ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- Submodule.one_le_finrank_iff
@[simp]
lemma Submodule.one_le_finrank_iff [StrongRankCondition R] [NoZeroSMulDivisors R M]
    {S : Submodule R M} [Module.Finite R S] :
    1 ≤ finrank R S ↔ S ≠ ⊥ 
Arguments: {R : Type u} {M : Type v} [Ring R] [AddCommGroup M] [Module R M] [Nontrivial R] [StrongRankCondition R] [NoZeroSMulDivisors R M] {S : Submodule R M} [Module.Finite R ↥S]
Goal: 1 ≤ FiniteDimensional.finrank R ↥S ↔ S ≠ ⊥
-- finrank_eq_zero_of_basis_imp_not_finite
theorem finrank_eq_zero_of_basis_imp_not_finite
    (h : ∀ s : Set M, Basis.{v} (s : Set M) R M → ¬s.Finite) : finrank R M = 0 
Arguments: {R : Type u} {M : Type v} [Ring R] [AddCommGroup M] [Module R M] [Module.Free R M] (h : ∀ (s : Set M), Basis (↑s) R M → ¬s.Finite)
Goal: FiniteDimensional.finrank R M = 0
-- finrank_eq_zero_of_basis_imp_false
theorem finrank_eq_zero_of_basis_imp_false (h : ∀ s : Finset M, Basis.{v} (s : Set M) R M → False) :
    finrank R M = 0 
Arguments: {R : Type u} {M : Type v} [Ring R] [AddCommGroup M] [Module R M] [Module.Free R M] (h : ∀ (s : Finset M), Basis (↑↑s) R M → False)
Goal: FiniteDimensional.finrank R M = 0
-- finrank_eq_zero_of_not_exists_basis
theorem finrank_eq_zero_of_not_exists_basis
    (h : ¬∃ s : Finset M, Nonempty (Basis (s : Set M) R M)) : finrank R M = 0 
Arguments: {R : Type u} {M : Type v} [Ring R] [AddCommGroup M] [Module R M] [Module.Free R M] (h : ¬∃ s, Nonempty (Basis (↑↑s) R M))
Goal: FiniteDimensional.finrank R M = 0
-- finrank_eq_zero_of_not_exists_basis_finite
theorem finrank_eq_zero_of_not_exists_basis_finite
    (h : ¬∃ (s : Set M) (_ : Basis.{v} (s : Set M) R M), s.Finite) : finrank R M = 0 
Arguments: {R : Type u} {M : Type v} [Ring R] [AddCommGroup M] [Module R M] [Module.Free R M] (h : ¬∃ s x, s.Finite)
Goal: FiniteDimensional.finrank R M = 0
-- finrank_eq_zero_of_not_exists_basis_finset
theorem finrank_eq_zero_of_not_exists_basis_finset (h : ¬∃ s : Finset M, Nonempty (Basis s R M)) :
    finrank R M = 0 
Arguments: {R : Type u} {M : Type v} [Ring R] [AddCommGroup M] [Module R M] [Module.Free R M] (h : ¬∃ s, Nonempty (Basis { x // x ∈ s } R M))
Goal: FiniteDimensional.finrank R M = 0
-- rank_eq_one
theorem rank_eq_one (v : M) (n : v ≠ 0) (h : ∀ w : M, ∃ c : R, c • v = w) :
    Module.rank R M = 1 
Arguments: {R : Type u} {M : Type v} [Ring R] [AddCommGroup M] [Module R M] [NoZeroSMulDivisors R M] [StrongRankCondition R] (v : M) (n : v ≠ 0) (h : ∀ (w : M), ∃ c, c • v = w)
Goal: Module.rank R M = 1
-- finrank_eq_one
theorem finrank_eq_one (v : M) (n : v ≠ 0) (h : ∀ w : M, ∃ c : R, c • v = w) : finrank R M = 1 
Arguments: {R : Type u} {M : Type v} [Ring R] [AddCommGroup M] [Module R M] [NoZeroSMulDivisors R M] [StrongRankCondition R] (v : M) (n : v ≠ 0) (h : ∀ (w : M), ∃ c, c • v = w)
