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Proof step 30 from proof less-mult-div-01 (previous step)

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

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 31:
0 < i0 * j + j0, 0 < j1, i1 < j, j0 < j, 0 < j, 0 < (j0 * i1) / j + i0 * i1, 0 ≤ (j0 * i1) / j + i0 * i1, 0 ≤ i0 * j + j0, i0 * j + j0 ≤ (j0 * i1) / j + i0 * i1, 0 ≤ i0, 0 ≤ j0, i0 = 0 ∨ 0 ≤ i1, j0 = 0 ∨ 0 ≤ i1 ⊦

Rule argument

Additional information

Used simplifier rules: mod-mult-04: j ≠ 0 ⊦ (i * j) % j = 0;%div-mult-03: j ≠ 0 ⊦ (i * j) / j = i;%forward-le-05: ⊦ i < j ∧ 0 ≤ i → 0 < j; (forward) from specification int-abs% ⊦ 0 ≤ i → (0 ≤ i * j ↔ i = 0 ∨ 0 ≤ j); (s)% ⊦ 0 < i → (0 ≤ i * j ↔ 0 ≤ j); (s)% ⊦ i < j → (j ≤ i ↔ false); (as)%add-c: ⊦ i + j = j + i; from specification int-basic2%add-a: ⊦ i + j + i0 = (i + j) + i0; from specification int-basic2% ⊦ i + 0 = i; (s)% ⊦ i < 0 ∧ 0 ≤ j → i ≤ j; (s)% ⊦ i < 0 → (0 ≤ i * j ↔ j ≤ 0); (s)%commutativity: ⊦ i * j = j * i; from specification int-mult%associativity: ⊦ i * j * j0 = (i * j) * j0; from specification int-mult% ⊦ i * 0 = 0; (s)