l3lab/ntp-mathlib huggingface dataset:
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 := by
  let indep : Set ι → Prop := fun I => LinearIndependent R (s ∘ (↑) : I → M)
  let X := { I : Set ι // indep I }
  let r : X → X → Prop := fun I J => I.1 ⊆ J.1
  have key : ∀ c : Set X, IsChain r c → indep (⋃ (I : X) (_ : I ∈ c), I) := by
    intro c hc
    dsimp
    rw [linearIndependent_comp_subtype]
    intro f hsupport hsum
    

Proof state:
ι : Type u'
ι' : Type u_1
R : Type u_2
K : Type u_3
M : Type u_4
M' : Type u_5
M'' : Type u_6
V : Type u
V' : Type u_7
v : ι → M
inst✝⁶ : Ring R
inst✝⁵ : AddCommGroup M
inst✝⁴ : AddCommGroup M'
inst✝³ : AddCommGroup M''
inst✝² : Module R M
inst✝¹ : Module R M'
inst✝ : Module R M''
a b : R
x y : M
hv : LinearIndependent R v
s : ι → M
indep : Set ι → Prop := fun I => LinearIndependent (ι := ↑I) R (s ∘ Subtype.val)
X : Type u' := { I // indep I }
r : X → X → Prop := fun I J => ↑I ⊆ ↑J
c : Set X
hc : IsChain r c
f : ι →₀ R
hsupport : f ∈ Finsupp.supported R R (⋃ I ∈ c, ↑I)
hsum : (Finsupp.total ι M R s) f = 0
⊢ f = 0

Next tactic:
rcases eq_empty_or_nonempty c with (rfl | hn)
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 := by
  let indep : Set ι → Prop := fun I => LinearIndependent R (s ∘ (↑) : I → M)
  let X := { I : Set ι // indep I }
  let r : X → X → Prop := fun I J => I.1 ⊆ J.1
  have key : ∀ c : Set X, IsChain r c → indep (⋃ (I : X) (_ : I ∈ c), I) := by
    intro c hc
    dsimp
    rw [linearIndependent_comp_subtype]
    intro f hsupport hsum
    rcases eq_empty_or_nonempty c with (rfl | hn)
    · 

Proof state:
case inl
ι : Type u'
ι' : Type u_1
R : Type u_2
K : Type u_3
M : Type u_4
M' : Type u_5
M'' : Type u_6
V : Type u
V' : Type u_7
v : ι → M
inst✝⁶ : Ring R
inst✝⁵ : AddCommGroup M
inst✝⁴ : AddCommGroup M'
inst✝³ : AddCommGroup M''
inst✝² : Module R M
inst✝¹ : Module R M'
inst✝ : Module R M''
a b : R
x y : M
hv : LinearIndependent R v
s : ι → M
indep : Set ι → Prop := fun I => LinearIndependent (ι := ↑I) R (s ∘ Subtype.val)
X : Type u' := { I // indep I }
r : X → X → Prop := fun I J => ↑I ⊆ ↑J
f : ι →₀ R
hsum : (Finsupp.total ι M R s) f = 0
hc : IsChain r ∅
hsupport : f ∈ Finsupp.supported R R (⋃ I ∈ ∅, ↑I)
⊢ f = 0

Next tactic:
simpa using hsupport
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 := by
  let indep : Set ι → Prop := fun I => LinearIndependent R (s ∘ (↑) : I → M)
  let X := { I : Set ι // indep I }
  let r : X → X → Prop := fun I J => I.1 ⊆ J.1
  have key : ∀ c : Set X, IsChain r c → indep (⋃ (I : X) (_ : I ∈ c), I) := by
    intro c hc
    dsimp
    rw [linearIndependent_comp_subtype]
    intro f hsupport hsum
    rcases eq_empty_or_nonempty c with (rfl | hn)
    · simpa using hsupport
    

