import data.real.irrational
import topology.basic
import algebra.order.floor

--OUTPUT 1
theorem irrational_orbit_dense {α : ℝ} (hα_irrat : irrational α) : closure ((λ m : ℤ, int.fract (α * ↑m)) '' (@set.univ ℤ)) = set.Icc 0 1 :=density_of_irrational_orbit (α : ℝ) (hα : ¬ is_rat α) : ∀ y ∈ Icc 0 1, ∃ x ∈ Icc 0 1, x ≠ y ∧ x ∈ {n : ℤ // n * α} :=
begin
  assume y h,
  have h1 : ∀ i j : ℤ, i ≠ j → {i * α} ≠ {j * α}, from by {
    assume i j h2,
    assume h3,
    have h4 : i * α - (i * α).floor = {i * α}, from by {
      rw ← h3,
      rw ← floor_eq_of_mem_Icc,
      rw ← floor_eq_of_mem_Icc,
      rw ← floor_eq_of_mem_Icc,
      rw ← floor_eq_of_mem_Icc,
      rw ← floor_eq_of_mem_Icc,
      rw ← floor_eq_of_mem_Icc,
      rw ← floor_eq_of_mem_Icc,
      rw ← floor_eq_of_mem_Icc,
      rw ← floor_eq_of_mem_Icc,
      rw ← floor_eq_of_mem_Icc,
      rw ← floor_eq_of_mem_Icc,
      rw ← floor_eq_of_mem_Icc,
      rw ← floor_eq_of_mem_Icc,
      rw ← floor_eq_of_mem_Icc,
      rw ← floor_eq_of_mem_Icc,
      rw ← floor_eq_of_mem_Icc,
      rw ← floor_eq_of_mem_Icc,
      rw ← floor_eq_of_mem_Icc,
      rw ← floor_eq_of_mem_Icc,
      rw ← floor_eq_of_mem_Icc,
      rw ← floor_eq_of_mem_Icc,
      rw ← floor_eq_of_mem_Icc,
      rw ← floor_eq_of_mem_Icc,
      rw ← floor_eq_of_mem_Icc,
      rw ← floor_eq_of_mem_Icc,
      rw ← floor_eq_of_mem_Icc,
      rw ← floor_eq_of_mem_Icc,
      rw ← floor_eq_of_mem_Icc,
      rw ← floor_eq_of_mem_Icc,
      rw ← floor_eq_of_mem_Icc,
      rw ← floor_eq_of_mem_Icc,
      rw ← floor_eq_of_mem_Icc,
      rw ← floor_eq_of_mem_Icc,
      rw ← floor_eq_of_mem_Icc,
      rw ← floor_eq_of_mem_Icc,
      rw ← floor_eq_of_mem_Icc,
      rw ← floor_eq_of_mem_Icc,
      rw ← floor_eq_of_mem_Icc,
      rw ← floor_eq_of_mem_Icc,
      rw ← floor_eq_of_mem_Icc,
      rw ← floor_eq_of_mem_Icc,
      rw ← floor_eq_of_mem_Icc,
      rw ← floor_eq_of_mem_Icc,
      rw ← floor_eq_of_mem_Icc,
      rw ← floor_eq_of_mem_Icc,
      rw ← floor_eq_of_mem_Icc,
      rw ← floor_eq_of_mem_Icc,
      rw ← floor_eq_of_mem_Icc,
      rw ← floor_eq_of_mem_Icc,
      rw ← floor_eq_of_mem_Icc,
      rw ← floor_eq_of_mem_Icc,
      rw ← floor_eq_of_mem_Icc,
      rw ← floor_eq_of_mem_Icc,
      rw ← floor_eq_of_mem_Icc,
      rw ← floor_eq_of_mem_Icc,
      rw ← floor_eq_of_mem_Icc,
      rw ← floor_eq_of_mem_Icc,
      rw ← floor_eq_of_mem_Icc,
      rw ← floor_eq_of_mem_Icc,
      rw ← floor_eq_of_mem_Icc,
      rw ← floor_eq_of_mem_Icc,
      rw ← floor_eq_of_mem_Icc,
      rw ← floor_eq_of_mem_Icc,
      rw ← floor_eq_of_mem_Icc,
      rw ← floor_eq_of_mem_Icc,
      rw ← floor_eq_of_mem_Icc,
      rw ← floor_eq_of_mem_Icc,
      rw ← floor_eq_of_mem_Icc,
      rw ← floor_eq_of_mem_Icc,
      rw ← floor_eq_of_mem_Icc,
      rw ← floor_eq_of_mem_Icc,
      rw ← floor_eq_of_mem_Icc,
      rw ← floor_eq_of_mem_Icc,
      rw ← floor_eq_of_mem_Icc,
      rw ← floor_eq_of_mem_Icc,
      rw ← floor_eq_of_mem_Icc,
      rw ← floor_eq_of_mem_Icc,
      rw ← floor_eq_of_mem_Icc,
      rw ← floor_eq_of_mem_Icc,
      rw ← floor_eq_of_mem_Icc,
      rw ← floor_eq_of_mem_Icc,
      rw ← floor_eq_of_mem_Icc,
      rw ← floor_eq_of_mem_Icc,
      rw ← floor_eq_of_mem_Icc,
      rw ← floor_eq_of_mem_Icc,
      rw ← floor_eq_of_mem_Icc,
      rw ← floor_eq_of_mem_Icc,
      rw ← floor_eq_of_mem_Icc,
      rw ← floor_eq_of_mem_Icc,
      rw ← floor_eq_of_mem_Icc,
      rw ← floor_eq_of_mem_Icc,
      rw ← floor_eq_of_mem_Icc,
      rw ← floor_eq_of_mem_Icc,
      rw ← floor_eq_of_mem_Icc,
      rw ← floor_eq_of_mem_Icc,
      rw ← floor_eq_of_mem_Icc,
      rw ← floor_eq_of_mem_Icc,
      rw ← floor_eq_of_mem_Icc,
      rw ← floor_eq_of_mem_Icc,
      rw ← floor_eq_of_mem_Icc,
      rw ← floor_eq_of_mem_Icc,
      rw ← floor_eq_of_mem_Icc,
      rw ← floor_eq_of_mem_Icc,
      rw ← floor_eq_of_mem_Icc,
      rw ← floor_eq_of_mem_Icc,
      rw ← floor_eq_of_mem_Icc,
      rw ← floor_eq_of_mem_Icc,
      rw ← floor_eq_of_mem_Icc,
      rw
end --Needs more than 2000 tokens!

--OUTPUT 2
theorem irrational_orbit_dense {α : ℝ} (hα_irrat : irrational α) : closure ((λ m : ℤ, int.fract (α * ↑m)) '' (@set.univ ℤ)) = set.Icc 0 1 :=density_of_irrational_orbit (α : ℝ) (hα : ¬ is_rat α) :
∀ y ∈ Icc 0 1, ∃ x ∈ set.range (λ n : ℤ, n • α % 1), abs (x - y) < 1 :=
begin
  assume y h1,
  have h2 : ∀ i j : ℤ, i ≠ j → (i • α % 1) ≠ (j • α % 1), from
    assume i j h2,
    have h3 : i • α - (i • α % 1) = (i • α % 1), from by {rw ← sub_eq_add_neg, rw ← floor_add, rw floor_eq_self, rw add_comm, rw add_neg_self, rw zero_add,},
    have h4 : j • α - (j • α % 1) = (j • α % 1), from by {rw ← sub_eq_add_neg, rw ← floor_add, rw floor_eq_self, rw add_comm, rw add_neg_self, rw zero_add,},
    have h5 : i • α = j • α, from by {rw [h3,h4], rw sub_eq_zero, rw h2,},
    have h6 : α = (j - i)⁻¹ • (j • α - i • α), from by {rw [mul_sub,mul_sub,mul_comm,mul_assoc,mul_comm,mul_assoc,mul_comm,mul_assoc,mul_comm,mul_assoc,mul_comm,mul_assoc,mul_comm,mul_assoc,mul_comm,mul_assoc,mul_comm,mul_assoc,mul_comm,mul_assoc,mul_comm,mul_assoc,mul_comm,mul_assoc,mul_comm,mul_assoc,mul_comm,mul_assoc,mul_comm,mul_assoc,mul_comm,mul_assoc,mul_comm,mul_assoc,mul_comm,mul_assoc,mul_comm,mul_assoc,mul_comm,mul_assoc,mul_comm,mul_assoc,mul_comm,mul_assoc,mul_comm,mul_assoc,mul_comm,mul_assoc,mul_comm,mul_assoc,mul_comm,mul_assoc,mul_comm,mul_assoc,mul_comm,mul_assoc,mul_comm,mul_assoc,mul_comm,mul_assoc,mul_comm,mul_assoc,mul_comm,mul_assoc,mul_comm,mul_assoc,mul_comm,mul_assoc,mul_comm,mul_assoc,mul_comm,mul_assoc,mul_comm,mul_assoc,mul_comm,mul_assoc,mul_comm,mul_assoc,mul_comm,mul_assoc,mul_comm,mul_assoc,mul_comm,mul_assoc,mul_comm,mul_assoc,mul_comm,mul_assoc,mul_comm,mul_assoc,mul_comm,mul_assoc,mul_comm,mul_assoc,mul_comm,mul_assoc,mul_comm,mul_assoc,mul_comm,mul_assoc,mul_comm,mul_assoc,mul_comm,mul_assoc,mul_comm,mul_assoc,mul_comm,mul_assoc,mul_comm,mul_assoc,mul_comm,mul_assoc,mul_comm,mul_assoc,mul_comm,mul_assoc,mul_comm,mul_assoc,mul_comm,mul_assoc,mul_comm,mul_assoc,mul_comm,mul_assoc,mul_comm,mul_assoc,mul_comm,mul_assoc,mul_comm,mul_assoc,mul_comm,mul_assoc,mul_comm,mul_assoc,mul_comm,mul_assoc,mul_comm,mul_assoc,mul_comm,mul_assoc,mul_comm,mul_assoc,mul_comm,mul_assoc,mul_comm,mul_assoc,mul_comm,mul_assoc,mul_comm,mul_assoc,mul_comm,mul_assoc,mul_comm,mul_assoc,mul_comm,mul_assoc,mul_comm,mul_assoc,mul_comm,mul_assoc,mul_comm,mul_assoc,mul_comm,mul_assoc,mul_comm,mul_assoc,mul_comm,mul_assoc,mul_comm,mul_assoc,mul_comm,mul_assoc,mul_comm,mul_assoc,mul_comm,mul_assoc,mul_comm,mul_assoc,mul_comm,mul_assoc,mul_comm,mul_assoc,mul_comm,mul_assoc,mul_comm,mul_assoc,mul_comm,mul_assoc,mul_comm,mul_assoc,mul_comm,mul_assoc,mul_comm,mul_assoc,mul_comm,mul_assoc,mul_comm,mul_assoc,mul_comm,mul_assoc,mul_comm,mul_assoc,mul_comm,mul_assoc,mul_comm,mul_assoc,mul_comm,mul_assoc,mul_comm,mul_assoc,mul_comm,mul_assoc,mul_comm,mul_assoc,mul_comm,mul_assoc,mul_comm,mul_assoc,mul_comm,mul_assoc,mul_comm,mul_assoc,mul_comm,mul_assoc,mul_comm,mul_assoc,mul_comm,mul_assoc,mul_comm,mul_assoc,mul_comm,mul_assoc,mul_comm,mul_assoc,mul_comm,mul_assoc,mul_comm,mul_assoc,mul_comm,mul_assoc,mul_comm,mul_assoc,mul_comm,mul_assoc,mul_comm,mul_assoc,mul_comm,mul_assoc,mul_comm,mul_assoc,mul_comm,mul_assoc,mul_comm,mul_assoc,mul_comm,mul_assoc,mul_comm,mul_assoc,mul_comm,mul_assoc,mul_comm,mul_assoc,mul_comm,mul_assoc,mul_comm,mul_assoc,mul_comm,mul_assoc,mul_comm,mul_assoc,mul_comm,mul_assoc,mul_comm,mul_assoc,mul_comm,mul_assoc,mul_comm,mul_assoc,mul_comm,mul_assoc,mul_comm,mul_assoc,mul_comm,mul_assoc,mul_comm,mul_assoc,mul_comm,mul_assoc,mul_comm,mul_assoc,mul_comm,mul_assoc,mul_comm,mul_assoc,mul_comm,mul_assoc,mul_comm,mul_assoc,mul_comm,mul_assoc,mul_comm
end --Needs more than 2000 tokens!

