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

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

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: modzero-02: ⊦ i < 0 ∧ j0 < j ∧ 0 < j → i * j + j0 < 0; from specification int-mult%ls-dist-03: ⊦ i * (j + 1) + j1 < i ↔ i * j + j1 < 0; from specification int-mult%commutativity: ⊦ i * j = j * i; from specification int-mult%associativity: ⊦ i * j * j0 = (i * j) * j0; from specification int-mult%add-c: ⊦ i + j = j + i; from specification int-basic2%add-a: ⊦ i + j + i0 = (i + j) + i0; from specification int-basic2%zero-gels-01: ⊦ i < 0 ∧ 0 ≤ j → i < j; from specification int-abs%numls-03: ⊦ i + (* j) < (* j0) ↔ i < j0 - j; from specification int-abs%ls-same: ⊦ 0 < i → (i * j < i ↔ j < 1); from specification int-mult%ls-01: ⊦ i + j < i + i0 ↔ j < i0; from specification int-basic2%mod-less-01: ⊦ 0 < j → i % j < j; from specification int-div%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]%lel: ⊦ i < i0 ∧ i = i1 → i1 < i0; (forward) from specification oelem[intlista]%mod-ngez: ⊦ 0 < j ∧ i < 0 → (0 ≤ i % j ↔ i % j = 0); from specification int-div%lels: ⊦ ¬ i < j ↔ j ≤ i; from specification int-abs