Proof state:
case inr
ι : Type u'
ι' : Type u_1
R : Type u_2
K : Type u_3
M : Type u_4
M' : Type u_5
M'' : Type u_6
V : Type u
V' : Type u_7
v : ι → M
inst✝⁶ : Ring R
inst✝⁵ : AddCommGroup M
inst✝⁴ : AddCommGroup M'
inst✝³ : AddCommGroup M''
inst✝² : Module R M
inst✝¹ : Module R M'
inst✝ : Module R M''
a b : R
x y : M
hv : LinearIndependent R v
s : ι → M
indep : Set ι → Prop := fun I => LinearIndependent (ι := ↑I) R (s ∘ Subtype.val)
X : Type u' := { I // indep I }
r : X → X → Prop := fun I J => ↑I ⊆ ↑J
c : Set X
hc : IsChain r c
f : ι →₀ R
hsupport : f ∈ Finsupp.supported R R (⋃ I ∈ c, ↑I)
hsum : (Finsupp.total ι M R s) f = 0
hn : Set.Nonempty c
⊢ f = 0

Next tactic:
haveI : IsRefl X r := ⟨fun _ => Set.Subset.refl _⟩
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 := by
  let indep : Set ι → Prop := fun I => LinearIndependent R (s ∘ (↑) : I → M)
  let X := { I : Set ι // indep I }
  let r : X → X → Prop := fun I J => I.1 ⊆ J.1
  have key : ∀ c : Set X, IsChain r c → indep (⋃ (I : X) (_ : I ∈ c), I) := by
    intro c hc
    dsimp
    rw [linearIndependent_comp_subtype]
    intro f hsupport hsum
    rcases eq_empty_or_nonempty c with (rfl | hn)
    · simpa using hsupport
    haveI : IsRefl X r := ⟨fun _ => Set.Subset.refl _⟩
    

Proof state:
case inr
ι : Type u'
ι' : Type u_1
R : Type u_2
K : Type u_3
M : Type u_4
M' : Type u_5
M'' : Type u_6
V : Type u
V' : Type u_7
v : ι → M
inst✝⁶ : Ring R
inst✝⁵ : AddCommGroup M
inst✝⁴ : AddCommGroup M'
inst✝³ : AddCommGroup M''
inst✝² : Module R M
inst✝¹ : Module R M'
inst✝ : Module R M''
a b : R
x y : M
hv : LinearIndependent R v
s : ι → M
indep : Set ι → Prop := fun I => LinearIndependent (ι := ↑I) R (s ∘ Subtype.val)
X : Type u' := { I // indep I }
r : X → X → Prop := fun I J => ↑I ⊆ ↑J
c : Set X
hc : IsChain r c
f : ι →₀ R
hsupport : f ∈ Finsupp.supported R R (⋃ I ∈ c, ↑I)
hsum : (Finsupp.total ι M R s) f = 0
hn : Set.Nonempty c
this : IsRefl X r
⊢ f = 0

Next tactic:
obtain ⟨I, _I_mem, hI⟩ : ∃ I ∈ c, (f.support : Set ι) ⊆ I :=
      hc.directedOn.exists_mem_subset_of_finset_subset_biUnion hn hsupport
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 := by
  let indep : Set ι → Prop := fun I => LinearIndependent R (s ∘ (↑) : I → M)
  let X := { I : Set ι // indep I }
  let r : X → X → Prop := fun I J => I.1 ⊆ J.1
  have key : ∀ c : Set X, IsChain r c → indep (⋃ (I : X) (_ : I ∈ c), I) := by
    intro c hc
    dsimp
    rw [linearIndependent_comp_subtype]
    intro f hsupport hsum
    rcases eq_empty_or_nonempty c with (rfl | hn)
    · simpa using hsupport
    haveI : IsRefl X r := ⟨fun _ => Set.Subset.refl _⟩
    obtain ⟨I, _I_mem, hI⟩ : ∃ I ∈ c, (f.support : Set ι) ⊆ I :=
      hc.directedOn.exists_mem_subset_of_finset_subset_biUnion hn hsupport
    

Proof state:
case inr.intro.intro
ι : Type u'
ι' : Type u_1
R : Type u_2
K : Type u_3
M : Type u_4
M' : Type u_5
M'' : Type u_6
V : Type u
V' : Type u_7
v : ι → M
inst✝⁶ : Ring R
inst✝⁵ : AddCommGroup M
inst✝⁴ : AddCommGroup M'
inst✝³ : AddCommGroup M''
inst✝² : Module R M
inst✝¹ : Module R M'
inst✝ : Module R M''
a b : R
x y : M
hv : LinearIndependent R v
s : ι → M
indep : Set ι → Prop := fun I => LinearIndependent (ι := ↑I) R (s ∘ Subtype.val)
X : Type u' := { I // indep I }
r : X → X → Prop := fun I J => ↑I ⊆ ↑J
c : Set X
hc : IsChain r c
f : ι →₀ R
hsupport : f ∈ Finsupp.supported R R (⋃ I ∈ c, ↑I)
hsum : (Finsupp.total ι M R s) f = 0
hn : Set.Nonempty c
this : IsRefl X r
I : X
_I_mem : I ∈ c
hI : ↑f.support ⊆ ↑I
⊢ f = 0

