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)), by {
    assume (i j : ℤ) (h1 : i ≠ j),
    have h2 : (α * ↑i) - int.floor (α * ↑i) = int.fract (α * ↑i), from sorry,
    have h3 : (α * ↑j) - int.floor (α * ↑j) = int.fract (α * ↑j), from sorry,
    have h4 : (α * ↑i) - int.floor (α * ↑i) = (α * ↑j) - int.floor (α * ↑j), from sorry,
    have h5 : (α * ↑i) = (α * ↑j), from sorry,
    have h6 : (i : ℝ) = (j : ℝ), from sorry,
    have h7 : i = j, from sorry,
    show (int.fract (α * ↑i)) ≠ (int.fract (α * ↑j)), from sorry,
  },
  have h2 : ∃ s : set ℝ, ∀ (i j : ℤ), i ≠ j → int.fract (α * ↑i) ∈ s ∧ int.fract (α * ↑j) ∈ s, from sorry,
  have h3 : ∃ s : set ℝ, ∀ (i j : ℤ), i ≠ j → (int.fract (α * ↑i) ∈ s) ∨ (int.fract (α * ↑j) ∈ s), from sorry,
  have h4 : ∃ s : set ℝ, ∀ (i j : ℤ), i ≠ j → int.fract (α * ↑i) ∈ s, from sorry,
  have h5 : ∃ s : set ℝ, ∀ (i j : ℤ), i ≠ j → int.fract (α * ↑j) ∈ s, from sorry,
  have h6 : ∃ s : set ℝ, ∀ (i j : ℤ), i ≠ j → int.fract (α * ↑i) ∈ s ∧ int.fract (α * ↑j) ∈ s, from sorry,
  have h7 : ∃ s : set ℝ, ∀ i : ℤ, int.fract (α * ↑i) ∈ s, from sorry,
  have h8 : ∃ s : set ℝ, ∀ i : ℤ, int.fract (α * ↑i) ∈ s ∧ int.fract (α * ↑j) ∈ s, from sorry,
  have h9 : ∃ s : set ℝ, ∀ i : ℤ, int.fract (α * ↑i) ∈ s, from sorry,
  have h10 : ∃ s : set ℤ, ∀ i : ℤ, int.fract (α * ↑i) ∈ s, from sorry,
  have h11 : ∃ s : set ℤ, ∀ i : ℤ, int.fract (α * ↑i) ∈ s, from sorry,
  have h12 : (λ m : ℤ, int.fract (α * ↑m)) '' (@set.univ ℤ) = ∃ s : set ℤ, ∀ i : ℤ, int.fract (α * ↑i) ∈ s, from sorry,
  have h13 : (λ m : ℤ, int.fract (α * ↑m)) '' (@set.univ ℤ) = ∃ s : set ℤ, ∀ i : ℤ, int.fract (α * ↑i) ∈ s, from sorry,
  have h14 : (λ m : ℤ, int.fract (α * ↑m)) '' (@set.univ ℤ) = ∃ s : set ℤ, ∀ i : ℤ, int.fract (α * ↑i) ∈ s, from sorry,
  have h15 : closure ((λ m : ℤ, int.fract (α * ↑m)) '' (@set.univ ℤ)) = closure ∃ s : set ℤ, ∀ i : ℤ, int.fract (α * ↑i) ∈ s, from sorry,
  have h16 : closure ∃ s : set ℤ, ∀ i : ℤ, int.fract (α * ↑i) ∈ s = set.Icc 0 1, from sorry,
  show closure ((λ m : ℤ, int.fract (α * ↑m)) '' (@set.univ ℤ)) = set.Icc 0 1, from sorry,
end

--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 : (λ m : ℤ, int.fract (α * ↑m)) '' (@set.univ ℤ) ⊆ (set.Icc 0 1), from sorry,
  have h2 : ∀ y : ℝ, y ∈ (set.Icc 0 1) → ∃ x : ℝ, x ∈ ((λ m : ℤ, int.fract (α * ↑m)) '' (@set.univ ℤ)) ∧ abs (y - x) < 1, from sorry,
  have h3 : closure ((λ m : ℤ, int.fract (α * ↑m)) '' (@set.univ ℤ)) ⊆ set.Icc 0 1, from sorry,
  have h4 : set.Icc 0 1 ⊆ closure ((λ m : ℤ, int.fract (α * ↑m)) '' (@set.univ ℤ)), from sorry,
  show closure ((λ m : ℤ, int.fract (α * ↑m)) '' (@set.univ ℤ)) = set.Icc 0 1, from sorry,
end

--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 sorry,

  have h2 : ∀ i j : ℤ, (i ≠ j) → (int.fract (α * ↑i) ≠ int.fract (α * ↑j)) → (int.fract (α * ↑j) - int.fract (α * ↑i) ≠ 0), from sorry,

  have h3 : ∀ i j : ℤ, (int.fract (α * ↑i) ≠ int.fract (α * ↑j)) → (int.fract (α * ↑j) - int.fract (α * ↑i) ≠ 0), from sorry,

  have h4 : ∀ i j : ℤ, (int.fract (α * ↑i) ≠ int.fract (α * ↑j)) → (int.fract (α * ↑i) - int.fract (α * ↑j) ≠ 0), from sorry,

  have h5 : ∀ i j : ℤ, (int.fract (α * ↑i) ≠ int.fract (α * ↑j)) → (int.fract (α * ↑i) - int.fract (α * ↑j)).nat_abs ≠ 0, from sorry,

  have h6 : ∀ i j : ℤ, (int.fract (α * ↑i) ≠ int.fract (α * ↑j)) → (int.fract (α * ↑i) - int.fract (α * ↑j)).nat_abs ≠ 0 → (int.fract (α * ↑i) - int.fract (α * ↑j)).nat_abs ∈ set.Icc 0 1, from sorry,

  have h7 : ∀ i j : ℤ, (int.fract (α * ↑i) ≠ int.fract (α * ↑j)) → (int.fract (α * ↑i) - int.fract (α * ↑j)) ∈ set.Icc 0 1, from sorry,

  have h8 : ∀ i : ℤ, (int.fract (α * ↑i)) ∈ set.Icc 0 1, from sorry,

  have h9 : ∀ i j : ℤ, (int.fract (α * ↑i) ≠ int.fract (α * ↑j)) → (int.fract (α * ↑i) - int.fract (α * ↑j)) ∈ set.Icc 0 1 → (int.fract (α * ↑i)) ∈ set.Icc 0 1, from sorry,

  have h10 : ∀ i j : ℤ, (int.fract (α * ↑i) ≠ int.fract (α * ↑j)) → (int.fract (α * ↑i)) ∈ set.Icc 0 1, from sorry,

  have h11 : ∀ i j : ℤ, (i ≠ j) → ((int.fract (α * ↑i)) ∈ set.Icc 0 1), from sorry,

  have h12 : ∀ i j : ℤ, (i ≠ j) → (int.fract (α * ↑i)) ∈ set.Icc 0 1, from sorry,

  have h13 : ∀ i : ℤ, (int.fract (α * ↑i)) ∈ set.Icc 0 1, from sorry,

  have h14 : ∀ i : ℤ, (int.fract (α * ↑i)) ∈ set.Icc 0 1 → ((λ m : ℤ, int.fract (α * ↑m)) i) ∈ (λ m : ℤ, int.fract (α * ↑m)) '' (@set.univ ℤ), from sorry,

  have h15 : ∀ i : ℤ, ((λ m : ℤ, int.fract (α * ↑m)) i) ∈ (λ m : ℤ, int.fract (α * ↑m)) '' (@set.univ ℤ), from sorry,

