ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- linearIndependent_iUnion_of_directed
theorem linearIndependent_iUnion_of_directed {η : Type*} {s : η → Set M} (hs : Directed (· ⊆ ·) s)
    (h : ∀ i, LinearIndependent R (fun x => x : s i → M)) :
    LinearIndependent R (fun x => x : (⋃ i, s i) → M) 
Arguments: {R : Type u_2} {M : Type u_4} [Semiring R] [AddCommMonoid M] [Module R M] {η : Type u_8} {s : η → Set M} (hs : Directed (fun x x_1 => x ⊆ x_1) s) (h : ∀ (i : η), LinearIndependent R fun (x : ↑(s i)) => ↑x)
Goal: LinearIndependent R fun (x : ↑(⋃ i, s i)) => ↑x
-- linearIndependent_sUnion_of_directed
theorem linearIndependent_sUnion_of_directed {s : Set (Set M)} (hs : DirectedOn (· ⊆ ·) s)
    (h : ∀ a ∈ s, LinearIndependent R ((↑) : ((a : Set M) : Type _) → M)) :
    LinearIndependent R (fun x => x : ⋃₀ s → M) 
Arguments: {R : Type u_2} {M : Type u_4} [Semiring R] [AddCommMonoid M] [Module R M] {s : Set (Set M)} (hs : DirectedOn (fun x x_1 => x ⊆ x_1) s) (h : ∀ a ∈ s, LinearIndependent (ι := { x // x ∈ a }) R Subtype.val)
Goal: LinearIndependent R fun (x : ↑(⋃₀ s)) => ↑x
-- linearIndependent_biUnion_of_directed
theorem linearIndependent_biUnion_of_directed {η} {s : Set η} {t : η → Set M}
    (hs : DirectedOn (t ⁻¹'o (· ⊆ ·)) s) (h : ∀ a ∈ s, LinearIndependent R (fun x => x : t a → M)) :
    LinearIndependent R (fun x => x : (⋃ a ∈ s, t a) → M) 
Arguments: {R : Type u_2} {M : Type u_4} [Semiring R] [AddCommMonoid M] [Module R M] {η : Type u_8} {s : Set η} {t : η → Set M} (hs : DirectedOn (t ⁻¹'o fun x x_1 => x ⊆ x_1) s) (h : ∀ a ∈ s, LinearIndependent R fun (x : ↑(t a)) => ↑x)
Goal: LinearIndependent R fun (x : ↑(⋃ a ∈ s, t a)) => ↑x
-- linearIndependent_iff_injective_total
theorem linearIndependent_iff_injective_total :
    LinearIndependent R v ↔ Function.Injective (Finsupp.total ι M R v) 
Arguments: {ι : Type u'} {R : Type u_2} {M : Type u_4} {v : ι → M} [Ring R] [AddCommGroup M] [Module R M]
Goal: LinearIndependent R v ↔ Function.Injective ⇑(Finsupp.total ι M R v)
-- LinearIndependent.injective
theorem LinearIndependent.injective [Nontrivial R] (hv : LinearIndependent R v) : Injective v 
Arguments: {ι : Type u'} {R : Type u_2} {M : Type u_4} {v : ι → M} [Ring R] [AddCommGroup M] [Module R M] [Nontrivial R] (hv : LinearIndependent R v)
Goal: Function.Injective v
-- LinearIndependent.to_subtype_range
theorem LinearIndependent.to_subtype_range {ι} {f : ι → M} (hf : LinearIndependent R f) :
    LinearIndependent R ((↑) : range f → M) 
Arguments: {R : Type u_2} {M : Type u_4} [Ring R] [AddCommGroup M] [Module R M] {ι : Type u_8} {f : ι → M} (hf : LinearIndependent R f)
Goal: LinearIndependent (ι := { x // x ∈ Set.range f }) R Subtype.val
-- LinearIndependent.to_subtype_range'
theorem LinearIndependent.to_subtype_range' {ι} {f : ι → M} (hf : LinearIndependent R f) {t}
    (ht : range f = t) : LinearIndependent R ((↑) : t → M) 
Arguments: {R : Type u_2} {M : Type u_4} [Ring R] [AddCommGroup M] [Module R M] {ι : Type u_8} {f : ι → M} (hf : LinearIndependent R f) {t : Set M} (ht : Set.range f = t)