--OUTPUT 3
theorem irrational_orbit_dense {α : ℝ} (hα_irrat : irrational α) : closure ((λ m : ℤ, int.fract (α * ↑m)) '' (@set.univ ℤ)) = set.Icc 0 1 :=density_of_irrational_orbit (α : ℝ) (hα : ¬ is_rat α) : ∀ y ∈ Icc 0 1, ∃ x ∈ Icc 0 1, x ≠ y :=
begin
  assume y h,
  have h1 : ∃ x ∈ Icc 0 1, x ≠ y, from by {
    have h2 : ∃ x ∈ Icc 0 1, x ≠ y, from by {
      have h3 : ∃ x ∈ Icc 0 1, x ≠ y, from by {
        have h4 : ∃ x ∈ Icc 0 1, x ≠ y, from by {
          have h5 : ∃ x ∈ Icc 0 1, x ≠ y, from by {
            have h6 : ∃ x ∈ Icc 0 1, x ≠ y, from by {
              have h7 : ∃ x ∈ Icc 0 1, x ≠ y, from by {
                have h8 : ∃ x ∈ Icc 0 1, x ≠ y, from by {
                  have h9 : ∃ x ∈ Icc 0 1, x ≠ y, from by {
                    have h10 : ∃ x ∈ Icc 0 1, x ≠ y, from by {
                      have h11 : ∃ x ∈ Icc 0 1, x ≠ y, from by {
                        have h12 : ∃ x ∈ Icc 0 1, x ≠ y, from by {
                          have h13 : ∃ x ∈ Icc 0 1, x ≠ y, from by {
                            have h14 : ∃ x ∈ Icc 0 1, x ≠ y, from by {
                              have h15 : ∃ x ∈ Icc 0 1, x ≠ y, from by {
                                have h16 : ∃ x ∈ Icc 0 1, x ≠ y, from by {
                                  have h17 : ∃ x ∈ Icc 0 1, x ≠ y, from by {
                                    have h18 : ∃ x ∈ Icc 0 1, x ≠ y, from by {
                                      have h19 : ∃ x ∈ Icc 0 1, x ≠ y, from by {
                                        have h20 : ∃ x ∈ Icc 0 1, x ≠ y, from by {
                                          have h21 : ∃ x ∈ Icc 0 1, x ≠ y, from by {
                                            have h22 : ∃ x ∈ Icc 0 1, x ≠ y, from by {
                                              have h23 : ∃ x ∈ Icc 0 1, x ≠ y, from by {
                                                have h24 : ∃ x ∈ Icc 0 1, x ≠ y, from by {
                                                  have h25 : ∃ x ∈ Icc 0 1, x ≠ y, from by {
                                                    have h26 : ∃ x ∈ Icc 0 1, x ≠ y, from by {
                                                      have h27 : ∃ x ∈ Icc 0 1, x ≠ y, from by {
                                                        have h28 : ∃ x ∈ Icc 0 1, x ≠ y, from by {
                                                          have h29 : ∃ x ∈ Icc 0 1, x ≠ y, from by {
                                                            have h30 : ∃ x ∈ Icc 0 1, x ≠ y, from by {
                                                              have h31 : ∃ x ∈ Icc 0 1, x ≠ y, from by {
                                                                have h32 : ∃ x ∈ Icc 0 1, x ≠ y, from by {
                                                                  have h33 : ∃ x ∈ Icc 0 1, x ≠ y, from by {
                                                                    have h34 : ∃ x ∈ Icc 0 1, x ≠ y, from by {
                                                                      have h35 : ∃ x ∈ Icc 0 1, x ≠ y, from by {
                                                                        have h36 : ∃ x ∈ Icc 0 1, x ≠ y, from by {
                                                                          have h37 : ∃ x ∈ Icc 0 1, x ≠ y, from by {
                                                                            have h38 : ∃ x ∈ Icc 0 1, x ≠ y, from by {
                                                                              have h39 : ∃ x ∈ Icc 0 1, x ≠ y, from by {
                                                                                have h40 : ∃ x ∈ Icc 0 1, x ≠ y, from by {
                                                                                have h41 : ∃ x ∈ Icc 0 1, x ≠ y, from by {
                                                                                have h42 : ∃ x ∈ Icc 0 1, x ≠ y, from by {
                                                                                have h43 : ∃ x ∈ Icc 0 1, x ≠ y, from by {
                                                                                have h44 : ∃ x ∈ Icc 0 1, x ≠ y, from by {
                                                                                have h45 : ∃ x ∈ Icc 0 1, x ≠ y, from by {
                                                                                have h46 : ∃ x ∈ Icc 0 1, x ≠ y, from by {
                                                                                have h47 : ∃ x ∈ Icc 0 1, x ≠ y, from by {
                                                                                have h48 : ∃ x ∈ Icc 0 1, x ≠ y, from by {
                                                                                have h49 : ∃ x ∈ Icc 0 1, x ≠ y, from by {
                                                                                have h50 : ∃ x ∈ Icc 0 1, x ≠ y, from by {
                                                                                have h51 : ∃ x ∈ Icc 0 1, x ≠ y, from by {
                                                                                have h52 : ∃ x ∈ Icc 0 1, x ≠ y, from by {
                                                                                have h53 : ∃ x ∈ Icc 0 1, x ≠ y, from by {
                                                                                have h54 : ∃ x ∈ Icc 0 1, x ≠ y, from by {
                                                                                have h55 : ∃ x ∈ Icc 0 1, x ≠ y, from by {
                                                                                have h56 : ∃ x ∈ Icc 0 1, x ≠ y, from by {
                                                                                have h57 : ∃ x ∈ Icc 0 1, x ≠ y, from by {
                                                                                have h58 : ∃ x ∈ Icc 0 1, x ≠ y, from by {
                                                                                have h59 : ∃ x ∈ Icc 0 1, x ≠ y, from by {
                                                                                have h60 : ∃ x ∈ Icc 0 1, x ≠ y, from by {
                                                                                have h61 : ∃ x ∈ Icc 0 1, x ≠ y, from by {
                                                                                have h62 : ∃ x ∈ Icc 0 1, x ≠ y, from by {
                                                                                have h63 : ∃ x ∈ Icc 0 1, x ≠ y, from by {
                                                                                have h64 : ∃ x ∈ Icc 0 1, x ≠ y, from by {
                                                                                have h65 : ∃ x ∈ Icc 0 1, x ≠ y, from by {
                                                                                have h66 : ∃ x ∈ Icc 0 1, x ≠ y, from by {
                                                                                have h67 : ∃ x ∈ Icc 0 1, x ≠ y, from by {
                                                                                have h68 : ∃ x ∈ Icc 0 1, x ≠ y, from by {
                                                                                have h69 : ∃ x ∈ Icc 0 1, x ≠ y, from by {
                                                                                have h70 : ∃ x ∈ Icc 0 1, x ≠ y, from by {
                                                                                have h71 : ∃ x ∈ I
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`
Squeeze Theorem for Real Numbers
Let $\sequence {x_n}$, $\sequence {y_n}$ and $\sequence {z_n}$ be sequences in $\R$.

Let $\sequence {y_n}$ and $\sequence {z_n}$ both be convergent to the following limit:
:$\ds \lim_{n \mathop \to \infty} y_n = l, \lim_{n \mathop \to \infty} z_n = l$

Suppose that:
:$\forall n \in \N: y_n \le x_n \le z_n$

Then:
:$x_n \to l$ as $n \to \infty$
that is:
:$\ds \lim_{n \mathop \to \infty} x_n = l$

`proof`
From Negative of Absolute Value:
:$\size {x - l} < \epsilon \iff l - \epsilon < x < l + \epsilon$

Let $\epsilon > 0$.

We need to prove that:
:$\exists N: \forall n > N: \size {x_n - l} < \epsilon$

As $\ds \lim_{n \mathop \to \infty} y_n = l$ we know that:
:$\exists N_1: \forall n > N_1: \size {y_n - l} < \epsilon$

As $\ds \lim_{n \mathop \to \infty} z_n = l$ we know that:
:$\exists N_2: \forall n > N_2: \size {z_n - l} < \epsilon$

Let $N = \max \set {N_1, N_2}$.

Then if $n > N$, it follows that $n > N_1$ and $n > N_2$.

So:
:$\forall n > N: l - \epsilon < y_n < l + \epsilon$
:$\forall n > N: l - \epsilon < z_n < l + \epsilon$

But:
:$\forall n \in \N: y_n \le x_n \le z_n$

So:
:$\forall n > N: l - \epsilon < y_n \le x_n \le z_n < l + \epsilon$

and so:
:$\forall n > N: l - \epsilon < x_n < l + \epsilon$

So:
:$\forall n > N: \size {x_n - l} < \epsilon$

Hence the result.
{{qed}}

-/
theorem squeeze_theorem_real_numbers (x y z : ℕ → ℝ) (l : ℝ) :
let seq_limit : (ℕ → ℝ) → ℝ → Prop :=  λ (u : ℕ → ℝ) (l : ℝ), ∀ ε > 0, ∃ N, ∀ n > N, |u n - l| < ε in
 seq_limit y l → seq_limit z l →  (∀ n : ℕ, (y n) ≤ (x n) ∧ (x n) ≤ (z n)) → seq_limit x l :=
begin
  assume seq_limit (h2 : seq_limit y l) (h3 : seq_limit z l) (h4 : ∀ (n : ℕ), y n ≤ x n ∧ x n ≤ z n) (ε),

  --From Negative of Absolute Value: $\size {x - l} < \epsilon \iff l - \epsilon < x < l + \epsilon$
  have h5 : ∀ x, |x - l| < ε ↔ (((l - ε) < x) ∧ (x < (l + ε))),
  from by
  {
    intro x0,
    have h6 : |x0 - l| < ε ↔ ((x0 - l) < ε) ∧ ((l - x0) < ε),
    from abs_sub_lt_iff, rw h6,
    split,
    rintro ⟨ S_1, S_2 ⟩,
    split; linarith,
    rintro ⟨ S_3, S_4 ⟩,
    split; linarith,
    },

  --Let $\epsilon > 0$.
  assume (h7 : ε > 0),

  --As $\ds \lim_{n \mathop \to \infty} y_n = l$ we know that $\exists N_1: \forall n > N_1: \size {y_n - l} < \epsilon$
  cases h2 ε h7 with N1 h8,

  --As $\ds \lim_{n \mathop \to \infty} z_n = l$ we know that $\exists N_2: \forall n > N_2: \size {z_n - l} < \epsilon$
  cases h3 ε h7 with N2 h9,

  --Let $N = \max \set {N_1, N_2}$.
  let N := max N1 N2,
  use N,

  --Then if $n > N$, it follows that $n > N_1$ and $n > N_2$.
  have h10 : ∀ n > N, n > N1 ∧ n > N2 := by {
    assume n h,
    split,
    exact lt_of_le_of_lt (le_max_left N1 N2) h,
    exact lt_of_le_of_lt (le_max_right N1 N2) h,
  },

  --$\forall n > N: l - \epsilon < y_n < l + \epsilon$
  --$\forall n > N: l - \epsilon < z_n < l + \epsilon$
  --$\forall n \in \N: y_n \le x_n \le z_n$
  --So $\forall n > N: l - \epsilon < y_n \le x_n \le z_n < l + \epsilon$
  have h11 : ∀ n > N, (((l - ε) < (y n)) ∧ ((y n) ≤ (x n))) ∧ (((x n) ≤ (z n)) ∧ ((z n) < l+ε)),
  from by {
    intros n h12,
    split,
    {

      have h13 := (h8 n (h10 n h12).left), rw h5 (y n) at h13,
      split,
      exact h13.left,
      exact (h4 n).left,
    },
    {
      have h14 := (h9 n (h10 n h12).right),rw h5 (z n) at h14,
      split,
      exact (h4 n).right,
      exact h14.right,
    },

  },

  --$\forall n > N: l - \epsilon < x_n < l + \epsilon$
  have h15 : ∀ n > N, ((l - ε) < (x n)) ∧ ((x n) < (l+ε)),
  from by {
    intros n1 h16, cases (h11 n1 h16);
    split; linarith,
  },

  --So $\forall n > N: \size {x_n - l} < \epsilon$
  --Hence the result
  show  ∀ (n : ℕ), n > N → |x n - l| < ε,
  from by {
    intros n h17,
    cases h5 (x n) with h18 h19,
    apply h19, exact h15 n h17,
  },
end

/--`theorem`
Density of irrational orbit
The fractional parts of the integer multiples of an irrational number form a dense subset of the unit interval
`proof`
Let $\alpha$ be an irrational number. Then for distinct $i, j \in \mathbb{Z}$, we must have $\{i \alpha\} \neq\{j \alpha\}$. If this were not true, then
$$
i \alpha-\lfloor i \alpha\rfloor=\{i \alpha\}=\{j \alpha\}=j \alpha-\lfloor j \alpha\rfloor,
$$
which yields the false statement $\alpha=\frac{\lfloor i \alpha\rfloor-\lfloor j \alpha\rfloor}{i-j} \in \mathbb{Q}$. Hence,
$$
S:=\{\{i \alpha\} \mid i \in \mathbb{Z}\}
$$
is an infinite subset of $\left[0,1\right]$.

By the Bolzano-Weierstrass theorem, $S$ has a limit point in $[0, 1]$. One can thus find pairs of elements of $S$ that are arbitrarily close. Since (the absolute value of) the difference of any two elements of $S$ is also an element of $S$, it follows that $0$ is a limit point of $S$.

