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

j2 < j0, j1 < j, 0 < j, 0 < j0, 0 ≤ j1, 0 ≤ j2, 0 ≤ i1 ⊦ ((i1 * j0 + j2) * j + j1) % (j * j0) = j1 + j2 * j

User applied rule cut formula

And obtained the following new goals

Proof step 9:
(i1 * j0 + j2) * j + j1 = (j2 * j + j1) + i1 * j * j0, j2 < j0, j1 < j, 0 < j, 0 < j0, 0 ≤ j1, 0 ≤ j2, 0 ≤ i1 ⊦ ((i1 * j0 + j2) * j + j1) % (j * j0) = j1 + j2 * j

Rule argument

fma: (i1 * j0 + j2) * j + j1 = (j2 * j + j1) + i1 * j * j0

Additional information

Used simplifier rules: add-in: ⊦ i + j = i + i0 ↔ j = i0; from specification int-basic2%eq-dist: ⊦ (i + j) * i0 = j0 + j * i0 ↔ i * i0 = j0; from specification int-mult%associativity: ⊦ i * j * j0 = (i * j) * j0; from specification int-mult%commutativity: ⊦ i * j = j * i; from specification int-mult%add-a: ⊦ i + j + i0 = (i + j) + i0; from specification int-basic2%add-c: ⊦ i + j = j + i; from specification int-basic2 Used formulas: (i1 * j0 + j2) * j + j1 = (j2 * j + j1) + i1 * j * j0