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Proof step 17 from proof modmod-mult (previous step)

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

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 18:
¬ i < 0, ¬ j < 0 ⊦ ¬ i + j < 0
Proof step 19:
j1 < j, j0 < j, 0 < j, (i1 * (j1 + i2 * j) + j0 * i2) * j + j0 * j1 < 0, 0 ≤ i1, 0 ≤ j1, 0 ≤ j0, 0 ≤ i2 ⊦ (j0 * j1) % j = ((i1 * (j1 + i2 * j) + j0 * i2) * j + j0 * j1) % j, ¬ (i1 * (j1 + i2 * j) + j0 * i2) * j < 0 ∧ ¬ j0 * j1 < 0

Rule argument

spec: int-basic2 name: pls-01 seq: ¬ i < 0, ¬ j < 0 ⊦ ¬ i + j < 0 subst: [j, i] → [j0 * j1, (i1 * (j1 + i2 * j) + j0 * i2) * j] flags: <all paths>

Additional information

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