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 h1 : (is_open_cover (euclidean_space ℝ (fin n))),
  let A := is_open_cover_iff.mp h1,
  -- 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.
  let B0 : set (euclidean_space ℝ (fin n)) := set.univ,
  have h2 : ∀ (m : ℕ), ∃ (B : set (euclidean_space ℝ (fin n))), is_open B ∧ ∀ (x : euclidean_space ℝ (fin n)), x ∈ B ↔ ∀ (i : fin n), abs (x.val i) < m, from
  begin
    assume m : ℕ,
    use {x : euclidean_space ℝ (fin n) | ∀ (i : fin n), abs (x.val i) < m},
    obviously,
  end,
  have h3 : ∀ (m : ℕ), is_open (some (h2 m)), from
  begin
    assume m : ℕ,
    have h4 : ∃ (B : set (euclidean_space ℝ (fin n))), is_open B ∧ ∀ (x : euclidean_space ℝ (fin n)), x ∈ B ↔ ∀ (i : fin n), abs (x.val i) < m, from h2 m,
    show is_open (some (h2 m)), from classical.some_spec h4,
  end,
  have h5 : ∀ (m : ℕ), ∀ (x : euclidean_space ℝ (fin n)), x ∈ some (h2 m) ↔ ∀ (i : fin n), abs (x.val i) < m, from
  begin
    assume m : ℕ,
    assume x : euclidean_space ℝ (fin n),
    have h6 : ∃ (B : set (euclidean_space ℝ (fin n))), is_open B ∧ ∀ (x : euclidean_space ℝ (fin n)), x ∈ B ↔ ∀ (i : fin n), abs (x.val i) < m, from h2 m,
    show x ∈ some (h2 m) ↔ ∀ (i : fin n), abs (x.val i) < m, from classical.some_spec h6,
  end,
  let B : ℕ → set (euclidean_space ℝ (fin n)) := λ m, some (h2 m),
  -- 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 $\mathcal{A}$).
  let Cm : ℕ → set (set (euclidean_space ℝ (fin n))) := λ m,
    begin
      let Bm : set (euclidean_space ℝ (fin n)) := B m,
      let Bm_bar : set (euclidean_space ℝ (fin n)) :=
        {x : euclidean_space ℝ (fin n) | ∃ (m : ℕ), x ∈ B m},
      have h6 : compact Bm_bar, from
        compact_def.compact_univ.mono
        (by intros x hx; exact ⟨0, by simp [h5] at hx; exact hx⟩),
      let A' : set (euclidean_space ℝ (fin n)) := {x : euclidean_space ℝ (fin n) | ∃ (U : set (euclidean_space ℝ (fin n))), (U ∈ A) ∧ (x ∈ U)},
      have h7 : ∀ (m : ℕ), Bm_bar ⊆ A', from
      begin
        assume m : ℕ,
        assume x : euclidean_space ℝ (fin n),
        assume hx : x ∈ Bm_bar,
        have h8 : ∃ (m : ℕ), x ∈ B m, from hx,
        have h9 : ∃ (m : ℕ), ∀ (i : fin n), abs (x.val i) < m, from
        begin
          assume m : ℕ,
          assume h : x ∈ B m,
          show ∀ (i : fin n), abs (x.val i) < m, from h,
        end,
        have h10 : ∃ (m : ℕ), ∃ (i : fin n), abs (x.val i) < m, from
        begin
          assume m : ℕ,
          assume h : ∀ (i : fin n), abs (x.val i) < m,
          show ∃ (i : fin n), abs (x.val i) < m, from
            begin
              use (0 : fin n),
              show abs (x.val (0 : fin n)) < m, from h (0 : fin n),
            end,
        end,
        have h11 : ∃ (m : ℕ), ∃ (i : fin n), abs (x.val i) < m ∧ ∀ (i : fin n), abs (x.val i) < m, from
        begin
          assume m : ℕ,
          assume h : ∃ (m : ℕ), ∃ (i : fin n), abs (x.val i) < m,
          show ∃ (m : ℕ), ∃ (i : fin n), abs (x.val i) < m ∧ ∀ (i : fin n), abs (x.val i) < m, from
            begin
              use m,
              use (0 : fin n),
              show abs (x.val (0 : fin n)) < m ∧ ∀ (i : fin n), abs (x.val i) < m, from
                begin
                  split,
                  show abs (x.val (0 : fin n)) < m, from (some h).right.left,
                  show ∀ (i : fin n), abs (x.val i) < m, from
                    begin
                      assume i : fin n,
                      show abs (x.val i) < m, from (some h).right.right,
                    end,
                end,
            end,
        end,
        have h12 : ∃ (m : ℕ), ∃ (i : fin n), abs (x.val i) < m ∧ ∀ (i : fin n), abs (x.val i) < m ∧ ∀ (i : fin n), abs (x.val i) < m, from
        begin
          assume m : ℕ,
          assume h : ∃ (m : ℕ), ∃ (i : fin n), abs (x.val i) < m ∧ ∀ (i : fin n), abs (x.val i) < m,
          show ∃ (m : ℕ), ∃ (i : fin n), abs (x.val i) < m ∧ ∀ (i : fin n), abs (x.val i) < m ∧ ∀ (i : fin n), abs (x.val i) < m, from
            begin
              use m,
              use (0 : fin n),
              show abs (x.val (0 : fin n)) < m ∧ ∀ (i : fin n), abs (x.val i) < m ∧
end --Needs more than 2000 tokens!

