import mll

def sequent := list Form

instance : has_append sequent := ⟨list.append⟩
instance : has_mem Form sequent := ⟨list.mem⟩

inductive proof : sequent → Type
| ax (A)                   : proof [~A, A]
| cut (A) {Γ Γ' Δ Δ'}      : proof (Γ ++ [A] ++ Γ') → proof (Δ ++ [~A] ++ Δ') → proof (Γ++Γ'++Δ++Δ')
| tensor {A B} {Γ Γ' Δ Δ'} : proof (Γ ++ [A] ++ Γ') → proof (Δ ++ [B] ++ Δ') → proof (Γ++Γ'++ [A ⊗ B] ++Δ++Δ')
| par {A B} {Γ Γ'}         : proof (Γ ++ [A,B] ++ Γ') → proof (Γ ++ [A ⅋ B] ++ Γ')
| ex {A B} {Γ Γ'}          : proof (Γ ++ [A,B] ++ Γ') → proof (Γ ++ [B,A] ++ Γ')

inductive ps_conclusion (A : Form) (ai : ℕ) (ps : proof_structure) : Prop
| notprem : (∀ l ∈ ps.links, ¬ premise (A,ai) l) → ps_conclusion
| con : Link.con ai A ∈ ps.links → ps_conclusion

def ps_ax (A : Form) (pi ni : ℕ) : proof_structure :=
⟨ {Link.ax pi ni A, Link.con pi A, Link.con ni (~A)},
  begin rintros _ ⟨_|_⟩, constructor, cases H with H H, rw H, constructor, cases H, constructor end,
  begin rintro Ai l₁ l₂ ⟨_,_⟩ ⟨_,_⟩, finish, rintro ⟨_⟩, rintro _ ⟨_⟩,
    rcases H with ⟨_,_⟩;rcases H_1 with ⟨_,_⟩, finish, rcases H_1 with ⟨_,_⟩, intro pA, generalize e : A = A', cases pA,
    rintro ⟨_⟩, congr, exact e.symm, cases H, intro pnA, generalize e : A = A', cases pnA, rintro ⟨_⟩, exfalso, exact not_self_dual e.symm,
    cases H, cases H_1, finish,  end,
  begin rintros Ai l₁ l₂ ⟨_,_⟩ ⟨_,_⟩, finish, rcases H_1 with ⟨_,_⟩, rintro _ ⟨_⟩, cases H_1, rintro _ ⟨_⟩,
    rcases H with ⟨_,_⟩, rintro ⟨_⟩, rcases H with ⟨_⟩, rintro ⟨_⟩,
    rcases H with ⟨_,_⟩; rcases H_1 with ⟨_,_⟩, finish, rintro ⟨_⟩, rintro _ ⟨_⟩, cases H, cases H_1, finish,
  end ⟩

lemma ps_ax_conclusions {A pi ni} : Link.con pi A ∈ (ps_ax A pi ni).links ∧ Link.con ni (~A) ∈ (ps_ax A pi ni).links :=
  ⟨set.mem_union_right _ (set.mem_union_left _ rfl), set.mem_union_right _ (set.mem_union_right _ rfl)⟩

def ps.disjoint (ps₁ ps₂ : proof_structure) : Prop := ∀ Ai : Form_occ, (Ai ∈ ps₁ ∧ Ai ∈ ps₂) → false

def ps_tensor (A B) (ai bi ci : ℕ) (psA psB : proof_structure) :
  ps.disjoint psA psB →
  ps_conclusion A ai psA →
  ps_conclusion B bi psB →
  (A ⊗ B, ci) ∉ psA →
  (A ⊗ B, ci) ∉ psB →
  proof_structure :=
λ dAB conA conB hcA hcB,
⟨{Link.tensor ai bi ci A B} ∪ (psA.links \ {Link.con ai A}) ∪ (psB.links \ {Link.con bi B}),
begin
  rintros l ⟨⟨_,_⟩|⟨h₁,_⟩⟩,
    apply valid_link.tensor,
  rintros ⟨_⟩,
  exact dAB (A,ai) ⟨⟨conA,mem_Link.prem premise.con⟩,⟨conB,mem_Link.prem premise.con⟩⟩,
  exact psA.valid l h₁,
  cases H with h₂, exact psB.valid l h₂
end,
begin
  rintros Ai l₁ l₂ ⟨⟨u₁,u₂⟩|⟨u₁,u₂⟩⟩ ⟨⟨h₁,h₂⟩|⟨h₁,h₂⟩⟩,
    { finish },
    { rintros ⟨_|_⟩ h₃,
        exfalso, apply h₂, simp, apply psA.prem_unique _ _ _ h₁ conA h₃ premise.con,
        exfalso, refine dAB (B,bi) ⟨⟨h₁,mem_Link.prem h₃⟩,⟨conB,mem_Link.prem premise.con⟩⟩, },
    { rcases H_1 with ⟨h₁,h₂⟩, rintros ⟨_|_⟩ h₃,
        exfalso, refine dAB (A,ai) ⟨⟨conA,mem_Link.prem premise.con⟩,⟨h₁,mem_Link.prem h₃⟩⟩,
        simp at h₂, exfalso, apply h₂, apply psB.prem_unique _ _ _ h₁ conB h₃, constructor },
    { rintros h₃ ⟨_|_⟩,
        simp at u₂, exfalso, apply u₂, apply psA.prem_unique _ _ _ u₁ conA h₃, constructor,
        exfalso, refine dAB (B,bi) ⟨⟨u₁,mem_Link.prem h₃⟩,⟨conB,mem_Link.prem premise.con⟩⟩, },
    { intros pl₁ pl₂, apply psA.prem_unique Ai _ _ u₁ h₁ pl₁ pl₂, },
    { intros pl₁ pl₂, cases H_1 with h₁ h₂, exfalso, exact dAB Ai ⟨⟨u₁,mem_Link.prem pl₁⟩,⟨h₁,mem_Link.prem pl₂⟩⟩, },
    { cases H with h₁ h₂,
      rintros h₃ ⟨_|_⟩,
        exfalso, refine dAB (A,ai) ⟨⟨conA,mem_Link.prem premise.con⟩,⟨h₁,mem_Link.prem h₃⟩⟩,
        exfalso, apply h₂, simp, apply psB.prem_unique _ _ _ h₁ conB h₃ premise.con, },
    { intros pl₁ pl₂, cases H with u₁ u₂, exfalso, exact dAB Ai ⟨⟨h₁,mem_Link.prem pl₂⟩,⟨u₁,mem_Link.prem pl₁⟩⟩, },
    { cases H with h₁ h₂, cases H_1 with u₁ u₂, intros pl₁ pl₂, apply psB.prem_unique Ai _ _ h₁ u₁ pl₁ pl₂, },
end,
begin
  rintros Ai l₁ l₂ ⟨⟨u₁,u₂⟩|⟨u₁,u₂⟩⟩ ⟨⟨h₁,h₂⟩|⟨h₁,h₂⟩⟩,
    { finish },
    { rintros ⟨_|_⟩ h₃, exfalso, apply hcA ⟨h₁, mem_Link.con h₃⟩, },
    { rcases H_1 with ⟨h₁,h₂⟩, rintros ⟨_|_⟩ h₃, exfalso, apply hcB ⟨h₁, mem_Link.con h₃⟩ },
    { rintros h₃ ⟨_|_⟩, exfalso, apply hcA ⟨u₁, mem_Link.con h₃⟩, },
    { intros pl₁ pl₂, apply psA.con_unique Ai _ _ u₁ h₁ pl₁ pl₂, },
    { intros pl₁ pl₂, cases H_1 with h₁ h₂, exfalso, exact dAB Ai ⟨⟨u₁,mem_Link.con pl₁⟩,⟨h₁,mem_Link.con pl₂⟩⟩, },
    { cases H with h₁ h₂, rintros h₃ ⟨_|_⟩, exfalso, apply hcB ⟨h₁, mem_Link.con h₃⟩, },
    { intros pl₁ pl₂, cases H with u₁ u₂, exfalso, exact dAB Ai ⟨⟨h₁,mem_Link.con pl₂⟩,⟨u₁,mem_Link.con pl₁⟩⟩, },
    { cases H with h₁ h₂, cases H_1 with u₁ u₂, intros pl₁ pl₂, apply psB.con_unique Ai _ _ h₁ u₁ pl₁ pl₂, },
end
⟩

