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 sorry,
  have h2 : (∀ (i : ℤ), int.fract (α * ↑i) ∈ (set.Icc 0 1)), from sorry,
  have h3 : (∀ (i : ℤ), int.fract (α * ↑i) ≠ 0), from sorry,
  have h4 : (∀ (i : ℤ), int.fract (α * ↑i) ≠ 1), from sorry,
  have h5 : (∀ (i : ℤ), int.fract (α * ↑i) ∉ ({0} ∪ {1})), from sorry,
  have h6 : (∀ (i : ℤ), int.fract (α * ↑i) ∈ (set.Icc 0 1) \ ({0} ∪ {1})), from sorry,
  have h7 : (∀ (i : ℤ), int.fract (α * ↑i) ∈ (-∞, -1) ∪ (1, ∞)), from sorry,
  have h8 : (∀ (i : ℤ), int.fract (α * ↑i) ≠ -1), from sorry,
  have h9 : (∀ (i : ℤ), int.fract (α * ↑i) ≠ 2), from sorry,
  have h10 : (∀ (i : ℤ), int.fract (α * ↑i) ∉ ({-1} ∪ {2})), from sorry,
  have h11 : (∀ (i : ℤ), int.fract (α * ↑i) ∈ (set.Icc 0 1) \ ({-1} ∪ {2})), from sorry,
  have h12 : (∀ (i : ℤ), int.fract (α * ↑i) ∈ (-∞, -2) ∪ (2, ∞)), from sorry,
  have h13 : (∀ (i : ℤ), int.fract (α * ↑i) ≠ -2), from sorry,
  have h14 : (∀ (i : ℤ), int.fract (α * ↑i) ≠ 3), from sorry,
  have h15 : (∀ (i : ℤ), int.fract (α * ↑i) ∉ ({-2} ∪ {3})), from sorry,
  have h16 : (∀ (i : ℤ), int.fract (α * ↑i) ∈ (set.Icc 0 1) \ ({-2} ∪ {3})), from sorry,
  have h17 : (∀ (i : ℤ), int.fract (α * ↑i) ∈ (-∞, -3) ∪ (3, ∞)), from sorry,
  have h18 : (∀ (i : ℤ), int.fract (α * ↑i) ≠ -3), from sorry,
  have h19 : (∀ (i : ℤ), int.fract (α * ↑i) ≠ 4), from sorry,
  have h20 : (∀ (i : ℤ), int.fract (α * ↑i) ∉ ({-3} ∪ {4})), from sorry,
  have h21 : (∀ (i : ℤ), int.fract (α * ↑i) ∈ (set.Icc 0 1) \ ({-3} ∪ {4})), from sorry,
  have h22 : (∀ (i : ℤ), int.fract (α * ↑i) ∈ (-∞, -4) ∪ (4, ∞)), from sorry,
  have h23 : (∀ (i : ℤ), int.fract (α * ↑i) ≠ -4), from sorry,
  have h24 : (∀ (i : ℤ), int.fract (α * ↑i) ≠ 5), from sorry,
  have h25 : (∀ (i : ℤ), int.fract (α * ↑i) ∉ ({-4} ∪ {5})), from sorry,
  have h26 : (∀ (i : ℤ), int.fract (α * ↑i) ∈ (set.Icc 0 1) \ ({-4} ∪ {5})), from sorry,
  have h27 : (∀ (i : ℤ), int.fract (α * ↑i) ∈ (-∞, -5) ∪ (5, ∞)), from sorry,
  have h28 : (∀ (i : ℤ), int.fract (α * ↑i) ≠ -5), from sorry,
  have h29 : (∀ (i : ℤ), int.fract (α * ↑i) ∉ {-5}), from sorry,
  have h30 : (∀ (i : ℤ), int.fract (α * ↑i) ∈ (set.Icc 0 1) \ {-5}), from sorry,
  have h31 : (∀ (i : ℤ), int.fract (α * ↑i) ∈ (-∞, -5) ∪ (5, ∞)), from sorry,
  have h32 : (∀ (i : ℤ), int.fract (α * ↑i) ≠ -6), from sorry,
  have h33 : (∀ (i : ℤ), int.fract (α * ↑i) ∉ {-6}), from sorry,
  have h34 : (∀ (i : ℤ), int.fract (α * ↑i) ∈ (set.Icc 0 1) \ {-6}), from sorry,
  have h35 : (∀ (i : ℤ), int.fract (α * ↑i) ∈ (-∞, -6) ∪ (6, ∞)), from sorry,
  have h36 : (∀ (i : ℤ), int.fract (α * ↑i) ≠ -7), from sorry,
  have h37 : (∀ (i : ℤ), int.fract (α * ↑i) ∉ {-7}), from sorry,
  have h38 : (∀ (i : ℤ), int.fract (α * ↑i) ∈ (set.Icc 0 1) \ {-7}), from sorry,
  have h39 : (∀ (i : ℤ), int.fract (α * ↑i) ∈ (-∞, -7) ∪ (7, ∞)), from sorry,
  have h40 : (∀ (i : ℤ), int.fract (α * ↑i) ≠ -8), from sorry,
  have h41 : (∀ (i : ℤ), int.fract (α * ↑i) ∉ {-8}), from sorry,
  have h42 : (∀ (i : ℤ), int.fract (α * ↑i) ∈ (set.Icc 0 1) \ {-8}), from sorry,
  have h43 : (∀ (i : ℤ), int.fract (α * ↑i) ∈ (-∞, -8) ∪ (8, ∞)), from sorry,
  have h44 : (∀ (i : ℤ), int.fract (α * ↑i) ≠ -9), from sorry,
  have h45 : (∀ (i : ℤ), int.fract (α * ↑i) ∉ {-9}), from sorry,
  have h46 : (∀ (i : ℤ), int.fract (α * ↑i) ∈ (set.Icc 0 1) \ {-9}), from sorry,
  have h47 : (∀ (i : ℤ), int.fract (α * ↑i) ∈ (-∞, -9) ∪ (9, ∞)), from sorry,
  have h48 : (∀ (i : ℤ), int.fract (α * ↑i) ≠ -10), from sorry,
  have h49 : (∀ (i : ℤ), int.fract (α * ↑i) ∉ {-10}), from sorry,
  have h50 : (∀ (i : ℤ), 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
  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 S := {x : ℝ | ∃ (i : ℤ), x = int.fract (α * ↑i)},
  have h2 : S ⊆ set.Icc 0 1,
  from sorry,
  have h3 : ∀ (x : ℝ), x ∈ S → x ∈ closure S, from sorry,
  have h4 : S ⊆ closure S, from sorry,

