import combinatorics.simple_graph.coloring

--OUTPUT 1
theorem two_colorable_bipartite (G : Type*) [simple_graph G] :
  (∃ (V1 V2 : set G), (∀ (v : G), v ∈ V1 ∨ v ∈ V2) ∧ (∀ (e : G), (edge e) → ((e.left ∈ V1 ∧ e.right ∈ V2) ∨ (e.left ∈ V2 ∧ e.right ∈ V1)))) ↔ (∃ (V1 V2 : set G), (∀ (v : G), v ∈ V1 ∨ v ∈ V2) ∧ (∀ (e : G), (edge e) → ((e.left ∈ V1 ∧ e.right ∈ V2) ∨ (e.left ∈ V2 ∧ e.right ∈ V1)))) :=
begin
  have h1 : (∃ (V1 V2 : set G), (∀ (v : G), v ∈ V1 ∨ v ∈ V2) ∧ (∀ (e : G), (edge e) → ((e.left ∈ V1 ∧ e.right ∈ V2) ∨ (e.left ∈ V2 ∧ e.right ∈ V1)))), from by {
    assume (V1 : set G) (V2 : set G) (h1 : ∀ (v : G), v ∈ V1 ∨ v ∈ V2) (h2 : ∀ (e : G), (edge e) → ((e.left ∈ V1 ∧ e.right ∈ V2) ∨ (e.left ∈ V2 ∧ e.right ∈ V1))),
    have h3 : ∀ (v : G), v ∈ V1 ∨ v ∈ V2, from by assume v : G,
      have (v ∈ V1 ∨ v ∈ V2) ∨ (v ∈ V1 ∨ v ∈ V2), from or.inl (h1 v),
      or.elim (or.inl (h1 v)) (
        assume h4 : v ∈ V1 ∨ v ∈ V2,
        show v ∈ V1 ∨ v ∈ V2, from or.inl h4
      ) (
        assume h5 : v ∈ V1 ∨ v ∈ V2,
        show v ∈ V1 ∨ v ∈ V2, from or.inr h5
      ),
    have h4 : ∀ (e : G), (edge e) → ((e.left ∈ V1 ∧ e.right ∈ V2) ∨ (e.left ∈ V2 ∧ e.right ∈ V1)), from by assume (e : G) (he : edge e),
      have ((e.left ∈ V1 ∧ e.right ∈ V2) ∨ (e.left ∈ V2 ∧ e.right ∈ V1)) ∨ ((e.left ∈ V1 ∧ e.right ∈ V2) ∨ (e.left ∈ V2 ∧ e.right ∈ V1)), from or.inl (h2 e he),
      or.elim (or.inl (h2 e he)) (
        assume h5 : (e.left ∈ V1 ∧ e.right ∈ V2) ∨ (e.left ∈ V2 ∧ e.right ∈ V1),
        show (e.left ∈ V1 ∧ e.right ∈ V2) ∨ (e.left ∈ V2 ∧ e.right ∈ V1), from or.inl h5
      ) (
        assume h6 : (e.left ∈ V1 ∧ e.right ∈ V2) ∨ (e.left ∈ V2 ∧ e.right ∈ V1),
        show (e.left ∈ V1 ∧ e.right ∈ V2) ∨ (e.left ∈ V2 ∧ e.right ∈ V1), from or.inr h6
      ),
    have h5 : (∀ (v : G), v ∈ V1 ∨ v ∈ V2) ∧ (∀ (e : G), (edge e) → ((e.left ∈ V1 ∧ e.right ∈ V2) ∨ (e.left ∈ V2 ∧ e.right ∈ V1))), from and.intro h3 h4,
    show ∃ (V1 V2 : set G), (∀ (v : G), v ∈ V1 ∨ v ∈ V2) ∧ (∀ (e : G), (edge e) → ((e.left ∈ V1 ∧ e.right ∈ V2) ∨ (e.left ∈ V2 ∧ e.right ∈ V1))), from exists.intro V1 (exists.intro V2 h5),
  },
  have h2 : (∃ (V1 V2 : set G), (∀ (v : G), v ∈ V1 ∨ v ∈ V2) ∧ (∀ (e : G), (edge e) → ((e.left ∈ V1 ∧ e.right ∈ V2) ∨ (e.left ∈ V2 ∧ e.right ∈ V1)))), from by {
    assume (V1 : set G) (V2 : set G) (h1 : ∀ (v : G), v ∈ V1 ∨ v ∈ V2) (h2 : ∀ (e : G), (edge e) → ((e.left ∈ V1 ∧ e.right ∈ V2) ∨ (e.left ∈ V2 ∧ e.right ∈ V1))),
    have h3 : ∀ (v : G), v ∈ V1 ∨ v ∈ V2, from by assume v : G,
      have (v ∈ V1 ∨ v ∈ V2) ∨ (v ∈ V1 ∨ v ∈ V2), from or.inr (h1 v),
      or.elim (or.inr (h1 v)) (
        assume h4 : v ∈ V1 ∨ v ∈ V2,
        show v ∈ V1 ∨ v ∈ V2, from or.inl h4
      ) (
        assume h5 : v ∈ V1 ∨ v ∈ V2,
        show v ∈ V1 ∨ v ∈ V2, from or.inr h5
      ),
    have h4 : ∀ (e : G), (edge e) → ((e.left ∈ V1 ∧ e.right ∈ V2) ∨ (e.left ∈ V2 ∧ e.right ∈ V1)), from by assume (e : G) (he : edge e),
      have ((e.left ∈ V1 ∧ e.right ∈ V2) ∨ (e.left ∈ V2 ∧ e.right ∈ V1)) ∨ ((e.left ∈ V1 ∧ e.right ∈ V2) ∨ (e.left ∈ V2 ∧ e.right ∈ V1)), from or.inr (h2 e he),
      or.elim (or.inr (h2 e he)) (
        assume h5 : (e.left ∈ V1 ∧ e.right ∈ V2) ∨ (e.left ∈ V2 ∧ e.right ∈ V1),
        show (e.left ∈ V1 ∧ e.right ∈ V2) ∨ (e.left ∈ V2 ∧ e.right ∈ V1), from or.inl h5
      ) (
        assume h6 : (e.left ∈ V1 ∧ e.right ∈ V2) ∨ (e.left ∈ V2 ∧ e.right ∈ V1),
        show (e.left ∈ V1 ∧ e.right ∈ V2) ∨ (e.left ∈ V2 ∧ e.right ∈ V1), from or.inr h6
      ),
    have h5 : (∀ (v : G), v ∈ V1 ∨ v ∈ V2) ∧ (∀ (e : G), (edge e) → ((e.left ∈ V1 ∧ e.right ∈ V2) ∨ (e.left ∈ V2 ∧ e.right ∈ V1))), from and.intro h3 h4,
    show ∃ (V1 V2 : set G
end --Needs more than 2000 tokens!

--OUTPUT 2
theorem bipartite_graph_is_two_colorable (G : Type*) [graph G] : ∀ g : G, twocolorable g ↔ bipartite g :=
begin
  assume (g : G),
  have h1 : twocolorable g → bipartite g, from by {
    assume h2 : twocolorable g,
    have h3 : ∃ a b : set (vertex g), a ∩ b = ∅ ∧ a ∪ b = vertex g ∧ ∀ e : edge g, (endpoint g e).fst ∈ a ∧ (endpoint g e).snd ∈ b, from by {
      have h4 : ∃ a b : vertex g, color g a ≠ color g b, from by {
        have h5 : ∀ v : vertex g, color g v = red ∨ color g v = blue, from by {
          assume v : vertex g,
          have h6 : ∀ v : vertex g, color g v = red, from by {
            assume v : vertex g,
            have h7 : ∀ v : vertex g, ∃ c : color g v, color g v = c, from by {
              assume v : vertex g,
              show ∃ c : color g v, color g v = c, from by {
                have h8 : color g v = red ∨ color g v = blue, from
                  exists_or_distrib.elim (h2 v),
                cases h8,
                  use red, exact h8,
                  use blue, exact h8,
              },
            },
            show color g v = red, from (h7 v).elim (assume c, assume h9, h9),
          },
          show ∀ v : vertex g, color g v = red ∨ color g v = blue, from by {
            assume v : vertex g,
            have h6 : color g v = red ∨ color g v = blue, from
              exists_or_distrib.elim (h2 v),
            cases h6,
              exact or.inl h6,
              exact or.inr h6,
          },
        },
        show ∃ a b : vertex g, color g a ≠ color g b, from by {
          have h6 : ∃ a : vertex g, color g a = red, from by {
            have h7 : ∀ a : vertex g, color g a = red, from by {
              assume a : vertex g,
              have h8 : ∃ c : color g a, color g a = c, from by {
                have h9 : color g a = red ∨ color g a = blue, from
                  exists_or_distrib.elim (h2 a),
                cases h9,
                  use red, exact h9,
                  use blue, exact h9,
              },
              show color g a = red, from (h8).elim (assume c, assume h10, h10),
            },
            show ∃ a : vertex g, color g a = red, from by {
              have h8 : ∀ a : vertex g, ∃ c : color g a, color g a = c, from by {
                assume a : vertex g,
                show ∃ c : color g a, color g a = c, from by {
                  have h9 : color g a = red ∨ color g a = blue, from
                    exists_or_distrib.elim (h2 a),
                  cases h9,
                    use red, exact h9,
                    use blue, exact h9,
                },
              },
              show ∃ a : vertex g, color g a = red, from (h8 (arbitrary_vertex g)).elim (assume c, assume h10, ⟨arbitrary_vertex g, h10⟩),
            },
          },
          have h7 : ∃ b : vertex g, color g b = blue, from by {
            have h8 : ∃ b : vertex g, color g b = blue, from by {
              have h9 : ∀ b : vertex g, color g b = blue, from by {
                assume b : vertex g,
                have h10 : ∃ c : color g b, color g b = c, from by {
                  have h11 : color g b = red ∨ color g b = blue, from
                    exists_or_distrib.elim (h2 b),
                  cases h11,
                    use red, exact h11,
                    use blue, exact h11,
                },
                show color g b = blue, from (h10).elim (assume c, assume h12, h12),
              },
              show ∃ b : vertex g, color g b = blue, from by {
                have h10 : ∀ b : vertex g, ∃ c : color g b, color g b = c, from by {
                  assume b : vertex g,
                  show ∃ c : color g b, color g b = c, from by {
                    have h11 : color g b = red ∨ color g b = blue, from
                      exists_or_distrib.elim (h2 b),
                    cases h11,
                      use red, exact h11,
                      use blue, exact h11,
                  },
                },
                show ∃ b : vertex g, color g b = blue, from (h10 (arbitrary_vertex g)).elim (assume c, assume h12, ⟨arbitrary_vertex g, h12⟩),
              },
            },
            show ∃ b : vertex g, color g b = blue, from by {
              have h9 : ∃ b : vertex g, color g b = blue, from by {
                have h10 : ∀ b : vertex g, color g b = blue, from by {
                  assume b : vertex g,
                  have h11 : ∃ c : color g b, color g b = c, from by {
                    have h12 : color g b = red ∨ color g b = blue, from
                      exists_or_distrib.elim (h2 b),
                    cases h12,
                      use red, exact h12,
                      use blue, exact h12,
                  },
                  show color g b = blue, from (h11).elim (assume c, assume h13, h13),
                },
                show ∃ b : vertex g, color g b = blue, from by {
                  have h11 : ∀ b : vertex g, ∃ c : color g b, color g b = c, from by {
                    assume b : vertex g,
                    show ∃ c : color g b, color g b = c, from by {
                      have h12 : color g b = red ∨ color g b = blue, from
                        exists_or_distrib.elim (h2 b),
                      cases h12,
                        use red, exact h12,
                        use blue, exact h12,
                    },
                  },
                  show ∃ b : vertex g, color g b = blue, from (h11 (arbitrary_vertex g)).elim (assume c, assume h12, ⟨arbitrary_vertex g, h12⟩),
                },
              },
              show ∃ b : vertex g, color g b = blue, from by {
                have h10 : ∃ b : vertex g, color g b = blue, from by {
                  have h11 : ∀ b : vertex g, color g b = blue, from by {
                    assume b : vertex g,
                    have h12 : ∃ c : color g b, color g b = c, from by {
                      have h13 : color g b = red ∨ color g b = blue, from
                        exists_or_distrib.elim (h2 b),
                      cases h13,
                        use red, exact h13,
                        use blue, exact h13,
                    },
                    show color g b = blue, from (h12).elim (assume c, assume h14, h14),
                  },
                  show ∃ b : vertex g, color g b = blue, from by {
                    have h12 : ∀ b : vertex g, ∃ c : color g b, color g b = c, from by {
                      assume b : vertex g,
                      show ∃ c : color g b, color g b = c, from by {
                        have h13 : color g b = red ∨ color g b = blue, from
                          exists_or_distrib.elim (h2 b
end --Needs more than 2000 tokens!

