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Proof step 45 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), ~1 ≤ ((i - 1) * (i0 - 1)) * i1 ⊦ i1 = 0, j = 0, i3 = 0, i = i0, j = j ^ (((i - 1) * (i0 - 1)) * i1 + 1) |%| (i * i0), ≡mod(j ^ (((i - 1) * (i0 - 1)) * i1 + 1), j, i0) ∧ ≡mod(j ^ (((i - 1) * (i0 - 1)) * i1 + 1), j, i)

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 46:
⊦ 0 ≤ i ∧ 0 ≤ i0 ∧ j ≠ 0 → (≡mod(i, i0, j) ↔ i |%| j = i0 |%| j)
Proof step 47:
0 ≤ j ^ (((i - 1) * (i0 - 1)) * i1 + 1) ∧ 0 ≤ j ∧ i0 ≠ 0, 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), ~1 ≤ ((i - 1) * (i0 - 1)) * i1 ⊦ i1 = 0, j = 0, i3 = 0, i = i0, j = j ^ (((i - 1) * (i0 - 1)) * i1 + 1) |%| (i * i0), j ^ (((i - 1) * (i0 - 1)) * i1 + 1) |%| i0 = j |%| i0 ∧ ≡mod(j ^ (((i - 1) * (i0 - 1)) * i1 + 1), j, i)

Rule argument

spec: inst: name: congmod-def-mod-01 seq: ⊦ 0 ≤ i ∧ 0 ≤ i0 ∧ j ≠ 0 → (≡mod(i, i0, j) ↔ i |%| j = i0 |%| j) subst: [j, i0, i] → [i0, j, j ^ (((i - 1) * (i0 - 1)) * i1 + 1)] flags: rewrite, current

Additional information

Used simplifier rules: lels-02: ⊦ i ≠ j → (i ≤ j ↔ i < j); from specification int-abs%lels-01: ⊦ ¬ i ≤ j ↔ j < i; from specification int-abs%notneg-02: ⊦ 0 < i → ¬ i ^ j < 0; from specification int-pot%minus-one-01: ⊦ i * ~1 = ~ i; from specification int-mult%minus-02: ⊦ ~(i - 1) = ~ i + 1; from specification int-abs%neg-15: ⊦ i + ~ j = i - 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%Mult-zero: ⊦ i * 0 = 0; from specification int-mult%sub-le-01: ⊦ i ≤ j - i0 ↔ i + i0 ≤ j; from specification int-abs%ax-04: ⊦ i + 0 = i; from specification int-basic2%bet: ⊦ 0 < i → (i ≤ 1 ↔ i = 1); from specification int-abs%sub-eq-01: ⊦ i = j → i - j = 0; from specification int-abs%gcd-zero-04: ⊦ gcd(i, 0) = abs(i); from specification int-gcd%abs-16: ⊦ 0 < j → abs(j) = j; from specification int-abs%one: ⊦ i ^ 1 = i; from specification int-pot%ls-01: ⊦ i < j → ¬ j < i; from specification int-basic1 Used terms: i0, j, j ^ (((i - 1) * (i0 - 1)) * i1 + 1)