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Proof step 18 from proof mod-rec-neg (previous step)

j ≠ 0, ∃ j1. (i0 + 1) * j + j0 = j1 * j, (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

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 19:
(i0 + 1) * j + j0 = j1 * j, j0 < j, j0 < 0, i0 < j + j0, i0 < j, 0 < j, 0 < j + j0, i0 < 0, j1 * j < j + j0, j1 < 0, j1 < j, j1 < j + j0 ⊦ j = 0, j0 + j = 0

Rule argument

Additional information

Used simplifier rules: mod-mult-01: ⊦ 0 < j → (i * j) |%| j = 0;%zero-gels-03: ⊦ i < 0 ∧ 0 ≤ j → i ≤ j; from specification int-abs%zero-ls: ⊦ 0 < i → (i * j < 0 ↔ j < 0); from specification int-mult%lsone: ⊦ i < 1 ↔ i ≤ 0; from specification int-abs%ls-same: ⊦ 0 < i → (i * j < i ↔ j < 1); from specification int-mult%commutativity: ⊦ i * j = j * i; from specification int-mult%associativity: ⊦ i * j * j0 = (i * j) * j0; from specification int-mult%forward-le-02: ⊦ j < i ∧ i ≤ 0 → j < 0; (forward) from specification int-abs%transitivity: ⊦ i < i0 ∧ i0 < i1 → i < i1; (forward) from specification oelem[intlista]