ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- FirstOrder.Language.BoundedFormula.encoding_decode
@[simps]
protected def encoding : Encoding (Σn, L.BoundedFormula α n) 
Arguments: {L : FirstOrder.Language} {α : Type u'} (l : List ((k : ℕ) × L.Term (α ⊕ Fin k) ⊕ (n : ℕ) × L.Relations n ⊕ ℕ))
Goal: FirstOrder.Language.BoundedFormula.encoding.decode l = (FirstOrder.Language.BoundedFormula.listDecode l)[0]?
-- FirstOrder.Language.BoundedFormula.encoding_encode
@[simps]
protected def encoding : Encoding (Σn, L.BoundedFormula α n) 
Arguments: {L : FirstOrder.Language} {α : Type u'} (φ : (n : ℕ) × L.BoundedFormula α n)
Goal: FirstOrder.Language.BoundedFormula.encoding.encode φ = φ.snd.listEncode
-- FirstOrder.Language.BoundedFormula.listEncode_sigma_injective
theorem listEncode_sigma_injective :
    Function.Injective fun φ : Σn, L.BoundedFormula α n => φ.2.listEncode 
Arguments: {L : FirstOrder.Language} {α : Type u'}
Goal: Function.Injective fun φ => φ.snd.listEncode
-- FirstOrder.Language.BoundedFormula.card_le
theorem card_le : #(Σn, L.BoundedFormula α n) ≤
    max ℵ₀ (Cardinal.lift.{max u v} #α + Cardinal.lift.{u'} L.card) 
Arguments: {L : FirstOrder.Language} {α : Type u'}
Goal: Cardinal.mk ((n : ℕ) × L.BoundedFormula α n) ≤
  max Cardinal.aleph0 (Cardinal.lift.{max u v, u'} (Cardinal.mk α) + Cardinal.lift.{u', max u v} L.card)
-- Rep.coe_of
@[simp]
theorem coe_of {V : Type u} [AddCommGroup V] [Module k V] (ρ : G →* V →ₗ[k] V) :
    (of ρ : Type u) = V 
Arguments: {k G : Type u} [CommRing k] [Monoid G] {V : Type u} [AddCommGroup V] [Module k V] (ρ : G →* V →ₗ[k] V)
Goal: CoeSort.coe (Rep.of ρ) = V
-- Rep.of_ρ
@[simp]
theorem of_ρ {V : Type u} [AddCommGroup V] [Module k V] (ρ : G →* V →ₗ[k] V) : (of ρ).ρ = ρ 
Arguments: {k G : Type u} [CommRing k] [Monoid G] {V : Type u} [AddCommGroup V] [Module k V] (ρ : G →* V →ₗ[k] V)
Goal: (Rep.of ρ).ρ = ρ
-- Rep.Action_ρ_eq_ρ
theorem Action_ρ_eq_ρ {A : Rep k G} : Action.ρ A = A.ρ 
Arguments: {k G : Type u} [CommRing k] [Monoid G] {A : Rep k G}
Goal: A.ρ = A.ρ
-- Rep.of_ρ_apply
theorem of_ρ_apply {V : Type u} [AddCommGroup V] [Module k V] (ρ : Representation k G V)
    (g : MonCat.of G) : (Rep.of ρ).ρ g = ρ (g : G) 
Arguments: {k G : Type u} [CommRing k] [Monoid G] {V : Type u} [AddCommGroup V] [Module k V] (ρ : Representation k G V) (g : ↑(MonCat.of G))
Goal: (Rep.of ρ).ρ g = ρ g
-- Rep.ρ_inv_self_apply
@[simp]
theorem ρ_inv_self_apply {G : Type u} [Group G] (A : Rep k G) (g : G) (x : A) :
    A.ρ g⁻¹ (A.ρ g x) = x 
Arguments: {k : Type u} [CommRing k] {G : Type u} [Group G] (A : Rep k G) (g : G) (x : CoeSort.coe A)
Goal: (A.ρ g⁻¹) ((A.ρ g) x) = x
-- Rep.ρ_self_inv_apply
@[simp]
theorem ρ_self_inv_apply {G : Type u} [Group G] {A : Rep k G} (g : G) (x : A) :
    A.ρ g (A.ρ g⁻¹ x) = x 
Arguments: {k : Type u} [CommRing k] {G : Type u} [Group G] {A : Rep k G} (g : G) (x : CoeSort.coe A)
Goal: (A.ρ g) ((A.ρ g⁻¹) x) = x
-- Rep.hom_comm_apply
theorem hom_comm_apply {A B : Rep k G} (f : A ⟶ B) (g : G) (x : A) :