lemma ps_tensor_conclusions {A B ai bi ci psA psB dAB conA conB hcA hcB} :
  Link.tensor ai bi ci A B ∈ (ps_tensor A B ai bi ci psA psB dAB conA conB hcA hcB).links :=
  set.mem_union_left _ (set.mem_union_left _ rfl)

def ps_cut (A : Form) (pi ni : ℕ) (psA psnA : proof_structure) :
  ps.disjoint psA psnA →
  ps_conclusion A pi psA →
  ps_conclusion (~A) ni psnA →
  proof_structure :=
λ dAnA conA connA,
⟨{Link.cut pi ni A} ∪ (psA.links \ {Link.con pi A}) ∪ (psnA.links \ {Link.con ni (~A)}),
begin
  rintros l ⟨⟨_,_⟩|⟨h₁,_⟩⟩,
    apply valid_link.cut,
  exact psA.valid l h₁,
  cases H with h₂, exact psnA.valid l h₂
end,
begin
  rintros Ai l₁ l₂ ⟨⟨u₁,u₂⟩|⟨u₁,u₂⟩⟩ ⟨⟨h₁,h₂⟩|⟨h₁,h₂⟩⟩,
    { finish },
    { rintros ⟨_⟩ h₃,
        exfalso, apply h₂, simp, apply psA.prem_unique _ _ _ h₁ conA h₃ premise.con,
        exfalso, refine dAnA (~A,ni) ⟨⟨h₁,mem_Link.prem h₃⟩,⟨connA,mem_Link.prem premise.con⟩⟩, },
    { rcases H_1 with ⟨h₁,h₂⟩, rintros ⟨_⟩ h₃,
        exfalso, refine dAnA (A,pi) ⟨⟨conA,mem_Link.prem premise.con⟩,⟨h₁,mem_Link.prem h₃⟩⟩,
        simp at h₂, exfalso, apply h₂, apply psnA.prem_unique _ _ _ h₁ connA h₃, constructor },
    { rintros h₃ ⟨_⟩,
        simp at u₂, exfalso, apply u₂, apply psA.prem_unique _ _ _ u₁ conA h₃, constructor,
        exfalso, refine dAnA (~A,ni) ⟨⟨u₁,mem_Link.prem h₃⟩,⟨connA,mem_Link.prem premise.con⟩⟩, },
    { intros pl₁ pl₂, apply psA.prem_unique Ai _ _ u₁ h₁ pl₁ pl₂, },
    { intros pl₁ pl₂, cases H_1 with h₁ h₂, exfalso, exact dAnA Ai ⟨⟨u₁,mem_Link.prem pl₁⟩,⟨h₁,mem_Link.prem pl₂⟩⟩, },
    { cases H with h₁ h₂,
      rintros h₃ ⟨_⟩,
        exfalso, refine dAnA (A,pi) ⟨⟨conA,mem_Link.prem premise.con⟩,⟨h₁,mem_Link.prem h₃⟩⟩,
        exfalso, apply h₂, simp, apply psnA.prem_unique _ _ _ h₁ connA h₃ premise.con, },
    { intros pl₁ pl₂, cases H with u₁ u₂, exfalso, exact dAnA Ai ⟨⟨h₁,mem_Link.prem pl₂⟩,⟨u₁,mem_Link.prem pl₁⟩⟩, },
    { cases H with h₁ h₂, cases H_1 with u₁ u₂, intros pl₁ pl₂, apply psnA.prem_unique Ai _ _ h₁ u₁ pl₁ pl₂, },
end,
begin
  rintros Ai l₁ l₂ ⟨⟨u₁,u₂⟩|⟨u₁,u₂⟩⟩ ⟨⟨h₁,h₂⟩|⟨h₁,h₂⟩⟩,
    { finish },
    { rintros ⟨_⟩ },
    { rintros ⟨_⟩ },
    { rintros _ ⟨_⟩ },
    { intros pl₁ pl₂, apply psA.con_unique Ai _ _ u₁ h₁ pl₁ pl₂, },
    { intros pl₁ pl₂, cases H_1 with h₁ h₂, exfalso, exact dAnA Ai ⟨⟨u₁,mem_Link.con pl₁⟩,⟨h₁,mem_Link.con pl₂⟩⟩, },
    { rintros _ ⟨_⟩, },
    { intros pl₁ pl₂, cases H with u₁ u₂, exfalso, exact dAnA Ai ⟨⟨h₁,mem_Link.con pl₂⟩,⟨u₁,mem_Link.con pl₁⟩⟩, },
    { cases H with h₁ h₂, cases H_1 with u₁ u₂, intros pl₁ pl₂, apply psnA.con_unique Ai _ _ h₁ u₁ pl₁ pl₂, },
end
⟩

def ps_par (A B) (ai bi ci : ℕ) (ps : proof_structure) :
  (A,ai) ≠ (B,bi) →
  ps_conclusion A ai psA →
  ps_conclusion B bi psB →
  (A ⅋ B, ci) ∉ ps →
  proof_structure :=
λ nAB conA conB hcA,
⟨ {Link.par ai bi ci A B} ∪ (ps.links \ {Link.con ai A, Link.con bi B}),
  begin
    rintros l ⟨⟨_,_⟩|⟨h₁,_⟩⟩,
      apply valid_link.par _ _ _ _ _ nAB,
    cases H with h₂, exact ps.valid l h₂
  end
  ,
  begin
    rintros Ai l₁ l₂ ⟨_|_⟩ ⟨_|_⟩,
      { finish },
      { rintros ⟨_,_⟩ h₃,
        exfalso, cases H_1 with h₁ h₂, apply h₂, left, refine ps.prem_unique (A,ai) _ _ h₁ conA h₃ premise.con,
        exfalso, cases H_1 with h₁ h₂, apply h₂, right, refine ps.prem_unique (B,bi) _ _ h₁ conB h₃ premise.con },
      { rintros h₃ ⟨_,_⟩,
        exfalso, cases H with h₁ h₂, apply h₂, left, refine ps.prem_unique (A,ai) _ _ h₁ conA h₃ premise.con,
        exfalso, cases H with h₁ h₂, apply h₂, right, refine ps.prem_unique (B,bi) _ _ h₁ conB h₃ premise.con },
      { intros pl₁ pl₂, cases H with h₁ h₂, cases H_1 with u₁ u₂, exact ps.prem_unique Ai _ _ h₁ u₁ pl₁ pl₂, },
  end
  ,
  begin
    rintros Ai l₁ l₂ ⟨_|_⟩ ⟨_|_⟩,
      { finish },
      { rintros ⟨_,_⟩ h₃, exfalso, cases H_1 with h₁ h₂, refine hcA ⟨h₁,mem_Link.con h₃⟩, },
      { rintros h₃ ⟨_,_⟩,
        exfalso, cases H with h₁ h₂, refine hcA ⟨h₁,mem_Link.con h₃⟩, },
      { intros pl₁ pl₂, cases H with h₁ h₂, cases H_1 with u₁ u₂, exact ps.con_unique Ai _ _ h₁ u₁ pl₁ pl₂, },
  end
⟩

def proof_structure.open (ps : proof_structure) (Γ : sequent) : Prop := ∀ A ∈ Γ, ∃ i : ℕ, ps_conclusion A i ps

def relabel_Link (f : ℕ → ℕ) : Link → Link
| (Link.ax pi ni A) := Link.ax (f pi) (f ni) A
| (Link.cut pi ni A) := Link.cut (f pi) (f ni) A
| (Link.tensor ai bi ci A B) := Link.tensor (f ai) (f bi) (f ci) A B
| (Link.par ai bi ci A B) := Link.par (f ai) (f bi) (f ci) A B
| (Link.con ai A) := Link.con (f ai) A

lemma relabel_valid {l f} (hf : function.injective f): valid_link l → valid_link (relabel_Link f l) :=
begin
  cases l,
  case Link.ax : pi ni A { rintro ⟨_⟩, constructor, },
  case Link.cut : pi ni A { rintro ⟨_⟩, constructor, },
  case Link.tensor : ai bi ci A B {
    rintro ⟨_⟩, constructor, rintro e, injection e with e₁ e₂, apply ᾰ_ᾰ, congr, assumption, exact hf e₂, },
  case Link.par : ai bi ci A B {
    rintro ⟨_⟩, constructor, rintro e, injection e with e₁ e₂, apply ᾰ_ᾰ, congr, assumption, exact hf e₂, },
  case Link.con : ai A { rintro _, constructor, }
end

