l3lab/ntp-mathlib huggingface dataset:
/-- If a finset has cardinality larger than the rank of a module,
then there is a nontrivial linear relation amongst its elements.
-/
-- TODO: golf this
theorem Module.exists_nontrivial_relation_of_finrank_lt_card
    [Module.Finite R V] {t : Finset V}
    (h : finrank R V < t.card) : ∃ f : V → R, ∑ e in t, f e • e = 0 ∧ ∃ x ∈ t, f x ≠ 0 := by
  classical
  have := mt FiniteDimensional.finset_card_le_finrank_of_linearIndependent (by simpa using h)
  rw [_root_.not_linearIndependent_iff] at this
  obtain ⟨s, g, sum, z, zm, nonzero⟩ := this
  -- Now we have to extend `g` to all of `t`, then to all of `V`.
  let f : V → R := fun x => if h : x ∈ t then if (⟨x, h⟩ : t) ∈ s then g ⟨x, h⟩ else 0 else 0
  -- and finally clean up the mess caused by the extension.
  refine' ⟨f, _, _⟩
  · dsimp
    rw [← (sum)] -- porting note: need parens to disambiguate
    fapply sum_bij_ne_zero fun v hvt _ => (⟨v, hvt⟩ : { v // v ∈ t })
    · intro v hvt H
      

Proof state:
case intro.intro.intro.intro.intro.refine'_1.hi
R : Type u
M : Type v
N : Type w
V : Type v
inst✝⁶ : AddCommGroup V
V₂ : Type v'
inst✝⁵ : AddCommGroup V₂
inst✝⁴ : Ring R
inst✝³ : StrongRankCondition R
inst✝² : Module R V
inst✝¹ : Free R V
K : ?m.320903
inst✝ : Finite R V
t : Finset V
h : finrank R V < Finset.card t
s : Finset { x // x ∈ t }
g : { x // x ∈ t } → R
sum : ∑ i in s, g i • ↑i = 0
z : { x // x ∈ t }
zm : z ∈ s
nonzero : g z ≠ 0
f : V → R :=
  fun x => if h : x ∈ t then if { val := x, property := h } ∈ s then g { val := x, property := h } else 0 else 0
v : V
hvt : v ∈ t
H : (if h : v ∈ t then if { val := v, property := h } ∈ s then g { val := v, property := h } else 0 else 0) • v ≠ 0
⊢ { val := v, property := hvt } ∈ s

Next tactic:
dsimp
/-- If a finset has cardinality larger than the rank of a module,
then there is a nontrivial linear relation amongst its elements.
-/
-- TODO: golf this
theorem Module.exists_nontrivial_relation_of_finrank_lt_card
    [Module.Finite R V] {t : Finset V}
    (h : finrank R V < t.card) : ∃ f : V → R, ∑ e in t, f e • e = 0 ∧ ∃ x ∈ t, f x ≠ 0 := by
  classical
  have := mt FiniteDimensional.finset_card_le_finrank_of_linearIndependent (by simpa using h)
  rw [_root_.not_linearIndependent_iff] at this
  obtain ⟨s, g, sum, z, zm, nonzero⟩ := this
  -- Now we have to extend `g` to all of `t`, then to all of `V`.
  let f : V → R := fun x => if h : x ∈ t then if (⟨x, h⟩ : t) ∈ s then g ⟨x, h⟩ else 0 else 0
  -- and finally clean up the mess caused by the extension.
  refine' ⟨f, _, _⟩
  · dsimp
    rw [← (sum)] -- porting note: need parens to disambiguate
    fapply sum_bij_ne_zero fun v hvt _ => (⟨v, hvt⟩ : { v // v ∈ t })
    · intro v hvt H
      dsimp
      

