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Proof step 179 from proof int2bytes-rec-pos-any (previous step)

i→n(i * 8) % 8 = 0, b2i(bits2bytes(firstn(# int2bits(j % 2 ^ (i * 8)) % 8, int2bits(j % 2 ^ (i * 8)))).first) < 0, 0 < i, 1 < i, 8 < # int2bits(j % 2 ^ (i * 8)), # int2bits(j % 2 ^ (i * 8)) < i→n(i * 8), 2 ^ (i * 8) ≤ j, 0 ≤ j % 2 ^ (i * 8) ⊦ bits2bytes(mklist(false, i→n(i * 8) - # int2bits(j % 2 ^ (i * 8))) + int2bits(j % 2 ^ (i * 8))) = fix (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))))) .rest, mklist(false, i→n(i * 8) - # int2bits(j % 2 ^ (i * 8))) + int2bits(j % 2 ^ (i * 8)) = [], i→n(i * 8) + 1 = # int2bits(j % 2 ^ (i * 8)), # int2bits(j % 2 ^ (i * 8)) = i→n(i * 8), i = 0, # int2bits(j % 2 ^ (i * 8)) % 8 = 0

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 180:
⊦ # bits ≤ 8 → bits2bytes(bits) = i2b(bits2int(bits)) '
Proof step 181:
# firstn(# int2bits(j % 2 ^ (i * 8)) % 8, int2bits(j % 2 ^ (i * 8))) ≤ 8, i→n(i * 8) % 8 = 0, b2i(i2b(bits2int(firstn(# int2bits(j % 2 ^ (i * 8)) % 8, int2bits(j % 2 ^ (i * 8))))) '.first) < 0, 0 < i, 1 < i, 8 < # int2bits(j % 2 ^ (i * 8)), # int2bits(j % 2 ^ (i * 8)) < i→n(i * 8), 2 ^ (i * 8) ≤ j, 0 ≤ j % 2 ^ (i * 8) ⊦ bits2bytes(mklist(false, i→n(i * 8) - # int2bits(j % 2 ^ (i * 8))) + int2bits(j % 2 ^ (i * 8))) = fix (i→n(i) + 1, i2b(bits2int(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, mklist(false, i→n(i * 8) - # int2bits(j % 2 ^ (i * 8))) + int2bits(j % 2 ^ (i * 8)) = [], i→n(i * 8) + 1 = # int2bits(j % 2 ^ (i * 8)), # int2bits(j % 2 ^ (i * 8)) = i→n(i * 8), i = 0, # int2bits(j % 2 ^ (i * 8)) % 8 = 0

Rule argument

name: bits2bytes-base rotate: F subst: [bits] → [firstn (# int2bits(j % 2 ^ (i * 8)) % 8, int2bits(j % 2 ^ (i * 8)))]

Additional information

Used simplifier rules: ⊦ i * j = j * i;% ⊦ # firstn(n, bits) = n;%n ≠ 0 ⊦ m % n ≤ n; Used terms: firstn(# int2bits(j % 2 ^ (i * 8)) % 8, int2bits(j % 2 ^ (i * 8)))