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Proof step 88 from proof int2bytes-rec-pos-any (previous step)

# int2bits(j % 2 ^ (i * 8)).rest % 8 = 0, # int2bits(j % 2 ^ (i * 8)) = i→n(i * 8) + 1, i→n(i) < # bits2bytes(int2bits(j % 2 ^ (i * 8))), 2 ^ (i * 8) ≤ j, 0 < i, i→n(i) + 1 < # bits2bytes(int2bits(j % 2 ^ (i * 8))) ⊦ int2bits(j % 2 ^ (i * 8)).rest = [], bits2bytes(int2bits(j % 2 ^ (i * 8)).rest) = lastn(i→n(i) + 1, bits2bytes(int2bits(j % 2 ^ (i * 8)))).rest, i→n(i) + 1 = # bits2bytes(int2bits(j % 2 ^ (i * 8)))

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 89:
⊦ # bits % 8 ≠ 0 → # bits2bytes(bits) = # bits / 8 + 1
Proof step 90:
¬ # int2bits(j % 2 ^ (i * 8)) % 8 ≠ 0, # int2bits(j % 2 ^ (i * 8)).rest % 8 = 0, # int2bits(j % 2 ^ (i * 8)) = i→n(i * 8) + 1, i→n(i) < # bits2bytes(int2bits(j % 2 ^ (i * 8))), 2 ^ (i * 8) ≤ j, 0 < i, i→n(i) + 1 < # bits2bytes(int2bits(j % 2 ^ (i * 8))) ⊦ int2bits(j % 2 ^ (i * 8)).rest = [], bits2bytes(int2bits(j % 2 ^ (i * 8)).rest) = lastn(i→n(i) + 1, bits2bytes(int2bits(j % 2 ^ (i * 8)))).rest, i→n(i) + 1 = # bits2bytes(int2bits(j % 2 ^ (i * 8)))
Proof step 93:
# int2bits(j % 2 ^ (i * 8)) % 8 ≠ 0, # int2bits(j % 2 ^ (i * 8)).rest % 8 = 0, # int2bits(j % 2 ^ (i * 8)) = i→n(i * 8) + 1, i→n(i) < # int2bits(j % 2 ^ (i * 8)) / 8 + 1, 2 ^ (i * 8) ≤ j, 0 < i, i→n(i) + 1 < # int2bits(j % 2 ^ (i * 8)) / 8 + 1 ⊦ int2bits(j % 2 ^ (i * 8)).rest = [], bits2bytes(int2bits(j % 2 ^ (i * 8)).rest) = lastn(i→n(i) + 1, bits2bytes(int2bits(j % 2 ^ (i * 8)))).rest, i→n(i) + 1 = # int2bits(j % 2 ^ (i * 8)) / 8 + 1

Rule argument

name: bits2bytes-len-02 rotate: F subst: [bits] → [int2bits(j % 2 ^ (i * 8))]

Additional information

Used terms: int2bits(j % 2 ^ (i * 8))