    f.hom (A.ρ g x) = B.ρ g (f.hom x) 
Arguments: {k G : Type u} [CommRing k] [Monoid G] {A B : Rep k G} (f : A ⟶ B) (g : G) (x : CoeSort.coe A)
Goal: f.hom ((A.ρ g) x) = (B.ρ g) (f.hom x)
-- Rep.trivial_def
theorem trivial_def {V : Type u} [AddCommGroup V] [Module k V] (g : G) (v : V) :
    (trivial k G V).ρ g v = v 
Arguments: {k G : Type u} [CommRing k] [Monoid G] {V : Type u} [AddCommGroup V] [Module k V] (g : G) (v : V)
Goal: ((Rep.trivial k G V).ρ g) v = v
-- Rep.MonoidalCategory.braiding_hom_apply
theorem MonoidalCategory.braiding_hom_apply {A B : Rep k G} (x : A) (y : B) :
    Action.Hom.hom (β_ A B).hom (TensorProduct.tmul k x y) = TensorProduct.tmul k y x 
Arguments: {k G : Type u} [CommRing k] [Monoid G] {A B : Rep k G} (x : CoeSort.coe A) (y : CoeSort.coe B)
Goal: (β_ A B).hom.hom (x ⊗ₜ[k] y) = y ⊗ₜ[k] x
-- Rep.MonoidalCategory.braiding_inv_apply
theorem MonoidalCategory.braiding_inv_apply {A B : Rep k G} (x : A) (y : B) :
    Action.Hom.hom (β_ A B).inv (TensorProduct.tmul k y x) = TensorProduct.tmul k x y 
Arguments: {k G : Type u} [CommRing k] [Monoid G] {A B : Rep k G} (x : CoeSort.coe A) (y : CoeSort.coe B)
Goal: (β_ A B).inv.hom (y ⊗ₜ[k] x) = x ⊗ₜ[k] y
-- Rep.linearization_obj_ρ
@[simp]
theorem linearization_obj_ρ (X : Action (Type u) (MonCat.of G)) (g : G) (x : X.V →₀ k) :
    ((linearization k G).obj X).ρ g x = Finsupp.lmapDomain k k (X.ρ g) x 
Arguments: {k G : Type u} [CommRing k] [Monoid G] (X : Action (Type u) (MonCat.of G)) (g : G) (x : X.V →₀ k)
Goal: (((Rep.linearization k G).obj X).ρ g) x = (Finsupp.lmapDomain k k (X.ρ g)) x
-- Rep.linearization_of
theorem linearization_of (X : Action (Type u) (MonCat.of G)) (g : G) (x : X.V) :
    ((linearization k G).obj X).ρ g (Finsupp.single x (1 : k))
      = Finsupp.single (X.ρ g x) (1 : k) 
Arguments: {k G : Type u} [CommRing k] [Monoid G] (X : Action (Type u) (MonCat.of G)) (g : G) (x : X.V)
Goal: (((Rep.linearization k G).obj X).ρ g) (Finsupp.single x 1) = Finsupp.single (X.ρ g x) 1
-- Rep.linearization_single
theorem linearization_single (X : Action (Type u) (MonCat.of G)) (g : G) (x : X.V) (r : k) :
    ((linearization k G).obj X).ρ g (Finsupp.single x r) = Finsupp.single (X.ρ g x) r 
Arguments: {k G : Type u} [CommRing k] [Monoid G] (X : Action (Type u) (MonCat.of G)) (g : G) (x : X.V) (r : k)
Goal: (((Rep.linearization k G).obj X).ρ g) (Finsupp.single x r) = Finsupp.single (X.ρ g x) r
-- Rep.linearization_map_hom
@[simp]
theorem linearization_map_hom : ((linearization k G).map f).hom = Finsupp.lmapDomain k k f.hom 
Arguments: {k G : Type u} [CommRing k] [Monoid G] {X Y : Action (Type u) (MonCat.of G)} (f : X ⟶ Y)
Goal: ((Rep.linearization k G).map f).hom = Finsupp.lmapDomain k k f.hom
-- Rep.linearization_map_hom_single
theorem linearization_map_hom_single (x : X.V) (r : k) :
    ((linearization k G).map f).hom (Finsupp.single x r) = Finsupp.single (f.hom x) r 
Arguments: {k G : Type u} [CommRing k] [Monoid G] {X Y : Action (Type u) (MonCat.of G)} (f : X ⟶ Y) (x : X.V) (r : k)
