| 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 |
| {f : Fin (k + 1) → Fin n |
| (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 |
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|
