| import OAI.NumberTheory.Ostmann.Tree.PartitionMatching |
| import OAI.NumberTheory.Ostmann.Tree.ComponentFactorial |
| import OAI.NumberTheory.Ostmann.Tree.PartitionProducts |
|
|
| namespace OAI |
|
|
| /-! # Uniform counting bound for paired leaf partitions -/ |
|
|
| namespace Ostmann |
|
|
| open scoped BigOperators Classical |
|
|
| abbrev BadPartitionPair (r t : ℕ) := |
| {fg : (Fin r → Fin t) × (Fin r → Fin t) // |
| Function.Surjective fg.1 ∧ Function.Surjective fg.2 ∧ |
| (∀ c, Fintype.card {l : Fin r // fg.1 l = c} = |
| Fintype.card {l : Fin r // fg.2 l = c}) ∧ 4 * (r - t) ≤ r} |
|
|
| abbrev BadPartitionWitness (r : ℕ) := Σ t : Fin (r + 1), BadPartitionPair r t |
|
|
| abbrev BadPartitionWitness.Matching {r : ℕ} (w : BadPartitionWitness r) (m : ℕ) := |
| PartitionMatching (fun x : Fin r × Fin m => w.2.1.1 x.1) |
| (fun x : Fin r × Fin m => w.2.1.2 x.1) |
|
|
| theorem card_badPartitionPair_le (r t : ℕ) : |
| Fintype.card (BadPartitionPair r t) ≤ t ^ (2 * r) := by |
| calc |
| _ ≤ Fintype.card ((Fin r → Fin t) × (Fin r → Fin t)) := |
| Fintype.card_le_of_injective Subtype.val Subtype.val_injective |
| _ = _ := by simp [Fintype.card_prod, pow_mul, pow_two, mul_pow] |
|
|
| theorem card_badPartitionWitness_le (r : ℕ) : |
| Fintype.card (BadPartitionWitness r) ≤ (r + 1) * r ^ (2 * r) := by |
| rw [Fintype.card_sigma] |
| calc |
| _ ≤ ∑ t : Fin (r + 1), r ^ (2 * r) := by |
| apply Finset.sum_le_sum |
| intro t _ |
| exact (card_badPartitionPair_le r t).trans |
| (Nat.pow_le_pow_left (Nat.le_of_lt_succ t.isLt) _) |
| _ = _ := by simp |
|
|
| noncomputable def badPartitionCost (r m : ℕ) : ℝ := |
| (m.factorial : ℝ) ^ r * Real.exp ((m : ℝ) * r * (Real.log r + 1) / 4) |
|
|
| theorem badPartitionCost_nonneg (r m : ℕ) : 0 ≤ badPartitionCost r m := by |
| unfold badPartitionCost |
| positivity |
|
|
| theorem badPartition_matching_bound {r : ℕ} (hr : 1 ≤ r) |
| (w : BadPartitionWitness r) (m : ℕ) : |
| (Fintype.card (w.Matching m) : ℝ) ≤ badPartitionCost r m := by |
| let a : Fin w.1 → ℕ := fun c => Fintype.card {l : Fin r // w.2.1.1 l = c} |
| have ha (c : Fin w.1) : 1 ≤ a c := by |
| obtain ⟨l, hl⟩ := w.2.2.1 c |
| exact Fintype.card_pos_iff.mpr ⟨⟨l, hl⟩⟩ |
| have hs : ∑ c, a c = r := by |
| simpa only [Fintype.card_fin] using sum_card_partition w.2.1.1 |
| have hc : (Fintype.card (w.Matching m) : ℝ) ≤ |
| ∏ c, ((a c * m).factorial : ℝ) := by |
| have hbal := fun c => w.2.2.2.2.1 c |
| have h := card_slotMatching_le w.2.1.1 w.2.1.2 m |
| (fun c => by simpa only [Fintype.card_eq_nat_card] using hbal c) |
| have hnat : Fintype.card (w.Matching m) ≤ ∏ c, (a c * m).factorial := by |
| simpa only [a, Fintype.card_eq_nat_card] using h |
| exact_mod_cast hnat |
| have ht : (w.1 : ℕ) ≤ r := Nat.le_of_lt_succ w.1.isLt |
| have hd : 4 * ((r : ℝ) - (w.1 : ℕ)) ≤ r := by |
| have h := w.2.2.2.2.2 |
| have h' : (4 : ℝ) * ((r - (w.1 : ℕ) : ℕ) : ℝ) ≤ r := by exact_mod_cast h |
| simpa only [Nat.cast_sub ht] using h' |
| have hl : 0 ≤ Real.log (r : ℝ) + 1 := by |
| have := Real.log_nonneg (show (1 : ℝ) ≤ r by exact_mod_cast hr) |
| linarith |
| apply hc.trans |
| apply (component_factorial_product_bound a r m ha hs).trans |
| unfold badPartitionCost |
| apply mul_le_mul_of_nonneg_left _ (by positivity) |
| apply Real.exp_le_exp.mpr |
| simp only [Fintype.card_fin] |
| have hh := mul_le_mul_of_nonneg_left hd |
| (mul_nonneg (show (0 : ℝ) ≤ m by positivity) hl) |
| nlinarith |
|
|
| / |
| depends only on r, and not on the number m of slots at a leaf. -/ |
| theorem badPartition_total_bound {r : ℕ} (hr : 1 ≤ r) (m : ℕ) : |
| (Fintype.card (Σ w : BadPartitionWitness r, w.Matching m) : ℝ) ≤ |
| ((r + 1) * r ^ (2 * r) : ℕ) * badPartitionCost r m := by |
| rw [Fintype.card_sigma, Nat.cast_sum] |
| calc |
| _ ≤ ∑ _w : BadPartitionWitness r, badPartitionCost r m := |
| Finset.sum_le_sum fun w _ => badPartition_matching_bound hr w m |
| _ = (Fintype.card (BadPartitionWitness r) : ℝ) * badPartitionCost r m := by simp |
| _ ≤ _ := mul_le_mul_of_nonneg_right |
| (by exact_mod_cast card_badPartitionWitness_le r) (badPartitionCost_nonneg r m) |
|
|
| end Ostmann |
|
|
| end OAI |
|
|
