openAI-math-dataset / lean /OAI /NumberTheory /Ostmann /Arithmetic /ArithmeticHistorySupportSum.lean
| 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 | |