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Proof step 7 from proof div-sum-ls (previous step)

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

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 8:
⊦ i ≤ j ↔ ¬ j < i
Proof step 9:
0 < ((i + i1) * (j - 1)) / j, j2 < j, j1 < j, 1 < j, 0 < j, 0 < j2, 0 < j1, 0 < i + i1, 0 < (i * j1 + i1 * j2) / j, (i * j1 + i1 * j2) / j ≤ ((i + i1) * (j - 1)) / j, i + i1 ≤ (i * j1 + i1 * j2) / j, 0 ≤ i1, 0 ≤ i ⊦ ((i + i1) * (j - 1)) / j < i + i1

Rule argument

spec: int-abs name: le-def seq: ⊦ i ≤ j ↔ ¬ j < i subst: [j, i] → [((i + i1) * (j - 1)) / j, i + i1] flags: <all paths>

Additional information

Used terms: ((i + i1) * (j - 1)) / j, i + i1