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Proof step 4 from proof mod-add-neg (previous step)

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

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 5:
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 = i % j

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%lf-01: ⊦ i < j ∧ j < i0 → i +1 < i0; (forward) from specification int-basic1%zero-ls: ⊦ 0 < i → (i * j < 0 ↔ j < 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%lsone: ⊦ i < 1 ↔ i ≤ 0; from specification int-abs%ls-same: ⊦ 0 < i → (i * j < i ↔ j < 1); 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%add-zero: ⊦ i + 0 = i; from specification int-basic2