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Proof step 26 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, 0, j)

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 27:
⊦ 0 ≤ j ∧ 0 ≤ j0 → i ^ (j + j0) = i ^ j * i ^ j0
Proof step 28:
0 ≤ i0 / 2 ∧ 0 ≤ i0 / 2, 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 + i0 / 2) - 1, 0, j)

Rule argument

spec: int-pot inst: name: pot-dist-pos seq: ⊦ 0 ≤ j ∧ 0 ≤ j0 → i ^ (j + j0) = i ^ j * i ^ j0 subst: [j0, j, i] → [i0 / 2, i0 / 2, i] flags: rewrite

Additional information

Used simplifier rules: div-two-zero-01: ⊦ 0 ≤ i / 2 ↔ ~1 ≤ i; from specification int-div%numleledel-01: i0 ≠ 0, i0 ≤ i ⊦ (* i) ≤ j → (* i0) ≤ j; from specification int-abs Used terms: i0 / 2, i0 / 2, i