| import OAI.NumberTheory.Ostmann.Arithmetic.ArithmeticSplitData |
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| namespace OAI |
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| /-! # Multiplying the actual split probabilities at fixed frequencies -/ |
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|
| namespace Ostmann |
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| open scoped BigOperators Classical |
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|
| structure NodeFrequencies where |
| root : ℤ |
| left : ℤ |
| right : ℤ |
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| def NodeFrequencies.reducedModulus (f : NodeFrequencies) : ℕ := |
| reducedFrequency (f.left.natAbs.gcd f.right.natAbs) f.root.natAbs |
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| def ArithmeticSplitData.hasFrequencies {Q : ℕ} (d : ArithmeticSplitData Q) |
| (f : NodeFrequencies) : Prop := |
| d.frequency = f.root ∧ d.leftFrequency = f.left ∧ d.rightFrequency = f.right |
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|
| theorem ArithmeticSplitData.reducedModulus_eq {Q : ℕ} (d : ArithmeticSplitData Q) |
| (f : NodeFrequencies) (h : d.hasFrequencies f) : d.reducedModulus = f.reducedModulus := by |
| simp only [ArithmeticSplitData.reducedModulus, NodeFrequencies.reducedModulus, h.1, h.2.1, h.2.2] |
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| noncomputable def pairFrequencySupportBound (D : ℝ) (f : NodeFrequencies × NodeFrequencies) : ℝ := |
| 2 * D ^ 2 / f.1.reducedModulus.lcm f.2.reducedModulus |
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| theorem pairFrequencySupportBound_nonneg (D : ℝ) (f : NodeFrequencies × NodeFrequencies) : |
| 0 ≤ pairFrequencySupportBound D f := by unfold pairFrequencySupportBound; positivity |
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| / |
| and unit data may still vary with the previously exposed split products. -/ |
| theorem arithmetic_support_fixed_frequencies {Q : ℕ} [NeZero Q] |
| (data : List (ZMod Q)ˣ → Option (ArithmeticSplitData Q × ArithmeticSplitData Q)) |
| (frequencies : ℕ → NodeFrequencies × NodeFrequencies) (D : ℝ) (_hD : 0 ≤ D) |
| (hmatch : ∀ past d e, data past = some (d, e) → |
| d.hasFrequencies (frequencies past.length).1 ∧ |
| e.hasFrequencies (frequencies past.length).2) |
| (hdiv : ∀ j, (((frequencies j).1.reducedModulus.lcm |
| (frequencies j).2.reducedModulus).divisors.card : ℝ) ≤ D) |
| (n : ℕ) : |
| (Fintype.card (ZMod Q)ˣ : ℝ)⁻¹ ^ n * |
| (∑ x : SplitSamples (ZMod Q)ˣ n, |
| sequentialSupport (fun past y => arithmeticPairSplitTest (data past) y) [] n x) ≤ |
| ∏ j ∈ Finset.range n, pairFrequencySupportBound D (frequencies j) := by |
| apply arithmetic_sequential_support_le data _ (fun _ => pairFrequencySupportBound_nonneg _ _) _ n |
| intro past d e hd |
| obtain ⟨hdmatch, hematch⟩ := hmatch past d e hd |
| rw [d.reducedModulus_eq _ hdmatch, e.reducedModulus_eq _ hematch] |
| unfold pairFrequencySupportBound |
| apply div_le_div_of_nonneg_right _ (Nat.cast_nonneg _) |
| exact mul_le_mul_of_nonneg_left |
| (pow_le_pow_left₀ (Nat.cast_nonneg _) (hdiv past.length) 2) (by norm_num) |
|
|
| / |
| tuples, conditional only on their total product. -/ |
| theorem arithmetic_leaf_support_fixed_frequencies {Q : ℕ} [NeZero Q] |
| (data : List (ZMod Q)ˣ → Option (ArithmeticSplitData Q × ArithmeticSplitData Q)) |
| (frequencies : ℕ → NodeFrequencies × NodeFrequencies) (D : ℝ) (hD : 0 ≤ D) |
| (hmatch : ∀ past d e, data past = some (d, e) → |
| d.hasFrequencies (frequencies past.length).1 ∧ |
| e.hasFrequencies (frequencies past.length).2) |
| (hdiv : ∀ j, (((frequencies j).1.reducedModulus.lcm |
| (frequencies j).2.reducedModulus).divisors.card : ℝ) ≤ D) |
| (n : ℕ) (P : (ZMod Q)ˣ) : |
| (Fintype.card (TreeLeafFiber (ZMod Q)ˣ n P) : ℝ)⁻¹ * |
| (∑ x : TreeLeafFiber (ZMod Q)ˣ n P, |
| sequentialSupport (fun past y => arithmeticPairSplitTest (data past) y) [] |
| (2 ^ n - 1) (treeLeafSplitEquiv n P x)) ≤ |
| ∏ j ∈ Finset.range (2 ^ n - 1), pairFrequencySupportBound D (frequencies j) := by |
| rw [treeLeafSplit_average] |
| exact arithmetic_support_fixed_frequencies data frequencies D hD hmatch hdiv (2 ^ n - 1) |
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| end Ostmann |
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| end OAI |
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