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Proof step 36 from proof Shor-algorithm (previous step)

i0 % 2 ≠ 1 ∧ 0 < i0, 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 - 1, 0, j)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 37:
i0 % 2 = 0, gcd(i ^ (i0 / 2) - 1, j) = 1, gcd(i, j) = 1, 0 < i0, 1 < j, 0 ≤ i0, even(i0), ≡mod(i ^ i0, 1, j), i < j, 1 < i, 0 < i, 0 < j, 2 ≤ i0 ⊦ ≡mod(i ^ i0 - 1, 0, j), prime(j), ≡mod(i ^ (i0 / 2), ~1, j)

Rule argument

Additional information

Used simplifier rules: mod-ls-same-01: j ≠ 0 ⊦ i % j < i ↔ 0 < i ∧ (abs(j) = 1 ∨ abs(j) ≤ i); from specification int-div%forward-le-05: ⊦ i < j ∧ 0 ≤ i → 0 < j; (forward) from specification int-abs%lel: ⊦ i < i0 ∧ i = i1 → i1 < i0; (forward) from specification oelem[intlista]%mod-two-06: ⊦ 0 < i → (i % 2 ≠ 1 ↔ i % 2 = 0); from specification int-div