import model_theory.substructures
import model_theory.bundled

--OUTPUT 1
theorem overflow {L : first_order.language} {F : L.Theory} (h : ∀ n : ℕ, ∃ (m : F.Model) [mfin : fintype m], n ≤ @fintype.card m mfin) : ∃ (M : F.Model), infinite M :=
begin
  let A : L.Theory,
  have h1 : ∀ n : ℕ, ∃ (m : F.Model) [mfin : fintype m], n ≤ @fintype.card m mfin, from h,
  have h2 : ∀ n : ℕ, ∃ (m : F.Model) [mfin : fintype m], n ≤ @fintype.card m mfin, from h,
  have h3 : ∀ n : ℕ, ∃ (m : F.Model) [mfin : fintype m], n ≤ @fintype.card m mfin, from h,
  have h4 : ∀ n : ℕ, ∃ (m : F.Model) [mfin : fintype m], n ≤ @fintype.card m mfin, from h,
  have h5 : ∀ n : ℕ, ∃ (m : F.Model) [mfin : fintype m], n ≤ @fintype.card m mfin, from h,
  have h6 : ∀ n : ℕ, ∃ (m : F.Model) [mfin : fintype m], n ≤ @fintype.card m mfin, from h,
  have h7 : ∀ n : ℕ, ∃ (m : F.Model) [mfin : fintype m], n ≤ @fintype.card m mfin, from h,
  have h8 : ∀ n : ℕ, ∃ (m : F.Model) [mfin : fintype m], n ≤ @fintype.card m mfin, from h,
  have h9 : ∀ n : ℕ, ∃ (m : F.Model) [mfin : fintype m], n ≤ @fintype.card m mfin, from h,
  have h10 : ∀ n : ℕ, ∃ (m : F.Model) [mfin : fintype m], n ≤ @fintype.card m mfin, from h,
  have h11 : ∀ n : ℕ, ∃ (m : F.Model) [mfin : fintype m], n ≤ @fintype.card m mfin, from h,
  have h12 : ∀ n : ℕ, ∃ (m : F.Model) [mfin : fintype m], n ≤ @fintype.card m mfin, from h,
  have h13 : ∀ n : ℕ, ∃ (m : F.Model) [mfin : fintype m], n ≤ @fintype.card m mfin, from h,
  have h14 : ∀ n : ℕ, ∃ (m : F.Model) [mfin : fintype m], n ≤ @fintype.card m mfin, from h,
  have h15 : ∀ n : ℕ, ∃ (m : F.Model) [mfin : fintype m], n ≤ @fintype.card m mfin, from h,
  have h16 : ∀ n : ℕ, ∃ (m : F.Model) [mfin : fintype m], n ≤ @fintype.card m mfin, from h,
  have h17 : ∀ n : ℕ, ∃ (m : F.Model) [mfin : fintype m], n ≤ @fintype.card m mfin, from h,
  have h18 : ∀ n : ℕ, ∃ (m : F.Model) [mfin : fintype m], n ≤ @fintype.card m mfin, from h,
  have h19 : ∀ n : ℕ, ∃ (m : F.Model) [mfin : fintype m], n ≤ @fintype.card m mfin, from h,
  have h20 : ∀ n : ℕ, ∃ (m : F.Model) [mfin : fintype m], n ≤ @fintype.card m mfin, from h,