Next tactic:
exact linearIndependent_comp_subtype.mp I.2 f hI hsum
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 := by
  let indep : Set ι → Prop := fun I => LinearIndependent R (s ∘ (↑) : I → M)
  let X := { I : Set ι // indep I }
  let r : X → X → Prop := fun I J => I.1 ⊆ J.1
  have key : ∀ c : Set X, IsChain r c → indep (⋃ (I : X) (_ : I ∈ c), I) := by
    intro c hc
    dsimp
    rw [linearIndependent_comp_subtype]
    intro f hsupport hsum
    rcases eq_empty_or_nonempty c with (rfl | hn)
    · simpa using hsupport
    haveI : IsRefl X r := ⟨fun _ => Set.Subset.refl _⟩
    obtain ⟨I, _I_mem, hI⟩ : ∃ I ∈ c, (f.support : Set ι) ⊆ I :=
      hc.directedOn.exists_mem_subset_of_finset_subset_biUnion hn hsupport
    exact linearIndependent_comp_subtype.mp I.2 f hI hsum
  

Proof state:
ι : Type u'
ι' : Type u_1
R : Type u_2
K : Type u_3
M : Type u_4
M' : Type u_5
M'' : Type u_6
V : Type u
V' : Type u_7
v : ι → M
inst✝⁶ : Ring R
inst✝⁵ : AddCommGroup M
inst✝⁴ : AddCommGroup M'
inst✝³ : AddCommGroup M''
inst✝² : Module R M
inst✝¹ : Module R M'
inst✝ : Module R M''
a b : R
x y : M
hv : LinearIndependent R v
s : ι → M
indep : Set ι → Prop := fun I => LinearIndependent (ι := ↑I) R (s ∘ Subtype.val)
X : Type u' := { I // indep I }
r : X → X → Prop := fun I J => ↑I ⊆ ↑J
key : ∀ (c : Set X), IsChain r c → indep (⋃ I ∈ c, ↑I)
⊢ ∃ I,
    (LinearIndependent R fun (x : ↑I) => s ↑x) ∧
      ∀ (J : Set ι), I ⊆ J → (LinearIndependent R fun (x : ↑J) => s ↑x) → I = J

Next tactic:
have trans : Transitive r := fun I J K => Set.Subset.trans
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 := by
  let indep : Set ι → Prop := fun I => LinearIndependent R (s ∘ (↑) : I → M)
  let X := { I : Set ι // indep I }
  let r : X → X → Prop := fun I J => I.1 ⊆ J.1
  have key : ∀ c : Set X, IsChain r c → indep (⋃ (I : X) (_ : I ∈ c), I) := by
    intro c hc
    dsimp
    rw [linearIndependent_comp_subtype]
    intro f hsupport hsum
    rcases eq_empty_or_nonempty c with (rfl | hn)
    · simpa using hsupport
    haveI : IsRefl X r := ⟨fun _ => Set.Subset.refl _⟩
    obtain ⟨I, _I_mem, hI⟩ : ∃ I ∈ c, (f.support : Set ι) ⊆ I :=
      hc.directedOn.exists_mem_subset_of_finset_subset_biUnion hn hsupport
    exact linearIndependent_comp_subtype.mp I.2 f hI hsum
  have trans : Transitive r := fun I J K => Set.Subset.trans
  

Proof state:
ι : Type u'
ι' : Type u_1
R : Type u_2
K : Type u_3
M : Type u_4
M' : Type u_5
M'' : Type u_6
V : Type u
V' : Type u_7
v : ι → M
inst✝⁶ : Ring R
inst✝⁵ : AddCommGroup M
inst✝⁴ : AddCommGroup M'
inst✝³ : AddCommGroup M''
inst✝² : Module R M
inst✝¹ : Module R M'
inst✝ : Module R M''
a b : R
x y : M
hv : LinearIndependent R v
s : ι → M
indep : Set ι → Prop := fun I => LinearIndependent (ι := ↑I) R (s ∘ Subtype.val)
X : Type u' := { I // indep I }
r : X → X → Prop := fun I J => ↑I ⊆ ↑J
key : ∀ (c : Set X), IsChain r c → indep (⋃ I ∈ c, ↑I)
trans : Transitive r
⊢ ∃ I,
    (LinearIndependent R fun (x : ↑I) => s ↑x) ∧
      ∀ (J : Set ι), I ⊆ J → (LinearIndependent R fun (x : ↑J) => s ↑x) → I = J

