l3lab/ntp-mathlib huggingface dataset:
theorem linearIndependent_iff_not_smul_mem_span :
    LinearIndependent R v ↔ ∀ (i : ι) (a : R), a • v i ∈ span R (v '' (univ \ {i})) → a = 0 :=
  ⟨fun hv i a ha => by
    rw [Finsupp.span_image_eq_map_total, mem_map] at ha
    rcases ha with ⟨l, hl, e⟩
    rw [sub_eq_zero.1 (linearIndependent_iff.1 hv (l - Finsupp.single i a) (by simp [e]))] at hl
    by_contra hn
    exact (not_mem_of_mem_diff (hl <| by simp [hn])) (mem_singleton _), fun H =>
    linearIndependent_iff.2 fun l hl => by
      ext i; simp only [Finsupp.zero_apply]
      by_contra hn
      refine' hn (H i _ _)
      refine' (Finsupp.mem_span_image_iff_total R).2 ⟨Finsupp.single i (l i) - l, _, _⟩
      · rw [Finsupp.mem_supported']
        intro j hj
        

Proof state:
case h.refine'_1
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
H : ∀ (i : ι) (a : R), a • v i ∈ span R (v '' (univ \ {i})) → a = 0
l : ι →₀ R
hl : (Finsupp.total ι M R v) l = 0
i : ι
hn : ¬l i = 0
j : ι
hj : j ∉ univ \ {i}
⊢ ((fun₀ | i => l i) - l) j = 0

Next tactic:
have hij : j = i :=
          Classical.not_not.1 fun hij : j ≠ i =>
            hj ((mem_diff _).2 ⟨mem_univ _, fun h => hij (eq_of_mem_singleton h)⟩)
theorem linearIndependent_iff_not_smul_mem_span :
    LinearIndependent R v ↔ ∀ (i : ι) (a : R), a • v i ∈ span R (v '' (univ \ {i})) → a = 0 :=
  ⟨fun hv i a ha => by
    rw [Finsupp.span_image_eq_map_total, mem_map] at ha
    rcases ha with ⟨l, hl, e⟩
    rw [sub_eq_zero.1 (linearIndependent_iff.1 hv (l - Finsupp.single i a) (by simp [e]))] at hl
    by_contra hn
    exact (not_mem_of_mem_diff (hl <| by simp [hn])) (mem_singleton _), fun H =>
    linearIndependent_iff.2 fun l hl => by
      ext i; simp only [Finsupp.zero_apply]
      by_contra hn
      refine' hn (H i _ _)
      refine' (Finsupp.mem_span_image_iff_total R).2 ⟨Finsupp.single i (l i) - l, _, _⟩
      · rw [Finsupp.mem_supported']
        intro j hj
        have hij : j = i :=
          Classical.not_not.1 fun hij : j ≠ i =>
            hj ((mem_diff _).2 ⟨mem_univ _, fun h => hij (eq_of_mem_singleton h)⟩)
        

Proof state:
case h.refine'_1
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
H : ∀ (i : ι) (a : R), a • v i ∈ span R (v '' (univ \ {i})) → a = 0
l : ι →₀ R
hl : (Finsupp.total ι M R v) l = 0
i : ι
hn : ¬l i = 0
j : ι
hj : j ∉ univ \ {i}
hij : j = i
⊢ ((fun₀ | i => l i) - l) j = 0

Next tactic:
simp [hij]
theorem linearIndependent_iff_not_smul_mem_span :
    LinearIndependent R v ↔ ∀ (i : ι) (a : R), a • v i ∈ span R (v '' (univ \ {i})) → a = 0 :=
  ⟨fun hv i a ha => by
    rw [Finsupp.span_image_eq_map_total, mem_map] at ha
    rcases ha with ⟨l, hl, e⟩
    rw [sub_eq_zero.1 (linearIndependent_iff.1 hv (l - Finsupp.single i a) (by simp [e]))] at hl
    by_contra hn
    exact (not_mem_of_mem_diff (hl <| by simp [hn])) (mem_singleton _), fun H =>
    linearIndependent_iff.2 fun l hl => by
      ext i; simp only [Finsupp.zero_apply]
      by_contra hn
      refine' hn (H i _ _)
      refine' (Finsupp.mem_span_image_iff_total R).2 ⟨Finsupp.single i (l i) - l, _, _⟩
      · rw [Finsupp.mem_supported']
        intro j hj
        have hij : j = i :=
          Classical.not_not.1 fun hij : j ≠ i =>
            hj ((mem_diff _).2 ⟨mem_univ _, fun h => hij (eq_of_mem_singleton h)⟩)
        simp [hij]
      · 

