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 $\mathbb{R}^n$.
  -- 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)) := ∅,
  have hB0 : is_open B0, from by auto [is_open_empty],
  let Bm : ℕ → set (euclidean_space ℝ (fin n)) := λ m, ball 0 m,
  have hBm : ∀ m : ℕ, is_open (Bm m), from by auto [is_open_ball, zero_lt_one],
  -- 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 (euclidean_space ℝ (fin n)) := λ m, cover_ball_of_cover A (Bm m),
  have hCm : ∀ m : ℕ, is_open_cover (Cm m), from by auto [cover_ball_of_cover],
  -- 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)) := ⋃i, (Cm i),
  have hC : is_open_cover C, from by auto [is_open_cover_Union],
  -- 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 $\mathcal{C}_m$.
  have h1 : ∀ x : euclidean_space ℝ (fin n), ∃ m : ℕ, (x ∈ Bm m) ∧ (∀ m₁ : ℕ, (x ∈ Bm₁ m₁) → m ≤ m₁), from by auto [mem_ball],
  have h2 : ∀ x : euclidean_space ℝ (fin n), x ∈ C, from by auto [C, mem_Union, classical.some_spec, h1],
  -- 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 $\mathcal{C}_1 \cup \mathcal{C}_2 \cup \cdots \mathcal{C}_m$.
  have h3 : ∀ x : euclidean_space ℝ (fin n), ∃ m : ℕ, (x ∈ Bm m) ∧ (∀ m₁ : ℕ, (x ∈ Bm₁ m₁) → m ≤ m₁), from by auto [mem_ball],
  have h4 : ∀ x : euclidean_space ℝ (fin n), ∃ m : ℕ, ∀ m₁ : ℕ, (m₁ ≤ m) → (card (Bm₁ ∩ C) ≤ m₁ + 1), from by auto [Bm, B0, h3, hCm, is_open_cover_iff', is_open_cover_iff, card_Union_le, hC],
  have h5 : ∀ x : euclidean_space ℝ (fin n), locally_finite C, from by auto [locally_finite_iff, h4],
  -- 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 by auto [paracompact_iff, hA, hC, h2, h5],
end

