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 :=density_of_irrational_orbit (α : ℝ) (hα : ¬ (α ∈ ℚ)) : ∀ ε > 0, ∃ N, ∀ n > N, |n*α - ⌊n*α⌋| < ε :=
begin
  assume ε (hε : ε > 0),
  have h1 : ∀ i j : ℤ, i ≠ j → (i*α - ⌊i*α⌋) ≠ (j*α - ⌊j*α⌋), from by auto [ne.def, floor_eq_iff],
  have h2 : ∀ i j : ℤ, i ≠ j → (i*α - ⌊i*α⌋) ≠ (j*α - ⌊j*α⌋), from by auto [ne.def, floor_eq_iff],
  have h3 : ∀ i j : ℤ, i ≠ j → (i*α - ⌊i*α⌋) ≠ (j*α - ⌊j*α⌋), from by auto [ne.def, floor_eq_iff],
  have h4 : ∀ i j : ℤ, i ≠ j → (i*α - ⌊i*α⌋) ≠ (j*α - ⌊j*α⌋), from by auto [ne.def, floor_eq_iff],
  have h5 : ∀ i j : ℤ, i ≠ j → (i*α - ⌊i*α⌋) ≠ (j*α - ⌊j*α⌋), from by auto [ne.def, floor_eq_iff],
  have h6 : ∀ i j : ℤ, i ≠ j → (i*α - ⌊i*α⌋) ≠ (j*α - ⌊j*α⌋), from by auto [ne.def, floor_eq_iff],
  have h7 : ∀ i j : ℤ, i ≠ j → (i*α - ⌊i*α⌋) ≠ (j*α - ⌊j*α⌋), from by auto [ne.def, floor_eq_iff],
  have h8 : ∀ i j : ℤ, i ≠ j → (i*α - ⌊i*α⌋) ≠ (j*α - ⌊j*α⌋), from by auto [ne.def, floor_eq_iff],
  have h9 : ∀ i j : ℤ, i ≠ j → (i*α - ⌊i*α⌋) ≠ (j*α - ⌊j*α⌋), from by auto [ne.def, floor_eq_iff],
  have h10 : ∀ i j : ℤ, i ≠ j → (i*α - ⌊i*α⌋) ≠ (j*α - ⌊j*α⌋), from by auto [ne.def, floor_eq_iff],
  have h11 : ∀ i j : ℤ, i ≠ j → (i*α - ⌊i*α⌋) ≠ (j*α - ⌊j*α⌋), from by auto [ne.def, floor_eq_iff],
  have h12 : ∀ i j : ℤ, i ≠ j → (i*α - ⌊i*α⌋) ≠ (j*α - ⌊j*α⌋), from by auto [ne.def, floor_eq_iff],
  have h13 : ∀ i j : ℤ, i ≠ j → (i*α - ⌊i*α⌋) ≠ (j*α - ⌊j*α⌋), from by auto [ne.def, floor_eq_iff],
  have h14 : ∀ i j : ℤ, i ≠ j → (i*α - ⌊i*α⌋) ≠ (j*α - ⌊j*α⌋), from by auto [ne.def, floor_eq_iff],
  have h15 : ∀ i j : ℤ, i ≠ j → (i*α - ⌊i*α⌋) ≠ (j*α - ⌊j*α⌋), from by auto [ne.def, floor_eq_iff],
  have h16 : ∀ i j : ℤ, i ≠ j → (i*α - ⌊i*α⌋) ≠ (j*α - ⌊j*α⌋), from by auto [ne.def, floor_eq_iff],
  have h17 : ∀ i j : ℤ, i ≠ j → (i*α - ⌊i*α⌋) ≠ (j*α - ⌊j*α⌋), from by auto [ne.def, floor_eq_iff],
  have h18 : ∀ i j : ℤ, i ≠ j → (i*α - ⌊i*α⌋) ≠ (j*α - ⌊j*α⌋), from by auto [ne.def, floor_eq_iff],
  have h19 : ∀ i j : ℤ, i ≠ j → (i*α - ⌊i*α⌋) ≠ (j*α - ⌊j*α⌋), from by auto [ne.def, floor_eq_iff],
  have h20 : ∀ i j : ℤ, i ≠ j → (i*α - ⌊i*α⌋) ≠ (j*α - ⌊j*α⌋), from by auto [ne.def, floor_eq_iff],
  have h21 : ∀ i j : ℤ, i ≠ j → (i*α - ⌊i*α⌋) ≠ (j*α - ⌊j*α⌋), from by auto [ne.def, floor_eq_iff],
  have h22 : ∀ i j : ℤ, i ≠ j → (i*α - ⌊i*α⌋) ≠ (j*α - ⌊j*α⌋), from by auto [ne.def, floor_eq_iff],
  have h23 : ∀ i j : ℤ, i ≠ j → (i*α - ⌊i*α⌋) ≠ (j*α - ⌊j*α⌋), from by auto [ne.def, floor_eq_iff],
  have h24 : ∀ i j : ℤ, i ≠ j → (i*α - ⌊i*α⌋) ≠ (j*α - ⌊j*α⌋), from by auto [ne.def, floor_eq_iff],
  have h25 : ∀ i j : ℤ, i ≠ j → (i*α - ⌊i*α⌋) ≠ (j*α - ⌊j*α⌋), from by auto [ne.def, floor_eq_iff],
  have h26 : ∀ i j : ℤ, i ≠ j → (i*α - ⌊i*α⌋) ≠ (j*α - ⌊j*α⌋), from by auto [ne.def, floor_eq_iff],
  have h27 : ∀ i j : ℤ, i ≠ j → (i*α - ⌊i*α⌋) ≠ (j*α - ⌊j*α⌋), from by auto [ne.def, floor_eq_iff],
  have h28 : ∀ i j : ℤ, i ≠ j → (i*α - ⌊i*α⌋) ≠ (j*α - ⌊j*α⌋), from by auto [ne.def, floor_eq_iff],
  have h29 : ∀ i j : ℤ, i ≠ j → (i*α - ⌊i*α⌋) ≠ (j*α - ⌊j*α⌋), from by auto [ne.def, floor_eq_iff],
  have h
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 :=density_irrational_orbit (α : ℝ) (hα : ¬ is_rational α) : ∀ x ∈ set.Ioo 0 1, ∃ n : ℤ, n ≠ 0 ∧ ∃ y ∈ set.Ioo 0 1, y ∈ {i | i ∈ ℤ} ∧ |x - y| < (1 / (abs n)) :=
begin
  assume x hx,
  have h1 : ∀ i j : ℤ, i ≠ j → ¬ (i * α - (i * α).floor = j * α - (j * α).floor), from by auto [hα, is_rational.def, floor_eq_iff_rat_mul_nat],
  have h2 : ∀ i j : ℤ, i ≠ j → ¬ (i * α - (i * α).floor = j * α - (j * α).floor), from by auto [hα, is_rational.def, floor_eq_iff_rat_mul_nat],
  have h3 : ∀ i j : ℤ, i ≠ j → ¬ (i * α - (i * α).floor = j * α - (j * α).floor), from by auto [hα, is_rational.def, floor_eq_iff_rat_mul_nat],
  have h4 : ∀ i j : ℤ, i ≠ j → ¬ (i * α - (i * α).floor = j * α - (j * α).floor), from by auto [hα, is_rational.def, floor_eq_iff_rat_mul_nat],
  have h5 : ∀ i j : ℤ, i ≠ j → ¬ (i * α - (i * α).floor = j * α - (j * α).floor), from by auto [hα, is_rational.def, floor_eq_iff_rat_mul_nat],
  have h6 : ∀ i j : ℤ, i ≠ j → ¬ (i * α - (i * α).floor = j * α - (j * α).floor), from by auto [hα, is_rational.def, floor_eq_iff_rat_mul_nat],
  have h7 : ∀ i j : ℤ, i ≠ j → ¬ (i * α - (i * α).floor = j * α - (j * α).floor), from by auto [hα, is_rational.def, floor_eq_iff_rat_mul_nat],
  have h8 : ∀ i j : ℤ, i ≠ j → ¬ (i * α - (i * α).floor = j * α - (j * α).floor), from by auto [hα, is_rational.def, floor_eq_iff_rat_mul_nat],
  have h9 : ∀ i j : ℤ, i ≠ j → ¬ (i * α - (i * α).floor = j * α - (j * α).floor), from by auto [hα, is_rational.def, floor_eq_iff_rat_mul_nat],
  have h10 : ∀ i j : ℤ, i ≠ j → ¬ (i * α - (i * α).floor = j * α - (j * α).floor), from by auto [hα, is_rational.def, floor_eq_iff_rat_mul_nat],
  have h11 : ∀ i j : ℤ, i ≠ j → ¬ (i * α - (i * α).floor = j * α - (j * α).floor), from by auto [hα, is_rational.def, floor_eq_iff_rat_mul_nat],
  have h12 : ∀ i j : ℤ, i ≠ j → ¬ (i * α - (i * α).floor = j * α - (j * α).floor), from by auto [hα, is_rational.def, floor_eq_iff_rat_mul_nat],
  have h13 : ∀ i j : ℤ, i ≠ j → ¬ (i * α - (i * α).floor = j * α - (j * α).floor), from by auto [hα, is_rational.def, floor_eq_iff_rat_mul_nat],
  have h14 : ∀ i j : ℤ, i ≠ j → ¬ (i * α - (i * α).floor = j * α - (j * α).floor), from by auto [hα, is_rational.def, floor_eq_iff_rat_mul_nat],
  have h15 : ∀ i j : ℤ, i ≠ j → ¬ (i * α - (i * α).floor = j * α - (j * α).floor), from by auto [hα, is_rational.def, floor_eq_iff_rat_mul_nat],
  have h16 : ∀ i j : ℤ, i ≠ j → ¬ (i * α - (i * α).floor = j * α - (j * α).floor), from by auto [hα, is_rational.def, floor_eq_iff_rat_mul_nat],
  have h17 : ∀ i j : ℤ, i ≠ j → ¬ (i * α - (i * α).floor = j * α - (j * α).floor), from by auto [hα, is_rational.def, floor_eq_iff_rat_mul_nat],
  have h18 : ∀ i j : ℤ, i ≠ j → ¬ (i * α - (i * α).floor = j * α - (j * α).floor), from by auto [hα, is_rational.def, floor_eq_iff_rat_mul_nat],
  have h19 : ∀ i j : ℤ, i ≠ j → ¬ (i * α - (i * α).floor = j * α - (j * α).floor), from by auto [hα, is_rational.def, floor_eq_iff_rat_mul_nat],
  have h20 : ∀ i j : ℤ, i ≠ j → ¬ (i * α - (i * α).floor = j * α - (j * α).floor), from by auto [hα, is_rational.def, floor_eq_iff_rat_mul_nat],
  have h21 : ∀ i j : ℤ, i ≠ j → ¬ (i * α - (i * α).floor = j * α - (j * α).floor), from by auto [hα, is_rational.def, floor_eq_iff_rat_mul_nat],
  have h22 : ∀ i j : ℤ, i ≠ j → ¬ (i * α - (i * α).floor = j * α - (j * α).floor), from by auto [hα, is_rational.def, floor_eq_iff_rat_mul_nat],
  have h23 : ∀ i j : ℤ, i ≠ j → ¬ (i * α - (i * α).floor = j * α - (j * α).floor), from by auto [hα, is_rational.def, floor_eq_iff_rat_mul_nat],
  have h24 : ∀ i j : ℤ, i ≠ j → ¬ (i * α - (i * α).floor = j * α - (j * α).floor), from by auto [hα, is_rational.def, floor_eq_iff_rat_mul_nat],
  have h25 : ∀ i j : ℤ, i ≠ j → ¬ (i * α - (i * α).floor = j * α - (j * α).floor), from by auto [hα, is_rational.def, floor_eq_iff_rat_mul_nat],
  have h26 : ∀ i j : ℤ, i ≠ j → ¬ (i * α - (i * α).floor = j * α - (j * α).floor), from by auto [hα, is_rational.def, floor_eq_iff_rat_mul_nat],
  have h27 : ∀ i j : ℤ, i ≠ j → ¬ (i * α - (i * α).floor = j * α - (j * α).floor), from by auto [hα, is
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 :=irrational_orbit_dense {α : Type*} [linear_ordered_field α] (α_irrational : ¬ is_rat α) : ∀ y : α, ∃ x : α, 0 ≤ x ∧ x < y :=
begin
  assume y : α,
  have h1 : ∀ x : α, ∃ n : ℕ, (n : α) * x > y, from by auto [mul_lt_mul_of_pos_right],
  have h2 : ∃ n : ℕ, (n : α) * (1 : α) > y, from h1 1,
  cases h2 with n h3,
  use (n : α),
  show 0 ≤ (n : α) ∧ (n : α) < y, from by auto [h3, one_mul],
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 :=density_of_irrational_orbit (α : ℝ) (h1 : ¬ (α ∈ ℚ)) : ∀ y ∈ Icc 0 1, ∃ x ∈ Icc 0 1, x ≠ y ∧ |y - x| < 1 :=
begin
  assume (y : ℝ) (h2 : y ∈ Icc 0 1),
  have h3 : ∀ y ∈ Icc 0 1, ∃ x ∈ Icc 0 1, x ≠ y ∧ |y - x| < 1 := by auto [dense_iff_open_of_irrational],
  show ∃ x ∈ Icc 0 1, x ≠ y ∧ |y - x| < 1, from by auto [h3],
end

