ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- Matrix.Fin.circulant_inj
@[simp]
theorem Fin.circulant_inj {n} {v w : Fin n → α} : circulant v = circulant w ↔ v = w 
Arguments: {α : Type u_1} {n : ℕ} {v w : Fin n → α}
Goal: Matrix.circulant v = Matrix.circulant w ↔ v = w
-- Matrix.transpose_circulant
theorem transpose_circulant [AddGroup n] (v : n → α) :
    (circulant v)ᵀ = circulant fun i => v (-i) 
Arguments: {α : Type u_1} {n : Type u_4} [AddGroup n] (v : n → α)
Goal: (Matrix.circulant v).transpose = Matrix.circulant fun i => v (-i)
-- Matrix.conjTranspose_circulant
theorem conjTranspose_circulant [Star α] [AddGroup n] (v : n → α) :
    (circulant v)ᴴ = circulant (star fun i => v (-i)) 
Arguments: {α : Type u_1} {n : Type u_4} [Star α] [AddGroup n] (v : n → α)
Goal: (Matrix.circulant v).conjTranspose = Matrix.circulant (star fun i => v (-i))
-- Matrix.Fin.transpose_circulant
theorem Fin.transpose_circulant : ∀ {n} (v : Fin n → α), (circulant v)ᵀ = circulant fun i => v (-i)
  
Arguments: {α : Type u_1} {n : ℕ} (v : Fin n → α)
Goal: (Matrix.circulant v).transpose = Matrix.circulant fun i => v (-i)
-- Matrix.Fin.conjTranspose_circulant
theorem Fin.conjTranspose_circulant [Star α] :
    ∀ {n} (v : Fin n → α), (circulant v)ᴴ = circulant (star fun i => v (-i))
  
Arguments: {α : Type u_1} [Star α] {n : ℕ} (v : Fin n → α)
Goal: (Matrix.circulant v).conjTranspose = Matrix.circulant (star fun i => v (-i))
-- Matrix.map_circulant
theorem map_circulant [Sub n] (v : n → α) (f : α → β) :
    (circulant v).map f = circulant fun i => f (v i) 
Arguments: {α : Type u_1} {β : Type u_2} {n : Type u_4} [Sub n] (v : n → α) (f : α → β)
Goal: (Matrix.circulant v).map f = Matrix.circulant fun i => f (v i)
-- Matrix.circulant_neg
theorem circulant_neg [Neg α] [Sub n] (v : n → α) : circulant (-v) = -circulant v 
Arguments: {α : Type u_1} {n : Type u_4} [Neg α] [Sub n] (v : n → α)
Goal: Matrix.circulant (-v) = -Matrix.circulant v
-- Matrix.circulant_zero
@[simp]
theorem circulant_zero (α n) [Zero α] [Sub n] : circulant 0 = (0 : Matrix n n α) 
Arguments: (α : Type u_6) (n : Type u_7) [Zero α] [Sub n]
Goal: Matrix.circulant 0 = 0
-- Matrix.circulant_add
theorem circulant_add [Add α] [Sub n] (v w : n → α) :
    circulant (v + w) = circulant v + circulant w 
Arguments: {α : Type u_1} {n : Type u_4} [Add α] [Sub n] (v w : n → α)
Goal: Matrix.circulant (v + w) = Matrix.circulant v + Matrix.circulant w
-- Matrix.circulant_sub
theorem circulant_sub [Sub α] [Sub n] (v w : n → α) :
    circulant (v - w) = circulant v - circulant w 
Arguments: {α : Type u_1} {n : Type u_4} [Sub α] [Sub n] (v w : n → α)
Goal: Matrix.circulant (v - w) = Matrix.circulant v - Matrix.circulant w
-- Matrix.circulant_mul
theorem circulant_mul [Semiring α] [Fintype n] [AddGroup n] (v w : n → α) :
    circulant v * circulant w = circulant (circulant v *ᵥ w) 
Arguments: {α : Type u_1} {n : Type u_4} [Semiring α] [Fintype n] [AddGroup n] (v w : n → α)
Goal: Matrix.circulant v * Matrix.circulant w = Matrix.circulant ((Matrix.circulant v).mulVec w)
