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Proof step 11 from proof div-le-pos (previous step)

i1 * j + j1 < 0, i0 / j < 0, i0 / j < i1, j1 < j, i1 < j, 0 < j, i0 / j < j0, abs(j1) < j, j0 < j, j0 ≤ 0, j0 ≤ i1, i1 * j + j1 ≤ i0, j1 ≤ 0, i1 ≤ 0 ⊦

User applied rule insert elim lemma

And obtained the following new goals

Proof step 12:
⊦ 0 < j ∧ 0 ≤ i → (i0 = i / j ∧ j0 = i % j ↔ i = i0 * j + j0 ∧ 0 ≤ j0 ∧ j0 < j ∧ 0 ≤ i0)
Proof step 13:
j0 < j, abs(j1) < j, i0 / j < j0, 0 < j, i1 < j, j1 < j, i0 / j < i1, i0 / j < 0, i1 * j + j1 < 0, i1 ≤ 0, j1 ≤ 0, i1 * j + j1 ≤ i0, j0 ≤ i1, j0 ≤ 0 ⊦ 0 < j ∧ 0 ≤ i0

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] → [i0, j, i, j2] flags: <rotated> <all paths>

Additional information

Used simplifier rules: ⊦ i < j → (j ≤ i ↔ false); (as) Used terms: i0, j, i, j2