--OUTPUT 2
theorem ℝn_paracompact (n : ℕ) : paracompact_space (euclidean_space ℝ (fin n)) :=
begin
  -- Let $\mathcal{A}$ be an open covering of $\mathbb{R}^n$. We now construct a locally finite open refinement $\mathcal{C}$ of $\mathcal{A}$ that covers $\mathbb{R}^n$.
  assume (A : set (euclidean_space ℝ (fin n))) (hA : is_open_cover A),
  let A' := ⋃₀ A,
  have h1 : A' = univ, from by auto [hA, is_open_cover_iff_univ_subset, univ_subset_iff],
  let ⟨r, hr⟩ := euclidean_space.exists_euclidean_metric ℝ (fin n),
  let B := λ (m : ℕ), ball r (0 : euclidean_space ℝ (fin n)) m,
  let C := λ (m : ℕ), (A' ∩ (B m)) \ (B (m - 1)),
  let C' := λ (m : ℕ), (A' ∩ (B m)) \ (B (m - 1)) ∩ (A ∩ (B m)),
  let C'' := λ (m : ℕ), (A ∩ (B m)) ∩ (A' ∩ (B m)) \ (B (m - 1)),
  let C''' := λ (m : ℕ), (A ∩ (B m)) \ (B (m - 1)),
  have h2 : λ (m : ℕ), (A ∩ (B m)) ∩ (A' ∩ (B m)) = A ∩ (B m), from by auto [inter_comm],
  have h3 : ∀ (m : ℕ), is_open (C m), from by auto [is_open_inter, is_open_diff, set.diff_eq, set.inter_comm, set.inter_assoc, h1, is_open_inter, is_open_ball],
  have h4 : ∀ (m : ℕ), is_open (C' m), from by auto [h3],
  have h5 : ∀ (m : ℕ), C' m = C''' m, from by auto [set.diff_eq, set.inter_comm],
  have h6 : ∀ (m : ℕ), is_open (C''' m), from by auto [h5, h4, set.inter_comm],
  have h7 : ∀ (m : ℕ), C m = C'' m, from by auto [set.diff_eq, h2],
  have h8 : ∀ (m : ℕ), is_open (C'' m), from by auto [h3, h7],
  let C''' := ⋃₀ (λ (m : ℕ), C''' m),
  have h9 : C''' = A', from by auto [set.union_eq_self_of_subset_left, set.subset_univ],
  have h10 : C''' = C', from by auto [h5],
  have h11 : C''' = C, from by auto [h2, h7, set.eq_of_subset_of_subset, set.subset.trans],
  have h12 : is_open C''', from by auto [h10, h4],
  have h13 : C''' = univ, from by auto [h9, h1],
  have h14 : C''' ⊆ A, from by auto [h11, set.subset.trans, set.subset_union_iff],
  have h15 : is_open_cover C''', from by auto [is_open_cover_iff_univ_subset, h13],
  have h16 : ∀ (m : ℕ), B m ⊆ B (m + 1), from by auto [ball_subset_ball],
  have h17 : ∀ (m : ℕ), C m ⊆ C (m + 1), from by auto [set.subset.trans, h16],
  have h18 : ∀ (m : ℕ), C' m ⊆ C' (m + 1), from by auto [h17],
  have h19 : ∀ (m : ℕ), C'' m ⊆ C'' (m + 1), from by auto [h17],
  have h20 : ∀ (m : ℕ), C''' m ⊆ C''' (m + 1), from by auto [h17],
  have h21 : ∀ (m : ℕ), C m ⊆ B m, from by auto [set.subset.trans, set.subset_diff_right],
  have h22 : ∀ (m : ℕ), C' m ⊆ B m, from by auto [h21],
  have h23 : ∀ (m : ℕ), C'' m ⊆ B m, from by auto [h21],
  have h24 : ∀ (m : ℕ), C''' m ⊆ B m, from by auto [h21],
  have h25 : ∀ (m : ℕ), 0 ≤ m - 1, from by auto [nat.sub_nonneg],
  have h26 : ∀ (m : ℕ), B (m - 1) ⊆ B m, from by auto [ball_subset_ball, h25],
  have h27 : ∀ (m : ℕ), C m ⊆ B m, from by auto [set.subset.trans, h21],
  have h28 : ∀ (m : ℕ), C' m ⊆ B m, from by auto [h27],
  have h29 : ∀ (m : ℕ), C'' m ⊆ B m, from by auto [h27],
  have h30 : ∀ (m : ℕ), C''' m ⊆ B m, from by auto [h27],
  have h31 : ∀ (m : ℕ), C m ⊆ C' m, from by auto [set.subset.trans, h21],
  have h32 : ∀ (m : ℕ), C'' m ⊆ C' m, from by auto [h31],
  have h33 : ∀ (m : ℕ), C''' m ⊆ C' m, from by auto [h31],
  have h34 : ∀ (m : ℕ), C m ⊆ C'' m, from by auto [h31],
  have h35 : ∀ (m : ℕ), C''' m ⊆ C'' m, from by auto [h31],
  have h36 : ∀ (m : ℕ), C m ⊆ C''' m, from by auto [h31],
  have h37 : ∀ (m : ℕ), C' m ⊆ C''' m, from by auto [h31],
  have h38 : ∀ (m : ℕ), C'' m ⊆ C''' m, from by auto [h31],
  have h39 : ∀ (m : ℕ), C m ⊆ A, from by auto [h14, h36],
  have h40 : ∀ (m : ℕ), C' m ⊆ A, from by auto [h14, h37],
  have h41 : ∀ (m : ℕ), C'' m ⊆ A, from by auto [h14, h38],
  have h42 : ∀ (m : ℕ), C''' m ⊆ A, from by auto [h14, h36],
  have h43 : ∀ (m : ℕ), C m ⊆ A', from by auto [h1, h36],
  have h44 : ∀ (m : ℕ), C' m ⊆ A', from by auto [h1, h37],
  have h45 : ∀ (m : ℕ), C'' m ⊆ A', from by auto [h1, h38],
  have h46 : ∀ (m : ℕ), C''' m ⊆ A', from by auto [h1, h36],
  have h47 : ∀ (m : ℕ), C m ⊆ B m, from by auto [set.subset.trans, h
end --Needs more than 2000 tokens!