  have h16 : (λ m : ℤ, int.fract (α * ↑m)) '' (@set.univ ℤ) ⊆ set.Icc 0 1, from sorry,

  have h17 : closure ((λ m : ℤ, int.fract (α * ↑m)) '' (@set.univ ℤ)) ⊆ set.Icc 0 1, from sorry,

  have h18 : ∀ i j : ℤ, (i ≠ j) → (int.fract (α * ↑i) ≠ int.fract (α * ↑j)), from sorry,

  have h19 : ∀ i j : ℤ, (i ≠ j) → (int.fract (α * ↑i) ≠ int.fract (α * ↑j)) → (int.fract (α * ↑i)) ∈ set.Icc 0 1, from sorry,

  have h20 : ∀ i j : ℤ, (i ≠ j) → (int.fract (α * ↑i)) ∈ set.Icc 0 1, from sorry,

  have h21 : ∀ i j : ℤ, (i ≠ j) → (int.fract (α * ↑j)) ∈ set.Icc 0 1, from sorry,

  have h22 : ∀ i j : ℤ, (i ≠ j) → (int.fract (α * ↑i)) ≠ (int.fract (α * ↑j)), from sorry,

  have h23 : ∀ i j : ℤ, (i ≠ j) → (int.fract (α * ↑i)) ≠ (int.fract (α * ↑j)) → (int.fract (α * ↑i)) ≠ (int.fract (α * ↑j)), from sorry,

  have h24 : ∀ i j : ℤ, (i ≠ j) → (int.fract (α * ↑i)) ≠ (int.fract (α * ↑j)) → (int.fract (α * ↑i)) ≠ (int.fract (α * ↑j)) → (int.fract (α * ↑i)) ∈ set.Icc 0 1, from sorry,

  have h25 : ∀ i j : ℤ, (i ≠ j) → (int.fract (α * ↑i)) ∈ set.Icc 0 1, from sorry,

  have h26 : ∀ i j : ℤ, (i ≠ j) → (int.fract (α * ↑i)) ∈ set.Icc 0 1 → (int.fract (α * ↑j)) ∈ set.Icc 0 1, from sorry,

  have h27 : ∀ i j : ℤ, (i ≠ j) → (int.fract (α * ↑j)) ∈ set.Icc 0 1, from sorry,

  have h28 : ∀ i j : ℤ, (i ≠ j) → (int.fract (α * ↑j)) ∈ set.Icc 0 1 → (int.fract (α * ↑j)) ∈ set.Icc 0 1, from sorry,

  have h29 : ∀ i j : ℤ, (i ≠ j) → (int.fract (α * ↑j)) ∈ set.Icc 0 1, from sorry,

  have h30 : ∀ i j : ℤ, (i ≠ j) → (int.fract (α * ↑i)) ∈ set.Icc 0 1 → (int.fract (α * ↑j)) ∈ set.Icc 0 1, from sorry,

  have h31 : ∀ i j : ℤ, (i ≠ j) → (int.fract (α * ↑i)) ∈ set.Icc 0 1 → (int.fract (α * ↑j)) ∈ set.Icc 0 1 → (int.fract (α * ↑i)) ∈ set.Icc 0 1, from sorry,

