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Proof step 1 from proof Shor-algorithm (conclusion)

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

Heuristic simplifier applied rule simplifier

And obtained the following new goals

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

Rule argument

Additional information

Used simplifier rules: forward-pos: ⊦ i < j ∧ 0 < i → 0 < j ∧ 1 < j; (forward) from specification int-basic1%forward-le-05: ⊦ i < j ∧ 0 ≤ i → 0 < j; (forward) from specification int-abs%transitivity: ⊦ i < i0 ∧ i0 < i1 → i < i1; (forward) from specification oelem[intlista]%lel: ⊦ i < i0 ∧ i = i1 → i1 < i0; (forward) from specification oelem[intlista]%forward-le-01: ⊦ i ≤ j ∧ 0 ≤ i → 0 ≤ j; (forward) from specification int-abs%forward-lels: ⊦ i ≤ j ∧ j < j0 → i < j0; (forward) from specification int-abs%lels: ⊦ ¬ i < j ↔ j ≤ i; from specification int-abs%gcd-div2: ⊦ gcd(i, j) | j; from specification int-gcd