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Proof step 5 from proof mod-add-neg (previous step)

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

Heuristic elimination applied rule insert elim lemma

And obtained the following new goals

Proof step 6:
⊦ 0 < j ∧ i < 0 → (i0 = i / j ∧ j0 = i % j ↔ i = i0 * j + j0 ∧ j0 ≤ 0 ∧ abs(j0) < j)
Proof step 7:
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

Rule argument

spec: int-div inst: name: div-elim2 seq: ⊦ 0 < j ∧ i < 0 → ( i0 = i / j ∧ j0 = i % j ↔ i = i0 * j + j0 ∧ j0 ≤ 0 ∧ abs(j0) < j) subst: [i, j, i0, j0] → [i, j, i0, j0] flags: rewrite, current

Additional information

Used terms: i, j, i0, j0