| import OAI.Probability.ClassicalON.LatticeModes |
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| namespace OAI |
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| noncomputable section |
| open scoped BigOperators |
| namespace ClassicalON |
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| theorem dyadic_inverse_square_sum (k : ℕ) : |
| (∑ p : DyadicFreq k, (1/(dyadicRadius p:ℝ))^2) ≤ 3*(k:ℝ) := by |
| rw [Fintype.sum_sigma] |
| calc _ ≤ ∑ j : Fin k, (3:ℝ) := by |
| apply Finset.sum_le_sum |
| intro j _ |
| simp only [dyadicRadius, Finset.sum_const, Finset.card_univ, nsmul_eq_mul] |
| have hr : (0:ℝ) < (2^j.val:ℕ) := by positivity |
| have hc : (Fintype.card (Fin (2^j.val) × Fin (2*2^j.val+1)):ℝ) ≤ 3*((2^j.val:ℕ):ℝ)^2 := by |
| exact_mod_cast (dyadic_shell_card_bounds j.val).2 |
| calc _ ≤ (3*((2^j.val:ℕ):ℝ)^2)*(1/((2^j.val:ℕ):ℝ))^2 := |
| mul_le_mul_of_nonneg_right hc (sq_nonneg _) |
| _ = 3 := by field_simp |
| _ = _ := by simp; ring |
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| theorem sum_dyadic_ratio_lower (k i : ℕ) : |
| (∑ j ∈ Finset.range k, if j ≤ i then (2:ℝ)^j/2^i else 0) ≤ 2 := by |
| classical |
| rw [← Finset.sum_filter] |
| have hs : (Finset.range k).filter (fun j => j ≤ i) ⊆ Finset.range (i+1) := by |
| intro j hj |
| simp only [Finset.mem_filter, Finset.mem_range] at hj ⊢ |
| omega |
| calc _ ≤ ∑ j ∈ Finset.range (i+1), (2:ℝ)^j/2^i := |
| Finset.sum_le_sum_of_subset_of_nonneg hs (fun _ _ _ => by positivity) |
| _ ≤ 2 := by |
| rw [← Finset.sum_div] |
| have hh := geom_sum_mul (2:ℝ) (i+1) |
| have hp : (0:ℝ) < 2^i := by positivity |
| rw [div_le_iff₀ hp] |
| simp only [show (2:ℝ)-1=1 by norm_num, mul_one, |
| pow_succ] at hh |
| linarith |
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| theorem dyadic_min_ratio_sum (k : ℕ) : |
| (∑ i : Fin k, ∑ j : Fin k, min ((2:ℝ)^i.val/2^j.val) (2^j.val/2^i.val)) ≤ 4*(k:ℝ) := by |
| classical |
| have point (i j : Fin k) : min ((2:ℝ)^i.val/2^j.val) (2^j.val/2^i.val) ≤ |
| (if j.val ≤ i.val then (2:ℝ)^j.val/2^i.val else 0)+ |
| (if i.val ≤ j.val then (2:ℝ)^i.val/2^j.val else 0) := by |
| by_cases h : j.val ≤ i.val |
| · rw [ite_eq_left h] |
| exact (min_le_right _ _).trans (le_add_of_nonneg_right (by split_ifs <;> positivity)) |
| · have hh : i.val ≤ j.val := by omega |
| rw [ite_eq_right h, ite_eq_left hh, zero_add] |
| exact min_le_left _ _ |
| have hs : (∑ i : Fin k, ∑ j : Fin k, if j.val ≤ i.val then (2:ℝ)^j.val/2^i.val else 0) ≤ 2*(k:ℝ) := by |
| calc _ ≤ ∑ _i : Fin k, (2:ℝ) := by |
| apply Finset.sum_le_sum |
| intro i _ |
| rw [← Finset.sum_range (fun j => if j ≤ i.val then (2:ℝ)^j/2^i.val else 0)] |
| exact sum_dyadic_ratio_lower k i.val |
| _ = _ := by simp; ring |
| calc _ ≤ ∑ i : Fin k, ∑ j : Fin k, |
| ((if j.val ≤ i.val then (2:ℝ)^j.val/2^i.val else 0)+ |
| (if i.val ≤ j.val then (2:ℝ)^i.val/2^j.val else 0)) := |