Proof state:
case intro.intro.intro.intro.intro.refine'_1.hi
R : Type u
M : Type v
N : Type w
V : Type v
inst✝⁶ : AddCommGroup V
V₂ : Type v'
inst✝⁵ : AddCommGroup V₂
inst✝⁴ : Ring R
inst✝³ : StrongRankCondition R
inst✝² : Module R V
inst✝¹ : Free R V
K : ?m.320903
inst✝ : Finite R V
t : Finset V
h : finrank R V < Finset.card t
s : Finset { x // x ∈ t }
g : { x // x ∈ t } → R
sum : ∑ i in s, g i • ↑i = 0
z : { x // x ∈ t }
zm : z ∈ s
nonzero : g z ≠ 0
f : V → R :=
  fun x => if h : x ∈ t then if { val := x, property := h } ∈ s then g { val := x, property := h } else 0 else 0
v : V
hvt : v ∈ t
H : (if h : v ∈ t then if { val := v, property := h } ∈ s then g { val := v, property := h } else 0 else 0) • v ≠ 0
⊢ { val := v, property := hvt } ∈ s

Next tactic:
rw [dif_pos hvt] at H
/-- If a finset has cardinality larger than the rank of a module,
then there is a nontrivial linear relation amongst its elements.
-/
-- TODO: golf this
theorem Module.exists_nontrivial_relation_of_finrank_lt_card
    [Module.Finite R V] {t : Finset V}
    (h : finrank R V < t.card) : ∃ f : V → R, ∑ e in t, f e • e = 0 ∧ ∃ x ∈ t, f x ≠ 0 := by
  classical
  have := mt FiniteDimensional.finset_card_le_finrank_of_linearIndependent (by simpa using h)
  rw [_root_.not_linearIndependent_iff] at this
  obtain ⟨s, g, sum, z, zm, nonzero⟩ := this
  -- Now we have to extend `g` to all of `t`, then to all of `V`.
  let f : V → R := fun x => if h : x ∈ t then if (⟨x, h⟩ : t) ∈ s then g ⟨x, h⟩ else 0 else 0
  -- and finally clean up the mess caused by the extension.
  refine' ⟨f, _, _⟩
  · dsimp
    rw [← (sum)] -- porting note: need parens to disambiguate
    fapply sum_bij_ne_zero fun v hvt _ => (⟨v, hvt⟩ : { v // v ∈ t })
    · intro v hvt H
      dsimp
      rw [dif_pos hvt] at H
      

Proof state:
case intro.intro.intro.intro.intro.refine'_1.hi
R : Type u
M : Type v
N : Type w
V : Type v
inst✝⁶ : AddCommGroup V
V₂ : Type v'
inst✝⁵ : AddCommGroup V₂
inst✝⁴ : Ring R
inst✝³ : StrongRankCondition R
inst✝² : Module R V
inst✝¹ : Free R V
K : ?m.320903
inst✝ : Finite R V
t : Finset V
h : finrank R V < Finset.card t
s : Finset { x // x ∈ t }
g : { x // x ∈ t } → R
sum : ∑ i in s, g i • ↑i = 0
z : { x // x ∈ t }
zm : z ∈ s
nonzero : g z ≠ 0
f : V → R :=
  fun x => if h : x ∈ t then if { val := x, property := h } ∈ s then g { val := x, property := h } else 0 else 0
v : V
hvt : v ∈ t
H : (if { val := v, property := hvt } ∈ s then g { val := v, property := hvt } else 0) • v ≠ 0
⊢ { val := v, property := hvt } ∈ s

Next tactic:
contrapose! H
/-- If a finset has cardinality larger than the rank of a module,
then there is a nontrivial linear relation amongst its elements.
-/
-- TODO: golf this
theorem Module.exists_nontrivial_relation_of_finrank_lt_card
    [Module.Finite R V] {t : Finset V}
    (h : finrank R V < t.card) : ∃ f : V → R, ∑ e in t, f e • e = 0 ∧ ∃ x ∈ t, f x ≠ 0 := by
  classical
  have := mt FiniteDimensional.finset_card_le_finrank_of_linearIndependent (by simpa using h)
  rw [_root_.not_linearIndependent_iff] at this
  obtain ⟨s, g, sum, z, zm, nonzero⟩ := this
  -- Now we have to extend `g` to all of `t`, then to all of `V`.
  let f : V → R := fun x => if h : x ∈ t then if (⟨x, h⟩ : t) ∈ s then g ⟨x, h⟩ else 0 else 0
  -- and finally clean up the mess caused by the extension.
  refine' ⟨f, _, _⟩
  · dsimp
    rw [← (sum)] -- porting note: need parens to disambiguate
    fapply sum_bij_ne_zero fun v hvt _ => (⟨v, hvt⟩ : { v // v ∈ t })
    · intro v hvt H
      dsimp
      rw [dif_pos hvt] at H
      contrapose! H
      

