| import OAI.NumberTheory.Ostmann.Arithmetic.ArithmeticLineIntegration |
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| namespace OAI |
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| /-! # Signed symbolic replacement for the actual integrated line factors -/ |
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| namespace Ostmann |
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| open scoped BigOperators Classical |
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| 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)) |
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| 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) |
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| 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] |
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| /-- 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 |
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|
| /-- 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 |
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| end Ostmann |
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| end OAI |
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