Proof state:
case h.refine'_2
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
H : ∀ (i : ι) (a : R), a • v i ∈ span R (v '' (univ \ {i})) → a = 0
l : ι →₀ R
hl : (Finsupp.total ι M R v) l = 0
i : ι
hn : ¬l i = 0
⊢ (Finsupp.total ι M R v) ((fun₀ | i => l i) - l) = l i • v i

Next tactic:
simp [hl]
/-- See also `CompleteLattice.independent_iff_linearIndependent_of_ne_zero`. -/
theorem LinearIndependent.independent_span_singleton (hv : LinearIndependent R v) :
    CompleteLattice.Independent fun i => R ∙ v 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✝ hv : LinearIndependent R v
⊢ CompleteLattice.Independent fun i => span R {v i}

Next tactic:
refine' CompleteLattice.independent_def.mp fun i => _
/-- See also `CompleteLattice.independent_iff_linearIndependent_of_ne_zero`. -/
theorem LinearIndependent.independent_span_singleton (hv : LinearIndependent R v) :
    CompleteLattice.Independent fun i => R ∙ v i := by
  refine' CompleteLattice.independent_def.mp fun i => _
  

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✝ hv : LinearIndependent R v
i : ι
⊢ Disjoint ((fun i => span R {v i}) i) (⨆ j, ⨆ (_ : j ≠ i), (fun i => span R {v i}) j)

Next tactic:
rw [disjoint_iff_inf_le]
/-- See also `CompleteLattice.independent_iff_linearIndependent_of_ne_zero`. -/
theorem LinearIndependent.independent_span_singleton (hv : LinearIndependent R v) :
    CompleteLattice.Independent fun i => R ∙ v i := by
  refine' CompleteLattice.independent_def.mp fun i => _
  rw [disjoint_iff_inf_le]
  

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✝ hv : LinearIndependent R v
i : ι
⊢ (fun i => span R {v i}) i ⊓ ⨆ j, ⨆ (_ : j ≠ i), (fun i => span R {v i}) j ≤ ⊥

Next tactic:
intro m hm
/-- See also `CompleteLattice.independent_iff_linearIndependent_of_ne_zero`. -/
theorem LinearIndependent.independent_span_singleton (hv : LinearIndependent R v) :
    CompleteLattice.Independent fun i => R ∙ v i := by
  refine' CompleteLattice.independent_def.mp fun i => _
  rw [disjoint_iff_inf_le]
  intro m hm
  

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✝ hv : LinearIndependent R v
i : ι
m : M
hm : m ∈ (fun i => span R {v i}) i ⊓ ⨆ j, ⨆ (_ : j ≠ i), (fun i => span R {v i}) j
⊢ m ∈ ⊥

Next tactic:
simp only [mem_inf, mem_span_singleton, iSup_subtype', ← span_range_eq_iSup] at hm
/-- See also `CompleteLattice.independent_iff_linearIndependent_of_ne_zero`. -/
theorem LinearIndependent.independent_span_singleton (hv : LinearIndependent R v) :
    CompleteLattice.Independent fun i => R ∙ v i := by
  refine' CompleteLattice.independent_def.mp fun i => _
  rw [disjoint_iff_inf_le]
  intro m hm
  simp only [mem_inf, mem_span_singleton, iSup_subtype', ← span_range_eq_iSup] at hm
  

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✝ hv : LinearIndependent R v
i : ι
m : M
hm : (∃ a, a • v i = m) ∧ m ∈ span R (range fun i_1 => v ↑i_1)
⊢ m ∈ ⊥