--OUTPUT 3
theorem bipartite_iff_two_colorable {G : Type*} [graph G] : bipartite G ↔ two_colorable G :=
begin
  split,
  {
    assume h : bipartite G,
    show two_colorable G, from by {
      -- Let $G$ be a 2-colorable graph, which means we can color every vertex either red or blue, and no edge will have both endpoints colored the same color.
      use {colors := {0,1}, color_map := λ v, if v ∈ h.left then 0 else 1},
      -- this is a valid coloring
      have hval : ∀ v, v ∈ (vertex_set G) → (color_map v) ∈ (colors), from by {
        assume v hv,
        have hl : v ∈ h.left → (color_map v) ∈ (colors), from by {
          assume hvl,
          unfold color_map,
          rw dif_pos hvl,
          apply set.mem_singleton_iff.mp,
          apply set.mem_insert,
          apply set.mem_singleton_iff.mpr,
          apply eq.refl 0,
        },
        have hr : v ∈ h.right → (color_map v) ∈ (colors), from by {
          assume hvr,
          unfold color_map,
          rw dif_neg hvr,
          apply set.mem_singleton_iff.mp,
          apply set.mem_insert,
          apply set.mem_singleton_iff.mpr,
          apply eq.refl 1,
        },
        exact or.elim (set.mem_or_mem_of_mem_union hv) hl hr,
      },
      -- and no edge has both endpoints colored the same color.
      have hne : ∀ e, e ∈ (edge_set G) → (color_map e.x) ≠ (color_map e.y), from by {
        assume e he,
        have hx : e.x ∈ h.left ∨ e.x ∈ h.right, from by apply set.mem_or_mem_of_mem_union,
        have hy : e.y ∈ h.left ∨ e.y ∈ h.right, from by apply set.mem_or_mem_of_mem_union,
        unfold color_map,
        have hnx : e.x ∉ h.left → e.x ∈ h.right, from by {
          assume hnx,
          apply set.mem_of_mem_diff hx,
          exact hnx,
        },
        have hny : e.y ∉ h.left → e.y ∈ h.right, from by {
          assume hny,
          apply set.mem_of_mem_diff hy,
          exact hny,
        },
        exact or.elim (em (e.x ∈ h.left)) (λ hl, or.elim (em (e.y ∈ h.left)) (λ hyl, (mt dif_pos hl) hyl) (λ hyr, (mt dif_pos hl) hyr)) (λ hr, or.elim (em (e.y ∈ h.left)) (λ hyl, (mt dif_neg (hnx hr)) hyl) (λ hyr, (mt dif_neg (hnx hr)) hyr))
      },
      exact ⟨hval,hne⟩,
    },
  },
  {
    assume h : two_colorable G,
    show bipartite G, from by {
      -- Let $A$ denote the subset of vertices colored red, and let $B$ denote the subset of vertices colored blue.
      use {left := {v | ((h.color_map v) = 0) }, right := {v | ((h.color_map v) = 1) }},
      -- Since all vertices of $A$ are red, there are no edges within $A$, and similarly for $B$.
      have hne : ∀ e, e ∈ (edge_set G) → e.x ∈ (h.left) → e.y ∈ (h.right), from by {
        assume e he hx,
        have hx' : (h.color_map e.x) = 0, from by {
          unfold h.left,
          rw set.mem_set_of_eq,
          exact hx,
        },
        have hy' : (h.color_map e.y) = 1, from by {
          unfold h.right,
          have hy : (h.color_map e.y) ≠ 0, from by {
            rw h.ne e he,
            exact hx',
          },
          rw set.mem_set_of_eq,
          exact hy,
        },
        unfold h.right,
        rw set.mem_set_of_eq,
        exact hy',
      },
      -- This implies that every edge has one endpoint in $A$ and the other in $B$, which means $G$ is bipartite.
      exact ⟨hne⟩,
    }
  }
end

--OUTPUT 4
theorem two_colorable_graph_is_bipartite (G : Type*) [simple_graph G] :
  ∀ (V1 V2 : set G) (h1 : disjoint V1 V2) (h2 : ∀ v w : G, v ≠ w → (v ∈ V1 ∧ w ∈ V2) ∨ (v ∈ V2 ∈ w ∈ V1)),
    ∀ v : G, ∃! c : color, color_graph G v c :=
begin
  -- Let $G$ be a 2-colorable graph, which means we can color every vertex either red or blue, and no edge will have both endpoints colored the same color.
  assume (V1 V2 : set G) (h1 : disjoint V1 V2) (h2 : ∀ v w : G, v ≠ w → (v ∈ V1 ∧ w ∈ V2) ∨ (v ∈ V2 ∧ w ∈ V1)),
  -- Let $A$ denote the subset of vertices colored red, and let $B$ denote the subset of vertices colored blue.
  let A := {v : G | ∃ c : color, color_graph G v c ∧ c = red},
  let B := {v : G | ∃ c : color, color_graph G v c ∧ c = blue},

  -- Since all vertices of $A$ are red, there are no edges within $A$, and similarly for $B$.
  have h3 : ∀ v w : G, (v ∈ A ∧ w ∈ A) → ¬ edge_graph G v w, from assume v w : G, assume h3 : (v ∈ A ∧ w ∈ A),
    have h4 : ∃ c1 : color, color_graph G v c1 ∧ c1 = red, from h3.left,
    have h5 : ∃ c2 : color, color_graph G w c2 ∧ c2 = red, from h3.right,
    have h6 : color_graph G v red ∧ color_graph G w red, from and.intro (h4.left) (h5.left),
    show ¬ edge_graph G v w, from by {apply two_color_graph_no_edge,exact h6},
  have h4 : ∀ v w : G, (v ∈ B ∧ w ∈ B) → ¬ edge_graph G v w, from assume v w : G, assume h4 : (v ∈ B ∧ w ∈ B),
    have h5 : ∃ c1 : color, color_graph G v c1 ∧ c1 = blue, from h4.left,
    have h6 : ∃ c2 : color, color_graph G w c2 ∧ c2 = blue, from h4.right,
    have h7 : color_graph G v blue ∧ color_graph G w blue, from and.intro (h5.left) (h6.left),
    show ¬ edge_graph G v w, from by {apply two_color_graph_no_edge,exact h7},

  -- This implies that every edge has one endpoint in $A$ and the other in $B$, which means $G$ is bipartite.
  have h5 : ∀ v w : G, edge_graph G v w → (v ∈ A ∧ w ∈ B) ∨ (v ∈ B ∧ w ∈ A), from
    assume v w : G, assume h5 : edge_graph G v w,
    have h6 : ¬ color_graph G v red ∨ ¬ color_graph G w red, from by {apply two_color_graph_no_edge,exact h5},
    have h7 : ¬ color_graph G v blue ∨ ¬ color_graph G w blue, from by {apply two_color_graph_no_edge,exact h5},
    or.elim h6 (assume h8 : ¬ color_graph G v red, or.elim h7 (assume h9 : ¬ color_graph G w blue,
      begin
        have h10 : ∃ c1 : color, color_graph G v c1 ∧ c1 = blue, from by {use blue, exact and.intro h9 rfl},
        have h11 : ∃ c2 : color, color_graph G w c2 ∧ c2 = red, from by {use red, exact and.intro h8 rfl},
        show (v ∈ B ∧ w ∈ A) ∨ (v ∈ A ∧ w ∈ B), from by {right,exact ⟨h10,h11⟩},
      end
    )
    (assume h9 : color_graph G w blue,
      begin
        have h10 : ∃ c1 : color, color_graph G v c1 ∧ c1 = blue, from by {use blue, exact and.intro h9 rfl},
        have h11 : ∃ c2 : color, color_graph G w c2 ∧ c2 = blue, from by {use blue, exact h9},
        show (v ∈ B ∧ w ∈ A) ∨ (v ∈ A ∧ w ∈ B), from by {left,exact ⟨h10,h11⟩},
      end
    )
    )
    (assume h8 : color_graph G v red, or.elim h7 (assume h9 : ¬ color_graph G w blue,
      begin
        have h10 : ∃ c1 : color, color_graph G v c1 ∧ c1 = red, from by {use red, exact h8},
        have h11 : ∃ c2 : color, color_graph G w c2 ∧ c2 = red, from by {use red, exact and.intro h9 rfl},
        show (v ∈ B ∧ w ∈ A) ∨ (v ∈ A ∧ w ∈ B), from by {right,exact ⟨h10,h11⟩},
      end
    )
    (assume h9 : color_graph G w blue,
      begin
        have h10 : ∃ c1 : color, color_graph G v c1 ∧ c1 = red, from by {use red, exact h8},
        have h11 : ∃ c2 : color, color_graph G w c2 ∧ c2 = blue, from by {use blue, exact h9},
        show (v ∈ B ∧ w ∈ A) ∨ (v ∈ A ∧ w ∈ B), from by {left,exact ⟨h10,h11⟩},
      end
    )
    ),
  have h6 : ∀ v w : G, edge_graph G v w → v ∈ A ∧ w ∈ B, from assume v w : G, assume h6 : edge_graph G v w,
    or.elim (h5 v w h6) (assume h7 : v ∈ B ∧ w ∈ A,
      begin
        have h8 : v ∈ A ∧ w ∈ B, from by {split,
          show v ∈ A, from by {rw A,existsi red,exact ⟨h7.right.left,rfl⟩},
          show w ∈ B, from by {rw B,existsi blue,exact ⟨h7.left.left,rfl⟩},},
        exact h8,
      end
    )
    (assume h7 : v ∈ A ∧ w ∈ B, h7),

  -- Conversely, suppose $G$ is bipartite, that is, we can partition the vertices into two subsets $V_{1}, V_{2}$ every edge has one endpoint in $V_{1}$ and the other in $V_{2}$.
  have h7 : ∀ v w : G, edge_graph G v w → v ∈ V1 ∧ w ∈ V2 ∨ v ∈ V2 ∧ w ∈ V1, from
    assume v w : G, assume h7 : edge_graph G v w, h6 v w h7,
  have h8 : ∀ v w : G, v ∈ V1 ∧ w ∈ V2 → edge_graph G v w, from
    assume v w : G, assume h8 : v ∈ V1 ∧ w ∈ V2, or.elim (h7 v
end --Needs more than 2000 tokens!

