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Proof step 22 from proof congmod-inverse-pos (previous step)

(i * i1) % j = 1, gcd(i, j) = 1, i0 < 0, ≡mod(i * i0, 1, j), 0 ≤ j, 0 ≤ i, 0 < i1, i0 < i1 ⊦ ∃ i0. ≡mod(i * i0, 1, j) ∧ 0 ≤ i0

User applied rule insert rewrite lemma

And obtained the following new goals

Rule argument

spec: int-mod inst: name: rem-mod-pos seq: ⊦ 0 ≤ i ∧ 0 < j → i % j = i |%| j subst: [j, i] → [j, i * i1] flags: rewrite, current

Additional information

Used simplifier rules: cong-zero: ⊦ ≡mod(i, j, 0) ↔ i = j;%zero-ge: ⊦ 0 < i → (0 ≤ i * j ↔ 0 ≤ j); from specification int-mult%zero-gels-04: ⊦ i ≤ 0 ∧ 0 ≤ j → i ≤ j; from specification int-abs%associativity: ⊦ i * j * j0 = (i * j) * j0; from specification int-mult%commutativity: ⊦ i * j = j * i; from specification int-mult%lels: ⊦ ¬ i < j ↔ j ≤ i; from specification int-abs%anti: ⊦ i ≤ j → (j ≤ i ↔ i = j); from specification int-abs%lel: ⊦ i < i0 ∧ i = i1 → i1 < i0; (forward) from specification oelem[intlista]%gcd-zero-04: ⊦ gcd(i, 0) = abs(i); from specification int-gcd%absle: ⊦ 0 ≤ i → abs(i) = i; from specification int-abs%one-01: ⊦ i = 1 → i * j = j; from specification int-mult Used terms: j, i * i1