ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- MultilinearMap.map_add_univ
theorem map_add_univ [DecidableEq ι] [Fintype ι] (m m' : ∀ i, M₁ i) :
    f (m + m') = ∑ s : Finset ι, f (s.piecewise m m') 
Arguments: {R : Type uR} {ι : Type uι} {M₁ : ι → Type v₁} {M₂ : Type v₂} [Semiring R] [(i : ι) → AddCommMonoid (M₁ i)] [AddCommMonoid M₂] [(i : ι) → Module R (M₁ i)] [Module R M₂] (f : MultilinearMap R M₁ M₂) [DecidableEq ι] [Fintype ι] (m m' : (i : ι) → M₁ i)
Goal: f (m + m') = ∑ s : Finset ι, f (s.piecewise m m')
-- MultilinearMap.map_sum_finset_aux
theorem map_sum_finset_aux [DecidableEq ι] [Fintype ι] {n : ℕ} (h : (∑ i, (A i).card) = n) :
    (f fun i => ∑ j ∈ A i, g i j) = ∑ r ∈ piFinset A, f fun i => g i (r i) 
Arguments: {R : Type uR} {ι : Type uι} {M₁ : ι → Type v₁} {M₂ : Type v₂} [Semiring R] [(i : ι) → AddCommMonoid (M₁ i)] [AddCommMonoid M₂] [(i : ι) → Module R (M₁ i)] [Module R M₂] (f : MultilinearMap R M₁ M₂) {α : ι → Type u_1} (g : (i : ι) → α i → M₁ i) (A : (i : ι) → Finset (α i)) [DecidableEq ι] [Fintype ι] {n : ℕ} (h : ∑ i : ι, (A i).card = n)
Goal: (f fun i => ∑ j ∈ A i, g i j) = ∑ r ∈ Fintype.piFinset A, f fun i => g i (r i)
-- MultilinearMap.map_sum_finset
theorem map_sum_finset [DecidableEq ι] [Fintype ι] :
    (f fun i => ∑ j ∈ A i, g i j) = ∑ r ∈ piFinset A, f fun i => g i (r i) 
Arguments: {R : Type uR} {ι : Type uι} {M₁ : ι → Type v₁} {M₂ : Type v₂} [Semiring R] [(i : ι) → AddCommMonoid (M₁ i)] [AddCommMonoid M₂] [(i : ι) → Module R (M₁ i)] [Module R M₂] (f : MultilinearMap R M₁ M₂) {α : ι → Type u_1} (g : (i : ι) → α i → M₁ i) (A : (i : ι) → Finset (α i)) [DecidableEq ι] [Fintype ι]
Goal: (f fun i => ∑ j ∈ A i, g i j) = ∑ r ∈ Fintype.piFinset A, f fun i => g i (r i)
-- MultilinearMap.map_sum
theorem map_sum [DecidableEq ι] [Fintype ι] [∀ i, Fintype (α i)] :
    (f fun i => ∑ j, g i j) = ∑ r : ∀ i, α i, f fun i => g i (r i) 
Arguments: {R : Type uR} {ι : Type uι} {M₁ : ι → Type v₁} {M₂ : Type v₂} [Semiring R] [(i : ι) → AddCommMonoid (M₁ i)] [AddCommMonoid M₂] [(i : ι) → Module R (M₁ i)] [Module R M₂] (f : MultilinearMap R M₁ M₂) {α : ι → Type u_1} (g : (i : ι) → α i → M₁ i) [DecidableEq ι] [Fintype ι] [(i : ι) → Fintype (α i)]
Goal: (f fun i => ∑ j : α i, g i j) = ∑ r : (i : ι) → α i, f fun i => g i (r i)
-- MultilinearMap.map_update_sum
theorem map_update_sum {α : Type*} [DecidableEq ι] (t : Finset α) (i : ι) (g : α → M₁ i)
    (m : ∀ i, M₁ i) : f (update m i (∑ a ∈ t, g a)) = ∑ a ∈ t, f (update m i (g a)) 
Arguments: {R : Type uR} {ι : Type uι} {M₁ : ι → Type v₁} {M₂ : Type v₂} [Semiring R] [(i : ι) → AddCommMonoid (M₁ i)] [AddCommMonoid M₂] [(i : ι) → Module R (M₁ i)] [Module R M₂] (f : MultilinearMap R M₁ M₂) {α : Type u_2} [DecidableEq ι] (t : Finset α) (i : ι) (g : α → M₁ i) (m : (i : ι) → M₁ i)
Goal: f (Function.update m i (∑ a ∈ t, g a)) = ∑ a ∈ t, f (Function.update m i (g a))
-- MultilinearMap.codRestrict_apply_coe
@[simps]
def codRestrict (f : MultilinearMap R M₁ M₂) (p : Submodule R M₂) (h : ∀ v, f v ∈ p) :
    MultilinearMap R M₁ p 