Next tactic:
obtain ⟨⟨r, rfl⟩, hm⟩ := hm
/-- See also `CompleteLattice.independent_iff_linearIndependent_of_ne_zero`. -/
theorem LinearIndependent.independent_span_singleton (hv : LinearIndependent R v) :
    CompleteLattice.Independent fun i => R ∙ v i := by
  refine' CompleteLattice.independent_def.mp fun i => _
  rw [disjoint_iff_inf_le]
  intro m hm
  simp only [mem_inf, mem_span_singleton, iSup_subtype', ← span_range_eq_iSup] at hm
  obtain ⟨⟨r, rfl⟩, hm⟩ := hm
  

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✝ hv : LinearIndependent R v
i : ι
r : R
hm : r • v i ∈ span R (range fun i_1 => v ↑i_1)
⊢ r • v i ∈ ⊥

Next tactic:
suffices r = 0 by simp [this]
/-- See also `CompleteLattice.independent_iff_linearIndependent_of_ne_zero`. -/
theorem LinearIndependent.independent_span_singleton (hv : LinearIndependent R v) :
    CompleteLattice.Independent fun i => R ∙ v i := by
  refine' CompleteLattice.independent_def.mp fun i => _
  rw [disjoint_iff_inf_le]
  intro m hm
  simp only [mem_inf, mem_span_singleton, iSup_subtype', ← span_range_eq_iSup] at hm
  obtain ⟨⟨r, rfl⟩, hm⟩ := hm
  suffices r = 0 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✝ hv : LinearIndependent R v
i : ι
r : R
hm : r • v i ∈ span R (range fun i_1 => v ↑i_1)
this : r = 0
⊢ r • v i ∈ ⊥

Next tactic:
simp [this]
/-- See also `CompleteLattice.independent_iff_linearIndependent_of_ne_zero`. -/
theorem LinearIndependent.independent_span_singleton (hv : LinearIndependent R v) :
    CompleteLattice.Independent fun i => R ∙ v i := by
  refine' CompleteLattice.independent_def.mp fun i => _
  rw [disjoint_iff_inf_le]
  intro m hm
  simp only [mem_inf, mem_span_singleton, iSup_subtype', ← span_range_eq_iSup] at hm
  obtain ⟨⟨r, rfl⟩, hm⟩ := hm
  suffices r = 0 by simp [this]
  

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✝ hv : LinearIndependent R v
i : ι
r : R
hm : r • v i ∈ span R (range fun i_1 => v ↑i_1)
⊢ r = 0

Next tactic:
apply linearIndependent_iff_not_smul_mem_span.mp hv i
/-- See also `CompleteLattice.independent_iff_linearIndependent_of_ne_zero`. -/
theorem LinearIndependent.independent_span_singleton (hv : LinearIndependent R v) :
    CompleteLattice.Independent fun i => R ∙ v i := by
  refine' CompleteLattice.independent_def.mp fun i => _
  rw [disjoint_iff_inf_le]
  intro m hm
  simp only [mem_inf, mem_span_singleton, iSup_subtype', ← span_range_eq_iSup] at hm
  obtain ⟨⟨r, rfl⟩, hm⟩ := hm
  suffices r = 0 by simp [this]
  apply linearIndependent_iff_not_smul_mem_span.mp hv i
  -- Porting note: The original proof was using `convert hm`.
  

Proof state:
case intro.intro.a
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✝ hv : LinearIndependent R v
i : ι
r : R
hm : r • v i ∈ span R (range fun i_1 => v ↑i_1)
⊢ r • v i ∈ span R (v '' (univ \ {i}))

Next tactic:
suffices v '' (univ \ {i}) = range fun j : { j // j ≠ i } => v j by rwa [this]
/-- See also `CompleteLattice.independent_iff_linearIndependent_of_ne_zero`. -/
theorem LinearIndependent.independent_span_singleton (hv : LinearIndependent R v) :
    CompleteLattice.Independent fun i => R ∙ v i := by
  refine' CompleteLattice.independent_def.mp fun i => _
  rw [disjoint_iff_inf_le]
  intro m hm
  simp only [mem_inf, mem_span_singleton, iSup_subtype', ← span_range_eq_iSup] at hm
  obtain ⟨⟨r, rfl⟩, hm⟩ := hm
  suffices r = 0 by simp [this]
  apply linearIndependent_iff_not_smul_mem_span.mp hv i
  -- Porting note: The original proof was using `convert hm`.
  suffices v '' (univ \ {i}) = range fun j : { j // j ≠ i } => v 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✝ hv : LinearIndependent R v
i : ι
r : R
hm : r • v i ∈ span R (range fun i_1 => v ↑i_1)
this : v '' (univ \ {i}) = range fun j => v ↑j
⊢ r • v i ∈ span R (v '' (univ \ {i}))

