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
      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
      rw [dif_pos 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 { val := a, property := h₁ } ∈ s then g { val := a, property := h₁ } else 0) • a = 0 →
    (if { val := a, property := h₁ } ∈ s then g { val := a, property := h₁ } else 0) • a =
      g { val := a, property := h₁ } • a

Next tactic:
intro h₂
/-- 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
      rw [dif_pos h₁]
      intro h₂
      

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
h₂ : ¬(if { val := a, property := h₁ } ∈ s then g { val := a, property := h₁ } else 0) • a = 0
⊢ (if { val := a, property := h₁ } ∈ s then g { val := a, property := h₁ } else 0) • a =
    g { val := a, property := h₁ } • a

Next tactic:
rw [if_pos]
/-- 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
      rw [dif_pos h₁]
      intro h₂
      rw [if_pos]
      

Proof state:
case intro.intro.intro.intro.intro.refine'_1.h.hc
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
h₂ : ¬(if { val := a, property := h₁ } ∈ s then g { val := a, property := h₁ } else 0) • a = 0
⊢ { val := a, property := h₁ } ∈ s

Next tactic:
contrapose! h₂
/-- 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
      rw [dif_pos h₁]
      intro h₂
      rw [if_pos]
      contrapose! h₂
      

Proof state:
case intro.intro.intro.intro.intro.refine'_1.h.hc
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
h₂ : { val := a, property := h₁ } ∉ s
⊢ (if { val := a, property := h₁ } ∈ s then g { val := a, property := h₁ } else 0) • a = 0

Next tactic:
rw [if_neg h₂, 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]
    · 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
      rw [dif_pos h₁]
      intro h₂
      rw [if_pos]
      contrapose! h₂
      rw [if_neg h₂, zero_smul]
  · 

Proof state:
case intro.intro.intro.intro.intro.refine'_2
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
⊢ ∃ x ∈ t, f x ≠ 0

Next tactic:
refine' ⟨z, z.2, _⟩
/-- 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
      rw [dif_pos h₁]
      intro h₂
      rw [if_pos]
      contrapose! h₂
      rw [if_neg h₂, zero_smul]
  · refine' ⟨z, z.2, _⟩
    

Proof state:
case intro.intro.intro.intro.intro.refine'_2
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
⊢ f ↑z ≠ 0

Next tactic:
dsimp only
/-- 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
      rw [dif_pos h₁]
      intro h₂
      rw [if_pos]
      contrapose! h₂
      rw [if_neg h₂, zero_smul]
  · refine' ⟨z, z.2, _⟩
    dsimp only
    

Proof state:
case intro.intro.intro.intro.intro.refine'_2
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
⊢ (if h : ↑z ∈ t then if { val := ↑z, property := h } ∈ s then g { val := ↑z, property := h } else 0 else 0) ≠ 0

Next tactic:
erw [dif_pos z.2, if_pos]
/-- 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
      rw [dif_pos h₁]
      intro h₂
      rw [if_pos]
      contrapose! h₂
      rw [if_neg h₂, zero_smul]
  · refine' ⟨z, z.2, _⟩
    dsimp only
    erw [dif_pos z.2, if_pos] <;> 

Proof state:
case intro.intro.intro.intro.intro.refine'_2
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
⊢ g { val := ↑z, property := (_ : ↑z ∈ t) } ≠ 0

Next tactic:
rwa [Subtype.coe_eta]
/-- 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
      rw [dif_pos h₁]
      intro h₂
      rw [if_pos]
      contrapose! h₂
      rw [if_neg h₂, zero_smul]
  · refine' ⟨z, z.2, _⟩
    dsimp only
    erw [dif_pos z.2, if_pos] <;> 

Proof state:
case intro.intro.intro.intro.intro.refine'_2.hc
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
⊢ { val := ↑z, property := (_ : ↑z ∈ t) } ∈ s

Next tactic:
rwa [Subtype.coe_eta]
/-- If a finset has cardinality larger than `finrank + 1`,
then there is a nontrivial linear relation amongst its elements,
such that the coefficients of the relation sum to zero.
-/
-- TODO: golf this
theorem Module.exists_nontrivial_relation_sum_zero_of_finrank_succ_lt_card [Module.Finite R V]
    {t : Finset V} (h : finrank R V + 1 < t.card) :
    ∃ f : V → R, ∑ e in t, f e • e = 0 ∧ ∑ e in t, f e = 0 ∧ ∃ x ∈ t, f x ≠ 0 := by
  classical
  -- Pick an element x₀ ∈ t,
  