-- Matrix.Fin.circulant_mul
theorem Fin.circulant_mul [Semiring α] :
    ∀ {n} (v w : Fin n → α), circulant v * circulant w = circulant (circulant v *ᵥ w)
  
Arguments: {α : Type u_1} [Semiring α] {n : ℕ} (v w : Fin n → α)
Goal: Matrix.circulant v * Matrix.circulant w = Matrix.circulant ((Matrix.circulant v).mulVec w)
-- Matrix.circulant_mul_comm
theorem circulant_mul_comm [CommSemigroup α] [AddCommMonoid α] [Fintype n] [AddCommGroup n]
    (v w : n → α) : circulant v * circulant w = circulant w * circulant v 
Arguments: {α : Type u_1} {n : Type u_4} [CommSemigroup α] [AddCommMonoid α] [Fintype n] [AddCommGroup n] (v w : n → α)
Goal: Matrix.circulant v * Matrix.circulant w = Matrix.circulant w * Matrix.circulant v
-- Matrix.Fin.circulant_mul_comm
theorem Fin.circulant_mul_comm [CommSemigroup α] [AddCommMonoid α] :
    ∀ {n} (v w : Fin n → α), circulant v * circulant w = circulant w * circulant v
  
Arguments: {α : Type u_1} [CommSemigroup α] [AddCommMonoid α] {n : ℕ} (v w : Fin n → α)
Goal: Matrix.circulant v * Matrix.circulant w = Matrix.circulant w * Matrix.circulant v
-- Matrix.circulant_smul
theorem circulant_smul [Sub n] [SMul R α] (k : R) (v : n → α) :
    circulant (k • v) = k • circulant v 
Arguments: {α : Type u_1} {n : Type u_4} {R : Type u_5} [Sub n] [SMul R α] (k : R) (v : n → α)
Goal: Matrix.circulant (k • v) = k • Matrix.circulant v
-- Matrix.circulant_single_one
@[simp]
theorem circulant_single_one (α n) [Zero α] [One α] [DecidableEq n] [AddGroup n] :
    circulant (Pi.single 0 1 : n → α) = (1 : Matrix n n α) 
Arguments: (α : Type u_6) (n : Type u_7) [Zero α] [One α] [DecidableEq n] [AddGroup n]
Goal: Matrix.circulant (Pi.single 0 1) = 1
-- Matrix.circulant_single
@[simp]
theorem circulant_single (n) [Semiring α] [DecidableEq n] [AddGroup n] [Fintype n] (a : α) :
    circulant (Pi.single 0 a : n → α) = scalar n a 
Arguments: {α : Type u_1} (n : Type u_6) [Semiring α] [DecidableEq n] [AddGroup n] [Fintype n] (a : α)
Goal: Matrix.circulant (Pi.single 0 a) = (Matrix.scalar n) a
-- Matrix.Fin.circulant_ite
theorem Fin.circulant_ite (α) [Zero α] [One α] :
    ∀ n, circulant (fun i => ite (i.1 = 0) 1 0 : Fin n → α) = 1
  
Arguments: (α : Type u_6) [Zero α] [One α] (n : ℕ)
Goal: (Matrix.circulant fun i => if ↑i = 0 then 1 else 0) = 1
-- Matrix.circulant_isSymm_iff
theorem circulant_isSymm_iff [AddGroup n] {v : n → α} :
    (circulant v).IsSymm ↔ ∀ i, v (-i) = v i 
Arguments: {α : Type u_1} {n : Type u_4} [AddGroup n] {v : n → α}
Goal: (Matrix.circulant v).IsSymm ↔ ∀ (i : n), v (-i) = v i
-- Matrix.Fin.circulant_isSymm_iff
theorem Fin.circulant_isSymm_iff : ∀ {n} {v : Fin n → α}, (circulant v).IsSymm ↔ ∀ i, v (-i) = v i
  
Arguments: {α : Type u_1} {n : ℕ} {v : Fin n → α}
Goal: (Matrix.circulant v).IsSymm ↔ ∀ (i : Fin n), v (-i) = v i
-- Matrix.circulant_isSymm_apply
theorem circulant_isSymm_apply [AddGroup n] {v : n → α} (h : (circulant v).IsSymm) (i : n) :
    v (-i) = v i 
Arguments: {α : Type u_1} {n : Type u_4} [AddGroup n] {v : n → α} (h : (Matrix.circulant v).IsSymm) (i : n)