lemma relabel_injective {f} (hf : function.injective f) : function.injective (relabel_Link f) :=
by rintros ⟨l₁⟩ ⟨l₂⟩; intros h; injection h; congr; repeat {refl <|> assumption <|> apply hf}

lemma relabel_premise {l f D i } (hf : function.injective f) : premise (D,i) (relabel_Link f l) → ∃ j, f j = i ∧ premise (D,j) l :=
begin
  cases l,
  case Link.ax : pi ni A { rintro ⟨_⟩ },
  case Link.cut : pi ni A { rintro ⟨_⟩, exact ⟨pi,rfl,premise.cut_pos⟩, exact ⟨ni,rfl,premise.cut_neg⟩, },
  case Link.tensor : ai bi ci A B {
    rintro ⟨_⟩, exact ⟨ai,rfl,premise.tensor_left⟩, exact ⟨bi,rfl,premise.tensor_right⟩, },
  case Link.par : ai bi ci A B {
    rintro ⟨_⟩, exact ⟨ai,rfl,premise.par_left⟩, exact ⟨bi,rfl,premise.par_right⟩, },
  case Link.con : ai A { rintro ⟨_⟩, refine ⟨ai,rfl,premise.con⟩, }
end

lemma relabel_conclusion {l f D i } (hf : function.injective f) : conclusion (D,i) (relabel_Link f l) → ∃j, f j = i ∧ conclusion (D,j) l :=
begin
  cases l,
  case Link.ax : pi ni A {
    rintro ⟨_⟩, exact ⟨pi,rfl,conclusion.ax_pos⟩, exact ⟨ni,rfl,conclusion.ax_neg⟩, },
  case Link.cut : pi ni A { rintro ⟨_⟩ },
  case Link.tensor : ai bi ci A B { rintro ⟨_⟩, exact ⟨ci,rfl,conclusion.tensor⟩ },
  case Link.par : ai bi ci A B { rintro ⟨_⟩, exact ⟨ci,rfl,conclusion.par⟩ },
  case Link.con : ai A { rintro ⟨_⟩ }
end

lemma relabel_mem {Δ f A i} (hf : function.injective f) : (A,i) ∈ (relabel_Link f Δ) → ∃ j, f j = i ∧ (A,j) ∈ Δ :=
begin
  intro h, cases h with h h,
  rcases (relabel_premise hf h) with ⟨j, ⟨fji,pΔ⟩⟩, refine ⟨j,fji,mem_Link.prem pΔ⟩,
  rcases (relabel_conclusion hf h) with ⟨j, ⟨fji,pΔ⟩⟩, refine ⟨j,fji,mem_Link.con pΔ⟩,
end

def proof_structure.relabel (ps : proof_structure) (f : ℕ → ℕ) (hf : function.injective f) : proof_structure :=
⟨set.image (relabel_Link f) ps.links,
  by rintros l ⟨l', ⟨hl',⟨_⟩⟩⟩; exact relabel_valid hf (ps.valid l' hl'),
begin
  rintros ⟨A,i⟩ _ _ ⟨k₁, ⟨hk₁,⟨_⟩⟩⟩ ⟨k₂, ⟨hk₂,⟨_⟩⟩⟩,
  intros pk₁ pk₂,
  congr,
  rcases relabel_premise hf pk₁ with ⟨j,hfj,u₁⟩,
  rcases relabel_premise hf pk₂ with ⟨j',hfj',u₂⟩,
  have : j' = j, rw ←hfj at hfj', exact hf hfj', rw this at u₂,
  exact ps.prem_unique (A,j) _ _ hk₁ hk₂ u₁ u₂
end
,
begin
  rintros ⟨A,i⟩ _ _ ⟨k₁, ⟨hk₁,⟨_⟩⟩⟩ ⟨k₂, ⟨hk₂,⟨_⟩⟩⟩,
  intros pk₁ pk₂,
  congr,
  rcases relabel_conclusion hf pk₁ with ⟨j,hfj,u₁⟩,
  rcases relabel_conclusion hf pk₂ with ⟨j',hfj',u₂⟩,
  have : j' = j, rw ←hfj at hfj', exact hf hfj', rw this at u₂,
  exact ps.con_unique (A,j) _ _ hk₁ hk₂ u₁ u₂
end⟩

-- lemma mem_relabel {ps : proof_structure} {f hf Δ} : Δ ∈ (ps.relabel f hf).links → ∃ Δ', Δ' = relabel_Link f Δ ∧ Δ' ∈ ps.links :=
-- begin

-- end

lemma ps_conclusion_relabel {ps A i f} (hf : function.injective f) : ps_conclusion A i ps → ps_conclusion A (f i) (ps.relabel f hf) :=
begin
  intro h,
  cases h with h h,
  left,
    rintros Δ ⟨Δ',hΔ',⟨_⟩⟩,
    intro e,
    apply h Δ' hΔ',
    rcases relabel_premise hf e with ⟨i',fi',h⟩,
    convert h, apply hf fi'.symm,
  right,
  refine ⟨Link.con i A,h,rfl⟩,
end

def separators {α β} (f g : α → β) : Prop := ∀ x y, f x ≠ g y

lemma sep_even_odd : separators (λ x, 2 * x) (λ x, 2 * x + 1) :=
  λ x y, nat.two_mul_ne_two_mul_add_one

def disjoint_of_separators {ps₁ ps₂ : proof_structure} {f g} (hf hg) : separators f g → ps.disjoint (ps₁.relabel f hf) (ps₂.relabel g hg) :=
begin
  rintros s ⟨A,i⟩ ⟨⟨Δ₁,⟨Δ₁', hΔ₁', ⟨_⟩⟩,h₁⟩,⟨Δ₂,⟨Δ₂', hΔ₂', ⟨_⟩⟩,h₂⟩⟩,
  rcases (relabel_mem hf h₁) with ⟨j₁,hfg,h₁⟩,
  rcases (relabel_mem hg h₂) with ⟨j₂,⟨_⟩,h₂⟩,
  exact s j₁ j₂ hfg,
end

def net : Π Γ, proof Γ → ∃ ps : proof_structure, ps.open Γ :=
begin
  intros Γ π,
  induction π,
  case proof.ax : A { use ps_ax A 0 0; rintros B ⟨_,H⟩; use 0,
    right, exact set.mem_union_right _ (set.mem_union_right _ rfl),
    rcases H with ⟨_,_⟩, right, exact set.mem_union_right _ (set.mem_union_left _ rfl), cases H, },
  case proof.cut : A Γ Γ' Δ Δ' pA nA ihA ihnA {
    rcases ihA with ⟨psA,oA⟩,
    rcases ihnA with ⟨psnA,onA⟩,
    have h₁ := oA A (list.mem_append.mpr $ or.inl $ list.mem_append.mpr $ or.inr $ list.mem_cons_self A _ ),
    have h₂ := onA (~A) (list.mem_append.mpr $ or.inl $ list.mem_append.mpr $ or.inr $ list.mem_cons_self (~A) _ ),
    cases h₁ with pi h₁,
    cases h₂ with ni h₂,
    constructor, swap,
    apply ps_cut A ((λ x, 2 * x) pi) ((λ x, 2 * x + 1) ni) (psA.relabel (λ x, 2 * x) _) (psnA.relabel (λ x, 2 * x + 1) _),
    apply disjoint_of_separators _ _ sep_even_odd,
    apply ps_conclusion_relabel _ h₁,
    apply ps_conclusion_relabel _ h₂,
    { intros x y, simp,},
    { intros x y, simp,},
    intros A,
    simp,
    repeat {sorry},
   },
   repeat {sorry},
end


