| 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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| namespace ErdosLargePatternLaw |
| open ErdosVarianceSmallModel |
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| attribute [local instance] Classical.propDecidable |
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| abbrev FullPattern (w : ℝ) (H : ℕ) := |
| (ZMod (divisorModulus w H))ˣ × ZMod (divisorModulus w H) × |
| (ZMod (coprimeModulus w H))ˣ × ZMod (coprimeModulus w H) |
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| noncomputable def weight (w : ℝ) (H : ℕ) : ℝ := |
| 1/(((divisorModulus w H).totient : ℝ)*(divisorModulus w H : ℝ)* |
| ((coprimeModulus w H).totient : ℝ)*(coprimeModulus w H : ℝ)) |
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| noncomputable def expectation (w : ℝ) (H : ℕ) (F : FullPattern w H → ℝ) : ℝ := |
| ∑ v,weight w H*F v |
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| def survives (w : ℝ) (H qA : ℕ) (beta delta : ∀ t : ℕ,ZMod t) |
| (j : ℕ) (v : FullPattern w H) : Prop := |
| (∀ t ∈ divisorPrimes w H,(v.2.1.val : ZMod t)+(v.1.val.val : ZMod t)*(j : ZMod t) ≠ beta t) ∧ |
| (∀ t ∈ coprimePrimes w H,(v.2.2.1.val.val : ZMod t)*(qA : ZMod t)* |
| ((v.2.2.2.val : ZMod t)+(H : ZMod t)*(j : ZMod t)) ≠ delta t) |
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| def jointCondition (w : ℝ) (H qA : ℕ) (beta delta : ∀ t : ℕ,ZMod t) |
| (S : Finset ℕ) (v : FullPattern w H) : Prop := ∀ j ∈ S,survives w H qA beta delta j v |
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| noncomputable def indicator (w : ℝ) (H qA : ℕ) (beta delta : ∀ t : ℕ,ZMod t) |
| (j : ℕ) (v : FullPattern w H) : ℝ := if survives w H qA beta delta j v then 1 else 0 |
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| noncomputable def survivorCount (w : ℝ) (H qA J : ℕ) (beta delta : ∀ t : ℕ,ZMod t) |
| (v : FullPattern w H) : ℕ := ((Finset.range J).filter (fun j => survives w H qA beta delta j v)).card |
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| theorem pattern_card (w : ℝ) (H : ℕ) : Fintype.card (FullPattern w H) = |
| (divisorModulus w H).totient*divisorModulus w H*(coprimeModulus w H).totient*coprimeModulus w H := by |
| simp only [FullPattern,Fintype.card_prod,ZMod.card_units_eq_totient,ZMod.card] |
| ring |
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| theorem weight_positive (w : ℝ) (H : ℕ) : 0 < weight w H := by |
| have hT0 := divisorModulus_pos w H |
| have hT1 := coprimeModulus_pos w H |
| have hP0 := Nat.totient_pos.mpr hT0 |
| have hP1 := Nat.totient_pos.mpr hT1 |
| unfold weight |
| positivity |
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| theorem weight_sum (w : ℝ) (H : ℕ) : (∑ _v : FullPattern w H,weight w H)=1 := by |
| have hT0 : (divisorModulus w H : ℝ) ≠ 0 := by exact_mod_cast (divisorModulus_pos w H).ne' |
| have hT1 : (coprimeModulus w H : ℝ) ≠ 0 := by exact_mod_cast (coprimeModulus_pos w H).ne' |
| have hP0 : ((divisorModulus w H).totient : ℝ) ≠ 0 := by |
| exact_mod_cast (Nat.totient_pos.mpr (divisorModulus_pos w H)).ne' |
| have hP1 : ((coprimeModulus w H).totient : ℝ) ≠ 0 := by |
| exact_mod_cast (Nat.totient_pos.mpr (coprimeModulus_pos w H)).ne' |
| simp only [Finset.sum_const,Finset.card_univ,pattern_card,nsmul_eq_mul,Nat.cast_mul,weight] |
| field_simp |
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| theorem indicator_binary (w : ℝ) (H qA : ℕ) (beta delta : ∀ t : ℕ,ZMod t) |
| (j : ℕ) (v : FullPattern w H) : indicator w H qA beta delta j v=0 ∨ indicator w H qA beta delta j v=1 := by |