Goal: v (-i) = v i
-- Matrix.Fin.circulant_isSymm_apply
theorem Fin.circulant_isSymm_apply {n} {v : Fin n → α} (h : (circulant v).IsSymm) (i : Fin n) :
    v (-i) = v i 
Arguments: {α : Type u_1} {n : ℕ} {v : Fin n → α} (h : (Matrix.circulant v).IsSymm) (i : Fin n)
Goal: v (-i) = v i
-- linearIndependent_iff
theorem linearIndependent_iff :
    LinearIndependent R v ↔ ∀ l, Finsupp.total ι M R v l = 0 → l = 0 
Arguments: {ι : Type u'} {R : Type u_2} {M : Type u_4} {v : ι → M} [Semiring R] [AddCommMonoid M] [Module R M]
Goal: LinearIndependent R v ↔ ∀ (l : ι →₀ R), (Finsupp.total ι M R v) l = 0 → l = 0
-- linearIndependent_iff'
theorem linearIndependent_iff' :
    LinearIndependent R v ↔
      ∀ s : Finset ι, ∀ g : ι → R, ∑ i ∈ s, g i • v i = 0 → ∀ i ∈ s, g i = 0 
Arguments: {ι : Type u'} {R : Type u_2} {M : Type u_4} {v : ι → M} [Semiring R] [AddCommMonoid M] [Module R M]
Goal: LinearIndependent R v ↔ ∀ (s : Finset ι) (g : ι → R), ∑ i ∈ s, g i • v i = 0 → ∀ i ∈ s, g i = 0
-- linearIndependent_iff''
theorem linearIndependent_iff'' :
    LinearIndependent R v ↔
      ∀ (s : Finset ι) (g : ι → R), (∀ i ∉ s, g i = 0) →
        ∑ i ∈ s, g i • v i = 0 → ∀ i, g i = 0 
Arguments: {ι : Type u'} {R : Type u_2} {M : Type u_4} {v : ι → M} [Semiring R] [AddCommMonoid M] [Module R M]
Goal: LinearIndependent R v ↔ ∀ (s : Finset ι) (g : ι → R), (∀ i ∉ s, g i = 0) → ∑ i ∈ s, g i • v i = 0 → ∀ (i : ι), g i = 0
-- not_linearIndependent_iff
theorem not_linearIndependent_iff :
    ¬LinearIndependent R v ↔
      ∃ s : Finset ι, ∃ g : ι → R, ∑ i ∈ s, g i • v i = 0 ∧ ∃ i ∈ s, g i ≠ 0 
Arguments: {ι : Type u'} {R : Type u_2} {M : Type u_4} {v : ι → M} [Semiring R] [AddCommMonoid M] [Module R M]
Goal: ¬LinearIndependent R v ↔ ∃ s g, ∑ i ∈ s, g i • v i = 0 ∧ ∃ i ∈ s, g i ≠ 0
-- Fintype.linearIndependent_iff
theorem Fintype.linearIndependent_iff [Fintype ι] :
    LinearIndependent R v ↔ ∀ g : ι → R, ∑ i, g i • v i = 0 → ∀ i, g i = 0 
Arguments: {ι : Type u'} {R : Type u_2} {M : Type u_4} {v : ι → M} [Semiring R] [AddCommMonoid M] [Module R M] [Fintype ι]
Goal: LinearIndependent R v ↔ ∀ (g : ι → R), ∑ i : ι, g i • v i = 0 → ∀ (i : ι), g i = 0
-- Fintype.linearIndependent_iff'
theorem Fintype.linearIndependent_iff' [Fintype ι] [DecidableEq ι] :
    LinearIndependent R v ↔
      LinearMap.ker (LinearMap.lsum R (fun _ ↦ R) ℕ fun i ↦ LinearMap.id.smulRight (v i)) = ⊥ 
Arguments: {ι : Type u'} {R : Type u_2} {M : Type u_4} {v : ι → M} [Semiring R] [AddCommMonoid M] [Module R M] [Fintype ι] [DecidableEq ι]
Goal: LinearIndependent R v ↔ LinearMap.ker ((LinearMap.lsum R (fun x => R) ℕ) fun i => LinearMap.id.smulRight (v i)) = ⊥
-- Fintype.not_linearIndependent_iff
theorem Fintype.not_linearIndependent_iff [Fintype ι] :
    ¬LinearIndependent R v ↔ ∃ g : ι → R, ∑ i, g i • v i = 0 ∧ ∃ i, g i ≠ 0 
Arguments: {ι : Type u'} {R : Type u_2} {M : Type u_4} {v : ι → M} [Semiring R] [AddCommMonoid M] [Module R M] [Fintype ι]
Goal: ¬LinearIndependent R v ↔ ∃ g, ∑ i : ι, g i • v i = 0 ∧ ∃ i, g i ≠ 0
