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Proof step 97 from proof int2bytes-rec-pos-any (previous step)

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

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 98:
⊦ # bits % 8 ≠ 0 ∧ 8 < # bits → bits2bytes(bits) = bits2bytes(firstn(# bits % 8, bits)) + bits2bytes(restn(# bits % 8, bits))
Proof step 99:
¬(# int2bits(j % 2 ^ (i * 8)) % 8 ≠ 0 ∧ 8 < # int2bits(j % 2 ^ (i * 8))), i→n(i) + 1 = # bits2bytes(int2bits(j % 2 ^ (i * 8))), # int2bits(j % 2 ^ (i * 8)).rest % 8 = 0, # int2bits(j % 2 ^ (i * 8)) = i→n(i * 8) + 1, 2 ^ (i * 8) ≤ j, 0 < i ⊦ int2bits(j % 2 ^ (i * 8)).rest = [], bits2bytes(int2bits(j % 2 ^ (i * 8)).rest) = bits2bytes(int2bits(j % 2 ^ (i * 8))).rest
Proof step 102:
# int2bits(j % 2 ^ (i * 8)) % 8 ≠ 0 ∧ 8 < # int2bits(j % 2 ^ (i * 8)), i→n(i) + 1 = # ( bits2bytes(firstn(# int2bits(j % 2 ^ (i * 8)) % 8, int2bits(j % 2 ^ (i * 8)))) + bits2bytes(restn(# int2bits(j % 2 ^ (i * 8)) % 8, int2bits(j % 2 ^ (i * 8))))), # int2bits(j % 2 ^ (i * 8)).rest % 8 = 0, # int2bits(j % 2 ^ (i * 8)) = i→n(i * 8) + 1, 2 ^ (i * 8) ≤ j, 0 < i ⊦ int2bits(j % 2 ^ (i * 8)).rest = [], bits2bytes(int2bits(j % 2 ^ (i * 8)).rest) = ( bits2bytes(firstn(# int2bits(j % 2 ^ (i * 8)) % 8, int2bits(j % 2 ^ (i * 8)))) + bits2bytes(restn(# int2bits(j % 2 ^ (i * 8)) % 8, int2bits(j % 2 ^ (i * 8)))) ).rest

Rule argument

name: bits2bytes-rec-01 rotate: F subst: [bits] → [int2bits(j % 2 ^ (i * 8))]

Additional information

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