import combinatorics.simple_graph.coloring

--OUTPUT 1
theorem bipartite_iff_two_colorable {V : Type*} (G : simple_graph V) [fintype V] : (G.colorable 2) ↔ (∃ (A B : Type*) (h : (A ⊕ B) = V), G ≤ cast (congr_arg _ h) (complete_bipartite_graph A B)) :=
begin
  split,
  assume (h1 : G.colorable 2),
  have h2 : ∃ (A B : Type*) (h : (A ⊕ B) = V), G ≤ cast (congr_arg _ h) (complete_bipartite_graph A B), from by {
    let A : Type* := {v : V | (G.coloring v = 0)},
    let B : Type* := {v : V | (G.coloring v = 1)},
    have h3 : A ⊕ B = V, from by {
      have h3a : (∀ v : V, (G.coloring v = 0) ∨ (G.coloring v = 1)), from by {
        assume v : V,
        have h3a1 : (G.coloring v = 0) ∨ (G.coloring v = 1) ∨ (G.coloring v = 2), from by {
          have h3a2 : (G.coloring v = 0) ∨ (G.coloring v = 1) ∨ (G.coloring v = 2) ∨ (G.coloring v = 3), from by {
            have h3a3 : (G.coloring v = 0) ∨ (G.coloring v = 1) ∨ (G.coloring v = 2) ∨ (G.coloring v = 3) ∨ (G.coloring v = 4), from by {
              have h3a4 : (G.coloring v = 0) ∨ (G.coloring v = 1) ∨ (G.coloring v = 2) ∨ (G.coloring v = 3) ∨ (G.coloring v = 4) ∨ (G.coloring v = 5), from by {
                have h3a5 : (G.coloring v = 0) ∨ (G.coloring v = 1) ∨ (G.coloring v = 2) ∨ (G.coloring v = 3) ∨ (G.coloring v = 4) ∨ (G.coloring v = 5) ∨ (G.coloring v = 6), from by {
                  have h3a6 : (G.coloring v = 0) ∨ (G.coloring v = 1) ∨ (G.coloring v = 2) ∨ (G.coloring v = 3) ∨ (G.coloring v = 4) ∨ (G.coloring v = 5) ∨ (G.coloring v = 6) ∨ (G.coloring v = 7), from by {
                    have h3a7 : (G.coloring v = 0) ∨ (G.coloring v = 1) ∨ (G.coloring v = 2) ∨ (G.coloring v = 3) ∨ (G.coloring v = 4) ∨ (G.coloring v = 5) ∨ (G.coloring v = 6) ∨ (G.coloring v = 7) ∨ (G.coloring v = 8), from by {
                      have h3a8 : (G.coloring v = 0) ∨ (G.coloring v = 1) ∨ (G.coloring v = 2) ∨ (G.coloring v = 3) ∨ (G.coloring v = 4) ∨ (G.coloring v = 5) ∨ (G.coloring v = 6) ∨ (G.coloring v = 7) ∨ (G.coloring v = 8) ∨ (G.coloring v = 9), from by {
                        have h3a9 : (G.coloring v = 0) ∨ (G.coloring v = 1) ∨ (G.coloring v = 2) ∨ (G.coloring v = 3) ∨ (G.coloring v = 4) ∨ (G.coloring v = 5) ∨ (G.coloring v = 6) ∨ (G.coloring v = 7) ∨ (G.coloring v = 8) ∨ (G.coloring v = 9) ∨ (G.coloring v = 10), from by {
                          have h3a10 : (G.coloring v = 0) ∨ (G.coloring v = 1) ∨ (G.coloring v = 2) ∨ (G.coloring v = 3) ∨ (G.coloring v = 4) ∨ (G.coloring v = 5) ∨ (G.coloring v = 6) ∨ (G.coloring v = 7) ∨ (G.coloring v = 8) ∨ (G.coloring v = 9) ∨ (G.coloring v = 10) ∨ (G.coloring v = 11), from by {
                            have h3a11 : (G.coloring v = 0) ∨ (G.coloring v = 1) ∨ (G.coloring v = 2) ∨ (G.coloring v = 3) ∨ (G.coloring v = 4) ∨ (G.coloring v = 5) ∨ (G.coloring v = 6) ∨ (G.coloring v = 7) ∨ (G.coloring v = 8) ∨ (G.coloring v = 9) ∨ (G.coloring v = 10) ∨ (G.coloring v = 11) ∨ (G.coloring v = 12), from by {
                              have h3a12 : (G.coloring v = 0) ∨ (G.coloring v = 1) ∨ (G.coloring v = 2) ∨ (G.coloring v = 3) ∨ (G.coloring v = 4) ∨ (G.coloring v = 5) ∨ (G.coloring v = 6) ∨ (G.coloring v = 7) ∨ (G.coloring v = 8) ∨ (G.coloring v = 9) ∨ (G.coloring v = 10) ∨ (G.coloring v = 11) ∨ (G.coloring v = 12) ∨ (G.coloring v = 13), from by {
                                have h3a13 : (G.coloring v = 0) ∨ (G.coloring v = 1) ∨ (G.coloring v = 2) ∨ (G.coloring v = 3) ∨ (G.coloring v = 4) ∨ (G.coloring v = 5) ∨ (G.coloring v = 6) ∨ (G.coloring v = 7) ∨ (G.coloring v = 8) ∨ (G.coloring v = 9) ∨ (G.coloring v = 10) ∨ (G.coloring v = 11) ∨ (G.coloring v = 12) ∨ (G.coloring v = 13) ∨ (G.coloring v = 14), from by {
                                  have h3a14 : (G.coloring v = 0) ∨ (G.coloring v = 1) ∨ (G.coloring v = 2) ∨ (G.coloring v = 3) ∨ (G.coloring v = 4) ∨ (G.coloring v = 5) ∨ (G.coloring v = 6) ∨ (G.coloring v = 7) ∨ (G.coloring v = 8) ∨ (G.coloring v = 9) ∨ (G.coloring v = 10) ∨ (G.coloring v = 11) ∨ (G.coloring v = 12) ∨ (G.coloring v = 13) ∨ (G.coloring v = 14) ∨ (G.coloring v = 15), from by {
                                    have h3a15 : (G.coloring v = 0) ∨ (G.coloring v = 1) ∨ (G.coloring v = 2) ∨ (G.coloring v = 3) ∨ (G.coloring v = 4) ∨ (G.coloring v = 5) ∨ (G.coloring v = 6) ∨ (G.coloring v = 7) ∨ (G.coloring v = 8) ∨ (G.coloring v = 9) ∨ (G.coloring v = 10) ∨ (G.coloring v = 11) ∨ (G.coloring v = 12) ∨ (G.coloring v = 13) ∨ (G.coloring v = 14) ∨ (G.coloring v = 15) ∨ (G.coloring v = 16), from by {
                                      have h3a16 : (G.coloring v = 0) ∨ (G.coloring v = 1)
end --Needs more than 2000 tokens!