--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 :=density_of_irrational_orbit (α : ℝ) [irrational α] : ∀ ε > 0, ∃ N : ℤ, ∀ n : ℤ, |n*α - n*α%ℤ| < ε :=
begin
  assume ε,
  assume h1 : ε > 0,
  have h2 : ∀ x y : ℤ, x ≠ y → x*α%ℤ ≠ y*α%ℤ, from by auto [irrational.irrational],
  have h3 : ∀ x : ℤ, ∃ y : ℤ, y ≠ x, from by auto [exists_ne],
  have h4 : ∀ x : ℤ, ∃ y : ℤ, y ≠ x ∧ y*α%ℤ ≠ x*α%ℤ, from by auto [h2, h3],
  have h5 : ∀ x : ℤ, ∃ y : ℤ, y ≠ x ∧ y*α%ℤ ≠ x*α%ℤ ∧ y*α ≠ x*α, from by auto [irrational.irrational],
  have h6 : ∀ x : ℤ, ∃ y : ℤ, y ≠ x ∧ y*α%ℤ ≠ x*α%ℤ ∧ y*α ≠ x*α ∧ |y*α - y*α%ℤ| < ε, from by auto [h5],
  have h7 : ∀ x : ℤ, ∃ y : ℤ, y ≠ x ∧ y*α%ℤ ≠ x*α%ℤ ∧ y*α ≠ x*α ∧ |y*α - y*α%ℤ| < ε ∧ |x*α - x*α%ℤ| < ε, from by auto [h6],
  have h8 : ∀ x : ℤ, ∃ y : ℤ, y ≠ x ∧ y*α%ℤ ≠ x*α%ℤ ∧ y*α ≠ x*α ∧ |y*α - y*α%ℤ| < ε ∧ |x*α - x*α%ℤ| < ε ∧ |y*α - x*α| < ε, from by auto [h7],
  have h9 : ∀ x : ℤ, ∃ y : ℤ, y ≠ x ∧ y*α%ℤ ≠ x*α%ℤ ∧ y*α ≠ x*α ∧ |y*α - y*α%ℤ| < ε ∧ |x*α - x*α%ℤ| < ε ∧ |y*α - x*α| < ε ∧ |y*α%ℤ - x*α%ℤ| < ε, from by auto [h8],
  have h10 : ∀ x : ℤ, ∃ y : ℤ, y ≠ x ∧ y*α%ℤ ≠ x*α%ℤ ∧ y*α ≠ x*α ∧ |y*α - y*α%ℤ| < ε ∧ |x*α - x*α%ℤ| < ε ∧ |y*α - x*α| < ε ∧ |y*α%ℤ - x*α%ℤ| < ε ∧ |y*α - y*α%ℤ - (x*α - x*α%ℤ)| < ε, from by auto [h9],
  have h11 : ∀ x : ℤ, ∃ y : ℤ, y ≠ x ∧ y*α%ℤ ≠ x*α%ℤ ∧ y*α ≠ x*α ∧ |y*α - y*α%ℤ| < ε ∧ |x*α - x*α%ℤ| < ε ∧ |y*α - x*α| < ε ∧ |y*α%ℤ - x*α%ℤ| < ε ∧ |y*α - y*α%ℤ - (x*α - x*α%ℤ)| < ε ∧ |y*α%ℤ - x*α%ℤ - (y*α - x*α)| < ε, from by auto [h10],
  have h12 : ∀ x : ℤ, ∃ y : ℤ, y ≠ x ∧ y*α%ℤ ≠ x*α%ℤ ∧ y*α ≠ x*α ∧ |y*α - y*α%ℤ| < ε ∧ |x*α - x*α%ℤ| < ε ∧ |y*α - x*α| < ε ∧ |y*α%ℤ - x*α%ℤ| < ε ∧ |y*α - y*α%ℤ - (x*α - x*α%ℤ)| < ε ∧ |y*α%ℤ - x*α%ℤ - (y*α - x*α)| < ε ∧ |y*α%ℤ - x*α%ℤ| < ε, from by auto [h11],
  have h13 : ∀ x : ℤ, ∃ y : ℤ, y ≠ x ∧ y*α%ℤ ≠ x*α%ℤ ∧ y*α ≠ x*α ∧ |y*α - y*α%ℤ| < ε ∧ |x*α - x*α%ℤ| < ε ∧ |y*α - x*α| < ε ∧ |y*α%ℤ - x*α%ℤ| < ε ∧ |y*α - y*α%ℤ - (x*α - x*α%ℤ)| < ε ∧ |y*α%ℤ - x*α%ℤ - (y*α - x*α)| < ε ∧ |y*α%ℤ - x*α%ℤ| < ε ∧ |y*α%ℤ - x*α%ℤ| < ε, from by auto [h12],
  have h14 : ∀ x : ℤ, ∃ y : ℤ, y ≠ x ∧ y*α%ℤ ≠ x*α%ℤ ∧ y*α ≠ x*α ∧ |y*α - y*α%ℤ| < ε ∧ |x*α - x*α%ℤ| < ε ∧ |y*α - x*α| < ε ∧ |y*α%ℤ - x*α%ℤ| < ε ∧ |y*α - y*α%ℤ - (x*α - x*α%ℤ)| < ε ∧ |y*α%ℤ - x*α%ℤ - (y*α - x*α)| < ε ∧ |y*α%ℤ - x*α%ℤ| < ε ∧ |y*α%ℤ - x*α%ℤ| < ε ∧ |y*α%ℤ - x*α%ℤ| < ε, from by auto [h13],
  have h15 : ∀ x : ℤ, ∃ y : ℤ, y ≠ x ∧ y*α%ℤ ≠ x*α%ℤ ∧ y*α ≠ x*α ∧ |y*α - y*α%ℤ| < ε ∧ |x*α - x*α%ℤ| < ε ∧ |y*α - x*α| < ε ∧ |y*α%ℤ - x*α%ℤ| < ε ∧ |y*α - y*α%ℤ - (x*α - x*α%ℤ)| < ε ∧ |y*α%ℤ - x*α%ℤ - (y*α - x*α)| < ε ∧ |y*α%ℤ - x*α%ℤ| < ε ∧
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 :=density_of_irrational_orbit (α : ℝ) (hα : ¬ (α ∈ ℚ)) : ∀ y ∈ Icc 0 1, ∃ x ∈ Icc 0 1, ∃ n : ℤ, |y - (n •α)%R| < 1 :=
begin
  assume (y : ℝ) (h1 : y ∈ Icc 0 1),
  have h2 : ∀ (x : ℝ) (hx : x ∈ Icc 0 1), ∃ n : ℤ, n •α ∈ Icc 0 1 ∧ |y - n •α| < 1,
  from by auto [hα, exists_nat_one_lt_of_lt, exists_nat_one_lt_of_lt, exists_nat_one_lt_of_lt, exists_nat_one_lt_of_lt, exists_nat_one_lt_of_lt, exists_nat_one_lt_of_lt, exists_nat_one_lt_of_lt, exists_nat_one_lt_of_lt, exists_nat_one_lt_of_lt, exists_nat_one_lt_of_lt, exists_nat_one_lt_of_lt, exists_nat_one_lt_of_lt, exists_nat_one_lt_of_lt, exists_nat_one_lt_of_lt, exists_nat_one_lt_of_lt, exists_nat_one_lt_of_lt, exists_nat_one_lt_of_lt, exists_nat_one_lt_of_lt, exists_nat_one_lt_of_lt, exists_nat_one_lt_of_lt, exists_nat_one_lt_of_lt, exists_nat_one_lt_of_lt, exists_nat_one_lt_of_lt, exists_nat_one_lt_of_lt, exists_nat_one_lt_of_lt, exists_nat_one_lt_of_lt, exists_nat_one_lt_of_lt, exists_nat_one_lt_of_lt, exists_nat_one_lt_of_lt, exists_nat_one_lt_of_lt, exists_nat_one_lt_of_lt, exists_nat_one_lt_of_lt, exists_nat_one_lt_of_lt, exists_nat_one_lt_of_lt, exists_nat_one_lt_of_lt, exists_nat_one_lt_of_lt, exists_nat_one_lt_of_lt, exists_nat_one_lt_of_lt, exists_nat_one_lt_of_lt, exists_nat_one_lt_of_lt, exists_nat_one_lt_of_lt, exists_nat_one_lt_of_lt, exists_nat_one_lt_of_lt, exists_nat_one_lt_of_lt, exists_nat_one_lt_of_lt, exists_nat_one_lt_of_lt, exists_nat_one_lt_of_lt, exists_nat_one_lt_of_lt, exists_nat_one_lt_of_lt, exists_nat_one_lt_of_lt, exists_nat_one_lt_of_lt, exists_nat_one_lt_of_lt, exists_nat_one_lt_of_lt, exists_nat_one_lt_of_lt, exists_nat_one_lt_of_lt, exists_nat_one_lt_of_lt, exists_nat_one_lt_of_lt, exists_nat_one_lt_of_lt, exists_nat_one_lt_of_lt, exists_nat_one_lt_of_lt, exists_nat_one_lt_of_lt, exists_nat_one_lt_of_lt, exists_nat_one_lt_of_lt, exists_nat_one_lt_of_lt, exists_nat_one_lt_of_lt, exists_nat_one_lt_of_lt, exists_nat_one_lt_of_lt, exists_nat_one_lt_of_lt, exists_nat_one_lt_of_lt, exists_nat_one_lt_of_lt, exists_nat_one_lt_of_lt, exists_nat_one_lt_of_lt, exists_nat_one_lt_of_lt, exists_nat_one_lt_of_lt, exists_nat_one_lt_of_lt, exists_nat_one_lt_of_lt, exists_nat_one_lt_of_lt, exists_nat_one_lt_of_lt, exists_nat_one_lt_of_lt, exists_nat_one_lt_of_lt, exists_nat_one_lt_of_lt, exists_nat_one_lt_of_lt, exists_nat_one_lt_of_lt, exists_nat_one_lt_of_lt, exists_nat_one_lt_of_lt, exists_nat_one_lt_of_lt, exists_nat_one_lt_of_lt, exists_nat_one_lt_of_lt, exists_nat_one_lt_of_lt, exists_nat_one_lt_of_lt, exists_nat_one_lt_of_lt, exists_nat_one_lt_of_lt, exists_nat_one_lt_of_lt, exists_nat_one_lt_of_lt, exists_nat_one_lt_of_lt, exists_nat_one_lt_of_lt, exists_nat_one_lt_of_lt, exists_nat_one_lt_of_lt, exists_nat_one_lt_of_lt, exists_nat_one_lt_of_lt, exists_nat_one_lt_of_lt, exists_nat_one_lt_of_lt, exists_nat_one_lt_of_lt, exists_nat_one_lt_of_lt, exists_nat_one_lt_of_lt, exists_nat_one_lt_of_lt, exists_nat_one_lt_of_lt, exists_nat_one_lt_of_lt, exists_nat_one_lt_of_lt, exists_nat_one_lt_of_lt, exists_nat_one_lt_of_lt, exists_nat_one_lt_of_lt, exists_nat_one_lt_of_lt, exists_nat_one_lt_of_lt, exists_nat_one_lt_of_lt, exists_nat_one_lt_of_lt, exists_nat_one_lt_of_lt, exists_nat_one_lt_of_lt, exists_nat_one_lt_of_lt, exists_nat_one_lt_of_lt, exists_nat_one_lt_of_lt, exists_nat_one_lt_of_lt, exists_nat_one_lt_of_lt, exists_nat_one_lt_of_lt, exists_nat_one_lt_of_lt, exists_nat_one_lt_of_lt, exists_nat_one_lt_of_lt, exists_nat_one_lt_of_lt, exists_nat_one_lt_of_lt, exists_nat_one_lt_of_lt, exists_nat_one_lt_of_lt, exists_nat_one_lt_of_lt, exists_nat_one_lt_of_lt, exists_nat_one_lt_of_lt, exists_nat_one_lt_of_lt, exists_nat_one_lt_of_lt, exists_nat_one_lt_of_lt, exists_nat_one_lt_of_lt, exists_nat_one_lt_of_lt, exists_nat_one_lt_of_lt, exists_nat_one_lt_of_lt, exists_nat_one_lt_of_lt, exists_nat_one_lt_of_lt, exists_nat_one_lt_of_lt, exists_nat_one_lt_of_lt, exists_nat_one_lt_of_lt, exists_nat_one_lt_of_lt, exists_nat_one_lt_of_lt, exists_nat_one_lt_of_lt, exists_nat_one_lt_of_lt, exists_nat_one_lt_of_lt, exists_nat_one_lt_of_lt, exists_nat_one_lt_of_lt, exists_nat
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
FEW SHOT PROMPTS TO CODEX(END)-/


import Lbar.ext_aux3
import Lbar.iota

noncomputable theory

universes v u u'

open opposite category_theory category_theory.limits category_theory.preadditive
open_locale nnreal zero_object

variables (r r' : ℝ≥0)
variables [fact (0 < r)] [fact (0 < r')] [fact (r < r')] [fact (r < 1)] [fact (r' < 1)]

open bounded_homotopy_category

variables {r'}
variables (BD : breen_deligne.package)
variables (κ κ₂ : ℝ≥0 → ℕ → ℝ≥0)
variables [∀ (c : ℝ≥0), BD.data.suitable (κ c)] [∀ n, fact (monotone (function.swap κ n))]
variables [∀ (c : ℝ≥0), BD.data.suitable (κ₂ c)] [∀ n, fact (monotone (function.swap κ₂ n))]
variables (M : ProFiltPseuNormGrpWithTinv₁.{u} r')

section preps

variables (V : SemiNormedGroup.{u}) [complete_space V] [separated_space V]
variables (ι : ulift.{u+1} ℕ → ℝ≥0) (hι : monotone ι)

set_option pp.universes true

lemma homotopy_category.colimit_cofan_bdd {A : Type u} [category.{v} A] [abelian A]
[has_coproducts.{v} A] {α : Type v} (X : α → bounded_homotopy_category A)
  [uniformly_bounded X] : homotopy_category.is_bounded_above
  (homotopy_category.colimit_cofan $ λ a : α, (X a).val).X :=
begin
    obtain ⟨n,hn⟩ := homotopy_category.is_uniformly_bounded_above.cond (val ∘ X),
      use n, intros i hi,
    dsimp [homotopy_category.colimit_cofan],
    let e : (∐ λ (a : α), (X a).val.as).X i ≅
      (∐ λ (a : α), (X a).val.as.X i) := homotopy_category.coproduct_iso _ _,
    refine is_zero_of_iso_of_zero _ e.symm,
    apply category_theory.is_zero_colimit,
    rintros ⟨j⟩,
    apply hn j _ hi,
  end

def Tinv2_iso_of_bicartesian_aux_1
  (i : ℤ) : commsq.{u+2 u+1}
  (shift_sub_id.{u+1}
     ((QprimeFP.{u} r' BD.data κ₂ M).op ⋙
        (Ext.{u+1 u+2} i).flip.obj ((single.{u+1 u+2} (Condensed.{u u+1 u+2} Ab.{u+1}) 0).obj V.to_Cond))
     ι
     hι)
  (pi_Ext_iso_Ext_sigma.{u} BD κ₂ M V (λ (k : ulift.{u+1 0} ℕ), ι k) i).hom
  (pi_Ext_iso_Ext_sigma.{u} BD κ₂ M V (λ (k : ulift.{u+1 0} ℕ), ι k) i).hom
  (((Ext.{u+1 u+2} i).map
      (of_hom.{u+1 u+2} (QprimeFP.shift_sub_id.{u u+2 u+1} ι hι (QprimeFP_int.{u} r' BD.data κ₂ M))).op).app
     ((single.{u+1 u+2} (Condensed.{u u+1 u+2} Ab.{u+1}) 0).obj (Condensed.of_top_ab.{u} ↥V))) :=
begin
    apply commsq.of_eq,
    dsimp only [shift_sub_id, QprimeFP.shift_sub_id],
    simp only [sub_comp, comp_sub, homological_complex.of_hom_sub, category_theory.op_sub,
      functor.map_sub, op_id, category_theory.functor.map_id, of_hom_id,
      nat_trans.app_sub, nat_trans.id_app, category.comp_id, category.id_comp],
    apply congr_arg2 _ _ rfl,
    rw ← iso.eq_comp_inv,
    dsimp only [pi_Ext_iso_Ext_sigma, iso.trans_hom, iso.trans_inv,
      iso.symm_hom, iso.symm_inv, functor.map_iso_hom,
      iso.op_hom, op_comp, functor.flip_obj_map, functor.map_iso_inv],
    simp only [category.assoc, ← nat_trans.comp_app_assoc, ← functor.map_comp_assoc,
      ← functor.map_comp, iso.op_inv, ← op_comp],
    rw cofan_point_iso_colimit_conj_eq_desc,
    rw iso.eq_inv_comp,
    have := Ext_coproduct_iso_naturality_shift _
      (λ (k : ulift ℕ), (QprimeFP r' BD.data κ₂ M).obj (ι k))
      (λ k, (QprimeFP r' BD.data κ₂ M).map (hom_of_le $ hι $
        by exact_mod_cast k.down.le_succ)) i ((single (Condensed Ab) 0).obj V.to_Cond),
    exact this.symm,
    { apply homotopy_category.colimit_cofan_bdd },
end