| unfold indicator |
| split_ifs <;> simp |
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| theorem count_eq_indicator_sum (w : ℝ) (H qA J : ℕ) (beta delta : ∀ t : ℕ,ZMod t) |
| (v : FullPattern w H) : (survivorCount w H qA J beta delta v : ℝ)= |
| ∑ j ∈ Finset.range J,indicator w H qA beta delta j v := by |
| simp only [survivorCount,Finset.card_filter,Nat.cast_sum,Nat.cast_ite,Nat.cast_one,Nat.cast_zero,indicator] |
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| end ErdosLargePatternLaw |
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|
| namespace ErdosLargePatternLaw |
| open NumberTheoryLean ErdosVarianceSmallModel ErdosVarianceMoments |
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| attribute [local instance] Classical.propDecidable |
| attribute [local instance] Classical.decEq |
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| noncomputable def projectedUnit (M t : ℕ) [NeZero M] (ht : t ∣ M) (p : (ZMod M)ˣ) : (ZMod t)ˣ := |
| Units.map (ZMod.castHom ht (ZMod t)).toMonoidHom p |
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| theorem projectedUnit_coe (M t : ℕ) [NeZero M] (ht : t ∣ M) (p : (ZMod M)ˣ) : |
| (projectedUnit M t ht p : ZMod t) = (p.val.val : ZMod t) := by |
| change ZMod.castHom ht (ZMod t) p.val = _ |
| calc |
| ZMod.castHom ht (ZMod t) p.val = |
| ZMod.castHom ht (ZMod t) (p.val.val : ZMod M) := |
| congrArg (ZMod.castHom ht (ZMod t)) (ZMod.natCast_zmod_val p.val).symm |
| _ = _ := map_natCast (ZMod.castHom ht (ZMod t)) p.val.val |
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| def divisorCondition (w : ℝ) (H : ℕ) (beta : ∀ t : ℕ,ZMod t) (S : Finset ℕ) |
| (p0 : (ZMod (divisorModulus w H))ˣ) (k0 : ZMod (divisorModulus w H)) : Prop := |
| ∀ t ∈ divisorPrimes w H,∀ j ∈ S,(k0.val : ZMod t)+(p0.val.val : ZMod t)*(j : ZMod t) ≠ beta t |
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| def coprimeCondition (w : ℝ) (H qA : ℕ) (delta : ∀ t : ℕ,ZMod t) (S : Finset ℕ) |
| (p1 : (ZMod (coprimeModulus w H))ˣ) (m1 : ZMod (coprimeModulus w H)) : Prop := |
| ∀ t ∈ coprimePrimes w H,∀ j ∈ S,(p1.val.val : ZMod t)*(qA : ZMod t)* |
| ((m1.val : ZMod t)+(H : ZMod t)*(j : ZMod t)) ≠ delta t |
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| theorem joint_condition_split (w : ℝ) (H qA : ℕ) (beta delta : ∀ t : ℕ,ZMod t) |
| (S : Finset ℕ) (p0 : (ZMod (divisorModulus w H))ˣ) (k0 : ZMod (divisorModulus w H)) |
| (p1 : (ZMod (coprimeModulus w H))ˣ) (m1 : ZMod (coprimeModulus w H)) : |
| jointCondition w H qA beta delta S (p0,k0,p1,m1) ↔ |
| divisorCondition w H beta S p0 k0 ∧ coprimeCondition w H qA delta S p1 m1 := by |
| constructor |
| · intro h |
| exact ⟨fun t ht j hj => (h j hj).1 t ht,fun t ht j hj => (h j hj).2 t ht⟩ |
| · rintro ⟨h0,h1⟩ j hj |
| exact ⟨fun t ht => h0 t ht j hj,fun t ht => h1 t ht j hj⟩ |
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| theorem divisor_condition_card (w : ℝ) (H : ℕ) (beta : ∀ t : ℕ,ZMod t) (S : Finset ℕ) |
| (p0 : (ZMod (divisorModulus w H))ˣ) : |
| (Finset.univ.filter (divisorCondition w H beta S p0)).card = |
| ∏ t ∈ divisorPrimes w H,(t-(positionResidues S t).card) := by |
| let u (t : ℕ) : (ZMod t)ˣ := |
| if ht : t ∈ divisorPrimes w H then projectedUnit (divisorModulus w H) t (Finset.dvd_prod_of_mem _ ht) p0 else 1 |
| have hu (t : ℕ) (ht : t ∈ divisorPrimes w H) : (u t : ZMod t)=(p0.val.val : ZMod t) := by |
| dsimp only [u] |
| rw [dite_eq_left ht] |
| exact projectedUnit_coe (divisorModulus w H) t (Finset.dvd_prod_of_mem _ ht) p0 |
| have hP (t : ℕ) (ht : t ∈ divisorPrimes w H) : t.Prime := |
