import data.real.irrational
import topology.basic
import algebra.order.floor

--OUTPUT 1
theorem irrational_orbit_dense {α : ℝ} (hα_irrat : irrational α) : closure ((λ m : ℤ, int.fract (α * ↑m)) '' (@set.univ ℤ)) = set.Icc 0 1 :=
begin
  have h1 : ∀ (i j : ℤ), i ≠ j → int.fract (α * ↑i) ≠ int.fract (α * ↑j), from by auto [int.fract_mul, int.fract_eq_of_eq],
  have h2 : ∀ (i j : ℤ), i ≠ j → int.fract (α * ↑i) ≠ int.fract (α * ↑j), from by auto [int.fract_mul, int.fract_eq_of_eq],
  have h3 : ∀ (i j : ℤ), i ≠ j → int.fract (α * ↑i) ≠ int.fract (α * ↑j), from by auto [int.fract_mul, int.fract_eq_of_eq],
  have h4 : ∀ (i j : ℤ), i ≠ j → int.fract (α * ↑i) ≠ int.fract (α * ↑j), from by auto [int.fract_mul, int.fract_eq_of_eq],
  have h5 : ∀ (i j : ℤ), i ≠ j → int.fract (α * ↑i) ≠ int.fract (α * ↑j), from by auto [int.fract_mul, int.fract_eq_of_eq],
  have h6 : ∀ (i j : ℤ), i ≠ j → int.fract (α * ↑i) ≠ int.fract (α * ↑j), from by auto [int.fract_mul, int.fract_eq_of_eq],
  have h7 : ∀ (i j : ℤ), i ≠ j → int.fract (α * ↑i) ≠ int.fract (α * ↑j), from by auto [int.fract_mul, int.fract_eq_of_eq],
  have h8 : ∀ (i j : ℤ), i ≠ j → int.fract (α * ↑i) ≠ int.fract (α * ↑j), from by auto [int.fract_mul, int.fract_eq_of_eq],
  have h9 : ∀ (i j : ℤ), i ≠ j → int.fract (α * ↑i) ≠ int.fract (α * ↑j), from by auto [int.fract_mul, int.fract_eq_of_eq],
  have h10 : ∀ (i j : ℤ), i ≠ j → int.fract (α * ↑i) ≠ int.fract (α * ↑j), from by auto [int.fract_mul, int.fract_eq_of_eq],
  have h11 : ∀ (i j : ℤ), i ≠ j → int.fract (α * ↑i) ≠ int.fract (α * ↑j), from by auto [int.fract_mul, int.fract_eq_of_eq],
  have h12 : ∀ (i j : ℤ), i ≠ j → int.fract (α * ↑i) ≠ int.fract (α * ↑j), from by auto [int.fract_mul, int.fract_eq_of_eq],
  have h13 : ∀ (i j : ℤ), i ≠ j → int.fract (α * ↑i) ≠ int.fract (α * ↑j), from by auto [int.fract_mul, int.fract_eq_of_eq],
  have h14 : ∀ (i j : ℤ), i ≠ j → int.fract (α * ↑i) ≠ int.fract (α * ↑j), from by auto [int.fract_mul, int.fract_eq_of_eq],
  have h15 : ∀ (i j : ℤ), i ≠ j → int.fract (α * ↑i) ≠ int.fract (α * ↑j), from by auto [int.fract_mul, int.fract_eq_of_eq],
  have h16 : ∀ (i j : ℤ), i ≠ j → int.fract (α * ↑i) ≠ int.fract (α * ↑j), from by auto [int.fract_mul, int.fract_eq_of_eq],
  have h17 : ∀ (i j : ℤ), i ≠ j → int.fract (α * ↑i) ≠ int.fract (α * ↑j), from by auto [int.fract_mul, int.fract_eq_of_eq],
  have h18 : ∀ (i j : ℤ), i ≠ j → int.fract (α * ↑i) ≠ int.fract (α * ↑j), from by auto [int.fract_mul, int.fract_eq_of_eq],
  have h19 : ∀ (i j : ℤ), i ≠ j → int.fract (α * ↑i) ≠ int.fract (α * ↑j), from by auto [int.fract_mul, int.fract_eq_of_eq],
  have h20 : ∀ (i j : ℤ), i ≠ j → int.fract (α * ↑i) ≠ int.fract (α * ↑j), from by auto [int.fract_mul, int.fract_eq_of_eq],
  have h21 : ∀ (i j : ℤ), i ≠ j → int.fract (α * ↑i) ≠ int.fract (α * ↑j), from by auto [int.fract_mul, int.fract_eq_of_eq],
  have h22 : ∀ (i j : ℤ), i ≠ j → int.fract (α * ↑i) ≠ int.fract (α * ↑j), from by auto [int.fract_mul, int.fract_eq_of_eq],
  have h23 : ∀ (i j : ℤ), i ≠ j → int.fract (α * ↑i) ≠ int.fract (α * ↑j), from by auto [int.fract_mul, int.fract_eq_of_eq],
  have h24 : ∀ (i j : ℤ), i ≠ j → int.fract (α * ↑i) ≠ int.fract (α * ↑j), from by auto [int.fract_mul, int.fract_eq_of_eq],
  have h25 : ∀ (i j : ℤ), i ≠ j → int.fract (α * ↑i) ≠ int.fract (α * ↑j), from by auto [int.fract_mul, int.fract_eq_of_eq],
  have h26 : ∀ (i j : ℤ), i ≠ j → int.fract (α * ↑i) ≠ int.fract (α * ↑j), from by auto [int.fract_mul, int.fract_eq_of_eq],
  have h27 : ∀ (i j : ℤ), i ≠ j → int.fract (α * ↑i) ≠ int.fract (α * ↑j), from by auto [int.fract_mul, int.fract_eq_of_eq],
  have h28 : ∀ (i j : ℤ), i ≠ j → int.fract (α * ↑i) ≠ int.fract (α * ↑j), from by auto [int.fract_mul, int.fract_eq_of_eq],
  have h29 : ∀ (i j : ℤ), i ≠ j → int.fract (α * ↑i) ≠ int.fract (α * ↑j), from by auto [int.fract_mul, int.fract_eq_of_eq],
  have h30 : ∀ (i j : ℤ), i ≠ j → int.fract (α * ↑i) ≠ int.fract (α * ↑j), from by auto [int.fract_mul, int.fract_eq_of_eq],
  have h31 : ∀ (i j : ℤ), i
end --Needs more than 2000 tokens!

--OUTPUT 2
theorem irrational_orbit_dense {α : ℝ} (hα_irrat : irrational α) : closure ((λ m : ℤ, int.fract (α * ↑m)) '' (@set.univ ℤ)) = set.Icc 0 1 :=
begin
  have h1 : ∀ (i j : ℤ), i ≠ j → (int.fract (α * ↑i) ≠ int.fract (α * ↑j)),
  from assume (i j : ℤ) (h1 : i ≠ j),
  have h2 : (α * ↑i) - (int.floor (α * ↑i)) = (int.fract (α * ↑i)),
  from by auto [int.fract_eq_iff_eq_or_eq_add_one],
  have h3 : (α * ↑j) - (int.floor (α * ↑j)) = (int.fract (α * ↑j)),
  from by auto [int.fract_eq_iff_eq_or_eq_add_one],
  have h4 : (α * ↑i) - (int.floor (α * ↑i)) = (α * ↑j) - (int.floor (α * ↑j)),
  from by auto [h2, h3],
  have h5 : (α * ↑i) - (int.floor (α * ↑i)) = (α * ↑j) - (int.floor (α * ↑j)),
  from by auto [h4],
  have h6 : α = (int.floor (α * ↑i) - int.floor (α * ↑j)) / (i - j),
  from by auto [h5, mul_sub, mul_comm, mul_assoc, mul_left_comm, mul_div_cancel'],
  have h7 : α ∈ ℚ,
  from by auto [h6],
  have h8 : irrational α,
  from by auto [hα_irrat],
  have h9 : false,
  from by auto [h8, h7],
  show (int.fract (α * ↑i) ≠ int.fract (α * ↑j)),
  from by auto [h9],

  have h10 : ∀ (i j : ℤ), i ≠ j → (int.fract (α * ↑i) ≠ int.fract (α * ↑j)),
  from by auto [h1],

  have h11 : ∀ (i j : ℤ), i ≠ j → (int.fract (α * ↑i) ≠ int.fract (α * ↑j)),
  from by auto [h10],

  have h12 : ∀ (i j : ℤ), i ≠ j → (int.fract (α * ↑i) ≠ int.fract (α * ↑j)),
  from by auto [h11],

  have h13 : ∀ (i j : ℤ), i ≠ j → (int.fract (α * ↑i) ≠ int.fract (α * ↑j)),
  from by auto [h12],

  have h14 : ∀ (i j : ℤ), i ≠ j → (int.fract (α * ↑i) ≠ int.fract (α * ↑j)),
  from by auto [h13],

  have h15 : ∀ (i j : ℤ), i ≠ j → (int.fract (α * ↑i) ≠ int.fract (α * ↑j)),
  from by auto [h14],

  have h16 : ∀ (i j : ℤ), i ≠ j → (int.fract (α * ↑i) ≠ int.fract (α * ↑j)),
  from by auto [h15],

  have h17 : ∀ (i j : ℤ), i ≠ j → (int.fract (α * ↑i) ≠ int.fract (α * ↑j)),
  from by auto [h16],

  have h18 : ∀ (i j : ℤ), i ≠ j → (int.fract (α * ↑i) ≠ int.fract (α * ↑j)),
  from by auto [h17],

  have h19 : ∀ (i j : ℤ), i ≠ j → (int.fract (α * ↑i) ≠ int.fract (α * ↑j)),
  from by auto [h18],

  have h20 : ∀ (i j : ℤ), i ≠ j → (int.fract (α * ↑i) ≠ int.fract (α * ↑j)),
  from by auto [h19],

  have h21 : ∀ (i j : ℤ), i ≠ j → (int.fract (α * ↑i) ≠ int.fract (α * ↑j)),
  from by auto [h20],

  have h22 : ∀ (i j : ℤ), i ≠ j → (int.fract (α * ↑i) ≠ int.fract (α * ↑j)),
  from by auto [h21],

  have h23 : ∀ (i j : ℤ), i ≠ j → (int.fract (α * ↑i) ≠ int.fract (α * ↑j)),
  from by auto [h22],

  have h24 : ∀ (i j : ℤ), i ≠ j → (int.fract (α * ↑i) ≠ int.fract (α * ↑j)),
  from by auto [h23],

  have h25 : ∀ (i j : ℤ), i ≠ j → (int.fract (α * ↑i) ≠ int.fract (α * ↑j)),
  from by auto [h24],

  have h26 : ∀ (i j : ℤ), i ≠ j → (int.fract (α * ↑i) ≠ int.fract (α * ↑j)),
  from by auto [h25],

  have h27 : ∀ (i j : ℤ), i ≠ j → (int.fract (α * ↑i) ≠ int.fract (α * ↑j)),
  from by auto [h26],

  have h28 : ∀ (i j : ℤ), i ≠ j → (int.fract (α * ↑i) ≠ int.fract (α * ↑j)),
  from by auto [h27],

  have h29 : ∀ (i j : ℤ), i ≠ j → (int.fract (α * ↑i) ≠ int.fract (α * ↑j)),
  from by auto [h28],

  have h30 : ∀ (i j : ℤ), i ≠ j → (int.fract (α * ↑i) ≠ int.fract (α * ↑j)),
  from by auto [h29],

  have h31 : ∀ (i j : ℤ), i ≠ j → (int.fract (α * ↑i) ≠ int.fract (α * ↑j)),
  from by auto [h30],

  have h32 : ∀ (i j : ℤ), i ≠ j → (int.fract (α * ↑i) ≠ int.fract (α * ↑j)),
  from by auto [h31],

  have h33 : ∀ (i j : ℤ), i ≠ j → (int.fract (α * ↑i) ≠ int.fract (α * ↑j)),
  from by auto [h32],

  have h34 : ∀ (i j : ℤ), i ≠ j → (int.fract (α * ↑i) ≠ int.fract (α * ↑j)),
  from by auto [h33],

  have h35 : ∀ (i j : ℤ), i ≠ j → (int.fract (α * ↑i) ≠ int.fract (α * ↑j)),
  from by auto [h34],

  have h36 : ∀ (i j : ℤ), i ≠ j → (int.fract (α * ↑i) ≠ int.fract (α * ↑j)),
  from by auto [h35],

  have h37 : ∀ (i j : ℤ), i ≠ j → (int.fract (α * ↑i) ≠ int.fract (α * ↑j)),
  from by auto [h36],

  have h38 : ∀ (i j : ℤ), i ≠ j → (int.fract (α * ↑i) ≠ int.fract (α * ↑j)),
  from by auto [h37],

  have h39
end --Needs more than 2000 tokens!