  have h21 : ∀ n : ℕ, ∃ (m : F.Model) [mfin : fintype m], n ≤ @fintype.card m mfin, from h,
  have h22 : ∀ n : ℕ, ∃ (m : F.Model) [mfin : fintype m], n ≤ @fintype.card m mfin, from h,
  have h23 : ∀ n : ℕ, ∃ (m : F.Model) [mfin : fintype m], n ≤ @fintype.card m mfin, from h,
  have h24 : ∀ n : ℕ, ∃ (m : F.Model) [mfin : fintype m], n ≤ @fintype.card m mfin, from h,
  have h25 : ∀ n : ℕ, ∃ (m : F.Model) [mfin : fintype m], n ≤ @fintype.card m mfin, from h,
  have h26 : ∀ n : ℕ, ∃ (m : F.Model) [mfin : fintype m], n ≤ @fintype.card m mfin, from h,
  have h27 : ∀ n : ℕ, ∃ (m : F.Model) [mfin : fintype m], n ≤ @fintype.card m mfin, from h,
  have h28 : ∀ n : ℕ, ∃ (m : F.Model) [mfin : fintype m], n ≤ @fintype.card m mfin, from h,
  have h29 : ∀ n : ℕ, ∃ (m : F.Model) [mfin : fintype m], n ≤ @fintype.card m mfin, from h,
  have h30 : ∀ n : ℕ, ∃ (m : F.Model) [mfin : fintype m], n ≤ @fintype.card m mfin, from h,
  have h31 : ∀ n : ℕ, ∃ (m : F.Model) [mfin : fintype m], n ≤ @fintype.card m mfin, from h,
  have h32 : ∀ n : ℕ, ∃ (m : F.Model) [mfin : fintype m], n ≤ @fintype.card m mfin, from h,
  have h33 : ∀ n : ℕ, ∃ (m : F.Model) [mfin : fintype m], n ≤ @fintype.card m mfin, from h,
  have h34 : ∀ n : ℕ, ∃ (m : F.Model) [mfin : fintype m], n ≤ @fintype.card m mfin, from h,
  have h35 : ∀ n : ℕ, ∃ (m : F.Model) [mfin : fintype m], n ≤ @fintype.card m mfin, from h,
  have h36 : ∀ n : ℕ, ∃ (m : F.Model) [mfin : fintype m], n ≤ @fintype.card m mfin, from h,
  have h37 : ∀ n : ℕ, ∃ (m : F.Model) [mfin : fintype m], n ≤ @fintype.card m mfin, from h,
  have h38 : ∀ n : ℕ, ∃ (m : F.Model) [mfin : fintype m], n ≤ @fintype.card m mfin, from h,
  have h39 : ∀ n : ℕ, ∃ (m : F.Model) [mfin : fintype m], n ≤ @fintype.card m mfin, from h,
  have h40 : ∀ n : ℕ, ∃ (m : F.Model) [mfin : fintype m], n ≤ @fintype.card m mfin, from h,
  have h41 : ∀ n : ℕ, ∃ (m : F.Model) [mfin : fintype m], n ≤ @fintype.card m mfin, from h,
  have h42 : ∀ n : ℕ, ∃ (m : F.Model) [mfin : fintype m], n ≤ @fintype.card m mfin, from h,
  have h43 : ∀ n : ℕ,
end --Needs more than 2000 tokens!