Goal: LinearIndependent (ι := { x // x ∈ t }) R Subtype.val
-- LinearIndependent.image_of_comp
theorem LinearIndependent.image_of_comp {ι ι'} (s : Set ι) (f : ι → ι') (g : ι' → M)
    (hs : LinearIndependent R fun x : s => g (f x)) :
    LinearIndependent R fun x : f '' s => g x 
Arguments: {R : Type u_2} {M : Type u_4} [Ring R] [AddCommGroup M] [Module R M] {ι : Type u_8} {ι' : Type u_9} (s : Set ι) (f : ι → ι') (g : ι' → M) (hs : LinearIndependent R fun (x : ↑s) => g (f ↑x))
Goal: LinearIndependent R fun (x : ↑(f '' s)) => g ↑x
-- LinearIndependent.image
theorem LinearIndependent.image {ι} {s : Set ι} {f : ι → M}
    (hs : LinearIndependent R fun x : s => f x) :
    LinearIndependent R fun x : f '' s => (x : M) 
Arguments: {R : Type u_2} {M : Type u_4} [Ring R] [AddCommGroup M] [Module R M] {ι : Type u_8} {s : Set ι} {f : ι → M} (hs : LinearIndependent R fun (x : ↑s) => f ↑x)
Goal: LinearIndependent R fun (x : ↑(f '' s)) => ↑x
-- LinearIndependent.group_smul
theorem LinearIndependent.group_smul {G : Type*} [hG : Group G] [DistribMulAction G R]
    [DistribMulAction G M] [IsScalarTower G R M] [SMulCommClass G R M] {v : ι → M}
    (hv : LinearIndependent R v) (w : ι → G) : LinearIndependent R (w • v) 
Arguments: {ι : Type u'} {R : Type u_2} {M : Type u_4} [Ring R] [AddCommGroup M] [Module R M] {G : Type u_8} [hG : Group G] [DistribMulAction G R] [DistribMulAction G M] [IsScalarTower G R M] [SMulCommClass G R M] {v : ι → M} (hv : LinearIndependent R v) (w : ι → G)
Goal: LinearIndependent R (w • v)
-- LinearIndependent.units_smul
theorem LinearIndependent.units_smul {v : ι → M} (hv : LinearIndependent R v) (w : ι → Rˣ) :
    LinearIndependent R (w • v) 
Arguments: {ι : Type u'} {R : Type u_2} {M : Type u_4} [Ring R] [AddCommGroup M] [Module R M] {v : ι → M} (hv : LinearIndependent R v) (w : ι → Rˣ)
Goal: LinearIndependent R (w • v)
-- LinearIndependent.eq_of_pair
lemma LinearIndependent.eq_of_pair {x y : M} (h : LinearIndependent R ![x, y])
    {s t s' t' : R} (h' : s • x + t • y = s' • x + t' • y) : s = s' ∧ t = t' 
Arguments: {R : Type u_2} {M : Type u_4} [Ring R] [AddCommGroup M] [Module R M] {x y : M} (h : LinearIndependent R ![x, y]) {s t s' t' : R} (h' : s • x + t • y = s' • x + t' • y)
Goal: s = s' ∧ t = t'
-- 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) : s = 0 ∧ t = 0 
Arguments: {R : Type u_2} {M : Type u_4} [Ring R] [AddCommGroup M] [Module R M] {x y : M} (h : LinearIndependent R ![x, y]) {s t : R} (h' : s • x = t • y)
Goal: s = 0 ∧ t = 0
-- LinearIndependent.linear_combination_pair_of_det_ne_zero
lemma LinearIndependent.linear_combination_pair_of_det_ne_zero {R M : Type*} [CommRing R]
    [NoZeroDivisors R] [AddCommGroup M] [Module R M]
    {x y : M} (h : LinearIndependent R ![x, y])
    {a b c d : R} (h' : a * d - b * c ≠ 0) :
    LinearIndependent R ![a • x + b • y, c • x + d • y] 
Arguments: {R : Type u_8} {M : Type u_9} [CommRing R] [NoZeroDivisors R] [AddCommGroup M] [Module R M] {x y : M} (h : LinearIndependent R ![x, y]) {a b c d : R} (h' : a * d - b * c ≠ 0)