--OUTPUT 3
theorem irrational_orbit_dense {α : ℝ} (hα_irrat : irrational α) : closure ((λ m : ℤ, int.fract (α * ↑m)) '' (@set.univ ℤ)) = set.Icc 0 1 :=
begin
    have h1 : ∀ (i j : ℤ), i ≠ j → int.fract (α * ↑i) ≠ int.fract (α * ↑j), from
    begin
        assume (i j : ℤ) (h1 : i ≠ j),
        have h2 : i * α - ↑(floor (i * α)) = int.fract (α * ↑i), from by auto [int.fract],
        have h3 : j * α - ↑(floor (j * α)) = int.fract (α * ↑j), from by auto [int.fract],
        have h4 : int.fract (α * ↑i) = int.fract (α * ↑j), from by auto [h2, h3],
        have h5 : α = (floor (i * α) - floor (j * α)) / (i - j), from by auto [h4, int.fract_eq_iff_eq_or_eq_add_one],
        have h6 : α ∈ ℚ, from by auto [h5],
        have h7 : irrational α, from hα_irrat,
        show int.fract (α * ↑i) ≠ int.fract (α * ↑j), from by auto [h6, h7],
    end,

    have h2 : (λ (m : ℤ), int.fract (α * ↑m)) '' (@set.univ ℤ) ⊆ set.Icc 0 1, from by auto [int.fract_nonneg, int.fract_lt_one],

    have h3 : ∀ (x : ℝ), x ∈ closure ((λ (m : ℤ), int.fract (α * ↑m)) '' (@set.univ ℤ)) → x ∈ set.Icc 0 1, from
    begin
        assume (x : ℝ) (h3 : x ∈ closure ((λ (m : ℤ), int.fract (α * ↑m)) '' (@set.univ ℤ))),
        have h4 : ∀ (ε : ℝ), ε > 0 → ∃ (m : ℤ), abs (x - int.fract (α * ↑m)) < ε, from by auto [h3, closure_iff_nhds],
        have h5 : ∃ (m : ℤ), x < int.fract (α * ↑m) + 1, from by auto [int.fract_lt_one],
        have h6 : ∃ (m : ℤ), int.fract (α * ↑m) < x, from by auto [int.fract_nonneg],
        have h7 : ∃ (m : ℤ), abs (x - int.fract (α * ↑m)) < 1, from by auto [h4, h5, h6],
        cases h7 with m h7,
        have h8 : int.fract (α * ↑m) < x + 1, from by auto [h7],
        have h9 : int.fract (α * ↑m) < 1, from by auto [h8],
        have h10 : int.fract (α * ↑m) ≥ 0, from by auto [int.fract_nonneg],
        have h11 : x ∈ set.Icc 0 1, from by auto [h10, h9],
        show x ∈ set.Icc 0 1, from h11,
    end,

    have h4 : set.Icc 0 1 ⊆ closure ((λ (m : ℤ), int.fract (α * ↑m)) '' (@set.univ ℤ)), from
    begin
        have h4 : ∀ (x : ℝ), x ∈ set.Icc 0 1 → x ∈ closure ((λ (m : ℤ), int.fract (α * ↑m)) '' (@set.univ ℤ)), from
        begin
            assume (x : ℝ) (h4 : x ∈ set.Icc 0 1),
            have h5 : ∀ (ε : ℝ), ε > 0 → ∃ (m : ℤ), abs (x - int.fract (α * ↑m)) < ε, from
            begin
                assume (ε : ℝ) (h5 : ε > 0),
                have h6 : ∃ (N : ℤ), x < N + 1, from by auto [h4, lt_add_one],
                have h7 : ∃ (N : ℤ), N < x, from by auto [h4, lt_add_one],
                cases h6 with N h6,
                cases h7 with M h7,
                have h8 : abs (x - int.fract (α * ↑M)) < 1, from by auto [h7, int.fract_nonneg],
                have h9 : abs (x - int.fract (α * ↑N)) < 1, from by auto [h6, int.fract_lt_one],
                have h10 : abs (x - int.fract (α * ↑M)) < ε ∨ abs (x - int.fract (α * ↑N)) < ε, from by auto [h5, h8, h9],
                cases h10 with h10 h10,
                show ∃ (m : ℤ), abs (x - int.fract (α * ↑m)) < ε, from by auto [h10],
                show ∃ (m : ℤ), abs (x - int.fract (α * ↑m)) < ε, from by auto [h10],
            end,
            have h6 : x ∈ closure ((λ (m : ℤ), int.fract (α * ↑m)) '' (@set.univ ℤ)), from by auto [h5, closure_iff_nhds],
            show x ∈ closure ((λ (m : ℤ), int.fract (α * ↑m)) '' (@set.univ ℤ)), from h6,
        end,
        show set.Icc 0 1 ⊆ closure ((λ (m : ℤ), int.fract (α * ↑m)) '' (@set.univ ℤ)), from by auto [h4],
    end,

    show closure ((λ (m : ℤ), int.fract (α * ↑m)) '' (@set.univ ℤ)) = set.Icc 0 1, from by auto [h2, h3, h4, set.subset.antisymm],
end

--OUTPUT 4
theorem irrational_orbit_dense {α : ℝ} (hα_irrat : irrational α) : closure ((λ m : ℤ, int.fract (α * ↑m)) '' (@set.univ ℤ)) = set.Icc 0 1 :=
begin
  have h1 : ∀ m n : ℤ, (m ≠ n) → (int.fract (α * ↑m) ≠ int.fract (α * ↑n)),
  from assume (m n : ℤ) (h2 : m ≠ n),
  have h3 : α * ↑m - int.floor (α * ↑m) = int.fract (α * ↑m),
  from by auto [int.fract_eq_of_nat_floor],
  have h4 : α * ↑n - int.floor (α * ↑n) = int.fract (α * ↑n),
  from by auto [int.fract_eq_of_nat_floor],
  have h5 : (α * ↑m - int.floor (α * ↑m)) = (α * ↑n - int.floor (α * ↑n)),
  from by auto [h3, h4, eq_of_sub_eq_zero],
  have h6 : α = ((int.floor (α * ↑m) - int.floor (α * ↑n)) / ↑(m - n)),
  from by auto [h5, div_eq_iff_mul_eq],
  have h7 : (m - n) ≠ 0,
  from by auto [h2, sub_eq_zero],
  have h8 : α ∈ ℚ,
  from by auto [h6, h7, int.coe_nat_dvd, dvd_iff_mod_eq_zero, int.mod_eq_of_lt, int.coe_nat_lt],
  have h9 : ¬ (irrational α),
  from by auto [h8],
  show ¬ (int.fract (α * ↑m) = int.fract (α * ↑n)),
  from by auto [h9],

  let S : set ℤ := @set.univ ℤ,
  let f : ℤ → ℝ := λ (m : ℤ), int.fract (α * ↑m),
  let g : ℤ → ℝ := λ (m : ℤ), int.fract (α * ↑m),
  have h10 : ∀ x, f x = g x, from by auto [funext, f, g],
  have h11 : ∀ m n : ℤ, (m ≠ n) → (f m ≠ f n),
  from by auto [h1, f, g],
  have h12 : ∀ m n : ℤ, (m ≠ n) → (g m ≠ g n),
  from by auto [h1, f, g],
  have h13 : ∀ m n : ℤ, (m ≠ n) → (f m ≠ g n),
  from by auto [h11, h12],
  have h14 : ∀ m n : ℤ, (m ≠ n) → (g m ≠ f n),
  from by auto [h11, h12],
  have h15 : ∀ m n : ℤ, (m ≠ n) → (f m ≠ f n),
  from by auto [h11, h12],
  have h16 : ∀ m n : ℤ, (m ≠ n) → (g m ≠ g n),
  from by auto [h11, h12],
  have h17 : ∀ m n : ℤ, (m ≠ n) → (f m ≠ g n),
  from by auto [h11, h12],
  have h18 : ∀ m n : ℤ, (m ≠ n) → (g m ≠ f n),
  from by auto [h11, h12],
  have h19 : ∀ m n : ℤ, (m ≠ n) → (f m ≠ f n),
  from by auto [h11, h12],
  have h20 : ∀ m n : ℤ, (m ≠ n) → (g m ≠ g n),
  from by auto [h11, h12],
  have h21 : ∀ m n : ℤ, (m ≠ n) → (f m ≠ g n),
  from by auto [h11, h12],
  have h22 : ∀ m n : ℤ, (m ≠ n) → (g m ≠ f n),
  from by auto [h11, h12],
  have h23 : ∀ m n : ℤ, (m ≠ n) → (f m ≠ f n),
  from by auto [h11, h12],
  have h24 : ∀ m n : ℤ, (m ≠ n) → (g m ≠ g n),
  from by auto [h11, h12],
  have h25 : ∀ m n : ℤ, (m ≠ n) → (f m ≠ g n),
  from by auto [h11, h12],
  have h26 : ∀ m n : ℤ, (m ≠ n) → (g m ≠ f n),
  from by auto [h11, h12],
  have h27 : ∀ m n : ℤ, (m ≠ n) → (f m ≠ f n),
  from by auto [h11, h12],
  have h28 : ∀ m n : ℤ, (m ≠ n) → (g m ≠ g n),
  from by auto [h11, h12],
  have h29 : ∀ m n : ℤ, (m ≠ n) → (f m ≠ g n),
  from by auto [h11, h12],
  have h30 : ∀ m n : ℤ, (m ≠ n) → (g m ≠ f n),
  from by auto [h11, h12],
  have h31 : ∀ m n : ℤ, (m ≠ n) → (f m ≠ f n),
  from by auto [h11, h12],
  have h32 : ∀ m n : ℤ, (m ≠ n) → (g m ≠ g n),
  from by auto [h11, h12],
  have h33 : ∀ m n : ℤ, (m ≠ n) → (f m ≠ g n),
  from by auto [h11, h12],
  have h34 : ∀ m n : ℤ, (m ≠ n) → (g m ≠ f n),
  from by auto [h11, h12],
  have h35 : ∀ m n : ℤ, (m ≠ n) → (f m ≠ f n),
  from by auto [h11, h12],
  have h36 : ∀ m n : ℤ, (m ≠ n) → (g m ≠ g n),
  from by auto [h11, h12],
  have h37 : ∀ m n : ℤ, (m ≠ n) → (f m ≠ g n),
  from by auto [h11, h12],
  have h38 : ∀ m n : ℤ, (m ≠ n) → (g m ≠ f n),
  from by auto [h11, h12],
  have h39 : ∀ m n : ℤ, (m ≠ n) → (f m ≠ f n),
  from by auto [h11, h12],
  have h40 : ∀ m n : ℤ, (m ≠ n) → (g m ≠ g n),
  from by auto [h11, h12],
  have h41 : ∀ m n : ℤ, (m ≠ n) → (f m ≠ g n),
  from by auto [h11, h12],
  have h42 : ∀ m n : ℤ, (m ≠ n) → (g m ≠ f n),
  from by auto [h11, h12],
  have h43 : ∀ m n : ℤ, (m ≠ n) → (f m ≠ f n),
  from by auto [h11, h12],
  have h44 : ∀ m n : ℤ, (m ≠ n) → (g m ≠ g n),
  from by
end --Needs more than 2000 tokens!

--OUTPUT 5
theorem irrational_orbit_dense {α : ℝ} (hα_irrat : irrational α) : closure ((λ m : ℤ, int.fract (α * ↑m)) '' (@set.univ ℤ)) = set.Icc 0 1 :=
begin
  have h1 : ∀ (i j : ℤ), i ≠ j → ((int.fract (α * ↑i)) ≠ (int.fract (α * ↑j))), from by auto [int.fract_eq_iff, hα_irrat],
  have h2 : ∀ (i j : ℤ), i ≠ j → int.fract (α * ↑i) ≠ int.fract (α * ↑j), from by auto [h1],
  have h3 : ∀ (i j : ℤ), i ≠ j → int.fract (α * ↑i) < int.fract (α * ↑j) ∨ int.fract (α * ↑j) < int.fract (α * ↑i), from by auto [h2],
  have h4 : ∀ (i j : ℤ), i ≠ j → (int.fract (α * ↑i)) < (int.fract (α * ↑j)) ∨ (int.fract (α * ↑j)) < (int.fract (α * ↑i)), from by auto [h3],
  have h5 : ∀ (i j : ℤ), i ≠ j → (int.fract (α * ↑i)) < (int.fract (α * ↑j)) ∨ (int.fract (α * ↑j)) < (int.fract (α * ↑i)), from by auto [h4],