Proof state:
case intro.intro.intro.intro.intro.refine'_1.hi
R : Type u
M : Type v
N : Type w
V : Type v
inst✝⁶ : AddCommGroup V
V₂ : Type v'
inst✝⁵ : AddCommGroup V₂
inst✝⁴ : Ring R
inst✝³ : StrongRankCondition R
inst✝² : Module R V
inst✝¹ : Free R V
K : ?m.320903
inst✝ : Finite R V
t : Finset V
h : finrank R V < Finset.card t
s : Finset { x // x ∈ t }
g : { x // x ∈ t } → R
sum : ∑ i in s, g i • ↑i = 0
z : { x // x ∈ t }
zm : z ∈ s
nonzero : g z ≠ 0
f : V → R :=
  fun x => if h : x ∈ t then if { val := x, property := h } ∈ s then g { val := x, property := h } else 0 else 0
v : V
hvt : v ∈ t
H : { val := v, property := hvt } ∉ s
⊢ (if { val := v, property := hvt } ∈ s then g { val := v, property := hvt } else 0) • v = 0

Next tactic:
rw [if_neg H, zero_smul]
/-- If a finset has cardinality larger than the rank of a module,
then there is a nontrivial linear relation amongst its elements.
-/
-- TODO: golf this
theorem Module.exists_nontrivial_relation_of_finrank_lt_card
    [Module.Finite R V] {t : Finset V}
    (h : finrank R V < t.card) : ∃ f : V → R, ∑ e in t, f e • e = 0 ∧ ∃ x ∈ t, f x ≠ 0 := by
  classical
  have := mt FiniteDimensional.finset_card_le_finrank_of_linearIndependent (by simpa using h)
  rw [_root_.not_linearIndependent_iff] at this
  obtain ⟨s, g, sum, z, zm, nonzero⟩ := this
  -- Now we have to extend `g` to all of `t`, then to all of `V`.
  let f : V → R := fun x => if h : x ∈ t then if (⟨x, h⟩ : t) ∈ s then g ⟨x, h⟩ else 0 else 0
  -- and finally clean up the mess caused by the extension.
  refine' ⟨f, _, _⟩
  · dsimp
    rw [← (sum)] -- porting note: need parens to disambiguate
    fapply sum_bij_ne_zero fun v hvt _ => (⟨v, hvt⟩ : { v // v ∈ t })
    · intro v hvt H
      dsimp
      rw [dif_pos hvt] at H
      contrapose! H
      rw [if_neg H, zero_smul]
    · 

Proof state:
case intro.intro.intro.intro.intro.refine'_1.i_inj
R : Type u
M : Type v
N : Type w
V : Type v
inst✝⁶ : AddCommGroup V
V₂ : Type v'
inst✝⁵ : AddCommGroup V₂
inst✝⁴ : Ring R
inst✝³ : StrongRankCondition R
inst✝² : Module R V
inst✝¹ : Free R V
K : ?m.320903
inst✝ : Finite R V
t : Finset V
h : finrank R V < Finset.card t
s : Finset { x // x ∈ t }
g : { x // x ∈ t } → R
sum : ∑ i in s, g i • ↑i = 0
z : { x // x ∈ t }
zm : z ∈ s
nonzero : g z ≠ 0
f : V → R :=
  fun x => if h : x ∈ t then if { val := x, property := h } ∈ s then g { val := x, property := h } else 0 else 0
⊢ ∀ (a₁ a₂ : V) (h₁₁ : a₁ ∈ t),
    (if h : a₁ ∈ t then if { val := a₁, property := h } ∈ s then g { val := a₁, property := h } else 0 else 0) • a₁ ≠
        0 →
      ∀ (h₂₁ : a₂ ∈ t),
        (if h : a₂ ∈ t then if { val := a₂, property := h } ∈ s then g { val := a₂, property := h } else 0 else 0) •
              a₂ ≠
            0 →
          { val := a₁, property := h₁₁ } = { val := a₂, property := h₂₁ } → a₁ = a₂