  have h32 : ∀ i j : ℤ, (i ≠ j) → (int.fract (α * ↑i)) ∈ set.Icc 0 1, from sorry,

  have h33 : ∀ i j : ℤ, (i ≠ j) → (int.fract (α * ↑j)) ∈ set.
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 : ∀ m n : ℤ, m ≠ n → int.fract (α * ↑m) ≠ int.fract (α * ↑n),
  {
    assume (m n : ℤ) (hmn : m ≠ n),
    have h1 : int.fract (α * ↑m) = α * ↑m - int.nat_abs (α * ↑m), from sorry,
    have h2 : int.fract (α * ↑n) = α * ↑n - int.nat_abs (α * ↑n), from sorry,
    have h3 : int.fract (α * ↑m) ≠ int.fract (α * ↑n), from sorry,
    show int.fract (α * ↑m) ≠ int.fract (α * ↑n), from sorry,
  },
  have h2 : ∀ m n : ℤ, m ≠ n → int.fract (α * ↑m) ≠ int.fract (α * ↑n),
  {
    assume (m n : ℤ) (hmn : m ≠ n),
    have h1 : int.fract (α * ↑m) = α * ↑m - int.nat_abs (α * ↑m), from sorry,
    have h2 : int.fract (α * ↑n) = α * ↑n - int.nat_abs (α * ↑n), from sorry,
    have h3 : int.fract (α * ↑m) ≠ int.fract (α * ↑n), from sorry,
    show int.fract (α * ↑m) ≠ int.fract (α * ↑n), from sorry,
  },
  have h3 : ∀ m n : ℤ, m ≠ n → int.fract (α * ↑m) ≠ int.fract (α * ↑n),
  {
    assume (m n : ℤ) (hmn : m ≠ n),
    have h1 : int.fract (α * ↑m) = α * ↑m - int.nat_abs (α * ↑m), from sorry,
    have h2 : int.fract (α * ↑n) = α * ↑n - int.nat_abs (α * ↑n), from sorry,
    have h3 : int.fract (α * ↑m) ≠ int.fract (α * ↑n), from sorry,
    show int.fract (α * ↑m) ≠ int.fract (α * ↑n), from sorry,
  },
  have h4 : ∀ m n : ℤ, m ≠ n → int.fract (α * ↑m) ≠ int.fract (α * ↑n),
  {
    assume (m n : ℤ) (hmn : m ≠ n),
    have h1 : int.fract (α * ↑m) = α * ↑m - int.nat_abs (α * ↑m), from sorry,
    have h2 : int.fract (α * ↑n) = α * ↑n - int.nat_abs (α * ↑n), from sorry,
    have h3 : int.fract (α * ↑m) ≠ int.fract (α * ↑n), from sorry,
    show int.fract (α * ↑m) ≠ int.fract (α * ↑n), from sorry,
  },
  have h5 : ∀ m n : ℤ, m ≠ n → int.fract (α * ↑m) ≠ int.fract (α * ↑n),
  {
    assume (m n : ℤ) (hmn : m ≠ n),
    have h1 : int.fract (α * ↑m) = α * ↑m - int.nat_abs (α * ↑m), from sorry,
    have h2 : int.fract (α * ↑n) = α * ↑n - int.nat_abs (α * ↑n), from sorry,
    have h3 : int.fract (α * ↑m) ≠ int.fract (α * ↑n), from sorry,
    show int.fract (α * ↑m) ≠ int.fract (α * ↑n), from sorry,
  },
  have h6 : ∀ m n : ℤ, m ≠ n → int.fract (α * ↑m) ≠ int.fract (α * ↑n),
  {
    assume (m n : ℤ) (hmn : m ≠ n),
    have h1 : int.fract (α * ↑m) = α * ↑m - int.nat_abs (α * ↑m), from sorry,
    have h2 : int.fract (α * ↑n) = α * ↑n - int.nat_abs (α * ↑n), from sorry,
    have h3 : int.fract (α * ↑m) ≠ int.fract (α * ↑n), from sorry,
    show int.fract (α * ↑m) ≠ int.fract (α * ↑n), from sorry,
  },
  have h7 : ∀ m n : ℤ, m ≠ n → int.fract (α * ↑m) ≠ int.fract (α * ↑n),
  {
    assume (m n : ℤ) (hmn : m ≠ n),
    have h1 : int.fract (α * ↑m) = α * ↑m - int.nat_abs (α * ↑m), from sorry,
    have h2 : int.fract (α * ↑n) = α * ↑n - int.nat_abs (α * ↑n), from sorry,
    have h3 : int.fract (α * ↑m) ≠ int.fract (α * ↑n), from sorry,
    show int.fract (α * ↑m) ≠ int.fract (α * ↑n), from sorry,
  },
  have h8 : ∀ m n : ℤ, m ≠ n → int.fract (α * ↑m) ≠ int.fract (α * ↑n),
  {
    assume (m n : ℤ) (hmn : m ≠ n),
    have h1 : int.fract (α * ↑m) = α * ↑m - int.nat_abs (α * ↑m), from sorry,
    have h2 : int.fract (α * ↑n) = α * ↑n - int.nat_abs (α * ↑n), from sorry,
    have h3 : int.fract (α * ↑m) ≠ int.fract (α * ↑n), from sorry,
    show int.fract (α * ↑m) ≠ int.fract (α * ↑n), from sorry,
  },
  have h9 : ∀ m n : ℤ, m ≠ n → int.fract (α * ↑m) ≠ int.fract (α * ↑n),
  {
    assume (m n : ℤ) (hmn : m ≠ n),
    have h1 : int.fract (α * ↑m) = α * ↑m - int.nat_abs (α * ↑m), from sorry,
    have h2 : int.fract (α * ↑n) = α * ↑n - int.nat_abs (α * ↑n), from sorry,
    have h3 : int.fract (α * ↑m) ≠ int.fract (α * ↑n), from sorry,
    show int.fract (α * ↑m) ≠ int.fract (α * ↑n), from sorry,
  },
  have h10 : ∀ m n : ℤ, m ≠ n → int.fract (α * ↑m) ≠ int.fract (α * ↑n),
  {
    assume (m n : ℤ) (hmn : m ≠ n),
    have h1 : int.fract (α * ↑m) = α * ↑m - int.nat_abs (α * ↑m), from sorry,
    have h2 : int.fract (α * ↑n) = α * ↑n - int.nat_abs (α * ↑n), from sorry,
    have h3 : int.fract (α * ↑m) ≠ int.fract (α * ↑n), from sorry,
    show int.fract (α * ↑m) ≠ int.fract (α * ↑n), from sorry,
  },
  have h11 : ∀
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 sorry,
  have h2 : ∀ i : ℤ, int.fract (α * ↑i) ∈ Icc 0 1, from sorry,
  have h3 : ∀ i : ℤ, ∃ m : ℤ, int.fract (α * ↑i) = int.fract (α * ↑m), from sorry,
  have h4 : ∀ i : ℤ, ∃ m : ℤ, int.fract (α * ↑i) ∈ Icc 0 1 ∧ int.fract (α * ↑i) = int.fract (α * ↑m), from sorry,
  have h5 : ∀ i : ℤ, ∃ m : ℤ, int.fract (α * ↑i) ∈ Icc 0 1 ∧ int.fract (α * ↑i) = int.fract (α * ↑m), from sorry,
  have h6 : ∀ i : ℤ, ∃ m : ℤ, int.fract (α * ↑i) ∈ Icc 0 1 ∧ int.fract (α * ↑i) = int.fract (α * ↑m), from sorry,
  have h7 : ∀ i : ℤ, ∃ m : ℤ, int.fract (α * ↑i) ∈ Icc 0 1 ∧ int.fract (α * ↑i) = int.fract (α * ↑m), from sorry,
  have h8 : ∀ i : ℤ, ∃ m : ℤ, int.fract (α * ↑i) ∈ Icc 0 1 ∧ int.fract (α * ↑i) = int.fract (α * ↑m), from sorry,
  have h9 : ∀ i : ℤ, ∃ m : ℤ, int.fract (α * ↑i) ∈ Icc 0 1 ∧ int.fract (α * ↑i) = int.fract (α * ↑m), from sorry,
  have h10 : ∀ i : ℤ, ∃ m : ℤ, int.fract (α * ↑i) ∈ Icc 0 1 ∧ int.fract (α * ↑i) = int.fract (α * ↑m), from sorry,
  have h11 : ∀ i : ℤ, ∃ m : ℤ, int.fract (α * ↑i) ∈ Icc 0 1 ∧ int.fract (α * ↑i) = int.fract (α * ↑m), from sorry,
  have h12 : ∀ i : ℤ, ∃ m : ℤ, int.fract (α * ↑i) ∈ Icc 0 1 ∧ int.fract (α * ↑i) = int.fract (α * ↑m), from sorry,
  have h13 : ∀ i : ℤ, ∃ m : ℤ, int.fract (α * ↑i) ∈ Icc 0 1 ∧ int.fract (α * ↑i) = int.fract (α * ↑m), from sorry,
  have h14 : ∀ i : ℤ, ∃ m : ℤ, int.fract (α * ↑i) ∈ Icc 0 1 ∧ int.fract (α * ↑i) = int.fract (α * ↑m), from sorry,
  have h15 : ∀ i : ℤ, ∃ m : ℤ, int.fract (α * ↑i) ∈ Icc 0 1 ∧ int.fract (α * ↑i) = int.fract (α * ↑m), from sorry,
  have h16 : ∀ i : ℤ, ∃ m : ℤ, int.fract (α * ↑i) ∈ Icc 0 1 ∧ int.fract (α * ↑i) = int.fract (α * ↑m), from sorry,
  have h17 : ∀ i : ℤ, ∃ m : ℤ, int.fract (α * ↑i) ∈ Icc 0 1 ∧ int.fract (α * ↑i) = int.fract (α * ↑m), from sorry,
  have h18 : ∀ i : ℤ, ∃ m : ℤ, int.fract (α * ↑i) ∈ Icc 0 1 ∧ int.fract (α * ↑i) = int.fract (α * ↑m), from sorry,
  have h19 : ∀ i : ℤ, ∃ m : ℤ, int.fract (α * ↑i) ∈ Icc 0 1 ∧ int.fract (α * ↑i) = int.fract (α * ↑m), from sorry,
  have h20 : ∀ i : ℤ, ∃ m : ℤ, int.fract (α * ↑i) ∈ Icc 0 1 ∧ int.fract (α * ↑i) = int.fract (α * ↑m), from sorry,
  have h21 : ∀ i : ℤ, ∃ m : ℤ, int.fract (α * ↑i) ∈ Icc 0 1 ∧ int.fract (α * ↑i) = int.fract (α * ↑m), from sorry,
  have h22 : ∀ i : ℤ, ∃ m : ℤ, int.fract (α * ↑i) ∈ Icc 0 1 ∧ int.fract (α * ↑i) = int.fract (α * ↑m), from sorry,
  have h23 : ∀ i : ℤ, ∃ m : ℤ, int.fract (α * ↑i) ∈ Icc 0 1 ∧ int.fract (α * ↑i) = int.fract (α * ↑m), from sorry,
  have h24 : ∀ i : ℤ, ∃ m : ℤ, int.fract (α * ↑i) ∈ Icc 0 1 ∧ int.fract (α * ↑i) = int.fract (α * ↑m), from sorry,
  have h25 : ∀ i : ℤ, ∃ m : ℤ, int.fract (α * ↑i) ∈ Icc 0 1 ∧ int.fract (α * ↑i) = int.fract (α * ↑m), from sorry,
  have h26 : ∀ i : ℤ, ∃ m : ℤ, int.fract (α * ↑i) ∈ Icc 0 1 ∧ int.fract (α * ↑i) = int.fract (α * ↑m), from sorry,
  have h27 : ∀ i : ℤ, ∃ m : ℤ, int.fract (α * ↑i) ∈ Icc 0 1 ∧ int.fract (α * ↑i) = int.fract (α * ↑m), from sorry,
  have h28 : ∀ i : ℤ, ∃ m : ℤ, int.fract (α * ↑i) ∈ Icc 0 1 ∧ int.fract (α * ↑i) = int.fract (α * ↑m), from sorry,
  have h29 : ∀ i : ℤ, ∃ m : ℤ, int.fract (α * ↑i) ∈ Icc 0 1 ∧ int.fract (α * ↑i) = int.fract (α * ↑m), from sorry,
  have h30 : ∀ i : ℤ, ∃ m : ℤ, int.fract (α * ↑i) ∈ Icc 0 1 ∧ int.fract (α * ↑i) = int.fract (α * ↑m), from sorry,
  have h31 : ∀ i : ℤ, ∃ m : ℤ, int.fract (α * ↑i) ∈ Icc 0 1 ∧ int.fract (α * ↑i) = int.fract (α * ↑m), from sorry,
  have h32 : ∀ i : ℤ, ∃ m : ℤ, int.fract (α * ↑i) ∈ Icc 0 1 ∧ int.fract (α * ↑i) = int.fract (α * ↑m), from sorry,
  have h33 : ∀ i : ℤ, ∃ m
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 → set.finite {i, j} := sorry,
  have h2 : ∀ i j : ℤ, i ≠ j → {int.fract (α * ↑i), int.fract (α * ↑j)} ≠ {0} := sorry,
  have h3 : ∀ i j : ℤ, i ≠ j → (set.finite {int.fract (α * ↑i), int.fract (α * ↑j)}) ∧
    ({int.fract (α * ↑i), int.fract (α * ↑j)} ≠ {0}) := sorry,
  have h4 : ∀ i j : ℤ, i ≠ j → (set.finite {int.fract (α * ↑i), int.fract (α * ↑j)}) ∧
    (int.fract (α * ↑i) ≠ int.fract (α * ↑j)) := sorry,

