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Proof step 11 from proof modmod-mult (previous step)

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

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 12:
0 ≤ i0 * j + j0, 0 < j, 0 ≤ j0 ⊦ (i0 * j + j0) % j = j0 % j
Proof step 13:
0 < j, j0 < j, j1 < j, 0 ≤ i2, 0 ≤ j0, 0 ≤ j1, 0 ≤ i1 ⊦ (j0 * j1) % j = ((i1 * i2 * j + i1 * j1 + j0 * i2) * j + j0 * j1) % j, 0 ≤ (i1 * i2 * j + i1 * j1 + j0 * i2) * j + j0 * j1 ∧ 0 < j ∧ 0 ≤ j0 * j1

Rule argument

spec: <toplevel> name: modeq-pos-02 seq: 0 ≤ i0 * j + j0, 0 < j, 0 ≤ j0 ⊦ (i0 * j + j0) % j = j0 % j subst: [j0, j, i0] → [j0 * j1, j, i1 * i2 * j + i1 * j1 + j0 * i2] flags: <all paths>

Additional information

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