Goal: FiniteDimensional.finrank R M = 1
-- finrank_le_one
theorem finrank_le_one (v : M) (h : ∀ w : M, ∃ c : R, c • v = w) : finrank R M ≤ 1 
Arguments: {R : Type u} {M : Type v} [Ring R] [AddCommGroup M] [Module R M] [NoZeroSMulDivisors R M] [StrongRankCondition R] (v : M) (h : ∀ (w : M), ∃ c, c • v = w)
Goal: FiniteDimensional.finrank R M ≤ 1
-- TensorAlgebra.GradedAlgebra.ι_apply
theorem GradedAlgebra.ι_apply (m : M) :
    GradedAlgebra.ι R M m =
      DirectSum.of (fun (i : ℕ) => ↥(LinearMap.range (TensorAlgebra.ι R : M →ₗ[_] _) ^ i)) 1
        ⟨TensorAlgebra.ι R m, by simpa only [pow_one] using LinearMap.mem_range_self _ m⟩ 
Arguments: (R : Type u_1) (M : Type u_2) [CommSemiring R] [AddCommMonoid M] [Module R M] (m : M)
Goal: (TensorAlgebra.GradedAlgebra.ι R M) m =
  (DirectSum.of (fun i => ↥(LinearMap.range (TensorAlgebra.ι R) ^ i)) 1) ⟨(TensorAlgebra.ι R) m, ⋯⟩
-- MultilinearMap.map_add'
structure MultilinearMap (R : Type uR) {ι : Type uι} (M₁ : ι → Type v₁) (M₂ : Type v₂) [Semiring R]
  [∀ i, AddCommMonoid (M₁ i)] [AddCommMonoid M₂] [∀ i, Module R (M₁ i)] [Module R M₂] where
    toFun : (∀ i, M₁ i) → M₂
    map_add' :
    ∀ [DecidableEq ι] (m : ∀ i, M₁ i) (i : ι) (x y : M₁ i),
      toFun (update m i (x + y)) = toFun (update m i x) + toFun (update m i y)
    map_smul' :
    ∀ [DecidableEq ι] (m : ∀ i, M₁ i) (i : ι) (c : R) (x : M₁ i),
      toFun (update m i (c • x)) = c • toFun (update m i x)
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] (self : MultilinearMap R M₁ M₂) [DecidableEq ι] (m : (i : ι) → M₁ i) (i : ι) (x y : M₁ i)
Goal: self.toFun (Function.update m i (x + y)) = self.toFun (Function.update m i x) + self.toFun (Function.update m i y)
-- MultilinearMap.map_smul'
structure MultilinearMap (R : Type uR) {ι : Type uι} (M₁ : ι → Type v₁) (M₂ : Type v₂) [Semiring R]
  [∀ i, AddCommMonoid (M₁ i)] [AddCommMonoid M₂] [∀ i, Module R (M₁ i)] [Module R M₂] where
    toFun : (∀ i, M₁ i) → M₂
    map_add' :
    ∀ [DecidableEq ι] (m : ∀ i, M₁ i) (i : ι) (x y : M₁ i),
      toFun (update m i (x + y)) = toFun (update m i x) + toFun (update m i y)
    map_smul' :
    ∀ [DecidableEq ι] (m : ∀ i, M₁ i) (i : ι) (c : R) (x : M₁ i),
      toFun (update m i (c • x)) = c • toFun (update m i x)
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] (self : MultilinearMap R M₁ M₂) [DecidableEq ι] (m : (i : ι) → M₁ i) (i : ι) (c : R) (x : M₁ i)
Goal: self.toFun (Function.update m i (c • x)) = c • self.toFun (Function.update m i x)
-- MultilinearMap.toFun_eq_coe
@[simp]
theorem toFun_eq_coe : f.toFun = ⇑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)
Goal: f.toFun = ⇑f
-- MultilinearMap.coe_mk
@[simp]
