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f4b3ad1 | 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 | 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)
/-- Both the inserted integer and its compensation product inherit all required coprimalities. -/
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⟩
/-- A prime occurring once in `u` occurs once in the numerator exactly when it misses `p`. -/
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]
/-- Divisibility by the frequency and all distinct new prime factors gives integer reconstruction. -/
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
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