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Proof step 50 from proof bytes2int-int2bytes-len-eq (previous step)

# bytes1 = # int2bytes((bytes2int(bytes1) * 256 + band(b2i(by), 255)) / 256 - 1), band(b2i(by), 255) < 256, bytes2int(bytes1) * 256 + band(b2i(by), 255) < ~256, bytes2int(bytes1) < 0, ∀ bytes0. # bytes0 < #(bytes1 + by ') → true → # bytes0 = # int2bytes(bytes2int(bytes0)) → bytes0 = int2bytes(bytes2int(bytes0)) ⊦ bytes1 + by ' = int2bytes((bytes2int(bytes1) * 256 + band(b2i(by), 255)) / 256 - 1) + int2bytes((bytes2int(bytes1) * 256 + band(b2i(by), 255)) % 256).last ', bytes1 = [], bytes2int(bytes1) * 256 + band(b2i(by), 255) = ~256, (bytes2int(bytes1) * 256 + band(b2i(by), 255)) % 256 = 0, b2i(by) = 0

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 51:
⊦ i % j = 0 ∧ i + i0 < 0 ∧ i < 0 ∧ 0 ≤ i0 ∧ i0 % j ≠ 0 ∧ 0 < j → (i + i0) % j = i0 % j - j
Proof step 52:
¬ ( (bytes2int(bytes1) * 256) % 256 = 0 ∧ bytes2int(bytes1) * 256 + band(b2i(by), 255) < 0 ∧ bytes2int(bytes1) * 256 < 0 ∧ 0 ≤ band(b2i(by), 255) ∧ band(b2i(by), 255) % 256 ≠ 0 ∧ 0 < 256), # bytes1 = # int2bytes((bytes2int(bytes1) * 256 + band(b2i(by), 255)) / 256 - 1), band(b2i(by), 255) < 256, bytes2int(bytes1) * 256 + band(b2i(by), 255) < ~256, bytes2int(bytes1) < 0, ∀ bytes0. # bytes0 < #(bytes1 + by ') → true → # bytes0 = # int2bytes(bytes2int(bytes0)) → bytes0 = int2bytes(bytes2int(bytes0)) ⊦ bytes1 + by ' = int2bytes((bytes2int(bytes1) * 256 + band(b2i(by), 255)) / 256 - 1) + int2bytes((bytes2int(bytes1) * 256 + band(b2i(by), 255)) % 256).last ', bytes1 = [], bytes2int(bytes1) * 256 + band(b2i(by), 255) = ~256, (bytes2int(bytes1) * 256 + band(b2i(by), 255)) % 256 = 0, b2i(by) = 0
Proof step 55:
(bytes2int(bytes1) * 256) % 256 = 0 ∧ bytes2int(bytes1) * 256 + band(b2i(by), 255) < 0 ∧ bytes2int(bytes1) * 256 < 0 ∧ 0 ≤ band(b2i(by), 255) ∧ band(b2i(by), 255) % 256 ≠ 0 ∧ 0 < 256, # bytes1 = # int2bytes((bytes2int(bytes1) * 256 + band(b2i(by), 255)) / 256 - 1), band(b2i(by), 255) < 256, bytes2int(bytes1) * 256 + band(b2i(by), 255) < ~256, bytes2int(bytes1) < 0, ∀ bytes0. # bytes0 < #(bytes1 + by ') → true → # bytes0 = # int2bytes(bytes2int(bytes0)) → bytes0 = int2bytes(bytes2int(bytes0)) ⊦ bytes1 + by ' = int2bytes((bytes2int(bytes1) * 256 + band(b2i(by), 255)) / 256 - 1) + int2bytes(band(b2i(by), 255) % 256 - 256).last ', bytes1 = [], bytes2int(bytes1) * 256 + band(b2i(by), 255) = ~256, band(b2i(by), 255) % 256 - 256 = 0, b2i(by) = 0

Rule argument

seq: ⊦ i % j = 0 ∧ i + i0 < 0 ∧ i < 0 ∧ 0 ≤ i0 ∧ i0 % j ≠ 0 ∧ 0 < j → (i + i0) % j = i0 % j - j subst: [j, i0, i] → [256, band(b2i(by), 255), bytes2int(bytes1) * 256] rotate: F spec: int-div inst: name: mod-add1-negpos-neg

Additional information

Used terms: 256, band(b2i(by), 255), bytes2int(bytes1) * 256