File size: 3,372 Bytes
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 | import OAI.NumberTheory.Ostmann.Arithmetic.ArithmeticLineComparison
import OAI.NumberTheory.Ostmann.Arithmetic.ArithmeticHistoryLines
namespace OAI
/-! # The quantitative line comparison from coefficient size and degree -/
namespace Ostmann
open scoped BigOperators Classical
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
end Ostmann
end OAI
|