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Proof step 5 from proof modeq-neg-02 (previous step)

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

User 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:
j * (i1 + i0) + j1 = i2 * j + j0 ∧ j0 ≤ 0 ∧ abs(j0) < j, j * (i1 + i0) + j1 < 0, 0 ≤ i1 * j + j1, 0 < j, 1 < j, 0 < j1, i0 < 0, j1 < j ⊦ j1 = j0 + j, j1 = 0

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] → [j * (i1 + i0) + j1, j, i, j0] flags: current, []

Additional information

Used simplifier rules: zero-gels-01: ⊦ i < 0 ∧ 0 ≤ j → i < j; from specification int-abs Used terms: j * (i1 + i0) + j1, j, i, j0