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Proof step 4 from proof modeq-neg-02 (previous step)

j0 = i1 * j + j1 ∧ 0 ≤ j1 ∧ j1 < j, i0 < 0, 0 < j, 0 ≤ j0, i0 * j + j0 < 0 ⊦ j1 = 0, j1 = (i0 * j + j0) % j + j

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 5:
j * (i1 + i0) + j1 < 0, 0 ≤ i1 * j + j1, 0 < j, 1 < j, 0 < j1, i0 < 0, j1 < j ⊦ j1 = (j * (i1 + i0) + j1) % j + j, j1 = 0

Rule argument

Additional information

Used simplifier rules: zero-gels-01: ⊦ i < 0 ∧ 0 ≤ j → i < j; from specification int-abs%ls: ⊦ i + j < j ↔ i < 0; from specification int-basic2%ls-pos: ⊦ 0 < i → (i * j < i * j0 ↔ j < j0); from specification int-mult%ls-01: ⊦ i + j < i + i0 ↔ j < i0; from specification int-basic2%forward-le-05: ⊦ i < j ∧ 0 ≤ i → 0 < j; (forward) from specification int-abs%forward-pos: ⊦ i < j ∧ 0 < i → 0 < j ∧ 1 < j; (forward) from specification int-basic1%forward-le-01: ⊦ i ≤ j ∧ 0 ≤ i → 0 ≤ j; (forward) from specification int-abs%forward-le-02: ⊦ j < i ∧ i ≤ 0 → j < 0; (forward) from specification int-abs%forward-lsle: ⊦ i < j ∧ j ≤ j0 → i < j0; (forward) from specification int-abs%lels-02: ⊦ i ≠ j → (i ≤ j ↔ i < j); from specification int-abs%add-c: ⊦ i + j = j + i; from specification int-basic2%add-a: ⊦ i + j + i0 = (i + j) + i0; from specification int-basic2%commutativity: ⊦ i * j = j * i; from specification int-mult%associativity: ⊦ i * j * j0 = (i * j) * j0; from specification int-mult%dist-rmerge: ⊦ i * j + i * j0 = i * (j + j0); from specification int-mult