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Proof step 3 from proof div-sum-ls (previous step)

(i * j1 + i1 * j2) / j ≤ (i * (j - 1) + i1 * (j - 1)) / j, j1 < j, j2 < j, i + i1 ≤ (i * j1 + i1 * j2) / j, 0 ≤ i, 0 ≤ i1, 0 < i + i1, 0 < j1, 0 < j2, 0 < j, 1 < j, 0 < (i * j1 + i1 * j2) / j ⊦

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 4:
⊦ (i + i0) * j = i * j + i0 * j
Proof step 5:
(i * j1 + i1 * j2) / j ≤ ((i + i1) * (j - 1)) / j, j1 < j, j2 < j, i + i1 ≤ (i * j1 + i1 * j2) / j, 0 ≤ i, 0 ≤ i1, 0 < i + i1, 0 < j1, 0 < j2, 0 < j, 1 < j, 0 < (i * j1 + i1 * j2) / j ⊦

Rule argument

spec: int-mult inst: name: dist-lsplit seq: ⊦ (i + i0) * j = i * j + i0 * j subst: [i0, j, i] → [i1, j - 1, i] flags: rewrite

Additional information

Used terms: i1, j - 1, i