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Proof step 6 from proof modmod-ma-negpos-neg (previous step)

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

User applied rule cut formula

And obtained the following new goals

Proof step 7:
j0 + i0 * (i1 * j + j1) = i0 * i1 * j + j0 + i0 * j1, j1 < j, j0 + i0 * (i1 * j + j1) < 0, j0 < 0, 0 < j, 0 ≤ i1, 0 ≤ j1, 0 ≤ i0 ⊦ (j0 + i0 * j1) % j = (j0 + i0 * (i1 * j + j1)) % j

Rule argument

fma: j0 + i0 * (i1 * j + j1) = i0 * i1 * j + j0 + i0 * j1

Additional information

Used simplifier rules: ⊦ i + j = i + i0 ↔ j = i0; (s)% ⊦ (i + j) * i0 = j0 + j * i0 ↔ i * i0 = j0; (s)%commutativity: ⊦ i * j = j * i; from specification int-mult%associativity: ⊦ i * j * j0 = (i * j) * j0; 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: j0 + i0 * (i1 * j + j1) = i0 * i1 * j + j0 + i0 * j1