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Proof step 88 from proof ushort-band-shift (previous step)

abs(j0) < 256, ~65536 ≤ (i0 * 256 + j0) * 256 + j, ~256 < j, j < 0, ~256 ≤ j, ~255 ≤ i0 * 256 + j0, j0 ≤ 0, i0 ≤ 0, i0 * 256 + j0 < 0, (i0 * 256 + j0) * 256 + j < 0 ⊦ j = 0, 255 + j0 = (256 * (i0 + 1) + (j0 - 1)) % 256

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 89:
⊦ 0 < j ∧ i ≤ 0 ∧ i0 ≤ 0 ∧ abs(i0) < j → (j * i + i0) % j = i0
Proof step 90:
¬(0 < 256 ∧ i0 + 1 ≤ 0 ∧ j0 - 1 ≤ 0 ∧ abs(j0 - 1) < 256), abs(j0) < 256, ~65536 ≤ (i0 * 256 + j0) * 256 + j, ~256 < j, j < 0, ~256 ≤ j, ~255 ≤ i0 * 256 + j0, j0 ≤ 0, i0 ≤ 0, i0 * 256 + j0 < 0, (i0 * 256 + j0) * 256 + j < 0 ⊦ j = 0, 255 + j0 = (256 * (i0 + 1) + (j0 - 1)) % 256
Proof step 102:
0 < 256 ∧ i0 + 1 ≤ 0 ∧ j0 - 1 ≤ 0 ∧ abs(j0 - 1) < 256, abs(j0) < 256, ~65536 ≤ (i0 * 256 + j0) * 256 + j, ~256 < j, j < 0, ~256 ≤ j, ~255 ≤ i0 * 256 + j0, j0 ≤ 0, i0 ≤ 0, i0 * 256 + j0 < 0, (i0 * 256 + j0) * 256 + j < 0 ⊦ j = 0, 255 + j0 = j0 - 1

Rule argument

seq: ⊦ 0 < j ∧ i ≤ 0 ∧ i0 ≤ 0 ∧ abs(i0) < j → (j * i + i0) % j = i0 subst: [i0, i, j] → [j0 - 1, i0 + 1, 256] rotate: F spec: int-div inst: name: mod-mult-neg

Additional information

Used terms: j0 - 1, i0 + 1, 256