--OUTPUT 5
theorem two_coloring_bipartite (G : Type*) [graph G] :
  (∀ A B : G, ∀ x y : G, (x ∈ A ∧ y ∈ B) ∨ (x ∈ B ∧ y ∈ A)) ↔ (∃ A B : G, ∀ x y : G, (x ∈ A ∧ y ∈ B) ∨ (x ∈ B ∧ y ∈ A)) :=
begin
  -- A graph G is 2-colorable iff G is bipartite
  split,
  -- A graph G is 2-colorable implies G is bipartite
  {
    assume h2col : ∀ A B : G, ∀ x y : G, (x ∈ A ∧ y ∈ B) ∨ (x ∈ B ∧ y ∈ A),
    show ∃ A B : G, ∀ x y : G, (x ∈ A ∧ y ∈ B) ∨ (x ∈ B ∧ y ∈ A), from
    begin
      use {A : G | true},
      use {B : G | true},
      assume x y : G,
      have h3 : ∃! C : G, x ∈ C, from by {
        use {x : G | true},
        rw mem_set_of,
        exact ⟨trivial, by {obviously}⟩,
      },
      have h4 : ∃! D : G, y ∈ D, from by {
        use {y : G | true},
        rw mem_set_of,
        exact ⟨trivial, by {obviously}⟩,
      },
      have h5 : ∃! A : G, x ∈ A, from by {
        have h6 : ∃! A : G, (∃ z : G, z ∈ A), from by {
          use {A : G | true},
          obviously,
        },
        have h7 : ∃! A : G, (∃ z : G, z ∈ A) ∧ x ∈ A, from by {
          use {A : G | x ∈ A},
          obviously,
        },
        exact h7,
      },
      have h8 : ∃! B : G, y ∈ B, from by {
        have h9 : ∃! B : G, (∃ z : G, z ∈ B), from by {
          use {B : G | true},
          obviously,
        },
        have h10 : ∃! B : G, (∃ z : G, z ∈ B) ∧ y ∈ B, from by {
          use {B : G | y ∈ B},
          obviously,
        },
        exact h10,
      },
      have h11 : ∃! C : G, x ∈ C ∧ y ∈ C, from by {
        use {C : G | x ∈ C ∧ y ∈ C},
        obviously,
      },
      have h12 : ∃! A : G, x ∈ A ∧ y ∈ A, from by {
        have h13 : ∃! A : G, (∃ z : G, z ∈ A) ∧ (∃ w : G, w ∈ A), from by {
          use {A : G | true},
          obviously,
        },
        have h14 : ∃! A : G, (∃ z : G, z ∈ A) ∧ (∃ w : G, w ∈ A) ∧ x ∈ A ∧ y ∈ A, from by {
          use {A : G | x ∈ A ∧ y ∈ A},
          obviously,
        },
        exact h14,
      },
      have h15 : ∃! B : G, x ∈ B ∧ y ∈ B, from by {
        use {B : G | x ∈ B ∧ y ∈ B},
        obviously,
      },
      have h16 : ∃! A : G, x ∈ A ∧ y ∈ B, from by {
        have h17 : ∃! A : G, (∃ z : G, z ∈ A) ∧ (∃ w : G, w ∈ B), from by {
          use {A : G | true},
          obviously,
        },
        have h18 : ∃! A : G, (∃ z : G, z ∈ A) ∧ (∃ w : G, w ∈ B) ∧ x ∈ A ∧ y ∈ B, from by {
          use {A : G | x ∈ A ∧ y ∈ B},
          obviously,
        },
        exact h18,
      },
      have h19 : ∃! B : G, x ∈ B ∧ y ∈ A, from by {
        have h20 : ∃! B : G, (∃ z : G, z ∈ A) ∧ (∃ w : G, w ∈ B), from by {
          use {B : G | true},
          obviously,
        },
        have h21 : ∃! B : G, (∃ z : G, z ∈ A) ∧ (∃ w : G, w ∈ B) ∧ x ∈ B ∧ y ∈ A, from by {
          use {B : G | x ∈ B ∧ y ∈ A},
          obviously,
        },
        exact h21,
      },
      have h22 : ∃! A : G, x ∈ A ∧ y ∈ B ∧ x ∈ B ∧ y ∈ A, from by {
        have h23 : ∃! A : G, (∃ z : G, z ∈ A) ∧ (∃ w : G, w ∈ B) ∧ (∃ z : G, z ∈ B) ∧ (∃ w : G, w ∈ A), from by {
          use {A : G | true},
          obviously,
        },
        have h24 : ∃! A : G, (∃ z : G, z ∈ A) ∧ (∃ w : G, w ∈ B) ∧ (∃ z : G, z ∈ B) ∧ (∃ w : G, w ∈ A) ∧ x ∈ A ∧ y ∈ B ∧ x ∈ B ∧ y ∈ A, from by {
          use {A : G | x ∈ A ∧ y ∈ B ∧ x ∈ B ∧ y ∈ A},
          obviously,
        },
        exact h24,
      },
      have h25 : ∃! B : G, x ∈ B ∧ y ∈ A ∧ x ∈ A ∧ y ∈ B, from by {
        have h26 : ∃! B : G, (∃ z : G, z ∈ A) ∧ (∃ w : G, w ∈ B) ∧ (∃ z : G, z ∈ A) ∧ (∃ w : G, w ∈ B), from by {
          use {B : G | true},
          obviously,
        },
        have h27 : ∃! B : G, (∃ z : G, z ∈ A) ∧ (∃ w : G, w ∈ B) ∧ (∃ z : G, z ∈ A) ∧ (∃ w : G, w ∈ B) ∧ x ∈ B ∧ y ∈ A ∧ x ∈ A ∧ y ∈ B, from by {
          use {B : G | x ∈ B ∧ y ∈ A ∧ x ∈ A ∧ y ∈ B},
          obviously,
        },
        exact h27,
      },
      have h28 : ∃! A : G, x ∈ A ∧ y ∈ B ∧ x ∈ B ∧ y ∈ A ∧ x ∈ A ∧ y ∈ B ∧ x ∈ B ∧ y ∈ A, from by {
        have h29 : ∃! A : G, (∃ z : G, z ∈ A) ∧ (∃ w : G, w ∈ B) ∧ (∃ z : G, z ∈ B) ∧ (∃ w : G, w ∈ A) ∧ (∃ z : G, z ∈ A) ∧ (∃ w : G, w ∈ B) ∧ (∃ z : G, z ∈ B
end --Needs more than 2000 tokens!

--OUTPUT 6
theorem graph_bipartite_iff_two_colorable (G : Type*) [graph G] :
∀ g : G, bipartite g ↔ two_colorable g :=
begin
  -- Let $G$ be a 2-colorable graph
  assume g : G,
  have h1 : ∀ g : G, two_colorable g → bipartite g, from by {
    assume g : G,
    assume h2c : two_colorable g,
    -- which means we can color every vertex either red or blue, and no edge will have both endpoints colored the same color
    have hcoloring : ∃ coloring : coloring g, coloring.is_valid coloring, from h2c,
    -- Let $A$ denote the subset of vertices colored red, and let $B$ denote the subset of vertices colored blue.
    let coloring : coloring g := classical.some hcoloring,
    let A : set (vertex g) := coloring.red coloring,
    let B : set (vertex g) := coloring.blue coloring,
    -- Since all vertices of $A$ are red, there are no edges within $A$, and similarly for $B$.
    have hA : ∀ v : vertex g, (v ∈ A) → (∀ w : vertex g, (v ≠ w) → (v,w) ∉ edge g), from by {
      assume (v : vertex g) (hv : v ∈ A),
      assume (w : vertex g) (hvw : v ≠ w),
      have hvred : coloring.is_red v coloring, from by {
        have hv : v ∈ vertex g, from by apply set.mem_univ v,
        show coloring.is_red v coloring, from coloring.is_valid coloring v hv,
      },
      have hwred : coloring.is_red w coloring, from by {
        have hw : w ∈ vertex g, from by apply set.mem_univ w,
        show coloring.is_red w coloring, from coloring.is_valid coloring w hw,
      },
      have hredred : (v,w) ∉ edge g, from
        two_colorable.no_adjacent_colored_same h2c v hvred w hwred hvw,
      show (v,w) ∉ edge g, from hredred,
    },
    have hB : ∀ v : vertex g, (v ∈ B) → (∀ w : vertex g, (v ≠ w) → (v,w) ∉ edge g), from by {
      assume (v : vertex g) (hv : v ∈ B),
      assume (w : vertex g) (hvw : v ≠ w),
      have hvblue : coloring.is_blue v coloring, from by {
        have hv : v ∈ vertex g, from by apply set.mem_univ v,
        show coloring.is_blue v coloring, from coloring.is_valid coloring v hv,
      },
      have hwblue : coloring.is_blue w coloring, from by {
        have hw : w ∈ vertex g, from by apply set.mem_univ w,
        show coloring.is_blue w coloring, from coloring.is_valid coloring w hw,
      },
      have hblueblue : (v,w) ∉ edge g, from
        two_colorable.no_adjacent_colored_same h2c v hvblue w hwblue hvw,
      show (v,w) ∉ edge g, from hblueblue,
    },
    -- This implies that every edge has one endpoint in $A$ and the other in $B$, which means $G$ is bipartite.
    have hbipartite : bipartite g, from by {
      show ∀ v w : vertex g, (v ≠ w) → (v,w) ∈ edge g → (v ∈ A ∧ w ∈ B) ∨ (v ∈ B ∧ w ∈ A), from by {
        assume (v : vertex g) (w : vertex g) (hvw : v ≠ w),
        assume (hvwedge : (v,w) ∈ edge g),
        have hvred : coloring.is_red v coloring, from by {
          have hv : v ∈ vertex g, from by apply set.mem_univ v,
          show coloring.is_red v coloring, from coloring.is_valid coloring v hv,
        },
        have hwred : coloring.is_red w coloring, from by {
          have hw : w ∈ vertex g, from by apply set.mem_univ w,
          show coloring.is_red w coloring, from coloring.is_valid coloring w hw,
        },
        have hvblue : coloring.is_blue v coloring, from by {
          have hv : v ∈ vertex g, from by apply set.mem_univ v,
          show coloring.is_blue v coloring, from coloring.is_valid coloring v hv,
        },
        have hwblue : coloring.is_blue w coloring, from by {
          have hw : w ∈ vertex g, from by apply set.mem_univ w,
          show coloring.is_blue w coloring, from coloring.is_valid coloring w hw,
        },
        or.elim (em (hvred ∧ hwblue)) (assume hvredblue : hvred ∧ hwblue,
          have hvA : v ∈ A, from by {
            apply set.mem_of_mem_of_mem_coloring hvredblue.left,
            show v ∈ vertex g, from by apply set.mem_univ v,
          },
          have hwB : w ∈ B, from by {
            apply set.mem_of_mem_of_mem_coloring hvredblue.right,
            show w ∈ vertex g, from by apply set.mem_univ w,
          },
          show (v ∈ A ∧ w ∈ B) ∨ (v ∈ B ∧ w ∈ A), from or.inl ⟨hvA,hwB⟩)
          (assume hvbluered : ¬ (hvred ∧ hwblue),
            have hredblue : hvblue ∧ hwred, from by {
              show hvblue ∧ hwred, from ⟨hvbluered.left, hvbluered.right.left,⟩
            },
            have hvB : v ∈ B, from by {
              apply set.mem_of_mem_of_mem_coloring hredblue.left,
              show v ∈ vertex g, from by apply set.mem_univ v,
            },
            have hwA : w ∈ A, from by {
              apply set.mem_of_mem_of_mem_coloring hredblue.right,
              show w ∈ vertex g, from by apply set.mem_univ w,
            },
            show (v ∈ A ∧ w ∈ B) ∨ (v ∈ B ∧ w ∈ A), from or.inr ⟨hvB, hwA⟩)
      },
      show ∀ v w : vertex g, (v ≠ w) → (v,w) ∉ edge g → (v ∈ A ∧ w ∈ A) ∨ (v ∈ B ∧ w ∈ B), from by {
        assume (v : vertex g) (w : vertex g) (hvw : v ≠ w),
        assume (hvwnotedge : (v,w) ∉ edge g),
        have hvA : v ∈ A, from set.mem_of_mem_of_mem_coloring hvwnotedge (h1 g).left hvw,
        have hwA : w ∈ A, from set.mem_of_mem_of_mem_coloring hvwnotedge (h1 g).left hvw,
        show (v ∈ A ∧ w ∈ A) ∨ (v ∈ B ∧ w ∈ B), from or.inl ⟨hvA,hwA⟩,
      },
    },
    show bipartite g, from hbipartite,
  },
  have h2 : ∀ g : G, bipartite g → two_colorable g, from by {
end --Needs more than 2000 tokens!

--OUTPUT 7
theorem graph_is_bipartite_iff_two_colorable (G : Type*) [graph G] : ∃! color : G → (color_type G), ∀ e : G, e.edges.forall (λ e, e.color ≠ e.other.color) :=
begin
  -- Assume $G$ is a 2-colorable graph
  assume color : G → (color_type G),
  assume e : G, assume h : e.edges.forall (λ e, e.color ≠ e.other.color),
  -- Let $A$ denote the subset of vertices colored red, and let $B$ denote the subset of vertices colored blue.
  let A : set G := {x : G | x.color = color_type.red},
  let B : set G := {x : G | x.color = color_type.blue},
  -- Since all vertices of $A$ are red, there are no edges within $A$, and similarly for $B$.
  have h1 : ∀ a ∈ A, a.edges.forall (λ e, e.other ∉ A), from
    assume a : G, assume hA : a ∈ A,
    begin
      assume e : G, assume h2 : e ∈ a.edges,
      have h3 : e.other.color = color_type.blue, from
        by {rw ← h, apply h2,},
      have h4 : a.color = color_type.red, from by {rw ← hA, rw A,},
      show e.other ∉ A, from by {rw h3, rw h4, rw A, intro h5, cases h5,},
    end,
  have h2 : ∀ b ∈ B, b.edges.forall (λ e, e.other ∉ B), from
    assume b : G, assume hB : b ∈ B,
    begin
      assume e : G, assume h2 : e ∈ b.edges,
      have h3 : e.other.color = color_type.red, from
        by {rw ← h, apply h2,},
      have h4 : b.color = color_type.blue, from by {rw ← hB, rw B,},
      show e.other ∉ B, from by {rw h3, rw h4, rw B, intro h5, cases h5,},
    end,
  -- This implies that every edge has one endpoint in $A$ and the other in $B$, which means $G$ is bipartite.
  have h3 : ∀ e : G, e ∈ G.edges → e.other ∈ A → e.other ∈ B, from
    assume e : G, assume h4 : e ∈ G.edges, assume h5 : e.other ∈ A,
    begin
      have h6 : e ∈ e.other.edges, from by {rw G.edges, exact or.inr h4,},
      have h7 : e.other ∉ A, from by {apply h1 e.other h5, exact h6,},
      show e.other ∈ B, from by {rw B, rw h7,},
    end,
  --
  have h4 : ∀ e : G, e ∈ G.edges → e.other ∈ B → e.other ∈ A, from
    assume e : G, assume h4 : e ∈ G.edges, assume h5 : e.other ∈ B,
    begin
      have h6 : e ∈ e.other.edges, from by {rw G.edges, exact or.inr h4,},
      have h7 : e.other ∉ B, from by {apply h2 e.other h5, exact h6,},
      show e.other ∈ A, from by {rw A, rw h7,},
    end,
  show G.is_bipartite, from ⟨A, B, h3, h4⟩,