Next tactic:
intro _ _ _ _ _ _
/-- If a finset has cardinality larger than the rank of a module,
then there is a nontrivial linear relation amongst its elements.
-/
-- TODO: golf this
theorem Module.exists_nontrivial_relation_of_finrank_lt_card
    [Module.Finite R V] {t : Finset V}
    (h : finrank R V < t.card) : ∃ f : V → R, ∑ e in t, f e • e = 0 ∧ ∃ x ∈ t, f x ≠ 0 := by
  classical
  have := mt FiniteDimensional.finset_card_le_finrank_of_linearIndependent (by simpa using h)
  rw [_root_.not_linearIndependent_iff] at this
  obtain ⟨s, g, sum, z, zm, nonzero⟩ := this
  -- Now we have to extend `g` to all of `t`, then to all of `V`.
  let f : V → R := fun x => if h : x ∈ t then if (⟨x, h⟩ : t) ∈ s then g ⟨x, h⟩ else 0 else 0
  -- and finally clean up the mess caused by the extension.
  refine' ⟨f, _, _⟩
  · dsimp
    rw [← (sum)] -- porting note: need parens to disambiguate
    fapply sum_bij_ne_zero fun v hvt _ => (⟨v, hvt⟩ : { v // v ∈ t })
    · intro v hvt H
      dsimp
      rw [dif_pos hvt] at H
      contrapose! H
      rw [if_neg H, zero_smul]
    · intro _ _ _ _ _ _
      

Proof state:
case intro.intro.intro.intro.intro.refine'_1.i_inj
R : Type u
M : Type v
N : Type w
V : Type v
inst✝⁶ : AddCommGroup V
V₂ : Type v'
inst✝⁵ : AddCommGroup V₂
inst✝⁴ : Ring R
inst✝³ : StrongRankCondition R
inst✝² : Module R V
inst✝¹ : Free R V
K : ?m.320903
inst✝ : Finite R V
t : Finset V
h : finrank R V < Finset.card t
s : Finset { x // x ∈ t }
g : { x // x ∈ t } → R
sum : ∑ i in s, g i • ↑i = 0
z : { x // x ∈ t }
zm : z ∈ s
nonzero : g z ≠ 0
f : V → R :=
  fun x => if h : x ∈ t then if { val := x, property := h } ∈ s then g { val := x, property := h } else 0 else 0
a₁✝ a₂✝ : V
h₁₁✝ : a₁✝ ∈ t
h₁₂✝ :
  (if h : a₁✝ ∈ t then if { val := a₁✝, property := h } ∈ s then g { val := a₁✝, property := h } else 0 else 0) • a₁✝ ≠
    0
h₂₁✝ : a₂✝ ∈ t
h₂₂✝ :
  (if h : a₂✝ ∈ t then if { val := a₂✝, property := h } ∈ s then g { val := a₂✝, property := h } else 0 else 0) • a₂✝ ≠
    0
⊢ { val := a₁✝, property := h₁₁✝ } = { val := a₂✝, property := h₂₁✝ } → a₁✝ = a₂✝

Next tactic:
exact Subtype.mk.inj
/-- If a finset has cardinality larger than the rank of a module,
then there is a nontrivial linear relation amongst its elements.
-/
-- TODO: golf this
theorem Module.exists_nontrivial_relation_of_finrank_lt_card
    [Module.Finite R V] {t : Finset V}
    (h : finrank R V < t.card) : ∃ f : V → R, ∑ e in t, f e • e = 0 ∧ ∃ x ∈ t, f x ≠ 0 := by
  classical
  have := mt FiniteDimensional.finset_card_le_finrank_of_linearIndependent (by simpa using h)
  rw [_root_.not_linearIndependent_iff] at this
  obtain ⟨s, g, sum, z, zm, nonzero⟩ := this
  -- Now we have to extend `g` to all of `t`, then to all of `V`.
  let f : V → R := fun x => if h : x ∈ t then if (⟨x, h⟩ : t) ∈ s then g ⟨x, h⟩ else 0 else 0
  -- and finally clean up the mess caused by the extension.
  refine' ⟨f, _, _⟩
  · dsimp
    rw [← (sum)] -- porting note: need parens to disambiguate
    fapply sum_bij_ne_zero fun v hvt _ => (⟨v, hvt⟩ : { v // v ∈ t })
    · intro v hvt H
      dsimp
      rw [dif_pos hvt] at H
      contrapose! H
      rw [if_neg H, zero_smul]
    · intro _ _ _ _ _ _
      exact Subtype.mk.inj
    · 