Arguments: {R : Type uR} {ι : Type uι} {M₁ : ι → Type v₁} {M₂ : Type v₂} [Semiring R] [(i : ι) → AddCommMonoid (M₁ i)] [AddCommMonoid M₂] [(i : ι) → Module R (M₁ i)] [Module R M₂] (f : MultilinearMap R M₁ M₂) (p : Submodule R M₂) (h : ∀ (v : (i : ι) → M₁ i), f v ∈ p) (v : (i : ι) → M₁ i)
Goal: ↑((f.codRestrict p h) v) = f v
-- MultilinearMap.coe_restrictScalars
@[simp]
theorem coe_restrictScalars (f : MultilinearMap A M₁ M₂) : ⇑(f.restrictScalars R) = f 
Arguments: (R : Type uR) {ι : Type uι} {M₁ : ι → Type v₁} {M₂ : Type v₂} [Semiring R] [(i : ι) → AddCommMonoid (M₁ i)] [AddCommMonoid M₂] [(i : ι) → Module R (M₁ i)] [Module R M₂] {A : Type u_1} [Semiring A] [SMul R A] [(i : ι) → Module A (M₁ i)] [Module A M₂] [∀ (i : ι), IsScalarTower R A (M₁ i)] [IsScalarTower R A M₂] (f : MultilinearMap A M₁ M₂)
Goal: ⇑(MultilinearMap.restrictScalars R f) = ⇑f
-- MultilinearMap.domDomCongr_apply
@[simps apply]
def domDomCongr (σ : ι₁ ≃ ι₂) (m : MultilinearMap R (fun _ : ι₁ => M₂) M₃) :
    MultilinearMap R (fun _ : ι₂ => M₂) M₃ 
Arguments: {R : Type uR} {M₂ : Type v₂} {M₃ : Type v₃} [Semiring R] [AddCommMonoid M₂] [AddCommMonoid M₃] [Module R M₂] [Module R M₃] {ι₁ : Type u_1} {ι₂ : Type u_2} (σ : ι₁ ≃ ι₂) (m : MultilinearMap R (fun x => M₂) M₃) (v : ι₂ → M₂)
Goal: (MultilinearMap.domDomCongr σ m) v = m fun i => v (σ i)
-- MultilinearMap.domDomCongr_trans
theorem domDomCongr_trans (σ₁ : ι₁ ≃ ι₂) (σ₂ : ι₂ ≃ ι₃)
    (m : MultilinearMap R (fun _ : ι₁ => M₂) M₃) :
    m.domDomCongr (σ₁.trans σ₂) = (m.domDomCongr σ₁).domDomCongr σ₂ 
Arguments: {R : Type uR} {M₂ : Type v₂} {M₃ : Type v₃} [Semiring R] [AddCommMonoid M₂] [AddCommMonoid M₃] [Module R M₂] [Module R M₃] {ι₁ : Type u_1} {ι₂ : Type u_2} {ι₃ : Type u_3} (σ₁ : ι₁ ≃ ι₂) (σ₂ : ι₂ ≃ ι₃) (m : MultilinearMap R (fun x => M₂) M₃)
Goal: MultilinearMap.domDomCongr (σ₁.trans σ₂) m = MultilinearMap.domDomCongr σ₂ (MultilinearMap.domDomCongr σ₁ m)
-- MultilinearMap.domDomCongr_mul
theorem domDomCongr_mul (σ₁ : Equiv.Perm ι₁) (σ₂ : Equiv.Perm ι₁)
    (m : MultilinearMap R (fun _ : ι₁ => M₂) M₃) :
    m.domDomCongr (σ₂ * σ₁) = (m.domDomCongr σ₁).domDomCongr σ₂ 
Arguments: {R : Type uR} {M₂ : Type v₂} {M₃ : Type v₃} [Semiring R] [AddCommMonoid M₂] [AddCommMonoid M₃] [Module R M₂] [Module R M₃] {ι₁ : Type u_1} (σ₁ σ₂ : Equiv.Perm ι₁) (m : MultilinearMap R (fun x => M₂) M₃)
Goal: MultilinearMap.domDomCongr (σ₂ * σ₁) m = MultilinearMap.domDomCongr σ₂ (MultilinearMap.domDomCongr σ₁ m)
-- MultilinearMap.domDomCongrEquiv_apply
@[simps apply symm_apply]
def domDomCongrEquiv (σ : ι₁ ≃ ι₂) :
    MultilinearMap R (fun _ : ι₁ => M₂) M₃ ≃+ MultilinearMap R (fun _ : ι₂ => M₂) M₃ 
Arguments: {R : Type uR} {M₂ : Type v₂} {M₃ : Type v₃} [Semiring R] [AddCommMonoid M₂] [AddCommMonoid M₃] [Module R M₂] [Module R M₃] {ι₁ : Type u_1} {ι₂ : Type u_2} (σ : ι₁ ≃ ι₂) (m : MultilinearMap R (fun x => M₂) M₃)
Goal: (MultilinearMap.domDomCongrEquiv σ) m = MultilinearMap.domDomCongr σ m
-- MultilinearMap.domDomCongrEquiv_symm_apply
@[simps apply symm_apply]
def domDomCongrEquiv (σ : ι₁ ≃ ι₂) :