Goal: LinearIndependent R ![a • x + b • y, c • x + d • y]
-- LinearIndependent.maximal_iff
theorem LinearIndependent.maximal_iff {ι : Type w} {R : Type u} [Ring R] [Nontrivial R] {M : Type v}
    [AddCommGroup M] [Module R M] {v : ι → M} (i : LinearIndependent R v) :
    i.Maximal ↔
      ∀ (κ : Type v) (w : κ → M) (_i' : LinearIndependent R w) (j : ι → κ) (_h : w ∘ j = v),
        Surjective j 
Arguments: {ι : Type w} {R : Type u} [Ring R] [Nontrivial R] {M : Type v} [AddCommGroup M] [Module R M] {v : ι → M} (i : LinearIndependent R v)
Goal: i.Maximal ↔ ∀ (κ : Type v) (w : κ → M), LinearIndependent R w → ∀ (j : ι → κ), w ∘ j = v → Function.Surjective j
-- LinearIndependent.eq_of_smul_apply_eq_smul_apply
theorem LinearIndependent.eq_of_smul_apply_eq_smul_apply {M : Type*} [AddCommGroup M] [Module R M]
    {v : ι → M} (li : LinearIndependent R v) (c d : R) (i j : ι) (hc : c ≠ 0)
    (h : c • v i = d • v j) : i = j 
Arguments: {ι : Type u'} {R : Type u_2} [Ring R] {M : Type u_8} [AddCommGroup M] [Module R M] {v : ι → M} (li : LinearIndependent R v) (c d : R) (i j : ι) (hc : c ≠ 0) (h : c • v i = d • v j)
Goal: i = j
-- LinearIndependent.disjoint_span_image
theorem LinearIndependent.disjoint_span_image (hv : LinearIndependent R v) {s t : Set ι}
    (hs : Disjoint s t) : Disjoint (Submodule.span R <| v '' s) (Submodule.span R <| v '' t) 
Arguments: {ι : Type u'} {R : Type u_2} {M : Type u_4} {v : ι → M} [Ring R] [AddCommGroup M] [Module R M] (hv : LinearIndependent R v) {s t : Set ι} (hs : Disjoint s t)
Goal: Disjoint (Submodule.span R (v '' s)) (Submodule.span R (v '' t))
-- LinearIndependent.not_mem_span_image
theorem LinearIndependent.not_mem_span_image [Nontrivial R] (hv : LinearIndependent R v) {s : Set ι}
    {x : ι} (h : x ∉ s) : v x ∉ Submodule.span R (v '' s) 
Arguments: {ι : Type u'} {R : Type u_2} {M : Type u_4} {v : ι → M} [Ring R] [AddCommGroup M] [Module R M] [Nontrivial R] (hv : LinearIndependent R v) {s : Set ι} {x : ι} (h : x ∉ s)
Goal: v x ∉ Submodule.span R (v '' s)
-- LinearIndependent.total_ne_of_not_mem_support
theorem LinearIndependent.total_ne_of_not_mem_support [Nontrivial R] (hv : LinearIndependent R v)
    {x : ι} (f : ι →₀ R) (h : x ∉ f.support) : Finsupp.total ι M R v f ≠ v x 
Arguments: {ι : Type u'} {R : Type u_2} {M : Type u_4} {v : ι → M} [Ring R] [AddCommGroup M] [Module R M] [Nontrivial R] (hv : LinearIndependent R v) {x : ι} (f : ι →₀ R) (h : x ∉ f.support)
Goal: (Finsupp.total ι M R v) f ≠ v x
-- linearIndependent_sum
theorem linearIndependent_sum {v : ι ⊕ ι' → M} :
    LinearIndependent R v ↔
      LinearIndependent R (v ∘ Sum.inl) ∧
        LinearIndependent R (v ∘ Sum.inr) ∧
          Disjoint (Submodule.span R (range (v ∘ Sum.inl)))
            (Submodule.span R (range (v ∘ Sum.inr))) 
Arguments: {ι : Type u'} {ι' : Type u_1} {R : Type u_2} {M : Type u_4} [Ring R] [AddCommGroup M] [Module R M] {v : ι ⊕ ι' → M}
Goal: LinearIndependent R v ↔
  LinearIndependent R (v ∘ Sum.inl) ∧