Proof state:
case intro.intro.intro.intro.intro.refine'_1.i_surj
R : Type u
M : Type v
N : Type w
V : Type v
inst✝⁶ : AddCommGroup V
V₂ : Type v'
inst✝⁵ : AddCommGroup V₂
inst✝⁴ : Ring R
inst✝³ : StrongRankCondition R
inst✝² : Module R V
inst✝¹ : Free R V
K : ?m.320903
inst✝ : Finite R V
t : Finset V
h : finrank R V < Finset.card t
s : Finset { x // x ∈ t }
g : { x // x ∈ t } → R
sum : ∑ i in s, g i • ↑i = 0
z : { x // x ∈ t }
zm : z ∈ s
nonzero : g z ≠ 0
f : V → R :=
  fun x => if h : x ∈ t then if { val := x, property := h } ∈ s then g { val := x, property := h } else 0 else 0
⊢ ∀ b ∈ s,
    g b • ↑b ≠ 0 →
      ∃ a,
        ∃ (h₁ : a ∈ t) (_ :
          (if h : a ∈ t then if { val := a, property := h } ∈ s then g { val := a, property := h } else 0 else 0) • a ≠
            0),
          b = { val := a, property := h₁ }

Next tactic:
intro b hbs hb
/-- If a finset has cardinality larger than the rank of a module,
then there is a nontrivial linear relation amongst its elements.
-/
-- TODO: golf this
theorem Module.exists_nontrivial_relation_of_finrank_lt_card
    [Module.Finite R V] {t : Finset V}
    (h : finrank R V < t.card) : ∃ f : V → R, ∑ e in t, f e • e = 0 ∧ ∃ x ∈ t, f x ≠ 0 := by
  classical
  have := mt FiniteDimensional.finset_card_le_finrank_of_linearIndependent (by simpa using h)
  rw [_root_.not_linearIndependent_iff] at this
  obtain ⟨s, g, sum, z, zm, nonzero⟩ := this
  -- Now we have to extend `g` to all of `t`, then to all of `V`.
  let f : V → R := fun x => if h : x ∈ t then if (⟨x, h⟩ : t) ∈ s then g ⟨x, h⟩ else 0 else 0
  -- and finally clean up the mess caused by the extension.
  refine' ⟨f, _, _⟩
  · dsimp
    rw [← (sum)] -- porting note: need parens to disambiguate
    fapply sum_bij_ne_zero fun v hvt _ => (⟨v, hvt⟩ : { v // v ∈ t })
    · intro v hvt H
      dsimp
      rw [dif_pos hvt] at H
      contrapose! H
      rw [if_neg H, zero_smul]
    · intro _ _ _ _ _ _
      exact Subtype.mk.inj
    · intro b hbs hb
      

Proof state:
case intro.intro.intro.intro.intro.refine'_1.i_surj
R : Type u
M : Type v
N : Type w
V : Type v
inst✝⁶ : AddCommGroup V
V₂ : Type v'
inst✝⁵ : AddCommGroup V₂
inst✝⁴ : Ring R
inst✝³ : StrongRankCondition R
inst✝² : Module R V
inst✝¹ : Free R V
K : ?m.320903
inst✝ : Finite R V
t : Finset V
h : finrank R V < Finset.card t
s : Finset { x // x ∈ t }
g : { x // x ∈ t } → R
sum : ∑ i in s, g i • ↑i = 0
z : { x // x ∈ t }
zm : z ∈ s
nonzero : g z ≠ 0
f : V → R :=
  fun x => if h : x ∈ t then if { val := x, property := h } ∈ s then g { val := x, property := h } else 0 else 0
b : { v // v ∈ t }
hbs : b ∈ s
hb : g b • ↑b ≠ 0
⊢ ∃ a,
    ∃ (h₁ : a ∈ t) (_ :
      (if h : a ∈ t then if { val := a, property := h } ∈ s then g { val := a, property := h } else 0 else 0) • a ≠ 0),
      b = { val := a, property := h₁ }