    MultilinearMap R (fun _ : ι₁ => M₂) M₃ ≃+ MultilinearMap R (fun _ : ι₂ => M₂) M₃ 
Arguments: {R : Type uR} {M₂ : Type v₂} {M₃ : Type v₃} [Semiring R] [AddCommMonoid M₂] [AddCommMonoid M₃] [Module R M₂] [Module R M₃] {ι₁ : Type u_1} {ι₂ : Type u_2} (σ : ι₁ ≃ ι₂) (m : MultilinearMap R (fun x => M₂) M₃)
Goal: (MultilinearMap.domDomCongrEquiv σ).symm m = MultilinearMap.domDomCongr σ.symm m
-- MultilinearMap.domDomCongr_eq_iff
@[simp]
theorem domDomCongr_eq_iff (σ : ι₁ ≃ ι₂) (f g : MultilinearMap R (fun _ : ι₁ => M₂) M₃) :
    f.domDomCongr σ = g.domDomCongr σ ↔ f = g 
Arguments: {R : Type uR} {M₂ : Type v₂} {M₃ : Type v₃} [Semiring R] [AddCommMonoid M₂] [AddCommMonoid M₃] [Module R M₂] [Module R M₃] {ι₁ : Type u_1} {ι₂ : Type u_2} (σ : ι₁ ≃ ι₂) (f g : MultilinearMap R (fun x => M₂) M₃)
Goal: MultilinearMap.domDomCongr σ f = MultilinearMap.domDomCongr σ g ↔ f = g
-- MultilinearMap.domDomRestrict_aux
lemma domDomRestrict_aux [DecidableEq ι] (P : ι → Prop) [DecidablePred P]
    [DecidableEq {a // P a}]
    (x : (i : {a // P a}) → M₁ i) (z : (i : {a // ¬ P a}) → M₁ i) (i : {a : ι // P a})
    (c : M₁ i) : (fun j ↦ if h : P j then Function.update x i c ⟨j, h⟩ else z ⟨j, h⟩) =
    Function.update (fun j => if h : P j then x ⟨j, h⟩ else z ⟨j, h⟩) i c 
Arguments: {ι : Type uι} {M₁ : ι → Type v₁} [DecidableEq ι] (P : ι → Prop) [DecidablePred P] [DecidableEq { a // P a }] (x : (i : { a // P a }) → M₁ ↑i) (z : (i : { a // ¬P a }) → M₁ ↑i) (i : { a // P a }) (c : M₁ ↑i)
Goal: (fun j => if h : P j then Function.update x i c ⟨j, h⟩ else z ⟨j, h⟩) =
  Function.update (fun j => if h : P j then x ⟨j, h⟩ else z ⟨j, h⟩) (↑i) c
-- MultilinearMap.domDomRestrict_aux_right
lemma domDomRestrict_aux_right [DecidableEq ι] (P : ι → Prop) [DecidablePred P]
    [DecidableEq {a // ¬ P a}]
    (x : (i : {a // P a}) → M₁ i) (z : (i : {a // ¬ P a}) → M₁ i) (i : {a : ι // ¬ P a})
    (c : M₁ i) : (fun j ↦ if h : P j then x ⟨j, h⟩ else Function.update z i c ⟨j, h⟩) =
    Function.update (fun j => if h : P j then x ⟨j, h⟩ else z ⟨j, h⟩) i c 
Arguments: {ι : Type uι} {M₁ : ι → Type v₁} [DecidableEq ι] (P : ι → Prop) [DecidablePred P] [DecidableEq { a // ¬P a }] (x : (i : { a // P a }) → M₁ ↑i) (z : (i : { a // ¬P a }) → M₁ ↑i) (i : { a // ¬P a }) (c : M₁ ↑i)
Goal: (fun j => if h : P j then x ⟨j, h⟩ else Function.update z i c ⟨j, h⟩) =
  Function.update (fun j => if h : P j then x ⟨j, h⟩ else z ⟨j, h⟩) (↑i) c
-- MultilinearMap.domDomRestrict_apply
@[simp]
lemma domDomRestrict_apply (f : MultilinearMap R M₁ M₂) (P : ι → Prop)
    [DecidablePred P] (x : (i : {a // P a}) → M₁ i) (z : (i : {a // ¬ P a}) → M₁ i) :
    f.domDomRestrict P z x = f (fun j => if h : P j then x ⟨j, h⟩ else z ⟨j, h⟩) 
Arguments: {R : Type uR} {ι : Type uι} {M₁ : ι → Type v₁} {M₂ : Type v₂} [Semiring R] [(i : ι) → AddCommMonoid (M₁ i)] [AddCommMonoid M₂] [(i : ι) → Module R (M₁ i)] [Module R M₂] (f : MultilinearMap R M₁ M₂) (P : ι → Prop) [DecidablePred P] (x : (i : { a // P a }) → M₁ ↑i) (z : (i : { a // ¬P a }) → M₁ ↑i)
Goal: (f.domDomRestrict P z) x = f fun j => if h : P j then x ⟨j, h⟩ else z ⟨j, h⟩
