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Proof step 36 from proof ushort-band-shift (previous step)

1 + i * 256 + (j0 - 1) = i * 256 + j0, s2i(sho) = (i * 256 + (j0 - 1)) * 256, i * 256 + (j0 - 1) < 0, ~256 ≤ i * 256 + (j0 - 1), abs(j0) < 256, i * 256 + (j0 - 1) < ~1, j0 ≤ 0, i ≤ 0 ⊦ 255 + j0 = (256 * (i + 1) + (j0 - 1)) % 256

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 37:
⊦ 0 < j ∧ i ≤ 0 ∧ i0 ≤ 0 ∧ abs(i0) < j → (j * i + i0) % j = i0
Proof step 38:
¬(0 < 256 ∧ i + 1 ≤ 0 ∧ j0 - 1 ≤ 0 ∧ abs(j0 - 1) < 256), 1 + i * 256 + (j0 - 1) = i * 256 + j0, s2i(sho) = (i * 256 + (j0 - 1)) * 256, i * 256 + (j0 - 1) < 0, ~256 ≤ i * 256 + (j0 - 1), abs(j0) < 256, i * 256 + (j0 - 1) < ~1, j0 ≤ 0, i ≤ 0 ⊦ 255 + j0 = (256 * (i + 1) + (j0 - 1)) % 256
Proof step 61:
0 < 256 ∧ i + 1 ≤ 0 ∧ j0 - 1 ≤ 0 ∧ abs(j0 - 1) < 256, 1 + i * 256 + (j0 - 1) = i * 256 + j0, s2i(sho) = (i * 256 + (j0 - 1)) * 256, i * 256 + (j0 - 1) < 0, ~256 ≤ i * 256 + (j0 - 1), abs(j0) < 256, i * 256 + (j0 - 1) < ~1, j0 ≤ 0, i ≤ 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, i + 1, 256] rotate: F spec: int-div inst: name: mod-mult-neg

Additional information

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