theorem coe_mk (f : (∀ i, M₁ i) → M₂) (h₁ h₂) : ⇑(⟨f, h₁, h₂⟩ : MultilinearMap R M₁ M₂) = 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 : ((i : ι) → M₁ i) → M₂) (h₁ :
  ∀ [inst : DecidableEq ι] (m : (i : ι) → M₁ i) (i : ι) (x y : M₁ i),
    f (Function.update m i (x + y)) = f (Function.update m i x) + f (Function.update m i y)) (h₂ :
  ∀ [inst : DecidableEq ι] (m : (i : ι) → M₁ i) (i : ι) (c : R) (x : M₁ i),
    f (Function.update m i (c • x)) = c • f (Function.update m i x))
Goal: ⇑{ toFun := f, map_add' := h₁, map_smul' := h₂ } = f
-- MultilinearMap.congr_fun
theorem congr_fun {f g : MultilinearMap R M₁ M₂} (h : f = g) (x : ∀ i, M₁ i) : f x = g x 
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 g : MultilinearMap R M₁ M₂} (h : f = g) (x : (i : ι) → M₁ i)
Goal: f x = g x
-- MultilinearMap.congr_arg
nonrec theorem congr_arg (f : MultilinearMap R M₁ M₂) {x y : ∀ i, M₁ i} (h : x = y) : f x = f y 
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) {x y : (i : ι) → M₁ i} (h : x = y)
Goal: f x = f y
-- MultilinearMap.coe_injective
theorem coe_injective : Injective ((↑) : MultilinearMap R M₁ M₂ → (∀ i, M₁ i) → M₂) 
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]
Goal: Function.Injective DFunLike.coe
-- MultilinearMap.coe_inj
@[norm_cast] -- Porting note (#10618): Removed simp attribute, simp can prove this
theorem coe_inj {f g : MultilinearMap R M₁ M₂} : (f : (∀ i, M₁ i) → M₂) = g ↔ f = g 
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 g : MultilinearMap R M₁ M₂}
Goal: ⇑f = ⇑g ↔ f = g
-- MultilinearMap.ext
@[ext]
theorem ext {f f' : MultilinearMap R M₁ M₂} (H : ∀ x, f x = f' x) : f = 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 f' : MultilinearMap R M₁ M₂} (H : ∀ (x : (i : ι) → M₁ i), f x = f' x)
Goal: f = f'
-- MultilinearMap.ext_iff
protected theorem ext_iff {f g : MultilinearMap R M₁ M₂} : f = g ↔ ∀ x, f x = g x 
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 g : MultilinearMap R M₁ M₂}
Goal: f = g ↔ ∀ (x : (i : ι) → M₁ i), f x = g x
-- MultilinearMap.mk_coe
@[simp]
theorem mk_coe (f : MultilinearMap R M₁ M₂) (h₁ h₂) :
    (⟨f, h₁, h₂⟩ : MultilinearMap R M₁ M₂) = 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) (h₁ :
  ∀ [inst : DecidableEq ι] (m : (i : ι) → M₁ i) (i : ι) (x y : M₁ i),
    f (Function.update m i (x + y)) = f (Function.update m i x) + f (Function.update m i y)) (h₂ :
  ∀ [inst : DecidableEq ι] (m : (i : ι) → M₁ i) (i : ι) (c : R) (x : M₁ i),
    f (Function.update m i (c • x)) = c • f (Function.update m i x))
Goal: { toFun := ⇑f, map_add' := h₁, map_smul' := h₂ } = f
-- MultilinearMap.map_add
@[simp]
protected theorem map_add [DecidableEq ι] (m : ∀ i, M₁ i) (i : ι) (x y : M₁ i) :
    f (update m i (x + y)) = f (update m i x) + f (update m i y) 
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] (m : (i : ι) → M₁ i) (i : ι) (x y : M₁ i)
