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Proof step 8 from proof div-le-pos (previous step)

i0 / j < j0, 0 < j, i0 / j < i / j, i < 0, i ≤ i0, j0 ≤ i / j ⊦

Heuristic elimination applied rule insert elim lemma

And obtained the following new goals

Proof step 9:
⊦ 0 < j ∧ i < 0 → (i0 = i / j ∧ j0 = i % j ↔ i = i0 * j + j0 ∧ j0 ≤ 0 ∧ abs(j0) < j ∧ i0 ≤ 0)
Proof step 10:
i = i1 * j + j1, i < 0, i0 / j < i1, 0 < j, i0 / j < j0, abs(j1) < j, j0 ≤ i1, i ≤ i0, j1 ≤ 0, i1 ≤ 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] → [i, j, i1, j1] flags: <rotated> <all paths>

Additional information

Used simplifier rules: ⊦ i < 0 ∧ 0 ≤ j → i < j; (s) Used terms: i, j, i1, j1