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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 96 97 | import Mathlib
import OAI.Probability.BinarySweep.Analytic.OneAxisPerturbation
import OAI.Probability.BinarySweep.FiniteLaws.QuantitativeMoment
import OAI.Probability.BinarySweep.Conditional.MainEasyCases
namespace OAI
noncomputable section
open scoped BigOperators
namespace BinaryCoordinateSweeps
def oneAxisRadius (r : ℕ) : ℝ :=
(2 * (((2^(2*r) : ℕ).factorial : ℝ))^2)⁻¹
lemma oneAxisRadius_pos (r : ℕ) : 0 < oneAxisRadius r := by
unfold oneAxisRadius
positivity
lemma oneAxisRadius_le_half (r : ℕ) : oneAxisRadius r ≤ 1/2 := by
have hB : (1 : ℝ) ≤ ((2^(2*r) : ℕ).factorial : ℝ) := by
exact_mod_cast Nat.factorial_pos (2^(2*r))
unfold oneAxisRadius
calc
_ ≤ (2 : ℝ)⁻¹ := inv_anti₀ (by norm_num) (by nlinarith)
_ = 1/2 := by norm_num
lemma oneAxis_size_le {r : ℕ} {bits : Fin 1 → ℕ} (hd : ∀ j, bits j ≤ 2*r) :
gridSize bits ≤ 2^(2*r) := by
simp only [gridSize, Fin.prod_univ_one]
exact Nat.pow_le_pow_right (by omega) (hd 0)
lemma oneAxis_norm_le_inv_dimension {r h D : ℕ} {bits : Fin 1 → ℕ}
(hd : ∀ j, bits j ≤ 2*r) (H : PathFamily bits h)
{z : ℝ} (hz : 0 ≤ z) (hzr : z ≤ oneAxisRadius r)
(ρ : Representation ℂ (Equiv.Perm (FreeSlot H 0)) (RepSpace D))
[ρ.IsIrreducible] (hρ : IsUnitaryRep ρ) (hD : 1 < D) :
‖conditionalOperator H z ρ‖ ≤ (D : ℝ)⁻¹ := by
let B : ℝ := ((2^(2*r) : ℕ).factorial : ℝ)
have hB : 0 < B := by dsimp [B]; positivity
have hfac : ((gridSize bits).factorial : ℝ) ≤ B := by
dsimp [B]
exact_mod_cast Nat.factorial_le (oneAxis_size_le hd)
have hdim : D ≤ (gridSize bits).factorial := by
have hbound := Irrep.finrank_le_card ρ
simp only [finrank_euclideanSpace_fin, Fintype.card_perm, card_freeSlot] at hbound
exact hbound.trans (Nat.factorial_le (Nat.sub_le _ _))
have hDB : (D : ℝ) ≤ B := (by exact_mod_cast hdim :
(D : ℝ) ≤ ((gridSize bits).factorial : ℝ)).trans hfac
have hD0 : (0 : ℝ) < D := by exact_mod_cast (by omega : 0 < D)
calc
_ ≤ 2 * ((gridSize bits).factorial : ℝ) * z :=
oneAxis_conditional_norm_le H hz (hzr.trans (oneAxisRadius_le_half r)) ρ hρ hD
_ ≤ 2 * B * oneAxisRadius r := by gcongr
_ = B⁻¹ := by dsimp [oneAxisRadius, B]; field_simp
_ ≤ (D : ℝ)⁻¹ := inv_anti₀ hD0 hDB
theorem conditional_moment_one_axis {r q : ℕ}
(hr : (400000000 : ℝ) ≤ (r : ℝ) * Real.log 2) (hq : 0 < q)
(bits : Fin 1 → ℕ) (hd : ∀ j, r ≤ bits j ∧ bits j ≤ 2*r)
{h D : ℕ} (H : PathFamily bits h) {z : ℝ} (hz : 0 ≤ z)
(hzr : z ≤ oneAxisRadius r)
(ρ : Representation ℂ (Equiv.Perm (FreeSlot H 0)) (RepSpace D))
[ρ.IsIrreducible] (hρ : IsUnitaryRep ρ) :
logMoment (traceMoment q (conditionalOperator H z ρ)) ≤
((-cExponent (gridSize bits) * Real.log D +
eExponent (gridSize bits) * h * Real.log (gridSize bits) - pathCost H : ℝ) : EReal) := by
have ha := grid_parameter_allowance (by omega : 1 ≤ 1) (fun j => (hd j).1) hr H
have hl0 : 0 ≤ Real.log (gridSize bits) := Real.log_nonneg (by
exact_mod_cast gridSize_pos bits)
have hallow : 0 ≤ eExponent (gridSize bits) * h * Real.log (gridSize bits) - pathCost H := by
apply le_trans _ ha.2.2
unfold e0
positivity
by_cases hD : 1 < D
· have hn := oneAxis_norm_le_inv_dimension (fun j => (hd j).2) H hz hzr ρ hρ hD
apply (logMoment_le_neg_log_dimension (by omega) hq _ hn).trans
have hc : cExponent (gridSize bits) ≤ 1 := by
have := ha.1
unfold c0 at this
linarith
have hlog : 0 ≤ Real.log D := Real.log_nonneg (by exact_mod_cast (by omega : 1 ≤ D))
exact_mod_cast (show -Real.log D ≤ -cExponent (gridSize bits) * Real.log D +
eExponent (gridSize bits) * h * Real.log (gridSize bits) - pathCost H by nlinarith)
· have hlog : Real.log D = 0 := by
have hD' : D = 0 ∨ D = 1 := by omega
rcases hD' with rfl | rfl <;> norm_num
apply conditional_moment_of_automatic_regime H hz
(lt_of_le_of_lt (hzr.trans (oneAxisRadius_le_half r)) (by norm_num)) q ρ hρ
simpa only [hlog, mul_zero] using hallow
end BinaryCoordinateSweeps
end
end OAI
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