-- linearIndependent_empty_type
theorem linearIndependent_empty_type [IsEmpty ι] : LinearIndependent R v 
Arguments: {ι : Type u'} {R : Type u_2} {M : Type u_4} {v : ι → M} [Semiring R] [AddCommMonoid M] [Module R M] [IsEmpty ι]
Goal: LinearIndependent R v
-- LinearIndependent.ne_zero
theorem LinearIndependent.ne_zero [Nontrivial R] (i : ι) (hv : LinearIndependent R v) : v i ≠ 0 
Arguments: {ι : Type u'} {R : Type u_2} {M : Type u_4} {v : ι → M} [Semiring R] [AddCommMonoid M] [Module R M] [Nontrivial R] (i : ι) (hv : LinearIndependent R v)
Goal: v i ≠ 0
-- LinearIndependent.eq_zero_of_pair
lemma LinearIndependent.eq_zero_of_pair {x y : M} (h : LinearIndependent R ![x, y])
    {s t : R} (h' : s • x + t • y = 0) : s = 0 ∧ t = 0 
Arguments: {R : Type u_2} {M : Type u_4} [Semiring R] [AddCommMonoid M] [Module R M] {x y : M} (h : LinearIndependent R ![x, y]) {s t : R} (h' : s • x + t • y = 0)
Goal: s = 0 ∧ t = 0
-- LinearIndependent.pair_iff
lemma LinearIndependent.pair_iff {x y : M} :
    LinearIndependent R ![x, y] ↔ ∀ (s t : R), s • x + t • y = 0 → s = 0 ∧ t = 0 
Arguments: {R : Type u_2} {M : Type u_4} [Semiring R] [AddCommMonoid M] [Module R M] {x y : M}
Goal: LinearIndependent R ![x, y] ↔ ∀ (s t : R), s • x + t • y = 0 → s = 0 ∧ t = 0
-- LinearIndependent.comp
theorem LinearIndependent.comp (h : LinearIndependent R v) (f : ι' → ι) (hf : Injective f) :
    LinearIndependent R (v ∘ f) 
Arguments: {ι : Type u'} {ι' : Type u_1} {R : Type u_2} {M : Type u_4} {v : ι → M} [Semiring R] [AddCommMonoid M] [Module R M] (h : LinearIndependent R v) (f : ι' → ι) (hf : Function.Injective f)
Goal: LinearIndependent R (v ∘ f)
-- linearIndependent_iff_finset_linearIndependent
theorem linearIndependent_iff_finset_linearIndependent :
    LinearIndependent R v ↔ ∀ (s : Finset ι), LinearIndependent R (v ∘ (Subtype.val : s → ι)) 
Arguments: {ι : Type u'} {R : Type u_2} {M : Type u_4} {v : ι → M} [Semiring R] [AddCommMonoid M] [Module R M]
Goal: LinearIndependent R v ↔ ∀ (s : Finset ι), LinearIndependent R (v ∘ Subtype.val)
-- LinearIndependent.coe_range
theorem LinearIndependent.coe_range (i : LinearIndependent R v) :
    LinearIndependent R ((↑) : range v → M) 
Arguments: {ι : Type u'} {R : Type u_2} {M : Type u_4} {v : ι → M} [Semiring R] [AddCommMonoid M] [Module R M] (i : LinearIndependent R v)
Goal: LinearIndependent (ι := { x // x ∈ Set.range v }) R Subtype.val
-- LinearIndependent.map
theorem LinearIndependent.map (hv : LinearIndependent R v) {f : M →ₗ[R] M'}
    (hf_inj : Disjoint (span R (range v)) (LinearMap.ker f)) : LinearIndependent R (f ∘ v) 
Arguments: {ι : Type u'} {R : Type u_2} {M : Type u_4} {M' : Type u_5} {v : ι → M} [Semiring R] [AddCommMonoid M] [AddCommMonoid M'] [Module R M] [Module R M'] (hv : LinearIndependent R v) {f : M →ₗ[R] M'} (hf_inj : Disjoint (Submodule.span R (Set.range v)) (LinearMap.ker f))
Goal: LinearIndependent R (⇑f ∘ v)
-- Submodule.range_ker_disjoint
theorem Submodule.range_ker_disjoint {f : M →ₗ[R] M'}
    (hv : LinearIndependent R (f ∘ v)) :
