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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 68 69 70 71 72 73 74 75 76 77 78 79 | import OAI.NumberTheory.Ostmann.Arithmetic.ArithmeticSplitData
namespace OAI
/-! # Multiplying the actual split probabilities at fixed frequencies -/
namespace Ostmann
open scoped BigOperators Classical
structure NodeFrequencies where
root : ℤ
left : ℤ
right : ℤ
def NodeFrequencies.reducedModulus (f : NodeFrequencies) : ℕ :=
reducedFrequency (f.left.natAbs.gcd f.right.natAbs) f.root.natAbs
def ArithmeticSplitData.hasFrequencies {Q : ℕ} (d : ArithmeticSplitData Q)
(f : NodeFrequencies) : Prop :=
d.frequency = f.root ∧ d.leftFrequency = f.left ∧ d.rightFrequency = f.right
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]
noncomputable def pairFrequencySupportBound (D : ℝ) (f : NodeFrequencies × NodeFrequencies) : ℝ :=
2 * D ^ 2 / f.1.reducedModulus.lcm f.2.reducedModulus
theorem pairFrequencySupportBound_nonneg (D : ℝ) (f : NodeFrequencies × NodeFrequencies) :
0 ≤ pairFrequencySupportBound D f := by unfold pairFrequencySupportBound; positivity
/-- The reduced moduli depend only on the fixed frequencies. All pivot
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)
/-- The same product estimate on the original uniformly sampled leaf
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)
end Ostmann
end OAI
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