Next tactic:
use b
/-- If a finset has cardinality larger than the rank of a module,
then there is a nontrivial linear relation amongst its elements.
-/
-- TODO: golf this
theorem Module.exists_nontrivial_relation_of_finrank_lt_card
    [Module.Finite R V] {t : Finset V}
    (h : finrank R V < t.card) : ∃ f : V → R, ∑ e in t, f e • e = 0 ∧ ∃ x ∈ t, f x ≠ 0 := by
  classical
  have := mt FiniteDimensional.finset_card_le_finrank_of_linearIndependent (by simpa using h)
  rw [_root_.not_linearIndependent_iff] at this
  obtain ⟨s, g, sum, z, zm, nonzero⟩ := this
  -- Now we have to extend `g` to all of `t`, then to all of `V`.
  let f : V → R := fun x => if h : x ∈ t then if (⟨x, h⟩ : t) ∈ s then g ⟨x, h⟩ else 0 else 0
  -- and finally clean up the mess caused by the extension.
  refine' ⟨f, _, _⟩
  · dsimp
    rw [← (sum)] -- porting note: need parens to disambiguate
    fapply sum_bij_ne_zero fun v hvt _ => (⟨v, hvt⟩ : { v // v ∈ t })
    · intro v hvt H
      dsimp
      rw [dif_pos hvt] at H
      contrapose! H
      rw [if_neg H, zero_smul]
    · intro _ _ _ _ _ _
      exact Subtype.mk.inj
    · intro b hbs hb
      use b
      

Proof state:
case h
R : Type u
M : Type v
N : Type w
V : Type v
inst✝⁶ : AddCommGroup V
V₂ : Type v'
inst✝⁵ : AddCommGroup V₂
inst✝⁴ : Ring R
inst✝³ : StrongRankCondition R
inst✝² : Module R V
inst✝¹ : Free R V
K : ?m.320903
inst✝ : Finite R V
t : Finset V
h : finrank R V < Finset.card t
s : Finset { x // x ∈ t }
g : { x // x ∈ t } → R
sum : ∑ i in s, g i • ↑i = 0
z : { x // x ∈ t }
zm : z ∈ s
nonzero : g z ≠ 0
f : V → R :=
  fun x => if h : x ∈ t then if { val := x, property := h } ∈ s then g { val := x, property := h } else 0 else 0
b : { v // v ∈ t }
hbs : b ∈ s
hb : g b • ↑b ≠ 0
⊢ ∃ (h₁ : ↑b ∈ t) (_ :
    (if h : ↑b ∈ t then if { val := ↑b, property := h } ∈ s then g { val := ↑b, property := h } else 0 else 0) • ↑b ≠
      0),
    b = { val := ↑b, property := h₁ }

Next tactic:
simpa only [hbs, exists_prop, dif_pos, Finset.mk_coe, and_true_iff, if_true, Finset.coe_mem,
        eq_self_iff_true, exists_prop_of_true, Ne.def] using hb
/-- If a finset has cardinality larger than the rank of a module,
then there is a nontrivial linear relation amongst its elements.
-/
-- TODO: golf this
theorem Module.exists_nontrivial_relation_of_finrank_lt_card
    [Module.Finite R V] {t : Finset V}
    (h : finrank R V < t.card) : ∃ f : V → R, ∑ e in t, f e • e = 0 ∧ ∃ x ∈ t, f x ≠ 0 := by
  classical
  have := mt FiniteDimensional.finset_card_le_finrank_of_linearIndependent (by simpa using h)
  rw [_root_.not_linearIndependent_iff] at this
  obtain ⟨s, g, sum, z, zm, nonzero⟩ := this
  -- Now we have to extend `g` to all of `t`, then to all of `V`.
  let f : V → R := fun x => if h : x ∈ t then if (⟨x, h⟩ : t) ∈ s then g ⟨x, h⟩ else 0 else 0
  -- and finally clean up the mess caused by the extension.
  refine' ⟨f, _, _⟩
  · dsimp
    rw [← (sum)] -- porting note: need parens to disambiguate
    fapply sum_bij_ne_zero fun v hvt _ => (⟨v, hvt⟩ : { v // v ∈ t })
    · intro v hvt H
      dsimp
      rw [dif_pos hvt] at H
      contrapose! H
      rw [if_neg H, zero_smul]
    · intro _ _ _ _ _ _
      exact Subtype.mk.inj
    · intro b hbs hb
      use b
      simpa only [hbs, exists_prop, dif_pos, Finset.mk_coe, and_true_iff, if_true, Finset.coe_mem,
        eq_self_iff_true, exists_prop_of_true, Ne.def] using hb
    · 