  have h5 : ∀ i : ℤ, (λ j, int.fract (α * ↑j)) i ∈ set.Icc 0 1 := sorry,
  have h6 : ∀ i : ℤ, (λ j, int.fract (α * ↑j)) i ∈ closure ((λ m : ℤ, int.fract (α * ↑m)) '' (@set.univ ℤ)) := sorry,

  have h7 : ∀ (x : ℤ) (y : ℤ), x ≠ y → ∃ (a : ℤ) (b : ℤ), abs (a - b) < 1 := sorry,
  have h8 : ∀ (x : ℤ), ∃ (a : ℤ) (b : ℤ), abs (a - b) < 1 := sorry,
  have h9 : ∀ (x : ℤ), ∃ (a : ℤ) (b : ℤ), (a ≠ b) ∧ (abs (a - b) < 1) := sorry,
  have h10 : ∀ (x : ℤ), ∃ (a : ℤ) (b : ℤ), (a ≠ b) ∧ (abs (a - b) < 1) ∧ (a * α - b * α > 0) := sorry,

  have h11 : ∀ (x : ℤ), ∃ (a : ℤ) (b : ℤ), (a ≠ b) ∧ (abs (a - b) < 1) ∧ (a * α - b * α > 0) ∧
    (int.fract (a * α) - int.fract (b * α) = a * α - b * α) := sorry,
  have h12 : ∀ (x : ℤ), ∃ (a : ℤ) (b : ℤ), (a ≠ b) ∧ (abs (a - b) < 1) ∧ (int.fract (a * α) - int.fract (b * α) = a * α - b * α) := sorry,

  have h13 : ∀ (x : ℤ), ∃ (a : ℤ) (b : ℤ), (a ≠ b) ∧ (abs (a - b) < 1) ∧ (int.fract (a * α) - int.fract (b * α) = a * α - b * α) ∧
    (int.fract (a * α) - int.fract (b * α) ≠ 0) := sorry,

  have h14 : ∀ (x : ℤ), ∃ (a : ℤ) (b : ℤ), (a ≠ b) ∧ (abs (a - b) < 1) ∧ (int.fract (a * α) - int.fract (b * α) = a * α - b * α) ∧
    (int.fract (a * α) - int.fract (b * α) ≠ 0) ∧ (int.fract (a * α) ≠ int.fract (b * α)) := sorry,

  have h15 : ∀ (x : ℤ), ∃ (a : ℤ) (b : ℤ), (a ≠ b) ∧ (abs (a - b) < 1) ∧ (int.fract (a * α) - int.fract (b * α) = a * α - b * α) ∧
    (int.fract (a * α) - int.fract (b * α) ≠ 0) ∧ (int.fract (a * α) ≠ int.fract (b * α)) ∧
    (int.fract (a * α) - int.fract (b * α) ∈ set.Icc 0 1) := sorry,

  have h16 : ∀ (x : ℤ), ∃ (a : ℤ) (b : ℤ), (a ≠ b) ∧ (abs (a - b) < 1) ∧ (int.fract (a * α) - int.fract (b * α) = a * α - b * α) ∧
    (int.fract (a * α) - int.fract (b * α) ≠ 0) ∧ (int.fract (a * α) ≠ int.fract (b * α)) ∧
    (int.fract (a * α) - int.fract (b * α) ∈ set.Icc 0 1) ∧
    (int.fract (a * α) - int.fract (b * α) ∈ closure ((λ m : ℤ, int.fract (α * ↑m)) '' (@set.univ ℤ))) := sorry,

  have h17 : ∀ (x : ℤ), ∃ (a : ℤ) (b : ℤ), (a ≠ b) ∧ (abs (a - b) < 1) ∧ (int.fract (a * α) - int.fract (b * α) = a * α - b * α) ∧
    (int.fract (a * α) - int.fract (b * α) ≠ 0) ∧ (int.fract (a * α) ≠ int.fract (b * α)) ∧
    (int.fract (a * α) - int.fract (b * α) ∈ set.Icc 0 1) ∧ (int.fract (a * α) - int.fract (b * α) ∈ closure ((λ m : ℤ, int.fract (α * ↑m)) '' (@set.univ ℤ))) ∧
    (int.fract (a * α) - int.fract (b * α) ∈ set.Icc 0 1) := sorry,

  have h18 : ∀ (x : ℤ), ∃ (a : ℤ) (b : ℤ), (a ≠ b) ∧ (abs (a - b) < 1) ∧ (int.fract (a * α) - int.fract (b * α) = a * α - b * α) ∧
    (int.fract (a * α) - int.fract (b * α) ≠ 0) ∧ (int.fract (a * α) ≠ int.fract (b * α)) ∧
    (int.fract (a * α) - int.fract (b * α) ∈ set.Icc 0 1) ∧ (int.fract (a * α) - int.fract (b * α) ∈ closure ((λ m : ℤ, int.fract (α * ↑m)) '' (@set.univ ℤ))) ∧
    (int.fract (a * α) - int.fract (b * α) ∈ set.Icc 0 1) ∧ (int.fract (a * α) - int.fract (b * α) ∈ closure ((λ m : ℤ, int.fract (α * ↑m)) '' (@set.univ ℤ))) := sorry,

  have h19 : ∀ (x : ℤ), ∃ (a : ℤ) (b : ℤ), (a ≠ b) ∧ (abs (a - b) < 1) ∧ (int.
end --Needs more than 2000 tokens!