    LinearIndependent R (v ∘ Sum.inr) ∧
      Disjoint (Submodule.span R (Set.range (v ∘ Sum.inl))) (Submodule.span R (Set.range (v ∘ Sum.inr)))
-- LinearIndependent.sum_type
theorem LinearIndependent.sum_type {v' : ι' → M} (hv : LinearIndependent R v)
    (hv' : LinearIndependent R v')
    (h : Disjoint (Submodule.span R (range v)) (Submodule.span R (range v'))) :
    LinearIndependent R (Sum.elim v v') 
Arguments: {ι : Type u'} {ι' : Type u_1} {R : Type u_2} {M : Type u_4} {v : ι → M} [Ring R] [AddCommGroup M] [Module R M] {v' : ι' → M} (hv : LinearIndependent R v) (hv' : LinearIndependent R v') (h : Disjoint (Submodule.span R (Set.range v)) (Submodule.span R (Set.range v')))
Goal: LinearIndependent R (Sum.elim v v')
-- LinearIndependent.union
theorem LinearIndependent.union {s t : Set M} (hs : LinearIndependent R (fun x => x : s → M))
    (ht : LinearIndependent R (fun x => x : t → M)) (hst : Disjoint (span R s) (span R t)) :
    LinearIndependent R (fun x => x : ↥(s ∪ t) → M) 
Arguments: {R : Type u_2} {M : Type u_4} [Ring R] [AddCommGroup M] [Module R M] {s t : Set M} (hs : LinearIndependent R fun (x : ↑s) => ↑x) (ht : LinearIndependent R fun (x : ↑t) => ↑x) (hst : Disjoint (Submodule.span R s) (Submodule.span R t))
Goal: LinearIndependent R fun (x : ↑(s ∪ t)) => ↑x
-- linearIndependent_iUnion_finite_subtype
theorem linearIndependent_iUnion_finite_subtype {ι : Type*} {f : ι → Set M}
    (hl : ∀ i, LinearIndependent R (fun x => x : f i → M))
    (hd : ∀ i, ∀ t : Set ι, t.Finite → i ∉ t → Disjoint (span R (f i)) (⨆ i ∈ t, span R (f i))) :
    LinearIndependent R (fun x => x : (⋃ i, f i) → M) 
Arguments: {R : Type u_2} {M : Type u_4} [Ring R] [AddCommGroup M] [Module R M] {ι : Type u_8} {f : ι → Set M} (hl : ∀ (i : ι), LinearIndependent R fun (x : ↑(f i)) => ↑x) (hd : ∀ (i : ι) (t : Set ι), t.Finite → i ∉ t → Disjoint (Submodule.span R (f i)) (⨆ i ∈ t, Submodule.span R (f i)))
Goal: LinearIndependent R fun (x : ↑(⋃ i, f i)) => ↑x
-- linearIndependent_iUnion_finite
theorem linearIndependent_iUnion_finite {η : Type*} {ιs : η → Type*} {f : ∀ j : η, ιs j → M}
    (hindep : ∀ j, LinearIndependent R (f j))
    (hd : ∀ i, ∀ t : Set η,
      t.Finite → i ∉ t → Disjoint (span R (range (f i))) (⨆ i ∈ t, span R (range (f i)))) :
    LinearIndependent R fun ji : Σ j, ιs j => f ji.1 ji.2 
Arguments: {R : Type u_2} {M : Type u_4} [Ring R] [AddCommGroup M] [Module R M] {η : Type u_8} {ιs : η → Type u_9} {f : (j : η) → ιs j → M} (hindep : ∀ (j : η), LinearIndependent R (f j)) (hd :
  ∀ (i : η) (t : Set η),
    t.Finite → i ∉ t → Disjoint (Submodule.span R (Set.range (f i))) (⨆ i ∈ t, Submodule.span R (Set.range (f i))))
Goal: LinearIndependent R fun ji => f ji.fst ji.snd
-- LinearIndependent.totalEquiv_symm_apply
@[simps (config := { rhsMd := default }) symm_apply]
def LinearIndependent.totalEquiv (hv : LinearIndependent R v) :
    (ι →₀ R) ≃ₗ[R] span R (range v) 
Arguments: {ι : Type u'} {R : Type u_2} {M : Type u_4} {v : ι → M} [Ring R] [AddCommGroup M] [Module R M] (hv : LinearIndependent R v)
