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9b6e014 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 | 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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