| 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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|
