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Proof step 9 from proof div-sum-ls (previous step)

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

User applied rule insert lemma

And obtained the following new goals

Rule argument

spec: <toplevel> name: less-mult-div-01 seq: ⊦ 0 < i ∧ 0 < j ∧ 0 < j1 ∧ i1 < j → (i * i1) / j < i subst: [i, j, j1, i1] → [i + i1, j, 1, j - 1]

Additional information

Used simplifier rules: ⊦ i < j ↔ ¬ j ≤ i; (ss)% ⊦ i - j < i0 ↔ i < j + i0; (s)% ⊦ i < j + 1 ↔ i ≤ j; (s)%add-a: ⊦ i + j + i0 = (i + j) + i0; from specification int-basic2%add-c: ⊦ i + j = j + i; from specification int-basic2% ⊦ i ≤ j → (j < i ↔ false); (as)% ⊦ i ≤ j ↔ ¬ j < i; (ss)% ⊦ ¬ i < i; (s) Used terms: i + i1, j, 1, j - 1