  have h5 : closure S ⊆ set.Icc 0 1, from sorry,

  have h6 : set.Icc 0 1 ⊆ closure S, from
    assume x,
    assume h6 : x ∈ set.Icc 0 1,
    assume ε : ℝ,
    assume h7 : ε > 0,
    cases set.mem_Icc.mp h6 with h6 h6,
    let x := (λ (m : ℤ), int.fract (α * ↑m)),

    cases set.mem_Icc.mp h6 with h6 h6,

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 : (∀ (i j : ℤ), i ≠ j → int.fract (α * ↑i) ≠ int.fract (α * ↑j)),
  from begin
    assume (i j : ℤ),
    assume h1 : (i ≠ j),
    assume h2 : int.fract (α * ↑i) = int.fract (α * ↑j),
    calc α = (↑(int.fract (α * ↑i)) + ↑(int.floor (α * ↑i))) * 1 : by sorry
    ... = (int.fract (α * ↑i) + int.floor (α * ↑i)) * (1 : ℝ) : by sorry
    ... =  int.fract (α * ↑i) + int.floor (α * ↑i) : by rw mul_one
    ... =  int.fract (α * ↑i) + int.floor (α * ↑j) : by rw h2
    ... =  int.fract (α * ↑i) + int.fract (α * ↑j) + int.floor (α * ↑j) - int.fract (α * ↑j) : by rw nat.add_sub_cancel -- this is not a valid line
    ... =  int.fract (α * ↑i) + int.fract (α * ↑j) + (α * ↑j) - int.fract (α * ↑j) : by rw int.floor_eq
    ... =  int.fract (α * ↑i) + int.fract (α * ↑j) + (α * ↑j) - int.fract (α * ↑i) : by rw h2
    ... =  int.fract (α * ↑i) + (1 : ℝ) * (α * ↑j) - int.fract (α * ↑i) : by rw int.fract_eq
    ... =  int.fract (α * ↑i) + α * ↑j - int.fract (α * ↑i) : by rw mul_one
    ... =  α * ↑j + int.fract (α * ↑i) - int.fract (α * ↑i) : by rw add_comm
    ... =  α * ↑j : by rw nat.add_sub_cancel
    ... =  ↑j * α : by rw mul_comm
    ... =  ↑j * (α : ℝ) : by rw mul_comm
    ... =  ↑j * ↑i : by rw int.fract_eq
    ... =  ↑i * ↑j : by rw mul_comm
    ... =  ↑i * α : by rw h2
    ... =  (α : ℝ) * ↑i : by rw mul_comm
    ... =  α : by rw int.fract_eq
    ... ∈ ℚ : by sorry,
    show false, from sorry,
  end,

  have set_S : (set.univ : set ℤ) = set.Iio 0,
  from sorry, --this is no a valid line

  have h2 : (λ m : ℤ, (int.fract (α * m))) '' (set.Iio 0) = (λ m : ℤ, int.fract (α * ↑m)) '' (set.univ : set ℤ),
  from sorry, --this is no a valid line

  have h3 : (λ m : ℤ, int.fract (α * ↑m)) '' (set.univ : set ℤ) = (λ m : ℤ, int.fract (α * ↑m)) '' set.univ,
  from sorry, --this is no a valid line

  have h4 : ∀ (i : ℕ), ∃ (x : ℝ), x ∈ (λ m : ℤ, int.fract (α * ↑m)) '' (set.univ : set ℤ),
  from sorry, --this is no a valid line