  -- Conversely, suppose $G$ is bipartite
  assume h1 : G.is_bipartite,
  let A : set G := h1.left,
  let B : set G := h1.right,
  let color : G → color_type G :=
    begin
      assume e : G,
      have h2 : ∃! x ∈ A, e ∈ x.edges, from by {
        use e,
        have h3 : e ∈ G.edges, from by {rw G.edges, exact or.inl e.edges.left,},
        have h4 : e.other ∈ A, from by {apply h1.left_right e h3,},
        have h5 : e ∈ e.other.edges, from by {rw G.edges, exact or.inr e.edges.left,},
        show e ∈ A, from by {apply exists_unique.unique (h2 e) h4 h5,},
      end,
      have h3 : ∃! x ∈ B, e ∈ x.edges, from by {
        use e,
        have h4 : e ∈ G.edges, from by {rw G.edges, exact or.inl e.edges.left,},
        have h5 : e.other ∈ B, from by {apply h1.right_left e h4,},
        have h6 : e ∈ e.other.edges, from by {rw G.edges, exact or.inr e.edges.left,},
        show e ∈ B, from by {apply exists_unique.unique (h3 e) h5 h6,},
      end,
      show color_type G, from classical.some (h2).property,
    end,
  have h2 : ∀ e : G, e.edges.forall (λ e, e.color ≠ e.other.color), from
    assume e : G,
    begin
      assume e : G, assume h2 : e ∈ e.edges,
      have h3 : e.color = color e, from by {rw e,},
      have h4 : e.other.color = color e.other, from by {rw e.other,},
      show e.color ≠ e.other.color, from
        by {rw ← h3, rw ← h4, intro h5, cases h5,},
    end,
  show ∃! color : G → color_type G, from ⟨color, h2⟩,
end

--OUTPUT 8
theorem two_colorable_iff_bipartite (G : Type*) [graph G] :
  (∃ (c : G → Prop), ∀ (a b : G), (a ≠ b) → (edge a b) → (c a ≠ c b)) ↔
  (∃ (V₁ V₂ : set G), ∀ (a : G), a ∈ V₁ ∨ a ∈ V₂ ∧ ∀ (a b : G), (edge a b) → ((a ∈ V₁ ∧ b ∈ V₂) ∨ (a ∈ V₂ ∧ b ∈ V₁))) :=
begin
  --Let $G$ be a 2-colorable graph, which means we can color every vertex either red or blue, and no edge will have both endpoints colored the same color.
  assume h1 : ∃ (c : G → Prop), ∀ (a b : G), (a ≠ b) → (edge a b) → (c a ≠ c b),
  --Let $A$ denote the subset of vertices colored red, and let $B$ denote the subset of vertices colored blue.
  cases h1 with c h2,
  have h3 : ∃ (V₁ V₂ : set G), ∀ (a : G), a ∈ V₁ ∨ a ∈ V₂ ∧ ∀ (a b : G), (edge a b) → ((a ∈ V₁ ∧ b ∈ V₂) ∨ (a ∈ V₂ ∧ b ∈ V₁)), from by {
    --Since all vertices of $A$ are red, there are no edges within $A$, and similarly for $B$.
    let V₁ : set G := {a : G | c a},
    let V₂ : set G := {a : G | ¬(c a)},
    have h4 : ∀ (a b : G), (a ≠ b) → (edge a b) → (a ∈ V₁ ∧ b ∈ V₂) ∨ (a ∈ V₂ ∧ b ∈ V₁), from by {
      assume a b : G, assume h5 : (a ≠ b), assume h6 : (edge a b),
      --This implies that every edge has one endpoint in $A$ and the other in $B$, which means $G$ is bipartite.
      have h7 : c a ≠ c b, from h2 a b h5 h6,
      have h8 : a ∈ V₁ ∨ a ∈ V₂, from by {
        cases (c a),
        {exact or.inl ⟨_, h⟩,},
        {exact or.inr ⟨_, h⟩,},
      },
      have h9 : b ∈ V₁ ∨ b ∈ V₂, from by {
        cases (c b),
        {exact or.inl ⟨_, h⟩,},
        {exact or.inr ⟨_, h⟩,},
      },
      cases h8 with h10 h11,
      {exact or.inl ⟨h10, h9⟩,},
      {exact or.inr ⟨h11, h8⟩,},
    },
    exact ⟨V₁, V₂, h4⟩,
  },
  show ∃ (V₁ V₂ : set G), ∀ (a : G), a ∈ V₁ ∨ a ∈ V₂ ∧ ∀ (a b : G), (edge a b) → ((a ∈ V₁ ∧ b ∈ V₂) ∨ (a ∈ V₂ ∧ b ∈ V₁)), from h3,

  --Conversely, suppose $G$ is bipartite, that is, we can partition the vertices into two subsets $V_{1}, V_{2}$ every edge has one endpoint in $V_{1}$ and the other in $V_{2}$.
  assume h1 : ∃ (V₁ V₂ : set G), ∀ (a : G), a ∈ V₁ ∨ a ∈ V₂ ∧ ∀ (a b : G), (edge a b) → ((a ∈ V₁ ∧ b ∈ V₂) ∨ (a ∈ V₂ ∧ b ∈ V₁)),
  --Then coloring every vertex of $V_{1}$ red and every vertex of $V_{2}$ blue yields a valid coloring, so $G$ is 2-colorable.
  cases h1 with V₁ h2, cases h2 with V₂ h3,
  have h4 : ∃ (c : G → Prop), ∀ (a b : G), (a ≠ b) → (edge a b) → (c a ≠ c b), from by {
    let c : G → Prop := λ (a : G), (a ∈ V₁),
    have h5 : ∀ (a b : G), (a ≠ b) → (edge a b) → (c a ≠ c b), from by {
      assume a b : G, assume h5 : (a ≠ b), assume h6 : (edge a b),
      have h7 : a ∈ V₁ ∨ a ∈ V₂, from h3 a,
      have h8 : b ∈ V₁ ∨ b ∈ V₂, from h3 b,
      cases h7 with h9 h10,
      {
        cases h8 with h11 h12,
        {
          have h13 : (a ∈ V₁ ∧ b ∈ V₂) ∨ (a ∈ V₂ ∧ b ∈ V₁), from h3 a b h6,
          cases h13 with h14 h15,
          {
            have h16 : c a ∧ c b, from ⟨h9, h11⟩,
            cases h16 with h17 h18,
            {
              have h19 : c a = c b, from by {rw ← h17,rw ← h18},
              show c a ≠ c b, from h19 ▸ h5,
            },
          },
          {
            have h16 : c a ∧ ¬c b, from ⟨h9, h12⟩,
            cases h16 with h17 h18,
            {
              have h19 : c a = ¬c b, from by {rw ← h17,rw ← h18},
              show c a ≠ c b, from h19 ▸ h5,
            },
          },
        },
        {
          have h13 : (a ∈ V₁ ∧ b ∈ V₂) ∨ (a ∈ V₂ ∧ b ∈ V₁), from h3 a b h6,
          cases h13 with h14 h15,
          {
            have h16 : c a ∧ c b, from ⟨h9, h11⟩,
            cases h16 with h17 h18,
            {
              have h19 : c a = c b, from by {rw ← h17,rw ← h18},
              show c a ≠ c b, from h19 ▸ h5,
            },
          },
          {
            have h16 : c a ∧ ¬c b, from ⟨h9, h12⟩,
            cases h16 with h17 h18,
            {
              have h19 : c a = ¬c b, from by {rw ← h17,rw ← h18},
              show c a ≠ c b, from h19 ▸ h5,
            },
          },
        },
      },
      {
        cases h8 with h11 h12,
        {
          have h13 : (a ∈ V₁ ∧ b ∈ V₂) ∨ (a ∈ V₂ ∧ b ∈ V₁), from h3 a b h6,
          cases h13 with h14 h15,
          {
            have h16 : c
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`
Bipartite Graph is two colorable
Let $G$ be a graph. Then $G$ is 2-colorable if and only if $G$ is bipartite.
`proof`
Let $G$ be a 2-colorable graph, which means we can color every vertex either red or blue, and no edge will have both endpoints colored the same color. Let $A$ denote the subset of vertices colored red, and let $B$ denote the subset of vertices colored blue. Since all vertices of $A$ are red, there are no edges within $A$, and similarly for $B$. This implies that every edge has one endpoint in $A$ and the other in $B$, which means $G$ is bipartite.

Conversely, suppose $G$ is bipartite, that is, we can partition the vertices into two subsets $V_{1}, V_{2}$ every edge has one endpoint in $V_{1}$ and the other in $V_{2}$. Then coloring every vertex of $V_{1}$ red and every vertex of $V_{2}$ blue yields a valid coloring, so $G$ is 2-colorable.

QED

-/
theorem
FEW SHOT PROMPTS TO CODEX(END)-/


/-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies

! This file was ported from Lean 3 source module category_theory.category.Twop
! leanprover-community/mathlib commit c8ab806ef73c20cab1d87b5157e43a82c205f28e
! Please do not edit these lines, except to modify the commit id
! if you have ported upstream changes.
-/
import Mathbin.CategoryTheory.Category.Bipointed
import Mathbin.Data.TwoPointing

/-!
# The category of two-pointed types

This defines `Twop`, the category of two-pointed types.

## References

* [nLab, *coalgebra of the real interval*]
  (https://ncatlab.org/nlab/show/coalgebra+of+the+real+interval)
-/

open CategoryTheory Option

universe u

variable {α β : Type _}

#print TwoP /-
/-- The category of two-pointed types. -/
structure TwoP : Type (u + 1) where
  pt : Type u
  toTwoPointing : TwoPointing X
#align Twop TwoP
-/

namespace TwoP

instance : CoeSort TwoP (Type _) :=
  ⟨X⟩

attribute [protected] TwoP.X

#print TwoP.of /-
/-- Turns a two-pointing into a two-pointed type. -/
def of {X : Type _} (to_two_pointing : TwoPointing X) : TwoP :=
  ⟨X, to_two_pointing⟩
#align Twop.of TwoP.of
-/

#print TwoP.coe_of /-
@[simp]
theorem coe_of {X : Type _} (to_two_pointing : TwoPointing X) : ↥(of to_two_pointing) = X :=
  rfl
#align Twop.coe_of TwoP.coe_of
-/

alias of ← _root_.two_pointing.Twop
#align two_pointing.Twop TwoPointing.TwoP

instance : Inhabited TwoP :=
  ⟨of TwoPointing.bool⟩

#print TwoP.toBipointed /-
/-- Turns a two-pointed type into a bipointed type, by forgetting that the pointed elements are
distinct. -/
def toBipointed (X : TwoP) : Bipointed :=
  X.toTwoPointing.toProd.Bipointed
#align Twop.to_Bipointed TwoP.toBipointed
-/

#print TwoP.coe_toBipointed /-
@[simp]
theorem coe_toBipointed (X : TwoP) : ↥X.toBipointed = ↥X :=
  rfl
#align Twop.coe_to_Bipointed TwoP.coe_toBipointed
-/

#print TwoP.largeCategory /-
instance largeCategory : LargeCategory TwoP :=
  InducedCategory.category toBipointed
#align Twop.large_category TwoP.largeCategory
-/

#print TwoP.concreteCategory /-
instance concreteCategory : ConcreteCategory TwoP :=
  InducedCategory.concreteCategory toBipointed
#align Twop.concrete_category TwoP.concreteCategory
-/

#print TwoP.hasForgetToBipointed /-
instance hasForgetToBipointed : HasForget₂ TwoP Bipointed :=
  InducedCategory.hasForget₂ toBipointed
#align Twop.has_forget_to_Bipointed TwoP.hasForgetToBipointed
-/

#print TwoP.swap /-
/-- Swaps the pointed elements of a two-pointed type. `two_pointing.swap` as a functor. -/
@[simps]
def swap : TwoP ⥤ TwoP where
  obj X := ⟨X, X.toTwoPointing.symm⟩
  map X Y f := ⟨f.toFun, f.map_snd, f.map_fst⟩
#align Twop.swap TwoP.swap
-/

/- warning: Twop.swap_equiv -> TwoP.swapEquiv is a dubious translation:
lean 3 declaration is
  CategoryTheory.Equivalence.{u1, u1, succ u1, succ u1} TwoP.{u1} TwoP.largeCategory.{u1} TwoP.{u1} TwoP.largeCategory.{u1}
but is expected to have type
  CategoryTheory.Equivalence.{u1, u1, succ u1, succ u1} TwoP.{u1} TwoP.{u1} TwoP.largeCategory.{u1} TwoP.largeCategory.{u1}
Case conversion may be inaccurate. Consider using '#align Twop.swap_equiv TwoP.swapEquivₓ'. -/
/-- The equivalence between `Twop` and itself induced by `prod.swap` both ways. -/
@[simps]
def swapEquiv : TwoP ≌ TwoP :=
  Equivalence.mk swap swap
    (NatIso.ofComponents
      (fun X =>
        { Hom := ⟨id, rfl, rfl⟩
          inv := ⟨id, rfl, rfl⟩ })
      fun X Y f => rfl)
    (NatIso.ofComponents
      (fun X =>
        { Hom := ⟨id, rfl, rfl⟩
          inv := ⟨id, rfl, rfl⟩ })
      fun X Y f => rfl)
#align Twop.swap_equiv TwoP.swapEquiv

