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Proof step 42 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, 0 < i1, 0 ≤ (i - 1) * (i0 - 1) ⊦ i1 = 0, j = 0, i3 = 0, i = i0, j = j ^ (((i - 1) * (i0 - 1)) * i1 + 1) |%| (i * i0)

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 43:
≡mod(i ^ i0, i, j), ≡mod(i ^ i0, i, j0), prime(j), prime(j0), 0 < i, j ≠ j0, 0 ≤ i0 ⊦ i ^ i0 |%| (j * j0) = i |%| (j * j0)
Proof step 44:
¬ ( ≡mod(j ^ (((i - 1) * (i0 - 1)) * i1 + 1), j, i) ∧ ≡mod(j ^ (((i - 1) * (i0 - 1)) * i1 + 1), j, i0) ∧ prime(i) ∧ prime(i0) ∧ 0 < j ∧ i ≠ i0 ∧ 0 ≤ ((i - 1) * (i0 - 1)) * i1 + 1), 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, 0 < i1, 0 ≤ (i - 1) * (i0 - 1) ⊦ i1 = 0, j = 0, i3 = 0, i = i0, j = j ^ (((i - 1) * (i0 - 1)) * i1 + 1) |%| (i * i0)
Proof step 58:
≡mod(j ^ (((i - 1) * (i0 - 1)) * i1 + 1), j, i) ∧ ≡mod(j ^ (((i - 1) * (i0 - 1)) * i1 + 1), j, i0) ∧ prime(i) ∧ prime(i0) ∧ 0 < j ∧ i ≠ i0 ∧ 0 ≤ ((i - 1) * (i0 - 1)) * i1 + 1, 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, 0 < i1, 0 ≤ (i - 1) * (i0 - 1) ⊦ i1 = 0, j = 0, i3 = 0, i = i0, j = j |%| (i * i0)

Rule argument

spec: inst: name: mod-crt seq: ≡mod(i ^ i0, i, j), ≡mod(i ^ i0, i, j0), prime(j), prime(j0), 0 < i, j ≠ j0, 0 ≤ i0 ⊦ i ^ i0 |%| (j * j0) = i |%| (j * j0) subst: [j0, j, i0, i] → [i0, i, ((i - 1) * (i0 - 1)) * i1 + 1, j] flags: rewrite, current

Additional information

Used terms: i0, i, ((i - 1) * (i0 - 1)) * i1 + 1, j