-- MultilinearMap.linearDeriv_apply
@[simp]
lemma linearDeriv_apply [DecidableEq ι] [Fintype ι] (f : MultilinearMap R M₁ M₂)
    (x y : (i : ι) → M₁ i) :
    f.linearDeriv x y = ∑ i, f (update x i (y i)) 
Arguments: {R : Type uR} {ι : Type uι} {M₁ : ι → Type v₁} {M₂ : Type v₂} [Semiring R] [(i : ι) → AddCommMonoid (M₁ i)] [AddCommMonoid M₂] [(i : ι) → Module R (M₁ i)] [Module R M₂] [DecidableEq ι] [Fintype ι] (f : MultilinearMap R M₁ M₂) (x y : (i : ι) → M₁ i)
Goal: (f.linearDeriv x) y = ∑ i : ι, f (Function.update x i (y i))
-- LinearMap.coe_compMultilinearMap
@[simp]
theorem coe_compMultilinearMap (g : M₂ →ₗ[R] M₃) (f : MultilinearMap R M₁ M₂) :
    ⇑(g.compMultilinearMap f) = g ∘ f 
Arguments: {R : Type uR} {ι : Type uι} {M₁ : ι → Type v₁} {M₂ : Type v₂} {M₃ : Type v₃} [Semiring R] [(i : ι) → AddCommMonoid (M₁ i)] [AddCommMonoid M₂] [AddCommMonoid M₃] [(i : ι) → Module R (M₁ i)] [Module R M₂] [Module R M₃] (g : M₂ →ₗ[R] M₃) (f : MultilinearMap R M₁ M₂)
Goal: ⇑(g.compMultilinearMap f) = ⇑g ∘ ⇑f
-- LinearMap.compMultilinearMap_apply
@[simp]
theorem compMultilinearMap_apply (g : M₂ →ₗ[R] M₃) (f : MultilinearMap R M₁ M₂) (m : ∀ i, M₁ i) :
    g.compMultilinearMap f m = g (f m) 
Arguments: {R : Type uR} {ι : Type uι} {M₁ : ι → Type v₁} {M₂ : Type v₂} {M₃ : Type v₃} [Semiring R] [(i : ι) → AddCommMonoid (M₁ i)] [AddCommMonoid M₂] [AddCommMonoid M₃] [(i : ι) → Module R (M₁ i)] [Module R M₂] [Module R M₃] (g : M₂ →ₗ[R] M₃) (f : MultilinearMap R M₁ M₂) (m : (i : ι) → M₁ i)
Goal: (g.compMultilinearMap f) m = g (f m)
-- LinearMap.subtype_compMultilinearMap_codRestrict
@[simp]
theorem subtype_compMultilinearMap_codRestrict (f : MultilinearMap R M₁ M₂) (p : Submodule R M₂)
    (h) : p.subtype.compMultilinearMap (f.codRestrict p h) = f 
Arguments: {R : Type uR} {ι : Type uι} {M₁ : ι → Type v₁} {M₂ : Type v₂} [Semiring R] [(i : ι) → AddCommMonoid (M₁ i)] [AddCommMonoid M₂] [(i : ι) → Module R (M₁ i)] [Module R M₂] (f : MultilinearMap R M₁ M₂) (p : Submodule R M₂) (h : ∀ (v : (i : ι) → M₁ i), f v ∈ p)
Goal: p.subtype.compMultilinearMap (f.codRestrict p h) = f
-- LinearMap.compMultilinearMap_codRestrict
@[simp]
theorem compMultilinearMap_codRestrict (g : M₂ →ₗ[R] M₃) (f : MultilinearMap R M₁ M₂)
    (p : Submodule R M₃) (h) :
    (g.codRestrict p h).compMultilinearMap f =
      (g.compMultilinearMap f).codRestrict p fun v => h (f v) 
Arguments: {R : Type uR} {ι : Type uι} {M₁ : ι → Type v₁} {M₂ : Type v₂} {M₃ : Type v₃} [Semiring R] [(i : ι) → AddCommMonoid (M₁ i)] [AddCommMonoid M₂] [AddCommMonoid M₃] [(i : ι) → Module R (M₁ i)] [Module R M₂] [Module R M₃] (g : M₂ →ₗ[R] M₃) (f : MultilinearMap R M₁ M₂) (p : Submodule R M₃) (h : ∀ (c : M₂), g c ∈ p)
Goal: (LinearMap.codRestrict p g h).compMultilinearMap f = (g.compMultilinearMap f).codRestrict p ⋯
-- LinearMap.compMultilinearMap_domDomCongr
@[simp]
theorem compMultilinearMap_domDomCongr (σ : ι₁ ≃ ι₂) (g : M₂ →ₗ[R] M₃)
    (f : MultilinearMap R (fun _ : ι₁ => M') M₂) :
    (g.compMultilinearMap f).domDomCongr σ = g.compMultilinearMap (f.domDomCongr σ) 