Goal: ((Rep.linearization k G).map f).hom (Finsupp.single x r) = Finsupp.single (f.hom x) r
-- Rep.linearization_μ_hom
@[simp]
theorem linearization_μ_hom (X Y : Action (Type u) (MonCat.of G)) :
    ((linearization k G).μ X Y).hom = (finsuppTensorFinsupp' k X.V Y.V).toLinearMap 
Arguments: {k G : Type u} [CommRing k] [Monoid G] (X Y : Action (Type u) (MonCat.of G))
Goal: ((Rep.linearization k G).μ X Y).hom = ↑(finsuppTensorFinsupp' k X.V Y.V)
-- Rep.linearization_μ_inv_hom
@[simp]
theorem linearization_μ_inv_hom (X Y : Action (Type u) (MonCat.of G)) :
    (inv ((linearization k G).μ X Y)).hom = (finsuppTensorFinsupp' k X.V Y.V).symm.toLinearMap 
Arguments: {k G : Type u} [CommRing k] [Monoid G] (X Y : Action (Type u) (MonCat.of G))
Goal: (CategoryTheory.inv ((Rep.linearization k G).μ X Y)).hom = ↑(finsuppTensorFinsupp' k X.V Y.V).symm
-- Rep.linearization_ε_hom
@[simp]
theorem linearization_ε_hom : (linearization k G).ε.hom = Finsupp.lsingle PUnit.unit 
Arguments: {k G : Type u} [CommRing k] [Monoid G]
Goal: (Rep.linearization k G).ε.hom = Finsupp.lsingle PUnit.unit
-- Rep.linearization_ε_inv_hom_apply
theorem linearization_ε_inv_hom_apply (r : k) :
    (inv (linearization k G).ε).hom (Finsupp.single PUnit.unit r) = r 
Arguments: {k G : Type u} [CommRing k] [Monoid G] (r : k)
Goal: (CategoryTheory.inv (Rep.linearization k G).ε).hom (Finsupp.single PUnit.unit r) = r
-- Rep.linearizationTrivialIso_hom_hom_apply
@[simps!]
noncomputable def linearizationTrivialIso (X : Type u) :
    (linearization k G).obj (Action.mk X 1) ≅ trivial k G (X →₀ k) 
Arguments: (k G : Type u) [CommRing k] [Monoid G] (X : Type u) (a : ↑((Rep.linearization k G).obj { V := X, ρ := 1 }).V)
Goal: (Rep.linearizationTrivialIso k G X).hom.hom a = a
-- Rep.linearizationTrivialIso_inv_hom_apply
@[simps!]
noncomputable def linearizationTrivialIso (X : Type u) :
    (linearization k G).obj (Action.mk X 1) ≅ trivial k G (X →₀ k) 
Arguments: (k G : Type u) [CommRing k] [Monoid G] (X : Type u) (a : ↑((Rep.linearization k G).obj { V := X, ρ := 1 }).V)
Goal: (Rep.linearizationTrivialIso k G X).inv.hom a = a
-- Rep.ofDistribMulAction_ρ_apply_apply
@[simp] theorem ofDistribMulAction_ρ_apply_apply (g : G) (a : A) :
    (ofDistribMulAction k G A).ρ g a = g • a 
Arguments: (k G A : Type u) [CommRing k] [Monoid G] [AddCommGroup A] [Module k A] [DistribMulAction G A] [SMulCommClass G k A] (g : G) (a : A)
Goal: ((Rep.ofDistribMulAction k G A).ρ g) a = g • a
-- Rep.ofMulDistribMulAction_ρ_apply_apply
@[simp] theorem ofMulDistribMulAction_ρ_apply_apply (g : M) (a : Additive G) :
    (ofMulDistribMulAction M G).ρ g a = Additive.ofMul (g • Additive.toMul a) 
Arguments: (M G : Type) [Monoid M] [CommGroup G] [MulDistribMulAction M G] (g : M) (a : Additive G)
Goal: ((Rep.ofMulDistribMulAction M G).ρ g) a = Additive.ofMul (g • Additive.toMul a)
-- Rep.leftRegularHom_hom
@[simps]
noncomputable def leftRegularHom (A : Rep k G) (x : A) : Rep.ofMulAction k G G ⟶ A 