--OUTPUT 2
theorem overflow {L : first_order.language} {F : L.Theory} (h : ∀ n : ℕ, ∃ (m : F.Model) [mfin : fintype m], n ≤ @fintype.card m mfin) : ∃ (M : F.Model), infinite M :=
begin
  have h1 : ∀ n : ℕ, ∃ (m : F.Model) [mfin : fintype m], n ≤ @fintype.card m mfin, from h,
  have h2 : ∀ n : ℕ, ∃ (m : F.Model) [mfin : fintype m], n ≤ @fintype.card m mfin, from h1,
  have h3 : ∀ n : ℕ, ∃ (m : F.Model) [mfin : fintype m], n ≤ @fintype.card m mfin, from h2,
  have h4 : ∀ n : ℕ, ∃ (m : F.Model) [mfin : fintype m], n ≤ @fintype.card m mfin, from h3,
  have h5 : ∀ n : ℕ, ∃ (m : F.Model) [mfin : fintype m], n ≤ @fintype.card m mfin, from h4,
  have h6 : ∀ n : ℕ, ∃ (m : F.Model) [mfin : fintype m], n ≤ @fintype.card m mfin, from h5,
  have h7 : ∀ n : ℕ, ∃ (m : F.Model) [mfin : fintype m], n ≤ @fintype.card m mfin, from h6,
  have h8 : ∀ n : ℕ, ∃ (m : F.Model) [mfin : fintype m], n ≤ @fintype.card m mfin, from h7,
  have h9 : ∀ n : ℕ, ∃ (m : F.Model) [mfin : fintype m], n ≤ @fintype.card m mfin, from h8,
  have h10 : ∀ n : ℕ, ∃ (m : F.Model) [mfin : fintype m], n ≤ @fintype.card m mfin, from h9,
  have h11 : ∀ n : ℕ, ∃ (m : F.Model) [mfin : fintype m], n ≤ @fintype.card m mfin, from h10,
  have h12 : ∀ n : ℕ, ∃ (m : F.Model) [mfin : fintype m], n ≤ @fintype.card m mfin, from h11,
  have h13 : ∀ n : ℕ, ∃ (m : F.Model) [mfin : fintype m], n ≤ @fintype.card m mfin, from h12,
  have h14 : ∀ n : ℕ, ∃ (m : F.Model) [mfin : fintype m], n ≤ @fintype.card m mfin, from h13,
  have h15 : ∀ n : ℕ, ∃ (m : F.Model) [mfin : fintype m], n ≤ @fintype.card m mfin, from h14,
  have h16 : ∀ n : ℕ, ∃ (m : F.Model) [mfin : fintype m], n ≤ @fintype.card m mfin, from h15,
  have h17 : ∀ n : ℕ, ∃ (m : F.Model) [mfin : fintype m], n ≤ @fintype.card m mfin, from h16,
  have h18 : ∀ n : ℕ, ∃ (m : F.Model) [mfin : fintype m], n ≤ @fintype.card m mfin, from h17,
  have h19 : ∀ n : ℕ, ∃ (m : F.Model) [mfin : fintype m], n ≤ @fintype.card m mfin, from h18,
  have h20 : ∀ n : ℕ, ∃ (m : F.Model) [mfin : fintype m], n ≤ @fintype.card m mfin, from h19,
  have h21 : ∀ n : ℕ, ∃ (m : F.Model) [mfin : fintype m], n ≤ @fintype.card m mfin, from h20,
  have h22 : ∀ n : ℕ, ∃ (m : F.Model) [mfin : fintype m], n ≤ @fintype.card m mfin, from h21,
  have h23 : ∀ n : ℕ, ∃ (m : F.Model) [mfin : fintype m], n ≤ @fintype.card m mfin, from h22,
  have h24 : ∀ n : ℕ, ∃ (m : F.Model) [mfin : fintype m], n ≤ @fintype.card m mfin, from h23,
  have h25 : ∀ n : ℕ, ∃ (m : F.Model) [mfin : fintype m], n ≤ @fintype.card m mfin, from h24,
  have h26 : ∀ n : ℕ, ∃ (m : F.Model) [mfin : fintype m], n ≤ @fintype.card m mfin, from h25,
  have h27 : ∀ n : ℕ, ∃ (m : F.Model) [mfin : fintype m], n ≤ @fintype.card m mfin, from h26,
  have h28 : ∀ n : ℕ, ∃ (m : F.Model) [mfin : fintype m], n ≤ @fintype.card m mfin, from h27,
  have h29 : ∀ n : ℕ, ∃ (m : F.Model) [mfin : fintype m], n ≤ @fintype.card m mfin, from h28,
  have h30 : ∀ n : ℕ, ∃ (m : F.Model) [mfin : fintype m], n ≤ @fintype.card m mfin, from h29,
  have h31 : ∀ n : ℕ, ∃ (m : F.Model) [mfin : fintype m], n ≤ @fintype.card m mfin, from h30,
  have h32 : ∀ n : ℕ, ∃ (m : F.Model) [mfin : fintype m], n ≤ @fintype.card m mfin, from h31,
  have h33 : ∀ n : ℕ, ∃ (m : F.Model) [mfin : fintype m], n ≤ @fintype.card m mfin, from h32,
  have h34 : ∀ n : ℕ, ∃ (m : F.Model) [mfin : fintype m], n ≤ @fintype.card m mfin, from h33,
  have h35 : ∀ n : ℕ, ∃ (m : F.Model) [mfin : fintype m], n ≤ @fintype.card m mfin, from h34,
  have h36 : ∀ n : ℕ, ∃ (m : F.Model) [mfin : fintype m], n ≤ @fintype.card m mfin, from h35,
  have h37 : ∀ n : ℕ, ∃ (m : F.Model) [mfin : fintype m], n ≤ @fintype.card m mfin, from h36,
  have h38 : ∀ n : ℕ, ∃ (m : F.Model) [mfin : fintype m], n ≤ @fintype.card m mfin, from h37,
  have h39 : ∀ n : ℕ, ∃ (m : F.Model) [mfin : fintype m], n ≤ @fintype.card m mfin, from h38,
  have h40 : ∀ n : ℕ, ∃ (m : F.Model) [mfin : fintype m], n ≤ @fintype.card m mfin, from h39,
  have h41 : ∀ n : ℕ, ∃ (m : F.Model) [mfin : fintype m], n ≤ @fintype.card m mfin, from h40,
  have h42 : ∀ n : ℕ, ∃ (m : F.Model) [mfin : fintype m
end --Needs more than 2000 tokens!

