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Proof step 24 from proof RSA-decrypt-ij-vars (previous step)

gcd(i2, (i - 1) * (i0 - 1)) = 1, i2 < i * i0, 2 < i0, 2 < i, prime(i0), prime(i), 0 < j, 0 < i0, 0 < i3, 0 ≤ i2, 1 < i * i0, 0 < i2, 0 < i, j < i * i0, (i - 1) * (i0 - 1) | i2 * i3 - 1 ⊦ j = 0, i3 = 0, i = i0, j = j ^ (i2 * i3) |%| (i * i0)

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 25:
⊦ i | j ↔ (∃ i0. i * i0 = j)
Proof step 26:
gcd(i2, (i - 1) * (i0 - 1)) = 1, i2 < i * i0, 2 < i0, 2 < i, prime(i0), prime(i), 0 < j, 0 < i0, 0 < i3, 0 ≤ i2, 1 < i * i0, 0 < i2, 0 < i, j < i * i0, ∃ i1. ((i - 1) * (i0 - 1)) * i1 = i2 * i3 - 1 ⊦ j = 0, i3 = 0, i = i0, j = j ^ (i2 * i3) |%| (i * i0)

Rule argument

spec: int-gcd inst: name: divides seq: ⊦ i | j ↔ (∃ i0. i * i0 = j) subst: [j, i] → [i2 * i3 - 1, (i - 1) * (i0 - 1)] flags: rewrite

Additional information

Used terms: i2 * i3 - 1, (i - 1) * (i0 - 1)