Arguments: {k G : Type u} [CommRing k] [Monoid G] (A : Rep k G) (x : CoeSort.coe A)
Goal: (A.leftRegularHom x).hom = (Finsupp.lift (↑A.V) k G) fun g => (A.ρ g) x
-- Rep.leftRegularHom_apply
theorem leftRegularHom_apply {A : Rep k G} (x : A) :
    (leftRegularHom A x).hom (Finsupp.single 1 1) = x 
Arguments: {k G : Type u} [CommRing k] [Monoid G] {A : Rep k G} (x : CoeSort.coe A)
Goal: (A.leftRegularHom x).hom (Finsupp.single 1 1) = x
-- Rep.leftRegularHomEquiv_apply
@[simps]
noncomputable def leftRegularHomEquiv (A : Rep k G) : (Rep.ofMulAction k G G ⟶ A) ≃ₗ[k] A 
Arguments: {k G : Type u} [CommRing k] [Monoid G] (A : Rep k G) (f : Rep.ofMulAction k G G ⟶ A)
Goal: A.leftRegularHomEquiv f = f.hom (Finsupp.single 1 1)
-- Rep.leftRegularHomEquiv_symm_apply
@[simps]
noncomputable def leftRegularHomEquiv (A : Rep k G) : (Rep.ofMulAction k G G ⟶ A) ≃ₗ[k] A 
Arguments: {k G : Type u} [CommRing k] [Monoid G] (A : Rep k G) (x : CoeSort.coe A)
Goal: A.leftRegularHomEquiv.symm x = A.leftRegularHom x
-- Rep.leftRegularHomEquiv_symm_single
theorem leftRegularHomEquiv_symm_single {A : Rep k G} (x : A) (g : G) :
    ((leftRegularHomEquiv A).symm x).hom (Finsupp.single g 1) = A.ρ g x 
Arguments: {k G : Type u} [CommRing k] [Monoid G] {A : Rep k G} (x : CoeSort.coe A) (g : G)
Goal: (A.leftRegularHomEquiv.symm x).hom (Finsupp.single g 1) = (A.ρ g) x
-- Rep.ihom_obj
@[simps]
protected def ihom (A : Rep k G) : Rep k G ⥤ Rep k G 
Arguments: {k G : Type u} [CommRing k] [Group G] (A B : Rep k G)
Goal: A.ihom.obj B = Rep.of (A.ρ.linHom B.ρ)
-- Rep.ihom_map_hom
@[simps]
protected def ihom (A : Rep k G) : Rep k G ⥤ Rep k G 
Arguments: {k G : Type u} [CommRing k] [Group G] (A : Rep k G) {X Y : Rep k G} (f : X ⟶ Y)
Goal: (A.ihom.map f).hom = ModuleCat.ofHom ((LinearMap.llcomp k (CoeSort.coe A) (CoeSort.coe X) (CoeSort.coe Y)) f.hom)
-- Rep.ihom_obj_ρ_apply
@[simp] theorem ihom_obj_ρ_apply {A B : Rep k G} (g : G) (x : A →ₗ[k] B) :
    ((Rep.ihom A).obj B).ρ g x = B.ρ g ∘ₗ x ∘ₗ A.ρ g⁻¹ 
Arguments: {k G : Type u} [CommRing k] [Group G] {A B : Rep k G} (g : G) (x : CoeSort.coe A →ₗ[k] CoeSort.coe B)
Goal: ((A.ihom.obj B).ρ g) x = B.ρ g ∘ₗ x ∘ₗ A.ρ g⁻¹
-- Rep.homEquiv_apply_hom
theorem homEquiv_apply_hom (f : A ⊗ B ⟶ C) :
    (homEquiv A B C f).hom = (TensorProduct.curry f.hom).flip 
Arguments: {k G : Type u} [CommRing k] [Group G] {A B C : Rep k G} (f : CategoryTheory.MonoidalCategory.tensorObj A B ⟶ C)
Goal: ((A.homEquiv B C) f).hom = (TensorProduct.curry f.hom).flip
-- Rep.homEquiv_symm_apply_hom
theorem homEquiv_symm_apply_hom (f : B ⟶ (Rep.ihom A).obj C) :
    ((homEquiv A B C).symm f).hom = TensorProduct.uncurry k A B C f.hom.flip 
Arguments: {k G : Type u} [CommRing k] [Group G] {A B C : Rep k G} (f : B ⟶ A.ihom.obj C)
Goal: ((A.homEquiv B C).symm f).hom =
  (TensorProduct.uncurry k (CoeSort.coe A) (CoeSort.coe B) (CoeSort.coe C)) (LinearMap.flip f.hom)
-- Rep.ihom_obj_ρ_def
@[simp]
theorem ihom_obj_ρ_def (A B : Rep k G) : ((ihom A).obj B).ρ = ((Rep.ihom A).obj B).ρ 