@[reassoc]
lemma Ext_coproduct_iso_π
  (A : Type u) [category.{v} A] [abelian A] [enough_projectives A] [has_coproducts.{v} A] [AB4 A]
  (X : ulift.{v} ℕ → bounded_homotopy_category A) [uniformly_bounded X] (i : ℤ) (Y) (k) :
  (Ext_coproduct_iso X i Y).hom ≫ pi.π _ k =
  ((Ext i).map $ quiver.hom.op $ sigma.ι _ _).app Y :=
begin
  dsimp only [Ext_coproduct_iso, iso.trans_hom, pi_iso, preadditive_yoneda_coproduct_iso,
    as_iso_hom, preadditive_yoneda_coproduct_to_product],
  simp only [category.assoc, limit.lift_π, limit.lift_π_assoc, fan.mk_π_app],
  dsimp only [Ext_iso, iso.symm_hom, functor.map_iso_hom, functor.map_iso_inv],
  simp only [← functor.map_comp, iso.op_hom, iso.op_inv, ← op_comp],
  dsimp only [Ext, Ext0, functor.comp_map, whiskering_left_obj_map, whisker_left_app,
    functor.flip_map_app, replacement_iso],
  congr' 2,
  simp only [category.assoc, iso.inv_comp_eq, quiver.hom.unop_op, unop_op, op_unop],
  apply lift_unique,
  simp only [category.assoc, iso.inv_comp_eq, quiver.hom.unop_op, unop_op, op_unop],
  erw lift_lifts,
  simp only [uniform_π, colimit.ι_desc, cofan.mk_ι_app, lift_lifts_assoc],
  refl,
end

lemma Tinv2_iso_of_bicartesian_aux_2
  [∀ c n, fact (κ₂ c n ≤ r' * κ c n)]
  (j) {e : (homotopy_category.colimit_cofan.{u+1 u+2}
     (λ (a : ulift.{u+1 0} ℕ),
        ((λ (k : ulift.{u+1 0} ℕ), (QprimeFP.{u} r' BD.data κ₂ M).obj (ι k)) a).val)).X.is_bounded_above } :
  ((cofan.{u+1 u+2} (λ (k : ulift.{u+1 0} ℕ), (QprimeFP.{u} r' BD.data κ₂ M).obj (ι k))).ι.app j ≫
     of_hom.{u+1 u+2} (sigma_map.{u u+2 u+1} ι (QprimeFP_int.Tinv.{u} BD.data κ₂ κ M))) ≫
  (cofan_point_iso_colimit.{u} (λ (k : ulift.{u+1 0} ℕ), (QprimeFP.{u} r' BD.data κ M).obj (ι k))).hom =
  (QprimeFP.Tinv _ _ _ _).app _ ≫
  sigma.ι (λ (k : ulift.{u+1 0} ℕ), (QprimeFP.{u} r' BD.data κ M).obj (ι k)) j.1 :=
begin
  rw [← iso.eq_comp_inv], simp only [category.assoc, cofan_point_iso_colimit,
    colimit.comp_cocone_point_unique_up_to_iso_inv],
  dsimp only [bounded_homotopy_category.cofan, cofan.mk_ι_app, of_hom,
    homotopy_category.colimit_cofan, QprimeFP.Tinv, whisker_right_app,
    chain_complex.to_bounded_homotopy_category, functor.comp_map],
  erw [← (homotopy_category.quotient.{u+1 u+2 0} (Condensed.{u u+1 u+2} Ab.{u+1}) (complex_shape.up.{0} ℤ)).map_comp],
  erw [← (homotopy_category.quotient.{u+1 u+2 0} (Condensed.{u u+1 u+2} Ab.{u+1}) (complex_shape.up.{0} ℤ)).map_comp],
  congr' 1,
  dsimp only [sigma_map],
  erw [colimit.ι_desc],
  refl,
end

lemma Tinv2_iso_of_bicartesian_aux_3
  [∀ c n, fact (κ₂ c n ≤ κ c n)]
  [∀ c n, fact (κ₂ c n ≤ r' * κ c n)]
  (j)
  {e : (homotopy_category.colimit_cofan.{u+1 u+2}
     (λ (a : ulift.{u+1 0} ℕ),
        ((λ (k : ulift.{u+1 0} ℕ), (QprimeFP.{u} r' BD.data κ₂ M).obj (ι k)) a).val)).X.is_bounded_above} :
  (cofan.{u+1 u+2} (λ (k : ulift.{u+1 0} ℕ), (QprimeFP.{u} r' BD.data κ₂ M).obj (ι k))).ι.app j ≫
  of_hom.{u+1 u+2} (sigma_map.{u u+2 u+1} ι (QprimeFP_int.ι.{u} BD.data κ₂ κ M)) ≫
    (cofan_point_iso_colimit.{u} (λ (k : ulift.{u+1 0} ℕ), (QprimeFP.{u} r' BD.data κ M).obj (ι k))).hom =
  (QprimeFP.ι _ κ₂ κ M).app _ ≫
  sigma.ι ((λ (k : ulift.{u+1 0} ℕ), (QprimeFP.{u} r' BD.data κ M).obj (ι k))) j.1 :=
begin
  simp only [← category.assoc], rw [← iso.eq_comp_inv],
  simp only [category.assoc, cofan_point_iso_colimit, colimit.comp_cocone_point_unique_up_to_iso_inv],
  dsimp only [bounded_homotopy_category.cofan, cofan.mk_ι_app, of_hom,
    homotopy_category.colimit_cofan, QprimeFP.ι, whisker_right_app,
    chain_complex.to_bounded_homotopy_category, functor.comp_map],
  erw [← (homotopy_category.quotient.{u+1 u+2 0} (Condensed.{u u+1 u+2} Ab.{u+1}) (complex_shape.up.{0} ℤ)).map_comp],
  erw [← (homotopy_category.quotient.{u+1 u+2 0} (Condensed.{u u+1 u+2} Ab.{u+1}) (complex_shape.up.{0} ℤ)).map_comp],
  congr' 1,
  dsimp only [sigma_map],
  erw [colimit.ι_desc],
  refl,
end

lemma Tinv2_iso_of_bicartesian_aux [normed_with_aut r V]
  [∀ c n, fact (κ₂ c n ≤ κ c n)] [∀ c n, fact (κ₂ c n ≤ r' * κ c n)]
  (i : ℤ)
  (H1 : (shift_sub_id.commsq (ExtQprime.Tinv2 r r' BD.data κ κ₂ M V i) ι hι).bicartesian) :
  (Ext_Tinv2_commsq (of_hom (sigma_map (λ (k : ulift ℕ), ι k) (QprimeFP_int.Tinv BD.data κ₂ κ M)))
  (of_hom (sigma_map (λ (k : ulift ℕ), ι k) (QprimeFP_int.ι BD.data κ₂ κ M)))
  (of_hom (sigma_map (λ (k : ulift ℕ), ι k) (QprimeFP_int.Tinv BD.data κ₂ κ M)))
  (of_hom (sigma_map (λ (k : ulift ℕ), ι k) (QprimeFP_int.ι BD.data κ₂ κ M)))
  (of_hom (QprimeFP.shift_sub_id ι hι (QprimeFP_int r' BD.data κ₂ M)))
  (of_hom (QprimeFP.shift_sub_id ι hι (QprimeFP_int r' BD.data κ M)))
  (auux $ commsq_shift_sub_id_Tinv _ _ _ _ _ _)
  (auux $ commsq_shift_sub_id_ι _ _ _ _ _ _)
  ((single _ 0).map (Condensed.of_top_ab_map (normed_add_group_hom.to_add_monoid_hom (normed_with_aut.T.inv : V ⟶ V)) (normed_add_group_hom.continuous _)))
  i).bicartesian :=
begin
  have h1 := _, have h2 := _, have h3 := _,
  refine commsq.bicartesian.of_iso
    (pi_Ext_iso_Ext_sigma _ _ _ _ _ _) (pi_Ext_iso_Ext_sigma _ _ _ _ _ _)
    (pi_Ext_iso_Ext_sigma _ _ _ _ _ _) (pi_Ext_iso_Ext_sigma _ _ _ _ _ _)
    h1 h2 h2 h3 H1,
  apply Tinv2_iso_of_bicartesian_aux_1,
  { clear h1, apply commsq.of_eq, rw ← iso.eq_comp_inv,
    apply limit.hom_ext, rintros ⟨j⟩, rw lim_map_π,
    dsimp [pi_Ext_iso_Ext_sigma],
    simp only [category.assoc],
    have := Ext_coproduct_iso_π _
      (λ (k : ulift.{u+1 0} ℕ), (QprimeFP.{u} r' BD.data κ₂ M).obj (ι k))
      i ((single.{u+1 u+2} (Condensed.{u u+1 u+2} Ab.{u+1}) 0).obj V.to_Cond) j,
    rw [this, ← nat_trans.comp_app, ← functor.map_comp, ← op_comp],
    clear this,
    erw colimit.ι_desc,
    dsimp [Ext_Tinv2, ExtQprime.Tinv2],
    simp only [sub_comp, comp_sub],
    refine congr_arg2 _ _ _,
    { simp only [← nat_trans.comp_app, ← functor.map_comp, ← op_comp],
      rw Tinv2_iso_of_bicartesian_aux_2,
      swap,
      { apply homotopy_category.colimit_cofan_bdd },
      simp only [functor.map_comp, op_comp, nat_trans.comp_app, category.assoc],
      have := Ext_coproduct_iso_π _
        (λ (k : ulift.{u+1 0} ℕ), (QprimeFP.{u} r' BD.data κ M).obj (ι k))
        i ((single.{u+1 u+2} (Condensed.{u u+1 u+2} Ab.{u+1}) 0).obj V.to_Cond) j,
      rw ← iso.eq_inv_comp at this,
      rw ← reassoc_of this, refl },
    { simp only [category.assoc, nat_trans.naturality, ← nat_trans.comp_app_assoc,
        ← functor.map_comp_assoc, ← functor.map_comp, ← nat_trans.comp_app, ← op_comp],
      rw Tinv2_iso_of_bicartesian_aux_3,
      simp only [functor.map_comp, op_comp, nat_trans.comp_app, category.assoc],
      have := Ext_coproduct_iso_π _
        (λ (k : ulift.{u+1 0} ℕ), (QprimeFP.{u} r' BD.data κ M).obj (ι k))
        i ((single.{u+1 u+2} (Condensed.{u u+1 u+2} Ab.{u+1}) 0).obj V.to_Cond) j,
      rw ← iso.eq_inv_comp at this,
      rw ← reassoc_of this,
      refl,
      { apply homotopy_category.colimit_cofan_bdd } } },
  apply Tinv2_iso_of_bicartesian_aux_1,
end

lemma Tinv2_iso_of_bicartesian [normed_with_aut r V]
  [∀ c n, fact (κ₂ c n ≤ κ c n)] [∀ c n, fact (κ₂ c n ≤ r' * κ c n)]
  (hκ : Lbar.sufficiently_increasing κ ι)
  (hκ₂ : Lbar.sufficiently_increasing κ₂ ι)
  (i : ℤ)
  (H1 : (shift_sub_id.commsq (ExtQprime.Tinv2 r r' BD.data κ κ₂ M V i) ι hι).bicartesian)
  (H2 : (shift_sub_id.commsq (ExtQprime.Tinv2 r r' BD.data κ κ₂ M V (i+1)) ι hι).bicartesian) :
  is_iso (((Ext (i+1)).map ((BD.eval freeCond'.{u}).map M.Tinv_cond).op).app
    ((single (Condensed Ab) 0).obj V.to_Cond) -
    ((Ext (i+1)).obj ((BD.eval freeCond').op.obj (op (M.to_Condensed)))).map
      ((single (Condensed Ab) 0).map
        (Condensed.of_top_ab_map
          (normed_add_group_hom.to_add_monoid_hom normed_with_aut.T.inv) (normed_add_group_hom.continuous _)))) :=
begin
  let Vc := (single (Condensed Ab) 0).obj V.to_Cond,
  have SES₁ := QprimeFP.short_exact BD κ₂ M ι hι hκ₂,
  have SES₂ := QprimeFP.short_exact BD κ M ι hι hκ,
  have := Ext_iso_of_bicartesian_of_bicartesian SES₁ SES₂
    (sigma_map _ (QprimeFP_int.Tinv BD.data _ _ M))
    (sigma_map _ (QprimeFP_int.Tinv BD.data _ _ M))
    (category_theory.functor.map _ M.Tinv_cond)
    (sigma_map _ (QprimeFP_int.ι BD.data _ _ M))
    (sigma_map _ (QprimeFP_int.ι BD.data _ _ M))
    (commsq_shift_sub_id_Tinv BD.data _ _ M ι hι)
    (commsq_sigma_proj_Tinv BD _ _ M ι)
    (commsq_shift_sub_id_ι BD.data _ _ M ι hι)
    (commsq_sigma_proj_ι BD _ _ M ι)
    Vc ((single _ _).map $ Condensed.of_top_ab_map
      (normed_add_group_hom.to_add_monoid_hom normed_with_aut.T.inv) (normed_add_group_hom.continuous _))
    _
    (Tinv2_iso_of_bicartesian_aux _ _ _ _ _ _ _ _ _ H1)
    (Tinv2_iso_of_bicartesian_aux _ _ _ _ _ _ _ _ _ H2),
  delta Ext_Tinv2 at this,
  simpa only [op_id, category_theory.functor.map_id, category.id_comp, nat_trans.id_app],
end

lemma Tinv2_iso_of_bicartesian' [normed_with_aut r V]
  [∀ c n, fact (κ₂ c n ≤ κ c n)] [∀ c n, fact (κ₂ c n ≤ r' * κ c n)]
  (H : ∀ i, ∃ (ι) (hι),
    Lbar.sufficiently_increasing κ ι ∧
    Lbar.sufficiently_increasing κ₂ ι ∧
    (shift_sub_id.commsq (ExtQprime.Tinv2 r r' BD.data κ κ₂ M V i) ι hι).bicartesian ∧
    (shift_sub_id.commsq (ExtQprime.Tinv2 r r' BD.data κ κ₂ M V (i+1)) ι hι).bicartesian)
  (i : ℤ) :
  is_iso (((Ext i).map ((BD.eval freeCond'.{u}).map M.Tinv_cond).op).app
    ((single (Condensed Ab) 0).obj V.to_Cond) -
    ((Ext i).obj ((BD.eval freeCond').op.obj (op (M.to_Condensed)))).map
      ((single (Condensed Ab) 0).map
        (Condensed.of_top_ab_map
          (normed_add_group_hom.to_add_monoid_hom normed_with_aut.T.inv) (normed_add_group_hom.continuous _)))) :=
begin
  obtain ⟨i, rfl⟩ : ∃ k, k+1 = i := ⟨i-1, sub_add_cancel _ _⟩,
  obtain ⟨ι, hι, hκ, hκ₂, H1, H2⟩ := H i,
  apply Tinv2_iso_of_bicartesian _ _ _ _ _ _ ι hι hκ hκ₂ i H1 H2,
end

end preps


/-
Copyright (c) 2019 Seul Baek. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Author: Seul Baek
-/
import Mathlib.PrePort
import Mathlib.Lean3Lib.init.default
import Mathlib.tactic.omega.prove_unsats
import Mathlib.tactic.omega.nat.dnf
import Mathlib.tactic.omega.nat.neg_elim
import Mathlib.tactic.omega.nat.sub_elim
import Mathlib.PostPort

namespace Mathlib

/-
Main procedure for linear natural number arithmetic.
-/

namespace omega

namespace nat

theorem univ_close_of_unsat_neg_elim_not (m : ℕ) (p : preform) :
    preform.unsat (neg_elim (preform.not p)) → univ_close p (fun (_x : ℕ) => 0) m :=
  sorry

end Mathlib


import mcl.defs
import mcl.rhl
import mcl.lemmas
import mcl.compute_list
import mcl.ts_updates
import syncablep

open parlang
open parlang.thread_state
open parlang.state
open mcl
open mcl.rhl

/-- Copies *var* from tlocal of the nth thread into index n of *m* (forall n). Generally used as an assertion language for Hoare proofs -/
def from_tlocal {sig : signature} {n} (var) (s : state n (memory $ parlang_mcl_tlocal sig) (parlang_mcl_shared sig)) (m : memory (parlang_mcl_shared sig)) (h : (((sig.val var).type).dim) = 1) :=
((list.range_fin n).foldl (λ (m : parlang.memory (parlang_mcl_shared sig)) tid,
    m.update ⟨var, eq.mpr (by rw h) v[tid.val]⟩ ((s.threads.nth tid).tlocal.get ⟨var, eq.mpr (by rw h) v[tid.val]⟩))) m

lemma from_tlocal_comm_update {sig : signature} {n} (var₁ var₂) (s : state n (memory $ parlang_mcl_tlocal sig) (parlang_mcl_shared sig))
(m : memory (parlang_mcl_shared sig)) {h₁} {idx val} :
from_tlocal var₁ s (m.update ⟨var₂, idx⟩ val) h₁ = memory.update (from_tlocal var₁ s m h₁) ⟨var₂, idx⟩ val := begin
    unfold from_tlocal,
    induction n,
    { refl, },
    {
        rw [list.foldl_range_fin_succ],
        sorry, -- complicated with dependent type fin
    }
end

lemma from_tlocal_comm {sig : signature} {n} (var₁ var₂) (s : state n (memory $ parlang_mcl_tlocal sig) (parlang_mcl_shared sig))
(s' : state n (memory $ parlang_mcl_tlocal sig) (parlang_mcl_shared sig)) (m : memory (parlang_mcl_shared sig)) {h₁ h₂} :
from_tlocal var₁ s (from_tlocal var₂ s' m h₂) h₁ = from_tlocal var₂ s' (from_tlocal var₁ s m h₁) h₂ := begin
    unfold from_tlocal,
    induction n,
    {
        refl,
    }, {
        rw [list.foldl_range_fin_succ],
        rw [list.foldl_range_fin_succ],
        repeat { rw ← from_tlocal },
        sorry,
    }
end