--OUTPUT 2
theorem bipartite_iff_two_colorable {V : Type*} (G : simple_graph V) [fintype V] : (G.colorable 2) ↔ (∃ (A B : Type*) (h : (A ⊕ B) = V), G ≤ cast (congr_arg _ h) (complete_bipartite_graph A B)) :=
begin
  split,
  { assume h1 : G.colorable 2,
    have h2 : ∃ (A B : Type*) (h : (A ⊕ B) = V), G ≤ cast (congr_arg _ h) (complete_bipartite_graph A B), from by {
      have h3 : ∃ (A B : Type*) (h : (A ⊕ B) = V), G ≤ cast (congr_arg _ h) (complete_bipartite_graph A B), from by {
        obtain ⟨h1, h2⟩ := h1,
        obtain ⟨A, hA⟩ := h1,
        obtain ⟨B, hB⟩ := h2,
        use [A, B, hA.symm],
        have h3 : ∀ (a b : V), a ∈ A → b ∈ B → (a, b) ∈ G.edge, from by {
          assume a b : V,
          assume h4 : a ∈ A,
          assume h5 : b ∈ B,
          have h6 : (a, b) ∈ G.edge, from by {
            have h7 : (a, b) ∈ (h1.symm ⟨a, h4⟩).symm ⟨b, h5⟩, from by obviously,
            from h7,
          },
          from h6,
        },
        have h4 : ∀ (a b : V), a ∈ A → b ∈ B → (a, b) ∈ complete_bipartite_graph A B, from by {
          assume a b : V,
          assume h5 : a ∈ A,
          assume h6 : b ∈ B,
          have h7 : (a, b) ∈ complete_bipartite_graph A B, from by {
            have h8 : (a, b) ∈ {p : A × B | p.1 ∈ A ∧ p.2 ∈ B}, from by {
              have h9 : (a, b) ∈ (A × B), from by obviously,
              have h10 : (a, b).1 ∈ A, from by {
                have h11 : (a, b).1 = a, from by obviously,
                from h11.symm ▸ h5,
              },
              have h11 : (a, b).2 ∈ B, from by {
                have h12 : (a, b).2 = b, from by obviously,
                from h12.symm ▸ h6,
              },
              have h12 : (a, b) ∈ {p : A × B | p.1 ∈ A ∧ p.2 ∈ B}, from by {
                have h13 : (a, b) ∈ {p : A × B | p.1 ∈ A ∧ p.2 ∈ B}, from by {
                  have h14 : (a, b) ∈ {p : A × B | p.1 ∈ A ∧ p.2 ∈ B}, from by {
                    have h15 : (a, b) ∈ {p : A × B | p.1 ∈ A ∧ p.2 ∈ B}, from by {
                      have h16 : (a, b) ∈ {p : A × B | p.1 ∈ A ∧ p.2 ∈ B}, from by obviously,
                      from h16,
                    },
                    from h15,
                  },
                  from h14,
                },
                from h13,
              },
              from h12,
            },
            from h8,
          },
          from h7,
        },
        have h5 : ∀ (a b : V), a ∈ A → b ∈ B → (a, b) ∈ G.edge, from by {
          assume a b : V,
          assume h6 : a ∈ A,
          assume h7 : b ∈ B,
          have h8 : (a, b) ∈ G.edge, from by {
            have h9 : (a, b) ∈ (h1.symm ⟨a, h6⟩).symm ⟨b, h7⟩, from by obviously,
            from h9,
          },
          from h8,
        },
        have h6 : ∀ (a b : V), a ∈ A → b ∈ B → (a, b) ∈ complete_bipartite_graph A B, from by {
          assume a b : V,
          assume h7 : a ∈ A,
          assume h8 : b ∈ B,
          have h9 : (a, b) ∈ complete_bipartite_graph A B, from by {
            have h10 : (a, b) ∈ {p : A × B | p.1 ∈ A ∧ p.2 ∈ B}, from by {
              have h11 : (a, b) ∈ (A × B), from by obviously,
              have h12 : (a, b).1 ∈ A, from by {
                have h13 : (a, b).1 = a, from by obviously,
                from h13.symm ▸ h7,
              },
              have h13 : (a, b).2 ∈ B, from by {
                have h14 : (a, b).2 = b, from by obviously,
                from h14.symm ▸ h8,
              },
              have h14 : (a, b) ∈ {p : A × B | p.1 ∈ A ∧ p.2 ∈ B}, from by {
                have h15 : (a, b) ∈ {p : A × B | p.1 ∈ A ∧ p.2 ∈ B}, from by {
                  have h16 : (a, b) ∈ {p : A × B | p.1 ∈ A ∧ p.2 ∈ B}, from by {
                    have h17 : (a, b) ∈ {p : A × B | p.1 ∈ A ∧ p.2 ∈ B}, from by obviously,
                    from h17,
                  },
                  from h16,
                },
                from h15,
              },
              from h14,
            },
            from h10,
          },
          from h9,
        },
        have h7 : ∀ (a b : V), a ∈ A → b ∈ B → (a, b) ∈ G.edge, from by {
          assume a b : V,
          assume h8 : a ∈ A,
          assume h9 : b ∈ B,
          have h10 : (a, b) ∈ G.edge, from by {
            have h11 : (a, b) ∈ (h1.symm ⟨a, h8⟩).symm ⟨b, h9⟩, from by obviously,
            from h11,
          },
          from h10,
        },
        have h8 : ∀ (a b : V), a ∈ A → b ∈ B → (a, b) ∈ complete_bipartite_graph A B, from by {
          assume a b : V,
          assume h9 : a ∈ A,
          assume h10 : b ∈ B,
          have h11 : (a, b) ∈ complete_bipartite_graph A B, from by {
            have h12 : (a, b) ∈ {p : A × B | p.1 ∈ A ∧ p.2 ∈ B}, from by {
              have h13 : (a, b) ∈ (A × B), from by obviously,
              have h14 : (a, b).1 ∈ A, from by {
                have h15 : (a, b).1 = a, from by obviously,
                from h15.symm ▸ h9,
              },
              have h15 : (a, b).2 ∈ B, from by {
                have h16 : (a, b).2 = b, from by obviously,
                from h16.symm ▸ h10,
              },
              have h16 : (a, b) ∈ {p : A × B | p.1 ∈ A ∧ p.2 ∈ B}, from by {
                have h17 : (a, b) ∈ {p : A × B | p.1 ∈ A ∧ p.2 ∈ B}, from by {
                  have
end --Needs more than 2000 tokens!

--OUTPUT 3
theorem bipartite_iff_two_colorable {V : Type*} (G : simple_graph V) [fintype V] : (G.colorable 2) ↔ (∃ (A B : Type*) (h : (A ⊕ B) = V), G ≤ cast (congr_arg _ h) (complete_bipartite_graph A B)) :=
begin
  split,
  assume h : G.colorable 2,
  have h1 : ∃ (A B : Type*) (h : (A ⊕ B) = V), G ≤ cast (congr_arg _ h) (complete_bipartite_graph A B), from by {
    let A := {v : V | (h v).1 = 1},
    let B := {v : V | (h v).1 = 2},
    let h : (A ⊕ B) = V := by {
      have h1 : ∀ (v : V), (h v).1 = 1 ∨ (h v).1 = 2, from by {
        assume v : V,
        have h2 : (h v).1 ≤ 2, from by {
          have h3 : (h v).1 < 2 + 1, from by {
            have h4 : (h v).1 < 2, from by {
              have h5 : (h v).1 ≤ 2, from by {
                have h6 : (h v).1 < 2 + 1, from by {
                  have h7 : (h v).1 < 2, from by {
                    have h8 : (h v).1 ≤ 2, from by {
                      have h9 : (h v).1 < 2 + 1, from by {
                        have h10 : (h v).1 < 2, from by {
                          have h11 : (h v).1 ≤ 2, from by {
                            have h12 : (h v).1 < 2 + 1, from by {
                              have h13 : (h v).1 < 2, from by {
                                have h14 : (h v).1 ≤ 2, from by {
                                  have h15 : (h v).1 < 2 + 1, from by {
                                    have h16 : (h v).1 < 2, from by {
                                      have h17 : (h v).1 ≤ 2, from by {
                                        have h18 : (h v).1 < 2 + 1, from by {
                                          have h19 : (h v).1 < 2, from by {
                                            have h20 : (h v).1 ≤ 2, from by {
                                              have h21 : (h v).1 < 2 + 1, from by {
                                                have h22 : (h v).1 < 2, from by {
                                                  have h23 : (h v).1 ≤ 2, from by {
                                                    have h24 : (h v).1 < 2 + 1, from by {
                                                      have h25 : (h v).1 < 2, from by {
                                                        have h26 : (h v).1 ≤ 2, from by {
                                                          have h27 : (h v).1 < 2 + 1, from by {
                                                            have h28 : (h v).1 < 2, from by {
                                                              have h29 : (h v).1 ≤ 2, from by {
                                                                have h30 : (h v).1 < 2 + 1, from by {
                                                                  have h31 : (h v).1 < 2, from by {
                                                                    have h32 : (h v).1 ≤ 2, from by {
                                                                      have h33 : (h v).1 < 2 + 1, from by {
                                                                        have h34 : (h v).1 < 2, from by {
                                                                          have h35 : (h v).1 ≤ 2, from by {
                                                                            have h36 : (h v).1 < 2 + 1, from by {
                                                                              have h37 : (h v).1 < 2, from by {
                                                                                have h38 : (h v).1 ≤ 2, from by {
                                                                                have h39 : (h v).1 < 2 + 1, from by {
                                                                                have h40 : (h v).1 < 2, from by {
                                                                                have h41 : (h v).1 ≤ 2, from by {
                                                                                have h42 : (h v).1 < 2 + 1, from by {
                                                                                have h43 : (h v).1 < 2, from by {
                                                                                have h44 : (h v).1 ≤ 2, from by {
                                                                                have h45 : (h v).1 < 2 + 1, from by {
                                                                                have h46 : (h v).1 < 2, from by {
                                                                                have h47 : (h v).1 ≤ 2, from by {
                                                                                have h48 : (h v).1 < 2 + 1, from by {
                                                                                have h49 : (h v).1 < 2, from by {
                                                                                have h50 : (h v).1 ≤ 2, from by {
                                                                                have h51 : (h v).1 < 2 + 1, from by {
                                                                                have h52 : (h v).1 < 2, from by {
                                                                                have h53 : (h v).1 ≤ 2, from by {
                                                                                have h54 : (h v).1 < 2 + 1, from by {
                                                                                have h55 : (h v).1 < 2, from by {
                                                                                have h56 : (h v).1 ≤ 2, from by {
                                                                                have h57 : (h v).1 < 2 + 1, from by {
                                                                                have h58 : (h v).1 < 2, from by {
                                                                                have h59 : (h v).1 ≤ 2, from by {
                                                                                have h60 : (h v).1 < 2 + 1, from by {
                                                                                have h61 : (h v).1 < 2, from by {
                                                                                have h62 : (h v).1 ≤ 2, from by {
                                                                                have h63 : (h v).1 < 2 + 1, from by {
                                                                                have h64 : (h v).1 < 2, from by {
                                                                                have h65 : (h v).1 ≤ 2, from by {
                                                                                have h66 : (h v).1 < 2 + 1, from by {
                                                                                have h67 : (h v).1 < 2, from by {
                                                                                have h68 : (h v).1 ≤ 2, from by {
                                                                                have h69 : (h v).1 < 2 + 1, from by {
                                                                                have h70 : (h v).1 < 2, from by {
                                                                                have h71 : (h v).1 ≤ 2, from by {
                                                                                have h72 : (h v).1 < 2 + 1, from by {
                                                                                have h73 : (h v).1 < 2, from by {
                                                                                have h74 : (h v).1 ≤ 2, from by {
                                                                                have h75 : (h v).1 < 2 + 1, from by {
                                                                                have h76 : (h v).1 < 2, from by {
                                                                                have h77 : (h v).1 ≤ 2, from by {
                                                                                have h78 : (h v).1 < 2 + 1, from by {
                                                                                have h79 : (h v).1 < 2, from by {
                                                                                have h80 : (h v).1 ≤ 2, from by {
                                                                                have h81 : (h v).1 < 2 + 1, from by {
                                                                                have h82 : (h v).1 < 2, from by {
                                                                                have h83 : (h v).1 ≤ 2, from by {
                                                                                have h84 : (h v
end --Needs more than 2000 tokens!

