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Proof step 2 from proof modeq-neg-02 (previous step)

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

User applied rule insert elim lemma

And obtained the following new goals

Proof step 3:
⊦ 0 < j ∧ 0 ≤ i → (i0 = i / j ∧ j0 = i % j ↔ i = i0 * j + j0 ∧ 0 ≤ j0 ∧ j0 < j)
Proof step 4:
j0 = i1 * j + j1 ∧ 0 ≤ j1 ∧ j1 < j, i0 < 0, 0 < j, 0 ≤ j0, i0 * j + j0 < 0 ⊦ j1 = 0, j1 = (i0 * j + j0) % j + j

Rule argument

spec: int-div inst: name: div-elim-pos seq: ⊦ 0 < j ∧ 0 ≤ i → ( i0 = i / j ∧ j0 = i % j ↔ i = i0 * j + j0 ∧ 0 ≤ j0 ∧ j0 < j) subst: [i, j, i0, j0] → [j0, j, i, j1] flags: current, []

Additional information

Used simplifier rules: zero-gels-04: ⊦ i ≤ 0 ∧ 0 ≤ j → i ≤ j; from specification int-abs Used terms: j0, j, i, j1