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Proof step 20 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) * (i ^ (i0 / 2) + 1), 0, j), ≡mod(i ^ (i0 / 2), ~1, j), prime(j)

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 21:
⊦ (i + i0) * j = i * j + i0 * j
Proof step 22:
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) * (i ^ (i0 / 2) - 1) + 1 * (i ^ (i0 / 2) - 1), 0, j), ≡mod(i ^ (i0 / 2), ~1, j), prime(j)

Rule argument

spec: int-mult inst: name: dist-lsplit seq: ⊦ (i + i0) * j = i * j + i0 * j subst: [j, i0, i] → [i ^ (i0 / 2) - 1, 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, 1, i ^ (i0 / 2)