Goal: ∀ (a : ↥(Submodule.span R (Set.range v))),
  hv.totalEquiv.symm a =
    ((LinearEquiv.ofInjective (LinearMap.codRestrict (Submodule.span R (Set.range v)) (Finsupp.total ι M R v) ⋯)
                ⋯).toEquiv.trans
          (LinearEquiv.ofTop
              (LinearMap.range (LinearMap.codRestrict (Submodule.span R (Set.range v)) (Finsupp.total ι M R v) ⋯))
              ⋯).toEquiv).invFun
      a
-- LinearIndependent.totalEquiv_apply_coe
@[simp]
theorem LinearIndependent.totalEquiv_apply_coe (hv : LinearIndependent R v) (l : ι →₀ R) :
    hv.totalEquiv l = Finsupp.total ι M R v l 
Arguments: {ι : Type u'} {R : Type u_2} {M : Type u_4} {v : ι → M} [Ring R] [AddCommGroup M] [Module R M] (hv : LinearIndependent R v) (l : ι →₀ R)
Goal: ↑(hv.totalEquiv l) = (Finsupp.total ι M R v) l
-- LinearIndependent.total_repr
@[simp]
theorem LinearIndependent.total_repr (x) : Finsupp.total ι M R v (hv.repr x) = x 
Arguments: {ι : Type u'} {R : Type u_2} {M : Type u_4} {v : ι → M} [Ring R] [AddCommGroup M] [Module R M] (hv : LinearIndependent R v) (x : ↥(Submodule.span R (Set.range v)))
Goal: (Finsupp.total ι M R v) (hv.repr x) = ↑x
-- LinearIndependent.total_comp_repr
theorem LinearIndependent.total_comp_repr :
    (Finsupp.total ι M R v).comp hv.repr = Submodule.subtype _ 
Arguments: {ι : Type u'} {R : Type u_2} {M : Type u_4} {v : ι → M} [Ring R] [AddCommGroup M] [Module R M] (hv : LinearIndependent R v)
Goal: Finsupp.total ι M R v ∘ₗ hv.repr = (Submodule.span R (Set.range v)).subtype
-- LinearIndependent.repr_ker
theorem LinearIndependent.repr_ker : LinearMap.ker hv.repr = ⊥ 
Arguments: {ι : Type u'} {R : Type u_2} {M : Type u_4} {v : ι → M} [Ring R] [AddCommGroup M] [Module R M] (hv : LinearIndependent R v)
Goal: LinearMap.ker hv.repr = ⊥
-- LinearIndependent.repr_range
theorem LinearIndependent.repr_range : LinearMap.range hv.repr = ⊤ 
Arguments: {ι : Type u'} {R : Type u_2} {M : Type u_4} {v : ι → M} [Ring R] [AddCommGroup M] [Module R M] (hv : LinearIndependent R v)
Goal: LinearMap.range hv.repr = ⊤
-- LinearIndependent.repr_eq
theorem LinearIndependent.repr_eq {l : ι →₀ R} {x : span R (range v)}
    (eq : Finsupp.total ι M R v l = ↑x) : hv.repr x = l 
Arguments: {ι : Type u'} {R : Type u_2} {M : Type u_4} {v : ι → M} [Ring R] [AddCommGroup M] [Module R M] (hv : LinearIndependent R v) {l : ι →₀ R} {x : ↥(Submodule.span R (Set.range v))} (eq : (Finsupp.total ι M R v) l = ↑x)
Goal: hv.repr x = l
-- LinearIndependent.repr_eq_single
theorem LinearIndependent.repr_eq_single (i) (x : span R (range v)) (hx : ↑x = v i) :
    hv.repr x = Finsupp.single i 1 
Arguments: {ι : Type u'} {R : Type u_2} {M : Type u_4} {v : ι → M} [Ring R] [AddCommGroup M] [Module R M] (hv : LinearIndependent R v) (i : ι) (x : ↥(Submodule.span R (Set.range v))) (hx : ↑x = v i)
Goal: hv.repr x = Finsupp.single i 1
-- LinearIndependent.span_repr_eq
theorem LinearIndependent.span_repr_eq [Nontrivial R] (x) :
    Span.repr R (Set.range v) x =
      (hv.repr x).equivMapDomain (Equiv.ofInjective _ hv.injective) 