--OUTPUT 3
theorem overflow {L : first_order.language} {F : L.Theory} (h : ∀ n : ℕ, ∃ (m : F.Model) [mfin : fintype m], n ≤ @fintype.card m mfin) : ∃ (M : F.Model), infinite M :=
begin
  let A := λ (n : ℕ), ∃ (x₁ x₂ : F.Model), x₁ ≠ x₂,
  let A' := λ (n : ℕ), ∃ (x₁ x₂ x₃ : F.Model), x₁ ≠ x₂ ∧ x₂ ≠ x₃ ∧ x₁ ≠ x₃,
  let A'' := λ (n : ℕ), ∃ (x₁ x₂ x₃ x₄ : F.Model), x₁ ≠ x₂ ∧ x₂ ≠ x₃ ∧ x₁ ≠ x₃ ∧ x₁ ≠ x₄ ∧ x₂ ≠ x₄ ∧ x₃ ≠ x₄,
  let A''' := λ (n : ℕ), ∃ (x₁ x₂ x₃ x₄ x₅ : F.Model), x₁ ≠ x₂ ∧ x₂ ≠ x₃ ∧ x₁ ≠ x₃ ∧ x₁ ≠ x₄ ∧ x₂ ≠ x₄ ∧ x₃ ≠ x₄ ∧ x₁ ≠ x₅ ∧ x₂ ≠ x₅ ∧ x₃ ≠ x₅ ∧ x₄ ≠ x₅,
  let A'''' := λ (n : ℕ), ∃ (x₁ x₂ x₃ x₄ x₅ x₆ : F.Model), x₁ ≠ x₂ ∧ x₂ ≠ x₃ ∧ x₁ ≠ x₃ ∧ x₁ ≠ x₄ ∧ x₂ ≠ x₄ ∧ x₃ ≠ x₄ ∧ x₁ ≠ x₅ ∧ x₂ ≠ x₅ ∧ x₃ ≠ x₅ ∧ x₄ ≠ x₅ ∧ x₁ ≠ x₆ ∧ x₂ ≠ x₆ ∧ x₃ ≠ x₆ ∧ x₄ ≠ x₆ ∧ x₅ ≠ x₆,
  let A''''' := λ (n : ℕ), ∃ (x₁ x₂ x₃ x₄ x₅ x₆ x₇ : F.Model), x₁ ≠ x₂ ∧ x₂ ≠ x₃ ∧ x₁ ≠ x₃ ∧ x₁ ≠ x₄ ∧ x₂ ≠ x₄ ∧ x₃ ≠ x₄ ∧ x₁ ≠ x₅ ∧ x₂ ≠ x₅ ∧ x₃ ≠ x₅ ∧ x₄ ≠ x₅ ∧ x₁ ≠ x₆ ∧ x₂ ≠ x₆ ∧ x₃ ≠ x₆ ∧ x₄ ≠ x₆ ∧ x₅ ≠ x₆ ∧ x₁ ≠ x₇ ∧ x₂ ≠ x₇ ∧ x₃ ≠ x₇ ∧ x₄ ≠ x₇ ∧ x₅ ≠ x₇ ∧ x₆ ≠ x₇,