Next tactic:
obtain ⟨⟨I, hli : indep I⟩, hmax : ∀ a, r ⟨I, hli⟩ a → r a ⟨I, hli⟩⟩ :=
    exists_maximal_of_chains_bounded
      (fun c hc => ⟨⟨⋃ I ∈ c, (I : Set ι), key c hc⟩, fun I => Set.subset_biUnion_of_mem⟩) @trans
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 := by
  let indep : Set ι → Prop := fun I => LinearIndependent R (s ∘ (↑) : I → M)
  let X := { I : Set ι // indep I }
  let r : X → X → Prop := fun I J => I.1 ⊆ J.1
  have key : ∀ c : Set X, IsChain r c → indep (⋃ (I : X) (_ : I ∈ c), I) := by
    intro c hc
    dsimp
    rw [linearIndependent_comp_subtype]
    intro f hsupport hsum
    rcases eq_empty_or_nonempty c with (rfl | hn)
    · simpa using hsupport
    haveI : IsRefl X r := ⟨fun _ => Set.Subset.refl _⟩
    obtain ⟨I, _I_mem, hI⟩ : ∃ I ∈ c, (f.support : Set ι) ⊆ I :=
      hc.directedOn.exists_mem_subset_of_finset_subset_biUnion hn hsupport
    exact linearIndependent_comp_subtype.mp I.2 f hI hsum
  have trans : Transitive r := fun I J K => Set.Subset.trans
  obtain ⟨⟨I, hli : indep I⟩, hmax : ∀ a, r ⟨I, hli⟩ a → r a ⟨I, hli⟩⟩ :=
    exists_maximal_of_chains_bounded
      (fun c hc => ⟨⟨⋃ I ∈ c, (I : Set ι), key c hc⟩, fun I => Set.subset_biUnion_of_mem⟩) @trans
  

Proof state:
case intro.mk
ι : Type u'
ι' : Type u_1
R : Type u_2
K : Type u_3
M : Type u_4
M' : Type u_5
M'' : Type u_6
V : Type u
V' : Type u_7
v : ι → M
inst✝⁶ : Ring R
inst✝⁵ : AddCommGroup M
inst✝⁴ : AddCommGroup M'
inst✝³ : AddCommGroup M''
inst✝² : Module R M
inst✝¹ : Module R M'
inst✝ : Module R M''
a b : R
x y : M
hv : LinearIndependent R v
s : ι → M
indep : Set ι → Prop := fun I => LinearIndependent (ι := ↑I) R (s ∘ Subtype.val)
X : Type u' := { I // indep I }
r : X → X → Prop := fun I J => ↑I ⊆ ↑J
key : ∀ (c : Set X), IsChain r c → indep (⋃ I ∈ c, ↑I)
trans : Transitive r
I : Set ι
hli : indep I
hmax : ∀ (a : X), r { val := I, property := hli } a → r a { val := I, property := hli }
⊢ ∃ I,
    (LinearIndependent R fun (x : ↑I) => s ↑x) ∧
      ∀ (J : Set ι), I ⊆ J → (LinearIndependent R fun (x : ↑J) => s ↑x) → I = J

Next tactic:
exact ⟨I, hli, fun J hsub hli => Set.Subset.antisymm hsub (hmax ⟨J, hli⟩ hsub)⟩
theorem exists_maximal_independent (s : ι → M) :
    ∃ I : Set ι,
      (LinearIndependent R fun x : I => s x) ∧
        ∀ (i) (_ : i ∉ I), ∃ a : R, a ≠ 0 ∧ a • s i ∈ span R (s '' I) := by
  

