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Proof step 5 from proof modeq-pos-03 (previous step)

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

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 6:
⊦ i * j = j * i
Proof step 7:
j1 < i1 + i0, i1 * j + j1 < j, i1 * j + j1 < i0, i1 ≤ 0, 0 < i1 + i0, j1 < j, j1 < i0, j1 + 1 < j, 1 < j, j1 + 1 < i0, i1 * j + j1 < 0, (i1 * j + j1) + 1 < i0, j1 < 0, 0 < j + j1, (i1 * j + j1) + 1 < j, 0 ≤ (i1 + i0) * j + j1, i1 < i0, 0 < i0, i1 < j, i1 < j + j1, i1 ≤ (i1 + i0) * j + j1, 0 ≤ j, 0 < j, (i1 * j + j1) + 1 < i1 + i0, i1 * j + j1 < i1 + i0, j1 + 1 < i1 + i0, i1 ≤ j ⊦ j1 = 0, ((i1 + i0) * j + j1) % j = j1 + j

Rule argument

spec: int-mult inst: name: commutativity seq: ⊦ i * j = j * i subst: [j, i] → [i1 + i0, j] flags: rewrite

Additional information

Used terms: i1 + i0, j