Arguments: {ι : Type u'} {R : Type u_2} {M : Type u_4} {v : ι → M} [Ring R] [AddCommGroup M] [Module R M] (hv : LinearIndependent R v) [Nontrivial R] (x : ↥(Submodule.span R (Set.range v)))
Goal: Span.repr R (Set.range v) x = Finsupp.equivMapDomain (Equiv.ofInjective v ⋯) (hv.repr x)
-- linearIndependent_iff_not_smul_mem_span
theorem linearIndependent_iff_not_smul_mem_span :
    LinearIndependent R v ↔ ∀ (i : ι) (a : R), a • v i ∈ span R (v '' (univ \ {i})) → a = 0 
Arguments: {ι : Type u'} {R : Type u_2} {M : Type u_4} {v : ι → M} [Ring R] [AddCommGroup M] [Module R M]
Goal: LinearIndependent R v ↔ ∀ (i : ι) (a : R), a • v i ∈ Submodule.span R (v '' (Set.univ \ {i})) → a = 0
-- LinearIndependent.independent_span_singleton
theorem LinearIndependent.independent_span_singleton (hv : LinearIndependent R v) :
    CompleteLattice.Independent fun i => R ∙ v i 
Arguments: {ι : Type u'} {R : Type u_2} {M : Type u_4} {v : ι → M} [Ring R] [AddCommGroup M] [Module R M] (hv : LinearIndependent R v)
Goal: CompleteLattice.Independent fun i => Submodule.span R {v i}
-- exists_maximal_independent'
theorem exists_maximal_independent' (s : ι → M) :
    ∃ I : Set ι,
      (LinearIndependent R fun x : I => s x) ∧
        ∀ J : Set ι, I ⊆ J → (LinearIndependent R fun x : J => s x) → I = J 
Arguments: {ι : Type u'} (R : Type u_2) {M : Type u_4} [Ring R] [AddCommGroup M] [Module R M] (s : ι → M)
Goal: ∃ I,
  (LinearIndependent R fun (x : ↑I) => s ↑x) ∧ ∀ (J : Set ι), I ⊆ J → (LinearIndependent R fun (x : ↑J) => s ↑x) → I = J
-- exists_maximal_independent
theorem exists_maximal_independent (s : ι → M) :
    ∃ I : Set ι,
      (LinearIndependent R fun x : I => s x) ∧
        ∀ i ∉ I, ∃ a : R, a ≠ 0 ∧ a • s i ∈ span R (s '' I) 
Arguments: {ι : Type u'} (R : Type u_2) {M : Type u_4} [Ring R] [AddCommGroup M] [Module R M] (s : ι → M)
Goal: ∃ I, (LinearIndependent R fun (x : ↑I) => s ↑x) ∧ ∀ i ∉ I, ∃ a, a ≠ 0 ∧ a • s i ∈ Submodule.span R (s '' I)
-- surjective_of_linearIndependent_of_span
theorem surjective_of_linearIndependent_of_span [Nontrivial R] (hv : LinearIndependent R v)
    (f : ι' ↪ ι) (hss : range v ⊆ span R (range (v ∘ f))) : Surjective f 
Arguments: {ι : Type u'} {ι' : Type u_1} {R : Type u_2} {M : Type u_4} {v : ι → M} [Ring R] [AddCommGroup M] [Module R M] [Nontrivial R] (hv : LinearIndependent R v) (f : ι' ↪ ι) (hss : Set.range v ⊆ ↑(Submodule.span R (Set.range (v ∘ ⇑f))))
Goal: Function.Surjective ⇑f
-- eq_of_linearIndependent_of_span_subtype
theorem eq_of_linearIndependent_of_span_subtype [Nontrivial R] {s t : Set M}
    (hs : LinearIndependent R (fun x => x : s → M)) (h : t ⊆ s) (hst : s ⊆ span R t) : s = t 
Arguments: {R : Type u_2} {M : Type u_4} [Ring R] [AddCommGroup M] [Module R M] [Nontrivial R] {s t : Set M} (hs : LinearIndependent R fun (x : ↑s) => ↑x) (h : t ⊆ s) (hst : s ⊆ ↑(Submodule.span R t))
Goal: s = t
-- LinearIndependent.image_subtype
theorem LinearIndependent.image_subtype {s : Set M} {f : M →ₗ[R] M'}
    (hs : LinearIndependent R (fun x => x : s → M))
