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Proof step 29 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 ^ (2 * i0 / 2) - 1, 0, j)

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 30:
⊦ i % 2 ≠ 1 ∧ 0 < i → i / 2 * 2 = i
Proof step 31:
¬(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 ^ (2 * i0 / 2) - 1, 0, j)
Proof step 36:
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)

Rule argument

spec: int-div inst: name: div2zero seq: ⊦ i % 2 ≠ 1 ∧ 0 < i → i / 2 * 2 = i subst: [i] → [i0] flags: rewrite

Additional information

Used simplifier rules: commutativity: ⊦ i * j = j * i; from specification int-mult Used terms: i0