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Proof step 18 from proof less-mult-div-01 (previous step)

j0 < j, i1 < j, 0 < j1, 0 < j, 0 < i0 * j + j0, 0 < (j0 * i1 + (i0 * i1) * j) / j, 0 ≤ j0, 0 ≤ i0, i0 * j + j0 ≤ (j0 * i1 + (i0 * i1) * j) / j, 0 ≤ i0 * j + j0, 0 ≤ (j0 * i1 + (i0 * i1) * j) / j ⊦ ((i0 * i1) * j) % j = 0 ∧ 0 ≤ j0 * i1 ∧ 0 ≤ (i0 * i1) * j ∧ 0 < j

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 19:
0 < i0 * j + j0, 0 < j, 0 < j1, i1 < j, j0 < j, 0 < (i1 * (i0 * j + j0)) / j, 0 ≤ (i1 * (i0 * j + j0)) / j, 0 ≤ i0 * j + j0, i0 * j + j0 ≤ (i1 * (i0 * j + j0)) / j, 0 ≤ i0, 0 ≤ j0 ⊦ (j0 = 0 ∨ 0 ≤ i1) ∧ (i0 = 0 ∨ 0 ≤ i1)

Rule argument

Additional information

Used simplifier rules: mod-mult-04: j ≠ 0 ⊦ (i * j) % j = 0;% ⊦ 0 < i → (0 ≤ i * j ↔ 0 ≤ j); (s)% ⊦ 0 ≤ i → (0 ≤ i * j ↔ i = 0 ∨ 0 ≤ j); (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% ⊦ i * j + i * j0 = i * (j + j0); (s)