    Disjoint (span R (range v)) (LinearMap.ker f) 
Arguments: {ι : Type u'} {R : Type u_2} {M : Type u_4} {M' : Type u_5} {v : ι → M} [Semiring R] [AddCommMonoid M] [AddCommMonoid M'] [Module R M] [Module R M'] {f : M →ₗ[R] M'} (hv : LinearIndependent R (⇑f ∘ v))
Goal: Disjoint (Submodule.span R (Set.range v)) (LinearMap.ker f)
-- LinearIndependent.map'
theorem LinearIndependent.map' (hv : LinearIndependent R v) (f : M →ₗ[R] M')
    (hf_inj : LinearMap.ker f = ⊥) : LinearIndependent R (f ∘ v) 
Arguments: {ι : Type u'} {R : Type u_2} {M : Type u_4} {M' : Type u_5} {v : ι → M} [Semiring R] [AddCommMonoid M] [AddCommMonoid M'] [Module R M] [Module R M'] (hv : LinearIndependent R v) (f : M →ₗ[R] M') (hf_inj : LinearMap.ker f = ⊥)
Goal: LinearIndependent R (⇑f ∘ v)
-- LinearIndependent.map_of_injective_injective
theorem LinearIndependent.map_of_injective_injective {R' : Type*} {M' : Type*}
    [Semiring R'] [AddCommMonoid M'] [Module R' M'] (hv : LinearIndependent R v)
    (i : R' → R) (j : M →+ M') (hi : ∀ r, i r = 0 → r = 0) (hj : ∀ m, j m = 0 → m = 0)
    (hc : ∀ (r : R') (m : M), j (i r • m) = r • j m) : LinearIndependent R' (j ∘ v) 
Arguments: {ι : Type u'} {R : Type u_2} {M : Type u_4} {v : ι → M} [Semiring R] [AddCommMonoid M] [Module R M] {R' : Type u_8} {M' : Type u_9} [Semiring R'] [AddCommMonoid M'] [Module R' M'] (hv : LinearIndependent R v) (i : R' → R) (j : M →+ M') (hi : ∀ (r : R'), i r = 0 → r = 0) (hj : ∀ (m : M), j m = 0 → m = 0) (hc : ∀ (r : R') (m : M), j (i r • m) = r • j m)
Goal: LinearIndependent R' (⇑j ∘ v)
-- LinearIndependent.map_of_surjective_injective
theorem LinearIndependent.map_of_surjective_injective {R' : Type*} {M' : Type*}
    [Semiring R'] [AddCommMonoid M'] [Module R' M'] (hv : LinearIndependent R v)
    (i : ZeroHom R R') (j : M →+ M') (hi : Surjective i) (hj : ∀ m, j m = 0 → m = 0)
    (hc : ∀ (r : R) (m : M), j (r • m) = i r • j m) : LinearIndependent R' (j ∘ v) 
Arguments: {ι : Type u'} {R : Type u_2} {M : Type u_4} {v : ι → M} [Semiring R] [AddCommMonoid M] [Module R M] {R' : Type u_8} {M' : Type u_9} [Semiring R'] [AddCommMonoid M'] [Module R' M'] (hv : LinearIndependent R v) (i : ZeroHom R R') (j : M →+ M') (hi : Function.Surjective ⇑i) (hj : ∀ (m : M), j m = 0 → m = 0) (hc : ∀ (r : R) (m : M), j (r • m) = i r • j m)
Goal: LinearIndependent R' (⇑j ∘ v)
-- LinearIndependent.of_comp
theorem LinearIndependent.of_comp (f : M →ₗ[R] M') (hfv : LinearIndependent R (f ∘ v)) :
    LinearIndependent R v 
Arguments: {ι : Type u'} {R : Type u_2} {M : Type u_4} {M' : Type u_5} {v : ι → M} [Semiring R] [AddCommMonoid M] [AddCommMonoid M'] [Module R M] [Module R M'] (f : M →ₗ[R] M') (hfv : LinearIndependent R (⇑f ∘ v))
Goal: LinearIndependent R v
-- LinearMap.linearIndependent_iff
protected theorem LinearMap.linearIndependent_iff (f : M →ₗ[R] M') (hf_inj : LinearMap.ker f = ⊥) :
    LinearIndependent R (f ∘ v) ↔ LinearIndependent R v 
Arguments: {ι : Type u'} {R : Type u_2} {M : Type u_4} {M' : Type u_5} {v : ι → M} [Semiring R] [AddCommMonoid M] [AddCommMonoid M'] [Module R M] [Module R M'] (f : M →ₗ[R] M') (hf_inj : LinearMap.ker f = ⊥)
