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Proof step 23 from proof modmod-mult (previous step)

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

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 24:
⊦ (i + i0) * j = i * j + i0 * j
Proof step 25:
0 < j, j0 < j, j1 < j, 0 ≤ i2, 0 ≤ j0, 0 ≤ j1, 0 ≤ i1 ⊦ (j0 * j1) % j = ((i1 * j) * (i2 * j + j1) + j0 * (i2 * j + j1)) % j, (i1 * j) * (i2 * j + j1) + j0 * (i2 * j + j1) = (i1 * (j1 + i2 * j) + j0 * i2) * j + j0 * j1

Rule argument

spec: int-mult name: dist-lsplit seq: ⊦ (i + i0) * j = i * j + i0 * j subst: [j, i0, i] → [i2 * j + j1, j0, i1 * j] flags: <all paths>

Additional information

Used terms: i2 * j + j1, j0, i1 * j