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Proof step 5 from proof mod-mult-neg-02 (previous step)

j + i0 ≤ 0, i + i0 < 0, i0 < 0, 0 < j, i ≤ 0, i < j, i0 < j ⊦ i0 = 0, (j * i + i0) % j = i0 % 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 ∧ i0 ≤ 0)
Proof step 7:
i0 = i1 * j + j0 ∧ j0 ≤ 0 ∧ abs(j0) < j ∧ i1 ≤ 0, j + i0 ≤ 0, i + i0 < 0, i0 < 0, 0 < j, i ≤ 0, i < j, i0 < j ⊦ i0 = 0, (j * i + i0) % j = j0

Rule argument

spec: int-div inst: name: div-elim-negpos seq: ⊦ 0 < j ∧ i < 0 → ( i0 = i / j ∧ j0 = i % j ↔ i = i0 * j + j0 ∧ j0 ≤ 0 ∧ abs(j0) < j ∧ i0 ≤ 0) subst: [i, j, i0, j0] → [i0, j, i1, j0] flags: rewrite, current

Additional information

Used terms: i0, j, i1, j0