--lemma : from_tlocal "b" (map_active_threads ac (ts_updates [op.compute_list (... :: coms)]) s = from_tlocal "b" (map_active_threads ac (ts_updates [op.compute_list (... :: coms)]) s

lemma from_tlocal_eq {sig : signature} {n}
{s s' : state n (memory $ parlang_mcl_tlocal sig) (parlang_mcl_shared sig)}
{m m' : memory (parlang_mcl_shared sig)} {var} {h : ((sig.val var).type).dim = 1} :
(∀ tid, (s.threads.nth tid).tlocal.get ⟨var, begin rw h, exact v[tid] end⟩ = (s'.threads.nth tid).tlocal.get ⟨var, begin rw h, exact v[tid] end⟩) →
m = m' →
from_tlocal var s m h = from_tlocal var s' m' h := begin
    intros hveq hmeq,
    subst hmeq,
    unfold from_tlocal,
    induction n,
    { refl, },
    {
        rw list.foldl_range_fin_succ,
        sorry,
    }
end

lemma syncable'_compute_list_syncable {sig : signature} {n} {ac : vector bool n} {computes} {shole lhole : set $ mcl_address sig}
{s : state n (memory $ parlang_mcl_tlocal sig) (parlang_mcl_shared sig)}
{m : memory (parlang_mcl_shared sig)} :
s.syncable m →
(∀ tid : fin n, (s.threads.nth tid).stores = ∅) →
(∀ tid : fin n, (s.threads.nth tid).loads = ∅) →
syncable' shole lhole (map_active_threads ac (ts_updates [op.compute_list computes]) s) m := begin
    intros syncable no_stores no_loads,
    unfold syncable' state.syncable,
    split,
    {
        simp only [accesses, compute_list_stores', compute_list_loads', compute_list_shared'],
        exact syncable,
    }, {
        intros i tid,
        simp [no_stores tid, no_loads tid],
    }
end

instance deciable_exists_nat (p) : decidable (@Exists ℕ p) := sorry
instance deciable_exists_fin (n p) : decidable (@Exists (fin n) p) := sorry

/-- Processes a store
Which thread accesses which index doesn't matter
-/
lemma syncable'_store {sig : signature} {n} {ac : vector bool n} {computes} {shole lhole : set $ mcl_address sig}
{dim} {idx : vector (expression sig type.int) dim} {var t} {h₁ : type_of (sig.val var) = t} {h₂}
{updates : list $ op sig}
{s : state n (memory $ parlang_mcl_tlocal sig) (parlang_mcl_shared sig)}
{m : memory (parlang_mcl_shared sig)}
(idx_1 : (((sig.val var).type).dim) = 1) :
(∀ idx, (⟨var, idx⟩ : mcl_address sig) ∉ shole) →
(∀ idx, (⟨var, idx⟩ : mcl_address sig) ∉ lhole) →
(∀ tid₁ tid₂, tid₁ ≠ tid₂ → idx.map (λ ind, eval (s.threads.nth tid₁).tlocal ind) ≠ idx.map (λ ind, eval (s.threads.nth tid₂).tlocal ind)) →
syncable' (shole ∪ array_address_range var) (lhole ∪ array_address_range var) (map_active_threads ac (ts_updates $ op.compute_list computes :: updates) s) m →
syncable' shole lhole (map_active_threads ac (ts_updates $ op.compute_list computes :: op.store var idx h₁ h₂ :: updates) s) (from_tlocal var (map_active_threads ac (ts_updates [op.compute_list computes]) s) m idx_1)
| var_not_in_shole var_not_in_lhole distinct_idx (and.intro syncable holes_constraint) := begin
    clear syncable'_store,
    unfold syncable',
    -- proof: syncable
    split, {
        intros i,
        by_cases i_is_var : i.fst = var,
        {
            subst i_is_var,
            specialize var_not_in_shole i.snd,
            specialize var_not_in_lhole i.snd,
            -- cases distinct out-of-bound
            -- by_cases i_is_oob : (∃ (tid : fin n), i.snd = eq.mpr _ (idx.map (λ ind, eval (s.threads.nth tid).tlocal ind))),
            sorry,
        },
        sorry,
    }, {
        -- proof: store hole
        intros i tid,
        have : i ∈ shole ∪ array_address_range var := sorry, --trivial

        by_cases i_is_var : i.fst = var,
        {
            -- if i is var we store into hole -> contradiction
            subst i_is_var,
            specialize var_not_in_shole i.snd,
            specialize var_not_in_lhole i.snd,
            cases i,
            split,
            {
                intros i_in_store,
                contradiction,
            }, {
                intros i_in_loads,
                contradiction,
            }
        }, {
            by_cases tid_activeness : ac.nth tid = tt,
            {
                rw map_active_threads_nth_ac tid_activeness,
                specialize holes_constraint i tid,

                rw map_active_threads_nth_ac tid_activeness at holes_constraint,
                rw [ts_updates] at holes_constraint,
                /- LARGE PROOF STARTS HERE -/
                clear syncable,
                rw [ts_updates, ts_updates],
                revert holes_constraint,
                generalize eq : compute_list computes (vector.nth (s.threads) tid) = s',
                rw ← list.reverse_reverse updates,
                generalize eq' : list.reverse updates = ups,
                intro holes_constraint,
                -- we do induction on the reverse of the list, such that we "append" elements to the end of updates (i.e. later)
                -- afterwards cases on the update (either store or compute)
                induction ups generalizing updates,
                {
                    simp [ts_updates, thread_state.tlocal_to_shared, store],
                    simp [ts_updates, thread_state.tlocal_to_shared, store] at holes_constraint,
                    cases holes_constraint with shole_constraint lhole_constraint,
                    split, {
                        intros i_in_shole i_in_stores,
                        cases i_in_stores, {
                            subst i_in_stores,
                            apply i_is_var,
                            refl,
                        }, {
                            specialize shole_constraint (or.inl i_in_shole),
                            contradiction,
                        },
                    }, {
                        intro i_in_lhole,
                        apply lhole_constraint (or.inl i_in_lhole),
                    }
                }, {
                    rw [ts_update_split],
                    simp,
                    cases ups_hd,
                    {
                        simp only [ts_updates],
                        simp only [ts_update_split] at holes_constraint,
                        simp [ts_updates, -set.mem_union_eq] at holes_constraint,
                        specialize @ups_ih _ (list.reverse ups_tl),
                        swap,
                        {
                            split, {
                                intro,
                                apply store_stores,
                                apply holes_constraint.left a,
                            }, {
                                intro,
                                apply store_loads,
                                apply holes_constraint.right a,
                            },
                        },
                        simp [thread_state.tlocal_to_shared, store],
                        simp [thread_state.tlocal_to_shared, store] at ups_ih,
                        split,
                        {
                            intros i_in_shole,
                            rw not_or_distrib,
                            split, {
                                -- proof that the new store doesn't store in i
                                cases holes_constraint with shole_constraint lhole_constraint,
                                simp [thread_state.tlocal_to_shared, store] at shole_constraint,
                                rw not_or_distrib at shole_constraint,
                                cases shole_constraint (or.inl i_in_shole),
                                rw ts_updates_tlocal s'.shared s'.loads s'.stores,
                                simp,
                                have : s' = {tlocal := s'.tlocal, shared := s'.shared, loads := s'.loads, stores := s'.stores} := begin
                                    cases s',
                                    simp,
                                end,
                                rw ← this,
                                assumption,
                            }, {
                                apply ups_ih.left i_in_shole,
                            }
                        }, {
                            intros i_in_lhole,
                            apply ups_ih.right i_in_lhole,
                        }
                    }, {
                        -- the head element is compute_list
                        simp [ts_updates],
                        apply ups_ih,
                        swap 3,
                        exact list.reverse ups_tl,
                        rw [ts_update_split] at holes_constraint,
                        simp [ts_updates] at holes_constraint,
                        simp,
                        exact holes_constraint,
                        simp,
                    }
                },
            }, {
                specialize holes_constraint i tid,
                rw ← map_active_threads_nth_inac tid_activeness,
                rw ← map_active_threads_nth_inac tid_activeness at holes_constraint,
                simp *,
                intro a,
                apply holes_constraint.right (or.inl a),
            }
        }
    },
end


import topology.sheaves.sheaf
import algebra.category.Group.abelian
import algebra.category.Group.colimits
import algebra.category.Group.limits
import topology.sheaves.sheaf_condition.sites
import group_epi_mono

noncomputable theory

section Ab

open Top category_theory opposite
open category_theory.limits

universe u

namespace AddCommGroup

def range_to_image {A B : Ab} (f : A ⟶ B) : mono_factorisation f :=
{ I := ⟨f.range⟩,
  m :=
  { to_fun := λ y, y.1,
    map_add' := λ _ _, rfl,
    map_zero' := rfl },
  m_mono := { right_cancellation := λ C g h eq1, begin
    ext1 x,
    replace eq1 := add_monoid_hom.congr_fun eq1 x,
    simpa only [comp_apply, add_monoid_hom.coe_mk, set_like.coe_eq_coe, subtype.val_eq_coe] using eq1,
  end },
  e :=
  { to_fun := λ a, ⟨f a, ⟨_, rfl⟩⟩,
    map_add' := λ _ _, by { simp only [map_add, subtype.ext_iff_val], refl, },
    map_zero' := by simp only [map_zero, subtype.ext_iff_val, show (0 : f.range).1 = 0, from rfl] } }.

lemma range_is_image {A B : Ab} (f : A ⟶ B) : is_image (range_to_image f) :=
{ lift := λ F,
  { to_fun := λ x, F.e (classical.some x.2), -- x ∈ f.range so x.2 says that ∃ y, f y = x
    map_zero' := begin
      have h : (0 : B) ∈ f.range := ⟨0, by rw map_zero⟩,
      have eq1 := classical.some_spec h,
      have eq2 := add_monoid_hom.congr_fun F.fac' (classical.some h),
      erw eq1 at eq2,
      have h2 : function.injective F.m,
      { apply add_monoid_hom.inj_of_mono F.m, },
      apply h2,
      rw map_zero,
      convert eq2,
    end,
    map_add' := λ ⟨_, ⟨x, rfl⟩⟩ ⟨_, ⟨y, rfl⟩⟩, begin
      rw ← map_add,
      apply_fun F.m using (add_monoid_hom.inj_of_mono F.m),
      have : ∀ x, F.m (F.e x) = f x := add_monoid_hom.congr_fun F.fac',
      rw [this, this],
      have t1 : ((⟨f x, ⟨x, rfl⟩⟩ : f.range) + (⟨f y, ⟨y, rfl⟩⟩ : f.range)).1 ∈ f.range := ⟨x + y, by simpa only [map_add, subtype.ext_iff_val]⟩,
      change ∃ _, _ at t1,
      have  := classical.some_spec t1,
      erw this,
      change f x + f y = _,
      rw map_add,
      have t2 : (⟨f x, ⟨_, rfl⟩⟩ : f.range).1 ∈ f.range := ⟨x, rfl⟩,
      have t3 : (⟨f y, ⟨_, rfl⟩⟩ : f.range).1 ∈ f.range := ⟨y, rfl⟩,
      change ∃ _, _ at t2,
      change ∃ _, _ at t3,
      have := classical.some_spec t2,
      erw this,
      have := classical.some_spec t3,
      erw this,
    end },
  lift_fac' := λ F, begin
    ext,
    change F.m (F.e _) = x.1,
    have eq1 : ∀ y, F.m (F.e y) = f y := add_monoid_hom.congr_fun F.fac',
    rw eq1,
    have t1 : x.1 ∈ f.range := x.2,
    change ∃ _, _ at t1,
    have := classical.some_spec t1,
    erw this,
  end }

end AddCommGroup

end Ab

section sheaf_has_image

open Top category_theory opposite
open category_theory.limits

universe u

variables {T : Top.{u}}

namespace Top.presheaf

section presheaf

open Top.presheaf

def presheaf.image' {F G : presheaf Ab T} (f : F ⟶ G) : presheaf Ab T :=
{ obj := λ U, image (f.app U),
  map := λ U V inc, begin
    refine (is_image.iso_ext (AddCommGroup.range_is_image (f.app U)) (image.is_image (f.app U))).inv ≫ _ ≫
      (is_image.iso_ext (AddCommGroup.range_is_image (f.app V)) (image.is_image (f.app V))).hom,
    refine
    { to_fun := λ x, ⟨f.app V (F.map inc (classical.some x.2)), ⟨_, rfl⟩⟩,
      map_add' := sorry,
      map_zero' := sorry },
  end,
  map_id' := sorry,
  map_comp' := sorry }

def presheaf.image'_ι {F G : presheaf Ab T} (f : F ⟶ G) : presheaf.image' f ⟶ G :=
{ app := λ U, image.ι _,
  naturality' := sorry }

def presheaf.image'_e {F G : presheaf Ab T} (f : F ⟶ G) : F ⟶ presheaf.image' f :=
{ app := λ U, factor_thru_image (f.app U),
  naturality' := sorry }

def presheaf.mono_factorisation {F G : presheaf Ab T} (f : F ⟶ G) : mono_factorisation f :=
{ I := presheaf.image' f,
  m := presheaf.image'_ι f,
  m_mono := sorry,
  e := presheaf.image'_e f,
  fac' := begin
    ext U x,
    simp only [comp_apply, nat_trans.comp_app],
    change (image.ι (f.app U)) (factor_thru_image (f.app U) x) = _,
    erw add_monoid_hom.congr_fun (image.fac (f.app U)) x,
  end }

def presheaf.image_factorisation {F G : presheaf Ab T} (f : F ⟶ G) : image_factorisation f :=
{ F := presheaf.mono_factorisation f,
  is_image := sorry }

instance {F G : presheaf Ab T} (f : F ⟶ G) : has_image f :=
{ exists_image := ⟨presheaf.image_factorisation f⟩ }
instance : has_images (presheaf Ab T) :=
{ has_image := λ F G f, by apply_instance }

end presheaf

section sheaf

open Top.presheaf category_theory.grothendieck_topology Top topological_space

variable [Π (X : opens T), preserves_colimits_of_shape ((opens.grothendieck_topology T).cover X)ᵒᵖ (forget Ab.{u})]

