| import OAI.NumberTheory.Jacobsthal.Estimates.SmallModelSourceScales |
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| namespace OAI |
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| namespace Erdos970 |
| open scoped _root_.Erdos970 |
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| section |
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| namespace ErdosVarianceSmallModel |
| attribute [local instance] Classical.propDecidable |
| attribute [local instance] Classical.decEq |
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| noncomputable def modelUnitIntegers (R xi : ℝ) (H : ℕ) (C0 : ℤ) : Finset ℤ := |
| (modelIntegers R xi (H : ℝ) (C0 : ℝ)).filter (fun m => m.natAbs.Coprime H) |
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| noncomputable def modelUniverse (R xi w : ℝ) (H : ℕ) (C0 : ℤ) : Finset (ModelPoint w H) := |
| (modelUnitIntegers R xi H C0).product Finset.univ |
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| noncomputable def atomWeight (R xi w : ℝ) (H : ℕ) (C0 : ℤ) : ℝ := |
| (H : ℝ)/((H.totient : ℝ)*modelLength R xi (H : ℝ) (C0 : ℝ)* |
| (divisorModulus w H : ℝ)*((coprimeModulus w H).totient : ℝ)) |
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| noncomputable def modelMass (R xi w : ℝ) (H : ℕ) (C0 : ℤ) (E : ModelPoint w H → Prop) : ℝ := |
| ∑ _v ∈ (modelUniverse R xi w H C0).filter E,atomWeight R xi w H C0 |
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| theorem mem_modelUniverse (R xi w : ℝ) (H : ℕ) (C0 : ℤ) (v : ModelPoint w H) : |
| v ∈ modelUniverse R xi w H C0 ↔ |
| modelLeft R xi (C0 : ℝ) ≤ (v.1 : ℝ) ∧ (v.1 : ℝ) < modelRight R xi (H : ℝ) (C0 : ℝ) ∧ |
| v.1.natAbs.Coprime H := by |
| simp only [modelUniverse,Finset.product_eq_sprod,Finset.mem_product,Finset.mem_univ,and_true, |
| modelUnitIntegers,Finset.mem_filter,mem_modelIntegers,and_assoc] |
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| theorem atomWeight_pos (R xi w : ℝ) (H : ℕ) (C0 : ℤ) (hR : 0 < R) (hxi : 0 ≤ xi) (hH : 0 < H) : |
| 0 < atomWeight R xi w H C0 := by |
| have hHR : (0 : ℝ) < H := by exact_mod_cast hH |
| have hphi : (0 : ℝ) < H.totient := by exact_mod_cast Nat.totient_pos.mpr hH |
| have hT0 : (0 : ℝ) < divisorModulus w H := by exact_mod_cast divisorModulus_pos w H |
| have hT1 : (0 : ℝ) < (coprimeModulus w H).totient := by |
| exact_mod_cast Nat.totient_pos.mpr (coprimeModulus_pos w H) |
| have hY : 0 < modelLength R xi (H : ℝ) (C0 : ℝ) := by |
| dsimp [modelLength,interceptScale] |
| positivity |
| unfold atomWeight |
| positivity |
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|
| theorem modelMass_card (R xi w : ℝ) (H : ℕ) (C0 : ℤ) (E : ModelPoint w H → Prop) : |
| modelMass R xi w H C0 E = |
| (((modelUniverse R xi w H C0).filter E).card : ℝ)*atomWeight R xi w H C0 := by |
| simp only [modelMass,Finset.sum_const,nsmul_eq_mul] |
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|
| theorem modelUniverse_card (R xi w : ℝ) (H : ℕ) (C0 : ℤ) : |
| (modelUniverse R xi w H C0).card = |
| (modelUnitIntegers R xi H C0).card*(divisorModulus w H*(coprimeModulus w H).totient) := by |
| simp only [modelUniverse,Finset.product_eq_sprod,Finset.card_product,Finset.card_univ,pattern_card] |
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| theorem model_total_mass (R xi w : ℝ) (H : ℕ) (C0 : ℤ) (hR : 0 < R) (hxi : 0 ≤ xi) (hH : 0 < H) : |
| modelMass R xi w H C0 (fun _ => True) = |
| ((modelUnitIntegers R xi H C0).card : ℝ)*(H : ℝ)/ |
| ((H.totient : ℝ)*modelLength R xi (H : ℝ) (C0 : ℝ)) := by |
| have hphi : (0 : ℝ) < H.totient := by exact_mod_cast Nat.totient_pos.mpr hH |
| have hT0 : (0 : ℝ) < divisorModulus w H := by exact_mod_cast divisorModulus_pos w H |
| have hT1 : (0 : ℝ) < (coprimeModulus w H).totient := by |
| exact_mod_cast Nat.totient_pos.mpr (coprimeModulus_pos w H) |
| have hY : 0 < modelLength R xi (H : ℝ) (C0 : ℝ) := by |
| have hHR : (0 : ℝ) < H := by exact_mod_cast hH |
| dsimp [modelLength,interceptScale] |
| positivity |
| rw [modelMass_card,Finset.filter_true,modelUniverse_card] |
| simp only [Nat.cast_mul,atomWeight] |
| field_simp |
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|
| end ErdosVarianceSmallModel |
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| end |
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| end Erdos970 |
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| end OAI |
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