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Proof step 15 from proof modmod-add-zero (previous step)

(i + j0) % j = 0, abs(j0) < j, 0 < j, j0 ≤ 0, j0 < j, i1 ≤ 0, i1 < j, i1 * j + j0 < 0 ⊦ 0 = j, (i1 * j + i + j0) % j = 0, 0 ≤ i1 * j + j0, i ≤ 0 ∧ i1 * j + j0 ≤ 0

User applied rule insert rewrite lemma

And obtained the following new goals

Rule argument

spec: int-div inst: name: modex seq: j ≠ 0 ⊦ i % j = 0 ↔ (∃ j0. i = j0 * j) subst: [j, i] → [j, i + j0] flags: rewrite, current

Additional information

Used simplifier rules: mod-mult-04: j ≠ 0 ⊦ (i * j) % j = 0;%dist-rmerge: ⊦ i * j + i * j0 = i * (j + j0); from specification int-mult%associativity: ⊦ i * j * j0 = (i * j) * j0; from specification int-mult%commutativity: ⊦ i * j = j * i; from specification int-mult%add-a: ⊦ i + j + i0 = (i + j) + i0; from specification int-basic2%add-c: ⊦ i + j = j + i; from specification int-basic2 Used terms: j, i + j0