| import Mathlib.Data.Nat.Choose.Multinomial |
| import Mathlib.Analysis.SpecialFunctions.Log.Basic |
| import Mathlib.Tactic.Linarith |
|
|
| namespace OAI |
|
|
| /-! # Factorial costs of components in the arrangement graph -/ |
|
|
| namespace Ostmann |
|
|
| open scoped BigOperators |
|
|
| / |
| inequality used in the count of bad arrangements. -/ |
| theorem component_factorial_le (a m : ℕ) : |
| (a * m).factorial ≤ m.factorial ^ a * a ^ (a * m) := by |
| classical |
| let s : Finset (Fin a) := Finset.univ |
| let f : Fin a → ℕ := fun _ => m |
| have hmem : f ∈ Finset.piAntidiag s (a * m) := by |
| simp [Finset.mem_piAntidiag, s, f] |
| have hsum : a ^ (a * m) = |
| ∑ k ∈ s.piAntidiag (a * m), Nat.multinomial s k := by |
| simpa [s] using |
| (Finset.sum_pow_eq_sum_piAntidiag (R := ℕ) s (fun _ => (1 : ℕ)) (a * m)) |
| have hb := Finset.single_le_sum (fun k (_ : k ∈ Finset.piAntidiag s (a * m)) => |
| Nat.zero_le (Nat.multinomial s k)) hmem |
| rw [← hsum] at hb |
| have hs := Nat.multinomial_spec s f |
| simp only [s, f, Finset.prod_const, Finset.card_univ, Fintype.card_fin, |
| Finset.sum_const, smul_eq_mul] at hs |
| rw [← hs] |
| exact Nat.mul_le_mul_left _ hb |
|
|
| theorem component_log_cost (a r : ℝ) (ha : 1 ≤ a) (har : a ≤ r) : |
| a * Real.log a ≤ (a - 1) * (Real.log r + 1) := by |
| have ha0 : 0 < a := by linarith |
| have hlog : Real.log a ≤ Real.log r := Real.log_le_log ha0 har |
| have hlog' : Real.log a ≤ a - 1 := Real.log_le_sub_one_of_pos ha0 |
| have hm := mul_le_mul_of_nonneg_left hlog (sub_nonneg.mpr ha) |
| nlinarith |
|
|
| / |
| of components, so the entropy cost is controlled by this deficit. -/ |
| theorem component_entropy_bound {I : Type*} [Fintype I] |
| (a : I → ℕ) (r : ℕ) (ha : ∀ i, 1 ≤ a i) (hsum : ∑ i, a i = r) : |
| (∑ i, (a i : ℝ) * Real.log (a i)) ≤ |
| ((r : ℝ) - Fintype.card I) * (Real.log r + 1) := by |
| have hle (i : I) : a i ≤ r := by |
| rw [← hsum] |
| exact Finset.single_le_sum (fun _ _ => Nat.zero_le _) (Finset.mem_univ i) |
| have hcost (i : I) := component_log_cost (a i) r |
| (by exact_mod_cast ha i) (by exact_mod_cast hle i) |
| calc |
| _ ≤ ∑ i, ((a i : ℝ) - 1) * (Real.log r + 1) := Finset.sum_le_sum fun i _ => hcost i |
| _ = ((∑ i, (a i : ℝ)) - Fintype.card I) * (Real.log r + 1) := by |
| rw [← Finset.sum_mul, Finset.sum_sub_distrib] |
| simp only [Finset.sum_const, Finset.card_univ, nsmul_eq_mul, mul_one] |
| _ = _ := by rw [← Nat.cast_sum, hsum] |
|
|
| / |
| through the deficit in the number of components. -/ |
| theorem component_factorial_product_bound {I : Type*} [Fintype I] |
| (a : I → ℕ) (r m : ℕ) (ha : ∀ i, 1 ≤ a i) (hsum : ∑ i, a i = r) : |
| (∏ i, ((a i * m).factorial : ℝ)) ≤ |
| (m.factorial : ℝ) ^ r * |
| Real.exp ((m : ℝ) * ((r : ℝ) - Fintype.card I) * (Real.log r + 1)) := by |
| classical |
| have hp (i : I) : (a i : ℝ) ^ (a i * m) = |
| Real.exp ((m : ℝ) * ((a i : ℝ) * Real.log (a i))) := by |
| have hi : 0 < (a i : ℝ) := by exact_mod_cast lt_of_lt_of_le Nat.zero_lt_one (ha i) |
| rw [show (m : ℝ) * ((a i : ℝ) * Real.log (a i)) = |
| ((a i * m : ℕ) : ℝ) * Real.log (a i) by push_cast; ring, |
| Real.exp_nat_mul, Real.exp_log hi] |
| calc |
| _ ≤ ∏ i, (m.factorial : ℝ) ^ a i * (a i : ℝ) ^ (a i * m) := by |
| exact_mod_cast (Finset.prod_le_prod (fun i (_ : i ∈ (Finset.univ : Finset I)) => |
| component_factorial_le (a i) m)) |
| _ = (m.factorial : ℝ) ^ r * Real.exp ((m : ℝ) * ∑ i, (a i : ℝ) * Real.log (a i)) := by |
| simp_rw [hp] |
| rw [Finset.prod_mul_distrib, Finset.prod_pow_eq_pow_sum, hsum, |
| ← Real.exp_sum, ← Finset.mul_sum] |
| _ ≤ _ := by |
| apply mul_le_mul_of_nonneg_left _ (by positivity) |
| apply Real.exp_le_exp.mpr |
| simpa only [mul_assoc] using mul_le_mul_of_nonneg_left |
| (component_entropy_bound a r ha hsum) (show 0 ≤ (m : ℝ) by positivity) |
|
|
| end Ostmann |
|
|
| end OAI |
|
|
