openAI-math-dataset / lean /OAI /NumberTheory /Ostmann /Arithmetic /ArithmeticIntegerComparison.lean
| import OAI.NumberTheory.Ostmann.Characters.IntegerPolynomialBounds | |
| import OAI.NumberTheory.Ostmann.Arithmetic.SymbolicLineComparison | |
| namespace OAI | |
| /-! # Signed arithmetic comparison for actual integer polynomial tests -/ | |
| namespace Ostmann | |
| open scoped BigOperators | |
| noncomputable def integerTestValue {A : Type*} {n : ℕ} (value : A → ℤ) | |
| (F : MvPolynomial (Fin n) ℤ) (x : Fin n → A) : ℤ := | |
| MvPolynomial.eval₂Hom (RingHom.id ℤ) (fun j => value (x j)) F | |
| /-- The signed replacement estimate with all zero and divisor-count bridges | |
| proved for the actual integer polynomial evaluations. -/ | |
| theorem integer_polynomial_symbolic_comparison_le {A I : Type*} [Fintype A] [Fintype I] | |
| {n : ℕ} (value : A → ℤ) (hinj : Function.Injective value) | |
| (F : I → MvPolynomial (Fin n) ℤ) | |
| (μ : Fin n → A → ℝ) (hμ : ∀ i a, 0 ≤ μ i a) | |
| (hmass : ∀ i, ∑ a, μ i a = 1) | |
| (α : ℝ) (hα : 0 ≤ α) (hmax : ∀ i a, μ i 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 : ∀ i x, |(integerTestValue value (F i) x : ℝ)| ≤ H) | |
| (Ψ : (Fin n → A) → ℕ → (I → Bool) → ℂ) | |
| (B : ℝ) (hB : 0 ≤ B) (hΨ : ∀ x p v, ‖Ψ x p v‖ ≤ B) : | |
| ‖∑ x, ∑ p ∈ P, ((productPrior μ x * ν p : ℝ) : ℂ) * | |
| (Ψ x p (fun i => arithmeticTestFlag (p ∣ (integerTestValue value (F i) x).natAbs)) - | |
| Ψ x p (fun i => arithmeticTestFlag (F i = 0)))‖ ≤ | |
| 2 * B * ∑ i, (((F i).totalDegree : ℝ) * α + (Real.log H / V) * β) := by | |
| classical | |
| let Q : I → MvPolynomial (Fin n) ℚ := fun i => MvPolynomial.map (Int.castRingHom ℚ) (F i) | |
| let N : I → (Fin n → A) → ℕ := fun i x => (integerTestValue value (F i) x).natAbs | |
| have hzero (i : I) (x : Fin n → A) : N i x = 0 ↔ | |
| MvPolynomial.eval (fun j => (value (x j) : ℚ)) (Q i) = 0 := by | |
| dsimp only [N, Q, integerTestValue] | |
| rw [integerPolynomial_rational_eval] | |
| simp only [Int.natAbs_eq_zero, Int.cast_eq_zero] | |
| have hcard (i : I) (x : Fin n → A) (hx : N i x ≠ 0) : | |
| ((P.filter fun p => p ∣ N i x).card : ℝ) ≤ Real.log H / V := by | |
| have hz : integerTestValue value (F i) x ≠ 0 := by | |
| intro hz | |
| apply hx | |
| simp only [N, hz, Int.natAbs_zero] | |
| exact int_prime_divisor_card_le P _ hz hprime V H hV hsize (hvalue i x) | |
| have hk : 0 ≤ Real.log H / V := div_nonneg (Real.log_nonneg hH) hV.le | |
| have h := polynomial_symbolic_comparison_le (fun a => (value a : ℚ)) | |
| (Int.cast_injective.comp hinj) Q μ hμ hmass α hα hmax P ν hν hνmass N hzero | |
| β (Real.log H / V) hβ hk hνmax hcard Ψ B hB hΨ | |
| have hflag (i : I) : arithmeticTestFlag (Q i = 0) = arithmeticTestFlag (F i = 0) := | |
| congrArg arithmeticTestFlag (propext (integerPolynomial_rational_zero_iff (F i))) | |
| have hdegree (i : I) : (Q i).totalDegree = (F i).totalDegree := | |
| integerPolynomial_rational_degree (F i) | |
| simpa only [hflag, hdegree, N] using h | |
| end Ostmann | |
| end OAI | |