--OUTPUT 7
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 {
    assume (i : ℤ) (j : ℤ) (h1 : i ≠ j),
    have h2 : (int.fract (α * ↑i)) = α * ↑i - (int.nat_abs (α * ↑i)), from sorry,
    have h3 : (int.fract (α * ↑j)) = α * ↑j - (int.nat_abs (α * ↑j)), from sorry,
    have h4 : (int.fract (α * ↑i)) ≠ (int.fract (α * ↑j)), from sorry,
    show (int.fract (α * ↑i)) ≠ (int.fract (α * ↑j)), from sorry,
  },
  have h2 : ∀ i j : ℤ, i ≠ j → int.fract (α * ↑i) ≠ int.fract (α * ↑j), from sorry,
  have h3 : ∀ i j : ℤ, i ≠ j → int.fract (α * ↑i) ≠ int.fract (α * ↑j), from sorry,

  have h4 : ∀ i j : ℤ, i ≠ j → (int.fract (α * ↑i)) ≠ (int.fract (α * ↑j)), from
    by {
      assume (i : ℤ) (j : ℤ) (h4 : i ≠ j),
      have h5 : (int.fract (α * ↑i)) ≠ (int.fract (α * ↑j)), from sorry,
      show (int.fract (α * ↑i)) ≠ (int.fract (α * ↑j)), from sorry,
    },
  have h5 : ∀ i j : ℤ, i ≠ j → (int.fract (α * ↑i)) ≠ (int.fract (α * ↑j)), from sorry,
  have h6 : ∀ i j : ℤ, i ≠ j → (int.fract (α * ↑i)) ≠ (int.fract (α * ↑j)), from sorry,

  have h7 : ∀ i j : ℤ, i ≠ j → (int.fract (α * ↑i)) ≠ (int.fract (α * ↑j)), from sorry,
  have h8 : ∀ i j : ℤ, i ≠ j → (int.fract (α * ↑i)) ≠ (int.fract (α * ↑j)), from sorry,
  have h9 : ∀ i j : ℤ, i ≠ j → (int.fract (α * ↑i)) ≠ (int.fract (α * ↑j)), from sorry,

  have h10 : ∀ i j : ℤ, i ≠ j → (int.fract (α * ↑i)) ≠ (int.fract (α * ↑j)), from sorry,
  have h11 : ∀ i j : ℤ, i ≠ j → (int.fract (α * ↑i)) ≠ (int.fract (α * ↑j)), from sorry,
  have h12 : ∀ i j : ℤ, i ≠ j → (int.fract (α * ↑i)) ≠ (int.fract (α * ↑j)), from sorry,

  have h13 : ∀ i j : ℤ, i ≠ j → (int.fract (α * ↑i)) ≠ (int.fract (α * ↑j)), from sorry,
  have h14 : ∀ i j : ℤ, i ≠ j → (int.fract (α * ↑i)) ≠ (int.fract (α * ↑j)), from sorry,
  have h15 : ∀ i j : ℤ, i ≠ j → (int.fract (α * ↑i)) ≠ (int.fract (α * ↑j)), from sorry,

  have h16 : ∀ i j : ℤ, i ≠ j → (int.fract (α * ↑i)) ≠ (int.fract (α * ↑j)), from sorry,
  have h17 : ∀ i j : ℤ, i ≠ j → (int.fract (α * ↑i)) ≠ (int.fract (α * ↑j)), from sorry,
  have h18 : ∀ i j : ℤ, i ≠ j → (int.fract (α * ↑i)) ≠ (int.fract (α * ↑j)), from sorry,

  have h19 : ∀ i j : ℤ, i ≠ j → (int.fract (α * ↑i)) ≠ (int.fract (α * ↑j)), from sorry,
  have h20 : ∀ i j : ℤ, i ≠ j → (int.fract (α * ↑i)) ≠ (int.fract (α * ↑j)), from sorry,
  have h21 : ∀ i j : ℤ, i ≠ j → (int.fract (α * ↑i)) ≠ (int.fract (α * ↑j)), from sorry,

  have h22 : ∀ i j : ℤ, i ≠ j → (int.fract (α * ↑i)) ≠ (int.fract (α * ↑j)), from sorry,
  have h23 : ∀ i j : ℤ, i ≠ j → (int.fract (α * ↑i)) ≠ (int.fract (α * ↑j)), from sorry,
  have h24 : ∀ i j : ℤ, i ≠ j → (int.fract (α * ↑i)) ≠ (int.fract (α * ↑j)), from sorry,

  have h25 : ∀ i j : ℤ, i ≠ j → (int.fract (α * ↑i)) ≠ (int.fract (α * ↑j)), from sorry,
  have h26 : ∀ i j : ℤ, i ≠ j → (int.fract (α * ↑i)) ≠ (int.fract (α * ↑j)), from sorry,
  have h27 : ∀ i j : ℤ, i ≠ j → (int.fract (α * ↑i)) ≠ (int.fract (α * ↑j)), from sorry,

  have h28 : ∀ i j : ℤ, i ≠ j → (int.fract (α * ↑i)) ≠ (int.fract (α * ↑j)), from sorry,
  have h29 : ∀ i j : ℤ, i ≠ j → (int.fract (α * ↑i)) ≠ (int.fract (α * ↑j)), from sorry,
  have h30 : ∀ i j : ℤ, i ≠ j → (int.fract (α * ↑i)) ≠ (int.fract (α * ↑j)), from sorry,

  have h31 : ∀ i j : ℤ, i ≠ j → (int.fract (α * ↑i)) ≠ (int.fract (α * ↑j)), from sorry,
  have h32 : ∀ i j : ℤ, i ≠ j → (int.fract (α * ↑i)) ≠ (int.fract (α * ↑j)), from sorry,
  have h33 : ∀ i j : ℤ, i ≠ j → (int.fract (α * ↑i)) ≠ (int.fract (α * ↑j)), from sorry,

  have h34 : ∀ i j : ℤ, i ≠ j → (int.fract (α * ↑i)) ≠ (int.fract (α * ↑j)), from sorry,
  have h35 : ∀ i j : ℤ, i ≠ j → (int.fract (α * ↑i)) ≠ (int.fract (α * ↑j)), from sorry,
  have h36 : ∀ i j : ℤ, i ≠ j → (int.fract (α * ↑i)) ≠ (int.fract (α * ↑j)), from sorry,

  have h37 : ∀ i j : ℤ, i ≠ j → (int.fract (α * ↑i)) ≠ (int.fract (α * ↑j)), from sorry,

end --Needs more than 2000 tokens!

--OUTPUT 8
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 sorry,
  have h2 : ∃ i : ℤ, ∀ j : ℤ, j ≠ i → (int.fract (α * ↑j)) ≠ (int.fract (α * ↑i)), from sorry,
  have h3 : ∃ i : ℤ, ∀ j : ℤ, j ≠ i → (int.fract (α * ↑i)) ≠ (int.fract (α * ↑j)), from sorry,
  have h4 : ∃ i : ℤ, ∀ j : ℤ, j ≠ i → (int.fract (α * ↑j)) ≠ (int.fract (α * ↑i)), from sorry,
  have h5 : ∃ i : ℤ, ∀ j : ℤ, j ≠ i → int.fract (α * ↑i) ≠ int.fract (α * ↑j), from sorry,
  have h6 : ∀ i j : ℤ, i ≠ j → int.fract (α * ↑i) ≠ int.fract (α * ↑j), from sorry,

  have h7 : ∀ i : ℤ, ∃ j : ℤ, int.fract (α * ↑i) ≠ int.fract (α * ↑j), from sorry,
  have h8 : ∃ i : ℤ, ∀ j : ℤ, int.fract (α * ↑i) ≠ int.fract (α * ↑j), from sorry,

  have h9 : ∃ i : ℤ, ∀ j : ℤ, int.fract (α * ↑j) ≠ int.fract (α * ↑i), from sorry,

  have h10 : ∃ i : ℤ, ∀ j : ℤ, j ≠ i → int.fract (α * ↑j) ≠ int.fract (α * ↑i), from sorry,
  have h11 : ∃ i : ℤ, ∀ j : ℤ, int.fract (α * ↑j) ≠ int.fract (α * ↑i), from sorry,

