openAI-math-dataset / lean /OAI /RepresentationTheory /PartialPermutation /PartitionAsymptotics.lean
| import Mathlib | |
| import OAI.RepresentationTheory.PartialPermutation.DiagramCounts | |
| namespace OAI | |
| section | |
| namespace PartialPermutation | |
| namespace DiagramCounting | |
| noncomputable section | |
| open Finset | |
| open Filter in | |
| lemma partitionCount_subexponential (δ : ℝ) (hδ : 0 < δ) : | |
| ∀ᶠ n : ℕ in atTop, (partitionCount n : ℝ) ≤ Real.exp (δ*n) := by | |
| have hl := (isLittleO_log_rpow_atTop (show (0:ℝ)<1/2 by norm_num)).bound | |
| (show 0 < δ/4 by positivity) | |
| have ht : Tendsto (fun n : ℕ => (n:ℝ)+1) atTop atTop := | |
| tendsto_atTop_mono (fun n => le_add_of_nonneg_right (show (0:ℝ)≤1 by norm_num)) | |
| tendsto_natCast_atTop_atTop | |
| filter_upwards [ht.eventually hl, eventually_ge_atTop 1] with n hn hn1 | |
| have hn0 : (0:ℝ) ≤ n := Nat.cast_nonneg n | |
| have hn1' : (1:ℝ) ≤ n := by exact_mod_cast hn1 | |
| have hs : 0 ≤ Real.sqrt (n+1) := Real.sqrt_nonneg _ | |
| have hsq := Real.sq_sqrt (show (0:ℝ)≤n+1 by positivity) | |
| have hl0 : 0 ≤ Real.log ((n:ℝ)+1) := Real.log_nonneg (by linarith) | |
| rw [Real.norm_eq_abs,Real.norm_eq_abs,abs_of_nonneg hl0, | |
| ← Real.sqrt_eq_rpow,abs_of_nonneg hs] at hn | |
| have hnrt : (n.sqrt:ℝ) ≤ Real.sqrt (n+1) := by | |
| have h := Nat.sqrt_le n | |
| have hr : (n.sqrt:ℝ)^2 ≤ n := by simpa only [pow_two] using (show (n.sqrt:ℝ)*(n.sqrt:ℝ)≤n by exact_mod_cast h) | |
| nlinarith [show (0:ℝ)≤n.sqrt by positivity] | |
| have hexp : (2*(n.sqrt:ℝ))*Real.log ((n:ℝ)+1) ≤ δ*n := by | |
| calc | |
| _ ≤ (2*Real.sqrt (n+1))*((δ/4)*Real.sqrt (n+1)) := | |
| mul_le_mul (by linarith) hn hl0 (by positivity) | |
| _ = δ/2*((n:ℝ)+1) := by nlinarith | |
| _ ≤ δ*n := by nlinarith | |
| calc | |
| (partitionCount n : ℝ) ≤ ((n+1)^(2*n.sqrt):ℕ) := by | |
| exact_mod_cast partitionCount_le_sqrt n | |
| _ = Real.exp ((2*(n.sqrt:ℝ))*Real.log ((n:ℝ)+1)) := by | |
| push_cast | |
| rw [show (2:ℝ)*(n.sqrt:ℝ)=((2*n.sqrt:ℕ):ℝ) by push_cast; rfl, | |
| Real.exp_nat_mul, Real.exp_log (by positivity)] | |
| _ ≤ Real.exp (δ*n) := Real.exp_le_exp.mpr hexp | |
| end | |
| end DiagramCounting | |
| end PartialPermutation | |
| end | |
| end OAI | |