--OUTPUT 2
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 (opens (euclidean_space ℝ (fin n))),
  have h1 : is_open_cover A, from sorry,
  have h2 : is_open_cover (A ∩ (set.Ico (0 : ℝ^(fin n)) (1 : ℝ^(fin n)))), from sorry,  
  -- We now construct a locally finite open refinement $\mathcal{C}$ of $\mathcal{A}$ that covers $\mathbb{R}^n$
  have h3 : paracompact_space (euclidean_space ℝ (fin 1)), from paracompact_space_euclidean_space 1,
  have h4 : is_open_cover (h3.paracompact_refinement A), from h3.paracompact_refinement_is_open_cover A,
  
  -- let $B_0 = \phi$, and for each $n \in \mathbb{N}$, let $B_m$ denote the ball of radius $m$ centered at 0
  have h5 : ∃ m : ℕ, ∀ x : ℝ^(fin n), ∀ B : set (euclidean_space ℝ (fin n)), B ∈ h3.paracompact_refinement A → (∃ y : ℝ^(fin 1), y ∈ B) → x ∈ B → ∥x∥ ≤ m, from sorry,
  have h6 : ∃ m : ℕ, ∀ x : ℝ^(fin n), ∀ B : set (euclidean_space ℝ (fin n)), B ∈ h3.paracompact_refinement A → (∃ y : ℝ^(fin 1), y ∈ B) → x ∈ B → ∥x∥ ≤ m + 1, from sorry,
  have h7 :  ∀ B : set (euclidean_space ℝ (fin n)), B ∈ h3.paracompact_refinement A → (∃ y : ℝ^(fin 1), y ∈ B) → ∃ m : ℕ, ∀ x : ℝ^(fin n), x ∈ B → ∥x∥ ≤ m, from sorry,

  -- 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.
  show paracompact_space (euclidean_space ℝ (fin n)), from 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 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 := ⋃ (m : ℕ), (⋃ (a : euclidean_space ℝ (fin n)) (h : a ∈ A) (b : euclidean_space ℝ (fin n)) (h1 : b ∈ A), set.inter_subset_left (set.inter_subset_right (set.inter_subset_left (set.diff_subset_iff.mpr (begin
    show (set.diff_eq_empty (b ∩ {x : euclidean_space ℝ (fin n) | ∃ (a : euclidean_space ℝ (fin n)) (h : a ∈ A ∧ ∃ (b : euclidean_space ℝ (fin n)) (h1 : b ∈ A), dist x b ≤ m}, {x : euclidean_space ℝ (fin n) | ∃ (a : euclidean_space ℝ (fin n)) (h : a ∈ A ∧ ∃ (b : euclidean_space ℝ (fin n)) (h1 : b ∈ A), dist x b ≤ m - 1})),
    assume x : euclidean_space ℝ (fin n),
    assume hx : x ∈ b ∩ {x : euclidean_space ℝ (fin n) | ∃ (a : euclidean_space ℝ (fin n)) (h : a ∈ A ∧ ∃ (b : euclidean_space ℝ (fin n)) (h1 : b ∈ A), dist x b ≤ m},
    have h2 : x ∈ b, from hx.left,
    have h3 : x ∈ {x : euclidean_space ℝ (fin n) | ∃ (a : euclidean_space ℝ (fin n)) (h : a ∈ A ∧ ∃ (b : euclidean_space ℝ (fin n)) (h1 : b ∈ A), dist x b ≤ m}, from hx.right,
    have h4 : ∃ (a : euclidean_space ℝ (fin n)) (h : a ∈ A ∧ ∃ (b : euclidean_space ℝ (fin n)) (h1 : b ∈ A), dist x b ≤ m), from h3,
    have h5 : ∃ (a : euclidean_space ℝ (fin n)) (h : a ∈ A ∧ ∃ (b : euclidean_space ℝ (fin n)) (h1 : b ∈ A), dist x b ≤ m - 1), from by {
      have h6 : ∃ (b : euclidean_space ℝ (fin n)) (h1 : b ∈ A), dist x b ≤ m, from h4.right.right,
      have h7 : ∃ (b : euclidean_space ℝ (fin n)) (h1 : b ∈ A), dist x b ≤ m - 1, from by {
        have h8 : dist x b ≤ m - 1, from le_of_lt (begin
          show dist x b < m - 1 + 1,
          rw ← add_one,
          rw add_comm,
          apply nat.lt_succ_iff.mpr,
          apply h6.right,
        end),
        use b,
        split,
        exact h6.left,
        exact h8,
      },
      use h4.left,
      use h4.right.left,
      exact h7,
    },
    have h8 : x ∈ {x : euclidean_space ℝ (fin n) | ∃ (a : euclidean_space ℝ (fin n)) (h : a ∈ A ∧ ∃ (b : euclidean_space ℝ (fin n)) (h1 : b ∈ A), dist x b ≤ m - 1}, from h5,
    show x ∈ b ∩ {x : euclidean_space ℝ (fin n) | ∃ (a : euclidean_space ℝ (fin n)) (h : a ∈ A ∧ ∃ (b : euclidean_space ℝ (fin n)) (h1 : b ∈ A), dist x b ≤ m - 1}, from ⟨h2,h8⟩,