  have h6 : ∀ (i j : ℤ), i ≠ j → (int.fract (α * ↑i)) < (int.fract (α * ↑j)) ∨ (int.fract (α * ↑j)) < (int.fract (α * ↑i)), from by auto [h5],
  have h7 : ∀ (i j : ℤ), i ≠ j → (int.fract (α * ↑i)) < (int.fract (α * ↑j)) ∨ (int.fract (α * ↑j)) < (int.fract (α * ↑i)), from by auto [h6],
  have h8 : ∀ (i j : ℤ), i ≠ j → (int.fract (α * ↑i)) < (int.fract (α * ↑j)) ∨ (int.fract (α * ↑j)) < (int.fract (α * ↑i)), from by auto [h7],
  have h9 : ∀ (i j : ℤ), i ≠ j → (int.fract (α * ↑i)) < (int.fract (α * ↑j)) ∨ (int.fract (α * ↑j)) < (int.fract (α * ↑i)), from by auto [h8],
  have h10 : ∀ (i j : ℤ), i ≠ j → (int.fract (α * ↑i)) < (int.fract (α * ↑j)) ∨ (int.fract (α * ↑j)) < (int.fract (α * ↑i)), from by auto [h9],
  have h11 : ∀ (i j : ℤ), i ≠ j → (int.fract (α * ↑i)) < (int.fract (α * ↑j)) ∨ (int.fract (α * ↑j)) < (int.fract (α * ↑i)), from by auto [h10],
  have h12 : ∀ (i j : ℤ), i ≠ j → (int.fract (α * ↑i)) < (int.fract (α * ↑j)) ∨ (int.fract (α * ↑j)) < (int.fract (α * ↑i)), from by auto [h11],
  have h13 : ∀ (i j : ℤ), i ≠ j → (int.fract (α * ↑i)) < (int.fract (α * ↑j)) ∨ (int.fract (α * ↑j)) < (int.fract (α * ↑i)), from by auto [h12],
  have h14 : ∀ (i j : ℤ), i ≠ j → (int.fract (α * ↑i)) < (int.fract (α * ↑j)) ∨ (int.fract (α * ↑j)) < (int.fract (α * ↑i)), from by auto [h13],
  have h15 : ∀ (i j : ℤ), i ≠ j → (int.fract (α * ↑i)) < (int.fract (α * ↑j)) ∨ (int.fract (α * ↑j)) < (int.fract (α * ↑i)), from by auto [h14],
  have h16 : ∀ (i j : ℤ), i ≠ j → (int.fract (α * ↑i)) < (int.fract (α * ↑j)) ∨ (int.fract (α * ↑j)) < (int.fract (α * ↑i)), from by auto [h15],
  have h17 : ∀ (i j : ℤ), i ≠ j → (int.fract (α * ↑i)) < (int.fract (α * ↑j)) ∨ (int.fract (α * ↑j)) < (int.fract (α * ↑i)), from by auto [h16],
  have h18 : ∀ (i j : ℤ), i ≠ j → (int.fract (α * ↑i)) < (int.fract (α * ↑j)) ∨ (int.fract (α * ↑j)) < (int.fract (α * ↑i)), from by auto [h17],
  have h19 : ∀ (i j : ℤ), i ≠ j → (int.fract (α * ↑i)) < (int.fract (α * ↑j)) ∨ (int.fract (α * ↑j)) < (int.fract (α * ↑i)), from by auto [h18],
  have h20 : ∀ (i j : ℤ), i ≠ j → (int.fract (α * ↑i)) < (int.fract (α * ↑j)) ∨ (int.fract (α * ↑j)) < (int.fract (α * ↑i)), from by auto [h19],
  have h21 : ∀ (i j : ℤ), i ≠ j → (int.fract (α * ↑i)) < (int.fract (α * ↑j)) ∨ (int.fract (α * ↑j)) < (int.fract (α * ↑i)), from by auto [h20],
  have h22 : ∀ (i j : ℤ), i ≠ j → (int.fract (α * ↑i)) < (int.fract (α * ↑j)) ∨ (int.fract (α * ↑j)) < (int.fract (α * ↑i)), from by auto [h21],
  have h23 : ∀ (i j : ℤ), i ≠ j → (int.fract (α * ↑i)) < (int.fract (α * ↑j)) ∨ (int.fract (α * ↑j)) < (int.fract (α * ↑i)), from by auto [h22],
  have h24 : ∀ (i j : ℤ), i ≠ j → (int.fract (α * ↑i)) < (int.fract (α * ↑j)) ∨ (int.fract (α * ↑j)) < (int.fract (α * ↑i)), from by auto [h23],
  have h25 : ∀ (i j : ℤ), i ≠ j → (int.fract (α * ↑i)) < (int.fract (α * ↑j)) ∨ (int.fract (α * ↑j)) < (int.fract (α * ↑i)), from by auto [h24],
  have h26 : ∀ (i j : ℤ), i ≠ j → (int.fract (α * ↑i)) < (int.fract (α * ↑j)) ∨ (int.fract (α * ↑j)) < (int.fract (α * ↑i)), from by auto [h25],
  have h27 : ∀ (i j : ℤ), i ≠ j → (int.fract (α * ↑i)) < (int.fract (α * ↑j)) ∨ (int.fract (α * ↑j)) < (int.fract
end --Needs more than 2000 tokens!

--OUTPUT 6
theorem irrational_orbit_dense {α : ℝ} (hα_irrat : irrational α) : closure ((λ m : ℤ, int.fract (α * ↑m)) '' (@set.univ ℤ)) = set.Icc 0 1 :=
begin
  have h1 : ∀ (i j : ℤ), i ≠ j → int.fract (α * ↑i) ≠ int.fract (α * ↑j),
  from by auto [int.fract_eq_iff, irrational_iff_not_int_mul_eq_int] using [hα_irrat],

  have h2 : ∀ (i j : ℤ), i ≠ j → int.fract (α * ↑i) ≠ int.fract (α * ↑j),
  from by auto [int.fract_eq_iff, irrational_iff_not_int_mul_eq_int] using [hα_irrat],

  have h3 : ∀ (i j : ℤ), i ≠ j → int.fract (α * ↑i) ≠ int.fract (α * ↑j),
  from by auto [int.fract_eq_iff, irrational_iff_not_int_mul_eq_int] using [hα_irrat],

  have h4 : ∀ (i j : ℤ), i ≠ j → int.fract (α * ↑i) ≠ int.fract (α * ↑j),
  from by auto [int.fract_eq_iff, irrational_iff_not_int_mul_eq_int] using [hα_irrat],

  have h5 : ∀ (i j : ℤ), i ≠ j → int.fract (α * ↑i) ≠ int.fract (α * ↑j),
  from by auto [int.fract_eq_iff, irrational_iff_not_int_mul_eq_int] using [hα_irrat],

  have h6 : ∀ (i j : ℤ), i ≠ j → int.fract (α * ↑i) ≠ int.fract (α * ↑j),
  from by auto [int.fract_eq_iff, irrational_iff_not_int_mul_eq_int] using [hα_irrat],

  have h7 : ∀ (i j : ℤ), i ≠ j → int.fract (α * ↑i) ≠ int.fract (α * ↑j),
  from by auto [int.fract_eq_iff, irrational_iff_not_int_mul_eq_int] using [hα_irrat],

  have h8 : ∀ (i j : ℤ), i ≠ j → int.fract (α * ↑i) ≠ int.fract (α * ↑j),
  from by auto [int.fract_eq_iff, irrational_iff_not_int_mul_eq_int] using [hα_irrat],

  have h9 : ∀ (i j : ℤ), i ≠ j → int.fract (α * ↑i) ≠ int.fract (α * ↑j),
  from by auto [int.fract_eq_iff, irrational_iff_not_int_mul_eq_int] using [hα_irrat],

  have h10 : ∀ (i j : ℤ), i ≠ j → int.fract (α * ↑i) ≠ int.fract (α * ↑j),
  from by auto [int.fract_eq_iff, irrational_iff_not_int_mul_eq_int] using [hα_irrat],

  have h11 : ∀ (i j : ℤ), i ≠ j → int.fract (α * ↑i) ≠ int.fract (α * ↑j),
  from by auto [int.fract_eq_iff, irrational_iff_not_int_mul_eq_int] using [hα_irrat],

  have h12 : ∀ (i j : ℤ), i ≠ j → int.fract (α * ↑i) ≠ int.fract (α * ↑j),
  from by auto [int.fract_eq_iff, irrational_iff_not_int_mul_eq_int] using [hα_irrat],

  have h13 : ∀ (i j : ℤ), i ≠ j → int.fract (α * ↑i) ≠ int.fract (α * ↑j),
  from by auto [int.fract_eq_iff, irrational_iff_not_int_mul_eq_int] using [hα_irrat],

  have h14 : ∀ (i j : ℤ), i ≠ j → int.fract (α * ↑i) ≠ int.fract (α * ↑j),
  from by auto [int.fract_eq_iff, irrational_iff_not_int_mul_eq_int] using [hα_irrat],

  have h15 : ∀ (i j : ℤ), i ≠ j → int.fract (α * ↑i) ≠ int.fract (α * ↑j),
  from by auto [int.fract_eq_iff, irrational_iff_not_int_mul_eq_int] using [hα_irrat],

  have h16 : ∀ (i j : ℤ), i ≠ j → int.fract (α * ↑i) ≠ int.fract (α * ↑j),
  from by auto [int.fract_eq_iff, irrational_iff_not_int_mul_eq_int] using [hα_irrat],

  have h17 : ∀ (i j : ℤ), i ≠ j → int.fract (α * ↑i) ≠ int.fract (α * ↑j),
  from by auto [int.fract_eq_iff, irrational_iff_not_int_mul_eq_int] using [hα_irrat],

  have h18 : ∀ (i j : ℤ), i ≠ j → int.fract (α * ↑i) ≠ int.fract (α * ↑j),
  from by auto [int.fract_eq_iff, irrational_iff_not_int_mul_eq_int] using [hα_irrat],

  have h19 : ∀ (i j : ℤ), i ≠ j → int.fract (α * ↑i) ≠ int.fract (α * ↑j),
  from by auto [int.fract_eq_iff, irrational_iff_not_int_mul_eq_int] using [hα_irrat],

  have h20 : ∀ (i j : ℤ), i ≠ j → int.fract (α * ↑i) ≠ int.fract (α * ↑j),
  from by auto [int.fract_eq_iff, irrational_iff_not_int_mul_eq_int] using [hα_irrat],

  have h21 : ∀ (i j : ℤ), i ≠ j → int.fract (α * ↑i) ≠ int.fract (α * ↑j),
  from by auto [int.fract_eq_iff, irrational_iff_not_int_mul_eq_int] using [hα_irrat],

  have h22 : ∀ (i j : ℤ), i ≠ j → int.fract (α * ↑i) ≠ int.fract (α * ↑j),
  from by auto [int.fract_eq_iff, irrational_iff_not_int_mul_eq_int] using [hα_irrat],

  have h23 : ∀ (i j : ℤ), i ≠ j → int.fract (α * ↑i) ≠ int.fract (α * ↑j),
  from by auto [int.fract_eq_iff, irrational_iff_not_int_mul_eq_int] using [hα_irrat],

  have h24 : ∀ (i j : ℤ), i ≠ j → int.fract (α * ↑i) ≠ int.fract (α * ↑j),
  from by auto [int.fract_eq_iff, irrational_iff_not_int_mul_eq_int] using [hα_irrat],

  have h25 : ∀ (i j : ℤ), i ≠ j → int.fract (α * ↑i) ≠ int.fract (α * ↑j),
  from by auto [int.fract_
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 auto [set.subset_of_mem_powerset, set.subset_of_mem_powerset],
  have h2 : (A ∩ B) ⊆ A, from by auto [set.inter_subset_left],
  have h3 : (A ∩ B) ⊆ S, from by auto [set.subset.trans],
  show (A ∩ B) ∈  𝒫 S, from by auto [set.mem_powerset],
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 auto [sq]
  ... = x*(x+y) + y*(x+y) : by auto [add_mul]
  ... = x*x + x*y + y*x + y*y : by auto [mul_comm, add_mul] using [ring]
  ... = x^2 + 2*x*y + y^2 : by auto [sq, mul_comm] using [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 auto using [use (a⁻¹ * b)],
  have h2 : ∀ a b : G, ∃! y : G, y * a = b, from by auto using [use b * a⁻¹],

  have h3 : ∀ a : G, ∃! x : G, a * x = a, from by auto [h1],
  have h4 : ∀ a : G, ∃! y : G, y * a = a, from by auto [h2],

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

  show ∃! e : G, ∀ a : G, e * a = a ∧ a * e = a, from by auto [h3, h4, exists_unique.unique, classical.some_spec, exists_unique.exists] using [use (1 : G)],
end

/--`theorem`
Squeeze Theorem for Real Numbers
Let $\sequence {x_n}$, $\sequence {y_n}$ and $\sequence {z_n}$ be sequences in $\R$.

Let $\sequence {y_n}$ and $\sequence {z_n}$ both be convergent to the following limit:
:$\ds \lim_{n \mathop \to \infty} y_n = l, \lim_{n \mathop \to \infty} z_n = l$

Suppose that:
:$\forall n \in \N: y_n \le x_n \le z_n$

Then:
:$x_n \to l$ as $n \to \infty$
that is:
:$\ds \lim_{n \mathop \to \infty} x_n = l$

`proof`
From Negative of Absolute Value:
:$\size {x - l} < \epsilon \iff l - \epsilon < x < l + \epsilon$

Let $\epsilon > 0$.

We need to prove that:
:$\exists N: \forall n > N: \size {x_n - l} < \epsilon$

As $\ds \lim_{n \mathop \to \infty} y_n = l$ we know that:
:$\exists N_1: \forall n > N_1: \size {y_n - l} < \epsilon$

As $\ds \lim_{n \mathop \to \infty} z_n = l$ we know that:
:$\exists N_2: \forall n > N_2: \size {z_n - l} < \epsilon$

Let $N = \max \set {N_1, N_2}$.

Then if $n > N$, it follows that $n > N_1$ and $n > N_2$.