Proof state:
ι : Type u'
ι' : Type u_1
R : Type u_2
K : Type u_3
M : Type u_4
M' : Type u_5
M'' : Type u_6
V : Type u
V' : Type u_7
v : ι → M
inst✝⁶ : Ring R
inst✝⁵ : AddCommGroup M
inst✝⁴ : AddCommGroup M'
inst✝³ : AddCommGroup M''
inst✝² : Module R M
inst✝¹ : Module R M'
inst✝ : Module R M''
a b : R
x y : M
hv : LinearIndependent R v
s : ι → M
⊢ ∃ I, (LinearIndependent R fun (x : ↑I) => s ↑x) ∧ ∀ i ∉ I, ∃ a, a ≠ 0 ∧ a • s i ∈ span R (s '' I)

Next tactic:
classical
    rcases exists_maximal_independent' R s with ⟨I, hIlinind, hImaximal⟩
    use I, hIlinind
    intro i hi
    specialize hImaximal (I ∪ {i}) (by simp)
    set J := I ∪ {i} with hJ
    have memJ : ∀ {x}, x ∈ J ↔ x = i ∨ x ∈ I := by simp [hJ]
    have hiJ : i ∈ J := by simp
    have h := by
      refine mt hImaximal ?_
      · intro h2
        rw [h2] at hi
        exact absurd hiJ hi
    obtain ⟨f, supp_f, sum_f, f_ne⟩ := linearDependent_comp_subtype.mp h
    have hfi : f i ≠ 0 := by
      contrapose hIlinind
      refine' linearDependent_comp_subtype.mpr ⟨f, _, sum_f, f_ne⟩
      simp only [Finsupp.mem_supported, hJ] at supp_f ⊢
      rintro x hx
      refine' (memJ.mp (supp_f hx)).resolve_left _
      rintro rfl
      exact hIlinind (Finsupp.mem_support_iff.mp hx)
    use f i, hfi
    have hfi' : i ∈ f.support := Finsupp.mem_support_iff.mpr hfi
    rw [← Finset.insert_erase hfi', Finset.sum_insert (Finset.not_mem_erase _ _),
      add_eq_zero_iff_eq_neg] at sum_f
    rw [sum_f]
    refine' neg_mem (sum_mem fun c hc => smul_mem _ _ (subset_span ⟨c, _, rfl⟩))
    exact (memJ.mp (supp_f (Finset.erase_subset _ _ hc))).resolve_left (Finset.ne_of_mem_erase hc)
theorem exists_maximal_independent (s : ι → M) :
    ∃ I : Set ι,
      (LinearIndependent R fun x : I => s x) ∧
        ∀ (i) (_ : i ∉ I), ∃ a : R, a ≠ 0 ∧ a • s i ∈ span R (s '' I) := by
  classical
    

Proof state:
ι : Type u'
ι' : Type u_1
R : Type u_2
K : Type u_3
M : Type u_4
M' : Type u_5
M'' : Type u_6
V : Type u
V' : Type u_7
v : ι → M
inst✝⁶ : Ring R
inst✝⁵ : AddCommGroup M
inst✝⁴ : AddCommGroup M'
inst✝³ : AddCommGroup M''
inst✝² : Module R M
inst✝¹ : Module R M'
inst✝ : Module R M''
a b : R
x y : M
hv : LinearIndependent R v
s : ι → M
⊢ ∃ I, (LinearIndependent R fun (x : ↑I) => s ↑x) ∧ ∀ i ∉ I, ∃ a, a ≠ 0 ∧ a • s i ∈ span R (s '' I)

Next tactic:
rcases exists_maximal_independent' R s with ⟨I, hIlinind, hImaximal⟩
theorem exists_maximal_independent (s : ι → M) :
    ∃ I : Set ι,
      (LinearIndependent R fun x : I => s x) ∧
        ∀ (i) (_ : i ∉ I), ∃ a : R, a ≠ 0 ∧ a • s i ∈ span R (s '' I) := by
  classical
    rcases exists_maximal_independent' R s with ⟨I, hIlinind, hImaximal⟩
    

Proof state:
case intro.intro
ι : Type u'
ι' : Type u_1
R : Type u_2
K : Type u_3
M : Type u_4
M' : Type u_5
M'' : Type u_6
V : Type u
V' : Type u_7
v : ι → M
inst✝⁶ : Ring R
inst✝⁵ : AddCommGroup M
inst✝⁴ : AddCommGroup M'
inst✝³ : AddCommGroup M''
inst✝² : Module R M
inst✝¹ : Module R M'
inst✝ : Module R M''
a b : R
x y : M
hv : LinearIndependent R v
s : ι → M
I : Set ι
hIlinind : LinearIndependent R fun (x : ↑I) => s ↑x
hImaximal : ∀ (J : Set ι), I ⊆ J → (LinearIndependent R fun (x : ↑J) => s ↑x) → I = J
⊢ ∃ I, (LinearIndependent R fun (x : ↑I) => s ↑x) ∧ ∀ i ∉ I, ∃ a, a ≠ 0 ∧ a • s i ∈ span R (s '' I)

