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Proof step 15 from proof mod-neg (previous step)

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

Heuristic elimination applied rule insert elim lemma

And obtained the following new goals

Rule argument

spec: int-div inst: name: div-elim-negpos seq: ⊦ 0 < j ∧ i < 0 → ( i0 = i / j ∧ j0 = i % j ↔ i = i0 * j + j0 ∧ j0 ≤ 0 ∧ abs(j0) < j ∧ i0 ≤ 0) subst: [i, j, i0, j0] → [i0 * j + j0, j, i, j1] flags:

Additional information

Used simplifier rules: zero-gels-01: ⊦ i < 0 ∧ 0 ≤ j → i < j; from specification int-abs%⊦ 0 < j ∧ i < 0 → ( i0 = i / j ∧ j0 = i % j ↔ i = i0 * j + j0 ∧ j0 ≤ 0 ∧ abs(j0) < j ∧ i0 ≤ 0); (elim) Used terms: i0 * j + j0, j, i, j1