| import Mathlib |
| import OAI.Probability.BinarySweep.FiniteLaws.MomentBounds |
| import OAI.Probability.BinarySweep.Trajectories.PathExtension |
|
|
| namespace OAI |
|
|
| noncomputable section |
| open scoped BigOperators |
|
|
| namespace BinaryCoordinateSweeps |
|
|
| lemma grid_log_lower {b r : ℕ} {bits : Fin b → ℕ} (hd : ∀ j, r ≤ bits j) : |
| (b : ℝ) * ((r : ℝ) * Real.log 2) ≤ Real.log (gridSize bits) := by |
| calc |
| _ = ∑ _j : Fin b, (r : ℝ) * Real.log 2 := by simp |
| _ ≤ ∑ j : Fin b, (bits j : ℝ) * Real.log 2 := by |
| apply Finset.sum_le_sum |
| intro j _ |
| apply mul_le_mul_of_nonneg_right (by exact_mod_cast hd j) |
| exact (Real.log_pos (by norm_num : (1 : ℝ) < 2)).le |
| _ = _ := by |
| rw [log_gridSize] |
| simp only [Nat.cast_pow, Nat.cast_ofNat, Real.log_pow] |
|
|
| lemma grid_parameter_allowance {b h r : ℕ} {bits : Fin b → ℕ} |
| (hb : 1 ≤ b) (hd : ∀ j, r ≤ bits j) |
| (hr : (400000000 : ℝ) ≤ (r : ℝ) * Real.log 2) (H : PathFamily bits h) : |
| cExponent (gridSize bits) ≤ 2*c0 ∧ |
| e0/2 ≤ eExponent (gridSize bits) ∧ |
| (e0/4) * h * Real.log (gridSize bits) ≤ |
| eExponent (gridSize bits) * h * Real.log (gridSize bits) - pathCost H := by |
| have hlogb := grid_log_lower hd |
| have hbc : (1 : ℝ) ≤ b := by exact_mod_cast hb |
| have hbn : (0 : ℝ) ≤ b := by positivity |
| have hm := mul_le_mul_of_nonneg_left hr hbn |
| have hbl : (400000000 : ℝ) * b ≤ Real.log (gridSize bits) := by nlinarith |
| have hl : (400000000 : ℝ) ≤ Real.log (gridSize bits) := by nlinarith |
| have hs : (20000 : ℝ) ≤ Real.sqrt (Real.log (gridSize bits)) := by |
| apply Real.le_sqrt_of_sq_le |
| norm_num |
| exact hl |
| have hs' : 1 / Real.sqrt (Real.log (gridSize bits)) ≤ (1 / 20000 : ℝ) := |
| one_div_le_one_div_of_le (by norm_num) hs |
| have hc : cExponent (gridSize bits) ≤ 2*c0 := by |
| unfold cExponent c0 |
| linarith |
| have he : e0/2 ≤ eExponent (gridSize bits) := by |
| unfold eExponent e0 |
| linarith |
| refine ⟨hc, he, ?_⟩ |
| have hh : (0 : ℝ) ≤ h := by positivity |
| have hl0 : 0 ≤ Real.log (gridSize bits) := by linarith |
| have he' := mul_le_mul_of_nonneg_right he (mul_nonneg hh hl0) |
| have hbb : (b : ℝ) ≤ (e0/4) * Real.log (gridSize bits) := by |
| unfold e0 |
| nlinarith |
| have hcost := (pathCost_le H).trans (mul_le_mul_of_nonneg_right hbb hh) |
| nlinarith |
|
|
| lemma exists_main_scale : ∃ r : ℕ, 0 < r ∧ (400000000 : ℝ) ≤ (r : ℝ) * Real.log 2 := by |
| have hl : 0 < Real.log 2 := Real.log_pos (by norm_num) |
| obtain ⟨r, hr⟩ := exists_nat_gt ((400000000 : ℝ) / Real.log 2) |
| have hr' : (400000000 : ℝ) < (r : ℝ) * Real.log 2 := (div_lt_iff₀ hl).mp hr |
| refine ⟨r, ?_, hr'.le⟩ |
| have hp : (0 : ℝ) < r := by nlinarith |
| exact_mod_cast hp |
|
|
| theorem conditional_moment_one_dimensional : |
| ∃ r : ℕ, 0 < r ∧ ∃ zStar : ℝ, 0 < zStar ∧ |
| ∀ b : ℕ, 1 ≤ b → ∀ bits : Fin b → ℕ, (∀ j, r ≤ bits j ∧ bits j ≤ 2*r) → |
| ∀ h : ℕ, ∀ H : PathFamily bits h, ∀ z : ℝ, 0 ≤ z → z ≤ zStar → |
| ∀ q : ℕ, ∀ ρ : Representation ℂ (Equiv.Perm (FreeSlot H 0)) (RepSpace 1), |
| IsUnitaryRep ρ → |
| logMoment (traceMoment q (conditionalOperator H z ρ)) ≤ |
| ((-cExponent (gridSize bits) * Real.log (1 : ℕ) + |
| eExponent (gridSize bits) * h * Real.log (gridSize bits) - pathCost H : ℝ) : EReal) := by |
| obtain ⟨r, hr0, hr⟩ := exists_main_scale |
| refine ⟨r, hr0, 1/2, by norm_num, ?_⟩ |
| intro b hb bits hd h H z hz hzr q ρ hρ |
| have hz1 : z < 1 := by linarith |
| apply conditional_moment_of_automatic_regime H hz hz1 q ρ hρ |
| have ha := (grid_parameter_allowance hb (fun j => (hd j).1) hr H).2.2 |
| have hl0 : 0 ≤ Real.log (gridSize bits) := Real.log_nonneg (by |
| exact_mod_cast gridSize_pos bits) |
| have hn : 0 ≤ (e0/4) * h * Real.log (gridSize bits) := by |
| unfold e0 |
| positivity |
| simpa only [Nat.cast_one, Real.log_one, mul_zero] using hn.trans ha |
|
|
| end BinaryCoordinateSweeps |
|
|
| end |
|
|
| end OAI |
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|