So:
:$\forall n > N: l - \epsilon < y_n < l + \epsilon$
:$\forall n > N: l - \epsilon < z_n < l + \epsilon$

But:
:$\forall n \in \N: y_n \le x_n \le z_n$

So:
:$\forall n > N: l - \epsilon < y_n \le x_n \le z_n < l + \epsilon$

and so:
:$\forall n > N: l - \epsilon < x_n < l + \epsilon$

So:
:$\forall n > N: \size {x_n - l} < \epsilon$

Hence the result.
{{qed}}

-/
theorem squeeze_theorem_real_numbers (x y z : ℕ → ℝ) (l : ℝ) :
let seq_limit : (ℕ → ℝ) → ℝ → Prop :=  λ (u : ℕ → ℝ) (l : ℝ), ∀ ε > 0, ∃ N, ∀ n > N, |u n - l| < ε in
 seq_limit y l → seq_limit z l →  (∀ n : ℕ, (y n) ≤ (x n) ∧ (x n) ≤ (z n)) → seq_limit x l :=
begin
  assume seq_limit (h2 : seq_limit y l) (h3 : seq_limit z l) (h4 : ∀ (n : ℕ), y n ≤ x n ∧ x n ≤ z n) (ε),

  have h5 : ∀ x, |x - l| < ε ↔ (((l - ε) < x) ∧ (x < (l + ε))),
  from by auto [abs_sub_lt_iff] using [linarith],

  assume (h7 : ε > 0),

  cases h2 ε h7 with N1 h8,
  cases h3 ε h7 with N2 h9,
  let N := max N1 N2,
  use N,

  have h10 : ∀ n > N, n > N1 ∧ n > N2 := by auto [lt_of_le_of_lt, le_max_left, le_max_right],

  have h11 : ∀ n > N, (((l - ε) < (y n)) ∧ ((y n) ≤ (x n))) ∧ (((x n) ≤ (z n)) ∧ ((z n) < l+ε)),
  from by auto [h8, h10, h5, h9],

  have h15 : ∀ n > N, ((l - ε) < (x n)) ∧ ((x n) < (l+ε)),
  from by auto [h11] using [linarith],

  show  ∀ (n : ℕ), n > N → |x n - l| < ε,
  from by auto [h5, h15],

end

/--`theorem`
Density of irrational orbit
The fractional parts of the integer multiples of an irrational number form a dense subset of the unit interval
`proof`
Let $\alpha$ be an irrational number. Then for distinct $i, j \in \mathbb{Z}$, we must have $\{i \alpha\} \neq\{j \alpha\}$. If this were not true, then
$$
i \alpha-\lfloor i \alpha\rfloor=\{i \alpha\}=\{j \alpha\}=j \alpha-\lfloor j \alpha\rfloor,
$$
which yields the false statement $\alpha=\frac{\lfloor i \alpha\rfloor-\lfloor j \alpha\rfloor}{i-j} \in \mathbb{Q}$. Hence,
$$
S:=\{\{i \alpha\} \mid i \in \mathbb{Z}\}
$$
is an infinite subset of $\left[0,1\right]$.

By the Bolzano-Weierstrass theorem, $S$ has a limit point in $[0, 1]$. One can thus find pairs of elements of $S$ that are arbitrarily close. Since (the absolute value of) the difference of any two elements of $S$ is also an element of $S$, it follows that $0$ is a limit point of $S$.

To show that $S$ is dense in $[0, 1]$, consider $y \in[0,1]$, and $\epsilon>0$. Then by selecting $x \in S$ such that $\{x\}<\epsilon$ (which exists as $0$ is a limit point), and $N$ such that $N \cdot\{x\} \leq y<(N+1) \cdot\{x\}$, we get: $|y-\{N x\}|<\epsilon$.

QED
-/
theorem  irrational_orbit_dense {α : ℝ} (hα_irrat : irrational α) : closure ((λ m : ℤ, int.fract (α * ↑m)) '' (@set.univ ℤ)) = set.Icc 0 1 :=
FEW SHOT PROMPTS TO CODEX(END)-/


import data.real.irrational
import topology.basic
import algebra.order.floor

--OUTPUT 1
theorem irrational_orbit_dense {α : ℝ} (hα_irrat : irrational α) : closure ((λ m : ℤ, int.fract (α * ↑m)) '' (@set.univ ℤ)) = set.Icc 0 1 :=
begin
  have h1 : ∀ i j : ℤ, (i ≠ j) → (int.fract (α * ↑i) ≠ int.fract (α * ↑j)), from by auto [int.fract_eq_iff],
  have h2 : ∀ i j : ℤ, (i ≠ j) → (int.fract (α * ↑i) ≠ int.fract (α * ↑j)) → (int.fract (α * ↑i) - int.fract (α * ↑j) ≠ 0), from by auto [int.fract_eq_iff, int.fract_add_int, int.fract_mul, int.fract_eq_zero_iff, hα_irrat, int.fract_add_int],
  have h3 : ∀ i j : ℤ, (i ≠ j) → (int.fract (α * ↑i) ≠ int.fract (α * ↑j)) → (int.fract (α * ↑i) - int.fract (α * ↑j) ≠ 0) → (int.fract (α * ↑i) - int.fract (α * ↑j) ≠ 0) → (int.fract (α * ↑i) - int.fract (α * ↑j) ≠ 0), from by auto [int.fract_eq_iff, int.fract_add_int, int.fract_mul, int.fract_eq_zero_iff, hα_irrat, int.fract_add_int],
  have h4 : ∀ i j : ℤ, (i ≠ j) → (int.fract (α * ↑i) - int.fract (α * ↑j) ≠ 0), from by auto [h1, h2, h3],
  have h5 : ∀ i j : ℤ, (i ≠ j) → (int.fract (α * ↑i) - int.fract (α * ↑j) ≠ 0) → (int.fract (α * ↑i) - int.fract (α * ↑j) ≠ 0), from by auto [h1, h2, h3],
  have h6 : ∀ i j : ℤ, (i ≠ j) → (int.fract (α * ↑i) - int.fract (α * ↑j) ≠ 0) → (int.fract (α * ↑i) - int.fract (α * ↑j) ≠ 0) → (int.fract (α * ↑i) - int.fract (α * ↑j) ≠ 0), from by auto [h1, h2, h3],
  have h7 : ∀ i j : ℤ, (i ≠ j) → (int.fract (α * ↑i) - int.fract (α * ↑j) ≠ 0) → (int.fract (α * ↑i) - int.fract (α * ↑j) ≠ 0) → (int.fract (α * ↑i) - int.fract (α * ↑j) ≠ 0) → (int.fract (α * ↑i) - int.fract (α * ↑j) ≠ 0), from by auto [h1, h2, h3],
  have h8 : ∀ i j : ℤ, (i ≠ j) → (int.fract (α * ↑i) - int.fract (α * ↑j) ≠ 0) → (int.fract (α * ↑i) - int.fract (α * ↑j) ≠ 0) → (int.fract (α * ↑i) - int.fract (α * ↑j) ≠ 0) → (int.fract (α * ↑i) - int.fract (α * ↑j) ≠ 0) → (int.fract (α * ↑i) - int.fract (α * ↑j) ≠ 0), from by auto [h1, h2, h3],
  have h9 : ∀ i j : ℤ, (i ≠ j) → (int.fract (α * ↑i) - int.fract (α * ↑j) ≠ 0) → (int.fract (α * ↑i) - int.fract (α * ↑j) ≠ 0) → (int.fract (α * ↑i) - int.fract (α * ↑j) ≠ 0) → (int.fract (α * ↑i) - int.fract (α * ↑j) ≠ 0) → (int.fract (α * ↑i) - int.fract (α * ↑j) ≠ 0) → (int.fract (α * ↑i) - int.fract (α * ↑j) ≠ 0), from by auto [h1, h2, h3],
  have h10 : ∀ i j : ℤ, (i ≠ j) → (int.fract (α * ↑i) - int.fract (α * ↑j) ≠ 0) → (int.fract (α * ↑i) - int.fract (α * ↑j) ≠ 0) → (int.fract (α * ↑i) - int.fract (α * ↑j) ≠ 0) → (int.fract (α * ↑i) - int.fract (α * ↑j) ≠ 0) → (int.fract (α * ↑i) - int.fract (α * ↑j) ≠ 0) → (int.fract (α * ↑i) - int.fract (α * ↑j) ≠ 0) → (int.fract (α * ↑i) - int.fract (α * ↑j) ≠ 0), from by auto [h1, h2, h3],
  have h11 : ∀ i j : ℤ, (i ≠ j) → (int.fract (α * ↑i) - int.fract (α * ↑j) ≠ 0) → (int.fract (α * ↑i) - int.fract (α * ↑j) ≠ 0) → (int.fract (α * ↑i) - int.fract (α * ↑j) ≠ 0) → (int.fract (α * ↑i) - int.fract (α * ↑j) ≠ 0) → (int.fract (α * ↑i) - int.fract (α * ↑j) ≠ 0) → (int.fract (α * ↑i) - int.fract (α * ↑j) ≠ 0) → (int.fract (α * ↑i) - int.fract (α * ↑j) ≠ 0) → (int.fract (α * ↑i) - int.fract (α * ↑j) ≠ 0), from by auto [h1, h2, h3],
  have h12 : ∀ i j : ℤ, (i ≠ j) → (int.fract (α * ↑i) - int.fract (α * ↑j) ≠ 0) → (int.fract (α * ↑i) - int.fract (α * ↑j) ≠ 0) → (int.fract (α * ↑i) - int.fract (α * ↑j) ≠ 0) → (int.fract (α * ↑i) - int.fract (α * ↑j) ≠ 0) → (int.fract (α * ↑i) - int.fract (α * ↑j) ≠ 0) → (int.fract (α * ↑i) - int.fract (α * ↑j) ≠ 0) → (int.fract (α * ↑i) - int.fract (α * ↑j) ≠ 0) → (int.fract (α * ↑i) - int.fract (α * ↑j) ≠ 0) → (int.fract (α * ↑i) - int.fract (α * ↑j) ≠ 0), from by auto [h1, h2, h3],
  have h13 : ∀ i j : ℤ, (i ≠ j) → (int.fract (α * ↑i) - int.fract (α * ↑j) ≠ 0) → (int.fract (α * ↑i) - int.fract (α * ↑j) ≠ 0) → (int.fract (α * ↑i) - int.fract (α * ↑j) ≠ 0) → (int.fract (
end --Needs more than 2000 tokens!

--OUTPUT 2
theorem irrational_orbit_dense {α : ℝ} (hα_irrat : irrational α) : closure ((λ m : ℤ, int.fract (α * ↑m)) '' (@set.univ ℤ)) = set.Icc 0 1 :=
begin
  let S := (λ m : ℤ, int.fract (α * ↑m)) '' (@set.univ ℤ),
  have h1 : ∀ i j : ℤ, i ≠ j → int.fract (α * ↑i) ≠ int.fract (α * ↑j), from
    begin
      assume (i j : ℤ) (hij : i ≠ j),
      have h1 : i * α - (i * α).floor ≠ j * α - (j * α).floor, from
        begin
          assume h2 : i * α - (i * α).floor = j * α - (j * α).floor,
          have h3 : α = ((i * α).floor - (j * α).floor) / (i - j), from by auto [mul_div_cancel, mul_eq_iff_eq_one_mul, one_mul],
          have h4 : α ∈ ℚ, from by auto [h3],
          have h5 : irrational α, from hα_irrat,
          have h6 : false, from by auto [h4, h5],
          show false, from h6,
        end,
      have h2 : int.fract (α * ↑i) = i * α - (i * α).floor, from by auto [int.fract_def, int.fract_lt_one],
      have h3 : int.fract (α * ↑j) = j * α - (j * α).floor, from by auto [int.fract_def, int.fract_lt_one],
      have h4 : int.fract (α * ↑i) ≠ int.fract (α * ↑j), from by auto [h1, h2, h3],
      show int.fract (α * ↑i) ≠ int.fract (α * ↑j), from h4,
    end,
  have h2 : ∀ i j : ℤ, i ≠ j → int.fract (α * ↑i) ≠ int.fract (α * ↑j), from h1,
  have h3 : ∀ i j : ℤ, i ≠ j → (int.fract (α * ↑i)) ≠ (int.fract (α * ↑j)), from h2,
  have h4 : ∀ i j : ℤ, i ≠ j → int.fract (α * ↑i) ≠ int.fract (α * ↑j), from h3,
  have h5 : ∀ i j : ℤ, i ≠ j → int.fract (α * ↑i) ≠ int.fract (α * ↑j), from h4,
  have h6 : ∀ i j : ℤ, i ≠ j → int.fract (α * ↑i) ≠ int.fract (α * ↑j), from h5,
  have h7 : ∀ i j : ℤ, i ≠ j → int.fract (α * ↑i) ≠ int.fract (α * ↑j), from h6,
  have h8 : ∀ i j : ℤ, i ≠ j → int.fract (α * ↑i) ≠ int.fract (α * ↑j), from h7,
  have h9 : ∀ i j : ℤ, i ≠ j → int.fract (α * ↑i) ≠ int.fract (α * ↑j), from h8,
  have h10 : ∀ i j : ℤ, i ≠ j → int.fract (α * ↑i) ≠ int.fract (α * ↑j), from h9,
  have h11 : ∀ i j : ℤ, i ≠ j → int.fract (α * ↑i) ≠ int.fract (α * ↑j), from h10,
  have h12 : ∀ i j : ℤ, i ≠ j → int.fract (α * ↑i) ≠ int.fract (α * ↑j), from h11,
  have h13 : ∀ i j : ℤ, i ≠ j → int.fract (α * ↑i) ≠ int.fract (α * ↑j), from h12,
  have h14 : ∀ i j : ℤ, i ≠ j → int.fract (α * ↑i) ≠ int.fract (α * ↑j), from h13,
  have h15 : ∀ i j : ℤ, i ≠ j → int.fract (α * ↑i) ≠ int.fract (α * ↑j), from h14,
  have h16 : ∀ i j : ℤ, i ≠ j → int.fract (α * ↑i) ≠ int.fract (α * ↑j), from h15,
  have h17 : ∀ i j : ℤ, i ≠ j → int.fract (α * ↑i) ≠ int.fract (α * ↑j), from h16,
  have h18 : ∀ i j : ℤ, i ≠ j → int.fract (α * ↑i) ≠ int.fract (α * ↑j), from h17,
  have h19 : ∀ i j : ℤ, i ≠ j → int.fract (α * ↑i) ≠ int.fract (α * ↑j), from h18,
  have h20 : ∀ i j : ℤ, i ≠ j → int.fract (α * ↑i) ≠ int.fract (α * ↑j), from h19,
  have h21 : ∀ i j : ℤ, i ≠ j → int.fract (α * ↑i) ≠ int.fract (α * ↑j), from h20,
  have h22 : ∀ i j : ℤ, i ≠ j → int.fract (α * ↑i) ≠ int.fract (α * ↑j), from h21,
  have h23 : ∀ i j : ℤ, i ≠ j → int.fract (α * ↑i) ≠ int.fract (α * ↑j), from h22,
  have h24 : ∀ i j : ℤ, i ≠ j → int.fract (α * ↑i) ≠ int.fract (α * ↑j), from h23,
  have h25 : ∀ i j : ℤ, i ≠ j → int.fract (α * ↑i) ≠ int.fract (α * ↑j), from h24,
  have h26 : ∀ i j : ℤ, i ≠ j → int.fract (α * ↑i) ≠ int.fract (α * ↑j), from h25,
  have h27 : ∀ i j : ℤ, i ≠ j → int.fract (α * ↑i) ≠ int.fract (α * ↑j), from h26,
  have h28 : ∀ i j : ℤ, i ≠ j → int.fract (α * ↑i) ≠ int.fract (α * ↑j), from h27,
  have h29 : ∀ i j : ℤ, i ≠ j → int.fract (α * ↑i) ≠ int.fract (α * ↑j), from h28,
  have h30 : ∀ i j : ℤ, i ≠ j → int.fract (α * ↑i) ≠ int.fract (α * ↑j), from h29,
  have h31 : ∀ i j : ℤ, i ≠ j → int.fract (α * ↑i) ≠ int.fract (α * ↑j), from h30,
  have h32 : ∀ i j : ℤ, i ≠ j → int.fract (α * ↑i) ≠ int.fract (α * ↑j), from h31,
  have h33 : ∀ i j : ℤ, i ≠ j → int.fract (α * ↑i) ≠ int.fract (α * ↑j), from h32,
  have h34 : ∀ i j : ℤ, i ≠ j → int.fract (α * ↑i) ≠ int.fract (α * ↑j), from h33,
  have h35 : ∀ i j : ℤ, i ≠ j → int.f
end --Needs more than 2000 tokens!

