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Proof step 3 from proof div-le-modnotzero (previous step)

0 ≤ i, 0 < j, 0 ≤ i ⊦ i0 ≤ i / j ↔ i0 * j < i, i % j = 0

Heuristic elimination applied rule insert elim lemma

And obtained the following new goals

Proof step 4:
⊦ 0 < j ∧ 0 ≤ i → (i0 = i / j ∧ j0 = i % j ↔ i = i0 * j + j0 ∧ 0 ≤ j0 ∧ j0 < j ∧ 0 ≤ i0)
Proof step 5:
i = i1 * j + j0 ∧ 0 ≤ j0 ∧ j0 < j ∧ 0 ≤ i1, 0 ≤ i, 0 < j, 0 ≤ i ⊦ i0 ≤ i1 ↔ i0 * j < i, j0 = 0

Rule argument

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

Additional information

Used simplifier rules: zero-gels-04: ⊦ i ≤ 0 ∧ 0 ≤ j → i ≤ j; from specification int-abs Used terms: i, j, i1, j0