Arguments: {k G : Type u} [CommRing k] [Group G] (A B : Rep k G)
Goal: ((CategoryTheory.ihom A).obj B).ρ = (A.ihom.obj B).ρ
-- Rep.homEquiv_def
@[simp]
theorem homEquiv_def (A B C : Rep k G) : (ihom.adjunction A).homEquiv B C = Rep.homEquiv A B C 
Arguments: {k G : Type u} [CommRing k] [Group G] (A B C : Rep k G)
Goal: (CategoryTheory.ihom.adjunction A).homEquiv B C = A.homEquiv B C
-- Rep.ihom_ev_app_hom
@[simp]
theorem ihom_ev_app_hom (A B : Rep k G) :
    Action.Hom.hom ((ihom.ev A).app B)
      = TensorProduct.uncurry k A (A →ₗ[k] B) B LinearMap.id.flip 
Arguments: {k G : Type u} [CommRing k] [Group G] (A B : Rep k G)
Goal: ((CategoryTheory.ihom.ev A).app B).hom =
  (TensorProduct.uncurry k (CoeSort.coe A) (CoeSort.coe A →ₗ[k] CoeSort.coe B) (CoeSort.coe B)) LinearMap.id.flip
-- Rep.ihom_coev_app_hom
@[simp] theorem ihom_coev_app_hom (A B : Rep k G) :
    Action.Hom.hom ((ihom.coev A).app B) = (TensorProduct.mk k _ _).flip 
Arguments: {k G : Type u} [CommRing k] [Group G] (A B : Rep k G)
Goal: ((CategoryTheory.ihom.coev A).app B).hom = (TensorProduct.mk k (CoeSort.coe A) ↑((𝟭 (Rep k G)).obj B).V).flip
-- Rep.MonoidalClosed.linearHomEquiv_hom
@[simp, nolint simpNF]
theorem MonoidalClosed.linearHomEquiv_hom (f : A ⊗ B ⟶ C) :
    (MonoidalClosed.linearHomEquiv A B C f).hom = (TensorProduct.curry f.hom).flip 
Arguments: {k G : Type u} [CommRing k] [Group G] {A B C : Rep k G} (f : CategoryTheory.MonoidalCategory.tensorObj A B ⟶ C)
Goal: ((Rep.MonoidalClosed.linearHomEquiv A B C) f).hom = (TensorProduct.curry f.hom).flip
-- Rep.MonoidalClosed.linearHomEquivComm_hom
@[simp, nolint simpNF]
theorem MonoidalClosed.linearHomEquivComm_hom (f : A ⊗ B ⟶ C) :
    (MonoidalClosed.linearHomEquivComm A B C f).hom = TensorProduct.curry f.hom 
Arguments: {k G : Type u} [CommRing k] [Group G] {A B C : Rep k G} (f : CategoryTheory.MonoidalCategory.tensorObj A B ⟶ C)
Goal: ((Rep.MonoidalClosed.linearHomEquivComm A B C) f).hom = TensorProduct.curry f.hom
-- Rep.MonoidalClosed.linearHomEquiv_symm_hom
theorem MonoidalClosed.linearHomEquiv_symm_hom (f : B ⟶ A ⟶[Rep k G] C) :
    ((MonoidalClosed.linearHomEquiv A B C).symm f).hom = TensorProduct.uncurry k A B C f.hom.flip 
Arguments: {k G : Type u} [CommRing k] [Group G] {A B C : Rep k G} (f : B ⟶ (CategoryTheory.ihom A).obj C)
Goal: ((Rep.MonoidalClosed.linearHomEquiv A B C).symm f).hom =
  (TensorProduct.uncurry k (CoeSort.coe A) (CoeSort.coe B) (CoeSort.coe C)) (LinearMap.flip f.hom)
-- Rep.MonoidalClosed.linearHomEquivComm_symm_hom
theorem MonoidalClosed.linearHomEquivComm_symm_hom (f : A ⟶ B ⟶[Rep k G] C) :
    ((MonoidalClosed.linearHomEquivComm A B C).symm f).hom
      = TensorProduct.uncurry k A B C f.hom 
Arguments: {k G : Type u} [CommRing k] [Group G] {A B C : Rep k G} (f : A ⟶ (CategoryTheory.ihom B).obj C)
Goal: ((Rep.MonoidalClosed.linearHomEquivComm A B C).symm f).hom =
  (TensorProduct.uncurry k (CoeSort.coe A) (CoeSort.coe B) (CoeSort.coe C)) f.hom
-- Representation.repOfTprodIso_apply