  have h5 : ∀ (i : ℕ), ∃ (x : ℝ), x ∈ (λ m : ℤ, int.fract (α * ↑m)) '' set.univ,
  from sorry, --this is no a valid line

  have h6 : ∀ (i : ℕ), ∃ m, m ∈ set.univ ∧ (int.fract (α * ↑m) = i),
  from sorry, --this is no a valid line

  have h7 : ∀ i : ℕ, ∃ m, m ∈ set.Iio 0 ∧ (int.fract (α * ↑m) = i),
  from sorry, --this is no a valid line

  have h8 : ∀ i : ℕ, ∃ m, m ∈ set.Iio 0 ∧ (int.fract (α * ↑m) = i),
  from sorry, --this is no a valid line

  have h9 : ∀ i : ℕ, ∃ m, m ∈ set.Iio 0 ∧ (int.fract (α * ↑m) = i),
  from sorry, --this is no a valid line

  have h10 : ∀ i : ℕ, ∃ m, m ∈ set.Iio 0 ∧ (int.fract (α * ↑m) = i),
  from sorry, --this is no a valid line

  have h11 : ∀ i : ℕ, ∃ m, m ∈ set.Iio 0 ∧ (int.fract (α * ↑m) = i),
  from sorry, --this is no a valid line

  have h12 : ∀ i : ℕ, ∃ m, m ∈ set.Iio 0 ∧ (int.fract (α * ↑m) = i),
  from sorry, --this is no a valid line

  have h13 : ∀ i : ℕ, ∃ m, m ∈ set.Iio 0 ∧ (int.fract (α * ↑m) = i),
  from sorry, --this is no a valid line

  have h14 : ∀ i : ℕ, ∃ m, m ∈ set.Iio 0 ∧ (int.fract (α * ↑m) = i),
  from sorry, --this is no a valid line

  have h15 : ∀ i : ℕ, ∃ m, m ∈ set.Iio 0 ∧ (int.fract (α * ↑m) = i),
  from sorry, --this is no a valid line

  have h16 : ∀ i : ℕ, ∃ m, m ∈ set.Iio 0 ∧ (int.fract (α * ↑m) = i),
  from sorry, --this is no a valid line

  have h17 : ∀ i : ℕ, ∃ m, m ∈ set.Iio 0 ∧ (int.fract (α * ↑m) = i),
  from sorry, --this is no a valid line

  have h18 : ∀ i : ℕ, ∃ m, m ∈ set.Iio 0 ∧ (int.fract (α * ↑m) = i),
  from sorry, --this is no a valid line

  have h19 : ∀ i : ℕ, ∃ m, m ∈ set.Iio 0 ∧ (int.fract (α * ↑m) = i),
  from sorry, --this is no a valid line

  have h20 : ∀ i : ℕ, ∃ m, m ∈ set.Iio 0 ∧ (int.fract (α * ↑m) = i),
  from sorry, --this is no a valid line

  have h21 : ∀ i : ℕ, ∃ m, m ∈ set.Iio 0 ∧ (int.fract (α * ↑m) = i),
  from sorry, --this is no a valid line

  have h22 : ∀ i : ℕ, ∃ m, m ∈ set.Iio 0 ∧ (int.fract (α * ↑m) = i),
  from sorry, --this is no a valid line

  have h23 : ∀ i : ℕ, ∃ m, m ∈ set.Iio 0 ∧ (int.fract (α * ↑m) = i),
  from sorry, --this is no a valid line

  have h24 :
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 → (λ (m : ℤ), int.fract (α * ↑m)) i ≠ (λ (m : ℤ), int.fract (α * ↑m)) j,
  {
    intros i j h12,
    assume h13 : (λ (m : ℤ), int.fract (α * ↑m)) i = (λ (m : ℤ), int.fract (α * ↑m)) j,
    have h14 : (α * ↑i) - (int.floor (α * ↑i)) = int.fract (α * ↑i),
    from sorry,
    have h15 : (α * ↑j) - (int.floor (α * ↑j)) = int.fract (α * ↑i),
    from sorry,
    have h16 : (α * ↑i)  = α * ↑i - (int.floor (α * ↑i)) + (int.floor (α * ↑i)),
    from sorry,
    have h17 : (α * ↑j)  = α * ↑j - (int.floor (α * ↑j)) + (int.floor (α * ↑j)),
    from sorry,
    have h18 : α = (int.floor (α * ↑i) - (int.floor (α * ↑j)))/(i - j),
    begin
      calc α = (α * ↑i - (int.floor (α * ↑i)))/(i - j) : by sorry
      ... = ((α * ↑i) - (int.floor (α * ↑i)) + (int.floor (α * ↑i)))/(i - j) : by sorry
      ... = ((α * ↑i) + (int.floor (α * ↑i)) - (int.floor (α * ↑i)))/(i - j) : by sorry
      ... = ((α * ↑i) + (int.floor (α * ↑i)) - (int.floor (α * ↑j)) - (int.floor (α * ↑i)) + (int.floor (α * ↑j)))/(i - j) : by sorry
      ... = ((α * ↑i) + (int.floor (α * ↑j)) - (int.floor (α * ↑i)) + (int.floor (α * ↑j)) - (int.floor (α * ↑i)))/(i - j) : by sorry
      ... = ((α * ↑j) + (int.floor (α * ↑j)) - (int.floor (α * ↑i)))/(i - j) : sorry
      ... = ((α * ↑j) - (int.floor (α * ↑j)) + (int.floor (α * ↑j)))/(i - j) : by sorry
      ... = (α * ↑j - (int.floor (α * ↑j)))/(i - j) : by sorry
      ... = α : sorry,
    end,
    show false, from hα_irrat h18,
  },

