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Proof step 29 from proof int2bytes-rec-neg-06 (previous step)

0 < i, j + 2 ^ (i * 8 - 1) < 0 ⊦ j % 2 ^ (i * 8) = 0, int2bytes(j / 2 ^ (i * 8) - 1) = int2bytes(((j + 1) + ~1 * 2 ^ (i * 8)) / 2 ^ (i * 8))

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 30:
⊦ i0 % j = 0 ∧ i < 0 ∧ i0 < 0 ∧ 0 < j → (i + i0) / j = i / j + i0 / j
Proof step 31:
¬((~1 * 2 ^ (i * 8)) % 2 ^ (i * 8) = 0 ∧ j + 1 < 0 ∧ ~1 * 2 ^ (i * 8) < 0 ∧ 0 < 2 ^ (i * 8)), 0 < i, j + 2 ^ (i * 8 - 1) < 0 ⊦ j % 2 ^ (i * 8) = 0, int2bytes(j / 2 ^ (i * 8) - 1) = int2bytes(((j + 1) + ~1 * 2 ^ (i * 8)) / 2 ^ (i * 8))
Proof step 41:
(~1 * 2 ^ (i * 8)) % 2 ^ (i * 8) = 0 ∧ j + 1 < 0 ∧ ~1 * 2 ^ (i * 8) < 0 ∧ 0 < 2 ^ (i * 8), 0 < i, j + 2 ^ (i * 8 - 1) < 0 ⊦ j % 2 ^ (i * 8) = 0, int2bytes(j / 2 ^ (i * 8) - 1) = int2bytes((j + 1) / 2 ^ (i * 8) + (~1 * 2 ^ (i * 8)) / 2 ^ (i * 8))

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), ~1 * 2 ^ (i * 8), j + 1] rotate: F spec: int-div inst: name: div-add-neg

Additional information

Used terms: 2 ^ (i * 8), ~1 * 2 ^ (i * 8), j + 1