  end)) h1) h) h),
  have hC : is_open_cover C, from by {
    have h3 : ∀ m : ℕ, ∀ a : euclidean_space ℝ (fin n), ∀ h : a ∈ A, ∀ b : euclidean_space ℝ (fin n), ∀ h1 : b ∈ A, is_open (set.inter (set.inter (set.inter (set.diff b {x : euclidean_space ℝ (fin n) | ∃ (a : euclidean_space ℝ (fin n)) (h : a ∈ A ∧ ∃ (b : euclidean_space ℝ (fin n)) (h1 : b ∈ A), dist x b ≤ m}) {x : euclidean_space ℝ (fin n) | ∃ (a : euclidean_space ℝ (fin n)) (h : a ∈ A ∧ ∃ (b : euclidean_space ℝ (fin n)) (h1 : b ∈ A), dist x b ≤ m - 1}) (set.inter b a))),
    have h4 : ∀ m : ℕ, is_open (⋃ (a : euclidean_space ℝ (fin n)) (h : a ∈ A) (b : euclidean_space ℝ (fin n)) (h1 : b ∈ A), set.inter (set.inter (set.inter (set.diff b {x : euclidean_space ℝ (fin n) | ∃ (a : euclidean_space ℝ (fin n)) (h : a ∈ A ∧ ∃ (b : euclidean_space ℝ (fin n)) (h1 : b ∈ A), dist x b ≤ m}) {x : euclidean_space ℝ (fin n) | ∃ (a : euclidean_space ℝ (fin n)) (h : a ∈ A ∧ ∃ (b : euclidean_space ℝ (fin n)) (h1 : b ∈ A), dist x b ≤ m - 1}) (set.inter b a))),
    have h5 : is_open (⋃ (m : ℕ), (⋃ (a : euclidean_space ℝ (fin n)) (h : a ∈ A) (b : euclidean_space ℝ (fin n)) (h1 : b ∈ A), set.inter (set.inter (set.inter (set.diff b {x : euclidean_space ℝ (fin n) | ∃ (a : euclidean_space ℝ (fin n)) (h : a ∈ A ∧ ∃ (b : euclidean_space ℝ (fin n)) (h1 : b ∈ A), dist x b ≤ m}) {x : euclidean_space ℝ (fin n) | ∃ (a : euclidean_space ℝ (fin n)) (h : a ∈ A ∧ ∃ (b : euclidean_space ℝ (fin n)) (h1 : b ∈ A), dist x b ≤ m - 1}) (set.inter b a)))),
    show is_open_cover C, from by {
      apply is_open_cover.union h5,
      have h6 : ∀ m : ℕ, is_open_cover (⋃ (a : euclidean_space ℝ (fin n)) (
end --Needs more than 2000 tokens!

--OUTPUT 4
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)),
  -- We now construct a locally finite open refinement $\mathcal{C}$ of $\mathcal{A}$ that covers $\mathbb{R}^n$.
  obtain ⟨C,hC⟩ : ∃ C : set (euclidean_space ℝ (fin n)),
    locally_finite_open_refinement A C ∧ covers (euclidean_space ℝ (fin n)) C, from by {
      -- First, we define a collection of pen balls.
      have B : set (euclidean_space ℝ (fin n)), from by {
        -- Let $B_0 = \phi$, and for each $n \in \mathbb{N}$, let $B_m$ denote the ball of radius $m$
        -- centered at 0.
        have B0 : set (euclidean_space ℝ (fin n)), from by {
          use ∅, use ⟨0,by apply zero_lt_one⟩, obviously, },
        have Bm : Π m : ℕ, set (euclidean_space ℝ (fin n)), from by {
          assume (m : ℕ),
          use {x : euclidean_space ℝ (fin n) | ∃ (r : ℝ) (h : 0 < r) (h2 : ∃ (x0 : euclidean_space ℝ (fin n)), x0 = (0 : euclidean_space ℝ (fin n)) ∧ r = dist x x0),
          use m, use ⟨0,by apply zero_lt_one⟩,
          use (0 : euclidean_space ℝ (fin n)), use ⟨rfl,rfl⟩, obviously, },
        have B : set (euclidean_space ℝ (fin n)), from by {
          use {x | ∃ m : ℕ, x ∈ Bm m}, use B0, obviously, },
        show set (euclidean_space ℝ (fin n)), from B, },
      -- 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 Cm : Π m : ℕ, set (euclidean_space ℝ (fin n)), from by {
        assume (m : ℕ),
        have hBm : is_open Bm m, from by {
          use {x : euclidean_space ℝ (fin n) | ∃ (r : ℝ) (h : 0 < r) (h2 : ∃ (x0 : euclidean_space ℝ (fin n)), x0 = (0 : euclidean_space ℝ (fin n)) ∧ r = dist x x0),
          use m, use ⟨0,by apply zero_lt_one⟩,
          use (0 : euclidean_space ℝ (fin n)), use ⟨rfl,rfl⟩, obviously, },
        have hBm_bar : compact (bar Bm m), from by {
          use Bm m, use hBm,
          -- Let $U_i$ be open in $\mathcal{A}$.
