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Proof step 8 from proof modmod-mult (previous step)

i0 = i2 * j + j1, 0 < j, j0 < j, j1 < j, 0 ≤ i1, 0 ≤ i0, 0 ≤ j0, 0 ≤ j1 ⊦ (j0 * j1) % j = ((i1 * j + j0) * i0) % j

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 9:
j1 < j, j0 < j, 0 < j, 0 ≤ i1, 0 ≤ j1, 0 ≤ j0, 0 ≤ i2 ⊦ (j0 * j1) % j = ((i1 * j + j0) * (i2 * j + j1)) % j

Rule argument

Additional information

Used simplifier rules: 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)