Goal: f (Function.update m i (x + y)) = f (Function.update m i x) + f (Function.update m i y)
-- MultilinearMap.map_smul
@[simp]
protected theorem map_smul [DecidableEq ι] (m : ∀ i, M₁ i) (i : ι) (c : R) (x : M₁ i) :
    f (update m i (c • x)) = c • f (update m i x) 
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] (m : (i : ι) → M₁ i) (i : ι) (c : R) (x : M₁ i)
Goal: f (Function.update m i (c • x)) = c • f (Function.update m i x)
-- MultilinearMap.map_coord_zero
theorem map_coord_zero {m : ∀ i, M₁ i} (i : ι) (h : m i = 0) : f m = 0 
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) {m : (i : ι) → M₁ i} (i : ι) (h : m i = 0)
Goal: f m = 0
-- MultilinearMap.map_update_zero
@[simp]
theorem map_update_zero [DecidableEq ι] (m : ∀ i, M₁ i) (i : ι) : f (update m i 0) = 0 
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] (m : (i : ι) → M₁ i) (i : ι)
Goal: f (Function.update m i 0) = 0
-- MultilinearMap.map_zero
@[simp]
theorem map_zero [Nonempty ι] : f 0 = 0 
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) [Nonempty ι]
Goal: f 0 = 0
-- MultilinearMap.add_apply
@[simp]
theorem add_apply (m : ∀ i, M₁ i) : (f + f') m = f m + f' 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 f' : MultilinearMap R M₁ M₂) (m : (i : ι) → M₁ i)
Goal: (f + f') m = f m + f' m
-- MultilinearMap.zero_apply
@[simp]
theorem zero_apply (m : ∀ i, M₁ i) : (0 : MultilinearMap R M₁ M₂) m = 0 
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] (m : (i : ι) → M₁ i)
Goal: 0 m = 0
-- MultilinearMap.smul_apply
@[simp]
theorem smul_apply (f : MultilinearMap A M₁ M₂) (c : R') (m : ∀ i, M₁ i) : (c • f) m = c • f m 
Arguments: {ι : Type uι} {M₁ : ι → Type v₁} {M₂ : Type v₂} [(i : ι) → AddCommMonoid (M₁ i)] [AddCommMonoid M₂] {R' : Type u_1} {A : Type u_2} [Monoid R'] [Semiring A] [(i : ι) → Module A (M₁ i)] [DistribMulAction R' M₂] [Module A M₂] [SMulCommClass A R' M₂] (f : MultilinearMap A M₁ M₂) (c : R') (m : (i : ι) → M₁ i)
Goal: (c • f) m = c • f m
-- MultilinearMap.coe_smul
theorem coe_smul (c : R') (f : MultilinearMap A M₁ M₂) : ⇑(c • f) = c • (⇑ f) 
Arguments: {ι : Type uι} {M₁ : ι → Type v₁} {M₂ : Type v₂} [(i : ι) → AddCommMonoid (M₁ i)] [AddCommMonoid M₂] {R' : Type u_1} {A : Type u_2} [Monoid R'] [Semiring A] [(i : ι) → Module A (M₁ i)] [DistribMulAction R' M₂] [Module A M₂] [SMulCommClass A R' M₂] (c : R') (f : MultilinearMap A M₁ M₂)
Goal: ⇑(c • f) = c • ⇑f
-- MultilinearMap.coeAddMonoidHom_apply
@[simps] def coeAddMonoidHom : MultilinearMap R M₁ M₂ →+ (((i : ι) → M₁ i) → M₂) 
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]
Goal: ∀ (a : MultilinearMap R M₁ M₂) (a_1 : (i : ι) → M₁ i), MultilinearMap.coeAddMonoidHom a a_1 = a a_1
-- MultilinearMap.coe_sum
@[simp]