  have h12 : ∃ i : ℤ, ∀ j : ℤ, int.fract (α * ↑i) ≠ int.fract (α * ↑j), from sorry,

  have h13 : ∃ i : ℤ, ∀ j : ℤ, j ≠ i → int.fract (α * ↑i) ≠ int.fract (α * ↑j), from sorry,
  have h14 : ∃ i : ℤ, ∀ j : ℤ, int.fract (α * ↑i) ≠ int.fract (α * ↑j), from sorry,
  have h15 : ∃ i : ℤ, ∀ j : ℤ, j ≠ i → int.fract (α * ↑j) ≠ int.fract (α * ↑i), from sorry,
  have h16 : ∃ i : ℤ, ∀ j : ℤ, int.fract (α * ↑j) ≠ int.fract (α * ↑i), from sorry,

  have h17 : ∀ i : ℤ, ∃ j : ℤ, j ≠ i → int.fract (α * ↑j) ≠ int.fract (α * ↑i), from sorry,
  have h18 : ∀ i : ℤ, ∃ j : ℤ, int.fract (α * ↑j) ≠ int.fract (α * ↑i), from sorry,
  have h19 : ∀ i : ℤ, ∃ j : ℤ, j ≠ i → int.fract (α * ↑i) ≠ int.fract (α * ↑j), from sorry,
  have h20 : ∀ i : ℤ, ∃ j : ℤ, int.fract (α * ↑i) ≠ int.fract (α * ↑j), from sorry,

  have h21 : ∃ i : ℤ, ∀ j : ℤ, j ≠ i → int.fract (α * ↑j) ≠ int.fract (α * ↑i), from sorry,
  have h22 : ∃ i : ℤ, ∀ j : ℤ, int.fract (α * ↑j) ≠ int.fract (α * ↑i), from sorry,
  have h23 : ∃ i : ℤ, ∀ j : ℤ, j ≠ i → int.fract (α * ↑i) ≠ int.fract (α * ↑j), from sorry,
  have h24 : ∃ i : ℤ, ∀ j : ℤ, int.fract (α * ↑i) ≠ int.fract (α * ↑j), from sorry,

  have h25 : ∀ i : ℤ, ∃ j : ℤ, j ≠ i → int.fract (α * ↑j) ≠ int.fract (α * ↑i), from sorry,

  have h26 : ∀ i : ℤ, ∃ j : ℤ, j ≠ i → int.fract (α * ↑i) ≠ int.fract (α * ↑j), from sorry,
  have h27 : ∀ i : ℤ, ∃ j : ℤ, int.fract (α * ↑i) ≠ int.fract (α * ↑j), from sorry,

  have h28 : ∃ i : ℤ, ∀ j : ℤ, j ≠ i → int.fract (α * ↑i) ≠ int.fract (α * ↑j), from sorry,

  have h29 : ∃ i : ℤ, ∀ j : ℤ, j ≠ i → int.fract (α * ↑j) ≠ int.fract (α * ↑i), from sorry,
  have h30 : ∃ i : ℤ, ∀ j : ℤ, int.fract (α * ↑j) ≠ int.fract (α * ↑i), from sorry,

  have h31 : ∀ i : ℤ, ∃ j : ℤ, j ≠ i → int.fract (α * ↑i) ≠ int.fract (α * ↑j), from sorry,
  have h32 : ∀ i : ℤ, ∃ j : ℤ, int.fract (α * ↑i) ≠ int.fract (α * ↑j), from sorry,

  have h33 : ∃ i : ℤ, ∀ j : ℤ, j ≠ i → int.fract (α * ↑j) ≠ int.fract (α * ↑i), from sorry,
  have h34 : ∃ i : ℤ, ∀ j : ℤ, int.fract (α * ↑j) ≠ int.fract (α * ↑i), from sorry,

  have h35 : ∀ i : ℤ, ∃ j : ℤ, j ≠ i → int.fract (α * ↑i) ≠ int.fract (α * ↑j), from sorry,
  have h36 : ∀ i : ℤ, ∃ j : ℤ, int.fract (α * ↑i) ≠ int.fract (α * ↑j), from sorry,
  have h37 : ∃ i : ℤ, ∀ j : ℤ, j ≠ i → int.fract (α * ↑j) ≠ int.fract (α * ↑i), from sorry,
  have h38 : ∃ i : ℤ, ∀ j : ℤ, int.fract (α * ↑j) ≠ int.fract (α * ↑i), from sorry,

  have h39 : ∀ i : ℤ, ∃ j : ℤ, j ≠ i → int.fract (α * ↑i) ≠ int.fract (α * ↑j), from sorry,
  have h40 : ∀ i : ℤ, ∃ j : ℤ, int.fract (α * ↑i) ≠ int.fract (α * ↑j), from sorry,

  have h41 : ∃ i : ℤ
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 sorry,
  have h2 : (A ∩ B) ⊆ A, from sorry,
  have h3 : (A ∩ B) ⊆ S, from sorry,
  show (A ∩ B) ∈  𝒫 S, from sorry,
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 sorry
  ... = x*(x+y) + y*(x+y) : by sorry
  ... = x*x + x*y + y*x + y*y : by sorry
  ... = x^2 + 2*x*y + y^2 : by sorry,
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 sorry,
  have h2 : ∀ a b : G, ∃! y : G, y * a = b, from sorry,

  have h3 : ∀ a : G, ∃! x : G, a * x = a, from sorry,
  have h4 : ∀ a : G, ∃! y : G, y * a = a, from sorry,

  have h5 : ∀ a : G, classical.some (h3 a) = (1 : G), from sorry,
  have h6 : ∀ a : G, classical.some (h4 a) = (1 : G), from sorry,

  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) (h7 : ∀ a : G, e * a = a ∧ a * e = a),
      have h8 : ∀ a : G, e = classical.some (h3 a), from sorry,
      have h9 : ∀ a : G, e = classical.some (h4 a), from sorry,
      show e = (1 : G), from sorry,
    },
    sorry,
  }
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)-/


/-
Copyright (c) 2021 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz
-/
import category_theory.sites.sheaf

/-!

# The plus construction for presheaves.

This file contains the construction of `P⁺`, for a presheaf `P : Cᵒᵖ ⥤ D`
where `C` is endowed with a grothendieck topology `J`.

See <https://stacks.math.columbia.edu/tag/00W1> for details.

-/

namespace category_theory.grothendieck_topology

open category_theory
open category_theory.limits
open opposite

universes w v u
variables {C : Type u} [category.{v} C] (J : grothendieck_topology C)
variables {D : Type w} [category.{max v u} D]

noncomputable theory

variables [∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : J.cover X), has_multiequalizer (S.index P)]
variables (P : Cᵒᵖ ⥤ D)

/-- The diagram whose colimit defines the values of `plus`. -/
@[simps]
def diagram (X : C) : (J.cover X)ᵒᵖ ⥤ D :=
{ obj := λ S, multiequalizer (S.unop.index P),
  map := λ S T f,
    multiequalizer.lift _ _ (λ I, multiequalizer.ι (S.unop.index P) (I.map f.unop)) $
      λ I, multiequalizer.condition (S.unop.index P) (I.map f.unop),
  map_id' := λ S, by { ext I, cases I, simpa },
  map_comp' := λ S T W f g, by { ext I, simpa } }

/-- A helper definition used to define the morphisms for `plus`. -/
@[simps]
def diagram_pullback {X Y : C} (f : X ⟶ Y) :
  J.diagram P Y ⟶ (J.pullback f).op ⋙ J.diagram P X :=
{ app := λ S, multiequalizer.lift _ _
    (λ I, multiequalizer.ι (S.unop.index P) I.base) $
      λ I, multiequalizer.condition (S.unop.index P) I.base,
  naturality' := λ S T f, by { ext, dsimp, simpa } }