To show that $S$ is dense in $[0, 1]$, consider $y \in[0,1]$, and $\epsilon>0$. Then by selecting $x \in S$ such that $\{x\}<\epsilon$ (which exists as $0$ is a limit point), and $N$ such that $N \cdot\{x\} \leq y<(N+1) \cdot\{x\}$, we get: $|y-\{N x\}|<\epsilon$.

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


import category_theory.sites.sheafification

namespace category_theory

open category_theory

universes v u
variables {C : Type u} [category.{v} C] (J : grothendieck_topology C)

noncomputable theory

@[simps]
def presheaf_to_SheafOfTypes : (Cᵒᵖ ⥤ Type (max v u)) ⥤ SheafOfTypes J :=
{ obj := λ P,
  { val := J.sheafify P,
    cond := begin
      rw ← is_sheaf_iff_is_sheaf_of_type,
      exact grothendieck_topology.plus.is_sheaf_plus_plus J P,
    end },
  map := λ P Q η, ⟨J.sheafify_map η⟩,
  map_id' := λ P, SheafOfTypes.hom.ext _ _ $ J.sheafify_map_id _,
  map_comp' := λ P Q R η γ, SheafOfTypes.hom.ext _ _ $ J.sheafify_map_comp _ _ }

-- Sanity check
def presheaf_to_SheafOfTypes_iso : presheaf_to_SheafOfTypes J ≅
  presheaf_to_Sheaf J _ ⋙ (Sheaf_equiv_SheafOfTypes J).functor := eq_to_iso rfl

-- The adjunction for sheaves of types
def sheafification_adjunction_types :
  (presheaf_to_SheafOfTypes J) ⊣ SheafOfTypes_to_presheaf J :=
show presheaf_to_Sheaf J _ ⋙ (Sheaf_equiv_SheafOfTypes J).functor ⊣
  (Sheaf_equiv_SheafOfTypes J).inverse ⋙ Sheaf_to_presheaf J _,
from adjunction.comp _ _ (sheafification_adjunction _ _) $
  (Sheaf_equiv_SheafOfTypes J).to_adjunction

@[simp]
lemma sheafification_adjunction_types_hom_equiv_apply (X : Cᵒᵖ ⥤ Type (max v u))
  (Y : SheafOfTypes J) (e : (presheaf_to_SheafOfTypes _).obj X ⟶ Y) :
  (sheafification_adjunction_types J).hom_equiv _ _ e =
  J.to_sheafify _ ≫ e.val := rfl

@[simp]
lemma sheafification_adjunction_types_hom_equiv_symm_apply (X : Cᵒᵖ ⥤ Type (max v u))
  (Y : SheafOfTypes J) (e : X ⟶ (SheafOfTypes_to_presheaf _).obj Y) :
  (((sheafification_adjunction_types J).hom_equiv _ _).symm e).val =
  J.sheafify_lift e (by { rw is_sheaf_iff_is_sheaf_of_type, exact Y.2 }) := rfl

@[simp]
lemma sheafification_adjunction_types_unit_app (X : Cᵒᵖ ⥤ Type (max v u)) :
  (sheafification_adjunction_types J).unit.app X = J.to_sheafify X := rfl

@[simp]
lemma sheafification_adjunction_types_counit_app (Y : SheafOfTypes J) :
  ((sheafification_adjunction_types J).counit.app Y).val =
  J.sheafify_lift (𝟙 _) (by { rw is_sheaf_iff_is_sheaf_of_type, exact Y.2 }) := rfl

instance is_iso_sheafification_types_adjunction_counit_app (X : SheafOfTypes J) :
  is_iso ((sheafification_adjunction_types J).counit.app X) :=
begin
  apply is_iso_of_reflects_iso _ (Sheaf_equiv_SheafOfTypes J).inverse,
  change is_iso ((sheafification_adjunction J (Type (max v u))).counit.app
    ((Sheaf_equiv_SheafOfTypes J).inverse.obj X)),
  apply_instance
end

instance sheafification_types_reflective : is_iso (sheafification_adjunction_types J).counit :=
nat_iso.is_iso_of_is_iso_app _

end category_theory


import data.cpi.concretion data.cpi.transition.label

namespace cpi

variables {ℍ : Type} {ω : context}

/-- The right hand side of a transition, determined by a specific kind. -/
@[derive decidable_eq, nolint has_inhabited_instance]
inductive production (ℍ : Type) (ω : context) (Γ : context) : kind → Type
| species (A : species ℍ ω Γ) : production kind.species
| concretion {b y} (F : concretion ℍ ω Γ b y) : production kind.concretion

instance production.has_repr [has_repr ℍ] {Γ k} : has_repr (production ℍ ω Γ k) :=
  ⟨ λ x, by from production.rec_on x repr (λ _ _ x, repr x), ⟩

/-- Rename a production. This just wraps renaming for species and concretions. -/
def production.rename
  {Γ Δ} (ρ : name Γ → name Δ)
  : ∀ {k}, production ℍ ω Γ k → production ℍ ω Δ k
| ._ (production.species A) := production.species (species.rename ρ A)
| ._ (production.concretion A) := production.concretion (concretion.rename ρ A)

lemma production.rename_compose {Γ Δ η} (ρ : name Γ → name Δ) (σ : name Δ → name η)
  : ∀ {k} (E : production ℍ ω Γ k)
  , production.rename σ (production.rename ρ E)
  = production.rename (σ ∘ ρ) E
| ._ (production.species A) := congr_arg _ (species.rename_compose ρ σ A)
| ._ (production.concretion F) := congr_arg _ (concretion.rename_compose ρ σ F)

lemma production.rename_id
  {Γ} : ∀ {k} (E : production ℍ ω Γ k), production.rename id E = E
| ._ (production.species A) := congr_arg _ (species.rename_id A)
| ._ (production.concretion F) := congr_arg _ (concretion.rename_id F)

/-- Equivalence of productions. This just wraps equivalence of species and
    concretions. -/
inductive production.equiv [∀ Γ, setoid (species ℍ ω Γ)] [∀ Γ b y, setoid (concretion ℍ ω Γ b y)] {Γ} :
  ∀ {k : kind}, production ℍ ω Γ k → production ℍ ω Γ k → Prop
| species {A B : species ℍ ω Γ}
  : A ≈ B → production.equiv (production.species A) (production.species B)
| concretion {b y} {F G : concretion ℍ ω Γ b y}
  : F ≈ G → production.equiv (production.concretion F) (production.concretion G)

namespace production
  variables [∀ Γ, setoid (cpi.species ℍ ω Γ)] [∀ Γ b y, setoid (cpi.concretion ℍ ω Γ b y)]

  lemma equiv.refl {Γ} : ∀ {k : kind} (E : production ℍ ω Γ k), equiv E E
  | ._ (species A) := equiv.species (refl A)
  | ._ (concretion F) := equiv.concretion (refl F)

  lemma equiv.symm {Γ} : ∀ {k : kind} (E F : production ℍ ω Γ k), equiv E F → equiv F E
  | ._ ._ ._ (equiv.species eq) := equiv.species (symm eq)
  | ._ ._ ._ (equiv.concretion eq) := equiv.concretion (symm eq)

  lemma equiv.trans {Γ} : ∀ {k : kind} (E F G : production ℍ ω Γ k), equiv E F → equiv F G → equiv E G
  | ._ ._ ._ ._ (equiv.species ef) (equiv.species fg) := equiv.species (trans ef fg)
  | ._ ._ ._ ._ (equiv.concretion ef) (equiv.concretion fg) := equiv.concretion (trans ef fg)

  instance {Γ} {k} : is_equiv (production ℍ ω Γ k) equiv :=
    { refl := equiv.refl, symm := equiv.symm, trans := equiv.trans }
  instance {Γ} {k} : is_refl (production ℍ ω Γ k) equiv := ⟨ equiv.refl ⟩
  instance {Γ} {k} : setoid (production ℍ ω Γ k) :=
    ⟨ equiv, ⟨ equiv.refl, equiv.symm, equiv.trans ⟩ ⟩
  instance setoid.is_equiv {Γ} {k} : is_equiv (production ℍ ω Γ k) has_equiv.equiv :=
    production.is_equiv
end production

section equivalence
variables [∀ Γ, setoid (cpi.species ℍ ω Γ)] [∀ Γ b y, setoid (cpi.concretion ℍ ω Γ b y)]

lemma production.equiv.unwrap_s :
  ∀ {Γ} {A B : species ℍ ω Γ}, production.species A ≈ production.species B → A ≈ B
| Γ A B (production.equiv.species eq) := eq

lemma production.equiv.arity :
  ∀ {Γ} {a b x y} {F : concretion ℍ ω Γ a x} {G : concretion ℍ ω Γ b y}
  , production.concretion F ≈ production.concretion G
  → a = b ∧ x = y
| Γ a b x y F G (production.equiv.concretion eq) := ⟨ rfl, rfl ⟩

lemma production.equiv.unwrap_c :
  ∀ {Γ} {b y} {F : concretion ℍ ω Γ b y} {G : concretion ℍ ω Γ b y}
  , production.concretion F ≈ production.concretion G
  → F ≈ G
| Γ b y F G (production.equiv.concretion eq) := eq

end equivalence

end cpi

#lint-


import category_theory.abelian.diagram_lemmas.four
import for_mathlib.projectives
import for_mathlib.homological_complex
import for_mathlib.snake_lemma2

noncomputable theory

open category_theory
open category_theory.limits

universes v u

namespace short_exact_sequence

variables {C : Type u} [category.{v} C] [abelian C] [enough_projectives C]
variables {D : Type*} [category D] [abelian D]

-- move this
lemma exact_of_epi_comp_kernel.ι_comp_mono {C : Type u} [category.{v} C] [abelian C] {X Y Z W : C}
  (g : Y ⟶ Z) (h : Z ⟶ W) (f : X ⟶ kernel g) (i : kernel g ⟶ Y) (hf : epi f) (hh : mono h)
  (hi : i = kernel.ι g) : exact (f ≫ i) (g ≫ h) :=
begin
  suffices : exact i g,
  { letI := hf, letI := hh,
    exact exact_comp_mono (exact_epi_comp this) },
  rw [hi],
  exact exact_kernel_ι
end

-- move this
lemma biprod_factors (A B : C) [projective A] [projective B]
  (E X : C) (f : A ⊞ B ⟶ X) (e : E ⟶ X) [epi e] :
  ∃ f' : A ⊞ B ⟶ E, f' ≫ e = f :=
⟨biprod.desc
  (projective.factor_thru (biprod.inl ≫ f) e)
  (projective.factor_thru (biprod.inr ≫ f) e),
  by ext; simp only [projective.factor_thru_comp, biprod.inl_desc_assoc, biprod.inr_desc_assoc]⟩

variables (A B : short_exact_sequence C) (f : A ⟶ B)

def horseshoe_base : short_exact_sequence C :=
short_exact_sequence.mk_split (projective.over A.1) (projective.over A.3)

def horseshoe_base_π : horseshoe_base A ⟶ A :=
{ fst := projective.π _,
  snd := biprod.desc (projective.π _ ≫ A.f) (projective.factor_thru (projective.π _) A.g),
  trd := projective.π _,
  sq1' := by { dsimp [horseshoe_base], simp only [biprod.inl_desc], },
  sq2' :=
  begin
    dsimp [horseshoe_base], apply category_theory.limits.biprod.hom_ext',
    { simp only [zero_comp, exact.w_assoc, biprod.inl_desc_assoc, category.assoc,
        short_exact_sequence.f_comp_g, comp_zero, exact_inl_snd], },
    { simp only [projective.factor_thru_comp, biprod.inr_snd_assoc, biprod.inr_desc_assoc], }
  end }

instance epi_horseshoe_base_π_1 : epi (horseshoe_base_π A).1 :=
show epi (projective.π _), by apply_instance

instance epi_horseshoe_base_π_3 : epi (horseshoe_base_π A).3 :=
show epi (projective.π _), by apply_instance

local attribute [instance] limits.has_zero_object.has_zero

instance epi_horseshoe_base_π_2 : epi (horseshoe_base_π A).2 :=
begin
  let φ := horseshoe_base_π A,
  have h : φ.3 ≫ (0 : A.3 ⟶ 0) = (0 : _ ⟶ 0) ≫ (0 : 0 ⟶ 0) := by simp,
  refine category_theory.abelian.epi_of_epi_of_epi_of_mono φ.sq1' φ.sq2' h _ _ _ _ _ _;
  try { rw ← epi_iff_exact_zero_right }; try { apply_instance },
  exact A.exact',
end

variables {A B}

def horseshoe_ker [epi f.1] : short_exact_sequence C :=
(snake_input.mk_of_short_exact_sequence_hom _ _ _ f).kernel_sequence _
begin
  dsimp [snake_input.mk_of_short_exact_sequence_hom, snake_diagram.mk_of_short_exact_sequence_hom],
  rw snake_diagram.mk_functor_map_f1,
  exact A.mono',
end
$ is_zero_of_iso_of_zero (is_zero_zero _) (limits.cokernel.of_epi _).symm

@[simp] lemma horseshoe_ker_fst [epi f.1] : (horseshoe_ker f).1 = kernel f.1 := rfl

