| import OAI.NumberTheory.Ostmann.Characters.PivotArithmeticTransfer |
|
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| namespace OAI |
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| /-! # Retaining the complex off-diagonal through the outside average -/ |
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| namespace Ostmann |
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| open scoped BigOperators |
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| /-- The real-part form must be averaged before taking the norm of the next |
| amplitude. The original outside law may be a subprobability law. -/ |
| theorem averaged_arithmetic_transfer {Y : Type*} [Fintype Y] |
| (μ : Y → ℝ) (hμ : ∀ y, 0 ≤ μ y) (hmass : ∑ y, μ y ≤ 1) |
| (η θ : Y → ℂ) (D : Y → ℝ) (C : ℝ) (hC : 0 ≤ C) |
| (hlocal : ∀ y, ‖η y‖ ^ 2 ≤ C * (D y + (θ y).re)) : |
| ‖∑ y, (μ y : ℂ) * η y‖ ^ 2 ≤ |
| C * ((∑ y, μ y * D y) + ‖∑ y, (μ y : ℂ) * θ y‖) := by |
| have hRe : (∑ y, (μ y : ℂ) * θ y).re = ∑ y, μ y * (θ y).re := by |
| simp only [Complex.re_sum, Complex.mul_re, Complex.ofReal_re, Complex.ofReal_im, |
| zero_mul, sub_zero] |
| calc |
| _ ≤ ∑ y, μ y * ‖η y‖ ^ 2 := subprobability_weighted_norm_sq_le μ hμ hmass η |
| _ ≤ ∑ y, μ y * (C * (D y + (θ y).re)) := |
| Finset.sum_le_sum fun y _ => mul_le_mul_of_nonneg_left (hlocal y) (hμ y) |
| _ = ∑ y, C * (μ y * D y + μ y * (θ y).re) := by |
| apply Finset.sum_congr rfl |
| intro y _ |
| ring |
| _ = C * ((∑ y, μ y * D y) + ∑ y, μ y * (θ y).re) := by |
| rw [← Finset.mul_sum, Finset.sum_add_distrib] |
| _ = C * ((∑ y, μ y * D y) + (∑ y, (μ y : ℂ) * θ y).re) := by rw [hRe] |
| _ ≤ _ := mul_le_mul_of_nonneg_left (add_le_add le_rfl (Complex.re_le_norm _)) hC |
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|
| /-- A concrete harmonic-prior loss bounded by exp(d) gives the normalized |
| transfer inequality used by the iteration theorem. -/ |
| theorem normalized_arithmetic_transfer (η θ : ℂ) (D C d : ℝ) |
| (hD : 0 ≤ D) (hC : C ≤ Real.exp d) |
| (htransfer : ‖η‖ ^ 2 ≤ C * (D + ‖θ‖)) : |
| Real.exp (-d) * ‖η‖ ^ 2 ≤ D + ‖θ‖ := by |
| have hCD : Real.exp (-d) * C ≤ 1 := by |
| calc |
| _ ≤ Real.exp (-d) * Real.exp d := mul_le_mul_of_nonneg_left hC (Real.exp_pos _).le |
| _ = 1 := by rw [← Real.exp_add]; simp |
| calc |
| _ ≤ Real.exp (-d) * (C * (D + ‖θ‖)) := |
| mul_le_mul_of_nonneg_left htransfer (Real.exp_pos _).le |
| _ = (Real.exp (-d) * C) * (D + ‖θ‖) := by ring |
| _ ≤ 1 * (D + ‖θ‖) := mul_le_mul_of_nonneg_right hCD (add_nonneg hD (norm_nonneg _)) |
| _ = _ := one_mul _ |
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| end Ostmann |
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| end OAI |
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