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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 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 | import OAI.NumberTheory.Ostmann.Arithmetic.ArithmeticLineIntegration
namespace OAI
/-! # Signed symbolic replacement for the actual integrated line factors -/
namespace Ostmann
open scoped BigOperators Classical
noncomputable def integerLineReduction {A : Type*} {n : ℕ} (value : A → ℤ)
(x : Fin n → A) (p : ℕ) : MvPolynomial (Fin n) ℤ →+* ZMod p :=
MvPolynomial.eval₂Hom (Int.castRingHom (ZMod p)) (fun j => (value (x j) : ZMod p))
noncomputable def sampledLineProbability {A J : Type*} {n : ℕ} (value : A → ℤ)
(L : J → PolynomialGiantLine (Fin n)) (external : Bool)
(P : Finset ℕ) (hprime : ∀ p ∈ P, p.Prime) (x : Fin n → A) (p : P) : ℝ :=
let _ : Fact p.val.Prime := ⟨hprime p p.property⟩
internalLineProbability external (fun j => ((L j).normalized (integerLineReduction value x p)).1)
(fun j => ((L j).normalized (integerLineReduction value x p)).2)
theorem sampledLineProbability_eq_flagWeight {A J : Type*} {n : ℕ} (value : A → ℤ)
(L : J → PolynomialGiantLine (Fin n)) (i : J) (external : Bool)
(P : Finset ℕ) (hprime : ∀ p ∈ P, p.Prime) (x : Fin n → A) (p : P)
(hd : ∀ j, integerLineReduction value x p (L j).denominator ≠ 0)
(hrow : integerLineReduction value x p (L i).a ≠ 0 ∨
integerLineReduction value x p (L i).b ≠ 0) :
sampledLineProbability value L external P hprime x p =
internalLineFlagWeight external p (fun s => arithmeticTestFlag
(p.val ∣ (integerTestValue value (lineTestPolynomials L i s) x).natAbs)) := by
let _ : Fact p.val.Prime := ⟨hprime p p.property⟩
have hr : ((L i).normalized (integerLineReduction value x p)).1 ≠ 0 ∨
((L i).normalized (integerLineReduction value x p)).2 ≠ 0 := by
have hz := (L i).normalized_zero_iff (integerLineReduction value x p) (hd i)
exact hrow.imp (fun h h₀ => h (hz.1.mp h₀)) (fun h h₀ => h (hz.2.mp h₀))
unfold sampledLineProbability
rw [internalLineProbability_eq_flags external L i _ hd hr]
change internalLineFlagWeight external p (fun s => arithmeticTestFlag
(MvPolynomial.eval₂Hom (Int.castRingHom (ZMod p.val))
(fun j => (value (x j) : ZMod p.val)) (lineTestPolynomials L i s) = 0)) = _
rw [line_divisibility_flags_eq value L i x p.val]
/-- Keep the amplitude, including the spectator factors, signed while replacing
the coefficient and rank flags of one internal prime. -/
theorem integer_line_flag_comparison_le {A J : Type*} [Fintype A] [Fintype J]
{n : ℕ} (value : A → ℤ) (hinj : Function.Injective value)
(L : J → PolynomialGiantLine (Fin n)) (i : J) (external : Bool)
(μ : Fin n → A → ℝ) (hμ : ∀ j a, 0 ≤ μ j a)
(hmass : ∀ j, ∑ a, μ j a = 1)
(α : ℝ) (hα : 0 ≤ α) (hmax : ∀ j a, μ j a ≤ α)
(P : Finset ℕ) (hprime : ∀ p ∈ P, p.Prime)
(ν : ℕ → ℝ) (hν : ∀ p ∈ P, 0 ≤ ν p) (hνmass : ∑ p ∈ P, ν p = 1)
(β V H : ℝ) (hβ : 0 ≤ β) (hV : 0 < V) (hH : 1 ≤ H)
(hνmax : ∀ p ∈ P, ν p ≤ β) (hsize : ∀ p ∈ P, V ≤ Real.log (p : ℝ))
(hvalue : ∀ s x, |(integerTestValue value (lineTestPolynomials L i s) x : ℝ)| ≤ H)
(F : (Fin n → A) → ℕ → ℂ) (B : ℝ) (hB : 0 ≤ B) (hF : ∀ x p, ‖F x p‖ ≤ B) :
‖∑ x, ∑ p ∈ P, ((productPrior μ x * ν p : ℝ) : ℂ) * F x p *
((internalLineFlagWeight external p (fun s => arithmeticTestFlag
(p ∣ (integerTestValue value (lineTestPolynomials L i s) x).natAbs)) : ℂ) -
(internalLineFlagWeight external p
(fun s => arithmeticTestFlag (lineTestPolynomials L i s = 0)) : ℂ))‖ ≤
2 * B * ∑ s, (((lineTestPolynomials L i s).totalDegree : ℝ) * α +
(Real.log H / V) * β) := by
let Ψ : (Fin n → A) → ℕ → (Bool ⊕ J → Bool) → ℂ :=
fun x p flags => F x p * (internalLineFlagWeight external p flags : ℂ)
have hΨ (x : Fin n → A) (p : ℕ) (flags : Bool ⊕ J → Bool) : ‖Ψ x p flags‖ ≤ B := by
dsimp only [Ψ]
rw [norm_mul, Complex.norm_real, Real.norm_of_nonneg
(internalLineFlagWeight_nonneg external p flags)]
calc
_ ≤ ‖F x p‖ * 1 := mul_le_mul_of_nonneg_left
(internalLineFlagWeight_le_one external p flags) (norm_nonneg _)
_ ≤ B := by simpa using hF x p
have h := integer_polynomial_symbolic_comparison_le value hinj (lineTestPolynomials L i)
μ hμ hmass α hα hmax P hprime ν hν hνmass β V H hβ hV hH hνmax hsize hvalue
Ψ B hB hΨ
simpa only [Ψ, mul_sub, mul_assoc] using h
/-- The same estimate expressed using the actual simultaneous line counts.