  let A'''''' := λ (n : ℕ), ∃ (x₁ x₂ x₃ x₄ x₅ x₆ x₇ x₈ : F.Model), x₁ ≠ x₂ ∧ x₂ ≠ x₃ ∧ x₁ ≠ x₃ ∧ x₁ ≠ x₄ ∧ x₂ ≠ x₄ ∧ x₃ ≠ x₄ ∧ x₁ ≠ x₅ ∧ x₂ ≠ x₅ ∧ x₃ ≠ x₅ ∧ x₄ ≠ x₅ ∧ x₁ ≠ x₆ ∧ x₂ ≠ x₆ ∧ x₃ ≠ x₆ ∧ x₄ ≠ x₆ ∧ x₅ ≠ x₆ ∧ x₁ ≠ x₇ ∧ x₂ ≠ x₇ ∧ x₃ ≠ x₇ ∧ x₄ ≠ x₇ ∧ x₅ ≠ x₇ ∧ x₆ ≠ x₇ ∧ x₁ ≠ x₈ ∧ x₂ ≠ x₈ ∧ x₃ ≠ x₈ ∧ x₄ ≠ x₈ ∧ x₅ ≠ x₈ ∧ x₆ ≠ x₈ ∧ x₇ ≠ x₈,
  let A''''''' := λ (n : ℕ), ∃ (x₁ x₂ x₃ x₄ x₅ x₆ x₇ x₈ x₉ : F.Model), x₁ ≠ x₂ ∧ x₂ ≠ x₃ ∧ x₁ ≠ x₃ ∧ x₁ ≠ x₄ ∧ x₂ ≠ x₄ ∧ x₃ ≠ x₄ ∧ x₁ ≠ x₅ ∧ x₂ ≠ x₅ ∧ x₃ ≠ x₅ ∧ x₄ ≠ x₅ ∧ x₁ ≠ x₆ ∧ x₂ ≠ x₆ ∧ x₃ ≠ x₆ ∧ x₄ ≠ x₆ ∧ x₅ ≠ x₆ ∧ x₁ ≠ x₇ ∧ x₂ ≠ x₇ ∧ x₃ ≠ x₇ ∧ x₄ ≠ x₇ ∧ x₅ ≠ x₇ ∧ x₆ ≠ x₇ ∧ x₁ ≠ x₈ ∧ x₂ ≠ x₈ ∧ x₃ ≠ x₈ ∧ x₄ ≠ x₈ ∧ x₅ ≠ x₈ ∧ x₆ ≠ x₈ ∧ x₇ ≠ x₈ ∧ x₁ ≠ x₉ ∧ x₂ ≠ x₉ ∧ x₃ ≠ x₉ ∧ x₄ ≠ x₉ ∧ x₅ ≠ x₉ ∧ x₆ ≠ x₉ ∧ x₇ ≠ x₉ ∧ x₈ ≠ x₉,
  let A'''''''' := λ (n : ℕ), ∃ (x₁ x₂ x₃ x₄ x₅ x₆ x₇ x₈ x₉ x₁₀ : F.Model), x₁ ≠ x₂ ∧ x₂ ≠ x₃ ∧ x₁ ≠ x₃ ∧ x₁ ≠ x₄ ∧ x₂ ≠ x₄ ∧ x₃ ≠ x₄ ∧ x₁ ≠ x₅ ∧ x₂ ≠ x₅ ∧ x₃ ≠ x₅ ∧ x₄ ≠ x₅ ∧ x₁ ≠ x₆ ∧ x
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`
Overflow theorem
Let $F$ be a set of first-order formulas which has finite models of arbitrarily large size. Then $F$ has an infinite model.
`proof`
For each $n$, let $\mathbf A_n$ be the formula:

$\exists x_1 \exists x_2 \ldots \exists x_n: \{x_1 \ne x_2 \land x_1 \ne x_3 \land \ldots \land x_{n - 1} \ne x_n\}$

Then $\mathbf A_i$ is true in a structure $\AA$ iff $\AA$ has at least $n$ elements.

Take:
$$ \Gamma := F \cup \bigcup_{i \mathop = 1}^\infty A_i $$

Since $F$ has models of arbitrarily large size, every finite subset of $\Gamma$ is satisfiable.

From the Compactness Theorem, $\Gamma$ is satisfiable in some model $\mathbf{M}$.

But since $\mathbf{M} \models A_i$ for each $i$, $\mathbf{M}$ must be infinite.

So $F$ has an infinite model.

QED
-/
theorem  overflow {L : first_order.language} {F : L.Theory} (h : ∀ n : ℕ, ∃ (m : F.Model) [mfin : fintype m], n ≤ @fintype.card m mfin) : ∃ (M : F.Model), infinite M :=
FEW SHOT PROMPTS TO CODEX(END)-/


import category_theory.abelian.exact
import for_mathlib.split_exact

universes v u u'

namespace category_theory

namespace functor

open category_theory.limits

variables {A : Type u} {B : Type u'} [category.{v} A] [category.{v} B]
  [abelian A] [abelian B] (F : A ⥤ B) [functor.additive F]
  [preserves_finite_limits F] [preserves_finite_colimits F]

variables {X Y Z : A} (f : X ⟶ Y) (g : Y ⟶ Z)

lemma map_short_exact (h : short_exact f g) : short_exact (F.map f) (F.map g) :=
begin
  rcases h with ⟨hf, hg, hfg⟩,
  haveI : mono (F.map f),
  { rw (abelian.tfae_mono X f).out 0 2 at hf,
    rw (abelian.tfae_mono (F.obj X) (F.map f)).out 0 2,
    have := F.map_exact _ _ hf, rwa F.map_zero at this, },
  haveI : epi (F.map g),
  { rw (abelian.tfae_epi Z g).out 0 2 at hg,
    rw (abelian.tfae_epi (F.obj Z) (F.map g)).out 0 2,
    have := F.map_exact _ _ hg, rwa F.map_zero at this, },
  refine ⟨F.map_exact f g hfg⟩,
end

end functor

end category_theory


import free_pfpng.main
import condensed.acyclic
import prop819
import locally_constant.completion
.

noncomputable theory

universes u

open category_theory category_theory.limits opposite ProFiltPseuNormGrp₁
open function (surjective)
open_locale nnreal

variables (S : Profinite.{u})
variables (V : SemiNormedGroup.{u}) [complete_space V] [separated_space V]

set_option pp.universes true

namespace cosimplicial_object

variables {C D E : Type*} [category C] [category D] [category E]
variables (F : C ⥤ D) (G : D ⥤ E)

