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Proof step 23 from proof Shor-algorithm (previous step)

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

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 24:
⊦ (i - j) * j0 = i * j0 - j * j0
Proof step 25:
gcd(i, j) = 1, gcd(i ^ (i0 / 2) - 1, j) = 1, 1 < i, i < j, ≡mod(i ^ i0, 1, j), even(i0), 0 < j, 0 < i, 0 ≤ i0, 1 < j ⊦ ≡mod(i ^ (i0 / 2), ~1, j), prime(j), ≡mod((i ^ (i0 / 2) * i ^ (i0 / 2) - 1 * i ^ (i0 / 2)) + i ^ (i0 / 2) - 1, 0, j)

Rule argument

spec: int-mult inst: name: dist-sub-split seq: ⊦ (i - j) * j0 = i * j0 - j * j0 subst: [j0, j, i] → [i ^ (i0 / 2), 1, i ^ (i0 / 2)] flags: rewrite

Additional information

Used simplifier rules: commutativity: ⊦ i * j = j * i; from specification int-mult Used terms: i ^ (i0 / 2), 1, i ^ (i0 / 2)