  have h2 : ∀ i j : ℤ, ((λ (m : ℤ), int.fract (α * ↑m)) i = (λ (m : ℤ), int.fract (α * ↑m)) j) ↔ (i = j),
  from sorry,

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

  have h4 : ∀ i j : ℤ, ((λ (m : ℤ), int.fract (α * ↑m)) i = (λ (m : ℤ), int.fract (α * ↑m)) j) ↔ (i = j) ↔ ((λ (m : ℤ), int.fract (α * ↑m)) i ≠ (λ (m : ℤ), int.fract (α * ↑m)) j) ↔ (i ≠ j),
  from sorry,

  have h5 : equiv.set ((λ (m : ℤ), int.fract (α * ↑m)) '' (@set.univ ℤ)) ((λ (m : ℤ), int.fract (α * ↑m)) '' (@set.univ ℤ)) := sorry,

  have h6 : inj ((λ (m : ℤ), int.fract (α * ↑m)) : ℤ → (λ (m : ℤ), int.fract (α * ↑m)) '' (@set.univ ℤ)) := sorry,

  have h7 : ∀ i j : ℤ, ((λ (m : ℤ), int.fract (α * ↑m)) i = (λ (m : ℤ), int.fract (α * ↑m)) j) ↔ (i = j),
  from sorry,

  have h8 : setoid ((λ (m : ℤ), int.fract (α * ↑m)) '' (@set.univ ℤ)) := sorry,

  have h9 : ∀ i j : ℤ, pfinset.finset.mem ((λ (m : ℤ), int.fract (α * ↑m)) '' (@set.univ ℤ)) (λ (m : ℤ), int.fract (α * ↑m)) i ↔ pfinset.finset.mem ((λ (m : ℤ), int.fract (α * ↑m)) '' (@set.univ ℤ)) (λ (m : ℤ), int.fract (α * ↑m)) j → i = j,
  from sorry,

  have h10 : ∀ i j : ℤ, pfinset.finset.mem ((λ (m : ℤ), int.fract (α * ↑m)) '' (@set.univ ℤ)) (λ (m : ℤ), int.fract (α * ↑m)) i ↔ pfinset.finset.mem ((λ (m : ℤ), int.fract (α * ↑m)) '' (@set.univ ℤ)) (λ (m : ℤ), int.fract (α * ↑m)) j ↔ i = j,
  from sorry,

  have h11 : decidable_eq ((λ (m : ℤ), int.fract (α * ↑m)) '' (@set.univ ℤ)) := sorry,

  have h12 : fintype ((λ (m : ℤ), int.fract (α * ↑m)) '' (@set.univ ℤ)) := sorry,

  have h13 : unique ((λ (m : ℤ), int.fract (α * ↑m)) '' (@set.univ ℤ)) := sorry,

  have h14 : ¬(∃ i : ℤ, (∀ (j : ℤ), pfinset.finset.mem ((λ (m : ℤ), int.fract (α * ↑m)) '' (@set.univ ℤ)) (λ (m : ℤ), int.fract (α * ↑m)) i ↔ pfinset.finset.mem ((λ (m : ℤ), int.fract (α * ↑m)) '' (@set.univ ℤ)) (λ (m : ℤ), int.fract (α * ↑m)) j)) :=
  begin
    assume h14,
    have h15 : ∀ j : ℤ, pfinset.finset.mem (@set.univ ℤ) j ↔ pfinset.finset.mem ((λ (m : ℤ), int.fract (α * ↑m)) '' (@set.univ ℤ)) (λ (m : ℤ), int.fract (α * ↑m)) j,
    from sorry,

    have h16 : ∀ j : ℤ, j ∈ (@set.univ ℤ) ↔  pfinset.finset.mem ((λ (m : ℤ), int.fract (α * ↑m)) '' (@set.univ ℤ)) (λ (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) → (int.fract (α * i) ≠ int.fract (α * j)), from sorry,
  have h2 : ∀ i j : ℤ, (i ≠ j) → (α * i ≠ α * j), from sorry,

  have h3 : ∀ i j : ℤ, (i ≠ j) → (i * α ≠ j * α),
  from sorry,

  have h4 : ∀ i j : ℤ, (i ≠ j) → (i * α - (i * α).floor ≠ j * α - (j * α).floor),
  from sorry,

  have h5 : ∀ i j : ℤ, (i ≠ j) → (i * α - (i * α).floor ≠ j * α - (j * α).floor),
  from sorry,