/- warning: Twop.swap_equiv_symm -> TwoP.swapEquiv_symm is a dubious translation:
lean 3 declaration is
  Eq.{succ (succ u1)} (CategoryTheory.Equivalence.{u1, u1, succ u1, succ u1} TwoP.{u1} TwoP.largeCategory.{u1} TwoP.{u1} TwoP.largeCategory.{u1}) (CategoryTheory.Equivalence.symm.{u1, u1, succ u1, succ u1} TwoP.{u1} TwoP.largeCategory.{u1} TwoP.{u1} TwoP.largeCategory.{u1} TwoP.swapEquiv.{u1}) TwoP.swapEquiv.{u1}
but is expected to have type
  Eq.{succ (succ u1)} (CategoryTheory.Equivalence.{u1, u1, succ u1, succ u1} TwoP.{u1} TwoP.{u1} TwoP.largeCategory.{u1} TwoP.largeCategory.{u1}) (CategoryTheory.Equivalence.symm.{u1, u1, succ u1, succ u1} TwoP.{u1} TwoP.largeCategory.{u1} TwoP.{u1} TwoP.largeCategory.{u1} TwoP.swapEquiv.{u1}) TwoP.swapEquiv.{u1}
Case conversion may be inaccurate. Consider using '#align Twop.swap_equiv_symm TwoP.swapEquiv_symmₓ'. -/
@[simp]
theorem swapEquiv_symm : swapEquiv.symm = swapEquiv :=
  rfl
#align Twop.swap_equiv_symm TwoP.swapEquiv_symm

end TwoP

#print TwoP_swap_comp_forget_to_Bipointed /-
@[simp]
theorem TwoP_swap_comp_forget_to_Bipointed :
    TwoP.swap ⋙ forget₂ TwoP Bipointed = forget₂ TwoP Bipointed ⋙ Bipointed.swap :=
  rfl
#align Twop_swap_comp_forget_to_Bipointed TwoP_swap_comp_forget_to_Bipointed
-/

#print pointedToTwoPFst /-
/-- The functor from `Pointed` to `Twop` which adds a second point. -/
@[simps]
def pointedToTwoPFst : Pointed.{u} ⥤ TwoP
    where
  obj X := ⟨Option X, ⟨X.point, none⟩, some_ne_none _⟩
  map X Y f := ⟨Option.map f.toFun, congr_arg _ f.map_point, rfl⟩
  map_id' X := Bipointed.Hom.ext _ _ Option.map_id
  map_comp' X Y Z f g := Bipointed.Hom.ext _ _ (Option.map_comp_map _ _).symm
#align Pointed_to_Twop_fst pointedToTwoPFst
-/

#print pointedToTwoPSnd /-
/-- The functor from `Pointed` to `Twop` which adds a first point. -/
@[simps]
def pointedToTwoPSnd : Pointed.{u} ⥤ TwoP
    where
  obj X := ⟨Option X, ⟨none, X.point⟩, (some_ne_none _).symm⟩
  map X Y f := ⟨Option.map f.toFun, rfl, congr_arg _ f.map_point⟩
  map_id' X := Bipointed.Hom.ext _ _ Option.map_id
  map_comp' X Y Z f g := Bipointed.Hom.ext _ _ (Option.map_comp_map _ _).symm
#align Pointed_to_Twop_snd pointedToTwoPSnd
-/

#print pointedToTwoPFst_comp_swap /-
@[simp]
theorem pointedToTwoPFst_comp_swap : pointedToTwoPFst ⋙ TwoP.swap = pointedToTwoPSnd :=
  rfl
#align Pointed_to_Twop_fst_comp_swap pointedToTwoPFst_comp_swap
-/

#print pointedToTwoPSnd_comp_swap /-
@[simp]
theorem pointedToTwoPSnd_comp_swap : pointedToTwoPSnd ⋙ TwoP.swap = pointedToTwoPFst :=
  rfl
#align Pointed_to_Twop_snd_comp_swap pointedToTwoPSnd_comp_swap
-/

#print pointedToTwoPFst_comp_forget_to_bipointed /-
@[simp]
theorem pointedToTwoPFst_comp_forget_to_bipointed :
    pointedToTwoPFst ⋙ forget₂ TwoP Bipointed = pointedToBipointedFst :=
  rfl
#align Pointed_to_Twop_fst_comp_forget_to_Bipointed pointedToTwoPFst_comp_forget_to_bipointed
-/

#print pointedToTwoPSnd_comp_forget_to_bipointed /-
@[simp]
theorem pointedToTwoPSnd_comp_forget_to_bipointed :
    pointedToTwoPSnd ⋙ forget₂ TwoP Bipointed = pointedToBipointedSnd :=
  rfl
#align Pointed_to_Twop_snd_comp_forget_to_Bipointed pointedToTwoPSnd_comp_forget_to_bipointed
-/

#print pointedToTwoPFstForgetCompBipointedToPointedFstAdjunction /-
/-- Adding a second point is left adjoint to forgetting the second point. -/
def pointedToTwoPFstForgetCompBipointedToPointedFstAdjunction :
    pointedToTwoPFst ⊣ forget₂ TwoP Bipointed ⋙ bipointedToPointedFst :=
  Adjunction.mkOfHomEquiv
    { homEquiv := fun X Y =>
        { toFun := fun f => ⟨f.toFun ∘ Option.some, f.map_fst⟩
          invFun := fun f => ⟨fun o => o.elim Y.toTwoPointing.toProd.2 f.toFun, f.map_point, rfl⟩
          left_inv := fun f => by
            ext
            cases x
            exact f.map_snd.symm
            rfl
          right_inv := fun f => Pointed.Hom.ext _ _ rfl }
      homEquiv_naturality_left_symm := fun X' X Y f g =>
        by
        ext
        cases x <;> rfl }
#align Pointed_to_Twop_fst_forget_comp_Bipointed_to_Pointed_fst_adjunction pointedToTwoPFstForgetCompBipointedToPointedFstAdjunction
-/

#print pointedToTwoPSndForgetCompBipointedToPointedSndAdjunction /-
/-- Adding a first point is left adjoint to forgetting the first point. -/
def pointedToTwoPSndForgetCompBipointedToPointedSndAdjunction :
    pointedToTwoPSnd ⊣ forget₂ TwoP Bipointed ⋙ bipointedToPointedSnd :=
  Adjunction.mkOfHomEquiv
    { homEquiv := fun X Y =>
        { toFun := fun f => ⟨f.toFun ∘ Option.some, f.map_snd⟩
          invFun := fun f => ⟨fun o => o.elim Y.toTwoPointing.toProd.1 f.toFun, rfl, f.map_point⟩
          left_inv := fun f => by
            ext
            cases x
            exact f.map_fst.symm
            rfl
          right_inv := fun f => Pointed.Hom.ext _ _ rfl }
      homEquiv_naturality_left_symm := fun X' X Y f g =>
        by
        ext
        cases x <;> rfl }
#align Pointed_to_Twop_snd_forget_comp_Bipointed_to_Pointed_snd_adjunction pointedToTwoPSndForgetCompBipointedToPointedSndAdjunction
-/


/-
Copyright (c) 2021 Yourong Zang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yourong Zang
-/
import topology.separation
import topology.opens

/-!
# The Alexandroff Compactification
We construct the Alexandroff compactification of an arbitrary topological space `X` and prove
some properties inherited from `X`.

## Main defintion
* `alexandroff`: the Alexandroff compactification
* `of`: the inclusion map defined by `option.some`. This map requires the argument
        `topological_space X`
* `infty`: the extra point

## Main results
* The topological structure of `alexandroff X`
* The connectedness of `alexandroff X` for noncompact, preconnected `X`
* `alexandroff X` is `T₁` for a T₁ space `X`
* `alexandroff X` is Hausdorff if `X` is locally compact and Hausdorff
-/

noncomputable theory
open set
open_locale classical topological_space filter

section option_topology

/-- The one-point extension of a topological space -/
@[reducible]
def one_point_extension (X : Type*) [topological_space X] :
  topological_space (option X) :=
{ is_open := λ s, if none ∈ s then is_compact (some⁻¹' s)ᶜ ∧ is_open (some⁻¹' s)
    else is_open (some⁻¹' s),
  is_open_univ := by simp,
  is_open_inter :=
  λ s t hs ht, begin
    split_ifs at hs ht with h h' h' h' h,
    { simpa [h, h', compl_inter] using and.intro (hs.1.union ht.1) (hs.2.inter ht.2) },
    { simpa [h, h'] using hs.inter ht.2 },
    { simpa [h, h'] using hs.2.inter ht },
    { simpa [h, h'] using hs.inter ht }
  end,
  is_open_sUnion :=
  λ S ht, begin
    suffices : is_open (some⁻¹' ⋃₀S),
    { split_ifs with h,
      { obtain ⟨(a : set (option X)), ha, ha'⟩ := mem_sUnion.mp h,
        specialize ht a ha,
        rw if_pos ha' at ht,
        refine ⟨compact_of_is_closed_subset ht.left this.is_closed_compl _, this⟩,
        rw [compl_subset_compl, preimage_subset_iff],
        intros y hy,
        refine ⟨a, ha, hy⟩ },
      { exact this } },
    rw is_open_iff_forall_mem_open,
    simp only [and_imp, exists_prop, mem_Union, preimage_sUnion, mem_preimage, exists_imp_distrib],
    intros y s hs hy,
    refine ⟨some⁻¹' s, subset_subset_Union _ (subset_subset_Union hs (subset.refl _)), _,
      mem_preimage.mpr hy⟩,
    specialize ht s hs,
    split_ifs at ht,
    { exact ht.right },
    { exact ht }
  end }

local attribute [instance] one_point_extension

namespace one_point_extension

variables {X : Type*} {s : set (option X)}

lemma some_preimage_none : (some⁻¹' {none} : set X) = ∅ :=
by { ext, simp }

lemma some_mem_range_some (x : X) : some x ∈ (some '' (univ : set X)) :=
by simp

lemma none_not_mem_range_some : none ∉ some '' (univ : set X) .

@[simp] lemma none_not_mem_image_some {s : set X} : none ∉ some '' s :=
not_mem_subset (image_subset _ $ subset_univ _) none_not_mem_range_some

lemma union_none_eq_univ : (some '' univ ∪ {none}) = (univ : set (option X)) :=
begin
  refine le_antisymm (subset_univ _) _,
  rintros ⟨_|x⟩;
  simp
end

lemma inter_none_eq_empty : (some '' univ) ∩ {none} = (∅ : set (option X)) :=
by { ext ⟨_|x⟩; simp }

variables [topological_space X]

lemma is_open_alexandroff_iff_aux :
  is_open s ↔ if none ∈ s then is_compact (some⁻¹' s)ᶜ ∧ is_open (some⁻¹' s)
  else is_open (some⁻¹' s) :=
iff.rfl

lemma is_open_iff_of_mem (h : none ∈ s) :
  is_open s ↔ is_compact (some⁻¹' s)ᶜ ∧ is_closed (some⁻¹' s)ᶜ :=
by simp [is_open_alexandroff_iff_aux, h, is_closed_compl_iff]

lemma is_open_iff_of_not_mem (h : none ∉ s) :
  is_open s ↔ is_open (some⁻¹' s) :=
by simp [is_open_alexandroff_iff_aux, h]

lemma is_open_of_is_open (h : is_open s) :
  is_open (some⁻¹' s) :=
begin
  by_cases H : none ∈ s,
  { simpa using ((is_open_iff_of_mem H).mp h).2 },
  { exact (is_open_iff_of_not_mem H).mp h }
end

lemma is_open_map_some : is_open_map (@some X) :=
λ s hs, begin
  rw [← preimage_image_eq s (option.some_injective X)] at hs,
  rwa is_open_iff_of_not_mem none_not_mem_image_some
end

lemma continuous_some : continuous (@some X) :=
continuous_def.mpr (λ s hs, is_open_of_is_open hs)

/-- An open set of the extension constructed from a closed compact set in `X`-/
def opens_of_compl {s : set X} (h : is_compact s ∧ is_closed s) :
  topological_space.opens (option X) :=
⟨(some '' s)ᶜ, by { rw [is_open_iff_of_mem ((mem_compl_iff _ _).mpr none_not_mem_image_some),
  preimage_compl, compl_compl, (option.some_injective X).preimage_image _], assumption' }⟩

lemma none_mem_opens_of_compl {s : set X} (h : is_compact s ∧ is_closed s) :
  none ∈ (opens_of_compl h) :=
by { simp only [opens_of_compl, topological_space.opens.coe_mk],
     exact mem_compl none_not_mem_image_some }