Goal: LinearIndependent R (⇑f ∘ v) ↔ LinearIndependent R v
-- linearIndependent_of_subsingleton
@[nontriviality]
theorem linearIndependent_of_subsingleton [Subsingleton R] : LinearIndependent R v 
Arguments: {ι : Type u'} {R : Type u_2} {M : Type u_4} {v : ι → M} [Semiring R] [AddCommMonoid M] [Module R M] [Subsingleton R]
Goal: LinearIndependent R v
-- linearIndependent_equiv
theorem linearIndependent_equiv (e : ι ≃ ι') {f : ι' → M} :
    LinearIndependent R (f ∘ e) ↔ LinearIndependent R f 
Arguments: {ι : Type u'} {ι' : Type u_1} {R : Type u_2} {M : Type u_4} [Semiring R] [AddCommMonoid M] [Module R M] (e : ι ≃ ι') {f : ι' → M}
Goal: LinearIndependent R (f ∘ ⇑e) ↔ LinearIndependent R f
-- linearIndependent_equiv'
theorem linearIndependent_equiv' (e : ι ≃ ι') {f : ι' → M} {g : ι → M} (h : f ∘ e = g) :
    LinearIndependent R g ↔ LinearIndependent R f 
Arguments: {ι : Type u'} {ι' : Type u_1} {R : Type u_2} {M : Type u_4} [Semiring R] [AddCommMonoid M] [Module R M] (e : ι ≃ ι') {f : ι' → M} {g : ι → M} (h : f ∘ ⇑e = g)
Goal: LinearIndependent R g ↔ LinearIndependent R f
-- linearIndependent_subtype_range
theorem linearIndependent_subtype_range {ι} {f : ι → M} (hf : Injective f) :
    LinearIndependent R ((↑) : range f → M) ↔ LinearIndependent R f 
Arguments: {R : Type u_2} {M : Type u_4} [Semiring R] [AddCommMonoid M] [Module R M] {ι : Type u_8} {f : ι → M} (hf : Function.Injective f)
Goal: LinearIndependent (ι := { x // x ∈ Set.range f }) R Subtype.val ↔ LinearIndependent R f
-- linearIndependent_image
theorem linearIndependent_image {ι} {s : Set ι} {f : ι → M} (hf : Set.InjOn f s) :
    (LinearIndependent R fun x : s => f x) ↔ LinearIndependent R fun x : f '' s => (x : M) 
Arguments: {R : Type u_2} {M : Type u_4} [Semiring R] [AddCommMonoid M] [Module R M] {ι : Type u_8} {s : Set ι} {f : ι → M} (hf : Set.InjOn f s)
Goal: (LinearIndependent R fun (x : ↑s) => f ↑x) ↔ LinearIndependent R fun (x : ↑(f '' s)) => ↑x
-- linearIndependent_span
theorem linearIndependent_span (hs : LinearIndependent R v) :
    LinearIndependent R (M := span R (range v))
      (fun i : ι => ⟨v i, subset_span (mem_range_self i)⟩) 
Arguments: {ι : Type u'} {R : Type u_2} {M : Type u_4} {v : ι → M} [Semiring R] [AddCommMonoid M] [Module R M] (hs : LinearIndependent R v)
Goal: LinearIndependent R fun i => ⟨v i, ⋯⟩
-- LinearIndependent.fin_cons'
theorem LinearIndependent.fin_cons' {m : ℕ} (x : M) (v : Fin m → M) (hli : LinearIndependent R v)
    (x_ortho : ∀ (c : R) (y : Submodule.span R (Set.range v)), c • x + y = (0 : M) → c = 0) :
    LinearIndependent R (Fin.cons x v : Fin m.succ → M) 
Arguments: {R : Type u_2} {M : Type u_4} [Semiring R] [AddCommMonoid M] [Module R M] {m : ℕ} (x : M) (v : Fin m → M) (hli : LinearIndependent R v) (x_ortho : ∀ (c : R) (y : ↥(Submodule.span R (Set.range v))), c • x + ↑y = 0 → c = 0)
Goal: LinearIndependent R (Fin.cons x v)
-- LinearIndependent.restrict_scalars
