| import Mathlib.RingTheory.Coprime.Lemmas |
| import Mathlib.Data.Nat.Prime.Int |
| import Mathlib.Tactic.LinearCombination |
|
|
| namespace OAI |
|
|
| /-! |
| # Arithmetic support at a reversing node |
|
|
| These are the elementary support facts preceding the progression estimate |
| in Section 8. The numerator is automatically coprime to inherited slots; |
| the remaining condition at a new prime is exactly nondivisibility by its square. |
| -/ |
|
|
| namespace Ostmann |
|
|
| theorem reversal_numerator_coprime (v w HL HR : ℤ) |
| (hLR : IsCoprime HL HR) (hv : IsCoprime v HL) (hw : IsCoprime w HR) : |
| IsCoprime (v * HR - w * HL) HL ∧ IsCoprime (v * HR - w * HL) HR := by |
| constructor |
| · exact IsCoprime.sub_mul_right_left_iff.mpr (hv.mul_left hLR.symm) |
| · exact IsCoprime.mul_sub_right_left_iff.mpr (hw.mul_left hLR) |
|
|
| / |
| theorem reversal_inherited_coprime (v w HL HR s u p : ℤ) |
| (hLR : IsCoprime HL HR) (hv : IsCoprime v HL) (hw : IsCoprime w HR) |
| (hrel : v * HR - w * HL = s * u * p) : |
| IsCoprime p (HL * HR) ∧ IsCoprime u (HL * HR) := by |
| obtain ⟨hL, hR⟩ := reversal_numerator_coprime v w HL HR hLR hv hw |
| have hboth := hL.mul_right hR |
| rw [hrel] at hboth |
| exact ⟨hboth.of_mul_left_right, hboth.of_mul_left_left.of_mul_left_right⟩ |
|
|
| / |
| theorem reversal_square_test (b s u p : ℕ) (hb : b.Prime) |
| (hbs : ¬ b ∣ s) (hbu : b ∣ u) (hbu2 : ¬ b ^ 2 ∣ u) : |
| (¬ b ^ 2 ∣ s * u * p) ↔ ¬ b ∣ p := by |
| obtain ⟨v, rfl⟩ := hbu |
| have hbv : ¬ b ∣ v := by |
| intro hv |
| apply hbu2 |
| simpa only [pow_two] using mul_dvd_mul_left b hv |
| have heq : s * (b * v) * p = b * (s * v * p) := by ring |
| rw [heq, pow_two, mul_dvd_mul_iff_left hb.ne_zero] |
| simp only [hb.dvd_mul, hbs, hbv, or_self, false_or] |
|
|
| / |
| theorem reversal_integral_of_prime_tests (s N : ℤ) (U : Finset ℕ) |
| (hprime : ∀ b ∈ U, b.Prime) (hs : s ∣ N) |
| (hsu : ∀ b ∈ U, IsCoprime s (b : ℤ)) |
| (hdiv : ∀ b ∈ U, (b : ℤ) ∣ N) : |
| (s * ∏ b ∈ U, (b : ℤ)) ∣ N := by |
| classical |
| have hpair : (U : Set ℕ).Pairwise (fun b c => IsCoprime (b : ℤ) (c : ℤ)) := by |
| intro b hb c hc hbc |
| exact Int.isCoprime_iff_gcd_eq_one.mpr (by |
| simpa only [Int.gcd_natCast_natCast] using (hprime b hb).coprime_iff_not_dvd.mpr |
| (fun h => hbc ((hprime c hc).eq_one_or_self_of_dvd b h |>.resolve_left (hprime b hb).ne_one))) |
| have hprod : (∏ b ∈ U, (b : ℤ)) ∣ N := Finset.prod_dvd_of_coprime hpair hdiv |
| have hcop : IsCoprime s (∏ b ∈ U, (b : ℤ)) := IsCoprime.prod_right hsu |
| exact hcop.mul_dvd hs hprod |
|
|
| end Ostmann |
|
|
| end OAI |
|
|
