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Proof step 33 from proof RSA-decrypt-ij-vars (previous step)

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

User applied rule insert rewrite lemma

And obtained the following new goals

Rule argument

spec: int-mult inst: name: zero-le seq: ⊦ 0 < i → (i * j ≤ 0 ↔ j ≤ 0) subst: [j, i] → [i0 - 1, i - 1] flags: rewrite

Additional information

Used simplifier rules: sub-ls-01: ⊦ 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%sub-le: ⊦ i - j ≤ i0 ↔ i ≤ j + i0; from specification int-abs%bet: ⊦ 0 < i → (i ≤ 1 ↔ i = 1); from specification int-abs%sub-eq-01: ⊦ i = j → i - j = 0; from specification int-abs%Mult-zero: ⊦ i * 0 = 0; from specification int-mult%associativity: ⊦ i * j * j0 = (i * j) * j0; from specification int-mult%commutativity: ⊦ i * j = j * i; from specification int-mult%eq-one: ⊦ i * j = 1 ↔ abs(i) = 1 ∧ i = j; from specification int-mult%abs-16: ⊦ 0 < j → abs(j) = j; from specification int-abs%gcd-zero-04: ⊦ gcd(i, 0) = abs(i); from specification int-gcd%one-01: ⊦ i = 1 → i * j = j; from specification int-mult%numlslsdel-01: i0 ≠ 0, i0 < i ⊦ (* i) < j → (* i0) < j; from specification int-basic1%lel: ⊦ i < i0 ∧ i = i1 → i1 < i0; (forward) from specification oelem[intlista]%one: ⊦ i ^ 1 = i; from specification int-pot%numlelsdel-01: i0 ≠ 0, i0 ≤ i ⊦ (* i) < j → (* i0) < j; from specification int-abs Used terms: i0 - 1, i - 1