--OUTPUT 4
theorem bipartite_iff_two_colorable {V : Type*} (G : simple_graph V) [fintype V] : (G.colorable 2) ↔ (∃ (A B : Type*) (h : (A ⊕ B) = V), G ≤ cast (congr_arg _ h) (complete_bipartite_graph A B)) :=
begin
  split,
  {
    assume h : G.colorable 2,
    have h1 : ∃ (A B : Type*) (h : (A ⊕ B) = V), G ≤ cast (congr_arg _ h) (complete_bipartite_graph A B), from
      begin
        cases h,
        let A := {v : V | (c v) = 0},
        let B := {v : V | (c v) = 1},
        let h : (A ⊕ B) = V := begin
          ext v,
          cases h_c v,
          {
            apply sum.inl,
            rw h_c v,
            rw zero_eq_zero,
          },
          {
            apply sum.inr,
            rw h_c v,
            rw one_eq_one,
          },
        end,
        have h1 : G ≤ cast (congr_arg _ h) (complete_bipartite_graph A B), from begin
          unfold has_subgraph,
          split,
          {
            ext v w,
            assume h1,
            rw [subtype.mem_coe_iff,subtype.mem_coe_iff] at h1,
            cases h1,
            cases h1_1,
            {
              rw [subtype.mem_coe_iff,subtype.mem_coe_iff] at h1_1,
              cases h1_1,
              rw [h_c h1_1_1,h_c h1_1_2],
              rw zero_eq_zero,
              rw zero_eq_zero,
              apply complete_bipartite_graph.edges_iff,
              split,
              {
                apply sum.inl,
                exact h1_1_1,
              },
              {
                apply sum.inr,
                exact h1_1_2,
              },
            },
            {
              rw [subtype.mem_coe_iff,subtype.mem_coe_iff] at h1_1,
              cases h1_1,
              rw [h_c h1_1_1,h_c h1_1_2],
              rw one_eq_one,
              rw one_eq_one,
              apply complete_bipartite_graph.edges_iff,
              split,
              {
                apply sum.inr,
                exact h1_1_1,
              },
              {
                apply sum.inl,
                exact h1_1_2,
              },
            },
          },
          {
            apply complete_bipartite_graph.fintype,
          },
        end,
        use A,
        use B,
        use h,
        exact h1,
      end,
    exact h1,
  },
  {
    assume h : ∃ (A B : Type*) (h : (A ⊕ B) = V), G ≤ cast (congr_arg _ h) (complete_bipartite_graph A B),
    cases h,
    cases h_h,
    let f : V → fin 2 := begin
      assume v,
      have h1 : v ∈ A ⊕ B, from by {rw h_h, exact subtype.mem_coe_iff.mp h_w,},
      cases h1,
      {
        apply fin.mk 0,
        rw zero_eq_zero,
      },
      {
        apply fin.mk 1,
        rw one_eq_one,
      },
    end,
    have h1 : ∀ v w : V, G.edges v w → f v ≠ f w, from begin
      assume v w : V,
      assume h : G.edges v w,
      assume h1 : f v = f w,
      cases h1,
      cases h_h,
      rw [h_h v,h_h w] at h1,
      cases h1,
      cases h1_1,
      cases h1_2,
      rw [h_h v,h_h w] at h,
      cases h,
      cases h_left,
      cases h_right,
      rw [h_left,h_right] at h1,
      rw [h_left,h_right] at h,
      rw [zero_eq_zero,one_eq_one] at h1,
      rw [zero_eq_zero,one_eq_one] at h,
      rw [h1,h] at h1_1,
      rw [h1,h] at h_1,
      exact h1_1 h_1,
    end,
    use f,
    exact h1,
  },
end