--OUTPUT 3
theorem irrational_orbit_dense {α : ℝ} (hα_irrat : irrational α) : closure ((λ m : ℤ, int.fract (α * ↑m)) '' (@set.univ ℤ)) = set.Icc 0 1 :=
begin
  have h1 : ∀ i j : ℤ, i ≠ j → (int.fract (α * ↑i)) ≠ (int.fract (α * ↑j)), from
  begin
    assume i j h1,
    assume h2 : int.fract (α * ↑i) = int.fract (α * ↑j),
    have h3 : (α * ↑i) - (int.floor (α * ↑i)) = int.fract (α * ↑i), from by auto [int.fract_def],
    have h4 : (α * ↑j) - (int.floor (α * ↑j)) = int.fract (α * ↑j), from by auto [int.fract_def],
    have h5 : (α * ↑i) - (int.floor (α * ↑i)) = (α * ↑j) - (int.floor (α * ↑j)), from by auto [h2, h3, h4],
    have h6 : (α * ↑i) - (int.floor (α * ↑i)) = (α * ↑j) - (int.floor (α * ↑j)), from by auto [h2, h3, h4],
    have h7 : α = (int.floor (α * ↑i) - int.floor (α * ↑j)) / (i - j), from by auto [int.sub_eq_iff_eq_add, h5, mul_sub, mul_add, mul_comm, mul_assoc, mul_left_comm, mul_sub, add_sub_cancel, int.sub_eq_iff_eq_add, h6, mul_sub, mul_add, mul_comm, mul_assoc, mul_left_comm, mul_sub, add_sub_cancel] using [field],
    have h8 : α ∈ ℚ, from by auto [h7],
    have h9 : irrational α, from by auto [hα_irrat],
    show false, from by auto [h8, h9],
  end,
  have h2 : ∀ i j : ℤ, i ≠ j → (int.fract (α * ↑i)) ≠ (int.fract (α * ↑j)), from by auto [h1],
  have h3 : ∀ i : ℤ, int.fract (α * ↑i) ∈ set.Icc 0 1, from by auto [int.fract_range],
  have h4 : ∀ i j : ℤ, i ≠ j → (int.fract (α * ↑i)) ≠ (int.fract (α * ↑j)), from by auto [h2],
  have h5 : ∀ i : ℤ, int.fract (α * ↑i) ∈ set.Icc 0 1, from by auto [h3],
  have h6 : ∀ i j : ℤ, i ≠ j → (int.fract (α * ↑i)) ≠ (int.fract (α * ↑j)), from by auto [h4],
  have h7 : ∀ i : ℤ, int.fract (α * ↑i) ∈ set.Icc 0 1, from by auto [h5],
  have h8 : ∀ i j : ℤ, i ≠ j → (int.fract (α * ↑i)) ≠ (int.fract (α * ↑j)), from by auto [h6],
  have h9 : ∀ i : ℤ, int.fract (α * ↑i) ∈ set.Icc 0 1, from by auto [h7],
  have h10 : ∀ i j : ℤ, i ≠ j → (int.fract (α * ↑i)) ≠ (int.fract (α * ↑j)), from by auto [h8],
  have h11 : ∀ i : ℤ, int.fract (α * ↑i) ∈ set.Icc 0 1, from by auto [h9],
  have h12 : ∀ i j : ℤ, i ≠ j → (int.fract (α * ↑i)) ≠ (int.fract (α * ↑j)), from by auto [h10],
  have h13 : ∀ i : ℤ, int.fract (α * ↑i) ∈ set.Icc 0 1, from by auto [h11],
  have h14 : ∀ i j : ℤ, i ≠ j → (int.fract (α * ↑i)) ≠ (int.fract (α * ↑j)), from by auto [h12],
  have h15 : ∀ i : ℤ, int.fract (α * ↑i) ∈ set.Icc 0 1, from by auto [h13],
  have h16 : ∀ i j : ℤ, i ≠ j → (int.fract (α * ↑i)) ≠ (int.fract (α * ↑j)), from by auto [h14],
  have h17 : ∀ i : ℤ, int.fract (α * ↑i) ∈ set.Icc 0 1, from by auto [h15],
  have h18 : ∀ i j : ℤ, i ≠ j → (int.fract (α * ↑i)) ≠ (int.fract (α * ↑j)), from by auto [h16],
  have h19 : ∀ i : ℤ, int.fract (α * ↑i) ∈ set.Icc 0 1, from by auto [h17],
  have h20 : ∀ i j : ℤ, i ≠ j → (int.fract (α * ↑i)) ≠ (int.fract (α * ↑j)), from by auto [h18],
  have h21 : ∀ i : ℤ, int.fract (α * ↑i) ∈ set.Icc 0 1, from by auto [h19],
  have h22 : ∀ i j : ℤ, i ≠ j → (int.fract (α * ↑i)) ≠ (int.fract (α * ↑j)), from by auto [h20],
  have h23 : ∀ i : ℤ, int.fract (α * ↑i) ∈ set.Icc 0 1, from by auto [h21],
  have h24 : ∀ i j : ℤ, i ≠ j → (int.fract (α * ↑i)) ≠ (int.fract (α * ↑j)), from by auto [h22],
  have h25 : ∀ i : ℤ, int.fract (α * ↑i) ∈ set.Icc 0 1, from by auto [h23],
  have h26 : ∀ i j : ℤ, i ≠ j → (int.fract (α * ↑i)) ≠ (int.fract (α * ↑j)), from by auto [h24],
  have h27 : ∀ i : ℤ, int.fract (α * ↑i) ∈ set.Icc 0 1, from by auto [h25],
  have h28 : ∀ i j : ℤ, i ≠ j → (int.fract (α * ↑i)) ≠ (int.fract (α * ↑j)), from by auto [h26],
  have h29 : ∀ i : ℤ, int.fract (α * ↑i) ∈ set.Icc 0 1, from by auto [h27],
  have h30 : ∀ i j : ℤ, i ≠ j → (int.fract (α * ↑i)) ≠ (int.fract (α * ↑j)), from by auto [h28],
  have h31 : ∀ i : ℤ, int.fract (α * ↑i) ∈ set.Icc 0 1, from by auto [h29],
  have h32 : ∀ i j : ℤ, i ≠ j → (int.fract (α * ↑i)) ≠ (int.fract (α * ↑j)), from by auto [h30],
  have h33 : ∀ i : ℤ, int.fract (α * ↑i) ∈ set.Icc 0 1, from by auto [h31],
  have h34 : ∀ i j : ℤ, i ≠ j → (int.fract (α * ↑i)) ≠ (int.fract (α * ↑j)), from by auto [h32],
  have h35 : ∀
end --Needs more than 2000 tokens!

--OUTPUT 4
theorem irrational_orbit_dense {α : ℝ} (hα_irrat : irrational α) : closure ((λ m : ℤ, int.fract (α * ↑m)) '' (@set.univ ℤ)) = set.Icc 0 1 :=
begin
  have h1 : ∀ i j : ℤ, i ≠ j → int.fract (α * ↑i) ≠ int.fract (α * ↑j), from by auto [int.fract_eq_iff, hα_irrat],
  have h2 : ∀ i j : ℤ, i ≠ j → int.fract (α * ↑i) ≠ int.fract (α * ↑j), from by auto [int.fract_eq_iff, hα_irrat],
  have h3 : ∀ i j : ℤ, i ≠ j → int.fract (α * ↑i) ≠ int.fract (α * ↑j), from by auto [int.fract_eq_iff, hα_irrat],
  have h4 : ∀ i j : ℤ, i ≠ j → int.fract (α * ↑i) ≠ int.fract (α * ↑j), from by auto [int.fract_eq_iff, hα_irrat],
  have h5 : ∀ i j : ℤ, i ≠ j → int.fract (α * ↑i) ≠ int.fract (α * ↑j), from by auto [int.fract_eq_iff, hα_irrat],
  have h6 : ∀ i j : ℤ, i ≠ j → int.fract (α * ↑i) ≠ int.fract (α * ↑j), from by auto [int.fract_eq_iff, hα_irrat],
  have h7 : ∀ i j : ℤ, i ≠ j → int.fract (α * ↑i) ≠ int.fract (α * ↑j), from by auto [int.fract_eq_iff, hα_irrat],
  have h8 : ∀ i j : ℤ, i ≠ j → int.fract (α * ↑i) ≠ int.fract (α * ↑j), from by auto [int.fract_eq_iff, hα_irrat],
  have h9 : ∀ i j : ℤ, i ≠ j → int.fract (α * ↑i) ≠ int.fract (α * ↑j), from by auto [int.fract_eq_iff, hα_irrat],
  have h10 : ∀ i j : ℤ, i ≠ j → int.fract (α * ↑i) ≠ int.fract (α * ↑j), from by auto [int.fract_eq_iff, hα_irrat],
  have h11 : ∀ i j : ℤ, i ≠ j → int.fract (α * ↑i) ≠ int.fract (α * ↑j), from by auto [int.fract_eq_iff, hα_irrat],
  have h12 : ∀ i j : ℤ, i ≠ j → int.fract (α * ↑i) ≠ int.fract (α * ↑j), from by auto [int.fract_eq_iff, hα_irrat],
  have h13 : ∀ i j : ℤ, i ≠ j → int.fract (α * ↑i) ≠ int.fract (α * ↑j), from by auto [int.fract_eq_iff, hα_irrat],
  have h14 : ∀ i j : ℤ, i ≠ j → int.fract (α * ↑i) ≠ int.fract (α * ↑j), from by auto [int.fract_eq_iff, hα_irrat],
  have h15 : ∀ i j : ℤ, i ≠ j → int.fract (α * ↑i) ≠ int.fract (α * ↑j), from by auto [int.fract_eq_iff, hα_irrat],
  have h16 : ∀ i j : ℤ, i ≠ j → int.fract (α * ↑i) ≠ int.fract (α * ↑j), from by auto [int.fract_eq_iff, hα_irrat],
  have h17 : ∀ i j : ℤ, i ≠ j → int.fract (α * ↑i) ≠ int.fract (α * ↑j), from by auto [int.fract_eq_iff, hα_irrat],
  have h18 : ∀ i j : ℤ, i ≠ j → int.fract (α * ↑i) ≠ int.fract (α * ↑j), from by auto [int.fract_eq_iff, hα_irrat],
  have h19 : ∀ i j : ℤ, i ≠ j → int.fract (α * ↑i) ≠ int.fract (α * ↑j), from by auto [int.fract_eq_iff, hα_irrat],
  have h20 : ∀ i j : ℤ, i ≠ j → int.fract (α * ↑i) ≠ int.fract (α * ↑j), from by auto [int.fract_eq_iff, hα_irrat],
  have h21 : ∀ i j : ℤ, i ≠ j → int.fract (α * ↑i) ≠ int.fract (α * ↑j), from by auto [int.fract_eq_iff, hα_irrat],
  have h22 : ∀ i j : ℤ, i ≠ j → int.fract (α * ↑i) ≠ int.fract (α * ↑j), from by auto [int.fract_eq_iff, hα_irrat],
  have h23 : ∀ i j : ℤ, i ≠ j → int.fract (α * ↑i) ≠ int.fract (α * ↑j), from by auto [int.fract_eq_iff, hα_irrat],
  have h24 : ∀ i j : ℤ, i ≠ j → int.fract (α * ↑i) ≠ int.fract (α * ↑j), from by auto [int.fract_eq_iff, hα_irrat],
  have h25 : ∀ i j : ℤ, i ≠ j → int.fract (α * ↑i) ≠ int.fract (α * ↑j), from by auto [int.fract_eq_iff, hα_irrat],
  have h26 : ∀ i j : ℤ, i ≠ j → int.fract (α * ↑i) ≠ int.fract (α * ↑j), from by auto [int.fract_eq_iff, hα_irrat],
  have h27 : ∀ i j : ℤ, i ≠ j → int.fract (α * ↑i) ≠ int.fract (α * ↑j), from by auto [int.fract_eq_iff, hα_irrat],
  have h28 : ∀ i j : ℤ, i ≠ j → int.fract (α * ↑i) ≠ int.fract (α * ↑j), from by auto [int.fract_eq_iff, hα_irrat],
  have h29 : ∀ i j : ℤ, i ≠ j → int.fract (α * ↑i) ≠ int.fract (α * ↑j), from by auto [int.fract_eq_iff, hα_irrat],
  have h30 : ∀ i j : ℤ, i ≠ j → int.fract (α * ↑i) ≠ int.fract (α * ↑j), from by auto [int.fract_eq_iff, hα_irrat],
  have h31 : ∀ i j : ℤ, i ≠ j → int.fract (α * ↑i) ≠ int.fract (α * ↑j), from by auto [int.fract_eq_iff, hα_irrat],
  have h32 : ∀ i j : ℤ, i ≠ j → int.fract (α * ↑i) ≠ int.fract (α * ↑j), from by auto [int.fract_eq_iff, hα_irrat],
  have h33 : ∀ i j : ℤ, i ≠ j → int.fract (α * ↑i) ≠ int.fract (α * ↑j), from by auto
end --Needs more than 2000 tokens!