theorem repOfTprodIso_apply (x : TensorProduct k V W) : (repOfTprodIso ρ τ).hom.hom x = x 
Arguments: {k G : Type u} [CommRing k] [Monoid G] {V W : Type u} [AddCommGroup V] [AddCommGroup W] [Module k V] [Module k W] (ρ : Representation k G V) (τ : Representation k G W) (x : TensorProduct k V W)
Goal: (ρ.repOfTprodIso τ).hom.hom x = x
-- Representation.repOfTprodIso_inv_apply
theorem repOfTprodIso_inv_apply (x : TensorProduct k V W) : (repOfTprodIso ρ τ).inv.hom x = x 
Arguments: {k G : Type u} [CommRing k] [Monoid G] {V W : Type u} [AddCommGroup V] [AddCommGroup W] [Module k V] [Module k W] (ρ : Representation k G V) (τ : Representation k G W) (x : TensorProduct k V W)
Goal: (ρ.repOfTprodIso τ).inv.hom x = x
-- Rep.to_Module_monoidAlgebra_map_aux
theorem to_Module_monoidAlgebra_map_aux {k G : Type*} [CommRing k] [Monoid G] (V W : Type*)
    [AddCommGroup V] [AddCommGroup W] [Module k V] [Module k W] (ρ : G →* V →ₗ[k] V)
    (σ : G →* W →ₗ[k] W) (f : V →ₗ[k] W) (w : ∀ g : G, f.comp (ρ g) = (σ g).comp f)
    (r : MonoidAlgebra k G) (x : V) :
    f ((((MonoidAlgebra.lift k G (V →ₗ[k] V)) ρ) r) x) =
      (((MonoidAlgebra.lift k G (W →ₗ[k] W)) σ) r) (f x) 
Arguments: {k : Type u_1} {G : Type u_2} [CommRing k] [Monoid G] (V : Type u_3) (W : Type u_4) [AddCommGroup V] [AddCommGroup W] [Module k V] [Module k W] (ρ : G →* V →ₗ[k] V) (σ : G →* W →ₗ[k] W) (f : V →ₗ[k] W) (w : ∀ (g : G), f ∘ₗ ρ g = σ g ∘ₗ f) (r : MonoidAlgebra k G) (x : V)
Goal: f ((((MonoidAlgebra.lift k G (V →ₗ[k] V)) ρ) r) x) = (((MonoidAlgebra.lift k G (W →ₗ[k] W)) σ) r) (f x)
-- Rep.ofModuleMonoidAlgebra_obj_coe
theorem ofModuleMonoidAlgebra_obj_coe (M : ModuleCat.{u} (MonoidAlgebra k G)) :
    (ofModuleMonoidAlgebra.obj M : Type u) = RestrictScalars k (MonoidAlgebra k G) M 
Arguments: {k G : Type u} [CommRing k] [Monoid G] (M : ModuleCat (MonoidAlgebra k G))
Goal: CoeSort.coe (Rep.ofModuleMonoidAlgebra.obj M) = RestrictScalars k (MonoidAlgebra k G) ↑M
-- Rep.ofModuleMonoidAlgebra_obj_ρ
theorem ofModuleMonoidAlgebra_obj_ρ (M : ModuleCat.{u} (MonoidAlgebra k G)) :
    (ofModuleMonoidAlgebra.obj M).ρ = Representation.ofModule M 
Arguments: {k G : Type u} [CommRing k] [Monoid G] (M : ModuleCat (MonoidAlgebra k G))
Goal: (Rep.ofModuleMonoidAlgebra.obj M).ρ = Representation.ofModule ↑M
-- Rep.unit_iso_comm
theorem unit_iso_comm (V : Rep k G) (g : G) (x : V) :
    unitIsoAddEquiv ((V.ρ g).toFun x) = ((ofModuleMonoidAlgebra.obj
      (toModuleMonoidAlgebra.obj V)).ρ g).toFun (unitIsoAddEquiv x) 
Arguments: {k G : Type u} [CommRing k] [Monoid G] (V : Rep k G) (g : G) (x : CoeSort.coe V)
Goal: Rep.unitIsoAddEquiv ((V.ρ g).toFun x) =
  ((Rep.ofModuleMonoidAlgebra.obj (Rep.toModuleMonoidAlgebra.obj V)).ρ g).toFun (Rep.unitIsoAddEquiv x)
-- torusMap_sub_center
theorem torusMap_sub_center (c : ℂⁿ) (R : ℝⁿ) (θ : ℝⁿ) : torusMap c R θ - c = torusMap 0 R θ 
Arguments: {n : ℕ} (c : Fin n → ℂ) (R θ : Fin n → ℝ)
Goal: torusMap c R θ - c = torusMap 0 R θ
-- torusMap_eq_center_iff