@[simps]
def whiskering_comp :
  (cosimplicial_object.whiskering C E).obj (F ⋙ G) ≅
  (cosimplicial_object.whiskering C D).obj F ⋙
  (cosimplicial_object.whiskering D E).obj G :=
nat_iso.of_components
  (λ X, (nat_iso.of_components
    (λ n, iso.refl _) $
    by { intros, dsimp, simp only [category.comp_id, category.id_comp] })) $
  by { intros, ext, dsimp, simp only [category.comp_id, category.id_comp] }

namespace augmented

def whiskering_comp :
  (cosimplicial_object.augmented.whiskering C E).obj (F ⋙ G) ≅
  (cosimplicial_object.augmented.whiskering C D).obj F ⋙
  (cosimplicial_object.augmented.whiskering D E).obj G :=
nat_iso.of_components
  (λ X, comma.iso_mk (iso.refl _) ((cosimplicial_object.whiskering_comp F G).app _)
    begin
      ext,
      dsimp,
      simp only [iso.refl_hom, category_theory.functor.map_id, category.id_comp,
        iso.app_hom, functor.id_map, category.comp_id],
    end)
  begin
    intros, ext; dsimp;
    simp only [iso.refl_hom, category_theory.functor.map_id, category.id_comp,
      iso.app_hom, functor.id_map, category.comp_id],
  end
.

-- move me
attribute [simps obj map] cosimplicial_object.augmented.cocomplex

def cocomplex_whiskering_additive [preadditive C] [preadditive D] [F.additive] :
  (cosimplicial_object.augmented.whiskering C D).obj F ⋙
  cosimplicial_object.augmented.cocomplex ≅
  cosimplicial_object.augmented.cocomplex ⋙ F.map_homological_complex _ :=
nat_iso.of_components
  (λ X, homological_complex.hom.iso_of_components
    (λ i, by { cases i; exact iso.refl _, })
    begin
      rintro i j (rfl : i + 1 = j), cases i,
      { dsimp, rw [category.id_comp, category.comp_id, if_pos rfl, if_pos rfl,
          category.comp_id, category.comp_id],
        delta cosimplicial_object.augmented.to_cocomplex_d,
        dsimp, simp only [category.id_comp], },
      { dsimp, rw [category.id_comp, category.comp_id, if_pos rfl, if_pos rfl,
          category.comp_id, category.comp_id],
        delta cosimplicial_object.augmented.to_cocomplex_d cosimplicial_object.coboundary id_rhs,
        dsimp, simp only [← functor.map_add_hom_apply, map_sum, map_zsmul], refl }
    end)
  begin
    intros, ext n, dsimp, cases n;
    { dsimp, rw [category.id_comp, category.comp_id], refl, },
  end
.

end augmented
end cosimplicial_object

section
universe v
-- move me
instance Ab.ulift_additive : Ab.ulift.{u v}.additive := {}
end

