| import Mathlib.Analysis.SpecialFunctions.Log.Basic |
| import OAI.NumberTheory.Ostmann.Tree.CellGroupPigeonhole |
|
|
| namespace OAI |
|
|
| /-! # The exact cell-list pigeonhole costs only a fixed exponential in m -/ |
| namespace Ostmann |
| open Filter |
| open scoped Classical BigOperators |
|
|
| theorem eventual_cell_group_cost (N r : ℕ) (C z : ℝ) |
| (hC : 0 ≤ C) (hz : 1 ≤ z) (hsize : (N + r : ℕ) ≤ z) : |
| ∀ᶠ L : ℝ in atTop, ∀ (B : ℕ) (m : ℝ), |
| (B + 1 : ℕ) ≤ Real.exp (C * L) → z * L ≤ 2 * m → |
| (((N + r + 1) * (max N B + 1) ^ (N + r) : ℕ) : ℝ) ≤ |
| Real.exp (2 * (C + 2) * m) := by |
| filter_upwards [eventually_ge_atTop (max 0 (max (Real.log (N + r + 1 : ℕ)) |
| (Real.log (N + 1 : ℕ))))] with L hL |
| intro B m hB hm |
| have hL0 : 0 ≤ L := (le_max_left _ _).trans hL |
| have hLr : Real.log (N + r + 1 : ℕ) ≤ L := |
| (le_max_left _ _).trans ((le_max_right _ _).trans hL) |
| have hLN : Real.log (N + 1 : ℕ) ≤ L := |
| (le_max_right _ _).trans ((le_max_right _ _).trans hL) |
| have hpref : (N + r + 1 : ℕ) ≤ Real.exp L := by |
| rw [← Real.exp_log (by positivity : (0 : ℝ) < (N + r + 1 : ℕ))] |
| exact Real.exp_le_exp.mpr hLr |
| have hN : (N + 1 : ℕ) ≤ Real.exp L := by |
| rw [← Real.exp_log (by positivity : (0 : ℝ) < (N + 1 : ℕ))] |
| exact Real.exp_le_exp.mpr hLN |
| have hmax : (max N B + 1 : ℕ) ≤ Real.exp ((C + 1) * L) := by |
| rw [Nat.cast_add, Nat.cast_one, Nat.cast_max, ← max_add_add_right] |
| apply max_le |
| · exact (show (N : ℝ) + 1 ≤ Real.exp L from by simpa only [Nat.cast_add, Nat.cast_one] using hN).trans |
| (Real.exp_le_exp.mpr (by nlinarith only [mul_nonneg hC hL0])) |
| · exact (show (B : ℝ) + 1 ≤ Real.exp (C * L) from by simpa only [Nat.cast_add, Nat.cast_one] using hB).trans |
| (Real.exp_le_exp.mpr (by nlinarith only [hL0])) |
| calc |
| _ ≤ Real.exp L * (Real.exp ((C + 1) * L)) ^ (N + r) := by |
| rw [Nat.cast_mul, Nat.cast_pow] |
| exact mul_le_mul hpref (pow_le_pow_left₀ (Nat.cast_nonneg _) hmax _) |
| (by positivity) (Real.exp_nonneg _) |
| _ = Real.exp ((1 + (C + 1) * (N + r : ℕ)) * L) := by |
| rw [← Real.exp_nat_mul, ← Real.exp_add] |
| congr 1 |
| ring |
| _ ≤ Real.exp ((C + 2) * z * L) := by |
| apply Real.exp_le_exp.mpr |
| have hh := mul_le_mul_of_nonneg_left hsize (by linarith : 0 ≤ C + 1) |
| have hcoef : 1 + (C + 1) * (N + r : ℕ) ≤ (C + 2) * z := by nlinarith only [hh, hz] |
| exact mul_le_mul_of_nonneg_right hcoef hL0 |
| _ ≤ _ := Real.exp_le_exp.mpr (by |
| have hh := mul_le_mul_of_nonneg_left hm (by linarith : 0 ≤ C + 2) |
| nlinarith only [hh]) |
|
|
| / |
| exponential rate, with no dependence on the number of fixed labels in it. -/ |
| theorem cell_group_endpoint_rate (e s : ℕ) (F X A B m : ℝ) |
| (hcount : (e : ℝ) ≤ F * s) (hF : F ≤ Real.exp (B * m)) |
| (he : Real.sqrt X * Real.exp (-A * m) ≤ e) : |
| Real.sqrt X * Real.exp (-(A + B) * m) ≤ s := by |
| have hb : Real.sqrt X * Real.exp (-A * m) ≤ Real.exp (B * m) * s := |
| he.trans (hcount.trans (mul_le_mul_of_nonneg_right hF (Nat.cast_nonneg s))) |
| have hh := mul_le_mul_of_nonneg_left hb (Real.exp_nonneg (-B * m)) |
| have hid : Real.exp (-B * m) * (Real.exp (B * m) * (s : ℝ)) = s := by |
| rw [← mul_assoc, ← Real.exp_add, show -B * m + B * m = 0 by ring, Real.exp_zero, one_mul] |
| rw [hid] at hh |
| calc |
| _ = Real.exp (-B * m) * (Real.sqrt X * Real.exp (-A * m)) := by |
| calc |
| _ = Real.sqrt X * (Real.exp (-B * m) * Real.exp (-A * m)) := by |
| rw [← Real.exp_add] |
| congr 1 |
| congr 1 |
| ring |
| _ = _ := by ring |
| _ ≤ _ := hh |
|
|
| end Ostmann |
|
|
| end OAI |
|
|