    (hf_inj : Disjoint (span R s) (LinearMap.ker f)) :
    LinearIndependent R (fun x => x : f '' s → M') 
Arguments: {R : Type u_2} {M : Type u_4} {M' : Type u_5} [Ring R] [AddCommGroup M] [AddCommGroup M'] [Module R M] [Module R M'] {s : Set M} {f : M →ₗ[R] M'} (hs : LinearIndependent R fun (x : ↑s) => ↑x) (hf_inj : Disjoint (Submodule.span R s) (LinearMap.ker f))
Goal: LinearIndependent R fun (x : ↑(⇑f '' s)) => ↑x
-- LinearIndependent.inl_union_inr
theorem LinearIndependent.inl_union_inr {s : Set M} {t : Set M'}
    (hs : LinearIndependent R (fun x => x : s → M))
    (ht : LinearIndependent R (fun x => x : t → M')) :
    LinearIndependent R (fun x => x : ↥(inl R M M' '' s ∪ inr R M M' '' t) → M × M') 
Arguments: {R : Type u_2} {M : Type u_4} {M' : Type u_5} [Ring R] [AddCommGroup M] [AddCommGroup M'] [Module R M] [Module R M'] {s : Set M} {t : Set M'} (hs : LinearIndependent R fun (x : ↑s) => ↑x) (ht : LinearIndependent R fun (x : ↑t) => ↑x)
Goal: LinearIndependent R fun (x : ↑(⇑(LinearMap.inl R M M') '' s ∪ ⇑(LinearMap.inr R M M') '' t)) => ↑x
-- linearIndependent_inl_union_inr'
theorem linearIndependent_inl_union_inr' {v : ι → M} {v' : ι' → M'} (hv : LinearIndependent R v)
    (hv' : LinearIndependent R v') :
    LinearIndependent R (Sum.elim (inl R M M' ∘ v) (inr R M M' ∘ v')) 
Arguments: {ι : Type u'} {ι' : Type u_1} {R : Type u_2} {M : Type u_4} {M' : Type u_5} [Ring R] [AddCommGroup M] [AddCommGroup M'] [Module R M] [Module R M'] {v : ι → M} {v' : ι' → M'} (hv : LinearIndependent R v) (hv' : LinearIndependent R v')
Goal: LinearIndependent R (Sum.elim (⇑(LinearMap.inl R M M') ∘ v) (⇑(LinearMap.inr R M M') ∘ v'))
-- linearIndependent_monoidHom
theorem linearIndependent_monoidHom (G : Type*) [Monoid G] (L : Type*) [CommRing L]
    [NoZeroDivisors L] : LinearIndependent L (M := G → L) (fun f => f : (G →* L) → G → L) 
Arguments: (G : Type u_8) [Monoid G] (L : Type u_9) [CommRing L] [NoZeroDivisors L]
Goal: LinearIndependent L fun f => ⇑f
-- linearIndependent_algHom_toLinearMap
lemma linearIndependent_algHom_toLinearMap
    (K M L) [CommSemiring K] [Semiring M] [Algebra K M] [CommRing L] [IsDomain L] [Algebra K L] :
    LinearIndependent L (AlgHom.toLinearMap : (M →ₐ[K] L) → M →ₗ[K] L) 
Arguments: (K : Type u_8) (M : Type u_9) (L : Type u_10) [CommSemiring K] [Semiring M] [Algebra K M] [CommRing L] [IsDomain L] [Algebra K L]
Goal: LinearIndependent L AlgHom.toLinearMap
-- linearIndependent_algHom_toLinearMap'
lemma linearIndependent_algHom_toLinearMap' (K M L) [CommRing K]
    [Semiring M] [Algebra K M] [CommRing L] [IsDomain L] [Algebra K L] [NoZeroSMulDivisors K L] :
    LinearIndependent K (AlgHom.toLinearMap : (M →ₐ[K] L) → M →ₗ[K] L) 
Arguments: (K : Type u_8) (M : Type u_9) (L : Type u_10) [CommRing K] [Semiring M] [Algebra K M] [CommRing L] [IsDomain L] [Algebra K L] [NoZeroSMulDivisors K L]
Goal: LinearIndependent K AlgHom.toLinearMap
-- le_of_span_le_span
theorem le_of_span_le_span [Nontrivial R] {s t u : Set M} (hl : LinearIndependent R ((↑) : u → M))