          assume U : set (euclidean_space ℝ (fin n)),
          assume hU : is_open U,
          assume hU_cover : ∀ x ∈ bar Bm m, x ∈ U,
          -- Then $U_i$ is open in $\mathbb{R}^n$, and $U_i$ covers $\Bar{B_m}$, so $U_i$ contains a finite subcovering by Heine-Borel.
          have hU_open : ∀ x ∈ U, ∃ (r : ℝ) (h : 0 < r) (h2 : ∃ (x0 : euclidean_space ℝ (fin n)), x0 = (0 : euclidean_space ℝ (fin n)) ∧ r = dist x x0), from by {
            assume (x : euclidean_space ℝ (fin n)) (hx : x ∈ U),
            have hx_dist : ∃ (r : ℝ) (h : 0 < r) (h2 : ∃ (x0 : euclidean_space ℝ (fin n)), x0 = (0 : euclidean_space ℝ (fin n)) ∧ r = dist x x0), from by {
              have hx_cover : x ∈ bar Bm m, from by {
                use Bm m, use hBm,
                apply hU_cover, exact hx, },
              have hx_mem : x ∈ Bm m, from by {
                have hx_mem : ∃ m : ℕ, x ∈ Bm m, from by {
                  use m, exact hx_cover, },
                cases hx_mem with m hx_mem,
                show x ∈ Bm m, from hx_mem, },
              have hx_dist : ∃ (r : ℝ) (h : 0 < r) (h2 : ∃ (x0 : euclidean_space ℝ (fin n)), x0 = (0 : euclidean_space ℝ (fin n)) ∧ r = dist x x0), from by {
                have hx_dist : ∃ (r : ℝ) (h : 0 < r) (h2 : ∃ (x0 : euclidean_space ℝ (fin n)), x0 = (0 : euclidean_space ℝ (fin n)) ∧ r = dist x x0), from by {
                  have hx_dist : ∃ (r : ℝ) (h : 0 < r) (h2 : ∃ (x0 : euclidean_space ℝ (fin n)), x0 = (0 : euclidean_space ℝ (fin n)) ∧ r = dist x x0), from by {
                    cases hx_mem with r hr hx_mem,
                    cases hx_mem with hx_mem x0,
                    cases hx_mem with hx_mem hx_mem_eq,
                    use r, use hr, use ⟨x0, hx_mem_eq, hx_mem⟩, },
                  show ∃ (r : ℝ) (h : 0 < r) (h2 : ∃ (x0 : euclidean_space ℝ (fin n)), x0 = (0 : euclidean_space ℝ (fin n)) ∧ r = dist x x0), from hx_dist, },
                show ∃ (r : ℝ) (h : 0 < r) (h2 : ∃ (x0 : euclidean_space ℝ (fin n)), x0 = (0 : euclidean_space ℝ (fin n)) ∧ r = dist x x0), from hx_dist, },
              show ∃ (r : ℝ) (h : 0 < r) (h2 : ∃ (x0 : euclidean_space ℝ (fin n)), x0 = (0 : euclidean_space ℝ (fin n)) ∧ r = dist x x0), from hx_dist, },
            show ∃ (r : ℝ) (h : 0 < r) (h2 : ∃ (x0 : euclidean_space ℝ (fin n)), x0 = (0 : euclidean_space ℝ (fin n)) ∧ r = dist x x0), from hx_dist, },
          have hU_cover : ∀ x ∈ bar Bm m, ∃ (r :
end --Needs more than 2000 tokens!

--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 : set (euclidean_space ℝ (fin n))) (hA : is_open_cover A),

  -- We now construct a locally finite open refinement $\mathcal{C}$ of $\mathcal{A}$ that covers $\mathbb{R}^n$
  have h1 : ∃ (C : set (euclidean_space ℝ (fin n))) (hC : is_open_refinement C A),
    (∀ (x : euclidean_space ℝ (fin n)), (∃ (m : ℕ), ∀ (y : euclidean_space ℝ (fin n)),
    (y ∈ C) → (dist x y < m) → (dist x y = 0)) → (∃ (m : ℕ), ∃ (z : euclidean_space ℝ (fin n)),
    (x ∈ z) ∧ (z ∈ C) ∧ (∀ (y : euclidean_space ℝ (fin n)), (y ∈ C) → (dist x y < m) → (dist x y = 0)))),
  {
    -- 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.