-- sheafify `image f`
def sheaf.image' {F G : sheaf Ab T} (f : F ⟶ G) : sheaf Ab T :=
let f' : (F.1 : presheaf Ab T) ⟶ (G.1 : presheaf Ab T) := f in
(Sheaf_sites_to_sheaf_spaces Ab T).obj ((presheaf_to_Sheaf (opens.grothendieck_topology T) _).obj  (image f'))

def sheaf.image'_ι {F G : sheaf Ab T} (f : F ⟶ G) : sheaf.image' f ⟶ G := sorry
def sheaf.image'_e {F G : sheaf Ab T} (f : F ⟶ G) : F ⟶ sheaf.image' f := sorry

def sheaf.mono_factorisation {F G : sheaf Ab T} (f : F ⟶ G) : mono_factorisation f :=
let f' : (F.1 : presheaf Ab T) ⟶ (G.1 : presheaf Ab T) := f in
{ I := sheaf.image' f,
  m := sheaf.image'_ι f,
  m_mono := sorry,
  e := sheaf.image'_e f,
  fac' := sorry }

#check Top.presheaf.category_theory.limits.has_image
def sheaf.image_factorisation {F G : sheaf Ab T} (f : F ⟶ G) : image_factorisation f :=
{ F := sheaf.mono_factorisation f,
  is_image := sorry }

instance sheaf.has_image {F G : sheaf Ab T} (f : F ⟶ G) : has_image f :=
{ exists_image := ⟨sheaf.image_factorisation f⟩ }

instance : has_images (sheaf Ab T) :=
{ has_image := λ F G f, sheaf.has_image f }

end sheaf
end Top.presheaf

end sheaf_has_image


import ClausalExtraction.Context

open Lean
open Lean.Meta

namespace ClausalExtraction

section ExpressionUtils

private def propExpr : Expr := mkSort levelZero

def exprHead : Expr → Option Name
| Expr.bvar .. => none
-- FIXME: Figure out if we want to see through free variables with let bindings in local context.
| Expr.fvar .. => none
| Expr.mvar .. => none
| Expr.sort .. => none
| Expr.const nm .. => nm
| Expr.app f a .. => exprHead f
| Expr.lam .. => none
| Expr.forallE .. => none
-- FIXME: Figure out if we want to see through let bindings.
| Expr.letE .. => none
| Expr.lit .. =>  none
-- FIXME: Figure out if ignoring metadata is always ok.
| Expr.mdata  _ x .. => exprHead x
 -- FIXME: Figure out if we want to see through projections.
| Expr.proj n i s .. => none

end ExpressionUtils

def falseExpr : Expr := mkConst ``False

-- | Assertion added
structure Assertion where
  -- Identifies the free variable in the initial context used to generate this one.
  -- Used to allow deleting original proposition from local context.
  origin : Option FVarId
  -- The free variable used to identify the assertion in the local context
  -- generated for the assertion.
  name : Name
  -- Predicate that was asserted.
  pred : Pred
  -- A proof of the predicate in the context of the original goal.
  --
  -- N.B. It may be ideal to explore if we can defer producing this until needed.
  proof : Expr

namespace Assertion

instance : Inhabited Assertion := ⟨
     { origin := arbitrary,
       name := arbitrary,
       pred := arbitrary,
       proof := arbitrary,
     }
  ⟩

end Assertion

structure State where
  -- Predicates asserted to be true.
  assertions : Array Assertion := #[]

namespace State

-- | Return proofs associated with assertions.
def proofs (s:State) : Array Expr := s.assertions.map Assertion.proof

end State

/--
This contains the inferred satisfiability problem from a lean proof.
-/
structure SatProblem where
  -- Identifier of original goal to prove
  goalId : MVarId
  -- Context used to convert to a decision proedure
  context : Context
  -- Assumptions inferred
  state : State
  -- Given a proof from the solver this constructs a proof of the goal.
  onMkProof : Expr → Expr

namespace SatProblem

-- | Add assertions to local context.
-- Return new local context along with array of free variables identifying assertions.
def adjustLocalContext (ctx:Context) (s:State) (lctx:LocalContext) : MetaM (LocalContext × Array Expr) := do
  let mut lctx := lctx
  let mut fvars : Array Expr := #[]
  for a in s.assertions do
    let newVar ← mkFreshFVarId
    fvars := fvars.push (mkFVar newVar)
    match a.origin with
    | some prevVar =>
      -- FIXME: Make sure erasing the previous declaration does not break things.
      lctx := lctx.erase prevVar
    | none =>
      pure ()
    let e ← ctx.predExpr a.pred
    lctx := lctx.mkLocalDecl newVar a.name e
  pure (lctx, fvars)

/-
This does the work of actually applying the normalization procedure to the problem.

It is setup as an extra step so that a decision procedure could be tried before
committing to using the normalization procedure.
-/
def apply (g:SatProblem) : MetaM (List MVarId) := do
  let tag   ← getMVarTag g.goalId
  let lctx ← getLCtx
  let (lctx, fvars) ← adjustLocalContext g.context g.state lctx
  withReader (fun ctx => { ctx with lctx := lctx }) $ do
    let solverSatProblem ← mkFreshExprSyntheticOpaqueMVar falseExpr tag
    -- Turn goal into lambda with free variables from assertions
    let fn ← Meta.mkLambdaFVars fvars solverSatProblem
    -- Assign goal
    assignExprMVar g.goalId (g.onMkProof (mkAppN fn g.state.proofs))
    pure [solverSatProblem.mvarId!]

end SatProblem

def matchNot (e:Expr) : MetaM (Option Expr) := do
  let mvar ← mkFreshExprMVar propExpr MetavarKind.natural `a
  let p := mkApp (mkConst ``Not) mvar
  pure $ if ← isDefEq p e then mvar else none

-- | See if we can add the local declaration to the arithmetic unit.
-- proof is a term of type prop.
def tryAddPred (ctx:Context)
                 (r:IO.Ref State)
                 (origin : Option FVarId)
                 (name:Name)
                 (proof:Expr)
                 (prop:Expr) : MetaM Bool := do
  let rules ← ctx.propTheoryMap.getMatch prop
  for t in rules do
    match ← t.action prop ctx.services with
    | none => pure ()
    | some (p, proofFn) => do
      let pred := Pred.mkPred t.theoryRef p
      let a := { origin := origin, name := name, proof := mkApp proofFn proof, pred := pred }
      r.modify $ λs => { s with assertions := s.assertions.push a }
      return true
  return false

theorem decidable_by_contra {P:Prop} [h:Decidable P] (q : ¬¬P) : P :=
  match h with
  | isFalse h => False.elim (q h)
  | isTrue h => h

def processDecls (ctx:Context) (r:IO.Ref State) (lctx:LocalContext) : MetaM Unit := do
  let mut seenFirst := false
  for md in lctx.decls do
    -- Skip first declaration (as it corresponds to initial goal for
    -- recursive purposes)
    unless seenFirst do
      seenFirst := true
      continue
    -- Skip if declaration has been deleted.
    let some d ← pure md
         | continue
    -- FIXME: Figure out how to support let declarations
    if d.isLet then
      continue
    -- If this is a proposition then try assuming it.
    if ← isDefEq (← inferType d.type) propExpr then do
      let _ ← tryAddPred ctx r (some d.fvarId) d.userName (mkFVar d.fvarId) d.type

-- | This analyze the goal of the problem.
def analyzeSatProblem (ctx:Context) (r : IO.Ref State) (tactic:Name) (goalId:MVarId) (target:Expr)
   : MetaM (Expr → Expr) := do
  -- If goal is already false, then just negate it.
  if ← isDefEq target falseExpr then
    return id
  -- If goal has form `Not p` then we can diretly process property.
  match ← matchNot target with
  | some prop => do
    if ← tryAddPred ctx r none `negGoal (mkBVar 0) prop then
      return (mkLambda Name.anonymous BinderInfo.default prop)
  -- Goal is not a negation, so we seeing if we can extract a predicate from
  -- negated goal and then use decidable_by_contra to get proof of negation
  -- of target.
  | none =>
    let prop := mkApp (mkConst ``Not) target
    if ← tryAddPred ctx r none `negGoal (mkBVar 0) prop then
      let some classVal ← synthInstance? (mkApp (mkConst ``Decidable) target)
          | throwTacticEx tactic goalId "Could not synthesize Decidable instance for proposition:."
      let decideByContraExpr := mkAppN (mkConst ``decidable_by_contra) #[target, classVal]
      return (fun goalProof => mkApp decideByContraExpr (mkLambda Name.anonymous BinderInfo.default prop goalProof))

  -- In final case, we just ignore the goal and try to prove a contradiction from assumptions
  -- alone.
  match ← inferType target with
  | Expr.sort lvl _ =>
    pure (mkApp (mkApp (mkConst ``False.elim [lvl]) target))
  | _ =>
    throwTacticEx tactic goalId "Expected a type."

def mkGoal (ctx:Context) (tactic:Name) (goalId:MVarId) : MetaM SatProblem := do
  withMVarContext goalId do
    let md ← getMVarDecl goalId
    let stateRef ← IO.mkRef {}
    processDecls ctx stateRef md.lctx
    let fn ← analyzeSatProblem ctx stateRef tactic goalId md.type
    let s ← stateRef.get
    pure { goalId := goalId, context := ctx, state := s, onMkProof := fn }

end ClausalExtraction


import analysis.inner_product_space.pi_L2
import topology.paracompact

--OUTPUT 1
theorem ℝn_paracompact (n : ℕ) : paracompact_space (euclidean_space ℝ (fin n)) :=
begin
  -- Let $\mathcal{A}$ be an open covering of $\mathbb{R}^n$
  assume A : set (euclidean_space ℝ (fin n)),
  assume hA : is_open_cover A,

  -- We now construct a locally finite open refinement $\mathcal{C}$ of $\mathcal{A}$ that covers $\mathbb{R}^n$.
  let C := {s : euclidean_space ℝ (fin n) | ∃ t ∈ A, ∃ m : ℕ, ∃ r : ℝ, s = ball t m r ∩ (euclidean_space ℝ (fin n)) \ {x | ∃ s ∈ C, ∃ m : ℕ, ball s m 1 ⊂ ball x m 1}},

  -- First, we define a collection of pen balls
  have h1 : {s : euclidean_space ℝ (fin n) | ∃ m : ℕ,ball s m 1 ⊂ ball 0 m 1} = ⋃ r : ℕ, {s : euclidean_space ℝ (fin n) | ∃ m : ℕ,ball s m 1 ⊂ ball 0 m 1} ∩ (euclidean_space ℝ (fin n)) \ {x | ∃ s ∈ ⋃ r : ℕ, {s : euclidean_space ℝ (fin n) | ∃ m : ℕ,ball s m 1 ⊂ ball 0 m 1} ∩ (euclidean_space ℝ (fin n)) \ {x | ∃ s ∈ ⋃ r : ℕ, {s : euclidean_space ℝ (fin n) | ∃ m : ℕ,ball s m 1 ⊂ ball 0 m 1} ∩ (euclidean_space ℝ (fin n)) \ {x | ∃ s ∈ ⋃ r : ℕ, {s : euclidean_space ℝ (fin n) | ∃ m : ℕ,ball s m 1 ⊂ ball 0 m 1} ∩ (euclidean_space ℝ (fin n)) \ {x | ∃ s ∈ ⋃ r : ℕ, {s : euclidean_space ℝ (fin n) | ∃ m : ℕ,ball s m 1 ⊂ ball 0 m 1} ∩ (euclidean_space ℝ (fin n)) \ {x | ∃ s ∈ ⋃ r : ℕ, {s : euclidean_space ℝ (fin n) | ∃ m : ℕ,ball s m 1 ⊂ ball 0 m 1} ∩ (euclidean_space ℝ (fin n)) \ {x | ∃ s ∈ ⋃ r : ℕ, {s : euclidean_space ℝ (fin n) | ∃ m : ℕ,ball s m 1 ⊂ ball 0 m 1} ∩ (euclidean_space ℝ (fin n)) \ {x | ∃ s ∈ ⋃ r : ℕ, {s : euclidean_space ℝ (fin n) | ∃ m : ℕ,ball s m 1 ⊂ ball 0 m 1} ∩ (euclidean_space ℝ (fin n)) \ {x | ∃ s ∈ ⋃ r : ℕ, {s : euclidean_space ℝ (fin n) | ∃ m : ℕ,ball s m 1 ⊂ ball 0 m 1} ∩ (euclidean_space ℝ (fin n)) \ {x | ∃ s ∈ ⋃ r : ℕ, {s : euclidean_space ℝ (fin n) | ∃ m : ℕ,ball s m 1 ⊂ ball 0 m 1} ∩ (euclidean_space ℝ (fin n)) \ {x | ∃ s ∈ ⋃ r : ℕ, {s : euclidean_space ℝ (fin n) | ∃ m : ℕ,ball s m 1 ⊂ ball 0 m 1} ∩ (euclidean_space ℝ (fin n)) \ {x | ∃ s ∈ ⋃ r : ℕ, {s : euclidean_space ℝ (fin n) | ∃ m : ℕ,ball s m 1 ⊂ ball 0 m 1} ∩ (euclidean_space ℝ (fin n)) \ {x | ∃ s ∈ ⋃ r : ℕ, {s : euclidean_space ℝ (fin n) | ∃ m : ℕ,ball s m 1 ⊂ ball 0 m 1} ∩ (euclidean_space ℝ (fin n)) \ {x | ∃ s ∈ ⋃ r : ℕ, {s : euclidean_space ℝ (fin n) | ∃ m : ℕ,ball s m 1 ⊂ ball 0 m 1} ∩ (euclidean_space ℝ (fin n)) \ {x | ∃ s ∈ ⋃ r : ℕ, {s : euclidean_space ℝ (fin n) | ∃ m : ℕ,ball s m 1 ⊂ ball 0 m 1} ∩ (euclidean_space ℝ (fin n)) \ {x | ∃ s ∈ ⋃ r : ℕ, {s : euclidean_space ℝ (fin n) | ∃ m : ℕ,ball s m 1 ⊂ ball 0 m 1} ∩ (euclidean_space ℝ (fin n)) \ {x | ∃ s ∈ ⋃ r : ℕ, {s : euclidean_space ℝ (fin n) | ∃ m : ℕ,ball s m 1 ⊂ ball 0 m 1} ∩ (euclidean_space ℝ (fin n)) \ {x | ∃ s ∈ ⋃ r : ℕ, {s : euclidean_space ℝ (fin n) | ∃ m : ℕ,ball s m 1 ⊂ ball 0 m 1} ∩ (euclidean_space ℝ (fin n)) \ {x | ∃ s ∈ ⋃ r : ℕ, {s : euclidean_space ℝ (fin n) | ∃ m : ℕ,ball s m 1 ⊂ ball 0 m 1} ∩ (euclidean_space ℝ (fin n)) \ {x | ∃ s ∈ ⋃ r : ℕ, {s : euclidean_space ℝ (fin n) | ∃ m : ℕ,ball s m 1 ⊂ ball 0 m 1} ∩ (euclidean_space ℝ (fin n)) \ {x | ∃ s ∈ ⋃ r : ℕ, {s : euclidean_space ℝ (fin n) | ∃ m : ℕ,ball s m 1 ⊂ ball 0 m 1} ∩ (euclidean_space ℝ (fin n)) \ {x | ∃ s ∈ ⋃ r : ℕ, {s : euclidean_space ℝ (fin n) | ∃ m : ℕ,ball s m 1 ⊂ ball 0 m 1} ∩ (euclidean_space ℝ (fin n)) \ {x | ∃ s ∈ ⋃ r : ℕ, {s : euclidean_space ℝ (fin n) | ∃ m : ℕ,ball s m 1 ⊂ ball 0 m 1} ∩ (euclidean_space ℝ (fin n)) \ {x | ∃ s ∈ ⋃ r : ℕ, {s : euclidean_space ℝ (fin n) | ∃ m : ℕ,ball s m 1 ⊂ ball 0 m 1} ∩ (euclidean_space ℝ (fin n)) \ {x | ∃ s ∈ ⋃ r : ℕ, {s : euclidean_space ℝ (fin n) | ∃ m : ℕ,ball s m 1 ⊂ ball 0 m
end --Needs more than 2000 tokens!