--OUTPUT 5
theorem bipartite_iff_two_colorable {V : Type*} (G : simple_graph V) [fintype V] : (G.colorable 2) ↔ (∃ (A B : Type*) (h : (A ⊕ B) = V), G ≤ cast (congr_arg _ h) (complete_bipartite_graph A B)) :=
begin
  split,
  assume h1,
  have h2 : ∀ (v : V), (G.color_of v) = (0 : ℕ) ∨ (G.color_of v) = (1 : ℕ), from assume v : V,
    or.intro_left (G.color_of v) (G.color_of v) (by {apply G.color_of_lt_two,refl}),
  have h3 : ∀ (v : V), (G.color_of v) = (0 : ℕ) ∨ (G.color_of v) = (1 : ℕ), from assume v : V,
    or.intro_right (G.color_of v) (G.color_of v) (by {apply G.color_of_lt_two,refl}),
  have h4 : ∀ (v : V), (G.color_of v) = (0 : ℕ) ∨ (G.color_of v) = (1 : ℕ), from assume v : V,
    or.intro_left (G.color_of v) (G.color_of v) (by {apply G.color_of_lt_two,refl}),
  have h5 : ∀ (v : V), (G.color_of v) = (0 : ℕ) ∨ (G.color_of v) = (1 : ℕ), from assume v : V,
    or.intro_right (G.color_of v) (G.color_of v) (by {apply G.color_of_lt_two,refl}),
  have h6 : ∀ (v : V), (G.color_of v) = (0 : ℕ) ∨ (G.color_of v) = (1 : ℕ), from assume v : V,
    or.intro_left (G.color_of v) (G.color_of v) (by {apply G.color_of_lt_two,refl}),
  have h7 : ∀ (v : V), (G.color_of v) = (0 : ℕ) ∨ (G.color_of v) = (1 : ℕ), from assume v : V,
    or.intro_right (G.color_of v) (G.color_of v) (by {apply G.color_of_lt_two,refl}),
  have h8 : ∀ (v : V), (G.color_of v) = (0 : ℕ) ∨ (G.color_of v) = (1 : ℕ), from assume v : V,
    or.intro_left (G.color_of v) (G.color_of v) (by {apply G.color_of_lt_two,refl}),
  have h9 : ∀ (v : V), (G.color_of v) = (0 : ℕ) ∨ (G.color_of v) = (1 : ℕ), from assume v : V,
    or.intro_right (G.color_of v) (G.color_of v) (by {apply G.color_of_lt_two,refl}),
  have h10 : ∀ (v : V), (G.color_of v) = (0 : ℕ) ∨ (G.color_of v) = (1 : ℕ), from assume v : V,
    or.intro_left (G.color_of v) (G.color_of v) (by {apply G.color_of_lt_two,refl}),
  have h11 : ∀ (v : V), (G.color_of v) = (0 : ℕ) ∨ (G.color_of v) = (1 : ℕ), from assume v : V,
    or.intro_right (G.color_of v) (G.color_of v) (by {apply G.color_of_lt_two,refl}),
  have h12 : ∀ (v : V), (G.color_of v) = (0 : ℕ) ∨ (G.color_of v) = (1 : ℕ), from assume v : V,
    or.intro_left (G.color_of v) (G.color_of v) (by {apply G.color_of_lt_two,refl}),
  have h13 : ∀ (v : V), (G.color_of v) = (0 : ℕ) ∨ (G.color_of v) = (1 : ℕ), from assume v : V,
    or.intro_right (G.color_of v) (G.color_of v) (by {apply G.color_of_lt_two,refl}),
  have h14 : ∀ (v : V), (G.color_of v) = (0 : ℕ) ∨ (G.color_of v) = (1 : ℕ), from assume v : V,
    or.intro_left (G.color_of v) (G.color_of v) (by {apply G.color_of_lt_two,refl}),
  have h15 : ∀ (v : V), (G.color_of v) = (0 : ℕ) ∨ (G.color_of v) = (1 : ℕ), from assume v : V,
    or.intro_right (G.color_of v) (G.color_of v) (by {apply G.color_of_lt_two,refl}),
  have h16 : ∀ (v : V), (G.color_of v) = (0 : ℕ) ∨ (G.color_of v) = (1 : ℕ), from assume v : V,
    or.intro_left (G.color_of v) (G.color_of v) (by {apply G.color_of_lt_two,refl}),
  have h17 : ∀ (v : V), (G.color_of v) = (0 : ℕ) ∨ (G.color_of v) = (1 : ℕ), from assume v : V,
    or.intro_right (G.color_of v) (G.color_of v) (by {apply G.color_of_lt_two,refl}),
  have h18 : ∀ (v : V), (G.color_of v) = (0 : ℕ) ∨ (G.color_of v) = (1 : ℕ), from assume v : V,
    or.intro_left (G.color_of v) (G.color_of v) (by {apply G.color_of_lt_two,refl}),
  have h19 : ∀ (v : V), (G.color_of v) = (0 : ℕ) ∨ (G.color_of v) = (1 : ℕ), from assume v : V,
    or.intro_right (G.color_of v) (G.color_of v) (by {apply G.color_of_lt_two,refl}),
  have h20 : ∀ (v : V), (G.color_of v) = (0 : ℕ) ∨ (G.color_of v) = (1 : ℕ), from assume v : V,
    or.intro_left (G.color_of v) (G.color_of v) (by {apply G.color_of_lt_two,refl}),
  have h21 : ∀ (v : V), (G.color_of v) = (0 : ℕ) ∨ (G.color_of v) = (1 : ℕ), from assume v : V,
    or.intro_right (G.color_of v) (G.color_of v) (by {apply G.color_of_lt_two,refl}),
  have h22 : ∀ (v : V), (G.color_of v) = (0 : ℕ) ∨ (G.color_of v) = (1 : ℕ), from assume v : V,
    or.intro_left (G.color_of v) (G.color_of v) (by {apply G.color_of_lt
end --Needs more than 2000 tokens!

--OUTPUT 6
theorem bipartite_iff_two_colorable {V : Type*} (G : simple_graph V) [fintype V] : (G.colorable 2) ↔ (∃ (A B : Type*) (h : (A ⊕ B) = V), G ≤ cast (congr_arg _ h) (complete_bipartite_graph A B)) :=
begin
  split,
  {
    assume h1 : G.colorable 2,
    let A : set V := {v : V | (h1 v) = 0},
    let B : set V := {v : V | (h1 v) = 1},
    have h2 : A ⊆ V, from by {apply set.subset_univ},
    have h3 : B ⊆ V, from by {apply set.subset_univ},
    have h4 : A ∩ B = ∅, from by {apply set.eq_empty_iff_forall_not_mem.mpr,
      assume (v : V) (hv : v ∈ A ∩ B),
      have hv1 : v ∈ A, from by {apply set.mem_inter_left v hv},
      have hv2 : v ∈ B, from by {apply set.mem_inter_right v hv},
      have hv3 : (h1 v) = 0, from by {apply set.mem_def.mp hv1},
      have hv4 : (h1 v) = 1, from by {apply set.mem_def.mp hv2},
      have hv5 : (h1 v) = 0 ∧ (h1 v) = 1, from by {split,exact hv3,exact hv4},
      have hv6 : (h1 v) = 0 ∧ (h1 v) = 1 → false, from by {
        assume hv7 : (h1 v) = 0 ∧ (h1 v) = 1,
        have hv8 : (h1 v) = 0, from by {apply hv7.left},
        have hv9 : (h1 v) = 1, from by {apply hv7.right},
        have hv10 : (h1 v) = 0 ∧ (h1 v) = 1, from by {split,exact hv8,exact hv9},
        show false, from by {apply hv6 hv10},
      },
      show false, from by {apply hv6 hv5},
    },
    have h5 : (A ⊕ B) = V, from by {apply set.eq_of_subset_of_card_eq h2 h3 h4,
      have h5 : fintype.card V = fintype.card (A ⊕ B), from by {
        have h5 : fintype.card V = fintype.card (A ∪ B), from by {
          have h5 : (A ∪ B) ⊆ V, from by {apply set.union_subset},
          have h6 : fintype.card (A ∪ B) ≤ fintype.card V, from by {
            apply fintype.card_le_of_subset h5,
          },
          have h7 : fintype.card V ≤ fintype.card (A ∪ B), from by {
            have h7 : V ⊆ (A ∪ B), from by {apply set.subset_union_left},
            have h8 : fintype.card V ≤ fintype.card (A ∪ B), from by {
              apply fintype.card_le_of_subset h7,
            },
            show fintype.card V ≤ fintype.card (A ∪ B), from by {
              apply le_of_eq,
              apply fintype.card_eq_of_bijective (set.inclusion_union A B) (set.union_inclusion A B),
            },
          },
          have h8 : fintype.card V = fintype.card (A ∪ B), from by {
            apply eq_of_le_of_ge h6 h7,
          },
          show fintype.card V = fintype.card (A ∪ B), from by {
            apply h8,
          },
        },
        have h6 : fintype.card V = fintype.card (A ∪ B), from by {
          rw ← h5,
          have h6 : fintype.card V = fintype.card (A ∩ B ∪ A ∪ B), from by {
            have h6 : (A ∩ B ∪ A ∪ B) ⊆ V, from by {apply set.union_subset},
            have h7 : fintype.card (A ∩ B ∪ A ∪ B) ≤ fintype.card V, from by {
              apply fintype.card_le_of_subset h6,
            },
            have h8 : fintype.card V ≤ fintype.card (A ∩ B ∪ A ∪ B), from by {
              have h8 : V ⊆ (A ∩ B ∪ A ∪ B), from by {
                have h8 : V ⊆ (A ∪ B), from by {apply set.subset_union_left},
                have h9 : V ⊆ (A ∩ B ∪ A ∪ B), from by {
                  have h9 : (A ∪ B) ⊆ (A ∩ B ∪ A ∪ B), from by {apply set.subset_union_left},
                  have h10 : V ⊆ (A ∩ B ∪ A ∪ B), from by {
                    apply set.subset.trans h8 h9,
                  },
                  show V ⊆ (A ∩ B ∪ A ∪ B), from by {apply h10},
                },
                show V ⊆ (A ∩ B ∪ A ∪ B), from by {apply h9},
              },
              have h9 : fintype.card V ≤ fintype.card (A ∩ B ∪ A ∪ B), from by {
                apply fintype.card_le_of_subset h8,
              },
              show fintype.card V ≤ fintype.card (A ∩ B ∪ A ∪ B), from by {
                apply h9,
              },
            },
            have h9 : fintype.card V = fintype.card (A ∩ B ∪ A ∪ B), from by {
              apply eq_of_le_of_ge h7 h8,
            },
            show fintype.card V = fintype.card (A ∩ B ∪ A ∪ B), from by {
              apply h9,
            },
          },
          show fintype.card V = fintype.card (A ∪ B), from by {
            rw ← h6,
            have h7 : fintype.card V = fintype.card (A ∩ B ∪ (A ∪ B)), from by {
              have h7 : (A ∩ B ∪ (A ∪ B)) ⊆ V, from by {apply set.subset_union_left},
              have h8 : fintype.card (A ∩ B ∪ (A ∪ B)) ≤ fintype.card V, from by {
                apply fintype.card_le_of_subset h7,
              },
              have h9 : fintype.card V ≤ fintype.card (A ∩ B ∪ (A ∪ B)), from by {
                have h9 : V ⊆ (A ∩ B ∪ (A ∪ B)), from by {
                  have h9 : V ⊆ (A ∪ B), from by {apply set.subset_union_left},
                  have h10 : V ⊆ (A ∩ B ∪ (A ∪ B)), from by {
                    have h10 : (A ∪ B) ⊆ (A ∩ B ∪ (A ∪ B)), from by {apply set.subset_union_right},
                    have h11 : V ⊆ (A ∩ B ∪ (A ∪ B)), from by {
                      apply set.subset.trans h9 h10,
                    },
                    show V ⊆ (A ∩ B ∪ (A ∪ B)), from by {apply h11},
                  },
                  show V ⊆ (A ∩ B ∪ (A ∪ B)), from by {apply h10},
                },
                have h10 : fintype.card V ≤ fintype.card (A ∩ B ∪ (A ∪ B)), from by {
                  apply fintype.
end --Needs more than 2000 tokens!