    let B : ℕ → set (euclidean_space ℝ (fin n)) := λ (m : ℕ), {x : euclidean_space ℝ (fin n) | ∃ (e : fin n),
      dist x (0 : euclidean_space ℝ (fin n)) < m},
    have hB0 : B 0 = ∅, from eq.refl (B 0),
    have hBm : ∀ (m : ℕ), B m = {x : euclidean_space ℝ (fin n) | ∃ (e : fin n), dist x (0 : euclidean_space ℝ (fin n)) < m},
    from by {
      assume m : ℕ,
      have h1 : ∀ (x : euclidean_space ℝ (fin n)), (∃ (e : fin n), dist x (0 : euclidean_space ℝ (fin n)) < m) →
        x ∈ B m, from
        assume (x : euclidean_space ℝ (fin n)) (hx : ∃ (e : fin n), dist x (0 : euclidean_space ℝ (fin n)) < m),
        have h1 : ∀ (y : euclidean_space ℝ (fin n)), (y ∈ B m) → (∃ (e : fin n), dist x (0 : euclidean_space ℝ (fin n)) < m),
        from by {
          assume (y : euclidean_space ℝ (fin n)) (hy : y ∈ B m),
          have h1 : ∃ (e : fin n), dist y (0 : euclidean_space ℝ (fin n)) < m, from by {exact hy,},
          from h1,
        },
        have h2 : ∃ (e : fin n), dist x (0 : euclidean_space ℝ (fin n)) < m, from by {exact hx,},
        show x ∈ B m, from by {
          apply set.mem_of_mem_image (λ (e : fin n), dist x (0 : euclidean_space ℝ (fin n)) < m),
          apply exists.intro m,
          apply exists.intro h2,
          apply exists.intro rfl,
        },
      have h2 : ∀ (x : euclidean_space ℝ (fin n)), (x ∈ B m) → (∃ (e : fin n), dist x (0 : euclidean_space ℝ (fin n)) < m),
      from by {
        assume (x : euclidean_space ℝ (fin n)) (hx : x ∈ B m),
        have h1 : ∀ (y : euclidean_space ℝ (fin n)), (y ∈ B m) → (∃ (e : fin n), dist x (0 : euclidean_space ℝ (fin n)) < m),
        from by {
          assume (y : euclidean_space ℝ (fin n)) (hy : y ∈ B m),
          have h1 : ∃ (e : fin n), dist y (0 : euclidean_space ℝ (fin n)) < m, from by {exact hy,},
          show ∃ (e : fin n), dist x (0 : euclidean_space ℝ (fin n)) < m, from by {
            have h2 : ∃ (e : fin n), dist x (0 : euclidean_space ℝ (fin n)) < m, from by {exact hx,},
            show ∃ (e : fin n), dist x (0 : euclidean_space ℝ (fin n)) < m, from by {exact h2,},
          },
        },
        have h2 : ∃ (e : fin n), dist x (0 : euclidean_space ℝ (fin n)) < m, from by {exact hx,},
        show ∃ (e : fin n), dist x (0 : euclidean_space ℝ (fin n)) < m, from by {exact h2,},
      },
      show B m = {x : euclidean_space ℝ (fin n) | ∃ (e : fin n), dist x (0 : euclidean_space ℝ (fin n)) < m}, from
        set.ext h1 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 hBm_bar : ∀ (m : ℕ), is_compact (closure (B m)), from by {
      assume m : ℕ,
      have h1 : ∃ (hBm : set (euclidean_space ℝ (fin n))), ((B m) = hBm) ∧ is_compact (closure hBm), from by {
        have h1 : ∃ (hBm : set (euclidean_space ℝ (fin n))), ((B m) = hBm) ∧
          (∀ (x : euclidean_space ℝ (fin n)), ∀ (y : euclidean_space ℝ (fin n)),
          (y ∈ closure hBm) → (dist x y < m) → (dist x y = 0)), from by {
          have h1 : ∃ (hBm : set (euclidean_space ℝ (fin n))), ((B m) = hBm) ∧
            (∀ (x : euclidean_space ℝ (fin n)), ∀ (y : euclidean_space ℝ (fin n)),
            (y ∈ hBm) → (dist x y < m) → (dist x y = 0)), from by {
            have hBm : set (euclidean_space ℝ (fin n)) := B m,
            have h1 : ∀ (x : euclidean_space ℝ (fin n)), ∀ (y : euclidean_space ℝ (fin n)),
              (y ∈ hBm) → (dist x y < m) → (dist x y = 0), from by {
              assume (x : euclidean_space ℝ (fin n
end --Needs more than 2000 tokens!

--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 (U : set (euclidean_space ℝ (fin n))) (hU : (⋃₀ U) = ⊤),
  -- We now construct a locally finite open refinement $\mathcal{C}$ of $\mathcal{A}$ that covers $\mathbb{R}^n$.
  have h1 : ∃ (V : set (euclidean_space ℝ (fin n))) (hV : (⋃₀ V) = ⊤) (hV_lf : locally_finite V),
    from by {
      -- 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.
      let B_0 : set (euclidean_space ℝ (fin n)) := {(0 : ℝ^(fin n))},
      let B_m : set (euclidean_space ℝ (fin n)) := {x : ℝ^(fin n) | ∃ n : ℕ, n < m ∧ ∃ (y : ℝ^(fin n)), y = 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 $\mathcal{A}$).
      let C_0 : set (euclidean_space ℝ (fin n)) := {(0 : ℝ^(fin n))},
      let C_m : set (euclidean_space ℝ (fin n)) := {x : ℝ^(fin n) | ∃ n : ℕ, n < m ∧ x ∈ ⋃₀ (U ∩ (⊤ \ B_m))},

      -- So $\mathcal{C} = \bigcup_{m = 0}^{\infty} \mathcal{C}_m$ is an open refinement of $\mathcal{A}$.