/-- The one-point extension is compact -/
@[reducible, nolint def_lemma]
def compact_space (X : Type*) [topological_space X] :
  compact_space (option X) :=
{ compact_univ :=
  begin
    refine is_compact_of_finite_subcover (λ ι Z h H, _),
    simp only [univ_subset_iff] at H ⊢,
    rcases Union_eq_univ_iff.mp H none with ⟨K, hK⟩,
    have minor₁ : is_compact (some⁻¹' Z K)ᶜ,
    { specialize h K, rw is_open_iff_of_mem hK at h, exact h.1 },
    let p : ι → set X := λ i, some⁻¹' Z i,
    have minor₂ : ∀ i, is_open (p i) := λ i, is_open_of_is_open (h i),
    have minor₃ : (some⁻¹' Z K)ᶜ ⊆ ⋃ i, p i :=
      by simp only [p, ← preimage_Union, H, preimage_univ, subset_univ],
    rcases is_compact_iff_finite_subcover.mp minor₁ p minor₂ minor₃ with ⟨ι', H'⟩,
    refine ⟨insert K ι', _⟩,
    rw ← preimage_compl at H',
    simp only [Union_eq_univ_iff],
    intros x,
    by_cases hx : x ∈ Z K,
    { exact ⟨K, mem_Union.mpr ⟨finset.mem_insert_self _ _, hx⟩⟩ },
    { have triv₁ : x ≠ none := (ne_of_mem_of_not_mem hK hx).symm,
      rcases option.ne_none_iff_exists.mp triv₁ with ⟨y, hy⟩,
      have triv₂ : some y ∈ {x} := mem_singleton_of_eq hy,
      rw [← mem_compl_iff, ← singleton_subset_iff] at hx,
      have : some⁻¹' {x} ⊆ some⁻¹' (Z K)ᶜ := λ y hy, hx hy,
      have key : y ∈ ⋃ (i : ι) (H : i ∈ ι'), p i := this.trans H' (mem_preimage.mpr triv₂),
      rcases mem_bUnion_iff'.mp key with ⟨i, hi, hyi⟩,
      refine ⟨i, mem_Union.mpr ⟨finset.subset_insert _ ι' hi, _⟩⟩,
      simpa [hy] using hyi }
  end }

/-- The one-point extension of a T₁ space `X` is T₁-/
@[reducible, nolint def_lemma]
def t1_space [t1_space X] : t1_space (option X) :=
{ t1 :=
  λ z, begin
    cases z,
    { rw [← is_open_compl_iff, compl_eq_univ_diff, ← union_none_eq_univ,
          union_diff_cancel_right (subset.antisymm_iff.mp inter_none_eq_empty).1],
      exact is_open_map_some _ is_open_univ },
    { have : none ∈ ({some z}ᶜ : set (option X)) :=
        mem_compl (λ w, (option.some_ne_none z).symm (mem_singleton_iff.mp w)),
      rw [← is_open_compl_iff, is_open_iff_of_mem this],
      rw [preimage_compl, compl_compl, ← image_singleton,
          (option.some_injective X).preimage_image _],
      exact ⟨is_compact_singleton, is_closed_singleton⟩ }
  end }

/-- The one-point extension of a Hausdorff `X` is Hausdorff -/
@[reducible, nolint def_lemma]
def t2_space [locally_compact_space X] [t2_space X] : t2_space (option X) :=
{ t2 :=
  λ x y hxy, begin
    have key : ∀ (z : option X), z ≠ none →
      ∃ (u v : set (option X)), is_open u ∧ is_open v ∧ none ∈ u ∧ z ∈ v ∧ u ∩ v = ∅ :=
    λ z h, begin
      rcases option.ne_none_iff_exists.mp h with ⟨y', hy'⟩,
      rcases exists_open_with_compact_closure y' with ⟨u, hu, huy', Hu⟩,
      have minor₁ : _ ∧ is_closed (closure u) := ⟨Hu, is_closed_closure⟩,
      refine ⟨opens_of_compl minor₁, some '' u, _⟩,
      refine ⟨(opens_of_compl minor₁).2, is_open_map_some _ hu,
        none_mem_opens_of_compl minor₁, ⟨y', huy', hy'⟩, _⟩,
      simp only [opens_of_compl, topological_space.opens.coe_mk],
      have minor₂ : (some '' closure u)ᶜ ∩ some '' u ⊆ (some '' u)ᶜ ∩ some '' u,
      { apply inter_subset_inter_left,
        simp only [compl_subset_compl, image_subset _ (subset_closure)] },
      rw compl_inter_self at minor₂,
      exact eq_empty_of_subset_empty minor₂
    end,
    cases x; cases y,
    { simpa using hxy },
    { simpa using key y hxy.symm },
    { rcases key x hxy with ⟨u, v, hu, hv, hxu, hyv, huv⟩,
      exact ⟨v, u, hv, hu, hyv, hxu, (inter_comm u v) ▸ huv⟩ },
    { have hxy' : x ≠ y := λ w, hxy ((option.some.inj_eq _ _).mpr w),
      rcases t2_separation hxy' with ⟨u, v, hu, hv, hxu, hyv, huv⟩,
      refine ⟨some '' u, some '' v, is_open_map_some _ hu, is_open_map_some _ hv,
        ⟨x, hxu, rfl⟩, ⟨y, hyv, rfl⟩, _⟩,
      simp only [image_inter (option.some_injective X), huv, image_empty] }
  end }

lemma dense_range_some (h : ¬ is_compact (univ : set X)) : dense (some '' (univ : set X)) :=
begin
  refine dense_iff_inter_open.mpr (λ s hs Hs, _),
  by_cases H : none ∈ s,
  { rw is_open_iff_of_mem H at hs,
    have minor₁ : s ≠ {none},
    { by_contra w,
      rw [not_not.mp w, some_preimage_none, compl_empty] at hs,
      exact h hs.1 },
    have minor₂ : some⁻¹' s ≠ ∅,
    { by_contra w,
      rw [not_not, eq_empty_iff_forall_not_mem] at w,
      simp only [mem_preimage] at w,
      have : ∀ z ∈ s, z = none := λ z hz,
        by_contra (λ w', let ⟨x, hx⟩ := option.ne_none_iff_exists'.mp w' in
          by rw hx at hz; exact (w x) hz),
      exact minor₁ (eq_singleton_iff_unique_mem.mpr ⟨H, this⟩) },
    rcases ne_empty_iff_nonempty.mp minor₂ with ⟨x, hx⟩,
    exact ⟨some x, hx, x, mem_univ _, rfl⟩ },
  { rcases Hs with ⟨z, hz⟩,
    rcases option.ne_none_iff_exists'.mp (ne_of_mem_of_not_mem hz H) with ⟨x, hx⟩,
    rw hx at hz,
    exact ⟨some x, hz, x, mem_univ _, rfl⟩ }
end

lemma connected_space [preconnected_space X] (h : ¬ is_compact (univ : set X)) :
  connected_space (option X) :=
{ is_preconnected_univ :=
  begin
    rw ← dense_iff_closure_eq.mp (dense_range_some h),
    exact is_preconnected.closure
      (is_preconnected_univ.image some continuous_some.continuous_on)
  end,
  to_nonempty := ⟨none⟩ }

end one_point_extension

end option_topology

section basic

/-- The Alexandroff extension of an arbitrary topological space `X` -/
@[nolint unused_arguments]
def alexandroff (X : Type*) [topological_space X] := option X

variables {X : Type*} [topological_space X]

/-- The embedding of `X` to its Alexandroff extension -/
def of : X → alexandroff X := some

/-- The range of the embedding -/
def range_of (X : Type*) [topological_space X] : set (alexandroff X) := of '' (univ : set X)

lemma of_apply {x : X} : of x = some x := rfl

lemma of_injective : function.injective (@of X _) :=
option.some_injective X

/-- The extra point in the extension -/
def infty : alexandroff X := none

local notation `∞` := infty

namespace alexandroff

instance : has_coe_t X (alexandroff X) := ⟨of⟩

instance : inhabited(alexandroff X) := ⟨∞⟩

@[norm_cast]
lemma coe_eq_coe {x y : X} : (x : alexandroff X) = y ↔ x = y :=
of_injective.eq_iff

@[simp] lemma coe_ne_infty (x : X) : (x : alexandroff X) ≠ ∞  .
@[simp] lemma infity_ne_coe (x : X) : ∞ ≠ (x : alexandroff X) .
@[simp] lemma of_eq_coe {x : X} : (of x : alexandroff X) = x := rfl

protected lemma prop_infty_of_prop_none {p : option X → Prop} (h : p none) : p infty :=
by simpa [infty] using h

/-- Recursor for `alexandroff` using the preferred forms `∞` and `↑x`. -/
@[elab_as_eliminator]
def rec_infty_coe (C : alexandroff X → Sort*) (h₁ : C infty) (h₂ : Π (x : X), C x) :
  Π (z : alexandroff X), C z :=
option.rec h₁ h₂

lemma ne_infty_iff_exists {x : alexandroff X} :
  x ≠ infty ↔ ∃ (y : X), x = y :=
by { induction x using alexandroff.rec_infty_coe; simp }

@[simp] lemma coe_mem_range_of (x : X) : (x : alexandroff X) ∈ (range_of X) :=
one_point_extension.some_mem_range_some x

lemma union_infty_eq_univ : (range_of X ∪ {∞}) = univ :=
one_point_extension.union_none_eq_univ

@[simp] lemma infty_not_mem_range_of : ∞ ∉ range_of X :=
one_point_extension.none_not_mem_image_some

@[simp] lemma not_mem_range_of_iff (x : alexandroff X) :
  x ∉ range_of X ↔ x = ∞ :=
by { induction x using alexandroff.rec_infty_coe; simp }

@[simp] lemma infty_not_mem_image_of {s : set X} : ∞ ∉ of '' s :=
one_point_extension.none_not_mem_image_some

lemma inter_infty_eq_empty : (range_of X) ∩ {∞} = ∅ :=
one_point_extension.inter_none_eq_empty

lemma of_preimage_infty : (of⁻¹' {∞} : set X) = ∅ :=
one_point_extension.some_preimage_none

end alexandroff

end basic

section topology
open alexandroff

variables {X : Type*} [topological_space X]

instance : topological_space (alexandroff X) := one_point_extension X

variables {s : set (alexandroff X)} {s' : set X}

lemma is_open_alexandroff_iff_aux :
  is_open s ↔ if infty ∈ s then is_compact (of⁻¹' s)ᶜ ∧ is_open (of⁻¹' s)
  else is_open (of⁻¹' s) :=
iff.rfl

lemma is_open_iff_of_mem' (h : infty ∈ s) :
  is_open s ↔ is_compact (of⁻¹' s)ᶜ ∧ is_open (of⁻¹' s) :=
by simp [is_open_alexandroff_iff_aux, h]

lemma is_open_iff_of_mem (h : infty ∈ s) :
  is_open s ↔ is_compact (of⁻¹' s)ᶜ ∧ is_closed (of⁻¹' s)ᶜ :=
by simp [is_open_alexandroff_iff_aux, h, is_closed_compl_iff]

lemma is_open_iff_of_not_mem (h : infty ∉ s) :
  is_open s ↔ is_open (of⁻¹' s) :=
by simp [is_open_alexandroff_iff_aux, h]

lemma is_open_of_is_open (h : is_open s) :
  is_open (of⁻¹' s) :=
one_point_extension.is_open_of_is_open h

end topology

section topological
open alexandroff

variables {X : Type*} [topological_space X]

@[continuity] lemma continuous_of : continuous (@of X _) :=
one_point_extension.continuous_some

/-- An open set in `alexandroff X` constructed from a closed compact set in `X` -/
def opens_of_compl {s : set X} (h : is_compact s ∧ is_closed s) :
  topological_space.opens (alexandroff X) :=
⟨(of '' s)ᶜ, (one_point_extension.opens_of_compl h).2⟩

lemma infty_mem_opens_of_compl {s : set X} (h : is_compact s ∧ is_closed s) :
  infty ∈ (opens_of_compl h : set (alexandroff X)) :=
one_point_extension.none_mem_opens_of_compl h

lemma is_open_map_of : is_open_map (@of X _) :=
one_point_extension.is_open_map_some

lemma is_open_range_of : is_open (@range_of X _) :=
one_point_extension.is_open_map_some _ is_open_univ

instance : compact_space (alexandroff X) := one_point_extension.compact_space X

lemma dense_range_of (h : ¬ is_compact (univ : set X)) : dense (@range_of X _) :=
one_point_extension.dense_range_some h

lemma connected_space_alexandroff [preconnected_space X] (h : ¬ is_compact (univ : set X)) :
  connected_space (alexandroff X) :=
one_point_extension.connected_space h

instance [t1_space X] : t1_space (alexandroff X) :=
one_point_extension.t1_space

instance [locally_compact_space X] [t2_space X] : t2_space (alexandroff X) :=
one_point_extension.t2_space

end topological

#lint


import number_theory.bernoulli

--OUTPUT 1
theorem bernoulli_eval_one_add (n : ℕ) (x : ℚ) :
  (polynomial.bernoulli n).eval (1 + x) = (polynomial.bernoulli n).eval x + n * x^(n - 1) := sorry

-- https://ncatlab.org/nlab/show/Yoneda+lemma#statement

/-!

# Yoneda Lemma

In category theory, the Yoneda lemma gives an equivalence between two functors, known as a
Yoneda embedding and its left Kan extension. The Yoneda lemma is fundamental to category theory
and is used to prove many more advanced results.

The Yoneda lemma shows that each functor from a small category $C$ to the category of sets is
completely determined by the set of values it takes when evaluated at each object of $C$, along
with the morphisms between them. More precisely, it states that, for each category $C$, the
functor $y_C : C \to [C^{op}, Set]$ is fully faithful and essentially surjective.

Let $\mathcal{C}$ be a category, and let $A,B \in \mathcal{C}$. The Yoneda lemma gives a natural
bijection between the set of morphisms $Mor(A, B)$ and the set of natural transformations
$A^{op} \to \mathcal{C}(B, −)$.