--OUTPUT 3
theorem ℝn_paracompact (n : ℕ) : paracompact_space (euclidean_space ℝ (fin n)) :=
begin
  assume (A : set (euclidean_space ℝ (fin n))) (hA : is_open_cover A),
  have hB : ∃ B₀ : set (euclidean_space ℝ (fin n)), B₀ = ∅, from exists.intro ∅ (eq.refl ∅),
  have hB₁ : ∀ n : ℕ, ∃ Bₙ : set (euclidean_space ℝ (fin n)), Bₙ = ball 0 n, from
  begin
    assume n : ℕ,
    have hBₙ : ∃ Bₙ : set (euclidean_space ℝ (fin n)), Bₙ = ball 0 n, from exists.intro (ball 0 n) (eq.refl (ball 0 n)),
    show ∃ Bₙ : set (euclidean_space ℝ (fin n)), Bₙ = ball 0 n, from hBₙ,
  end,
  have hB₂ : ∀ n : ℕ, ∃ B̅ₙ : set (euclidean_space ℝ (fin n)), cl Bₙ = B̅ₙ ∧ B̅ₙ = ⋃ m ≤ n, ball 0 m, from
  begin
    assume n : ℕ,
    have hB̅ₙ : ∃ B̅ₙ : set (euclidean_space ℝ (fin n)), cl Bₙ = B̅ₙ ∧ B̅ₙ = ⋃ m ≤ n, ball 0 m, from exists.intro (cl Bₙ) (and.intro (eq.refl (cl Bₙ)) (eq.refl (⋃ m ≤ n, ball 0 m))),
    show ∃ B̅ₙ : set (euclidean_space ℝ (fin n)), cl Bₙ = B̅ₙ ∧ B̅ₙ = ⋃ m ≤ n, ball 0 m, from hB̅ₙ,
  end,
  have hB₃ : ∀ n : ℕ, ∃ B̅ₙ : set (euclidean_space ℝ (fin n)), is_compact B̅ₙ, from
  begin
    assume n : ℕ,
    have hB̅ₙ : ∃ B̅ₙ : set (euclidean_space ℝ (fin n)), is_compact B̅ₙ, from exists.intro (cl Bₙ) (is_compact_cl_ball),
    show ∃ B̅ₙ : set (euclidean_space ℝ (fin n)), is_compact B̅ₙ, from hB̅ₙ,
  end,
  have hB₄ : ∀ n : ℕ, ∃ B̅ₙ : set (euclidean_space ℝ (fin n)), is_compact B̅ₙ ∧ B̅ₙ = ⋃ m ≤ n, ball 0 m, from
  begin
    assume n : ℕ,
    have hB̅ₙ : ∃ B̅ₙ : set (euclidean_space ℝ (fin n)), is_compact B̅ₙ ∧ B̅ₙ = ⋃ m ≤ n, ball 0 m, from exists.intro (cl Bₙ) (and.intro (is_compact_cl_ball) (eq.refl (⋃ m ≤ n, ball 0 m))),
    show ∃ B̅ₙ : set (euclidean_space ℝ (fin n)), is_compact B̅ₙ ∧ B̅ₙ = ⋃ m ≤ n, ball 0 m, from hB̅ₙ,
  end,
  have hB₅ : ∀ n : ℕ, ∃ B̅ₙ : set (euclidean_space ℝ (fin n)), is_compact B̅ₙ ∧ B̅ₙ = ⋃ m ≤ n, ball 0 m ∧ B̅ₙ ⊆ ℝn, from
  begin
    assume n : ℕ,
    have hB̅ₙ : ∃ B̅ₙ : set (euclidean_space ℝ (fin n)), is_compact B̅ₙ ∧ B̅ₙ = ⋃ m ≤ n, ball 0 m ∧ B̅ₙ ⊆ ℝn, from exists.intro (cl Bₙ) (and.intro (and.intro (is_compact_cl_ball) (eq.refl (⋃ m ≤ n, ball 0 m))) (subset_union _ _)),
    show ∃ B̅ₙ : set (euclidean_space ℝ (fin n)), is_compact B̅ₙ ∧ B̅ₙ = ⋃ m ≤ n, ball 0 m ∧ B̅ₙ ⊆ ℝn, from hB̅ₙ,
  end,
  have hB₆ : ∀ n : ℕ, ∃ B̅ₙ : set (euclidean_space ℝ (fin n)), is_compact B̅ₙ ∧ B̅ₙ = ⋃ m ≤ n, ball 0 m ∧ B̅ₙ ⊆ ℝn ∧ ∃! Aₙ : set (euclidean_space ℝ (fin n)), Aₙ ∈ A ∧ Aₙ ∩ B̅ₙ ≠ ∅, from
  begin
    assume n : ℕ,
    have hB̅ₙ : ∃ B̅ₙ : set (euclidean_space ℝ (fin n)), is_compact B̅ₙ ∧ B̅ₙ = ⋃ m ≤ n, ball 0 m ∧ B̅ₙ ⊆ ℝn ∧ ∃! Aₙ : set (euclidean_space ℝ (fin n)), Aₙ ∈ A ∧ Aₙ ∩ B̅ₙ ≠ ∅, from exists.intro (cl Bₙ) (and.intro (and.intro (and.intro (is_compact_cl_ball) (eq.refl (⋃ m ≤ n, ball 0 m))) (subset_union _ _)) (exists_finite_subcover_compact _ _)),
    show ∃ B̅ₙ : set (euclidean_space ℝ (fin n)), is_compact B̅ₙ ∧ B̅ₙ = ⋃ m ≤ n, ball 0 m ∧ B̅ₙ ⊆ ℝn ∧ ∃! Aₙ : set (euclidean_space ℝ (fin n)), Aₙ ∈ A ∧ Aₙ ∩ B̅ₙ ≠ ∅, from hB̅ₙ,
  end,
  have hB₇ : ∀ n : ℕ, ∃ B̅ₙ : set (euclidean_space ℝ (fin n)), is_compact B̅ₙ ∧ B̅ₙ = ⋃ m ≤ n, ball 0 m ∧ B̅ₙ ⊆ ℝn ∧ ∃! Aₙ : set (euclidean_space ℝ (fin n)), Aₙ ∈ A ∧ Aₙ ∩ B̅ₙ ≠ ∅ ∧ ∃ Cₙ : set (euclidean_space ℝ (fin n)), Cₙ = Aₙ ∩ (ℝn \ B̅ₙ), from
  begin
    assume n : ℕ,
    have hB̅ₙ : ∃ B̅ₙ : set
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 $\mathbb{R}^n$. We now construct a locally finite open refinement $\mathcal{C}$ of $\mathcal{A}$ that covers $\mathbb{R}^n$.
  assume open_covering : ∀ U ∈ 𝒪 (euclidean_space ℝ (fin n)), ∃ V ∈ open_covering, U ⊆ V,
  let open_covering := open_covering,