Arguments: {R : Type uR} {M₂ : Type v₂} {M₃ : Type v₃} {M' : Type v'} [Semiring R] [AddCommMonoid M₂] [AddCommMonoid M₃] [AddCommMonoid M'] [Module R M₂] [Module R M₃] [Module R M'] {ι₁ : Type u_1} {ι₂ : Type u_2} (σ : ι₁ ≃ ι₂) (g : M₂ →ₗ[R] M₃) (f : MultilinearMap R (fun x => M') M₂)
Goal: MultilinearMap.domDomCongr σ (g.compMultilinearMap f) = g.compMultilinearMap (MultilinearMap.domDomCongr σ f)
-- MultilinearMap.ofSubsingletonₗ_symm_apply
@[simps (config := { simpRhs := true })]
def ofSubsingletonₗ [Subsingleton ι] (i : ι) :
    (M₂ →ₗ[R] M₃) ≃ₗ[S] MultilinearMap R (fun _ : ι ↦ M₂) M₃ 
Arguments: (R : Type uR) (S : Type uS) {ι : Type uι} (M₂ : Type v₂) (M₃ : Type v₃) [Semiring R] [AddCommMonoid M₂] [Module R M₂] [Semiring S] [AddCommMonoid M₃] [Module S M₃] [Module R M₃] [SMulCommClass R S M₃] [Subsingleton ι] (i : ι)
Goal: ∀ (a : MultilinearMap R (fun x => M₂) M₃),
  (MultilinearMap.ofSubsingletonₗ R S M₂ M₃ i).symm a = (MultilinearMap.ofSubsingleton R M₂ M₃ i).symm a
-- MultilinearMap.ofSubsingletonₗ_apply
@[simps (config := { simpRhs := true })]
def ofSubsingletonₗ [Subsingleton ι] (i : ι) :
    (M₂ →ₗ[R] M₃) ≃ₗ[S] MultilinearMap R (fun _ : ι ↦ M₂) M₃ 
Arguments: (R : Type uR) (S : Type uS) {ι : Type uι} (M₂ : Type v₂) (M₃ : Type v₃) [Semiring R] [AddCommMonoid M₂] [Module R M₂] [Semiring S] [AddCommMonoid M₃] [Module S M₃] [Module R M₃] [SMulCommClass R S M₃] [Subsingleton ι] (i : ι)
Goal: ∀ (a : M₂ →ₗ[R] M₃), (MultilinearMap.ofSubsingletonₗ R S M₂ M₃ i) a = (MultilinearMap.ofSubsingleton R M₂ M₃ i) a
-- MultilinearMap.domDomCongrLinearEquiv'_apply
@[simps apply symm_apply]
def domDomCongrLinearEquiv' {ι' : Type*} (σ : ι ≃ ι') :
    MultilinearMap R M₁ M₂ ≃ₗ[S] MultilinearMap R (fun i => M₁ (σ.symm i)) M₂ 
Arguments: (R : Type uR) (S : Type uS) {ι : Type uι} (M₁ : ι → Type v₁) (M₂ : Type v₂) [Semiring R] [(i : ι) → AddCommMonoid (M₁ i)] [(i : ι) → Module R (M₁ i)] [AddCommMonoid M₂] [Module R M₂] [Semiring S] [Module S M₂] [SMulCommClass R S M₂] {ι' : Type u_1} (σ : ι ≃ ι') (f : MultilinearMap R M₁ M₂)
Goal: (MultilinearMap.domDomCongrLinearEquiv' R S M₁ M₂ σ) f =
  { toFun := ⇑f ∘ ⇑(Equiv.piCongrLeft' M₁ σ).symm, map_add' := ⋯, map_smul' := ⋯ }
-- MultilinearMap.domDomCongrLinearEquiv'_symm_apply
@[simps apply symm_apply]
def domDomCongrLinearEquiv' {ι' : Type*} (σ : ι ≃ ι') :
    MultilinearMap R M₁ M₂ ≃ₗ[S] MultilinearMap R (fun i => M₁ (σ.symm i)) M₂ 
Arguments: (R : Type uR) (S : Type uS) {ι : Type uι} (M₁ : ι → Type v₁) (M₂ : Type v₂) [Semiring R] [(i : ι) → AddCommMonoid (M₁ i)] [(i : ι) → Module R (M₁ i)] [AddCommMonoid M₂] [Module R M₂] [Semiring S] [Module S M₂] [SMulCommClass R S M₂] {ι' : Type u_1} (σ : ι ≃ ι') (f : MultilinearMap R (fun i => M₁ (σ.symm i)) M₂)
Goal: (MultilinearMap.domDomCongrLinearEquiv' R S M₁ M₂ σ).symm f =
  { toFun := ⇑f ∘ ⇑(Equiv.piCongrLeft' M₁ σ), map_add' := ⋯, map_smul' := ⋯ }
-- MultilinearMap.constLinearEquivOfIsEmpty_apply
@[simps]
def constLinearEquivOfIsEmpty [IsEmpty ι] : M₂ ≃ₗ[S] MultilinearMap R M₁ M₂ 