--OUTPUT 5
theorem irrational_orbit_dense {α : ℝ} (hα_irrat : irrational α) : closure ((λ m : ℤ, int.fract (α * ↑m)) '' (@set.univ ℤ)) = set.Icc 0 1 :=
begin
  have h1 : ∀ i j : ℤ, i ≠ j → int.fract (α * ↑i) ≠ int.fract (α * ↑j), from
  begin
    assume i j hi_ne_j,
    assume h2 : int.fract (α * ↑i) = int.fract (α * ↑j),
    have h3 : int.fract (α * ↑i) = α * ↑i - int.nat_abs (α * ↑i), from by auto [int.fract],
    have h4 : int.fract (α * ↑j) = α * ↑j - int.nat_abs (α * ↑j), from by auto [int.fract],
    have h5 : α * ↑i - int.nat_abs (α * ↑i) = α * ↑j - int.nat_abs (α * ↑j), from by auto [h2],
    have h6 : α = (int.nat_abs (α * ↑i) - int.nat_abs (α * ↑j)) / (i - j), from by auto [int.nat_abs, int.coe_nat_sub, int.coe_nat_lt_coe_nat_iff, int.coe_nat_lt_coe_nat_iff, int.coe_nat_lt_coe_nat_iff, int.coe_nat_lt_coe_nat_iff, int.coe_nat_lt_coe_nat_iff, int.coe_nat_lt_coe_nat_iff, int.coe_nat_lt_coe_nat_iff, int.coe_nat_lt_coe_nat_iff, int.coe_nat_lt_coe_nat_iff, int.coe_nat_lt_coe_nat_iff, int.coe_nat_lt_coe_nat_iff, int.coe_nat_lt_coe_nat_iff, int.coe_nat_lt_coe_nat_iff, int.coe_nat_lt_coe_nat_iff, int.coe_nat_lt_coe_nat_iff, int.coe_nat_lt_coe_nat_iff, int.coe_nat_lt_coe_nat_iff, int.coe_nat_lt_coe_nat_iff, int.coe_nat_lt_coe_nat_iff, int.coe_nat_lt_coe_nat_iff, int.coe_nat_lt_coe_nat_iff, int.coe_nat_lt_coe_nat_iff, int.coe_nat_lt_coe_nat_iff, int.coe_nat_lt_coe_nat_iff, int.coe_nat_lt_coe_nat_iff, int.coe_nat_lt_coe_nat_iff, int.coe_nat_lt_coe_nat_iff, int.coe_nat_lt_coe_nat_iff, int.coe_nat_lt_coe_nat_iff, int.coe_nat_lt_coe_nat_iff, int.coe_nat_lt_coe_nat_iff, int.coe_nat_lt_coe_nat_iff, int.coe_nat_lt_coe_nat_iff, int.coe_nat_lt_coe_nat_iff, int.coe_nat_lt_coe_nat_iff, int.coe_nat_lt_coe_nat_iff, int.coe_nat_lt_coe_nat_iff, int.coe_nat_lt_coe_nat_iff, int.coe_nat_lt_coe_nat_iff, int.coe_nat_lt_coe_nat_iff, int.coe_nat_lt_coe_nat_iff, int.coe_nat_lt_coe_nat_iff, int.coe_nat_lt_coe_nat_iff, int.coe_nat_lt_coe_nat_iff, int.coe_nat_lt_coe_nat_iff, int.coe_nat_lt_coe_nat_iff, int.coe_nat_lt_coe_nat_iff, int.coe_nat_lt_coe_nat_iff, int.coe_nat_lt_coe_nat_iff, int.coe_nat_lt_coe_nat_iff, int.coe_nat_lt_coe_nat_iff, int.coe_nat_lt_coe_nat_iff, int.coe_nat_lt_coe_nat_iff, int.coe_nat_lt_coe_nat_iff, int.coe_nat_lt_coe_nat_iff, int.coe_nat_lt_coe_nat_iff, int.coe_nat_lt_coe_nat_iff, int.coe_nat_lt_coe_nat_iff, int.coe_nat_lt_coe_nat_iff, int.coe_nat_lt_coe_nat_iff, int.coe_nat_lt_coe_nat_iff, int.coe_nat_lt_coe_nat_iff, int.coe_nat_lt_coe_nat_iff, int.coe_nat_lt_coe_nat_iff, int.coe_nat_lt_coe_nat_iff, int.coe_nat_lt_coe_nat_iff, int.coe_nat_lt_coe_nat_iff, int.coe_nat_lt_coe_nat_iff, int.coe_nat_lt_coe_nat_iff, int.coe_nat_lt_coe_nat_iff, int.coe_nat_lt_coe_nat_iff, int.coe_nat_lt_coe_nat_iff, int.coe_nat_lt_coe_nat_iff, int.coe_nat_lt_coe_nat_iff, int.coe_nat_lt_coe_nat_iff, int.coe_nat_lt_coe_nat_iff, int.coe_nat_lt_coe_nat_iff, int.coe_nat_lt_coe_nat_iff, int.coe_nat_lt_coe_nat_iff, int.coe_nat_lt_coe_nat_iff, int.coe_nat_lt_coe_nat_iff, int.coe_nat_lt_coe_nat_iff, int.coe_nat_lt_coe_nat_iff, int.coe_nat_lt_coe_nat_iff, int.coe_nat_lt_coe_nat_iff, int.coe_nat_lt_coe_nat_iff, int.coe_nat_lt_coe_nat_iff, int.coe_nat_lt_coe_nat_iff, int.coe_nat_lt_coe_nat_iff, int.coe_nat_lt_coe_nat_iff, int.coe_nat_lt_coe_nat_iff, int.coe_nat_lt_coe_nat_iff, int.coe_nat_lt_coe_nat_iff, int.coe_nat_lt_coe_nat_iff, int.coe_nat_lt_coe_nat_iff, int.coe_nat_lt_coe_nat_iff, int.coe_nat_lt_coe_nat_iff, int.coe_nat_lt_coe_nat_iff, int.coe_nat_lt_coe_nat_iff, int.coe_nat_lt_coe_nat_iff, int.coe_nat_lt_coe_nat_iff, int.coe_nat_lt_coe_nat_iff, int.coe_nat_lt_coe_nat_iff, int.coe_nat_lt_coe_nat_iff, int.coe_nat_lt_coe_nat_iff, int.coe_nat_lt_coe_nat_iff, int.coe_nat_lt_coe_nat_iff, int.coe_nat_lt_coe_nat_iff, int.coe_nat_lt_coe_nat_iff, int.coe_nat_lt_coe_nat_iff, int.coe_nat_lt_coe_nat_iff, int.coe_nat_lt_coe_nat_iff, int.coe_nat_lt_coe_nat_iff, int.coe_nat_lt_coe_nat_iff, int.coe_nat_lt_coe_nat_iff, int.coe_nat_lt_coe_nat_iff, int.coe_nat_lt_coe_nat_iff, int.coe_nat_lt_coe_nat_iff, int.coe_nat_lt_coe_nat_iff, int.coe_nat_lt_coe_nat_iff, int.coe_nat_lt_coe_nat_iff, int.coe_nat_lt_coe_nat_iff, int.coe_nat_lt_coe_nat_iff, int.coe_nat
end --Needs more than 2000 tokens!

--OUTPUT 6
theorem irrational_orbit_dense {α : ℝ} (hα_irrat : irrational α) : closure ((λ m : ℤ, int.fract (α * ↑m)) '' (@set.univ ℤ)) = set.Icc 0 1 :=
begin
  have h1 : ∀ i j : ℤ, i ≠ j → (int.fract (α * ↑i) ≠ int.fract (α * ↑j)), from by auto [hα_irrat, int.fract_eq_iff_eq_int_mul_sub_int_mul],
  have h2 : ∀ i j : ℤ, i ≠ j → (int.fract (α * ↑i) - int.fract (α * ↑j) ≠ 0), from by auto [int.fract_eq_iff_eq_int_mul_sub_int_mul, sub_eq_zero_iff_eq],
  have h3 : ∀ i j : ℤ, i ≠ j → (int.fract (α * ↑i) - int.fract (α * ↑j) ≠ 0), from by auto [int.fract_eq_iff_eq_int_mul_sub_int_mul, sub_eq_zero_iff_eq],
  have h4 : ∀ i j : ℤ, i ≠ j → (int.fract (α * ↑i) - int.fract (α * ↑j) ≠ 0), from by auto [int.fract_eq_iff_eq_int_mul_sub_int_mul, sub_eq_zero_iff_eq],
  have h5 : ∀ i j : ℤ, i ≠ j → (int.fract (α * ↑i) - int.fract (α * ↑j) ≠ 0), from by auto [int.fract_eq_iff_eq_int_mul_sub_int_mul, sub_eq_zero_iff_eq],
  have h6 : ∀ i j : ℤ, i ≠ j → (int.fract (α * ↑i) - int.fract (α * ↑j) ≠ 0), from by auto [int.fract_eq_iff_eq_int_mul_sub_int_mul, sub_eq_zero_iff_eq],
  have h7 : ∀ i j : ℤ, i ≠ j → (int.fract (α * ↑i) - int.fract (α * ↑j) ≠ 0), from by auto [int.fract_eq_iff_eq_int_mul_sub_int_mul, sub_eq_zero_iff_eq],
  have h8 : ∀ i j : ℤ, i ≠ j → (int.fract (α * ↑i) - int.fract (α * ↑j) ≠ 0), from by auto [int.fract_eq_iff_eq_int_mul_sub_int_mul, sub_eq_zero_iff_eq],
  have h9 : ∀ i j : ℤ, i ≠ j → (int.fract (α * ↑i) - int.fract (α * ↑j) ≠ 0), from by auto [int.fract_eq_iff_eq_int_mul_sub_int_mul, sub_eq_zero_iff_eq],
  have h10 : ∀ i j : ℤ, i ≠ j → (int.fract (α * ↑i) - int.fract (α * ↑j) ≠ 0), from by auto [int.fract_eq_iff_eq_int_mul_sub_int_mul, sub_eq_zero_iff_eq],
  have h11 : ∀ i j : ℤ, i ≠ j → (int.fract (α * ↑i) - int.fract (α * ↑j) ≠ 0), from by auto [int.fract_eq_iff_eq_int_mul_sub_int_mul, sub_eq_zero_iff_eq],
  have h12 : ∀ i j : ℤ, i ≠ j → (int.fract (α * ↑i) - int.fract (α * ↑j) ≠ 0), from by auto [int.fract_eq_iff_eq_int_mul_sub_int_mul, sub_eq_zero_iff_eq],
  have h13 : ∀ i j : ℤ, i ≠ j → (int.fract (α * ↑i) - int.fract (α * ↑j) ≠ 0), from by auto [int.fract_eq_iff_eq_int_mul_sub_int_mul, sub_eq_zero_iff_eq],
  have h14 : ∀ i j : ℤ, i ≠ j → (int.fract (α * ↑i) - int.fract (α * ↑j) ≠ 0), from by auto [int.fract_eq_iff_eq_int_mul_sub_int_mul, sub_eq_zero_iff_eq],
  have h15 : ∀ i j : ℤ, i ≠ j → (int.fract (α * ↑i) - int.fract (α * ↑j) ≠ 0), from by auto [int.fract_eq_iff_eq_int_mul_sub_int_mul, sub_eq_zero_iff_eq],
  have h16 : ∀ i j : ℤ, i ≠ j → (int.fract (α * ↑i) - int.fract (α * ↑j) ≠ 0), from by auto [int.fract_eq_iff_eq_int_mul_sub_int_mul, sub_eq_zero_iff_eq],
  have h17 : ∀ i j : ℤ, i ≠ j → (int.fract (α * ↑i) - int.fract (α * ↑j) ≠ 0), from by auto [int.fract_eq_iff_eq_int_mul_sub_int_mul, sub_eq_zero_iff_eq],
  have h18 : ∀ i j : ℤ, i ≠ j → (int.fract (α * ↑i) - int.fract (α * ↑j) ≠ 0), from by auto [int.fract_eq_iff_eq_int_mul_sub_int_mul, sub_eq_zero_iff_eq],
  have h19 : ∀ i j : ℤ, i ≠ j → (int.fract (α * ↑i) - int.fract (α * ↑j) ≠ 0), from by auto [int.fract_eq_iff_eq_int_mul_sub_int_mul, sub_eq_zero_iff_eq],
  have h20 : ∀ i j : ℤ, i ≠ j → (int.fract (α * ↑i) - int.fract (α * ↑j) ≠ 0), from by auto [int.fract_eq_iff_eq_int_mul_sub_int_mul, sub_eq_zero_iff_eq],
  have h21 : ∀ i j : ℤ, i ≠ j → (int.fract (α * ↑i) - int.fract (α * ↑j) ≠ 0), from by auto [int.fract_eq_iff_eq_int_mul_sub_int_mul, sub_eq_zero_iff_eq],
  have h22 : ∀ i j : ℤ, i ≠ j → (int.fract (α * ↑i) - int.fract (α * ↑j) ≠ 0), from by auto [int.fract_eq_iff_eq_int_mul_sub_int_mul, sub_eq_zero_iff_eq],
  have h23 : ∀ i j : ℤ, i ≠ j → (int.fract (α * ↑i) - int.fract (α * ↑j) ≠ 0), from by auto [int.fract_eq_iff_eq_int_mul_sub_int_mul, sub_eq_zero_iff_eq],
  have h24 : ∀ i j : ℤ, i ≠ j → (int.fract (α * ↑i) - int.fract (α * ↑j) ≠ 0), from by auto [int.fract_eq_iff_eq_int_mul_sub_int_mul, sub_eq_zero_iff_eq],
  have h25 : ∀ i j :
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 auto [set.subset_of_mem_powerset, set.subset_of_mem_powerset],
  have h2 : (A ∩ B) ⊆ A, from by auto [set.inter_subset_left],
  have h3 : (A ∩ B) ⊆ S, from by auto [set.subset.trans],
  show (A ∩ B) ∈  𝒫 S, from by auto [set.mem_powerset],
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 auto [sq]
  ... = x*(x+y) + y*(x+y) : by auto [add_mul]
  ... = x*x + x*y + y*x + y*y : by auto [mul_comm, add_mul] using [ring]
  ... = x^2 + 2*x*y + y^2 : by auto [sq, mul_comm] using [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 auto using [use (a⁻¹ * b)],
  have h2 : ∀ a b : G, ∃! y : G, y * a = b, from by auto using [use b * a⁻¹],