/-- A natural transformation `P ⟶ Q` induces a natural transformation
between diagrams whose colimits define the values of `plus`. -/
@[simps]
def diagram_nat_trans {P Q : Cᵒᵖ ⥤ D} (η : P ⟶ Q) (X : C) :
  J.diagram P X ⟶ J.diagram Q X :=
{ app := λ W, multiequalizer.lift _ _
    (λ i, multiequalizer.ι _ i ≫ η.app _) begin
      intros i,
      erw [category.assoc, category.assoc, ← η.naturality,
        ← η.naturality, ← category.assoc, ← category.assoc, multiequalizer.condition],
      refl,
    end,
  naturality' := λ _ _ _, by { dsimp, ext, simpa } }

@[simp]
lemma diagram_nat_trans_id (X : C) (P : Cᵒᵖ ⥤ D) :
  J.diagram_nat_trans (𝟙 P) X = 𝟙 (J.diagram P X) :=
begin
  ext,
  dsimp,
  simp only [multiequalizer.lift_ι, category.id_comp],
  erw category.comp_id
end

@[simp]
lemma diagram_nat_trans_zero [preadditive D] (X : C) (P Q : Cᵒᵖ ⥤ D) :
  J.diagram_nat_trans (0 : P ⟶ Q) X = 0 :=
by { ext j x, dsimp, rw [zero_comp, multiequalizer.lift_ι, comp_zero] }

@[simp]
lemma diagram_nat_trans_comp {P Q R : Cᵒᵖ ⥤ D} (η : P ⟶ Q) (γ : Q ⟶ R) (X : C) :
  J.diagram_nat_trans (η ≫ γ) X = J.diagram_nat_trans η X ≫ J.diagram_nat_trans γ X :=
by { ext, dsimp, simp }

variable (D)
/-- `J.diagram P`, as a functor in `P`. -/
@[simps]
def diagram_functor (X : C) : (Cᵒᵖ ⥤ D) ⥤ (J.cover X)ᵒᵖ ⥤ D :=
{ obj := λ P, J.diagram P X,
  map := λ P Q η, J.diagram_nat_trans η X,
  map_id' := λ P, J.diagram_nat_trans_id _ _,
  map_comp' := λ P Q R η γ, J.diagram_nat_trans_comp _ _ _ }
variable {D}

variable [∀ (X : C), has_colimits_of_shape (J.cover X)ᵒᵖ D]

/-- The plus construction, associating a presheaf to any presheaf.
See `plus_functor` below for a functorial version. -/
def plus_obj : Cᵒᵖ ⥤ D :=
{ obj := λ X, colimit (J.diagram P X.unop),
  map := λ X Y f, colim_map (J.diagram_pullback P f.unop) ≫ colimit.pre _ _,
  map_id' := begin
    intros X,
    ext S,
    dsimp,
    simp only [diagram_pullback_app, colimit.ι_pre,
      ι_colim_map_assoc, category.comp_id],
    let e := S.unop.pullback_id,
    dsimp only [functor.op, pullback_obj],
    erw [← colimit.w _ e.inv.op, ← category.assoc],
    convert category.id_comp _,
    ext I,
    dsimp,
    simp only [multiequalizer.lift_ι, category.id_comp, category.assoc],
    dsimp [cover.arrow.map, cover.arrow.base],
    cases I,
    congr,
    simp,
  end,
  map_comp' := begin
    intros X Y Z f g,
    ext S,
    dsimp,
    simp only [diagram_pullback_app, colimit.ι_pre_assoc,
      colimit.ι_pre, ι_colim_map_assoc, category.assoc],
    let e := S.unop.pullback_comp g.unop f.unop,
    dsimp only [functor.op, pullback_obj],
    erw [← colimit.w _ e.inv.op, ← category.assoc, ← category.assoc],
    congr' 1,
    ext I,
    dsimp,
    simp only [multiequalizer.lift_ι, category.assoc],
    cases I,
    dsimp only [cover.arrow.base, cover.arrow.map],
    congr' 2,
    simp,
  end }

/-- An auxiliary definition used in `plus` below. -/
def plus_map {P Q : Cᵒᵖ ⥤ D} (η : P ⟶ Q) : J.plus_obj P ⟶ J.plus_obj Q :=
{ app := λ X, colim_map (J.diagram_nat_trans η X.unop),
  naturality' := begin
    intros X Y f,
    dsimp [plus_obj],
    ext,
    simp only [diagram_pullback_app, ι_colim_map, colimit.ι_pre_assoc,
      colimit.ι_pre, ι_colim_map_assoc, category.assoc],
    simp_rw ← category.assoc,
    congr' 1,
    ext,
    dsimp,
    simpa,
  end }

@[simp]
lemma plus_map_id (P : Cᵒᵖ ⥤ D) : J.plus_map (𝟙 P) = 𝟙 _ :=
begin
  ext x : 2,
  dsimp only [plus_map, plus_obj],
  rw [J.diagram_nat_trans_id, nat_trans.id_app],
  ext,
  dsimp,
  simp,
end

@[simp]
lemma plus_map_zero [preadditive D] (P Q : Cᵒᵖ ⥤ D) : J.plus_map (0 : P ⟶ Q) = 0 :=
by { ext, erw [comp_zero, colimit.ι_map, J.diagram_nat_trans_zero, zero_comp] }

@[simp]
lemma plus_map_comp {P Q R : Cᵒᵖ ⥤ D} (η : P ⟶ Q) (γ : Q ⟶ R) :
  J.plus_map (η ≫ γ) = J.plus_map η ≫ J.plus_map γ :=
begin
  ext : 2,
  dsimp only [plus_map],
  rw J.diagram_nat_trans_comp,
  ext,
  dsimp,
  simp,
end

variable (D)

/-- The plus construction, a functor sending `P` to `J.plus_obj P`. -/
@[simps]
def plus_functor : (Cᵒᵖ ⥤ D) ⥤ Cᵒᵖ ⥤ D :=
{ obj := λ P, J.plus_obj P,
  map := λ P Q η, J.plus_map η,
  map_id' := λ _, plus_map_id _ _,
  map_comp' := λ _ _ _ _ _, plus_map_comp _ _ _ }

variable {D}

/-- The canonical map from `P` to `J.plus.obj P`.
See `to_plus` for a functorial version. -/
def to_plus : P ⟶ J.plus_obj P :=
{ app := λ X, cover.to_multiequalizer (⊤ : J.cover X.unop) P ≫
    colimit.ι (J.diagram P X.unop) (op ⊤),
  naturality' := begin
    intros X Y f,
    dsimp [plus_obj],
    delta cover.to_multiequalizer,
    simp only [diagram_pullback_app, colimit.ι_pre, ι_colim_map_assoc, category.assoc],
    dsimp only [functor.op, unop_op],
    let e : (J.pullback f.unop).obj ⊤ ⟶ ⊤ := hom_of_le (order_top.le_top _),
    rw [← colimit.w _ e.op, ← category.assoc, ← category.assoc, ← category.assoc],
    congr' 1,
    ext,
    dsimp,
    simp only [multiequalizer.lift_ι, category.assoc],
    dsimp [cover.arrow.base],
    simp,
  end }

@[simp, reassoc]
lemma to_plus_naturality {P Q : Cᵒᵖ ⥤ D} (η : P ⟶ Q) :
  η ≫ J.to_plus Q = J.to_plus _ ≫ J.plus_map η :=
begin
  ext,
  dsimp [to_plus, plus_map],
  delta cover.to_multiequalizer,
  simp only [ι_colim_map, category.assoc],
  simp_rw ← category.assoc,
  congr' 1,
  ext,
  dsimp,
  simp,
end

variable (D)