Proof state:
case intro.intro.intro.intro.intro.refine'_1.h
R : Type u
M : Type v
N : Type w
V : Type v
inst✝⁶ : AddCommGroup V
V₂ : Type v'
inst✝⁵ : AddCommGroup V₂
inst✝⁴ : Ring R
inst✝³ : StrongRankCondition R
inst✝² : Module R V
inst✝¹ : Free R V
K : ?m.320903
inst✝ : Finite R V
t : Finset V
h : finrank R V < Finset.card t
s : Finset { x // x ∈ t }
g : { x // x ∈ t } → R
sum : ∑ i in s, g i • ↑i = 0
z : { x // x ∈ t }
zm : z ∈ s
nonzero : g z ≠ 0
f : V → R :=
  fun x => if h : x ∈ t then if { val := x, property := h } ∈ s then g { val := x, property := h } else 0 else 0
⊢ ∀ (a : V) (h₁ : a ∈ t),
    (if h : a ∈ t then if { val := a, property := h } ∈ s then g { val := a, property := h } else 0 else 0) • a ≠ 0 →
      (if h : a ∈ t then if { val := a, property := h } ∈ s then g { val := a, property := h } else 0 else 0) • a =
        g { val := a, property := h₁ } • ↑{ val := a, property := h₁ }

Next tactic:
intro a h₁
/-- If a finset has cardinality larger than the rank of a module,
then there is a nontrivial linear relation amongst its elements.
-/
-- TODO: golf this
theorem Module.exists_nontrivial_relation_of_finrank_lt_card
    [Module.Finite R V] {t : Finset V}
    (h : finrank R V < t.card) : ∃ f : V → R, ∑ e in t, f e • e = 0 ∧ ∃ x ∈ t, f x ≠ 0 := by
  classical
  have := mt FiniteDimensional.finset_card_le_finrank_of_linearIndependent (by simpa using h)
  rw [_root_.not_linearIndependent_iff] at this
  obtain ⟨s, g, sum, z, zm, nonzero⟩ := this
  -- Now we have to extend `g` to all of `t`, then to all of `V`.
  let f : V → R := fun x => if h : x ∈ t then if (⟨x, h⟩ : t) ∈ s then g ⟨x, h⟩ else 0 else 0
  -- and finally clean up the mess caused by the extension.
  refine' ⟨f, _, _⟩
  · dsimp
    rw [← (sum)] -- porting note: need parens to disambiguate
    fapply sum_bij_ne_zero fun v hvt _ => (⟨v, hvt⟩ : { v // v ∈ t })
    · intro v hvt H
      dsimp
      rw [dif_pos hvt] at H
      contrapose! H
      rw [if_neg H, zero_smul]
    · intro _ _ _ _ _ _
      exact Subtype.mk.inj
    · intro b hbs hb
      use b
      simpa only [hbs, exists_prop, dif_pos, Finset.mk_coe, and_true_iff, if_true, Finset.coe_mem,
        eq_self_iff_true, exists_prop_of_true, Ne.def] using hb
    · intro a h₁
      