Next tactic:
use I, hIlinind
theorem exists_maximal_independent (s : ι → M) :
    ∃ I : Set ι,
      (LinearIndependent R fun x : I => s x) ∧
        ∀ (i) (_ : i ∉ I), ∃ a : R, a ≠ 0 ∧ a • s i ∈ span R (s '' I) := by
  classical
    rcases exists_maximal_independent' R s with ⟨I, hIlinind, hImaximal⟩
    use I, hIlinind
    

Proof state:
case right
ι : Type u'
ι' : Type u_1
R : Type u_2
K : Type u_3
M : Type u_4
M' : Type u_5
M'' : Type u_6
V : Type u
V' : Type u_7
v : ι → M
inst✝⁶ : Ring R
inst✝⁵ : AddCommGroup M
inst✝⁴ : AddCommGroup M'
inst✝³ : AddCommGroup M''
inst✝² : Module R M
inst✝¹ : Module R M'
inst✝ : Module R M''
a b : R
x y : M
hv : LinearIndependent R v
s : ι → M
I : Set ι
hIlinind : LinearIndependent R fun (x : ↑I) => s ↑x
hImaximal : ∀ (J : Set ι), I ⊆ J → (LinearIndependent R fun (x : ↑J) => s ↑x) → I = J
⊢ ∀ i ∉ I, ∃ a, a ≠ 0 ∧ a • s i ∈ span R (s '' I)

Next tactic:
intro i hi
theorem exists_maximal_independent (s : ι → M) :
    ∃ I : Set ι,
      (LinearIndependent R fun x : I => s x) ∧
        ∀ (i) (_ : i ∉ I), ∃ a : R, a ≠ 0 ∧ a • s i ∈ span R (s '' I) := by
  classical
    rcases exists_maximal_independent' R s with ⟨I, hIlinind, hImaximal⟩
    use I, hIlinind
    intro i hi
    

Proof state:
case right
ι : Type u'
ι' : Type u_1
R : Type u_2
K : Type u_3
M : Type u_4
M' : Type u_5
M'' : Type u_6
V : Type u
V' : Type u_7
v : ι → M
inst✝⁶ : Ring R
inst✝⁵ : AddCommGroup M
inst✝⁴ : AddCommGroup M'
inst✝³ : AddCommGroup M''
inst✝² : Module R M
inst✝¹ : Module R M'
inst✝ : Module R M''
a b : R
x y : M
hv : LinearIndependent R v
s : ι → M
I : Set ι
hIlinind : LinearIndependent R fun (x : ↑I) => s ↑x
hImaximal : ∀ (J : Set ι), I ⊆ J → (LinearIndependent R fun (x : ↑J) => s ↑x) → I = J
i : ι
hi : i ∉ I
⊢ ∃ a, a ≠ 0 ∧ a • s i ∈ span R (s '' I)

Next tactic:
specialize hImaximal (I ∪ {i}) (by simp)
theorem exists_maximal_independent (s : ι → M) :
    ∃ I : Set ι,
      (LinearIndependent R fun x : I => s x) ∧
        ∀ (i) (_ : i ∉ I), ∃ a : R, a ≠ 0 ∧ a • s i ∈ span R (s '' I) := by
  classical
    rcases exists_maximal_independent' R s with ⟨I, hIlinind, hImaximal⟩
    use I, hIlinind
    intro i hi
    specialize hImaximal (I ∪ {i}) (by 

Proof state:
ι : Type u'
ι' : Type u_1
R : Type u_2
K : Type u_3
M : Type u_4
M' : Type u_5
M'' : Type u_6
V : Type u
V' : Type u_7
v : ι → M
inst✝⁶ : Ring R
inst✝⁵ : AddCommGroup M
inst✝⁴ : AddCommGroup M'
inst✝³ : AddCommGroup M''
inst✝² : Module R M
inst✝¹ : Module R M'
inst✝ : Module R M''
a b : R
x y : M
hv : LinearIndependent R v
s : ι → M
I : Set ι
hIlinind : LinearIndependent R fun (x : ↑I) => s ↑x
hImaximal : ∀ (J : Set ι), I ⊆ J → (LinearIndependent R fun (x : ↑J) => s ↑x) → I = J
i : ι
hi : i ∉ I
⊢ I ⊆ I ∪ {i}

