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Proof step 9 from proof Shor-algorithm (previous step)

gcd(i, j) = 1, gcd(i ^ (i0 / 2) - 1, j) = 1, ≡mod((i ^ (i0 / 2) - 1) * (i ^ (i0 / 2) + 1), 0, j), 1 < i, i < j, ≡mod(i ^ i0, 1, j), even(i0), 0 < j, 0 < i, 0 ≤ i0, 1 < j ⊦ prime(j), j | i ^ (i0 / 2) + 1

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 10:
⊦ ≡mod(i, i0, j) ↔ j | abs(i - i0)
Proof step 11:
gcd(i, j) = 1, gcd(i ^ (i0 / 2) - 1, j) = 1, j | abs((i ^ (i0 / 2) - 1) * (i ^ (i0 / 2) + 1) - 0), 1 < i, i < j, ≡mod(i ^ i0, 1, j), even(i0), 0 < j, 0 < i, 0 ≤ i0, 1 < j ⊦ prime(j), j | i ^ (i0 / 2) + 1

Rule argument

spec: inst: name: congmod-def seq: ⊦ ≡mod(i, i0, j) ↔ j | abs(i - i0) subst: [j, i0, i] → [j, 0, (i ^ (i0 / 2) - 1) * (i ^ (i0 / 2) + 1)] flags: rewrite, current

Additional information

Used terms: j, 0, (i ^ (i0 / 2) - 1) * (i ^ (i0 / 2) + 1)