| import OAI.NumberTheory.Ostmann.Arithmetic.ArithmeticLineComparison |
| import OAI.NumberTheory.Ostmann.Arithmetic.ArithmeticHistoryLines |
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|
| namespace OAI |
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|
| /-! # The quantitative line comparison from coefficient size and degree -/ |
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|
| namespace Ostmann |
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|
| open scoped BigOperators Classical |
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|
| theorem bounded_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) |
| (D : ℕ) (U : ℝ) (hU : 1 ≤ U) |
| (hdegree : ∀ j, (L j).a.totalDegree ≤ D ∧ (L j).b.totalDegree ≤ D) |
| (hcoeff : ∀ (x : Fin n → A) j, |
| |MvPolynomial.eval₂Hom (Int.castRingHom ℝ) (fun k => (value (x k) : ℝ)) (L j).a| ≤ U ∧ |
| |MvPolynomial.eval₂Hom (Int.castRingHom ℝ) (fun k => (value (x k) : ℝ)) (L j).b| ≤ U) |
| (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 : ℝ) (hβ : 0 ≤ β) (hV : 0 < V) |
| (hνmax : ∀ p ∈ P, ν p ≤ β) (hsize : ∀ p ∈ P, V ≤ Real.log (p : ℝ)) |
| (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 * (Fintype.card J + 2 : ℝ) * |
| (2 * D * α + (Real.log (2 * U ^ 2) / V) * β) := by |
| have hU' : 1 ≤ 2 * U ^ 2 := by nlinarith |
| have hv (s : Bool ⊕ J) (x : Fin n → A) : |
| |(integerTestValue value (lineTestPolynomials L i s) x : ℝ)| ≤ 2 * U ^ 2 := by |
| rw [integerTestValue_real_cast] |
| exact lineTestPolynomials_value_le L i _ U hU (hcoeff x) s |
| have h := sampled_line_probability_comparison_le value hinj L i external μ hμ hmass |
| α hα hmax P hprime ν hν hνmass β V (2 * U ^ 2) hβ hV hU' hνmax hsize hv hd hrow F B hB hF |
| refine h.trans ?_ |
| have hsum : (∑ s : Bool ⊕ J, |
| (((lineTestPolynomials L i s).totalDegree : ℝ) * α + Real.log (2 * U ^ 2) / V * β)) ≤ |
| (Fintype.card J + 2 : ℝ) * (2 * D * α + Real.log (2 * U ^ 2) / V * β) := by |
| calc |
| _ ≤ ∑ _s : Bool ⊕ J, (2 * D * α + Real.log (2 * U ^ 2) / V * β) := by |
| apply Finset.sum_le_sum |
| intro s _ |
| have hd' : ((lineTestPolynomials L i s).totalDegree : ℝ) ≤ 2 * D := by |
| exact_mod_cast lineTestPolynomials_degree_le L i D hdegree s |
| have hm := mul_le_mul_of_nonneg_right hd' hα |
| linarith |
| _ = _ := by simp [Fintype.card_sum, add_comm]; ring |
| calc |
| _ ≤ 2 * B * ((Fintype.card J + 2 : ℝ) * |
| (2 * D * α + Real.log (2 * U ^ 2) / V * β)) := |
| mul_le_mul_of_nonneg_left hsum (by positivity) |
| _ = _ := by ring |
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|
| end Ostmann |
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| end OAI |
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