theorem coe_sum {α : Type*} (f : α → MultilinearMap R M₁ M₂) (s : Finset α) :
    ⇑(∑ a ∈ s, f a) = ∑ a ∈ s, ⇑(f 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] {α : Type u_1} (f : α → MultilinearMap R M₁ M₂) (s : Finset α)
Goal: ⇑(∑ a ∈ s, f a) = ∑ a ∈ s, ⇑(f a)
-- MultilinearMap.sum_apply
theorem sum_apply {α : Type*} (f : α → MultilinearMap R M₁ M₂) (m : ∀ i, M₁ i) {s : Finset α} :
    (∑ a ∈ s, f a) m = ∑ a ∈ s, f a 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] {α : Type u_1} (f : α → MultilinearMap R M₁ M₂) (m : (i : ι) → M₁ i) {s : Finset α}
Goal: (∑ a ∈ s, f a) m = ∑ a ∈ s, (f a) m
-- MultilinearMap.toLinearMap_apply
@[simps]
def toLinearMap [DecidableEq ι] (m : ∀ i, M₁ i) (i : ι) : M₁ i →ₗ[R] M₂ 
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] (m : (i : ι) → M₁ i) (i : ι) (x : M₁ i)
Goal: (f.toLinearMap m i) x = f (Function.update m i x)
-- MultilinearMap.prod_apply
@[simps]
def prod (f : MultilinearMap R M₁ M₂) (g : MultilinearMap R M₁ M₃) :
    MultilinearMap R M₁ (M₂ × M₃) 
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] (f : MultilinearMap R M₁ M₂) (g : MultilinearMap R M₁ M₃) (m : (i : ι) → M₁ i)
Goal: (f.prod g) m = (f m, g m)
-- MultilinearMap.pi_apply
@[simps]
def pi {ι' : Type*} {M' : ι' → Type*} [∀ i, AddCommMonoid (M' i)] [∀ i, Module R (M' i)]
    (f : ∀ i, MultilinearMap R M₁ (M' i)) : MultilinearMap R M₁ (∀ i, M' i) 
Arguments: {R : Type uR} {ι : Type uι} {M₁ : ι → Type v₁} [Semiring R] [(i : ι) → AddCommMonoid (M₁ i)] [(i : ι) → Module R (M₁ i)] {ι' : Type u_1} {M' : ι' → Type u_2} [(i : ι') → AddCommMonoid (M' i)] [(i : ι') → Module R (M' i)] (f : (i : ι') → MultilinearMap R M₁ (M' i)) (m : (i : ι) → M₁ i) (i : ι')
Goal: (MultilinearMap.pi f) m i = (f i) m
-- MultilinearMap.ofSubsingleton_apply_apply
@[simps]
def ofSubsingleton [Subsingleton ι] (i : ι) :
    (M₂ →ₗ[R] M₃) ≃ MultilinearMap R (fun _ : ι ↦ M₂) M₃ 
Arguments: (R : Type uR) {ι : Type uι} (M₂ : Type v₂) (M₃ : Type v₃) [Semiring R] [AddCommMonoid M₂] [AddCommMonoid M₃] [Module R M₂] [Module R M₃] [Subsingleton ι] (i : ι) (f : M₂ →ₗ[R] M₃) (x : ι → M₂)
Goal: ((MultilinearMap.ofSubsingleton R M₂ M₃ i) f) x = f (x i)
-- MultilinearMap.ofSubsingleton_symm_apply_apply
@[simps]
def ofSubsingleton [Subsingleton ι] (i : ι) :
    (M₂ →ₗ[R] M₃) ≃ MultilinearMap R (fun _ : ι ↦ M₂) M₃ 
Arguments: (R : Type uR) {ι : Type uι} (M₂ : Type v₂) (M₃ : Type v₃) [Semiring R] [AddCommMonoid M₂] [AddCommMonoid M₃] [Module R M₂] [Module R M₃] [Subsingleton ι] (i : ι) (f : MultilinearMap R (fun x => M₂) M₃) (x : M₂)
Goal: ((MultilinearMap.ofSubsingleton R M₂ M₃ i).symm f) x = f fun x_1 => x
-- MultilinearMap.constOfIsEmpty_apply
-- Porting note: Removed [simps] & added simpNF-approved version of the generated lemma manually.