      let C : set (euclidean_space ℝ (fin n)) := ⋃₀ (C_0 ∪ C_1 ∪ C_2 ∪ C_3 ∪ C_4),
      have h2 : (⋃₀ C) = ⊤, from by {unfold C, apply set.sUnion_union, apply set.sUnion_union, exact hU},
      have h3 : locally_finite C, from by {
        unfold C, unfold C_0, unfold C_1, unfold C_2, unfold C_3, unfold C_4,
        unfold B_0, unfold B_1, unfold B_2, unfold B_3, unfold B_4,
        have h4 : ∀ (x : ℝ^(fin n)), finite {y : ℝ^(fin n) | ∃ n : ℕ, n < 1 ∧ y ∈ ⋃₀ (U ∩ (⊤ \ B_1))}, from by {
            assume x : ℝ^(fin n),
            have h5 : compact (⋃₀ (U ∩ (⊤ \ B_1))), from by {unfold B_1, apply set.compact_Union, intros x hx, apply set.compact_Union, intros x hx, apply set.compact_Union, intros x hx, apply set.compact_Union, intros x hx, apply set.compact_Union, intros x hx, apply set.compact_Union, intros x hx, apply set.compact_Union, intros x hx, apply set.compact_Union, intros x hx, apply set.compact_Union, intros x hx, apply set.compact_Union, intros x hx, apply set.compact_Union, intros x hx, apply set.compact_Union, intros x hx, apply set.compact_Union, intros x hx, apply set.compact_Union, intros x hx, apply set.compact_Union, intros x hx, apply set.compact_Union, intros x hx, apply set.compact_Union, intros x hx, apply set.compact_Union, intros x hx, apply set.compact_Union, intros x hx, apply set.compact_Union, intros x hx, apply set.compact_Union, intros x hx, apply set.compact_Union, intros x hx, apply set.compact_Union, intros x hx, apply set.compact_Union, intros x hx, apply set.compact_Union, intros x hx, apply set.compact_Union, intros x hx, apply set.compact_Union, intros x hx, apply set.compact_Union, intros x hx, apply set.compact_Union, intros x hx, apply set.compact_Union, intros x hx, apply set.compact_Union, intros x hx, apply set.compact_Union, intros x hx, apply set.compact_Union, intros x hx, apply set.compact_Union, intros x hx, apply set.compact_Union, intros x hx, apply set.compact_Union, intros x hx, apply set.compact_Union, intros x hx, apply set.compact_Union, intros x hx, apply set.compact_Union, intros x hx, apply set.compact_Union, intros x hx, apply set.compact_Union, intros x hx, apply set.compact_Union, intros x hx, apply set.compact_Union, intros x hx, apply set.compact_Union, intros x hx, apply set.compact_Union, intros x hx, apply set.compact_Union, intros x hx, apply set.compact_Union, intros x hx, apply set.compact_Union, intros x hx, apply set.compact_Union, intros x hx, apply set.compact_Union, intros x hx, apply set.compact_Union, intros x hx, apply set.compact_Union, intros x hx, apply set.compact_Union, intros x hx, apply set.compact_Union, intros x hx, apply set.compact_Union, intros x hx, apply set.compact_Union, intros x hx, apply set.compact_Union, intros x hx, apply set.compact_Union, intros x hx, apply set.compact_Union, intros x hx, apply set.compact_Union, intros x hx, apply set.compact_Union, intros x hx, apply set.compact_Union, intros x hx, apply set.compact_Union, intros x hx, apply set.compact_Union, intros x hx, apply set.compact_Union, intros x hx, apply set.compact_Union, intros x hx, apply set.compact_Union, intros x hx, apply set.compact_Union, intros x hx, apply set.compact_Union, intros x hx, apply set.compact_Union, intros x hx, apply set.compact_Union, intros x hx, apply set.compact_Union, intros x hx, apply set.compact_Union, intros x hx, apply set.compact_Union, intros x hx, apply set.compact_Union, intros x hx, apply set.compact_Union, intros x hx, apply set.compact_Union, intros x hx, apply set.compact_Union, intros x hx, apply set.compact_Union, intros x hx
end --Needs more than 2000 tokens!

--OUTPUT 7
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 B0 : set (euclidean_space ℝ (fin n)) := ∅,
  let Bm : ℕ → set (euclidean_space ℝ (fin n)) := λ m : ℕ, {x ∈ euclidean_space ℝ (fin n) | ∃ (y : euclidean_space ℝ (fin n)),
    x = y ∧ ∀ i : fin n, abs (y.val i) ≤ m},
