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Proof step 53 from proof int2bytes-rec-neg-03 (previous step)

i0 * 2 ^ (i * 8) + j0 < ~1, 0 < i, i0 * 2 ^ (i * 8) + j0 < 0, j0 < 0, 0 < 2 ^ (i * 8) + j0, i0 ≤ 0, i0 < 2 ^ (i * 8) + j0, i0 < i, 2 ^ (i * 8) * (i0 + 1) + j0 < 0 ⊦ 2 ^ (i * 8) = 0, j0 = 0, int2bytes(i0 - 1) = int2bytes(((j0 + 1) + i0 * 2 ^ (i * 8)) / 2 ^ (i * 8) - 1)

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 54:
⊦ i0 % j = 0 ∧ i < 0 ∧ i0 < 0 ∧ 0 < j → (i + i0) / j = i / j + i0 / j
Proof step 55:
(i0 * 2 ^ (i * 8)) % 2 ^ (i * 8) = 0 ∧ j0 + 1 < 0 ∧ i0 * 2 ^ (i * 8) < 0 ∧ 0 < 2 ^ (i * 8), i0 * 2 ^ (i * 8) + j0 < ~1, 0 < i, i0 * 2 ^ (i * 8) + j0 < 0, j0 < 0, 0 < 2 ^ (i * 8) + j0, i0 ≤ 0, i0 < 2 ^ (i * 8) + j0, i0 < i, 2 ^ (i * 8) * (i0 + 1) + j0 < 0 ⊦ 2 ^ (i * 8) = 0, j0 = 0, int2bytes(i0 - 1) = int2bytes((j0 + 1) / 2 ^ (i * 8) + (i0 * 2 ^ (i * 8)) / 2 ^ (i * 8) - 1)

Rule argument

seq: ⊦ i0 % j = 0 ∧ i < 0 ∧ i0 < 0 ∧ 0 < j → (i + i0) / j = i / j + i0 / j subst: [j, i0, i] → [2 ^ (i * 8), i0 * 2 ^ (i * 8), j0 + 1] rotate: F spec: int-div inst: name: div-add-neg

Additional information

Used simplifier rules: int2bytes-byte: ~128 ≤ i, i < 128 ⊦ int2bytes((* i)) = i2b(i) ';% ⊦ i + j + i0 = (i + j) + i0;% ⊦ i + j = j + i;% ⊦ i * j = j * i;%j ≠ 0 ⊦ (i * j) % j = 0;% ⊦ i * j * j0 = (i * j) * j0;% ⊦ i + (* j) < (* j0) ↔ i < j0 - j;% ⊦ ¬ i < j ↔ j ≤ i;% ⊦ i < 0 → (~1 ≤ i ↔ i = ~1);% ⊦ 0 < i → (i * j < 0 ↔ j < 0);% ⊦ i ≤ j → (j ≤ i ↔ i = j);% ⊦ 0 < 2 ^ i ↔ 0 ≤ i;% ⊦ 0 < i → (0 ≤ i * j ↔ 0 ≤ j);% ⊦ i + 0 = i;%j ≠ 0 ⊦ (i * j) / j = i;% ⊦ i * 0 = 0;% ⊦ 1 * i = i;% ⊦ i < j → ¬ j < i; Used terms: 2 ^ (i * 8), i0 * 2 ^ (i * 8), j0 + 1