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Proof step 32 from proof Shor-algorithm (previous step)

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

User applied rule cut formula

And obtained the following new goals

Proof step 33:
gcd(i ^ (i0 / 2) - 1, j) = 1, gcd(i, j) = 1, 1 < j, 0 ≤ i0, 0 < i, 0 < j, even(i0), ≡mod(i ^ i0, 1, j), i < j, 1 < i ⊦ i0 = 0, ≡mod(i ^ (2 * i0 / 2) - 1, 0, j), prime(j), ≡mod(i ^ (i0 / 2), ~1, j), i0 % 2 ≠ 1 ∧ 0 < i0

Rule argument

fma: i0 = 0

Additional information

Used simplifier rules: cong-same: ⊦ ≡mod(i, i, j);%Pot-zero: ⊦ j ^ 0 = 1; from specification int-pot%gcd-zero-04: ⊦ gcd(i, 0) = abs(i); from specification int-gcd%abs: ⊦ 1 < i → abs(i) = i; from specification int-abs%gcd-com: ⊦ gcd(i, j) = gcd(j, i); from specification int-gcd%associativity: ⊦ i * j * j0 = (i * j) * j0; from specification int-mult%commutativity: ⊦ i * j = j * i; from specification int-mult Used formulas: i0 = 0