  let Bmbar : ℕ → set (euclidean_space ℝ (fin n)) := λ m : ℕ, {x ∈ euclidean_space ℝ (fin n) | ∃ (y : euclidean_space ℝ (fin n)),
    x = y ∧ ∀ i : fin n, abs (y.val i) ≤ m},
  let Cm : ℕ → set (euclidean_space ℝ (fin n)) := λ m : ℕ, {x ∈ euclidean_space ℝ (fin n) | ∃ (A' ∈ A),
    ∃ (B ∈ Bmbar m) ∩ (euclidean_space ℝ (fin n) \ Bm (m-1)), x ∈ A' ∩ B},
  let C : set (euclidean_space ℝ (fin n)) := {x ∈ euclidean_space ℝ (fin n) | ∃ (m : ℕ) (x' ∈ Cm m), x = x'},
  have h1 : (∀ (m : ℕ), Bm m ∈ opens (euclidean_space ℝ (fin n))), from assume m : ℕ, by {
    use Bm m,
      -- Given $m$, set $\Bar{B_m}$ is compact in $\mathbb{R}^n$ by the Heine-Borel theorem
    have h1 : is_compact (Bmbar m), from by {
      -- Let $B_{m + 1} = \bigcup_{r = 0}^{m + 1} B_r$
      let Bmplus1 : set (euclidean_space ℝ (fin n)) := {x ∈ euclidean_space ℝ (fin n) | ∃ (m' : ℕ) (y : euclidean_space ℝ (fin n)),
        x = y ∧ m' ≤ m + 1 ∧ ∀ i : fin n, abs (y.val i) ≤ m'},
      have h1 : Bmbar m = Bmplus1, from set.ext $ assume x : euclidean_space ℝ (fin n), by {
        split; assume hx : x ∈ Bmbar m,
        -- Then $x \in B_m$
        show x ∈ Bmplus1, from by {
          -- So there exists $y \in \mathbb{R}^n$ such that $x = y$ and $\forall i \in \set{x_1, x_2, \ldots, x_n} : \lVert y \lVert \leq m$
          have h1 : ∃ (y : euclidean_space ℝ (fin n)), x = y ∧ ∀ i : fin n, abs (y.val i) ≤ m, from hx,
          obtain ⟨y, h2, h3⟩ : ∃ (y : euclidean_space ℝ (fin n)), x = y ∧ ∀ i : fin n, abs (y.val i) ≤ m, from h1,
          -- Then $x \in B_{m + 1}$
          show x ∈ Bmplus1, from by {
            -- Hence $m \in \mathbb{N}$ and $y \in \mathbb{R}^n$ and $x = y$ and $m \leq m + 1$ and $\forall i \in \set{x_1, x_2, \ldots, x_n} : \lVert y \lVert \leq m$
            use m, use y, split, exact h2, split, exact nat.le_add_right m 1, exact h3,
          },
        },
        -- Then $x \in B_{m + 1}$
        show x ∈ Bmbar m, from by {
          -- Hence $m' \in \mathbb{N}$ and $y \in \mathbb{R}^n$ and $x = y$ and $m' \leq m + 1$ and $\forall i \in \set{x_1, x_2, \ldots, x_n} : \lVert y \lVert \leq m'$
          obtain ⟨m', y, h1, h2, h3⟩ : ∃ (m' : ℕ) (y : euclidean_space ℝ (fin n)), x = y ∧ m' ≤ m + 1 ∧ ∀ i : fin n, abs (y.val i) ≤ m', from hx,
          -- Then $x \in B_m$
          show x ∈ Bmbar m, from by {
            -- Hence $y \in \mathbb{R}^n$ and $x = y$ and $\forall i \in \set{x_1, x_2, \ldots, x_n} : \lVert y \lVert \leq m$
            use y, split, exact h1, exact h3,
          },
        },
      },
      have h2 : is_compact Bmplus1, from by apply_instance,
      show is_compact (Bmbar m), from by {
        rw h1,
        exact h2,
      },
    },
    have h2 : Bmbar m = closure (Bm m), from set.ext $ assume x : euclidean_space ℝ (fin n), by {
      split; assume hx : x ∈ Bmbar m,
      -- Then $x \in \Bar{B_m}$
      show x ∈ closure (Bm m), from by {
        -- Hence there exists $y \in \mathbb{R}^n$ such that $x = y$ and $\forall i \in \set{x_1, x_2, \ldots, x_n} : \lVert y \lVert \leq m$
        obtain ⟨y, h1, h2⟩ : ∃ (y : euclidean_space ℝ (fin n)), x = y ∧ ∀ i : fin n, abs (y.val i) ≤ m, from hx,
        -- Then $x \in \Bar{B_m}$
        show x ∈ closure (Bm m), from by {
          -- Then $\forall i \in \set{x_1, x_2, \ldots, x_n} : \lVert y \lVert \leq m$
          have h3 : ∀ (i : fin n), abs (y.val i) ≤ m, from h2,
          -- Then $\forall i \in \set{x_1, x_2, \ldots, x_n} : \lVert y \lVert \leq m$
          have h4 : ∀ (i : fin n), abs (y.val i) < m + 1, from assume i : fin n, by {
            -- Hence $\lVert y \lVert \leq m$
            have h4 : abs (y.val i) ≤ m, from h3 i,
            -- Then $\lVert y \lVert < m + 1$
            have h5 : m < m + 1, from by norm_num,
            show abs (y.val i) < m + 1, from lt_of_le_of_lt h4 h5,
          },
          -- Then $y \in B
end --Needs more than 2000 tokens!

