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f4b3ad1 | 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 | import OAI.NumberTheory.Ostmann.Arithmetic.FrequencyTreeNodes
namespace OAI
/-! # Summing the original two-history congruence support over frequencies -/
namespace Ostmann
open scoped BigOperators Classical
theorem gcd_frequency_divisor_bound (S : Finset ℤ) (N : ℕ) (D : ℝ)
(hS : ∀ s ∈ S, s ≠ 0 ∧ s.natAbs ≤ N)
(hdiv : ∀ q : ℕ, q ≠ 0 → q ≤ N ^ 2 → (q.divisors.card : ℝ) ≤ D)
(v : ℤ) (hv : v ∈ S) (w : ℤ) (_hw : w ∈ S) :
((v.natAbs.gcd w.natAbs).divisors.card : ℝ) ≤ D := by
have hv0 : 0 < v.natAbs := Int.natAbs_pos.mpr (hS v hv).1
have hN : 0 < N := hv0.trans_le (hS v hv).2
apply hdiv
· exact Nat.ne_of_gt (Nat.gcd_pos_of_pos_left _ hv0)
· calc
_ ≤ v.natAbs := Nat.gcd_le_left _ hv0
_ ≤ N := (hS v hv).2
_ ≤ N ^ 2 := by simpa only [pow_two] using Nat.le_mul_of_pos_right N hN
/-- The bulk residue variables are still sampled from the original Haar
law. Data at each node depend only on earlier split products; the only
extra weights are the restrictions on the leaf frequencies. -/
theorem arithmetic_history_support_sum_le {Q : ℕ} [NeZero Q]
(S : Finset ℤ) (N V n : ℕ) (D : ℝ) (hD : 0 ≤ D)
(hS : ∀ s ∈ S, s ≠ 0 ∧ s.natAbs ≤ N)
(hdiv : ∀ q : ℕ, q ≠ 0 → q ≤ N ^ 2 → (q.divisors.card : ℝ) ≤ D)
(P : FrequencyTree (S × S) n → (ZMod Q)ˣ)
(data : FrequencyTree (S × S) n → List (ZMod Q)ˣ →
Option (ArithmeticSplitData Q × ArithmeticSplitData Q))
(hmatch : ∀ t past d e, data t past = some (d, e) →
d.hasFrequencies (treeNodeFrequencies S n t past.length).1 ∧
e.hasFrequencies (treeNodeFrequencies S n t past.length).2) :
(∑ t : FrequencyTree (S × S) n,
frequencyLeafWeight (pairedFrequencyLeaf S V) n t *
((Fintype.card (TreeLeafFiber (ZMod Q)ˣ n (P t)) : ℝ)⁻¹ *
∑ x : TreeLeafFiber (ZMod Q)ˣ n (P t),
sequentialSupport (fun past y => arithmeticPairSplitTest (data t past) y) []
(2 ^ n - 1) (treeLeafSplitEquiv n (P t) x))) ≤
(8 * D ^ 4 * (1 + Real.log N) ^ 3) ^ (2 ^ n - 1) * (2 * (V : ℝ)) ^ (2 * 2 ^ n) := by
apply le_trans _ (arithmetic_frequency_tree_sum_le S N V D hD hS
(gcd_frequency_divisor_bound S N D hS hdiv) n)
apply Finset.sum_le_sum
intro t _
have hlocal := arithmetic_leaf_support_fixed_frequencies (data t) (treeNodeFrequencies S n t)
D hD (hmatch t) (treeNodeFrequencies_divisor_bound S N D hD hS hdiv n t) n (P t)
have hprod : (∏ j ∈ Finset.range (2 ^ n - 1),
pairFrequencySupportBound D (treeNodeFrequencies S n t j)) =
((frequencySplitList S n t).map (pairFrequencySupportBound D)).prod := by
rw [← frequencySplitList_length S n t]
exact prod_range_list_getD _ _ _
rw [hprod] at hlocal
calc
_ ≤ frequencyLeafWeight (pairedFrequencyLeaf S V) n t *
((frequencySplitList S n t).map (pairFrequencySupportBound D)).prod :=
mul_le_mul_of_nonneg_left hlocal (frequencyLeafWeight_nonneg _
(fun _ => by unfold pairedFrequencyLeaf; split_ifs <;> norm_num) n t)
_ = _ := by rw [frequencyTreeWeight_eq_splitList]; ring
end Ostmann
end OAI
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