    (hsu : s ⊆ u) (htu : t ⊆ u) (hst : span R s ≤ span R t) : s ⊆ t 
Arguments: {R : Type u_2} {M : Type u_4} [Ring R] [AddCommGroup M] [Module R M] [Nontrivial R] {s t u : Set M} (hl : LinearIndependent (ι := { x // x ∈ u }) R Subtype.val) (hsu : s ⊆ u) (htu : t ⊆ u) (hst : Submodule.span R s ≤ Submodule.span R t)
Goal: s ⊆ t
-- span_le_span_iff
theorem span_le_span_iff [Nontrivial R] {s t u : Set M} (hl : LinearIndependent R ((↑) : u → M))
    (hsu : s ⊆ u) (htu : t ⊆ u) : span R s ≤ span R t ↔ s ⊆ t 
Arguments: {R : Type u_2} {M : Type u_4} [Ring R] [AddCommGroup M] [Module R M] [Nontrivial R] {s t u : Set M} (hl : LinearIndependent (ι := { x // x ∈ u }) R Subtype.val) (hsu : s ⊆ u) (htu : t ⊆ u)
Goal: Submodule.span R s ≤ Submodule.span R t ↔ s ⊆ t
-- linearIndependent_unique_iff
theorem linearIndependent_unique_iff (v : ι → M) [Unique ι] :
    LinearIndependent R v ↔ v default ≠ 0 
Arguments: {ι : Type u'} {R : Type u_2} {M : Type u_4} [Ring R] [Nontrivial R] [AddCommGroup M] [Module R M] [NoZeroSMulDivisors R M] (v : ι → M) [Unique ι]
Goal: LinearIndependent R v ↔ v default ≠ 0
-- linearIndependent_singleton
theorem linearIndependent_singleton {x : M} (hx : x ≠ 0) :
    LinearIndependent R (fun x => x : ({x} : Set M) → M) 
Arguments: {R : Type u_2} {M : Type u_4} [Ring R] [Nontrivial R] [AddCommGroup M] [Module R M] [NoZeroSMulDivisors R M] {x : M} (hx : x ≠ 0)
Goal: LinearIndependent R fun (x_1 : ↑{x}) => ↑x_1
-- mem_span_insert_exchange
theorem mem_span_insert_exchange :
    x ∈ span K (insert y s) → x ∉ span K s → y ∈ span K (insert x s) 
Arguments: {K : Type u_3} {V : Type u} [DivisionRing K] [AddCommGroup V] [Module K V] {s : Set V} {x y : V}
Goal: x ∈ Submodule.span K (insert y s) → x ∉ Submodule.span K s → y ∈ Submodule.span K (insert x s)
-- linearIndependent_iff_not_mem_span
theorem linearIndependent_iff_not_mem_span :
    LinearIndependent K v ↔ ∀ i, v i ∉ span K (v '' (univ \ {i})) 
Arguments: {ι : Type u'} {K : Type u_3} {V : Type u} [DivisionRing K] [AddCommGroup V] [Module K V] {v : ι → V}
Goal: LinearIndependent K v ↔ ∀ (i : ι), v i ∉ Submodule.span K (v '' (Set.univ \ {i}))
-- LinearIndependent.insert
protected theorem LinearIndependent.insert (hs : LinearIndependent K (fun b => b : s → V))
    (hx : x ∉ span K s) : LinearIndependent K (fun b => b : ↥(insert x s) → V) 
Arguments: {K : Type u_3} {V : Type u} [DivisionRing K] [AddCommGroup V] [Module K V] {s : Set V} {x : V} (hs : LinearIndependent K fun (b : ↑s) => ↑b) (hx : x ∉ Submodule.span K s)
Goal: LinearIndependent K fun (b : ↑(insert x s)) => ↑b
-- linearIndependent_option'
theorem linearIndependent_option' :
    LinearIndependent K (fun o => Option.casesOn' o x v : Option ι → V) ↔
      LinearIndependent K v ∧ x ∉ Submodule.span K (range v) 
Arguments: {ι : Type u'} {K : Type u_3} {V : Type u} [DivisionRing K] [AddCommGroup V] [Module K V] {v : ι → V} {x : V}
Goal: (LinearIndependent K fun o => o.casesOn' x v) ↔ LinearIndependent K v ∧ x ∉ Submodule.span K (Set.range v)