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Proof step 6 from proof div-add1-posneg-pos-01 (previous step)

i % j = 0, j0 < 0, 0 < j, i1 * j + j0 < 0, 0 < j + j0, i1 < j + j0, i1 < j, i1 < i, 0 < i, 0 ≤ i, 0 ≤ i + i1 * j + j0, i1 ≤ 0 ⊦ i / j + i1 = (i + i1 * j + j0) / j + 1, j0 = 0

Heuristic elimination applied rule insert elim lemma

And obtained the following new goals

Proof step 7:
⊦ 0 < j ∧ 0 ≤ i → (i0 = i / j ∧ j0 = i % j ↔ i = i0 * j + j0 ∧ 0 ≤ j0 ∧ j0 < j ∧ 0 ≤ i0)
Proof step 8:
i = i0 * j + j1, j1 = 0, 0 < i, i1 < i, i1 < j, i1 < j + j0, 0 < j + j0, i1 * j + j0 < 0, 0 < j, j0 < 0, j1 < j, i1 ≤ 0, 0 ≤ i + i1 * j + j0, 0 ≤ i, 0 ≤ j1, 0 ≤ i0 ⊦ i0 + i1 = (i + i1 * j + j0) / j + 1, j0 = 0

Rule argument

spec: <toplevel> 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, i0, j1] flags: <rotated> <all paths>

Additional information

Used terms: i, j, i0, j1