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Proof step 8 from proof mod-add1-posneg-neg (previous step)

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

User applied rule insert elim lemma

And obtained the following new goals

Proof step 9:
⊦ 0 < j ∧ i < 0 → (i0 = i / j ∧ j0 = i % j ↔ i = i0 * j + j0 ∧ j0 ≤ 0 ∧ abs(j0) < j ∧ i0 ≤ 0)
Proof step 10:
j * (i1 + i0) + j0 = i2 * j + j1 ∧ j1 ≤ 0 ∧ abs(j1) < j ∧ i2 ≤ 0, j * (i1 + i0) + j0 < 0, 0 ≤ i0, abs(j0) < j, 0 < j, j0 ≤ 0, j0 < j, i1 ≤ 0, i1 < j, i1 * j + j0 < 0 ⊦ j1 = 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] → [j * (i1 + i0) + j0, j, i, j1] 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) + j0, j, i, j1