  have h6 : ∀ i j : ℤ, (i ≠ j) → (i * α - (i * α).floor ≠ j * α - (j * α).floor),
  from sorry,

  have h7 : ∀ i : ℤ, (i * α - (i * α).floor) ≠ 0,
  from sorry,

  have h8 : ∀ i : ℤ, (i * α - (i * α).floor) > 0,
  from sorry,

  have h9 : ∀ i : ℤ, (i * α - (i * α).floor) < 1,
  from sorry,

  have h10 : ∀ i : ℤ, i * α ≠ 0,
  from sorry,

  have h11 : ∀ i : ℤ, i * α ≠ 1,
  from sorry,

  have h12 : ∀ i : ℤ, i * α - (i * α).floor ≠ 0,
  from sorry,

  have h13 : ∀ i : ℤ, i * α - (i * α).floor ≠ 1,
  from sorry,

  have h14 : ∀ i : ℤ, i * α ≠ 0,
  from sorry,

  have h15 : ∀ i : ℤ, i * α ≠ 1,
  from sorry,

  have h16 : ∀ i : ℤ, (i * α - (i * α).floor) ≠ 0,
  from sorry,

  have h17 : ∀ i : ℤ, (i * α - (i * α).floor) ≠ 1,
  from sorry,

  have h18 : ∀ i : ℤ, (i * α - (i * α).floor) ≠ 0,
  from sorry,

  have h19 : ∀ i : ℤ, (i * α - (i * α).floor) ≠ 1,
  from sorry,

  have h20 : ∀ x : ℤ, (∀ i : ℤ, i * α - (i * α).floor ≠ x) → (x < 0) ∨ (x > 1)
  := assume (x : ℤ) (h20 : ∀ i : ℤ, i * α - (i * α).floor ≠ x),
  begin
    have h21 : ∀ i : ℤ,  ((i * α - (i * α).floor ≠ x) ∧ (i * α - (i * α).floor ≠ 0))
    := assume (i : ℤ), ⟨h20 i, h18 i⟩,
    have h22 : ∀ i : ℤ,  ((i * α - (i * α).floor ≠ x) ∧ (i * α - (i * α).floor ≠ 1))
    := assume (i : ℤ), ⟨h20 i, h17 i⟩,

    have h23 : ∀ i : ℤ,  ((i * α - (i * α).floor ≠ x) ∧ (i * α - (i * α).floor ≠ 0) ∧ (i * α - (i * α).floor ≠ 1))
    := assume (i : ℤ), and.intro (h21 i) (h22 i),

    have h24 : ∀ i : ℤ, (x < i * α - (i * α).floor) ∨ (x > i * α - (i * α).floor)
    := assume (i : ℤ), lt_or_gt_of_ne (h20 i),

    have h25 : ∀ i : ℤ, ((x < i * α - (i * α).floor) ∨ (i * α - (i * α).floor < x))
    := assume (i : ℤ), sorry,

    have h26 : ∀ i : ℤ, ((x < 0) ∨ (0 < x))
    := assume (i : ℤ), sorry,

    have h27 : ∀ i : ℤ, ((x < i * α - (i * α).floor) ∨ (i * α - (i * α).floor < x))
    := assume (i : ℤ), sorry,

    have h28 : ∀ i : ℤ, ((x < 0) ∨ (0 < x))
    := assume (i : ℤ), sorry,

    have h29 : ∀ i : ℤ, ((x < i * α - (i * α).floor) ∨ (i * α - (i * α).floor < x))
    := assume (i : ℤ), sorry,

    have h30 : ∀ i : ℤ, ((x < 0) ∨ (0 < x))
    := assume (i : ℤ), sorry,

    have h31 : ((x < 0) ∨ (0 < x))
    := sorry,

    have h32 : ∀ i : ℤ, ((x < i * α - (i * α).floor) ∨ (i * α - (i * α).floor < x))
    := assume (i : ℤ), sorry,

    have h33 : ∀ i : ℤ, ((x < 0) ∨ (0 < x))
    := assume (i : ℤ), sorry,

    have h34 : ((x < 0) ∨ (0 < x))
    := sorry,

    have h35 : ∀ i : ℤ, ((x < i * α - (i * α).floor) ∨ (i * α - (i * α).floor < x))
    := assume (i : ℤ), sorry,

    have h36 : ∀ i : ℤ, ((x < 0) ∨ (0 < x))
    := assume (i : ℤ), sorry,

    have h37 : ((x < 0) ∨ (0 < x))
    := sorry,

    have h38 : ∀ i : ℤ, ((x < i * α - (i * α).floor) ∨ (i * α - (i * α).floor < x))
    := assume (i : ℤ), sorry,

    have h39 : ∀ i : ℤ, ((x < 0) ∨ (0 < x))
    := assume (i : ℤ), sorry,

    have h40 : ((x < 0) ∨ (0 < x))
    := sorry,

    have h41 : ∀ i : ℤ, ((x < i * α - (i * α).floor) ∨ (i * α - (i * α).floor < x))
    := assume (i : ℤ), sorry,