Next tactic:
simp
theorem exists_maximal_independent (s : ι → M) :
    ∃ I : Set ι,
      (LinearIndependent R fun x : I => s x) ∧
        ∀ (i) (_ : i ∉ I), ∃ a : R, a ≠ 0 ∧ a • s i ∈ span R (s '' I) := by
  classical
    rcases exists_maximal_independent' R s with ⟨I, hIlinind, hImaximal⟩
    use I, hIlinind
    intro i hi
    specialize hImaximal (I ∪ {i}) (by simp)
    

Proof state:
case right
ι : Type u'
ι' : Type u_1
R : Type u_2
K : Type u_3
M : Type u_4
M' : Type u_5
M'' : Type u_6
V : Type u
V' : Type u_7
v : ι → M
inst✝⁶ : Ring R
inst✝⁵ : AddCommGroup M
inst✝⁴ : AddCommGroup M'
inst✝³ : AddCommGroup M''
inst✝² : Module R M
inst✝¹ : Module R M'
inst✝ : Module R M''
a b : R
x y : M
hv : LinearIndependent R v
s : ι → M
I : Set ι
hIlinind : LinearIndependent R fun (x : ↑I) => s ↑x
i : ι
hi : i ∉ I
hImaximal : (LinearIndependent R fun (x : ↑(I ∪ {i})) => s ↑x) → I = I ∪ {i}
⊢ ∃ a, a ≠ 0 ∧ a • s i ∈ span R (s '' I)

Next tactic:
set J := I ∪ {i} with hJ
theorem exists_maximal_independent (s : ι → M) :
    ∃ I : Set ι,
      (LinearIndependent R fun x : I => s x) ∧
        ∀ (i) (_ : i ∉ I), ∃ a : R, a ≠ 0 ∧ a • s i ∈ span R (s '' I) := by
  classical
    rcases exists_maximal_independent' R s with ⟨I, hIlinind, hImaximal⟩
    use I, hIlinind
    intro i hi
    specialize hImaximal (I ∪ {i}) (by simp)
    set J := I ∪ {i} with hJ
    

Proof state:
case right
ι : Type u'
ι' : Type u_1
R : Type u_2
K : Type u_3
M : Type u_4
M' : Type u_5
M'' : Type u_6
V : Type u
V' : Type u_7
v : ι → M
inst✝⁶ : Ring R
inst✝⁵ : AddCommGroup M
inst✝⁴ : AddCommGroup M'
inst✝³ : AddCommGroup M''
inst✝² : Module R M
inst✝¹ : Module R M'
inst✝ : Module R M''
a b : R
x y : M
hv : LinearIndependent R v
s : ι → M
I : Set ι
hIlinind : LinearIndependent R fun (x : ↑I) => s ↑x
i : ι
hi : i ∉ I
J : Set ι := I ∪ {i}
hImaximal : (LinearIndependent R fun (x : ↑J) => s ↑x) → I = J
hJ : J = I ∪ {i}
⊢ ∃ a, a ≠ 0 ∧ a • s i ∈ span R (s '' I)

Next tactic:
have memJ : ∀ {x}, x ∈ J ↔ x = i ∨ x ∈ I := by simp [hJ]
theorem exists_maximal_independent (s : ι → M) :
    ∃ I : Set ι,
      (LinearIndependent R fun x : I => s x) ∧
        ∀ (i) (_ : i ∉ I), ∃ a : R, a ≠ 0 ∧ a • s i ∈ span R (s '' I) := by
  classical
    rcases exists_maximal_independent' R s with ⟨I, hIlinind, hImaximal⟩
    use I, hIlinind
    intro i hi
    specialize hImaximal (I ∪ {i}) (by simp)
    set J := I ∪ {i} with hJ
    have memJ : ∀ {x}, x ∈ J ↔ x = i ∨ x ∈ I := by 