--OUTPUT 2
theorem ℝn_paracompact (n : ℕ) : paracompact_space (euclidean_space ℝ (fin n)) := sorry

theorem  ℝ_paracompact : paracompact_space euclidean : paracompact_space (euclidean_space ℝ (fin 1)) := sorry

/-`theorem`
Twelve Fold Symmetry of Rhombic Dodecahedron
Let $\struct {D, \mathcal{E}}$ be a rhombic dodecahedron. Then $D$ has a twelve-fold symmetry.
`proof`
Let the twelve edges of $D$ be denoted by $\mathcal{E} = \anset{E_1, E_2, \ldots, E_{12}}$. Since every vertex has valence 3, every edge is incident to two vertices. Let $\mathcal{V}$ denote the set of all vertices of $D$. Then it follows that $\mathcal{E} \subseteq \mathcal{V} \times \mathcal{V}$. We now define a binary relation $R$ on $\mathcal{V}$ such that for all $v, w \in \mathcal{V}$:

- $v R w$ if and only if $(v, w) \in \mathcal{E}$

Let $p$ be a permutation of $\anset{E_1, E_2, \ldots, E_{12}}$. We now define a map $f_p$ on $\mathcal{V}$ such that for any $v \in \mathcal{V}$:

- $f_p(v) = w$ such that $(v, w) \in \mathcal{E}$ and $(v, w) \in p(\mathcal{E})$

We now prove $\mathcal{V}$ has a partition $\mathcal{V} = \bigcup_{v \in \mathcal{V}} C_v$ such that for every $v \in \mathcal{V}$:

- $v \in C_v$,
- $C_v \cap C_w = \phi$ if $v \not= w$, and
- $f_p(x) = y$ for some $x \in C_v$ and for some $y \in C_w$ if and only if $x = y$ and $v = w$.

To prove that $D$ has a twelve-fold symmetry, we will then prove that for every $x \in \mathcal{V}$, there exists a $y \in \mathcal{V}$ such that $x R y$. ${pf}$
{{qed}}
-/
theorem rhombic_dodecahedron_12_fold_symmetry : ∃ f : perm (fin 12) → perm (fin 12), ∀ x : fin 12, f x = x := sorry

end

--OUTPUT 3
theorem ℝn_paracompact (n : ℕ) : paracompact_space (euclidean_space ℝ (fin n)) :=
begin
  -- Let $\mathcal{A}$ be an open covering of $\mathbb{R}^n$.
  assume A : set (euclidean_space ℝ (fin n)),
  assume h1 : is_open A,
  assume h2 : ∀ x : euclidean_space ℝ (fin n), ∃ U ∈ A, x ∈ U,

  -- We now construct a locally finite open refinement $\mathcal{C}$ of $\mathcal{A}$ that covers $\mathbb{R}^n$.
  let C1 : set (euclidean_space ℝ (fin n)) := {B : set (euclidean_space ℝ (fin n)) | ∃ (b : euclidean_space ℝ (fin n)) (m : ℝ),
    is_open B ∧ is_ball ℝ ℝ ((b : euclidean_space ℝ (fin n)).to_fun) m ∧ ⋂ (x : euclidean_space ℝ (fin n)) (h3 : x ∈ B),
    ∃ (U : euclidean_space ℝ (fin n)) (h4 : U ∈ A) (h5 : x ∈ U), B ⊆ U},
  have h3 : ∀ B : set (euclidean_space ℝ (fin n)), ∃ B : set (euclidean_space ℝ (fin n)) (m : ℝ),
    is_open B ∧ is_ball ℝ ℝ (0 : ℝ^(fin n)) m ∧ ⋂ (x : euclidean_space ℝ (fin n)) (h3 : x ∈ B),
    ∃ (U : euclidean_space ℝ (fin n)) (h4 : U ∈ A) (h5 : x ∈ U), B ⊆ U, from by {
      assume B : set (euclidean_space ℝ (fin n)),
      have h4 : ∃ U ∈ A, B ⊆ U, from by {
        have h5 : ∅ ∈ A, from by {
          by_contradiction h6,
          have h7 : ∃ x : euclidean_space ℝ (fin n), x ∉ A, from by {
            let f : ℕ → (euclidean_space ℝ (fin n)), from by {
              assume z : ℕ,
              use (z : ℝ) 0,
            },
            have h8 : ∀ (z : ℕ) (h9 : z ∈ f ⁻¹' A), false, from by {
              assume (z : ℕ) (h9 : z ∈ f ⁻¹' A),
              have h10 : f z ∈ A, from by  {
                simp at h9,
                exact h9,
              },
              have h11 : (0 : ℝ) 0 ∈ A, from by {
                simp at h10,
                exact h10,
              },
              have h12 : ∃ U ∈ A, (0 : ℝ) 0 ∈ U, from by {
                have h13 : ∃ U ∈ A, (0 : ℝ) 0 ∈ U, from by {
                  assume h14,
                  have h15 : (0 : ℝ) 0 ∉ A, from by {
                    assume h16,
                    have h17 : ∅ ∈ A, from by {
                      apply set.subset.subset_singleton,
                      assume x : euclidean_space ℝ (fin n),
                      assume h18 : x ∈ ∅,
                      exact h16,
                    },
                    show false, from h14 h17,
                  },
                  show false, from h15 h16,
                },
                show ∃ U ∈ A, (0 : ℝ) 0 ∈ U, from h13,
              },
              rw show (0 : ℝ) 0 = f z, from rfl,
              exact h12,
            },
            have h14 : f '' A = ∅, from by {
              apply set.subset.antisymm,
              {
                assume x : ℕ,
                assume h15 : x ∈ f '' A,
                show false, from h8 x h15,
              },
              {
                assume x : ℕ,
                assume h15 : x ∈ f '' A,
                show false, from h8 x h15,
              },
            },
            have h16 : A ≠ ∅, from by {
              assume h17,
              have h18 : ∅ = f ''  A, from by {
                rw show ∅ = f '' A, from eq.symm h17,
              },
              show false, from h14 h18,
            },
            show ∃ x : euclidean_space ℝ (fin n), x ∉ A, from ⟨f 0,h16⟩,
          },
          have h8 : ∅ ∉ A, from by {
            assume h9,
            show false, from h5 h9,
          },
          exact h8,
        },
        have h6 : ∃ (U : euclidean_space ℝ (fin n)) (h7 : U ∈ A), ∅ ⊆ U, from ⟨h5,univ_subset_iff.mpr (λ x, true.intro)⟩,
        exact h6,
      },
      let d : ℝ, from by {
        have h7 : ∃ d : ℝ, ∀ (x : ℝ) (h8 : x ∈ B), d < dist x 0, from by {
          have h9 : ∃ (x : ℝ) (h10 : x ∈ B), d < dist x 0, from by {
            have h11 : ∃ (x : ℝ) (h12 : x ∈ B), d < dist x 0, from by {
              have h13 : ∃ (x : ℝ) (h14 : x ∈ B), dist x 0 < d + 1, from by {
                let f : (euclidean_space ℝ (fin n)) → ℝ, from by {
                  assume x : euclidean_space ℝ (fin n),
                  use x.to_fun.sum,
                },
                have h16 : ∀ (g : (euclidean_space ℝ (fin n)) → ℝ), ∃ (x : ℝ) (h15 : x ∈ B), dist x 0 < d + 1, from by {
                  assume (g : (euclidean_space ℝ (fin n)) → ℝ),
                  have h17 : ∃ (x : ℝ) (h15 : x ∈ B), dist x 0 < d + 1, from by {
                    have h18 : ∃ (x : ℝ) (h15 : x ∈ B), dist x 0 < d + 1, from by {
                      have h19 : ∃ (x : ℝ) (h15 : x ∈ B), dist x 0 < d + 1, from by {
                        have h21 : ∃ (x : ℝ) (h15 : x ∈ B), dist x 0 < d + 1, from by {
                          have h23 : ∃ (x : ℝ) (h15 : x ∈ B), dist x 0 < d + 1, from by {
                            have h24 : ∃ (x : ℝ) (h15 : x ∈ B), dist x 0 < d + 1, from by {
                              use f 0,
                              have h25 : ∃ (U : euclidean_space ℝ (fin n)) (h26 : U ∈ A) (h27 : f 0 ∈ U), B ⊆ U, from by {
                                use h4.left,
                                use h4.right.left,
                                use h4.right.right.left,
                                use h4.right.right.right,
                              },
                              have h28 : f 0 ∈ B, from by {
                                simp at h25,
                                exact h25,
                              },
end --Needs more than 2000 tokens!

--OUTPUT 4
theorem ℝn_paracompact (n : ℕ) : paracompact_space (euclidean_space ℝ (fin n)) :=
begin
  let A := λ x : fin n, Ioo x x,
  let K := λ m : ℕ, { x : fin n | 0 ≤ (fintype.card x).val ∧ (fintype.card x).val < m+1},

  have memA_K : ∀ m : ℕ, A (K m) ∈ 𝓝 (0 : fin n), from
    by {
      assume (m : ℕ),
      use (λ x : fin n, x ∈ K m),
      exact ⟨by obviously, by obviously⟩,
    },

  have emptyset_in_memA : ∅ ∈ A ∅, from begin
    use (λ x : fin n, x = 0),
    have h1 : (λ x : fin n, x = 0) 0 = tt, from by obviously,
    have h2 : (λ x : fin n, x = 0) 0 ∈ 𝓝 0, from by {
      use (λ x : fin n, x = 0),
      show ∀ x, x = 0 → x ∈ 𝓝 0, from id,
      exact ⟨by obviously, by obviously⟩,
    },
    exact ⟨h1, h2⟩,
  end,

  have set_emptyset_in_memA_in_Km : ∀ m : ℕ, (A ∅) ∩ K m = ∅, from by {
    assume (m : ℕ),
    show (A ∅) ∩ K m = ∅, from begin
      rw set.inter_eq_self_of_subset_left,
      rw set.inter_eq_self_of_subset_left,
      intro x,
      exact ⟨assume h1, fintype.card_pos_iff.mpr h1.right.left, assume h2, h2.elim $ by {unfold_coes ∅, apply empty_ne_univ}⟩,
    end
  },

  have memA_Km_in_Km : ∀ m : ℕ, A (K m) ∈ A (K m), from by {
    assume m,
    show A (K m) ∈ A (K m), from begin
      use (λ x : fin n, x ∈ K m),
      have h1 : (λ x : fin n, x ∈ K m) ∈ 𝓝 (K m), from by {
        use (λ x : fin n, x ∈ K m),
        show ∀ x, x ∈ K m → x ∈ 𝓝 (K m), from by assume x h1, from memA_K m,
        exact ⟨by obviously, by obviously⟩
      },
      exact ⟨by obviously, h1⟩
    end
  },

  show paracompact_space (euclidean_space ℝ (fin n)), from paracompact_space.intro ⟨emptyset_in_memA, set_emptyset_in_memA_in_Km, memA_Km_in_Km⟩,
end

--OUTPUT 5
theorem ℝn_paracompact (n : ℕ) : paracompact_space (euclidean_space ℝ (fin n)) :=
begin

/-
  -- Let $\mathcal{A}$ be an open covering of $\mathbb{R}^n$.
  assume 𝒜 : opens (euclidean_space ℝ (fin n)),
  -- We now construct a locally finite open refinement $\mathcal{C}$ of $\mathcal{A}$ that covers $\mathbb{R}^n$.
  have h1 : ∃ 𝒞 : opens (euclidean_space ℝ (fin n)), euclidean_space ℝ (fin n).is_open_cover 𝒞 ∧
  -- refinement
  euclidean_space ℝ (fin n).is_open_refinement 𝒜 𝒞 ∧
  -- locally finite
  locally_finite_family 𝒞 ∧
  -- covers $\mathbb{R}^n$
  euclidean_space ℝ (fin n).is_open_cover 𝒞,
  from
  begin
    have h2 : ∃! (B₀ : opens (euclidean_space ℝ (fin n))), B₀ = ∅, from
      by {use (∅ : opens (euclidean_space ℝ (fin n))),
        obviously, obviously,},
    let B₀ := classical.some h2,
    have h3 : B₀ = ∅, from classical.some_spec (exists_unique.exists h2),

    -- Given $m$, set $\Bar{B_m}$ is compact in $\mathbb{R}^n$ by the Heine-Borel theorem, so choose finitely many elements of $\mathcal{A}$ that cover $\Bar{B_m}$ and intersect each one with the open set $\mathbb{R}^n \setminus \Bar{B_{m - 1}}$, and let $\mathcal{C}_{m}$ denote this collection of open sets (each an open subset of an element of $\mathcal{A}$).
    have h4 : ∀ (m : ℕ), ∃ ∃ (Cm : opens (euclidean_space ℝ (fin n))),
    have h5 : ∃ ∃ (Cm : opens (euclidean_space ℝ (fin n))),
      euclidean_space ℝ (fin n).is_open_refinement ∅ Cm ∧
      euclidean_space ℝ (fin n).is_open_refinement 𝒜 Cm ∧
      ∃ (m : ℕ),
      comp 𝒜 Cm ∧
      is_compact (euclidean_space ℝ (fin n) ∪ 𝒜 ∪ ∅ ∪ Cm),
      from
      begin
        use B₀,
        obviously, obviously,
        have h6 : 𝒜 = (∅ : opens (euclidean_space ℝ (fin n))), from 𝒜,
        split,
        obviously, obviously,
        obviously, obviously,
        obviously, obviously,
        obviously, obviously,
        obviously, obviously,
        obviously, obviously,
      end,
  end,
-/
end

--OUTPUT 6
theorem ℝn_paracompact (n : ℕ) : paracompact_space (euclidean_space ℝ (fin n)) :=
begin
  -- Let $\mathcal{A}$ be an open covering of $\mathbb{R}^n$
  assume (h : open_cover (euclidean_space ℝ (fin n))) (U : set (euclidean_space ℝ (fin n))),
  have h1 : ∃ (A : set (fin n → ℝ)), open_cover A, from h,
  have h2 : ∃ (A : set (fin n → ℝ)), is_open A ∧ (∀ (x : fin n → ℝ), x ∈ U → ∃ (a ∈ A), x ∈ a), from h1,
  have h3 : ∃ (A : set (fin n → ℝ)), (∀ (x : fin n → ℝ), x ∈ U → ∃ (a ∈ A), x ∈ a), from h2.left,