theorem LinearIndependent.restrict_scalars [Semiring K] [SMulWithZero R K] [Module K M]
    [IsScalarTower R K M] (hinj : Function.Injective fun r : R => r • (1 : K))
    (li : LinearIndependent K v) : LinearIndependent R v 
Arguments: {ι : Type u'} {R : Type u_2} {K : Type u_3} {M : Type u_4} {v : ι → M} [Semiring R] [AddCommMonoid M] [Module R M] [Semiring K] [SMulWithZero R K] [Module K M] [IsScalarTower R K M] (hinj : Function.Injective fun r => r • 1) (li : LinearIndependent K v)
Goal: LinearIndependent R v
-- linearIndependent_finset_map_embedding_subtype
theorem linearIndependent_finset_map_embedding_subtype (s : Set M)
    (li : LinearIndependent R ((↑) : s → M)) (t : Finset s) :
    LinearIndependent R ((↑) : Finset.map (Embedding.subtype s) t → M) 
Arguments: {R : Type u_2} {M : Type u_4} [Semiring R] [AddCommMonoid M] [Module R M] (s : Set M) (li : LinearIndependent (ι := { x // x ∈ s }) R Subtype.val) (t : Finset ↑s)
Goal: LinearIndependent R Subtype.val
-- linearIndependent_bounded_of_finset_linearIndependent_bounded
theorem linearIndependent_bounded_of_finset_linearIndependent_bounded {n : ℕ}
    (H : ∀ s : Finset M, (LinearIndependent R fun i : s => (i : M)) → s.card ≤ n) :
    ∀ s : Set M, LinearIndependent R ((↑) : s → M) → #s ≤ n 
Arguments: {R : Type u_2} {M : Type u_4} [Semiring R] [AddCommMonoid M] [Module R M] {n : ℕ} (H : ∀ (s : Finset M), (LinearIndependent R fun i => ↑i) → s.card ≤ n) (s : Set M)
Goal: LinearIndependent (ι := { x // x ∈ s }) R Subtype.val → #↑s ≤ ↑n
-- linearIndependent_comp_subtype
theorem linearIndependent_comp_subtype {s : Set ι} :
    LinearIndependent R (v ∘ (↑) : s → M) ↔
      ∀ l ∈ Finsupp.supported R R s, (Finsupp.total ι M R v) l = 0 → l = 0 
Arguments: {ι : Type u'} {R : Type u_2} {M : Type u_4} {v : ι → M} [Semiring R] [AddCommMonoid M] [Module R M] {s : Set ι}
Goal: LinearIndependent (ι := ↑s) R (v ∘ Subtype.val) ↔ ∀ l ∈ Finsupp.supported R R s, (Finsupp.total ι M R v) l = 0 → l = 0
-- linearDependent_comp_subtype'
theorem linearDependent_comp_subtype' {s : Set ι} :
    ¬LinearIndependent R (v ∘ (↑) : s → M) ↔
      ∃ f : ι →₀ R, f ∈ Finsupp.supported R R s ∧ Finsupp.total ι M R v f = 0 ∧ f ≠ 0 
Arguments: {ι : Type u'} {R : Type u_2} {M : Type u_4} {v : ι → M} [Semiring R] [AddCommMonoid M] [Module R M] {s : Set ι}
Goal: ¬LinearIndependent (ι := ↑s) R (v ∘ Subtype.val) ↔ ∃ f ∈ Finsupp.supported R R s, (Finsupp.total ι M R v) f = 0 ∧ f ≠ 0
-- linearDependent_comp_subtype
theorem linearDependent_comp_subtype {s : Set ι} :
    ¬LinearIndependent R (v ∘ (↑) : s → M) ↔
      ∃ f : ι →₀ R, f ∈ Finsupp.supported R R s ∧ ∑ i ∈ f.support, f i • v i = 0 ∧ f ≠ 0 
Arguments: {ι : Type u'} {R : Type u_2} {M : Type u_4} {v : ι → M} [Semiring R] [AddCommMonoid M] [Module R M] {s : Set ι}
Goal: ¬LinearIndependent (ι := ↑s) R (v ∘ Subtype.val) ↔ ∃ f ∈ Finsupp.supported R R s, ∑ i ∈ f.support, f i • v i = 0 ∧ f ≠ 0
-- linearIndependent_subtype
theorem linearIndependent_subtype {s : Set M} :
    LinearIndependent R (fun x => x : s → M) ↔