Proof state:
ι : Type u'
ι' : Type u_1
R : Type u_2
K : Type u_3
M : Type u_4
M' : Type u_5
M'' : Type u_6
V : Type u
V' : Type u_7
v : ι → M
inst✝⁶ : Ring R
inst✝⁵ : AddCommGroup M
inst✝⁴ : AddCommGroup M'
inst✝³ : AddCommGroup M''
inst✝² : Module R M
inst✝¹ : Module R M'
inst✝ : Module R M''
a b : R
x y : M
hv : LinearIndependent R v
s : ι → M
I : Set ι
hIlinind : LinearIndependent R fun (x : ↑I) => s ↑x
i : ι
hi : i ∉ I
J : Set ι := I ∪ {i}
hImaximal : (LinearIndependent R fun (x : ↑J) => s ↑x) → I = J
hJ : J = I ∪ {i}
⊢ ∀ {x : ι}, x ∈ J ↔ x = i ∨ x ∈ I

Next tactic:
simp [hJ]
theorem exists_maximal_independent (s : ι → M) :
    ∃ I : Set ι,
      (LinearIndependent R fun x : I => s x) ∧
        ∀ (i) (_ : i ∉ I), ∃ a : R, a ≠ 0 ∧ a • s i ∈ span R (s '' I) := by
  classical
    rcases exists_maximal_independent' R s with ⟨I, hIlinind, hImaximal⟩
    use I, hIlinind
    intro i hi
    specialize hImaximal (I ∪ {i}) (by simp)
    set J := I ∪ {i} with hJ
    have memJ : ∀ {x}, x ∈ J ↔ x = i ∨ x ∈ I := by simp [hJ]
    

Proof state:
case right
ι : Type u'
ι' : Type u_1
R : Type u_2
K : Type u_3
M : Type u_4
M' : Type u_5
M'' : Type u_6
V : Type u
V' : Type u_7
v : ι → M
inst✝⁶ : Ring R
inst✝⁵ : AddCommGroup M
inst✝⁴ : AddCommGroup M'
inst✝³ : AddCommGroup M''
inst✝² : Module R M
inst✝¹ : Module R M'
inst✝ : Module R M''
a b : R
x y : M
hv : LinearIndependent R v
s : ι → M
I : Set ι
hIlinind : LinearIndependent R fun (x : ↑I) => s ↑x
i : ι
hi : i ∉ I
J : Set ι := I ∪ {i}
hImaximal : (LinearIndependent R fun (x : ↑J) => s ↑x) → I = J
hJ : J = I ∪ {i}
memJ : ∀ {x : ι}, x ∈ J ↔ x = i ∨ x ∈ I
⊢ ∃ a, a ≠ 0 ∧ a • s i ∈ span R (s '' I)

Next tactic:
have hiJ : i ∈ J := by simp
theorem exists_maximal_independent (s : ι → M) :
    ∃ I : Set ι,
      (LinearIndependent R fun x : I => s x) ∧
        ∀ (i) (_ : i ∉ I), ∃ a : R, a ≠ 0 ∧ a • s i ∈ span R (s '' I) := by
  classical
    rcases exists_maximal_independent' R s with ⟨I, hIlinind, hImaximal⟩
    use I, hIlinind
    intro i hi
    specialize hImaximal (I ∪ {i}) (by simp)
    set J := I ∪ {i} with hJ
    have memJ : ∀ {x}, x ∈ J ↔ x = i ∨ x ∈ I := by simp [hJ]
    have hiJ : i ∈ J := by 

Proof state:
ι : Type u'
ι' : Type u_1
R : Type u_2
K : Type u_3
M : Type u_4
M' : Type u_5
M'' : Type u_6
V : Type u
V' : Type u_7
v : ι → M
inst✝⁶ : Ring R
inst✝⁵ : AddCommGroup M
inst✝⁴ : AddCommGroup M'
inst✝³ : AddCommGroup M''
inst✝² : Module R M
inst✝¹ : Module R M'
inst✝ : Module R M''
a b : R
x y : M
hv : LinearIndependent R v
s : ι → M
I : Set ι
hIlinind : LinearIndependent R fun (x : ↑I) => s ↑x
i : ι
hi : i ∉ I
J : Set ι := I ∪ {i}
hImaximal : (LinearIndependent R fun (x : ↑J) => s ↑x) → I = J
hJ : J = I ∪ {i}
memJ : ∀ {x : ι}, x ∈ J ↔ x = i ∨ x ∈ I
⊢ i ∈ J

Next tactic:
simp