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Proof step 171 from proof bytes2int-int2bytes-len-eq (previous step)

# bytes = # int2bytes((bytes2int(bytes.butlast) * 256 + band(b2i(bytes.last), 255)) / 256) + 1, 0 < bytes2int(bytes.butlast) * 256 + band(b2i(bytes.last), 255), 255 < bytes2int(bytes.butlast) * 256 + band(b2i(bytes.last), 255), ∀ bytes0. # bytes0 < # bytes → true → # bytes0 = # int2bytes(bytes2int(bytes0)) → bytes0 = int2bytes(bytes2int(bytes0)) ⊦ bytes = [], # bytes = 1, bytes = int2bytes((bytes2int(bytes.butlast) * 256 + band(b2i(bytes.last), 255)) / 256) + int2bytes((bytes2int(bytes.butlast) * 256 + band(b2i(bytes.last), 255)) % 256).last '

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 172:
⊦ i % j = 0 ∧ 0 ≤ i ∧ 0 ≤ i0 ∧ 0 < j → (i + i0) / j = i / j + i0 / j
Proof step 173:
¬((bytes2int(bytes.butlast) * 256) % 256 = 0 ∧ 0 ≤ bytes2int(bytes.butlast) * 256 ∧ 0 ≤ band(b2i(bytes.last), 255) ∧ 0 < 256), # bytes = # int2bytes((bytes2int(bytes.butlast) * 256 + band(b2i(bytes.last), 255)) / 256) + 1, 0 < bytes2int(bytes.butlast) * 256 + band(b2i(bytes.last), 255), 255 < bytes2int(bytes.butlast) * 256 + band(b2i(bytes.last), 255), ∀ bytes0. # bytes0 < # bytes → true → # bytes0 = # int2bytes(bytes2int(bytes0)) → bytes0 = int2bytes(bytes2int(bytes0)) ⊦ bytes = [], # bytes = 1, bytes = int2bytes((bytes2int(bytes.butlast) * 256 + band(b2i(bytes.last), 255)) / 256) + int2bytes((bytes2int(bytes.butlast) * 256 + band(b2i(bytes.last), 255)) % 256).last '
Proof step 176:
(bytes2int(bytes.butlast) * 256) % 256 = 0 ∧ 0 ≤ bytes2int(bytes.butlast) * 256 ∧ 0 ≤ band(b2i(bytes.last), 255) ∧ 0 < 256, # bytes = # int2bytes((bytes2int(bytes.butlast) * 256) / 256 + band(b2i(bytes.last), 255) / 256) + 1, 0 < bytes2int(bytes.butlast) * 256 + band(b2i(bytes.last), 255), 255 < bytes2int(bytes.butlast) * 256 + band(b2i(bytes.last), 255), ∀ bytes0. # bytes0 < # bytes → true → # bytes0 = # int2bytes(bytes2int(bytes0)) → bytes0 = int2bytes(bytes2int(bytes0)) ⊦ bytes = [], # bytes = 1, bytes = int2bytes((bytes2int(bytes.butlast) * 256) / 256 + band(b2i(bytes.last), 255) / 256) + int2bytes((bytes2int(bytes.butlast) * 256 + band(b2i(bytes.last), 255)) % 256).last '

Rule argument

seq: ⊦ i % j = 0 ∧ 0 ≤ i ∧ 0 ≤ i0 ∧ 0 < j → (i + i0) / j = i / j + i0 / j subst: [j, i0, i] → [256, band(b2i(bytes.last), 255), bytes2int(bytes.butlast) * 256] rotate: F spec: int-div inst: name: div-add1-pospos

Additional information

Used terms: 256, band(b2i(bytes.last), 255), bytes2int(bytes.butlast) * 256