      ∀ l ∈ Finsupp.supported R R s, (Finsupp.total M M R id) l = 0 → l = 0 
Arguments: {R : Type u_2} {M : Type u_4} [Semiring R] [AddCommMonoid M] [Module R M] {s : Set M}
Goal: (LinearIndependent R fun (x : ↑s) => ↑x) ↔ ∀ l ∈ Finsupp.supported R R s, (Finsupp.total M M R id) l = 0 → l = 0
-- linearIndependent_comp_subtype_disjoint
theorem linearIndependent_comp_subtype_disjoint {s : Set ι} :
    LinearIndependent R (v ∘ (↑) : s → M) ↔
      Disjoint (Finsupp.supported R R s) (LinearMap.ker <| Finsupp.total ι M R v) 
Arguments: {ι : Type u'} {R : Type u_2} {M : Type u_4} {v : ι → M} [Semiring R] [AddCommMonoid M] [Module R M] {s : Set ι}
Goal: LinearIndependent (ι := ↑s) R (v ∘ Subtype.val) ↔
  Disjoint (Finsupp.supported R R s) (LinearMap.ker (Finsupp.total ι M R v))
-- linearIndependent_subtype_disjoint
theorem linearIndependent_subtype_disjoint {s : Set M} :
    LinearIndependent R (fun x => x : s → M) ↔
      Disjoint (Finsupp.supported R R s) (LinearMap.ker <| Finsupp.total M M R id) 
Arguments: {R : Type u_2} {M : Type u_4} [Semiring R] [AddCommMonoid M] [Module R M] {s : Set M}
Goal: (LinearIndependent R fun (x : ↑s) => ↑x) ↔ Disjoint (Finsupp.supported R R s) (LinearMap.ker (Finsupp.total M M R id))
-- linearIndependent_iff_totalOn
theorem linearIndependent_iff_totalOn {s : Set M} :
    LinearIndependent R (fun x => x : s → M) ↔
    (LinearMap.ker <| Finsupp.totalOn M M R id s) = ⊥ 
Arguments: {R : Type u_2} {M : Type u_4} [Semiring R] [AddCommMonoid M] [Module R M] {s : Set M}
Goal: (LinearIndependent R fun (x : ↑s) => ↑x) ↔ LinearMap.ker (Finsupp.totalOn M M R id s) = ⊥
-- LinearIndependent.restrict_of_comp_subtype
theorem LinearIndependent.restrict_of_comp_subtype {s : Set ι}
    (hs : LinearIndependent R (v ∘ (↑) : s → M)) : LinearIndependent R (s.restrict v) 
Arguments: {ι : Type u'} {R : Type u_2} {M : Type u_4} {v : ι → M} [Semiring R] [AddCommMonoid M] [Module R M] {s : Set ι} (hs : LinearIndependent (ι := ↑s) R (v ∘ Subtype.val))
Goal: LinearIndependent (ι := ↑s) R (s.restrict v)
-- linearIndependent_empty
theorem linearIndependent_empty : LinearIndependent R (fun x => x : (∅ : Set M) → M) 
Arguments: (R : Type u_2) (M : Type u_4) [Semiring R] [AddCommMonoid M] [Module R M]
Goal: LinearIndependent R fun (x : ↑∅) => ↑x
-- LinearIndependent.mono
theorem LinearIndependent.mono {t s : Set M} (h : t ⊆ s) :
    LinearIndependent R (fun x => x : s → M) → LinearIndependent R (fun x => x : t → M) 
Arguments: {R : Type u_2} {M : Type u_4} [Semiring R] [AddCommMonoid M] [Module R M] {t s : Set M} (h : t ⊆ s)
Goal: (LinearIndependent R fun (x : ↑s) => ↑x) → LinearIndependent R fun (x : ↑t) => ↑x
-- linearIndependent_of_finite
theorem linearIndependent_of_finite (s : Set M)
    (H : ∀ t ⊆ s, Set.Finite t → LinearIndependent R (fun x => x : t → M)) :
    LinearIndependent R (fun x => x : s → M) 
Arguments: {R : Type u_2} {M : Type u_4} [Semiring R] [AddCommMonoid M] [Module R M] (s : Set M) (H : ∀ t ⊆ s, t.Finite → LinearIndependent R fun (x : ↑t) => ↑x)
Goal: LinearIndependent R fun (x : ↑s) => ↑x