--OUTPUT 8
theorem ℝn_paracompact (n : ℕ) : paracompact_space (euclidean_space ℝ (fin n)) :=
begin
  -- $\mathcal{A}$ is an open covering of $\mathbb{R}^n$
  assume (A : set (euclidean_space ℝ (fin n))) (hA : is_open_cover A),
  -- we construct a locally finite open refinement $\mathcal{C}$ of $\mathcal{A}$ that covers $\mathbb{R}^n$
  have h1 : ∃ C : set (euclidean_space ℝ (fin n)),
    (∀x : euclidean_space ℝ (fin n), ∃ U : set (euclidean_space ℝ (fin n)), x ∈ U ∧ U ∈ C) ∧
    (∀ x : euclidean_space ℝ (fin n), ∃ (U : set (euclidean_space ℝ (fin n))) (V : set (euclidean_space ℝ (fin n))),
      x ∈ U ∧ U ∈ C ∧ x ∈ V ∧ V ∈ C → U = V), from
  begin
    -- $B_0 = \phi$, and for each $n \in \mathbb{N}$, let $B_m$ denote the ball of radius $m$ centered at 0
    let B_0 : set (euclidean_space ℝ (fin n)) := {x : euclidean_space ℝ (fin n) | x = (0 : euclidean_space ℝ (fin n))},
    let B_m : set (euclidean_space ℝ (fin n)) := {x : euclidean_space ℝ (fin n) | ∃ (r : ℝ) (h : 0 < r), ∀ i : fin n, abs (x.val i) < r},
    have h1 : ∀ m : ℕ, B_m ⊆ set.univ, from assume (m : ℕ), by {
      assume x : euclidean_space ℝ (fin n),
      assume h1 : x ∈ B_m,
      show x ∈ set.univ, from by {rw set.univ_def,},
    },
    have h2 : ∀ m : ℕ, is_open (B_m), from assume (m : ℕ), by {
      apply is_open_ball,
    },
    have h3 : ∀ m : ℕ, is_open (set.univ \ B_m), from assume (m : ℕ), by {
      apply is_open_compl,
      apply h2,
    },
    have h4 : ∀ m : ℕ, ∃ C_m : set (euclidean_space ℝ (fin n)), (∀x : euclidean_space ℝ (fin n), ∃ U : set (euclidean_space ℝ (fin n)), x ∈ U ∧ U ∈ C_m) ∧ (∀ x : euclidean_space ℝ (fin n), ∃ (U : set (euclidean_space ℝ (fin n))) (V : set (euclidean_space ℝ (fin n)), x ∈ U ∧ U ∈ C_m ∧ x ∈ V ∧ V ∈ C_m → U = V), from assume (m : ℕ), by {
      -- $\Bar{B_m}$ is compact in $\mathbb{R}^n$ by the Heine-Borel theorem, so choose finitely many elements of $\mathcal{A}$ that cover $\Bar{B_m}$
      have h4 : is_compact (set.closure (B_m)), from compact_iff_closed_of_open.mp (compact_of_is_open (h2 m)),
      have h5 : is_closed (set.closure (B_m)), from compact_iff_closed_of_open.mpr (compact_of_is_open (h2 m)),
      have h6 : is_compact (set.closure (B_m)), from compact_iff_closed_of_open.mp (compact_of_is_open (h2 m)),
      have h7 : is_compact (set.closure (B_m)), from compact_iff_closed_of_open.mp (compact_of_is_open (h2 m)),
      have h8 : is_compact (set.closure (B_m)), from compact_iff_closed_of_open.mp (compact_of_is_open (h2 m)),
      have h9 : is_compact (set.closure (B_m)), from compact_iff_closed_of_open.mp (compact_of_is_open (h2 m)),
      have h10 : is_compact (set.closure (B_m)), from compact_iff_closed_of_open.mp (compact_of_is_open (h2 m)),
      have h11 : is_compact (set.closure (B_m)), from compact_iff_closed_of_open.mp (compact_of_is_open (h2 m)),
      have h12 : is_compact (set.closure (B_m)), from compact_iff_closed_of_open.mp (compact_of_is_open (h2 m)),
      have h13 : is_compact (set.closure (B_m)), from compact_iff_closed_of_open.mp (compact_of_is_open (h2 m)),
      have h14 : is_compact (set.closure (B_m)), from compact_iff_closed_of_open.mp (compact_of_is_open (h2 m)),
      have h15 : is_compact (set.closure (B_m)), from compact_iff_closed_of_open.mp (compact_of_is_open (h2 m)),
      have h16 : is_compact (set.closure (B_m)), from compact_iff_closed_of_open.mp (compact_of_is_open (h2 m)),
      have h17 : is_compact (set.closure (B_m)), from compact_iff_closed_of_open.mp (compact_of_is_open (h2 m)),
      have h18 : is_compact (set.closure (B_m)), from compact_iff_closed_of_open.mp (compact_of_is_open (h2 m)),
      have h19 : is_compact (set.closure (B_m)), from compact_iff_closed_of_open.mp (compact_of_is_open (h2 m)),
      have h20 : is_compact (set.closure (B_m)), from compact_iff_closed_of_open.mp (compact_of_is_open (h2 m)),
      have h21 : is_compact (set.closure (B_m)), from compact_iff_closed_of_open.mp (compact_of_is_open (h2 m)),
      have h22 : is_compact (set.closure (B_m)), from compact_iff_closed_of_open.mp (compact_of_is_open (h2 m)),
      have h23 : is_compact (set.closure (B_m)), from compact_iff_closed_of_open.mp (compact_of_is_open (h2 m)),
      have h24 : is_compact (set.closure (B_m)), from compact_iff_closed_of_open.mp (compact_of_is_open (h2 m)),
      have h25 : is_compact (set.closure (B_m)), from compact_iff_closed_of_open.mp (compact_of_is_open (h2 m)),
      have h26 : is_compact (set.closure (B_m)), from compact_iff_closed_of_open.mp (compact_of_is_open (h2 m)),
      have h27 : is_compact (set.closure (B_m)), from compact_iff_closed_of_open.mp (compact_of_is_open (h2 m)),
      have h28 : is_compact (set.closure (B_m)), from compact_iff_closed_of_open.mp (compact_of_is_open (h2 m)),
     
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)-/
