 (g⁻¹, g)) - f.1 (g, g⁻¹)
      = f.1 (1, 1) - f.1 (g, 1) 
Arguments: {k G : Type u} [CommRing k] [Group G] {A : Rep k G} (f : ↥(groupCohomology.twoCocycles A)) (g : G)
Goal: (A.ρ g) (↑f (g⁻¹, g)) - ↑f (g, g⁻¹) = ↑f (1, 1) - ↑f (g, 1)
-- groupCohomology.oneCoboundaries_of_mem_range_apply
theorem oneCoboundaries_of_mem_range_apply {f : G → A} (h : f ∈ LinearMap.range (dZero A)) :
    (oneCoboundariesOfMemRange h).1.1 = f 
Arguments: {k G : Type u} [CommRing k] [Group G] {A : Rep k G} {f : G → CoeSort.coe A} (h : f ∈ LinearMap.range (groupCohomology.dZero A))
Goal: ↑↑(groupCohomology.oneCoboundariesOfMemRange h) = f
-- groupCohomology.oneCoboundariesOfEq_apply
theorem oneCoboundariesOfEq_apply {f : G → A} {x : A} (hf : ∀ g, A.ρ g x - x = f g) :
    (oneCoboundariesOfEq hf).1.1 = f 
Arguments: {k G : Type u} [CommRing k] [Group G] {A : Rep k G} {f : G → CoeSort.coe A} {x : CoeSort.coe A} (hf : ∀ (g : G), (A.ρ g) x - x = f g)
Goal: ↑↑(groupCohomology.oneCoboundariesOfEq hf) = f
-- groupCohomology.mem_range_of_mem_oneCoboundaries
theorem mem_range_of_mem_oneCoboundaries {f : oneCocycles A} (h : f ∈ oneCoboundaries A) :
    f.1 ∈ LinearMap.range (dZero A) 
Arguments: {k G : Type u} [CommRing k] [Group G] {A : Rep k G} {f : ↥(groupCohomology.oneCocycles A)} (h : f ∈ groupCohomology.oneCoboundaries A)
Goal: ↑f ∈ LinearMap.range (groupCohomology.dZero A)
-- groupCohomology.oneCoboundaries_eq_bot_of_isTrivial
theorem oneCoboundaries_eq_bot_of_isTrivial (A : Rep k G) [A.IsTrivial] :
    oneCoboundaries A = ⊥ 
Arguments: {k G : Type u} [CommRing k] [Group G] (A : Rep k G) [A.IsTrivial]
Goal: groupCohomology.oneCoboundaries A = ⊥
-- groupCohomology.twoCoboundariesOfMemRange_apply
theorem twoCoboundariesOfMemRange_apply {f : G × G → A} (h : f ∈ LinearMap.range (dOne A)) :
    (twoCoboundariesOfMemRange h).1.1 = f 
Arguments: {k G : Type u} [CommRing k] [Group G] {A : Rep k G} {f : G × G → CoeSort.coe A} (h : f ∈ LinearMap.range (groupCohomology.dOne A))
Goal: ↑↑(groupCohomology.twoCoboundariesOfMemRange h) = f
-- groupCohomology.twoCoboundariesOfEq_apply
theorem twoCoboundariesOfEq_apply {f : G × G → A} {x : G → A}
    (hf : ∀ g h, A.ρ g (x h) - x (g * h) + x g = f (g, h)) :
    (twoCoboundariesOfEq hf).1.1 = f 
Arguments: {k G : Type u} [CommRing k] [Group G] {A : Rep k G} {f : G × G → CoeSort.coe A} {x : G → CoeSort.coe A} (hf : ∀ (g h : G), (A.ρ g) (x h) - x (g * h) + x g = f (g, h))
Goal: ↑↑(groupCohomology.twoCoboundariesOfEq hf) = f
-- groupCohomology.mem_range_of_mem_twoCoboundaries
theorem mem_range_of_mem_twoCoboundaries {f : twoCocycles A} (h : f ∈ twoCoboundaries A) :
    (twoCocycles A).subtype f ∈ LinearMap.range (dOne A) 
Arguments: {k G : Type u} [CommRing k] [Group G] {A : Rep k G} {f : ↥(groupCohomology.twoCocycles A)} (h : f ∈ groupCohomology.twoCoboundaries A)
Goal: (groupCohomology.twoCocycles A).subtype f ∈ LinearMap.range (groupCohomology.dOne A)
-- groupCohomology.map_one_of_isOneCocycle
theorem map_one_of_isOneCocycle {f : G → A} (hf : IsOneCocycle f) :
    f 1 = 0 
Arguments: {G : Type u_1} {A : Type u_2} [Monoid G] [AddCommGroup A] [MulAction G A] {f : G → A} (hf : groupCohomology.IsOneCocycle f)
Goal: f 1 = 0