Arguments: (R : Type uR) (S : Type uS) {ι : Type uι} (M₁ : ι → Type v₁) (M₂ : Type v₂) [Semiring R] [(i : ι) → AddCommMonoid (M₁ i)] [(i : ι) → Module R (M₁ i)] [AddCommMonoid M₂] [Module R M₂] [Semiring S] [Module S M₂] [SMulCommClass R S M₂] [IsEmpty ι] (m : M₂)
Goal: (MultilinearMap.constLinearEquivOfIsEmpty R S M₁ M₂) m = MultilinearMap.constOfIsEmpty R M₁ m
-- MultilinearMap.constLinearEquivOfIsEmpty_symm_apply
@[simps]
def constLinearEquivOfIsEmpty [IsEmpty ι] : M₂ ≃ₗ[S] MultilinearMap R M₁ M₂ 
Arguments: (R : Type uR) (S : Type uS) {ι : Type uι} (M₁ : ι → Type v₁) (M₂ : Type v₂) [Semiring R] [(i : ι) → AddCommMonoid (M₁ i)] [(i : ι) → Module R (M₁ i)] [AddCommMonoid M₂] [Module R M₂] [Semiring S] [Module S M₂] [SMulCommClass R S M₂] [IsEmpty ι] (f : MultilinearMap R M₁ M₂)
Goal: (MultilinearMap.constLinearEquivOfIsEmpty R S M₁ M₂).symm f = f 0
-- MultilinearMap.domDomCongrLinearEquiv_apply
@[simps apply symm_apply]
def domDomCongrLinearEquiv {ι₁ ι₂} (σ : ι₁ ≃ ι₂) :
    MultilinearMap R (fun _ : ι₁ => M₂) M₃ ≃ₗ[S] MultilinearMap R (fun _ : ι₂ => M₂) M₃ 
Arguments: (R : Type uR) (S : Type uS) (M₂ : Type v₂) (M₃ : Type v₃) [Semiring R] [AddCommMonoid M₂] [Module R M₂] [Semiring S] [AddCommMonoid M₃] [Module R M₃] [Module S M₃] [SMulCommClass R S M₃] {ι₁ : Type u_1} {ι₂ : Type u_2} (σ : ι₁ ≃ ι₂)
Goal: ∀ (a : MultilinearMap R (fun x => M₂) M₃),
  (MultilinearMap.domDomCongrLinearEquiv R S M₂ M₃ σ) a = (MultilinearMap.domDomCongrEquiv σ).toFun a
-- MultilinearMap.domDomCongrLinearEquiv_symm_apply
@[simps apply symm_apply]
def domDomCongrLinearEquiv {ι₁ ι₂} (σ : ι₁ ≃ ι₂) :
    MultilinearMap R (fun _ : ι₁ => M₂) M₃ ≃ₗ[S] MultilinearMap R (fun _ : ι₂ => M₂) M₃ 
Arguments: (R : Type uR) (S : Type uS) (M₂ : Type v₂) (M₃ : Type v₃) [Semiring R] [AddCommMonoid M₂] [Module R M₂] [Semiring S] [AddCommMonoid M₃] [Module R M₃] [Module S M₃] [SMulCommClass R S M₃] {ι₁ : Type u_1} {ι₂ : Type u_2} (σ : ι₁ ≃ ι₂)
Goal: ∀ (a : MultilinearMap R (fun x => M₂) M₃),
  (MultilinearMap.domDomCongrLinearEquiv R S M₂ M₃ σ).symm a = (MultilinearMap.domDomCongrEquiv σ).invFun a
-- MultilinearMap.iteratedFDeriv_aux
lemma iteratedFDeriv_aux {α : Type*} [DecidableEq α]
    (s : Set ι) [DecidableEq { x // x ∈ s }] (e : α ≃ s)
    (m : α → ((i : ι) → M₁ i)) (a : α) (z : (i : ι) → M₁ i) :
    (fun i ↦ update m a z (e.symm i) i) =
      (fun i ↦ update (fun j ↦ m (e.symm j) j) (e a) (z (e a)) i) 
Arguments: {ι : Type uι} {M₁ : ι → Type v₁} {α : Type u_2} [DecidableEq α] (s : Set ι) [DecidableEq { x // x ∈ s }] (e : α ≃ ↑s) (m : α → (i : ι) → M₁ i) (a : α) (z : (i : ι) → M₁ i)
Goal: (fun i => Function.update m a z (e.symm i) ↑i) = fun i => Function.update (fun j => m (e.symm j) ↑j) (e a) (z ↑(e a)) i
-- MultilinearMap.compLinearMapₗ_apply
@[simps] def compLinearMapₗ (f : Π (i : ι), M₁ i →ₗ[R] M₁' i) :
    (MultilinearMap R M₁' M₂) →ₗ[R] MultilinearMap R M₁ M₂ 
Arguments: {R : Type uR} {ι : Type uι} {M₁ : ι → Type v₁} {M₂ : Type v₂} [CommSemiring R] [(i : ι) → AddCommMonoid (M₁ i)] [AddCommMonoid M₂] [(i : ι) → Module R (M₁ i)] [Module R M₂] {M₁' : ι → Type u_1} [(i : ι) → AddCommMonoid (M₁' i)] [(i : ι) → Module R (M₁' i)] (f : (i : ι) → M₁ i →ₗ[R] M₁' i) (g : MultilinearMap R M₁' M₂)
