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Proof step 6 from proof diveq-pos (previous step)

i0 * j + j0 < ((i0 * j + j0) / j + 1) * j, 0 < j, j0 < j, (i0 * j + j0) / j * j ≤ i0 * j + j0, 0 ≤ i0, 0 ≤ j0 ⊦ (i0 * j + j0) / j = i0

User applied rule insert spec-lemma

And obtained the following new goals

Proof step 7:
j * i + j0 < j * i0, abs(j0) < j ⊦ i ≤ i0
Proof step 8:
j0 < j, 0 < j, i0 * j + j0 < ((i0 * j + j0) / j + 1) * j, 0 ≤ j0, 0 ≤ i0, (i0 * j + j0) / j * j ≤ i0 * j + j0, i0 ≤ (i0 * j + j0) / j + 1 ⊦ (i0 * j + j0) / j = i0

Rule argument

spec: int-mult name: modlsp-02 seq: j * i + j0 < j * i0, abs(j0) < j ⊦ i ≤ i0 subst: [j, i, j0, i0] → [j, i0, j0, (i0 * j + j0) / j + 1]

Additional information

Used simplifier rules: commutativity: ⊦ i * j = j * i; from specification int-mult% ⊦ 0 ≤ i → abs(i) = i; (s) Used terms: j, i0, j0, (i0 * j + j0) / j + 1