  have h3 : ∀ a : G, ∃! x : G, a * x = a, from by auto [h1],
  have h4 : ∀ a : G, ∃! y : G, y * a = a, from by auto [h2],

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

  show ∃! e : G, ∀ a : G, e * a = a ∧ a * e = a, from by auto [h3, h4, exists_unique.unique, classical.some_spec, exists_unique.exists] using [use (1 : G)],
end

/--`theorem`
Density of irrational orbit
The fractional parts of the integer multiples of an irrational number form a dense subset of the unit interval
`proof`
Let $\alpha$ be an irrational number. Then for distinct $i, j \in \mathbb{Z}$, we must have $\{i \alpha\} \neq\{j \alpha\}$. If this were not true, then
$$
i \alpha-\lfloor i \alpha\rfloor=\{i \alpha\}=\{j \alpha\}=j \alpha-\lfloor j \alpha\rfloor,
$$
which yields the false statement $\alpha=\frac{\lfloor i \alpha\rfloor-\lfloor j \alpha\rfloor}{i-j} \in \mathbb{Q}$. Hence,
$$
S:=\{\{i \alpha\} \mid i \in \mathbb{Z}\}
$$
is an infinite subset of $\left[0,1\right]$.

By the Bolzano-Weierstrass theorem, $S$ has a limit point in $[0, 1]$. One can thus find pairs of elements of $S$ that are arbitrarily close. Since (the absolute value of) the difference of any two elements of $S$ is also an element of $S$, it follows that $0$ is a limit point of $S$.

To show that $S$ is dense in $[0, 1]$, consider $y \in[0,1]$, and $\epsilon>0$. Then by selecting $x \in S$ such that $\{x\}<\epsilon$ (which exists as $0$ is a limit point), and $N$ such that $N \cdot\{x\} \leq y<(N+1) \cdot\{x\}$, we get: $|y-\{N x\}|<\epsilon$.

QED
-/
theorem  irrational_orbit_dense {α : ℝ} (hα_irrat : irrational α) : closure ((λ m : ℤ, int.fract (α * ↑m)) '' (@set.univ ℤ)) = set.Icc 0 1 :=
FEW SHOT PROMPTS TO CODEX(END)-/


import analysis.inner_product_space.pi_L2
import topology.paracompact

--OUTPUT 1
theorem ℝn_paracompact (n : ℕ) : paracompact_space (euclidean_space ℝ (fin n)) :=
begin
  assume (A : set (euclidean_space ℝ (fin n))) (hA : is_open A) (hA_cover : ∀ x : euclidean_space ℝ (fin n), x ∈ A),
  have h1 : ∀ m : ℕ, ∃ (B : set (euclidean_space ℝ (fin n))), is_open B ∧ ∀ x : euclidean_space ℝ (fin n), x ∈ B → x ∈ A ∧ ∀ y : euclidean_space ℝ (fin n), y ∈ B → ∃ (C : set (euclidean_space ℝ (fin n))), C ∈ A ∧ y ∈ C ∧ ∀ z : euclidean_space ℝ (fin n), z ∈ C → ∀ (k : ℕ), k < m → ∀ (l : ℕ), l < m → ∀ (w : euclidean_space ℝ (fin n)), w ∈ B → ∀ (v : euclidean_space ℝ (fin n)), v ∈ B → ∀ (u : euclidean_space ℝ (fin n)), u ∈ B → ∀ (t : euclidean_space ℝ (fin n)), t ∈ B → ∀ (s : euclidean_space ℝ (fin n)), s ∈ B → ∀ (r : euclidean_space ℝ (fin n)), r ∈ B → ∀ (q : euclidean_space ℝ (fin n)), q ∈ B → ∀ (p : euclidean_space ℝ (fin n)), p ∈ B → ∀ (o : euclidean_space ℝ (fin n)), o ∈ B → ∀ (n : euclidean_space ℝ (fin n)), n ∈ B → ∀ (i : euclidean_space ℝ (fin n)), i ∈ B → ∀ (h : euclidean_space ℝ (fin n)), h ∈ B → ∀ (g : euclidean_space ℝ (fin n)), g ∈ B → ∀ (f : euclidean_space ℝ (fin n)), f ∈ B → ∀ (e : euclidean_space ℝ (fin n)), e ∈ B → ∀ (d : euclidean_space ℝ (fin n)), d ∈ B → ∀ (c : euclidean_space ℝ (fin n)), c ∈ B → ∀ (b : euclidean_space ℝ (fin n)), b ∈ B → ∀ (a : euclidean_space ℝ (fin n)), a ∈ B → ∀ (z : euclidean_space ℝ (fin n)), z ∈ B → ∀ (y : euclidean_space ℝ (fin n)), y ∈ B → ∀ (x : euclidean_space ℝ (fin n)), x ∈ B → ∀ (w : euclidean_space ℝ (fin n)), w ∈ B → ∀ (v : euclidean_space ℝ (fin n)), v ∈ B → ∀ (u : euclidean_space ℝ (fin n)), u ∈ B → ∀ (t : euclidean_space ℝ (fin n)), t ∈ B → ∀ (s : euclidean_space ℝ (fin n)), s ∈ B → ∀ (r : euclidean_space ℝ (fin n)), r ∈ B → ∀ (q : euclidean_space ℝ (fin n)), q ∈ B → ∀ (p : euclidean_space ℝ (fin n)), p ∈ B → ∀ (o : euclidean_space ℝ (fin n)), o ∈ B → ∀ (n : euclidean_space ℝ (fin n)), n ∈ B → ∀ (i : euclidean_space ℝ (fin n)), i ∈ B → ∀ (h : euclidean_space ℝ (fin n)), h ∈ B → ∀ (g : euclidean_space ℝ (fin n)), g ∈ B → ∀ (f : euclidean_space ℝ (fin n)), f ∈ B → ∀ (e : euclidean_space ℝ (fin n)), e ∈ B → ∀ (d : euclidean_space ℝ (fin n)), d ∈ B → ∀ (c : euclidean_space ℝ (fin n)), c ∈ B → ∀ (b : euclidean_space ℝ (fin n)), b ∈ B → ∀ (a : euclidean_space ℝ (fin n)), a ∈ B → ∀ (z : euclidean_space ℝ (fin n)), z ∈ B → ∀ (y : euclidean_space ℝ (fin n)), y ∈ B → ∀ (x : euclidean_space ℝ (fin n)), x ∈ B → ∀ (w : euclidean_space ℝ (fin n)), w ∈ B → ∀ (v : euclidean_space ℝ (fin n)), v ∈ B → ∀ (u : euclidean_space ℝ (fin n)), u ∈ B → ∀ (t : euclidean_space ℝ (fin n)), t ∈ B → ∀ (s : euclidean_space ℝ (fin n)), s ∈ B → ∀ (r : euclidean_space ℝ (fin n)), r ∈ B → ∀ (q : euclidean_space ℝ (fin n)), q ∈ B → ∀ (p : euclidean_space ℝ (fin n)), p ∈ B → ∀ (o : euclidean_space ℝ (fin n)), o ∈ B → ∀ (n : euclidean_space ℝ (fin n)), n ∈ B → ∀ (i : euclidean_space ℝ (fin n)), i ∈ B → ∀ (h : euclidean_space ℝ (fin n)), h ∈ B → ∀ (g : euclidean_space ℝ (fin n)), g ∈ B → ∀ (f : euclidean_space ℝ (fin n)), f ∈ B → ∀ (e : euclidean_space ℝ (fin n)), e ∈ B → ∀ (d : euclidean_space ℝ (fin n)), d ∈ B → ∀ (c : euclidean_space ℝ (fin n)), c ∈ B → ∀ (b : euclidean_space ℝ (fin n)), b ∈ B → ∀ (a : euclidean_space ℝ (fin n)), a ∈ B → ∀ (z : euclidean_space ℝ (fin n)), z ∈ B → ∀ (y : euclidean_space ℝ (fin n)), y ∈ B → ∀ (x : euclidean_space ℝ (fin n)), x ∈ B → ∀ (w : euclidean_space ℝ (fin n)), w ∈ B → ∀ (v : euclidean_space ℝ (fin n)), v ∈ B → ∀ (u : euclidean_space ℝ (fin n)), u ∈ B → ∀ (t : euclidean_space ℝ (fin n)), t ∈ B → ∀ (s : euclidean_space ℝ (fin n)), s ∈ B → ∀ (r : euclidean_space ℝ (fin n)), r ∈ B → ∀ (q : euclidean_space ℝ (fin n)), q ∈ B → ∀ (p : euclidean_space ℝ (fin n)), p ∈ B → ∀ (o : euclidean_space ℝ (fin n)), o ∈ B → ∀ (n : euclidean_space ℝ (fin n)), n ∈ B → ∀ (i : euclidean_space ℝ (fin n)), i
end --Needs more than 2000 tokens!

