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Proof step 5 from proof diveq-pos (previous step)

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

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 6:
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

Rule argument

Additional information

Used simplifier rules: ⊦ i ≤ i; (s)% ⊦ i < j + 1 ↔ i ≤ j; (s)% ⊦ 0 < i → (i * j < i * j0 ↔ j < j0); (s)%forward-lels: ⊦ i ≤ j ∧ j < j0 → i < j0; (forward) from specification int-abs% ⊦ i +1 = i + 1; (s)%add-c: ⊦ i + j = j + i; from specification int-basic2%add-a: ⊦ i + j + i0 = (i + j) + i0; from specification int-basic2%commutativity: ⊦ i * j = j * i; from specification int-mult%associativity: ⊦ i * j * j0 = (i * j) * j0; from specification int-mult% ⊦ 0 ≤ j1 ∧ j1 < j → (0 ≤ i0 * j + j1 ↔ 0 ≤ i0); (s)