Goal: (MultilinearMap.compLinearMapₗ f) g = g.compLinearMap f
-- MultilinearMap.compLinearMapMultilinear_apply
@[simps] def compLinearMapMultilinear :
  @MultilinearMap R ι (fun i ↦ M₁ i →ₗ[R] M₁' i)
    ((MultilinearMap R M₁' M₂) →ₗ[R] MultilinearMap R M₁ M₂) _ _ _
      (fun i ↦ LinearMap.module) _ 
Arguments: {R : Type uR} {ι : Type uι} {M₁ : ι → Type v₁} {M₂ : Type v₂} [CommSemiring R] [(i : ι) → AddCommMonoid (M₁ i)] [AddCommMonoid M₂] [(i : ι) → Module R (M₁ i)] [Module R M₂] {M₁' : ι → Type u_1} [(i : ι) → AddCommMonoid (M₁' i)] [(i : ι) → Module R (M₁' i)] (f : (i : ι) → M₁ i →ₗ[R] M₁' i)
Goal: MultilinearMap.compLinearMapMultilinear f = MultilinearMap.compLinearMapₗ f
-- MultilinearMap.piLinearMap_apply_apply_apply
@[simps!] def piLinearMap :
    MultilinearMap R M₁' M₂ →ₗ[R]
    MultilinearMap R (fun i ↦ M₁ i →ₗ[R] M₁' i) (MultilinearMap R M₁ M₂) 
Arguments: {R : Type uR} {ι : Type uι} {M₁ : ι → Type v₁} {M₂ : Type v₂} [CommSemiring R] [(i : ι) → AddCommMonoid (M₁ i)] [AddCommMonoid M₂] [(i : ι) → Module R (M₁ i)] [Module R M₂] {M₁' : ι → Type u_1} [(i : ι) → AddCommMonoid (M₁' i)] [(i : ι) → Module R (M₁' i)] (g : MultilinearMap R M₁' M₂)
Goal: ∀ (a : (i : ι) → (fun i => M₁ i →ₗ[R] M₁' i) i) (m : (i : ι) → M₁ i),
  ((MultilinearMap.piLinearMap g) a) m = g fun i => (a i) (m i)
-- MultilinearMap.map_piecewise_smul
theorem map_piecewise_smul [DecidableEq ι] (c : ι → R) (m : ∀ i, M₁ i) (s : Finset ι) :
    f (s.piecewise (fun i => c i • m i) m) = (∏ i ∈ s, c i) • f m 
Arguments: {R : Type uR} {ι : Type uι} {M₁ : ι → Type v₁} {M₂ : Type v₂} [CommSemiring R] [(i : ι) → AddCommMonoid (M₁ i)] [AddCommMonoid M₂] [(i : ι) → Module R (M₁ i)] [Module R M₂] (f : MultilinearMap R M₁ M₂) [DecidableEq ι] (c : ι → R) (m : (i : ι) → M₁ i) (s : Finset ι)
Goal: f (s.piecewise (fun i => c i • m i) m) = (∏ i ∈ s, c i) • f m
-- MultilinearMap.map_smul_univ
theorem map_smul_univ [Fintype ι] (c : ι → R) (m : ∀ i, M₁ i) :
    (f fun i => c i • m i) = (∏ i, c i) • f m 
Arguments: {R : Type uR} {ι : Type uι} {M₁ : ι → Type v₁} {M₂ : Type v₂} [CommSemiring R] [(i : ι) → AddCommMonoid (M₁ i)] [AddCommMonoid M₂] [(i : ι) → Module R (M₁ i)] [Module R M₂] (f : MultilinearMap R M₁ M₂) [Fintype ι] (c : ι → R) (m : (i : ι) → M₁ i)
Goal: (f fun i => c i • m i) = (∏ i : ι, c i) • f m
-- MultilinearMap.map_update_smul
@[simp]
theorem map_update_smul [DecidableEq ι] [Fintype ι] (m : ∀ i, M₁ i) (i : ι) (c : R) (x : M₁ i) :
    f (update (c • m) i x) = c ^ (Fintype.card ι - 1) • f (update m i x) 
Arguments: {R : Type uR} {ι : Type uι} {M₁ : ι → Type v₁} {M₂ : Type v₂} [CommSemiring R] [(i : ι) → AddCommMonoid (M₁ i)] [AddCommMonoid M₂] [(i : ι) → Module R (M₁ i)] [Module R M₂] (f : MultilinearMap R M₁ M₂) [DecidableEq ι] [Fintype ι] (m : (i : ι) → M₁ i) (i : ι) (c : R) (x : M₁ i)
Goal: f (Function.update (c • m) i x) = c ^ (Fintype.card ι - 1) • f (Function.update m i x)
-- MultilinearMap.mkPiAlgebra_apply
@[simp]
