| 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 |
| open ErdosHyperbolaError |
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| noncomputable def coordinateUnit (w : ℝ) (H : ℕ) (m : ℤ) (hm : m.natAbs.Coprime H) : |
| (ZMod (H*divisorModulus w H))ˣ := |
| ZMod.unitOfIsCoprime m (by |
| have hgm : Int.gcd m (H : ℤ) = 1 := by |
| simpa only [Int.gcd_def,Int.natAbs_natCast] using hm.gcd_eq_one |
| have hh := source_product_coprime H (divisorModulus w H) (divisorModulus_dvd_H w H) m 0 hgm |
| simpa only [mul_zero,add_zero] using hh) |
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| noncomputable def liftedNumeratorUnit (w : ℝ) (H : ℕ) (C0 : ℤ) |
| (hC0 : Int.gcd C0 (H : ℤ) = 1) (k : ZMod (divisorModulus w H)) : |
| (ZMod (H*divisorModulus w H))ˣ := |
| ZMod.unitOfIsCoprime (C0+(H : ℤ)*(k.val : ℤ)) |
| (source_product_coprime H (divisorModulus w H) (divisorModulus_dvd_H w H) C0 k.val hC0) |
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| noncomputable def restrictionUnit (w : ℝ) (H : ℕ) (C0 : ℤ) |
| (hC0 : Int.gcd C0 (H : ℤ) = 1) (m : ℤ) (hm : m.natAbs.Coprime H) |
| (k : ZMod (divisorModulus w H)) : (ZMod (H*divisorModulus w H))ˣ := |
| liftedNumeratorUnit w H C0 hC0 k*(coordinateUnit w H m hm)⁻¹ |
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| noncomputable def patternClass (w : ℝ) (H : ℕ) [NeZero H] (C0 : ℤ) |
| (hC0 : Int.gcd C0 (H : ℤ) = 1) (m : ℤ) (hm : m.natAbs.Coprime H) (s : Pattern w H) : ℕ := |
| (Nat.chineseRemainder (effective_pattern_coprime w H) |
| (restrictionUnit w H C0 hC0 m hm s.2).val.val s.1.val.val).val |
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| theorem patternClass_components (w : ℝ) (H : ℕ) [NeZero H] (C0 : ℤ) |
| (hC0 : Int.gcd C0 (H : ℤ) = 1) (m : ℤ) (hm : m.natAbs.Coprime H) (s : Pattern w H) : |
| Nat.ModEq (H*divisorModulus w H) (patternClass w H C0 hC0 m hm s) |
| (restrictionUnit w H C0 hC0 m hm s.2).val.val ∧ |
| Nat.ModEq (coprimeModulus w H) (patternClass w H C0 hC0 m hm s) s.1.val.val := |
| (Nat.chineseRemainder (effective_pattern_coprime w H) |
| (restrictionUnit w H C0 hC0 m hm s.2).val.val s.1.val.val).property |
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| theorem patternClass_coprime (w : ℝ) (H : ℕ) [NeZero H] (C0 : ℤ) |
| (hC0 : Int.gcd C0 (H : ℤ) = 1) (m : ℤ) (hm : m.natAbs.Coprime H) (s : Pattern w H) : |
| (patternClass w H C0 hC0 m hm s).Coprime (H*divisorModulus w H*coprimeModulus w H) := by |
| have hh := patternClass_components w H C0 hC0 m hm s |
| have h0 : (patternClass w H C0 hC0 m hm s).Coprime (H*divisorModulus w H) := by |
| apply (ZMod.isUnit_iff_coprime _ _).mp |
| rw [(ZMod.natCast_eq_natCast_iff _ _ _).mpr hh.1,ZMod.natCast_zmod_val] |
| exact (restrictionUnit w H C0 hC0 m hm s.2).isUnit |
| have h1 : (patternClass w H C0 hC0 m hm s).Coprime (coprimeModulus w H) := by |
| apply (ZMod.isUnit_iff_coprime _ _).mp |