No rank, feasibility, or probability formula is assumed. -/
theorem sampled_line_probability_comparison_le {A J : Type*} [Fintype A] [Fintype J]
{n : ℕ} (value : A → ℤ) (hinj : Function.Injective value)
(L : J → PolynomialGiantLine (Fin n)) (i : J) (external : Bool)
(μ : Fin n → A → ℝ) (hμ : ∀ j a, 0 ≤ μ j a)
(hmass : ∀ j, ∑ a, μ j a = 1)
(α : ℝ) (hα : 0 ≤ α) (hmax : ∀ j a, μ j a ≤ α)
(P : Finset ℕ) (hprime : ∀ p ∈ P, p.Prime)
(ν : ℕ → ℝ) (hν : ∀ p ∈ P, 0 ≤ ν p) (hνmass : ∑ p ∈ P, ν p = 1)
(β V H : ℝ) (hβ : 0 ≤ β) (hV : 0 < V) (hH : 1 ≤ H)
(hνmax : ∀ p ∈ P, ν p ≤ β) (hsize : ∀ p ∈ P, V ≤ Real.log (p : ℝ))
(hvalue : ∀ s x, |(integerTestValue value (lineTestPolynomials L i s) x : ℝ)| ≤ H)
(hd : ∀ (x : Fin n → A) (p : P) j, integerLineReduction value x p (L j).denominator ≠ 0)
(hrow : ∀ (x : Fin n → A) (p : P),
integerLineReduction value x p (L i).a ≠ 0 ∨
integerLineReduction value x p (L i).b ≠ 0)
(F : (Fin n → A) → ℕ → ℂ) (B : ℝ) (hB : 0 ≤ B) (hF : ∀ x p, ‖F x p‖ ≤ B) :
‖∑ x, ∑ p : P, ((productPrior μ x * ν p : ℝ) : ℂ) * F x p *
((sampledLineProbability value L external P hprime x p : ℂ) -
(internalLineFlagWeight external p
(fun s => arithmeticTestFlag (lineTestPolynomials L i s = 0)) : ℂ))‖ ≤
2 * B * ∑ s, (((lineTestPolynomials L i s).totalDegree : ℝ) * α +
(Real.log H / V) * β) := by
have heq (x : Fin n → A) (p : P) :=
sampledLineProbability_eq_flagWeight value L i external P hprime x p (hd x p) (hrow x p)
simp_rw [heq]
let term := fun x p => ((productPrior μ x * ν p : ℝ) : ℂ) * F x p *
((internalLineFlagWeight external p (fun s => arithmeticTestFlag
(p ∣ (integerTestValue value (lineTestPolynomials L i s) x).natAbs)) : ℂ) -
(internalLineFlagWeight external p
(fun s => arithmeticTestFlag (lineTestPolynomials L i s = 0)) : ℂ))
change ‖∑ x, ∑ p : P, term x p‖ ≤ _
have hs : (∑ x, ∑ p : P, term x p) = ∑ x, ∑ p ∈ P, term x p := by
apply Finset.sum_congr rfl
intro x _
exact (Finset.sum_subtype P (fun _ => Iff.rfl) (term x)).symm
rw [hs]
exact integer_line_flag_comparison_le value hinj L i external μ hμ hmass α hα hmax
P hprime ν hν hνmass β V H hβ hV hH hνmax hsize hvalue F B hB hF
end Ostmann
end OAI
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