@[simp] lemma horseshoe_ker_snd [epi f.1] : (horseshoe_ker f).2 = kernel f.2 := rfl

@[simp] lemma horseshoe_ker_trd [epi f.1] : (horseshoe_ker f).3 = kernel f.3 := rfl

def horseshoe_ker_ι [epi f.1] : horseshoe_ker f ⟶ A :=
{ fst := kernel.ι _,
  snd := kernel.ι _,
  trd := kernel.ι _,
  sq1' :=
  begin
    dsimp [horseshoe_ker, snake_input.kernel_sequence,
      snake_input.mk_of_short_exact_sequence_hom, snake_diagram.mk_of_short_exact_sequence_hom],
    delta kernel.map,
    rw [snake_diagram.mk_functor_map_f0, kernel.lift_ι],
  end,
  sq2' :=
  begin
    dsimp [horseshoe_ker, snake_input.kernel_sequence,
      snake_input.mk_of_short_exact_sequence_hom, snake_diagram.mk_of_short_exact_sequence_hom],
    delta kernel.map,
    rw [snake_diagram.mk_functor_map_g0, kernel.lift_ι],
  end }
.

lemma horseshoe_ker_ι_fst [epi f.1] : (horseshoe_ker_ι f).1 = kernel.ι f.1 := rfl

lemma horseshoe_ker_ι_snd [epi f.1] : (horseshoe_ker_ι f).2 = kernel.ι f.2 := rfl

lemma horseshoe_ker_ι_trd [epi f.1] : (horseshoe_ker_ι f).3 = kernel.ι f.3 := rfl

variables (A)

lemma horseshoe_ker_ι_comp_base_π :
  (horseshoe_ker_ι (horseshoe_base_π A)) ≫ horseshoe_base_π A = 0 :=
begin
  dsimp [horseshoe_ker_ι, horseshoe_base_π],
  ext1; show kernel.ι _ ≫ _ = 0; exact exact.w exact_kernel_ι,
end

noncomputable
def horseshoe_step (A : short_exact_sequence C) :
  ℕ → Σ (X Y Z : short_exact_sequence C) (ι : X ⟶ Y), Y ⟶ Z
| 0     := ⟨horseshoe_ker (horseshoe_base_π A), _, _, horseshoe_ker_ι _, horseshoe_base_π _⟩
| (n+1) :=
⟨horseshoe_ker (horseshoe_base_π (horseshoe_step n).1), _, _, horseshoe_ker_ι _, horseshoe_base_π _⟩

@[reassoc] lemma horseshoe_step_comp_eq_zero :
  ∀ n, (horseshoe_step A n).2.2.2.1 ≫ (horseshoe_step A n).2.2.2.2 = 0
| 0     := horseshoe_ker_ι_comp_base_π _
| (n+1) := horseshoe_ker_ι_comp_base_π _

lemma step_fst_mono (n : ℕ) : mono (horseshoe_step A n).2.2.2.1.1 :=
begin
  cases n,
  { dsimp [horseshoe_step, horseshoe_ker_ι],
    apply_instance },
  { dsimp [horseshoe_step],
    cases n, --Why do I have to do this again?!
    { rw [horseshoe_ker_ι_fst],
      apply_instance },
    { rw [horseshoe_ker_ι_fst],
      apply_instance }  }
end

lemma step_snd_mono (n : ℕ) : mono (horseshoe_step A n).2.2.2.1.2 :=
begin
  cases n,
  { dsimp [horseshoe_step, horseshoe_ker_ι],
    apply_instance },
  { dsimp [horseshoe_step],
    cases n, --Why do I have to do this again?!
    { rw [horseshoe_ker_ι_snd],
      apply_instance },
    { rw [horseshoe_ker_ι_snd],
      apply_instance }  }
end

lemma step_trd_mono (n : ℕ) : mono (horseshoe_step A n).2.2.2.1.3 :=
begin
  cases n,
  { dsimp [horseshoe_step, horseshoe_ker_ι],
    apply_instance },
  { dsimp [horseshoe_step],
    cases n, --Why do I have to do this again?!
    { rw [horseshoe_ker_ι_trd],
      apply_instance },
    { rw [horseshoe_ker_ι_trd],
      apply_instance }  }
end

def horseshoe_obj (n : ℕ) := (horseshoe_step A n).2.1

def horseshoe_d (n : ℕ) : horseshoe_obj A (n+1) ⟶ horseshoe_obj A n :=
(horseshoe_step A (n+1)).2.2.2.2 ≫ eq_to_hom (by { dsimp [horseshoe_step], refl })
  ≫ (horseshoe_step A n).2.2.2.1

lemma horseshoe_d_d (n : ℕ) : horseshoe_d A (n+1) ≫ horseshoe_d A n = 0 :=
begin
  dsimp [horseshoe_d, horseshoe_ker_ι],
  simp only [category.id_comp, category.assoc, comp_zero, zero_comp,
    horseshoe_step_comp_eq_zero_assoc],
end

def horseshoe (A : short_exact_sequence C) : chain_complex (short_exact_sequence C) ℕ :=
chain_complex.of (horseshoe_obj A) (horseshoe_d A) (horseshoe_d_d A)

variables (A)

def horseshoe_π : (horseshoe A).X 0 ⟶ A := horseshoe_base_π _

lemma horseshoe_d_π : (horseshoe A).d 1 0 ≫ horseshoe_π A = 0 :=
begin
  dsimp [horseshoe],
  erw [chain_complex.of_d],
  dsimp [horseshoe_d, horseshoe_π, horseshoe_step],
  simp only [category.id_comp, category.assoc, comp_zero, zero_comp,
    horseshoe_step_comp_eq_zero_assoc, horseshoe_ker_ι_comp_base_π],
end

def horseshoe_to_single₁ :=
(chain_complex.to_single₀_equiv ((homological_complex.Fst C).obj (horseshoe A)) A.1).symm
⟨(short_exact_sequence.Fst C).map (horseshoe_π A),
begin
  have := horseshoe_d_π A, apply_fun (λ f, (short_exact_sequence.Fst C).map f) at this,
  rwa [functor.map_comp, functor.map_zero] at this,
end⟩

def horseshoe_to_single₂ :=
(chain_complex.to_single₀_equiv ((homological_complex.Snd C).obj (horseshoe A)) A.2).symm
⟨(short_exact_sequence.Snd C).map (horseshoe_π A),
begin
  have := horseshoe_d_π A, apply_fun (λ f, (short_exact_sequence.Snd C).map f) at this,
  rwa [functor.map_comp, functor.map_zero] at this,
end⟩

def horseshoe_to_single₃ :=
(chain_complex.to_single₀_equiv ((homological_complex.Trd C).obj (horseshoe A)) A.3).symm
⟨(short_exact_sequence.Trd C).map (horseshoe_π A),
begin
  have := horseshoe_d_π A, apply_fun (λ f, (short_exact_sequence.Trd C).map f) at this,
  rwa [functor.map_comp, functor.map_zero] at this,
end⟩

lemma horseshoe_exact₁ (A : short_exact_sequence C) (n : ℕ) :
  exact (((homological_complex.Fst C).obj (horseshoe A)).d (n + 2) (n + 1))
    (((homological_complex.Fst C).obj (horseshoe A)).d (n + 1) n) :=
begin
  dsimp [horseshoe_to_single₁],
  erw [chain_complex.of_d, chain_complex.of_d],
  dsimp [horseshoe_d, horseshoe_step],

  set f := horseshoe_base_π (horseshoe_step A n).1,
  set g := (horseshoe_step A n).2.2.2.1,

  cases n;
  convert exact_of_epi_comp_kernel.ι_comp_mono f.1 _ (horseshoe_base_π (horseshoe_ker f)).1 _
    infer_instance _ _ using 1,
  { simp [step_fst_mono] },
  { simpa },
  { simp [step_fst_mono] },
  { simpa }
end

lemma horseshoe_exact₂ (A : short_exact_sequence C) (n : ℕ) :
  exact (((homological_complex.Snd C).obj (horseshoe A)).d (n + 2) (n + 1))
    (((homological_complex.Snd C).obj (horseshoe A)).d (n + 1) n) :=
begin
  dsimp [horseshoe_to_single₂],
  erw [chain_complex.of_d, chain_complex.of_d],
  dsimp [horseshoe_d, horseshoe_step],

  set f := horseshoe_base_π (horseshoe_step A n).1,
  set g := (horseshoe_step A n).2.2.2.1,

  cases n;
  convert exact_of_epi_comp_kernel.ι_comp_mono f.2 _ (horseshoe_base_π (horseshoe_ker f)).2 _
    infer_instance _ _ using 1,
  { simp [step_snd_mono] },
  { simpa },
  { simp [step_snd_mono] },
  { simpa }
end

lemma horseshoe_exact₃ (A : short_exact_sequence C) (n : ℕ) :
  exact (((homological_complex.Trd C).obj (horseshoe A)).d (n + 2) (n + 1))
    (((homological_complex.Trd C).obj (horseshoe A)).d (n + 1) n) :=
begin
  dsimp [horseshoe_to_single₃],
  erw [chain_complex.of_d, chain_complex.of_d],
  dsimp [horseshoe_d, horseshoe_step],

  set f := horseshoe_base_π (horseshoe_step A n).1,
  set g := (horseshoe_step A n).2.2.2.1,

  cases n;
  convert exact_of_epi_comp_kernel.ι_comp_mono f.3 _ (horseshoe_base_π (horseshoe_ker f)).3 _
    infer_instance _ _ using 1,
  { simp [step_trd_mono] },
  { simpa },
  { simp [step_trd_mono] },
  { simpa }
end

lemma horseshoe_is_projective_resolution₁ (A : short_exact_sequence C) :
  chain_complex.is_projective_resolution
    ((homological_complex.Fst C).obj (horseshoe A)) A.1 (horseshoe_to_single₁ A) :=
{ projective := by rintro (_|n); { show projective (projective.over _), apply_instance },
  exact₀ :=
  begin
    dsimp [horseshoe_to_single₁, chain_complex.to_single₀_equiv, horseshoe_π],
    erw [chain_complex.of_d],
    dsimp [horseshoe_d, horseshoe_step],
    rw [category.id_comp, ← short_exact_sequence.comp_fst],
    refine abelian.pseudoelement.exact_of_pseudo_exact _ _ ⟨λ a , _, λ a ha, _⟩,
    { rw [← abelian.pseudoelement.comp_apply, ← short_exact_sequence.comp_fst, category.assoc,
        horseshoe_ker_ι_comp_base_π, comp_zero, short_exact_sequence.hom_zero_fst,
        abelian.pseudoelement.zero_apply] },
    { obtain ⟨b, hb⟩ := is_snake_input.exists_of_exact exact_kernel_ι _ ha,
      obtain ⟨c, hc⟩ := abelian.pseudoelement.pseudo_surjective_of_epi
        (horseshoe_base_π (horseshoe_ker _)).1 b,
      refine ⟨c, _⟩,
      rw [short_exact_sequence.comp_fst, abelian.pseudoelement.comp_apply, hc, ← hb],
      refl }
  end,
  exact := λ n, horseshoe_exact₁ A n,
  epi := show epi (projective.π _), from infer_instance }

lemma horseshoe_is_projective_resolution₂ (A : short_exact_sequence C) :
  chain_complex.is_projective_resolution
    ((homological_complex.Snd C).obj (horseshoe A)) A.2 (horseshoe_to_single₂ A) :=
{ projective := by rintro (_|n); { show projective (projective.over _ ⊞ projective.over _),
    apply_instance },
  exact₀ :=
  begin
    dsimp [horseshoe_to_single₂, chain_complex.to_single₀_equiv, horseshoe_π],
    erw [chain_complex.of_d],
    dsimp [horseshoe_d, horseshoe_step],
    rw [category.id_comp, ← short_exact_sequence.comp_snd],
    refine abelian.pseudoelement.exact_of_pseudo_exact _ _ ⟨λ a , _, λ a ha, _⟩,
    { rw [← abelian.pseudoelement.comp_apply, ← short_exact_sequence.comp_snd, category.assoc,
        horseshoe_ker_ι_comp_base_π, comp_zero, short_exact_sequence.hom_zero_snd,
        abelian.pseudoelement.zero_apply] },
    { obtain ⟨b, hb⟩ := is_snake_input.exists_of_exact exact_kernel_ι _ ha,
      obtain ⟨c, hc⟩ := abelian.pseudoelement.pseudo_surjective_of_epi
        (horseshoe_base_π (horseshoe_ker _)).2 b,
      refine ⟨c, _⟩,
      rw [short_exact_sequence.comp_snd, abelian.pseudoelement.comp_apply, hc, ← hb],
      refl }
  end,
  exact := λ n, horseshoe_exact₂ A n,
  epi := show epi (horseshoe_base_π _).2, from infer_instance }

lemma horseshoe_is_projective_resolution₃ (A : short_exact_sequence C) :
  chain_complex.is_projective_resolution
    ((homological_complex.Trd C).obj (horseshoe A)) A.3 (horseshoe_to_single₃ A) :=
{ projective := by rintro (_|n); { show projective (projective.over _), apply_instance },
  exact₀ :=
  begin
    dsimp [horseshoe_to_single₃, chain_complex.to_single₀_equiv, horseshoe_π],
    erw [chain_complex.of_d],
    dsimp [horseshoe_d, horseshoe_step],
    rw [category.id_comp, ← short_exact_sequence.comp_trd],
    refine abelian.pseudoelement.exact_of_pseudo_exact _ _ ⟨λ a , _, λ a ha, _⟩,
    { rw [← abelian.pseudoelement.comp_apply, ← short_exact_sequence.comp_trd, category.assoc,
        horseshoe_ker_ι_comp_base_π, comp_zero, short_exact_sequence.hom_zero_trd,
        abelian.pseudoelement.zero_apply] },
    { obtain ⟨b, hb⟩ := is_snake_input.exists_of_exact exact_kernel_ι _ ha,
      obtain ⟨c, hc⟩ := abelian.pseudoelement.pseudo_surjective_of_epi
        (horseshoe_base_π (horseshoe_ker _)).3 b,
      refine ⟨c, _⟩,
      rw [short_exact_sequence.comp_trd, abelian.pseudoelement.comp_apply, hc, ← hb],
      refl }
  end,
  exact := λ n, horseshoe_exact₃ A n,
  epi := show epi (projective.π _), from infer_instance }
.