  -- We now construct a locally finite open refinement $\mathcal{C}$ of $\mathcal{A}$ that covers $\mathbb{R}^n$
  -- First, we define a collection of pen balls.
  -- Let $B_0 = \phi$, and for each $n \in \mathbb{N}$, let $B_m$ denote the ball of radius $m$
  -- centered at 0.
  have h4 : ∀ (m : ℕ), ∃ (Bm : set (fin n → ℝ)), (∀ (x : fin n → ℝ), (∃ (a ∈ Bm), x ∈ a) ↔ ∀ (y : fin n → ℝ), dist x y < m), from
    by {
      assume (m : ℕ),
      use {x | ∀ (y : fin n → ℝ), ∑ i, (x i - y i) ^ 2 =  m},
      assume (x : fin n → ℝ) (h : (∃ (a ∈ {x | ∀ (y : fin n → ℝ), ∑ i, (x i - y i) ^ 2 =  m}), x ∈ a) ↔
        ∀ (y : fin n → ℝ), dist x y < m),
      split,
      assume h1 : ∃ (a ∈ {x | ∀ (y : fin n → ℝ), ∑ i, (x i - y i) ^ 2 =  m}), x ∈ a,
      assume (y : fin n → ℝ),
      --have h2 : ∀ (y : fin n → ℝ), dist x y < m ↔ ∑ i, (x i - y i) ^ 2 = m, from cauchy_swartz_squared,
      have h2 : dist x y < m ↔ ∑ i, (x i - y i) ^ 2 = m, from cauchy_swartz_squared,
      have h3 : dist x y < m ↔ ∑ i, (x i - y i) ^ 2 < m, from by ring,
      have h4 : dist x y < m ↔ ∑ i, (x i - y i) ^ 2 < m ↔ x ∈ {x | ∀ (y : fin n → ℝ), ∑ i, (x i - y i) ^ 2 =  m},
        from h,
      rw [h4,h1.right],
      assume h1 : ∀ (y : fin n → ℝ), dist x y < m,
      rw h,
      rw h1,
      split,
      exact set.mem_set_of_eq (eq.symm (dist_self x)),
      assume (y : fin n → ℝ),
      rw eq.symm (dist_self x),
      rw h1,
      exact eq.symm (dist_self x),
    },

  -- Given $m$, set $\Bar{B_m}$ is compact in $\mathbb{R}^n$ by the Heine-Borel theorem, so choose finitely many elements of $\mathcal{A}$ that cover $\Bar{B_m}$
  -- and intersect each one with the open set $\mathbb{R}^n \setminus \Bar{B_{m - 1}}$, and let $\mathcal{C}_{m}$ denote this collection of open sets (each an open subset of an element of $\mathcal{A}$).
  have h5 : ∀ (m : ℕ), ∃ (Cm : set (fin n → ℝ)), (∀ (x : fin n → ℝ), (∃ (a ∈ Cm), x ∈ a) ↔ ∀ (y : fin n → ℝ), dist x y < m) ∧ (∀ (x : fin n → ℝ), ∃ (z ∈ Cm), x ∈ z ↔ x ∈ {x | ∃ (y : fin n → ℝ), ∑ i, (x i - y i) ^ 2 =  m}),
    from by {
      assume (m : ℕ),
      have h6 : ∃ (Bm : set (fin n → ℝ)), (∀ (x : fin n → ℝ), (∃ (a ∈ Bm), x ∈ a) ↔ ∀ (y : fin n → ℝ), dist x y < m), from h4 m,
      have h7 : ∃ (Bm : set (fin n → ℝ)), (∀ (x : fin n → ℝ), ∃ (z ∈ Bm), x ∈ z ↔ x ∈ {x | ∀ (y : fin n → ℝ), ∑ i, (x i - y i) ^ 2 =  m}), from by {exact subset_finite_intersection_union (h6.left) (h3.left.left)},
      -- use A ∩ B
      use {x | ∃ (y : fin n → ℝ), ∑ i, (x i - y i) ^ 2 =  m} ∩ {x | ∀ (y : fin n → ℝ), dist x y < m},
      -- split
      split,
      assume (x : fin n → ℝ) (h : (∃ (a ∈ {x | ∃ (y : fin n → ℝ), ∑ i, (x i - y i) ^ 2 =  m} ∩ {x | ∀ (y : fin n → ℝ), dist x y < m}), x ∈ a) ↔ ∀ (y : fin n → ℝ), dist x y < m),
        by {
          split,
          assume h1 : ∃ (a ∈ {x | ∃ (y : fin n → ℝ), ∑ i, (x i - y i) ^ 2 =  m} ∩ {x | ∀ (y : fin n → ℝ), dist x y < m}), x ∈ a,
          rw h,
          rw h6,
          rw h1.right,
          assume h1 : ∀ (y : fin n → ℝ), dist x y < m,
          rw h,
          rw h6,
          rw h1,
          exact set.mem_set_of_eq (eq.symm (dist_self x)),
        },
      assume (x : fin n → ℝ) (h : ∃ (z ∈ {x | ∃ (y : fin n → ℝ), ∑ i, (x i - y i) ^ 2 =  m} ∩ {x | ∀ (y : fin n → ℝ), dist x y < m}), x ∈ z ↔ x ∈ {x | ∃ (y : fin n → ℝ), ∑ i, (x i - y i) ^ 2 =  m}),
        by {
          split,
          assume h1 : ∃ (z ∈ {x | ∃ (y : fin n → ℝ), ∑ i, (x i - y i) ^ 2 =  m} ∩ {x | ∀ (y : fin n → ℝ), dist x y < m}), x ∈ z,
          rw h3,
          rw h,
          rw h1.right,
          rw set.inter_def,
          assume h1 : x ∈ {x | ∃ (y : fin n → ℝ), ∑ i
end --Needs more than 2000 tokens!

--OUTPUT 7
theorem ℝn_paracompact (n : ℕ) : paracompact_space (euclidean_space ℝ (fin n)) :=
begin
  assume A : set (euclidean_space ℝ (fin n)),
  assume ha : (is_open_set_cover A),

  have h1 : ∀ m : ℕ, ∀ x : ℝ, ∃! n : ℕ, ∀ j : ℕ, n ≤ j → ∥x - 0∥ < j :=
    assume m : ℕ, assume x : ℝ, exists_unique.intro m
    (exists_unique.intro (le_of_lt (lt_add_one (abs x)))
    (exists_unique.intro rfl rfl)),

  let B_0 := {0 : ℝ^(fin n)},
  let B_m : ℕ → set (euclidean_space ℝ (fin n)) := assume m : ℕ,
    {x : ℝ^(fin n) | ∃ y : ℝ, ∃ m' : ℕ, ∥y∥ = ∥x∥ ∧ y ∈ B(0,m) ∧ m' = m ∧ m' ≤ m + 1 ∧ ∥y∥ < m'},
  let Bar_B_m_n : set (euclidean_space ℝ (fin n)) := λ m : ℕ, closure (B_m m),
  let Bar_B_m_n_plus_1 : set (euclidean_space ℝ (fin n)) := λ m : ℕ, closure (B_m (m+1)),
  let C_m_n : set (euclidean_space ℝ (fin n)) := λ m : ℕ, {(X ∩ (univ \ Bar_B_m_n m)) | X ∈ A ∧ X ⊆ (univ \ Bar_B_m_n_plus_1 m)},
  have h2 : ∀ m : ℕ, ∃ C : set (euclidean_space ℝ (fin n)), is_cover C A ∧ is_locally_finite C :=
    assume m : ℕ, let p := closure (B_m (m+1)) in exists.intro (λ m : ℕ, {(X ∩ (univ \ Bar_B_m_n m)) | X ∈ A ∧ X ⊆ (univ \ p)})
    (and.intro (is_cover_of_subcover (λ m : ℕ, {(X ∩ (univ \ Bar_B_m_n m)) | X ∈ A ∧ X ⊆ (univ \ p)}) (λ m : ℕ, (λ X : set (euclidean_space ℝ (fin n)), (X ∩ (univ \ Bar_B_m_n m)) ∈ A ∧ (X ∩ (univ \ Bar_B_m_n m)) ⊆ (univ \ p)) A)) (show is_locally_finite (λ m : ℕ, {(X ∩ (univ \ Bar_B_m_n m)) | X ∈ A ∧ X ⊆ (univ \ p)}), from
    (is_locally_finite_inter_compact_open (λ m : ℕ, {(X ∩ (univ \ Bar_B_m_n m)) | X ∈ A ∧ X ⊆ (univ \ p)}) (is_locally_finite_of_subcover (λ m : ℕ, {(X ∩ (univ \ Bar_B_m_n m)) | X ∈ A ∧ X ⊆ (univ \ p)}) (λ m : ℕ, {X ∈ A | X ⊆ (univ \ p)}) (is_cover_of_subcover (λ m : ℕ, {X ∈ A | X ⊆ (univ \ p)}) (λ m : ℕ, {X ∈ A | X ⊆ (univ \ p)} A)) (λ m : ℕ, is_open_set.inter_open_set (is_open_set.univ) (is_open_set_set.diff (is_open_set.univ) (B_m (m+1))))) (show is_compact_set (λ m : ℕ, closure (B_m (m+1))), from
      is_compact_set_union (is_compact_set (closure (B_m 0))) (show is_compact_set (λ m : ℕ, closure (B_m (m + 1))), from
        @is_compact_iff_closed_of_heine_borel_is_closed_of_uniform_continuity_of_order_is_compact_of_iota_nat_is_order (λ m : ℕ, closure (B_m (m + 1))) (is_compact_set (closure (B_m 0)))) (is_closed_set_closure (B_m 0)) (show ∀ m : ℕ, is_closed_set (closure (B_m (m + 1))), from
        assume m : ℕ, is_closed_set_closure (B_m (m+1))) (show ∀ m : ℕ, continuous (λ x : ℝ^(fin n), (λ m : ℕ, closure (B_m m)) (m + 1)) x, from
        assume m : ℕ, continuous_at_continuous_on_of_continuous_on_subset (by {apply continuous_closure_of_ball_at_point,} ) (show continuous_on (B_m (m + 1)) (λ x : ℝ^(fin n), (λ m : ℕ, closure (B_m m)) (m + 1)), from
        continuous_on_const)) (show uniform_continuity_on (λ x : ℝ^(fin n), (λ m : ℕ, closure (B_m m)) (m + 1)) (closure (B_m 0)), from
       uniform_continuity_on_of_uniform_continuity_of_subset_of_subset (show uniform_continuity (λ x : ℝ^(fin n), (λ m : ℕ, closure (B_m m)) (m + 1)), from
        (uniform_continuous_on.const (λ m : ℕ, closure (B_m m))).comp (uniform_continuous_on.fst (by {apply uniform_continuous_on_nat,}) (show uniform_continuous_on (λ n : ℕ, closure (B_m (n + 1))) (λ n : ℕ, n + 1), from
        continuous_on.comp (uniform_continuous_on.id (univ)) (show continuous_on (univ) (λ x : ℕ, x + 1), from
        continuous_on_add_one)))) (show (λ x : ℝ^(fin n), (λ m : ℕ, closure (B_m m)) (m + 1)) '' closure (B_m 0) ⊆ closure (B_m 0), from by {apply image_subset_closure,} ) (show ∀ x : ℝ^(fin n), (λ m : ℕ, closure (B_m m)) (m + 1) x ∈ closure (B_m 0), from
        assume x : ℝ^(fin n), by {apply closure_mono, rw iota_succ, apply set.mem_union_left,} )) (is_order_nat (λ m : ℕ, closure (B_m m))))) (is_order_nat (λ m : ℕ, B_m (m + 1))) (show ∀ n x : ℕ, B_m (n + 1) x ⟶ B_m (n + 1) x, from
        assume n x : ℕ, trivial) (show ∀ n x : ℕ, classically (∀ m : ℕ, (λ m : ℕ, closure (B_m m)) m x ⟶ (λ m : ℕ, closure (B_m m)) (m + 1) x), from
        assume n x :
end --Needs more than 2000 tokens!

--OUTPUT 8
theorem ℝn_paracompact (n : ℕ) : paracompact_space (euclidean_space ℝ (fin n)) :=
begin
  -- Let $\mathcal{A}$ be an open covering of $\mathbb{R}^n$.
  assume (U : set (euclidean_space ℝ (fin n))),
  assume (hU : is_open_cover U),

  -- We now construct a locally finite open refinement $\mathcal{C}$ of $\mathcal{A}$ that covers $\mathbb{R}^n$.
  show ∃ (W : set (euclidean_space ℝ (fin n))), is_open_cover W ∧ is_locally_finite W, from
  begin
    let W := set.univ,
    have hW : is_open_cover W, by obviously,
    have hW' : is_locally_finite W, by {sorry},
    have hW'' : W ⊆ U, by obviously,
    have hW_is_refinement : is_refinement W U, from is_refinement_intro hW'' hW,

    have hW'_is_refinement : is_refinement W U ∧ is_locally_finite W, from ⟨hW_is_refinement,hW'⟩,
    show is_locally_finite W ∧ ∃ V : set (euclidean_space ℝ (fin n)), is_open_cover V ∧ is_refinement W V, from hW'_is_refinement,
  end,
end

--OUTPUT 9
theorem ℝn_paracompact (n : ℕ) : paracompact_space (euclidean_space ℝ (fin n)) :=
begin
  /-
    Part 1:
    Goal: For any $x \in \mathbb{R}^n$, construct a ball around it.
  -/
  -- Let $x$ be a variable we chose from $\mathbb{R}^n$
  variable x : euclidean_space ℝ (fin n),
  -- Let $B_0$ be empty set
  def B_0 := ∅,
  -- Construct a ball centered at 0 with radius $m$
  def B_m : set (euclidean_space ℝ (fin n)) := {y : euclidean_space ℝ (fin n) | ∥y - 0∥ ≤ (↑m : ℝ) },
  -- Let $B_m$ be compact in $\mathbb{R}^n$, by Heine-Borel Theorem
  def Bar_B_m : compact_space (euclidean_space ℝ (fin n)) := by apply_instance,
  -- Let $\{B_m\}$ be the collection of all balls centered at 0 with radius $m \in \mathbb{N}$
  def f : ℕ → set (euclidean_space ℝ (fin n)) := λ (m : ℕ), B_m,

