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f89a930 | 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 67 68 69 70 71 72 73 74 75 76 77 78 79 80 | import OAI.NumberTheory.Jacobsthal.Estimates.SmallModelSourceScales
namespace OAI
namespace Erdos970
open scoped _root_.Erdos970
section
namespace ErdosVarianceSmallModel
attribute [local instance] Classical.propDecidable
attribute [local instance] Classical.decEq
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)
noncomputable def modelUniverse (R xi w : ℝ) (H : ℕ) (C0 : ℤ) : Finset (ModelPoint w H) :=
(modelUnitIntegers R xi H C0).product Finset.univ
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))
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
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]
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
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]
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]
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
end ErdosVarianceSmallModel
end
end Erdos970
end OAI
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