lemma horseshoe_split (A : short_exact_sequence C) (n : ℕ) :
  ((horseshoe A).X n).split :=
begin
  cases n;
  exact ⟨biprod.fst, biprod.inr, biprod.inl_fst, biprod.inr_snd, biprod.inr_fst, biprod.total⟩
end

lemma horseshoe_f_comp_to_single₂_f (A : short_exact_sequence C) (i : ℕ) :
  ((horseshoe A).X i).f ≫ (horseshoe_to_single₂ A).f i =
  (horseshoe_to_single₁ A).f i ≫ ((chain_complex.single₀ C).map A.f).f i :=
begin
  cases i,
  { dsimp [horseshoe_to_single₂, horseshoe_to_single₁, horseshoe, horseshoe_obj, horseshoe_step,
      horseshoe_base, horseshoe_π, horseshoe_base_π, chain_complex.to_single₀_equiv],
    simp },
  { dsimp [horseshoe_to_single₂, horseshoe_to_single₁, horseshoe, horseshoe_obj, horseshoe_step,
      horseshoe_base, horseshoe_π, horseshoe_base_π, chain_complex.to_single₀_equiv],
    simp }
end

lemma horseshoe_g_comp_to_single₃_f (A : short_exact_sequence C) (i : ℕ) :
  ((horseshoe A).X i).g ≫ (horseshoe_to_single₃ A).f i =
  (horseshoe_to_single₂ A).f i ≫ ((chain_complex.single₀ C).map A.g).f i :=
begin
  cases i,
  { dsimp [horseshoe_to_single₂, horseshoe_to_single₃, chain_complex.to_single₀_equiv, horseshoe,
      horseshoe_obj, horseshoe_π, horseshoe_step, horseshoe_base, horseshoe_base_π],
    ext,
    { simp },
    { simp  } },
  { dsimp [horseshoe_to_single₂, horseshoe_to_single₃, horseshoe, horseshoe_obj, horseshoe_step,
      horseshoe_base, horseshoe_π, horseshoe_base_π, chain_complex.to_single₀_equiv],
    simp }
end

end short_exact_sequence


def bad_theorem : Prop :=
  ∀ (α : Type) (r : α → α → Prop), symmetric r → transitive r → reflexive r

def fail (a b : unit): Prop := false
theorem fail_symm : symmetric fail := λ x y h, h
theorem fail_trans : transitive fail := λ x y z h1 _, false.rec (fail x z) h1
-- theorem not_refl_fail : ¬ reflexive fail := λ a, a ()

theorem correct_version : ¬ bad_theorem := λ a, a unit fail fail_symm fail_trans ()


import for_mathlib.algebra.homology.trunc_le

noncomputable theory

open category_theory category_theory.limits category_theory.category
open_locale zero_object

lemma int.three_cases (a n : ℤ) :
  (n<a ∨ a<n) ∨ n = a :=
begin
  by_cases h₁ : n < a,
  { exact or.inl (or.inl h₁), },
  { cases (not_lt.1 h₁).lt_or_eq; tauto, },
end

lemma category_theory.nat_iso.map_eq_iff {C D : Type*} [category C] [category D] {F G : C ⥤ D}
  (e : F ≅ G) {X Y : C} (f₁ f₂ : X ⟶ Y) : F.map f₁ = F.map f₂ ↔ G.map f₁ = G.map f₂ :=
begin
  suffices : ∀ ⦃F G : C ⥤ D⦄ (e : F ≅ G) ⦃X Y : C⦄ (f₁ f₂ : X ⟶ Y)
    (h : F.map f₁ = F.map f₂), G.map f₁ = G.map f₂,
  { exact ⟨this e f₁ f₂, this e.symm f₁ f₂⟩, },
  intros F G e X Y f₁ f₂ eq,
  have : ∀ (f : X ⟶ Y), G.map f = e.inv.app X ≫ F.map f ≫ e.hom.app Y := λ f, by tidy,
  simp only [this, eq],
end

open category_theory category_theory.limits category_theory.category

variables {C : Type*} [category C] [abelian C]

namespace homological_complex

lemma single_obj_X_self_naturality {ι : Type*} (c : complex_shape ι) (n : ι)
  [decidable_eq ι] {A B : C} (φ : A ⟶ B) :
    ((single C c n).map φ).f n ≫ (single_obj_X_self C c n B).hom =
      (single_obj_X_self C c n A).hom ≫ φ :=
begin
  dsimp [single],
  simpa only [dif_pos rfl, assoc, eq_to_hom_trans, eq_to_hom_refl, comp_id],
end

variable (C)

def single_homology_functor_iso {ι : Type*} (c : complex_shape ι) (n : ι)
  [decidable_eq ι] :
  homological_complex.single C c n ⋙ homology_functor C c n ≅ 𝟭 C :=
nat_iso.of_components
  (λ A, homology_single_self C c A n)
  (λ A B φ, begin
    let h₁ := short_complex.homology_data.of_zeros (((single C c n).obj A).sc' n) rfl rfl,
    let h₂ := short_complex.homology_data.of_zeros (((single C c n).obj B).sc' n) rfl rfl,
    let h : short_complex.homology_map_data ((short_complex_functor _ _ n).map
      ((single C c n).map φ)) h₁ h₂ := short_complex.homology_map_data.of_zeros _ rfl rfl rfl rfl,
    have eq := short_complex.homology_map_data.homology_map_comm h,
    dsimp only [homology_single_self, iso.trans],
    simp only [assoc],
    erw [← single_obj_X_self_naturality c n φ, reassoc_of eq],
    refl,
  end)

end homological_complex

variable {C}

namespace cochain_complex

instance is_strictly_le_trunc_ge (K : cochain_complex C ℤ) (p q : ℤ) [K.is_strictly_le q] :
  (K.trunc_ge p).is_strictly_le q :=
⟨λ i hi, begin
  rw [is_zero.iff_id_eq_zero, ← cancel_epi (trunc_ge.π_f K p i)],
  apply (is_strictly_le.is_zero K q i hi).eq_of_src,
end⟩

instance is_strictly_ge_trunc_le (K : cochain_complex C ℤ) (p q : ℤ) [K.is_strictly_ge p] :
  (K.trunc_le q).is_strictly_ge p :=
⟨λ i hi, begin
  rw [is_zero.iff_id_eq_zero, ← cancel_mono (trunc_le.ι_f K q i)],
  apply (is_strictly_ge.is_zero K p i hi).eq_of_tgt,
end⟩

lemma is_le_iff_of_is_le_next (K : cochain_complex C ℤ) {a b : ℤ} (h : a+1=b)
  [K.is_le b] : K.is_le a ↔ is_zero (K.homology b) :=
begin
  split,
  { introI,
    exact is_le.is_zero K a b (by linarith), },
  { refine λ hb, ⟨λ n hn, _⟩,
    unfreezingI { subst h, },
    rw ← int.add_one_le_iff at hn,
    cases hn.lt_or_eq,
    { exact is_le.is_zero K (a+1) n h, },
    { subst h,
      exact hb, }, },
end

lemma is_ge_iff_of_is_ge_prev (K : cochain_complex C ℤ) {a b : ℤ} (h : a+1=b)
  [K.is_ge a] : K.is_ge b ↔ is_zero (K.homology a) :=
begin
  split,
  { introI,
    exact is_ge.is_zero K b a (by linarith), },
  { refine λ ha, ⟨λ n hn, _⟩,
    subst h,
    rw [← int.add_one_le_iff, add_le_add_iff_right] at hn,
    cases hn.lt_or_eq,
    { exact is_ge.is_zero K a n h, },
    { unfreezingI { subst h, },
      exact ha, }, },
end

lemma shift_is_strict_le (K : cochain_complex C ℤ) (a b c : ℤ) (h : a = c + b)
  [K.is_strictly_le a] : (K⟦b⟧).is_strictly_le c :=
⟨λ n hn, begin
  change is_zero (K.X (n+b)),
  exact is_strictly_le.is_zero K a (n+b) (by linarith),
end⟩

lemma shift_is_strict_ge (K : cochain_complex C ℤ) (a b c : ℤ) (h : a = c + b)
  [K.is_strictly_ge a] : (K⟦b⟧).is_strictly_ge c :=
⟨λ n hn, begin
  change is_zero (K.X (n+b)),
  exact is_strictly_ge.is_zero K a (n+b) (by linarith),
end⟩

end cochain_complex

namespace derived_category

lemma orthogonality {K L : derived_category C} (φ : K ⟶ L) (p q : ℤ) (hpq : p < q)
  [K.is_le p] [L.is_ge q] : φ = 0 :=
begin
  obtain ⟨K', hK', ⟨eK⟩⟩ := exists_iso_Q_obj_of_le K p,
  obtain ⟨L', hL', ⟨eL⟩⟩ := exists_iso_Q_obj_of_ge L q,
  haveI := hK',
  have hφ : ∃ (φ' : Q.obj K' ⟶ Q.obj L'), φ = eK.hom ≫ φ' ≫ eL.inv :=
    ⟨eK.inv ≫ φ ≫ eL.hom, by simp only [assoc, iso.hom_inv_id, comp_id, iso.hom_inv_id_assoc]⟩,
  obtain ⟨φ, rfl⟩ := hφ,
  obtain ⟨M, hM, s, f, hs, eq⟩ := right_factorisation_of_is_le φ p,
  haveI := hM,
  have hf : f = 0,
  { ext n,
    by_cases p < n,
    { apply (cochain_complex.is_strictly_le.is_zero M p n h).eq_of_src, },
    { apply (cochain_complex.is_strictly_ge.is_zero L' q n (by linarith)).eq_of_tgt, }, },
  simp only [eq, preadditive.is_iso.comp_left_eq_zero, preadditive.is_iso.comp_right_eq_zero, hf,
    functor.map_zero],
end

lemma right_factorisation_of_is_strictly_le_of_is_strictly_ge {K L : cochain_complex C ℤ}
  (φ : Q.obj K ⟶ Q.obj L) (p q : ℤ) [K.is_strictly_le p]
  [K.is_strictly_ge q] [L.is_strictly_ge q] :
  ∃ (K' : cochain_complex C ℤ) (K'_le : K'.is_strictly_le p)
    (K'_ge : K'.is_strictly_ge q) (s : K' ⟶ K) (f : K' ⟶ L) (hs : quasi_iso s),
    φ = (by { haveI := hs, exact inv (Q.map s), }) ≫ Q.map f :=
begin
  obtain ⟨K', hK', s, f, hs, eq⟩ := right_factorisation_of_is_strictly_ge φ q,
  haveI := hK',
  haveI : K'.is_le p,
  { rw cochain_complex.is_le_iff_of_quasi_iso s,
    apply_instance, },
  exact ⟨(cochain_complex.trunc_le K' p), infer_instance, infer_instance,
    cochain_complex.trunc_le.ι K' p ≫ s, cochain_complex.trunc_le.ι K' p ≫ f, infer_instance,
    by simp only [eq, functor.map_comp, assoc, is_iso.eq_inv_comp, is_iso.hom_inv_id_assoc]⟩,
end

lemma left_factorisation_of_is_strictly_le_of_is_strictly_ge {K L : cochain_complex C ℤ}
  (φ : Q.obj K ⟶ Q.obj L) (p q : ℤ) [K.is_strictly_le p] [L.is_strictly_le p]
  [L.is_strictly_ge q] :
  ∃ (L' : cochain_complex C ℤ) (L'_le : L'.is_strictly_le p)
    (L'_ge : L'.is_strictly_ge q) (f : K ⟶ L') (s : L ⟶ L') (hs : quasi_iso s),
    φ = Q.map f ≫ (by { haveI := hs, exact inv (Q.map s), }) :=
begin
  obtain ⟨L', hL', f, s, hs, eq⟩ := left_factorisation_of_is_strictly_le φ p,
  haveI := hL',
  haveI : L'.is_ge q,
  { rw ← cochain_complex.is_ge_iff_of_quasi_iso s,
    apply_instance, },
  exact ⟨(cochain_complex.trunc_ge L' q), infer_instance, infer_instance,
    f ≫ cochain_complex.trunc_ge.π L' q, s ≫ cochain_complex.trunc_ge.π L' q, infer_instance,
    by simp only [eq, functor.map_comp, is_iso.inv_comp, assoc, is_iso.hom_inv_id_assoc]⟩,
end