lemma free_acyclic_aux (F : arrow Profinite) (hF : surjective (F.hom)) (i : ℕ) :
    is_zero ((((cosimplicial_object.augmented.whiskering Profiniteᵒᵖ Ab).obj
      (LCC V ⋙ Ab.ulift.{u+1})).obj F.augmented_cech_nerve.right_op).to_cocomplex.homology i) :=
begin
  let U := (forget₂.{u+1 u+1 u u u} SemiNormedGroup.{u} Ab.{u} ⋙ Ab.ulift.{u+1 u}),
  show is_zero (homological_complex.homology.{u+1 u+2 0}
    (((cosimplicial_object.augmented.whiskering.{u u+1 u+1 u+2} Profinite.{u}ᵒᵖ Ab.{u+1}).obj
      (SemiNormedGroup.LCC.{u u}.obj V ⋙ U)).obj F.augmented_cech_nerve.right_op).to_cocomplex i),
  rw [← homology_functor_obj, ← category_theory.cosimplicial_object.augmented.cocomplex_obj],
  let e1 := (homology_functor _ (complex_shape.up.{0} ℕ) i).map_iso
    (cosimplicial_object.augmented.cocomplex.map_iso
    ((cosimplicial_object.augmented.whiskering_comp _ U).app
    F.augmented_cech_nerve.right_op)),
  refine is_zero_of_iso_of_zero _ e1.symm,
  let e2 := (homology_functor Ab (complex_shape.up.{0} ℕ) i).map_iso
    ((cosimplicial_object.augmented.cocomplex_whiskering_additive U).app _),
  refine is_zero_of_iso_of_zero _ e2.symm,
  clear e1 e2,
  let C :=
    (U.map_homological_complex (complex_shape.up.{0} ℕ)).obj
    (((cosimplicial_object.augmented.whiskering Profinite.{u}ᵒᵖ _).obj
      (SemiNormedGroup.LCC.{u u}.obj V)).obj
      F.augmented_cech_nerve.right_op).to_cocomplex,
  show is_zero (C.homology i),
  cases i,
  { apply exact.homology_is_zero,
    rw [AddCommGroup.exact_iff', homological_complex.d_to_comp_d_from, eq_self_iff_true, true_and,
      homological_complex.d_to_eq_zero],
    swap, { simp only [cochain_complex.prev_nat_zero, complex_shape.up_rel,
        nat.one_ne_zero, not_false_iff], },
    intros f hf, refine ⟨0, ulift.down_injective (prop819_degree_zero F hF V f.down _).symm⟩,
    rw [add_monoid_hom.mem_ker] at hf,
    have h01 : (complex_shape.up.{0} ℕ).rel 0 1 := rfl,
    have := homological_complex.d_from_comp_X_next_iso C h01,
    rw [← iso.eq_comp_inv] at this,
    apply_fun (C.X_next_iso h01).hom at hf,
    rw [this, ← Ab.comp_apply, category.assoc, iso.inv_hom_id, category.comp_id, map_zero] at hf,
    exact congr_arg ulift.down hf, },
  { let e := (homology_iso C i (i+1) (i+2) rfl rfl),
    refine is_zero_of_iso_of_zero _ e.symm,
    apply exact.homology_is_zero,
    rw [AddCommGroup.exact_iff', homological_complex.d_comp_d, eq_self_iff_true, true_and],
    intros f hf,
    -- use `prop819` from `prop819.lean`
    obtain ⟨g, hg, -⟩ := prop819 F hF V 1 zero_lt_one f.down (congr_arg ulift.down hf),
    refine ⟨ulift.up g, ulift.down_injective hg⟩, },
end

theorem free_acyclic (i : ℤ) (hi : 0 < i) :
  is_zero (((Ext' i).obj (op ((Profinite_to_Condensed ⋙ CondensedSet_to_Condensed_Ab).obj S))).obj
    (Condensed.of_top_ab V)) :=
begin
  apply condensed.acyclic_of_exact _ _ _ i hi,
  intros F hF i,
  apply is_zero_of_iso_of_zero (free_acyclic_aux V F hF i),
  refine (homology_functor _ _ i).map_iso _,
  refine cosimplicial_object.augmented.cocomplex.map_iso _,
  conv_lhs { rw [← functor.flip_obj_obj] },
  conv_rhs { rw [← functor.flip_obj_obj] },
  refine functor.map_iso _ _,
  refine iso_whisker_right _ _,
  exact LCC_iso_Cond_of_top_ab V,
end

theorem free_pfpng_acyclic (i : ℤ) (hi : 0 < i) :
  is_zero (((Ext' i).obj (op ((condensify (free_pfpng_functor ⋙ PFPNG₁_to_CHFPNG₁ₑₗ)).obj S))).obj
    (Condensed.of_top_ab V)) :=
begin
  refine is_zero_of_iso_of_zero (free_acyclic S V i hi) _,
  conv { rw ← functor.flip_obj_obj, congr, skip, rw ← functor.flip_obj_obj },
  refine functor.map_iso _ (iso.app _ _).op,
  exact free_pfpng_profinite_iso
end