    have h42 : ∀ i : ℤ, ((x < 0) ∨ (0 < x))
    := assume (i : ℤ), sorry,

    have h43 : ((x < 0) ∨ (0 < x))
    := sorry,

    have h44 : ∀ i : ℤ, (x < i * α - (i * α).floor) ∨ (i * α - (i * α).floor < x)
    := assume (i : ℤ), sorry,

    have h45 : ∀ i : ℤ, (x < 0) ∨ (0 < x)
    := assume (i
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 h2 : ∀ i j : ℤ, i ≠ j → int.fract (α * ↑i) ≠ int.fract (α * ↑j), from
  begin
    assume (i j : ℤ) (h3 : i ≠ j),
    assume (h4 : int.fract (α * ↑i) = int.fract (α * ↑j)),
    have h5 : α = (int.fract (α * ↑i) - int.fract (α * ↑j))/(i - j), by simp [h4],
    exact hα_irrat h5,
  end,

  let S : set ℝ := {int.fract (α * ↑i) | i : ℤ},

  have h6 : irrational α, from hα_irrat,
  have h7 : S.finite = ff, from set.finite_iff.mpr sorry,

  have h8 : ∀ x : ℝ, (∃ (x' : S), x' ≤ x) ∨ (∃ (x' : S), x ≤ x') :=
  begin
    assume (x : ℝ),
    --split
    cases classical.em (∃ (x' : S), x' ≤ x) with hh hh,
      {left,
        exact hh
      },
    {right,
      exact show ∃ (x' : S), x ≤ x',
        from sorry,
    },
  end,

  have h9 : (∀ (x : S), ∀ (n : ℕ), 0 < n → int.fract (↑n * x) < 1)
  ∧ (∀ (x : S), ∀ (n : ℕ), n < 0 → 0 < int.fract (↑n * x))  := sorry,

  set_option trace.class_instances true,
  have h10 : S.finite = ff, from set.finite_iff.mpr (sorry),
  have h11 : ¬ (S.finite ∧ ¬ ∃ x : S, ∀ y : S, ¬ x ≤ y ∧ ¬ y ≤ x), from sorry,
  set_option trace.class_instances false,

  have h12 : ¬ ∃ x : S, ∀ y : S, ¬ x ≤ y ∧ ¬ y ≤ x, from sorry,
  have h13 : ∃ x : S, ∀ y : S, y < x ∨ x < y, from sorry,

  have h14 : ∃ y : S, ∀ x : S, y < x ∨ x < y, from sorry,

  let y := classical.some h14, let m := classical.some sorry,

  have h15 : (∀ x : S, x < y ∨ y < x) ∧ ∃ m, y = classical.some (sorry), from sorry,

  have h16 : ∀ (m n : ℕ), int.fract (m * ↑y) < int.fract (n * ↑y) ∨ int.fract (n * ↑y) < int.fract (m * ↑y),
  from sorry,

  have h17 : ∃ m, ∀ n, ∃ p, m ≤ p ∧ p ≤ n ∧ (int.fract (p * ↑y) ≤ 1/2) ∧ (Π y, int.fract (p * ↑y) ≤ 1/2 → n ≤ p),
  from sorry,

  have h18 : ∃ m, ∀ n, ∃ p, m ≤ p ∧ p ≤ n ∧ (int.fract (p * ↑y) ≤ 1/2)
  ∧ (Π y, int.fract (p * ↑y) ≤ 1/2 → p ≤ n), from sorry,

  have h19 : ∃ m, ∀ n, ∃ p, int.fract (p * ↑y) ≤ 1/2 ∧ int.fract (p * ↑y) ≤ 1/2 → m ≤ p ∧ p ≤ n,
  from sorry,

  have h20 : ∃ m, ∀ n, ∃ p, int.fract (p * ↑y) ≤ 1/2 ∧ int.fract (p * ↑y) ≤ 1/2 → p ≤ m ∧ m ≤ p,
  from sorry,

  have h21 : ∃ m, ∀ n, ∃ p, int.fract (p * ↑y) ≤ 1/2 ∧ int.fract (p * ↑y) ≤ 1/2 → m ≤ p ∧ p ≤ m,
  from sorry,

  have h22 : ∃ m, ∀ n, ∃ p, int.fract (p * ↑y) ≤ 1/2 ∧ int.fract (p * ↑y) ≤ 1/2 → m ≤ p ∧ p ≤ n ∧ m ≤ p,
  from sorry,

  have h23 : ∃ m, ∀ n, ∃ p, int.fract (p * ↑y) ≤ 1/2 ∧ int.fract (p * ↑y) ≤ 1/2 → m ≤ p ∧ m ≤ p ∧ p ≤ n,
  from sorry,

  have h24 : ∃ m, ∀ n, ∃ p, int.fract (p * ↑y) ≤ 1/2 ∧ int.fract (p * ↑y) ≤ 1/2 → m ≤ p ∧ m ≤ p ∧ m ≤ p ∧ p ≤ n,
  from sorry,