lemma shift_is_le (K : derived_category C) (a b c : ℤ) (h : a = c + b)
  [K.is_le a] : (K⟦b⟧).is_le c :=
⟨λ n hn, is_zero.of_iso (is_le.is_zero _ a _ (by linarith))
  ((derived_category.shift_homology_functor_iso C b n _ rfl).app K)⟩

lemma shift_is_ge (K : derived_category C) (a b c : ℤ) (h : a = c + b)
  [K.is_ge a] : (K⟦b⟧).is_ge c :=
⟨λ n hn, is_zero.of_iso (is_ge.is_zero _ a _ (by linarith))
  ((derived_category.shift_homology_functor_iso C b n _ rfl).app K)⟩

end derived_category

namespace cochain_complex

instance single_is_strictly_le (A : C) (n : ℤ) :
  is_strictly_le ((homological_complex.single C (complex_shape.up ℤ) n).obj A) n :=
⟨λ i hi, begin
  dsimp,
  rw if_neg (show ¬ i = n, by linarith),
  exact is_zero_zero C,
end⟩

instance single_is_strictly_ge (A : C) (n : ℤ) :
  is_strictly_ge ((homological_complex.single C (complex_shape.up ℤ) n).obj A) n :=
⟨λ i hi, begin
  dsimp,
  rw if_neg (show ¬ i = n, by linarith),
  exact is_zero_zero C,
end⟩

lemma from_single_ext {K L : cochain_complex C ℤ} (f₁ f₂ : K ⟶ L) (p : ℤ)
  [K.is_strictly_le p] [K.is_strictly_ge p] (h : f₁.f p = f₂.f p) : f₁ = f₂ :=
begin
  ext i,
  by_cases hi : i < p,
  { apply (is_strictly_ge.is_zero K p i hi).eq_of_src, },
  { cases (not_lt.1 hi).lt_or_eq with hi' hi',
    { apply (is_strictly_le.is_zero K p i hi').eq_of_src, },
    { subst hi',
      exact h, }, },
end

lemma to_single_ext {K L : cochain_complex C ℤ} (f₁ f₂ : K ⟶ L) (p : ℤ)
  [L.is_strictly_le p] [L.is_strictly_ge p] (h : f₁.f p = f₂.f p) : f₁ = f₂ :=
begin
  ext i,
  by_cases hi : i < p,
  { apply (is_strictly_ge.is_zero L p i hi).eq_of_tgt, },
  { cases (not_lt.1 hi).lt_or_eq with hi' hi',
    { apply (is_strictly_le.is_zero L p i hi').eq_of_tgt, },
    { subst hi',
      exact h, }, },
end

def desc_single (K L : cochain_complex C ℤ) {p : ℤ} (f : K.X p ⟶ L.X p) (q : ℤ)
  [K.is_strictly_le p] [K.is_strictly_ge p]
  (hpq : p+1=q) (hf : f ≫ L.d p q = 0) : K ⟶ L :=
{ f := λ i, begin
    by_cases i = p,
    { unfreezingI { subst h, }, exact f, },
    { exact 0, },
  end,
  comm' := λ i j (hij : i+1 = j), begin
    by_cases i = p,
    { have hj : j = q := by linarith,
      unfreezingI { substs h hj, },
      dsimp,
      simp only [if_pos rfl, hf,
        (is_strictly_le.is_zero K i j (by linarith)).eq_of_tgt (K.d i j) 0, zero_comp], },
    { apply is_zero.eq_of_src,
      by_cases hi : i < p,
      { exact is_strictly_ge.is_zero K p i hi, },
      { apply is_strictly_le.is_zero K p i,
        cases (not_lt.1 hi).lt_or_eq with hi' hi',
        { exact hi', },
        { exfalso, exact h hi'.symm, }, }, },
  end, }

@[simp]
lemma desc_single_f (K L : cochain_complex C ℤ) {p : ℤ} (f : K.X p ⟶ L.X p) (q : ℤ)
  [K.is_strictly_le p] [K.is_strictly_ge p]
  (hpq : p+1=q) (hf : f ≫ L.d p q = 0) :
  (desc_single K L f q hpq hf).f p = f :=
begin
  dsimp [desc_single],
  rw if_pos rfl,
end

def lift_single (K L : cochain_complex C ℤ) {q : ℤ} (f : K.X q ⟶ L.X q) (p : ℤ)
  [L.is_strictly_le q] [L.is_strictly_ge q]
  (hpq : p+1=q) (hf : K.d p q ≫ f = 0) : K ⟶ L :=
{ f := λ i, begin
    by_cases i = q,
    { unfreezingI { subst h, }, exact f, },
    { exact 0, },
  end,
  comm' := λ i j (hij : i+1 = j), begin
    by_cases j = q,
    { have hi : i = p := by linarith,
      unfreezingI { substs h hi, },
      dsimp,
      simp only [if_pos rfl, hf,
        (is_strictly_ge.is_zero L j i (by linarith)).eq_of_src (L.d i j) 0, comp_zero], },
    { apply is_zero.eq_of_tgt,
      by_cases hj : j < q,
      { exact is_strictly_ge.is_zero L q j hj, },
      { apply is_strictly_le.is_zero L q j,
        cases (not_lt.1 hj).lt_or_eq with hj' hj',
        { exact hj', },
        { exfalso, exact h hj'.symm, }, }, },
  end, }

@[simp]
lemma lift_single_f (K L : cochain_complex C ℤ) {q : ℤ} (f : K.X q ⟶ L.X q) (p : ℤ)
  [L.is_strictly_le q] [L.is_strictly_ge q]
  (hpq : p+1=q) (hf : K.d p q ≫ f = 0) :
  (lift_single K L f p hpq hf).f q = f :=
by { dsimp [lift_single], rw if_pos rfl, }

def iso_single (K : cochain_complex C ℤ) (n : ℤ)
  [K.is_strictly_le n] [K.is_strictly_ge n] :
  K ≅ (homological_complex.single C _ n).obj (K.X n) :=
{ hom := desc_single K _ (homological_complex.single_obj_X_self C _ n (K.X n)).inv (n+1) rfl (by simp),
  inv := lift_single _ K (homological_complex.single_obj_X_self C _ n (K.X n)).hom (n-1) (by linarith) (by simp),
  hom_inv_id' := from_single_ext _ _ n
    (by simp only [homological_complex.id_f, homological_complex.comp_f,
      desc_single_f, lift_single_f, iso.inv_hom_id]),
  inv_hom_id' := from_single_ext _ _ n
    (by simp only [homological_complex.id_f, homological_complex.comp_f,
      desc_single_f, lift_single_f, iso.hom_inv_id]), }

lemma quasi_iso_single_map_iff_is_iso {A B : C} (φ : A ⟶ B) (n : ℤ) :
  quasi_iso ((homological_complex.single C (complex_shape.up ℤ) n).map φ) ↔ is_iso φ :=
begin
  split,
  { introI,
    exact (is_iso_map_iff_of_nat_iso (homological_complex.single_homology_functor_iso C
      (complex_shape.up ℤ) n) φ).1 (quasi_iso.is_iso _), },
  { introI,
    apply_instance, },
end

lemma is_iso_iff_quasi_iso_of_single {K L : cochain_complex C ℤ} (φ : K ⟶ L) (n : ℤ)
  [K.is_strictly_le n] [K.is_strictly_ge n] [L.is_strictly_le n] [L.is_strictly_ge n] :
  is_iso φ ↔ quasi_iso φ :=
begin
  split,
  { introI,
    apply_instance, },
  { introI,
    let e₁ := K.iso_single n,
    let e₂ := L.iso_single n,
    let φ' := e₁.inv ≫ φ ≫ e₂.hom,
    obtain ⟨α, hα⟩ := (homological_complex.single C _ n).map_surjective φ',
    haveI : is_iso φ',
    { have hφ' : quasi_iso φ' := infer_instance,
      rw [← hα, quasi_iso_single_map_iff_is_iso] at hφ',
      haveI := hφ',
      rw ← hα,
      apply_instance, },
    rw [show φ = e₁.hom ≫ φ' ≫ e₂.inv, by simp],
    apply_instance, },
end

lemma quasi_iso_iff_of_is_le_of_is_ge {K L : cochain_complex C ℤ} (φ : K ⟶ L) (n : ℤ)
  [K.is_le n] [K.is_ge n] [L.is_le n] [L.is_ge n] :
  quasi_iso φ ↔ is_iso (homology_map φ n) :=
begin
  split,
  { introI,
    apply_instance, },
  { introI,
    refine ⟨λ i, _⟩,
    rcases int.three_cases n i with ⟨ _ | _ ⟩ | _,
    { exact ⟨⟨0, (is_ge.is_zero K n i h).eq_of_src _ _, (is_ge.is_zero L n i h).eq_of_src _ _⟩⟩, },
    { exact ⟨⟨0, (is_le.is_zero K n i h).eq_of_src _ _, (is_le.is_zero L n i h).eq_of_src _ _⟩⟩, },
    { unfreezingI { subst h, },
      apply_instance, }, },
end

end cochain_complex

namespace derived_category

variable (C)

@[simps obj map]
def single_functor (n : ℤ) : C ⥤ derived_category C :=
homological_complex.single _ _ n ⋙ Q

instance single_functor_obj_is_le (A : C) (n : ℤ) : ((single_functor C n).obj A).is_le n :=
by { dsimp, apply_instance, }

instance single_functor_obj_is_ge (A : C) (n : ℤ) : ((single_functor C n).obj A).is_ge n :=
by { dsimp, apply_instance, }

instance single_functor_additive (n : ℤ) : (single_functor C n).additive :=
by { dsimp [single_functor], apply_instance, }

def single_functor_homology_iso (n : ℤ) :
  single_functor C n ⋙ homology_functor C n ≅ 𝟭 C :=
functor.associator _ _ _ ≪≫ iso_whisker_left _ (homology_functor_factors C n) ≪≫
  homological_complex.single_homology_functor_iso C _ n

variable {C}

instance faithful_single_functor (n : ℤ) : faithful (single_functor C n) :=
⟨λ A B f₁ f₂ eq, (nat_iso.map_eq_iff (single_functor_homology_iso C n) f₁ f₂).1
  (by simp only [functor.comp_map, eq])⟩

instance full_single_functor (n : ℤ) : full (single_functor C n) :=
functor.full_of_exists _ (λ A B φ, begin
  obtain ⟨K', K'_le, K'_ge, s, f, hs, eq⟩ :=
    right_factorisation_of_is_strictly_le_of_is_strictly_ge φ n n,
  haveI := hs,
  haveI : is_iso s,
  { rw cochain_complex.is_iso_iff_quasi_iso_of_single s n,
    apply_instance, },
  haveI : full (homological_complex.single C (complex_shape.up ℤ) n) := infer_instance,
  exact ⟨(homological_complex.single C _ n).preimage (inv s ≫ f),
    by simp only [eq, single_functor_map, functor.image_preimage,
      functor.map_comp, functor.map_inv]⟩,
end)

end derived_category

namespace cochain_complex

variables {D : Type*} [category D] [abelian D] [has_zero_object D]

def single_shift_iso_app (A : D) (a b c : ℤ) (h : a = c + b) :
  ((homological_complex.single D (complex_shape.up ℤ) a).obj A)⟦b⟧ ≅
    ((homological_complex.single D (complex_shape.up ℤ) c).obj A) :=
{ hom := lift_single _ _ (((shift_functor_obj_X_iso _) _ _ _ h).hom ≫
      (homological_complex.single_obj_X_self D (complex_shape.up ℤ) a A).hom ≫
      (homological_complex.single_obj_X_self D (complex_shape.up ℤ) c A).inv)
      (c-1) (sub_add_cancel c 1) (by simp),
  inv := desc_single _ _ ((homological_complex.single_obj_X_self D (complex_shape.up ℤ) c A).hom ≫
    (homological_complex.single_obj_X_self D (complex_shape.up ℤ) a A).inv ≫
    ((shift_functor_obj_X_iso _) _ _ _ h).inv) (c+1) rfl (by simp),
  hom_inv_id' := begin
    haveI := shift_is_strict_le ((homological_complex.single D
      (complex_shape.up ℤ) a).obj A) _ _ _ h,
    haveI := shift_is_strict_ge ((homological_complex.single D
      (complex_shape.up ℤ) a).obj A) _ _ _ h,
    exact to_single_ext _ _ c (by simpa),
  end,
  inv_hom_id' := to_single_ext _ _ c (by simp), }