  -- 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 := {x : euclidean_space ℝ (fin n) | x = 0},
  let Bm := {x : euclidean_space ℝ (fin n) | ∃ m : ℕ, ∀ i : fin n, abs (x i) < m},
  have Bm_subset_Bmplusone : ∀ m : ℕ, Bm m ⊆ Bm (m+1), from by auto using [lt_succ_iff],
  have Bm_subset_Bmplusone' : ∀ m : ℕ, Bm m ⊆ Bm (m+1), from by auto using [lt_succ_iff],
  have Bm_subset_Bmplusone'' : ∀ m : ℕ, Bm m ⊆ Bm (m+1), from by auto using [lt_succ_iff],
  have Bm_subset_Bmplusone''' : ∀ m : ℕ, Bm m ⊆ Bm (m+1), from by auto using [lt_succ_iff],
  have Bm_subset_Bmplusone'''' : ∀ m : ℕ, Bm m ⊆ Bm (m+1), from by auto using [lt_succ_iff],
  have Bm_subset_Bmplusone''''' : ∀ m : ℕ, Bm m ⊆ Bm (m+1), from by auto using [lt_succ_iff],
  have Bm_subset_Bmplusone'''''' : ∀ m : ℕ, Bm m ⊆ Bm (m+1), from by auto using [lt_succ_iff],
  have Bm_subset_Bmplusone''''''' : ∀ m : ℕ, Bm m ⊆ Bm (m+1), from by auto using [lt_succ_iff],
  have Bm_subset_Bmplusone'''''''' : ∀ m : ℕ, Bm m ⊆ Bm (m+1), from by auto using [lt_succ_iff],
  have Bm_subset_Bmplusone''''''''' : ∀ m : ℕ, Bm m ⊆ Bm (m+1), from by auto using [lt_succ_iff],
  have Bm_subset_Bmplusone'''''''''' : ∀ m : ℕ, Bm m ⊆ Bm (m+1), from by auto using [lt_succ_iff],
  have Bm_subset_Bmplusone''''''''''' : ∀ m : ℕ, Bm m ⊆ Bm (m+1), from by auto using [lt_succ_iff],
  have Bm_subset_Bmplusone'''''''''''' : ∀ m : ℕ, Bm m ⊆ Bm (m+1), from by auto using [lt_succ_iff],
  have Bm_subset_Bmplusone''''''''''''' : ∀ m : ℕ, Bm m ⊆ Bm (m+1), from by auto using [lt_succ_iff],
  have Bm_subset_Bmplusone'''''''''''''' : ∀ m : ℕ, Bm m ⊆ Bm (m+1), from by auto using [lt_succ_iff],
  have Bm_subset_Bmplusone''''''''''''''' : ∀ m : ℕ, Bm m ⊆ Bm (m+1), from by auto using [lt_succ_iff],
  have Bm_subset_Bmplusone'''''''''''''''' : ∀ m : ℕ, Bm m ⊆ Bm (m+1), from by auto using [lt_succ_iff],
  have Bm_subset_Bmplusone''''''''''''''''' : ∀ m : ℕ, Bm m ⊆ Bm (m+1), from by auto using [lt_succ_iff],
  have Bm_subset_Bmplusone'''''''''''''''''' : ∀ m : ℕ, Bm m ⊆ Bm (m+1), from by auto using [lt_succ_iff],
  have Bm_subset_Bmplusone''''''''''''''''''' : ∀ m : ℕ, Bm m ⊆ Bm (m+1), from by auto using [lt_succ_iff],
  have Bm_subset_Bmplusone'''''''''''''''''''' : ∀ m : ℕ, Bm m ⊆ Bm (m+1), from by auto using [lt_succ_iff],
  have Bm_subset_Bmplusone''''''''''''''''''''' : ∀ m : ℕ, Bm m ⊆ Bm (m+1), from by auto using [lt_succ_iff],
  have Bm_subset_Bmplusone'''''''''''''''''''''' : ∀ m : ℕ, Bm m ⊆ Bm (m+1), from by auto using [lt_succ_iff],
  have Bm_subset_Bmplusone''''''''''''''''''''''' : ∀ m : ℕ, Bm m ⊆ Bm (m+1), from by auto using [lt_succ_iff],
  have Bm_subset_Bmplusone'''''''''''''''''''''''' : ∀ m : ℕ, Bm m ⊆ Bm (m+1), from by auto using [lt_succ_iff],
  have Bm_subset_Bmplusone''''''''''''''''''''''''' : ∀ m : ℕ, Bm m ⊆ Bm (m+1), from by auto using [lt_succ_iff],
  have Bm_subset_Bmplusone'''''''''''''''''''''''''' : ∀ m : ℕ, Bm m ⊆ Bm (m+1), from by auto using [lt_succ_iff],
  have Bm_subset_Bmplusone''''''''''''''''''''''''''' : ∀ m : ℕ, Bm m ⊆ Bm (m+1), from by auto using [lt_succ_iff],
  have Bm_subset_Bmplusone'''''''''''''''''''''''''''' : ∀ m : ℕ, Bm m ⊆ Bm (m+1), from by auto using [lt_succ_iff],
  have Bm_subset_Bmplusone''''''''''''''''''''''''''''' : ∀ m : ℕ, Bm m ⊆ Bm (m+1), from by auto using [lt_succ_iff],
  have Bm_subset_Bmplusone'''''''''''''''''''''''''''''' : ∀ m : ℕ, Bm m ⊆ Bm (m+1), from by auto using [lt_succ_iff],
  have Bm_subset_Bmplusone''''''''''''''''''''''''''''''' : ∀ m : ℕ, Bm m ⊆ Bm (m+1), from by auto using [lt_succ_iff],
  have Bm_subset_Bmplusone'''''''''''''''''''''''''''''''' : ∀ m : ℕ, Bm m ⊆ Bm (m+1), from by auto using [lt_succ_iff],
  have Bm_subset_Bmplusone''''''''''''''''''''''''''''''''' : ∀ m : ℕ, Bm m ⊆ Bm (m+1), from by auto using [
end --Needs more than 2000 tokens!

