import Mathlib

namespace OAI

/-!
# Counting ordered matrix copies through a constrained cell

The counting injection is independent of the host construction.
-/

namespace Problem348.CopyBound

/-- If one specified row-column pair in every member of a finite family belongs
 to `L`, the family is encoded by that cell and all remaining coordinates. -/
theorem card_le_card_mul_pow {k n : ℕ}
    (S : Finset
      ({f : Fin (k + 1) → Fin n // StrictMono f} ×
       {f : Fin (k + 1) → Fin n // StrictMono f}))
    (i j : Fin (k + 1)) (L : Finset (Fin n × Fin n))
    (hL : ∀ rc ∈ S, (rc.1.val i, rc.2.val j) ∈ L) :
    S.card ≤ L.card * n ^ (2 * k) := by
  classical
  let encode : S → L × ((Fin k → Fin n) × (Fin k → Fin n)) :=
    fun rc =>
      (⟨(rc.val.1.val i, rc.val.2.val j), hL rc.val rc.property⟩,
       (fun a => rc.val.1.val (i.succAbove a),
        fun b => rc.val.2.val (j.succAbove b)))
  have hinj : Function.Injective encode := by
    intro x y hxy
    apply Subtype.ext
    apply Prod.ext
    · apply Subtype.ext
      funext a
      rcases Fin.eq_self_or_eq_succAbove i a with ha | ⟨b, hb⟩
      · subst a
        exact congrArg (fun z => z.1.val.1) hxy
      · subst a
        exact congrArg (fun z => z.2.1 b) hxy
    · apply Subtype.ext
      funext a
      rcases Fin.eq_self_or_eq_succAbove j a with ha | ⟨b, hb⟩
      · subst a
        exact congrArg (fun z => z.1.val.2) hxy
      · subst a
        exact congrArg (fun z => z.2.2 b) hxy
  have hc := Fintype.card_le_of_injective encode hinj
  simpa only [Fintype.card_coe, Fintype.card_prod, Fintype.card_fun,
    Fintype.card_fin, two_mul, pow_add, mul_assoc] using hc

/-- Normalize the integer count bound from the leaf-diagonal argument. -/
theorem normalized_count_le {count d m : ℕ} (hd : 0 < d) (hm : 0 < m)
    (hcount : count ≤ m * (d * m) ^ 130) :
    (count : ℝ) / (((d * m : ℕ) : ℝ) ^ 132) ≤
      (1 / ((d : ℝ) ^ 2)) * (1 / (m : ℝ)) := by
  have hd0 : (d : ℝ) ≠ 0 := by exact_mod_cast Nat.ne_of_gt hd
  have hm0 : (m : ℝ) ≠ 0 := by exact_mod_cast Nat.ne_of_gt hm
  have hcount' : (count : ℝ) ≤ (m : ℝ) * ((d : ℝ) * (m : ℝ)) ^ 130 := by
    exact_mod_cast hcount
  push_cast
  calc
    (count : ℝ) / ((d : ℝ) * (m : ℝ)) ^ 132 ≤
        ((m : ℝ) * ((d : ℝ) * (m : ℝ)) ^ 130) /
          ((d : ℝ) * (m : ℝ)) ^ 132 :=
      div_le_div_of_nonneg_right hcount' (by positivity)
    _ = (1 / ((d : ℝ) ^ 2)) * (1 / (m : ℝ)) := by
      field_simp

end Problem348.CopyBound

end OAI