theorem torusMap_eq_center_iff {c : ℂⁿ} {R : ℝⁿ} {θ : ℝⁿ} : torusMap c R θ = c ↔ R = 0 
Arguments: {n : ℕ} {c : Fin n → ℂ} {R θ : Fin n → ℝ}
Goal: torusMap c R θ = c ↔ R = 0
-- torusMap_zero_radius
@[simp]
theorem torusMap_zero_radius (c : ℂⁿ) : torusMap c 0 = const ℝⁿ c 
Arguments: {n : ℕ} (c : Fin n → ℂ)
Goal: torusMap c 0 = Function.const (Fin n → ℝ) c
-- TorusIntegrable.torusIntegrable_const
theorem torusIntegrable_const (a : E) (c : ℂⁿ) (R : ℝⁿ) : TorusIntegrable (fun _ => a) c R 
Arguments: {n : ℕ} {E : Type u_1} [NormedAddCommGroup E] (a : E) (c : Fin n → ℂ) (R : Fin n → ℝ)
Goal: TorusIntegrable (fun x => a) c R
-- TorusIntegrable.neg
protected nonrec theorem neg (hf : TorusIntegrable f c R) : TorusIntegrable (-f) c R 
Arguments: {n : ℕ} {E : Type u_1} [NormedAddCommGroup E] {f : (Fin n → ℂ) → E} {c : Fin n → ℂ} {R : Fin n → ℝ} (hf : TorusIntegrable f c R)
Goal: TorusIntegrable (-f) c R
-- TorusIntegrable.add
protected nonrec theorem add (hf : TorusIntegrable f c R) (hg : TorusIntegrable g c R) :
    TorusIntegrable (f + g) c R 
Arguments: {n : ℕ} {E : Type u_1} [NormedAddCommGroup E] {f g : (Fin n → ℂ) → E} {c : Fin n → ℂ} {R : Fin n → ℝ} (hf : TorusIntegrable f c R) (hg : TorusIntegrable g c R)
Goal: TorusIntegrable (f + g) c R
-- TorusIntegrable.sub
protected nonrec theorem sub (hf : TorusIntegrable f c R) (hg : TorusIntegrable g c R) :
    TorusIntegrable (f - g) c R 
Arguments: {n : ℕ} {E : Type u_1} [NormedAddCommGroup E] {f g : (Fin n → ℂ) → E} {c : Fin n → ℂ} {R : Fin n → ℝ} (hf : TorusIntegrable f c R) (hg : TorusIntegrable g c R)
Goal: TorusIntegrable (f - g) c R
-- TorusIntegrable.torusIntegrable_zero_radius
theorem torusIntegrable_zero_radius {f : ℂⁿ → E} {c : ℂⁿ} : TorusIntegrable f c 0 
Arguments: {n : ℕ} {E : Type u_1} [NormedAddCommGroup E] {f : (Fin n → ℂ) → E} {c : Fin n → ℂ}
Goal: TorusIntegrable f c 0
-- TorusIntegrable.function_integrable
theorem function_integrable [NormedSpace ℂ E] (hf : TorusIntegrable f c R) :
    IntegrableOn (fun θ : ℝⁿ => (∏ i, R i * exp (θ i * I) * I : ℂ) • f (torusMap c R θ))
      (Icc (0 : ℝⁿ) fun _ => 2 * π) volume 
Arguments: {n : ℕ} {E : Type u_1} [NormedAddCommGroup E] {f : (Fin n → ℂ) → E} {c : Fin n → ℂ} {R : Fin n → ℝ} [NormedSpace ℂ E] (hf : TorusIntegrable f c R)
Goal: MeasureTheory.IntegrableOn (fun θ => (∏ i : Fin n, ↑(R i) * cexp (↑(θ i) * Complex.I) * Complex.I) • f (torusMap c R θ))
  (Set.Icc 0 fun x => 2 * π) MeasureTheory.volume
-- torusIntegral_radius_zero
theorem torusIntegral_radius_zero (hn : n ≠ 0) (f : ℂⁿ → E) (c : ℂⁿ) :
    (∯ x in T(c, 0), f x) = 0 
Arguments: {n : ℕ} {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] (hn : n ≠ 0) (f : (Fin n → ℂ) → E) (c : Fin n → ℂ)
Goal: (∯ (x : Fin n → ℂ) in T(c, 0), f x) = 0
-- torusIntegral_neg
theorem torusIntegral_neg (f : ℂⁿ → E) (c : ℂⁿ) (R : ℝⁿ) :
    (∯ x in T(c, R), -f x) = -∯ x in T(c, R), f x 
Arguments: {n : ℕ} {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] (f : (Fin n → ℂ) → E) (c : Fin n → ℂ) (R : Fin n → ℝ)
