| import Mathlib.Data.Nat.Totient |
| import OAI.NumberTheory.Ostmann.Characters.PolynomialLineFlags |
|
|
| namespace OAI |
|
|
| /-! # Actual line integrals as bounded functions of the cleared polynomial tests -/ |
|
|
| namespace Ostmann |
|
|
| open scoped BigOperators Classical |
|
|
| noncomputable def internalLineScalar (external : Bool) (p : ℕ) : ℝ := |
| ((if external then p else p - 1 : ℕ) : ℝ)⁻¹ |
|
|
| theorem internalLineScalar_nonneg (external : Bool) (p : ℕ) : |
| 0 ≤ internalLineScalar external p := by |
| unfold internalLineScalar |
| positivity |
|
|
| theorem internalLineScalar_le_one (external : Bool) (p : ℕ) : |
| internalLineScalar external p ≤ 1 := by |
| let n : ℕ := if external then p else p - 1 |
| change (n : ℝ)⁻¹ ≤ 1 |
| rcases Nat.eq_zero_or_pos n with h | h |
| · simp [h] |
| · exact inv_le_one_of_one_le₀ (by exact_mod_cast h) |
|
|
| noncomputable def internalLineFlagWeight {J : Type*} (external : Bool) (p : ℕ) |
| (flags : Bool ⊕ J → Bool) : ℝ := |
| if lineFeasibleFlags external flags then internalLineScalar external p else 0 |
|
|
| theorem internalLineFlagWeight_nonneg {J : Type*} (external : Bool) (p : ℕ) |
| (flags : Bool ⊕ J → Bool) : 0 ≤ internalLineFlagWeight external p flags := by |
| unfold internalLineFlagWeight |
| split_ifs |
| · exact internalLineScalar_nonneg external p |
| · rfl |
|
|
| theorem internalLineFlagWeight_le_one {J : Type*} (external : Bool) (p : ℕ) |
| (flags : Bool ⊕ J → Bool) : internalLineFlagWeight external p flags ≤ 1 := by |
| unfold internalLineFlagWeight |
| split_ifs |
| · exact internalLineScalar_le_one external p |
| · norm_num |
|
|
| / |
| noncomputable def internalLineProbability {J : Type*} {p : ℕ} [Fact p.Prime] |
| (external : Bool) (a b : J → ZMod p) : ℝ := |
| if external then |
| (Fintype.card {z : ZMod p × (ZMod p)ˣ // ∀ j, a j * z.1 + b j * z.2 = 0} : ℝ) / |
| Fintype.card (ZMod p × (ZMod p)ˣ) |
| else |
| (Fintype.card {z : (ZMod p)ˣ × (ZMod p)ˣ // ∀ j, a j * z.1 + b j * z.2 = 0} : ℝ) / |
| Fintype.card ((ZMod p)ˣ × (ZMod p)ˣ) |
|
|
| theorem internalLineProbability_nonneg {J : Type*} {p : ℕ} [Fact p.Prime] |
| (external : Bool) (a b : J → ZMod p) : 0 ≤ internalLineProbability external a b := by |
| unfold internalLineProbability |
| split_ifs <;> positivity |
|
|
| theorem external_lineSystem_integral_eq_flags {σ J : Type*} {p : ℕ} [Fact p.Prime] |
| (L : J → PolynomialGiantLine σ) (i : J) (φ : MvPolynomial σ ℤ →+* ZMod p) |
| (hd : ∀ j, φ (L j).denominator ≠ 0) |
| (hrow : ((L i).normalized φ).1 ≠ 0 ∨ ((L i).normalized φ).2 ≠ 0) : |
| (Fintype.card {z : ZMod p × (ZMod p)ˣ // |
| ∀ j, ((L j).normalized φ).1 * z.1 + ((L j).normalized φ).2 * z.2 = 0} : ℝ) / |
| Fintype.card (ZMod p × (ZMod p)ˣ) = |
| internalLineFlagWeight true p |
| (fun s => arithmeticTestFlag (φ (lineTestPolynomials L i s) = 0)) := by |
| rw [external_lineSystem_probability _ _ i hrow] |
| unfold internalLineFlagWeight |
| rw [external_polynomialLine_flags_iff L i φ hd] |
| rfl |
|
|
| theorem unit_lineSystem_integral_eq_flags {σ J : Type*} {p : ℕ} [Fact p.Prime] |
| (L : J → PolynomialGiantLine σ) (i : J) (φ : MvPolynomial σ ℤ →+* ZMod p) |
| (hd : ∀ j, φ (L j).denominator ≠ 0) |
| (hrow : ((L i).normalized φ).1 ≠ 0 ∨ ((L i).normalized φ).2 ≠ 0) : |
| (Fintype.card {z : (ZMod p)ˣ × (ZMod p)ˣ // |
| ∀ j, ((L j).normalized φ).1 * z.1 + ((L j).normalized φ).2 * z.2 = 0} : ℝ) / |
| Fintype.card ((ZMod p)ˣ × (ZMod p)ˣ) = |
| internalLineFlagWeight false p |
| (fun s => arithmeticTestFlag (φ (lineTestPolynomials L i s) = 0)) := by |
| rw [unit_lineSystem_probability _ _ i hrow] |
| unfold internalLineFlagWeight |
| rw [unit_polynomialLine_flags_iff L i φ hd] |
| have hc : Fintype.card (ZMod p)ˣ = p - 1 := by |
| rw [ZMod.card_units_eq_totient, Nat.totient_prime (Fact.out : p.Prime)] |
| simp only [internalLineScalar, Bool.false_eq_true, ite_false, hc] |
|
|
| theorem internalLineProbability_eq_flags {σ J : Type*} {p : ℕ} [Fact p.Prime] |
| (external : Bool) (L : J → PolynomialGiantLine σ) (i : J) |
| (φ : MvPolynomial σ ℤ →+* ZMod p) (hd : ∀ j, φ (L j).denominator ≠ 0) |
| (hrow : ((L i).normalized φ).1 ≠ 0 ∨ ((L i).normalized φ).2 ≠ 0) : |
| internalLineProbability external (fun j => ((L j).normalized φ).1) |
| (fun j => ((L j).normalized φ).2) = |
| internalLineFlagWeight external p |
| (fun s => arithmeticTestFlag (φ (lineTestPolynomials L i s) = 0)) := by |
| cases external |
| · exact unit_lineSystem_integral_eq_flags L i φ hd hrow |
| · exact external_lineSystem_integral_eq_flags L i φ hd hrow |
|
|
| end Ostmann |
|
|
| end OAI |
|
|