Proof state:
case intro.intro.intro.intro.intro.refine'_1.h
R : Type u
M : Type v
N : Type w
V : Type v
inst✝⁶ : AddCommGroup V
V₂ : Type v'
inst✝⁵ : AddCommGroup V₂
inst✝⁴ : Ring R
inst✝³ : StrongRankCondition R
inst✝² : Module R V
inst✝¹ : Free R V
K : ?m.320903
inst✝ : Finite R V
t : Finset V
h : finrank R V < Finset.card t
s : Finset { x // x ∈ t }
g : { x // x ∈ t } → R
sum : ∑ i in s, g i • ↑i = 0
z : { x // x ∈ t }
zm : z ∈ s
nonzero : g z ≠ 0
f : V → R :=
  fun x => if h : x ∈ t then if { val := x, property := h } ∈ s then g { val := x, property := h } else 0 else 0
a : V
h₁ : a ∈ t
⊢ (if h : a ∈ t then if { val := a, property := h } ∈ s then g { val := a, property := h } else 0 else 0) • a ≠ 0 →
    (if h : a ∈ t then if { val := a, property := h } ∈ s then g { val := a, property := h } else 0 else 0) • a =
      g { val := a, property := h₁ } • ↑{ val := a, property := h₁ }

Next tactic:
dsimp
/-- If a finset has cardinality larger than the rank of a module,
then there is a nontrivial linear relation amongst its elements.
-/
-- TODO: golf this
theorem Module.exists_nontrivial_relation_of_finrank_lt_card
    [Module.Finite R V] {t : Finset V}
    (h : finrank R V < t.card) : ∃ f : V → R, ∑ e in t, f e • e = 0 ∧ ∃ x ∈ t, f x ≠ 0 := by
  classical
  have := mt FiniteDimensional.finset_card_le_finrank_of_linearIndependent (by simpa using h)
  rw [_root_.not_linearIndependent_iff] at this
  obtain ⟨s, g, sum, z, zm, nonzero⟩ := this
  -- Now we have to extend `g` to all of `t`, then to all of `V`.
  let f : V → R := fun x => if h : x ∈ t then if (⟨x, h⟩ : t) ∈ s then g ⟨x, h⟩ else 0 else 0
  -- and finally clean up the mess caused by the extension.
  refine' ⟨f, _, _⟩
  · dsimp
    rw [← (sum)] -- porting note: need parens to disambiguate
    fapply sum_bij_ne_zero fun v hvt _ => (⟨v, hvt⟩ : { v // v ∈ t })
    · intro v hvt H
      dsimp
      rw [dif_pos hvt] at H
      contrapose! H
      rw [if_neg H, zero_smul]
    · intro _ _ _ _ _ _
      exact Subtype.mk.inj
    · intro b hbs hb
      use b
      simpa only [hbs, exists_prop, dif_pos, Finset.mk_coe, and_true_iff, if_true, Finset.coe_mem,
        eq_self_iff_true, exists_prop_of_true, Ne.def] using hb
    · intro a h₁
      dsimp
      

Proof state:
case intro.intro.intro.intro.intro.refine'_1.h
R : Type u
M : Type v
N : Type w
V : Type v
inst✝⁶ : AddCommGroup V
V₂ : Type v'
inst✝⁵ : AddCommGroup V₂
inst✝⁴ : Ring R
inst✝³ : StrongRankCondition R
inst✝² : Module R V
inst✝¹ : Free R V
K : ?m.320903
inst✝ : Finite R V
t : Finset V
h : finrank R V < Finset.card t
s : Finset { x // x ∈ t }
g : { x // x ∈ t } → R
sum : ∑ i in s, g i • ↑i = 0
z : { x // x ∈ t }
zm : z ∈ s
nonzero : g z ≠ 0
f : V → R :=
  fun x => if h : x ∈ t then if { val := x, property := h } ∈ s then g { val := x, property := h } else 0 else 0
a : V
h₁ : a ∈ t
⊢ ¬(if h : a ∈ t then if { val := a, property := h } ∈ s then g { val := a, property := h } else 0 else 0) • a = 0 →
    (if h : a ∈ t then if { val := a, property := h } ∈ s then g { val := a, property := h } else 0 else 0) • a =
      g { val := a, property := h₁ } • a

Next tactic:
rw [dif_pos h₁]