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Proof step 52 from proof int2bytes-rec-neg-06 (previous step)

0 < i, i0 * 2 ^ (i * 8) + j0 < ~1, i0 ≤ 0, i0 < i, i0 < 2 ^ (i * 8) + j0, (i0 * 2 ^ (i * 8) + j0) + 2 ^ (i * 8 - 1) < 0, i0 * 2 ^ (i * 8) + j0 < 0, j0 < 0, 0 < 2 ^ (i * 8) + j0 ⊦ 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 53:
⊦ i0 % j = 0 ∧ i < 0 ∧ i0 < 0 ∧ 0 < j → (i + i0) / j = i / j + i0 / j
Proof step 54:
¬((i0 * 2 ^ (i * 8)) % 2 ^ (i * 8) = 0 ∧ j0 + 1 < 0 ∧ i0 * 2 ^ (i * 8) < 0 ∧ 0 < 2 ^ (i * 8)), 0 < i, i0 * 2 ^ (i * 8) + j0 < ~1, i0 ≤ 0, i0 < i, i0 < 2 ^ (i * 8) + j0, (i0 * 2 ^ (i * 8) + j0) + 2 ^ (i * 8 - 1) < 0, i0 * 2 ^ (i * 8) + j0 < 0, j0 < 0, 0 < 2 ^ (i * 8) + j0 ⊦ 2 ^ (i * 8) = 0, j0 = 0, int2bytes(i0 - 1) = int2bytes(((j0 + 1) + i0 * 2 ^ (i * 8)) / 2 ^ (i * 8) - 1)
Proof step 57:
(i0 * 2 ^ (i * 8)) % 2 ^ (i * 8) = 0 ∧ j0 + 1 < 0 ∧ i0 * 2 ^ (i * 8) < 0 ∧ 0 < 2 ^ (i * 8), 0 < i, i0 * 2 ^ (i * 8) + j0 < ~1, i0 ≤ 0, i0 < i, i0 < 2 ^ (i * 8) + j0, (i0 * 2 ^ (i * 8) + j0) + 2 ^ (i * 8 - 1) < 0, i0 * 2 ^ (i * 8) + j0 < 0, j0 < 0, 0 < 2 ^ (i * 8) + j0 ⊦ 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 terms: 2 ^ (i * 8), i0 * 2 ^ (i * 8), j0 + 1