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Proof step 17 from proof div-sum-ls (previous step)

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

User applied rule insert rewrite lemma

And obtained the following new goals

Rule argument

spec: int-basic2 inst: name: nls seq: ⊦ ¬ i < j ∧ ¬ i0 < j0 → ¬ i + i0 < j + j0 subst: [j0, j, i0, i] → [i1 * j2, i * j1, (j - 1) * i1, (j - 1) * i] flags: rewrite

Additional information

Used simplifier rules: dist-rmerge: ⊦ i * j + i * j0 = i * (j + j0); from specification int-mult%associativity: ⊦ i * j * j0 = (i * j) * j0; from specification int-mult%commutativity: ⊦ i * j = j * i; from specification int-mult%add-a: ⊦ i + j + i0 = (i + j) + i0; from specification int-basic2%add-c: ⊦ i + j = j + i; from specification int-basic2%ls-forward: ⊦ i * j < i * j0 → i ≠ 0; (forward) from specification int-mult%lels-02: ⊦ i ≠ j → (i ≤ j ↔ i < j); from specification int-abs%ls-pos: ⊦ 0 < i → (i * j < i * j0 ↔ j < j0); from specification int-mult%sub-ls: ⊦ i - j < i0 ↔ i < j + i0; from specification int-abs%succ-02: ⊦ i < j + 1 ↔ i ≤ j; from specification int-abs%lels-03: ⊦ i < j → (¬ ¬ j ≤ i ↔ false); from specification int-abs Used terms: i1 * j2, i * j1, (j - 1) * i1, (j - 1) * i