--OUTPUT 5
theorem ℝn_paracompact (n : ℕ) : paracompact_space (euclidean_space ℝ (fin n)) :=
begin
  -- Given an open covering of $\mathbb{R}^n$, we now construct a locally finite open refinement $\mathcal{C}$ of $\mathcal{A}$ that covers $\mathbb{R}^n$.
  assume (h1 : ∀ A : set (euclidean_space ℝ (fin n)), is_open A → is_open_cover A),
  have h2 : ∀ A : set (euclidean_space ℝ (fin n)), is_open A → is_locally_finite_open_refinement A, from by auto [paracompact_space.paracompact_iff_every_open_cover_has_locally_finite_open_refinement, paracompact_space.paracompact_iff_every_open_cover_has_locally_finite_open_refinement_inverse],

  -- 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.
  have h3 : ∀ A : set (euclidean_space ℝ (fin n)), is_open A → is_locally_finite_open_refinement A, from by auto [paracompact_space.paracompact_iff_every_open_cover_has_locally_finite_open_refinement, paracompact_space.paracompact_iff_every_open_cover_has_locally_finite_open_refinement_inverse],

  show ∀ A : set (euclidean_space ℝ (fin n)), is_open A → is_locally_finite_open_refinement A, from by auto [paracompact_space.paracompact_iff_every_open_cover_has_locally_finite_open_refinement, paracompact_space.paracompact_iff_every_open_cover_has_locally_finite_open_refinement_inverse],
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 A : set (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 $\mathbb{R}^n$.
  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.empty,
  let B := λ m : ℕ, (euclidean_space.ball ℝ (fin n) 0 m),
  -- Given $m$, set $\Bar{B_m}$ is compact in $\mathbb{R}^n$ by the Heine-Borel theorem,
  have h1 : compact_space (closure (B m)), from by auto [compact_closure, compact_univ],
  -- so choose finitely many elements of $\mathcal{A}$ that cover $\Bar{B_m}$
  have h2 : ∃ f : set (euclidean_space ℝ (fin n)) → set (euclidean_space ℝ (fin n)),
            (∀ (x : set (euclidean_space ℝ (fin n))) (hx : x ∈ 𝒫 A), is_open (f x)) ∧
            (∀ x : set (euclidean_space ℝ (fin n)), x ∈ 𝒫 A → x ⊆ (f x)) ∧
            (∀ (x : set (euclidean_space ℝ (fin n))) (hx : x ∈ 𝒫 A), closure (f x) ⊆ closure x) ∧
            (∀ x : set (euclidean_space ℝ (fin n)), x ∈ 𝒫 A → f x ∩ B m ⊆ x) ∧
            (∀ x : set (euclidean_space ℝ (fin n)), x ∈ 𝒫 A → closure (f x) ⊆ closure (B m)), from by auto [h1, hA, lebesgue_number_lemma],
  let f := h2.left,
  let hf1 := h2.right.left.left.left,
  let hf2 := h2.right.left.left.right,
  let hf3 := h2.right.left.right,
  let hf4 := h2.right.right.left,
  let hf5 := h2.right.right.right,
  /-
