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Proof step 7 from proof modmod-ma-negpos-neg (previous step)

i0 * i1 * j + j0 + i0 * j1 < j, i0 * i1 * j + j0 + i0 * j1 < 0, 0 ≤ i1, 0 ≤ j1, (i0 * i1 * j + j0 + i0 * j1) + 1 < j, j0 < 0, j0 < j, j0 + 1 < j, 0 ≤ i0, 0 < j, j1 < j ⊦ (j0 + i0 * j1) % j = (i0 * i1 * j + j0 + i0 * j1) % j

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 8:
⊦ i % j = 0 ∧ 0 ≤ i ∧ 0 ≤ i0 ∧ 0 < j → (i + i0) % j = i0 % j
Proof step 9:
¬((i0 * i1 * j) % j = 0 ∧ 0 ≤ i0 * i1 * j ∧ 0 ≤ j0 + i0 * j1 ∧ 0 < j), i0 * i1 * j + j0 + i0 * j1 < j, i0 * i1 * j + j0 + i0 * j1 < 0, 0 ≤ i1, 0 ≤ j1, (i0 * i1 * j + j0 + i0 * j1) + 1 < j, j0 < 0, j0 < j, j0 + 1 < j, 0 ≤ i0, 0 < j, j1 < j ⊦ (j0 + i0 * j1) % j = (i0 * i1 * j + j0 + i0 * j1) % j

Rule argument

spec: int-div inst: name: mod-add1-pospos seq: ⊦ i % j = 0 ∧ 0 ≤ i ∧ 0 ≤ i0 ∧ 0 < j → (i + i0) % j = i0 % j subst: [j, i0, i] → [j, j0 + i0 * j1, i0 * i1 * j] flags: rewrite, current

Additional information

Used terms: j, j0 + i0 * j1, i0 * i1 * j