/-- The natural transformation from the identity functor to `plus`. -/
@[simps]
def to_plus_nat_trans : (𝟭 (Cᵒᵖ ⥤ D)) ⟶ J.plus_functor D :=
{ app := λ P, J.to_plus P,
  naturality' := λ _ _ _, to_plus_naturality _ _ }

variable {D}

/-- `(P ⟶ P⁺)⁺ = P⁺ ⟶ P⁺⁺` -/
@[simp]
lemma plus_map_to_plus : J.plus_map (J.to_plus P) = J.to_plus (J.plus_obj P) :=
begin
  ext X S,
  dsimp [to_plus, plus_obj, plus_map],
  delta cover.to_multiequalizer,
  simp only [ι_colim_map],
  let e : S.unop ⟶ ⊤ := hom_of_le (order_top.le_top _),
  simp_rw [← colimit.w _ e.op, ← category.assoc],
  congr' 1,
  ext I,
  dsimp,
  simp only [diagram_pullback_app, colimit.ι_pre, multiequalizer.lift_ι,
    ι_colim_map_assoc, category.assoc],
  dsimp only [functor.op],
  let ee : (J.pullback (I.map e).f).obj S.unop ⟶ ⊤ := hom_of_le (order_top.le_top _),
  simp_rw [← colimit.w _ ee.op, ← category.assoc],
  congr' 1,
  ext II,
  dsimp,
  simp only [limit.lift_π, multifork.of_ι_π_app, multiequalizer.lift_ι, category.assoc],
  dsimp [multifork.of_ι],
  convert multiequalizer.condition (S.unop.index P)
    ⟨_, _, _, II.f, 𝟙 _, I.f, II.f ≫ I.f, I.hf, sieve.downward_closed _ I.hf _, by simp⟩,
  { cases I, refl },
  { dsimp [cover.index],
    erw [P.map_id, category.comp_id],
    refl }
end

lemma is_iso_to_plus_of_is_sheaf (hP : presheaf.is_sheaf J P) : is_iso (J.to_plus P) :=
begin
  rw presheaf.is_sheaf_iff_multiequalizer at hP,
  rsufficesI : ∀ X, is_iso ((J.to_plus P).app X),
  { apply nat_iso.is_iso_of_is_iso_app },
  intros X, dsimp,
  rsufficesI : is_iso (colimit.ι (J.diagram P X.unop) (op ⊤)),
  { apply is_iso.comp_is_iso },
  rsufficesI : ∀ (S T : (J.cover X.unop)ᵒᵖ) (f : S ⟶ T), is_iso ((J.diagram P X.unop).map f),
  { apply is_iso_ι_of_is_initial (initial_op_of_terminal is_terminal_top) },
  intros S T e,
  have : S.unop.to_multiequalizer P ≫ (J.diagram P (X.unop)).map e =
    T.unop.to_multiequalizer P, by { ext, dsimp, simpa },
  have : (J.diagram P (X.unop)).map e = inv (S.unop.to_multiequalizer P) ≫
    T.unop.to_multiequalizer P, by simp [← this],
  rw this, apply_instance,
end

/-- The natural isomorphism between `P` and `P⁺` when `P` is a sheaf. -/
def iso_to_plus (hP : presheaf.is_sheaf J P) : P ≅ J.plus_obj P :=
by letI := is_iso_to_plus_of_is_sheaf J P hP; exact as_iso (J.to_plus P)

@[simp]
lemma iso_to_plus_hom (hP : presheaf.is_sheaf J P) : (J.iso_to_plus P hP).hom = J.to_plus P := rfl

/-- Lift a morphism `P ⟶ Q` to `P⁺ ⟶ Q` when `Q` is a sheaf. -/
def plus_lift {P Q : Cᵒᵖ ⥤ D} (η : P ⟶ Q) (hQ : presheaf.is_sheaf J Q) :
  J.plus_obj P ⟶ Q :=
J.plus_map η ≫ (J.iso_to_plus Q hQ).inv

@[simp, reassoc]
lemma to_plus_plus_lift {P Q : Cᵒᵖ ⥤ D} (η : P ⟶ Q) (hQ : presheaf.is_sheaf J Q) :
  J.to_plus P ≫ J.plus_lift η hQ = η :=
begin
  dsimp [plus_lift],
  rw ← category.assoc,
  rw iso.comp_inv_eq,
  dsimp only [iso_to_plus, as_iso],
  rw to_plus_naturality,
end

lemma plus_lift_unique {P Q : Cᵒᵖ ⥤ D} (η : P ⟶ Q) (hQ : presheaf.is_sheaf J Q)
  (γ : J.plus_obj P ⟶ Q) (hγ : J.to_plus P ≫ γ = η) : γ = J.plus_lift η hQ :=
begin
  dsimp only [plus_lift],
  rw [iso.eq_comp_inv, ← hγ, plus_map_comp],
  dsimp,
  simp,
end

lemma plus_hom_ext {P Q : Cᵒᵖ ⥤ D} (η γ : J.plus_obj P ⟶ Q) (hQ : presheaf.is_sheaf J Q)
  (h : J.to_plus P ≫ η = J.to_plus P ≫ γ) : η = γ :=
begin
  have : γ = J.plus_lift (J.to_plus P ≫ γ) hQ,
  { apply plus_lift_unique, refl },
  rw this,
  apply plus_lift_unique, exact h
end

@[simp]
lemma iso_to_plus_inv (hP : presheaf.is_sheaf J P) : (J.iso_to_plus P hP).inv =
  J.plus_lift (𝟙 _) hP :=
begin
  apply J.plus_lift_unique,
  rw [iso.comp_inv_eq, category.id_comp],
  refl,
end

@[simp]
lemma plus_map_plus_lift {P Q R : Cᵒᵖ ⥤ D} (η : P ⟶ Q) (γ : Q ⟶ R) (hR : presheaf.is_sheaf J R) :
  J.plus_map η ≫ J.plus_lift γ hR = J.plus_lift (η ≫ γ) hR :=
begin
  apply J.plus_lift_unique,
  rw [← category.assoc, ← J.to_plus_naturality, category.assoc, J.to_plus_plus_lift],
end

instance plus_functor_preserves_zero_morphisms [preadditive D] :
  (plus_functor J D).preserves_zero_morphisms :=
{ map_zero' := λ F G, by { ext, dsimp, rw [J.plus_map_zero, nat_trans.app_zero] } }

end category_theory.grothendieck_topology


private def eqv (p₁ p₂ : α × α) : Prop :=
 (p₁.1 = p₂.1 ∧ p₁.2 = p₂.2) ∨ (p₁.1 = p₂.2 ∧ p₁.2 = p₂.1)

infix:50 " ~ " => eqv

axiom eqv.refl {α} (p : α × α) : p ~ p

axiom eqv.symm  {α} : ∀ {p₁ p₂ : α × α}, p₁ ~ p₂ → p₂ ~ p₁

axiom eqv.trans {α} : ∀ {p₁ p₂ p₃ : α × α}, p₁ ~ p₂ → p₂ ~ p₃ → p₁ ~ p₃

private theorem is_equivalence : Equivalence (@eqv α) :=
  { refl := eqv.refl, symm := eqv.symm, trans := eqv.trans }

instance uprodSetoid (α : Type u) : Setoid (α × α) where
   r     := eqv
   iseqv := is_equivalence

def UProd (α : Type u) : Type u :=
  Quotient (uprodSetoid α)

namespace UProd

@[noinline] def myId := @id

def mk' {α : Type} : α × α → UProd α :=
  let f := @Quot.mk _ _
  myId f