## Examples

## References

https://ncatlab.org/nlab/show/Yoneda+lemma#statement

## Tags

Yoneda lemma
-/

namespace yoneda

open category_theory

universes v u -- declare the `v`'s first; see `category_theory.category` for an explanation

variable (C : Type u) -- Note that this changes the universe levels to `(C : Type u₁)`, where `u₁ > u`

/--
`ob C` is the type of objects of the category C.
-/
def ob : Type v := C

variables {C}

/--
`hom C` is the type of morphisms of the category C.
-/
def hom : C → C → Type v
| a b := a ⟶ b

/--
`comp C` is the composition of morphism of the category C.
-/
def comp {C : Type u} [category.{v} C] (a b c : C) : hom C a b → hom C b c → hom C a c
| f g := f ≫ g  -- same as `g ≫ f`, because `direct` and `inverse` are symmetric

/--
`id C` is the identity of morphism of the category C.
-/
def id {C : Type u} [category.{v} C] (a : C) : hom C a a
| a := 𝟙 a

attribute [simp] comp id

instance hom_setoid {C : Type u} [category.{v} C] (a b : C) : setoid (hom C a b) := ⟨λ f g, f = g⟩

/--
The Yoneda embedding is the functor that takes each object $x$ of $C$ to the covariant
hom-functor $\hom_C(−,x)$ and each morphism $f : x → y$ to the function $α → f∘α$ that sends
each morphism $α : z → x$ to the composite $f∘α : z → y$.

The Yoneda embedding is fully faithful:
Given objects $x,y\in C$, there is a natural isomorphism
`natural_equiv (yoneda.obj C x) (yoneda.obj C y)`

Proof:
Let $x,y\in Ob(C)$.
Then the naturality square of the Yoneda embedding yields:

{{begin-eqn}}
{{eqn | l = \hom_{[C^{op},Set]}(Hom_C(-,x),Hom_C(-,y))
      | r = \hom_{[C^{op},Set]}(Hom_C(-,y),Hom_C(-,x)))^{op}
      | c = \hom_{[C^{op},Set]}(Hom_C(-,y),Hom_C(-,x))
}}
{{eqn | r = \hom_{[C^{op},Set]}(Hom_C(-,y),Hom_C(-,x))
      | c = natural_equiv
}}
{{eqn | r = \hom_{C^{op}}(y,x)
      | c = \hom_{[C^{op},Set]}(Hom_C(-,y),Hom_C(-,x))
}}
{{eqn | r = \hom_C(x,y)
      | c = \hom_{C^{op}}(y,x)
}}
{{end-eqn}}

`natural_equiv (yoneda.obj C x) (yoneda.obj C y) ≅ 𝟙 (hom C y x)`
-/
instance yoneda_embedding_fully_faithful [small_category C] : fully_faithful (yoneda.obj C) :=
begin
  intros a b,
  apply set_category.functor_iso,
  apply set_category.natural_equiv_iso,
  simp,
  /-
  This does not work without some "cleanup"
  apply inverse, -- must be the inverse because there are two symmetries
  exact opposite_functor_iso_preimage _ _ _,
  --/
  ext, -- Let $f : a → b$ be a morphism in $C$.
  rw function.funext_iff, -- It suffices to prove that
  intros α, -- for all $α : x → a$, $f∘α = α'∘f$
  apply eq_of_inv_eq_inv, -- Since the inverse is unique,
  rw [←cancel_epi f, ←cancel_epi α], -- $g∘f = id_a$ and $f'∘α$
  exact opposite_functor_iso_preimage _ _ _, -- follows from `nat_iso_inv_comp_eq_comp_inv_nat_iso`
  split; ext; refl,
end

--OUTPUT 2
theorem bernoulli_eval_one_add (n : ℕ) (x : ℚ) :
  (polynomial.bernoulli n).eval (1 + x) = (polynomial.bernoulli n).eval x + n * x^(n - 1) :=
begin
  sorry
end

--OUTPUT 3
theorem bernoulli_eval_one_add (n : ℕ) (x : ℚ) :
  (polynomial.bernoulli n).eval (1 + x) = (polynomial.bernoulli n).eval x + n * x^(n - 1) :=
begin
  assume (n : ℕ) (x : ℚ),
  assume h1 : ∀ m : ℕ, m < n → (polynomial.bernoulli m).eval (1 + x) = (polynomial.bernoulli m).eval x + m * x^(m - 1),
  have h2 : ∀ l : ℕ, (n + 1 : ℚ) * ((1 + x) : ℚ)^l = (n + 1 : ℚ) * (x : ℚ)^l + l * ∑ k = 0 to l - 1, ((nat.choose (l - 1) k) : ℚ) * (x : ℚ)^(l - 1 - k)  :=
  begin
    assume l : ℕ,
    show ((n + 1) * ((1 + x) : ℚ)^l) = ((n + 1) * x^l) + l * (∑ k = 0 to l - 1, (nat.choose (l - 1) k) * x^(l - 1 - k)),
    have h3 : (n + 1 : ℚ) * ((1 + x) : ℚ)^l = (n + 1 : ℚ) * (∑ k = 0 to l - 1, nat.choose l k * x^k), from sorry,
    have h4 : ∀ k : ℕ, (n + 1 : ℚ) * nat.choose l k * x^k = nat.choose (l - 1) k * x^(l - 1 - k) + l * nat.choose (l - 1) k * x^(l - 1 - k),
    begin
      assume k : ℕ,
      show (n + 1) * nat.choose l k * x^k = nat.choose (l - 1) k * x^(l - 1 - k) + l * nat.choose (l - 1) k * x^(l - 1 - k),
      sorry,
    end,
    have h5 : ∀ k : ℕ, nat.choose (l - 1) k * x^(l - 1 - k) + l * nat.choose (l - 1) k * x^(l - 1 - k) = nat.choose (l - 1) k * x^(l - 1 - k) + (l * x^(l - 1) * (x^(k - l + 1) * (nat.choose (l - 1) k))), from sorry,
    have h6 : ∀ k : ℕ, k < l → (l * nat.choose (l - 1) k * x^(l - 1 - k) = l * x^(l - 1) * (x^(k - l + 1) * (nat.choose (l - 1) k))), from sorry,
    have h7 : ∀ k : ℕ, k < l → (n + 1 : ℚ) * (nat.choose l k : ℚ) * (x : ℚ)^k = (nat.choose (l - 1) k : ℚ) * (x : ℚ)^(l - 1 - k) + (l * x^(l - 1) * (x^(k - l + 1) * (nat.choose (l - 1) k))), from sorry,
    have h8 : ∀ k : ℕ, k < l → ∃! (n + 1) * (nat.choose l k) * x^k = (nat.choose (l - 1) k) * x^(l - 1 - k) + l * x^(l - 1) * (x^(k - l + 1) * nat.choose (l - 1) k), from sorry,
    have h9 : ∀ k : ℕ, k < l → (n + 1) * (nat.choose l k) * x^k = (nat.choose (l - 1) k) * x^(l - 1 - k) + l * x^(l - 1) * (x^(k - l + 1) * nat.choose (l - 1) k), from sorry,
    have h10 : ∀ k : ℕ, k < l → ∃! (n + 1) * nat.choose l k * x^k = nat.choose (l - 1) k * x^(l - 1 - k) + l * x^(l - 1) * (x^(k - l + 1) * nat.choose (l - 1) k), from sorry,
    have h11 : (n + 1 : ℚ) * (∑ k = 0 to l - 1, (nat.choose l k : ℚ) * (x : ℚ)^k) = (∑ k = 0 to l - 1, (nat.choose (l - 1) k : ℚ) * (x : ℚ)^(l - 1 - k) + (l * (x : ℚ)^(l - 1) * (x^(k - l + 1) * (nat.choose (l - 1) k)))), from sorry,
    have h12 : ∀ k : ℕ, k < l → ∃! (n + 1) * (nat.choose l k)* x^k = (nat.choose (l - 1) k) * x^(l - 1 - k) + l * x^(l - 1) * (x^(k - l + 1) * nat.choose (l - 1) k), from sorry,
    have h13 : ∀ k : ℕ, ∃! ((n + 1) * (nat.choose l k) * x^k) = (nat.choose (l - 1) k) * x^(l - 1 - k) + l * x^(l - 1) * (x^(k - l + 1) * nat.choose (l - 1) k), from sorry,
    have h14 : ∀ k : ℕ, nat.choose (l - 1) k * x^(l - 1 - k) + (l * x^(l - 1) * (x^(k - l + 1) * (nat.choose (l - 1) k))) = (n + 1 : ℚ) * (nat.choose l k) * x^k, from sorry,
    sorry,
  end,
  rw (n + 1 : ℚ) * ((1 + x) : ℚ)^n = (n + 1 : ℚ) * (x : ℚ)^n + n * (∑ k : ℕ, (nat.choose n k : ℚ) * (x : ℚ)^(n - 1 - k)) at h2,
  have h3 : ∀ k : ℕ, ∃! (n + 1 : ℚ) * (nat.choose n k : ℚ) * (x : ℚ)^k = (nat.choose (n - 1) k : ℚ) * (x : ℚ)^(n - 1 - k) + n * (nat.choose (n - 1) k : ℚ) * (x : ℚ)^(n - 1 - k), from sorry,
  have h4 : ∀ k : ℕ, (n + 1 : ℚ) * (nat.choose n k : ℚ) * (x : ℚ)^k = (nat.choose (n - 1) k : ℚ) * (x : ℚ)^(n - 1 - k) + n * (nat.choose (n - 1) k : ℚ) * (x : ℚ)^(n - 1 - k), from sorry,
  have h5 : ∀ k : ℕ, (n + 1) * (nat.choose n k) * x^k = (nat.choose (n - 1) k) * x^(n - 1 - k) + n * (nat.choose (n - 1) k) * x^(n - 1 - k), from sorry,
  have h6 : (n + 1) * (polynomial.bernoulli n).eval ((1 + x) : ℝ) = (n + 1 : ℚ) * (polynomial.bernoulli n).eval x + n * (
end --Needs more than 2000 tokens!

--OUTPUT 4
theorem bernoulli_eval_one_add (n : ℕ) (x : ℚ) :
  (polynomial.bernoulli n).eval (1 + x) = (polynomial.bernoulli n).eval x + n * x^(n - 1) :=
begin
  sorry,
end

--OUTPUT 5
theorem bernoulli_eval_one_add (n : ℕ) (x : ℚ) :
  (polynomial.bernoulli n).eval (1 + x) = (polynomial.bernoulli n).eval x + n * x^(n - 1) :=
begin
  sorry,
end

--OUTPUT 6
theorem bernoulli_eval_one_add (n : ℕ) (x : ℚ) :
  (polynomial.bernoulli n).eval (1 + x) = (polynomial.bernoulli n).eval x + n * x^(n - 1) :=
begin
  have h1 : ∀ (l : ℕ), ∑ k in range (l + 1), ((l + 1) choose k) * (polynomial.bernoulli k).eval x = (l + 1) * x^l, from
  begin
    assume (l : ℕ),
    have h2 : polynomial.degree ((∑ k in range (l + 1), ((l + 1) choose k) * (polynomial.bernoulli k)) : polynomial ℚ) ≤ l, from sorry,
    rw polynomial.sum_degree_eq_max_of_degree_le at h2,
    have h3 : ∀ (k : ℕ), k ∈ range (l + 1) → (polynomial.degree (polynomial.const ((l + 1 : ℕ) choose k) * polynomial.bernoulli k)) < l, from sorry,
    rw polynomial.sum_degree_eq_max_of_degree_lt at h3,
    have h4 : ∀ (k : ℕ), k ∈ range (l + 1) → (polynomial.degree (polynomial.const ((l + 1 : ℕ) choose k))) < l, from sorry,
    rw polynomial.sum_degree_eq_max_of_degree_lt at h4,

    calc (∑ k in range (l + 1), ((l + 1) choose k) * (polynomial.bernoulli k).eval x)
        = (∑ k in range (l + 1), ((l + 1) choose k) * (polynomial.bernoulli k).eval x) : by sorry
    ... = (∑ k in range (l + 1), ((l + 1) choose k) * x^k) : by sorry
    ... = (∑ k in range (l + 1), ((l + 1) choose k) * x^k) : by sorry
    ... = (l + 1) * x^l : by sorry
  end,

  have h3 : ∀ (m : ℕ) (x : ℚ), m < n → polynomial.bernoulli m).eval (1 + x) = polynomial.bernoulli (m).eval (x) + m * x^(m - 1), from
  begin
    assume (m : ℕ) (x : ℚ) (h1 : m < n),
    induction m with m hm,
    show polynomial.bernoulli 0).eval (1 + x) = polynomial.bernoulli 0).eval x + 0 * x^(-1), from by sorry,
    assume (m : ℕ) (ihm : m < n → (polynomial.bernoulli m).eval (1 + x) = (polynomial.bernoulli m).eval (x) + m * x^(m - 1)),
    assume (h2 : succ m < n),
    have h3 : ((succ m) + 1 : ℕ) = succ succ m, from rfl,
    rw [h3, polynomial.eval_pow] at *,
    have h4 : ((succ m) + 1 : ℕ) = succ succ m, from rfl,
    rw [h4, polynomial.eval_pow] at *,
    have h5 : polynomial.bernoulli n).eval (1 + x) = polynomial.bernoulli n).eval (x) + n * x^(n - 1), from sorry,
    assume (h6 : (m : ℕ) < n → (polynomial.bernoulli m).eval (1 + x) = (polynomial.bernoulli m).eval (x) + m * x^(m - 1)),
    assume (h7 : (n : ℕ) < n),
  end
end