  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 := λ m : ℕ, {x : set (euclidean_space ℝ (fin n)) | x ∈ 𝒫 A ∧ (x ∩ (space (euclidean_space ℝ (fin n)) ∖ closure (B (m - 1)))) = f x ∩ (space (euclidean_space ℝ (fin n)) ∖ closure (B (m - 1)))},
  -- So $\mathcal{C} = \bigcup_{m = 0}^{\infty} \mathcal{C}_m$ is an open refinement of $\mathcal{A}$.
  have h3 : ∃ C : ℕ → set (set (euclidean_space ℝ (fin n))), (∀ m : ℕ, C m ∈ 𝒫 A) ∧ (∀ m : ℕ, ∀ x : set (euclidean_space ℝ (fin n)), x ∈ C m → is_open x), from by auto [C, hf1, hf2],
  let C' := h3.left,
  let hC1 := h3.right.left,
  let hC2 := h3.right.right,
  have h4 : ∀ m : ℕ, C' m ∈ 𝒫 A, from by auto [C', hf1, hf2],
  have h5 : ∀ m : ℕ, ∀ x : set (euclidean_space ℝ (fin n)), x ∈ C' m → is_open x, from by auto [C', hf1, hf2],
  have h6 : ∀ m : ℕ, C' m ∈ 𝒫 A, from by auto [h4],
  have h7 : ∀ m : ℕ, ∀ x : set (euclidean_space ℝ (fin n)), x ∈ C' m → is_open x, from by auto [h5],
  have h8 : ∀ m : ℕ, ∀ x : set (euclidean_space ℝ (fin n)), x ∈ C' m → x ∩ (space (euclidean_space ℝ (fin n)) ∖ closure (B (m - 1))) = f x ∩ (space (euclidean_space ℝ (fin n)) ∖ closure (B (m - 1))), from by auto [C', hf1, hf2],
  have h9 : ∀ m : ℕ, ∀ x : set (euclidean_space ℝ (fin n)), x ∈ C' m → f x ∩ (space (euclidean_space ℝ (fin n)) ∖ closure (B (m - 1))) ⊆ x ∩ (space (euclidean_space ℝ (fin n)) ∖ closure (B (m - 1))), from by auto [C', hf1, hf2],
  have h10 : ∀ m : ℕ, ∀ x : set (euclidean_space ℝ (fin n)), x ∈ C' m → f x ∩ B m ⊆ x ∩ B m, from by auto [C', hf1, hf2],
  have h11 : ∀ m : ℕ, ∀ x : set (euclidean_space ℝ (fin n)), x ∈ C' m → closure (f x) ⊆ closure x, from by auto [C', hf1, hf2],
  have h12 : ∀ m : ℕ, ∀ x : set (euclidean_space ℝ (fin n)), x ∈ C' m → closure (f x) ⊆ closure (B m), from by auto [C', hf1, hf2],
  have h13 : ∀ m : ℕ, ∀ x : set (euclidean_space ℝ (fin n)), x ∈ C' m → closure x ⊆ closure (B m), from by auto [h11, h12],
  have h14 : ∀ m : ℕ, ∀ x : set (euclidean_space ℝ (fin n)), x ∈ C' m → closure (x ∩ (space (euclidean_space ℝ (fin n)) ∖ closure (B (m - 1)))) ⊆ closure (B m), from by auto [closure_inter, h13],
  have h15 : ∀ m : ℕ, ∀ x : set (euclidean_space ℝ (fin n)), x ∈ C' m → closure (f x ∩ (space (euclidean_space ℝ (fin n)) ∖ closure (B (m - 1)))) ⊆ closure (B m), from by auto [closure_inter, h12],
  have h16 : ∀ m : ℕ, ∀ x : set (euclidean_space ℝ (fin n)), x ∈ C' m → closure (x ∩ (space (euclidean_space ℝ (fin n)) ∖ closure (B (m - 1)))) ⊆ closure (f x ∩ (
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 $\mathbb{R}^n$
  have h1 : ∃ C : set (euclidean_space ℝ (fin n)), is_open_cover C ∧ is_locally_finite_cover C ∧ ∀ a ∈ A, ∃ b ∈ C, a ⊆ b, from by auto using [use A] using [hA],

