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Proof step 16 from proof mod-rec-neg (previous step)

((i0 + 1) * j + j0) % j = 0, (i0 + 1) * j + j0 < 0, i0 < 0, 0 < j + j0, 0 < j, i0 < j, i0 < j + j0, j0 < 0, j0 < j ⊦ j0 + j = ((i0 + 1) * j + j0) |%| j

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, (i0 + 1) * j + j0] flags: rewrite

Additional information

Used simplifier rules: 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%ax-04: ⊦ i + 0 = i; from specification int-basic2%add-a: ⊦ i + j + i0 = (i + j) + i0; from specification int-basic2%add-c: ⊦ i + j = j + i; from specification int-basic2%zero-gels-01: ⊦ i < 0 ∧ 0 ≤ j → i < j; from specification int-abs%ls-01: ⊦ i < j → ¬ j < i; from specification int-basic1 Used terms: j, (i0 + 1) * j + j0