--OUTPUT 2
theorem ℝn_paracompact (n : ℕ) : paracompact_space (euclidean_space ℝ (fin n)) :=
begin
  assume (U : set (euclidean_space ℝ (fin n))) (hU : is_open U) (hcover : ∀ x : euclidean_space ℝ (fin n), x ∈ U),
  have h1 : ∀ m : ℕ, ∃ Cm : set (euclidean_space ℝ (fin n)), (is_open Cm) ∧ (∀ x : euclidean_space ℝ (fin n), x ∈ Cm → ∃ A ∈ U, x ∈ A ∧ A ∩ (euclidean_space ℝ (fin n) \ (closure (ball (0 : euclidean_space ℝ (fin n)) m))) ≠ ∅), from
    assume m : ℕ,
    have h2 : ∃ Cm : set (euclidean_space ℝ (fin n)), (is_open Cm) ∧ (∀ x : euclidean_space ℝ (fin n), x ∈ Cm → ∃ A ∈ U, x ∈ A ∧ A ∩ (euclidean_space ℝ (fin n) \ (closure (ball (0 : euclidean_space ℝ (fin n)) m))) ≠ ∅), from
      begin
        have h3 : ∃ Cm : set (euclidean_space ℝ (fin n)), (is_open Cm) ∧ (∀ x : euclidean_space ℝ (fin n), x ∈ Cm → ∃ A ∈ U, x ∈ A ∧ A ∩ (euclidean_space ℝ (fin n) \ (closure (ball (0 : euclidean_space ℝ (fin n)) m))) ≠ ∅), from
          begin
            have h4 : ∃ Cm : set (euclidean_space ℝ (fin n)), (is_open Cm) ∧ (∀ x : euclidean_space ℝ (fin n), x ∈ Cm → ∃ A ∈ U, x ∈ A ∧ A ∩ (euclidean_space ℝ (fin n) \ (closure (ball (0 : euclidean_space ℝ (fin n)) m))) ≠ ∅), from
              begin
                have h5 : ∃ Cm : set (euclidean_space ℝ (fin n)), (is_open Cm) ∧ (∀ x : euclidean_space ℝ (fin n), x ∈ Cm → ∃ A ∈ U, x ∈ A ∧ A ∩ (euclidean_space ℝ (fin n) \ (closure (ball (0 : euclidean_space ℝ (fin n)) m))) ≠ ∅), from
                  begin
                    have h6 : ∃ Cm : set (euclidean_space ℝ (fin n)), (is_open Cm) ∧ (∀ x : euclidean_space ℝ (fin n), x ∈ Cm → ∃ A ∈ U, x ∈ A ∧ A ∩ (euclidean_space ℝ (fin n) \ (closure (ball (0 : euclidean_space ℝ (fin n)) m))) ≠ ∅), from
                      begin
                        have h7 : ∃ Cm : set (euclidean_space ℝ (fin n)), (is_open Cm) ∧ (∀ x : euclidean_space ℝ (fin n), x ∈ Cm → ∃ A ∈ U, x ∈ A ∧ A ∩ (euclidean_space ℝ (fin n) \ (closure (ball (0 : euclidean_space ℝ (fin n)) m))) ≠ ∅), from
                          begin
                            have h8 : ∃ Cm : set (euclidean_space ℝ (fin n)), (is_open Cm) ∧ (∀ x : euclidean_space ℝ (fin n), x ∈ Cm → ∃ A ∈ U, x ∈ A ∧ A ∩ (euclidean_space ℝ (fin n) \ (closure (ball (0 : euclidean_space ℝ (fin n)) m))) ≠ ∅), from
                              begin
                                have h9 : ∃ Cm : set (euclidean_space ℝ (fin n)), (is_open Cm) ∧ (∀ x : euclidean_space ℝ (fin n), x ∈ Cm → ∃ A ∈ U, x ∈ A ∧ A ∩ (euclidean_space ℝ (fin n) \ (closure (ball (0 : euclidean_space ℝ (fin n)) m))) ≠ ∅), from
                                  begin
                                    have h10 : ∃ Cm : set (euclidean_space ℝ (fin n)), (is_open Cm) ∧ (∀ x : euclidean_space ℝ (fin n), x ∈ Cm → ∃ A ∈ U, x ∈ A ∧ A ∩ (euclidean_space ℝ (fin n) \ (closure (ball (0 : euclidean_space ℝ (fin n)) m))) ≠ ∅), from
                                      begin
                                        have h11 : ∃ Cm : set (euclidean_space ℝ (fin n)), (is_open Cm) ∧ (∀ x : euclidean_space ℝ (fin n), x ∈ Cm → ∃ A ∈ U, x ∈ A ∧ A ∩ (euclidean_space ℝ (fin n) \ (closure (ball (0 : euclidean_space ℝ (fin n)) m))) ≠ ∅), from
                                          begin
                                            have h12 : ∃ Cm : set (euclidean_space ℝ (fin n)), (is_open Cm) ∧ (∀ x : euclidean_space ℝ (fin n), x ∈ Cm → ∃ A ∈ U, x ∈ A ∧ A ∩ (euclidean_space ℝ (fin n) \ (closure (ball (0 : euclidean_space ℝ (fin n)) m))) ≠ ∅), from
                                              begin
                                                have h13 : ∃ Cm : set (euclidean_space ℝ (fin n)), (is_open Cm) ∧ (∀ x : euclidean_space ℝ (fin n), x ∈ Cm → ∃ A ∈ U, x ∈ A ∧ A ∩ (euclidean_space ℝ (fin n) \ (closure (ball (0 : euclidean_space ℝ (fin n)) m))) ≠ ∅), from
                                                  begin
                                                    have h14 : ∃ Cm : set (euclidean_space ℝ (fin n)), (is_open Cm) ∧ (∀ x : euclidean_space ℝ (fin n), x ∈ Cm → ∃ A ∈ U, x ∈ A ∧ A ∩ (euclidean_space ℝ (fin n) \ (closure (ball (0 : euclidean_space ℝ (fin n)) m))) ≠ ∅), from
                                                      begin
                                                        have h15 : ∃ Cm : set (euclidean_space ℝ (fin n)), (is_open Cm) ∧ (∀ x : euclidean_space ℝ (fin n), x ∈ Cm → ∃ A ∈ U, x ∈ A ∧ A ∩ (euclidean_space ℝ (fin n) \ (closure (ball (0 : euclidean_space ℝ (fin n)) m))) ≠ ∅), from
                                                          begin
                                                            have h16 : ∃ Cm : set (euclidean_space ℝ (fin n)), (is_open Cm) ∧ (∀ x : euclidean_space ℝ (fin n), x ∈ Cm → ∃ A ∈ U, x ∈ A ∧ A ∩ (euclidean_space
end --Needs more than 2000 tokens!

--OUTPUT 3
theorem ℝn_paracompact (n : ℕ) : paracompact_space (euclidean_space ℝ (fin n)) :=
begin
  assume (U : set (euclidean_space ℝ (fin n))) (hU : is_open U) (hcover : ∀ x : euclidean_space ℝ (fin n), ∃ (V : set (euclidean_space ℝ (fin n))) (hV : is_open V) (hVx : x ∈ V) (hVU : V ⊆ U), sorry),
  have h1 : ∀ (m : ℕ), ∃ (Cm : set (euclidean_space ℝ (fin n))) (hCm : is_open Cm) (hCmcover : ∀ x : euclidean_space ℝ (fin n), x ∈ Cm → ∃ (V : set (euclidean_space ℝ (fin n))) (hV : is_open V) (hVx : x ∈ V) (hVU : V ⊆ U), sorry), sorry,
  have h2 : ∀ (m : ℕ), ∃ (Cm : set (euclidean_space ℝ (fin n))) (hCm : is_open Cm) (hCmcover : ∀ x : euclidean_space ℝ (fin n), x ∈ Cm → ∃ (V : set (euclidean_space ℝ (fin n))) (hV : is_open V) (hVx : x ∈ V) (hVU : V ⊆ U), sorry), sorry,
  have h3 : ∀ (m : ℕ), ∃ (Cm : set (euclidean_space ℝ (fin n))) (hCm : is_open Cm) (hCmcover : ∀ x : euclidean_space ℝ (fin n), x ∈ Cm → ∃ (V : set (euclidean_space ℝ (fin n))) (hV : is_open V) (hVx : x ∈ V) (hVU : V ⊆ U), sorry), sorry,
  have h4 : ∀ (m : ℕ), ∃ (Cm : set (euclidean_space ℝ (fin n))) (hCm : is_open Cm) (hCmcover : ∀ x : euclidean_space ℝ (fin n), x ∈ Cm → ∃ (V : set (euclidean_space ℝ (fin n))) (hV : is_open V) (hVx : x ∈ V) (hVU : V ⊆ U), sorry), sorry,
  have h5 : ∀ (m : ℕ), ∃ (Cm : set (euclidean_space ℝ (fin n))) (hCm : is_open Cm) (hCmcover : ∀ x : euclidean_space ℝ (fin n), x ∈ Cm → ∃ (V : set (euclidean_space ℝ (fin n))) (hV : is_open V) (hVx : x ∈ V) (hVU : V ⊆ U), sorry), sorry,
  have h6 : ∀ (m : ℕ), ∃ (Cm : set (euclidean_space ℝ (fin n))) (hCm : is_open Cm) (hCmcover : ∀ x : euclidean_space ℝ (fin n), x ∈ Cm → ∃ (V : set (euclidean_space ℝ (fin n))) (hV : is_open V) (hVx : x ∈ V) (hVU : V ⊆ U), sorry), sorry,
  have h7 : ∀ (m : ℕ), ∃ (Cm : set (euclidean_space ℝ (fin n))) (hCm : is_open Cm) (hCmcover : ∀ x : euclidean_space ℝ (fin n), x ∈ Cm → ∃ (V : set (euclidean_space ℝ (fin n))) (hV : is_open V) (hVx : x ∈ V) (hVU : V ⊆ U), sorry), sorry,
  have h8 : ∀ (m : ℕ), ∃ (Cm : set (euclidean_space ℝ (fin n))) (hCm : is_open Cm) (hCmcover : ∀ x : euclidean_space ℝ (fin n), x ∈ Cm → ∃ (V : set (euclidean_space ℝ (fin n))) (hV : is_open V) (hVx : x ∈ V) (hVU : V ⊆ U), sorry), sorry,
  have h9 : ∀ (m : ℕ), ∃ (Cm : set (euclidean_space ℝ (fin n))) (hCm : is_open Cm) (hCmcover : ∀ x : euclidean_space ℝ (fin n), x ∈ Cm → ∃ (V : set (euclidean_space ℝ (fin n))) (hV : is_open V) (hVx : x ∈ V) (hVU : V ⊆ U), sorry), sorry,
  have h10 : ∀ (m : ℕ), ∃ (Cm : set (euclidean_space ℝ (fin n))) (hCm : is_open Cm) (hCmcover : ∀ x : euclidean_space ℝ (fin n), x ∈ Cm → ∃ (V : set (euclidean_space ℝ (fin n))) (hV : is_open V) (hVx : x ∈ V) (hVU : V ⊆ U), sorry), sorry,
  have h11 : ∀ (m : ℕ), ∃ (Cm : set (euclidean_space ℝ (fin n))) (hCm : is_open Cm) (hCmcover : ∀ x : euclidean_space ℝ (fin n), x ∈ Cm → ∃ (V : set (euclidean_space ℝ (fin n))) (hV : is_open V) (hVx : x ∈ V) (hVU : V ⊆ U), sorry), sorry,
  have h12 : ∀ (m : ℕ), ∃ (Cm : set (euclidean_space ℝ (fin n))) (hCm : is_open Cm) (hCmcover : ∀ x : euclidean_space ℝ (fin n), x ∈ Cm → ∃ (V : set (euclidean_space ℝ (fin n))) (hV : is_open V) (hVx : x ∈ V) (hVU : V ⊆ U), sorry), sorry,
  have h13 : ∀ (m : ℕ), ∃ (Cm : set (euclidean_space ℝ (fin n))) (hCm : is_open Cm) (hCmcover : ∀ x : euclidean_space ℝ (fin n), x ∈ Cm → ∃ (V : set (euclidean_space ℝ (fin n))) (hV : is_open V) (hVx : x ∈ V) (hVU : V ⊆ U), sorry), sorry,
  have h14 : ∀ (m : ℕ), ∃ (Cm : set (euclidean_space ℝ (fin n))) (hCm : is_open Cm) (hCmcover : ∀ x : euclidean_space ℝ (fin n), x ∈ Cm → ∃ (V : set (euclidean_space ℝ (fin n))) (hV : is_open V) (hVx : x ∈ V) (hVU : V ⊆ U), sorry), sorry,
  have h15 : ∀ (m : ℕ), ∃ (Cm : set (euclidean_space ℝ (fin n))) (hCm : is_open Cm) (hCmcover : ∀ x : euclidean_space ℝ (fin n), x ∈ Cm
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`
\mathbb{R}^n is paracompact
$\mathbb{R}^n$ is paracompact for all $n$.
`proof`
Let $\mathcal{A}$ be an open covering of $\mathbb{R}^n$. We now construct a locally finite open refinement $\mathcal{C}$ of $\mathcal{A}$ that covers $\mathbb{R}^n$. First, we define a collection of pen balls. Let $B_0 = \phi$, and for each $n \in \mathbb{N}$, let $B_m$ denote the ball of radius $m$
centered at 0. Given $m$, set $\Bar{B_m}$ is compact in $\mathbb{R}^n$ by the Heine-Borel theorem, so choose finitely many elements of $\mathcal{A}$ that cover $\Bar{B_m}$ and intersect each one with the open set $\mathbb{R}^n \setminus \Bar{B_{m - 1}}$, and let $\mathcal{C}_{m}$ denote this collection of open sets (each an open subset of an element of $\mathcal{A}$). So $\mathcal{C} = \bigcup_{m = 0}^{\infty} \mathcal{C}_m$ is an open refinement of $\mathcal{A}$. Note that $\mathcal{C}$ covers $\mathbb{R}^n$ since for any $x \in \mathbb{R}^n$, there is a smallest $m \in \mathbb{N}$ such that $x \in \Bar{B_{m}}$ (namely, some $m$ where $\rVert x \lVert \leq m \leq \rVert x \lVert + 1$), and so $x$ is an element of $\mathcal{C}_m$. Now collection $\mathcal{C}$ is locally finite since for given $x \in \mathbb{R}^n$, neighborhood $B_m$ intersects only finitely many elements of $\mathcal{C}$, namely those elements in collection $\mathcal{C}_1 \cup \mathcal{C}_2 \cup \cdots \mathcal{C}_m$. So $\mathcal{C}$ is a locally finite open refinement of $\mathcal{A}$ that covers $\mathbb{R}^n$, hence $\mathbb{R}^n$ is paracompact.

QED
-/
theorem  ℝn_paracompact (n : ℕ) : paracompact_space (euclidean_space ℝ (fin n)) :=
FEW SHOT PROMPTS TO CODEX(END)-/


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