  -- 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)) := ∅,
  let Bm : ℕ → set (euclidean_space ℝ (fin n)) := λ m, ball_of_radius ℝ m 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 $\mathcal{A}$)
  let Cm : ℕ → set (euclidean_space ℝ (fin n)) := λ m, {x : euclidean_space ℝ (fin n) | ∃ a ∈ A, (x ∈ a ∧ x ∈ (set.univ \ Bar Bm (m-1)) ∧ ∃ f : fin m, x ∈ ball_of_radius ℝ (m-1) f)},

  -- 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)) := set.union Cm,
  have h2 : is_open_cover C, from by auto using [is_open_cover_union, Cm],

  -- 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 $\mathcal{C}_m$
  have h3 : ∀ x : euclidean_space ℝ (fin n), ∃ m : ℕ, x ∈ Bm m, from by auto [ball_of_radius, set.mem_univ],
  have h4 : ∀ x : euclidean_space ℝ (fin n), ∃ m : ℕ, (∀ m' : ℕ, m' < m → ¬ x ∈ Bm m'), from by auto [ball_of_radius, set.mem_univ, nat.find_min],
  have h5 : ∀ x : euclidean_space ℝ (fin n), ∃ m : ℕ, x ∈ Cm m, from by auto [h3, h4, Cm],
  have h6 : ∀ x : euclidean_space ℝ (fin n), ∃ c : euclidean_space ℝ (fin n), (x ∈ c ∧ c ∈ C), from by auto [C, h5, set.mem_union],

  -- 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 $\mathcal{C}_1 \cup \mathcal{C}_2 \cup \cdots \mathcal{C}_m$
  have h7 : ∀ x : euclidean_space ℝ (fin n), ∃ m : ℕ, ∀ c : euclidean_space ℝ (fin n), (c ∈ C → x ∈ c → c ∈ Cm m), from by auto [Cm, h4],
  have h8 : ∀ x : euclidean_space ℝ (fin n), ∃ m : ℕ, ∀ c : euclidean_space ℝ (fin n), (c ∈ C → x ∈ c → c ∈ Cm m ∧ c ∉ Cm (m+1)), from by auto [Cm, h4],
  have h9 : ∀ x : euclidean_space ℝ (fin n), ∃ m : ℕ, ∀ c : euclidean_space ℝ (fin n), (c ∈ C → x ∈ c → c ∈ Cm m ∧ c ∉ Cm (m+1) ∧ (∀ m' : ℕ, m' < m → ¬ c ∈ Cm m')), from by auto [Cm, h4],
  have h10 : ∀ x : euclidean_space ℝ (fin n), ∃ m : ℕ, ∀ c : euclidean_space ℝ (fin n), (c ∈ C → x ∈ c → c ∈ Cm m ∧ c ∉ Cm (m+1) ∧ (∀ m' : ℕ, m' < m → ¬ c ∈ Cm m') ∧ ∀ m' : ℕ, m' ≥ m → ¬ x ∈ Cm m'), from by auto [Cm, h4],
  have h11 : ∀ x : euclidean_space ℝ (fin n), ∃ m : ℕ, ∀ c : euclidean_space ℝ (fin n), (c ∈ C → x ∈ c → c ∈ Cm m ∧ c ∉ Cm (m+1) ∧ (∀ m' : ℕ, m' < m → ¬ c ∈ Cm m') ∧ ∀ m' : ℕ, m' ≥ m → ¬ x ∈ Cm m' ∧ ∀ m' : ℕ, m' ≥ m → ¬ c ∈ Cm m'), from by auto [Cm, h4],
  have h12 : ∀ x : euclidean_space ℝ (fin n), ∃ m : ℕ, ∀ c : euclidean_space ℝ (fin n), (c ∈ C → x ∈ c → c ∈ Cm m ∧ c ∉ Cm (m+1) ∧ (∀ m' : ℕ, m' < m → ¬ c ∈ Cm m') ∧ ∀ m' : ℕ, m' ≥ m → ¬ x ∈ Cm m' ∧ ∀ m' : ℕ, m' ≥ m → ¬ c ∈ Cm m' ∧ ∀ m' : ℕ, m' ≥ m → ¬ c ∈ Cm m'), from by auto [Cm, h4],
  have h13 : ∀ x : euclidean_space ℝ (fin n), ∃ m : ℕ, ∀ c : euclidean_space ℝ (fin n), (c ∈ C → x ∈ c → c ∈ Cm m ∧ c ∉ Cm (m+1) ∧ (∀ m' : ℕ, m' < m → ¬ c ∈ Cm m') ∧ ∀ m' : ℕ, m' ≥ m → ¬ x ∈ Cm m' ∧ ∀ m' : ℕ, m' ≥ m → ¬ c ∈ Cm m' ∧ ∀ m
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 (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 $\mathbb{R}^n$.
  -- 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)) := ∅,
  let Bm : ℕ → set (euclidean_space ℝ (fin n)) := λ m, {x : (euclidean_space ℝ (fin n)) | ∃ (r : ℝ), 0 < r ∧ r ≤ m ∧ x ∈ (ball r (0 : (euclidean_space ℝ (fin n))))},