| Finset.sum_le_sum fun i _ => Finset.sum_le_sum fun j _ => point i j |
| _ = 2*∑ i : Fin k, ∑ j : Fin k, if j.val ≤ i.val then (2:ℝ)^j.val/2^i.val else 0 := by |
| simp only [Finset.sum_add_distrib] |
| rw [Finset.sum_comm (f := fun i j : Fin k => if i.val ≤ j.val then (2:ℝ)^i.val/2^j.val else 0)] |
| ring |
| _ ≤ _ := by linarith |
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| theorem weighted_dyadic_min_pair (a b : ℝ) (ha : 0<a) (hb : 0<b) : |
| a^2*b^2*min ((1/a)^3*(1/b)) ((1/b)^3*(1/a)) = min (b/a) (a/b) := by |
| rw [mul_min_of_nonneg _ _ (by positivity : (0:ℝ) ≤ a^2*b^2)] |
| congr 1 <;> field_simp |
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| theorem dyadic_min_cost_sum (k : ℕ) : |
| (∑ p : DyadicFreq k, ∑ q : DyadicFreq k, |
| min ((1/(dyadicRadius p:ℝ))^3*(1/(dyadicRadius q:ℝ))) |
| ((1/(dyadicRadius q:ℝ))^3*(1/(dyadicRadius p:ℝ)))) ≤ 36*(k:ℝ) := by |
| simp only [Fintype.sum_sigma] |
| have he : (∑ i : Fin k, ∑ a : Fin (2^i.val) × Fin (2*2^i.val+1), |
| ∑ j : Fin k, ∑ b : Fin (2^j.val) × Fin (2*2^j.val+1), |
| min ((1/(dyadicRadius ⟨i,a⟩:ℝ))^3*(1/(dyadicRadius ⟨j,b⟩:ℝ))) |
| ((1/(dyadicRadius ⟨j,b⟩:ℝ))^3*(1/(dyadicRadius ⟨i,a⟩:ℝ)))) = |
| ∑ i : Fin k, ∑ j : Fin k, |
| (Fintype.card (Fin (2^i.val) × Fin (2*2^i.val+1)):ℝ)* |
| (Fintype.card (Fin (2^j.val) × Fin (2*2^j.val+1)):ℝ)* |
| min ((1/(2:ℝ)^i.val)^3*(1/(2:ℝ)^j.val)) |
| ((1/(2:ℝ)^j.val)^3*(1/(2:ℝ)^i.val)) := by |
| apply Finset.sum_congr rfl |
| intro i _ |
| rw [Finset.sum_comm] |
| apply Finset.sum_congr rfl |
| intro j _ |
| simp only [dyadicRadius, Nat.cast_pow, Nat.cast_ofNat, Finset.sum_const, Finset.card_univ, nsmul_eq_mul] |
| ring |
| rw [he] |
| calc _ ≤ ∑ i : Fin k, ∑ j : Fin k, 9*min ((2:ℝ)^i.val/2^j.val) (2^j.val/2^i.val) := by |
| apply Finset.sum_le_sum; intro i _ |
| apply Finset.sum_le_sum; intro j _ |
| have hi : (Fintype.card (Fin (2^i.val) × Fin (2*2^i.val+1)):ℝ) ≤ 3*((2:ℝ)^i.val)^2 := by |
| exact_mod_cast (dyadic_shell_card_bounds i.val).2 |
| have hj : (Fintype.card (Fin (2^j.val) × Fin (2*2^j.val+1)):ℝ) ≤ 3*((2:ℝ)^j.val)^2 := by |
| exact_mod_cast (dyadic_shell_card_bounds j.val).2 |
| calc _ ≤ (3*((2:ℝ)^i.val)^2)*(3*((2:ℝ)^j.val)^2)* |
| min ((1/(2:ℝ)^i.val)^3*(1/(2:ℝ)^j.val)) ((1/(2:ℝ)^j.val)^3*(1/(2:ℝ)^i.val)) := by |
| gcongr |
| _ = _ := by |
| rw [show (3*((2:ℝ)^i.val)^2)*(3*((2:ℝ)^j.val)^2) = 9*(((2:ℝ)^i.val)^2*((2:ℝ)^j.val)^2) by ring, |
| mul_assoc, weighted_dyadic_min_pair _ _ (by positivity) (by positivity), min_comm] |
| _ ≤ _ := by |
| simp only [← Finset.mul_sum] |
| have hs := dyadic_min_ratio_sum k |
| linarith |
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| end ClassicalON |
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| end |
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| end OAI |
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