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Proof step 7 from proof mod-add-neg (previous step)

i = i0 * j + j0 ∧ j0 ≤ 0 ∧ abs(j0) < j, i1 < j, i1 < 0, i1 ≤ 0, 1 + i1 * j < j, i < 0, i < j, 0 < j, i + 1 < j, i1 + 1 < j ⊦ (i + i1 * j) % j = j0

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 8:
i0 * j + j0 < j, j0 < j, i1 + 1 < j, 0 < j, i0 * j + j0 < 0, 1 + i1 * j < j, i1 ≤ 0, i1 < 0, i1 < j, abs(j0) < j, j0 ≤ 0, (i0 * j + j0) + 1 < j ⊦ (j * (i1 + i0) + j0) % j = j0

Rule argument

Additional information

Used simplifier rules: succ-01: ⊦ i +1 = i + 1; from specification int-basic2%forward-lels: ⊦ i ≤ j ∧ j < j0 → i < j0; (forward) from specification int-abs%trans: ⊦ i < j ∧ j < j0 → i < j0; (forward) from specification int-basic1%lf-01: ⊦ i < j ∧ j < i0 → i +1 < i0; (forward) from specification int-basic1%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