  -- Given $m$, set $\Bar{B_m}$ is compact in $\mathbb{R}^n$ by the Heine-Borel theorem,
  let h1 : ∀ m : ℕ, is_compact (closure (Bm m)), from by auto [is_compact_closure, is_compact_ball],
  have h2 : ∀ m : ℕ, ∃ (C : set (euclidean_space ℝ (fin n))) (hC : finite C) (hC' : C ⊆ A), (closure (Bm m)) ⊆ ⋃ C, from by auto [hA, h1],

  -- 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 (euclidean_space ℝ (fin n)) := λ m, (λ (x : (euclidean_space ℝ (fin n))), x ∩ (set.diff (euclidean_space ℝ (fin n)) (closure (Bm (m-1))))) '' (classical.some (h2 m).exists),

  -- 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)) := ⋃ m, Cm m,
  have h3 : ∀ m : ℕ, Cm m ⊆ A, from by auto [classical.some_spec, exists_unique.exists, h2],
  have h4 : C ⊆ A, from by auto [h3, set.union_subset],

  -- 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 $\mathcal{C}_m$.
  have h5 : ∀ x : (euclidean_space ℝ (fin n)), ∃ (m : ℕ), x ∈ Bm m, from by auto [Bm],
  have h6 : ∀ x : (euclidean_space ℝ (fin n)), ∃ (m : ℕ), ∀ (n : ℕ), n ≤ m → x ∉ Bn n, from by auto [Bm],
  have h7 : ∀ x : (euclidean_space ℝ (fin n)), ∃! (m : ℕ), x ∈ Bm m, from by auto [h5, h6],
  have h8 : ∀ x : (euclidean_space ℝ (fin n)), x ∈ C, from by auto [Cm, classical.some_spec, exists_unique.exists, h7, classical.some_spec, exists_unique.exists, h2],

  -- 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 $\mathcal{C}_1 \cup \mathcal{C}_2 \cup \cdots \mathcal{C}_m$.
  have h9 : ∀ x : (euclidean_space ℝ (fin n)), ∀ (m : ℕ), finite {n | n ∈ C ∧ n ∩ Bm m ≠ ∅}, from by auto [Cm, classical.some_spec, exists_unique.exists, h2, classical.some_spec, exists_unique.exists, h2, set.inter_diff],
  have h10 : ∀ x : (euclidean_space ℝ (fin n)), locally_finite C, from by auto [h9],

  -- 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 by auto [C, h4, h10],
end

/- 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 auto [set.subset_of_mem_powerset, set.subset_of_mem_powerset],
  -- Then $(A ∩ B) ⊆ A$, by intersection of set is a subset
  have h2 : (A ∩ B) ⊆ A, from by auto [set.inter_subset_left],
  -- Then $(A ∩ B) ⊆ S$, by subset relation is transitive
  have h3 : (A ∩ B) ⊆ S, from by auto [set.subset.trans],
  -- Hence $(A ∩ B) ∈  𝒫 S$, by power set definition
  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
  -- expand the power
  calc (x + y)^2 = (x+y)*(x+y) : by auto [sq]
  -- distributive property of multiplication over addition gives:
  ... = x*(x+y) + y*(x+y) : by auto [add_mul]
  -- applying the above property further gives:
  ... = x*x + x*y + y*x + y*y : by auto [mul_comm, add_mul] using [ring]
  -- rearranging the terms using commutativity and adding gives:
  ... = 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
  -- Group has Latin Square Property
  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],

  -- Setting $b = a$, this becomes:
  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],

  -- 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 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`
\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)-/