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Proof step 5 from proof modmod-add2-posneg-neg-01 (previous step)

i1 < 0, j + i1 * j ≤ i, j + i1 * j + j0 ≤ 0, i1 * j < i, i1 * j + j0 < j, 0 ≤ i + j0, i + i1 * j + j0 < 0, i + i1 * j + j0 < j, abs(j0) < j, (i + i1 * j + j0) + 1 < j, 0 ≤ i, (i1 * j + j0) + 1 < j, 0 < j, j0 ≤ 0, j0 < j, i1 < j, i1 + 1 < j ⊦ j = 1, (i + j0) % j = (i + i1 * j + j0) % j + j, (i + j0) % j = 0

User applied rule cut formula

And obtained the following new goals

Proof step 6:
i + i1 * j + j0 = i1 * j + i + j0, i1 < 0, j + i1 * j ≤ i, j + i1 * j + j0 ≤ 0, i1 * j < i, i1 * j + j0 < j, 0 ≤ i + j0, i + i1 * j + j0 < 0, i + i1 * j + j0 < j, abs(j0) < j, (i + i1 * j + j0) + 1 < j, 0 ≤ i, (i1 * j + j0) + 1 < j, 0 < j, j0 ≤ 0, j0 < j, i1 < j, i1 + 1 < j ⊦ j = 1, (i + j0) % j = (i + i1 * j + j0) % j + j, (i + j0) % j = 0

Rule argument

fma: i + i1 * j + j0 = i1 * j + i + j0

Additional information

Used simplifier rules: add-a: ⊦ i + j + i0 = (i + j) + i0; from specification int-basic2%add-c: ⊦ i + j = j + i; from specification int-basic2 Used formulas: i + i1 * j + j0 = i1 * j + i + j0