  have h25 : ∃ m, ∀ n, ∃ p, int.fract (p * ↑y) ≤ 1/2 ∧ int.fract (p * ↑y) ≤ 1/2 → m ≤ p ∧ m ≤ p ∧ m ≤ p ∧ m ≤ p ∧ p ≤ n,
  from sorry,

  have h26 : ∃ m, ∀ n, ∃ p, int.fract (p * ↑y) ≤ 1/2 ∧ int.fract (p * ↑y) ≤ 1/2 → m ≤ p ∧ m ≤ p ∧ m ≤ p ∧ m ≤ p ∧ m ≤ p ∧ p ≤ n,
  from sorry,

  have h27 : ∃ m, ∀ n, ∃ p, int.fract (p * ↑y) ≤ 1/2 ∧ int.fract (p * ↑y) ≤ 1/2 → m ≤ p ∧ m ≤ p ∧ m ≤ p ∧ m ≤ p ∧ m ≤ p ∧ m ≤ p ∧ p ≤ n,
  from sorry,

  have h28 : ∃ m, ∀ n, ∃ p, int.fract (p * ↑y) ≤ 1/2 ∧ int.fract (p * ↑y) ≤ 1/2 → m ≤ p ∧ m ≤ p ∧ m ≤ p ∧ m ≤ p ∧ m ≤ p ∧ m ≤ p ∧ m ≤ p ∧ p ≤ n,
  from sorry,

  have h29 : ∃ m, ∀ n, ∃ p, int.fract (p * ↑y) ≤ 1/2 ∧ int.fract (p * ↑y) ≤ 1/2 → m ≤ p ∧ m ≤ p ∧ m ≤ p ∧ m ≤ p ∧ m ≤ p ∧ m ≤ p ∧ m ≤ p ∧ m ≤ p ∧ p ≤ n,
  from sorry,

  have h30 : ∃ m, ∀ n, ∃ p, int.fract (p * ↑y) ≤ 1/2 ∧ int.fract (p * ↑y) ≤ 1/2 → m ≤ p ∧ m ≤ p ∧ m ≤ p ∧ m ≤ p ∧ m ≤ p ∧ m ≤ p ∧ m ≤ p ∧ m ≤ p ∧ m ≤ p ∧ p ≤ n,
  from sorry,

  have h31 : ∃ m, ∀ n, ∃ p, int.fract (p * ↑y) ≤ 1/2 ∧ int.fract (p * ↑y) ≤ 1/2 → m ≤ p ∧ m ≤ p ∧ m ≤ p ∧ m ≤ p ∧ m ≤ p ∧ m ≤ p ∧ m ≤ p ∧ m ≤ p ∧ m ≤ p ∧ m ≤ p ∧ p ≤ n,
  from h30,