theorem mkPiAlgebra_apply (m : ι → A) : MultilinearMap.mkPiAlgebra R ι A m = ∏ i, m i 
Arguments: {R : Type uR} {ι : Type uι} [CommSemiring R] {A : Type u_1} [CommSemiring A] [Algebra R A] [Fintype ι] (m : ι → A)
Goal: (MultilinearMap.mkPiAlgebra R ι A) m = ∏ i : ι, m i
-- MultilinearMap.mkPiAlgebraFin_apply
@[simp]
theorem mkPiAlgebraFin_apply (m : Fin n → A) :
    MultilinearMap.mkPiAlgebraFin R n A m = (List.ofFn m).prod 
Arguments: {R : Type uR} {n : ℕ} [CommSemiring R] {A : Type u_1} [Semiring A] [Algebra R A] (m : Fin n → A)
Goal: (MultilinearMap.mkPiAlgebraFin R n A) m = (List.ofFn m).prod
-- MultilinearMap.mkPiAlgebraFin_apply_const
theorem mkPiAlgebraFin_apply_const (a : A) :
    (MultilinearMap.mkPiAlgebraFin R n A fun _ => a) = a ^ n 
Arguments: {R : Type uR} {n : ℕ} [CommSemiring R] {A : Type u_1} [Semiring A] [Algebra R A] (a : A)
Goal: ((MultilinearMap.mkPiAlgebraFin R n A) fun x => a) = a ^ n
-- MultilinearMap.smulRight_apply
@[simp]
theorem smulRight_apply (f : MultilinearMap R M₁ R) (z : M₂) (m : ∀ i, M₁ i) :
    f.smulRight z m = f m • z 
Arguments: {R : Type uR} {ι : Type uι} {M₁ : ι → Type v₁} {M₂ : Type v₂} [CommSemiring R] [(i : ι) → AddCommMonoid (M₁ i)] [AddCommMonoid M₂] [(i : ι) → Module R (M₁ i)] [Module R M₂] (f : MultilinearMap R M₁ R) (z : M₂) (m : (i : ι) → M₁ i)
Goal: (f.smulRight z) m = f m • z
-- MultilinearMap.mkPiRing_apply
@[simp]
theorem mkPiRing_apply [Fintype ι] (z : M₂) (m : ι → R) :
    (MultilinearMap.mkPiRing R ι z : (ι → R) → M₂) m = (∏ i, m i) • z 
Arguments: {R : Type uR} {ι : Type uι} {M₂ : Type v₂} [CommSemiring R] [AddCommMonoid M₂] [Module R M₂] [Fintype ι] (z : M₂) (m : ι → R)
Goal: (MultilinearMap.mkPiRing R ι z) m = (∏ i : ι, m i) • z
-- MultilinearMap.mkPiRing_apply_one_eq_self
theorem mkPiRing_apply_one_eq_self [Fintype ι] (f : MultilinearMap R (fun _ : ι => R) M₂) :
    MultilinearMap.mkPiRing R ι (f fun _ => 1) = f 
Arguments: {R : Type uR} {ι : Type uι} {M₂ : Type v₂} [CommSemiring R] [AddCommMonoid M₂] [Module R M₂] [Fintype ι] (f : MultilinearMap R (fun x => R) M₂)
Goal: MultilinearMap.mkPiRing R ι (f fun x => 1) = f
-- MultilinearMap.mkPiRing_eq_iff
theorem mkPiRing_eq_iff [Fintype ι] {z₁ z₂ : M₂} :
    MultilinearMap.mkPiRing R ι z₁ = MultilinearMap.mkPiRing R ι z₂ ↔ z₁ = z₂ 
Arguments: {R : Type uR} {ι : Type uι} {M₂ : Type v₂} [CommSemiring R] [AddCommMonoid M₂] [Module R M₂] [Fintype ι] {z₁ z₂ : M₂}
Goal: MultilinearMap.mkPiRing R ι z₁ = MultilinearMap.mkPiRing R ι z₂ ↔ z₁ = z₂
-- MultilinearMap.mkPiRing_zero
theorem mkPiRing_zero [Fintype ι] : MultilinearMap.mkPiRing R ι (0 : M₂) = 0 
Arguments: {R : Type uR} {ι : Type uι} {M₂ : Type v₂} [CommSemiring R] [AddCommMonoid M₂] [Module R M₂] [Fintype ι]
Goal: MultilinearMap.mkPiRing R ι 0 = 0
-- MultilinearMap.mkPiRing_eq_zero_iff
theorem mkPiRing_eq_zero_iff [Fintype ι] (z : M₂) : MultilinearMap.mkPiRing R ι z = 0 ↔ z = 0 
Arguments: {R : Type uR} {ι : Type uι} {M₂ : Type v₂} [CommSemiring R] [AddCommMonoid M₂] [Module R M₂] [Fintype ι] (z : M₂)
Goal: MultilinearMap.mkPiRing R ι z = 0 ↔ z = 0