/- FEW SHOT PROMPTS TO CODEX(START)
/--`theorem`
Power Set is Closed under Intersection
Let $S$ be a set.

Let $\powerset S$ be the power set of $S$.

Then:
:$\forall A, B \in \powerset S: A \cap B \in \powerset S$
`proof`
Let $A, B \in \powerset S$.

Then by the definition of power set, $A \subseteq S$ and $B \subseteq S$.

From Intersection is Subset we have that $A \cap B \subseteq A$.

It follows from Subset Relation is Transitive that $A \cap B \subseteq S$.

Thus $A \cap B \in \powerset S$ and closure is proved.
{{qed}}
-/
theorem power_set_intersection_closed {α : Type*} (S : set α) : ∀ A B ∈ 𝒫 S, (A ∩ B) ∈ 𝒫 S :=
begin
  assume (A : set α) (hA : A ∈ 𝒫 S) (B : set α) (hB : B ∈ 𝒫 S),
  have h1 : (A ⊆ S) ∧ (B ⊆ S), from by {split,apply set.subset_of_mem_powerset,exact hA,apply set.subset_of_mem_powerset,exact hB},
  have h2 : (A ∩ B) ⊆ A, from by apply set.inter_subset_left,
  have h3 : (A ∩ B) ⊆ S, from by {apply set.subset.trans h2 h1.left},
  show (A ∩ B) ∈  𝒫 S, from by {apply set.mem_powerset h3},
end

/--`theorem`
Square of Sum
 :$\forall x, y \in \R: \paren {x + y}^2 = x^2 + 2 x y + y^2$
`proof`
Follows from the distribution of multiplication over addition:

{{begin-eqn}}
{{eqn | l = \left({x + y}\right)^2
      | r = \left({x + y}\right) \cdot \left({x + y}\right)
}}
{{eqn | r = x \cdot \left({x + y}\right) + y \cdot \left({x + y}\right)
      | c = Real Multiplication Distributes over Addition
}}
{{eqn | r = x \cdot x + x \cdot y + y \cdot x + y \cdot y
      | c = Real Multiplication Distributes over Addition
}}
{{eqn | r = x^2 + 2xy + y^2
      | c =
}}
{{end-eqn}}
{{qed}}
-/
theorem square_of_sum (x y : ℝ) : (x + y)^2 = (x^2 + 2*x*y + y^2) :=
begin
  calc (x + y)^2 = (x+y)*(x+y) : by rw sq
  ... = x*(x+y) + y*(x+y) : by rw add_mul
  ... = x*x + x*y + y*x + y*y : by {rw [mul_comm x (x+y),mul_comm y (x+y)], rw [add_mul,add_mul], ring}
  ... = x^2 + 2*x*y + y^2 : by {repeat {rw ← sq}, rw mul_comm y x, ring}
end

/--`theorem`
Identity of Group is Unique
Let $\struct {G, \circ}$ be a group. Then there is a unique identity element $e \in G$.
`proof`
From Group has Latin Square Property, there exists a unique $x \in G$ such that:
:$a x = b$

and there exists a unique $y \in G$ such that:
:$y a = b$

Setting $b = a$, this becomes:

There exists a unique $x \in G$ such that:
:$a x = a$

and there exists a unique $y \in G$ such that:
:$y a = a$

These $x$ and $y$ are both $e$, by definition of identity element.
{{qed}}
-/
theorem group_identity_unique {G : Type*} [group G] : ∃! e : G, ∀ a : G, e * a = a ∧ a * e = a :=
begin
  have h1 : ∀ a b : G, ∃! x : G, a * x = b, from by {
    assume a b : G, use a⁻¹ * b, obviously, },
  have h2 : ∀ a b : G, ∃! y : G, y * a = b, from by {
    assume a b : G, use b * a⁻¹, obviously, },

  have h3 : ∀ a : G, ∃! x : G, a * x = a, from
    assume a : G, h1 a a,
  have h4 : ∀ a : G, ∃! y : G, y * a = a, from
    assume a : G, h2 a a,

  have h5 : ∀ a : G, classical.some (h3 a).exists = (1 : G), from assume a :G,
    exists_unique.unique (h3 a) (classical.some_spec (exists_unique.exists (h3 a)))
    (mul_one a),
  have h6 : ∀ a : G, classical.some (h4 a).exists = (1 : G), from assume a : G,
    exists_unique.unique (h4 a) (classical.some_spec (exists_unique.exists (h4 a))) (one_mul a),

  show ∃! e : G, ∀ a : G, e * a = a ∧ a * e = a, from by {
    use (1 : G),
    have h7 : ∀ e : G, (∀ a : G, e * a = a ∧ a * e = a) → e = 1, from by {
      assume (e : G) (hident : ∀ a : G, e * a = a ∧ a * e = a),
      have h8 : ∀ a : G, e = classical.some (h3 a).exists, from assume (a : G),
        exists_unique.unique (h3 a) (hident a).right
        (classical.some_spec (exists_unique.exists (h3 a))),
      have h9 : ∀ a : G, e = classical.some (h4 a).exists, from assume (a : G),
        exists_unique.unique (h4 a) (hident a).left
        (classical.some_spec (exists_unique.exists (h4 a))),
      show e = (1 : G), from eq.trans (h9 e) (h6 _),
    },
    exact ⟨by obviously, h7⟩,
  }
end

/--`theorem`
Bipartite Graph is two colorable
Let $G$ be a graph. Then $G$ is 2-colorable if and only if $G$ is bipartite.
`proof`
Let $G$ be a 2-colorable graph, which means we can color every vertex either red or blue, and no edge will have both endpoints colored the same color. Let $A$ denote the subset of vertices colored red, and let $B$ denote the subset of vertices colored blue. Since all vertices of $A$ are red, there are no edges within $A$, and similarly for $B$. This implies that every edge has one endpoint in $A$ and the other in $B$, which means $G$ is bipartite.

Conversely, suppose $G$ is bipartite, that is, we can partition the vertices into two subsets $V_{1}, V_{2}$ every edge has one endpoint in $V_{1}$ and the other in $V_{2}$. Then coloring every vertex of $V_{1}$ red and every vertex of $V_{2}$ blue yields a valid coloring, so $G$ is 2-colorable.

QED

-/
theorem  bipartite_iff_two_colorable {V : Type*} (G : simple_graph V) [fintype V] : (G.colorable 2) ↔ (∃ (A B : Type*) (h : (A ⊕ B) = V), G ≤ cast (congr_arg _ h) (complete_bipartite_graph A B)) :=
FEW SHOT PROMPTS TO CODEX(END)-/