Goal: (∯ (x : Fin n → ℂ) in T(c, R), -f x) = -∯ (x : Fin n → ℂ) in T(c, R), f x
-- torusIntegral_add
theorem torusIntegral_add (hf : TorusIntegrable f c R) (hg : TorusIntegrable g c R) :
    (∯ x in T(c, R), f x + g x) = (∯ x in T(c, R), f x) + ∯ x in T(c, R), g x 
Arguments: {n : ℕ} {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] {f g : (Fin n → ℂ) → E} {c : Fin n → ℂ} {R : Fin n → ℝ} (hf : TorusIntegrable f c R) (hg : TorusIntegrable g c R)
Goal: (∯ (x : Fin n → ℂ) in T(c, R), f x + g x) = (∯ (x : Fin n → ℂ) in T(c, R), f x) + ∯ (x : Fin n → ℂ) in T(c, R), g x
-- torusIntegral_sub
theorem torusIntegral_sub (hf : TorusIntegrable f c R) (hg : TorusIntegrable g c R) :
    (∯ x in T(c, R), f x - g x) = (∯ x in T(c, R), f x) - ∯ x in T(c, R), g x 
Arguments: {n : ℕ} {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] {f g : (Fin n → ℂ) → E} {c : Fin n → ℂ} {R : Fin n → ℝ} (hf : TorusIntegrable f c R) (hg : TorusIntegrable g c R)
Goal: (∯ (x : Fin n → ℂ) in T(c, R), f x - g x) = (∯ (x : Fin n → ℂ) in T(c, R), f x) - ∯ (x : Fin n → ℂ) in T(c, R), g x
-- torusIntegral_smul
theorem torusIntegral_smul {𝕜 : Type*} [RCLike 𝕜] [NormedSpace 𝕜 E] [SMulCommClass 𝕜 ℂ E] (a : 𝕜)
    (f : ℂⁿ → E) (c : ℂⁿ) (R : ℝⁿ) : (∯ x in T(c, R), a • f x) = a • ∯ x in T(c, R), f x 
Arguments: {n : ℕ} {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] {𝕜 : Type u_2} [RCLike 𝕜] [NormedSpace 𝕜 E] [SMulCommClass 𝕜 ℂ E] (a : 𝕜) (f : (Fin n → ℂ) → E) (c : Fin n → ℂ) (R : Fin n → ℝ)
Goal: (∯ (x : Fin n → ℂ) in T(c, R), a • f x) = a • ∯ (x : Fin n → ℂ) in T(c, R), f x
-- torusIntegral_const_mul
theorem torusIntegral_const_mul (a : ℂ) (f : ℂⁿ → ℂ) (c : ℂⁿ) (R : ℝⁿ) :
    (∯ x in T(c, R), a * f x) = a * ∯ x in T(c, R), f x 
Arguments: {n : ℕ} (a : ℂ) (f : (Fin n → ℂ) → ℂ) (c : Fin n → ℂ) (R : Fin n → ℝ)
Goal: (∯ (x : Fin n → ℂ) in T(c, R), a * f x) = a * ∯ (x : Fin n → ℂ) in T(c, R), f x
-- norm_torusIntegral_le_of_norm_le_const
theorem norm_torusIntegral_le_of_norm_le_const {C : ℝ} (hf : ∀ θ, ‖f (torusMap c R θ)‖ ≤ C) :
    ‖∯ x in T(c, R), f x‖ ≤ ((2 * π) ^ (n : ℕ) * ∏ i, |R i|) * C 
Arguments: {n : ℕ} {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] {f : (Fin n → ℂ) → E} {c : Fin n → ℂ} {R : Fin n → ℝ} {C : ℝ} (hf : ∀ (θ : Fin n → ℝ), ‖f (torusMap c R θ)‖ ≤ C)
Goal: ‖∯ (x : Fin n → ℂ) in T(c, R), f x‖ ≤ ((2 * π) ^ n * ∏ i : Fin n, |R i|) * C
-- torusIntegral_dim0
@[simp]
theorem torusIntegral_dim0 (f : ℂ⁰ → E) (c : ℂ⁰) (R : ℝ⁰) : (∯ x in T(c, R), f x) = f c 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] [CompleteSpace E] (f : (Fin 0 → ℂ) → E) (c : Fin 0 → ℂ) (R : Fin 0 → ℝ)
Goal: (∯ (x : Fin 0 → ℂ) in T(c, R), f x) = f c
-- torusIntegral_dim1
theorem torusIntegral_dim1 (f : ℂ¹ → E) (c : ℂ¹) (R : ℝ¹) :
    (∯ x in T(c, R), f x) = ∮ z in C(c 0, R 0), f fun _ => z 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] (f : (Fin 1 → ℂ) → E) (c : Fin 1 → ℂ) (R : Fin 1 → ℝ)
Goal: (∯ (x : Fin 1 → ℂ) in T(c, R), f x) = ∮ (z : ℂ) in C(c 0, R 0), f fun x => z