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Proof step 9 from proof mod-rec-neg (previous step)

i % j < 0, i + j < 0, 0 < j, i < 0, i < j ⊦ i % j + j = (i + j) |%| j

Heuristic elimination applied rule insert elim lemma

And obtained the following new goals

Proof step 10:
i = i0 * j + j0 ∧ j0 ≤ 0 ∧ abs(j0) < j ∧ i0 ≤ 0, j0 < 0, i + j < 0, 0 < j, i < 0, i < j ⊦ j0 + j = (i + j) |%| j

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] → [i, j, i0, j0] 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: i, j, i0, j0