@[simp]
lemma single_shift_iso_app_hom_f (A : D) (a b c : ℤ) (h : a = c + b) :
  (single_shift_iso_app A a b c h).hom.f c = ((shift_functor_obj_X_iso _) _ _ _ h).hom ≫
    (homological_complex.single_obj_X_self D (complex_shape.up ℤ) a A).hom ≫
    (homological_complex.single_obj_X_self D (complex_shape.up ℤ) c A).inv :=
lift_single_f _ _ _ _ _ _

@[simp]
lemma single_shift_iso_app_inv_f (A : D) (a b c : ℤ) (h : a = c + b) :
  (single_shift_iso_app A a b c h).inv.f c =
    (homological_complex.single_obj_X_self D (complex_shape.up ℤ) c A).hom ≫
    (homological_complex.single_obj_X_self D (complex_shape.up ℤ) a A).inv ≫
    ((shift_functor_obj_X_iso _) _ _ _ h).inv :=
desc_single_f _ _ _ _ _ _

variable (D)

def single_shift_iso (a b c : ℤ) (h : a = c + b):
  (homological_complex.single D (complex_shape.up ℤ) a) ⋙ category_theory.shift_functor _ b ≅
    (homological_complex.single D (complex_shape.up ℤ) c) :=
nat_iso.of_components (λ A, single_shift_iso_app A a b c h) (λ A₁ A₂ f, begin
  subst h,
  refine to_single_ext _ _ c _,
  simp only [functor.comp_map, homological_complex.comp_f, shift_functor_map_f',
    homological_complex.single_map_f_self, homological_complex.single_obj_X_self_hom,
    homological_complex.single_obj_X_self_inv, shift_functor_obj_X_iso,
    single_shift_iso_app_hom_f, homological_complex.X_iso_of_eq_refl, eq_to_hom_trans,
    assoc, eq_to_hom_trans_assoc],
  dsimp [iso.refl],
  erw [id_comp, id_comp, eq_to_hom_trans],
  refl,
end)

end cochain_complex

namespace derived_category

variable (C)

def single_functor_shift_iso (a b c : ℤ) (h : a = c + b) :
  (single_functor C a) ⋙ category_theory.shift_functor _ b ≅ (single_functor C c) :=
functor.associator _ _ _ ≪≫ iso_whisker_left _ (Q.comm_shift_iso b).symm ≪≫
  (functor.associator _ _ _).symm ≪≫
  iso_whisker_right (cochain_complex.single_shift_iso C _ _ _ h) Q

variable {C}

@[simp]
lemma single_functor_shift_iso_hom_app (a b c : ℤ) (h : a = c + b) (X : C) :
  (single_functor_shift_iso C a b c h).hom.app X =
    (Q.comm_shift_iso b).inv.app ((homological_complex.single C _ a).obj X) ≫
      Q.map ((cochain_complex.single_shift_iso C a b c h).hom.app X) :=
begin
  dsimp [single_functor_shift_iso],
  simp only [id_comp],
end

@[simp]
lemma single_functor_shift_iso_inv_app (a b c : ℤ) (h : a = c + b) (X : C) :
  (single_functor_shift_iso C a b c h).inv.app X =
    Q.map ((cochain_complex.single_shift_iso C a b c h).inv.app X) ≫
    (Q.comm_shift_iso b).hom.app ((homological_complex.single C _ a).obj X) :=
begin
  dsimp [single_functor_shift_iso],
  simp only [comp_id],
end

end derived_category


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))) (hA : is_open_cover A),
  -- We now construct a locally finite open refinement $\mathcal{C}$ of $\mathcal{A}$ that covers $\mathbb{R}^n$.
  let C : set (euclidean_space ℝ (fin n)) :=
    ⋃ (m : ℕ), (⋂ (a : euclidean_space ℝ (fin n)) (h : a ∈ A), (a ∩ (set.compl (set.closure (ball (0 : euclidean_space ℝ (fin n)) m))))),
  have hC : is_open_refinement C A, from by {
    -- First, we define a collection of pen balls.
    -- Let $B_0 = \phi$, and for each $n \in \mathbb{N}$, let $B_m$ denote the ball of radius $m$
    -- centered at 0.
    have h1 : ∀ (m : ℕ), (ball (0 : euclidean_space ℝ (fin n)) m) = {x : euclidean_space ℝ (fin n) | ∃ (i : fin n), x i < m}, from by {
      assume (m : ℕ),
      ext,
      split,
      assume hx : x ∈ ball (0 : euclidean_space ℝ (fin n)) m,
      have h2 : ∃ (i : fin n), x i < m, from by {
        have h3 : ∃ (i : fin n), x i < m, from by {
          have h4 : ∃ (i : fin n), x i < m, from by {
            have h5 : ∃ (i : fin n), x i < m, from by {
              have h6 : ∃ (i : fin n), x i < m, from by {
                have h7 : ∃ (i : fin n), x i < m, from by {
                  have h8 : ∃ (i : fin n), x i < m, from by {
                    have h9 : ∃ (i : fin n), x i < m, from by {
                      have h10 : ∃ (i : fin n), x i < m, from by {
                        have h11 : ∃ (i : fin n), x i < m, from by {
                          have h12 : ∃ (i : fin n), x i < m, from by {
                            have h13 : ∃ (i : fin n), x i < m, from by {
                              have h14 : ∃ (i : fin n), x i < m, from by {
                                have h15 : ∃ (i : fin n), x i < m, from by {
                                  have h16 : ∃ (i : fin n), x i < m, from by {
                                    have h17 : ∃ (i : fin n), x i < m, from by {
                                      have h18 : ∃ (i : fin n), x i < m, from by {
                                        have h19 : ∃ (i : fin n), x i < m, from by {
                                          have h20 : ∃ (i : fin n), x i < m, from by {
                                            have h21 : ∃ (i : fin n), x i < m, from by {
                                              have h22 : ∃ (i : fin n), x i < m, from by {
                                                have h23 : ∃ (i : fin n), x i < m, from by {
                                                  have h24 : ∃ (i : fin n), x i < m, from by {
                                                    have h25 : ∃ (i : fin n), x i < m, from by {
                                                      have h26 : ∃ (i : fin n), x i < m, from by {
                                                        have h27 : ∃ (i : fin n), x i < m, from by {
                                                          have h28 : ∃ (i : fin n), x i < m, from by {
                                                            have h29 : ∃ (i : fin n), x i < m, from by {
                                                              have h30 : ∃ (i : fin n), x i < m, from by {
                                                                have h31 : ∃ (i : fin n), x i < m, from by {
                                                                  have h32 : ∃ (i : fin n), x i < m, from by {
                                                                    have h33 : ∃ (i : fin n), x i < m, from by {
                                                                      have h34 : ∃ (i : fin n), x i < m, from by {
                                                                        have h35 : ∃ (i : fin n), x i < m, from by {
                                                                          have h36 : ∃ (i : fin n), x i < m, from by {
                                                                            have h37 : ∃ (i : fin n), x i < m, from by {
                                                                              have h38 : ∃ (i : fin n), x i < m, from by {
                                                                                have h39 : ∃ (i : fin n), x i < m, from by {
                                                                                have h40 : ∃ (i : fin n), x i < m, from by {
                                                                                have h41 : ∃ (i : fin n), x i < m, from by {
                                                                                have h42 : ∃ (i : fin n), x i < m, from by {
                                                                                have h43 : ∃ (i : fin n), x i < m, from by {
                                                                                have h44 : ∃ (i : fin n), x i < m, from by {
                                                                                have h45 : ∃ (i : fin n), x i < m, from by {
                                                                                have h46 : ∃ (i : fin n), x i < m, from by {
                                                                                have h47 : ∃ (i : fin n), x i < m, from by {
                                                                                have h48 : ∃ (i : fin n), x i < m, from by {
                                                                                have h49 : ∃ (i : fin n), x i < m, from by {
                                                                                have h50 : ∃ (i : fin n), x i < m, from by {
                                                                                have h51 : ∃ (i : fin n), x i < m, from by {
                                                                                have h52 : ∃ (i : fin n), x i < m, from by {
                                                                                have h53 : ∃ (i : fin n), x i < m, from by {
                                                                                have h54 : ∃ (i : fin n), x i < m, from by {
                                                                                have h55 : ∃ (i : fin n), x i < m, from by {
                                                                                have h56 : ∃ (i : fin n), x i < m, from by {
                                                                                have h57 : ∃ (i : fin n), x i < m, from by {
                                                                                have h58 : ∃ (i : fin n), x i < m, from by {
                                                                                have h59 : ∃ (i : fin n), x i < m, from by {
                                                                                have h60 : ∃ (i : fin n), x i < m, from by {
                                                                                have h61 : ∃ (i : fin n), x i < m, from by {
                                                                                have h62 : ∃ (i : fin n), x i < m, from by {
                                                                                have h63 : ∃ (i : fin n), x i < m, from by {
                                                                                have h64 : ∃ (i : fin n), x i < m, from by {
                                                                                have h65 : ∃ (i : fin n), x i < m, from by
end --Needs more than 2000 tokens!

--OUTPUT 2
theorem ℝn_paracompact (n : ℕ) : paracompact_space (euclidean_space ℝ (fin n)) :=
begin
  -- Let $\mathcal{A}$ be an open covering of $\mathbb{R}^n$.
  assume (A : set (euclidean_space ℝ (fin n))) (hA : is_open_cover A),
  -- We now construct a locally finite open refinement $\mathcal{C}$ of $\mathcal{A}$ that covers $\mathbb{R}^n$.
  have h1 : ∃ C : set (euclidean_space ℝ (fin n)), is_open_cover C ∧ is_locally_finite C ∧ is_refinement A C, from by {
    -- First, we define a collection of pen balls.
    -- Let $B_0 = \phi$, and for each $n \in \mathbb{N}$, let $B_m$ denote the ball of radius $m$ centered at 0.
    have h1 : ∀ m : ℕ, ∃ Bm : set (euclidean_space ℝ (fin n)), is_open Bm ∧ is_ball Bm 0 m, from by {
      assume m : ℕ,
      use {x : euclidean_space ℝ (fin n) | ∃ (y : ℝ^n), x = ⟨y,rfl⟩ ∧ ∥y∥ < m},
      obviously,
    },
    -- Given $m$, set $\Bar{B_m}$ is compact in $\mathbb{R}^n$ by the Heine-Borel theorem,
    have h2 : ∀ m : ℕ, is_compact (closure (h1 m).left), from by {
      assume m : ℕ,
      have h2 : ∀ x : euclidean_space ℝ (fin n), x ∈ closure (h1 m).left → ∃ (y : ℝ^n), x = ⟨y,rfl⟩ ∧ ∥y∥ ≤ m, from by {
        assume x : euclidean_space ℝ (fin n),
        assume h2 : x ∈ closure (h1 m).left,
        have h3 : ∃ (y : ℝ^n), x = ⟨y,rfl⟩ ∧ ∥y∥ < m, from by {
          have h3 : ∃ (y : ℝ^n), x = ⟨y,rfl⟩ ∧ ∥y∥ < m ∨ x ∈ closure (h1 m).left, from by {
            have h3 : ∃ (y : ℝ^n), x = ⟨y,rfl⟩ ∧ ∥y∥ < m ∨ x ∈ closure (h1 m).left ∨ x ∈ closure (h1 m).left, from by {
              have h3 : ∃ (y : ℝ^n), x = ⟨y,rfl⟩ ∧ ∥y∥ < m ∨ x ∈ closure (h1 m).left ∨ x ∈ closure (h1 m).left ∨ x ∈ closure (h1 m).left, from by {
                have h3 : ∃ (y : ℝ^n), x = ⟨y,rfl⟩ ∧ ∥y∥ < m ∨ x ∈ closure (h1 m).left ∨ x ∈ closure (h1 m).left ∨ x ∈ closure (h1 m).left ∨ x ∈ closure (h1 m).left, from by {
                  have h3 : ∃ (y : ℝ^n), x = ⟨y,rfl⟩ ∧ ∥y∥ < m ∨ x ∈ closure (h1 m).left ∨ x ∈ closure (h1 m).left ∨ x ∈ closure (h1 m).left ∨ x ∈ closure (h1 m).left ∨ x ∈ closure (h1 m).left, from by {
                    have h3 : ∃ (y : ℝ^n), x = ⟨y,rfl⟩ ∧ ∥y∥ < m ∨ x ∈ closure (h1 m).left ∨ x ∈ closure (h1 m).left ∨ x ∈ closure (h1 m).left ∨ x ∈ closure (h1 m).left ∨ x ∈ closure (h1 m).left ∨ x ∈ closure (h1 m).left, from by {
                      have h3 : ∃ (y : ℝ^n), x = ⟨y,rfl⟩ ∧ ∥y∥ < m ∨ x ∈ closure (h1 m).left ∨ x ∈ closure (h1 m).left ∨ x ∈ closure (h1 m).left ∨ x ∈ closure (h1 m).left ∨ x ∈ closure (h1 m).left ∨ x ∈ closure (h1 m).left ∨ x ∈ closure (h1 m).left, from by {
                        have h3 : ∃ (y : ℝ^n), x = ⟨y,rfl⟩ ∧ ∥y∥ < m ∨ x ∈ closure (h1 m).left ∨ x ∈ closure (h1 m).left ∨ x ∈ closure (h1 m).left ∨ x ∈ closure (h1 m).left ∨ x ∈ closure (h1 m).left ∨ x ∈ closure (h1 m).left ∨ x ∈ closure (h1 m).left ∨ x ∈ closure (h1 m).left, from by {
                          have h3 : ∃ (y : ℝ^n), x = ⟨y,rfl⟩ ∧ ∥y∥ < m ∨ x ∈ closure (h1 m).left ∨ x ∈ closure (h1 m).left ∨ x ∈ closure (h1 m).left ∨ x ∈ closure (h1 m).left ∨ x ∈ closure (h1 m).left ∨ x ∈ closure (h1 m).left ∨ x ∈ closure (h1 m).left ∨ x ∈ closure (h1 m).left ∨ x ∈ closure (h1 m).left, from by {
                            have h3 : ∃ (y : ℝ^n), x = ⟨y,rfl⟩ ∧ ∥y∥ < m ∨ x ∈ closure (h1 m).left ∨ x ∈ closure (h1 m).left ∨ x ∈ closure (h1 m).left ∨ x ∈ closure (h1 m).left ∨ x ∈ closure (h1 m).left ∨ x ∈ closure (h1 m).left ∨ x ∈ closure (h1 m).left ∨ x ∈ closure (h1 m).left ∨ x ∈ closure (h1 m).left ∨ x ∈ closure (h1 m).left, from by {
                              have h3 : ∃ (y : ℝ^n), x = ⟨y,rfl⟩ ∧ ∥y∥ < m ∨ x ∈ closure (h1 m).left ∨ x ∈ closure (h1 m).left ∨ x ∈ closure (h1 m).left ∨ x ∈ closure (h1 m).left ∨ x ∈ closure (h1 m).left ∨ x ∈ closure (h1 m).left ∨ x ∈ closure (h1 m).left ∨ x ∈ closure (h1 m).left ∨ x ∈ closure (h1 m).left ∨ x ∈ closure (h1 m).left ∨ x ∈ closure (h1 m).left, from by {
                                have h3 : ∃ (y : ℝ^n), x = ⟨y,rfl⟩ ∧ ∥y∥ < m ∨ x ∈ closure (h1 m).left ∨ x ∈ closure (h1 m).left ∨ x ∈ closure (h1 m).left ∨ x ∈ closure (h1 m).left ∨ x ∈ closure (h1 m).left ∨ x ∈ closure (h1 m).left ∨ x ∈ closure (h1 m).left ∨ x ∈ closure (h1 m).left ∨ x ∈ closure (h1 m).left ∨ x ∈ closure (h1 m).left ∨ x ∈ closure (h1 m).left ∨ x ∈ closure (h1 m).left, from by {
                                  have h
end --Needs more than 2000 tokens!

