| import OAI.Geometry.HeilbronnTriangle.AuxiliarySamplingLaw |
| import OAI.Geometry.HeilbronnTriangle.ParameterSequence |
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| namespace OAI |
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| noncomputable section |
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| namespace Problem355.AuxiliarySampling |
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| open scoped BigOperators |
| open Parameters Parameters.PrimeParameterData |
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| namespace Law |
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| variable {q H : ℕ} [Fact q.Prime] (L : Law q H) |
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| theorem density_ratio_le : |
| (q : ℝ) ^ 3 / L.size ≤ 2000 ^ 3 * (q : ℝ) * (H : ℝ) ^ 6 := by |
| have hs : (0 : ℝ) < L.size := by exact_mod_cast L.size_pos |
| have hsize : (q : ℝ) ^ 2 ≤ 2000 ^ 3 * (H : ℝ) ^ 6 * L.size := by |
| exact_mod_cast L.size_lower |
| apply (div_le_iff₀ hs).mpr |
| calc |
| (q : ℝ) ^ 3 = (q : ℝ) * (q : ℝ) ^ 2 := by ring |
| _ ≤ (q : ℝ) * (2000 ^ 3 * (H : ℝ) ^ 6 * L.size) := |
| mul_le_mul_of_nonneg_left hsize (Nat.cast_nonneg q) |
| _ = (2000 ^ 3 * (q : ℝ) * (H : ℝ) ^ 6) * L.size := by ring |
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| theorem density_scale_le : |
| (q : ℝ) ^ 3 ≤ (2000 ^ 3 * (q : ℝ) * (H : ℝ) ^ 6) * L.size := by |
| exact (div_le_iff₀ (by exact_mod_cast L.size_pos : (0 : ℝ) < L.size)).mp |
| L.density_ratio_le |
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| end Law |
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| theorem nonempty_parameterLaw {k r : ℕ} (P : PrimeParameterData k r) : |
| Nonempty (@Law P.q (P.h ^ 2) ⟨P.auxiliary_prime⟩) := by |
| let : Fact P.q.Prime := ⟨P.auxiliary_prime⟩ |
| apply nonempty_law |
| · have hh : 1 ≤ P.h := by have := P.h_ge_three; omega |
| exact one_le_pow₀ hh |
| · exact P.auxiliary_large |
| |
| def parameterLaw {k r : ℕ} (P : PrimeParameterData k r) : |
| @Law P.q (P.h ^ 2) ⟨P.auxiliary_prime⟩ := |
| Classical.choice (nonempty_parameterLaw P) |
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| section PrimeParameters |
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| variable {k r : ℕ} (P : PrimeParameterData k r) [Fact P.q.Prime] |
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| theorem density_ratio_le_parameter (L : Law P.q (P.h ^ 2)) : |
| (P.q : ℝ) ^ 3 / L.size ≤ 2000 ^ 3 * (P.q : ℝ) * (P.h : ℝ) ^ 12 := by |
| have h := L.density_ratio_le |
| push_cast at h |
| convert h using 1 ; ring |
| |
| theorem sets_nonempty_parameter (L : Law P.q (P.h ^ 2)) (ω : L.Outcome) : |
| (L.sets ω).Nonempty := by |
| apply Finset.card_pos.mp |
| rw [L.sets_card] |
| exact L.size_pos |
|
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| theorem weight_le_four_parameter (L : Law P.q (P.h ^ 2)) |
| (p : Fin 3 → (Fin 3 → ZMod P.q)) (i j : Fin 3) (hp : p i ≠ p j) : |
| LiftingProbability.auxiliaryWeight P.q L.size L.weight L.sets p ≤ |
| (4 * 2000 ^ 3) * (P.q : ℝ) * (P.h : ℝ) ^ 12 := by |
| calc |
| LiftingProbability.auxiliaryWeight P.q L.size L.weight L.sets p ≤ |
| 4 * ((P.q : ℝ) ^ 3 / L.size) := L.weight_le_four p i j hp |
| _ ≤ 4 * (2000 ^ 3 * (P.q : ℝ) * (P.h : ℝ) ^ 12) := |
| mul_le_mul_of_nonneg_left (density_ratio_le_parameter P L) (by norm_num) |
| _ = (4 * 2000 ^ 3) * (P.q : ℝ) * (P.h : ℝ) ^ 12 := by ring |
|
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| theorem weight_le_two_parameter (L : Law P.q (P.h ^ 2)) |
| (p : Fin 3 → (Fin 3 → ZMod P.q)) : |
| LiftingProbability.auxiliaryWeight P.q L.size L.weight L.sets p ≤ |
| (2 * 2000 ^ 6) * (P.q : ℝ) ^ 2 * (P.h : ℝ) ^ 24 := by |
| have hratio := density_ratio_le_parameter P L |
| have hsq : ((P.q : ℝ) ^ 3 / L.size) ^ 2 ≤ |
| (2000 ^ 3 * (P.q : ℝ) * (P.h : ℝ) ^ 12) ^ 2 := by |
| gcongr |
| calc |
| LiftingProbability.auxiliaryWeight P.q L.size L.weight L.sets p ≤ |
| 2 * ((P.q : ℝ) ^ 3 / L.size) ^ 2 := L.weight_le_two p |
| _ ≤ 2 * (2000 ^ 3 * (P.q : ℝ) * (P.h : ℝ) ^ 12) ^ 2 := |
| mul_le_mul_of_nonneg_left hsq (by norm_num) |
| _ = (2 * 2000 ^ 6) * (P.q : ℝ) ^ 2 * (P.h : ℝ) ^ 24 := by ring |
|
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| end PrimeParameters |
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| theorem parameterLaw_density_ratio_le {k r : ℕ} (P : PrimeParameterData k r) : |
| (P.q : ℝ) ^ 3 / |
| (@Law.size P.q (P.h ^ 2) ⟨P.auxiliary_prime⟩ (parameterLaw P) : ℝ) ≤ |
| 2000 ^ 3 * (P.q : ℝ) * (P.h : ℝ) ^ 12 := by |
| let : Fact P.q.Prime := ⟨P.auxiliary_prime⟩ |
| exact density_ratio_le_parameter P (parameterLaw P) |
|
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| end Problem355.AuxiliarySampling |
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| end |
|
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| end OAI |
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|