Proof state:
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.334640
inst✝ : Finite R V
t : Finset V
h : finrank R V + 1 < Finset.card t
⊢ ∃ f, ∑ e in t, f e • e = 0 ∧ ∑ e in t, f e = 0 ∧ ∃ x ∈ t, f x ≠ 0

Next tactic:
have card_pos : 0 < t.card := lt_trans (Nat.succ_pos _) h
/-- If a finset has cardinality larger than `finrank + 1`,
then there is a nontrivial linear relation amongst its elements,
such that the coefficients of the relation sum to zero.
-/
-- TODO: golf this
theorem Module.exists_nontrivial_relation_sum_zero_of_finrank_succ_lt_card [Module.Finite R V]
    {t : Finset V} (h : finrank R V + 1 < t.card) :
    ∃ f : V → R, ∑ e in t, f e • e = 0 ∧ ∑ e in t, f e = 0 ∧ ∃ x ∈ t, f x ≠ 0 := by
  classical
  -- Pick an element x₀ ∈ t,
  have card_pos : 0 < t.card := lt_trans (Nat.succ_pos _) h
  

Proof state:
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.334640
inst✝ : Finite R V
t : Finset V
h : finrank R V + 1 < Finset.card t
card_pos : 0 < Finset.card t
⊢ ∃ f, ∑ e in t, f e • e = 0 ∧ ∑ e in t, f e = 0 ∧ ∃ x ∈ t, f x ≠ 0

Next tactic:
obtain ⟨x₀, m⟩ := (Finset.card_pos.1 card_pos).bex
/-- If a finset has cardinality larger than `finrank + 1`,
then there is a nontrivial linear relation amongst its elements,
such that the coefficients of the relation sum to zero.
-/
-- TODO: golf this
theorem Module.exists_nontrivial_relation_sum_zero_of_finrank_succ_lt_card [Module.Finite R V]
    {t : Finset V} (h : finrank R V + 1 < t.card) :
    ∃ f : V → R, ∑ e in t, f e • e = 0 ∧ ∑ e in t, f e = 0 ∧ ∃ x ∈ t, f x ≠ 0 := by
  classical
  -- Pick an element x₀ ∈ t,
  have card_pos : 0 < t.card := lt_trans (Nat.succ_pos _) h
  obtain ⟨x₀, m⟩ := (Finset.card_pos.1 card_pos).bex
  -- and apply the previous lemma to the {xᵢ - x₀}
  

Proof state:
case intro
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.334640
inst✝ : Finite R V
t : Finset V
h : finrank R V + 1 < Finset.card t
card_pos : 0 < Finset.card t
x₀ : V
m : x₀ ∈ t
⊢ ∃ f, ∑ e in t, f e • e = 0 ∧ ∑ e in t, f e = 0 ∧ ∃ x ∈ t, f x ≠ 0

Next tactic:
let shift : V ↪ V := ⟨fun x => x - x₀, sub_left_injective⟩
/-- If a finset has cardinality larger than `finrank + 1`,
then there is a nontrivial linear relation amongst its elements,
such that the coefficients of the relation sum to zero.
-/
-- TODO: golf this
theorem Module.exists_nontrivial_relation_sum_zero_of_finrank_succ_lt_card [Module.Finite R V]
    {t : Finset V} (h : finrank R V + 1 < t.card) :
    ∃ f : V → R, ∑ e in t, f e • e = 0 ∧ ∑ e in t, f e = 0 ∧ ∃ x ∈ t, f x ≠ 0 := by
  classical
  -- Pick an element x₀ ∈ t,
  have card_pos : 0 < t.card := lt_trans (Nat.succ_pos _) h
  obtain ⟨x₀, m⟩ := (Finset.card_pos.1 card_pos).bex
  -- and apply the previous lemma to the {xᵢ - x₀}
  let shift : V ↪ V := ⟨fun x => x - x₀, sub_left_injective⟩
  

Proof state:
case intro
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.334640
inst✝ : Finite R V
t : Finset V
h : finrank R V + 1 < Finset.card t
card_pos : 0 < Finset.card t
x₀ : V
m : x₀ ∈ t
shift : V ↪ V := { toFun := fun x => x - x₀, inj' := (_ : Injective fun a => a - x₀) }
⊢ ∃ f, ∑ e in t, f e • e = 0 ∧ ∑ e in t, f e = 0 ∧ ∃ x ∈ t, f x ≠ 0

Next tactic:
let t' := (t.erase x₀).map shift