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Proof step 32 from proof RSA-decrypt-ij-vars (previous step)

i1 < 0, gcd(i2, (i - 1) * (i0 - 1)) = 1, i2 * i3 = ((i - 1) * (i0 - 1)) * i1 + 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)) * i1 < 0, j = 0, i3 = 0, i = i0, j = j ^ (i2 * i3) |%| (i * i0)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 33:
i2 * i3 = ((i - 1) * (i0 - 1)) * i1 + 1, gcd(i2, (i - 1) * (i0 - 1)) = 1, (i - 1) * (i0 - 1) < i, (i - 1) * (i0 - 1) < i2, (i - 1) * (i0 - 1) < i3, (i - 1) * (i0 - 1) < i0, (i - 1) * (i0 - 1) < j, (i - 1) * (i0 - 1) ≤ 0, i1 < i, i1 < i3, i1 < i0, i1 < j, i1 < 0, j < i * i0, 0 < i, 0 < i2, 1 < i * i0, 0 ≤ i2, 0 < i3, 0 < i0, 0 < j, prime(i), prime(i0), 2 < i, 2 < i0, i2 < i * i0, (i - 1) * (i0 - 1) < i * i0, i1 < i * i0 ⊦ j = j ^ (i2 * i3) |%| (i * i0), i = i0, i3 = 0, j = 0

Rule argument

Additional information

Used simplifier rules: zero-gels-01: ⊦ i < 0 ∧ 0 ≤ j → i < j; from specification int-abs%zero-gels-04: ⊦ i ≤ 0 ∧ 0 ≤ j → i ≤ j; from specification int-abs%forward-le-02: ⊦ j < i ∧ i ≤ 0 → j < 0; (forward) from specification int-abs%transitivity: ⊦ i < i0 ∧ i0 < i1 → i < i1; (forward) from specification oelem[intlista]%forward-le-04: ⊦ j ≤ i ∧ i ≤ 0 → j ≤ 0; (forward) from specification int-abs%forward-lele: ⊦ j ≤ j0 ∧ j0 ≤ j1 → j ≤ j1; (forward) from specification int-abs%forward-lels: ⊦ i ≤ j ∧ j < j0 → i < j0; (forward) from specification int-abs%commutativity: ⊦ i * j = j * i; from specification int-mult%associativity: ⊦ i * j * j0 = (i * j) * j0; from specification int-mult%lels: ⊦ ¬ i < j ↔ j ≤ i; from specification int-abs%zero-ls-01: ⊦ i < 0 → (i * j < 0 ↔ 0 < j); from specification int-mult