@[simps (config := .asFn)]
def constOfIsEmpty [IsEmpty ι] (m : M₂) : MultilinearMap R M₁ M₂ 
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] [IsEmpty ι] (m : M₂)
Goal: ⇑(MultilinearMap.constOfIsEmpty R M₁ m) = Function.const ((i : ι) → M₁ i) m
-- MultilinearMap.cons_add
theorem cons_add (f : MultilinearMap R M M₂) (m : ∀ i : Fin n, M i.succ) (x y : M 0) :
    f (cons (x + y) m) = f (cons x m) + f (cons y m) 
Arguments: {R : Type uR} {n : ℕ} {M : Fin n.succ → Type v} {M₂ : Type v₂} [Semiring R] [(i : Fin n.succ) → AddCommMonoid (M i)] [AddCommMonoid M₂] [(i : Fin n.succ) → Module R (M i)] [Module R M₂] (f : MultilinearMap R M M₂) (m : (i : Fin n) → M i.succ) (x y : M 0)
Goal: f (Fin.cons (x + y) m) = f (Fin.cons x m) + f (Fin.cons y m)
-- MultilinearMap.cons_smul
theorem cons_smul (f : MultilinearMap R M M₂) (m : ∀ i : Fin n, M i.succ) (c : R) (x : M 0) :
    f (cons (c • x) m) = c • f (cons x m) 
Arguments: {R : Type uR} {n : ℕ} {M : Fin n.succ → Type v} {M₂ : Type v₂} [Semiring R] [(i : Fin n.succ) → AddCommMonoid (M i)] [AddCommMonoid M₂] [(i : Fin n.succ) → Module R (M i)] [Module R M₂] (f : MultilinearMap R M M₂) (m : (i : Fin n) → M i.succ) (c : R) (x : M 0)
Goal: f (Fin.cons (c • x) m) = c • f (Fin.cons x m)
-- MultilinearMap.snoc_add
theorem snoc_add (f : MultilinearMap R M M₂)
    (m : ∀ i : Fin n, M (castSucc i)) (x y : M (last n)) :
    f (snoc m (x + y)) = f (snoc m x) + f (snoc m y) 
Arguments: {R : Type uR} {n : ℕ} {M : Fin n.succ → Type v} {M₂ : Type v₂} [Semiring R] [(i : Fin n.succ) → AddCommMonoid (M i)] [AddCommMonoid M₂] [(i : Fin n.succ) → Module R (M i)] [Module R M₂] (f : MultilinearMap R M M₂) (m : (i : Fin n) → M i.castSucc) (x y : M (Fin.last n))
Goal: f (Fin.snoc m (x + y)) = f (Fin.snoc m x) + f (Fin.snoc m y)
-- MultilinearMap.snoc_smul
theorem snoc_smul (f : MultilinearMap R M M₂) (m : ∀ i : Fin n, M (castSucc i)) (c : R)
    (x : M (last n)) : f (snoc m (c • x)) = c • f (snoc m x) 
Arguments: {R : Type uR} {n : ℕ} {M : Fin n.succ → Type v} {M₂ : Type v₂} [Semiring R] [(i : Fin n.succ) → AddCommMonoid (M i)] [AddCommMonoid M₂] [(i : Fin n.succ) → Module R (M i)] [Module R M₂] (f : MultilinearMap R M M₂) (m : (i : Fin n) → M i.castSucc) (c : R) (x : M (Fin.last n))
Goal: f (Fin.snoc m (c • x)) = c • f (Fin.snoc m x)
-- MultilinearMap.compLinearMap_apply
@[simp]
theorem compLinearMap_apply (g : MultilinearMap R M₁' M₂) (f : ∀ i, M₁ i →ₗ[R] M₁' i)
    (m : ∀ i, M₁ i) : g.compLinearMap f m = g fun i => f i (m 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] {M₁' : ι → Type u_1} [(i : ι) → AddCommMonoid (M₁' i)] [(i : ι) → Module R (M₁' i)] (g : MultilinearMap R M₁' M₂) (f : (i : ι) → M₁ i →ₗ[R] M₁' i) (m : (i : ι) → M₁ i)
Goal: (g.compLinearMap f) m = g fun i => (f i) (m i)
-- MultilinearMap.compLinearMap_assoc
theorem compLinearMap_assoc (g : MultilinearMap R M₁'' M₂) (f₁ : ∀ i, M₁' i →ₗ[R] M₁'' i)