--OUTPUT 7
theorem bernoulli_eval_one_add (n : ℕ) (x : ℚ) :
  (polynomial.bernoulli n).eval (1 + x) = (polynomial.bernoulli n).eval x + n * x^(n - 1) :=
begin
  -- The proof is by strong induction
  apply nat.strong_induction_on n,
  assume (d : ℕ) (h_ind : ∀ m : ℕ, m < d →
    (polynomial.bernoulli m).eval (1 + x) = (polynomial.bernoulli m).eval x + m * x^(m - 1)),
  show (polynomial.bernoulli d).eval (1 + x) = (polynomial.bernoulli d).eval x + d * x^(d - 1),
  begin
    -- we want to multiply both sides by d+1
    have h_mult : (polynomial.bernoulli d).eval (1 + x) * (d + 1) =
                  (polynomial.bernoulli d).eval x * (d + 1) + d * x^d,
    begin
      have h1 : (polynomial.bernoulli d).eval (1 + x) * (d + 1) =
                  (1 + x)^d * (d + 1),
      {
        rw polynomial.eval_pow_bernoulli,
      },
      have h2 : (polynomial.bernoulli d).eval x * (d + 1) + d * x^d =
                  x^d * (d + 1) + d * x^d,
      {
        rw polynomial.eval_pow_bernoulli,
      },
      -- we are going to use the fact that for all l in N, sum from k=0 to l of
      -- choose(l+1,k)*Bk = (l+1)*X^l
      have h_sum_bernoulli : ∀ l : ℕ,
                             (sum (polynomial.bernoulli l) (finset.range (l + 1))).eval (1 + x) =
                             (l + 1) * (1 + x)^l,
      {
        assume (l : ℕ),
        have h_sum_polynomial : ∀ (l : ℕ) (P : polynomial ℚ),
        (sum P (finset.range (l + 1))).eval x = ∑ (i : finset (l + 1)), P.eval x,
        {
          assume (l : ℕ) (P : polynomial ℚ),
          rw finset.sum_hom,
          apply finset.sum_congr rfl,
          assume (i : finset (l + 1)),
          apply polynomial.eval_C,
        },
        have h_bernoulli_eval_C : ∀ (l : ℕ),
                                  (sum (polynomial.bernoulli l) (finset.range (l + 1))).eval x =
                                  ∑ (i : finset (l + 1)), (polynomial.bernoulli l).eval x,
        {
          assume (l : ℕ),
          rw h_sum_polynomial,
          apply finset.sum_congr rfl,
          assume (i : finset (l + 1)),
          apply polynomial.eval_bernoulli,
        },
        have h_sum_bernoulli_number : ∀ (l : ℕ),
                                      (sum (polynomial.bernoulli l) (finset.range (l + 1))).eval x =
                                      ∑ (i : finset (l + 1)), polynomial.bernoulli_number i,
        {
          -- I will use another corollary of the definition of Bernoulli polynomial
          -- for all l in N, sum from k=0 to l of
          -- choose(l+1,k)*Bk = (l+1)*X^l
          assume (l : ℕ),
          have h_sum_bernoulli_poly : ∀ (l : ℕ) (n : ℕ),
                                       sum (polynomial.bernoulli l)
                                           (finset.range (l + 1)) =
                                       ∑ (i : finset (l + 1)), (polynomial.bernoulli l).eval x,
          {
            assume (l : ℕ) (n : ℕ),
            rw polynomial.sum_hom,
            apply finset.sum_congr rfl,
            assume (i : finset (l + 1)),
            apply polynomial.eval_bernoulli,
          },
          have h_sum_bernoulli_poly_one : ∀ (l : ℕ) (n : ℕ),
                                       sum (polynomial.bernoulli l)
                                           (finset.range (l + 1)) =
                                       ∑ (i : finset (l + 1)), polynomial.bernoulli_number i,
          {
            assume (l : ℕ) (n : ℕ),
            have h_sum_bernoulli_poly : ∀ (l : ℕ) (n : ℕ),
                                       sum (polynomial.bernoulli l)
                                           (finset.range (l + 1)) =
                                       ∑ (i : finset (l + 1)), (polynomial.bernoulli l).eval x,
            {
              assume (l : ℕ) (n : ℕ),
              rw polynomial.sum_hom,
              apply finset.sum_congr rfl,
              assume (i : finset (l + 1)),
              apply polynomial.eval_bernoulli,
            },
            have h_sum_bernoulli_poly_one : ∀ (l : ℕ) (n : ℕ),
                                       sum (polynomial.bernoulli l)
                                           (finset.range (l + 1)) =
                                       ∑ (i : finset (l + 1)), polynomial.bernoulli_number i,
            {
              assume (l : ℕ) (n : ℕ),
              by {
                apply polynomial.bernoulli_eval_one,
              },
            },
            have h_sum_bernoulli_poly_one : ∀ (l : ℕ) (n : ℕ),
                                       sum (polynomial.bernoulli l)
                                           (finset.range (l + 1)) =
                                       ∑ (i : finset (l + 1)), polynomial.bernoulli_number i,
            {
              assume (l : ℕ) (n : ℕ),
              by {
                apply polynomial.bernoulli_eval_one,
              },
            },
            have h_sum_bernoulli_poly_one : ∀ (l : ℕ) (n : ℕ),
                                       sum (polynomial.bernoulli l)
                                           (finset.range (l + 1)) =
                                       ∑ (i : finset (l + 1)), polynomial.bernoulli_number i,
            {
              assume (l : ℕ) (n : ℕ),
              by {
                apply polynomial.bernoulli_eval_one,
              },
            },
             sorry,
          },
          rw h_bernoulli_eval_C,
          rw h_sum_bernoulli_poly_one,
        },
        have h_sum_bernoulli_number_one : ∀ (l : ℕ),
                                          ∑ (i : finset (l + 1)), polynomial.bernoulli_number i =
                                           (l + 1) * x^l,
        {
          assume (l : ℕ),
          have h_sum_bernoulli_number : ∀ (l : ℕ),
                                      (sum (polynomial.bernoulli l) (finset.range (l + 1))).eval x =
                                      ∑ (i : finset (l + 1)), polynomial.bernoulli_number i,
          {
            -- I will use another corollary of the definition of Bernoulli polynomial
            -- for all l in N, sum from k=0 to l of
            -- choose(l+1
end --Needs more than 2000 tokens!

--OUTPUT 8
theorem bernoulli_eval_one_add (n : ℕ) (x : ℚ) :
  (polynomial.bernoulli n).eval (1 + x) = (polynomial.bernoulli n).eval x + n * x^(n - 1) :=
begin
  sorry,
end

--OUTPUT 9
theorem bernoulli_eval_one_add (n : ℕ) (x : ℚ) :
  (polynomial.bernoulli n).eval (1 + x) = (polynomial.bernoulli n).eval x + n * x^(n - 1) :=
begin
  sorry
end

--OUTPUT 10
theorem bernoulli_eval_one_add (n : ℕ) (x : ℚ) :
  (polynomial.bernoulli n).eval (1 + x) = (polynomial.bernoulli n).eval x + n * x^(n - 1) :=
begin
  have h1 : ∀ n : ℕ, ∀ x : ℚ, n < n → (polynomial.bernoulli n).eval (1 + x) = (polynomial.bernoulli n).eval x + n * x^(n - 1), from begin
    assume (n : ℕ) (h1 : n < n),
    show (polynomial.bernoulli n).eval (1 + x) = (polynomial.bernoulli n).eval x + n * x^(n - 1), from sorry,
  end,
  have h2 : ∀ m : ℕ, ∀ d : ℕ, m < d →
    ∀ x : ℚ,
    (polynomial.bernoulli m).eval (1 + x) = (polynomial.bernoulli m).eval x + m * x^(m - 1) →
    (polynomial.bernoulli d).eval (1 + x) = (polynomial.bernoulli d).eval x + d * x^(d - 1), from begin
    assume (m : ℕ) (d : ℕ) (h2 : m < d) (x : ℚ) (h3 : (polynomial.bernoulli m).eval (1 + x) = (polynomial.bernoulli m).eval x + m * x^(m - 1)),
    have h4 : ∀ l : ℕ, (polynomial.bernoulli l).eval (1 + x) = (polynomial.bernoulli l).eval x + l * x^(l - 1), from begin
      assume (l : ℕ),
      cases l,
      simp at h4,
      have h5 : 0 < (l+1), from sorry,
      have h6 : l < (l+1), from sorry,
      have h7 : (polynomial.bernoulli l).eval (1 + x) = (polynomial.bernoulli l).eval x + l * x^(l - 1), from sorry,
      show (polynomial.bernoulli (l+1)).eval (1 + x) = (polynomial.bernoulli (l+1)).eval x + (l+1) * x^((l+1) - 1), from sorry,
    end,
    show (polynomial.bernoulli d).eval (1 + x) = (polynomial.bernoulli d).eval x + d * x^(d - 1), from sorry,
  end,
  sorry,
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
  assume (A : set α) (hA : A ∈ 𝒫 S) (B : set α) (hB : B ∈ 𝒫 S),
  have h1 : (A ⊆ S) ∧ (B ⊆ S), from sorry,
  have h2 : (A ∩ B) ⊆ A, from sorry,
  have h3 : (A ∩ B) ⊆ S, from sorry,
  show (A ∩ B) ∈  𝒫 S, from sorry,
end

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

{{begin-eqn}}
{{eqn | l = \left({x + y}\right)^2
      | r = \left({x + y}\right) \cdot \left({x + y}\right)
}}
{{eqn | r = x \cdot \left({x + y}\right) + y \cdot \left({x + y}\right)
      | c = Real Multiplication Distributes over Addition
}}
{{eqn | r = x \cdot x + x \cdot y + y \cdot x + y \cdot y
      | c = Real Multiplication Distributes over Addition
}}
{{eqn | r = x^2 + 2xy + y^2
      | c =
}}
{{end-eqn}}
{{qed}}
-/
theorem square_of_sum (x y : ℝ) : (x + y)^2 = (x^2 + 2*x*y + y^2) :=
begin
  calc (x + y)^2 = (x+y)*(x+y) : by sorry
  ... = x*(x+y) + y*(x+y) : by sorry
  ... = x*x + x*y + y*x + y*y : by sorry
  ... = x^2 + 2*x*y + y^2 : by sorry,
end

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

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

Setting $b = a$, this becomes:

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

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

These $x$ and $y$ are both $e$, by definition of identity element.
{{qed}}
-/
theorem group_identity_unique {G : Type*} [group G] : ∃! e : G, ∀ a : G, e * a = a ∧ a * e = a :=
begin
  have h1 : ∀ a b : G, ∃! x : G, a * x = b, from sorry,
  have h2 : ∀ a b : G, ∃! y : G, y * a = b, from sorry,

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

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

  show ∃! e : G, ∀ a : G, e * a = a ∧ a * e = a, from by {
    use (1 : G),
    have h7 : ∀ e : G, (∀ a : G, e * a = a ∧ a * e = a) → e = 1, from by {
      assume (e : G) (h7 : ∀ a : G, e * a = a ∧ a * e = a),
      have h8 : ∀ a : G, e = classical.some (h3 a), from sorry,
      have h9 : ∀ a : G, e = classical.some (h4 a), from sorry,
      show e = (1 : G), from sorry,
    },
    sorry,
  }
end

/--`theorem`
Bernoulli polynomial evaluation
Given a natural number $n$ and a rational $x$, let $B_n (x)$ denote the $n$-th Bernoulli polynomial evaluated at $x$. Then, $$B_n (1 + x) = B_n (x) + n x^{n - 1}$$
`proof`
We apply strong induction on $n$. So, for all $m < d$, we have $B_{m} (1 + x) = B_{m} (x) + m x^{m - 1}$ and we want to show that $$B_{d} (1 + x) = B_{d} (x) + d x^{d - 1}$$
Multiplying both sides by $d + 1$, and using the fact that, for all $l \in \mathbb{N}$, $\sum_{k = 0}^{l} {l + 1 \choose k} B_k = (l + 1) X^l$ (where $B_k$ is the $k$-th Bernoulli number), we get that
$$ (d + 1) (1 + x)^d - (d + 1) x^d = \sum_{l = 0}^{d} {d + 1 \choose l} l x^{l - 1} $$
The conclusion then follows easily.

QED

-/
theorem  bernoulli_eval_one_add (n : ℕ) (x : ℚ) :
  (polynomial.bernoulli n).eval (1 + x) = (polynomial.bernoulli n).eval x + n * x^(n - 1) :=
FEW SHOT PROMPTS TO CODEX(END)-/