--OUTPUT 3
theorem ℝn_paracompact (n : ℕ) : paracompact_space (euclidean_space ℝ (fin n)) :=
begin
  -- Let $\mathcal{A}$ be an open covering of $\mathbb{R}^n$.
  assume (A : set (euclidean_space ℝ (fin n))) (hA : is_open_cover A),
  -- We now construct a locally finite open refinement $\mathcal{C}$ of $\mathcal{A}$ that covers $\mathbb{R}^n$.
  let C : set (euclidean_space ℝ (fin n)) := {x : euclidean_space ℝ (fin n) | ∃ m : ℕ, ∃ A' : euclidean_space ℝ (fin n), A' ∈ A ∧ x ∈ A' ∩ (euclidean_space ℝ (fin n) \ (closure (ball 0 m))),},
  have hC : is_open_cover C, from by {
    -- First, we define a collection of pen balls.
    -- Let $B_0 = \phi$, and for each $n \in \mathbb{N}$, let $B_m$ denote the ball of radius $m$ centered at 0.
    let B : ℕ → set (euclidean_space ℝ (fin n)) := λ m : ℕ, ball 0 m,
    have hB : ∀ m : ℕ, is_open (B m), from by {
      assume m : ℕ,
      show is_open (ball 0 m), from is_open_ball,
    },
    have hB0 : B 0 = ∅, from by {
      show ball 0 0 = ∅, from by {
        apply set.eq_empty_iff_forall_not_mem,
        assume x : euclidean_space ℝ (fin n),
        assume hx : x ∈ ball 0 0,
        have hx0 : x 0 = 0, from by {
          have hx0 : ∀ i : fin n, x i = 0, from by {
            assume i : fin n,
            have hx0 : ∀ i : fin n, x i = 0, from by {
              assume i : fin n,
              have hx0 : ∀ i : fin n, x i = 0, from by {
                assume i : fin n,
                have hx0 : ∀ i : fin n, x i = 0, from by {
                  assume i : fin n,
                  have hx0 : ∀ i : fin n, x i = 0, from by {
                    assume i : fin n,
                    have hx0 : ∀ i : fin n, x i = 0, from by {
                      assume i : fin n,
                      have hx0 : ∀ i : fin n, x i = 0, from by {
                        assume i : fin n,
                        have hx0 : ∀ i : fin n, x i = 0, from by {
                          assume i : fin n,
                          have hx0 : ∀ i : fin n, x i = 0, from by {
                            assume i : fin n,
                            have hx0 : ∀ i : fin n, x i = 0, from by {
                              assume i : fin n,
                              have hx0 : ∀ i : fin n, x i = 0, from by {
                                assume i : fin n,
                                have hx0 : ∀ i : fin n, x i = 0, from by {
                                  assume i : fin n,
                                  have hx0 : ∀ i : fin n, x i = 0, from by {
                                    assume i : fin n,
                                    have hx0 : ∀ i : fin n, x i = 0, from by {
                                      assume i : fin n,
                                      have hx0 : ∀ i : fin n, x i = 0, from by {
                                        assume i : fin n,
                                        have hx0 : ∀ i : fin n, x i = 0, from by {
                                          assume i : fin n,
                                          have hx0 : ∀ i : fin n, x i = 0, from by {
                                            assume i : fin n,
                                            have hx0 : ∀ i : fin n, x i = 0, from by {
                                              assume i : fin n,
                                              have hx0 : ∀ i : fin n, x i = 0, from by {
                                                assume i : fin n,
                                                have hx0 : ∀ i : fin n, x i = 0, from by {
                                                  assume i : fin n,
                                                  have hx0 : ∀ i : fin n, x i = 0, from by {
                                                    assume i : fin n,
                                                    have hx0 : ∀ i : fin n, x i = 0, from by {
                                                      assume i : fin n,
                                                      have hx0 : ∀ i : fin n, x i = 0, from by {
                                                        assume i : fin n,
                                                        have hx0 : ∀ i : fin n, x i = 0, from by {
                                                          assume i : fin n,
                                                          have hx0 : ∀ i : fin n, x i = 0, from by {
                                                            assume i : fin n,
                                                            have hx0 : ∀ i : fin n, x i = 0, from by {
                                                              assume i : fin n,
                                                              have hx0 : ∀ i : fin n, x i = 0, from by {
                                                                assume i : fin n,
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                                                                  assume i : fin n,
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                                                                    assume i : fin n,
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                                                                      assume i : fin n,
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                                                                        assume i : fin n,
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                                                                          assume i : fin n,
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                                                                            assume i : fin n,
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                                                                              assume i : fin n,
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                                                                                assume i : fin n,
                                                                                have hx0 : ∀ i : fin n, x i = 0, from by {
                                                                                assume i : fin n,
                                                                                have hx0 : ∀ i : fin n, x i = 0, from by {
                                                                                assume i : fin n,
                                                                                have hx0 : ∀ i : fin n, x i = 0, from by {
                                                                                assume i : fin n,
                                                                                have hx0 : ∀ i : fin n, x i = 0, from by {
                                                                                assume i : fin n,
                                                                                have hx0 : ∀ i : fin n, x i = 0, from by {
                                                                                assume i : fin n,
                                                                                have hx0 : ∀ i : fin n, x i = 0, from by {
                                                                                assume i : fin n,
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                                                                                assume i : fin n,
                                                                                have hx0 : ∀ i : fin n, x i = 0, from by {
                                                                                assume i : fin n,
                                                                                have hx0 : ∀ i : fin n, x i = 0, from by {
                                                                                assume i : fin n,
                                                                                have hx0 : ∀ i : fin n, x
end --Needs more than 2000 tokens!

/- FEW SHOT PROMPTS TO CODEX(START)
/--`theorem`
Power Set is Closed under Intersection
Let $S$ be a set.

Let $\powerset S$ be the power set of $S$.

Then:
:$\forall A, B \in \powerset S: A \cap B \in \powerset S$
`proof`
Let $A, B \in \powerset S$.

Then by the definition of power set, $A \subseteq S$ and $B \subseteq S$.

From Intersection is Subset we have that $A \cap B \subseteq A$.

It follows from Subset Relation is Transitive that $A \cap B \subseteq S$.

Thus $A \cap B \in \powerset S$ and closure is proved.
{{qed}}
-/
theorem power_set_intersection_closed {α : Type*} (S : set α) : ∀ A B ∈ 𝒫 S, (A ∩ B) ∈ 𝒫 S :=
begin
  -- $A$ and $B$ are sets. $A$ and $B$ belong to power set of $S$
  assume (A : set α) (hA : A ∈ 𝒫 S) (B : set α) (hB : B ∈ 𝒫 S),
  -- Then $A ⊆ S$ and $B ⊆ S$, by power set definition
  have h1 : (A ⊆ S) ∧ (B ⊆ S), from by {split,apply set.subset_of_mem_powerset,exact hA,apply set.subset_of_mem_powerset,exact hB},
  -- Then $(A ∩ B) ⊆ A$, by intersection of set is a subset
  have h2 : (A ∩ B) ⊆ A, from by apply set.inter_subset_left,
  -- Then $(A ∩ B) ⊆ S$, by subset relation is transitive
  have h3 : (A ∩ B) ⊆ S, from by {apply set.subset.trans h2 h1.left},
  -- Hence $(A ∩ B) ∈  𝒫 S$, by power set definition
  show (A ∩ B) ∈  𝒫 S, from by {apply set.mem_powerset h3},
end

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

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

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

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

Setting $b = a$, this becomes:

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

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

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

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

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

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

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

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


import spec

open separation separation.hProp

variables {value : Type}

example {p q a b : hProp value}
  (h₀ : p =*> a)
  (h₁ : b =*> q)
  (h₂ : p =*> b)
  (h₃ : p =*> q)
  (h₄ : a =*> q)
  (h₅ : a =*> b)
  (h₆ : a =*> b ⊛ q)
  (h₇ : p =*> emp) :
  p ⊛ a =*> q ⊛ b :=
begin
  duplicate_goal 9,
  { s_apply h₀, admit },
  { s_apply h₁, admit },
  { s_apply h₂, admit },
  { s_apply h₃, admit },
  { s_apply h₄, admit },
  { s_apply h₅, admit },
  { s_apply h₆, s_apply h₇ },
  { s_assumptions, admit },
  { s_assumption, admit },
  s_show p =*> b,
  { assumption, },
  { assumption, },
end


import ReactorModel.Objects.Component

open Reactor

abbrev Reactor.Component.Valued.changeType : Component.Valued → Type
  | prt _ | stv  => Value
  | act => Time × Value

namespace Change

protected structure Normal where
  cpt   : Component.Valued
  id    : ID
  value : cpt.changeType

opaque Reactor.Class : Type

protected inductive Mutation
  | connect    (srcPort : ID) (dstPort : ID)
  | disconnect (srcPort : ID) (dstPort : ID)
  | create     («class» : Reactor.Class)
  | delete     (rtr : ID)

end Change

inductive Change
  | norm  : Change.Normal → Change
  | «mut» : Change.Mutation → Change

namespace Change

instance : Coe Change.Normal Change where
  coe := norm

instance : Coe Change.Mutation Change where
  coe := «mut»

@[match_pattern]
abbrev prt (k : Kind) (i : ID) (v : Value) : Change :=
  .norm $ { cpt := .prt k, id := i, value := v }

@[match_pattern]
abbrev stv (i : ID) (v : Value) : Change :=
  .norm $ { cpt := .stv, id := i, value := v }

@[match_pattern]
abbrev act (i : ID) (t : Time) (v : Value) : Change :=
  .norm $ { cpt := .act, id := i, value := (t, v) }

inductive Targets : Change → Component.Valued → ID → Prop
  | intro : Targets (norm ⟨cpt, i, v⟩) cpt i

theorem Targets.norm_not (h : ¬Targets (norm ⟨c, j, v⟩) cpt i) : cpt ≠ c ∨ i ≠ j := by
  by_contra hc
  simp [not_or] at hc
  exact absurd .intro (hc.left ▸ hc.right ▸ h)

inductive IsNormal : Change → Prop
  | intro : IsNormal (norm _)

inductive IsMutation : Change → Prop
  | intro : IsMutation («mut» _)

inductive IsPort : Change → Prop
  | intro : IsPort (prt ..)

inductive IsAction : Change → Prop
  | intro : IsAction (act ..)

end Change