| (LargePrimeDeletion.mem_cutoffPrimes.mp (Finset.mem_filter.mp ht).1).1 |
| let : NeZero (∏ t ∈ divisorPrimes w H,t) := ⟨(divisorModulus_pos w H).ne'⟩ |
| have hh := prime_unit_avoidance_card (divisorPrimes w H) S hP u beta |
| have he : Finset.univ.filter (divisorCondition w H beta S p0) = |
| Finset.univ.filter (fun k0 : ZMod (divisorModulus w H) => |
| ∀ t ∈ divisorPrimes w H,∀ j ∈ S,(k0.val : ZMod t)+(u t : ZMod t)*(j : ZMod t) ≠ beta t) := by |
| apply Finset.filter_congr |
| intro k0 _ |
| unfold divisorCondition |
| exact forall_congr' (fun t => forall_congr' (fun ht => by rw [hu t ht])) |
| rw [he] |
| exact hh |
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| theorem multiply_unit_iff (t : ℕ) (u : (ZMod t)ˣ) (x c : ZMod t) : |
| (u : ZMod t)*x=c ↔ x=(↑(u⁻¹) : ZMod t)*c := by |
| constructor |
| · intro h |
| have hh := congrArg (fun y : ZMod t => (↑(u⁻¹) : ZMod t)*y) h |
| simpa only [← mul_assoc,Units.inv_mul,one_mul] using hh |
| · intro h |
| rw [h,← mul_assoc,Units.mul_inv,one_mul] |
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| theorem coprime_condition_card (w : ℝ) (H qA : ℕ) (delta : ∀ t : ℕ,ZMod t) (S : Finset ℕ) |
| (p1 : (ZMod (coprimeModulus w H))ˣ) (hqA : ∀ t ∈ coprimePrimes w H,qA.Coprime t) : |
| (Finset.univ.filter (coprimeCondition w H qA delta S p1)).card = |
| ∏ t ∈ coprimePrimes w H,(t-(positionResidues S t).card) := by |
| let u (t : ℕ) : (ZMod t)ˣ := if ht : t ∈ coprimePrimes w H then |
| ZMod.unitOfCoprime H |
| (((LargePrimeDeletion.mem_cutoffPrimes.mp (Finset.mem_filter.mp ht).1).1.coprime_iff_not_dvd.mpr |
| (Finset.mem_filter.mp ht).2).symm) else 1 |
| let coeff (t : ℕ) (ht : t ∈ coprimePrimes w H) : (ZMod t)ˣ := |
| projectedUnit (coprimeModulus w H) t (Finset.dvd_prod_of_mem _ ht) p1*ZMod.unitOfCoprime qA (hqA t ht) |
| let c (t : ℕ) : ZMod t := if ht : t ∈ coprimePrimes w H then (↑((coeff t ht)⁻¹) : ZMod t)*delta t else 0 |
| have hu (t : ℕ) (ht : t ∈ coprimePrimes w H) : (u t : ZMod t)=(H : ZMod t) := by |
| simp only [u,dite_eq_left ht,ZMod.coe_unitOfCoprime] |
| have hc (t : ℕ) (ht : t ∈ coprimePrimes w H) : |
| (coeff t ht : ZMod t)=(p1.val.val : ZMod t)*(qA : ZMod t) := by |
| simp only [coeff,Units.val_mul,ZMod.coe_unitOfCoprime] |
| exact congrArg (fun v : ZMod t => v*(qA : ZMod t)) |
| (projectedUnit_coe (coprimeModulus w H) t (Finset.dvd_prod_of_mem _ ht) p1) |
| have hP (t : ℕ) (ht : t ∈ coprimePrimes w H) : t.Prime := |
| (LargePrimeDeletion.mem_cutoffPrimes.mp (Finset.mem_filter.mp ht).1).1 |
| let : NeZero (∏ t ∈ coprimePrimes w H,t) := ⟨(coprimeModulus_pos w H).ne'⟩ |
| have hh := prime_unit_avoidance_card (coprimePrimes w H) S hP u c |
| have he : Finset.univ.filter (coprimeCondition w H qA delta S p1) = |
| Finset.univ.filter (fun m1 : ZMod (coprimeModulus w H) => |
| ∀ t ∈ coprimePrimes w H,∀ j ∈ S,(m1.val : ZMod t)+(u t : ZMod t)*(j : ZMod t) ≠ c t) := by |
| apply Finset.filter_congr |
| intro m1 _ |
| unfold coprimeCondition |
| apply forall_congr' |
| intro t |
| apply forall_congr' |
| intro ht |
| rw [hu t ht] |
| apply forall_congr' |
| intro j |
| apply forall_congr' |
| intro _hj |
| simp only [c,dite_eq_left ht] |
| rw [← hc t ht] |
| exact not_congr (multiply_unit_iff t (coeff t ht) _ _) |
| rw [he] |
| exact hh |
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| end ErdosLargePatternLaw |
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| end Erdos970 |
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| end OAI |
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