| rw [(ZMod.natCast_eq_natCast_iff _ _ _).mpr hh.2,ZMod.natCast_zmod_val] |
| exact s.1.isUnit |
| exact h0.mul_right h1 |
|
|
| end ErdosVarianceSmallModel |
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| end |
|
|
| section |
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|
| namespace ErdosVarianceSmallModel |
|
|
| theorem actual_pattern_class (w : ℝ) (H : ℕ) [NeZero H] (C0 : ℤ) |
| (hC0 : Int.gcd C0 (H : ℤ) = 1) (m : ℤ) (hm : m.natAbs.Coprime H) |
| (s : Pattern w H) (p k : ℕ) |
| (hEquation : (p : ℤ)*m = C0+(H : ℤ)*(k : ℤ)) |
| (hp1 : (p : ZMod (coprimeModulus w H)) = (s.1 : ZMod (coprimeModulus w H))) |
| (hk0 : (k : ZMod (divisorModulus w H)) = s.2) : |
| Int.ModEq ((H*divisorModulus w H*coprimeModulus w H : ℕ) : ℤ) |
| (p : ℤ) (patternClass w H C0 hC0 m hm s : ℤ) := by |
| have hkNat : Nat.ModEq (divisorModulus w H) k s.2.val := by |
| apply (ZMod.natCast_eq_natCast_iff _ _ _).mp |
| rw [ZMod.natCast_zmod_val] |
| exact hk0 |
| have hkInt := Int.natCast_modEq_iff.mpr hkNat |
| have hlift : Int.ModEq ((H : ℤ)*(divisorModulus w H : ℤ)) |
| ((H : ℤ)*(k : ℤ)) ((H : ℤ)*(s.2.val : ℤ)) := hkInt.mul_left' |
| have hpm : Int.ModEq ((H*divisorModulus w H : ℕ) : ℤ) ((p : ℤ)*m) |
| (C0+(H : ℤ)*(s.2.val : ℤ)) := by |
| simpa only [Nat.cast_mul,← hEquation] using hlift.add_left C0 |
| have hpmZ : (p : ZMod (H*divisorModulus w H))*(m : ZMod (H*divisorModulus w H)) = |
| (liftedNumeratorUnit w H C0 hC0 s.2 : ZMod (H*divisorModulus w H)) := by |
| simpa only [liftedNumeratorUnit,ZMod.coe_unitOfIsCoprime,Int.cast_mul,Int.cast_natCast] using |
| (ZMod.intCast_eq_intCast_iff _ _ _).mpr hpm |
| have hminv : (m : ZMod (H*divisorModulus w H))* |
| (↑((coordinateUnit w H m hm)⁻¹) : ZMod (H*divisorModulus w H)) = 1 := by |
| simpa only [coordinateUnit,ZMod.coe_unitOfIsCoprime] using (coordinateUnit w H m hm).mul_inv |
| have hmul := congrArg (fun x : ZMod (H*divisorModulus w H) => |
| x*(↑((coordinateUnit w H m hm)⁻¹) : ZMod (H*divisorModulus w H))) hpmZ |
| have hpUnit : (p : ZMod (H*divisorModulus w H)) = |
| (restrictionUnit w H C0 hC0 m hm s.2 : ZMod (H*divisorModulus w H)) := by |
| simpa only [mul_assoc,hminv,mul_one,restrictionUnit,Units.val_mul] using hmul |
| have hp0 : Nat.ModEq (H*divisorModulus w H) p (restrictionUnit w H C0 hC0 m hm s.2).val.val := by |
| apply (ZMod.natCast_eq_natCast_iff _ _ _).mp |
| rw [ZMod.natCast_zmod_val] |
| exact hpUnit |
| have hp1' : Nat.ModEq (coprimeModulus w H) p s.1.val.val := by |
| apply (ZMod.natCast_eq_natCast_iff _ _ _).mp |
| rw [ZMod.natCast_zmod_val] |
| exact hp1 |
| exact Int.natCast_modEq_iff.mpr (Nat.chineseRemainder_modEq_unique (effective_pattern_coprime w H) hp0 hp1') |
|
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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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