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Proof step 3 from proof div-add1-posneg-pos-01 (previous step)

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

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

Rule argument

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

Additional information

Used terms: i0, j, i1, j0