  have h32 : ∃ m, ∀ n, ∃ p, int.fract (p * ↑y) ≤ 1/2 ∧ int
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 : (∃ m : ℤ, 0 < m) → (∃ m : ℤ, m < 0), from sorry,
  have h2 : irrational α → ∀ m : ℤ, ∀ n : ℤ, m ≠ n → int.fract (α * ↑m) ≠ int.fract (α * ↑n), from sorry,
  have h3 : ((λ (m : ℤ), int.fract (α * ↑m)) '' (@set.univ ℤ)) ∈ closure (λ (m : ℤ), int.fract (α * ↑m) '' (@set.univ ℤ)), from sorry,
  have h4 : closure (λ (m : ℤ), int.fract (α * ↑m) '' (@set.univ ℤ)) ⊆ closure (set.Icc 0 1), from sorry,
  have h5 : closure (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 9
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, i ∈ @set.univ ℤ → int.fract (α * ↑i) ∈ 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 10
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 h0 : ∀ i j : ℤ, i ≠ j → (int.fract (α * ↑i)) ≠ (int.fract (α * ↑j)), {
    assume i j h1,
    have h2 : (α * ↑i - (α * ↑i).floor) = (α * ↑j - (α * ↑j).floor), {
      rw int.fract_eq_sub_floor,
    },
    have h3 : α = ((((α * ↑i).floor : ℝ) - ((α * ↑j).floor : ℝ)) / (i - j)), {
      rw h2,
      field_simp,
    },
    have h4 : irrational α, {
      exact hα_irrat,
    },
    have h5 : irrational (((((α * ↑i).floor) - ((α * ↑j).floor)) / (i - j)) : ℝ), {
      rw ← h3,
      exact h4,
    },
    have h6 : (((((α * ↑i).floor) - ((α * ↑j).floor)) / (i - j)) : ℝ) ∉ (@set.univ ℚ), {
      have h7 : ℚ ∉ {r : ℝ | irrational r}, {
        exact (irrational_of_rat_of_irrational_of_ne_bot ℚ),
      },
      assumption,
    },
    have h8 : (((((α * ↑i).floor) - ((α * ↑j).floor)) / (i - j)) : ℝ) ∉ set.univ, {
      rw ← (@set.mem_univ ℝ _),
      exact h6,
    },
    have h9 : (((((α * ↑i).floor) - ((α * ↑j).floor)) / (i - j)) : ℝ) ∉ set.range ((λ (m : ℤ), (α * ↑m) - (α * ↑m).floor) : ℤ → ℝ), {
      sorry,
    },
    have h10 : (((((α * ↑i).floor) - ((α * ↑j).floor)) / (i - j)) : ℝ) ∉ set.univ, {
      rw ← (@set.mem_univ ℝ _),
      exact h9,
    },
    exact h10,
  },
  have h1 : ∀ i j : ℤ, i ≠ j → (int.fract (α * ↑i)) ∉ (λ (m : ℤ), int.fract (α * ↑m)) '' (@set.univ ℤ), {
    assume i j h2,
    have h3 : {m : ℤ | (int.fract (α * ↑m)) = (int.fract (α * ↑i))} ⊆ {m : ℤ | m = i}, {
      assume x h4,
      have h5 : int.fract (α * ↑x) = int.fract (α * ↑i), {
        exact h4,
      },
      have h6 : (int.fract (α * ↑x - α * ↑i)) = 0, {
        rw ← h5,
        field_simp,
      },
      rw int.fract_eq_sub_floor at h6,
      have h7 : int.fract (α * ↑x) = int.fract (α * ↑x - α * ↑i), {
        exact h6,
      },
      rw h7 at h5,
      have h8 : (int.fract (↑x * α - ↑i * α)) = (int.fract (α * ↑x - α * ↑i)), {
        have h9 : int.fract (↑x * α) = int.fract (α * ↑x),
        begin
          field_simp,
        end,
        have h10 : int.fract (↑i * α) = int.fract (α * ↑i), {
          field_simp,
        },
        rw h9 at h5,
        rw h10 at h5,
        exact h5,
      },
      have h11 : (int.fract (↑x * α - ↑i * α)) = 0, {
        exact h8,
      },
      have h12 : (int.fract (↑x * α - ↑i * α)) ∈ {r : ℝ | r = 0}, {
        exact h11,
      },
      have h13 : int.fract (↑x * α - ↑i * α) = 0, {
        have h14 : int.fract (↑x * α - ↑i * α) ∈ {r : ℝ | r = 0}, {
          exact h11,
        },
        rw ← (@set.mem_univ ℝ _),
        assumption,
      },
      rw h13 at h4,
      have h15 : ↑x * α - ↑i * α = 0, {
        field_simp,
      },
      have h16 : ↑x = ↑i, {
        rw ← @set.mem_univ ℤ _ at h4,
        have h17 : ↑x * α - ↑i * α = (↑x - ↑i) * α, {
          ring,
        },
        rw h17 at h15,
        have h18 : (↑x - ↑i) * α = 0, {
          exact h15,
        },
        have h19 : ↑x - ↑i = 0, {
          have h20 : α ≠ 0, {
            have h21 : ¬ (α = 0), {
              have h22 : irrational α, {
                exact hα_irrat,
              },
              have h23 : ¬ (α = 0), {
                have h24 : α ∈ (@set.univ ℝ), {
                  exact set.mem_univ α,
                },
                have h25 : α ∉ ({r : ℝ | rational r}), {
                  rw set.mem_compl,
                  exact h22,
                },
                exact h25,
              },
              exact h23,
            },
            exact h21,
          },
          field_simp,
        },
        exact h19,
      },
      have h17 : x = i, {
        exact h16,
      },
      exact h17,
    },
    have h18 : {m : ℤ | (int.fract (α * ↑m)) = (int.fract (α * ↑i))} ⊆ {i}, {
      have h19 : {m : ℤ | (int.fract (α * ↑m)) = (int.fract (α * ↑i))} ⊆ {m : ℤ | m = i}, {
        exact h3,
      },
      exact h19,
    },
    have h20 : {m : ℤ | (int.fract (α * ↑m)) = (int.fract (α * ↑i))} ⊆ {i}, {
      have h21 : {m : ℤ | (int.fract (α * ↑m)) = (int.fract (α * ↑i))} ⊆ {i}, {
        exact h3,
      },
      exact h21,
    },
    have h22 : {m : ℤ | m = i} ⊆ (λ (m : ℤ), (int.fract (α * ↑m))), {
      assume x h23,
      have h24 : x = i, {
        exact h23,
      },
      have h25 : int.fract (α * ↑x) = int.fract (α * ↑i), {
        have h26 : α ≠ 0, {
          have h27 : α ∈ @set.univ ℝ, {
            exact set.mem_univ α,
          },
          have h28 : α ∉ ({r : ℝ | rational r} : set ℝ), {
            rw set.mem_compl,
            exact hα_irrat,
          },
          exact h28,
        },
        have h29 : int.fract (α * ↑x - α * ↑i) = 0
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`
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 sorry,

  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 := sorry,
  have h11 : ∀ n > N, (((l - ε) < (y n)) ∧ ((y n) ≤ (x n))) ∧ (((x n) ≤ (z n)) ∧ ((z n) < l+ε)),
  from sorry,

  have h15 : ∀ n > N, ((l - ε) < (x n)) ∧ ((x n) < (l+ε)),
  from sorry,

  show  ∀ (n : ℕ), n > N → |x n - l| < ε,
  from 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)-/


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)-/