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Proof step 3 from proof zero-04 (previous step)

i / i0 + j / i0 < i0, i / i0 + j / i0 < i + j, i / i0 + j / i0 < 0, 0 < i0, (i / i0 + j / i0) + 1 < i0, 0 ≤ i + j ⊦

User applied rule insert elim lemma

And obtained the following new goals

Proof step 4:
⊦ 0 < j ∧ i < 0 → (i0 = i / j ∧ j0 = i % j ↔ i = i0 * j + j0 ∧ j0 ≤ 0 ∧ abs(j0) < j)
Proof step 5:
(i / i0 + j / i0) + 1 < i0, 0 < i0, i / i0 + j / i0 < 0, i / i0 + j / i0 < i + j, i / i0 + j / i0 < i0, 0 ≤ i + j ⊦ 0 < i0 ∧ i < 0
Proof step 14:
i = i1 * i0 + j0, (i1 + j / i0) + 1 < i0, 0 < i0, i1 + j / i0 < 0, i1 + j / i0 < i + j, i1 + j / i0 < i0, abs(j0) < i0, i < 0, 0 ≤ i + j, j0 ≤ 0 ⊦

Rule argument

spec: <toplevel> 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] → [i, i0, i1, j0] flags: <rotated> <all paths>

Additional information

Used terms: i, i0, i1, j0