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Proof step 81 from proof ushort-band-shift (previous step)

0 ≤ (256 * (i + 255) + 256 + j) / 256, i * 256 + j < 0, ~256 < i, ~65536 ≤ i * 256 + j, ~256 < j, i ≤ 0, j < 0 ⊦ j = 0, 255 + i % 256 = ((256 * (i + 255) + 256 + j) / 256) % 256

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 82:
⊦ 0 < j ∧ 0 ≤ i ∧ 0 ≤ i0 ∧ i0 < j → (j * i + i0) / j = i
Proof step 83:
0 < 256 ∧ 0 ≤ i + 255 ∧ 0 ≤ 256 + j ∧ 256 + j < 256, 0 ≤ i + 255, i * 256 + j < 0, ~256 < i, ~65536 ≤ i * 256 + j, ~256 < j, i ≤ 0, j < 0 ⊦ j = 0, 255 + i % 256 = (i + 255) % 256

Rule argument

seq: ⊦ 0 < j ∧ 0 ≤ i ∧ 0 ≤ i0 ∧ i0 < j → (j * i + i0) / j = i subst: [i0, i, j] → [256 + j, i + 255, 256] rotate: F spec: int-div inst: name: div-mult-pos

Additional information

Used simplifier rules: ⊦ i * j + i = i * (j + 1);% ⊦ i * j * j0 = (i * j) * j0;% ⊦ i * j = j * i;% ⊦ i + j + i0 = (i + j) + i0;% ⊦ i + j = j + i;% ⊦ (* j) ≤ i0 + (* j0) ↔ j - j0 ≤ i0;% ⊦ ¬ i ≤ j ↔ j < i;% ⊦ i < j → ¬ j < i;% ⊦ i + (* j) < (* j0) ↔ i < j0 - j;% ⊦ ¬ i < j ↔ j ≤ i;% ⊦ i ≠ j → (i ≤ j ↔ i < j);%i + 1 = i0 ⊦ j < (* i0) → ¬ (* i) < j; Used terms: 256 + j, i + 255, 256