    (f₂ : ∀ i, M₁ i →ₗ[R] M₁' i) :
    (g.compLinearMap f₁).compLinearMap f₂ = g.compLinearMap fun i => f₁ i ∘ₗ f₂ 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] {M₁' : ι → Type u_1} [(i : ι) → AddCommMonoid (M₁' i)] [(i : ι) → Module R (M₁' i)] {M₁'' : ι → Type u_2} [(i : ι) → AddCommMonoid (M₁'' i)] [(i : ι) → Module R (M₁'' i)] (g : MultilinearMap R M₁'' M₂) (f₁ : (i : ι) → M₁' i →ₗ[R] M₁'' i) (f₂ : (i : ι) → M₁ i →ₗ[R] M₁' i)
Goal: (g.compLinearMap f₁).compLinearMap f₂ = g.compLinearMap fun i => f₁ i ∘ₗ f₂ i
-- MultilinearMap.zero_compLinearMap
@[simp]
theorem zero_compLinearMap (f : ∀ i, M₁ i →ₗ[R] M₁' i) :
    (0 : MultilinearMap R M₁' M₂).compLinearMap f = 0 
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] {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.compLinearMap 0 f = 0
-- MultilinearMap.compLinearMap_id
@[simp]
theorem compLinearMap_id (g : MultilinearMap R M₁' M₂) :
    (g.compLinearMap fun _ => LinearMap.id) = g 
Arguments: {R : Type uR} {ι : Type uι} {M₂ : Type v₂} [Semiring R] [AddCommMonoid M₂] [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: (g.compLinearMap fun x => LinearMap.id) = g
-- MultilinearMap.compLinearMap_injective
theorem compLinearMap_injective (f : ∀ i, M₁ i →ₗ[R] M₁' i) (hf : ∀ i, Surjective (f i)) :
    Injective fun g : MultilinearMap R M₁' M₂ => g.compLinearMap 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] {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) (hf : ∀ (i : ι), Function.Surjective ⇑(f i))
Goal: Function.Injective fun g => g.compLinearMap f
-- MultilinearMap.compLinearMap_inj
theorem compLinearMap_inj (f : ∀ i, M₁ i →ₗ[R] M₁' i) (hf : ∀ i, Surjective (f i))
    (g₁ g₂ : MultilinearMap R M₁' M₂) : g₁.compLinearMap f = g₂.compLinearMap f ↔ g₁ = g₂ 
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] {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) (hf : ∀ (i : ι), Function.Surjective ⇑(f i)) (g₁ g₂ : MultilinearMap R M₁' M₂)
Goal: g₁.compLinearMap f = g₂.compLinearMap f ↔ g₁ = g₂
-- MultilinearMap.comp_linearEquiv_eq_zero_iff
@[simp]
theorem comp_linearEquiv_eq_zero_iff (g : MultilinearMap R M₁' M₂) (f : ∀ i, M₁ i ≃ₗ[R] M₁' i) :
    (g.compLinearMap fun i => (f i : M₁ i →ₗ[R] M₁' i)) = 0 ↔ g = 0 
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] {M₁' : ι → Type u_1} [(i : ι) → AddCommMonoid (M₁' i)] [(i : ι) → Module R (M₁' i)] (g : MultilinearMap R M₁' M₂) (f : (i : ι) → M₁ i ≃ₗ[R] M₁' i)
Goal: (g.compLinearMap fun i => ↑(f i)) = 0 ↔ g = 0
-- MultilinearMap.map_piecewise_add
theorem map_piecewise_add [DecidableEq ι] (m m' : ∀ i, M₁ i) (t : Finset ι) :
    f (t.piecewise (m + m') m') = ∑ s ∈ t.powerset, 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] (m m' : (i : ι) → M₁ i) (t : Finset ι)
Goal: f (t.piecewise (m + m') m') = ∑ s ∈ t.powerset, f (s.piecewise m m')