  /-
    Part 2:
    Goal: construct a nested collection of sets.
    Let $B_0 = \phi$ be empty set
  -/
  have h1 : B_0 ∪ (B_1 \ B_0) = B_1, from by obviously,
  have h2 : B_1 ∪ (B_2 \ B_1) = B_2, from by obviously,
  have h3 : B_2 ∪ (B_3 \ B_2) = B_3, from by obviously,
  have h4 : B_3 ∪ (B_4 \ B_3) = B_4, from by obviously,
  have h5 : B_4 ∪ (B_5 \ B_4) = B_5, from by obviously,
  -- ... and for any other $B_m, m \in \mathbb{N}$, we have this sequence
  have h6 : B_m ∪ (B_{m+1} \ B_m) = B_{m+1}, from by obviously,
  -- Let $C_0$ be empty set
  def C_0 := ∅,
  /-
    Construct all $C_m$ as follows.
    We have $C_m$ to be the ball centered at 0 with radius $m$ and an open set in $\mathbb{R}^n$
  -/
  def C_m : set (euclidean_space ℝ (fin n)) := {y : euclidean_space ℝ (fin n) | (∥y - 0∥ ≤ (m : ℝ)) ∧ (y ∈ set.univ)},
  -- Let $\{C_m\}$ be the collection of all $C_m$ for any $m \in \mathbb{N}$
  def g : ℕ → set (euclidean_space ℝ (fin n)) := λ (m : ℕ), C_m,
  -- Hence we have a nested sequence of open sets.
  have h7 : B_0 ⊂ B_1, from by rw [set.subset_empty_iff],
  have h8 : B_1 ⊂ B_2, from by rw [set.subset_empty_iff],
  have h9 : B_2 ⊂ B_3, from by rw [set.subset_empty_iff],
  have h10 : B_3 ⊂ B_4, from by rw [set.subset_empty_iff],
  have h11 : B_4 ⊂ B_5, from by rw [set.subset_empty_iff],
  -- ... and for any other $B_m, m \in \mathbb{N}$, we have this sequence
  have h12 : B_m ⊂ B_{m+1}, from by rw [set.subset_empty_iff],

  /-
    Part 3:
    Goal: Give the open cover $\mathcal{A}$, and construct a locally finite open refinement $\mathcal{C}$ of $\mathcal{A}$ that covers $\mathbb{R}^n$.
    Let $\mathcal{A}$ be an open covering of $\mathbb{R}^n$
  -/
  variable A : set (set (euclidean_space ℝ (fin n))),
  -- $\mathcal{A}$ is an open covering of $\mathbb{R}^n$
  assume hA : is_open_cover (euclidean_space ℝ (fin n)) A,
  -- By Heine-Borel theorem, $\Bar{B_m}$ is compact in $\mathbb{R}^n$
  -- Hence we pick finitely many elements of $\mathcal{A}$ that cover $\Bar{B_m}$
  have h13 : ∃ (F : set (set (euclidean_space ℝ (fin n)))), (F ⊆ A) ∧ is_open_cover (euclidean_space ℝ (fin n)) F ∧ compact_space.compact (euclidean_space ℝ (fin n)) F,
     from compact_space.exists_finite_open_cover Bar_B_m (by {apply_instance}),
  -- and intersect each one with the open set $\mathbb{R}^n \setminus \Bar{B_{m - 1}}$
  -- Let $C_m$ denote this collection of open sets (each an open subset of an element of $\mathcal{A}$)
  have h14 : ∃ (C_m : set (set (euclidean_space ℝ (fin n)))), (C_m ⊆ A) ∧ is_open_cover (euclidean_space ℝ (fin n)) C_m ∧ (∀ (x : euclidean_space ℝ (fin n)), x ∈ ⋃₀ C_m → (x ∈ B_m) → (x ∉ Bar_B_m)) ∧ (∀ (x : euclidean_space ℝ (fin n)), x ∈ ⋃₀ C_m → (x ∉ B_m) → (x ∈ Bar_B_m)),
    from exists.elim h13
    (assume (F : set (set (euclidean_space ℝ (fin n)))),
     assume (h15 : (F ⊆ A) ∧ is_open_cover (euclidean_space ℝ (fin n)) F ∧ compact_space.compact (euclidean_space ℝ (fin n)) F),
     assume (h16 : (∀ (x : euclidean_space ℝ (fin n)), x ∈ ⋃₀ F → (∃ (A : set (euclidean_space ℝ (fin n))), (A ∈ F) ∧ (x ∈ A)))),
     use {U : set (euclidean_space ℝ (fin n)) | (U ∈ F) ∧ (U ⊆ B_m)},
     show (∃ (C_m : set (set (euclidean_space ℝ (fin n)))), (C_m ⊆ A) ∧ is_open_cover (euclidean_space ℝ (fin n)) C_m ∧ (∀ (x : euclidean_space ℝ (fin n)), x ∈ ⋃₀ C_m → (x ∈ B_m) → (x ∉ Bar_B_m)) ∧ (∀ (x : euclidean_space ℝ (fin n)), x ∈ ⋃₀ C_m → (x ∉ B_m) → (x ∈ Bar_B_m))), from
     and.intro
     (and.intro (by obviously)
                (and.intro (by {rw [set.forall_mem_iff],
                             intros j jmem,
                             simp [set.mem_
end --Needs more than 2000 tokens!

--OUTPUT 10
theorem ℝn_paracompact (n : ℕ) : paracompact_space (euclidean_space ℝ (fin n)) :=
begin
  -- Let $\mathcal{A}$ be an open covering of $\mathbb{R}^n$. We now construct a locally finite open refinement $\mathcal{C}$ of $\mathcal{A}$ that covers $\mathbb{R}^n$. First, we define a collection of pen balls. Let $B_0 = \phi$, and for each $n \in \mathbb{N}$, let $B_m$ denote the ball of radius $m$
  -- centered at 0.
  fix (A : set (euclidean_space ℝ (fin n))) (hA : is_open A) (hA_cover : ∀ p ∈ 𝕜, ∃ U ∈ A, p ∈ U),

  let Bm := ball n 0 m,
  let j : ℕ → set (euclidean_space ℝ (fin n)) := λ i, Bm ⊓ A,
  let Πm := (∀ U ∈ j m, ∀ V ∈ j m, V ≠ U → V ∩ U ≠ ∅ → ∃ W ∈ A, W ⊆ U ∩ V),
  let ⟨U, hU⟩ := exists_is_open_ball 0 m,

  have h1 : ∀ x : euclidean_space ℝ (fin n), x ∈ U → T(x) < m, from begin
    assume (x : euclidean_space ℝ (fin n)) (hx : x ∈ U),
    by {
      unfold U Bm T,
      unfold ball,
      simp,
      assume hmx,
      exact lt_add_of_pos_of_le (show 0 < m, from nat.pos_of_ne_zero H) hmx,
    }
  end,

  have h2 : ∀ x : euclidean_space ℝ (fin n), x ∈ Bm → x ∈ U, from begin
    assume (x : euclidean_space ℝ (fin n)) (hx : x ∈ Bm),
    show x ∈ U, by {
      unfold U Bm T,
      unfold ball,
      simp,
      assume hmx,
      exact le_add_of_nonneg_of_le hmx (show 0 ≤ m, from nat.zero_le _),
    },
  end,

  have h3 : ∀ x : euclidean_space ℝ (fin n), x ∈ U → x ∈ Bm, from begin
    assume (x : euclidean_space ℝ (fin n)) (hx : x ∈ U),
    have h4 : ∀ x : euclidean_space ℝ (fin n), x ∈ U → T(x) < m, from h1 x hx,
    have h5 : ∀ x : euclidean_space ℝ (fin n), x ∈ U → T(x) ≤ m, from begin
      assume (x : euclidean_space ℝ (fin n)) (hx : x ∈ U),
      show T(x) ≤ m, from le_of_lt (h1 x hx),
    end,
    have h6 : ∀ x : euclidean_space ℝ (fin n), x ∈ U → T(x) = m, from begin
      assume (x : euclidean_space ℝ (fin n)) (hx : x ∈ U),
      show T(x) = m, from eq_of_le_of_ge (h5 x hx) (h5 x hx),
    end,
    have h7 : ∀ x : euclidean_space ℝ (fin n), x ∈ Bm → T(x) ≤ m, from begin
      assume (x : euclidean_space ℝ (fin n)) (hx : x ∈ Bm),
      unfold Bm T,
      unfold ball,
      simp,
      assume hmx,
      exact le_add_of_nonneg_of_le hmx (show 0 ≤ m, from nat.zero_le _),
    end,
    have h8 : ∀ x : euclidean_space ℝ (fin n), x ∈ Bm → T(x) = m, from begin
      assume (x : euclidean_space ℝ (fin n)) (hx : x ∈ Bm),
      show T(x) = m, from eq_of_le_of_ge (h7 x hx) (h7 x hx),
    end,
    have h9 : ∀ x : euclidean_space ℝ (fin n), x ∈ U → T(x) ≤ m, from begin
      assume (x : euclidean_space ℝ (fin n)) (hx : x ∈ U),
      show T(x) ≤ m, from eq.symm (h6 x hx) ▸ (h7 x hx) ,
    end,
    have h10 : ∀ x : euclidean_space ℝ (fin n), x ∈ U → T(x) = m, from begin
      assume (x : euclidean_space ℝ (fin n)) (hx : x ∈ U),
      show T(x) = m, from eq_of_le_of_ge (h9 x hx) (h9 x hx),
    end,
    show x ∈ Bm, from begin
      unfold Bm T,
      unfold ball,
      simp,
      assume hmx,
      exact eq.symm (h10 x hx) ▸ hmx,
    end,
  end,

  have h4 : ∀ x : euclidean_space ℝ (fin n), x ∈ Bm → x ∈ U, from begin
    assume (x : euclidean_space ℝ (fin n)) (hx : x ∈ Bm),
    have h5 : ∀ x : euclidean_space ℝ (fin n), x ∈ U → T(x) < m, from h1 x hx,
    have h6 : ∀ x : euclidean_space ℝ (fin n), x ∈ U → T(x) ≤ m, from begin
      assume (x : euclidean_space ℝ (fin n)) (hx : x ∈ U),
      show T(x) ≤ m, from le_of_lt (h1 x hx),
    end,
    have h7 : ∀ x : euclidean_space ℝ (fin n), x ∈ U → T(x) = m, from begin
      assume (x : euclidean_space ℝ (fin n)) (hx : x ∈ U),
      show T(x) = m, from eq_of_le_of_ge (h6 x hx) (h6 x hx),
    end,
    have h8 : ∀ x : euclidean_space ℝ (fin n), x ∈ Bm → T(x) ≤ m, from begin
      assume (x : euclidean_space ℝ (fin n)) (hx : x ∈ Bm),
      unfold Bm T,
      unfold ball,
      simp,
      assume hmx,
      exact le_add_of_nonneg_of_le hmx (show 0 ≤ m, from nat.zero_le _),
    end,
    have h9 : ∀ x : euclidean_space ℝ (fin n), x ∈ Bm → T(x) = m, from begin
      assume (x : euclidean_space ℝ (fin n)) (hx : x ∈ Bm),
      show T(x) = m, from eq_of_le_of_ge (h8 x hx) (h8 x hx),
    end,
    have h10 : ∀ x : euclidean_space ℝ (fin n), x ∈ U → T(x) ≤ m, from begin
      assume (x : euclidean_space ℝ (fin n)) (hx : x ∈ U),
      show T(x) ≤ m, from eq.symm (h7 x hx) ▸ (h8 x hx),
    end,
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
  -- $A$ and $B$ are sets. $A$ and $B$ belong to power set of $S$
  assume (A : set α) (hA : A ∈ 𝒫 S) (B : set α) (hB : B ∈ 𝒫 S),
  -- Then $A ⊆ S$ and $B ⊆ S$, by power set definition
  have h1 : (A ⊆ S) ∧ (B ⊆ S), from by {split,apply set.subset_of_mem_powerset,exact hA,apply set.subset_of_mem_powerset,exact hB},
  -- Then $(A ∩ B) ⊆ A$, by intersection of set is a subset
  have h2 : (A ∩ B) ⊆ A, from by apply set.inter_subset_left,
  -- Then $(A ∩ B) ⊆ S$, by subset relation is transitive
  have h3 : (A ∩ B) ⊆ S, from by {apply set.subset.trans h2 h1.left},
  -- Hence $(A ∩ B) ∈  𝒫 S$, by power set definition
  show (A ∩ B) ∈  𝒫 S, from by {apply set.mem_powerset h3},
end

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

{{begin-eqn}}
{{eqn | l = \left({x + y}\right)^2
      | r = \left({x + y}\right) \cdot \left({x + y}\right)
}}
{{eqn | r = x \cdot \left({x + y}\right) + y \cdot \left({x + y}\right)
      | c = Real Multiplication Distributes over Addition
}}
{{eqn | r = x \cdot x + x \cdot y + y \cdot x + y \cdot y
      | c = Real Multiplication Distributes over Addition
}}
{{eqn | r = x^2 + 2xy + y^2
      | c =
}}
{{end-eqn}}
{{qed}}
-/
theorem square_of_sum (x y : ℝ) : (x + y)^2 = (x^2 + 2*x*y + y^2) :=
begin
  -- expand the power
  calc (x + y)^2 = (x+y)*(x+y) : by rw sq
  -- distributive property of multiplication over addition gives:
  ... = x*(x+y) + y*(x+y) : by rw add_mul
  -- applying the above property further gives:
  ... = x*x + x*y + y*x + y*y : by {rw [mul_comm x (x+y),mul_comm y (x+y)], rw [add_mul,add_mul], ring}
  -- rearranging the terms using commutativity and adding gives:
  ... = x^2 + 2*x*y + y^2 : by {repeat {rw ← sq}, rw mul_comm y x, ring}
end

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

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

Setting $b = a$, this becomes:

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

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

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

  -- Setting $b = a$, this becomes:
  have h3 : ∀ a : G, ∃! x : G, a * x = a, from
    assume a : G, h1 a a,
  have h4 : ∀ a : G, ∃! y : G, y * a = a, from
    assume a : G, h2 a a,

  -- These $x$ and $y$ are both $(1 : G)$, by definition of identity element
  have h5 : ∀ a : G, classical.some (h3 a).exists = (1 : G), from assume a :G,
    exists_unique.unique (h3 a) (classical.some_spec (exists_unique.exists (h3 a)))
    (mul_one a),
  have h6 : ∀ a : G, classical.some (h4 a).exists = (1 : G), from assume a : G,
    exists_unique.unique (h4 a) (classical.some_spec (exists_unique.exists (h4 a))) (one_mul a),

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

/--`theorem`
\mathbb{R}^n is paracompact
$\mathbb{R}^n$ is paracompact for all $n$.
`proof`
Let $\mathcal{A}$ be an open covering of $\mathbb{R}^n$. We now construct a locally finite open refinement $\mathcal{C}$ of $\mathcal{A}$ that covers $\mathbb{R}^n$. First, we define a collection of pen balls. Let $B_0 = \phi$, and for each $n \in \mathbb{N}$, let $B_m$ denote the ball of radius $m$
centered at 0. Given $m$, set $\Bar{B_m}$ is compact in $\mathbb{R}^n$ by the Heine-Borel theorem, so choose finitely many elements of $\mathcal{A}$ that cover $\Bar{B_m}$ and intersect each one with the open set $\mathbb{R}^n \setminus \Bar{B_{m - 1}}$, and let $\mathcal{C}_{m}$ denote this collection of open sets (each an open subset of an element of $\mathcal{A}$). So $\mathcal{C} = \bigcup_{m = 0}^{\infty} \mathcal{C}_m$ is an open refinement of $\mathcal{A}$. Note that $\mathcal{C}$ covers $\mathbb{R}^n$ since for any $x \in \mathbb{R}^n$, there is a smallest $m \in \mathbb{N}$ such that $x \in \Bar{B_{m}}$ (namely, some $m$ where $\rVert x \lVert \leq m \leq \rVert x \lVert + 1$), and so $x$ is an element of $\mathcal{C}_m$. Now collection $\mathcal{C}$ is locally finite since for given $x \in \mathbb{R}^n$, neighborhood $B_m$ intersects only finitely many elements of $\mathcal{C}$, namely those elements in collection $\mathcal{C}_1 \cup \mathcal{C}_2 \cup \cdots \mathcal{C}_m$. So $\mathcal{C}$ is a locally finite open refinement of $\mathcal{A}$ that covers $\mathbb{R}^n$, hence $\mathbb{R}^n$ is paracompact.

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