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Proof step 7 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, 1 < i * i0, 0 < i2, 0 < i, j < i * i0, (i - 1) * (i0 - 1) | 1 ⊦ i = i0, j = 1

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 8:
⊦ 0 < i → (i | 1 ↔ i = 1)
Proof step 9:
¬ 0 < (i - 1) * (i0 - 1), gcd(i2, (i - 1) * (i0 - 1)) = 1, i2 < i * i0, 2 < i0, 2 < i, prime(i0), prime(i), 0 ≤ j, 0 < i0, 1 < i * i0, 0 < i2, 0 < i, j < i * i0, (i - 1) * (i0 - 1) | 1 ⊦ i = i0, j = 1

Rule argument

spec: int-gcd inst: name: div-one-pos seq: ⊦ 0 < i → (i | 1 ↔ i = 1) subst: [i] → [(i - 1) * (i0 - 1)] flags: rewrite

Additional information

Used simplifier rules: eq-one: ⊦ i * j = 1 ↔ abs(i) = 1 ∧ i = j; from specification int-mult%abs-09: ⊦ 0 < i → abs(i - 1) = abs(i) - 1; from specification int-abs%abs-16: ⊦ 0 < j → abs(j) = j; from specification int-abs%sub-eq: ⊦ i - j = i0 ↔ i = i0 + j; from specification int-abs%add-a: ⊦ i + j + i0 = (i + j) + i0; from specification int-basic2%add-c: ⊦ i + j = j + i; from specification int-basic2%associativity: ⊦ i * j * j0 = (i * j) * j0; from specification int-mult%commutativity: ⊦ i * j = j * i; from specification int-mult Used terms: (i - 1) * (i0 - 1)