Next tactic:
rwa [this]
/-- See also `CompleteLattice.independent_iff_linearIndependent_of_ne_zero`. -/
theorem LinearIndependent.independent_span_singleton (hv : LinearIndependent R v) :
    CompleteLattice.Independent fun i => R ∙ v i := by
  refine' CompleteLattice.independent_def.mp fun i => _
  rw [disjoint_iff_inf_le]
  intro m hm
  simp only [mem_inf, mem_span_singleton, iSup_subtype', ← span_range_eq_iSup] at hm
  obtain ⟨⟨r, rfl⟩, hm⟩ := hm
  suffices r = 0 by simp [this]
  apply linearIndependent_iff_not_smul_mem_span.mp hv i
  -- Porting note: The original proof was using `convert hm`.
  suffices v '' (univ \ {i}) = range fun j : { j // j ≠ i } => v j by rwa [this]
  

Proof state:
case intro.intro.a
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✝ hv : LinearIndependent R v
i : ι
r : R
hm : r • v i ∈ span R (range fun i_1 => v ↑i_1)
⊢ v '' (univ \ {i}) = range fun j => v ↑j

Next tactic:
ext
/-- See also `CompleteLattice.independent_iff_linearIndependent_of_ne_zero`. -/
theorem LinearIndependent.independent_span_singleton (hv : LinearIndependent R v) :
    CompleteLattice.Independent fun i => R ∙ v i := by
  refine' CompleteLattice.independent_def.mp fun i => _
  rw [disjoint_iff_inf_le]
  intro m hm
  simp only [mem_inf, mem_span_singleton, iSup_subtype', ← span_range_eq_iSup] at hm
  obtain ⟨⟨r, rfl⟩, hm⟩ := hm
  suffices r = 0 by simp [this]
  apply linearIndependent_iff_not_smul_mem_span.mp hv i
  -- Porting note: The original proof was using `convert hm`.
  suffices v '' (univ \ {i}) = range fun j : { j // j ≠ i } => v j by rwa [this]
  ext
  

Proof state:
case intro.intro.a.h
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✝ hv : LinearIndependent R v
i : ι
r : R
hm : r • v i ∈ span R (range fun i_1 => v ↑i_1)
x✝ : M
⊢ x✝ ∈ v '' (univ \ {i}) ↔ x✝ ∈ range fun j => v ↑j

Next tactic:
simp
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
  

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) ∧
      ∀ (J : Set ι), I ⊆ J → (LinearIndependent R fun (x : ↑J) => s ↑x) → I = J

Next tactic:
let indep : Set ι → Prop := fun I => LinearIndependent R (s ∘ (↑) : I → M)
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)
  

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)
⊢ ∃ I,
    (LinearIndependent R fun (x : ↑I) => s ↑x) ∧
      ∀ (J : Set ι), I ⊆ J → (LinearIndependent R fun (x : ↑J) => s ↑x) → I = J

Next tactic:
let X := { I : Set ι // indep I }
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 }
  

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 }
⊢ ∃ I,
    (LinearIndependent R fun (x : ↑I) => s ↑x) ∧
      ∀ (J : Set ι), I ⊆ J → (LinearIndependent R fun (x : ↑J) => s ↑x) → I = J

Next tactic:
let r : X → X → Prop := fun I J => I.1 ⊆ J.1
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
  

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
⊢ ∃ 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 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
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
    

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), IsChain r c → indep (⋃ I ∈ c, ↑I)

Next tactic:
intro c hc
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
    

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
⊢ indep (⋃ I ∈ c, ↑I)

Next tactic:
dsimp
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
    

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
⊢ LinearIndependent (ι := ↑(⋃ I ∈ c, ↑I)) R (s ∘ Subtype.val)

Next tactic:
rw [linearIndependent_comp_subtype]
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]
    

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
⊢ ∀ l ∈ Finsupp.supported R R (⋃ I ∈ c, ↑I), (Finsupp.total ι M R s) l = 0 → l = 0

Next tactic:
intro f hsupport hsum